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

Hierarchical screening for capacity allocation in distribution systems Ying-Ju Chen Stern School of Business, New York University, [email protected]

Mingcherng Deng Columbia Business School, [email protected]

Ke-Wei Huang Stern 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

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Chen, Deng, and Huang: Hierarchical screening in distribution systems 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 information 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 technical processes, and establishing external supplier-retailer specific domain knowledge and business


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

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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 technology. 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 optimal 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.


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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 information 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 Section 3, we introduce the model setting. We then proceed to solve the optimal contracting mechanisms. 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 management literature, including priority pricing (Afeche (2006)), manufacturing/marketing compensations (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.


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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 perspective. 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 replenishment 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 predetermined 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 policy 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


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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 geographicallydispersed 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., d 1−F (θ) dθ 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 limited prior belief of the demand distribution f (θ), the type-n distributor can pinpoint the interval , k ] with k ∈ {1, ..., 2n } being the unique integer to which the retailer’s type belongs. The θ ∈ ( k−1 2n 2n , k ], k = 1, ..., 2n can be viewed as a partition of the interval θ ∈ [0, 1]; as n becomes interval ( k−1 2n 2n larger, the partition gets finer. Thus, we denote the value n as the precision of information technology 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 ) N such that an ≥ 0, ∀n = 0, ..., N , and an = 1. Although we consider multiple distributors, the n=0 1

Under this assumption, we bypass the possible competition over customer demand among downstream retailers and concentrate on strategic interaction entirely for capacity allocation. This assumption is also adopted in, e.g.,Cachon and 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 and Laplace. See Bagnoli and Bergstrom (2005) for a more complete list. It is adopted in the screening literature to rule out the possibility of bunching phenomenon. The population is common knowledge when the supplier knows aggregate 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 by their information sets. It is appropriate for scenarios when distributors receive discrete forecasts and are able to segment retailers into different groups based on demand forecasting. Practical examples include customer relationship management systems or business intelligence systems. These systems help managers to efficiently collect business information and provide analysis tools for segmenting markets. Our analysis goes through as long as the partition exhibits the nested manner. Equal-length assumption is made for ease of presentation.


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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, pˆ(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 − q 2 − p(q), where p(q) is the lump-sum price paid to the distributor. Similarly, while directly contracting with ˆ (q, θ) = θq − q 2 − pˆ(q). the supplier, a type-θ retailer’s payoff becomes U 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


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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 − q 2 − 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 ˜ − qˆ(θ) ˜2− ˆ (θ˜|θ) = θqˆ(θ) use qˆ(θ) and pˆ(θ) to denote the quantity and price schedule and denote U ˜ θ ∈ [0, 1]. The supplier’s maximization problem is as follows: ˜ ∀θ, pˆ(θ), 1 ˆ Π = max [ˆ p(θ) − cˆ q (θ)]f (θ)dθ, {ˆ q (θ), p(θ)} ˆ

0

˜ θ ∈ [0, 1], and U (θ|θ) ≥ 0, ∀θ ∈ [0, 1]. Following a standard procedure subject to U (θ|θ) ≥ U (θ˜|θ), ∀θ, (Laffont and Martimort (2002)) to solve this problem, we can obtain the solution as: 1 1 − F (θ) (θ − qˆ(θ) = max − c), 0 2 f (θ) ˆ = 1 qˆ(θ) θ − qˆ(θ) − 1−F (θ) − c f (θ)dθ, where θ := inf {θ : θ − 1−F (θ) = and the supplier’s payoff is Π f (θ) f (θ) 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

ˆ is common across regions if the supplier accepts retailers’ orders directly and can asymmetry. Π 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 distributor. 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 contract (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 concepts, 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 saves transportation cost, and hence we ignore this effect. For the discussions on the impact of introducing a secondary market, see Lee and Whang (2002) in a two-echelon setting.


