Electronic Commerce, Price Discrimination and Mass Customisation David Ulph* and Nir Vulkan** November 2007 Abstract As consumer e-commerce matures, on-line retailers are adopting personalisation technologies, enabling them to exploit information about individual consumers in order to tailor both products and prices to individual requirements. Thus firms can engage in two new competitive strategies – first degree price discrimination and mass customisation. In this first part of this paper we present a model of duopolistic competition between firms capable of using first-degree price discrimination. These firms are able to extract more surplus from their customers, but also face more intense competition, because they compete, in effect, for each and every consumer in the market. Whether or not firms will choose to use this new technology will depend on whether or not the enhanced surplus extraction effect dominates the intensified competition effect. We present a model that makes these ideas precise, and characterise the conditions under which firms will choose to employ first-degree price discrimination technologies. In the second part of the paper, we extend the analysis to include mass customisation and develop a simple framework within which to understand firms’ incentives to adopt the strategies of first-degree price discrimination and mass customisation in a competitive environment. In particular we examine the interaction between the two decisions. We show that the incentives to use one technology increase with the use of the other, but also that firms are in a prisoners dilemma since they would be better off if they neither customised nor engaged in first-degree price discrimination. Key Words:
e-commerce, first-degree price discrimination; competition
JEL Classification: D43, L13, O30 * University of St Andrews, School of Economics and Finance, St. Andrews, Fife, KY16 9AL, Scotland, UK ** Saїd Business School, University of Oxford, Park End Street, Oxford OX1 1HP. Tel: +44-1865-288529; Fax: +44-(0)1865-288805; E-Mail:
[email protected]
Acknowledgements We are grateful to our RAs, Karen Croxson and Ingrid Boxall and to John Beath and Avner Shaked for useful discussions, and to Hal Varian for helpful comments. Both authors thank the ESRC for financial support through grant no. R000222642.
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1.
Introduction
In recent years an enormous amount of consumer-specific data has been collected by retailers and marketing companies – loyalty cards and air mileage programmes are used to collect data on the shopping patterns of individual consumers, Internet Service Providers (ISPs) like British Telecom (BT), as well as companies like the Post Office, promise commercial users the opportunity to “treat all your customers as individuals”, and so on. With the growth of e-commerce, there are increasing opportunities for firms to employ personalisation technologies that allow them to use information collected about the customer to offer individually tailored goods and services at individually tailored prices. This information can be collected by the firm itself or, more likely, be provided by independent data companies. One example of such a data agency is Experian, which provides a software package that enables websites to recognise customers instantly and sends the customer’s profile to the retailer. The same technology can then be used to develop customer profiles and relationships that are appropriate to each client individually. There are two separate and independent phenomena at work here: (i) The first is first degree price discrimination - offering the same good at different prices to different customers. (ii) The second is mass customisation1. We define this as a situation in which firms can offer a whole range of finely differentiated products at the same constant marginal costs without having to incur additional fixed costs on every differentiated brand they offer. Thus firms can reap the benefits of customisation without foregoing the benefits of scale economies. These two technologies can, in principle, be used by sellers to undertake first-degree price discrimination by offering consumers the product they want at a price which they are likely to be willing to pay. However, until recently, such discrimination has been too costly. For example, a physical catalogue that is individually tailored is hardly likely to be cost effective. With the advent of consumer e-commerce, this needs no longer be the case. In particular, personalisation technologies, such as agents2, significantly increase the ability of firms to undertake first-degree price discrimination. Using agents, an online catalogue can be individually customised, for the agent can identify the shopper and automatically redesign the company’s Web site to match the user’s likely requirements. 1
The terms mass customisation and product customisation are used interchangeably in this paper. An agent is a program that is authorised to act independently on behalf of its user. This can range from automatic search (for example, an agent searching for airline tickets on the Web can match preferred dates, price-range, class of travel, etceteras, without consulting its user) to automatic negotiations (for example, agents can place binding bids in a Web auction). Personalisation can be carried out according to Web site demographics (the user profile for a given site); individual domains (attributes inferred from the user’s browser, for example by using cookies or agents); learning (the agent monitors and learns about the preferences of the user), and correlation (where the agent compares the user’s preferences with other users with similar interests and suggests possible contents based on this correlation. For example, Barnes & Noble and Firefly created a system that recommends books based on this method). An example of personalisation technology in use is MyYahoo! (my.yahoo.com), an agent that allows the construction a individually tailored news web page, while at the same time customising the banner advertising on this page according to the user’s profile. 2
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In particular the technology can be used to offer different prices to different consumers. Since on-line menu costs are practically zero, on-line retailers can change their prices to match what they expect the individual to be willing to pay for whatever they are selling. And there is some data to suggest that they indeed do so. As Shapiro and Varian (1999) note, the on-line data provider Lexis-Nexis “sells to virtually every user at a different price”. Furthermore, in Brynjolfsson and Smith (1999), Internet retailers’ prices adjustments over time were found to be up to 100 times smaller than conventional retailers’ price adjustments. Similarly, in a recent article in the Washington Post3, it was reported that a regular customer of Amazon.com was quoted a price of $26.24 for a certain DVD, while that same consumer, when contacting Amazon as a new user, was quoted a price of $22.74 for the exactly the same DVD. As suggested, the technical background needed to support the strategies of price discrimination and customisation is vital. In this regard, highly targeted marketing initiatives are made possible by so-called Customer Relationship Management (CRM) technology. Typically, CRM systems collect data from any point where a customer "touches" a company (such as a store, a call centre or a website), and perhaps combine these with information from other sources (e.g. credit ratings). A host of firms serve as providers of CRM technology, related services and expertise. Siebel Systems leads the CRM industry, but significant others include Oracle, PeopleSoft and SAP. Established niche technology providers include Clarify (now owned by Amdocs), E.piphany (now owned by SSA Global) and Kana. Meanwhile others such as Dunn Humby, specialize in provision of CRM-related expertise.4 The technology these firms and others provide supports client firms in determining how profitable individual customers are, and how recently and frequently they've made purchases, so that the best customers can be identified and retained, the firm can “up-sell” to some of the others, and the less profitable clients can be either shifted over to less expensive service channels, making them more profitable, or effectively “fired.” A manifestation of this approach might well be that customers are offered a different quality of service depending on their perceived value to the business. For instance, a highly profitable client might have their call to a customer services desk answered straightaway whilst their low value counterpart is left to wait some time. In its most advanced implementation, CRM is used to generate a highly personalized offer involving a product or service recommendation so well targeted that the individual is unlikely not to be tempted. A concrete example of CRM technology is XACCT’s mobile software solution suite, designed for telecommunications service providers. The package constitutes a comprehensive, integrated data management platform. Included in the suite since 2002 is a new capability designed to enable operators to “launch highly personalized advanced mobile data services that exploit the full capabilities of 2.5G and 3G mobile devices.”5 3
“On the Web, Price Tags Blue”, The Washington Post, 27 September, 2000. Top 10 CRM Predictions for 2002 by Bob Thompson, Founder, CRMGuru.com. Full text available at: http://www.crmguru.com/features/2002a/0110bt.html 4
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Information about XACCT’s mobile CRM solution is available at: the provider’s website: www.amdocs.com
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The question we wish to address in the first part of this paper is whether, if a technology is available that allows firms to undertake perfect first-degree price discrimination6, firms will necessarily choose to use it. At first sight this may seem an odd question, since conventional theory tells us that the ability of a firm to employ first-degree price discrimination always raises its profits, since it can extract greater surplus from consumers. Call this the enhanced surplus extraction effect. However, like virtually all the analysis of price discrimination, this conclusion is drawn in the context of price discrimination by a monopolist. A key feature of the environment in which firms are operating using e-commerce is that it is highly competitive. Intuitively one suspects that this will introduce a second important consequence of the decision by firms to use firstdegree price discrimination – namely that it will intensify competition between firms, since they will now be competing consumer-by-consumer. Call this the intensified competition effect. This will naturally lower firms’ profits. Thus whether or not firms will choose to use this new technology will depend on whether or not the enhanced surplus extraction effect dominates the intensified competition effect The aim of the first part of this paper is to make these ideas precise, and to characterise conditions under which firms might choose to employ first-degree price discrimination in a competitive environment. We use the Hotelling framework to model price competition in differentiated goods between two firms. Consumers are located along a line representing the degree of product differentiation. The two firms are located at either end of the line. The firms have the same technology with marginal costs independent of output. Firms can either set mill prices, or offer each consumer an individually tailored price. We obtain the following major results. 1. 2.
3.
4.
If neither firm price discriminates, both firms will set price above marginal cost. If a firm chooses to use first-degree price discrimination, then the prices it sets and the profits it obtains are independent of whether or not the other firm chooses to use first-degree price discrimination. If at least one firm operates first-degree price discrimination, the consumer located half-way between the two firms will be offered a price equal to marginal cost by both firms – the conventional Bertrand conclusion. Whether or not profits are higher under first-degree price discrimination depends crucially on the nature of the transport cost function.
To elaborate on this latter point, consider the class of functions t ( z ) = t.z β , β ≥ 1 , where z is distance travelled and β takes integer values. If this function is linear or quadratic (i.e. β = 1, 2 ) then, the prices that a firm offers consumers will be almost everywhere7 lower than if neither firm discriminates. When β ≥ 3 then a firm that price discriminates will charge higher prices to those consumers located close to it, than it would if neither 6 7
That is, there is no scope for consumers to disguise their identity. The exception is the consumer located at the same end of the line as the firm.
