Serving Fraudulent Consumers? The Impact of Return Policies on Retailer's Profitability

SERVICE SCIENCEVol. 5, No. 4, December 2013, pp. 296–309ISSN 2164-3962 (print) � ISSN 2164-3970 (online) http://dx.doi.org/10.1287/serv.2013.0051

© 2013 INFORMS

Serving Fraudulent Consumers? The Impact ofReturn Policies on Retailer’s Profitability

M. Ali Ülkü, Lynn C. DaileySchool of Management and Leadership, Capital University, Columbus, Ohio 43209

{[email protected], [email protected]}

H. Müge Yayla-KüllüLally School of Management and Technology, Rensselaer Polytechnic Institute, Troy, New York 12180, [email protected]

Competition in the current marketplace requires businesses to provide consumers with the utmost convenience in purchas-ing services and goods. Buyers expect that, as an aftersales service and risk reliever, they can “return” goods if they are

not satisfied with them. Such product returns significantly influence retailers’ profits not only because of a reduction in netsales but also because of increased costs. Despite its substantial impact, research focused on retailer–consumer return policieshas been limited. To fill this gap, we study a retailer that has to set the product return policy parameters—specifically, theprice and the return period. The impact of these parameters on consumers’ valuation function is also taken into account.We study the model analytically and provide insights through various numerical examples. We find that even with fraudulentconsumers’ negative effect on sales, a retailer could expect both its profits and prices to increase with an optimally determinedreturn policy. We also find that a retailer that operates in an environment where consumers are not sensitive to return policiesshould be cautious when setting the return period.

Key words : service management; pricing; risk reliever; consumer behavior; retailing; return periodHistory : Received October 22, 2012; Received in final revised form April 17, 2013; Accepted May 18, 2013 by Paul

Maglio. Published online in Articles in Advance August 26, 2013.

1. IntroductionProduct returns are on the rise. A 2011 survey reports that product returns account for approximately 8.9%of total sales, up from 8.1% in 2010 (National Retail Federation’s Return Fraud Survey 2011). According tothe survey, consumers returned an estimated $217 billion worth of merchandise; retailers’ profits are negativelyaffected not only because of the reduction in net sales but also because of the increased costs involved withreverse logistics (Guide et al. 2006, Anderson et al. 2009). A fraudulent return is defined as an illegitimateproduct return, including returns of used or consumer-damaged products (Harris 2008). Such fraudulent returnscan significantly hurt retailers’ bottom lines. In a Bloomberg BusinessWeek article, Winter (2012) writes that“almost 5 percent of all retail returns during the holidays are fraudulent. Return fraud cost merchants an estimated$8.9 billion in 2012.” It is also reported that such fraudulent behavior accounts for almost 50% of all returns inthe clothing industry (King et al. 2008).

Despite the significant impact product returns have on profitability, there is a wide range of retailer returnpolicies (RRPs). At one end are retailers who offer an unlimited return period (e.g., L.L. Bean). At the other endare retailers who do not accept any returns; for example, Apple’s App Store has an “all sales final” policy. Mostretailers fall between these two extremes. Best Buy, for example, has a 30-day return policy for the majority ofits products but offers only 14 days for some products and up to 45 days for others. Moreover, retailers oftenvary their return policies, as evidenced by the 2011 National Retail Federation survey of retailers; it is foundthat 17.5% of retailers intended to alter their return policies for the holiday season. Walmart had an unlimitedreturn policy through the early 1990s. Currently, Walmart has a 90-day return policy for many products but hasseveral restrictions on various product categories. These changes suggest that there is a need to study retailers’return policies in general.

Interestingly, research on RRPs has been limited: “The literature on product returns is sparse, especially inrelation to analyzing individual customer1 product return behavior” (Petersen and Kumar 2009, p. 35). We aim tofill that gap with this paper. In particular, we address the following research questions: How are optimal product

1 In the remainder of this paper, we use the term “consumer” referring to an individual customer to emphasize the consumer decision-makingprocess.

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Ülkü, Dailey, and Yayla-Küllü: Serving Fraudulent Consumers?Service Science 5(4), pp. 296–309, © 2013 INFORMS 297

return policies characterized in a retailer–consumer setting?2 What is the impact of consumers’ fraudulentbehavior on RRPs? How do different market characteristics such as product depreciation and consumer sensitivityaffect optimal product return policies?

In our analytical model, we study a retailer that has to set the return policy parameters for a product: the priceand the return period. We use the price and the return period as decision variables in the design of RRPs becausepractice and extant research provide evidence of their importance in a consumer’s decision to purchase: pricenaturally is a central factor in consumer’s valuation and buying behavior, and the length of the return periodperforms as an influential risk reliever for the consumer. Moreover, our consumer valuation function captures therelationship between price and return period. We also take into account the impact of these parameters on theconsumers’ valuation function. A consumer may be willing to purchase and pay more for a product when thereis the option of longer return period. We formally incorporate the cost of fraudulent returns into our model.

We find that even with the negative effect of fraudulent consumers on sales, a retailer could expect its profit toincrease with an optimally determined return policy. We also find that a retailer that operates in an environmentwhere consumers are not sensitive to return policies should be careful in setting return periods. In addition, weshow that if the product depreciates quickly, it may not be profitable for the retailer to operate with a returnpolicy. We also observe that for those cases when the retailer may have to give a lifetime guarantee, setting amaximum consumption level before accepting a return may be a good idea.

