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Warranty pricing with consumer learning
Accepted Manuscript
Warranty Pricing with Consumer Learning Yong Lei, Qian Liu, Stephen Shum PII: DOI: Reference:
S0377-2217(17)30563-5 10.1016/j.ejor.2017.06.024 EOR 14506
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
29 October 2016 13 April 2017 6 June 2017
Please cite this article as: Yong Lei, Qian Liu, Stephen Shum, Warranty Pricing with Consumer Learning, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.06.024
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Highlights • The dynamic warranty pricing with consumer learning is studied. • Warranty sales do not generate profit directly though are profitable overall. • Both consumers’ beliefs and the firm’s warranty policy are stable in the long run.
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• The firm only induces consumer learning when the true failure rate is high.
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Warranty Pricing with Consumer Learning Yong Lei∗,a , Qian Liub , Stephen Shumc a
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School of International Business Southwestern University of Finance and Economics, China 611130, China b Department of Industrial Engineering and Logistics Management The Hong Kong University of Science and Technology, Hong Kong c College of Business City University of Hong Kong, Hong Kong
Abstract
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We consider a problem in which a firm dynamically prices a product and its warranty service over time. Consumers can learn about the reliability of products based on warranty prices. A firm’s optimal product and warranty pricing policies are characterized. We find that a warranty should be priced lower than the marginal warranty service cost, which implies that warranty sales will not generate profits directly. However, offering a modest warranty still benefits the firm’s overall profits. We also show that consumers’ beliefs and the firm’s warranty policy converge in the long run. In a steady state, either a fraction of consumers will purchase a warranty or no consumer will purchase a warranty. Comparative statics analysis is conducted to show how factors such as a firm’s warranty service cost, consumers’ learning speed, and the heterogeneity of consumers’ handling costs determine consumers’ beliefs, the firm’s warranty policy and profitability in a steady state. Lastly, we note that a firm benefits from consumer learning by hiding the information about the true product reliability only when the true product failure rate is relatively high.
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Key words: Revenue management; Consumer learning; Pricing; Warranty management
Introduction
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Firms change warranty provisions for their products from time to time. For example, in October 2011, before formally introducing the iPhone 4S, Apple introduced a new Applecare+ service plan for iPhones at $99 to replace the previous $69 Applecare program (Slivka 2011). In September 2013, before formally ∗
Corresponding author. Tel.: +86 13540609462. E-mail addresses: [email protected] (Y. Lei), [email protected] (Q. Liu), [email protected] (S. Shum).
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introducing the iPhone 5C and iPhone 5S, Apple increased the accidental damage service fee from $49 to $79 per incident, but the price for Applecare+ remained at $99 (Allen 2013). The most recent change took place in September 2015. Before formally introducing the iPhone 6S and iPhone 6S Plus, Apple increased the price of Applecare+ for the iPhone 6S and iPhone 6S Plus from $99 to $129. Apple also increased the service fee from $79 to $99 for the new iPhone 6S and iPhone 6S Plus (Whitney 2015).
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Apple is not the only example in practice of a firm changing its warranty provision. In the automobile
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industry, for example, Ford increased its powertrain warranty from 3 years or 36,000 miles to 5 years or 60,000 miles on its 2007 Ford Lincoln and Mercury models in 2006, and Kia and General Motors made similar moves in 2001 and 2006, respectively (see, Warranty Week 2006, Krisher 2006, and Guajardo et al. 2015). In contrast, Chrysler dropped its lifetime warranty in 2009 two years after its introduction (Wernle 2009).
Changing the warranty policy can drive a company’s profit in opposite ways. On one hand, offering
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a more generous warranty policy increases the cost to the firm due to a higher cost of honouring the warranty. On the other hand, a more generous warranty usually means that the warranty coverage is better or the warranty price will be lower. Thus, it increases the demand for the product by directly reducing consumers’ cost of owning it.
A more subtle yet important impact is that the warranty terms, such as the service length, price and service coverage, are considered important indicators influencing consumers’ beliefs about product
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reliability. As noted by extensive empirical studies, including Wiener (1985), Boulding and Kirmani (1993), Pilon (2009) and Jindal (2015), consumers tend to perceive warranty terms as an accurate
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and informative reflection of product reliability, and hence rely on the warranty terms and scope to make inferences about product reliability. This is also consistent with the beliefs of practitioners. As mentioned in Warranty Week (2006) and Krisher (2006), Ford and General Motors increased their
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warranties “in an effort to help it sell more autos by boosting its reputation for quality”, which indicates the consumers are treating the warranty as a reliable indicator of a firm’s product reliability.
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In this paper, we study a dynamic model of a firm selling a single product with an optional warranty. The firm dynamically optimizes the price of the product and the price of the optional warranty. Consumers perceive the warranty price as an indicator of product reliability, and in particular, consumers
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perceive the product as more reliable when the warranty price is lower. Consumers have heterogeneous handling costs in case of product failures. Each consumer decides whether to buy both the product and warranty, the product only, or neither the product nor the warranty. Our goal is to consider how a firm should dynamically manage its warranty provision when warranty provision affects consumers’ beliefs 1
The price and service fee of Applecare+ remained the same with the introduction of the iPhone 7 and iPhone 7 Plus in September 2016. Apple cut the screen damage service fee from $99 to $29. Information can be found via the official website of Apple.
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about the product reliability. For example, how should a firm price its warranty policy? What is the optimal warranty policy in the long run? How does it depend on different factors, such as warranty cost and customer heterogeneity? Our results suggest that the warranty should not be priced higher than its marginal cost; hence warranty sales by themselves should not make a profit for the firm. While this contradicts the results of some analytic models (e.g., Padmanabhan and Rao 1993, Lutz and Padmanabhan 1995, and Pad-
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manabhan 1995) that state the warranties are a highly profitable service, there is no consensus among practitioners on the profitability of warranties due to some accounting issues. For example, Warranty Week (2014) questions the profitability of Applecare and Applecare+. Moreover, our results are consistent with the findings of the empirical study of Jindal (2015) that warranties should not be priced higher than its marginal costs when the warranty is sold simultaneously with the product. Thus, to some extent, our paper complements the results of this empirical study by providing support from a
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theoretical modelling perspective.
Our steady state analysis shows that consumers’ beliefs and the firm’s warranty policies converge. In the long run, it is possible that a fraction of consumers will purchase a warranty or no consumer purchases warranty depending on the firm’s warranty cost, consumers’ learning speed and heterogeneity of consumers’ handling costs. The long-run warranty policy and consumers’ beliefs depend on these parameters.
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Lastly, we further find that the firm prefers to hide the true product reliability and manipulate consumers’ perceived beliefs about the product failure rate through learning rather than disclosing
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information about the true product reliability to the consumers only when the true product failure rate is relatively high.
The remainder of the paper is organized as follows. Section 2 reviews the relevant literature. Section
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3 introduces the base model, and Section 4 studies the firm’s optimal dynamic product and warranty pricing problem. Section 5 compares the results with when information about true product failure rate
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is disclosed to the consumers to illustrate the impact of consumer learning. Section 6 extends our model to the case when there is some uncertainty in consumers’ belief updating about the product failure rate in order to validate the model robustness. Finally, Section 7 provides concluding remarks, a discussion
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of the limitations of this paper, and possible future research.
Literature Review
As pointed by Emons (1989), it is generally recognized that warranties have four economic roles: insurance, screening, signalling, and incentive. For example, Heal (1977) shows that warranties provide
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consumers with insurance and build up a risk-sharing mechanism based on the assumption that consumers are risk averse. Chun and Tang (1995) and Zhou et al. (2009) further examine how consumers’ risk preferences affect the optimal warranty price. Based on the idea that a firm with less information about buyers can use a menu of offerings and prices to screen buyers through self-selection (see Rothschild and Stiglitz 1976 and Stiglitz and Weiss 1981 for screening in general settings), some papers consider the use of warranty menus to differentiate different types of consumers (e.g., Kubo 1986,
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Matthews and Moore 1987, Padmanabhan and Rao 1993, Padmanabhan 1995, and Lutz and Padmanabhan 1998). Consumers’ efforts in maintaining the product have an impact on the firm’s warranty servicing cost, whereas this effort is costly to customers and unobservable to firm. Firms must take consumers’ moral hazard into account when designing warranty terms. Thus, some works consider the design of warranty provision to provide consumers incentives for better product maintenance (e.g., Lutz and Padmanabhan 1995). From the firm’s prospective, some other works investigate the firm’s incen-
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tive to signal product quality to consumers (e.g., Cooper and Ross 1985 and Lutz 1989) and supply high-quality products (e.g., Heal 1977 and Priest 1981). Gallego et al. (2014b) study both drivers by considering the use of residual value warranties, a practice of refunding part of the warranty price to customers who have zero or few claims according to a predetermined refund schedule. Residual value warranties can price discriminate customers, and they also affect customers’ strategic behaviour in using warranty services. The authors show that the total warranty service cost, which includes repair costs
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and refunds, is lower for more risk-averse customers under residual value warranties, and they identify the conditions when residual value warranties are more profitable than traditional warranties.
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The problem studied in our paper is more related to the papers that study cases when consumers do not have perfect information about product reliability, and the firm uses warranty policies to signal product reliability. As noted by Akerlof (1970), markets can break down when potential buyers cannot
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verify the quality of the product they are offered, and as a result, sellers of bad quality products may crowd out sellers of good quality products. In contrast, firms may have incentives to take observable and
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costly actions to signal potential buyers about their superior product quality when compared with those of other sellers (Spence 1973, 1974). In particular, a warranty may serve as a credit signal; hence better warranty terms may signal better product quality (e.g., Spence 1977 and DeCroix 1999). Because better
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product quality incurs higher production cost, in equilibrium, it is optimal for manufacturers to offer a better quality product with a higher warranty price to more failure-sensitive consumers and a lower quality product with a lower warranty price to less failure-sensitive consumers. However, some papers also suggest that it is possible that poorer warranty may be associated with higher product reliability due to consumers’ moral hazard on product maintenance (e.g., Cooper and Ross 1985 and Lutz 1989) and the differences in consumers’ knowledge about the reliability of established and newer products
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(Balachander 2001).
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Nevertheless, most of the empirical studies have shown that customers typically
perceive better product reliability when warranty provisions are better (e.g., Wiener 1985, Kelley 1988, and Boulding and Kirmani 1993). Readers may refer to Emons (1989), Blischke and Murthy (1992), Murthy and Blischke (1992a,b), Murthy and Djamaludin (2002), and Shafiee and Chukova (2013) for more extensive reviews of warranties. Similar to some of the aforementioned papers, our paper focuses on analytic modelling to identify
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the optimal warranty provision for the firm. The key difference is that these papers consider a singleperiod case, whereas our paper studies a dynamic model in which the firm can change the product and warranty prices over time, which resembles the practice in the industry, where firms change warranty provision from time to time. In addition, the study of a dynamic setting allows us to consider the impact of not only static factors, such as costs, but also dynamic consumer characteristics, such as their learning speed.
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Our paper is also related to the operations management literature on consumer learning behaviour. Popescu and Wu (2007) consider consumers’ reference price learning and study a firm’s multi-period dynamic pricing strategies with reference price effects. Based on Popescu and Wu (2007), Yuan and Hwarng (2016) further investigate the dynamic pricing issue when customers form reference prices adaptively through social interactions. Liu and van Ryzin (2011) study the firm’s optimal capacity decision when consumers learn about product availability via repeated purchasing experiences. To
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the best of our knowledge, the only paper that considers the relationship between warranty provision and consumer learning is Gallego et al. (2014a). They consider the case of the firm offering a flexible
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warranty policy with a monthly premium and consumers may discontinue the warranty anytime. Their focus is consumers’ decisions to continue or discontinue the warranty coverage and how the firm should design the optimal flexible warranty policy at the beginning of the planning horizon. Our paper differs
Model
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from theirs in that we focus on the case when a firm changes its warranty provision over time.
A monopolist firm sells a product over repeated sales periods in an infinite horizon. At the beginning
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of each period, say period t, the firm sells the product at price pt and offers an optional warranty with price wt to consumers. The product and warranty prices, pt and wt , are the firm’s decisions made at the beginning of period t. The firm’s unit procurement cost of the product is c and the discounted expected warranty service cost on each unit of product is cw . Notice that cw is simply regarded as the discounted expected repair cost that the firm incurs from providing a warranty service for each warranty 2 Indeed, the aforementioned papers Cooper and Ross (1985), Lutz (1989)), and Balachander (2001) have proven that both positive and negative correlations between warranty provisions and product quality are possible.
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sold. Obviously, cw is related to the true product failure rate because a higher warranty service cost is incurred when a product fails and is repaired more frequently. Moreover, we assume that the true product failure rate will not change over time; thus, the warranty service cost cw is also fixed over time. There is a mass of infinitesimal consumers with a deterministic size in the market. We normalize the market size to 1 without loss of generality. Consumers have a homogeneous individual valuation of the product, denoted by v. We assume v − c − cw ≥ 0; otherwise, a warranty should be never offered. together with a warranty.
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In each period, consumers will decide whether to buy the product only or purchase the product If consumers do not purchase the warranty, they will incur some utility loss
when the product fails during the warranty period because they will have to handle the failed product all by themselves. We assume that each consumer i has a different utility loss when the product ¯ fails during the warranty period, denoted by h, follows a uniform distribution on the support of [0, h]. Suppose that in period t, all consumers will share the same beliefs about the product’s failure rate,
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denoted by λt , which may not be the same as the true failure rate of the product. We assume both consumers’ beliefs and the corresponding updating rule are known to the firm and will describe how the consumers update their beliefs λt later. Thus, a consumer with the utility loss of product failure h will incur a cost λt h if he does not purchase the warranty and handles the failed product by himself. However, if the consumer buys the warranty, the firm will take care of product failures for the entire product life free of charge. Moreover, notice in this paper we only consider that repairs both by the firm
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and consumers’ self-handling are “good-as-new”. It implies both the product and the warranty policy remains the same as the new product after each repair. Readers are referred to Blischke and Murthy
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(1992) for a detailed taxonomy of warranty policies. Hence, a consumer obtains an expected surplus, v − pt − min{wt , λt h}, when he purchases the product. We assume that he will buy the product as long
as he gains a nonnegative surplus, i.e., v − pt − min{wt , λt h} ≥ 0. One can easily see that there is a
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threshold value of the individual handling cost, which is equal to
wt λt ,
such that consumers with a utility
loss greater than this value will prefer to purchase the warranty over self-repair, and consumers with a 3
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utility loss smaller than it will not purchase the warranty.
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In industrial practices, consumers may either purchase a warranty after a long period of product usage or be required to purchase a warranty simultaneously with the product within a short time period after buying the product (Jindal 2015). For example Apple changed its policy from buying Applecare anytime to the simultaneous purchasing of Applecare+ with an iPhone, which is now “within 60 days of your iPhone purchase” (see Slivka 2011 and http://www.apple.com/ shop/product/APP_IPHONE_6S_PLUS_AUTO-133580/applecare-for-iphone-6s-and-iphone-6s-plus). In our paper, we assume consumers need to buy the warranty simultaneously with the product, which is commonly assumed in the warranty literature.
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3.1
Consumer Learning Model
Consumers have an initial belief about product failure rate, which is updated over time. Specifically, at the beginning of period t, consumers update their beliefs about the product’s failure rate in period t, denoted by λt , via the following learning equation: λt−1 |{z}
+θ
P rior belief
(αwt + β) | {z }
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Conversion f unction
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λt = (1 − θ)
In (1), θ is consumers’ learning speed, and it measures how fast the consumers adjust their beliefs about the product failure rate with the firm’s warranty price. We use a linear function αwt + β to convert the warranty price into the failure rate of the product. Specifically, α is the conversion factor reflecting the sensitivity of consumers’ beliefs about the product failure rate on the firm’s warranty
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price. As mentioned in the literature, abundant results show that a generous warranty is positively correlated with a lower product failure rate (e.g., Spence 1977, Wiener 1985, Kelley 1988, and Boulding and Kirmani 1993). Therefore, we assume that α > 0, implying that a higher warranty price indicates a higher warranty service cost and thus a higher belief about the product’s failure rate. Literally, β is the converted consumers’ belief of the failure rate when the firm offers a free warranty, and β > 0. Otherwise, the firm can manipulate the consumers’ beliefs about the failure rate to zero by offering a
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free warranty for a sufficiently large number of periods. On the other hand, as revealed in the literature of product quality signalling, some factors, such as advertising, brand reputation, product certification and specialized function, can also influence the consumers’ beliefs about the product failure rate (e.g.,
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Kirmani and Rao 2000, Terlaak and King 2006, and Kalra and Li 2008). Hence, β captures these factors to some extent. We assume that, in our model, β is exogenously given and fixed over time.
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Note that consumers’ beliefs about the failure rate λt are not necessarily directly related to the true failure rate of the product. This is because consumers are not able to observe the product reliability before they make purchases. The firm may manipulate consumers’ beliefs about the failure rate via
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its warranty policy or social networks. Abundant evidence shows that some firms make efforts to manipulate consumers’ beliefs about firms’ product/service, for example, hiring people to post positive
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reviews on online forums, which is largely received negatively by consumers. We also assume that all consumers share homogeneous beliefs about the product failure rate. It
does not necessarily mean that consumers are buying the product repeatedly in every period. Instead, it refers to some market beliefs commonly shared by all the consumers who make purchase decisions. In each period, this belief is updated based on the most updated warranty price. This approach is also used in the literature on consumer learning such as Popescu and Wu (2007), Liu and van Ryzin (2011), and Yuan and Hwarng (2016). 8
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When an individual consumer decides whether to buy a product and/or warranty service in a period, say period t, he will have a prior belief about product failure rate, which is integrated market information carried over from the previous periods. After the warranty price is observed in the current period, the market belief about product failure rate will be updated accordingly via the learning function (1), which becomes the prior belief about the product failure rate for the purchasers in the next period t + 1. Hence, the learning model (1) describes the mechanism of how the market information of consumers’
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beliefs about product failure rate is updated over periods, and consumers do not have to make purchases repeatedly over time to learn the product failure rate.
3.2
Firm’s Profit Function
In period t, given the consumers’ prior belief of the product failure rate λt−1 , which is the state variable of the system, the firm determines the product price pt and the warranty price wt . Consumers then
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instantly update their beliefs about the product failure rate according to (1). For a given state λt−1 and the firm’s product and warranty decision (pt , wt ), the firm’s expected profit gained in period t, denoted by π(pt , wt |λt−1 ), can be expressed as follows:
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wt wt (pt − c) [(1−θ)λt−1 +θ(αwt +β)]h¯ + (pt − c + wt − cw ){1 − [(1−θ)λt−1 +θ(αwt +β)]h¯ }, π(pt , wt |λt−1 ) = , if 0 ≤ wt < w(λ ¯ t−1 ) and pt + wt ≤ v; v−pt (pt − c) min{ , 1} , otherwise. ¯ [(1−θ)λt−1 +θ(αwt +β)]h
(2)
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¯ That is, when the warranty price In (2), w(λ ¯ t−1 ) satisfies w(λ ¯ t−1 ) = [(1−θ)λt−1 +θ(αw(λ ¯ t−1 )+β)]h. ¯ will be indifferent between purchasing the is set at w(λ ¯ t−1 ), consumers with the highest utility loss h
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warranty from the firm and handling the failed product on their own. This implies that the consumers ¯ will always prefer to address product failure by themselves, rather with a utility loss strictly less than h
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than purchase a warranty. One can then easily derive that w(λ ¯ t−1 ) =
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¯ [(1 − θ)λt−1 + θβ]h . ¯ + (1 − θαh)
(3)
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We assume that consumers will not buy a warranty when they are indifferent to the two options (buying a warranty or handling failed products themselves). Furthermore, notice that consumers purchase the product and warranty only when they obtain the nonnegative surplus, that is, v − pt − wt ≥ 0. Hence,
when wt < w(λ ¯ t−1 ) and pt + wt ≤ v, some or all consumers with a relatively high utility loss of product
failure will purchase both the product and warranty service from the firm, whereas the rest will buy 4
In this paper, [∗]+ = max{∗, 0}, and
1 [∗]+
= +∞ if ∗ ≤ 0. Specifically,
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¯ [(1−θ)λ+θβ]h ¯ + (1−θαh)
¯ ≤ 0. = +∞ if 1 − θαh
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only the product. Otherwise, there are no actual sales of the warranty, and consumers will consider only buying the product from the firm and handle the product failure on their own. In each period t, the firm makes decisions on the product and warranty prices. The firm’s profit in period t is then determined by (2). Note that once consumers observe the warranty price, their beliefs about the product failure rate is updated according to the learning function (1). Hence, the learning function links the firm’s warranty decision in two consecutive periods together. When a firm makes
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its warranty decision in each period, it considers not only the impact on the current profits but also the impact on consumers’ future beliefs about product reliability, which in turn affect the firm’s future profit. Let V (λt−1 ) be the firm’s maximum discounted total profit when consumers hold a prior belief of the product failure rate λt−1 . The future profit is discounted by a discount factor of δ per period. The value function V (λt−1 ) for a given state λt−1 satisfies the following Bellman equation:
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max {π(pt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β))} .
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V (λt−1 ) =
pt ≥0,wt ≥0
Model Analysis
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In Section 4.1, we study the properties of a firm’s optimal product and warranty pricing decisions when consumers learn about the product failure rate through warranty prices. We then focus on characterizing
Optimal Product and Warranty Pricing
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4.1
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the steady state in Section 4.2 and conduct comparative statics of the steady state in Section 4.3.
We first present the monotone property of the firm’s discounted profit function regarding the state, which is established in Proposition 1 below.
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Proposition 1. The firm’s discounted expected profit V (λt−1 ) given in (4) is continuously decreasing in λt−1 .
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This result is consistent with our intuition; a firm would prefer consumers to hold a good belief about product reliability (i.e., a lower estimate of product failure rate). This is simply because the firm does not have to offer warranty services at a very low price to improve the belief about product
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reliability held by the consumers. We now investigate the properties of the firm’s optimal warranty pricing policy. Let p∗ (λt−1 ) and
w∗ (λt−1 ) denote the optimal product and warranty prices for any given consumers’ prior belief of the failure rate λt−1 . That is, (p∗ (λt−1 ), w∗ (λt−1 )) is the solution to problem (4): (p∗ (λt−1 ), w∗ (λt−1 )) = arg max {π(pt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β))} . pt ≥0,wt ≥0
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Further, denote λ∗ (λt−1 ) = (1 − θ)λt−1 + θ(αw∗ (λt−1 ) + β), which is the optimal updated consumer
belief for the given state λt−1 . We can show that a firm’s optimal product pricing and warranty pricing
decisions have the following properties: Proposition 2. Give a prior belief of the product failure rate λt−1 , the firm’s optimal product price and warranty price always satisfy p∗ (λt−1 ) + w∗ (λt−1 ) = v and w∗ (λt−1 ) ≤ w(λ ¯ t−1 ).
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Recall that, for a given prior belief λt−1 , when the firm charges the warranty price at w(λ ¯ t−1 ) given in (3), the consumers may buy the product only and no one will purchase the warranty from the firm. When the firm charges the warranty price to strictly smaller than w(λ ¯ t−1 ), there always exists a positive fraction of consumers who will purchase both the product and the warranty service from the firm as long as an appropriate price is charged to induce the purchase of the base product. Furthermore, Proposition 2 implies that the firm is able to fully extract a consumer surplus in this case because
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p∗ (λt−1 ) + w∗ (λt−1 ) = v.
Regardless of whether consumers purchase a warranty service from the firm, the optimal product price and warranty service price always satisfy p∗ (λt−1 ) + w∗ (λt−1 ) = v for any given prior belief about the product failure rate. This result enables us to simplify the expressions of the one-period profit function (2) and the Bellman equation of the firm’s expected profit (4). Specifically, the firm’s profit price, and (4) is reduced to the following: max
0≤wt ≤w(λ ¯ t−1 )
π(v − wt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β)) ,
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V (λt−1 ) =
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optimization problem can be formulated as a function of a single decision about the warranty service
where
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π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt )
(6)
wt ¯ . [(1 − θ)λt−1 + θ(αwt + β)]h
We further find that whenever there are warranty service sales (i.e., some consumers purchase the
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warranty service from the firm), the optimal warranty service price will be smaller than the expect service cost. This result is established in the following proposition.
