trade-in remanufacturing, strategic customer behavior, and

Trade-in Remanufacturing, StrategicCustomer Behavior, and Government

SubsidiesFuqiang Zhang, Renyu Zhang

Trade-in Remanufacturing, Strategic Customer

Behavior, and Government Subsidies

Abstract

This paper studies the impact of remanufacturing and the associated trade-in program

on a firm’s profit, the environment, and the social welfare. The firm sells a product in

two periods, and the used products from the first period can be recycled for remanufac-

turing in the second period. To collect used products, the firm offers a trade-in program

that allows repeat customers to trade in used products for upgraded ones at a discounted

price. Customers are forward-looking and choose the optimal timing to purchase the prod-

uct. We analyze the game between the firm and the customers and report several major

findings. First, trade-in remanufacturing essentially provides early purchase rewards to cus-

tomers, so it helps mitigate strategic customer behavior and may significantly improve the

firm’s profit. Second, contrary to the commonly-held belief, the adoption of remanufac-

turing may have a negative impact on the environment. This is because remanufacturing

leads to higher production quantity (higher quantity means greater environmental impact),

which may outweigh the environmental benefit of remanufacturing. Similarly, some widely

applied government subsidies for promoting remanufactured products may also aggravate

the detrimental impact of production on the environment. Finally, we study how a social

planner (e.g., the government) should design the incentive mechanism to maximize social

welfare. We find that the socially optimal outcome can be achieved by using a simple linear

subsidy and tax scheme.

Key words: remanufacturing, trade-in program, strategic customer behavior, environment,

subsidization

1 Introduction

Remanufacturing is the rebuilding of a product to specifications of the original manufactured

product using a combination of reused, repaired, and new parts (Johnson and McCarthy, 2014).

The initial purpose of remanufacturing was to recover the residual value of used products by

reusing components that are still functioning well (see, e.g., Guide and Van Wassenhove, 2009;

Debo et al., 2005). More recently, with increasing awareness of sustainability, the environmental

advantages of remanufacturing have also been widely recognized. Giutini and Gaudette (2003)

document that remanufacturing annually saves 16 million barrels of crude oil and reduces 28

million tons of carbon dioxide emission worldwide. In addition, remanufacturing may serve as a

1

Trade-in Remanufacturing 2

marketing tool that helps meet different customer expectations and moderate competition from

second-hand markets (see, e.g., Atasu et al., 2008). Due to all these benefits, remanufacturing

has been increasingly adopted in industry as an important strategy. According to the U.S.

International Trade Commission (2012), the economic value of remanufactured products in the

U.S. was more than $43 billion in 2011. Yu (2012) estimates that the total output value of

remanufactured products in China would reach $24 billion by 2015.

Remanufacturing requires a firm to collect used products from consumers. The trade-in

program is a common practice that encourages consumers to return their used products in

exchange for new ones. For example, Apple offers the in-store iPhone trade-in program, which

allows customers to trade in their used iPhones for credits to purchase new ones (Jones, 2013).

Analogously, Amazon allows Kindle owners to trade in their old products for newer versions at

a lower price (Copy, 2011). Xerox, which partly bases its remanufacturing on trade-in returns,

has generated cost savings of several hundred million dollars each year (Ray et al., 2005). The

trade-in program not only helps the firm collect used products from consumers, but also grants

price discounts to repeat customers who return their used products, thus enabling the firm to

price discriminate the new and repeat customers (Van Ackere and Reyniers, 1995). How does

this trade-in program affect consumers’ purchasing behavior? There is an emerging stream of

research on strategic customer behavior in the operations management literature (e.g., Aviv

and Pazgal, 2008; Su and Zhang, 2008; Cachon and Swinney, 2009). While few studies in

the literature take into account the interactions between the trade-in program and strategic

customer behavior, such a program naturally changes the purchasing behavior of a forward-

looking customer, because she can anticipate a possible price discount in the future if making

a purchase now. Therefore, it is important for firms to understand the interactions between

trade-in remanufacturing and strategic customer behavior.

Being aware of the economic, environmental, and social benefits of remanufacturing, many

governments have made legislative efforts to promote remanufacturing as well as the associated

trade-in program. For example, it is required in Europe that no more than 15 percent of a scrap

vehicle can be discarded (the rest has to be recycled and remanufactured) as of 2002, with that

percentage dropping to 5 percent by 2015 (Giutini and Gaudette, 2003). More recently, policy

makers have been providing economic incentives to further encourage trade-in remanufacturing.

In 2009, to stimulate domestic consumption, curb pollution, and promote circular economy,

the Chinese government offered a subsidy program for home appliance trade-ins, under which

customers receive subsidies for trading in five kinds of used appliances for new ones: TVs, refrig-

erators, washing machines, air-conditioners, and PCs (Ma et al., 2013). Despite its prevalence

in practice, government subsidization for trade-in remanufacturing has received little attention

in the literature. How does government subsidization affect the firm, the customers, and the

environment? What is the optimal subsidization policy that can maximize the social welfare?

The answers to these questions have not yet been thoroughly explored.

2


The primary goal of this paper is threefold: (a) to characterize the impact of trade-in re-

manufacturing on a firm under strategic customer behavior, (b) to analyze the environmental

and social impact of remanufacturing under the trade-in program, and (c) to identify the gov-

ernment policy that can induce the socially optimal outcome. For this purpose, we develop a

two-period model in which a profit-maximizing manufacturing firm sells two generations of a

product to an ex-ante uncertain number of customers. Customers are strategic in the sense that

they make their purchasing decisions based not only on current utilities, but also on anticipated

future utilities. In the first period, the firm sells the first-generation product to the customers.

In the second period, the firm sells the second-generation product to new customers (who do

not purchase in the first period); meanwhile the firm offers a trade-in program, through which

repeat customers trade in used products for remanufactured ones at a discounted price. We

also model the government as a policy maker whose subsidy/tax policy may affect the firm’s

pricing and production strategy and the customers’ purchasing decisions. The objective of the

government is to maximize the social welfare, i.e., the sum of firm profit and customer surplus

less environmental impact.

There are three major findings in this paper. First, from the firm’s perspective, we charac-

terize the interactions between strategic customer behavior and the trade-in program. We find

that, with trade-in remanufacturing, the product price for new customers is higher, whereas for

repeat customers it is lower than what the firm would charge without the trade-in program.

As a consequence, the trade-in program increases the customers’ willingness-to-pay and induces

more purchases in the first period. This reveals an interesting insight: trade-in remanufac-

turing is an effective mechanism in dealing with strategic customer behavior, because it offers

early purchase rewards to customers. Through numerical experiments, we further show that

the value of trade-in remanufacturing as a mechanism to mitigate strategic customer behavior

is most significant when the innovation of the second-generation product is low, the product

durability is moderate, and the demand variability is low.

The second finding is about the social implications of trade-in remanufacturing. Remanu-

facturing has been widely lauded for its benefits to environment. However, our analysis shows

that remanufacturing might result in a higher (negative) environmental impact. This counter-

intuitive result is because trade-in remanufacturing induces higher willingness-to-pay among

customers, which leads to higher demand and production quantities. Therefore, although re-

manufacturing is greener than new product manufacturing, such environmental benefit might

be outweighed by the increased production quantities. In addition, the firm could use its

first-period pricing strategy to extract all the cost savings from remanufacturing, so the total

customer surplus remains the same regardless of the use of remanufacturing. That is, the sole

beneficiary of trade-in remanufacturing is the firm; customers may not necessarily prefer the

adoption of remanufacturing, and it may not be better for the environment.

Given the above two findings, we proceed to study how government intervention can help

3


achieve the socially optimal outcome. We find that some widely applied government subsidies

to encourage the adoption of remanufacturing can give rise to worsened environmental impact.

This cautions the policy makers about how to promote remanufacturing through subsidization.

We characterize the socially optimal outcome and demonstrate that, in order to induce this out-

come, it suffices for the government to use a simple linear subsidy/tax scheme for the sales of

all three product versions: (i) the new first-generation product, (ii) the new second-generation

product, and (iii) the remanufactured second-generation product. In particular, merely sub-

sidizing the firm/customers for remanufactured products may not be sufficient to achieve the

socially optimal outcome. If the environmental impact of one specific product version is low

(high), the firm should subsidize for (tax on) this version.

The rest of the paper is organized as follows. In Section 2, we position this paper in the

related literature. The base model is introduced in Section 3, and the equilibrium analysis is

presented in Section 4. In Section 5, we analyze the impact of remanufacturing from the firm’s

perspective. Section 6 characterizes the social impact of remanufacturing and demonstrates

how government intervention on remanufacturing can achieve the social optimum. We explore

some extensions of the base model in Section 7 and demonstrate the robustness of our main

results. This paper concludes with Section 8. All proofs are given in Appendix A.

2 Literature Review

This paper is built upon two streams of research in the literature: (a) remanufacturing and

closed-loop supply chain management, and (b) strategic customer behavior.

There is a rapidly growing stream of literature on remanufacturing and closed-loop supply

management. Comprehensive reviews of this literature are given by Guide and Van Wassenhove

(2009) and Souza (2013). Several papers study the optimal inventory policy with return flows

of used products; see, e.g., Van der Laan et al. (1999); Toktay et al. (2000), and Gong and Chao

(2013). These papers focus on characterizing the cost-minimizing inventory policy in a system

with exogenously given demand rate, price, and remanufacturability. More recently, researchers

start to explicitly model some strategic issues related to remanufacturing, such as used product

acquisition, demand segmentation, product cannibalization, and competition. Savaskan et al.

(2004) study the optimal reverse channel structure for the collection of used products from cus-

tomers. Ferguson and Toktay (2005) analyze the competition between new and remanufactured

products (i.e., the cannibalization effect) and characterize the optimal recovery strategy. When

remanufacturability is an endogenous decision, Debo et al. (2005) solve a joint pricing and pro-

duction technology selection problem of a manufacturer who sells a remanufacturable product to

heterogeneous customers. Under the cannibalization effect of remanufactured products, Ferrer

and Swaminathan (2006) investigate the competition between an original equipment manufac-

turer (OEM) and an independent operator who only sells remanufactured products. Atasu et al.

(2008) show that remanufacturing could serve as a marketing strategy to target the customers

4


in the green segment and, hence, enhance the profitability of the OEM. Oraiopoulos et al. (2012)

characterize the optimal relicensing strategy of an OEM to mitigate the cannibalization effect

in the secondary market. There are papers that address behaviorial issues related to remanufac-

turing such as how the remanufactured products affect the customer valuation of new products

(Agrawal et al., 2015). Government regulations on remanufacturing have also been studied in

the literature; see, e.g., Ma et al. (2013). Cohen et al. (2015) study the impact of demand

uncertainty on government subsidies for green technology adoption. The impact of trade-in

programs has also received some attention in the remanufacturing literature (e.g., Ray et al.,

2005) and the durable goods literature (e.g., Van Ackere and Reyniers, 1995). Our contribution

to this line of research is that we demonstrate that trade-in remanufacturing can effectively

mitigate strategic customer behavior by offering customers early purchase rewards, and identify

how government subsidization/taxation can help achieve the socially optimal outcome.

The impact of strategic customer behavior has received an increasing amount of attention in

the operations management literature. Shen and Su (2007) provide a comprehensive review on

customer behavior models in revenue management and auctions. Bensako and Winston (1990)

shows that rational customers drive a monopolist firm to charge a lower price for any given state

in each period. Su (2007) characterizes the optimal pricing strategy with a heterogenous group of

strategic and myopic customers. When customers are forward-looking, Aviv and Pazgal (2008)

study the optimal single mark-down timing with finite inventories. In a newsvendor model where

customers anticipate the likelihood of stockout before deciding whether to make a purchase,

Dana and Petruzzi (2001); Su and Zhang (2008, 2009) study the impact of strategic customer

behavior on newsvendor profit, supply chain performance, and the role of product availability

in inducing demand, respectively. Liu and Van Ryzin (2008) propose the effective capacity

rationing strategy to induce early purchases with strategic customers. Cachon and Swinney

(2009, 2011); Swinney (2011) demonstrate how quick response can be employed to mitigate

strategic customer behavior. Jerath et al. (2010) study opaque selling and last-minute selling

with strategic customers in a revenue management framework. In a cheap talk framework, Allon

et al. (2011) show that, though nonverifiable, the availability information improves the profits

of a service firm and the expected utility of its customers. Allon and Bassamboo (2011) further

demonstrate that a single retailer providing availability information on its own cannot create

any credibility with homogeneous customers. Chu and Zhang (2011) investigate the integrated

information and pricing strategy with strategic customers and the customer preorders before

product release. Parlakturk (2012) demonstrates how vertical product differentiability helps

mitigate strategic customer behavior. Our paper involves strategic customer behavior and is

therefore related to the above studies. However, we study a different setting with trade-in

remanufacturing and government subsidies. We find that the presence of strategic customers

will increase the value of trade-in remanufacturing, which is not the focus of the aforementioned

customer behavior studies.

5


3 Model

We consider a monopoly firm (he) in the market who sells a product to customers (she) in a two-

period sales horizon. In the first period, the firm produces the first-generation product at a unit

production cost c. The market demand X, which consists of a mass of infinitesimal customers,

is ex-ante unknown, with a distribution function F (·) and density function f(·) = F ′(·). Each

customer requests one unit of the product. The valuation V for the first-generation product

of each customer is independently drawn from a continuous distribution with a distribution

function G(·) supported on [v, v] (0 ≤ v < v). We assume that the product is a brand new

one in the market so that each customer only knows the distribution of her own valuation V ,

but not the realization, at the beginning of the horizon. The valuation distribution G(·) has

an increasing failure rate, i.e., g(v)/G(v) is increasing in v, where g(·) = G′(·) is the density

function and G(·) = 1 − G(·). This is a standard assumption in the literature and can be

satisfied by most commonly used distribution functions. For convenience, we call the customer

with product valuation V the type-V customer. Let µ := E(V ) > c, i.e., in expectation a

customer’s valuation exceeds the production cost.

In the second period, the market uncertainty is resolved so the realized demand X becomes

known to the firm; moreover, each individual customer observes her own valuation through

either personal experience or social learning. Note that in our model setting, the customers are

homogeneous ex ante (before period 1) but heterogeneous ex post (after period 1). The firm may

offer an upgraded version of the product in period 2. This practice is quite common for product

categories like consumer electronics, home appliances, and furniture. Let the production cost

of the second-generation product be (1 + α)c, where α ≥ 0 is exogenously given and captures

the innovation level (e.g., the improved features) of the upgraded product. Accordingly, the

type-V customer has a valuation of (1 + α)V for the second-generation upgraded product. If

a customer with valuation V has already bought the product in period 1, her valuation of

consuming the used product in period 2 is (1 − k)V , where k ∈ [0, 1] refers to the depreciation

factor. Specifically, if k = 0, the product is completely durable; if k = 1, the product is

completely useless after the first period (either the product is worn out or the technology is

obsolete); and, if k ∈ (0, 1), the product is partially durable. Therefore, the willingness-to-pay

of the type-V customer in period 2 is (1 + α)V if she did not get the product in period 1, and

is (1 + α)V − (1 − k)V = (k + α)V if she got the product in period 1.

