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Scrap or Sell: The Decision on Production Yield Loss

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Many production processes not only produce desired quality products (high‐end products), but also generate yield loss or Not‐Quite‐Perfect Products (NQPPs) that do not fully meet the quality standards. In practice, a manufacturer may choose to (1) scrap all NQPPs at a cost and carry the high‐end products only, or (2) sell some or all NQPPs to a value‐conscious low‐end market and carry both high‐end products and low‐end products (NQPPs). This research studies the optimal decision on production yield loss (scrap or sell) and the corresponding pricing and operational strategies under different practical situations. Building upon a standard marketing model for two separated markets, i.e., the high‐end and the low‐end markets, we model the manufacturer's profit maximization problem as a nonlinear programming problem. We characterize the optimal yield‐loss decision and the corresponding optimal pricing for each market and production quantity. We also consider the situation that the NQPPs may face competition in the low‐end market with products designed and produced specifically for that market. In contrast to the common belief that selling NQPPs to a low‐end market can recover some of the cost and hence lead to a higher profit, we show that when the yield rate is small or large enough, selling NQPPs may hurt the manufacturer due to the loss of full control over both markets. This is especially true when competition exists in the low‐end market. This research provides practitioners with detailed guidelines on when and how a specific yield loss (product line or marketing) strategy should be adopted. Managerial insights are generated for the optimal yield loss strategies; numerical tests further demonstrate our results. This article is protected by copyright. All rights reserved.
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Scrap or Sell: The Decision on Production Yield Loss
Rong Li
Martin J. Whitman School of Management, Syracuse University, Syracuse, NY 13244
rli138@syr.edu
Yu Xia
Raymon A. Mason School of Business, College of William & Mary, Williamsburg, VA 23187
amy.xia@mason.wm.edu
Xiaohang Yue
Lubar School of Business, University of Wisconsin Milwaukee, Milwaukee, WI 53201
xyue@uwm.edu
Abstract
Many production processes not only produce desired quality products (high-end products), but
also generate yield loss or Not-Quite-Perfect Products (NQPPs) that do not fully meet the quality
standards. In practice, a manufacturer may choose to (1) scrap all NQPPs at a cost and carry the
high-end products only, or (2) sell some or all NQPPs to a value-conscious low-end market and
carry both high-end products and low-end products (NQPPs). This research studies the optimal
decision on production yield loss (scrap or sell) and the corresponding pricing and operational
strategies under different practical situations. Building upon a standard marketing model for two
separated markets, i.e., the high-end and the low-end markets, we model the manufacturer’s profit
maximization problem as a nonlinear programming problem. We characterize the optimal yield-loss
decision and the corresponding optimal pricing for each market and production quantity. We also
consider the situation that the NQPPs may face competition in the low-end market with products
designed and produced specifically for that market. In contrast to the common belief that selling
NQPPs to a low-end market can recover some of the cost and hence lead to a higher profit, we show
that when the yield rate is small or large enough, selling NQPPs may hurt the manufacturer due
to the loss of full control over both markets. This is especially true when competition exists in the
low-end market. This research provides practitioners with detailed guidelines on when and how a
specific yield loss (product line or marketing) strategy should be adopted. Managerial insights are
generated for the optimal yield loss strategies; numerical tests further demonstrate our results.
1
Key words: yield loss; product line; not-quite-perfect products; vertically differentiated products;
high-end market; low-end market; pricing; low-end market competition
1. Introduction
Production processes often generate yield loss, or Not-Quite-Perfect products (NQPPs), that do
not fully meet the desired quality standards. According to IBM Researchi, around three million
wafers are rejected every year in the semiconductor industry when new generations of technology
are introduced. Such a yield loss phenomenon is observed not only in the semiconductor industry,
but also in other industries such as apparel and food. In the apparel industry, garments with minor
defects have been a common production challenge (Lee et al. 2014), the situation is particularly
true when manufacturers request a short lead time for fabric supplies (Choi and Cai 2018). In
the food industry, around six billion pounds of fruits and vegetables are wasted every year in the
United States according to the Natural Resources Defense Council; much edible food is wasted
simply because it is aesthetically imperfect or unsightly (Bratskeir 2015). Evidently, how to deal
with the production yield loss or the NQPPs is a seemingly important and urgent operational and
marketing decision.
The traditional practice to deal with the yield loss in the semiconductor industry is to scrap
it during the production at a cost. However, with recent technological advancements, NQPPs
offer a reasonably good quality and attract a serious amount of demand in a value-conscious low-
end market. For example, most chip makers sell high-quality branded DRAM (Dynamic Random
Access Memory) chips in a high-end market at a premium price to large computer manufacturers
for making computers and servers. Those NQPPs generated from the same production process,
the low-quality unbranded memory chips, can be sold at a much lower price to memory brokers,
who then trade with other electronic product manufacturers for making toys and DVD recorders.
To protect the high-end brand image, the low-quality memory chips are often sold under different
brand names, such as Spectek (for Micron), Elixir (for Nanya), and Aeneon (for Infineon).
It is important to note that these two markets (high-end and low-end) are often separated
(Kirsch 2005,Tian 2011); for example, toys and DVD recorders do not need high-end computer
chips, and computers and servers cannot use low-end toy chips. In other words, the high-end
market consumes only the high-end products, while the low-end market consumes only the low-end
ihttp://www.zurich.ibm.com/pdf/isl/infoportal/research-flyer-150x300-v4.pdf
2
products. Also, with technological advancements, the semiconductor production can grow quickly
after the yield is stable and high enough (Bohn and Terwiesch 1997 and Franssila 2010). Therefore,
the semiconductor industry is no longer focused on yield uncertainty, but rather on how to deal
with yield loss or NQPPs: scrap or sell?
In practice, some companies sell some or all NQPPs to a separate low-end market, while others
choose to scrap all their NQPPs. In the food industry, McDonald’s opened its first Not-Quite-
Perfect outlet store in Illinois in 2005 selling imperfect and irregular itemsii. Walmart is now selling
imperfect fruits and vegetables that are still fresh and tasty (NPR 2016). In the furniture industry,
high-end furniture with minor scratches and wear can be sold at huge reductions in discount stores
(Torabi 2011). In the apparel industry, Lands’ End sells its own clothes with minor problems in
chosen outlet venues with deep discounts. American Apparel used to sell its own NQPPs on its
website, but recently stopped doing soiii.
At first glance, selling NQPPs to the low-end market does sound like a creative idea of turning
waste and costs into sales profits. What a thrilling and sustainable idea! However, if we examine
the decision on yield loss closely, we may realize that the links between the high-end and low-end
markets are overlooked. There are indeed two critical links. The first is the supply link that the
potential supply quantities of the high-end and low-end (NQPP) products from the same production
run are proportional to the yield rate. The second is the price link that the price difference of the
two markets must be large enough to justify the product quality difference. It is possible that as
the manufacturer tries to extract profit from the low-end market by supplying the right quantity
of NQPPs, the supply to the high-end market is affected due to these two links, which may hurt
the profit earned in the high-end market. Therefore, the total profit earned from selling both high-
end and low-end products might be worse than that from selling the high-end product only and
scrapping the NQPPs.
In addition to the above concerns, recall that the low-end market is not the focal market, but
a possible remedy for the yield loss. Most likely, the low-end market does not depend on the
NQPPs to meet its demand. It is quite possible that low-end products designed for the low-end
market currently exist. Those products tend to have much lower and competitive production costs.
Therefore, the manufacturer should investigate the competition in the low-end market before relying
on the market as a viable outlet for its NQPPs.
In this research, we consider a manufacturer whose production of an item has a stable yield
iihttp://www.theonion.com/article/not-quite-perfect-mcdonalds-opens-in-illinois-outl-1328
iiihttps://www.landsend.com/aboutus/company/?company=retail
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loss, we investigate how this manufacturer should deal with the yield loss: scrap or sell. The
regular yield (the high-quality product) and the yield loss (the low-quality product) can be sold
to exclusive markets, the high-end market and the low-end market, respectively. Building upon a
standard marketing model for such vertically differentiated products, we are able to characterize a
manufacturer’s optimal production quantity and pricing for each strategy (scrapping, selling, and
partial-scrapping-partial-selling) and the optimal operational strategies in dealing with the yield
loss. Under the scrapping strategy, the manufacturer carries only the high-end product and sells it
to the high-end market. For the partial-scrapping-partial-selling strategy and the selling strategy,
the manufacturer carries both the high-end and low-end products and sells them to the high-end
and low-end markets, respectively. We also investigate the situation that NQPPs compete with
low-end products in the low-end market.
Our model reflects the industry practice of using different brands or selling in different stores
and maintaining exclusive markets for products with different quality levels as we discussed earlier.
We are particularly interested in understanding when and why a certain strategy is better in the
production process with different yield rates and scrapping costs. We obtain closed-form solutions
for the optimal production quantity, pricing and profits and offer some counterintuitive but inspiring
insights. The study of the competition of the low-end market also generates some interesting results,
which can lead to an enriched future project.
The rest of this paper is organized as follows. We review the related literature in Section 2
and introduce the model formulation in Section 3. We then analyze the scrapping strategy and
the selling strategy in Section 4and Section 5, respectively. These two strategies are compared
analytically and numerically in Section 6. The general case, the partial-scrapping-partial-selling
strategy, is studied in Section 7.1. The competition in the low-end market is discussed in Section
7.2, followed by the conclusion in Section 8. All the proofs are included in the online supplement.
2. Literature Review
Our research shares a common ground with the literature on managing yield uncertainty within
production and inventory systems. It is also related to the literature on product line design, pricing
strategy and market competition for differentiated products.
Production management with yield uncertainty has been an important topic for both practice
and academia. There has been extensive research on this topic (Lee and Yano 1988,Bitran and
Gilbert 1994, and Yano and Lee 1995) with emphasis on deciding the optimal production and
4
inventory policies to meet the demand of a single product in the presence of yield uncertainty.