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Proposition 3. For a given prior belief of product failure rate λt−1 , the firm sets the optimal warranty price such that either w∗ (λt−1 ) ≤ cw or w∗ (λt−1 ) = w(λ ¯ t−1 ). Proposition 3 implies that warranty sales cannot make direct profits for the firm, which is a rather
surprising result. It is believed that a warranty is a highly profitable service, as shown in Padmanabhan and Rao (1993), Lutz and Padmanabhan (1995), and Padmanabhan (1995). However, in their works, a menu of price and warranty bundles are offered, heterogeneous consumers self-select and thus the 11
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market is segmented. In this sense, warranty generates profits because of the price discrimination of warranty service. Nevertheless, in our work, we show that the warranty is not a direct profit-making part when a single warranty price is employed. The result can be explained as follows. An important fact in our problem is that all consumers purchase the product regardless of whether they purchase the warranty service from the firm. Although a higher warranty price increases the profit margin of a warranty service, it reduces the profit margin
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of product sales by the same amount. This is because the optimal price for the product for a given consumers’ belief about the product failure rate λt−1 is equal to v − w∗ (λt−1 ), as shown in Proposition 2. When the warranty price is raised, the product price has to be lowered such that the firm can fully extract the surplus from consumers who purchase both the product and warranty service from the firm. Hence, the firm actually earns a lower profit margin on average for both the product and warranty sales with a higher warranty price as only a fraction of the consumers purchases a warranty whereas all of
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them buy the product. On the other hand, if the firm charges a higher warranty price, fewer consumers will want to buy the warranty service from the firm. The profit on warranty sales alone will also shrink if the firm tries to make money from warranty sales directly by setting the warranty price higher than the corresponding warranty service cost. Then, we can easily show that the firm’s one-period profit decreases in the warranty price once it exceeds the service cost cw . In addition, the future expected profit also decreases in the firm’s warranty price (i.e., V ((1 − θ)λt−1 + θ(αwt + β)) decreases in wt
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based on Proposition 1). All these lead to the result that the firm will set a lower warranty price, even below the service cost, when it is optimal for the firm to induce some consumers to purchase a warranty 5
Although the firm does not make profits from selling the warranty service directly, the higher
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service.
profits from product sales can more than compensate for the cost of warranty sales. In fact, the question of whether the warranty itself generates profit is not so clear in the industry
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due to certain accounting issues. Warranty Week (2014) doubts the profitability of Applecare and Applecare+, especially with its fast growth in warranty expenses. An empirical study by Jindal (2015) also
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notes that a firm should sell its warranty at the marginal cost when the warranty is sold simultaneously with the product. The result indicates that warranty does not generate any extra profit in this case. Our result provides some theoretical support for widely applied warranty services. Although a warranty In fact, w∗ (λt−1 ) ≤ cw is true for any general distribution of consumers’ individual handling cost of product failure h. For a general distribution, let g(·) and G(·) denote the p.d.f. and c.d.f. of h. The one-period profit function is t now formulated as π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt )G( (1−θ)λt−1w+θ(αw ). Notice here pt = v − wt . Then t +β)
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(1−θ)λt−1 +θβ ∂ t t π(v −wt , wt |λt−1 ) = −G( (1−θ)λt−1w+θ(αw )+(cw −wt )g( (1−θ)λt−1w+θ(αw ) 2 . It is easy to check ∂wt t +β) t +β) [(1−θ)λt−1 +θ(αwt +β)] ∂ that ∂wt π(v − wt , wt |λt−1 ) ≤ 0 when wt ≥ cw . Since V ((1 − θ)λt−1 + θ(αwt + β)) is decreasing in wt by Proposition 1 (this result also holds for a general distribution of h), we can conclude that w∗ (λt−1 ) ≤ cw . Moreover, when h follows uniform ¯ as assumed, we can further prove that w∗ (λt−1 ) ≤ cw , which is shown in the proof of Proposition 3 in distribution U[0, h] 2
the appendix.
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is not a direct profit-generating tool for the firm, it helps the firm to obtain a higher profit in total.
4.2
Steady State Analysis
When a firm dynamically changes its product and warranty service prices, consumers adjust their beliefs of product failure rate based on the observed warranty service prices. It is interesting to investigate the long-run behaviours of the firm and consumers. An important question is whether the firm’s pricing
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policy and consumers’ beliefs about the product failure rate converge in the long term. Namely, we would like to see for problem (6), whether there exists a unique state such that the firm’s pricing decisions as well as the consumers’ beliefs stay stable. Before we answer this question, we provide the formal definition of steady state below.
A steady state of problem (6) is defined to be a fixed point of the state updating function:
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f (λ) = (1 − θ)λ + θ(αw∗ (λ) + β) ,
(7)
where w∗ (λ) is the optimal warranty price to problem (4) for a given state λ as defined in (5). Hence, λ is the steady state if f (λ) = λ, which we denote as λ∞ . At the steady state, the updated consumers’ belief based on the firm’s optimal pricing decision remains the same as the prior belief. Moreover, the product price p∗ (λ) and warranty price w∗ (λ) determined by the firm also converge to constants, denoted by p∞ = v − w∞ by Proposition 2.
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p∞ and w∞ , respectively. We can easily show that w∞ =
1 α (λ∞
− β) by the learning equation (1) and
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To show the existence of a steady state, we first analyse the pattern of the state path. For a given
initial belief of the product failure rate, the optimal state path is monotonically increasing. Proposition 4. For a given belief of product failure rate λt−1 , the consumers’ updated belief according
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to the state updating function (7), denoted by λ∗ (λt−1 ), is increasing in λt−1 . Because a bounded monotone path will eventually converge, the result that there exists a steady
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state for problem (6) follows immediately from Proposition 4. ˜ ∞ as the positive Proposition 5. There exists a unique steady state λ∞ for problem (6). Denote λ
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solution to:
− [2 − 2(1 − θ)δ − θ]λ2∞ + (1 − δ)(1 − θ)(αcw + 2β)λ∞ + θβ(αcw + β) = 0.
(8)
β ˜∞, Then λ∞ = min{λ ¯ + }. (1−αh)
˜∞ < When λ
β ¯ +, (1−αh)
the steady state is the positive solution (there is only one positive solution) to ˜ ∞ − β) equation (8). The firm’s stationary warranty and product prices are determined by w∞ = α1 (λ 13
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¯ Hence, consumers with relatively high individual handling and p∞ = v − w∞ . In this case, w∞ < λ∞ h.
cost of product failure, i.e. h ≥
w∞ λ∞ ,
buy both the product and the warranty from the firm, while
consumers with relatively low individual handling cost of product failure, i.e. h <
w∞ λ∞ ,
buy the product
only. In the extreme case that cw = 0, the consumers’ beliefs converges to λ∞ = β and the firm sells the β ˜∞ ≥ product at price p∞ = v and offers a free warranty w∞ = 0 in the steady state. When λ ¯ +, (1−αh)
In this instance, the consumers’ beliefs converges to λ∞ = (1−αβ h) ¯ + ¯ and the firm chooses a relatively expensive warranty price w∞ = λ∞ h and sells the product at price β ¯ +. (1−αh)
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the steady state is equal to
p∞ = v − w∞ such that no consumer purchases the warranty from the firm in the long run.
Furthermore, notice the steady state consumers’ belief of product failure rate is independent of
the initial belief. Based on Proposition 4, starting from any initial belief of the product failure rate, consumers’ beliefs converge monotonically to the same steady state as specified in Proposition 5. This indicates if the initial product failure rate belief is low (below λ∞ ), the firm should gradually increase
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warranty prices to increase the beliefs; if the initial belief is high (above λ∞ ), the firm should gradually decrease warranty prices to reduce the beliefs; or if the initial belief equals to the steady state belief, the firm should keep the warranty price at w∞ such that the belief remains at λ∞ . In all the three cases, consumers’ beliefs, warranty prices, and product prices all eventually converge to λ∞ , w∞ , and
4.3
6
Figure 1 provides an example to demonstrate the above results.
Comparative Statics
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p∞ , respectively, in the long run.
In this section, we conduct comparative statics analysis of the steady state. We examine how the firm’s
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stationary policy of product price and warranty price and consumers’ long-run beliefs about the product failure rate change with important system parameters, such as service cost, consumers’ learning speed,
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and the heterogeneity of consumers’ individual handling costs. ˜ ∞ when cw ≤ c¯w Proposition 6. There exists a threshold warranty service cost c¯w such that λ∞ = λ β ¯ + (1−αh)
otherwise.
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and λ∞ =
The result is intuitive. When the warranty service cost gets higher, the firm will increase the warranty price in order to reduce the loss from warranty sales. The higher the warranty service cost,
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the higher the warranty price, leading to a smaller fraction of consumers purchases the warranty. Indeed, consumers’ beliefs about the product failure rate increase in warranty service cost. Once the service cost is high enough, though the firm offers warranty services, no consumer will purchase the warranty as it is too costly. As a result, all the consumers only purchase the products. We find that the threshold value ¯ and δ, while it is independent of v and c. In other words, when consumers c¯w is increasing in θ, α, β, h, 6
Notice the expression of λ∞ is given in Proposition 5; the expressions of w∞ and p∞ are given below (7).
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0.1
0.06
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λt
0.08
0.04
0.02 0
5
10
15
20
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t
¯ = 0.2, θ = 0.3, α = 1, β = 0.05, Figure 1: Optimal belief updating paths for v = 1, c = 0.2, cw = 0.01, h and δ = 0.9. adjust their beliefs about the product failure rate more quickly with the recently observed warranty (θ
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increases), and/or consumers are more sensitive to the firm’s warranty price (α increases), it becomes less costly for the firm to influence consumers’ beliefs via its warranty price decisions. Therefore, the firm has more incentive to offer a relatively modest warranty price to induce purchases of warranty
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services.
The long-run profit of the firm decreases in warranty service cost. The profit from warranty sales in the long run, determined by (w∞ − cw )
¯ w∞ )+ (h− λ∞ , ¯ h
is first decreasing then increasing in cw . The result is a
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bit surprising at the first sight as one may expect that it decreases with cw . The reason is the following: a higher service cost cw reduces the warranty sales, which is
¯ w∞ )+ (h− λ∞ . ¯ h
Meanwhile, it also reduces the
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“profit margin” w∞ − cw of selling warranty services. Note that warranty sales do not generate any
profit in our setting, it leads to profit loss to the contrary. Therefore, a higher service cost increases
the direct profit loss from warranty sales. When the warranty service cost cw is relatively small, the
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direct profit loss from warranty sales is also small. As cw increases, the direct profit loss in warranty sales increases as well. However, when cw becomes relatively large, the demand for warranty becomes small. It diminishes to almost zero when the service cost is close to the threshold. Because a higher cw significantly reduces warranty sales, it brings down the direct profit loss in warranty sales. Therefore, the profit from warranty sales decreases and then increases in cw . 7
7
A detailed result and the proof of the comparative studies on cw can be found in Proposition 15 and the following
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˜ ∞ when θ > θ¯ and λ∞ = Proposition 7. There exists a threshold value θ¯ such that λ∞ = λ
β ¯ + (1−αh)
otherwise. Learning speed θ measures how fast the consumers update their beliefs about the product failure rate according to warranty prices. Proposition 7 implies that when consumers adjust their beliefs about the product failure rate faster with the recent warranty price, the firm will offer a modest warranty price such that some consumers will purchase both the product and warranty. As the learning speed
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becomes faster, the firm incurs less cost to change consumers’ beliefs via the warranty pricing policies. We can show that the stationary warranty price and consumers’ beliefs decrease with the learning speed in the long term. As expected, the firm’s long-run profit increases with the learning speed; that is, the firm benefits from fast-learning consumers. The profit loss from warranty sales increases with the learning speed, which, however, can be more than offset by the increased profit from product sales. The overall profit turns out to be increasing in the learning speed. Proposition 7 also explains that when
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the learning speed of consumers is below some certain level, it is too costly for the firm to use a lower warranty price to improve consumers’ beliefs. Under this circumstance, the firm would rather choose a relatively expensive warranty price w∞ =
¯ βh ¯ + (1−αh)
such that there are no warranty sales at all.
¯ u such that λ∞ = λ ˜ ∞ when h ¯>h ¯ u and λ∞ = Proposition 8. There exists a threshold value h otherwise.
β ¯ + (1−αh)
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¯ βh ¯ u such that the firm chooses a relatively expensive warranty price w∞ = There exists a h ¯ + (1−αh) ¯ ¯ where no actual warranty sales exist in the steady state when h ≤ hu and chooses an interior warranty ˜ ∞ −β λ α
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¯ >h ¯ u . Furthermore both the steady state where warranty has actual sales when h ¯ The total profit V (λ∞ ), per period consumers’ belief λ∞ and warranty price w∞ are increasing in h.
price w∞ =
¯ w∞ )+ (h− λ∞ ¯ h
are all
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profit on product sales v − c − w∞ , and per period profit on warranty sales (w∞ − cw ) ¯ decreasing in h.
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Note that the consumers’ individual handling costs of product failure are assumed uniformly dis¯ Thus, h ¯ reflects the average value and heterogeneity of handling costs if consumers tributed on [0, h]. deal with the failed products on their own. The results suggest that when consumers incur highly heterogenous handling costs for failed products, the firm should always induce some consumers to purchase
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the warranty service. As such heterogeneity increases, the stationary warranty price increases as well. Since a higher warranty price reduces the profit margin of the product sales, the firm will earn a smaller ¯ increases. According to Proposition 8, the firm should charge a relatively profit in the long run when h ¯ expensive warranty price once consumers do not differ much in their individual handling costs (i.e., h ¯ u ). When this is the case, no one purchases the warranty. is smaller than h proof in the appendix. Besides, all the proofs of the detailed comparative studies for the other parameters in the following Proposition 7 to 9 are included in the corresponding proofs in the appendix.
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˜ ∞ when α > α Proposition 9. There exists a threshold value α ¯ such that λ∞ = λ ¯ and λ∞ =
β ¯ + (1−αh)
otherwise. The parameter α measures how sensitive the consumers are to the warranty price when they update their beliefs of product failure rate. Intuitively, all else being equal, a larger α results in a higher belief of failure rate, thus a higher demand for warranties and a greater loss in warranty sales. Finally, there
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will be a lower profit in the steady state. An interesting result here is that the firm’s warranty price is not monotone decreasing in α. When α is small enough such that α ≤ α ¯ , consumers’ beliefs of the product failure rate are affected by the current warranty price at a very low degree. This means it is more important for the firm to choose a relatively expensive warranty price w∞ =
¯ βh ¯ + (1−αh)
to cover the
service cost because consumers’ beliefs will not be worsen too much when consumers are insensitive to warranty price decisions. This explains why the warranty price increases with α. However, as α becomes
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larger and reaches α > α ¯ , consumers are more sensitive to the warranty price changes when they update their beliefs. The firm has to offer a lower warranty price to reduce consumers’ beliefs. We can also show that the direct profit loss in warranty sales is always increasing in α as both the unit profit loss in warranty sales and the demand for warranties are increasing in α as long as some consumers purchase warranties from the firm.
Comparison to Full Information about Reliability
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5
We have studied how consumers’ behaviour and a firm’s pricing policies are affected by the learning
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mechanism in the previous sections. In this section, we are going to study the impact of consumer learning by comparing it with the case of when consumers know the true reliability. All else is the same as before. The only difference is that the firm discloses the true reliability
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information and consumers have full information about the true failure rate λa . In this case, the consumer’s surplus of purchasing is v − p − min{w, λa h}. Notice because there is no consumer learning,
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both the product price p and warranty price w are simply chosen to maximize the firm’s one period
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profit; thus we eliminate the subscript index t. Then, the firm’s problem is formulated as follows: πF = max (p + w − c − cw ) + (cw − w) min{ p≥0,w≥0
w ¯ , 1} . λa h
(9)
¯ The optimal product price is It is easy to derive the optimal warranty price: wF = min{ c2w , λa h}.
pF = v − wF . Correspondingly, the associated one period profit is πF = v − c − cw + (cw − wF ) 17
wF ¯ . λa h
(10)
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Recall that the one-period profit in the steady state in the consumer learning case is determined by π∞ = v − c − cw + (cw − w∞ ) where w∞ =
w∞ ¯ , λ∞ h
(11)
λ∞ −β α .
Notice that the discounted expected warranty service cost cw is simply regarded as the discounted
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expected repair cost per unit of warranty sold as introduced in Section 3. Naturally, cw depends on the product true failure rate λa . Hence hereafter, we define cw as a function of λa , which is cw (λa ). We further assume that cw (λa ) is proportional to λa . This can be explained in the following.
Each time when the product fails and is fixed based on the warranty provision, the firm incurs a fixed repairing cost c0 . As we have assumed the repair is “good-as-new” in Section 3, the repaired product is returned to the consumer like a new one for free. Thus, we reasonably assume the product
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failure follows a Poisson process with a failure rate λa , where the memoryless feature of the Poisson process captures the “good-as-new” property of the repair. In addition, recall that the warranty covers the entire product life once it is purchased. As δ is the firm’s profit discount factor for each discrete ˜ period, we denote δ˜ = − ln δ as the continuous time discount factor, which satisfies δ = e−δ·1 . Suppose
the product fails for the first time after x units of time of purchase. Then, the firm spends c0 to fix it right away. As product failure follows a Poisson process which is memoryless, the firm still bears ˜
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the same future repairing cost cw (λa ) after the first product failure. The firm’s total repairing cost is ˜
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e−δx [cw (λa ) + c0 ], where e−δx is the corresponding discount factor. Conditioning on the time to the R∞ ˜ first product failure from the purchase, we obtain cw (λa ) = 0 e−δx [cw (λa ) + c0 ]λa e−λa x dx. Based on this equation, we obtain cw (λa ) =
c0 λ . δ˜ a
This model of warranty service cost is also used in Karmarkar
(1978). Readers can refer to Blischke (1990), Murthy and Blischke (1992b), and Murthy and Djamaludin function λa (cw ) =
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(2002) for more general discussions on warranty service cost evaluation. In addition, we use the inverse δ˜ c0 cw
of cw (λa ) to represent the correspondence of λa to cw .
Based on (10) and (11), we have the following proposition to compare the average profits with and
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without consumer learning.
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˜ a such that π∞ ≥ πF if and only if λa ≥ λ ˜a. Proposition 10. There exists a λ Proposition 10 suggests that it is better for the firm to hide the information about the true product
reliability and induce consumers to learn when λa (or cw equivalently) is relatively large. Oppositely, it is better for the firm to disclose the true product reliability to consumers and prevent consumers from learning. The basic logic is that the firm would like to hide the true reliability and manipulate consumers’ beliefs about the product reliability through learning only when the firm is efficient enough to induce 18
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Failure rate λa ˜a λ
•
0
Repairing cost cw
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λ∞
Figure 2: Illustration of Product Failure Rate v.s. Repairing Cost
a relatively low belief about the product’s failure rate, compared with the case of when true product
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reliability is disclosed.
As illustrated in Figure 2, the solid and dashed lines represent the steady state consumer belief about product failure rate with learning λ∞ and true failure rate λa with respect to warranty service cost cw respectively. The dash-dotted line represents the corresponding true failure rate such that hiding and disclosing the information of true product reliability yield the same profit for any given warranty service cost cw . According to Proposition 5, λ∞ is concave increasing in cw . That is, the marginal increment of
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consumers’ beliefs about the product failure rate is decreasing in cw with learning, while the marginal increment of true failure rate λa is constant in cw when true reliability is disclosed.
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According to (10) and (11), when true reliability is disclosed, the consumers’ perceptions of the product failure rate λa are independent of the firm’s decision. The firm simply chooses the warranty ¯ to maximize its profit. However, when true reliability is not known by the price wF = min{ cw , λa h} 2
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consumers, the consumers’ perceived steady state product failure rate through learning λ∞ is determined by the firm’s warranty decision w∞ . In other words, the warranty price with true reliability disclosed
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wF directly optimizes the firm’s profit, while the warranty price with learning w∞ cannot be freely chosen due to the additional constraint on warranty decision with learning (i.e., w∞ =
λ∞ −β α ).
Therefore, when warranty service cost cw (or equivalently true failure rate λa ) is small, the manipu-
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lated belief about product failure rate through learning λ∞ is relatively large as demonstrated in Figure 2. Specifically, the firm is worse off by hiding the true reliability (i.e., π∞ ≤ πF ) when λ∞ ≥ λa , because w∞ cannot be chosen without any constraint based on the above argument. Nevertheless, λ∞ < λa
doesn’t always guarantee that hiding the information about the true reliability is more beneficial for the ˜ a , the firm firm. As demonstrated in Figure 2, when λ∞ is only slightly lower than λa , i.e. λ∞ < λa < λ still cannot benefit from hiding the information about the true reliability. The firm may only achieve
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a higher profit by hiding the information about the true reliability when the perceived belief about product failure rate through learning λ∞ is sufficiently low compared with the intrinsic true failure rate λa , which only happens when the true failure rate λa (or cw equivalently) is relatively high. Hence, ˜ a as described in the firm benefits from hiding the information about the true reliability when λa ≥ λ Proposition 10.
˜ a changes with the system parameters. The following proposition describes how λ
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˜ a is decreasing in θ; and increasing in α, β, and h. ¯ Proposition 11. λ
˜ a means that it is better for the firm to disclose the true reliability rather than hide it and Larger λ let the consumers adaptively learn through the warranty prices. For any fixed true failure rate λa (or any fixed cw equivalently), the firm would prefer to induce learning for a relatively small steady state belief λ∞ and disclose the true reliability for a relatively large λ∞ .
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Based on the comparative statics in Section 4.3, when the learning speed θ increases, it is less costly for the firm to manipulate consumers’ beliefs via the warranty prices, namely the steady state belief λ∞ is decreasing in learning speed θ. Thus, hiding the true reliability and inducing consumers to learn become more beneficial for the firm compared with disclosing when the learning speed increases. ¯ characterizes both the average cost and heterogeneity A consumer’s highest value of handling cost h of consumer individual handling. When consumers bear higher individual handling cost on average,
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the firm offers a higher warranty price because consumers are more willing to buy warranties when handling by themselves becomes more costly. This eventually results in a higher steady state belief of
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product failure rate when true reliability is hidden. In this case, the firm would rather disclose the true ¯ becomes larger. reliability when h Lastly, both the conversion factors α and β measure the degree to which the consumers take warranty
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price into account when they update their beliefs through learning. As larger α or β results in a higher belief of product failure rate for any given warranty price, the firm is better to disclose the true reliability
Extension: Uncertainty in Consumers’ Belief Updating
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6
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when α or β is large.
In our basic model, we assume that consumers’ belief updating process about product failure rate, as given by equation (1), is deterministic. In reality, there are stochastic factors that can affect consumers’ beliefs on product reliability. For example, a firm may face uncertainties in online reviews, which also affect consumers’ perception on reliability and their purchasing decisions. We model this type of effect, on an abstract level, by a stochastic factor Xt , t = 1, 2, · · · . We assume that {Xt } is a Markov chain
and independent of the firm’s product price pt , warranty price wt , and consumers’ beliefs about the 20
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product failure rate λt−1 . For a fair comparison with the basic model, suppose that E(Xt ) = β and V ar(Xt ) = σ 2 for all t. At the beginning of each period t, consumers observe the realized value of the random factor Xt , denoted by xt , and Xt+1 stochastically increases in xt . In period t, when consumers observe the realized value xt , they will update their beliefs about the product failure rate in the following: (12)
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λt = (1 − θ)λt−1 + θ(αwt + xt ).
In (12), xt can be interpreted as some other information that reveals the true product failure rate directly, such as consumer reviews. Note that consumers update their beliefs based on not only the warranty service price but also additional information sources, such as consumer reviews. Hence, the state variable in each period now becomes (λt−1 , xt ). Accordingly, the firm’s single-period profit function
π R (pt , wt |λt−1 , xt )
wt (pt − c) [(1−θ)λt−1 +θ(αwt +xt )]h¯ + (pt − c + wt − cw ){1 − , if 0 ≤ wt < w ¯ R (λt−1 , xt ) and pt + wt ≤ v; v−pt (pt − c) min{ ¯ , 1}, otherwise. [(1−θ)λ +θ(αw +x )]h t−1
In (13) above,
w ¯ R (λ
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under uncertainty in the updating of consumers’ beliefs, denoted by π R (pt , wt |λt−1 , xt ), is as follows:
t−1 , xt )
=
¯ [(1−θ)λ+θxt ]h ¯ + . (1−θαh)
t
wt ¯} [(1−θ)λt−1 +θ(αwt +xt )]h
t
When the firm’s warranty price is equal to
(13) w ¯ R (λ
t−1 , xt ),
the
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¯ are indifferent between purchasing the warranty from the firm consumers with the highest utility loss h and handling failed products themselves. Again, we can see that no consumer will buy the warranty in
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this case because consumers with lower utility loss prefer individual handling even more. We use V R (λt−1 , xt ) to denote the firm’s total discounted profit when consumers have a prior belief about the product failure rate λt−1 and observe the realized value generated from other information
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sources xt . Then it satisfies the following Bellman equation:
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V R (λt−1 , xt ) =
max
{π R (pt , wt |λt−1 , xt ) + δE[V (λt , Xt+1 )]}
s.t.
λt = (1 − θ)λt−1 + θ(αwt + xt ) .
pt ≥0,wt ≥0
(14)
Similar to the basic model, the total discounted profit decreases in the state variables, which is
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established in the following proposition. Proposition 12. The firm’s total discounted profit V R (λt−1 , xt ) given in (14) is continuously decreasing in λt−1 and xt .
Let pR (λt−1 , xt ) and wR (λt−1 , xt ) be the optimal product and warranty prices for any given consumers’ prior belief of product failure rate λt−1 and a realized random factor xt . That is, 21
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(pR (λt−1 , xt ), wR (λt−1 , xt )) is the solution to problem (14). Further, denote λR (λt−1 , xt ) = (1−θ)λt−1 + θ(αwR (λt−1 , xt ) + xt ), which is the updated consumers’ belief for the given state (λt−1 , xt ) under the optimal warranty price wR (λt−1 , xt ). The firm’s optimal warranty pricing decision satisfies the similar property as in the basic model. Proposition 13. For a given state (λt−1 , xt ), the firm’s optimal warranty price wR (λt−1 , xt ) is either
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smaller than the expected warranty service cost cw or equal to w ¯ R (λt−1 , xt ). When wR (λt−1 , xt ) ≤ cw , there are always some consumers purchasing both the product and war-
ranty. When wR (λt−1 , xt ) = w ¯ R (λt−1 , xt ), no one will purchase the warranty from the firm. Moreover, the optimal product price pR (λt−1 , xt ) is always equal to v − wR (λt−1 , xt ). Similar to Proposition 4, we can show that the updated belief under the optimal warranty decision is increasing in consumers’ prior
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belief of product failure rate and the realized value of the random factor.
