Supply Chain Intermediation When Retailers Lead*faculty.london.edu/avmiguel/ABD-2012-12-11.pdf2 Adida, Bakshi, and DeMiguel: Supply chain intermediation base the ﬂrms required to

Supply Chain Intermediation When Retailers Lead*

Elodie AdidaSchool of Business Administration, University of California at Riverside, [email protected]

Nitin BakshiManagement Science and Operations, London Business School, [email protected]

Victor DeMiguelManagement Science and Operations, London Business School, [email protected]

To understand the joint impact of horizontal and vertical competition on supply chain intermediation,

we propose a model of competition in a three-tier supply chain where intermediaries compete to mediate

between competing retailers and capacity-constrained suppliers. We argue that in many important contexts

it is more realistic to model the retailers as Stackelberg leaders, and show that as a result intermediaries

should focus on products for which the supplier base (existing production capacity) is neither too narrow

nor too broad. We also find that the right balance of horizontal and vertical competition can entirely offset

double marginalization and lead to supply chain efficiency. Finally, on accounting for intermediaries’ private

information about the supply side, we find that surprisingly, competing retailers could potentially be better

off, relative to the case with complete information.

Key words : Supply chain, intermediation, vertical and horizontal competition, Stackelberg leader.

History : December 11, 2012

1. Introduction

The objective of this paper is to study the joint impact of horizontal and vertical competition

on supply chains with intermediation firms such as Li & Fung Limited, Global Sources, and The

Connor Group, which have enjoyed substantial success over the years. For instance, Li & Fung

Limited (herein L&F) doubled its revenue between 2004 and 2007 (Einhorn 2010), and its sales

almost tripled between 2002 and 2011 (The Telegraph 2012); see also Fung et al. [2008]. These

firms deal with many suppliers and offer intermediation services to numerous retailers (McFarlan

et al. 2007). When an intermediary receives an order from a retailer, it identifies from its supplier

*We are grateful for comments from Long Gao, Serguei Netessine, Guillaume Roels, Robert Swinney, Song (Alex)Yang, and seminar participants at the 2011 INFORMS Annual Meeting in Charlotte, the 2012 POMS Conference inChicago, the 2012 MSOM Conference in New York, the 2012 INFORMS Annual Meeting in Phoenix, the AndersonGraduate School of Management at UC Riverside, ESSEC Business School, the department of Management Scienceand Innovation at University College London, the Stuart School of Business at Illinois Institute of Technology, andSan Jose State University.

2 Adida, Bakshi, and DeMiguel: Supply chain intermediation

base the firms required to fulfill the order, and charges a margin to the retailer for its mediation.

For example, L&F acts as an intermediary between apparel suppliers across Asia and retailers such

as Walmart, Liz Claiborne, and Zara—see Cheng [2010] and Einhorn [2010].

Clearly, competition is a prominent aspect of supply chain intermediation, as described above.

However, the literature on intermediation, both in Economics and Operations, has essentially

focused on identifying rationales for the existence of intermediaries. We aim to address this gap

by improving our understanding of the joint impact of horizontal and vertical competition on

supply chain intermediation. To do so, we model a three-tier supply chain, where the middle tier

consists of a set of intermediaries who compete in quantities to mediate between the other two

tiers, which consist of quantity-competing retailers and capacity-constrained suppliers. A critical

difference from most existing models of multi-tier supply-chain competition is that our model

portrays the retailers as Stackelberg leaders with respect to the intermediaries. This is consistent

with the business model of firms such as L&F that orchestrate order-specific supply chains to

satisfy incidental demand from large retailers; see Knowledge@Wharton [2007].

We provide a complete analytical characterization of the symmetric supply chain equilibrium, and

use the closed-form expressions to answer three research questions. First, how does the competitive

environment affect the performance of retailers and intermediaries? Second, does the presence of

intermediaries necessarily lead to inefficiency in the supply chain? Third, can intermediaries exploit

their private information about the supply side to improve their bargaining position with respect

to the retailers?

With respect to the performance of intermediaries, we find that the intermediary profits are

unimodal with respect to the number of suppliers in its base. This is in direct contrast with the

insight from existing models of supply chain competition, in which suppliers lead. Based on that

literature, one might have expected that the larger the supplier base, the larger the market power of

the intermediaries and thus the larger their profits. This intuition does bear out when the size of the

supplier base is “small”. However, in a world where retailers lead, when the supplier base is “large

enough”, we show that the weakness of the suppliers becomes the weakness of the intermediaries,

and the retailers exploit their leadership position to increase their market power and retain greater

supply chain profits. A crucial implication of this result is that supply chain intermediaries should

focus on products for which the supplier base available is neither too narrow nor too broad; that is,

intermediaries should avoid specialty products as well as large-volume commodity products, and

focus on products in the middle range in terms of existing production capacity.

With respect to supply chain efficiency, we observe that the presence of intermediaries in the

supply chain does not necessarily result in inefficient supply chains; that is, the aggregate supply

chain profit in the decentralized chain with intermediaries is not necessarily smaller than that in

the centralized supply chain. The classic result on double marginalization would have suggested

otherwise (Spengler 1950). However, the differentiating feature of our analysis is that, along with

Adida, Bakshi, and DeMiguel: Supply chain intermediation 3

vertical competition, we simultaneously account for horizontal competition. It is well known that

the relative bargaining strength of players in a vertical relationship significantly determines the

extent of double marginalization. Further, increasing the number of within-tier competitors reduces

a specific player’s bargaining power in the vertical interaction. Such adjustments to competitive

intensity may be carried out at each tier. We find that there always exists an appropriate balance

between horizontal and vertical competition that completely offsets the effect of double marginal-

ization, and leads to supply chain efficiency.

Finally, we mentioned earlier that the retailers exploit their leadership position to increase their

market power with respect to the intermediaries. This leads one to wonder whether intermediaries

can exploit their private information about the supply side to extract greater rents. Indeed, it is

reasonable to assume that intermediaries will know the low-cost suppliers that they regularly deal

with better than retailers. We find that when intermediaries possess private information about

the supply sensitivity, its impact depends on the direction and extent of misperception in the

retailers’ belief about it. In particular, when the retailers underestimate the supply sensitivity, not

surprisingly the intermediary profits are lower than in the case with complete information because

the retailers believe that the supply sensitivity is smaller than it actually is and, as a result, they

offer a low price to the intermediary. However, the impact on the retailer profits is less obvious

because it depends on how much the retailers underestimate the supply sensitivity. When the

retailers grossly underestimate the sensitivity, their profits are smaller, but when the extent of

their misperception is moderate, the retailer profits are larger. The reason for this is that the

underestimation of supply sensitivity by the retailers has a dual effect on their profits: a negative

incomplete information effect, and a positive competition mitigation effect. The negative effect is

that the retailers believe the supply is less sensitive than it really is, and thus they offer a lower

than optimal (for retailers) price to the intermediary. The positive effect, however, is that the

missing information attenuates the intensity of competition among retailers—because the retailers

believe supply sensitivity is smaller than it actually is. Summarizing, when the retailers grossly

underestimate the supply sensitivity, the incomplete information effect dominates, and otherwise

the competition mitigation effect dominates.1

We make two contributions. First, we propose a model of supply chain intermediation (with

and without complete information) that incorporates both horizontal and vertical competition

and portrays the retailers as leaders. This representation is appropriate for the business model

of intermediary firms such as L&F. Second, we use this framework to shed new light on aspects

of supply chain intermediation such as intermediary product portfolios, supply chain efficiency,

and the impact of asymmetric information. In the process, we synthesize and extend two parallel

streams of literature: one on supply chain competition and the other on intermediation.

1 The results for the case when the retailers overestimate the supply sensitivity proceed along expected lines, and we

elaborate on them in Section 5.


The remainder of this manuscript is organized as follows. Section 2 discusses how our work relates

to the existing literature. Section 3 describes the three-tier model of supply chain competition.

Section 4 characterizes the equilibrium, discusses the properties of the equilibrium in terms of

intermediary performance and supply chain efficiency, and compares the equilibrium to those for

other models in the literature. Section 5 studies the effect of asymmetric information, and Section 6

concludes. Appendix A contains proofs for all results, Appendix B contains tables, Appendix C

contains figures, and a supplemental file gives Appendix D containing the analysis of the robustness

of our results to the use of a nonlinear marginal cost function.

2. Relation to the literature

2.1. Literature on supply chain competition

One of our contributions is to propose a model of multi-tier supply-chain competition in which

retailers lead. Most of the existing models of multi-tier supply-chain competition assume the retail-

ers are followers. A prominent example is Corbett and Karmarkar [2001], herein C&K, who consider

entry in a multi-tier supply chain with vertical competition across tiers and horizontal quantity-

competition within each tier. C&K assume the retailers face a deterministic linear demand function,

and they are followers with respect to the suppliers who face constant marginal costs.2 Portraying

retailers as followers is reasonable for many real-world supply chains (e.g., Dell, Coca-Cola, and

suppliers of luxury goods may well lead the supply chains for distribution of their products), but

may not be realistic in the context of supply chain intermediation firms like L&F. These firms

orchestrate demand-driven supply chains as a response to a specific order from a retailer facing

incidental demand; see Knowledge@Wharton [2007]. Chronologically, the retailer order precedes

the orchestration of the supply chain. Interestingly, we find that there is a fundamental difference

between the equilibrium for the model with retailers as followers (as in C&K) and the equilibrium

for our model with retailers as leaders: whereas C&K find that the profit of the intermediaries is

increasing in the number of suppliers, we find that the profit of the intermediaries is unimodal

in the number of suppliers. The unimodality of intermediary profits in the number of suppliers

results in the insight that intermediaries should avoid specialty as well as commodity products.

2 Several papers use variants of the multi-tier supply chain proposed by C&K with quantity competition at every tier

and where the retailers are followers: Carr and Karmarkar [2005] consider the case where there is assembly, Adida

and DeMiguel [2011] consider the case with multiple differentiated products and risk-averse retailers facing uncertain

demand, and Cho [2011] uses the framework by C&K to study the effect of horizontal mergers on consumer prices.

Even for two-tier supply chains, several papers consider models in which a single supplier leads several competing

retailers: Bernstein and Federgruen [2005] consider one manufacturer and multiple retailers who compete by choosing

their retail prices (they assume that the demand faced by each retailer is stochastic with a distribution that depends

on the retail prices of all retailers), Netessine and Zhang [2005] consider a supply chain with one manufacturer

and quantity-competing retailers who face an exogenously determined retail price and a stochastic demand whose

distribution depends on the order quantities of all retailers, Cachon and Lariviere [2005] consider one supplier who

leads competing retailers (their results hold both for the case where the retailers are competitive newsvendors and

when the retailers compete a la Cournot).


