How Does Product Recovery Affect Quality Choice? Atalay Atasu College of Management, Georgia Institute of Technology, Atlanta, GA 30332. Phone: (404)894-4928, Fax: (404) 894-6030 e-mail: [email protected]. Gilvan C. Souza Kelley School of Business, Indiana University, Bloomington, IN 47405. Phone: (812)855-3491, Fax: (812)856-5222 e-mail: [email protected]. July 18, 2011

1

Abstract We study the impact of product recovery on a firm’s product quality choice, where quality is defined as an observable performance measure that increases a consumer’s valuation for the product. We consider three general forms of product recovery: (i) when product recovery reuses (after reprocessing) quality inducing components or material (e.g., remanufacturing), (ii) when product recovery does not reuse quality inducing components or material but it is overall profitable (e.g., cell phone recycling), and (iii) when product recovery is costly (but mandated by legislation, e.g., recycling of small appliances in the European Union). Using a stylized economic model, we show that the form of product recovery, recovery cost structure and the presence of product take-back legislation play an important role in quality choice. Generally speaking, product recovery increases the firm’s quality choice, except for some instances of recovery form (ii). In addition, we find that product take-back legislation can lead to higher quality choice as opposed to voluntary take-back. We further demonstrate that both the firm and the consumers benefit from recovery form (ii), while both are worse off with recovery form (iii). However, environmental implications of the three recovery modes differ from their impact on consumer surplus and firm profit. While recovery forms (i) and (iii) reduce consumption and increase environmental benefits, the same is not true with recovery form (ii), which can increase consumption, potentially resulting in higher environmental impact. (Keywords: Quality Choice, Remanufacturing, Recycling, Take-Back Legislation.) Submitted: August 2010; Revisions Submitted: February 2011, and May 2011; Accepted: May 2011.

1

Introduction

In their book “Green to Gold”, Esty and Winston (2006) offer a list of the top ten environmental issues facing humanity: climate change, energy, water, biodiversity and land use, chemicals, toxins and heavy metals, air pollution, waste management, ozone layer depletion, oceans and fisheries, and deforestation. Anecdotal evidence suggests that the top two issues receive significant global coverage in the popular press, whereas landfill and depletion of natural resources only indirectly make Esty and Winston’s top 10 list, under the general category of “waste management”, even though they are critical to the sustainability efforts of many manufacturing firms. In practice, many firms implement waste management programs that consist of taking back and processing used products in an environmentally friendly manner, which we call “product recovery” in broader terms. There are two basic reasons for such efforts: Firms take back used

1

products either because it is profitable, or in order to comply with environmental legislation. When there is high residual value in used products (e.g., industrial copiers, container products etc.), firms may increase their profitability by recycling materials, remanufacturing products or components, or reselling used products as-is. However, when residual value is limited (e.g., old technology TV sets), the only reasons for taking back used products are legislative compliance requirements (e.g., recycling mandates) or public pressure from environmental groups (ETC 2010). Environmental legislation that mandates product take-back is a reality today. Twenty-seven countries in Europe have enacted product take-back legislation for e-waste (Waste Electrical and Electronics Equipment (WEEE) Directive 2002/96/EC), twenty three states in the U.S. have passed similar e-waste laws (ETC 2010) and Japan has enacted recycling laws for home appliances and computers (Tojo 2004). Such legislation has a direct impact on manufacturing firms, who must be prepared to collect and recover a significant volume of products post-consumer use (Atasu and Van Wassenhove 2010). It is clear that the presence of product recovery would have a direct impact on the profitability of firms. What is less clear, however, is how product recovery—be it a result of environmental legislation, economic opportunity, or both—impacts the firm’s choices with respect to the design of new products. This paper aims to address this question by studying how firm’s product design, in particular quality choice, is impacted by product recovery, where quality is interpreted as an observable performance measure, which increases market valuation of the product, as in Moorthy (1988) and Desai (2001). Product recovery occurs in three main forms in our framework. It depends on the firm’s ability to recover quality-inducing components or quality-inducing materials from used products, and overall recovery profitability. • Under quality recovery, the firm is able to recover certain quality inducing components that can be used, after some reprocessing, in the production of new products. Reprocessing can be in the form of remanufacturing, which preserves the geometry of the component, or recycling, which does not preserve the geometry of the component, and essentially means material recovery. Note that one can have both remanufacturing and recycling coexisting under quality recovery. For example, quality inducing components can be remanufactured, whereas obsolete components or modules can be recycled for material recovery. One example of this type of recovery is Xerox copiers, which have a modular design and are typically

2

leased to customers. Products are designed such that a new generation of copiers may differ from an older generation in certain modules, but many modules are identical to the older generation’s. Xerox collects end-of-lease equipment, disassembles and remanufactures modules that can be used in the newer generation of copiers, and blends in remanufactured modules with new modules (specifically those modules with major technological upgrades), into machines that are called “newly manufactured”. Quality of a module directly increases its variable cost when produced new, due to the use of higher quality materials and more expensive, higher performance technology. If that module can be remanufactured and used in a newly manufactured machine, Xerox can save significant amounts on production costs. Thus, product recovery in the form of remanufacturing directly affects Xerox’ product design and quality choices. Another example is that of Kodak single-use cameras, where some components of recovered used cameras are fed back into the production of new cameras, so that a newly manufactured camera has, on average, a high percentage of remanufactured components. Other examples abound, including many cases where remanufactured products are differentiated from new products (Hauser and Lund 2003, Souza 2008). • Profitable material recovery represents scenarios where the firm cannot reuse recovered components or materials in production, but can recycle them at a net profit. In a number of practical examples, revenues from recycled materials can outweigh recovery costs. As an example, consider the recycling of cell phones. Compared to other products (e.g., industrial copiers), it can be relatively cheaper to collect used cell phones, considering their size and weight—it is not uncommon, for example, for firms to use pre-paid mailers to collect used cell phones. In addition to relatively low reverse logistics costs, cell phones also contain a significant amount of precious metals, so that recycling of cell phones is, in many instances, a profitable operation (Geyer and Doctori Blass 2009). Thus, once it has collected used phones, a cell phone manufacturer can generate a positive net revenue from recycling the recovered devices. Similar examples can be observed for large appliances, where component reuse is not common but recycling of steel and copper can generate net profits. • Under costly recovery, reusing components or materials is not economical, and recovery overall is unprofitable because of excessive recovery and processing costs. The main reason for a firm to engage in costly recovery is legislative take-back requirements, as discussed above. Consider the examples of Hewlett-Packard, Sony, Braun, and Electrolux, who recycle 3

small appliances in the European Union due to the WEEE Directive. According to Shao and Lee (2009), these firms paid a net recycling cost of about 7.5 cents per kilogram for the recycling of small appliances in the first quarter of 2007. In all of the three recovery cases, there is an important link between product quality and recovery choices: the sales volume. Intuitively, a higher quality product has a higher production cost, and as a result is sold at a higher price. A higher price can be afforded by fewer customers, which decreases the sales volume and implies a lower number of products that can be recovered. Thus, if recovery is profitable (costly), the firm may have an incentive to increase (decrease) sales by changing the quality of the new product to increase (decrease) recovery benefits (costs). At the same time, the recovery of a higher quality product can also mean higher cost savings, which can result in lower prices and higher sales volumes. Hence, there is an inherent dependence between product recovery and a firm’s quality choice. Based on this observation, we study the economic trade-offs firms face when choosing product quality in the presence of product recovery. The starting point for our analysis is the model of Moorthy (1988), who studies the optimal quality choice without recovery, which we extend to account for different forms of product recovery as discussed above. We consider the quality of the product as a decision variable, in addition to price, and investigate how these decisions are affected under each recovery scenario. We focus on a monopolistic setting and consider two basic environments in terms of achievable and required recovery rates: (i) The recovery rate is beyond the control of the firm, e.g., recovery is constrained by exogenous factors or mandated by legislation, and (ii) The recovery rate is a decision variable for the firm. In general, we find that the impact of product recovery on quality choice depends on the form of product recovery, recovery cost structure and legislative environment. In particular, we show that with profitable material recovery (e.g., with cell phone recycling), a firm may design products of lower quality, whereas quality choice is higher if there is quality recovery (e.g., with Xerox copiers) or costly recovery (e.g., WEEE recycling). We further demonstrate that both the firm and the consumers benefit from profitable recovery, while both are worse off with costly recovery. However, environmental implications of the three forms of product recovery differ from their impact on consumer surplus and firm profit. While quality recovery and costly recovery reduce consumption and increase environmental benefits, the same is not true with profitable material recovery, which can increase consumption and result in higher environmental impact.

4

The paper is organized as follows. We first provide an overview of the related literature in §2. We then formulate our monopoly model in §3 and analyze scenarios where the recovery rate is exogenously or endogenously determined, in §4. We identify the optimal quality and pricing choices analytically, and compare with our benchmark scenario without product recovery. We then investigate welfare implications of product recovery in §5. We conclude with a summary of our findings, their practical implications, and discuss limitations of our research in §6.

2

Related Literature

Recent operations management literature on closed-loop supply chains has highlighted the importance of product take-back and economic advantages associated with remanufacturing or recycling end-of-use products (Guide and Van Wassenhove 2009, Atasu et al. 2008b). A number of papers in this literature demonstrated that there are many reasons for product recovery with remanufacturing: reaching new market segments (Atasu et al. 2008a, Debo et al. 2005), deterring the entry of third-party remanufacturers (Ferguson and Toktay 2006), expanding the product line (Pierce 2009), brand protection (providing a “certified” remanufactured product), value recovery, and environmental concerns (Hauser and Lund 2003). We contribute to this stream of literature by bringing a product quality perspective and exploring how different forms of product recovery, e.g., remanufacturing or recycling, can impact product design through quality choice. To our knowledge, there is only one paper in this stream of literature that explicitly incorporates product design as an endogenous decision variable in the context of product recovery. Debo et al. (2005) considers a firm that sells remanufactured and new products in a differentiated remanufacturing case. The firm can impact the profitability of remanufacturing by choosing a production technology 0 ≤ q ≤ 1, which impacts marginal cost and product remanufacturability (a product cannot be remanufactured if q = 0), such that marginal cost of producing a new product is convex increasing in q, whereas marginal cost of remanufacturing a product is decreasing in q. In their model, q can be viewed as a production technology choice that does not impact a consumer’s willingness to pay for the product; in our model we have instead quality choice s, which directly impacts a consumer’s willingness-to-pay for the product, in addition to manufacturing and recovery costs. There is also a growing segment of closed-loop supply chain literature that focuses on the economic impact of environmental legislation for stimulating greener technologies (e.g., Ovchinnikov

5

et al. 2010), or more related to this paper, take-back legislation, which stimulates recovery of end-of-use products. Research in the latter stream investigates the design of welfare maximizing take-back legislation from a policy maker’s perspective (Atasu et al. 2009a, 2009b), or its impact on recycling markets (Toyasaki et al. 2011), while there is limited research on how firms react to the existence of product take back legislation, from a product design standpoint, which is a theme on this paper. In this context, Atasu and Subramanian (2009) investigate how recyclability choices of firms depend on the structural form of take-back legislation, and Plambeck and Wang (2009) investigate how incremental innovations and the new product introduction frequency of firms are affected by such legislation. Our paper differs from these in that we capture the design decision using quality as a decision variable, which impacts how consumers value the product and we investigate how quality choices of firms can be impacted by different forms of product recovery, with or without product take-back legislation. Finally, we contribute to the marketing literature by following the traditional quality choice models such as Mussa and Rosen (1978), Shaked and Sutton (1982), Moorthy (1984, 1988), and Desai (2001), however, we explicitly consider product recovery. We focus on quality as an observable performance measure, which impacts a consumer’s willingness to pay for the product while increasing variable production cost, as in Moorthy (1988) and Desai (2001). Our model follows Moorthy (1988), where we focus on the choice of quality in a monopolistic environment. To the best of our knowledge, this is the first paper to analyze the impact of product recovery on quality choice under voluntary and mandated product take-back environments.

