J ' Espère

Eric Danan

Ani Guerdjikova

Laurence Jacquet

David Ettinger

Huang, Tan, Wu, Xu Chen

Zeng

Je tiens tout d'abord à remercier chaleureusement mon directeur de thèse, Régis Renault.

Il a ouvert la porte et m'a conduit à la théorie économique. Quelle que soit la pertinence de ma question, il les m'a répondu avec égale patience. En même temps, Régis est assez généreux pour toujours payer notre repas quand on s'est rendu dans les restaurants. Je rends hommage à Gorkem Celik, en raison de son aide gracieuse à la fois dans la recherche et dans le nancement.

Sans lui, je n'aurais pas pu vivre. Je remercie Thomas Tregouët ensuite pour son aide dans mes travaux, sa disponibilité et son caractère exigeant.

A mes parents

INTRODUCTION

A common observation is that price dispersion exists for homogeneous products. In Stigler (1961)'s example, the average asking price of an identical model of Chevrolet (in 1959) is 2436 dollars, ranging from 2350 to 2515, with a standard deviation of 42 dollars. The same model of Asus laptop is asked 349 euros in an on-line store, 399 euros in an electronic store and 499 euros in Auchan. These price variations are not coincidences. Other examples are food, clothing and airline tickets. A well-known fact is that informational structure (dierent consumers know dierent things about the market) is one cause of price dispersion. The 349 euros is targeted at internet shoppers, the 399 euros is targeted at consumers knowing electronic products, and the 499 euros is targeted mainly at consumers who do not know much about the market.

Consumers can learn the products through two channels, by searching the products and by directly receiving product information.

Consumers can search products in a sequential way or in a parallel way. If he searches in a sequential way, each time of search reveals information about one piece of product to the consumer. If he searches in a parallel way, the consumer commits a priori to search a number of rms and picks up the one maximizing his utility. On the other hand, consumers can search either in a random way or in an ordered way. By random search, each consumer has his own searching path, which is independent from others. By ordered search, all consumers search in the same path.

Products must be heterogeneous in certain dimensions ; otherwise the outcome is predicted by [START_REF] Diamond | A model of price adjustment[END_REF] (known as the diamond paradox). In some markets (train tickets, stocks), products are homogeneous. Firms must employ mixed strategies in prices in order to motivate consumers to search. When products are dierentiated, consumers search for lower prices, higher quality and better match.

Meanwhile, consumers are heterogeneous in their knowledge about the market. Some consumers have zero search cost and are labelled as shoppers. Other consumers are labeled as global searchers or local searchers. Recent attempts to estimate consumer search cost [START_REF] Hong | Using price distributions to estimate search costs[END_REF], los [START_REF] Santos | Consumer search on the internet[END_REF] and [START_REF] Moraga-Gonzalez | Maximum likelihood estimation of search costs[END_REF]) use structural methodologies to show that search cost is actually quite high (for instance, around 29 dollars for a textbook of around 65 dollars) and varies among the searched products and dierent education levels. [START_REF] Wilson | Market frictions : A unied model of search costs and switching costs[END_REF] nds a low search cost but a high switching cost : search cost is estimated to be around 0.01-0.3% of the maximum gain from search, while switching cost is estimated to be around 48-73%. Some models allow the search cost to be chosen by rms, named obfuscation. Most search models assume free recall, meaning that once the product is searched, the consumer can pick it up at any time at no cost.

Consumers can also learn the products through the mass media or other consumers, either immediately or through a learning process. Prices can serve as incentives for better dissemination of product information. Group-buying, a recent shopping strategy originated in Asia is such an example. Consumers on the internet form groups in order to bargain for lower prices. Since individual consumer typically does not need multiples of one item or do not have the resources to buy in bulk, group buys allow (encourage) people to invite others to purchase in bulk jointly.

The rest of this chapter is organized as the following : section two elaborates certain theoretical models related to the thesis. Section three discusses contribution of the thesis to the search literatures.

Review on Theoretical Models

Search models date back to [START_REF] Diamond | A model of price adjustment[END_REF] which predicts the market price to be the monopoly price if both consumers and rms are homogeneous. No consumers search under his context, which is far from real-world market. [START_REF] Salop | Bargains and ripos : A model of monopolistically competitive price dispersion[END_REF] and [START_REF] Varian | A model of sales[END_REF] show that the diamond paradox can be avoided by assuming two types of consumes : the informed and the uninformed. As a result, there is price dispersion ; some rms sell to informed consumers, others sell to uninformed consumers. [START_REF] Burdett | Equilibrium price dispersion[END_REF] constructs a parallel search model, where search cost is the same for every consumer. [START_REF] Stahl | Oligopolistic pricing with sequential consumer search[END_REF] bridges the gap between the Diamond and Bertrand. In his model, consumers either have zero search cost, or positive search cost, and they conduct sequential search. In [START_REF] Chen | Equilibrium price dispersion with heterogeneous searchers[END_REF], there are informed consumers, global searchers and local searchers, where global searchers conduct sequential search and local searchers search only once. [START_REF] Janssen | Sequential search with incompletely informed consumers : Theory and evidence from retail gasoline markets[END_REF] looks into the case where consumers can not observe rms' marginal cost. [START_REF] Janssen | Consumer search markets with costly second visits[END_REF] analyses the sequential search model with costly recall.

Consumers search products in the hope of getting lower prices. When consumers search sequentially, they observe the price dispersion, and decide at which point to stop. Behind these decisions, they compare the expected incremental surplus from searching once more with search cost. If the expected incremental surplus is smaller, the optimal choice is to stop and make a purchase. If the expected incremental surplus is larger than search cost, the optimal choice is to search the next rm. Consumers with low search cost on average search more rms than consumers with high search cost, that is to say that, their demand is more elastic. When rms sell products to consumers with varying search cost, it is as if they sell products to consumers with varying price elasticity of demand.

The optimal searching process takes into account the expected price dispersion. The price dispersion in equilibrium takes into account of consumers' optimal searching process. One main result of these papers is that, prices increase in search cost. The more dicult to obtain product information, consumers tend to search less and each rm has larger local monopoly power.

The equilibrium price distribution in [START_REF] Stahl | Oligopolistic pricing with sequential consumer search[END_REF] (and models based on that) has a larger probability mass in the high prices if the number of rms in the market increases. The reason is that as the number of rms increases, the probability of being the lowest-price rm observed by consumers decreases exponentially, which weakens the rm's incentive to lower its price.

When the products are horizontally dierentiated, consumers search not only for lower prices, but also for better match. [START_REF] Perlo | Equilibrium with product dierentiation[END_REF]'s spatial model shows that the equilibrium price is bounded away from marginal cost even if search cost is zero. [START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF] obtains a sequential search model where consumers search for horizontally dierentiated products. [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF] studies the eects of product diversity on prices. In [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF], the optimal number of rms for the society decreases in search cost, while in a free-entry equilibrium, the number of rms increases in search cost. Therefore, there is overentry of rms. [START_REF] Zhou | Multiproduct search[END_REF] looks into the case where consumers search for several products. [START_REF] Armstrong | Exploding orders and buy-now discounts[END_REF] analyses the buy-now discount. [START_REF] Moraga-Gonzalez | Search costs, demand-side economies and the incentives to merge under bertrand competition[END_REF] discusses the incentive and welfare implications of horizontal mergers.

In [START_REF] Armstrong | Prominence and consumer search[END_REF], one rm is prominent in the sense that each consumer will visit the prominent rm rst. In [START_REF] Zhou | Ordered search in dierentiated markets[END_REF], each consumer searches a line of rms in the same ordering (as in [START_REF] Arbatskaya | Ordered search[END_REF]). Contrary to [START_REF] Arbatskaya | Ordered search[END_REF], [START_REF] Armstrong | Prominence and consumer search[END_REF] and [START_REF] Zhou | Ordered search in dierentiated markets[END_REF] nd that the rms searched by consumers rst usually charge lower prices than rest of the rms. The conicting results lie in the fact that consumers search only for lower prices in [START_REF] Arbatskaya | Ordered search[END_REF], but both prices and match in the context of [START_REF] Armstrong | Prominence and consumer search[END_REF] and [START_REF] Zhou | Ordered search in dierentiated markets[END_REF]. In [START_REF] Arbatskaya | Ordered search[END_REF], subsequent rms can use only lower prices to attract consumers. However, behind the price dierence in [START_REF] Armstrong | Prominence and consumer search[END_REF] and [START_REF] Zhou | Ordered search in dierentiated markets[END_REF], there are the prominent rms who lower the prices in order to prevent consumers from further search, and the subsequent rms, who infer from consumers' optimal search rule that their visitors are those who do not obtain high surplus from the products of the prominent rms.

Three factors can be endogenized. First, even though rms are horizontally dierentiated, rm can choose their extent of dierentiation before choosing prices. [START_REF] Johnson | On the simple economics of advertising, marketing, and product design[END_REF] nd that rms choose extreme designs. [START_REF] Larson | Niche products, generic products, and consumer search[END_REF] demonstrates that search cost is one factor aecting rms' decisions about dierentiation. When search cost is low enough, rms maximize their product dierentiation. When search cost is quite large, rms minimize their product dierentiation. [START_REF] Bar-Isaac | Search, design, and market structure[END_REF] nds that a larger fraction of niche rms arises as search cost drops.

Second, rms can choose the cost for consumers to obtain product information. [START_REF] Wilson | Ordered search and equilibrium obfuscation[END_REF] shows that there is no equilibrium of full transparency. If one rm chooses obfuscation and the other one chooses no obfuscation, consumers are directed to the one without obfuscation, who raises the price. [START_REF] Ellison | A search cost model of obfuscation[END_REF] develops a similar model where the level of obfuscation is not observable, and search cost is a non-linear function of obfuscation. [START_REF] Petrikaite | Consumer obfuscation by a multiproduct rm[END_REF] nds that obfuscation is protable for a multi-product rm in order to internalize the externalities that one product imposes on others.

Third, the search order can be endogenized. The intermediary can choose an order to present the products to the consumers. In [START_REF] Hagiu | Why do intermediaries divert search[END_REF], the intermediary observes the preferences of consumers, and deliberately divert each consumer to the one less preferred. The reason to divert consumers is to reduce the prot dierence between the two rms while maintaining the participation of both.

Summary of This Thesis

If consumers search in the same ordering, [START_REF] Arbatskaya | Ordered search[END_REF] nds that prices decrease along consumers' searching path. In the other model, [START_REF] Armstrong | Prominence and consumer search[END_REF] shows that the rst rm charges a lower price than the rest of the rms. Most aforementioned models study homogeneous rms or heterogeneous rms where the valuation of each rm's product is under the same distribution. However, product ordering matters only when the products are dierent. Otherwise, their rankings have no implications on welfare. Chapter 2 looks for the optimal product ordering from an industry perspective and a social perspective. First, the chapter introduces two types of goods : niche goods and generic goods.

In practice, the distributions of valuations of products can dier : some products attract the mass population, whereas some products attract only a market niche. For instance, road bikes are generic goods designed for travelers on paved roads (although one can also ride it on rocks.).

Mountain bikes are niche goods designed for o-road cycling (although one can also ride it on paved roads.)

Niche products do not have on average better quality than generic products. For instance, people riding bikes mostly on roads often nd mountain bikes heavy and bulky. I start with a duopoly model (from [START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF]) with two competing rms : a niche rm and a generic rm. When consumer search is frictionless, the unique price equilibrium has that both rms charge the same price.

A non-innitesimal search cost somehow prevents consumers from searching the subsequent rm. I show that this pricing pattern is solid. The underlying mechanism is similar to [START_REF] Armstrong | Prominence and consumer search[END_REF] and [START_REF] Zhou | Ordered search in dierentiated markets[END_REF] : the rst rm lowers down the price in order to further prevent consumers from searching the second rm ; the second rm infers from the optimal search rule that its visitors are those who have obtained poor match from the rst rm.

By charging a lower price, the rst rm actually always gains higher prot than the second rm, which does not automatically imply that each rm of each type gains higher prot when it is rstly searched by consumers than when it is searched by consumers after. Imagine that consumers see the generic rm rst. Consumers immediately infer that the next rm is the niche rm, without knowing which niche the rm is actually in. As a result, more consumers wonder what is behind and search the second rm, resulting in intensied competition, and therefore lower price levels. By the same logic, the niche goods rst, generic goods second ordering is optimal for the industry. The same ordering is preserved by consumers if consumers are believed to search the niche rm rst, according to which rms set prices. The same ordering is also social optimal.

Firms agglomerates in dierent ways on the Internet and in real markets. In real markets, the clusters are formed more naturally and often without a planer : restaurants, hotels choose to be near the train station. On the Internet, wherever there are clusters, there are platforms behind controlling the fees. More precisely, the platforms choose the number of rms on the Internet.

Chapter 3 provides an explanation to a subtle question : why do rms agglomerate on the Internet ? Firms agglomerate in order to obtain larger prot than those who do not. They obtain higher prot by facing larger demand from those consumers visiting the cluster rst. And the consumers visit the clusters rst for lower prices. If this is the case, two rms are enough to attract all consumers. Then arises the following question : why do platforms on the Internet have thousands of rms ?

In this chapter, I develop a search model à la [START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF]. In the model, there is a monopoly platform, innitely many rms and one unit mass of consumers. Consumers can obtain any product information in the platform without cost, and outside the platform only through sequential search. I nd an equilibrium where all consumers enter the platform rst, no matter the platform imposes them a subscription fee or not. In the equilibrium, prices are lower in the platform due to competition intensifying eect inside the platform and the incentive for inside rms to prevent consumers form searching outside rms.

The answer to the last question lies in search cost. When search cost is large enough, consumers do not search and the optimal size of the platform is 2. When search cost is low, not only lower prices, but also more varieties are needed to hold these consumers. When several platforms (stores) compete, in a symmetric equilibrium, each platform (store) has the same number of rms, and the number of rms decreases in search cost.

Observations are also in real markets. A usual store in a centralized market for electronic products (called computer city in developing countries) often oers almost every brand. It is rare to see such a store selling only one type of product, since consumers will easily ow away.

The last chapter is aimed at discussing the protability of a pricing strategy, referred as the group-buying. I use a location model with a monopolist and several consumers. Consumers, depending on their types, choose among informing another consumer, waiting or rejecting the oer. Dierent from the earlier chapters, no consumers search for product information, but some of them receive it from other consumers.

Two models are proposed. The rst model assumes that consumers can not invite others. I nd that the monopolist resorts to inter-temporal price discriminations (an early price and a late price), which suggests that the group-buying strategy is not optimal. The reason is that the monopolist is facing the same number of consumers, for whom group-buying is equivalent to a pure strategy in pricing in the rst period, and a mixed strategy in pricing in the second period.

The second model allows each consumer to invite at most one more consumer. In equilibrium, the monopolist resorts to a bait and switchstrategy. The monopolist gains less from successful deals than from unsuccessful ones. Low discounted prices are oered in order to motivate consumers to make invitations, and the monopolist places hope on that consumers arrive in insucient number and some of them choose to purchase at the normal price. If the inviting cost is large enough, it is optimal to set the discounted price to zero.

Compared with monopoly pricing, the model with invitations decreases surplus of the rstcohort consumers and increases surplus of the second-cohort consumers. Overall it increases consumer surplus as long as inviting cost is not too large, and thereby, it can increase social welfare.

ORDERED SEARCH WITH ASYMMETRIC PRODUCT DESIGN

Introduction

It is common that consumers see through products in the same ordering. While visiting a supermarket or a web-store, most observe the products at the entrance or in the homepage before looking into a specic category. Even when products are listed in the same category, shoppers see through them in a top-down manner. Some experiments show that the click-through rate (CTR) of the rst item can be 17 times higher than the eighth item.

1 This huge dierence implies that choosing the ordering is important, as dierent orderings can give rise to very dierent outcomes.

The consumers' anticipation about their match with dierent types of products may be very dierent. The present paper focuses on the dierence between generic products and niche products, in the terminology of [START_REF] Larson | Niche products, generic products, and consumer search[END_REF]. A generic product serves the mass population, while a niche product serves some small population better than the generic product, and others worse than the generic product. Dierent product types do not imply any superior (or inferior) quality. Take standard jeans as examples of generic products. Then niche products are either colourful jeans or ripped jeans. There is no evidence that standard jeans have on average higher quality than ripped jeans, besides the fact that ripped jeans usually target the young population.

The paper uses a duopoly model to study how the ordering of product presentation aects 1. [START_REF] Granka | The determinants of web page viewing behavior : An eye-tracking study[END_REF] employed eye-tracking analysis to explore both the click and attention distribution from Google. Evidence has shown that levels of attention depends on their locations. the market. One rm (the niche rm) sells a niche product and the other rm (the generic rm) sells a generic product. The products can be arranged in two manners. In the niche-generic ordering, each consumer sees the niche product at the beginning. In the generic-niche ordering, each consumer sees the generic product at the beginning. In either ordering, consumers can always visit the other product after incurring a search cost.

In equilibrium, the rm that consumers visit rst (the rst rm) always sets a lower price than the second rm from the fact that the rst rm has more elastic demand. Reasons are as follows. All consumers always observe the price of the rst rm, but only those having received poor match from the rst rm observe the price of the second rm. Therefore price deviations of the rst rm aect the decisions of all consumers, while price deviations of the second rm aect only the decisions of a fraction of them. Behind this price pattern, the rst rm prevents consumers from searching the second rm, who in turn charges a higher price to those failing to obtain good enough match from the rst rm.

While charging a lower price, the rst rm actually earns larger prots than the second rm. However, this does not imply that both types of rms earn higher prots being rst than being second. This property, which would hold if both rms had identical types, remains valid in our setting for the rm selling the niche product. However, However, the generic rm may earn lower prots in the generic-niche ordering than in the niche-generic ordering, for some parameter congurations. After comparing the price equilibrium in both orderings, we nd that the generic-niche ordering induces a larger proportion of consumers to search the second rm than the niche-generic ordering. More search intensies competition, which brings down prices and industry prot. Therefore, the niche-generic ordering is optimal for the industry.

The framework is then extended. In section 5, consumers are free to choose the order in which they search. According to [START_REF] Weitzman | Optimal search for the best alternative[END_REF], consumers optimally visit rms in the descending order of reservation prices. In equilibrium, this order of search is correctly anticipated by rms and consumers anticipate that rms price accordingly. The exists an equilibrium, where consumers search in the niche-generic manner thus yielding the largest industry prot. An equilibrium with the reverse order of search does not necessarily exist.

We then introduce an outside option by assuming that each consumer buys the product only if the product oers a positive surplus. Again, industry prots are always higher in the niche-generic ordering, whether we consider the duopoly setting or the case of a multiproduct rm selling both products.

Last, we analyze welfare. If the market is covered, simulations show that the niche-generic ordering decreases consumer surplus and increases social welfare when the dierence between the two types of rms is large enough. The same ordering increases consumer surplus and decreases welfare if the dierence is very small. If the market is not covered, the niche-generic ordering increases consumer surplus when search cost is large because prices increase faster in search cost in the generic-niche ordering.

The niche-generic ordering is socially ecient, where less price distortion is at presence. The literature on search provides a theoretical basis for the paper. [START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF] introduces sequential search in [START_REF] Perlo | Equilibrium with product dierentiation[END_REF] and obtained a model of true monopolistic competition. [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF] provide some general comparative statics for the oligopoly model and show that search costs exacerbate over entry in the market.

The paper is most closely related to the ordered search literature. [START_REF] Arbatskaya | Ordered search[END_REF] nds that prices for homogeneous products decline from the rst to the last. However, [START_REF] Armstrong | Prominence and consumer search[END_REF] obtains the opposite result by considering dierentiated product : one prominent rm charges a lower price than the other rms. [START_REF] Zhou | Ordered search in dierentiated markets[END_REF] further conrms that prices are indeed increasing along the searching path by considering ordered search with product dierentiation and an arbitrary number of rms. [START_REF] Armstrong | Prominence and consumer search[END_REF] further discusses products with varying quality and nds that the highest-quality rm has the greatest incentive to become prominent.

Moraga-Gonzalez and Petrikaite (2013) looks into the incentives for horizontally dierentiated rms to merger and nd that in the short run the merged entity is searched last because it is expected to charge higher prices.

Finally, the paper draws on the literatures on product design. [START_REF] Johnson | On the simple economics of advertising, marketing, and product design[END_REF] study families of "variance-ordered" distributions and show that a monopolist's prot is maximized with either minimum or maximum preference diversity. [START_REF] Kuksov | Buyer search costs and endogenous product design[END_REF] uses a duopoly model to study the impact of search cost on the endogenous level of product dierentiation.

Lower search cost leads rms to dierentiate themselves more, which leads to higher prices and lower social welfare. [START_REF] Larson | Niche products, generic products, and consumer search[END_REF] and [START_REF] Bar-Isaac | Search, design, and market structure[END_REF] use sequential search models, in which rms are identical ex-ante and choose dierent levels of product dierentiation according to other parameters. [START_REF] Hagiu | Why do intermediaries divert search[END_REF] discuss intermediaries' decisions to divert some consumers to the unfavourable rm even after consumers have learned their types. [START_REF] Wang | Informative advertising, consumer search and transparent policy[END_REF] uses a monopoly model and looks at the impact of search cost on information disclosure.

Section 2 develops the basic model. Section 3 uses the basic model to nd out the equilibrium in three cases : perfect information, generic-niche ordering and niche-generic ordering. After that, some comparative statics are done with respect to various parameters. Section 4 answers the main question of the paper : which ordering is optimal for the industry. Prot is rst compared at the rm level and then at the industry level. Section 5 analyses consumers' incentives to deviate to another searching order. Section 6 introduces the outside option to the original model and studies two related cases. Section 7 provides numerical examples of consumer surplus and social welfare from each specic ordering. The last sections (8-9) concludes our paper and discuss possible future works.

The Model

Consumers. There is a large number of consumers with measure normalized to one. Each consumer looks for one unit of good in the market. We assume that if a consumer does not purchase any good, his surplus is low enough so that the market is covered. If consumer i purchases the product from rm j, the utility derived from consuming the product consists of the price of the product j ∈ {N, G} and a random value εij = µ j ij . There are two components in εij : µ j is a rm-specic term that we will interpret in the next paragraph ; ij is the match between consumer i and rm j. We assume that ij is randomly distributed according to c.d.f. F (x). The match is independent across consumers and rms. Overall, consumer i's net utility when purchasing product j at price p j is given by :

u ij = µ j ij match -p j (1)
We make the following assumption on c.d.f. F (.) : Assumption 1 (Symmetry). The density function f ( ) exists and is continuous, symmetric, with support [-a , a] with a > 0 and has zero mean.

Firms. The market has a niche rm (Firm N) and a generic rm (Firm G). Firms are producing products that are weak substitutes to each other. Firms incur neither xed nor marginal cost.

The rm-specic term µ j is a measure of the importance of match in a consumer's overall utility function.

2 We assume that

µ N > µ G
Given assumption 1, having a larger µ does not imply a superior or inferior quality but rather the importance of quirkiness on buyers' purchase decisions. With this interpretation in mind (and following [START_REF] Johnson | On the simple economics of advertising, marketing, and product design[END_REF], [START_REF] Larson | Niche products, generic products, and consumer search[END_REF] and Bar-Isaac et al. ( 2012)), we call rm N's product the niche good and rm G's product the generic good. The niche good (with a larger consumer heterogeneity) is more likely to provide extreme valuations which satisfy the needs of some small population, while missing the needs of others. On the other hand, the generic good has a smaller consumer heterogeneity and is aimed at the mass population.

Consumers know the type of each rm, but not its price and match. In order to learn the price and match information, consumers conduct product search in a sequential ordering. Each search costs c, which is identical for all consumers. Searching rm j informs the consumer of the true price and the value of ij . After being informed about one rm, the consumer always has the free recall option for that rm.

3 Firms cannot distinguish between the consumers coming for the rst time from those choosing the recall option.

Firms are not only dierent in types, but also regarding the order in which they are searched.

Use ≺ to denote the spatial order. For example, Firm N ≺ Firm G indicates that rms are in the niche-generic ordering. Denote rm j's demand function by D N ≺G j (p N , p G ) when rms are in the niche-generic ordering, and its prot function by Π N ≺G j (p N , p G ). In the niche-generic ordering, each consumer visits the niche rm rst. In the generic-niche ordering, each consumer visits the generic rm rst. We call the rm that consumers visit rst (second), the rst rm (the second rm).

Game Structure : Each rm observes the type of the rival rm, and the product ordering, but not any match information. Consumers can acquire price and match information of the second rm only through searching that rm and paying search cost c. The game has the following timeline. In stage 1, rms choose prices simultaneously. In stage 2, consumers observe the price and match information about the rst rm for free. Each individual consumer then either purchases the rst product immediately or searches the second rm, correctly anticipating its type. In the latter case, he purchases the one oering higher surplus. In equilibrium, consumers hold correct beliefs about the prices set by the second rm, and form an optimal search rule. We skip the uninteresting equilibrium where no consumers search when they expect high prices and rms set prices to innity. A consumer's pay-o is the surplus from the product minus the sum of search cost. The rm

j's pay-o is the expected prot, either E[Π G≺N j (p N , p G , c)] or E[Π N ≺G j (p N , p G , c)].

Equilibrium Analysis

Start from the last stage and by backward inductions, we solve for the prot-maximizing prices and the most protable ordering for the industry with two active rms. The benchmark case assumes zero search cost and later parts compare the benchmark case with the cases with positive search cost.

Perfect Information

If price and match information is costless to obtain, each consumer weakly prefers sampling two rms to sampling only one.

4 For a consumer holding exact information about two products, he purchases the generic product if the following is satised :

µ N N -p N < µ G G -p G
When the generic rm deviates to a new price p, the new price pair becomes (p N , p), and the probability of the above inequity becomes F ((µ G G + p N -p)/µ N ) for each realization of G .

The demand function of the generic rm is the probability of oering larger surplus than the niche rm to a particular consumer :

D G (p, p N ; c = 0) = a max(-a, -µ N a-p N +p µ G ) F µ G + p N -p µ N f ( )∂ (2)
Suppose that the generic rm sticks to p G , and the niche rm deviates to a new price p. Since the market is covered, the demand share of the niche rm is the entire market minus the share of the generic rm. The below expression is the demand for the niche rm.

