An Analysis of Recycled Content Claims Under Demand Benefit

and Supply Uncertainty

Aditya Vedantam, Krannert School of Management, Purdue University

Ananth Iyer, Krannert School of Management, Purdue University

Paul Lacourbe, CEU Business School, Budapest, Hungary

Abstract

We study how a manufacturer’s recycled content claim decision is impacted by demand for

recycled content through environmentally preferable purchasing (EPP) programs and supply

uncertainty. We model a manufacturer making a recycled content commitment that can com-

mit to a minimum recycled content that holds for every period (product specific claim), or an

average recycled content across periods (time aggregated claim).

Our main contributions are as follows: (a) We establish conditions under which the optimal

product specific claim is larger than the time aggregated claim; (b) we find that simultaneously

easing supply constraints and creating demand side incentives for recycled content can create

win-win conditions; (c) we find that lowering variability in the municipal stream decreases the

recycled content claim if the manufacturer’s collection cost is low, and (d) show that EPP de-

creases the recycled content if the collection cost is sufficiently high. Based on this we suggest

that regulators should adopt calibrated strategies to incentivize higher recycled content and

greater manufacturer profits.

Keywords: Recycled Content Claims, Supply Uncertainty, Green Procurement, Sustainable

Product Design

1

An Analysis of Recycled Content Claims Under Demand Benefit

and Supply Uncertainty

Abstract

We study how a manufacturer’s recycled content claim decision is impacted by demand for

recycled content through Environmentally Preferable Purchasing (EPP) programs and supply

uncertainty. We model a manufacturer making a recycled content commitment that can commit

to a minimum recycled content that holds for every batch (batch specific claim) or as recycled

content averaged across batches (batch average claim).

Our main contributions are as follows: (a) We establish conditions under which the optimal

batch specific claim is larger than the batch average claim; (b) we find that simultaneously

easing supply constraints and creating demand side incentives for recycled content can create

win-win conditions; (c) we find that lowering variability in the municipal stream decreases the

recycled content claim if the manufacturer’s collection cost is low, and (d) show that EPP de-

creases the recycled content if the collection cost is sufficiently high. Based on this we suggest

that regulators should adopt calibrated strategies to incentivize higher recycled content and

greater manufacturer profits. We parameterize our model to the fiberglass insulation industry

and suggest insights.

Keywords: Recycled Content Claims, Supply Uncertainty, Green Procurement, Sustainable

Product Design

1

1 Introduction

1.1 Recycled Content Claims:

Manufacturers often highlight the environmentally friendly features of their products through vol-

untary recycled content claims. We focus on two types of claims: a ‘batch specific claim’ in which

the manufacturer declares a minimum recycled content that is met for each batch, and a ‘batch

average claim’ in which the average recycled content is met across a number of batches. If the

recycled content claim is at 100% then both claims are equivalent. One way consumers can identify

batch specific claims on products is by checking whether the claim is qualified as a ‘minimum’.

Claims not qualified as a minimum (provided the FTC Green Guides allow it) can be regarded as

batch averages.

With an annual purchasing footprint exceeding $500 billion, the green purchasing decisions

of the federal government can significantly impact the demand for products containing recycled

content (Coggburn and Rahm (2005)).The US Environmental Protection Agency (EPA) in its

Comprehensive Procurement Guidelines gives preference to higher recycled content products as

long as price, quality and other factors are competitive, and leaves it to the discretion of the state

agency to require whether the recovered materials content of a product be certified on a batch-

by-batch basis or as an average over a calendar quarter or some other “appropriate averaging

period” (EPA CPG (2014)). Batch-by-batch recycled content levels are preferred, but recycled

content averaged over three months or some other appropriate averaging period is also considered

acceptable1. Most governmental buy recycled programs require some kind of certification to validate

the recycled content in products and many third-party certification agencies like SCS Certified offer

batch-by-batch certification of products.2. We next provide several examples of recycled content

claims which are either batch specific or batch average:

1Recycled Content: What is it and What it is Worth? https://greenspec.buildinggreen.com/article/

recycled-content-what-it-and-what-it-worth-02Recycled Content Certification, Frequently Asked Questions:http://www.scsglobalservices.com/recycled-

content-certification

2

• California’s State Agency Buy Recycled Campaign (SABRC) is a state mandated program

that requires agencies to purchase recycled content products within eleven product categories

including printed paper, fiberglass insulation and plastic products, such that the percentage

of postconsumer3 recycled content in the products is greater than or equal to a recommended

minimum. For example, glass products (windows, test tubes, fiberglass insulation, glass

containers) must contain a minimum of 10 percent postconsumer cullet by weight certified on

a batch-by-batch basis; 4.

• Minimum Content Legislation: California state law mandates that any fiberglass made or sold

in California should contain at least 30 percent postconsumer recycled content (Fiberglass

Recycled Content Act of 1991) where the recycled content is calculated as an annual batch

average.

• The new LEED interpretation ruling states that if the fiberglass content used in a build-

ing construction contains at least 10 percent postconsumer content, where the percent in

expressed as an annual batch average in a specific plant, then the building gets one credit

towards LEED certification and two credits if it contains at least 20 percent PCC 5.

Despite EPP and buy recycled procurement, manufacturers face uncertainty in supply of recy-

cled input. Most post-consumer recycled input is collected by municipalities under single stream

collection, where contamination due to mixing of different waste streams causes the yield of recy-

cled input recovered downstream to be uncertain. The uncertainty in municipal supply and cost of

collection outside the municipal stream can impact the recycled content in products. For example,

despite being committed to make their packaging with 100% recycled PET (rPET), Nestle Waters

North America (NWNA) was only able to incorporate 50% recycled content in all their bottled

water packaging due to inadequate supply of quality rPET (Washburn (2013)). To understand the

dual impact of demand benefit and supply uncertainty, we consider a stylized model of a manu-

3Post-consumer content refers to recycled input obtained from products discarded by consumers post-use.4www.calrecycle.ca.gov/BuyRecycled/StateAgency/Forms/CalRecycle074.pdf5Source:United States Green Building Council & LEED

3

facturer that can impact demand for a single product, for a given final price, by improving on a

single environmental attribute - the fraction of recycled content. We then apply the model to the

glass wool insulation industry and suggest managerial insights. Next, we describe the key research

issues that we investigate in this paper.

1.2 Research Goals:

Given an uncertain availability of recycled material, and associated demand side impact of recycled

content claims, our research goal is to develop a model that answers the following questions: (a)

What are the profit maximizing recycled content claims under a batch specific and a batch average

commitment?, (b) Are there parameter values under which making a batch specific or batch average

claim leads to both increased manufacturer profits and greater environmental impact?, and (c) How

do approaches to stabilize recycled content availability and schemes to increase the demand impact

of recycling claims through EPP impact the recycled content claim?

2 Literature Review

Our paper draws upon literature in research related to sustainable product design, environmental

claims, and inventory management. Also relevant is literature on supply uncertainty and recent

literature on extended producer responsibility involving mandates on recovery and collection. Here,

we discuss the most relevant literature at the intersection of sustainable product design, extended

producer responsibility and inventory management.

Several papers in the extant literature have looked at the impact on product design from a

sustainability perspective. The design attributes considered have varied from material recyclability

(Subramanian et al. (2009)), product durability (Runkel (2003)), modularity of the components

(Agrawal and Ulku (2013)), component commonality (Subramanian et al. (2013)), design for envi-

ronment (Raz et al. (2013)) and rate of introduction of new products (Plambeck and Wang (2009)).

4

We contribute to this literature by considering the amount of recycled content a manufacturer in-

corporates in its product and how it is impacted when supply of recycled input is variable. We thus

focus on recycled content supply as one of the factors influencing the design of sustainable products.

In the marketing literature, the impact of consumer’s increasing appreciation for environmentally

friendly products and the marketing impact of sustainability claims made by corporations have been

studied by Straughan and Roberts (1999), Arora and Henderson (2007) and Sen and Bhattacharya

(2001). Peattie (2001) reviews the evolution of “green marketing” and divides the process into three

stages: ecological marketing, environmental marketing, and sustainable marketing. Other studies

find that as the ‘green’ consumer segment becomes more sophisticated they examine environmental

attributes and related claims more carefully (Ginsberg and Bloom (2004), Mohr and Webb (2005)).

Several papers have looked at various types of extended producer responsibility legislation and

its impact. Atasu et al. (2009) focus on the design of efficient take back legislation when producers

are responsible for end-of-life recycling of their products, while Jacobs and Subramanian (2012)

model the impact of sharing responsibility for product recovery between various supply chain part-

ners. In both Atasu et al. (2009) and Jacobs and Subramanian (2012), the regulator sets mandated

collection/recycling targets, while in our model the manufacturer voluntarily chooses a recycled

content claim. Atasu and Souza (2011) study the impact of product recovery on a firm’s product

quality choice and find that product take-back legislation can lead to higher quality choice as op-

posed to voluntary take-back.

Our paper studies the problem of managing recyclables inventory under two supply sources

with one of sources (municipal supply) being random. As such, our paper draws from literature in

managing inventory under random yield (Zhou et al. (2011), Tao et al. (2012)) and is similar in

structure to papers in the inventory rationing literature (Topkis (1968), Ha (1997), Benjaafar et al.

(2004)), and the revenue management literature (Talluri and Van Ryzin (2006)). Our paper is also

related to Demeester et al. (2013), who find that companies benefit by investing in colocation of

5

recycling and production plants. To the best of our knowledge, the link between design attributes

like recycled content and recycled input supply when manufacturers making voluntary recycling

content claims see a demand benefit has not been explored in literature. Our model will target this

gap. We next state our assumptions and formulate the model.

3 Notation & Assumptions

• Assumption 1: Municipal supply of recycled input every period is independent and identically

distributed with a probability density function (pdf) φ() > 0 and distribution function (cdf)

Φ on a closed set [A, A] with A < Di(r) ∀i, r .

Our model of municipal supply involves a manufacturer that contracts with a municipality to

purchase all the collected recyclables at a pre-agreed upon market price. Prices of cullet in

recycled glass markets have been reported to be stable with little variation over time (Kusa

et al. (2001); also see the appendix for historical prices of cullet). Our assumption is that

the largest realization of supply from the municipal stream is less than the demand. This

assumption is reasonable since imperfect sorting at the consumer end, mixing with other

waste streams leading to contamination, and manual sorting at recyclers causes a significant

quantity of used products to end up in landfills, so the yield from the municipal recyclables

stream is low.

