sustainability in the fast fashion industry...long and nasiry: sustainability in fast fashion 2...

Sustainability in the Fast Fashion Industry

Xiaoyang LongWisconsin School of Business, University of Wisconsin-Madison, Madison, WI 53706, [email protected]

Javad NasiryDesautels Faculty of Management, McGill University, Montreal, Quebec H3A 1G5, [email protected]

Problem Definition: A fast fashion system allows firms to react quickly to changing consumer demand by

replenishing inventory (via quick response) and introducing more fashion styles. In this paper, we study the

environmental impact of the fast fashion business model by analyzing its implications for product quality,

variety, and inventory decisions. Relevance: Our work establishes a much-needed understanding of the link

between the fast fashion business model and its environmental consequences. Methodology: We consider

a two-period model in which a firm sells to fashion-sensitive consumers whose preferences are influenced

by a random fashion trend. We analyze the effect of fast fashion capabilities (quick response and design

flexibility) on the firm’s quality decision and leftover inventory. Results: We find that a key driver of low

product quality in the fast fashion industry is the firm’s incentive to offer variety to hedge against uncer-

tain fashion trends. When variety is endogenous, quality decreases as consumers become more sensitive to

fashion or as the cost of introducing new styles decreases. We also identify conditions under which expected

leftover inventory increases as the firm’s fast fashion capabilities increase. Managerial Implications: We

assess the effectiveness of three environmental initiatives (waste disposal regulations, consumer education,

and post-consumer recycling programs) in countering the environmental impact of fast fashion. We show

that waste disposal policies are effective in reducing the firm’s leftover inventory—but may have the unin-

tended consequence of lowering product quality. We also compare firm-owned versus third-party recycling

programs and propose a revenue-sharing scheme which could induce higher levels of recycling rate and

product quality—while increasing the system’s total profit.

Key words : fast fashion, quick response, sustainability, quality, product variety

1. Introduction

The fast fashion business model relies on quick reaction to consumer demand changes. The model

deploys a ‘quick response’ production system to replenish the inventory of hit styles and introduce

new ones. This business model has become increasingly popular in recent years following the success

of companies such as Zara and H&M, and aided by the rise of e-commerce platforms. For example,

Boohoo—a UK-based online retailer—operates by ordering a broad range of products in small

quantities from local factories; the retailer then “requests more of the products that sell and stops

1

Long and Nasiry: Sustainability in Fast Fashion2

buying those that don’t” (Business of Fashion 2018a)1. Quick response boosts a firm’s profitability

by effectively matching supply with demand (e.g., Cachon and Swinney 2011, Caro and Martınez-

de-Albeniz 2010). Little is known, however, about the fast fashion business model’s environmental

consequences, which is the focus of our study.

We study the environmental implications of the fast fashion business model by analyzing its

implications for product quality, variety, and inventory decisions. Despite their economic success,

fast fashion companies are criticized for having a negative environmental impact (Saicheua et al.

2012), particularly due to poor product quality and waste generation. Currently, nearly three-fifths

of all clothing ends up in incinerators or landfills within a year of being produced (McKinsey 2016).

Recycling options for fast fashion items are limited, as many secondhand stores will reject items

from fast-fashion chains, citing low quality and poor resale value (Newsweek 2016). Surveys show

that consumers typically classify clothes from fast fashion retailers as “disposable” (Fisher et al.

2008). From 2000 to 2014, the US textiles and clothing waste increased 70%, to 16 million tonnes

(EPA 2014). Companies such as H&M have also come under fire for incinerating leftover stock,

highlighting the over-production problem in the industry (Environmental Audit Committee 2019).

In this paper, we seek to answer three research questions. First, what are the firm’s optimal

product variety, quality and inventory decisions in the fast fashion business model? Second, how do

key fast fashion elements (such as quick response efficiency and design flexibility) affect the firm’s

product quality decision and leftover inventory? Third, what are the effects of common sustain-

ability policies such as waste disposal regulations, consumer education, and recycling programs on

the fast fashion system?

Our work establishes a much-needed understanding of the link between the fast fashion business

model and its environmental consequences. Cachon and Swinney (2011) identify two key compo-

nents of a fast fashion system: quick response (i.e., short production/distribution lead times), and

highly fashionable product design. While existing studies typically model “fashionable” design as

an exogenous increase in consumer willingness-to-pay, we consider the firm’s variety decision—a

firm may introduce a larger variety to increase its likelihood of being on trend, thus endogenizing

the value of “fashion” that consumers perceive from the firm’s products. Our results show that it

is the variety decision (rather than quick response) which is the key driver behind the low product

quality observed in the fast fashion industry.

Specifically, we consider a two-period model in which a firm sells to fashion-sensitive consumers.

In the first period, the firm decides the quality of its products, the number of styles to introduce to

the market, as well as how many units of each style to produce. In the second period, the fashion

1 For more examples of online retailers that have adopted the fast fashion business model, see Business of Fashion(2018b) and Women’s Wear Daily (2018).


trend is realized and the consumers choose the product which is closest to the trend. Based on the

realized demand, the firm can then choose to replenish its inventory using quick response.

In our model quality is a vertical attribute of the product which we interpret as apparel durability.

Low quality would increase the disposal frequency and worsen the environmental impact of fast

fashion as low quality garments are less likely to have a “second life” and are more likely to end

up in landfills (Environmental Audit Committee 2019). Our basic model does not account for

repeated consumer purchase and disposal decisions (or how consumers manage their wardrobe)

and implicitly assumes low quality garments have a larger environmental footprint. However, we

discuss a dynamic model in which the firm’s current quality decisions affect demand in future

selling seasons in Section 3.2.

Firms may have different incentives to introduce product variety. In traditional monopolistic

models, a representative consumer has an inherent preference for variety or there is taste het-

erogeneity among consumers (Gaur and Honhon 2006, Lancaster 1990). Product variety in these

models helps increase the purchase likelihood or the market size as each variety tailors to the needs

of a consumer segment. In our basic model, consumers are homogeneous and do not have a prefer-

ence for variety. They do however have a willingness-to-pay for fashion. The firm introduces variety

to have a better chance of matching the uncertain fashion trend. This implies that the demand is

high for “hits” (a subset of styles) while other styles do not excite demand and are discontinued,

salvaged, or discarded. This left-over inventory is the source of environmental impact we capture

explicitly in our work.

We analyze and compare two cases in our model: the benchmark case where the firm does not

have the design flexibility to introduce multiple styles (e.g., a traditional firm which introduces one

collection per season), and the fast fashion case where the firm can choose the number of styles to

introduce.

First consider the benchmark where design is not flexible. In this case, we find that when the

cost of production via quick response decreases, the firm chooses higher product quality and lower

inventory levels. Intuitively, a higher product quality increases consumers’ willingness-to-pay, but

also incurs a larger unit production cost. When the cost of quick response is lower, the firm produces

less inventory in the first period and relies more on quick response to satisfy demand, which reduces

the risk of leftover inventory. Since the firm is now more likely to sell the products that it produces,

it is willing to invest more in product quality. Our result thus shows that quick response—by

itself—does not result in the lower product quality that we observe in fast fashion.

When the variety decision is endogenous, however, we find that the firm chooses a lower product

quality as consumers become more sensitive to fashion or as the firm’s cost of introducing new

styles decreases. In these situations, the firm prefers to introduce a larger variety of products


in order to better match the fashion trend and increase the consumers’ willingness-to-pay. Since

fashion trends favor only a small subset of products, the firm’s risk of leftover inventory increases

as variety increases. To offset the larger production costs due to more leftover inventory and/or

more reliance on quick response, the firm reduces product quality. Surprisingly, our results show

that a lower cost of quick response could also lead to lower product quality. As discussed above,

if the number of styles is fixed, then lower cost of quick response leads to higher product quality.

However, we find that if variety is a decision, then as quick response becomes cheaper, the firm

finds it optimal to introduce more styles, which leads to an overall decrease in product quality. Put

together, our results imply that as the firm’s fast fashion capabilities increase (i.e., as the costs of

quick response and of designing new styles decrease), the product quality almost always declines.

We also show that the overall production (and leftover inventory) may increase or decrease under

the fast fashion model, depending on the firm’s costs of quick response and design.

To counter the environmental impact of the fast fashion business model due to low quality and

overproduction, we look into the effectiveness of three sustainability initiatives—waste disposal

regulations, consumer education, and post-purchase recycling programs. First consider government-

imposed waste disposal policies (e.g., banning firms from sending leftover stock to landfills) which

would increase the firm’s cost of disposing leftover inventory. We find that these policies—while

effective in reducing the firm’s leftover inventory (and total production)—may have the unintended

consequence of a lower product quality. Specifically, as the firm tries to avoid the costs associated

with leftover inventory, it produces less inventory in the first period and relies more on quick

response to satisfy demand. Since quick response is more costly than production in the first period,

the firm now faces a higher unit production cost, which it offsets by reducing product quality. In

a way, the waste disposal problem is now shifted from the firm to the consumers—who generate

more waste due to the declining quality of the clothes they purchase. Therefore, our results imply

that, unless effective measures are devised to control waste on the consumer side, waste disposal

policies may not achieve their intended purpose.

Second, we analyze the effect of consumer education (e.g., raising awareness about the environ-

mental impact of low-quality clothing, or emphasizing the value of quality clothes on the second-

hand market), which may increase consumers’ sensitivity to quality. We establish that increasing

consumers’ sensitivity to quality leads to higher product quality and lower variety. However, in

shifting from a low-quality/high-variety business model to a high-quality/low-variety one, the firm’s

expected leftover inventory (and total production) first increases before it decreases. This is because

as product variety decreases, the product’s likelihood of hitting a trend decreases, and it becomes

less profitable for the firm to use quick response. As a result, the firm produces more inventory

of each style in the first period, which may lead to larger total production and expected leftover


inventory (even though there are fewer styles to produce). As variety continues to decrease, how-

ever, the total production decreases as the firm finds it optimal to produce a very limited amount

of high-quality products. Our results thus suggest that, as fast fashion firms shift towards a more

sustainable business model with higher-quality products, it is important for governments not to

overly penalize inventory disposal—which is a necessary part of the transformation process.

Third, we examine recycling (“take-back”) programs for post-consumer products, and compare

the effectiveness of firm-owned versus third-party recycling programs. We find that if the firm’s

cost disadvantage of recycling is not too large, then the firm-owned recycling program is preferred

because it leads to higher product quality and a higher recycling rate. If the firm’s cost disad-

vantage is large, however, then there is a trade-off between quality and efficiency—a firm-owned

recycling program leads to higher quality but a lower recycling rate. To address this inefficiency in

recycling, we propose a revenue-sharing scheme which—combined with the appropriate government

fees/subsidies—can effectively induce higher levels of recycling rate and product quality, while also

increasing the system’s total profit. Overall, our results highlight the importance of considering

multiple aspects of the firm’s business model when implementing sustainability policies in the fast

fashion industry.

Our work contributes to the sustainable operations literature by developing a deeper understand-

ing of the environmental implications of the fast fashion business model. There is a growing number

of papers on sustainable product design, waste management, and the effectiveness of various envi-

ronmental regulations (see, e.g., Atasu and Subramanian 2012, Esenduran and Kemahlıoglu-Ziya

2015, Gui et al. 2018, Huang et al. 2019, Plambeck and Wang 2009). However, existing studies

typically assume deterministic demand since in the industries they focus on (e.g., the electronic or

automobile industries), uncertain demand—and the strategies the firm employs to match supply

with demand such as quick response and design flexibility—are not as critical as in the fashion

industry. Our study highlights the complex relationship between a firm’s production/design capa-

bilities and its product quality decision and leftover inventory. We show that policies which affect

the firm’s cost of leftover inventory disposal may have unintended consequences on product quality.

We also take a step in investigating the design of “take-back” policies for the fashion industry.

Such policies—while well-established for electronic waste—are at a nascent stage for textile waste

(Bukhari et al. 2018, Environmental Audit Committee 2019). In particular, our study yields new

insights in terms of how firm-owned versus third-party recycling programs affect product quality

and the recycling rate.

The rest of the paper proceeds as follows. After reviewing the literature in Section 2, in Section

3 we describe the model set-up. Section 4 presents the analyses for both the benchmark case

(with limited variety) and the fast fashion case (with endogenous variety decision). In Section


5 we discuss the policy implications. We conclude in Section 6 by summarizing our results and

suggesting avenues for future research.

