COMPETITIVE PRODUCT VERSIONING IN THE PRESENCE OF A NETWORK EXTERNALITY

Transcription

1 COMPETITIVE PRODUCT VERSIONING IN THE PRESENCE OF A NETWORK EXTERNALITY PETER HERMAN Abstract. The existence of firms that offer multiple versions of the same product represent a curious occurrence. These firms not only compete against other firms in the market but the use of multiple product versions by the same firm implies that the firms compete against themselves as well. Sales cannibalization, in which a low cost version detracts sales from a higher cost version, is well documented in monopoly markets but remains mostly unaddressed in competitive markets. In order to study the issue of product versioning in competitive markets, this paper develops a model of catalog competition in which firms compete within a single market by offering multiple versions of the same product. The presence of a network externality is employed to motivate the use of multiple versions. It can be shown that under some basic assumptions, the set of stable catalog profiles of product versions is nonempty when acting firms endogenously select both prices and the characteristics of multiple product versions. Using a simulation, properties of these stable profiles are demonstrated in a simplified model. The results of these simulations suggest that these markets feature high price competition, considerable cannibalization effects, and either complete product specialization or monopoly emergence. Current Version: August 2014 Department of Economics, Indiana University, Wylie Hall 105, 100 S. Woodlawn, Bloomington IN prherman@indiana.edu. 1

2 1. Introduction Product versioning has become a very common market structure. It occurs when a firm sells multiple differentiated but competing products within one market. Each version of the good satisfies the same basic demand but features some characteristics that are unique to that product. A typical consumer in the market is interested in purchasing a product but will select only one of the versions being sold so that total market demand is divided among all the different available versions. Because the versions compete with one another, firms engaged in product versioning face several unique trade offs. The first of these trade offs relates to the fact that product versions provide a price discrimination mechanism to firms. The sale of multiple versions allows firms to charge multiple prices within the market but doing so creates a risk of revenue cannibalization. Suppose a firms decides to sell an inexpensive version of a good and a premium version of the good. This tactic allows the firm to charge two different prices and generate greater revenue from consumers with a relativity high willingness to pay. However, the presence of an inexpensive version my cause some buyers to purchase that version instead of a more expensive product that they would have bought had only one version been offered. This behavior is known as cannibalization. The second tradeoff relates to market control. Suppose a market features a network externality related to firms. That is, all consumers purchasing from a particular firm benefit from the other consumers purchasing from the same firm. In such a market, firms may have an additional incentive to increase sales by introducing multiple versions because a larger network causes consumers willingness to pay to increase. Again, however, doing so may cannibalize sales of high profit versions as before. It is easy to find markets that fit this description quite well. Product versioning is particularly common within the software and entertainment industry or among information goods and subscription services. Consider, for example, the online chess market. The basic market is an platform upon which people can play chess against other players online. Firms frequently compete by offering multiple versions of this same basic game platform. The table below presents some of the different subscription offerings from two of the most popular of these websites 1 : chess.com and Chess Tempo. In both cases, all versions provide the ability to play chess with other people online. However, each 1 The subscription offerings were taken from the websites chess.com and chesstempo.com, respectively. 2

3 version also includes a catalog of additional features that are unique to that particular version. For example, some versions include tactics puzzles, game analysis, game databases, or numerous other features in varying levels. Free versions typically feature few or no tactics puzzles, intermediary versions offer a modest number, and the premium versions offer an unlimited number. In the case of online chess (or any online game platform), the network externality is clear. A chess platform is only as good as the players using it. The premium features are less attractive if the player base is weak. Additionally, firms have an easier time selling premium memberships if the player base is strong and active. Thus, the network externality provides a tremendous incentive to version the goods and, in this case, offer a free version. Table 1. Chess website catalogs. Chess.com Chess Tempo Version Price Version Price Basic free Basic free Gold $29/yr Silver $20/yr Diamond $99/yr Gold $35/yr We can observe similar behavior in many other markets. Software is frequently versioned into student, home, and professional versions under the belief that cheap student versions create familiarity with the product that carries over into more expensive home version sales and often very expensive professional version sales. Music streaming services fit this market closely as well. Firms like Spottily and Pandora both offer free and premium versions. Additionally, both firms include social features like user generated playlists or ratings that represent network externalities. In each of these examples, firms compete with one another by selecting a catalog of versions and prices subject to the two tradeoffs above. Of these two tradeoffs, the first has been studied in various settings. The original description of versioning can be attributed to Varian (1997) (see also Shapiro and Varian (1999)). Varian describes the prevalence of product versioning as a price discrimination mechanism, particularly among software and information goods. The modeling is relatively simple but provides some descriptions of optimal pricing strategies in a monopoly setting where versions and product differentiation are characterized by a single level of quality. Following Varian s work, Bhargava and Choudhary (2008) use a more formal model to determine criteria under which a monopoly selling a good defined by a single quality dimension sells multiple versions. They find that versioning is optimal if selling a low quality good alone generates higher 3