• Assumption 2: Outside the municipal supply, the firm faces a (high) linear cost for collecting

recycled input i.e., ce > cm.

Municipal collection is subsidized by consumers who pay for pick up of recyclables along with

their trash (e.g., pay-as-you-throw programs) or voluntarily drop off recyclables in a curbside

bin. In such cases, the manufacturer does not bear the burden of collection and does not

incur the collection cost. However, outside the municipal stream, the manufacturer faces a

cost to collect recyclables.

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• Assumption 3: Quality of product is unaffected by the amount of recycled content.

In other words, we focus our attention on a single environmental attribute without considering

effects of multiple related attributes like quality. An example is container glass and fiberglass

manufacturing, where the process involves batch melting of cullet along with raw material to

form the final product and the quality of the batch is unaffected by the proportion of cullet

used as long as the input cullet used is as per quality specifications (Cook (1978)).

• Assumption 4: Demand is monotonically increasing in the recycled content claim.

We do not place any restriction on the form of the demand function, only that it be monotone

increasing in the recycled content. In addition to increasing in the recycled content, the

demand also depends on type of claim being made. The federal government’s EPP programs

give purchasing preference to batch specific claims while green building standards e.g., LEED

certification accept batch average claims.

Under these set of assumptions we formulate our model.

3.1 Model Formulation

We formulate the model in two stages:

Stage One (Design Stage): In stage one, the manufacturer declares whether the commitment

will be met each batch (batch specific claim), or on average across batches (batch average claim).

Let Ψi(r) denote the expected stage one profit for a claim type i and recycled content fraction r.

Then the stage one problem is,

max0≤r≤r

Ψi(r), i ∈ {p, c} (1)

where r is an upper bound on the proportion of recycled content that can be used in a batch subject

to technical constraints.

Stage Two (Procurement of recycled input): Stage two is modeled over a finite time horizon

consisting of T − periods where a period is defined as time to manufacture a batch of the product.

We model stage two as a stochastic dynamic program with recourse over a T− period time horizon.

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Each period the manufacturer sources recycled input to meet the recycled content commitment

and satisfy demand, which depends on the recycled content claim in stage one. The manufacturer

observes the realization of the random municipal supply at the beginning of each period and can

take recourse to non-municipal collection to meet the commitment. While traditional inventory

models model carry over finished good inventory, in our model the manufacturer carries recycled

input between periods to meet recycled input commitments and demand. The timeline of events is

as shown in Figure 1.

Manufacturer chooses recycled content r and claim type i

Demand is determined as a function of r and i

𝑇 𝑇 − 1 𝑡 1

𝜉𝑇𝜉𝑇−1

𝜉𝑡

𝜉1

Stage TwoStage One

Initial recycled input inventory 𝑥𝑡

Realization of municipal supply

Collect from non-municipal sources to meet claim

Satisfy periods demand and carry over left over recycled input to next period

𝜉𝑡

Figure 1: Timeline of events

Problem formulation for batch specific claim

Let Πt(xt) = Profit to go function for periods t to 1 with initial recycled input inventory xt. Then,

the stage one expected profit can be written as,

Ψp(r) = ΠT (0), (2)

8

Table 1: Notation.Cost and Revenue Parameters

s price per unit of finished product,cm cost of recycled input from municipal sources,ce cost of self-collection of recycled input,cv cost of procuring raw material,h holding cost per unit of recycled input,

Decision Variables

r fraction of recycled content in product (0 ≤ r ≤ r),r technical limitation on percentage of recycled content in product,

i ∈ {p, c} type of claim - (p) batch specific or (c) batch average,yt recycled input used in production in period t,zt recycled input carried over to the next period to period t+ 1,

Other Notation

xt initial inventory of recycled input in period t,ξt Municipal recycled input supply in period t,

[A, A] support of ξt (A < A),µ mean of ξt,

Di(r) demand per period for claim type i ∈ {p, c},Ψi(r) stage one expected profit for claim type i,Πt(xt) maximum expected profit in period t of stage two

i.e., xT = 0, the manufacturer starts stage two with no initial inventory of recycled input where,

Πt(xt) = EξtΠt(xt, ξt), (3)

Πt(xt, ξt) = maxyt

[sDp(r)− cmξt − ce(yt − xt − ξt)+ − cv(Dp(r)− yt)

− h(xt + ξt − yt)+ + Πt−1(xt−1)], (4)

where xt−1 = (xt + ξt − yt)+ ∀t,

subject to rDp(r) ≤ yt ≤ Dp(r) and x+ = max(x, 0). The first term in (4) is the revenue from

selling Dp(r) units of the finished product at price s in period t. The second term is the cost

of recycled input received from municipal sources; the third term is the cost if recycled input is

collected from non-municipal sources, i.e., if yt is greater than the available supply xt + ξt; the

remaining demand Dp(r) − yt is met with raw material; the fifth term is the cost of holding the

remaining recycled input for the next period, and the last term is the optimal profit for the t − 1

9

period problem with initial recycled input inventory xt−1 = (xt+ ξt−yt)+. The constraints require

that amount of recycled input used each period yt ≥ rDp(r), but not exceed demand i.e, yt ≤ Dp(r).

The terminal value function is Π0(x) = cmx ∀x ≥ 0, where leftover recycled input is salvaged at

cost.

Problem formulation for batch average claim

For the batch average claim, we use an additional state variable, χt = remaining commitment in

period t. If Πt(xt, χt) = maximum expected profit for the t period problem with initial inventory

xt and remaining commitment χt, then we can write the stage one expected profit as,

Ψc(r) = ΠT (0, T rDc(r)). (5)

The manufacturer commits to using χT = TrDc(r) of recycled content across T periods, i.e., rDc(r)

on average, and starts stage two with no initial recycled input inventory where,

Πt(xt, χt) = EξtΠt(xt, χt, ξt), and Πt(xt, χt, ξt) (6)

= maxyt

[sDc(r)− cmξt − ce(yt − xt − ξt)+ − h(xt + ξt − yt)+

− cv(Dc(r)− yt) + Πt−1(xt−1, χt−1)], (7)

where xt−1 = (xt + ξt − yt)+, χt−1 = (χt − yt)+ ∀t,

subject to 0 ≤ yt ≤ Dc(r) and χ1 = 0. The terminal value function is Π0(x, .) = cmx ∀x ≥ 0. The

difference from the batch specific claim is in the presence of an additional state variable, and the

last term of (7), where the minimum commitment for the next period is decreased by the amount of

recycled input used, i.e., (χt − yt)+. The constraint 0 ≤ yt ≤ Dc(r) requires that the amount used

is non negative, and not exceed demand, while χ1 = 0 requires that the commitment be satisfied

by the end of T periods. In the next section, we provide the optimal actions for the stage two

problem for both claim types. The recycled content fraction r is known at this stage. We will solve

the stage one problem later. For ease of exposition, we drop the dependence on r and the claim

type to write demand Dp(r) and Dc(r) as D wherever the meaning is unambiguous.

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4 Model Analysis

4.1 Optimal Policy of the Batch Specific Claim

Lemma 1. The manufacturer collects recycled input from non-municipal sources in period t only

if the available supply xt + ξt is less than or equal to the minimum commitment rD.

Proof. All proofs are provided in the Appendix.

The manufacturer’s commitment requires that a fraction r of demand each period be manu-

factured using recycled input. Lemma 1 states that if the available total supply from on hand

inventory and municipal sources in period t is less than the minimum commitment rD, then the

manufacturer resorts to self collection to meet the remaining commitment.

If collection cost is lower than cost of raw material then we expect the manufacturer to collect

and satisfy all demand with recycled inputs. If there is no demand benefit (bp = 0) the manufacturer

declares no recycled content. Moreover, if the cost of virgin material is greater than municipal

recycled input (cv > cm), then the manufacturer uses all available municipal recycled input in its

production and does not self-collect. Our focus in this paper, is on the more interesting case when

collection cost ce > cv, and there is a demand benefit to recycled content (bp > 0).

Proposition 1. For a recycled content r, there exist a threshold zt(r), in period t, such that the

optimal amount of recycled input used (y∗t ) and carried over (z∗t ) are given by

(y∗t , z∗t ) =

(rD, 0) if xt + ξt < rD (Region R1)

(rD, xt + ξt − rD) if rD ≤ xt + ξt < rD + zt(r) (Region R2)

(xt + ξt − zt(r), zt(r)) if rD + zt(r) ≤ xt + ξt < D + zt(r) (Region R3)

(D,xt + ξt −D) if xt + ξt ≥ D + zt(r) (Region R4),

(8)

if ce > cv else (y∗t , z∗t ) = (D,xt + ξt − D), where zt(r) depends only on r, i.e., independent of ξt,

and zT (r) ≥ ... ≥ zt(r) ≥ zt−1(r) ≥ ...z2(r) ≥ z1(r) = 0.

11

Proposition 1 identifies the amount of recycled input used and the amount carried over each

period. In Region R1, the total supply is less than the minimum commitment, and the manufac-

turer collects to meet the commitment exactly, but does not exceed it. In Region R2, the supply

exceeds the commitment, and the manufacturer carries over what is left after meeting the mini-

mum commitment to the next period. In Region R3, the manufacturer carries over a threshold

amount zt(r) that depends only on the recycled content r and the time index t. The remaining

recycled input is used in production. Finally, in Region R4, the total supply exceeds demand.

The manufacturer satisfies all demand with recycled input, and carries over the leftover recycled

input to the next period. The proposition provides insight into when the manufacturer meets the

commitment exactly (Regions R1 and R2), and when it might choose to exceed the commitment

(Regions R3 and R4). Whether the manufacturer equals or exceeds the commitment depends on

realization of municipal supply in the period. The thresholds are decreasing, which means that the

manufacturers holds less recycled input near the end of the time horizon. This is intuitive since

near the end of the time horizon the manufacturer has fewer periods left to meet the claim and less

recycled input is needed for meeting future claim requirements.

4.2 Optimal Policy for the Batch Average Claim

Proposition 2. The optimal actions for a batch average recycled content claim are as follows: if

ξt < yt = χt − (t − 1)D then y∗t = yt else y∗t = ξt for t ≥ 2, and y∗1 = max(ξ1, χ1), where ξt is the

realization of municipal supply, and χt the remaining commitment in period t respectively.