The next section of our paper provides a longitudinal review of the consumer research literature that highlightsthe important characteristics of consumer behavior in responding to RRPs. In §3, we develop our mathematicalmodel. Next, we solve and use sensitivity analyses on key parameters to better understand RRPs. Finally, wegive our conclusions in §4.

2. Literature ReviewFirst of all, we note that this paper focuses exclusively on retailer–consumer return policies as opposed to man-ufacturer warranties (see Marvel and Peck 1995). Whereas the reason for returns in the manufacturer warrantiesliterature is often conformance quality failure (e.g., Hsiao and Chen 2012), the reason for returns in this paper isthe mismatch between consumers’ tastes and product characteristics. We assume that the product in our settingis perfect in terms of conformance to quality issues.

The literature on retailer–consumer returns is not extensive. Below we give a longitudinal review of RRPresearch. As noted previously, “The literature on product returns is sparse, especially in relation to analyzingindividual customer product return behavior” (Petersen and Kumar 2009, p. 35). By listing all relevant papershere, we also aim to provide the foundations of our modeling assumptions that are presented in §3.

Davis et al. (1995) examined the impact of retailers’ money-back guarantees, taking into account salvagevalue, consumer value of product trial, and probability of mismatch. They also considered the important impactof the transactions costs associated with return, including travel costs, lost time, and anxiety. They found thatmoney-back guarantees are more profitable than selling as-is when the retailer has a salvage value advantageover consumers, and it is less profitable than selling as-is when the consumer transaction cost is too smallrelative to the trial value. Their research specifically calls for an investigation of the impact of the return period.Our paper sheds light on this.

Che (1996) presented an exploratory study that developed a model for product returns. He argued that returnpolicies are a method of risk balancing between retailers and consumers because some goods require experiencebefore determining their value. His model highlighted consumers’ risk aversion and its impact on retailer profits.Hess et al. (1996) were among the first to examine the impact of nonrefundable charges in RRPs. They usedempirical observations from a mail-order clothing company to support their assertion that nonrefundable chargescan be used to profitably control inappropriate returns. Hess and Mayhew (1997) investigated consumers andtheir propensity to return and suggested that simply examining the overall return rate is not sufficient for anaccurate analysis. They argued that direct marketers have access to detailed information on consumers and theirreturn behavior; thus, they should use this information to predict a consumer’s propensity to return merchandiseand when this return will occur. They further showed support that a split-adjusted hazard model outperforms aregression model when modeling product return times.

2 Manufacturer–retailer return policies have been studied extensively as they fall into the general category of “warranty design,” where thequality of the product is a determinant of the optimal return policy (Gurnani et al. 2010). However, in a retailer–consumer environment,assuming that the product is of acceptable quality, the uncertainty comes from the possibility of a mismatch between the product and theconsumer’s taste.

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Ülkü, Dailey, and Yayla-Küllü: Serving Fraudulent Consumers?298 Service Science 5(4), pp. 296–309, © 2013 INFORMS

Davis et al. (1998) studied 133 retailers and found that none had a partial refund policy; instead, most imposedsome type of hassle on consumers. They defined “hassle” as restrictions on returns; such hassles includingrefusing a return without receipt, requiring original packaging, refusing returns on opened merchandise, refusingproducts that appeared to be used, and limiting return time. Their study found that when product benefits cannotbe consumed during a short period of time, when there is an opportunity for cross-selling, and when a highsalvage value can be obtained for returned merchandise, retailers were more likely to offer a low-hassle RRP.These researchers developed a novel “hassle index” by combining the hassle variables stated previously. Thesevariables were dichotomous. Thus, return time was considered a hassle; however, it was evaluated as havingeither an unlimited time for returns or a limited one. In practice, return periods vary widely among retailers andacross products, as demonstrated in the preceding section.

Wood (2001) examined return policies in remote purchase environments. She found that lenient RRPsincrease purchase probability. Furthermore, they also decrease prepurchase deliberation time while not increasingpostpurchase deliberation regarding whether or not to keep the product.

Anderson et al. (2009) developed a product return model that quantifies consumers’ valuation of an RRP.They found that this valuation varies across product categories and consumers. Interestingly, for their sampledretailer, they found that consumers’ ability to return footwear ($15) was more valuable than their ability to returnmen’s tops ($3). Their model was used to balance the product depreciation and reverse logistics cost associatedwith RRPs against the benefit of expected increase in demand for the product.

Returns are more likely when a purchase is made online. At the junction of marketing and supply chainmanagement and from the perspective of end consumers, Mollenkopf et al. (2007) constructed a structuralequation model that explores the impact of an Internet product returns management system on customer loyaltyintentions. Kang and Johnson (2009) investigated the relationship between apparel return behavior and fashioninnovativeness, buying impulsiveness, and consideration of return policies of the U.S. consumers. Their empiricalfindings suggest that apparel return behavior is positively correlated to impulsive buying and consideration ofreturn policies. Moreover, they found that these two variables are significant predictors of the frequency ofapparel returns. Petersen and Kumar (2009) addressed the issue of the negative assumption associated withproduct returns. They identified several factors that help explain situations when consumers are more likely toreturn products. It is worth noting that they found that people who return moderate amounts of product actuallypurchase more in the future. Their study stressed the importance of evaluating both the cost and future benefitsassociated with product returns.

