Should an Online Retailer Penalize its Independent

Sellers for Stockout?

Wenqiang Xiao Yi Xu

Stern School of Business Smith School of Business

New York University University of Maryland

January 8, 2018

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1 Introduction and Literature

Online retailers need to constantly expand their product o®erings into new categories to

grow their businesses. Such product and category expansions can be highly challenging

for online retailers because of their limited experience in the new categories. For example,

Amazon's initial expansion into toys by itself in 1999 was unsuccessful. Amazon even had

trouble in identifying hot-selling toys (Goldman 2000). Thus, many online retailers choose

to partner with or allow third-party sellers, who already have extensive experience and

knowledge in the new categories, to sell products on their websites or online platforms.

In August 2000, unsatis¯ed with its performance in toys, Amazon.com decided to form

a 10-year exclusive partnership with Toys\R"Us to create a co-branded online toy store.

Amazon was responsible for site development, order processing and customer service, and

relied on Toys\R"Us' expertise to \identify, buy and manage inventory" (Amazon.com

news release 2000). However, Amazon soon found that Toys\R"Us could not keep up with

the sales volume, and its performance in ful¯llment and delivery was deteriorating. The

once promising partnership terminated in 2004 after a series of lawsuits between the two

¯rms in which Amazon mainly alleged that its business had been harmed by Toys\R"Us'

failure to maintain su±cient stock which resulted in stockout on more than 20% of its

most-popular products, and seek more than $750m in damages (Claburn 2004).

The quick rise and fall of the partnership between Amazon and Toys\R"Us reveals

an important yet challenging task for online retailers: How to incentivize a independent

third-party seller to install proper inventory/capacity to achieve the desired service level for

order ful¯llment? This is challenging for several reasons as the Amazon and Toys"R"Us

partnership illustrated. First, it is infeasible to enforce a third-party seller to install a

certain level of capacity, because the online retailer often lacks an e®ective monitoring

system that can accurately assess and verify the seller's capacity level. Second, relative to

the seller, the online retailer may have less information about the demand for the products

that the seller sells (e.g., toys sold by Toy\R"Us) (Jiang et al. 2011), and consequently, the

online retailer is not in the ideal position to determine what should be the right capacity

level for a product sold by a third party seller.

To address this issue, many online retailers have started to set speci¯c targets on order

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ful¯lment and other performance measures for third-party sellers, and to penalize third-

party sellers who fail to meet the targets. Amazon currently tracks three seller performance

measures: pre-ful¯lment cancel rate (which closely relates to inventory and ¯ll rate),

order defect rate, and late shipment rate. The targets Amazon sets on these performance

measures are: pre-ful¯lment cancel rate < 2:5%, order defect rate < 1%, and late shipment

rate < 4%. Amazon clearly indicates that it may terminate the selling privileges of sellers

who fail to meet these targets (Amazon Seller Performance Measurement, 2017). Staples,

Inc. publishes a third-party seller guideline with speci¯c targets or standards on detailed

performance measures such as inventory, ¯ll rate, data interchange, among others (see

Staples Vendor Guidelines, 2015). A seller's failure to meet any standards or targets

speci¯ed in the guideline will result in a ¯nancial penalty called Customer Service Charge.

Particularly, Staples strictly requires all sellers to maintain a ¯ll rate of 100%. In other

words, any stockout by a seller will not be tolerated and will result in a ¯nancial penalty

charged by Staples. In this paper, we will provide theoretical support for the e®ectiveness

of the simple and implementable lost-sale penalty-based contracts for online retailers to

incentivize their third-party sellers' capacity installations.

We consider an online retailer-third-party seller relationship over a single sales season.

We capture the two important features of the relationship as we discussed above: the

seller privately determines the capacity level that is not contractible; the seller possesses

better demand information than the online retailer. We ¯rst study the commonly-used

commission contracts under which the seller pays the online retailer a commission for

every unit sold via her online platform together with a ¯xed fee. Our analysis reveals

the con°icting roles of the commission in motivating the seller to install capacity and

in extracting surplus from the seller with private demand information. On one hand,

to motivate the seller to install desired level of capacity, the retailer needs to charge a

lower commission. On the other hand, to extract more surplus from the seller with higher

demand potential, the retailer needs to charge a higher commission. We ¯nd that this

tension between incentive provision for capacity installation and surplus extraction leads

to ine±ciency of the commission contracts.

To improve the contractual e±ciency, we then study the lost-sale penalty contracts

which have also appeared in the online retailer/seller relationships as illustrated in the

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Amazon and Staples examples discussed above. Under the lost-sale penalty contracts,

the seller needs to pay the online retailer not only a commission but also a penalty if a

stockout event occurs. We show that the optimal lost-sale penalty contracts can help the

online retailer to achieve the ¯rst-best pro¯t, i.e., inducing the seller to install the ¯rst-best

capacity and extracting all surplus from the seller.

Under a lost-sale penalty contract, both reducing the commission and increasing the

lost-sale penalty strengthen the seller's incentive to install capacity, because the former

acts like a carrot rewarding the seller for installing higher capacity and the latter acts

like a stick penalizing the seller for his insu±cient amount of capacity that more likely

results in lost-sales. This implies that for any given lost-sale penalty, the online retailer can

always properly adjust the commission so that these two incentive instruments together

can induce the seller to install the ¯rst-best capacity level. It remains to extract the

full surplus from the seller who has private demand information. This goal often fails in

many other contracts because the seller with optimistic demand can make more pro¯ts

than the seller with pessimistic demand under the contract intended for the low-type

seller. However, this is no longer true under the lost-sale penalty, because the seller with

optimistic demand is more likely to experience stockout and hence incur higher penalties in

expectation. Therefore, the online retailer can tailor the lost-sales penalty for every seller

type to fully extract the surplus. By adjusting the commission accordingly for each type,

the retailer can also provide the right amount of incentives for capacity installation. In

summary, combining carrot (commission) and stick (lost-sale penalty) can e®ectively allow

the online retailer to accomplish the two goals at once to achieve the ¯rst best outcome.