This insight seems to be in concordance with the type of products that command most of L&F

demand. These include apparel, toys and sporting goods (see Alberts [2010]), but not products

with ample production capacity such as agricultural commodities, or products with very limited

production capacity such as luxury goods and sophisticated electronic items.

Very few papers in the existing literature model the retailers as leaders. For example, Choi [1991]

considers a model in which one retailer leads two suppliers. His model assumes that the suppliers

possess complete information about the demand function facing the retailer, and that they exploit

this information strategically when making their production decisions. This assumption imposes

a level of sophistication on the suppliers’ strategic capabilities, and endows them with a degree

of information, that does not seem appropriate for the context of low-cost international suppliers

interacting with intermediaries such as L&F. Moreover, we show in Section 4.4 that the equilibrium

in Choi’s model with the retailer as leader is equivalent to that of the model by C&K, where

the retailers are followers, in the sense that the equilibrium quantity, retail price, and supply

chain aggregate profits are identical for both models. Overall, we think that assuming suppliers

can strategically exploit their complete knowledge of the retailers demand function, or for that

matter even strategically compete with numerous other similar suppliers, is not realistic in our

setting. This is crucial, because as we demonstrate in this paper, incorporating a more apt model

for suppliers, along with retailers leading the interaction, results in substantially different insights

than those suggested by the literature on supply chain competition.

Majumder and Srinivasan [2008], herein M&S, consider a model where any of the firms in a net-

work supply chain could be the leader, and study the effect of leadership on supply chain efficiency

as well as the effect of competition between network supply chains. Their model is closely related

to C&K’s, but the two models differ in three important aspects. First, while C&K consider a

serial multi-tier supply chain, M&S consider a network supply chain. Second, while C&K consider

both vertical competition across tiers and horizontal competition within tiers, M&S consider only

vertical competition within networks, and they consider horizontal competition only between net-

works. Third, while C&K assume constant marginal cost of manufacturing, M&S assume increasing

marginal cost of manufacturing, and they argue that, with wholesale price contracts, this is the

only assumption that results in equilibrium when suppliers follow.3

3 Perakis and Roels [2007] study efficiency in supply chains with price-only contracts, and consider a comprehensive

range of models, including both a push and a pull supply chain where the retailer leads. For the push chain (where

the retailer keeps the inventory) they assume that the retailer decides both the wholesale price and the quantity,

which results in the retailer keeping all the profits. In contrast, our model allows the intermediary to keep a positive

margin, as does L&F; see Fung et al. [2008]. Their pull supply chain does not capture the business model of firms such

as L&F since it requires inventory to be held by the intermediaries, something that is not observed in practice (Fung

et al. 2008). Another key difference between our model and the push and pull models by Perakis and Roels [2007] is

that they model horizontal competition in only a single tier, while we consider simultaneous horizontal competition

in multiple tiers of the supply chain.


Our model combines elements from the models of C&K and M&S and is tailored to the context

of supply chain intermediation. Similar to C&K we consider vertical and horizontal competition in

a multi-tier supply chain, but unlike C&K we model the retailers as leaders. As in M&S, we assume

increasing marginal costs of supply, but our motivation to do this, as we explain in §3.1.1, is not

only that this is the sole assumption that results in equilibrium for the case where retailers lead,

but also that this is realistic in the context of intermediary firms like L&F that use the existing

production capacity of their base of suppliers to satisfy incidental demand from retailers. Because

the suppliers use existing capacity, they do not have to make any additional capacity investments,

which are the main rationale for the existence of decreasing marginal costs of supply, and they are

likely to be subject to capacity constraints that will result in increasing marginal costs.

2.2. Literature on intermediation

Our work is related to the Economics literature on intermediation, which according to Wu [2004]

“studies the economic agents who coordinate and arbitrage transactions in between a group of

suppliers and customers.” One difference between our work and the work in this area is that

while the Economics literature has focused on justifying the existence of middlemen through their

ability to reduce fixed transaction costs (Rubinstein and Wolinsky [1987] and Biglaiser [1993]), we

focus on understanding the joint impact of horizontal and vertical competition on supply chain

intermediation and its operational aspects such as the quantity competition and supply chain

efficiency. Our modeling framework differs in several respects from the modeling frameworks used

in this literature. First, the intermediation literature generally models competition via bargaining,

whereas we assume quantity competition a la Cournot-Stackelberg. Second, the intermediation

literature generally assumes each buyer and seller is interested in a single unit and they each

have their own valuation, whereas we assume there is a consumer demand function and suppliers

and retailers are interested in multiple units. Finally, the Economics literature on intermediation

generally models the intermediaries as price-setting firms and focuses on characterizing the bid-

ask spread. In contrast, we simplify the price determination process by assuming a consumer

demand function and a Stackelberg framework, and track not only the intermediary margin but

also the overall efficiency of the supply chain. In summary, our model is closer to the multi-tier

competition models developed in the recent Operations Management literature, such as C&K, with

the difference that we model retailers as leaders, and our focus is on the operational aspects of

supply chain intermediation.4

Our work is also related to Belavina and Girotra [2011], who study intermediation in a supply

chain with two suppliers, one intermediary, and two buyers, with players in the same tier not

4 Although this is not the focus of our manuscript, it is easy to show that fixed transaction costs can also be used to

justify the existence of intermediaries within a supply chain competition context. Furthermore, in analysis available

from the authors upon request, we find that an operational rationale for the existence of intermediaries is that they

help diversify supply disruption risk.


competing directly. They provide a new rationale for the existence of intermediaries. Specifically,

they show that in a multi-period setting the intermediary is more effective in inducing efficient

decisions from the suppliers (e.g., quality related), because the intermediary has access to the

pooled demand of both buyers, and therefore superior ability to commit to future business with

each supplier. Again, a key difference between our work and theirs is that they focus on explaining

the existence of intermediaries, while we focus on understanding the joint impact of horizontal and

vertical competition on supply chain intermediation.

3. A model of multi-tier supply-chain competition where retailers lead

Modeling simultaneous horizontal and vertical competition in a multi-tier supply chain is a chal-

lenging problem. Fortunately, the seminal paper by C&K and the more recent paper by M&S have

led the way on this front and come up with a parsimonious framework to model supply chain com-

petition. We build on these well-established models and carefully justify any departure warranted

by our specific context of supply chain intermediation.

We consider a three-tier supply chain, where the first tier consists of a set of retailers who

compete in quantities to satisfy the consumer demand modeled by a deterministic linear demand

function. The retailers act as Stackelberg leaders with respect to the second tier consisting of a

set of intermediaries who compete in quantities to serve the retailers. The intermediaries act as

leaders with respect to a third tier consisting of a set of capacity-constrained suppliers.

As discussed in Section 2, we believe it is realistic to portray the retailers as leaders in the

context of intermediation firms such as L&F, which orchestrate demand-driven supply chains as a

response to a specific order from a retailer. For instance, the CEO and President of L&F, Bruce

Rockowitz describes their supply chains as being market-driven. Rather than selling what factories

make, their supply chains are matched to demand. He further points out that L&F orchestrates

supply chains on a customer-by-customer basis; that is, they assemble suppliers, manufacturers

and logistics service providers together and tailor a supply chain for each customer based on their

specific requirements (Knowledge@Wharton 2007). This sort of interaction is naturally captured

by a model in which retailers lead.

Our model is a static (one-shot) model of competition. Our motivation to focus on a static

game is that we study situations where a large retailer such as Walmart contacts an intermediary

firm such as L&F to satisfy incidental demand for certain new products (e.g., fashion apparel or

toys). Indeed, we believe large retailers are more likely to use intermediation services to satisfy

such incidental demand needs. For the case in which retailers such as Walmart are interested in

satisfying a recurrent demand for a specific item, they would be more likely to deal directly with the

suppliers rather than using the intermediation services. Moreover, as discussed above, in response

to the product request from the retailer, intermediaries orchestrate a one-time order-specific supply

chain.


Finally, we focus on wholesale price contracts between firms. We do so not only because there is

empirical support for their widespread usage and popularity (Lafontaine and Slade 2012), but also

because there is ample precedence in the supply chain literature (see, for instance, Lariviere and

Porteus [2001] and Perakis and Roels [2007]).5 An advantage of considering a static model with

wholesale price contracts is that these are standard assumptions in the literature on supply chain

competition (C&K, Choi 1991, M&S, Perakis and Roels 2007), and therefore a direct comparison

with that literature is possible. In the remainder of this section we give a detailed description of

our proposed model.

3.1. The suppliers

We consider S symmetric suppliers for a homogeneous product. Like M&S we assume that each

supplier has the following linearly increasing marginal cost function:

ps(q) = s1 + s2q, (1)

where s1 > 0 is the intercept, and s2 ≥ 0 is the sensitivity. The total cost of producing qs,j units is

thus increasing convex quadratic: c(qs,j) =∫ qs,j0

(s1 + s2q)dq= s1qs,j +(s2/2)q2s,j.

The profit of the jth supplier when supplying qs,j units at a price ps is simply the difference

between the revenue and the cost of producing qs,j units πs,j = psqs,j −s1qs,j − (s2/2)q2s,j. Moreover,

when the suppliers are offered a supply price ps by an intermediary, they optimally choose to

produce the quantity such that their marginal cost equals the supply price; that is, the quantity

qs,j such that ps = s1 + s2qs,j. This implies that the supplier profit can be rewritten as6

πs,j =s2q

2s,j

2. (2)

A few comments are in order. First, although we do not explicitly include the supplier capacity

constraints in our model, they are implicitly considered because in equilibrium a supplier would

never produce a quantity larger than (d1 − s1)/s2, where d1(> s1) is the intercept of the demand

function. Second, we assume that when the suppliers are faced with a price offer from an intermedi-

ary, they simply supply the quantity such that their marginal cost of supply equals the price offered

by the intermediary. This implies that our suppliers are not explicitly and strategically competing

with each other, but we believe this is an accurate representation of the decision process followed

by the type of low-cost suppliers that intermediary firms like L&F deal with.7 Third, although we

5 We have also considered a model where the retailers offer a two-part tariff to the intermediaries, but we find

that as the retailers set the contract terms to maximize their own profits while guaranteeing a reservation profit to

intermediaries, the intermediaries earn exactly their reservation profit, leaving all surplus to the retailers. Thus, this

is not adequate to model intermediation because we know that intermediaries do keep a margin.