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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 considering 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’ ˆ − qk (θ) ˆ 2 − pk (θ) ˆ as the type-θ retailer’s payoff if she private information. Denote U (θˆ|θ) = θqk (θ) ˆ where θ, θˆ ∈ ( k−1 reports her type as θ, , k ]. Furthermore, we define U (θ) = U (θ|θ). 2n 2n We now characterize the optimal quantity discount contracts. The maximization problem for the type-n distributor is

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n

πn (K) ≡

max

{qk (θ), pk (θ)}

k=1

k 2n k−1 2n

pk (θ)f (θ)dθ,

k−1 k k−1 k U (θ|θ) ≥ U (θˆ|θ), ∀ θˆ ∈ ( n , n ], ∀ θ ∈ ( n , n ], ∀k ∈ {1, ..., 2n } , 2 2 2 2 k−1 k U (θ) ≥ 0, ∀ θ ∈ ( n , n ], ∀k ∈ {1, ..., 2n } , 2 2 n k 2

2n qk (θ)f (θ)dθ ≤ K, k=1

(IC-R) (IR-R)

k−1 2n

(CC)

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

F ( 2kn ) − F (θ) 1 k−1 k ∗ q(θ|K, n) = max [θ − − λ (K, n)], 0 , ∀ θ ∈ ( n , n ], ∀k ∈ {1, ..., 2n } , 2 f (θ) 2 2

(1)


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where λ∗ (K, n) ≥ 0 is the shadow price for the capacity constraint, i.e., it satisfies 2n kn

2 λ∗ (K, n) K − q(θ|K, n)f (θ)dθ ≥ 0, k=1 ∗

and λ (K, n) = 0 if

2n k=1

k 2n k−1 2n

k−1 2n

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 − q 2 − 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 information, the distributor intentionally distorts the quantity ( F ( 2kn )−F (θ) f (θ)

≤

1−F (θ) , f (θ)

F ( 2kn )−F (θ) ) f (θ)

to induce truth telling. Since

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


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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. ˆ N such that πn (K) = π0 (K), ∀K ≤ K ˆ N , ∀n ∈ {0, 1, ..., N }. Theorem 1. There exists a capacity K 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 maximization 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 distributor 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 distributor. Hence, the first-best capacity allocations {KnF B }’s are given by the first order condition: ∂πn (K) |K=KnF B ∂K

= 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 F ( 2kn ) − F (θ) 1 k−1 k [θ − − c], 0 , ∀ θ ∈ ( n , n ]. (2) qn (θ) = max 2 f (θ) 2 2 Moreover, qn1 (θ) ≤ qn2 (θ), ∀θ ∈ [0, 1], ∀n1 ≤ n2 , n1 , n2 ∈ {0, ..., N }, and KnF1B ≤ KnF2B .


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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

an (Tn − cKn ) ,

(3)

n=0

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., ∂πn1 (K) , ∀n2 ∂K

∂πn2 (K) ∂K

>

> 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


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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 distributor. 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 ( 12 , 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


15

(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 ≥ 18 , q(θ|K, 0) = max θ − 12 , 0 , θ ∈ [0, 1]. The type-0 distributor optimal revenue is

π0 (K) =

1 3

√ 3 − 4 2K K, 0 < K < 18 . 1 1 , ≤K 24 8

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

⎧ ⎪ ⎨

π1 (K) =

1

⎪ ⎩ 384

10 . 64

The type-1 distributor’s profit is √ 1 1 3 − 4 2K K, 0 < K < 32 3 3 [288K − (64K − 1) 2 ], 1 ≤ K < 10 . 3 , 64

32 10 64

≤K

64

Fig. 2 illustrates how the distributor’s profits are affected by capacity. First, when K is sufficiently small (K
12 [θ −

k

F ( n22 )−F (θ) F ( n11 )−F (θ) k2 k1 2 2 ≤ , and therefore ≤ . n n 2 2 2 1 f (θ) f (θ) k1 F ( n1 )−F (θ) 2 − λ∗ (K, n1 )], and we obtain that for all θ’s f (θ)

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

such that q(θ|K, n1 ) and q(θ|K, n2 ) are nonzero: F ( 2kn22 ) − F (θ) F ( 2kn11 ) − F (θ) 1 1 ∗ − λ (K, n2 )], 0} > max{ [θ − − λ∗ (K, n1 )], 0}, q(θ|K, n2 ) = max{ [θ − 2 f (θ) 2 f (θ) and hence q(θ|K, n2 ) > q(θ|K, n1 ). When the capacity constraint is binding for both n1 and n2 , 2 2

n

K= >

k 2n2

2 2

k=1 n

k−1 2n2 k 2n2

k=1

k−1 2n2

2 2

n

q(θ|K, n2 )f (θ)dθ =

k=1

k 2n2 k−1 2n2

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

n

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

k=1

k 2n2 k−1 2n2

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 ≥ [0,

2N −1 ], ∀n 2N

= {0, ..., N }. Therefore, when c >

2N −1 2N

2N −1 , 2N

qn (θ) = 0, ∀θ ∈

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 {KnF B }’s are all identical

28


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 −1 , 2N

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 ) − Tn1 and 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.