The firm can collect the used products from first-period buyers and recycle the components

for the second-period remanufacturing. The unit remanufacturing cost is (1 + α)c− (1 − k)c =

(k+α)c, where (1+α)c is the production cost for a new second-generation product and (1−k)c

is the residual value of a used first-generation product. Hence, higher durability (i.e., a lower

k) corresponds to greater residual value. The used products are collected through a trade-

in program, i.e., customers who bought the product in period 1 can return it to the firm in

exchange for a remanufactured second-generation one at a discounted price in period 2. The

6


collection cost (e.g., logistics and administration) is normalized to zero; a non-zero collection

cost can be included in the unit remanufacturing cost without affecting the analysis. For the

ease of exposition, we make two assumptions in the base model: First, all customers treat new

and remanufactured products equally in the second period. That is, we focus on situations

where either the customers are unable to distinguish between these products, or, they only care

about the quality of a product but not its source. Second, the repeat customers do not have the

option to return the product without purchasing a new one in period 2 (see, also, Ray et al.,

2005). We will relax these assumptions in Section 7 and show that the main results remain

unchanged.

In our model, the firm seeks to maximize his total expected profit whereas each customer

aims to maximize her total expected surplus over the two-period horizon. For simplicity, we

assume there is a common discount factor for the firm and customers in the second period,

denoted by δ ∈ (0, 1]. The sequence of events unfolds as follows. At the beginning of period 1,

the firm announces the price p1 and decides the production quantityQ1. Each customer observes

p1, but not Q1, and strategically makes her decision whether to order a product immediately

or to wait until the second period. The first-period demand X1 ≤ X is then realized, the firm

collects his first-period revenue, and all the customers stay in the market. Note that X1 is

determined by the collective effect of all customers’ purchasing behaviors. If X1 ≤ Q1, any

customer who requests a product can get one in the first period. Otherwise, X1 > Q1, then

the Q1 products are randomly allocated to the demand and X1 − Q1 customers have to wait

due to the limited availability. We assume that at the end of period 1, the firm salvages all

the leftover inventory with a unit salvage value s < c. This assumption is for tractability and

appropriate in situations where inventory holding cost is sufficiently high or the firm does not

want to dilute the sales of the newer version of the product. At the beginning of the second

period, the firm learns the realized total market demand X, and each individual customer learns

her type V . The firm then announces the price pn2 for new customers as well as the trade-in

price pr2 ≤ pn

2 ; all customers decide whether to purchase or trade in for the second-generation

product. Finally, the firm produces the new and remanufactured second-generation products

and collects the second-period revenue.

For notational convenience, we will use E[·] to denote the expectation operation, x ∧ y to

denote the minimum of two numbers x and y, and ϵ1d= ϵ2 to denote that two random variables

ϵ1 and ϵ2 follow the same distribution.

4 Equilibrium Analysis

This section presents the equilibrium analysis of the game between the firm and the customers.

We adopt the rational expectation (RE) equilibrium concept to characterize the game outcome.

The RE equilibrium concept was proposed by Muth (1961) and has been widely used in the

operations management literature (e.g., Su and Zhang, 2008, 2009; Cachon and Swinney, 2009,

7


2011). Using backward induction, we start with the decisions of the two parties in period

2. Since the customers already know their individual valuations at the beginning of period

2, a type-V new (repeat) customer would purchase the product if and only if (1 + α)V ≥ pn2

((k+α)V ≥ pr2). Thus, from the firm’s perspective, the probability that a new (repeat) customer

would purchase the second-generation product in period 2 is G(pn2

1+α)(G(

pr2

k+α)). Note that there

are Xr2 = X1 ∧ Q1 repeat customers and Xn

2 = X − (X1 ∧ Q1) new customers in the market.

Therefore, in period 2, the firm’s objective is to maximize his expected profit

Π2(pn2 , p

r2|Xn

2 , Xr2) := Xn

2 (pn2 − (1 + α)c)G

(pn2

1 + α

)+Xr

2(pr2 − (k + α)c)G

(pr2

k + α

). (1)

We use (pn2 (Xn

2 , Xr2), pr

2(Xn2 , X

r2)) := argmax{(pn

2 ,pr2),pr

2≤pn2 }Π2(p

n2 , p

r2|Xn

2 , Xr2) to denote the opti-

mal pricing strategy of the firm in period 2. Moreover, let π2(Xn2 , X

r2) := max{Π2(p

n2 , p

r2|Xn

2 , Xr2) :

0 ≤ pr2 ≤ pn

2} denote the corresponding optimal profit.

We analyze the decision-making behaviors of the firm and the customers in period 1 sep-

arately. We begin with the customers’ purchasing behavior. Since the market demand X is

uncertain and the production quantity Q1 is unobservable, a customer expects to obtain the

product with probability ξa if she purchases now. Analogously, a customer perceives the second

period price for new customers as a random variable pn2 and the trade-in price as a random

variable pr2. Recall that in period 2 the market is known to the firm, and all customers can get

their requested products in that period. So a new customer in period 2 can get a discounted

expected utility δE[(1+α)V −pn2 ]+, whereas a repeat customer in period 2 can get a discounted

expected utility δE[(k+α)V −pr2]

+. Therefore, given the announced price p1 and the customers’

expectations (ξa, pn2 , p

r2), the expected utility of purchasing the product in period 1 is given by:

Uvisit := ξa(µ − p1 + δE[(k + α)V − pr2]

+) + (1 − ξa)δE[(1 + α)V − pn2 ]+, where µ = E(V ) is

the expected customer valuation. On the other hand, the expected utility of waiting until the

second period is given by Uwait := δE[(1 + α)V − pn2 ]+. Thus, the number of customers who

request a product in period 1, X1, is given by X1 = X ·1{Uvisit≥Uwait}, and the reservation price

of these customers in period 1, r1, satisfies

ξa(µ− r1 + δE[(k + α)V − pr2]

+) + (1 − ξa)δE[(1 + α)V − pn2 ]+ = δE[(1 + α)V − pn

2 ]+.

If ξa > 0, r1 = µ + δE[(k + α)V − pr2]

+ − δE[(1 + α)V − pn2 ]+. Otherwise, ξa = 0, customers

anticipate zero product availability in the first period, so r1 can be any nonnegative number.

Thus, without loss of generality, let r1 = µ+ δE[(k + α)V − pr2]

+ − δE[(1 + α)V − pn2 ]+ for all

ξa ≥ 0.

Next, we consider the firm’s problem in period 1. The firm does not know the exact reserva-

tion price r1, but forms a belief ξr about it. To maximize his expected profit, the firm sets the

first period price p1 equal to the expected reservation price ξr, which is the highest price (the

firm believes) at which customers are willing to pay in the first period. Thus, the firm believes

that first-period demand X1 = X. Therefore, we can write the expected total profit of the firm

8


as

Πf (p1, Q1) := p1E(X1 ∧Q1) − cQ1 + sE(Q1 −X1)+ + δE{π2(X

n2 , X

r2)}, (2)

where X1 = X, Xn2 = (X −Q1)

+, and Xr2 = X ∧Q1.

Under the RE equilibrium, players independently maximize their own utilities based on

their rational expectations (two binary choices for the customers, and a joint pricing and

production problem for the firm). Note that the rational expectations (pn2 , p

r2, ξa, ξr) must

be consistent with the actual outcome. More specifically, (pn2 , p

r2) must follow the same dis-

tribution as (pn2 (Xn

2 , Xr2), pr

2(Xn2 , X

r2)), which is a random vector contingent on the realiza-

tion of the random variable X. The customers’ belief on the product availability ξa must

agree with the actual in-stock probability under the first-period production quantity Q1, i.e.,

ξa = A(Q1) := E(X1 ∧Q1)/E(X1) (see, e.g., Su and Zhang, 2009). Finally, the firm’s belief on

customers’ willingness-to-pay ξr must coincide with the customer’s reservation price r1.

We summarize the above discussions in the following definition of the RE equilibrium:

Definition 1 An RE equilibrium consists of (p∗1, Q

∗1, p

n2 (·, ·), pr

2(·, ·), pn∗2 , pr∗

2 , ξ∗a, ξ

∗r ) satis-

fying

(a) Given (Xn2 , X

r2), (pn

2 (Xn2 , X

r2), pr

2(Xn2 , X

r2)) = argmax{(pn

2 ,pr2),pr

2≤pn2 }Π2(p

n2 , p

r2|Xn

2 , Xr2);

(b) r∗1 = µ+ δE[(k + α)V − pr∗

2 ]+ − δE[(1 + α)V − pn∗2 ]+;

(c) p∗1 = ξ∗

r , Q∗1 = argmaxQ1≥0Πf (p∗

1, Q1);

(d) ξ∗a = A(Q∗

1), (pn∗2 , pr∗

2 )d= (pn

2 (Xn2 , X

r2), pr

2(Xn2 , X

r2)), where Xr

2 = X ∧ Q∗1 and Xn

2 =

(X −Q∗1)

+;

(e) ξ∗r = r∗

1.

Condition (a) follows from the optimal pricing policy of the firm in period 2. Conditions (b)

and (c), respectively, are due to the optimal decision of the customers given beliefs (pn∗2 , pr∗

2 , ξ∗a),

and that of the firm given the belief ξ∗r in period 1. Conditions (d) and (e) represent the

consistency between belief and outcome for customers and the firm, respectively.

We now characterize the RE equilibrium in our model. To begin with, we characterize

the optimal pricing strategy of the firm in period 2. Define p∗ := argmaxp≥0(p − c)G(p) and

R∗ := max{(p − c)G(p) : p ≥ 0}, i.e., p∗ is the optimal price in the single period model if

customers know their own types upfront and without demand uncertainty, and R∗ is the optimal

profit per customer in this scenario. Note that, under the increasing failure rate assumption,

p∗ is unique. Because µ = E(V ) > c, we have p∗ > c and R∗ > 0.

Lemma 1 (a) For any (Xn2 , X

r2), Π2(p

n2 , p

r2|Xn

2 , Xr2) is continuously differentiable and

quasiconcave in (pn2 , p

r2).

(b) For any (Xn2 , X

r2), pn

2 (Xn2 , X

r2) ≡ pn∗

2 = (1 + α)p∗ and pn2 (Xn

2 , Xr2) ≡ pr∗

2 = (k + α)p∗.

9


Lemma 1 shows that the optimal pricing strategy (pn2 (·, ·), pr

2(·, ·)) in period 2 is a constant

price vector independent of the realized market demands (Xn2 , X

r2). The proof of Lemma 1

implies that the price for new customers and that for repeat customers can be determined

separately. By the definition of the RE equilibrium, the customers’ beliefs satisfy (pn∗2 , pr∗

2 ) =

((1 + α)p∗, (k + α)p∗) with probability 1.

To characterize the RE equilibrium, we define an auxiliary variable m∗1 := µ− δ(1−k)(R∗ +

E(V −p∗)+). As will be clear in our subsequent analysis, m∗1 is the first-period effective marginal

revenue, which summarizes the impact of second-period prices on the first-period profit. Based

on Lemma 1, we can characterize the RE equilibrium market outcome in the following theorem.

Theorem 1 In the base model:

(a) An RE equilibrium (p∗1, Q

∗1, p

n2 (·, ·), pr

2(·, ·), pn∗2 , pr∗

2 , ξ∗a, ξ

∗r ) exists with

(i) p∗1 = µ− δ(1 − k)E(V − p∗)+;

(ii) If m∗1 > c, Q∗

1 = F−1( c−sm∗

1−s) > 0; otherwise, m∗1 ≤ c, Q∗

1 = 0;

(iii) pn2 (·, ·) ≡ (1 + α)p∗ and pr

2(·, ·) ≡ (k + α)p∗.

(b) Under any RE equilibrium, the expected total profit of the firm is identical and given by

Π∗f = (m∗

1 − s)E(X ∧Q∗1) − (c− s)Q∗

1 + δ(1 + α)R∗E(X),

and the expected total customer surplus is identical and given by

S∗c = δ(1 + α)E(V − p∗)+E(X).

Theorem 1 implies that, though there may exist multiple RE equilibria, they all lead to the

same profit for the firm and the same surplus for the customers. Thus, any RE equilibrium in

our model is essentially equivalent and we will use the one characterized by Theorem 1(a) in the

subsequent analysis. Another implication from Theorem 1 is that, under the RE equilibrium,

the first-period production quantity Q∗1 is the solution to a standard newsvendor problem with

stochastic demand X, marginal revenue m∗1, marginal cost c, and salvage value s. As a conse-

quence, the firm may either supply the market with two generations of the product (Q∗1 > 0)

or with the second-generation product only (Q∗1 = 0), depending on the relative magnitude of

m∗1 and c. By the definition of m∗

1, this dichotomy is essentially determined by the discount

factor δ and the depreciation factor k. Specifically, it is more profitable to produce and sell two

generations of the product if and only if δ(1 − k) < µ−cR∗+E(V −p∗)+

. As a corollary of Theorem

1, the following proposition characterizes the impact of model parameters on the equilibrium

outcome.

Proposition 1 In the RE equilibrium:

(a) pn∗2 is increasing in α and independent of k. pr∗

2 is increasing in α and k.

10


(b) m∗1, p

∗1, Q

∗1, and ξ∗

a are decreasing in δ, increasing in k, and independent of α.

(c) Π∗f is increasing in k and α.

(d) S∗c is increasing in δ and α, and independent of k.

Proposition 1(a) implies that the optimal new product price pn∗2 only depends on α, the

innovation level of the second-generation product, whereas the optimal trade-in price pr∗2 is

increasing in both α and the depreciation factor k. Part (b) proves that, in period 1, the effective

marginal revenue m∗1, the equilibrium price p∗

1, and the equilibrium production quantity Q∗1 are

all decreasing in the discount factor δ. This is because, a higher discount induces a higher relative

utility of purchasing a new second-generation product against joining in the trade-in program

for a remanufactured one, thus motivating customers to wait in the first period. Analogously,

if k increases, customers can obtain more utility from joining the trade-in program in period

2, so they are more willing to make a purchase in period 1. Thus, m∗1, p

∗1, and Q∗

1 are all

increasing in k. What is interesting is that m∗1, p

∗1, and Q∗

1 are independent of α. On one hand,

a higher α increases the customers’ willingness to join the trade-in program in period 2, thus

increasing their willingness-to-pay in period 1 as well. On the other hand, higher innovation of

the second-generation product prompts the customers to wait, instead of purchasing the first-

generation product immediately. The above two effects cancel out each other so that m∗1, p

∗1

and Q∗1 are independent of α. Part (c) states that the equilibrium total profit of the firm Π∗

f

is increasing in the discount factor, the depreciation factor, and the innovation level. Higher

depreciation or innovation increases customers’ willingness-to-pay, thus giving rise to a higher

profit of the firm. Part (d) shows that the total expected customer surplus S∗c is increasing in

δ and α, and is independent of k. Higher discount or innovation boosts the customers’ utility

in period 2 and, thus, the total expected customer surplus. Higher depreciation increases the

utility a customer gains from a remanufactured second-generation product, but it also gives rise

to a higher first-period price and a higher trade-in price. The above two effects cancel out each

other, so S∗c is independent of k.

5 Impact of Trade-in Remanufacturing: Firm’s Perspective

In this section, we analyze the impact of remanufacturing and the associated trade-in program

from the firm’s perspective. More specifically, we first examine who will benefit from the cost

savings from remanufacturing; then we focus on understanding the interactions between trade-in

remanufacturing and strategic customer behavior.