Under a batch average claim, the manufacturer has greater flexibility to comply with the com-

mitment and can postpone collection until the final period. Proposition 2 states that the manufac-

turer defers collection until the final period unless ξt < χt − (t − 1)D. If ξt < χt − (t − 1)D, the

manufacturer will fall short of the commitment if it defers, even if it uses only recycled input to

meet demand in each of the remaining periods. Thus, given the flexibility afforded by aggregation,

12

the manufacturer uses municipal recycled input each period, and only resorts to collection towards

the end of the time horizon if it cannot meet the commitment. A batch average claim provides

sourcing flexibility by allowing aggregation between periods, whereas in the batch specific claim

holding recycled input between periods buffers against supply variability.

4.3 Stage One problem

We now turn to the stage one problem, where the manufacturer chooses the recycled content

fraction, and type of claim to maximize the expected profit. Since we do not assume a specific

functional form for demand, Ψi(r) need not be concave. To gain insight, we solve the stage one

problem for T = 2, and use numerical results to explore the impact of many (T > 2) batches. In

Proposition 3, we first provide conditions for existence of a unique recycled content decision (r) for

the batch average claim. In Proposition 4, we show that analogous conditions hold for the batch

specific claim.

Proposition 3. For the batch average claim, if Dc(r) is log-concave in r, and 1+Dc(1)

D′c(1)>

s− cvce − cv

,

then Ψc(r) has an optimum that is given by the unique root of

Ψ′c(r) = 2(s− cv)D

′c(r)− (ce − cv)2(rD

′c(r) +Dc(r))Φ12[2rDc(r)], (9)

where Φ12 is the cumulative distribution function of the random variable representing total municipal

supply ξ1 + ξ2.

Under a batch average claim, the manufacturer commits to using a quantity 2rDc(r) of recycled

input across the two periods. The manufacturer chooses the optimum recycled content fraction to

balance the marginal revenue with the expected future marginal cost. The first term is the product

of the marginal demand, 2D′c(r), and the base margin s− cv, i.e., the profit margin if raw material

was used to meet the marginal demand. The marginal cost is composed of three terms: the first term

is the difference between the unit cost of collection and purchasing raw material; the second term is

the marginal commitmentd(2rDc(r))

dr= 2(rD

′c(r) +Dc(r)); the third term is the lookahead proba-

bility that the manufacturer will resort to collection in stage two, which occurs if the total municipal

13

supply is less than the commitment, i.e., with probability P [ξ1 + ξ2 ≤ 2rDc(r)] = Φ12[2rDc(r)].

Proposition 3 thus gives the tradeoffs in incorporating a greater recycled content in the design stage

(stage one): (1) an increase in demand leading to an increase in revenue, (2) an increase in the

recycled content commitment, and (3) a greater likelihood of collection in stage two. The first order

condition in (9) confirms the intuition that changes at the product design stage should account for

the impact on expected future revenues and costs.

��𝑟1

𝑧𝑧2∗

Amount carried over

𝑟𝑟

)𝑧𝑧2(𝑟𝑟

𝜉𝜉2 − 𝑟𝑟𝑟𝑟(𝑟𝑟)

)𝑟𝑟(𝑟𝑟

)𝑟𝑟𝑟𝑟(𝑟𝑟

10 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅1

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅2

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅3

Figure 2: Plot of the amount carried over z∗2 versus r for a fixed ξ2 under the batch specific claim(r2 > 1).

14

Corollary 1. For a batch specific recycled content claim r, and realization of municipal supply ξ2

the optimum amount carried over to period one is given by

z∗2 =

0 if ξ2 < rD (Region R1)

ξ2 − rD if rD ≤ ξ2 < rD + z2(r) (Region R2)

z2(r) if rD + z2(r) ≤ ξ2 (Region R3),

(10)

For a batch specific claim, Corollary 1 gives the optimum amount carried over for a fixed ξ2, and

Figure 2 shows how it varies with the recycled content. The regions are the same as that defined in

Proposition 1 except that R4 = {∅}, since the starting inventory x2 = 0, and by assumption we have

ξ2 ≤ A < D. In Region R1, the manufacturer does not carry over since all municipal supply has

been used to satisfy the claim; in Region R2, the manufacturer carries over all remaining municipal

supply after meeting the minimum commitment ξ2 − rD; in Region R3, the manufacturer carries

over z2(r), i.e., the amount it would carry over if the municipal supply was not constrained. Figure

2 depicts the fundamental trade off in our model of a recycled content decision and the concomitant

supply constraint. Using a linear demand, for different r, the figure shows D(r), rD(r), ξ2− rD(r),

and z2(r). The curve in bold shows how the amount carried over z∗2 varies with r. For a low value

of r (close to zero), the municipal supply is sufficient to meet the commitment, and there is no carry

over. For a high value of r (close to one), the municipal supply is insufficient, and again there is no

carry over. For a value in between, the manufacturer carries recycled input till it is constrained by

available supply. Figure 2 shows how the recycled content choice impacts downstream operations

through the amount carried between periods. It also identifies the Regions R1, R2, and R3 defined

in Proposition 1. We next provide the uniqueness result for the batch specific claim.

Proposition 4. If Dp(r) is log-concave in r, (rDp(r))′> D

′p(r) for all r and 1 +

Dp(1)

D′p(1)>

s− cvce − cv

then Ψp(r) has a unique optimum.

15

Proposition 4 provides conditions under which the stage one expected profit has a unique

maximum for a batch specific claim. We find that conditions on the demand function, specifically

that the demand is log-concave in r, and that (rDp(r))′> D

′p(r) is sufficient to ensure a unique

optimum. The latter condition requires that for an increase in recycled content, the recycled input

commitment increases at a faster rate than the demand and guarantees that the amount of virgin

material consumed ((1− r)Dp(r)) is decreasing in the recycled content claim. For a linear demand

function Dp(r) = a+ bp · r, this condition is true if a > bp i.e., the base demand exceeds the slope

of the demand function. Note that we do not require conditions on the supply distribution, except

that it be continuous and differentiable. Propositions 3 and 4 allow us to compare the optimum

batch specific and batch average recycled content claims.

4.4 Comparing the Batch Specific and Batch Average Claims

In this section, we will compare the profits and recycled content for both claims and examine

how these change with demand benefit for batch specific claims through EPP. To simplify the

discussion and obtain insights, we use a linear relationship between the demand and recycled

content Dc(r) = a + bcr, and Dp(r) = a + bpr. Further, for illustrative purposes, we demonstrate

our results when the manufacturer does not carry recyclables between periods i.e., mathematically

we let h→∞. The implications of not allowing carry over of supply is relevant when manufacturers

declare claims over larger number of batches i.e., annual, half-yearly or quarterly batch averages.

Moreover, some manufacturers may not wish to hold recyclables for long periods of time due to

contamination concerns or space restrictions.

In Lemma 2, we first establish the directional results for the base case when bp = bc. We

then look at the how the profits and recycled content claim change as bp increases in Proposition

5. Finally, we give a sufficient condition when EPP increases profits and recycled content in

Proposition 6.

16

Lemma 2. If bp = bc then Ψ∗c ≥ Ψ∗p and the ordering of the claims is r∗p ≥ r∗c (and r∗pDp(r∗p) ≥

r∗cDc(r∗c ) ≥ µ) if ce ≤ c = cv +

bc(s− cv)√a2 + 4bcµ

and r∗p ≤ r∗c (and r∗pDp(r∗p) ≤ r∗cDc(r

∗c ) ≤ µ) if

ce ≥ c = cv +bc(s− cv)√a2 + 4bcµ

where µ is the mean of the municipal supply.

The proposition states that the greater flexibility in using recycled input makes the batch

average claim more profitable in the absence of any revenue side benefits. However, it is not

straightforward that the manufacturer’s recycled content commitment will also be greater. The

manufacturer’s commitment under a batch specific claim can be greater under a lower collection

cost. This can be seen by noting that since the marginal revenue from recycled content is same for

both claims, the direction of the claims depends only on the marginal cost.

If the municipal supply were deterministic, the manufacturer would declare a recycled content

at the mean supply i.e., rD(r) = µ. With variability in supply, the manufacturer declares a claim

that is around the mean. If the collection cost is lower (higher) than a threshold, then the recycled

content commitment is greater (lower) than the mean. Further, for the batch average claim, pooling

the municipal supply across the two periods to meet the claim reduces the variability in supply as

compared to a batch specific claim. The net effect is that if the cost of collection is higher than

the threshold i.e., the manufacturer commits lower than the mean, a lower variability around the

mean benefits the manufacturer by increasing the amount of recycled input supply. On the other

hand, if the cost of collection is lower than the threshold i.e., the manufacturer commits greater

than the mean, a higher variability in municipal supply benefits the manufacturer by meeting a

greater share of the commitment from municipal supply.

A batch specific claim thus, results in a lower (higher) marginal cost when collection cost is

lower (higher) than the threshold and is greater (lower) than the batch average. Put differently,

the batch specific claim benefits from the upside of a higher than usual municipal supply when the

collection cost is lower than the threshold, which the time averaged claim forgoes, and this increases

it in comparison. Lemma 2 thus highlights that a tighter batch specific claim leads to a greater

17

(lower) commitment and recycled content if the collection cost is lower (higher) than a threshold.

Having shown that the profits and recycled content for the claims need not be ordered in the same

direction, we next ask: What effect does increasing bp through EPP have on profits and recycled

content claims?

Proposition 5. If the demand benefit increases from bp = b1 to bp = b2 (b2 > b1 ≥ bc), then there

exists a threshold ce(b1, b2) such that if ce > ce(b1, b2) then the optimum recycled content claim is

decreasing, i.e., r∗p(b2) < r∗p(b1), else it is increasing. Further, Ψ∗p is increasing in bp.

In the presence of demand side benefits for batch specific claims like EPP, a manufacturer

making a batch specific claim sees higher profits. However, EPP does not always increase the

recycled content claim. Proposition 5 states that if the cost of collection is sufficiently high then

demand side incentives like EPP may cause the manufacturer to reduce the recycled content. This

happens due to an increased demand benefit also entailing a greater commitment (for the same r,

if Dp(r) increases then rDp(r) also increases), which can decrease profits. Thus, EPP alone may

not be sufficient to encourage products with greater recycled content - supply issues also matter.

We now give a sufficient condition when EPP increases profits and recycled content levels.

