The Impac

Decision fo

Faculty & Research

t of Outsourcing on the Timing

r Entry into Uncertain Markets

by

S. Ülkü B. Toktay

and E. Yücesan

2003/82/TM

Working Paper Series

The Impact of Outsourcing on the Timing

Decision for Entry into Uncertain Markets

Sezer Ulku

105G Old North, McDonough School of Business

Georgetown University, 20057, Washington, DC

Telephone: (202) 687 0377

Fax: (202) 687 1366

e-mail: su8@georgetown.edu

Beril Toktay

INSEAD, 77305 Fontainebleau, France

Telephone: +33 1 60 72 44 96

e-mail: beril.toktay@insead.edu

Enver Yucesan

INSEAD, 77305 Fontainebleau, France

Telephone: +33 1 60 72 40 17

e-mail: enver.yucesan@insead.edu

The Impact of Outsourcing on the Timing

Decision for Entry into Uncertain Markets

Abstract

A new production technology becomes available, enabling the introduction of new

products in various end-markets. The opportunity windows in the end-markets are

short in the sense that delaying introduction leads to a loss in expected profits. How-

ever, early entry is risky: Initially, the precision of the demand forecasts for the new

products is low, and the firms learn about the market over time. We consider two

scenarios where the investments into the new technology are made by an original

equipment manufacturer (OEM) selling to the end-market and by a contract manu-

facturer (CM) serving M OEMs, respectively. The decision maker chooses the time

of entry and the capacity that maximize its expected profit. We compare the profits,

time-to-market and expected market size in the two scenarios. We identify time-to-

market as a reason for outsourcing production in certain cases. However, contrary to

the assertion in popular press, outsourcing does not always guarantee faster time to

market.

Key Words: Time-to-market, Outsourcing, Technology Adoption, Uncer-

tainty

1 Introduction

In this paper, we consider the impact of outsourcing on the time-to-market and profit

performance of a firm facing a short opportunity window to introduce a new product

with uncertain demand. The introduction of the new product requires the adoption

of a new process technology. We compare original equipment manufacturer (OEM)

performance under two scenarios where the investments into the new technology are

made by an original equipment manufacturer (OEM) selling to the market and by a

contract manufacturer (CM) serving several OEMs, respectively. The decision maker

chooses the time of entry and the capacity that maximize its own expected profits

over the life-cycle of the technology.

Let us start with two motivating examples. The first example is from the electron-

ics industry. This industry is characterized by “short windows of opportunity, rapid

obsolescence of products, and decreasing prices over time” (Solectron Website). In

the last few years, there has been a major move towards outsourcing production to the

so called Electronics Manufacturing Services (EMS) firms. Among other advantages,

EMS firms propose improved time-to-market and “access to the latest equipment,

process knowledge and manufacturing expertise without making substantial capital

investments.” (Solectron Website)

The second example is from the semiconductor industry. This industry is char-

acterized by product life cycles measured in months and rapidly decreasing prices

in the order of 25-30 percent per year (Smith and Reinertsen 1991, Macher 2001,

Leachman et al. 1999). Therefore, time-to-market is crucial for profitability. Intro-

ducing a new/superior product often requires an investment into new manufacturing

facilities and equipment, and the industry migrates to a new process technology every

2-3 years (Cataldo 2002). For instance, in the microprocessor market,“...the faster

a company can move to 0.1 micron [technology], the better it can scale up its pro-

cessor clock-speeds and introduce chips that consume less power than its previous

generation.”(Electronic News)

Due to short life-cycles of the production technologies and high variability in

1

demand, there is a significant risk associated with such investments. For example,

the 0.13 micron technology is considered as a lost generation: After investments by

many chip companies into the 0.13 micron technology, the demand for chips dipped.

By the time it picked up again, there was a new process technology to replace the

0.13 micron technology (Cataldo 2002).

Integrated manufacturers (OEM) and foundries (CM) co-exist in the semiconduc-

tor industry as well; the difference between the two is that the latter focus only on

wafer production and they are not involved in product design. Integrated manufac-

turers and foundries differ in the timing of the adoption of new process technologies

(Manners 2002, Cataldo 2002).

Time-to-market is defined by Mahajan and Muller (1996) as “the time elapsed

between making the decision to start product development and introduction of the

product into the marketplace.” In this paper, “time-to-market” is the time of adop-

tion of the new process technology. We focus on the delays in the introduction of

new products resulting from the strategic timing of this investment. In particular, we

focus on how the time-to-market and resulting OEM profits are impacted depending

on whether it is the OEM or the CM that invests in building capacity with the new

process technology.

We examine a product market that is characterized by a short window of oppor-

tunity at the end of which superior new products replace previous ones. The OEM

may benefit from early entry because of three reasons: First, a fraction of sales would

be lost if the entry into the market were delayed. Second, competitive intensity is

low during the early phases, and the market becomes more competitive later with an

increasing number of competitors to share the total market volume with. (Increasing

competition may also lead to falling prices over time, but we limit our scope to de-

creasing sales quantities in this paper.) Finally, first mover advantages can accrue to

early entrants (Lieberman and Montgomery 1988, Kerin et al. 1992).

By introducing its product early, the OEM avoids lost sales, benefits from low

competitive intensity and possibly pioneering advantages. However, early entry is

more risky: Initially, the level of uncertainty concerning the market demand for new

2

product is high, and the OEM has the opportunity to learn about the market by

delaying entry. Learning can occur by observing the sales of competitors (Chatterjee

and Sugita 1990, Hoppe 2000), by market research (McCardle 1985) or by following

trade journals (Jensen 1982). The decision maker decides on the entry time consid-

ering both the economic benefits of early introduction and the associated risks.

We consider two scenarios to study the impact of outsourcing on time-to-market

and OEM profit. In the first scenario, the OEM decides on the timing of adoption

and the investment level. At each point in time, it can invest or it can postpone the

investment and learn more about the market. In the second scenario, production is

outsourced to a CM who has invested in the process technology and who serves M

OEMs. In this case, the OEM’s ability to introduce the new products depends on the

CM’s timing and capacity decisions. We compare time-to-market and profits in these

two scenarios and answer the following questions: Does a CM always adopt a new

production technology earlier than an OEM? Under what conditions does outsourcing

lead to better time to market for the outsourcing OEM? How does outsourcing impact

OEM profits? What are drivers other then time-to-market that favor outsourcing?

We identify time-to-market as a reason for outsourcing production in certain cases.

The paper is structured as follows. After an overview of the relevant literature

in §2, we describe our model in §3. §4 provides our results for general functional

forms for the evolution of the expected market potential and the market uncertainty.

A specific functional form is subsequently analyzed for additional insight. Next, in

§5, the implications of relaxing some of the assumptions are discussed. We conclude

in §6 with a summary of results, managerial implications, and a discussion of future

research directions.