D N (p G , p; c = 0) = 1 - a max(-a, -µ N a-p+p G µ G ) F µ G + p -p G µ N f ( )∂ (3)
Clearly, both D G and D N can be rewritten into functions of the price dierence. Stemming 4. Some consumers may skip the rm behind in case they expect zero gain from the search. For example, if goods are arranged in the niche-generic ordering, consumers who obtain values of match higher than (µ G ap G + p N )/µ N will skip the generic rm. To get around this, we assume that every consumer searches every rm.

from D G = 1 -D N , the derivatives of D G and D N with respect to p are equal.

∂ ∂p D G (p, p N ; c = 0) = ∂ ∂p D N (p G , p; c = 0) = - 1 µ N a max(-a, -µ N a-p N +p µ G ) f µ G + p N -p µ N f ( )∂
In equilibrium, rms maximize their expected prots and do not deviate to other prices. By the rst order conditions, we obtain the equilibrium p N and p G :

p N = p G = µ N a -a F (µ G /µ N )f ( )∂ a -a f (µ G /µ N )f ( )∂ (4)
The unique existing equilibrium satises p N -p G + a(µ N -µ G ) > 0, therefore the consumer receiving (-a, -a) purchases the generic good.

Proposition 1. Under perfect information, rms set identical prices and earn equal amounts of prot.

p N = p G = µ N 2 a -a f (µ G /µ N )f ( )∂
In Figure 1, the horizontal axis represents the match with the generic product, and the vertical axis represents the match with the niche product. The grey curve represents those consumers indierent between two products. Then the consumers above it choose the niche product, and consumers below it choose the generic product. If prices are equal, this grey curve must cross the original point. If a consumer at ( G , N ) purchases at Firm N, then the consumer at (-G , -N ) will purchase at Firm G, and vice versa.

Since f ( ) is symmetric, the integral above the grey curve is always as large as the integral below it. It is clear that the market is equally split. From equal prices and sizes of demand, rms always earn equal amounts of prot. On the other hand if one rm earns higher prot than the

G N µ G µ N Firm N Firm G u iN = u iG (-a,-a) (a,a)
other rm, then the other rm can always set the same price as the rst rm and obtain equal demand shares.

Particularly if F is uniform[-1, 1], the common equilibrium price is µ N . In the polar case of µ N → 0 (then µ G → 0), the equilibrium price approaches the marginal cost, and the outcome becomes Bertrand :

lim µ N →0 p N , p G = 0
When two products have an identical level of consumer heterogeneity, (4) becomes the Perlo and [START_REF] Perlo | Equilibrium with product dierentiation[END_REF] price, where the equilibrium prices rise in consumer heterogeneity. While "consumer heterogeneity" levels dier, the equilibrium prices dier according to the shape of the distribution.

Imperfect Information

To simplify the analysis, we assume the followings in the remainder of the paper : Assumption 2. is uniformly distributed on [-1, 1].

Consider very large search cost (c ≥ µ G ). If consumer i searches the generic rm, the surplus that he gains is µ G iG -p N -c ≤ 0. To get around this situation, we make the third assumption in the following. Assumption 3. Search cost is not too large, namely c < µ G .

In the subsection where we discuss the niche-generic ordering, the above is to ensure that the price elasticity of demand for the generic rm is negative.

The Optimal Stopping Rule

Suppose that rm j is the rst rm. After his match to the rst rm (e.g. (p j , j )), a consumer can either make a purchase from the current rm or sample the next rm k at a cost c > 0.

Suppose that the consumer expects the next rm k to charge the price p k . The expected gain in surplus from searching rm k is

1 x µ k -µ j j + p j -p k ∂ 2 -c, x = µ j j -p j + p k µ k
The integral represents the expected incremental benet from searching another time given his current oer u ij , and the second term is the search cost. The consumer will sample rm k i the benet exceeds the cost, that is, the value of the above is larger than zero, and make a purchase of the current good otherwise. Let ˆ j solve the following equation :

1 x µ k -µ j ˆ j + p j -p k ∂ 2 = c, x = µ j ˆ j -p j + p k µ k (5)
The integral of ( 5) decreases in ˆ j , so the solution for ˆ j exists uniquely and is a threshold determining the searching rule. If j > ˆ j , the expected incremental surplus is less than the search cost so that the consumer purchases the good and leaves the market. If j < ˆ j , the expected incremental surplus exceeds the search cost so that the consumer searches the second rm. Solve (5) for the threshold :

(

5) ⇒ (µ k -µ j ˆ j + p j -p k ) 2 4µ k 1 x = c ⇒ ˆ j (p j , p k ) = µ k + p j -p k -2 √ µ k c µ j (6)
First, every time consumers have seen a higher price posted by the rst rm, they are more willing to search for a lower price from the second rm. For this reason, (6) increases in current price and decreases in the expected price of the second rm. Second, (6) decreases in consumer heterogeneity with respect to the current product and increases in the others' under a moderate search cost, because more consumers are likely to receive very good match and less are attracted by the other if the current oer is more consumer heterogeneous. In case the second product is more consumer heterogeneous, it leaves consumers a larger room (possibility) to gain from searching that rm and more consumers will move on. Last, (6) decreases in search cost.

Suppose there is a level of search cost that even the consumer receiving the poorest match from the rst rm is indierent between searching and not searching. Dene this search cost by c max , and solve for c max .

c max = min j µ k + µ j + p j -p k 2 2 /µ k , j ∈ {N, G} and j = k
Suppose there is the other level of search cost that even the consumer receiving the best match from the rst rm is indierent between searching and not searching. Dene this search cost by c min , and solve for c min .

c min = max j µ k -µ j + p j -p k 2 2 /µ k , j ∈ {N, G} and j = k
We are not interested in any equilibrium satisfying c > c max or c < c min . The reasons are as follows. When c > c max , no consumers search the second rm. 5 When c < c min , each consumer searches each rm, then product orderings do not play any role. In the two following sub-cases, we look for equilibria (if they exist) when some positive proportion (less than proportion one) of consumers search.

Generic-Niche Ordering

According to the optimal stopping rule, those having received better match than ˆ G purchase at the rst rm immediately. These consumers compose the fresh demand for the rst rm. Other consumers have received worse match than ˆ G , so they check out the second rm (niche rm).

After these consumers have sampled the second rm, some return to the rst rm because they nd out that goods from the generic rm actually oer them larger surplus, and the others 5. c max is very large. In the polar case, the rst rm can charge price zero in order to prevent consumers from searching the second rm, who in turn also charges the price zero in order to attract consumers. Then c > c max is equivalent to c > min j ((µ k + µ j )/2) 2 /µ k , j = k. purchase at the niche rm. As in many ordered search models, the demand of the rst rm (generic rm) is composed of two parts : fresh demand and returning demand. And the demand of the last rm (niche rm) consists of only the fresh demand.

To illustrate the two sources of demand, we depict the demand shares in the following gure : the vertical line splits the whole demand into two parts. The fraction on the right-hand side represents the fraction of consumers purchasing at Firm G immediately (fresh demand of the rst rm). On the left-hand side are those consumers who decide to search, which are further divided into two parts by the curve u iN = u iG . Above the curve, consumers choose Firm N.

Below the curve, consumers choose Firm G.

6 If Firm G deviates to a price p while Firm N sets 6. The intercept of u iN = u iG on the N axis can be either greater or smaller than -1. Therefore the demand functions have dierent forms according to

p G -p N -µ N µ G . In case max( p G -p N -µ N µ G , -1) = p G -p N -µ N µ G , the consumer receiving ( N , G ) = (-1, -1) purchases the niche product. In case p G -p N -µ N µ G < -1, consumers who receive poor match (between -1 and p N -p G -µ G µ N
) from the niche product switch to the generic product. We look for equilibria satisfying respectively

p G -p N -µ N µ G > -1, and p G -p N -µ N µ G < -1. Since the unique existing equilibrium satises p G -p N -µ N µ G
< -1, we only draw this case in Figure 2 for simplicity.

the price right to the expectation, its demand function is given by :

D G≺N G (p N , p) = 1 2 1 -min ˆ G (p, p N ), 1 f resh demand + 1 4 min(ˆ G (p,p N ),1) max -1, p-p N -µ N µ G 1 + p N -p + µ G µ N ∂ returning demand (7)
From the above demand function, the rst term corresponds to fresh demand, and the second term corresponds to returning demand. And if the niche rm deviates to another price p while the generic rm sets the price to the expectation, its demand is given by,

D G≺N N (p, p G ) = 1 2 min(ˆ G (p N ,p G ),1) max -1, p G -p-µ N µ G 1 - 1 2 1 + p -p G + µ G µ N ∂ + max 1 2 1 + p G -p -µ N µ G , 0 (8)
Both ( 7) and (8) depend on max(-1, p G -p N -µ N µ G

). In the appendix, we show that there is no

equilibrium satisfying p G -p N -µ N µ G > -1.
Then we try to nd out whether there is any equilibrium

satisfying p G -p N -µ N µ G < -1. The derivative of D G≺N G (p N , p) in p is, ∂ ∂p D G≺N G (p N , p) = - 1 2µ G + 1 4µ G 1 + p N -p -µ G µ N And the derivative of D G≺N N (p G , p) in p is, ∂ ∂p D G≺N N (p G , p) = - µ N + µ G -p N + p G -2 √ µ N c 4µ G
In any price equilibrium, each rm should have no incentive to deviate from the price expectations. From the rst-order conditions, we nd equilibrium prices. Let ∆ = p N -p G . For the generic rm,

p G = (4µ N (µ G + c + ∆) -((µ N -µ G ) + ∆) 2 ) 2(µ N + µ G -∆) (9)
For the niche rm,

p N = (µ N + µ G -∆ + 2 √ µ N c) 2 (10)
Consider the generic rm's prot function. When prices are set according to ( 9) and (10),

we have, Lemma 1. In the generic-niche ordering, the prot of the generic rm increases in search cost.

Proof. To see this, rewrite the niche rm's best response (10) into the following form.

3 2 p N = µ N + µ G + p G + 2 √ µ N c 2
Suppose that search cost increases. If the generic rm does not change its price p G , the above suggests that the niche rm's best response is to increase its price p N . If p N increases and p G remains at the same level, the generic rm faces larger demand according to the optimal search rule, and thereafter higher prot. Therefore, the generic rm can at least keeps its price at the same level and still earns larger prot. Any change of the price is intended only to further increase its prot.

When search cost is small enough, the generic rm's prot can fall below its prot in the perfect information case. We may nd a unique level of search cost, where the generic rm gets the same prot as if in the perfect information case. The curves in the following gure draw the prot when prices are set according to (9) and (10). It is clear that such a level of search cost exists.

Fig. 2.3: Prot patterns in the generic-niche ordering (µ G /µ N = 1/2).

Dene two levels of search cost : c , c , where c is the maximum level of search cost at which the generic rm earns weaker lower prot than the perfect information case, and the use of c will be explained in the following text. The gure draws both search cost levels. Proof. When search cost is below c , the generic rm prefers the "perfect information" prot.

c = max{c|Π G≺N G (∆ * (c ), c ) ≤ µ N /2, c ≤ µ G }, c = 1 µ N µ N -µ G 2 2 (11)
The niche rm prefers the "perfect information" prot as well, because otherwise its prot cannot exceed the prot of the generic rm. The remaining question are : once the generic rm charges p G = µ N as in the "perfect information" case, 1) whether it is also optimal for the niche rm to charge p N = µ N ; 2) whether each consumer searches each rm under this price setting.

The answers lie in whether c < min{c , c }. Suppose that both rms set p N = p G = µ N .

According to the optimal search rule, each consumer visits the niche rm if

ˆ G (µ N , µ N ) > 1 ⇒ c < c (12)
Then consider that the generic rm sets p G = µ N and the niche rm sets the price according to (10

) (p N = 2µ N +µ G +2 √ µ N c 3
). The optimal search rule shows that,

ˆ G µ N , p N > 1 ⇒ c < c (13)
Therefore, when search cost is lower than min{c , c } and the generic rm sets price to µ N , each consumer visits the niche rm no matter whether the niche rm sets the price at µ N or according to (10). Then it is logically to think that the niche rm will set the price to µ N . This price pattern is illustrated in the following gure. Proposition 2. In the generic-niche ordering, two types of equilibria exist and depend on the parametrizations.

When search cost is higher than min{c , c }, prices are set from ( 9) and (10). The rst rm sets a lower price than the second rm.

When search cost is lower than min{c , c }, equilibrium prices are given by p N = p G = µ N .

Prices p j (µ N , µ G , c) and price dierences ∆(µ N , µ G , c) are homogeneous of degree one.

Proof. See the appendix.

There are at least three ways to explain why the second rm sets a higher price than the rst rm : rst, the demand of the rst rm is more sensitive to price change, because each consumer observes price deviations of the rst rm, but only a proportion of them observe price deviations of the second rm. Second, the second rm knows that consumers follow the optimal stopping rule. Therefore, those coming to the second rm must have received match lower than ˆ G from the rst rm. They are willing to pay more, in order not to return to the rst rm. Third, more consumers are likely to obtain good match from niche products than generic products.

Niche-Generic Ordering

When rms are arranged in the niche-generic manner, consumers must visit the niche rm at the beginning. The search pattern is similar to the previous section. From the optimal stopping rule, the second rm (Firm G) knows that those who get to them must have received poorer match than ˆ N from the niche rm. As in the previous case, each demand function can be written in dierent forms according to the sign of p N -p G + µ N -µ G . We employ the same trick by assuming the sign of p N -p G + µ N -µ G and verifying the sign in the equilibrium. An increase (1,1) of p N can be observed by each consumer immediately : more consumers search the generic rm and more consumers choose the generic product after having seen the two products. However, consumers cannot observe Firm G's price directly. Hence a deviation by the generic rm changes only the decisions of those who observe it. If the generic rm deviates to a new price p while the other rm keeps its price to the expectation, its demand function is the conditional probability of oering a consumer higher surplus than Firm N :

D N ≺G G (p N , p) = 1 2 ˆ N (p N ,p G ) max -1, p N -p-µ G µ N 1 - 1 2 1 + p -p N + µ N µ G ∂ + max 1 2 1 + p N -p -µ G µ N , 0 The derivative of D N ≺G G (p N , p) with respect to p is, for p N -p-µ G µ N < -1, ∂ ∂p D N ≺G G (p N , p) = - µ N + µ G + p N -p G -2 √ µ G c 4µ N For p N -p-µ G µ N > -1, ∂ ∂p D N ≺G G (p N , p) = - 1 2µ N + p G -p + 2 √ µ G c 4µ N µ G At p = p G , the above is - √ µ G ( √ µ G - √ c) 2µ N µ G
. To ensure that the price elasticity of demand is negative, we need c < µ G . Accordingly, the demand function of Firm N is 1 minus the demand of Firm G :

D N ≺G N (p, p G ) = 1 2 1 -ˆ N (p G , p) f resh demand + 1 4 ˆ N (p G ,p) max -1, p-p G -µ G µ N 1 + p G -p + µ N µ G ∂ returning demand The derivative of D N ≺G N (p, p G ) with respect to p is, for p-p G -µ G µ N < -1, ∂ ∂p D N ≺G N (p, p G ) = - 1 2µ N + 1 4µ N 1 + p G -p -µ N µ G For p-p G -µ G µ N > -1, D N ≺G N (p, p G ) = -1 2µ N
. By the rst order conditions we obtain the following proposition.

Proposition 3. In the niche-generic ordering, two types of price equilibria exist and depend on the parametrizations. Let ∆ = p N -p G .

Equilibrium

1 satises p N = µ N + c -∆, p G = µ G (µ N + ∆ -c)/(µ G - √ µ G c) and (p N - p G ) + (µ N -µ G ) > 0. Equilibrium 2 satises p N = (4µ G (µ N + c -∆) -(∆ + (µ N -µ G )) 2 )/2(µ G + µ N + ∆), p G = (µ N + µ G + ∆ + 2 √ µ G c)/2 and (p N -p G ) + (µ N -µ G ) < 0.
The rst rm sets a lower price than the second rm. Prices p j (µ N , µ G , c) and price dierences

∆(µ N , µ G , c
) are homogeneous of degree one.

Proof. See the appendix.

We illustrate the equilibria and their parameter ranges in the following gure. and c, there exist two dierent sets of equilibrium prices. The second rm always charges a higher price than the rst rm.

7 By increasing (reducing) para- meters in the same proportion, the equilibrium prices (price dierence) and prot change again in the same proportion. Hence, we say that these functions are homogeneous of degree one.

Henceforth in this section, we focus the discussion on the properties of equilibrium 1, which is relevant when taste intensities for the two products are dierent enough. Larger search cost discourages consumers from searching and grants each rm more market power. By lowering its price the rst rm actually further discourages consumers from searching the subsequent rm.

The second rm learns from the optimal stopping rule, that the consumers searching the second rm have obtained match lower than ˆ N (p N , p G ), which is decreasing in the search cost. Hence the upward pressure in the generic product price, which is searched second, is very strong and,

as we now see it is stronger than the upward pressure on the niche product's price. Solving for

∆ = p N -p G we have ∆ = -µ N √ c -c √ c + 2c √ µ G 3 √ µ G -2 √ c
Its derivative with respect to c is,

-(3µ N + 13c + 12 √ µ G c) √ µ G + 4c √ c 2(3 √ µ G -2 √ c) 2 √ c ,
Corollary 1. In the niche-generic ordering, the impact of consumer heterogeneity and search cost on prices and prot can be summarized from the below table.

Tab. 2.2: Eects on prices and prot in the nichegeneric ordering.

p niche p generic Π niche Π generic µ N + + + + µ G - - - - search cost + + + ambiguous
The above results show that an increase in taste heterogeneity for the niche product is consistent with the standard market power eect, whereby more heterogeneity leads to higher prices for both rms. However, more taste heterogeneity for the generic product has the opposite impact on prices. If the second product is more consumer heterogeneous, a consumer is more likely to visit the second rm because by doing that he expects a larger probability of obtaining a better match than with the rst rm. As a result, more consumers compare the two prices.

Demands for the two rms become more elastic. In short, more consumer heterogeneity regarding the second product may lead to more intense competition. This is analogous to the result in [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF], who nd that, in a random search setting, more taste heterogeneity may lead to lower prices. Although the additional incentive to search for the second product is also present in the generic/niche ordering, it is outweighed by the increased market power aorded by more taste heterogeneity pertaining to the niche product.

A related argument that also relies on the consumers' incentive to search the second rm can be made by comparing the generic-niche ordering with the niche-generic ordering in terms of the percentages of searchers. The prediction is that consumers have higher reservation match if rms are in a generic-niche ordering, that is when the second product exhibits more consumer taste heterogeneity : (ˆ G > ˆ N ).

Lemma 3. The generic-niche ordering induces more searching than the reverse order if search cost is not too large (µ G c) and µ G /µ N < 12/13. 8

Proof. See the appendix.

Once more consumers sample both products, they make comparisons and buy the better oer. The direct outcome is more intense competition, which should be seen from the prices.

p N ≺G N -µ N = (µ N + c -∆ N ≺G ) -µ N = c -∆ N ≺G > 0 8
. By simulations, we nd that more consumers search in the generic-niche ordering for µ G /µ N > 12/13. We also nd that the reverse can hold when µ G is large and search cost is very large. For example, µ G = 0.8, c = 0.75.

N

in the niche-generic ordering. We next show that the higher price level leads to higher joint prots.

Analysis of Prot

From the analysis above, it should be expected that having the niche rm in front of the other will give rise to higher industry prot than the other way around. The following results conrms that this is the case.

Proposition 4. There exists a cut-o value μ such that the industry prot is higher in niche-

generic ordering if µ G µ N < μ. 9
Proof. The main proof is in the appendix. The idea is as follows. Previous sections have found that ∆ G≺N > 0 and ∆ N ≺G < 0. Provided that the market is covered, consumers pay at least p N ≺G N in the niche-generic setting and p G≺N N in the reverse setting. In case p N ≺G N > p G≺N N , the joint prot will be larger in the niche-generic ordering. Therefore which ordering induces higher joint prot can be seen on which ordering induces higher price of the niche rm.

In the previous literatures on ordered search, the rm obtains larger prot if it is visited by consumers rst. In [START_REF] Armstrong | Prominence and consumer search[END_REF], the prominent rm earns more than the nonprominent rms. In [START_REF] Zhou | Ordered search in dierentiated markets[END_REF], rm k receives a higher prot than rm k + 1. If all rms are identical, they predict that the rm is willing to pay more to be either prominent or placed in a higher rank. This is also our result for the niche rm.

9. We focus on two most common equilibria with µ G /µ N < 12/13, which requires that rms always choose extreme designs as in [START_REF] Johnson | On the simple economics of advertising, marketing, and product design[END_REF] and [START_REF] Larson | Niche products, generic products, and consumer search[END_REF].

Proposition 5. The niche rm receives a higher prot in the niche-generic ordering than the reverse ordering if µ G /µ N < 12/13. Proof. See the appendix.

The result is intuitive. In the niche-generic ordering, less consumers search the second rm according to the optimal stopping rule. Therefore, prices are higher. Meanwhile, the niche rm captures a larger market share as opposed to its demand share in the generic-niche ordering. This result remains valid for µ G /µ N ≥ 12/13.

When search cost is large enough, we nd that the generic rm receives larger prot in the generic-niche ordering. However the locational preference of the generic rm is not so straightforward if search cost is small. To illustrate this, we show the generic rm's prot in the gure.

When µ G µ N , the generic rm prefers the rst position, as in most ordered search literatures because the dierence in price level from two orderings is dominated by the fact that rm G takes larger market share in the generic-niche ordering. When µ G is small, less consumers purchase the generic product in the niche-generic ordering as opposed to the reverse ordering, but prices are high enough so that Firm G actually gains more from the subordinate position.

Rational Consumers

Suppose that consumers can actually choose the order of search. In the new game, rms have some prior beliefs about the manners in which consumers search rms. In stage 1, rms set prices. In stage 2, consumers choose which rm to visit rst, and then make their search and purchase decisions as in previous sections.

We focus on two situations : either rms believe that each consumer visits the niche rm rst, or rms believe that each consumer visits the generic rm rst. We assume throughout, µ G /µ N < 12/13.

If rms believe that consumers search in the niche-generic manner, prices are set according

to p N = µ N + c -∆ and p G = µ G (µ N + ∆ -c) µ G - √ µ G c
The optimal stopping rule is determined by the threshold ˆ N (p N , p G ) from ( 6). Such an equilibrium exists as we now show.

Proposition 6. There exists an equilibrium where consumers search the niche rm rst.

Proof. We prove the above result by applying [START_REF] Weitzman | Optimal search for the best alternative[END_REF]. In [START_REF] Weitzman | Optimal search for the best alternative[END_REF], the optimal strategy is to open the box with the highest reservation price r j , j ∈ {N, G}, where r j satises,

∞ r j (U j -r j )dF j (U j ) = c, ( 14
)
where c is search cost as before. U j = µ j j -p j is the utility of the consumer, a random variable under distribution function F j (U j ). Since j is uniformly distributed on [-1, 1], U j is uniformly distributed on [-µ j -p j , µ j -p j ] with density 1/2µ j . Solve the above for r j .

r j = µ j -2 √ µ j c -p j r j increases in µ j because dr j dµ j = 1 - c µ j > 0
Knowing that p N < p G from proposition 3, it is clear that r N > r G . We apply the results in [START_REF] Weitzman | Optimal search for the best alternative[END_REF], and conclude that any consumer will search the niche rm rst.

When rms believe that consumers search in the generic-niche manner, prices are set according to the following

p G = (4µ N (µ G + c + ∆) -((µ N -µ G ) + ∆) 2 ) 2(µ N + µ G -∆) and p N = (µ N + µ G -∆ + 2 √ µ N c)/2, for c > min{c , c } (c , c are dened in (11)). For c ≤ min{c , c }, p N = p G = µ N .
According to [START_REF] Weitzman | Optimal search for the best alternative[END_REF], the optimal strategy for consumers is that, they visit the generic rm rst if r G > r N , and visit the niche rm rst if otherwise. So far we know µ N - deviations. We assume that consumers receive two units of xed surplus from each good and c = 1/5. We assume that all consumers are expected to search in the niche-generic way. Then we have noticed a 3.3%-27% drop in consumer surplus if one chooses the other way round (generic-niche). We now assume that all consumers are expected to search in a generic-niche way. Expected consumer surplus will change between 0% and 3.7% after choosing the other way round (for µ G µ N ≤ 0.85).

2 √ µ N c > µ G -2 √ µ G c and p G < p N . It
Take one step further. Since every consumer have no prior information about the products besides the expected prices, it is obvious that they will visit the same rm rst. In case the parameters are in the blue region in the above gure, rms will not expect consumers to search in the generic-niche way, because rms should have expected consumers to deviate. Finally if consumers search in the niche-generic way, then it is clearly not optimal for each rm to set the price as if in the generic-niche ordering.

The Outside Option

Suppose that rms in the duopoly case charge the price pair (p N , p G ). For rm j, the probability of receiving negative surplus from its good is Pr( j < p j /µ j ) = F (p j /µ j ). In the following model, we assume that zero surplus is the outside option for any consumer, so he does not buy the good from rm j if j < p j /µ j . Further, the proportion of consumers receiving negative surplus from both goods is the multiplication of two probabilities :

F (p N /µ N )F (p G /µ G ).
These consumers quit the market without buying anything and the surplus is zero minus the search cost. Hence, the lowest possible surplus for any consumer is -c. The below reiterates the outside option in detail.

Assumption 4. any consumer purchases a good only if the good oers him positive surplus :

u ij ≥ 0 for j ∈ {N, G}. (outside option)
The question remains when for consumers to drop out. There are two possibilities : rst, some consumers sample the rst rm and decides to drop out immediately. Second, the same consumers decide to drop out only after sampling both rms. We can show that the existence of the rst possibility depends essentially on the level of search cost (in 6.11-6.12). If the search cost is too large, even those having received bad match decide to quit the market since otherwise they have to pay the large sampling cost only to nd out if the second good ts them. To simplify our analysis, we assume away the rst possibility. So the present model has three new features : rst, market is not always covered. Second, prices are lower to avoid negative surplus on the consumers' part. Therefore, consumer surplus is more often a positive number unless he drops out without any purchase after searching. Third, those consumers receiving negative surplus from the rst rm will search the second rm so that we avoid the existence of consumers who do not purchase any good and do not search the next rm. To summarize the new restriction on the model, we impose the following condition.