Recently, Dailey and Ülkü (2012) studied the psychological aspects of restrictive product return policies byspecifically addressing how and why consumers react to restrictive RRPs as well as how the variation betweenand within RRPs affects consumers. Building on the theory of psychological reactance, they examined theconsumers’ expectations of product return control and how management can mitigate possible negative outcomesrelated to such reactance. In the context of distance purchasing, Foscht et al. (2013) studied the return behavior ofconsumers through factors such as buying experience, perceived risk, and return frequency, and they segmentedconsumers into different groups to better reveal the characteristics of “returners.”

We contribute to the above stream of articles by considering both the consumer behavior and the optimalsetting of an RRP. None of these articles takes into account fraudulent behavior when identifying optimal pricingand return period decisions, which is the key driver of our results.

3. Model Development and AnalysisIn this section, we first present the model assumptions and provide the decision timeline. Then, we providethe details of the analysis. Figure 1 is a graphical representation of the decisions in our model, where a squarerepresents a decision node and a circle represents a chance node. Similarly, the dark color represents the retailerand the light color represents the consumer. The notation we use is shown in Table 1. (Some less frequentlyused notation is explained in the relevant section.)

3.1. Decision Timeline and Model Assumptions

Our model considers a retailer that offers a particular product. In Stage I, the retailer makes two decisions: theprice of the product P and the allowable return period T . To evaluate the product, the consumer must purchaseand try it during this return period (e.g., 60 days). For example, a retailer may set a return period of six monthsfor a baby crib.

In Stage II, the consumer decides whether or not to buy the product based on her expected value from thepurchase. Each consumer considers buying only one unit, and the repeat purchase of an identical item does notoccur. The product can be fully judged only after purchase.

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Figure 1. Sequence of Decisions

Stage I

Buy

RetailerannouncesP and T

Don’t buy

Stage II Stage III Utilityv – P

vc – h

s – P

–h

0

Retailer Consumer Consumer

Return

Mismatch

Match

Return

1 – m

m

Keep

Keep

In Stage III, the consumer tries the product. The product may satisfy (good match) or dissatisfy (mismatch)the consumer’s taste and fit. The probability of a good match m is identical for all consumers and is known bythe retailer. If a poor match occurs, she may decide to keep or return the product. As is common practice, theretailer fully refunds a returned product, yet a hassle cost of h dollars for returning the product is accrued to theconsumer. If she decides to keep the product, she can salvage the product for s dollars. For example, a misfitbaby crib could be salvaged in a garage sale if the return hassle is burdensome for the consumer.

If there is a match between the product and the consumer’s tastes, she derives a value of v. We assume thatthe consumer base is heterogeneous in their tastes and that this valuation is uniformly distributed over 601 �7.Yet how can we explicitly integrate the effect of return period into consumer’s valuation of the product? Theconsumer behavior and perceived risk literature helps explain the impact of the return period on the consumerpurchase decision (e.g., Bettman 1973, Mitchell 1999, Zeithaml and Bitner 2003, Dailey and Ülkü 2012). On onehand, the perceived risk of a purchase has been conceptualized as having two components: (1) the likelihoodof adverse consequences and (2) the magnitude of those adverse consequences. Such adverse consequences

Table 1. Model Nomenclature

Decision variablesP Selling price (in dollars per unit)T Length of return period (in time units)

Parametersc Consumption proportion of the product during the

return period 40 < c < 15d Depreciation sensitivity to salvage (dollars per time unit)h Hassle cost of returning a product, both psychological

and physical (in dollars)l Length of product life (in time units)m Probability of a good match between the consumer’s

taste/needs and the productr Return period sensitivity to valuation (in dollars per

time unit (r ≥ 0))� Consumer’s maximum valuation of the product without

an RRP (in dollars)s Salvage value to the consumer (dollars per unit)sT Salvage value to the retailer for a return period of

length T (in time units)� Maximum valuation as a function of the return period,

� = �+ rT (in dollars)v Product valuation of a consumer (a realization of a

random variable)w Wholesale purchase cost (in dollars per unit)

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include, for example, consumer dissatisfaction and switching brands as a result of not being able to return themismatched product. On the other hand, risk relievers (i.e., brand reputation, information sources, product trial,etc.) are ways to reduce consumers’ perceived risk (Dowling and Staelin 1994). Lwin and Williams (2006) foundthat allowing consumers the ability to return products can be viewed as a risk reliever. The authors suggestthat the likelihood of adverse consequences is also reduced by the length of the return period. We capture thisproperty, as seen in Table 1, by modeling the upper bound of the valuation range � as a function of the returnperiod T and the “intrinsic” valuation of the product �; that is, � = �+ rT . In other words, the longer the returnperiod, the higher the probability that the consumer will buy the product. However, longer return periods implymore damage or less salvage value to the retailer sT 1 which we model as a linear function of the return period Tand the product life time l as

sT =w41 −d4T /l551 0 ≤ d < 1 and 0 ≤ Tmin ≤ T ≤ l0 (1)

