Li and Sundararajan

The Rise of Recommerce

The Rise of Recommerce: Ownership and Sustainability with Overlapping
Generations

Rubing Li New York University<EMAIL_ADDRESS>Arun Sundararajan New York
University<EMAIL_ADDRESS>
The emergence of the branded recommerce channel — digitally enabled and
branded marketplaces that facilitate purchasing pre-owned items directly from
a manufacturer’s e-commerce site — leads to new variants of classic IS and
economic questions relating to secondary markets. Such branded recommerce is
increasingly platform-enabled, creating opportunities for greater
sustainability and stronger brand experience control but posing a greater risk
of cannibalization of the sales of new items. We model the effects that the
sales of pre-owned items have on market segmentation and product durability
choices for a monopolist facing heterogeneous customers, contrasting outcomes
when the trade of pre-owned goods takes place through a third-party
marketplace with outcomes under branded recommerce. We show that the direct
revenue benefits of branded recommerce are not their primary source of value
to the monopolist, and rather, there are three indirect effects that alter
profits and sustainability. Product durability increases, a seller finds it
optimal to forgo marketplace fees altogether, and there are greater seller
incentives to lower the quality uncertainty associated with pre-owned items.
We establish these results for a simple two-period model as well as developing
a new infinite horizon model with overlapping generations. Our paper sheds new
insight into this emerging digital channel phenomenon, underscoring the
importance of recommerce platforms in aligning seller profits with
sustainability goals.

analytical model, platform, secondary market, pricing, versioning

##  1 Introduction

The market for pre-owned items has seen a significant growth and acceleration
of digital innovation over the last few years. Third-party digital markets
such as Depop, Poshmark, StockX, and The RealReal have contributed to this
recent explosion of “recommerce” — the sale or trading of pre-owned items. In
parallel, new digital platforms like Trove and Recurate have made it possible
for companies ranging from Canada Goose and Patagonia to Zara and lululemon to
set up their own branded marketplaces that facilitate purchasing pre-owned
items directly from the brand, a practice we refer to as _branded recommerce_.
Figure 1 presents examples of two branded recommerce sites.

Sales of pre-owned apparel, over $100 billion globally in 2022, are projected
to grow at five times the rate of overall commerce, doubling to $218 billion
by 2026 and making up 23% of apparel retail sales (Thredup, 2022).

Figure 1: Examples of two “branded recommerce” sites. (The Lululemon Like New
site is powered by the Trove recommerce platform; the Zara Pre-Owned site is
powered by the Recurate recommerce platform.)

The emergence of platform-based branded recommerce leads to new variants of
classic economics and IS questions relating to secondary markets. For example,
manufacturers who see branded recommerce as an opportunity to shape the
initial brand experience of customers who begin their brand relationship by
buying a pre-owned item have to weigh these benefits against the possible
increased threat of cannibalization of the sales of new items. A manufacturer
who may not have previously encouraged the purchasing of pre-owned items may
find their economic incentives are altered by the prospect of value capture
from these transactions under branded recommerce, or may see branded
recommerce as an “entry deterring” move against specialty third-party
marketplaces for pre-owned items. The ability to capture value from recommerce
may increase the incentives of a manufacturer to create more durable products
better aligned with sustainability goals.

In this paper, our focus is on the interplay of some of these factors with a
brand’s choice of product durability. Intuitively, a more durable product can
be used (and reused) more. A higher potential level of reusability can be
thought of as being more sustaiable. As an expert on sustainable fashion
recently observed in a Bloomberg article (Woody (2024)):

“Durability and reparability affect environmental impact,” says Veronica Bates
Kassatly, a London-based fashion industry analyst focused on sustainability.
She cites the example of wearing a shirt 500 times. If a cheaply made shirt is
replaced after being worn 10 times, 50 shirts will need to be produced and
discarded. But if one better-made and easily mended shirt can be worn 500
times, the environmental cost is significantly lower, even if making the
garment uses more water and energy upfront.

A tradeoff for a brand considering this kind of move towards sustainability
stems from the threat of cannibalizing the sales of new products; products
that degrade faster help drive more frequent replacement or sales of new
products. Perhaps the emergence of a channel that allows a brand to
internalize some of the value created by resale alters this tradeoff.

We develop an economic model of recommerce that sheds light on some of these
issues. We begin by examining the effects that the sales of pre-owned items
have on market segmentation and product durability choices in a two-period
model, contrasting outcomes when the trade of pre-owned goods takes place
through a third-party marketplace with outcomes under branded recommerce. Our
model includes stylized measures that capture third-party market power (the
extent to which value from pre-owned goods trade is captured by the
marketplace owner) and quality uncertainty (the gap between the valuation of a
pre-owned item by the original owner who knows its actual quality and
condition, and a potential buyer who does not). We then extend this analysis
to a more general infinte-horizon model in which there are overlapping
generations of customers.

Our results show that under fairly general assumptions, adopting branded
recommerce is profitable for a monopolist who faces a third-party secondary
marketplace. Furthermore, the monopolist’s profit-maximizing durability
choices are more aligned with sustainability goals under branded recommerce.
We show how this “sustainability bump” from branded recommerce is higher when
third-party markets are more powerful, and when pre-owned product quality
uncertainty is higher, even though the move to branded recommerce by itself
may not resolve any of the underlying quality uncertainty. Counterintuitively,
the direct revenue potential of branded recommerce is not its equilibrium
source of value. Rather, we demonstrate a set of indirect economic effects
which benefit manufacturers and increase sustainability. Product durability
increases, there are greater seller incentives to invest in lowering the
quality uncertainty associated with pre-owned items, and it is actually
optimal for manufacturers to forgo marketplace fees altogether.

##  2 Literature Review

At a time when planned obsolescence is both widespread and lowering product
durability is under fire as being environmentally irresponsible, our results
suggest that branded recommerce could offer an additional economic lever to
sellers balancing profit maximization with sustainability considerations. We
build on the theory of the optimal pricing of a product line (Mussa and Rosen
(1978), Maskin and Riley (1984), Moorthy and Png (1992), Sundararajan (2004),
Bhargava and Choudhary (2008)), Mantena and Saha (2012), Pang and Etzion
(2012)). For example, Waldman (1996) shows that a monopoly seller has an
incentive to lower product durability because at higher levels of durability,
the ensuing higher quality of the pre-owned item in future periods limits
pricing power. We follow Waldman (1996) and Hendel and Lizzeri (1999) in
modeling durability as determining the quality gap between new and pre-owned
goods, in contrast with Johnson (2011) and others who treat new and pre-owned
goods as having equal quality and model durability as the survival probability
of a product.

Following the literature in economics and IS that has shown the economic
importance of product uncertainty (Pavlou et al. (2007), Dimoka et al. (2012),
Ghose (2009), Hong and Pavlou (2014)) and transaction cost (Bakos (1997)), we
explicitly incorporate simple representations of these two factors into our
model. A more detailed model of how the nature of secondary market frictions
affect market outcomes (Kannan et al. (2023)) may lead to further insight, and
this represents a direction for future researchers.

