Should an OEM Retain Component Procurement when the CM

Produces Competing Products?∗

Abstract

We consider a large original equipment manufacturer (OEM) who relies on a contract manufac-

turer (CM) to produce her product. In addition to the OEM’s product, the CM also produces for

a smaller OEM. The two products not only are engaged in Cournot competition in the consumer

market, but also require a common critical component. Both the larger OEM and the CM can

purchase the component from the supplier, but their purchase prices may differ and are unknown

to each other. In contrast, the smaller OEM has to rely on the CM to procure components for her.

The main question we address is whether the larger OEM should retain component procurement

by purchasing components from the supplier and reselling to the CM (buy-sell), or outsource

component procurement by letting the CM purchase directly from the supplier (turnkey).

We show that, under buy-sell, the larger OEM’s optimal strategy is to resell components at

the highest possible component purchase price of the CM (i.e., the street price). This result of

masking to the street price is consistent with recent industry practices by HP and Motorola. By

comparing buy-sell and turnkey, we find that a CM with low component price is better off under

turnkey, even though under buy-sell he receives more profits through the products sold to the

smaller OEM. Furthermore, the larger OEM’s preference between buy-sell and turnkey depends

on her component price, the volatility of the CM’s component price and substitutability between

the two products.

Keywords: buy-sell, price masking, turnkey, information asymmetry

∗We thank the review team for their detailed comments and many valuable suggestions that have significantlyimproved the quality of the paper. All remaining errors are our own.

1

1 Introduction

While outsourcing production to contract manufacturers (CMs), many original equipment manufac-

turers (OEMs) retain in-house component procurement by using the so called buy-sell process, under

which an OEM buys components from a supplier and resells to her CM (Amaral et al. (2006)). Buy-

sell is one way to accomplish price masking, where the OEM charges her CM for the component at a

higher price than her privileged price offered by the supplier (Carbone (2004); Purchasing (2003))1.

As recognized by practitioners, the CM, armed with the OEM’s pricing information, could lower the

CM’s costs of components that are used to produce products competing with the OEM’s2. Conse-

quently, the OEM loses her competitive advantage of her privileged component pricing (Jorgensen

(2004)). As a countermeasure, price masking allows the OEM to keep her component cost at her

privileged price while ensuring such price information is not disclosed to her CM. While buy-sell is a

common practice in industries, there are also many OEMs who leave procurement responsibility to

the contract manufacturers. For example, Dell decided to delegate procurement of components for

its laptops PCs to its Taiwanese contract manufacturers (CENS (2007)). In this scenario, referred

to as turnkey, the OEM delegates control of upstream activities to the CM. Under turnkey, the

CM is entirely responsible for all activities prior to final delivery of the end product to the OEM,

including component procurement and product assembly. Many smaller OEMs collaborate with the

CM through turnkey (Sullivan (2003)).

In this paper, we focus on the study of buy-sell and turnkey. We study a model in which a

large OEM relies on a CM to assemble her product. At the same time, the CM also assembles a

competitive product for a smaller OEM. In practice, there are numerous examples where large OEMs

such as HP or Motorola outsource production to a CM that also produces competitive products for

smaller OEMs. For example, Flextronics was producing cellular phones for Motorola (Dignan (2000),

Flextronics (2009), VentureOutsource (2008)) when it started producing cellular phones for Kyocera

Wireless Corporation (Kyocera (2006), VentureOutsource (2008)).

It is important to note that there are other approaches widely adopted in the procurement prac-

tice. For example, the supplier rebate is an alternative approach through which OEMs collaborate1A flow chart of the transaction flow in the buy-sell price-masking process can be found on page 9 of CSCMP (2006):

The OEM inflates the component price to a higher price which is often the street price; The CM places an order ofcomponents at the inflated price to the OEM (payment is made from the CM to the OEM), who then sends the orderto the supplier at her negotiated price; The components are shipped directly from the supplier to the CM; Finally, theOEM pays the CM for the final product.

2To illustrate this point, the CM could use the OEM’s pricing information as his baseline to shop around for lowestpossible price that could force the component suppliers to compete with each other (Jorgensen (2003)).

2

with CMs. Under supplier rebates, the supplier charges the CM the market price and rebates to

the OEM the difference between the street price and the OEM’s preferential price. This approach

allows the OEM to hide her privileged price from the CM, but it also presents some disadvantages

(Amaral et al. (2006); Metty (2006)). The supplier rebate requires extra accounting effort in mon-

itoring and processing rebates. A sophisticated information system is needed to track the rebate

when the CM purchases the same component for the products of different OEMs. In addition, since

the supplier pays the rebate back to the OEM some time after the OEM and the CM pay for the

components, the supplier rebate essentially results in free loans from the OEM to the supplier and

improves the supplier’s cash flow at the expense of the OEM’s. Because of these reasons, buy-sell

was found to be more appealing, and several large OEMs, such as HP and Motorola, widely adopted

buy-sell in their procurement processes. Another approach for OEMs to collaborate with CMs is

consignment, under which the OEM buys and owns components that are used by the CM. However,

this approach is considered risky by practitioners since the CM will not have the incentive to reduce

excess inventories.

We use the Cournot model to study the competition between the products of the two OEMs

in the consumer market: the price of each product is influenced by both its quantity and the other

product’s quantity. In particular, the larger OEM is the market leader and she decides her quantity

before the smaller OEM. We assume that every product (regardless of the brand) requires a common

critical component that can be purchased from a supplier. The larger OEM and the CM can purchase

components from the supplier, but they do not know each other’s procurement cost. On the other

hand, the smaller OEM has to rely on the CM to purchase components for her products. We first

study the case in which the larger OEM adopts buy-sell strategy. We characterize her optimal

contract with the CM and identify closed form solution for the profits of her and the CM under

buy-sell. We will primarily take the larger OEM’s standpoint and in the sequel refer to her as “the

OEM” while articulating the OEM’s preferences and incentives.

We show that if the products of the two OEMs do not compete, the larger OEM can achieve

the same profit by selling components to the CM at her privileged price or at any inflated price.

However, as long as the larger OEM faces competition from the smaller OEM’s products, the former

should inflate the component price when reselling to the CM. Specifically, according to our results,

it is optimal for the larger OEM to resell components to the CM at the CM’s highest possible

procurement cost. Under this policy, despite a higher CM’s cost in producing products of the larger

3

OEM, the quantity is not reduced because the larger OEM still controls the quantity through her

contract with the CM. On the other hand, by increasing the cost of the CM’s product, reselling

components to the CM at an inflated price increases not only the total but also the larger OEM’s

share of profits from her product. This result is consistent with HP’s and Motorola’s practice of

price masking through buy-sell.

After studying the OEM’s optimal component price when selling to the CM, the next question is

when the OEM is better off under buy-sell than under turnkey. To address this question, we continue

to study the turnkey scenario in which the CM procures components directly from the supplier for

the products of not just the smaller OEM but also the larger OEM. We observe an interesting

two-fold feature of the CM’s component procurement cost in the turnkey scenario from the OEM’s

perspective. First, it represents the cost of the OEM’s products. Second, it also represents the cost

of the competitive products for the OEM. Thus, the CM’s procurement cost provides information

about both the supply and demand of the OEM’s products. We characterize the larger OEM’s

optimal contract with the CM. We also identify closed form solutions for the profits of her and the

CM under turnkey, and compare with those under buy-sell.

The comparison between the turnkey and buy-sell scenarios generates insights regarding the

incentives of the larger OEM and the CM to commit to buy-sell and turnkey. Our results show

that buy-sell is always more preferable when volatility of the CM’s cost increases. This is driven

by the benefit of cost transparency in buy-sell over turnkey. Our results also suggest that the

impact of substitutability between the two products depends on the OEM’s negotiated price with

the component supplier. When the larger OEM’s own procurement cost is low, substitutability

between the two products amplifies the advantage of buy-sell. However, when her own procurement

cost is high, substitutability also amplifies the disadvantage of buy-sell, making turnkey even more

preferable. In other words, competition from the smaller OEM makes the choice between buy-sell

and turnkey more significant for the larger OEM. Thus, while it is important for an OEM who can

negotiate a good price with the supplier to maintain the preferential status using buy-sell, turnkey

can be more favorable for an OEM who cannot negotiate a good price. Whether to maintain the

OEM’s preferential status with the use of buy-sell becomes a more critical decision when her product

is more substitutable with the competitive product.

On the methodological side, this paper is distinct from the classical principal/agent model

in two aspects. First, the CM’s ability to sell his own branded product gives rise to an outside

4

option that depends on his own type (purchasing cost of the critical component). Such a type-

dependent outside option has also been investigated in a few papers, including Jullien (2000), Lewis

and Sappington (1989), and Maggi and Rodriguez-Clare (1995). Nevertheless, unlike our paper, the

outside options in these papers are mostly originated from the trading costs, the competing offers

given by the other principal, or the status quo from earlier transactions/negotiations. Second, since

the agent (CM) determines the quantity of the competitive product, his effective payoff function

after optimizing over this quantity decision is no longer differentiable. Due to this non-smooth

payoff function, the systematic approach in the standard principal-agent models cannot be directly

applied. Consequently, we develop a technique with an exhaustive use of the indicator function; it

allows us to get around with the issues arising from the non-smoothness. This technique may find

its applicability in other principal-agent contexts in which the agents possess peculiar payoffs.

The rest of this paper is organized as follows. Section 2 reviews related literature. In Section

3, we describe the model. We analyze the buy-sell scenario in Section 4 and the turnkey scenario in

Section 5. Section 6 provides some comparisons between the turnkey and buy-sell scenarios, Section

7 analyzes several extensions of the basic model, and Section 8 concludes. All the proofs are relegated

to the Appendix.

2 Literature Review

Our paper is closely related to the stream of literature that studies the impact of private cost

information in supply chain contracting. For example, Corbett and de Groote (2000) investigate

how the supplier can induce the buyer to reveal her private inventory holding cost by offering a

quantity discount contract. Ha (2001) studies the supplier’s optimal contracting problem when the

demand is price-sensitive and the buyer has private variable cost. Corbett et al. (2004) study a

similar setting and focus on the value of information under different contract types. Lutze and Ozer

(2008) consider whether a supplier should charge a retailer based on the promised lead times when

the retailer attempts to inflate the service level. In these papers, the upstream party cannot observe

the cost information of the downstream party. Some papers study the case when the upstream party

has private information. For example, Cachon and Zhang (2006) study a buyer who has to procure

components from a supplier with private capacity cost. They analyze the buyer’s optimal contract

and evaluate the performance of some simple contracts when there is only one potential supplier and

multiple potential suppliers. Bolandifar et al. (2010) show that for procuring from a supplier with

5

a private cost, there exist multiple contract forms that allow different allocation of demand risks.

They also investigate the efficiency of simple mechanisms using numerical studies.

Different from the aforementioned papers, we consider a supply chain setting where the upstream

party (the CM) is also a competitor of the downstream party (the OEM). In this case, private cost

information also represents private demand information. Second, we focus on the case where a

critical component has to be purchased from a supplier. Hence, the contracting problem of the OEM

is different depending on whether she procures from the supplier herself or through the CM.

Our paper falls within the stream of literature on production outsourcing (see, for example, Ca-

chon and Harker (2002), Plambeck and Taylor (2005), Feng and Lu (2010)). Operations management

researchers have recently started to pay attention to the delegation of procurement responsibility in

the OEM-CM relationship. Kayis et al. (2007) study the OEM’s contracting problem under turnkey

and buy-sell. They show that the OEM’s profits in the two scenarios are the same with the use of

arbitrarily complex contracts but are different under simple price-only contracts. Guo et al. (2006)

analyze the case when the OEM faces price-dependent demand. They show that delegation of pro-

curement always reduces the OEM’s profit in a one-period setting, but it may be profitable for the

supplier in a two-period repeated setting. Although the model setting of these two papers are similar

to ours, there are several distinctions. First, the two papers study the case when the procurement

contracts with the supplier are negotiated only after the OEM contracts with the CM. We consider

the case where the OEM and CM have existing long-term contracts with the supplier. Thus, their

focus is on how delegation of procurement affects the procurement contracts with the supplier, while

we focus on the impact of competition and pre-existing competitive advantage (or disadvantage) on

the OEM-CM relationship. In particular, we provide insights on how the existing relationship with

the supplier affects the OEM’s optimal choice of production outsourcing model. Second, they assume

that the OEM does not face any competition in the product market. In our model, the CM competes

with the OEM in the consumer market. Thus, our model provides insights on how competition in the

final product market affects the OEM’s optimal choice between buy-sell and turnkey. In particular,

we show that competition magnifies the relative advantage of buy-sell and turnkey, and makes the

choice between the two procurement models more significant.

Guo et al. (2010) study a supply chain in which an OEM outsources its production to a CM that

also produces a competing product. In terms of quantity competition games, they consider three

scenarios: simultaneous move, OEM-leadership, and CM-leadership games. By comparing across

6

the three scenarios, they identify some temptation of OEM or CM being leader or follower, and

thus any of the three scenarios can emerge as an equilibrium. The primary differences between their

paper and ours are that we introduce information asymmetry in this outsourcing context and that

we allow the OEM to pass the negotiated price to the CM endogenously through the contractual

transactions. A recent paper by Feng and Lu (2010) studies the implications of design outsourcing on

market competition in a dynamic game setting between an ODM and two competing OEMs. They

show that design outsourcing may boost the negotiated wholesale prices upwards, thereby softening

the market competition between the OEMs. They further find that strong OEMs tend to outsource

their designs when the bargaining powers between OEMs and the ODM are relatively symmetric. In

contrast, our paper investigates the competition between the OEM and the CM in a vertical supply

chain, and focuses exclusively on (production) outsourcing in the presence of information asymmetry.