Proposition 14. If {Xt }s are independent and identically distributed (i.i.d.), the consumers’ updated belief of product failure rate λR (λt−1 , xt ) associated with the firm’s optimal warranty price is increasing in λt−1 and xt .
Notice that in Proposition 14, an additional i.i.d. assumption is made on {Xt }s. This is required
to show the joint supermodularity of the firm’s profit function in state variables (λt−1 , xt ), which is
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sufficient to show that λR (λt−1 , xt ) increases in λt−1 and xt . Nevertheless, our numerical studies will show that this result still holds when the random factor {Xt }s are not i.i.d. and affected by the past
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realized values.
We numerically study how the firm’s optimal warranty policy changes with system parameters.
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In numerical examples, the basic setup of parameters is as follows: v = {1, 2}, θ = {0.1, 0.3, ...0.9}, ¯ = {0.1, 0.2, 0.3}, c = {0.1, 0.2}, cw = {0.01, 0.02, 0.03}, α = {0.5, 1.0, 1.5}, β = {0.05, 0.10, 0.15}, h
and δ = {0.8, 0.9}. Suppose the {Xt }s are modelled as an order 1 autoregressive model; that is,
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Xt = (1 − η)Xt−1 + ηt with 0 < η ≤ 1 where {t }s are i.i.d. with E(t ) = β and V ar(t ) =
2−η 2 η σ .
Note
that {Xt }s are i.i.d. when η = 1. We set σ = {0.015, 0.030} and η = {0.3, 0.7, 1}. The distributions of t we have tried are uniform, exponential and truncated normal.
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We use a standard value iteration approach to numerically solve the basic model problem (with a
deterministic β) in (4) and the problem with uncertainty in consumers’ belief updating given in (14), respectively. We find that all the results regarding comparative statics hold as well when there is uncertainty in consumers’ belief updating. Indeed, we obtain all the results similar to problem (4).
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7
Conclusion
In this paper, we consider the problem of a firm dynamically pricing a product and its warranty service in a multi-period setting when the warranty price affects consumers’ beliefs about product reliability. We show that the warranty should not be priced higher than its marginal cost, which is consistent with the findings of some empirical studies. In the long run, both a firm’s optimal warranty price
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and consumers’ beliefs about product failure rate converge. In particular, when the firm’s warranty service cost is higher, consumers’ learning speed is slower, or heterogeneity in consumers’ handling cost in the case of product failure is lower, the long-run warranty price will be higher and long-run consumers’ perception of product reliability will be poorer. Moreover, it is better for the firm to hide the information about true product reliability and let the consumers learn about it adaptively when the product intrinsically has a relatively high failure rate.
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In this paper, we consider a model with stationary parameters. In other words, we assume that, other than the warranty price, other factors that affect consumers’ beliefs of product reliability are time independent. In reality, these other factors may also change over time. This can be captured by using non-stationary rather than stationary parameters in the learning function. While this will complicate the analysis of the optimal policy, we expect that the main insights and the structure of the long-run optimal policy will remain the same.
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We also assume that all consumers hold the same beliefs about product failure rate. While this assumption is reasonable when consumers for a product come from the same market segment, it may
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not be valid in cases where consumers come from very diverse backgrounds. In those cases, consumers from different segments may have different initial beliefs, learning speed, and sensitivities to warranty prices. This may require a different learning function for each individual market segment; hence a state
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variable is needed to capture customers’ beliefs in each segment. The increase in the dimension of the state space will significantly increase the challenge. Thus, whether the dimension of state space can be reduced is by itself a challenging question and deserves a separate study.
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A monopolistic setting is a good starting point to consider the impact of consumer learning on dynamic product and warranty pricing, but another interesting extension is to consider the case of
AC
when competition is involved. In this case, a firm’s warranty price will affect customers’ beliefs about the reliability of competitive products. In addition to increasing the state space of the problem, this will also lead to a dynamic game that may not be tractable under existing dynamic programming tools. The first challenging question for this extension is whether a subgame perfect equilibrium exists.
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Acknowledgments The authors thank two anonymous reviewers for their comments and suggestions that helped to improve the paper.
Proof of Proposition 1. Proposition 1 will be proved by induction.
CR IP T
Appendix: Proofs First, let’s look at the one period problem, which is formulated as V1 (λt−1 ) =
max
pt ≥0,wt ≥0
where π(pt , wt |λt−1 ) is defined in (2).
π(pt , wt |λt−1 ) ,
¯
(A.1)
AN US
t−1 +θβ]h As defined below (2), w(λ ¯ t−1 ) = [(1−θ)λ , which is the warranty price satisfies w(λ ¯ t−1 ) = ¯ + (1−θαh) ¯ It implies that w(λ [(1 − θ)λt−1 + θ(αw(λ ¯ t−1 ) + β)]h. ¯ t−1 ) is the highest warranty price such that
consumers possibly take buying warranty into account, instead of all consumers handling failed products on their own.
Therefore, let’s first consider the simpler case when wt ≥ w(λ ¯ t−1 ) such that consumers may only buy
the base product and warranty is never considered as an option, which is formulated in the second case
M
in (2). In this case, the firm’s one period profit is π(pt , wt |λt−1 ) = (pt − c) min{ [(1−θ)λ
when wt ≥ w(λ ¯ t−1 ) by (2). It is easy to see
t (pt − c) min{ [(1−θ)λ v−p ¯ , 1} t−1 +θ(αwt +β)]h
v−pt
¯ , 1}
t−1 +θ(αwt +β)]h
is decreasing in wt for
ED
any given pt as higher wt increases consumers’ beliefs about the product failure rate and brings down the demand for product. Thus, wt > w(λ ¯ t−1 ) is never optimal and the optimal warranty price for the case of when wt ≥ w(λ ¯ t−1 ) is achieved on wt = w(λ ¯ t−1 ).
PT
As the firm’s profit function in (2) is continuous in wt at wt = w(λ ¯ t−1 ) for any given pt , we only
need to further consider the case of 0 ≤ wt ≤ w(λ ¯ t−1 ).
CE
Now let’s consider the case of when 0 ≤ wt ≤ w(λ ¯ t−1 ). In this instance, consumers with high losses
in product failures h buy the product and warranty together if the total cost is acceptable, which is pt + wt ≤ v as formulated in the first case in (2).
AC
profit is π(pt , wt |λt−1 ) =
wt (pt − c) [(1−θ)λ +θ(αw ¯ t +β)]h t−1
0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt ≤ v. It is easy to see wt ¯} [(1−θ)λt−1 +θ(αwt +β)]h
8
Under this circumstance, the firm’s one period
+ (pt − c + wt − cw ){1 − [(1−θ)λ
wt (pt − c) [(1−θ)λ +θ(αw ¯ t +β)]h t−1
wt
¯}
t−1 +θ(αwt +β)]h
when
+ (pt − c + wt − cw ){1 −
is increasing in pt for any given wt as higher pt increases the unit profit for product
sales while maintaining the product and warranty sales segmentation. Thus, the firm’s optimal decision 8 Here we emphasise that there are still no warranty sales when wt = w(λ ¯ t−1 ) even with pt + wt ≤ v. In the following we do not explain the case of wt = w(λ ¯ t−1 ) when warranty has no sales separately and wt ≤ w(λ ¯ t−1 ) is always interpreted as having warranty sales as long as pt + wt ≤ v.
24
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should be on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt . Notice in this case, warranty
has actually sales and the corresponding boundary profit is (v − c − cw ) + (cw − wt ) [(1−θ)λ
wt
¯.
t−1 +θ(αwt +β)]h
However, even with a relatively low warranty price 0 ≤ wt ≤ w(λ ¯ t−1 ), consumers do not buy warranties
if the total cost exceeds the total valuation, which is pt + wt > v as formulated in the second case in (2). They may only buy the product for an acceptable product price. In this case, the firm’s one period Again, as (pt −
v−pt
¯ , 1}
t−1 +θ(αwt +β)]h
t c) min{ [(1−θ)λ v−p ¯ , 1} t−1 +θ(αwt +β)]h
when 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt > v.
is decreasing in wt for any given pt , the firm’s optimal
CR IP T
profit is π(pt , wt |λt−1 ) = (pt − c) min{ [(1−θ)λ
decision should still be on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt + , where is
some infinitely small positive value.
9
Notice in this case, there are no actually sales of warranty and the
t + corresponding boundary profit is (v − c − wt + ) min{ [(1−θ)λv−c−w +θ(αw t−1
¯ , 1}.
t +β)]h
Then the limited profit
on the boundary 0 ≤ wt < min{w(λ ¯ t−1 ), v} and pt = v − wt from the region of 0 ≤ wt ≤ w(λ ¯ t−1 ) and
t + pt +wt > v is lim (v−c−wt +) min{ [(1−θ)λv−c−w +θ(αw
→0+
t−1
¯ , 1}
t +β)]h
t = (v−c−wt ) min{ [(1−θ)λ v−c−w +θ(αw t−1
¯ , 1},
t +β)]h
AN US
as is infinitely small and positive. We need to mention that π(pt , wt |λt−1 ) is discontinuous on 0 ≤
wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt as it separates the cases of having warranty sales when 0 ≤
wt ≤ w(λ ¯ t−1 ) and pt + wt ≤ v and not having warranty sales when 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt > v.
10
As mentioned above, the optimal decisions for both sub-cases are on the boundary of
0 ≤ wt < min{w(λ ¯ t−1 ), v} and pt = v − wt and the corresponding one period profits are (v − c − cw ) + wt
¯
t−1 +θ(αwt +β)]h
t and (v − c − wt ) min{ [(1−θ)λ v−c−w +θ(αw
(v − c − cw ) + (cw − wt ) [(1−θ)λ
t−1
wt
¯ , 1}
t +β)]h
respectively. Furthermore,
t is larger than (v − c − wt ) min{ [(1−θ)λ v−c−w +θ(αw
M
(cw − wt ) [(1−θ)λ
¯
t−1 +θ(αwt +β)]h
t−1
¯ , 1}
t +β)]h
for
any given warranty price 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v}, as we have assumed v > c + cw and considering
ED
selling warranty makes extra profit. It means the firm’s optimal choice could only be achieved on 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt , which is the boundary of considering selling warranty with choosing 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt ≤ v as illustrated in the first case in (2).
PT
Therefore, the one period problem can be simplified as
CE
V1 (λt−1 ) =
max
0≤wt ≤min{w(λ ¯ t−1 ),v}
π(v − wt , wt |λt−1 ) ,
where π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt ) [(1−θ)λ
wt
¯.
t−1 +θ(αwt +β)]h
(A.2)
Notice here warranty only has
AC
actual sales when wt < min{w(λ ¯ t−1 ), v} and still has no actual sales when wt = min{w(λ ¯ t−1 ), v}. First we calculate the first and second order partial derivatives with respect to wt for the profit
9
Notice here wt ≤ v. Suppose the firm chooses warranty price wt > v, then pt + wt > v and consumers do not buy warranty. Then the firm’s one period profit is decreasing in wt based on the second case in (2). Thus, it is never optimal to choose wt > v and the firm only considers wt ≤ v. 10 Notice π(pt , wt |λt−1 ) is continuous at wt = min{w(λ ¯ t−1 ), v} and pt = v − wt as there are no warranty sales for both cases.
25
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function of the one period problem, which are 1 ∂ wt (1 − θ)λt−1 + θβ π(v −wt , wt |λt−1 ) = − +(cw −wt ) (A.3) 2 ¯ ¯ ∂wt [(1 − θ)λt−1 + θ(αwt + β)]h [(1 − θ)λt−1 + θ(αwt + β)] h and
2[(1 − θ)λt−1 + θβ][(1 − θ)λt−1 + θ(αcw + β)] 1 ∂2 π(v − wt , wt |λt−1 ) = − ¯ . ∂wt2 h [(1 − θ)λt−1 + θ(αwt + β)]3
CR IP T
By (A.4), π(v − wt , wt |λt−1 ) is concave in wt . Let
(A.4)
p [(1 − θ)λt−1 + θβ] · [(1 − θ)λt−1 + θ(αcw + β)] − [(1 − θ)λt−1 + θβ] w (λt−1 ) = θα 0
(A.5)
be solution to (A.3), which is constrained on 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v}. Based on (A.3), it is easy to see
∂ ∂wt π(v
− wt , wt |λt−1 ) ≤ 0 when wt > cw . Therefore we have the following claim.
AN US
Claim. w0 (λt−1 ) ≤ cw .
The claim above is straightforward by (A.3). Moreover, w∗ (λt−1 ) ≤ cw is true for any general distri-
butions on h. For any general distribution, let g(·) and G(·) denote the p.d.f. and c.d.f. of consumers’
individual handling cost of product failure respectively. The one period profit function is formulated t as π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt )G( (1−θ)λt−1w+θ(αw ). Notice here pt = v − wt . Then t +β)
(1−θ)λt−1 +θβ t t − wt , wt |λt−1 ) = −G( (1−θ)λt−1w+θ(αw ) + (cw − wt )g( (1−θ)λt−1w+θ(αw ) . t +β) t +β) [(1−θ)λ +θ(αw +β)]2
M
∂ ∂wt π(v
∂ ∂wt π(v
t−1
t
− wt , wt |λt−1 ) ≤ 0 when wt ≥ cw . Thus, it is easy to have ¯ especially with lower bound 0 w0 (λt−1 ) ≤ cw . Furthermore, for h following uniform distribution U[0, h]
Based on this, it is easy to see
ED
as assumed, we can further prove that w0 (λt−1 ) ≤
cw 2
by the formulation of w0 (λt−1 ) in (A.5).
w0 (λt−1 ) is the solution to the one period problem only when it is an interior solution. It is easy to see ¯
2
¯ ¯
PT
(1−θαh) θβ(2−θαh)h there exists a λE = max{ (1−θ)(2−θα ¯ h ¯ [cw − (1−θαh) ¯ 2 ], 0}, such that h)
if and only if λt−1 < λE . Besides, as it is always wt , wt |λt−1 )|wt =v < 0, it indicates 0 ≤
w0 (λt−1 )
∂ ∂wt π(v
−
∂ ¯ t−1 ) > 0 ∂wt π(v−wt , wt |λt−1 )|wt =w(λ ∂ wt , wt |λt−1 )|wt =0 > 0 and ∂w π(v − t
≤ v is always satisfied. Thus an interior solution
CE
w0 (λt−1 ) exists, or equivalently w0 (λt−1 ) < w(λ ¯ t−1 ) if and only if λt−1 > λE . By the concavity of
AC
π(v − wt , wt |λt−1 ), the optimal decision of warranty price for the one period problem is w1∗ (λt−1 ) =
(
w(λ ¯ t−1 )
, if λt−1 ≤ λE
w0 (λt−1 ) , if λt−1 > λE
,
(A.6)
and the corresponding product price is p∗1 (λt−1 ) = v − w1∗ (λt−1 ). Furthermore, we can show that
w(λ ¯ t−1 ) ≤ v when λt−1 ≤ λE and w0 (λt−1 ) ≤ v when λt−1 > λE . Thus, the constraint wt ≤ v is actually redundant for the one period problem. Besides as w1∗ (λt−1 ) is also equivalently defined as
26
ACCEPTED MANUSCRIPT
w1∗ (λt−1 ) = min{w(λ ¯ t−1 ), w0 (λt−1 )}, it is easy to show w1∗ (λt−1 ) ≤ cw based on the claim above. Plus
the corresponding one period profit is V1 (λt−1 ) =
(
v − c − w(λ ¯ t−1 )
(v − c − cw ) + [cw −
w0 (λt−1 ) w0 (λt−1 )] [(1−θ)λ +θ(αw 0 (λ ¯ t−1 t−1 )+β)]h
, if λt−1 ≤ λE
, if λt−1 > λE
.
(A.7)
In (A.7), first it is easy to see when λt−1 ≤ λE , v − c − w(λ ¯ t−1 ) is decreasing in λt−1 . When
=
CR IP T
λt−1 > λE , as
wt ∂ ¯} ∂λt−1 {(v − c − cw ) + (cw − wt ) [(1−θ)λt−1 +θ(αwt +β)]h cw −wt wt ∂ ( ∂λt−1 (1−θ)λt−1 +θ(αwt +β) ) , ¯ h
by Envelop Theorem we have
As
wt ∂ ∂λt−1 (1−θ)λt−1 +θ(αwt +β)
t = − [(1−θ)λ (1−θ)w +θ(αw t−1
AN US
w0 (λt−1 ) d 0 ¯} dλt−1 {(v − c − cw ) + [cw − w (λt−1 )] [(1−θ)λt−1 +θ(αw0 (λt−1 )+β)]h wt ∂ = ∂λt−1 {(v − c − cw ) + (cw − wt ) [(1−θ)λ +θ(αw +β)]h¯ }|wt =w0 (λt−1 ) t t−1 cw −wt wt ∂ = { h¯ ( ∂λt−1 (1−θ)λt−1 +θ(αwt +β) )}|wt =w0 (λt−1 ) .
t +β)]
2
(A.8)
(A.9)
≤ 0 and w0 (λt−1 ) ≤ cw , we have
w0 (λt−1 ) d 0 ¯ } ≤ 0. Therefore V1 (λt−1 ) is still dλt−1 {(v − c − cw ) + [cw − w (λt−1 )] [(1−θ)λt−1 +θ(αw0 (λt−1 )+β)]h in λt−1 when λt−1 > λE . Together we can see V1 (λt−1 ) is continuous decreasing in λt−1 .
decreasing
M
Then, we are going to proceed with induction. Suppose we have proved that for any n − 1 period
problem, the maximum discounted total value Vn−1 (·) is a decreasing function. Then let’s consider the
Vn (λt−1 ) =
ED
n period problem, which is formulated as max
pt ≥0,wt ≥0
π(pt , wt |λt−1 ) + δVn−1 ((1 − θ)λt−1 + θ(αwt + β)) .
(A.10)
PT
First we still separate the n period problem into two cases, which are 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt +w≤ v,
and otherwise. As Vn−1 ((1 − θ)λt−1 + θ(αwt + β)) is independent of pt and continuous decreasing in wt
CE
for any given prior belief λt−1 by the induction condition, with the similar arguments, we can also show the optimal decision is also achieved on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt .
AC
Hence, the problem can be simplified as Vn (λt−1 ) =
max
0≤wt ≤min{w(λ ¯ t−1 ),v}
π(v − wt , wt |λt−1 ) + δVn−1 ((1 − θ)λt−1 + θ(αwt + β)) .
(A.11)
Because π(v − wt , wt |λt−1 ) is concave in wt when 0 ≤ wt ≤ w(λ ¯ t−1 ) and Vn−1 ((1 − θ)λt−1 + θ(αwt + β))
is decreasing in wt , we can see the optimal warranty price for the n period problem should be obtained
in the region of wt ∈ [0, w1∗ (λt−1 )]. By the characterisation of w1∗ (λt−1 ) in (A.6) and (A.8), we have 27
ACCEPTED MANUSCRIPT
π(v − wt , wt |λt−1 ) is decreasing in λt−1 for any wt ∈ [0, w1∗ (λt−1 )]. Besides, we also have Vn−1 ((1 −
θ)λt−1 + θ(αwt + β)) decreasing in λt−1 . Hence, we have proved that Vn (λt−1 ) is also decreasing in λt−1 .
Plus, as both π(v − wt , wt |λt−1 ) and Vn−1 ((1 − θ)λt−1 + θ(αwt + β)) are continuous in λt−1 and wt , we
can show Vn (·) is also continuous.
11
CR IP T
Then we have shown V (·) is continuous decreasing.
Proof of Proposition 2. As V (·) is a continuous decreasing function based on Proposition 1 and by the similar arguments in Proof of Proposition 1, we have the optimal decision should be achieved on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt . First by the Proof of Proposition
1, the optimal warranty price for the one period problem w1 (λt−1 ) characterized in (A.6) satisfies
0 ≤ w1 (λt−1 ) ≤ w(λ ¯ t−1 ) and the constraint w1 ≤ v is redundant. Besides, it is also easy to see that
w∗ (λt−1 ) ≤ w1 (λt−1 ) as V ((1 − θ)λt−1 + θ(αwt + β)) is decreasing in wt by Proposition 1. Then we have
AN US
0 ≤ w∗ (λt−1 ) ≤ w(λ ¯ t−1 ) and wt ≤ v is still redundant for the infinite horizon problem. Thus, we have
proved that the optimal product price p∗ (λt−1 ) and warranty price w∗ (λt−1 ) satisfy w∗ (λt−1 ) ≤ w(λ ¯ t−1 )
and p∗ (λt−1 ) = v − w∗ (λt−1 ).
Proof of Proposition 3. When w∗ (λt−1 ) < w(λ ¯ t−1 ) and the warranty has actual sales in the current period, as we have already shown that w1∗ (λt−1 ) ≤ cw where w1∗ (λt−1 ) is the optimal warranty price for
M
the one period problem in Proof of Proposition 1 and we also have w∗ (λt−1 ) ≤ w1∗ (λt−1 ), it is easy to
see w∗ (λt−1 ) ≤ cw . Otherwise, w∗ (λt−1 ) = w(λ ¯ t−1 ) and warranty has no actual sales in the current
ED
period. Hereby we finish the proof.
Moreover, as illustrated in the Proof of Proposition 1, we further have w1∗ (λt−1 ) ≤ c2w when h follows ¯ as in the base assumption. Thus we further have the uniform distribution with 0 lower bound U[0, h] cw 2 .
PT
result of w∗ (λt−1 ) ≤
Proof of Proposition 4. Because λt has a one-to-one correspondence to wt based on the learning equation
CE
(1), to finish the proof for Proposition 4, we need to consider another equivalent problem in which the decision variables are changed from (pt , wt ) to (pt , λt ). The problem is reformulated in the following
AC
equation:
V (λt−1 ) =
max
pt ≥0 λt ≥(1−θ)λt−1 +θβ
π ˜ (pt , λt |λt−1 ) + δV (λt ) ,
11
(A.12)
Proposition 1 could be easily shown by Theorem 4.7 in Stokey (1989) if π(v − wt , wt |λt−1 ) decreasing in λt−1 for any given wt is guaranteed. However, π(v − wt , wt |λt−1 ) is decreasing in λt−1 only when wt ≤ cw . As w∗ (λt−1 ) ≤ cw cannot be proved directly, though is true by Proposition 3, Theorem 4.7 in Stokey (1989) cannot be employed directly and we prove Proposition 1 in the above induction way.
28
ACCEPTED MANUSCRIPT
where the one period profit is λt −[(1−θ)λt−1 +θβ] t−1 +θβ] t−1 +θβ] + (pt − c + λt −[(1−θ)λ } − cw ){1 − λt −[(1−θ)λ ¯ ¯ θα (pt − c) θαλt h θαλt h ¯ t−1 ) and λt ≤ (1 − θ)λt−1 + θ[α(v − pt ) + β] π ˜ (pt , λt |λt−1 ) = . , if (1 − θ)λt−1 + θβ ≤ λt < λ(λ (v−pt ) (pt − c) min{ λ h¯ , 1} , otherwise t (A.13)
Here wt =
λt −[(1−θ)λt−1 +θβ] θα
is derived from the learning equation (1). The expression in (A.12) and
h
CR IP T
(A.13) are equivalent to (4) and (2) respectively, where pt and wt are the original decision variables. In ¯ t−1 ) ¯ t−1 ) is equivalent to w(λ ¯ t−1 ) = w(λ (A.13), λ(λ ¯ t−1 ) and λ(λ . Plus, λt ≥ (1 − θ)λt−1 + θβ is equivalent ¯
to wt ≥ 0 and λt ≤ (1 − θ)λt−1 + θ[α(v − pt ) + β] is equivalent to wt ≤ v − pt .
Similarly, we can first prove the optimal decision is achieved on the boundary of (1 − θ)λt−1 + ¯ t−1 ) and pt = v − λt −[(1−θ)λt−1 +θβ] . Then the problem is simplified to a probθβ ≤ λt ≤ λ(λ θα
AN US
λt −[(1−θ)λt−1 +θβ] λt −[(1−θ)λt−1 +θβ] . } ¯ t θα θαhλ
λt −[(1−θ)λt−1 +θβ] , λt |λt−1 ) θα
= (v − c − cw ) + {cw − ¯ t−1 ). Notice warranty may still have no actual sales when λt = λ(λ
lem of only considering the following case: π ˜ (v −
Notice there may exist multiple solutions to (6). In this work, we are interested in the steady state of
the dynamic programming problem. When multiple solutions exist for (6), we choose the corresponding solution with the highest warranty price:
0≤wt ≤w(λ ¯ t−1 )
π(v − wt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β))} .