2. Literature Review

Our work is related to the literature in operations management which models the key elements of

fast fashion systems. There is extensive research on the effect of quick response in supply chain

systems with uncertain demand. For example, Fisher and Raman (1996) study a two-period model

and show that quick response can increase a firm’s profit by as much as 60%. Cachon and Swin-

ney (2009) find that the value of quick response is generally greater in the presence of strategic

consumers than without them. Caro and Martınez-de-Albeniz (2010) analyze the impact of quick

response when firms compete with each other on inventory levels. More recently, Cachon and

Swinney (2011) develop a model which combines quick response with fashionable product design—

assuming that the latter increases consumers’ utility by a fixed amount. The authors identify a

complementary effect between quick response and fashionable product design. That is, the incre-

mental profit gain from using both strategies together is higher than the sum of profit gains from

using each strategy in isolation. Caro and Gallien (2007) also consider a fast fashion system, but

model it as a dynamic assortment problem, where the firm learns about consumer preferences by

offering a fixed number of products each period. Similarly, Ulu et al. (2012) assume that consumer

tastes are distributed on a Hotelling line, and consider how the firm should dynamically update the

locations of its products over time. Our work contributes to this literature by studying the firm’s

product quality decision in a fast fashion system. Furthermore, while existing studies typically

assume that the firm introduces a fixed number of products to the market, we explicitly model the

firm’s variety decision.

Traditional models of product variety typically assume that a representative consumer has an

inherent preference for variety or that tastes are heterogeneous across consumers (for a review,

see Lancaster 1990). For example, McAlister (1982) suggests that consumers may exhibit variety-

seeking behavior if they get bored or satiated with repeat purchases of the same product. Caro and

Martınez-de-Albeniz (2012) model the dynamic purchasing decisions of variety-seeking consumers

and identify the optimal satiation levels that firms should choose for their products. Caro et al.

(2014) extend this model to study how a firm should introduce its products over time. Researchers

have also studied variety decisions when consumer tastes are heterogeneous—mostly using loca-

tional choice models (Hotelling 1929, Lancaster 1966, Salop 1979). For instance, Gaur and Honhon

(2006) consider a continuous and possibly nonuniform distribution of consumer tastes (represented

by locations) on a Hotelling line, and analyze the firm’s optimal inventory, product location, and

variety decisions. Using a similar consumer choice model, Alptekinoglu and Corbett (2008) study


competition between a mass customizer that can offer any variety on a Hotelling line and a mass

producer that can offer only a finite set of products in the same space. Alptekinoglu and Corbett

(2010) further extend this model to study a monopoly firm’s trade-off between increased variety

and longer leadtime. In contrast to the papers discussed above, our model does not assume that

consumers are variety-seeking or have heterogeneous tastes. Instead, we assume that consumer

tastes are influenced by an uncertain fashion trend and the firm may increase variety to better

match this trend. This model allows us to gain new insights into the link between a firm’s leftover

inventory and its variety and quality decisions.

In modelling the firm’s quality decision, our work draws from the economics and marketing

literature on vertical product differentiation. Most existing research study the firm’s quality decision

with deterministic demand (Jeuland and Shugan 1983, Moorthy 1984, Moorthy 1988, Motta 1993).

A notable exception is Jerath et al. (2017), who consider the firm’s joint quality and inventory

decisions in a newsvendor setting without quick response. They find that as demand uncertainty

increases, quality decreases. This result is similar to our finding that, when variety is fixed, more

efficient quick response leads to higher quality. However, we also show that the relationship between

quick response and quality is no longer monotonic when variety is endogenous.

Finally, our research contributes to the growing literature on sustainability issues in OM (for

detailed reviews, see Agrawal et al. 2018, Girotra and Netessine 2013, Lee and Tang 2017). In

particular, a number of papers have studied the environmental implications of product design and

introduction decisions. For example, Plambeck and Wang (2009) build a model of the electronics

industry, and study a firm’s dynamic new product introduction decisions over an infinite horizon.

They find that e-waste regulations which reduce the frequency of new production introduction may

fail to motivate manufacturers to design for recyclability. Agrawal and Ulku (2012) consider the

effect of modular upgradability on a firm’s product development and introduction decisions. They

find that modular upgradability can increase the environmental impact by leading to more fre-

quent introduction and replacement. Atasu and Souza (2013) study the effect of different product

recovery schemes on the firm’s quality decision and the system’s environmental impact. Bhaskaran

and Gilbert (2015) find that channel structure and mode of operations (leasing vs. selling) can

significantly affect the manufacturer’s product durability choice. Huang et al. (2019) consider the

effect of take-back legislation on a firm’s product design decision, in which the firm can make

products more recyclable or more durable. Other papers that study sustainable design incentives

and recycling outcomes under government regulations include Alev et al. (2018), Atasu and Sub-

ramanian (2012), Cohen et al. (2018), Esenduran and Kemahlıoglu-Ziya (2015), Gui et al. (2018),

Subramanian et al. (2009).


Our work complements these papers by studying a firm’s quality, variety and inventory decisions

in a fast fashion system. In the fashion industry, uncertain demand—and the strategies that the

firm employs to match supply with demand such as quick response and flexible design—are essen-

tial drivers of the firm’s decisions. To the extent of our knowledge, we are the first to model these

elements and analyze the link between the firm’s production/design capabilities and its environ-

mental consequences, as reflected by product quality and leftover inventory. Furthermore, while

existing papers mostly focus on regulations for post-consumer products, we also consider policies

which affect the firm’s cost of leftover inventory disposal, and show that such policies can have

unintended consequences on product quality. Finally, our study takes a step in investigating the

optimal design of “take-back” policies for the fashion industry. We show how firm-owned versus

third-party recycling programs affect the recycling rate and product quality. We also propose a

“win-win” regulation scheme which leads to high quality and collection rates while increasing profit

for both the fast fashion firm and the third-party recycler.

3. Model

In this section, we first describe the model, and then present the firm’s profit maximization problem.

3.1. Model Set-up

We consider a firm that sells an assortment of fashion products over a season. Before the selling

season starts, the firm decides the product quality, the number (and design characteristics) of

products to introduce, as well as the inventory level of each product. The consumer demand is

a priori uncertain and may be affected by a random fashion trend, which is realized during the

selling season. After the demand uncertainties are resolved, the firm chooses the price for each

product and—if demand exceeds supply—can produce additional units at extra cost (i.e., use quick

response).

We model product quality as a vertical attribute which contributes to the lifespan of the product

(e.g., fabric sturdiness, stitching, quality of the dying process).2 Consumers’ willingness-to-pay

increases in product quality, but higher quality requires a greater unit production cost (Moorthy

1988, Villas-Boas 1998).

We also assume that products can be horizontally differentiated in their designs. Specifically,

following the “circular city” model (Salop 1979), we think of each product’s design (or “style”) as a

location on a circle of unit circumference. The circle represents the space of potential designs. Such

2 In the durable goods literature, researchers have sometimes modeled product quality and durability separately(Fishman et al. 1993, Waldman 1996). For example, Fishman et al. (1993) propose a model where quality representsthe amount of technology in a product and durability captures how long the physical product lasts. In the apparelindustry, however, quality typically reflects durability (see Laitala et al. 2015 for more detailed discussions of whatdetermines apparel quality).


location models have often been used to model firms’ variety decisions (Alptekinoglu and Corbett

2008, 2010, Dewan et al. 2000, Gaur and Honhon 2006). In making the product variety decision,

the firm chooses the number of products (m≥ 1) as well as their locations ti, i ∈ {1, · · · ,m}. Let

the firm’s cost of designing m products be βm. Here the parameter β captures the firm’s design

capability/flexibility. For example, if the firm has invested in an in-house team of designers and

“trend-spotters”, then it can quickly produce new styles based on budding trends and introduce

many styles in a season at a low variable cost. In contrast, if the firm follows a more traditional

design process, then it may only be able to introduce one style or collection per season.

Each consumer’s valuation for product i is Vi = a + θu + γ( 12− |ti − x|). Here a is the base

value of the product, u is the product quality, and x is the consumer’s preferred location on the

circular space. The parameters θ and γ capture the consumer’s sensitivities to quality and fashion,

respectively. The term |ti−x| denotes the shortest arc distance between the product’s location and

the consumer’s preferred location on the circle. Therefore, the farther away the product’s style is

from the consumer’s preferred fashion style, the lower the consumer’s valuation for the product.3

In our basic model, consumers are homogeneous and do not have an intrinsic preference for

variety. However, they do have a preference for fashion which is uncertain before the selling season

begins. This motivates the firm to introduce variety to have a better chance of matching the fashion

trend. We assume the consumers’ preferred fashion style x is drawn from a uniform distribution on

the circular space. Once x is realized, the firm chooses the product prices and consumers choose

to purchase the product which yields the highest (nonnegative) surplus. Since consumer valuation

is highest for the product with location closest to x (denote this location as t∗), the firm prefers to

sell this product and does so by charging pi = Vi for ti = t∗ and pi ≥ Vi for all other products (i.e.,

ti 6= t∗). Consumer demand for product i= 1, · · · ,m is

Di =

{0, if ti 6= t∗,

n, if ti = t∗,

where n∼U(0, n) is the number of consumers in the market. The random market size n captures the

non-fashion related factors which affect demand. For example, weather conditions would influence

the total demand for winter jackets, regardless of the fashion trend. The parameter n represents the

upper bound on the market size. The “lumpiness” of demand is a direct consequence of consumer

homogeneity and the role of variety in our model. Once the fashion trend is realized, the demand

is concentrated on the product with the best match. We consider cases where demand could be

3 We can also model consumer valuation as Vi = a+ θu− γ|ti − x| and derive similar insights. However, the currentformulation captures the fact that consumers who are more sensitive to fashion are willing to pay more for fashionableclothes.


positive for all products, due to heterogeneous consumer preferences in Appendix B or dynamic

substitution effects in Appendix D.

Before the selling season starts, the unit cost of production is k2u2. During the selling season,

however, the unit cost is k′

2u2, with k′ >k reflecting the higher cost of quick response. Throughout

the paper, we assume the consumer’s base value (i.e., the parameter a) is sufficiently high (or

k′ sufficiently low) so that it is always optimal for the firm to use quick response when demand

exceeds supply.

The sequence of events is as follows. There are two periods. In the first period, the firm decides

the quality of its products u, the number of products/styles to introduce m (and their locations

ti), and the quantity to produce for each style q. In the second period, the trend (x) and market

size (n) are realized. The firm decides the price of each product and can produce additional units

of the trendy style at extra cost (i.e., using quick response). Then the consumers make purchasing

decisions.

3.2. Discussion of the Model

In this section we discuss the model and how it relates to the fashion industry as well as to the

literature.

In this paper, we use the quality decision u as a proxy for the long-term environmental impact

of the product. Specifically, we assume that lower quality clothes have shorter life-spans and thus

lead to more post-consumption waste. However, by considering only one selling season, we do not

explicitly model consumers’ disposal behavior over time. Our approach differs from the durable

goods literature, where the consumer’s purchase decision is typically modeled as a replacement

decision between new and previously purchased products (see, e.g., Fishman and Rob 2000, Lobel

et al. 2015, Plambeck and Wang 2009). For example, Plambeck and Wang (2009) model an infinite-

horizon setting and assume that when new products are introduced by a firm, consumers purchase

the new products and throw away the old ones they own; the authors then estimate the annual

quantity of electronic waste based on the average rate of new product introduction. We do not

incorporate such dynamic behavior in our model for the following reasons.

First, for apparel products, consumers’ disposal decisions are not necessarily driven by new

purchases. That is, purchasing a new sweater need not imply that the consumer will stop using

the old one—unlike the case with typical durable goods such as cellphones or automobiles. In

fact, a number of studies show that the major reasons for consumers to stop using clothing are

quality-related (e.g., wear and tear, pilling, color fading) or fit-related, rather than fashion changes

or a desire to purchase new items (Collett et al. 2013, Laitala 2014, Laitala et al. 2015). For

example, Laitala et al. (2015) report that the “material properties of the clothes dominate when


the informants describe their reasons to stop using clothing” and that only “4% of garments are

disposed of because they are out of fashion or otherwise outdated.” Therefore, we do not assume

that new purchases prompt disposal behavior.

Second, even when consumers stop using a piece of clothing, they need not throw the clothes

away and may instead prefer to donate them to charitable organizations or give them to friends

and family (Collett et al. 2013). From this perspective, lower quality garments lead to more waste

(i.e., more products going to landfill) because “the opportunity for [these clothes] to have a second

hand opportunity is quite limited.” (Environmental Audit Committee 2019, page 7).