However, with technological advancement, the production for many products can grow quickly after
the yield becomes stable and high enough (Bohn and Terwiesch 1997,Franssila 2010). Therefore,
in the semiconductor industry as well as in many other industries, yield rate is often stable and
high, and yield uncertainty is no longer a primary focus as technology advances. Motivated by the
recently observed practice of selling yield loss, our work moves beyond the production decision for
a single product. We focus on marketing trade-offs of 1) selling the regular yield (the high-quality
product) and having the NQPPs scrapped, and 2) selling the regular yield and the NQPPs to the
high-end and the low-end market, respectively.
In the literature of product line design, varied products are introduced to market to meet
diversification needs of customers. Each product may or may not target a different market or
market segment. Typical problems of product line design include decisions on the number and
physical characteristics of products in the line, the price charged by a manufacturer and/or a
retailer for each product, etc.(Villas-Boas 1998,Luchs et al. 2016). In recent years, literature in this
field studies various specific issues in different operational circumstances. Heese and Swaminathan
(2006) determine the component quality levels to reduce total production cost for the product
line with two products for two market segments. Ng et al. (2012) investigate the co-production
of a set of output products of different quality levels with a permitted downward substitution
between the levels. Chen et al. (2013) examine how the production cost and output distribution
of the vertical co-product technology influence product line design. Zhang et al. (2014) study
probabilistic selling quality-differentiated markets, with an offer consisting of a lottery between two
products with different features. Aydin et al. (2015) propose a novel methodology to simultaneously
consider re-manufactured and new products to maximize profit and market share of the product line.
Paul et al. (2015) determine the optimal stocking level of modular end-products while considering
multiple assembly option choices for a modular product family. Bertsimas and Misic (2016) assume
uncertainty in customers’ responses to the new product line and optimize the worst case expected
revenue. Zhu and He (2017) address how the green quality levels of the products are affected by
supply chain structures, product types and competition types.
The topic we study in this paper can also be viewed as a special product line design case.
Two type of products, the high-end products (regular yield) and the low-end products (yield loss
or NQPPs), are co-produced in a same production process with a stable yield rate. We then
investigate whether NQPPs should be introduced to a separated low-end market and if so, with
what amount. The scrapping cost of NQPPs is considered if NQPPs are not introduced to the
5
market.
In the literature of market competition, different pricing and marketing strategies are investi-
gated under various scenarios. Ferrer and Swaminathan (2006) study the joint pricing of new and
remanufactured products for a monopolist in a multi-period setting. Debo et al. (2005) investigate
the joint pricing and technology decision for new and remanufactured products. Atasu et al. (2008)
characterize the conditions on market share, cost savings, and consumer preference between new
and remanufactured products under which remanufacturing is profitable for a monopolist. Ru et
al. (2015) study the competition between store brands and national brands. Li et al. (2016) de-
scribe the cost structure of a decentralized bandit supply chain and a centralized mainstream firm,
and they show that a bandit supply chain’s inefficiency and fierce competition actually leads to its
higher product price and quality. Li et al. (2017) propose pricing strategies for a reliable channel
and a unreliable channel. Our work differs from this literature in a number of ways due to the
special connection (the regular yield and its yield loss) between the two product quality levels we
consider. First, we consider a product with two quality levels (high- and low-end) that are sold
in separate or exclusive markets as observed in practice, and thus do not consider any direct com-
petition between them. Second, the products with two quality levels in our model have the same
production cost if the manufacturer decides to offer both of them. When some or all NQPPs are
scrapped, the manufacturer would incur an additional scrapping cost. Finally, the potential supply
of the two product quality levels we consider is proportional. We determine the actual supply to
each market to maximize the manufacturer’s profit. In the low-end market, we also discuss the
competition between the NQPPs and the existing low-end products in the market.
The literature on vertical differentiated products studies the marketing and operational strate-
gies of a product with multiple levels of quality. Their topics vary from monopoly pricing (Mussa
and Rosen 2009,Moorthy and Png 1992,Debo et al. 2005) to competitive pricing (Choi and Shin
1992,Wauthy 1996,Jing 2006,Ishibashi and Matsushima 2009,Xia 2011,Amaldoss and Shin 2011).
Monopoly pricing in marketing literature generally refers to the market-clearing pricing, which usu-
ally assumes market price responds to supply quantity and is appropriate for commodity-like prod-
ucts. Competitive pricing, on the other hand, assumes that prices are determined by competitors
in the market to actively influence the demand of the markets. The focus of the study on monopoly
pricing includes optimal pricing for continuous quality (Mussa and Rosen 2009), cannibalization
effect on choice of simultaneous or sequential product launch (Moorthy and Png 1992), and single
versus multiple quality. Moorthy and Png (1992) show that when cannibalization is too signifi-
cant, the monopolist should offer a single quality under simultaneous product launch. Considering
6
non-commodity products, monopoly setting both price and quantity—a different type of monopoly
pricing—is assumed. Specifically, Chen et al. (2013) develop a comprehensive model to study prod-
uct line design, price setting and production decisions for a co-production system of non-commodity
products such as microprocessors; Bansal and Transchel (2014) study two non-commodity vertically
differentiated co-products sold at two connected markets with supply-independent prices (through
downward substitution) and partially characterize the optimal substitution and/or withholding
strategy. For the literature on competitive price setting, one well-known result is the high-quality
advantage, which claims that high-quality products generate higher profit margin than low-quality
products. This result assumes zero variable cost for improving quality. The opposite result—the
low-quality advantage—is shown in recent works with either supply-side reasons (e.g., increasing
variable cost for improving quality as in Jing 2006) or demand-side reasons (e.g., large size of low-
end market as in Amaldoss and Shin 2011). Most of the work in this line of research assumes that
consumers are indistinguishable, always preferring high quality product, and that each manufac-
turer chooses a single but different quality. One exception is Ishibashi and Matsushima (2009),
whose work assumes distinguishable customers (high-end and low-end markets as in our paper).
It is found that when there are infinitely many low-end manufacturers competing in the low-end
market, high-end manufacturers should only sell to the high-end market. This result does not apply
to the single high-end manufacturer case.
In contrast, our work considers monopoly pricing (market-clearing price) for two commodity
products that are sold to distinguishable customers in two separate or exclusive markets. The two
product quality levels we consider are specially linked: the regular yield and the yield loss of the
same production. Specifically, we examine the effect of the supply link (the proportional supply
to the two markets) and the price link (the minimum price-difference between the two markets to
signal/justify the quality difference) on the manufacturer’s choice of single level of product quality
(i.e., the scrapping strategy) or two levels of product quality (i.e., the selling and partial-scrapping-
partial-selling strategies) and the corresponding optimal pricing and profit. In short, we add to the
literature the impact of an important supply-side factor, the supply link, on market segmentation
which changes the traditional belief for monopolists with distinguishable customers.
3. Model Formulation
This research studies the optimal yield loss (scrap or sell) and product line (one or two levels of
product quality) strategy. We consider a manufacturer producing a single type of product with two
7
different quality levels of outputs: the regular yield with high-quality, referred to as h-product, and
the yield loss or NQPP with low-quality, referred to as l-product. The h-product (e.g., high-quality
memory chips) is sold exclusively to a high-end market (with customers like large PC makers),
referred to as h-market; l-product (e.g., low-quality memory chips) is sold exclusively to a low-end
market (with customers like toy and DVD makers), referred to as l-market under a different brand
label.
The h-product and l-product come out of the same production process with capacity Q > 0
and a stable yield rate ρ(0,1). Thus, they are assumed to hold the same unit production cost
cp0 and their supply quantities are ρQ and (1 ρ)Q, respectively. Without loss of generality,
we normalize the unit production cost cpto 0 and denote the cost of scrapping each unit of l-
product as cs. Note that, since scrapping may recover some raw materials that can be reused for
production, cscan be negative. The h-product is sold in h-market; l-product is either scrapped at
a cost or sold in l-market. For simplicity and a sharper contrast, we first consider two base models:
scrapping all the l-product as the scrapping strategy and sell all the l-product as the selling strategy.
Later we analyze the general case, the partial-scrapping-partial-selling strategy, in Section 7.1. If
the manufacturer adopts the scrapping (Sf) strategy, it carries a single level of product quality
(h-product) and operates in a single market (h-market). On the other hand, if the manufacturer
adopts the selling ( ¯
S) strategy or partial-scrapping-partial-selling strategy (Sp), it carries both
products and operates in both markets. Figures 1and 2illustrate the manufacturer’s operational
setup under Sfstrategy and ¯
Sstrategy, respectively.
Capacity
Q
high-quality
ρ
Q
low-quality
(
1ρ
)Q
high-end market
Dh
Scrap
$cs
$cp=0
$cp=0
Figure 1: Scrapping (Sf) Strategy
Capacity
Q
high-quality
ρ
Q low-quality
(
1ρ
)Q
high-end market
Dh
low-end market
Dl
$cp=0 $cp=0
Figure 2: Selling ( ¯
S) Strategy
We model the quality difference between h-product and l-product via the difference of their
prices, i.e., phpl> δ(>0), where piis the price of the i-product, i=h, l. To model the
8
demand and price for each market, we follow the standard marketing model on high-end and low-
end markets (see Ishibashi and Matsushima 2009). Indeed, we consider the market-clearing price
for both markets (which is appropriate for the markets of commodities like memory chips). We
assume that the customers’ willingness-to-pay above the production cost is uniformly distributed
on [0,1] in h-market (with market size 1) and on [0, a] in l-market (with market size b > 0 and
willingness-to-pay range for l-product a < 1). Thus, the demand in the i-market, Di, is the number
of customers whose willingness-to-pay will exceed the i-market price, i.e., the selling price of the
i-product, pi,i=h, l. That is, the demand functions for h-market (Dh) and l-market (Dl) are
Dh=(1phif ph<1,
0 otherwise, (1)
Dl=(b(1 pl
a) if pl< a,
0 otherwise, (2)
where phplδ, which we referred to as the minimum price-difference constraint, should hold
under the ¯
Sand Spstrategies when the manufacturer sells to both markets. Note that this con-
straint along with the constraint that a < 1 guarantees that h-product is sold at a higher price
than l-product and consumers are willing to pay more for h-product than for l-product, to reflect
common business practice.
