Product development and pricing strategy for information goods under heterogeneous participation constraints

Transcription

1 Product development and pricing strategy for information goods under heterogeneous participation constraints Ying-Ju Chen and Sridhar Seshadri Stern School of Business, New York University Abstract This paper considers a two-stage development problem for information goods with costless quality degradation. In our model, a seller of information goods faces customers that are heterogeneous with regard to both the marginal willingness to pay for quality and the outside opportunity. The seller first determines the quality limit in the development phase, and then in the production stage designs the optimal monopolistic pricing schedule given this technically feasible quality level. We show that versioning is optimal for the seller when customers exhibit heterogeneous participation constraints. At optimality, the seller discards both the low-end and the high-end customers. Among those that are served, the seller offers a continuum of (inferior) versions to customers with relatively low willingness to pay, and extracts full information rent from each of them. A common version with the quality limit is offered to the rest, and in its interior sits the customer that enjoys the maximum rent. We then investigate the cases where the quality limit is exogeneously given. The seller earns strictly more with a higher quality limit under both types of discrimination. However, compared to first-degree price discrimination, the seller excludes more low-end customers and high-end customers in the case of second-degree discrimination. Keywords: software versioning, quality degradation, price discrimination, information goods, heterogeneous participant constraints 1 Introduction The value of a digital good is measured by its information content rather than its physical content, and hence it is usually hard to produce the first copy, but easy to reproduce and We thank Anindya Ghose, Ke-Wei Huang, and Arun Sundararajan for many stimulating discussions. Address for correspondence: 44 W4 St, NY, NY 10012; {ychen0,sseshadr}@stern.nyu.edu 1

2 distribute (Bhargava and Choudhary [2001]). These characteristics, as the defining property of information goods, imply that they have a fairly specific cost structure: high fixed costs but zero or near zero marginal costs. For example, the cost of establishing a database such as the Wharton Research Data Services is considerably high, but once it has been established, more subscriptions do not incur high cost to service (Varian [1995]). Music recording, fullproduction movies, internet search engines, on-line content providers, and journals like Wall Street Journal and NY Times all share the same characteristics, since sellers do not have to pay significantly more for making one more copy or allowing one more download (Jones and Mendelson [1998], and Riggins [2002]). Software is probably the most canonical product that represents and justifies the nomenclature of information good (Riggins [2002] and Sundararajan [2004b]). The specific cost structure near zero variable cost results in two important phenomena: (1) the traditional cost-based pricing policy fails since competitive pricing behavior ends the hope of recovering the large fixed cost; (2) the degradation (i.e., reduction of the product s quality) is relatively easy and no significant cost is incurred. To exploit these features in the production of information goods, a versioning strategy (selling different versions at different prices) is often adopted as a common tool to price discriminate amongst customers in practice. This strategy, which Shapiro and Varian [1998] call the smart way to sell information, finds its profitability in both quality and quantity differentiated/segmented markets. In strict contrast with industrial goods (small development cost but high marginal cost), software development bears a different cost structure and hence different strategic concerns for R&D and marketing. Since the degradation can be easily performed on digitized goods, information providers usually develop a high-end product in the development phase that depends on advance market information. After the technological quality limit has been established, in the production phase the software sellers then downgrade the product and provide different versions/price bundles (Riggins [2002] and Varian [1997]). Similar strategic quality degradation has been observed in the case of Intel processors, IBM Laser Printers, automobiles, airlines, and cable television (Varian [1997]). 2

3 This two-stage problem for information goods development is the focus of this paper. In our model, a seller of information goods faces heterogeneous customers with unobservable preferences (types) with regard to the quality. The seller incurs a fixed cost for each sales transaction, independent of the product s quality. This fixed expenditure may arise due to the setup cost for establishing a new phone line, the fixed expense of providing technical support, the maintenance expense while getting a new subscription, the production/delivery cost of the first copy, or any other administrative cost. The seller is a profit maximizer, and her goal is to first find the optimal quality limit in the development phase, where the associated cost is convex in quality, and then design the optimal monopolistic pricing schedule given this technically feasible quality level. Following the literature on pricing information goods, we assume that customers differ in their marginal willingness to pay, see Bhargava and Choudhary [2001], Jing [2002], and Riggins [2002]. This inherent heterogeneity among customers allows the seller to offer a variety of options, and consequently customers accept different deals due to their different preferences. This heterogeneity is also observed for a variety of non-information goods. For example, customers exhibit different degrees of aversion to waiting, and for changing their schedules. Hence, airline/railroad seats are sold at different fares, and restrictions such as advance purchase, specific day/time, penalty on refunding are imposed for different fare classes. Washers, dryers, bicycles, television sets all can be differentiated by color, size, functions, and speed (Bhargava and Choudhary [2001]), since when these options co-exist, customers will self-select the items that most closely correspond to their requirements. Regarding the pricing schedule in the production stage, we assume that the seller offers a menu and customers self-select their own quality-price bundles. This is labelled as seconddegree price discrimination, which is popular among information-goods and non-information goods sellers. Mathematical software sellers (e.g. Mathematica, ILOG, Matlab) usually provide enterprise, professional, and student versions at different prices. The enterprise edition contains all functions, part of which are de-activated in the professional edition, and further more packages (for example, symbolic operations or simulation macros) are purposely 3