5.1 Impact of Remanufacturing Cost Advantage

It has been widely recognized that under remanufacturing the firm is able to recover the residual

value of used products (e.g., Guide and Van Wassenhove, 2009). Thus, remanufacturing helps

11


the firm reduce his production cost in period 2. Who will benefit from the cost reduction

through remanufacturing, the firm or the customers? To answer this question, we introduce a

benchmark model that is identical to the base model except that now remanufacturing has no

cost advantage: In period 2, the unit production cost of remanufactured products is (1 + α)c,

the same as that of new products. We call this the No Cost Advantage (NCA) model. By

comparing the NCA model and the base model in Section 4, we can single out the impact

of cost advantage from remanufacturing. Later we will also utilize the NCA model to study

other benefits of remanufacturing (see Section 5.2). In the NCA model, the firm is indifferent

between using new and remanufactured products to serve demand in the second period; however,

to facilitate comparison, we assume that the firm adopts remanufacturing and the associated

trade-in program as in the base model. In the remainder of this paper, we use “˜” to denote the

NCA model. The following lemma shows that in the NCA model, the optimal second-period

prices (pn2 (·, ·), pr

2(·, ·)) are independent of the realized market demands (Xn2 , X

r2).

Lemma 2 For any (Xn2 , X

r2), pn

2 (Xn2 , X

r2) ≡ pn

2 (Xn2 , X

r2) = (1+α)p∗ and pr

2(Xn2 , X

r2) ≡ pr∗

2 ,

where

pr∗2 = argmaxpr

2≥0(pr2 − (1 + α)c)G

(pr2

k + α

). (3)

Because the anticipated prices must be consistent with the actual outcome, (pn∗2 , pr∗

2 )d=

((1 + α)p∗, pr∗2 ). The same argument from the base model suggests that the equilibrium first-

period price is p∗1 = r∗

1 = µ + δE((k + α)V − pr∗2 )+ − δ(1 + α)E(V − p∗)+. Thus, the firm’s

optimal production quantity in period 1 follows from maximizing Πf (p∗1, Q1) = p∗

1E(X1 ∧Q1)−cQ1 + sE(Q1 − X1)

+ + δE{π2(Xn2 , X

r2)}, where X1 = X, Xn

2 = (X − Q1)+, Xr

2 = X ∧ Q1,

and π2(Xn2 , X

r2) = max{Xn

2 (pn2 − (1 + α)c)G(

pn2

1+α) +Xr2(pr

2 − (1 + α)c)G(pr2

k+α) : 0 ≤ pr2 ≤ pn

2}.

Similar to the base model, we denote m∗1 := µ+δ[(pr∗

2 −(1+α)c)G(pr∗2

k+α)+E((k+α)V − pr∗2 )+ −

(1+α)(R∗ +E(V − p∗)+)] as the first-period effective marginal revenue. The following theorem

characterizes the equilibrium market outcome in the NCA model.

Theorem 2 In the NCA model:


∗1, p

n2 (·, ·), pr

2(·, ·), pn∗2 , pr∗

2 , ξ∗a, ξ

∗r ) exists, under which

(i) p∗1 = µ+ δ[E((k + α)V − pr∗

2 )+ − (1 + α)E(V − p∗)+];

(ii) If m∗1 > c, Q∗

1 = F−1( c−sm∗


1 = 0;

(iii) pn2 (·, ·) ≡ (1 + α)p∗ and pr

2(·, ·) ≡ pr∗2 .

(b) Under any RE equilibrium, the expected profit of the firm is identical and given by

Π∗f = (m∗

1 − s)E(X ∧ Q∗1) − (c− s)Q∗

1 + δ(1 + α)R∗E(X),

and the expected total customer surplus is identical and given by

S∗c = δ(1 + α)E(V − p∗)+E(X).

12


Theorem 2 reveals that the RE equilibrium outcome in the NCA model shares the same

structure as that in the base model. Hence, a direct comparison of Theorems 1 and 2 demon-

strates the impact of remanufacturing cost advantage, as shown in the following theorem.

Theorem 3 The following statements hold:

(a) pn2 (·, ·) ≡ pn

2 (·, ·) and pr2(X

n2 , X

r2) ≥ pr

2(Xn2 , X

r2) for all (Xn

2 , Xr2).

(b) m∗1 ≤ m∗

1, p∗1 ≤ p∗

1, and Q∗1 ≤ Q∗

1. In particular, if Q∗1 > 0, Q∗

1 > 0 as well.

(c) Π∗f ≤ Π∗

f and S∗c = S∗

c .

Theorem 3(a) confirms our intuition that the optimal price for repeat customers in period

2 is higher in the NCA model. Theorem 3(b) further implies that, when remanufacturing

has a cost advantage, the customers anticipate a lower second-period trade-in price and, thus,

have a higher willingness-to-pay in the first period. Thus, remanufacturing cost advantage

increases the effective marginal revenue and drives the firm to price higher and produce more in

period 1. The increased first-period price and service level, together with the cost advantage of

remanufacturing, leads to a higher expected profit, as proved in Theorem 3(c). The interesting

aspect of Theorem 3 is that the expected customer surplus is not affected by the remanufacturing

cost advantage. In other words, the cost benefit from remanufacturing will be completely

extracted by the firm.

5.2 Value of Trade-in Remanufacturing under Strategic Customer Behavior

Trade-in remanufacturing not only allows the firm to recover the residual value of recycled prod-

ucts, but also enables the firm to tailor the second-period prices to new and repeat customers.

It is clear that this may improve the firm’s profit by exploiting customer segmentation. Are

there any other benefits from differentiated pricing? How does the trade-in program affect the

customers’ purchasing behavior? We address these questions in this subsection.

First we introduce a benchmark in which the firm does not adopt the trade-in remanu-

facturing. That is, the firm does not use the trade-in program to recycle used products for

remanufacturing; accordingly, the firm announces a single second-period price pu2 to both new

and repeat customers. We call this the No Trade-in Remanufacturing (NTR) model. In this

model, the optimal second-period pricing strategy pu2(·, ·) is contingent on the realized market

demands (Xn2 , X

r2), because new and repeat customers have different willingness-to-pay in pe-

riod 2. The following lemma characterizes the optimal second-period pricing policy in the NTR

model.

Lemma 3 (a) pu2(Xn

2 , Xr2) is determined by the market size ratio λ2 := Xn

2 /Xr2 . More

specifically, pu2(Xn

2 , Xr2) is continuously increasing in λ2, with pu

2(Xn2 , X

r2) = pr∗

2 if λ2 = 0,

and

limλ2→+∞

pu2(Xn

2 , Xr2) = pn∗

2 = (1 + α)p∗.

13


(b) For any fixed Xr2 , p

u2(·, Xr

2) is continuously increasing in Xn2 ; for any fixed Xn

2 , pu2(Xn

2 , ·)is continuously decreasing in Xr

2 .

Let pu2 be the random variable that represents the customers’ anticipated second-period price.

By our previous argument, the customers’ first-period reservation price is ru1 = µ+δ[E((k+α)V −

pu2)+−E((1+α)V −pu

2)+], which also equals the equilibrium first-period price pu1 . Meanwhile, the

expectations should be consistent with the actual outcome, i.e, pu2

d= pu

2(Xn2 , X

r2), where Xn

2 =

(X −Q1)+, and Xr

2 = X ∧Q1. Thus, the firm’s first-period production quantity is determined

by maximizing Πuf (Q1) := pu

1(Q1)E(X1 ∧Q1) − cQ1 + sE(X1 −Q1)+ + δE{πu

2 (Xn2 , X

r2)}, where

pu1(Q1) := µ + δ[E((α + k)V − pu

2(Xn2 , X

r2))+ − E((1 + α)V − pu

2(Xn2 , X

r2))+], X1 = X, Xn

2 =

(X − Q1)+, Xr

2 = X ∧Q1, and πu2 (Xn

2 , Xr2) = max{Xn

2 (pu2 − (1 + α)c)G(

pu2

1+α) +Xr2(pu

2 − (1 +

α)c)G(pu2

k+α) : pu2 ≥ 0}. We define the first-period quantity-dependent marginal revenue

mu1(Q1) = µ+ δ{E[(pu

2(Xn2 , X

r2) − (1 + α)c)G

(pu2(Xn

2 , Xr2)

k + α

)] + E((k + α)V − pu

2(Xn2 , X

r2))+

−E[(pu2(Xn

2 , Xr2) − (1 + α)c)G

(pu2(Xn

2 , Xr2)

1 + α

)] − E((1 + α)V − pu

2(Xn2 , X

r2))+},(4)

where Xn2 = (X − Q1)

+ and Xr2 = X ∧ Q1. We have the following lemma that evaluates the

total expected profit of the firm.

Lemma 4 The total expected profit of the firm in the NTR model is given by

Πuf (Q1) := (mu

1(Q1)−s)E(X∧Q1)−(c−s)Q1+δE[(pu2(Xn

2 , Xr2)−(1+α)c)G

(pu2(Xn

2 , Xr2)

1 + α

)X],

where Xn2 = (X −Q1)

+, Xr2 = X ∧Q1, and mu

1(Q1) is defined by (4).

Based on Lemma 4, we characterize the RE equilibrium outcome of the NTR model in the

following theorem.

Theorem 4 In the NTR model:

(a) An RE equilibrium (pu∗1 , Qu∗

1 , pu2(·, ·), pu∗

2 , ξu∗a , ξu∗

r ) exists, under which

(i) Qu∗1 = argmaxQ1≥0Π

uf (Q1), with Qu∗

1 < +∞;

(ii) pu∗1 = pu

1(Q∗1);

(iii) pu2(·, ·) is characterized by Lemma 3.

(b) Under any RE equilibrium, the expected profit of the firm is identical and given by Πu∗f =

Πuf (Qu∗

1 ), and the expected total customer surplus is identical and given by

Su∗c = E{X≤Qu∗

1 }X[µ− pu∗1 + δEV ((k + α)V − pu

2(Xn∗2 , Xr∗

2 ))+]

+E{X>Qu∗1 }{Qu∗

1 [µ− pu∗1 + δEV ((k + α)V − pu

2(Xn∗2 , Xr∗

2 ))+]

+(X −Qu∗1 )δEV ((1 + α)V − pu

2(Xn∗2 , Xr∗

2 ))+},

where Xn∗2 = (X −Qu∗

1 )+ and Xr∗2 = X ∧Qu∗

1 .

14


Lemma 3 and Theorem 4 imply that, in the NTR model, the optimal production quantity

in the first period is no longer determined by the solution to a newsvendor problem. This is

because, in this model, the first-period production quantity will influence the second-period

pricing strategy, which, in turn, impacts the willingness-to-pay of the customers in period 1

and, thus, the first-period price.

To investigate the value of differentiated pricing caused by trade-in remanufacturing, we com-

pare the NTR model (i.e., the model without trade-in remanufacturing) with the NCA model

(i.e., the model with trade-in remanufacturing, but without remanufacturing cost advantage).

Notice that remanufacturing is not used at all in the NTR model, so the remanufacturing cost

advantage is irrelevant. Thus the performance difference between these two models is purely due

to the pricing strategy rather than the remanufacturing cost advantage. Recall the equilibrium

outcomes of the two models are presented in Theorem 4 and Theorem 2, respectively.

Theorem 5 Comparing the NTR model with the NCA model, we have:

(a) pr2(X

n2 , X

r2) ≤ pu

2(Xn2 , X

r2) ≤ pn

2 (Xn2 , X

r2) for all (Xn

2 , Xr2).

(b) pu∗1 ≤ p∗

1, and mu1(Q1) < m∗

1 for all Q1 ≥ 0. If Qu∗1 > 0, Q∗

1 > 0 as well. In particular, if

mu1(Q1) is decreasing in Q1, Q

u∗1 ≤ Q∗

1.

(c) Πu∗f ≤ Π∗

f .

As shown in Theorem 5(a), the optimal second-period price without trade-in remanufac-

turing is between the optimal second-period trade-in price, and the optimal second-period new

product price in the model with trade-in remanufacturing (i.e., pr2(X

n2 , X

r2) ≤ pu

2(Xn2 , X

r2) ≤

pn2 (Xn

2 , Xr2) for all (Xn

2 , Xr2)). Since pr

2(Xn2 , X

r2) ≤ pu

2(Xn2 , X

r2), E[(k + α)V − pr

2(Xn2 , X

r2)]+ ≥

E[(k+α)V − pu2(Xn

2 , Xr2)]+, i.e., the expected utility of purchasing the first-generation product

is higher with the trade-in program. On the other hand, pn2 (Xn

2 , Xr2) ≥ pu

2(Xn2 , X

r2) implies that

E[(1 + α)V − pn2 (Xn

2 , Xr2)]+ ≤ E[(1 + α)V − pu

2(Xn2 , X

r2)]+, i.e., the trade-in program makes

waiting less attractive. Therefore, with trade-in remanufacturing, customers are more willing

to make an immediate purchase rather than wait until period 2. As a consequence, with the

trade-in program, the equilibrium first-period price is higher and the firm is more likely to serve

the market with two generations of the product. Another implication of Theorem 5 is that

trade-in remanufacturing gives rise to higher expected profit of the firm. Since there is not

remanufacturing cost advantage for both models, trade-in remanufacturing benefits the firm by

charging differentiated prices to new and repeat customers.

The trade-in program boosts the profit of the firm in two ways: (a) It exploits customer

segmentation with the price discrimination strategy in period 2, and (b) it exploits the forward-

looking behavior of customers by offering them early purchase rewards. While the former has

already been studied in the literature (see, e.g., Ray et al., 2005), the latter is one of the key new

insights of our paper. More specifically, trade-in remanufacturing offers customers an option to

15


buy the second-generation product at a discounted price in period 2. Since customers believe

that discounts will be offered to repeat customers in period 2, they are more willing to make an

immediate purchase in the first period. Therefore, our model demonstrates another important

value of trade-in remanufacturing: It offers customers early purchase rewards that effectively

mitigate strategic customer behavior.

5.3 Numerical Study

In this subsection, we conduct a comprehensive numerical study to quantify the value of trade-in

remanufacturing and derive additional insights. The design of the numerical study is as follows.

Let the customer valuation V follow a uniform distribution on [0, 1] (µ = E(V ) = 0.5). The

discount factor is δ = 0.95, and the salvage value of the leftover inventory in period 1 is s = 0.

The innovation level of the second-generation product is α ∈ {0, 0.1, 0.2, 0.3, 0.4} and the depre-

ciation factor is k ∈ {0.1, 0.3, 0.5, 0.7, 0.9}. The demand X follows a gamma distribution with

mean 100 and coefficient of variation CV (X) taking values from the set {0.1, 0.3, 0.5, 0.7, 0.9}.

The unit production cost of the first-generation product is c ∈ {0.05, 0.15, 0.25, 0.35, 0.45} (note

we need c < µ = 0.5). Thus, we have a total of 625 parameter combinations that cover a wide

range of reasonable problem scenarios.