2 Literature Review

There is a large body of literature on time-to-market, considering such issues as timing

of innovation, first-mover advantages and impact of uncertainty. Our contribution is

to study the impact of outsourcing on the time-to-market and profit for a firm that

3

faces an uncertain and short-lived market opportunity to adopt a process innovation.

To our knowledge, this is the first paper to consider time-to-market in a two-level

supply chain. In developing our model, we draw on some elements from models in

the literature, as described below.

The literature on R&D races examines “how the expected benefits, R&D costs, and

interaction among firms determine the pattern of expenditure, date of introduction,

and identity of the innovating firm” (Reinganum 1989). Kamien and Schwartz (1982)

and Reinganum (1989) provide excellent surveys for decision theoretical and game

models in the timing of innovation. Once the innovation is produced, there may be

delays in its adoption due to insufficient demand, decreasing cost of adoption over

time, high uncertainty (Tirole 1988) and network externalities (Reinganum 1989).

We assume that innovations are given (a process innovation provided by an external

supplier, and product innovations to be launched at the OEM). The question of

interest to us is when to adopt a process innovation in the presence of economic

benefits to moving early and high initial uncertainty.

The literature on first mover (dis)advantages examines the impact of the order of

entry on firm performance (Lieberman and Montgomery 1988, 1998, Robinson and

Min 2002, Bowman and Gatignon 1996, Lilien and Yoon 1990, Moore et al. 1991,

Kerin et al. 1992, Tellis and Golder 1996). Apart from initial monopoly status,

first movers can benefit from leadership in product and process technology, preempt

resources, and develop buyer switching costs (Lieberman and Montgomery 1988).

However, first movers are subject to some disadvantages. Among these, the resolution

of market and technology uncertainty is “... the major factor affecting the timing of

entry in practice” (Lieberman and Montgomery 1988). Hence, the pioneer’s business

is a high risk and high return one. Typically, the pioneer’s temporary monopoly

and its first mover advantages amount to more than the risks associated with market

and technological uncertainties (Robinson and Min 2002). The pioneer survival rates

increase with the lead time between pioneer entry and the followers (Robinson and

Min 2002, Lieberman and Montgomery 1998). Similarly, fast followers have a higher

chance of survival (Robinson and Min 2002). In our model, we capture first mover

4

advantages with a decreasing sales potential based on the time of entry, without

reference to specific sources of advantage. Similarly, the associated risk decreases

over time.

The uncertainty that surrounds the market potential of a new product is important

in the determination the timing of its introduction (Lieberman and Montgomery 1998,

Lilien and Yoon 1990). Jensen (1982) studies the adoption decision of a single firm

regarding a new technology of uncertain profitability. The success probability of the

firm is updated at zero cost in a Bayesian manner as external information becomes

available. Sources of information can be the innovator itself or trade press. It is shown

that the firm stops collecting information when the posterior probability of success is

above a certain threshold. Firms with different priors adopt at different times, leading

to diffusion, and all firms eventually adopt. In McCardle (1985) information gathering

is costly. The sources of information can be testing of the process, the product or the

market. In the optimal solution, the firm takes one of three possible actions at any

point of time: If estimated profitability is high, the firm adopts innovation. If it is low,

it is rejected. More information is collected if the probability takes an intermediate

value. The firm can end up adopting a bad innovation or rejecting a good one. Hoppe

(2000) and Chatterjee and Sugita (1990) consider adoption and market entry when

learning from the leader is possible. Hoppe (2000) shows that, in a duopoly, firms can

play a preemption game or a waiting game depending on the probability of success

for the adoption. When the probability of success is low, it is preferable to wait and

learn the true value of the innovation by observing the leader. In these papers, and

in the economics literature in general, the firm adopting an innovation incurs a fixed

cost; however, there is no consideration of a constraint on the firm’s ability to fulfill

demand. Only the last two papers take early mover advantages into account. In

our model, we assume that uncertainty concerning the market demand is resolved

costlessly over time and is observed by both the OEM and the CM.

The pertinent papers in the operations management literature focus on quality,

time and cost trade-offs, and the design of processes for product development. Cohen

et al. (1996), Bayus et al. (1997) and Morgan et al. (2001) examine the time-

5

to-market and product performance trade-off. Rushing to the market may reduce

the chances of success, or the product performance may be very different from the

customer preferences. Therefore, the entry time should take into account the firm

characteristics, competitors in the market, profit margins, and the length of the time

window. Furthermore, when the introduction time is fixed, there is a trade-off be-

tween fine-tuning the product to customer preferences and reducing unit costs (Bhat-

tacharya et al. 1998). MacCormack et al. (2001), Eisendhardt and Tabrizi (1995)

and Ward et al. (1995) focus on the design of development processes and suggest the

use of flexible processes to shorten time-to-market in dynamic and highly uncertain

markets.

Finally, an important element of our model is the possibility of outsourcing. The

decision to vertically integrate or outsource has been studied extensively in the eco-

nomics and strategy literatures. Three recent papers in the operations management

literature examine reasons for outsourcing. Van Mieghem (1999) considers a two-

firm setting where an OEM faces the choices of manufacturing in house, or fully or

partly outsourcing. He evaluates the option value of subcontracting and examines the

ability of price-only contracts, state-dependent contracts and incomplete contracts to

coordinate the channel. Plambeck and Taylor (2001) consider the trade-off between

pooling effects and the incentive to innovate when manufacturing and innovation take

place in different organizations, at the CM and OEM, respectively. They conclude

that outsourcing production to CMs does not improve profits, unless the OEM has

higher bargaining power. Otherwise, the gains from production efficiency are more

than offset by the decrease in the investment into innovation, and it may be better

to share capacity with other OEMs. Finally, Cachon and Harker (2002) examine

competition between firms in the presence of economies of scale. In such a context,

competition is fiercer as firms can lower their costs with a higher throughput. The

authors identify economies of scale as a motivation for outsourcing, since it mitigates

competition by eliminating the dependence of costs on volume. In the same spirit,

our paper explores time-to-market as a reason for outsourcing.

6

3 Model Description

This paper examines the impact of outsourcing on the time-to-market and profit per-

formance of a firm facing an uncertain market opportunity that decreases over time.

The precondition for entry is investment into a process innovation which becomes

available at time t = 0 and enables the introduction of new products in a variety of

markets. Time-to-market is defined as the adoption time of the process technology.

We do not consider other time-related advantages such as shorter production cycle

times or zero/shorter lead-times for capacity building and process perfection.

Two scenarios are considered. In the first scenario, an OEM performs production

in-house, and considers adopting the process innovation in order to introduce a series

of new products. In the second scenario, a CM has M ′ OEM customers, and M of

them can use the innovation. The OEMs can introduce their products to the market

only after the CM adopts the innovation. We assume that all the OEMs are identical,

that is, they all have the same cost parameters and their demands are drawn from

independent and identical normal distributions. The OEMs served by the CM are

not in direct competition with each other, even though they operate in competitive

product-markets. This assumption is valid when the CM has clients operating in

different industries or geographic markets.