4 closed in student editions. Limited functionality, intentional delay, and restrictive technical support are all methods for achieving quality differentiation in the software industry. Internet service also possesses the same feature. Many services contain free-sponsored sites that provide daily news and fee-based sites that convey more specific information, for example, AOL, CNN.com, Yahoo, and Classmates.com (Riggins [2002]). Versioning may also be implemented by the timing choice, (e.g., extra charges for real-time financial information by PAWWS Financial Network and InterQuote (interquote.com)), or by introducing banner advertisement to discomfort users (Silicon investor membership, and peer-to-peer software like Kazaa), see Hui and Chau [2002] and Shapiro and Varian [1998] for more examples. The main deviation in our paper from the existing literature on pricing information goods is that customers outside opportunities are type-dependent. Throughout this paper it is assumed that the reservation utility of customers is increasing and strictly convex in type. This specific structure of outside opportunities is motivated by either that the customers can develop the information good by themselves or that there exists a competing company to which customers have access. The heterogeneous outside opportunities are observed in both general economics scenarios and information goods markets. For general economics scenarios, Maggi and Rodriguez-Clare [1995] suggest that different participation constraints are a natural reflection of users different fixed costs. Jullien [2000] argues that potential competition and renegotiation may also result in effectively different reservation utilities for different types of users. In Agrawal and Seshadri [2000] where they consider the possibility of risk intermediation in supply chain channels, the newsvendors possess heterogeneous status quos due to difference in risk aversion. For markets of information goods, Sundararajan [2004a] considers the piracy and protection problem of information goods, and he introduces the outside opportunity as the chance of obtaining the pirated version of software. Huang and Sundararajan [2005] interpret the reservation utility as the level if the customer has to pay a fixed cost and develop the product by themselves. The model we develop here can also be applied to incorporate quantity differentiation for information goods. It provides the rationale for usage-based pricing in spite of the near zero 4

5 variable cost characteristic. Price discrimination has also been observed in various industries for information goods based upon quantity differentiation. In the E-trade industry, online brokers charge the customers by the quantity/frequency at which they trade via the intermediacy. Firms that offer corporate software (for example, Oracle) and cellphone companies (for example, AT&T and Verizon) adopt a usage-based pricing as well (Sundararajan [2004b]). Modem users are charged at different prices for different minute-based plans of household internet access. On-demand computing is another up-to-date example that charges companies based on the amount of CPU usage. Even though the cost of establishing databases is a one-time expenditure, research databases charge colleges, institutions, and individual users by the number of inquiries. Several other pricing schemes have been proposed for information goods. Two examples are personalized/target pricing (first-degree price discrimination) and group pricing (thirddegree discrimination). The former is possible when the seller has information regarding users preferences, e.g., via past purchase history or through in-person negotiation (Villas- Boas [1999]). The latter is justified based on grounds such as extremeness aversion, see Varian [1997]. There are other pricing strategies for information goods in which price discrimination is costly or practically impossible to induce. For example, Dewan and Mendelson [1990] derive the optimal common price for services when users possess nonlinear delay cost. Another example is Bakos and Brynjolfsson [1999], who show that firms have the incentive to bundle digital goods because this aggregates the customers willingness to pay, and hence it yields a higher profit. Finally, when usage-based pricing requires costly monitoring of the individuals usage, a fixed-fee contract may be favored (Sundararajan [2004b]). Our paper contributes to the literature on information goods versioning, which has received attention since the rapid growth of digitalized markets. With the constant marginal willingness to pay assumption, many papers suggest no versioning should be observed when the cost structure presents certain properties such as concavity and linearity, see Bhargava and Choudhary [2001], Jones and Mendelson [1998], Raghunathan [2000], and Salant [1989]. When quality levels are exogeneously given, Bhargava and Choudhary [2001] shows that if the 5

6 cost-quality ratio is downward sloping, which is particularly true for information goods, the seller s optimal strategy is to offer a single version. Jing and Radner [2005] consider a more general setting, and they show that the monopolistic seller will offer only the quality-cost bundles that lie in the lower convex envelope, which immediately implies that no versioning occurs when the marginal cost is the same for all quality levels. To our knowledge, versioning is reported as a profitable pricing strategy with linear separable utilities only when the information goods convey network effects (Jing [2002]), and when the cost of producing different quality levels is sufficiently convex (Riggins [2002]). Otherwise, one has to assume that users possess nonlinear utilities to induce versioning (Sundararajan [2004a]). We show that versioning could be profitable because customers exhibit heterogeneity with regard to outside opportunities, even though they have constant marginal willingness to pay and the network effects do not contribute to the utilities. As summarized in Theorem 2, the seller will discard both the low-end and the high-end customers. Among those served, the seller extracts full information rent from customers with relatively low willingness to pay, and offers a common version to the rest. We also provide a simple rule for selecting the optimal quality limits to achieve either first-degree or second-degree discrimination. When the quality limit is pre-determined, the information asymmetry forces the seller to give up some transactions that are efficient in the first-best scenario. As the quality limit is raised, the seller gathers a strictly higher profit under both price discrimination. The rest of this paper is organized as follows. In Section 2, we introduce the model. Section 3 considers the scenario where the seller is able to observe the customers willingnessto-pay, and in Section 4 this becomes customers private information, and hence the seller has to offer a menu to induce self-selection. In Section 5, we discuss some comparative statics for both informational scenarios, and finally conclude in Section 6. 6