4 market share than selling a high quality good alone. Jing (2000) considers a similar monopoly market in which a network externality is present. The inclusion of the network externality causes firms to offer multiple versions but never more than two. Krishnan and Zhu (2006) relax the single quality dimension assumption and consider a product line in which gods have two dimensions of differentiation. This allows them to consider cases in which versions may not be simply higher or lower quality versions of one another. They find similar criteria under which firms offer multiple versions and when these versions differ along multiple quality dimensions. In all of these papers, the models consider only monopoly markets. The versioning criteria determine the profit maximizing behavior of the firm when product differentiation is one dimensional (two dimensional) and explains the cannibalization trade off faced by these firms. However, by only considering monopoly markets, the second trade off concerning market share is not considered. Without competition, the versioning decision need only consider the issue of cannibalization. If we consider the examples covered above, cannibalization cannot be the only consideration of firms. The existence of free versions that certainly cannibalize sales but generate zero revenue do not fit the monopoly criteria. Thus, the second trade off concerning market share must be of critical importance in an oligopoly framework. Research studying the use of product versioning in a competitive framework is extremely limited. To our knowledge, no papers have been written on this exact topic. Page Jr and Monteiro (2003), Monteiro and Page Jr (2008), and Page Jr (2008) consider a class of games referred to as nonlinear pricing and catalog games. Their framework allows for the consideration of markets in which multiple firms compete by choosing a catalog of goods and prices to submit to a market. They establish the existence of stable profiles of catalogs Page Jr (2008) and nash profiles of catalogs?. While they do not address product versioning specifically, the framework they develop is well suited for it. This paper seeks to fill the gap between the monopolistic product versioning present in previous literature and the competitive product versioning observed in real markets. The paper proceeds in the following manner. Section 2 defines a formal model based on nonlinear pricing and catalog methodology. The existence of a stable set of catalogs is established. Section 3 describes characteristics of the stable profiles using a simplified simulation. Section 4 concludes. 4

5 2. Model The model presented here builds upon the foundations of catalog competitions presented in Page Jr (2008). Firms compete over consumers by selecting a catalog of versions and associated prices to sell in the market. Let the sets of firms and consumers be given by J and I, respectively, such that both sets are finite. Each firm j J offers a catalog C j consisting of m j many versions where m j is assumed to be finite as well. Each product version is defined as a version-price pair (s, p) where s is a vector of l attributes describing the version and p is the chosen price for that version. Specifically, define the vector of attributes as s = (a 1, a 2,..., a l ) such that a k = {0, 1} for all k = 1,..., l. If a k = 1 for some attribute k then the version is said to include that attribute. If a k = 0, the attribute is absent. In what follows, it will be assumed that all firms face the same set of possible attributes 2 l and that this set is finite. Let S R l denote the set of possible versions. Prices are chosen from a price set P = {0.00, 0.01, 0.02,..., P } where P is an arbitrarily large upper bound on prices. A catalog is a subset of version-price pairs such that (s, p) C j S P. A catalog profile, denoted C, is collection of catalogs offered by all firms such that C = (C 0, C 1, C 2,...C J ) (S P)... (S P) := C. In this expression, catalog C 0 is taken to be a special catalog offered by a non-real firm such that C 0 = ((0, 0,..., 0), 0). This catalogs allows consumers the option of abstaining from the market by insuring the existence of a free option with no attributes or network externality. Given the assumptions above, the set of possible catalog profiles (C) is finite. Firms select catalogs in order to maximize profits given the decisions of other firms and the preferences of consumers. To study this selection, the game is divided into periods composed of two stages 3. In the first stage, firms select a catalog of versions and prices to sell to the market. In the second stage, consumers select and purchase a single version price pair from among all the offered catalogs. After each sequence of the two stages, the game repeats. Firms are allowed to modify their catalogs and consumers are allowed to reselect their product. The sequential nature of the problem allows for the use of backwards induction to analyze it with the goal of establishing the existence of stable profiles of catalogs The Consumer Problem. Each consumer i I derives utility from consuming a version of the product from some firm in each period. Consumers differ in their preferences over attributes. 2 This assumption could be relaxed without difficulty and is not important for the existence result. For example, it could be alternatively be assumed that firm j faces attribute set l j l. 3 In the simulation presented in section 3, the game is further subdivided into 3 stages for computational reasons. 5