Our paper relates to the stream of literature that studies the strategic incentive issues

in the platform based Internet retailing. Jiang et al. (2011) consider a setting where a

third-party seller with superior demand information pays an online retailer a commission

for every unit sold via the online retailer's platform. They focus on the third-party seller's

service provision decision based on its private demand information, with the anticipation

of the online platform's strategic response in whether or not o®ering similar products to

compete with the seller. Abhishek et al. (2016) examine how the channel competition and

the competition among online platforms in°uence the online platforms' strategic choice of

the agency selling or the conventional reselling. Kwark et al. (2017) examines how the

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information about the product quality and the product ¯tness in°uences the online plat-

form's performance under the agency selling format and the reselling format, respectively.

We complement this line of research by providing theoretical support for the simple and

implementable stockout penalty-based contracts for the online platform in coping with the

third-party seller's service provision.

Our paper is also related to work on supply chain contracting with asymmetric infor-

mation. A majority of this literature studies variations of the adverse selection problem

with hidden information in which the party who accepts contracts (agent) has private or

better information on a relevant operational factor than the party who o®ers contracts

(principal). Asymmetric information on cost (e.g., Ha 2001, Corbett et al. 2004, Cachon

and Zhang 2006), quality (e.g., Lim 2001), market demand and forecasts (e.g., Li et al.

2008, Li and Zhang 2013), and demand-in°uencing e®orts (e.g., Plambeck and Taylor

2006, Kim et al. 2007) have been extensively studied. The main focus of this line of

research is to examine the e®ectiveness of various incentive contracts in mitigating the in-

e±ciencies caused by information asymmetry. Our paper di®ers from these papers in that

we study the platform based online retailing setting that is distinct from theirs and we

show that the simple and implementable penalty-based contracts can completely eliminate

the ine±ciencies arising from both hidden information and hidden action.

A stream of literature in economics has shown that contracting on a single ex-post

signal that is correlated with the agent's private information may enable the principal

to extract full surplus from the agent to achieve the ¯rst-best. Riordan and Sappington

(1988) establish conditions for an ex-post signal to fully extract the surplus from every

agent type for the principal. Bose and Zhao (2006) identify situations where such a full

surplus extraction is not possible, and study how to optimally use ex-post signals in the

principal's contract design under these situations. This stream of literature suggests that

it is possible to achieve the ¯rst-best result for the principal by properly designing payment

schemes that are contingent on some ex-post signals. Nevertheless, it remains to be an

open question as to what ex-post signals and how to use them to achieve the ¯rst-best

result for a speci¯c contractual relationship. In our online retailing setting, we show

that the sales, an ex-post signal containing information about the seller's private demand

knowledge, alone fails to achieve the ¯rst-best, whereas the sales information coupled with

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the lost-sale information can achieve the ¯rst-best result. Moreover, we show that the

lost-sale penalty contract achieves the ¯rst-best for the online retailer.

Interestingly, our core ¯nding that a simple and implementable lost-sale penalty con-

tract achieves the ¯rst-best result for the online platform owner is similar in spirit to the

recent stream of literature that investigates the e®ectiveness of simple and implementable

contracts in various supply chain settings. Dai et al. (2016) show that a combination of

simple and implementable contractual structures (i.e., a late rebate term and a buyback

term) can coordinate the in°uenza vaccine supply chain. Chen and Lee (2016) show that a

delivery-schedule-based contract with a targeted delivery date and a bonus/penalty term

contingent on the supplier's delivery date relative to the targeted date can coordinate a

project supply chain. Hwang et al. (2016) show that a unit-penalty (i.e., a penalty for

each unit of shortage) contract coordinates the supply chain with random yield on supply.

Our work di®ers from the above literature in that the online retailer in our context needs

to cope with both the hidden demand information and the hidden capacity decision of the

third-party seller.

The rest of the paper is organized as follows. We present our model in Section 2.

Section 3 analyzes the commission contracts, while Section 4 studies the lost-sale penalty

contract. Section 5 concludes the paper.

2 Model

We consider an online retailer (she) who allows an independent third-party seller (he)

to sell a product over a single selling season on her online platform. The seller possesses

private information about the product potential and installs capacity of the product before

the selling season to ful¯ll potential orders on the product. The product is sold at a ¯xed

retail price p. The ¯xed retail price assumption is reasonable for products sold by multiple

sellers such as many household products, whose prices are kept stable by competition,

or products sold directly by highly visible and popular brands such as Dyson vacuums,

which rarely has price discounts. The demand for the product in the selling season d is

determined by the product's demand potential µ and a random noise " in an additive form:

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d = µ + ". The demand potential µ re°ects the extent of product popularity perceived

by end consumers. Assume that µ has a ¯nite support on the interval [µ; µ] with density

function f( ) and distribution function F ( ), and " has a zero mean with density function

g( ) and distribution function G( ). While the distributions of µ and ", and the functional

form of demand d are common knowledge of both parties, only the seller perfectly observes

the realized value of µ before the selling season. This information asymmetry in the two

parties' knowledge about the demand potential µ may be due to the seller's expertise

in knowing the product attributes and in understanding how the product attributes are

perceived by the consumers. For example, Toys\R"Us (the independent seller) may have

a better idea about the demand potential of a toy selected by itself than Amazon (the

online retailer) does.