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firm discriminates. However, since prices are lower for consumers in the middle (result 3), profits are higher if neither firm discriminates. Only when β ≥ 5 do firms make more profits if they can price discriminate than if neither discriminates. The intuition is straightforward. When transport costs do not increase very fast with distance, then competition between products is intense, and so the intensified competition effect is the dominant consideration. However, when transport costs rise very fast with distance then consumers located close to firms are effectively locked in, giving firms considerable powers to extract surplus from them, just as in the traditional monopoly analysis. The extra profits extracted from these consumers can offset the reduced profits made on the consumers located in the middle where competition is intense. Our model abstracts from two features of current e-commerce markets. First, in most industries firms are still establishing their identity. Companies such as Amazon, for example, spend millions on building consumer loyalty. This effectively implies that the location of consumers is still endogenous Second, at least for now, consumers can still hide their identity by logging in as new consumers (possibly from a different IP location), hence reducing the degree of price discrimination that firms can undertake8. Taking these assumptions on board will clearly complicate our analysis considerably, but it is easy to see that considering either of these features will in fact strengthen our results. If firms cannot gain from perfect first-degree price discrimination technologies when locations are fixed, they have even less incentives to adopt these technologies when locations are endogenous or when they can only partially discriminate. In addition to the actual investigation into first degree price discrimination, the first part of this paper facilitates a smooth progression into an analysis of mass customisation, the second main phenomena of interest. Mass customisation, as it is defined, offers firms the possibility of what we will call second degree price discrimination where different brands of the same product are sold at different prices. In a similar way to the opportunities for first degree price discrimination, the possibility of mass customisation is closely related to developments in computer technology; for example, information goods can be customised and sold over the Internet at virtually zero marginal cost. On-line news providers, like Reuters, can offer any bundle of news and can sell different bundles at different prices. For other goods, such as cars, it is other technological developments that have brought the possibility of mass customisation. Ecommerce, however, still enhances mass customisation as it provides a useful means by which consumers can view the range of alternatives and prices on offer and communicate their choice. It is important to note that the availability of these different technologies does not necessarily mean that firms will choose to use them. As is shown in the first part of this paper, price discrimination has two separate effects: it allows firms to extract more 8
However, if this is the only way in which consumers can hide their identity, then one can show that in equilibrium firms will just charge anonymous consumers a price equal to the price of their most loyal consumer, thus inducing truthful revelation of consumer location, and all our conclusions will go through.
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surplus, but it also intensifies price competition. In a wide class of cases the second effect dominates and firms would choose not to first-degree price discriminate. In this second part of the paper we extend the analysis to the case where firms also have the possibility of introducing mass customisation. Because there are two independent processes at work here, firms have a considerable degree of choice over their combined differentiation/customisation and pricing strategies. Thus a firm could choose not to customise and so produce just a single product, but could still choose whether or not to employ first-degree price discrimination. On the other hand if it chooses to customise, it has three pricing strategies9. (i) It can engage in no price discrimination and so sell all products at the same price irrespective of brand or consumer. (ii) It could second degree price discriminate and so sell different brands at different prices – though different consumers buying the same brand will pay the same price. (iii) It could engage in first degree price discrimination and so sell at prices that are in principle differentiated both by consumer and by brand. It is easy to see, however, that in this case the firm will discriminate solely by consumer. This is because the firm wants to extract the maximum surplus from each consumer, and to do this it wants the consumer to choose the brand that comes closest to its taste. It does this by having prices independent of the brand. The goal of this part of the paper is to provide a framework in which we can understand what choices firms will make along each of these two dimensions, and, in particular, the interaction between these decisions. The emphasis will be on the case where firms are making these decisions in a competitive environment like the internet. There are three sets of issues we wish to explore. (a) How does the degree of mass customisation chosen by firms affect their choice of price discrimination strategy? (b) How does their price discrimination strategy affect their choice of customisation? (c) What combination of the two strategies will firms choose? To address these issues we use an extension of the Hotelling (1929) framework. As before, consumers are located along a line of length 1 representing the degree of product differentiation, with the two firms located at either end of the line. The firms have the same technology with marginal costs independent of output. Firms can choose either: (i) not to mass customise – in which case they just produce the single product at their end of the line; or (ii) to mass customise – in which case they can produce at the same constant marginal 1 costs all products over an interval of length m < from their end of the line. 2 9 It turns out, however, that if firms mass customise, they will always use second-degree price discrimination in preference to no discrimination. Also if firms do not mass customise, second-degree price discrimination is formally identical to no price discrimination. So in all circumstances the choice is effectively between first-degree and second –degree discrimination.
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Firms also choose what price discrimination technology (i.e. first or second) to use. We prove the following results. 1.
2.
3.
4.
5.
The profits of the firm that uses first degree price discrimination technology are independent of the pricing strategy used by its competitors (this generalises the result found in the first part of the paper). The greater the degree of mass customisation chosen by firms the more likely they are to choose to first-degree price discriminate. More precisely, profits under both first and second-degree price discrimination fall as the degree of customisation increases. However they fall faster under second-degree discrimination. A firm that uses first-degree price discrimination technology is strictly better off if it also mass customises (i.e. this is true whatever the behaviour of its competitor). By mass customising it is able to extract more surplus from its most loyal customers by producing for each of them exactly the good which it wants to buy. If firms do not use first-degree price discrimination technology then they are better off if they mass customise, ceteris paribus, because they are better able to segment the market and extract surplus from their most loyal customers. It follows from the above results that firms are locked in a prisoners’ dilemma situation, where they would be better off not adopting any of the two technologies. However, because mass customisation dominates a single price setting, firms will mass customise. And once mass customisation is adopted, the incentives to use first degree price discrimination increase and firms will end up adopting both technologies and making lower profits.
The rest of the paper is organised in the following way: Section 2 presents a discussion on the empirical evidence associated with price discrimination and customisation. Section 3 then provides a brief discussion on some of the relevant existing literature. Section 4 focuses on first degree price discrimination and discusses the model we use along with a number of interesting examples. Section 5 focuses on mass customisation, initially describing the competition between firms that are symmetric with respect to their ability to mass customise and then analysing the scenario when these firm abilities are asymmetric. Section 5 also considers the interactions between firms that endogenously and simultaneously choose whether to mass customise and what price discrimination technology to adopt. Finally, section 5 goes on to summarise the key results and section 6 concludes.
2.
Empirical Evidence and Motivation
The theory under investigation in this paper utilizes the assumption that firms focus on their most loyal customers when customizing offers. Evidence from several important industries lends support to this modelling approach. For instance, using sophisticated
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analysis of customer data US casino operator Harrah’s Entertainment Inc found that over a quarter of its revenues came from just 5% of gamers, and subsequently undertook initiatives to target marketing efforts at these “high value” individuals. Each such “VIP customer” was assigned a dedicated salesforce, or “personal host,” and these hosts were expected to know a great deal about their guests---their preferred room at a particular property, for instance---and to create a personal bond with them. The host would arrange the guest’s booking and ensure the room the person would be staying in was well stocked with items to suit personal tastes. A Player Contact System (PCS) was implemented to support this personalized approach, providing information about guest preferences, visit histories, worth, relationships and contact information. Similar attempts to focus customization efforts on high-worth customers can be seen in the mobile telecoms market. For instance, BT Mobile Netherlands, a subsidiary of BT Mobile International which uses AMDOCS CRM technology as the single interface to all customer information, says its adoption of the solution has allowed its customer service division to segment customers (based on Average Revenue per User, ARPU), personalize offers and grow sales: “Amdocs CRM tracks average revenue per customer so we can segment customers based on their ARPU. Our agents see ARPU information on their screens for each customer whose contract expires. We personalize our promotional offers for customers depending on ARPU and target high-value customers. It's not a matter of attracting new customers; it's keeping the existing customer base so we don't have to invest money in attracting new customers. To increase margins, the most important thing is to cross-sell to existing customers, with an eye to giving the customers what they want.”10 Examples of this sort of practice are far from uncommon elsewhere in mobile telecoms. Consider the approach taken by O2 Ireland, which has 40% of the Irish mobile communications market, and relies on Amdocs technology for customer care and billing:11 “With the Amdocs system, all customer data is in one place. Whether a customer contact comes through our dealers, self service through our portal, IVR or through the agent, it's not only in one database, it's in one system. Everybody is looking at one system - the same Amdocs e-care system. Whenever a customer calls O2 Ireland through the Amdocs environment, we get a full picture of how valuable the customer is to O2 Ireland and we deliver a graded level of service based on that value. We manage customer complaints better because of the follow-up functionality of the Amdocs customer care system. All those things have helped us improve our relationship management with our customers.”12
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Excerpt from Amdocs’ website. See: http://www.amdocs.com/Site/Success/T-Mobile.htm for full text. 11 “Meteor is Eircom's mobile target,” The Post I.E. Sunday, March 27, 2005 - By Eamon Quinn Full text available at http://archives.tcm.ie/businesspost/2005/03/27/story3423.asp. 12 Excerpt from Amdocs’ website. See: http://www.amdocs.com/Site/Success/02+Ireland.htm for full text.
