hierarchical screening for capacity allocation in...

Submitted to Management Sciencemanuscript (Please, provide the mansucript number!)

Hierarchical screening for capacity allocation indistribution systems

Ying-Ju ChenStern School of Business, New York University, [email protected]

Mingcherng DengColumbia Business School, [email protected]

Ke-Wei HuangStern School of Business, New York University, [email protected]

We consider the capacity allocation in a decentralized distribution system with a continuum of retailers, a

number of distributors, and a supplier. Retailers privately observe their local demands, and distributors are

equipped with information technology that facilitates finer prediction of retailers’ demands. The supplier

builds capacity, but observes neither local demands nor the precision of distributors’ technology.

We show that a distributor’s profit weakly increases in capacity and precision, but she cannot capitalize

on her technology with sufficiently small capacity. High production cost may induce the supplier to accept

retailers’ orders directly. With low production cost, the supplier delegates more capacity to distributors who

know local demands better, and no distributor is excluded.

We then numerically investigate scenarios with two types of distributors and uniform distribution of

retailers’ type. With moderate production costs, the supplier should offer a wholesale price contract, even

though some distributors do utilize technology to screen retailers. Quantity discount contracts are offered to

distributors when production cost is low, and the supplier may extract full surplus from all distributors when

technology is relatively common. Retailers need not receive more capacity when distributors have better

technology, and quantity allocation among retailers may be non-monotonic in local demand.

Key words : capacity allocation, distribution systems, multi-echelon, mechanism design

History :

1. Introduction

The objective of a supply chain is to create values for end consumers. To this end, it is essential

that capacity is delivered to the right channel participant, and this efficiency often requires demand

information from downstream retailers. Facing geographically dispersed, heterogeneous markets,

a supplier may not be able to monitor the demand of each local market, and hence must rely on

the reports of local retailers. This demand information is most valuable to supply chains when

products have short life cycles and long leadtimes (e.g., food, fashion, and electronics) or those

whose success heavily depends on local expertise (e.g., the automobile industry). Without sufficient

knowledge of the local markets, capacity cannot be allocated properly. It might result in excess

1

Chen, Deng, and Huang: Hierarchical screening in distribution systems2 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)

inventory leftover, lost profit margins and goodwill, and future demand reduction.

Recent advances in information technology (IT) have made it possible for channel participants

to share information accurately and in a timely manner. Appropriate use of IT may accelerate

information exchange, so that suppliers can improve their prediction of local demands. This in turn

allows them to allocate capacity to those retailers who can extract the most benefit out of the scarce

resource, thereby increasing supply chain efficiency. For instance, Electronic Data Interchange

(EDI) and Internet have successfully made it less expensive to capture Point-of-Sale (POS) data

and transmit these data to regional distributors in real time. This practice has provided managers

invaluable information to better predict future demand and manage raw materials, inventories, and

merchandise much more efficiently. SAP enabled Sony Marketing Asia Pacific to reduce inventory

costs by 40% and helped OfficeMax to reduce inventory by $390 million and improve in-stock

rates from 89% to 98% (www.sap.com). Diageo, a premium drinks company, implemented demand

planning software and expects to reduce inventory by $1 million (Albright (2004)). Cognos Business

Intelligence system helped the exclusive importer in Belgium of Volkswagen, Audi, and Porsche

vehicles to get a daily picture of sales and demand figures (www.cognos.com).

When a supplier delegates the role of monitoring retailers’ demand to distributors who are

equipped with IT, supply chain performance critically depends on how effectively these distributors

utilize the technology. Apart from the successful examples reported above, SAP also had many

failure cases. The drug wholesaler FoxMeyer Corp. accumulated $5 billion in revenues but went into

bankruptcy after adopting SAP products. The downfall of K-Mart has been blamed on outdated

SCM systems, which caused it to fall far behind rivals such as Wal-Mart (Taylor (2003)). Hendricks

and Singhal (2005) provided a sample of 885 SCM glitches announced by publicly traded firms.

Overall, firms that experience glitches report on average 6.92% lower sales growth, 10.66% higher

growth in cost, and 13.88% higher growth in inventories. “Performance differences among those

utilizing information technology” is more the norm than the exception.

While the benefit of boosting information exchange has been well documented, information

technology might create further sophistication of incentive problems. An additional source of infor-

mation asymmetry arises between distributors and suppliers due to the performance differences

among distributors. First, these performance differences may result from different functionalities

of SCM systems or intangible organizational capabilities. To capitalize on information technology,

companies might need to invest in training employees, redesigning internal organizational and tech-

nical processes, and establishing external supplier-retailer specific domain knowledge and business

Chen, Deng, and Huang: Hierarchical screening in distribution systemsArticle submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 3

processes (Rai et al. (2006) and Subramani (2004)). Second, even if suppliers can identify those fac-

tors, it is impractical and illegal to price discriminate the distributors based on these factors (e.g.,

Robinson-Patman Act). Hence, the upstream suppliers can resolve the information asymmetry only

via incentive contracts.

The challenge of optimal contract design for capacity allocation mainly results from information

asymmetry, and from the heterogeneity, inextricably intertwined among the participants in a supply

chain. We have to design two different truth-inducing mechanisms to achieve information exchange:

(1) for aligning the incentives of intermediaries to truthfully reveal their precision on monitoring

their retailers, and (2) for each retailer to truthfully reveal her local demand at her own will.

Coordination schemes for multi-echelon (distribution) systems with heterogeneous downstream

parties are well studied. Nevertheless, researchers typically focus on scenarios where complete

information is available (e.g., Cachon (2003), Chen et al. (2001), and Munson and Rosenblatt

(2001)). Mechanism designs for supply chains with information asymmetry have also been studied

for decades, but the majority adopt simplified two-echelon settings where either upstream suppliers

or downstream retailers possess superior information, see, e.g., Cachon and Lariviere (1999b),

Corbett and de Groote (2000), and Porteus and Whang (1991). Two very recent papers (Erhun

et al. (2006) and Ozer and Raz (2006)) consider multi-layer screening problems, but they focus

on component sourcing issues where one retailer attempts to purchase from two suppliers and in

their model the private information is on the cost structure. In contrast, we include two sources

of information asymmetry, namely the precision of distributors’ technology and retailers’ local

demands. In addition, we consider three kinds of channel participants who specialize in production,

information technology, and selling to end customers, respectively. Because of the difference in

research focus, none of the papers aforementioned provides sufficient insights into how to design

optimal contracting mechanisms in this capacity allocation problem.

This paper attempts to analyze how contracting parties in supply chains respond to the infor-

mation technology via these informational effects. Specifically, we consider a stylized single-period

model with a three-echelon, decentralized distribution system, which comprises of a supplier, a

number of distributors, and a continuum of retailers. All channel participants are self-interested

profit maximizers. The supplier has the ability to build capacity, the distributors have access

to information technology, and the retailers specialize in their local markets with price-sensitive

demands. The inverse demand is linearly downward sloping with a market-specific intercept, which

is privately known to the corresponding retailer. Each distributor controls a region with a pool of

retailers, and these pools are assumed to have identical population and identical diversity of local


demand. This assumption bypasses other underlying differences on aggregate demands of these

regions, thereby allowing us to concentrate on the informational effects. Each distributor privately

knows how precise her technology is in monitoring retailers’ demand. A higher precision allows the

distributor to get a finer partition of the space of retailers’ demands. The supplier knows neither

the local demand of any particular retailer, nor the precision of the distributor’s information tech-

nology. The supplier has two options to meet local demand: (1) She may leave the distributor aside

and accept the orders from retailers directly. (2) she could also make take-it-or-leave-it offers to

distributors, and delegate to the distributor if the distributor accepts it. This setting most suits a

supply chain system consists of a dominant manufacturer (e.g., auto and premium drinks industry),

several state (national) exclusive dealers (importers), and many local retailers.

We assume that quantity discount contracts can be implemented between any two layers. That

is, the supplier can offer quantity discount contracts to the distributors or retailers, and each

distributor in turn is allowed to offer a quantity discount contract to retailers as well. Quantity

discount contracts are commonly proposed to coordinate individuals’ incentives in the literature of

many fields, including economics, marketing, and operations management, see Chen et al. (2001),

Corbett and de Groote (2000), Jeuland and Shugan (1983), and Weng (1995). The quantity discount

contracts are also commonly used by practitioners, most notably in food industry (e.g., Barilla SpA

distribution system Hammond (1994)) and high-tech industries such as CPU (Kanellos (2001)),

DRAM, and personal computers (Vizard (2004)).

Basing on the model characteristics aforementioned, we obtain the following results:

• More effective information technology leads to higher revenue for distributors when they are

awarded adequate capacity. A distributor may not capitalize on her technology when allocated

sufficiently small capacity, because it suffices to serve the local retailers in the highest segment,

regardless of the precision in monitoring local demands. In addition, a distributor’s profit may

plateau when capacity allocation exceeds a certain threshold, suggesting that there exists an opti-

mal capacity needed for such capacity allocation.

• Whether or not the supplier should delegate to the distributors depends critically on pro-

duction costs. When the production costs are sufficiently high, the supplier should not delegate

capacity allocation to any distributor, in which case the supply chain structure is relatively flat.

With relatively low production costs, the supplier delegates more capacity to the distributors who

can segment retailers better. Surprisingly, the supplier never excludes any distributor, irrespective

of the distributors’ population and the heterogeneity of their information technology.


We then numerically investigate the capacity allocation based on two assumptions. First, there

are two types of distributors–one owns information technology and the other does not. Second, the

distribution of retailers’ type is uniform. The numerical results suggest that

• When the production cost is moderate and the information technology is pervasively adopted

among distributors, the supplier offers a wholesale price contract, even though some distributors do

utilize the technology to screen retailers. With low production costs, the distributors are offered a

quantity discount contract. This might explain why quantity discount contracts are more common

for high-tech industries. Moreover, when only a few distributors adopt information technology, the

distributors with superior technology enjoy information rents (which might allow them to recover

the investment on technology); however, when the technology becomes commonly adopted, the

supplier extracts full surplus from all distributors, even if she cannot observe the precision of the

technology. Supply chain structure is completely independent of the distributors’ population.