Third, our model can be extended to account for the dynamic that consumers who purchased

high quality products in previous seasons may be less likely to purchase again. Specifically, we may

consider a finite horizon model of selling seasons (with each season consisting of the two periods in

our current model) and assume that the market size n is stochastically decreasing in the quality

of the product sold in the previous season. However, such a dynamic model—while significantly

more complicated than the current model—does not yield additional insights other than to provide

an explicit relationship between product quality and future demand/quantity sold.4 Therefore, for

expositional reasons, we focus on the one-season model in this paper which allows us to derive

explicit solutions and elucidate the effects of fast fashion capabilities on a firm’s inventory, variety

and quality decisions, which are new insights to the literature.

In our model, variety can be interpreted as the number of styles the firm introduces throughout

the selling season. For example, this decision could be related to how many runway styles the fast

fashion firm adopts, or how frequently the firm introduces new styles that mimic emerging trends

on social media. Fast fashion is usually associated with both larger and more frequently updated

assortments (Financial Times 2017, Caro and Martınez-de-Albeniz 2015). Previous papers have

studied the optimal schedule for introducing product assortments dynamically over a selling sea-

son (Caro et al. 2014, Cınar and Martınez-de-Albeniz 2013). While these papers yield interesting

insights, the models are typically NP-complete even without considering inventory decisions. There-

fore, to shed light on inventory and quality issues in the fast fashion business model, we take an

aggregate approach to model variety; that is, we do not distinguish between products introduced

at different times in the selling season. Our model—albeit stylized—captures the key elements

of demand in fashion: that consumer demand concentrates disproportionately on a subset of the

styles, and that the firm cannot anticipate which styles would draw the most demand in advance.

In Appendix B, we show that if consumers do not chase fashion and demand is evenly distributed

among styles, then our key results do not arise.

4 Details are available upon request.


Finally, we discuss two assumptions in our model. First, we assume “static substitution” (Gaur

and Honhon 2006, van Ryzin and Mahajan 1999). That is, consumers make their purchase decisions

without considering the inventory level, and do not purchase another product upon encountering a

stock-out. In Appendix D, we consider the case of dynamic substitution, in which consumers may

purchase other products if their preferred product is sold out. Second, by allowing the firm to make

pricing decisions after demand uncertainties are realized, we posit that pricing is a more flexible

decision than product design or inventory. In practice, firms often adjust product prices as they

observe demand. This practice is referred to as “responsive pricing”, or “pricing postponement”

(Aviv et al. 2019, Chod and Rudi 2005, Jerath et al. 2017, Van Mieghem and Dada 1999). This

assumption is not critical to our results. In Appendix A, we analyze the case where the pricing

decision is made before demand realization and show that the key insights are robust.

3.3. The Firm’s Problem

In the second period, given the location of fashion x and the number of consumers n, the firm’s

profit is:

Π = pn−mk

2u2q− k

′

2u2(n− q)+−βm.

Here mq is the initial production in period 1, and (n− q)+ is the amount of production in period

2 using quick response. The second and third terms of Π represent the total production costs for

the firm, while the last term represents the total design cost. The price is p= a+θu+ γ2−γ|t∗−x|,

which is the consumers’ maximum willingness-to-pay for the trendy product.

In period 1, the firm’s expected profit is

EΠ =E[(a+ θu+γ

2− γ|t∗−x|)]E[n]−mk

2u2q− k

′

2u2E[(n− q)+]−βm.

We first show that the firm’s design decision simplifies to choosing the number of products to

introduce into the market.

Lemma 1. Given any number of products m, it is optimal for the firm to locate the products at

equal distances to each other.

By Lemma 1, given that the firm introduces m products, the distance between any two neighbor-

ing styles is 1m

, and we have max |t∗−x|= 12m

. That is, the trendy location x is at most a distance of

12m

away from the nearest product location. We can now calculate E[p] =E[a+θu+ γ2−γ|t∗−x|] =

2m∫ 1

2m

0(a+θu+ γ

2−γz)dz = a+θu+ γ

2− γ

4m. Note that the expected price increases as the number

of styles m increases. This is because the more styles the firm introduces, the “closer” its products

are likely to be to the fashion trend, and the more consumers are willing to pay. After substituting

E(n) = n2

and E[(n− q)+] = (n−q)22n

, the firm’s maximization problem in period 1 becomes

maxm,u,q

EΠ = maxm,u,q

[(a+ θu+γ

2− γ

4m)n

2−mk

2u2q− k

′

2u2 (n− q)2

2n−βm]. (1)


4. Analysis and Results

In this section, we first establish the optimal inventory and quality decisions in the benchmark case

where the firm’s design capabilities are limited (i.e., m= 1). We then investigate the model with

endogenous product variety decision and compare the results to those in the benchmark model.

4.1. Benchmark with Quick Response and Limited Variety

We first consider the benchmark case of m= 1. This represents the traditional model in which the

firm does not have the design flexibility to introduce multiple styles in a short time horizon. The

firm chooses inventory q and quality u to maximize its expected profit in period 1:

maxu,q

EΠ = (a+ θu+γ

4)n

2− k

2u2q− k

′

2u2 (n− q)2

2n−β. (2)

Solving the firm’s maximization problem in (2) yields the optimal quality and inventory decisions:

q1 = n(1− kk′ ) and u1 = θk′

k(2k′−k). The following proposition summarizes the effect of the quick

response cost on the firm’s decisions.

Proposition 1. Suppose m= 1. As quick response becomes more efficient (i.e., k′ decreases),

(i) the inventory q1 decreases and the quality level u1 increases; (ii) the expected leftover inventory

E[(q1−n)+] decreases.

As quick response becomes less expensive, the firm produces less inventory in the first period and

relies more on quick response to satisfy demand in period 2. This implies that the risk of having

leftover inventory is smaller. Thus, the firm is more willing to invest in a higher quality product

(because the likelihood of selling the product and recouping the production cost is higher). There-

fore, we establish that when product variety is limited, as the firm becomes more efficient at quick

response, product quality increases and the expected leftover inventory decreases. Our findings are

similar to those in Jerath et al. (2017), who model the firm’s quality decision in a newsvendor

setting without quick response. They find that quality is higher when demand uncertainty is lower.

In our setting, quick response moderates the effect of demand uncertainty, and therefore more

efficient quick response leads to higher quality decisions.

Fast fashion companies are often recognized for being particularly efficient at quick response. For

example, Zara owns its factories in Europe and therefore has a lower marginal cost for quick inven-

tory replenishment than companies sourcing from outside suppliers. Other fast fashion companies—

which may not have their own factories—emphasize the importance of maintaining good rela-

tionships with their local suppliers so that they can receive replenishment orders in a timely and

cost-efficient manner (Bruce and Daly 2006). Therefore, our results would seem to imply that

the fast fashion business model is more “sustainable” than traditional retail models—by inducing

higher quality products and lower leftover inventory. However, we show next that these results may

be reversed when the firm has the design flexibility to introduce multiple styles in a short period

of time.


4.2. Fast Fashion System with Quick Response and Variety Decision

We now allow the firm to introduce multiple styles, i.e., m≥ 1. The firm’s problem is to choose the

number of products m to introduce, the quality u, and the inventory q to maximize its expected

profit in (1). Since there is nothing to differentiate the styles from each other in period 1, the firm

produces equal quantities of each style. Here q denotes the production quantity of one style.

We solve the problem by backward induction. First, for given variety m and quality u, the

optimal inventory decision is

q=

{n(1− mk

k′ ), if k′ ≥mk,0, if k′ <mk.

We observe that the firm’s inventory decision q decreases in m. That is, the more styles the

firm introduces, the less it produces of each style in period 1. This is because, of the m styles that

the firm introduces, only one style will become a hit, while the other styles will become leftover

inventory. Thus, to avoid too much leftover inventory, as m increases, the firm relies more on quick

response in period 2 to satisfy consumer demand. If m is large, then the firm does not produce

anything5 in period 1 and instead relies entirely on quick response in period 2 to meet consumer

demand.

When the firm introduces multiple products (m≥ 1), the firm’s total production is mq+ (n−q)22n

,

where mq represents the total production in period 1, and E[(n− q)+] = (n−q)22n

is the expected

amount of total production in period 2 (using quick response). The expected amount of leftover

inventory is Q≡ (m− 1)q+E(q−n)+ = (m− 1)q+ q2/(2n). In the next lemma, we summarize the

effect of product variety m on the inventory outcomes.

Lemma 2. As m increases, (i) the inventory per style q decreases; (ii) the expected total pro-

duction (and the expected leftover inventory) first increases and then decreases.

As the firm introduces more styles, there are two effects on the expected leftover inventory. First,

the number of styles that end up unsold (i.e., m− 1) increases. Second, the firm produces fewer

units of each style in period 1. When m is small, the former effect dominates and the expected

leftover inventory increases in m. When m is large, however, the latter effect dominates, in which

case the expected leftover inventory decreases in m.

To explore the relationship between the firm’s quality and variety decisions, we now solve for

the optimal quality u given m. The solution is given in Proposition 2.

Proposition 2. Suppose m is given. (i) For m> k′

k, the optimal quality and inventory levels

are u= θk′ and q= 0. The firm’s expected profit is EΠ = n

2(a+ θ2

2k′ + γ2− γ

4m)−βm.

5 In practice, factories may require minimum batch sizes for each style it produces. Here we normalize the minimumorder size to zero.


(ii) For m≤ k′

k, the optimal quality and inventory levels are u= θk′

mk(2k′−mk)and q = n(1− mk

k′ ).

The firm’s expected profit is EΠ = n(a2

+ θ2k′

4mk(2k′−mk)− γ

8m+ γ

4)− βm. (iii) The optimal quality u

decreases in k′ and m.

Proposition 2(i) and (ii) show that, when product variety is large (or the cost of quick response

is low), the firm relies solely on quick response to satisfy consumer demand. In contrast, when

product variety is small (or the cost of quick response is high), the firm uses quick response but

also produces some inventory in period 1. Proposition 2(iii) states that, for given m, the optimal

quality is lower when the cost of quick response is higher or product variety is larger. In this case,

similar to the discussion in Section 4.1, as quick response becomes less expensive, the firm is better

at matching supply with demand, and thus invests in a higher-quality product. To explain the

effect of product variety, note that as the number of styles increases, the total costs of production

increase (due to more unsold inventory and/or more reliance on quick response). In response, the

firm lowers the product quality to reduce the unit production cost. Figure 1 illustrates the optimal

quality decision and expected leftover inventory as functions of m.

u

2 3 4 5

0.4

0.6

0.8

1.0

1.2

m

(a) Optimal quality

Q

2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

m

(b) Expected leftover inventory

Figure 1 Optimal quality decision and expected leftover inventory as functions of m. In this figure, We let

k= 1/2, k′ = 5/2, θ= 1, n= 1.

Now we solve for the firm’s endogenous variety decision. The optimal number of products m∗ is

characterized as follows:

Proposition 3. (i) When γ ≤ 2(k′−k)2θ2

kk′(2k′−k), then m∗ = 1.


Table 1 Firm’s Equilibrium Decisions.

m∗ u∗ q∗

Region I 1 θk′

k(2k′−k)n(1− k

k′ )

Region II m∗ =mE ∈ (1,√

nγ8β

) θk′

m∗k(2k′−m∗k)n(1− m∗k

k′ )

Region III√

nγ8β

θk′ 0

(ii) When 2(k′−k)2θ2

kk′(2k′−k)< γ < (k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3, then m∗ =

√nγ8β

if β ≤ n(√

γ8− (k′−k)θ√

4k(2k′−k)k′

)2

and

m∗ = 1 otherwise.

(iii) When γ ≥ (k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3, then there exists thresholds β1 and β2 such that m∗ =

√nγ8β

if β <

β1, m∗ =mE if β1 ≤ β < β2, and m∗ = 1 if β ≥ β2. Here mE is the smallest root of − θ2k′(k′−km)

2km2(2k′−km)2+

γ8m2 − β

n= 0 on [1, k′/k].

To explain Proposition 3, we first make the following observation regarding the optimal variety

decision.

Corollary 1. (i) If γ = 0, then the optimal number of styles to introduce is m∗ = 1. (ii) If

γ > 0 and β = 0, then the optimal number of styles to introduce is m∗→∞ if γ ≥ 2(k′−k)2θ2

kk′(2k′−k), and

m∗ = 1 otherwise.

