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Is Leasing Green? The Environmental Impact ofProduct Recovery, Remarketing and Disposal under

Leasing and SellingVishal Agrawal

College of Management, Georgia Institute of Technology, Atlanta, GA 30332, [email protected]

Mark FergusonCollege of Management, Georgia Institute of Technology, Atlanta, GA 30332, [email protected]

Valerie ThomasSchool of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332,

[email protected]

L. Beril ToktayCollege of Management, Georgia Institute of Technology, Atlanta, GA 30332, [email protected]

The growth in environmental regulations governing the landfilling of used products and in the environmental

awareness of consumers poses two challenges for manufacturers as they go to market with their products.

First, landfilling fees and bans increase the cost of disposing of used products that they maintain ownership

of, such as products coming off operating leases or previously sold products recovered via trade-in programs.

Second, it is becoming important for firms to have a better understanding of the total environmental impact

of their decisions. In this paper, we characterize the optimal integrated pricing, recovery, remarketing and

disposal strategy of a monopolist manufacturer under leasing and selling. Our contributions are three-fold.

First, we provide insights to manufacturers about how production cost, disposal cost and durability should

shape their decisions. Second, while the question of whether a firm should lease or sell its product has

been studied, little is known about which strategy dominates when disposal is costly and when a firm can

recover previously sold products through trade-ins. We find that leasing performs at least as well as selling

with respect to profitability. Third, to investigate claims in the industrial ecology literature that leasing is

environmentally preferable to selling, we characterize under what conditions each strategy dominates from an

environmental impact perspective. We find that while leasing dominates in the presence of high production

cost, environmental impact depends on disposal cost when production cost is low: For low disposal cost,

selling is environmentally superior for products such as computers and carpets that have significantly higher

environmental impact during production and disposal, while for high disposal cost, selling is better for

products such as automobiles that have high impact during use. These results are driven by differences in

production volume and in average duration of use, dimensions not considered by proponents of leasing.

Key words : Closed-Loop Supply Chains, Durable Goods, Environmental Impact, Post-Use Strategies

1

Agrawal et al.: Is Leasing Green?2 Article submitted to Management Science; manuscript no.

1. Introduction

The growth in industrial production and increased consumption by a growing population are

increasing the strain on the environment. Many statistics point to the need to find solutions for

reducing waste. For example, in 2006, municipal solid waste amounted to more than 251 million

tons (U.S. EPA 2007). To reduce waste, the U.S. Environmental Protection Agency recommends

adopting a reduce-reuse-recycle hierarchy and resorting to combustion or landfilling only as a last

resort (U.S. EPA 2008b). Despite this recommendation, 67.5% of the municipal waste went directly

to land-fills or incineration facilities in 2006 (U.S. EPA 2007) as only a subset of the waste stream

can be recycled for profit (e.g. equipment with high metal content or containing precious metals).

For the rest, in response to increasing community concerns and lack of landfill space in some regions,

the trend today is increasing landfill fees or landfilling bans that require costly recycling. In this

climate, it is important to take end-of-life economics into consideration when choosing how to bring

a product to market, in particular, whether to lease or sell the product. For example, an operating

lease ensures that the manufacturer maintains ownership of the product, and therefore is either

forced to incur the end-of-life cost, or to remarket the product1. With selling, the manufacturer

typically avoids end-of-life costs, but has to offer monetary incentives to recover the product if it

wants to benefit from remarketing.

Leasing versus selling is an interesting question from an environmental impact perspective. Leas-

ing durable goods rather than selling them has been proposed as an environmentally beneficial

business model that promotes reuse (Hawken et al. 1999, Lifset and Lindhqvist 2000). This is moti-

vated by the fact that under an operating lease, the firm must maintain ownership of the product

(this is a requirement of the Financial Accounting Standards Board). The argument is that if a

firm retains ownership of the product, it can efficiently remarket the used product at the end of the

lease duration, thus extending the effective life of the product, diverting waste from landfills and

1 In this paper, the term “remarketing” refers to offering the used product on the market.

Agrawal et al.: Is Leasing Green?Article submitted to Management Science; manuscript no. 3

reducing the environmental impact related to manufacturing new products (Fishbein et al. 2000,

Mont 2002, Robert et al. 2002). Since $199 billion out of $722 billion spent by U.S. businesses on

assets in 2004, or 28%, was through leasing (ELA 2005), the environmental benefits of leasing can

be significant if this proposition holds. This point of view has gained traction in practice. Interface

Inc., a carpet manufacturer, introduced the Evergreen Lease with the express purpose of reducing

the environmental impact of its products (Olivia and Quinn 2003). The state of Minnesota recom-

mends leasing as an environmentally preferred option (Minnesota Pollution Control Agency 2006)

and the New York City Government states on its web page that “most leasing companies strip

equipment for reusable parts, or refurbish the used item so that it can be re-leased, or donated to

a nonprofit organization” (New York City Government 2007).

There are others, however, who argue that the environmental benefits of leasing rather than

selling are far from clear (Ruth 1998), and some even argue it is a “fallacy” (Lawn 2001). The

reason is the direct control that the firm exerts on used equipment coupled with lease durations

that are typically shorter than average use durations of sold products. Remarketing a used product

may benefit the firm by reaching consumers who cannot afford the new product, but can hurt the

demand for the firm’s new product. This cannibalization effect may lead the firm not only to not

exercise the remarketing option and remove used equipment from the market, but to also do so

earlier than it would happen under a selling strategy. Thus, leasing may lead to environmentally

inferior outcomes than selling.

We note that both these arguments – for and against leasing relative to selling– ignore the control

that firms who choose to sell have over the secondary market. In practice, firms who produce

durable goods (copiers, printers, servers, refrigerators, etc) often offer trade-ins when they sell their

products, whereby the customers who return their used product get a price discount on the new

product they buy. For example, Xerox, HP and Pitney Bowes offer trade-ins to customers who own

one of their products (Fishbein et al. 2000, HP 2007). A firm might offer trade-ins to control the

secondary market competition by removing used units from the market. Thus, under selling, the

firm has the same options as under leasing for recovered units (although the firm needs to offer


monetary incentives to recover the used units): to sell them to consumers for reuse, or to remove

them from the market (via recycling, incineration or landfilling). For these reasons, it is important

to consider the recovery strategy of the selling firm in any analysis of leasing versus selling.

We also note that the arguments for the environmental benefits of leasing have primarily focused

on reducing the impact of disposal. However, other phases of the life cycle, such as production and

use, also contribute to the total environmental impact of a product. For example, manufacturing

a single personal computer requires 1.7 tonnes of materials and consumes 10 times its own weight

in fossil fuels (Kuehr and Williams 2003). Environmental life-cycle analysis incorporating these

phases typically measures the environmental impact of only one unit of the product, and does

not take into account the difference in the total quantities produced, used, and disposed under

different strategies. For a meaningful comparison of leasing and selling, it is important to consider

the entire life-cycle impact (production, use, and disposal), and differences in product volume and

use duration between the two strategies.

Motivated by these issues, this paper provides insights into environmental considerations of

the lease versus sell decision by incorporating the economics of disposal and comparing the total

environmental impact of the two strategies. The existing research on durable goods has studied

selling and leasing with respect to pricing power, market segmentation and competitive positioning

but does not take into account disposal or the relative environmental impact of the two strategies.

With the growth in environmental regulations governing the landfilling of used products and in the

environmental awareness of consumers, both are important issues to consider for manufacturers in

their strategy choice.

In our model, the leasing firm owns the product at the end of the lease period, and its post-

use strategy consists of remarketing and disposal2. The selling firm can choose to remain inactive

post-use, or deploy a post-use strategy that consists of recovering products via trade-ins, and then

either remarketing or disposing of them. We find the optimal new product sales or lease quantities

2 In this paper, we use the term “disposal” to refer to taking the used product off the market via recycling, incinerationor landfilling.


and the optimal integrated recovery, remarketing and disposal strategy under both leasing and

selling. We focus on the total volume of units produced, used, and disposed to evaluate the total

environmental impact of each strategy when deployed with a profit-maximization objective. With

this analysis, we answer the following questions:

1. What is the optimal integrated remarketing and disposal strategy of the leasing firm?

2. Under what conditions is it optimal for the selling firm to recover products through a trade-in

program? What is its integrated optimal recovery, remarketing and disposal strategy?

3. What are the environmental implications of leasing and selling? Under what conditions does

the profit-maximizing strategy also lead to lower environmental impact?

The rest of the paper is organized as follows: In the next section, we position our research in

the context of the relevant literature. We present our model in §3. In §4, we determine the optimal

strategies of a monopolist under both leasing and selling. In §5, we compare leasing and selling

with respect to their relative profitability and their total environmental impact. Finally, in §6,

we conclude by summarizing our results and discussing the managerial insights derived from our

analysis. All proofs are included in the online companion.

2. Literature Review

In this paper, we draw on and contribute to three streams of literature: i) the durable goods liter-

ature on when and how (e.g. via leasing, trade-ins, etc) a firm should interfere with the secondary

market; ii) literature on a firm’s strategic product line decisions regarding forms of reuse such as

remanufacturing; and iii) the industrial ecology literature.

The research on durable goods has a long history. A significant portion of the industrial organi-

zation and marketing literature on durable goods, starting with Coase (1972), has focused on the

time-inconsistency problem: A selling monopolist cannot commit to a static price path because once

having sold a durable product to high-valuation consumers in the first period, he has the incentive

to reduce his price to capture the residual demand. Rational consumers, expecting this drop, post-

pone their purchases, leading the monopolist to lose pricing power. Stokey (1981), Bulow (1982),


Bond and Samuelson (1984) and the subsequent literature formalize this argument and identify

scenarios where the monopolist would retain pricing power. In particular, leasing the durable good

instead of selling it solves the time-inconsistency problem by giving the monopolist total control

over the used products and impeding the formation of second-hand markets. Consequently, the

leasing firm makes more profit than the selling firm.

Subsequent literature in this stream has focused on analyzing the relative profitability of leasing

and selling in different settings: interfering with secondary markets (Waldman 1997, Hendel and

Lizzeri 1999), differences in depreciation under leasing and selling (Desai and Purohit 1998), com-

petition (Desai and Purohit 1999), the presence of complementary goods (Bhaskaran and Gilbert

2005) and channel structure (Purohit 1995, Bhaskaran and Gilbert 2007). The paper closest to our

work is Desai and Purohit (1998), which compares leasing versus selling for a monopolist firm with

zero production cost that puts all off-lease products on the market.

This literature assumes the selling firm does not interfere with the secondary market via trade-

ins or other mechanisms despite their prevalence in practice. Fudenberg and Tirole (1998) study

a monopolist firm that sells improving generations of a durable product and analyze the optimal

pricing, trade-in and upgrade-discount strategy. However, they do not allow for remarketing and do

not consider disposal. Ray et al. (2005) study the same problem when the firm can remanufacture

the recovered old products to produce new ones at a lower cost. Assuming that remanufactured

and new products are perfect substitutes, they analyze the optimal trade-in decisions of the firm.

Another relevant stream of literature studies firms that recover and remarket remanufactured

products, which compete with the new products on the market. Motivated by an example from

the cellular phone industry, Guide et al. (2003) determine the optimal quality-dependent take-

back price schedule for a remanufacturer. Groenevelt and Majumder (2001), Debo et al. (2005),

Ferrer and Swaminathan (2006), Ferguson and Toktay (2006), Oraiopoulos et al. (2007), Jin et al.

(2007), Atasu et al. (2008) and Esenduran et al. (2008) study the effect of internal and external

competition from a remanufactured product on an OEM’s joint pricing strategies for new and

remanufactured products. These papers assume that the new product has a useful life of only one


period and the firm has to remanufacture the product before it can provide a positive utility in

the second period. Thus, unlike the case with durable goods, customers who purchase the product

in the first period obtain zero utility from the product beyond the end of the first period. This

characterization eliminates the time-inconsistency effect normally observed with selling, and blurs

the distinction between leasing and selling.

Our research is at the intersection of these streams of literature. As in the basic durable goods

literature, we analyze and compare leasing and selling, but enrich the modeling by allowing the

selling firm to recover products via trade-ins, and incorporating a remarketing versus disposal

decision for both the leasing and the selling firm. Motivated by current trends, we incorporate

a cost for disposal. We also allow for a positive production cost to capture the impact of the

production stage on the integrated firm strategy. While we use a simpler construct than in the

remanufacturing literature by normalizing the remarketing cost to zero, we do not limit customers

to one period of use. This helps us to isolate the difference between leasing and selling regarding the

firm’s first-period pricing power, and makes the distinction between the two strategies particularly

salient.

Our research provides guidelines for the integrated management of recovery, remarketing and

disposal. We show that it is optimal for the selling firm to recover products through a trade-in

program even in the presence of costly disposal, except for the case when the production cost is

very high. This is consistent with, and provides theoretical support for, the prevalence of trade-in

programs in practice. Concerning remarketing, we show that the optimal strategy depends on both

the production and disposal cost. Both the selling firm and the leasing firm prefer to remarket only

a fraction of the recovered products when the production and disposal costs are low; otherwise,

they fully remarket the recovered products.

An interesting finding is that in the presence of a disposal cost, trade-ins moderate the time-

inconsistency effect - the selling firm restricts new product sales to be able to charge a higher price

in the future for used products recovered through the trade-in program. For low enough production

cost, this can result in a lower original sales volume than lease volume. Another impact of disposal


cost is that it can penalize the leasing firm more since all off-lease products belong to the firm by

default. We find that while there are cases where leasing only generates as much profit as selling,

leasing is typically more profitable. We conclude that leasing continues to be a good strategy even

in the presence of disposal cost.

A novel aspect of our research is its focus on total environmental impact, inspired by the indus-

trial ecology literature, where conventional life-cycle analysis (LCA) is typically used to evaluate

the environmental impact of a product throughout its life-cycle. LCAs have been performed on

a variety of products (see Gloria et al. (1995), Jensen et al. (1997) for examples). Bennett and

Graedel (2000) use an LCA to compare the environmental impact of selling with that of a product

service system (which includes leasing), but they only model a single unit of the product and do

not take into account the environmental impact of the volume effects driven by consumer choice.