Our paper is also related to the literature on the incentives of monopoly
sellers of durable goods that stems from the celebrated Coase conjecture
(Coase (1972)) — that absent commitment devices, the price charged by a
durable goods monopolist is driven down to marginal cost by a time
inconsistency problem, thus leading to monopoly choices of durability that are
lower than the socially optimal levels, and, as shown by earlier work (Levhari
and Srinivasan (1969), Schmalensee (1970)), inducing a monopolist to pursue a
strategy of planned obsolescence. A series of papers that followed have either
reinforced and refuted the Coase conjecture. For example, in a series of
papers, Swan (1970), Swan (1971), Swan (1972) and Sieper and Swan (1973)
assert using a model that endogenizes the monopolist’s choice of durability
that this will actually be at the socially optimal level. In contrast, Bulow
(1982), Bulow (1986) reaffirms the Coase results, strengthening the
theoretical case for leasing and planned obsolescence. While we do not aim to
exhaustively summarize this literature, to the best of our knowledge, ours is
the first paper that connects the _ownership_ of the secondary market to this
discussion on optimal durability.

Our model with overlapping generations of consumers aims to capture the effect
of evolving consumer preferences that Johnson (2011) and others like Chen et
al. (2013) and Fraiberger and Sundararajan (2017) models as idiosyncratic
preference shocks. Our approach follows a standard infinite horizon setting in
which consumers live for two periods (see, for example, Villas-Boas (2004)).
Rather than incorporating explicitly changing preferences like some of the
prior work on secondary markets, in our model, changes in the “age” of a
consumer alter their preferences because their horizon is different (two
periods versus one).

Fundamentally, our paper addresses a variant of the central question in Chen
et al. (2013) — whether secondary markets help or hurt producers — albeit with
a focus specifically on the marketplace ownership choice (recommerce versus
third-party). While simpler, our model captures elements of all three key
effects of this literature — the substitution effect (pre-owned items may
cannibalize new product sales), the allocative effect (sellers indirectly earn
higher rents from higher valuation and forward-looking customers if these
customers can sell their products in the future), and aspects of the time
consistency effect (this increase in willingness to pay of the higher
valuation consumers is affected by their future price expectations, which in
turn are shaped by the seller’s output choices, and the seller does not have
the ability to commit to future prices).

More saliently, our paper adds to an extensive literature in economics,
marketing and information systems on secondary markets (see, for example,
Bapna et al. (2004), Ghose et al. (2006), Ghose (2009), Rao et al. (2009),
Gavazza et al. (2014), Chen et al. (2013), Rao (2015), Pilehvar et al. (2017),
Ishihara and Ching (2019)). We also add to the digital platform and sharing
economy literature (Bardhi and Eckhardt (2012), Cramer and Krueger (2016),
Bapna et al. (2016), Sundararajan (2016), Zervas et al. (2017), Li and Wang
(2024)). Over the last decade, while true sharing economy platforms that
facilitate product sharing and reuse have promised to lead to more sustainable
and circular usage, behavior changes that lead to widespread durable goods
renting and sharing has been limited outside of the automobile sector.
Meanwhile, the trade of pre-owned items has exploded, leading us to believe
that perhaps the real impact of platforms on sustainable consumption will come
not from sharing but from the digital trade and consumption of pre-owned
items. Our efforts to provide a more robust theoretical basis for the
ownership structure and economic factors that aid or abet this evolution is
aligned with this belief.

##  3 Overview of Model

###  3.1 Firm and customers

We model a monopoly seller whose products last 2 periods. We begin with a two-
period world, and then extend it to an infinite period model with overlapping
generations. In the first period, the seller chooses how much durability $D$
to build into its products. This choice is fixed across both periods. 111We
have also analyzed the case where the firm makes a separate durability choice
in each period. It is of comparable analytical complexity. Since there is no
“next period” in the second period, this artificially distorts durability
downwards. Further, in practice, it may not be realistic to imagine a firm
continuously varying its durability, either for technological reasons, or
because of brand perception concerns. The firm bears a constant marginal cost
of production $c(D)$ and no fixed costs. We assume that the function $c(D)$ is
positive, increasing and strictly convex for all $D>0$, and thus
$c^{\prime}(0)=0,c^{\prime}(D)>0,c^{\prime\prime}(D)>0$ for all $D>0$. We also
assume that $c(0)=0$.

New products are of a fixed quality level that is normalized to 1. In the
second period, units that are 1 period old are of quality $s(D)$, where
$s(0)=0,s(D)<1$ for all $D,s^{\prime}(D)>0,s^{\prime\prime}(D)<0$ for all
$D\geq 0$. Put simply, the more durable a product, the higher its value as a
pre-owned item.

Customers are heterogeneous, indexed by type $j=L,H$. Each consumer lives 2
periods. There are $n_{j}$ identical customers of type $\mathrm{j}$. To ensure
that the trade of pre-owned goods actually occurs in what follows, we assume
that $n_{L}>n_{H}$. In each period, type-H customers derive a gross benefit
$v_{H}s$ from consumption of one unit of good of quality $s$, while the
corresponding gross benefit derived by the low valuation customers is
$v_{L}s$, $v_{H}>v_{L}>0$. Customers use either zero units or one unit of the
seller’s output in each period. The firm and customers discount the future
using the discount factor $\delta,0<\delta<1$.

###  3.2 Interaction Between the Firm and its Customers

We assume that the firm cannot explicitly distinguish between consumer
types.222Thus, we ignore the customer recognition information the firm may
have in the second period from the choices consumers make in the first period
(Villas-Boas (2004)). The firm offers one product in each period. Consumers in
the second period may also have the option of buying a pre-owned item. Thus,
in the second period, the firm faces a design problem analogous to one faced
by a multiproduct monopolist, but where the quality level of the inferior
version in period 2 is determined by the durability choice made in period 1
and whether the secondary market is run by a third party (more on this later).

The sequence of interaction between the firm and its customers is as follows:

  * •

Period 1: The firm first announces a durability choice $D$, and the price for
a new unit of output $p_{1}^{N}$. All consumers then simultaneously decide
what, if anything, to purchase from the firm.

  * •

Period 2: First, the firm announces the price $p_{2}^{N}$ for a new unit of
output while maintaining its durability level $D$. Second, all consumers then
decide what, if anything, to purchase directly from the firm. Third, a
secondary market opens up where prices on the secondhand marketplace equate
supply and demand.

Trade in the secondary market is affected by the following economic factors:

  * •

Quality uncertainty: Potential buyers of a pre-owned item may not be able to
ascertain its true quality prior to purchase. We model this by assuming that
in a third-party marketplace, potential buyers of pre-owned goods have a
willingness to pay that is deflated by a factor $\alpha,0<\alpha<1$, relative
to existing owners of these pre-owned goods who know the actual quality. We
later assume that when the seller operates the secondary market (the
”recommerce case”), this quality uncertainty may change.