The only reason for the firm to introduce multiple styles is to better match the consumers’

preference for fashionable designs. So if consumers are not sensitive to fashion (γ = 0), then the

firm introduces only one product. In contrast, if γ is sufficiently large and introducing new styles is

costless (β = 0), then the firm introduces as many styles as possible so that the fashion trend can be

matched perfectly. Finally, we note that even when product introduction is costless and consumers

are sensitive to fashion, the firm may find it optimal to refrain from introducing multiple styles if γ

is small. This is because, as discussed above, larger product variety implies higher production costs

(due to more leftovers and/or higher reliance on quick response), which is only worth incurring

when consumer valuation of fashion is sufficiently high.

Table 1 presents the firm’s optimal strategy in terms of product quality, inventory, and variety

when β > 0 and γ > 0. There are three regions, depending on the values of γ and β, as illustrated

in Figure 2. If consumers are highly sensitive to fashion and the cost of introducing new styles is

low (i.e., Region III), then the firm only uses quick response (q∗ = 0), introduces a large number of

styles (m∗ =√

nγ8β

) and chooses a low product quality. Otherwise, the firm produces some inventory

in period 1, and introduces fewer styles (m∗ <√

nγ8β

). In particular, if consumers are not very

sensitive to fashion or the cost of introducing new styles is high (i.e., Region I), then the firm only

introduces one product (m∗ = 1).


Region I

Region II

Region III

0.5 0.6 0.7 0.8 0.9 1.0

0.002

0.004

0.006

0.008

0.010

γ

β

Figure 2 Optimal number of styles (k= 1/2, k′ = 5/2, θ= 0.6, n= 1). The three regions are as described in

Table 1.

We next investigate the sensitivity of the firm’s decisions to key factors in fast fashion such as

consumer sensitivity to fashion, the cost of new product design, the cost of quick response and

potential market size.

Proposition 4. (i) The optimal number of styles m∗ increases in γ, decreases in β, decreases

in k′, and increases in n. (ii) The optimal product quality u∗ decreases in γ, increases in β, and

decreases in n. (iii) The optimal inventory per product q∗ decreases in γ, increases in β, and

increases in k′.

Recall from Proposition 2 that when m is exogenous, the parameters γ, β and n do not affect the

firm’s quality decision. Proposition 4 shows that this is no longer the case when m is endogenous.

Specifically, as consumers become more sensitive to fashion (i.e., γ increases), the firm has incentives

to introduce a larger variety of products in order to increase the “match” between its products

and the fashion trend. Correspondingly, the firm balances its production costs by reducing the

inventory per product and also reducing product quality. Similarly, a cheaper cost of design (i.e.,

smaller β) or a larger potential market size (i.e., larger n) pushes the firm to introduce more styles

and lower the product quality.

Interestingly, Proposition 4 states that as quick response becomes more efficient (i.e., k′

decreases), the firm introduces more styles. Intuitively, as the firm becomes more efficient at quick

response, it relies more on quick response to satisfy demand and thus produces less inventory of

each product in period 1. Therefore, the firm’s risk of having leftovers decreases and so it can

afford to introduce a larger variety of products. This result implies that, with endogenous m, a


lower k′ now has two opposite effects on the firm’s product quality decision. First, quality could

increase because quick response (lower k′) reduces the risk of having unsold inventory for given m

(see Proposition 2). Second, quality could decrease due to a larger m when quick response is more

efficient. The next result provides sufficient conditions for quality to increase in k′.

Corollary 2. The optimal product quality decision u∗ increases in k′ if (k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3≤ γ <

2k′θ2

k(2k′−k)and β1 ≤ β < β2.

Figures 3(a) and (b) illustrate a numerical example where as k′ decreases, the number of styles

increases, and the product quality first decreases and then increases. That is, when quick response

becomes more efficient (but not too much so), the firm prefers to lower product quality so that

it can introduce more styles. Figure 3(c) shows that the leftover inventory could also increase as

quick response becomes cheaper. These results contrast with the case of fixed m, where cheaper

quick response always leads to higher quality and lower leftover inventory.

m*

2 4 6 8 10

0

1

2

3

4

5

6

k′

(a) Optimal number of styles

u*

2 4 6 8 10

0.5

1.0

1.5

2.0

k′

(b) Optimal quality

Q*

2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

k′

(c) Expected leftover inventory

Figure 3 The dashed line represents m= 1. We let k= 1/2, β = 0.009, θ= 1, γ = 2.1, n= 1.

Finally, we compare the fast fashion model with the benchmark case where the firm only intro-

duces one product (i.e., m= 1). The next corollary describes the effect of fast fashion on the firm’s

product quality and leftover inventory.

Corollary 3. Compared to the case of m= 1:

(i) The fast fashion firm chooses a lower product quality (i.e., u∗ < u1) if γ > γ0, and the same

quality otherwise; the threshold γ0 increases in β and k′.

(ii) The fast fashion firm’s expected leftover inventory is higher when γ is moderate or β is

moderate, and lower (or the same) otherwise.


Corollary 3(i) shows that the firm’s ability to introduce multiple styles always leads to lower (or

the same) product quality, and that the region for lower quality is larger when consumers are more

sensitive to fashion, the cost of product design is lower, or quick response is more efficient. Figure

4(a) illustrates the quality comparison as a function of β and k′. By Corollary 3(ii), the fast fashion

firm may incur a higher or lower level of expected leftover inventory (compared to the traditional

firm with m = 1). This is due to the non-monotone relationship between product variety m and

the firm’s expected leftover inventory. Figure 4(b) illustrates the comparison in expected leftover

inventory. Observe that for moderate β, fast fashion may lead to higher or lower expected leftover

inventory, depending on the cost of quick response k′. In particular, the expected leftover inventory

is lower (higher) when k′ is low (high). This matches our observation that fast fashion companies—

while universally criticized for selling low quality products—vary in their leftover inventory levels.

For example, Zara is known for having high turnover and low inventory levels, while H&M is

recently reported to have $4.3 billion in unsold clothes (The New York Times 2018b). Our results

suggest that the difference is due to the companies’ different quick response capabilities.

Lower quality

Same quality

0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.01

0.02

0.03

0.04

k′

β

(a) Quality comparison

Less Inv

Same Inv

More Inv

0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.01

0.02

0.03

0.04

k′

β

(b) Expected leftover inventory comparison

Figure 4 Quality and leftover inventory comparisons. We let k= 1/2, θ= 1, γ = 2.1, n= 1.

5. Policy Implications

To counter the environmental impact of the fast fashion business model, we now look into the

effectiveness of three common sustainability programs/policies.


5.1. Disposal of Leftover Inventory

With the increasing environmental impact of the fast fashion industry, a common response by the

government is to implement stricter rules on the firm’s waste disposal practices. For example, a

recent UK parliament investigation into the sustainability of the fast fashion industry discusses

bans on “dumping clothes in landfill” and on “incineration of unsold stock” (Environmental Audit

Committee 2019). After drawing public criticism for burning leftover inventory, some fashion firms

are abandoning the practice and adopting alternative methods of inventory disposal (The New

York Times 2018a). Supposedly, imposing a higher cost for leftover inventory will induce firms to

produce less. However, we will show that these policies may have the unintended consequence of

lowering the firm’s product quality.

Suppose that the government’s waste disposal policies increase the firm’s disposal cost of its

leftover inventory by cr. The firm’s expected profit in period 1 now becomes

EΠ = (a+ θu+γ

2− γ

4m)n

2−mk

2u2q− k

′

2u2 (n− q)2

2n− cr

((m− 1)q+

q2

2n

)−βm. (3)

Proposition 5. For given m, as cr increases, the firm’s inventory decision q and product quality

u both decrease.

As the cost of disposing leftover inventory increases, the firm produces less inventory in the first

period to reduce the risk of having leftovers. Therefore, q decreases in cr. However, this also implies

that the firm relies more on quick response to satisfy consumer demand in the second period,

which implies a higher cost of production per product. To maximize its profit, the firm then tries

to balance its costs by reducing the quality of the product. Therefore, increasing the disposal

cost could be an effective policy in reducing the firm’s total leftover inventory, but may have the

unintended consequence of lowering the quality of the product. Given the short lifetime, limited

recycling options, and higher environmental impact that are typically associated with low-quality

garments, our results suggest that the net environmental effect of waste disposal policies is not

necessarily positive.

Now consider endogenous m. When inventory disposal is costly, the firm can reduce leftover

inventory by either reducing production q or reducing variety m. If m is reduced, then product

quality may be higher (by Proposition 2). However, our next result shows that even with endogenous

m, quality could decrease in cr. For illustrative purposes, we restrict our attention to m ∈ {1,2},

and compare the firm’s optimal profits when m= 1 and m= 2, respectively.

Proposition 6. Let m∈ {1,2} and suppose γ is not too large. Compared to the case of cr = 0,

the firm chooses a lower product quality when the cost of waste disposal cr is sufficiently high.


Figure 5 illustrates the comparison for different values of cr, and three levels of β. First consider

moderate β. Figures 5(a) and (b) show that in this case, when there is no disposal cost (i.e., cr = 0),

the firm prefers to introduce multiple products (m= 2) and chooses quality u2. As disposal becomes

more costly, the firm switches to introducing only one product (m= 1), and correspondingly chooses

a higher product quality (u1). However, when the cost of disposal is large (i.e., cr ≥ 0.137 in Figure

5a), then the firm switches back to m= 2 and low product quality u2. This is because when cr is

large, leftover inventory becomes very costly for the firm, so the firm relies almost entirely on quick

response to satisfy demand. In this situation, introducing more styles does not increase expected

leftover inventory very much (since very few units of each style are produced in period 1), while still

increasing consumers’ willingness-to-pay. Therefore, the firm prefers to introduce multiple styles

(at a low quality).

Figures 5(c) and (d) illustrate the profit comparisons for low and high beta, respectively. In both

cases, the quality decisions are as presented in Figure 5(b). When β is low, Figure 5(c) shows that

the firm always introduces multiple styles (m= 2), and thus quality u2 decreases in cr. When β

is high, as cr increases, the firm switches from introducing one product to introducing multiple

products, which leads to the product quality dropping from u1 to u2. In this case, quality is always

decreasing in cr.

Overall, our results suggest that increasing the firm’s inventory disposal cost cr could lead to

lower total production and lower leftover inventory, but may also result in a lower product quality.

5.2. Education and Consumer Awareness

One way to push for higher quality products is to increase customers’ awareness of the environ-

mental issues associated with low-quality products. For example, not-for-profit organizations like

“Sustain Your Style” and Fashion Revolution educate consumers about the environmental impact

of fashion, and encourage consumers to buy higher-quality clothes. With the rise of online second-

hand stores, customers can become increasingly aware that higher-quality clothes are more valuable

on the second-hand market (Fisher et al. 2008). When customers are more appreciative of quality

(i.e., θ is higher), it is intuitive that the firm would introduce higher quality products. However, the

effects on the firm’s variety and inventory decisions are less clear. On the one hand, higher quality

products are more costly to produce, and the firm may avoid the high cost of leftover inventory by

introducing fewer styles in the first period. On the other hand, to increase the price it can charge

for the high-quality product, the firm may wish to increase variety in order to make the product

as fashionable as possible. The next result follows directly from our analysis in Section 4.2.

Corollary 4. As customers become more sensitive to quality (i.e., θ increases), quality u∗

increases, variety m∗ decreases, and expected leftover inventory first increases and then decreases.


Π1

Π2

0.00 0.05 0.10 0.15 0.20 0.252.02

2.04

2.06

2.08

2.10

2.12

cr

(a) Profit comparison (β = 0.05)

u1

u2

0.05 0.10 0.15 0.20 0.25

0.2

0.4

0.6

0.8

1.0

1.2

cr

(b) Quality comparison (β = 0.05)

Π1

Π2

0.00 0.05 0.10 0.15 0.20 0.25

2.10

2.15

2.20

cr

(c) Profit comparison (β = 0.01)

Π1

Π2

0.0 0.1 0.2 0.3 0.4

1.95

2.00

2.05

cr

(d) Profit comparison (β = 0.1)

Figure 5 We let k= 1/2, k′ = 5/2, θ= 1, γ = 3, n= 1. The subscripts 1 and 2 correspond to m= 1 and m= 2,

respectively.