M
w∗ (λt−1 ) = max{ arg max
Notice here we still use the subscript “*” to denote the highest optimal warranty price compared with
ED
(5). Correspondingly, the optimal product price is still p∗ (λt−1 ) = v − w∗ (λt−1 ) and the updated consumers’ belief is denoted by
PT
λ∗ (λt−1 ) = (1 − θ)λt−1 + θ(αw∗ (λt−1 ) + β)
(A.14)
based on (1). In the following proofs, terms with subscript “*” is corresponding to the highest optimal
CE
warranty price if not further specified. Correspondingly, here we also only consider the firm’s maximum optimal decision of future state in case of multiple solutions, which is defined as
AC
λ∗ (λt−1 )
λ∗ (λt−1 ) = max{
arg max
¯ t−1 ) (1−θ)λt−1 +θβ≤λt ≤λ(λ
π ˜ (v −
λt − [(1 − θ)λt−1 + θβ] , λt |λt−1 ) + δV (λt )} . θα
Here we need to mention this definition is equivalent to the definition in (A.14) above, which is corresponding to the base framework of the main body in which (pt , wt ) are the decision variables. ¯ t−1 ) are increasing in λt−1 , which In (A.12), first it is easy to see both (1 − θ)λt−1 + θβ and λ(λ 29
ACCEPTED MANUSCRIPT
¯ t−1 )] is an increasing set in λt−1 (The definition of “induced set ordering” implies [(1 − θ)λt−1 + θβ, λ(λ ¯ t−1 ) and warranty strategy is employed, the is defined in Section 2.4 in Topkis (1998)). When λt ≤ λ(λ partial derivatives with respect to λt and λt−1 of the total profit function is
∂2 λt − [(1 − θ)λt−1 + θβ] (1 − θ){2[(1 − θ)λt−1 + θβ] + θαcw } π ˜ (v − . , λt |λt−1 ) = ¯ ∂λt ∂λt−1 θα θ2 α2 λ2t h
0. Together we can
And it is easy to see
∂2 ∂λt ∂λt−1 δV
CR IP T
∂2 t−1 +θβ] ˜ (v− λt −[(1−θ)λ , λt |λt−1 ) ≥ 0. ∂λt ∂λt−1 π θα λt −[(1−θ)λt−1 +θβ] see π ˜ (v − , λt |λt−1 ) + δV (λt ) θα
By (A.15), we have
(A.15) (λt ) =
is supermodular in (λt , λt−1 ) within
¯ t−1 ) , λt−1 ≥ 0} . By Theorem 2.8.1 in the optimization region {(λt , λt−1 ) : (1 − θ)λt−1 + θβ ≤ λt ≤ λ(λ Topkis (1998), we can prove λ∗ (λt−1 ) is increasing in λt−1 .
¯ t−1 ) Proof of Proposition 5. First, we have λ∗ (λt−1 ) is bounded by (1 − θ)λt−1 + θβ ≤ λ∗ (λt−1 ) ≤ λ(λ by definition. As wt =
λ∗ (λt )−[(1−θ)λ∗ (λt−1 )+θβ] θα
≤ v must be satisfied, which is to say λ∗ (λt ) ≤ (1 −
AN US
θ)λ∗ (λt−1 ) + θ(αv + β) must be satisfied, we have λ∗ (λt−1 ) ≤ αv + β as long as t is large enough. Thus,
we further have λ∗ (λt−1 ) is globally smaller than αv + β. By Proposition 4, as λ∗ (λt−1 ) is increasing
and bounded, there exists at least one steady state. Then we need to prove the uniqueness of the steady state.
By treating the single period profit function π(v − wt , wt |λt−1 ) as a function of λt−1 and λt , the
M
Euler equation can be derived from
ED
∂ λt − [(1 − θ)λt−1 + θβ] ∂ λt+1 − [(1 − θ)λt + θβ] π(v − , λt |λt−1 ) + δ π(v − , λt+1 |λt ) = 0 (A.16) ∂λt θα ∂λt−1 θα by setting λt+1 = λt = λt−1 = λ∞ . It is equivalent to
PT
−[2 − 2(1 − θ)δ − θ]λ2∞ + (1 − δ)(1 − θ)(αcw + 2β)λ∞ + θβ(αcw + β) = 0 , which has already been presented in (8).
CE
It is easy to see the above equation is a concave quadratic function in λ∞ with zero order coefficient being positive. It is also easy to verify the discriminant is positive. Thus, it has one positive and one ˜ ∞ denote the only negative solutions exactly. Thus, (8) has one and only one positive solution. Let λ
AC
positive solution to (8). Based on this result, we are going to show the uniqueness and details of the steady state in the following. ¯ t−1 ) = [(1−θ)λ+θβ] First notice the fixed point of (1 − θ)λt−1 + θβ and λ(λ are β and ¯ 1−θαh ¯ tively. Besides, both (1 − θ)λt−1 + θβ ≥ 0 and λ(λt−1 ) ≥ 0 when λt−1 = 0.
β ¯ + (1−αh)
respec-
˜ ∞ = β, which is not an interior solution. When cw = 0, we can show that the solution to (8) is λ ˜ ∞ = β. In this case, it is easy to have w∞ = 0 and Then the unique and stable steady state is λ∞ = λ 30
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p∞ = v as w∞ =
λ∞ −β α
by learning equation (1) and p∞ = v − w∞ by Proposition 2.
12
Otherwise, when cw > 0, as α > 0, β > 0, and δ < 1, we can show the solution to (8) is strictly larger than β, which indicates β is not the steady state and λ∞ > β. Then we can show the solution to ¯ ¯ w − {2 − [2(1 − θ)δ + θαh]}β ¯ h ¯ ≤ 0. (8) is an interior solution if and only if (1 − αh){1 − [(1 − θ)δ + θαh]}c
is not an interior solution, the unique and stable steady state is w∞ =
¯ βh ¯ + (1−αh)
and p∞ = v −
¯ βh ¯ +. (1−αh)
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¯ ¯ w − {2 − [2(1 − θ)δ + θαh]}β ¯ h ¯ ≤ 0, which implies λ ˜ ∞ is Then if (1 − αh){1 − [(1 − θ)δ + θαh]}c ˜ ∞ < λ(λ ¯ t−1 ), we can show λ ˜ ∞ is the unique steady an interior solution, i.e. (1 − θ)λt−1 + θβ < λ ˜ ∞ , w∞ = λ˜ ∞ −β and p∞ = v − w∞ . Otherwise, if state and it is stable, which is to say λ∞ = λ α ¯ ¯ ¯ ¯ ˜ ∞ ≥ λ(λ ¯ t−1 ) and λ ˜∞ (1 − αh){1 − [(1 − θ)δ + θαh]}cw − {2 − [2(1 − θ)δ + θαh]}β h > 0, which implies λ β ¯ +, (1−αh)
which is to say λ∞ =
β ¯ +, (1−αh)
c¯w =
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Proof of Proposition 6. From Proof of Proposition 5, we can see there exists a ¯ h ¯ {2 − [2(1 − θ)δ + θαh]}β ¯ + {1 − [(1 − θ)δ + θαh]}} ¯ + {(1 − αh)
such that in the steady state, free warranty is offered and all the consumers who buy the product purchase the warranty when cw = 0, part of the consumers who buy the product purchases the warranty
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when 0 < cw < c¯w and no actual warranty sales when cw ≥ c¯w . Based on the characterisation on c¯w in ¯ and δ, while it is independent of the previous formula, it is easy to show c¯w is increasing in θ, α, β, h,
ED
v and c.
Proposition 15 (Impact of cw ). Both the steady state consumers’ belief λ∞ and warranty price w∞ are increasing in cw . Plus, both the total profit V (λ∞ ) and the per period profit on product sales v − c − w∞
then increasing in cw .
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are decreasing in cw , while the per period profit on warranty sales (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
is first decreasing
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Proof of Proposition 15 (Impact of cw ). Let EL(λ) = −[2 − 2(1 − θ)δ − θ]λ2 + (1 − δ)(1 − θ)(αcw +
2β)λ + θβ(αcw + β), which is the Euler function to facilitate the proof. By the Proof of Proposition 5, ˜ ∞ as the only positive solution to EL(λ) = 0. Actually it is expressed as we have defined λ
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p (1 − δ)2 (1 − θ)2 (acw + 2β)2 + 4θβ[2(1 − θ)(1 − δ) + θ](αcw + β) ˜∞ = λ . 2[2(1 − θ)(1 − δ) + θ] (A.17) p Furthermore, it is easy to show that EL( β(αcw + β)) ≥ 0. Together with Proposition 3, we have p ˜ ∞ ≤ α cw + β. Notice here we have used the special result of w∗ (λt−1 ) ≤ cw as β(αcw + β) ≤ λ (1 − δ)(1 − θ)(acw + 2β) +
2
12
2
∗
∗
p∞ = v − w∞ is derived from p (λt−1 ) = v − w (λt−1 ) by Proposition 2.
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demonstrated in the Proof of Proposition 3, which only holds when h follows uniform distribution with ¯ 0 lower bound U[0, h].
First by (6), as π(v − wt , wt |λt−1 ) is decreasing in cw , it is easy to have the discounted total profit
V (·) is decreasing in cw for any given state by coupling method.
(i) When cw = 0, free warranty
is employed in the steady state, which firstly implies the steady state warranty price and belief are w∞ = 0 and λ∞ = β respectively and constant in cw . Besides, we have the steady state total profit 1 1−δ (v
− c − cw ) decreasing in cw , one period demand for warranty
direct profit of warranty sales (w∞ −
¯ w∞ )+ (h− λ∞ cw ) ¯ h
¯ w∞ )+ (h− λ∞ ¯ h
= 1, one period
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V (λ∞ ) =
= −cw decreasing in cw , and one period profit of
product sales v − c − w∞ = v − c constant in cw . (ii) When 0 < cw < c¯w , part of the consumers buys
the warranty in the steady state. Under this circumstance, λ∞ is the solution to the Euler equation
∂ EL(λ∞ ) ∂cw ∂ ∂ d dcw λ∞ = − ∂ EL(λ∞ ) . As − ∂λ EL(λ∞ ) ≥ 0 and ∂cw EL(λ∞ ) ≥ 0, we can see λ∞ ∂λ w∞ = λ∞α−β , we have w∞ is also increasing in cw and v−c−w∞ is decreasing in cw .
EL(λ) = 0 and we have is increasing in cw . As
¯ w∞ + d (h− λ∞ ) ¯ dcw h
in cw . Besides, by we have
¯ w∞ )+ (h− λ∞ ¯ h
= ( ∂λ∂∞
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Based on V (λt ) decreasing in cw and λt and λ∞ increasing in cw , it is easy to have V (λ∞ ) is decreasing ¯ w∞ )+ (h− λ∞ ) ¯ h
· ( dcdw λ∞ ) +
¯ w∞ + ∂ (h− λ∞ ) ¯ ∂cw h
= − αλβ2 · ( dcdw λ∞ ) + 0 ≤ 0, ∞
decreasing in cw . Besides, by the similar approach we can show (w∞ − cw ) ¯ w∞ )+ (h− λ∞ = 0 when cw = 0 ¯ h w∞ + ¯ (h− λ ) ∞ cw ) is decreasing in cw ¯ h
is convex in cw . Plus, notice (w∞ − cw ) there exists a c˜w such that (w∞ −
¯ w∞ )+ (h− λ∞ ¯ h
or cw = c¯w . Hence, we can show when cw ≤ c˜w and increasing in
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cw when cw > c˜w . (iii) When cw ≥ c¯w , expensive warranty is employed and there are no actual sales
of warranty. The total profit is only consisted as the profit of product sales and there is no profit for ¯ β βh ¯ + and λ∞ = (1−αh) ¯ + are (1−αh) w ∞ )+ ¯ ¯ w∞ )+ (h− ( h− λ∞ λ∞ = 0, and (w∞ − cw ) ¯ ¯ h h
have V (λ∞ ) = in cw .
ED
warranty sales. In this instance, w∞ = 1 1−δ (v − c − w∞ ),
constant in cw . Accordingly, we = 0, which are also all constant
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Proof of Proposition 7 (Impact of θ). In the following proof, despite of showing Proposition 7, we are also going to show the comparative statics on θ, which is: the steady state consumers’ belief λ∞ and
CE
warranty price w∞ are decreasing in θ; besides, both the total profit V (λ∞ ) and per period profit on product sales v − c − w∞ are increasing in θ, whereas the profit on warranty sales (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
is decreasing in θ.
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First, V (·) is not monotone in θ and how V (·) changes with θ is state dependent. Second, by Proposition 6, c¯w is increasing in θ. Thus, there exists a θ¯ such that expensive warranty is employed ¯ (i) When θ ≤ θ, ¯ expensive warranty is when θ ≤ θ¯ and partial warranty is employed when θ > θ.
¯ βh β ¯ + and λ∞ = (1−αh) ¯ + are constant in θ. Accordingly, we have V (λ∞ ) = (1−αh) w ∞ )+ ¯ ¯ w∞ )+ ( h− (h− λ∞ λ∞ 1 ¯ (v − c − w ), = 0, and (w − c ) = 0 are all constant in θ. (ii) When θ > θ, ∞ ∞ w ¯ ¯ 1−δ h h ∂ partial warranty is employed, where λ∞ is the solution to EL(λ) = 0. First we can show ∂θ EL(λ∞ ) ≤ 0
employed, where w∞ =
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based on the fact that
p β(αcw + β) ≤ λ∞ ≤ α c2w + β. Therefore, λ∞ as well as w∞ is decreasing in θ.
¯ w∞ )+ (h− w∞ λ∞ 1 [(v − c − c ) + (c − w ) ] decreasing in λ , w w ∞ ∞ ¯ ¯ 1−δ λ∞ h h w + ∞) ¯ (h− λ∞ increasing in λ∞ ; while they are all independent of θ (despite decreasing in in λ∞ , and (w∞ −cw ) ¯ h 1 λ∞ is a function of θ). As λ∞ is decreasing in θ, we have V (λ∞ ) = 1−δ [(v − c − cw ) + (cw − w∞ ) λw∞h¯ ] ∞ ¯ w∞ )+ ¯ w∞ )+ (h− (h− λ∞ λ∞ increasing in θ, increasing in θ, and (w − c ) decreasing in θ. Besides, it is easy to ∞ w ¯ ¯ h h
Accordingly, it is easy to show V (λ∞ ) =
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see v − c − w∞ increasing in θ.
¯ Again, despite of showing Proposition 8, we are also going to Proof of Proposition 8 (Impact of h). ¯ which is: both the steady state consumers’ belief λ∞ and warranty show the comparative statics on h, ¯ besides, the total profit V (λ∞ ) and per period profit on product sales price w∞ are increasing in h; ¯ w∞ )+ (h− λ∞ ¯ are all decreasing in h. v − c − w∞ and warranty sales (w∞ − cw ) ¯ h
Note as we have shown that free warranty is employed if and only if cw = 0 in Proof of Proposition
Proof of Proposition 15.
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15, we only consider cw > 0 in the following proofs. Plus, the function EL(λ) is the same as defined in ¯ for any given state again. Second, by First, by coupling method, we can show V (·) is decreasing in h ¯ Therefore, there exists a h ¯ u such that expensive warranty is employed Proposition 6, c¯w is increasing in h. ¯≤h ¯ u and partial warranty is employed when h ¯>h ¯ u . (i) When h ¯≤h ¯ u , expensive warranty is when h ¯ β ¯ employed, where w∞ = β h¯ + and λ∞ = ¯ + are increasing in h. Accordingly, we have V (λ∞ ) = ¯ − c − w∞ ) decreasing in h;
¯ w∞ )+ (h− λ∞ ¯ h
(1−αh)
= 0 and (w∞ − cw )
M
(1−αh)
1 1−δ (v
¯ w∞ )+ (h− λ∞ ¯ h
¯ (ii) When = 0 constant in h.
¯
ED
¯>h ¯ u , partial warranty is employed, where λ∞ is the solution to EL(λ) = 0. As EL(λ) is independent h ¯ λ∞ and w∞ are independent of h. ¯ Accordingly, we have V (λ∞ ) = 1 [(v − c − cw ) +(cw − w∞ ) w∞¯ ] of h, w∞ +
¯
w∞ +
1−δ
λ∞ h
PT
¯ (h− λ¯∞ ) increasing in h, ¯ (w∞ − cw ) (h− λ¯∞ ) decreasing in h ¯ (w∞ − cw is constant and decreasing in h, h h ¯ and v − c − w∞ constant in h. ¯ negative in h),
Proof of Proposition 9 (Impact of α). Here we are still going to show the comparative statics on α in addition, which is: the steady state consumers’ belief λ∞ is increasing in α while the warranty price
CE
w∞ is first increasing then decreasing in α; besides, both the total profit V (λ∞ ) and per period profit on warranty sales (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
are decreasing in α, while the per period profit on product sales
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v − c − w∞ is first decreasing then increasing in α.
First, by coupling method, V (·) is decreasing in α. Second, by Proposition 6, c¯w is increasing in α,
which implies there exists a α ¯ such that expensive warranty is employed when α ≤ α ¯ and partial warranty is employed when α > α ¯.
and λ∞ =
β ¯ + (1−αh)
α; and
¯ h
¯ w∞ )+ (h− λ∞
(i) When α ≤ α ¯ , expensive warranty is employed, where w∞ =
are increasing in α. Accordingly, we have V (λ∞ ) =
= 0 and (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
1 1−δ (v
¯ βh ¯ + (1−αh)
− c − w∞ ) decreasing in
= 0 constant in α. (ii) When α > α ¯ , partial warranty is
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employed, where λ∞ is the solution to EL(λ) = 0. As we can show w∞ =
˜ ∞ −β λ α
∂ ∂λ∞
further show
∂ EL(λ∞ ) ∂α ∂ − ∂λ EL(λ∞ ) ¯ w∞ )+ (h− λ∞
·
≥ 0, λ∞ is increasing in α. Besides,
is decreasing in α. As V (λt ) is decreasing in α and λt and λ∞ is increasing
in α, we have V (λ∞ ) is also decreasing in α. As ¯ w∞ )+ (h− λ∞ ¯ h
∂ ∂α EL(λ∞ )
+
¯ w∞ + ∂ (h− λ∞ ) , ¯ ∂α h
¯ w∞ + d (h− λ∞ ) ¯ dα h
=
¯ w∞ + ∂ (h− λ∞ ) ¯ ∂λ∞ h
·
d dα λ∞
+
¯ w∞ + ∂ (h− λ∞ ) ¯ ∂α h
and based on the expression of λ∞ shown in (A.17), we can
is increasing in α. By similar approach, we can show (w∞ − cw )
¯ h
=
¯ w∞ )+ (h− λ∞ ¯ h
is
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decreasing in α. Besides, it is easy to see v − c − w∞ is increasing in α.
¯ a , where Proof of Proposition 10. By (10) and (11), it is easy to see that π∞ ≥ πF if and only if λa ≥ λ ¯a = λ
(
α2 c2w 4(λ∞ −β)(αcw +β−λ∞ ) λ∞ ¯ λ2∞ −[α(1−αh)+2β]λ ∞ +β(αcw +β) ¯ 2 λ∞ α2 h
, if (cw − w∞ ) λw∞h¯ ≤ , otherwise.
∞
cw 2 ;
(A.18)
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The threshold value above is presented in terms of λa , which is demonstrated as the dash-dotted ¯ a is represented in cw and λ∞ in (A.18) . Nevertheless, λ∞ is a function line in Figure 2. Also notice λ of the system parameters including cw which is given in Proposition 5. ¯ a ≥ λ∞ and λ ¯ a is concave increasing in Based on Proposition 5 and (A.18), we can further verify that λ
¯ a = λ∞ = β when cw = 0; and λ ¯ a = λ∞ = cw . Besides, λ
β ¯ + (1−αh)
when cw > c¯w and (cw −w∞ ) λw∞h¯ > ∞
cw 2 .
˜
Furthermore, notice λa is linear increasing in cw and λa = 0 when cw = 0, which is λa (cw ) = cδ0 cw as ˜ a such that λa < λ ¯ a when λa < λ ˜a described in Section 5. Then, it is easy to see there exists a unique λ
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¯ a when λa ≥ λ ˜ a . Based on the definition of λ ¯ a in (A.18), we finish the proof of Proposition and λa ≥ λ
ED
10.
˜ a is the value such that hiding the information about the true Proof of Proposition 11. First, notice λ
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product reliability to inducing consumer learning and disclosing the information about the true product ˜ a . The expressions of π∞ and πF are reliability yield the same profit, which is π∞ = πF when λa = λ illustrated in (11) and (10) respectively. Second, the system parameters θ, α, and β are only related
CE
to learning, while they do not affect the consumer behaviours as well as the firm’s decisions in the disclosing case. In addition, π∞ is decreasing in λ∞ . Then, based on (11), (10), and the comparative
AC
statics in Section 4.3, as λ∞ is decreasing in θ and increasing in α and β, while πF is independent from ˜ a is decreasing in θ, and increasing in α and β. θ, α, and β, it is easy to see that λ ¯ is the largest value of consumers’ individual handling costs, which characterizes both the average h
cost and heterogeneity of consumer individual handling. It affects the consumer behaviours as well as the firm’s profits in both cases of adaptive learning and disclosing. We cannot prove the comparative ¯ on λ ˜ a directly. Nevertheless, based on the comparative statics in Section 4.3 which include statics of h ¯ and the expression of λ ¯ a in (A.18), we can show that λ ¯ a is increasing in h ¯ while λa is λ∞ increasing in h 34
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¯ for any given cw . As both λ ¯ a and λa are increasing in cw and λ ˜ a is the intersection independent from h ¯ a and λa as functions of cw , it is easy to prove that λ ˜ a is increasing in h. ¯ of λ
Proof of Proposition 12. Proposition 12 can be proved by induction which is similar to the Proof for Proposition 1, where the property of Xt+1 stochastically increasing in xt is used.
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Proof of Proposition 13. In this proof, we show pR (λt−1 , xt ) = v − wR (λt−1 , xt ) as well.
First, as pt is not directly involved in consumer learning, which means pt only affects the current
period profit π R (pt , wt |λt−1 , xt ) and has no impact on the future profit E[V R ((1 − θ)λt−1 + θ(αwt +
xt ), Xt+1 )] for any given wt and (λt−1 , xt ). Then, it is easy to show pR (λt−1 , xt ) = v − wR (λt−1 , xt ).
Moreover, by Proposition 12, as V R (λt−1 , xt ) is decreasing in λt−1 and xt , we can show wR (λt−1 , xt ) ≤
w ¯ R (λt−1 , xt ) by the argument which is similar to Proof of Proposition 2.
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Then, the proof of either wR (λt−1 , xt ) ≤ cw or wR (λt−1 , xt ) = w ¯ R (λt−1 , xt ) is similar to Proof of ¯ Proposition 3. Besides, we can still further prove that wR (λt−1 , xt ) ≤ cw when h follows U[0, h]. 2
Proof of Proposition 14. The proof is similar to Proof of Proposition 4. First, we again reformulate the problem into the following problem in which the decision variable is the consumers’ belief in the
λR (λt−1 , xt ) = max{
arg max
π ˜ R (v −
λt −[(1−θ)λt−1 +θxt ] , λt |λt−1 , xt ) θα
+ δE[V R (λt , Xt+1 )]} . (A.19)
ED
¯ R (λt−1 ,xt ) (1−θ)λt−1 +θxt ≤λt ≤λ
M
following period λt , while the decision variable in the origin problem in (14) is the warranty price wt .
Notice here we have already use the results from Proposition 13 and let pt = v − ¯ R (λt−1 , xt ). λt constrained on λt ≤ λ
λt −[(1−θ)λt−1 +θxt ] θα
and
CE
PT
∂2 t−1 +θxt ] ˜ R (v − λt −[(1−θ)λ , λt |λt−1 , xt ) ≥ 0 and ∂λt ∂λt−1 π θα λt −[(1−θ)λt−1 +θxt ] ∂2 ∂2 R R (λ , X ˜ (v− , λt |λt−1 , xt ) ≥ 0. Besides, it is also easy to see ∂λt ∂λ δE[Vn−1 t t+1 )] = ∂λt ∂xt π θα t−1 2 0 because δE[V R (λt , Xt+1 )] is independent of λt−1 and ∂λ∂t ∂xt δE[V R (λt , Xt+1 )] = 0 because Xt+1 is in-
It is easy to show
dependent of λt as assumed in Section 6 and Xt+1 is independent of xt by the i.i.d. assumption specified
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t−1 +θxt ] in Proposition 14. Hence, we have shown that π ˜ R (v − λt −[(1−θ)λ , λt |λt−1 , xt ) + δE[V R (λt , Xt+1 )] θα ¯ R (λt−1 , xt ) are both increashas increasing differences in (λt , (λt−1 , xt )). Plus, as (1 − θ)λt−1 + θxt and λ
¯ R (λt−1 , xt )] is an increasing set in (λt−1 , xt ). Finally by ing in (λt−1 , xt ), it implies [(1 − θ)λt−1 + θxt , λ Theorem 2.8.1 in Topkis (1998), we can prove λR (λt−1 , xt ) is increasing in (λt−1 , xt ).
