Price Adjustment Policy with Partial Refunds

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Price Adjustment Policy with Partial Refunds Dinah Cohen-Vernik, Amit Pazgal ∗ Department of Marketing, Jones Graduate School of Business, Rice University, 6100 Main St, Houston, TX 77005, United States

Abstract Have you ever purchased an item only to notice a short while later that its price was reduced? Many retailers offer to refund customers the full price difference as long as the discount occurred within a short period of time after the original purchase. Such policy looks attractive to consumers as it shields them from future price fluctuations. But can this policy be advantageous for the retailer? In this paper we investigate the price difference refund policy (commonly referred to as price adjustment) and demonstrate how it can result in a higher profit even if all consumers request and receive a price adjustment. Further, the existing literature and practice both assume that if a price adjustment policy is employed, the consumers should get the full price difference refunded. In this paper we endogenize the refund amount, allowing the retailer to determine the optimal percentage of the price difference to be returned to consumers. We fully characterize the conditions under which it is optimal to offer a partial or greater than full price adjustment to customers as well as the optimal refund percentage. Additionally, we demonstrate that the practice of limiting the price adjustment option to a short period after the purchase incident is not necessarily in the best interest of retailers. © 2017 New York University. Published by Elsevier Inc. All rights reserved. Keywords: Price adjustment; Price discrimination; Price protection; Best-price provision; Game theory

Introduction “Buy with confidence from Sears. If an item you have purchased from Sears goes on sale for a lower price within 14 days of your purchase, Sears will refund the difference” [Sears] “If we lower our in-store or online price during the return and exchange period [15-45 days], we will match our lower price, upon request.” [Best Buy] The above quotes represent examples of an increasingly common price adjustment policy employed by many retailers. Under such policy the retailer promises to refund all consumers the full price difference, should the product’s price drop within a specified period of time after a consumer purchases it. In order to get the refund, the consumers need to actively request the price adjustment, produce a valid receipt and are limited to a single request in a short period of time after their original purchase.1 The abovementioned price adjustment strategy is related to the familiar price matching policy, where a retailer declares that it will “not be undersold” and promises that if consumers find the same product for a cheaper price in a competing store, they can request and get the product at the same price as offered by the competitors. Such price matching policies started gaining popularity in the early 1980 (Arbatskaya, Hviid, and Shaffer 2004; Lai, Debo, and Sycara 2010), and are common and well known to most consumers today. Many stores actively advertise their policy of matching competitors’ prices (e.g., Walmart’s ad campaign (2014–2015), Lowes, Best Buy, etc.). In contrast, the price adjustment policy, which is similar to the price matching policy in that the retailer promises to match prices of its future self (as opposed to matching a competitor), is a newer policy that is clearly stated but rarely advertised. ∗

Corresponding author. E-mail addresses: [email protected] (D. Cohen-Vernik), [email protected] (A. Pazgal). 1 While the majority of retailers refer to this policy as “price adjustment,” it is sometimes also called “price protection,” “price matching,” “favorite customer clause,” or “best price provision.” http://dx.doi.org/10.1016/j.jretai.2017.08.002 0022-4359/© 2017 New York University. Published by Elsevier Inc. All rights reserved.

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A similar policy, usually referred to as “price protection policy” is sometimes also offered by credit card companies, which at their own expense issue a price difference refund when the price of a recently purchased item drops. This policy is used to incentivize consumers to use a specific credit card. It is important to note that this is a completely different policy from the one we are focusing on in this paper: a credit-card-offered price protection policy does not cost the retailer anything, and the terms of such policy are not in the retailer’s control. In contrast, retailer-offered price adjustment policy, which we investigate here, is a sellers’ tool to improve their own profits. Price adjustment policies can be observed in a variety of industries: ranging from fashion to sporting goods, electronics, home improvement, financial services, and even in many B2B settings.2 A survey conducted on 162 consumers3 between the ages of 25 and 55 has demonstrated that a full 89% of them were aware of a retailer-offered price adjustment policy, naming more than a dozen different large retailers with such policies in place. Common wisdom suggests that some consumers will not request (and hence will not obtain) the promised price difference refund, and that this is the only driver for the profitability and common use of the price adjustment policy. Contrary to this belief, we show that optimal price adjustment policies may lead to higher profit even when the retailer guarantees automatic unsolicited price difference refunds. That is, the price adjustment policies increase the retailers’ profit by changing the purchasing incentives of sophisticated forward-looking consumers rather than by preying on their forgetfulness. Thus it is important for retailers to realize that as price adjustments become easier for consumers to obtain, for example, via online interactions with a retailer and price-checking services provided by third-parties, it will not necessarily hurt their profitability. We further show that the profitability of requiring consumers to request the price adjustment depends on whether consumers are optimistic (overestimate their likelihood of requesting the refund) or skeptical (underestimate the likelihood). Consumer optimism and forgetfulness can indeed improve retailer profits as compared to offering automatic price adjustment. Whereas if consumers underestimate their likelihood of asking for the refund, then profitability is hurt, and the automatic price adjustment policy would do better. Further, if the seller requires consumers to request the price adjustment, then it may be optimal to promise even more than 100% price difference refund. A well-known solution to the durable goods monopoly (Coase) problem is to offer a full price adjustment in the second period to all consumers who bought the good in the first period (Tirole 1988). This solution effectively results in the monopolist not lowering the price of the product in the second period. The monopoly price is charged in both periods, all the consumers buy the product in period 1, and there are no production or sales in period 2. In this paper we generalize this result by suggesting a new potential modification of the commonly practiced price adjustment policy to improve its profitability. Specifically, we suggest a that in some situations, the retailer should advertise and offer only a partial price difference refund (i.e., refunding less than 100% of the price difference) when prices are dropped, and sometimes a higher than 100% price difference refund should be offered, as opposed to the standard full price difference refund. As a result, in our model the price can actually drop in the second period (in contrast to the Coase result), sales occur in the second period, and the price difference refund is actually given to consumers. We show that when consumers and retailers discount the future at different rates,4 sometimes no price adjustment should be offered, and sometimes the price adjustment should be only partial (we characterize the optimal fraction of refund for this partial price adjustment policy). Further, when some consumers do not request the price adjustment, the retailer should offer a higher than 100% price difference refund to optimize the profit. The intuition for the original Coase result is that full price adjustment is meant to discourage consumers from opportunistically waiting for a price drop in the next period. Indeed, offering full price difference refund ensures that consumers buy early. However, it also discourages the retailer from lowering the price in the future (thus serving as a price-commitment mechanism) and hence prevents profitable price discrimination. Now consider the opposite situation: when no price adjustment is offered. In this case the retailer is able to price discriminate by charging a higher price in the first period, then lowering it in subsequent periods to sell to consumers with lower valuation. However, in this situation there is a group of consumers who value the product above the first period price but anticipate the price reduction and thus strategically wait for it. These consumers are costly to the retailer because they could have bought earlier and at a higher price. In this paper we show that offering partial price adjustment allows the retailer to price discriminate, while at the same time mitigating the impact of strategically waiting consumers (by decreasing the number of waiting consumers and the amount of refund the buying consumers get). Thus, we argue, that considering a new suggested policy of partial price difference refund (in addition to a full price adjustment or none, which are currently observed in practice), can improve the retailer’s profit even further, especially in a situation where the retailer is more patient than the consumers.