We find that as consumers become more sensitive to quality, the firm will find it optimal to

introduce a higher quality product along with less variety. There is a non-monotone relationship

between the expected leftover inventory and θ. Overall, if the fast fashion firm is currently relying

mostly on quick response to satisfy consumer demand, then increasing customers’ sensitivity to

quality would increase the product’s quality but also increase the firm’s leftover inventory. This is

because as variety decreases, the products’ “trendiness” decreases, and it becomes less profitable

for the firm to use quick response. However, as θ continues to increase, the quality would increase

while the leftover inventory would decrease, as the firm successfully shifts to a business model


with very high quality, low variety, and few unsold clothes. This represents a shift away from the

fast fashion model. Our results show that consumer education can be effective in pushing the firm

towards a more sustainable model. However, during this process, firms may experience high leftover

inventory for some time. Therefore, to facilitate a successful shift to more sustainable business

models, governments should be careful to not over-penalize firms on leftover inventory.

5.3. Recycling of Post-Consumer Products: Firm-owned versus Third-partyRecycling Programs

Recently, firms have started to engage in the collection and recycling of post-purchase clothing from

customers, either due to regulation or corporate social responsibility initiatives. These recycling

programs may vary in their effectiveness. In this section we consider and compare the product

quality and recycling effort under firm-owned versus third-party recycling programs. For example,

H&M partners with the recycling company “I:CO” to recycle discarded clothing from customers.

According to the company’s website, “I:CO” sorts the clothing, sells the wearable items to second-

hand stores, and breaks down the unwearable items into raw materials (such as fiber) that are

then reused in apparel supply chains or other industries. Patagonia, on the other hand, runs its

own recycling program. Similarly, Eileen Fisher accepts old clothes from its customers and repairs

them to be resold on its own website.

Suppose the salvage value of a product in the post-consumer stage is increasing in its quality.

That is, let the salvage value per unit of product be u. When the firm owns its recycling program,

its expected profit is

EΠ = (a+ θu+γ

2− γ

4m)n

2−mk

2u2q− k

′

2u2 (n− q)2

2n−βm+

sun

2− s

2

2. (4)

Here s ∈ [0,1] is the firm’s effort decision for recycling, and captures the proportion of sold items

that are recycled. The expected value of recycled clothes for the firm is suE[n] = sun2

. The last

term s2

2represents the cost of recycling logistics, such as collecting, sorting and processing the

clothes, which is increasing and convex in the proportion of clothes recycled. Note that the recycling

program may also involve a set-up cost, which—if high—could mean that the firm runs the recycling

program at a loss. Since the set-up cost does not affect the firm’s decisions, we normalize it to

zero. We first show that the existence of such a recycling program leads the firm to choose a higher

quality for its products.

Proposition 7. Under a firm-owned recycling program: (i) The optimal quality decision is

higher than that under no recycling program; (ii) The total production (and leftover inventory) may

be higher or lower than that under no recycling program.


When the firm runs a recycling program, it is willing to invest in a higher-quality product because

the higher the quality, the larger the salvage value from recycling. Proposition 7(ii) shows that a

recycling program for post-consumer products does not necessarily reduce the firm’s production

or leftover inventory. This is because the recycling program pushes the firm to introduce a higher-

quality product and lower variety. As we discussed in Section 5.2, as the firm shifts from a low-

quality and high-variety business model to one with high-quality and low-variety, the expected

leftover inventory first increases and then decreases.

Now, if the recycling program is run by a third party, we can assume that the firm pays the

third party a lump sum for managing the logistics of the program. The firm’s expected profit is

simply as given in (1), minus the lump sum (which we normalize to zero). The third party’s profit,

as a function of its effort sr is: EΠr = srun2− c′s2r

2. The third party decides the recycling effort sr

given the cost of recycling c′ and the quality of the product u (which is decided by the firm). Since

third parties like I:CO are usually professional recycling companies and therefore more efficient in

managing the logistics of recycling, we let c′ < 1. It is easy to see that the optimal recycling effort

in this case is s∗r = nu∗

2c′ , where u∗ is as given in Table 1. We have the following result.

Proposition 8. Compared to the third-party recycling program: (i) The quality decision under

a firm-owned recycling program is always higher; (ii) The recycling effort under a firm-owned

recycling program is higher if and only if the firm’s recycling cost disadvantage 1− c′ is sufficiently

small.

Overall, in comparing the firm-owned and third-party recycling programs, we find that if the

firm’s recycling cost disadvantage is small, then a firm-owned recycling program will lead to higher-

quality products and a higher recycling rate. If the cost disadvantage is large, however, then there is

a trade-off between quality and recycling efficiency. That is, if the firm owns its recycling program,

then it has more incentive to increase product quality, but may exert less recycling effort due to its

higher recycling cost. Ideally, to achieve high levels of quality and recycling effort, we would like

a centralized decision-maker to choose the decisions that maximize the sum of the firm’s and the

third party’s profits. That is, the central planner solves

maxm,u,q,s

EΠc = maxm,u,q,s

[(a+ θu+γ

2− γ

4m)n

2−mk

2u2q− k

′

2u2 (n− q)2

2n−βm+

sun

2− c

′s2

2]. (5)

Denote the quality and effort decisions which solve (5) as uc and sc. We now show that with the

correct scheme, the central planner can indeed induce the centralized solution while increasing

everyone’s profit. In particular, we consider a revenue-sharing scheme where the revenue and costs

of the recycling program are separated—the third party incurs the cost of recycling while the

retailer firm receives the revenue. The third-party recycling firm receives a fixed payment A from


the central planner for achieving the target recycling rate sc, so its total expected profit is EΠr =

A− c′s2c2

. For the retailer firm, it receives the revenue from recycling but pays the central planner

a fixed rate δ for each unit of product sold6. Therefore, its expected profit becomes

EΠ = (a+ θu+γ

2− γ

4m)n

2−mk

2u2q− k

′

2u2 (n− q)2

2n−βm+

scun

2− δn

2. (6)

It is easy to see that in this scheme, the centralized levels of recycling effort and product quality

are induced. By adjusting δ and A, the central planner can determine any split of profit between

the third-party recycler and the firm.

Proposition 9. By encouraging a revenue-sharing scheme and setting a target recycling rate

sc for the third party, the central planner can induce the centralized levels of recycling effort and

product quality (i.e., uc and sc) while increasing both the firm’s and the third party’s profits.

Some elements of our proposed scheme already exist in practice. For example, in France, the

government requires textile producers to either introduce a state-approved recycling program for

post-consumer products, or pay a fee proportional to the amount of clothing sold. In the latter case,

the government hires private recycling organizations who are required to meet certain recycling

targets (Bukhari et al. 2018). However, the recycling organizations typically keep the revenue from

recycling. Our results show that in the absence of any revenue-sharing, such policies can induce

a high recycling rate, but do not motivate the firm to increase the quality of its products. In

contrast, if there is revenue-sharing, then the central planner can induce a higher product quality

while maintaining the same high recycling rate and increase the system’s total profit. Our results

highlight the potential benefit of revenue-sharing contracts. In terms of implementability, we note

that such contracts are quite common in the electronic waste recycling industry (Cascade 2017,

Greenbiz 2019).

6. Conclusion

In this paper, we analyze a firm’s product design and inventory decisions in a fast fashion model

with quick response and flexible design, and assess the model’s environmental implications. We

show that a key driver of low product quality in the fast fashion industry is the firm’s incentive to

introduce multiple styles in the presence of uncertain fashion trends. When the variety decision is

endogenous, quality decreases as consumers become more sensitive to fashion, as the cost of design

becomes cheaper, and (in certain scenarios) as the cost of quick response becomes cheaper. Our

6 We focus on scenarios where the central planner can monitor the amount of clothing sold and recycled, but cannotdirectly monitor the quality of the clothes. Otherwise, the solution is trivial—the central planner only needs to imposea quality requirement on the firm.


analysis also identifies the conditions under which fast fashion leads to higher (or lower) leftover

inventory.

An in-depth understanding of the fast fashion model is key to developing effective sustainabil-

ity policies for the industry. Our results show that policies which focus on improving one envi-

ronmental measure (e.g., reducing production/leftover inventory) could have unintended negative

consequences on another (e.g., product quality). We also investigate the optimal design of post-

consumer recycling programs and suggest that, by imposing a revenue-sharing scheme between the

fast fashion firm and a third-party recycler, the central planner can induce system-optimal product

quality and recycling rates while increasing both parties’ profits.

There are several limitations to our study. First, we focus on product quality and do not consider

alternative design elements such as recyclable design. There is growing interest in the fashion

industry in using recycled fiber to produce clothes or improving the design for easier recycling (Ellen

MacArthur Foundation 2017). Such changes do not necessarily increase the quality/durability of

the product, but may have a positive environmental impact by lowering the cost of recycling and

increasing material reuse. It would be interesting to investigate the firm’s incentive to develop

recyclable designs, and how the decision interacts with other elements of the business model such

as variety and inventory strategies. Second, we consider a single firm in our model. In practice,

competition could play an important role in inducing high variety and low quality in the fast fashion

industry. Competitor dynamics—such as competition between a trend maker and a follower—may

also affect the equilibrium outcome in terms of product quality and total production. We leave

these issues for future research.

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Appendix A: Pricing Before Demand Realization

In this appendix, we consider the case where the pricing decision is made before the trend x and the market

size n are realized. We will show that the insights from the main model still hold.

In the first period, the firm chooses the variety (m), the product quality (u), the inventory per product

(q) and the price (p). The price is the same for all products since they are ex ante identical. In the second

period, after observing x, the consumers prefer to purchase the product with the highest surplus—which

is the product with location closest to the trend—and do so if p≤ a+ θu+ γ( 12− |t∗ − x|). Therefore, the

demand for product i is:

Di =

{n, if ti = t∗ and |t∗−x| ≤ a+θu−p

γ+ 1

2,

0, otherwise,

and the firm’s profit is

Π =

{pn− k′

2u2(n− q)+−m k

2u2q−βm, if |t∗−x| ≤ a+θu−p

γ+ 1

2,

0−m k2u2q−βm, otherwise.

If the price is lower than the consumers’ willingness-to-pay for the trendy product (i.e., |t∗−x| ≤ a+θu−pγ

+ 12),

then the total demand is n, and the firm’s profit is the same as that in the main model (see Section 3.3). On the

other hand, if the price is too high or the trendy product is located too far from x (i.e., |t∗−x|> a+θu−pγ

+ 12),

then the consumers do not purchase anything in period 2. In this case, the firm earns zero revenue and does

not use quick response in period 2.

In period 1, the firm chooses m, u, q and p to maximize its expected profit:

EΠ =( np

2− k′

2u2 · (n− q)

2

2n

)Prob

(|t∗−x| ≤ a+ θu− p

γ+

1

2

)−mk

2u2q−βm.

Here Prob

(|t∗−x| ≤ a+θu−p

γ+ 1

2

)is the probability that consumers purchase the trendy product in period

2. Recall that |t∗−x| is uniformly distributed on [0, 12m

]. There are two possible cases. In the first case, the

price is relatively high (i.e., a+θu−pγ

+ 12< 1

2m) so that the probability of purchase is 2m(a+θu−p

γ+ 1

2)< 1. In

this case, there is a positive probability that consumers do not purchase anything in period 2. In the second

case, the firm charges a low enough price (i.e., a+θu−pγ

+ 12≥ 1

2m) so that the probability of purchase in the

second period is always 1.

To generate non-trivial results, we assume that the base value of the product (i.e., the parameter a) is

sufficiently large so that the firm always prefers to sell its products with probability 1.7 Therefore, the firm

solves

maxm,u,q,p

EΠ =np

2− k′

2u2 · (n− q)

2

2n−mk

2u2q−βm,

subject to the constraint a+θu−pγ

+ 12≥ 1

2m. Since EΠ is strictly increasing in p, the firm’s optimal pricing

decision is p= a+ θu+ γ

2− γ

2m. The firm’s maximization problem in period 1 becomes

maxm,u,q

EΠ = maxm,u,q

[(a+ θu+γ

2− γ

2m)n

2−mk

2u2q− k′

2u2 (n− q)2

2n−βm], (7)

7 If the probability of purchase is smaller than 1, then the firm’s expected profit is EΠ =(np2− k′

2u2 ·

(n−q)22n

)2m(a+θu−p

γ+ 1

2)−m k

2u2q − βm ≡m · Π, which is strictly increasing in m unless Π ≤ 0, in which case the

optimal expected profit is nonpositive.


which is very similar to the maximization problem in (1). The only difference is a lower average price (i.e.,

a+ θu+ γ

2− γ

2m< a+ θu+ γ

2− γ

4m). This reflects the fact that when the pricing decision is made before

demand realization, the firm can no longer extract all of the consumer’s surplus.