Before the selling season, the seller privately installs capacity k which will be used to

satisfy demand for the product during the selling season. Such capacity installation may

include investments in equipment, personnel training, and procurement of raw materials.

Capacity installation is costly to the seller, with the cost denoted by V (k). It is reasonable

to assume that V (k) is increasing and convex in k (i.e., V 0(k) 0 and V 00(k) 0).

We assume that the seller's installed capacity k is not observable to the retailer, and

thus not contractible. However, the e®ort cost function V is common knowledge of both

parties. The assumption that the seller's capacity is not contractible re°ects the constraint

commonly faced by online retailers selling products for independent third-party sellers.

In such business relationship, the online retailer provides online platform for displaying

product information and collecting orders from consumers, whereas the seller takes full

ownership of his capacity throughout the selling season and ful¯lls the orders up to his

capacity.

If the realized demand is less than the seller's capacity, i.e., d k, all demand will be

satis¯ed, and the leftover capacity k d would be salvaged at a value normalized to zero

without loss of generality. If the realized demand is more than the seller's capacity, i.e.,

d > k, the seller is only able to satisfy k units of demand, and the unsatis¯ed demand

would be lost. The retailer will collect revenue from all satis¯ed orders, and will then

make a transfer payment to the seller according to the contractual terms agreed.

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T h e o n l in e r e ta i l e r o ffe r s c o n t ra c t s

T h e s e l le r o b s e r v e s th e d e m a n d p o te n t ia l o f th e p ro d u c t

T h e s e l le r d e c id e s w h e th e r to p a r t ic ip a te ; I f y e s , h e c h o o s e s a c o n t ra c t

T h e s e l le r in s ta l l s c a p a c i ty, k

S e l l in g s e a s o n b e g in s . D e m a n d , d i s r e a l iz e d a n d fu l f i l l e d u s in g in s ta l le d c a p a c i ty , k

T h e t ra n s a c t io n s a r e m a d e a c c o r d in g to th e c o n tra c t c h o s e n

T im e

Figure 1: The Sequence of Events

The retailer moves ¯rst to o®er contractual terms to the seller to maximize her own

expected pro¯t. The seller will only accept the retailer's contractual terms if and only if his

expected pro¯t under these terms is no less than his reservation pro¯t which is normalized

to zero. The sequence of events is as follows (See Figure 1): (1) The seller observes µ; (2)

The retailer announces the contract (which can be a menu of contracts); (3) The seller

decides whether or not to participate; If he participates, then he chooses a contract if there

is a menu; If he rejects all the contracts, the game ends and each party gets zero pro¯t; (4)

After choosing a contract, the seller installs capacity k to maximize his expected pro¯t;

(4) The selling season starts; Demand d is realized, the ¯nal sales is s = min(d; k), and

transfer payments are made according to the chosen contract.

As a useful benchmark, we consider the centralized system with a single decision maker.

For any given demand potential µ and capacity k, the expected pro¯t of the centralized

system is pE"min(µ+"; k) V (k), where the ¯rst term represents the expected sales revenue

and the second term is the cost of capacity installation. Hence, for any given demand

potential µ, the ¯rst-best capacity, denoted by kI(µ), is kI(µ) = argmaxk pE"min(µ +

"; k) V (k) . It is straightforward to verify that the function inside the maximization is

strictly concave in k. Therefore, the ¯rst-best capacity kI(µ) is uniquely determined from

the following ¯rst-order condition:

pG(kI(µ) µ) V 0(kI(µ)) = 0: (1)

The retailer's ¯rst-best expected pro¯t, denoted by RI , is

RI = Eµ pE"min(µ + "; kI(µ)) V (kI(µ)) :

For analytical tractability, the following standard and mild assumptions are needed.

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(A1). The demand potential µ has an increasing failure rate (IFR), or equivalently, the

reciprocal of hazard rate H(µ) = F (µ)=f(µ) is decreasing in µ;

(A2). " has an increasing hazard rate and a decreasing reverse hazard rate, i.e.,

g(")=G(") is increasing in " and g(")=G(") is decreasing in ";

(A3). V 000( ) 0 and V 00( )=V 0( ) is a decreasing function.

Among these assumptions, (A1) and (A2) are widely adopted in the screening literature

and satis¯ed by many common distributions such as normal, uniform, exponential, gamma,

etc. (A3) is satis¯ed for a wide class of convex functions, e.g., V (k) = kn, where n 1.

3 Commission Contracts

Consider a menu of commission contracts under which the total transfer payment from

the seller to the online retailer is T (µ; s) = ®(µ)s+ t(µ), where s = min(d; k) is the realized

sales of the product, ®(µ) is the per unit commission rate, and t(µ) is the ¯xed lump-sum

fee over the sales season. It is convenient to denote a menu of commission contracts by

®( ); t( ) .