7 2 Model 2.1 The setting In our model, a seller of information goods faces customers that possess heterogeneous willingness to pay on the quality. The willingness to pay is assumed to be of the linear, separable format u(q, θ) = θq p(q), where q is the quality level, p(q) is the money transfer between the seller and the customer, and θ is the user s marginal willingness to pay (type) with distribution function F (θ) and its density f(θ) over a finite support [0, R]. The value of R captures the maximum marginal willingness to pay for quality and the extent of market heterogeneity of customers preferences on quality. Besides purchasing the product from the seller, each customer also has an outside opportunity which guarantees a reservation utility r(θ). We assume that r(θ) is increasing and strictly convex in θ. Moreover, r ( ), r ( ) exist, and r(0) is normalized to zero. The seller knows the utility function u(q, θ), the entire distribution F (θ) and r(θ), but she is unable to observe customers types. The product development takes place sequentially in two stages: the development stage and the production stage. In the development stage, the seller chooses the quality limit q by devoting a deterministic convex cost C( q). We assume that degradation is costless, and hence the seller can provide any quality level q [0, q] in the production stage, without incurring any extra cost of reengineering/redeveloping. There is a fixed cost c(q) c if a product is sold to a customer, independent of the product s quality. That is, if the seller sells a product with quality q to a customer, her net payoff will be π(q) = p(q) c, q [0, q]. The seller s problem is to first find an optimal target quality level q, and then propose a (possibly nonlinear) pricing schedule to these customers. 2.2 Discussion of our model The linear utility format corresponds to the assumption of constant marginal willingness to pay for quality. This is in fact adopted widely in the literature, including that related to non- 7

8 linear pricing (Mussa and Rosen [1978]), marketing research (Moorthy [1988]), and pricing digital goods (e.g., Bhargava and Choudhary [2001] and Jing [2002]). The zero marginal cost assumption captures the characteristics of information goods (Jones and Mendelson [1998]). Our model can be regarded as the dual problem of Maggi and Rodriguez-Clare [1995], which specifies a contract design problem. In their model, the principal is the buyer, and the agents are sellers with different marginal costs. They require that the utility of the buyer be strictly concave, which is equivalent to assuming in our model that the cost c(q) is strictly convex. However, we consider a constant cost for each sales transaction, and hence convexity does not hold here. Jullien [2000] provides a general framework for designing the optimal price-quality schedule when users possess type-dependent participation constraints, but he assumes the strict quasi-concavity of the virtual surplus (the maximum profit one can extract from a customer under the incentive compatibility), which fails in our case. 1 As we will mention in Section 4, the assumption on the distribution of types (Assumption 1) is different from the standard condition when the participation constraints are type- d F (θ) dependent, i.e., dθ f(θ) 0 d dθ 1 F (θ) f(θ) standard distributions satisfy our assumption. (Jullien [2000] and Sundararajan [2004a]). Several The production problem can also be regarded as an alternative problem with quantity differentiation: customers generate more utility while possessing or consuming more units of information goods. Since information goods are easy to reproduce, it is appropriate to assume that the variable cost of quantity is zero. The finite quantity limit that a seller can offer to a customer is plausible in a variety of settings. For example, in cellphone plans, a user can talk on the air for at most minutes per month. For internet service providers, the maximum bandwidth a household can get through a cable is restricted by the current technology of fiber optics. A subscribed customer of an online music provider can 1 More specifically, in his Assumption 2, σ(γ, θ, q) u(q, θ) c(q) + F (θ) γ f(θ) u(q,θ) θ is assumed to be strictly quasi-concave, γ [0, 1]. We find this assumption fairly restrictive. For example, if θ is uniformly distributed over [0, 1], then this assumption requires (2θ γ)q 2 > min{(2θ γ)q 1, (2θ γ)q 3 }, q 1 < q 2 < q 3, θ, γ [0, 1]. However, for every θ [0, 1 2 ], we can find γ = 2θ such that the above inequality fails. 8

9 download at most a finite number of copies determined by either the maximum affordable workload of the remote server or the limitation of transmission rate via the internet On the reservation utilities We now justify the use of type-dependent reservation utilities by introducing two possible scenarios, i.e., in both scenarios the reservation utility is increasing and strictly convex. The differentiability of the reservation utility is assumed for technical convenience. We close this section by studying some structural properties of the reservation utilities. We first assume that each customer can develop the information good by herself with the same cost function C( ), but they have no opportunity to coordinate amongst themselves and hence no transaction between any pair of customers takes place. The reservation utilities are their payoffs under self-development, which has the desired properties as shown below: Lemma 1. If customers can develop the information good by themselves and their reservation utility follow from this alternative, r(θ) is increasing and strictly convex in θ, and r(0) = 0. All proofs are in the appendix. We now outline our second scenario. Suppose that customers cannot develop the information good by themselves, but they are free to purchase from an incumbent company. The incumbent company is endowed with a strictly convex cost function s(q) and she has to pay the cost for each unit of the product she sells. The production problem has the flavor of make-to-order, i.e., the seller produces only when she expects an order. This is a standard setting in the nonlinear pricing literature, and it is adopted in Riggins [2002] to describe the pricing problem of information goods. We shall assume that this incumbent company behaves naively, i.e., she assumes the full monopoly power and does not consider the interaction between her and our seller. This can 2 Sundararajan [2004b] assumes an infinite usage of internet service, but he at the same time introduces the saturation of utility to avoid an unlimited charge from the service provider. This produces the similar effect of a finite quantity limit. 9