To focus on the effect of trade-in remanufacturing, we assume there is no remanufacturing

cost advantage in the numerical study (i.e., the second-period unit remanufacturing cost is

(1+α)c). We evaluate the firm’s optimal profits in four cases: (i) Customers are myopic and the

firm adopts trade-in remanufacturing (the profit is denoted Πm∗f ); (ii) customers are myopic and

the firm does not adopt trade-in remanufacturing (Πmu∗f ); (iii) customers are strategic and the

firm adopts trade-in remanufacturing (Π∗f ); and (iv) customers are strategic and the firm does

not adopt trade-in remanufacturing (Πu∗f ). The RE equilibrium outcomes and profits for cases

(iii) and (iv) are presented in Section 5.2, while the analysis for cases (i) and (ii) can be found

in Appendix B. The metrics of interest are γm :=Πm∗

f −Πmu∗f

Πmu∗f

× 100% and γs :=Π∗

f −Πu∗f

Πu∗f

× 100%,

which quantify the profit improvements of adopting the trade-in remanufacturing under myopic

and strategic customers, respectively.

We evaluate γm and γs under the 625 parameter combinations and find that, under each

parameter combination, γs is higher than γm. Moreover, if the product depreciation is not too

low (e.g., k > 0.1 in our experiments), γs is significantly higher than γm. Specifically, γs is

at least 5.7% and can be as high as 122.7%, with an average value 32.8%; whereas γm ranges

from 0.1% to 17.1%, with an average value 6.3%. Thus, our model delivers the new message

to firms that the value of trade-in remanufacturing is more pronounced under the presence of

strategic customers. Note that, with myopic customers, the trade-in program only has the value

of price discrimination, while, with strategic customers, this program has the value of both price

discrimination and early purchase rewards. Therefore, our numerical experiments further reveal

that, when the product depreciation is not too small, the value of trade-in remanufacturing to the

16


firm mainly comes from its early purchase rewards effect to mitigate strategic customer behavior

rather than from its price discrimination effect to exploit customer segmentation. When the

product depreciation is very low (k = 0.1 in our experiments), however, remanufacturing adds

little value to a repeat customer, so the trade-in program only offers marginal early purchase

rewards. In this case, the effect of mitigating strategic behavior is less significant, and the value

of trade-in remanufacturing mainly comes from the price discrimination effect.

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

10

20

30

40

50

60

70

80

90

α

Pro

fit Im

prov

emen

ts o

f Tra

de−

in R

eman

ufac

turin

g (%

)

Myopic customers (γ

m)

Strategic customers (γs)

Figure 1: Value of Trade-in Remanu-

facturing (k = 0.5, c = 0.25, CV (X) =

0.5)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

90

k

Pro

fit Im

prov

emen

ts o

f Tra

de−

in R

eman

ufac

turin

g (%

)


m)



facturing (α = 0.2, c = 0.25, CV (X) =

0.5)

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

CV(X)

Pro

fit Im

prov

emen

ts o

f Tra

de−

in R

eman

ufac

turin

g (%

)


m)



facturing (α = 0.2, k = 0.5, c = 0.25)

We also examine how the value of trade-in remanufacturing depends on problem parameters.

Figure 1 shows that such a value is decreasing in the second-generation innovation level α under

both strategic and myopic customers. As α increases, the relative difference between the optimal

trade-in price pr∗2 and the optimal price for new customers pn∗

2 (i.e.,pn∗2 −pr∗

2pr∗2

) decreases. Thus,

when α is higher, both the price discrimination effect and the early purchase rewards effect of

trade-in remanufacturing decrease. Hence, γm and γs are both decreasing in α.

Figure 2 indicates that γm is decreasing in the depreciation factor k, while γs is increasing

in k when k is small, and decreasing when k is large. When k increases (equivalently, the

product durability decreases), there are two opposing effects: (i) pr∗2 converges to pn∗

2 ; and

(ii) remanufacturing delivers higher value to repeat customers. Effect (i) implies that higher

product depreciation leads to lower price discrimination and lower early purchase rewards, so γm

is decreasing in k. On the other hand, effect (ii) suggests that an increase in product depreciation

gives rise to higher early purchase rewards and, thus, higher value of trade-in remanufacturing.

Our numerical results suggest that when the product depreciation is low (i.e., k ≤ 0.5), effect (ii)

dominates, so that γs is increasing k when k is small. However, when the product depreciation

is high (i.e., k ≥ 0.5), effect (i) outweighs effect (ii) and, thus, γs is decreasing k when k is large.

Figure 3 suggests that γm is increasing, while γs is decreasing in demand variability (mea-

sured by CV (X)). Higher demand variability leads to more new customers and fewer repeat

customers in the second period. Thus, if the demand is more variable, the second-period mar-

ket becomes more segmented and the price discrimination effect of trade-in remanufacturing is

higher. On the other hand, more variable demand results in fewer repeat customers in period

17


2, so trade-in remanufacturing offers less early purchase rewards to customers. Therefore, γm

is increasing, whereas γs is decreasing in CV (X).

From the γs curves in Figures 1-3, we can see that trade-in remanufacturing is most effective

in mitigating strategic customer behavior and improving the firm’s profit when (i) second-

generation innovation is low, (ii) product durability is moderate, and (iii) demand variability

is low. It is worth noting that Cachon and Swinney (2011) identify enhanced design/product

innovation as an effective strategy in mitigating strategic customer behavior. The mechanism

is different in their framework: Enhanced design gives customers a more attractive product so

that they are less willing to wait for a sale under the stock-out risk. In contrast, innovation is

for the second-generation product in our problem, and therefore higher innovation reduces the

early purchase rewards that mitigate strategic waiting.

To summarize, besides the benefits due to production cost advantage and price discrimi-

nation, trade-in remanufacturing can effectively mitigate strategic customer behavior and may

substantially improve the firm’s profit. Thus, the presence of strategic customers may signifi-

cantly increase the value of trade-in remanufacturing. Moreover, the effect of trade-in reman-

ufacturing as a mechanism to mitigate strategic customer behavior is most prominent when

second-generation innovation is low, product durability is moderate, and demand variability is

low.

6 Impact of Trade-in Remanufacturing: Social Perspective

In this section, we analyze the impact of trade-in remanufacturing from the social perspective.

For this purpose, we introduce the environmental consideration into the base model. Moreover,

we consider the role of government, which may design incentive policies such as subsidiza-

tion to influence the market equilibrium. Specifically, we study whether remanufacturing and

government subsidization for remanufacturing are beneficial to the society under the strategic

interactions between the firm and customers. Moreover, we demonstrate how the government

should design a regulatory mechanism to induce the socially optimal outcome.

6.1 Environmental Impact

We start with the environmental impact of remanufacturing. A commonly-held belief is that

remanufacturing is beneficial to the environment (see, e.g., Guide and Van Wassenhove, 2009).

It would be useful to investigate how remanufacturing actually affects the environment. For

simplicity, we assume that the environmental impact takes place only at the production, re-

manufacturing, and disposal stages, but not at the consumption stage. This assumption is ap-

propriate for product categories such as electronics, furniture, and carpets (see Agrawal et al.,

2012). A relaxation of this assumption will be discussed in Section 7.

Let κ1 > 0 denote the unit environmental impact of the first-generation product, κn2 > 0

18


the unit environmental impact of the new second-generation product, and κr2 > 0 as the unit

environmental impact of the remanufactured second-generation product. Such impact may refer

to the use of natural resources, emission of harmful gases, and generation of solid wastes. Since

remanufacturing does not have to make the product from scratch, we assume that κr2 = θκn

2 ,

where θ ∈ (0, 1]. The ratio θ captures the environmental advantage of remanufacturing, i.e.,

the lower the θ, the greener the remanufacturing as opposed to new product manufacturing.

Moreover, we assume κ1 ≥ κn2 to capture the fact that the second-generation product is more

environmentally friendly than the first-generation. Hence, the equilibrium total environmental

impact in the base model is given by

I∗e = κ1Q

∗1 + δ(κn

2 G(p∗)E(X −Q∗1)

+ + κr2G(p∗)E(X ∧Q∗

1))

= κ1Q∗1 + δκn

2 (G(p∗)E(X −Q∗1)

+ + θG(p∗)E(X ∧Q∗1)).

Next we examine whether adopting remanufacturing could lead to a more environmentally

friendly outcome. To single out the effect of remanufacturing, we consider a benchmark model

without remanufacturing. In this model, however, the firm may still offer different second-period

prices to new and repeat customers. We call this the No Remanufacturing (NR) model. This

treatment ensures that the price discrimination effect is removed from the later comparison. It

can be readily seen that the equilibrium outcome in this benchmark model will be the same as

that in the NCA model in Section 5.1 (i.e., the model without remanufacturing cost advantage).

Thus, let I∗e denote the equilibrium environmental impact in the NR model (recall we use “˜”

to denote the NCA model), which can be written as

I∗e = κ1Q

∗1 + δκn

2 (G(p∗)E(X − Q∗1)

+ + G

(pr∗2

k + α

)E(X ∧ Q∗

1)). (5)

By comparing the NR model to the base model with remanufacturing, the following theorem

sheds light on the environmental impact of adopting remanufacturing:

Theorem 6 There exists a threshold θ < 1, such that I∗e ≥ I∗

e if θ ≥ θ. In particular, if

θ ≥ θ and Q∗1 > Q∗

1, then I∗e > I∗

e .

Theorem 6 shows that if the environmental advantage of remanufactured products over

new products is not sufficiently large (i.e., θ ≥ θ), remanufacturing will generate a greater

environmental impact when it induces a higher production quantity in the first period (i.e.,

Q∗1 > Q∗

1). In other words, remanufacturing does not necessarily lead to a greener outcome

from an environmental perspective. This result contrasts with the conventional belief that

adopting remanufacturing helps reduce the negative impact of production on the environment.

The driving force behind Theorem 6 is that, as shown in Theorem 3, remanufacturing gives rise

to a higher production quantity in period 1 and a higher probability that a repeat customer

would purchase the second-generation product in period 2. These two indirect effects lead to

a greater environmental impact caused by increased total production, which may offset the

19


environmental benefit from remanufactured products. Thus, our analysis delivers an important

message for practitioners and policy makers: The impact of remanufacturing should be carefully

evaluated because it may not always benefit the environment.

6.2 Impact of Government Subsidization

As discussed in Section 1, governments around the world have been active in creating en-

vironmentally friendly industry policies. For example, subsidization for the production and

consumption of remanufactured products is a widely-employed incentive scheme to promote

remanufacturing. From a government’s perspective, a well-designed subsidization policy should

maximize the social welfare, which includes the firm’s profit, the customers’ surplus, and the

environmental impact. There are two commonly-used subsidy forms in practice: (i) per-unit

subsidy to the firm, under which the government subsidizes the firm srg,2 per unit remanufactured

product sold; and (ii) per-unit subsidy to customers, under which the government subsidizes

customers srg,2 per unit remanufactured product purchased. How do these subsidization policies

affect the firm, the customers, and the environment? What is the socially optimal subsidization

policy? We address these two questions in the rest of this section.

It can be shown that the above two subsidy forms are essentially equivalent: They give rise

to identical firm profit, customer surplus, and environmental impact. This is because the firm

could use its pricing power to fully exploit the government subsidies to the customers. Thus,

without loss of generality, we focus on the analysis of the per-unit subsidy to the firm. The

analysis and results for the other subsidy form are similar.

In the remainder of this paper, we use “ ˆ ” to denote the model with government subsi-

dization for remanufactured products. As in our base model, the optimal second-period pricing

strategy in the model with government subsidization (pn2 (·, ·), pr

2(·, ·)) is independent of the re-

alized market demands (Xn2 , X

r2), as shown in the following lemma.

Lemma 5 For any subsidy rate srg,2 and realized demand (Xn

2 , Xr2), pn

2 (Xn2 , X

r2) ≡ pn

2 (Xn2 , X

r2) =

(1 + α)p∗ and pr2(X

n2 , X

r2) ≡ pr∗

2 , where

pr∗2 = argmaxpr

2≥0(pr2 + sr

g,2 − (k + α)c)G

(pr2

k + α

).

Because the anticipated second-period prices must be consistent with the actual outcome

(i.e., (pn∗2 , pr∗

2 )d= ((1 + α)p∗, pr∗

2 )), the equilibrium first-period price is p∗1 = r∗

1 = µ + δE((k +

α)V − pr∗2 )+ − δ(1 + α)E(V − p∗)+. Thus, the firm’s optimal production quantity in period 1

follows from maximizing Πf (p∗1, Q1) := p∗

1E(X1 ∧Q1)− cQ1 + sE(Q1 −X1)+ + δE{π2(X

n2 , X

r2)},

where X1 = X, Xn2 = (X − Q1)

+, Xr2 = X ∧ Q1, and π2(X

n2 , X

r2) = max{Xn

2 (pn2 − (1 +

α)c)G(pn2

1+α) +Xr2(pr

2 + srg,2 − (k+α)c)G(

pr2

k+α) : 0 ≤ pr2 ≤ pn

2}. Denote m∗1 := µ+ δ[(pr∗

2 + srg,2 −

(k+α)c)G(pr∗2

k+α) + E((k+α)V − pr∗2 )+ − (1 +α)(R∗ + E(V − p∗)+)] as the first-period effective

marginal revenue. The following theorem characterizes the RE equilibrium market outcome in

the model with government subsidization.

20


Theorem 7 In the model with government subsidization for remanufacturing:


∗1, p

n2 (·, ·), pr

2(·, ·), pn∗2 , pr∗

2 , ξ∗a, ξ

∗r ) exists, under which

(i) p∗1 = µ+ δ[E((k + α)V − pr∗

2 )+ − (1 + α)E(V − p∗)+];

(ii) If m∗1 > c, Q∗

1 = F−1( c−sm∗


1 = 0;

(iii) pn2 (·, ·) ≡ (1 + α)p∗ and pr

2(·, ·) ≡ pr∗2 .

(b) Under any RE equilibrium, the expected profit of the firm is identical and given by

Π∗f = (m∗

1 − s)E(X ∧ Q∗1) − (c− s)Q∗

1 + δ(1 + α)R∗E(X),

the expected total customer surplus is identical and given by

S∗c = δ(1 + α)E(V − p∗)+E(X),

and the expected total environmental impact is identical and given by

I∗e = κ1Q

∗1 + δκn

2 (G(p∗)E(X − Q∗1)

+ + θG

(pr∗2

k + α

)E(X ∧ Q∗

1)).

Moreover, there exists a threshold θ < 1, such that I∗e ≥ I∗

e if θ ≥ θ.

In Theorem 7, we show that the RE equilibrium outcome in the model with government

subsidization for remanufacturing has the same structure as that in the base model. In par-

ticular, the equilibrium first-period production quantity is determined by solving a newsvendor

problem. Moreover, consistent with Theorem 6, Theorem 7(b) suggests that, under government

subsidization for remanufacturing, the equilibrium environmental impact is higher than that

in the model without remanufacturing, as long as the environmental advantage of remanufac-

turing is not sufficiently large. The following theorem characterizes the impact of government

subsidization for remanufacturing.

Theorem 8 (a) pr∗2 is continuously decreasing in sr

g,2. In particular, for any srg,2 > 0,

we have pr∗2 ≤ (k + α)p∗ ≤ pr∗

2 + srg,2.

(b) m∗1, p

∗1, and Q∗

1 are continuously increasing in srg,2. In particular, for any sr

g,2 > 0, we

have m∗1 ≥ m∗

1, p∗1 ≥ p∗

1, and Q∗1 ≥ Q∗

1.

(c) Π∗f and I∗

e are continuously increasing in srg,2, whereas θ is continuously decreasing in sr

g,2.

In particular, for any srg,2 > 0, we have Π∗

f ≥ Π∗f , S∗

c = S∗c , I

∗e ≥ I∗

e , and θ ≤ θ.