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As will be discussed in section 5 of this paper, the theory predicts that when one firm uses customisation and/or price discrimination, the incentives for other firms to do both increase. The empirical evidence provides strong support for this theory. For example, following Harrah’s foray into personalised marketing, rivals have been quick to respond.13 Foxwoods Resort Casino moved quickly to establish a relationship with All Star Incentive Marketing, a specialist provider of personalized marketing services. Providing loyalty-based reward systems to the Casino industry has since become a major line of business for All Star, and Foxwood’s new capability has been held up elsewhere as an example of cutting-edge CRM implementation. As CNN reported in 2001, “Native American-owned Foxwoods Resort Casino can parse its 200GB customer database, match it against third-party demographic data and tell whether a patron has kids or how much he makes per year. If he spends $100 or more daily at Foxwoods, he gets the redcarpet treatment. […].“We know who these people are and cater to them. We make sure they have flowers in the room, a drink in the hand and reservations at the restaurant," said Brian Charette, director of gaming systems at the $1.2 billion Foxwoods casino complex in Mashantucket, Connecticut.” 14 Elsewhere in the same industry, another key rival to Harrah’s, MGM Mirage, which All Star now also lists as a client, moved to develop a similarly impressive CRM capability: “MGM Mirage can sort its 6TB of data on Microsoft Corp. SQL Server databases to tell you which of its 9 million customers are poker players who also like onions on their hamburgers. […]“Customers, meanwhile, collect card points as they gamble, eat, shop or see shows, which they redeem for prizes, such as free hotel rooms or tickets to hot shows. MGM Mirage last year gave out $286.3 million in such comps, or complimentary items”15 Evidence that when one firm deploys customization or price discrimination, or both, others wish to follow does not seem confined to the gaming industry. In 1995, following successful trials at selected stores, the UK’s largest supermarket chain, Tesco, launched “Tesco Clubcard,” a scheme designed to reward loyalty and enable the group to connect on a personal basis with its millions of customers. Shoppers nationwide were invited to register for a card, which would be presented to the cashier and swiped each time a purchase was made. In return for allowing Tesco to link data on category of product purchased with name and address records, customers would receive vouchers representing 1% of the amount recently spent, along with a variety of targeted promotions. At the time of Tesco’s Clubcard launch, its biggest rival Sainsbury’s had already begun trialling its own reward card (thought to have been a much less successful trial than the Clubcard), and others in the industry (such as Safeway, whose trial scheme was rumoured to have gone well) were believed to be about to take their own schemes 13
General evidence for this appeared in The Economist in 2004, which noted that: “Behind the scenes, in the past two years casino companies have been busily applying airline-style "yield management" techniques to extract the maximum dosh from each customer.” Excerpt from: “Wedding in Vegas?”, Economist, 6/12/2004, Vol. 371, Issue 8379. 14 “Casinos hit jackpot with customer data,” By Kim Nash, CNN. Full text available at: http://archives.cnn.com/2001/TECH/industry/07/03/casinos.crm.idg 15
“Casinos hit jackpot with customer data,” By Kim Nash, CNN. Full text available at:
http://archives.cnn.com/2001/TECH/industry/07/03/casinos.crm.idg
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nationwide.16 Sainsbury’s responded to Tesco’s initiative by pushing on with its own Reward scheme. In 2002, Sainsbury’s took the decision to join a new UK-based consumer loyalty alliance called Nectar, whose membership includes BP, Amex and Debenhams. Nectar was designed to operate independently of its members (though subject to tight agreements with each), with the objective of promoting cross-selling at the organizational level. Towards this end, for instance, a customer currently shopping with Debenhams and BP but not with Sainsbury’s would be offered promotions by Nectar to encourage this, and spending at any one of the member firms would give rise to loyalty points redeemable anywhere else within the Nectar community. Beyond this, individual member firms would be able to exploit Nectar’s centralised administration facility to have own promotions sent out to own customers under the Nectar banner. Although the Nectar project is regarded a reasonable success, the overall sophistication of Sainsbury’s customer data analysis remained far behind that of Tesco. Losing market share to Tesco and Asda, Sainsbury’s prioritised data analysis and relevant marketing in a new marketing initiative. This has helped Sainsbury’s design far more effective direct marketing campaigns based on customers' actual purchasing habits. As well as conventional campaigns, such as those designed to promote new in-store brands based on past purchases, marketeers have been able to experiment. For instance, in one campaign Nectar customers were sent birthday cards offering discounts on frequently-purchased items. In another campaign the firm identified the product category from which each customer purchased most frequently, then sent out a coupon for that category, along with five other coupons for other product areas. At 26% the response rate was highly impressive for retail. Sainsburys annual direct marketing (DM) revenues have risen from £35m to more than £400m since the initiative began.17 Further evidence of one firm responding in kind to another’s customization of products and services comes from the mobile phone market. Take the UK market, which is led by Vodafone (around 30% market share). This is a concentrated market; not far behind Vodafone are O2 (around 25%), and Orange (also nearly 25%). T-mobile captures some 15% of the market’s total quarterly revenues.18 In 2002 Vodafone announced the implementation of Amdocs ClarifyCRM solutions throughout all of its UK Call Centers. The Amdocs system allows Vodafone to generate a single view of the customer and tightens integration between front and back-end applications, improving customer loyalty and retention and enhancing customer service. Amdocs integrated CRM and billing solution fully supports all inbound customer service and a new lifecycle management call center for outbound campaigns to strategic 16
Humby, Clive and Terry Hunt (with Tim Phillips) (2003): “Scoring Points: How Tesco is Winning Customer Loyalty,” Kogan Page Limited. 17 “Sainsbury's scrutinises buying habits to woo customers,” David Braue, 24 June 2005 Silicon.com 18 Presentation by Tim Miles, Vodafone CEO, 19th September 2005, at the Vodafone Analyst and Investor Day 2005. Full presentation available online at: http://www.vodafone.com/assets/files/en/Vodafone_UK_Tim_Miles.pdf
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customers.19 In response, O2 is in the process of adopting CRM and aims to align itscustomer service, billing, etc with individual customer needs. Online bill viewing is now operational and other market initiatives are being explored in an attempt to increase customer loyalty, a particularly important consideration in a market with such high competition. Adoption of CRM by key rivals can be seen from analysis of other geographical mobile phone markets. Examples include the Irish, Dutch and Austrian markets, where in each case major players have launched CRM offensives, following which competitors appear to have responded with CRM initiatives of their own, often drawing on the same key CRM technology providers for support. For instance, Vodafone Netherlands has been using Amdocs’ end-to-end customer care and billing system since 1999.20 And in 2003 it was reported by the European Association of Database and Directory Publishers, that Vodafone’s Dutch rival T-Mobile Netherlands had deployed Amdocs CRM technology with a view to attain “…improved operational efficiencies and enhanced customer satisfaction.”21 Another key result discussed in section 5 of this paper is that discrimination and customisation are complements. The empirical support for this theoretical result is less clear-cut than the above-mentioned case studies, although a look across several industries does suggest the two are often used together. For instance, credit card company Capital One offers its customers credit products that are highly customized to their needs. Among the features tailored to the individual are: annual APR, card fee, credit line, introductory rate, co-branding, affinity partnerships etc. When someone calls to ask for an improved credit line, the firm instantaneously makes an actuarial calculation of the customer’s lifetime NPV and assesses her likely response. It does this using statistical models generated from field experiments previously carried out on samples of customers. The customer services representative sees an instant recommendation, such as to negotiate APR down to a specific rate. Here, price, i.e. APR, is an element of the customization the firm undertakes.22 Returning to the gaming industry, the following excerpt from SAS’s website might be taken as evidence that customization and discrimination are used together: “SAS also allows Harrah’s to customize its predictions for each regional market, which is important since frequent customers in Las Vegas can look quite different from frequent customers in St. Louis or New Orleans. Plus, customers who live within driving distance
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“Vodafone UK Implements Amdocs ClarifyCRM Solutions,” report by CRM2day, Thursday, 23rd May, 2002. Full article available at: http://www.crm2day.com/news/crm/EpkkEVpuFpbUfWYGOh.php 20
Reported at amdocs’ website: http://www.amdocs.com/Site/Success/Vodafone+Netherlands.htm “T-Mobile Netherlands uses the Amdocs CRM system, ” Tuesday April 1, 2pm. Extract taken from full article available at: 22 “Capital One Financial Corporation,” Harvard Business School Case Study prepared by Christopher H Paige, 1st May 2001. 21
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of a casino receive different marketing promotions than customers who live farther away and plan their casino trips as vacations.”23 Meanwhile, in the IT hardware industry, Dell appears to use customization and discrimination in tandem. The firm invites customers to customize their computer by adding-on accessories to suit personal needs and tastes: “They (…) offer a completely differentiated product because it is made-to-order”24 At the same time, the firm also utilizes segmented prices, selling the same product to different customer segments for different prices. As Gary McWilliams, a reporter from The Wall Street Journal, found out:25 “One day recently, the Dell Latitude L400 ultralight laptop was listed at $2,307 on the company's Web page catering to small businesses. On the Web page for sales to healthcare companies, the same machine was listed at $2,228, or 3% less. For state and local governments, it was priced at $2,072.04, or 10% less than the price for small businesses.” One final result for which a diversion into the empirical evidence is important is that concerning the prisoners’ dilemma, again discussed in section 5 of the paper. Specifically, the theory shows that prices are potentially lower in a discrimination/customisation regime. One example from the marketplace is found within the UK Grocery Industry. The market as a whole is characterized by some opacity in pricing (due to price flexing, local pricing, some below-cost selling on selected lines, etc.), so that pricing trends are not straightforward to discern, suggests that prices are falling, with some evidence of increasing choice and improving quality. Elsewhere, a recent report by Portio Research (www.portioresearch.com) entitled: “Understanding the Evolution of Pricing Trends in the Mobile Services” provides evidence that prices have been falling steeply in the mobile phone industry:26 “One well-documented and significant trend in the mobile industry is the ongoing decline in the prices of various mobile services. Voice prices have declined steeply over the years and a similar trend has been observed in the more popular data services, including SMS and other non-voice services, such as monotone or polyphonic ringtones, mobile games, etc.”