10 be justified if the participation of the seller is unexpected to the incumbent company, and the entry into the market can take place in a short period of time. We obtain that Lemma 2. Suppose an incumbent company is endowed with cost s(q) and adopts a nonlinear pricing schedule with full participation. If θ+ F (θ) f(θ) is strictly increasing in θ and s(q) is strictly convex with s(0) = 0, then r(θ) is increasing and strictly convex, and r(0) = 0. 3 From the above discussion, in both scenarios the reservation utilities do preserve the pattern we assume in the beginning. We in the sequel abstract the rationale behind the assumption and use the general form r(θ) to denote the reservation utilities. We now introduce a function: G(θ) θr (θ) r(θ) c, which we show later is the virtual surplus associated with type-θ customer when she is offered a specific version. Its structural properties are used in the subsequent analysis. Let θ > 0 denote the solution to G(θ) = 0. Lemma 3. G(θ) is strictly increasing for θ > 0. Moreover, c 0, θ is unique. 3 First-degree price discrimination We first assume the seller can observe customers types. This benchmark case is not only illustrative but facilitates why our model stands alone from all others in the existing literature. Following the technique of backward induction, we start with the production stage. Proposition 1. Let q denote the quality level chosen in the development stage. Then if q r (θ ), then no customer is served. If q > r (θ ), then for a given q, there exists a unique pair (θ( q), θ( q)) with θ( q) < θ < θ( q) such that the seller provides q F B (θ) = q to customers with θ [θ( q), θ( q)] and no other customer purchases the product. Moreover, q 1, q 2 s.t. q 1 q 2, we have θ( q 1 ) θ( q 2 ) and θ( q 1 ) θ( q 2 ). Under first-degree price discrimination, every customer that is offered a version receives the same quality level, but is charged a different price. Whenever the transaction is efficient, 3 The monotonicity of θ + F (θ) f(θ) is commonly adopted in the nonlinear pricing literature (Salanie [1998]). It indicates that the virtual values that the seller can extract from customers are nondecreasing in θ. 10

11 trade always occurs, and the reservation utilities of those excluded customers are so high that the seller finds it unprofitable to even offer the highest possible quality. Moreover, as the quality limit q increases, the set of customers served enlarges from both ends, and the low-end customers also benefit from the technology shift. In particular, when θ( q) hits the upper bound R of θ s support, the seller s incentive to increase the quality limit arises due to (1) the ability to charge a higher price for high-end customers; (2) the ability to include more low-end customers. Note also that when q > r (θ ), the type-θ customer is always served under first-degree price discrimination. Figure 1 illustrates the total surplus, the positions of boundary points, and the enlargement of the interval of served customers when the quality limit is increased from q 1 to q 2 for cases where θ < R. As the quality limit increases, the total surplus grows at a higher rate, and the total surplus catches up with the reservation utility sooner and stays above it longer. The portion of customers whose participation constraints are satisfied becomes larger. Figure 1: An example to demonstrate the total surplus and the positions of boundary points. The fact that every customer who is served receives the same quality level is in strict contrast with the majority of results in the nonlinear pricing literature. In that literature, it is common to assume the strict concavity of the social surplus s(q, θ) u(q, θ) c(q), see, e.g., Jullien [2000], Salanie [1998], and Sundararajan [2004b]. 4 With this assumption 4 They may assume either the strict concavity on utility u(q, θ) (in the relevant part) or the strict convexity on cost c(q). 11

12 and the single-crossing condition (u qθ (q, θ) > 0, q, θ), we can show that the firstbest quality level q F B (θ) is strictly increasing in θ, whose proof is briefly given as follows. The strict concavity implies that a unique solution q SB (θ) can be obtained from the first-order condition, i.e., s q (q F B (θ), θ) = 0. Differentiating this equality by θ, we have s qq (q F B (θ), θ) d qf B (θ) + s dθ qθ (q F B (θ), θ) = 0. Note that s qq (q F B (θ), θ) < 0 < s qθ (q F B (θ), θ), we conclude that d qf B (θ) must be strictly positive, and hence every served customer receives a dθ version specific for her. In our information good pricing framework, especially the software versioning scenario, this seems to be implausible, since it implies that some customers strictly prefer technologically inferior versions. If the price is not a concern, does a customer really prefer a student edition of Mathematica that cannot perform a huge number of functions/macros to the enterprise edition? Do people feel excited when they realize that some functions of the software they just obtained are intentionally disabled? In this context, assuming every customer prefers the best quality makes better sense. The next step is to consider the quality selection problem in the development stage. Theorem 1. Let q denote the unique solution to the equation θ( q) θ( q) θf(θ)dθ = Eθ, and q F B denote the optimal quality limit in the first-best scenario. Then q F B can be obtained by searching over points that satisfy θ( q) θ( q) θf(θ)dθ = C ( q), provided that q r (θ ). If the above equation has no solution, then q F B = 0. In particular, if C (r (θ )) > Eθ, then q F B = 0; if C ( q) > Eθ, then q F B < q. Moreover, in all cases q F B > r (θ ), and choosing any quality limit less than r (θ ) is a strictly dominated strategy, independent of the structure of the development cost. This theorem characterizes the optimal level of quality limit in the first-best scenario, and has a clear economics intuition. Any choice below the critical level r (θ ) is suboptimal since by offering it no transaction is efficient but the seller pays the development cost. If the development cost is fairly high (i.e., if C (r (θ )) > Eθ), then the seller finds it unprofitable to develop the information goods, and no transaction occurs due to the inefficiency. When 12