Deshpande et al. (2011) approach the problem from a mechanism design perspective and propose a

mechanism between an OEM and the CM to ensure that the procurement role is always assigned to

the party who has the lowest procurement cost and that the procurement costs will not be disclosed

to the other party. On the other hand, we focus on two commonly used contractual arrangements,

buy-sell and turnkey.

Finally, our work is also related to supply chain structure issues regarding competition with a

supply chain partner. Chiang et al. (2003) and Tsay and Agrawal (2004) consider a manufacturer

who can sell through a direct channel in addition to an independent retailer. These models assume

that the channel structure has no impact on the supply side because the independent retailer is a

downstream partner and downstream competitor from the manufacturer’s perspective. In our model,

the CM is both an upstream partner of the OEM and a competitor in the product market. Hence,

supply chain structure and contracting decisions affect both the supply and demand sides of the

OEM.

3 Model

An OEM (called OEM 1) sells a product (indexed by 1) to the consumer market and relies on a CM

to produce this product. Besides OEM 1’s product, the CM also also produces a product (indexed

by 2) for another OEM (called OEM 2). The two products require a common critical component,

which can be purchased from a supplier. For simplicity we assume that the cost of procuring other

components and the assembly cost are negligible, i.e., the cost of producing a unit of product is equal

7

to the cost of purchasing the critical component3. The CM can purchase the component directly

from the supplier at a unit cost c, which is unobservable to the OEMs due to the confidential bilateral

agreement between the CM and the supplier. We assume that the CM knows the realized value of

c while the OEMs only know the distribution function of c, denoted by F (·) (with f(·) being the

density function). Assume that c ∈ [c, c] where c is referred to as the street price.

The two products compete in the consumer market; consequently, the price of each product is

determined by both its quantity and the other product’s quantity. For tractability, we adopt the

commonly-used linear inverse demand form to model how the selling prices {p1, p2} are determined

by the produced quantities {q1, q2}, where the subscript corresponds to the product index:

pi = m− aqi − bqj ,

for i, j ∈ {1, 2} and i 6= j, where a > b ≥ 0. The linear (inverse) demand functions are commonly

adopted in the literature of economics, marketing, and operations to model the competition among

differentiated products; see, e.g., Bernstein and Federgruen (2004), Farahat and Perakis (2008), and

Jeuland and Shugan (1983). To rule out trivial cases where no production occurs, we assume m is

sufficiently large relative to the street price c. In particular, we assume

(A1). m ≥ max{ 8a2 − b2

8a2 − b2 − 4ab,

4a

4a− b}c.

Because a > b, a simpler condition that ensures (A1) is m > (7/5)c.

Motivated by the examples mentioned earlier, we consider two scenarios: buy-sell and turnkey,

depending on whether or not OEM 1 is involved in the component procurement. Under buy-sell,

OEM 1 buys the component from the supplier and then resells to the CM. Under turnkey, OEM 1 is

not involved in the component procurement and the CM purchases the component directly from the

supplier. We focus on the case where OEM 2 is a small OEM and always rely on the CM to purchase

components. This may be the case when, for example, OEM 2 cannot bargain a low component cost

with the supplier while the CM can leverage a lower component cost by aggregating purchases for

different OEMs (Sullivan (2003)).3If the CM’s assembly cost is not negligible, then there are two cases. First, if the assembly cost is known by the

OEMs, then all of our subsequent results continue to hold with only minor adjustments on the transfer payments.Second, if the assembly cost is the CM’s private information, then the optimal contracts can be derived analogously,with the exception that the CM’s private type is defined to be the sum of the assembly cost and the component cost.Most of our qualitative results remain valid.

8

Because our main objective is to examine OEM 1’s performance under buy-sell and turnkey, we

assume that OEM 1 is the Stackelberg leader determining the contractual terms to maximize her

expected profit, and that the CM is the follower who accepts OEM 1’s contract if and only if his

expected profit is no less than his reservation profit. On the other hand, OEM 2 is weaker than the

CM in terms of size and market power; thus, she determines her order quantity from the CM only

after the CM signs the contract with OEM 1. We restrict the contract form between the CM and

OEM 2 to a simple wholesale price contract with the wholesale price w determined by the CM. We

have verified that all of our qualitative results also hold under the two-part tariff contract. Every

firm is risk neutral and maximizes the expected profit.

To model the CM’s reservation profit, we assume that if the CM rejects OEM 1’s contract, then

the production quantity of product 1 is zero and OEM 2 becomes a monopolist in selling her own

product. This assumption is reasonable when the CM is the sole source for OEM 1 to produce product

1 due to highly specialized skills and equipment. In Online Supplement, we relax this assumption by

considering a general reservation profit function. Specifically, we refer to the CM with purchase cost

c as the type-c CM; the type-c CM’s reservation profit is equal to his profit by refusing to produce

product 1 for OEM 1, i.e., q1 = 0. In this case, given the CM’s wholesale price w, OEM 2 solves the

following problem to determine her optimal order quantity:

maxq2≥0

{(m− aq2)q2 − wq2},

where OEM 2’s optimal order quantity is q2(w) = (m− w)+/(2a), where x+ ≡ max{x, 0}. Accord-

ingly, the type-c CM’s reservation profit is:

R(c) = maxw≥0

{(w − c)q2(w)} = (m− c)2/(8a).

Note that OEM 2’s optimal order decision is independent of the CM’s component purchase cost c.

Thus, although the game between the CM and OEM 2 becomes a signaling game (Fudenberg and

Tirole (1994)) where an informed player (the CM) moves first and the uninformed player (OEM 2)

must infer the unknown facts from the observable actions. Here, the signaling problem is rather easy

to solve because OEM 2’s payoff is independent of the CM’s private component cost c.

In the next two sections, we characterize the optimal mechanisms (from OEM 1’s perspective)

under buy-sell and turnkey, respectively. Comparisons between buy-sell and turnkey are made in

9

Section 6. For tractability, we need the following mild assumptions on the distribution of c, which

are satisfied by many common distributions such as exponential, uniform, normal, gamma, etc.

(A2). G(c) ≡ F (c)/f(c) is increasing in c;

(A3). H(c) ≡ (1− F (c))/f(c) is decreasing in c.

These monotonicity conditions are adopted to exclude the possibility of bunching (i.e., multiple

types of CMs select the same contract), see Laffont and Martimort (2002). Assumption (A3) is the

monotone hazard rate, or increasing failure rate (IFR) property. Furthermore, if a distribution is

IFR, it also has the increasing generalized failure rate property (IGFR), namely, cf(c)/(1− F (c)) is

increasing in c, see Lariviere and Porteus (2001). It can be verified that Assumption (A3), and more

generally IGFR, implies the unimodality of this “revenue function.”

4 Buy-sell

In this section, we consider the buy-sell scenario in which OEM 1 retains the procurement responsi-

bility for her own product. Specifically, OEM 1 buys the component at a unit cost co (unknown to

the CM) and resells it to the CM at a unit price p, where p ≥ co. We restrict our attention to p ≥ co

because the OEM normally resells components to the CM at a price higher than her purchase cost

(Amaral et al. (2006)).

The CM is obliged to purchase from OEM 1 the components for product 1. To produce product

2, the CM purchases components directly from a supplier. The resale price p charged by OEM 1

greatly impacts the component price that the CM can obtain for producing product 2. There are two

cases. First, if OEM 1 sells the components to the CM at a price that is higher than the negotiated

price between the supplier and the CM, i.e., p ≥ c, then OEM 1’s resale price p has no impact on

the CM’s component purchase cost. In this case, the CM purchases components at per unit cost c

to produce product 2. Second, if p < c, the CM will try to lower the component price for his own

product. A phenomenon observed in practice is that “[t]he most a CM will pay is the least it knows

someone else is paying”(Amaral et al. (2004, 2006)).

We assume that if p < c, the CM’s cost of obtaining a unit of component from the supplier

is equal to p. For example, if the CM has not come across OEM 1’s supplier (thus the CM has a

different component supplier), the CM is likely to reduce the component price for product 2 to p.

This is made possible as the CM goes back and forth between his original supplier and OEM 1’s

10

supplier if his original supplier bears a marginal cost that is lower than p. If his original supplier

has a marginal cost between p and c, it is possible that the negotiated price for product 2 may

lie somewhere between p and c. These illustrative arguments are based on the observations made

in Jorgensen (2003). Our analytical results hold in this general setting where the final negotiated

price is k(p, c), where p ≤ k(p, c) < c. Even if the CM has come across OEM 1’s supplier before,

it is mentioned by practitioners that “knowing what its OEM customer pays gives a CM additional

leverage for driving prices down”. To summarize the two cases, the CM pays OEM 1 p for every

unit of component used to produce product 1, and pays the supplier min{p, c} for every unit of

component used to produce product 2.

Our objective is to examine how OEM 1, under buy-sell, should design a mechanism to maximize

her expected profit. By the revelation principle (Laffont and Martimort (2002)), we can without loss

of generality restrict to the truth-telling menu {p, q1(·), t(·)}. Following the standard argument in

the mechanism design literature (see Laffont and Martimort (2002)), this menu of contracts can be

implemented as a quantity discount contract. Quantity discount contracts are commonly proposed

to coordinate individuals’ incentives in the literature (see Cachon and Kok (2010) and Chen et al.

(2001)), and are widely used in a variety of industries such as food (Hammond (1994)) and high-tech

products (e.g., CPU (Kanellos (2001)), DRAM, and personal computers (Vizard (2004))). Under

such a menu, the resale price is p; it is the best interest of the type-c CM to report his true cost c

and to produce q1(c) units of product 1 for OEM 1 in return for a transfer payment t(c) from OEM

1.

The sequence of events is as follows. 1) OEM 1 announces the menu {p, q1(·), t(·)}; 2) The CM

decides whether or not to participate in the contract. If the CM rejects the contract, he earns the

reservation profit by producing product 2 only. Otherwise, the CM chooses a contract {p, q1(c), t(c)}from the menu; 3) The CM purchases q1(c) units of components through OEM 1 at price p (OEM 1

pays co per unit to the supplier), and then produces q1(c) units of product 1; 4) The CM offers the

wholesale price w to OEM 2; afterwards, OEM 2 decides the number of units to produce for product

2, denoted by q2; 5) The CM purchases the components for product 2 from the supplier at unit cost

min{c, p}; 6) OEM 1 obtains from the CM q1(c) units of product 1 and pays the CM t(c); 7) Both

OEMs sell their products in the market and realize the revenues.

While the details of the contract between OEM 1 and the CM are not observable, we assume

that the final quantity q1(c) is observable by OEM 2. Theoretically, this assumption is consistent

11

with the leader-follower relationship in the Stackelberg duopoly model. In practice, this assumption

is valid in many cases where certain information regarding deals between large OEMs and CMs is

made public. For example, in 2009, Digitimes reported that HP and Flextronics signed a contract

in which the latter would produce two million netbooks for the former at a price of forty-five dollars

each (Tsai (2009)). This kind of information may be available because of reports by independent

media (see, for example, Dignan (2000)), or due to news release by either the OEM or the CM (see,

for example, Celestica (1996)).

We follow the backward analysis. We start with OEM 2’s problem. Given the wholesale price

w and the product 1’s quantity q1, OEM 2 solves the following problem to determine her optimal

order quantity:

maxq2≥0

{(m− aq2 − bq1)q2 − wq2}

Because the function inside maximization is a concave quadratic function of q2, OEM 2’s optimal

order quantity as a function of w and q1 is

q2(w, q1) = (m− w − bq1)+/(2a). (1)

Now we turn the CM’s problem in determining his optimal wholesale price. Given that the

CM’s procurement cost is c and that the CM has chosen the contract (q1(c), t(c)), the CM solves the

following problem to determine her optimal wholesale price:

maxw≥0

{(w −min(p, c))q2(w, q1(c))}+ t(c)− pq1(c)}.

By (1), the function inside maximization is a concave quadratic function of w, and thus the CM’s

optimal wholesale price is w(c, c) = [m− bq1(c)+min(p, c)]+/2 and her maximum profit, denoted by

R(c, c), is

R(c, c) =[m− bq1(c)−min(p, c)]+2

8a+ t(c)− pq1(c).

Further, by (1), under the CM’s optimal wholesale price w(c, c), OEM 2’s optimal order quantity is

q2(w(c, c), q1) = [m− bq1(c)−min(p, c)]+/(4a). (2)

Let R(c) ≡ R(c, c), which is the type-c CM’s profit under truth-telling.

Next we turn OEM 1’s problem. Given that the type-c CM truthfully reports his type, OEM

12

1’s profit is

M(c) = (p− co)q1(c) + q1(c)[m− aq1(c)− b[m− bq1(c)−min(p, c)]+/(4a)

]− t(c),

where the three parts at the right hand side represent the profit from selling components to the CM,

the revenues from the market (recall from (2) that [m − bq1(c) − min(p, c)]+/(4a) is the product

2’s quantity under the truth-telling of the type-c CM), and the payment to the CM, respectively.

Therefore, the OEM’s contract design problem, denoted by PBS , is:4

(PBS) maxp≥co,q1(·)≥0,t(·)

EcM(c)

s.t. (IC) R(c) ≥ R(c, c), ∀c, c ∈ [c, c]

(IR) R(c) ≥ R(c), ∀c ∈ [c, c]

The incentive compatibility constraint (IC) ensures that truth-telling is the CM’s best response. The

individual rationality constraint (IR) ensures the CM’s participation.