  * •

Marketplace transaction costs: Sellers of used goods pay a fraction $\beta$,
$0<\beta<1$, of the transaction price as a commission. This revenue is
received by whoever operates the secondhand marketplace.

The firm chooses durability and prices to maximize total anticipated profits
in all remaining periods. The consumers have the following choices.

  * •

In period 2, a consumer who did not purchase a product in the first period
chooses between purchasing a new product, purchasing a pre-owned product (if
such products are offered) or not participating. A consumer who purchased a
product in the first period chooses whether or not to purchase a new product,
and whether or not to sell their pre-owned product in the secondary market.

  * •

In period 1, a consumer chooses whether or not to purchase a new product.

##  4 A two-period model

We analyze the two-period model in some detail in what follows, since many of
the expressions and results from this analysis carry over to the model with
overlapping generations. The following situation —which we refer to as the
active pre-owned marketplace case — is the one of interest in our analysis,
since it is what leads to activity in the marketplace for pre-owned items:

  * •

Period 1: Consumers of type-H purchase a new product from the firm.

  * •

Period 2: Consumers of type-H purchase a new product from the firm, selling
their pre-owned products in the secondhand marketplace, and consumers of
type-L purchase a pre-owned product.

It is straightforward to show that this case leads to higher firm profits than
any case under which the firm’s choices lead to type-H consumers wanting to
keep their goods in the second period rather than selling them in the
secondhand marketplace. This is because the profits from the latter,
$n_{H}\left[v_{H}(1+\delta s(D))-c(D)\right]$, are strictly less than the
profits $(1+\delta)n_{H}v_{H}$ that the firm could earn from offering a
product of durability zero in each period priced at $v_{H}$. There is yet
another possibility - that the firm chooses a strategy that induces type-L
consumers to purchase new goods in one or both periods. However, under any
strategy of this kind, there will be no trade in the secondhand marketplace:
if type-L consumers purchase a new good in period 1, there is no difference
between the pre-owned goods being offered for sale and the ones they hold in
period 2 as a consequence of their period 1 purchase, and if type-L consumers
do not purchase pre-owned goods in period 2 , there are no potential buyers
for any pre-owned goods being offered for sale in period 2. We examine this in
greater detail in our overlapping generations model in Section 4 .

###  4.1 Socially optimal product choices

We benchmark the socially optimal level of durability under this active pre-
owned marketplace setting — where type-H customers consume a new good in each
period, and where type-L customers do not participate in the market in the
first period, while buying a pre-owned good in the second period.333This may
seem like a somewhat unusual socially optimal benchmark. However, it is the
one that is meaningful in our context. In contrast, if we consider the
situation under which both consumer types consumed a new good in each period,
durability is irrelevant, and the optimal level of durability is zero. To
motivate this benchmark, imagine that the seller can identify types, creates a
single period rental contract for the new good offered to type-H in each
period, and in the second period, creates a single-period rental contract that
offers the pre-owned goods returned by the type-H consumer at the end of
period 1 to type-L customers in period 2. This situation yields what is akin
to the “first best” durability choice.

Under this benchmark setting, viewing things from the first period, suppose
the firm chooses a durability level $D$. The gross surplus from the
consumption of the type-H customers is $n_{H}v_{H}$ in each period. The total
cost borne by the seller from this consumption is $c(D)$ in each period. Since
$n_{L}>n_{H}$, only a fraction $n_{H}/n_{L}$ of type-L customers will have a
pre-owned good available for their consumption in period 2. The gross surplus
generated from this consumption is $n_{H}v_{L}s(D)$. Thus, the total surplus
from this consumption and production activity, viewing things from the first
period is:

$(1+\delta)n_{H}v_{H}+\delta n_{H}v_{L}s(D)-(1+\delta)n_{H}c(D).$ (1)

Maximizing (1) with respect to $D$ yields the following expression for the
socially optimal durability level $D^{**}$:

$c^{\prime}\left(D^{**}\right)=\frac{\delta}{1+\delta}v_{L}s^{\prime}(D^{**}).$
(2)

###  4.2 Prices under the active digital pre-owned marketplace case

Let $p_{2}^{n}$ denote the price for a new good in the second period, and
$p_{2}^{u}$ denote the price for a pre-owned good in the second period. One
can formulate the seller’s problem in period 2 in terms of incentive-
compatibility (IC) and individual rationality (IR) constraints that induce
each type to choose the product “designed” for them. (This is not essential,
but we present it this way so that the parallel with the standard price
screening problem is clearer to those readers familiar with that literature.)

  * •

In period 2, type-H customers choose to sell their pre-owned product and buy a
new product rather than holding on to their pre-owned product (akin to an IC
condition for type-H)

$v_{H}-p_{2}^{n}+(1-\beta)p_{2}^{u}\geq v_{H}s(D).$ (3)

  * •

In period 2, type-L customers choose to buy a pre-owned product rather than a
new product (akin to an IC condition for type-L)

$\alpha v_{L}s(D)-p_{2}^{u}\geq v_{L}-p_{2}^{n}.$ (4)

  * •

The surplus of each type in period 2 is non-negative (akin to the IR
conditions)

$v_{H}-p_{2}^{n}\geq 0,$ (5) $\alpha v_{L}s(D)-p_{2}^{u}\geq 0.$ (6)

It is straightforward to see that the IR condition for type-L will bind, and
thus:

$p_{2}^{u}=\alpha v_{L}s(D).$ (7)

Next, we can use (7) to rewrite equation (3) as

$v_{H}-p_{2}^{n}\geq v_{H}s(D)-\alpha(1-\beta)v_{L}s(D),$ (8)

which yields the maximum price the firm can charge for a new good of
durability $D$ in period 2 444This is akin to the IC condition for type-H
binding.:

$p_{2}^{n}=\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D)).$ (9)

###  4.3 Durability choices

Let $D^{*}$ denote the firm’s profit-maximizing durability choice. We can now
characterize the optimal choices of the seller under two regimes: when the
secondhand marketplace is run by a third-party (“third-party marketplace”) and
when the secondhand marketplace is run by the seller (“branded recommerce”).

###### Lemma 4.1

(Equilibrium outcomes with a third-party marketplace):

If $v_{L}>v_{H}/2\alpha(1-\beta)$, then

  1. (a)

The profit maximizing level of durability $D^{*}$ satisfies:

$c^{\prime}\left(D^{*}\right)=\frac{\delta}{1+\delta}\left[2\alpha(1-\beta)v_{L}-v_{H}\right]s\left(D^{*}\right).$
(10)

  2. (b)

The price of new goods in period 1 is

$p_{1}^{n}=v_{H}+\delta\alpha(1-\beta)v_{L}s(D^{*}).$ (11)

  3. (c)

The price of new goods in period 2 is

$p_{2}^{n}=\alpha(1-\beta)v_{L}s(D^{*})+v_{H}(1-s(D^{*})).$ (12)

Note that if $v_{L}\leq v_{H}/2\alpha(1-\beta)$, there is no positive value of
$\mathrm{D}$ that supports an active pre-owned marketplace, since it is more
profitable for the seller to choose a durability level of zero, setting prices
$p_{1}^{n}=p_{2}^{n}=v_{H}$ and shutting the lower types out of the market.