13 the development cost is moderate, the optimal quality limit falls in the region [r (θ ), q]. 4 Second-degree price discrimination We now consider the optimal strategy to achieve second-degree price discrimination. We first take the quality limit q as given, and derive the optimal quality-price schedule assuming that the seller offers versions to only an interval of customers. We next allow arbitrary exclusions of customers, and show that it is in the seller s best interest to serve only an interval of customers. Finally, we consider the optimal quality limit in the development problem. We make the following assumption regarding the distribution of θ in the sequel. Let F c ( ) = 1 F ( ) be the complementary cdf of θ. Assumption 1. θf c (θ) is unimodal and has a unique maximum at k (0, R). In particular, Assumption 1 implies that the function F c (θ) θf(θ) is initially positive and then becomes and stays negative. Note that if we interpret θ as the price and the complementary cdf as the effective demand, θf c (θ) represents the revenue as a function of price. Its unimodality is commonly assumed in many papers on revenue management, e.g., Lariviere and Porteus [2001] Optimal schedule when an interval of customers are served We start with the case when the seller offers versions to an interval of customers. We will first take the interval as given and characterize the optimal quality-price schedule under 5 A sufficient condition for unimodality is when the distribution has the increasing generalized failure rate property (IGFR), namely, θf(θ)/(1 F (θ)) is increasing in θ. This is satisfied for the beta and the lognormal distributions (Lariviere [2004]). Ziya et al. [2004] compare three conditions that induce revenue unimodality, and mention some common distributions that satisfy these conditions such as normal, uniform, and gamma. It can be shown that this function is unimodal for the triangular distribution too. 13

14 such an assumption. We then allow the seller to choose one interval arbitrarily, and find the optimal boundary points that maximize the seller s profit. We will assume that the seller offers a menu of versions to customers with θ [θ, τ), customers with θ [τ, θ] accept the same version with quality limit q q(τ) and price p(τ), and customers in [0, θ) ( θ, R] are excluded, where 0 θ τ θ R. We further assume that by accepting ( q, p(τ)), the type-τ customer receives her reservation utility, i.e., p(τ) = τ q r(τ), independent of the quality-price schedule give for customers with θ [θ, τ). We will verify later that this is a necessary condition for optimality. 6 Suppose the customers with θ [θ, τ) are offered versions with (q(θ), p(θ)) being the quality and price. The seller s problem is to find a quality-price schedule that solves : { τ } (p(τ) c)(f ( θ) F (τ)) + (p(θ) c)f(θ)dθ, max q( ),p( ) s.t. (IC-1) θ argmax z [θ,τ) θq(z) p(z), θ [θ, τ), θ (IC-2) θ q p(τ) max θq(z) p(z), θ [τ, θ], z [θ,τ) (IC-3) r(θ) max θq(z) p(z), θ [0, θ), z [θ,τ] (1) (IC-4) r(θ) max θq(z) p(z), θ [ θ, R], z [θ,τ] (IR-1) θq(θ) p(θ) r(θ) 0, θ [θ, τ), (IR-2) θ q p(τ) r(θ), θ [τ, θ]. In Eq. (1), the first four inequalities are incentive compatibility (IC) conditions, where (IC-1) is for a customer that receives a version specific for herself, (IC-2) is for those customers 6 Note that for a given quality limit q, these thresholds θ, τ, θ shall be functions of q, but for notational ease we suppress this dependence in the analysis. Notice also that we do not exclude the possibilities of θ = 0, θ = τ, τ = θ, or θ = R, which represent respectively the cases when no low-end customer is excluded, no versioning occurs, only one customer receives the first-best quality level, and no high-end customer is excluded. Therefore, this is without loss of generality whenever the seller serves an interval of customers and at least one customer offered the efficient quality level receives no surplus. As some of these intervals degenerate, the corresponding constraints become inactive. 14