Note that the linear demand function we consider is commonly used in the marketing literature
and contains no uncertainty (which makes our analysis tractable in a complex setting). It differs
significantly from the demand functions used in the Operations Management (OM) literature which
often consider demand uncertainty () using either an additive demand model (D=y(p) + ) or a
multiplicative demand model (D=y(p)), as reviewed by Petruzzi and Dada (1999). Our demand
model simultaneously captures customers’ evaluations on the product and the change of market
size with respect to price. It also allows us to perceptibly illustrate the impact of the price link
(phplδ) on the choice of yield loss strategy. Thus, our model can provide more and clearer
managerial insights than traditional OM demand models. Meeting the demand, the manufacturer
decides the supply quantity to each market, which implicitly fixes the market price (market-clearing
price) and thus the profit earned in each market. Before that, the manufacturer should first choose
an optimal strategy to deal with yield loss: scrapping, selling, or partial-scrapping-partial-selling.
Since the use of different strategies will lead to different quantities of h-product and l-product, the
manufacturer’s total profit should differ. We next analyze the optimal production quantity under
each strategy and then compare them based on the profit.
9
4. The Scrapping (Sf) Strategy
In this section, we study the (full) scrapping Sfstrategy, under which the manufacturer scraps all
the l-product at unit cost csduring the production process immediately upon inspection. When
adopting the scrapping strategy, the manufacturer carries the products with a single level of product
quality (h-product) and sells it to h-market. That is, the manufacturer with production capacity Q
will supply ρQ units of the regular yield (h-product) to h-market. Matching this supply quantity
(ρQ) to the demand (Dh= 1 ph) and thus selling all of the h-product, the resulting market-
clearing price for h-product is: ph= 1 ρQ (ρQ = 1 ph). Note that phshould not exceed 1 to
attract any h-market demand. Thus, the manufacturer’s profit maximization problem under the
Sfstrategy is:
max ΠSf(Q) = ph(ρQ)cs(1 ρ)Q=phcs(1 ρ)
ρ(ρQ) (3)
s.t. ph= 1 ρQ [0,1].(h-market price)
Recall that we have normalized the (direct) production cost to 0 and for each unit of h-product,
the manufacturer will have 1ρ
ρunits of yield loss needed to scrap. As (3) shows, for each unit
of h-product sold, the manufacturer earns phas the revenue (less the production cost) and pays
cs(1ρ)
ρfor scrapping the corresponding yield loss, which is thus referred as the indirect production
cost. Note that the higher the yield rate (ρ), the less the units of yield loss needed to scrap for a
unit of h-product, and thus the lower the indirect production cost. Since the manufacturer’s profit
function, ΠSf(Q), is quadratic and concave in Q, we can easily characterize the optimal production
quantity and the corresponding market-clearing price and profit, all denoted with superscript Sf.
Proposition 1 When the manufacturer chooses the scrapping Sfstrategy,
If ρ > cs
1+cs, the manufacturer’s optimal production quantity is QSf=1
2ρcs(1ρ)
2ρ2with profit
ΠSf= (1
2cs(1ρ)
2ρ)2and the resulting market-clearing price is pSf
h=1
2+cs(1ρ)
2ρ;
If ρcs
1+cs, the manufacturer should not produce at all (QSf= 0 and ΠSf= 0).
As mentioned above, for each unit of h-product, the manufacturer needs to earn enough rev-
enue (ph1) to cover the indirect production cost ( cs(1ρ)
ρ) in order to justify production. This
requires 1 >cs(1ρ)
ρρ > cs
1+csor, intuitively the yield rate ρmust be high enough to make the
indirect production cost (for scrapping) low enough, which we refer to as the minimum yield rate
requirement. Note that when the scrapping cost (cs) is lower, the minimum yield rate requirement
is lower. As mentioned in Section 3, since scrapping may help recover some raw materials that can
10
be reused for production, cscan be negative and in this case, the minimum yield rate requirement
disappears and the manufacturer will always produce under the scrapping strategy.
Corollary 1 As the yield rate, ρ, increases, pSf
hdecreases, but QSfand ΠSfincrease. The man-
ufacturer’s maximum profit, supρ(0,1) ΠSf=1
4, occurs at ρ= 1.
Intuitively, as the yield rate, ρ, increases, fewer units of yield loss will be scrapped for each
unit of h-product and thus the indirect production cost, cs(1ρ)
ρ, drops. This will clearly benefit
the manufacturer and induce him to produce more and thus supply more, which in turn will drive
down the market price. Thus, the manufacturer’s maximum possible profit under the scrapping
(Sf) strategy is 1
4, which is achieved at ρ= 1.
5. The Selling (
¯
S) Strategy
In this section, we study the selling ( ¯
S) strategy, under which the manufacturer carries both prod-
ucts and sells them to their respective markets. As illustrated in Figure 2, once the manufacturer
determines its production quantity Q, it will supply ρQ units of the regular yield (h-product) to
h-market and (1 ρ)Qunits of the yield loss (l-product) to l-market. Thus, the resulting market-
clearing prices in these two markets, phand pl, are determined as:
ρQ =Dh= 1 phph= 1 ρQ,
(1 ρ)Q=Dl=b1pl
apl=a1(1 ρ)Q
b.(4)
In contrast to the scrapping (Sf) strategy, under the selling ( ¯
S) strategy, the manufacturer’s profit
comes from both h-market and l-market, although the profit still depends solely on the production
quantity Q. Recall that phplδ(the minimum price-difference constraint) is required to reflect
the product quality difference. The manufacturer’s profit maximization problem under the selling
(¯
S) strategy is:
max Π ¯
S(Q) = ph(ρQ) + pl(1 ρ)Q(5)
s.t. ph= 1 ρQ [0,1],(h-market price)
pl=a1(1 ρ)Q
b[0, a],(l-market price)
phplδ. (min price difference)
Since phand plare both linear in Q, the manufacturer’s profit function is quadratic and concave
in Q. Due to the additional constraints (for the l-market price and the minimum price-difference),
11
characterizing the optimal production quantity is much more complex under the selling ( ¯
S) strategy
than under the scrapping (Sf) strategy. Note that the standard method of solving such a nonlinear
program is solving the KKT conditions and proving the required properties. This method may be
difficult to apply to our problem above, i.e., problem (5), due to the large number of constraints
(5 of them). Therefore, we use an equivalent alternative method which requires only the concave
property of the objective function. We start by solving the relaxed version of the above problem, i.e.,
ignoring constraints (l-market price) and (min price difference). We define its optimal production
quantity solution and associated prices and profit, all with superscript u, as: Qu:= b
2
a(1ρ)+ρ
a(1ρ)2+2,
pu
h:= 1 b
2
a(1ρ)ρ+ρ2
a(1ρ)2+2,pu
l:= a11
2
a(1ρ)2+ρ(1ρ)
a(1ρ)2+2, and Πu:= b
4
[a(1ρ)+ρ]2
a(1ρ)2+2, where Quis referred
to as the unconstrained optimal solution.
If the unconstrained optimal solution, Qu, is feasible to problem (5), it is therefore optimal.
Otherwise, one of the boundary solutions will be optimal because the objective function, Π ¯
S(Q),
is concave. It is easy to see that the boundary solutions, ph= 1 and pl=a, will never be reached
as long as Q > 0. Hence we need only to consider the other two boundary solutions, pl= 0 and
phpl=δ, and have the following results:
The boundary solution at pl= 0, denoted with superscript 0, is Q0:= b
1ρ; the associated
prices and profit are p0
h:= 1
1ρ,p0
l:= 0, and Π0:=
1ρ1
1ρ.
The boundary solution at phpl=δ, denoted with superscript δ, is Qδ:= 1aδ
ρa
b(1ρ); the
associated prices and profit are pδ
h:= (a+δ)ρa
b(1ρ)
ρa
b(1ρ),pδ
l:= (a+δ)ρa
b(1ρ)
ρa
b(1ρ)δ, and Πδ:=
pδ
h(ρQδ) + pδ
l(1 ρ)Qδ.
When the unconstrained optimal solution, Qu, violates one or both of the constraints, pl0
and phplδ, the problem is infeasible to problem (5). Intuitively, such violations happen when
the use of Quresults in a loss in l-market and/or insufficient price difference for h-product and l-
product. Note that the profit function, Π ¯
S(Q), is concave in Q; the l-product price, pl, decreases in
Q; and the price difference, phpl, increases or decreases in Qfor ρ < a
a+bor ρ > a
a+b, respectively.
This means if Quviolates constraint pl0, we should reduce Qto avoid loss in l-market; if Qu
violates constraint phplδ, we should reduce the production quantity Qfor ρ > a
a+b, but
increase the production quantity Qfor ρ < a
a+b, to warrant a sufficient price difference. Thus,
for ρ < a
a+b, even if Quviolates only the latter constraint, adjusting Qup may lead to violation
of the former constraint, thus resulting an infeasible problem. Therefore, we make the following
necessary, but reasonable, assumptions on the model parameters, a, b and δ, to make problem (5)
feasible and analytically tractable.
12
Assumption 1 The market parameters, a, b and δ, should satisfy: b1,δ1a
2,(pu
h
pu
l)|ρ=a
a+bδ, and 8ab 1.