Note that Equation (1) guarantees that the salvage value to the retailer of the returned product is less thanits wholesale price. The return period is bounded by the product life time, and the “trial period,” if one exists,which we denote by Tmin. The determination of the parameter d (depreciation sensitivity to the length of thereturn period) essentially depends on the product specifics and the contract (i.e., rebate) between the retailer andthe manufacturer. In reality, products differ in their probability of mismatch because they vary with regard to fit(Davis et al. 1995). For example, clothing would likely have a greater likelihood of mismatch compared withlightbulbs. Thus, the value assigned to the return period will vary depending on the product and the likelihoodfor mismatch. An increased likelihood of mismatch will increase the likelihood of adverse consequences and,hence, perceived risk. In these situations, consumers will put a higher value on the length of the return period T ;the lower the probability of mismatch, the less value the consumer would place on T . Because reduction inperceived risk could be viewed as a benefit for consumers (Wood 2001), it would therefore positively impactconsumers’ perceived value. To that end, the return period should have a measurable value (Anderson et al.2009). We measure this value by benchmarking the length of the return period by the product’s useful life.

If, however, there is a match between the product and consumer’s tastes and she returns it to the retailer afterextracting some value, we identify this decision as fraudulent behavior. For example, a consumer who purchaseda crib may use it for six months and then return it to claim a full refund.

3.2. Analysis

We solve the problem with backward induction. In Stage III, once the purchase is made and the level of matchis determined, the consumer decides whether or not to keep the product. If a mismatch occurs and she keepsthe product, she will receive a utility of s − P . If a poor match occurs and she returns the product, her utilitywill be the full refund less the price paid and the hassle cost of returning the product h. Our model assumesthat the consumer will return the product when there is mismatch. This translates into the participation-incentiveconstraint P ≥ h+ s. We note that this type of return is legitimate because the consumer cannot derive any valuefrom the mismatch.

By contrast, if a match occurs and the consumer decides to keep the product, her utility would be v − P .However, as noted in Figure 1, even when a match occurs, the consumer might decide to return the product;this return would be deemed fraudulent. The utility of a fraudulent returner is the value consumed during thereturn period cv less the hassle cost h. Then, the fraudulent behavior will materialize whenever the gain frommaking a return P −h is higher than or equal to the residual value of keeping the product 41 − c5v. Hence, afraudulent consumer is indifferent when these two quantities are equal. Denote by v1 the boundary valuationof that fraudulent consumer. A consumer will behave opportunistically whenever her valuation is less than orequal to

v1 = 4P −h5/41 − c50 (2)

Next, in Stage II, the consumer will purchase the product if her expected surplus is positive. Let v0 bethe value of the boundary fraudulent consumer3 who is indifferent between purchasing the product. Note thatthe fraudulent consumer will return the product regardless of its match. Therefore, the utility calculation for

3 We emphasize that v0 is determined particularly for the fraudulent consumer; hence it does not consider the utility of keeping the product.On the other hand, if only legitimate returns were allowed, the boundary consumer valuation—say, vb—would be calculated by solvingm4vb − P5 − 41 − m5h = 0, yielding vb = P − h + h/m. One can easily show that vb ≥ v01 implying that the probability for a possiblefraudulent return is decreased. Legitimate returns can be imposed by stricter scrutiny as opposed to a “no questions asked”–type of returnpolicy. It is also important to note that the condition v1 > v0 enables us to capture the expected amount of fraudulent returns.

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such a consumer will be based on the differences of returning the product rather than keeping it. Explicitly,if she makes the purchase and there is a good match, she gains a value of cv0 yet loses h dollars whenthe product is returned. On the other hand, if there is no match and she returns the product, it will cost heronly h dollars. Hence the expected utility of a boundary fraudulent consumer EU4v05 can be calculated byEU4v05=m4cv0 −h5− 41 −m5h. Now, solving EU4v05= 0 for v0, we get

v0 = h/4mc50 (3)

Thus, the consumer will buy the product only if her valuation is greater than v0. Employing Equation (3), wefind that the probability that a consumer will buy the product pB is

pB = 1 − v0/� = 1 −h

mc4�+ rT 50 (4)

Equation (4) suggests that the probability of purchase is decreasing in hassle cost and increasing in theprobability of match, the consumption rate, the product valuation without RRP, and the return period. Our modelincludes the impact of the return period on the consumer’s purchase decision, which has not been done inprevious literature.

Finally, in Stage I of our model, the retailer determines and announces the profit-maximizing selling priceand the return period. Hence, our focus is on the direct and joint optimization of the price P and the length ofreturn period T . We note that because the demand size in our model is normalized to 1, the expected number ofbuyers, for example, has the same value as the probability of a consumer buying the product. Similar analogieshold for other probabilities we define later.

Figure 2 displays how consumers are segmented in the three-stage decision process displayed in Figure 1.For ease of further analysis, we now introduce the following definitions.

Definition 1. pL is the probability that the return is legitimate (e.g., as a result of consumer–productmismatch).