3

Model

We consider a monopolist who faces demand for a durable product in a single period. The single period here can be viewed as the maturity stage of the product’s life cycle, so that prices and recovery rates are stable1 . Our modeling of consumer preferences follows that of Moorthy (1988). There is a heterogeneous customer base, whose valuations θ for quality are uniformly distributed over a support, which we normalize to be [0, 1]. Quality level is based on observable features of the product, such as performance, and higher quality is valued more by consumers. For example, a single use camera with water resistance, better lens quality or a flash would have higher customer 1

The single-period model considered in this paper can be derived as a steady state characterization of a multi-

period infinite horizon dynamic programming model, which we omit for brevity (see Agrawal et al. 2010).

6

willingness to pay. Similarly, customers would also be willing to pay more for a copier/printer with color printing capabilities, higher printing speed or larger trays. Given the quality of the product s and its price p, a consumer with valuation θ gets a net utility u = θs − p from buying the product. Thus, the firm can increase a consumer’s willingness-to-pay by providing a higher quality product. Normalizing market size to one, demand comprises those consumers with a positive net utility, and given the uniform distribution for consumer types θ, it can be characterized as q = 1 − p/s. We assume that a fraction τ (which can be exogenous or a decision variable) of products sold in the market in a period are recovered. In our steady state single period context, that means that τ q products sold in the previous period (not explicitly modeled) are recovered at the beginning of the current period. With quality recovery, we further assume that products with all new components and products with recovered components cannot be distinguished by consumers (similar to the cases of Xerox and Kodak as described earlier). This assumption simplifies the exposition of our model and results. In Appendix B, we relax this assumption and demonstrate that our results do not change when consumers differentiate between products with all new components (new) and products with recovered components (remanufactured). Thus, this assumption is without loss of generality for the purposes of this model. The total cost of meeting demand q with a product of quality s, if there is product recovery at a rate τ is equal to: (

) (c − rτ )s2 q + n(τ q) + | {z } production cost

α(τ q)2 | {z }

(1)

collection cost

where c = the unit variable cost for a product with unit quality, or quality cost, r = the per unit cost saving from recovery and reuse of components, and n = the per unit processing cost plus (minus) the cost (benefit) of recycling components that are not quality inducing (or cannot be reused). If there is no recovery (τ = 0), then the total cost of meeting demand is (cs2 )q, which is identical to Moorthy (1988) . With recovery (τ > 0), the firm recovers τ q products, and the first and second components of the total cost represent the variable cost of production, which captures all three forms of recovery in our model: • Under quality recovery, when a component that affects the quality of the product is recovered (remanufactured or recycled), and used in manufacturing a product, the cost to produce that component is saved. Put it differently, r > 0 can be interpreted as the fraction of quality inducing components that can be recovered (0 < r ≤ c). When r → c, all quality 7

inducing components in recovered products can be reused to satisfy demand. The parameter n is a unit cost incurred per recovered product and is independent of the quality choice. Thus, reuse and recycling can co-exist in our quality recovery model. • Under profitable material recovery and costly recovery, there is no quality recovery, meaning that r = 0. If the cost of processing exceeds the value of recycled materials, then n > 0 and the costly recovery scenario takes place. If, on the other hand, the cost of processing is below the value of recycled materials, then n < 0 and the profitable material recovery scenario takes place. The last term α(τ q)2 , α ≥ 0, represents the total collection cost. We borrow this collection cost form from the closed-loop supply chain literature (Ferguson and Toktay 2006, Ovchinnikov 2011), which suggests that achieving larger recovery volumes requires additional effort, due to the fact that collecting from increasingly distant locations becomes more and more expensive. We also provide empirical support for this functional form from two data sets in two different industries (Appendix C).2 Thus, the firm’s problem with product recovery is: ( ) max ΠR = q p − (c − rτ )s2 − nτ − α(τ q)2 p,s,τ ( ( ( ) p ))2 p) ( p − (c − rτ )s2 − nτ − α τ 1 − = 1− , s s s.t.

(2)

p, q ≥ 0, 0 ≤ τ ≤ 1.

4

Analysis

In this section, we analyze our model to generate insights on how product recovery affects quality choice. We first consider a benchmark scenario (§4.1) without product recovery. Next, we consider a scenario with an exogenously determined recovery rate τ (§4.2) and discuss how different forms of product recovery affect quality choice. After that, we consider a scenario (§4.3) where the 2

There can be economies of scale in collection and recovery, that is, the recovery cost is concave in recovery

volume as opposed to convex (as in our model). This could be the case where the are significant scale investments for recovery capacity, and collection is confined mostly to a small geographical area. If recovery cost is concave in volume, then profits (and recovery rates) will be higher than our model predicts, but our structural results and insights do not change.

8

firm can choose the recovery rate τ . A comparison between these two scenarios allows us to understand how the presence of product take-back legislation affects quality choice, since an exogenous recovery rate could represent a case where it is dictated by government regulation.

4.1

Benchmark: No-recovery (τ = 0)

When there is no product recovery, our model (2) reduces to ( ) p) ( max Π = 1 − p − cs2 p,s s

(3)

Sequentially solving for p and s, assuming the firm makes the quality (design) decision first, ( ) 1 and then the pricing decision, we obtain: p∗N R = 12 cs∗ 2 + s∗ and s∗N R = 3c , which results in p∗N R =

4.2

2 9c

and Π∗N R =

1 27c

(see Moorthy 1988).

Quality Choice with Exogenous Recovery Rates

In this section, we assume that the recovery rate τ is exogenously determined. This scenario is particularly relevant when the firm faces take-back legislation, which mandates a certain level of product recovery. Consider costly recovery (e.g., for recycling of small electronic appliances as discussed earlier), where firms have no economic incentive to recover products; the incentive to recover is created by legislation. As an example, the WEEE Directive of the European Commission requires electronics firms to collect and recycle products that they previously put on the market, where minimum recovery rates range between 50 and 80 percent depending on the product type. Thus, τ in our model can be interpreted as the minimum recovery rate. If recovery is profitable, however, the firm may decide to recover more than the mandated minimum; in that case τ is a decision variable and one should consider the model in Section 4.3. Note also that while our discussion in this section builds on an exogenous recovery rate determined by legislation, other externalities may restrict achievable recovery rates as well. In the context of quality recovery or profitable material recovery, a firm may not be able to increase the recovery rate beyond a certain maximum, even though it would be profitable to do so. Consider the case of Kodak single-use cameras for instance. The fraction of demand for single use cameras that Kodak can satisfy with remanufactured cameras is at most 60% (Geyer et al 2007). This is because (i) product life cycles are short while used products have a certain positive collection leadtime; (ii) there are leakages in the collection of used products; and (iii) single use cameras are not perfectly durable, i.e., some used cameras may be damaged beyond recovery (see Geyer 9

et al. 2007 for further discussion). Similarly, in the context of cell phone recycling, recovery rates can be bounded from above. For instance, Sprint (2010) has reported a recovery rate of 42% for their cell phone recycling program in 2009 in the United States. Our experiences with cell phone manufacturers suggest that accessing used cell phones can be even more difficult for them, with typical product recovery rates being less than 5% (Castren 2011). One possible reason for the difference between the recovery rates of cell phone manufacturers and service providers is the higher control that service providers exert on their customers (e.g., through service plans). Therefore, in this section we consider environments with a feasible exogenous recovery target τ , be it a result of take-back legislation or constrained recovery, and investigate how quality choice is affected by product recovery in such environments. The solution to our analytical model is obtained by sequentially solving for p and s, assuming the firm makes the design decision first and then the pricing decision. The optimal solution to (2) is given below, which represents all three forms of product recovery we consider. All proofs are provided in Appendix A. For the . . remainder of this paper, we denote cm = cm (τ ) = c − rτ < c as the effective cost of quality, when there is product recovery. Proposition 1 For a given recovery rate τ , s∗ (s∗ +cm s∗2 +τ (n+2ατ )) a. the optimal price is p∗ (s∗ ) = , where the optimal quality choice is given 2(s∗ +ατ 2 ) √ 2+ 2 )) 1−4αc τ 1+4c τ (3n+4ατ (1+c ατ m m m by s∗ = . 6cm

b. s∗ increases in the cost of recovery n, decreases in the cost of quality c, and increases in the quality recovery savings r. Proposition 1 explains how the recovery cost structure affects quality choice. First, it confirms our intuition that a lower effective quality cost (cm = c − rτ ) implies higher quality choice. This suggests that companies such as Xerox and Kodak, whose modules are designed for reuse, should design superior quality products under product recovery. Furthermore, the higher the savings from recovery of quality components or modules, the higher the quality choice. Second, it explains the effect of the recovery cost n on quality choice. While one could intuitively argue that a higher cost implies lower quality, this is not the case for processing related costs such as n that are independent of product quality. The quality choice increases in n. The intuition behind this result is as follows. When n is higher, the average cost to satisfy a unit of demand is higher, thus the firm would like to charge a higher price for the product and decrease the total demand. 10

To be able to charge a higher price, the firm needs to increase product quality. These results have two implications. First, if a firm conducting quality recovery (e.g., remanufacturing) faces high collection and processing costs, then it should design its new products with superior quality. Second, the higher the cost of recycling under take-back legislation, the higher the quality choice. An important technical observation relates to the impact of recovery cost n under the profitable material recovery case, with n < 0. When the recovery benefit −n increases, quality choice decreases, approaching zero as −n approaches n ¯ = (16c2 α2 τ 3 + 16cατ + 1/τ )/(12c). At this limiting case, the firm sells an extremely low quality product at zero price, reaching the entire market q = 1, so that it can collect the products later and recover them, where it obtains all of its profits. Practically, this means that the value of recovered material in a product is worth more than the product itself, which is unrealistic. Hence, we restrict our attention to the case where −n < n ¯. Proposition 2 For a given recovery rate τ , a. the optimal quality level and price with quality recovery (r > 0, n ≥ 0) or costly recovery (r = 0, n ≥ 0) are higher than in the case with no-recovery, i.e., s∗ > s∗N R and p∗ > p∗N R . b. The optimal quality level and price with profitable material recovery (r = 0, n < 0) are higher (s∗ > s∗N R , p∗ > p∗N R ) than in the case without recovery if τ > τ¯ = (if τ ≤ τ¯ =

−3n 2α ),

−3n 2α .

Otherwise

they are weakly lower (s∗ ≤ s∗N R , p∗ ≤ p∗N R ).