6 An alternative model would involve a choice of ps that leaves the suppliers with zero profits. This does not strike

us as being realistic. Besides, the alternate formulation would not affect the qualitative nature of our insights.

7 The alternative would be to model suppliers as being cognizant of their strategic interaction with numerous other

suppliers and possessing knowledge of the retailers’ demand function. Neither of these assumptions seem very palatable


choose a linear marginal cost function for tractability and clarity of exposition, in Appendix D we

study the robustness of our results to the use of a nonlinear marginal opportunity cost function,

and we show that the insight that the intermediary profits are unimodal with respect to the num-

ber of suppliers holds also for a convex monomial marginal cost function. Finally, as we discuss

at the end of Section 3.1.1 and in Footnote 8, the assumption that suppliers are symmetric is not

fundamental to our analysis as all results can be derived from an aggregate marginal cost function.

3.1.1. On increasing marginal costs of supply. In addition to M&S, other authors who

have assumed increasing marginal costs include Anand and Mendelson [1997], Correa et al. [2011],

and Ha et al. [2011]. In particular, Ha et al. [2011] explain that this assumption: “is commonly used

in the inventory (Porteus 2002) and queuing (e.g., Kalai et al. [1992]) literatures. It is supported

by empirical evidence in industries such as petroleum refining (Griffin 1972) and auto making

(Mollick 2004)”.

M&S motivate their assumption of increasing marginal cost arguing that this assumption is

required to achieve an equilibrium when the suppliers are followers in the supply chain. Specifically,

[Majumder and Srinivasan 2008, p. 1190] claim: “ Since we have models in which the manufacturer

can be at the receiving end of a wholesale price contract, if she had a constant marginal cost, she

would choose to either not produce (if the wholesale price is lower than his marginal cost), produce

an arbitrarily large quantity (if the wholesale price is higher) or produce an indeterminate quantity

(if they are equal).”

While we agree with M&S that this is the only assumption that results in equilibrium, we also

believe this is the most realistic assumption in the context of supply chain intermediation, and

we give three different motivations for using this assumption. First, we focus on the case in which

intermediary firms such as Li & Fung use the existing production capacity of their network of

suppliers to satisfy incidental demand from retailers such as Walmart. Because the suppliers use

existing capacity, they do not make any additional capacity investments and thus they do not

incur any additional fixed costs. One of the key rationales for modeling decreasing marginal cost

of supply (economies of scale) is that any fixed cost of capacity investment can be defrayed over

multiple units. In the absence of incremental fixed costs, it is sufficient for the purposes of decision

making to capture the variable costs. In this setting, although marginal variable costs could be

constant for small quantities, they will inevitably increase as the order quantities approach the

capacity constraint of the suppliers. Citing a popular Economics textbook [Varian 1992, Section

5.2]: “When we are near to capacity, we need to use more than a proportional amount of the variable

inputs to increase output. Thus, the average variable cost function should eventually increase as

output increases.” Summarizing, we believe in the context of intermediaries using existing supplier

in our context. We also note that although we do not model explicit competition between suppliers, our model has the

appealing feature that as the number of suppliers increases, the resulting order allocation to each supplier decreases,

and therefore the marginal cost of supply decreases. This is qualitatively consistent with what one might expect if

the suppliers were competing.


capacity to meet retailers incidental demand, it is sufficient for decision making to consider only the

variable costs, and due to the capacity constraints, the associated marginal costs will be increasing

at least in the proximity of the capacity constraint.

Our second motivation for increasing marginal costs is to assume suppliers base their production

quantity decisions on their best outside option; that is, a supplier accepts a request to produce

one unit if the price offered by the retailer or intermediary is higher than the best outside offer

for that unit of capacity. For instance, assume a supplier has a capacity of 300 units and a unit

variable cost of 1 cent per unit. Assume also that the supplier has received two production offers

from other retailers or intermediaries. The first offer to produce 100 units at 10 cents per unit, and

the second to produce 100 units at 5 cents per unit. Then the supplier would accept a new offer to

produce 100 units at any price on or above its unit cost of 1 cent per unit, it would accept an offer

to produce 200 units provided the price offered was above 5 cents per unit, and would accept to

produce all 300 units if the price offered was above 10 cents. In other words, the supplier marginal

opportunity cost in cents per unit is the following increasing staircase function:

ps(q) =

1 if q≤ 100,

5 if 100< q≤ 200,

10 if 200< q≤ 300,

∞ if q > 300.

We believe that for the type of suppliers in Asia that intermediary firms such as L&F work with,

it is realistic to assume that the suppliers will make their production decisions based on how the

price offered by L&F compares to the offers from other clients.

Finally, the third motivation for increasing marginal costs is to relax the assumption that sup-

pliers are symmetric, and adopt an asymmetric aggregate view instead. Specifically, confronted

with a heterogeneous (in cost) supply base, the intermediaries would first order from the cheapest

supplier (up to its maximum capacity) and then would engage with progressively more expensive

suppliers. Such a supplier selection procedure would result in an increasing aggregate marginal

cost function (common to all intermediaries) that could be approximated with a linear function:

pS(Q) = s1 + s2 ∗Q with s2 > 0. As we argue below in Footnote 8, it is easy to see that the equi-

librium and the insights from our model would not change much if we used this aggregate supply

function.

3.2. The intermediaries

We consider I symmetric intermediaries. The lth intermediary chooses its order quantity qi,l to

maximize its profit; that is,

maxqi,l

[pi − ps

(qi,l +Qi,−l

S

)]qi,l,

where pi is the price offered by the retailer, Qi,−l is the total quantity ordered by the rest of the

intermediaries at equilibrium, and ps((qi,l+Qi,−l)/S) is the price that the intermediaries must pay


to acquire a total quantity qi,l +Qi,−l from the S suppliers.8 Note that here we are assuming that

the intermediaries are followers with respect to the retailers, but they are Stackelberg leaders with

respect to the suppliers. Moreover, from the definition of the supplier marginal cost function in

Equation (1), we can rewrite the intermediaries decision problem as

maxqi,l

[pi −

(s1 + s2

qi,l +Qi,−l

S

)]qi,l. (3)

3.3. The retailers

We consider R symmetric retailers. Similar to C&K, we assume retailers face a linear inverse

demand function9:

pr = d1 − d2Q. (4)

Then the kth retailer chooses its order quantity qr,k to maximize its profit anticipating the reaction

of the intermediaries; that is the retailer is a Stackelberg leader with respect to the intermediaries:

maxqr,k

[d1 − d2(qr,k +Qr,−k)− pi(qr,k +Qr,−k)] qr,k,

where the kth retailer as a Stackelberg leader anticipates the price pi(qr,k +Qr,−k) that it must

pay the intermediaries as a function of the total quantity Q= qr,k +Qr,−k, where Qr,−k is the total

quantity ordered by all retailers other than the kth retailer.

4. The equilibrium

In this section we characterize the supply chain equilibrium and analyze its properties. Section 4.1

gives closed-form expressions for the equilibrium quantities, Sections 4.2–4.3 give the interpretation

of these results, and Section 4.4 compares the equilibrium for our model to that for the models by

Corbett and Karmarkar [2001] and Choi [1991].

8 Note that given S suppliers, the aggregate supplier marginal opportunity cost function is pS(Q) = s1 + (s2/S)Q.

Moreover, the equilibrium can be determined from the aggregate supply function, and therefore the impact on the

equilibrium of an increase in the number of suppliers S is equivalent to the impact of a certain decrease in the supply

sensitivity s2, which is equivalent to an increase in the total production capacity in the supply chain. Finally, if we

were to consider a model based on an aggregate supply function pS(Q) = s1 + s2 ∗Q, as discussed at the end of

Section 3.1.1, the insights from our analysis would not change provided that s2 = s2/S.

9 Linear demand models have been widely used both in the Economics literature (see, for instance, Singh and Vives

[1984] and Hackner [2003]) as well as in the Operations Management literature (Goyal and Netessine [2007], Farahat

and Perakis [2011], and Farahat and Perakis [2009]). In addition to C&K, other authors have also used it in the

context of supply chain competition. Cachon and Lariviere [2005], for instance, mention that the results for their

model with competing retailers can be applied for the particular case of “Cournot competition with deterministic

linear demand”, and they give as an example a deterministic linear inverse demand function for a single homogeneous

product with retailer differentiation.


4.1. Closed-form expressions

Theorem 4.1 shows that the equilibrium quantities are as given in Table 1, Theorem 4.2 shows that

the monotonicity properties of the equilibrium quantities are as given in Table 3, and Theorem 4.4

gives the closed-form expression for the supply chain efficiency and characterizes its monotonicity

properties.

Theorem 4.1. Let d1 ≥ s1, then there exists a unique equilibrium for the symmetric supply chain,

the equilibrium is symmetric, and the equilibrium quantities are given by Table 1.

Theorem 4.2. Let d1 ≥ s1, the monotonicity properties of the equilibrium quantities are as stated

in Table 3. Moreover, the intermediary profit πi achieves a maximum with respect to the number

of suppliers for S = s2(I +1)/(d2I).

We now characterize the efficiency of the supply chain in the presence of intermediaries, defined

as the ratio of the aggregate supply chain profits in the decentralized and centralized supply chains.

The following proposition gives the optimal quantity and aggregate profit in the centralized supply

chain.

Proposition 4.3. The optimal production quantity and aggregate profit in the centralized supply

chain are Q= (d1 − s1)/(2d2 + s2/S) and πc = (d1 − s1)2/(2(2d2 + s2/S)).

The following proposition gives closed-form expressions of the supply chain efficiency, and char-

acterizes its monotonicity properties.

Theorem 4.4. The supply chain efficiency is

Efficiency =RI [s2R(I +2)+2(d2SI + s2(I +1))]

(d2SI + s2(I +1))2

2d2S+ s2(R+1)2

. (5)

Moreover the monotonicity properties of the efficiency with respect to the number of retailers,

intermediaries, and suppliers are as follows:

1. The efficiency is unimodal with respect to the number of suppliers S and reaches a maximum

equal to one for S = s2(R+ I +1)/(d2I(R− 1)) provided that R> 1. If R= 1, then the efficiency

is monotonically increasing in the number of suppliers, and tends to one as S →∞.

2. The efficiency is unimodal with respect to the number of intermediaries I and reaches a

maximum equal to one for I = s2(R + 1)/(d2S(R − 1) − s2) provided that d2S(R − 1) − s2 > 0.