An OEM entering the market at time t faces a market potential that is normally

distributed with mean S(t) and standard deviation σ(t). S(t) is the total expected

demand for the product from the time of entry, t, until the end of the time window

TD, whereas σ(t) is the standard deviation of the total demand during [t, TD]. The

coefficient of variation at time t is ν(t) = σ(t)S(t)

. Both the CM and the OEM have

access to the same information about the market.

We assume that the expected demand for the firm’s product strictly decreases with

the time of entry (S ′(t) < 0) until it drops to zero at time TD due to three reasons.

First, there is a finite opportunity window for commercializing the product, at the

end of which new products replace it, irrespective of the time of introduction. As

the product introduction is delayed, the sales opportunity for some customers is lost.

7

Second, competition is milder in the beginning, hence early entrants command higher

market shares, whereas competition becomes increasingly fiercer with new entries into

the market. Finally, there may be advantages that accrue to early-movers.

We allow S(t) to take different forms. A concave decreasing S(t) implies that the

demand rate is growing over time, and the initial periods are not very important,

since the sales are slow. A convex decreasing S(t) would be observed in a market

where the loss due to delays in introduction decreases over time. This is the case in

a market with pioneering advantages, where delays at early stages are costly. S(t)

can also take the shape of an inverted S-curve, if cumulative sales follow a diffusion

curve.

Even though there may be a high market potential at the early stages of the

time window, the firm knows very little about the new market opportunity; entry

at this stage is more risky. Therefore, the firm may delay entry and expand its

knowledge set prior to making a decision (Jensen 1982, McCardle 1985, Mamer and

McCardle 1987). However, there are diminishing returns to each additional piece

of information (McCardle 1985, Griffin and Hauser 1993), and the firm may stop

collecting information at some point and enter the market. We do not explicitly

model the acquisition of information. Rather, we assume that the standard deviation

for the sales potential is convex decreasing (σ′(t) < 0 and σ′′(t) > 0) to reflect the

fact that the market uncertainty is resolved over time, but at a diminishing rate.

The decision maker knows the past values of σ and S and that both are mono-

tonically decreasing in t (the market uncertainty will decrease and the expected total

sales will decrease as time progresses). However, the trajectories of S(t) and σ(t) are

not known to the decision maker. Therefore, it can not find the optimal t using the

first order conditions over t. We assume that the entry decision is taken based on

local information: At each point in time, the decision maker evaluates entry, given

the market uncertainty at that point of time. To this end, it solves a Newsboy prob-

lem, and finds the capacity K that balances the costs of over- and under-investment,

namely, excess capacity and lost sales. The resulting expected profit is compared to

the previous expected profit value. The entry decision is taken when the change in

8

profits is no longer positive. The investment level is the one that maximizes profits,

given the market information at that point of time. As we will show, this myopic deci-

sion rule is optimal when the expected profits are concave, convex or concave-convex

in t. In other cases, the entry time can be suboptimal.

Upon entry, the firm makes an irreversible commitment with a one-shot investment

into the process technology. Hence, there is an up-front fixed cost at entry that

depends on the capacity level. Any other fixed cost independent of the capacity level

is normalized to zero. Furthermore, the lead-time for building production capacity is

assumed to be equal for all OEMs and the CM, and is normalized to 0.

We assume that OEMs and the CM are risk-neutral and maximize expected prof-

its. In Scenarios I and II, the OEM and the CM, respectively, maximize their expected

profit as a function of capacity and time.

The economic parameters in the model are unit revenue (r), up-front investment

and subsequent production costs for the OEM and the CM ((cI , cp) and (c′I , c′p) re-

spectively), and the unit wholesale price charged to the OEM by the CM w. All the

economic parameters are assumed to be given. We subsequently perform a sensitivity

analysis to study the impact of these parameters.

The OEM earns a fixed unit revenue r for each item sold. Investment (cI , c′I)

and production (cp, c′p) costs are incurred by the party performing production. The

investment cost is incurred before demand realization and is linear in the capacity.

Production cost includes labor and material and is also linear in the realized output.

The OEM and the CM are assumed to have different unit costs. The source of the

difference in investment costs may be economies of scale at the CM due to higher

total capacity (cI > c′I). Similarly, the CM may have lower production costs (c′p < cp)

because of economies of scale in purchasing and economies of learning resulting from

a higher total volume of production.

When production is outsourced, a wholesale price w is charged by the CM for

each unit of production. Finally, the salvage value is normalized to zero.

As the capacity at the CM is allocated among multiple OEMs, an allocation rule

needs to be defined. Since allocation mechanisms are not of primary interest here, we

9

assume that a fair allocation rule is used, described in Netessine and Rudi (2001) as

“any inventory rule (deterministic or probabilistic) that does not give a preference to

any particular OEM”. Homogeneous OEMs achieve identical expected profits under

a fair allocation rule.

In §5, we discuss how our results would be impacted by relaxing some of these

assumptions.

4 Analysis

Several issues are analyzed in this section. We start with the timing decision of a firm

facing a short opportunity window of uncertain potential (§4.1). The OEM has the

option of outsourcing production, and thereby avoiding the investment in the face of

uncertainty. However in this case, it is dependent on the CM to invest into capacity,

and product introduction does not take place until the CM makes its investment.

We compare time-to-market in these two scenarios, and study their dependence on

costs, revenues, learning rate, decay rate for the market potential and the scale of the

contract manufacturer. Next we examine the value of outsourcing to the OEM, and

underline different motivations for outsourcing (§4.2). Finally, for additional insight,

we consider an example where S(t) and σ(t) have exponential functional forms (§4.3).

All proofs are provided in the appendix.

4.1 The Optimal Time of Investment

The optimization problem of the OEM and the CM in Scenarios I and II, respectively,

are similar. In Scenario I (II), the OEM (CM) maximizes its expected profit as a

function of capacity and time. This corresponds to solving a Newsboy problem at

each point in time and choosing the time that results in the highest Newsboy profit.

At time t∗I (t∗II), each OEM (CM) invests into the new process technology and starts

producing and selling the product.

Two trade-offs are faced by the decision makers, which are resolved by choosing

the time of entry and the capacity respectively. The market potential is higher with

10

earlier entry, however the variability is also higher, whereas both are lower if the firm

postpones investment. Time of entry is chosen to balance the profit improvements re-

sulting from early entry with the associated risks. Furthermore, demand is uncertain:

demand realization can exceed or fall short of the level of investment. This trade-off

is solved by choosing capacity to maximize the expected profit function, which is

concave over capacity for each t. The optimal capacity at time t can be found by

solving the first order condition for K, using the market information (S(t), σ(t)).

Integrated OEM

In Scenario I, the OEM is integrated into production. Hence, the OEM itself

decides on the time of investment and the capacity to be built. The expected profit

for an OEM that invests at time t into capacity K is as follows:

πI(KI , t) = −cIKI + S(t)(r − cp) − (r − cp)

∫ ∞

KI

(D(t) − KI)dF (D(t)).