6 These preferences are governed by their type t i which is drawn from the type space T = R l and are common knowledge after the draw 4. Types are l-dimensional vectors that can be viewed as specifying preferences over the l-possible attributes. For example, utility may be specified such that a high value of the kth element of t i implies strong preferences towards the kth attribute. Let t = (t 1, t 2,..., t I ) denote a profile of consumer types. In the second stage of each period, consumer i selects a firm and a version-price pair offered by that firm from the current profile of catalogs in order to maximize the value function max j, (s,p) C j V i (t i, j, (p, s)) := max j, (s,p) C j U i (t i, (s, p)) + Ψ(N(j)) (1) N(j) denotes the the number of consumers purchasing some version from firm j. We will make the following assumptions about the value function. Let the value function V i (t i, j, (p, s)) be (i) continuous in t i, (ii) decreasing in p, and (iii) increasing in N(j) with dψ(n(j)) dn(j) > 0. Assumption (iii) represents the positive network externality and is increasing in value as more consumers purchase from consumer i s chosen firm. Note that the network externality is common to all consumers purchasing from a particular firm and does not depend on the selected version. Given a profile of catalogs, it can be shown that there exist equilibrium outcomes for the consumers problem in the second stage. Konishi et al. (1997) provide the following proposition that we will apply here. Proposition 1. (Konishi, Le Breton, and Weber (1997), Proposition 4.1) Suppose that C is finite and the payoff function for each player satisfies positive externality, anonymity, and order preservation. Then a no spillover game admits a (pure strategy) Nash equilibrium. The proposition relies on three assumptions. The positive externality assumption requires that payoffs be increasing in the number of players that select a particular action. This assumption holds with respect to the selected firm due to to assumption (iii) above. The anonymity assumption requires that the utility derived from the positive externality be independent of the specific consumers buying from a particular firm, only the number of consumers. As before, anonymity holds due to assumption (iii); the consumer only considers the number of players buying from their chosen firm. Order preservation requires a certain degree of consistency across products with 4 While the assumption of common knowledge is considerable, it is not entirely unreasonable. Firms often expand tremendous effort eliciting the preferences of consumers through market research. 6

7 an externality. Specifically, if the triple [j, (s, p)] generates a higher value for a consumer i than [j, (s, p )], then order preservation requires that this remain true if an additional consumer enters the network of both j and j. That is, N(j) and N(j ) both increase by one. In the present model, the additive separability of the value function ensures that order preservation holds. Establishing the existence of a solution for the consumers problem is relatively straight forward. Because the profile of catalogs C is finite, the existence of at least one solution guaranteed by conventional knowledge. The benefit of applying the Konishi, Le Breton, and Weber proposition is that it guarantees a solution in pure strategies. While we will not make use of this property in the formal derivation of the model, it will be of value in the simulation in section 3. Let X = (σ 1 (C),..., σ I (C))) denote a solution to the consumers problem where X is a profile of consumer actions and each σ i (C) is a mixed strategy (possibly degenerate) over the version-price pairs in C. Additionally, let Φ(C) denote the set of equilibrium consumer action profiles for a given catalog so that X Φ(C). As established above, the set Φ(C) is non-empty for each catalog profile The Firm Problem. In the first stage of each period, firms select catalogs composed of version-price pairs in order to maximize expected profits. Firm profits are contingent on the actions of consumers and therefore depend on the anticipated Nash equilibria of the consumers game. A complication arrises at this point in determining consumer demand for products. If the consumer game presented above is degenerate, meaning that at least one consumer is indifferent between two products, the set Φ(C) can be infinite. In this case, firms are typically unable to determine expected demand for different price-catalog offerings. In order to avoid this problem, we must include a special mechanism that limits the size of Φ(C. Define a mechanism M : Φ(t, C) Φ(t, C) that selects a finite subset of the consumer solution set. Denote that subset by M(Φ(t, C)). While the reliance on such a mechanism is a strong assumption, it does have the benefit of being completely arbitrary. Because all elements of Φ(C) are Nash profiles for the consumer game, any selection M( ) is necessarily both individually rational and incentive compatible. A possible example of such a mechanism would be one that directly targets the issue of degeneracy. Any time a consumer is indifferent between two or more products, she is willing to mix between them with any probability and Φ(C) is infinite. Let M be a rule that specifies a particular product choice or probability for the indifferent consumer. In this case, M 7