3.
Existing Literature
23
“Harrah’s Hits Customer Loyalty Jackpot” full text available at SAS’ website: http://www.sas.com/success/harrahs.html 24 “Dell: It’s Time to Diversify Dude,” published August 28, 2002 in Knowledge@Wharton. Full text available at: http://knowledge.wharton.upenn.edu/article.cfm?articleid=613 25 “Dell Fine-Tunes Its PC Pricing To Gain an Edge in Slow Market,” by Gary McWilliams, June 8, 2001, The Wall Street Journal. 26 A brochure outlining the report from which this quote is extracted is available online at http://www.portioresearch.com/Pricing_trends_brochure.pdf.
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While much of the literature on price discrimination has focused on the case of monopoly27, a number of papers have analysed how price competition operates in a competitive environment - Katz (1984), Borenstein (1985), Thisse and Vives (1988), Holmes (1989), Corts (1998) and Armstrong and Vickers (1999). The Armstrong and Vickers (1999) paper, in particular, provides an elegant framework that incorporates much of the earlier work. This analysis, however, concentrates on the case of third-degree price discrimination in a competitive environment. As the authors show, the analysis of this form of price discrimination can essentially be split into two parts. First, each firm determines how much profit it can make for any given level of gross utility that a typical consumer will obtain by buying from that firm. Devices such as price-discrimination and commodity bundling are just ways of extracting a larger surplus from consumers and so making higher profits for any given level of gross utility. Firms then compete in terms of the amount of gross utility they offer their customers. Consumer specific preference characteristics will determine the net utility they obtain from any gross utility provided by a particular firm. The distribution of these characteristics will determine the demand for each firm as a function of the gross utilities offered by both firms. Firms cannot observe these consumer specific characteristics and so cannot influence the net utility a consumer obtains from any gross utility they get from buying from that firm. In contrast, in the model of competitive first-degree price discrimination considered here, firms will face a trivial problem of extracting surplus from consumers once they have chosen to buy from them. However, firms can observe individual consumer characteristics and can use their pricing policy to operate directly on consumers net utility and hence on the decision to buy from one firm rather than the other. It is this that gives rise to the more intense competition between firms. The paper that comes closest to ours is that by Thisse and Vives (1988). In this paper firms can charge different consumers different prices, but only by offering them different goods. They model this by having firms absorb the transport costs. However, this means that there are two different processes at work – first-degree price discrimination and product customisation. There are two weaknesses of their work. (a) As indicated, they are conflating the process of price discrimination with that of product customisation, when benefit could be gained by studying the effects of price discrimination and product customisation separately. (b) Consumers are offered a particular product at a particular price tailored to their location. Consumers exercise no choice over which product they get. This is why, in their framework second degree price discrimination is equivalent to firstdegree. This makes sense if one interprets the Thisse and Vives model as a pure location model. While this is one interpretation that they place on their model, they also indicate that it can be interpreted as a model of product differentiation where consumers differ in taste. In this case, however, consumers offered a range of goods at different prices will be free to choose which product to consume. This 27
The survey by Varian (1989) has a small section 3.6 covering the analysis for the case of monoplistic competition.
13
makes the problem of determining the firm’s optimal second-degree pricing rule more complicated. One of the achievements of this paper is to determine this pricing rule. As mentioned, one of the powerful features of e-commerce is that it provides a technology whereby the processes of first degree price discrimination and product customisation can be undertaken separately. To fully understand the effects of this technology, it is therefore useful to begin an analysis focused exclusively on price discrimination, which then broadens to incorporate product customisation and investigate the relationships between the processes.
4.
Price Discrimination – The Model
Suppose there are two firms, A and B, located at either end-point of a line of length L. Consumers are uniformly distributed along this line, and each consumer buys a fixed amount of the good. Choose units so that there is 1 unit of demand at each point on the line. The firms have identical constant marginal costs of production c>0. Consumers have to incur transport costs to visit firms. Let t ( z ) be the transport costs of travelling distance x. Assume t (0) = 0; t ′( z ) > 0; t ′′( z ) ≥ 0 . z
For later purposes let T ( z ) be the integral of t(z). So T ( z ) = ∫ t ( y )dy . 0
Case I: No-Discrimination Suppose firm A charges a price p while firm B charges a price q. The consumer that is just indifferent between buying from A and B lies a distance x from A where p + t ( x ) = q + t ( L − x) .
(1)
Given the assumption on how we measure units, x is also the demand for firm A, which is defined through (1) as an implicit function of p and q. We have
∂x 1 =− < 0. ∂p t ′( x) + t ′( L − x) For any given q firm A chooses p to MAX p
( p − c).x ( p, q )
The first order condition is
14
(2)
x ( p, q) + ( p − c).
∂x = 0. ∂p
In a symmetric Bertrand equilibrium p = q, which implies x =
(3) L . 2
Substitute this into
(2) and (3), and then substitute (2) into (3) to get
L p e = c + L.t ′ . 2 But then the equilibrium profits that each firm makes under no-discrimination is
πn =
L2 L .t ′ . 2 2
(4)
Case II: Price Discrimination: Only One Firm Discriminates Suppose now that firm A can price discriminate, but firm B cannot. A can now set a price schedule p(x), while firm B sets a single price q that it charges to all consumers wherever they are located. To understand the nature of the equilibrium in this case, we will first characterise the best response by firm A to any price set by B. Suppose then that B sets a price q, c ≤ q < c + t ( L) . The total cost of buying from firm B to a consumer located at x is therefore q + t ( L − x), 0 ≤ x ≤ L . We need to determine the best response to this by firm A. Let x(q ), 0 < x(q ) < L be determined by c + t x(q ) = q + t L − x (q) .
Notice that
x(q)
is a strictly increasing function of q with
x (c ) =
L 2
and
x(q ) → L as q → c + t ( L) . Consider first consumers located a distance x from A, where 0 ≤ x < x(q ) . For these consumers c + t ( x ) < q + t ( L − x) , so A can set a price p ( x ) = q + t ( L − x ) − t ( x ) − ε , pick up all the consumers at x and still make a profit. Now consider consumers located a distance x from A, where x(q ) < x ≤ L . For these consumers c + t ( x) > q + t ( L − x) , so if A set a price p ( x ) ≤ q + t ( L − x) − t ( x) it would pick up some or all of the consumers but
15
make a loss. So the best response by A is to set any price p ( x) > q + t ( L − x) − t ( x) , which will guarantee that it does not sell to these consumers at all. Thus we can take A’s best response to any price q to be to set a price schedule p ( x ) = q + t ( L − x) − t ( x), 0 ≤ x ≤ x(q)
(5)
p ( x ) ≥ q + t ( L − x) − t ( x), x(q ) ≤ x ≤ L It is now straightforward to show the following result.
Proposition 1 The only Bertrand equilibrium is p e ( x ) = c + t ( L − x) − t ( x), 0 ≤ x ≤ L; q e = c 28 Proof: The proof is in two parts. We first prove that the equilibrium involves setting q e = c , and then that p e ( x ) = c + t ( L − x ) − t ( x ), 0 ≤ x ≤ L . (i) q e = c . Suppose that this were not the case and that B’s equilibrium price were q > c , and that, following (5) the corresponding price schedule set by A, p ( x ) , satisfied the properties p ( x ) = q + t ( L − x) − t ( x), 0 ≤ x ≤ x(q) p ( x ) ≥ q + t ( L − x) − t ( x), x(q ) ≤ x ≤ L Notice that the cost of buying from A for any consumer located at a distance x from A, is p ( x) + t ( x ) . This is illustrated in Figure 1 below for the case of a linear transport cost function. The heavy dark line indicates the cost p ( x ) + t ( x) of buying from firm A. For
()
the purposes of illustration we have assumed that on the interval x q , L that p ( x ) + t ( x) is strictly increasing.
p ( x) is such
We adopt the shorthand convention of saying p ( x ) = c + t ( L − x) − t ( x) , when we mean more precisely that, for some positive but arbitrarily small value of ε 28
p ( x ) = c + t ( L − x) − t ( x) − ε , 0 ≤ x
q − ε > c then ∀x, 0 ≤ x ≤ L : q − ε + t ( L − x) < q + t ( L − x) ≤ p( x) + t ( x ) = MAX q + t ( L − x ), c + t ( x ) .
Thus B would pick up the entire market and make a profit
( q − ε − c ) .L .
This is
illustrated in Figure 1 by the broken line that represents the cost q − ε + t ( L − x) to a consumer located at x of buying from firm B if it has charged a price q − ε . For sufficiently small ε, L > ( q − c ).( L − x ) 2 Hence q is not the profit-maximising response by B to the price schedule p ( x ) chosen by A.
( q − ε − c ) .L > ( q − c ) .
Hence the equilibrium must involve q e = c .
