Ram Bala Scott Carr July 1, 005 Abstract We analyze and compare ixed-ee and usage-ee sotware pricing schemes in ixed-ee pricing, all users pay the same price; in usage-ee pricing, the users ees depend on the amount that they use the sotware (e.g., the user o an online-database service might be charged or each data query). We employ a two-dimensional model o customer heterogeneity speciically, we assume that customers vary in the amount that they will use the sotware (usage heterogeneity) and also in their per-use valuation o the sotware. To understand the perormance o these pricing schemes and their sensitivity to the competitive environment in which they are used, we look at a number o dierent scenarios: a monopolist oering just one o these schemes, a monopolist oering a choice o pricing schemes, and several duopoly scenarios. We characterize and compare the equilibria that arise in these scenarios and provide insights into optimal pricing strategies. CLA Anderson School o Management, Los Angeles, CA
1 Introduction In this paper we analyze and compare ixed-ee and usage-ee sotware pricing schemes in ixed-ee pricing all users pay the same price; in usage-ee pricing the users ees depend on the amount that the use the sotware (e.g., the user o a dat abase-based system might be charged or each data query). Fixed-ee pricing is today s dominant pricing scheme, and our interest in usage-ee pricing stems rom the ollowing: 1. The sotware services industry is currently acquiring scale in the orm o Application Service Providers (ASPs) that oer industrial sotware such as CRM (e.g., Salesorce.com) to small and medium sized irms. The technical easibility o usage monitoring, a prerequisite or usageee pricing, is assured by the act that the World Wide Web is the primary delivery platorm used by ASPs, and irms are currently oering usage-ee pricing (e.g., KnowledgePoint). Nonetheless, usage-ee pricing in this industry is still in a nascent stage. 1. In contrast to 1, usage-ee pricing is relatively common among computing inrastructure service providers. Some notable examples o irms that have pursued such a strategy in this space are Jamcracker and HP. In other areas such as media licensing, irms oer quite dierent pricing schemes; or example, Apple itunes charges customers 99 cents or every song downloaded while Realplayer s Rhapsody charges a ixed monthly ee with an unlimited number o downloads 3. In a sense, usage-ee pricing is a rebirth o an old business model that was once prevalent in the sotware industry. Back in the days when all computers were huge and expensive, time-sharing on large IBM mainrames was common practice, so irms implemented usage-ee pricing schemes. This business model became less important as computer hardware got a lot cheaper and aster. Recently, the spread o complex enterprise sotware has again increased the costs o sotware deployment and maintenance, so enterprise sotware irmssuchassap are turning to usage-ee pricing to reach small and medium size customers. An interesting question is how do consumers behave in this setting? nlike many other service industries in which consumers pay a price based on resource utilization (also reerred to in the literature as access industries), two consumers may use a sotware package or the same amount o 1 http://www.comptia.org/sections/ssg/research.asp 1
time but may dier in the relative importance o their work. Compare a long but rivolous web suring session by one consumer versus a short search on a high quality industrial database. While a usage meter shows a low number in the ormer case, the latter consumer is probably willing to pay much more. Similarly, some music lovers value a small collection o speciic songswhileothers preer quantity and selection over speciic tastes. These consumer heterogeneity issues introduce several modeling and analytical issues. First, we believe that this setting begs or a -dimensional representation o consumer heterogeneity and this is in contrast to the preponderance o existing literature. Second, the perormance o these pricing schemes are sensitive to the competitive environment in which they are used; to better understand this, we look at a number o dierent scenarios: a monopolist oering just one o these schemes, a monopolist oering a choice o pricing schemes, and several duopoly scenarios. We characterize and compare the equilibria that arise in these scenarios and provide insights into optimal pricing strategies. Our models assume customer usage heterogeneity meaning that some customers will use the product more than others. The literature on pricing in the ace o this orm o heterogeneity has roots in economics via Oi(1971) who shows that a truly discriminating two part tari globally maximizes monopoly proits by extracting all consumer surpluses. A stream o literature that analyzes dierent pricing mechanisms under dierent conditions lowsromthisbasicsource.the Oi model is extended by Schmalensee(1981) who analyzes the case o proit constrained welare maximization. Phillips and Battalio(1983) investigate the situation where buyers can substitute between visits and also consumption between visits. Hayes(1987) shows that two part taris act as a orm o insurance in environments with uncertainty and hence is oered by irms even in a competitive setting. Other orms o pricing can also be mapped into this ramework, most notably quantity discounts (see Dolan(1987)). Fixed-ee pricing in the ace o customer usage heterogeneity is also known as buet pricing as in Nahata et al(1999). Also related is Bashyam(000) who models competition in business inormation services markets between a ixed-price two-part tari, Sundararajan (00) who compares a monopolist s ixed-ee versus a usage-ee in the presence o network externalities, and Wu et al(00). In comparing ixed- and usage-ees it becomes important to recognize that customers may also be heterogeneous with respect to their per use valuation o the product and that this likely exists in combination with customer usage heterogeneity. We thus incorporate both types o heterogeneity, and this two-dimensional model o customer heterogeneity is a distinctive eature o this paper.