The first term in this expression is the cost of investment into the new technology;

the second term is the total revenues minus the production costs in the absence of

uncertainty; the final term is the expected loss due to uncertainty. The overage cost

for the OEM is cI , the cost of a unit of excess capacity investment, and the underage

cost is r − cp − cI , the opportunity cost of a unit of lost sale.

Lemma 1 At time t, the profit maximizing capacity is K∗I (t) = S(t) + zoσ(t).

Furthermore, the expected profit on the optimal trajectory is given by

πI(K∗I (t), t) = mo (S(t) − ξ(zo)σ(t)) where mo = r − cp − cI , zo = Φ−1(1 − cI

r−cp) and

ξ(zo) = φ(zo)Φ(zo)

.

Using Lemma 1, the maximization problem can be rewritten as

t∗I = arg maxt≥0

(S(t) − ξ(zo)σ(t)). (1)

While the profit function may have many local maxima in the general case, the

optimal entry time can be easily characterized in certain cases.

11

Proposition 1 If S(t) and σ(t) are three-times differentiable; S ′(t) < 0, S(TD) = 0,

σ′(t) < 0 and σ′′(t) > 0, the optimal time of entry t∗I is given by one of the following

mutually exclusive cases:

(i) t∗I = ∞ if S(t) < ξ(zo)σ(t)∀t ∈ (0, TD).

(ii) t∗I = 0 if S ′(t) < ξ(zo)σ′(t)∀t ∈ (0, TD); or if πI(0) > 0 and S ′′(t) > ξ(zo)σ

′′(t)∀t ∈(0, TD); or S ′′′|π′′

I=0 > ξ(zo)σ

′′′|π′′

I=0 and πI(0) > πI(t0).

(iii) t∗I = t0 ∈ (0, TD) if S ′′ < ξ(zo)σ′′ ∀t ∈ (0, TD) and ∃to ∈ (0, TD) such that

π′I(to) = 0 ; or if S ′′′|π′′

I=0 < ξ(zo)σ

′′′|π′′

I=0; or if S ′′′|π′′

I=0 > ξ(zo)σ

′′′|π′′

I=0 and

πI(0) < πI(t0).

(iv) For more general S(t) and σ(t), the optimal t∗I can be found through line-search.

Proposition 1 characterizes the optimal time of entry for various conditions on

S(t) and σ(t). The first case states that if the expected profit is never positive, the

firm should never enter. The second case says that entry should be immediate if

the profit function is strictly decreasing, or convex, or convex-concave with the profit

at time 0 greater than the profit at the interior maximum. Stated in terms of the

primitives S(t) and σ(t), when πI(0) > 0, the firm should enter immediately if the

ratio of time derivatives of market potential and uncertainty is always less than ξ(zo),

or if the market potential is more convex than the uncertainty scaled by ξ(zo). In this

case, information that can be obtained about the market does not justify the delay

in entry. The third case indicates that it is optimal to enter after a delay if the profit

function is concave, or concave-convex, or convex-concave with the expected profit at

time 0 less than the profit at the interior maximum.

The condition on the third derivative allows the profit function to switch between

concavity and convexity at most once. This condition is useful, for instance, when

the difference of the two convex curves inherits convexity for some ranges of t and

concavity for some others. The resulting profit function has at most one interior

maximum. Some diffusion type S(t) specifications satisfy this condition. For instance,

when S(t) is specified with the Bass diffusion model S(t) = S0(a + 1) e−γt

1+ae−γt (Bass

12

1969) and σ(t) = σ0e−βt with β > γ, the profit function is concave-convex (case iii)

and the optimal time of investment is the smaller root of the derivative of the profit

function.

For all but one of the specific cases described above, the myopic decision rule yields

the optimal entry time. Entry is premature in the case where the profit function is

convex-concave and the profit at time 0 is lower than at the interior maximum. In

this case, the decision maker is misled by the decrease in the profits at t > 0 and

decides to enter at t = 0 even though the optimal time is yet to come.

Virtual OEM

In the second scenario, the CM invests into production capacity while no invest-

ment is made by the OEM. The timing and capacity are chosen to maximize the CM

profits. In this scenario, the OEM is dependent on the CM, since it can commercialize

its products only after an investment is made by the CM.

The CM may have two advantages over OEMs: A CM may have a lower cost of

delivery for deterministic demand; that is, it may benefit from deterministic efficiency

(cI > c′I , cp > c′p). Furthermore, a CM serving multiple customers has stochastic

efficiency, arising from risk pooling over a broader customer base (M > 1).

The CM profit function is similar to that of the OEM in Scenario I, only with

different parameters:

πcmII (K, t) = −c′IK + MS(t)(w − c′p) − (w − c′p)

∫ ∞

K

(D(t) − K)dFcm(D(t)).

Lemma 2 At time t, the profit maximizing capacity is K∗(t) = MS(t) + zc

√Mσ(t).

Furthermore, the expected profit on the optimal trajectory is given by

πcmII (K∗(t), t) = Mmc(S(t) − ξ(zc)√

Mσ(t)) where mc = w − c′I − c′p, zc = Φ−1(1 − c′

I

w−c′p)

and ξ(zc) = φ(zc)Φ(zc)

.

Using Lemma 2, the maximization problem can be written as

t∗II = arg maxt≥0

(S(t) − ξ(zc)√M

σ(t)).

13

This is the same expression as Equation (1), only with a different multiplier for

σ(t). Therefore, when ξ(zo) is replaced by ξ(zc)√M

, Proposition 1 characterizes the opti-

mal investment time of the CM for different conditions on S(t) and σ(t).

The expected OEM profit in this scenario is

πoemII = (r − w)(S(t) − 1√

Mσ(t)L(zc)), (2)

where L(z) is the standard loss function. The first term in this expression is the OEM

profit in the absence of uncertainty. The second term reflects the expected profits

lost due to smaller CM capacity than the total demand. Since capacity is rationed

fairly among OEMs, expected loss is shared equally. On the other hand, OEMs do

not bear any portion of the over-investment risk in this scenario.

The following property will be useful in understanding the impact of the economic

environment on the time of entry and comparing the entry times of different parties.

Lemma 3 ξ(z) ∈ (0,∞) and it decreases in z. Equivalently, ξ(z) decreases in the

unit revenue (dξ(z)∂r

< 0) and increases in costs (dξ(z)∂cp

> 0 and dξ(z)∂cI

> 0).

Lemma 3 states that lower values of ξ(z) correspond to a more desirable economic

climate with higher unit revenues, lower unit investment or production costs.

Let us assume that the optimal time of entry is 0 < t < TD. The following

proposition summarizes the results regarding the time of entry.

Proposition 2 If OEM (CM) profit πoemI (t) (πcm

II (t)) is maximized at t∗ ∈ (0, TD),

the following is true for the optimal investment time t∗:

(i) t∗ increases with ξ(z0) and ξ(zc), and decreases with M .

(ii) t∗ increases with S ′(t), the rate of decrease in market potential.