We now turn to the branded recommerce situation under which the seller
operates the pre-owned marketplace. To aid in comparability with the prior
case, we retain the possibility of some quality uncertainty remaining even if
the seller operates the pre-owned marketplace. We discuss the case of
$\alpha=1$ as a special case later in the paper.

###### Lemma 4.2

(Equilibrium outcomes with branded recommerce):

If $v_{L}>v_{H}/\alpha(2-\beta)$, then

  1. (a)

The profit maximizing level of durability $D^{*}$ satisfies:

$c^{\prime}\left(D^{*}\right)=\frac{\delta}{1+\delta}\left[\alpha(2-\beta)v_{L}-v_{H}\right]s\left(D^{*}\right)$
(13)

  2. (b)

The price of new goods in period 1 is

$p_{1}^{n}=v_{H}+\delta\alpha(1-\beta)v_{L}s(D^{*})$ (14)

  3. (c)

The price of new goods in period 2 is

$p_{2}^{n}=\alpha(1-\beta)v_{L}s(D^{*})+v_{H}(1-s(D^{*}))$ (15)

Akin to the case of a third-party marketplace, if $v_{L}\leq
v_{H}/\alpha(2-\beta)$, then it is more profitable for the seller to choose a
durability level of zero, setting prices $p_{1}^{n}=p_{2}^{n}=v_{H}$ and
shutting the lower types out of the market. Since our focus is on examining
the difference between third-party and branded recommerce, we do not analyze
this case further.

###  4.4 Contrasting branded recommerce with third-party pre-owned markets

We now turn to analyzing how the possibility of branded recommerce changes a
firm’s choices. First, we examine how the pre-owned marketplace commissions
and quality uncertainty distort the durability choices of the seller,
independent of who runs the pre-owned marketplace. Next, we establish how
branded recommerce leads to more sustainable choices by inducing an increase
in the durability of products (and as a consequence, the quality of pre-owned
goods) and separate out three different drivers of this increase.

###### Proposition 4.3

Independent of whether the seller or a third-party operates the pre-owned
marketplace:

  1. (a)

As quality uncertainty goes down (and thus, $\alpha$ increases), the optimal
level of durability $D^{*}$ chosen by the seller increases, and the seller
profit increases.

  2. (b)

As the commission $\beta$ charged by the operator of the pre-owned marketplace
increases, the optimal level of durability $D^{*}$ chosen by the seller
decreases, and the seller profit decreases.

The proof of the proposition is provided in Appendix. Proposition 4.3
indicates that the emergence of robust third-party pre-owned marketplace can
have countervailing effects on a seller’s sustainability incentives. On the
one hand, the emergence of a pre-owned marketplace where none existed in the
past has a strictly positive effect on the durability, raising it above the
baseline level of 0 in our model 555One might ask whether we are ignoring, in
the absence of a pre-owned marketplace, the possibility of the seller creating
a product with positive durability aimed at getting higher type consumers to
purchase the product and hold it for 2 periods. This option is strictly
dominated by the options of offering a product of durability 0 in both
periods, priced at $v_{H}$.. However, an increase in the power of a third-
party marketplace operator (leading to an increase in the commission rate )
lowers the seller’s incentives to create a more durable (and thus sustainable)
product.

We now show how these potentially countervailing incentives are all aligned
when the firm runs its own pre-owned marketplace.

###### Proposition 4.4

Relative to the case of a third-party pre-owned marketplace, under the case of
branded recommerce:

  1. (a)

Optimal durability is higher at any third-party commission rate $\beta$ and
quality uncertainty $\alpha$.

  2. (b)

The seller will choose a commission rate $\beta=0.$

  3. (c)

The incentives of a seller to invest in reducing quality uncertainty $\alpha$
are higher.

Proposition 4.4 establishes a main result — that branded recommerce has a
direct effect of inducing higher durability, or put differently, increases the
incentives of a firm that makes something to design it to last longer. It also
proves a set of indirect economic effects which benefit manufacturers and
increase sustainability. Part (b) shows that it is optimal for manufacturers
to forgo marketplace fees altogether. Counterintuitively, therefore, the
direct revenue potential of branded recommerce is not its equilibrium source
of value. Part (c) shows that there are greater seller incentives to invest in
lowering the quality uncertainty associated with pre-owned items. The
“branding” of recommerce by itself may reduce quality uncertainty — after all,
a customer is buying the pre-owned item directly from the manufacturer rather
than from an unknown individual. However, a key implication of part (a) of the
proposition is that the welfare gains of branded recommerce from an increase
in durability do not have to assume any change in perceived quality, and the
incentive to lower quality uncertainty is a separate positive economic effect.
In what follows, we examine whether the insights gained from our two-period
model carry over to a more general setting.

##  5 A model with overlapping generations

In this section, we develop a model of infinitely many periods in which a
fixed number of customers enter in each period, and each generation of
customers lives for two periods. We continue to assume that the firm sets a
level of durability $D$ that remains fixed across periods. In each period $t$,
the seller chooses its price $p^{n}(t)$, bearing a constant marginal cost of
production $c(D)$ that is positive, increasing and strictly convex for all
$D>0$. Products last two periods. In each period $t$, new units of output are
of a fixed quality level that is normalized to 1 , and pre-owned units (those
produced in period t-1) are of quality $s(D)$, where, as before,
$s(0)=0,s(D)<1,s^{\prime}(D)>0,s^{\prime\prime}(D)<0$ for all $D\geq 0$. The
price of pre-owned items in period $\mathrm{t}$ is $p^{u}(t)$.

As before, the fraction of type-H (L) customers that enter in each period is
$n_{H}\left(n_{L}\right)$, where $n_{H}+n_{L}=1$. Type-H customers derive a
gross benefit per period of $v_{H}s$ from consuming a good of perceived
quality $s$, while the corresponding gross benefit derived by type-L customers
is $v_{L}s$, and $v_{H}>v_{L}>0$. In any ”current” period, the customers who
entered in the previous period are referred to as ”age-2” customers, and those
customers who enter in the current period are referred to as ”age-1”
customers. Customers and sellers discount the future at the rate $\delta$.

The payoff-relevant state variable $x(t)$ for period $t$ is the fraction of
age-2 customers who bought the new product in the previous period $t-1$. We
focus on deterministic consumer and seller choices. As before, the seller
cannot observe customer types, which means that if type-L customers purchase
in any period, then so do type-H customers. Thus, the possible states are
$\mathrm{x}(\mathrm{t})\in\left\\{0,n_{H},1\right\\}$.

An Illustration of the overlapping generation model It illustrates the
overlapping of generations in each period. In period t, the prior generation
that was born in period (t — 1) is of age 2, and the new generation that is
born in period t is of age 1. In every period, the seller thus faces both
“legacy” customers and “fresh” customers.