Other than Bashyam(1996), which ocuses on a technology choice problem, we are unaware o any other papers that employ such a model. Recently emerging is research into pricing o various kinds o access services. Jain et al(1999) develop a model or cellular phonecall prices and ind the optimal pricing strategy over time. Essegaier et al(00) compare two part taris withixed-ee pricing and usage pricing or access service industries under conditions o customer usage heterogeneity and limited capacity. Danaher(00) conducts a market experiment to compare dierent two part pricing packages or new subscription services. There is also a growing literature around economics o sotware which studies various economic issues related to sotware, highlighting its distinctive eatures and unique production process. Boehm(1981) and Kemerer(1987) address sotware cost estimation. Richmond et al(199) and Whang(199) tackle contract design or custom sotware development. Whang(1995) addresses pricing and bidding or custom sotware projects in the presence o learning eects or the developers. Snir and Hitt(003) and Carr(003) investigate bidding behavior o sotware vendors in online auctions or sotware development projects. Conner and Rumelt(1991) analyze piracy protection strategies or a sotware irm. In short, this paper adds to the literature on sotware pricing by comparing these dierent pricing schemes in a richer model o customer heterogeneity than previously seen. It contributes to the access industry pricing literature since the sotware services industry is part o the online access industry. It also adds to the sotware economics literature by tackling economic issues related to the latest developments in the sotware industry. The paper is organized as ollows. Section deines the model. Sections 3 and 4 apply the model to monopoly and duopoly settings respectively, and the body o the paper concludes with section 5. Section 6 contains proos and derivations. Model deinition Again, we analyze equilibrium behavior in scenarios in which one (section 3) or two (section 4) irms oer combinations o ixed and usage-based pricing schemes to heterogenous customers who 3
sel-select which price (i any) to pay. This section deines the general model and notation. Customers and product: To model the two types o heterogeneity discussed earlier, each customer is described by a two dimensional vector (α, β) in which α denotes the utility that the customer will derive rom a single use o the product and β represents the requency with which the customer will use the product. A scalar parameterizes the products quality or unctionality, so an (α, β)-customer enjoys utility o αβ rom purchasing the product. The set o potential customers is modelled as an atomless spread o (α, β) pairs distributed evenly over the [0, 1] [0, 1] square. Thus, i M is the size o the potential market, then any market o size x M corresponds to a raction x o the area o this square. To simpliy however, the market is normalized by setting M equal to one; this is just scaling and does not sacriice generality. Purchasing decisions ollow naturally; customers are assumed to sel-select whatever purchasing option maximizes this utility net o price. Fixed-ee pricing: This is today s ubiquitous pricing scheme; buyers pays a common price P, and receive unlimited use o the product. Disregarding any other pricing options, purchasing is worthwhile or an (α, β)-customer i αβ P > 0. (1) Figure 1 illustrates. Customers who share the same value o α β derive the same utility rom the product, so hyperbolae on the unit square become lines o iso-utility. These are the curves in igure 1(a); each iso-utility line is a locus o customers with identical purchasing behavior under ixed-ee pricing. The P price chosen by the irm then segments the customers along one o these hyperbolae as shown in igure 1(b). In that igure, the boundary between the do not purchase segment and the pay ixed-ee segment is the curve α β = P. sage-ee pricing: In this pricing scheme, the customer pays P u or each use. Since she uses the product with requency β, her total payments are β P u versus utility o αβ, sothisscheme is worthwhile to her i β (α P u ) > 0. This is analogous to (1) but results in a very dierent structure as illustrated by igure. Iso-utility lines are now vertical, and the irm s selection o a particular value o P u segments the market along 4
1 HaL Iso-utility curves 1 HbL Segmentation α β= P sage requenc yhbl 0.8 0.6 0.4 0. sage requenc yhbl 0.8 0.6 0.4 0. Do not purchase Pay ixed-ee 0 0 0. 0.4 0.6 0.8 1 tility per use HaL 0 0 0. 0.4 0.6 0.8 1 tility per use HaL Figure 1: Fixed pricing: (a) iso-utility curves, and (b) segmentation α = P u. Costs and revenues: With ixed-ee pricing, a irm s proits are P times the area above the segmenting hyperbola. With usage-ee pricing, revenues accrue on a per-use basis and the irm additionally incurs costs o c u dollars per-use to cover, or example, costs o metering and monitoring. Equilibrium: The irst stage o this game is the pricing decision; customer purchases then ollow. The solution concept employed is a standard Nash equilibrium between the set o potential customers and the irm or irms supplying the sotware. However, to better distinguish between the models with and without inter-irm competition, we use the term optimal or the price that maximizes a monopolist s proits and reserve equilibrium or the duopoly cases. This is admitedly an abuse o the standard taxonomy more precisely, the price establishes an equilibrium between the monopolist irm and the customers. 5
1 HaL Iso-utility lines 1 HbL Segmentation 0.8 0.8 0.6 0.6 Payusageee 0.4 0.4 Do not purchase 0. 0. 0 0 0. 0.4 0.6 0.8 1 α= P u 0 0 0. 0.4 0.6 0.8 1 Figure : Iso-utility lines and segmentation: metered pricing 3 Monopoly analysis In this section we consider a single irm in isolation that is, a monopolist. For this irm, we analyze three pricing scenarios: ixed-ee pricing (section 3.1), usage-ee pricing (3.), and ixedand usage-ee pricing oered simultaneously (section 3.3). 3.1 The ixed-ee monopolist: Reerring to igure 1(a), given ixed-ee P and product quality, all customers with (α, β) above the αβ = P hyperbola have αβ P > 0, the size o the ixed-ee segment is and the irm s proits π are q =(1 P ) Z 1 P P β dβ =(1 P )+P ln(p ), π, P q. The ollowing lemma is a handy result that acilitates optimization o π and is also useul later. Lemma 1 I a non-negative unction g has g (a) =g (b) =0and g 000 > 0. Then, on the interval (a, b), theunctiong has a maximum and no (other) local maxima. 6