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Stokey, N. L. (1989). Recursive Methods in Economic Dynamics. Harvard University Press. Terlaak, A. and A. A. King (2006). The effect of certification with the iso 9000 quality management standard: A signaling approach. Journal of Economic Behavior & Organization 60 (4), 579–602. Topkis, D. M. (1998). Supermodularity and complementarity. Princeton university press. Warranty Week (2006). Ford’s powertrain warranties. Warranty Week. July 25. 38
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Warranty Week (2014). Apple’s warranty & applecare programs. Warranty Week. October 30. Wernle, B. (2009). Chrysler drops lifetime warranty. Automotive News. August 24. Whitney, L. (2015). Apple raises price of applecare+ support for new iphones. CNET . September 10. Wiener, J. L. (1985). Are warranties accurate signals of product reliability?
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lifetime. European Journal of Operational Research 196 (3), 940–948.
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Warranty Pricing with Consumer Learning Yong Lei, Qian Liu, Stephen Shum PII: DOI: Reference:
S0377-2217(17)30563-5 10.1016/j.ejor.2017.06.024 EOR 14506
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
29 October 2016 13 April 2017 6 June 2017
Please cite this article as: Yong Lei, Qian Liu, Stephen Shum, Warranty Pricing with Consumer Learning, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.06.024
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights • The dynamic warranty pricing with consumer learning is studied. • Warranty sales do not generate profit directly though are profitable overall. • Both consumers’ beliefs and the firm’s warranty policy are stable in the long run.
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• The firm only induces consumer learning when the true failure rate is high.
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Warranty Pricing with Consumer Learning Yong Lei∗,a , Qian Liub , Stephen Shumc a
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School of International Business Southwestern University of Finance and Economics, China 611130, China b Department of Industrial Engineering and Logistics Management The Hong Kong University of Science and Technology, Hong Kong c College of Business City University of Hong Kong, Hong Kong
Abstract
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We consider a problem in which a firm dynamically prices a product and its warranty service over time. Consumers can learn about the reliability of products based on warranty prices. A firm’s optimal product and warranty pricing policies are characterized. We find that a warranty should be priced lower than the marginal warranty service cost, which implies that warranty sales will not generate profits directly. However, offering a modest warranty still benefits the firm’s overall profits. We also show that consumers’ beliefs and the firm’s warranty policy converge in the long run. In a steady state, either a fraction of consumers will purchase a warranty or no consumer will purchase a warranty. Comparative statics analysis is conducted to show how factors such as a firm’s warranty service cost, consumers’ learning speed, and the heterogeneity of consumers’ handling costs determine consumers’ beliefs, the firm’s warranty policy and profitability in a steady state. Lastly, we note that a firm benefits from consumer learning by hiding the information about the true product reliability only when the true product failure rate is relatively high.
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Key words: Revenue management; Consumer learning; Pricing; Warranty management
Introduction
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Firms change warranty provisions for their products from time to time. For example, in October 2011, before formally introducing the iPhone 4S, Apple introduced a new Applecare+ service plan for iPhones at $99 to replace the previous $69 Applecare program (Slivka 2011). In September 2013, before formally ∗
Corresponding author. Tel.: +86 13540609462. E-mail addresses: [email protected] (Y. Lei), [email protected] (Q. Liu), [email protected] (S. Shum).
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introducing the iPhone 5C and iPhone 5S, Apple increased the accidental damage service fee from $49 to $79 per incident, but the price for Applecare+ remained at $99 (Allen 2013). The most recent change took place in September 2015. Before formally introducing the iPhone 6S and iPhone 6S Plus, Apple increased the price of Applecare+ for the iPhone 6S and iPhone 6S Plus from $99 to $129. Apple also increased the service fee from $79 to $99 for the new iPhone 6S and iPhone 6S Plus (Whitney 2015).
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Apple is not the only example in practice of a firm changing its warranty provision. In the automobile
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industry, for example, Ford increased its powertrain warranty from 3 years or 36,000 miles to 5 years or 60,000 miles on its 2007 Ford Lincoln and Mercury models in 2006, and Kia and General Motors made similar moves in 2001 and 2006, respectively (see, Warranty Week 2006, Krisher 2006, and Guajardo et al. 2015). In contrast, Chrysler dropped its lifetime warranty in 2009 two years after its introduction (Wernle 2009).
Changing the warranty policy can drive a company’s profit in opposite ways. On one hand, offering
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a more generous warranty policy increases the cost to the firm due to a higher cost of honouring the warranty. On the other hand, a more generous warranty usually means that the warranty coverage is better or the warranty price will be lower. Thus, it increases the demand for the product by directly reducing consumers’ cost of owning it.
A more subtle yet important impact is that the warranty terms, such as the service length, price and service coverage, are considered important indicators influencing consumers’ beliefs about product
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reliability. As noted by extensive empirical studies, including Wiener (1985), Boulding and Kirmani (1993), Pilon (2009) and Jindal (2015), consumers tend to perceive warranty terms as an accurate
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and informative reflection of product reliability, and hence rely on the warranty terms and scope to make inferences about product reliability. This is also consistent with the beliefs of practitioners. As mentioned in Warranty Week (2006) and Krisher (2006), Ford and General Motors increased their
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warranties “in an effort to help it sell more autos by boosting its reputation for quality”, which indicates the consumers are treating the warranty as a reliable indicator of a firm’s product reliability.
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In this paper, we study a dynamic model of a firm selling a single product with an optional warranty. The firm dynamically optimizes the price of the product and the price of the optional warranty. Consumers perceive the warranty price as an indicator of product reliability, and in particular, consumers
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perceive the product as more reliable when the warranty price is lower. Consumers have heterogeneous handling costs in case of product failures. Each consumer decides whether to buy both the product and warranty, the product only, or neither the product nor the warranty. Our goal is to consider how a firm should dynamically manage its warranty provision when warranty provision affects consumers’ beliefs 1
The price and service fee of Applecare+ remained the same with the introduction of the iPhone 7 and iPhone 7 Plus in September 2016. Apple cut the screen damage service fee from $99 to $29. Information can be found via the official website of Apple.
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about the product reliability. For example, how should a firm price its warranty policy? What is the optimal warranty policy in the long run? How does it depend on different factors, such as warranty cost and customer heterogeneity? Our results suggest that the warranty should not be priced higher than its marginal cost; hence warranty sales by themselves should not make a profit for the firm. While this contradicts the results of some analytic models (e.g., Padmanabhan and Rao 1993, Lutz and Padmanabhan 1995, and Pad-
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manabhan 1995) that state the warranties are a highly profitable service, there is no consensus among practitioners on the profitability of warranties due to some accounting issues. For example, Warranty Week (2014) questions the profitability of Applecare and Applecare+. Moreover, our results are consistent with the findings of the empirical study of Jindal (2015) that warranties should not be priced higher than its marginal costs when the warranty is sold simultaneously with the product. Thus, to some extent, our paper complements the results of this empirical study by providing support from a
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theoretical modelling perspective.
Our steady state analysis shows that consumers’ beliefs and the firm’s warranty policies converge. In the long run, it is possible that a fraction of consumers will purchase a warranty or no consumer purchases warranty depending on the firm’s warranty cost, consumers’ learning speed and heterogeneity of consumers’ handling costs. The long-run warranty policy and consumers’ beliefs depend on these parameters.
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Lastly, we further find that the firm prefers to hide the true product reliability and manipulate consumers’ perceived beliefs about the product failure rate through learning rather than disclosing
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information about the true product reliability to the consumers only when the true product failure rate is relatively high.
The remainder of the paper is organized as follows. Section 2 reviews the relevant literature. Section
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3 introduces the base model, and Section 4 studies the firm’s optimal dynamic product and warranty pricing problem. Section 5 compares the results with when information about true product failure rate
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is disclosed to the consumers to illustrate the impact of consumer learning. Section 6 extends our model to the case when there is some uncertainty in consumers’ belief updating about the product failure rate in order to validate the model robustness. Finally, Section 7 provides concluding remarks, a discussion
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of the limitations of this paper, and possible future research.
Literature Review
As pointed by Emons (1989), it is generally recognized that warranties have four economic roles: insurance, screening, signalling, and incentive. For example, Heal (1977) shows that warranties provide
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consumers with insurance and build up a risk-sharing mechanism based on the assumption that consumers are risk averse. Chun and Tang (1995) and Zhou et al. (2009) further examine how consumers’ risk preferences affect the optimal warranty price. Based on the idea that a firm with less information about buyers can use a menu of offerings and prices to screen buyers through self-selection (see Rothschild and Stiglitz 1976 and Stiglitz and Weiss 1981 for screening in general settings), some papers consider the use of warranty menus to differentiate different types of consumers (e.g., Kubo 1986,
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Matthews and Moore 1987, Padmanabhan and Rao 1993, Padmanabhan 1995, and Lutz and Padmanabhan 1998). Consumers’ efforts in maintaining the product have an impact on the firm’s warranty servicing cost, whereas this effort is costly to customers and unobservable to firm. Firms must take consumers’ moral hazard into account when designing warranty terms. Thus, some works consider the design of warranty provision to provide consumers incentives for better product maintenance (e.g., Lutz and Padmanabhan 1995). From the firm’s prospective, some other works investigate the firm’s incen-
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tive to signal product quality to consumers (e.g., Cooper and Ross 1985 and Lutz 1989) and supply high-quality products (e.g., Heal 1977 and Priest 1981). Gallego et al. (2014b) study both drivers by considering the use of residual value warranties, a practice of refunding part of the warranty price to customers who have zero or few claims according to a predetermined refund schedule. Residual value warranties can price discriminate customers, and they also affect customers’ strategic behaviour in using warranty services. The authors show that the total warranty service cost, which includes repair costs
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and refunds, is lower for more risk-averse customers under residual value warranties, and they identify the conditions when residual value warranties are more profitable than traditional warranties.
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The problem studied in our paper is more related to the papers that study cases when consumers do not have perfect information about product reliability, and the firm uses warranty policies to signal product reliability. As noted by Akerlof (1970), markets can break down when potential buyers cannot
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verify the quality of the product they are offered, and as a result, sellers of bad quality products may crowd out sellers of good quality products. In contrast, firms may have incentives to take observable and
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costly actions to signal potential buyers about their superior product quality when compared with those of other sellers (Spence 1973, 1974). In particular, a warranty may serve as a credit signal; hence better warranty terms may signal better product quality (e.g., Spence 1977 and DeCroix 1999). Because better
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product quality incurs higher production cost, in equilibrium, it is optimal for manufacturers to offer a better quality product with a higher warranty price to more failure-sensitive consumers and a lower quality product with a lower warranty price to less failure-sensitive consumers. However, some papers also suggest that it is possible that poorer warranty may be associated with higher product reliability due to consumers’ moral hazard on product maintenance (e.g., Cooper and Ross 1985 and Lutz 1989) and the differences in consumers’ knowledge about the reliability of established and newer products
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(Balachander 2001).
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Nevertheless, most of the empirical studies have shown that customers typically
perceive better product reliability when warranty provisions are better (e.g., Wiener 1985, Kelley 1988, and Boulding and Kirmani 1993). Readers may refer to Emons (1989), Blischke and Murthy (1992), Murthy and Blischke (1992a,b), Murthy and Djamaludin (2002), and Shafiee and Chukova (2013) for more extensive reviews of warranties. Similar to some of the aforementioned papers, our paper focuses on analytic modelling to identify
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the optimal warranty provision for the firm. The key difference is that these papers consider a singleperiod case, whereas our paper studies a dynamic model in which the firm can change the product and warranty prices over time, which resembles the practice in the industry, where firms change warranty provision from time to time. In addition, the study of a dynamic setting allows us to consider the impact of not only static factors, such as costs, but also dynamic consumer characteristics, such as their learning speed.
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Our paper is also related to the operations management literature on consumer learning behaviour. Popescu and Wu (2007) consider consumers’ reference price learning and study a firm’s multi-period dynamic pricing strategies with reference price effects. Based on Popescu and Wu (2007), Yuan and Hwarng (2016) further investigate the dynamic pricing issue when customers form reference prices adaptively through social interactions. Liu and van Ryzin (2011) study the firm’s optimal capacity decision when consumers learn about product availability via repeated purchasing experiences. To
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the best of our knowledge, the only paper that considers the relationship between warranty provision and consumer learning is Gallego et al. (2014a). They consider the case of the firm offering a flexible
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warranty policy with a monthly premium and consumers may discontinue the warranty anytime. Their focus is consumers’ decisions to continue or discontinue the warranty coverage and how the firm should design the optimal flexible warranty policy at the beginning of the planning horizon. Our paper differs
Model
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from theirs in that we focus on the case when a firm changes its warranty provision over time.
A monopolist firm sells a product over repeated sales periods in an infinite horizon. At the beginning
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of each period, say period t, the firm sells the product at price pt and offers an optional warranty with price wt to consumers. The product and warranty prices, pt and wt , are the firm’s decisions made at the beginning of period t. The firm’s unit procurement cost of the product is c and the discounted expected warranty service cost on each unit of product is cw . Notice that cw is simply regarded as the discounted expected repair cost that the firm incurs from providing a warranty service for each warranty 2 Indeed, the aforementioned papers Cooper and Ross (1985), Lutz (1989)), and Balachander (2001) have proven that both positive and negative correlations between warranty provisions and product quality are possible.
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sold. Obviously, cw is related to the true product failure rate because a higher warranty service cost is incurred when a product fails and is repaired more frequently. Moreover, we assume that the true product failure rate will not change over time; thus, the warranty service cost cw is also fixed over time. There is a mass of infinitesimal consumers with a deterministic size in the market. We normalize the market size to 1 without loss of generality. Consumers have a homogeneous individual valuation of the product, denoted by v. We assume v − c − cw ≥ 0; otherwise, a warranty should be never offered. together with a warranty.
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In each period, consumers will decide whether to buy the product only or purchase the product If consumers do not purchase the warranty, they will incur some utility loss
when the product fails during the warranty period because they will have to handle the failed product all by themselves. We assume that each consumer i has a different utility loss when the product ¯ fails during the warranty period, denoted by h, follows a uniform distribution on the support of [0, h]. Suppose that in period t, all consumers will share the same beliefs about the product’s failure rate,
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denoted by λt , which may not be the same as the true failure rate of the product. We assume both consumers’ beliefs and the corresponding updating rule are known to the firm and will describe how the consumers update their beliefs λt later. Thus, a consumer with the utility loss of product failure h will incur a cost λt h if he does not purchase the warranty and handles the failed product by himself. However, if the consumer buys the warranty, the firm will take care of product failures for the entire product life free of charge. Moreover, notice in this paper we only consider that repairs both by the firm
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and consumers’ self-handling are “good-as-new”. It implies both the product and the warranty policy remains the same as the new product after each repair. Readers are referred to Blischke and Murthy
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(1992) for a detailed taxonomy of warranty policies. Hence, a consumer obtains an expected surplus, v − pt − min{wt , λt h}, when he purchases the product. We assume that he will buy the product as long
as he gains a nonnegative surplus, i.e., v − pt − min{wt , λt h} ≥ 0. One can easily see that there is a
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threshold value of the individual handling cost, which is equal to
wt λt ,
such that consumers with a utility
loss greater than this value will prefer to purchase the warranty over self-repair, and consumers with a 3
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utility loss smaller than it will not purchase the warranty.
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In industrial practices, consumers may either purchase a warranty after a long period of product usage or be required to purchase a warranty simultaneously with the product within a short time period after buying the product (Jindal 2015). For example Apple changed its policy from buying Applecare anytime to the simultaneous purchasing of Applecare+ with an iPhone, which is now “within 60 days of your iPhone purchase” (see Slivka 2011 and http://www.apple.com/ shop/product/APP_IPHONE_6S_PLUS_AUTO-133580/applecare-for-iphone-6s-and-iphone-6s-plus). In our paper, we assume consumers need to buy the warranty simultaneously with the product, which is commonly assumed in the warranty literature.
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3.1
Consumer Learning Model
Consumers have an initial belief about product failure rate, which is updated over time. Specifically, at the beginning of period t, consumers update their beliefs about the product’s failure rate in period t, denoted by λt , via the following learning equation: λt−1 |{z}
+θ
P rior belief
(αwt + β) | {z }
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Conversion f unction
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λt = (1 − θ)
In (1), θ is consumers’ learning speed, and it measures how fast the consumers adjust their beliefs about the product failure rate with the firm’s warranty price. We use a linear function αwt + β to convert the warranty price into the failure rate of the product. Specifically, α is the conversion factor reflecting the sensitivity of consumers’ beliefs about the product failure rate on the firm’s warranty
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price. As mentioned in the literature, abundant results show that a generous warranty is positively correlated with a lower product failure rate (e.g., Spence 1977, Wiener 1985, Kelley 1988, and Boulding and Kirmani 1993). Therefore, we assume that α > 0, implying that a higher warranty price indicates a higher warranty service cost and thus a higher belief about the product’s failure rate. Literally, β is the converted consumers’ belief of the failure rate when the firm offers a free warranty, and β > 0. Otherwise, the firm can manipulate the consumers’ beliefs about the failure rate to zero by offering a
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free warranty for a sufficiently large number of periods. On the other hand, as revealed in the literature of product quality signalling, some factors, such as advertising, brand reputation, product certification and specialized function, can also influence the consumers’ beliefs about the product failure rate (e.g.,
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Kirmani and Rao 2000, Terlaak and King 2006, and Kalra and Li 2008). Hence, β captures these factors to some extent. We assume that, in our model, β is exogenously given and fixed over time.
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Note that consumers’ beliefs about the failure rate λt are not necessarily directly related to the true failure rate of the product. This is because consumers are not able to observe the product reliability before they make purchases. The firm may manipulate consumers’ beliefs about the failure rate via
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its warranty policy or social networks. Abundant evidence shows that some firms make efforts to manipulate consumers’ beliefs about firms’ product/service, for example, hiring people to post positive
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reviews on online forums, which is largely received negatively by consumers. We also assume that all consumers share homogeneous beliefs about the product failure rate. It
does not necessarily mean that consumers are buying the product repeatedly in every period. Instead, it refers to some market beliefs commonly shared by all the consumers who make purchase decisions. In each period, this belief is updated based on the most updated warranty price. This approach is also used in the literature on consumer learning such as Popescu and Wu (2007), Liu and van Ryzin (2011), and Yuan and Hwarng (2016). 8
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When an individual consumer decides whether to buy a product and/or warranty service in a period, say period t, he will have a prior belief about product failure rate, which is integrated market information carried over from the previous periods. After the warranty price is observed in the current period, the market belief about product failure rate will be updated accordingly via the learning function (1), which becomes the prior belief about the product failure rate for the purchasers in the next period t + 1. Hence, the learning model (1) describes the mechanism of how the market information of consumers’
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beliefs about product failure rate is updated over periods, and consumers do not have to make purchases repeatedly over time to learn the product failure rate.
3.2
Firm’s Profit Function
In period t, given the consumers’ prior belief of the product failure rate λt−1 , which is the state variable of the system, the firm determines the product price pt and the warranty price wt . Consumers then
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instantly update their beliefs about the product failure rate according to (1). For a given state λt−1 and the firm’s product and warranty decision (pt , wt ), the firm’s expected profit gained in period t, denoted by π(pt , wt |λt−1 ), can be expressed as follows:
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wt wt (pt − c) [(1−θ)λt−1 +θ(αwt +β)]h¯ + (pt − c + wt − cw ){1 − [(1−θ)λt−1 +θ(αwt +β)]h¯ }, π(pt , wt |λt−1 ) = , if 0 ≤ wt < w(λ ¯ t−1 ) and pt + wt ≤ v; v−pt (pt − c) min{ , 1} , otherwise. ¯ [(1−θ)λt−1 +θ(αwt +β)]h
(2)
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¯ That is, when the warranty price In (2), w(λ ¯ t−1 ) satisfies w(λ ¯ t−1 ) = [(1−θ)λt−1 +θ(αw(λ ¯ t−1 )+β)]h. ¯ will be indifferent between purchasing the is set at w(λ ¯ t−1 ), consumers with the highest utility loss h
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warranty from the firm and handling the failed product on their own. This implies that the consumers ¯ will always prefer to address product failure by themselves, rather with a utility loss strictly less than h
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than purchase a warranty. One can then easily derive that w(λ ¯ t−1 ) =
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¯ [(1 − θ)λt−1 + θβ]h . ¯ + (1 − θαh)
(3)
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We assume that consumers will not buy a warranty when they are indifferent to the two options (buying a warranty or handling failed products themselves). Furthermore, notice that consumers purchase the product and warranty only when they obtain the nonnegative surplus, that is, v − pt − wt ≥ 0. Hence,
when wt < w(λ ¯ t−1 ) and pt + wt ≤ v, some or all consumers with a relatively high utility loss of product
failure will purchase both the product and warranty service from the firm, whereas the rest will buy 4
In this paper, [∗]+ = max{∗, 0}, and
1 [∗]+
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¯ [(1−θ)λ+θβ]h ¯ + (1−θαh)
¯ ≤ 0. = +∞ if 1 − θαh
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only the product. Otherwise, there are no actual sales of the warranty, and consumers will consider only buying the product from the firm and handle the product failure on their own. In each period t, the firm makes decisions on the product and warranty prices. The firm’s profit in period t is then determined by (2). Note that once consumers observe the warranty price, their beliefs about the product failure rate is updated according to the learning function (1). Hence, the learning function links the firm’s warranty decision in two consecutive periods together. When a firm makes
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its warranty decision in each period, it considers not only the impact on the current profits but also the impact on consumers’ future beliefs about product reliability, which in turn affect the firm’s future profit. Let V (λt−1 ) be the firm’s maximum discounted total profit when consumers hold a prior belief of the product failure rate λt−1 . The future profit is discounted by a discount factor of δ per period. The value function V (λt−1 ) for a given state λt−1 satisfies the following Bellman equation:
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max {π(pt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β))} .
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V (λt−1 ) =
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Model Analysis
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In Section 4.1, we study the properties of a firm’s optimal product and warranty pricing decisions when consumers learn about the product failure rate through warranty prices. We then focus on characterizing
Optimal Product and Warranty Pricing
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4.1
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the steady state in Section 4.2 and conduct comparative statics of the steady state in Section 4.3.
We first present the monotone property of the firm’s discounted profit function regarding the state, which is established in Proposition 1 below.
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Proposition 1. The firm’s discounted expected profit V (λt−1 ) given in (4) is continuously decreasing in λt−1 .
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This result is consistent with our intuition; a firm would prefer consumers to hold a good belief about product reliability (i.e., a lower estimate of product failure rate). This is simply because the firm does not have to offer warranty services at a very low price to improve the belief about product
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reliability held by the consumers. We now investigate the properties of the firm’s optimal warranty pricing policy. Let p∗ (λt−1 ) and
w∗ (λt−1 ) denote the optimal product and warranty prices for any given consumers’ prior belief of the failure rate λt−1 . That is, (p∗ (λt−1 ), w∗ (λt−1 )) is the solution to problem (4): (p∗ (λt−1 ), w∗ (λt−1 )) = arg max {π(pt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β))} . pt ≥0,wt ≥0
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Further, denote λ∗ (λt−1 ) = (1 − θ)λt−1 + θ(αw∗ (λt−1 ) + β), which is the optimal updated consumer
belief for the given state λt−1 . We can show that a firm’s optimal product pricing and warranty pricing
decisions have the following properties: Proposition 2. Give a prior belief of the product failure rate λt−1 , the firm’s optimal product price and warranty price always satisfy p∗ (λt−1 ) + w∗ (λt−1 ) = v and w∗ (λt−1 ) ≤ w(λ ¯ t−1 ).
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Recall that, for a given prior belief λt−1 , when the firm charges the warranty price at w(λ ¯ t−1 ) given in (3), the consumers may buy the product only and no one will purchase the warranty from the firm. When the firm charges the warranty price to strictly smaller than w(λ ¯ t−1 ), there always exists a positive fraction of consumers who will purchase both the product and the warranty service from the firm as long as an appropriate price is charged to induce the purchase of the base product. Furthermore, Proposition 2 implies that the firm is able to fully extract a consumer surplus in this case because
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p∗ (λt−1 ) + w∗ (λt−1 ) = v.
Regardless of whether consumers purchase a warranty service from the firm, the optimal product price and warranty service price always satisfy p∗ (λt−1 ) + w∗ (λt−1 ) = v for any given prior belief about the product failure rate. This result enables us to simplify the expressions of the one-period profit function (2) and the Bellman equation of the firm’s expected profit (4). Specifically, the firm’s profit price, and (4) is reduced to the following: max
0≤wt ≤w(λ ¯ t−1 )
π(v − wt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β)) ,
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V (λt−1 ) =
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optimization problem can be formulated as a function of a single decision about the warranty service
where
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π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt )
(6)
wt ¯ . [(1 − θ)λt−1 + θ(αwt + β)]h
We further find that whenever there are warranty service sales (i.e., some consumers purchase the
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warranty service from the firm), the optimal warranty service price will be smaller than the expect service cost. This result is established in the following proposition.