(iii) t∗ decreases with σ′(t), the rate of resolution of market uncertainty.

In addition, there exists a ξ0 such that t∗ = 0 if π(0) > 0 and ξ < ξ0.

14

To summarize Proposition 2, the time of entry depends on margins (ξ(z)), the

rate of decrease in the sales potential over time (S ′(t)) and the rate of resolution of

the market uncertainty (σ′(t)). In addition to these factors, the number of OEMs

(M) also affects the decision when it is taken at the CM. Finally, if the margin is

high enough, entry is immediate (t = 0).

By Lemma 3, increasing ξ(z) implies decreasing margins for the decision maker.

In a market with low margins, the willingness of the decision maker to take risks

is limited. The gains from early entry are not high enough to offset the associated

risks. Therefore, investment into the new technology is delayed until there is more

information available about the demand. On the other hand, the decision maker has

a lot to lose with late entry in a market with high margins (low ξ(z)); thus, the

investment is done earlier.

The adoption time of the CM is accelerated with a larger number of OEMs served.

This is due to pooling at the CM. With increasing M, the variance of the market

potential for each t decreases. Therefore, t∗ decreases with increasing M .

Everything else being equal, the optimal investment time is earlier in a market

with a rapidly decreasing potential. Such a case might correspond to having a shorter

life-cycle with the same initial potential. In such a market, products become obsolete

quickly and firms have a short time interval within which to commercialize products.

Similarly, a firm is expected to enter earlier if competitive intensity is expected to

increase quickly, eroding the market potential for the firm. On the other hand, if

the firm is a monopoly and can control the obsolescence time of the product to be

introduced, the entry can be delayed until more information is gathered about the

market.

The rate at which uncertainty is resolved also affects the time of entry. Entry is

earlier in a market where the rate of improvement for demand information is slow

(σ′(t) is close to zero). In this case, delaying entry leads to losses in sales. However, the

increase in the accuracy of demand information obtained in exchange is not significant.

Therefore there is little reason for delaying entry. Finally, beyond a threshold ξ0, all

ξ lead to immediate adoption.

15

Even though the market characteristics are the same for both, the OEM and

the CM differ in two ways. First, the OEM and the CM have different economic

parameters, summarized by ξ(zo) and ξ(zc). Second, the CM has the ability to pool

over different customers, hence faces lower overall demand variability. Because of

these reasons, the OEM and the CM make different decisions regarding entry time

and capacity. The following proposition compares the two in terms of their chosen

time of investment.

Proposition 3 OEM and CM entry times can be compared as follows:

(i) If ξ(zo) < ξI0 and ξ(zc) < ξII

0 , t∗I = t∗II = 0.

(ii) Otherwise, if ξ(zo) < ξ(zc)√M

, then t∗I < t∗II (and if ξ(zo) ≥ ξ(zc)√M

, then t∗I ≥ t∗II).

Since time-to-market cannot be reduced below zero, any ξ(zo) (ξ(zc)) below the

threshold ξI0 (ξII

0 ) leads to immediate entry. Beyond these threshold values, there is

no difference between the OEM and the CM in terms of time-to-market, but there is

still a profit differential as the capacity decision is not the same for fixed t.

Given that the OEM and the CM have access to the same market information, the

difference in the times of investment arises from differences in economic parameters

and pooling. When the CM serves a single customer (M = 1), the pooling effect dis-

appears and the one with a better newsvendor ratio invests earlier. If the investment

and production costs at the OEM and the CM are equal (c′I = cI , c′p = cp), this party

is the OEM unless w = r. This changes as M increases. As the pooling advantage

becomes available to the CM, t∗I > t∗II becomes possible for w < r, even with identical

unit cost parameters.

A CM may have cost advantages due to larger volumes and experience in produc-

tion (c′I < cI , c′p < cp). First of all, the capacity cost can be lower at the CM. Due to

higher total capacity (possibly including facilities other than the one under study),

the CM may benefit from economies of scale in building capacity. Second, the CM

may have lower production costs (c′p < cp) due to economies of scale in purchasing,

and economies of learning resulting from a higher aggregate volume of production.

16

Any of these unit cost advantages leads to an improvement in the CM margins, and

therefore to earlier investment.

Our first conclusion is that outsourcing does not always lead to a better time

to market for an OEM as compared to in-house production. Faster time to market

depends on the market characteristics (S(t), σ(t)), unit margins at the CM and the

OEM and finally the scale of the CM.

Let us now compare two CMs in the light of Propositions 2 and 3. Higher degrees

of deterministic and stochastic efficiency, that is, lower (c′I , c′p) and higher M lead to

faster adoption of new technologies at the CM. Thus, a faster time to market can

be expected from an efficient CM, in addition to the more obvious cost advantage.

Therefore, power considerations aside, a CM of larger scale is a better choice for the

OEM seeking faster time to market.

4.2 Other Motives for Outsourcing

In the previous section, it was shown that outsourcing does not necessarily lead to

faster time-to market. In this section, we identify other motives for outsourcing by

dissecting OEM profits.

OEM profits in the two scenarios are given by

πoemI = (r − cp − cI)S(t∗I)[1 − ν(t∗I)ξ(zo)] and

πoemII = (r − w)S(t∗II)[1 − ν(t∗II)

L(zc)√M

],

where ν(t) = σ(t)/S(t). The OEM chooses to outsource production if the expected

profits are improved as compared to in-house levels, that is, if there is a gain from

outsourcing, where gain is defined as Gπ =πoem

II

πoemI

. The two scenarios differ in four

dimensions: the unit costs faced by the OEM for each unit delivered to the market,

the time of entry and consequently the market potential, the variability faced at the

time of entry, and finally the opportunity cost of unit uncertainty to the OEM for unit

value. Outsourcing gain can be written as a combination of these four components:

Gπ =πoem

II

πoemI

= GmGS

(

1 − GRGνν(t∗I)ξ(zo)

1 − ν(t∗I)ξ(zo)

)

,

17

where Gm = r−wr−cI−cp

, GS =S(t∗

II)

S(t∗I), Gν =

ν(t∗II

)

ν(t∗I)√

Mand GR = L(zc)

ξ(zo).

The first component, Gm, is the ratio of the OEM margins in the two scenarios for

each unit of product delivered to the market. The wholesale price w can be written as

the sum of CM margin and costs: w = c′I + c′p + mc. Everything else being equal, Gm

is higher for lower costs and lower margin at the CM. That is, higher deterministic

efficiency and lower margins at the CM improve the OEM profits.

The second component, GS, is the relative change in the expected market poten-

tial due to differences in the timing of entry. An early investment by the contract

manufacturer allows the OEM to introduce its products earlier, and achieve a higher

expected market potential. GS depends on the relative newsvendor ratios, the num-

ber of CM clients, and the functional forms of S(t) and σ(t). As opposed to Gm, GS

increases in mc.