Assumption 5. c ≤ min{µ j /16},j ∈ {N, G}. Therefore anyone receiving negative surplus from the rst rm searches the second rm. (search cost)

Compared with our previous assumption that c < µ G , assumption 5 is very restrictive but intuitive. If the threshold ˆ is lower than the point that the consumer obtains zero surplus, then any consumer receiving positive surplus from the rst rm will purchase immediately and a fraction of those receiving negative surplus do not buy and do not search. The second rm, can decrease the price to attract this relatively inelastic demand since these consumers do not buy from the rst rm because of negative surplus, until the the matching threshold rises above to the point the consumer obtains zero surplus. Then follows the assumption 5.

Competing Firms

For the most of this paper, we have made an assumption that sellers are competing : shops leasing spaces from commercial centres, rms selling goods on EBay. A third party determines the ordering of rms in the rst step, then competing rms simultaneously choose prices. The market is divided into ve parts from each ordering (see details in appendix c).

Generic-Niche Ordering

In the generic-niche order, if the niche rm keeps its price to the price expectation p N and the generic rm deviates to a new price p, the demand function of the generic rm is given by,

DG≺N G (p N , p) = (1 -min(ˆ G , 1))/2 + min(ˆ G ,1) p/µ G P r(u iG > u iN )f ( G )∂ G /4 (15)
The demand function DG≺N G (p N , p) consists of two parts : the rst part (1/2)(1-min(ˆ G (p, p N ), 1))

corresponds to the fraction of consumers who stop and purchase immediately at Firm G, and the second part corresponds to the fraction of consumers who have sampled both rms and decide to return to Firm G. Deviations by Firm G have impacts on both parts because it is observed by each consumer.

DG≺N

N (p, p G ) = ˜ N p/µ N P r(u iN > u iG )f ( N )d N /4 + (1 -˜ N )(1 + min(ˆ G , 1))/4 (16)
The demand DG≺N N (p, p G ) consists of merely visitors for the rst time : the rst part of DG≺N

N (p, p G ) corresponds to demand for < ˜ N ; the second part of DG≺N N (p, p G ) is F (ˆ G )(1 -F (˜ N )) repre-
senting those more into the niche good, where

˜ i (p, p i , p j ) = µ j min(ˆ j (p i , p j ), 1) + p -p j µ i
and its value should be smaller than or equal to the upper bound of the distribution. Any deviation by Firm N can only be observed by those who search it. In an equilibrium, no rms have incentives to deviate. The FOC deliver the equilibrium prices :

p G = (µ G + p N -p G + c) - (p N + µ N ) 2 4µ N and p N = µ N + µ G -p N + p G + 2 √ µ N c 2 - (p G + µ G ) 2 2(µ N + µ G + p G -p N -2 √ µ N c)
Even after dening ∆ = p N -p G , there is no closed-formed expression for ∆, which is the main reason that our analysis stops at the numerical stage. If c = µ N 16 , the above yields the following solution : each rm behaves like a monopolist.

p N = µ N 2 and p G = µ G 2 If p N = µ N 2
, the optimal stopping rule becomes

ˆ G (p N , p G ) = µ N + p G -p N -2 √ µ N c µ G = p G µ G

Niche-Generic Ordering

In the niche-generic order, if the generic rm keeps its price at the price expectation p G and the niche rm deviates to a new price p, the demand function of the niche rm is given by,

DN≺G N (p, p G ) = (1 -min(ˆ N , 1))/2 + min(ˆ N ,1) p/µ N P r(u iN > u iG )f ( N )∂ N /4 (17)
The demand function DN≺G

N (p, p G ) consists of : the rst part (1/2)(1 -min(ˆ N (p, p G ), 1)) cor-
responds to fresh demand of Firm N, and the second part corresponds to returning demand.

DN≺G

G (p N , p) = ˜ G p/µ G P r(u iG > u iN )f ( G )d G /4 + (1 -˜ G )(1 + min(ˆ N , 1))/4 (18)
The demand DN≺G G (p N , p) consists of fresh demand. Deviations by the niche rm can be observed directly by each consumer, but any deviation by the generic rm can be observed only by those who search that generic rm. In an equilibrium, no rms deviate. The FOC deliver the equilibrium prices :

p N = (µ N -p N + p G + c) - (p G + µ G ) 2 4µ G and p G = (µ N + µ G + p N -p G + 2 √ µ G c) 2 + (p N + µ N ) 2 2(µ N + µ G + p N -p G -2 √ µ G c) If c = µ G 16 , p N = µ N 2 and p G = µ G
2 are the solutions of the above. The optimal stopping rule becomes,

ˆ N (p N , p G ) = µ G + p N -p G -2 √ µ G c µ N = p N µ N When search cost is larger than µ G 16 , ˆ N (p N , p G ) < p N µ N
. There are some consumers who do not purchase at the rst rm, and who do not search the second rm.

The Merged Firm

Occasionally rms do not make price decisions independently. The rst possibility is that, the rm is a two-product rm. In practice, lots of rms develop more than one product. For instance, a restaurant typically has plenty of dishes to oer. The second possibility is that, rms can cooperate on their price decisions and share the joint prot. There are two stages in the game. In the rst stage, the rms (or the multi-product rm) choose the ordering. In the second stage, the rms (or the multi-product rm) choose prices. From either ordering, consumers observe the price of the rst rm directly and make their search decisions after. In the generic-niche ordering, the expected joint prot is a function of the price pair (p G , p) given consumers' belief about p N :

ΠG≺N col (p G , p) = p G DG≺N G (p G , p) + p DG≺N N (p G , p) (19)
In the niche-generic ordering, the expected joint prot is a function of the price pair (p, p N )

given consumers' belief about p G :

ΠN≺G col (p, p N ) = p DN≺G G (p, p N ) + p N DN≺G N (p, p N ) (20)
Provided a price pair (p N , p G ), the market is divided in the same way as Figure 16. However, the two forms, along with the previous cases, are found to have quite dierent pricing patterns.

Prices

Firstly, prices are lower (more than 50%) if we apply assumption 4 to the model. This result is not surprising because with high prices only few consumers buy ; one rm can deviate to a lower price and increases its prot.

Secondly, rms should have charged the same price since the expectation of the match is identical for each rm and each consumer. However from some ordered search models [START_REF] Armstrong | Prominence and consumer search[END_REF] and Zhou ( 2011)), the rm that is searched at the beginning charges a lower price than the rest. We call this pricing pattern distortionary pricing. We observe similar patterns from our duopoly model without the outside option. The distortion is primarily caused by search cost because it enforces the ordered search and prevents a consumer from getting better match. If the search cost is larger, the price dierence becomes larger, which further discourages consumers from searching. The distortion can also be caused by competition because the rst rm intentionally lowers down the price to capture fresh demand. Accordingly the generic-niche ordering which induces more intense competition results in more distortion than the reverse ordering.

If the model considers the non-covered market, the price distortion either softens or completely disappears. Under duopoly competition, we still observe that p N -p G is smaller under the niche-generic ordering at small search cost. It completely disappears for the merged rm, which is from the fact that the multi-product rm internalizes the competition. From our numerical results, the price of the second good is not necessarily larger than the price of the rst good.

For example, in the niche-generic ordering where µ G /µ N = 0.8, p N ≈ 0.54 and p G ≈ 0.42. In the perfect information case, this price pattern (rms charge the same price) completely disappears.

Percentage of Searchers

Price distortion results in the distortion of consumer search. In the perfect information case, there is clearly no distortion because price distortion does not play any role except lowering down its own prot. In many random search models [START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF] and [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF]), rms are identical and there is no distortion. When rms dier in order, they charge dierent prices, consumers no longer search only for what match them better, but they consider also the price dierence.

When µ G /µ N is too small and c < c min , all consumers sample every rm as in the perfect information case. In general, percentage of searchers decreases in µ f irst /µ second as (6) predicts, taking into account of the price dierence. As a summary, the niche-generic ordering induces less searching activities.

Here is a reference with huge search cost and no outside options. When µ G /µ N = 3/5 and

Industry Prot

If consumers consistently search more intensively in the generic-niche ordering, then products must be also more compared. Firstly, more comparison results in more intense competition and lowers down prices and industry prot. Secondly, more search infers better match quality, then more surplus can be extracted by rms through prices. Thirdly, lower prices actually contribute to more purchases (less drop-os). As for industry prot, the rst eect is a direct reduction of prot, and the rest two are indirect and increase prot. Choosing product ordering is a trade-o between price and quantity.

Figure 12 compares industry prot for models with the outside option. The horizontal axis stands for the consumer heterogeneity ratio. The vertical axis stands for the joint prot.

Even though more consumers purchase goods and the market for each rm can be sometimes Fig. 2.12: Industrial prot comparison under models with the outside option.

better targeted in the generic-niche ordering, Figure 12 shows that the niche-generic ordering does lead to a higher joint prot. Therefore the direct eect from softened competition dominates the rest two eects.

First, this result holds for the duopoly model, and the model for a multi-product rm. The numerical results are solid wherever c min < c < µ j /16 for j ∈ {N, G}. Second, curves are upward sloping, which suggests that the joint prot increases in the general heterogeneity of consumer taste. Last but not least, joint prot is slightly higher under collusion.

Welfare Consequences

Even though the previous model suggests to us that the niche-generic ordering increases industry prot, the primary issue is that the market is covered therefore rms tend to set very high prices and from equilibrium prices consumers typically obtain negative surplus. Moreover, the previous parts are silent about consumer surplus and welfare. We try other models to test the solidarity of our previous results. The following subsections concentrate on two types of models : in the rst type, two rms are competing and the market is covered. We focus on three sub-cases : perfect information, generic-niche and niche-generic. The second family discusses the same set of sub-cases under the uncovered market : perfect information, generic-niche and niche-generic. And for each sub-case, we briey discuss two situations. In the rst situation, two rms are competing. In the second situation, there is a multi-product rm. The total nine cases shed light on how surplus is allocated between rms and consumers, and how product orderings have impacts on social welfare.

Consumer Surplus

Recall that a consumer's surplus is its utility minus the search cost he has incurred.

u ij -kc (21)
k is the number of times the consumer has searched. After visiting the rst rm, he decides whether to invest some search cost to visit the next rm. If so and the match of the new product is higher, he takes the new one. If not, he sticks to the old one and pays the search cost. The search cost is sunk. If both products deliver negative utilities to the consumer, he chooses not to buy anything.

Changing the ordering causes at least three eects from prices, product match and search cost. Under the niche-generic ordering, on average less consumers visit the next rm, suggesting less incurred search cost. Fewer searches dampen competition and both rms set higher prices.

Consumers pay more and more consumers drop o. On the contrary under the generic-niche ordering, more consumers search. Prices are lower and consumers pay more search cost, which is as if some parts of search cost are converted into prices and two orderings produce trade-os between prices, product match and search cost : either one searches more and pays less, or one searches less and pays more.

Fig. 2.13: Surplus comparison under covered market

If the market is covered, the consumer surplus shows one pattern. Consumer surplus is higher in the genetic-niche ordering. The reasons are that prices are much more higher in the niche-generic ordering, and on average, more consumers search and obtain better match in the genetic-niche ordering.

The duopoly model with the outside option shows two patterns. First, consumer surplus increases in the overall consumer heterogeneity because the probability of obtaining a high match is larger when products are more consumer heterogeneous.

10 Second, the niche-generic ordering can induce higher consumer surplus than the other ordering when the search cost is 10. When we do not consider the prices, the probability of receiving lower matches than η from both rms is (1. Duopoly competition. 2. The merged rm)

(1 + η/µ N )(1 + η/µ G )/4, which decreases in both µ N and µ G .
In any case when welfare comes mainly from consumers' satisfactions from their match, prices are transfers and the only playing factor is how consumers are matched under dierent circumstances. And the two main factors aecting product match are the search cost and the price distortion. In our original model without the outside option, both the search cost and the price distortion aect consumers' search decision in a detrimental way for the social welfare. For instance under Firm j ≺ Firm m , consumers should have chosen Firm j in case j > (µ m /µ j ) m but not j > ˆ j (p j , p m ). The search cost and the price distortion enter the searching threshold in a way that distorts the optimal search rule and prevents consumers to obtain their best match.

From this reasoning, we are not surprised to see that the product ordering (the generic-niche ordering) which induces more distortion lowers down the social welfare. And this pattern is consistent in both the competition and collusion models, with and without the outside option.

Discussion

Implementation

In practice, product ordering is usually set by a third party. The third party can be an on-line store or a multi-product rm. In these cases, the third party chooses the product ordering which maximizes its prot, which is usually a share of the industry prot. From proposition 4 and some numerical results, we know that products are more often arranged in the niche-generic manner. If the third party is a social planner, it chooses the niche-generic ordering. Last, the product ordering can be determined by an auction.

Niche Market

According to the denition, a niche market is a small subset of market. Hence the niche product normally satises merely the needs of a certain minority group. However in our model, the niche rm usually accounts for the majority of demand. Then the niche rm becomes actually the mass rm and vice-versa.

One solution is allowing varying qualities. Then the niche product is the one with lower quality but higher preference intensity, and a generic product is the one with higher quality but lower preference intensity. Then which ordering implies larger industry prot depends on the trade-o between anticipated quality dierence and the willingness to search for better match.

But the new model becomes less tractable.

Since the niche rm gains more than the generic rm in the original model, perhaps an easier alternative is to let more niche rms enter the market and keep the number of generic rms small.

In practice, it is dicult for rms to target a broad population and increasing competition drives many rms into niche markets, partitions the market into many segments, makes the long-tail.

Conclusion

This paper tries to address the problem of optimal product ordering. There are two types of rms in the market, the niche rms and the generic rms. If the search cost is zero, rms are setting the same price and obtain the same amount of prot. If the search cost is positive, the second rm always sets a price larger than the rst rm. The niche-generic ordering outcome augments industrial prot. The same ordering is preserved by the strategic consumers if rms believe them to visit in the niche-generic manner. Numerical examples show that the nichegeneric ordering delivers higher consumer surplus only under a large search cost in a duopoly model with the outside option. Finally, numerical examples show that social welfare is almost always higher under the niche-generic way, because the same way has the least price distortion.

The major limitation is the uniform distribution used to simplify our analysis. Search models without assumptions about the distribution function are often-times not tractable. A usual assumption is the log-concavity of the density function. So far, there have been no papers applying this condition in search models with product heterogeneity.

In section 6, the model assumes that search cost is low enough that the situation never occurs when some consumers do not search the second product and do not buy the rst product. If we relax this assumption for the two rms (c > max{µ j /16}), the model degenerates and the two rms set prices to monopoly prices. If only one assumption is relaxed, then there are monopoly prices only in the niche-generic ordering. It remains to see which ordering implies larger industry prot.

The paper restricts each consumer to unit demand. The model does not capture the fact that consumers are tempted to purchase more than one product upon dierent search paths. The impulsive purchase, arising from a mixture of socio-economic and psychological eects, not only triggers new incentives for rearrangements in the usual sense, but also opens up the possibility of bundling.

A Appendix A

A.1 Proof of Proposition 1

Proof. We rst look for equilibrium satisfying c=0) , we obtain the equilibrium prices as the following,

p N -p G +a(µ N -µ G ) > 0. The derivative of D G (p N , p; c = 0) with respect to p is : D G (p N , p; c = 0) = - 1 µ N a -a f µ G G + p N -p µ N f ( G )d G (A.1) From the FOC conditions p G = -D G (p N ,p G ;c=0) D G (p N ,p G ;c=0) and p N = -D N (p N ,p G ;c=0) D N (p N ,p G ;
p G = µ N a -a F ( µ G G +∆ µ N )f ( G )∂ G a -a f ( µ G G +∆ µ N )f ( G )∂ G , p N = µ N 1 - a -a F ( µ G G +∆ µ N )f ( G )∂ G a -a f ( µ G G +∆ µ N )f ( G )∂ G Since f is symmetric, we have F (x) = 1 -F (-x). Change the integrating variable from G to -G .
p N can be rewritten as,

p N = µ N a -a F (-µ G G +∆ µ N )f ( G )∂ G a -a f ( µ G G +∆ µ N )f ( G )∂ G = µ N a -a F ( µ G G -∆ µ N )f ( G )∂ G a -a f ( µ G G +∆ µ N )f ( G )∂ G
Subtract the above two equations. Then ∆ = 0 is an evident solution of the below equation.

∆ = µ N a -a {F ( µ G G -∆ µ N ) -F ( µ G G + ∆ µ N )}f ( G )∂ G / a -a f ( µ G G + ∆ µ N )f ( G )∂ G Second, we look equilibrium satisfying p N -p G + a(µ N -µ G ) < 0.
The demand faced by the generic rm is,

D G (p, p N ; c = 0) = a p-p N -µ N a µ G F p N -p + µ G G µ N f ( G )∂ G
The derivative of the above with respect to p is,

∂ ∂p D G (p, p N ; c = 0) = - 1 µ N a p-p N -µ N a µ G f p N -p + µ G G µ N f ( G )∂ G
The rst-order condition states that,

p G = µ N a p G -p N -µ N a µ G F p N -p G +µ G G µ N f ( G )∂ G a p G -p N -µ N a µ G f p N -p G +µ G G µ N f ( G )∂ G
On the other hand, we have,

p N = µ N 1 - a p G -p N -µ N a µ G F p N -p G +µ G G µ N f ( G )∂ G a p G -p N -µ N a µ G f p N -p G +µ G G µ N f ( G )∂ G For p N -p G + a(µ N -µ G ) < 0, p N -p G < -a(µ G -µ N ) < 0.
From the above equations, this is to say that,

p N -p G = µ N 1 -2 a p G -p N -µ N a µ G F p N -p G +µ G G µ N f ( G )∂ G a p G -p N -µ N a µ G f p N -p G +µ G G µ N f ( G )∂ G < 0 which is equivalent to, a p G -p N -µ N a µ G F p N -p G + µ G G µ N f ( G )∂ G > 1 2
The above is exactly the expression of D G . Since p G > p N , it is not possible that rm G faces larger demand than rn N, which poses a contradiction.

A.2 Proof of Proposition 2

Proof. On one hand, we look for price equilibrium satisfying p N -p G + µ N -µ G > 0. Substitute ( 9) and ( 10) into each other and we obtain the following

Λ(∆) = 2∆ 2 -(3µ N + 3µ G + √ µ N c)∆ + (µ N -µ G ) 2 -2µ N c + (µ N + µ G ) √ µ N c
The price equilibrium satises Λ(∆) = 0. Note rst that the vertical axis of symmetry of this parabola is,

∆ = 3µ N + 3µ G + √ µ N c 4 > 0 Second, Λ(0) = (µ N -µ G ) 2 + (µ N + µ G ) √ µ N c -2µ N c > 0.
Therefore, the two roots of Λ(∆) = 0 are positive.

On the other hand, we look for price equilibrium satisfying p N -p G + µ N -µ G < 0, the solutions are the following equations :

p N = µ N (µ G -∆ -c)/(µ N - √ µ N c)
and p G = c + µ G + ∆. Substitute one into the other, we obtain

∆ = (c + √ µ G ) √ c -2c √ µ N 3 √ µ N -2 √ c Then p N -p G + µ N -µ G = (c + √ µ G ) √ c -2c √ µ N + (3 √ µ N -2 √ c)(µ N -µ G ) 3 √ µ N -2 √ c Normalize µ N → 1. Then, p N -p G + µ N -µ G = 3 + c -2µ G + 2µ G √ c -4 √ c 3 -2 √ c Let h(µ G , c) = 3 + c -2µ G + 2µ G √ c -4 √ c. It is clear that h(µ G , c) decreases in µ G . Therefore h(1, c) > 0 is a sucient condition for h(µ G , c) > 0. h(1, c) = (1 - √ c) 2 ≥ 0. Therefore we have p N -p G + µ N -µ G > 0, which is a contradiction. A.3 Proof of Proposition 3 Proof. Let ∆ = p N ≺G N -p N ≺G G .
Equilibrium 1 : We have the root satisfying

∆ 1 = 2c √ µ G -µ N √ c -c √ c 3 √ µ G -2 √ c For p N -p G + µ N -µ G > 0, we have (2c √ µ G -µ N √ c -c √ c) + (µ N -µ G )(3 √ µ G -2 √ c) > 0 which is equivalent to, ( √ µ G - √ c)(c - √ µ G c + 3µ N -3µ G ) > 0 For c - √ µ G c + 3µ N -3µ G > 0, we need either 13µ G < 12µ N or 13µ G ≥ 12µ N and c ∈ [0, c 1 ]∪[c 2 , µ G ), c 1 = √ µ G - √ 13µ G -12µ N 2 2 , c 2 = √ µ G + √ 13µ G -12µ N 2 2
Equilibrium 2 : We nd out that the price dierence of the rst equilibrium satises

∆ = (4µ G (µ N -∆ + c) -(µ G -∆ -µ N ) 2 ) 2(∆ + µ N + µ G ) - (µ N + µ G + ∆ + 2 √ µ G c) 2
Arranging it gives λ(∆) = 0, where

λ(∆) = 4µ G (µ N -∆ + c) -(µ G -µ N -∆) 2 -(µ N + µ G + ∆ + 2 √ µ G c)(∆ + µ N + µ G ) -2∆(∆ + µ N + µ G )
which is a quadratic function in ∆, with axis ∆ = -(6µ N + 6µ G + 2 √ µ G c)/8 < 0. Therefore, to prove both roots of λ(∆) = 0 to be negative, it suces to prove that λ(0) < 0.

λ(0) = -2(µ 2 N -2µ N µ G + µ 2 G ) -2 √ µ G c(µ N -2µ G c + µ G ) < 0
Hence the equation λ(∆) = 0 has only negative roots and ∆ = p N -p G < 0. Denote the roots of

λ(∆) = 0 as ∆ 11 and ∆ 12 (let ∆ 11 < ∆ 12 ). From λ(-(µ N + µ G )) = 4µ G (2µ N + c) > 0 we know that ∆ 11 < -(µ N + µ G ), in which case p N = 4µ G (µ N + c -∆ 11 ) -(∆ 11 + µ N -µ G ) 2 2(∆ 11 + µ N + µ G ) < 0
Therefore, we exclude ∆ 11 . For ∆ 12 to satisfy p

N -p G + µ N -µ G < 0, we must have λ(µ G -µ N ) = 4µ G (c - √ µ G c + 3µ N -3µ G ) ≤ 0, which is equivalent to 13µ G ≥ 12µ N and c ∈ [c 1 , c 2 ]
A.4 Proof of Corollary 1

Proof. Substitute the equations from proposition 3 into each other and we obtain that :

p G = √ µ G (3µ N - c)/(3 √ µ G -2 √ c
). Take its partial derivative in µ N , µ G and c, we obtain the following,

∂p G /∂µ N = 3µ G /(3µ G -2 √ c) > 0, ∂p G /∂µ G = - √ c(3µ N -c) √ µ G (3 √ µ G -2 √ c) 2 < 0 ∂p G /∂c = √ µ G (3µ N -3 √ µ G c + c)/ √ c(3 √ µ G -2 √ c) 2 > 0 Firm N's price, ∂p N /∂µ N = 1 + √ c/(3 √ µ G -2 √ c) > 0, ∂p N /∂µ G = - √ c(3µ N -c) 2 √ µ G (3 √ µ G -2 √ c) 2 < 0 ∂p N /∂c = (4c √ c + 3µ N √ µ G + 6µ G √ c -11c √ µ G )/2 √ c(3 √ µ G -2 √ c) 2 > 0 Firm G's prot, Π N ≺G G = √ µ G ( √ µ G - √ c)(3µ N -c) 2 2µ N (3 √ µ G -2 √ c) 2
. Its derivatives in µ G and µ N are given by the below.

∂Π N ≺G G /∂µ G = - √ c(3µ N -c) 2 ( √ µ G -2 √ c) 4µ N √ µ G (3 √ µ G -2 √ c) 3 < 0 ∂Π N ≺G G /∂µ N = µ G (9µ 2 N -c 2 )( √ µ G - √ c) 2µ 2 N (3 √ µ G -2 √ c) 2 > 0 Firm N's prot, Π N ≺G N = 1 2µ N (c + µ N - (2c √ µ G -µ N √ c-c √ c) (3 √ µ G -2 √ c)
)2 . Its derivatives in µ G and µ N are given by the below.

∂Π N ≺G N /∂µ N = 9µ G µ 2 N + µ 2 N c + 2c 2 √ µ G c -6µ 2 N √ µ G c -µ G c 2 -c 3 2µ 2 N (2 √ c -3 √ µ G ) 2 > 0
The above is positive and the below is negative for c < µ G . The niche rm's prot increases in the consumer heterogeneity of the niche rm, and decreases in the consumer heterogeneity of the generic rm.

∂Π N ≺G N /∂µ G = √ c(c -3µ N )(c √ c + µ N √ c -c √ µ G -3µ N √ µ G ) 2µ N √ µ G (2 √ c -3 √ µ G ) 3
In the last case, if the search cost is larger, more consumers are stuck by the rst rm, which further raises the market power of both rms and both prices. Therefore the prot of the rst rm increases, conrmed by the below.

A.5 Proof of Proposition 4

Proof. there exists a unique equilibrium for the generic-niche ordering, but two sets of equilibria for the niche-generic ordering. We focus exclusively on the case that rms always choose extreme designs :

µ G /µ N <
> p G≺N N . Let ϕ(µ N , µ G , c) = 2(3 √ µ G -2 √ c)(p N ≺G N -p G≺N N )
The expression of p G≺N N decreases in ∆ G≺N . Use 0 to replace ∆ G≺N in the expression of p G≺N N in ϕ(µ N , µ G , c) and obtain

ψ(µ N , µ G , c) = (3 √ µ G -2 √ c)(µ N -µ G + 2c -2 √ µ N c) -2 √ c(2 √ µ G c -µ N -c) (A.2) Then ϕ(µ N , µ G , c) > ψ(µ N , µ G , c). Take the second derivative of ψ(µ N , µ G , c) in µ G ψ µ G µ G (µ N , µ G , c) = -2c -3µ N -9µ G + 6 √ µ N c 4µ G √ µ G < 0
11. By increasing the values of µ N , µ G and c by a certain percentage, the equilibrium price and prot will increase by the same percentage. Therefore, the equilibrium prices and prots are homogeneous of degree one in search cost and consumer heterogeneity.