8 can be thought of as a sort of tie-breaking rule. In the simulation in section three, a specific form of this type of mechanism is considered in which the tie is broken in favor of the consumer welfare maximizing equilibrium. Now, given mechanism M, producers face a finite set of Nash consumer responses to a given catalog profile and maximize expected profits equal to Π j = (s,p) C j X M(Φ) i I (p z)θ (X, [j, (s, p)], i) M(Φ) F (C j ). (2) z represents a version-specific marginal cost and F (C j ) is a fixed cost associated of offering catalog C j. Θ is an indicator function that takes the value 1 if version (s, p) is purchased from firm j by consumer i in profile X and zero otherwise. In the case where M( ) contains more than one outcome, the firm considers each outcome equally likely and weighs the profit generated form each entry equally. This is the motivation for the division by M( ). Firms select catalogs in order to maximize profits. max Π j(t, C) = Π j (t, C j, C j ). C j S P Recall that the set of possible catalogs S P if finite so that the firms game is finite. By conventional methods (see Nash (1951)), there exists an equilibrium for the stage 1 game. Proposition 2. [Nash (1951), Theorem 1] Every finite game has an equilibrium point. This result allows us to finally state our key proposition. Proposition 3. The set of equilibria of the two-stage catalog game is non-empty. Proof. An equilibrium of the game constitutes a profile of catalogs selected by firms C = (C j, C j ) and a profile of actions selected by consumers X = (σ i (C), σ i (C)) such that (1) for all consumers i I, V i (σ i (C), σ i (C)) V i (σ i (C), σ i(c)) for any σ i (C) Σ i and (2) for all firms j J, Π j (C j, C j ) Π j (C j, C j) for all C j S P. Proposition 1 insures the existence of an equilibrium to (1) and Proposition 2 ensures an equilibrium to (2) given (1). Taken together, there exists an equilibrium to the sequential game. The equilibrium ensured by Proposition 3 describes a stable profile of catalogs. That profile is a collection of version-price pairs offered by firms such that no firm has any incentive to modify 8

9 their offerings. In order to better understand what these stable profiles look like, we consider a simulation of the game. 3. Simulation By making a small change to the timing of the game, simulations of the catalog game can be generated for an example market. In the simulations, I consider a market featuring two firms, three consumers, and two possible product attributes. To better fit with the examples presented in the introduction, we add some addition stipulations. Each firm is assumed to offer a catalog featuring two version-price pairs. One version is a free version and the other is a premium version with a (potentially) positive price. Given that each firm has two potential attributes that can be included, there are 16 possible catalogs that each firm can offer without considering price. These catalogs are described and labeled in table 2. Although not shown in the table, consumers always have the option of choosing a free catalog that offers no attributes and generates a payout of zero (i.e. C 0 is available but unlisted). In equilibrium, this catalog is never considered as any other choice yields payouts greater than or equal to zero and strictly so if there exists a positive network externality. Table 2. The set of possible versions. Label Free Premium Label Free Premium 1 (0,0) (0,0) 9 (1,1) (0,0) 2 (0,0) (0,1) 10 (1,1) (0,1) 3 (0,0) (1,0) 11 (1,1) (1,0) 4 (0,0) (1,1) 12 (1,1) (1,1) 5 (0,1) (1,0) 13 (0,1) (0,0) 6 (0,1) (1,1) 14 (0,1) (0,1) 7 (1,0) (0,1) 15 (1,0) (0,0) 8 (1,0) (1,1) 16 (1,0) (1,0) The simulated game proceeds in a similar way as the model above with slight change to the timing of the game. In the simulated game, version selection and pricing occur sequentially. That is, firms first select the versions they are to offer and observe the versions offered by the other firm. They then select an optimal pricing strategy given the versions offered by both firms. The end result of this sequential selection is a set of version-price pairs in which neither firm has an incentive to modify either selection. In the case in which there are multiple Nash price profiles, the profile that maximizes producer surplus is selected. The resulting set of equilibria represents a subset of the prices and catalog decisions that are immune to deviations by either firm but some equilibria 9

