strategic management of r&d pipelines with co...

Strategic Management of R&D Pipelines with Co-Specialized Investments and Technology Markets

Tat Chan*

John M. Olin School of Business Washington University in St. Louis

Campus Box 1133 One Brookings Drive

St. Louis, MO 63130-4899 [email protected]

Jack A. Nickerson John M. Olin School of Business

Washington University in St. Louis Campus Box 1133

One Brookings Drive St. Louis, MO 63130-4899 [email protected]

Hideo Owan John M. Olin School of Business

Washington University in St. Louis Campus Box 1133

One Brookings Drive St. Louis, MO 63130-4899

[email protected]

September 9, 2004

*We wish to thank Ganesh Kishore for helpful discussions about R&D thresholds in agricultural biotechnology. We also thank Zeynep Hansen, Bart Hamilton, Michael Leiblein, Glenn MacDonald, Brian McManus, Ambar Rao, Anjolein Schmeits, and Todd Zenger and seminar participants at Georgetown and The Ohio State University for their comments. We also thank Daniel Levinthal and two anonymous reviewers for encouraging up to consider the effects technology markets in our model.

Strategic Management of R&D Pipelines with

Co-Specialized Investments and Technology Markets

Abstract The theoretical literature on managing R&D pipeline is largely based on real option theory, which has been the workhorse for assessing R&D projects and making decisions about undertaking, continuing, or terminating projects. The theory typically assumes that each project or causally related set of projects is independent. Yet, casual observation suggests that this assumption is not valid in many circumstances. For instance, firms expend much effort on “managing” its R&D project pipeline, where managing appears to be related to the choice of R&D selection rules. Not only do these rule appear to differ across firms, they also appear to vary over time for the same firm. Such managing suggests there may be some type of interdependency among R&D projects and that the choice of R&D selection rules is a strategic decision. In this paper we develop a model using dynamic programming techniques that explains why firms vary in their R&D project selection rules. The novelty and value of our model derives from the central insight that some firms invest in downstream co-specialized activities that would incur substantial adjustment costs if R&D efforts are unsuccessful while other firms have no such investment. If transaction costs in technology markets are positive, which implies that accessing the market for projects is costly, these investments lead to state-contingent project selection rules that create a dynamic interdependency among R&D activities and product mix. The existence of a dynamic interdependency fundamentally implies that a firm's current project selection decision will affect future state conditions and future profit and that the project selection prescription for assessing each project independently is never correct. We describe how strategic management of the R&D pipeline optimally changes in response to a variety of state conditions. We conclude that our model can be extended to address several additional research questions associated with R&D portfolio management, joint ventures, and mergers and acquisitions.

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Strategic Management of R&D Pipelines with Co-Specialized Investments and Technology Markets

Introduction

Research and development is the life-blood of many firms. Deciding on which projects

to undertake, continue, or terminate is central to firm success since R&D is costly and poor

project selection can lead to low profitability and firm failure. Since Roberts and Weitzman

(1981), real option theory has been the theoretical workhorse for assessing R&D projects and

making decisions about undertaking, continuing, or terminating projects.1 This workhorse,

however, has largely hinged on assessing each project independently, which we will refer to as

the “independent project evaluation prescription”. Even if projects are in some way

interdependent, technology markets for projects would eliminate such interdependences if firms

could always and easily evaluate and purchase the projects they need from this market. Yet,

casual observation suggests such technology markets involve transaction costs and that firms

expend much effort on “managing” its R&D project pipeline, where managing appears to be

related to the choice of R&D selection rules for the portfolio, not for individual projects. Indeed,

some scholars argue that “balancing” the portfolio is one of the most important goals of portfolio

management (Cooper et al. 2001). Such managing suggests there is some type of

interdependency among project selection rules in the R&D pipeline and that the choice is a

strategic decision in the sense that the associated pattern of resource allocation decisions are

critical to a firm’s performance (Barney 1997, 27). 1 Based on option pricing theory (Black and Scholes 1973, Merton 1973), a voluminous theoretical literature on real options and investment under uncertainty has emerged that explores not only R&D project decisions but also any project requiring sunk investment with uncertain outcomes (e.g., Schwartz and Trigeorgis 2001).