• The retailers under a distributor with superior information technology may receive less capac-

ity than those reporting to an uninformed distributor. This observation may explain why in practice

some downstream retailers resist adopting advanced technology, but others are willing to cooperate

with upstream divisions. It is also worth noting that when the distributors are equipped with infor-

mation technology, the quantity allocation among retailers may not be monotonic in the retailers’

demand.

The rest of this paper is organized as follows. Section 2 reviews the relevant literature. In Sec-

tion 3, we introduce the model setting. We then proceed to solve the optimal contracting mech-

anisms. Section 4 characterizes the optimal contracts offered by the distributors, and Section 5

investigates the supplier’s problem. In Section 6, we numerically investigate capacity allocation,

and Section 7 concludes. All proofs are in the Appendix.

2. Literature review

Our paper belongs to the literature on screening (adverse selection) problem, which refers to a

principal-agent problem where agents possess private information. A principal is endowed with the

bargaining power and aims at designing a set of (possibly different) take-it-or-leave-it offers for

agents to self-select. This framework has been applied to studying optimal taxation, government

regulation, product design, managerial compensations, and auctions, see Laffont and Martimort

(2002) for comprehensive discussions. It is also studied extensively in the operations manage-

ment literature, including priority pricing (Afeche (2006)), manufacturing/marketing compensa-

tions (Porteus and Whang (1991)), and kidney allocation (Su and Zenios (2006)), to name a few.

See Chen (2003) for more papers along this research stream.


Inventory control theorists have discussed the impact of information technology, which facilitates

the information sharing within supply chains. Their emphasis is heavily on the non-strategic per-

spective. In a centralized multi-echelon system, information sharing makes it possible to acquire

the real-time downstream inventory status and demand information. As a result, a better replen-

ishment policy may be achieved to mitigate bullwhip effect, reduce holding costs, or increase fill

rates, see, e.g., Cachon and Fisher (2000), Gavirneni et al. (1999), and Lee et al. (2000). Apart from

the benefit of coordination, the adoption of information technology is also widely recognized as

mainly an engineering success. For example, IT could reduce operating costs, shorten the leadtime,

and lower the lot sizes (Kulp et al. (2004)). We show that the technological improvement may also

significantly change the business models and the nature of optimal contracting mechanisms.

Our work contributes to the literature on capacity allocation under information asymmetry.

Researchers typically focus on the strategic interaction among channel participants by allowing

informed players to bid for their desired quantities (see Harris and Raviv (1981) and Maskin

and Riley (1989)). Cachon and Lariviere (1999b) examine retailers’ incentives under certain pre-

determined allocation rules in a two-echelon supply chain. In his model, retailers are privately

informed regarding their local demands; a supplier builds a fixed capacity upfront and accepts the

retailers’ bids to determine how she allocates the scarce capacity. They prove that many simple

allocation rules may lead to manipulation, and suppliers may adopt these manipulable allocation

rules to amplify competition among retailers. Our paper assumes that there exists a continuum of

retailers, so that the aggregate distribution is known ex ante. This avoids the micro-level allocations

for every instance of type realization. Nevertheless, in our model the capacity constraint affects the

allocation among heterogeneous retailers in a qualitatively similar manner.

Finally, there has been a vast literature on multi-echelon inventory management for distribution

systems–e.g., the “one-warehouse-multi-retailer” problem. Though the optimal replenishment pol-

icy remains unknown, many researchers have successfully characterized nearly optimal policies and

proposed useful heuristics to minimize the long-run average inventory cost, see, e.g., Cachon and

Fisher (2000), Chan and Simchi-Levi (1998), and Roundy (1983). Our stylized single-period model

bypasses the inventory management problem and therefore does not aim at proposing heuristics

or algorithms following this literature.

3. Model

We consider a single-period model with a three-echelon distribution system that consists of a

supplier, a number of distributors, and a continuum of retailers indexed by θ. The supplier has the


ability to build capacity at a constant production cost c, the distributors have access to information

technology, and the retailers specialize on their local markets. The retailers serve geographically-

dispersed markets, and hence each retailer is considered as a local monopolist.1 A type-θ retailer

faces a deterministic downward sloping demand. We focus on the linear case, i.e., the inverse

demand function for the type-θ retailer is P (q) = θ− q, where q is the quantity in her local market.

We briefly discuss how to incorporate nonlinear demand functions in the appendix.2 The intercept

θ is the retailer’s private information, representing the demand of a local market. The population

of retailers is identical across regions and is represented by F (θ) with its associated density f(θ).

Both terms are common knowledge to all players and have monotone hazard rate property, i.e.,ddθ

1−F (θ)

f(θ)≤ 0.3 The intercept θ has a finite support and is normalized to [0,1].

There are N + 1 types of distributors, indexed by 0,1,2, ...,N . While trading with the retailers,

a type-n distributor is able to locate the intercept θ of the local market into one of 2n mutually

exclusive segments with equal length. In other words, compared with the supplier, who has a lim-

ited prior belief of the demand distribution f(θ), the type-n distributor can pinpoint the interval

θ ∈ (k−12n , k

2n ] with k ∈ {1, ...,2n} being the unique integer to which the retailer’s type belongs. The

interval{(k−1

2n , k2n ], k = 1, ...,2n

}can be viewed as a partition of the interval θ ∈ [0,1]; as n becomes

larger, the partition gets finer. Thus, we denote the value n as the precision of information technol-

ogy owned by the type-n distributor.4 When n = 0, the above union of intervals degenerates to the

grand partition. The type-0 distributor thus is uninformed regarding retailers’ private information

(the intercept θ). The precision of information technology n is not observable by the supplier,

but it is common knowledge that the proportion (or population) of distributors is (a0, a1, ..., aN)

such that an ≥ 0, ∀n = 0, ...,N , andN∑

n=0

an = 1. Although we consider multiple distributors, the

1 Under this assumption, we bypass the possible competition over customer demand among downstream retailers andconcentrate on strategic interaction entirely for capacity allocation. This assumption is also adopted in, e.g.,Cachonand Lariviere (1999a) and Chen et al. (2001).

2 The deterministic demand setting shares the same spirit with the classical EOQ (Economic Order Quantity) model,which allows us to characterize closed-form solutions.

3 This condition is satisfied by most usual distributions–uniform, normal, logistic, chi-squared, exponential andLaplace. See Bagnoli and Bergstrom (2005) for a more complete list. It is adopted in the screening literature to ruleout the possibility of bunching phenomenon. The population is common knowledge when the supplier knows aggre-gate demand for the entire market, but cannot distinguish which retailer gets high demand locally. The aggregate(macro-level) information is usually obtained via the supplier’s market investigation or forecasting system.

4 This representation follows from Celik (2006) and Liu and Serfes (2004), and models distributors’ knowledge bytheir information sets. It is appropriate for scenarios when distributors receive discrete forecasts and are able tosegment retailers into different groups based on demand forecasting. Practical examples include customer relationshipmanagement systems or business intelligence systems. These systems help managers to efficiently collect businessinformation and provide analysis tools for segmenting markets. Our analysis goes through as long as the partitionexhibits the nested manner. Equal-length assumption is made for ease of presentation.


model can also be interpreted as if there is a representative distributor with uncertain precision of

technology, where {an}’s are the corresponding probabilities.

Given that the supplier observes neither the retailer’s private information θ nor the distributor’s

type n, she has two choices to allocate the capacity: (1) She may leave the distributor aside and

accept the orders from retailers directly, which bypasses one source of information asymmetry; (2)

She could make a take-it-or-leave-it offer to the distributor, delegating the allocation right to the

distributor if the distributor accepts. Once the contract is signed by both parties, the supplier is

prohibited from accepting the retailers’ orders. If the distributor accepts the contract, she pays

the supplier for capacity and then redistributes it to downstream retailers. The supplier accepts

retailers’ orders directly if the delegation does not lead to profit gain. Figure 1 shows the structures

of the supply chain under these two scenarios.

Figure 1 Supply chain structures under different scenarios.

We assume that quantity discount contracts can be implemented between any two layers of supply

chain. Let (K,T (K)) and (q, p(q)) denote the quantity discount contracts offered to the distributor

by the supplier, and to retailers by a regional distributor, respectively. We denote (q, p(q)) as the

contracts if the supplier deals with retailers’ orders directly.

As a type-θ retailer orders q units from the distributor, her profit is

U(q, θ) = q(θ− q)− p(q) = θq− q2 − p(q),

where p(q) is the lump-sum price paid to the distributor. Similarly, while directly contracting with

the supplier, a type-θ retailer’s payoff becomes U(q, θ) = θq− q2 − p(q).

Since each retailer faces a deterministic demand, the quantity discount contract is the most

general format to screen these retailers (as seen from U(q, θ)). To highlight the informational effects

and incentive problems, all operational costs (shipping, holding costs, etc.) are assumed to be


negligible, and no transshipment or resale between retailers is allowed. 5 Given the demand function

of each local market, the surplus of supply chain in the market θ is θq − q2 − cq. We assume that

c < 1 to avoid the trivial case where no transaction is profitable.