We now partition the range of ρbased on whether pu
hpu
lδholds. Below we explain, for
the different ranges of ρ, how the optimal solution, Q¯
S, is derived, along with the explanation for
each assumption stated in Assumption 1. We can prove that the first two assumptions (b1 and
δ1a
2) imply that the price difference pu
hpu
lis unimodal—it increases and then decreases
in ρ(0,1), as shown in Figure 3. Note that b1 means l-market is smaller than or equal to
h-market, which is observed in practice; δ1a
2means the price difference between the two
markets must be practical, and no h-market price is feasible if the l-market price is a
2or above.
We thus define ρ0:= sup{ρ(0,1)|pu
hpu
lδ}to represent the turning point of ρto violate the
minimum price-difference constraint, where ρ0= 1 means that pu
hpu
lδalways holds for any
yield rate ρ. The third assumption, (pu
hpu
l)|ρ=a
a+bδ, guarantees that ρ0<1 or the feasibility of
problem (5); the last assumption, 8ab 1, assures no loss in l-market when using Qu, i.e., pu
l0
for any ρ(0,1). The third assumption means that the price difference of the two markets cannot
be so high that it is never satisfied. The last assumption means that the l-market price is not too
high.
!
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!"%
!"&
!"'
!"(
!")
!"*
!"+
!!"$ !" & !"( !"* #
r ,-./012345/6
d
ph
upl
u
r
0
Figure 3: The price difference pu
hpu
lwith respect to ρ(a= 0.5, b = 0.25, δ = 0.25)
In summary, we partition the range of ρat ρ=ρ0. We find that for ρ(0, ρ0], pu
hpu
lδ
and pu
l0 are always met; for ρ(ρ0,1), pu
hpu
lδis violated, and thus we should reduce
the production quantity Qto Q0or Qδ, whichever is smaller, to satisfy both constraints and reach
the optimal solution Q¯
S. The detailed expression of the optimal solution, Q¯
S, together with the
resulting prices and profit, is presented below.
13
Proposition 2 The optimal production quantity as well as the resulting market-clearing prices and
profit under the selling ( ¯
S) strategy are:
If Q¯
Sp¯
S
hp¯
S
lΠ¯
S
ρ(0, ρ0]Qupu
hpu
lΠu
ρ(ρ0,1δ
1δ+b]Q0p0
h0 Π0
ρ(1δ
1δ+b,1) Qδpδ
hpδ
lΠδ
where ρ0= sup{ρ(0,1)|pu
hpu
lδ},Qu>max{Q0, Qδ}, and Q0and Qδincreases and decreases
in ρ, respectively.
The above results show that, upon applying the selling ( ¯
S) strategy, only when the yield rate
is small (ρ(0, ρ0]), the manufacturer will use the unconstrained optimal solution (Qu) and earn
a global-optimaliv profit from both markets. When the yield rate is medium (ρ(ρ0,1δ
1δ+b]), the
manufacturer will use the boundary solution (Q0such that pl= 0) with a lesser supply (Q0< Qu)
to both markets, and earn only a global-suboptimal profit from h-market and no profit from l-
market. Moreover, the higher the yield rate, the more the total supply (Q0). When the yield rate
is large (ρ(1δ
1δ+b,1)), the manufacturer will use the boundary solution (Qδsuch that phpl=δ)
with a further reduced supply (Qδ< Q0< Qu) to both markets, and earn a global-suboptimal
profit from both markets. The total supply (Qδ) decreases in the yield rate.
Corollary 2 Comparing the maximum profits earned under the selling ( ¯
S) and scrapping (Sf)
strategies, we find that if ρ01
1+b, the manufacturer earns more profit under ¯
Sstrategy, i.e.,
maxρ(0,1) Π¯
S= maxρ(0,1) Πu= Πu|ρ=1
1+b(= 1+ab
4)>supρ(0,1) ΠSf(= 1
4).
The above results show that if the two constraints, pl0 and phpl0, are not violated
at the ideal yield rate ρi=1
1+b, the manufacturer should adopt the selling strategy to extract the
global-optimal profit from both markets. Note that at the ideal yield rate ρi=1
1+b, the supply
ratio to the two markets is ρi: (1 ρi) = 1 : b. This means the supply ratio ρi: (1 ρi) perfectly
matches with the ratio of the two markets (in size) 1 : b. Thus, the manufacturer can earn the
global-optimal profit from both markets by selling all the output (h-product and l-product) to
supply half of their respective markets (ρQ =1
2and (1 ρ)Q=b
2). We also refer to 1 : bas the
perfect supply ratio.
ivThe global-optimal profit refers to the optimal profit earned under the relaxed version of problem (5).
14
6. Comparison of Scrapping (Sf) and Selling (
¯
S) Strategies
In this section, we analyze how the manufacturer should choose between the Sf(scrapping) and
¯
S(selling) strategies using the results from the previous two sections. Recall that under the
scrapping strategy, the manufacturer sells h-product to h-market and scraps all NQPPs. Under
the selling strategy, the manufacturer sells h-product to h-market and all NQPPs (l-product) to
l-market. Note that for both strategies, the demand function of the h-product is assumed to be
the number of customers whose willingness-to-pay exceeds the market price, which is determined
by the production quantity as 1ρQ. Although the demand function remains the same, the exact
optimal demand sizes and selling prices of the h-market can be different for scrapping and selling
strategies at various situations, as discussed in the previous two sections.
When examining how to choose between the scrapping and selling strategies, we identify two
drivers: the supply link and price link between the two markets (h-market and l-market). Recall
that the supply link in Section 1is defined as a fixed supply ratio to the two markets (ρ: (1 ρ)).
The price link is defined as the price of the two markets maintaining a minimum difference (i.e.,
phplδ) to signal the quality difference.
We start with the supply link. Note that the actual supply ratio ρ: (1 ρ) may be far away
from the perfect supply ratio 1 : bin instances where there is too little supply to h-market and too
much supply to l-market. In this situation, the manufacturer under the selling strategy, bounded
by the supply link, tries to gain profit from both markets, but loses full control over either market;
hence, the manufacturer earns only the global-suboptimal profit from both markets. Scrapping
all NQPPs, however, allows the manufacturer to have full control over h-market (through supply
quantity) and earn the global-optimal profit from h-market. This explains why the manufacturer
should choose to scrap all NQPPs. The price link (the minimum price-difference) has a similar
effect. The enforcement of the minimum price-difference causes the manufacturer under the selling
strategy to lose full control over either market and leads to only the global-suboptimal profit earned
in both markets; therefore, scrapping all NQPPs can help the manufacturer regain full control over
h-market and earn the global-optimal profit. In other words, not selling (scrapping) the l-products
may enhance the demand/profit for h-products due to the supply link and price link, making the
scrapping strategy favorable for yield loss in certain situations, as we will further discuss in this
section. In summary, due to the supply and price links, the manufacturer under the selling strategy
may lose full control over both markets; thus, the manufacturer should choose to scrap all NQPPs
to regain full control over h-market (its focal market) and earn the most profit.
15
We find that, for both high and low yield rates (ρ) meeting the minimum yield rate requirement
(ρ > cs
1+csrequired to justify production under the scrapping strategy), the manufacturer is better
off with the scrapping (Sf) strategy. In this section, we first illustrate and discuss in detail how the
supply and price links work via a numerical example. We then summarize the comparison results
analytically.
6.1 Numerical Illustration
We choose the following parameters for our numerical example: a= 0.5 (half of the maximum
willingness-to-pay in h-market), cs= 0.2 (40% of the maximum willingness-to-pay in l-market),
b= 0.1 (10% of h-market), and δ= 0.25 (= 1
2a
2, the price difference of the unconstrained
optimized markets).
As shown in Proposition 1, if ρcs
1+cs= 0.17, the scrapping (Sf) strategy is not worth consider-
ing as the manufacturer would not even produce. Thus, the valid range of ρfor comparing the scrap-
ping (Sf) and selling ( ¯
S) strategies is between 0.17 and 1. We find that if ρ[0.24,0.76] S[0.94,1),
which counts 70% of the valid range, the manufacturer earns more profit when choosing the scrap-
ping strategy. The details of how much more profit can be earned are shown for the following cases,
L (Low), M-L (Medium-Low), and H (High) range of ρ:
(L) If ρ[0.24,0.36], ΠSfexceeds Π ¯
Sby up to 96%. Under the selling strategy, the unconstrained
optimal solution, Qu, is optimal.
(M-L) If ρ(0.36,0.76], ΠSfexceeds Π ¯
Sby up to 97%. Under the selling strategy, the boundary
solution Q0is optimal, i.e., pl= 0.
(H) If ρ[0.94,1), ΠSfexceeds Π ¯
Sby up to 33%. Under the selling strategy, the boundary
solution Qδis optimal, i.e., phpl=δ.
In cases (L) and (M-L), the yield rate ρis either low or medium-low, and we observe that the
supply link is the sole driver. Recall that at the ideal yield rate (ρi=1
1+b), we introduced the
perfect supply ratio to the two markets, ρi: (1 ρi)=1:b, which is 10 : 1 in this example. The
manufacturer can reach the global-optimal profit with this perfect supply ratio. However, in the
cases of (L) and (M-L), the corresponding supply ratio is below 3.17 : 1, which is far below the
perfect supply ratio. Such imperfect supply ratios will make the manufacturer supply too little to
h-market (resulting in a high h-price) and too much to l-market (resulting in a low l-price); the
minimum price-difference is naturally secured (i.e., the price link is ineffective or non-binding). The
16
manufacturer earns global-suboptimal profits from both markets, and the total profit is even less
than the optimal profit earned only in h-market under the scrapping strategy. Note that under the
scrapping strategy, despite of the scrapping cost, the manufacturer has full control over the supply
quantity to h-market, and thus can extract the maximum profit from h-market.
In case (H), the yield rate is high, and we observe that the price link dominates the effect.