Following the standard Blackwell/Bellman approach, we look for a stationary
policy which maps each state $\left\\{0,n_{H},1\right\\}$ to a period price
$p^{n}(t)$ for the new good. We begin with the case of $\mathrm{D}=0$. The
state will always be $\mathrm{x}=0$, and the seller’s choices here are
therefore to price the product at either $v_{L}$ or $v_{H}$. Thus,the optimal
steady-state policy will be to price at $v_{L}$ if $v_{L}>n_{H}v_{H}$ and at
$v_{H}$ otherwise. Now consider $\mathrm{D}>0$. In each period apart from the
first period, the seller faces a setting somewhat similar to what was faced in
period 2 of the two-period model. However, there are, essentially, four types
of customers rather than two - age-1 type-L, age-1 type-H, age-2 type-L and
age-2 type-H.

The set of possible choices that each type-age pair has available to them are
summarized below.

Table 1: Set of possible choices when $x=n_{H}$ | Type-L | Type-H
---|---|---
Age 2 | {buy new, buy used, do nothing} | {sell used+buy new, buy new, keep used}
Age 1 | {buy new, buy used, do nothing} | {buy new, buy used, do nothing}
Table 2: Set of possible choices when $x=0$ | Type-L | Type-H
---|---|---
Age 2 | {buy new, buy used, do nothing} | {buy new, buy used, do nothing}
Age 1 | {buy new, buy used, do nothing} | {buy new, buy used, do nothing}
Table 3: Set of possible choices when $x=1$ | Type-L | Type-H
---|---|---
Age 2 | {sell used+buy new, buy new, keep used} | {sell used+buy new, buy new, keep used}
Age 1 | {sell used+buy new, buy new, keep used} | {sell used+buy new, buy new, keep used}

We next characterize the properties of an optimal steady-state policy. Under
such a policy, the seller prices the new goods at the same level in each
period, and this price leads to choices by the customers that maintain the
same state.

###### Lemma 5.1

The only state that can be part of a steady-state policy is one in which all
customers with higher willingness to pay (Type-H) purchase in each period,
that is, $x=n_{H}$

We present the proof in the main body of the paper since it is simple and aids
in understanding what follows. Clearly, a steady-state in which $x=0$ is
suboptimal for the seller, since it yields no profits. Consider next a steady
state in which $x=1$. Referring to Table 3 , it is clear that both type-L and
type-H customers of age-1 must choose to buy new, since the state must remain
at $x=1$. However, if this is in fact a steady-state, then both type-L and
type-H customers of age-2 already own a pre-owned item. As a consequence,
there are no potential buyers for the pre-owned items in any period. The
profit-maximizing price that sustains this steady-state is $v_{L}(1+\delta
s(D))$ which yields a demand level of 1 in each period. Following an argument
similar to one made in Section 4, the profit stream from this price is
strictly lower than the profit stream from a choice of $D=0$ and a price
$v_{L}$ which yields a demand level of 2 in each period. This completes the
proof.

Next, let us consider the set of consumer actions that are consistent with a
steady-state policy. We show in our next result that under any steady-state
policy in which there is any kind of pre-owned market activity, customers with
lower willingness to pay will always buy pre-owned items while the choices of
customers with higher willingness to pay depends on their decision horizon.

###### Lemma 5.2

Under any steady-state policy

  * •

All customers with lower willingness-to-pay (type-L customer) will purchase a
pre-owned item that lasts one period.

  * •

Customers with a higher willingess-to-pay who have just entered the market
(Type-H, age 1) will buy a new product

  * •

Customers with a higher willingess-to-pay who were in the market in the
previous period (Type-H, age 2) will sell their existing product and buy a new
product

Again, rather than relegating this proof to the Appendix, we present it in the
main body of the paper because it aids in understanding the implications of
the result.

First, to maintain the steady-state $x=n_{H}$, it is clear that type-H
customers of age-1 must choose to buy new. The choices of type-H customers of
age 2 can be partitioned as follows:

  * •

If type-H customers of age-2 discard their existing product and buy new, the
steady-state then involves type-$L$ customers not participating, and type-H
customers consuming their product for one period. In this case, any durability
level $D>0$ is strictly less profitable for the seller than the choice of
$D=0$.

  * •

If type-H customers of age 2 keep their existing product, then the steady-
state again involves type-L customers not participating, and type-H customers
consuming their product for both periods of their life. The profit-maximizing
price that sustains this steady-state is $v_{H}(1+\delta s(D))$ which yields a
demand level of $n_{H}$ per period. However, this is strictly less profitable
for the seller than choosing a durability $\mathrm{D}=0$, pricing at $v_{H}$,
and realizing a demand of $2n_{H}$ (from both all type-H customers).

  * •

If type-H customers of age 2 sell their existing product and buy a new
product, then one must characterize the choices of type-L customers (since
someone has to buy these products). Since the product lasts 2 periods, a pre-
owned product has a remaining life of 1 period. As a consequence, its value is
identical to both age 1 and age 2 customers of type-L.

    * –

The steady-state cannot be maintained if type-L customers of age 1 prefer to
buy the new product.

    * –

The steady-state can be maintained if type-L customers of age 2 prefer to buy
the new product at whatever price levels the new and pre-owned products are at
(since they are no longer in the economy after the current period, their
choices do not affect the state in the next period), while type-L customers of
age 1 prefer to purchase the pre-owned product. However, since type-L
customers of age 1 have an additional period of life and thus have a strictly
higher consumption value from the new product, which in turn means that the
set of actions (type-L customers of age 2 buy new, type-L customers of age 1
buy pre-owned) is not individually rational.

    * –

The steady state can also be maintained if type-L customers of both age 1 and
age 2 prefer to buy the used product over buying the new product. This is
possible if there are sufficient type-L customers, that is, if $2n_{L}>n_{H}$,
or if $n_{H}<2/3$

This completes the proof.

The choices implied by Lemma 4 are summarized in Table 4

Table 4: Choices made by each type under any steady-state policy for which $D>0$ | Type-L | Type-H
---|---|---
Age 2 | buy used | sell used+buy new
Age 1 | buy used | buy new

Next, we characterize the prices $p^{n},p^{u}$ that are consistent with these
choices. We reintroduce the marketplace commission $\beta$ and quality
uncertainty $\alpha$ parameters defined in Section 2. A subset of the
conditions that define the consumer choices are summarized below.