10 are potentially omitted. For example, in a case when one firm offers a catalog in which no version is ever purchased, any price selected by that firm supports a Nash outcome and is equally desirable. In the simulation, only one of those prices is included. In each simulation, consumer utility is determined using the function v i (t i, j, (s, p)) = t i s p + m(n(j) 1). Firm profits are determined using the mechanism selection method in which ties are broken in favor of the consumer welfare maximizing outcome and equation (2) with the added specification of a fixed cost function. The fixed cost function is given by 0 if no attributes are included in either version F (C j ) = if only one attribute is developed in either version f j 2f j if at least one of the catalogs features each attribute That is, a firm pays a fixed cost if it offers an attribute in either version but once that fixed cost is incurred, it may use it in other versions without additional cost. For example, catalogs 4, 5, 6, 7, and 8 all cost 2f j. The employed mechanism selects a finite set of consumer equilibria based on two rules. First, it considers only the set of pure strategy Nash equilibria. This restriction is primarily due to computational feasibility. Fortunately, Proposition 1 endures that at least one pure strategy equilibrium exists. The second criteria selects only consumer surplus maximizing equilibria. When the surplus maximizing equilibria is not unique, all maximizing outputs are assigned equal weight and firms maximizes expected profits under the belief that each outcome is equally likely. Again, this mechanism is both individually rational and incentive compatible. Different simulations were run for varying parameterizations of the model. Specifically, we look at consumer types (t 1, t 2, t 3 ), the network externality (m), and firm fixed costs (f 1,f 2 ). A common observation among versioned goods is that the variable cost of producing them tends to be negligible. This is certainly the case in the chess, music streaming, and software examples described above. As such, we will not consider the role of marginal costs and let z = 0 for all versions. As in the theoretical model, the price set will be bounded above and below. The lower bound is zero and the upper bound is chosen to be sufficiently high that no agent would willingly pay that price for any version. The rate at which the price set is incremented is included as a parameter. However, for 10

11 sufficiently small increments (e.g. 0.01, 0.02) the equilibrium behavior is unaffected by the size of the increment but the computational speed is dramatically altered. For each parameterization, a directed graph of catalogs can generated that illustrates the equilibria of the game. Each node in the graph depicts a catalog profile. For example, the node labeled 4.6 indicates that firm 1 has chosen catalog 4 and firm 2 has chosen catalog 6. A black (red) arc from one node to another indicates that firm 1 (firm 2) can and will deviate from the original node to the new node by altering their catalog. A solution to the game is a catalog profile for which there is no deviation from either firm. In the graph, this is equivalent to a node without a path leading away from it. These graphs can be viewed in exactly the same way as the supernetworks described in Page and Wooders (2009). In the network language, an equilibrium catalog profile is a node without descendants. For the catalog game considered here, there are 256 possible catalog profiles. Constructing graphs with that many nodes produces an image with so many nodes and arcs that any details are obscured. In order to avoid this problem, we omit some catalog profiles. Specifically, catalogs 9-16 all include free versions that are at least as good as or superior to the premium versions. A firm would never reasonable select any of these catalogs 5 because doing so would trivially result in complete cannibalization and guarantee zero profits or losses. As such, these catalogs can be safely omitted in the graphs presented here as they will never be equilibria. Let us now consider a few simulations. Simulation 1. Suppose we parameterize the simulation in the following way. Let consumer types be given by t 1 = [1, 0], t 2 = [0.5, 0.5], and t 3 = [[0.1], the externality be given by m =.25, costs be given by f 1 = 0 and f 2 = 0, and maximum price and increment be given by 1.43 and 0.05, respectively. The zero fixed costs can be though of as being low enough relative to the market to be insignificant in the firms decisions. A graph indicating the deviations made by both firms at any profile in the game is presented in figure 1. We can see that there exists 9 equilibria for this parameterization. Both firms are identical in this case so that the equilibria are symmetric and are composed of profiles (4,4), (6,6), (6,8), (8,6), and (8,8). In all five equilibrium profiles the equilibrium price charged by both firms is Given the price increment and the way in which the simulation randomizes over indifference, a price of assuming fixed costs are greater than zero. 11