Definition 2. pF is the probability that the return is fraudulent.Definition 3. pK is the probability that the consumer will keep the product.We can now obtain the probability of a consumer’s decision type by simply calculating the corresponding

area in Figure 2. For example, the probability that the consumer will return the product despite it being a goodmatch pF is the rectangular area with height m/� and width v1 − v0. Similar geometric interpretations enable usto explicitly write the expressions of those probabilities as

pL = 41 −m5pB = 1 −m−h41 −m5

m4�+ rT 51 (5)

pF =m4v1 − v05/� =cm4P −h5−h41 − c5

c41 − c54�+ rT 51 and (6)

pK =m41 − v1/�5= 1 −4P −h5

41 − c54�+ rT 50 (7)

Note that the expected number of buyers (i.e., the probability that the consumer will buy the product, pB) isequal to the sum of the expected number (i.e., the probability) of legitimate and fraudulent returns and thosethat keep the product. That is,

pB = pL +pF +pK 0 (8)

Figure 2. Breakdown of Consumer Types (Based on Chu et al. 1998)

Legitimate returns pL

Fraudulentreturns Keep

No buy

P – hv1 =1 – cmc

hv0 =

1

Utility

Con

sum

er d

ensi

ty

Buy the product

1 – pB

v

v

pKpF

1 – mv

mv

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We highlight that the values of the probabilities in Equation (8) directly impact the formation of the RRPand, therefore, the retailer’s profit. It would be best to minimize the probability of fraudulent returns whilemaximizing the probability of a consumer buying the product and keeping it. (Note that a legitimate returncould be the result of other reasons such as defective quality rather than a mismatch. A proper analyticalmodel for that case should consider the cost of quality as well, which is not in the domain of this study.)Proposition 1 how those probabilities are affected by some key parameters. All proofs are provided in theappendix.

Proposition 1. Let the symbols −, →, ↑, ←, and ↓ represent the directional derivative terms noneffective,nondecreasing, strictly increasing, nonincreasing, and strictly decreasing, respectively. Sensitivities of proba-bilities of different consumer decisions for changes in each and all of the variables and their correspondingcondition (if it exists), follow.

Variable pB pK pL pF

P − ↓ − ↑

T → → ↑ ←(i)c ↑ ← ↑ ↑

h ↓ ↑ ↓ ↓

m ↑ →(ii) ←(iii) →

r ↑ → ↑ ←(i)� ↑ → ↑ ←(i)

Condition. (i) is where P ≥ 4h61 − c41 −m575/4cm5, (ii) is where P ≤ h +

41 − c54�+ rT 5, and (iii) is where T ≥ 4h/cm2 −�5/r .

Proposition 1 summarizes, by comparative statistics, how the buying behavior (whether to keep or return)changes with respect to some key variables and provides retailers with valuable insights for designing theirRRPs. Our results imply that the total expected number of buyers (i.e., the probability that a consumer willpurchase the product) increases as the consumption proportion, return period effect on valuation, probabilityof match, and intrinsic valuation of the product increases. However, it decreases with the hassle cost and isnot affected by price. The latter result is because product returns, as set in our model, are fully refunded. It isinteresting to note that legitimate returns tend to increase with the return period. Although this may sound likea mismatch increase as a result of longer trial periods, it is in fact due to the inflation of buyers as a result theattractiveness of the returns opportunity. Note also that the probability of a legitimate return is nonincreasing inthe probability of a match after a threshold value of time during the return period. We believe that this interestingbehavior arises for two reasons: First, the higher the match with the consumer’s needs or tastes, the less likelythe product will be returned. Second, “non-fraudulent” consumers might want to return a mismatched productas soon as possible before the end of the return period to receive their refund quickly. This also provides thema relief in “thinking cost.” Note that increased time in return period also increases fraudulent returns when theproduct price is below a certain threshold. More interestingly, although an increase in the hassle costs reducesthe total number of buyers, eventually, the number of consumers keeping the product increases. We also findthat although the probability of match increases the number of buyers, it eventually reduces the number ofkeepers if the price is above a certain threshold. More fraudulent returns are expected in that case. Proposition 1emphasizes that retailers should take into account all the parameters (and most importantly, the price and thelength of the return period for a particular type of product) when designing their RRPs, as all these variableshave an effect on their profit.

Employing the expressions in Equations (5)–(7), the retailer’s total expected profit, Eç, can be calculated asfollows:

Eç= P4pK5+ sT 4pL +pF 5−w0

Including the regularity conditions (0 ≤ pB1 pK1 pL1 pF ≤ 15 and those on the length of the return period anddemand generation, we get the following nonlinear mathematical programming model, which we name ORRP.

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Optimization Model for RRP (ORRP):

maxP1T

Eç= l−1

(

m6P −h+ 4c− 154�+ rT 576l4P −w5+dwT 7

4c− 154�+ rT 5−

hw4L−dT 5

cm4�+ rT 5−dwT

)

s.t. max{

w1h+ s1h41 − c+ cm5

cm

}

≤ P ≤ min{

�+ rT 141 − c54�+ rT 5

m+h41 − c+ cm5

}

1

max{

Tmin1h− cm�

rcm1P − 4h+ 41 − c5�5

r41 − c5

}

≤ T ≤ min{

l1l4w−h5

dw

}

0

(9)

Proposition 2. The optimization model ORRP is jointly concave in its decision variables P and T ifc <ì1/ì2, where ì1 = 4rlwh4rl+d�5 and ì2 = 4rhlm52 − 2rhlw4m2 − 254rl+d�5+ 4mw4rl+d�552.