15 that accept the same version with quality q, and (IC-3) and (IC-4) are for respectively customers whose types are excluded from below and above. In (IC-1), the menu is said to be incentive compatible since the utility of customer θ is maximized if she chooses the bundle (q(θ), p(θ)); similarly, (IC-2) ensures that accepting the version with quality limit q leads to a higher utility for a customer θ [τ, θ] than choosing any other version. On the other hand, by staying out customer θ receives her reservation utility r(θ), and hence (IC-3) and (IC-4) guarantee that customers with θ [0, θ) [ θ, R] prefer staying out to accepting any other bundle (q(z), p(z)) with z [θ, τ]. The last two inequalities in Eq. (1) represent individual rationality (IR) conditions, i.e., each customer should get at least her reservation utility. Note that in the seller s plan, customers with θ [0, θ) [ θ, R] obtain their reservation utilities, and hence their individual rationality conditions are automatically satisfied. The optimal quality price schedule is summarized below, where θ solves G(θ ) = 0. Theorem 2. Suppose that q is given and the seller wishes to obtain second-degree price discrimination. Then customers with θ [0, θ ) are not served, independent of q. Transactions occur if and only if R > θ and r (θ ) < q, in which case τ (θ, k] and θ(τ) such that Customers with θ [0, θ ) ( θ(τ), R] are not served. Each customer with θ [θ, τ) receives a specific version with q(θ) = r (θ), p(θ) = θr (θ) r(θ). No information rent is left for any customer in this region. Customers in [τ, θ(τ)] accept the same version with quality q and price τ q r(τ), and everybody in the interior of this region receives a nonzero surplus. The seller gets positive profit from every customer she serves. θ(τ) = R if r(r) (R τ) q + r(τ); otherwise, r( θ(τ)) = ( θ(τ) τ) q + r(τ). The value of τ is then determined by the exhaustive search of local maxima on points in [0, k] that satisfy [τ q r(τ) c]f( θ(τ)) d θ(τ) dτ + ( q r (τ))[f ( θ(τ)) F (τ) τf(τ)] = 0. 15

16 When either q r (θ ) or R θ, the seller is unable to make any profit by offering versions and maintaining customers incentive compatibility, and therefore no transaction occurs. Transactions are efficient when θ q c as we have seen under the first-degree price discrimination, but the information asymmetry drives out the possibility of transactions. Now consider q > r (θ ) and R > θ. In this case, the seller finds it profitable to offer different versions to customers. Note that this is in strict contrast with the scenario where customers are endowed with a common reservation utility (Bhargava and Choudhary [2001], Jing and Radner [2005], and Salant [1989]), where versioning is known to be suboptimal for the seller of information goods when the product cost is not sufficiently convex in quality. 7 Our result uncovers an incentive for the seller to provide different versions. As customers possess heterogeneous reservation utility that is strictly convex, versioning helps the seller to extract more profits from these customers even if the production cost is independent of the quality level, the customers possess constant marginal willingness to pay, and the products do not exhibit network effects. The inclination to provide versioning is fairly strong since the production cost does not change as a different quality level is provided. Figure 2: An example of the optimal quality schedule under the second-degree price discrimination. 7 In the absence of heterogeneous outside opportunities, it can be shown that without interaction between θ and q in the utility function, e.g., u(θ, q) is separable and multiplicative, versioning is not profitable (private communication with Arun Sundararajan). 16

17 Furthermore, Theorem 2 characterizes the optimal quality-price schedule, whereas a generic shape of the quality levels offered to customers is presented in Fig. 2. At optimality, the seller discards both the low-end and high-end customers. For those served, the seller extracts full information rent for customers with relatively low willingness to pay, and offers a common version to the rest. The rationale to exclude low-end customers is clear in the standard nonlinear pricing literature: the seller is unable to extract positive rent from a low-end customer, and therefore the seller should not serve her. The cutoff point is one at which the virtual surplus turns positive. In the region [θ, τ), each customer is offered a version specific for her. Nevertheless, by accepting it the customer receives exactly her reservation utility. The seller is able to fully extract the information rent from customers in this region, but has to distort the quality levels away from the first-best levels to maintain incentive compatibility. Inefficiency occurs due to this, because the seller cannot observe customers types. Moreover, the offered quality level is strictly increasing in the type, i.e., customers with higher marginal willingness to pay receive products of better quality. Prices are chosen such that the versions are incentive compatible and they are automatically monotonic to avoid dominance among versions. The customers in [τ, θ(τ)] are offered a common version that makes the type-τ customer receives her reservation utility. The seller earns no information rent from these customers. This result is labelled as the bunching or pooling phenomenon in the nonlinear pricing literature (Salanie [1998]), which may occur when the monotone hazard rate property fails, see also Maggi and Rodriguez-Clare [1995] for more discussions on the pooling results that arise in the optimal contract design. The upper bound of this region is determined by the critical customer who is indifferent to accepting this version and staying with her outside opportunity if such a critical customer exists; otherwise, the upper bound is R, i.e., no highend customer is excluded. Moreover, at the critical point τ we see a clear discontinuity in the version specification: both the quality and the price have jumps at θ = τ. This implies that the profit the seller collects is also discontinuous at τ since the marginal cost is independent of the quality. Although the quality schedule exhibits discontinuity, the utility generated 17