12 represents a Bertrand outcome. By dropping the price to 0.05, the firm increases its probability of selling from 1/2 to 1 but does so by dropping its price by 1/2. The loss of revenue exactly offsets the increased likelihood of selling so that a firm has no incentive to do so. As such, we find that the market is highly competitive with respect to prices. Another interesting observation that can be made is which consumers are buying which version. When catalogs 6 and 8 are offered, two consumers buy the premium version while one consumer simply selects the free version. For example, when catalog 6 is offered, consumer s 1 and 2 are willing to buy the premium version because they both value attribute one, which is only included in the premium version, more than the price. Consumer 3 places no value on attribute one and only selects the free version. Interestingly, this reasoning explains why profile (4,4) is the only equilibrium profile containing catalog 4. When catalog profile (8,4) is offered, both firms are willing to alter their catalog. Catalog 8 is more attractive to consumers because consumer one is perfectly happy with the free version so that the firm offering catalog 4 cannot sell any premium versions. This implies that the firm would rather offer either catalog 6 or 8 for which premium versions can be sold. Alternatively, if we consider the firm offering catalog 8, they prefer to offer catalog 4 instead. The free version canibalizes a sale to consumer 1. Prices are such that the firm would rather offer catalog 4 which can be sold to all three consumers even though there is a 50 percent chance the other firm will capture the market. As such, there are two distinct groups of catalog profiles. One group in which a free version is offered ((6,6), (6,8), (8,6), and (8,8)) and a group in which no free catalog is offered ((4,4)). To determine more robust effects, simulations of this sort were run over ranges of different parameters, simulating series of comparative statics exercises. Simulation 2 considers changes in the fixed costs for both firms while maintaining identical costs across both firms. Simulation 3 considers changes in relative fixed costs wherein the costs of one firm are constant while the other s varies. Simulation 4 considers changes in the value of the network externality. Simulation 2. In this simulation, the fixed cost of developing and offering an attribute is considered under the assumption that both firms face the same cost of doing so, Table 3 presents the outcomes for a variety of cost levels. In all cases presented here, the equilibria are symmetric but may only be listed once. For each outcome, there exists a second equilibrium outcome in which firm 1 and firm 2 are interchanged. Over the presented range of costs, the set of stable catalogs changes. For 12

13 zero fixed costs, firms face several possible catalog profiles: (4,4), (6,6), (6,8), and (8,8). In the profiles composed of catalogs 6 or 8, the free version is offered to one of the consumer that derives utility from only one amenity. The firms benefits from drawing the consumer into the network and increasing the willingness to pay of the other consumers. As the cost of including amenities increases, many of these catalogs are lost such that for even low costs, firms are no longer willing to offer free versions that feature any amenities. This suggests that versioning is closely tied to the negligible cost of developing the free version. Additionally, for any costs greater than zero, price competition forces one of the two firms out of the market resulting in a single firm offering catalog 4 and supplying all three consumers. As a result, the surviving firm becomes a monopolist and sets prices much higher than the oligopoly firms. This offering remains stable until costs rise sufficiently to force both firms out of the market, in which case both firms offer catalog 1. As costs increase from zero, the firm that survives in the market generates much higher profits than before but these profits diminish as costs rise. Surplus for consumers is high when costs are zero due to strong price competition among the competing firms but decreases substantially as costs rise. The monopoly firm is able to capture much of the consumer surplus as long as it is the sole provider by pricing just low enough to make consumers prefer consuming over foregoing consumption. This causes a large reappropriation of surplus across participants. When both firms exit the market, consumers generate a minimum level of utility. As would be expected, total surplus is monotonically decreasing. Simulation 3. Simulation 3 considers a case in which firms do not face the same fixed costs. Outcomes for this simulation are presented in Table 4. The fixed costs per attribute of firm 2 are set at 0.55, which is sufficiently low to allow for participation in the market, while the fixed costs for firm 1 range between 0 and 2, as in the previous simulation. For all cost pairs, the catalog profiles (1,4) or (4,1) in which one firm operates as a monopoly are stable. There is no equilibrium in which two firms remain active. As before, the monopolist selects the price 0.95 which is just low enough to induce consumption from all three consumers. Surplus is constant for consumers due to the fact that the equilibrium price is the same monopolist price in all outcomes. When firm 1 faces costs low enough to allow for participation, each equilibrium is symmetric except for the fact that producer surplus, and thus total surplus, is lower for the firm facing higher costs. For this reason, total surplus is monotonically decreasing in costs among the equilibria in which firm 1 survives as the monopolist. 13