nder this model o customer heterogeneity and ater disallowing negative prices, the optimal P is guaranteed to all in the interval [0,]. Evaluatingπ over this interval, π P =0 = π P = =0, and π 0<P < > 0. Dierentiating π gives dπ dp =1 P +P ln(p ), d π dp = 1 (1 + ln(p )), and d 3 π dp 3 = P. This strictly positive third derivative (over (0,)) together with (15) allows use o the lemma which guarantees a unique and π -optimizing solution to the irst order optimality condition dπ dp =0. The irst derivative dπ dp has roots o the orm P = 1 1 y where y is a solution to = y ln (y); only e the root in the open interval (0,) is relevant, and it identiies this monopolist s optimal price, proits, and segment size: P =0.85, π =0.10, q =0.357 () 3. The usage-ee monopolist Illustrated by igure, when a monopolist oers a usage-ee (instead o a ixed-ee) the customer segment that uses the product is the area to the right o α = P u. The size o this segment is 1 P u, and it is the total number o uses (or transactions) by this segment that determines the irm s costs and proits. Speciically, the usage-ee monopolist s proit (denoted π u )is Z 1 (P u c u ) (1 P u ) ( βdβ) which equals 1 0 (P u c u ) (1 P u ) This is concave, and irst order optimality conditions give optimal price and proits: P u = 1 (1 + c u ) and π u = 1 8 (1 c u ). (3) The simple orms o () and (3) acilitate comparison o the two pricing schemes. As given in the proposition below, the monitoring/metering cost determines which scheme is most proitable. For simplicity o exposition, denote c to be the monitoring cost normalized by product value; that is, c, c u. Proposition 1 A monopolist optimally selects the usage-ee pricing scheme i and only i the monitoring cost c is less than 0.097; above this threshold the ixed-price scheme is optimal. 7
1 0.8 Do not purchase αβ= P Payixed-ee sage requenc yhbl 0.6 0.4 0. Do not purchase β= P P u Payusageee α= P u 0. 0.4 0.6 0.8 1 tility per use HaL Figure 3: Segmentation or dual-scheme monopoly Intuitively, usage-ee pricing dominates when monitoring costs are low, and this is relected in many real world settings, some ar removed rom the traditional sotware arena. For example, at a ski resort it would be costly and diicult (not to mention cold and miserable) to manually sell and collect a ticket each time a skier mounts a chairlit, so it is unsurprising that resorts have mostly relied on per-day pricing. However, inormation systems now enable automated monitoring and billing, so a number o resorts now oer customers a choice o paying a ixed-ee or paying on a per-ride basis. The question then becomes, when it is easible to oer both schemes simultaneously, what prices should the irm choose, and what are the beneits o oering a second pricing scheme? 3.3 The dual-ee monopolist We now consider a monopolist who oers customers a choice o ixed- or usage-ees. The model is otherwise unchanged the cost c u is incurred or all usage-ee transactions, and the same value service is oered under both schemes (there is no versioning). Figure 3 illustrates the segmentation that results when both options are oered although there is just one irm, and thus just one proit unction to be maximized, the two pricing mechanisms compete in the sense that each truncates the other s segment along the horizontal line β = P P u this is the locus o indierence between the ixed- and usage-ee schemes. 8
P is chosen by the pay ixed-ee segment o igure 3, andq, the size o this segment, is q = Z 1 P µ 1 P Z 1 µ P dα P dα =1 P + P µ α Pu P u α P u ln P. P u P u is chosen by the pay usage-ee segment, and this then number o times this segment uses the sotware is Z 1 Pu P Z Pu 0 βdβdα = µ P P u µ 1 P u Altogether, the dual-ee monopolists proits, denoted π d,are P (1 P P π d = u )+ P ³ ³ ln P P u + 1 (P u c u ) (1 Pu ) P P u 1 (P u c u ) (1 P u ) i P <P u otherwise The objective o the irm is to set prices P and P u in order to maximize proit: max P,P u {π d } (4) Proposition In a dual-ee monopoly: (i) Auniquepricepair ³ P,P u maximizes proits. (ii) P u = (1 c)+ (1 c) +16c 4 (iii) P u >P ³ The optimal usage-ee (Pu) is given in the proposition above, but the optimal ixed-ee P is not available in closed orm. Proceeding numerically, proits by segment and in total are shown in igure 4. An immediate observation is the optimality o oering a choice o pricing schemes. Intuitively, oering two schemes simultaneously increases the size o the pie through market segmentation, and this has the greatest beneit when monitoring costs are roughly at the c =.097 level that appeared in proposition 1. Looking at the limiting cases, as c goes to 1 the equilibrium degenerates to the ixed-ee monopoly. On the other hand, as c goes to 0, the equilibrium degenerates to a usage-ee pricing monopoly. 9
0.1 0.1 dual-schemehp d L ixed-ee Hp L 0.08 Proit 0.06 0.04 usage-ee H L 0.0 0 0 0. 0.4 0.6 0.8 1 Monitoring cost HcL Figure 4: Proit or monopoly strategies 4 Duopoly analysis We now consider scenarios with competition between two irms, and we again consider three scenarios: (1) One irm oers ixed pricing, the other irm oers usage-ee pricing, and the irms products are undierentiated. () Same as the previous scenario except that the products are now dierentiated. (3) Both irms oer usage-ee pricing; products are again dierentiated. 4.1 Dual-ee Duopoly, undierentiated products Here, one irm oers ixed-ee pricing and the other oers usage-ee pricing, and both oer the same value product (i.e., both oer the same ). From the customers perspective, the model remains basically unchanged or a given price pair (P,P u ) the customers will make the same selection as in the previous section, so segmentation remains as shown in igure 3. What does change is that the model is no longer proit optimization by a single irm with a single objective unction. Rather, the competitors each have their own objectives, and we solve or equilibrium rather than optimal prices. The proit unctions or the two irms are: 10