2

For a comprehensive list see http://lifehacker.com/all-the-stores-that-will-give-you-a-refund-if-a-price-d-1661273299. Full details of the survey are available from the authors upon request. 4 Fudenberg and Levine (1989) and Lehrer and Pauzner (1999) argued similarly that economic players usually discount the future differently leading to a more general and interesting sets of equilibria. 3

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Our paper offers several important managerial insights. First, we demonstrate that a price adjustment policy should be used by retailers as it leads to higher profits for a wide range of parameters. Second, when offering a price adjustment policy, the retailers should be mindful that offering a full price difference refund (which is commonly observed in practice) is not always optimal. Instead, we propose they consider the full range of possibilities: from only a partial refund, to a more than 100% price difference refund. The optimal refund amount depends on consumer patience and the likelihood of them in fact requesting the refund. Third, contrary to common belief, the profitability of the price adjustment policy does not solely hinge on consumers intending to obtain the refund but forgetting to request it. Rather, the promise of a refund influences the timing of the purchase by alleviating consumer concerns about future price drops. Moreover, when consumers tend to underestimate their likelihood of requesting the refund, offering automatic (unrequested) price difference refund to consumers improves retailer’s profits, and may also offer additional marketing and promotional benefits. Finally, in practice, most retailers guarantee the price adjustment only within a very limited time horizon (e.g., 14 days for Target and Sears, 30 days for Walmart). Again, we recommend that the retailer should consider a wider range of possibilities: for example extended refund windows, and multiple price adjustments. We specify conditions under which giving the refund at any time during a sales horizon leads to higher profits than limiting the refund to a short period of time after the original purchase takes place. Moreover, we show that refunding once as opposed to refunding each time the price drops is not always better for the retailer, especially when consumers are patient. The rest of the paper is organized as follows: Section “Related literature” provides a review of the relevant literature, Section “The Base Model” presents the base model and main results, Section “Model Extensions” considers extensions of the main model and outlines the additional results, Section “Three-Period Scenario” considers a model with a longer time horizon, and finally Section “Conclusions” concludes. Related Literature This paper is positioned at the intersection of three research streams: price protection policies offered by retailers have been studied in similar settings in economics, operations management, and marketing. For example, there are several papers that investigate the price adjustment policy in a monopolistic setting. Butz (1990) considers an infinite time horizon and demand uncertainty. He demonstrates that best-price provision are dynamically consistent and can benefit the seller by allowing him to postpone the price determination until after the demand uncertainty had been resolved. Without uncertainty, (as shown by Stokey (1979)) this result no longer holds as the monopoly retailer is always better off offering and selling the product only once in the first period. Png (1991) requires demand uncertainty to demonstrate the benefits of price adjustment. He shows that capacity constraints in combination with demand uncertainty is what makes the price protection policy beneficial for retailers. Levin, McGill, and Nediak (2007) study a revenue management problem where sellers may offer, for a fee, the option of price adjustment. They characterize the optimal price path and option pricing structure but assume consumers to be non-strategic. Further, Lai, Debo, and Sycara (2010), introduce a capacity constraint and declining product valuation to the mix. Their results – the optimality of the price protection policy and the optimal inventory level – depend on the ratio of strategic consumers to non-strategic. Similarly, Xu (2011) studies price adjustment in monopolistic settings in continuous time with strategic consumers whose product valuation stochastically drops once. She characterizes the optimal policy in terms of policy length and refund scale. There are two main features that distinguish the model presented here from the existing price protection literature in monopolistic setting. Specifically, we demonstrate that neither capacity constraints nor demand uncertainty are necessary to generate higher retail profits under the price adjustment policy. In a stylized model with monopolistic seller and forward looking consumers, without demand uncertainty or capacity constraint not only do we demonstrate the benefits of the price adjustment policy to the retailers, but also offer a set of new results. We show that promising a full price drop refund is not always optimal for the retailer. Instead we derive the optimal refund fraction as a function of the model parameters. Further, there are several papers that require a competitive setting to demonstrate the benefits of the price protection policy. For example, Cooper (1986), and Nelson and Winter (1993) suggest that price protection policies works as a commitment mechanism in a competitive environment, leading to higher prices and profits. Further, Salop (1986), and Holt and Scheffman (1987) suggest that price matching with competitors mitigates price competition between retailers and lowers the customers’ defection probability. On the other hand, Chen, Narasimhan, and Zhang (2001) show that in the presence of loyal consumers as well as price shoppers, price-matching policies enhance the price search, leading to lower overall retailer’s profits. Further, Hviid and Shaffer (2010) analyze the combination of both price matching and price adjustments and show that there are complementarities between them, which can lead to even higher prices and profits for the retailers. The intuition is similar — these policies reinforce each other and together act as a stronger price commitment mechanism inducing the retailers to not lower prices in the future. Png and Hirshleifer (1987) and Corts (1997) highlight the fact that price matching policies signal low prices and allow for discrimination between informed and uninformed consumers. In a behavioral study, Srivastava and Lurie (2001) confirm that price matching policies indeed change consumer perceptions about the expected generalized level of prices at a store. Please cite this article in press as: Cohen-Vernik, Dinah, and Pazgal, Amit, Price Adjustment Policy with Partial Refunds, Journal of Retailing (xxx, 2017), http://dx.doi.org/10.1016/j.jretai.2017.08.002

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In contrast to the above literature, we demonstrate that a competitive environment is by no means necessary to ensure the profitability of the price adjustment policy. Additionally, the existing literature on price protection (both in monopolistic and competitive settings) assumes that the retailer offers a single price adjustment. In this paper we suggest that the retailer might benefit from offering multiple price adjustments as the price decreases over several periods. The Base Model We begin with the base scenario where a single retailer is selling a certain product over two periods to a unit mass of potential consumers. Consumers require at most a single unit of the product and can purchase it either in the first or second period. Consumers are heterogeneous in their valuations (willingness to pay), v, for the product and the seller knows only the distribution of consumers’ valuations and not the specific willingness to pay of each shopper. For the sake of simplicity we assume that consumer valuations are uniformly distributed between zero and one: v ∼ U[0, 1]. Further, the retailer may offer a price adjustment policy: a policy that guarantees consumers that if the price drops in the second period, then a fraction f ≥ 0 of the price difference will be refunded to them. In the base model we assume that the price adjustments are done automatically, that is the consumers get price difference refunds without having to explicitly request it from the seller. Note that f = 0 means the retailer offers no refund, f = 1 means the full price difference is refunded to consumers and f > 1 (f < 1) means that the retailer offers to compensate the consumer with an amount larger (smaller) than the actual price drop. At the beginning of each period the retailer announces the product price p1 or p2 respectively, and then utility maximizing forward looking consumers make their purchase decisions. In the first period, the consumers need to decide if they want to buy the product immediately, or wait until the next period and consider buying the product then at a potentially lower price. The net present utility of each option for a consumer with valuation v is given by: U1 = v − p1 + δC f (p1 − p2 )+ U2 = δC (v − p2 )

(2.1)

Where 0 < δC ≤ 1 represents the consumer’s inter-temporal discount factor and (x)+ = Max (0, x). Note that at the beginning of the first period the consumers do not actually know the second period price. Hence, when making their decision whether to buy or wait, forward looking consumers need to rationally expect and infer the price that will be charged by the seller in the second period. Since all the parameters of the environment are known to them, sophisticated consumers can correctly anticipate the second period prices by solving the (full information) seller’s optimization problem. We look for a sub-game perfect Nash equilibrium of this game5 by first considering pricing and purchasing decisions in the second period. It is straight forward to see that if a consumer with valuation v decides to buy the product in the first period, all consumers with higher valuations would find it even more beneficial to buy in the first period. Let v denote the smallest valuation of the consumers who have purchased the product in period 1: in the second period, only consumers with lower valuations, v < v, are left in the market. Then, for any given v, the seller’s profit maximization problem in the second period is thus given by:   maxΠ2 (p2 ) = max (v − p2 )p2 − f (1 − v)(p1 − p2 )+ p2 p2 . (2.2) s.t. p2 ≤ p1 Note that the first term in the maximization represents the profit from consumers who buy the product in the second period while the second term represents the partial refund of the (potential) price drop to consumers who bought in the first period. The second period price that maximizes the seller’s second period profit given that only consumers with valuation v < v are left in the market in the second period is thus:   1 (f + v(1 − f )), p1 . p∗2 (v) = min (2.3) 2 Now, substitute this optimal second period price (2.3) into the utilities given by expression (2.1), and equate consumer utilities from buying in the first (U1 ) and in the second period (U2 ): 2p1 (1 − fδC ) − v(2 − (1 − f 2 + 2f )δC ) = (1 − f )fδC .