An exception is Thomas (2008), who analyzes the environmental impact of reuse on secondary

markets and acknowledges the volume effect. In this paper, we formalize and analyze the volume

effect, both in total production (and disposal) volume and in total use volume over the product

life-cycle. We show that leasing and selling can differ significantly along these dimensions and that

either strategy can dominate depending on production cost, disposal cost, and the environmental

impact profile, i.e. whether the product has more environmental impact in the production, use or

disposal phases. These results have important implications for environmental groups and policy

makers in terms of which strategy to promote as the environmentally preferred option.

3. Model Development

We study a profit-maximizing monopolist that produces a durable good and can either use a leasing

or a selling strategy. We develop a two-period model similar to Waldman (1997) and Desai and

Purohit (1998) that captures the key problem dynamics while still maintaining tractability. The

product has a useful lifetime of two periods. A consumer uses at most one product in a period and

the size of the market is normalized to one and remains constant across the two periods. Consumers

are heterogeneous in their willingness to pay. Consumer type θ has a valuation of θ for a new


product, where θ is uniformly distributed in the interval [0,1]. Products undergo depreciation with

use, where more durable products depreciate less. The durability of the product is parametrized

by δ ∈ (0,1], where consumer type θ has a valuation of δθ for a used product. Thus, this is a

vertical differentiation model where, ceteris paribus, every consumer prefers a new product to a

used product. To this model, we introduce a non-negative unit production cost c and a non-negative

unit disposal cost s to capture the impact of the production and disposal stages on the optimal

integrated firm strategy.

Under both leasing and selling, the firm can only market new units in the first period. If a firm

chooses to sell its products, the units remain in the ownership of the customers at the end of the

first period. At this point, the firm may offer a trade-in program where customers can choose to

return their used unit and buy a new unit. In addition, some existing customers may choose to

continue using their product and not participate in the trade-in program. Any recovered units that

are not remarketed incur a disposal cost of s per unit. At the end of the second period, customers

dispose of their products. We assume that in the second period, the firm only offers new products

through the trade-in program. If we allow the firm to also sell new products outside the trade-

in program, it increases the complexity of the analysis without offering any fundamentally new

insights.

In the leasing case, the lease duration is one period and the manufacturer maintains ownership

of the product. Thus, all off-lease units become available to the manufacturer at the end of the

first period and have to be either disposed of at a cost of s or remarketed as a used product in

the second period. To offer a fair comparison to the selling case, we assume that the leasing firm

only sells (not leases) used or new products in the second period and that customers dispose of

their products at the end of the second period. If, instead, the firm offered leases in the terminal

(second) period, the firm would be obligated to take the products back at the end of the period

and incur a disposal cost for each unit since there is no opportunity to remarket them, whereas the

selling firm would not face such a burden. These assumptions allow us to isolate the effect of the


direct control that a leasing firm has on the recovered off-lease products in the first-period analysis

without penalizing it in the terminal period.

Since the quantity disposed by individual consumers is small, landfill bans and fees typically

either do not apply or are lower than they are for firms. For example, only non-household entities

generating more than 220 lbs. of hazardous electronics waste per month are regulated under the

federal law (U.S. EPA 2008a). Where regulations apply to neither consumers nor firms, firms may

nevertheless recycle due to pressure from environmental groups and incur a cost that individual

consumer do not. We therefore assume that the disposal cost for consumers is less than that for

the firm and normalize it to zero.

3.1. Consumer Demand

We derive the inverse demand functions from consumer utility functions as in Desai and Purohit

(1998) and Desai et al. (2004). The detailed derivation of the functions presented below can be

found in the Appendix. Table 1 summarizes the notation used for our model parameters and

decision variables.

Table 1 Notation for Parameters and Decision Variables

Parameters Symbol Definitionc Unit production costδ Durability of the products Unit disposal cost

Decision Variables Symbol DefinitionLeasing: Second Period q2n, p2n Quantity and price of new products

q2u, p2u Quantity and price of used productsSelling: Second Period q2t, p2t Quantity and net price of a new product

sold through the trade-in programq2u, p2u Quantity and price of a used product

Leasing: First Period q1, p1 Quantity and price of new products leasedSelling: First Period q1, p1 Quantity and price of new products sold

3.1.1. Leasing. Under leasing, no consumer enters the second period with a product and the

two periods decouple. In the first period, the consumer options are to remain inactive or buy a


new product while in the second period, there is the third option of buying a used product from

the manufacturer. The inverse demand functions are as follows:

p1 = 1− q1 (1)

p2u = δ(1− q2n− q2u) (2)

p2n = 1− q2n− δq2u. (3)

The second-period functions capture the cannibalization of new products by used products. As the

durability of the product increases, the used and new products become closer substitutes and the

cannibalization effect intensifies.

3.1.2. Selling. When the firm sells the product, customers can choose to keep the product in

the second period. Thus, we need to model the consumer strategy over both periods. There are

four two-period strategies available to the consumer: i) Buy new, trade-in for new product (NT), ii)

Buy new, hold (NH), iii) Remain inactive, buy used (XU) or iv) Remain inactive, remain inactive

(XX).

At the beginning of the first period, the consumers evaluate their total utility over the two

periods. We assume that consumer have perfect expectations of the future, a common assumption

in the durable goods literature. If the quantity and net price of products sold through the trade-in

program are denoted by q2t and p2t, respectively, then the inverse demand functions are

p1 = (1 + δ)(1− q1)− δq2u (4)

p2t = (1− δ)(1− q2t) (5)

p2u = δ(1− q1− q2u). (6)

Note that as the durability of the product increases, the firm has to offer a lower price for units

sold through the trade-in program and can charge a higher price for the used units.


3.2. Evaluating Total Environmental Impact

A product has an environmental impact at the production, use and disposal stages of its life cycle.

Let the subscripts p, u and d denote these three phases, respectively, so that the total environmental

impact under leasing and selling is given by ε= εp + εu + εd and ε= εp + εu + εd, respectively.

We assume that the environmental impact from the use of the product is constant over time,

and the unit environmental impact during production and at disposal are independent of whether

the product is leased or sold. We further assume that all units are disposed of, irrespective of their

age, at the end of the second period. These assumptions help us focus on the effects of consumer

choice and the lease vs. sell decision made by the firm on the total environmental impact. The

environmental impact in each of the life-cycle phases can be calculated as follows using the first-

and second-period demand levels:

Production Phase: Let ip denote the environmental impact due to the production of one unit.

The total environmental impact due to production is given by ip(q1 + q2n) when the firm leases and

by ip(q1 + q2t) when the firm sells.

Use Phase: Let iu denote the environmental impact during use of one unit for the duration of one

period. Then the total environmental impact due to use is given by εu = iu(q1 + q2n + q2u) under

leasing and by εu = iu(q1 + q2t + (q1− q2t) + q2u) = iu(2q1 + q2u) under selling. Here, q1− q2t is the

quantity of units held onto by customers in the second period.

Disposal Phase: In our model, either the firm or the customers can dispose of a unit. We assume

the environmental impact of disposal per unit product is the same, independent of who disposes

of it. Let id be the environmental impact per unit disposed. In either case, the number of units

disposed equals the total production volume over the two periods. Thus, the environmental impact

due to disposal is εd = id(q1 + q2n) under leasing and εd = id(q1 +q2t) under selling. Combining these

three elements, the total environmental impact under leasing and selling is given by

ε=εp + εu + εd = (ip + iu + id)(q1 + q2n) + iu(q2u)

ε=εp + εu + εd = (ip + iu + id)(q1 + q2t) + iu(2q1 + q2u).


In the next section, we find the profit-maximizing decisions under leasing and selling, setting up

the stage for determining the optimal strategy and its relative total environmental impact in §5.

4. Profit-Maximizing Decisions under Leasing and Selling

In this section, we determine the firm’s optimal quantity to lease or sell in the first period, the

amount to recover through trade-ins (if selling), and the quantities of new and used units to sell

in the second period. We begin by solving the firm’s problem under leasing.

4.1. Leasing

Recall that in the first period, the leasing firm offers one-period operating leases. At the end of the

first period, the firm gains ownership of all the off-lease units and may choose to offer them for sale

in the second period as depreciated used units in parallel with the firm’s new units. The remaining

units are disposed of at a cost s at the end of the first period. We determine the subgame-perfect

equilibrium using backward induction. In the second period, the key decisions are the quantity of

new and used units to sell, where the number of used units is limited by how many units were

leased in the first period. The inverse demand functions for new and used products in the second

period are given by (2) and (3) respectively. The firm’s second-period optimization problem, which

is jointly concave in q2u and q2n, is given by

maxq2n,q2u

Π2(q2n, q2u|q1) = (p2n− c)q2n + p2uq2u− s(q1− q2u)

s.t. q2u ≤ q1, q2n, q2u ≥ 0.

In the first period, the firm only determines the lease quantity, where the inverse demand function is

given by (1). The firm’s objective is to maximize the total two-period profit Π(q1) = Π1(q1)+Π∗2(q1)

with respect to q1, where Π1(q1) = (p1 − c)q1. The following proposition summarizes the firm’s

optimal decisions and Figure 1 depicts the second-period strategy in the optimal solution.

Proposition 1. The firm’s optimal integrated strategy under leasing is as follows, where

s(c, δ) .= δ(1−2c−δ+δc)1+δ−δ2 and c(δ) .= 1− δ2.

Condition 1: If c < c(δ) (low production cost), the firm markets new units in both periods, and


either partially or fully remarkets off-lease products in the second period depending on the disposal

cost:Case q∗1 q∗2n(q∗1) q∗2u(q∗1)

(low disposal cost) 0≤ s < s(c, δ) 1−c−s2

1−c−δ−s2(1−δ)

cδ+s2δ(1−δ)

(high disposal cost) s(c, δ)≤ s 1−c+cδ2(1+δ−δ2)

1−c−2δq∗12

q∗1

Condition 2: If c(δ) ≤ c (high production cost), the firm markets new products only in the first

period and remarkets all of the off-lease units in the second period (q∗2n = 0 and q∗2u = q∗1 = 1−c+δ2(1+δ)

).

Figure 1 Second-Period Remarketing Strategy under Leasing in the Optimal Solution. All off-lease products are

recovered by default. The firm remarkets only a portion of these products if s < s(c, δ), and remarkets all of them

otherwise. It markets new products in the second period only if c < c(δ).

s

c

0

1

1

New & Full

Remarketing

New & Partial

Remarketing

Only Full

Remarketing

(full recovery by default)

Proposition 1 summarizes the joint effect of the disposal cost and production cost on the opti-

mal integrated strategy under leasing. In particular, the firm’s incentive to offer new products in

the second period is driven by the production cost. If the production cost is low (c < c(δ)), the

margin earned by selling new products is high and the firm prefers to offer new products. When

the firm offers new units in conjunction with remarketing off-lease units, cannibalization occurs.

Consequently, the firm’s remarketing strategy depends on the disposal cost: If the cost of disposal

is low, the firm only remarkets a fraction of the off-lease units to reduce cannibalization of the sale

of new units. On the other hand, if the disposal cost is high, the cannibalization effect is outweighed

by the potential cost of disposal and the firm remarkets all of the off-lease products.


If the production cost is high (c(δ)≤ c), the potential margin from selling new products is low

and the firm does not sell any new units in the second period. Thus, even if the disposal cost is

low, the firm remarkets all of the off-lease units.

These results highlight the importance of considering production and disposal costs while adopt-

ing a remarketing strategy under leasing. Desai and Purohit (1998) assume the firm always remar-

kets all off-lease products despite assuming c = 0. They point out that this strategy would be

optimal for high enough c, as confirmed by our results. At the same time, we show that when pro-

duction and disposal costs are below certain thresholds, it is optimal for the firm to only partially

remarket off-lease products and dispose of the rest. In this case, the firm exploits the direct control

it has over the off-lease units to prematurely dispose of them and sell new products instead.

4.2. Selling

Recall that in the first period, the firm offers new units for sale, while in the second period, the firm

can recover used units and sell new units through a trade-in program and may choose to remarket

some or all of them. We determine the subgame-perfect equilibrium using backward induction. In

the second period, the inverse demand functions are given by (5) and (6). The firm’s second period

objective function, which is jointly concave in q2t and q2u, is given by

maxq2t,q2u

Π2(q2t, q2u|q1) = (p2t− c)q2t + p2uq2u− s(q2t− q2u)

s.t. q2u ≤ q2t, q2t ≤ q1, q2t, q2u ≥ 0.

The first constraint requires the quantity of used units sold to be constrained by the supply of

units recovered through trade-ins. The second constraint ensures that the firm cannot recover

more products than it sold in the first period. In the first period, there are only new units on

the market and the inverse demand function is given by (4). The firm chooses the optimal first-

period quantity of units to sell that maximizes its two-period profit Π(q1) = Π1(q1)+Π∗2(q1), where

Π1(q1) = (p1 − c)q1. The following proposition summarizes the firm’s profit maximizing decisions

under selling and Figure 2 depicts the second-period strategy in the optimal solution.


Proposition 2. The firm’s optimal integrated strategy is as follows, where c1(δ) .=√

2− δ− 1,

s1(c, δ) .= δ(2+3δ)(8−16c−5δ)−δ(2−2c+δ)√

(8−3δ)(2+3δ)

(2+3δ)(8+13δ)and c2(δ) .= 4+2δ−3δ2

4+2δ−δ2 .

Condition 1: If c < c1(δ) (low production cost), the firm recovers the entire quantity of used units

through the trade-in program and either partially or fully remarkets them depending on the disposal

cost:

Case q∗1 q∗2t(q∗1) q∗2u(q∗1)(low disposal cost) 0≤ s < s1(c, δ) 2(2−2c−δ−2s)

8−3δq∗1

δ+s−δq∗12δ

(high disposal cost) s1(c, δ)≤ s 2−2c+δ4+6δ

q∗1 q∗2t(q∗1)

Condition 2: If c1(δ) ≤ c < c2(δ) (medium production cost), the firm only recovers a fraction of

the used units through trade-ins and remarkets all of them. The optimal decisions are given by

q∗1 = 2(1−c+δ)(2+3δ)(2−δ) and q∗2u(q∗1) = q∗2t(q∗1) = 1−c−δq∗1

2.

Condition 3: If c2(δ)≤ c (high production cost), the firm does not recover any used units through

the trade-in program. The optimal decisions are given by q∗1 = max{ 1−cδ, 1−c+δ

2(1+δ)} and q∗2u = q∗2t = 0.