Otherwise, the efficiency is monotonically increasing in the number of intermediaries, and tends to

R(2d2S+ s2)(2d2S+ s2(R+2))/((R+1)2(d2S+ s2)2) as I →∞.

3. The efficiency is unimodal with respect to the number of retailers R and reaches a maximum10

equal to one for R = (d2SI + s2(I + 1))/(d2SI − s2) provided that d2SI − s2 > 0. Otherwise, the

10 Note that the values of R,S and I respectively that maximize the efficiency may not be integers, in which case the

maximum would be reached at the integer value just below or above it. To keep the exposition simple, we approximate

the true integer maximum with the possibly non integer expressions above.


efficiency is monotonically increasing in the number of retailers, and tends to s2I(I + 2)(2d2S +

s2)/(d2SI + s2(I +1))2 as R→∞.

4.2. Monotonicity properties

Table 3 summarizes the monotonicity properties of the equilibrium quantities. The table gives sev-

eral intuitive results: (i) the total quantity produced in the supply chain is increasing in the number

of retailers, intermediaries, and suppliers, and decreasing in the demand and supply sensitivities,

(ii) the prices at each tier are decreasing in the number of players in following tiers, increasing in

the number of players in leading tiers, increasing in the supply function sensitivity, and decreasing

in the demand function sensitivity, and (iii) the retailer profits are increasing in the number of

intermediaries and suppliers, but decreasing in the number of retailers, and decreasing in both the

supply and demand sensitivities.

Compared to the above findings, the results about the intermediary profits are arguably more

interesting. Although expectedly the intermediary margin decreases in the number of intermediaries

and increases in the number of retailers, surprisingly the intermediary margin decreases in the

number of suppliers. Moreover, since the overall order quantity is monotonically increasing in the

number of suppliers, therefore the intermediary profits are unimodal with respect to the number of

suppliers, reaching their maximum for a finite number of suppliers. Based on the results in C&K

and Choi [1991], one might have expected that the larger the supplier base, the larger the market

power of the intermediaries and thus the larger their margin and profits. However, in a world where

retailers lead, when the supplier base is “large enough”, we show that the weakness of the suppliers

becomes the weakness of the intermediaries, and the retailers exploit their leadership position to

increase their market power and retain greater supply chain profits. Specifically, in the presence

of an infinite number of suppliers, the retailers know that the intermediaries can get an unlimited

quantity of the product at a price of s1 per unit. As a result, the retailers can exploit their leader

advantage to drive the intermediary price to the suppliers’ marginal cost s1. In this limiting case,

the retailers have all the market power and keep all profits. The crucial implication of this result is

that supply chain intermediaries should focus on products for which the supplier base available is

neither too narrow nor too broad; that is, intermediaries should avoid specialty products as well as

large-volume commodity products, and focus on products in the middle range in terms of available

production capacity.11

4.3. Efficiency

Theorem 4.4 gives the closed-form expression for the supply chain efficiency and characterizes its

monotonicity properties. Our main observation is that a supply chain efficiency equal to one (that

is, an efficient supply chain) can always be achieved in the presence of intermediaries provided there

11 Note that even though, as pointed out in the introduction, firms such as L&F deal with a large number of suppliers

across their portfolio; for any specific product, the relevant supplier base will be substantially smaller.


is the right balance of competition at the three tiers. To see this, note from Point 1 in Theorem 4.4

that provided the number of retailers is greater than one, we can always adjust the number of

suppliers so that efficiency one is achieved. Also, note that provided that there is a sufficiently large

number of suppliers and retailers, the condition d2S(R−1)−s2 > 0 will be satisfied, and thus from

Point 2 we have that there exists a number of intermediaries for which the supply chain efficiency

is equal to one. Finally, from Point 3 we see that provided there is a sufficiently large number of

suppliers and intermediaries we can always satisfy the condition d2SI − s2 > 0 and thus there will

exist a number of retailers for which the efficiency of the supply chain is equal to one.

The main takeaway from this analysis is that the presence of intermediaries in the supply chain

does not necessarily result in inefficient supply chains; that is, the aggregate supply chain profit in

the decentralized chain with intermediaries is not necessarily smaller than that in the centralized

supply chain. The classic result on double marginalization would have suggested otherwise (Spen-

gler 1950). However, accounting for competition in our model is the differentiating feature. It is

well known that the relative bargaining strength of players in a vertical relationship significantly

determines the extent of double marginalization. Further, increasing the number of within-tier

competitors reduces a specific player’s bargaining power in the vertical interaction. Such adjust-

ments to competitive intensity may be carried out at each tier. We find that there always exists an

appropriate balance between horizontal and vertical competition that completely offsets the effect

of double marginalization, and leads to supply chain efficiency.

4.4. Comparison to other models in the literature

We now give a detailed comparison of the equilibria for our proposed model and for the models

proposed by C&K and Choi [1991]. To do so, we first briefly state three-tier variants of the models

of C&K and Choi that are similar to our model, and then we compare the equilibria of the three

models.

As discussed in Section 2.1, our model also shares some common elements with that proposed

by M&S. Their model, however, does not consider horizontal competition within tiers, and instead

focuses on competition between supply networks. Moreover, M&S focus on how the equilibrium

depends on the position of the leader within the supply network, whereas we fix the leader position

to be at the retailer, which is the situation faced by the supply chain intermediary firms that we

are interested in. For these reasons, the equilibrium for M&S’s model is not comparable to that for

our model, and thus we focus in this section on the comparison with the equilibria for the models

by C&K and Choi, who consider serial multi-tier supply chain models with vertical and horizontal

competition similar to our model.

4.4.1. A Corbett-and-Karmarkar-type model. We first consider a three-tier version of

C&K’s model. The first tier consists of S suppliers who lead the second tier consisting of I inter-

mediaries who lead the third tier consisting of R retailers. There is quantity competition at all


three tiers. We focus on the case where only the first tier of suppliers face production costs, which

is the closest to our proposed model of intermediation.

The kth retailer chooses its order quantity qr,k to maximize its profit given an inverse demand

function pr = d1 − d2Q and for a given price requested by the intermediary pi:

maxqr,k

[d1 − d2(qr,k +Qr,−k)− pi)] qr,k.

The lth intermediary chooses its order quantity qi,l to maximize its profit for a given price

requested by the suppliers ps, and anticipating the price that the retailers are willing to pay for a

total quantity qi,l +Qi,−l:

maxqi,l

[pi (qi,l +Qi,−l)− ps] qi,l.

Finally, the jth supplier chooses its production quantity qs,j to maximize its profit anticipating

the price that the intermediaries are willing to pay for a total quantity qs,j +Qs,−j and given its

unit variable cost is s1:

maxqs,j

[ps (qs,j +Qs,−j)− s1] qs,j.

4.4.2. A Choi-type model. We now consider a three-tier version of Choi’s model with the

retailers as leaders. Choi assumes that suppliers know the demand function and exploit this knowl-

edge strategically when making production decisions. Moreover, Choi assumes that the suppliers

are margin takers with respect to the intermediaries. Thus the jth supplier’s decision problem is

maxqs,j

(d1 − d2 (qs,j +Qs,−j)−mr −mi − s1)qs,j,

where s1 is the unit production cost, and mr and mi are the retailer and intermediary margins,

respectively. Note that because pi =mr+mi+s1 it is apparent that the supplier’s decision problem

for the Choi-type model is exactly equivalent to the retailer’s decision problem for the C&K-type

model.

Following the spirit of Choi’s model with retailers as leaders, we assume that the intermediary

also knows the demand function and exploits this knowledge strategically when making order

quantity decisions. On the other hand, the intermediary is a follower with respect to the retailers

and thus is a margin taker with respect to the retailers, but the intermediary is a leader with

respect to the suppliers and thus anticipates the price requested by the suppliers to deliver a given

quantity. Therefore the lth intermediary decision can be written as

maxqi,l

(d1 − d2 (qi,l +Qi,−l)−mr − ps(qi,l +Qi,−l))qi,l.

Finally, the kth retailer in Choi’s model chooses its order quantity to maximize the profit given

the demand function and anticipating the price required by the intermediaries to supply a total

quantity qk,r +Qr,−k:

maxqk,r

(d1 − d2 (qk,r +Qr,−k)− pi(qk,r +Qr,−k))qk,r.


4.4.3. Comparing the C&K and Choi models. C&K give closed-form expressions for the

equilibrium quantities for their model. Choi also gives closed-form expressions for the equilibrium

quantities for his two-tier model with retailers as leaders, and it is straightforward to extend these

closed-form expressions for the three-tier variant of his model that we consider. The resulting

closed-form expressions are collected in the second and third columns of Table 2.

The striking realization when comparing the second and third columns in Table 2 is that the

equilibrium total order quantities in the C&K and Choi models coincide. Moreover, the aggregate

intermediary profits also coincide. Furthermore, a careful look at the expressions for the aggregate

profits of the retailers and suppliers reveals that the aggregate retailer profit in C&K’s model

coincides with the aggregate supplier profit in Choi’s model if one replaces the number of retailers

by the number of suppliers. Likewise, the aggregate supplier profit in C&K’s model coincides with

the aggregate retailer profit in Choi’s model if one replaces the number of suppliers by the number

of retailers. In other words, the equilibria of the C&K and Choi model are equivalent. We believe

the reason for this is the assumption in Choi’s model that the suppliers have perfect information

about the retailer demand function and exploit this strategically when making decisions. This is

not a realistic assumption for the intermediation context that we study.

4.4.4. Comparing our model to the C&K and Choi models. The most important

difference between the equilibrium to our model and those to the C&K and Choi models is in the

intermediary margins and profits. For all three models, the suppliers market power decreases in the

number of suppliers, and their profits become zero in the limit when there is an infinite number

of suppliers. The intermediaries’ margin and profit, however, behave quite differently for the three

models. In C&K’s model, the intermediaries’ margin increases with the number of suppliers; in

Choi’s model, it remains constant; and in our model, it decreases. As a result, for the C&K and Choi

models, the intermediary profits increase in the number of suppliers, because order quantities also

increase. In our model, on the other hand, the increase in the order quantities is not sufficient to

offset the decrease in margin and, as a result, the intermediary profits are unimodal in the number of

suppliers. The reason for this is that as the number of suppliers grows large, the retailers know that

intermediaries and suppliers will agree to produce at any price above s1, and therefore the retailers

can take advantage of their leading position to extract higher rents, leaving the intermediaries with

zero margin and profit for the limiting case where the number of suppliers is infinite.