Fixed price irm: P (1 P P π = u )+ P ³ ln P P u i P <P u 0 otherwise (5) sage-ee irm: π u = 1 (P u c u ) (1 Pu ) ³ P P u 1 (P u c u ) (1 P u )} i P u >P otherwise (6) An equilibrium is a (P u,p ) pair that simultaneously satisies the irms respective objectives: max{π u } and max{π } P u P The next lemma characterizes the best response prices and is used to develop the equilibrium results given in the immediately ollowing proposition. Lemma At equilibrium: (i) For any usage-ee P u, the ixed price irm selects a strictly lower price (ii) For any ixed price P, the usage-ee irm selects: P u = c 1+c i P < c 1+c P i c 1+c P (1+c) (1+c) i P > (1+c) Proposition 3 (i) There exists a unique equilibrium in pure strategies. At equilibrium, (ii) P u = c 1+c (iii) Fixed-price proit increases with the monitoring cost. (iv) There exists an interval bounded below by c =0or which the usage-ee proit increases with the monitoring cost. 11
(v) There exists an interval bounded above by c =1or which the usage-ee proit decreases with the monitoring cost. (vi) In the limit as monitoring cost approaches zero: the market is ully covered, each irm gets exactly hal o the market, and π = π u =0. Noting that the irms compete through price-setting and that the irms products are identical, one might expect to see the Bertrand result that the irms are unproitableatequilibrium.itis thus interesting to observe that both irms equilibrium proits are actually strictly positive in this model (or all c (0, 1)). The key here is that the Bertrand result depends on a complete lack o dierentiation. In this duopoly however, the act that the two irms oer dierent pricing schemes provides a orm o dierentiation that results in a proitable equilibrium. Equilibrium results or the dual-ee duopoly are illustrated in igure 5. 3 In this igure, we see that both irms proits are lower than their monopolist counterparts (c. igure 4). This is not surprising; we should expect competition to reduce proits. What is more notable is that the usage-ee irm s duopolists are so drastically reduced rom those o the usage-ee monopolist. Also changed rom the previous section is that the usage-ee proits (π u ) are no longer monotonic in the monitoring cost c. Rather, that irm s proits irst increase and then decrease with c (as anticipated by proposition 3(iv) and (v)). The act that π u can increase as its costs increase is somewhat surprising, but it has a straightorward explanation the increase in c has the eect o reducing the degree o competition between the two irms, and this is beneicial to both competitors. This does not continue indeinitely however; or c greater than about 0. thenegativeeects o increasing c dominates and π u begins to all. As c gets large, π u disappears and the ixed-ee irm becomes essentially a monopolist. Figure 6 compares the dual-ee monopolist s proits to the combined proits o the dual-ee duopolists. As is inevitable 4, the monopoly outperorms the duopoly. Another observation is that, as c increases, the monopolist s proits always decrease, 5 but the duopoly proits actually increase. 3 As or the dual-scheme monopolist, the equilibrium ixed price is unavailable in closed orm. Numerical analysis provides the equilibrium prices and proitsshownintheseigures. 4 The monopolist sets P and P v to jointly optimize π d thereby guaranteeing this result. 5 Proo that the monopoly π d decreases with c is straightorward since π d decreases with c or any (P,P v ) pair, it must decrease when the prices are optimally chosen. 1
0.1 0.08 ixed-ee competitorhp L Proit 0.06 0.04 0.0 0 usage-eecompetitorh L 0 0. 0.4 0.6 0.8 1 Monitoring costhcl Figure 5: Competitor s proits: dual scheme duopoly 0.1 dual-schememonopoly 0.1 Proits 0.08 0.06 0.04 dual-schemeduopoly HaggregateproitsL 0.0 0 0 0. 0.4 0.6 0.8 1 Monitoring costhcl Figure 6: Aggregate duopoly proit vs dual scheme monopoly proit 13
0.1 ixed-ee monopoly 0.08 dual-schememonopoly, p m only Proit 0.06 0.04 dual-schemeduopoly, ixed-ee competitor 0.0 0 0 0. 0.4 0.6 0.8 1 Monitoring cost HcL Figure 7: Fixed price proit under dierent settings Figure 7 compares the perormance o ixed-ee pricing in the three scenarios in which is has thus ar appeared. The ixed-ee generates the highest proits when there is no competition and it is the only alternative oered. It generates lower proits when oered in combination with a usage-ee by a dual-ee monopolist, and it perorms worst in the competitive duopoly. When the cost to monitor usage-ee transactions is large however, the situation essentially becomes a ixed-ee monopoly in each o these scenarios. Analogous to igure 7, igure 8 compares the perormance o ixed-ee pricing in dierent scenarios, andcomparisonothesetwoigures highlights the dierences between the two pricing schemes. The ordering o these curves in igure 7 is the same as the analogous curves in igure 7, but a key dierence is the relatively poorer perormance o the usage-ee in the duopoly setting since proit is low not only at high monitoring cost (because the ixed-ee duopolist acquires monopoly power) but also at low monitoring cost (because intensity o competition increases). While this numerical study reveals important details, a broader conclusion emerges regarding the usage-ee pricing scheme. While the usage-ee pricing scheme is optimal or a monopoly irm at low monitoring cost, it is highly sensitive to competition against a ixed-ee competitor. 14
Proit 0.1 0.1 0.08 0.06 usage-ee monopoly dual-schemeduopoly, usage-eecompetitor dual-schememonopoly, only 0.04 0.0 0 0 0. 0.4 0.6 0.8 1 Monitoring cost HcL Figure 8: Metered price proit under dierent settings 4. Dual-ee duopoly, dierentiated products In several real world contexts in the sotware industry a sotware vendor oers a higher value product at a ixed-ee while an ASP oers a lower value service with a usage-ee pricing scheme. For instance, in the CRM (customer relationship management) space, Siebel is an established sotware vendor oering a product o higher value geared towards large irms while Salesorce.com is a online service provider targeting small and medium sized irms. To model such a scenario, we relax the assumption o undierentiated products; now, and u denote the values o the services oered by the ixed- and usage-ee irms respectively. With a dierentiated product, the boundary between the two market segments is no longer a constant β; instead the line o indierence is a P curve β = α( u )+P u. nlike the previous scenarios, two cases are now possible: Case A: P < P u u This case gives segmentation illustrated by the irst graph in igure 9, and we reer to this as structure A. In structure A, the set o ixed-ee customers is adjacent to the set o non-purchasing customer types. Case B: P P u u This case gives segmentation illustrated by the second graph o igure 9, and we reer to this as structure B. In structure B, the ixed-ee and the do-not purchase sets are strictly disjoint. 15