Proposition 2 indicates that product quality and price are always higher under quality recovery or costly recovery than under no-recovery. More importantly, this result holds for any range of cost parameters r, c and n as long as it is profitable for the firm to serve the market. This has an important implication for firms: two recovery environments that appear completely different (i.e., quality recovery, such as remanufacturing, and costly recycling) result in the same effect on their quality choice. Both with quality recovery and costly recovery, firms should design products with superior quality, allowing them to charge higher prices. With profitable material recovery, the opposite happens when the recovery rate τ is low. The firm chooses a lower quality level (and price) with profitable material recovery than under norecovery. The intuition behind this result is as follows. With a low τ , the effect of the collection cost term (α(τ q)2 ) is small and the firm can obtain a net benefit from recovery. In this case, the exact opposite of the costly recovery scenario takes place. With a higher recovery benefit 11

−n, the average cost to satisfy a unit of demand is lower; thus the firm can charge a lower price for the product. As such, the firm chooses to decrease both quality level and price to maximize profits. With a high recovery rate τ on the other hand, (i.e., τ >

−3n 2α ),

the firm still chooses

a higher quality (and price) than in the no-recovery case. This is because, although product recovery (excluding collection cost) is profitable, the effect of the collection cost term (α(τ q)2 ) is significant and optimal decisions of the firm resemble the costly recovery scenario. Proposition 2 also indicates that take-back legislation affects welfare in two directions. It not only reduces the environmental impact of production by avoiding landfilling of end-of-use products, but can also affect product design, i.e., quality choices of firms and the market segments their products are targeted for. When take-back legislation mandates costly product recovery (n > 0), firms will find incentives to design better quality products, which will result in higher priced products targeted at higher-end customers who value product quality more than others. A similar result takes place when profitable material recovery is possible (n < 0), but the mandated recovery rates are high. On the other hand, when the mandated take-back rates are not as high under profitable material recovery, firms will be better off designing lower quality products targeting a broader customer base. This implies that firms can benefit from carefully evaluating the impact of take-back legislation. For instance, their product design choices can account for possible future changes in recovery targets imposed by take-back legislation. Consider cell phone firms in Europe for instance, for whom recycling is a profitable option as discussed above. With the current legislative WEEE target in Europe that requires EU member countries to ensure collection and recovery of 4 kg of electronics waste per capita, cell phone manufacturers do not face a significant challenge in their recycling operations. However, with recent considerations for the revision of the WEEE directive that are likely to mandate higher recovery rates (EC 2008), these firms may be facing an important challenge. Given that most used cell phones end up in developing countries beyond the reach of the original manufacturers, achieving high recovery rates may result in a significant increase in collection costs. In this case, the impact of product recovery on these firms’ product quality decisions may be reversed.

4.3

Quality Choice with Endogenous Recovery Rate Choice

In the previous section, we considered the case where the recovery rate τ was exogenous, such as is the case, for example, with take-back legislation. If there is no take-back requirement (e.g., as 12

in nearly 30 American states to-date), then the firm can choose a recovery rate τ that maximizes its profits. We study this scenario in this section. The firm now solves the problem stated in (2), with τ as a decision variable. As before, the solution to this problem is found sequentially: the firm makes the strategic quality decision s first, and then sets the price p and recovery rate τ to maximize profit. Mathematically, we find p and τ that solves (2), assuming s is given. We then substitute the resulting optimal p∗ (s) and τ ∗ (s) into (2), and find the optimal s. The solution is given in Proposition 3 below. Proposition 3 Consider the scenario where the recovery rate τ is a decision variable and the solutions listed in Table 1. a. With quality recovery (r > 0, n ≥ 0), the optimal solution is (τ ∗ , p∗ , s∗ ) = (τi , pi , si ), where i = arg maxj ΠRj . b. With costly recovery (r = 0, n > 0), the optimal solution is (τ ∗ , p∗ , s∗ ) = (0, p3 , s3 ), i.e., no-recovery. c. With profitable material recovery (r = 0, n < 0) the optimal solution is   (1, p1 , s1 ), (τ ∗ , p∗ , s∗ ) =  (−3n/2α, p , s ), 3 3

if − n > 2α/3 otherwise.

Solution j

1

2

3

τj

1

(rs2int − n)/ (α(1 − csint ))

0

pj

s1 (n+s1 +(c−r)s21 +2α) 2(s1 +α)

sint (1 + csint )/2

2/(9c)

sint

1/(3c)

sj

1−4α(c−r)+

ΠRj

√

1+4(c−r)(3n+4α(1+(c−r)α)) 6(c−r)

(rs2int −n)2 +αsint (1−csint )2

(s1 −cm s21 −n)2 4(s1 +α)

4α

1/(27c)

Table 1: Manufacturer’s optimal decisions with endogenous τ . sint is defined in Appendix A for the sake of brevity. We note that Proposition 3 assumes an unrestricted τ ∗ . When there is take-back legislation mandating a minimum recovery rate (say, τ ) and/or there is a maximum feasible recovery rate that can be achieved by the firm (say, τ¯), and it is optimal for the firm to choose a recovery rate τ ∗ ∈ [τ , τ¯], Proposition 3 continues to apply. If, however, τ ∗ < τ or τ ∗ > τ¯, then the analysis of the previous section (§4.2) applies with τ = τ or τ = τ¯, respectively. 13

Proposition 3 indicates that with quality recovery, there are three possible outcomes: norecovery (τ ∗ = 0), full recovery (τ ∗ = 1), and partial recovery (0 < τ ∗ < 1), depending on the profitability of product recovery. With costly recovery, as one would expect, τ ∗ = 0. With profitable material recovery, there is either full recovery, or partial recovery, depending on the profitability of product recovery −n relative to the collection cost α. In Proposition 4 we investigate how the quality choice is affected in these potential outcomes. Proposition 4 Consider the scenario where the recovery rate τ is a decision variable. a. With quality recovery (r > 0, n ≥ 0), the optimal quality choice and price are weakly higher than the no-recovery scenario, i.e., s∗ ≥ s∗N R , and p∗ ≥ p∗N R . b. With costly recovery (r = 0, n > 0) the optimal quality choice and price do not change, i.e., s∗ = s∗N R and p∗ = p∗N R . c. With profitable material recovery (n < 0), the optimal quality choice and price are lower than the no-recovery scenario, i.e., s∗ < s∗N R and p∗ < p∗N R if −n > 2α/3. Otherwise (if −n ≤ 2α/3), they do not change (s∗ = s∗N R , p∗ = p∗N R ). A comparison between Propositions 2 and 4 demonstrates the impact of an endogenous recovery rate choice. With quality recovery and costly recovery (n > 0), the firm designs a product with no lower quality and price, compared to the no-recovery scenario, when the recovery rate decision is endogenous. The only difference from Proposition 2 (with exogenous τ ) is that under some circumstances, the optimal recovery rate is zero, and in that case the optimal quality choice and price coincides with those under no-recovery (s∗N R ). When there is profitable material recovery (n < 0), the result in Proposition 4 parallels that of Proposition 2, with the exception that when τ is endogenous, the firm never chooses higher product quality or price. These results allow us to compare the firm’s quality and pricing choices under endogenous and exogenous (i.e, mandated) recovery rates. It is clear that firms may find no incentive for product recovery in the absence of take-back legislation, even though product recovery can provide additional environmental and social benefits. In such cases, take-back legislation may be needed to induce firms to undertake such activities and reduce environmental impact by landfill diversion. Our analysis however, demonstrates that take-back legislation does more than landfill diversion. The presence of take-back legislation (i.e., an exogenous τ ) can change quality and price choices of firms significantly and impact welfare as we illustrate in the next section. 14

5

Welfare Implications

A comparison of our results in §4.2 and §4.3 suggests that take-back legislation can induce better product quality choice. Whether this is a desirable outcome, however, is an important question. This is because product recovery not only influences quality choice, but also the prices in the market, and consequently consumption. This has direct welfare implications through the impact on the firm’s profit, the consumer surplus, or the environmental impact of production and consumption. As such, we now investigate how mandated product recovery (with exogenous τ ) influences these welfare components. We focus on α = 0 for ease of exposition, however our results in this section can be extended for α > 0.

5.1

Firm Profit and Consumer Surplus

We first investigate the impact of product recovery on firm profits and consumer surplus. The firm’s profit follows our definition in §3, and we adopt the traditional definition of consumer ∫1 2 surplus (CS), that is CSR = θ=p/s (θs − p) dθ = 12 s − p + p2s . The impact of recovery on profit and consumer surplus is formalized in Proposition 5 below. Proposition 5 Assume α = 0. Then, CSR = Π∗R /2. In addition, there exists n ˆ (r, c, τ ) ≥ 0 such that if n ≤ n ˆ (r, c, τ ), then Π∗R ≥ Π∗N R and CSR ≥ CSN R for any recovery scenario and recovery rate τ . With r = 0 (i.e., costly or profitable material recovery), this condition reduces to n ≤ 0. Proposition 5 indicates that the impacts of costly and profitable material recovery on firm profit are quite intuitive. The firm’s profit is lower with costly recovery, and higher with profitable material recovery. Under quality recovery, however, the outcome is not as straightforward. Relative to no-recovery, the firm’s profit is higher with quality recovery as long as recovery costs are sufficiently low. This happens when the quality cost savings (i.e., rτ ) outweigh the recovery cost n, i..e, when n ≤ n ˆ (r, c, τ ). Otherwise, firm profit may be lower than the no-recovery case. A closer look at the behavior of consumer surplus reveals the consumption effect of product recovery. Consumer surplus increases (decreases) with profitable material (costly) recovery because the price decrease (increase) under profitable material (costly) recovery enlarges (shrinks) the segment of the market that can afford a lower (higher) quality product. Under quality recovery, although the price is always higher, the increase in product value to consumers can be achieved at a lower cost because of lower effective cost of quality (c − rτ ). Thus, the price increase 15

can be relatively smaller than the increase in product value from a consumer’s perspective. This, however, can only be achieved when the impact of n is not too high (n ≤ n ˆ (r, c, τ )) such that it results in a limited price increase. Otherwise, the price increase can outweigh the increase in product value to a consumer.

5.2

Environmental Impact

An investigation of the environmental impact requires a life cycle perspective. Environmental impact could be measured in numerous ways, such as energy consumption, non-renewable raw material consumption, carbon emissions (which is highly correlated to energy consumption), impact on water, toxicity, and so on. We use a single, aggregate measure of environmental impact per unit at each stage of the life cycle, for simplicity, as in Atasu et al. (2009a) and others. Consistent with Life Cycle Assessment (LCA), we denote the environmental impact per unit of product during production (of a new product), use by consumers, recovery process, and end-of-life (for products not recovered, e.g., landfilling), by ep , eu , er , and eeol , respectively. Environmental impact thus depends on quantities processed at each stage of the life cycle, as well as their respective environmental impacts per unit. • If there is no recovery, then all products sold are landfilled after use by consumers. As a ∗ , where result, the life cycle environmental impact of the firm for the period is IN R = EqN R ∗ E , ep + eu + eeol . Given that qN R = 1/3, then IN R = E/3. ∗ are made • In the quality recovery scenario, a fraction 1 − τ of products sold in the period qR

from new components only, which means that a fraction 1−τ of products are not recovered at the end of life in steady state. It follows that with quality recovery a fraction τ of products sold in the period are made from recovered components. The life cycle environmental ∗ (1 − τ ) + e q ∗ τ + e q ∗ + e q ∗ (1 − τ ) = q ∗ (E − eτ ), where impact of the firm is IR = ep qR r R u R eol R R

e , ep + eeol − er > 0, because eeol > er , as recovery most likely reduces environmental impact compared to landfilling. • With profitable material recovery and costly recovery, a fraction τ of products sold are ∗ (E − e′ τ ), where 0 < e′ , e recovered and not landfilled, resulting in IR = qR eol − er < e.