$\displaystyle\text{[Type-H, Age 2]: }v_{H}-p^{n}+(1-\beta)p^{u}\geq
v_{H}s(D).$ (16) $\displaystyle\text{[Type-H, Age 1]:
}v_{H}-p^{n}+\delta\left(\max\left[v_{H}-p^{n}+(1-\beta)p^{u},v_{H}s(D)\right]\right)\geq
v_{H}s(D)-p^{u}.$ (17) $\displaystyle\text{[Type-L, both ages]: }\alpha
v_{L}s(D)-p^{u}\geq 0.$ (18) $\displaystyle\text{[Type-L, Age 1]: }\alpha
v_{L}s(D)-p^{u}\geq
v_{L}-p^{n}+\delta\left(\max\left[v_{L}-p^{n}+(1-\beta)p^{u},v_{L}s(D)\right]\right).$
(19) $\displaystyle\text{[Type-L, Age 2]: }\alpha v_{L}s(D)-p^{u}\geq
v_{L}-p^{n}.$ (20)

(16) and (17) are analogous to the IC conditions for the type-H customers.
Since $v_{H}>v_{L}>0$, (16), (17) and (18) also imply that type-H consumers of
both ages get non-negative surplus from their choice. (18) ensures that type-L
customers get non-negative surplus, and (19) and (20) are analogous to their
IC conditions. It is straightforward to see that if (19) is satisfied, (20)
can be ignored.

Our final lemma establish the prices consistent with the steady-state
described in Lemma 4 and summarized in Table 4.

###### Lemma 5.3

The prices of the new and pre-owned item consistent with a steady-state are
identical to those obtained in the two-period model:

$p^{u}=\alpha v_{L}s(D)$ (21) $p^{n}=\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))$
(22)

We are now in a position to contrast the outcomes under branded recommerce
with those obtained when there is a third-party marketplace. By Lemma 5.3,
under the steady-state policy, the firm’s per-period payoffs with a third-
party marketplace are:

$n_{H}\left[\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))-c(D)\right],$ (23)

and the firm’s per-period payoffs under branded recommerce after taking
quality undertainty and transaction fees into consideration are

$n_{H}\left[\alpha v_{L}s(D)+v_{H}(1-s(D))-c(D)\right].$ (24)

Notice the close parallel between these expressions and those derived in the
two-period model. Happily, we can now establish that under the optimal steady-
state policy, a set of results in Proposition 5.4 that parallel those
established for the two-period model continue to hold.

###### Proposition 5.4

In a model with overlapping generations, under the unique optimal steady-state
policy:

  1. (a)

Independent of the ownership structure of the third-party marketplace, as
quality uncertainty goes down (and thus, $\alpha$ increases), the optimal
level of durability $D^{*}$ chosen by the seller increases, and the seller
profit increases.

  2. (b)

Independent of the ownership structure of the third-party marketplace, as the
commission $\beta$ charged by the operator of the pre-owned marketplace
increases, the optimal level of durability $D^{*}$ chosen by the seller
decreases, and the seller profit decreases.

  3. (c)

Under branded recommerce: (i) the durability level chosen by the seller is
higher, (ii) the seller will choose a commission rate $\beta=0$ , and the
incentives of a seller to invest in reducing quality uncertainty are higher.

Proposition 3 establishes that the main results of Proposition 2 – that
choosing _branded recommerce_ has the direct effect of inducing the
manufacturer of a product to make more durable products – is not a modeling
artifact of a two-period setting, but generalizes to an infinite horizon
setting with overlapping generations Broadly, this result underscores what
could be a key societal benefit of branded recommerce. Granted, sellers may be
drawn to branded recommerce by the prospect of greater control of the customer
experience for those buyers who begin their brand interaction by purchasing
pre-owned items, or by the promise of profits from secondary market
transactions. Our result establishes that there is a path to greater profits
from sacrificing the latter, and that in addition to whatever firm-specific
benefits that may accrue from branded recommerce, it shifts incentives in a
manner that makes it more desirable for a seller to make goods that last
longer, rather than resorting to the less sustainable path of more perishable
durables in pursuit of repeat sales.

##  6 Conclusion

The promise of digitally-enabled branded recommerce is clear to a brand —
running one’s own pre-owned marketplace allows value capture from the trade of
one’s used items, signals support for circular consumption and allows stronger
brand experience control for the growing subset of consumers who favor buying
pre-owned items. Brands, however, may initially balk at the prospect of
placing pre-owned items aside new products on their ecommerce sites, fearing
profit reductions from cannibalization.

Our analysis presents reassuring news: that branded recommerce has a number of
positive effects on both durability and profitability. Interestingly, we find
that a brand’s focus should not be on the value capture promise of branded
recommerce — in our stylized setting, a seller finds it optimal to forgo
marketplace fees altogether, suggesting that direct revenue impacts of branded
recommerce should not be seen as their primary economic benefit. Rather, we
uncover more subtle benefits that stem from the ability to set higher prices,
the ability to profitably raise product durability, and greater seller
incentives to lower quality uncertainty associated with pre-owned items.

While formal empirical validation of our theoretical results remains in the
future, it has been reassuring to observe anecdotal evidence of a relationship
between the adoption of branded recommerce and investments into raising
product durability. For example, Patagonia, which pioneered its ”Worn Wear”
effort to sell pre-owned items in stores in 2013, and was of the early
adopters of branded recommerce in 2017, has since made significant efforts to
increase the durability and reusability of the products it manufactures (Woody
(2024)).

Our future work aims to extend our model in a few directions. We are working
on an extension that considers the potential simultaneous presence of branded
recommerce and a third-party marketplace. Our preliminary findings suggest
that under certain conditions, the threat of a third-party marketplace can
cause a firm to introduce branded recommerce in settings under which it would
not have customarily encouraged the trade of pre-owned items – this move can
deter the entry of a third-party marketplace that would have a more adverse
effect on seller outcomes. Our model extension in which the two pre-owned
marketplace options are differentiated by both quality uncertainty and product
availability is in progress, but is one we hope will take us closer to having
richer testable results.

We are also exploring extensions to the baseline model that add additional
details that reflect what we have observed in practice. For example, in the
implementation of recommerce, there may be differing levels of search
frictions (Gavazza and Lizzeri (2021), Rhodes et al. (2021)) associated with
the pre-owned item when compared to new items, depending on whether the pre-
owned item is listed alongside the new item (like it is on Amazon), in a
separate tab on the brand’s primary e-commerce site (like it is on Lululemon),
or on an entirely different web site. While we currently model this as being
subsumed by the reduction in perceived value, a more detailed model could shed
new insight and lead to testable hypotheses, given the variations in these
frictions we have observed across recommerce sites. The choices on these front
may also have quality signaling implications (Nian and Sundararajan (2022),
Ananthakrishnan et al. (2023)), as firms that highlight their recommerce sites
more visibly could be perceived as standing more firmly behind the quality of
their durable goods. There are often unexpected and subtle economic effects
from changes to market participant visibility (Bapna et al. (2016)). Many of
these will eventually be empirical questions, and we have partnered with a
leading recommerce platform to test some of our current and ongoing theory
results.

While research into digital secondary markets has been widespread in
economics, marketing and IS over the last two decades, the phenomenon of
branded recommerce is new, and, powered by new recommerce platforms, will
account for tens of billions of dollars of ecommerce in the coming years.
Branded recommerce is the most significant new digital “channel,” as important
today as ecommerce was two decades ago. Our paper is the first we are aware of
that sheds new insight into this important new digital development,
underscoring its importance in aligning seller profits with sustainability
goals.