The third component, Gν , is the ratio of the demand variability per OEM in the

two scenarios. Like GS, Gν depends on the relative newsvendor ratios, the number

of CM clients, and the functional forms of S(t) and σ(t). At the time of entry, the

variability for each individual market can be higher. Gν < 1 is possible due to pooling

over multiple OEMs.

Finally, GR captures the relative cost of uncertainty in the two scenarios, for unit

margin and unit variability. When the production is in-house, OEM incurs both

underage and overage costs due to uncertainty. When production is outsourced, the

excess capacity risk is fully transferred to the CM. However, the OEM still faces the

risk of not being able to fulfill demand. If total demand is higher than capacity set

by the CM, it is rationed to OEMs; as a result, OEMs may lose sales. GR increases

in zo and decreases in zc. In addition, for zo = zc, GR < 1 since L(z) < ξ(z).

Outsourcing results in improved profits when Gπ > 1. The value of outsourcing

increases in Gm and GS. Higher margins are achieved for each unit delivered, and

the market served is larger due to earlier entry. On the other hand, the value of

outsourcing decreases in GR. In this case, the transfer of risk is not significant due to

insufficient investment at the CM. Even though the cost of excess capacity is borne

by the CM, the OEM faces a higher cost of lost sales. Finally, Gπ > 1 decreases in

18

Gν : due to a lower number of customers, the ability to achieve statistical efficiency is

lower.

4.3 Example: Exponential S(t) and σ(t)

For further insight, let us assume that S(t) and σ(t) are exponentially decreasing

functions of t: S(t) = S0e−γt; σ(t) = σ0e

−βt and the corresponding coefficient of

variation ν(t) = σ(t)S(t)

= σ0

S0

e(γ−β)t. The parameter of the market potential γ can be

interpreted as the rate of obsolescence: the market potential decreases over time with

parameter γ. On the other hand, β is the learning parameter: learning is faster at a

higher β compared to a lower one.

Consider a CM that does not have any cost superiority over the OEM (cI = c′I ,

cp = c′p). Nevertheless, it earns a non-zero profit margin (w − cI − cp > 0). Figure

1 illustrates the gain in profits, Gπ, and the difference in times of entry, ∆t for two

parameter sets. ∆t < 0 and Gπ > 1 correspond to improvements in entry time and

OEM profits due to outsourcing.

Our first observation is that ∆t is non-increasing in w: At low wholesale prices,

the OEM entry is earlier than the CM entry (∆t > 0), and the opposite is true at

high wholesale prices. Above a threshold dependent on M , a further increase in w

does not provide any improvement in ∆t, as t∗II = 0 already. This threshold is lower

for a higher M .

Our second observation is that outsourcing does not always improve OEM profits.

For instance, when M = 1, profits for any w under outsourcing are lower than those

with in-house production, and for M = 5, profits are marginally improved using

outsourcing in a narrow range of w.

Third, there is a wholesale price w∗ that maximizes the gain from outsourc-

ing, and its value is higher than the unit cost of delivery with in-house production

(w∗ > cI + cp). Therefore the OEM benefits from paying a premium to the contract

manufacturer. This premium decreases in the number of OEMs served at the CM

(M).

Fourth, the OEM benefits from a larger scale CM (high M) both in terms time-

19

(a) (b)

(c) (d)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

60 70 80 90 100

w

M=1

M=5

M=10

M=50

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

70 75 80 85 90 95 100

w

M=1

M=5

M=10

M=50

II Itt t

II Itt t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

60 65 70 75 80 85 90 95 100

w

M=1

M=5

M=10

M=50

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

70 75 80 85 90 95 100

w

M=1

M=5

M=10

M=50

II

I

GII

I

G

Figure 1: The impact of outsourcing on time-to-market and profits. (Parameters: σ0 = 40,

S0 = 100, β = 1.3, γ = 0.4, r = 100, c′p = cp = 30; c′I = cI = 40 for (a) and (c), c′I = cI = 30

for (b) and (d).)

to-market and total profit gains. The CM does not provide any value to the OEMs

unless it has a critical number of customers.

Finally, even without time (t∗o = t∗c) and unit price (w > cI + cp) advantages

(Gm < 1 and GS < 1), it is possible to have a gain from outsourcing (Gπ > 1). In

this case, the motivation for outsourcing is the risk transfer to the CM. This happens

for higher values of M , where the CM can better pool risks.

5 Discussion

In this section, we discuss the implications of relaxing some of the earlier assumptions.

Fixed Cost of Entry. Any fixed cost of entry independent of the capacity level is

20

normalized to zero in this paper. The effect of such a cost is to lower the profit curves

by a constant amount, possibly making outsourced production the only alternative

for entry. This is the case, for instance, for the fabless start-ups that are supplied

by semiconductor foundries. In 2001, there were about 1300 IC design companies

that did not have production capabilities (Business Week 2001). If the profit after

the fixed cost is still positive, the presence of a fixed cost does not impact the timing

of investment. Therefore the high fixed cost of building a fab does not explain why

Intel, an integrated manufacturer, and TSMC, a foundry, differ in their timing of new

process technology adoption.

Nonzero Salvage Value. In this paper salvage values are normalized to 0. The

CM may have a higher salvage value for capacity that is originally built to serve a

specific market, since it can be put into other uses afterwards, for instance to produce

lower margin products. Since the risks faced are lower, one can expect such a party

to invest earlier and at a higher level.

Substitution. We did not consider substitution effects. On the product side, poten-

tial cannibalization of the markets for old generation products may result in delays

in the introduction of the new product. On the capacity side, it may be possible to

use old process technology for introducing the new products with lower efficiency. In

both cases, investment into new process technology would be delayed further.

Margins Decreasing over Time. Throughout the paper, we have assumed that the

OEM margins are constant. In reality, product price may decrease through the life-

cycle (r′(t) < 0). There may also be an accompanying decrease in the production costs

over time. As long as the difference r(t)−cp(t) is constant, the current results remain

unchanged. If the profit margins decrease over time, entry by an OEM is earlier, as

there is a higher penalty for postponing investment. Unless there is a change in the

wholesale prices, w, the entry time for the CM does not change. Therefore, the time

advantage provided by the CM decreases. In this case, outsourcing is motivated more

by lower risks and increased efficiency, and less by an improvement in time-to-market.

Nonidentical OEMs. The OEMs need not be identical in terms of their unit costs

and revenues. In this case, by Proposition 2, the OEMs with better economics (lower

21

ξ(zo)) invest earlier. If the CM has identical costs across all OEMs and charges

uniform prices, the party that benefits the most from outsourcing, both in terms of

time-to-market and profits, is the one with the worst in-house economics (highest

ξ(zo)).

Asymmetric Allocation. With non-identical OEMs, allocation need not be ‘fair’:

The CM may favor high-margin, high-volume or low-variability customers. An OEM

facing an asymmetric allocation mechanism is dependent on the demands of the higher

priority customers of the CM. Therefore, even if outsourcing production to a CM

can improve time-to-market for such a firm, its profits may decrease due to limited

allocated capacity.