It is easy to confirm that all results from the main model still hold. The firm’s optimal decisions are given

in Table 2 (similar to Table 1). Here mE is the smallest root of − θ2k′(k′−km)

2km2(2k′−km)2+ γ

4m2 − β

n= 0 on [1, k′/k].

Table 2 Firm’s Equilibrium Decisions.

m∗ u∗ q∗

Region I 1 θk′

k(2k′−k)n(1− k

k′ )

Region II m∗ =mE ∈ (1,√

nγ4β

) θk′

m∗k(2k′−m∗k)n(1− m∗k

k′ )

Region III√

nγ4β

θk′ 0

Figure 6 illustrates the optimal variety, quality and leftover inventory outcomes when pricing happens

before and after demand realization, respectively. Comparing the decisions, we find that when the pricing

decision is made before demand realization, the optimal variety is larger (i.e., m∗ >m0 in Figure 6a). This

is because the average price is more sensitive to m in the new model (i.e., ∂p

∂m= γ

2m2 ) than in the original

model (i.e., ∂Ep∂m

= γ

4m2 , see (1)). Intuitively, in the new model, the price is set at the “border”—it is equal

to the consumers’ willingness-to-pay when the product is exactly 12m

away from x. In the original model,

the price could be the willingness-to-pay when the distance is from 0 to 12m

—that is, the price does not

always depend on m. So increasing m has a stronger marginal effect on the optimal price (and the firm’s

expected revenue) in the new model. In other words, when price is made before demand realization, the

variety decision becomes a more important lever for the firm. Figure 6(b) shows that, corresponding to a

larger variety, the firm chooses a lower product quality when the pricing decision is made before demand

realization. Finally, Figure 6(c) shows that the total leftover inventory could be either higher or lower in the

new model, depending on the consumers’ sensitivity to fashion (γ).

m0

m*

1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

γ

(a) Optimal number of styles

u0

u*

1 2 3 4 5 6 7 8

0.2

0.4

0.6

0.8

1.0

γ

(b) Optimal quality

Q0

Q*

1 2 3 4 5 6 7 8

0.0

0.5

1.0

1.5

2.0

2.5

γ


Figure 6 The subscript “0” represents the decisions in the original model. We let k= 1/2, k′ = 5, β = 0.009,

θ= 1, n= 1.


Appendix B: Consumer Heterogeneity

In the main model, we have assumed that consumers are homogeneous in their fashion tastes, and therefore

are only interested in purchasing the product whose location is closest to x. This implies that all other

products become leftover inventory. In practice, however, consumers may be heterogeneous in their prefer-

ences and firms may observe demand for more than one product. This raises the question of whether our

main model overestimates the amount of leftover inventory in the fast fashion system. In this appendix, we

extend our model by considering two types of consumers: ones who are influenced by trend and ones who are

not. The latter type of consumers have predetermined, heterogeneous style preferences and may purchase

the non-trendy products. We show that although demand is more evenly distributed across products in

the new model, total leftover inventory is not necessarily lower. We also demonstrate that the existence of

trend-driven consumers is a key driver to our results.

Specifically, of the n consumers in the market, suppose ρn are trend-driven (i.e., their preferred location

is x), and (1 − ρ)n are “basic”, 0 ≤ ρ ≤ 1. The basic consumers’ preferred locations are heterogeneous—

uniformly distributed on the circular design space—and not influenced by trend x. Note that if ρ= 1, then

all consumers are trend-driven and we revert to the original model. On the other hand, if ρ = 0, then we

have a more traditional model with heterogeneous consumers and no fashion shock (see, e.g., Alptekinoglu

and Corbett 2008 and de Groote 1994).

For simplicity, let the pricing decision be made before uncertainties are realized so that the firm charges

the same price for all products8. A basic consumer with preferred location z would buy the nearest product

(with location ti) if and only if p≤ a+θu+ γ

2−γ|ti−z|, or equivalently, |ti−z| ≤ a+θu−p

γ+ 1

2. Recall that the

furthest that any consumer’s preferred location can be from the nearest product location is 12m

. Therefore,

the basic consumers’ demand for product i= 1, ...,m is

DBi = (1− ρ)n ·min{2(a+ θu− p)

γ+ 1,

1

m}.

As in the original model, the ρn trendy-driven consumers only buy the trendy product (with location t∗),

and purchase if and only if p ≤ a+ θu+ γ

2− γ|t∗ − x|. Overall, consumer demand for product i = 1, ...,m

becomes

Di =

DBi , if ti 6= t∗,

DBi + ρn, if ti = t∗ and p≤ a+ θu+ γ

2− γ|ti−x|,

DBi , if ti = t∗ and p > a+ θu+ γ

2− γ|ti−x|.

We note that if a+θu−pγ

+ 12< 1

2m, then the market is uncovered (i.e.,

∑m

i=1Di < n), and a larger m implies

a larger total demand∑m

i=1Di. On the other hand, if a+θu−pγ

+ 12≥ 1

2m, then the market is covered and m

does not affect total demand size.

The firm’s profit in period 2 is

Π = p

m∑i=1

Di−mk

2u2q− k′

2u2

m∑i=1

[(Di− q)+]−βm.

8 Similar results can be obtained if pricing happens after uncertainties are realized and the firm is restricted to chargethe same price for all products.


In period 1, the firm chooses price, quality, inventory and variety to maximize its expected profit:

maxp,u,q,m

EΠ. (8)

We first show that if ρ= 0 (i.e., consumers are not trend-driven), then the firm’s quality decision does not

depend on the number of styles it introduces.

Lemma B.1 Suppose consumers are not trend-driven (ρ= 0) and m is given. The optimal quality decision

is independent from m, and the firm’s expected leftover inventory is first increasing and then constant in m.

It follows from Lemma B.1 that when consumers are not trend-driven, the fast fashion system (as defined in

the paper) does not lead to lower quality. We can further show that in this case, fast fashion also does not

alter the expected amount of leftover inventory.

Proposition B.1 Suppose consumers are not trend-driven (ρ= 0). Compared to the case of m= 1, the fast

fashion firm chooses the same quality and incurs the same expected leftover inventory.

Recall that in the original model with trend-driven consumers, a larger variety leads to higher production

costs for the firm because of higher leftover inventory risk and/or more reliance on quick response. When

consumers are not trend-driven, however, this dynamic does not arise—increasing variety simply results in

a redistribution of demand proportionally across all styles. As a result, the firm can produce less inventory

of each style without increasing its use of quick response for “trendy” products. This analysis clarifies that

the key driver of our results in the paper lies in the assumptions that consumer demand disproportionately

concentrates on a subset of styles, and that the firm cannot anticipate which styles would draw the most

demand before the selling season starts.

We now analyze the case of ρ> 0. Consistent with the original model (see also the discussion in Appendix

A), we assume that the base value of the product (i.e., a) is sufficient high such that the firm prefers to cover

the market (i.e., sell to everyone). It follows that the firm’s pricing decision is p= a+θu+ γ

2− γ

2m. The firm’s

profit in period 2 is

Π = (a+ θu+γ

2− γ

2m)n−mk

2u2q− (m− 1)

k′

2u2(

(1− ρ)n

m− q)+− k′

2u2(

(1− ρ)n

m+ ρn− q)+−βm.

For given m, the optimal inventory and quality decisions in period 1 are

q=

{n(k′−km)(1−ρ+mρ)

k′m, if m≤ k′

k− 1−ρ

ρ,

n(k′−k)(1−ρ)(1−ρ+mρ)k′m(1−2ρ+mρ)

, if m> k′

k− 1−ρ

ρ,

and

u=

{k′θ

k(2k′−km)(1−ρ+mρ) , if m≤ k′

k− 1−ρ

ρ,

kθ(1−2ρ+mρ)

k′2(m−1)ρ2+k(2k′−k)(1−ρ)(1−ρ+mρ) , if m> k′

k− 1−ρ

ρ.

Figure 7 illustrates the firm’s optimal variety and quality decisions, as well as its expected leftover inventory,

for ρ= 12

and ρ= 1 respectively. Interestingly, Figure 7(c) shows that total expected leftover inventory may

be lower or higher when consumers are heterogeneous (ρ= 12). Recall that when consumers are homogeneous

and trend-driven (i.e., ρ= 1), as variety increases, leftover inventory first increases and then decreases. In


particular, when consumers are highly sensitive to fashion (so variety is large), the firm relies solely on quick

response to satisfy all demand, resulting in zero leftover inventory. With heterogeneous consumers, however,

demand is more “dispersed”, so the firm always produces some inventory in the first period, regardless of

how large γ is. As a result, expected leftover inventory may be higher when consumers are heterogeneous

than when they are homogeneous. Proposition B.2 formalizes this result.

m2m*

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

2

4

6

8

10

12

γ

(a) Optimal variety

u2

u*

1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

γ

(b) Optimal quality

Q2

Q*

1 2 3 4 5 60.0

0.5

1.0

1.5

γ


Figure 7 The subscript “2” and the superscript “*” represent the solutions under ρ= 12

and ρ= 1, respectively.

We let k= 1, k′ = 8, β = 0.008, θ= 1, n= 1.

Proposition B.2 When γ is sufficiently large, the total expected leftover inventory is always higher under

ρ< 1 than under ρ= 1.


Appendix C: Proofs

Proof of Lemma 1. We will show that, for any given m, q, and u, it is always optimal for the firm to

choose the products’ locations (t1, ..., tm) at equal distances from each other on the circular space. Recall

that the firm wishes to maximize its expected profit, which is given by

EΠ = E[(a+ θu+γ

2− γ|t∗−x|)]E[n]−mk

2u2q− k′

2u2E[(n− q)+]−βm.

It suffices to prove that E|t∗−x| is minimized when the products are located at equal distances to each other.

Consider the location of any product (say t2), and denote the locations of its two neighboring products by

t1 and t3 (if m= 2, then t1 = t3). Take the arc between t1 and t3 (with t2 on it), and define T0 = |t1− t2| and

T = |t1− t3|. Given that x is realized on this arc, the expected distance between x and the nearest product

is 2T

∫ T0/2

0zdz+ 2

T

∫ (T−T0)/2

0zdz =

T2−2TT0+2T20

4T, which is minimized at T0 = T/2. Therefore, the firm always

locates t2 at equal distances to t1 and t3. �

Proof of Proposition 1. (i) The results hold because dq1dk′

= nk(k′)2

> 0 and du1

dk′=− θ

(2k′−k)2< 0. (ii) The

expected leftover inventory E[(q1−n)+] =q212n

increases in k′ because q1 increases in k′. �

Proof of Lemma 2. (i) follows directly from the expression for q. To prove (ii), observe that both the

expected production and the expected unsold inventory increase in m for m≤ (k′)2

k(2k′−k)and decrease in m

otherwise. �

Proof of Proposition 2. We substitute q into equation (1) to obtain

EΠ(u,m) =

{n2(a+ θu+ γ

2− γ

4m)− nk(2k′−mk)mu2

4k′−βm, if k′ ≥mk,

n2(a+ θu+ γ

2− γ

4m)− nk′u2

4−βm, if k′ <mk.

To find the optimal quality decision, it suffices to solve ∂EΠ(u,m)

∂u= 0 for the cases of k′ >mk and k′ ≤mk,

respectively. To derive the sensitivity results, observe that for m> k′

k, u= θ

k′is constant in m and decreasing

in k′. For m≤ k′

k, we get du

dk′=− θ

(2k′−mk)2< 0 and du

dm= 2k′(mk−k′)θ

km2(2k′−mk)2≤ 0. �

Proof of Proposition 3. The firm’s expected profit as a function of u and m (EΠ(u,m)) is given in

proof of Proposition 2. We find the quality u and variety m decisions that jointly maximize EΠ(u,m) by

solving two constrained maximization problems separately and then comparing the solutions.

Problem (I): We first solve maxu,mEΠ(u,m) =maxu,m(n2(a+ θu+ γ

2− γ

4m)− nk′u2

4−βm

)such that u≥ 0

and m≥ k′/k. The Lagrangian of this constrained optimization problem is

L=n

2(a+ θu+

γ

2− γ

4m)− nk′u2

4−βm+λ(m− k′

k),

where λ≥ 0 is the Lagrange multiplier. The critical points satisfy the Karush-Kuhn-Tucker (KKT) conditions:

∂L

∂u=nθ

2− nk′u

2≤ 0,

∂L

∂m=

nγ

8m2−β+λ≤ 0,

∂L

∂λ=m− k′

k≥ 0,

u∂L

∂u= 0, m

∂L

∂m= 0, λ

∂L

∂λ= 0, u≥ 0, λ≥ 0.