To characterize the OEM’s optimal menu, our analysis proceeds in two steps. First, take any

value p ≥ c0, we characterize in Proposition 1 the OEM’s optimal quantity-payment menu, denoted

by {qBS1 (c, p), tBS(c, p)} (which depends on p). Second, we characterize in Proposition 2 the optimal

component resale price p that maximizes the OEM’s expected profit.

Proposition 1. In the buy-sell scenario with OEM 1’s component resale price p being fixed, under

the optimal menu that maximizes OEM 1’s expected profit, the product 1’s quantity for type-c CM is

qBS1 (c, p) =

4a8a2−3b2

[m− c0 − b

2a(m− c)− b4aH(c)

]+, if c < p,

4a8a2−3b2

[m− c0 − b

2a(m− p)]+

, if c ≥ p.

Under this menu, the type-c CM’s profit, denoted by RBS(c, p), is

RBS(c, p) =

R(c) +∫ cc

b4aqBS

1 (x)dx, if c < p;

R(p) +∫ pc

b4aqBS

1 (x)dx, if c ≥ p.

4Alternatively, suppose that the CM is able to reject the menu of contracts while still using the invoice to renegotiatethe purchase price with the supplier. In such a scenario, the CM’s reservation profit becomes R(min(p, c)). It can beverified that all our analysis and results remain valid under this alternative setting.

13

and the OEM’s expected profit, denoted by ΠBS(p), is

ΠBS(p) =∫ p

c

2a

8a2 − 3b2

[m− co − b

2a(m− c)− b

4aH(c)

]+2

f(c)dc+2a[1− F (p)]8a2 − 3b2

[m− co − b

2a(m− p)

]+2

.

Several observations are noteworthy. First, qBS1 (c, p) increases in c for c < p. The intuition for

this result is as follows. When c increases for c < p, the CM’s cost of producing product 2 increases

and thus q2 decreases. This gives rise to a higher demand for product 1 since the two products

are substitutes. In response, OEM 1 increases the quantity of product 1. When c exceeds p, the

quantity of product 1 remains constant, because the CM’s actual procurement cost remains at p.

Second, if OEM 1 could observe the CM’s procurement cost c, then her optimal quantity of product

1 for the type-c CM, denoted by q1(c, p), is q1(c, p) = 4a8a2−3b2

[m− c0 − b

2a(m− c)]+

for c ≤ p and

q1(c, p) = q1(p) for c > p. We call q1(c, p) the efficient quantity. It is interesting to see how the

quantity qBS1 (c, p) in the optimal menu are distorted from the efficient quantity q1(c, p) when the

OEM lacks the CM’s cost information: qBS1 (c, p) ≤ q1(c, p) for c < p (where the equality holds only

in the uninteresting case where q1(c, p) = 0) and qBS1 (c, p) = q1(c, p) for c ≥ p. In other words,

there is downward distortion in the production quantities intended for the CMs whose bargaining

power (relative to the supplier) is strong (i.e., c < p) and no distortion for the CMs whose bargaining

power is weak (i.e., c ≥ p). This result is in contrast with the standard result in adverse selection

that distortions occur in the contracts intended for the inefficient types (e.g., the high-production-

cost type when the production cost information is private). Such a contrast emerges because in our

setting, the efficient type (i.e., the CM with lower c) also demands a higher reservation profit. In

order to ensure the participation from the efficient type whose reservation profit is high, the transfer

payment must be generous. Then such a generous transfer payment becomes attractive for the

inefficient type. To deter the inefficient type’s mimicking behavior, a viable approach is to distort

downward the quantity intended for the efficient type, because the inefficient type values less about

the reduction of quantity than the efficient type.

Third, while the CM’s profit RBS(c, p) is nonincreasing in c, the CM’s information rent (in

excess of his reservation profit), i.e., RBS(c, p) − R(c) is increasing in c for c < p. This is because

OEM 1 must leave more surplus to the inefficient type in order to prevent him from choosing the

contract intended for the efficient type whose reservation profit is higher. Fourth, under the optimal

contract menu, the CM is compensated by OEM 1 the difference between OEM 1’s negotiated price

and the inflated price. Similar transaction flow was used in the buy-sell process at HP (see the flow

14

chart on page 9 of CSCMP (2006)). Any business unit in HP that outsources to a CM notified

the CM and the HP Buy-Sell Operation Department the inflated component price. After the CM

sent an order of components to the HP Buy-Sell Department, the latter billed the former using the

inflated price and sent an order to HP’s supplier at HP’s negotiated price. The difference between

HP’s negotiated price and the inflated price was allocated back to the corresponding business unit,

who then reimbursed the CM through a higher payment.

Proposition 2. If b = 0, ΠBS(p) remains constant for all p ∈ [co, c]. If b > 0, ΠBS(p) strictly

increases in p for p ∈ [co, c]; thus the OEM’s optimal component resale price, denoted by p∗, is

p∗ = c. Under the optimal menu, the type-c CM’s profit, denoted by RBS(c), is

RBS(c) =∫ c

c

2b

8a2 − 3b2

[m− co − b

2a(m− x)− b

4aH(x)

]+

dx +(m− c)2

8a, (3)

and OEM 1’s expected profit, denoted by ΠBS, is

ΠBS =∫ c

c

2a

8a2 − 3b2

[m− co − b

2a(m− c)− b

4aH(c)

]+2

f(c)dc. (4)

Proposition 2 shows that OEM 1 can achieve the same expected profit by selling components

to the CM at her negotiated price or at any higher price if the products of OEM 1 and OEM 2 do

not compete. This implies that OEM 1 can choose any transfer price for convenience; in particular,

it does not hurt OEM 1 by disclosing her true purchasing cost, i.e., p = co. However, as long as

OEM 1 faces the competition from OEM 2’s product, her best strategy under buy-sell is to set

the component cost to be the highest possible value. The intuition behind this result is as follows.

One one hand, a high component cost will not reduce the quantity for OEM 1’s product, because

OEM 1 can enforce the CM’s production for her products through contractual agreement. On the

other hand, by preventing the CM from procuring components for OEM 2’s product at a reduced

cost, a high component price benefits OEM 1 in two aspects. First, it avoids an increase in the

quantity of OEM 2’s product and hence maintains the profitability of OEM 1’s product when OEM

1’s negotiated price is lower than that of the CM. Second, it increases OEM 1’s share of profit from

her own product. Although a high component price reduces the CM’s profit from selling to OEM 2,

the reduction is more if he does not contract with the OEM. Thus, by preserving her competitive

advantage, OEM 1 improves not only the overall profit of her products but also her share of it. The

strategic role of price-masking is thus clear.

15

It is interesting to see how OEM 1’s performance depends on the CM’s bargaining power relative

to the supplier. To do that, we let the CM’s cost be c = c′+k where c′ is a random variable and k is

a constant. Then, the CM’s procurement cost stochastically increases when k increases. Specifically,

when OEM 1’s procurement cost, co, does not depend on k, a smaller k means a larger bargaining

power for the CM (when negotiating with the supplier). From (4), ΠBS is increasing in k. Intuitively,

when the CM’s bargaining power (relative to the supplier) decreases, he will produce less because

of a higher procurement cost. That increases the demand for OEM 1’s products. Hence, OEM 1 is

better off.

Suppose OEM 1’s procurement cost also increases with k. Specifically, let co = c′o + k. In

this case, k can be interpreted as the component supplier’s capability: a larger k represents a lower

capability for the component supplier. There are two effects on OEM 1 as k increases. First, her

procurement cost increases. Second, the CM’s procurement cost increases, which reduces the product

2’s quantity and increases the demand for OEM 1’s product. The first effect has a negative impact

on OEM 1’s profit while the second effect has a positive impact on OEM 1’s profit. From (4), ΠBS

is decreasing in k in this case. Thus, the first effect overrides the second one, and OEM 1’s profit

decreases when the component supplier’s capability decreases.

5 Turnkey

Now, we turn to the turnkey scenario where OEM 1 delegates the purchasing responsibility to the

CM. OEM 1 offers to the CM a menu of contracts {q1(c), t(c)}, where the CM, by reporting c,

commits to producing q1(c) units of product 1 and receives a transfer payment t(c) from OEM 1.

Again, we without loss of generality focus on mechanisms that induce the CM to truthfully report

his cost.

In this case, the modified sequence of events proceeds as follows. 1) OEM 1 announces the

contract {q1(c), t(c)}; 2) The CM decides whether to participate in OEM 1’s contract; and if yes,

he reports his procurement cost c; 3) The CM decides the wholesale price w; 4) OEM 2 chooses the

order quantity q2, and pays wq2 to the CM; 5) The CM purchases components for both the products

for OEM 2 and OEM 1’s products from the supplier at a cost c; 6) OEM 1 obtains the committed

quantity, q1(c), and pays the CM the fixed payment, t(c); 7) Both OEMs sell their products in the

product market and realize the revenues.

16

The subsequent analysis is similar to that in the previous section. Given the wholesale price

w and the product 1’s quantity q1, OEM 2’s optimal order quantity is the solution to the following

problem

maxq2≥0

{(m− aq2 + bq1)q2 − wq2},

where the maximizer is q2(w, q1) = [m− w − bq1]+ /(2a). Thus, if the type-c CM chose the contract

(q1(c), t(c)), her optimal wholesale price can be determined by solving the following problem:

maxw≥0

{wq2(w, q1(c)) + t(c)− c(q1(c) + q2(w, q1(c)))}.

where the maximizer is w(c, c) = [m− bq1(c) + c]+/2. Thus, the corresponding maximum profit is

R(c, c) =[m− bq1(c)− c]+2

8a− cq1(c) + t(c).

Let R(c) ≡ R(c, c).

OEM 1’s profit facing a type-c CM is

M(c) = q1(c)[m− aq1(c)− b

[m− bq1(c)− c]+

4a

]− t(c).

Thus, OEM 1’s optimal contracts can be derived by solving the following problem, denoted by PTK ,

(PTK) maxq1(·)≥0,t(·)

EcM(c)

s.t. (IC) R(c) ≥ R(c, c), ∀c, c ∈ [c, c]

(IR) R(c) ≥ R(c), ∀c ∈ [c, c].

Proposition 3. In the turnkey scenario, under the optimal menu that maximizes OEM 1’s expected

profit, the product 1’s quantity for type-c CM is

qTK1 (c) =

2a

8a2 − 3b2

[(m− c)(1− b

2a)− (1− b

4a)G(c)

]+

.

Under this menu, the type-c CM’s profit is

RTK(c) =∫ c

c(1− b

4a)q1(x)dx +

(m− c)2

8a,

17

and OEM 1’s expected profit is

ΠTK =∫ c

c

2a

8a2 − 3b2

[(m− c)(1− b

2a)− (1− b

4a)G(c)

]+2

f(c)dc. (5)

In contrast to the buy-sell scenario where the quantity of product 1 increases in c, qTK1 (c) is

decreasing in c. This contrast is due to a major difference in the meaning of the CM’s procurement

cost from OEM 1’s perspective in the two scenarios. In the buy-sell scenario, the CM’s procurement

cost represents the intensity of competition in the product market: The lower the CM’s procurement

cost, the more competition OEM 1 faces from OEM 2’s products. In the turnkey scenario, both

OEM 1’s products and OEM 2’s products are produced at the same cost (the CM’s procurement

cost), and hence the intensity of competition in the product market does not change with the CM’s

procurement cost. In this case, the CM’s procurement cost represents the profitability of OEM 1’s

products: The lower the CM’s procurement cost, the lower the cost of production and therefore the

higher the profit margin for OEM 1’s products. Thus, under turnkey, the quantity for OEM 1’s

product increases when the CM’s procurement cost decreases.

The conclusion regarding the information rent in the turnkey scenario is also different from

that in the buy-sell scenario. Recall that the information rent accounts for the excess surplus a CM

obtains beyond his reservation payoff. Since the quantity for OEM 1’s products is decreasing in the

CM’s procurement cost in the turnkey scenario, there is incentive for the CM to report an inflated

cost rather than a deflated cost. First, by pretending to be a CM with high procurement cost,

the CM with a low cost can soften the competition from OEM 1’s products and gain more residual

demand for OEM 2’s products. Second, by reporting a higher procurement cost, the CM can capture

savings in the procurement cost. In this sense, there is an unambiguous dominance between the two

forces that contribute to the countervailing incentives. To induce the CM to truthfully report his

cost, OEM 1 has to leave surplus to the CM whose cost is low. Thus, the information rent to the

CM, i.e., RTK(c)−R(c), increases when the CM’s procurement cost decreases.

The relationship between OEM 1’s profit and the CM’s procurement cost is also different.

Under turnkey, OEM 1’s expected profit increases as the CM’s procurement cost becomes lower.

This is because the CM’s procurement cost is also shared by OEM 1 who delegates the procurement

responsibility to the CM. In contrast, under buy-sell, OEM 1 suffers from interacting with a CM

with a lower procurement cost (according to (4)) due to the intensified competition.

18

6 Comparison

In this section, we illustrate how OEM 1’s preference between buy-sell and turnkey depends on OEM

1’s procurement cost co, the competition factor b, and the uncertainty of the CM’s procurement cost

c. Suppose c is a two-sided truncated normal random variable defined over the interval [0, 2] with

mean 1 and standard deviation σ. We test in total 1539 instances by fixing m = 8, a = 1 and varying

b ∈ {0.1, 0.2, ..., 0.9}, σ ∈ {0.1, 0.2, ..., 0.9}, co ∈ {0.1, 0.2, ..., 1.9}. For each instance, we compute the

percentage gap

∆ ≡ 100 ∗ ΠBS −ΠTK

max(ΠBS , ΠTK),

which measures the relative advantage (disadvantage) of buy-sell compared with turnkey when ∆ is

positive (negative).