Under such a menu, the type-µ seller (i.e., whose product potential is µ)'s optimal

expected pro¯t by reporting µ, denoted by M(µ; µ), is

M(µ; µ) = maxk

[p ®(µ)]E"min(µ + "; k) t(µ) V (k) :

Let M(µ) M(µ; µ) and k(µ) be the type-µ seller's optimal capacity under the contract

®(µ); t(µ) . According to the Revelation Principle (e.g., Myerson 1983), we can without

loss of generality restrict to the class of truth-telling mechanisms under which it is the best

interest of the seller to truthfully report his product potential. Therefore, the retailer's

contract design problem is

max®(¢);t(¢)

Eµ[pE"min(µ + "; k(µ)) V (k(µ)) M(µ)]

s.t. M(µ) M(µ; µ); µ = µ; (IC)

M(µ) 0; µ [µ; µ]: (IR)

The ¯rst two terms inside the square brackets are the supply chain's total pro¯t. Sub-

tracting the seller's expected pro¯t under his truth-telling is the retailer's expected pro¯t.

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The (IC) constraint ensures that it is the best interest of the seller to truthfully report his

product potential and the (IR) constraint ensures that the seller earns at least his reser-

vation pro¯t so that he is willing to participate. The following proposition characterizes

the optimal solution to the above problem.

Proposition 1. The optimal commission contracts are

®cf(µ) = p V 0(kcf (µ))=G(kcf(µ) µ);

tcf(µ) = [p ®cf(µ)]E"min(µ + "; kcf (µ)) V (kcf(µ))µ

µ

[p ®cf(x)]G(kcf (x) x)dx

where kcf (µ) is uniquely determined by

pG(kcf (µ) µ) V 0(kcf(µ))

=g(kcf(µ) µ)

G2(kcf(µ) µ)

V 0(kcf(µ)) +G(kcf(µ) µ)

G(kcf(µ) µ)V 00(kcf (µ)) H(µ);

Under the optimal commission contracts, the type-µ seller chooses the contract ®cf (µ); tcf (µ)

and installs capacity kcf (µ), and his expected pro¯t is

M cf(µ)=µ

µ

[p ®cf (x)]G(kcf(x) x)dx.

The retailer's expected pro¯t, denoted by Rcf , is

Rcf = Eµ pE"min(µ + "; kcf (µ)) V (kcf (µ)) M cf(µ) :

Corollary 1. The optimal commission rate decreases in the seller's demand potential,

i.e., [®cf(µ)]0 0.

We make two observations from the analytical results in Proposition 1. First, the seller

under-invests in capacity relative to the ¯rst-best capacity level except the highest type,

i.e., kcf(µ) kI(µ), where the inequality binds only at µ = µ. Second, the retailer is unable

to extract the full surplus from the seller except the lowest type, i.e., M cf (µ) 0, where

the inequality is strict for µ > µ. These two observations together imply that neither

goal (i.e., incentivizing the seller to install the ¯rst-best capacity and extracting the full

surplus from the seller) is achieved under the optimal commission contracts, and therefore

the retailer earns strictly less than the ¯rst-best pro¯t, i.e., Rcf < RI .

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The ine±ciency of the commission contracts can be explained from the con°icting roles

of the commission in motivating the seller to install capacity and in extracting full surplus

from the seller. To motivate the seller to install the ¯rst-best capacity level, the retailer

needs to set the commission at zero. This leaves the lump-sum fee alone to di®erentiate

the seller's type. Because the seller regardless of his type always strictly prefers to pay the

retailer a lower lump-sum fee, it is in the best interest of the seller to choose the commission

contract with the lowest lump-sum fee when he is given a menu of commission contracts

with zero commission rate. Consequently, any menu of commission contracts with zero

commission rate essentially reduces to a single commission contract, under which a seller

with higher demand potential would earn strictly higher pro¯ts than a seller with lower

demand potential. Therefore, under the commission contracts, the goal of incentivizing

¯rst-best capacity installation con°icts with the goal of extracting full surplus from the

seller, implying that the commission contracts are unable to achieve the ¯rst-best pro¯t

for the retailer.

To see the magnitude of the e±ciency loss and how the e±ciency loss depends on various

factors, we conduct a numerical study. The model parameters are set as follows: p = 1;

V (k) = ck, where c = 0:1; 0:2; 0:3; :::; 0:9 ; " is uniformly distributed on the interval

[ "; "] and µ is uniformly distributed on the interval [1; µ], where " = 0:1; 0:2; 0:3; :::; 0:9

and µ = 1:1; 1:2; 1:3; :::; 1:9 . For each instance, we compute the percentage deviation

of the retailer's pro¯t under the optimal commission contracts and the ¯rst-best pro¯t,

i.e., ¢ = (RI Rcf )=RI . We ¯nd that among these 729 instances, the average value

of ¢ is 9.74% and the maximum value of ¢ is 25.76%. Our numerical study suggests

that signi¯cant improvements in the retailer's pro¯ts can be made if the retailer replaces

the commission contracts with contracts that can achieve the ¯rst-best result. Figure 2

illustrate how the e±ciency loss of the optimal commission contracts relative to the ¯rst

best pro¯t changes with di®erent model parameters.

As we can see in Figure 2, the e±ciency loss of the optimal commission contracts relative

to the ¯rst best pro¯t is low when the capacity cost c is either very low or very high. When

c is very low (high), both the ¯rst best capacity and the optimal capacity the seller would

install under the optimal commission contract would be very high (low). So, in both

cases, the impact of the distortions from the ¯rst best capacity under the commission

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Figure 2: The E±ciency Loss of the Optimal Commission Contracts Relative to the First

Best

contracts (i.e., the distortions) is relatively insigni¯cant. The e±ciency loss of the optimal

commission contracts increases in " which can be viewed as a measure of the degree of

demand uncertainty. The higher the ", the higher the demand uncertainty is, thereby the

higher the distortion in capacity. The e±ciency loss of the optimal commission contracts

also increases in µ which can be considered as a measure of the degree of information

asymmetry. As µ increases, the seller earns higher information rents and distorts more on

the capacity. Therefore, the commission contract would perform well as relative to the

¯rst best when the cost of capacity is extreme, the demand uncertainty is low and the

degree of information asymmetry is mild.