18 from accepting versions for any customer has to be continuous to prevent any profitable deviation (verified in the proof of Theorem 2). Finally, the exclusion of high-end customers is due to the pre-determined quality limit in the development stage, and hence with the linear utility format and strictly convex reservation utility, the seller must give up those high-end customers because their outside opportunities are too high. This was observed in the first-best scenario, even though the determination of the cutoff type is based on a different criterion. Figure 3: An example to show the net utility under the second-degree price discrimination. Figure 4: changing the value of τ. The influence of profits while The shape of net utilities is also worth noting. Assuming that some high-end customers are excluded by the optimal quality-price schedule, we draw in Fig. 3 the received utilities of customers and the net utilities (the received utility net the reservation utility). Except θ (τ, θ(τ)), customers receive their reservation utilities in the end, regardless of whether she accepts a version or stays unserved. Inside the region (τ, θ(τ)), the net utility is unimodal, and the customer that receives the maximal rent is located in the interior. This bell-shaped net utility is also reported in Maggi and Rodriguez-Clare [1995, Figure 3] in the contract design. If no high-end customer is excluded at optimality, the gross utility will end up in the affine part, and the net utility stops inside the bell rather than in the flat region. The optimal schedule can be interpreted as follows. Sundararajan [2004a] reports that the quality-price schedule can be decomposed into two parts: one is driven by the outside 18

19 opportunity (reservation utility) and the other is determined by the heterogeneity of utility (possibly price shifted to differentiate with the other schedule). In our model, the first one corresponds to the case θ [θ, τ), where the optimal quality schedule is determined by customers outside opportunities. Nevertheless, the seller obtains positive profits from these customers, in contrast to Jullien [2000]. The second part corresponds to the region θ [τ, θ]. If r( ) were constant, a common version with q would be offered to all customers but at a lower price. Thus, we see the price shift alluded to above (c.f. Sundararajan [2004a]). Fig. 4 demonstrates the trade-off the seller faces while choosing the value of τ. As a seller increases τ from τ 1 to τ 2, i.e., she increases the starting point of offering a common version, the price for that common version increases even though the quality remains q. This change influences the profit in three ways: First, the seller loses some profits on customers with θ between τ 1 and τ 2, since the quality levels offered to them are {r (θ)} s rather than q. Second, the shift of τ increases the profit gained from those who accept the common version; the seller gets (τ 2 τ 1 ) q (r(τ 2 ) r(τ 1 )) more in the region [τ 2, θ(τ 2 )]. Third, since for a given quality limit q this version is priced higher (from τ 1 q r(τ 1 ) to τ 2 q r(τ 2 )), fewer customers are willing to purchase, and hence this shift excludes more high-end customers. The optimal value of τ balances the gains and losses. We now discuss the condition on τ in the theorem. If f( ) is widely spread-out, i.e., f( θ(τ)) is relatively small, the second term ( q r (τ))[f ( θ(τ)) F (τ) τf(τ)] dominates. Since q r (τ) > 0, the sign of the derivative depends only on F ( θ(τ)) F (τ) τf(τ), which approximately is the derivative of τ(1 F (τ)) as θ R, and hence it turns negative right at τ = k. This ties in with the unimodality assumption (Assumption 1). 4.2 Optimal schedule and target quality with arbitrary exclusion Theorem 2 provides the optimal quality-price schedule when an interval of customers is served. An immediate question is whether this schedule remains optimal if the seller can exclude customers arbitrarily, e.g., she excludes customers with θ [0.23, 0.31] [0.45, 0.79]. 19

20 The set of excluded customers can be even more sophisticated, namely any measurable set with respect to the probability space ([0, R], B, F ( )), where B is the collection of Borel measurable sets over [0, R]. Nevertheless, our proposed quality schedule is indeed optimal. Theorem 3. Given the quality limit q, the quality-price schedule proposed in Theorem 2 is optimal even if arbitrary exclusion is allowed. The proof follows the approach of Jullien [2000], where we show that no intermediate exclusion is profitable, and hence at optimality the seller must offer versions to an interval of customers. Since the schedule proposed in Theorem 2 is optimal if customers in an interval are served, its optimality continues to hold in this broader class of schedules. Note that the structure of our optimal schedule is labelled as bunching with exclusion in Jullien [2000], since the seller offers a pool of customers a common version and excludes some customers. Our characterization and verification of optimal quality-price schedule is now complete. The seller s problem in the development stage is as follows. Let V ( q) be the optimal value of Eq. (1) when quality-price schedule is optimally chosen. The optimal quality limit can be found through exhaustive search of the local maxima: q SB = argmax q {V ( q) C( q)}. 5 Comparative statics In this section we discuss the comparative statics of our model. This includes (1) given a fixed quality limit, how does second-degree price discrimination differ from the first-degree price discrimination? (2) How does the profit change as the quality limit varies? We first compare these two informational scenarios while assuming a fixed quality limit q. Theorem 4. Suppose that the quality limit q is given. Then If q > r (θ ), let θ F B ( q), θ SB ( q), θ F B ( q), θ SB ( q) denote respectively the lowest and highest type of customers that are offered a version under the two price discrimination. Then for all q, θ F B ( q) < θ SB ( q) < θ SB ( q) < θ F B ( q). In particular, customers with 20