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Proposition 3. For a given prior belief of product failure rate λt−1 , the firm sets the optimal warranty price such that either w∗ (λt−1 ) ≤ cw or w∗ (λt−1 ) = w(λ ¯ t−1 ). Proposition 3 implies that warranty sales cannot make direct profits for the firm, which is a rather
surprising result. It is believed that a warranty is a highly profitable service, as shown in Padmanabhan and Rao (1993), Lutz and Padmanabhan (1995), and Padmanabhan (1995). However, in their works, a menu of price and warranty bundles are offered, heterogeneous consumers self-select and thus the 11
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market is segmented. In this sense, warranty generates profits because of the price discrimination of warranty service. Nevertheless, in our work, we show that the warranty is not a direct profit-making part when a single warranty price is employed. The result can be explained as follows. An important fact in our problem is that all consumers purchase the product regardless of whether they purchase the warranty service from the firm. Although a higher warranty price increases the profit margin of a warranty service, it reduces the profit margin
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of product sales by the same amount. This is because the optimal price for the product for a given consumers’ belief about the product failure rate λt−1 is equal to v − w∗ (λt−1 ), as shown in Proposition 2. When the warranty price is raised, the product price has to be lowered such that the firm can fully extract the surplus from consumers who purchase both the product and warranty service from the firm. Hence, the firm actually earns a lower profit margin on average for both the product and warranty sales with a higher warranty price as only a fraction of the consumers purchases a warranty whereas all of
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them buy the product. On the other hand, if the firm charges a higher warranty price, fewer consumers will want to buy the warranty service from the firm. The profit on warranty sales alone will also shrink if the firm tries to make money from warranty sales directly by setting the warranty price higher than the corresponding warranty service cost. Then, we can easily show that the firm’s one-period profit decreases in the warranty price once it exceeds the service cost cw . In addition, the future expected profit also decreases in the firm’s warranty price (i.e., V ((1 − θ)λt−1 + θ(αwt + β)) decreases in wt
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based on Proposition 1). All these lead to the result that the firm will set a lower warranty price, even below the service cost, when it is optimal for the firm to induce some consumers to purchase a warranty 5
Although the firm does not make profits from selling the warranty service directly, the higher
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service.
profits from product sales can more than compensate for the cost of warranty sales. In fact, the question of whether the warranty itself generates profit is not so clear in the industry
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due to certain accounting issues. Warranty Week (2014) doubts the profitability of Applecare and Applecare+, especially with its fast growth in warranty expenses. An empirical study by Jindal (2015) also
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notes that a firm should sell its warranty at the marginal cost when the warranty is sold simultaneously with the product. The result indicates that warranty does not generate any extra profit in this case. Our result provides some theoretical support for widely applied warranty services. Although a warranty In fact, w∗ (λt−1 ) ≤ cw is true for any general distribution of consumers’ individual handling cost of product failure h. For a general distribution, let g(·) and G(·) denote the p.d.f. and c.d.f. of h. The one-period profit function is t now formulated as π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt )G( (1−θ)λt−1w+θ(αw ). Notice here pt = v − wt . Then t +β)
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5
(1−θ)λt−1 +θβ ∂ t t π(v −wt , wt |λt−1 ) = −G( (1−θ)λt−1w+θ(αw )+(cw −wt )g( (1−θ)λt−1w+θ(αw ) 2 . It is easy to check ∂wt t +β) t +β) [(1−θ)λt−1 +θ(αwt +β)] ∂ that ∂wt π(v − wt , wt |λt−1 ) ≤ 0 when wt ≥ cw . Since V ((1 − θ)λt−1 + θ(αwt + β)) is decreasing in wt by Proposition 1 (this result also holds for a general distribution of h), we can conclude that w∗ (λt−1 ) ≤ cw . Moreover, when h follows uniform ¯ as assumed, we can further prove that w∗ (λt−1 ) ≤ cw , which is shown in the proof of Proposition 3 in distribution U[0, h] 2
the appendix.
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is not a direct profit-generating tool for the firm, it helps the firm to obtain a higher profit in total.
4.2
Steady State Analysis
When a firm dynamically changes its product and warranty service prices, consumers adjust their beliefs of product failure rate based on the observed warranty service prices. It is interesting to investigate the long-run behaviours of the firm and consumers. An important question is whether the firm’s pricing
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policy and consumers’ beliefs about the product failure rate converge in the long term. Namely, we would like to see for problem (6), whether there exists a unique state such that the firm’s pricing decisions as well as the consumers’ beliefs stay stable. Before we answer this question, we provide the formal definition of steady state below.
A steady state of problem (6) is defined to be a fixed point of the state updating function:
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f (λ) = (1 − θ)λ + θ(αw∗ (λ) + β) ,
(7)
where w∗ (λ) is the optimal warranty price to problem (4) for a given state λ as defined in (5). Hence, λ is the steady state if f (λ) = λ, which we denote as λ∞ . At the steady state, the updated consumers’ belief based on the firm’s optimal pricing decision remains the same as the prior belief. Moreover, the product price p∗ (λ) and warranty price w∗ (λ) determined by the firm also converge to constants, denoted by p∞ = v − w∞ by Proposition 2.
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p∞ and w∞ , respectively. We can easily show that w∞ =
1 α (λ∞
− β) by the learning equation (1) and
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To show the existence of a steady state, we first analyse the pattern of the state path. For a given
initial belief of the product failure rate, the optimal state path is monotonically increasing. Proposition 4. For a given belief of product failure rate λt−1 , the consumers’ updated belief according
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to the state updating function (7), denoted by λ∗ (λt−1 ), is increasing in λt−1 . Because a bounded monotone path will eventually converge, the result that there exists a steady
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state for problem (6) follows immediately from Proposition 4. ˜ ∞ as the positive Proposition 5. There exists a unique steady state λ∞ for problem (6). Denote λ
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solution to:
− [2 − 2(1 − θ)δ − θ]λ2∞ + (1 − δ)(1 − θ)(αcw + 2β)λ∞ + θβ(αcw + β) = 0.
(8)
β ˜∞, Then λ∞ = min{λ ¯ + }. (1−αh)
˜∞ < When λ
β ¯ +, (1−αh)
the steady state is the positive solution (there is only one positive solution) to ˜ ∞ − β) equation (8). The firm’s stationary warranty and product prices are determined by w∞ = α1 (λ 13
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¯ Hence, consumers with relatively high individual handling and p∞ = v − w∞ . In this case, w∞ < λ∞ h.
cost of product failure, i.e. h ≥
w∞ λ∞ ,
buy both the product and the warranty from the firm, while
consumers with relatively low individual handling cost of product failure, i.e. h <
w∞ λ∞ ,
buy the product
only. In the extreme case that cw = 0, the consumers’ beliefs converges to λ∞ = β and the firm sells the β ˜∞ ≥ product at price p∞ = v and offers a free warranty w∞ = 0 in the steady state. When λ ¯ +, (1−αh)
In this instance, the consumers’ beliefs converges to λ∞ = (1−αβ h) ¯ + ¯ and the firm chooses a relatively expensive warranty price w∞ = λ∞ h and sells the product at price β ¯ +. (1−αh)
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the steady state is equal to
p∞ = v − w∞ such that no consumer purchases the warranty from the firm in the long run.
Furthermore, notice the steady state consumers’ belief of product failure rate is independent of
the initial belief. Based on Proposition 4, starting from any initial belief of the product failure rate, consumers’ beliefs converge monotonically to the same steady state as specified in Proposition 5. This indicates if the initial product failure rate belief is low (below λ∞ ), the firm should gradually increase
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warranty prices to increase the beliefs; if the initial belief is high (above λ∞ ), the firm should gradually decrease warranty prices to reduce the beliefs; or if the initial belief equals to the steady state belief, the firm should keep the warranty price at w∞ such that the belief remains at λ∞ . In all the three cases, consumers’ beliefs, warranty prices, and product prices all eventually converge to λ∞ , w∞ , and
4.3
6
Figure 1 provides an example to demonstrate the above results.
Comparative Statics
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p∞ , respectively, in the long run.
In this section, we conduct comparative statics analysis of the steady state. We examine how the firm’s
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stationary policy of product price and warranty price and consumers’ long-run beliefs about the product failure rate change with important system parameters, such as service cost, consumers’ learning speed,
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and the heterogeneity of consumers’ individual handling costs. ˜ ∞ when cw ≤ c¯w Proposition 6. There exists a threshold warranty service cost c¯w such that λ∞ = λ β ¯ + (1−αh)
otherwise.
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and λ∞ =
The result is intuitive. When the warranty service cost gets higher, the firm will increase the warranty price in order to reduce the loss from warranty sales. The higher the warranty service cost,
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the higher the warranty price, leading to a smaller fraction of consumers purchases the warranty. Indeed, consumers’ beliefs about the product failure rate increase in warranty service cost. Once the service cost is high enough, though the firm offers warranty services, no consumer will purchase the warranty as it is too costly. As a result, all the consumers only purchase the products. We find that the threshold value ¯ and δ, while it is independent of v and c. In other words, when consumers c¯w is increasing in θ, α, β, h, 6
Notice the expression of λ∞ is given in Proposition 5; the expressions of w∞ and p∞ are given below (7).
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0.1
0.06
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λt
0.08
0.04
0.02 0
5
10
15
20
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t
¯ = 0.2, θ = 0.3, α = 1, β = 0.05, Figure 1: Optimal belief updating paths for v = 1, c = 0.2, cw = 0.01, h and δ = 0.9. adjust their beliefs about the product failure rate more quickly with the recently observed warranty (θ
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increases), and/or consumers are more sensitive to the firm’s warranty price (α increases), it becomes less costly for the firm to influence consumers’ beliefs via its warranty price decisions. Therefore, the firm has more incentive to offer a relatively modest warranty price to induce purchases of warranty
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services.
The long-run profit of the firm decreases in warranty service cost. The profit from warranty sales in the long run, determined by (w∞ − cw )
¯ w∞ )+ (h− λ∞ , ¯ h
is first decreasing then increasing in cw . The result is a
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bit surprising at the first sight as one may expect that it decreases with cw . The reason is the following: a higher service cost cw reduces the warranty sales, which is
¯ w∞ )+ (h− λ∞ . ¯ h
Meanwhile, it also reduces the
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“profit margin” w∞ − cw of selling warranty services. Note that warranty sales do not generate any
profit in our setting, it leads to profit loss to the contrary. Therefore, a higher service cost increases
the direct profit loss from warranty sales. When the warranty service cost cw is relatively small, the
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direct profit loss from warranty sales is also small. As cw increases, the direct profit loss in warranty sales increases as well. However, when cw becomes relatively large, the demand for warranty becomes small. It diminishes to almost zero when the service cost is close to the threshold. Because a higher cw significantly reduces warranty sales, it brings down the direct profit loss in warranty sales. Therefore, the profit from warranty sales decreases and then increases in cw . 7
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A detailed result and the proof of the comparative studies on cw can be found in Proposition 15 and the following
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˜ ∞ when θ > θ¯ and λ∞ = Proposition 7. There exists a threshold value θ¯ such that λ∞ = λ
β ¯ + (1−αh)
otherwise. Learning speed θ measures how fast the consumers update their beliefs about the product failure rate according to warranty prices. Proposition 7 implies that when consumers adjust their beliefs about the product failure rate faster with the recent warranty price, the firm will offer a modest warranty price such that some consumers will purchase both the product and warranty. As the learning speed
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becomes faster, the firm incurs less cost to change consumers’ beliefs via the warranty pricing policies. We can show that the stationary warranty price and consumers’ beliefs decrease with the learning speed in the long term. As expected, the firm’s long-run profit increases with the learning speed; that is, the firm benefits from fast-learning consumers. The profit loss from warranty sales increases with the learning speed, which, however, can be more than offset by the increased profit from product sales. The overall profit turns out to be increasing in the learning speed. Proposition 7 also explains that when
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the learning speed of consumers is below some certain level, it is too costly for the firm to use a lower warranty price to improve consumers’ beliefs. Under this circumstance, the firm would rather choose a relatively expensive warranty price w∞ =
¯ βh ¯ + (1−αh)
such that there are no warranty sales at all.
¯ u such that λ∞ = λ ˜ ∞ when h ¯>h ¯ u and λ∞ = Proposition 8. There exists a threshold value h otherwise.
β ¯ + (1−αh)
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¯ βh ¯ u such that the firm chooses a relatively expensive warranty price w∞ = There exists a h ¯ + (1−αh) ¯ ¯ where no actual warranty sales exist in the steady state when h ≤ hu and chooses an interior warranty ˜ ∞ −β λ α
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¯ >h ¯ u . Furthermore both the steady state where warranty has actual sales when h ¯ The total profit V (λ∞ ), per period consumers’ belief λ∞ and warranty price w∞ are increasing in h.
price w∞ =
¯ w∞ )+ (h− λ∞ ¯ h
are all
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profit on product sales v − c − w∞ , and per period profit on warranty sales (w∞ − cw ) ¯ decreasing in h.
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Note that the consumers’ individual handling costs of product failure are assumed uniformly dis¯ Thus, h ¯ reflects the average value and heterogeneity of handling costs if consumers tributed on [0, h]. deal with the failed products on their own. The results suggest that when consumers incur highly heterogenous handling costs for failed products, the firm should always induce some consumers to purchase
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the warranty service. As such heterogeneity increases, the stationary warranty price increases as well. Since a higher warranty price reduces the profit margin of the product sales, the firm will earn a smaller ¯ increases. According to Proposition 8, the firm should charge a relatively profit in the long run when h ¯ expensive warranty price once consumers do not differ much in their individual handling costs (i.e., h ¯ u ). When this is the case, no one purchases the warranty. is smaller than h proof in the appendix. Besides, all the proofs of the detailed comparative studies for the other parameters in the following Proposition 7 to 9 are included in the corresponding proofs in the appendix.
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˜ ∞ when α > α Proposition 9. There exists a threshold value α ¯ such that λ∞ = λ ¯ and λ∞ =
β ¯ + (1−αh)
otherwise. The parameter α measures how sensitive the consumers are to the warranty price when they update their beliefs of product failure rate. Intuitively, all else being equal, a larger α results in a higher belief of failure rate, thus a higher demand for warranties and a greater loss in warranty sales. Finally, there
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will be a lower profit in the steady state. An interesting result here is that the firm’s warranty price is not monotone decreasing in α. When α is small enough such that α ≤ α ¯ , consumers’ beliefs of the product failure rate are affected by the current warranty price at a very low degree. This means it is more important for the firm to choose a relatively expensive warranty price w∞ =
¯ βh ¯ + (1−αh)
to cover the
service cost because consumers’ beliefs will not be worsen too much when consumers are insensitive to warranty price decisions. This explains why the warranty price increases with α. However, as α becomes
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larger and reaches α > α ¯ , consumers are more sensitive to the warranty price changes when they update their beliefs. The firm has to offer a lower warranty price to reduce consumers’ beliefs. We can also show that the direct profit loss in warranty sales is always increasing in α as both the unit profit loss in warranty sales and the demand for warranties are increasing in α as long as some consumers purchase warranties from the firm.
Comparison to Full Information about Reliability
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5
We have studied how consumers’ behaviour and a firm’s pricing policies are affected by the learning
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mechanism in the previous sections. In this section, we are going to study the impact of consumer learning by comparing it with the case of when consumers know the true reliability. All else is the same as before. The only difference is that the firm discloses the true reliability
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information and consumers have full information about the true failure rate λa . In this case, the consumer’s surplus of purchasing is v − p − min{w, λa h}. Notice because there is no consumer learning,
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both the product price p and warranty price w are simply chosen to maximize the firm’s one period
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profit; thus we eliminate the subscript index t. Then, the firm’s problem is formulated as follows: πF = max (p + w − c − cw ) + (cw − w) min{ p≥0,w≥0
w ¯ , 1} . λa h
(9)
¯ The optimal product price is It is easy to derive the optimal warranty price: wF = min{ c2w , λa h}.
pF = v − wF . Correspondingly, the associated one period profit is πF = v − c − cw + (cw − wF ) 17
wF ¯ . λa h
(10)
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Recall that the one-period profit in the steady state in the consumer learning case is determined by π∞ = v − c − cw + (cw − w∞ ) where w∞ =
w∞ ¯ , λ∞ h
(11)
λ∞ −β α .
Notice that the discounted expected warranty service cost cw is simply regarded as the discounted
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expected repair cost per unit of warranty sold as introduced in Section 3. Naturally, cw depends on the product true failure rate λa . Hence hereafter, we define cw as a function of λa , which is cw (λa ). We further assume that cw (λa ) is proportional to λa . This can be explained in the following.
Each time when the product fails and is fixed based on the warranty provision, the firm incurs a fixed repairing cost c0 . As we have assumed the repair is “good-as-new” in Section 3, the repaired product is returned to the consumer like a new one for free. Thus, we reasonably assume the product
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failure follows a Poisson process with a failure rate λa , where the memoryless feature of the Poisson process captures the “good-as-new” property of the repair. In addition, recall that the warranty covers the entire product life once it is purchased. As δ is the firm’s profit discount factor for each discrete ˜ period, we denote δ˜ = − ln δ as the continuous time discount factor, which satisfies δ = e−δ·1 . Suppose
the product fails for the first time after x units of time of purchase. Then, the firm spends c0 to fix it right away. As product failure follows a Poisson process which is memoryless, the firm still bears ˜
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the same future repairing cost cw (λa ) after the first product failure. The firm’s total repairing cost is ˜
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e−δx [cw (λa ) + c0 ], where e−δx is the corresponding discount factor. Conditioning on the time to the R∞ ˜ first product failure from the purchase, we obtain cw (λa ) = 0 e−δx [cw (λa ) + c0 ]λa e−λa x dx. Based on this equation, we obtain cw (λa ) =
c0 λ . δ˜ a
This model of warranty service cost is also used in Karmarkar
(1978). Readers can refer to Blischke (1990), Murthy and Blischke (1992b), and Murthy and Djamaludin function λa (cw ) =
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(2002) for more general discussions on warranty service cost evaluation. In addition, we use the inverse δ˜ c0 cw
of cw (λa ) to represent the correspondence of λa to cw .
Based on (10) and (11), we have the following proposition to compare the average profits with and
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without consumer learning.
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˜ a such that π∞ ≥ πF if and only if λa ≥ λ ˜a. Proposition 10. There exists a λ Proposition 10 suggests that it is better for the firm to hide the information about the true product
reliability and induce consumers to learn when λa (or cw equivalently) is relatively large. Oppositely, it is better for the firm to disclose the true product reliability to consumers and prevent consumers from learning. The basic logic is that the firm would like to hide the true reliability and manipulate consumers’ beliefs about the product reliability through learning only when the firm is efficient enough to induce 18
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Failure rate λa ˜a λ
•
0
Repairing cost cw
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λ∞
Figure 2: Illustration of Product Failure Rate v.s. Repairing Cost
a relatively low belief about the product’s failure rate, compared with the case of when true product
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reliability is disclosed.
As illustrated in Figure 2, the solid and dashed lines represent the steady state consumer belief about product failure rate with learning λ∞ and true failure rate λa with respect to warranty service cost cw respectively. The dash-dotted line represents the corresponding true failure rate such that hiding and disclosing the information of true product reliability yield the same profit for any given warranty service cost cw . According to Proposition 5, λ∞ is concave increasing in cw . That is, the marginal increment of
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consumers’ beliefs about the product failure rate is decreasing in cw with learning, while the marginal increment of true failure rate λa is constant in cw when true reliability is disclosed.
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According to (10) and (11), when true reliability is disclosed, the consumers’ perceptions of the product failure rate λa are independent of the firm’s decision. The firm simply chooses the warranty ¯ to maximize its profit. However, when true reliability is not known by the price wF = min{ cw , λa h} 2
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consumers, the consumers’ perceived steady state product failure rate through learning λ∞ is determined by the firm’s warranty decision w∞ . In other words, the warranty price with true reliability disclosed
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wF directly optimizes the firm’s profit, while the warranty price with learning w∞ cannot be freely chosen due to the additional constraint on warranty decision with learning (i.e., w∞ =
λ∞ −β α ).
Therefore, when warranty service cost cw (or equivalently true failure rate λa ) is small, the manipu-
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lated belief about product failure rate through learning λ∞ is relatively large as demonstrated in Figure 2. Specifically, the firm is worse off by hiding the true reliability (i.e., π∞ ≤ πF ) when λ∞ ≥ λa , because w∞ cannot be chosen without any constraint based on the above argument. Nevertheless, λ∞ < λa
doesn’t always guarantee that hiding the information about the true reliability is more beneficial for the ˜ a , the firm firm. As demonstrated in Figure 2, when λ∞ is only slightly lower than λa , i.e. λ∞ < λa < λ still cannot benefit from hiding the information about the true reliability. The firm may only achieve
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a higher profit by hiding the information about the true reliability when the perceived belief about product failure rate through learning λ∞ is sufficiently low compared with the intrinsic true failure rate λa , which only happens when the true failure rate λa (or cw equivalently) is relatively high. Hence, ˜ a as described in the firm benefits from hiding the information about the true reliability when λa ≥ λ Proposition 10.
˜ a changes with the system parameters. The following proposition describes how λ
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˜ a is decreasing in θ; and increasing in α, β, and h. ¯ Proposition 11. λ
˜ a means that it is better for the firm to disclose the true reliability rather than hide it and Larger λ let the consumers adaptively learn through the warranty prices. For any fixed true failure rate λa (or any fixed cw equivalently), the firm would prefer to induce learning for a relatively small steady state belief λ∞ and disclose the true reliability for a relatively large λ∞ .
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Based on the comparative statics in Section 4.3, when the learning speed θ increases, it is less costly for the firm to manipulate consumers’ beliefs via the warranty prices, namely the steady state belief λ∞ is decreasing in learning speed θ. Thus, hiding the true reliability and inducing consumers to learn become more beneficial for the firm compared with disclosing when the learning speed increases. ¯ characterizes both the average cost and heterogeneity A consumer’s highest value of handling cost h of consumer individual handling. When consumers bear higher individual handling cost on average,
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the firm offers a higher warranty price because consumers are more willing to buy warranties when handling by themselves becomes more costly. This eventually results in a higher steady state belief of
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product failure rate when true reliability is hidden. In this case, the firm would rather disclose the true ¯ becomes larger. reliability when h Lastly, both the conversion factors α and β measure the degree to which the consumers take warranty
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price into account when they update their beliefs through learning. As larger α or β results in a higher belief of product failure rate for any given warranty price, the firm is better to disclose the true reliability
Extension: Uncertainty in Consumers’ Belief Updating
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6
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when α or β is large.
In our basic model, we assume that consumers’ belief updating process about product failure rate, as given by equation (1), is deterministic. In reality, there are stochastic factors that can affect consumers’ beliefs on product reliability. For example, a firm may face uncertainties in online reviews, which also affect consumers’ perception on reliability and their purchasing decisions. We model this type of effect, on an abstract level, by a stochastic factor Xt , t = 1, 2, · · · . We assume that {Xt } is a Markov chain
and independent of the firm’s product price pt , warranty price wt , and consumers’ beliefs about the 20
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product failure rate λt−1 . For a fair comparison with the basic model, suppose that E(Xt ) = β and V ar(Xt ) = σ 2 for all t. At the beginning of each period t, consumers observe the realized value of the random factor Xt , denoted by xt , and Xt+1 stochastically increases in xt . In period t, when consumers observe the realized value xt , they will update their beliefs about the product failure rate in the following: (12)
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λt = (1 − θ)λt−1 + θ(αwt + xt ).
In (12), xt can be interpreted as some other information that reveals the true product failure rate directly, such as consumer reviews. Note that consumers update their beliefs based on not only the warranty service price but also additional information sources, such as consumer reviews. Hence, the state variable in each period now becomes (λt−1 , xt ). Accordingly, the firm’s single-period profit function
π R (pt , wt |λt−1 , xt )
wt (pt − c) [(1−θ)λt−1 +θ(αwt +xt )]h¯ + (pt − c + wt − cw ){1 − , if 0 ≤ wt < w ¯ R (λt−1 , xt ) and pt + wt ≤ v; v−pt (pt − c) min{ ¯ , 1}, otherwise. [(1−θ)λ +θ(αw +x )]h t−1
In (13) above,
w ¯ R (λ
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under uncertainty in the updating of consumers’ beliefs, denoted by π R (pt , wt |λt−1 , xt ), is as follows:
t−1 , xt )
=
¯ [(1−θ)λ+θxt ]h ¯ + . (1−θαh)
t
wt ¯} [(1−θ)λt−1 +θ(αwt +xt )]h
t
When the firm’s warranty price is equal to
(13) w ¯ R (λ
t−1 , xt ),
the
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¯ are indifferent between purchasing the warranty from the firm consumers with the highest utility loss h and handling failed products themselves. Again, we can see that no consumer will buy the warranty in
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this case because consumers with lower utility loss prefer individual handling even more. We use V R (λt−1 , xt ) to denote the firm’s total discounted profit when consumers have a prior belief about the product failure rate λt−1 and observe the realized value generated from other information
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sources xt . Then it satisfies the following Bellman equation:
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V R (λt−1 , xt ) =
max
{π R (pt , wt |λt−1 , xt ) + δE[V (λt , Xt+1 )]}
s.t.
λt = (1 − θ)λt−1 + θ(αwt + xt ) .
pt ≥0,wt ≥0
(14)
Similar to the basic model, the total discounted profit decreases in the state variables, which is
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established in the following proposition. Proposition 12. The firm’s total discounted profit V R (λt−1 , xt ) given in (14) is continuously decreasing in λt−1 and xt .