According to Theorem 8, government subsidization will give rise to a lower price but a higher

margin for remanufactured products in period 2. As a consequence, in period 1, customers have

higher willingness-to-pay and the firm can charge a higher price and produce more with govern-

ment subsidization. Theorem 8(c) demonstrates that the benefit of government subsidization

21


has been exclusively extracted by the firm, but the customer surplus remains the same and

the environmental impact is actually higher with government subsidization. This result follows

from the rationale that pricing flexibility enables the firm to fully exploit government subsidies,

whereas the environment suffers from the increased production levels under subsidization. As a

result, under government subsidization for remanufacturing, the environmental impact is more

likely to dominate that without remanufacturing (i.e., θ ≤ θ for any srg,2 > 0). The environmen-

tal advantage is one of the main reasons why government has made tremendous legislative and

economic efforts to promote remanufacturing (see, e.g., Giutini and Gaudette, 2003). Theorem

8(c), however, suggests that the government should rethink how to design the subsidization

policy, because haphazard subsidization for remanufacturing may result in the opposite of the

desired outcome.

6.3 Socially Optimal Government Intervention

Theorem 6 and Theorem 8(c) imply that although remanufactured products are greener than

new products, simply adopting remanufacturing or providing subsidies may actually harm the

environment. Hence, a proper design of the incentive scheme for remanufacturing is crucial to

achieve the desired outcome. The purpose of this subsection is to derive the optimal regulatory

policy that can maximize the social welfare through remanufacturing. Let Ws denote the social

welfare, which is defined by the expected profit of the firm Πf , plus the expected customer

surplus Sc, net the expected environmental impact Ie, i.e.,

Ws = Πf + Sc − Ie.

We first characterize the socially optimal outcome by assuming that the government can set the

prices and production levels for the firm, with an objective to maximize the social welfare. By

backward induction, we start with the second-period pricing problem. For any given realized

demands (Xn2 , X

r2), the social welfare is given by

W2(pn2 , p

r2|Xn

2 , Xr2) := Xn

2 [(pn2 − (1 + α)c− κn

2 )G

(pn2

1 + α

)+ E((1 + α)V − pn

2 )+]

+Xr2 [(pr

2 − (k + α)c− κr2)G

(pr2

k + α

)+ E((k + α)V − pr

2)+].

We use (pns,2(X

n2 , X

r2), pr

s,2(Xn2 , X

r2)) := argmax{(pn

2 ,pr2),pr

2≤pn2 }W2(p

n2 , p

r2|Xn

2 , Xr2) to denote the

optimal social welfare maximizing pricing strategy in period 2. Moreover, let w2(Xn2 , X

r2) :=

max{W2(pn2 , p

r2|Xn

2 , Xr2) : 0 ≤ pr

2 ≤ pn2} denote the corresponding optimal social welfare with

realized demands (Xn2 , X

r2). We have the following lemma that characterizes the optimal second-

period pricing strategy.

Lemma 6 For any (Xn2 , X

r2), pn

s,2(Xn2 , X

r2) ≡ pn∗

s,2 = (1 + α)c + κn2 and pr

s,2(Xn2 , X

r2) ≡

pr∗s,2 = (k + α)c+ κ2

r.

22


Under the RE equilibrium, the anticipated prices must be consistent with the outcome (i.e.,

(pn∗s,2, p

r∗s,2)

d= (pn∗

s,2, pr∗s,2)). The same argument from the base model implies that the equilibrium

first-period reservation price is r∗s,1 = µ + δE((k + α)V − pr∗

s,2)+ − δE((1 + α)V − pn∗

s,2)+ =

µ + δ[E((k + α)V − (k + α)c − κr2)

+ − E((1 + α)V − (1 + α)c − κn2 )+]. Clearly, as long as

p1 ≤ r∗s,1, the total expected social welfare can be maximized. Without loss of generality and to

be consistent with the other models in this paper, we assume that p∗s,1 = r∗

s,1. Thus, the total

expected social welfare is given by

Ws(p∗s,1, Q1) = p∗

s,1E(X1∧Q1)+(µ−p∗s,1)E(X1∧Q∗

1)−(c+κ1)Q1+sE(Q1−X1)++δE{w2(X

n2 , X

r2)},

where X1 = X, Xn2 = (X −Q1)

+ and Xr2 = X ∧Q1. Note that the term (µ− p∗

s,1)E(X1 ∧Q∗1)

refers to the expected total customer surplus in period 1. As in the base model, we introduce

the first-period effective marginal welfare,

m∗s,1 := µ+ δ[(pr∗

s,2 − (k + α)c− κr2)G

(pr∗

s,2

k + α

)+ E((k + α)V − pr∗

s,2)+

−(pn∗s,2 − (1 + α)c− κn

2 )G

(pn∗

s,2

1 + α

)− E((1 + α)V − pn∗

s,2)+]

= µ+ δ[E((k + α)V − (k + α)c− κr2)

+ − E((1 + α)V − (1 + α)c− κn2 )+].

The following theorem characterizes the social welfare maximizing market outcome.

Theorem 9 (a) An RE equilibrium (p∗s,1, Q

∗s,1, p

ns,2(·, ·), pr

s,2(·, ·), pn∗s,2, p

r∗s,2, ξ

∗s,a, ξ

∗s,r) ex-

ists with

(i) p∗s,1 = µ+ δ[E((k + α)V − (k + α)c− κr

2)+ − E((1 + α)V − (1 + α)c− κn

2 )+];

(ii) If m∗s,1 > c+ κ1, Q

∗s,1 = F−1( c+κ1−s

m∗s,1−s ) > 0; otherwise, m∗

s,1 ≤ c+ κ1, Q∗s,1 = 0;

(iii) pns,2(X

n2 , X

r2) ≡ pn∗

s,2 = (1 + α)c+ κn2 and pr

s,2(Xn2 , X

r2) ≡ pr∗

s,2 = (k + α)c+ κr2.

(b) Under any RE equilibrium, the expected social welfare is identical and given by

W ∗s = (m∗

s,1 − s)E(X ∧Q∗s,1) − (c+ κ1 − s)Q∗

s,1 + δE[(1 + α)V − (1 + α)c− κn2 ]+E(X).

Theorem 9 has several important implications: (1) The social planner and the firm may have

conflicting incentives, because the social-welfare-maximizing equilibrium outcome may be quite

different from the profit-maximizing one (i.e., Theorem 1). In particular, we can show that,

if the unit environmental impacts, κ1, κn2 , and κr

2, are sufficiently large, the socially optimal

equilibrium will induce lower production quantities and thus have smaller total environmental

impact. (2) The socially optimal second-period pricing strategy takes the form that the prices

for new and repeat customers are equal to the respective unit production cost plus the unit

environmental impact (i.e., pn∗s,2 = (1 + α)c + κn

2 and pr∗s,2 = (k + α)c + κr

2). (3) The socially

optimal first-period production quantity is also the solution to a newsvendor problem.

Now we analyze how the government, whose objective is to maximize the expected social

welfare Ws, could induce the firm, whose objective is to maximize his expected profit Πf , to set

23


socially optimal prices and production quantities (i.e., Theorem 9). Theorems 7 and 8 imply that

subsidization is an effective approach to adjusting the equilibrium outcome, because it controls

the margin of the firm and the willingness-to-pay of the customers. With this observation, we

may design a subsidy/tax scheme that aligns the interests of both parties and thus induces the

socially optimal outcome. Note that the firm offers essentially three versions of the product in

the market: (i) the new first-generation product, (ii) the new second-generation product, and

(iii) the remanufactured second-generation product. Let sg = (sg,1, sng,2, s

rg,2) be the subsidy/tax

scheme the government adopts. That is, the government offers a per-unit subsidy to the firm for

each product version sold (a negative subsidy is a tax). Observe that the comparative statics

results in Theorem 8 can be generalized to the case where srg,2 < 0. Analogously, such a linear

subsidy/tax scheme can control the price and production quantity of the new second-generation

product as well. In fact, this subsidy/tax scheme can also induce the desired first-period effective

marginal revenue and production quantity. We have the following theorem.

Theorem 10 (a) There exists a linear subsidy/tax scheme s∗g = (s∗

g,1, sn∗g,2, s

r∗g,2), under

which the RE equilibrium outcome of the game played by the firm and the customers is

(p∗s,1, Q

∗s,1, p

ns,2(·, ·), pr

s,2(·, ·), pn∗s,2, p

r∗s,2, ξ

∗s,a, ξ

∗s,r).

(b) If s∗g induces the socially optimal outcome, we have sn∗

g,2 is the unique solution to pn∗s,2 =

argmaxpn2 ≥0{(pn

2+sng,2−(1+α)c)G(

pn2

1+α)}, sr∗g,2 is the unique solution to pr∗

s,2 = argmaxpr2≥0{(pr

2+

srg,2 − (k + α)c)G(

pr2

k+α)}, and s∗g,1 is the unique solution to c+κ1−s

m∗s,1−s = c−s

ms1(sg,1)−s , where

ms1(sg,1) := sg,1 +m∗

s,1 − δ[(κn2 + sn∗

g,2)G((1+α)c+κn

21+α ) − (κr

2 + sr∗g,2)G(

(k+α)c+κr2

k+α )].

(c) We have the following characterizations on the sign of s∗g. s

r∗g,2

> 0, if κr2 < (k + α)(p∗ − c),

= 0, if κr2 = (k + α)(p∗ − c),

< 0, if κr2 > (k + α)(p∗ − c).

sn∗g,2

> 0, if κn2 < (1 + α)(p∗ − c),

= 0, if κn2 = (1 + α)(p∗ − c),

< 0, if κn2 > (1 + α)(p∗ − c).

s∗g,1

> 0, if κ1 <A

m∗s,1−A−s(c− s),

= 0, if κ1 = Am∗

s,1−A−s(c− s),

< 0, if κ1 >A

m∗s,1−A−s(c− s),

where

A := δ

[(1 + α)

G2((1+α)c+κn

21+α )

g((1+α)c+κn

21+α )

− (k + α)G2(

(k+α)c+κr2

k+α )

g((k+α)c+κr

2k+α )

].

Theorem 10 demonstrates that, given the firm and the customers are both rational and

self-interested, the government can use a simple linear subsidy/tax scheme to induce the so-

cially optimal outcome. This result, on one hand, corroborates the commonly used government

subsidization strategy for remanufacturing (e.g., in 2009, Chinese government offered a subsidy

program for the trade-in remanufacturing of home appliances; see Ma et al., 2013). On the other

hand, Theorem 10 suggests that subsidizing for remanufactured products alone is not sufficient

to achieve the social optimum. Instead, the government should provide a combined subsidy/tax

24


scheme for all three product versions. Moreover, some components in s∗g may be negative, i.e.,

it is possible that the government taxes the firm on some product versions to discourage their

sales. This phenomenon results from the government’s goal of balancing the tradeoff between

firm profit, customer surplus, and environmental impact. In particular, we show, by Theorem

10(c), that whether the government should subsidize for or tax on one product version depends

on the magnitude of its environmental impact. For example, the government should subsi-

dize for the remanufactured second-generation product when its environmental impact is small

(i.e., κr2 < (k + α)(p∗ − c)), and should tax on it when its environmental impact is big (i.e.,

κr2 > (k + α)(p∗ − c)). The same is true for the new products of both generations.

In summary, remanufacturing and government subsidization for remanufacturing may not

always lead to a greener outcome. Thus, the government should exert caution when creating

regulatory policies to promote remanufacturing. To induce the socially optimal outcome, the

government needs to use a combined subsidy/tax scheme that is tailored to all three versions

of the product.

7 Extensions and Discussion

In this section, we extend our base model to three different settings to demonstrate the ro-

bustness of our main findings. In the first extension, we assume that customers may strictly

prefer a new product to a remanufactured one. In the second extension, we consider a model

where customers could return the used products for cash in period 2. In the third extension,

the environmental impact not only occurs at the production and disposal stages, but also at

the consumption stage.

7.1 Different Valuations for New and Remanufactured Products

Even though using recycled components should not affect product quality, some customers may

still prefer a new product to a remanufactured one. To model this preference, we assume that

the market consists of two segments of customers: (i) the regular customers who value new

and remanufactured products equally, and (ii) the special customers who value a new product

more than a remanufactured one (see, also, Atasu et al., 2008). We assume that the fraction of

regular customers is β, and that of special customers is 1−β (our base model represents the case

with β = 1). Although each customer does not know the realization of her type V , she knows

the segment she belongs to at the beginning of the sales horizon. For special customers, let ω

denote the disutility of consuming a remanufactured product relative to a new one. Thus the

willingness-to-pay of a type-V special customer in period 2 to obtain a new second-generation

product is (k + α)V , and that to obtain a remanufactured one is (k + α)V − ω. For simplicity,

when the utilities tie, we assume a special customer will purchase a new product instead of

joining the trade-in program for a remanufactured one.

25


To characterize the RE equilibrium in this model, we observe that there are two pricing

options for the firm in the first period: (i) charge the reservation price of the regular customers,

and (ii) charge the reservation price of the special customers. Because the regular customers

can receive higher expected utilities regardless of the firm’s second-period pricing strategy, in

the first period, the firm sells to regular customers only with option (i), and to both customer

segments with option (ii). Thus, with option (i), all repeat customers are regular ones, so

the firm adopts the same pricing strategy as that in the base model. Hence, the equilibrium

first-period production quantity is the solution to a newsvendor problem.

If the firm adopts pricing option (ii), the repeat customers will consist of customers from

both segments, with proportion β from the regular segment and proportion 1 − β from the

special segment. Using backward induction, we now characterize the optimal second-period

pricing strategy under option (ii). The firm announces three prices in period 2, (pn2 , p

rn,2, p

rr,2),

where pn2 is the price for new customers, pr

n,2 is the price for the repeat customers who purchase

a new second-generation product, and prr,2 is the price for the repeat customers who trade in the

old product for a remanufactured one. Note here we implicitly assume that if a repeat customer

purchases a new product, she has no incentive to return the old product. Under equilibrium, we

must have prr,2 ≤ pr

n,2. Thus, a type-V repeat customer from the regular segment will join the

trade-in program if and only if (k + α)V ≥ prr,2, and will not purchase anything in the second

period otherwise. A type-V repeat customer from the special segment will purchase a new

product if and only if (k+α)V − prn,2 ≥ max{(k+α)V − ω− pr

r,2, 0}, and will join the trade-in

program if and only if (k+ α)V − ω− prr,2 > max{(k+ α)V − pr

n,2, 0}. We can characterize the

optimal second-period pricing strategy under option (ii) in the following lemma.

Lemma 7 Assume that β ∈ (0, 1). There exists a threshold ω(β) ∈ [pr∗2 −(k+α)p∗, (1−k)c],

such that the following dichotomy holds.

(a) If ω < ω(β), the optimal second-period pricing policy is ((1+α)p∗, (1+α)p∗, pr∗r1,2), where

and pr∗r,2 = argmaxp≥0{β(p− (k+α)c)G( p

k+α) + (1 − β)(p− (k+α)c)G( p+ωk+α)}. Moreover,

(1 + α)p∗ > pr∗r,2 + ω. In the second period, the repeat regular customers will join the

trade-in program if and only if (k + α)V ≥ pr∗r,2; the repeat special customers will join the

trade-in program if and only if (k + α)V ≥ pr∗r,2 + ω.