Can the retailer possibly achieve the ¯rst-best pro¯ts under such a setting where the

seller with private demand information installs capacity that is not contractible? If so,

how? We address these questions in the subsequent section.

4 Lost-Sale Penalty Contracts

We now consider lost-sale penalty contracts. There are three components in a lost-sale

penalty contract: a ¯xed fee, a commission rate and a lost-sale penalty which will be

charged to the seller if a lost sale or stockout occurs. Let l 0; 1 be an indicator on

the occurrence of lost-sale, i.e., l = 1 if lost-sale occurs, and l = 0; otherwise. So, l is

stochastically increasing in µ and decreasing in k. Consider a menu of lost-sale penalty

contracts ®(µ); °(µ); t(µ) based on the realized sales s = min(d; k) and the indicator of

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lost-sale event l. The transfer payment the seller would pay to the retailer is T (µ; s; l) =

®(µ)s+°(µ)l+t(µ), where ®(µ) is the commission rate, °(µ) is the lost-sale penalty, and t(µ)

is the lump-sum payment. Under the menu of lost-sale penalty contracts ®(µ); °(µ); t(µ) ,

if the seller of type µ falsely reports his demand potential as µ by choosing the contract

®(µ); °(µ); t(µ) , his maximum expected pro¯t is

M(µ; µ) = maxk

[p ®(µ)]E"min(µ + "; k) °(µ)Pr(µ + " > k) t(µ) V (k) ;

which consists of the sales revenue allocated to the seller less the expected lost-sales

penalty, the lump-sum payment, and the cost of capacity installation. Consequently,

the retailer's optimal lost-sale penalty contracts is the solution to the following problem:

max®(¢);°(¢);t(¢)

Eµ [pE"min(µ + "; k(µ)) V (k(µ)) M(µ)]

s.t. M(µ) M(µ; µ); µ = µ; (IC)

M(µ) 0; µ [µ; µ]: (IR)

The following proposition characterizes the optimal lost-sale penalty contracts.

Proposition 2. The optimal lost-sale penalty contracts are

®¤(µ) = pG(kI(µ) µ);

°¤(µ) = [p ®¤(µ)]G(kI(µ) µ)=g(kI(µ) µ);

t¤(µ) = [p ®¤(µ)]E"min(µ + "; kI(µ)) °¤(µ)G(kI(µ) µ) V (kI(µ));

which achieve the ¯rst-best pro¯t RI for the retailer.

Figure 3 provides a numerical illustration of the optimal lost-sale penalty contracts for

µ [1; 2]. The model parameters used in the numerical illustration in Figure 3 are set as

follows: p = 1, " follows a uniform distribution with support on [ 0:5; 0:5], cost function

of capacity V (k) = 0:5k2=2.

Corollary 2. The optimal commission rate decreases in the seller's demand potential,

i.e., [®¤(µ)]0 0.

Proposition 2 not only provides a de¯nitive answer to the question of whether or not the

¯rst-best pro¯t can be achieved, but also shows that the ¯rst-best pro¯t can be achieved

by the simple contracts that have been used in practice. In particular, the only structural

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Figure 3: The optimal lost-sale penalty contracts

change that the retailer needs to make to the commission contracts is to penalize the

seller if the lost-sale event occurs. In the platform based online retailing partnerships, order

information is normally recorded and shared through computer-based information systems,

which makes it possible to observe and verify the occurrence of lost-sale. For example,

Amazon can accurately calculate and track the three seller performance measures for all

sellers. Among the three seller performance measures, the pre-ful¯llment cancel rate is

directly related to lost-sale or stockout. According to Amazon, \the pre-ful¯llment cancel

rate is the number of orders cancelled by a seller prior to ship-con¯rmation, divided by

the number of orders in the time period of interest." (Amazon Customer Metrics, 2017)

Amazon suggests that \It is very important that when an order comes that the item is

in stock and available to ship. Pre-ful¯llment order cancellations that are not in response

to buyer requests can point to areas of improvement in your inventory management."

(Amazon Customer Metrics, 2017) Amazon requires sellers to maintain a pre-ful¯llment

cancel rate of less than 2.5%, and will penalize sellers who fail to meet the targets by

terminating their selling privileges on Amazon.com, which can be essentially viewed as a

lump sum lost-sale penalty.

To see the intuition of why the commission coupled with the lost-sale penalty can

achieve the ¯rst-best pro¯t for the retailer, we examine how these two instruments work

together to achieve the two goals at the same time. The ¯rst is to ensure that a higher

type seller does not gain any surplus by choosing the contract intended for a lower type,

i.e., the full surplus extraction. The second is to ensure that every type of seller under

his intended contract is incentivized to install the ¯rst-best capacity level. Note that if

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a higher type seller chooses the contract intended for a lower type seller, the higher type

would earn higher revenue in expectation than the lower type under the same contract due

to his higher demand potential. This is the key reason of why the menu of commission

contracts is unable to fully extract surplus from the seller. However, with the presence of

lost-sale penalty, by mimicking the lower type, the higher type seller would incur higher

penalty cost in expectation than the lower type because the higher type seller is more likely

to run into stockout than the lower type under the same contract. Therefore, the retailer

can then set the stockout penalty °(µ) based on the commission rate ®(µ) (according to

the second equation in Proposition 2) so that any seller with his type higher than µ would

incur more penalty cost than what he would gain in revenue relative to the type µ seller by

choosing the type-µ's contract. Consequently, every type of seller would not make more

pro¯ts in expectation than any lower type of seller by mimicking. This, together with the

fact that the retailer can set the lump-sum payment (according to the third equation in

Proposition 2) to ensure that every type of seller under truth-telling earns his reservation

pro¯t, implies that the full surplus extraction is achieved. After achieving the ¯rst goal,

the retailer can then set the commission rate ®(µ) (according to the ¯rst equation in

Proposition 2) to ensure that the type µ seller would install the ¯rst-best capacity level.