Proposition 6. If bp > bc and ce < cv +bc(s− cv)√a2 + 4bcA

then r∗p ≥ r∗c and Ψ∗p ≥ Ψ∗c .

Proposition 6 suggests that demand incentives under low collection costs can create win-win

situations. Note that we have stated a sufficient condition where a small increase in bp over bc

guarantees win-win. This is because the condition requires ce to be quite low. In our model

parametrization, we find that bp has to be sufficiently higher than bc to ensure greater profits and

higher recycled content. Thus, EPP can incentivize manufacturers to make batch specific claims

that lead to higher profits. However, to ensure that the resultant recycled content is also greater,

policy makers should address supply issues involving the cost of collection (ce) and the municipal

supply.

18

Effect of municipal supply variability on recycled content

We next examine the another aspect of supply, i.e., effect of variability of municipal supply. Many

municipalities consolidate the recyclables stream, i.e, recyclables like plastics, paper and glass are

collected in a single bin. It is argued that by reducing the sorting effort on part of the residents, a

municipality can encourage residents to divert greater amount of recyclables from the regular waste

stream (trash) that ends up in landfill. However, commingling of recyclables into a single bin also

leads to significant contamination. In Proposition 7, we examine whether efforts to decrease supply

variability in the municipal stream and ensure a more consistent supply to the manufacturer can

have a favorable impact on the manufacturer’s recycled content decision. We do this by letting the

variability reduction be a mean preserving spread.

Proposition 7. Let X be a random variable with mean µ and Xα = αX + (1 − α)µ, a mean

preserving transformation of X, represent the municipal supply each period and bp = bc = b. Then

the manufacturer’s expected profit decreases in α. The optimal recycled content claim increases

(decreases) in α if ce− cv < (>)b(s− cv)√a2 + 4bµ

. Moreover, the magnitude of the increase (decrease) is

greater (smaller) for the batch specific recycled content claim than the batch average claim.

If α = 0, then X0 = µ. As α increases the random variable Xα “spreads out” around the

mean. Proposition 7 states that decreasing α, i.e., lowering the variability of the supply stream,

keeping mean the same, increases both manufacturer profits and the recycled content claim only if

the cost of non-municipal collection is higher than a threshold. If instead, the cost is lower than

the threshold then reducing variability decreases the recycled content claim. The result can be

explained in the same way as Lemma 2 by noting that the when the collection cost is lower (higher)

than the threshold, the recycled input commitment is greater (lower) than the mean, and lowering

the variability increases (decreases) the marginal cost, by reducing (increasing) the share of the

municipal supply in the commitment.

19

5 Application to the Glass Wool Industry & Managerial Insights

5.1 Industry Application

Data for this analysis was obtained from a glass recycling study performed by the Division of

Recycling & Litter Prevention in Ohio6. We used collection cost estimates provided in the report

and supplement it with demand for recycled content fiberglass from LEED standards. We used

this data to explore the dual impact of a supply side intervention through separate collection of

glass (dual-stream) and demand side through Environmentally Preferable Purchasing. In what

follows, we first describe use of recycled glass in fiberglass manufacturing, explain how we derived

the parameters for our study and parameterize the model.

Use of Cullet in Container Glass and Fiberglass manufacturing:

The North American Insulation Manufacturers Association (NAIMA) reports that U.S. manufac-

turers used almost 1.7 billion pounds of recycled glass in the production of residential, commercial,

and industrial insulation in 2014 (Insulation (2014)). The largest fiberglass manufacturer in the

United States, Owens Corning, is headquartered in Ohio and has a history of consistently increasing

recycled content in its building insulation products from 35% in 2008, 40% in 2009, to 50% recycled

content in 20107.

In Ohio, the fiberglass (glass wool) insulation industry is one of the largest users of glass

cullet. The fiberglass industry utilizes cullet obtained from container and plate glass (e.g. from

windshields). The use of glass cullet in fiberglass this represents an example of downcycling, which

is the recovery and reuse of materials for products other than their original use. In our model,

this decouples supply of cullet (which comes from various sources) with demand for fiberglass

insulation8.

6http://epa.ohio.gov/Portals/41/recycling/OhioGlassRecyclingStudy.pdf7Owens Corning Glass Summit

http://www.ecy.wa.gov/programs/swfa/commingled/pdf/GSFiberglassIndustry.pdf8Fiberglass typically has a long lifetime (10-20 years) and used fiberglass is generally not recycled into new

constructionhttp://epa.gov/epawaste/conserve/tools/warm/pdfs/Fiberglass.pdf

20

Fiberglass is manufactured in batches in a process similar to container glass, where finely ground

raw materials which includes silica, sand, limestone, boron, and cullet are weighted according to

the recipe, mixed and spun into fibers. The EPA’s Waste Reduction Model (WARM) states that

fiberglass insulation from recycled cullet requires less energy - For every 10 percent of recycled

content in fiberglass insulation, the manufacturing energy needs decrease by 3.25 percent (warm

(2014)).

The current maximum recycled content in fiberglass is 58% which includes 36% post-consumer

content, though industry representatives state that fiberglass insulation containing upwards of 90%

recycled glass is technically feasible9. The main issue seems to be supply of uncontaminated cullet.

Cullet obtained as source separated is usually very clean “furnace ready cullet”, where as cullet

obtained from commingled curbside collection is contaminated from metals, organic material and

ceramics10. Moreover, optical sorting and removal of contamination requires material size of re-

covered cullet to be no lower than 38

thof an inch11. To be used in manufacturing, cullet needs to

be cleaned of contaminants (e.g., metals, ceramics, and organic material) and ground to a specific

size. Next, we calibrate the demand and supply parameters of our model.

Demand for recycled content:

Fiberglass insulation is the most widely used insulation material worldwide and accounted for over

half of the US market in 2011. Manufacturers of fiberglass insulation containing recycled content see

demand from state governments (EPP), large commercial building projects (LEED) and residential

buildings (NAHB), all having varying levels of recycled content requirements. We calibrate the de-

mand side with LEED certification data. In 2011, the United States Mid-west region saw close to

100 million square feet of LEED certified construction12, which consisted of LEED Certified (16%),

9Source:http://www.doi.gov/greening/buildings/upload/iEnvironmental-Considerations-of-Building-

Insulation-National-Park-Service-insulation.pdf

10Maximum tolerance is roughly 500 parts per million for organic contamination11In single-stream systems, it is virtually impossible to prevent glass from breaking, as it goes to the curb, is

loaded in the truck where it gets compacted, and offloaded onto conveyor belts to be processed by the MRF Source:

ContainerRecyclingInstitute.12http://www.usgbc.org/articles/list-top-10-states-leed-green-buildings-released

21

LEED Silver (34%), LEED Gold (35%) and LEED Platinum (15%)13. Von Paumgartten (2003)

reports that a campus building in University of California in Santa Barbara, which was LEED Gold

certified, used insulation containing between 35−100% recycled content. Construction specific data

can be used to estimate the recycled content levels in fiberglass used in various LEED ratings and

buildings. For the purpose of our analysis, we assume that LEED certified buildings used building

insulation with post-consumer recycled content uniformly distributed between 0 and 100 percent14.

An average construction uses 0.85 pounds of fiberglass insulation per square foot, and with 50%

market share this represents potential recycled-content fiberglass demand of 19.25 thousand tons

in our analysis15. Green Construction (2014) estimates that federal, state and local governments

contributed to over 27% of LEED-registered projects i.e., 5.2 thousand tons, with the rest 14.05

thousand tons going toward private LEED-certified projects. We used this to set the elasticity

parameter for demand bc = 1.4 thousand tons (certified as annual average), and for EPP bp = 1.92.

We estimated the green building market to be 30% of total non-residential construction16, and we

used this to set the base “non-green” demand at 230 million square feet of construction space. With

a fiberglass market share of 50%, this gives a base “non-green” demand of a = 45 thousand tons.

Supply:

The per capita generation of glass in Ohio (consisting of beverage, food container glass and plate

glass) from residential and commercial sources was 68 pounds in 2011. At 2011 scrap prices for

post-consumer container glass (sorted clear) of $25 per ton (see appendix), the glass recovery rate

13LEED points are awarded based on a recycled credit calculation - for each building material, the recycled contentvalue, which equals the recycled content weight of the material multiplied by the cost, is calculated and the weightedsum is evaluated, and the LEED certification awarded on the weighted sum. Clearly, the contribution of recycledcontent fiberglass varies depending on the overall composition of the building material and can vary widely betweenprojects. Since, we did not have data related to the exact composition of projects, we used estimates.

14We assume building insulation for obtaining LEED rating, were chosen primarily for recycled content levels, andassuming cost/structural properties is not a factor. Plastic, Non-Woven Batt is known to contain 100% recycled con-tent Source: http://www.doi.gov/greening/buildings/upload/iEnvironmental-Considerations-of-Building-

Insulation-National-Park-Service-insulation.pdf

15100 million square ft * 0.85lbs

sqft*0.453

kgs

lb∗ 0.5/1000 = 19, 250 tons.

Our demand estimates are conservative since we consider only LEED construction. However, non-LEED residentialconstruction (NAHB certified) also represents a large demand, but we did not have access to this data

16www.usgbc.org/Docs/Archive/General/Docs18693.pdf

22

was 11% 17 18. This gives a total mean of 54, 395 thousand tons of glass, recovered annually primar-

ily through single-stream. A recent study by the Container Recycling Institute on single-stream

collection, estimated that the average processing yield for curbside single-stream and dual-stream

is δs = 0.4 and δd = 0.9 respectively19. Accordingly, the mean supply of municipally collected

cullet available for use in manufacturing (post-processing) with only single stream collection is 21.6

thousand tons and the mean supply if dual stream collection was used in 20% of the municipali-

ties, is 27 thousand tons. We used a coefficient of variation = 0.36 obtained from the Streets and

Sanitation department in West Lafayette for single stream to obtain N(21.6, 7.34) and a coefficient

of variation of 0.32 for dual stream collection to obtain N(27, 8.26).

Table 2: Non-municipal sources of cullet

Projected annual recoverablequantity (in thousand tons)

Estimated average costper ton 20

Source separate collection of glassusing drop-off points

14.5 $63-$110

Source separation of glass frombars and restaurants

53 $30-$60

The manufacturer can self-collect cullet or source from outside the municipal stream. Table 2

lists the non-municipal sources and the per-ton costs that can be used to supplement manufacturer

supply of cullet and the cost estimates provided from the report. Table 3 summarizes the parameter

values used. Next, we present the results for the parameterized model.