Because,

-3µ N -9µ G + 6 √ µ N c = -3(µ N + 3µ G -2 √ µ N c) < -3(µ N + 3µ G -2 √ µ N µ G ) < 0 Therefore ψ(µ N , µ G , c) is concave in µ G .
For each value of c and each µ G ∈ (c, μ], there exists a γ that

µ G = γc + (1 -γ)μ.
From the concavity of ψ(µ N , µ G , c), there is automatically

ψ(µ N , µ G , c) > γψ(µ N , c, c) + (1 -γ)ψ(µ N , μ, c) Normalize µ N → 1. Therefore, to prove ψ(µ N , µ G , c) > 0 as {µ G , c} ∈ (c, μ] × [c, c],
it is sucient to verify the signs of two polynomials :

ψ(µ N , c, c) = - √ c( √ c + 1)( √ c -3) > 0 and ψ(µ N , μ = 0.7, c) = 0.75 -3.62 √ c + 5.67c -2c √ c > 0.
Then we reach our conclusion that the niche-generic ordering induces higher industry prot than the reverse for 0 < µ G ≤ 0.7.

A.6 Proof of Lemma 1

Proof. this corollary is equivalent to saying that ˆ G > ˆ N . From (6), we dene a function,

ψ(µ N , µ G , c) = (µ N + p G≺N G -p G≺N N -2 √ µ N c)/µ G -(µ G + p N ≺G N -p N ≺G G -2 √ µ G c)/µ N
The above decreases in ∆ G≺N . Dene another function

ψ(µ N , µ G , c) = µ G (3 √ µ G -2 √ c)ψ(µ N , µ G , c)
We replace ∆ G≺N with η(µ N , µ G , c), which is the upper bound function for ∆ G≺N , and obtain

ϕ(µ N , µ G , c) = (3 √ µ G -2 √ c)(µ N (µ N -η -2 √ µ N c) -µ G µ G + 2µ G √ µ G c) -µ G √ c(2 √ µ G c -µ N -c) Then ϕ(µ N , µ G , c) > 0 is the sucient condition for ψ(µ N , µ G , c) > 0. First, ϕ(µ N , µ G , c) is a concave function in µ G because ϕ µ G µ G (µ N , µ G , c) = 16 √ c - 9(5µ G + 2c) 4 √ µ G - 3(µ N (4µ N -8 √ µ N c -3) + (9µ G -4.94 √ µ N c)) 16µ G √ µ G < 0 in case µ G c. Normalize µ N → 1. Let μ = 0.7. Then for each c, and each µ G ∈ (c, μ]. There must exist a γ that µ G = γc + (1 -γ)μ.
From the strict concavity of ϕ, we have that

ϕ(1, µ G , c) > γϕ(1, c, c) + (1 -γ)ϕ(1, μ, c) Therefore it suces to prove that ϕ(1, c, c) > 0 and ϕ(1, μ, c) > 0. Second, ϕ(1, c, c) = √ c(1 + c + √ 35 c -9 √ c)/4 > 0 while c ∈ [0, 0.135]. Third, ϕ(1, μ, c) ≈ 1.18 - √ c(2.95 -0.98 √ c -0.7 c) > 0 while c ∈ [0, 0.28].
We reach our conclusion that the niche-generic ordering induces less consumers to check out both rms.

A.7 Proof of Proposition 5

Proof. There are three parts in the proof.

(i) From p N ≺G N < p N ≺G G
, the niche rm's demand share is more than half. Hence the niche rm's prot is larger than (ii) Next, we will look for conditions for the below to hold when µ G ∈ [ 7 10 , 12 13 ] :

p N ≺G
Π N ≺G N -Π G≺N N = (µ N + c -∆ N ≺G ) 2 2µ N - (µ N + µ G -∆ G≺N + 2 √ µ N c) 2 (µ N + µ G -∆ G≺N -2 √ µ N c) 16µ G µ N > 0 (A.3) Since Π G≺N N
decreases in ∆ G≺N , we use zero to replace ∆ G≺N . Dene a function

Γ(µ G , c) = 8µ G (µ N + c -∆ N ≺G ) 2 -(µ N + µ G + 2 √ µ N c) 2 (µ N + µ G -2 √ µ N c) Hence Γ > 0 is a sucient condition for Π N ≺G N -Π G≺N N > 0.
First Γ is a concave function in µ G for c < 4 25 . Take the second derivative in µ G of the rst term and we obtain,

∂ 2 ∂µ 2 G 8µ G (µ N + c -∆ N ≺G ) 2 = 4 √ c(c -3)(6c 2 -24 √ µ G c -8c √ µ G c + 9µ G + 6c + 3µ G c) √ µ G (3 √ µ G -2 √ c) 4 < 0 Further - ∂ 2 ∂µ 2 G (µ N + µ G + 2 √ µ N c) 2 (µ N + µ G -2 √ µ N c) = -(6 + 6µ G + 4 √ c) < 0
Therefore it suces to show that Γ( 7 10 , c) > 0 and Γ( 12 13 , c) > 0 for c ∈ [0, 4 25 ]. 12 For the limit of space, we will show it in the on-line appendix.

(iii) Finally we consider the case for c > 4 25 and µ G ∈ [ 7 10 , 12 13 ]. Dene,

Γ(µ G , c) = 8µ G (µ N + c) 2 -(µ N + µ G + 2 √ µ N c) 2 (µ N + µ G -2 √ µ N c)
12. Then for each µ G ∈ ( 7 10 , 12 13 ]. There must exist a γ that µ G = 7γ 10 + 12(1-γ)

13

. From the concavity of Γ, we have that Γ(µ G , c) > γΓ(7/10, c) + (1 -γ)Γ(12/13, c).

Normalize µ

N → 1. Since ∆ N ≺G < 0, it is clear that Γ < Γ. Hence Γ > 0 is a sucient condition. ∂ Γ ∂c = 4 √ c(1 + 3 √ c + 4µ G c + 5µ G ) -(1 + µ 2 G ) √ c Since √ c > 2 5 , it is easy to show that ∂ Γ ∂c > 0. It can be shown that, for µ G ∈ [ 7 10 , 12 13 ], Γ(µ G , c; c = 0) = 32µ G -(1 + µ G ) 3 > 0 B Bound Functions of Price Dierence
From the generic-niche order, we suppose that ∆ G≺N satises the below :

∆ G≺N < η(µ N , µ G , c)
where the bound function is dened as below :

η(µ N , µ G , c) = 3µ N + 3µ G + √ µ N c -35 µ N µ G /4 (B.1) Proof. Assume c < µ G .
From transformations, it suces to prove that

35µ N µ G < µ 2 G + µ N (17c + µ N -2 √ µ N c) + 2µ G (17µ N - √ c)
Square both sides, the above can be reduced to,

µ 2 G + 17µ N c + (µ N -µ G )(µ N + 2 √ µ N c) > 0
The above is clearly positive, since (µ N -µ G ) > 0. When the search cost is small, the dierence Fig. 2.17: Demand share comparison from both orderings.

G ˆ G p G µ G p N µ N ˜ N N D 1 D 2 D 3 D 4 D 5 G N ˆ N p N µ N p G µ G ˜ G T 1 T 2 T 4 T 3 T 5

C.2 The Merged Firm

The generic-niche ordering : p N is the expected price of the niche product. When the merged rm charges (p, p G ), the prot function of the industry is,

Π G≺N (p, p G ) = p G 1 -ˆ G 2 + 1 4 ˆ G p G µ G (1 + p -p G + µ G G µ N )∂ G + p 4 p-p G +µ G ˆ G µ N p µ N (1 + µ N N -p + p G µ G )∂ N + (1 + ˆ G )( 2 √ µ N c -p + p N µ N )
The rst-order conditions are,

∂ ∂p G Π G≺N (p N , p G ) = - 3p 2 N -4µ N p N -µ N (4c + 4µ G -µ N -8p G ) 8µ N µ G = 0 and lim p→p N ∂ ∂p Π G≺N (p, p G ) = 3p 2 N -4µ N c + 4(p N -p G ) √ µ N c + µ N (4p G + µ N + 2µ G ) -2p N (3p G + 2(µ N + µ G )) = 0.
The niche-generic ordering : p G is the expected price of the generic product. When the merged rm charges (p N , p), the prot function of the industry is,

Π N ≺G (p N , p) = p N 1 -ˆ N 2 + 1 4 ˆ N p N µ N (1 + p -p N + µ N N µ G )∂ N + p 4 p-p N +µ N ˆ N µ G p µ G (1 + µ G G -p + p N µ N )∂ G + (1 + ˆ N )( 2 √ µ G c -p + p G µ G )
The rst-order conditions are,

∂ ∂p N Π N ≺G (p N , p G ) = - 3p 2 G -4p G µ G -µ G (4c -8p N + 4µ N -µ G ) 8µ N µ G = 0 and lim p→p G ∂ ∂p Π N ≺G (p N , p) = 3p 2 G -4µ G c + 4(p G -p N ) √ µ G c + µ G (4p N + µ G + 2µ N ) -2p G (3p N + 2(µ G + µ N )) = 0 D Perfect Information w. Outside Option D.

The Duopoly Model

Under perfect information, it is a weakly dominant strategy for each consumer to visit each rm.

Hence each price and match information is perfectly observable by each consumer. Consider the case that ∆ < µ N -µ G . From the price pair (p N , p G ), Firm N receives the demand given by,

D N (p N , p G ; c = 0) = (1 -(µ G + p N -p G )/µ N ) 2 + (µ G +p N -p G )/µ N p N /µ N P r(u iN > u iG )f ( N )d N
And provided the same price pair, Firm G receives the demand given by,

D G (p N , p G ; c = 0) = 1 p G /µ G P r(u iG > u iN )f ( G )d G
Under the price equilibrium, none of them has incentives to change their prices. Therefore FOC deliver price equilibrium characterized by

p N = µ G (µ N -∆) -(µ G + p G ) 2 4µ G and p G = (µ N + µ G + p N -p G ) 2 -(µ N + p N ) 2 2(µ N + µ G + p N -p G )
Consider the case that ∆ > µ N -µ G , we get the price equilibrium,

p N = (µ N + µ G + p G -p N ) 2 -(µ G + p G ) 2 2(µ N + µ G + p G -p N )
and

p G = (µ N + p N ) 2 -4µ N (µ G + p N -p G ) 4µ N
From simulations we nd that the equilibrium prices from above never satisfy ∆ > µ N -µ G , which is a contradiction.

The equivalent prices and demand shares do not hold any more under this setting, because each of them faces a dierent outside option. Even by charging the same price, rms would not necessarily receive the same demand. Apart from the duopoly case provided above, the multi-product rm case is similar in that it borrows the same form of demand functions except having the objective function maximizing industry prot.

D.2 The Merged Firm

We consider rst the case ∆ < µ N -µ G . The rm receives prot from two products. From the generic product, the rm receives the rst term of the following. From the niche product, the rm receives the second term of the following. The merged rm's prot is,

Π(p G , p N ) = p G 1 4 1 p G µ G (1 + µ G G + p N -p G µ N )∂ G f rom the generic product + p N 1 2 (1 - µ G + p N -p G µ N ) + 1 4 µ G +p N -p G µ N p N µ N (1 + µ N N + p G -p N µ G )∂ N f rom the niche product
The rst-order conditions are 4µ G (µ

N + p G -2p N ) + 2p G (µ G -p G ) -(µ G + p G ) 2 = 0 and µ N (µ G - p G )(2µ N + 2p N + µ G -p G ) -2p G (µ N + µ G + p N -p G ) -2p N (p G -µ G ) = 0.
Then consider the case ∆ > µ N -µ G . The merged rm's prot is,

Π(p G , p N ) = p G 1 2 (1 - p G -p N + µ N µ G ) + 1 4 p G -p N +µ N µ G p G µ G (1 + p N -p G + µ G G µ N )∂ G f rom the generic product + p N 1 4 1 p N µ N (1 + p G -p N + µ N N µ G )∂ N f rom the niche product
The rst-order conditions are 4µ

N (µ G + p N -2p G ) + 2p N (µ N -p N ) -(µ N + p N ) 2 = 0 and µ G (µ N - p N )(2µ G + 2p G + µ N -p N ) -2p N (µ G + µ N + p G -p N ) -2p G (p N -µ N ) = 0.
From simulations, we could not nd equilibrium prices satisfying ∆ > µ N -µ G . The only equilibrium prices (if they exist), satisfy ∆ < µ N -µ G .

3. SEARCH COST AND FIRM AGGLOMERATION : THE CASE OF ON-LINE PLATFORMS

Introduction

Firm agglomeration is ubiquitous. Hotels, restaurants and malls form clusters near central business districts and train stations (and airports). For on-line shopping, these clusters are platforms (EBay, Amazon and Taobao.com). Many consumers prefer visiting the clusters to stand-alone rms. They visit these on-line stores because going through these goods takes less time. In case there is not any good match, they will consider search engines or rms around the corner and look for surprises. One observation is that, the market in the cluster is far more competitive than local markets. Take mountain bikes as the example, there are 58,828 listings in EBay and 69,070 listings in Amazon, but a couple of them in a sports shop. The other observation is that, each store in the platform typically lists many brands. For example, Asus, Lenovo and Sony are usually found in the same laptop shop, but not in separate shops.

What causes rm agglomeration ? The usual explanation is that competition implies lower prices which attract consumers to the cluster rst. However intuitively, more than two rms in the cluster are enough to form lower prices. Each rm in the cluster gains larger prot than a rm outside the cluster. If the cost to search a rm outside the cluster is larger, more rms enter the cluster, leading to greater agglomeration.

However, this can not explain the large-scale agglomeration on the Internet where search cost is small. Firm agglomeration on the Internet is dierent in that the cost for rms to join the cluster is always endogenous and easy to change. Firms are either advertised in the platform, or reached only through a search engine by consumers. This paper tries to ll the gap by delving into the relationship between search cost and rm agglomeration in Internet markets with and without platforms.

We use a sequential search model and borrows product dierentiation from [START_REF] Perlo | Equilibrium with product dierentiation[END_REF]. Consumers with unit demand can either look for price and match information from the platform, or search it outside the platform. Each search entails an identical cost for each consumer. The platform sets a membership fee for rms who make simultaneous decisions to join the platform and provides consumers with product information in the platform for free.

The model exhibits some rst results with an exogenous number of inside rms : prices inside the platform are lower than prices outside the platform and drop in the number inside rms.

Prices inside and outside the platform increase in search cost because rms have more monopoly power if it is harder to obtain price and product information. When there are nitely many rms in the market, all prices are higher than the previous case with innitely many rms because consumers have less choices so that each rm has more market power. Particularly in this case, prices of outside rms increase in the number of inside rms.

Lower prices are charged inside the platform for two reasons : 1) inside rms face more intense competition and their demand is more elastic. 2) Outside rms hold more match information of consumers than inside rms. Accordingly, outside rms set even higher prices after having observed more inside rms because more match information can be inferred from the visitors.

Having expected this price pattern, consumers will indeed visit the platform before going elsew-here. And expecting consumers' searching path, rms do have incentives to pay some amount to join the platform.

The platform can choose the number of rms it hosts by adjusting the membership fee. The monopoly platform under-supplies product information to avoid too much inside competition which reduces the market performance. We show that this under-provision can be worsened because of larger search cost : the platform reduces its capacity if consumers nd it harder to sample products. On the contrary, the platform expands its capacity if search cost is small, which increases the size of agglomeration. This relationship is solid when we consider a nite total number of rms in the market. In the presence of multiple platforms (or multiple shops in a single platform), each platform (shop) lists the same number of brands, and the number decreases in search cost. This relationship can explain the large-scale rm agglomeration on the Internet.

This paper is rst related to search literatures : [START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF] presents a sequential search model in which consumers search for price and match information. [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF] proves that prices increase in search cost and decrease in number of rms. Since rms do not always receive the same amount of attention, this paper is more closely related to [START_REF] Armstrong | Prominence and consumer search[END_REF] in which the prominent rm charges the same price as the rest if the market has innitely many rms, and a lower price if the market has nitely many rms. Armstrong et al.

(2009) also nds out that the prominent rm earns more than a non-prominent rm, and earns more than it would with random search. For that reason, rms have incentives to be visited rst. [START_REF] Zhou | Prominence and consumer search : The case with multiple prominent rms[END_REF] provides a more generalized framework, where nitely many rms are into two groups, and consumers incur dierent cost to search the two groups.

Two papers [START_REF] Arbatskaya | Ordered search[END_REF] and [START_REF] Zhou | Ordered search in dierentiated markets[END_REF] investigate ordered search. In case many rms are located one after each other and consumers follow the same searching path, both nd declining prot along the search path. [START_REF] Arbatskaya | Ordered search[END_REF] shows decreasing prices while [START_REF] Zhou | Ordered search in dierentiated markets[END_REF] nds the opposite price pattern. A simple explanation is that goods in Arbatskaya ( 2007) are homogeneous but are horizontally dierentiated in [START_REF] Zhou | Ordered search in dierentiated markets[END_REF]. In [START_REF] Zhou | Ordered search in dierentiated markets[END_REF], some consumers search even when they expect higher and higher prices, which would never happen in [START_REF] Arbatskaya | Ordered search[END_REF] where consumers look only for lower prices. [START_REF] Fischer | Product variety and rm agglomeration[END_REF] is the closest paper by far, which extends [START_REF] Wolinsky | Retail trade concentration due to consumers' imperfect information[END_REF] and [START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF] to study the incentives for rms to agglomerate. An agglomeration also takes place in our paper, yet only when chosen by a platform. The two papers are dierent in that the cost to join the cluster is exogenous in [START_REF] Fischer | Product variety and rm agglomeration[END_REF], but is set by the platform in our work. This dierence in setting provides some dierent properties. From some parameter ranges, [START_REF] Fischer | Product variety and rm agglomeration[END_REF] predicts a larger cluster while our work predicts the opposite. Further, [START_REF] Konishi | Concentration of competing retail stores[END_REF] analyses a similar case of geographical concentration of stores.

Moraga-Gonzalez and Petrikaite (2013) studies conditions under which rms selling horizontally dierentiated products have incentives to merge.

On two-sided markets, [START_REF] Baye | Information gatekeepers on the internet and the competitiveness of homogeneous product markets[END_REF] studies an internet gatekeeper gathering all price information and charging fees on both sellers and buyers. Firms and buyers choose whether to subscribe to the platform. In [START_REF] Galeotti | Platform intermediation in a market for dierentiated products[END_REF], rms and buyers play mixed strategies on entering the platform or staying outside.

The rest of the paper is structured as the following : Section 2 shows the model with a monopoly platform. In Section 3, the prices in equilibrium are found along with the prot maximizing number of rms in the platform. In section 4, we briey discuss the competition among stores in the platform. In section 5, the platform can also charge the buyers who become platform and buy a good. 2) Leave the platform and search outside rms. 3) Leave without buying anything. A consumer searches not only for better prices, but also for the goods tting his own tastes. Before searching outside rms, he has formed expectations about their prices.

Each round of search oers him price and match information of one rm. Having searched one rm, the consumer can freely recall that rm at any time. The consumer purchases at one rm if the three following conditions are satised : 1) the product oers him a positive utility (i.e.

u ij > 0).
2) The same product oers him the highest utility hitherto. 3) Searching another rm yields lower expected surplus. If he has visited all rms and all of them oer him a negative utility, he chooses not to make any purchase.

We denote rm j's strategy as σ j = {I j , p j }. A rm makes two decisions : the rst decision is whether to join the platform. I j = in indicates that rm j joins the platform. The opposite (I j = out) indicates that it chooses not to join the platform. The second decision is the price.

Let p I j be the expected symmetric equilibrium price set by rm j. Therefore, inside rms and outside rms are expected to charge p in and p out . Let ∆ be the expected price dierence p in -p out .

The demand function D I j (p j , p -j ; m) indicates rm j's expected demand when it charges p j and other rms charge p -j provided m inside rms. Denote Π I j (p j , p -j , w; m) as rm j's expected prot. All inside rms must pay membership fees w to the platform for entrance, which is the only source of revenue for the platform (i.e. no per transaction fees). Suppose there are already m entries. Having observed the m entries, the marginal rm compares its expected future prot after his entrance with the entrance fee w. If its expected future prot exceeds the fee, it decides to join the platform. Otherwise, it stays outside.

Timing

The pay-o of a consumer i is simply his surplus denoted by u ij -kc from purchasing a good,

where k is the number of times he has searched, and zero by walking away. The pay-o of rm j is its expected prot, denoted by

Π I j (p j , p -j , w; m) = p j D I j (p j , p -j ; m) -w × I j
The pay-o of a platform is its collection of entrance fees :

Π p (w) = m(w)w

Equilibrium Analysis

Optimal Search Rule

Since rms inside the platform are freely accessible, every consumer then samples all inside rms to nd out the best oer. Suppose that a consumer has his current best oer from rm j, which is a pair {¯ , p}. Then his next step is to choose between taking the current best oer immediately or searching for a better oer. If he samples an additional rm and the newly sampled product oers { , p }, he will prefer the new product if -p > ¯ -p. Based on his expectation about prices outside the platform p out and the search cost c, the expected incremental surplus from sampling another rm is,

v ¯ +pout-p ( -p out -(¯ -p))d F ( ) -c
The rst (second) term above is the expected incremental benet (cost) from another search.

From the cost-benet analysis, he will sample another outside rm if the above is positive, and take the current best oer if otherwise. The optimal search rule is for the consumer to decide at which points he should stop and purchase the current best oer. Since the expected incremental benet from another search decreases in ¯ -p + p out , there exists at most one point where the consumer is indierent between both options (i.e. (2) holds).

v ¯ +pout-p ( -p out -(¯ -p))d F ( ) = c (2)
In a symmetric equilibrium (p = p out ), denote the unique solution for ¯ by a, so that a satises v a ( -a)d F ( ) = c. When the current best match is larger than a, the consumer purchases the best match. Otherwise, he moves on to another rm. From [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF], this stopping rule is stationary and does not depend on the number of rms left. Therefore consumers face the same threshold while deciding whether to search the 1 st outside rm, and the N th (N > 1) outside rm.

If rms charge dierent prices (p = p out ), the corresponding threshold becomes a -p out + p.

The threshold increases in current price and decreases in expected price of outside rms. When a solution for (2) exists, the optimal search rule can be summarized as the following :

a -p out          > ¯ -p search again < ¯ -p

buy the best match

The threshold further decreases in search cost if the search cost is not too large. Therefore consumers are more likely to take the current oer if it is costly to sample another product.

In case the search cost becomes too large, consumers do not necessarily participate in searching rms. While a < p out , a -p out is even below the outside option. Each consumer is then content with the outside option and will not perform any search.

Prot Maximization by Firms

Suppose other inside (outside) rms are charging the same price p in (p out ) and rm j is freely choosing price p. After checking out all inside rms at the beginning, a consumer faces three choices : choose rm j to buy a good, search randomly outside the platform or drop out without any purchase. The rst choice is better if the following two conditions are satised : Firm j's good is the best match for him in the platform : j -p ≥ max{ k -p in , 0}, ∀ k ≤ m.

A random search will not give him a higher incremental utility than the search cost :

j > a -p out + p.
For a given realization of j , the probability for rm j to be a better match than any other inside rm is F m-1 ( j + p in -p). For the second condition to be satised, j must be larger than a -p out + p, which partly gives the lower bound for the integral of (3). So the joint probability is Pr( j -p > max k∈1,...,m , k =j { k -p in , a -p out }) for a ≥ p out , and Pr( j -p > max k∈1,...,m , k =j { k -p in , 0}) for a < p out . In case a + p -p out < v or a -p out < 0, no consumers leave the platform and outside rms receive no demand. We have assumed that v < 0, which has two implications : 1) v -p < 0. 2) a + p -p out < v then implies a -p out < 0. 4 Here is the demand function of an inside rm :

D In (p, p out , p in ; m) = v max{p,a+p-pout,v} F m-1 ( + p in -p)dF ( ) (3)
If a ≥ p out , fraction F m (a -p out + p in ) of them are not satised by any good in the platform after checking out all inside rms. And they leave to search outside the platform. An outside 4. In case where a < p out , the rms set prices to p out if some consumers search. We learn from the previous example that no consumers conduct search after considering their expected surplus. Hence p > a -p out + p. If ˜ < p, the consumer chooses not to buy anything. The lower bound of the integration needs to be adjusted to p. rm could be the consumer's 1 st visit (2 nd , 3 rd , 4 th ...∞ th ). The probability that it is his S th visit is a S N , where N is the total number of rms. Assume that the ratio of the number of consumers to the number of rms is 1. By adding up the probabilities, we obtain the term ∞ k=0 a k . An outside rm's demand is :

D Out (p, p out , p in ; m) =        F m (a -p out + p in )( ∞ k=0 a k )(1 -F (a -p out + p)), for a ≥ p out 0, otherwise (4)
From p out = -D Out (pout,pout,p in ;m) ∂D Out (p out ,p out ,p in ;m) ∂p , it is easy to obtain the equilibrium price for outside rms :

5

p out = 1 -F (a) f (a) , f or a ≥ p out (5)
Outside rms are targeting only their fresh demand because everyone visits inside rms rst and once a consumer passes an outside rm, he never returns. The equilibrium price increases in search cost, which is in line with [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF].

For a < p out , outside rms are not active because they receive no visitors. Fraction F m (p in )

of consumers drop out without any purchase.

Consider a ≥ p out for inside rms, max(p, a -

p out + p, v) = a -p out + p. The derivative of rm j's expected demand in p is, ∂ ∂p D In (p, p out , p in ; m) = (6) -F m-1 (a + p in -p out )f (a + p -p out ) -(m -1) v a+p-pout F m-2 ( + p in -p)f ( + p in -p)f ( )d
5. While a ≥ p out , the only required condition for the consumer to purchase at an inside rm is > a-p out +p. Because once he stops there, the surplus he obtain is ¯ -p > a -p out > 0. While a < p out , consumers choose not to search since from searching they expect a negative surplus.