Figure 2 Second-Period Recovery & Remarketing Strategy under Selling in the Optimal Solution. Used

products are recovered through a trade-in program that sells a new product to each customer who returns a used

product. A trade-in program is used only if c < c2(δ), and all previously sold products are recovered via this

program if c < c1(δ). The firm remarkets only a portion of the recovered products if s < s1(c, δ), and remarkets all

of them otherwise.

s

c

0

1

1

Full Recovery and

Full Remarketing

Full

Recovery

& Partial

Remarketing

No

Recovery

Partial Recovery

and Full

Remarketing

(recovery through trade-ins)


These results show the effect of the production cost, disposal cost and durability on the recovery

and remarketing strategy under selling. In particular,

Observation 1. The firm recovers all used units through a trade-in program for low values of the

production cost. The remarketing strategy depends on the disposal cost.

For c < c1(δ), the production cost is low enough to be the dominating factor - in the second

period, the firm sells new units to all of its first-period customers through the trade-in program

regardless of the disposal cost. The remarket-versus-dispose decision is then driven by the disposal

cost. Note that used products are remarketed to lower willingness-to-pay consumers who do not

already own a product. Thus, remarketing all recovered units requires setting a low used-product

price. If the disposal cost is low (s ≤ s1(c, δ)), the firm prefers to dispose of a fraction of the

recovered units at a cost, but benefit from the higher price it can charge to sell the remaining units.

For high values of disposal cost, this strategy is not profitable - the firm prefers to price low and

sell all used units to avoid disposal cost altogether.

Observation 2. For moderate levels of production cost, partial recovery with full remarketing is

optimal.

For moderate values of the production cost (c1(δ)≤ c < c2(δ)), the potential margin from offering

a trade-in is lower and the firm should only recover some of the existing units through the trade-in

program. Since the quantity of recovered products is smaller, the price that can be charged for

the recovered units is higher. This implies that the cost of disposal can be profitably avoided by

remarketing all of the recovered units and the remarketing strategy does not depend on the disposal

cost.

Observation 3. For high enough production cost, the firm does not find it optimal to offer a

trade-in program.

In particular, offering the trade-in program is not optimal if c > c2(δ). The value of c2(δ) lies

between 0.6 and 1 for δ ∈ [0,1], which implies a low profit potential on new units. Each trade-

in requires the firm to produce a new unit. For high enough production cost, this becomes less

profitable than selling the unit once and not interfering with the market in the second period.


Note that c2(δ) is decreasing in δ: As durability increases, the utility the consumer obtains from

the product in the second period increases. This means the firm can charge a higher price for the

product in the first-period. Therefore, the cost threshold above which offering the trade-in program

is unprofitable decreases with durability.

These observations demonstrate the power of a trade-in program as a mechanism for a selling firm

to interfere with the secondary market of durable goods. They are consistent with the prevalence

of trade-ins in practice and highlight the importance of considering the ability to offer trade-ins in

an analysis of the selling strategy.

5. Comparing Profit and Environmental Impact under Leasing andSelling

In this section, we compare leasing and selling under the profit-maximizing decisions for each along

two dimensions: profit and total environmental impact. We first summarize the differences between

leasing and selling that will be helpful in understanding the results.

Nature of recovery: The selling firm does not have to take back any units, but can do so

at a cost via the trade-in program, while the leasing firm has to recover all off-lease products

involuntarily, albeit at no cost. The greater control enjoyed by the leasing firm may translate into

higher levels of disposal and production.

Potential market for recovered units: Under selling, if a consumer purchases a product in

the first period, she can either choose to avail of the trade-in program or hold on to the product

in the second period. The firm can offer used units only to the consumer segment with a lower

willingness-to-pay that did not purchase a product in the first period. Thus, under selling, the

firm’s first-period quantity decision directly affects the revenue potential from used products. In

contrast, under leasing, the target segment for used products is only affected by the second-period

new product sales quantity.

Time inconsistency: As discussed in the literature review, it is well-known that with durable

goods, the selling firm suffers from time inconsistency, which leads to a loss of pricing power relative


to leasing. Consequently, the selling firm sells a larger quantity of new products in the first period

than the leasing firm.

The effect of trade-ins on time inconsistency: As discussed above, with selling, the potential

revenue from used products is directly influenced by the first-period sales quantity. When recovering

used products via the trade-in program is optimal, the disposal cost affects first-period sales: With

a very high disposal cost, the firm remarkets all recovered products. As these products are sold

to the lower valuation segment that did not buy in the first period, the firm restricts first-period

sales to be able to charge a higher price for the used product and the time inconsistency effect is

moderated. For lower values of disposal cost, the firm only remarkets a fraction of the recovered

products, so the time inconsistency effect is moderated to a lesser extent.

These effects interact to determine the relative profitability and environmental impact of the

two strategies, as summarized below.

Proposition 3. The relative profitability and environmental impact of leasing and selling under

the profit-maximizing decisions can be summarized as follows, where c1(δ) .=√

2− δ − 1, c2(δ) .=

4+2δ−3δ2

4+2δ−δ2 , c3(δ) .= 2+δ−δ22+δ

, s(c, δ) .= δ(2−4c−3δ+3δc)

(1−δ)(2+3δ)and s(c, δ) .= δ(4−8c−4δ+3δc)

(2−δ)(2+3δ).

Condition 1: If c < c1(δ), leasing is more profitable than selling and whether the environmental

impact is lower under leasing or selling depends on the disposal cost as follows:

a. If 0≤ s < s(c, δ), the environmental impact due to production (or disposal) is lower under selling

and the environmental impact due to use is lower under leasing.

b. If s(c, δ)≤ s < s(c, δ), the total environmental impact is lower under selling.

c. If s(c, δ) ≤ s, the environmental impact due to production (or disposal) is lower under leasing

and the environmental impact due to use is lower under selling.

Condition 2: If c1(δ)≤ c < c3(δ), leasing is more profitable than selling and the total environmen-

tal impact is strictly lower under leasing.

Condition 3: If c3(δ) ≤ c, the profits and environmental impact under leasing and selling are

equal.


For convenience, we note that c1(δ) < c(δ) < c2(δ) ≤ c3(δ) ∀δ ∈ (0,1], and s1(c, δ) < s(c, δ) <

s(c, δ)< s(c, δ) ∀δ ∈ (0,1] and c∈ [0,1]. We first discuss the relative profitability of the two scenarios.

Despite the disposal cost that would at first sight appear to penalize leasing more due to involuntary

take-back, leasing dominates or performs the same as selling.

If the marginal production cost is high enough (c3(δ) ≤ c), the selling firm mimics the leasing

firm’s strategy (full remarketing and no new product sales) by not offering a trade-in program and

letting its customers keep their products for two periods. The selling firm charges a first-period

price equal to the sum of the first- and second-period prices charged by the leasing firm, and

the same volume is sold with both strategies. Neither firm incurs the disposal cost. Consequently,

profits (and environmental impact) are identical under the two strategies.

Under low and moderate levels of production cost (c < c3(δ)), the strategic nature of the consumer

purchase decision hurts the selling firm’s pricing power. This effect is especially apparent for c <

c2(δ) where the firm remarkets products in the second period, but is also at work in the range

c2(δ) < c < c3(δ), where despite not having a trade-in program and not remarketing products in

the second period, the selling firm optimally reduces its sales price and increases its sales volume

relative to leasing in the first period. In addition, when it is profitable to sell new products in the

second period, the leasing firm has the advantage of owning all off-lease products, while the selling

firm must offer a trade-in program. Disposal cost might be expected to hurt the leasing firm when

it does not fully remarket off-lease products, but needs to dispose of them. This happens only for

low production and low disposal cost (New and Partial Remarketing region in Figure 1), but in

this region, the selling firm wants to benefit from the low production cost and recovers all used

products via the trade-in program, even when it disposes of some of them. When the disposal

cost is high, both firms fully remarket to avoid this cost. Thus, disposal cost, while it constitutes

an absolute liability, does not adversely affect the relative profitability of leasing as compared to

selling.

We now turn to the relative environmental impact of the two strategies. As discussed above,

for high levels of the production cost (c3(δ)≤ c), the environmental impact of the two strategies


is the same. For moderate levels of the production cost (c1(δ)≤ c < c3(δ)), when selling, the firm

finds either partial or no recovery optimal. At these recovery levels, the moderating effect of trade-

ins on time-inconsistency is not strong, so the firm produces a larger quantity of products than

under a leasing strategy. Since no units are prematurely disposed of under either strategy, the

total quantities used and disposed are higher under selling. Therefore, for moderate levels of the

production cost, leasing leads to a lower total environmental impact than selling.

For low production cost (c < c1(δ)), whether the total environmental impact is lower under leasing

or selling depends on the disposal cost. The typical behavior of the relevant quantities is depicted in

Figures 3 and 4. Let us first focus on the total production quantity (also equal to the total disposal

quantity). With selling, the presence of trade-ins reverses what the time-inconsistency effect would

predict - the total quantity sold in the first period is actually lower with selling (Figure 3a). All

customers take advantage of the trade-in program (q∗2t = q∗1), so the total quantity produced is 2q∗1 .

Used units have to be offered to the lower willingness to pay consumers who did not buy a product

in the first period. With leasing, since no consumer owns a product at the end of the first period,

the firm has more flexibility in the second period about how to manage the market segmentation.

When the disposal cost is low, the firm disposes of a significant portion of used products (Figure 3c)

with two goals: to sell new products instead and to keep the price of remarketed products as high

as possible. Above the disposal cost threshold s(c, δ), the leasing firm remarkets all used products

at the expense of selling a small volume of new products due to cannibalization (Figures 3b,d).

Overall, the net effect is that the total production (and disposal) volume over the two periods is

lower with selling for s < s(c, δ) and lower with leasing for s > s(c, δ) (Figure 4a).

Next, we compare the environmental impact based on product use. Recall that the total use

volume over the two periods is q1 + q2n + q2u under leasing and q1 + q2t + (q1− q2t) + q2u = 2q1 + q2u

under selling. As discussed above, q∗1 + q∗2t = 2q∗1 in this production cost region. Since we already

compared q∗1 + q∗2n and q∗1 +q∗2t and concluded the latter is lower for s < s(c, δ) and higher otherwise,

we only need to focus on q∗2u and q∗2u. As also discussed above, for lower levels of disposal cost,

the leasing firm prefers to prematurely dispose of used products, resulting in a significantly lower


Figure 3 All plots are for c= 0.2 and δ= 0.2 (Condition 1 holds). Dashed lines represent leasing and solid lines

represent selling. The values of s at which leasing and selling change regimes are s(0.2,0.2) = 0.076 and

s1(0.2,0.2) = 0.014, respectively.

0.02 0.04 0.06 0.08 0.10 0.12 0.14s

0.34

0.36

0.38

0.40

qè

1*, q1

*

Quantity of new products in first period

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14s

0.34

0.36

0.38

0.40

qè

2 n*

, q2 t*

Quantity of new products in second period

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14s

0.1

0.2

0.3

0.4

qè

1*-qè

2 u*

,q2 t*-q2 u

*

Quantity of products disposed prematurely

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14s

0.1

0.2

0.3

0.4

qè

2 u*

, q2 u*

Quantity of used products remarketed

(a) (b)

(c) (d)

remarketing volume relative to selling (Figures 3c,d). This difference is large enough below the

threshold s(c, δ) so as to make the environmental impact due to use lower with leasing (Figure 4b).

We conclude that under the condition c < c1(δ), the total environmental impact of selling versus

leasing for a given product depends on the relative environmental impact of the different phases. For

example, automobiles and refrigerators consume much more energy during the use phase relative

to the production phase (White et al. 1999), while PCs consume much more energy during the

production phase relative to the use phase (Williams and Sasaki 2003).

6. Conclusions

Is leasing green? To answer this question, we develop an integrated model of pricing, recovery,

remarketing and disposal for durable goods put on the market via leasing or selling. Important

features of our model are the incorporation of production and disposal costs, and the existence of a


Figure 4 All plots are for c= 0.2 and δ= 0.2 (Condition 1 holds). Dashed lines represent leasing and solid lines

represent selling. s and s1 are the same as in Figure 3. The disposal cost at which the total quantity

produced/disposed are equal for leasing and selling is s(0.2,0.2) = 0.074, while that at which the total quantity

used is equal for leasing and selling is s(0.2,0.2) = 0.069.

0.02 0.04 0.06 0.08 0.10 0.12 0.14

s

0.70

0.72

0.74

0.76

qè

1*+qè

2 n*

,q1*+q2 t

*

Total quantity producedêdisposed

0.02 0.04 0.06 0.08 0.10 0.12 0.14s

0.90

0.95

1.00

1.05

1.10

qè

1*+qè

2 n*

+qè

2 u*

,2q1*+q2 u

*

Total quantity used(b) (a)

trade-in option for the selling firm. The analysis contributes to three streams of literature - durable

goods, remanufacturing and industrial ecology. The conclusions are of relevance to manufacturers,

environmental groups and policy makers.

The durable goods literature has compared leasing and selling under a variety of settings. The

dimensions we add are the consideration of a disposal cost, the recovery/remarketing/disposal mix

and the trade-in option under selling. The dominant paradigm in the durable goods literature is

the time inconsistency effect whereby the selling firm loses pricing power and sells a larger quantity

at a lower price. We show that when the selling firm’s trade-in option is modeled, it is optimal for

this firm to employ trade-ins unless the production cost is high. In the presence of costly disposal,

the existence of trade-ins moderates the time inconsistency effect - the firm restricts new product

sales to be able to charge a higher price for used products recovered through the trade-in program

in the future. In fact, below a production cost threshold, the trade-in effect is so strong that the

initial sales volume under selling is lower than under leasing.

A subset of the remanufacturing literature has studied pricing, recovery, remanufacturing and

remarketing of remanufacturable products. Remanufacturable products are by definition durable.


However, for tractability, these papers (including some of our own) assume that after one period

of use, the product has to be remanufactured to continue delivering value to customers, so the

time inconsistency effect inherent in selling durable goods is not captured. By not imposing this

restriction, we capture the differences in the recovery and remarketing strategies between selling

and leasing firms. We also take a life-cycle perspective and quantify the total environmental impact

resulting from each strategy choice.

For manufacturers, this analysis provides insights about the optimal integrated strategy under

either leasing or selling, and an assessment of these strategies’ relative profit and environmental

impact. We find that:

• With leasing, since the firm owns all off-lease products, the only decision is the remarketing

versus disposal decision. When production and disposal costs are low, the firm should limit can-

nibalization by only partially remarketing off-lease products in conjunction with new products. If

the disposal cost is high, the firm should remarket all off-lease products, regardless of whether it

sells new products (optimal for low production cost) or not (optimal for high production cost).