Let pR (λt−1 , xt ) and wR (λt−1 , xt ) be the optimal product and warranty prices for any given consumers’ prior belief of product failure rate λt−1 and a realized random factor xt . That is, 21
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(pR (λt−1 , xt ), wR (λt−1 , xt )) is the solution to problem (14). Further, denote λR (λt−1 , xt ) = (1−θ)λt−1 + θ(αwR (λt−1 , xt ) + xt ), which is the updated consumers’ belief for the given state (λt−1 , xt ) under the optimal warranty price wR (λt−1 , xt ). The firm’s optimal warranty pricing decision satisfies the similar property as in the basic model. Proposition 13. For a given state (λt−1 , xt ), the firm’s optimal warranty price wR (λt−1 , xt ) is either
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smaller than the expected warranty service cost cw or equal to w ¯ R (λt−1 , xt ). When wR (λt−1 , xt ) ≤ cw , there are always some consumers purchasing both the product and war-
ranty. When wR (λt−1 , xt ) = w ¯ R (λt−1 , xt ), no one will purchase the warranty from the firm. Moreover, the optimal product price pR (λt−1 , xt ) is always equal to v − wR (λt−1 , xt ). Similar to Proposition 4, we can show that the updated belief under the optimal warranty decision is increasing in consumers’ prior
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belief of product failure rate and the realized value of the random factor.
Proposition 14. If {Xt }s are independent and identically distributed (i.i.d.), the consumers’ updated belief of product failure rate λR (λt−1 , xt ) associated with the firm’s optimal warranty price is increasing in λt−1 and xt .
Notice that in Proposition 14, an additional i.i.d. assumption is made on {Xt }s. This is required
to show the joint supermodularity of the firm’s profit function in state variables (λt−1 , xt ), which is
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sufficient to show that λR (λt−1 , xt ) increases in λt−1 and xt . Nevertheless, our numerical studies will show that this result still holds when the random factor {Xt }s are not i.i.d. and affected by the past
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realized values.
We numerically study how the firm’s optimal warranty policy changes with system parameters.
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In numerical examples, the basic setup of parameters is as follows: v = {1, 2}, θ = {0.1, 0.3, ...0.9}, ¯ = {0.1, 0.2, 0.3}, c = {0.1, 0.2}, cw = {0.01, 0.02, 0.03}, α = {0.5, 1.0, 1.5}, β = {0.05, 0.10, 0.15}, h
and δ = {0.8, 0.9}. Suppose the {Xt }s are modelled as an order 1 autoregressive model; that is,
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Xt = (1 − η)Xt−1 + ηt with 0 < η ≤ 1 where {t }s are i.i.d. with E(t ) = β and V ar(t ) =
2−η 2 η σ .
Note
that {Xt }s are i.i.d. when η = 1. We set σ = {0.015, 0.030} and η = {0.3, 0.7, 1}. The distributions of t we have tried are uniform, exponential and truncated normal.
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We use a standard value iteration approach to numerically solve the basic model problem (with a
deterministic β) in (4) and the problem with uncertainty in consumers’ belief updating given in (14), respectively. We find that all the results regarding comparative statics hold as well when there is uncertainty in consumers’ belief updating. Indeed, we obtain all the results similar to problem (4).
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7
Conclusion
In this paper, we consider the problem of a firm dynamically pricing a product and its warranty service in a multi-period setting when the warranty price affects consumers’ beliefs about product reliability. We show that the warranty should not be priced higher than its marginal cost, which is consistent with the findings of some empirical studies. In the long run, both a firm’s optimal warranty price
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and consumers’ beliefs about product failure rate converge. In particular, when the firm’s warranty service cost is higher, consumers’ learning speed is slower, or heterogeneity in consumers’ handling cost in the case of product failure is lower, the long-run warranty price will be higher and long-run consumers’ perception of product reliability will be poorer. Moreover, it is better for the firm to hide the information about true product reliability and let the consumers learn about it adaptively when the product intrinsically has a relatively high failure rate.
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In this paper, we consider a model with stationary parameters. In other words, we assume that, other than the warranty price, other factors that affect consumers’ beliefs of product reliability are time independent. In reality, these other factors may also change over time. This can be captured by using non-stationary rather than stationary parameters in the learning function. While this will complicate the analysis of the optimal policy, we expect that the main insights and the structure of the long-run optimal policy will remain the same.
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We also assume that all consumers hold the same beliefs about product failure rate. While this assumption is reasonable when consumers for a product come from the same market segment, it may
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not be valid in cases where consumers come from very diverse backgrounds. In those cases, consumers from different segments may have different initial beliefs, learning speed, and sensitivities to warranty prices. This may require a different learning function for each individual market segment; hence a state
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variable is needed to capture customers’ beliefs in each segment. The increase in the dimension of the state space will significantly increase the challenge. Thus, whether the dimension of state space can be reduced is by itself a challenging question and deserves a separate study.
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A monopolistic setting is a good starting point to consider the impact of consumer learning on dynamic product and warranty pricing, but another interesting extension is to consider the case of
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when competition is involved. In this case, a firm’s warranty price will affect customers’ beliefs about the reliability of competitive products. In addition to increasing the state space of the problem, this will also lead to a dynamic game that may not be tractable under existing dynamic programming tools. The first challenging question for this extension is whether a subgame perfect equilibrium exists.
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Acknowledgments The authors thank two anonymous reviewers for their comments and suggestions that helped to improve the paper.
Proof of Proposition 1. Proposition 1 will be proved by induction.
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Appendix: Proofs First, let’s look at the one period problem, which is formulated as V1 (λt−1 ) =
max
pt ≥0,wt ≥0
where π(pt , wt |λt−1 ) is defined in (2).
π(pt , wt |λt−1 ) ,
¯
(A.1)
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t−1 +θβ]h As defined below (2), w(λ ¯ t−1 ) = [(1−θ)λ , which is the warranty price satisfies w(λ ¯ t−1 ) = ¯ + (1−θαh) ¯ It implies that w(λ [(1 − θ)λt−1 + θ(αw(λ ¯ t−1 ) + β)]h. ¯ t−1 ) is the highest warranty price such that
consumers possibly take buying warranty into account, instead of all consumers handling failed products on their own.
Therefore, let’s first consider the simpler case when wt ≥ w(λ ¯ t−1 ) such that consumers may only buy
the base product and warranty is never considered as an option, which is formulated in the second case
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in (2). In this case, the firm’s one period profit is π(pt , wt |λt−1 ) = (pt − c) min{ [(1−θ)λ
when wt ≥ w(λ ¯ t−1 ) by (2). It is easy to see
t (pt − c) min{ [(1−θ)λ v−p ¯ , 1} t−1 +θ(αwt +β)]h
v−pt
¯ , 1}
t−1 +θ(αwt +β)]h
is decreasing in wt for
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any given pt as higher wt increases consumers’ beliefs about the product failure rate and brings down the demand for product. Thus, wt > w(λ ¯ t−1 ) is never optimal and the optimal warranty price for the case of when wt ≥ w(λ ¯ t−1 ) is achieved on wt = w(λ ¯ t−1 ).
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As the firm’s profit function in (2) is continuous in wt at wt = w(λ ¯ t−1 ) for any given pt , we only
need to further consider the case of 0 ≤ wt ≤ w(λ ¯ t−1 ).
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Now let’s consider the case of when 0 ≤ wt ≤ w(λ ¯ t−1 ). In this instance, consumers with high losses
in product failures h buy the product and warranty together if the total cost is acceptable, which is pt + wt ≤ v as formulated in the first case in (2).
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profit is π(pt , wt |λt−1 ) =
wt (pt − c) [(1−θ)λ +θ(αw ¯ t +β)]h t−1
0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt ≤ v. It is easy to see wt ¯} [(1−θ)λt−1 +θ(αwt +β)]h
8
Under this circumstance, the firm’s one period
+ (pt − c + wt − cw ){1 − [(1−θ)λ
wt (pt − c) [(1−θ)λ +θ(αw ¯ t +β)]h t−1
wt
¯}
t−1 +θ(αwt +β)]h
when
+ (pt − c + wt − cw ){1 −
is increasing in pt for any given wt as higher pt increases the unit profit for product
sales while maintaining the product and warranty sales segmentation. Thus, the firm’s optimal decision 8 Here we emphasise that there are still no warranty sales when wt = w(λ ¯ t−1 ) even with pt + wt ≤ v. In the following we do not explain the case of wt = w(λ ¯ t−1 ) when warranty has no sales separately and wt ≤ w(λ ¯ t−1 ) is always interpreted as having warranty sales as long as pt + wt ≤ v.
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should be on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt . Notice in this case, warranty
has actually sales and the corresponding boundary profit is (v − c − cw ) + (cw − wt ) [(1−θ)λ
wt
¯.
t−1 +θ(αwt +β)]h
However, even with a relatively low warranty price 0 ≤ wt ≤ w(λ ¯ t−1 ), consumers do not buy warranties
if the total cost exceeds the total valuation, which is pt + wt > v as formulated in the second case in (2). They may only buy the product for an acceptable product price. In this case, the firm’s one period Again, as (pt −
v−pt
¯ , 1}
t−1 +θ(αwt +β)]h
t c) min{ [(1−θ)λ v−p ¯ , 1} t−1 +θ(αwt +β)]h
when 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt > v.
is decreasing in wt for any given pt , the firm’s optimal
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profit is π(pt , wt |λt−1 ) = (pt − c) min{ [(1−θ)λ
decision should still be on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt + , where is
some infinitely small positive value.
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Notice in this case, there are no actually sales of warranty and the
t + corresponding boundary profit is (v − c − wt + ) min{ [(1−θ)λv−c−w +θ(αw t−1
¯ , 1}.
t +β)]h
Then the limited profit
on the boundary 0 ≤ wt < min{w(λ ¯ t−1 ), v} and pt = v − wt from the region of 0 ≤ wt ≤ w(λ ¯ t−1 ) and
t + pt +wt > v is lim (v−c−wt +) min{ [(1−θ)λv−c−w +θ(αw
→0+
t−1
¯ , 1}
t +β)]h
t = (v−c−wt ) min{ [(1−θ)λ v−c−w +θ(αw t−1
¯ , 1},
t +β)]h
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as is infinitely small and positive. We need to mention that π(pt , wt |λt−1 ) is discontinuous on 0 ≤
wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt as it separates the cases of having warranty sales when 0 ≤
wt ≤ w(λ ¯ t−1 ) and pt + wt ≤ v and not having warranty sales when 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt > v.
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As mentioned above, the optimal decisions for both sub-cases are on the boundary of
0 ≤ wt < min{w(λ ¯ t−1 ), v} and pt = v − wt and the corresponding one period profits are (v − c − cw ) + wt
¯
t−1 +θ(αwt +β)]h
t and (v − c − wt ) min{ [(1−θ)λ v−c−w +θ(αw
(v − c − cw ) + (cw − wt ) [(1−θ)λ
t−1
wt
¯ , 1}
t +β)]h
respectively. Furthermore,
t is larger than (v − c − wt ) min{ [(1−θ)λ v−c−w +θ(αw
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(cw − wt ) [(1−θ)λ
¯
t−1 +θ(αwt +β)]h
t−1
¯ , 1}
t +β)]h
for
any given warranty price 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v}, as we have assumed v > c + cw and considering
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selling warranty makes extra profit. It means the firm’s optimal choice could only be achieved on 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt , which is the boundary of considering selling warranty with choosing 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt + wt ≤ v as illustrated in the first case in (2).
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Therefore, the one period problem can be simplified as
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V1 (λt−1 ) =
max
0≤wt ≤min{w(λ ¯ t−1 ),v}
π(v − wt , wt |λt−1 ) ,
where π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt ) [(1−θ)λ
wt
¯.
t−1 +θ(αwt +β)]h
(A.2)
Notice here warranty only has
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actual sales when wt < min{w(λ ¯ t−1 ), v} and still has no actual sales when wt = min{w(λ ¯ t−1 ), v}. First we calculate the first and second order partial derivatives with respect to wt for the profit
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Notice here wt ≤ v. Suppose the firm chooses warranty price wt > v, then pt + wt > v and consumers do not buy warranty. Then the firm’s one period profit is decreasing in wt based on the second case in (2). Thus, it is never optimal to choose wt > v and the firm only considers wt ≤ v. 10 Notice π(pt , wt |λt−1 ) is continuous at wt = min{w(λ ¯ t−1 ), v} and pt = v − wt as there are no warranty sales for both cases.
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function of the one period problem, which are 1 ∂ wt (1 − θ)λt−1 + θβ π(v −wt , wt |λt−1 ) = − +(cw −wt ) (A.3) 2 ¯ ¯ ∂wt [(1 − θ)λt−1 + θ(αwt + β)]h [(1 − θ)λt−1 + θ(αwt + β)] h and
2[(1 − θ)λt−1 + θβ][(1 − θ)λt−1 + θ(αcw + β)] 1 ∂2 π(v − wt , wt |λt−1 ) = − ¯ . ∂wt2 h [(1 − θ)λt−1 + θ(αwt + β)]3
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By (A.4), π(v − wt , wt |λt−1 ) is concave in wt . Let
(A.4)
p [(1 − θ)λt−1 + θβ] · [(1 − θ)λt−1 + θ(αcw + β)] − [(1 − θ)λt−1 + θβ] w (λt−1 ) = θα 0
(A.5)
be solution to (A.3), which is constrained on 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v}. Based on (A.3), it is easy to see
∂ ∂wt π(v
− wt , wt |λt−1 ) ≤ 0 when wt > cw . Therefore we have the following claim.
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Claim. w0 (λt−1 ) ≤ cw .
The claim above is straightforward by (A.3). Moreover, w∗ (λt−1 ) ≤ cw is true for any general distri-
butions on h. For any general distribution, let g(·) and G(·) denote the p.d.f. and c.d.f. of consumers’
individual handling cost of product failure respectively. The one period profit function is formulated t as π(v − wt , wt |λt−1 ) = (v − c − cw ) + (cw − wt )G( (1−θ)λt−1w+θ(αw ). Notice here pt = v − wt . Then t +β)
(1−θ)λt−1 +θβ t t − wt , wt |λt−1 ) = −G( (1−θ)λt−1w+θ(αw ) + (cw − wt )g( (1−θ)λt−1w+θ(αw ) . t +β) t +β) [(1−θ)λ +θ(αw +β)]2
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∂ ∂wt π(v
∂ ∂wt π(v
t−1
t
− wt , wt |λt−1 ) ≤ 0 when wt ≥ cw . Thus, it is easy to have ¯ especially with lower bound 0 w0 (λt−1 ) ≤ cw . Furthermore, for h following uniform distribution U[0, h]
Based on this, it is easy to see
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as assumed, we can further prove that w0 (λt−1 ) ≤
cw 2
by the formulation of w0 (λt−1 ) in (A.5).
w0 (λt−1 ) is the solution to the one period problem only when it is an interior solution. It is easy to see ¯
2
¯ ¯
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(1−θαh) θβ(2−θαh)h there exists a λE = max{ (1−θ)(2−θα ¯ h ¯ [cw − (1−θαh) ¯ 2 ], 0}, such that h)
if and only if λt−1 < λE . Besides, as it is always wt , wt |λt−1 )|wt =v < 0, it indicates 0 ≤
w0 (λt−1 )
∂ ∂wt π(v
−
∂ ¯ t−1 ) > 0 ∂wt π(v−wt , wt |λt−1 )|wt =w(λ ∂ wt , wt |λt−1 )|wt =0 > 0 and ∂w π(v − t
≤ v is always satisfied. Thus an interior solution
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w0 (λt−1 ) exists, or equivalently w0 (λt−1 ) < w(λ ¯ t−1 ) if and only if λt−1 > λE . By the concavity of
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π(v − wt , wt |λt−1 ), the optimal decision of warranty price for the one period problem is w1∗ (λt−1 ) =
(
w(λ ¯ t−1 )
, if λt−1 ≤ λE
w0 (λt−1 ) , if λt−1 > λE
,
(A.6)
and the corresponding product price is p∗1 (λt−1 ) = v − w1∗ (λt−1 ). Furthermore, we can show that
w(λ ¯ t−1 ) ≤ v when λt−1 ≤ λE and w0 (λt−1 ) ≤ v when λt−1 > λE . Thus, the constraint wt ≤ v is actually redundant for the one period problem. Besides as w1∗ (λt−1 ) is also equivalently defined as
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w1∗ (λt−1 ) = min{w(λ ¯ t−1 ), w0 (λt−1 )}, it is easy to show w1∗ (λt−1 ) ≤ cw based on the claim above. Plus
the corresponding one period profit is V1 (λt−1 ) =
(
v − c − w(λ ¯ t−1 )
(v − c − cw ) + [cw −
w0 (λt−1 ) w0 (λt−1 )] [(1−θ)λ +θ(αw 0 (λ ¯ t−1 t−1 )+β)]h
, if λt−1 ≤ λE
, if λt−1 > λE
.
(A.7)
In (A.7), first it is easy to see when λt−1 ≤ λE , v − c − w(λ ¯ t−1 ) is decreasing in λt−1 . When
=
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λt−1 > λE , as
wt ∂ ¯} ∂λt−1 {(v − c − cw ) + (cw − wt ) [(1−θ)λt−1 +θ(αwt +β)]h cw −wt wt ∂ ( ∂λt−1 (1−θ)λt−1 +θ(αwt +β) ) , ¯ h
by Envelop Theorem we have
As
wt ∂ ∂λt−1 (1−θ)λt−1 +θ(αwt +β)
t = − [(1−θ)λ (1−θ)w +θ(αw t−1
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w0 (λt−1 ) d 0 ¯} dλt−1 {(v − c − cw ) + [cw − w (λt−1 )] [(1−θ)λt−1 +θ(αw0 (λt−1 )+β)]h wt ∂ = ∂λt−1 {(v − c − cw ) + (cw − wt ) [(1−θ)λ +θ(αw +β)]h¯ }|wt =w0 (λt−1 ) t t−1 cw −wt wt ∂ = { h¯ ( ∂λt−1 (1−θ)λt−1 +θ(αwt +β) )}|wt =w0 (λt−1 ) .
t +β)]
2
(A.8)
(A.9)
≤ 0 and w0 (λt−1 ) ≤ cw , we have
w0 (λt−1 ) d 0 ¯ } ≤ 0. Therefore V1 (λt−1 ) is still dλt−1 {(v − c − cw ) + [cw − w (λt−1 )] [(1−θ)λt−1 +θ(αw0 (λt−1 )+β)]h in λt−1 when λt−1 > λE . Together we can see V1 (λt−1 ) is continuous decreasing in λt−1 .
decreasing
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Then, we are going to proceed with induction. Suppose we have proved that for any n − 1 period
problem, the maximum discounted total value Vn−1 (·) is a decreasing function. Then let’s consider the
Vn (λt−1 ) =
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n period problem, which is formulated as max
pt ≥0,wt ≥0
π(pt , wt |λt−1 ) + δVn−1 ((1 − θ)λt−1 + θ(αwt + β)) .
(A.10)
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First we still separate the n period problem into two cases, which are 0 ≤ wt ≤ w(λ ¯ t−1 ) and pt +w≤ v,
and otherwise. As Vn−1 ((1 − θ)λt−1 + θ(αwt + β)) is independent of pt and continuous decreasing in wt
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for any given prior belief λt−1 by the induction condition, with the similar arguments, we can also show the optimal decision is also achieved on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt .
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Hence, the problem can be simplified as Vn (λt−1 ) =
max
0≤wt ≤min{w(λ ¯ t−1 ),v}
π(v − wt , wt |λt−1 ) + δVn−1 ((1 − θ)λt−1 + θ(αwt + β)) .
(A.11)
Because π(v − wt , wt |λt−1 ) is concave in wt when 0 ≤ wt ≤ w(λ ¯ t−1 ) and Vn−1 ((1 − θ)λt−1 + θ(αwt + β))
is decreasing in wt , we can see the optimal warranty price for the n period problem should be obtained
in the region of wt ∈ [0, w1∗ (λt−1 )]. By the characterisation of w1∗ (λt−1 ) in (A.6) and (A.8), we have 27
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π(v − wt , wt |λt−1 ) is decreasing in λt−1 for any wt ∈ [0, w1∗ (λt−1 )]. Besides, we also have Vn−1 ((1 −
θ)λt−1 + θ(αwt + β)) decreasing in λt−1 . Hence, we have proved that Vn (λt−1 ) is also decreasing in λt−1 .
Plus, as both π(v − wt , wt |λt−1 ) and Vn−1 ((1 − θ)λt−1 + θ(αwt + β)) are continuous in λt−1 and wt , we
can show Vn (·) is also continuous.
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Then we have shown V (·) is continuous decreasing.
Proof of Proposition 2. As V (·) is a continuous decreasing function based on Proposition 1 and by the similar arguments in Proof of Proposition 1, we have the optimal decision should be achieved on the boundary of 0 ≤ wt ≤ min{w(λ ¯ t−1 ), v} and pt = v − wt . First by the Proof of Proposition
1, the optimal warranty price for the one period problem w1 (λt−1 ) characterized in (A.6) satisfies
0 ≤ w1 (λt−1 ) ≤ w(λ ¯ t−1 ) and the constraint w1 ≤ v is redundant. Besides, it is also easy to see that
w∗ (λt−1 ) ≤ w1 (λt−1 ) as V ((1 − θ)λt−1 + θ(αwt + β)) is decreasing in wt by Proposition 1. Then we have
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0 ≤ w∗ (λt−1 ) ≤ w(λ ¯ t−1 ) and wt ≤ v is still redundant for the infinite horizon problem. Thus, we have
proved that the optimal product price p∗ (λt−1 ) and warranty price w∗ (λt−1 ) satisfy w∗ (λt−1 ) ≤ w(λ ¯ t−1 )
and p∗ (λt−1 ) = v − w∗ (λt−1 ).
Proof of Proposition 3. When w∗ (λt−1 ) < w(λ ¯ t−1 ) and the warranty has actual sales in the current period, as we have already shown that w1∗ (λt−1 ) ≤ cw where w1∗ (λt−1 ) is the optimal warranty price for
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the one period problem in Proof of Proposition 1 and we also have w∗ (λt−1 ) ≤ w1∗ (λt−1 ), it is easy to
see w∗ (λt−1 ) ≤ cw . Otherwise, w∗ (λt−1 ) = w(λ ¯ t−1 ) and warranty has no actual sales in the current
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period. Hereby we finish the proof.
Moreover, as illustrated in the Proof of Proposition 1, we further have w1∗ (λt−1 ) ≤ c2w when h follows ¯ as in the base assumption. Thus we further have the uniform distribution with 0 lower bound U[0, h] cw 2 .
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result of w∗ (λt−1 ) ≤
Proof of Proposition 4. Because λt has a one-to-one correspondence to wt based on the learning equation
CE
(1), to finish the proof for Proposition 4, we need to consider another equivalent problem in which the decision variables are changed from (pt , wt ) to (pt , λt ). The problem is reformulated in the following
AC
equation:
V (λt−1 ) =
max
pt ≥0 λt ≥(1−θ)λt−1 +θβ
π ˜ (pt , λt |λt−1 ) + δV (λt ) ,
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(A.12)
Proposition 1 could be easily shown by Theorem 4.7 in Stokey (1989) if π(v − wt , wt |λt−1 ) decreasing in λt−1 for any given wt is guaranteed. However, π(v − wt , wt |λt−1 ) is decreasing in λt−1 only when wt ≤ cw . As w∗ (λt−1 ) ≤ cw cannot be proved directly, though is true by Proposition 3, Theorem 4.7 in Stokey (1989) cannot be employed directly and we prove Proposition 1 in the above induction way.
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where the one period profit is λt −[(1−θ)λt−1 +θβ] t−1 +θβ] t−1 +θβ] + (pt − c + λt −[(1−θ)λ } − cw ){1 − λt −[(1−θ)λ ¯ ¯ θα (pt − c) θαλt h θαλt h ¯ t−1 ) and λt ≤ (1 − θ)λt−1 + θ[α(v − pt ) + β] π ˜ (pt , λt |λt−1 ) = . , if (1 − θ)λt−1 + θβ ≤ λt < λ(λ (v−pt ) (pt − c) min{ λ h¯ , 1} , otherwise t (A.13)
Here wt =
λt −[(1−θ)λt−1 +θβ] θα
is derived from the learning equation (1). The expression in (A.12) and
h
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(A.13) are equivalent to (4) and (2) respectively, where pt and wt are the original decision variables. In ¯ t−1 ) ¯ t−1 ) is equivalent to w(λ ¯ t−1 ) = w(λ (A.13), λ(λ ¯ t−1 ) and λ(λ . Plus, λt ≥ (1 − θ)λt−1 + θβ is equivalent ¯
to wt ≥ 0 and λt ≤ (1 − θ)λt−1 + θ[α(v − pt ) + β] is equivalent to wt ≤ v − pt .
Similarly, we can first prove the optimal decision is achieved on the boundary of (1 − θ)λt−1 + ¯ t−1 ) and pt = v − λt −[(1−θ)λt−1 +θβ] . Then the problem is simplified to a probθβ ≤ λt ≤ λ(λ θα
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λt −[(1−θ)λt−1 +θβ] λt −[(1−θ)λt−1 +θβ] . } ¯ t θα θαhλ
λt −[(1−θ)λt−1 +θβ] , λt |λt−1 ) θα
= (v − c − cw ) + {cw − ¯ t−1 ). Notice warranty may still have no actual sales when λt = λ(λ
lem of only considering the following case: π ˜ (v −
Notice there may exist multiple solutions to (6). In this work, we are interested in the steady state of
the dynamic programming problem. When multiple solutions exist for (6), we choose the corresponding solution with the highest warranty price:
0≤wt ≤w(λ ¯ t−1 )
π(v − wt , wt |λt−1 ) + δV ((1 − θ)λt−1 + θ(αwt + β))} .