21 θ θ are never served under second-degree price discrimination, whereas an interval around the type-θ customer is included in first-degree price discrimination. Under first-degree price discrimination, each customer either is not served or receives q. However, under the second-degree price discrimination, a continuum of versions may be offered. If at optimality the seller chooses τ = (r ) 1 ( q), then only the customer with θ = (r ) 1 ( q) receives the efficient quality level q. When R θ, where θ is the critical customer whose virtual surplus just turns positive, under the second-degree price discrimination the seller will not develop the information goods, regardless of the cost structure C( ). Nevertheless, there exist situations where the seller does use first-degree discrimination over customers. The first comparison shows that the set of customers offered under the second-degree price discrimination is a proper subset of that under the first-degree price discrimination. The information asymmetry does prevent the seller from serving some customers although the transactions are efficient. The second comparison demonstrates the inefficiency on the quality levels offered under the second-degree price discrimination. Except possibly a subset of customers, a continuum of customers receive versions that have inferior quality levels. By intentionally shading the quality levels, the seller gains against the information asymmetry. Finally, to induce development of information goods, the second-best scenario requires a larger maximal marginal willingness to pay of customers. If we interpret value of R as a measure of customers heterogeneity, a higher degree of heterogeneity among customers is needed to overcome the information asymmetry faced by the seller. Now we discuss the impact of different quality limits on the quality-price schedule under both first- and second-degree price discrimination. Theorem 5. Suppose the two quality limits q 1, q 2 are pre-determined, and q 2 > q 1. Then in the production stage, the seller obtains strictly greater profits with q 2 compared to the case with q 1 under both first- and second-degree price discrimination. 21

22 In the first-best scenario, both the prices are higher and the set of served customers is larger when a higher quality limit is set in the development stage. Under the second-degree price discrimination, as a higher quality limit is chosen, the seller can always choose the same starting point of offering a common version. By doing so she gains in two ways: (1) the price of this common version is strictly higher; (2) more high-end customers are willing to purchase this version compared to the case with q 1. Hence, the seller collects a strictly higher profit in the production stage with information asymmetry as well. 6 Conclusion In this paper we consider a two-stage problem for information goods production. We show that versioning is profitable when customers possess heterogeneous participation constraints, and characterize the optimal quality-price schedule by using optimal control theory. In the optimal strategy, the seller discards both the low-end and high-end customers. For those served, the seller extracts full information rent from customers with relatively low willingness to pay, but offers a common version to the rest. We also provide a simple rule for selecting the optimal quality limit in both cases. When the quality limit is pre-determined, the information asymmetry forces the seller to give up some transactions in the production stage even though they are efficient in the first-best scenario. As the quality limit is raised, the seller gathers a strictly higher profit under both types of discrimination. Several extensions arise naturally, a particular one being the incorporation of network externalities. When network effects are measured by the total usage of the product across different versions, the seller always has an incentive to include more customers. However, if products of different quality levels share only part of the benefit, 8 the seller faces an intriguing trade-off: should she offer user-specific versions to fully extract low-end customers rent or should she offer only a limited number of versions to induce higher network effects? 8 For example, customers of enterprise versions may not benefit from the increased sale of the student edition, because technical help for advanced functions is granted only after customers have experienced them. 22

23 Another direction is the analysis of the dynamic setting of our model with generations of customers when the distribution of reservation utilities varies over time. Since the choice of quality limit is irrevertible, the seller faces a constrained optimization problem in the development stage with respect to the current quality limit. While developing a new (and higher) quality limit is costly, the dynamic setting may allow us to predict the optimal timing of investing in new product development for such an industry. Introducing competition between sellers is another avenue for research. As sellers choose quality limits upfront, they may distinguish themselves by selecting different levels, and therefore adopt different quality-price schedules given such quality limits. This differentiation bypasses the head-to-head price competition that could potentially drive away all the profit. Finally, our approach is also applicable to nonlinear pricing problems for non-information goods such as airline/railroad seats, washers, bicycles, and television sets. More generally, this paper also sheds light on the nonlinear pricing problems with heterogeneous participation constraints when the total surplus fails to be strictly quasi-concave, as assumed in the majority of the existing literature. Further explorations on this class of problems are needed. Appendix. Proofs Proof of Lemma 1 By the convexity of C( ), the first-order condition is sufficient to characterize the optimal quality level. Hence θ = C (q(θ)), and her net profit is r(θ) = θq(θ) C(q(θ)). Differentiating r(θ) twice, we obtain r (θ) = 2 dq +θ d2 q C (q)( dq dθ dθ 2 dθ )2 C (q) d2 q dθ 2 = dq (2 C (q) dq ), where the dθ dθ second equality follows from the first-order condition. By strict convexity of C( ) and the first-order condition, q(θ) is strictly increasing, and hence dq dθ > 0. Moreover, performing the full differentiation w.r.t. θ on θ = C (q(θ)), we obtain C (q) dq dθ = 1. Thus, r (θ) = dq dθ > 0, which gives us the strict convexity of r( ). To see the monotonicity of r(θ), let θ 1 < θ 2. If type-θ 1 customer uses type-θ 2 s optimal quality level q(θ 1 ), she gets θ 2 q(θ 1 ) C(q(θ 1 )) r(θ 1 ) because θ 1 is less than θ 2. If she optimizes the quality, then her utility can only be higher. 23