• With selling, the firm determines the recovery level in addition to remarketing versus disposal.

Production and disposal costs interact in the same manner as in leasing to determine whether

full or partial remarketing is optimal. Whether it is optimal to offer trade-ins depends on the

production cost and durability - for a given durability level, trade-ins are profitable for low to

moderate production costs. The cost threshold up to which trade-ins are profitable increases as

the durability decreases, as low durability makes it easier to induce existing customers to trade in

their used product for a new one.

• As in much of the durable goods literature on a monopolist selling directly to customers,

leasing dominates selling from a profit perspective. Disposal cost, which penalizes leasing, at worst

makes the leasing firm’s profit equal to the selling firm’s profit.

• Life-cycle environmental impact is a function of production and disposal cost. For low pro-

duction cost, whether leasing or selling is better environmentally depends on the disposal cost

and the relative environmental impact arising from the disposal, production and use phases. For


intermediate production cost, leasing dominates, while for high production cost, the two strategies

have the same total impact.

The industrial ecology literature has traditionally focused on the life-cycle environmental impact

of a single unit. This approach disregards the volume effect, which we show is critical in evaluating

effective environmental performance, and is quite subtle. There are two interacting components

to the volume effect - production (and disposal) volume and total use volume over the product

life-cycle. We discuss each of these below.

Relative production (and disposal) volumes are driven primarily by the strength of the time

inconsistency effect. As previously discussed, the durable goods literature predicts a higher pro-

duction volume under selling due to this effect. The environmental implication is that selling would

fare worse than leasing in the total quantity produced and disposed. In contrast, we show that

when production cost and disposal cost are low, selling does better from this perspective in the

presence of the trade-in option. With a high disposal cost or a high production cost, leasing does

at least as well as selling.

The total use volume is tied to the level of premature disposal in addition to the production

volume. The premise that leasing is better for the environment rests on the assumption that the

firm will use the greater control it has over used products to promote reuse. In contrast, our results

show that the leasing firm prematurely disposes of as many or more products than the selling

firm. Ironically, when leasing dominates at low production and disposal costs, it is despite having

a higher premature disposal volume. The reason is that the firm prematurely disposes of products

not with the purpose of selling a large volume of new products, but to limit the remarketing volume

to keep the used product price high. Thus, it fares better than selling in total volume used over the

product life-cycle, and this effect is strong enough to result in a lower environmental impact due

to use. In all other cases, recovered products are fully remarketed in both strategies; leasing has

lower environmental impact only because it avoids time inconsistency and produces (and hence

uses) less.


For environmental groups and policy makers, our analysis sheds light on the appropriateness of

promoting leasing as an environmentally preferred alternative. We conclude that care must be taken

to avoid blanket statements about the dominance of one strategy over the other. For moderate or

high levels of production cost, the arguments for leasing being environmentally superior to selling

are validated by our analysis. At the same time, we find that for low production cost, this is not

uniformly true. The dominant strategy depends on the disposal cost and the relative environmental

impact of the production, use and disposal phases. In particular, with low production and disposal

costs, selling is environmentally superior for products that have significantly higher environmental

impact in their production and/or disposal phases such as computers and carpets. The opposite

occurs in the presence of low production cost and high disposal cost. Here, selling is environmentally

superior for products such as automobiles and refrigerators that have higher environmental impact

during their use phases. These conclusions are derived from the volume effects discussed above.

We conclude with a discussion of the effect of relaxing some of our assumptions. First, we assume

that disposal is costly. A firm may be able to dispose of some products profitably by recycling them

to recover value. If disposal is sufficiently profitable, the firm will recover and dispose of all used

units in either strategy. At the same time, under selling, the firm will still face time inconsistency

and will produce a higher quantity of products. Since no used products are remarketed, this will

imply higher quantities of products used and disposed. Thus, if disposal is sufficiently profitable,

selling will lead to a higher environmental impact. Second, we assume that there is no collection

cost associated with taking back off-lease or trade-in products. Such a cost would have a more

negative impact on the leasing firm that must assume ownership of all off-lease products. Finally,

we assume that product durability is exogenous, but in practice, the firm may have the ability to

determine product durability during the design stage. A numerical investigation of optimal profit

as a function of durability indicates that the leasing firm would optimally choose an equal or

higher durability than the selling firm. An interesting direction for future research is to analyze

the impact of optimal durability choice on these strategies’ relative financial and environmental


performance. Another interesting direction is to evaluate the optimal strategies developed in the

remanufacturing literature through the lens of total life-cycle impact.

Acknowledgments

This research was supported by NSF DMI Grant No. 0620763.

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Appendix: Consumer Demand Functions

Recall that consumer willingness-to-pay is distributed uniformly in the interval θ ∈ [0,1] and that

each consumer uses one unit per period. The net utility (NU) from using a unit in the second

period is given by NU = δmθ−p, where p is the price for the product and m is an indicator variable

with m= 0 for a new product and m= 1 for a used product.

Leasing

To derive the demand functions, we begin with the choices available to the consumer in the second

period. Since all of the leased products are returned at the end of the first period, all the consumers

enter the second period without a product. Therefore, under leasing, the firm does not face the

time inconsistency problem and the consumers have three possible strategies in the second period:

i) Buy a new product (N), ii) Buy a used product (U) or iii) Remain inactive (X). In terms of

consumer utility, if all three are observed in equilibrium, the consumers who follow the N strategy

have a higher willingness-to-pay than the ones who follow the U strategy, who have a higher

willingness-to-pay than the consumers who choose to remain inactive X. Now, consider the lowest

willingness-to-pay consumer who follows a U strategy and is indifferent between a used unit or

remaining inactive. This consumer is located at θ= 1− q2n− q2u. Equating the consumer’s utility

from a used unit δθ− p2u and from remaining inactive 0, we find the price for a used unit to be:

p2u = δ(1− q2n− q2u). (7)


Next, consider a marginal consumer who is indifferent between a new and used unit. This consumer

is located at θ= 1− q2n. Equating the consumer’s net utility from a new unit θ− p2n, and from a

used unit δθ− p2u, provides the price for a new unit:

p2n = 1− q2n− δq2u. (8)

Note that as the durability of the product increases, the price for a used unit increases to take

advantage of the higher quality level of the used product in the second period. As this param-

eter increases, the used and new units become closer substitutes and the cannibalization effect

intensifies.

Now consider the first-period problem. The consumer has only two available strategies: i) Lease

a new product (N) or ii) Remain inactive (X). At the end of the first period, all consumers who

own a product must return it. Since all consumers enter the second period without a product,

their choices are independent across the two periods and the consumer’s decision in the first period

does not depend on future actions. Consider a consumer who is indifferent between leasing a new

product and remaining inactive. This consumer is located at θ= 1− q1. The consumer’s net utility

from leasing a new product is θ− p1 and zero from remaining inactive. Equating these two provides

the first-period price for leasing a new product, given by

p1 = 1− q1. (9)

Selling

Let p2t and q2t denote the net price and quantity of products sold through the trade-in program,

respectively. Since consumers can choose to keep the product in the second period, we need to

consider two-period consumer strategies. There are four two-period strategies available to the

consumer: i) Buy new, trade-in (NT), ii) Buy new, hold on (NH), iii). Remain inactive, buy used

(XU) or iv) Remain inactive, remain inactive (XX). Therefore, at the beginning of the first period,

the consumers have to evaluate their strategy over the two periods.

The first-period sales are denoted by q1 so the size of the segment that does not own a product

is 1 − q1. This group of consumers chooses to remain inactive in the first period and have a


willingness-to-pay distributed uniformly in the interval [0,1− q1]. Consider a marginal consumer

who is indifferent between buying a used unit and remaining inactive. This consumer is located at

θ= 1− q1− q2u. Equating the consumer’s net utility from a used unit δθ−p2u, and from remaining

inactive 0, provides the price for a used unit:

p2u = δ(1− q1− q2u). (10)

We next focus on the segment that owns a used unit at the start of the second period. The size of

the segment is q1 and the willingness-to-pay of each of these consumers is distributed uniformly in

the interval [1− q1,1]. These consumers have two possible strategies: i) Trade-in for a new product

(T) or ii) Hold onto the product from the first period (H). In terms of utility, if both are observed

in equilibrium, the consumer who follows a T strategy has a higher willingness-to-pay than the one

who follows the H strategy. Consider a marginal consumer who is indifferent between following the

T and H strategies and is located at θ = 1− q2t. The net utility θ− p2t from the T strategy and

δθ from a H strategy. Equating these two provides the price for a new product with the trade-in

option:

p2t = (1− δ)(1− q2t). (11)

Since the consumers are forward looking, they make their first-period purchase while taking into

account the second-period offerings. A consumer who decides to purchase a product in the first

period has a higher willingness-to-pay than a consumer who chooses to remain inactive, so the

lowest willingness-to-pay consumer who would buy a new product in the first period would be

indifferent between the NH and XU strategies. This consumer is located at θ = 1− q1. The two

period utility from following the NH strategy is θ− p1 + δθ and from following the XU strategy is

δθ− p2u. Equating these two provides the price for a new product in the first period,

p1 = (1 + δ)(1− q1)− δq2u. (12)

e-companion to Agrawal et al.: Is Leasing Green? ec1

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ec2 e-companion to Agrawal et al.: Is Leasing Green?

Proofs

Proof of Proposition 1.

We solve the firm’s profit maximization problem using backward induction starting with the

second period problem. In the second period, the firm earns revenues from the sales of both new

and used units and has to incur disposal costs for recovered off-lease units which are not offered

for resale.

maxq2n,q2u

Π2(q2n, q2u|q1) = (p2n− c)q2n + p2uq2u− s(q1− q2u) (EC.1)

s.t q2u ≤ q1, q2n, q2u ≥ 0. (EC.2)

Here, p2n = 1− q2n− δq2u and p2u = δ(1− q2n− q2u). The first constraint represents the limit on the

supply of recovered off-lease products, which the firm can offer as used products to consumers in

the second period. The hessian of the objective function is(−2 −δ−δ −2δ

), whose leading coefficient

is negative and the determinant δ(4− δ) is positive. Thus, the hessian is negative definite and the

profit function is jointly concave in q2n and q2u). The Lagrangean is given by

L(q2n, q2u, λ) =(1− q2n− δq2u− c)q2n + δ(1− q2n− q2u)q2u− s(q1− q2u)−λ(q2u− q1) +µ1q2n +µ2q2u,

with first-order conditions

σ1(q2n, q2u, µ1) .=∂L∂q2n

=1− c− 2(q2n− δq2u) +µ1 = 0

σ2(q2n, q2u, µ2, λ) .=∂L∂q2u

=s+ δ− 2δ(q2n− q2u) +µ2−λ= 0

Since the profit function is concave, necessary and sufficient conditions for optimality (Kuhn-Tucker

conditions) are that the first order conditions (FOC) are satisfied and λ(q1 − q2u) = 0, µ1q2n = 0,

µ2q2u = 0, λ ≥ 0, µ1 ≥ 0 and µ2 ≥ 0. Since 0 ≤ q2u ≤ q1, λ and µ2 cannot both be positive. This

leaves six candidate solutions to the optimization problem defined as cases 1 - 6 below.

Case 1. q2n > 0 and q2u = 0. Then λ = 0, µ1 = 0 and µ2 ≥ 0. Solving σ1(q2n,0,0) = 0 and

σ2(q2n,0, µ2,0) = 0 gives q2n = 1−c2

and µ2 = −δc − s. µ2 ≥ 0 holds for s ≤ −δc. Since s ≤ −δc

contradicts s≥ 0, this case is ruled out.


Case 2. q2n = 0 and q2u = q1. Then λ ≥ 0, µ1 ≥ 0 and µ2 = 0. Solving σ1(0, q1, µ1) = 0 and

σ2(0, q1,0, λ) = 0 gives µ1 = c− 1 + 2δq1 and λ= δ + s− 2δq1. µ1 ≥ 0 and λ≥ 0 hold for 1−c2δ≤ q1

and q1 ≤ δ+s2δ

, respectively. These two conditions hold together only if 1− c− δ ≤ s holds. Since

1− δ− c≤ s holds, we also have δ+s2δ≤ cδ+s

2δ(1−δ) . Thus, the required conditions for this case to apply

are 1−c2δ≤ q1 ≤ δ+s

2δ

(≤ cδ+s

2δ(1−δ)

). The price of a used unit is p2u = δ(1− q1). Π2(q1) = δq1(1− q1).

Case 3. q2n > 0 and q2u = q1. Then λ ≥ 0, µ1 = 0 and µ2 = 0. Solving σ1(q2n, q1,0) = 0 and

σ2(q2n, q1,0, λ) = 0 gives q2n = 1−c2− δq1 and λ = cδ + s − 2δq1(1 − δ). q2n > 0 and λ ≥ 0 hold

for q1 <1−c2δ

and q1 ≤ cδ+s2δ(1−δ) , respectively. q1 ≤ cδ+s

2δ(1−δ) and q1 > 0 hold only if −δc < s, which is

satisfied for all s≥ 0. Thus, the required condition for this case to apply is q1 <min{ cδ+s2δ(1−δ) ,

1−c2δ}.

If s < 1− δ− c holds, then we have min{ cδ+s2δ(1−δ) ,

1−c2δ}= cδ+s

2δ(1−δ) , and if 1− δ− c≤ s holds, we have

min{ cδ+s2δ(1−δ) ,

1−c2δ}= 1−c

2δ. The prices are p2n = 1+c

2and p2u = δ(1+c−2q1(1−δ))

2. Π2(q1) = (1−c)2

4+ q1(δc−

q1(1− δ)).

Case 4. q2n = 0 and 0< q2u < q1. Then λ= 0, µ1 ≥ 0 and µ2 = 0. Solving σ1(0, q2u, µ1) = 0 and

σ2(0, q2u,0,0) = 0 gives q2u = δ+s2δ

and µ1 =−1 + δ+ s+ c. µ1 ≥ 0 and q2u < q1 hold for 1− c− δ≤ s

and δ+s2δ

< q1, respectively. If 1− c− δ ≤ s holds, then we also have 1−c2δ≤ δ+s

2δ. Thus, the required

condition for this case to apply is(

1−c2δ≤)δ+s2δ≤ q1. The price of a used unit is p2u = δ−s

2. Π2(q1) =

(δ+s)2

4δ− q1.