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w∗ (λt−1 ) = max{ arg max
Notice here we still use the subscript “*” to denote the highest optimal warranty price compared with
ED
(5). Correspondingly, the optimal product price is still p∗ (λt−1 ) = v − w∗ (λt−1 ) and the updated consumers’ belief is denoted by
PT
λ∗ (λt−1 ) = (1 − θ)λt−1 + θ(αw∗ (λt−1 ) + β)
(A.14)
based on (1). In the following proofs, terms with subscript “*” is corresponding to the highest optimal
CE
warranty price if not further specified. Correspondingly, here we also only consider the firm’s maximum optimal decision of future state in case of multiple solutions, which is defined as
AC
λ∗ (λt−1 )
λ∗ (λt−1 ) = max{
arg max
¯ t−1 ) (1−θ)λt−1 +θβ≤λt ≤λ(λ
π ˜ (v −
λt − [(1 − θ)λt−1 + θβ] , λt |λt−1 ) + δV (λt )} . θα
Here we need to mention this definition is equivalent to the definition in (A.14) above, which is corresponding to the base framework of the main body in which (pt , wt ) are the decision variables. ¯ t−1 ) are increasing in λt−1 , which In (A.12), first it is easy to see both (1 − θ)λt−1 + θβ and λ(λ 29
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¯ t−1 )] is an increasing set in λt−1 (The definition of “induced set ordering” implies [(1 − θ)λt−1 + θβ, λ(λ ¯ t−1 ) and warranty strategy is employed, the is defined in Section 2.4 in Topkis (1998)). When λt ≤ λ(λ partial derivatives with respect to λt and λt−1 of the total profit function is
∂2 λt − [(1 − θ)λt−1 + θβ] (1 − θ){2[(1 − θ)λt−1 + θβ] + θαcw } π ˜ (v − . , λt |λt−1 ) = ¯ ∂λt ∂λt−1 θα θ2 α2 λ2t h
0. Together we can
And it is easy to see
∂2 ∂λt ∂λt−1 δV
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∂2 t−1 +θβ] ˜ (v− λt −[(1−θ)λ , λt |λt−1 ) ≥ 0. ∂λt ∂λt−1 π θα λt −[(1−θ)λt−1 +θβ] see π ˜ (v − , λt |λt−1 ) + δV (λt ) θα
By (A.15), we have
(A.15) (λt ) =
is supermodular in (λt , λt−1 ) within
¯ t−1 ) , λt−1 ≥ 0} . By Theorem 2.8.1 in the optimization region {(λt , λt−1 ) : (1 − θ)λt−1 + θβ ≤ λt ≤ λ(λ Topkis (1998), we can prove λ∗ (λt−1 ) is increasing in λt−1 .
¯ t−1 ) Proof of Proposition 5. First, we have λ∗ (λt−1 ) is bounded by (1 − θ)λt−1 + θβ ≤ λ∗ (λt−1 ) ≤ λ(λ by definition. As wt =
λ∗ (λt )−[(1−θ)λ∗ (λt−1 )+θβ] θα
≤ v must be satisfied, which is to say λ∗ (λt ) ≤ (1 −
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θ)λ∗ (λt−1 ) + θ(αv + β) must be satisfied, we have λ∗ (λt−1 ) ≤ αv + β as long as t is large enough. Thus,
we further have λ∗ (λt−1 ) is globally smaller than αv + β. By Proposition 4, as λ∗ (λt−1 ) is increasing
and bounded, there exists at least one steady state. Then we need to prove the uniqueness of the steady state.
By treating the single period profit function π(v − wt , wt |λt−1 ) as a function of λt−1 and λt , the
M
Euler equation can be derived from
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∂ λt − [(1 − θ)λt−1 + θβ] ∂ λt+1 − [(1 − θ)λt + θβ] π(v − , λt |λt−1 ) + δ π(v − , λt+1 |λt ) = 0 (A.16) ∂λt θα ∂λt−1 θα by setting λt+1 = λt = λt−1 = λ∞ . It is equivalent to
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−[2 − 2(1 − θ)δ − θ]λ2∞ + (1 − δ)(1 − θ)(αcw + 2β)λ∞ + θβ(αcw + β) = 0 , which has already been presented in (8).
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It is easy to see the above equation is a concave quadratic function in λ∞ with zero order coefficient being positive. It is also easy to verify the discriminant is positive. Thus, it has one positive and one ˜ ∞ denote the only negative solutions exactly. Thus, (8) has one and only one positive solution. Let λ
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positive solution to (8). Based on this result, we are going to show the uniqueness and details of the steady state in the following. ¯ t−1 ) = [(1−θ)λ+θβ] First notice the fixed point of (1 − θ)λt−1 + θβ and λ(λ are β and ¯ 1−θαh ¯ tively. Besides, both (1 − θ)λt−1 + θβ ≥ 0 and λ(λt−1 ) ≥ 0 when λt−1 = 0.
β ¯ + (1−αh)
respec-
˜ ∞ = β, which is not an interior solution. When cw = 0, we can show that the solution to (8) is λ ˜ ∞ = β. In this case, it is easy to have w∞ = 0 and Then the unique and stable steady state is λ∞ = λ 30
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p∞ = v as w∞ =
λ∞ −β α
by learning equation (1) and p∞ = v − w∞ by Proposition 2.
12
Otherwise, when cw > 0, as α > 0, β > 0, and δ < 1, we can show the solution to (8) is strictly larger than β, which indicates β is not the steady state and λ∞ > β. Then we can show the solution to ¯ ¯ w − {2 − [2(1 − θ)δ + θαh]}β ¯ h ¯ ≤ 0. (8) is an interior solution if and only if (1 − αh){1 − [(1 − θ)δ + θαh]}c
is not an interior solution, the unique and stable steady state is w∞ =
¯ βh ¯ + (1−αh)
and p∞ = v −
¯ βh ¯ +. (1−αh)
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¯ ¯ w − {2 − [2(1 − θ)δ + θαh]}β ¯ h ¯ ≤ 0, which implies λ ˜ ∞ is Then if (1 − αh){1 − [(1 − θ)δ + θαh]}c ˜ ∞ < λ(λ ¯ t−1 ), we can show λ ˜ ∞ is the unique steady an interior solution, i.e. (1 − θ)λt−1 + θβ < λ ˜ ∞ , w∞ = λ˜ ∞ −β and p∞ = v − w∞ . Otherwise, if state and it is stable, which is to say λ∞ = λ α ¯ ¯ ¯ ¯ ˜ ∞ ≥ λ(λ ¯ t−1 ) and λ ˜∞ (1 − αh){1 − [(1 − θ)δ + θαh]}cw − {2 − [2(1 − θ)δ + θαh]}β h > 0, which implies λ β ¯ +, (1−αh)
which is to say λ∞ =
β ¯ +, (1−αh)
c¯w =
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Proof of Proposition 6. From Proof of Proposition 5, we can see there exists a ¯ h ¯ {2 − [2(1 − θ)δ + θαh]}β ¯ + {1 − [(1 − θ)δ + θαh]}} ¯ + {(1 − αh)
such that in the steady state, free warranty is offered and all the consumers who buy the product purchase the warranty when cw = 0, part of the consumers who buy the product purchases the warranty
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when 0 < cw < c¯w and no actual warranty sales when cw ≥ c¯w . Based on the characterisation on c¯w in ¯ and δ, while it is independent of the previous formula, it is easy to show c¯w is increasing in θ, α, β, h,
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v and c.
Proposition 15 (Impact of cw ). Both the steady state consumers’ belief λ∞ and warranty price w∞ are increasing in cw . Plus, both the total profit V (λ∞ ) and the per period profit on product sales v − c − w∞
then increasing in cw .
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are decreasing in cw , while the per period profit on warranty sales (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
is first decreasing
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Proof of Proposition 15 (Impact of cw ). Let EL(λ) = −[2 − 2(1 − θ)δ − θ]λ2 + (1 − δ)(1 − θ)(αcw +
2β)λ + θβ(αcw + β), which is the Euler function to facilitate the proof. By the Proof of Proposition 5, ˜ ∞ as the only positive solution to EL(λ) = 0. Actually it is expressed as we have defined λ
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p (1 − δ)2 (1 − θ)2 (acw + 2β)2 + 4θβ[2(1 − θ)(1 − δ) + θ](αcw + β) ˜∞ = λ . 2[2(1 − θ)(1 − δ) + θ] (A.17) p Furthermore, it is easy to show that EL( β(αcw + β)) ≥ 0. Together with Proposition 3, we have p ˜ ∞ ≤ α cw + β. Notice here we have used the special result of w∗ (λt−1 ) ≤ cw as β(αcw + β) ≤ λ (1 − δ)(1 − θ)(acw + 2β) +
2
12
2
∗
∗
p∞ = v − w∞ is derived from p (λt−1 ) = v − w (λt−1 ) by Proposition 2.
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demonstrated in the Proof of Proposition 3, which only holds when h follows uniform distribution with ¯ 0 lower bound U[0, h].
First by (6), as π(v − wt , wt |λt−1 ) is decreasing in cw , it is easy to have the discounted total profit
V (·) is decreasing in cw for any given state by coupling method.
(i) When cw = 0, free warranty
is employed in the steady state, which firstly implies the steady state warranty price and belief are w∞ = 0 and λ∞ = β respectively and constant in cw . Besides, we have the steady state total profit 1 1−δ (v
− c − cw ) decreasing in cw , one period demand for warranty
direct profit of warranty sales (w∞ −
¯ w∞ )+ (h− λ∞ cw ) ¯ h
¯ w∞ )+ (h− λ∞ ¯ h
= 1, one period
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V (λ∞ ) =
= −cw decreasing in cw , and one period profit of
product sales v − c − w∞ = v − c constant in cw . (ii) When 0 < cw < c¯w , part of the consumers buys
the warranty in the steady state. Under this circumstance, λ∞ is the solution to the Euler equation
∂ EL(λ∞ ) ∂cw ∂ ∂ d dcw λ∞ = − ∂ EL(λ∞ ) . As − ∂λ EL(λ∞ ) ≥ 0 and ∂cw EL(λ∞ ) ≥ 0, we can see λ∞ ∂λ w∞ = λ∞α−β , we have w∞ is also increasing in cw and v−c−w∞ is decreasing in cw .
EL(λ) = 0 and we have is increasing in cw . As
¯ w∞ + d (h− λ∞ ) ¯ dcw h
in cw . Besides, by we have
¯ w∞ )+ (h− λ∞ ¯ h
= ( ∂λ∂∞
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Based on V (λt ) decreasing in cw and λt and λ∞ increasing in cw , it is easy to have V (λ∞ ) is decreasing ¯ w∞ )+ (h− λ∞ ) ¯ h
· ( dcdw λ∞ ) +
¯ w∞ + ∂ (h− λ∞ ) ¯ ∂cw h
= − αλβ2 · ( dcdw λ∞ ) + 0 ≤ 0, ∞
decreasing in cw . Besides, by the similar approach we can show (w∞ − cw ) ¯ w∞ )+ (h− λ∞ = 0 when cw = 0 ¯ h w∞ + ¯ (h− λ ) ∞ cw ) is decreasing in cw ¯ h
is convex in cw . Plus, notice (w∞ − cw ) there exists a c˜w such that (w∞ −
¯ w∞ )+ (h− λ∞ ¯ h
or cw = c¯w . Hence, we can show when cw ≤ c˜w and increasing in
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cw when cw > c˜w . (iii) When cw ≥ c¯w , expensive warranty is employed and there are no actual sales
of warranty. The total profit is only consisted as the profit of product sales and there is no profit for ¯ β βh ¯ + and λ∞ = (1−αh) ¯ + are (1−αh) w ∞ )+ ¯ ¯ w∞ )+ (h− ( h− λ∞ λ∞ = 0, and (w∞ − cw ) ¯ ¯ h h
have V (λ∞ ) = in cw .
ED
warranty sales. In this instance, w∞ = 1 1−δ (v − c − w∞ ),
constant in cw . Accordingly, we = 0, which are also all constant
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Proof of Proposition 7 (Impact of θ). In the following proof, despite of showing Proposition 7, we are also going to show the comparative statics on θ, which is: the steady state consumers’ belief λ∞ and
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warranty price w∞ are decreasing in θ; besides, both the total profit V (λ∞ ) and per period profit on product sales v − c − w∞ are increasing in θ, whereas the profit on warranty sales (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
is decreasing in θ.
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First, V (·) is not monotone in θ and how V (·) changes with θ is state dependent. Second, by Proposition 6, c¯w is increasing in θ. Thus, there exists a θ¯ such that expensive warranty is employed ¯ (i) When θ ≤ θ, ¯ expensive warranty is when θ ≤ θ¯ and partial warranty is employed when θ > θ.
¯ βh β ¯ + and λ∞ = (1−αh) ¯ + are constant in θ. Accordingly, we have V (λ∞ ) = (1−αh) w ∞ )+ ¯ ¯ w∞ )+ ( h− (h− λ∞ λ∞ 1 ¯ (v − c − w ), = 0, and (w − c ) = 0 are all constant in θ. (ii) When θ > θ, ∞ ∞ w ¯ ¯ 1−δ h h ∂ partial warranty is employed, where λ∞ is the solution to EL(λ) = 0. First we can show ∂θ EL(λ∞ ) ≤ 0
employed, where w∞ =
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based on the fact that
p β(αcw + β) ≤ λ∞ ≤ α c2w + β. Therefore, λ∞ as well as w∞ is decreasing in θ.
¯ w∞ )+ (h− w∞ λ∞ 1 [(v − c − c ) + (c − w ) ] decreasing in λ , w w ∞ ∞ ¯ ¯ 1−δ λ∞ h h w + ∞) ¯ (h− λ∞ increasing in λ∞ ; while they are all independent of θ (despite decreasing in in λ∞ , and (w∞ −cw ) ¯ h 1 λ∞ is a function of θ). As λ∞ is decreasing in θ, we have V (λ∞ ) = 1−δ [(v − c − cw ) + (cw − w∞ ) λw∞h¯ ] ∞ ¯ w∞ )+ ¯ w∞ )+ (h− (h− λ∞ λ∞ increasing in θ, increasing in θ, and (w − c ) decreasing in θ. Besides, it is easy to ∞ w ¯ ¯ h h
Accordingly, it is easy to show V (λ∞ ) =
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see v − c − w∞ increasing in θ.
¯ Again, despite of showing Proposition 8, we are also going to Proof of Proposition 8 (Impact of h). ¯ which is: both the steady state consumers’ belief λ∞ and warranty show the comparative statics on h, ¯ besides, the total profit V (λ∞ ) and per period profit on product sales price w∞ are increasing in h; ¯ w∞ )+ (h− λ∞ ¯ are all decreasing in h. v − c − w∞ and warranty sales (w∞ − cw ) ¯ h
Note as we have shown that free warranty is employed if and only if cw = 0 in Proof of Proposition
Proof of Proposition 15.
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15, we only consider cw > 0 in the following proofs. Plus, the function EL(λ) is the same as defined in ¯ for any given state again. Second, by First, by coupling method, we can show V (·) is decreasing in h ¯ Therefore, there exists a h ¯ u such that expensive warranty is employed Proposition 6, c¯w is increasing in h. ¯≤h ¯ u and partial warranty is employed when h ¯>h ¯ u . (i) When h ¯≤h ¯ u , expensive warranty is when h ¯ β ¯ employed, where w∞ = β h¯ + and λ∞ = ¯ + are increasing in h. Accordingly, we have V (λ∞ ) = ¯ − c − w∞ ) decreasing in h;
¯ w∞ )+ (h− λ∞ ¯ h
(1−αh)
= 0 and (w∞ − cw )
M
(1−αh)
1 1−δ (v
¯ w∞ )+ (h− λ∞ ¯ h
¯ (ii) When = 0 constant in h.
¯
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¯>h ¯ u , partial warranty is employed, where λ∞ is the solution to EL(λ) = 0. As EL(λ) is independent h ¯ λ∞ and w∞ are independent of h. ¯ Accordingly, we have V (λ∞ ) = 1 [(v − c − cw ) +(cw − w∞ ) w∞¯ ] of h, w∞ +
¯
w∞ +
1−δ
λ∞ h
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¯ (h− λ¯∞ ) increasing in h, ¯ (w∞ − cw ) (h− λ¯∞ ) decreasing in h ¯ (w∞ − cw is constant and decreasing in h, h h ¯ and v − c − w∞ constant in h. ¯ negative in h),
Proof of Proposition 9 (Impact of α). Here we are still going to show the comparative statics on α in addition, which is: the steady state consumers’ belief λ∞ is increasing in α while the warranty price
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w∞ is first increasing then decreasing in α; besides, both the total profit V (λ∞ ) and per period profit on warranty sales (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
are decreasing in α, while the per period profit on product sales
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v − c − w∞ is first decreasing then increasing in α.
First, by coupling method, V (·) is decreasing in α. Second, by Proposition 6, c¯w is increasing in α,
which implies there exists a α ¯ such that expensive warranty is employed when α ≤ α ¯ and partial warranty is employed when α > α ¯.
and λ∞ =
β ¯ + (1−αh)
α; and
¯ h
¯ w∞ )+ (h− λ∞
(i) When α ≤ α ¯ , expensive warranty is employed, where w∞ =
are increasing in α. Accordingly, we have V (λ∞ ) =
= 0 and (w∞ − cw )
¯ w∞ )+ (h− λ∞ ¯ h
1 1−δ (v
¯ βh ¯ + (1−αh)
− c − w∞ ) decreasing in
= 0 constant in α. (ii) When α > α ¯ , partial warranty is
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employed, where λ∞ is the solution to EL(λ) = 0. As we can show w∞ =
˜ ∞ −β λ α
∂ ∂λ∞
further show
∂ EL(λ∞ ) ∂α ∂ − ∂λ EL(λ∞ ) ¯ w∞ )+ (h− λ∞
·
≥ 0, λ∞ is increasing in α. Besides,
is decreasing in α. As V (λt ) is decreasing in α and λt and λ∞ is increasing
in α, we have V (λ∞ ) is also decreasing in α. As ¯ w∞ )+ (h− λ∞ ¯ h
∂ ∂α EL(λ∞ )
+
¯ w∞ + ∂ (h− λ∞ ) , ¯ ∂α h
¯ w∞ + d (h− λ∞ ) ¯ dα h
=
¯ w∞ + ∂ (h− λ∞ ) ¯ ∂λ∞ h
·
d dα λ∞
+
¯ w∞ + ∂ (h− λ∞ ) ¯ ∂α h
and based on the expression of λ∞ shown in (A.17), we can
is increasing in α. By similar approach, we can show (w∞ − cw )
¯ h
=
¯ w∞ )+ (h− λ∞ ¯ h
is
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decreasing in α. Besides, it is easy to see v − c − w∞ is increasing in α.
¯ a , where Proof of Proposition 10. By (10) and (11), it is easy to see that π∞ ≥ πF if and only if λa ≥ λ ¯a = λ
(
α2 c2w 4(λ∞ −β)(αcw +β−λ∞ ) λ∞ ¯ λ2∞ −[α(1−αh)+2β]λ ∞ +β(αcw +β) ¯ 2 λ∞ α2 h
, if (cw − w∞ ) λw∞h¯ ≤ , otherwise.
∞
cw 2 ;
(A.18)
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The threshold value above is presented in terms of λa , which is demonstrated as the dash-dotted ¯ a is represented in cw and λ∞ in (A.18) . Nevertheless, λ∞ is a function line in Figure 2. Also notice λ of the system parameters including cw which is given in Proposition 5. ¯ a ≥ λ∞ and λ ¯ a is concave increasing in Based on Proposition 5 and (A.18), we can further verify that λ
¯ a = λ∞ = β when cw = 0; and λ ¯ a = λ∞ = cw . Besides, λ
β ¯ + (1−αh)
when cw > c¯w and (cw −w∞ ) λw∞h¯ > ∞
cw 2 .
˜
Furthermore, notice λa is linear increasing in cw and λa = 0 when cw = 0, which is λa (cw ) = cδ0 cw as ˜ a such that λa < λ ¯ a when λa < λ ˜a described in Section 5. Then, it is easy to see there exists a unique λ
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¯ a when λa ≥ λ ˜ a . Based on the definition of λ ¯ a in (A.18), we finish the proof of Proposition and λa ≥ λ
ED
10.
˜ a is the value such that hiding the information about the true Proof of Proposition 11. First, notice λ
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product reliability to inducing consumer learning and disclosing the information about the true product ˜ a . The expressions of π∞ and πF are reliability yield the same profit, which is π∞ = πF when λa = λ illustrated in (11) and (10) respectively. Second, the system parameters θ, α, and β are only related
CE
to learning, while they do not affect the consumer behaviours as well as the firm’s decisions in the disclosing case. In addition, π∞ is decreasing in λ∞ . Then, based on (11), (10), and the comparative
AC
statics in Section 4.3, as λ∞ is decreasing in θ and increasing in α and β, while πF is independent from ˜ a is decreasing in θ, and increasing in α and β. θ, α, and β, it is easy to see that λ ¯ is the largest value of consumers’ individual handling costs, which characterizes both the average h
cost and heterogeneity of consumer individual handling. It affects the consumer behaviours as well as the firm’s profits in both cases of adaptive learning and disclosing. We cannot prove the comparative ¯ on λ ˜ a directly. Nevertheless, based on the comparative statics in Section 4.3 which include statics of h ¯ and the expression of λ ¯ a in (A.18), we can show that λ ¯ a is increasing in h ¯ while λa is λ∞ increasing in h 34
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¯ for any given cw . As both λ ¯ a and λa are increasing in cw and λ ˜ a is the intersection independent from h ¯ a and λa as functions of cw , it is easy to prove that λ ˜ a is increasing in h. ¯ of λ
Proof of Proposition 12. Proposition 12 can be proved by induction which is similar to the Proof for Proposition 1, where the property of Xt+1 stochastically increasing in xt is used.
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Proof of Proposition 13. In this proof, we show pR (λt−1 , xt ) = v − wR (λt−1 , xt ) as well.
First, as pt is not directly involved in consumer learning, which means pt only affects the current
period profit π R (pt , wt |λt−1 , xt ) and has no impact on the future profit E[V R ((1 − θ)λt−1 + θ(αwt +
xt ), Xt+1 )] for any given wt and (λt−1 , xt ). Then, it is easy to show pR (λt−1 , xt ) = v − wR (λt−1 , xt ).
Moreover, by Proposition 12, as V R (λt−1 , xt ) is decreasing in λt−1 and xt , we can show wR (λt−1 , xt ) ≤
w ¯ R (λt−1 , xt ) by the argument which is similar to Proof of Proposition 2.
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Then, the proof of either wR (λt−1 , xt ) ≤ cw or wR (λt−1 , xt ) = w ¯ R (λt−1 , xt ) is similar to Proof of ¯ Proposition 3. Besides, we can still further prove that wR (λt−1 , xt ) ≤ cw when h follows U[0, h]. 2
Proof of Proposition 14. The proof is similar to Proof of Proposition 4. First, we again reformulate the problem into the following problem in which the decision variable is the consumers’ belief in the
λR (λt−1 , xt ) = max{
arg max
π ˜ R (v −
λt −[(1−θ)λt−1 +θxt ] , λt |λt−1 , xt ) θα
+ δE[V R (λt , Xt+1 )]} . (A.19)
ED
¯ R (λt−1 ,xt ) (1−θ)λt−1 +θxt ≤λt ≤λ
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following period λt , while the decision variable in the origin problem in (14) is the warranty price wt .
Notice here we have already use the results from Proposition 13 and let pt = v − ¯ R (λt−1 , xt ). λt constrained on λt ≤ λ
λt −[(1−θ)λt−1 +θxt ] θα
and
CE
PT
∂2 t−1 +θxt ] ˜ R (v − λt −[(1−θ)λ , λt |λt−1 , xt ) ≥ 0 and ∂λt ∂λt−1 π θα λt −[(1−θ)λt−1 +θxt ] ∂2 ∂2 R R (λ , X ˜ (v− , λt |λt−1 , xt ) ≥ 0. Besides, it is also easy to see ∂λt ∂λ δE[Vn−1 t t+1 )] = ∂λt ∂xt π θα t−1 2 0 because δE[V R (λt , Xt+1 )] is independent of λt−1 and ∂λ∂t ∂xt δE[V R (λt , Xt+1 )] = 0 because Xt+1 is in-
It is easy to show
dependent of λt as assumed in Section 6 and Xt+1 is independent of xt by the i.i.d. assumption specified
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t−1 +θxt ] in Proposition 14. Hence, we have shown that π ˜ R (v − λt −[(1−θ)λ , λt |λt−1 , xt ) + δE[V R (λt , Xt+1 )] θα ¯ R (λt−1 , xt ) are both increashas increasing differences in (λt , (λt−1 , xt )). Plus, as (1 − θ)λt−1 + θxt and λ
¯ R (λt−1 , xt )] is an increasing set in (λt−1 , xt ). Finally by ing in (λt−1 , xt ), it implies [(1 − θ)λt−1 + θxt , λ Theorem 2.8.1 in Topkis (1998), we can prove λR (λt−1 , xt ) is increasing in (λt−1 , xt ).
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