For these high yield rates, the corresponding supply ratio is above 15.7 : 1, which is far above the
perfect supply ratio (10 : 1). Such imperfect fixed supply ratio will make the manufacturer supply
too much to h-market (resulting in a low h-price) and too little to l-market (resulting in a high
l-price); the minimum price-difference between the two markets cannot be secured. Thus, both the
supply and price links are effective. The enforcement of the minimum price-difference makes the
manufacturer earns the global-suboptimal profits in both markets, and the total profit under the
selling strategy is less than the profit under the scrapping strategy. Although the supply link (the
imperfect supply ratios) also contributes to the global-suboptimal profits, it alone does not make
the scrapping (Sf) strategy better (as proved in Corollary 2, Πu
ρ(1
1+b,1) >maxρ(0,1) ΠSf).
6.2 Analytical Comparison Results
We now analytically compare the scrapping (Sf) and selling ( ¯
S) strategies at various yield rate (ρ)
using the results from Propositions 1and 2. Let ςdenote the optimal strategy (ς=Sfor ¯
Sin this
comparison). Recall that when ρcs
1+cs, the manufacturer will never produce under the scrapping
strategy, and thus the selling strategy is optimal, i.e., ς=¯
S. When ρ > cs
1+cs, however, the profit
comparison is quite complex as it requires solving a number of polynomial equations of degree up
to 5. We thus perform the comparison under some reasonable conditions, csaand G(ρ0)>0,
where
G(ρ) := ac2
s(1ρ
ρ)32acs(1ρ
ρ)2+ (a+bc2
sa2b)(1ρ
ρ)2b(a+cs).
Recall that ΠSf=(1cs(1ρ)
ρ)2
4(the optimal profit under the scrapping strategy), Πu=b
4
[a(1ρ)+ρ]2
a(1ρ)2+2
(the global-optimal profit) and Π0=
1ρ1
1ρ(the optimal profit under the selling strategy
when the optimal solution Q¯
Sis the boundary solution to pl= 0). We define two critical values of
the yield rate as:
ρ1:= inf{ρ(cs
1 + cs
, ρ0) : G(ρ)>0ΠSf>Πu},
ρ2:= inf{x(ρ0,1) : ρx, ΠSf>Π0},
which have the following properties:
17
Lemma 1 ρ1>cs
1+csand ρ2<1
1+b(which means ΠSfΠ0>0always holds for ρ1
1+b).
Using the above properties of ρ1and ρ2, we can generate the following comparison results.
Proposition 3 If csaand G(ρ0)>0,ΠSfΠ0is first negative and then positive as ρincreases
in (ρ0,1). The optimal yield loss strategy ς(ς=Sfor ¯
Sin this comparison) and the associated
results are:
If ς Qςpς
hpς
lΠς
ρ(0, ρ1]¯
S Qupu
hpu
lΠu
ρ(ρ1, ρ0]SfQSfpSf
hN/A ΠSf
ρ(ρ0, ρ2]¯
S Q0Qδp0
hpδ
h0pδ
lΠ0Πδ
ρ(ρ2,1) SfQSfpSf
hN/A ΠSf
Note that “N/A” stands for not applicable as l-market does not exist under the scrapping (Sf)
strategy. In the last two rows of the above table, Π ¯
S= Π0or Πδdepending on if ρ2<1δ
1δ+b,
following immediately from Proposition 2.
0
0.2
0.4
0.6
0.8
1
00.1 0.2 0. 3 0.4 0. 5
r (yield rate)
cs(scrapping cost)
Scrappi ng
(sup ply link)
Scrappi ng (both li nks) r2
r0
r1
Selling
Figure 4: Optimal strategy for any csand ρ
(a= 0.5, b = 0.1, δ = 0.25)
0
0.2
0.4
0.6
0.8
1
00.1 0.2 0. 3 0.4 0. 5
r (yield rate)
cs(scrapping cost)
Scrappi ng (both Li nks)
Scrappi ng
(sup ply link)
r1
r0
r2
Selling
Figure 5: Optimal strategy for any csand ρ
(a= 0.5, b = 0.25, δ = 0.25)
We next use Figures 4-7above to illustrate how the comparison of the scrapping and selling
strategies is affected by the scrapping cost (cs) and yield rate (ρ). Note that as csincreases, the
scrapping (Sf) strategy becomes more costly and thus less attractive. When csis not large, we find
that the optimal strategy alternates twice from the selling ( ¯
S) strategy to the scrapping strategy
(Sf) as ρincreases from 0 to 1. The first alternation is the result of the supply link alone, while the
18
0
0.2
0.4
0.6
0.8
1
00.05 0.1 0. 15 0.2 0. 25 0.3
r(yield rate)
cs(scrapping cost)
Scrappi ng (both li nks) r2
r0
r1
Scrappi ng
(sup ply link)
Selling
Selling
Figure 6: Optimal strategy for any csand ρ
(a= 0.3, b = 0.1, δ = 0.35)
0
0.2
0.4
0.6
0.8
1
00.05 0. 1 0.15 0.2 0. 25 0.3
r (yield rate)
cs(scrapping cost)
Scrappi ng (both li nks)
Scrappi ng
(sup ply link)
r0
r2
r1
Selling
Figure 7: Optimal strategy for any csand ρ
(a= 0.3, b = 0.25, δ = 0.35)
second alternation is the result of both the supply and price links. When csis large, however, the
optimal strategy alternates only once due to both links. Comparing Figures 4and 5or Figures 6and
7, we observe that as bincreases (from 0.1 to 0.25), the (cs, ρ) region in which the scrapping strategy
is optimal due to the supply link, shrinks, but the (cs, ρ) region in which the scrapping strategy is
optimal due to both links, expands. Intuitively, as l-market becomes more attractive in terms of
the market size, the manufacturer would have more incentive to sell l-product rather than scrap.
But at high yield rates, we find the opposite result. This is because the manufacturer would have a
small quantity of l-product to supply to the bigger l-market (b= 0.25), making l-product even more
expensive (compared to h-product) to justify its inferior quality. Thus, the manufacturer, under
the selling strategy, would have to increase the production quantity to supply more to l-market,
which hurts the h-market profit and the total profit even more. Thus, the manufacturer would
rather use the scrapping strategy and completely give up l-market.
7. Extensions
In this section, we extend our monopoly model with either full-scrapping (Sf) or selling ¯
Sstrategies
in two directions. First, we consider that the monopoly manufacturer adopts a general strategy:
partial-scrapping-partial-selling (Sp) strategy. Second, we study that the monopoly manufacturer
monopolizes the h-market only and faces competition in the l-market.
19
7.1 The Partial-Scrapping-Partial-Selling (Sp) Strategy
Note that the partial-scrapping-partial-selling (Sp) strategy a general case including both the full
scrapping (Sf) strategy and selling ( ¯
S) strategy as special cases. Under the Spstrategy, as illus-
trated in Figure 8, some of l-product is scrapped during the production at unit cost cs, whereas
others are sold to l-market at unit price pl. The manufacturer operates in both markets, selling all
of h-product to h-market and some of l-product to l-market and scrapping the rest of l-product. In
contrast to the selling ( ¯
S) strategy, when the production quantity Qis fixed, the h-market supply
quantity is still ρQ, but the l-market supply quantity is Ql(1ρ)Qand the scrapping quantity is
[(1ρ)QQl]. When Ql= (1ρ)Q, the scrapping quantity is 0 and the Spstrategy is equivalent to
the selling ( ¯
S) strategy; when Ql= 0, l-product is fully scrapped and the Spstrategy is equivalent
to the full scrapping (Sf) strategy.
Capacity
Q
Scrap
high-quality
ρ
Q
low-quality
(
1ρ
)Q
high-end market
Dh
low-end market
Dl
Ql
(1-
ρ
)Q-Ql
$cp=0
$cp=0
$cs
Figure 8: Partial-scrap (Sp) strategy
As under the selling strategy, we match the supply with the demand in each market under the
partial-scrapping-partial-selling (Sp) strategy to obtain the market-clearing price.
ρQ =Dh= 1 phph= 1 ρQ,
Ql=Dl=b1pl
apl=a1Ql
b.(6)
Recall that if the manufacturer chooses to sell to l-market (if Ql>0), we will need the minimum
price-difference constraint (phplδ) to justify and signal the product quality difference. The
20
manufacturer’s profit maximization problem under the Spstrategy is:
max ΠSp(Q, Ql) = ph(ρQ) + plQlcs[(1 ρ)QQl] (7)
=ρ2Q2+ (ρcs+ρcs)Q+a/bQ2
l+ (a+cs)Ql
s.t. Ql[0,(1 ρ)Q],(l-market supply)
ph= 1 ρQ, [0,1],(h-product price)
pl=a1Ql
b[0, a],(l-product price)
phplδif Ql>0.(min price difference)
Note that the manufacturer’s profit function, ΠSp(Q, Ql), is jointly concave in Qand Ql. As
we analyze the other strategies, we start by solving the relaxed version of the above problem,
i.e., ignoring the l-market supply and minimum price-difference constraints. We thus obtain the
unconstrained optimal solutions under the Spstrategy as Q=ρ(1+cs)cs
2ρ21{ρ> cs
1+cs}and Q
l=
b(a+cs)
2a>0, where 1{} is the indicator function which equals to 1 if the condition in {} is satisfied
and 0 otherwise.
We then analyze when this unconstrained optimal solution, (Q, Q
l>0), will satisfy the l-
market supply constraint, Ql[0,(1 ρ)Q].
Lemma 2 (Q, Q
l)satisfies Ql[0,(1 ρ)Q]if and only if cs
1+cs< ρ 1
1+band cs[1(1+b)ρ]
2+a(1ρ)2.