17
(ii)
p e ( x ) = c + t ( L − x) − t ( x), 0 ≤ x ≤ L
L L ; that B serves consumers in the interval , L 2 2 L and makes zero profits; and that, from (5), p e ( x ) = c + t ( L − x) − t ( x), 0 ≤ x ≤ . 2 L Suppose now that on some interval [ x1 , x2 ] < x1 < x2 ≤ L we had 2 If q = c it must be the case that x =
p e ( x ) > c + t ( L − x ) − t ( x) + ε . Then firm B could set a price q = c + ε , pick up all the consumers in this interval, and make a positive profit. Hence it must be the case that, effectively, L p e ( x) = c + t ( L − x) − t ( x) on the interval , L . 2 This completes the proof. From Theorem 1 it follows that once again each firm serves half the market. The profits made by firm B are clearly zero, while those made by A are L
L
π d = ∫ 2 [t ( L − x) − t ( x)] dx = T ( L) − 2T . 0 2
(6)
Case III: Price Discrimination: Both Firms Discriminate Here firms compete for consumers at each point on the line. Clearly the only outcome is that the firm that is at a distance disadvantage will have to set a price = c, while the other firm will set a price that just picks up all the consumers. On the other hand, at the midpoint of the line both firms are identical, so the conventional Bertrand conclusion holds and both firms’ prices are driven down to costs. This means that on the interval 0 ≤ x ≤
L the solution is as above, namely, 2
p e ( x) = MAX [ c + t ( L − x) − t ( x), c ] ; q e = c . On the other hand if we let y = L − x denote the distance between a consumer and firm L B, then, on the interval 0 ≤ y ≤ the solution is 2 q e ( y ) = MAX [ c + t ( L − y ) − t ( y ), c ] ;
18
pe = c .
But then, if both firms price discriminate they both make profits L π d = T ( L) − 2T . 2
(7)
From equations (6) and (7) we get our first result: L If a firm price discriminates it makes profits π d = T ( L) − 2T whether 2 or not the other firm price discriminates.
Lemma 1.
And by comparing equations (4) and (7) we get:
Price discrimination is profitable if and only if L2 L L d n π = T ( L) − 2T > π = .t ′ 2 2 2
Proposition 2.
The following section illustrates our result for a large class of transport costs.
4.1
Examples
Consider the class of simple transport cost functions t ( x) = t.x β ; t > 0, β ≥ 1 . Using (4) it is straightforward to show that π n =
π = d
t.L1+ β . ( 2 β − 1)
.
If we let ρ ( β ) ≡
πd πn
(1 + β ).2 β discrimination to non-discrimination, then we have ( 2β − 1) . ρ (β ) = β .(1 + β )
t.β 1+ β .L , while from (7) 2β
denote the ratio of profits under
It is straightforward to show that ρ ( β ) is monotonic increasing in β for β>1.5. It is less than 1 for integer values of β ≤ 4 , but greater than 1 for integers greater or equal to 5. To see what is going on in more detail consider what happens to the price schedules under discrimination and non-discrimination.
19
β
t.β .L L p e = c + L.t ′ = c + β −1 , while with 2 2 e β discrimination we have p (0) = c + t ( L) = c + t.L , and the price then falls with distance L until p e = c . 2 We know that with non-discrimination
Table 1 reports the values of these various variables for values of β ranging from 1 to 5. β
pe
p e (0)
ρ
1 2 3
c + t.L c + t.L2 3 c + t.L3 4 1 c + t.L4 2 5 c + t.L5 16
c + t.L c + t.L2 c + t.L3
1/2 1/2 7/12
c + t.L4
3/4
c + t.L5
31/30
4 5
Table 1 Figure 2 illustrates the price paths for the case of linear transport costs - β = 1 . Figure 2 p
p e = p e (0)
c 0
L/2
x
Here the broken line represents the equilibrium price when there is no discrimination, and the heavy line the equilibrium price path under discrimination. Figure 3 illustrates the situation when β = 5 .
20
Figure 3 p p (0) e
pe c 0
L/2
x
Once again the broken line represents the equilibrium price when neither firm discriminates. The curve represents the price path under discrimination. So when β is small - i.e. β ≤ 2 , then p e = p e (0) and so prices are almost everywhere lower under discrimination. When β = 3 the ability to discriminate means that prices are higher under discrimination for consumers close to firms. But, this is not enough to offset the fact that prices are lower for consumers around the middle so profits are lower under discrimination. As β gets larger so too does the gain from being able to discriminate amongst consumers close to the firms, until, eventually, price discrimination becomes possible. So, as we mention earlier, there are two effects at work: The intensification of competition between firms, and the ability to discriminate. Consider first the impact of these two effects on prices. The first effect certainly lowers the prices of consumers that are least loyal to any firm – those in the middle. When transport costs do not rise very fast with distance, then there are almost no loyal consumers and so the first effect means that prices are lower for all consumers. However, when transport costs rise more sharply with distance, then consumers close to a firm are locked in much more and so the firm can really discriminate. This offsets the competition effect and means that firms can now charge higher prices for those closer to them. Consider now the effect on profits. When prices are driven down for all consumers then profits obviously fall with discrimination. When firms can effectively raise prices for their most loyal consumers, this may or may not be enough to offset the effects of low prices in the middle. Only when the ability to discriminate is very strong because of sharply rising transport costs will discrimination actually be profitable. While this analysis of price discrimination yields informative results as to the effectiveness of such a technology, the choice to use discrimination either in place of, or in connection with, a technology such as customisation is important to investigate. The following section draws on the results found above and explores what happens when
21
these technologies are used together or in isolation, as well as the impact this has on the choices of the competing firms within the market. 5.
Mass Customisation – The Model
There are two firms located at either end of a line of length 1. Firm A is located at the left hand end of the line, and firm B at the right hand end. All distances are measured from the location of firm A. Mass customisation is captured by the idea that these firms can produce at the same constant marginal cost c all products up to a distance m < 1/2 from their location. Thus if firm A chooses to mass customise then it can produce all goods between 0 and m, while if firm B chooses to mass customise it can produce all goods between 1-m and 1. Thus, if both firms chose to mass customise, all consumers located up to a distance m from either firm can in principle get exactly the product they want. Consumers located between m and 1-m have to travel to only the nearest product at m or 1-m . Consumers are uniformly distributed on the line. Each consumer has an inelastic demand and will buy one and only one unit of this product. Let x denote the location of consumers and z the location of products. A consumer located at x who buys a product located at z will have to travel a distance d = x − z to buy this product. Let t (d ) be the transport costs of travelling distance d. Assume t (0) = 0; t ′(0) ≥ 0; ∀d > 0 t ′(d ) > 0, t ′′(d ) ≥ 0 . We will make use of the specific class of functions t (d ) = τ .d β , d > 0, β ≥ 1 to illustrate our results. In general we can think of firm A (respectively firm B) as offering a price schedule p ( x, z ), 0 ≤ x ≤ 1, 0 ≤ z ≤ m (respectively q( x, z ), 0 ≤ x ≤ 1, 1 − m ≤ z ≤ 1 ) to a consumer located at x who buys product z from its range. We will say that … • first-degree price discrimination is possible by a firm when it is able to offer a price schedule that varies with both x and z; • second-degree price discrimination is possible by a firm when it is constrained to offer a price schedule that is independent of x though it can vary with z; • no price discrimination is possible by a firm if it is constrained to offer a price schedule that is independent of both x and z.29 Assuming that consumers choose optimally (i.e. each consumer chooses the product which minimises its total cost from buying), we now analyse the equilibrium price 29
No price discrimination can be thought of as a special case of second-degree price discrimination where m=0. In what follows we therefore adopt the notion of only two price discrimination technologies, first and second.
22
schedules that emerge under alternative assumptions about what price discrimination possibilities are open to firms. For expositional purposes, we will conduct most of the analysis from the point of view of firm A. The conclusions for firm B will follow by analogy. We begin by analysing the case where firms are symmetric with respect to their ability to mass customise. Next we consider the case where only one of the firms mass customise. We then endogenise the choice of firms whether or not to mass customise, and what price discrimination technology to adopt. 5.1
m symmetry
We first consider the case where firms’ ability to mass customise is symmetric (i.e. they either both can or cannot mass customise). This advantage of this case, as we show below, is that we are able to find explicitly compute profits for all price discrimination technologies. 5.1.1
Both firms use First Degree Price Discrimination technology30
Firms compete for consumers at each point on the line. Clearly the only outcome is that the firm that is at a distance disadvantage will have to set a price = c, while the other firm will set a price that just picks up all the consumers. On the other hand, at the mid-point of the line both firms are identical, so the conventional Bertrand conclusion holds and both firms’ prices are driven down to costs. Assuming both firms mass customise, this implies the following price schedule for firm A (the schedule for B is symmetric): p e ( x, z ) = c + t (1 − m − x) e p ( x, z ) = c + t (1 − m − x) − t ( x − m) e p ( x, z ) = c
0≤ x≤m m≤x≤ x>
1 2
1 2
Notice that this implies that a consumer located at x < m will buy the good produced at z = x. In other words, the optimal schedule is independent of z. Consumers located between m and one half will buy the good produced at m. Similarly, consumers located between one half and 1-m will buy the good located at 1-m. By integrating the above expressions, we obtain that the profits of the firm with firstdegree price discrimination are therefore: 1 Π(m, m) = T (1 − m) − 2T − m 2 30
This relates to the first part of the paper – the model in section 4 did not allow firms to mass customise, and so m=0.
23
For the t = τx β we get: Π ( m) = 1
τ . 2 β .(1 − m)1+ β − (1 − 2m)1+ β 2 β .(1 + β )
(8)
If neither firm mass customise then A’s optimal price schedule is
e p ( x ) = c + t (1 − x ) − t ( x ) p e ( x) = c
0≤ x ≤ x>
1 2
1 2
1 And A’s profits are Π(0,0) = T (1) − 2T . 2 5.1.2
Both firms use Second Degree Price Discrimination technology
Under this regime, the firm is offering a tariff, p(z) of prices for its goods at 0 ≤ z ≤ m. If the firm chooses not to mass customise, then it offers a single price for its single good. If A mass customises, then consumers located to the right of m will buy model m at a price of p(m). All those consumers located between 0 and m will buy one of firm A’s models and pay at least p(m). However, since demand is inelastic, the amount of additional surplus the firm can extract from these consumers through suitable choice of the function α (.) is independent of p. This has two implications: (i)
All the competition between firm A and firm B is essentially over the setting of p(m) (which, in equilibrium, is identical to q(1-m)), and the marginal consumer, located between m and 1-m, will pay p(m). Hence we will begin by finding the value of p(m).