We first examine how OEM 1’s procurement cost co impacts the comparison between buy-sell

and turnkey. We compute the average value of ∆, denoted by ∆, among all the instances where

co is fixed. The average percentage gap as a function of co is depicted in Figure 1. The figure is,

not surprisingly, consistent with the analytical result in Proposition 5, suggesting that there exists

a threshold value c such that OEM 1 prefers buy-sell if co < c. Figure 1 further reveals that the

threshold value c ∈ [1.3, 1.4] which is larger than 1, the mean value of c.

Figure 1: Average percentage gap ∆ as function of the OEM’s procurement cost co.

We then focus on two cases: co = 0.5 and co = 1.5. The first (second) case represents the

scenario where OEM 1 has superior (inferior) procurement expertise relative to that of the CM. For

each case, we examine how the relative performance between buy-sell and turnkey is influenced by the

19

competition factor b and by the uncertainty of the CM’s procurement cost σ. Figure 2 suggests that

when co = 0.5, the relative advantage of buy-sell over turnkey is strengthened as the two products are

closer substitutes (b increases) or the CM’s procurement cost is more volatile (σ increases). Figure

3 suggests that when co = 0.5, the relative advantage of turnkey over buy-sell is strengthened as the

two products become closer substitutes (b increases) or the CM’s procurement cost is less volatile

(σ decreases).Thus, the volatility works in favor of buy-sell, whereas the substitutability between

the two products amplifies the advantage (disadvantage) of buy-sell versus turnkey when the OEM’s

purchasing cost is low (high).

We first discuss the impact of volatility of the CM’s purchasing cost. Volatility intensifies the

information asymmetry and strengthens the information advantage (disadvantage) of the CM (OEM

1). OEM 1 is thus forced to leave more information rent for the CM under both buy-sell and turnkey.

Nevertheless, under turnkey, high volatility in the CM’s purchasing cost has another negative impact

of OEM 1 because of the resulting high volatility in the cost of the OEM’s products. This negative

impact does not take place under buy-sell when OEM 1 purchases components directly from the

supplier using her own negotiated cost. In this sense, retaining the procurement decision is an

effective way to mitigate the impact of volatility.

Now we discuss the impact of substitutability between the two products. When the two products

are closer substitutes, there is an impact on the information rent paid to the CM and the total

profit of the supply chain; the former favors turnkey while the latter favors buy-sell from OEM 1’s

perspective. First, with a higher substitutability (larger b), OEM 1 has to pay more information

rent for the CM under buy-sell but less under turnkey (as seen in equations (4) and (5)). Recall that

under buy-sell the CM has an incentive to report a deflated procurement cost because OEM 1 will

believe there is a strong competition from OEM 2’s product and produce a smaller quantity for her

own products. When the two products are closer substitutes, OEM 1’s quantity is more sensitive

to the CM’s cost report and hence there is a larger incentive for the CM to report a deflated cost.

Under turnkey, the CM has an incentive to report an inflated cost to reduce quantity for OEM 1’s

products and to capture savings in procurement cost. When the two products are closer substitutes,

OEM 1’s quantity may be more or less sensitive to the CM’s procurement cost. However, the CM’s

savings in procurement cost definitely decreases because OEM 1’s production quantity is smaller

when compared to the case when b is small. This second driving force dominates when the two

driving forces are contradictory. Thus, under turnkey the CM always has a smaller incentive to

20

report an inflated cost when the two products are close substitutes.

On the other hand, substitutability hurts total profit of the supply chain more under turnkey

than under buy-sell. This is because both quantities are purchased at the same cost under turnkey,

whereas under buy-sell the OEM procures the critical components herself. In this sense, the quantity

competition is relatively head-to-head under turnkey but asymmetric (and differentiated) under buy-

sell. As the conventional wisdom suggests that the competition between relatively more homogeneous

firms drives down their aggregate profit in a greater magnitude, it is conceivable that the unified

procurement process (and the common purchasing cost) under turnkey leads to a larger decrease in

the total profit of the supply chain when substitutability increases.

Figure 2: Average percentage gap as function of competition factor b and volatility of CM’s purchasecost σ when co = 0.5.

Figure 3: Average percentage gap as function of competition factor b and volatility of CM’s purchasecost σ when co = 1.5.

The impact on the information rent and the impact on the supply chain total profit drive the

comparison between buy-sell and turnkey in different directions when substitutability between OEM

1’s and the CM’s products increases. Thus, whether the relative advantage of buy-sell and turnkey

is amplified critically depends on how dominant one effect is over the other. As demonstrated in

Figures 2 and 3, when OEM 1’s purchasing cost is low (and therefore occupies an advantageous

21

position in the quantity competition), the second (differentiation) effect outplays the first effect (on

the information rent). However, when OEM 1’s purchasing cost is high, mitigating the information

rent becomes a more important concern, thereby leading to an amplification of turnkey’s advantage.

7 Extensions

In this section, we study a few extensions of the basic model.

7.1 Asymmetric demand functions

First, we study the case when the two OEMs have different demand functions, and the inverse

demand function of OEM i is given as follows:

pi = mi − aiqi − biqj ,

for i, j ∈ {1, 2} and i 6= j, where ai > bi ≥ 0. To rule out trivial cases where no production

occurs, we assume m is sufficiently large relative to the street price c. In particular, we assume

that (1 + 4a2b28a1a2−2b1b2−b22

b12a2

)m2 − 4a2b28a1a2−2b1b2−b22

m1 ≥ 4a2b28a1a2−2b1b2−b22

[b12a2

c− c0

]+ c, 4a1m2 − b2m1 ≥

4a1c− b2c0, m1 − b12a2

m2 ≥ 0 and 4b1 > b2.

If the CM rejects the contract proposed by OEM 1, given the CM’s wholesale price w, the

optimal order quantity of OEM 2 is given by q2(w) = (m2 − w)+/(2a2), where x+ ≡ max{x, 0}.Accordingly, the reservation profit of the type-c CM is given by

R(c) = maxw≥0

{(w − c)q2(w)} = (m2 − c)2/(8a2).

Suppose the CM chooses the contract (q1(c), t(c)), the CM then chooses w to maximize her

profit, given by

(w −min(p, c))q2(w, q1(c))}+ t(c)− pq1(c),

where w, q1(c) = (m2 − w − b2q1(c))+/(2a2) is OEM 2’s optimal order quantity. Thus, the CM’s

optimal wholesale price is w(c, c) = [m2 − b2q1(c) + min(p, c)]+/2. Under this wholesale price, the

22

CM’s profit profit, denoted by R(c, c), is given by

R(c, c) =[m2 − b2q1(c)−min(p, c)]+2

8a2+ t(c)− pq1(c),

and the order quantity of OEM 2 is given by

q2(w(c, c), q1) = [m2 − b2q1(c)−min(p, c)]+/(4a2).

Let R(c) ≡ R(c, c), which is the type-c CM’s profit under truth-telling. If the type-c CM

truthfully reports his type, OEM 1’s profit is

M(c) = (p− co)q1(c) + q1(c)[m1 − a1q1(c)− b1[m2 − b2q1(c)−min(p, c)]+/(4a2)

]− t(c).

Therefore, the OEM faces the following contract design problem, denoted by PEXT1:

maxp≥co,q1(·)≥0,t(·)

EcM(c)

s.t. (IC) R(c) ≥ R(c, c), ∀c, c ∈ [c, c]

(IR) R(c) ≥ R(c), ∀c ∈ [c, c]

The incentive compatibility constraint (IC) ensures that truth-telling is the CM’s best response. The

individual rationality constraint (IR) ensures the CM’s participation.

The following proposition shows that the price masking to the highest possible CM’s cost is

optimal for OEM 1 under buy-sell.

Proposition 4. Suppose that the two OEMs are asymmetric in their market powers. If b1 > 0,

ΠBS(p) strictly increases in p for p ∈ [co, c]; thus the OEM’s optimal component resale price, denoted

by p∗, is p∗ = c.

7.2 Simultaneous contracting

In the basic model, we assume that CM offers a contract to OEM 2 after contracting with OEM 1.

In this section, we study the case when the contract between OEM 2 and the CM is offered by OEM

2 but not the CM. In particular, OEM 1 and OEM 2 offers contracts to the CM simultaneously.

The contract offered by OEM 1 is a menu of buy-sell contract, denoted by {p, q1(c), t1(c)}, and the

23

contract offered by OEM 2 is a menu of turnkey contract, denoted by {q2(c), t2(c)}.

Suppose OEM 1 offers a menu {p, q1(c), t1(c)}. The type-c CM’s expected profit by reporting c

is

R1(c, c) = −pq1(c) + t1(c).

Let R1(c) = R1(c, c). Given {q2(c), t2(c)}, OEM 1’s optimal contract design, denoted by (P1),

is

maxp,q1(·),t1(·)

Ec[(m− aq1(c)− bq2(c)− co + p)q1(c)− t1(c)]

s.t. R1(c) ≥ R1(c, c)

R1(c) ≥ 0

Suppose OEM 2 offers a menu {q2(c), t2(c)}. The type-c CM’s expected profit by reporting c is

R2(c, c) = −min(c, p)q2(c) + t2(c).

Let R2(c) = R2(c, c). Given {p, q1(c), t1(c)}, OEM 2’s optimal contract design, denoted by

PEXT2, is

maxq2(·),t2(·)

Ec[(m− aq2(c)− bq1(c))q2(c)− t2(c)]

s.t. R2(c) ≥ R2(c, c)

R2(c) ≥ 0

In the following proposition, we show that even when the two OEMs offer contracts simultane-

ously, the optimal strategy of OEM 1 is to sell components to the CM at the highest possible CM’s

cost.

Proposition 5. At the equilibrium to the problem of two OEMs simultaneously contracting with a

single CM with private component cost information, the buy-sell price is equal to the street price,

i.e., p∗ = c.

24

7.3 Contract between OEM 2 and CM

While the wholesale price contract is a commonly used contract form, it is natural to ask whether

our results are sensitive to this specific choice. To this end, we study an alternative scenario in which

the CM can offer a two-part tariff (w, T ) to OEM 2, where w corresponds to the variable per unit

cost and T is the fixed transfer. As we shall see below, this contract form provides the CM a greater

power to extract revenue from OEM 2 and all our results are qualitatively similar.

Given the two-part tariff (w, T ), we follow the backward analysis and start with OEM 2’s

problem. Given the wholesale price w and the product 1’s quantity q1, OEM 2’s optimal order

quantity is given by

q2(w, q1) = (m− w − bq1)+/(2a).

Now we turn to the CM’s problem in determining the optimal wholesale price. Given that the

CM’s procurement cost is c and that the CM has chosen the contract (q1(c), t(c)), the CM solves the

following problem to determine her optimal wholesale price:

maxw,T≥0

{(w −min(p, c))q2(w, q1(c))}+ T + t(c)− pq1(c)}.

The presence of fixed transfer allows the CM to first induce the quantity decision that maximizes

the joint profit between herself and OEM 2 and then use the fixed transfer to appropriate the entire

surplus. Thus, the CM’s optimal wholesale price is w(c, c) = min(p, c) and her maximum profit,

denoted by R(c, c), is

R(c, c) =[m− bq1(c)−min(p, c)]+2

4a+ t(c)− pq1(c).

Under the CM’s optimal wholesale price w(c, c), OEM 2’s optimal order quantity is

q2(w(c, c), q1) = [m− bq1(c)−min(p, c)]+/(2a).

Let R(c) ≡ R(c, c), which is the type-c CM’s profit under truth-telling. OEM 1 faces the

25

following contract design problem, denoted by PEXT3

maxp≥co,q1(·)≥0,t(·)

EcM(c)

s.t. (IC) R(c) ≥ R(c, c), ∀c, c ∈ [c, c]

(IR) R(c) ≥ R(c), ∀c ∈ [c, c]

The incentive compatibility constraint (IC) ensures that truth-telling is the CM’s best response. The

individual rationality constraint (IR) ensures the CM’s participation.

The following proposition characterizes OEM 1’s optimal strategy when reselling components

to the CM.

Proposition 6. Suppose that the contract between OEM 2 and the CM is a two-part tariff contract.

If b1 > 0, ΠBS(p) strictly increases in p for p ∈ [co, c]; thus the OEM’s optimal component resale

price, denoted by p∗, is p∗ = c.

7.4 General Reservation Profit

In our basic model, we assume that the CM becomes a monopolist upon refusing to work for the

OEM. This is admittedly restrictive, as in certain occasions the CM may still face competition from

the OEM. For example, the OEM may, after being rejected by the CM, collaborate with other CMs

and still compete in the consumer market. Another possibility is that OEM 1 may insource the

production and decides to produce products at its own production facility. In such a scenario, how

does the OEM determine her transfer price? To address this question, let us now consider the case

when the CM is endowed with a general reservation utility R(c). We find that reselling components

at the highest possible price remains optimal under buy-sell as long as certain regularity condition

holds.

Proposition 7. If R′(c) ≤ − 1

2a [m− bq∗1(c)− c]+ for all c, the OEM’s optimal component resale price

is p∗ = c.