We report the manufacturer profits, recycled content claim, recycled input used, and annual

diversion of recyclables from landfills due to manufacturer self-collection for an annual time horizon

with averaging over different number of batches. To do this, we start with T = 1 (annual batch

average), and increase T by splitting the year into smaller time intervals i.e, T = 2 (half-yearly

17‘Recovery Rate’ represents the percentage of generated glass that was recovered by the municipality for recycling(residential + commercial)

18Cook (1978) states that the municipality collects until it breaks even i.e., till the marginal cost of collection equalsthe scrap value of $25 per ton. In 2011, this number was $54, 395 tons

19The report also estimates the yield from container deposit system to be 0.9820The cost includes collection cost and processing the material (remove contaminants etc.) for use by the glass

industry. Bars and Restaurants are more concentrated users of glass products, hence the lower cost per ton, but barand restaurant glass also tends to be more contaminated from ceramics.

23

batch average), T = 4 (quarterly batch average) and T = 8 (batch specific)21. Correspondingly, we

also split the demand and supply appropriately each period. Thus, for T = 1, if the annual supply

is µ, standard deviation is σ, annual demand is D(r), then for each period of the T period problem,

the mean supply isµ

T, standard deviation is

σ√T

, and demand isD(r)

T.

Table 4 shows the manufacturer’s profits for each claim type when batch specific claims are given

purchasing preference by government EPP for bp = 1.92 and bp = 40. We note that at the current

levels of EPP, bp = 1.92, batch specific claims see lower profits that annual batch average claims.

However, with batch-specific claims the manufacturer also diverts more recyclables from the landfill

in addition to incorporating more recycled input leading to lower manufacturing emissions. It is

interesting to note that a manufacturer making a batch specific claim declares a greater recycled

content claim than a batch average claim. The direction of the claims mirrors that in Lemma 2 i.e,

the batch specific claim is greater than the batch average claim when collection cost is lower than

a threshold.

Table 4 also highlights that fiberglass manufacturers making a voluntary claim prefer an annual

average over batch specific claims for bp = 1.92. Our explanation that manufacturers prefer mak-

ing annual batch average claims under supply variability corroborates with practice. The North

American Insulation Manufacturers Association, NAIMA, stated in comments on the FTC Green

Guides, that it supports the FTC’s retention of the current recycled content calculation formula,

which is an annual batch average. The NAIMA offered explanation is that “[...] for fiber glass

manufacturers, suitable glass cullet is not always available on a consistent and continuous basis

from recyclers [...] low throughput and other technical limitations will cause a substantial variance

in the amount of glass cullet that can be used at any given time [...] For those manufacturing

reasons, calculated recycled content must be done on a longer timeframe, so the FTC’s annual

weighted basis is appropriate”

21In the glass industry, furnaces are run continuously and must be set for a given mix of cullet and other inputs.The plant operator must ensure there is sufficient cullet to maintain the feed rate and typically, furnaces are not shutdown in periods shorter than a month22. Accordingly, in our analysis batch specific claims must be met over every1.5 months i.e., T = 8.

24

Table 3: Numerical Analysiss 394 $/tona price per ton of fiberglass batt (not including installation

cost and material cost of borax and binder),

cm 25 $/ton Average price of post-consumer clear (flint) glass (2011) ,

ce ∈ [30, 110] $/ton Estimated collection and processing cost of non-municipalcullet,

cv 36 $/ton batch cost of soda ash, limestone and sand (can be substi-tuted by cullet in formulation),

h 2.5 $/ton holding cost per unit of recycled input with internal rate ofreturn = 10%,

r 90% technical limitation for proportion of cullet in batch ,

δs 0.4 Yield for Single Stream collection (tons),

δd 0.9 Yield for Dual Stream collection (tons),

ξs N(21.6, 7.34) Annual supply of uncontaminated municipality collectedcullet (post-processing) with single stream collection (inthousand tons),

ξd N(27, 8.26) Annual supply of uncontaminated municipality collectedcullet (post-processing) with dual stream collection (in thou-sand tons),

Dc(r) 45 + 1.4r Annual demand of fiberglass insulation for LEED-certifiednon-public construction in 2011 (in thousand tons),

Dp(r) 45 + 1.92r Annual demand for batch specific claims for fiberglass insu-lation for LEED-certified public construction (EPP) in 2011(in thousand tons),

aAverage price for the North American fiberglass insulation is $0.50 per pound i.e $1100 per ton in 2003(Source:U.S. Thermal Insulation Materials Market for the Building Industry, Frost & Sullivan Database). http://www.

insulation-guide.com/insulation-cost.html reports the installation cost of R-19 Fiberglass Batt at 0.45 $/squarefoot, which is approximately half of the product cost. Also, cullet in fiberglass replaces soda ash, limestone and sand,but not borax and binder, which are more expensive. Borax and Binder together constitute approximately 16%by weight of formulation and the per ton price of borax was $979 in 2011 (Source: USGS) which gives 156 $/ton.Subtracting these costs, gives the estimate fiberglass price net of installation and borax material cost as 394 $/tonhttp://ws680.nist.gov/bees/ProductListFiles/Generic%20Fiberglass.pdf

http://epa.gov/wastes/conserve/tools/warm/pdfs/Fiberglass.pdf

25

Table 4: With EPP under single stream collection with a collection cost ce = 60$ per ton anda = 45, bc = 1.4)

T=1Annual batch average

bc = 1.4

T=2Half-yearly batch average

bp = 1.4

T=4Quarterly batch average

bp = 1.4

T=8Batch Specificbp = 1.92

T=8Batch Specific

bp = 40

ManufacturerProfit (in million dollars)

16.49 16.18 14.47 10.35 17.45

Recycled contentClaim (r∗)

0.46 0.37 0.48 0.77 0.9

Recycled input (r∗D(r∗))(in thousand tons)

21 16.84 21.92 35.8 72.9

Expected Annual Diversiondue to manufacturer self-collection

(in thousand tons)0 0 0.32 14.2 51.3

A higher value of bp indicates greater share of public construction which can lead to a greater

demand benefit. Table 4 shows that for bp = 40 for batch specific claims, EPP increases the

manufacturer profits and recycled content claims to be greater than annual batch average claims.

The values in bold show that the batch specific claim is Pareto improving compared to the annual

batch average claim. When the magnitude of EPP is high, batch specific claims can be given

purchasing preference to increase manufacturer profits and maximum diversion from the landfill.

States with greater magnitude of public construction such as California and Washington 23 are good

candidates for EPP which gives purchasing preference to batch specific claims. In Ohio, current

levels of EPP are not sufficient to incentivize batch specific claims. However, several states are

requiring greater share of public construction be LEED certified. We explored the impact with

an increased bp = 10 on batch specific claims. Table 5 shows that with bp = 10, though batch

specific has lower profits than annual batch average, Pareto improving conditions can be achieved

if EPP gave preference to quarterly batch average claims. Hence, our recommendation is that

states where magnitude of EPP is large (e.g., California, Washington) batch specific claims can be

encouraged, whereas states will lower magnitudes of EPP, quarterly or half-yearly batch averages

can be considered.

23Highly populated states like New York and California registered over 72 million and 37 million square feet respec-tively in 2013. Source: www.usgbc.org/articles/usgbc-releases-top-10-states-nation-leed-green-building

26

Table 5: With EPP under single stream collection with a collection cost ce = 60$ per ton anda = 45, bc = 1.4, bp = 10)

T=1Annual batch average

bc = 1.4

T=2Half-yearly batch average

bp = 1.4

T=4Quarterly batch average

bp = 10

T=8Batch Specific

bp = 10

ManufacturerProfit (in million dollars)

16.49 16.18 16.57 11.82

Recycled contentClaim (r∗)

0.46 0.37 0.9 0.9

Recycled input (r∗D(r∗))(in thousand tons)

21 16.84 48.6 48.6

Expected Annual Diversiondue to manufacturer self-collection

(in thousand tons)0 0 27 27

Table 6: With single stream collection at a high collection cost (ce = 1000$ per ton, a = 45, bc =1.4, bp = 1.92)

T=1Annual batch average

bc = 1.4

T=2Half-yearly batch average

bp = 1.92

T=4Quarterly batch average

bp = 1.92

T=8Batch Specificbp = 1.92

ManufacturerProfit (in million dollars)

16.36 16.11 14.34 10.16

Recycled contentClaim (r∗)

0.11 0.06 0.04 0

Recycled input (r∗D(r∗))(in thousand tons)

4.96 2.7 1.8 0

Expected Annual Diversiondue to manufacturer self-collection

(in thousand tons)0 0 0 0

27

Table 6 shows that with a higher collection cost ce = 1000$/ton the recycled content claim,

recycled input and manufacturer profit is maximized for the annual batch average claim. Manu-

facturers voluntarily prefer to make annual batch average claims which also uses greater recycled

input leading to lowering of manufacturing emissions. Thus from a policy makers perspective,

when a manufacturer’s collection cost of recyclables is as above, then encouraging annual batch

average claims would ensure manufacturers profits are maximized and manufacturers voluntarily

use greater amount of recycled input. However, at this collection cost the manufacturer does not

self-collect i.e, there is no diversion from the landfill.

In Table 7 we look at the effect of the municipalities moving to dual stream collection of

recyclables. Dual stream collection increases the yield of the municipal stream (net of collection

and processing) compared to single stream. It is also expected to reduce variability in the supply

due to less commingling and contamination of recyclables. We chose a coefficient of variation for

dual stream ρd = 0.32 compared to single stream ρs = 0.36, so that dual stream increased mean

supply but the variance did not increase as much as single stream collection. Table 7 shows that

dual stream recycling increases manufacturer profits and recycled content claims due to greater

yield of cullet in the municipal stream. The manufacturer is less reliant on self-collection to meet

the claim hence the annual diversion from the landfills in lower. However, at current EPP levels in

Ohio, moving to dual stream does not lead to increased diversion. The manufacturers continue to

declare annual batch averages - however, the recycled content levels increase.

Table 7 also shows that supply side intervention through dual stream recycling combined with

EPP at bp = 40 can increase manufacturer profits and improve recycled content levels compared

with Table 4. This shows that combining of supply and demand intervention can be used to achieve

Pareto improving outcomes.