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Although our model applies to any industry, we motivate our paper by considering the

pharmaceutical industry. Although the technology market in this industry is quite active,

substantial transaction costs are incurred in evaluating and acquiring projects. Given this

backdrop, some pharmaceutical firms, typically the large established ones, have R&D project

pipelines with a mix of moderate-value low-variance rojects with a few high-value high-variance

projects. In contrast, biotechnology firms tend to invest in high-value high-uncertainty projects

with almost no investments in moderate- or small-value low-uncertainty projects. These

differences imply different project selection rules. Not only might firms differ in their project

selection rules, a firm may change its selection rules over time. For instance, in the early 1990s,

Zeneca, with its most promising pipeline in many years, implicitly raised its project selection

threshold and stopped several R&D projects (Pharmaceutical Business News, 1992). Zeneca is

not alone in changing its selection rules. Indeed, industry observers assert that large

pharmaceutical manufacturers recently lowered their R&D project selection rules in response to

weak pipelines (Tanouye and Langreth, 1997). What explains differences and changes in R&D

project selection rules? Why might firms change these selection rules?

Previous economic literature has asked whether incumbents or entrants engage in more

R&D investment (e.g. Schumpeter 1942, Arrow 1962, Reinganum 1983 1989, Gans and Stern

2000). However, early research does not directly address why incumbent or entrant or, in more

general, firms with different portfolios of R&D projects may choose different selection rules.

One explanation from more recent research by Gans and Stern (2000) implies biotechnology

start-ups’ have a preference for novel innovation in order to enhance their bargaining power in

possible post-innovation negotiations for licensing and acquisitions.2

2 Also, Kamien and Schwartz (1978) argue that liquidity constraints associated with ongoing operations may explain systematic differences among firms’ decision rules for R&D project selection. Such constraints themselves,

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We directly ask how and why the incumbent and the entrant differ in their R&D project

selection rules even with the existence of a technology market from which firms can buy and sell

projects. We further ask why these rules may change. Unlike the previous literature, we do not

specifically model product and technology market competition or cooperation. Rather, our focus

is on a particular type of interdependency that arises from the effort of managing the R&D

portfolio within the firm, conditional on the environment in the technology market and

downstream product market. We build a theory on our observation that some firms invest in

downstream co-specialized activities that would incur substantial adjustment costs if R&D

efforts are unsuccessful while others have no such investment.

Following Williamson (1985), we argue that co-specialized investments are strategic

because they represent sunk-cost commitments and because they have substantial implications

for innovative behavior and firm performance. For instance, some firms in the pharmaceutical

industry make co-specialized investments in “drug reps” that not only receive firm-specific

training but also invest in idiosyncratic relationships with the select physicians who are most

likely to prescribe the pharmaceutical products the firm sells. Should a firm fail to deliver new

pharmaceutical products to these reps, the firm not only would forego returns from these

investments (as personnel would leave for better opportunities or would be laid off) but also later

may have to reinvest in sunk costs in these activities should products be developed in the future.

Although these two types of costs differ in their timing, they both have the same economic

impact so we refer to them collectively as adjustment costs. In theory, such adjustment costs

should be included in the expected value calculation for each project. But, expected adjustment

costs are a function of R&D activities in the pipeline and are hard to measure in reality. For

however, may be endogenous to these decision rules rather than the cause of them. For example, a firm may have difficulty in borrowing and face liquidity constraints, because it targets only high-uncertainty project where asymmetric information is significant.

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example, an incumbent firm’s decision to terminate one project certainly raises the probability of

incurring such adjustment costs but the probability depends on the performance of other projects.