1 0.8 Do not purchase Structure A Payixed-ee 1 0.8 Structure B Payixed-ee sage requenc yhbl 0.6 P β = P u u α+α 0.4 Do not purchase 0. Payusageee sage requenc yhbl 0.6 0.4 0. Do not purchase P β= P u u α+α Payusage-ee α= P u 0. 0.4 0.6 0.8 1 tility per use HaL α= P u 0. 0.4 0.6 0.8 1 tility per use HaL Figure 9: Market segmentation in a dierentiated duopoly In order to capture the level o dierentiation, we introduce a parameter γ that is deined by γ, u. We use the same deinitions or π and π u as beore and derive the proit unctions or both irms under both cases. The exact derivation is provided in the appendix (page 7). The ixed-price irm s proit is: P (1 P )+ P ³ π = ln u P P u P ³³ ³ u ln u P u u +P u i P < P u γ P (1 P P u u ) P ³ ln u +P u P i P P u (7) γ The usage-ee irm s proit is: 1 π u = (P u c u ) ( u P u ) 1 (P u c u ) ³³ P P u u P u u + P P u (P u + u) ³ ³ P u +P u P u u +P u i P u >γp i P u γp (8) Next we state a proposition relating to the equilibrium o this duopoly. Proposition 4 Suppose (P,P u ) is an equilibrium price pair in pure strategies I P < P u u (case A): 16
(i) P u = cγ+ c(1 γ+cγ) 1+cγ u with c, cu u. (ii) ThereisnootherequilibriuminpurestrategieshavingstructureA I P P u u (case B): (iii) P = (1 γ)( 0.78γ)+0.39γ(1 γ γc)+γc 3.5( 0.78γ), P u = 0.39(1 γ γc)+c 0.78γ u (iv) ThereisnootherequilibriuminpurestrategieshavingstructureB This proposition greatly restricts the equilibrium possibilities but leaves open the question o whether there are 0, 1 or equilibria. Analytical conditions or uniqueness and existence are cumbersome and not available in closed orm. However, or any given c and γ, they can be computed numerically. Below we state the conditions or existence o the equilibria aided by the ollowing deinitions: ρ A (P u), arg max P π subject to P Pu γ ρ B (P u), arg max P π subject to P Pu γ ρ A u (P u ), arg max Pu π u subject to P u γp u ρ B (P u), arg max Pu π u subject to P u γp u nder these deinitions, necessary and suicient conditions or the existence o equilibrium with each structure are: 1. An A-equilibrium exists i and only i there exists a price pair (P eq,peq u ) with P eq Pu eq 1γ such that (c1) : P eq = ρ A eq (Pu ) (c) : π (P eq,peq u ) π (ρ B eq (Pu ),Pu eq ) (c3) : P eq u = ρ A u (P eq ) (c4) : π u (P eq,peq u ) π u (P eq,ρb u (P eq )). A B-equilibrium exists i and only i there exists a price pair (P eq,peq u ) with P eq such that (c1) : P eq = ρ B eq (Pu ) (c) : π (P eq,peq u ) π (ρ A eq (Pu ),Pu eq ) (c3) : P eq u And, in both these cases (P eq,peq u ) is the equilibrium price. = ρ B u (P eq ) (c4) : π u (P eq,peq u ) π u (P eq,ρa u (P eq )) P eq u 1 γ 17
0.1 0.08 monitoring costhcl 0.06 both A&B Structure Aonly 0.04 0.0 Structure Bonly none 0 0. 0.4 0.6 0.8 dierentiation parameter HgL Figure 10: Equilibria as a unction o monitoring cost and product dierentiation All equilibrium prices are o the orm g u or h where g and h are unctions o c and γ but not o or u. 6 It then ollows, because the model can be solved in terms o normalized prices p, P and, Pu u, that equilibrium existence with a particular structure depends on c and γ but is not aected by the and u values. Figure 10 illustrates. Figure 10 displays the equilibria or all values o monitoring cost c and product dierentiation γ. As observed, there is a threshold or structure A above which there exists an equilibrium with properties given by proposition 4. Similarly, there exists a threshold or structure B below which there exists an equilibrium with the structure given in proposition 4. The regions determined by the intersection o these thresholds correspond to the dierent results on the existence o equilibria: there exist two large regions where there is only one equilibrium, either A or B. There are also two thin regions where there are either two equilibria or no equilibria. 18
1 0.8 sage requency HbL 0.6 0.4 no purchase select lower-value service select higher-value service 0. a= Ph ÅÅÅÅÅÅÅÅÅ l Ph -Pl a= ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ h - l 0. 0.4 0.6 0.8 1 tility per usage HaL Figure 11: Market segmentation when both irms oer a metered price 4.3 sage-ee Duopoly, dierentiated products We now model a scenario in which two irms oer products o dierent quality and both oer usage-ee pricing. sing subscripted h toindicatetheirm with the higher quality product and l to indicate the irm with lower quality, the ollowing apply: The irms prices are denoted P h and P l and proits are π h and π l. The products quality levels are h and l with h > l. The cost o metering/monitoring are c h and c l, and we additionally assume that c h h this assumption is primarily or ease o exposition, and it is easily relaxed. = c l l Figure 11 illustrates the corresponding market segmentation it consists o two nested rectangles with the higher quality irm acquiring the higher-α customers. As always under usage-ee pricing, 6 For P u, this can be seen by inspection o proposition 4 and it can be shown analytically or P. 19
proits are determined by the number o transactions; the proit unctions are: 7 1 higher quality irm : π h = (P h c h ) (1 P h P l h l ) i P l l and lower quality irm : π l = 1 (P h c h ) (1 P h h ) < P h h otherwise 1 (P l c l ) ( P h P l h l P l ) i P l l 0 otherwise < P h h (9) (10) At equilibrium, both irms set prices in order to maximize their individual proits (max Ph {π h } and max Pl {π l }) as given in the next proposition. Proposition 5 There exists a unique equilibrium in pure strategies. The equilibrium prices are: P h = (1 γ)+(+γ)c 4 γ h and P l = (1 γ)+3c 4 γ l The proposition that ollows provides insight on the comparison between this duopoly situation and the previous one involving dierent pricing schemes, particularly at lower monitoring costs. Proposition 6 For the dierentiated duopoly in which both irms oer usage-ees: the total duopoly proit is a decreasing unction o the monitoring cost This proposition is in direct contrast to the dual-ee duopoly there, the aggregate duopoly proit increases with monitoring cost despite the act that both irms incur the cost o monitoring usage by doing so. 5 Discussion The rise o the sotware-as-a-service business model has led to the rebirth o usage-metering as a pricing mechanism. However, usage alone is not an adequate measure o the willingness-to-pay or consumers o sotware products and services. Consumers vary in the value they derive with 7 The constraints in (9) and (10) ensure non-negativity o prices and market segment areas. 0