Case 5. q2n > 0 and 0< q2u < q1. Then λ= 0, µ1 = 0 and µ2 = 0. Solving σ1(q2n, q2u,0) = 0 and

σ2(q2n, q2u,0,0) = 0 gives q2n = 1−c−δ−s2(1−δ) and q2u = cδ+s

2δ(1−δ) . q2n > 0, q2u < q1 and q2u > 0 hold for

s < 1− δ− c, cδ+s2δ(1−δ) < q1 and −δc < s, respectively. −δc < s is compatible with s≥ 0. The prices

are p2n = 1+c2

and p2u = δ−s2

. Π2(q1) = (1−c)(1+c−δ+δc)+s(c−q1(1−δ))+s24(1−δ) .

Case 6. q2n = 0 and q2u = 0. Then λ = 0, µ1 ≥ 0 and µ2 ≥ 0. Solving σ1(0,0, µ1) = 0 and

σ2(0,0, µ2,0) = 0 gives µ1 = c− 1 and µ2 =−δ− s. µ1 ≥ 0 holds for c≥ 1. This cannot happen by

definition and is ruled out.

Thus, there are a total of four possible cases, summarized in the table below:


Case q2n q2u p2n p2u Π2(q1)2 0 q1 NA δ(1− q1) δq1(1− q1)3 1−c

2− δq1 q1

1+c2

δ( 1+c2− q1(1− δ)) (1−c)2

4+ q1(δc− q1(1− δ))

4 0 δ+s2δ

NA δ−s2

(δ+s)2

4δ− q1

5 1−c−δ−s2(1−δ)

cδ+s2δ(1−δ)

1+c2

δ−s2

(1−c)(1+c−δ+δc)+s(c−q1(1−δ))+s24(1−δ)

Thus, the optimal solution can be summarized by two conditions, with mutually exclusive and

collectively exhaustive regions in q∗1 :

Condition 1. 0≤ s < 1− δ− c: Cases 3 and 5 apply for q1 ≤ cδ+s2δ(1−δ) and cδ+s

2δ(1−δ) < q1 respectively.

Condition 2. 1− c− δ ≤ s: Cases 3, 2 and 4 apply for q1 <1−c2δ

, 1−c2δ≤ q1 ≤ δ+s

2δand δ+s

2δ< q1,

respectively.

The firm’s profit in the first period is given by Π1(q1) = (p1 − c)q1. We now find the optimal

first-period q1 which maximizes the profits over both periods, i.e. Π(q1) = Π1(q1) + Π∗2(q1). It is

easy to show that Π is concave in q1 for all five cases. We denote Πi(q1) as the profit under Case i.

Condition 1. 0≤ s < 1−δ−c: Let qa= cδ+s2δ(1−δ) . Cases 3 and 5 apply for q1 ≤ qa and qa < q1, respec-

tively. Since Π3(qa) = Π5(qa), Π(q1) is continuous at the boundary q1 = qa. The derivatives of Π3(q1)

and Π5(q1) with respect to q1, evaluated at q1 = qa are equal and given by δ−2cδ−δ2+cδ2−s(1+δ−δ2)

δ(1−δ) .

Thus, Π(q1) is also continuously differentiable. If this derivative is non-positive ( δ(1−2c−δ+δc)1+δ−δ2

.=

s(c, δ) ≤ s), the optimum is reached at q1 ≤ qa and is given by q∗1 = 1−c+δc2(1+δ−δ2)

, which gives Π∗ =

(2−δ)(1+δ+c2)−2c(2−δ2)

4(1+δ−δ2). If it is positive (s < s(c, δ)), then the optimum is reached at qa < q1 and is

given by q∗1 = 1−c−s2

, which gives Π∗ = δ(2(1−c)2−δ(2−4c+c2))−2δs((1−δ)(1−c)−c)+s2(1+δ−δ2)

4δ(1−δ) .

Condition 2. 1− c−δ≤ s: Let qb= 1−c2δ

and qc= δ+s2δ

. Cases 3, 2 and 4 apply for q1 < qb, qb ≤ q1 ≤ qc

and qc < q1, respectively. Since Π3(qb) = Π2(qb), Π(q1) is continuous at the boundary q1 = qb. The

derivatives of Π3(q1) and Π2(q1) evaluated at q1 = qb are equal and given by −1+c+δ2

δ. Thus, Π(q1)

is also continuously differentiable at q1 = qb. Since Π2(qc) = Π4(qc) = (δ+s)(δ(1−2c+δ)−s(1+δ))

4δ2, Π(q1)

is continuous at the boundary q1 = qc. The derivatives of Π2(q1) and Π4(q1), with respect to q1

evaluated at q1 = qc are equal and given by −s(1+δ)−δcδ

. Thus, Π(q1) is also continuously differentiable

at q1 = qc. Since s ≥ 0, the derivative is negative and the optimum is reached at q1 ≤ qc. If the

derivative at qb is negative (c < c(δ) .= 1−δ2), the optimum is reached at q1 < qb. If it is non-negative


(c(δ)≤ c), the optimum is reached at qb ≤ q1 < qc. In the former case, q1 = 1−c+δc2(1+δ−δ2)

, which gives

Π∗ = (2−δ)(1+δ+c2)−2c(2−δ2)

4(1+δ−δ2). In the latter case, q∗1 = 1−c+δ

2+2δ, which gives Π∗ = (1−c+δ)2

4(1+δ).

Note that since 0≤ s < s(c, δ) holds only if c≤ 1−δ2−δ and 1−δ

2−δ < 1− δ2, if 0≤ s < s holds, then we

also have c < c(δ). Since 1− δ− s≤ 1− δ2 we can summarize the results in the table below. �

Condition q∗1 q∗2n(q∗1) q∗2u(q∗1)c < c(δ) & 0≤ s < s(c, δ) 1−c−s

21−c−δ−s

2(1−δ)cδ+s

2δ(1−δ)

c < c(δ) & s(c, δ)≤ s 1−c+cδ2(1+δ−δ2)

1−c−2δq∗12

q∗1c(δ)≤ c 1−c+δ

2(1+δ)0 q∗1

where, s(c, δ) .= δ(1−2c−δ+δc)1+δ−δ2 and c(δ) .= 1− δ2.

Proof of Proposition 2.

In the second period, the firm’s profit maximization problem is given by

maxq2t,q2u

Π2 = (p2t− c)q2t + p2uq2u− s(q2t− q2u)

s.t. q2u ≤ q2t

q2t ≤ q1

q2t, q2u ≥ 0

Here, p2u = δ(1−q1−q2u) and p2t = (1−δ)(1−q2t). The hessian of the objective function is given

by H =(−2(1− δ) 0

0 −2δ

), whose leading coefficient −2(1− δ) is negative and whose determinant

4δ(1−δ) is positive. Thus, the hessian is negative definite and the profit function is jointly concave

in (q2t and q2u). The Lagrangean is given by

L(q2t, q2u) = [(1− δ)(1− q2t)− c]q2t + δ(1− q1− q2u)q2u− s(q2t− q2u)−λ1(q2u− q2t)

−λ2(q2t− q1) +µ1q2t +µ2q2u

with first-order conditions,

σ1(q2t, λ1, λ2, µ1) .=∂L∂q2t

= 1− c− δ− s+λ1−λ2 +µ1− 2(1− δ)q2t = 0


σ2(q2u, λ1, µ2) .=∂L∂q2u

= δ+ s− δq1− 2δq2u−λ1 +µ2

Since the profit function is concave, necessary and sufficient conditions for optimality (Kuhn-

Tucker conditions) are that the first-order conditions (FOC) are satisfied and λ1(q2u − q2t) = 0,

λ2(q2t− q1) = 0, µ1q2t = 0, µ2q2u = 0 and λ1, λ2, µ1, µ2 ≥ 0. There are 16 possible cases but we can

rule out certain cases using the above conditions. If q2t = 0, then only q2u = 0 is possible. If q2t = q1

holds, then only q2t > 0 is possible. If q2t = q2u and q2u = 0 hold together, then only q2t = 0 is

possible. This leaves seven candidate solutions that satisfy the Kuhn-Tucker conditions and are

defined as cases 1-7 below.

Case 1. q2t = q2u = 0. Then λ2 = 0 and λ1, µ1, µ2 ≥ 0. Solving σ1(0, λ1,0, µ1) = 0 and σ2(0, λ1, µ2) =

0 gives µ1 = −1 + c + δ + s − λ1 and µ2 = λ1 − δ(1 − q1) − s. µ1 ≥ 0 and µ2 ≥ 0 hold for λ1 ≤

−1+c+δ+s and δ+s−δq1 ≤ λ1, respectively. These two conditions hold together only if 1−cδ≤ q1.

λ1 ≥ 0 and λ1 ≤−1 + c+ δ+ s hold together only if 1− c− δ≤ s. Thus, the required conditions for

this case to apply are 1− c− δ≤ s and 1−cδ≤ q1.

Case 2. 0 < q2t < q1 and q2u = 0. Then λ2 = λ1 = µ1 = 0 and µ2 ≥ 0. Solving σ1(q2t,0,0,0) = 0

and σ2(0,0, µ2) = 0 gives q2t = 1−c−δ−s2(1−δ) and µ2 =−δ− s+ δq1. q2t < q1, q2t > 0 and µ2 ≥ 0 hold for

1−δ−c−s2(1−δ) < q1, s < 1− δ− c and δ+s

δ≤ q1, respectively. Since s≥ 0 and 0< δ ≤ 1, we have 1≤ δ+s

δ.

This gives us 1≤ q1, which cannot happen and this case is ruled out.

Case 3. 0< q2t = q1 and q2u = 0. Then λ1 = µ1 = 0 and λ2, µ2 ≥ 0. Solving σ1(q1,0, λ2,0) = 0 and

σ2(0,0, µ2) = 0 gives µ2 =−δ(1− q1)− s and λ2 = 1− c− δ− s− 2q1(1− δ). λ2 ≥ 0 and µ2 ≥ 0 hold

for q1 ≤ 1−c−δ−s2(1−δ) and δ+s

δ≤ q1, respectively. Since s≥ 0 and 0< δ≤ 1, we have 1≤ δ+s

δ. Thus, 1≤ q1,

which cannot happen and this case is ruled out.

Case 4. 0 < q2u < q2t < q1. Then λ2 = λ1 = µ1 = µ2 = 0. Solving σ1(q2t,0,0,0) = 0 and

σ2(q2u,0,0) = 0 gives q2t = 1−c−δ−s2(1−δ) and q2u = δ−δq1+s

2δ. q2t > 0, q2u > 0, q2t < q1 and q2u < q2t hold

for s < 1− δ − c, q1 <δ+sδ

, 1−c−δ−s2(1−δ) < q1 and cδ+s

δ(1−δ) < q1, respectively. q1 ≤ δ+sδ

is always satisfied

for all s≥ 0 and δ > 0 as q1 ≤ 1. Thus, the required conditions for this case to apply are 0≤ s <

1− δ− c and max{ cδ+sδ(1−δ) ,

1−c−δ−s2(1−δ) }< q1. We have max{ cδ+s

δ(1−δ) ,1−c−δ−s

2(1−δ) }= cδ+sδ(1−δ) if δ(1−3c−δ)

2+δ≤ s and


max{ cδ+sδ(1−δ) ,

1−c−δ−s2(1−δ) }= 1−c−δ−s

2(1−δ) if s < δ(1−3c−δ)2+δ

.

Case 5. 0 < q2u < q2t = q1. Then λ1 = µ1 = µ2 = 0 and λ2 ≥ 0. Solving σ1(q1,0, λ2,0) = 0 and

σ2(q2u,0,0) = 0 gives q2u = δ+s−δq12δ

and λ2 = 1− c− δ − s− 2q1(1− δ). λ2 ≥ 0 and q2u < q1 hold

for q1 ≤ 1−c−δ−s2(1−δ) and δ+s

3δ< q1, respectively. Thus, we need δ+s

3δ< q1 ≤ 1−c−δ−s

2(1−δ) . δ+s3δ

< q1 and q1 ≤

1−c−δ−s2(1−δ) hold together only if s < δ(1−3c−δ)

2+δ. In addition, q1 ≤ 1−c−δ−s

2(1−δ) and q1 > 0 hold together only

if s < 1− δ− c. .

Case 6. 0 < q2u = q2t < q1. Then λ2 = µ1 = µ2 = 0 and λ1 ≥ 0. Solving σ1(q2t, λ1,0,0) = 0 and

σ2(q2t, λ1,0) = 0 gives q2t = q2u = 1−c−δq12

and λ1 = cδ+ s− δ(1− δ)q1. q2t > 0, q2t < q1 and λ1 ≥ 0

hold for q1 <1−cδ

, 1−c2+δ

< q1 and q1 ≤ cδ+sδ(1−δ) , respectively. Thus, the required condition for this

case to apply is 1−c2+δ≤ q1 ≤min{ 1−c

δ, cδ+sδ(1−δ)}.

1−c2+δ

< 1−cδ

always holds for c, δ ∈ [0,1]. 1−c2+δ≤ q1 and

q1 ≤ cδ+sδ(1−δ) hold together for δ(1−3c−δ)

1+δ≤ s. min{ 1−c

δ, cδ+sδ(1−δ)} = cδ+s

δ(1−δ) if s < 1− δ − c, since s ≥ 0.

min{ 1−cδ, cδ+sδ(1−δ)}= 1−c

δfor 1− δ− c≤ s.

Case 7. 0 < q2u = q2t = q1. Then µ1 = µ2 = 0 and λ2, λ1 ≥ 0. Solving σ1(q1, λ1, λ2,0) = 0 and

σ2(q1, λ1,0) = 0 gives λ1 = δ−3δq1 +s and λ2 = 1−c−q1(2+δ). λ1 ≥ 0 and λ2 ≥ 0 hold for q1 ≤ δ+s3δ

and q1 ≤ 1−c2+δ

, respectively. Thus, the required condition for this case to apply is q1 ≤min{ δ+s3δ, 1−c

2+δ}.

min{ δ+s3δ, 1−c

2+δ}= δ+s

3δif s < δ(1−3c−δ)

2+δand min{ δ+s

3δ, 1−c

2+δ}= 1−c

2+δif δ(1−3c−δ)

2+δ≤ s.