Note that the minimum yield rate requirement should be met ( cs
1+cs< ρ) to justify production
under the Spstrategy. To justify partial-scrapping, the scrapping cost (cs) should be lower than
the profit earned from l-market ([1(1+b)ρ]
2+a(1ρ)2) and the yield rate cannot be higher than 1
1+b. This is
because when ρ > 1
1+b, the potential supply ratio to the two markets (ρ: (1 ρ)) is higher than the
perfect supply ratio (1 : b), and partial-scrapping will make the actual supply ratio deviate further
away from the perfect ratio, thus hurting the manufacturer more.
We next use the above property of (Q, Q
l) to derive the optimal solution to problem (7),
denoted by (QSp, QSp
l), via the following observations:
If (Q, Q
l) satisfies Ql[0,(1 ρ)Q], the optimal solution (QSp, QSp
l) can be derived around
(Q, Q
l) to satisfy the price difference constraint (phplδ). In this case, the Spstrategy
is optimal.
If (Q, Q
l) does not satisfy Ql[0,(1 ρ)Q], since the profit function ΠSp(Q, Ql) is jointly
concave in Qand Ql, we know that the optimal solution (QSp, QSp
l) should satisfy either
Ql= (1 ρ)Q(the selling strategy) or Ql= 0 (the scrapping strategy), whichever gives the
21
manufacturer the higher profit. For the former case, it is possible that the minimum price-
difference constraint (phplδ) is violated; when this happens, (Q, Ql) should be adjusted
such that phpl=δ.
Proposition 4 If cs
1+cs< ρ 1
1+band cs[1(1+b)ρ]
2+a(1ρ)2, the Spstrategy is optimal; otherwise,
either the ¯
S(selling) strategy or Sf(scrapping) strategy is optimal.
0
0.2
0.4
0.6
0.8
1
00.1 0.2 0. 3 0.4 0. 5
r(yield rate)
cs(scrapping cost)
Partial-Scrappi ng-Partial-Selling
(supp ly link)
Selling
a
ρ
[1(1+b)
ρ
]
b
ρ
2+a(1
ρ
)2
r2
Scrappi ng (both li nks)
Selling
c/(1+c)
Figure 9: Optimal strategy for any csand ρ
(a= 0.5, b = 0.1, δ = 0.25)
0
0.2
0.4
0.6
0.8
1
00.1 0.2 0. 3 0.4 0. 5
r(yield rate)
cs(scrapping cost)
Selling
Scrappi ng (both links)
Partial-Scrappi ng-Partial-Selli ng
(supp ly link)
a
ρ
[1(1+b)
ρ
]
b
ρ
2+a(1
ρ
)2
r2
Selling
a
ρ
[1(1+b)
ρ
]
b
ρ
2+a(1
ρ
)2
Figure 10: Optimal strategy for any csand
ρ(a= 0.5, b = 0.25, δ = 0.25)
Intuitively, as shown in Figures 9and 10, when the scrapping cost (cs) is lower, both the partial-
scrapping-partial-selling strategy and scrapping strategies become more attractive. They may be
even more attractive in the cases where cs<0. The partial-scrapping-partial-selling strategy should
be adopted for low and medium yield rates, while the scrapping strategy should be adopted for
high yield rates. Our numerical examples show that switching to the scrapping strategy will help
raise the manufacturer’s profit by up to 33%. Also, comparing Figures 9and 10, we find that as the
l-market size increases, l-market becomes more profitable, but the scrapping strategy surprisingly
becomes more attractive. The reason is, as explained at the end of Section 6.2, that the growing
l-market actually hurts the manufacturer with a high yield rate (which implies a low supply of
l-product) via a forced production increase. This in turn sacrifices the h-market profit, thus the
manufacturer would rather use the scrapping strategy at a cost to protect the h-market profit. Since
all these phenomena—low or even negative scrapping costs, high and stable yield rates, and the
growing l-market—are observed in the semiconductor industry (Kirsch 2005 and Franssila 2010),
our comparison analysis on the different yield-loss strategies provides an important guidance for
the industry: scrapping (full or partial) can indeed be the best choice.
22
7.2 Competition in the Low-End Market
We have so far focused on a monopoly manufacturer who has exclusive control over both h-market
and l-market. This manufacturer primarily serves h-market with h-products and sells the yield loss
(NQPPs) to l-market. In practice, however, there may exist some manufacturers who primarily
serves l-market. That is, they design and produce l-products only to serve l-market. For example,
in the electronic toy industry, although NQPPs of high-bit (32-bit) processors are welcomed, low-bit
(4-bit and 8-bit) micro-controllers are widely used. Manufacturers such as Tung Wing Electronics
Ltd and Haiwang Sensor Ltd, both located in Shenzhen, China, focus on producing the low-bit
micro-controllers, and they are major suppliers in the market (www.made-in-china.com). Therefore,
if a manufacturer of high-bit processors (the high-end manufacturer) is interested in selling some
or all of its NQPPs in the micro-controller market (the l-market), the supply from the low-end
manufacturers such as Tung Wing and Haiwang must be considered, and the competition in l-
market needs to be investigated.
In this section, we briefly discuss how the l-market competition affects the high-end manufac-
turer’s yield-loss strategy. A much more intensive and focused study is needed to fully investigate
the competition issues in a future project. For convenience, we denote our focal manufacturer, the
high-end manufacturer who treats h-market as the primary market but may sell the yield-loss to
l-market if profitable, as manufacturer Mf. We aggregateithe existing low-end manufacturers, who
treat l-market as the primary market, as manufacturer Me. Since higher quality products are more
costly to produce, the difference of the unit production cost between manufacturers Mfand Meis
c > 0. Recall that the production cost of Mf,c, is normalized to zero (i.e., c= 0) in the earlier
sections of this paper. Thus, the normalized production cost of Meis cc=c < 0; the actual
production cost, however, is positive.
Under quantity competition, manufacturers Mfand Meneed to simultaneously decide their
supply quantities to l-market. Specifically, Medecides its production quantity Qe, all of which will
be supplied to l-market. Mfmakes two decisions: how much to produce overall (Q) and how much
yield-loss to sell to l-market (Ql(1 ρ)Q). Hence, the market-clearing prices in h-market and
l-market (phand pl) are as follows:
ρQ =Dh= 1 phph= 1 ρQ,
Ql+Qe=Dl=b1pl
apl=a1Ql+Qe
b.(8)
iWe aggregate the supply of the low-end manufacturers in this supply competition game and thus can treat them
as a single entity.
23
Note that the market prices need to satisfy the conditions such as ph[0,1], pl[0, a], and
phplδas we defined in Section 3.
We describe the l-market competition (C) problem of manufacturers Mfand Meby objective or
profit functions (9) and (10) and the commonii constraints in (11). Note that Mf’s profit function
ΠC
f(Q, Ql|Qe) is jointly concave in Qand Qlfor any given Qe; and Me’s profit function ΠC
e(Qe|Ql)
is concave in Qefor any given Ql. This means that Mfand Mehave a unique best response
function, responding to each other’s decisions. In addition, ΠC
fdecreases in Qeand ΠC
edecreases
in Ql; more supply of one manufacturer to l-market always hurts the other manufacturer. However,
since their decisions jointly affect the price constraints, the manufacturers may or may not reach
an equilibrium (on supply quantities).
max ΠC
f(Q, Ql|Qe) = phρQ +aplQl(QρQ Ql)cs(9)
= (1 ρQ)ρQ +a1Ql+Qe
bQl(QρQ Ql)cs;
max ΠC
e(Qe|Ql) = plQe+ ∆cQe(10)
=a1Ql+Qe
bQe+ ∆cQe.
s.t. Ql[0,(1 ρ)Q],(l-market supply from Mf)
Qe[0,),(l-market supply from Me)
ph=ph= 1 ρQ [0,1],(h-market price)
pl=a1Ql+Qe
b[0, a],(l-market price)
phplδif Ql>0.(min price difference)
(11)
As mentioned above, the equilibrium supply quantities, if exist, are subjected to the constraints
in (11), which greatly complicate the search of the equilibrium. First, parameters a,b,ρ,cs, ∆c
and δcan take any values and result in many different feasible regions. Second, since the l-market
supply comes from both Qland Qe, yield rate ρcannot represent the potential supply ratio of the
two markets; thus, we cannot use ρalone to enumerate various situations of the game as we did in
the earlier sections. Finally, the equilibrium search requires intensive work of enumerating various
conditions of the game while evaluating the connections between the two objective functions. It
may also requires a number of assumptions on the parameters to guarantee certain properties of the
iiCommon constraints mean that these constraints apply to both Mfand Me.
24
objective functions under the boundary solutions (e.g,. pl=aand phpl=δ). Therefore, in the
rest of the section, we discuss how the constraints influence the decisions of the two manufacturers
in the game and then present some partial results with managerial insights.
Recall that we have discussed, in the previous sections, the insights of the optimal yield-loss
strategy using the two links for our focal manufacturer (Mf): the supply link and the price link. The
constraints of the competition game presented in equations (11) reflect these two links. With the
l-market supply from Me, however, the supply ratio of the two markets (ρQ :QlρQ : (1 ρ)Q=
ρ: (1 ρ)) solely determined by ρbecomes (ρQ :Ql+QeρQ : (1 ρ)Q+Qe) determined by ρ,
Qand Qe. That is, the original supply link, ρ: (1 ρ), becomes ρQ :Ql+QeρQ : (1 ρ)Q+Qe
which can now be controlled by both Mfand Methrough their production quantities. Similarly,
the price link, which requires Mfto maintain the minimum price difference between h-products
and l-products (δ) through its h-market supply (ρQ), is also affected by Me. Indeed if willing to
enter l-market with its NQPPS, Mfshould control its production quantity and l-market supply to
maintain price difference δ. If, however, δcannot be maintained, Mfwill not enter l-market and
scrap all the NQPPs. We thus have the following necessary conditions for Mfto enter the l-market.