Proposition 6 The life cycle environmental impact (and consumption) under quality recovery and costly recovery is always lower than when there is no recovery. The life cycle environmental 16

impact with profitable material recovery is lower than under no-recovery if τ > τˆ(ep , eu , er , eeol ); otherwise it is higher. Proposition 6 provides a comparison of the environmental impact between the no-recovery scenario and the three forms of product recovery considered. It indicates that while quality recovery and costly recovery reduce the life cycle environmental impact, profitable material recovery can increase it, depending on the recovery rate τ and the magnitudes of ep , eu , er , and eeol .

5.3

Synopsis

The two sections above demonstrate that the form of product recovery is critically important to determine its economic and environmental impact. Table 2 summarizes our insights. Recovery Scenario

Effect on Profits and Consumer Surplus

Environmental Benefits

Quality Recovery

+ if n ≤ n ˆ (r, c, τ )

+

Profitable Material Recovery

+

+ if τ > τˆ

Costly Recovery

-

+

Table 2: The effect of product recovery on different welfare components

• The impact of higher quality choice under costly recovery is reduced consumption, which results in improved environmental benefits (i.e., lower life cycle environmental impact), and reduced firm profit and consumer surplus. Hence, there is an inherent trade-off between the economic and environmental impacts of costly product recovery, which needs to be balanced by the take-back legislation. • The impact of profitable material recovery on quality choice depends on the mandated recovery rate. When the recovery rate is low, a lower quality choice associated with lower price results in higher consumption and increases firm profit and consumer surplus. The increased consumption, however, results in higher landfilling and lower environmental benefits. Nevertheless, this negative impact can be overcome by mandating higher recovery rates, which effectively reduces the landfill volume despite a potential consumption increase. • Quality recovery, of which remanufacturing is a typical example, appears to be a desirable scenario. In this case, firm profit and consumer surplus are increased when the recovery 17

benefit is sufficiently higher than the recovery cost (i.e., n ≤ n ˆ (r, c, τ )). At the same time, the environmental benefits are guaranteed to improve. This is because quality recovery results in higher quality choice, which results in reduced consumption of new products. Coupled with previous studies (e.g., Quarigasi Frota Neto et al. 2007) which have indicated the superiority of remanufacturing over recycling from an energy consumption standpoint, this suggests that remanufacturing appears to be a preferred form of product recovery from both environmental and economic perspectives.

6

Discussion and Conclusions

This paper presents a stylized model, in order to capture the important trade-offs regarding how product recovery (driven by economic incentives or legislation) affects a firm’s product quality choice, where quality is interpreted as an observable performance measure that increases consumers’ valuation for the product. Our results demonstrate that product recovery can ameliorate or deteriorate product quality choice depending on the form of product recovery, while product take-back legislation can induce better quality choice by firms. The critical trade-off in our paper is driven by structural differences in cost or benefits of product recovery. In particular, we argue that if product recovery provides a means to benefit more from quality investments (e.g., by reusing quality inducing components to satisfy demand), then product quality improves. However, if benefits from product recovery do not recover quality investments (e.g., with profitable material recovery), then product quality may decrease. Remarkably, we find that product recovery results in improved product quality when it is costly to do so (e.g., under product take-back legislation).

Quality Recovery Profitable Material Recovery

Regulated Take-Back

Voluntary Take-Back

sR > s N R

sR ≥ sN R

τ > τ¯ ⇒ sR > sN R

sR ≤ sN R

τ ≤ τ¯ ⇒ sR ≤ sN R Costly Recovery

sR > s N R

sR = sN R

Table 3: The effect of product recovery on quality choice s

Table 3 summarizes the results from the analysis of our model and highlights the impacts

18

of different forms of recovery and product take-back legislation on product quality choice. In addition to the economic incentives described above, we find that product take-back legislation can result in improved product quality choices under any form of product recovery we discussed. The key message here is that take-back legislation does have quality choice implications and can induce better product designs by mandating higher product recovery targets. At the same time, although product quality improvement is typically not a policy objective, we show that potential quality choice improvements to be caused by take-back legislation have welfare implications. While profitable recovery modes (e.g., profitable material recovery) can improve profits and consumer surplus, they can also affect consumption and waste generation, ultimately affecting the environmental hazard. This means that the product design implications of take-back legislation has to be taken into account in policy making. A desirable scenario from all perspectives (firms, consumers, and the environment) is quality recovery, of which remanufacturing is the main example. This means that product take-back mandates can significantly benefit from considering remanufacturing as a preferred form of product recovery. Unfortunately however, most takeback legislation faced by the industry to-date, including the WEEE directive of the European Commission and e-waste laws in twenty three U.S. states (ETC 2010), focus on recycling as the preferred form of product recovery. In addition, mandating higher recovery rate requirements can be welfare improving when profitable material recovery is a feasible product recovery option. This can result in a win-win outcome both from environmental and economic perspectives. In closing, our paper, like all research, is not without its limitations. First, we focused on a monopolistic model in our analysis. An extension of our model to a duopoly setting is analytically intractable (even the duopoly model without recovery is analytically intractable, as demonstrated by Moorthy (1988)). However, we analyzed the duopoly model numerically and showed that most of our monopoly results carry over to a duopoly setting. That is, both firms choose a higher quality level when recovery is in the form of quality recovery or costly recovery, however, with profitable material recovery both firms choose a lower quality level. Second, we assumed that consumer valuations for quality are uniformly distributed, which is in keeping with the marketing literature, and necessary for analytical tractability. This assumption is equivalent to assuming linear demand curves, which has been shown to approximate reality for a narrow band of prices. Our demand model is also completely deterministic, which is appropriate for strategic-level models such as ours, however, demand uncertainty is a reality and could impact recovery. For example, lower than expected sales of new products in one period would constrain the amount of recovery 19

possible in the following period. Related to this, we assume a single period model, which can be thought of as a slice of an infinite horizon model when the market is stable. In reality, firms constantly introduce new products, and diffusion rates may differ for different products, which again, impacts the availability of used products for recovery in subsequent periods. Finally, our uni-dimensional measure of quality is performance based, and does not capture product durability; to that end a multi-period problem would be necessary (but analytically very challenging, as our single period model indicates.) Acknowledgements: We thank the editor and two reviewers for their helpful comments. We also thank professors Beril Toktay, Stelios Kavadias, Luk Van Wassenhove and Miklos Sarvary for valuable suggestions on earlier versions of this paper.

REFERENCES Agrawal V., A. Atasu, and K. Van Ittersum. 2010. The Effect of Remanufacturing on the Perceived Value of New Products. Working Paper, Georgia Institute of Technology, Atlanta, GA.

Atasu A., M. Sarvary, and L. N. Van Wassenhove. 2008a. Remanufacturing as a Marketing Strategy. Management Science 54(10), 1731-1746. Atasu, A., V. D. R. Guide Jr., and L. N. Van Wassenhove. 2008b. Product Reuse Economics in Closed-Loop Supply Chain Research. Production and Operations Management 17(5), 483-496. Atasu, A., M. Sarvary, and L. N. Van Wassenhove. 2009a. Efficient Take-Back Legislation. Production and Operations Management 18(3), 243-258. Atasu, A., O. Ozdemir, and L. N. Van Wassenhove. 2009b. Stakehoder Perspectives under TakeBack Legislation: Takes or Rates. Working Paper, Georgia Institute of Technology, Atlanta, GA.

Atasu, A., and R. Subramanian. 2009. Competition under Take-Back Laws: Individual or Collective Systems. Working Paper, Georgia Institute of Technology, Atlanta, GA. Atasu, A., and L. N. Van Wassenhove. 2010. Environmental Legislation on Product Take-Back and Recovery. M. Ferguson and G. Souza, eds. Closed Loop Supply Chains: New Developments

20

to Improve the Sustainability of Business Practices, CRC Press, Boca Ratton, 23-38. Castren, H. 2011. Personal communication with Helena Castren, Nokia. Brussels, Belgium. January 2011. Debo, L.G., Toktay, L. B., and L. N. Van Wassenhove. 2005. Market Segmentation and Production Technology Selection for Remanufacturable Products. Management Science 51(8), 11931205. Desai, P. 2001. Quality Segmentation in Spatial Markets: When Does Cannibalization Affect Product Line Design? Marketing Science 20(3), 265-283. Esty, D. C., and A. S. Winston. 2006. Green to Gold. Yale University Press, New Haven, CT. EC, European Commission. 2008. Proposal for a Directive of the European Parliament and of the Council on Waste Electrical and Electronic Equipment (WEEE). 2008/041 (COD). Brussels, Belgium. ETC, Electronics TakeBack Coalition. http://www.electronicstakeback.com/legislation. Accessed June 10, 2010. Ferguson, M. and B. Toktay. 2006. The Effect of Competition on Recovery Strategies. Production and Operations Management 15(3), 351-368. Ferrer, G. and J. Swaminathan. 2006. Managing New and Remanufactured Products. Management Science 52(1), 15-26. Geyer, R., L. N. Van Wassenhove and A. Atasu. 2007. The Economics of Remanufacturing under Limited Component Durability and Finite Life Cycles. Management Science 53(1), 88-100. Geyer, R., V. Doctori Blass. 2009. The Economics of Cell Phone Reuse and Recycling. International Journal of Advanced Manufacturing Technology. Online publication, accessed January 2010. Guide, Jr. V.D.R., and K. Li. 2010. The Potential for Cannibalization of New Product Sales by Remanufactured Products. Decision Sciences 41(3), 547-572. Guide, Jr. V.D.R. and L.N. Van Wassenhove. 2009. The Evolution of Closed-Loop Supply Chain 21

Research. Operations Research 27(1), 10-18. Hauser, W. M., and R. T. Lund. 2003. The Remanufacturing Industry: Anatomy of a Giant. Boston, MA, Boston University. Kandra A. 2002. Refurbished PCs: Sweet Deals of Lemons? PC World Magazine, April 2002. Lund, R. T., F.D. Skeels. 1983. Start-up Guidelines for the Independent Refirm. Center for Policy Alternatives, Massachusetts Institute of Technology, Boston, MA. Moorthy, K.S. 1984. Market Segmentation, Self Selection and Product Line Design. Marketing Science 3(4), 288. Moorthy, K.S. 1988. Product and Price Competition in a Duopoly. Marketing Science 7(2), 141-165. Mussa, M. and S. Rosen. 1978. Monopoly and Product Quality. Journal of Economic Theory 18(2), 301-317. Ovchinnikov, A. 2011. Revenue and Cost Management for Remanufactured Products. Production and Operations Management, DOI 10.1111/J.1937-5956.2010.01214.X. Ovchinnikov, A., D. Krass, and T. Nedorezov. 2010. Environmental Taxes and the Choice of Green Technology. Working Paper, Darden School, University of Virginia, Charlottesville, VA. Pierce, Allen (2009), Senior Manager for Remanufacturing, Cummins. Personal Communication with the Authors. Plambeck, E.L. and Q. Wang. 2009. Effects of E-Waste Regulation on New Product Introduction. Management Science 55(3), 333-347. Quarigasi Frota Neto, J., G. Walther, J. Bloemhof-Ruwaard, Nunen, J. A. E. E., and T. Spengler. 2007. From Closed-Loop to Sustainable Supply Chains: The WEEE Case. Working Paper, Erasmus University, Rotterdam, The Netherlands. Available at http://hdl.handle.net/1765/10176.