(2.4)

Solve this equation for v to find the location of the consumer indifferent between buying in period 1 and 2, v: v(p1 ) =

5

2p1 (1 − fδC ) − (1 − f )fδC . 2 − (1 + 2f − f 2 )δC

(2.5)

For a full formal treatment of a T-period model with no price adjustment (f = 0) see Besanko and Winston (1990).

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Fig. 1. Seller’s profit in a two-period model (δC = 0.55).

Substituting (2.5) into expression (2.3) yields the optimal second period price as a function of the first period price6 : p∗2 (p1 ) =

p1 (1 + f 2 δC ) + f (1 − δC + p1 (1 + δC )) . 2 − (1 + 2f − f 2 )δC

(2.6)

We now turn to solving the seller’s profit maximization problem in the first period:   maxΠ(f, p1 ) = max (1 − v(p1 )) p1 + δS Π2 (p∗2 (p1 )) p1 p1 , s.t v(p1 ) ≤ 1

(2.7)

where 0 < δS ≤ 1 is the retailer’s inter-temporal discount factor. The inter-temporal discount factors δC and δS are considered to be distinct in order to better study the role of each of them in the optimal pricing and price adjustment policy (see Villas-Boas (1999, 2015) for similar treatment). Lemma 1. For any given refund fraction f, the Sub-game Perfect Nash Equilibrium of the two-period game described above is as follows: The Seller’s prices:  p∗1 (f ) = min

2 − δC (1 + f ) 4 − 2f (1 + f )δS − 2δC (2 + f (3 − f − δS (1 + 3f ))) + δC 2 (1 + f (3 + f − f 2 − (1 + f (1 + (3 − f )f ))δS )) , 2(1 − fδC ) 2(1 − fδC )(4 − (2 + 2(2 − f )f )δC + (1 + f )(1 + f − (3 − f )fδC )δS )

 p∗2 (f ) = min p∗1 (f ),

(2 + f (2 − 4δC ) − δC + f 2 δC )(1 − fδS ) 2(4 − (1 + f )2 δS − δC (2 + f (4 − 3δS ) − f 3 δS + 2f 2 (1 + δS )))





  Consumers with valuation above v∗ (f ) buy in period 1, while consumers with valuation v ∈ p∗2 (f ), v∗ (f ) buy in period 2, where   2 − fδS − f 2 δS − δC (1 + f (3 − δS ) + f 3 δS − 2f 2 (1 + δS )) . v∗ (f ) = min 1, 4 − (1 + f )2 δS − δC (2 + f (4 − 3δS ) + f 3 δS − 2f 2 (1 + δS )) Proof. All proofs are relegated to Appendix A. The seller’s equilibrium profit is then given by  1 (1 − fδS )(4 − 4(1 + f )δC + δC 2 (1 + f (2 − δS ) − f 3 δS + f 2 (1 + 2δS ))) ∗ , Π (f ) = max 4 4(1 − fδC )(4 − (1 + f )2 δS − δC (2 + f (4 − 3δS ) + f 3 δS − 2f 2 (1 + δS )))

(2.8)

Fig. 1 depicts the seller’s equilibrium profit as a function of the price difference refund fraction f for three different values of the seller’s inter-temporal discount factor δS , demonstrating that offering a full refund is not always optimal. In fact, the profit function for δS = 1 is monotonically decreasing in f, which means that offering no price adjustment (f* = 0) is optimal for the seller. For δS = 0.85 the profit function is inverse-U-shaped, leading to a single optimal partial refund fraction f approximately equal to 0.31 6 Recall that forward looking consumers can reach Eq. (2.5) and thus by observing the first period price they can rationally anticipate the correct second period price.

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Fig. 2. Optimal fraction of price difference refund in a two-period model, f ∗ .

Finally, the profit function for δS = 0.7 is monotonically increasing in f, which means that offering to refund the full price difference (f* = 1) is optimal in this case. We proceed with the derivation of the Sub-game Perfect Equilibrium by characterizing the optimal refund fraction. In order to obtain the optimal fraction of the price difference refund, we need to maximize the retailer’s profit given by expression (2.8) with respect to f. The following proposition characterizes the optimal price difference refund fraction. Proposition 1. In the subgame perfect equilibrium of the two period game, the optimal price difference refund fraction is given by7

f∗ =

⎧ ⎪ ⎪ ⎪ ⎨

0

δC 4 (1 − δS )2 (7δS 2 − 2δC δS (4 + 3δS ) + δC 2 (1 + 6δS )) δC − 4δC δS + (5 − 2δS )δS + +ξ ⎪ 3δS (δC + δS (1 − 2δC )) ξ ⎪ ⎪ ⎩ 1

if δC ≤ g(δS ) if g(δS ) < δC < g(δS ) , if δC ≥ g(δS )

To obtain the full Sub-game Perfect Equilibrium of the game we need to substitute f* into the expressions presented in Lemma 1. For the rest of the analysis we highlight the equilibrium pricing strategies of the seller only. In each case, the corresponding equilibrium purchasing strategy for consumers is straightforward and can be described as follows: Consumers with valuation greater than the valuation of a consumer indifferent between buying in the first and second period purchase in the first period. Consumers with valuations below that but above second period price purchase in the second period. The rest do not buy the product at all. Fig. 2 depicts the optimal refund fraction for various values of the seller’s and consumer’s inter-temporal discount factors. One can see that the optimal refund fraction is increasing in δC , and that for small values of δC not offering the price adjustment at all is the optimal strategy. The critical values of δC where it becomes optimal to offer a partial refund (g(δS )) and where it becomes optimal to offer full refund (g(δS )) are increasing in δS . Also note that for a fixed δC the optimal refund fraction is decreasing as seller’s patience increases. First notice that the optimal fraction of price difference refund is never greater than 1 (see Fig. 2), which means that sellers should never offer to refund higher than a 100% of the price difference. In practice sometimes we do observe policies that offer refunds higher than price differences. In the next section we offer a potential explanation for when it might be optimal for the seller to do so. Next notice, that when consumers are patient enough (δC > g(δS )) the optimal refund fraction is equal to one. However, when f = 1, the optimal prices and profit for a retailer offering the full price adjustment policy are as follows8 : ∗

∗

pF1 = pF2 = ∗

F =

1 2

1 4

So the implementation of full price adjustment policy is used by the seller as a commitment mechanism to not dropping the price in period 2 and thus being able to charge the optimal single period price of ½. All consumers with valuations above ½ will buy in the first period and no sales will occur in the second period. So effectively in equilibrium price adjustment policy is announced but

7 8

Where ξ, g(δS ) and g(δS ) are defined in the Appendix A. Optimal outcomes corresponding to the case of full price adjustment policy (f = 1) are denoted with “F” superscript and are independent of the discount factors.