Proposition 2 concludes that finding the optimal solution to this model necessitates a complete search withinthe constraints. Once the values of the parameters are identified and fed into the model, ORRP can be directlysolved to optimality by using optimization software that employs gradient search techniques. Alternatively,because the objective function is jointly concave in the decision variables, and because in practice, and particu-larly in a competitive market, conditions may dictate the price (e.g., Ülkü and Bookbinder 2009) or the returnperiod, the optimal solution to ORRP can also be achieved by an iterative approach by fixing one variable andthen finding the optimal solution in the other one, and vice versa. We coded and solved ORRP to optimalityusing Mathematica 9, in which the solution procedure finds the interior solution and checks its feasibility whencompared with the constraints to ensure global optimality.

4. Numerical Example and Sensitivity AnalysisTo illustrate the mechanics of the optimization model ORRP in equation set (9), we use the base case parametersdisplayed in Table 2. In our numerical example, consumers are sensitive (r > 0) to the return period choice ofthe retailer such that the valuations’ range increases as the return period T increases. Conversely, the retailer isless sensitive to the depreciation of the product. Consistent with the earlier example of a baby crib, the lifetimeof the crib is approximately 50 weeks, and the consumption proportion of the product during the return period is60%. We set the probability of match at 80%, and the salvage value to the consumer is 5 monetary units whenthe maximum valuation in the market is 100. Note also that the product costs the retailer 20 monetary units.With this example, we are looking for the optimal price and the optimal return period. Using our optimizationsoftware, Mathematica 9, we get the following optimal solution:

P ∗= 41051 T ∗

= 501 p∗

B = 007501 p∗

K = 003761 p∗

L = 001501 p∗

F = 002241 and Eç∗= 203360

Figure 3 shows the objective function for a specified range of the decision variables. If there was no RRPapplied to this product, the optimal selling price and the optimal profit would be m�/2 and m�/4 −w, respec-tively. The optimal percentage of buyers in a “no refund” policy would thus be 50% (cf. Chu et al. 1998). Undersuch a policy, the optimal price would be 40, yielding zero profit for the retailer. However, according to ourmodel in which there is a lifetime return policy, the optimal price is higher than 40 and the percentage of buyersis 75%. Moreover, returns are salvaged at a positive price, yielding an overall positive net profit. With thisparticular numerical example, we have shown that a return policy relieves the risk of the consumer’s purchase

Table 2. Base Case Parameters

Parameter Value

r 0.5d 0.1c 0.6h 15l 50m 0.8s 5p 100w 20

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Figure 3. Graphical Exhibit of the Objective Function

3

2

1

030

40

P

EΠ

5010

20

30

T

40

50

decision, and even though fraudulent returns are allowed, the cost of such returns is mitigated by the increasein purchase volume.

We have also studied separate optimization problems as we change each parameter in our base case. Figures 4and 5 show how retailer’s profits and prices change as the consumers’ sensitivity to return period changes. Weobserve that a retailer could expect both its profits and its prices to increase with the sensitivity of consumers.Hence, either through advertising or through public relations a retailer should try to increase the sensitivity ofits consumers to the return policy.

We also observe that the optimal return period T ∗ may not be the maximum product lifetime. Below a certainthreshold value of r , we observe in Figure 6 that the optimal return period is smaller than the maximum lifetimeof the product. Hence, a retailer that operates in an environment where consumers are not sensitive to returnpolicies should be careful in setting the return policy parameters.

Figure 4. Effect on Optimal Profits as Consumer Return Period Sensitivity Changes

5

4

3

2

1

00.1 0.3 0.5

r0.7 0.9

EΠ

*

Figure 5. Effect on Optimal Prices as Consumer Return Period Sensitivity Changes

46

45

44

43

42

41

40

39

38

370.1 0.3 0.5

r0.7 0.9

P*

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Figure 6. Effect on Optimal Return Period Length as Consumer Return Period Sensitivity Changes

50

48

46

44

42

40

380.1 0.2

38.33

0.3 0.4 0.5 0.6 0.7 0.8 0.9

T*

r

Figures 7 and 8 depict the influence of depreciation on retailer’s decisions regarding profits and return periodlength, respectively. We observe that it is only profitable for the retailer to operate for lower levels of depreciationsensitivity. Moreover, as the depreciation sensitivity increases, the optimal return period decreases. This reductionis observed to be nonlinear and quite sharp beyond a certain threshold value. For example, at d = 003, the retailercan optimally offer a lifetime return period. However, at d = 004, the optimal return period is almost half of thelifetime of the product.

Figure 9 investigates the impact of the consumption proportion of the product at the end of the return periodon the retailer’s expected profit. We observe that this relationship is strictly concave. There might be cases wherethe retailer may have to give a lifetime guarantee because of competitive pressures. For such cases, setting amaximum consumption level (with a cutoff at the maximum profit point) may be a good idea, as suggested byour numerical example.