Solving these conditions yields two solutions, depending on the value of β.

Solution I.1: When β ≤ nγk2

8k′2, the constraint does not bind (λ= 0) and the optimal solution is

u=θ

k′, m=

√nγ

8β, EΠ(u,m) = n(

a

2+γ

4+θ2

4k′)−√nγβ

2≡ΠI.1.


Solution I.2: When β > nγk2

8k′2, the constraint binds (λ> 0) and the optimal solution is

u=θ

k′, m=

k′

k, EΠ(u,m) = n(

a

2+γ

4− γk

8k′+θ2

4k′)− βk′

k≡ΠI.2.

Problem (II): We then solve maxu,mEΠ(u,m) =maxu,m(n2(a+ θu+ γ

2− γ


4k′− βm

)such

that u≥ 0 and 1≤m≤ k′/k. The Lagrangian of this constrained optimization problem is

L=n

2(a+ θu+

γ

2− γ


4k′−βm+λ1(m− 1) +λ2(

k′

k−m),

where λ1, λ2 are the Lagrange multipliers. The critical points satisfy the KKT conditions:

∂L

∂u=nθ

2− nk(2k′−mk)mu

2k′≤ 0,

∂L

∂m=

nγ

8m2−β− nk(k′− km)u2

2k′+λ1−λ2 ≤ 0,

∂L

∂λ1

=m− 1≥ 0,

∂L

∂λ2

=k′

k−m≥ 0, u

∂L

∂u= 0, m

∂L

∂m= 0, λ1

∂L

∂λ1

= 0, λ2

∂L

∂λ2

= 0 u≥ 0, λ1 ≥ 0, λ2 ≥ 0.

Solving these conditions (and checking the second-order conditions) yields three possible solutions.

Solution II.1: When β > nγ

8− n(k′−k)k′θ2

2k(2k′−k)2, λ1 = 0 and λ2 > 0, the quality and variety decisions are

u=k′θ

k(2k′− k), m= 1, EΠ(u,m) = n(

a

2+γ

8)−β+

nθ2k′

4k(2k′− k)≡ΠII.1.

Solution II.2: When β < nγk2

8k′2, λ1 > 0 and λ2 = 0, the quality and variety decisions are

u=θ

k′, m=

k′

k, EΠ(u,m) = n(

a

2+γ

4+θ2

4k′− kγ

8k′)− βk′

k≡ΠII.2.

Solution II.3: When λ1 = 0 and λ2 = 0 (neither of the constraints binds), the quality and variety decisions

are

u=k′θ

mEk(2k′−mEk), m=mE, EΠ(u,m) = n(

a

2+

θ2k′

4mEk(2k′−mEk)− γ

8mE

+γ

4)−βmE ≡ΠII.3,

where mE ∈ [1, k′/k] satisfies

∂L(u,m)

∂m=− nk′(k′−mk)θ2

2km2(2k′−mk)2+

nγ

8m2−β = 0, (9)

and∂2L

∂u2

∂2L

∂m2− (

∂2L

∂u∂m)2 =

−2k′(θ2− kγ)((k′)2 + 3(km− k′)2)− k4m3γ

8k′m2(2k′− km)2/n2> 0. (10)

Note that this critical point (if it exists) satisfies the second-order conditions because ∂2L∂u2

∂2L∂m2 − ( ∂2L

∂u∂m)2 > 0

(by definition), ∂2L∂u2 =− nmk(2k′−mk)

2k′< 0, and ∂2L

∂m2 < 0 (this is implied by the two former inequalities).

Before comparing the solutions to find the global optimum, we establish two useful results.

Result 1 There exists a threshold β > 0 such that the solution m=mE exists (and is unique) if and only if

γ > 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3and β < β ≤ nγ

8− nk′(k′−k)θ2

2k(2k′−k)2.

Proof: Substitute λ1 = 0, λ2 = 0 and u= k′θmEk(2k′−mEk)

into L(u,m) to get the function F (m)≡ n(a2

+

θ2k′

4mk(2k′−mk)− γ

8m+ γ

4)−βm. The conditions (9) and (10) are equivalent to:

dF (m)

dm=− nk′(k′−mk)θ2

2km2(2k′−mk)2+

nγ

8m2−β = 0, and

d2F (m)

dm2=

2k′(θ2− kγ)((k′)2 + 3(km− k′)2) + k4m3γ

4km3(2k′− km)3/n< 0.


By definition, the solution mE ∈ [1, k′/k] exists if and only if it satisfies the above two conditions. We consider

four possible cases.

Case 1: If γ < θ2

k, then mE does not exist because d2F (m)

dm2 > 0 for all m∈ [1, k′

k].

Case 2: If θ2

k≤ γ ≤ 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3, then, again, mE does not exist because d2F (m)

dm2 > 0, which holds

because the numerator of d2F (m)

dm2 is increasing in m (since the slope is 3k4m2γ+12kk′(k′−mk)(kγ−θ2)> 0)

and is positive at m= 1 (since γ ≤ 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3).

Case 3: If 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3<γ ≤ 2θ2

k, then the numerator of d2F (m)

dm2 is increasing in m, negative at m= 1

and positive at m= k′/k. So there is a threshold m such that d2F (m)

dm2 is negative for m∈ [1, m) and positive

for m ∈ [m, k′/k]. Therefore, dF (m)

dmdecreases in m ∈ [1, m) and increases in m ∈ [m, k′/k]. We now identify

the conditions under which dF (m)

dmcrosses zero on m ∈ [1, m). The conditions are dF (1)

dm≥ 0 and dF (m)

dm< 0.

The former inequality simplifies to β ≤ nγ

8− nk′(k′−k)θ2

2k(2k′−k)2. It remains to prove that the latter inequality holds

if and only if β is greater than some threshold β ∈ (0, nγ8− nk′(k′−k)θ2

2k(2k′−k)2). We prove this by observing that

dF (m)

dmis decreasing in β (by the envelope theorem, d

dβ( dF (m)

dm) = ∂

∂β( dF (m)

dm) =−1< 0), positive at β = 0 (since

dF (m)

dm= k3m2γ+(kγ−θ2)4k′(k′−km)

8km2(2k′−km)2/n> 0) and negative at β = nγ

8− nk′(k′−k)θ2

2k(2k′−k)2.

Case 4: If γ > 2θ2

k, then the numerator of d2F (m)

dm2 is increasing in m and negative at both m = 1 and

m= k′/k. Therefore, the slope of dF (m)

dmis negative for m∈ [1, k′/k]. This implies that dF (m)

dmcrosses zero on

m∈ [1, k′/k] if and only if dF (1)

dm≥ 0 and dF (k′/k)

dm≤ 0. These conditions simplify to nγk2

8(k′)2≤ β ≤ nγ

8− nk′(k′−k)θ2

2k(2k′−k)2.

This completes the proof. �

Result 2 When both solutions I.1 (m =√

nγ

8β) and II.3 (m = mE) exist, the profit difference ΠII.3 −ΠI.1

increases in β.

Proof: By the proof of Result 1 and the envelope theorem, dΠII.3

dβ= dF (mE)

dβ= ∂F (mE)

∂β=−mE. It follows

that dΠII.3

dβ− dΠI.1

dβ= −mE +

√nγ

8β, which is positive because by definition (when both solutions exist),

mE <k′

k<√

nγ

8β. �

Comparison of Solutions: We now compare the solutions from Problems (I) and (II). We first observe

that the corner solution ΠII.2 (or equivalently, ΠI.2) is always dominated. We then divide the remaining

proof into four regions, depending on the value of γ.

Region I: γ ≤ 2θ2(k′−k)2

kk′(2k′−k). In this region, since γ ≤ 2θ2(k′−k)2

kk′(2k′−k)< 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3for all k′ ≥ k, mE does not

exist (see Result 1), and it suffices to compare ΠI.1 and ΠII.1. If β > nγk2

8(k′)2, then m∗ = 1 is the only solution

(since the other solutions either do not exist or are not locally optimal). If β ≤ nγk2

8(k′)2, then m∗ = 1 again

because ΠII.1 − ΠI.1 = nθ2(k′−k)2

4kk′(2k′−k)− nγ

8− β +

√nγβ

2> 0. The inequality holds because γ ≤ 2θ2(k′−k)2

kk′(2k′−k)and

−β+√

nγβ

2> 0 (since β ≤ nγk2

8(k′)2< nγ

2).

Region II: 2θ2(k′−k)2

kk′(2k′−k)< γ ≤ 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3. In this region, similar to Region I, if β > nγk2

8(k′)2, then m∗ = 1.

If β ≤ nγk2

8(k′)2, then analyzing the sign of ΠII.1 −ΠI.1 shows that m∗ = 1 if β > β ≡ n

(√γ

8− (k′−k)θ

2√k′k(2k′−k)

)2

and m∗ =√

nγ

8βif β ≤ β.

Region III: 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3< γ < (k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3. Note that this region always exists because

(k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3− 2k′(3k2−6kk′+4k′2)θ2

k(2k′−k)3= k(5k′2−6kk′+k2)θ2

2k′(2k′−k)3> 0. The inequality γ < (k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3implies β >

nγ

8− nk′(k′−k)θ2

2k(2k′−k)2(see definition of β in Region II). There are two cases to consider.


(1) If β > nγ

8− nk′(k′−k)θ2

2k(2k′−k)2, then mE does not exist by Result 1. Thus, as in Region II, m∗ = 1 if β > β and

m∗ =√

nγ

8βif β ≤ β.

(2) If β ≤ nγ

8− nk′(k′−k)θ2

2k(2k′−k)2, then the optimal m is m = mE (if it exists) or m =

√nγ

8β. However, because

m=√

nγ

8βdominates at β = nγ

8− nk′(k′−k)θ2

2k(2k′−k)2, and we know from Result 2 that ΠII.3−ΠI.1 increases in β, it

follows that ΠII.3 <ΠI.1 for all β ≤ nγ

8− nk′(k′−k)θ2

2k(2k′−k)2. Therefore, m∗ =

√nγ

8β.

Region IV: γ ≥ (k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3. In this region, β ≤ nγ

8− nk′(k′−k)θ2

2k(2k′−k)2. Similar to the analyses above, there are

three cases to consider.

(1) if β > nγ

8− nk′(k′−k)θ2

2k(2k′−k)2, then m∗ = 1.

(2) If β < β ≤ nγ

8− nk′(k′−k)θ2

2k(2k′−k)2, then m∗ = mE (because in this region, ΠI.1 < ΠII.1 and solution II.1 is

dominated by II.3).

(3) If β ≤ β, then the solution is either m=mE or m=√

nγ

8β. Since ΠII.3−ΠI.1 increases in β, there exists

a threshold on β such that m=mE dominates for β greater than this threshold, and m=√

nγ

8βdominates

otherwise.

The results follow by combining the regions. �

Proof of Corollary 1. The results follow directly from Proposition 3. �

Proof of Proposition 4. (i) It is easy to see that m=√

nγ

8βincreases in γ, decreases in β, increases in

n, and is constant in k′. Next we show that m=mE increases in γ and n, and decreases in β and k′. Define

G(m)≡ nk′(mk−k′)θ2

2km2(km−2k′)2+ nγ

8m2 −β. By the implicit theorem, dmE

dγ=− ∂G(mE)

∂γ/ ∂G(mE)

∂m, which is positive because

∂G(mE)

∂m< 0 (by definition, see proof of Proposition 3) and ∂G(mE)

∂γ> 0. Similarly, we can show dmE

dn> 0,

dmE

dβ< 0 and dmE

dk′< 0. Finally, we observe from proof of Proposition 3 that the optimal variety decision

changes from m∗ = 1 to m∗ =mE to m∗ =√

nγ

8βas γ increases, n increases, β decreases, or k′ decreases. Since

1≤mE <√

nγ

8β, this concludes the proof.

(ii) The proof follows from (i) and observing that the optimal quality u∗ decreases as m∗ increases.