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Fig. 3. Optimal price adjustment strategy: with partial refunds considered (solid) or not (dashed) (␦S = 0.9).

the refund is never actually given. This rational is very similar to the one offered for the profitability of price matching policies in a competitive scenario (see Tirole 1988).9 When consumers heavily discount the future (δC < g(δS )) offering price adjustment is not optimal. The intertemporal discount factor alone is enough to incentivize many consumers to purchase early, thus allowing for future price adjustments will influence very few additional consumers, which does not justify the expense of refunds. In this case, the retailer sells the product at a higher price in the first period, and lowers the price in the second to capture the consumers with lower valuation. Further, the critical value of δC where it becomes optimal to offer a partial refund (g(δS )) is increasing in δS . That is, the more patient the seller is, the larger is the set of parameters where offering no price adjustment is optimal. If the seller only considers two policies: either no price adjustment, or full price difference refund, then the former would be √ optimal for δC < 1 − 1 − δS , and for higher δC the full price adjustment policy would be optimal. Our novel suggestion, as depicted in Fig. 3, is for the seller to consider offering partial refunds, in addition to no or full price adjustment. The partial refunds are beneficial for the seller around the switching point: that is when consumers have intermediate levels of inter-temporal discount factor: g(δS ) < δC < g(δS ). In this intermediate range the expense of full refund would not be justifiable as few additional consumers would purchase the product early (while the refund must be given to all), and no refund means that the seller is not providing any incentive to consumers to purchase the product sooner. A partial refund policy mitigates these two considerations. In other words,10 the partial refunds allow the retailer to fine-tune its own incentive to offer price discounts in the second period. By adjusting its second-period pricing incentives, the retailer can alter consumers’ decisions to purchase in the early period (via both the refund fraction and expected price savings from waiting). When consumers are modestly patient, the retailer finds it beneficial to do some price discrimination and to offer some compensation for subsequent price drops (via partial refunds). In contrast, when consumers are very patient, price discrimination over time is prohibitively costly to the firm and, when consumers are very impatient, the firm does not need to incur the costs of compensating consumers for subsequent price drops since delaying purchase is not an attractive option to consumers anyway. The existing research suggests that retailers are often significantly more patient than consumers. The disparity between the discount factors stems from differences in access to capital and credit markets. For example, consider pay-day loans: a small, short-term unsecured loan taken by consumers. Most borrowers use payday loans to cover ordinary living expenses, not unexpected emergencies (Pew 2012). For two-week loans, the interest rates range from 390 to 780% APR demonstrating that some consumers heavily discount future revenue streams (even those in the near future). In contrast, retailers would never borrow money under such terms. Another example of consumers having lower discount factors than retailers is the high interest rates imposed by credit cards. At the average credit card’s interest rate of 15% only 35% of credit card users do not carry a balance even though many more could afford to pay the balance in full on a monthly basis (Fulford and Schuh 2015). Loewenstein and Thaler (1989) offer more examples as well as experimental results showing that consumers heavily discount even the short term future. See Yao et al. (2012) for further discussion and estimation of consumers’ discount rates. Specifically, they estimated consumer weekly discount factors and found them to be much lower than those typically assumed for firms.

9 In practice, however, we do observe full price difference refunds offered, prices drop in the second period, and refunds given to consumers. This seems contradictory to our findings. However, in practice the refunds are not usually given automatically. In Section “Price Adjustment upon Request” we extend our base model to cases where consumers must proactively request price adjustments, and we demonstrate that in such scenarios full refunds may be both offered and given in equilibrium. 10 We thank an anonymous referee for this interpretation.

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Fig. 4. Seller’s profits under full price adjustment policy (f = 1), No price adjustment policy (f = 0), and the price adjustment policy with optimal partial refund (δS = 1).

Most empirical studies assume the short term firms’ discount factors to typically be 0.995–1.0 (see Yao et al. (2012)). Thus consider the special case where the seller does not discount the future, that is where δS = 1. Under this condition, Eq. (2.6) is simplified and the seller’s profit is given by: ∗P (f ) =

(4 − 4(1 + f )δC + δC 2 (1 + f + 3f 2 − f 3 )) 4(1 − fδC )(3 + f − δC (2 + 3f − f 2 ))

(3.1)

And Proposition 1 reduces to: Corollary 1. When the seller does not discount future revenue, the optimal refund fraction is given by fP∗ =  √ 2 C) , so when δ < 2 − 2, disallowing price adjustments maximizes the seller’s profit, and offering full max 0, 1 − 3 2(1−δ C 2 δ C

price adjustment is only optimal if δC = 1. The optimal refund fraction for price adjustment policy with δS = 1 is depicted in Fig. 2. The corollary above suggests that a seller who does not discount future revenue facing consumers who do should never offer a full price difference refund policy. Indeed, as Fig. 4 below demonstrates, such policy results in profits lower than either the optimal partial refund or no refund at all. In other words, managers who offer optimal partial price adjustment instead of full can increase their profits. Observe that the partial price adjustment has the most impact on the seller’s profits when the consumer’s discount factor is about 0.9. Model Extensions Price Adjustment upon Request In the base model we have assumed that when a seller employs the price adjustment policy, it automatically refunds the price difference to all consumers who have purchased the product in the first period. In this section we consider a modification to the base model where instead of getting the price difference refund automatically, consumers have to request the refund in the second period. Such assumption is in line with the practice of price difference refunds: generally the consumers have to notice that the price of the product went down, and then contact the seller in order to get the price adjustment. The previously mentioned survey conducted by the authors demonstrated that although an overwhelming majority of consumers is aware of the price adjustment policies, not all of them will request the refund: some will forget or simply not get around to checking the price in the second period or contacting the seller about the refund. We assume that all the consumers are aware of the price adjustment policy, but each of them believes that she would get around to requesting the refund only with probability α11 (once requested, the refund is always given). At the same time, the retailer knows from experience that a fraction β of consumers asks for the price adjustment, where β might be different from α. As in the base model, consumers are assumed to be forward looking and rational: meaning that when making a decision whether to buy the product in the first period or wait until the second period, the consumers will anticipate that they might not get around to

11

Known to the retailer.

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Fig. 5. Seller’s profit in a two-period model (δC = 0.55, α = β = 0.8).

requesting the refund, that is they will only obtain it with probability α. Thus consumer’s expected utilities from buying in period one and two, originally given by Eq. (2.1) in the base model, now look as follows: U1 = v − p1 + αδC f (p1 − p2 )+ U2 = δC (v − p2 ) Further, the firm’s profit maximization problem in period 2 (given by Eq. (2.2) in the base model), now changes to:   max2 (p1 , p2 , f ) = max (v − p2 )p2 − βf (1 − v)(p1 − p2 )+ p2

s.t.p2 ≤ p1

p2

(4.1)

(4.2)

Solving the firm’s profit maximizations problem in backward induction fashion is similar to the solution process in the base model in Section “The Base Model”. ∗ = α f , where f is the The price difference refund fraction that maximizes retailer’s profit in this case is as follows12 : fαβ ∗ β2 ∗ optimal fraction of price difference refund in the base model, as defined in Proposition 1. Notice, that while in the base model the maximum refund fraction was 1, here the maximum refund fraction is βα2 , which can be greater than 1. Fig. 5 depicts the seller’s equilibrium profit as a function of the price difference refund fraction f when α = β = 0.8, that is only 80% of the customers get the refund and they correctly estimate their likelihood of requesting it. Just as in Fig. 1, different shapes of the profit function for various values of the seller’s intertemporal discount factor lead to different optimal price adjustment policies. For example, no refund (δS = 0.95), partial refund (δS = 0.8), or full refund (δS = 0.7). However, even a greater than 100% refund can be optimal when not all the consumers request the refund (for example, for δS = 0.6). Several key observations can be further made. ∗ = 1) is Proposition 2. When only fraction βˆ of consumers requests the price adjustment, the full price difference refund (fαβ offered, price drops in the second period, and the price difference refund is given. Further, if β < βˆ then the retailer offers a greater ∗ >1 than 100% refund of the price difference: that is fαβ √ Here βˆ = αf ∗ . Note that βˆ is a function of the intertemporal discount factors δS and δC . With automatic refunds (the base model), whenever the full price difference refund is optimal, the equilibrium first and second period prices are identical. Which means that even though full price difference refund is advertised, it is never actually given in equilibrium: the full price difference refund promise serves as a self-regulating mechanism that prevents the retailer from lowering the price in the second period; and since the price does not change in the second period, there is no reason to give the refund to consumers. The proposition above demonstrates that the full price difference refund is not only advertised, but actually given in equilibrium if only a fraction of consumers requests and obtains the price adjustment. Thus, Proposition 2 provides a plausible explanation for why in practice we often observe retailers giving the full price difference refund to consumers. Further, in practice some retailers advertise a higher than 100% refund of the price difference. For example, Lowes’s “Everyday Low Prices” policy reads: “We guarantee our prices for 30 days. Find a lower-advertised price on the same in-stock item, and we will beat it by 10%.” Another example is the online retailer Lamps Plus (lampsplus.com), who offers a 120% price difference refund for 60 days after purchase. Proposition 2 above sheds light on why and when such price adjustment policy might be optimal.