As a benchmark, we first consider the case in which the supplier directly contracts with the

retailers. The supplier first announces a quantity discount contract (q, p(q)) for the retailers in her

region to self-select. By the revelation principle (Laffont and Martimort (2002)), the retailers may

as well report their local demands and then the supplier assigns quantities accordingly. Thus, we

use q(θ) and p(θ) to denote the quantity and price schedule and denote U(θ|θ) = θq(θ)− q(θ)2 −p(θ),∀θ, θ ∈ [0,1]. The supplier’s maximization problem is as follows:

Π = max{q(θ), p(θ)}

∫ 1

0

[p(θ)− cq(θ)]f(θ)dθ,

subject to U(θ|θ)≥U(θ|θ),∀θ, θ ∈ [0,1], and U(θ|θ)≥ 0,∀θ ∈ [0,1]. Following a standard procedure

(Laffont and Martimort (2002)) to solve this problem, we can obtain the solution as:

q(θ) = max{

12(θ− 1−F (θ)

f(θ)− c),0

}

and the supplier’s payoff is Π =∫ 1

θq(θ)

[θ− q(θ)− 1−F (θ)

f(θ)− c]f(θ)dθ, where θ := inf{θ : θ− 1−F (θ)

f(θ)=

c} is the critical point at which the virtual surplus just becomes positive. Note that θ − c is

the efficient capacity, and the downward distortion 1−F (θ)

f(θ)is made in response to information

asymmetry. Π is common across regions if the supplier accepts retailers’ orders directly and can

be regarded as the supplier’s endogenous reservation value.

When the supplier contracts with the distributor, the sequence of events is as follows. At the

beginning, each retailer observes θ, and the distributor knows the precision of her information

technology n. The supplier first announces her quantity discount contract (K,T (K)) to the dis-

tributor. This menu of contract cannot be contingent on distributor’s choice of retail quantity

discount contracts. If a distributor accepts the delegation, she chooses a capacity K and pays the

money transfer T (K) to the supplier. This distributor then announces her quantity discount con-

tract (q, p(q)) for the retailers in her region to self-select. In contrast, the distributor receives zero

surplus if she refuses the supplier’s offering. In the end, each retailer selects a quantity q, pays the

lump-sum payment to either the distributor or the supplier, and realizes her profit.

We use Bayesian Nash equilibrium and subgame perfect Nash equilibrium as the solution con-

cepts, since our model involves incomplete information and multiple stages of actions (Fudenberg

5 The existing literature in operations management has well documented how introducing regional distributors savestransportation cost, and hence we ignore this effect. For the discussions on the impact of introducing a secondarymarket, see Lee and Whang (2002) in a two-echelon setting.


and Tirole (1994)). By backward induction, we study the subgame facing each distributor in the

next section, and in Section 5 we analyze the supplier’s problem.

4. The distributor’s problem

In this section we first analyze how the distributor designs contracts to retailers when the supplier

delegates capacity to them. Suppose that the type-n distributor, awarded by capacity K, is con-

sidering how to design the quantity discount contract (q, p(q)). The revelation principle allows us

to denote the quantity and price schedule by q(θ) and p(θ).

Since the distributor cannot produce products, she faces a capacity constraint pre-determined

from the transaction with the supplier. Moreover, recall that the type-n distributor can locate

the retailers’ demand θ into one of 2n mutually exclusive segments. Hence, she can offer different

menus (sets) of quantity discount contracts to the retailers whose intercepts θ fall into different

segments. Let {qk(θ), pk(θ), k = 1, ...,2n} denote these menus of quantity discount contracts. When

n = 0, only one quantity discount contract is offered since the distributor has no access to retailers’

private information. Denote U(θ|θ) = θqk(θ)− qk(θ)2 − pk(θ) as the type-θ retailer’s payoff if she

reports her type as θ, where θ, θ ∈ (k−12n , k

2n ]. Furthermore, we define U(θ) = U(θ|θ).We now characterize the optimal quantity discount contracts. The maximization problem for the

type-n distributor is

πn(K)≡ max{qk(θ), pk(θ)}

2n∑k=1

∫ k2n

k−12n

pk(θ)f(θ)dθ,

U(θ|θ)≥U(θ|θ),∀ θ ∈ (k− 12n

,k

2n],∀ θ ∈ (

k− 12n

,k

2n],∀k ∈ {1, ...,2n} , (IC-R)

U(θ)≥ 0,∀ θ ∈ (k− 12n

,k

2n],∀k ∈ {1, ...,2n} , (IR-R)

2n∑k=1

∫ k2n

k−12n

qk(θ)f(θ)dθ ≤K, (CC)

where (IC-R) is the incentive compatibility constraint for retailers, (IR-R) is their individual ratio-

nality constraint, and (CC) is the capacity constraint facing the distributor. Note that a retailer

can choose a quantity/price bundle only from the contract offered to her segment.

By studying this optimal control problem, we can derive the optimal quantity discount contracts

from the distributor’s perspective.

Proposition 1. Suppose a type-n distributor is awarded capacity K. Then the optimal quantity

discount contract is

q(θ|K,n) = max{

12[θ− F ( k

2n )−F (θ)f(θ)

−λ∗(K,n)],0}

,∀ θ ∈ (k− 12n

,k

2n],∀k ∈ {1, ...,2n} , (1)


where λ∗(K,n)≥ 0 is the shadow price for the capacity constraint, i.e., it satisfies

λ∗(K,n)

[K −

2n∑k=1

∫ k2n

k−12n

q(θ|K,n)f(θ)dθ

]≥ 0,

and λ∗(K,n) = 0 if2n∑

k=1

∫ k2n

k−12n

q(θ|K,n)f(θ)dθ ≤K. Moreover, q(θ|K,n) is decreasing in K.

Proposition 1 illustrates the quantity allocated to retailers has three components: (1) efficient

amount (if there were no production cost), (2) distortion due to information asymmetry, and (3)

the shadow price that originates from capacity constraint, see Eq. (1). Recall that a type-θ retailer

generates profit U(q, θ) = θq − q2 − p(q). If the distributor were to know the local demand, the

efficient quantity for this local market should be θ. When the retailers possess private informa-

tion, the distributor intentionally distorts the quantity (F ( k2n )−F (θ)

f(θ)) to induce truth telling. Since

F ( k2n )−F (θ)

f(θ)≤ 1−F (θ)

f(θ), the quantity is less distorted because the information of the distributor is more

precise. The capacity constraint translates to an endogenous variable cost λ∗(K,n), which depends

on both the capacity and the precision of information technology. When the distributor has less

capacity, each retailer receives a lower quantity, and all retailers suffer from that irrespective of

her type. Moreover, even if the capacity K is sufficiently large, the distributor may not allocate

capacity to all retailers. Some retailers may not receive any capacity simply because by doing so

the distributor avoids the cannibalization problem and extracts more revenue.

We next investigate how the distributor’s profit is affected by the information precision as well

as the capacity. The following proposition demonstrates that given any fixed capacity K, the

distributor with more accurate information of the retailers’ local markets extracts more revenue

from the retailers.

Proposition 2. The distributor’s expected revenue, πn(K), is (weakly) increasing in K and n,

∀K ≥ 0,∀n∈ {0,1, ...,N}.

Since the distributor with more accurate information can always replicate the same quantity

schedule as that offered by the distributor with less precise information, advance in information

technology does lead to a more efficient allocation among retailers. This revenue differential comes

entirely from the informational effect, since our setting rules out other benefits of information

technology such as cost saving and leadtime reduction. We also find that when the capacity is

sufficiently low, the distributor cannot capitalize on her information technology, so that distributors

collect exactly the same profit from retailers. Proposition 2 suggests an economic tension between

the distributor and the supplier. When allocated more capacity, a distributor always gains more


from retailers in her region, and hence she may incline to request more capacity. The supplier,

on the other hand, would like to allocate capacity to those distributors who are more informed

regarding the local markets (and hence are able to utilize the capacity appropriately).

We next show that the distributor cannot capitalize on her technology if the awarded capacity

is sufficiently small.

Theorem 1. There exists a capacity KN such that πn(K) = π0(K),∀K ≤ KN ,∀n∈ {0,1, ...,N}.

Theorem 1 can be rationalized as follows. When facing sufficiently low capacity, the distributor

cannot fulfill the local demands, thereby tending to give top priority to the retailers within the

highest segment. Consequently, all types of distributors incline to shut down all other segments

and a finer segmentation among retailers does not lead to a higher profit for the distributor. This

peculiar property implies that the distributors can be sorted naturally but full separation may not

be always possible, because their revenue functions are identical in certain situations. The strict

conflict between the supplier and the distributor drive the cannibalization problem in the first-stage

game, which we elaborate next.

5. The supplier’s problem

We in this section focus on how the supplier designs the quantity discount contract offered to the

distributors. Before analyzing this problem, let us first consider a benchmark scenario where the

supplier has full access to the distributors’ technology (i.e., the supplier knows n). By revelation

principle, we replace the quantity discount contract (K,T (K)) by {(Kn, Tn), n = 0,1, ...,N}.

5.1. The complete information scenario

When the supplier has complete information regarding the distributors’ type, her maximiza-

tion problem is max{Kn,Tn} {Tn − cKn : πn(Kn)−Tn ≥ 0, n = 0,1, ...,N} , where πn(Kn) is the profit

earned by the type-n distributor given capacity Kn, and the constraint ensures that each dis-

tributor receives at least a null profit. The efficient capacity allocation is obtained by equating

the supplier’s marginal cost of production with the marginal value of capacity for the distrib-

utor. Hence, the first-best capacity allocations {KFBn }’s are given by the first order condition:

∂πn(K)

∂K|K=KFB

n= c,∀n∈ {0,1, ...,N}, resulting in the optimal allocation as below.

Proposition 3. Suppose that the supplier can observe n. Then the optimal quantity schedule is

qn(θ) = max{

12[θ− F ( k

2n )−F (θ)f(θ)

− c],0}

,∀ θ ∈ (k− 12n

,k

2n]. (2)

Moreover, qn1(θ)≤ qn2

(θ),∀θ ∈ [0,1],∀n1 ≤ n2, n1, n2 ∈ {0, ...,N}, and KFBn1

≤KFBn2

.