Correlated Demands at the OEMs. If the OEM demands are positively corre-

lated, variability reduction at the CM due to pooling is less significant. The optimal

investment time for the CM increases compared to the no correlation case, resulting

in a smaller market potential. The advantages provided by the CM decrease with

positive correlation. The opposite is true if demand correlation is negative.

Information Asymmetry. Being closer to end markets, OEMs may be able to

reduce uncertainty faster than the CM through more frequent and higher quality

market input. In this case, the time advantage provided by a CM would be lower.

Sequential Investment. In some cases, the firm may have the opportunity to phase

the investments and build capacity sequentially. When this is possible, the firms can

probe the market through smaller-scale initial investments and expand capacity once

the market uncertainty has diminished. This would be possible when the fixed costs

of each additional investment is not prohibitive. The decision maker still makes a

risk-return trade-off, only at a smaller scale.

6 Conclusion

As noted in the introduction, time-to-market is one of the many benefits promised by

contract manufacturers. We find that when a new investment is required, as in the

case of adopting a new process technology, outsourcing does not necessarily result in

22

a time advantage compared to in-house production. In particular, we show that the

time (dis)advantage through outsourcing depends on relative economics at the CM

and the OEM, the scale of the CM, and the evolution of the market potential and

the associated risks.

Apart from time-to-market, several other potential benefits can be achieved through

outsourcing production. First, outsourcing may imply a transfer of risks: In our

model, the risk of excess capacity is transferred to the CM. Second, the CM can

reduce the demand losses due to uncertainty. This is because the CM benefits from

pooling over multiple OEMs, and is therefore in a better position to own capacity:

While the utilization of specialized OEM investment is dependent on the success of

a few products, a CM may allocate the capacity left idle by unsuccessful OEMs to

successful ones. Despite these potential advantages, the risk of lost demand still re-

mains at the OEM. It is possible that the overall risk exceeds that under in-house

production if the right incentives are not provided to the CM and the CM builds

insufficient capacity. A third advantage is that the cost to the OEM of each unit can

be reduced if the wholesale prices are sufficiently low. On the other hand, due to the

other advantages listed above, the OEM may benefit from outsourcing even when the

wholesale price quoted by the CM is higher than its total unit in-house capacity and

production cost.

In the semiconductor industry, anecdotal evidence suggests that there are indeed

cases where outsourcing leads to better time-to-market as defined by faster adoption

of new process technologies. For example, TSMC, the largest foundry (CM), leads the

integrated manufacturers and foundries alike. For the 90 nanometer technology, Intel

is expected lag behind TSMC by one year (Manners 2002, Cataldo 2002). TSMC’s

lead is made possible by an extensive customer base and high margins: TSMC serves

600 fabless semiconductor firms, and its profits in year 2000 were $1.9B over revenues

of $5.3B (Business Week 2001). This is consistent with our results concerning the

impact of CM margin and scale on time-to-market.

We also show that waiting for the CM to make an independent investment into

the process technology may not be advantageous for OEMs. In our model, where we

23

allow for either CM or OEM investment, it is the OEM that makes an investment

in such a setting. In reality, intermediate options are available and various forms of

risk sharing arrangements are observed in industry. While some OEMs, such as Intel

and Conexant, make equity investments into CMs (Cameron 2002), some others form

joint ventures with CMs (for instance AMD and UMC) (Dickie 2002) for investments

into new process technologies. One motivation for such arrangements is to accelerate

the adoption of new technologies by reducing the investment costs, and consequently

the risks for the CMs. In addition, these agreements reduce the delivery risks faced

by the OEMs as they ensure capacity and priority at the CM. Joint ventures are

also formed by OEM alliances, allowing OEMs to pool resources without relying on

contract manufacturers. For example, NEC, Toshiba, Fujitsu, Hitachi and Mitsubishi

Electric have recently announced a joint venture (Dickie 2002). These agreements

allow OEMs to share risks, as well as the fixed costs of new process technologies.

Finally, consolidation in the industry not only mitigates competition, but also allows

risk pooling (Dickie 2002).

Our results have several implications for managers. One reason for outsourcing

in practice is rapid access to leading edge production technology. It is seen that a

contract manufacturer may in fact prefer to adopt a new technology later than an

OEM. This results from inadequate compensation, inefficiency, or insufficient client

base at the CM. These factors should be carefully evaluated before an outsourcing

decision.

Another reason for outsourcing in practice is to achieve cost reduction. Our results

show that cost focus in outsourcing relations may in fact reduce the profit and time-to-

market performance of the OEM. It is very striking that when the time dimension is

taken into account, the OEM may even benefit from paying a wholesale price above in-

house costs. Therefore, when comparing unit costs of delivery between in-house and

outsourced production, the yardstick should not be unit production and investment

costs. These measures do not capture the effects of risk transfer and earlier market

entry.

Finally, our results highlight that the choice of CM is important. One would

24

expect an OEM to obtain better prices from a lower-cost CM, thus achieving cost

efficiency, one of the major drivers of outsourcing. It is a pleasant surprise that this

CM can also provide a better time-to-market: With higher margins and more to gain,

the CM invests early, taking a higher risk than a low-margin CM. The same is true for

a CM with a broad customer base. Benefiting from pooling effects, such a CM faces

a lower risk compared to a smaller CM, and can afford to enter earlier. On the other

hand, a large CM may also be very powerful and appropriate a large fraction of the

profits. Power issues aside, a large CM emerges as the best source of manufacturing

capacity, not only due to efficiency, but also because of better time to market.

An important limitation of our model is that it is based on a single CM serving

multiple OEMs, and does not consider competition between CMs. CMs competing for

capacity may be forced to invest earlier into new technologies to acquire customers or

keep existing ones. Alternatively, some CMs may choose to be “followers” and invest

later. We expect that an extended model capturing competition between CMs would

provide further insights into the time-to-market (dis)advantage of outsourcing; this

is left for future research.

Acknowledgements

We gratefully acknowledge Luk Van Wassenhove for his valuable input. We would

also like to thank Harold Clark, Joe Bellefeuille, Norbert Schmidt, Enrique Salas and

Carlos Nieva from Lucent Technologies for sharing their industry knowledge. This

research was partially funded by the Center for Integrated Manufacturing and Service

Operations and the INSEAD-PwC Initiative on High Performance Organizations.

Appendix

Proof of Lemma 1. The OEM profit is given by

πI(KI , t) = −cIKI+(r−cp) min(D(t), KI) = −cIKI+(r−cp)(D(t)−max(D(t)−KI , 0)),

where Dt ∼ N(S(t), σ(t)2). The OEM maximizes its expected profit over entry time

25

and capacity:

maxKI ,t

πI(KI , t) = maxKI ,t

{−cIKI + S(t)(r − cp) + (r − cp)

∫ ∞

KI

(D(t) − KI)dF (D(t))}.

For a fixed t, πI(KI , t) is strictly concave in KI . The unique optimal capacity at time

t is therefore found by solving the first order conditions and is K∗I (t) = S(t) + zoσ(t).