Comparing the aggregate retailer profits in all three models, we observe that the equilibrium

retailer margins for our proposed model and the model by Choi are equal and they are both larger

than the equilibrium retailer margin for the model by C&K. Moreover, because the equilibrium

total quantity in the models by C&K and Choi are identical, this implies that the aggregate retailer

profit in the model by Choi is larger than that in the model by C&K. This is not surprising as


it is well known that a leading position often results in larger profits for a given player; see Vives

[1999].12

The question of whether the aggregate retailer profits are larger in our model than in those

by C&K and Choi is a bit harder to answer. Note that there is an additional parameter in our

model: the marginal cost sensitivity s2. This parameter enters the closed-form expression for the

equilibrium quantity in our model and thus it is difficult to compare to the other two models.

However, assuming the marginal cost sensitivity s2 equals the demand sensitivity d2, it is easy to

show that the equilibrium quantity in our model is larger than in the C&K and Choi models. This

implies that the equilibrium aggregate retailer profits in our model are larger not only than those

in the C&K model, but also than those in Choi’s model (for the case s2 = d2). Two comments are

in order. First, since in our proposed model the retailers act as leaders, they are able to capture

greater profits than in the model by C&K. Second, while Choi also captures the retailers as leaders,

he assumes that intermediaries and suppliers know the retailer demand function and exploit this

knowledge strategically. This assumption leaves the retailers in a weaker position compared to our

model.

We conclude that there are significant differences between the models by C&K and Choi [1991]

and our proposed model, particularly the leadership positions and information available within the

game, which result in different insights. Our model fits best situations when retailers act as leaders,

while C&K’s model is more appropriate when suppliers can be considered leaders. Choi’s model

makes sense when the retail demand function can realistically be known by all players.

5. The case with asymmetric information

An important insight from our analysis is that in a supply chain where retailers lead, they take

advantage of their leading position to increase their market power with respect to the intermedi-

aries. This leads one to wonder whether the intermediaries can exploit their private information

about the supply side to improve their bargaining position. Indeed, it is reasonable to assume that

the intermediaries have better knowledge of the supply sensitivity than the retailers, and that they

may strategically choose to exploit their knowledge. To answer this question, in this section we

study how the presence of asymmetric information about the supply sensitivity alters the balance

of market power between the retailers and the intermediaries.

5.1. The model and equilibrium

To model asymmetric information, we assume that retailers know the number of suppliers S and

the intercept s1 of the supplier marginal cost function, but they do not know whether the supply

12 Note that the retailer margin and profits in Choi’s model coincide with the supplier margin and profits in C&K’s

model after replacing the number of retailers with the number of suppliers, so as we argue above both models are

essentially equivalent.


sensitivity is high s2 = sH2 or low s2 = sL2 . Instead, we assume that all retailers share the prior belief

that there is a probability ν that the sensitivity is low (s2 = sL2 ) and 1−ν that it is high (s2 = sH2 ).

Like in the case with complete information, we assume the retailers act as Stackelberg leaders

with respect to the intermediaries. Specifically, the retailers offer a single price pi to intermediaries,

and they anticipate the intermediary best response (in terms of quantity) for each scenario (with

low or high sensitivity).13 As in the case with complete information, the intermediary best response

for each scenario is characterized by Theorem 4.1. Then the retailers maximize their expected profit

E[πr,k

]= ν

(d1 − d2(q

Lr,k +QL

r,−k)− pi)qLr,k +(1− ν)

(d1 − d2(q

Hr,k +QH

r,−k)− pi)qHr,k, (6)

where the first (second) term on the right-hand side of Equation (6) corresponds to the profit for

the case with low (high) sensitivity, qLr,k (qHr,k) is the kth retailer order quantity when the supply

sensitivity is low (high), which is determined by the intermediary best-response, and QLr,−k (QH

r,−k)

is the aggregate order quantity of all retailers except the kth retailer when the supply sensitivity

is low (high).

The presence of asymmetric information complicates the analysis substantially, so to keep the

exposition simple we restrict our analysis to symmetric equilibria. The following proposition gives

closed-form expressions for the unique symmetric equilibrium.

Proposition 5.1. Let d1 ≥ s1, then there exists a unique symmetric equilibrium for the supply

chain with asymmetric information. Moreover, for the case where the realized sensitivity is low

(s2 = sL2 ), the realized aggregate quantity (Q=QL) is

QL =RSI [(1− ν)sL2 + νsH2 ]

(1− ν)sL2 [d2SIsL2 /s

H2 +(I +1)sL2 ] + νsH2 [d2SI +(I +1)sL2 ]

d1 − s1R+1

and for the case where the realized sensitivity is high (s2 = sH2 ), the realized aggregate quantity

(Q=QH) is QH = (sL2 /sH2 )Q

L. Moreover, the rest of the realized equilibrium quantities are given

by the last column of Table 1 as a function of the realized aggregate quantity Q and the realized

supply sensitivity s2.

5.2. The impact of asymmetric information

The following proposition characterizes how the presence of asymmetric information affects the

equilibrium. To simplify the exposition we focus on the case when the ratio of the high to low

sensitivity (sH2 /sL2 ) is sufficiently large.14

13 This is the equivalent of a pooling contract in the standard treatment of games with asymmetric information.

Screening of the different types of intermediary is not possible in the context of wholesale price contracts because

only one contracting variable (price) is available. Note that we restrict our analysis to wholesale price contracts for

the reasons argued in Section 3.

14 See the proof in Appendix A for the exact threshold value. We have also analyzed the case where this ratio is small,

and the insights from the analysis are similar.


Proposition 5.2. Let d1 ≥ s1 and let sH2 /sL2 be sufficiently large, then the comparison between

equilibrium quantities in the supply chains with asymmetric and complete information is as indi-

cated in Table 4.

We now interpret the results in Table 4. We consider two cases: (i) the case where the true

(realized) sensitivity is low (s2 = sL2 ), and (ii) the case where the true (realized) sensitivity is high

(s2 = sH2 ). Another way to think about these two cases is to note that, for the case where the

realized sensitivity is low, the retailers’ prior belief overestimates the supply sensitivity, whereas

for the case where the realzied sensitivity is high, the retailers’ prior belief underestimates the

sensitivity.

When retailers overestimate the supply sensitivity, they are willing to offer a higher price to

the intermediaries compared to the case with complete information, which results in lower retailer

margin and profits, together with higher intermediary margin and profits. In other words, interme-

diaries facing low-sensitivity suppliers can exploit their superior knowledge of the supply sensitivity

to increase their profits at the expense of the retailers.

When retailers underestimate the supply sensitivity, on the other hand, they offer the intermedi-

aries a lower price compared to the case with complete information. This lower intermediary price

results in lower order quantities and lower intermediary profits. The retailer profits, however, could

be larger or smaller depending on their prior beliefs. Specifically, Table 4 shows that, when the

true (realized) sensitivity is high, the retailer profit is lower if it believes that the probability of

low sensitivity (ν) is greater than a certain threshold (ν0):

ν > ν0 ≡ α(R− 1)

α(R− 1)+ (sH2 /sL2 )(η−Rα)

,

where α= d2 + sH2 (I +1)/(SI) and η= d2sH2 /s

L2 + sH2 (I +1)/(SI).

For the case with a single retailer we have that ν0 = 0, and thus monopolist retailer profits are

always lower in the case with asymmetric information. Indeed, a monopolist retailer with perfect

information on supply sensitivity must be able to extract larger profits from the supply chain

than in the absence of perfect information, regardless of whether its prior belief overestimates or

underestimates the true supply sensitivity.

A more interesting result occurs when there are several competing retailers. For this case, when

retailers severely underestimate the supply sensitivity (ν > ν0), the retailer profits are smaller than

in the case with complete information. When the retailers moderately underestimate the supply

sensitivity (ν < ν0), their profits are larger in the presence of asymmetric information. The expla-

nation for this is that in the case in which multiple competing retailers underestimate the supply

sensitivity, the presence of asymmetric information has a negative and a positive effect on retailer

profits. The negative effect is that the retailers lack information about the supply sensitivity that

would help to identify the optimal price to offer to intermediaries. The positive effect, however,

is that this missing information attenuates the intensity of competition among retailers because


the retailers believe on average that supply sensitivity is smaller than it actually is and thus offer

a lower price to intermediaries. For the case where they severely underestimate the supply sensi-

tivity, the negative effect (incomplete information) dominates; for the case where they moderately

underestimate the sensitivity, the positive effect of asymmetric information (competition mitiga-

tion) dominates. This result is illustrated in Figure 1, which depicts the realized prices and profits

for the case where two retailers underestimate the supply sensitivity as a function of the retailers

belief probability ν.

One interesting implication of our analysis is that it is not always optimal for intermediaries to

keep their private information private. Specifically, when the retailers prior belief underestimates

the supply sensitivity, the intermediaries would benefit from disclosing their private information

to the retailers, if they can do this in a credible manner.

Finally, the following proposition shows that, although the insight that intermediary profits are

unimodal with respect to the number of suppliers is robust to the presence of asymmetric infor-

mation, the number of suppliers that maximizes intermediary profits in the case with asymmetric

information is larger (smaller) than in the case with complete information depending on whether

the retailers overestimate (underestimate) the supply sensitivity.

Proposition 5.3. Let d1 ≥ s1, then the intermediary profits in the presence of asymmetric infor-

mation are unimodal with respect to the number of suppliers, and the number of suppliers that

maximizes intermediary profits is larger (smaller) in the presence of asymmetric information for

the case with s2 = sL2 (s2 = sH2 ).

The main insight from Proposition 5.3 is that the presence of asymmetric information can result

in alterations to the intermediaries’ preferred product portfolio. When the retailers overestimate

(underestimate) the supply sensitivity, the preferred product portfolio is biased towards products

with more (less) available production capacity.

6. Conclusion

To understand the impact of horizontal and vertical competition on supply chain intermediation,

we study a model of supply chain competition with retailers as leaders. This is an appropriate repre-

sentation for the business model of successful sourcing companies such as L&F, Global Sources, and

The Connor Group. Our focus on the role of competition distinguishes our work from the bulk of

the prior literature on intermediation, which provides rationales for the existence of intermediaries.

Our analysis predicts that intermediaries prefer products with moderate levels of available capac-

ity on the supply side, which is consistent with the observation that L&F typically deals with

products such as apparel and toys, as opposed to large-volume commodities (e.g., agricultural

produce) or specialty items (e.g., luxury goods).