(b) If ω ≥ ω(β), the optimal second-period pricing policy is ((1 + α)p∗, pr∗2 , (k + α)p∗). In the

second period, the repeat regular customers will join the trade-in program if V ≥ p∗; all

repeat special customers will not join the trade-in program.

Lemma 7(a) implies that if the disutility from the remanufactured products is relatively

small (i.e., ω < ω(β)), the firm will use a pricing strategy such that the repeat customers

from both segments will choose the trade-in program. However, as Lemma 7(b) shows, if the

disutility is large enough (i.e., ω ≥ ω(β)), the firm uses a pricing strategy to separate the

repeat customers: The regular customers will get remanufactured products through the trade-

26


in program, while the special customers will opt for the new products. Based on Lemma 7,

we can characterize the RE equilibrium with pricing option (ii). Since the second-period prices

do not depend on the realized demands (Xn2 , X

r2), the willingness-to-pay of the customers and,

thus, the equilibrium first-period price are independent of the first-period production quantity.

Therefore, the equilibrium production quantity in period 1 is also the solution to a newsvendor

problem under pricing option (ii).

The firm will select the pricing option that yields a higher expected profit. Thus, the RE

equilibrium outcome will be under one of the above two pricing options. Because both pricing

options result in equilibria with similar structure as in our base model, the qualitative results

from Sections 5-6 will continue to hold in this more general setting.

7.2 Old Product Buyback Program

In practice, many firms offer a product buyback program in addition to the trade-in program.

A customer may return the used products to the firm in period 2 without getting a replacement;

as a compensation, the firm pays a buyback price to the customer (see, also, Savaskan et al.,

2004). This offers another option to the customers and at the same time helps the firm collect

and recycle used products. We study how this buyback program may affect our main results.

Let rs denote the buyback price set by the firm in period 2. Clearly, only customers with

sufficiently low realized valuations (i.e., (1 − k)V ≤ rs) will take advantage of the buyback

program. Under equilibrium, rs must satisfy the following three conditions: The first con-

dition is rs ≤ (1 − k)c, i.e., rs should not exceed the per-unit cost saving of remanufactur-

ing. The second condition is Xr2G( rs

1−k ) ≤ Xn2 G(

pn2

1+α) for any realization of (Xn2 , X

r2), i.e.,

the used products recycled through the buyback program, Xr2G( rs

1−k ), should be bounded from

above by the demand from new customers in period 2, Xn2 G(

pn2

1+α). The third condition is

G( rs1−k ) + G(

pr2

k+α) ≤ 1 (or equivalently, rs1−k ≤ pr

2k+α), i.e., the sum of the customers who join

the buyback program, Xr2G( rs

1−k ), and the customers who join the trade-in program should

be bounded from above by the total number of repeat customers in the market, Xr2 . Let

(pn2 (Xn

2 , Xr2), pr

2(Xn2 , X

r2), rs(X

n2 , X

r2)) be the optimal second-period policy with the realized de-

mands (Xn2 , X

r2), where we use “¯” to denote the model with the buyback program. Moreover,

we define an auxiliary parameter r∗s := argmaxrs∈[0,(1−k)c]{((1−k)c− rs)G( rs

1−k )}. Then we can

characterize the optimal second-period pricing policy in the following lemma.

Lemma 8 For any realization of (Xn2 , X

r2), the following statements hold:

(a) pr2(X

n2 , X

r2) = pr

2(Xn2 , X

r2) ≡ (k + α)p∗.

(b) pn2 (Xn

2 , Xr2) and rs(X

n2 , X

r2) are determined by the market size ratio λ2 = Xn

2 /Xr2 .

(c) There exists a threshold λ =G(

r∗s

1−k)

G(p∗)> 0, such that:

27


(i) If λ2 < λ,

(pn2 (Xn

2 , Xr2), rs(X

n2 , X

r2))

= argmaxXr

2G( rs1−k

)=Xn2 G(

pn2

1+α)≤Xr

2G(c){Xn

2 (pn2 − (1 + α)c)G

(pn2

1 + α

)

+Xr2((1 − k)c− rs)G

(rs

1 − k

)}.

(ii) If λ2 ≥ λ, pn2 (Xn

2 , Xr2) = (1 + α)p∗ and rs(X

n2 , X

r2) = r∗

s .

Lemma 8 reveals that the optimal trade-in price, pr2(X

n2 , X

r2), is not affected by the buyback

program, but the optimal second-period price for new customers, pn2 (Xn

2 , Xr2) and the optimal

buyback price rs(Xn2 , X

r2) are dependent on the market size ratio λ2 = Xn

2 /Xr2 . Specifically, if

the realized market size ratio is low (i.e., λ2 ≤ λ) the optimal price for new products and the

optimal buyback price are dependent on the realized (Xn2 , X

r2). Otherwise, λ2 > λ, pn

2 (Xn2 , X

r2),

and rs(Xn2 , X

r2) are independent of the realized demands (Xn

2 , Xr2).

Recall that, under equilibrium, Xn2 = (X − Q1)

+ and Xr2 = X ∧ Q1. Thus, by Lemma

8, the willingness-to-pay of the customers in period 1 depends on the first-period production

quantity Q1. So, the equilibrium first-period production quantity cannot be characterized by

the solution to a newsvendor problem. The buyback program enhances the value of remanufac-

turing and improves the firm’s profit, because it further reduces the production cost in period

2 and intensifies the early purchase rewards effect of trade-in remanufacturing. Therefore, the

qualitative insights from Section 5 will remain valid in the model with the buyback program.

7.3 Environmental Impact at the Consumption Stage

In the base model, we assume that the environmental impact occurs at the production and

disposal stages, but not at the consumption stage. This assumption does not apply to product

categories whose consumption may have significant impact on the environment (e.g., printers

and photocopiers; see Agrawal et al., 2012). In this subsection, we relax this assumption and

consider products with non-negligible consumption impact.

We assume that the first-generation product has a life-long consumption impact of κu,1,

and the second-generation new product has a life-long consumption impact of κnu,2. If an old

first-generation product is used for remanufacturing, its consumption impact will reduce by

br2 ∈ [0, κu,1], because a remanufactured product is greener and has less consumption impact in

period 2. Thus, the total environmental impact under the RE equilibrium is given by:

I∗e = κ1Q

∗1 + κu,1E(X ∧Q∗

1) + δ((κn2 + κn

u,2)G(p∗)E(X −Q∗1)

+ + (κr2 − br2)G(p∗)E(X ∧Q∗

1))

= κ1Q∗1 + δ((κn

2 + κnu,2)G(p∗)E(X −Q∗

1)+ +

(κr

2 − br2 +κu,1

δG(p∗)

)G(p∗)E(X ∧Q∗

1)). (6)

Comparing (6) with (5), we can see that the model with consumption impact can be transformed

into the base model, with the (modified) unit impact of first-generation product κ1, the unit

28


impact of new second-generation product κn2 + κn

u,2, and the unit impact of remanufactured

second-generation product κr2 − br2 +

κu,1

δG(p∗). Therefore, the qualitative results in Section 6 will

continue to hold with the consumption environmental impact.

8 Conclusion

In this paper, we develop an analytical model to study the impact of trade-in remanufacturing

from both the firm’s and the society’s perspectives. Our emphasis is on strategic interactions

between the firm and customers under the trade-in program and on governmental intervention

through subsidization and taxation. From the firm’s perspective, we show that in addition to

production cost savings, trade-in remanufacturing is also an effective mechanism to mitigate

strategic customer behavior. This is because the trade-in program essentially offers early pur-

chase rewards to customers, which may significantly improve the firm’s profit by mitigating

strategic waiting. From the social perspective, trade-in remanufacturing benefits the firm, but

not the customers, and may harm the environment. This is because remanufacturing leads to a

higher production quantity, which may outweigh the environmental benefit of remanufacturing.

Moreover, providing government subsidies only to remanufactured products will also have an

negative impact on the environment. Thus, the government should carefully design regulatory

policies to promote the adoption of remanufacturing. To achieve the socially optimal outcome,

it is sufficient for the government to employ a simple incentive scheme that imposes either sub-

sidies or taxes to all three product versions: (i) the new first-generation product, (ii) the new

second-generation product, and (iii) the remanufactured second-generation product.

Our research can be extended in several directions. First, this paper considers a product

with only two generations. A natural extension is to study a multi-period model in which

the product can have three or more generations. Second, the market may consist of multiple

competing firms that offer partially substitutable products. It would be interesting to examine

how competition affects the adoption of remanufacturing and the associated trade-in program.

Finally, one may extend the current model to consider a supply chain setting. How trade-

in remanufacturing affects supply chain performance is also a promising direction for future

research.

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32


Online Supplement

Appendix A: Selected Proofs

1Proof of Lemma 1: Part (a). The continuous differentiability of Π2(·, ·|Xn2 , X

r2 ) (see Equation (1))

for any given (Xn2 , X

r2 ) follows immediately from that the distribution of V is continuous. Moreover,

for any given (Xn2 , X

r2 ), Π2(p

n2 , p

r2|Xn

2 , Xr2 ) = Xn

2 vn2 (pn

2 ) + Xr2v

r2(p

r2) is additively separable in pn

2 and

pr2, where vn

2 (pn2 ) := (pn

2 − (1 + α)c)G(pn2

1+α ) and vr2(p

r2) := (pr

2 − (k + α)c)G(pr2

k+α ). Thus, it suffices

to show that vn2 (·) is quasiconcave in pn

2 , and vr2(·) is quasiconcave in pr

2. Note that ∂pn2vn2 (pn

2 ) =

−(

pn2

1+α − c)g

(pn2

1+α

)+ G

(pn2

1+α

)and ∂pr

2vr2(p

r2) = −

(pr2

k+α − c)g

(pr2

k+α

)+ G

(pr2

k+α

). Because g(v)/G(v)

is continuously increasing in v, g(pn2

1+α )/G(pn2

1+α ) is continuously increasing in pn2 and g(

pr2

k+α )/G(pr2

k+α ) is

continuously increasing in pr2. Hence, ∂pn

2vn2 (pn

2 ) = 0 has a unique solution pn∗2 and ∂pr

2vr2(p

r2) = 0 has a

unique solution pr∗2 , where vn

2 (·) [vr2(·)] is strictly increasing on [0, pn∗

2 ) [[0, pr∗2 )] and strictly decreasing on

(pn∗2 ,+∞) [(pr∗

2 ,+∞)]. Therefore, for any given (Xn2 , X

r2 ), Xn

2 vn2 (·) is quasiconcave in pn

2 , and Xr2v

r2(·)

is quasiconcave in pr2. This completes the proof of Part (a).

Part (b). Given part (a), it suffices to show that, for any Xn2 ≥ 0, pn∗

2 = (1 + α)p∗, and, for any

Xr2 ≥ 0, pr∗

2 = (k + α)p∗, where p∗ = argmax[(p− c)G(p)]. By definition,

pn∗2 = argmaxpn

2 ≥0[(pn2 − (1 + α)c)G

(pn2

1 + α

)]

pn2 =(1+α)p

= (1 + α)argmaxp[(p− c)G(p)] = (1 + α)p∗,

and

pr∗2 = argmaxpr

2≥0[(pr2 − (k + α)c)G

(pr2

k + α

)]

pr2=(k+α)p

= (k + α)argmaxp[(p− c)G(p)] = (k + α)p∗.

Part (b) then follows immediately.

Proof of Theorem 1: Part (a). By Definition 1 and Lemma 1(b),

p∗1 = ξ∗

r = r∗1 = µ+ δE[(k + α)V − pr∗

2 ]+ − δE[(1 + α)V − pn∗2 ]+

= µ+ δE[(k + α)V − (k + α)p∗]+ − δE[(1 + α)V − (1 + α)p∗]+

= µ+ δ[(k + α) − (1 + α)]E(V − p∗)+

= µ− δ(1 − k)E(V − p∗)+.

To evaluate Πf (p∗1, Q1), we first compute π2(X

n2 , X

r2 ):

π2(Xn2 , X

r2 ) = Xn

2 ((1 + α)p∗ − (1 + α)c)G

((1 + α)p∗

1 + α

)+Xr

2 ((k + α)p∗ − (k + α)c)G

((k + α)p∗

k + α

)

= (1 + α)(X −Q1)+(p∗ − c)G(p∗) + (k + α)(X ∧Q1)(p

∗ − c)G(p∗)

= (1 + α)[X − (X ∧Q1)]R∗ + (k + α)(X ∧Q1)R

∗

= [(1 + α)X − (1 − k)(X ∧Q1)]R∗,

1Due to the page limit, we provide the selected proofs of our results in this Online Supplement. The proofs

of all other results are available from the authors upon request.

33


where the first equality follows from Lemma 1(b), the second from Xn2 = (X −Q1)

+ and Xr2 = X ∧Q1,

and the third from (X −Q1)+ = X − (X ∧Q1) and R∗ = (p∗ − c)G(p∗). Therefore, by (2),

Πf (p∗1, Q1) = p∗

1E(X ∧Q1) − cQ1 + sE(Q1 −X)+ + δE{π2(X − (X ∧Q1), X ∧Q1)}= (p∗

1 − s)E(X ∧Q1) − (c− s)Q1 + δR∗E[(1 + α)X − (1 − k)(X ∧Q1)]

= (p∗1 − δ(1 − k)R∗ − s)E(X ∧Q1) − (c− s)Q1 + δ(1 + α)R∗E(X)

= (m∗1 − s)E(X ∧Q1) − (c− s)Q1 + δ(1 + α)R∗E(X),

where the first inequality follows from (Q1 − X)+ = Q1 − (X ∧ Q1), and the last from the identity

m∗1 = p∗

1 − δ(1 − k)R∗. Therefore, Q∗1 is the solution to a newsvendor problem with marginal revenue

m∗1 − s, marginal cost c − s, and demand distribution F (·). Hence, Q∗

1 = F−1( c−sm∗

1−s ) if m∗1 > c, and

Q∗1 = 0 otherwise. This completes the proof of part (a).

Part (b). The value of Π∗f follows directly from Πf (p∗

1, Q1) = (m∗1 − s)E(X ∧ Q1) − (c − s)Q1 +

δ(1 + α)R∗E(X). To compute S∗c , we observe that if X ≤ Q∗

1, all X customers are repeat customers in

period 2. Thus, the total customer surplus is

X[µ−p∗1+δE[(k+α)V−(k+α)p∗]+] = X[µ−µ+δ(1−k)E(V−p∗)+δ(k+α)E(V−p∗)+] = Xδ(1+α)E(V−p∗)+.

On the other hand, if X > Q∗1, there are Q∗

1 repeat customers and X −Q∗1 new customers in period 2.

Thus, the total customer surplus in this case is

Q∗1δ(1 + α)E(V − p∗)+ + (X −Q∗

1)δE[(1 + α)V − (1 + α)p∗]+ = Xδ(1 + α)E(V − p∗)+.

Taking expectation over X, we have

S∗c = E{X≤Q∗

1}[Xδ(1 + α)E(V − p∗)+] + E{X>Q∗1}[Xδ(1 + α)E(V − p∗)+] = δ(1 + α)E(V − p∗)+E(X).

This completes the proof of Part (b).