Proposition 7 shows that price masking emerges as the optimal solution under buy-sell even if

we relax the assumption on the CM’s outside option. It can be verified that this assumption holds

when R(c) is sufficiently convex, e.g., R′(c) ≤ −m−c

2a . To see this, when m − bq∗1(c) − c ≤ 0, this is

26

obviously true since R′(c) ≤ 0. When m− bq∗1(c)− c > 0, we have

− 12a

[m− bq∗1(c)− c]+ =12a

bq∗1(c)−m− c

2a≥ R

′(c)

because that R′(c) ≤ −m−c

2a .

Moreover, the assumption R′(c) ≤ − 1

2a [m − bq∗1(c) − c]+ holds if after rejecting the OEM’s

contract, the CM still faces competition in selling product 2 but the competition is less intense

than that under accepting the OEM’s contract. Formally, suppose that after rejecting the OEM’s

contract, the OEM would seek another channel to produce product 1. Let qo be the expected

quantity of product 1 produced via the other channel if the CM rejects the OEM’s contract. As a

result, after rejecting the OEM’s contract, the CM faces Cournot competition in selling product 2,

with the expected quantity of the OEM’s product being qo. Then the condition stated in Corollary

1 is equivalent to qo ≤ q∗1(c), for all c, i.e., the OEM would produce fewer units of product 1 through

the alternative channel than that through the CM. This is quite likely to occur when it is more costly

to produce product 1 with the alternative channel than with the CM, where this alternative channel

can be another CM or OEM 1’s production facility.

8 Conclusion and Discussion

We consider a supply chain in which a CM assembles a product for a large OEM (OEM 1) and at

the same time produces a different product for a smaller OEM (OEM 2). Every unit of product

requires a critical component which can be purchased from a component supplier. In this supply

chain, OEM 1’s procurement decision affects not only the cost of her products, but also the quantity

of OEM 2’s products, thereby altering the nature of the competition in the product market. We

identify OEM 1’s optimal contract with the CM under two scenarios: the buy-sell scenario where

she purchases components for her products directly from the component supplier and resells to the

CM, and the turnkey scenario in which she delegates the responsibility to procure components to the

CM. In both scenarios, the negotiated cost between the CM and the component supplier is the CM’s

private information which is not known by the OEM. Thus, even in the buy-sell scenario, OEM 1

has only partial information about the cost of the competitive product. In each scenario, the OEM

offers a menu of contracts to the CM specifying the production quantity and the terms of payment.

We show that, in the buy-sell scenario, OEM 1 can achieve the same expected profit by selling

27

components to the CM at her privileged price or at any higher price if the products of the two OEMs

do not compete. However, as long as the OEM faces competition from the CM’s products, her best

strategy is to resell components to the CM at the highest possible negotiated price between the CM

and the supplier. In this way, OEM 1 can avoid the risk of reducing the cost of competing products

and at the same time enforce the quantity for her products through contractual agreement. Thus,

our results provide a rationale for reselling components at an inflated price under buy-sell, which is a

common practice in the industry, and further provide handy guidelines for determining the optimal

masked price.

Leakage of an OEM’s procurement cost can affect the OEM along dimensions other than the

cost of competing products, such as changing the future procurement cost of the OEM. This can

be studied by adding a second period to the model: If the OEM does not resell components at

an inflated price in the first period, her procurement cost in the second period will change to the

street price. One issue that will arise in this two-period model is that OEM 2’s quantity in the first

period will signal to OEM 1 information regarding CM’s procurement cost. This interaction between

ordering quantity and procurement cost signaling is an interesting but technically challenging issue

that presents an opportunity for future research.

Our analysis also generate some interesting insights OEM 1’s incentives in committing to buy-

sell and turnkey. The preference depends on the volatility of the CM’s cost and the substitutability

of the two products. We show that volatility in the CM’s cost always favors buy-sell because OEM 1

can reduce risk in the cost of her products by purchasing components directly from the supplier. Our

results also indicate that substitutability of OEM 1’s and the CM’s products intensifies the relative

advantage (disadvantage) of buy-sell over turnkey. Thus, OEM 1 should decide her procurement

strategy more carefully when facing a CM producing very similar products.

This paper focuses on two types of strategies of a large OEM, i.e., buy-sell with a reselling

price higher than the component cost, and turnkey. As mentioned in Section 4, the OEM may

also resell components to the CM at a price lower than her negotiated price with the component

supplier. However, since this is not a common practice in the industry, it is not clear how the CM’s

component cost will change according to the reselling price under this strategy. In addition, there are

other factors that may motivate an OEM to use buy-sell or turnkey. For example, a key motivator

of buy-sell is that the OEM and the CM can manipulate the tax basis that determines the input

duties (see Billington et al. (2008)). Another consideration is that the substitutability between the

28

products of the two OEMs may be different depending on whether OEM 1 uses buy-sell or turnkey.

These considerations can be incorporated into the model by having a different administrative cost

and/or using different substitutability factors b for buy-sell and turnkey. With the incorporation

of these ingredients, the comparison between buy-sell and turnkey will depend on the difference in

substitutability factors and the magnitude of the administrative cost. Thus, these can be interesting

extensions that may lead to new managerial implications.

Finally, our model studies the case when the OEM’s product consists of only one component,

and the OEM has to choose between buy-sell and turnkey. In reality an OEM’s product may require

multiple components, and the OEM may simultaneously use multiple production outsourcing model

(including buy-sell and turnkey) for different components. One may wonder which outsourcing model

the OEM should use for each of the different components. In order to answer this question, two types

of components shall be incorporated into the model, with the first type being common between the

two OEMs’ products and second type being specific to one product. This will be another interesting

extension that deserves a separate study. Generally speaking, since many OEMs now outsource their

product to CMs who either have their own branded products or produce products for other OEMs,

procurement decisions of the OEM under product market competition presents a fruitful direction

for research.

References

Amaral, J., C. Billington, A. Tsay. 2004. Outsourcing production without losing control. SupplyChain Management Review, November 1, 2004.

Amaral, J., C. Billington, A. Tsay. 2006. Safeguarding the promise of production outsourcing.Interfaces 36(3) 220–233.

Bernstein, F., A. Federgruen. 2004. Comparative statics, strategic complements and substitutes inoligopolies. Journal of Mathematical Economics 40(6) 713–746.

Billington, C., F. Jager, A. Tsay. 2008. Engineering a lower tax bill. CPO Agenda, winter 2008-09.

Bolandifar, E., T. Feng, F. Zhang. 2010. Asymmetric information and enforcement in supply contractdesign. Working paper, Washington University at St. Louis.

Cachon, G., P. Harker. 2002. Competition and outsourcing with scale economies. ManagementScience 48(10) 1314–1333.

Cachon, G., G. Kok. 2010. Competing manufacturers in a retail supply chain: On contractual formand coordination. Management Science 56(3) 571–589.

29

Cachon, G., F. Zhang. 2006. Procuring fast delivery: Sole sourcing with information asymmetry.Management Science 52(6) 881–896.

Carbone, J. 2004. HP sources globally to cut costs. Purchasing (http://www.purchasing.com), June17, 2004.

Celestica. 1996. Celestica awarded major contract by Cisco. Celestica News Release, November 5,1996.

CENS. 2007. Dell to delegate notebook pc component procurement to taiwanese makers. EMSNow(http://www.emsnow.com), May 11, 2007.

Chen, F., A. Federgruen, Y. Zheng. 2001. Coordination mechanisms for decentralized distributionsystems with one supplier and multiple retailers. Management Science 47 693–708.

Chiang, W. K., D. Chhajed, J. D. Hess. 2003. Direct marketing, indirect profits: A strategic analysisof dual-channel supply-chain design. Management Science 49(1) 1–20.

Corbett, C., X. de Groote. 2000. A supplier’s optimal quantity discount policy under asymmetricinformation. Management Science 46(3) 444–450.

Corbett, C., D. Zhou, C. Tang. 2004. Designing supply contracts: contract type and informationasymmetry. Management Science 50(4) 550–559.

CSCMP. 2006. 2006 CSCMP supply chain innovation award/case study: HP’s buy-sell process.Http://cscmp.org/downloads/public/education/innovation-06/HP.pdf.

Deshpande, V., L. Schwarz, M. Atallah, M. Blanton, K. Frikken. 2011. Outsourcing manufacturing:Secure price-masking mechanisms for purchasing component parts. Production and OperationsManagement 20(2) 165–180.

Dignan, L. 2000. Motorola, Flextronics ink 5-year, $30b outsourcing pact. CNET News(http://news.cnet.com), May 31, 2000.

Farahat, A., G. Perakis. 2008. A comparison of Bertrand and Cournot profits in oligopolies withdifferentiated products. Working Paper, MIT.

Feng, Q., L. Lu. 2010. Design outsourcing in a differentiated product market: The role of bargainingand scope economics. Working paper.

Flextronics. 2009. Our geographic, market and vertical integration diversification strategy is thecornerstone on which we have built a highly competitive company. Flextronics International LtdAnnual report, 2009, Flextronics International Ltd.

Fudenberg, D., J. Tirole. 1994. Game Theory . The MIT Press, Cambridge, Massachussetts.

Guo, P., B. Niu, Y. Wang. 2010. The advantage of quantity leadership when outsourcing productionto a competitive CM. Working paper, Hong Kong Polytechnic University.

30

Guo, P., J. Song, Y. Wang. 2006. Outsourcing structures and information flow in a three-tier supplychain. Working Paper.

Ha, A. 2001. Supplier-buyer contracting: Asymmetric cost information and the cut-off level policyfor buyer participation. Naval Research Logistics 48(1) 41–64.

Hammond, J. 1994. Barilla SpA (A). Harvard Business School Case 9–694–046.

Jeuland, A., S. Shugan. 1983. Managing channel profits. Marketing Science 2 239–272.

Jorgensen, B. 2003. Motorola to hide pricing. Electronic Business (http://www.edn.com), November3, 2003.

Jorgensen, B. 2004. One step at a time. Electronic Business (http://www.edn.com), April 12, 2004.

Jullien, B. 2000. Participation constraints in adverse selection models. Journal of Economic Theory93(1) 1–47.

Kanellos, M. 2001. Negotiations lead to lower AMD chip prices. CNET News.com March 6.

Kayis, E., F. Erhun, E. Plambeck. 2007. Delegation vs. control of component procurement. StanfordUniversity Working Paper.

Kyocera. 2006. Competitive responses to evolving market trends. Kyocera Corporation AnnualReport, 2006, Kyocera Corporation.

Laffont, J., D. Martimort. 2002. The Theory of Incentives: The Principal-Agent Model . PrincetonUniversity Press, USA.

Lariviere, M.A., E.L. Porteus. 2001. Selling to the newsvendor: An analysis of price-only contracts.Manufacturing & Service Operations Management 3(4) 293–305.

Lewis, T., D. Sappington. 1989. Countervailing incentives in agency problems. Journal of EconomicTheory 49(2) 294–313.

Lutze, H., O. Ozer. 2008. Promised lead time contracts under asymmetric information. OperationsResearch 56(4) 898–915.

Maggi, G., A. Rodriguez-Clare. 1995. On countervailing incentives. Journal of Economic Theory 66

238–263.

Metty, T. 2006. How to outsource your way to competitive advantage. Purchasing(http://www.purchasing.com), January 12, 2006.

Plambeck, E., T. Taylor. 2005. Sell the plant? the impact of contract manufacturing on innovation,capacity, and profitability. Management Science 51(1) 133–150.

Purchasing. 2003. Oems should outsource, but maintain control of sourcing. Purchasing(http://www.purchasing.com), December 11, 2003.

31

Sullivan, L. 2003. Motorola rewrites rules for EMS. EE Times Supply Network(http://www.eetimessupplynetwork.com), November 10, 2003.

Tsai, J. 2009. Flextronics lands 2 million netbook order from HP for 2010. Digitimes(http://www.digitimes.com), December 7, 2009.

Tsay, A., N. Agrawal. 2004. Channel conflict and coordination in the e-commerce age. Productionand Operations Management 13(1) 93–110.

VentureOutsource. 2008. Dissecting the new Flextronics. Deutsche Bank Equity Research, Venture-Outsource.com, January 2008.

Vizard, M. 2004. Demand transparency. CRN December 24.

32

Appendix: Proofs

Proof of Proposition 1. The proof consists of two main steps. First, we show that there exists an

optimal solution to PBS (for fixed p), in which both the quantity and payment remain constant for c ≥p, i.e., q1(c) = q1(p) and t(c) = t(p) for all c ≥ p. This property, together with a variable interchange

(a standard technique used to deal with the type-dependent reservation profit), transforms PBS (for

fixed p) to a standard adverse selection problem, denoted by P′BS . Second, we solve P′

BS by using the

standard first-order approach. Specifically, we use the incentive compatibility constraint to rewrite

the objective function as a function of q1(·); we then use the pointwise optimization to derive the

optimal solution, denoted by q∗1(·), that maximizes the rewritten objective; this clearly provides an

upper bound on OEM 1’s expected profit and a candidate solution {q∗1(·), t∗(·)}; finally we claim that

the candidate solution also satisfies the constraints of P′BS and thus is OEM 1’s optimal menu. For

ease of notation, we drop the superscript BS.

Step 1. Take any menu {q1(·), t(·)} satisfies (IC) and (IR). Recall that for c ≥ p,

M(c) = (p− co)q1(c) + q1(c)[m− aq1(c)− b[m− bq1(c)− p]+/(4a)

]− t(c).

Clearly, there exists c∗ ≥ p such that M(c∗) = maxc≥p M(c). We now construct an alternative

menu {q1(·), t(·)} as follows: for c < p, {q1(c), t(c)} = {q1(c), t(c)}; and for c ≥ p, {q1(c), t(c)} =

{q1(c∗), t(c∗)}. It follows from the definition of c∗ that OEM 1 is (weakly) better off under {q1(·), t(·)}relative to {q1(·), t(·)}. Now it remains to prove that the alternative menu also satisfies (IC) and

(IR).