Shaked, A. and J. Sutton. 1982. Relaxing Price Competition Through Product Differentiation. Review of Economic Studies XLIX, 3-13. 22

Shao, M., and H. Lee. 2009. The European Recycling Platform: Promoting Competition in E-Waste Recycling. Stanford Graduate School of Business case # GS-67. Souza, G. C. 2008. Closed-Loop Supply Chains with Remanufacturing. Z. L. Chen and R. Raghavan, eds. Tutorials in Operations Research, INFORMS, Hanover, MD, 130-153. Sprint. Sprint Partners with the U.S. EPA to Urge Americans to Recycle Their Unwanted Cell Phones. http://newsreleases.sprint.com. Retrieved August 5, 2010. Tojo, N. 2004. Extended Producer Responsibility as a Driver for Design Change – Utopia or Reality? IIIEE Dissertations 2004:2, Lund University. Toyasaki, F., T. Boyaci and V. Verter. 2011. An Analysis of Monopolistic and Competitive TakeBack Schemes for WEE Recycling, Production and Operations Management, DOI: 10.1111/J.19375956.2010.01207.X.

Appendix A: Proofs Proof of Proposition 1 In the second decision stage, the firm finds the optimal price given a previous choice of quality s in the s(s+cm s2 +τ (n+2ατ )) first stage. From (2), the first-order condition ∂Π/∂p = 0 results in p∗ (s) = . This price 2(s+ατ 2 ) 2

) maximizes profit (given s), because ∂ 2 Π/∂p2 = − 2(s+ατ , which is negative for all positive values of s, s2

and therefore Π(p) is concave in p for any given s. The quantity sold is then q(s) = 1 − p/s = To ensure q(s) ≥ 0, we need s0 ≤ s ≤ s1 , where s0 =

√ 1− 1−4cm nτ , 2cm

roots of q(s) = 0. Substituting p∗ (s) into (2), we obtain Π(s) =

and s1 =

(s−cm s −nτ ) 4(s+ατ 2 ) 2

2

s−cm s2 −nτ 2(s+ατ 2 ) .

√ 1+ 1−4cm nτ 2cm

are the

≥ 0. In addition, we

can rewrite Π(s) as Π(s) = (s + ατ 2 )q 2 (s) ≥ 0, and thus the roots of Π(s) are the same as the roots of q(s), s0 and s1 . The solution to the first-order condition ∂Π(s)/∂s = 0 yields four possibilities: s0 , s1 , √ √ 1−4αcm τ 2 + 1+4cm τ (3n+4ατ (1+cm ατ 2 )) 1−4αcm τ 2 − 1+4cm τ (3n+4ατ (1+cm ατ 2 )) s2 = , and s3 = . We now need 6cm 6cm to differentiate between the case where n ≥ 0 (quality recovery or costly recovery) and the case where n < 0 (profitable material recovery). Case 1: n ≥ 0. Notice that p∗ (s) ≥ 0 for all s. In addition, both s0 > 0 and s1 > 0. The function Π(s) is neither concave or convex, however it is unimodal in the interval of interest s ∈ (s0 , s1 ) that ensures non-negative quantities, as follows. First, notice that s3 < 0 as the term inside the square root √ 1−4αcm τ 2 + 1+12cm nτ +16cm ατ 2 +16c2m α2 τ 4 in the numerator is greater than one, and s2 > 0 because s2 = 6cm √ 1−4αcm τ 2 + (1+4cm ατ 2 )2 1 > = 3cm . Thus, the function Π(s), which is always non-negative has two roots at 6cm

23

s0 and s1 , which are also extreme points, and one positive extreme point s2 is between s0 and s1 , which means that the function is unimodal, and that s2 is the unique maximizer of Π(s). Case 2: n < 0 (here, necessarily r = 0, and thus cm = c). The condition p∗ (s) ≥ 0 results in two possibilities: • If −n ≤ 2ατ − •

1 4cτ ,

then p∗ (s) > 0 always.

1 If −n > 2ατ − 4cτ , then p∗ (s) ≥ 0 in the intervals s ≥ s5 √ −1+ 1+4cτ (−n−2ατ ) and s5 = . Notice that s6 < 0, and 2c

and s ≤ s6 , where s6 = −

1+

√

1+4cτ (−n−2ατ ) , 2c

thus p∗ (s) ≥ 0 for s ≥ s5 .

Regarding non-negative quantities, first, notice that s0 < 0. As a result, to satisfy the conditions of nonnegative prices and quantities, we require max{s5 , 0} ≤ s ≤ s1 . Regarding the solutions to the first-order conditions, notice that s2 and s3 are non-negative real numbers only if −n
Π(s5 ). Thus, s5 cannot be a solution. The solution s = 0 is optimal only if s2 and s3 are non-real numbers, i.e., for −n >

16c2 η 2 τ 3 +16cητ +1/τ , 12c

but that is ruled out by assumption. As a result, 3/2 [1+4cτ (3n+4ατ (1+cατ 2 ))] we only have two candidates, s2 and s3 . Now, Π(s2 ) − Π(s3 ) = > 0, and thus 27c the optimal solution is s∗ = s2 . The fact that s∗ increases with n comes directly from observing that n only appears inside the square root in the numerator of s∗ , with a positive coefficient. Now, ∂s∗ /∂c = − 6c12 − m

0. Further, ∂s∗ /∂r = ∗

Finally, ∂s /∂α =

2τ 2 3

1 6c2m

[

+

−1 +

6c2m

√

2

τ +2τ cm (3n+4ατ )

1+4cm τ (3n+4ατ (1+cm ατ 2 ))

2+4cm ατ 2 √ 1+4cm τ (3n+4ατ (1+cm ατ 2 ))

3 ≤ 0, then this last term is greater than or equal to

]

2τ 2 3

≥ 0. 2τ 2 3

[

6c2m

√

1+2τ cm (3n+4ατ ) 1+4cm τ (3n+4ατ (1+cm ατ 2 ))

2+4cm ατ 2 √ (2+4cm ατ 2 )2 +12ncm τ −3

]

= −1 + . If 12ncm τ − [ ] 2 −1 + √ 2+4cm ατ 2 2 = 0; otherwise it is ≤ 0.  (2+4cm ατ )

Proof of Proposition 2 Again, we need to differentiate between n ≥ 0, and n < 0. √ 1−4αcm τ 2 + 1+12cm nτ +16cm ατ 2 +16c2m α2 τ 4 Case 1: n ≥ 0. s∗ = s2 = > 6cm Since s∗ >

1 3cm ,

then s∗ >

1 3c

2 9c

=

N 9cm ,

where N =

√

1−4αcm τ 2 + (1+4cm ατ 2 )2 6cm

=

1 3cm .

= s∗N R , because cm = c − rτ ≤ c. Now, consider the optimal price.

We need to show that p∗ (s∗ ) > p∗N R , or that p∗ (s∗ ) − p∗ (s∗ ) −

2rτ c

2 9c

> 0. We can rewrite, after some algebra, √ − 1 + b + y(1 + y) + (1 − y) 1 + 2b + 4y + y 2 , with y = 4cm ατ 2 ,

and b = 6cm nτ . Thus, we only need to show that N > 0. We consider two cases: (i) y < 1, and (ii) √ 2 y ≥ 1. Consider case (i), y < 1. Because b > 0, then N > 2rτ c − 1 + b + y(1 + y) + (1 − y) 1 + 2y + y = 2rτ c

− 1 + b + y(1 + y) + (1 − y)(1 + y) =

2rτ c

+ b + y > 0. Now, consider case (ii), y ≥ 1. First note

24

≤

that, because s1 ≥ 0, then, 1 − 4cm nτ ≥ 0, which means that 12cm nτ ≤ 3, or 2b ≤ 3. As a result, √ √ 2rτ 2 2 N = 2rτ c − 1 + b + y(1 + y) − (y − 1) 1 + 2b + 4y + y ≥ c − 1 + b + y(1 + y) − (y − 1) 1 + 3 + 4y + y =

2rτ c

− 1 + b + y(1 + y) − (y − 1)(y + 2) =

+ 1 + b > 0. This concludes the proof that p∗ (s∗ ) > p∗N R

2rτ c

for n ≥ 0. 1 Case 2: n < 0 (here, necessarily r = 0, and thus cm = c). If we solve s∗ = 3c for α, we find α = [ ] [ ] 2 2 2 −3n 2τ 2+4cατ 2τ 2+4cατ 2 ∗ √ √ −1 + = 3 −1 + 2τ . Now, we note that ∂s /∂α = 3 1+4cτ (3n+4ατ (1+cατ 2 )) (2+4cατ 2 )2 +12ncτ −3 [ ] 2 2 > 2τ3 −1 + √ 2+4cατ 2 2 = 0. Thus, ∂s∗ /∂α > 0, and we know that s∗ = sN R for α = −3n 2τ . Thus, ∗

s ≤ sN R for

(2+4cατ ) α ≤ −3n 2τ , and

(equivalently −n ≥

2ατ 3 ),

s∗ > sN R for α >

−3n 2τ .

and s∗ > sN R for τ >

−3n 2α

This can also be rewritten as s∗ ≤ sN R for τ ≤ (equivalently, −n
0 (N ≤ 0) √ 2 2 implies p∗ (s2 ) > 9c (p∗ (s2 ) ≤ 9c ). Rewrite N = −(1 − y)(1 + y) + (1 − y) 1 + 2y + y 2 + 2(b + y) + (b + y), ( ) √ or N (x) = (y − 1) (1 + y) − (1 + y)2 + 2x + x, with x = b + y. Notice that N (0) = 0. Further,

x = b + y = 0 implies τ =

−3n 2α ,

x > 0 (x < 0) implies τ >

−3n 2α

(τ
0. Taking the derivative, we find N (x) = 1 − √ (y−1) . It suffices to show that (y+1)2 +2x √ (y − 1) < (y + 1)2 + 2x, For y ≤ 1, this holds trivially. Otherwise for y > 1, we need to show that (√ )2 (y − 1)2 < (y + 1)2 + 2x , or, after simplification, 2x + 4y ≥ 0. For x > 0, this holds trivially since y > 1. For x ≤ 0, we first show that x ≥ −2 + y. If x ≤ 0, then τ ≤ s ≤ sN R =

1 3c .

−3n 2α ,

and from the above proof,

The maximum willingness to pay by a consumer is s, thus the maximum price the

firm can charge that results in a non-negative sales quantity is also s; however, s ≤ p∗ (s2 ) ≤

1 3c .