(ii)
Once p(m) is set, firm A just has to choose the function α (.) so as to extract the maximum surplus from those consumers located between 0 and m. We will compute the optimal schedule, α (.) at the second part of this section.
Notice that the value of p(m) is essentially independent of α (.) , and that it corresponds to the price that the firm would charge if it would set a single price for all its products. We begin by solving for this p(m) (and of course for q(1-m)): Suppose firm A charges a price p while firm B charges a price q. The consumer that is just indifferent between buying from A and B lies a distance x from m where
24
p + t ( x − m) = q + t (1 − m − x )
(9)
If both firms choose not to mass customise, then
p + t ( x ) = q + t (1 − x )
(10)
Given the assumption on how we measure units, x is also the demand for firm A, which is defined through (9) as an implicit function of p and q. We have, for the case where both firms mass customise:
∂x 1 =− 0 or m = 0). Substitute this into (11) and (12) and then substitute (11) into (12) to get:
In a symmetric Bertrand equilibrium p = q and x =
1 p n ( m) = c + t ′ − m , 2 1 Notice that if the firms do not customise then the price is simply c + t ′ . 2 The equilibrium profits are: R(m, m) =
1 1 ⋅ t' − m 2 2
Once again, if firms do not mass customise then profits are just R(0,0) =
1 1 ⋅ t' . 2 2
Suppose now that both firms do mass customise, so that each firm A sets a price schedule for each of the products it has on offer. 25
In order to ensure that consumers have well-defined model choices, we assume from now on that t ′′(d ) > 0 ∀d ≥ 0 and that t ′(0) = 0 . So the transport cost function is everywhere strictly convex, thus ruling out linear transport costs. For the particular functional form t (d ) = τ .d β we are essentially assuming from now on that β ≥ 2 . We are then able to prove the following result. Proposition 3. The optimal price schedule for firm A takes the form M
p ( z ) = p (m) + ∫ σ ( y )dy .
(13)
z
Here σ ( y ) ≥ 0, 0 ≤ y ≤ m , and is the solution to the equation y = ϕ [σ ( y )] + σ ( y ).ϕ ′ [σ ( y )] ,
(14)
where the function ϕ (σ ) = t ′−1 (σ ) .31
Proof: See Appendix To see what is implied by (14), consider the case where t ( z ) = τ .z β , τ > 0, β > 1 . 1
σ β −1 Then the function ϕ (σ ; β ,τ ) = . It then follows from (14) that βτ
β −1 β
β −1
.z β −1 .
σ ( z ; β , τ ) = τ .β .
(15)
This has a number of implications. x ( z; β ,τ ) =
z
(16)
β
β −1 p ( z; β ,τ ) = τ . β
β −1
β −1 S (m; β ,τ ) = τ . β
(m
β −1
.
β
− zβ )
m β +1 β +1
(17)
(18)
If we combine (18) with (12) we find that the overall profits made by the firm which mass customise and uses second-degree price discrimination technology are S+R(m,m). 31
That is
ϕ (σ )
is defined through the identity
σ ≡ t ′ [ϕ (σ )] .
26
For the case where t ( z ) = τ .z β , τ > 0, β > 1 we get:
β .(1 − 2m) β −1 β − 1 β −1 m β +1 Π ( m) = τ . + . β + 1 2β β 2
(19)
5.1.3 One firm uses First Degree Price Discrimination technology while the other firm is using Second Degree Price Discrimination technology
Assume without loss of generality that A is using a first-degree price discrimination technology, and B is using a second-degree price discrimination technology. Assume first that neither firm can mass customise. Then B sets a unique price, q, while A sets a price schedule, p ( x), 0 ≤ x ≤ 1 . Section 4 of this paper showed that in this case, the only possible equilibrium is where q = c. To see why, suppose that B were to set a price q 0 > c . A’s optimal price schedule is then p ( x) = MAX q 0 + t (1 − x) − t ( x), c . Given this schedule by A, however, firm B can profitably deviate by slightly lower its price and serve the whole market. In the unique equilibrium, therefore firm B charges c and makes zero profits. A’s equilibrium price schedule is p ( x) = MAX [ c + t (1 − x) − t ( x), c ] , with 1 which it makes profits of T (1) − 2T . 2 When both firms mass customise, a similar argument applies. In this case, firm A’s optimal schedule depends on firm A’s price for its good at 1 - m. The exact same argument as above applies, and B can be shown to set q (1 − m) = c . A’s optimal schedule is identical to what it would have been if both firms used first-degree price discrimination technologies, namely p e ( x, z ) = c + t (1 − m − x) e p ( x, z ) = c + t (1 − m − x) − t ( x − m) e p ( x, z ) = c
0≤ x≤m m≤x≤ x>
1 2
1 2
By integrating the above expressions, A’s profits are: 1 Π(m, m) = T (1 − m) − 2T − m 2 While B’s profits come solely from what it is able to extract from its loyal customers using the price schedule α, described in section 5.1.2. B’s profits are therefore given by the quantity S described in equation (18).
27
5.2
m asymmetry
We now consider the case where only one firm is capable of mass customising its products. Without loss of generality we will assume that A is the firm which mass customise (the case where B mass customise follows by analogy).
5.2.1
Both firms use First Degree Price Discrimination technology
The argument here is identical to that made in section 5.1.1: Firms compete for every customer, only in this case the indifferent consumer lies not at ½ but at (1+m)/2 (because she is indifferent between buying the goods produced at z = m and at z = 1). Firm A’s will charge the following price schedule: p e ( x ) = c + t (1 − x ) 0≤ x≤m 1+ m e m≤ x≤ p ( x ) = c + t (1 − x ) − t ( x − m) 2 1+ m e x> p ( x ) = c 2 1 m And will make profits of Π(m,0) = T (1) − 2T − , while B charges: 2 2
1− m ≤ x ≤1 2 1− m x< 2
e q ( x) = c + t ( x − m) − t ( x) q e ( x) = c
1 m And will make profits of Π(0, m) = T (1 − m) − 2T − 2 2
5.2.2
Both firms use Second Degree Price Discrimination technology
Since B cannot mass customise, it is forced to set a single price, q, while A sets a schedule, p(z). Consumers located to the right of m will buy model m at a price of p(m). All those consumers located between 0 and m will buy one of firm A’s models and pay at least p(m). Once q and p(m) are chosen, A will use the same schedule, α(z) as in section 5.1.2 to extract maximum surplus from its customers. We now turn to the problem of finding the equilibrium values of q and p(m). We repeat the analysis of section 5.1.2 for this case. First, we identify the consumer indifferent between buying from A and B:
28
p + t ( x − m) = q + t (1 − x)
(9’)
Given the assumption on how we measure units, x is also the demand for firm A, which is defined through (9’) as an implicit function of p and q. So we can re-write equation (11) as: 1 ∂x (11’) =− 0, β ≥ 1 we get:
β .(2 x − 1). ( x − m) β −1 + (1 − x) β −1 = (1 − x) β − ( x − m) β And the equilibrium profits of the two firms will be: R (m,0) = β ⋅ x 2 [( x − m) β + (1 − x) β ] R (0, m) = β ⋅ (1 − x) 2 [( x − m) β + (1 − x) β ] So clearly A makes higher profits than B, even before considering the extra profits extracted using use the schedule α(z). Moreover, because the profit function is continuous with respect to m we get that: R(0,m) < R(m,m)
and
R(m,0) > R(m,m)
(20)
In summary, firm A will make profits of R(m,0)+S, where S is given by equation (18), and firm B will make profits of R(0,m).
5.2.3 Firm A uses First Degree Price Discrimination technology, firm B is using Second Degree Price Discrimination technology The analysis in this case is similar to that of section 5.1.3: Firm B sets a unique price, q, while A sets a price schedule. The same arguments as in 5.1.3 apply and so the only possible equilibrium is where q=c. A’s optimal schedule then becomes:
29
p e ( x, z ) = c + t (1 − x) e p ( x, z ) = c + t (1 − x) − t ( x − m) e p ( x, z ) = c
0≤ x≤m 1+ m 2 1+ m x> 2
m≤x≤
By integrating the above expressions, A’s profits are: 1− m T (1) − 2T 2 While B’s make zero profits.
5.2.4
Firm B uses First Degree Price Discrimination technology, firm A is using Second Degree Price Discrimination technology
In this case, firm B sets its schedule taking into account the nearest substitute, A’s product at m, and its price, p(m). The same arguments as in 5.1.3 apply and so the only possible equilibrium is where p(m)=c. B’s optimal schedule then becomes: e q ( x) = c q e ( x) = c + t ( x − m) − t (1 − x)
0≤x≤
1+ m 2
1+ m ≤ x1 2
1 − m By integrating the above expressions, B’s profits are now: T (1 − m) − 2T , while 2 firm A’s profits (which come solely from its schedule α) are equal to quantity S described in equation (18).
5.3
Analysis and Results
Table 2 summarises the payoffs for firm A for all possible choices of price discrimination technologies adopted by the two firms (the table for firm B is symmetric). The first row describes the payoffs when the firm chooses not to mass customise and not to first degree price differentiate. The second row describes the payoffs when the firm chooses not to mass customise but does use first degree price differentiation. The third row describes the payoffs when the firm mass customises and does not first degree price differentiate. The fourth and final row describes the payoffs when the firm mass customises and first degree price differentiate.