Lemma 3 For Mfto enter and compete in l-market,
if ρa
a+2b,Mf’s production should not exceed (b+bc)/(1 ρ), and Mfmay choose the
selling-strategy for its NQPPs;
if ρ > a
a+2b,Mf’s production should not exceed 2+ac2δa
2ρ, and Mfmay choose the selling
or partial-selling-partial-scrapping strategy for its NQPPs.
Besides ρ,Mfshould also check scrapping cost csbefore entering the l-market. In particular,
csneeds to be lower than the profit earned from l-market under the unconstrained equilibrium
(ue), cue
s:= 3(3a+2ab+2bc)ρ2
3a+(4b3a)ρ2, to justify scrapping. As shown in Proposition 5below, if cscue
s,
scrapping is too expensive to be justified and thus Mfwill not scrap and sell all the NQPPs to
l-market, i.e., Ql= (1 ρ)Que.
After understanding the necessary conditions for Mfto enter l-market, we want to know what
the supply quantities of Mfand Memay look like under equilibrium when Mfdoes enter l-market
and compete with Me. We define the unconstrained equilibrium decisions as
Que := ρ(1 ρ)cs
2ρ21{ρ> cs
1+cs}, Que
l:= b
3+b
3a(2csc), Que
e:= b
3+b
3a(2∆ccs),
where unconstrained means that ignoring the constraints in (11). Note that this unconstrained
equilibrium can be the equilibrium under certain conditions.
25
Proposition 5 Given that Mfis able to maintain the minimum price difference between the two
markets, the equilibrium decisions of Mfand Meare as follows:
1. If ca+ 2cs,Mfchooses the scrapping-strategy and Memonopolizes l-market, i.e., Q=
Que,Ql= 0, and Qe=1
2(b+bc/a).
2. If (cs+a)/2<c < a + 2cs,Mfand Mecoexist and compete in l-market.
If cs< cue
s, then Q=Que,Ql=Que
l, and Qe=Que
e.
If cscue
s, then Q=Que,Ql= (1 ρ)Que, and Qe=1
2(b+bc/a (1 ρ)Que).
3. If c(cs+a)/2,Meleaves l-market, and Mfmonopolizes l-market and chooses yield-loss
strategies according to the results in early sections.
Intuitively, in case 1 above, i.e., when the production cost difference of Mfand Me(∆c) is
high relative to the scrapping cost (∆ca+ 2cs), the cost advantage and hence the competitive
power of Meis too high such that Mfshould not enter l-market (Mfis forced out of the l-
market). In this case, Mfshould choose the scrapping-strategy: set production quantity at Que
and scrap all the NQPPs; Medominates l-market and produces the optimal quantity for l-market,
Qe=1
2(b+bc/a). In case 3 above, i.e., when the production cost difference is low relative to
the scrapping cost (∆c(cs+a)/2), Medoes not have enough cost advantage to compete and is
forced out of the l-market. In case 2 above, i.e., when the production cost difference is medium
((cs+a)/2<c<a+ 2cs), Me’s competitive power is medium so that Mfand Mecoexist and
compete in l-market. This is the case we are most interested in and have observed in practice.
We find that the equilibrium of this case depends on the scrapping cost. If the scrapping cost is
lower than the profit earned from l-market (cs< cue
s), scrapping is justified and the unconstrained
equilibrium is indeed the equilibrium. Otherwise, scrapping is not justified and Mfshould sell all
of its NQPPs to avoid scrapping. Thus, although the production quantity of Mfis still the same
as that under the unconstrained equilibrium (Q=Que), the l-market supply quantities of Mfand
Meare adjusted accordingly.
Note that the unconstrained equilibrium (Que,Que
land Que
e) has explicit expressions determined
by the operational and marketing parameters (a,b,ρ,cs, and ∆c). Our results show that as long as
competition exists in the low-end market, the production quantity of Mfremains at Que regardless
of its decision as well as Me’s decision for l-market. That is, the manufacturer is able to maintain
astable production quantity, or a stable production capacity, when l-market competition exists.
26
From the point view of the whole market, we find that when cs= 0 and Q=1
2ρ, the total profit of
both Mfand Meis maximized, i.e., the whole market reaches its optimal performance.
Comparing the results of this extension with the results from our main model, we can further
understand the impact of l-market competition. First, the l-market competition renders the effect
of supply link obscure. On the one hand, with the extra l-market supply from Me, in the case that
there is too little h-market supply but too much l-market supply from Mf(cases (L) and (M-L)
discussed in Section 6), the supply ratio between the two markets will be further away from the
perfect supply ratio. Thus, the supply-link effect is strengthened, resulting in a higher chance for
Mfto earn only global-suboptimal profits from both markets and thus choose scrapping NQPPs
over selling. On the other hand, the extra supply from Melowers the price of l-products and hence
Mfneeds to exert less effort to maintain the price-difference between the two markets. That is,
the supply-link effect is weakened with the competition. Second, with l-market competition, it is
the relative magnitude of production cost difference ∆cand scrapping cost csthat determines the
production and marketing strategies of the two competing manufacturers. Moreover, the production
cost difference is the key determinator measuring Me’s competitive power. Third, the existence of
low-end manufacturer Meand the potential competition in l-market may counter-intuitively benefit
the high-end manufacturer, Mf. This is because the competition may weakened the supply-link
effect which gives Mfmore indirect control over the supply ratio of the two markets and hence a
better profit. Overall, the l-market competition complicates the yield-loss management decisions of
our focal manufacturer (Mf). Future research to fully investigate the competition effect may lead
to productive results.
8. Conclusions
When dealing with yield loss, many industries are embracing the new idea of selling the NQPPs to a
low-end market. Building upon a standard marketing model for vertically differentiated products,
we analyzed the impact of the strategy choice (scrapping, selling, or partial-scrapping-partial-
selling) on the manufacturer’s profit. We identified two drivers for the strategy selection: the
supply link (the fixed supply ratio) and the price link (the minimum price-difference) between the
high-end and low-end markets. We found that as the yield rate improves (which is common due to
technological advancements), the selling and partial-scrapping-partial-selling strategies suffer from
both links, thus leading to overproduction and global-suboptimal profit. Although the manufacturer
earns profits from both markets, the total profit is still lower than that under the scrapping strategy.
27
Intuitively, without the supply and price links, the manufacturer should supply the markets with
the ideal supply ratio 1 : band the resulting ideal price difference is 1
2a
2, which will extract the
most profit from both markets. In our problem, however, the two markets’ supplies come from the
same production run and the supply link (the supply ratio is fixed by the yield rate) exists therein.
Therefore, unless this supply ratio (ρ: (1 ρ)) happens to be the same as the perfect supply ratio
(1 : b), the manufacturer cannot extract the optimal profit from both markets. Note that the supply
ratio also affects the (market-clearing) price in each market, and the minimum price-difference is
needed to justify the product quality difference. Therefore, the price link also exists between the
two markets. Thus, unless the ideal price difference ( 1
2a
2) exceeds the required minimum (δ), the
manufacturer cannot extract the most profit from both markets. We showed numerically that the
profit increase could be up to 33% or even higher if the manufacturer switches to the scrapping
strategy. The benefit of such a switch is more salient when the scrapping cost is lower or the
low-end market size is bigger, both of which are indeed observed in practice. Therefore, despite the
additional scrapping cost incurred, it may be vital for a manufacturer to regain full control of the
supply quantity to the high-end market.
In addition to the above findings, we also extended our study to consider the possible compe-
tition in the low-end market between the existing low-end product supply and the NQPPs of our
focal manufacturer, which may also be sold to the low-end market. We characterized the equilib-
rium supply quantities for the competing manufacturers under some conditions. We found that
this competition may intensify the effect of the supply link, but weaken the effect of the price link
between the two markets. Further investigation in this direction may generate productive results.
Finally, this research still has its limits. We have assumed that the manufacturer determines its
yield-loss strategy based on a given yield rate and scrapping cost. In practice, manufacturers may
invest in technologies to improve yield rate or reduce scrapping cost. We believe potential future
research could explore the optimal technological investment on yield rate and scrapping cost.
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31
ONLINE SUPPLEMENT:
Proofs for “Scrap or Sell: The Decision on Production Yield Loss”
Proof of Proposition 1:Since ph= 1 ρQ, we have ΠSf(Q) = 1ρQ cs(1ρ)
ρ(ρQ), which
is a quadratic function of Q. If 1 >cs(1ρ)
ρ, solving the first order condition, we obtain the optimal
production quantity QSf=1cs(1ρ)
ρ
2ρ, which is guaranteed positive if 1 >cs(1ρ)
ρ. Substituting this
expression of QSfinto to the expression for phand ΠSf(Q), we obtain that pSf
h=1+ cs(1ρ)
ρ
2and
ΠSf=(1cs(1ρ)
ρ)2
4. It is not difficult to see that QSfis feasible as ph[0,1] is satisfied at QSf.
If, however, 1 cs(1ρ)
ρ, ΠSf(Q) is negative for any Q0 and thus QSf= 0 and ΠSf= 0. This
completes the proof.
Proof of Proposition 2:Note that constraints ph1 and plaalways hold as long as Q0
and the profit function, Π ¯
S(Q), is concave. If the unconstrained optimal solution, Qu, satisfies all
the constraints in NLP (5), it is the optimal solution for the NLP. Otherwise, the optimal solution
will be a boundary solution, either Q0or Qδ, which satisfies all the constraints.