the same amount o use. The eect o this orm o consumer heterogeneity on the optimal pricing structure has not been discussed in previous literature. This paper analyzes this issue in both monopolistic and competitive settings. The basic parameters used to classiy the results are the transaction costs o monitoring usage and the level o dierentiation between irms. Given a choice between ixed-ee and usage-ee schemes, a monopolist would ind usage-ees to be optimal at low monitoring costs and a ixed-ee at higher monitoring costs. However, the monopolist would always ind it beneicial to oer both pricing schemes. This is because the nature o market segmentation enlarges the size o the market when both pricing schemes are oered. This act also enables irms to dierentiate themselves purely on the basis o pricing mechanisms even when their service quality values are not dierent. However, when irms compete with dierent pricing schemes, lower monitoring costs lead to intense price competition. In particular, the usageee scheme is highly sensitive to competition, speciically when the competitor oers a ixed-ee. This eect can be relieved i both irms oer a usage-ee even though this involves incurring a corresponding monitoring cost. Relating the results o this model to real world settings: a service provider should be cautious in oering usage-ee pricing i the competitor is a vendor oering ixed pricing. Such a strategy adversely aects both irms, particularly at low transaction costs o monitoring usage. In such cases, i the vendor chooses to oertheproductintheormoaservice,itmightwanttoconsider oering usage-ee pricing even though this means incurring a usage monitoring cost. 1
6 Appendix Proo o lemma 1 First, it must be that g is concave-convex on [a, b]. 8 That is, there is a value c such that g is strictly concave over subinterval [a, c) andstrictlyconvexover(c, b]. Tosee this: (1) g must be strictly concave at a otherwise,g 000 > 0 would imply that g is strictly convex everywhere, and this would make g (a) =g (b) =0impossible. () g 000 > 0 implies that i g is covex at c then g is strictly convex over all (c, b]. Next, g must have exactly one point in (a, b) at which g 0 =0. This is seen by contradiction: assume g 0 (x) =g 0 (y) =0or some x<yin (a, b). Then, (because g is concave-convex) g must be concave at x, convex at y, and increasing between y and b. This then implies that g (y) <g(b) which together with g (b) =0contradicts the premise o nonnegative g. Finally, the strict concavity o g over (a, c) together with g =0at endpoints a and b implies that the unique inlection point must be a maximum. Proooproposition1: The monopolist irm preers usage-ee pricing to ixed pricing when πu π. Substituting or these expressions rom equations () and (3), we have 1 8 (1 cu ) 0.1 which can be rewritten as c 0.097 QED Lemma 3 I the price pair (P,P u ) is optimal and P P u,thenp u = (1 + c). Proo: Since P P u, the dual-ee monopolist s proits are 9 π d = 1 µ (P u c u ) 1 P u and (11) π d = 1 P u (c u P u + ), (1) the optimality condition implies equation (1) equals 0, and this solves to P u = 1 (c u + ) = (1 + c). The lemma then ollows by contradiction i P u 6= (1 + c) then either: 8 Although the convex portion may be empty. 9 This is equation (4) given that the ixed-ee exceeds the usage-ee
π d P u 6=0 implying the prices are not actually optimal or π d does not take the orm given in (11) implyingp <P u. Either case contradicts the lemma s premise. QED Proooproposition: Please note: (1) the three parts o this proposition are proven in reverse order (iii) is irst and (i) is last, and () the proo relies on lemma 3 that is stated with proo just above. Recall that the irm s proit-maximization objective is max P,P u (π d ) with π d given by (4). Because π d is continuous in prices and is independent o P or P >P u,theirm s proit maximization problem can be rewritten as the constrained optimization program à max π = P (1 P )+ P µ P,P u P u ln P + 1 P u (P u c u ) (1 P µ! u ) P (13a) P u At optimality we must have 10 subject to P P u. (13b) 0= P u [π λ (P P u )] = π P u + λ. (14) with λ denoting the LaGrange multiplier or constraint (13b). Next is to show by contradiction that λ =0at optimality. We thus assume that λ is strictly positive at optimal prices. λ > 0 implies that constraint (13b) is binding, so P = P u and: (1) π simpliies to 1 (P u c u ) (1 P u ), () P u = (1 + c) (by lemma 3), and (3) P π u =0(ater dierentiating π and substituting P u as just given). This, together with λ > 0 shows that equation (14) is a contradiction implying that λ = 0at optimality. To complete veriication o (iii), λ>0 implies P <P u at optimality by complementary slackness. (ii): Solving P π u =0(with π given by (13a)) givesp u = (1 c)+ (1 c) +16c 4 as the only (positive) critical point or π. The act that this is independent o P together with π < 0 at this value Pu o P (easily veriied by inspection o this partial), is suicient to imply that this critical point identiies a maximum. 10 The irstequalityisbylagrange smethod;thesecondissimpliying. 3