Also, a start-up firm’s decision of pushing an R&D project into the stage of commercialization

implies that the firm in the future may incur adjustment costs for which the probability depends

on the strength of its R&D pipeline. These adjustment costs are impossible to evaluate in

general without investigating the dynamic interdependencies among project decisions.

When technology market transaction costs are positive, such state-contingent adjustment

costs, when present, require firms to manage the R&D pipeline to avoid the adjustment costs,

which implies strategic project selection decisions about developing a project within the firm,

abandoning internal efforts, or purchasing a project from the technology market. For instance, a

firm may decide to continue a low expected value project if it has a weak pipeline (i.e., few

projects that soon could be commercialized) and purchasing from a technology market incurs

large transaction costs. By not advancing such a project the firm would incur either adjustment

costs associated with its prior downstream investments or transaction costs associated with

acquiring a project in the technology market. On the other hand, with other projects in the

pipeline that have high probability of success, the firm may choose to terminate the same project

since it is not of high value, the likelihood of incurring adjustment costs due to its termination is

small, or selling the project in a technology market with large transaction costs does not pay off.

We model this dynamic interdependency by using dynamic programming techniques. This

method helps us to identify the optimal thresholds that a forward-looking rational agent would

choose, contingent on the existence of downstream co-specialized investments and the strength

of the R&D pipeline. Without co-specialized investment and, hence, adjustment costs, or

without transaction costs in the technology market, each R&D project becomes independent of

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the other projects and their management becomes less strategic; thus, Roberts and Weitzman

(1981) is special case of our model.

We begin by recalling that a large literature portrays R&D project selection decision-

making as a sequential series of stages with gates between each stage. At each gate, projects are

valued and a decision to continue to the next stage or terminate is made based on a comparison

of the project’s expected value to the gate’s threshold value—those projects with expected value

below the threshold are terminated while those projects that exceed the threshold value continue

to the next stage of development (i.e. optimal stopping rule). The extant literature does little to

inform how the optimal threshold for each gate is affected by the R&D pipeline (for an

overview, see Calantone and di Benedetto, 1990, 21). Building on this literature, our model

assumes that the pipeline is comprised of three development stages—research, development, and

commercialization—and two gates—advance-to-development and advance-to-

commercialization. We assume the possibility of six different firm states depending on the type

of projects they have in the development and commercialization stages, which we assert are

analogous to different types of firms we observe in the real world. Then we investigate the

optimal project selection thresholds and “buy or sell” price thresholds for each firm type.

We parameterize the transaction costs and market premium in the technology market and

derive the corresponding optimal R&D management rules. Comparative analysis of the optimal

thresholds yields three primary sets of conclusions about the economics of managing R&D

pipelines. Our general finding is that the optimal threshold for the advance-to-development or

advance-to-commercialization gate differs for each firm type. On one hand, when the pipeline is

weak and the firm has made the co-specialized downstream investments, the firm needs either to

lower the advance-to-development thresholds and hence fill up the pipeline, or to purchase a

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project from the technology market in order to avoid incurring future adjustment costs associated

with the loss of co-specialized investments. If transaction costs in the technology market are

non-negligible, it is more likely to take the former choice and accept moderate-valued projects.

In this case, if adjustment costs are not imminent, projects across the stages are “substitutes”, in

the sense that a weaker project in the later stage will lower the selection threshold for projects in

an earlier stage. On the other hand, when there is no need for a firm to fill up the pipeline, either

because the adjustment costs will be incurred for certain or because no downstream co-

specialized investments have been made, the firm chooses a higher threshold and hence fills its

pipeline on average with higher-valued projects. Interestingly, we find that projects across the

stages are “compliments” in this case: the firm will choose a higher selection threshold for

projects in an earlier stage when there is a weaker project in the later stage. Finally, the

interdependency across projects in the pipeline is stronger, and project selection rules are more

different across different types of firms, when adjustment costs or transaction costs in the

technology market are higher.