14 Table 3. Nash outcomes over varying costs assuming both firms face the same cost. Net. Eff. Fixed Cost (Firm) Total Cost (Firm) Catalogs (Firm) Profits (Firm) Prices (Firm) Utility (Cons.) Surplus 1 & Prod. Cons. Total

15 Table 4. Nash outcomes when the costs of one firm are varied. Net. Eff. Fixed Cost (Firm) Total Cost (Firm) Catalogs (Firm) Profits (Firm) Prices (Firm) Utility (Cons.) Surplus 1 & Prod. Cons. Total

16 Simulation 4. Simulation 4 considers changes in the value of the network effect. Outcomes from this simulation are presented in Table 5. As in Simulation 2, all outcomes are fully symmetric but only one is listed. At low values of the network effect (m < 0.1), a new combination of catalogs is stable. In addition to the profile (1,4), firms may also select (2,3). In the latter profile, firms divide the market and each specializes in a different attribute. As the network effect increases, it is eventually enough to induce the consumer that derives utility from both amenities to favor a larger network over a good that is more closely tied to their preferences. Specialized catalogs are no longer feasible and one firm reverts to catalog 4 which offers both attributes and the market becomes a monopoly. Interestingly, consumers actually benefit in terms of consumer surplus from the monopoly outcomes compared to the specialization outcomes. This is due to the second consumer s preferences being mixed across both attributes. Due to high price competition between the two firms, the second consumer receives free additional utility when firms offer catalog 4 because it contains both attributes but firms are not able to increase prices due to competition. For some levels of the network effect when firms divide the market and select a specialization profile, prices are asymmetric with firm one selecting a lower price. The lower price is selected in order to induce the second consumer, who is indifferent between the two firms, to buy their product. In each of the cases presented here, 0.04 m 0.08, firm 1 sells to two consumers while firm 2 sells to the remaining consumer. Note that as the value of the network effect increases, firm 2 must drop its price in order to keep consumer 1 as a customer due to the rising opportunity cost of forgoing the network externality. As would be expected, total surplus is increasing in the value of the externality. Across these four simulations, we observe several important characteristics of the model presented in this section regarding prices and product versions. The equilibrium outcomes in each parameterization are driven by strong price competition. When both firms remain in the market, prices are reduced to the lowest point at which neither firm benefits from reducing prices and increasing market share. When a single firm acts as a monopoly, prices remain low enough to prevent reentry from the competitor. In either case, neither firm is able to fully capture the surpluses. It is for this reason that product versioning is not more apparent in the simulations. A key motivation for offering free versions is that it induces consumers who would otherwise be unwilling to purchase the premium version to join the network. Adding consumers to the network increases the value of the premium version and the willingness to pay of other consumers. However, in the presence 16

17 Table 5. Nash outcomes over varried network effect values. Net. Eff. Fixed Cost (Firm) Total Cost (Firm) Catalogs (Firm) Profits (Firm) Prices (Firm) Utility (Cons.) Surplus 1 & Prod. Cons. Total