12

pendix A for derivation of the optimal refund fraction.

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Fig. 6. Seller’s profit in a two-period model (δS = 0.8, δC = 0.55, β = 0.8).

Specifically, offering more than a 100% refund, is beneficial when only a small enough fraction of consumers is likely to request the price adjustment. Common intuition suggests that the seller would be always better off if some consumers do not request the price adjustment than when it automatically refunds the price difference. However, Fig. 6 demonstrates that such intuition is not correct. Specifically, the firm indeed benefits from consumer optimism: when α > β (which means that consumers overestimate the likelihood of them asking for price adjustment) the firm’s profit is higher than if the retailer were to offer automatic refunds. But on the other hand, consumer skepticism, α < β (which means that consumers underestimate the likelihood of returning to request the refund) hurts the firm. Finally, if the consumers are correct about their likelihood of requesting the refund (α = β), then seller’s profit is the same as in the case of automatic refunds. The last two observations suggests a valuable managerial insight that sellers do not necessarily have to make it more difficult for consumers to obtain the price adjustments: indeed, if the seller knows that consumers are underestimating their probability of requesting the refund, the retailer should offer automatic price adjustment policy. And in fact, if the seller anticipates an additional (i.e., not explicitly modeled in this paper) benefit from advertising that their price adjustments are automatic, those automatic price difference refunds can result in even higher profit.

Diminishing Product Value In this section we consider a product that has a diminishing per-period utility for consumers. Consider, for example, a piece of fashionable clothing. Wearing it in the early months, while it is still at the peak of fashion, is more enjoyable, than wearing it in later periods, when it is not “all the rage,” or being on the way out, or even outdated. Or consider another example: the most recent model of an electronic device — for example an iPhone 8. Owning and using the iPhone 8 in the first few weeks after it comes out brings higher joy to users who value novelty than using it a later weeks or months when it is no longer the newest gadget on the market. Let ω denote the initial per-period product utility, and let it decrease over time at the rate of (1 − ) fraction of utility per period, where < 1. In other words, the product’s utility is ω in period 1, ω in period 2, ( ω) in period 3, . . . k−1 ω in period k, etc. (even though the product is only sold in periods 1 and 2, usage of it continues long past the sales horizon). Let v denote the lifetime value of a product purchased in period 1:

v = ω + δC ω + δ2C 2 ω + ... = ω

∞ 

δkC k =

k=0

ω , 1 − δC

where δC is consumer’s discount factor, as in the base model. Then the residual lifetime value of the product purchased in period 2 is as follows:  ω + δC ω 2

+ δ2C 3 ω...

= ω

∞ 

 δkC k

= v.

k=0

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Fig. 7. Optimal refund fractions in (δC , ) space.

In other words, the product purchased in period 1 has a higher lifetime value to the consumer than the same product purchased in period 2 (v > v) when the per-period utility of the product is declining over time.13 With that, the consumer compares the following net-present utilities when deciding whether to purchase the product in period 1 or 2: U1 = v − p1 + δC f (p1 − p2 )+

(4.3)

U2 = δC ( v − p2 )

Solving for the optimal first- and second-period prices and then for the optimal refund fraction, in the backwards induction fashion analogous to the base model, results in the optimal refund fraction given by14 :

f ∗ =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

√ 3 1−

0 2δC (1 − δC )(1 − δC ) δC

⎪  ⎪ ⎪ ⎪ (1 − )2 + 2δC 2 (2 − 1)2 − 2δC (2 − 7 + 6 2 ) − (1 − ) ⎪ ⎪ ⎩1 − δC (2 − 1)

δC < δ( ) δ( ) ≤ δC < δ( )

(4.4)

δ( ) ≤ δC

It is easy to see that with = 1 (i.e., the case when the per-period value of the product does not decline over time) the optimal refund fraction f ∗ is equal to that in the base model, fp∗ . Fig. 7 illustrates Eq. (4.4). The price adjustment policy encourages the consumers to buy early rather than wait for a future price drop. In addition, the decrease in product’s value over time also incentivizes the consumers to purchase the product sooner. As a results, the retailer can scale back on the price difference refund fraction. Indeed, it is straightforward to show that the smaller is , the lower is the optimal refund fraction. Further, diminishing product value increases consumer willingness to pay in the earlier period resulting in generally higher price and larger demand. Thus lower , combined with lower optimal refund fraction, increases retailers profits for any given δC , as illustrated in Fig. 8. It is interesting to note that in this case, there exist consumers who buy the product in the first period at a price higher than their valuation: v < p∗1 . In the base case ( = 1), these consumers would have waited until the second period to buy the product at a lower price. However, when product valuation diminishes over time, it is no longer optimal for them to wait. Instead, they pay a high price in the first period, enjoy the full product value, and obtain a price adjustment in the future. Fig. 9 illustrates this observation. The horizontal axis represents different values of intertemporal discount factor δC , while the vertical axis represents the corresponding ranges of consumer valuation. As one can see, when the consumer discount factor is high enough, there exist a set of such consumers.

13 Note that if the per-period product utility remains the same through the life of the product (i.e., if = 1), then the residual lifetime value of product purchased in period 2 is the same as the lifetime value of the product purchased in period 1: v, as in the base model. And the utilities presented in (4.3) are the same as utilities in the base model given by (2.1). 14 Subscript “ ” denotes the optimal solutions of the model with diminishing lifetime product value.

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Fig. 8. Optimal profit under price adjustment policy in case of diminishing product value.

Fig. 9. Valuation of consumers purchasing in the first period ( = 0.7).

Three-Period Scenario “Purchases made at any of our stores or at pier1.com can be returned or exchanged at any Pier 1 Imports store [. . .] Except for seasonal merchandise, we will offer a one-time sale price adjustment, limited to 14 days from purchase date.” [Pier 1]. “If within two weeks of your order date, we permanently reduce the price of an item you purchased, we’ll happily adjust the sale price for you. Please note that we can only make one price adjustment per item.” [Nordstrom] As demonstrated by the quote above, most retailers limit their price adjustment policy to a single price difference refund done within a relatively short period of time after the original purchase. In order to investigate the optimality of such strategies we need to extend our analysis and allow for more than two sales periods. Specifically we consider a three period selling scenario and compare and contrast the profitability of a price adjustment policy that offers refunds only in the period immediately following the purchase with the one that allows for a price adjustment in any future period.15 Price Adjustment with Refund Limited to the Next Period Only In the three period model the consumers have a more complicated decision: they can buy the product in either period one, or two, or even three. The net present utility of each purchase option for the consumer whose willingness to pay is v, is given by: U1 3P = v − p1 + δC f (p1 − p2 )+ , U2 3P = δC (v − p2 + δC f(p2 − p3 )+ ),

(5.1)

U3 3P = δC 2 (v − p3 ). In a backward induction fashion we start by solving the seller’s profit maximization problem in the third period:   maxπ3 3P (p3 ) = max (v2 − p3 )p3 − f (v1 − v2 )(p2 − p3 ) , p3

s.t.