The paper proceeds by providing some background from the practitioner literature on

project selection. Next, we introduce our dynamic model. We then discuss the implications of

our model and modeling approach for the strategic management of R&D pipelines. We conclude

by identifying the strengths and weaknesses of our model, which helps us identify several

additional research questions associated with R&D portfolio management. We then outline a

research program to address these questions.

Background

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Several aspects of the practitioner literature provide useful foundations for our model.

We first recall the results of a 1970 study on project selection by the Industrial Research Institute

(IRI) (Gee, 1971). The study concludes that all R&D can be classified into three categories. The

first category, exploratory research, represents projects “for advancing knowledge of phenomena

of general company interest”, and “for finding major new high-risk developments” (p.40). The

second two categories represent different types of development projects. High-risk business

development projects are “major programs aimed at new business or other development of

potentially high impact, and which involve higher-than-normal risk.” The report cites examples

of “the transition from animal testing of a new drug and to human testing in the pharmaceutical

industry; or getting into a new business where company has no marketing experience, or

gambling for high stakes on a totally new way of accomplishing an important function.”

Projects for the support of the existing business are “for maintaining or improving the

profitability of established lines of businesses.” “Examples of R&D in support of existing

business include the development of the new product for the same familiar marketing channels,

and process research to reduce cost of manufacture of an established product.” The IRI report

goes on to conclude that project selection criteria differ depending on whether the firm is

concerned with supporting existing business or generating new businesses. For instance, the

report states “the pressures of day-to-day business suggest that certain research projects are

required to keep the business healthy, or to head off competitive pressure” (p. 42).

We draw two implications for our model from this study and others like it.3 First,

decisions rules for project selection can differ depending on whether or not an “established

3 Scores of books and papers on managing new product development are available. Two of the better-known references are Cooper (1986) and Wheelwright and Clark (1992). See also Calantone and di Benedetto (1990) who provide a literature review of publications up through 1990. Our conclusions can be drawn from many of these other sources.

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business” is present. We infer that the establishment of a business affects a firm’s incentives for

selecting projects. More precisely, we assume that the establishment of a business means that a

firm has invested in downstream co-specialized activities. Co-specialized activities can include

manufacturing, distribution, marketing, and sales. With no new projects to commercialize, a

firm will incur substantial adjustment costs due to opportunity, carrying, and redeployment costs

from not utilizing the co-specialized activities.

A second aspect of the literature is that project selection typically involves three stages:

research, development, and commercialization. The practitioner literature offers great diversity

in its description of the development process. The number of stages varies across the literature

from a minimum of three (Wheelwright and Clark, 1992) to as many as seven (Cohen,

Kamienski, and Espino, 1998). The key difference lies in the extent to which research or

development is segmented into multiple stages. For instance, one seven-stage model divides

research and development each into three stages (Cohen, Kamienski, and Espino, 1998). Thus

while descriptions in the practitioner literature differ, all descriptions nonetheless map directly

onto the three stages of research, development, and commercialization.

A third and less commented on implication for our model is the idea that firms may

change their R&D project selection thresholds contingent on various state conditions. For

instance, consider the following WSJ article about pharmaceutical companies, which discusses

how pharmaceutical manufacturers responded to the thinning of their R&D pipelines:

And in a radical change, some companies are shedding their Hollywood-style focus on finding a few big hits—a longtime practice that left them vulnerable when patents expired—in favor of developing a broader range of drugs. Ultimately, that change could yield drugs aimed at smaller markets, and for rarer illnesses, which might have been shelved in the past (Tanouye and Langreth, 1997).

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Reports like this imply that at least some pharmaceutical firms strategically lowered their

R&D project selection thresholds in response to state conditions like an aging set of

commercialized products and few projects in their development pipelines. Other firms, like

Zeneca, have apparently raised thresholds in response to a full pipeline (Pharmaceutical Business

News, 1992). More generally, firms might change their thresholds in response to a variety of

state contingent conditions. We use these three implications as a foundation for the structure of

our model.