Note that the lump-sum lost-sale penalty is triggered under the lost-sale event, and is

a one-time charge to the seller. The advantage is that it does not require the system to

monitor orders received after stockout. One may argue that there could be a risk that the

retailer may have an incentive to in°ate orders by either purchasing some units or sending

false orders to try to trigger the lost-sale penalty, especially when the retailer believes that

the seller is about to run out of capacity. However, such manipulations can be e®ectively

mitigated in the online retailing context we study. In online retailing, an online retailer has

to provide detailed customer information of each order (e.g., name, address, credit card

number) on electronic record to a seller for ful¯llment, which makes it extremely hard to

arti¯cially in°ate the orders without leaving a trace. Furthermore, the seller's realized

capacity is not observable to the retailer in our model. As a result, in°ating orders is risky

for the retailer because it cannot guarantee to trigger the lost-sale penalty. So, how many

orders to in°ate would be a complicated decision for the retailer, which may discourage

her from this manipulation.

15

5 Conclusion

In this paper, we consider an online retailer who collects and processes order information,

and leaves the operations in ful¯lling demand to an independent third-party seller who

has superior information about the product demand and privately installs capacity. The

online retailer faces the challenging task of providing the seller incentives for to install the

right level of capacity and extracting maximum surplus from the seller. We show that

the commonly-used commission contracts fail to allow the online retailer to achieve the

two goals e®ectively and could potentially result in signi¯cant pro¯t loss relative to the

¯rst-best. This is because of the con°icting roles of the commission in motivating the seller

to install capacity and in extracting surplus from the seller: to motivate the seller to install

desired level of capacity, a lower commission is needed, while to extract more surplus from

the seller, a higher commission is needed.

In contrast, we show that the retailer can e®ectively accomplish the two goals to achieve

the ¯rst-best by incorporating a lost-sales penalty into the commission contracts. Under

the lost-sale penalty contract, both reducing the commission and increasing the lost-sale

penalty strengthen the seller's incentive to install capacity, because the former acts like

a carrot rewarding the seller for installing higher capacity to achieve higher sales and the

latter is a stick penalizing the seller for his insu±cient amount of capacity that is more likely

to result in lost sales. The online retailer can always ¯nd the right combination of a lost-

sale penalty and a commission to induce the seller to install the ¯rst-best capacity level. In

addition, a seller with optimistic demand is more likely to occur lost sales and hence incur a

higher penalty cost in expectation under lost-sale penalty contracts. Therefore, the online

retailer can tailor the lost-sales penalty for every seller type to fully extract the surplus.

By adjusting the commission accordingly for each type, the retailer can also induce the

sellers to install the ¯rst-best capacity. As a result, the online retailer can e®ectively

accomplish the two goals to achieve the ¯rst best using the lost-sale penalty contracts.

Our results suggest that including both a \carrot" and a \stick" in the contract is a

powerful approach for online retailers to incentivize their independent third-party sellers

with information advantage about product potential to install capacity appropriately.

One limitation of our study is the ¯xed retail price assumption. In reality, the retail

16

price of a product sold online can be adjusted dynamically over time based on both de-

mand strength and inventory availability. The price adjustment is more likely to happen

for products with high demand uncertainty such as fashion products or new products.

An online retailer would need to take this dynamic retail price adjustment into account

when designing the optimal contracts for products with high price volatility. The optimal

contract parameters such as the lost-sale penalty would be functions of the retail price. A

multi-period dynamic principle-agent model seems to be necessary to analyze the optimal

contracts, which could be a fruitful future research direction.

17

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19

AppendixProof of Proposition 1. Take any menu ®( ); t( ) that satis¯es both (IC) and (IR)

constraints. Recall that under this menu, the type-µ seller's optimal expected pro¯t by

reporting µ is

M(µ; µ) = maxk

[p ®(µ)]E"min(µ + "; k) t(µ) V (k) : (2)

It is veri¯able that the function inside the above maximization is strictly concave in k.

Therefore, if the type-µ seller truthfully reports µ = µ, then her optimal capacity level,

denoted by k(µ), is uniquely determined from the following ¯rst-order condition:

[p ®(µ)]G(k(µ) µ) = V 0(k(µ)): (3)

Recall that M(µ) = M(µ; µ). It follows from the Envelope Theorem and the (IC)

constraint that

M 0(µ) =@M(µ; µ)

@µ µ=µ

= [p ®(µ)]G(k(µ) µ);

where the last equality is due to (2) and the de¯nition of k(µ). Consequently,

M(µ) =µ

µ

M 0(x)dx +M(µ)

=µ

µ

[p ®(x)]G(k(x) x)dx+M(µ): (4)

Substituting M(µ) with the right hand side of the above equation, we can rewrite the

retailer's objective as

Eµ pE"min(µ + "; k(µ)) V (k(µ))µ

µ

[p ®(x)]G(k(x) x)dx M(µ)

= Eµ [pE"min(µ + "; k(µ)) V (k(µ)) [p ®(µ)]G(k(µ) µ)H(µ) M(µ)]

= Eµ pE"min(µ + "; k(µ)) V (k(µ))V 0(k(µ))

G(k(µ) µ)G(k(µ) µ)H(µ) M(µ)

where the ¯rst equality follows from integration by parts and the second is due to (3).