28

Table 7: With EPP and Dual stream collection (ce = 60$ per ton, bc = 1.4, a = 45, Single Stream:

N(21.6, 7.34) with ρs = 0.36, Dual Stream: N(27, 8.26) with ρd = 0.32 )

T=1Annual batch average

bc = 1.4

T=2Half-yearly batch average

bp = 1.4

T=4Quarterly batch average

bp = 1.4

T=8Batch Specific

(bp = 1.92)

T=8Batch Specific

(bp = 40)

ManufacturerProfit (in million dollars)

16.63 16.50 15.12 11.19 18.87

Recycled contentClaim (r∗)

0.57 0.46 0.56 0.77 0.9

Recycled input (r∗D(r∗))(in thousand tons)

26.1 21 25.63 35.8 72.9

Expected Annual Diversiondue to manufacturer self-collection

(in thousand tons)0 0 0 8.78 45.9

5.2 Managerial Insights

Environmentally Preferable Purchasing (EPP) under supply constraints:

When there is no preferential purchasing of products with batch specific claims, manufacturers

make batch average claims since averaging allows greater flexibility in meeting the commitment by

pooling supply across periods. Lack of batch specific claims thus indicates absence of downstream

demand for such claims. The US government’s Environmentally Preferable Purchasing encourages

manufacturers to voluntarily make batch specific claims exceeding LEED requirements.

From the policy perspective, it is an interesting question - which type of claims should the

government give purchasing preference to? Moreover, when can Pareto improving conditions which

ensure greater manufacturer profits and voluntarily higher recycled content claims be obtained?

We find that if the collection costs outside the municipal stream are extremely high and the state

practices single stream collection of recyclables, policy makers should encourage annual average

claims. Doing so, would allow manufacturers to retain higher profit and voluntarily use more

recycled input in production. In states, where the magnitude of EPP is very large due large public

construction e.g. California and Washington, encouraging batch specific claim might be Pareto

optimal. In states such as Ohio, where collection costs are relatively lower and there is plenty of

untapped supply from bars and restaurants, batch average claims may not lead to higher profits

29

compared to batch averages if the magnitude of EPP is not large enough. In such states, a quarterly

or half-yearly batch average for EPP might be more appropriate. Whether voluntary batch specific

claims lead to Pareto improving outcomes depends on the magnitude of the demand benefit due to

EPP.

Concerted effort of demand side and supply side (through moving to dual stream for at least

part of the municipalities) can improve manufacturer profits and use of recycled input in produc-

tion, thus reducing manufacturing emissions. Moving to dual stream collection of recyclables can

thus supplement the demand benefit provided by EPP and lead to Pareto improving outcomes.

However, the outcomes from single vs. dual stream also depend on the variability in each collection

scheme. More analysis should be done as to the supply situation in each states before adoption

either demand- or supply- intervention.

Reducing manufacturer collection cost through bottle bills:

Another way to achieve pareto improving outcomes as shown in Proposition 6 is to reduce collection

cost through legislation like container deposit laws, also known as ‘bottle bills’, which provide

financial incentives to consumers to divert recyclables from the municipal stream. Bottle bills are

deposit laws that charge a 5- or 10-cent deposit from consumers at the time of sale and refund it

when the consumer returns the used bottle to an authorized retailer or at designated reverse vending

locations. Encouraging consumers to return used bottles reduces manufacturer travel distances to

collect the used bottles than without a bottle bill (since consumers travel part of the way to deposit

the bottle and manufacturers can pick up the used bottle from retailers in their delivery return

route). Further, diverting recyclables from the municipal stream (lower A) makes the manufacturer

less dependent on the variable municipal stream.

(ce)bottle bill < (ce)w/o bottle bill

Abottle bill < Aw/o bottle bill

30

The EPP demand benefit drives manufacturer profitability and lower non-municipal collection

cost (and dependence on the variable municipal stream) through bottle bills results in greater

recycled content being used. The net effect (Proposition 6) is that manufacturers switch from

making batch average claims to batch specific claims that are higher and more profitable. The

discussion thus far looked at the impact of two levers - demand incentives for recycled content

and municipal collection mechanism - single vs. dual stream. We find that impact of these levers

depends on the manufacturer’s cost of collection from non-municipal sources. Finally, policy with

regard to demand incentives for recycled content should be tailored to supply limitations.

6 Extensions

Our model and assumptions are specific to the industry context of glass and fiberglass production,

since quality of glass is unaffected by the amount of recycled content. However, there are industries

where quality of the product might depend on the recycled content of the product for e.g., printing

& copy paper. An interesting extension is investigating whether the final price of the product

increases to maintain the same quality for the consumer. Many EPP programs give price ranges for

purchasing recycled products - usually prices 5-10% greater than virgin products are acceptable.

In this case, the demand curve is unchanged, but the firms revenue increases due to the higher

price. However, the cost effect of maintaining quality also needs to be explored. These are some

considerations a future extension can consider.

Further, while we have considered uncertainty in supply of recycled input over time, where a

firm produces in multiple locations, there could be more variation in the availability of recycled

materials across geographical regions than over time. The observation is even more pertinent in

light of the recent LEED Interpretation Ruling that disallowed making recycled content claims

averaged across several different plants i.e., claims made about a product must be specific to the

plant it is manufactured in. Considering the problem of geographical variation in availability and

supply represents a interesting extension to our paper.

31

Finally, though the supply and demand are uncorrelated for the glass wool industry, it may not

be true for other recyclables like container glass, paper products, aluminum etc. A future extension

to other recyclables should consider the closed loop from the demand for recyclables feeding the

supply. A future extension can also allow for uncertainty in the recyclables recovered from non-

municipal sources. In this case, the model might have to levy a penalty in case the firm violates

the claim in a period due to unavailable supply from both municipal and non-municipal sources.

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APPENDIX

6.1 Historical Price of Cullet

Figure 3: Cullet Prices (Source: Sustainable Glass Management Option Study, Dubuque Metropoli-

tan Area Solid Waste Agency

6.2 Sensitivity to holding cost

We show the sensitivity of the recycled content claims and result in Lemma 2 to finite holding cost

values.

35

(a) ce = 60$/ton

(b) ce = 200$/ton

Figure 4: Plot of recycled content vs. holding cost for s = 394$/ton, cv = 36$/ton,cm = 25$/ton,

a = 57.75 thousand tons, bp = bc = 2.81 thousand tons. The annual supply of uncontaminated

cullet from single-stream recycling is normally distributed with mean µ = 21.6 and σ = 7.3 thousand

tons.

We plot the optimal recycled content claims for different holding cost values and a high collection

cost (ce = 200 $/ton) and a lower collection cost (ce = 60 $/ton) (with bp = bc = 2.81) to show the

sensitivity of the results. The dashed lines show the optimal annual batch average claim, which does

not change with holding cost, since no cullet is carried over in the average claim. The direction of

results with a finite holding cost mirrors that for the infinite holding cost case: when the collection

cost is sufficiently high (low), the optimal batch specific claim is lower (greater) than the annual

batch average claim. We next provide the main managerial insights from the analysis thus far.

36

6.3 Proofs

Proof of Lemma 1

In period t, if xt + ξt < rD then we have,

Πt(xt, ξt) = maxrD≤yt≤D

[sD − cmξt − ce(yt − xt − ξt)− cv(D − yt) + Πt−1(0)

]= sD − cmξt − ce(rD − xt − ξt)− cv(D − rD) + Πt−1(0) if ce > cv (11)

= sD − cmξt − ce(D − xt − ξt) + Πt−1(0) if ce ≤ cv (12)

If rD ≤ xt + ξt then y∗t = D if ce ≤ cv. If ce > cv the manufacturer solves,

Πt(xt, ξt) = maxrD≤yt≤D

[sD − cmξt − ce(yt − xt − ξt)+ − cv(D − yt)− h(xt + ξt − yt)+ + Πt−1[(xt + ξt − yt)+]

](13)

Noting that the optimum y∗t ≤ xt + ξt (for yt > xt + ξt slope is negative), the problem can be

rewritten in terms of zt = xt + ξt − yt as,

Πt(xt, ξt) = maxzt

[sD − cmξt − cv(D − xt − ξt + zt)− hzt + Πt−1(zt)

](14)

subject to (xt + ξt −D)+ ≤ zt ≤ xt + ξt − rD. In other words, the manufacturer carries over z∗t to

the next period, where z∗t solves

maxzt

Gt(zt),

where Gt(zt) = −(h+ cv)zt + Πt−1(zt), (15)

subject to (xt + ξt −D)+ ≤ zt ≤ xt + ξt − rD, and y∗t = xt + ξt − z∗t . �

Proof of Proposition 1

It is trivial to note that y∗t = D if ce ≤ cv. We use induction to prove the proposition if ce > cv.

Clearly in period one the manufacturer will use up all the recycled input available x1 + ξ1 and

salvage leftover recycled input at end of the period. Setting z1(r) = 0 gives the corresponding

37

optimum actions for period one. We first show that the result is true in period two. The expected

profit in period one is,

Π1(x1) = Ex1+ξ1<rD

[sD − cmξ1 − ce(rD − x1 − ξ1)− cv(D − rD)

]+ ErD≤x1+ξ1<D

[sD − cmξ1 − cv(D − x1 − ξ1)

]+ ED≤x1+ξ1

[sD − cmξ1 − h(x1 + ξ1 −D) + cm(x1 + ξ1 −D)

](16)

which gives,

Π′1(x1) = cv + (ce − cv)Φ[rD − x1]− (h+ cv − cm)Φ[D − x1] (17)

Π1(x1) is concave in x1, since Π′1(x1) is decreasing in x1. Note that Π

′1(x1) ≤ ce. In period two, if

x2 + ξ2 < rD then from Lemma 1, y∗2 = rD. If x2 + ξ2 > rD the manufacturer solves (15). The

first order condition is,

G′2(z2) = −h− cv + Π

′1(z2) = −h+ (ce − cv)Φ[rD − z2]− (h+ cv − cm)Φ[D − z2] (18)

subject to (x2 + ξ2 −D)+ ≤ z2 ≤ x2 + ξ2 − rD. Since Π1 is concave, G2(z2) is concave in z2. Let

z2(r) be the unique positive root of G′2(z2) = 0. Note that the expression is independent of x2 and

ξ2. (If it does not does not have a positive root we set z2(r) = 0). Then the optimum actions in

period two can be written as,

(y∗2, z∗2) =

(rD, 0) if x2 + ξ2 < rD

(rD, x2 + ξ2 − rD) if rD ≤ x2 + ξ2 < rD + z2(r)

(x2 + ξ2 − z2(r), z2(r)) if rD + z2(r) ≤ x2 + ξ2 < D + z2(r)

(D,x2 + ξ2 −D) if x2 + ξ2 ≥ D + z2(r)

(19)

This proves the proposition for the two period problem. We now assume that the following is true

for period t− 1 (t ≥ 2) and prove for period t, i.e, assuming that,

A1) Πl(x) is concave and Π′l(x) ≤ ce for all x and 1 ≤ l ≤ t− 1,

38

A2) zt(r) ≥ zt−1(r) ≥ ... ≥ z2(r) ≥ z1(r) = 0,

A3) If zt−1(r) > 0 then G′t(x) ≥ G′t−1(x) ∀x ≤ zt−1(r),

we show,

B1)

(y∗t , z∗t ) =

(rD, 0) if xt + ξt < rD

(rD, xt + ξt − rD) if rD ≤ xt + ξt < rD + zt(r)

(xt + ξt − zt(r), zt(r)) if rD + zt(r) ≤ xt + ξt < D + zt(r)

(D,xt + ξt −D) if xt + ξt ≥ D + zt(r)

(20)

,

B2) Πt(x) is concave in x with Π′t(x) ≤ ce for all x,

B3) zt+1(r) ≥ zt(r) ≥ ... ≥ z2(r) ≥ z1(r) = 0,

B4) If zt(r) > 0 then G′t+1(x) ≥ G′t(x) ∀x ≤ zt(r).