1 3c .

As a result

In any realistic scenario, the maximum revenue from recycling one unit of product sold,

given by −nτ , cannot exceed the unit price of a new product, otherwise the firm would have no economic incentives to sell new products, and recovery would not be sustainable. Thus, −nτ ≤

1 3c ,

or, equivalently,

1 1 nτ ≥ − 3c . As a result b = 6cnτ ≥ 6c(− 3c ) = −2. Thus, x = b + y ≥ −2 + y for x ≤ 0. As a result,

2x + 4y ≥ 2(y − 2) + 4y = 6y − 4 > 0, where the first inequality comes from x ≥ y − 2, and the second inequality comes from y > 1. 

Proof of Proposition 3 Given the sequence of decisions, we first find, for a given quality s, the optimal p(s) and τ (s). We then substitute these optimal prices and recovery rates into the objective function and optimize over s. Given s, we first start analyzing the general cost structure for all three product recovery scenarios. The first order conditions ∂Π/∂τ = 0 and ∂Π/∂p = 0 result in two solutions: the first solution is p1 (s) = 21 s(cs + 1) and τ1 (s) =

rs2 −n α(1−cs) ,

whereas the second solution is p2 (s) = s, and τ2 (s) =

the determinant of its Hessian equal to − (n−rs s2

2 2

)

cs2 −s rs2 −n .

The second solution has

≤ 0, and as a result cannot be an interior maximizer.

25

In addition, substituting these solutions into the profit function, we find that the second solution results in a profit of zero. The first solution results in a profit of Π(s) =

(rs2 −n)2 +αs(1−cs)2 , 4α

which is positive for

positive values of s. We now check the second order conditions for this solution. We find that ∂ 2 Π/∂p2 = 2(rs2 −n)2 αs2 (1−cs)2

2

< 0, ∂ 2 Π/∂τ 2 = − α(1−cs) < 0, and ∂ 2 Π/∂p∂τ = rs s−n . The determinant of the Hessian s ( )2 2 is ∂ 2 Π/∂p2 ∗ ∂ 2 Π/∂τ 2 − ∂ 2 Π/∂p∂τ = α(1−cs) > 0, and thus the Hessian H is negative definite for this s − 2s −

2

solution ((∂ 2 Π/∂p2 < 0, ∂ 2 Π/∂τ 2 < 0, and det(H)> 0). Thus, for a given s, the solution p1 (s) = 12 s(cs + 1) and τ1 (s) =

rs2 −n α(1−cs)

is the only interior local maximizer. This solution however is not necessarily a global

maximizer. It is a global maximum if its objective function, when optimized over s, is higher than at any of the bounds τ = 0 and τ = 1. We now determine the optimal quality choice s for the three scenarios. Case 1: Quality Recovery (r > 0, n ≥ 0). In this case, the optimal solutions for the boundary conditions at τ = 0 and τ = 1 can be obtained as special cases of Proposition 1. For the interior solution at p1 (s) and τ1 (s), the optimization over s has to be performed only in the range of s that results in 0 < τ1 (s) < 1, otherwise the resulting s is not well defined. We first remind that, because the maximum willingness-to-pay for a product is s, the maximum price that can be charged is also s, and as a result the maximum variable cost for a product cs2 is also s, which means that s < 1c , and as a result 1 − cs > 0. √ Thus, the condition τ1 (s) > 0 results in s > s, where s = nr . The condition τ1 (s) < 1 results in s < s¯, √ −αc+ α2 c2 +4r(n+α) rs2 −n where s¯ = . Thus, the interior solution p1 (s) = 21 s(cs + 1) and τ1 (s) = α(1−cs) can only 2r be optimal it the optimization of its profit function Π(s) =

(rs2 −n)2 +αs(1−cs)2 4α

with respect to s results in

s < s < s¯. The first order condition ∂Π/∂s = 0 results in a third-degree polynomial equation in s, that is,

rs(rs2 −n) α

−

cs(4−3cs) 4

+

1 4

= 0. Although the roots of this equation can be given in closed-form, it is

unfortunately impossible to determine algebraically if it falls in the interval s ∈ (s, s¯). Moreover, each one of these roots can also be a minimum or maximum, since again it is not possible to determine the sign of the second derivative at each of the roots. As a result, the solution to the problem can be one of the three 2

∂ Π possible candidates: τ = 0, τ = 1, or at an interior solution sint , where {sint |s ∈ [s, s¯], ∂Π ∂s = 0, ∂s2 ≤ 0},

whichever provides a higher profit. A closed-form comparison between these potential maximizers however, is tedious and not insightful. Nevertheless, this characterization is quite powerful as it allows us to develop further insights regarding the quality choice, as illustrated in the proof of Proposition 4. Case 2: Costly recovery (r = 0, n > 0). In this case, a local interior maximizer does not exist, since τ1 (s) < 0 when r = 0 and n > 0. Hence, the only possible candidates for the global maximizers are at the τ = 0 and τ = 1 boundaries. The solution τ = 1 results in the firm solving (2), with r = 0, for p and s, whereas the solution τ = 0 results in the firm solving (3), or maxs ΠN R = q(p − cs2 ). We can rewrite (2) ( ) ( ) as ΠR = q p − cs2 − nq + αq 2 , that is, (3) minus a term that is guaranteed to be non-negative; thus (2) can never be higher than (3). Thus, the solution τ = 1 can also not be an optimal solution if r = 0, and n > 0. Thus, τ ∗ = 0. Case 3: Profitable material recovery (r = 0, n < 0). In this case, determining the optimal s

26

for an interior solution at p1 (s) and τ1 (s) is feasible and simplified because r = 0. Specifically, Π(s) = n2 +αs(1−cs)2 . 4α

Applying the first order condition ∂Π/∂s = 0, we obtain two possible solutions s1 =

s2 = 1/c. The second derivative is ∂ 2 Π/∂s2 = c(−2 + 3cs)/2, which is negative for s < for s >

2 3c .

2 3c ,

1 3c

and

and positive

That means, the profit function is concave for s < 2/3c, and convex otherwise. As a result, the

solution s2 = 1/c is an interior minimum and cannot be a maximizer of the problem. The solution s1 = is the unique interior local maximum. Substituting s∗ = −3n 2α ,

1 3c

1 3c

back into p∗ (s) and τ ∗ (s), we obtain 2/9c and

respectively. The resulting optimal profit for this interior maximizer is

n2 4α

+

1 27c .

It is straightforward to see that this objective value is always higher than that at the τ = 0 boundary (which is

1 27c ).

Hence, the τ = 0 boundary cannot be a global maximizer.

Now, notice that for this interior solution τ ∗ =

−3n 2α ,

which increases in −n and reaches one when

−n = 2α/3. Hence, for −n > 2α/3, the interior solution is no longer feasible. This means that when −n > 2α/3, the only solution is at the τ = 1 boundary, which happens to be the global maximizer. When −n ≤ 2α/3, however, we need to compare the interior solution with the one at the τ ∗ = 1 boundary solution (with corresponding solution in Proposition 1). It is straightforward to show that the interior solution with s1 =

1 3c

(and an objective value of

n2 4α

+

1 27c )

provides a higher profit than at the at the

∗

τ = 1 boundary solution (with corresponding solution in Proposition 1) when −n ≤ 2α/3. Therefore, the optimal solution in the (r = 0, n < 0) case can be summarized as follows: When −n ≤ 2α/3, s∗ = τ∗ =

−3n 2α .

1 3c ,

Otherwise, the optimal solution is characterized by Proposition 1 with τ = 1. 

Proof of Proposition 4 Case 1: Quality Recovery (r > 0, n ≥ 0). In this case, we have to evaluate the prices and quality choices for all potential maximizers, since we cannot determine the optimal quality choice in closed form. The solutions at the (τ ∗ = 1) and (τ ∗ = 0) boundaries are special cases of Proposition 2, and thus the proofs follow directly. The interior solution, however, needs further investigation. Note that an interior local maximizer sint satisfies

∂Π(s) ∂s (sint )

= 0. Applying standard comparative statics techniques, we can write this equation

as F (α, sint ) = 0, with F (·) =

∂Π(s) ∂s (·).

2

Taking the full derivative

∂F ∂α

+

∂F ∂sint

·

∂sint ∂α

2

= 0, or

∂sint ∂α

=

2

Π(s) Π(s) Π(s) = −( ∂∂s∂α )/( ∂ ∂s (sint )). But if sint is a local maximum, then ∂ ∂s (sint ) < 0, 2 2 2 2 rs n−rs ( ) ∂ Π(s) int and thus, the sign of ∂s∂α . From Proposition 3, this is negative is equal to the sign of ∂s∂α = α2 ∂F −( ∂F ∂α )/( ∂sint ), or

since τ1 (s) =

∂sint ∂α

rs2 −n α(1−cs)

> 0 and s < 1/c for any interior maximizer. Thus, as α increases, sint decreases. At

the same time, using Proposition 3 it is easy to see that limα→∞ Π(s) = 41 s(1 − cs)2 , which is maximized at s∗ =

1 3c

(equivalent to the solution at the τ = 0 boundary). Hence, any local interior maximizer sint

for a given α < ∞ is greater than

1 3c

, since sint decreases in α. The proof for the optimal price at the

interior solution immediately follows since the optimal interior price p(sint ) is given as which strictly increases in s. 

27

1 2 sint (csint

+ 1),

Case 2: Costly recovery (r = 0, n > 0). The proof for this case immediately follows from Propositions 2 and 3, because the optimal solution takes place at the τ = 0 boundary. Case 3: Profitable material recovery (r = 0, n < 0). The proof for this case immediately follows from Propositions 2 and 3, because the optimal solution can be at the τ = 1 boundary or at the interior with s∗ =

1 3c

and p∗ (s) = 2/9c . 

Proof of Proposition 5 Substituting the values of p∗ and s∗ from Proposition 1 under α = 0 into the profit function (2), the optimal ( ) √ firm profit under product recovery Π∗R is obtained as Π∗R =

12nτ (rτ −c)+ 12nτ (c−rτ )+1+1 (√ ) 54(c−rτ ) 12nτ (c−rτ )+1+1

2

. Following

the definition surplus (CS) in section 4.4, it is straightforward to show that CSR = ( )2 √ of the consumer 12nτ (rτ −c)+ 12nτ (c−rτ )+1+1 (√ ) . Hence, CSR = Π∗ /2. Note that it is straightforward to show that CSN R = R 108(c−rτ )

12nτ (c−rτ )+1+1

Π∗N R /2, since the no-recovery scenario can be obtained as a special case of the recovery scenario with τ = 0. Thus, consumer surplus proofs directly follow those for the firm profit. √ 3 −3B 2 +4 The firm profit can be rewritten as Π∗R = B54(c−rτ 1 + 12nτ (c − rτ ). Simple algebra ) , where B = yields: Π∗R ≥ Π∗N R =

1 27c

⇔ B 3 − 3B 2 + 2 + 2 rτ c ≥ 0.