(i): Part(ii) tell us that there is a unique optimal P u that is independent o P. Thus, it is only let to show that when P u takes this value a unique P maximizes the irm s proits. To simpliy the expressions that ollow, we use the ollowing deinitions: p, P,, P u, and c, c u. Substituting these into the proit unction π and dierentiating µ π =1 + (p u c) (1 ) p p 1 p +p ln u and π p = + (p µ u c) (1 ) p p 3+ln u µ p The [ ]-bracked term is positive (ater noting that P u >c u implies >c) and is constant in p, and the last term is negative (because 0 < p < by part (iii) o this proposition), so π p is unambiguously negative. π is thus concave in p and also in P. There is thus a unique proit-maximizing P or the given P u. QED Proo o lemma : (i) (by contradiction): I P P u then: (1) For every (α, β)-customer, the utility αβ P derived rom the ixed-ee option is β (α P u ), the utility derived rom the usage-ee option (because β 1). () Thus, q,salesbytheixed-ee irm are 0. (3) But, the continuity o π guarantees that the ixed-ee irm can always ind a price that will supply strictly positive proits. (4) Thus, a (P u,p ) pair with P P u violates this ixed irm s optimality criterion. Hence at optimal pricing or the ixed price irm, we have P <P u (ii) To simpliy the analysis, let: π 0 u = πu, p = P, = Pu and c = cu : The proit unction o the usage-ee irm in equation (6) becomes: ( c) (1 ) p πu 0 i p p = u >p u ( c) (1 ) otherwise Dierentiating the irst line o the proit unction in equation (??) with respect to : πu 0 µ = 1 p + c u p 3 c p u p u 4
π 0 u p u = µ 6c p +c p 3 u Setting the irst derivative to zero, we get p u(p )= 1+c c. Now πu 0 3c 0 or p p u [0, u 1+c ] and > 0 or ( 1+c 3c, 1] implying that π0 u is concave-convex in. Also, at = c, πu 0 =0and π 0 u = (1 c) p 0 at =1implying that the unction is concave with zero value at the lower limit o the domain and convex decreasing at the upper limit o the domain. Combining the above acts implies quasi-concavity o the objective unction over the given domain. Setting the irst derivative equal to zero provides a unique maximum. I p < c 1+c, the best response p u(p )= 1+c c >p. I p 1+c c, then the quasi-concavity o the proit unction dictates that p u(p )=p Dierentiating the second line o the proit unction in equation 6, we ind the optimal usage-ee to be p u(p )= 1+c.Ip 1+c,thenp u(p )= 1+c else p u(p )=p Combining the results above, the best response usage-ee or the entire range o ixed prices can be constructed as stated in the lemma. Proooproposition3: (i) sing lemma, we can restrict analysis to the case where p <. Similar to previous part o the proo, we set π 0 = π, p = P,and = P u. The irst line o the proit unction o the ixed price irm rom equation (5) is simpliied to give: π 0 = p (1 p µ )+p p ln p (15) u with: π 0 p =1+p p µ p +p ln and: π 0 p =3 µ p +ln 3 π 0 p 3 = > 0 p sing this act about the third derivative and that π =0at the endpoints p =0and p =, lemma 1 provides the result that there exists a unique ixed price response to any usage-ee. From 5
the earlier part o this proo, we know that when p <, the usage-ee irm has a unique response = c 1+c. Since there is a unique ixed price response to this price, the resulting unique set o prices maximizes the proit obothirms given the strategy o the competitor and hence constitutes an equilibrium in pure strategies. Let this price pair be denoted by (p e,pe u). We show uniqueness o this equilibrium by contradiction: suppose that (p 0,p0 u) represents an equilibrium price pair in addition to the price pair (p e,pe u). By lemma 3, p 0 <p0 u resulting in p 0 u = 1+c c (again rom proposition ) which equals pe u. From the earlier part o this proposition, there is a unique ixed price response to this price. Hence p 0 = pe. Hence (p0,p0 u)=(p 0,p0 u) and the equilibrium is unique. (ii) Follows rom i) (iii) The ixed-price irm s modiied proit unction is given by equation (15). This proit isan increasing unction o the usage-ee as shown by direct dierentiation: π 0 µ = p 1 1 > 0 This is true or every,soitisalsotrueortheequilibrium. Also, c does not appear in the ixed-price irm s proit unction. This implies that the ixed-price irm s equilibrium proit increases i and only i the equilibrium increases with c. By inspection o lemma, we ind that it does. (iv) &(v) Let πu(c) represent the equilibrium proit ortheirm oering the usage-ee at optimal prices as a unction o the monitoring cost. p u < 1 implies that customers will purchase a strictly positive number o usage-ee transactions, and p u >cimplies that these transactions are proitable. Thus, πu(c) > 0 whenever 0 <c<1. The stated results are then implied by the continuity o πu in c (which can be veriied by the implicit unction theorem). (vi) At equilibrium prices, the irst derivative o the ixed price irm s proit (c.. equation 15) as a unction o price must be zero. That is: π 0 p =1+p p +p ln( p )=0 where p and are the equilibrium prices. Taking the limit o the above equation as c 0 gives: 1+lim c 0 p L + lim c 0 p ln(l) =0 6
where L =lim c 0 ³ p We know that at equilibrium, 0 < p <. However, lim c 0 = 0 (using the closed orm expression o ). Taking limits on both sides o the inequality, we have lim c 0 p =0. sing this in the above equation: 1 L =0 µ p = 1 lim c 0 which is the same as: µ P lim = 1 c 0 P u At c 0, we have both prices: 0 and p 0. At zero prices, the market is ully covered. p is the line o indierence between the two segments and p 1 as c 0. Hence each irm gets exactly hal o the potential market. Derivation o proit unctions (??) and (??): For structure A ( P < Pu u ): we derive the number o customers who use the ixed-ee service, q (given by the corresponding area in igure 9) to be: q = (1 P Z 1 P Z ) dα α 1 Z P 1 P dα dα α( u )+P u α P =(1 P )+ P ln Pu u µ u P P u P u ln µµ u P u Pu u µ u + P u and the proit unction or the ixed pricing irm when P < P u u ollows. The usage-ee irm s objective again depends on the number o transactions, now Z 1 Pu u P α( u)+pu Z 0 βdβdα which equals P ( u P u ) P u (P u + u ), and the irm s proit unction ollows. 7
For structure B ( P P u u ): we derive the number o customers who use the ixed-ee service, q (given by the corresponding area in igure 9) to be: q =1 P P u u =1 P P u u Z 1 P Pu u P u ln P dα α( u )+P u µ u + P u P and the irm s proit unction when ( P P u u )ollows. The usage-ee irm s objective again depends on the number o transactions, now µ 1 P P u P u u u + µ 1 P P u P u + u u Z 1 P Pu u µ P α( u)+pu Z 0 P ( u ) βdβdα which equals µ u + P u P u + P u and the proit unction ollows. Proooproposition4: Again, we use the ollowing notation to simpliy the analysis: π 0 = π, πu 0 = π u u, p = P,and = P u u (i) Case A: ( >p ) For case 1, the proit unction o the irm oering usage-ee pricing given by equation (8) can be simpliied to: πu 0 = ( c) (1 ) p (1 γ + γ ) To ind a maximum, we dierentiate with respect to π u : πu 0 µ pu (1 γ + γ )(1 + c) ( c)(1 )(1 γ +γ ) p = p u(1 γ + γ ) Setting π0 u =0,wegetp u(p )= cγ± c(1 γ+cγ) 1+cγ. For γ<1, cγ c(1 γ+cγ) 1+cγ < 0 by inspection leaving us with cγ+ c(1 γ+cγ) 1+cγ as the only root in the domain [c, 1]. Dierentiating again with respect to, we can show that πu 0 p < 0 or c p u < cγ u 1+cγ + c 3 (γ(1 γ+cγ)) 1 3 1+cγ + c 1 3 (1 γ+cγ) 3 and γ 1 3 (1+cγ) 8