29

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 {Kn , Tn } where (Kn , Tn ) = ˆ πN (K)), ˆ n = 0, ..., n-1, where K ˆ is sufficiently small that (Kn , Tn ), ∀n ≥ n and (Kn , Tn ) = (K, ¯ ¯ ˆ = π0 (K), ˆ ∀n = 0, ..., N . By Theorem 1, such K ˆ must exist. Under this new allocation, we πn (K)

ˆ πN (K)). ˆ Distributors only need to check the incentive compatibility conditions associated with (K, ˆ because ˆ π0 (K)) with n = 0, ..., n-1 are willing to accept (K, ¯ ˆ − π0 (K) ˆ = 0 ≥ πn (Km ) − Tm = πn (Km πn (Kn ) − Tn = πn (K) ) − Tm , ∀m ≥ n, ¯

ˆ and the inequality is because type-n where the second equality follows from the choice of K, distributor is unwilling to participate under the original allocation. Therefore, their incentive compatibility and individual rationality constraints are all satisfied. For n ≥ n, when the distributor ¯ chooses the contract designed for her, she receives πn (Kn ) − Tn = πn (Kn ) − Tn ≥ 0. Nevertheless, if ˆ − π0 (K) ˆ = 0, which she accepts a contract for m ≤n-1, her payoff will become πn (Km ) − Tm = πn (K) ¯ implies that she will not deviate. Hence the new allocation {Kn , Tn } is both incentive compatible

and individually rational. Under the new allocation, the supplier collects more profit from the distributors since she offers ˆ from others. When c < 1, we can exactly the same to distributors with n ≥ n, and receives π0 (K) ¯ ˆ > cK. ˆ Thus, we conclude that the find a sufficiently small amount of capacity such that π0 (K) 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) θ represents higher demand. (2) on retailers’ payoffs. (3)

2

∂ Γ(q,θ) ∂θ 2

2

∂ Γ(q,θ) ∂q∂θ

+

≥ 0. (4)

∂Γ(q,θ) ∂θ ∂Γ(q,θ) ∂q

∂Γ(q,θ) ∂θ

> 0 is increasing in θ: high

≥ 0, which imposes the single-crossing condition

< 0 since it is downward sloping. Note that the

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

30


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 Proposition 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 distributor can simply use the same quantity schedule for the newly added interval. In Eq. (8), the local incentive compatibility condition requires [

∂Γ(˜ q , θ) ∂Γ(˜ q , θ) ∂Γ(˜ q , θ) ∂Γ(˜ q , θ) q˜ (θ)+ ]˜ q +Γ(˜ q , θ)˜ q (θ) − p˜ (θ) = 0 ⇔ p˜ (θ) = [ +Γ(˜ q , θ)]˜ q (θ)+ q˜(θ). ∂q ∂θ ∂q ∂θ

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

p˜ (θ) = [

∂ 2 Γ(q, θ) ∂Γ(q, θ) ∂ 2 Γ(q, θ) + ]˜ q + q˜ ≥ 0, ∂q∂θ ∂θ ∂θ2

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 sufN

ficiently large λ such that q(θ) = 0, ∀θ ∈ [0, 2 2N−1 ]. 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) ∂Γ(q, θ) 2N − 1 2N − 1 = q + Γ(q, θ) − λ ≤ Γ(q, θ) − λ ≤ Γ(0, ) − λ, ∀ θ ∈ [0, ], ∂q ∂q 2N 2N where we have used Condition (4) in the first inequality and Conditions (1) and (4) in the second N

inequality. Since Γ(0, 2 2N−1 ) 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.


31

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