Before we evaluate this condition, consider the following. Note that the maximum price that the monopolist can charge to stimulate demand is p = s. At the same time, we need s − (c − r)s2 − nτ ≥ 0 such that the monopolist can sell to make profit. Equivalently, this can be written as nτ ≤ s − (c − rτ )s2 . The function s − (c − rτ )s2 is maximized at s =

1 2(c−rτ ) ,

condition for the monopolist to sell is nτ ≤

1 4(c−rτ ) .

and its maximum value is

1 4(c−rτ ) .

Thus, a necessary

Now, consider the function h(B) = B 3 − 3B 2 + 2. We first determine an upper bound for B using the √ √ 1 condition above. Note that, B = 1 + 12nτ (c − rτ ) ≤ 1 + 12 4(c−rτ ) (c − rτ ) = 2. Thus, 0 ≤ B ≤ 2. The function h(B) = B 3 − 3B 2 + 2 decreases monotonically from h(0) = 2 to h(2) = −2 in the interval ˆ of interest [0,2], and h(1) = 0. Our condition is h(B) + 2 rτ c ≥ 0. As such, there exists a B such that ˆ B 3 −3B 2 +2+2 rτ c ≥ 0 if B ≤ B (since

rτ c

< 1). Equivalently, there exists a n ˆ (r, c, τ ), such that Π∗R ≥ Π∗N R

when n ≤ n ˆ (r, c, τ ) . For the profitable or costly material recovery (e.g., recycling) scenarios (r = 0), the condition above simplifies to h(B) ≥ 0, which implies B ≤ 1, or n ≤ 0. The proof for consumer surplus follows immediately since CSR = Π∗R /2. 

Proof of Proposition 6 ∗ In the quality recovery scenario, a fraction 1 − τ of products sold in the period qR are made from new

components only, which means that a fraction 1 − τ of products are not recovered at the end of life in steady state. It follows that with quality recovery a fraction τ of products sold in the period are made ∗ ∗ from recovered components. The life cycle environmental impact of the firm is IR = ep qR (1 − τ ) + er qR τ+

28

∗ ∗ ∗ (E − eτ ), where e , ep + eeol − er > 0, because eeol > er , as recovery most eu qR + eeol qR (1 − τ ) = qR

likely reduces environmental impact compared to landfilling. With profitable material recovery and costly recovery, 100% of products sold are made with new components, but a fraction τ of products sold are ∗ recovered and not landfilled, resulting in IR = qR (E − e′ τ ), where 0 < e′ , eeol − er < e. Thus, the

firm’s environmental impact will be reduced by any form of product recovery if consumption is reduced ∗ ∗ ∗ ∗ relative to the no-recovery case (i.e., qR < qN R ), for any given recovery rate. If, however, qR > qN R , then

the environmental impact under recovery can still be lower, depending on the recovery rate τ and relative environmental impacts at different stages of the life cycle. ∗ = Following Proposition 1, the equilibrium consumption is qR

1 3

(

2−

) √ 1 + 12nτ (c − rτ ) for any form

∗ ∗ < qN of product recovery. Note that qR R = 1/3 when n > 0. On the other hand, consumption is higher

under profitable material recovery (n < 0) than under no-recovery, which means that the environmental impact can be increased under profitable material recovery. In particular, the life cycle environmental ( √ ) impact is higher than the no-recovery case if τ < τˆ, where τˆ solves 2 − 1 + 12nˆ τ c (E − τˆe′ ) = 1. 

Appendix B: Differentiated Recovered and New Products In this section, we assume that recovery is in the form of quality recovery, however, products with recovered quality components (remanufactured or recycled), and products with all new components are differentiated, as stated in the assumption below. Further, for ease of exposition, we denote products with recovered components as ‘remanufactured”, and products with all new components as “new”. Assumption 1 If a product of quality s is remanufactured and offered at a price pr , then consumer θ’s net utility for this remanufactured product is (s − ϕ)θ − pr . Under this assumption, a consumer of type θ, who has a willingness-to-pay sθ for a new product, has a willingness-to-pay of (s − ϕ)θ for the remanufactured product. Thus, the nature of competition between new and remanufactured units is that of vertical differentiation (consumers prefer a new product to a remanufactured one for the same price). In many cases, consumers perceive remanufactured products to be of “lower quality” than new products, even if they belong to the same technological generation. We here represent the remanufactured product’s quality gap by ϕ. This “quality gap” perspective is reflected in a number of articles in the practitioner and academic literature (Lund and Skeels 1983, Hauser and Lund 2003, Kandra 2002, Debo et al. 2005, Ferrer and Swaminathan 2006, Atasu et al. 2008a), and has empirical validation for several categories of products. An empirical study using online auctions by Guide and Li (2010) indicates that customers’ willingness-to-pay for remanufactured routers is about 15% lower than for new routers, with both new and remanufactured routers from the same technological generation and carrying identical warranties from the OEM. Note that if ϕ = s, consumers are not willing to pay anything for the remanufactured unit; this is not an interesting case. If ϕ = 0, consumers view the new

29

and remanufactured units as being identical and are willing to pay the same amount for either unit; this is the case analyzed previously in Section 4.2. Many remanufactured products fall between the two extremes, and thus we assume in this section that 0 < ϕ < s. Consider a customer with a valuation of θ. If sθ < pn , the customer obtains a negative utility from the new unit, but a (s − ϕ)θ − pr net utility from the remanufactured unit, which is positive for θ > pr /(s − ϕ). On the other hand, if θ > pr /(s − ϕ), both products offer positive net utilities, and for high enough values of θ (specifically, for θ > (pn − pr )/ϕ, found by solving sθ − pn > (s − ϕ)θ − pr ), the net utility for the new unit is higher than that of the remanufactured unit. Summarizing, customers with a valuation higher than (pn − pr )/ϕ will buy a new unit while those with a valuation between pr /(s − ϕ) and (pn − pr )/ϕ will buy a remanufactured unit and those with a valuation less than pr /(s − ϕ) will not buy anything. The quantities of new and remanufactured units sold (qn and qr , respectively) are pn − pr , ϕ pn − pr pr qr (pn , pr ) = − . ϕ s−ϕ

qn (pn , pr ) = 1 −

(4) (5)

Given a fixed collection rate τ (say, mandated by legislation), the firm collects τ qn used products, at a total cost of n(τ qn ) + α(τ qn )2 . However, with differentiated remanufacturing, the firm’s remanufacturing quantity qr is a function of profit-maximizing prices, as seen above, and not necessarily equal to the number of collected products τ qn . As before, remanufacturing costs cr = (c − r)s2 per unit. As a result, the monopolist’s optimization problem is now: max Π

pn ,pr ,s

s.t.

=

(pn − cs2 − τ n)qn + (pr − (c − r)s2 )qr − α(τ qn )2

(6)

qr

≤ τ qn

(7)

qn

≥ 0

(8)

qr

≥ 0.

(9)

We assume that s > cs2 + nτ , which ensures a positive margin for the new product as the maximum willingness to pay for the new product is s, to rule out the uninteresting case qn = 0. Again, we first determine the optimal prices, assuming s is given. We substitute the optimal prices p∗n (s) and p∗r (s) into the profit function, and find the optimal quality level s∗ that maximizes Π(s). Proposition 7 With an exogenous collection rate τ , differentiated remanufacturing, and linear collection and remanufacturing costs, the optimal prices are:   cs∗3 +s∗2 +s∗ τ (n+2ατ )    2(s∗ +ατ 2 )   ∗ 2 ∗ ∗ +cs∗2 +τ (n+ατ )) p∗n (s∗ ) = s ατ (1+(c−r)s )+ϕ(s 2(ατ 2 +ϕ)     ∗3 ∗2 2 ∗ 2   s (1+τ )(c+(c−r)τ )+s [(1+τ ) −τ ϕ(c+(c−r)τ )]+s τ [n(1+τ )+2ατ −2ϕ]−τ ϕ(n+ϕ) 2(s∗ +ατ 2 )+2(s∗ −ϕ)τ (2+τ )

30

if r ≤ r0 if r0 < r < r1 if r ≥ r1 ,

   not offered     p∗r (s∗ ) = 1 (s∗ + (c − r)s∗2 − ϕ) 2     ∗ ∗2 ∗ 2   (s −ϕ)[s (1+τ )(c+(c−r)τ )+s (1+τ ) +τ (n(1+τ )+2ατ −(3+τ )ϕ)] 2(s∗ +ατ 2 )+2(s∗ −ϕ)τ (2+τ )

where the optimal quality level is given by  √ 1−4αcτ 2 + 1+4cτ (3n+4ατ (1+cατ 2 ))     6c   ∗ s = s∗int   √    −b2 + b22 −4b1 b3  2b1

if r ≤ r0 if r0 < r < r1 if r ≥ r1 ,

if r ≤ r0 if r0 < r < r1 if r ≥ r1 ,

with b1 = 3(1 + τ )3 (c + (c − r)τ ), b2 = −1 − τ [3 + τ (3 + τ − 4ατ (c − r) − 4cα) + 4ϕ(2 + τ )(c + (c − r)τ )], and b3 = τ (1 + τ ) [ϕ(3 + τ ) − n − τ (n + 2α)], and where s∗int is the optimal quality level for the unconstrained problem (i.e., when constraints (7)-(9) are not binding), and whose characterization is given in the proof below. Proposition 7 indicates that if the remanufacturing cost savings r are at or below a certain threshold r0 , then the firm does not remanufacture (qr∗ = 0); the solution to this no-remanufacturing case corresponds to the costly recovery scenario of Section 4.2 (Proposition 1), with cm = c. If the remanufacturing cost savings are above the threshold r0 but below another threshold r1 , then the firm remanufactures but not all collected used products (qr∗ < τ qn∗ ). Finally, if remanufacturing cost savings are at or above the threshold r1 , then the firm remanufactures all collected used products (qr∗ = τ qn∗ ). Note that the thresholds r0 and r1 are a function of ϕ, α, n and c. It can be shown that ∂p∗n /∂ϕ > 0, ∂p∗r /∂ϕ < 0, ∂qn∗ /∂ϕ > 0, and ∂qr∗ /∂ϕ < 0. This means that for lower consumers’ valuation for remanufactured products relative to new products (higher values of ϕ), the firm charges lower prices for remanufactured products, and higher prices for new products. This increases product differentiation in the product line, which is in line with intuition, and it increases demand for new products, while decreasing demand for remanufactured products. The sign of ∂s∗ /∂ϕ, however, is unclear, and thus the impact on the optimal quality choice depends on other parameters. Similarly to the undifferentiated remanufacturing case, the optimal quality level is higher than the no-recovery case, as shown below. Proposition 8 With an exogenous collection rate τ , and differentiated remanufacturing, the optimal quality choice is higher than in the no-recovery scenario, i.e., s∗ > s∗N R = 1/3c. Thus, our finding that the optimal quality choice is higher with remanufacturing than in the no-recovery scenario holds for all remanufacturing scenarios, differentiated and undifferentiated.