In the above analysis, we have used the ¯rst-order necessary condition of the (IC)

constraint to rewrite the retailer's objective function as a function of k( ). This, together

20

with the (IR) constraint, leads to the following relaxed problem:

maxk(¢);M(µ)

Eµ

pE"min(µ + "; k(µ)) V (k(µ))

V 0(k(µ))

G(k(µ)¡µ)G(k(µ) µ)H(µ) M(µ)

s.t. M(µ) 0:

The optimal objective value of the relaxed problem is an upper bound of that of the

original problem because the (IC) constraint is ignored. It is straightforward to solve the

relaxed problem as follows. First, note that at the optimal solution, M(µ) = 0. Second,

for any given µ, the function inside the square brackets in the objective is strictly concave

in k(µ) because its ¯rst-order derivative is

pG(k(µ) µ) V 0(k(µ))g(k(µ) µ)

G2(k(µ) µ)

V 0(k(µ)) +G(k(µ) µ)

G(k(µ) µ)V 00(k(µ)) H(µ)

which strictly decreases in k(µ) (due to V 00( ) 0, V 000( ) 0, and A2). Consequently, by

pointwise optimization, the optimal solution to the relaxed problem, denoted by kcf ( ), is

uniquely determined from the ¯rst-order condition. The optimal objective value, denoted

by Rcf , is

Rcf = Eµ pE"min(µ + "; kcf (µ)) V (kcf(µ))V 0(kcf(µ))

G(kcf (µ) µ)G(kcf(µ) µ)H(µ) :

Next, we construct based on kcf( ) a menu ®cf ( ); tcf( ) that not only satis¯es (IC)

and (IR) but also achieves for the retailer the same pro¯t as the upper bound Rcf . First,

®cf( ) is determined from (3) by letting k( ) = kcf( ), i.e.,

®cf(µ) = p V 0(kcf (µ))=G(kcf (µ) µ):

This ensures that the seller's optimal capacity decision under truth telling is indeed given

by kcf( ). Second, setting tcf( ) such that (4) is satis¯ed, i.e.,

tcf (µ) = [p ®cf(µ)]E"min(µ + "; kcf (µ)) V (kcf(µ))µ

µ

[p ®cf(x)]G(kcf (x) x)dx:

It is veri¯able that the retailer's expected pro¯t under the menu ®cf( ); tcf ( ) is equal

to Rcf and the (IR) constraint is satis¯ed. It remains to show that this menu also satis¯es

the (IC) constraint. By (2) and the de¯nition of tcf (µ), under the menu ®cf( ); tcf( ) ,

M(µ; µ) = maxk

[p ®cf(µ)]E"min(µ + "; k) V (k)

+µ

µ

[p ®cf(x)]G(kcf (x) x)dx [p ®cf(µ)]E"min(µ + "; kcf (µ)) + V (kcf(µ)):

21

Let kcf (µ; µ) be the maximizer, which is uniquely determined by the ¯rst-order condition:

[p ®cf(µ)]G(kcf(µ; µ) µ) = V 0(kcf(µ; µ)):

Clearly, kcf(µ; µ) is nondecreasing in µ. Note that kcf(µ) = kcf (µ; µ). It is veri¯able that

@M(µ; µ)

@µ= [®cf(µ)]0 E"min(µ + "; kcf (µ; µ)) E"min(µ + "; kcf (µ)) :

Because [®cf(µ)]0 0 (by Corollary 1) and kcf (µ; µ) is nondecreasing in µ, @M(µ; µ)=@µ 0

when µ µ and @M(µ; µ)=@µ 0 when µ µ. This implies that for ¯xed µ, M(µ; µ) is

maximized when µ = µ, which proves that the (IC) constraint is satis¯ed.

Proof of Corollary 1. From the de¯nition, kcf (µ) is uniquely determined from the

equation ¡(kcf (µ); µ) = 0 where

¡(k; µ) pG(k µ) V 0(k)g(k µ)

G2(k µ)

V 0(k) +G(k µ)

G(k µ)V 00(k) H(µ):

Therefore,dkcf(µ)

dµ=

@¡(k; µ)=@µ

@¡(k; µ)=@k k=kcf (µ):

It follows from the assumptions (A1) and (A2) that @¡(k; µ)=@µ 0; and from the as-

sumptions (A2) and (A3) that @¡(k; µ)=@k < 0. Therefore, dkcf (µ)=dµ 0, implying that

kcf(µ) is increasing in µ.

To prove that ®cf(µ) is decreasing in µ, we consider two cases. Take any µ. Case 1. If

d[kcf(µ) µ]=dµ 0, then it follows from the de¯nition of ®cf (µ) that d[®cf (µ)]=dµ 0.