15

p2 ≥ p3

p3

(5.2)

As before, we continue with the assumption common in the empirical literature: δS = 1.

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where vi is the smallest product valuation of a consumer purchasing in period i. Denote the solution to this optimization problem as p∗3 . Then we proceed with solving the second period maximization problem:     max π2 3P (p2 ) = max v1 − v2 p2 , p∗3 p2 − f (1 − v1 )(p1 − p2 ) + π3 (p∗3 )) , p2

s.t.

p1 ≥ p2

p2

The solution to which is denoted p∗2 . And then solve the first period maximization problem:     max π1 3P (p1 ) = max 11 − v1 p1 , p∗2 , p∗3 p1 + π2 (p∗2 , p∗3 ) . p1

p1

Lemma 2 summarizes the equilibrium strategies of the seller (retail prices and optimal refund fractions).16 Lemma 2. 3P 3P 1) When δC < δ1 , the seller charges three different prices in the three periods, p3P 1 > p2 > p3 , and offers no price difference ∗ 3P refund to consumers: f = 0. Here:

  2 3P 3P 3 p3P 1 = 6 − 6δC + δC θ, p2 = (2 − δC ) θ, p3 = (2 − δC )θ, where θ =

(6−6δC +δC 3 ) . 2(24−40δC +13δC 2 +8δC 3 −4δC 4 )

3P 3P 2) When δC ∈ [δ1 , δ2 ] the seller charges the same prices in periods 1 and 2, decreases the price in period three: p3P 1 = p2 > p3 , ∗ 1 and offers a partial price difference refund equal to f 3P = 1+δ . Here C

2

3P p3P 1 = p2 =

(2 − δC 2 ) 2 − δC 2 , p3P . 3 = 2 2(3 − 2δC ) 2(3 − 2δC 2 )

3P 3P 3) When δC > δ2 the seller charges the same price in periods 2 and 3: p3P 1 > p2 = p3 and offers a price difference refund of ∗ γ f 3P = 1 − δ 2 . Here C

γ − δC (4(1 − δC )δC + 161/3 γ) , γ − δC (6(1 − δC )δC + 261/3 γ)   2δC 3 (11 + 6δC ) − δC γ − 162/3 γ 2 − δC 2 (2 + 361/3 γ) 3P = p3 = 2δ2 (1 + δC )(17δC − 1)

p3P 1 = p3P 2

1/3

Where γ = 21/3 ((1 − δC )2 δC 4 ) , and δ1 ≈ 0.4762 and δ2 ≈ 0.9404. Note that for small and medium values of δC the seller does not have to refund any money. Indeed, for δC < δ1 the refund is not even offered, and for δC ∈ [δ1 , δ2 ] even though the refund is offered, the fact that prices are the same in the first and second periods means that no consumer will buy the product in the second period and thus the seller will not need to pay any refunds. Therefore, refund is only paid when δC > δ2 , but in this case the seller’s profit is the same as in the two period case (since prices are equal in the last two periods, and no consumer waits to purchase until the last period). Price Adjustment with Refund Given in Any Period In this scenario the retailer guarantees to refund a portion of the price difference to consumers in any period after the original purchase and even multiple times if needed. In other words the consumer who purchased in the first period will get the price difference refund in period 2: f(p1 − p2 ), and then again in period 3: f(p2 − p3 ) if the price dropped even further.

16

Details of profit maximization, optimal prices derivations, and equilibrium profits are presented in the Appendix A, proof of Lemma 2.

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Fig. 10. Comparison of seller’s equilibrium profits under price adjustment policy in a three period scenario.

The consumers’ net present utility from each of the purchase options is slightly different from those presented earlier by Eq. (5.1): U1 3PA = v − p1 + δC f (p1 − p2 ) + δC 2 f (p2 − p3 ) U2 3PA = δC (v − p2 + δC f (p2 − p3 ))

(5.3)

U3 3PA = δC 2 (v − p3 ) And the seller’s maximization problem in the last period is also slightly different — the seller now has to pay the refund to all the consumers who purchased previously (1 − v2 ) as opposed to only those who purchased in the second period as in the previous case of limited time refund. The maximization problem is:   max π3 3PA (p3 ) = max (v2 − p3 )p3 − f (1 − v2 )(p2 − p3 ) p3 p3 (5.4) s.t. p2 ≥ p3 The seller’s optimization problems for the first two periods remain the same, except for the need to use the new optimal third period price and profit. All three maximization problems are analytically solvable, and all optimal prices are internal solutions to the optimization problems — in other words the pricing constraints are never binding and the seller indeed lowers the price as time goes by.17 The seller’s profit as a function of the fraction f of the price drop that is refunded to consumers, is given below and is a ratio of two high-degree polynomials:

Given the discount factor δC , one can numerically solve the profit maximization problem for the optimal price drop return fraction, f* . Although we cannot analytically characterize it, we can still prove that there exists a range of discount values, δC , such that the price adjustment policy with refund in any period yields higher profits than the policy with refund limited to the next period after purchase. Proposition 3. When consumers are patient enough, δC > δ2 , a price adjustment policy with refund in any period results in strictly higher profits for the seller than a price adjustment policy where refund is limited to the next period only. Fig. 10 illustrates Proposition 3. Recall that if the price adjustment policy is limited to a single period after the purchase, then if the consumers are patient enough (δC > δ2 ) the retailer does not lower the price between periods 2 and 3. Whereas if the refund period is unlimited, and, moreover, the consumers can receive multiple refunds (for multiple price drops), the retailer does decrease the price in period 3. So when consumers are relatively patient, the retailer needs to provide an additional incentive for high value consumers to buy early, which,

17

The explicit expressions of the optimal prices are given in the Appendix A, in the proof of Proposition 3.