We thank Anindya Ghose, Silin Huang, Xiao Liu, Vandith Pemuru, Andy Ruben,
Joao Sedoc, Zhengyuan Zhou and seminar participants at New York University,
the 2024 International Industrial Organization Conference, and the 2023
Marketing Science, ICIS and WISE conferences for helpful comments and
discussion.

Technical Details and Proofs

Proof of Lemma 1. Recall that in the active pre-owned marketplace case, type-H
customers purchase new goods in period 1 and type-L customers purchase the
pre-owned goods from type-H customers via the third-party marketplace in
period 2. We derive the optimal level of durability and the optimal prices of
new goods in each period under this setting.

Consider any durability choice D. From (9), the corresponding price of new
goods in the second period is:

$p_{2}^{n}=\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D)).$

The firm’s profits in second period are therefore

$n_{H}\left[\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))-c(D)\right].$

If the firm and customers anticipate the second period, since the type-L
customers are not active in the first period, the maximum price the firm can
charge for new units in the first period is the value realized from the new
good in the first period added to the discounted price the consumer will
receive from selling the good in the pre-owned marketplace in period 2, or

$p_{1}^{n}=v_{H}+\delta\alpha(1-\beta)v_{L}s(D),$

and the firm’s profits from these first period sales are therefore

$n_{H}[v_{H}+\delta\alpha(1-\beta)v_{L}s(D)],$

Thus, the firm’s ex ante total expected profits at the time they choose $D$
is:

$\pi(D)=n_{H}\left[v_{H}+\delta\alpha(1-\beta)v_{L}s(D)-c(D)\right]+\delta
n_{H}\left[\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))-c(D)\right]$ (A1)

The firm’s optimal durability choice $D^{*}$ maximizes (A1). The first-order
condition yields:

$c^{\prime}\left(D^{*}\right)=\frac{\delta}{1+\delta}\left[2\alpha(1-\beta)v_{L}-v_{H}\right]s^{\prime}\left(D^{*}\right)$
(A2)

Now, since $v_{L}>v_{H}/2\alpha(1-\beta)$, it follows that
$2\alpha(1-\beta)v_{L}-v_{H}>0$, and since $c^{\prime}(0)=0$ and
$c^{\prime\prime}(D)>0$, there is a positive value of $D^{*}$ that satisfies
(A2). Comparing 2 and (A2), since $c^{\prime\prime}(D)>0$, it follows that
$D^{*}<D^{**}$ if

$\left[2\alpha(1-\beta)v_{L}-v_{H}\right]<v_{L}$

which simplifies to

$v_{H}>v_{L}[2\alpha(1-\beta)-1],$

which is true for all $0<\alpha<1,0<\beta<1$, since $v_{H}>v_{L}$. (Q.E.D.)

Proof of Lemma 2. This proof parallels that of Lemma 1. Following the same
logic, the first and second period price for new goods are, respectively

$\displaystyle p_{1}^{n}=v_{H}+\delta\alpha(1-\beta)v_{L}s(D)$ $\displaystyle
p_{2}^{n}=\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))$

As in Lemma 1, the firm’s first period profits are
$n_{H}\left[v_{H}+\delta\alpha(1-\beta)v_{L}s(D)-c(D)\right]$. However, in the
second period, as the operator of the pre-owned marketplace, the firm also
receives marketplace commissions $n_{H}\beta p_{2}{}^{u}$. (Note that only a
fraction $n_{H}/n_{L}$ of type-L customers end up purchasing the pre-owned
product owing to this being the total supply.) Recall from (7) that
$p_{2}{}^{u}=\alpha v_{L}s(D)$. The firm’s second period profits are
therefore:

$\pi=n_{H}\left[\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))-c(D)\right]+\alpha\beta
v_{L}s(D),$

which simplifies to:

$\pi=n_{H}\left[\alpha v_{L}s(D)+v_{H}(1-s(D))-c(D)\right].$

Thus, the firm’s ex ante expected profits in first period when choosing
durability D are

$\pi=n_{H}\left[\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))-c(D)\right]+\alpha\beta
v_{L}s(D),$

which simplifies to:

$\pi=n_{H}\left[\alpha v_{L}s(D)+v_{H}(1-s(D))-c(D)\right].$

Thus, the firm’s ex ante expected profits in first period when choosing
durability $D$ are

$\pi(D)=n_{H}\left[v_{H}+\delta\alpha(1-\beta)s(D)v_{L}-c(D)\right]+\delta
n_{H}\left[\alpha v_{L}s(D)+v_{H}(1-s(D))-c(D)\right].$ (A3)

The firm’s optimal durability choice $D^{*}$ therefore maximizes (A3). The
first-order condition yields:

$c^{\prime}\left(D^{*}\right)=\frac{\delta}{1+\delta}\left[\alpha(2-\beta)v_{L}-v_{H}\right]s^{\prime}\left(D^{*}\right).$
(A4)

Now, since $v_{L}>v_{H}/\alpha(2-\beta)$, it follows that
$\alpha(2-\beta)v_{L}-v_{H}>0$, and thus, there is a positive value of $D^{*}$
that satisfies (A4). As before, comparing (2) and (A4), since
$c^{\prime\prime}(D)>0,D^{*}<D^{**}$ if

$\left[\alpha(2-\beta)v_{L}-v_{H}\right]<v_{L},$

which simplifies to

$v_{H}>v_{L}(\alpha(2-\beta)-1),$

which is true for all $0<\alpha<1,0<\beta<1$ since $v_{H}>v_{L}$. The result
follows. (Q.E.D.)

Proof of Proposition 1.

  1. (a)

By the envelope theorem, and the profit expressions in (A1), it follows that
in the third-party marketplace case:

$\frac{d\pi\left(D^{*}(\alpha)\right)}{d\alpha}=n_{H}\delta\left[(2-2\beta)v_{L}s\left(D^{\prime}\right)\right]$

which is strictly positive for $\beta<1$. Also, rearranging (A2) yields

$\frac{c^{\prime}\left(D^{*}\right)}{s^{\prime}\left(D^{*}\right)}=\frac{\delta}{1+\delta}\left[2\alpha(1-\beta)v_{L}-v_{H}\right]$
(A5)

Since $c(D)$ is strictly convex and $s(D)$ is strictly concave, it follows
that $\frac{c^{\prime}(D)}{s^{\prime}(D)}$ is strictly increasing in $D$ for
all $D>0$. Notice that as $\alpha$ increases, the right-hand side of A5 is
higher, thus requires a higher value of
$\frac{c^{\prime}\left(D^{*}\right)}{s^{\prime}\left(D^{*}\right)}$ on the
left-hand side, and consequently, a higher value of $D^{*}$.