Proposition 3 shows that when the supplier knows the effectiveness of information technology

n, the optimal quantity discount is parallel with the scenario of directly contracting with local

retailers without any capacity constraint. The distributors who know the local demands better are

allocated more capacity. Recall that πn(K) increases in K and n from Proposition 2. The optimal

allocation balances the marginal benefit from expanding the capacity and the marginal cost of

production cost. As information regarding local demands gets more precise, the supplier should

build more capacity and each retailer receives a higher quantity.

5.2. The incomplete information scenario

We now suppose that the supplier cannot access the distributors’ technology and thus must induce

the distributors to truthfully reveal her private information. The supplier’s problem is

Π = max{Kn,Tn}

N∑n=0

an (Tn − cKn) , (3)

πn(Kn)−Tn ≥ πn(Km)−Tm,∀m,n∈ {0,1, ...,N}, (IC-D)

πn(Kn)−Tn ≥ 0,∀n∈ {0,1, ...,N}, (IR-D)

where (IC-D) represents the incentive compatibility of the distributors, and (IR-D) assures that

each distributor receives at least a null profit.

Since {πn(K)}’s are endogenously determined via the distributors’ rational behavior, we cannot

naively impose conditions such as the Spence-Mirrlees single-crossing condition, i.e., ∂πn2 (K)

∂K>

∂πn1 (K)

∂K,∀n2 > n1, n1, n2 ∈ {0,1, ...,N}. As shown in the Appendix, this condition is only weakly

satisfied, which gives rise to the following theorem.

Theorem 2. Suppose the supplier cannot observe n.

1. When c is sufficiently high, the supplier accepts retailers’ orders directly.

2. If the supplier opts to delegate to distributors, then

(a) Kn increases in the effectiveness of information technology n,

(b) All distributors are served, independent of {an}’s and N .

Theorem 2 shows that when the production costs are sufficiently high, the distributor cannot

capitalize on her technology. Thus, it is optimal for the supplier to bypass the distributors and

accept retailers’ orders directly. We also find that when the supplier delegates capacity allocation

to the distributors, she is willing to allocate more capacity to the distributors with more effective

technology, because more advanced technology allows the distributor to pinpoint a finer partition

of retailers’ demand and to allocate capacity more efficiently. Conventional wisdom suggests that


more effective information technology reduces demand uncertainty and hence safety stock. However,

in our model, because a finer segmentation among retailers allows the distributor to serve the

obscure markets without facing the cannibalization problem, a more-informed distributor orders

more quantity from the supplier.

An important implication of Theorem 2 is that the supplier shall allocate capacity based on

distributors’ knowledge regarding the local markets. Our results identify a previously ignored factor

for capacity allocation: information matters. The information effect might become more critical as

the improvement in information technology is evolving. A distributor’s knowledge of local markets

may change over time in response to technological advance, and leaders may become laggards

overnight. Allocating capacity based entirely on past performance, which is common practice in

various industries, might inevitably lack quick response to rapid technology advances.

Moreover, once delegating capacity to distributors, the supplier should never exclude any distrib-

utor. This assertion holds even when some distributors have completely no information advantage

over the supplier, and when the effectiveness of information technology among distributors may

be highly diverse. When the awarded capacity is sufficiently low, the distributors extract identical

profits from retailers regardless of how effective their technology is; that is, {πn(K)}’s are all equal

under scarce capacity from Proposition 1. Thus, the supplier can always induce distributors with

less effective technology to participate without giving up too much information rent. Difference in

information technology cannot justify the breakdown of contracting relationship in supply chains.

Since the single-crossing condition is only weakly satisfied by the distributors’ profit functions,

the standard procedure does not apply and fully separating equilibrium may not always occur. In

the next section we investigate optimal capacity allocation via numerical examples and demonstrate

that our solutions are significantly deviated from the standard solutions of screening problems.

6. Numerical examples

In this section, we provide some numerical examples to demonstrate how information technology

may affect the capacity allocation, the quantity discount contract and even the structure of a

supply chain. In this stylized model, we make the following assumptions. (1) Retailer’s type θ

follows a uniform distribution over [0,1]. (2) There are only two types of distributors (n = 0,1),

with a proportion of 1-a and a respectively. Type-1 distributor is able to distinguish whether a

retailer’s type θ is in the lower segment [0, 12] or in the upper segment (1

2,1]. Type-0 distributor is

as uninformed about retailers’ types as the supplier. Through our numerical studies, we are able to

investigate situations when products have high production costs (i.e., c is high) or are less costly


(c is low). We also examine how the capacity allocation as well as distributor’s profits are affected

when information technology is pervasively adopted (i.e., a is high) and rarely adopted (a is low).

From Proposition 1, we can obtain the optimal quantity schedule, as well as the distributor’s

profit function for a given capacity level. Since these solutions are obtained via straight-forward

algebra, we omit the details.

Proposition 4. Assume that θ follows a uniform distribution over [0,1], and let K be the

capacity allocated to the distributor.

1. Under the type-0 distributor, if 0 < K < 18, the optimal quantity schedule is q(θ|K,0) =

max{θ− 1 +

√2K,0

}, θ ∈ [0,1]. When K ≥ 1

8, q(θ|K,0) = max

{θ− 1

2,0}

, θ ∈ [0,1]. The type-0

distributor optimal revenue is

π0(K) =

{13

(3− 4

√2K)

K, 0 < K < 18

124

, 18≤K

.

2. Now we consider the type-1 distributor. When 0 < K < 132

, q(θ|K,1) = max{θ− 1+

√2K,0

}, θ ∈

[1/2,1], and q(θ|K,1) = 0 otherwise. If 132

≤K ≤ 1064

, the optimal quality schedule is

q(θ|K,1) ={

max{θ− 5

8+ 1

8

√64K − 1,0

}θ ∈ [0, 1

2)

max{θ− 7

8+ 1

8

√64K − 1,0

}θ ∈ [ 1

2,1]

.

Moreover, q(θ|K,1) = q(θ| 1064

,1),∀K > 1064

. The type-1 distributor’s profit is

π1(K) =

⎧⎪⎨⎪⎩

13

(3− 4

√2K)

K, 0 < K < 132

1384

[288K − (64K − 1)32 ], 1

32≤K < 10

64364

, 1064

≤K

.

Fig. 2 illustrates how the distributor’s profits are affected by capacity. First, when K is suffi-

ciently small (K < 132

in this case), both types of the distributors have identical profit functions,

as suggested in Theorem 2. Second, when the capacity exceeds certain thresholds, the distribu-

tor’s profit has begun to level off (K = 18, 10

64for n = 0,1, respectively). Thus, under the quantity

discount contract, there exists an optimal aggregate capacity for each region. Given π0(K), π1(K),

the first-best levels of capacity for type-0 and type-1 distributors are given by

KFB0 =

(1− c)2

8, c∈ (0,1); KFB

1 =

{(1−c)2

8, 1

2< c < 1

1+(3−4c)2

64, 0 < c≤ 1

2

, (4)

respectively from Proposition 3. Note that the upper bound of the production cost c equals to the

maximum demand intercept of a local market (i.e., the maximum value of θ). Thus, the production

cost c can be interpreted as a measure of potential payoff that the supply chain can extract from


Figure 2 Distributors’ profit functions. Figure 3 The four regimes of supply chain struc-

tures and contracts.

local markets. Eq. (4) implies that when the product cost is relatively large (i.e., 12

< c < 1),

information technology does not generate higher revenue from local markets and hence the first-best

capacity is the same.

We next characterize the supplier’s optimization problem under information asymmetry. Propo-

sition 2 shows that the optimal capacity allocation depends on both production cost c and the

population of distributors a. We characterize the second-best solution in four mutually exclusive

regimes as shown in Fig. 3: Direct contracting (Regime 1), Wholes price contract (Regime 2),

Quantity discount with pioneering IT adopters (Regime 3), and Quantity discount with pervasive

IT adopters (Regime 4).

First, when the production cost is sufficiently high, the information technology is useless and

hence, the supplier should directly accept retailers’ orders rather than delegating capacity to

distributors–labelled as Regime 1 in Fig. 3. This insight may explain why in commodity industries

such as steel and gas, information technology is seldom used, and their supply chain structures are

relatively flat (or the left case of Fig. 1). Second, in Regime 2 where production cost is moderate

and technology is relatively common, the supplier offers a single contract to the distributors, even

though the type-1 distributor may use information technology to generate a higher profit. Since

both types of distributors are given capacity KFB0 , the capacity allocated to the type-1 distributors

is downward distorted. This result is in sharp contrast with that in the standard screening litera-

ture in which the single-crossing condition is strictly satisfied, see, e.g., Chen (2003) and Laffont

and Martimort (2002). In particular, the distributors may utilize their information technology, but

the wholesale price contract suffices to maximize the suppliers’ profit.


Fig. 3 also demonstrates that with relatively low production cost, quantity discount contracts are

offered to the distributors. Moreover, type-1 distributor receives her first-best capacity but type-0

distributor’s capacity is distorted downwards. We also find that distributors’ profits depend on

their population when quantity discount contracts are offered. When the distributors are seldom

equipped with technology (i.e., Regime 3 in Fig. 3), more informed distributors earn strictly positive

information rent. Thus, the leaders of information technology within a supply chain receive higher

capacity and make profit from knowing their downstream divisions better. When a large proportion

of distributors have adopted the technology (i.e., Regime 4 in Fig. 3), the supplier can fully extract

surplus from these distributors even if she cannot observe their types.

Of particular interest is that although type-0 distributors are as uninformed as the supplier, they

are never excluded from the supply chain network, as predicted in Theorem 2. Moreover, the supply

chain structure depends only on production costs, independent of the distributors’ population.