Thus, there are five candidate solutions (as shown in Figure EC.1) which can be summarized by

three conditions as follows:

Condition 1. 0 ≤ s < δ(1−3c−δ)2+δ

: Cases 7, 5 and 4 apply for q1 ≤ δ+s3δ

, δ+s3δ

< q1 ≤ 1−c−δ−s2(1−δ) and

1−c−δ−s2(1−δ) < q1, respectively.

Condition 2. δ(1−3c−δ)2+δ

≤ s < 1− δ− c: Case 7, 6 and 4 apply for q1 ≤ 1−c2+δ

, 1−c2+δ

< q1 ≤ cδ+sδ(1−δ) and

cδ+sδ(1−δ) < q1, respectively.

Condition 3. 1− c− δ ≤ s: Case 7, 6 and 1 apply for q1 ≤ 1−c2+δ

and 1−c2+δ

< q1 <1−cδ

and 1−cδ≤ q1,

respectively.

We now proceed to solve the firm’s first-period decision. In period 1, the firm chooses q1 to

maximize Π(q1) = Π1(q1)+Π2(q1), where Π1(q1) = (p1−c)q1. We denote Πi(q1) as the profit function


Figure EC.1 Five candidate solutions summarized over three conditions on s

4Partial Recovery &

Remarketing

6Partial Recovery & Full

Remarketing

7 Full Recovery & Remarketing

5Full Recovery

& Partial

Remarketing

1Inactive

under Case i, the unconstrained optimum q1 for Case i as qiu1 and the constrained maximizer of Πi

over the range it is defined as qi∗1 .

Condition 1. 0≤ s < s0(c, δ) .= δ(1−3c−δ)2+δ

: Let qd = δ+s3δ

and qe = 1−c−δ−s2(1−δ) . Cases 7, 5 and 4 require

q1 ≤ qd, qd < q1 ≤ qe and qe < q1, respectively. Note that the profit function under all three cases is

concave in q1, Π7(qd) = Π5(qd) and Π5(qe) = Π4(qe). In addition, since we have s≥ 0, the condition

0≤ s < s0 will only hold if s0 > 0. This happens if and only if c < 1−δ3

. The derivatives of the profit

function evaluated at the thresholds is given by

Π′7(qd) =

2δ− 6δc− 3δ2− s(4 + 6δ)3δ

Π′5(qd) =

4δ− 12δc− 3δ2− s(8 + 9δ)6δ

Π′

5(qe) = Π′

4(qe) =δ(−1 + 5c+ δ+ 5s)

4(1− δ)

Let sa(c, δ).= δ(2−6c−3δ)

4+6δbe defined such that Π

′7(qd) = 0, sb(c, δ)

.= δ(4−12c−3δ)

8+9δsuch that Π

′5(qd) =

0 and sc(c, δ).= 1−5c−δ

5such that Π

′4(qe) = Π

′5(qe) = 0. These thresholds on s are such that the

unconstrained maximizer of the associated function is below or above the associated q1 threshold

depending on whether s is less than or more than the threshold on s. It is easy to show that ∀

δ ∈ [0,1] and ca.= 1−2δ

5, the following set of inequalities hold


c < ca⇔ sa ≤ sb < s0 < sc

ca ≤ c⇔ sa ≤ sc < s0 < sb

For the rest of this analysis, we refer to c < ca as Condition 1a and ca ≤ c < 1−δ3

as Condition 1b.

We begin our analysis with Condition 1a.

Condition 1a. c < ca: Under this condition, we have sa ≤ sb < s0 < sc.

A. If s≤ sa, we have Π′7(qd)≥ 0, Π

′5(qd)> 0 and Π

′5(qe),Π

′4(qe)< 0. This implies that q7∗

1 = qd,

q5∗1 = q5u

1 = 2(2−2c−δ−2s)

8−3δ, q4∗

1 = qe, Π(q1) is increasing over the range q1 ∈ [0, qd] and decreasing over

the range q1 ∈ (qe,1]. Thus, if s≤ sa, the optimum is reached under Case 5 (in the range q1 ∈ (qd, qe]).

B. If sa < s< sb, we have Π′5(qd)> 0 and Π

′7(qd),Π

′5(qe),Π


1 = q7u1 =

2−2c+δ4+6δ

, q5∗1 = q5u

1 = 2(2−2c−δ−2s)

8−3δ, q4∗

1 = qe and Π(q1) is decreasing over the range q1 ∈ (qe,1].

We need to compare the optimal profits under Case 5 and Case 7 which are given by Π5(q5∗1 ) =

δ(16c2+(4−δ)2−16c(2−δ))−2δs(8−16c−5δ)+s(8+13δ)

4δ(8−3δ)and Π7(q7∗

1 ) = (2−2c+δ)2

8+12δ. Let σ(s) .= Π5(q5∗

1 ) − Π7(q7∗1 ).

Since σ′′(s) = 8+13δ

16δ(1−δ) > 0, σ(sa) = 9δ2(2−2c+δ)2

16(8−3δ)(2+3δ)2> 0 and σ(sb) = −9δ2(2−2c+δ)2

4(8+9δ)2(2+3δ)< 0, σ(s) is convex in

c, positive at s= sa and negative at s= sb. Thus, there exists an unique value of s ∈ (sa, sb) such

that σ(s) = 0, given by s1(c, δ) .= δ(2+3δ)(8−16c−5δ)−δ(2−2c+δ)√

(8−3δ)(2+3δ)

(2+3δ)(8+13δ). Thus, if sa < s < s1, the

optimum lies under Case 5 (in the range q1 ∈ (qd, qe]) and if s1 ≤ s < sb, the optimum lies under

Case 7 (in the range q1 ∈ [0, qd]).

C. If sb ≤ s < s0, we have Π′5(qd)≤ 0 and Π

′7(qd),Π

′5(qe),Π


1 = q7u1 =

2−2c+δ4+6δ

, q5∗1 = qd and q4∗

1 = qe. Since Π(q1) is decreasing over the range q1 ∈ (qd,1], under this setting,

the optimum is reached under Case 7 (in the range q1 ∈ [0, qd]).

Condition 1b. ca ≤ c: Under this condition, we have sa ≤ sc < s0 < sb. This gives us s ≤ sb

under this setting, which implies that Π′5(qd)≥ 0.

A. If s≤ sa, we have Π′7(qd)> 0, Π

′5(qd)≥ 0 and Π

′5(qe),Π

′4(qe)< 0, which is same as A of 1a.

Thus, if s≤ sa, the optimum is reached under Case 5 (in the range q1 ∈ (qd, qe]).

B. If sa < s < sc, we have Π′5(qd)> 0 and Π

′7(qd),Π

′5(qe),Π

′4(qe)< 0, which is same as B of 1a.

Thus, we need to compare the optimal profits under Case 5 and Case 7. Let σ(s) .= Π5(q5∗1 )−Π7(q7∗

1 ).


Since σ′′(s) = 8+13δ

16δ(1−δ) > 0, σ(sa) = 9δ2(2−2c+δ)2

16(8−3δ)(2+3δ)2> 0 and σ(sc) = −9δ2(2−2c+δ)2

4(8+9δ)2(2+3δ)< 0, σ(s) is convex

in c, positive at s = sa and negative at s = sc. Thus, there exists an unique value of s ∈ (sa, sc)

such that σ(s) = 0, given by s1(c, δ) .= δ(2+3δ)(8−16c−5δ)−δ(2−2c+δ)√

(8−3δ)(2+3δ)

(2+3δ)(8+13δ). Thus, if s < s1, the

optimum lies under Case 5 (in the range q1 ∈ (qd, qe]) and if s1 ≤ s, the optimum lies under Case 7

(in the range q1 ∈ [0, qd]).

C. If sc ≤ s < s0, we have Π′5(qe),Π

′4(qe) ≥ 0, Π

′5(qd) > 0 and Π

′7(qd) < 0.

This implies that q7∗1 = q7u

1 = 2−2c+δ4+6δ

, q5∗1 = qe and q4∗

1 = q4u1 = 2(1−c−s)

4+δand Π(q1)

is increasing over the range q1 ∈ (qd,1]. Thus, we need to compare Π4(q4∗1 ) =

−2 s (4−c (8−3 δ)−4 δ) δ+s2 (4+5 δ−4 δ2)+δ (c2 (8−3 δ)+(1−δ) (8+δ)−2 c (1−δ) (8+δ))

4 (1−δ) δ (4+δ)and Π7(q7∗

1 ) = (2−2c+δ)2

8+12δto find

which region the optimum would lie in.

Let ω(s) .= Π7(q7∗1 )−Π4(q4∗

1 ). Since ω′′(s) = −(4+5δ−4δ2)

2δ(1−δ)(4+δ)< 0, ω(s) is concave in s. We cannot say

anything analytically about the sign of ω(sc) and ω(s0). Thus, we numerically find the minimum

values of ω(sc) and ω(s0) over the range they are defined by solving the following problems:

minc,δ

ω(sc) minc,δ

ω(s0)

s.t. ca ≤ c <1− δ

3s.t. ca ≤ c <

1− δ3

δ≤ 1 δ≤ 1

δ, c≥ 0 δ, c≥ 0

We find that both of these values are non-negative. Due to concavity of ω(s), we conclude that

ω(s)≥ 0 on s ∈ [sc, s0] or Π7(q7∗1 )≥Π4(q4∗

1 ) for all s ∈ [sc, s0]. Thus, if sc ≤ s < s0, the optimum is

reached under Case 7 (in the range q1 ∈ [0, qd]).

The results under both Condition 1a (c ≤ ca) and Condition 1b (ca < c) are exactly the same

and thus the results under Condition 1 (0≤ s < s0(c, δ)) can be summarized as follows:Range Case q∗1 q2t(q∗1) q2u(q∗1)

s < s1(c, δ) 5 2(2−2c−δ−2s)

8−3δq∗1

δ+s−δq∗12δ

s1(c, δ)≤ s 7 2−2c+δ4+6δ

q∗1 q2t(q∗1)

where s1(c, δ) .= δ(2+3δ)(8−16c−5δ)−δ(2−2c+δ)√

(8−3δ)(2+3δ)

(2+3δ)(8+13δ).


Condition 2. s0(c, δ) ≤ s < 1− δ − c: Let qf = 1−c2+δ

and qg = cδ+sd(1−δ) . Cases 7, 6 and 4 require

q1 ≤ qf , qf < q1 ≤ qg and qg < q1, respectively. Note that the profit function under all three cases

is concave in q1, Π7(qf ) = Π6(qf ) and Π6(qg) = Π4(qg). In addition, s ≥ 0 and s < 1− δ − c hold

together if c < 1− δ. The derivatives of the profit function evaluated at the thresholds is given by

Π′

7(qf ) =−δ(2− 4c− δ)

2 + δΠ′

6(qg) =δ(2− 6c− 2δ+ cδ2)− s(4 + 4δ− 3δ2)

2δ(1− δ)

Π′6(qf ) =

−δ(2− 6c− 3δ+ δc)2(2 + δ)

Π′4(qg) =

δ(2− 6c− 2δ+ cδ)− s(4 + 3δ− 2δ2)2δ(1− δ)

Let sd(c, δ).= δ(2−6c−2δ+cδ2)

(2−δ)(2+3δ)be defined such that Π

′6(qg) = 0, se(c, δ)

.= δ(2−6c−2δ+cδ)

4+3δ−2δ2such that

Π′4(qg) = 0, cb(δ)

.= 2−3δ6−δ such that Π

′6(qf ) = 0, and ce(δ)

.= 2−δ4

such that Π′7(qf ) = 0. Since c1− c2 =

−(2+δ)2

4(6−δ) < 0 and sd− se = −2δ2(1−δ)2(1−c+δc)16+28δ−8δ2−17δ3+6δ4

< 0, we have cb < ce and sd < se.

It is easy that

s0 ≤ sd⇔2− 3δ6− δ

.= cb ≤ c

s0 < se⇔ ca ≤ c

sd ≥ 0⇔ c < cc.=

2(1− δ)6− δ2

se ≥ 0⇔ c < cd.=

2(1− δ)6− δ

Using the above inequalities, ∀ δ ∈ [0,1], it is easy to show that the following set of inequalities

hold

ca < cb ≤ cc ≤ cd < ce

Based on the above conditions, we can summarize the following sub-cases to analyze based on

values of c and s,

Condition 2a. If c≤ ca, then sd < se ≤ s0 < 1− δ− c.

Condition 2b. If ca < c≤ cb, then sd < s0 < se < 1− δ− c.

Condition 2c. If cb < c< cc, then s0 < sd < se < 1− δ− c.

Condition 2d. If cc ≤ c < cd, then sd < s0 < se < 1− δ− c.


Condition 2e. If cd ≤ c <min(ce,1− δ), then se < s0 < 1− δ− c.

Condition 2f. If min(ce,1− δ)≤ c, then se < s0 < 1− δ− c.

Condition 2a. c ≤ ca: Under this condition we have se < s0 ≤ s < 1 − δ − c. Thus, we have

Π′7(qf ),Π

′6(qf ),Π

′6(qg),Π

′4(qg)< 0. This implies that Π(q1) is decreasing over the range q1 ∈ (qf ,1].

Thus, under this condition, the optimum lies under Case 7 (in the range q1 ∈ [0, qf ]). Since Π′7(qf )<

0, q7∗1 = q7u

1 = 2−2c+δ4+6δ

.

Condition 2b. ca < c≤ cb, then sd < s0 < se < 1− δ− c.

A. If s < se, we have Π′7(qf ),Π

′6(qf ),Π

′6(qg)< 0 and Π

′4(qg)> 0. This implies that q7∗

1 = q7u1 = 2−2c+δ

4+6δ,

q6∗1 = qf , q4∗

1 = q4u1 = 2(1−c−s)

4+δ, Π(q1) is decreasing over the range q1 ∈ (qf , qg] and there is a local

maximizer q7∗1 ∈ [0, , qf ] and q4∗

1 ∈ (qg,1]. This implies that we need to compare Π7(q7∗1 ) and Π4(q4∗

1 )

to find where the optimum is reached.

Let ω(s) .= Π7(q7∗1 )−Π4(q4∗

1 ). Note that ω′(s)> 0 if and only if s < δ(4−8c−4δ+3δc)

4+5δ−4δ2. It is easy to

show that se <δ(4−8c−4δ+3δc)

4+5δ−4δ2. Since in this case, s < se we have ω(s) is increasing in s on [s0, se).