Proof of Theorem 3: Part (a). pn2 (·, ·) ≡ pn

2 (·, ·) follows from Lemma 2. It remains to show

that pr∗2 ≥ (k + α)p∗. Note that

∂pr2vr2((k + α)p∗) = −

(p∗ − 1 + α

k + αc

)g(p∗) + G(p∗) ≥ 0,

where the inequality follows from g(v)/G(v) is increasing in v and −(p∗ − c)g(p∗) + G(p∗) = 0. Thus,

the quasiconcavity of vr2(·) implies that pr∗

2 ≥ (k + α)p∗. This concludes the proof of Part (a).

Part (b). Since vr2(·) ≤ vr

2(·) for all pr2 ≥ 0, vr

2(pr∗2 ) = maxpr

2≥0 vr2(p

r2) ≤ maxpr

2≥0 vr2(p

r2) =

vr2((k+α)p∗) = (k+α)R∗. Moreover, E((k+α)V −pr∗

2 )+ ≤ E((k+α)V −(k+α)p∗)+ = (k+α)E(V −p∗)+,

where the inequality follows from pr∗2 ≥ (k + α)p∗. Thus,

m∗1 = µ+ δ[(pr∗

2 − (1 + α)c)G

(pr∗2

k + α

)+ E((k + α)V − pr∗

2 )+ − (1 + α)(R∗ + E(V − p∗)+)]

≤ µ+ δ[(k + α)(R∗ + E(V − p∗)+) − (1 + α)(R∗ + E(V − p∗)+)]

= m∗1.

and

p∗1 = µ+δ[E((k+α)V − pr∗

2 )+ −(1+α)E(V −p∗)+] ≤ µ+δ[(k+α)E(V −p∗)+ −(1+α)E(V −p∗)+] = p∗1.

34


Hence, Q∗1 = F−1( c−s

m∗1−s ) ≤ F−1( c−s

m∗1−s ) = Q∗

1. Thus, Q∗1 > 0 implies Q∗

1 > 0. This completes the proof

of Part (b).

Part (c). Note that, since m∗1 ≤ m∗

1, (m∗1 − s)(X ∧Q1) − (c− s)Q1 ≤ (m∗

1 − s)(X ∧Q1) − (c− s)Q1

for each Q1 ≥ 0. Thus,

Π∗f = max

Q1≥0{(m∗

1 − s)(X ∧Q1) − (c− s)Q1 + δ(1 + α)R∗E(X)}

≤ maxQ1≥0

{(m∗1 − s)(X ∧Q1) − (c− s)Q1 + δ(1 + α)R∗E(X)}

= Π∗f .

The equality S∗c = S∗

c follows directly from Theorem 1(b) and Theorem 2(b), which completes the proof

of part (c).

Proof of Lemma 3: Part (a). The second period pricing policy pu2 (Xn

2 , Xr2 ) is the maximizer of

the second-period profit function, i.e., pu2 (Xn

2 , Xr2 ) = argmaxpu

2 ≥0Πu2 (pu

2 |Xn2 , X

r2 ), where

Πu2 (pu

2 |Xn2 , X

r2 ) = Xn

2 (pu2 − (1 + α)c)G

(pu2

1 + α

)+Xr

2 (pu2 − (1 + α)c)G

(pu2

k + α

)

= Xr2

[λ2(p

u2 − (1 + α)c)G

(pu2

1 + α

)+ (pu

2 − (1 + α)c)G

(pu2

k + α

)]

=: Xr2v

u2 (pu

2 |λ2).

Thus, pu2 (Xn

2 , Xr2 ) is the optimizer of vu

2 (pu2 |λ2) and determined by λ2. Note that since Π2(p

n2 , p

r2|Xn

2 , Xr2 )

is quasiconcave, so Πu2 (pu

2 |Xn2 , X

r2 ) = Π2(p

u2 , p

u2 |Xn

2 , Xr2 ) and vu

2 (pu2 |λ2) are also quasiconcave in pu

2 .

Observe that

∂pu2vu2 (pu

2 |λ2) = λ2

[G

(pu2

1 + α

)−

(pu2

1 + α− c

)g

(pu2

1 + α

)]+G

(pu2

k + α

)−

(pu2

k + α− 1 + α

k + αc

)g

(pu2

k + α

).

Since g(v)/G(v) is increasing in v, ∂pu2vu2 (pu

2 |λ2) < 0 if pu2 > (1+α)p∗, and ∂pu

2vu2 (pu

2 |λ2) > 0 if pu2 < pr∗

2 .

Thus, pu2 (Xn

2 , Xr2 ) ∈ [pr∗

2 , (1 + α)p∗]. When pu2 ∈ [pr∗

2 , (1 + α)p∗], G(pu2

1+α ) − (pu2

1+α − c)g(pu2

1+α ) ≥ 0 and

G(pu2

k+α ) − (pu2

k+α − 1+αk+αc)g(

pu2

k+α ) ≤ 0. Thus, ∂pu2vu2 (pu

2 |λ2) is increasing in λ2 if pu2 ∈ [pr∗

2 , (1 + α)p∗], i.e.,

vu2 (pu

2 |λ2) is supermodular in (pu2 , λ2) on the lattice [pr∗

2 , (1 + α)p∗] × [0,+∞). Therefore, pu2 (Xn

2 , Xr2 )

is continuously increasing in λ2. If λ2 = 0, ∂pu2vu2 (pu

2 |λ2) = 0 has a unique solution pu2 (Xn

2 , Xr2 ) = pr∗

2 .

Moreover, as λ2 → +∞, pu2 (Xn

2 , Xr2 ) converges to the maximizer of λ2(p

u2 − (1 + α)c)G(

pu2

1+α ), which

equals (1+α)p∗. Thus, limλ2→+∞ pu2 (Xn

2 , Xr2 ) = pr∗

2 = (1+α)p∗. This completes the proof of Part (a).

Part (b). Given Xr2 , pu

2 (·, Xr2 ) is continuously increasing in λ2 = Xn

2 /Xr2 and, thus, Xn

2 . Analo-

gously, given Xn2 , pu

2 (Xn2 , ·) is continuously increasing in λ2 = Xn

2 /Xr2 and, thus, continuously decreasing

in Xr2 . This concludes the proof of Part (b).

Proof of Theorem 4: Part (a). By the definition of RE equilibrium, Qu∗1 = argmaxQ1≥0Π

uf (Q1).

Note that limQ1→+∞ Πuf (Q1) = C− limQ1→+∞(c− s)Q1 = −∞ for some constant C. Thus, Qu∗

1 < +∞.

Therefore, by the definition of RE equilibrium, pu∗1 = pu

1 (Qu∗1 ). The second period pricing policy follows

immediately from Lemma 3 and the standard backward induction argument. This concludes the proof

of Part (a).

Part (b). Clearly, the expected profit of the firm is given by Πu∗f = maxQ1≥0 Πu

f (Q1) = Πuf (Qu∗

1 ).

To compute Su∗c , we observe that if X ≤ Qu∗

1 , all X customers are repeat customers in period 2. Thus,

the second-period price is pr∗2 and the total customer surplus is X[µ− pu∗

1 + δE((k + α)V − pr∗2 )+]. On

35


the other hand, if X > Qu∗1 , there are Qu∗

1 repeat customers and X − Qu∗1 new customers in period 2.

Thus, the total customer surplus in this case is

Qu∗1 δ[µ− pu∗

1 + EV ((k + α)V − pu2 (Xn

2 , Xr2 ))+] + (X −Qu∗

1 )δEV ((1 + α)V − pu2 (Xn

2 , Xr2 ))+.

Taking expectation over X, we have

Su∗c = E{X≤Qu∗

1 }X[µ− pu∗1 + δE((k + α)V − pr∗

2 )+]

+E{X>Qu∗1 }{Qu∗

1 δ[µ− pu∗1 + EV ((k + α)V − pu

2 (Xn∗2 , Xr∗

2 ))+] + (X −Qu∗1 )δEV ((1 + α)V − pu

2 (Xn∗2 , Xr∗

2 ))+},

where Xn∗2 = (X −Qu∗

1 )+ and Xr∗2 = X ∧Qu∗

1 . This completes the proof of Part (b).

Proof of Theorem 5: Part (a). By Lemma 3, pu2 (·, ·) is increasing in λ2 with pu

2 (Xn2 , X

r2 ) = pr

2(Xn2 , X

r2 )

if λ2 = 0 and pu2 (Xn

2 , Xr2 ) → pn

2 (Xn2 , X

r2 ) as λ2 → +∞. Thus, pr

2(Xn2 , X

r2 ) ≤ pu

2 (Xn2 , X

r2 ) ≤ pn

2 (Xn2 , X

r2 ).

Part (b). Note that

pu∗1 − p∗

1 = δ[E((k + α)V − pu2 (Xn∗

2 , Xr∗2 ))+ − E((k + α)V − pr∗

2 )+]

−δ[E((1 + α)V − pu2 (Xn∗

2 , Xr∗2 ))+ − E((1 + α)V − pn∗

2 )+]

< 0,

where the inequality follows from pr∗2 = pr

2(Xn2 , X

r2 ) ≤ pu

2 (Xn2 , X

r2 ) ≤ pn

2 (Xn2 , X

r2 ) = pn∗

2 . Similarly,

mu1 (Q1) − m∗

1 = δ[Ur(Q1) − Un(Q1)],

where

Ur(Q1) := E[(pu2 (Xn

2 , Xr2 ) − (1 + α)c)G

(pu2 (Xn

2 , Xr2 )

k + α

)] + E((k + α)V − pu

2 (Xn2 , X

r2 ))+

−[(pr∗2 − (1 + α)c)G

(pr∗2

k + α

)+ E((k + α)V − pr∗

2 )+]

and

Un(Q1) := E[(pu2 (Xn

2 , Xr2 ) − (1 + α)c)G

(pu2 (Xn

2 , Xr2 )

1 + α

)] + E((1 + α)V − pu

2 (Xn2 , X

r2 ))+

−[(pn∗2 − (1 + α)c)G

(pn∗2

1 + α

)+ E((1 + α)V − pn∗

2 )+].

Let un(p) := (p−(1+α)c)G( p1+α )+E((1+α)V −p)+ and ur(p) := (p−(1+α)c)G( p

k+α )+E((k+α)V −p)+.

Note that un(p) = (1 + α)E(V − c)1{(1+α)V ≥p} and ur(p) = E((k + α)V − (1 + α)c)1{(k+α)V ≥p}, so

un(·) and ur(·) are continuously decreasing in p if p ≥ (1 + α)c. Therefore, by (1 + α)c < pr∗2 =

pr2(X

n2 , X

r2 ) ≤ pu

2 (Xn2 , X

r2 ) ≤ pn

2 (Xn2 , X

r2 ) = pn∗

2 , Ur(Q1) = E[ur(pu2 (Xn

2 , Xr2 )) − ur(p

r∗2 )] ≤ 0 and

Un(Q1) = E[un(pu2 (Xn

2 , Xr2 )) − un(pn∗

2 )] ≥ 0 and one of the inequalities must be strict. Therefore,

mu1 (Q1) − m∗

1 < 0 for all Q1 ≥ 0.

Now we show, by contradiction, Qu∗1 > 0 implies that Q∗

1 > 0. If Q∗1 = 0, mu

1 (Q1) < m∗1 ≤ c. Thus,

(mu1 (Q1) − s)(X ∧Q1) − (c− s)Q1 < 0 for all Q1 > 0. Moreover, since pu

2 (Xn2 , X

r2 ) is increasing in Xn

2

and pu2 (Xn

2 , Xr2 ) ≤ (1+α)p∗, E[(pu

2 (Xn2 , X

r2 )− (1+α)c)G

(pu2 (Xn

2 ,Xr2 )

1+α

)X] is decreasing in Q1. Therefore,

Πuf (0) > Πu

f (Q1) for all Q1 > 0. Hence, Qu∗1 = 0. Thus, Qu∗

1 > 0 implies that Q∗1 > 0.

Now we show if mu1 (Q1) is decreasing in Q1, Q

u∗1 ≤ Q∗

1. Observe that

Πuf (Q1)−Πf (p∗

1, Q1) = (mu1 (Q1)−m∗

1)(X∧Q1)+δE[(pu2 (Xn

2 , Xr2 )−(1+α)c)G

(pu2 (Xn

2 , Xr2 )

1 + α

)]−δ(1+α)R∗E(X).

36


Let Π(Q1, 1) = Πf (p∗1, Q1) and Π(Q1, 0) = Πu

f (Q1). Then,

Π(Q1, 1) − Π(Q1, 0) = Πf (p∗1, Q1) − Πu

f (Q1)

= (m∗1 −mu

1 (Q1))(X ∧Q1) + δEX[(1 + α)R∗ − (pu2 (Xn

2 , Xr2 ) − (1 + α)c)G

(pu2 (Xn

2 , Xr2 )

1 + α

)]

Note that for any realization of X, pu2 (Xn

2 , Xr2 ) and, thus, (pu

2 (Xn2 , X

r2 ) − (1 + α)c)G(

pu2 (Xn

2 ,Xr2 )

1+α ) is

decreasing in Q1. Therefore, if mu1 (Q1) is decreasing in Q1, Π(Q1, 1) − Π(Q1, 0) is increasing in Q1.

Hence, Π(·, ·) is supermodular on the lattice [0,+∞) × {0, 1}. Hence, Qu∗1 = argmaxQ1≥0Π

uf (Q1) ≤

argmaxQ1≥0Πf (p∗1, Q1) = Q∗

1. This concludes the proof of Part (b).

Part (c). Because mu1 (Q1) − m∗

1 < 0 and E[(pu2 (Xn

2 , Xr2 ) − (1 + α)c)G

(pu2 (Xn

2 ,Xr2 )

1+α

)] ≤ (1 + α)R∗,

Πuf (Q1)−Πf (p∗

1, Q1) = (mu1 (Q1)−m∗

1)(X∧Q1)+δE[(pu2 (Xn

2 , Xr2 )−(1+α)c)G

(pu2 (Xn

2 , Xr2 )

1 + α

)]−δ(1+α)R∗E(X) ≤ 0

for all Q1 ≥ 0. Therefore, Πu∗f = maxQ1≥0 Πu∗

f (Q1) ≤ maxQ1≥0 Πf (p∗1, Q1) = Π∗

f .

Proof of Theorem 6: Let ϕ(Q1) := κ1Q1+δκn2 G(p∗)E(X−Q1)

+. Since ϕ′(Q1) = κ1−δκn2 G(p∗)F (Q1) >

0 for all Q1 ≥ 0, ϕ(Q∗1) ≥ ϕ(Q∗

1) and ϕ(Q∗1) > ϕ(Q∗

1) if Q∗1 > Q∗

1. Hence,

I∗e − I∗

e = ϕ(Q∗1) − ϕ(Q∗

1) + δκn2 [θG(p∗)E(X ∧Q∗

1) − G

(pr∗2

k + α

)E(X ∧ Q∗

1)]

≥ δκn2 [θG(p∗)E(X ∧Q∗

1) − G

(pr∗2

k + α

)E(X ∧ Q∗

1)],

where the inequality is strict if Q∗1 > Q∗

1. Define θ := min{θ : I∗e − I∗

e ≥ 0}. By Theorem 3(a),

pr∗2 > (k + α)p∗ = pr∗

2 . Thus, G(pr∗2

k+α )/G(p∗) < 1. If θ ≥ G(pr∗2

k+α )/G(p∗), δθG(p∗)E(X ∧ Q∗1) ≥

δG(pr∗2

k+α )E(X ∧ Q∗1). Therefore,

I∗e − I∗

e ≥ δκn2 [θG(p∗)E(X ∧Q∗

1) − G

(pr∗2

k + α

)E(X ∧ Q∗

1)] ≥ 0,

where the inequalities are strict if Q∗1 > Q∗

1. Thus, θ ≤ G(pr∗2

k+α )/G(p∗) < 1 and, as long as θ ≤ θ,

I∗e ≥ I∗

e , where the inequality is strict if Q∗1 > Q∗

1.