To corroborate our analytical predictions, we examine the financial reports of public firms within

the wholesale sector.6 We identify three financial measurements to represent the four mutual exclu-

sive regimes: profit margin (PM), return on assets (ROA) and standard deviation of ROA, which

are proxies for production cost, capacity allocation, and variance in capacity allocation within an

industry, respectively. Table 1 demonstrates the relation between the characteristics of industries

and the financial measurements.

Regime Industry Characteristics PM(%) ROA(%) Std(ROA)

1 Direct contracting Low Low Low2 Wholesale Price Moderate/High Moderate/High High3 Quantity Discount with

”Pioneering IT Adopters”Moderate/High Moderate/High High

4 Quantity Discount with”Pervasive IT Adopters”

Moderate/High Low Low

Table 1 Characteristics of the four regimes

Collectively, our analytical predictions are consistent with empirical evidence. Table 2 shows the

industries in the wholesale sector that satisfy these criteria listed in Table 1. Indeed, as shown in

table 2, information technology is infrequently adopted and the structure of supply chain is flat

6 Our sample includes all available firms within wholesale sector in the Compustat database from year 2000 to2005. Within the wholesale sector, firms are further classified into 41 industries according to the Standard IndustrialClassification system (SIC) by the U.S. Department of Labor. We restrict our analysis to the periods of year 2000to 2005 because of increasing popularity of Supply chain management (SCM) and to the industries with at least 20firm-year observations so as to eliminate the effect of outliers. We also conduct the analysis under different sampleperiods such as 2003-2005 and 2005 only; the results are qualitatively similar.


for the industries in Regime 1, such as machinery equipment and petroleum products, including

infamous Enron Company. In Regime 2, suppliers tend to offer a wholesale contract to distributors,

although some distributors may utilize information technology. Autos and parts industry is a

famous example of this case. We expect to observe that suppliers in Regime 3 and Regime 4 adopt

quantity discount contract. Table 2 confirms our expectation. Industries classified in Regime 4,

like tobacco, electrical, and hardware equipment, are mostly adopters of information technology

than those in Regime 3, such as medical equipment and paper products. However, suppliers in

Regime 4 may extract more rents from the distributors. For example, in Regime 4, Hughes Supply

Inc. (a subsidiary of Home Depot), known for its utilization of IT technology, report a ROA of

only 4.31% in 2006. In contrast, in Regime 3, only leading wholesalers (e.g., OfficeMax) adopt

information technology and benefit from it, whereas other smaller firms do not utilize IT much

and suffer mediocre returns.

Regime Industry SIC code Nobs PM(%) ROA(%) Std (ROA)1 Machinery equip. 5080 74 2.46 4.72 0.0991 Petroleum prods. 5170-5172 84 2.21 7.19 0.0662 Drugs & proprietary 5120-5122 111 2.86 7.98 0.3112 Autos and parts 5010-5015 52 4.36 9.23 0.1793 Medical equip. 5047 54 2.49 10.15 3.0493 Paper prods. 5110-5113 29 2.95 7.51 0.1443 Apparel 5130-5139 24 8.29 12.50 0.1354 Metal and minerals 5050-5059 53 3.05 6.80 0.1034 Electrical equip. 5063 28 3.15 8.03 0.1214 Hardware equip. 5070-5078 71 2.97 6.53 0.0894 Farm (tobacco) products 5150-5159 22 4.45 6.92 0.037All Wholesale 5000-5199 1180 3.01 7.76 0.112

Table 2 Empirical illustrations of supply chain’s structure and contracts

Finally, although type-1 distributor is awarded by a (weakly) higher aggregate capacity, not

all local markets benefit from this unambiguously. Figs. 4 and 5 demonstrate optimal quantity

allocation among retailers under different types of distributors in two representative cases. Fig.

4 represents the case where the production costs are moderate, i.e., in Regime 2. We find that

retailers in the upper segment ( 12,1] actually receive less capacity when the distributor uses more

advanced technology compared to what they would have received under a type-0 distributor. This is

because the distributor also allocates capacity to retailers in the lower segment (0, 12], which in turn

reduces the amount of capacity she leaves for those in the high segment. Nevertheless, in this case

the retailers in lower segment benefit from the distributor’s technology. This might explain why in


Figure 4 An example where retailers of high seg-

ment receive lower capacity under type-1

distributor’s region (a = 0.85, c = 0.2).

Figure 5 An example where all retailers receive

higher capacity under type-1 distributor’s

region (a = 0.5, c = 0.1).

practice some downstream retailers resist adopting collaborative supply chain management systems,

but others are willing to cooperate with upstream divisions. On the contrary, in Fig. 5 where

production costs are relatively low (Regimes 3 and 4), all retailers receive higher quantities under

the more informed distributor. It is also worth noting that under type-1 distributors, the quantity

allocation among retailers is not monotonic in θ. The distributor could intentionally allocate more

capacity to local markets with less demand when equipped with information technology.

7. Conclusions and extensions

In this paper, we investigate the optimal capacity allocation when the distributors are equipped

with information technology. We show that the supply chain structure and optimal contracting

mechanism crucially depend on production cost as well as the distributors’ population. Surprisingly,

once the supplier delegates to the distributors, she never excludes any distributor, regardless of

their population and the heterogeneity of their technology. Through our numerical investigations

over uniform distribution of the retailers’ types, we find that no delegation may occur when the

production costs are sufficiently high. And in some circumstances, the supplier may offer a single

wholesale price contract to the distributors and distort the quantity for the distributors with

technology, even though information technology does result in more efficient allocation among the

retailers. We also find that when information technology is less common, the supplier may extract

full rent from all distributors. Not all retailers receive more capacity under the distributors with

superior technology, and the distributors with advanced technology may not allocate more capacity

to retailers with higher demands.


Several extensions are in order. Our stylized model focuses on the informational effects of tech-

nology, ignoring other important issues in inventory management such as lot size, service level,

customers’ propensity to wait, set-up time, inventory cost, and variety planning. The assumption of

deterministic demand in the local markets allows us to obtain analytical results and is admittedly

a limitation. Channel participants may need to appropriately adjust the contracting mechanisms in

order to accommodate the inventory costs and demand uncertainty. Moreover, in many industries,

customized pricing may be possible at the retail level (e.g., zero financing from car dealers, person-

alized pricing via category promotions in the fashion industry). When local retailers are entitled

to make multiple decisions, achieving efficient capacity allocation may be a tough task.

Although in our model the heterogeneity among distributors is exogenous, there are situations

where the supplier may design appropriate incentive contracts to affect the investment decisions of

distributors. The effectiveness of information technology requires not only the acquisition of more

advanced technology, but also internal employee training and organizational reconfiguration. The

improvement of these internal factors is usually non-contractible, and hence more sophisticated

contracts are needed to coordinate the supply chain. To this end, an appropriate contracting

mechanism requires distributors to fully internalize the benefit of better demand monitoring and

a supplier to share appropriately the cost of technology investment, thereby inducing efficient

capacity allocation among retailers.

We may also consider voluntary information sharing among retailers. We have argued that

whether a retailer is willing to cooperate with the regional distributors is ambiguous. Apart from

these individual incentives, it is also possible to consider the scenario where retailers form a union

negotiating with the distributor. For example, the union may participate in determining how precise

the distributor is able to monitor the local demand (i.e., the value of n). Given the pre-announced

quantity discount contracts offered by a supplier, if the distributor is ineffective in monitoring the

local markets, the supplier may be unwilling to allocate capacity to that region, which eliminates

the possibility of meeting the local demand. On the other hand, if the information sharing is

perfect (i.e., the distributor can distinguish perfectly among these retailers), all the retailers are

left empty-handed even though capacity shortage becomes less severe. Thus, we conjecture that

there exists an endogenized optimal level of information sharing.

Finally, our focus on monopolies is clearly a bit excessive. In reality, it is common that dis-

tributors/wholesalers serve multiple suppliers. For example, a regional automobile dealer in US

may carry cars manufactured by Ford, GM, Hyundai, and Toyota, and can dynamically adjust her

inventory positions of these cars at her own. Distributors such as AmBev, National Distributing


Company, and U.S. Lumber all carry a variety of substitute products. This gives rise the “common

agency” problem. As we can expect, the distributor will allocate capacity to local retailers basing on

the incentive contracts offered by these competing suppliers. Since suppliers act non-cooperatively,

their bargaining power reduces. Consequently, the distributor may carry less inventory for each

supplier due to the substitutability among products, but the aggregate inventory level might be

higher because of demand stealing effect. The competitive behavior among upstream suppliers may

alter the predictions provided under the monopolistic setting, and is left for future research.

AppendixProof of Proposition 1

To simplify notation, we suppress the dependence on K and n, i.e.,

qk(θ)≡ qk(θ|K,n), pk(θ)≡ pk(θ|K,n),∀θ ∈ (k− 12n

,k

2n],∀k ∈ {1, ...,2n}.

Note that U(q, θ) = θq−q2−p(q) satisfies the single-crossing condition, i.e., ∂2U(q,θ)

∂q∂θ≥ 0. Therefore

the standard approach allows us to replace the incentive compatibility condition by the local

incentive compatibility (LOIC):

U′(θ) = qk(θ),∀θ ∈ (

k− 12n

,k

2n],∀k ∈ {1, ...,2n},

and monotonicity condition on qk(θ) (see, e.g., Laffont and Martimort (2002)). Recall that U(θ) =

U(θ|θ) = θqk(θ)− qk(θ)2 − pk(θ), and thus we can represent pk(θ) by θqk(θ)− qk(θ)2 −U(θ). After

these substitutions, the distributor’s maximization problem can be written as

πn(K) ≡ max{qk(θ)}, U(θ)

2n∑k=1

∫ k2n

k−12n

[θqk(θ)− qk(θ)2 −U(θ)]f(θ)dθ

s.t. U′(θ) = qk(θ),∀θ ∈ (

k− 12n

,k

2n],∀k ∈ {1, ...,2n},

q′k(θ)≥ 0,∀θ ∈ (

k− 12n

,k

2n],∀k ∈ {1, ...,2n},

U(θ)≥ 0,∀θ ∈ [0,1],2n∑

k=1

∫ k2n

k−12n

qk(θ)f(θ)dθ ≤K.