π 0 u p u cγ 0 or 1+cγ + c 3 (γ(1 γ+cγ)) 1 3 1+cγ + c 1 3 (1 γ+cγ) 3 p γ 1 u 1. Thisimpliesthatπu 0 is concave-convex 3 (1+cγ) ³ µ µ in. Also, at = c, π0 u 1 c p = c(1 γ+γc) > 0 and at =1, π0 p u = (1 c) < 0. So the proit unction is concave increasing at the lower limit o the domain and convex decreasing at the upper limit. Hence the root cγ+ c(1 γ+cγ) 1+cγ is the usage-ee irm s best response to p. The proit unction o the ixed price irm or case A in (c.. equation (7)) ater normalizing becomes: µ µ π 0 = p (1 p )+p ln p p 1 γ + 1 γ ln γpu Dierentiating with respect to p : By inspection, π 0 p =1 p 1 γ ln π 0 p =1 1 γ ln µ 1 γ + γpu µ 1 γ + γpu µ p p +p ln µ p +ln h ³ i ³ 1 1 γ ln 1 γ+γpu p is constant in p and ln is monotone increasing (note that ln(0) = and ln(1) = 0), so π 0 p is monotone increasing. Also at p =, π 0 p = ³ 1 1 γ ln 1 γ+γpu < 0 implying that the unction is decreasing at the upper limit o the domain. Hence the unction is strictly quasi-concave over the domain with its maximum in the interior. Thus, there is a unique proit maximizing ixed price response or any usage-ee in the region p [c, ]. The usage-ee irm has a unique best response price independent o the ixed price. The ixed price irm has a unique response to this price. I an equilibrium in pure strategies exists or case A, this unique price pair characterizes the equilibrium and there is no other equilibrium. (ii) Case B (p ): The proit unction o the ixed price irm or structure B given by equation (7) can be rewritten as: µ µ π 0 = p p γ p p 1 γ + 1 γ 1 γ ln γpu p To ind a maximum, compute the irst and second order derivatives: π 0 =1+ γ p 1 γ p 1 γ p µ 1 γ + 1 γ ln γpu p (16) 9
π 0 p = 1 1 γ µ 1 γ + 1 γ ln γpu p Setting the irst derivative to zero to get the best response or the ixed price irm: p ()= 1 γ + γ 3.5 This price gives a local maximum or the proit unction over the domain [, 1] or any usage-ee since π 0 p evaluated at p = 1 γ+γpu 3.5 is 1 ln(3.5) 1 γ < 0 implying concavity. The proit unction o the usage-ee irm or structure B given by equation (8) can be rewritten as: πu 0 = 1 µ µ µ (p p γ p 1 γ + γpu p u c) + 1 γ (1 γ) 1 γ + γ To ind a maximum, compute the irst and second derivatives: Ã πu 0 1 = c +3p (1 γ +! γc)p 4(1 γ) (1 γ + γ ) 4 π 0 u p u = Ã! 1 γ(1 γ + γc)p (1 γ) (1 γ + γ ) 3 < 0 by inspection, implying concavity It is tedious but straightorward to show that the irst derivative always has a unique real root in. Concavity o the unction implies that this price provides a local maximum over the domain [0,p ] or any given p. Thus, it only remains to simultaneously solve or the roots o equations (16) and 17. Doing so provides the values given in the proposition. Only one such solution exists implying the uniqueness o the equilibrium. QED (17) Proooproposition5: Again, we use the ollowing notation to simpliy the analysis: πh 0 = π h h, πu 0 = π u u, p = P,and = P u u The proit unction or the irm oering the higher value service given by equation (9) can be rewritten as: πh 0 = 1 (p h c) (1 p h γp l 1 γ ) Computing the irst and second derivatives with respect to p : πh 0 = 1 µ 1 p h p h 1 γ + γp l 1 γ + c 1 γ 30
πh 0 p = 1 h 1 γ < 0 The sign o the second derivative implies concavity o the proit unction and hence the existence o a unique maximum. Setting the irst derivative to zero, we get the best response price: p h (p l)= γp l +(1 γ)+c The proit unction or the irm oering the lower value service given by equation (10) can be rewritten as: πl 0 = 1 (p l c) ( p h γp l 1 γ p l) Computing the irst and second derivatives with respect to p l : π 0 l p l = 1 µ ph 1 γ γp l 1 γ p l + cγ 1 γ + c πl 0 p = 1 l 1 γ < 0 The sign o the second derivative implies concavity o the proit unction and hence the existence o a unique maximum. Setting the irst derivative to zero, we get the best response price: p l (p h)= p h + c Both response unctions are linear giving rise to a unique equilibrium with the stated equilibrium prices. Proooproposition6: proit atoptimalprices: sing the values derived in proposition 5, we write the total duopoly π 0 = π 0 h + π0 l = 1 (p h c) (1 p h γp l 1 γ )+1 (p l c) (p h γp l 1 γ p l ) Substituting or the optimal prices rom proposition 5, we dierentiate the proit unction with respect to the monitoring cost and make use o the envelope theorem to analyze the equilibrium proit as a unction o the monitoring cost parameter: π 0 c = π(p h,p l ) 5(1 γ)(1 c) = c (4 γ) < 0 implying that the total duopoly proit is decreasing with monitoring cost. QED 31
Reerences [1] Bashyam, T. C. A., 000. Service design and price competition in business inormation services. Operations Research 48(3) 36-375. C [] Bashyam, T. C. A., 1996. Service design and price competition in business inormation services. npublished Ph.D. dissertation, CLA [3] Boehm, B. H. 1981. Sotware engineering economics. Prentice Hall Inc., Englewood clis, N. J. [4] Carr, S. M. 003. Note on online auctions with costly bid evaluations. Management Science 49(11) 151-158 [5] Conner, Kathleen, and Rumelt, Richard. 1991. Sotware piracy: an analysis o protection strategies. Management Science 37() 15-139 [6] Danaher, Peter. 00. Optimal pricing o new subscription services: analysis o a market experiment. Marketing Science 1() 119-138 [7] Dolan, Robert. 1987. Quantity discounts: managerial issues and research opportunities. Marketing Science 6(1) 1- [8] Essegaier, S., Gupta, S. and Zhang, Z. J. 00. Pricing access services. Marketing Science 1() 139-159 [9] Oi, Walter. 1971. A disneyland dilemma: two-part taris or a mickey mouse monopoly. Quarterly Journal o Economics 85(1) 77-96 [10] Hayes, Beth. 1987. Competition and two-part taris. Journal o Business 60(1) 41-54 [11] Jain, D., Muller, E., and Vilcassim, N. 1999. Pricing patterns o cellular phones and phonecalls: a segment level analysis. Management Science 45() 131-141 [1] Kemerer, C. F. 1987. An empirical validation o sotware cost estimation models. Communications o the ACM 30(5) 416-49 [13] Nahata, B., Ostaszewski, K., and Sahoo, P. 1999. Buet pricing. Journal o Business 7() 15-8 3