A key strategy espoused by L&F over the years has been to acquire other sourcing companies

such as Inchcape, Swire & Maclaine, and Camberley Enterprises (Fung Group 2012), with recent

speculation about further acquisitions (Reuters 2012). A natural question then is what the impact


of these acquisitions is not only on L&F, but also on the efficiency of the entire supply chain. Our

competition model provides a framework to think about these issues. Specifically, we find that

intermediation is not necessarily detrimental to supply chain performance, but rather that the

right balance of horizontal and vertical competition can entirely offset the adverse effect of double

marginalization.

Finally, we study the impact of asymmetric information in the context of supply chain inter-

mediation, and find that surprisingly intermediaries’ private information about supply sensitivity

may indirectly benefit retailers by alleviating inter-retailer competition.

Appendix A: Proofs for the results in the paper

Proof of Theorem 4.1.We prove the result in three steps. First, we characterize the best response

of the intermediaries to the retailers. Second, we characterize the retailer equilibrium as leaders

with respect to the intermediaries. Third, simple substitution into the best response functions leads

to the closed form expressions.

Step 1. The intermediary best response. We first show that the intermediary equilibrium

best response exists, is unique, and symmetric. It is easy to see from equation (3) that the interme-

diary decision problem is a strictly concave problem that can be equivalently rewritten as a linear

complementarity problem (LCP); see Cottle et al. [2009] for an introduction to complementarity

problems. Hence, the intermediary order vector qi = (qi,1, . . . , qi,I) is an intermediary equilibrium

if and only if it solves the following LCP, which is obtained by concatenating the LCPs charac-

terizing the best response of the I intermediaries 0≤ (−pi + s1)e+ (s2/S)Miqi ⊥ qi ≥ 0, where e

is the I-dimensional vector of ones, and Mi ∈RI ×RI is a positive definite matrix whose diagonal

elements are all equal to one and its off-diagonal elements are all equal to two. Thus this LCP

has a unique solution which is the unique intermediary equilibrium best response. Because the

intermediary equilibrium is unique and the game is symmetric with respect to all intermediaries,

the intermediary equilibrium must be symmetric. Indeed, if the equilibrium was not symmetric,

because the game is symmetric with respect to all intermediaries, it would be possible to permute

the strategies among intermediaries and obtain a different equilibrium, hereby contradicting the

uniqueness of the equilibrium.

We now characterize the intermediary equilibrium best response. To avoid the trivial case where

the quantity produced equals zero, we assume the equilibrium production quantity is nonzero. In

this case, for the symmetric equilibrium, the first-order optimality conditions for the intermediary

are: pi − (s1 + s2Q/S) − s2Q/(SI) = 0, where Q is the aggregate intermediary order quantity,

aggregated over all intermediaries. Hence, the intermediary price can be written at equilibrium as

pi = s1 + s2I +1

SIQ. (7)

Note that the intermediary margin is therefore mi = pi − ps = s2Q/(SI), and the intermediary

profit is

πi =mi

Q

I=

s2Q2

SI2. (8)


Step 2. The retailer equilibrium. We first show that the retailer equilibrium exists, is unique,

and symmetric. The kth retailer decision is

maxqr,k

[d1 − d2(qr,k +Qr,−k)−

(s1 + s2

I +1

SI(qr,k +Qr,−k)

)]qr,k. (9)

It is easy to see from (9) that the retailer problem is a strictly concave problem that can be

equivalently rewritten as an LCP. Hence, the retailer order vector qr = (qr,1, . . . , qr,R) is a retailer

equilibrium if and only if it solves the following LCP, which is obtained by concatenating the LCPs

characterizing the optimal strategy of theR retailers: 0≤ (−d1+s1)e+(d2 + s2(I +1)/(SI))Mrqr ⊥qr ≥ 0, where e is the R-dimensional vector of ones, and Mr ∈RR×RR is a positive definite matrix

whose diagonal elements are all equal to two and whose off-diagonal elements are all equal to one.

Thus this LCP has a unique solution which is the unique retailer equilibrium. Using an argument

similar to the intermediary equilibrium, the retailer equilibrium must be symmetric.

We now characterize the retailer equilibrium. To avoid the trivial case where the quantity

produced equals zero, we focus on the more interesting case with non zero quantities. For the

symmetric equilibrium, the first-order optimality conditions for the retailer can be written as

d1 − s1 − (d2 + s2(I +1)/(SI)) (R+ 1)qr,k = 0, and therefore assuming d1 ≥ s1, we have that the

optimal retailer order quantity is

qr,k =d1 − s1

(R+1)(d2 + s2

I+1SI

) =SI

(d2SI + s2(I +1))

d1 − s1(R+1)

, (10)

the intermediary price is

pi = s1 + s2R(I +1)

d2SI + s2(I +1)

d1 − s1(R+1)

,

and the retailer profit is

πr =

[d1 − s1 − (d2 + s2

I +1

SI)

R

R+1

d1 − s1(d2 + s2

I+1SI

)]

1

R+1

d1 − s1(d2 + s2

I+1SI

) =SI(d1 − s1)

2

(d2SI + s2(I +1)) (R+1)2.

(11)

Step 3. Derivation of the final results. It follows from (10) that

Q=Rqr,k =RSI

(d2SI + s2(I +1))

d1 − s1(R+1)

. (12)

Since qs,j = Q/S, the expression for the supply price follows from (1). The expression for the

intermediary price follows from (7) and (12). The retailer price is obtained by substituting (12)

into (4). Expressions for mi = pi − ps and mr = pr − pi are obtained by direct substitution. Using

qs,j = Q/S, (2) and (12), we obtain the supplier profit. Substituting (12) into (8) leads to the

expression for the intermediary profit. The expression for πr was found in (11). Finally, the total

aggregate profit follows from straightforward algebra.

Proof of Theorem 4.2. The results follows from straightforward calculus from the expressions

in Table 1.

Proof of Proposition 4.3. The objective of a central planner is to maximize the sum of the

profits of all supply chain members:

max πc = S(pS − s1 − s22qS)qS + I(pi − pS)qI +R(pr − pi)qR


where qS, qI and qR are respectively the quantities selected by each supplier, intermediary and

retailer. Since Q= SqS = IqI =RqR, the central planner’s problem is equivalent to

max πc = (pr − s1 − s22

Q

S)Q= (d1 − d2Q− s1 − s2

2

Q

S)Q.

The first-order optimality conditions are d1 − s1 − 2 (d2 + s2/(2S))Q = 0, which result in Q =

(d1 − s1)/(2d2 + s2/S). Therefore the total supply chain profit in the centralized chain is

πc =

[d1 − s1 −

(d2 +

s22S

) d1 − s12d2 +

s2S

]d1 − s12d2 +

s2S

=(d1 − s1)

2

2(2d2 +s2S). (13)

Proof of Theorem 4.4. Equation (5) is obtained by direct substitution of the aggregate profit

given in Table 1 and (13). The monotonicity properties follow by applying straightforward algebra

to the partial derivatives of the efficiency with respect to R, S, and I.

Proof of Proposition 5.1. The proof consists of three steps. First, we use the intermediary equi-

librium property on the intermediary price to reformulate the kth retailer maximization problem

in terms of his quantity decision in one scenario (namely, the case s2 = sH2 ) only. Second, we use

first order conditions and symmetry to solve this optimization problem. Third, we use this solution

to determine all equilibrium quantities.

Step 1: Reformulation. We have

E[πr,k] = ν(d1 − d2(q

Lr,k +QL

r,−k)− pi)qLr,k +(1− ν)

(d1 − d2(q

Hr,k +QH

r,−k)− pi)qHr,k.

For each possible realization of the random variable s2, the retailers anticipate the intermediary

equilibrium as given by Theorem 4.1 and Table 1. Accordingly, they anticipate the intermediaries

best response to a price pi selected by the retailers by selecting aggregately quantity QL with

probability ν and QH with probability 1− ν such that pi = s1 + sL2 (I + 1)QL/(SI) and pi = s1 +

sH2 (I +1)QH/(SI). Note in particular that this implies sH2 QH = sL2Q

L. It follows that

E[πr,k] = ν

(d1 − d2(q

Lr,k +QL

r,−k)− s1 − sL2I +1

SI(qLr,k +QL

r,−k)

)qLr,k

+(1− ν)

(d1 − d2(q

Hr,k +QH

r,−k)− s1 − sH2I +1

SI(qHr,k +QH

r,−k)

)qHr,k.

Moreover, because qLr,k = sH2 (qHr,k+QH

r,−k)/sL2 −QL

r,−k, the retailers’ objective function can be written

as

maxqHr,k

ν

(d1 − d2

sH2sL2

(qHr,k +QHr,−k)− s1 − sH2

I +1

SI(qHr,k +QH

r,−k)

)(sH2sL2

(qHr,k +QHr,−k)−QL

r,−k

)

+(1− ν)

(d1 − d2(q

Hr,k +QH

r,−k)− s1 − sH2I +1

SI(qHr,k +QH

r,−k)

)qHr,k.

Step 2: Solving the optimization problem. It is easy to see that the retailer problem is a

strictly concave problem and hence it has a unique solution qHr,k such that the first order conditions

hold:

0 = νsH2sL2

(d1 − d2

sH2sL2

(qHr,k +QHr,−k)− s1 − sH2

I +1

SI(qHr,k +QH

r,−k)

)

− ν

(sH2sL2

(qHr,k +QHr,−k)−QL

r,−k

)(d2

sH2sL2

+ sH2I +1

SI

)

+(1− ν)

(d1 − d2(q

Hr,k +QH

r,−k)− s1 − sH2I +1

SI(qHr,k +QH

r,−k)

)− (1− ν)qHr,k

(d2 + sH2

I +1

SI

).


Using symmetry and the relation sL2QL = sH2 Q

H , which implies that at equilibrium QLr,−k =

(sH2 /sL2 )(R− 1)qHr,k, the first order conditions can be rewritten as

νsH2sL2

(d1 − s1 −

(d2

sH2sL2

+I +1

SIsH2

)(R+1)qHr,k

)+(1− ν)(d1− s1−

(d2 +

I +1

SIsH2

)(R+1)qHr,k = 0,

which results in

QH =R

R+1(d1 − s1)

1− ν+ νsH2sL2

νsH2sL2

(d2

sH2sL2

+ I+1SI

sH2

)+(1− ν)

(d2 +

I+1SI

sH2) .

Step 3: Determination of equilibrium quantities. It follows that

QL =R

R+1(d1 − s1)

ν+(1− ν)sL2sH2

(1− ν)sL2sH2

(d2

sL2sH2

+ I+1SI

sL2

)+ ν(d2 +

I+1SI

sL2 ).