We first verify that pu
l0 iff 8ab 1. We next show that the price difference, pu
hpu
l=
(1a)(a+b)ρ2(ab+3a2a2)ρ+(2aa2)
2a(1ρ)2+22, is unimodal in ρ(0,1). Indeed, we have:
∂ρ (pu
hpu
l) = 2a(a+b)(1 + b)ρ24a(a+b)ρ+ 2a2(1 b)
(2a(1 ρ)2+ 22)2,(12)
which is negative (and thus pu
hpu
lis decreasing in ρ) iff (a+b)(a+b)b(1+ab)
(a+b)(1+b)<ρ<(a+b)+(a+b)b(1+ab)
(a+b)(1+b),
where (a+b)+(a+b)b(1+ab)
(a+b)(1+b)1 because b1. This means that (pu
hpu
l) is unimodal—increases in
ρ(0,(a+b)(a+b)b(1+ab)
(a+b)(1+b)] and then decreases in ρ[(a+b)(a+b)b(1+ab)
(a+b)(1+b),1).
Note that it is assumed (pu
hpu
l)|ρ=0 = 1 a
2δand (pu
hpu
l)|ρ=a
a+bδ. These properties
of (pu
hpu
l) imply that ρ0= max{ρ(0,1)|pu
hpu
lδ} ≥ a
a+bis unique. Thus, the range of ρ
is partitioned at ρ=ρ0: in the lower range, (0, ρ0], pu
hpu
lδand pu
l0 are always met and
thus Quis feasible; in the upper range, (ρ0,1), pu
hpu
lδis not met and thus Quis infeasible.
Since when ρ > ρ0a
a+b, both phpland pldecrease in Q, in ρ(ρ0,1), Qcan be reduced to
reach feasibility and thus optimality. Indeed, due to the concavity of the profit function, Qshould
be lowered from Quto Q0or Qδ, whichever is smaller. Comparing the expressions of Q0and Qδ,
we find that Q0Qδiff ρ1δ
1δ+b. Therefore, we complete the proof for the characterization of
Q¯
Sshown in Proposition 2.
1
Proof of Corollary 2:It is not difficult to see that 1
4>ΠSf=(1cs(1ρ)
ρ)2
41{ρ> cs
1+cs}. We first prove
for maxρ(0,1) Πu= Πu|ρ=1
1+b=1+ab
4. Note that Π ¯
S(Q) = (1ρQ)(ρQ)+ a1(1ρ)Q
b(1ρ)Q
(1 x)x+a1y
by, where x, y > 0 are independent variables. It is not difficult to verify that
maxx,y>0(1 x)x+a1y
by=1+ab
4with x=1
2and y=b
2. Note that (x, y) coincides with
Quat ρ=1
1+b. This corollary result will then follow immediately from the result in Proposition 2
for the case with ρ(0, ρ0].
Proof of Proposition 3:We prove this proposition by comparing ΠSfto Π ¯
Sfor the various
ranges of ρgiven to describe Π ¯
Sin this proposition. Recall that ΠSf=(1cs(1ρ)
ρ)2
41{ρ> cs
1+cs},
Πu=b
4
[a(1ρ)+ρ]2
a(1ρ)2+2and Π0=
1ρ1
1ρ.
For ρ(0,cs
1+cs], as shown in Proposition 1, ΠSf= 0 <Π¯
S. Thus the best strategy ς=¯
Sin
this range of ρ.
For ρ(cs
1+cs, ρ0], Π ¯
S= Πu, comparing which to ΠSfreduces to checking the sign of function
G(ρ) = ac2
s(1ρ
ρ)32acs(1ρ
ρ)2+(a+bc2
sa2b)(1ρ
ρ)2b(a+cs). Specifically, a positive sign means
that ΠSf>Πuand a negative sign means otherwise. To study the behavior of G(ρ), we first note
that G(ρ=cs
1+cs) = a2b
csb(2a+cs)<0 and G(ρ= 1) = 2b(a+cs)<0. We then differentiate
G(ρ) and obtain:
G0(ρ)=(ρ2)3ac2
s(1ρ
ρ)24acs(1ρ
ρ)+(a+bc2
sa2b),
where the second term in the product is negative (and thus G(ρ) is increasing in ρ) if 1ρ
ρ
(1
cs
2q1+ 3b(a2c2
s)
a
3,1
cs
2+q1+ 3b(a2c2
s)
a
3) and is positive (and thus G(ρ) is decreasing in ρ) if 1ρ
ρ
(0,1
cs
2q1+ 3b(a2c2
s)
a
3). Since it is assumed csa, we have 1
cs
2+q1+ 3b(a2c2
s)
a
31
cs=1ρ
ρ|ρ=cs
1+cs.
Thus for ρ(cs
1+cs, ρ0), G(ρ) starts from a negative number, increases first and then may decrease
(definitely decreases if 1ρ
ρ|ρ=ρ0<1
cs
2q1+ 3b(a2c2
s)
a
3) as ρincreases. Assume G(ρ=ρ0)>0,
ρ1= inf{ρ(cs
1+cs, ρ0) : G(ρ)>0}exists and is unique. Moreover, ΠuΠSon ρ(cs
1+cs, ρ1] and
ΠS>Πuon ρ(ρ1, ρ0].
For ρ(ρ0,1), we need to compare ΠSfto Π ¯
S= Π0Πδ. We first compare ΠSfto Π0. It is
easy to verify that ΠSf>Π0on ρ1
1+b. Furthermore, since we assume ΠSfΠ0is negative and
then positive on ρ(ρ0,1), ρ2= inf{x(ρ0,1) : ρx, ΠSf>Π0}is unique and ρ1
1+b. In
summary, when ρ(ρ0, ρ2], ΠSfΠ0; when ρ(ρ2,1), ΠSf>Π0. This completes the proof.
2
Proof of Lemma 2:Note that
cs[1 (1 + b)ρ]
2+a(1 ρ)2H(ρ) = [(a+b)cs+a(1 + b)]ρ2(2acs+a)ρ+acs0Q
l(1 ρ)Q.
Moreover, ρ > cs
1+csand ρ1
1+bare used to guarantee Q>0 and (cs)[1(1+b)ρ]
2+a(1ρ)20, respec-
tively. This concludes the proof.
Proof of Proposition 4:Using Corollary 2and the joint concavity of ΠSp(Q, Ql), we modify
(Q, Q
l) to derive the optimal solution to the constrained problem (7), denoted by (QSp, QSp
l). We
prove for the following two cases.
If (Q, Q
l) satisfies Ql[0,(1 ρ)Q]: if (Q, Q
l) also satisfies phplδ, (QSp, QSp
l) =
(Q, Q
l) as it is easy to verify that Q[0,1] and Q
l[0, a]. Otherwise, due to the joint
concavity of ΠSp(Q, Ql), we know that optimal solution (QSp, QSp
l) is the optimal solution
to max ΠSp(Q=1a(1Ql
b)δ
ρ, Ql), noting that phpl=δat (Q=1a(1Ql
b)δ
ρ, Ql). In this
case, the Spstrategy is optimal.
If (Q, Q
l) does not satisfy Ql[0,(1 ρ)Q]: because of the joint concavity of ΠSp(Q, Ql),
we know that the optimal solution (QSp, QSp
l) should satisfy either Ql= (1 ρ)Q(the
selling strategy) or Ql= 0 (the scrapping strategy), whichever gives the manufacturer the
higher profit. For the former case, it is possible that the minimum price-difference constraint
(phplδ) is violated; when this happens, (Q, Ql) should be adjusted such that phpl=δ,
i.e., we have (QSp, QSp
l)=(b(1aδ)
(1+b)ρ1,b(1aδ)(1ρ)
(1+b)ρ1)>0, which is the joint solution to Ql=
(1 ρ)Qand phpl=δ.
This concludes the proof.
The proof of Lemma 3:The price difference condition, phplδ, can be rewritten as
a
b(Ql+Qe)ρQ (δ+a1).
For Meto respond to any Qlin the l-market, we set
∂QeΠC
e(Qe) = 0, Qe=1
2(b+bcQl) if
Qlb+bc. Consequently, the price difference condition can be re-written as
a
bQl2ρQ 2δ+a2ac
if Ql(1 ρ)Qand Qlb+bc.
3
Note that from Assumption 1, we know 2δ+a2ac < 0. Since Ql(1 ρ)Q,
a
bQ(a
b+2)ρQ > a
bQl2ρQ 2δ+a2ac. This condition always hold if a
b(a
b+2)ρ > 0 and
Q(b+bc)/(1 ρ). That is, if ρa
a+2b, as long as Q(b+bc)/(1 ρ), the price difference
condition always holds. However, if ρ > a
a+2b, for Ql>0, the price difference condition becomes
Q2+ac2δa
2ρ. Please note that above conditions are loose conditions.
The proof of Proposition 5:To solve the unconstrained competition problem, we simulta-
neously solve
∂Q ΠC
o(Q, Ql|Qe) = 0,
∂QlΠC
o(Q, Ql|Qe) = 0, and
∂QeΠC
e(Qe|Ql) = 0. That is, the
unconstrained equilibrium satisfies:
ρ2ρ2QCs+Csρ= 0,
a(2Ql+Qe) = ab +bcs,
a(Ql+ 2Qe) = ab +bc.
Jointly solving these equations, we obtain the unconstrained equilibrium:
Que =ρ(1 ρ)cs
2ρ2, Que
e=b
3+b
3a(2∆ccs), Que
l=b
3+b
3a(2csc).
The constraints in (11) can also be rewritten as 0 Ql(1 ρ)Q,Qe0, 0 Q1,
Qe+Qlb, and ρQ+a(1Ql+Qe
b)1δ. We note that Que
l>0 requires ∆c<a+ 2cs. Otherwise,
the production cost advantage of Meis too big for Mfto enter the l-market, and should choose
the scrapping strategy. On the other hand, Que
l<(1 ρ)Qrequires csρ[2b2abρ+3a(1ρ)]
3a(1ρ)2+4ρ2. In
other words, if csis too large, then the Mfshould choose selling strategy for all NQPPs. Qe>0
requires ∆c > (csa)/2. Otherwise, the cost advantage of Meis too low, the NQPPs from Mf
can force Meout of market. The rest of the proof is straightforward.
4
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