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0 & 2nd 0 & 1st m& 2nd m & 1st
0 & 2nd 1 1 t' 2 2
0 & 1st 0
m & 2nd G(0,m)
m & 1st 0
1 T (1) − 2T 2 G(m,0)+S
1 T (1) − 2T 2
1 m T (1 − m) − 2T − 2 2
1 m T (1 − m) − 2T − 2 2
S
1 1 t ' − m + S 2 2
S
1 T (1 − m) − 2T − m 2
1 T (1 − m) − 2T − m 2
1 m T (1) − 2T − 2 2
1 m T (1) − 2T − 2 2
Table 2 We are now able to prove our main results:
Lemma 2. If a firm uses first-degree price discrimination technology then it is strictly better off if it also mass customises. Proof. By comparing the payoffs in the second and fourth row in Table 2 (note that the function T is strictly increasing because it is the integral of the transport function, t, which is positive). Lemma 3. For the case t ( z ) = τ .z β , τ > 0, β > 1 , if a firm uses second-degree price discrimination technology then it is strictly better off if it also mass customises. Proof. Follows immediately from equation (20), and from comparing the first and third rows in Table 2. Existing empirical evidence supports these results. One example, mentioned in Section 2, is that of Capital One which offers not only a range of prices but also offers its customers particular credit products that are tailored to their needs. Similarly, Dell allows its customers to decide exactly what features they want on their computers while, at the same time, offering different prices to different consumers.
Proposition 4. As m→ ½, the only equilibrium of the total game, described in Table 3, is for both firms to choose to mass customise and use first-degree price discrimination technology. Proof. Using Lemmas 2 and 3, the strategies 0 & 2nd, and 0 & 1st are dominated. In equilibrium dominated strategies are never used, and the game is reduced to the following: Again, many empirical examples of this result were mentioned in Section 2. The gaming industry is one where the decision of one firm (Harrah) to customise its products and services led to competitors, such as Foxwoods and MGM Mirage, adopting these personalisation marketing techniques in the knowledge that they failure to do so would
31
almost certainly lead to lower market share and decreased profits. Similarly, the concept of consumer loyalty cards, adopted by many supermarket chains, and the personalisation of customer services within the mobile telephone market provide excellent examples of where all firms within the market adopt the technologies of mass customisation and/or price discrimination as best responses to the behaviour of the other firms.
m & 2nd m & 1st
m & 2nd 1 1 t ' − m + S 2 2
m & 1st S
1 T (1 − m) − 2T − m 2
1 T (1 − m) − 2T − m 2
Table 3 To find the equilibrium of this game, we need to compare the payoffs 1 t ' 1 − m + S with 2 2
1 T(1− m) − 2T − m . 2
For the case t ( z ) = τ .z β , τ > 0, β > 1 , by using equations (8) and (19), we now compute the ratio of profits under first-degree price discrimination to those under second degree price discrimination are:
ρ ( m) =
Π1 ( m ) = Π 2 ( m)
2 β .(1 − m)1+ β − (1 − 2m)1+ β
β .(1 + β )(1 − 2m)
β −1
β −1 + β
β −1
.2 β .m1+ β
β −1
1 β 1 Notice that when m = ½ then ρ ; β = . When β = 2, ρ = 2 . It is 2 β −1 2 straightforward to check that first degree price discrimination is always worthwhile for m ≈ 0.5 . When β = 2 , ρ is a strictly increasing function of m. When β takes values greater than 2 then ρ (m) is initially increasing, then decreasing as 1 m → 0.5 , although, as pointed out above ρ ≥ 2 . Section 4 of this paper shows that 2 for β ≥ 5, ρ (0) > 1 , so, for all these values of β first-degree price discrimination is profitable for all values of m, 0 ≤ m ≤ 0.5 .
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values of β table 4 indicates the critical value of 1 m, m( β ), 0 < m( β ) < at which first-degree price discrimination becomes profitable. 2
For
the
remaining
β
m( β )
2 3 4
0.47 0.31 0.15 Table 4
This completes the proof of Proposition 4. Finally, our last result shows why firms are locked-in a prisoners’ dilemma situation.
Proposition 5. Compare to the equilibrium outcome, where firms mass customise and use first-degree price discrimination technology, firms would have been better-off by not mass customising and not differentiating. Proof. Follows immediately from Table 2 (comparing the payoffs when both firms choose 0 & 2nd to the payoffs when both firms choose m & 1st) and Proposition 4. Once again, empirical evidence supports this result. Within the UK grocery industry, customisation and price discrimination appear to have led to falling prices, along with increased choice and, possibly, improved quality. A further example can be found in the mobile telephone industry where prices of the various mobile services (voice and sms services, ringtones, etc) have been following a decreasing trend.
5.
Conclusions
In recent years the Internet has become a popular shopping channel. The Internet offers consumers 24 hour shopping and the convenience of shopping from home. It is also much easier to compare products and prices on-line than it is in any other means of shopping. For homogenous goods, like CDs and books, consumers can use shopping agents (known as ShopBots), like BargainFinder.com to compare prices and buy from the seller which offer the cheapest price. Internet technologies offer greater opportunities for sellers too. On-line retailers face much lower entry and set-up costs and menu costs are practically zero (once the appropriate software is installed). Easy access to information about their users allows online retailers to better segment the market they are serving and to offer customers products which they are likely to want to buy. It is not clear at this early stage what effects these technologies will have on prices and market structure. Even in the simple case of homogenous goods, it is not clear what the
33
overall effect on prices will be32. There is clearly a need for rigorous models of electronic markets where intuitions can be put to the test. This paper can therefore be seen as a contribution to filling that gap. The first part of this paper, dealt with primarily in section 4, focuses on one particular feature of electronic markets - the ability of sellers to first degree price discriminate. Personalisation technologies have been in the centre of media attention in recent years. Many, including the US Justice Department, have voiced their concern about the possible violation of individual privacy rights through the exchange of information between the user’s agent and the host Website, which takes place automatically. In particular, the view that the more information the seller has about the user, the more surplus it is able to extract from her, remains highly popular. While this is always true in a monopoly context, this paper questions this intuition in the context of competitive markets. We have shown that in that context there are two effects at work: (i) the enhanced surplus extraction effect - which is the effect at work in the traditional monopoly analysis of first degree price discrimination; (ii) the intensified competition effect – the fact that firms end up competing consumer by consumer – with the traditional Bertrand result that at least one of them has its price driven down to marginal cost. We have used a simple duopoly setting with product differentiation to understand the interaction between these two effects, and to characterise the exact conditions under which firms would prefer to employ first-degree price discrimination. We showed that when the average and marginal transport costs rise very slowly with distance – more precisely when transport costs are linear or quadratic – then the intensified competition effect dominates the enhanced surplus extraction effect and almost all consumers face lower prices under price discrimination. As transport costs rise more steeply with distance, the surplus extraction effect is strengthened. Now consumers with the greatest brand loyalty will pay higher prices under first-degree price discrimination, though those with least loyalty will still face lower prices. Only when transport costs rise sufficiently steeply so that the increased profits extracted from the most loyal consumers exceeds the loss in profits from the least loyal consumers will first-degree price discrimination turn out to be profitable for firms. While this analysis into first degree price discrimination has been motivated by emerging e-commerce technology, it can be applied in any context where this type of discrimination is possible, and can be used to explain why, when there is competition, price discrimination may not always be used. The conclusion of this first section of the paper is therefore that even though electronic commerce may provide firms with the opportunity to engage in first degree price discrimination, in a wide class of cases the firms will choose not to use this technology 32
See Vulkan (1999) for a general discussion, and Ulph and Vaughan (1999) for an analysis that shows that increased price transparency can raise or lower prices, depending on whether producers or consumers can better exploit the technology.
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because of the extra competition that such discrimination will induce. Moreover, even when firms do choose to engage in price discrimination, precisely because of the increased competition, there are many consumers who benefit from this. Thus the case for prohibiting such discrimination is very weak, while there is a strong case for promoting competition. As mentioned in the introduction, firms have the opportunity to engage in two new competitive strategies – first degree price discrimination and mass customisation. While these are often conflated, we have argued that they are separate strategies, and firms will be able to decide independently which of them to adopt. The second part of this paper, contained mostly in section 5, focuses on the latter of these two strategies while also looking specifically at the interaction between the two strategic decisions; we have developed a simple framework within which to understand the incentives that firms would have to adopt each of these strategies. The model we have used in this section of the paper is a very simple extension of the basic Hotelling model in which we have allowed firms the possibility of producing a range of goods, and of employing first-degree price discrimination. There are many obvious extensions to this model: (a) endogenising firm location; increasing the number of firms; (b) (c) increasing the number of dimensions in which goods can be differentiated. Some of the key results from section 4 include the following: (i) the greater the degree of customisation adopted by firms the stronger will be their incentives to adopt first degree price discrimination; (ii) if firms use first degree price discrimination then mass customisation is a dominant strategy; (iii) however, firms are in a prisoners dilemma since they would be better off if they neither customised nor engaged in first-degree price discrimination.