It is analytically difficult to determine the sign of ω(s) at s = s0. Thus, we numerically find the

minimum value of ω(s0) by solving the following problem:

minc,δ

ω(s0)

s.t. ca < c≤ cb

δ≤ 1

δ, c≥ 0

We find that this value is non-negative and since ω(s) is increasing in s, we conclude that ω(s)≥ 0

on s ∈ [s0, se). This implies that Π4(q4∗1 )≤Π7(q7∗

1 ) for all s ∈ [s0, se), the optimum lies under Case

7 (in the range q1 ∈ [0, qf ]) and is q7∗1 = 2−2c+δ

4+6δ.

B. If se ≤ s < 1− δ− c, then Π′7(qf ),Π

′6(qf ),Π

′6(qg)< 0 and Π

′4(qg)≤ 0, which is same as 2a. Thus,

under this condition, the optimum lies under Case 7 (in the range q1 ∈ [0, qf ]) and is q7∗1 = 2−2c+δ

4+6δ.

Condition 2c. If cb < c< cc, then s0 < sd < se < 1− δ− c.


A. If s0 ≤ s ≤ sd, then Π′7(qf ) < 0, Π

′6(qg) ≥ 0, and Π

′6(qf ),Π

′4(qg) > 0. This implies that q7∗

1 =

q7u1 = 2−2c+δ

4+6δ, q6∗

1 = qg, q4∗1 = q4u

1 = 2(1−c−s)4+δ

, Π(q1) is increasing over the range q1 ∈ (qf , qg]. There is

a local maximizer q7∗1 ∈ [0, qf ] and q4∗

1 ∈ (qg,1]. Thus, to find which case the optimum lies under,

we need to compare Π4(q4∗1 ) and Π7(q7∗

1 ).

Let ω(s) .= Π7(q7∗1 )−Π4(q4∗

1 ). Note that ω′(s)> 0 if s < se. It is analytically difficult to determine

the sign of ω(s) at s= s0. Thus, we numerically find the minimum value of ω(s0) over the range it

is defined by solving the following problem:

minc,δ

ω(s0)

s.t. cb < c≤ cc

δ≤ 1

δ, c≥ 0

We find that this value is non-negative. Since ω(s) is increasing in s and s0 < sd, we have that

ω(s)≥ 0 or Π4(q4∗1 )≤Π7(q7∗

1 ) for all s ∈ [s0, sd]. Thus, under this setting, the optimum lies under

Case 7 (in the range q1 ∈ [0, qf ]).

B. If sd < s < se, then Π′7(qf ),Π

′6(qg)< 0 and Π

′6(qf ),Π

′4(qg)> 0. This implies that q7∗

1 = q7u1 =

2−2c+δ4+6δ

, q6∗1 = q6u

1 = 2(1−c+δc)(2−δ)(2+3δ)

, q4∗1 = q4u

1 = 2(1−c−s)4+δ

and we have three local maxima to compare.

Thus, to find which case the optimum lies under, we need to compare Π4(q4∗1 ), Π7(q7∗

1 ) and Π6(q6∗1 ).

We first begin by comparing Π4(q4∗1 ) and Π7(q7∗

1 ).

Let ω(s) .= Π7(q7∗1 )−Π4(q4∗

1 ). Recall that ω′(s)> 0 if s < se. It is analytically difficult to determine

the sign of ω(s) at s= sd. Thus, we numerically find the minimum value of ω(sd) over the range it

is defined by solving the following problem:

minc,δ

ω(sd)

s.t. cb < c< cc

δ≤ 1

δ, c≥ 0


We find that this value is non-negative and since ω(s) is increasing in s, we have ω(s)≥ 0 and

Π4(q4∗1 ) ≤ Π7(q7∗

1 ) for all s ∈ (sd, se]. We now need to compare Π7(q7∗1 ) and Π6(q6∗

1 ), which are

given by Π6(q6∗1 ) = 8(1−c)2+4δ(1−c2)−δ2(3−6c−c2)

4(2−δ)(2+3δ)and Π7(q7∗

1 ) = (2−2c+δ)2

8+12δ. Let ρ(c) .= Π6(q6∗

1 )−Π7(q7∗1 ).

ρ′′(c) = δ2

8+8δ−6δ2> 0, ρ(cb) = −δ2(2+δ)2

4(6−δ)2(2+3δ)< 0 and ρ(cc) = −δ2(1−δ)(8−4δ+δ2)

4(6−δ)2(2−δ)(2+3δ)< 0, i.e. ρ(c) is convex

in c, negative at c = cb and negative at c = cc. Thus, ρ(c) is negative for all c ∈ (cb, cc) and the

optimum lies in Case 7 (in the range q1 ∈ [0, qf ]).

C. If se ≤ s, then Π′6(qf )> 0, Π

′7(qf ),Π

′6(qg)< 0 and Π

′4(qg)≤ 0. Thus, q7∗

1 = q7u1 = 2−2c+δ

4+6δ, q6∗

1 =

q6u1 = 2(1−c+δc)

(2−δ)(2+3δ), q4∗

1 = qg, Π(q1) is decreasing over the range q1 ∈ (qg,1] and there are two local

maxima q7∗1 ∈ [0, qf ] and q6∗

1 ∈ (qf , qg]. Thus, we need to compare the optimal profits under Case

6 and Case 7. This comparison is the same as that under B of 2c. Thus, under this setting, the

optimum lies in Case 7 (in the range q1 ∈ [0, qf ]).

Condition 2d. cc ≤ c < cd, then s0 < se < 1− δ− c.

A. If s < se, then Π′7(qf ),Π

′6(qg)< 0 and Π

′6(qf ),Π

′4(qg)> 0, which is the same as B of Condition

2c. Thus, under the condition, the optimum lies in Case 7 (in the range q1 ∈ [0, qf ]).

B. If se ≤ s, then Π′6(qf )> 0, Π

′7(qf ),Π

′6(qg)< 0 and Π

′4(qg)≤ 0, which is the same as C of 2c.

Thus, under the condition, the optimum lies in the Case 7 (in the range q1 ∈ [0, qf ]).

Condition 2e. cd ≤ c <min(ce,1− δ). Under this setting we have se < s0 ≤ s < 1− δ− c. Thus,

we have Π′6(qf ) > 0 and Π

′7(qf ),Π

′6(qg),Π

′4(qg) < 0. This gives us q7∗

1 = q7u1 = 2−2c+δ

4+6δ, q6∗

1 = q6u1 =

2(1−c+δc)(2−δ)(2+3δ)

, q4∗1 = qg and Π(q1) is decreasing over the range q1 ∈ (qg,1]. Thus, we need to compare

the optimal profits under Case 6 and Case 7. Recall that ρ(c) .= Π6(q6∗1 )−Π7(q7∗

1 ) is convex in c

and negative at c= cd. Since ρ(ce) = δ2(2+δ)2

64(2−δ)(2+3δ)> 0, and ρ(1− δ) = δ2(1−δ)

8+12δ> 0, we have a unique

value of c ∈ [cd,min(ce,1− δ)) given by c1(δ) .=√

2− δ− 1 such that ρ(c) = 0. Thus, if cd < c < c1,

the optimum lies under Case 7 (in the range q1 ∈ [0, qf ]) and if c1 ≤ c <min(ce,1− δ), the optimum

lies under Case 6 (in the range q1 ∈ (qf , qg]).

Condition 2f. min(ce,1− δ)≤ c: Under this setting, we have se < s0 ≤ s < 1− δ− c. Thus, we

have Π′6(qg),Π

′4(qg)< 0 and Π

′7(qf ),Π

′6(qf )> 0. This implies that q6∗

1 = q6u1 = 2(1−c+δc)

(2+3δ)(2−δ) and Π(q1)


is increasing over the range q1 ∈ [0, qf ] and decreasing over the range q1 ∈ (qg,1], thus the optimum

lies under Case 6 (in the range q1 ∈ (qf , qg]) and is q6∗1 = 2(1−c+δc)

(2+3δ)(2−δ) .

The results under Condition 2 (s0(c, δ) ≤ s < 1 − δ − c) are summarized in the table below,

where c1(δ) .=√

2− δ− 1:

Region Case q∗1 q2t(q∗1) q2u(q∗1)c < c1(δ) 7 2−2c+δ

4+6δq∗1 q2t(q∗1)

c1(δ)≤ c 6 2(1−c+δ)(2+3δ)(2−δ)

1−c−δq∗12

q2t(q∗1)

Condition 3. 1− δ − c ≤ s : Let qf = 1−c2+δ

and qh = 1−cδ

. Case 7, 6 and 1 require q1 ≤ qf and

qf < q1 < qh and qh ≤ q1, respectively. Note that the profit function under all three cases is concave

in q1, Π7(qf ) = Π6(qf ) and Π6(qh) = Π1(qh). The derivatives of the profit function evaluated at the

thresholds is given by

Π′

7(qf ) =−δ(2− 4c− δ)

2 + δΠ′

6(qf ) =−δ(2− 6c− 3δ+ δc)

2(2 + δ)

Π′

6(qh) =−4− 2δ+ 3δ2 + c(4 + 2δ− δ2)

2δΠ′

1(qh) =−2− δ+ δ2 + c(2 + δ)

δ

Let cb(δ).= 2−3δ

6−δ be defined such that Π′6(qf ) = 0, ce(δ)

.= 2−δ4

such that Π′7(qf ) = 0, c2(δ) .= 4+2δ−3δ2

4+2δ−δ2

such that Π′6(qh) = 0 and cf (δ) .= 2+δ−δ2

2+δsuch that Π

′1(qh) = 0. It is easy to show that ∀ δ ∈ [0,1],

the following set of inequalities hold

cb < ce < c2 ≤ cf

A. If c ≤ cb, we have Π′6(qf ) ≤ 0 and Π

′7(qf ),Π

′6(qh),Π

′1(qh) < 0. This implies that q7∗

1 = q7u1 =

2−2c+δ4+6δ

, q6∗1 = qf , q1∗

1 = qh and Π(q1) is decreasing over the range q1 ∈ (qf ,1]. Thus, if c ≤ cb, the

optimum is reached under Case 7 (in the range q1 ∈ [0, qf ]), is given by q7∗1 = 2−2c+δ

4+6δand Π∗ =

(2−2c+δ)2

8+12δ.

B. If cb < c < ce, we have Π′6(qf ) > 0 and Π

′7(qf ),Π

′1(qh),Π

′6(qh) < 0. This implies that q7∗

1 =

q7u1 = 2−2c+δ

4+6δ, q6∗

1 = q6u1 = 2(1−c+δc)

(2+3δ)(2−δ) , q1∗1 = qh and Π(q1) is decreasing over the range q1 ∈ [qh,1].

Thus, we need to compare the optimal profits under Case 6 and Case 7 which are given by

Π6(q6∗1 ) = 8(1−c)2+4δ(1−c2)−δ2(3−6c−c2)

4(2−δ)(2+3δ)and Π7(q7∗

1 ) = (2−2c+δ)2

8+12δ. Let ρ(c) .= Π6(q6∗

1 ) − Π7(q7∗1 ). Since


ρ′′(c) = δ2

8+8δ−6δ2> 0, ρ(cb) = −δ2(2+δ)2

4(6−δ)2(2+3δ)< 0 and ρ(ce) = δ2(2+δ)2

64(2−δ)(2+3δ)> 0, ρ(c) is convex in c, neg-

ative at c = cb and positive at c = ce. Thus, there exists a unique value of c ∈ (cb, ce) given by

c1(δ) .=√

2− δ− 1 such that ρ(c) = 0. Thus, if c < c1, the optimum lies under Case 7 (in the range

q1 ∈ [0, qf ]) and if c1 ≤ c, the optimum lies under Case 6 (in the range q1 ∈ (qf , qh)).

C. If ce ≤ c < c2, we have Π′7(qf )≥ 0, Π

′6(qf )> 0 and Π

′6(qh),Π

′1(qh)< 0. This implies that q7∗

1 =

qf , q6∗1 = q6u

1 = 2(1−c+δc)(2+3δ)(2−δ) , q

1∗1 = qh, Π(q1) is increasing over the range q1 ∈ [0, qf ] and decreasing over

q1 ∈ [qh,1]. Thus, if ce ≤ c < c2, the optimum is reached under Case 6 (in the range q1 ∈ (qf , qh)).

D. If c2 ≤ c < cf , we have Π′7(qf ),Π

′6(qf )> 0, Π

′6(qh)≥ 0 and Π

′1(qh)< 0. This implies that q7∗

1 =

qf , q6∗1 = qh, q1∗

1 = qh, Π(q1) is non-decreasing over the range q1 ∈ [0, qh) and non-increasing over

q1 ∈ [qh,1]. Thus, we have Π1(q1∗1 ) = Π6(q6∗

1 ), where q6∗1 = q1∗

1 = qh, and without loss of generality,

we can say the optimum lies under Case 1 (in the range q1 ∈ [qe,1]).

E. If cf ≤ c, we have Π′7(qf ),Π

′6(qf ),Π

′6(qh) > 0 and Π

′1(qh) ≥ 0. This implies that q7∗

1 = qf ,

q6∗1 = qh, q1∗

1 = q1u1 = 1−c+δ

2(1+δ)and Π(q1) is increasing over the range q1 ∈ [0, qh). Thus, if cf ≤ c, the

optimum lies under Case 1 (in the range q1 ∈ [qe,1]).

The results under Condition 3 (1 − δ − c ≤ s) are summarized in the table below, where

c1(δ) .=√

2− δ− 1 and c2(δ) .= 4+2δ−3δ2

4+2δ−δ2 .