Facing the menu {q1(·), t(·)}, the type-c CM whose c < p would choose his intended contract

(q1(c), t(c)) (= (q1(c), t(c))) because otherwise it contradicts the fact that {q1(·), t(·)} satisfies (IC).

It follows that by choosing the same contract from the menu {q1(·), t(·)}, the type-c CM’s profit

remains constant for every c ≥ p. This, together with the fact that the type-c∗ prefers (q1(c∗), t(c∗))

over the rest of the contracts in the menu {q1(·), t(·)}, implies that it is the best interest of every

type-c CM to choose his intended contract (q1(c), t(c)) (= (q1(c∗), t(c∗))) for c ≥ p. Hence {q1(·), t(·)}satisfies (IC). The result that {q1(·), t(·)} satisfies (IR) follows directly from the fact that {q1(·), t(·)}satisfies (IR). Thus, there exists an optimal solution to PBS (for fixed p), in which both the quantity

and payment remain constant for c ≥ p, i.e., q1(c) = q1(p) and t(c) = t(p) for all c ≥ p.

i

To transform PBS into a problem in which the reservation profit is type independent, define

R(c, c) ≡ R(c, c)−R(c)

=[m− bq1(c)−min(p, c)]+2

8a− pq1(c) + t(c)− (m− c)2

8a, (6)

and R(c) ≡ R(c, c). This variable interchange, together with the property of the optimal solution

(proved above), allows us to rewrite PBS (for fixed p) as the following equivalent problem, denoted

by P′BS ,

(P′BS) max

{q1(·),t(·)}

∫ p

cM(c)dF (c) + (1− F (p))M(p)

s.t. (IC) R(c) ≥ R(c, c) ∀c, c ≤ p,

(IR) R(c) ≥ 0 ∀c ≤ p.

Step 2. Take any c ≤ p. It follows from the Envelope Theorem and (IC) thatR′(c) = ∂R(c,c)∂c |c=c,

which by (17) leads to

R′(c) =b

4aq1(c)1{m−bq1(c)−c≥0} +

m− c

4a1{m−bq1(c)−c<0},

where the indicator 1{x≥0} ≡ 1 if x ≥ 0, 0 otherwise. Thus, by integration,

R(c) = R(c) +∫ c

c

[b

4aq1(x)1{m−bq1(x)−x≥0} +

m− x

4a1{m−bq1(x)−x<0}

]dx. (7)

By (17) and (18),

t(c) = − [m− bq1(c)− c]+2

8a+ pq1(c) +

(m− c)2

8a

+∫ c

c

[b

4aq1(x)1{m−bq1(x)−x≥0} +

m− x

4a1{m−bq1(x)−x<0}

]dx +R(c). (8)

This, together with the fact that R(c) = 0 at the optimal solution, allows us to rewrite OEM 1’s

ii

expected profit as a function of q1(·):∫ p

cM(c)dF (c) + (1− F (p))M(p)

=∫ p

c

q21(c)[−a + 3b2

8a 1{m−bq1(c)−c≥0}] + q1(c)[m− c0 − b4a(m− c)1{m−bq1(c)−c≥0}]

− (m−c)2

8a 1{m−bq1(c)−c<0} −∫ cc

[b4aq1(x)1{m−bq1(x)−x≥0} + m−x

4a 1{m−bq1(x)−x<0}]dx

f(c)dc

+ [1− F (p)]

q21(p)[−a + 3b2

8a 1{m−bq1(p)−p≥0}] + q1(p)[m− c0 − b2a(m− p)1{m−bq1(p)−p≥0}]

− (m−p)2

8a 1{m−bq1(p)−p<0} −∫ pc

[b4aq1(x)1{m−bq1(x)−x≥0} + m−x

4a 1{m−bq1(x)−x<0}]dx

=∫ p

c

q21(c)[−a + 3b2

8a 1{m−bq1(c)−c≥0}] + q1(c)[m− c0 − b2a(m− c)1{m−bq1(c)−c≥0}]

− (m−c)2

8a 1{m−bq1(c)−c<0} −[

b4aq1(c)1{m−bq1(c)−c≥0} + m−c

4a 1{m−bq1(c)−c<0}]H(c)

f(c)dc

+ [1− F (p)]

q21(p)[−a + 3b2

8a 1{m−bq1(p)−p≥0}] + q1(p)[m− c0 − b2a(m− p)1{m−bq1(p)−p≥0}]

− (m−p)2

8a 1{m−bq1(p)−p<0}

,

where the last equality follows from integration by parts.

We now characterize by using pointwise optimization the optimal solution q∗1(·) that maximizes

the above objective function. We first consider any c < p. Because the objective function is concave

quadratic in q1(c) for m−bq1(c)−c ≥ 0 and for m−bq1(c)−c < 0, its maximizer q∗1(·) is of only three

possible values: ∆1(c) = 4a8a2−3b2

[m− c0 − b

2a(m− c)− b4aH(c)

]+, ∆2(c) = m−c0

4a , or ∆3(c) = m−cb .

We claim that q∗1(c) = ∆1(c). This is because

m− b∆1(c)− c ≥ 18a2 − 3b2

[(8a2 − b2 − 4ab)m− (8a2 − b2)c + 4abco + b2H(c)

]

≥ 18a2 − 3b2

[(8a2 − b2 − 4ab)m− (8a2 − b2)c

]

≥ 0 (by A1)

and

m− b∆2(c)− c =14a

[(4a− b)m− 4ac + bc0]

≥ 0 (by A1).

To summarize,

q∗1(c) =4a

8a2 − 3b2

[m− c0 − b

2a(m− c)− b

4aH(c)

]+

for c < p.

iii

Similarly, for the case c ≥ p, the maximizer is

q∗1(p) =4a

8a2 − 3b2

[m− c0 − b

2a(m− p)

]+

.

After obtaining the closed form expression for q∗1(c), we can then derive t∗(c) by (19).

Finally, it follows from (A3) that q∗1(c) is nondecreasing in c for c ≤ p. As standard in adverse

selection, it is verifiable that the monotonicity property of q∗1(·) ensures that the candidate solution

{q∗1(·), t∗(·)} satisfies the constraints of P′BS and thus is the optimal menu.

Proof of Proposition 2. The case of b = 0 is straightforward. Suppose b > 0. Note that[m− c0 − b

2a(m− p)]+

= m − c0 − b2a(m − p) > 0 because b ≤ a and c0 ≤ p. Differentiating Π(p)

with respect to p, we have

Π′(p) =

{[m− co − b

2a(m− p)− b

4aH(p)

]+2

−[m− co − b

2a(m− p)

]2}

2af(p)8a2 − 3b2

(9)

+2a[1− F (p)]8a2 − 3b2

2b

2a

[m− c0 − b

2a(m− p)

]

=

{[m− co − b

2a(m− p)− b

4aH(p)

]+2

−[m− co − b

2a(m− p)

]2}

2af(p)8a2 − 3b2

+2aH(p)f(p)8a2 − 3b2

2b

2a

[m− c0 − b

2a(m− p)

].

Case 1. m− co − b2a(m− p)− b

4aH(p) ≥ 0. Then,

Π′(p) ={− b

2aH(p)

[m− co − b

2a(m− p)

]+

b2

8a2H2(p)

}2af(p)

8a2 − 3b2

+2aH(p)f(p)8a2 − 3b2

2b

2a

[m− c0 − b

2a(m− p)

]

={

b

2aH(p)

[m− co − b

2a(m− p)

]+

b2

8a2H2(p)

}2af(p)

8a2 − 3b2

> 0.

Case 2. 0 > m− co − b2a(m− p)− b

4aH(p). Then

Π′(p) = −[m− co − b

2a(m− p)

]2 2af(p)8a2 − 3b2

+2aH(p)f(p)8a2 − 3b2

2b

2a

[m− c0 − b

2a(m− p)

]

=[m− co − b

2a(m− p)

]{b

aH(p)−

[m− co − b

2a(m− p)

]}2af(p)

8a2 − 3b2

> 0.

iv

Proof of Proposition 3. We again drop the superscript TK for ease of notation. First, we

transform OEM 1’s problem into a typical adverse selection problem where the reservation profit is

type independent. Define R(c, c) ≡ R(c, c) − R(c). OEM 1’s optimal menu is transformed into the

following problem, denoted as P′TK :

(P′TK) max

q1(·)≥0,t(·)EcM(c)

s.t. (IC) R(c) ≥ R(c, c) ∀c, c,

(IR) R(c) ≥ 0 ∀c.

By definition,

R(c, c) =[m− bq1(c)− c]+2

8a− cq1(c) + t(c)− (m− c)2

8a.

By the Envelope Theorem, (IC) suggests thatR′(c) = ∂R(c,c)∂c |c=c = −(1− b

4a)q1(c) if m−bq1(c)−c ≥ 0,

and R′(c) = −(q1(c)− m−c4a ) otherwise. Thus, by integrating,

R(c) = R(c) +∫ c

c

[(1− b

4a)q1(x)1{m−bq1(x)−x≥0} + (q1(x)− m− x

4a)1{m−bq1(x)−x<0}

]dx. (10)

Recall from the definition of R(c) that

R(c) =[m− bq1(c)− c]+2

8a− cq1(c) + t(c)− (m− c)2

8a. (11)

By (10) and 11), together with the fact that R(c) = 0 at the optimal menu, we can express the

transfer payment as

t(c) =∫ c

c

[(1− b

4a)q1(x)1{m−bq1(x)−x≥0} + (q1(x)− m− x

4a)1{m−bq1(x)−x<0}

]dx

+cq1(c)− [m− bq1(c)− c]+2

8a+

(m− c)2

8a. (12)

Hence, the profit of an OEM facing a type-c CM is

M(c) =q1(c)[m− aq1(c)− b

[m− bq1(c)− c]+

4a

]− cq1(c) +

[m− bq1(c)− c]+2

8a− (m− c)2

8a

−∫ c

c

[(1− b

4a)q1(x)1{m−bq1(x)−x≥0} + (q1(x)− m− x

4a)1{m−bq1(x)−x<0}

]dx,

v

and OEM 1’s expected profit is

EcM(c) =∫ c

c

q1(c)[m− aq1(c)− b [m−bq1(c)−c]+

4a

]− cq1(c) + [m−bq1(c)−c]+2

8a − (m−c)2

8a

−G(c)[(1− b

4a)q1(c)1{m−bq1(c)−c≥0} + (q1(c)− m−c4a )1{m−bq1(c)−c<0}

]

f(c)dc.

By using the pointwise optimization and similar analysis in deriving q∗1(c) in the proof of Propo-

sition 1, one can show that the above objective is maximized at

qTK1 (c) =

2a

8a2 − 3b2

[(m− c)(1− b

2a)− (1− b

4a)G(c)

]+

,

by verifying that m− bqTK1 (c)− c ≥ 0.

Now we can determine tTK(c) by substituting q1(c) with qTK1 (c) in (12). This gives a candidate

solution {qTK1 (·), tTK(·)}. It follows from (A2) that qTK

1 (c) is nonincreasing in c. Following the

standard arguments in adverse selection, the monotonicity property ensures that {qTK1 (·), tTK(·)}

satisfies (IC) and (IR), and thus is the optimal solution.

Proof of Proposition 4. From Propositions 2 and 3,

RBS(c)−RTK(c) =∫ c

c

b

4aq∗1(x)dx−

∫ c

c(1− b

4a)qTK

1 (x)dx.

Differentiating with respect to c, we have

∂[RBS(c)−RTK(c)]∂c

=b

4aq∗1(c) + (1− b

4a)qTK

1 (c) ≥ 0.

The proposition then follows by noting that RBS(c)−RTK(c) < 0 if c = c and RBS(c)−RTK(c) > 0

if c = c.

Proof of Proposition 4. As in the proofs of Propositions 1 and 2, we divide the analysis into

three main steps. First, we show that there exists an optimal solution to PEXT1 (for fixed p), in

which both the quantity and payment remain constant for c ≥ p, i.e., q1(c) = q1(p) and t(c) = t(p)

for all c ≥ p. This property, together with a variable interchange (a standard technique used to deal

with the type-dependent reservation profit), transforms PEXT1 (for fixed p) to a standard adverse

selection problem, denoted by P′EXT1. Second, we solve P′

EXT1 by using the standard first-order

approach. Specifically, we use the incentive compatibility constraint to rewrite the objective function

as a function of q1(·); we then use the pointwise optimization to derive the optimal solution, denoted

vi

by q∗1(·), that maximizes the rewritten objective; this clearly provides an upper bound on OEM 1’s

expected profit and a candidate solution {q∗1(·), t∗(·)}; finally we claim that the candidate solution

also satisfies the constraints of P′EXT1 and thus is OEM 1’s optimal menu. For ease of notation, we

drop the superscript BS. Finally, we show that ΠEXT1 is increasing in p as long as b1 > 0, thereby

establishing the optimality of price masking.

Step 1. Take any menu {q1(·), t(·)} satisfies (IC) and (IR). Recall that for c ≥ p,

M(c) = (p− co)q1(c) + q1(c)[m1 − a1q1(c)− b1[m2 − b2q1(c)− p]+/(4a2)

]− t(c).