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References Armstrong, M. and J. Vickers (1999), “Competitive price discrimination”, Working Paper, Nuffield College, Oxford University. Borenstein, S. (1985), “Price Discrimination in Free-Entry Markets”, Rand Journal of Economics, 16, pp. 380-397. Brynjolfsson, E. and M. Smith (1999), “Frictionless Commerce? A comparison of Internet and Conventional Retailers”, Working Paper MIT. Corts, K. (1998), “Third-Degree Price Discrimination in Oligopoly: All-Out Competition and Strategic Commitment”, Rand Journal of Economics, 29, pp. 306-323. Holms, T. (1989), “The Effects of Third-Degree Price Discrimination in Oligopoly”, American Economic Review, 79, 244-250. Hotteling, H. (1929), “Stability in Competition”, Economic Journal, 39, pp. 41-57. Katz, M. (1984), “Price Discrimination and Monopolistic Competition”, Econometrica, 52, pp. 1453-1471. Shapiro, C. and H. Varian, (1999), Information Rules, Boston: Harvard Business School Press. Thisse, J-F and X. Vives (1988), “On the Strategic Choice of Spatial Price Policy”, American Economic Review, 78, 122-137. Varian, H. (1989), “Price Discrimination”, in R. Schmalensee and R. Willig, eds., Handbook of Industrial Organization, Volume 1, Amsterdam: North Holland. Ulph, D. and R. Vaughan (1999), “Price Transparency and Market Equilibria”, mimeo. Vulkan, N. (1999), “Economic implications of agent technology and e-commerce”, The Economic Journal, 453, pp. 67-90.
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Appendix: Proof of Proposition 3. Begin by writing the price schedule as p ( z ) = p(m) + α ( z ), 0 ≤ z ≤ m where α ( m) = 0 . We first establish
Lemma A1 α ( z ) is concave. Proof: The argument proceeds by contradiction, by showing that IF the function were linear over a range, then profits could be increased by making it non-linear – and indeed by making it concave. The argument is developed in the context where the function is assumed to be strictly increasing, but can be completely re-cast to apply in the case where the function is decreasing. Suppose therefore that all models are bought in the neighbourhood of some model z0 , 0 < z0 < m and also that throughout this neighbourhood, the price function is linear. Suppose that p′( z0 ) = σ 0 > 0 .
Thus over a neighbourhood [ z0 − ε , z0 + ε ] we can write
the price function as p ( z ) = p( z0 ) + σ 0 ( z − z0 ) , and we can assume that all models in this neighbourhood are bought. Using the notation in the notes, the consumers who buy these models are located in the interval z0 + ϕ (σ 0 ) − ε , z0 + ϕ (σ 0 ) + ε . Consider now the following perturbation.
ε + ( z − z0 ) .δ p ( z ) + ε p% ( z ) = p ( z ) + ε + ( z0 − z ) .δ ε
z0 − ε ≤ z ≤ z0 z0 ≤ z ≤ z0 + ε
where δ > 0 . Notice that
δ σ 0 + ε , z0 − ε ≤ z ≤ z0 p% ′( z ) = σ − δ , z ≤ z ≤ z + ε 0 0 0 ε δ ε
Let ∆z = ϕ ′ (σ 0 ) . .
Choose δ to be sufficiently small that σ 0 −
δ > 0 and ∆z < ε . ε
Notice that consumers in the interval z0 + ϕ (σ 0 ) − ε , z0 + ϕ (σ 0 ) would now like to buy a model ∆z further away from their location than their original choice while those in the interval z0 + ϕ (σ 0 ) , z0 + ϕ (σ 0 ) + ε would now like to buy a model ∆z closer to their location than their original choice.
37
More precisely what will happen with this new price schedule is the following: (i) Consumers who originally bought models in the interval ( z0 − ε , z0 − ε + ∆z ] will now buy the model z0 − ε paying the price p ( z0 ) − σ 0 .ε rather than the price
p ( z0 ) + σ 0 .( z − z0 ) > p ( z0 ) − σ 0 .ε that they were originally paying. This generates a loss of revenue to the firm equal to
(ii)
σ 0 . ( ∆z )
2
. 2 Consumers who originally bought models in the interval ( z0 − ε + ∆z , z0 ) will buy a model that is ∆z further away from their location than their original choice . Thus a consumer that would have bought model z at a price p( z ) = p ( z0 ) + σ 0 ( z − z0 ) will now buy model z − ∆z at a price ε + ( z − ∆z − z0 ) p% ( z − ∆z ) = p( z0 ) + σ 0 ( z − ∆z − z0 ) + .δ . This generates a net
ε δ 2 change in revenue equal to . ( ε − ∆z ) − σ 0 .∆z.(ε − ∆z ) 2ε
(iii)
Consumers who originally bought models in the interval ( z0 , z0 + ε − ∆z ) will buy a model that is ∆z closer to their location than their original choice . Thus a consumer that would have bought model z at a price p ( z ) = p( z0 ) + σ 0 ( z − z0 ) will now buy model z + ∆z at a price z + ε − ( z + ∆z ) p% ( z + ∆z ) = p ( z0 ) + σ 0 ( z + ∆z − z0 ) + 0 .δ . This generates a net
ε δ 2 increase in revenue of . ( ε − ∆z ) + σ 0 .∆z.(ε − ∆z ) 2ε
(iv)
Consumers who originally bought models in the interval
[ z0 + ε − ∆z, z0 + ε )
will
now buy the model z0 + ε paying the price p ( z0 ) + σ 0 .ε rather than the price p ( z0 ) + σ 0 .( z − z0 ) < p ( z0 ) + σ 0 .ε that they were originally paying. This generates an increase in revenue to the firm equal to
σ 0 . ( ∆z )
2
. 2 If we add together all these changes in revenue we see that this perturbation is definitely profitable. Therefore it cannot be part of a profit-maximising price schedule to have a linear segment – and it pays to make the schedule concave. Continuous repetition of this argument will lead to a concave function. QED. Before proceeding it may help to see a more intuitive version of the above argument for the case where α ( z ) is initially assumed to be linear and decreasing. Suppose first of all that the price schedule were linear as illustrated by the line AC in Figure 4.
38
P A
B C
0
Z m/2
m
Figure 4 Suppose now we generate a piecewise linear price schedule by increasing the price on the model m/2, and on all other models except 0 and m, resulting in the new price schedule ABC. Consumers who bought models in the range [0,m/2] when the price schedule was AC will now buy models closer to zero, paying a higher price – both because of the new schedule and because they are moving up the AC schedule towards A. Consumers who bought models in the range [m/2,m] when the price schedule was AC will now buy models closer to m. There will be two effects on the price they pay – it will be lower because of the downward shift along AC towards C, but higher because of the shift to the new schedule ABC. It is intuitively obvious – and is confirmed in the proof of Lemma A1 - that the increase in prices paid by the first group of consumers dominates the reduction in prices paid by the second, resulting in an increase in profits. Lemma A2 α ( z ) is strictly decreasing. Proof: It is clear that in order to push consumers to buying the highest price, we want the marginal price at zero to be zero - i.e. we should have p′(0) = 0 - since there is no consumer to the left of zero to try to induce to pay a higher price. This is just the analogue of the familiar result that marginal tax rates at the top of the income distribution should be zero. Combining this second result with the concavity result shows that the price schedule should be strictly decreasing. QED m
Since we now know that α ( z ) is decreasing, we can write α ( z ) = ∫ σ ( y )dy z
where
σ ( y ) ≥ 0, 0 ≤ y ≤ m . Notice that, because the price is non-increasing, no consumer will buy a product to the left of their location. More formally, a consumer located at x, 0 ≤ x ≤ m has to solve the problem min α ( z ) + t ( z − x)
0≤ z ≤ m
Consider first the case of an interior solution. This is characterised by the condition:
σ ( z ) = t ′( z − x) .
(A1)
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For this to be a well-defined minimum we need to assume that the second-order condition is satisfied i.e. that σ ′( z ) < t ′′( z − x) . It is straightforward to see that in this case z is a strictly increasing function of x. Now define the function ϕ (σ ) through the identity:
σ ≡ t ′ [ϕ (σ )] . Clearly, ϕ (0) = 0, ϕ ′(σ ) > 0 . Notice that the function x( z ) = z − ϕ [σ ( z ) ] defines the location of the consumer who is willing to buy z as an unconstrained optimum. Note the following points. 1)
If σ (0) > 0 , then x(0) < 0 - so nobody will buy model 0. Indeed, if we define z0 by the condition z0 = ϕ ( z0 ) then consumer located at 0 will buy z0 , and none of the models in the interval [0, z0 ) will be purchased. We will show later that this cannot happen.
2)
If σ (m) > 0 then x (m) < m and all the consumers in the interval [x(m),m] will buy good m.
Now define 0, 0 ≤ z ≤ z0 G( z) = z − ϕ [σ ( z ) ] , z0 ≤ z ≤ m Let g ( z ) = G′( z ), z0 ≤ z ≤ m . For any given function σ ( y ) ≥ 0, 0 ≤ y ≤ m the total additional surplus extracted from consumers located between 0 and m is m
S = ∫ α ( z ).g ( z )dz z0
Integrate by parts and we get
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S = ∫ σ ( z ).{ z − ϕ [σ ( z )]} dz m
z0
If we choose σ ( y ) ≥ 0, 0 ≤ y ≤ m to maximise S then we have the following f.o.c. z − ϕ [σ ( z )] − σ ( z ).ϕ ′ [σ ( z )] =0 i.e.
z = ϕ [σ ( z ) ] + σ ( z ).ϕ ′ [σ ( z )]
(A2)
Equation (A2) implicitly defines the optimal solution σ ( z ) ≥ 0, 0 ≤ z ≤ m as a function of z. Notice right away that this implies σ (0) = 0 which in turn implies that z0 = 0 . QED.
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