The expressions for pi, ps, pr, mr, mi, πs, πi and πr follow similarly to the symmetric information

case by using the realized price sensitivity (sL2 or sH2 ) and realized cumulative quantity (QL or QH).

Proof of Proposition 5.2. In this proof, for each quantity in the table, we determine the sign of

the difference between the symmetric and asymmetric cases. Let QL,a be the aggregate quantity

produced with asymmetric information and QL,s the aggregate quantity produced with symmetric

information when s2 = sL2 . We have

QL,a −QL,s =R(d1 − s1)

R+1

ν+(1− ν)

sL2sH2

(1− ν)sL2sH2

(d2

sL2sH2

+ I+1SI

sL2

)+ ν(d2 +

I+1SI

sL2 )− 1

d2 + sk2I+1SI

=R(d1 − s1)

R+1

(1− ν)sL2sH2

d2

(1− sL2

sH2

)((1− ν)

sL2sH2

(d2

sL2sH2

+ I+1SI

sL2

)+ ν(d2 +

I+1SI

sL2 ))(

d2 + sL2I+1SI

) ,

and because sL2 < sH2 , it follows that QL,a −QL,s > 0. After substituting sH2 for sL2 and 1− ν for ν

and vice-versa, we obtain

QH,a −QH,s =R(d1 − s1)

R+1

νsH2sL2

d2

(1− sH2

sL2

)(ν

sH2sL2

(d2

sH2sL2

+ I+1SI

sH2

)+(1− ν)(d2 +

I+1SI

sH2 ))(

d2 + sH2I+1SI

) ,

thus QH,a −QH,s > 0.

The comparisons for ps, pr, pi, mi, πs and πi follow from the aggregate quantity comparison and

Table 1.

Let mL,ar be the retailer margin with asymmetric information and mL,s

r the retailer margin with

symmetric information when s2 = sL2 . We have

mL,ar −mL,s

r = d1 − s1 −(d2 + sL2

I +1

SI

)d1 − s1R+1

Rν+(1− ν)

sL2sH2

(1− ν)sL2sH2

(d2

sL2sH2

+ I+1SI

sL2

)+ ν(d2 +

I+1SI

sL2 )− d1 − s1

R+1

=d1 − s1R+1

R(1− ν)sL2sH2

d2

(sL2sH2

− 1)

(1− ν)sL2sH2

(d2

sL2sH2

+ I+1SI

sL2

)+ ν(d2 +

I+1SI

sL2 ),


and thus mL,ar <mL,s

r . Similarly, we obtain

mH,ar −mH,s

r =d1 − s1R+1

RνsH2sL2

d2

(sH2sL2

− 1)

νsH2sL2

(d2

sH2sL2

+ I+1SI

sH2

)+(1− ν)(d2 +

I+1SI

sH2 ),

and therefore mH,ar >mH,s

r .

Let πL,ar be the retailer profit with asymmetric information and πL,s

r the retailer profit with

symmetric information when s2 = sL2 . To ease exposition, let us denote γ = ν + (1 − ν)sL2 /sH2 ,

δ= sL2 /sH2 , α= d2 + sL2 (I +1)/(SI) and η= d2δ+ sL2 (I +1)/(SI). We have

πL,sr −πL,a

r =

(d1 − s1R+1

)21

α− γ

(1− ν)δη+ να

d1 − s1R+1

(d1 − s1 −α

γ

(1− ν)δη+ να

R(d1 − s1)

R+1

)

=

(d1 − s1R+1

)2

δ(1− ν)(α− η)(1− ν)δ(αR− η)+αν(R− 1)

α((1− ν)δη+ να)2,

after simplification. Note that α> η and therefore αR− η > 0 and it follows that πL,sr >πL,a

r .

For the case s2 = sH2 , it suffices to substitute sH2 for sL2 and 1− ν for ν and vice-versa, so we

redefine γ = (1− ν) + νsH2 /sL2 , δ = sH2 /s

L2 , α = d2 + sH2 (I + 1)/(SI) and η = d2δ + sH2 (I + 1)/(SI)

and we obtain

πH,sr −πH,a

r =

(d1 − s1R+1

)2

δν(α− η)νδ(αR− η)+α(1− ν)(R− 1)

α(νδη+(1− ν)α)2.

Note that now α< η and so we want to determine the sign of

∆πHr =−νδ(αR− η)−α(1− ν)(R− 1)

= ν(−δ(αR− η)+α(R− 1))−α(R− 1).

Because α< η, the sign of αR−η (and thus of ∆πHr ) varies depending on the input parameters. In

the case when the ratio δ= sH2 /sL2 is sufficiently large, then αR−η < 0. This occurs for sH2 /s

L2 > δ0,

where δ0 =R+ sH2 (I +1)(R− 1)/(d2SI). In this case, −δ(αR− η)+α(R− 1)> 0 and so ∆πHr < 0

when ν < ν0, and ∆πHr > 0 when ν > ν0, where

ν0 =α(R− 1)

−δ(αR− η)+α(R− 1)(∈ [0,1]).

Notice that if the ratio sH2 /sL2 is below the threshold δ0, then αR− η ≥ 0, ∆πH

r ≤ 0 and so πH,sr −

πH,ar ≤ 0 for all values of ν ∈ [0,1].

Proof of Proposition 5.3. If s2 = sL2 , the actual intermediary profit can be written as

πi =R2(d1 − s1)

2sL2SI2(R+1)2

(ν+(1− ν)δ

(1− ν)δ(d2δ+I+1SI

sL2 )+ ν(d2 +I+1SI

sL2 )

)2

,

where δ = sL2 /sH2 . It is easy to find that the derivative of the intermediary profit is, after simplifi-

cations,

∂πi

∂S=

R2(d1 − s1)2sL2

S2I2(R+1)2

(ν+(1− ν)δ


sL2 )+ ν(d2 +I+1SI

sL2 )

)2

×(−1+

2sL2 (I +1)

SI

ν+(1− ν)δ


sL2 )+ ν(d2 +I+1SI

sL2 )

).


It is clear that the intermediary profit is unimodal and reaches its maximum at (after simplifica-

tions)

SL =sL2 (I +1)

d2I

ν+(1− ν)δ

ν+(1− ν)δ2.

The result follows from the fact that for the case with symmetric information, Theorem 4.2 shows

that the intermediary profits are maximized for S = s2(I +1)/(d2I) and δ= sL2 /sH2 < 1.

The case when s2 = sH2 follows similarly after substituting sH2 for sL2 and ν for 1− ν (and vice-

versa).

Appendix B: Tables


Table

1Equilibrium

quantities

withcomplete

andasymmetricinform

ation

This

table

gives

theequilibrium

quantities

forthecaseswith

complete

and

asymmetricinform

ation.Thefirstcolumn

inthetable

liststhe

differen

tequilibrium

quantities

reported

:theaggregate

supply,thesupply

price,theinterm

ediary

price,theretailprice,theinterm

ediary

margin,

theretailer

margin,theaggregate

supplier

profit,theaggregate

interm

ediary

profit,theaggregate

retailer

profit,andtheaggregate

supply

chain

profit.

Thesecondcolumnreportsthedifferen

tsymbols

usedto

representtheequilibrium

quantities.Thethirdandfourthcolumnsgivethe

expressionfortheequilibrium

quantity

inthecase

withcomplete

inform

ation—

thethirdcolumnin

term

sofaggregate

quantity

andthefourth

columnafter

substitutingtheaggregate

quantity.Thefifthcolumngives

theexpressionoftherealizedequilibrium

quantity

forthecase

with

asymmetricinform

ationasafunctionofrealizedaggregate

quantity.Theseresultsare

proven

inTheorem

4.1

andProposition5.1.

Complete

inform

ation

Asymmetricinform

ation

Quantity

Symbol

Interm

sofQ

Closed-form

Interm

sofQ

Aggregate

quantity

QQ

RSI

(d2SI+s2(I

+1))

d1−s1

(R+1)

Q

Supply

price

ps

s 1+

s2 SQ

s 1+s 2

RI

d2SI+s2(I

+1)d1−s1

(R+1)

s 1+

s2 SQ

Interm

ediary

price

pi

s 1+

s2 S

I+1

IQ

s 1+s 2

R(I

+1)

d2SI+s2(I

+1)d1−s1

(R+1)

s 1+

s2 S

I+1

IQ

Retailprice

pr

d1−d2Q

d1−d2

RSI

(d2SI+s2(I

+1))

d1−s1

(R+1)

d1−d2Q

Interm

ediary

margin

mi

s2Q

SI

s 2R

d2SI+s2(I

+1)d1−s1

(R+1)

s2Q

SI

Retailer

margin

mr

d1−s1

R+1

d1−s1

R+1

d1−s 1

−( d

2+

s2 S

I+1

I

) Q

Agg.Supplier

profit

Sπs

s2Q

2

2S

s 2SR

2I2

2(d

2SI+s2(I

+1))

2(d

1−s1)2

(R+1)2

s2Q

2

2S

Agg.Interm

.profit

Iπi

s2Q

2

SI

s 2ISR

2

(d2SI+s2(I

+1))

2(d

1−s1)2

(R+1)2

s2Q

2

SI

Agg.Retailer

profit

Rπr

d1−s1

R+1Q

RSI

(d2SI+s2(I

+1))

(d1−s1)2

(R+1)2

( d1−s 1

−( d

2+

s2 S

I+1

I

) Q) Q

Agg.ch

ain

profit

πa

Sπs+Iπi+RπR

RSI[s

2(R

I+2R)+

2(d

2SI+s2(I

+1))]

2(d

2SI+s2(I

+1))

2(d

1−s1)2

(R+1)2

Sπs+Iπi+RπR


Table

2Comparisonoftheequilibrium

forthemodelsbyC&K,Choi[1991],andtheproposedmodel.

This

table

gives

theequilibrium

quantities

forvariants

ofthemodelsproposedbyC&K

andChoi[1991],aswellasforourproposedmodel.Thefirst

columnin

thetable

liststhedifferen

tequilibrium

quantities

reported

:theaggregate

supply

quantity,thesupply

price,theinterm

ediary

price,theretail

price,thesupplier

margin,theinterm

ediary

margin,theretailer

margin,theaggregate

supplier

profit,

theaggregate

interm

ediary

profit,

theaggregate

retailer

profit,theaggregate

supply

chain

profit,andtheeffi

cien

cy.Thesecond,third,andfourthcolumnsgivetheexpressionoftheequilibrium

quantity

fortheva

riants

ofthemodelsproposedbyC&K

andChoi[1991],andforourproposedmodel,resp

ectively.