We now apply optimal control theory to obtain its solution. We first ignore the monotonicity of

{qk(θ)}’s and obtain a candidate quantity schedule. We then verify that it is indeed monotonic

and hence is feasible. Letting {qk(θ)} be the control variable and U be the state variable, the

Lagrangian L(q1, ..., q2n , µ1, ..., µ2n , λ;θ) can be expressed as

2n∑k=1

{[θqk(θ)− q2

k(θ)−U(θ)]f(θ) +µk(θ)qk(θ)}

+λ[K −2n∑

k=1

∫ k2n

k−12n

qk(θ)f(θ)dθ],


where µk(θ) is the dual variable for the local incentive compatibility, ∀θ ∈ (k−12n , k

2n ],∀k ∈ {1, ...,2n},and λ is the dual variable for the capacity constraint. Let us first consider the interior solutions.

The necessary conditions are:

∂L

∂qk

= [θ− 2qk(θ)]f(θ) +µk(θ)−λ

∫ k2n

k−12n

f(θ)dθ = 0,∀θ ∈ (k− 12n

,k

2n],∀k ∈ {1, ...,2n}, (5)

− ∂L

∂U= f(θ) = µ

′k(θ),∀θ ∈ (

k− 12n

,k

2n],∀k ∈ {1, ...,2n}, (6)

∂L

∂λ= K −

2n∑k=1

∫ k2n

k−12n

qk(θ)f(θ)dθ ≥ 0. (7)

µ′k(θ) = f(θ) implies that µk(θ) = αk + F (θ), where αk is a constant. Note that in the inter-

val (k−12n , k

2n ], the state variable U is free of constraint at θ = k2n . Therefore, we obtain µk( k

2n ) =

αk +F ( k2n ) = 0, which leads to µk(θ) = F (θ)−F ( k

2n ),∀θ ∈ (k−12n , k

2n ],∀k ∈ {1, ...,2n}. The candidate

quantity schedule is therefore

qk(θ) = max{

12[θ− F ( k

2n )−F (θ)f(θ)

−λ∗],0}

,∀ θ ∈ (k− 12n

,k

2n],∀k ∈ {1, ...,2n},

where λ∗ is the corresponding dual variable that is common across k.

Finally, we verify that this quantity schedule is indeed monotonic. To see this, it suffices to show

that Hk(θ) = F ( k2n )−F (θ)

f(θ)is decreasing in θ, ∀θ ∈ (k−1

2n , k2n ], ∀k = 1, ...,2n. Consider two types θ1, θ2

such that k−12n ≤ θ1 ≤ θ2 ≤ k

2n . We have

Hk(θ2)−Hk(θ1) =F ( k

2n )−F (θ2)f(θ2)

− F ( k2n )−F (θ1)

f(θ1)

=1

f(θ1)f(θ2)

[F (

k

2n)(f(θ1)− f(θ2))−F (θ2)f(θ1) +F (θ1)f(θ2)

].

Suppose first that f(θ1) < f(θ2). Then we obtain

Hk(θ2)−Hk(θ1) =1

f(θ1)f(θ2)[(

F (k

2n)−F (θ2)

)f(θ1)−

(F (

k

2n)−F (θ1)

)f(θ2)]

≤ 1f(θ1)f(θ2)

[(

F (k

2n)−F (θ2)

)f(θ2)−

(F (

k

2n)−F (θ1)

)f(θ2)]

=1

f(θ1)(F (θ1)−F (θ2)) ,

where the inequality follows from f(θ1) < f(θ2) and F ( k2n ) − F (θ2) being positive. Therefore,

Hk(θ2)−Hk(θ1)≤ 0 because θ1 ≤ θ2.

If f(θ1)≥ f(θ2), we can rearrange the above equation as follows.

Hk(θ2)−Hk(θ1) =1

f(θ1)f(θ2)

[F (

k

2n)(f(θ1)− f(θ2))−F (θ2)f(θ1) +F (θ1)f(θ2)

]≤ 1

f(θ1)f(θ2)[f(θ1)− f(θ2)−F (θ2)f(θ1) +F (θ1)f(θ2)]


=1−F (θ2)

f(θ2)− 1−F (θ1)

f(θ1),

where we have applied that both f(θ1)−f(θ2)≥ 0 and F ( k2n )≤ 1 to obtain the inequality. From the

monotone hazard rate assumption on F, we obtain that Hk(θ2)−Hk(θ1) ≤ 0, k−12n ≤ θ1 ≤ θ2 ≤ k

2n ,

which implies that the quantity schedule is monotonic.

To show that ∂qk(θ)

∂K≥ 0, it suffices to show that ∂λ(K,n)

∂K≤ 0. We first consider the case in which the

capacity constraint is binding. Suppose that K1 < K2 but λ(K1, n) < λ(K2, n). Since the capacity

constraint is binding, we have

Ki =2n∑

k=1

∫ k2n

k−12n

qk(θ|Ki, n)f(θ)dθ =2n∑

k=1

∫ k2n

k−12n

max{

12[θ− F ( k

2n )−F (θ)f(θ)

−λ(Ki, n)],0}

f(θ)dθ, i = 1,2,

Since λ(K1, n) < λ(K2, n), we have

K1 =2n∑

k=1

∫ k2n

k−12n

max{

12[θ− F ( k

2n )−F (θ)f(θ)

−λ(K1, n)],0}

f(θ)dθ ≥2n∑

k=1

∫ k2n

k−12n

qk(θ|K2, n)f(θ)dθ = K2,

a contradiction. Moreover, when the constraint is not binding, we have dλdK

= 0. Thus the claim is

true. Q.E.D.

Proof of Proposition 2

Since increasing capacity K only relaxes the capacity constraint, the distributor’s expected payoff

is increasing in K. To prove that the distributor’s expected payoff is increasing in n, we recall

that πn(K) =2n∑

k=1

∫ k2n

k−12n

[q(θ|K,n)(θ − q(θ|K,n) − F ( k2n )−F (θ)

f(θ))]f(θ)dθ. Let us consider two integers

n1, n2, where n1 < n2, representing the effectiveness of information technology. A distributor with

the information set {⋃2n2

k=1[k−12n2

, k2n2

} can always replicate the schedule while her information set is

{⋃2n1

k=1[k−12n1

, k2n1

}. Thus, by optimality condition we have πn2(K)≥ πn1

(K). Q.E.D.

Proof of Theorem 1

We first prove a technical lemma, and then show that the theorem holds.

Lemma 1. Suppose a type-n distributor is given capacity K. Then there exists an integer

k∗(K,n) such that all retailers in [k1−12n , k1

2n ] are excluded for all k1 < k∗(K,n), and when k2 ≥k∗(K,n), the distributor will not exclude the entire segment of retailers in the intervals [k2−1

2n , k22n ].

Proof. We suppress the dependence on K and n, i.e.,

qk(θ) = qk(θ|K,n), pk(θ) = pk(θ|K,n),∀θ ∈ (k− 12n

,k

2n],∀k ∈ {1, ...,2n}.


To show that this lemma holds, it suffices to prove that there does not exist a pair of integers i,

j such that 1≤ i < j ≤ 2n and the distributor serves a positive measure of retailers in the interval

[ i−12n , i

2n ] but excludes the entire interval [ j−12n , j

2n ] at optimality.

We now prove this by contradiction. Suppose the above were not true. Let {q∗(θ), p∗(θ)} be the

optimal quantity price schedule, and π∗ is the associated profit of the distributor. Denote q∗(θ) =

p∗(θ) = 0 if type-θ retailer is excluded from the schedule. Hence by our assumption q∗(θ) = p∗(θ) =

0, ∀θ ∈ [ j−12n , j

2n ], since the entire interval of retailers are excluded. We further let θi represent the

retailer that sits in the cutoff point within the interval [ i−12n , i

2n ], i.e., the distributor starts to refuse

offering the quantity price schedule to the retailers whose types are below her.

Now consider another quantity price schedule {q(θ), p(θ)} which is identical to the original

schedule {q∗(θ), p∗(θ)}, except that we swap the quantity schedule between intervals [ i−12n , i

2n ] and

[ j−12n , j

2n ]. That is, we offer

q(θ) = q∗(θ− j − i

2), ∀θ ∈ [

j − 12n

,j

2n],

and set the price as p(θ) =∫ θ

θj

(τq

′(τ)− 2q(τ)q

′(τ))

dτ, where θj denotes the cutoff level for the

retailers in [ j−12n , j

2n ]. Through the transition to this new menu of offers, the distributor discards a

subset of low-end retailers and serves a subset of high-end retailers instead. The quantity schedule

offered to this new inclusion remains the same, but the price is modified as above. Since the quantity

schedule for [ j−12n , j

2n ] in the new menu {q(θ), p(θ)} is identical to that for [ i−12n , i

2n ] under the original

menu {q∗(θ), p∗(θ)}, it follows that θj = θi +j−i2

. Note also that the swap does not change the total

capacity consumed by the retailers.

We now verify that this new menu of quantity price schedules is again incentive compatible.

Since we do not modify the menu offered to retailers whose types are outside these two intervals,

and they are unable to access the quantity schedules offered to those in these two intervals, the

incentive compatibility should still hold. In the interval [ i−12n , i

2n ], all retailers are not served, and

hence the incentive compatibility is automatically true. It remains to check the interval [ j−12n , j

2n ].