Hassle costs, albeit under different names (e.g., restocking fee, return fee), are widely used in practice. Theappropriate determination of the hassle cost helps impose moral hazard on the consumer’s return decision (e.g.,Davis et al. 1995). Figure 10 shows the impact of the increase in hassle cost on the optimal probabilities of the

Figure 7. Effect on Optimal Profits as Depreciation Sensitivity Changes

3

2

1

00 0.2 0.4 0.6 0.8 1.0

EΠ

*

d

Figure 8. Effect on Optimal Return Period Length as Depreciation Sensitivity Changes

50

40

30

20

10

00 0.2 0.4 0.6 0.8 1.0

T*

d

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Figure 9. Effect on Optimal Profits as the Consumption Proportion During the Return Period Changes

6

5

4

3

2

1

00.2 0.3 0.4 0.5 0.6 0.7 0.8

EΠ

*

c

consumer’s decisions. (We use the term “optimal probability” in the sense that the corresponding probabilities(see Equations (4)–(7)) are calculated using the optimal values of the price and the return period.) We foremostobserve that as the hassle cost increases, the probability that the consumer will purchase the product decreases,as this cost has a direct effect on the utility. On the other hand, once the product is purchased, the probabilitythat it will not be returned increases in the hassle cost. We also notice that an increase in hassle costs discouragesfraudulent returns; the probability of a fraudulent return shows a steeper decrease compared with that of alegitimate return. Interestingly, we see that above a threshold value of a high hassle cost (in our numericalexample, this is about $19), the probability of a fraudulent return is less compared with that of a legitimatereturn. We also note that the findings in Figure 10 verify the static sensitivity analysis in Proposition 1. Finally,Figure 11 shows that the optimal expected profit is increasing in the hassle cost. We note that this result is validfor our model setting of a full refund. Of course, this should not imply that the optimal hassle cost should beset to its highest possible level, as this can reduce the probability of purchase and hence the sales volume.

Figure 10. Effect of Hassle Cost on the Optimal Probabilities of Consumer Decisions

1.0

0.9

0.8

0.7

0.6

0.5

0.4

Opt

imal

pro

babi

litie

s

0.3

0.2

0.1

06 9 12 15 18 21 24

h

pB*

pK*

pL*

pF*

Figure 11. Effect of Hassle Cost on Optimal Profits

EΠ

*

h

4

3

2

15 10 15 20 25

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5. Concluding RemarksRetailers have the ability to vary their return policies by altering their return periods, which can ultimatelyinfluence the product price. In this paper, we illustrated a three-stage decision process that included a retailer’sdetermination of the return policy and consumers’ decisions to purchase and return a product. The impact ofthe retailers’ return policies on the three stages and the consumers’ corresponding decisions are cast as anoptimization problem with two continuous prices and return period decision variables. The distinct features ofour model include the explicit incorporation of a fundamental purchase-risk reliever (i.e., a longer return period)into the consumer’s valuation function. The model shows that the RRP decision has very important implications.

From a consumer perspective, RRPs influence the purchase and return decisions. From a retailer perspective,product returns, in turn, influence retailer costs and profit. Previous research has empirically validated the effectof price and various aspects of the return policies on purchase probability and return behavior (e.g., Hess andMayhew 1997, Wood 2001, Kang and Johnson 2009, Petersen and Kumar 2009). But to develop optimal RRPs,our study finds that retailers must weigh the impact of the consumer’s return period the same as their ownprofits.

Our observations suggest that a retailer could expect its profits and prices to increase with return policysensitivity of consumers. Hence, either through advertising or through public relations, a retailer should put effortinto increasing consumer sensitivity to the RRPs. We also note that a retailer that operates in an environmentwhere consumers are not sensitive to return policies should be careful in setting the length of the return period.

Another finding implies that only for lower levels of depreciation sensitivity is it profitable for the retailer tooperate an RRP. As the depreciation sensitivity increases, the optimal return period decreases. We also examinedthe impact the consumption proportion of the product has at the end of the return period. Our numericalinvestigation suggests that for those cases when the retailer may have to give a lifetime guarantee, setting amaximum consumption level in the return policy may be a good idea.

For a multichannel retailer (see Metters and Walton 2007), we plan on studying the optimal design of anRRP. Also, an empirical determination of the relationships between the return times, the type of products, andthe length of the return period would be an interesting research project to pursue in the future.

AppendixProof of Proposition 1. The sensitivity conditions are obtained by taking the first derivative of the corresponding prob-

ability function (from Equations (5) to (8)) with respect to the key parameter of interest and then by checking the conditionsunder which that derivative is strictly positive or negative. Our plausible assumptions on the seven parameters and decisionvariables that apply to all of the calculations below are T ≥ 0, P − h ≥ 0, h > 0, r ≥ 0, � > 0, 0 < c < 1, and 0 <m< 1.Accompanying conditions for certain directional behaviors (i.e., nonincreasing) of the derivative function at hand are givenin the corresponding line.