18 of heavy price competition, firms are not able to take advantage of that increased willingness to pay. As a result, the cannibalization effect of the free versions are the only effect present in most parameterizations, resulting in firms failing to offer a free catalog with any amenities. The only situation in which firms are willing to offer a free version featuring any amenities occurs when fixed costs are zero. This is the only case in which the increase in consumer willingness to pay is sufficient to offset the cannibalization effect. Put simply, firms appear to be willing to offer amenities for free only when the cost of doing so is negligible. This observation is consistent with the tendency among real firms of offering free versions that are simply stripped down versions of their premium product. While this result appears robust to the three consumer model presented here 6, inclusion of more consumers may generate different results. 4. Conclusion In the previous sections, we have constructed a model of catalog competition in which firms compete over a single market by offering multiple versions of the same product. It can be shown that under some basic assumptions, the set of stable catalog profiles is nonempty. Using the simulation presented in section 3, properties of these stable profiles are demonstrated. We observe high price competition when similar versions are offered and considerable cannibalization effects. In the three consumer case, the cannibalization effect generally dominates any effect that the network externality may have on other consumers will to pay. Only when costs are negligible do firms elect to offer free versions that feature any amenities. Given the predictions in the model, markets featuring multiple providers of a single general good would be expected to feature firms specializing in different attributes. The presence of the network effect results in either the division of the market across amenities or the monopolization of the market by a single firm. Firms that tend to offer versioned goods would be expected to offer a good for which the cost of producing the lower cost version is negligibly low. Such types of versions are common such as software with locked features. The work presented here represents an initial foray into the realm of competitive product versioning, a market structure that is becoming increasingly popular. The simulation approach provides intuitive illustrations of behavior within the model but does not precisely allow for analytical study 6 Although not included, other simulations using different combinations of consumer preferences yield similar results regarding equilibrium catalog profiles. 18

19 of the equilibrium profiles of catalogs. A possible line of research could combine the notion of product versioning with the line of industrial organization literature studying competition through product differentiation. For example, introducing multiple products offered by a single firm to the models developed by Shaked and Sutton (1982), Choi and Shin (1992), or Motta (1993) could be effective in providing a framework more conducive to analytical analysis. A draw back of the simulation approach used here is the limitation placed on the consumer set. An analytical approach could allow for more consumers and possibly a stronger roll for multiple versions. 19

20 References Bhargava, H. K. and V. Choudhary (2008). Research Note-When Is Versioning Optimal for Information Goods. Management Science 54 (5), Choi, C. J. and S. H. Shin (1992). A Comment on a Model of Vertical Product Differentiation. The Journal of Industrial Economics 40 (2), Jing, B. (2000). Versioning information goods with network externalities. In Proceedings of the twenty first international conference on Information systems, pp Association for Information Systems. Konishi, H., M. Le Breton, and S. Weber (1997). Pure strategy Nash equilibrium in a group formation game with positive externalities. Games and Economic Behavior 21 (1), Krishnan, K. and W. Zhu (2006, June). Designing a Family of Developement-Intensive Products. Management Science 52 (6), Monteiro, P. K. and F. H. Page Jr (2008). Catalog competition and Nash equilibrium in nonlinear pricing games. Economic Theory 34 (3), Motta, M. (1993). Endogenous Quality Choice: Price vs. Quantity Competition. The Journal of Industrial EconomicsThe Journal of Industrial Economics 41 (2), Nash, J. (1951). Non-cooperative games. The Annals of Mathematics 54 (2), Page, F. H. J. and M. Wooders (2009). Strategic Basins of Attraction, the Path Dominance Core, and Network Formation Games. Games and Economic Behavior 66, Page Jr, F. H. (2008). Catalog competition and stable nonlinear prices. Journal of Mathematical Economics 44 (7), Page Jr, F. H. and P. K. Monteiro (2003). Three principles of competitive nonlinear pricing. Journal of Mathematical Economics 39 (1), Shaked, A. and J. Sutton (1982). Price Competition Through Product. The Review of Economic Studies 49 (1), Shapiro, C. and H. R. Varian (1999). Information Rules: A Strategic Guide to the Network Economy. Harvard Business School Press. Varian, H. R. (1997). Versioning information goods. University of California, Berkeley. 20

21 Figure 1. Graph of Simulation 1 (no cost) zerocost_ Externality = 0.25 Maximum Price = 1.3 Price Increment = 0.05 Firm 1 Cost = 0 Firm 2 Cost = 0 [1,0] Consumer 2 type = [0.5,0.5] Consumer Consumer 1 type = 3 type = [0,1]