(iii) This result follows from (i) and observing that q∗ decreases as m∗ increases. �

Proof of Corollary 2. Proposition 3 states that when γ ≥ (k2−3kk′+4k′2)2θ2

2kk′(2k′−k)3and β1 ≤ β < β2, then the

optimal variety and quality decisions are m∗ = mE and u∗ = θk′

mEk(2k′−mEk). In this region, we will derive

conditions under which du∗

dk′> 0. We first obtain du∗

dk′= ∂u∗

∂k′+ ∂u∗

∂m

dmE

dk′=− θ

(2k′−km)2− 2k′(k′−km)θ

km2(2k′−km)2· dmE

dk′(we

sometimes omit the subscript E in mE for ease of notation). Define G(m)≡ k′(mk−k′)θ2

2km2(2k′−km)2+ γ

8m2 − β

n, then by

the implicit theorem, dmE

dk′=− ∂G(mE)

∂k′/ ∂G(mE)

∂m= kθ2

2(2k′−km)3/(k′(4k′2−6kk′m+3k2m2)θ2

2km3(2k′−km)3− γ

4m3

). Substituting this

back into the expression yields du∗

dk′=− θ

(2k′−km)2− 2k′(k′−km)θ

km2(2k′−km)2· kθ2

2(2k′−km)3/(k′(4k′2−6kk′m+3k2m2)θ2

2km3(2k′−km)3− γ

4m3

).

Rearranging the terms shows that du∗

dk′> 0 if and only if γ < 2k′θ2

k(2k′−kmE). The right-hand-side of the inequality

is bounded from below by 2k′θ2

k(2k′−k)since mE ≥ 1, so a sufficient condition for the inequality to hold is

γ < 2k′θ2

k(2k′−k). �

Proof of Corollary 3. (i) The fast fashion firm chooses a lower product quality if and only if m∗ > 1.

Therefore, the result follows from Proposition 4 and observing that m∗ > 1 if and only if γ is sufficiently

large (see Proposition 3). The threshold γ0 increases in β and k′ because by Proposition 4(i), m∗ decreases in

β and k′. (ii) We observe that the leftover inventory is first increasing and then decreasing as m∗ increases.

Also by Proposition 4(i), as γ increases or β decreases, m∗ increases. �


Proof of Proposition 5. For given m, the firm solves maxu,q EΠ such that u ≥ 0 and q ≥ 0. Solving

the KKT conditions ( ∂EΠ∂u≤ 0, ∂EΠ

∂q≤ 0, q ∂EΠ

∂q= 0, q ∂EΠ

∂u= 0, q≥ 0, u≥ 0) yields two solutions, depending on

the value of cr.

Solution 1: If cr ≥ (k′−mk)θ2

2k′2(m−1), then u∗ = θ

k′and q∗ = 0.

Solution 2: If cr <(k′−mk)θ2

2k′2(m−1), then u∗ = θ

k′(1−qE)2+2kmqEand q∗ = qE, where the solution qE satisfies

∂EΠ(u∗(q), q)

∂q=

(u∗)2

2(k′(1− q/n)−mk)− cr(m− 1)− crq/n= 0 (11)

and the second-order condition ∂2EΠ(u∗,q)∂q2

· ∂2EΠ(u∗,q)∂u2 − ( ∂

2EΠ(u∗,q)∂u∂q

)2 ≥ 0.

We observe that equation (11) implies k′(1− qE/n)−mk > 0 for all cr > 0. Therefore, Solution 2 (u∗, q∗)

is the interior solution to maxu,q EΠ(u, q) such that u ≥ 0 and 0 ≤ q ≤ k′−kmk′/n

. In this region, EΠ(u, q) has

increasing differences in (q,−cr), (u,−cr) and (u, q), respectively. It follows that both u∗ and q∗ are decreasing

in cr (see Theorem 2.3 in Vives 2001). This completes the proof. �

Proof of Proposition 6. By proof of Proposition 5, when cr is sufficiently large, the optimal quality

decision is θk′

, which is always weakly lower than the optimal quality when cr = 0 (see the optimal solution

in Table 1). In particular, if γ is not too large, then the optimal quality when cr = 0 is strictly larger than

θk′

. �

Proof of Corollary 4. The proof is similar to that of Proposition 4 and hence omitted. �

Proof of Proposition 7. The firm’s problem is

maxm,u,q,s

EΠ = maxm,u,q,s

[(a+ θu+γ

2− γ

4m)n

2−mk

2u2q− k′

2u2 (n− q)2

2n+sun

2− s2

2−βm]. (12)

Denote the optimal recycling effort by s∗. Observe that the optimal quality decision in (12) is equivalent to

that when recycling effort is s= 0 and the sensitivity to fashion is θ′ = θ+ s∗. The results follow because we

know from Corollary 4 that when the consumers’ sensitivity to fashion is higher, u∗ is higher, and leftover

inventory may be higher or lower. �

Proof of Proposition 8. (i) The result follows immediately from Proposition 7(i) because the quality

under the third-party recycling program is the same as the one without any recycling program. (ii) The

optimal recycling effort under the firm-owned recycling program is sf = min{ nuf

2,1}, while the optimal

recycling effort under the third-party recycling program is s∗r = min{ nu∗2c′,1}. First, if c′ = 1, then sf ≥ s∗r

because uf > u∗ (by i). As c′ decreases, s∗r increases while sf is constant. As c′→ 0, sf ≤ s∗r = 1. The result

follows. �

Proof of Proposition 9. It is easy to see that under the proposed scheme, the centralized levels of

recycling effort and product quality (uc and sc) are induced. Further, the total expected profit EΠc is higher

than the sum of profits in the decentralized case (EΠ +EΠr). Therefore, the central planner can distribute

the additional profit (using σ and A) arbitrarily between the recycler and the firm to increase both parties’

profits. �

Proof of Lemma B.1. We separate the proof into two cases, depending on whether the market is

covered.


Case 1: if a+θu−pγ

+ 12< 1

2m, then the market is uncovered, and Di = n( 2(a+θu−p)

γ+ 1). The firm’s expected

profit in period 1 becomes

EΠ = E[mpn(2(a+ θu− p)

γ+ 1)−mk

2u2q− k′

2u2m

(n(

2(a+ θu− p)γ

+ 1)− q)+−βm]

≡m ·EΠ.

It is easy to see that the optimal solution (u∗, q∗, p∗) maximizes EΠ, which does not depend on m. Therefore,

u∗ is independent of m.

Case 2: if a+θu−pγ

+ 12≥ 1

2m, then the market is covered and Di = n

m. In this case, the optimal price is

p= a+ θu+ γ

2− γ

2m. The firm’s expected profit in period 1 becomes

EΠ = E[(a+ θu+γ

2− γ

2m)n−mk

2u2q− k′

2u2m(

n

m− q)+−βm]

= (a+ θu+γ

2− γ

2m)n

2−mk

2u2q− k′

2u2m · (n−mq)

2

2nm−βm.

We solve maxu,q EΠ to get q∗ = (k′−k)n

k′mand u∗ = k′θ

k(2k′−k). Again, the quality decision u∗ does not depend on

m.

By analyzing the shape of the firm’s optimal expected profit with respect to m in both cases, we can show

that case 1 dominates when m is sufficiently small and case 2 dominates otherwise. In case 1, the expected

leftover inventory per style (i.e., E[q∗−n( 2(a+θu−p∗)γ

+1)]+) is independent of m, so the total expected leftover

inventory is increasing in m. In case 2, the total expected leftover inventory is mE[q∗− nm

]+ =E[mq∗−n]+,

which is independent of m. This concludes the proof. �

Proof of Proposition B.1. By the proof of Lemma B.1, when m is endogenous, case 2 always domi-

nates (i.e., the market is always covered). In this case, both the quality decision and total expected leftover

inventory are independent of m. �

Proof of Proposition B.2. We observe that for ρ < 1, the firm always produces a positive amount

of inventory in the first period. In contrast, for ρ = 1 and sufficiently large γ, the firm does not produce

anything in the first period and relies solely on quick response to satisfy demand, leading to zero leftover

inventory. Therefore, it follows that when γ is large, the total expected leftover inventory is higher under

ρ< 1 than under ρ= 1. �


Appendix D: Dynamic Substitution

In our main model we assume static substitution—consumers make the purchasing decision before observing

inventory information. Therefore, consumers who decide to purchase the trendy product would leave the

system if the product is out of stock. In response, the firm always satisfies demand for the trendy product

using quick response. In reality, however, the consumer may be willing to purchase a less trendy product if

the one they want most is out of stock (i.e., applying dynamic substitution) and in some cases, the firm may

prefer to sell consumers the less trendy in-stock products (albeit at a lower price) rather than re-stock the

most popular product using quick response. We consider this possibility in this appendix.

The general case of this model is difficult to solve9. For illustrative purposes, we restrict our attention

to m ∈ {1,2}. If m = 1, then the firm has only one product and always uses quick response (see Section

4.1). If m= 2, then in period 2, given realized x and n> q, the firm chooses between restocking the trendy

style and selling the less trendy in-stock product. If the firm decides to restock the trendy style, then the

unit profit margin is p∗ − k′

2u2, where p∗ = a + θu + γ

2− γ|t∗ − x| is the price for the trendy product.

On the other hand, if the firm decides to sell the less trendy in-stock product, the unit profit margin is

p= a+ θu+ γ

2− γ( 1

2− |t∗ − x|), which equals the consumers’ highest willingness-to-pay for the less trendy

product. Comparing the margins, the firm should restock the trendy product to satisfy all demand if and

only if trendiness is sufficiently high, i.e., |t∗−x| ≤ 14− k′

4γu2. In contrast, if 1

4− k′

4γu2 < |t∗−x| ≤ 1

4, then there

are two outcomes depending on total demand n and total inventory 2q. (I) When n≤ 2q, then the firm sells

both styles and does not use quick response at all. (II) When n> 2q, then the firm sells out all inventory (of

both styles) and then re-stocks the popular style using quick response to satisfy remaining demand.

Figure 8 illustrates the firm’s quality and inventory decisions with and without dynamic substitution, as

functions of γ. In both cases, as consumers become more sensitive to fashion (i.e., γ increases), the firm

introduces more styles. Correspondingly, quality decreases and the expected amount of leftover inventory

increases. When it is optimal to introduce multiple styles (m= 2), i.e., when γ is relatively large, dynamic

substitution leads to higher quality and lower expected leftover inventory. This is because under dynamic

substitution, the risk of leftover inventory is reduced (as the firm may sell the less trendy in-stock product).

However, this lower risk of leftover inventory may lead the firm to introduce more products. Figure 8 shows

that the firm switches from m= 1 to m= 2 earlier when there is dynamic substitution. As a result, when γ is

moderate, the existence of dynamic substitution leads to worse product quality and more leftover inventory.

Figure 9 further illustrates the firm’s quality and inventory decisions as functions of k′, and compares them

with the benchmark case of m= 1. We find that the following results hold regardless of static or dynamic

substitution. When m is endogenous: (i) The product quality is lower; (ii) The expected leftover inventory is

lower when k′ is small, and higher when k′ is moderate. These results are in line with our findings in Section

4.2.

9 For detailed analyses on a firm’s optimal assortment and inventory decisions under dynamic customer substitutionin a traditional retail setting (with no quick response nor fashion-sensitive customers), see Mahajan and van Ryzin(2001) and Gaur and Honhon (2006).


uS

u*

0 1 2 3 4 5 6 7

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

γ

(a) Optimal quality

QS

Q*

0 1 2 3 4 5 6 7

0.2

0.3

0.4

0.5

0.6

0.7

0.8

γ


Figure 8 As γ increases, the firm switches from m= 1 to m= 2 (represented by the vertical lines). The

superscripts “S” and “*” represent the cases with and without dynamic substitution, respectively. We let k= 1/2,

k′ = 2, β = 0.08, θ= 1, n= 1.


u*

uS

0.5 1.0 1.5 2.0 2.5 3.0

0.6

0.8

1.0

1.2

1.4

1.6

k′

(a) Optimal quality (γ = 3.2)

Q*

QS

0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

k′


uS

u*

0.5 1.0 1.5 2.0 2.5 3.0

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

k′

(c) Optimal quality (γ = 3)

QS

Q*

0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

0.5

0.6

k′

(d) Expected leftover inventory

uS

u*

0.5 1.0 1.5 2.0 2.5

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

k′

(e) Optimal quality (γ = 2.8)

QS

Q*

0.5 1.0 1.5 2.0 2.50.0

0.1

0.2

0.3

0.4

k′

(f) Expected leftover inventory

Figure 9 As k′ increases, the firm switches from m= 2 to m= 1 (the switching points are represented by the

vertical lines). The superscripts “S” and “*” represent the cases with dynamic and static substitution,

respectively. The dotted line is the benchmark case with m= 1. We let k= 1/2, β = 0.08, θ= 1, n= 1.

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