1. Derivative analysis for P :

¡pB

¡P= 0 (i.e., noneffective)1

¡pK

¡P=

m

4c− 154�+ rT 5< 0 4strictly decreasing because c− 1 < 01 m> 01 and1 �+ rT > 051

¡pL

¡P= 0 (noneffective)1

¡pF

¡P=

−m

4c− 154�+ rT 5> 0 (strictly increasing)0

2. Derivative analysis for T :

¡pB

¡T=

hr

cm4�+ rT 52≥ 0 (nondecreasing)1

¡pK

¡T=

−mr4P −h5

4c− 154�+ rT 52≥ 0 (nondecreasing)1

¡pL

¡T=

hr41 −m5

cm4�+ rT 52≥ 0 (nondecreasing)1

¡pF

¡T=

r6h4c− 1 − cm5+ cmP7

c4c− 154�+ rT 52≤ 0

(

nonincreasing, if h4c− 1 − cm5+ cmP ≥ 01 implying that

P ≥h61 − c41 −m57

cm1 which is expressed as condition (i) in Proposition 1

)

0

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3. Derivative analysis for c:

¡pB

¡c=

h

c2m4�+ rT 5> 0 (strictly increasing)1

¡pK

¡c=

−m4P −h5

4c− 1524�+ rT 5≤ 0 4nonincreasing because P −h≥ 0 and 4c− 152 > 051

¡pL

¡c=

h41 −m5

c2m4�+ rT 5> 0 4strictly increasing because 0 < 41 −m5< 151

¡pF

¡c=

mh

c2m4�+ rT 5+

m4P −h5


4. Derivative analysis for h:

¡pB

¡h=

−1cm4�+ rT 5

< 0 (strictly decreasing)1

¡pK

¡h=

−m


¡pL

¡h=

m− 1cm4�+ rT 5

< 0 (strictly decreasing)1

¡pF

¡h=

1 + c4m− 15c4c− 154�+ rT 5

< 0 4strictly decreasing3 note also that − 1 < c4m− 15 < 050

5. Derivative analysis for m:

¡pB

¡m=

h

cm24�+ rT 5> 0 (strictly increasing)1

¡pK

¡m= 1 +

P −h

4c− 154�+ rT 5≥ 0 4if the given values of P and T satisfy P ≤ h+ 41 − c54�+ rT 51

which is expressed as condition (ii) in Proposition 1; otherwise, the directional derivative is nonincreasing51

¡pL

¡m= −1 +

h

cm24�+ rT 5≤ 0 4nonincreasing if T ≥ r−14h/cm2

−�51 which is given as condition (iii)

in Propositon 151

¡pF

¡m=

−4P −h5

4c− 154�+ rT 5≥ 0 (nondecreasing)0

6. Derivative analysis for r :

¡pB

¡r=

hT

cm4�+ rT 52≥ 0 4i.e., nondecreasing51

¡pK

¡r=

−mT 4P −h5

4c− 154�+ rT 52≥ 0 4nondecreasing51

¡pL

¡r=

hT 41 −m5

cm4�+ rT 52≥ 0 4nondecreasing51

¡pF

¡r=

T 6h4c− 1 − cm5+ cmP7

c4c− 154�+ rT 52≤ 0

(

this holds if P ≥h61 − c41 −m57

cm1 i.e., condition (i)

)

0

7. Derivative analysis for �:

¡pB

¡�=

h

cm4�+ rT 52> 0 4strictly increasing51

¡pK

¡�=

−m4P −h5

4c− 154�+ rT 52≥ 0 4nondecreasing51

¡pL

¡�=

h41 −m5

cm4�+ rT 52> 0 4strictly increasing51

¡pF

¡�=

h4c− 1 − cm5+ cmP

c4c− 154�+ rT 52≤ 0

(

if h4c− 1 − cm5+ cmP ≥ 0 or P ≤h61 − c41 −m57

cm3 that is,

if condition (i) in Proposition 1 holds)

0 �

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Proof of Proposition 2. Assuming an interior solution, we check the second-order conditions for the objective functionvia the Hessian matrix, 6H7, as follows:

6H7=

¡2Eç¡P 2

¡2Eç¡P¡T

¡2Eç¡T ¡P

¡2Eç¡T 2

1

where we find

¡2Eç¡P 2

=2m

4c− 154�+ rT 51

¡2Eç¡P¡T

=¡2Eç¡T ¡P

=rlm4h− 2P +w5+dm�w

l4c− 154�+ rT 521 and

¡2Eç¡T 2

=2r6−clr6m2P4w−P5+h4m24P −w5+w57+ cd�w4h4m2 − 15−m2P5+hw4rl+d�57

clm4c− 154�+ rT 530

The first principal minor of 6H7 by definition is �H1� = ¡2Eç/¡P 2. The second principal minor, or the determinant of theHessian matrix 6H7 is �H2� = 4¡2Eç/¡P 254¡2Eç/¡T 25 − 4¡2Eç/¡P¡T 52. After some algebra we obtain the expressions�H1� = 2m/44c− 154�+ rT 55 and �H2� = 4ì1 − cì25/4cl

24c− 1524�+ rT 545. Because 0 < c < 1, 0 < m < 1, r ≥ 0, andl > 0, �H1�< 0 and �H2�> 0 if c <ì1/ì2, which are the two conditions that suffice to show that Eç is jointly concave inboth P and T . (For joint concavity analysis, see, for example, Chiang 1974.) We note that if c ≥ ì1/ì2, then a boundarysolution is optimal. �

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Serving Fraudulent Consumers? The Impact of Return Policies on Retailer's Profitability

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