Clearly, there exists c∗ ≥ p such that M(c∗) = maxc≥p M(c). We now construct an alternative

menu {q1(·), t(·)} as follows: for c < p, {q1(c), t(c)} = {q1(c), t(c)}; and for c ≥ p, {q1(c), t(c)} =

{q1(c∗), t(c∗)}. It follows from the definition of c∗ that OEM 1 is (weakly) better off under {q1(·), t(·)}relative to {q1(·), t(·)}. It then follows the argument in the proof of Proposition 1 that the alternative

menu also satisfies (IC) and (IR). Thus, there exists an optimal solution to PEXT1 (for fixed p), in

which both the quantity and payment remain constant for c ≥ p, i.e., q1(c) = q1(p) and t(c) = t(p)

for all c ≥ p.

To transform PEXT1 into a problem in which the reservation profit is type independent, define

R(c, c) ≡ R(c, c)−R(c)

=[m2 − b2q1(c)−min(p, c)]+2

8a2− pq1(c) + t(c)− (m2 − c)2

8a2, (13)

and R(c) ≡ R(c, c). This variable interchange, together with the property of the optimal solution

(proved above), allows us to rewrite PEXT1 (for fixed p) as the following equivalent problem, denoted

by P′EXT1,

(P′EXT1) max

{q1(·),t(·)}

∫ p

cM(c)dF (c) + (1− F (p))M(p)

s.t. (IC) R(c) ≥ R(c, c) ∀c, c ≤ p,

(IR) R(c) ≥ 0 ∀c ≤ p.

Step 2. Take any c ≤ p. It follows from the Envelope Theorem and (IC) thatR′(c) = ∂R(c,c)∂c |c=c,

vii

which by (17) leads to

R′(c) =b2

4a2q1(c)1{m2−b2q1(c)−c≥0} +

m2 − c

4a21{m2−b2q1(c)−c<0},

where the indicator 1{x≥0} ≡ 1 if x ≥ 0, 0 otherwise. Thus, by integration,

R(c) = R(c) +∫ c

c

[b2

4a2q1(x)1{m2−b2q1(x)−x≥0} +

m2 − x

4a21{m2−b2q1(x)−x<0}

]dx. (14)

We can then replace t(c) by

t(c) = − [m2 − b2q1(c)− c]+2

8a2+ pq1(c) +

(m2 − c)2

8a2

+∫ c

c

[b2

4a2q1(x)1{m2−b2q1(x)−x≥0} +

m2 − x

4a21{m2−b2q1(x)−x<0}

]dx +R(c). (15)

and use the fact that R(c) = 0 to rewrite OEM 1’s expected profit as a function of q1(·). Since:

M(c) = (p− co)q1(c) + q1(c)[m1 − a1q1(c)− b1[m2 − b2q1(c)−min(p, c)]+/(4a2)

]− t(c).

t(c) = − [m2 − b2q1(c)− c]+2

8a2+ pq1(c) +

(m2 − c)2

8a2

+∫ c

c

[b2

4a2q1(x)1{m2−b2q1(x)−x≥0} +

m2 − x

4a21{m2−b2q1(x)−x<0}

]dx +R(c),

viii

we obtain that

∫ p

cM(c)dF (c) + (1− F (p))M(p)

=∫ p

c

q21(c)[−a1 + ( b22

8a2+ b1b2

4a2)1{m2−b2q1(c)−c≥0}] + q1(c)[m1 − c0 − b1

4a2(m2 − c)1{m2−b2q1(c)−c≥0}]

− (m2−c)2

8a21{m2−b2q1(c)−c<0} −

∫ cc

[b24a2

q1(x)1{m2−b2q1(x)−x≥0} + m2−x4a2

1{m2−b2q1(x)−x<0}]dx

f(c)dc

+ [1− F (p)]

q21(p)[−a1 + ( b22

8a2+ b1b2

4a2)1{m2−b2q1(p)−p≥0}] + q1(p)[m1 − c0 − b1

2a2(m2 − p)1{m2−b2q1(p)−p≥0}]

− (m2−p)2

8a21{m2−b2q1(p)−p<0} −

∫ pc

[b24a2

q1(x)1{m2−b2q1(x)−x≥0} + m2−x4a2

1{m2−b2q1(x)−x<0}]dx

=∫ p

c

q21(c)[−a1 + 2b1b2+b22

8a21{m2−b2q1(c)−c≥0}] + q1(c)[m1 − c0 − b1

2a2(m2 − c)1{m2−b2q1(c)−c≥0}]

− (m2−c)2

8a21{m2−b2q1(c)−c<0} −

[b24a2

q1(c)1{m2−b2q1(c)−c≥0} + m2−c4a2

1{m2−b2q1(c)−c<0}]H(c)

f(c)dc

+ [1− F (p)]

q21(p)[−a1 + 2b1b2+b22

8a21{m2−b2q1(p)−p≥0}] + q1(p)[m− c0 − b1

2a2(m− p)1{m2−b2q1(p)−p≥0}]

− (m2−p)2

8a21{m2−b2q1(p)−p<0}

,

where the last equality follows from integration by parts.

We now characterize by using pointwise optimization the optimal solution q∗1(·) that maximizes

the above objective function. We first consider any c < p. Because the objective function is concave

quadratic in q1(c) for m2− b2q1(c)− c ≥ 0 and for m2− b2q1(c)− c < 0, its maximizer q∗1(·) is of only

three possible values: ∆1(c) = 4a2

8a1a2−2b1b2−b22

[m1 − c0 − b1

2a2(m2 − c)− b2

4a2H(c)

]+, ∆2(c) = m1−c0

4a1,

or ∆3(c) = m2−cb2

. We claim that q∗1(c) = ∆1(c). This is because

m2 − b2∆1(c)− c ≥ m2 − 4a2b2

8a1a2 − 2b1b2 − b22

[m1 − c0 − b1

2a2(m2 − c)

]− c

≥ 0 (by A1-NEW)

and

m2 − b2∆2(c)− c = m2 − b2m1 − c0

4a1− c =

14a1

[4a1m2 − b2m1 − 4a1c + b2c0]

≥ 0 (by A1-NEW).

To summarize,

q∗1(c) =4a2

8a1a2 − 2b1b2 − b22

[m1 − c0 − b1

2a2(m2 − c)− b2

4a2H(c)

]+

for c < p.

ix

Similarly, for the case c ≥ p, the maximizer is

q∗1(p) =4a2

8a1a2 − 2b1b2 − b22

[m1 − c0 − b1

2a2(m2 − p)

]+

.

After obtaining the closed form expression for q∗1(c), we can then derive t∗(c), and OEM 1’s profit is

ΠEXT1(p) =∫ p

c

{q21(c)[−a1 +

2b1b2 + b22

8a2] + q1(c)[m1 − c0 − b1

2a2(m2 − c)]− b2

4a2q1(c)H(c)

}f(c)dc

+ [1− F (p)]{

q21(p)[−a1 +

2b1b2 + b22

8a2] + q1(p)[m− c0 − b1

2a2(m− p)]

}

=∫ p

c

2a2

8a1a2 − 2b1b2 − b22

[m1 − co − b1

2a2(m2 − c)− b2

4a2H(c)

]+2

f(c)dc

+2a2[1− F (p)]

8a1a2 − 2b1b2 − b22

[m1 − c0 − b1

2a2(m2 − p)

]+2

.

Finally, it follows from (A3) that q∗1(c) is nondecreasing in c for c ≤ p. As standard in adverse

selection, it is verifiable that the monotonicity property of q∗1(·) ensures that the candidate solution

{q∗1(·), t∗(·)} satisfies the constraints of P′EXT1 and thus is the optimal menu.

Step 3. Note that[m1 − c0 − b1

2a2(m2 − p)

]+= m1 − c0 − b1

2a2(m2 − p) > 0 by (A1-NEW).

Differentiating Π(p) with respect to p, we have

Π′(p) =

{[m1 − co − b1

2a2(m2 − p)− b2

4a2H(p)

]+2

−[m1 − co − b1

2a2(m2 − p)

]2}

2a2f(p)8a1a2 − 2b1b2 − b2

2

(16)

+2a2[1− F (p)]

8a1a2 − 2b1b2 − b22

2b1

2a2

[m1 − co − b1

2a2(m2 − p)

]

=

{[m1 − co − b1

2a2(m2 − p)− b2

4a2H(p)

]+2

−[m1 − co − b1

2a2(m2 − p)

]2}

2a2f(p)8a1a2 − 2b1b2 − b2

2

+2a2H(p)f(p)

8a1a2 − 2b1b2 − b22

2b1

2a2

[m1 − c0 − b1

2a2(m2 − p)

].

x

Case 1. m1 − co − b12a2

(m2 − p)− b24a2

H(p) ≥ 0. Then,

Π′(p) ={− b1

2a2H(p)

[m1 − c0 − b1

2a2(m2 − p)

]+

b22

8a22

H2(p)}

2a2f(p)8a1a2 − 2b1b2 − b2

2

+2a2H(p)f(p)

8a1a2 − 2b1b2 − b22

2b1

2a2

[m1 − c0 − b1

2a2(m2 − p)

]

={

b1

2a2H(p)

[m1 − c0 − b1

2a2(m− p)

]+

b22

8a22

H2(p)}

2a2f(p)8a1a2 − 2b1b2 − b2

2

> 0.

Case 2. 0 > m1 − co − b12a2

(m2 − p)− b24a2

H(p). Then

Π′(p) = −[m1 − c0 − b1

2a2(m− p)

]2 2a2f(p)8a1a2 − 2b1b2 − b2

2

+2a2H(p)f(p)

8a1a2 − 2b1b2 − b22

2b1

2a2

[m1 − c0 − b1

2a2(m2 − p)

]

=[m1 − c0 − b1

2a2(m− p)

]{b1

a2H(p)−

[m1 − c0 − b1

2a2(m2 − p)

]}2a2f(p)

8a1a2 − 2b1b2 − b22

> 0,

when 4b1 > b2.

Proof of Proposition 5. Consider PEXT2. By the Envelope Theorem, the incentive compati-

bility constraint implies that dR2(c)/dc = ∂R2(c, c)/∂c|c=c = −q2(c) for c ≤ p, and dR2(c)/dc = 0 for

c > p. Hence, R2(c) =∫ pc q2(x)dx + R2(p). This, together with the individual rationality constraint,

suggests that R2(p) = 0 and thus

R2(c) =

∫ pc q2(x)dx if c ≤ p

0 if c ≥ p,

which leads to t2(c) = cq2(c) +∫ pc q2(x)dx for c ≤ p, and t2(c) = pq2(c) for c > p. Consequently, the

objective function of PEXT2 can be written as

∫ p

c

[(m− aq2(c)− bq1(c))q2(c)− cq2(c)−

∫ p

cq2(x)dx

]f(c)dc +

∫ c

p[(m− aq2(c)− bq1(c))q2(c)− pq2(c)] f(c)dc

=∫ p

c[(m− aq2(c)− bq1(c)− c−G(c))q2(c)] f(c)dc

+∫ c

p[(m− aq2(c)− bq1(c)− p)q2(c)] f(c)dc.

xi

Note that the above objective function has an decreasing difference with respect to p and q2(c) for

every c. Hence, it follows from the paremetric monotonicity property that OEM 1’s best response of

production quantity q2(c) is nonincreasing in p for every c.

Now consider (P1). By the Envelope Theorem, the incentive compatibility constraint implies

that dR1(c)/dc = ∂R1(c, c)/∂c|c=c = 0. This, together with the individual rationality constraint,

suggests that R1(c) = 0 for every c ∈ [c, c]. Hence, t1(c) = pq1(c). Substituting t1(c) with pq1(c),

OEM 1’s objective function can be written as Ec[(m− aq1(c)− bq2(c)− co)q1(c)], which depends on

p only via q2(c). Recall that OEM 2’s best response q2(c) is nonincreasing in p. This together with

the fact that OEM 1’s objective function is decreasing in q2(c) and the fact that p has no impact on

the constraints of (P1), implies that OEM 1 should set p to be the highest possible value. That is,

at the equilibrium, p∗ = c.

Proof of Proposition 6.

The proof consists two main steps. First, we show that there exists an optimal solution to

PEXT3 (for fixed p), in which both the quantity and payment remain constant for c ≥ p, i.e.,

q1(c) = q1(p) and t(c) = t(p) for all c ≥ p. This property, together with a variable interchange (a

standard technique used to deal with the type-dependent reservation profit), transforms PEXT3 (for

fixed p) to a standard adverse selection problem, denoted by P′EXT3. Second, we solve P′

EXT3 by

using the standard first-order approach. Specifically, we use the incentive compatibility constraint

to rewrite the objective function as a function of q1(·); we then use the pointwise optimization to

derive the optimal solution, denoted by q∗1(·), that maximizes the rewritten objective; this clearly

provides an upper bound on OEM 1’s expected profit and a candidate solution {q∗1(·), t∗(·)}; finally

we claim that the candidate solution also satisfies the constraints of P′EXT3 and thus is OEM 1’s

optimal menu.

Step 1. Take any menu {q1(·), t(·)} satisfies (IC) and (IR). Recall that for c ≥ p,

M(c) = (p− co)q1(c) + q1(c)[m− aq1(c)− b[m− bq1(c)− p]+/(2a)

]− t(c).

Clearly, there exists c∗ ≥ p such that M(c∗) = maxc≥p M(c). We now construct an alternative

menu {q1(·), t(·)} as follows: for c < p, {q1(c), t(c)} = {q1(c), t(c)}; and for c ≥ p, {q1(c), t(c)} =

{q1(c∗), t(c∗)}. It follows from the definition of c∗ that OEM 1 is (weakly) better off under {q1(·), t(·)}relative to {q1(·), t(·)}. Now it remains to prove that the alternative menu also satisfies (IC) and

xii

(IR).