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in turn, allows the retailer to decrease the price later resulting in better segmentation and higher profits than with a standard policy of a single price adjustment and limited refund time. On the other hand, when consumers are impatient, no additional incentive (i.e., multiple adjustments) is necessary to effectively price discriminate — the high value consumers will buy early anyway, and therefore offering multiple refunds over longer period of time is costly to the retailer, leading to lower profits. As can be seen from the graph in Fig. 10, proposition 4 provides only a sufficient condition. The range of δC for which the refund any time strategy yields strictly higher profits is much larger (for δC approximately greater than 0.855, whereas δ2 in proposition 4 is ∼0.9404). Fig. 10 also illustrates the new and important managerial insight provided in this section: The currently used policy of single price adjustment and limited refund time is only beneficial when consumers are relatively impatient. When consumers are patient enough, the seller would benefit from extending the price adjustment period, and from allowing the consumers to obtain multiple price adjustments as the time goes by and the price decreases. In fact, our model shows that limiting the price difference refund to a single time within a short period after the purchase may lower seller’s profit by as much as 7–8%. Conclusions Nowadays consumers expect the price of most products to decline over time. Forward looking consumers may decide to delay their purchase decision until a later time to try and take advantage of these potentially lower prices. This purchase postponement may lead to lower profits for the retailers in the long run. To counteract the discount seeking behavior of consumers many retailers choose to implement a price adjustment policy. The guarantee of refunding the difference between the price the consumer paid and a lower future price gives such consumers appropriate incentives to buy earlier. In this paper we have shown that full price adjustment (i.e., full price difference refund) is only beneficial for the seller when consumers’ inter-temporal discount factor is high enough as compared to that of a seller. However, a more realistic assumption is that the sellers are significantly more patient than consumers. Thus we suggest a novel managerial approach: a policy under which the seller promises to refund a fraction (potentially greater than one) of any future price difference. We characterize the conditions under which such a policy results in strictly higher profits to the retailer than the traditional full price difference refund. We have extended our base model to account for the fact that in reality not all the consumers request the price adjustment. We demonstrate that in this case the full refund does not serve as a price commitment mechanist, but the prices do drop and the refund is actually given to consumers who request the price adjustment in the second period. Furthermore, in practice some retailers offer more than 100% refund of the price difference (e.g., Lowes, Lamps Plus), which is also in agreement with our extended model. Furthermore, we account for the possibility that consumers under- or overestimate the probability of them requesting the price adjustment. We demonstrate that requiring consumers to request the price adjustment provides an additional increase in profits when consumers are optimistic. However, when consumers are skeptical (underestimate the probability), then automatic price adjustment is optimal. In this paper we also considered a case where consumers have a decreasing per-period consumption utility, resulting in the residual lifetime product value diminishing over time. In that case, the retailer can increase profits by scaling down the price difference refund fraction because the decrease in product value itself already incentivizes the consumers to buy early. Furthermore, we show that in this scenario some consumers even buy the product at an initial price which is higher than their immediate willingness to pay. They enjoy the full product value, and request the price adjustment later. In cases where there are more than two periods, the seller can offer several variations of the price adjustment policy: refund limited to the next period only, refund in any future period, refund in the last period only, etc. In practice most retailers limit the refund to the “one refund in the next period only” option. However, such a policy is not always optimal. Our analysis offers a somewhat surprising managerial insight: allowing the consumers to get a partial price difference refund every time the price drops, can lead to an even higher profit when the consumers are patient enough. We demonstrated this result in a three period model, but in future research it can be generalized in a straight forward manner to any number of periods. Further, we have considered a model where some consumers are completely unaware of such policy, and our results still hold as long as there are other consumers who are aware. We chose not to include this extension in the paper because it did not provide any additional insights besides the obvious: the more consumers are aware of the policy, the higher is the offered refund fraction and the higher is the retailer’s profit. In this paper we have not presented the results for the case where the seller’s cost of implementing the price adjustment policy is explicitly accounted for. However it is not too complicated to show that qualitatively our results still hold. Accounting for such policy implementation cost results in lower optimal refund fraction as a function of consumers’ discount factor. The seller’s profits are also lower when accounting for policy implementation cost, but offering only a fraction refund of the future price drop is still beneficial.18

18

The exact calculations for these models are available from the authors upon request.

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We would like to emphasize that the profitability of the price adjustment policy does not rely on the premise that some consumers might not actually redeem the refund. In fact, even if the sellers automatically refund the price difference, as they do in our base model, the incentive for consumers to buy early is still strong enough to guarantee higher profits. Furthermore, in contrast to most of the existing literature, in the model presented here the seller is assumed to have enough inventory to satisfy all potential demand. That is we do not require limited stock in order for the price adjustment with partial refund policy to be beneficial. In the future, consumer behavior issues (e.g., fairness, regret, publicity and word of mouth), as well as the hassle cost of requesting the refund, could be investigated in the context of price protection policy. It will be also interesting for future studies to examine what version of the price adjustment policy would be optimal depending on the degree of consumer sophistication, heterogeneity, and market factors that impact the ease of predicting and identifying the price reductions, and requesting the price adjustments. Another potential avenue for research would be to analyze a combination of price adjustment with partial refunds as well as partial price-matching guarantees (the policy where sellers promise to match their competitor’s price). In a competitive setting, these two policies might complement each other, since price adjustment policy incentivizes the consumers to buy early, while the price matching policy provides incentives to buy from a particular retailer. Appendix A. Proof of Lemma 1 Solving maximization problem (2.7) results in the optimal first period price being p∗1 (f ) =

4 − 2f (1 + f )δS − 2δC (2 + f (3 − f − δS (1 + 3f ))) + δC 2 (1 + f (3 + f − f 2 − (1 + f (1 + (3 − f )f ))δS )) . 2(1 − fδC )(4 − (2 + 2(2 − f )f )δC + (1 + f )(1 + f − (3 − f )fδC )δS )

Substituting that into (2.5) results in the optimal second period price being p2 (f ) =

(2 + f (2 − 4δC ) − δC + f 2 δC )(1 − fδS ) 2(4 − (1 + f )2 δS − δC (2 + f (4 − 3δS ) − f 3 δS + 2f 2 (1 + δS )))

Statement of the lemma then follows from combining (A.1) and (A.2) with feasibility constraints 0 < v(p1 , p2 , f ) < 1 and p1 (f) ≥ p2 (f). (A.1) Proof of Proposition 1 Taking FOC with respect to f of the retailer’s profit given by (2.8) results in f ∗∗ =

δC − 4δC δS + (5 − 2δS )δS δC 4 (1 − δS )2 (7δS 2 − 2δC δS (4 + 3δS ) + δC 2 (1 + 6δS )) + +ξ 3δS (δC + δS (1 − 2δC )) ξ

(A.1)

where

2

Further, one can show that feasibility conditions p1 (f) ≥ p2 (f), v*(p1 , p2 , f) ≤ 1 and second order condition for maxima ∂∂2 (f ) < 0. f g(δS ) is defined as solution to f** = 0 equation solved with respect to δC . g(δS ) is defined as smaller of the two real roots of equation f** = 1 solved with respect to δC . Then the solution to seller’s profit maximization problem which satisfies the feasibility conditions outlines above can be written as follows: ⎧ 0 if δC < g(δS ) ⎪ ⎪ ⎪ ⎨ 2 4 2 2 f ∗ = δC − 4δC δS + (5 − 2δS )δS + δC (1 − δS ) (7δS − 2δC δS (4 + 3δS ) + δC (1 + 6δS )) + ξ if g(δS ) < δC < g(δS ) , ⎪ 3δS (δC + δS (1 − 2δC )) ξ ⎪ ⎪ ⎩ 1 if δC > g(δS )

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Proof of Proposition 2 Solving seller’s profit maximization in the second period (Eq. (4.2)) provides an optimal second period price:   p1 (1 − fβ)(1 − fαδC ) − fβ(1 − δC ) ∗ p2 (p1 , f ) = min p1 , . 2 − (1 − f 2 αβ + f (α + β))δC Substitute this into Eq. (4.2) to obtain second period profit (denoted 2 (p1 , p2 ∗ (p1 , f ), f )) and into the location of consumer −fαδ(p1 −p2 ) indifferent between buying in periods 1 and 2: v(p1 , p2 , f ) = p1 −p2 δC1−δ (obtained by equating U1 = U2 defined in (4.1)), C

C (fα−1)−2p1 (1−fαδC ) to obtain v(p1 , f ) = fβδ which is necessary for the feasibility condition in the next profit maximization problem. 2−(1−f 2 αβ+f (α+β))δC Next, the retailer solves the following profit maximization problem:   max (p1 , f ) = max (1 − v(p1 , f )) p1 + 2 (p1 , p2 ∗ (p1 , f ), f )

p1

p1

s.t.v(p1 , f ) ≤ 1

Solving it results in the optimal first period price p1 *(f), which we then substitute into p∗2 (p1 , f ) above to obtain the optimal second period price, both as functions of f:

p∗αβ2 (f )

 = min

p∗αβ1 (f ),

(2 − δC + f (−αδC + β (2 − (3 − fα)δC ))) (1 − fβδS ) (2 (1 4 − + f (α + β − fαβ)) δC ) − 2δS (1 + fβ) (1 + f (β − αδC − (2 − fα)βδC ))