Recall that a type-θ retailer’s profit is θq(θ) − q(θ)2 − p(θ) under the new menu, and hence a

necessary condition for the menu being incentive compatible is that

q(θ) + θq′(θ)− 2q(θ)q

′(θ)− p

′(θ) = 0. (8)

The price p(θ) can be obtained by integrating over θ from the above equation, which gives us

p(θ) =∫ θ

θj

(τq

′(τ) + q(τ)− 2q(τ)q

′(τ))

dτ . From the proof of Proposition 1, we know that q∗(θ) is

monotonic in [ i−12n , i

2n ], and hence q(θ) is monotonic as well under the new menu when θ ∈ [ j−12n , j

2n ]


according to the swap. The local incentive compatibility and monotonicity of q(θ) ensure that

{q(θ), p(θ)} is incentive compatible for all retailers.

Finally, let us examine the expected profit collected by the distributor under these two menus.

The only difference between the two menus is the swap, and hence the distributor earns exactly

the same amount from the retailers outside these two intervals. Consider the payment for a retailer

with θ ∈ [θj,j

2n ]. We have

p(θ) =∫ θ

θj

(τ q

′(τ) + q(τ)− 2q(τ)q

′(τ))

dτ =∫ θ− j−i

2

θi

((τ +

j − i

2)(q∗)

′(τ) + q(τ)− 2q∗(τ)(q∗)

′(τ))

dτ

=∫ θ− j−i

2

θi

(τ(q∗)

′(τ) + q(τ)− 2q∗(τ)(q∗)

′(τ))

dτ +j − i

2

∫ θ− j−i2

θi

(q∗)′(τ)dτ

≥∫ θ− j−i

2

θi

(τ(q∗)

′(τ) + q(τ)− 2q∗(τ)(q∗)

′(τ))

dτ = p∗(θ− j − i

2),

where the second equality follows from the change of variables and the identity of q(τ) and q∗(τ −j−i2

). From the monotonicity of q∗(θ),∫ θ− j−i

2

θi(q∗)

′(τ)dτ is positive, which leads to the inequality.

The last equality results from the way we construct {q(θ), p(θ)}. Henceforth, the distributor earns

at least as much from the type-θ retailer under the new menu as from the type-(θ− j−i

2

)retailer

under the original menu.

We conclude that the distributor earns at least as much from the swapped intervals under the

new menu, which in turn implies that under the new menu {q(θ), p(θ)} she gets a higher total

profit. This, however, would contradict the fact that {q∗(θ), p∗(θ)} is optimal. Q.E.D.

We now prove the theorem. Let us first define G(θ)≡ θ− 1−F (θ)

f(θ). From the monotone hazard rate

property of F (θ), we know that G(θ) is strictly increasing. Moreover, G(θ) is continuous since f(θ)

exists, and therefore G−1 exists.

Consider the case when type-N distributor serves only one segment of retailers. From Lemma 1,

we know that if this is the case, she must serve retailers with θ ∈ [ 2N−12N ,1]. Moreover, the optimality

condition requires that the capacity K and λ∗(K,n) must jointly satisfy

∫ 1

2N−1

2N

q2N−1(θ)f(θ)dθ =∫ 1

2N−1

2N

max{12[θ− 1−F (θ)

f(θ)−λ∗(K,n)],0}f(θ)dθ = K,

12

(2N − 1

2N− 1−F ( 2N−1

2N )

f( 2N−12N )

−λ∗(K,n)

)≤ 0⇔ λ∗(K,n)≥G(

2N − 12N

), (9)

where the first equation is the capacity constraint and the second inequality ensures that the

distributor serves at most the entire segment [ 2N−12N ,1] of retailers. The integral can be further

simplified as∫ 1

G−1(λ∗(K,n))12[G(θ)−λ∗(K,n)]f(θ)dθ.


Consider the function Γ(λ) =∫ 1

G−1(λ)12[G(θ)−λ]f(θ)dθ. Note that (λ,K) = (1,0) is a solution to

Γ(λ) = K. Moreover, Γ(λ) is strictly decreasing and continuous in λ since the integrand is decreasing

in λ, G−1 is increasing in λ, and they are both continuous in λ. Therefore, for a fixed constant

G( 2N−12N ), there exists a ξ such that whenever K1 < ξ, we can find a corresponding λ1 > G( 2N−1

2N )

such that Γ(λ1) = K1. This implies that all such pairs (λ1,K1) are feasible solutions to Eq. (9).

Now let KN = ξ. The above shows that when type-N distributor is allocated a capacity K1 less

than KN , she will serve only the retailers in the higher segment [ 2N−12N ,1]. Note also that the same

conditions will hold for any type-n distributors with n < N , since [ 2N−12N ,1] ⊂ [ 2

n−12n ,1], i.e., these

retailers also belong to the highest segment according to type-n distributor’s information set. This

implies that under capacity K1 < KN , type-n distributor will serve exactly the same set of retailers

and give them the same allocation, and consequently the same payment scheme. Thus, we conclude

that all types of distributors have identical profit functions under capacity K1 < KN . Q.E.D.

Proof of Proposition 3

The proof of optimal quantity is parallel with that of Proposition 1. The only modifications are

that there is no capacity constraint and that production cost is incurred. The problem becomes

πn(K)≡ max{qk(θ)}, U(θ)

2n∑k=1

∫ k2n

k−12n

[θqk(θ)− qk(θ)2 − cqk(θ)−U(θ)]f(θ)dθ,

subject to (IC-R) and (IR-R). After replacing (IC-R) by local incentive compatibility and mono-

tonicity of quantity schedule, we obtain the corresponding Lagrangian as

L(q1, ..., q2n , µ1, ..., µ2n ;θ) =2n∑

k=1

[θqk(θ)− q2k(θ)− cqk(θ)−U(θ)]f(θ) +µk(θ)qk(θ),

where {µk(θ)}’s are the dual variable for the local incentive compatibility. Following the argument

of Proposition 1, the optimal quantity schedule is

qn(θ) = max{

12[θ− F ( k

2n )−F (θ)f(θ)

− c],0}

,∀ θ ∈ (k− 12n

,k

2n], (10)

as stated in the lemma.

Now we show that when n1 ≤ n2, qn1(θ) ≤ qn2

(θ),∀θ ∈ [0,1], and KFBn1

≤ KFBn2

. From Eq. (2),

since the operator max{x,0} is monotonic with respect to x, it suffices to compare the terms inside

the parentheses in Eq. (10) for n1, n2. Consider a particular θ ∈ ( 2k2−12n2

, k22n2

] ⊂ ( 2k1−12n1

, k12n1

], where

( 2k2−12n2

, k22n2

] and (2k1−12n1

, k12n1

] are respectively the unique intervals that include θ given partitions

according to n2 and n1. Note that k22n2

≤ k12n1

, and therefore

F ( k22n2

)−F (θ)f(θ)

≤ F ( k12n1

)−F (θ)f(θ)

⇔ 12[θ− F ( k2

2n2)−F (θ)

f(θ)− c]≥ 1

2[θ− F ( k1

2n1)−F (θ)

f(θ)− c],

which implies that qn2(θ)≥ qn1

(θ),∀θ ∈ [0,1]. Finally, since the quantity for each retailer is weakly

higher under n2, the aggregate capacity needed is also higher. Q.E.D.


Proof of Theorem 2

We start with a technical lemma, and then provide the proof of this theorem. The following lemma

states that the marginal benefit of capacity is increasing in the distributor’s type.

Lemma 2. λ∗(K,n) is increasing in n, ∀K ≥ 0.

Proof. Recall that the optimal quantity schedule is given by

q(θ|K,n) = max{12[θ− F ( k

2n )−F (θ)f(θ)

−λ∗(K,n)],0},∀ θ ∈ (k− 12n

,k

2n],∀ k ∈ {1, ...,2n}, (11)

and the total capacity constraint is given by2n∑

k=1

∫ k2n

k−12n

q(θ|K,n)f(θ)dθ = K.

We prove this lemma by contradiction. Consider two integers n1 ≤ n2, and a particular θ ∈( 2k2−1

2n2, k2

2n2] ⊂ ( 2k1−1

2n1, k1

2n1], where ( 2k2−1

2n2, k2

2n2] and (2k1−1

2n1, k1

2n1] are respectively the unique intervals

that include θ given partitions according to n2 and n1.

Assume that λ∗(K,n1) > λ∗(K,n2) . Note that k22n2

≤ k12n1

, and thereforeF (

k22n2

)−F (θ)

f(θ)≤ F (

k12n1

)−F (θ)

f(θ).

Thus, 12[θ− F (

k22n2

)−F (θ)

f(θ)−λ∗(K,n2)] > 1

2[θ− F (

k12n1

)−F (θ)

f(θ)−λ∗(K,n1)], and we obtain that for all θ’s

such that q(θ|K,n1) and q(θ|K,n2) are nonzero:

q(θ|K,n2) = max{12[θ− F ( k2

2n2)−F (θ)

f(θ)−λ∗(K,n2)],0}> max{1

2[θ− F ( k1

2n1)−F (θ)

f(θ)−λ∗(K,n1)],0},

and hence q(θ|K,n2) > q(θ|K,n1). When the capacity constraint is binding for both n1 and n2,

K =2n2∑k=1

∫ k2n2

k−12n2

q(θ|K,n2)f(θ)dθ =2n2∑k=1

∫ k2n2

k−12n2

q(θ|K,n2)1{q(θ|K,n2) > 0}f(θ)dθ

>2n2∑k=1

∫ k2n2

k−12n2

q(θ|K,n1)1{q(θ|K,n1) > 0}f(θ)dθ =2n2∑k=1

∫ k2n2

k−12n2

q(θ|K,n1)f(θ)dθ,

which leads to K > K, a contradiction. Thus, λ∗(K,n1)≤ λ∗(K,n2) in this case.

When the capacity constraint is binding for n2 but not for n1, λ∗(K,n1) = 0≤ λ∗(K,n2). Finally,

we show that when the capacity constraint is binding for n1, it must be binding for n2 as well.