Case q∗1 q2t(q∗1) q2u(q∗1)7 c < c1(δ) 2−2c+δ

4+6δq∗1 q2t(q∗1)

6 c1(δ)≤ c < c2(δ) 2(1−c+δ)(2+3δ)(2−δ)

1−c−δq∗12

q2t(q∗1)1 c2(δ)≤ c max{ 1−c

δ, 1−c+δ

2(1+δ)} 0 0

Summary of Results: We can now summarize the results from the analysis for each of the three

conditions. Note that since δ ∈ [0,1], the following set of inequalities hold

0≤ c1(δ)< c2(δ)≤ 1.

where, c1(δ) .=√

2− δ − 1 and c2(δ) .= 4+2δ−3δ2

4+2δ−δ2 . The final results are summarized in the table

below:


Condition q∗1 q2t(q∗1) q2u(q∗1)5 c < c1(δ) & s < s1(c, δ) 2(2−2c−δ−2s)

8−3δq∗1

δ+s−δq∗12δ

7 c < c1(δ) & s1(c, δ)≤ s 2−2c+δ4+6δ

q∗1 q2t(q∗1)6 c1(δ)≤ c < c2(δ) 2(1−c+δ)

(2+3δ)(2−δ)1−c−δq∗1

2q2t(q∗1)

1 c2(δ)≤ c max{ 1−cδ, 1−c+δ

2(1+δ)} 0 0

where, s1(c, δ) .= δ(2+3δ)(8−16c−5δ)−δ(2−2c+δ)√

(8−3δ)(2+3δ)

(2+3δ)(8+13δ), c1(δ) .=

√2− δ− 1 and c2(δ) .= 4+2δ−3δ2

4+2δ−δ2 .

Proof of Proposition 3. Based on the optimal recovery and disposal strategies and using the

frame-work described in Section 3.2, we compare the relative profitability and environmental

impact under leasing and selling. Recall that since the environmental impact of disposal per unit

is the same whether the firm or the consumer disposes the unit, the total environmental impact

under production is equal to that under disposal. Since c1(δ)− c(δ) =−(2− δ2) +√

2− δ < 0 and

c(δ)− c2(δ) = −δ2(2+2δ−δ2)

4+2δ−δ2 < 0, for all δ ∈ [0,1], we have c1(δ)< c(δ)≤ c2(δ). In addition, since the

maximum value of s1(c, δ)− s(c, δ) over the range c < c(δ) is 0, we have that s1(c, δ)− s(c, δ)≤ 0

or s1(c, δ)< s(c, δ).

A. If c < c1(δ) and 0 ≤ s < s1(c, δ): The total profit under leasing

is Π∗ = δ(2+2c2−4c(1−s)−s(2−s))+s2−δ2(2+c2−(2c+s)(2−s))4δ(1−δ) and under selling is Π∗ =

δ(16c2+(4−δ)2−16c(2−δ))−2δs(8−16c−5δ)+(8+13δ)s2

4δ(8−3δ). The difference is ∆Πa(s, c)

.= Π∗ − Π∗ =

δ(δ2+c2(2+3δ)+2(1−s)2−c(4−4δ−4s−6δs)−δ(3−4s−3s2))

4(1−δ)(8−3δ). Since ∂∆Πa(s,c)

∂s= ∂Πa(s,c)

∂c= δ(2+3δ)

(16−6δ)(1−δ) > 0, the

minimum value of ∆Πa(s, c) will be ∆Πa(0,0) = δ(2−δ)4(8−3δ)

> 0 for all 0 ≤ δ ≤ 1. Thus, ∆Πa(s, c) is

positive for all c∈ [0, c1(δ)) and s∈ [0, s1(c, δ)) and leasing is more profitable.

In this setting, both the total quantity produced (and disposed) under leasing and sell-

ing are given by 2(1−c)+δc−2δ+δs−2s

2(1−δ) and 4(2(1−c)−δ−2s)

8−3δ, respectively. The difference is given by

δ(2−2δ−c(2+3δ)−s(2+3δ))

2(1−δ)(8−3δ), which takes a minimum value of 0 on the range of parameters for this set-

ting. Thus, the environmental impact due to production (or disposal) is higher under leasing. The

quantity of products used under leasing is s(1−δ)+δ(2−c)2δ

and under selling is 8s+δ(20−12c−9δ−15s)

2δ(8−3δ). The

difference is given by c(4+3δ)−4(1−s)+3δ(1+s)

16−6δ, whose maximum value on the range of parameters for

this setting is −0.1. Thus, this difference is negative and the environmental impact due to use is

lower under leasing.


B. c < c1(δ) and s1(c, δ) ≤ s < s(c, δ): The total profit under leasing is Π∗ =

δ(2+2c2−4c(1−s)−s(2−s))+s2−δ2(2+c2−(2c+s)(2−s))4δ(1−δ) and under selling is Π∗ = (2−2c+δ)2

8+12δ. If ∆Πb(s)

.= Π∗−Π∗,

then ∆Π′b(s) = s(1+δ−δ2)−δ2(1−c)−δ(1−2c)

2δ(1−δ) which is negative for s < s(c, δ). In addition, ∆Πb(s(c, δ)) =

δ2(c+δ)2

4(2+5δ+δ2−3δ3)> 0 and thus, ∆Πb(s) is positive for all s∈ [s1(c, δ), s(c, δ)) and c∈ [0, c1(c, δ)). Thus,

leasing is more profitable.

When the firm leases, the total quantity produced (and disposed) is 2−2c−δ(2−c)−s(2−δ)2(1−δ) and

under selling is 2−2c+δ2+3δ

. The difference is given by εpb.= −δ2(4−3c−3s)−4s+4δ(1−2c−s)

4+2δ−6δ2. Thus, if s < s

.=

δ(4−8c−4δ+3δc)

(2−δ)(2+3δ), then the environmental impact is lower under selling and if s(c, δ) ≤ s, then the

environmental impact due to production (or disposal) is lower under leasing.

Under leasing, the quantity used is s(1−δ)+2δ−δc2δ

and under selling is 3(2−2c+δ)

4+6δ. The difference is

given by εub.= 2s+3δ2(1−c−s)−δ(2−4c−s)

2δ(2+3δ), where ε

′ub = 1−δ

2δ> 0 and is negative if and only if s < s

.=

δ(2−4c−3δ+3δc)

(1−δ)(2+3δ). Thus, if s < s, the environmental impact due to use is higher under selling and if

s≤ s < s1, then the environmental impact is higher under leasing.

Since s(c, δ) − s(c, δ) = −δ2(c+δ)

(4−3δ2)(1−δ2)≤ 0, we have s(c, δ) ≤ s(c, δ). Thus, if s1(c, δ) ≤ s < s, the

environmental impact due to production (and disposal) is lower under selling and due to use is

lower under leasing. If s(c, δ) ≤ s < s(c, δ), the total environmental impact is lower under selling

and if s(c, δ) ≤ s < s(c, δ), the environmental impact due to production (and disposal) is higher

under selling and due to use is higher under leasing.

C. c < c1(δ) and s(c, δ)≤ s: The total profit under leasing is Π∗ = (2−δ)(1+δ+c2)−2c(2−δ2)

4(1+δ−δ2)and under

selling is Π∗ = (2−2c+δ)2

8+12δ. The difference is given by Π∗ − Π∗ = δ2(c+δ)2

4(2+5δ+δ2−3δ3)> 0, and leasing is

more profitable than selling. The total quantity produced (and disposed) under leasing and selling

is 2(1−c)+δ(c−δ)2(1+δ−δ2)

and 2−2c+δ2+3δ

respectively. The difference is −δ2(c+δ)

4+10δ+2δ2−6δ3> 0 and the environmental

impact under production (and disposal) is higher under selling. The total quantity used under

leasing and selling is given by 3(1−c)+δ(2c−δ)2(1+δ−δ2)

and 3(2−2c+δ)

4+6δrespectively. The difference is given by

δ(c+δ)

2(2+5δ+δ2−3δ3)> 0 and the environmental impact due to use is higher under leasing.

D. c1(δ) ≤ c < c(δ) and 0 ≤ s < s(c, δ): Total profits under leasing and selling are given by

Π∗ = δ(2+2c2−4c(1−s)−s(2−s))+s2−δ2(2+c2−(2c+s)(2−s))4δ(1−δ) and Π∗ =

8 (1−c)2+4(1−c2) δ−(3−6 c+c2) δ24 (2−δ) (2+3 δ)

respectively.


Let Πd(s) = Π∗ − Π∗. Since, Π′d(s) = s(1+δ−δ2)−δ(1−2c−δ+δc)

2δ(1−δ) which is negative under s < s1(c, δ) .=

δ(1−2c−δ+δc)1+δ−δ2 , which implies that Πd(s) is decreasing in s. In addition, Πd(s) = −δ(1+δ−c+δ2c)

4+10δ+2δ2−6δ3> 0.

Thus, Πd(s) is positive over the range 0≤ s < s(c, δ) and leasing is more profitable than selling.

The total quantity produced (and disposed) under leasing is 2(1−c)+δc−2δ+δs−2s

2(1−δ) and under selling

is 4(1−c)+δ(3−c)4+6δ

. The difference is εpd(s).= −δ2(3−2c−3s)−4s+δ(3−7c−4s)

4+2δ−6δ2. Since εpd(s)

′= −(2−δ)

2(1−δ) < 0, we

have εpd(s) is decreasing in s and εpd(0) = δ(3−7c−3δ+2δc)

4+2δ−6δ2, which is negative if and only if 3(1−δ)

7−2δ< c.

Since 3(1−δ)7−2δ

< c1(δ), we have that εpd(0) is negative for all c∈ [c1(δ), c(δ)). Thus, εpd(s) is negative

for all s∈ [0, s(c, δ)) and the environmental impact due to production (or disposal) is lower under

leasing.

The total quantity used under leasing is s+2δ−δc−δs2δ

and is 12(1−c)+δ(2−3δ)+δc(6+δ)

8+8δ−6δ2under selling.

The difference is εud(s).= −4δ+8δc+6δ2−10cδ2−3δ3+2cδ3+4s−7δ2s+3δ3s

2δ(2−δ)(2+3δ). Since ε

′ud(s) = 1−δ

2δ> 0, εud(s) is

increasing in s. In addition, εud(s) = −δ(2−δ2)(1−c+δc)2(4+8δ−3δ2−7δ3+3δ4)

< 0. εud(s) is negative for all s ∈ [0, s(c, δ))

and the environmental impact due to use is higher under selling. Thus, the total environmental

impact under leasing is lower.

E. c1(δ) ≤ c < c(δ) and s(c, δ) ≤ s: The total profit under leasing is Π∗ = (2−δ)(1+δ+c2)−2c(2−δ2)

4(1+δ−δ2)

and under selling is Π∗ = 8(1−c)2+4δ(1−c2)−δ2(3−6c−c2)

4(2−δ)(2+3δ). The difference is given by Π∗ − Π∗ =

δ2(1−c+δc)24(4+8δ−3δ2−7δ3+3δ4)

> 0 and leasing is more profitable. The total quantity produced (and disposed)

under leasing is 2(1−c)+δc−δ22(1+δ−δ2)

and under selling is 4(1−c)−3δ−δc4+6δ

. The difference is −δ(1+δ−c+cδ2)

4+10δ+2δ2−6δ3< 0.

Thus, the environmental impact due to production (and disposal) is strictly lower under leasing.

The total quantity used under leasing and selling is 3(1−c)−2δc+δ2

2(1+δ−δ2)and 12(1−c)+δ(2−3δ)+6δc+δ2c

8+8δ−6δ2, respec-

tively. The difference is given by −δ(1−c+δc)(2−δ2)

2(4+8δ−3δ2−7δ3+3δ4)< 0 and the environmental impact due to use

is lower under leasing. Thus, the total environmental impact is lower under leasing.

F. c(δ) ≤ c < c2(δ): The total profit under leasing and selling is Π∗ = (1−c+δ)24(1+δ)

and Π∗ =

8(1−c)2+4δ(1−c2)−δ2(3−6c−c2)

4(2−δ)(2+3δ), respectively. The difference is given by ∆Πf (c, δ) .= Π∗ − Π∗ =

−4(1−c)2+8δ2(1−c)+δ3(1−c2)−3δ4

4(2−δ)(1+δ)(2+3δ). Since ∆Π

′′f (c) = −(4+δ3)

2(4+8δ+δ2−3δ3)< 0, ∆Πf (c) = δ4(1+δ−δ2)

4(4+4δ−3δ2)> 0 and

∆Πf (c2) = δ6

4(1+δ)(4+2δ−δ2)2> 0, we have ∆Πf (c) is concave in c and positive over the range c ∈

[c(δ), c2(δ)), which implies that leasing is more profitable.


The total quantity produced (and disposed) under leasing and selling is 1−c+δ2(1+δ)

and 4(1−c)+δ(3−c)4+6δ

,

respectively and the difference is given by −2(1+δ)+c(2+2δ+δ2)

4+10δ+6δ2, which is negative for all c∈ [c(δ), c2(δ)).

Thus, the environmental impact due to production (and disposal) is lower under leasing. The total

quantity used under leasing and selling is given by 1+δ−c1+δ

and 12(1−c)+δ(2−3δ)+δc(6+δ)

8+8δ−6δ2, respectively.

The difference is given by−(2(1−c)(2−δ)−3δ2(1−δ)+cδ2(1+δ))

2(2−δ)(1+δ)(2+3δ). This difference is negative over the range

c ∈ [c(δ), c2(δ)) and the environmental impact due to use is lower under leasing. Thus, the total

environmental impact is lower under leasing.

G. c2(δ) ≤ c: Under this case, we have two subcases. If max{ 1−cδ, 1−c+δ

2(1+δ)} = 1−c

δ⇒ c < c3(δ) .=

2+δ−δ22+δ

, then we have that the total profit under leasing and selling is given by Π∗ = (1−c+δ)24(1+δ)

and Π∗ = (1−c)(c−1−δ2)

δ2. The difference is given by − ((2−δ)(1+δ)−c(2+δ))2

4δ2(1+δ)≤ 0. Thus, leasing is more

profitable. Since the total quantity produced (and disposed) and used under leasing are both given

by 1−c+δ2(1+δ)

and those under selling are both given by 1−cδ

, we have that the total environmental

impact is lower under leasing. Thus, the relationship between selling and leasing is the same as the

previous case.

If max{ 1−cδ, 1−c+δ

2(1+δ)}= 1−c+δ

2(1+δ)⇒ c3(δ) .= 2+δ−δ2

2+δ≤ c, the total profit under both leasing and selling

is given by Π∗ = Π∗ = (1−c+δ)24(1+δ)

so the firm should be indifferent between leasing and selling. When

the firm leases, the total quantity produced (and disposed) is 1−c+δ2(1+δ)

and the total quantity used is

1−c+δ(1+δ)

. Under selling, the quantity produced (and disposed) and the quantity used are 1−c+δ2(1+δ)

and

1−c+δ(1+δ)

, respectively. Thus, all these quantities are the same and the total environmental impact

under leasing and selling is the same.�

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