Similarly for a < p out and p in = -D In (pout,pout,p in ;m) ∂D In (p out ,p out ,p in ;m) ∂p we obtain the following :

p in = 1 -F m (max{p in , a + ∆}) m (f (v) -v max{p in ,a+∆} F m-1 ( )f ( )d ) (7)
Proposition 1. if m rms in the market are inside the platform, there exists an equilibrium where all inside rms charge identical prices and all outside rms charge identical prices under the log-concavity of f ( ). Equilibrium prices are :

(let ∆ = p in -p out ) a ≥ p out :        p out = 1-F (a) f (a) p in = 1-F m (a+∆) m (f (v)-v a+∆ F m-1 ( )f ( )d ) a < p out :        p out = undef ined p in = 1-F m (p in ) m (f (v)-v p in F m-1 ( )f ( )d ) (8)
The outside rms set a higher (or equal) price than inside rms, whose both prices decrease in number of inside rms if is under uniform distribution or f (v) -F m (a + ∆)f (a + ∆) < 0. 6 Both the inside price and the outside price increase in search cost.

Proof. See the appendix.

The existence of price equilibrium inside the platform needs the log-concavity of f ( ) 7 and implies the existence of price equilibrium outside the platform. The former suggests that each inside rm's prot function is quasi-concave in its own price according to [START_REF] Caplin | Aggregation and imperfect competition : On the existence of equilibrium[END_REF] and the latter is suggested by [START_REF] Anderson | Pricing, product diversity, and search costs : a bertrandchamerlin-diamond model[END_REF].

8

Another entrance can aect the inside prices in two directions : 1) competition becomes more intense so that each demand is more elastic. 2) Total inside demand becomes larger so 6. Besides the uniform case which we have proved separately. This list has included exponential, gamma, Laplace, normal and Weibull distribution.

7. The log-concavity of f ( ) means that the log of that function is concave. 8. This price equilibrium is also a natural extension of [START_REF] Armstrong | Prominence and consumer search[END_REF] except that competition inside the platform is similar to [START_REF] Perlo | Equilibrium with product dierentiation[END_REF]. that there is a small market-expansion eect. In our model, the rst eect outweighs the other so that inside prices drop in number of inside rms. The outside prices depend only on the level of search cost because changes can not have eects if not being observed. In short, inside rms compete with each other and the search option, and outside rms only compete with the search option. Last, consumers will indeed visit inside rms rst because lower prices are charged in the platform.

An inside rm's expected prot is :

Π In (p in , p out , w; m) = D in (p in , p in , p out ; m) p in -w (9)
w is the set-up cost of joining the platform. To simplify the analysis, we substitute F with the uniform distribution from [0, 1] from now on. In the uniform case, the condition for consumers to search rationally a ≥ p out becomes a ≥ 1 2 or c ≤ 1

8

. The equilibrium prices are :

a ≥ 1 2 (active search) :

       p out = 1 -a p in = 1-(a+∆) m m a < 1 2 (no search) :        p out = undefined p in = 1-p m in m (10)
We are about to show that the prot Π In (p in , p out , w; m) also decreases in m so that the demand for entrance drops in number of existing inside rms. The platform thus faces a downward sloping demand for entrance. On the eects from a higher search cost, the sign of the partial derivative of Π In in a can be seen from the sign of the partial derivative of p in in a :

∂Π In (p in , p out , w; m) ∂a = 2p in ∂p in ∂a , ∂p in ∂a = (a + ∆) m-1 1 + (a + ∆) m-1 ∂p out ∂a -1 , where ∂p out ∂a = -1 (11)
Because p in increases in c, both inside (outside) prices increase in search cost.

Proposition 2. under a Uniform(0,1) distribution, an inside rm receives higher demand and prot under higher search cost. Given a xed set-up cost w, an inside rm receives higher demand and prot under a lower number of inside rms.

Lower search cost means an easy access to outside goods. Consumers are therefore more tempted to visit outside rms, who charge also a lower price according to (11). To keep consumers in the platform, the inside rms have to lower prices which immediately lowers the threshold

Platform's Pricing

Having observed w, rms decide simultaneously whether to be "in" or "out". In this entry game, if one rm decides to be "in", it pays w to the platform and enters. Since an outside rm's prot is negligible compared to an inside rm's prot, the expected dierence in prot between an inside rm and an outside rm is actually approximate to Π In (p in , p out ; m).

Π In (p in , p out ; m) decreases in number of entrances and equals -w when there exist innite entrances. The only (pure strategy) SPNE is a number m * satisfying : D in (p in , p in , p out ; m * ) p in ≥ w and D in (p in , p in , p out ; m * + 1) p in < w. After the entrances, the outside rms are indierent bet-9. For instance, if a certain number of brands sell goods in the city center and the rest of them are far from both the city center and each other. Then, whether they are 20 miles or 30 miles away will aect none of the prices since nobody visit them.

ween entering the platform and not entering it because the entrance fee is too high ; the inside rms prefer to stay in the platform because they have already paid the fees.

On top of entry decisions and price decisions on rms' part, the platform sets the set-up cost w, which is the only way for the platform to determine its size m, under a relation indicated by

(3), ( 7) and w = D in (p in , p in , p out ; m * ) p in . Hence choosing an entrance fee is indeed equivalent to choosing a size of the platform. Since an inside rm's prot decreases in number of inside rms, the demand for entrance is downward sloping in w. Therefore the platform's prot could increase if each individual rm's prot does not fall too fast.

The platform does not take every rm since then the joint prot is zero. Nevertheless, taking too few rms makes more consumers leave the platform and search outside rms. There is a trade-o between in-house competition and attractiveness of the platform, essentially the tradeo between price and quantity. The platform chooses the optimal number of brands exposed to consumers. For instance, if it contains only one brand, then the total prot will be (1 -F (a)) 2 from (3) and ( 5). The platform's problem is to maximize its own prot, denoted as Π p :

max Π p (w) = m(w) w = mD In (p in , p in , p out ; m)p in , s.t. p in = 1 -(max(p in , a + ∆)) m m (12)
Dene the elasticity ε(m) = -m∂p in p in ∂m as the percentage of drop of inside price with regard to the increase of the platform's size. We assume rst that the platform's prot is quasi-concave in its size. Thus a globally optimal size exists. Take the rst derivative of ( 12) with regard to m and let it be zero.

∂ ∂m Π p (m) = p 2 in + 2mp in ∂p in ∂m = p 2 in 1 -2ε(m) = 0 ⇒ ε(m) = 1/2 (13)
The monopolist continues to take new rms until the elasticity reaches 1/2. While maintaining this elasticity at 1/2, we have the following proposition regarding dierent levels of search cost.

Proposition 3. in case the inside rms are charged for entrance fees and the buyers enter the platform for free : there exist a number of brands m maximizing the platform's prot satisfying :

(1 -(a + ∆) m (1 -log(a + ∆) m ))/(1 + (a + ∆) m-1 )(1 -(a + ∆) m ) = 1/2
In an active search equilibrium, the number of brands chosen by a monopoly platform decreases in search cost. In a no search equilibrium, search cost has no eects on its choice of brands.

Proof. See the appendix.

The below gure illustrates two cases : a = 1/2, a = 3/4. The search cost is higher when a = 1/2. The solid curves represent the decreasing relations between the equilibrium price and the number of inside rms. The dashed curves are iso-prot curves. The platform chooses the points where the isoprot curve ( 12) and the price curve (10) are tangent to each other. The gure makes two points : 1) the platform receives a higher prot when a = 1/2. 2) The platform nds a smaller number of inside rms more protable under a higher search cost. While a = 1/2, its most protable size is roughly around 2 ; while a = 3/4, the platform prefers a size at around 4 or 5. To put dierently, the platform oers two (4-5) brands, if by searching costly outside the platform, one naturally stops at an oer better than average (75th percentile). If the search Fig. 3.2: Impact of search cost on the optimal number of rms chosen by the platform cost is extremely high (c larger than 1/8), the platform accepts two rms (see claim 1-2 in the appendix), and each outside rm sets its price to the monopoly price (in our case the monopoly price is exactly 1 2

.). With an extremely low search cost (c ≈ 0), the platform sets an extremely low fee and provides consumers innitely many choices. The large size of a platform on the Internet is due to low fees which can be further explained by low search cost. For example, Amazon and Taobao.com charge $0 for each listing and EBay charges $0.50 for almost every listing.

From (11), the platform faces a larger demand for prominence under a larger c, and it is natural to oer more "slots" to meet the rising demand. However, more rms cause more competition, which might be counter-productive for the market performance. The usual "business stealing" eect from a new entrant is internalized because the monopoly platform will nally extract all prots from the inside rms. The platform can only pull out consumer surplus through the inside rms. Due to the "non-appropriability of social surplus ", it contains less than the ecient number of rms. From the standpoint of consumers, shops from one industry look more clustered than others if to look for product information in the former industry is easier (for example, electronic shops). This result is clearly dierent from [START_REF] Fischer | Product variety and rm agglomeration[END_REF] where the joining cost does not change corresponding to higher demand to join the cluster (platform).

From their framework, the number of inside rms will be larger at higher search cost where the agglomeration should more likely take place, whereas our work predicts the opposite pattern.

To see how large is the loss of consumer surplus, we illustrate the impact of search cost (horizontal axis) on the expected consumer surplus (vertical axis) in the gure below. The platform takes 4 rms when search cost is around 1 25 and the expected consumer surplus is denoted by the up-left corner. Along the horizontal axis, the search cost gets larger and larger. If the platform keeps its number of inside rms unchanged, expected consumer surplus slowly decreases from the solid line because searches are harder and rms then charge higher prices. However at certain points (c=0.053 and c=0.097), the platform reduces its size to 3 and 2, which de facto pushes more consumers towards outside rms. The dash line with two jumps is the actual curve of the expected consumer surplus after considering the choice by the platform. The gray region denotes the amount of surplus which the platform fails to appropriate from some higher search costs, which can be as large as total consumer surplus when the search cost is large.

The equilibrium fee charged by the platform is also unique. Suppose rst that ∂ε(m) ∂m is positive.

From ( 13), there must exist a m * where the value of p 2 in (1 -2ε(m)) is positive for m < m * , is quasi-concave. We take the FOC of ( 13) in a and obtain the following

∂ε(m) ∂a + ∂ε(m) ∂m ∂m ∂a = 0 ⇒ sgn ∂ε(m) ∂m = -sgn ∂ε(m) ∂a (1/ ∂m ∂a )
The sign of ∂ε(m) ∂m can be unveiled by the other two terms. The uniqueness of a maximum of Π p (m) requires that ε(m) falls in a, for which we have the following :

Corollary 1. the elasticity between the inside price and number of slots increases in search cost and number of rms. Therefore, the equilibrium membership fee is unique.

Proof. See the appendix.

Having more rms in the platform increases the elasticity. In order to keep ε(m) at 1 2 , the platform prefers choosing fewer rms under a higher search cost, where ε(m) is generally larger.

Particularly for a < 1 2 , we show in appendix a.1 that the platform only selects two rms.

From simulations, the result is solid while taking a more general form of consumer utility function, or in other words, when the platform is able to provide consumers some surplus ei-ther from some "consumer support" service or non-commercial contents exclusively provided to buyers.

10 The other reason we introduce α is to compare the number of rms taken by the platforms in the asymmetric duopoly case from section 4. To start, we use the alternative form of the utility function,

u ij = α 1 + ij -p i
where α 1 is the surplus oered by platform 1. Then the inside price is determined as the below.

p in = 1 -(max(p in -α 1 , a -α 1 + ∆)) m m
When we take α 1 = 0.25, the platform takes 5 rms when a = 0.75, and 11 rms when a = 0.9.

When we take the small surplus α 1 = 0.05, the platform takes 5 rms when a = 0.75, and 12 rms when a = 0.9.

The Oligopoly Case

Here we consider a new situation : rms agglomerate in stores, and then stores compete with each other. Each store is a platform in the previous section. Provided free entrance on consumers' part, each consumer typically checks out each store before making any decision.

Searching products across stores is seen as costless.

11 Since rms produce weak substitutes to each other according to (1), stores are homogeneous and dier in the number of inside rms.

Consider the following game : In the rst stage, each store chooses its respective fee for the 10. For example, some websites oers better delivery service, coupons, free product trials, free CDs and the access to some other on-line content to whoever made a purchase on the website.

11. Also since each store oers free entrance for consumers, there exists no multi-homing on the rms' side because no rms are willing to pay twice the entrance fee.

rms. In the second stage, rms observe the fees and decide which store (platform) to join. In the third stage, rms set prices. In the last stage, consumers visit the stores.

We look for a symmetric equilibrium of store sizes.

12 First consider the duopoly model. Store 1 takes the size of store 2 as given, and looks for its optimal size. Denote the expected sizes of the two stores respectively by m and k. Its maximization program is :

max Π p (w) = m(w)w = mp 2 in , s.t. p in = 1 -(max(p in , a + ∆)) m+ k m + k (14)
Proposition 4. there exists a symmetric equilibrium, where two stores have an equal size and the total size is larger than one store. The industry prot is lower with two stores than with the monopoly platform.

m∂p in (m + k) p in (m + k)∂m = m∂p in (m + k) p in (m + k)∂(m + k) = - 1 2 , and k∂p in (k + m) p in (k + m)∂(k + m) = - 1 2
If there exists an equilibrium, m = m and k = k. Summing up the above yields :

- ( k + m)∂p in ( k + m) p in ( k + m)∂( k + m) = ε( m + k) = 1
12. Two remarks here : rst, the competition between two stores is quantity competition rather than price competition due to the limit of space in the store (yellow pages are usually a few pages long). The conditions are similar to [START_REF] Kreps | Quantity precommitment and bertrand competition yield cournot outcomes[END_REF], where both choose their capacities in the rst stage then the price in the second stage. Their results coincide with the Cournot model. Second, this competition happens in the 1st stage of the game.

We already know that ε(m) increases in m from corollary 1. Therefore the above suggests that two stores take more rms than one, consistent with most previous duopoly literatures. Besides, a higher resulting elasticity leads to a lower joint prot than the monopoly case, because the joint prot of two stores containing respectively m and k is equivalent to the prot of the single store with m + k entries.

For the oligopoly model, we have found similar results. Proposition 5. there exists a symmetric equilibrium, where the stores' total size increases in number of existing stores, each existing store has the equal size and the size decreases in search cost.

Proof : suppose that there are H stores, with indices from 1 to H. i is here the index of a store. Each store i chooses its number of rms inside q i to maximize its prot function Π p i given that a certain store k is choosing qk . This is equivalent to adding more rms until

ε i (q i + Σ H k=1,k =i qk ) = 1 2 , which is, ε i (q i + Σ H k=1,k =i qk ) = - q i ∂p in (q i + Σ H k=1,k =i qk ) p in (q i + Σ H k=1,k =i qk )∂q i = - q i ∂p in (q i + Σ H k=1,k =i qk ) p in (Σ H k=1 q k )∂(q i + Σ H k=1,k =i qk ) = 1 2
If there exists an equilibrium, q i = qi . Aggregating the above with all H stores, we have

ε(Σ H k=1 qk ) = Σ H i=1 ε i (Σ H k=1 qk ) = - Σ H k=1 qk ∂p in (Σ H k=1 qk ) p in (Σ H k=1 qk )∂Σ H k=1 qk = H 2
From corollary 1, we know that ε(m) increases in m. From the above, ε(Σ H k=1 qk ) increases in

H.

Hence a larger H infers a larger Σ H k=1 qk , which is the total size of the stores. Again from corollary 1 we know that ε(m) increases in search cost. For the search cost c > c, the existence

of ε(Σ H k=1 q k ; c) = H/2 infers that ε(Σ H k=1 q k ; c ) ≥ H/2. Since ε(m) increases in m, the existence of ε(Σ H k=1 q k ; c ) = H/2 further infers that Σ H k=1 q k ≤ Σ H k=1 q k .
Since each store takes the same number of brands in a symmetric equilibrium, the number of brands taken by each store decreases in search cost.

Numerical results show that if m ≤ 2 is the size for store 1 with a < 1 2 , then it is most protable for store 2 to adjust its size to 2 (k = 2). However if store 1's size is m = n > 2, then it is optimal for store 2 to adjust its size to k = n. In equilibrium, two stores in the platform usually have the same size. The similar case can be shown under a > 1 2 , but generally with more inside rms than when a < 1 2 .

Observations can be seen from electronic stores in a few developing countries (China, India, etc.), where most are found in a centralized market (e.g. the computer city). Each store in the market corresponds to a platform in our model. Travelling between stores takes few seconds and is considered costless. In this centralized market, one typically nd each store sell almost every brand of cellphones and laptops. Few stores focus on one brand, because consumers will easily ow away.

More inside rms and lower prices suggest that consumers generally obtain better match, pay lower prices and search less, which further suggests that they pay less search cost in general.

This answers how multiple platforms imply larger consumer surplus and lower industry prot.

Charging Consumers

From the model in section 2-4, consumers are certainly willing to pay some amount for the entrance to the platform, which may in turn impose a fee on the consumers. In practice, consumers purchases membership from on-line music stores (e.g. Deezer) and on-line TV (e.g.

iTVmediaPlayer and Sky Go). While some platform contents are exclusively available through a hardware device (e.g. Amazon Kindle), some consumers purchases the devices. Because of this subscription fee, some consumers stay out of the platform.

Firms can charge consumers dierent prices, if they nd out whether the consumers are subscribers or not. A price discrimination strategy might include three prices.

13 Further, if the number of rms in the market is nite, this always leads to an ordered search and an unstationary search rule, because rational non-members can always identify inside (outside) rms by prices, and recall the number of them he has visited.

14 If the number of rms is innite, the chance to come across a particular inside rm is negligible, so that consumers would not take any past search into his optimal search rule. In order to have a stationary search rule, we only consider the case with innitely many rms. Henceforth price discrimination is no longer an issue.

Denote the fraction of subscribers by ρ, which is chosen by the platform by the subscription fee (for consumers) s and exogenous afterwards. pin and pout are the corresponding price expec- tations of an inside rm and an outside rm. For simplicity, we only consider the case where a ≥ 1/2. For a subscriber, his searching threshold is given by a + pin -pout in a similar way to

(2). For a non-subscriber, his searching threshold is always a.

13. The rst price is the price charged by the inside rms on members. The inside rms charge the non-members for the second price. And the outside rms charge all consumers for the third price.

14. For a non-member with perfect memory, the current search is never the same as the next search. If the rm he just visited is an inside (outside) rm, he expects an higher (lower) price for the next search. Thus the demand function of a rm must include all possible orders a consumer may have.

If an inside rm deviates to a new price p while other rms are keeping their prices to the expectation, the inside rm receives the demand given by :

DIn (p, pout , pin ; m) = ρ min{1,1+p-p in } a+p-pout ( -p + pin ) m-1 d + ρ max 0, pin -p (15)
The above expression shows that the inside rm faces demand approximately equal to a proportion ρ of the demand from (3). Let λ = n c /(n -m) be the ratio of number of consumers n c to number of outside rms (n-m). If an outside rm deviates to a new price p while other rms are keeping the same price, its demand is given as follows :

lim n→inf DOut (p, pout , pin ; m) = λ ρ(a + pin -pout ) m + (1 -ρ) n -m n n-m-1 k=0 a k 1 -a + pout -p (16)
Provided that ρ is a positive constant and there exist innitely many rms in the market, the demand from the non-subscribers is innitesimal compared with demand from subscribers so that inside rms consider only subscribers while making price decisions. The outside rms face a blend of subscribers and non-subscribers. In case there exists an equilibrium price pair, both types of rms have no incentive to deviate to other prices. From the FOC, the equilibrium prices are again given by ( 10).

Next we turn to the consumer i's surplus : let ˆ = a + pIn -pOut . We use ṼIn to denote the expected surplus of a subscriber and ṼOut for that of a non-subscriber.

E( ṼIn ) = Pr max ij,j≤m ≥ a + pIn -pOut E max ij,j≤m -pIn | max ij,j≤m ≥ a + pIn -pOut + Pr max ij,j≤m < a + pIn -pOut E ṼOut | max ij,j≤m < a + pIn -pOut -s (17) E( ṼOut ) = E i | i > a - ∞ k=0 a k-1 1 -a kc -pOut (18)
(17) and ( 18) can be understood by the fact that if the consumer is satised with one product in the platform, he obtains the conditional expected surplus given that that particular match is above the threshold ˆ , which corresponds to the 1 st term in ( 17). In case all inside rms provide match below ˆ , he obtains another expected quality conditioned on the fact that one particular match is above the threshold a after some product search, which corresponds to the 2 nd term in ( 17). If the consumer does not attend the platform and searches directly outside rms, then his expected surplus is given by ( 18). ( 17) is equivalent to the following :

E( ṼIn ) = 1 a+p In -p Out m( -pIn ) m-1 d + Pr max ij,j≤m < ˆ E ṼOut | max ij,j≤m < ˆ -s
Particularly,

E( ṼOut ) = E | > a -c(1 -a) ∞ k=0 ka k-1 -pOut = a -pOut
15 On the consumers' part, the subscription fee is the amount equating E( ṼIn ) with E( ṼOut ),

s = 1 a+p In -p Out m( -pIn ) m-1 d gain f rom platf orm +(a -pOut ) (a + pIn -pOut ) m -1 (19)
The above subscription fee is composed of two parts : the rst part is the gain from information in the platform. The second part comes from the fact that subscribers have smaller chances to visit outside rms. The dierence in prot between an inside rm and an outside rm is ρmp 2 1 .

15. The last equality comes from the relation between the threshold and the search cost. c = (1 -a) 2 /2.

Since the platform is a monopoly and can extracts the whole surplus from both inside rms and subscribers. The platform's prot is given by :

Πp (w, s) = ρ m DIn (p in , pout , pin ; m) pIn + ρ s ( 20
)
The above is monotonic in ρ. Therefore the platform will continuously increase ρ until it reaches 1. In the end, the platform's prot function is :

Πp (w, s) = (m/(m + 1)) 1 -(a + pIn -pOut ) m+1 -(a -pOut ) 1 -(a + pIn -pOut ) m (21)
Proposition 6. with an innite number of rms in the market, if the platform is able to charge the consumers for subscription fees, then regardless of the value of search cost, the platform will enrol all consumers ρ = 1 and its prot increases in m.

Proof. See the appendix.

From section 2-4, the platform charges only the rms, and an insucient number of rms enter. The subscription fee s allows the platform to internalize consumer surplus, partially from the product match, and partially from search cost. The platform decreases gradually the rms' fee and increases the consumers' fee until it captures the whole dierence in surplus between subscribers and non-subscribers. For this reason, consumers are indierent between joining the platform and staying outside. It thus makes no dierence to say "some consumers join the platform" and "the platform picks these consumers in". Social optimality is almost restored because the platform will include innitely many rms, consumers have innitely many choices and pay nearly zero for the product. The main dierence from section 2-4 is that here the rms will enter the platform for free and the platform charges mainly the consumers. Some comparable results are shown in [START_REF] Galeotti | Platform intermediation in a market for dierentiated products[END_REF]. In their paper with nitely many rms, all consumers know only in advance the expected gain from joining the platform and the platform charges this expected gain. In [START_REF] Rochet | Platform competition in two-sided markets[END_REF], both buyers and sellers know in advance their exact benets, thus there exist always some sellers (buyers) outside the platform.

Table 1 shows that, with an innite number of rms (sellers), a positive search cost, it is more protable for the platform to charge only the consumers rather than only the rms (when c > 0.02). max Π p (w; c) is calculated according to section 3. min lim Πp (w, s; c) can be found from ( 21) by taking c to the lower bound and m to innity. Since we assume innitely many rms, the example may also apply to cases with a large number of rms. The remaining issue is that, we assume always the number of inside rms to be negligible compared with the total number of rms in the proof. Yet the assumption is undermined if the platform accredits as many rms as possible. The problem can be eased if the entry of rms is costly to the platform.

Finitely Many Firms

In case there exists a xed cost of production, each rm, by entering the market, pays the xed cost. Still the rm pays the membership fee to be in the platform. Denote this xed cost by κ, and denote as before the membership fee by w. There would not be as many rms as under no xed cost. Therefore, this setting is more general to test the solidarity of our previous results.

Suppose under a free-entry equilibrium, n out of an innite number of rms are active, and m out of n rms become inside rms. If consumers expect the price charged by outside rms to be p out , then section 3.1 suggests that the optimal search rule is indicated by (2).

From the optimal search rule, both types of rms receive two sources of demand : fresh demand and returning demand. Denote the returning buyers towards an inside rm by rd in and the returning buyers towards an outside rm by rd out . For simplicity, we only consider the case a + ∆ ≥ 0. If one of inside rms deviates to a new price p and other rms keep the price at consumers' expectation, its demand becomes

D In (p, p out , p in ; n, m) = min{1,1-p in +p} a+p-pout ( -p+p in ) m-1 d +max 0, p in -p +rd in (p, p out , p in ; n, m) (22)
The rst term represents the fresh demand, the second term is a technical term and the third term represents the returning demand, which can be elaborated as the following,

rd in (p, p out , p in ; n, m) = a+p-pout p ( + p in -p) m-1 ( + p 0 -p) n-m d = a-pout 0 ( + p in ) m-1 ( + p out ) n-m d (23)
If an outside rm deviates to a new price p, it faces the demand as :

D Out (p, p out , p in ; n, m) = (a + ∆) m n -m n-m-1 k=0 a k (1 -a + p out -p) + rd out (p, p out , p in ; n, m) (24)
where the returning demand of an outside rm can be elaborated as,

rd out (p, p out , p in ; n, m) = a+p-pout p ( + p in -p) m ( + p out -p) n-m-1 d = a-pout 0 ( + p in ) m ( + p out ) n-m-1 d (25)
Notably, the determination of rd out and rd in aren't relevant to p. This is because all returning consumers have viewed all prices (including p) so that they take p into both search and purchase decisions and outside options in a linear form. In case the rm has no intention to deviate, we derive the inside price from p in = -D In (p in ,pout,p in ;n,m) D In (p in ,pout,p in ;n,m) :

p in = 1 -(a + ∆) m m + rd in (p in , p out , p in ; n, m) (26)
It is notable that the rst term is the same as (8) and the second term comes directly from the returning consumers. Further the returning demand enters the equilibrium price in a linear form. The same pattern appears also in the price of an outside rm. If an outside rm does not deviate, we then derive the price equilibrium p out from the rst order condition :

p out = (1 -a) + n -m (a + ∆) m 1 Σ n-m-1 k=0 a k rd out (p out , p out , p in ; n, m) (27)
Proposition 7. with a nite number of rms in the market, there exists a symmetric equilibrium given by ( 26) and ( 27), where each inside rm sets a lower price than each outside rm.