Proof of Theorem 8: Part (a). Let vr2(p

r2|sr

g,2) := (pr2 + sr

g,2 − (k + α)c)G(pr2

k+α ). The same ar-

gument from the proof of Lemma 1 yields that vr2(·|sr

g,2) is quasiconcave in pr2 for all sr

g,2. Note that

pr∗2 is the maximizer of vr

2(·|srg,2). Since ∂pr

2∂sr

g,2vr2(p

r2|sr

g,2) = − 1k+αg(

pr2

k+α ) ≤ 0, vr2(·|·) is supermodular

in (pr2, s

rg,2). Hence, pr∗

2 is continuously decreasing in srg,2. Thus, pr∗

2 ≤ (k + α)p∗. Note that, since

−(p∗ − c)g(p∗) − G(p∗) = 0,

∂pr2vr2((k + α)p∗ − sr

g,2|srg,2) = G

((k + α)p∗ − sr

g,2

k + α

)− (p∗ − c)g(p∗) > 0.

Therefore, pr∗2 ≥ (k + α)p∗ − sr

g,2, which completes the proof of Part (a).

Part (b). Clearly, vr2(p

r∗2 |sr

g,2) is continuously increasing in srg,2. Since pr∗

2 is continuously decreasing

in srg,2, E((k + α)V − pr∗

2 )+ is continuously increasing in srg,2. Thus, m∗

1 = µ + δ[vr2(p

r∗2 |sr

g,2) + E((k +

α)V − pr∗2 )+ − (1 + α)(R∗ + E(V − p∗)+)] and p∗

1 = µ+ δ[E((k + α)V − pr∗2 )+ − (1 + α)E(V − p∗)+] are

continuously increasing in srg,2. Thus, Q∗

1 = F ( c−sm∗

1−s ) is continuously increasing in m∗1 and, hence, sr

g,1.

Therefore, m∗1 ≥ m∗

1, p∗1 ≥ p∗

1, and Q∗1 ≥ Q∗

1 then follow immediately.

37


Part (c). Since m∗1 is continuously increasing in sr

g,2, Πf (p∗1, Q1) = (m∗

1 −s)E(X ∧Q1)− (c−s)Q1 +

δ(1 + α)R∗E(X) is continuously increasing in srg,2. Hence, Π∗

f = maxQ1≥0 Πf (p∗1, Q1) is continuously

increasing in srg,2.

By the proof of Theorem 6, ϕ(Q1) = κ1Q1 + δκn2 G(p∗)E(X −Q1)

+ is continuously increasing in Q1,

so ϕ(Q∗1) is continuously increasing in Q∗

1 and, thus, srg,2. Since pr∗

2 is continuously decreasing in srg,2 and

Q∗2 is continuously increasing in sr

g,2, δκr2G(

pr∗2

k+α )E(X ∧ Q∗1) is continuously increasing in sr

g,2. Therefore,

I∗2 = ϕ(Q∗

1) + δκr2G(

pr∗2

k+α )E(X ∧ Q∗1) is continuously increasing in sr

g,2. Hence,

I∗e − I∗

e = ϕ(Q∗1) − ϕ(Q∗

1) + δκn2 [θG

(pr∗2

k + α

)E(X ∧ Q∗

1) − G

(pr∗2

k + α

)E(X ∧ Q∗

1)]

is continuously increasing in srg,2 and θ. Thus, θ := min{θ : I∗

e − I∗e ≥ 0} is continuously decreasing in

srg,2. The inequalities Π∗

f ≥ Π∗f , I∗

e ≥ I∗e , and θ ≤ θ then follow immediately. Finally, S∗

c = S∗c follows

from Theorem 1(b) and Theorem 7(b), which concludes the proof of Part (c).

Proof of Theorem 10: Since Part (b) implies Part (a), we only show Parts (b) and (c).

Part (b). If sn∗g,2 is the solution to pn∗

s,2 = argmaxpn2 ≥0{(pn

2 + sng,2 − (1 + α)c)G(

pn2

1+α )} and sr∗g,2

is the solution to pr∗s,2 = argmaxpr

2≥0{(pr2 + sr

g,2 − (k + α)c)G(pr2

k+α )}, by Lemma 5, the firm would

adopt the second period pricing strategy (pns,2(·, ·), pr

s,2(·, ·)). Now, we show that sn∗g,2 and sr∗

g,2 exist. Let

vn2 (pn

2 |sng,2) := (pn

2 +sng,2−(1+α)c−κn

2 )G(pn2

1+α ), which is quasiconcave in pn2 for any sn

g,2. Observe that for

any (sng,2, s

rg,2), v

n2 (·|sn

g,2) and vr2(·|sn

g,2) have a unique maximizer and characterized by ∂pn2vn2 (pn

2 |sng,2) = 0

and ∂pr2vr2(p

r2|sn

g,2) = 0, respectively. Moreover,

∂pn2vn2 (pn∗

s,2|sng,2) = G

(pn∗

s,2

1 + α

)− pn∗

s,2 + sng,2 − (1 + α)c

1 + αg

(pn∗

s,2

1 + α

),

and

∂pr2vr2(p

r∗s,2|sn

g,2) = G

(pr∗

s,2

k + α

)− pr∗

s,2 + srg,2 − (k + α)c

k + αg

(pr∗

s,2

k + α

).

Therefore, there exists a unique (sn∗g,2, s

r∗g,2) such that ∂pn

2vn2 (pn∗

s,2|sn∗g,2) = 0 and ∂pr

2vr2(p

r∗s,2|sr∗

g,2) = 0, i.e.,

pn∗s,2 = argmaxpn

2 ≥0{(pn2 +sn∗

g,2−(1+α)c)G(

pn2

1+α

)} and pr∗

s,2 = argmaxpr2≥0{(pr

2+sr∗g,2−(k+α)c)G

(pr2

k+α

)}.

Given the subsidy/tax scheme (sg,1, sn∗g,2, s

r∗g,2), as shown above, the firm adopts the same pricing

policy as the social welfare maximizing one: (pns,2(·, ·), pr

s,2(·, ·)). Hence, the first-period price should

also be the same as the one which is socially optimal and characterized by Theorem 9(a): p∗s,1 =

µ+ δ[E((k + α)V − pr∗s,2)

+ − E((1 + α)V − pn∗s,2)

+]. Thus, the expected profit of the firm in period 1 is

Πsf (p∗

s,1, Q1) = (p∗s,1 + sg,1 − s)E(X ∧Q1) − (c− s)Q1 + δE[(X −X ∧Q1)(p

n∗s,2 + sn∗

g,2 − (1 + α)c)G

(pn∗

s,2

1 + α

)

+(X ∧Q1)(pr∗s,2 + sr∗

g,2 − (k + α)c)G

(pr∗

s,2

k + α

)]

= (ms1(sg,1) − s)E(X ∧Q1) − (c− s)Q1 + δ(pn∗

s,2 + sn∗g,2 − (1 + α)c)G

(pn∗

s,2

1 + α

)E(X),

where ms1(sg,1) = sg,1 + µ + δ[(pr∗

s,2 + sr∗g,2 − (k + α)c)G(

pr∗s,2

k+α ) + E((k + α)V − pr∗s,2)

+ − (pn∗s,2 + sn∗

g,2 −(1 + α)c)G(

pn∗s,2

1+α ) − E((1 + α)V − pn∗s,2)

+]. Thus, Πsf (p∗

s,1, Q1) has a unique optimizer F−1( c−sms

1(sg,1)−s ) if

ms1(sg,1) > c and 0 otherwise. Moreover, as shown in Theorem 9, Q∗

s,1 = F−1( c+κ1−sm∗

s,1−s ) if m∗s,1 > c+ κ1

and Q∗s,1 = 0 otherwise. Therefore, if s∗

g,1 is the unique solution to c−sms

1(sg,1)−s = c+κ1−sm∗

s,1−s , the optimal

production quantity is Q∗s,1, which is the socially optimal first-period production quantity. Since pr∗

s,2 =

38


(k + α)c+ κr2 and pn∗

s,2 = (1 + α)c+ κn2 ,

ms1(sg,1) = sg,1 + µ+ δ[E((k + α)V − (k + α)c− κr

2)+ − E((1 + α)V − (1 + α)c− κn

2 )+]

−δ[(κn2 + sn∗

g,2)G

((1 + α)c+ κn

2

1 + α

)− (κr

2 + sr∗g,2)G

((k + α)c+ κr

2

k + α

)]

= sg,1 +m∗s,1 − δ[(κn

2 + sn∗g,2)G

((1 + α)c+ κn

2

1 + α

)− (κr

2 + sr∗g,2)G

((k + α)c+ κr

2

k + α

)].

This concludes the proof of Part (b).

Part (c). Let vrs,2(p

r2|sr

g,2) := (pr2 + sr

g,2 − (k + α)c)G(pr2

k+α ). Thus, vrs,2(·|sr

g,2) is quasiconcave in pr2

and ∂pr2vr

s,2(pr2|sr

g,2) = 0 gives the unique optimal second-period price. We know

∂pr2vr

s,2((k + α)c+ κr2|sr

g,2) = −κr2 + sr

g,2

k + αg

((k + α)c+ κr

2

k + α

)+ G

((k + α)c+ κr

2

k + α

)= 0

implies that

(κr2 + sr

g,2)g((k+α)c+κr

2

k+α )

(k + α)G((k+α)c+κr

2

k+α )= 1.

In particular, sr∗g,2 > 0 if and only if ψ(κr

2) < 1, sr∗g,2 = 0 if and only if ψ(κr

2) = 1, and sr∗g,2 < 0 if and only

if ψ(κr2) > 1, where

ψ(κr2) :=

κr2g(

(k+α)c+κr2

k+α )

(k + α)G((k+α)c+κr

2

k+α ).

Because g(v)/G(v) is continuously increasing, ψ(·) is strictly continuously increasing. Recall that

ψ((k + α)(p∗ − c)) = (p∗ − c)g(p∗)

G(p∗)= 1,

by the fact that p∗ = argmaxp≥0[(p − c)G(p)]. Therefore, sr∗g,2 > 0 if and only if κr

2 < (k + α)(p∗ − c),

sr∗g,2 = 0 if and only if κr

2 = (k + α)(p∗ − c), and sr∗g,2 < 0 if and only if κr

2 > (k + α)(p∗ − c). The

characterization of the sign of sn∗g,2 follows from the same argument as that of sr∗

g,2, so we omit its proof

for brevity.

Finally, we characterize the sign of s∗g,1. Since ∂pn

2vn2 (pn∗

s,2|sn∗g,2) = 0 and ∂pr

2vr2(p

r∗s,2|sr∗

g,2) = 0,

κn2 + sn∗

s,2 = (1 + α)G(

(1+α)c+κn2

1+α )

g((1+α)c+κn

2

1+α )

and

κr2 + sr∗

s,2 = (k + α)G(

(k+α)c+κr2

k+α )

g((k+α)c+κr

2

k+α ).

Plug the above identities into the expression of ms1(sg,1) and we have

ms1(sg,1) = sg,1 +m∗

s,1 − δ[(κn2 + sn∗

g,2)G

((1 + α)c+ κn

2

1 + α

)− (κr

2 + sr∗g,2)G

((k + α)c+ κr

2

k + α

)]

= sg,1 +m∗s,1 − δ[(1 + α)

G2((1+α)c+κn

2

1+α )

g((1+α)c+κn

2

1+α )− (k + α)

G2((k+α)c+κr

2

k+α )

g((k+α)c+κr

2

k+α )]

= sg,1 +m∗s,1 −A,

andc− s

s∗g,1 +m∗

s,1 −A− s=c+ κ1 − s

m∗s,1 − s

. (7)

Clearly, s∗g,1 = 0 if and only if κ1 = A

m∗s,1−A−s (c−s). The left-hand side of (7) is strictly decreasing in sg,1

while the right-hand side is strictly increasing in κ1. Thus, s∗g,1 > 0, if and only if κ1 <

Am∗

s,1−A−s (c− s),

and s∗g,1 < 0, if and only if κ1 >

Am∗

s,1−A−s (c− s). This concludes the proof of Part (c).

39


Appendix B: Equilibrium Analysis with Myopic Customers

In this section, we consider the model in which all customers are nonstrategic/myopic, i.e., customers are

non-anticipative, so they do not take into account the second-period purchasing options when making

the purchasing decision in the first period. Therefore, in period 1, they do not form beliefs about product

availability and future prices. The reservation price and, thus, the equilibrium first-period price are the

expected valuation pm∗1 = rm∗

1 = µ = E(V ), where the superscript “m” refers to “myopic”. For the

second-period pricing strategy, we consider two cases: (a) the firm adopts trade-in remanufacturing, and

(b) the firm does not adopt remanufacturing and offers a single price to all customers in the second

period. In case (a), the optimal second-period pricing strategy is the same as that characterized in

Lemma 2; in case (b), the optimal second-period pricing strategy is the same as that characterized in

Lemma 3. Specifically, in case (a), we use pmr2 (·, ·) ≡ pr

2(·, ·) to denote the optimal trade-in price, and

pmn2 (·, ·) ≡ pn

2 (·, ·) to denote the optimal price for new customers; in case (b), we use pmu2 (·, ·) ≡ pu

2 (·, ·)to denote the optimal price for all customers in the model without trade-in remanufacturing. As in the

models with strategic customers, the profit functions for the firm in period 1 can be written as

Case (a): Πmf (Q1) = (mm∗

1 − s)E(X ∧Q1) − (c− s)Q1 + δ(1 + α)R∗E(X),

Case (b): Πmuf (Q1) = (mmu

1 (Q1) − s)E(X ∧Q1) − (c− s)Q1 + δE[(pm2 (Xn

2 , Xr2 ) − (1 + α)c)G

(pm2 (Xn

2 , Xr2 )

1 + α

)X],

where mm∗1 = µ+ δ(pr∗

2 − (1 + α)c)G

(pr∗2

k + α

)− δ(1 + α)R∗,

and mmu1 (Q1) = µ+ δ{E[(pu

2 (Xn2 , X

r2 ) − (1 + α)c)G

(pu2 (Xn

2 , Xr2 )

k + α

)]

−E[(pu2 (Xn

2 , Xr2 ) − (1 + α)c)G

(pu2 (Xn

2 , Xr2 )

1 + α

)]},

with Xn2 = (X − Q1)

+ and Xr2 = X ∧ Q1. Maximizing Πm

f (Q1) and Πmuf (Q1) over the region Q1 ≥ 0,

we can obtain the production quantity under the RE equilibrium Qm∗1 and Qmu∗

1 in case (a) and case

(b), respectively. The equilibrium profits are, respectively, given by Πm∗f := Πm

f (Qm∗1 ) and Πmu∗

f =

Πmuf (Qmu∗

1 ).

40

trade-in remanufacturing, strategic customer behavior, and

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