Facing the menu {q1(·), t(·)}, the type-c CM whose c < p would choose his intended contract

(q1(c), t(c)) (= (q1(c), t(c))) because otherwise it contradicts the fact that {q1(·), t(·)} satisfies (IC). It

follows that by choosing the same contract from the menu {q1(·), t(·)}, the type-c CM’s profit remains

constant for every c ≥ p. This, together with the fact that the type-c∗ prefers (q1(c∗), t(c∗)) over the

rest of the contracts in the menu {q1(·), t(·)}, implies that it is the best interest of every type-c CM

to choose his intended contract (q1(c), t(c)) (= (q1(c∗), t(c∗))) for c ≥ p. Hence {q1(·), t(·)} satisfies

(IC). The result that {q1(·), t(·)} satisfies (IR) follows directly from the fact that {q1(·), t(·)} satisfies

(IR). Thus, there exists an optimal solution to PEXT3 (for fixed p), in which both the quantity and

payment remain constant for c ≥ p, i.e., q1(c) = q1(p) and t(c) = t(p) for all c ≥ p.

To transform PEXT3 into a problem in which the reservation profit is type independent, define

R(c, c) ≡ R(c, c)−R(c)

=[m− bq1(c)−min(p, c)]+2

4a− pq1(c) + t(c)− (m− c)2

4a, (17)

and R(c) ≡ R(c, c). This variable interchange, together with the property of the optimal solution

(proved above), allows us to rewrite PEXT3 (for fixed p) as the following equivalent problem, denoted

by P′EXT3,

(P′EXT3) max

{q1(·),t(·)}

∫ p

cM(c)dF (c) + (1− F (p))M(p)

s.t. (IC) R(c) ≥ R(c, c) ∀c, c ≤ p,

(IR) R(c) ≥ 0 ∀c ≤ p.

Step 2. Take any c ≤ p. It follows from the Envelope Theorem and (IC) thatR′(c) = ∂R(c,c)∂c |c=c,

which by (17) leads to

R′(c) =b

2aq1(c)1{m−bq1(c)−c≥0} +

m− c

2a1{m−bq1(c)−c<0},

where the indicator 1{x≥0} ≡ 1 if x ≥ 0, 0 otherwise. Thus, by integration,

R(c) = R(c) +∫ c

c

[b

2aq1(x)1{m−bq1(x)−x≥0} +

m− x

2a1{m−bq1(x)−x<0}

]dx. (18)

xiii

By (17) and (18),

t(c) = − [m− bq1(c)− c]+2

4a+ pq1(c) +

(m− c)2

4a

+∫ c

c

[b

2aq1(x)1{m−bq1(x)−x≥0} +

m− x

2a1{m−bq1(x)−x<0}

]dx +R(c). (19)

This, together with the fact that R(c) = 0 at the optimal solution, allows us to rewrite OEM 1’s

expected profit as a function of q1(·):∫ p

cM(c)dF (c) + (1− F (p))M(p)

=∫ p

c

q21(c)[−a + 3b2

4a 1{m−bq1(c)−c≥0}] + q1(c)[m− c0 − ba(m− c)1{m−bq1(c)−c≥0}]

− (m−c)2

4a 1{m−bq1(c)−c<0} −∫ cc

[b2aq1(x)1{m−bq1(x)−x≥0} + m−x

2a 1{m−bq1(x)−x<0}]dx

f(c)dc

+ [1− F (p)]

q21(p)[−a + 3b2

4a 1{m−bq1(p)−p≥0}] + q1(p)[m− c0 − ba(m− p)1{m−bq1(p)−p≥0}]

− (m−p)2

4a 1{m−bq1(p)−p<0} −∫ pc

[b2aq1(x)1{m−bq1(x)−x≥0} + m−x

2a 1{m−bq1(x)−x<0}]dx

=∫ p

c

q21(c)[−a + 3b2

4a 1{m−bq1(c)−c≥0}] + q1(c)[m− c0 − ba(m− c)1{m−bq1(c)−c≥0}]

− (m−c)2

4a 1{m−bq1(c)−c<0} −[

b2aq1(c)1{m−bq1(c)−c≥0} + m−c

2a 1{m−bq1(c)−c<0}]H(c)

f(c)dc

+ [1− F (p)]

q21(p)[−a + 3b2

4a 1{m−bq1(p)−p≥0}] + q1(p)[m− c0 − ba(m− p)1{m−bq1(p)−p≥0}]

− (m−p)2

4a 1{m−bq1(p)−p<0}

,

where the last equality follows from integration by part.

We now characterize by using pointwise optimization the optimal solution q∗1(·) that maximizes

the above objective function. We first consider any c < p. Because the objective function is concave

quadratic in q1(c) for m−bq1(c)−c ≥ 0 and for m−bq1(c)−c < 0, its maximizer q∗1(·) is of only three

possible values: ∆1(c) = 2a4a2−3b2

[m− c0 − b

a(m− c)− b2aH(c)

]+, ∆2(c) = m−c0

2a , or ∆3(c) = m−cb .

We claim that q∗1(c) = ∆1(c). This is because

m− b∆1(c)− c ≥ 14a2 − 3b2

[(4a2 − b2 − 2ab)m− (4a2 − b2)c + 2abco + b2H(c)

]

≥ 14a2 − 3b2

[(4a2 − b2 − 2ab)m− (4a2 − b2)c

]

≥ 0 (by A1)

xiv

and

m− b∆2(c)− c =12a

[(2a− b)m− 2ac + bc0]

≥ 0 (by A1).

To summarize,

q∗1(c) =2a

4a2 − 3b2

[m− c0 − b

a(m− c)− b

2aH(c)

]+

for c < p.

Similarly, for the case c ≥ p, the maximizer is

q∗1(p) =2a

4a2 − 3b2

[m− c0 − b

a(m− p)

]+

.

After obtaining the closed form expression for q∗1(c), we can then derive t∗(c) by (19).

Finally, it follows from (A3) that q∗1(c) is nondecreasing in c for c ≤ p. As standard in adverse

selection, it is verifiable that the monotonicity property of q∗1(·) ensures that the candidate solution

{q∗1(·), t∗(·)} satisfies the constraints of P′EXT3 and thus is the optimal menu.

Suppose b > 0. Note that[m− c0 − b

a(m− p)]+

= m − c0 − ba(m − p) > 0 because b ≤ a and

c0 ≤ p. Differentiating Π(p) with respect to p, we have

Π′(p) =

{[m− co − b

a(m− p)− b

2aH(p)

]+2

−[m− co − b

a(m− p)

]2}

af(p)4a2 − 3b2

+a[1− F (p)]4a2 − 3b2

2b

a

[m− c0 − b

a(m− p)

]

=

{[m− co − b

a(m− p)− b

2aH(p)

]+2

−[m− co − b

a(m− p)

]2}

af(p)4a2 − 3b2

+aH(p)f(p)4a2 − 3b2

2b

a

[m− c0 − b

a(m− p)

].

xv

Case 1. m− co − ba(m− p)− b

2aH(p) ≥ 0. Then,

Π′(p) ={− b

aH(p)

[m− co − b

a(m− p)

]+

b2

4a2H2(p)

}af(p)

4a2 − 3b2

+aH(p)f(p)4a2 − 3b2

2b

a

[m− c0 − b

a(m− p)

]

={

b

aH(p)

[m− co − b

a(m− p)

]+

b2

4a2H2(p)

}af(p)

4a2 − 3b2

> 0.

Case 2. 0 > m− co − ba(m− p)− b

2aH(p). Then

Π′(p) = −[m− co − b

a(m− p)

]2 af(p)4a2 − 3b2

+aH(p)f(p)4a2 − 3b2

2b

a

[m− c0 − b

a(m− p)

]

=[m− co − b

a(m− p)

] {2b

aH(p)−

[m− co − b

a(m− p)

]}af(p)

4a2 − 3b2

> 0.

Proof of Proposition 7. We first characterize the optimal solution for OEM 1’s problem,

and then show that OEM 1’s maximum expected profit is increasing in p, thereby leading to the

conclusion p∗ = c.

Following the argument in the proof of Proposition 1, we can restrict to the solution in which

both the quantity and payment remain constant for c ≥ p, i.e., q1(c) = q1(p) and t(c) = t(p) for all

c ≥ p. Next, we redefine R(c, c) as

R(c, c) ≡ [m− bq1(c)−min(p, c)]+2

4a− pq1(c) + t(c)−R(c),

and R(c) ≡ R(c, c). This allows us to rewrite PBS (for fixed p) as the following equivalent problem:

max{q1(·),t(·)}

∫ p

cM(c)dF (c) + (1− F (p))M(p)

s.t. (IC) R(c) ≥ R(c, c) ∀c, c ≤ p,

(IR) R(c) ≥ 0 ∀c ≤ p.

For any c ≤ p, we can apply the Envelope Theorem and use (IC) to obtain that R′(c) =

xvi

∂R(c,c)∂c |c=c, and thus

R′(c) = − 12a

[m− bq1(c)− c]+ −R′(c).

Thus, by integration,

R(c) = R(c) +∫ c

c

[− 1

2a[m− bq1(x)− x]+ −R

′(x)

]dx,

and the transfer t(c) can be expressed as

t(c) = − [m− bq1(c)−min(p, c)]+2

4a+ pq1(c) + R(c)

+∫ c

c

[− 1

2a[m− bq1(x)− x]+ −R

′(x)

]dx +R(c).

Assume for a moment that − 12a [m − bq1(x) − x]+ − R

′(x) is positive. In this case, we obtain

that R(c) is increasing in c and at optimality R(c) = 0. We can then express OEM 1’s expected

profit as a function of q1(·):∫ p

cM(c)dF (c) + (1− F (p))M(p) (20)

=∫ p

c

(p− c0)q1(c) + q1(c)[m− aq1(c)− b 1

2a [m− bq1(c)− c]+]

+ [m−bq1(c)−c]+2

4a − pq1(c)−R(c)− ∫ cc

[− 1

2a [m− bq1(x)− x]+ −R′(x)

]dx

f(c)dc

+ [1− F (p)]

(p− c0)q1(p) + q1(p)[m− aq1(p)− b 1

2a [m− bq1(p)− p]+]

+ [m−bq1(p)−p]+2

4a − pq1(p) +∫ pc

[− 1

2a [m− bq1(x)− x]+ −R′(x)

]dx

=∫ p

c

(p− c0)q1(c) + q1(c)[m− aq1(c)− b 1

2a [m− bq1(c)− c]+]

+ [m−bq1(c)−c]+2

4a − pq1(c)−R(c)−H(c)[− 1

2a [m− bq1(c)− c]+ −R′(c)

] f(c)dc

+ [1− F (p)]

(p− c0)q1(p) + q1(p)[m− aq1(p)− b 1

2a [m− bq1(p)− p]+]

−R(c) + [m−bq1(p)−p]+2

4a − pq1(p)

,

where we have applied integration by parts. Note that − 12a [m− bq1(c)− c]+−R

′(c) can be rewritten

xvii

as [ b2aq1(c)−R

′(c)− m−c

2a ]1{m−bq1(c)−c≥0} −R′(c)1{m−bq1(c)−c<0}. Thus, OEM 1’s profit becomes:

∫ p

c

(p− c0)q1(c) + q1(c)[m− aq1(c)− b 1

2a [m− bq1(c)− c]+]

+ [m−bq1(c)−c]+2

4a − pq1(c)−R(c)

−H(c)[[ b2aq1(c)−R

′(c)− m−c

2a ]1{m−bq1(c)−c≥0} −R′(c)1{m−bq1(c)−c<0}

]

f(c)dc

+[1− F (p)]

(p− c0)q1(p) + q1(p)[m− aq1(p)− b 1

2a [m− bq1(p)− p]+]

−R(p) + [m−bq1(p)−p]+2

4a − pq1(p)

,

OEM 1’s problem is to find a quantity schedule that maximizes the above objective function

(20). The optimal solution that maximizes the above objective function can be characterized by

using pointwise optimization. Following the same argument in the proof of Proposition 1, since the

integrand in (20) is quadratic in q1(c), the optimal solution occurs in the interior. This gives rise to

the candidate solution:

q∗1(c) =2a

4a2 − 3b2

[m− c0 − b

a(m− c)− b

2aH(c)

]+

for c < p,

q∗1(p) =2a

4a2 − 3b2

[m− c0 − b

a(m− p)

]+

,

which coincides with the solution in Proposition 1. Since q∗1(c) is nondecreasing in c for c ≤ p, the

candidate solution {q∗1(·), t∗(·)} satisfies the (omitted) constraints and thus is the optimal menu. We

note that given this quantity schedule and the assumption in Corollary 1, R′(c) ≤ − 1

2a [m−bq∗1(c)−c]+

is indeed satisfied, thereby closing the loop.

We can now present OEM 1’s maximum expected profit below:

Π(p) =∫ p

c

a4a2−3b2

[m− co − b

a(m− c)− b2aH(c)

]+2

+ (m−c)2

4a −R(c)−H(c)[−m−c2a −R

′(c)]

f(c)dc

+[1− F (p)]

{a

4a2 − 3b2

[m− co − b

a(m− p)

]+2

+(m− p)2

4a−R(p)

}.

Note that[m− c0 − b

a(m− p)]+

> 0 because b ≤ a and c0 ≤ p. Differentiating Π(p) with respect to

p, we obtain exactly the same expression for Π′(p) as (16). Thus, Π′(p) > 0 as claimed in Proposition

2.

xviii