Substituting the optimal capacity for each time t, the OEM expected profit as a

function of the entry time t can be written as

πI(K∗I (t), t) = (r − cp − cI)[S(t) − ξ(zo)σ(t)], (3)

where ξ(zo) = φ(zo)Φ(zo)

.

Proof of Proposition 1. Lemma 1 reduces the problem of finding the optimal

t to the following:

maxt≥0

πI(K∗I (t), t) = (r − cp − cI) max

t≥0[S(t) − ξ(zo)σ(t)].

The proof proceeds by examining the two cases where there is no point satisfying

the first order condition π′ = 0 and where there is at least one such point. In the

second case, we examine in detail the subcases where π′ has one or two roots. The

results are collected under the cases in Proposition 1.

Case 1. There exists no t1 ∈ [0, TD] s.t. π′I(t1) = 0: In this case, the profit is

either monotonically increasing (π′I < 0) or decreasing in t (π′

I > 0).

(a) If π′I(t) < 0 ∀t and πI(0) > 0 then t∗I = 0. If πI(0) < 0, then t∗I = ∞.

(b) S ′ < 0 and S(TD) = 0. Therefore, if π′I > 0, then πI > 0 is not possible for any

t. Therefore, πI < 0 for all t ∈ [0, TD) and t∗I = ∞.

Case 2. There exists a t1 ∈ [0, TD] s.t. π′I(t1) = 0:

(a) S ′′′|π′′

I=0 > ξσ′′′|π′′

I=0. π′

I is convex and has at most two roots. Since the case

with no roots is proven above, only one and two root cases are examined below:

26

(i) π′I = 0 has one root t1. If π′

I > 0 for t ∈ [0, t1] and π′I < 0 t ∈ [t1, TD] then

there exists a unique maximum at t∗I = t1. If π′I < 0 for t ∈ [0, t1] and π′

I > 0

t ∈ [t1, TD] then there exists a unique minimum at t1. In the latter case, the

entry is at t∗I = 0 if πI(0) > 0, and t∗I = ∞ if πI(0) < 0 since πI(t) first decreases

and then increases up to πI(TD) = 0.

(ii) π′I = 0 has two roots t1 < t2. π′

I > 0 for t < t1, π′I < 0 for t1 < t < t2 and

π′I > 0 for t > t2. Therefore, πI(t1) is a maximum and πI(t2) is a minimum.

t∗I = t1.

(b) S ′′′|π′′

I=0 < ξσ′′′|π′′

I=0. π′

I is concave and has at most two roots. Again, only one

and two root cases are considered below:

(i) π′I = 0 has one root t1. If π′

I > 0 for t ∈ [0, t1] and π′I < 0 t ∈ [t1, TD] then

there exists a unique maximum at t∗I = t1. If π′I < 0 for t ∈ [0, t1] and π′

I > 0

t ∈ [t1, TD] then there exists a unique minimum at t1. πI(TD) = 0 and the profit

function has a single inflection point at t1. Therefore, if the profit function has

a minimum at t1, then the non-negative maximum can only be at t = 0. Thus,

t∗I = 0 if πI(t1) > 0 and t∗I = ∞ if πI(t1) < 0.

(ii) π′I = 0 has two roots t2 < t1. π′

I < 0 for t < t2, π′I > 0 for t2 < t < t1 and

π′I < 0 for t > t1. Therefore, πI(t2) is a minimum and πI(t1) is a maximum.

The global maximum is the larger of πI(0) and πI(t1): If πI(0) < πI(t1) <,

t∗I = t1, otherwise t∗I = 0.

(c) More general S and σ may have multiple local minima and maxima, and the

global maximum can be found by line search.

Proof of Lemma 2. Similar to the proof of Lemma 1.

Proof of Lemma 3.

ξ(z) =φ(z)

Φ(z)=

φ(−z)

1 − Φ(−z).

27

dξ(z)

dz=

d

dz

(

φ(−z)

1 − Φ(−z)

)

= − d

dz

(

φ(z)

1 − Φ(z)

)

.

The Normal distribution is an increasing failure rate distribution:

d

dz

(

φ(z)

1 − Φ(z)

)

> 0.

Therefore,dξ(z)

dz= − d

dz

(

φ(z)

1 − Φ(z)

)

< 0.

Proof of Proposition 2. Since t∗ is an interior maximum, it satisfies the first order

condition

S ′(t) − xσ′(t) = 0

and the second order condition

S ′′(t) − xσ′′(t) < 0,

where x = ξ(zo) for the OEM and x = ξ(zc)√M

for the CM.

(i) Define h(t, x).= S ′(t)− xσ′(t). Let t∗(x) denote the optimal time as a function of

the parameter x. Then t∗(x) satisfies

S ′(t∗(x)) − xσ′(t∗(x)) = 0.

Taking the derivative of this equality with respect to x, we obtain

S ′′(t∗(x))dt∗(x)

dx−

(

σ′(t∗(x)) + xσ′′(t∗(x))dt∗(x)

dx

)

= 0

dt∗(x)

dx(S ′′(t∗(x)) − xσ′′(t∗(x))) − σ′(t∗(x)) = 0.

The multiplier of dt∗(x)dx

is negative. Since σ′ is also negative by assumption, we

conclude that dt∗(x)dx

is positive. In our problem, we have x = ξ(zo) for the OEM and

x = ξ(zc)√M

for the CM, so t∗ increases in ξ(zo) and ξ(zc), and decreases in M .

(ii) Consider S1 and S2 such that S ′2(t) > S ′

1(t) ∀t. Let t∗1 and t∗2 be the optimal time

of investment under the two cases, respectively. Then S ′2(t

∗1) − xσ′(t∗1) > S ′

1(t∗1) −

xσ′(t∗1) = 0. Since S(t) − xσ(t) decreases in t, t∗2 > t∗1.

28

(iii) With a parallel argument we conclude that t∗2 < t∗1.

By assumption, S ′(t) < 0 and σ′(t) < 0. Hence, we can find a small ε s.t. S ′(t) <

εσ′(t)∀t. When x = ε, π′ < 0 ∀t, and t∗ = 0 if π(0) > 0.

Proof of Proposition 3. (i) From Proposition 2, there exists a threshold ξ0 for

ξ(·), below which t∗ = 0. If ξ(zo) < ξI0 and ξ(zc) < ξII

0 , then t∗o = t∗c = 0. (ii) The

optimal adoption times for the OEM and the CM are, respectively,

t∗I = arg maxt≥0

mo(S(t) − ξ(zo)σ(t))

t∗II = arg maxt≥0

mc(S(t) − ξ(zc)√M

σ(t))

For identical OEMs, the difference in timing is driven by the difference in the multi-

plier of the standard deviation in these expressions. Recall from the proof of Propo-

sition 2 that t∗ increases in x. Therefore, if ξ(zo) < ξ(zc)√M

, then t∗I < t∗II .

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