Substitute these into Eq. (4.2) to obtain total retailer’s profit as a function of f: αβ (f ) =

(2 − δC − fαδC )2 − 2fβδS (2 − δC (2 − δC + fα(2 − fαδC ))) + f 2 β2 δ2C δ2S (1 − fα)2 4(1 − fαδC ) (2 (2 − (1 + f (α + β − fαβ)) δC ) − δS (1 + fβ) (1 + f (β − αδC − (2 − fα)βδC )))

Next, seller chooses f to maximize this profit. When comparing this maximization problem to the one in the base model (Eq. ∗ = α f (where f maximizes seller’s profit in the base model, defined in Proposition 1) maximizes seller (2.8)) one can see that fαβ ∗ β2 ∗ profit αβ (f ) defined above. Notice, that while in the base model the maximum refund fraction was 1, here the maximum refund fraction is equal to βα2 . Finally, to find the range of parameter β for the optimal refund fraction to be greater than 1, one needs to solve βα2 f ∗ > 1 for β. √ √ ˆ the optimal refund fraction f ∗ is equal to 1. The solution is β < αf ∗ . Denote βˆ = αf ∗ . Obviously, when β = β, αβ

1) Proof of Lemma 2 1) First, solve the retailer’s profit maximization problems with respect to prices in a backward induction fashion disregarding the s.t. constraints p1 ≥ p2 and p2 ≥ p3 .The solution is (in the four expressions below δC is replaced with δ for the ease of exposition)

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With these prices, the seller’s profit is

One can show that this profit is convex in f for all δC ∈ [0, 1], and that (0) > (1). Thus the optimal fraction of price difference refund is f = 0.With this, the optimal prices and profit are, respectively: 2

(6 − 6δC + δC 3 ) , 2(24 − 40δC + 13δC 2 + 8δC 3 − 4δC 4 ) (2 − δC )2 (6 − 6δC + δC 3 ) , = 2(24 − 40δC + 13δC 2 + 8δC 3 − 4δC 4 ) (2 − δC )(6 − 6δC + δC 3 ) , = 2(24 − 40δC + 13δC 2 + 8δC 3 − 4δC 4 )

p3P 1 = p3P 2 p3P 3

2

3P 1 =

(6 − 6δ + δ3 ) 4(24 − 40δ + 13δ2 + 8δ3 − 4δ4 )

2) Second, solve the retailer’s profit maximization problems with respect to prices assuming that p1 ≥ p2 constraint is binding. Then (2−δC )2 2−δC 2 3P 3P the optimal prices are p3P 1 = p2 = 2(3−δ 2 ) , p3 = 2(3−2δ 2 ) . C

C

2

The optimal fraction of refund is f =

1 1+δC ,

and the optimal profit is 3P 2 =

3P It is easy to see that 3P 2 > 1 if f δC > δ1 = 0.4762.

(2 − δC 2 ) . 4(3 − 2δC 2 )

3) Finally, solving retailer’s profit maximization problems assuming that p2 ≥ p3 condition is binding. This is equivalent to solving two period model outlined above. It is easy to see that profit in this case is greater or equal to 3P 2 , and it is strictly greater iff δC > δ2 = 0.9404. Please cite this article in press as: Cohen-Vernik, Dinah, and Pazgal, Amit, Price Adjustment Policy with Partial Refunds, Journal of Retailing (xxx, 2017), http://dx.doi.org/10.1016/j.jretai.2017.08.002

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1) Proof of Proposition 3 1) First, we provide the solution to profit maximization problem with respect to prices. The optimal second period price (solution to (5.4)) is p∗3 (p2 ) =

2p2 (1 − fδC ) − (1 − f )fδC . 2 − (1 + 2f − f 2 )δC

(A.2)

Substitute this optimal price into the second period profit maximization problem:    max π2 3PA (p2 ) = max v1 − v2 p2 − f (1 − v1 )(p1 − p2 ) + π3 3PA (p∗3 ) p2

p2

p1 ≥ p2

s.t.

Solution to which is   −p1 (2 − δC )2 + f 5 p1 δC 3 + f 4 δC 2 (1 − p1 − δC (1 + 4p1 )) + f 3 δC 2 (2p1 (4 + δ) − 3(1 − δ)) +     −f 2 − 2(1 + 6p1 )δ − (1 − 8p1 )δ2 + (1 − p1 )δ3 + f 2 δC 4 − 5δ + δ2 + 2p1 (−2 − 5δ + 2δ2 ) ∗ p2 (p1 ) = (A.3) −6(1 − δC ) − δC 3 − f 4 δC 3 (5 − f ) + 2f 3 δC 2 (3 + 2δC ) − 2f 2 δC 2 (9 − 4δC ) − f (2 − 18δC + 12δC 2 − δC 3 ) Finally, substitute (A.2) and (A.3) into the first period retailer’s profit maximization problem:   max π1 3PA (p2 ) = max (1 − v1 )p1 + π2 3PA (p∗2 ) . p1

p1

Solving it results in the optimal first period price being

∗

∗

Substitute that into (A.3) and then further into (A.2) to obtain expressions for p3PA and p3PA . 2 1 2) Now let us consider the value of profit function 3PA (f) (price adjustment policy in a three-period scenario with refund in any period after purchase) when the refund fraction f is equal to δ2 = 0.9404: 

3PA



(f )|f =0.94

0.321 − 1.857δC + 4.47δC 2 − 5.737δC 3 + 4.14δC 4 − 1.593δC 5 + 0.255δC 6   = (1.064 − δC ) 1.113 − 4.339δC + 6.338δC 2 − 4.113δC 3 + δC 4

 (A.4)

∗

Further, recall that when P3 – t he profit with optimally chosen refund fraction for price adjustment policy in a 3-period scenario with refund in the next period only case – is equal to 1/3

δC (−3δC + 3δC 2 − 221/3 ((1 − δC )2 δC 4 )

∗

3P =

2(−6δC + 6δC + 21/3 ((1 − δC ) δC 2

3

2

4 )1/3

− 52

1/3

)

δC ((1 − δC )2 δC 4 )

1/3

, )

for

δC > δ2

(A.5)

∗

Straightforward comparison of expressions (A.4) and (A.5) shows that 3PA (f)|f = .94 > 3P , hence the profit with optimally ∗ chosen refund fraction in case of refund in any period (3PA ) is strictly greater than optimal profit with refund given in the next ∗ ∗ period only because the expression (A.4) represents a lower bound for 3PA : 3PA ≥ Π 3PA (f)|f = 0.94 . References Arbatskaya, Maria, Morten Hviid and Greg Shaffer (2004), “On the Incidence and Variety of Low Price Guarantees,” Journal of Law and Economics, 47, 307–32. Besanko, David and Wayne L. Winston (1990), “Optimal Price Skimming by a Monopolist Facing Rational Consumers,” Management Science, 36 (5), 555–67. Butz, David A. (1990), “Durable Goods Monopoly and Best-Price Provisions,” The American Economic Review, 80, 659–83. Chen, Yuxin, Chakravarthi Narasimhan and Z. John Zhang (2001), “Research Note: Consumer Heterogeneity and Competitive Price Matching Guarantees,” Marketing Science, 20, 300–14. Cooper, Thomas E. (1986), “Most-Favored-Customer Pricing and Tacit Collusion,” The Rand Journal of Economics, 17, 377–88. Corts, Kenneth S. (1997), “On the Competitive Effects of Price-Matching Policies,” International Journal of Industrial Organization, 15 (3), 283–99. Fudenberg, Drew and David Levine (1989), “Reputation and Equilibrium Selection in Games with a Patient Player,” Econometrica, 57, 578–9.

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