Induction assumption (A1) implies that Gt(zt) is concave in zt. Then, similar to period two,

(B1) can be directly inferred. We now show (B2) using (B1). The period-t expected profit is,

Πt(x) = Eξ[Πt(x, ξ)] = Ex+ξ<rD

[sD − cmξ − ce(rD − x− ξ)− cv(D − rD) + Πt−1(0)

]+ ErD≤x+ξ<rD+zt(r)

[sD − cmξ − h(x+ ξ − rD)− cv(D − rD) + Πt−1(x+ ξ − rD)

]+ ErD+zt(r)≤x+ξ<D+zt(r)

[sD − cmξ − hzt(r)− cv(D − x− ξ + zt(r)) + Πt−1(zt(r))

]+ Ex+ξ≥D+zt(r)

[sD − cmξ − h(x+ ξ −D) + Πt−1(x+ ξ −D)

](21)

39

To show Πt(x) is concave, we will show that Πt(x, ξ) is concave in x for any given ξ. Since a convex

combination of concave functions is concave, this implies that Πt(x) is concave in x. For a given ξ

we have,

dΠt(x, ξ)

dx=

ce if x+ ξ < rD

−h+dΠt−1(x+ ξ − rD)

dxif rD ≤ x+ ξ < rD + zt(r)

cv if rD + zt(r) ≤ x+ ξ < D + zt(r)

−h+dΠt−1(x+ ξ −D)

dxif x+ ξ ≥ D + zt(r)

(22)

Let zt(r) > 0 (if zt(r) = 0 the proof is similar). From (A1), since Π′t−1(x) ≤ ce we have,

−h+dΠt−1(x+ ξ − rD)

dx< ce. Furthermore, for the region rD ≤ x+ξ < rD+zt(r), since Πt−1(x) is

concave in x we have Π′t−1(x+ξ−rD) ≥ Π

′t−1(zt(r)) = h+cv. Thus, −h+

dΠt−1(x+ ξ − rD)

dx≥ cv.

Similarly, if x + ξ ≥ D + zt(r) then, Π′t−1(x + ξ − D) ≤ Π

′t−1(zt(r)) = h + cv =⇒ −h +

dΠt−1(x+ ξ −D)

dx≤ cv. Then since both Π

′t−1(x+ ξ − rD) and Π

′t−1(x+ ξ −D) are decreasing in

x we have shown thatdΠt(x, ξ)

dxis decreasing in x i.e. Πt(x, ξ) is concave in x for a given ξ. Noting

also that Π′t(x) ≤ ce we have shown (B2). We will show (B3) and (B4) using (A2) and (A3). If

zt(r) = 0 then the result is trivial. We will prove for zt(r) > 0. Using (22) we can write,

G′t+1(x) = −h− cv +

dΠt(x)

dx= −h+ Ex+ξ<rD(ce − cv) + ErD≤x+ξ<rD+zt(r)(−h− cv +

dΠt−1(x+ ξ − rD)

dx)

+ Ex+ξ≥D+zt(r)(−h− cv +dΠt−1(x+ ξ −D)

dx)

= −h+ Ex+ξ<rD(ce − cv) + ErD≤x+ξ<rD+zt(r)(G′t(x+ ξ − rD)) + Ex+ξ≥D+zt(r)(G

′t(x+ ξ −D))

= −h+ Ex+ξ<rD(ce − cv) + ErD≤x+ξ<rD+zt−1(r)(G′t(x+ ξ − rD))

+ ErD+zt−1(r)≤x+ξ<rD+zt(r)(G′t(x+ ξ − rD)) + Ex+ξ≥D+zt(r)(G

′t(x+ ξ −D)) (23)

40

where in the last expression the region rD ≤ x+ ξ < rD + zt(r) has been split into rD ≤ x+ ξ <

rD + zt−1(r) and rD + zt−1(r) ≤ x+ ξ < rD + zt(r) since by (A2) zt(r) ≥ zt−1(r). Also,

G′t(x) = −h− cv +

dΠt−1(x)

dx= −h+ Ex+ξ<rD(ce − cv) + ErD≤x+ξ<rD+zt−1(r)(G

′t−1(x+ ξ − rD))

+ Ex+ξ≥D+zt−1(r)(G′t−1(x+ ξ −D)) (24)

We will compare the above two expressions to show (B3) and (B4). The first two terms are the

same in both expressions. If x ≤ zt(r), the last term in the expression for G′t+1(x) is 0 (since

ξ < A < D) while the corresponding term in G′t(x) is negative (since by (A1), Πt−2(x) is concave

in x). Using (A3), the third term in G′t+1(x) is greater than or equal the corresponding term

in G′t(x). The fourth term in G

′t+1(x) is non negative (using (A1)). Comparing term by term

we have G′t+1(x) ≥ G

′t(x) for all x ≤ zt(r) if zt(r) > 0, which shows (B4). This implies that

G′t+1(zt(r)) ≥ G

′t(zt(r)) = 0 which along with (A2) gives (B3). Lastly, if zt−1(r) = 0 then the third

term in both G′t+1(x) and G

′t(x) are 0 and the same argument as before applies. This completes

the induction. �

Proof of Proposition 2

The proof follows by noting that the manufacturer can maximize profits by deferring collection

till period one when all uncertainty in supply has been resolved. In period one, the manufacturer

must meet any remaining commitment therefore, y∗1 = max(ξ1, χ1). In each of the other periods the

manufacturer uses up all the municipal supply y∗t = ξt, and defers collection unless ξt < χt−(t−1)D.

If ξt < χt−(t−1)D the manufacturer will fall short of the commitment if it defers, even if it uses only

recycled input to meet demand in each of the remaining t−1 periods. Thus, if ξt < χt−(t−1)D, the

manufacturer collects up to y∗t = χt−(t−1)D; if ξt ≥ χt−(t−1)D the manufacturer uses up all the

municipal supply, y∗t = ξt and updates the next periods minimum commitment i.e, χt−1 = (χt−ξt)+

and xt−1 = 0. Note that the manufacturer never carries over recycled input, since every unit of

recycled input used in the current period goes towards meeting the commitment unlike the product

specific case, where any recycled input that exceeds the periods minimum commitment does not

count towards future commitments. �

41

Proof of Proposition 3

We have,

Ψc(r) = Π2(0, 2rD) = Eξ2Π2(0, 2rD, ξ2) (25)

where χ2 = 2rD. From Proposition 2, y2 = χ2 − D = (2r − 1)D. We consider two cases: If (i)

0 ≤ r ≤ 1

2then y∗2 = ξ2. Substituting we get,

Π2(0, 2rD, ξ2) =

sD − cmξ2 − cv(D − ξ2) + Π1(0, 2rD − ξ2) if ξ2 ≤ 2rD

sD − cmξ2 − cv(D − ξ2) + Π1(0, 0) if ξ2 > 2rD

Similarly, if (ii)1

2≤ r ≤ 1 then,

Π2(0, 2rD, ξ2) =

sD − cmξ2 − ce(2rD −D − ξ2)− cv(2D − 2rD) + Π1(0, D) if ξ2 ≤ (2r − 1)D

sD − cmξ2 − cv(D − ξ2) + Π1(0, 2rD − ξ2) if 2rD ≥ ξ2 > (2r − 1)D

sD − cmξ2 − cv(D − ξ2) + Π1(0, 0) if ξ2 > 2rD

By Assumption 3, we know that ξ1 ≤ A < D which gives Π1(0, D) = sD − cmξ1 − ce(D − ξ1).

Also, Π1(0, 2rD − ξ2) = sD − cmξ1 − ce(2rD − ξ2 − ξ1)− cv(D − 2rD + ξ2) if ξ1 < 2rD − ξ2 else,

Π1(0, 2rD − ξ2) = Π1(0, 0) = sD − cmξ1 − cv(D − ξ1). Combining and simplifying we get,

Ψc(r) = Eξ1+ξ2≤2rD[2(s− cv)D + (cv − cm)(ξ1 + ξ2)− (ce − cv)(2rD − ξ2 − ξ1)]

+ Eξ1+ξ2>2rD[2(s− cv)D + (cv − cm)(ξ1 + ξ2)]

= 2(s− cv)D + 2(cv − cm)µ− (ce − cv)Eξ1+ξ2≤2rD[2rD − ξ2 − ξ1] (26)

which gives,

dΨc(r)

dr= 2(s− cv)D

′ − 2(ce − cv)(rD)′P [ξ1 + ξ2 ≤ 2rD]

= 2(ce − cv)D′[ s− cvce − cv

− (r +D

D′)Φ12[2rD]

](27)

42

If 0 ≤ rD ≤ A, the last term is 0 anddΨc(r)

dr= 2(s − cv)D

′> 0. For rD > A, Ψc(r) is strictly

quasiconcave if (r +D

D′)Φ12[2rD] is strictly increasing in r. Thus, if

D

D′is increasing in r, i.e., D

is log-concave in r, and if at r = 1, (r +D

D′)Φ12[2rD] = 1 +

D(1)

D′(1)>

s− cvce − cv

, then there exists a

unique maximizer less than or equal to 1. �

Proof of Proposition 4

We first derive the two period expected profit for the product specific claim. The steps of the

derivation can be provided upon request by the authors. Here, we directly state the first derivative

and give conditions under which there exists a unique optimum. We have,

dΨp(r)

dr= 2(s− cv)D

′ − (ce − cv)(rD)′p1(r) + (h+ cv − cm)D

′p2(r)− (rD)

′k(r) (28)

where,

p1(r) = Φ[rD] +

∫ξ2∈R1

Φ[rD]φ(ξ2)dξ2 +

∫ξ2∈R2

Φ[2rD − ξ2]φ(ξ2)dξ2 +

∫ξ2∈R3

Φ[rD − z2(r)]φ(ξ2)dξ2.