Proof. See the appendix.

In a symmetric equilibrium, inside rms charge one price and outside rms charge the other price. The prevailing price of an outside rm is the term in ( 5) plus a weighted returning demand. If their second terms rd out (p out , p out , p in , n, m) and rd in (p out , p out , p in , n, m) are positive, the equilibrium p out ,p in are larger than their counterparts in ( 5) and ( 7), because the returning demand provides the rms with more market power even when consumers may choose the outside option if prices are too high. The larger the returning demand, the higher both prices are. Further, the inside price is always lower than the outside price, which is the corresponding version of proposition 1 under nitely many rms. Finally when the total number of rms tends to innity, returning demand tends to zero and we return to the innite case. The below gure draws each price equilibrium.

If only one rm is prominent, the inside price and the outside price coincide, as in [START_REF] Armstrong | Prominence and consumer search[END_REF]. Provided more inside rms, the inside price is decreasing as in section 3. Interestingly, the outside rms raise gradually their prices if more inside rms are present. The probable reason is that with more inside rms and less outside rms, more information is revealed to the latter if the consumers turn to the rms outside the platform. The outside rms can not observe any match between consumers and inside rms, but can update their beliefs about the match as soon as they see a consumer come to visit an outside rm. The updated probability of having received poor matches among inside rms increases. And the larger the number of inside rms, the poorer match it must be. Therefore, the outside rms will increase their prices. From the decreasing inside price, the price gap between an inside rm and an outside rm enlarges more and more. Keeping the number of inside rms unchanged, this price gap is larger than the case with innitely many rms. To see this, we compare the inside price with that in section 3 :

The inside price with limited number of rms is larger than that with innite number of rms. This result might be due to the presence of returning buyers, which corresponds to rd in (p out , p out , p in , n, m) in ( 23). In the gure, with a smaller number of inside rms, the difference is more noticeable, because then their prominence is more notable. Besides, all inside rms have higher market power and higher prots.

In the usual case, an inside rm's prot has the form of Π In (p in , p out ; n, m) = p 2 in -w -κ and an outside rm's prot has the form of Π Out (p in , p out ; n, m)

= (a+∆) m n-m ( n-m-1 k=0 a k )p 2 out -κ.
From simulations, both Π In and Π Out decrease in n, number of active rms. Further both Π In and Π Out decrease in m, number of inside rms. Since it is a free entry equilibrium, rms will keep entering the market until Π Out = 0. Last, the prot dierence Π In -Π Out decreases in m.

Having observed the platform's fee, rms pay that membership fee and become inside rms until this prot dierence vanishes. The platform reaps the rest of prot dierence until Π In = Π Out .

Therefore in such an equilibrium, κ

= (a+∆) m n-m ( n-m-1 k=0 a k )p 2 out .
And the dierence in prot, turns out to be the platform's membership fee w.

w(m, n, a) = p 2 in -

(a + ∆) m n -m n-m-1 k=0 a k p 2 out ( 28
)
The platform has another option to take all active rms, so that there are no standalone rms in the market. In this case, consumers do not search any rm because rms outside the platform are not active. The prot of an inside rm still has the form of Π In = p 2 in -w -κ and the rest earns zero. Finally, the membership fee becomes w = p 2 in -κ.

An increase in κ rst drives some rms out of the market. Among the fewer rms in the market, the platform chooses a new number of them. The new size of the platform will also have an impact on rest of rms. In case the new platform size is smaller, the prot of an outside rm increases. Secondly several rms who dropped out of the market will be active again. And after observing the returning of rms, the platform will consider again a new size. This process goes on and on until both the platform and rms have no incentives to move. A similar process applies to a decrease in the xed cost. The table below summarizes the stable points after each impact of xed cost.

By leaving several rms alone, the platform can only charge the prot dierence between Tab. 3.2: Joint impact of search cost and xed cost on number of rms.

a\κ 1 × 10 -4 2 × 10 -4 5 × 10 -4 7.5 × 10 -4 1 × 10 -3 2 × 10 -3 3 × 10 -3 5 × 10 -3 0.75 two types of rms. And by taking all active rms, the platform can obtain industry prot. If the search cost is small, the prot dierence is usually less than the prot of a standalone rm.

Therefore sometimes, the platform opts to taking all standalone rms provided a large xed cost even though this engenders more intense in-house competition. If the xed cost drops and too many rms enter the market, then taking another rm eventually becomes more costly than letting it alone. From our numerical sample, the choice of number of rms can even drop in the market size. On the other hand, a large search cost can maintain a certain level of prot dierence, thereby the platform is less tempted to adding in all active rms. The above table

predicts that the agglomeration of rms will more likely take place when the xed cost is large and search cost is small (e.g. auto-mobiles, computers).

To compare this case with section 2-3, the prot of the platform is generally smaller with nitely many rms, from the reason that the outside rm receives a non-negligible prot and the platform charges the prot dierence. Therefore, the platform can not extract as large the dierence in prot as before. The optimal size of the platform is smaller with innite rms, which infers that the most protable size of the platform decreases in the total number of rms in the market. The ratio of Number of Inside firms Total Market Size is decreasing in market size.

Conclusion

The paper constructs an equilibrium where each consumer visits the Internet platform rst and standalone rms in case he can not nd any good match inside the platform. The equilibrium has price dispersion. Prices inside the platform decrease in the size of the platform, and are generally lower than prices outside the platform.

Consumer search and rm agglomeration have been discussed since [START_REF] Wolinsky | Retail trade concentration due to consumers' imperfect information[END_REF]. The idea is that rms in the cluster use lower prices to attract consumers to visit the cluster rst. The larger the search cost, the larger the cluster can be sustained. None of the papers endogenize cost of entry to the cluster, which is the case of most Internet platforms. The paper shows that the opposite relationship between size of the platform and search cost can be found after endogenizing cost of entry, which is due to the "non-appropriability of social surplus". To ease the information under-provision, we can either introduce more platforms or charge both consumers and rms.

The model has several major limitations. First, there is no heterogeneity among consumers in search cost. If there are high-cost and low-cost searchers, low-cost searchers are more likely to obtain better match. The price equilibrium normally depends on the percentages of each type of consumers. Second, there is no quality dierence among rms. If some rms provide products with higher quality, these rms may want to pay more to enter the platform. This improves industry prot and social welfare. Third, a dierent fee structure does not change the main result when the platform is a monopoly. When the platform instead charges per transaction fees, it still grabs all the prot.

Last, searching in the platform may be costly as well and the platform can control the search cost and number of inside rms. One sure thing is that the platform will not set search cost larger than the cost to search an outside rm. If search cost inside the platform is lower, it is logical to think that consumers still search inside rms rst. This requires the generalized framework of [START_REF] Zhou | Prominence and consumer search : The case with multiple prominent rms[END_REF].

A Appendix A

A.1 Proof of Claim 1 : Under the no-search equilibrium a < 1/2, the most protable size of the monopoly platform is two.

Denote the highest match value the consumer could obtain from inside goods as ¯ . Then either

¯ < a -p out + p in or ¯ > a -p out + p in . For ¯ < a -p out + p in , these consumers leave the platform if a -p out + p in > 0.
For ¯ > a -p out + p in , they stay in the platform with or without buying. Under the no-search equilibrium, the outside price is constantly 0.5 and the inside price is solely determined by (10). First, we compare three cases : m = 1, 2, 3. Then we prove that Π p decreases in m when m > 3.

For m = 1, p in = 0.5 and Π p = 0.25. For m = 2, p in ≈ 0.4142 and Π p ≈ 0.3431. For m = 3, p in ≈ 0.322 and Π p ≈ 0.311. Therefore 2 is the most protable size for the platform if m ≤ 3.

Take the derivative of Π p w.r.t. m and obtain :

∂Π p /∂m = p 2 in + 2m p in ∂p in /∂m (A.1)
We substitute ∂p in /∂m from (A.3), substitute p in with (1 -p m in )/m and expand the expression, we obtain :

∂Π p /∂m = p in (p m-1 in (1 + p in -p m in -p in log(p m in )) η(p in ,m) -1)/m(1 + p m-1 in )
It suces to prove that : η(p in , m) < 1. η(p in , m) increases in p in . Since p in < 1 m , we then substitute p in with 1 m to achieve its upper bound. We want to show that this upper bound is smaller than 1 when

m ≥ 3. (1/m) m + (1/m) m-1 f (m) -(1/m) 2m-1 -2(1/m) m log((1/m) m ) g(m)
f (m) decreases in m. The last term decreases in m when m > 3.

Let γ = (1/m) m , g (γ) γ = 2 + 2 log γ > 0, f or γ < 1/27.
In the end, it suces to examine the case with m=3, whose value is approximately 0.3881 < 1. Therefore when search cost is large, the platform never obtains higher prot than the case when m=2. Further, search cost does not play any role since consumers do not search as long as a < 0.5.

Hence if a result holds for any a < 0.5, it must hold for all a < 0.5. We arrive at the result.

A.2 Proof of Claim 2 : The monopoly platform always admit at least two rms under the active search equilibrium a ≥ 1/2.

When a ≥ 1 2 , all consumers who stay in the platform make a purchase. We compare Π p in two cases m = 1, 2.

Π p = :

       (1 -a) 2 , m = 1 2(2 √ a -2a) 2 , m = 2 (A.2)
Let Π p m denote the platform's prot by taking m inside rms :

Π p (m = 1) < Π p (m = 2) ⇔ 1 -a < 2 √ 2( √ a -a) ⇔ 1 + √ a < 2 √ 2 a ⇔ 1 2 √ 2 -1 2 ≈ 0.3 < a A.3 Proof of Claim 3 : ˆ * < 1 -1/m
Since ˆ decreases in m, the above means that at a given search cost, the monopoly platform adds rms until the threshold ˆ is below ˜ = 1 -1/m. From (13), it suces to show that 1 -2ε(m) > 0 at

(2m -1)(1 -1/m) m-1 ζ(m) < 2(1 -1/m) m-1 < 2/e ≈ 0.
74. The rest two terms are dened by φ(m) = 4m log(1-1/m)+ 2m/(m-1). These last two terms are decreasing in m because, ∂φ(m)/∂m = 4(log((m -1)/m) + 1/(m -1)) -2/(m -1) 2 < 0

The above can be proven by,

lim m→2 ∂φ(m)/∂m ≈ -0.7726, lim m→∞ ∂φ(m)/∂m ≈ 0 and, ∂ 2 φ(m)/∂m 2 = 4/m(m -1) -4/(m -1) 2 + 4/(m -1) 3 > 0. Since, lim m→2 φ(m) ≈ -1.545
The rst two terms have an upper bound of 2/e. The last two terms are decreasing and have an upper of -1.545. Therefore, the value of (2m

-1)(1 -1/m) m-1 + 2m 2 log(1 -1/m) is always decreasing and lim m→2 (2m -1)(1 -1/m) m-1 + 2m 2 log(1 -1/m) ≈ -4.04
We arrive at the conclusion.

(iii) To prove that Γ(˜ , m) < 0, we take some linear transformations and it could be simplied to the

proof that ϕ(m) = 1/m -2(1 -1/m) m -2(m -1) log(1 -1/m) > 0. Take the derivative of ϕ(m) in m, ∂ϕ(m)/∂m = -1/m 2 -2(1 -1/m) m (log(1 -1/m) + 1/(m -1)) + -(log(1 -1/m) + 2/m) + < 0
Therefore, ϕ(m) decreases in m. ϕ(2) ≈ 1.386 and lim m→∞ ϕ(m) ≈ 2 -2 e > 0. Therefore, the expression for ϕ(m) is always positive.

A.5 Proof of Proposition 1 Uniform Case

We focus on the uniform case for a -p out ≥ 0. For other cases, the way to prove this proposition is quite similar. First we look at the uniform case. From (12), we can get :

1 + (a -p out + p in ) m-1 f (v) f (v) ∂p in ∂m = (a -p out + p in ) m [1 -log(a -p out + p in ) m ] -1 m 2 (A.4)
Here we only consider the case where m ≥ 1.

Since 0 < x(1 -log[x]) < 1 as long as x ∈ [0, 1],
The right hand-side is positive. Thus one obtains that ∂p in ∂m < 0. Similarly we can prove the case where a < p out .

Non-Uniform Cases

p in = 1 -F m (∆ + a) m (f (v) -v ∆+a F m-1 ( )f ( )d ) , a -p out > 0 (A.5)
To proof the above decreases in m, which is equivalent to, 16

1 -F m (∆ + a) m (f (v) -v ∆+a F m-1 ( )f ( )d ) > 1 -F m+1 (∆ + a) (m + 1) (f (v) -v ∆+a F m ( )f ( )d )
Expand the above,

(m + 1) f (v) -(m + 1) v ˆ F m ( )f ( )d -(m + 1)F m (ˆ )f (v) + (m + 1)F m (ˆ ) v ˆ F m ( )f ( )d > m f (v) -m v ˆ F m-1 ( )f ( )d -m F m+1 (ˆ )f (v) + m F m+1 (ˆ ) v ˆ F m-1 ( )f ( )d
16. Normally, one should take the derivative on both sides and use implicit function theorem. Since we will provide the proof that ∂RHS ∂pin < 0 in A.4.3. We omit that proof in this part.

Since, 17

(m + 1) f (v) -(m + 1)F m (ˆ )f (v) > m f (v) -m F m+1 (ˆ )f (v)
it only suces to show that,

(m + 1)(1 -F m (ˆ )) v ˆ F m ( )f ( )d < m (1 -F m+1 (ˆ )) v ˆ F m-1 ( )f ( )d
which is further simplied to the below because

F (ˆ ) < 1, (m + 1) v ˆ F m ( )f ( )d < m v ˆ F m-1 ( )f ( )d
The left is expanded from integration by parts,

f (v) -F m (ˆ )f (ˆ ) < m v ˆ F m-1 ( )f ( )d -m v ˆ F m ( )f ( )d + m v ˆ F m-1 ( )f 2 ( )d
On the right side it becomes

m v ˆ F m-1 ( )(f ( )(1 -F (ˆ )) + f 2 ( ))d > 0
The above is met since F ( ) > 0 and 1 -F ( ) is log-concave.

Search Cost

Assume the log-concavity of 1 -F ,

p in increases in search cost c. 17. 1 + m F m+1 (ˆ ) -(m + 1)F m (ˆ ) > 0 for any 0 ≤ F (ˆ ) ≤ 1. First, m F m+1 (ˆ ) -(m + 1)F m (ˆ ) falls in ˆ . To see this, the derivative in ˆ is -m(m + 1)(1 -F (ˆ ))F (ˆ ) m-1 f (ˆ ) < 0. Then 1 + m F m+1 (ˆ ) -(m + 1)F m (ˆ ) > Let p in = 1 -F m (∆ + a) m (f (v) -v ∆+a F m-1 ( )f ( )d ) = λ
Using the chain rule, we obtain from the above :

∂p in ∂a - ∂λ ∂p in ∂p in ∂a = ∂λ ∂a
Denote ˆ = ∆ + a, indeed ∂λ ∂p in = ∂λ ∂a = ∂λ ∂ˆ assuming that p in is not linked with a while taking the derivative. Hence we focus on showing ∂λ ∂ˆ < 0, which is to show that :

m F m-1 (ˆ )f (ˆ )D In (p in ) -F m-1 (ˆ )f (ˆ )(1 -F m (ˆ )) < 0 which reduces to the following m f (ˆ )D In (p in ) -f (ˆ )(1 -F m (ˆ )) < 0
The above is met because

D In (p in ) = -f (v) + v ∆+a F m-1 ( )f ( )d .
From the log-concavity of 1 -F , we know that the hazard rate is increasing which suggests that f (ˆ ) > -f 2 (ˆ ) 1-F (ˆ ) . Substitute it to the above, it suces to show :

m D In (p in )(1 -F (ˆ )) + f (ˆ )(1 -F m (ˆ )) < 0 which reduces to the following because D in = 1-F m (ˆ ) m D In (p in ) 1 -F (ˆ ) f (ˆ ) + D In (p in ) < 0 ˆ = a -p out + p in .
From proposition 1, p in < p out , hence ˆ < a. From increasing hazard rate, we know that

1 -F (ˆ ) f (ˆ ) > 1 -F (a) f (a) = p out > p in
From FOC conditions on (3), we know that D In (p in )p in + D In (p in ) = 0 and D In (p in ) < 0. Hence we reach the conclusion.

A.6 Proof of Proposition 3

The problem of the platform is to maximize the total sum of prot by inside rms,

max Π p (w) = m p in D In (p in , p in , p out ; m(w))
FOC indicate that,

p 2 in + 2mp in (∂p in /∂m) = 0 ⇒ p in + 2m(∂p in /∂m) = 0
Replace p in alone with 1-(a-pout+p in ) m m , and ∂p in ∂m according to (A.1). Let ˆ = a -p out + p in . We obtain,

1 -ˆ m m + 2m 1 + ˆ m-1 -1 ˆ m (1 -log ˆ m ) -1 m 2 = 0 Multiply on two sides m (1 + ˆ m-1 ). Dene ψ(ˆ , m) = m (1 + ˆ m-1 ) (p in + 2m ∂p in ∂m ). Then ψ(ˆ , m) = (1 -ˆ m )(1 + ˆ m-1 ) + 2ˆ m (1 -log(ˆ m )) -2 = 0.
From the chain rule (m a = m ˆ ˆ a ) and 0 < ˆ a < 2, it suces to determine sgn(m ˆ ). From the implicit function theorem : must have at most one root between (0 , 1] for any m.

∂m ∂ˆ ψ(ˆ ,m)=0 = - ψ ˆ ψ m ψ(ˆ ,m)=0 (A.8) 1. First let ϕ = ψ ˆ and let φ = ˆ 2-m ϕ = (m -1) -mˆ -(2m -1)ˆ m -2m 2 ˆ log(ˆ ). φ(0, m) > 0 and φ(1, m) = -2m < 0. Since φ(ˆ , m) is a continuous function in [0, 1] × [1, ∞),
2. Second let Then (A.5) and the chain rule imply that equilibrium number of inside rms decreases in search cost. From numerical examples, the number of rms increases to twelve when a = 0.9, and decreases to two when a = 0.5. Finally the unique root of ψ(ˆ , m) = 0 between (0 , 1] implies that for each level of search cost, there will be only one size of the platform and a unique equilibrium membership fee optimizing its prot. We arrive at the result.

Γ(ˆ , m) = ψ m (ˆ , m) = ˆ m-1 -ˆ m -2ˆ 2m-1 -2mˆ m log(ˆ ) log(ˆ ) = Γ(ˆ , m) ˆ m-1 log(ˆ ) Let Γ ˆ (ˆ , m) = -1 -2m -2mˆ m-1 -2m log(ˆ ). Γ ˆ (ˆ ,

A.7 Proof of Corollary 1

Suppose that ε(m) decreases in m, then the term 1 -2ε(m) would increase in m. Then there would not exist any m where the above expression has the value zero unless at m = 0.

∂ ∂m Π p (m) = p 2 in + 2mp in ∂p in ∂m = p 2 in 1 -2ε(m)
From proposition 3 we know that, the unique equilibrium for the membership fee exists and is larger than zero, which contradicts the above. Therefore it can only be that ε(m) starts from a low value, and Therefore when the search cost is larger, by increasing a certain number of slots, the platform is obliged to give away a larger percentage of prot per slot.

A.8 Proof of Proposition 6

Instead of a rigorous proof which is available on-line, we try to prove this proposition by reasoning.

From the monotonicity of (20) in ρ, we know that on the top the platform will capture all consumers by setting the subscription fee exactly to the expected surplus gain from a subscription to the platform.

While prices are transfers, what matters to the platform is how much it can internalized the sum of consumer surplus (essentially from the match) rather than leave it to outside rms, which is essentially how much the optimal search rule prevents consumers from searching. Ideally, consumers should have searched a rm as long as a positive probability exists to obtain a better match. However, the search cost and the price distortion caused by the search cost, enter the searching threshold (as shown below)

and prevent consumers from a better match.

ˆ = a search cost + p in -p out price distortion
If more rms are in the platform, consumers clearly gain more from more variety in the platform and lower inside prices. Therefore less consumers will visit outside rms, which infers that less surplus will be wasted in transportation and less surplus will go to outside rms. Therefore the monopoly platform clearly prefers taking as many rms as possible. After taking all rms, prevailing prices inside the platform drop to zero. Consumers do not incur search cost and they visit a new rm as long as a positive probability exists of obtaining a better match. Distortionary pricing disappears, and the platform actually maximizes social surplus. Since there is no cost of taking another inside rm, it is straightforward to say that the platform will take as many rms as possible.

A.9 Proof of Proposition 7

We assume that p in > p out , and build a contradiction. Then from a + p in -p out > a, we have when

m > 1, (1 -(a -p out + p in ) m )/m < (1 -a m )/m < 1 -a
Therefore, it must be that the second term of ( 26) is larger than the second term of ( 27). Further we know that n-m

Σ n-m-1 k=0 a k > 1 and 1 (a+∆) m > 1.
Then it must be that, for each value of a, a-pout

0 ( + p in ) m-1 ( + p out ) n-m d > a-pout 0 ( + p in ) m ( + p out ) n-m-1 d
From p in > p out , for each ∈ [0, a -p out ], there is,

( + p in ) m-1 ( + p out ) n-m < ( + p in ) m ( + p out ) n-m-1
Taking all above and ( 26)-( 27) into consideration, we can only have

p in < p out
Finally, this is a contradiction to our assumption.

4. WHEN DOES GROUP-BUYING INCREASE PROFIT ?

Introduction

Group-buying or collectively buying is a phenomenon mostly seen on the Web. The mechanism works as follows. A rm posts a normal price, a discounted price, the minimum required number of participants and a date limit on the group-buying website. Upon the deadline, if the number of consumers having promised to make a purchase exceeds the minimum requirement (sometimes buyers are required to pledge some amount.), the rm sends a voucher to whoever promised to buy the product, and the consumer can use the voucher to buy the product at the discounted price. Otherwise, transactions still take place but only at the normal price. A consumer can either buy the product immediately at the normal price, or wait to see whether the price will be lower on the deadline. Furthermore, during the waiting period he can inform other consumers. This mechanism is similar to a referral reward program except that subsequent visitors are oered the same discounted price. Therefore, any invitation not only benets the inviter but also the invitees. From Groupon to "Google Oers" and LivingSocial, thousands of new deals are oered every day. A leader such as Groupon achieved a market capitalization of 17.4 billion dollars.

Even though such deals mostly take place on the Web, group-buying is not a new phenomenon and can be dated back to the pre-Internet era. For example, some airlines and railway companies oer group travel packages. KFC introduced the "family barrel" package. The other form used by Groupon is similar to an on-line promotion. Consumers not knowing preference information about other consumers choose whether to accept the deal simultaneously. Meanwhile the groupbuying websites oer tools (Email, Facebook, Twitter, etc.) to make invitations. Firms often use some heavily discounted prices (≥ 70%) to attract consumers. For example, a concert ticket costs 30 Euros for an individual, but merely 12 Euros if a minimum of 30 people commit to attend.

This practice has several features that set it apart from other popular pricing schemes.

First, dierent from quantity discount models, consumers usually have unit demand. Second, the rm is no longer the only messenger for product information. The group-buying strategy allows buyers themselves to be messengers and invite potential buyers. Consumers might have more information about consumers around than the rm. Group-buying aligns the incentives for consumers to look for lower prices, with the incentives for rms to look for consumers. It is for this reason, that I nd group-buying as an interesting and important mechanism, from which one may draw some market implications.

I rst analyse the model with no invitations, in which N consumers decide simultaneously whether to participate in the group buying option or to buy immediately at the undiscounted price. The results are rather unfavourable to the use of group buying. Given that the monopolist is can commit to a discounted price (that is, it is committed to not lowering it) group buying is dominated by a simple inter-temporal price discrimination scheme. It is optimal to price discriminate because consumers discount the future more than the rm. Intuitively, group buying would amount to committing to a mixed strategy on prices in the second period, which is shown to always be dominated by oering a discounted price with certainty.

In the second analysis, each consumer can not only wait, but also inform one more who are similar to them in terms of willingness to pay. The model is restricted to have two consumers in the rst cohort, therefore at most two more consumers in the second cohort. The monopolist sets prices and the minimum requirement and each consumer decides whether to inform one more consumer.

I nd that, when the inviting cost is low, the monopolist always nds it optimal to set a high minimum and heavily discounted prices. Heavily discounted prices are used to motivate the high-type consumers to invite other high type consumers, and a high minimum helps sorting out the high-type consumers from the rest.

1 This is protable because if the number of consumers turns out to be insucient for the discount, it is likely that these consumers will purchase the product at a large normal price. While setting the low minimum implies no sales at the normal price, setting the high minimum implies more than half of prot coming from selling the product at the normal price. When the inviting cost is large, the group-buying strategy is dominated by either a simple one-price setting, or a time-dependent price setting.