Similarly, by the envelope theorem, and the profit expressions in (A1), it
follows that in the branded recommerce case:

$\frac{d\pi\left(D^{*}(\alpha)\right)}{d\alpha}=n_{H}\delta\left[(2-\beta)v_{L}s\left(D^{\prime}\right)\right],$

which is also strictly positive for $\beta\leq 1$. Correspondingly,
rearranging (A4) yields

$\frac{c^{\prime}\left(D^{*}\right)}{s^{\prime}\left(D^{*}\right)}=\frac{\delta}{1+\delta}\left[\alpha(2-\beta)v_{L}-v_{H}\right]$
(A6)

and based on the same logic as above, a higher value of $\alpha$ implies a
higher value of $D^{*}$. The result follows.

  2. (b)

Inspecting (A5) and (A6) above establishes that a higher value of $\beta$
induces a lower value of $\mathrm{D}^{*}$, by the same argument involving
$\frac{c^{\prime}(D)}{s^{\prime}(D)}$ being strictly increasing in D. Applying
the envelope theorem to the profit expressions in (A1) and (A3) yields:

$\frac{d\pi\left(D^{*}(\beta)\right)}{d\beta}=-2n_{H}\delta\alpha
v_{L}s\left(D^{\prime}\right)<0$

in the case of the third-party marketplace, and

$\frac{d\pi\left(D^{*}(\beta)\right)}{d\beta}=-n_{H}\delta\alpha
v_{L}s\left(D^{\prime}\right)<0$ (A7)

(Q.E.D.)
Proof of Proposition 2.

  1. (a)

Recall from (A5) and (A6), optimal durability $D^{*}$ satisfies

$\frac{c^{\prime}\left(D^{*}\right)}{s^{\prime}\left(D^{*}\right)}=\frac{\delta}{1+\delta}\left[2\alpha(1-\beta)v_{L}-v_{H}\right],$

with a third-party marketplace and

$\frac{c^{\prime}\left(D^{*}\right)}{s^{\prime}\left(D^{*}\right)}=\frac{\delta}{1+\delta}\left[\alpha(2-\beta)v_{L}-v_{H}\right],$

with branded recommerce. Comparing the right-hand sides of (A5) and (A6), and
using the fact that $2\alpha(1-\beta)<\alpha(2-\beta)$ if $\alpha>0,\beta>0$,
it follows that:

$\left[\alpha(2-\beta)v_{L}-v_{H}\right]>\left[2\alpha(1-\beta)v_{L}-v_{H}\right].$

Since $\frac{c^{\prime}(D)}{s^{\prime}(D)}$ is strictly increasing in $D$ as
established in the proof of Proposition 1, the durability level that satisfies
(A5) must be higher than the durability level that satisfies (A6). The result
follows.

  2. (b)

Recall from (A7) that by the envelope theorem,

$\frac{d\pi\left(D^{*}(\beta)\right)}{d\beta}=-n_{H}\delta\alpha
v_{L}s\left(D^{\prime}\right)<0.$

Under branded recommerce, the seller can choose $\beta$, and will thus choose
the lowest feasible level $\beta=0$. The result follows.

  3. (c)

Compare the expressions for $\frac{d\pi\left(D^{*}(\alpha)\right)}{d\alpha}$
in (A5) and (A6). Since $\beta\geq 0$, it follows that

$n_{H}\delta\left[(2-\beta)v_{L}s\left(D^{\prime}\right)\geq
n_{H}\delta\left[(2-2\beta)v_{L}s\left(D^{\prime}\right)\right]\right.$ (A8)

However, part (b) establishes that under branded recommerce, $\beta=0$, and
thus the inequality in (A8) is strict for any third-party marketplace with a
non-zero commission. Thus, the seller gets a larger increase in profits under
branded recommerce from any fixed increase in $\alpha$ or equivalently, any
fixed reduction in quality uncertainty, than with a third-party marketplace.
The result follows.

Proof of Lemma 5.

Note that (16) implies that:

$\left.\left.\max\left[v_{H}-p^{n}+1-\beta\right)p^{u},v_{H}s(D)\right]=v_{H}-p^{n}+1-\beta\right)p^{u}.$
(25)

Rearranging (16) yields

$[1-s(D)]v_{H}+(1-\beta)p^{u}\geq p^{n}$ (26)

We can use (25) to rewrite (17) as

$v_{H}-p^{n}+\delta\left[v_{H}-p^{n}+(1-\beta)p^{u}\right]\geq\alpha
v_{H}s(D)-p^{u}$ (27)

Rearranging (27) yields

$\left[1-\frac{\alpha}{1+\delta}s(D)\right]v_{H}+\left(1-\frac{\delta}{1+\delta}\beta\right)p^{u}\geq
p^{n}$ (28)

Since $\frac{\alpha}{1+\delta}<1$ and $\frac{\delta}{1+\delta}<1$, if (26) is
satisfied, (28) can be ignored. It follows that (16) implies (17).

From (18), since there are more buyers than sellers in the pre-owned
marketplace, it follows that

$p^{u}=\alpha v_{L}s(D)$ (29)

(29) and (16) imply that the highest price the seller can charge for new goods
is

$p^{n}=\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))$ (30)

Note that these are identical to the second-period prices obtained in the two-
period model.

We can use (29) and (30) to derive that

$\max\left[v_{L}-p^{n}+(1-\beta)p^{u},v_{L}s(D)\right]=v_{L}s(D)$ (31)

We can use (29), (30), (31) together to simplify (19) as:

$\frac{v_{L}}{v_{H}}\leq\frac{1-s(D)}{1-[(1-\beta)\alpha-\delta]s(D)}$ (32)

If (32) is true, we can establish that (19) is satisfied and (20) is thus
implied. This completes the proof. (Q.E.D.)

Proof of Proposition 3. Imagine the optimization problem faced by the seller
when choosing their optimal durability level D. Let the first-period price be
g1(D). Define

$G_{T}(D)=\sum_{t=1}^{\infty}\delta^{t}n_{H}\left[\alpha(1-\beta)v_{L}s(D)+v_{H}(1-s(D))-c(D)\right]$
(33)

and

$G_{B}(D)=\sum_{t=1}^{\infty}\delta^{t}n_{H}\left[\alpha
v_{L}s(D)+v_{H}(1-s(D))-c(D)\right].$ (34)

These are the steady-state profit streams as viewed from period 1 , under the
third-party pre-owned marketplace $(\mathrm{T})$ and branded recommerce (B)
respectively. The seller therefore chooses $\mathrm{D}$ to maximize
$n_{H}g_{1}(D)+G_{T}(D)$ with a third-party marketplace, and
$n_{H}g_{1}(D)+G_{B}(D)$ with branded recommerce. Under the parameter
restrictions that ensure the seller will not want to shut out the type-L
customers, $G_{B}{}^{\prime}(D)>G_{T}{}^{\prime}(D)$ and
$G_{B}{}^{\prime\prime}(D)<G_{T}{}^{\prime\prime}(D)$, which is sufficient to
establish that durability is higher under branded recommerce. Standard
comparative statics methods establish the rest of the results. (Q.E.D.)

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