(29)

p2(r) =

∫ξ2∈R2

Φ[D(1 + r)− ξ2]φ(ξ2)dξ2 +

∫ξ2∈R3

Φ[D − z2(r)]φ(ξ2)dξ2, (30)

and k(r) =

∫ξ2∈R2

G′2(ξ2 − rD)φ(ξ2)dξ2.

The first order condition can then be written as,

dΨp(r)

dr= D

′[2(s− cv)−

((ce − cv)(rD)′p1(r) + (rD)

′k(r)− (h+ cv − cm)p2(r)D

′

D′

)︸ ︷︷ ︸

f(r)

](31)

where by assumption D′> 0 for all r. We first note that f(r = 0) = 0 < 2(s − cv) (if r = 0 then

R1 = R2 = {∅} since z2(r) = 0 =⇒ p1(0) = p2(0) = k(0) = 0). If r = 1 then R2 = R3 = {∅} =⇒

p1(1) = 2, p2(1) = 0, k(1) = 0 so that if 1 +D(1)

D′(1)>

s− cvce − cv

then f(r = 1) > 2(s − cv). We will

now find conditions so that f(r) is increasing in r, so that Ψp(r) is strictly quasiconcave and has a

unique maximum. We use the following (detailed steps can be provided upon request):

43

1. p′1(r) ≥ 0

2. p′2(r) ≤ 0

3. (ce − cv)p′1(r) + k

′(r) ≥ 0

We have,

f′(r) =

[D′(rD)

′′ −D′′(rD)′]((ce − cv)p1(r) + k(r))

(D′)2+ (rD)

′ (ce − cv)p′1(r) + k

′(r)

(D′)− (h+ cv − cm)p

′2(r)

Using (1),(2) and (3) the last two terms are non negative. Since D′(rD)

′′ −D′′(rD)′> 0 (D(r) is

log-concave) and p1(r), k(r) > 0, we have that f(r) is strictly increasing in r and Ψp(r) is strictly

quasiconcave, thus has a unique optimum. �

Proof of Lemma 2

As h → ∞, the amount carried over z∗2 → 0. Since Ψp(r) and Ψc(r) are unimodal, a sufficient

condition for r∗p > r∗c is that (Ψ′p(r)−Ψ

′c(r))

∣∣∣r=r∗c

> 0. Moreover, since bp = bc we have,

Ψ′p(r)−Ψ

′c(r) = 2(ce − cv)(a+ 2bcr)

(Φ12[2rD]− Φ[rD]

)= 2(ce − cv)(a+ 2bcr)

(P [ξ1 + ξ2

2< rD]− P [ξ1 < rD]

)(32)

At r(a + bcr) = µ, Ψ′p(r) − Ψ

′c(r) = 0. If r(a + bcr) > µ then Ψ

′p(r) − Ψ

′c(r) > 0 because

ξ1 + ξ2

2

has mean µ and lower variance since ξ1 are ξ2 independent. Thus, a sufficient condition for r∗p > r∗c

is Ψ′p(r∗c ) > 0 i.e., ce < c = cv +

bc(s− cv)√a2 + 4bcµ

and vice versa. It is straightforward to check that

Ψ∗c > Ψ∗p if bp = bc. �

Proof of Proposition 5

Applying the Envelope Theorem with the Kuhn-Tucker multiplier λ for the constraint bp ≥ 0

we get the first-order condition,

dΨ∗pdbp

=∂Ψp(r)

∂bp

∣∣∣r=r∗p

+ λ = 2(s− cv)r∗p − 2(ce − cv)(r∗p)2Φ[r∗p(a+ br∗p)] + λ

= 2r∗p[(s− cv)− (ce − cv)r∗pΦ[r∗p(a+ br∗p)]] + λ =2(s− cv)r∗p(a+ br∗p)

a+ 2bpr∗p+ λ (33)

44

with the complementary slackness condition λ ·bp = 0. If bp = 0, it can be verified that r∗p = 0 since

dΨp(r)

dr

∣∣∣bp=0

< 0. Thus, since λ > 0 by complementary slackness,dΨ∗pdbp

> 0. If bp > 0 then λ = 0 and

dΨ∗pdbp

> 0 sincedΨp(r)

dr

∣∣∣r=0

> 0 which implies r∗p > 0. Therefore, Ψ∗p is increasing in bp. From (??) it

can be verified that if ce−cv ≥bp(s− cv)√a2 + 4bpA

then the optimum lies in the region A ≤ r(a+bpr) ≤ A.

Consider two values of bp = b1, b2 such that b2 > b1 and ce − cv ≥b2(s− cv)√a2 + 4b2A

(>

b1(s− cv)√a2 + 4b1A

)

i.e., both the maximizers, say r1 = r∗p(b1) and r2 = r∗p(b2), lie in the region above. Then both

minimizers satisfy the first order conditions,

Ψ′p(r1, b1) = (s− cv)b1 − 2(ce − cv)(a+ 2b1r1)Φ[r1(a+ b1r1)] = 0 (34)

Ψ′p(r2, b2) = (s− cv)b2 − 2(ce − cv)(a+ 2b2r2)Φ[r2(a+ b2r2)] = 0 (35)

A sufficient condition for r2 < r1 is Ψ′p(r1, b2) < Ψ

′p(r1, b1) = 0. Simplifying we need,

s− cvce − cv

<(a+ 2b2r1)Φ[r1(a+ b2r1)]− (a+ 2b1r1)Φ[r1(a+ b1r1)]

b2 − b1(36)

From (34), as ce → ∞ we have r1(a + b1r1) → A (i.e., r1 → rA(b1)). The left hand side of the

inequality tends to 0 while the right hand side tends to(a+ 2b2rA(b1))

b2 − b1Φ[rA(b1)(a+ b2rA(b1))] > 0

since Φ[r1(a + b1r1)] → Φ[A] = 0 . Thus, there must exists a value of ce = ce(b1, b2) such that

r2 < r1 for all ce > ce(b1, b2) else r2 > r1. This completes the proof. �

Proof of Proposition 6

If r(a+ bcr) > A, the first order condition (??) simplifies to,

dΨc(r)

dr= 2(s− cv)bc − 2(ce − cv)(a+ 2bcr)

45

Equating above expression to 0 gives r =s− cv

2(ce − cv)− a

2bc. Substituting into the inequality above

and simplifying we that if ce < cv+bc(s− cv)√a2 + 4bcA

then r∗c = min( s− cv

2(ce − cv)− a

2bc, 1)

. The maximum

expected profit for the time aggregated claim then is,

Ψ∗c = 2(s− cv)(a+ bcr∗c ) + 2(cv − cm)µ− (ce − cv)

∫ξ1+ξ2≤2r∗c (a+bcr∗c )

(2r∗c (a+ bcr

∗c )− ξ1 − ξ2

)φ(ξ1)φ(ξ2)dξ2dξ1

=(s− cv)2bc2(ce − cv)

+a2(ce − cv)

2bc+ a(s− cv) + 2µ(ce − cm) if r∗c =

s− cv2(ce − cv)

− a

2bc(37)

and Ψ∗c = 2(s− ce)(a+ bc) + 2µ(ce − cm)µ if r∗c = 1.

Since bp > bc we have ce < cv+bc(s− cv)√a2 + 4bcA

< cv+bp(s− cv)√a2 + 4bpA

so that r∗p = min( s− cv

2(ce − cv)−

a

2bp, 1)

. The maximum expected profit for the product specific claim is,

Ψ∗p = 2(s− cv)(a+ bpr∗p) + 2(cv − cm)µ− (ce − cv)

∫ξ1≤r∗p(a+bpr∗p)

(r∗p(a+ bpr

∗p)− ξ1

)φ(ξ1)dξ1

− (ce − cv)∫ξ1≤r∗p(a+bpr∗p)

(r∗p(a+ bpr

∗p)− ξ2

)φ(ξ2)dξ2 =

(s− cv)2bp2(ce − cv)

+a2(ce − cv)

2bp+ a(s− cv) + 2µ(ce − cm)

(38)

if r∗p =s− cv

2(ce − cv)− a

2bpand Ψ∗p = 2(s− ce)(a+ bp) + 2µ(ce − cm)µ if r∗p = 1. Thus, if bp > bc and

ce − cv <bc(s− cv)√a2 + 4bcA

we have r∗p ≥ r∗c and Ψ∗p ≥ Ψ∗c . �

Proof of Proposition 7

We have,

dΨp(r)

dr= 2(s− cv)bp − 2(ce − cv)(a+ 2bpr)Φ[r(a+ bpr)] (39)

= 2(s− cv)bp − 2(ce − cv)(a+ 2bpr)P [Xα = αX + (1− α)µ ≤ r(a+ bpr)]

= 2(s− cv)bp − 2(ce − cv)(a+ 2bpr)P [X ≤ r(a+ bpr)− µα

+ µ] (40)

46

If ce− cv ≶bp(s− cv)√a2 + 4bpµ

then the optimum lies in the region r(a+ bpr) ≷ µ. If the optimum lies in

the region r(a + bpr) > µ then increasing α increases the claim and decreases it otherwise, which

gives the required result. The expected profit is,

Ψp(r) = 2(s− cv)D + 2(cv − cm)µ− 2(ce − cv)rD + 2(ce − cv)Emin(rD,Xα) (41)

For a fixed r, as α increases, Xα becomes (second order) stochastically smaller i.e., the last term

decreases. Thus, the manufacturer’s optimum expected profit decreases with increase in α. This

completes the proof. �

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