The group-buying phenomenon attracts recent attention from a number of researchers. The paper that is most closely related to my work is that of [START_REF] Chen | Group coupons : Interpersonal bundling on the internet[END_REF] who consider a similar setting but assume a discrete demand with some demand uncertainty and rule out inter-temporal price discrimination. They nd that group-buying may be a protable strategy, even without invitations. In [START_REF] Jing | Group buying : A new mechanism for selling through social interactions[END_REF], consumers are initially heterogeneous in their information about the product. The group-buying strategy is a contract that induces the informed consumer to disseminate information to the uninformed consumers. When the information gap between experts and novice consumers is neither too high or too low, this word-of-mouth 1. According to recent surveys, the average rate of discount is over 50% for deals in Groupon.

model dominates traditional individual-selling strategies. [START_REF] Chen | Group buying commitment and sellers' competitive advantages[END_REF] discusses a particular case when buyers form a group and commit to purchase exclusively from one of two rms who dier in their incentives to invest in quality improvement. In [START_REF] Hu | Simultaneous versus sequential group-buying mechanisms[END_REF], two cohorts of consumers observe the group-buying deal one after the other. Provided exogenous prices and sizes of two cohorts, their paper compares the simultaneous mechanism and the sequential mechanism and nds out that the deal success rate is higher under the sequential mechanism. [START_REF] Gerstner | Customer referral management : Optimal reward programs[END_REF] studies the referral rewards program.

The paper is also related to the price discrimination literatures. Stokey (1979), Landsberger and[START_REF] Landsberger | Intertemporal price discrimination and sales strategy under incomplete information[END_REF] show that the inter-temporal price discrimination is protable when consumers are heterogeneous in time preference. [START_REF] Sobel | The timing of sales[END_REF] studies the cyclical pricing patterns, where rms face continuous inows of consumers. [START_REF] Doyle | Intertemporal price discrimination, uncertainty and introductory oers[END_REF] presents a similar model where rms advertise introductory oers to hold new consumers. [START_REF] Chen | Learning buyers' valuation distribution in posted-price selling[END_REF] consider a seller facing a sequence of potential buyers whose valuations are drawn from the distribution unknown to the seller. The seller learns gradually the distribution of buyers' willingness to pay and lowers down the price over time.

On joint-buying and price discrimination, Alger (1999), using a two-type setting (highvaluation and low-valuation consumers), studies the optimal menu of price-quantity pairs when a continuum of consumers are able to purchase multiple times or jointly. Using a similar twotype setting, Jeon (2005) characterizes a complete (collusion-proof ) contract for the monopolist to achieve the same prot regardless of whether or not buyers can form a coalition. Prices and quality can sometimes vary from time to time, for the purpose of exploiting dierences in consumers' reservation prices.

I considers in turn two monopoly models. In section 2, I propose the N-consumer model where consumers in the rst cohort determine simultaneously whether to accept the deal. In section 3, I propose a two-consumer model where each consumer is able to invite another consumer to accept the deal. Some proofs are relegated to the appendix.

Without Invitations

The market consists of a durable-goods monopolist (M) and N ≥ 2 risk-neutral consumers indexed by i = 1, 2, ..., N , each having unit demand. There is a linear city from zero to ū. M is located at the point zero. Denote the locations of consumers by the vector Θ = (θ 1 , θ 2 ..., θ N ).

Each element of the vector is independently distributed under distribution function F (θ) on [0 , ū]. The reservation price for each consumer is ū, and the transportation cost per unit is 1.

The utility function for the consumer purchasing at price p is given by

U (θ, p) = ū -θ -p (1)
Only the distribution of consumer locations is common knowledge. The rm has no additional information whereas each consumer knows her location, her type, but not that of other consumers.

The game has four stages.

In the rst stage, nature chooses the consumers' locations.

In the second stage, M oers a package (p R , p G , κ) to the N consumers, characterized by the normal price p R , the group buying price p G and a minimum requirement on the number of participants for group buying to take place, κ.

In the third stage, each consumer observes her location and the rm's oer, and choose one of three options : 1) purchase the product immediately at the normal price. 2) register to participate in the group buying option ; participation costs w > 0 and the utility is discounted by a factor β ∈ (0, 1). 3) reject the oer.

In the last stage, if more than κ consumers have accepted to wait for the group buying opportunity, M oers p G to each of them. If not, consumers have two options : 1) buying

at p R . 2) Reject p R .
The rm is committed throughout to the prices announced in stage 2. The pay-o of M is the expected prot. The pay-o of a consumer is the (discounted) surplus over the two periods. I look for Perfect Bayesian Equilibria where the consumers' strategies are symmetric in the sense that, at any information set, all consumers with identical type make the same choice. Let m i be the number of consumers (excluding consumer i) accepting to wait in equilibrium. It is a random variable which, thanks to the symmetry restriction, is the same for all consumers, i. It is henceforth denoted m. Let Γ = P r(m ≥ κ -1) be the probability of a deal success if consumer i also accepts to wait. In equilibrium, this probability can be anticipated without bias by both M and consumers after having observed the oer.

Consider the four options of consumer i. Note that group buying is attractive only if it allows a consumer to purchase at a lower price than if she buys on her own. Hence we necessarily have p G < p R . The consumer's surplus when rejecting the oer is zero. If she purchases the product immediately, her utility is given by U A (θ i ) = ū-θ i -p R . If she waits and purchases the product at the lowest available price no matter whether the discounted price is realized or not, her expected utility is

U B (θ i ) = βΓ (ū -θ i -p G ) + β(1 -Γ)(ū -θ i -p R ) -w
Last, if she purchases only at the discounted price, her utility is U C (θ i ) = βΓ(ū-θ i -p R )-w.

Consumer Behaviour

Consider a consumer in the third stage. Let θG = ū -p G and θR = ū -p R . Hence, θR < θG .

A consumer i with type θ i > θG never buys and receives a pay-o of 0. If θ i ≤ θR , then the consumer necessarily buys and chooses between buying immediately at price p R , thus receiving utility U A or waiting and buying the product at either p G or p R , which yields pay-o U B . The marginal consumer type is at the cut-o θ1 solution to

U A (θ 1 ) = U B (θ 1 ) ⇔ (β -1) (ū -θ1 -p R ) -w loss f rom waiting + βΓ p R -p G discounted price gain = 0 ⇔ θ1 = (β -1) θR -w + βΓ( θG -θR ) β -1 (2)
The rst term in (2) refers to the full surplus loss from waiting until the last stage if it is not associated with a drop in price. The second term refers to the expected gain from the price dierence. High-type consumers (low θ i ) are more tempted to give up the group discount.

For θR < θ i ≤ θG , the consumer buys only if she benets from the group discount. Because there is a positive cost, w, of waiting around for the oer, she chooses between dropping out immediately thus obtaining zero surplus or waiting to see whether the group deal takes place with corresponding expected surplus U C (θ i ). The indierent type is at the cut-o location θ3 that satises Depending on prices, on the required minimum for group buying and on parameter values, a consumer's behaviour may be characterized by up to three cut-os, θ1 solution to (2), θ2 = θ R and θ3 solution to (3). If all three cut-os are relevant, they are ordered as follows.

U C ( θ3 ) = βΓ (ū -θ3 -p G ) -w = 0 (3)
θ1 < θ2 = θR < θ3 < θG < ū

If all regions exist : high valuation consumers, θ i ≤ θ1 , buy immediately at price p R ; consumers with intermediate/high valuations, θ1 < θ i ≤ θ2 choose to wait and buy at whichever lowest price is available in the last stage ; those with intermediate/low valuations, θ2 < θ i ≤ θ3 , choose to participate and buy in the last stage only if group buying is an option ; low valuation consumers, θ i > θ3 do not buy. When M sets a very large κ, the probability of a deal success is close to 0, θ3 approaches θ2 and the mechanism becomes similar to individual sales at p R .

On the other hand, setting a low κ implies that Γ is close to 1, meaning that θ3 approaches θGw β and the mechanism becomes similar to individual sales at p G + w β and p R (some form of inter-temporal price discrimination).

Equilibrium Prices and Prots

The rm's prot function can be written as, (Note that consumers whose types θ i ∈ [ θ1 , θ3 ] accept to wait. )

Π(p * R , p * G , κ * ) = max p R ,p G ,κ Π = p G N i=k i N i F θ3 -F θ1 i 1 -F θ3 -F θ1 N -i purchase at the discounted price + p R N F ( θ1 ) purchase immediately + F ( θ2 ) -F ( θ1 ) F ( θ3 ) -F ( θ1 ) k-1 i=0 i N i F θ3 -F θ1 i 1 -F θ3 -F θ1 N -i
return and purchase at the normal price

subject to θ2 ≤ θ3 , θ3 ≤ θG - w β (4)
Demand is composed of three parts. 1) High-type consumers θ i ≤ θ1 purchase the products immediately at the normal price. 2) Between ( θ1 , θ3 ], consumers purchase the products at the discounted price if the discounted price is realized. 3) If the discounted price is not realized, only those between ( θ1 , θ2 ] still purchase the products.

I now show that the prot function can be rewritten a function of the thresholds θ1 , θ2 , θ3

and θG , where θ1 is the function of the rest according to (2), as follows,

Π(p R , p G , κ) = N p R F ( θ1 ) + N p R F ( θ2 ) -F ( θ1 ) + N p G F ( θ3 ) -F ( θ1 ) -p R F ( θ2 ) -F ( θ1 ) w β ū -θ3 -p G -1 (5)
where p G =ū -θG , p R = ū -θ2

Indeed, the expected number of consumers, j, whose types are θ j ≤ θ1 , where the expectation as,

N i=k i N i F θ3 -F θ1 i 1 -F θ3 -F θ1 N -i =N N i=k N -1 i -1 F θ3 -F θ1 i 1 -F θ3 -F θ1 N -i =N N -1 i=k-1 N -1 i F θ3 -F θ1 i+1 1 -F θ3 -F θ1 N -i-1 = N F θ3 -F θ1 N -1 i=k-1 N -1 i F θ3 -F θ1 i 1 -F θ3 -F θ1 N -i-1 =N F θ3 -F θ1 Γ = N F θ3 -F θ1 w β(ū -θ3 -p G )
The last equation comes from (3). Seen from ( 5), the prot is proportional to the total number of consumers. M's problem is to maximize the expected prot per consumer. Its problem can be fully reduced to choosing the cutting points. To solve this non-linear programming problem, I write the Lagrangian as the following,

L ( θ2 , θ3 , θG ) =(ū -θ2 ) F ( θ2 ) + w β ( θG -θ3 ) (ū -θG ) F ( θ3 ) -(ū -θ2 ) F ( θ2 ) + ( θG -θ2 ) F ( θ1 ) -λ 1 ( θ3 -θG + w β ) -λ 2 ( θ2 -θ3 ) (7)
Appendix A.1 solves the Lagrangian problem in detail. The solution has either of the constraints binding. If θ2 ≤ θ3 turns out to be binding, the group-buying strategy has κ > N .

Otherwise if θ3 ≤ θGw β is binding, the group-buying strategy has κ = 1. After comparing the two congurations κ > N and κ = 1, I nd the following.

Proposition 1. If the discount factor 0 < β < 1, M sets κ = 1. The optimal group-buying strategy degenerates to an early price and a late price.

Proof. See the appendix.

The idea behind the result is as follows. The group buying strategy is a way for the rm to commit to a pricing scheme where the consumer may either pay p R immediately or wait in order to have access to a discounted price p G that he may obtain with probability σ. The proof shows that if the rm had to select such a random pricing scheme, it would select σ = 1.

In view of this negative result as to the protability of group buying for the rm, I now turn to an analysis of how it might be protable if consumers have the opportunity to invite other consumers who, otherwise, would not have been aware of the availability of the product.

With Invitations

There are two ways in practice in which consumers can make invitations. First, consumers can inform directly other consumers. Second, group-buying websites often oer tools (Facebook, Twitter, etc.) facilitating communication with potential buyers. Sharing deals takes time and eort. What prompts consumers to invite others is that, when more consumers are informed about the deal, more are likely to participate in the deal, and the minimum is more likely to be met.

Consider the following monopoly model with two consumers (A and B), Each of the two consumers observes the deal and can at most inform one more consumer about the deal. Inviting cost is c ≤ ū. Assume that the consumer invited by consumer A is dierent from the consumer invited by consumer B. Assume that the newly informed consumer has the same type as the consumer informing her.

For simplicity, I assume that future utility is not discounted (β = 1) and waiting is costless (w = 0). Taking I as the decision to inform the other consumer, the utility function of the consumer of type θ is,

U (θ, p, I) = ū -θ -p -I × c (8)
The distribution F (θ) can be observed by consumers and the monopolist, while the type of each consumer can be observed only by herself.

The game has three players : M and consumer A, B and the following timing.

In the rst stage, nature chooses locations of consumers.

In the second stage, M proposes the oer (p R , p G , κ) to consumer A and B.

In the third stage, consumers observe the oer and choose simultaneously one of the three options : 1) accepting the oer and informing another consumer. 2) Accepting the oer without informing anyone. 3) Rejecting the oer.

In the fourth stage, the demand is realized. If the minimum is met, the discounted price is oered to the consumers who accept the oer. Otherwise, only the normal price is available.

Again, the rm is committed throughout to the prices announced in stage 2. As in the model without invitations, I look for Perfect Bayesian Equilibria, assuming consumers' strategies are such that consumers with identical types make the same choice.

There are two consumers in the rst cohort, so that κ can be ranged from 2 to 4. Based on κ, optimal prices and consumers' best responses dier. The next few sections discuss the optimal pricing strategy for each value of κ.

Low Minimum (κ = 2)

When κ = 2, there are two manners for the minimum to be met. The rst manner is that one of the consumers informs the other consumer. By this manner, the discounted price is oered with probability 1. The second manner is that both consumers accept the oer and wait. Without losing generality, consider consumer A's choices. If she informs the other consumer, her utility is,

U 1 (θ A , p G , p R ) = ū -θ A -p G -c
If she simply accepts the oer without informing anyone, her expected utility is

U 2 (θ A , p G , p R ) =          ū-p G ū ū -θ A -p G + p G ū (ū -θ A -p R ), 0 ≤ θ A ≤ θR ū-p G ū ū -θ A -p G , θR ≤ θ A ≤ θG ū-p G ū
is the probability that the other consumer accepts the oer. p G ū is the probability that the other consumer rejects the oer. Suppose that there is a threshold of θ, say θ1 . If consumer A's type θ A = θ1 , she is indierent between informing the other and not informing.

For 0 ≤ θ A ≤ θR , the dierence U 1 (θ A , p G , p R ) -U 2 (θ A , p G , p R ) = (p R -p G ) p G ū -c
The dierence does not depend on θ A , implying that consumer A's optimal decision is the same for all 0 ≤ θ A ≤ θR . If prices are set such that (p R -p G ) p G ū -c < 0, no consumers make invitations, which is certainly not an interesting case. For this reason, I focus on the optimal price setting where

(p R -p G ) p G ū -c ≥ 0. For θR ≤ θ A ≤ θG , let U 1 ( θ1 , p G , p R ) = U 2 ( θ1 , p G , p R ) ⇔ θ1 (κ = 2) = ū -p G - c ū p G
When θ A is smaller than the threshold, accepting the oer and informing the other consumer dominates the other two options.

The normal price is set merely to ensure that consumers do not purchase at the normal price.

To see this, suppose that p R < p G + c. Consumers receiving positive surplus from accepting the oer and informing another consumer would rather purchase at the normal price. We are then back to the model without invitations.

If consumer A and B inform another consumer, the total demand is 4. When only one consumer makes the invitation and the other consumer participates without making any invitation, the total demand is 3. When only one consumer makes the invitation and the other consumer does not participate, the total demand is 2. When both consumers participate without inviting any other consumer, the total demand is 2. The following gure illustrates the composition of demand.

By changing the left panel into the right panel does not change the demand, but simplies the demand function. The demand function is,

D(p R , p G ; κ = 2) = 4 ū ū -p G - cū p G + 2c ūp G (ū -p G ) (9)
; κ = 2) with respect to p G is, ∂ ∂p G D(p R , p G ; κ = 2) = 2c p 2 G - 4 ū
From the rst-order condition,

p * G (κ = 2) = 2ū -c 4 (10)
The indirect prot function is given by,

Π(p * R , p * G ; κ = 2) = ū + c 2 4ū -3c (11)
As p G is set according to (10), the equilibrium θ1 can be zero if the inviting cost is large enough Similarly, there are two manners for the minimum to be met. By the rst manner, both consumer A and B inform the other consumer, and the total demand will be 4. By the second manner, one of them informs another consumer and the other consumer only accepts to wait.

. Solve θ1 = ū -p G -c ū p G = 0 for p G . I have p G = (ū - √ ū2 -4cū)/2.
In case consumer A informs the other consumer, her expected utility is given by

U 1 (θ A , p G , p R ) =          ū-p G ū ū -θ A -p G -c + p G ū (ū -θ A -p R -c), 0 ≤ θ A ≤ θR ū-p G ū ū -θ A -p G -c + p G ū (-c), θR ≤ θ A ≤ θG (12)
ū-p G ū refers to the probability that consumer B participates in the group-oer. p G ū refers to the probability that consumer B rejects the oer. If consumer A accepts the oer without informing anybody, her expected utility is given by,

U 2 (θ A , p G , p R ) =          θ1 ū ū -θ A -p G + ū-θ1 ū (ū -θ A -p R ), 0 ≤ θ A ≤ θR θ1 ū ū -θ A -p G , θR ≤ θ A ≤ θG (13)
where θ1 ū stands for the probability that consumer B accepts the oer and informs another consumer. Since U 1 (θ A , p G , p R ) decreases continuously in θ A and lim θ A ->ū U 1 (θ A , p G , p R ) < 0, 2. Substitute (10) into the expression of θ1 . I obtain,

θ1 = 1 4 2ū + c - 16 ū c 2ū -c , d θ1 d c = c 2 -4ūc -28ū 2 4(2ū -c) 2 < 0
there must be a cut-o point where consumer A is indierent between informing another consumer, and not.

Suppose that this cut-o point is θ1 . There is θ1 ≥ θR . To see this, suppose θ1 < θR . Calculate the dierence in utility for 0

≤ θ A ≤ θR , U 1 (θ A , p G , p R ) -U 2 (θ A , p G , p R ) = ( θG -θ1 ū )(p R -p G ) -c
If consumer A informs the other consumer, the chance of getting the discounted price is ( θG ū ).

If he simply participates in the group-oer without informing anyone, the chance is ( θ1 ū ). The dierence in chance is ( θG -θ1 ū ). Hence the price dierence must compensate for the cost, that is

p R -p G ≥ c ū θG -θ1
in order for consumer A to make the invitation. The utility dierence does not depend on θ A . If at some θ A < θR , consumer A is indierent between informing somebody and not informing, then consumer A must be indierent at any θ A < θR . Suppose that she is indierent at any θ A < θR , it follows that no consumers invite anyone, and we come back the model without invitations.

For this reason, the unique existing θ1 satises θ1 ≥ θR

. Solving U 1 ( θ1 , p G , p R ) = U 2 ( θ1 , p G , p R ) for θ1 yields. θ1 (κ = 3) = ū - √ ūc -p G
The demand is illustrated in the following gure.

The right panel demonstrates the same demand pattern as the left panel. The prot function

Π(p R , p G ; κ = 3) = 4p G ū2 ū -p G - √ ūc 2 selling four + 6p G ū2 ū -p G - √ ūc √ ūc selling three + 4p R ū -p R ū p G ū offer the normal price subject to θR ≤ θ1 and U (θ, p R ) ≤ U 1 (θ, p G , p R ) f or θ ≤ θ1 (14)
Consider the case where both constraints are inactive. The rst-order condition for p R is ∂Π(p R , p G ; κ = 3)/∂p R = 0, and apparently the optimal normal price is p * R = ū/2. To see this, suppose that consumer A informs another consumer and consumer B rejects the oer so that the minimum is not met in the last stage. Then clearly consumer A and the informed consumer have types θ ≤ θ1 . Setting the normal price to p R , M's prot is then

2p R (ū-p R ) ū . Solving ∂ 2p R (ū-p R ) ū /∂p R = 0 yields p * R = ū/2. Solving ∂Π(p R , p G ; κ = 3)/∂p G = 0 yields the following. p * G (κ = 3) = 1 6 4ū - √ ūc -ū2 + 7ūc -2ū √ ūc
The above contradicts θ1 > θR . Substitute the above into θ1 (κ = 3), I have θ1 (κ = 3) < ū/2

(shown in the appendix). Consider the binding case ( θ1 = θR ) where p R is set so that each consumer who did not reject the oer purchases at p R , that is p R = ū -θ1 (κ = 3). The prot function is then

Π(p R , p G ; κ = 3) = 4p G ū2 ū -p G - √ ūc 2 selling four + 6p G ū2 ū -p G - √ ūc √ ūc selling three + 4(p G + √ ūc) θ1 ū p G ū offer the normal price First-order condition in p G yields that p * G = ū - √ ūc 2 , p * R = ū + √ ūc 2 (15) Consider any θ < θ1 = θR . Examine the constraint U (θ, p R ) ≤ U 1 (θ, p G , p R ) for the optimal prices from above. U 1 (θ, p G , p R ) -U (θ, p R ) = ū -p G ū (ū -θ -p G -c) + p G ū (ū -θ -p R -c) -(ū -θ -p R ) = ( √ ūc -c)/2 > 0
Therefore no consumers purchase the product immediately and leave the market. The indirect prot function is,

Π(p * R , p * G ; κ = 3) = ( √ ū - √ c) 2 (2 √ ū + 3 √ c) 2 √ ū 4.3.3 High Minimum (κ = 4)
As opposed to the low (medium) minimum, there is only one manner by which the minimum can be met. Either both consumer A and B inform other consumers, or the minimum is not met. If consumer i does not inform anybody, either she buys at the normal price and receives ū -θ -p R or she rejects the oer and receives 0. Suppose that if a consumer's type is θ1 , she is indierent between informing somebody and not informing. Intuitively, for any consumer-type θ i < θ1 , she chooses to inform. Write consumer A's expected utility when he informs somebody as,

U 1 (θ A , p G , p R ) =          θ1 ū ū -θ A -p G -c + ū-θ1 ū (ū -θ A -p R -c), 0 ≤ θ A ≤ θR θ1 ū ū -θ A -p G -c + ū-θ1 ū (-c), θR ≤ θ A ≤ θG (16) If consumer A does not inform anyone, his utility is, U 2 (θ A , p G , p R ) =          ū -θ A -p R , 0 ≤ θ A ≤ θR 0, θR ≤ θ A ≤ ū
First, I show that there exists not a θ1 satisfying θ1 < θR . Suppose that such a θ1 exists.

Calculate the utility dierence for 0

≤ θ A ≤ θR , U 1 (θ A , p G , p R ) -U 2 (θ A , p G , p R ) = θ1 ū (p R -p G ) -c
The above is clearly positive. By denition, the chance of getting a discounted price is ( θ1 ū ) 2 .

Therefore, M has to oer a price dierence p R -p G ≥ ( ū θ1

) 2 c in order to get consumers to invite others. Hence, U 1

(θ A , p G , p R ) -U 2 (θ A , p G , p R ) = 0 has no solutions for 0 ≤ θ A ≤ θR .
θ1 ≥ θR , solve U 1 ( θ1 , p G , p R ) = 0 for θ1 . θ1 (κ = 4) = 1 2 ū -p G + (ū -p G ) 2 -4ūc (17)
The demand is illustrated in the following gure. Again, the right panel illustrates the same demand function as the left panel. The monopolist's problem is given by max

Π(p R , p G ; κ = 4) = 4p G θ1 ū 2 the minimum is met + 4p R ū -p R ū ū -θ1 ū the minimum is not met (18) subject to θR ≤ θ1 and U (θ, p R ) ≤ U 1 (θ, p G , p R ) f or θ ≤ θ1
First, consider that neither of the constraints is active. First-order condition in p R yields that p R = ū/2. Solving θR = θ1 yields that p G = ( ū-4c 2 ). To show that the optimal p G from rst-order conditions does not satisfy θR < θ1 , it suces to demonstrate that Π(p R , p G ; κ = 4) keeps increasing in p G even at p G = ( ū-4c 2 ), as long as c ≤ ū. Therefore M keeps increasing p G until the constraint θR ≤ θ1 is binding.

∂Π(ū/2, (ū -4c)/2; κ = 4)

∂p G = 2c ū2 ū2 + 24c 2 + 6c(|ū -4c| -ū) > 0
Then the constraint θR ≤ θ1 is binding, equivalent to that p R = ū -θ1 . p R is low enough that when the minimum is not met, then those informing the other consumer always make a purchase. The monopolist's problem is then, According to (21), the optimal discounted price p G = 0 for ( √ 5 -2)ū < c ≤ ū/4. Suppose that the optimal discounted price p G = 0, then we are back to ( 18) and ( 19). Fixing p G = 0, For c > ū/4, there are no solutions for U 1 ( θ1 , p G , p R ) = 0. No invitations take place and we are back to the model without invitations.

max Π(p R , p G ; κ = 4) = 4p G θ1 ū 2 the minimum is met + 4(ū -θ1 ) θ1 ū ū -θ1 ū the minimum is not met

Comparative Statics

The above pricing strategies can be summarized into the following proposition. Proof. See the appendix.

Intuitively, larger inviting cost forces M to set a lower discounted price (or a higher normal price) to motivate consumers to inform others, particularly when a higher minimum is set so that consumers faces a lower probability that the minimum is met in the last stage. Besides, larger inviting cost helps sort out consumers having high valuations in the sense that only those with higher valuations invite others. Thus, a higher normal price can be oered to each of them if the minimum is not met. The following illustrates the optimal pricing strategy at dierent minimums. The following gure shows that the expected prot decreases in inviting cost. Both the κ = 2 conguration and the κ = 3 conguration are dominated by the κ = 4 conguration. In the polar case when inviting cost is very large, the group-buying strategy is suboptimal.

Proposition 3. Setting the high minimum always dominates setting the low (and medium) minimum.

There exists a threshold of inviting cost c = ū/4. For c < c , the monopolist uses the groupbuying strategy. For c ≥ c , the monopolist uses a single-price setting.

Proof. See the appendix. Setting dierent prices is about encouraging high-type consumers to invite other consumers, and the trade-o between motivating more consumers to inform others and setting lower discounted prices. The best situation is when both consumer A and B invite others, as they invite only high-type consumers willing to pay a higher discounted price. A worse situation is when only one consumer invites another and the other does not and free-ride on the rst one. The discounted price must be low enough to let the free-rider accept the oer. The worst situation is when there are two free-riders.