Note

thatforourproposedmodel

thesupplier

marginalopportunitycost

isnotconstant,

andthuswereport

theav

eragesupplier

margin.

Quantity

Corbettan

dKarmarkar

Choi

Retailers

lead

Agg

regatequan

tity

RSI

d2(S

+1)(I+1)d1−s1

R+1

RSI

d2(S

+1)(I+1)d1−s1

R+1

RSI

(d2SI+s2(I

+1))

d1−s1

R+1

Supply

price

d1−

d2(R

+1)(I+1)

RI

Qs 1

+d2 SQ

s 1+

s2 SQ

Interm

ediary

price

d1−

d2(R

+1)

RQ

s 1+

d2 SS+I+1

IQ

s 1+

s2 SI+1

IQ

Retailprice

d1−d2Q

d1−d2Q

d1−d2Q

Supplier

margin

d1−s1

S+1

d2Q

Ss2Q

2S

Interm

ediary

margin

d2Q(R

+1)

RI

d2Q(S

+1)

SI

s2Q

SI

Retailermargin

d1−s1

R+1

IS

(I+1)(S+1)

d1−s1

R+1

d1−s1

R+1

Agg

.Supplier

profit

d1−s1

S+1Q

d2Q

2

Ss2Q

2

2S

Agg

.Interm

.profit

d2Q

2(R

+1)

RI

d2Q

2(S

+1)

SI

s2Q

2

SI

Agg

.Retailerprofit

d2Q

2

Rd1−s1

R+1Q

d1−s1

R+1Q

Agg

.supply

chainprofit

Sπs+Iπi+RπR

Sπs+Iπi+RπR

Sπs+Iπi+RπR

Efficien

cy4SIR(R

I+RS+SI+R+S+I+1)

(S+1)2

(I+1)2

(R+1)2

4SIR(R

I+RS+SI+R+S+I+1)

(S+1)2

(I+1)2

(R+1)2

RI[s

2R(I

+2)+

2(d

2SI+s2(I

+1))]

(d2SI+s2(I

+1))

22d2S+s2

(R+1)2


Table 3 Monotonicity properties

This table gives the monotonicity properties of the equilibrium quantities. The first column in the table lists thedifferent equilibrium quantities for which we report the monotonicity properties: the aggregate supply, the supplyprice, the intermediary price, the retail price, the intermediary margin, the retailer margin, the supplier profit, theintermediary profit, and the retailer profit. The second column reports the different symbols used to represent theequilibrium quantities. The next five columns give the monotonicity relation of each equilibrium quantity to thenumber of suppliers (S), the number of intermediaries (I), the number of retailers (R), the supply sensitivity (s2),and the demand sensitivity (d2), respectively, where the symbol “+” (“−”) indicates that the equilibrium quantityincreases (decreases) with respect to the parameter, and “∩” indicates that the equilibrium quantity is unimodal withrespect to the parameter. The results in this table are proven in Theorem 4.2.

Quantity Symbol S I R s2 d2

Aggregate quantity Q + + + − −Supply price ps − + + + −Intermediary price pi − − + + −Retail price pr − − − + −Intermediary margin mi − − + + −Retailer margin mr 0 0 − 0 0

Supplier profit πs − + + ∩ −Intermediary profit πi ∩ − + ∩ −Retailer profit πr + + − − −

Table 4 Effect of asymmetric information on equilibrium quantities

This table shows the effect of asymmetric information about supply sensitivity on the equilibrium quantities. The firstcolumn lists the different equilibrium quantities for which we report the relation: the aggregate supply, the supplyprice, the intermediary price, the retail price, the intermediary margin, the retailer margin, the supplier profit, theintermediary profit, and the retailer profit. The second column reports the different symbols used to represent theequilibrium quantities. The third column shows the effect for the case where the realized sensitivity is low (s2 = sL2 ).The fourth column for the case where the realized sensitivity is high and the prior probability of low sensitivity is low(s2 = sH2 and ν < ν0). The fifth column for the case where the realized sensitivity is high and the prior probability oflow sensitivity is high (s2 = sH2 and ν > ν0). The symbol “+” (“−”) indicates that the equilibrium quantity is larger(smaller) in the supply chain with asymmetric information. The results in this table are proven in Proposition 5.2.

Quantity Symbol Low sensitivity High sensitivity High sensitivity

ν < ν0 ν > ν0

Aggregate quantity Q + − −Supply price ps + − −Intermediary price pi + − −Retail price pr − + +

Intermediary margin mi + − −Retailer margin mr − + +

Supplier profit πs + − −Intermediary profit πi + − −Retailer profit πr − + −


Appendix C: Figures

Figure 1 Equilibrium prices and profits when retailers underestimate supply sensitivity

This figure depicts the equilibrium prices and profits in the presence of asymmetric information when retailersunderestimate the true (realized) sensitivity. We assume d2 = 1, s1 = 3, d1 = 5, I = 2, S = 3,R = 2, sH2 =6, sL2 = 1. The horizontal axis gives the retailers belief probability ν that the supply sensitivity is low (ν =0 corresponds to the case with complete information, and the larger ν, the more severely the retailersunderestimate the supply sensitivity). The left vertical axis gives the realized retail and intermediary prices,and the right vertical axis gives the realized aggregate retailer and intermediary profits. A vertical lineindicates the value of the probability such that the retailer profits with asymmetric information equal theretailer profits with complete information, and a horizontal line indicates the value of the retailer profitswith complete information.

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Supplemental file


Appendix D: The case with convex increasing marginal opportunity cost

We now show that πi is unimodal in S even for the following more general marginal opportunity

cost function:

ps(q) = s1 + s2qθ; for θ≥ 1.

Note that this is a convex increasing monomial function.

The supplier’s profit is:

πs,j = ps(qs,j)qs,j −∫ qs,j

0

(s1 + s2qθ)dq=

s2θqθ+1s,j

θ+1.

The intermediary’s decision is:

maxqi,l

[pi −

[s1 + s2

(qi,l +Qi,−l

S

)θ]]

qi,l.

The first-order conditions imply:

pi −[s1 + s2

(qi,l +Qi,−l

S

)θ]− s2θ

S

(qi,l +Qi,−l

S

)θ−1

qi,l = 0;

which implies:

pi = s1 + s2

(Q

S

)θ

+s2θ

I

(Q

S

)θ

;

and the margin, mi of the intermediary is:

mi = pi − ps =s2θ

I

(Q

S

)θ

;

and the profit, πi, of each intermediary is:

πi =mi

Q

I=

s2θS

I2

(Q

S

)θ+1

.

The retailer’s decision problem can be stated as:

maxqr,k

(pr − pi)qr,k =

[d1 − d2(qr,k +Qr,−k)−

[s1 + s2

(Q

S

)θ

+s2θ

I

(Q

S

)θ]]

qr,k.

The first order conditions then imply:

Φ(Q,S) = d2

(1+

1

R

)Q+ s2

(1+

θ

I

)(1+

θ

R

)(Q

S

)θ

− (d1 − s1) = 0.

Now we are ready to determine whether πi is unimodal in S. We first calculate dπi/dS:

dπi

dS=

(∂πi

∂Q

)dQ

dS+

∂πi

∂S=

(Q

S

)θs2θ

I2

[(θ+1)

dQ

dS− θ

Q

S

];


where dQ/dS can be determined using the following relationship:

dΦ

dS=

∂Φ

∂Q

dQ

dS+

∂Φ

∂S= 0;

which implies:

dQ

dS=

s2θ(1+ θ

I

) (1+ θ

R

)(Qθ

Sθ+1

)

d2(1+ 1

R

)+ s2θ

(1+ θ

I

) (1+ θ

R

)(Qθ−1

Sθ

) =QS

1+

(d2(R+1)

s2θR(1+ θI )(1+

θR)

)(Sθ

Qθ−1

) <Q

S. (14)

Substituting the above relationship in the expression for dπi/dS and evaluating the terms we

conclude that dπi/dS < 0 if and only if dQ/dS < θQ/((θ+1)S) or equivalently, if and only if:

1

1+

(d2(R+1)

s2θR(1+ θI )(1+

θR)

)(SQ

)θ−1

S

<θ

θ+1.

Also note that:

d(QS

)

dS=

1

S

(dQ

dS− Q

S

)< 0 =⇒

d(

SQ

)

dS> 0.

Hence, it is easy to verify that:

d

dS

1

1+

(d2(R+1)

s2θR(1+ θI )(1+

θR)

)(SQ

)θ−1

S

< 0.

This implies that, as S is increased, then dπi/dS can change sign from positive to negative at most

once. In other words πi is unimodal in S.

We also show that for this marginal opportunity cost function the retailer margin does depend on

the number of suppliers and of intermediaries, as opposed to the case of a linear function. Indeed,

mr = d1 − d2Q− s1 − s2

(1+

θ

I

)(Q

S

)θ

anddmr

dS=

∂mr

∂Q

dQ

dS+

∂mr

∂S,

where∂mr

∂S=

s2S

(1+

θ

I

)(Q

S

)θ

,

and∂mr

∂Q=−d2 − s2

S

(1+

θ

I

)(Q

S

)θ−1

.

Since both of the two expressions above are independent of R but dQ/dS does depend on R when

θ > 1, as is apparent from (14), it is clear that dmr/dS 6= 0 in general.

Similarly,dmr

dI=

∂mr

∂Q

dQ

dI+

∂mr

∂I,


where∂mr

∂I=

s2θ

I2

(Q

S

)θ

.

dQ/dI can be deduced from the first order conditions

Ψ(Q,I) = d2

(1+

1

R

)Q+ s2

(1+

θ

I

)(1+

θ

R

)(Q

S

)θ

− (d1 − s1) = 0

and the relationship:dΨ

dI=

∂Ψ

∂Q

dQ

dI+

∂Φ

∂I= 0;

which implies:

dQ

dI=

s2θI2

(1+ θ

R

) (QS

)θ

d2(1+ 1

R

)+ s2θ

(1+ θ

I

) (1+ θ

R

)(Qθ−1

Sθ

) =QI2

1+ θI+(

d2(R+1)

s2θ(R+θ)

)(Sθ

Qθ−1

) . (15)

Since both ∂mr/∂I and ∂mr/∂Q are independent of R but dQ/dI does depend on R when θ > 1,

as is apparent from (15), it is clear that dmr/dI 6= 0 in general.

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