Case 2. If d[kcf (µ) µ]=dµ < 0, then it follows from ¡(kcf(µ); µ) = 0 that

p

p ®cf(µ)1 =

g(kcf(µ) µ)

G2(kcf(µ) µ)

+G(kcf(µ) µ)

G(kcf(µ) µ)

V 00(kcf(µ))

V 0(kcf(µ))H(µ) (5)

It follows from d[kcf(µ) µ]=dµ < 0 and (A2) that the ¯rst term inside the above square

brackets is decreasing in µ. The second term is also decreasing in µ because d[kcf(µ)

µ]=dµ < 0, V 00( )=V 0( ) is decreasing (see A3), and kcf (µ) is increasing in µ. This, together

with the assumption that H(µ) is decreasing in µ, implies that the right hand side of (5)

is decreasing in µ. Therefore, d[®cf(µ)]=dµ 0.

Proof of Proposition 2. We will show that ®¤( ); °¤( ); t¤( ) satis¯es the (IC) and

(IR) constraints and achieves the ¯rst-best pro¯t for the retailer.

22

Recall that

M(µ; µ) = maxk

[p ®¤(µ)]E"min(µ + "; k) t¤(µ) °¤(µ)G(k µ) V (k) : (6)

Let k(µ; µ) be the maximizer and k(µ) k(µ; µ).

Claim 1. k(µ) = kI(µ).

Proof of Claim 1. By de¯nition, we have k(µ) = argmaxk ¡(k) and

¡(k) [p ®¤(µ)]E"min(µ + "; k) °¤(µ)G(k µ) V (k)

= V 0(kI(µ)) E"min(µ + "; k)G(kI(µ) µ)

g(kI(µ) µ)G(k µ) V (k);

where the last equality is due to the de¯nition of ®¤(µ) and °¤(µ). Note that

¡0(k) = V 0(kI(µ)) G(k µ) +G(kI(µ) µ)

g(kI(µ) µ)g(k µ) V 0(k): (7)

The claim that k(µ) = kI(µ) is proved in three steps. Step 1. It is straightforward to

verify from (7) that ¡0(kI(µ)) = 0. Step 2. For k > kI(µ), it follows from (7) that ¡0(k)

V 0(kI(µ)) G(k µ) +G(k µ) V 0(k) because G(kI (µ)¡µ)g(kI(µ)¡µ) g(k µ) G(k µ) (due to the

condition that k > kI(µ) and the assumption (A2) where the reverse hazard rate g( )=G( )

is nonincreasing). Therefore, ¡0(k) V 0(kI(µ)) V 0(k) < 0 for k > kI(µ) because V 0( ) is

strictly increasing. Step 3. Analogous to the arguments in Step 2, for k < kI(µ), it follows

from (7) that ¡0(k) V 0(kI(µ)) G(k µ) +G(k µ) V 0(k) because G(kI (µ)¡µ)g(kI(µ)¡µ) g(k µ)

G(k µ) (due to the condition that k < kI(µ) and the assumption (A2) where the reverse

hazard rate g( )=G( ) is nonincreasing). Therefore, ¡0(k) V 0(kI(µ)) V 0(k) > 0 for

k < kI(µ) because V 0( ) is strictly increasing. Combining the three steps, we have shown

that ¡(k) has a unique maximizer kI(µ). This completes the proof of Claim 1.

Claim 2. k(µ; µ) µ decreases in µ.

Proof of Claim 2. By de¯nition, k(µ; µ) = argmaxk ¤(k) where

¤(k) [p ®¤(µ)]E"min(µ + "; k) °¤(µ)G(k µ) V (k)

= [p ®¤(µ)][µ +E"min("; k µ)] °¤(µ)G(k µ) V (k):

We perform the following variable interchange k = k µ, and let k(µ; µ) = k(µ; µ) µ.

Therefore, k(µ; µ) = argmaxk ¤(k) where

¤(k) [p ®¤(µ)][µ +E"min("; k)] °¤(µ)G(k) V (k + µ):

23

Note that ¤0(k) = [p ®¤(µ)]G(k)+°¤(µ)g(k) V 0(k+ µ);which is strictly decreasing in µ,

implying that ¤ has an strictly decreasing di®erence with respect to k and µ. Therefore, the

maximizer of ¤(k), i.e., k(µ; µ), or equivalently k(µ; µ) µ, decreases in µ. This completes

the proof of Claim 2.

Now we proceed to prove that the (IC) constraint is satis¯ed. Take any µ. It follows

from (6) that

@M(µ; µ)

@µ= [p ®¤(µ)]G(k(µ; µ) µ) °¤(µ)g(k(µ; µ) µ)

= V 0(k(µ))g(k(µ; µ) µ)G(k(µ; µ) µ)

g(k(µ; µ) µ)

G(kI(µ) µ)

g(kI(µ) µ):

This, together with Claim 1, implies that (i) @M(µ; µ)=@µ = 0 when µ = µ; together with

Claim 2 and the assumption that G( )=g( ) is nondecreasing, implies that (ii) @M(µ; µ)=@µ

0 when µ µ and (iii) @M(µ; µ)=@µ 0 when µ µ. Combining (i), (ii), and (iii), we have

proved that for any ¯xed µ, M(µ; µ) is maximized at µ = µ. This result, together with the

fact that M(µ) = M(µ; µ) = 0 for every µ, the type-µ cannot do better than truth-telling,

implying that the (IC) constraint is satis¯ed. It is clear that (IR) is also satis¯ed because

M(µ) = 0 for every µ. We have also veri¯ed in Claim 1 that the seller's optimal capacity

under truth telling is equal to the ¯rst-best capacity, implying that the retailer achieves

the ¯rst-best pro¯t.

Proof of Corollary 2. The result follows directly from the de¯nition of kI(µ).

24