The minimum should be set in the same vein, and if possible, to sort out the high-type consumers. The κ = 2 conguration oers the discounted price even to the situation with two free-riders, while the κ = 3 conguration oers the discounted price to the situations allowing at most one free-rider. Both congurations indeed encourage high-type consumers, while they fails to keep the discounted price away from the low-type. In the equilibrium under large inviting cost, selling two only to the high-type is sometimes more protable than selling three to both the high-type and the low-type.

Here is a related example. Suppose that consumer A is located at x (therefore can pay at most ū -x), and consumer B is located at y x (therefore can pay at most ū -y). The monopolist observes the location information (without knowing who is who). Setting κ = 3 and selling 3 give the prot 3(ū -y -c), whereas setting κ = 4 and selling 2 to consumer A and the informed give 2(ū -x -c). ū -y being small enough, 2(ū -x -c) > 3(ū -y -c) ; selling 2 is more protable than selling 3.

The κ = 4 conguration keeps all low-type away because each consumer has to make the invitation for the possibility of getting the discounted price. No matter whether the minimum is not realized, only the high-type consumers purchase. In equilibrium, consumers having made invitations purchase at the normal price no matter the minimum is met or not in the last stage.

The share coming from failing to sell at the discounted price can be always above 50% for κ = 4 This percentage can decrease or increase in inviting cost. In the polar case, p G is set to zero.

Under the κ = 3 conguration, the chance that the "group oer" is carried out in equilibrium is between 0% and 33%. Under the κ = 4 conguration, the chance that the "group oer" is carried out in equilibrium is between 0% and 25%. M provides a low p G as the bait for invitations, gains nothing from the successful "group oer" and expects that the "group oer" cannot be carried out and some consumers purchase at p R .

Corollary 1. In the optimal conguration (κ = 4), the monopolist gains larger prot from the normal price than from the discounted price.

Proof. λ(p * R , p * G ; κ = 4) = (ū+c) 2 2ū(ū-c) > 1 2 ⇔ -2c(3ū + c) < 0.

Consumer Surplus

There are two measures of consumer surplus, respectively from the rst cohort and the second cohort. Take (8) as the form of utility function. It is dicult to measure consumer surplus of the second cohort because low-type consumers are never informed.

First Cohort

For the rst cohort of consumers, her expected consumer surplus when κ is set to 2 is given by, (here θ1 = θ1 (κ = 2))

CS (1) (κ = 2) = 1 ū θ1 0 (ū -θ -p G -c)dθ + 1 ū ū-p G θ1 ū -p G ū (ū -θ -p G )dθ (24)
For κ = 3, let θ1 = θ1 (κ = 3), Here I draw the comparison of consumer surplus when dierent minimums are set. The "κ = 2" case always delivers lower consumer surplus than other cases. The "κ = 3" case always delivers larger consumer surplus than other group-buying cases.

CS (1) (κ = 3) = 1 ū θR 0 ū -p G ū (ū -θ -p G -c) + p G ū (ū -θ -p R -c)dθ+ (
Proposition 4. in the optimal conguration, the group-buying strategy lowers down the surplus of the rst-generation consumers. Proof. I plug the respective optimal price into (24), and rearrange the terms. Under the κ = 3 (and κ = 4) conguration, the surplus function is illustrated as the blue curve in Figure 9. Again consumers of high types are worse o compared with monopoly pricing, while consumers of low-types are better o. Summing up these two eects yields lower expected surplus for a consumer of the rst generation.

Second Cohort

For the second cohort of consumers, I have to assume that there is a one-to-one relationship between each consumer of the second cohort and his counterpart in the rst cohort. For example, for each second-cohort consumer having θ < θ1 , there must exist a rst-cohort consumer having θ < θ1 who invites him. For the κ = 2 conguration, I have,

CS (1) (κ = 2) = 1 ū θ1 0 (ū -θ -p G )dθ (28)
For the κ = 3 conguration, let θ1 = θ1 (κ = 3),

CS (1) (κ = 3) = 1 ū θR 0 ū -p G ū (ū -θ -p G ) + p G ū (ū -θ -p R )dθ = 1 8 ( √ ū - √ c) 2 (1 -2 √ ūc) (29)
For the κ = 4 conguration, let θ1 = θ1 (κ = 4),

Two Cohorts

In the end, I compare the sum of consumer surplus of the rst cohort and the second cohort with the monopoly pricing case. It is clear that consumer surplus is larger using the group-buying strategy as long as inviting cost is not too large, which along with our result in industry prot, demonstrates that social welfare is larger using group-buying.

Conclusion

This paper builds two tractable models, both oering clear-cut results. In the model without invitation, the group-buying strategy is always suboptimal with respect to either a single-price setting or a time-dependent price setting. In the model with invitation, the monopolist uses the group-buying strategy with a high minimum when inviting cost is low, and comes back to the single-price setting when inviting cost is large. Consumers of high types are worse o, and consumers of lower types are better o.

The paper has three major limitations. From the most important to the least important, the rst limitation is that consumers are merely allowed to inform one more consumer. If this restriction is relaxed, then consumers with highest types are likely to inform more than one consumers, which may exacerbate the "free rider" problem because low-types consumers are more ready to wait.

The second limitation is that consumers merely invite other consumers with the same consumer-types. I have considered a similar model where each consumer is either a local or non-local, with probability ρ and 1 -ρ. A local informs another consumer with the same type, and a non-local informs another consumer whose type is randomly drawn from U [0, ū]. Some preliminary simulation results have suggested that prices are lower when ρ is smaller.

The third limitation is that this is a monopoly model. As soon as I introduce competition, the model becomes less tractable. For example, there are no longer closed-form solutions for the minimums under κ = 3 and κ = 4. Further, asymmetries can arise such that rms choose dierent minimums. In what follows, I abstract from the fact that in a group buying arrangement, the probability Γ should be given by the cumulative of a binomial distribution. so I solve a relaxed problem where Γ can take any value in [0, 1]. From (3), it can be rewritten as a function of thresholds, Γ = w β ( θG -θ3 )

I check ex post that the solution to the relaxed problem can be implemented by merely oering two prices without resorting to group buying.

Plug the above into (2), and rewrite θ1 into the expression of θ2 , θ3 and θG .

(β -1)(ū -θ1 -p R ) -w + w( θG -θ2 ) θG -θ3 = 0 ↔ θ1 = θ2 -w ( θ3 -θ2 )

(1 -β) ( θG -θ3 )

(A.1)
The three derivatives of θ1 in other variables are, ∂ θ1

∂ θ2 = 1 + w (1 -β)( θG -θ3 ) , ∂ θ1
∂ θ3 = -w ( θG -θ2 ) (1 -β)( θG -θ3 ) 2 , ∂ θ1
∂ θG = w( θ3 -θ2 ) (1 -β)( θG -θ3 ) 2
θ3 -θG + w β ≤ 0 is binding, (3) implies that Γ = 1, meaning that each consumer will be oered the discounted price if he waits until the last stage. From (A.1), we knows that θ1 = θ2 = θ3 . θ1 and θ3 are determined by

θ1 = ū + w -p R + βp G 1 -β , θ3 = ū -p G - w β
Note that p R = p G implies that θ1 = θ3 . M maximizes the joint prot, which is given by

Π(p G , p R ) = p G F (ū -p G - w β ) -F (ū + w -p R + βp G 1 -β ) + p R F (ū + w -p R + βp G 1 -β )
The prot function can be rewritten as the following by replacing p R , p G with θ1 , θ3 . Suppose that θ1 = θ3 . Then the above two conditions yield, F ( θ1 )

f ( θ1 ) = w β(1 -β) , F ( θ1 )
f ( θ1 ) = ū -θ1 -w β β
Clearly, there is not a single θ1 satisfying the above two. Dans le second modèle, sur invitations, nous avons montré que la stratégie de prix, caractérisée

Fig. 2

 2 Fig. 2.1: Market shares under perfect information

Fig

  Fig. 2.2: Market shares in the generic-niche ordering

Fig. 2

 2 Fig. 2.4: Levels of c (blue) and c (red).

Fig. 2. 5 :

 5 Fig. 2.5: Discontinuous prices in the generic-niche ordering (µ G /µ N = 1/2).

Fig. 2

 2 Fig. 2.6: Market shares in the niche-generic ordering

Fig. 2

 2 Fig. 2.7: Two existing equilibria in the niche-generic ordering

Fig. 2

 2 Fig. 2.8: Price levels for each ordering (location) at c = 1/20.

  have p N ≺G G > µ N . The market being covered, consumers pay at least p N ≺G

Fig. 2

 2 Fig. 2.9: Prot of the generic rm at each ordering at c = 1/20.

  is ambiguous which rm consumers will visit rst. For convenience, we draw consumers' decisions in the following graph. If the parameters are in the blue area, each consumer visits the niche rm rst. If parameters are in the grey area, consumers visit the generic rm rst.Some numerical examples can show us exactly how large is the loss (gain) surplus from consumer

Fig. 2 .

 2 Fig. 2.10: Consumers' decisions about product search.

Fig. 2 .

 2 Fig. 2.11: Percentage of searchers from the competition outcome

Fig. 2

 2 Fig. 2.14: Surplus comparison under models with the outside option

Fig. 2

 2 Fig. 2.15: Welfare comparison assuming the covered market.

Fig. 2 .

 2 Fig. 2.16: Welfare comparison with assumption 4.

  -generic ordering. In the generic-niche ordering, the niche rm's demand share is less than half, because p G≺N N it is clear the niche rm receives larger prot in the niche-generic ordering.

  between ∆ and η(µ N , µ G , c) is smaller. We use the bound function to simplify the price dierence function in the proof of Lemma 1. C Incomplete Information w. Outside Option C.1 Market Share In the below gure, demand shares in the generic-niche order are shown on the left-hand side, and demand shares in the niche-generic order are shown on the right-hand side. D 1,2 and T 3,4 represent the demand of Firm G. T 1,2 and D 3,4 represent the demand of Firm N. D 5 and T 5 represent those drop-os. Independent of the product order, D 5 and T 5 always have the same size. Assumption 4 ensures that D 5 and T 5 always have positive sizes. And, assumption 5 ensures that D 2,4 and T 2,4 always have positive sizes.

-

  -form game with imperfect information, three types of players are : the platform, rms and consumers. Match realization is decided by nature. The platform does not observe any match information (neither can rms) while setting w, although it knows the distribution of the match value. A rm can observe other rms' decisions of whether to enter the platform before setting its price. Firms set prices simultaneously. Consumers can observe the platform's size and prices set by inside rms. They learn price and match information of outside rms only through searching them. However, they have beliefs about the distribution of prices outside the platform.

Fig. 3

 3 Fig. 3.1: Share of demand with two inside rms and a ≥ p out

Fig. 3

 3 Fig. 3.3: Impact of search cost on consumer surplus

Proof:

  In section 3, the monopoly platform (store) is maximizing mp 2 in (m). The monopoly platform (store) takes new entrants until ε(m) = 1 2 . However, in the duopoly case, two stores are maximizing respectively mp 2 in (m + k) and kp 2 in ( m + k), and each takes new entrants until ε(m + k) = 1 2 , as the following :

Fig. 3

 3 Fig. 3.4: Impact of m on the prices at a = 3/4 and n = 30.

Fig. 3 . 5 :

 35 Fig. 3.5: Impact of m on inside prices at a = 3/4 and n = 10.

  there must exist at least one point where φ(ˆ , m) = 0.Its derivative φ ˆ = -(m + 2m 2 ) + (m -2m 2 )ˆ m-1 -2m 2 log(ˆ ). For m ≥ 2, φ ˆ (ˆ , m) declines from φ ˆ (0, m) > 0 to φ ˆ (1, m) < 0.Therefore we can say that φ increases in ˆ until some ˆ > 0 then becomes negative till φ(1, m) = -2m < 0. It becomes clear that φ ˆ (ˆ , m) and φ(ˆ , m) are quasi-concave and start from positive values. Hence ϕ(ˆ , m) = 0

  m) decreases from positive to negative and changes the sign only once. Particularly Γ(0, m) = 1 and Γ(1, m) = -2. Therefore both Γ(ˆ , m) and Γ ˆ (ˆ , m) are quasi-concave. Γ(ˆ , m) = 0 must have the unique root between [0 , 1].3. φ ˆ (0, m) > 0, φ ˆ (1, m) < 0, φ(0, m) > 0, φ(1, m) = -2m < 0, ψ(0, m) = -1 and ψ(1, m) = 0infer that ψ(ˆ , m) increases from negative to zero and ψ(ˆ , m) = 0 has the unique root between (0 , ¯ ). Since ϕ(ˆ , m) = 0 has at most one root between (0 , 1], ψ(ˆ , m) can not increase between ( ¯ , 1), which implies that ψ(ˆ , m) is quasi-concave and ψ(ˆ , m) = 0 has the unique root between (0 , 1). 4. Finally construct ˜ (m) = 1 -1 m . The platform chooses ψ(ˆ * , m) = 0. A.4, (i) proves that ψ(˜ , m) > 0, for which we have ˆ * < ˜ . Hence the platform never chooses thresholds above m-1 m . (ii) proves that ϕ(ˆ , m) > 0 for ˆ ∈ (0, ˜ ]. (iii) proves that Γ(˜ , m) < 0. Then Γ(ˆ , m) < 0 for each ˆ ∈ (0 , ˜ ]. Therefore ϕ and Γ do not change their signs until ˜ . Henceforth Γ(ˆ * , m) • ϕ(ˆ * , m) < 0.

  increases in m. The following expression gives us the negative sign of ∂ε(m)/∂a.

Fig. 4

 4 Fig. 4.1: Consumer i's Decision on the Linear Segment.

  whose locations are indicated by the vector Θ = (θ A , θ B ). Each consumer has unit demand and reservation price ū. θ i is distributed under the distribution function F (θ) for θ i ∈ [0, ū]. The monopolist (M) is located at the point zero, with zero marginal cost of production. Transportation cost per unit is 1.

Fig

  Fig. 4.2: Demand composition when κ = 2.

Fig

  Fig. 4.3: Demand composition when κ = 3.

Fig

  Fig. 4.4: Demand composition when κ = 4.

  subject to θR = θ1 and U (θ, p R ) ≤ U 1 (θ, p G , p R ) f or θ ≤ θ1 p G can be written into an expression of θ1 . Solve (17) for p G , I have p G (κ = 4) = ūθ 1 -ūcinto (19). Solving the rst-order conditions of (19) with respect to θ1 yields θ1 = ūexamine the constraintU (θ, p R ) ≤ U 1 (θ, p G , p R ). U (θ, p * R ) -U 1 (θ, p * G , p * R ) = (ū -θ -p * R ) -(ū -θ -p * G -c) θ1 ū + ū -θ1 ū (ū -θ -p * R -c) = 0The above means that prices are set such that consumers who make purchase in the last stage are always indierent between informing other consumers and not informing. The prot function is given by, for c ≤ (

  The monopolist's problem is given by,max Π(p R , p G ; κ = 4) = 4p R ū -p R ū ū -θ1 ū subject to θR ≤ θ1 and U (θ, p R ) ≤ U 1 (θ, p G , p R ) f or θ ≤ θ1If neither of the constraints are binding, rst-order conditions yield p R = ū/2, which vio-lates U (θ, p R ) ≤ U 1 (θ, p G , p R ) for θ ≤ θ1 (proof omitted). Therefore the constraint U (θ, p R ) ≤ U 1 (θ, p G , p R ) is binding. Solving the constraint U (θ, p R ) = U 1 (θ, p G , p R ) for p R yields p R = ūc θ1 .Replace p R in the above prot function with ūc θ1 . Rearranging the terms gives the prot function below. The optimal price also satises θR = θ1 .Π(p * R , p * G ; κ = 4) = 2c(1 + 1 -

5 -

 5 2)ū < c ≤ ū/4

Proposition 2 .p

 2 While setting κ = 2, the pricing functions and the prot function are, While setting κ = 4, price equilibrium satises, for c ≤ ( G decreases in inviting cost and p R increases in inviting cost. Industry prot decreases in inviting cost. All price and prot functions are homogeneous of degree one in ū and c.

Fig. 4 . 5 :

 45 Fig. 4.5: Optimal discounted prices under various κ (ū = 1).

Fig. 4

 4 Fig. 4.6: Measures under various κ (ū = 1).

(

  above 40% for κ = 3), illustrated in Figure7. To see this, consider the indirect prot function for κ

Fig. 4

 4 Fig. 4.7: Prot share coming from sales at p R (ū = 1).

  θ -p R -c)dθ = (ū -c) 2 /8ū < ū/8 (27)

Fig. 4

 4 Fig. 4.8: First-cohort consumer surplus under various κ (ū = 1).

  CS (1) (κ = 2) can be simplied as the below. CS(1) (κ = 2) > ū/8 i c < ( √ 1129 -31)ū/7. CS (1) (κ = 2) = 8ū 3 + 62ūc 2 -28ū 2 c + 7c 3 64ū 2 -32ūcComparing with monopoly pricing, consumers of types θ ∈ [0, ū/2) are worse o under the κ = 2 conguration because they pay more (p * G + c = (2ū + 3c)/4 > ū/2). Consumers of types θ ∈ (ū/2, (2ū + c)/4) are better o because chances are that they pay a lower price. Summing up these eects yields lower total surplus.

Fig. 4

 4 Fig. 4.9: Consumer surplus under group-buying and monopoly pricing (c = 0.15, ū = 1).

  increases, there are largely two eects. The larger the invitation cost, the lower the discounted price and the less chance consumers are invited. The rst eect increases the surplus of second-cohort consumers, and the second eect decreases it. The rst eect dominates the second in the κ = 4 conguration and the opposite holds in the κ = 3 conguration. Besides, the reason for the lowest surplus for the κ = 3 conguration is that, the chance for consumers to pay the normal price is too large.

Fig. 4

 4 Fig. 4.10: Second-cohort consumer surplus under various κ (ū = 1).

Fig. 4 .

 4 Fig. 4.11: Total consumer surplus under various κ (ū = 1).

  Solving the optimization problem by Lagrangian.

  β)F ( θ1 ) + ((1 -β)( θ3 -

A. 2

 2 Proof of Proposition 2Proof. First I show that for the κ = 3 conguration, by charging try to show that for the κ = 4 conguration, for inviting cost small enough, the normal price increases in the inviting cost. From (21), we know that, -4ūc -c 2 2(ū -c)+ ū2 -6ūc + c 2 2(ū -c)If the inviting cost is small enough such that ū2 -6ūc + c 2 > 0, the above can be simplied as, normal price increases in the inviting cost.Last I focus on the comparison between the κ = 4 conguration and a single-price setting. du consommateur plus grand sous certaines conditions. Ce même ordonnancement réduit généralement les distorsions de prix et sera de ce fait maximise le bien-être social.Dans le second chapitre, nous avons utilisé une variation basée sur le modèle de recherche de[START_REF] Wolinsky | True monopolistic competition as a result of imperfect information[END_REF] an de d'examiner de nouveau les relations entre les coûts de recherche et la concentration des entreprises. Les consommateurs recherchent un prix et recoupent les informations sur une plateforme de commerce électronique monopolistique ou bien par le biais de recherches séquentielles. Les entreprises proposants les produits peuvent ou non, adhérer à ce type de plateformes. Par ailleurs, la plateforme détermine le nombre d'entreprises qu'elle référence. Notre étude a montré que le degré de concentration voulu par la plateforme est inversement proportionnel à l'évolution du coût de recherche. Nous avons retrouvé le même schéma dans le cas d'un contexte concurrentiel entre plusieurs plateformes. Dans le dernier chapitre, nous nous sommes concentrés sur les achats groupés. Une stratégie d'achats groupés se réfère à un système de tarication basé sur trois variables : le prix normal ; le prix réduit et le nombre minimum requit de participants pour constituer un accord d'achats groupés. Alors que le prix normal est disponible à tout moment et pour tous, le prix réduit est réservé à un groupe de consommateurs atteignant le nombre minimum requis de participants. Cette stratégie s'est récemment fortement développée dans les ventes sur internet. Notre étude a proposé d'étudier deux modèles monopolistiques. Dans un premier modèle, sans invitation pré-requise, nous avons démontré que la stratégie d'achats groupés est toujours sous-optimale comparée à un système fondé sur la simple dépendance du prix au facteur temps.

  With an innitesimal search cost (a → v), lim a→v ∂p in ∂a = -1, that is, the inside price falls almost one for one in a. By comparison with p in , we have as well that lim a→v Under negligible search cost, the boundary between both types of rms is blurred. Price dispersion vanishes and the equilibrium prot for a rm drops to almost zero. On the contrary, when the search cost approaches innity (a ≤ p out ), no outside rms are then active. The inside rms' prots are then not connected to the search cost since no consumers search.

	∂pout ∂a =
	-1.

ˆ = a + ∆. 9 Turning to the threshold, dˆ /da = d(a + ∆)/da = 1 + d(p in -p out )/da. Again from (11), it is clear that ˆ decreases in search cost in the uniform case. Lower search cost makes consumers more choosy about goods even when price decisions are endogenized. Last but not least, the whole industrial prot increases in c, provided number of inside rms constant, because of less intense competition induced by fewer searching behaviours. However the platform might responds to a dierent search cost by dierent set-up costs, which is discussed in the next section.

  to informing other consumers and consumers at most are willing to accept the oer without informing others. If this happens, we are back to the model without invitations.

						2
	4.3.2 Medium Minimum (κ = 3)
						Solve the following
	for the inviting cost.
	p G =	ū -	√	ū2 -4cū 2	=	2ū -c 4

I obtain the solution c = 2( √ 17 -4)ū. For c > 2( √ 17 -4)ū, setting price according to (10) leads no consumers

In previous literature, this parameter is often termed preference intensity or consumer heterogeneity.

Free recall is a standard assumption in search literatures because costly recall can lead to unstationary prices.

2.7.2 Social WelfareThe gures below clearly support our view that the niche-generic ordering delivers more social surplus in general, except some rare cases with a huge search cost. Second, the social welfare increases in product heterogeneity of Firm G (when the other normalized to one) so it infers that more product heterogeneity increases social welfare. Last, the gures show that collusive outcome delivers less social welfare than the competitive outcome.

√ cµ N (3 √ µ G -2 √ c)

> 0

+ m F m+1 (v) -(m + 1)F m (v) = 0.

Remerciements

subscribers. As a result, the social optimum is restored. In section 6, we illustrate some simple results with nitely many rms. Section 7 concludes.

The Model

There exists one unit mass of consumers and an innite number of rms with zero marginal cost in the market. Consumers are restricted to purchase at most one product, as in other discrete choice models. As in [START_REF] Perlo | Equilibrium with product dierentiation[END_REF], consumer i's utility if she buys product j is :

ij is the match between rm j's product and consumer i, which is an i.i.d. random variable under a distribution function F with support [v, v] satisfying (v > 0 and v ≤ 0) 1 . p j is the price set by rm j. There is a platform in the market. Some rms are in the platform, which we later call "inside rms". The rest of the rms are called "outside rms". Denote the number of inside rms as m. Without loss of generality, these rms are indexed from 1 to m. The platform informs consumers on products and prices of inside rms. Consumers can either enter the platform and check out all inside goods for free or see through outside rms, each at the search cost c > 0.

The cost is identical for all consumers and for each search.

2 For simplicity, we assume that all rms expect consumers to visit inside rms rst.

3

Having visiting all rms in the platform, a consumer faces three choices : 1) stay in the 1. We assume these so that : rst, demand is always positive. Second, the market is not always covered. 2. A tourist nds that the restaurant is not interesting only after he travelled and visited it. A computer user may nd a software package unattractive only after 15 days' free trial.

3. Otherwise, rms can expect consumers to search goods in many possible paths : 1) consumers may run into the platform while searching goods randomly. 2) Consumers may refrain from the platform while expecting high prices there. We make this assumption on the consumers' part and show after that other strategies are weakly dominated.

, we arrive at the result.

A.4 Extension of A.6

We take its rst-order derivative in m :

(ii) The proof of ϕ(˜ , m) > 0 when m ≥ 2 can be reduced to the following : .

Its derivative in m is the following.

is taken over all realization of i, the number of consumers accepting the group deal.

For any binomial distribution,

And for the other term,

Substitute these two expressions into (6), the right-hand side becomes exactly N F ( θ1 ). Then N p R F ( θ1 ) is the expected prot from those purchasing the goods immediately. The last two terms in (5) refer to the expected prot from consumers postponing the decisions until the last stage. To see how they are obtained, note that the binomial combination in (4) can be rewritten Replace p R by ū -θ2 , and p G by ū -θG . The Lagrangian is,

where θ1 satises (A.1). Then the platform's choice satises L θ2 = 0, L θ3 = 0, L θG = 0 and two Kuhn-Tucker conditions. They are, after some rearrangements,

2) Look for solutions while both constraints are not binding.

If there are no solutions when both constraints are inactive, then I am done. If either one of constraints is binding, M resorts to an early price plus a late price. Suppose that λ 1 = 0 and λ 2 = 0. I will show that θ1 = θ2 = θ3 is the solution. Indeed when θ2 = θ3 , (A.1) suggests that θ1 = θ2 . Adding (A.4) to (A.5) yields the following after some transformations,

Then if I apply θ1 = θ2 = θ3 = θG , the above is reduced to,

Plug (A.7) into (A.3) and apply θ1 = θ2 = θ3 = θG . After cancelling out some terms, (A.3) can be rewritten into,

The second term of the above is clearly zero from (A.7). Since θ3 = θG , the above implies the monopoly price : (ū -θ2 ) f ( θ2 ) -F ( θ2 ) = 0, which along with θ1 = θ2 = θ3 and (A.7), is a solution in the admissible set. Hence, I nd a binding constraint by assuming that perhaps both constraints are not binding. In this solution, M sets a target which is not achievable. Then all the buyers purchases the products at the normal price determined by

3) Look for solutions while either of the constraints is binding.

On the other hand, the constraints are binding : either θ3 -θG + w β ≤ 0 or θ2 -θ3 ≤ 0. If the constraint θ2 -θ3 ≤ 0 is binding, we come back to the above monopoly price. If the constraint A.3 Proof of Proposition 3

Proof. First I focus on the comparison between the κ = 2 conguration and the κ = 3 conguration. .13)

Π(p