31

Proof of Propositions 7 and 8 In the second decision stage, the firm finds the optimal prices pn and pr given a previous choice of quality s in the first stage. We first note that the profit function (6) is jointly concave in pn and pr , because 2

2

2(ατ +ϕ) −2s 2 ∂ 2 Π/∂p2n = − 2(ατϕ2+ϕ) < 0, ∂ 2 Π/∂p2r = ϕ(s−ϕ) − 2ατ , which means that ϕ2 < 0, and ∂ Π/∂pn ∂pr = ϕ2 ( 2 )2 4(ατ 2 +ϕ) ∂2Π ∂2Π ∂ Π = ϕ2 (s−ϕ) > 0, ensuring a negative definite Hessian. Since the constraints (7)-(9) ∂p2 · ∂p2 − ∂pn ∂pr n

2

r

are linear in the decision variables, they together constitute a convex set. As a result, there is a single set of profit maximizing prices, assuming a given s. We construct the Lagrangian for the problem, which is given by L(pn , pr , λ, σ, η) = Π(pn , pr )+λ(τ qn −qr )+σqn +ηqr . The KKT conditions are: ∂L/∂pn = 0, ∂L/∂pr = 0, the complementary slackness conditions λ(τ qn − qr ) = 0, σqn = 0, and ηqr = 0, in addition to the constraints (7)-(9). The system of equations ∂L/∂pn = 0 and ∂L/∂pr = 0 yields the solution p∗n (λ, σ, η) = ( ) ατ 2 (s+(c−r)s2 −η+λ)+ϕ(s+cs2 −σ+τ (n−λ+ατ )) , and p∗r (λ, σ, η) = 21 s + (c − r)s2 − η + λ − ϕ . We have four 2(ατ 2 +ϕ) cases to consider: 1. qr = 0, qn = 0. In this case λ = 0. Substituting p∗n (0, σ, η) and p∗n (0, σ, η) into the expressions for qn and qr , given by (4) and (5), respectively, and solving the system qn = 0 and qr = 0 for σ and η, we obtain σ = −s + cs2 + nτ , which is negative due to our assumption that s > cs2 + nτ , and thus this case can be ruled out, as intended. 2. qr = 0 and qn > 0. In this case, by complementary slackness, λ = 0 and σ = 0. Substituting p∗n (0, 0, η) and p∗r (0, 0, η) into the expression for qr (5), and solving qr = 0, we obtain η = −rs3 −nsτ +sατ 2 (−1+(c−r)s)+ϕ(cs2 +τ (n+ατ )) . s+ατ 2

The condition η ≥ 0 results in r ≤

−sτ (n+(1−cs)ατ )+ϕ(cs2 +τ (n+ατ )) s2 (s+ατ 2 )

r0 . Substituting η into p∗n (0, 0, η) and p∗r (0, 0, η), and plugging these into qn (4), we obtain qn = s−cs2 −nτ 2(s+ατ 2 ) ,

which is non-negative by assumption. Plugging the prices into the profit function (6), we

obtain Π(s) =

(s−cs2 −nτ )2 4(s+ατ 2 ) ,

which is the case of costly recovery in the special case where cm = c,

whose solution is given by Proposition 1 . We know from Proposition 1 that the resulting quality choice is higher than in the no-recovery scenario. 3. qr = τ qn and qn > 0. In this case, by complementary slackness, σ = 0, and η = 0. Substituting p∗n (λ, 0, 0) and p∗r (λ, 0, 0) into the expression for qr (5) and qn (4), and solving qr = τ qn , we obtain sτ (n+τ (n+α−csα))−ϕ[cs2 +τ (n+s+τ (n+α))]+τ ϕ2 +rs2 (s+ατ 2 +τ (s−ϕ)) λ = . The condition λ ≥ 0 results in s+ατ 2 +(s−ϕ)τ (2+τ ) −nτ (1+τ )(s−ϕ)+τ ϕ(ατ −ϕ)+sτ (ϕ−ατ )+cs2 (ατ 2 +ϕ) , r1 . Substituting λ into p∗n (λ, 0, 0) and p∗r (λ, 0, 0), s2 (s+ατ 2 +τ (s−ϕ)) 3 2 2 s (1+τ )(c+(c−r)τ )+s [(1+τ ) −τ ϕ(c+(c−r)τ )] )+2ατ −2ϕ]−τ 2 ϕ(n+ϕ) we get p∗n (s) = + sτ [n(1+τ and p∗r (s) = 2(s+ατ 2 )+2(s−ϕ)τ (2+τ ) 2(s+ατ 2 )+2(s−ϕ)τ (2+τ ) (s−ϕ)[s2 (1+τ )(c+(c−r)τ )+s(1+τ )2 +τ (n(1+τ )+2ατ −(3+τ )ϕ)] , respectively. Substituting these prices into (4), 2(s+ατ 2 )+2(s−ϕ)τ (2+τ )

r≥

we get qn (s) =

−s2 (c+(c−r)τ )+s(1+τ )−τ (n+ϕ) . 2(s+ατ 2 )+2(s−ϕ)τ (2+τ )

The condition qn (s) > 0 implies s0 < s < s1 , where √ (1+τ )− (1+τ )2 −4(c+(c−r)τ )τ (n+ϕ) s0 and s1 are the roots of qn (s), given by s0 = > 0, and s1 = 2(c+τ (c−r)) √ (1+τ )+ (1+τ )2 −4(c+(c−r)τ )τ (n+ϕ) . Substituting the prices and quantities into the objective function, 2(c+τ (c−r))

we obtain

[ 2 ]2 s (c + (c − r)τ ) − s(1 + τ ) + τ (n + ϕ) Π(s) = 4(s + ατ 2 ) + 4(s − ϕ)τ (2 + τ )

32

,

[ ] which is non-negative. We can rewrite Π(s) as Π(s) = qn2 (s) s + ατ 2 + (s − ϕ)τ (2 + τ ) . The function Π(s) is neither concave or convex, however it is uni-modular in the interval of interest s ∈ (s0 , s1 ) that ensures positive quantities, as follows. The roots of Π(s) are the same as the roots of qn (s), s0 and s1 . The solution to the first-order condition ∂Π(s)/∂s = 0 yields four √ √ −b2 + b22 −4b1 b3 −b2 − b22 −4b1 b3 possibilities: s0 , s1 , s2 = , and s = , where b1 = 3(1 + τ )3 (c + 3 2b1 2b1 (c − r)τ ), b2 = −1 − τ [3 + τ (3 + τ − 4ατ (c − r) − 4cα) + 4ϕ(2 + τ )(c + (c − r)τ )], and b3 = τ (1 + τ ) [ϕ(3 + τ ) − n − τ (n + 2α)]. It can be verified numerically that s2 < s1 , and s3 < s0 . Thus, the function Π(s), which is always non-negative has two roots at s0 and s1 , which are also extreme points, and one positive extreme point s2 between s0 and s1 , which means that the function is unimodal in (s0 , s1 ), and that s2 is the unique maximizer of Π(s). It can also be shown, through tedious algebra (omitted here) that s2 >

1 3c

= s∗N R .

4. 0 < qr < τ qn and qn > 0. In this case, by complementary slackness, λ = 0, σ = 0, and η = 0, which is the unconstrained solution. This yields the prices p∗n (0, 0, 0) = and p∗r (0, 0, 0) =

1 2 (s

sατ 2 (1+(c−r)s)+ϕ(s+cs2 +τ (n+ατ )) 2(ατ 2 +ϕ)

+ (c − r)s2 − ϕ). Substituting p∗n (0, 0, 0) and p∗r (0, 0, 0) into the constraint

qr > 0, we obtain r >

−sτ (n+(1−cs)ατ )+ϕ(cs2 +τ (n+ατ )) s2 (s+ατ 2 )

into the constraint qr < τ qn , we obtain r


1 3c .



Appendix C: Convex Recovery Cost We use two data sets to test the recovery cost structure C(τ, q) = nτ q + α(τ q)2 (Ferguson and Toktay ¯ q) = 2006), resulting in a linear (in recovery volume τ q) average cost C(τ . ¯ r ) = n + αqr . recovery volume by qr = τ q, we can rewrite C(q

C(τ,q) τq

= n + α(τ q). Denoting the

Our first data set comes from the North-American branch of a chemical company that produces plastics sheets to be used in the automotive industry. The company takes back the trim residuals of plastic sheets and uses them to produce the same quality product. The data set consists of two values per observation: (i) the annual quantity of collected products qrj from customer j and (ii) the annual cost of collection from customer j, Cj . Figure 1 plots the cumulative amount collected from the firm’s customers and the total collection cost versus the cumulative amount collected. The data is presented in order of increasing average collection cost Cj /qrj , yielding a convex cost structure.

33

4

5

3

x 10

16

x 10

14 12 total cost of collection

cumulative collected quantity

2.5

2

1.5

1

10 8 6 4

0.5 2 0

0

5

10

15

20

0

25

0

company

(a) Cumulative quantity collected. Data point i is ∑ given by (i, ij=1 qrj ), where qrj is the annual quan-

0.5

1 1.5 2 cumulative collected quantity

2.5

3 5

x 10

(b) Total cost of collection versus the cumulative quantity collected. Data point i is given by ∑ ∑ ( ij=1 qrj , ij=1 Cj )

tity collected from firm j

Figure 1: Collection cost profile of a chemical company in 2006. The second data set is from the Central European branch of a leading global consumer electronics company. The company takes back used products from individuals using a pick-up system and brings them to a central warehouse, where remanufacturing takes place. Figure 2 below plots quantities qrj collected from different countries over two months and the total cost of collection as a function of the cumulative amount collected. The data is presented by country, in order of increasing average collection cost. To fit the two data sets to the linear average cost model (resulting in a quadratic cost model), we regress the cumulative weighed average collection cost as a function of the cumulative collected quantity. In other words, the regression model used is: C¯ = n + αqr + ϵ where ϵ represents the random error term. For the first data set, we obtain α = 0.09 and n = 0.09 with an R2 value of 0.972. The analysis of variance for the regression model shows significance at p < 0.0001. For the second data set, we obtain α = 0.054 and n = 3.61 with an R2 value of 0.798. The analysis of variance for the regression model shows significance at p < 0.0001. Since the data in the second set is already aggregated at a country level, there may be an ignored population impact, i.e. different countries have different populations. To take this into account, we ran another regression analysis with respect to the per capita collected quantity. The findings from this regression model also support the quadratic cost structure. The estimated parameters are α = 1.72 and n = 3.24 with an R2 value of 0.985. The analysis of variance for the regression model shows significance at p < 0.0001. A higher order polynomial regression model may give a marginally better fit, but it also

34

5

2.2

6

x 10

2.2 2

2

1.8 total cost of collection

cumulative collected quantity

1.8 1.6 1.4 1.2 1 0.8

1.6 1.4 1.2 1 0.8 0.6

0.6 0.4

x 10

0.4 0

2

4

6 country

8

10

0.2 0.4

12

(a) Cumulative quantity collected in two months. ∑ Data point i is given by (i, ij=1 qrj ), where qrj is

0.6

0.8 1 1.2 1.4 1.6 1.8 cumulative collected quantity from countries

2

2.2 5

x 10

(b) Total cost of collection versus the cumulative quantity collected. Data point i is given by ∑ ∑ ( ij=1 qrj , ij=1 Cj )

quantity collected from country j

Figure 2: Collection cost profile of Central European branch of a consumer electronics company in 2006. complicates the analysis of our analytic model . Given the high statistical significance of the quadratic cost model for both data sets, it appears to be appropriate for modeling increasing marginal recovery cost in recovery volume.

35