Suppose that this is not the case. Then it means the optimal capacity for n1 is higher than that for

n2 when both have unlimited capacity. This corresponds to a special case of Proposition 3 when

c = 0. Proposition 3 then rules out this possibility. Thus, λ∗(K,n1)≤ λ∗(K,n2). Q.E.D.

We now prove the theorem. We first prove that when the production cost is sufficiently high,

the supplier should not delegate to any distributor. From Eq. (2), when c ≥ 2N−12N , qn(θ) = 0,∀θ ∈

[0, 2N−12N ],∀n = {0, ...,N}. Therefore, when c > 2N−1

2N and the supplier knows the precision n, she

serves only the retailers in the highest segment independent of n. Given this, the allocation to

retailers is independent of n from Eq. (2). Hence, the first-best levels {KFBn }’s are all identical


and the information is useless. When the supplier accepts retailers’ orders directly, the supply

chain profit is maximized since quantities are not distorted and she does not have to leave rents to

distributors. Consequently, when c≥ 2N−12N , the supplier should not delegate to distributors.

Now we consider the case when quantity discount contract is used. Recall that λ∗(K,n) is the dual

variable associated with the capacity constraint, and therefore it corresponds to ∂πn(K)

∂K. Lemma 2

shows that this term is increasing in n, ∀K ≥ 0, and therefore we have the single-crossing condition

on distributors’ payoffs: ∂πn2 (K)

∂K≥ ∂πn1 (K)

∂K,∀K ≥ 0, whenever n1 ≤ n2.

We first prove the monotonicity of K. If n1 ≤ n2, from (IC-D), we have πn2(Kn2

) − Tn2≥

πn2(Kn1

)−Tn1and πn1

(Kn1)−Tn1

≥ πn1(Kn2

)−Tn2. Therefore,

πn2(Kn2

) +πn1(Kn1

)≥ πn2(Kn1

) +πn1(Kn2

)⇔ πn2(Kn2

)−πn2(Kn1

)≥ πn1(Kn2

)−πn1(Kn1

),

which implies that Kn2≥Kn1

according to Lemma 2.

Moreover, some constraints in the optimization problem can be removed. If n1 ≤ n2, we have

πn2(Kn2

) − Tn2≥ πn2

(Kn1) − Tn1

≥ πn1(Kn1

) − Tn1≥ 0, where the first inequality follows from

(IC-D) and the second inequality is because πn(K) is increasing in n by Proposition 2. The last

inequality is due to (IR-D) for type-n1. Thus, (IR-D) for any type higher than the lowest type

served is automatically satisfied. This also implies that when some distributors are excluded, the

exclusion starts from lowest types.

Now we show that it suffices to consider local incentive compatibility constraints. Suppose n1 <

n2 < n3, and assume that type-n3 and type-n2 are unwilling to report as type-n2 and type-n1

respectively. We claim that type-n3 distributor would not like to pretend as if she is type-n1. From

their incentive compatibility constraints, we have

πn3(Kn3

)−Tn3≥ πn3

(Kn2)−Tn2

= πn3(Kn2

)−πn2(Kn2

) +πn2(Kn2

)−Tn2,

and thus πn3(Kn3

) − Tn3≥ πn3

(Kn2) − πn2

(Kn2) + πn2

(Kn1) − Tn1

≥ πn3(Kn1

) − Tn1, where the

first inequality follows from the local incentive compatibility of type-n2, and the last inequality is

according to Lemma 2. A similar argument shows that when type-n1 and type-n2 are unwilling to

report as type-n2 and type-n3 respectively, it is unprofitable for type-n1 distributor to pretend as

if she is type-n3. Thus, it suffices to consider incentive compatibility constraints for adjacent types.

After removing redundant constraints, if type-n¯

is the lowest type that is served, the optimization

problem becomes

Π = max{Kn,Tn}

N∑n=n

¯

an (Tn − cKn) ,


s.t. πn(Kn)−Tn ≥ πn(Km)−Tm,m∈ {n− 1, n+1},∀n∈ {0,1, ...,N},πn

¯(Kn

¯)−Tn

¯≥ 0.

If πn¯(Kn

¯)− Tn

¯> 0, we can increase all {Tn}’s simultaneously by πn

¯(Kn

¯)− Tn

¯. This will not affect

the incentive compatibility constraints by strictly increase the supplier’s profit. Therefore, type-n¯

distributor receives no rent.

We now prove that the supplier will not exclude any distributor. Suppose that there exists an

optimal allocation {Kn, Tn} where type-0 distributor receives no capacity. Let n > 0 be the lowest

type of distributor served by the supplier. Consider another allocation {K ′n, T

′n} where (K

′n, T

′n) =

(Kn, Tn),∀n ≥ n¯

and (K′n, T

′n) = (K,πN(K)), n = 0, ..., n

¯-1, where K is sufficiently small that

πn(K) = π0(K),∀n = 0, ...,N . By Theorem 1, such K must exist. Under this new allocation, we

only need to check the incentive compatibility conditions associated with (K,πN(K)). Distributors

with n = 0, ..., n¯-1 are willing to accept (K,π0(K)) because

πn(K′n)−T

′n = πn(K)−π0(K) = 0≥ πn(Km)−Tm = πn(K

′m)−T

′m,∀m≥ n

¯,

where the second equality follows from the choice of K, and the inequality is because type-n

distributor is unwilling to participate under the original allocation. Therefore, their incentive com-

patibility and individual rationality constraints are all satisfied. For n ≥ n¯, when the distributor

chooses the contract designed for her, she receives πn(K′n)−T

′n = πn(Kn)−Tn ≥ 0. Nevertheless, if

she accepts a contract for m≤n¯-1, her payoff will become πn(K

′m)−T

′m = πn(K)−π0(K) = 0, which

implies that she will not deviate. Hence the new allocation {K ′n, T

′n} is both incentive compatible

and individually rational.

Under the new allocation, the supplier collects more profit from the distributors since she offers

exactly the same to distributors with n≥ n¯, and receives π0(K) from others. When c < 1, we can

find a sufficiently small amount of capacity such that π0(K) > cK. Thus, we conclude that the

supplier is better off, and at optimality she should not exclude any distributor. Q.E.D.

The case with nonlinear demand functions

Our results go through to scenarios with nonlinear demand functions if the following technical

conditions hold. Let Γ(q, θ) denote the inverse demand function, and a type-θ retailer’s net payoff

is Γ(q, θ)q−p(q) if she receives quantity q and pays price p(q). (1) ∂Γ(q,θ)

∂θ> 0 is increasing in θ: high

θ represents higher demand. (2) ∂2Γ(q,θ)

∂q∂θ+ ∂Γ(q,θ)

∂θ≥ 0, which imposes the single-crossing condition

on retailers’ payoffs. (3) ∂2Γ(q,θ)

∂θ2 ≥ 0. (4) ∂Γ(q,θ)

∂q< 0 since it is downward sloping. Note that the

linear demand function used in the main text satisfies all these conditions.


We only highlight the major modifications required for nonlinear demand functions. Since the

single-crossing condition is satisfied, the procedure to obtain optimal quantity schedule in Propo-

sition 1 remains unchanged, but quantity schedule is no longer separable. The capacity constraint

still translates to an endogenous variable cost for the distributor. There are two critical results for

establishing Theorem 1: Lemma 1 and that information is useless if capacity is sufficiently small.

We address both problems as follows.

For Lemma 1, the trick of swapping two segments of retailers continues to hold, and the distrib-

utor can simply use the same quantity schedule for the newly added interval. In Eq. (8), the local

incentive compatibility condition requires

[∂Γ(q, θ)

∂qq′(θ)+

∂Γ(q, θ)∂θ

]q+Γ(q, θ)q′(θ)− p

′(θ) = 0⇔ p

′(θ) = [

∂Γ(q, θ)∂q

+Γ(q, θ)]q′(θ)+

∂Γ(q, θ)∂θ

q(θ).

If q is a fixed monotonic quantity schedule, then p′(θ) is increasing in θ because

p′′(θ) = [

∂2Γ(q, θ)∂q∂θ

+∂Γ(q, θ)

∂θ]q

′+

∂2Γ(q, θ)∂θ2

q ≥ 0,

from Conditions (2) and (3). This implies that after swapping, the price increases at a higher

rate compared to that offered to the original interval. After swapping, it is still optimal for the

distributor to extract all the revenue from the lowest served retailer in the new interval, and hence

the price for her is p(θ) = Γ(q, θ)q, which by Condition (1) is higher than the lowest type in the

original interval. Thus, the entire price schedule for the new interval is higher, and the distributor

is better off by swapping the capacity between two intervals.

Now we deal with the second result. To this end, it suffices to prove that there exists a suf-

ficiently large λ such that q(θ) = 0,∀θ ∈ [0, 2N−12N ]. Consider the first-best quantity and take λ

as the endogenous marginal cost as in the proof of Theorem 1. The surplus of supply chain is

S(q|θ,λ) = Γ(q, θ)q−λq. The first-order condition yields

∂S(q|θ,λ)∂q

=∂Γ(q, θ)

∂qq +Γ(q, θ)−λ≤ Γ(q, θ)−λ≤ Γ(0,

2N − 12N

)−λ,∀θ ∈ [0,2N − 1

2N],

where we have used Condition (4) in the first inequality and Conditions (1) and (4) in the second

inequality. Since Γ(0, 2N−12N ) is bounded, and second-best quantity is lower than the first-best one,

there exists a sufficiently large λ such that the distributor wants to discard all retailers not in the

highest segment. Thus, distributors receive identical profits irrespective of their types.

Given that Theorem 1 holds for nonlinear demand functions, all the subsequent results remain

true since they are based on the property of {πn(K)}’s rather than retailers’ payoffs. Thus, it

merely requires some mild technical conditions for cases with nonlinear demand functions.


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