Model

We study the strategic management of R&D pipelines of a firm with co-specialized

investments and facing a technology market, with focus on identifying optimal R&D project

selection rules. We introduce a dynamic model in which the focal firm, in each period, initiates a

project that goes through two stages (in two periods) before generating a possible new product.

Stage I is the research stage in which a potential product is first identified while stage II is the

development stage in which the full characteristics of the product and its demand in the market

are examined. To simplify our analysis, we assume that, at any point of time, the focal firm has

at most one project in each stage, and can introduce at most one product to the market. Also,

R&D capability is homogeneous among firms in the industry, which implies that the prior

distributions for the market value of potential products are identical. We further assume that the

technology market for projects exists only in stage II; but, lack of a market for stage I projects

does not affect our results because no firm would trade projects at the end of stage I, as long as

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transaction costs in the stage are not significantly smaller than those in stage II.4 Therefore, the

firm makes decisions on (1) whether to continue or terminate the existing stage I project, and (2)

whether to commercialize, sell, terminate the stage II project if it exists, or to buy a technology

developed by another firm.

We assume that the demand for the product once it is introduced to the market lasts only

for two periods. The firm can have at most one new and/or old product in the market. When the

product expires in two periods, the firm will incur an adjustment cost c, unless it can introduce a

new product in the market during this time.5 Let xt be the value of the product generated from

the project started in period t. Although we do not know xt at the time when the project starts,

we do know the distribution f(x) that xt follows. We assume that f is continuous and f(x) > 0 for

all x. By the end of stage I, the firm receives a signal, st, about the products value: st = xt + εt

where εt is a noise and E[εt] = 0. The posterior expected mean is given by zt = E(xt | st). We

write f(xt | st) as the density function of xt conditional on signal st. By abusing notation, we also

denote f(xt | zt) ≡ f(xt | st).

The firm then decides whether or not to advance the product to stage II based on the

updated information zt. The research cost in stage I is assumed to be sunk and hence neglected

throughout this analysis. If the firm decides to advance a project to stage II, the firm incurs the

development cost d > 0. By the end of the stage, we assume that xt is fully revealed and the firm

may launch the product into the market. Without loss of generality, we assume that the firm

4 In a setting where some buyers have complementary technology for the seller, or the market is not thick enough to make technology available anytime, which we rule out in this paper, there will be trades for early stage projects. 5 In reality c can be an endogenous variable. For example, a pharmaceutical firm may still keep or rent out its “drug reps” to avoid the need of hiring and training again in future even if there is no new drug in the market currently. We assume that c is exogenous for the sake of simplicity. This is a case in the example when the costs of keeping or renting “drug reps” for a period are higher than the costs of re-investment in human capital if needed in future.

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earns the profit of xt in the first year of the sale and none in the second year. The firm will get

zero returns if it decides to terminate the project.

Instead of commercializing its own R&D project, the firm may sell or buy a project or,

more precisely, the right to sell a product from the technology in each period in the technology

market. It is assumed that this market is competitive, namely, there are many firms and each

firm takes the price as given. When firms trade projects in the market, a transaction cost will be

incurred. Let the buyer incur the transaction cost τB while the seller incurs the transaction cost

τS. Both values are independent of the value of a project x. Let τ = τB + τS. In order to keep the

consideration of transaction costs consistent with our perfect competition assumption, we assume

that the value of a project x is verifiable and a buyer and a seller can sign a contract that is

contingent on x.6

Let P(x) be the price of the project with value x. Because of the adjustment cost, firms

with different products and R&D portfolios may vary in their urgency for a new product. P(x)

can be higher or lower than x depending on the demand and supply in the technology market.

Let ε(x) = P(x) – x + τB, which is the market premium for the product. Then, the direct revenue

from the trade of a project is P(x) – τS = x + ε(x) − τ for the seller and x – P(x) – τB = – ε(x) for

the buyer. Note that the buyer’s revenue from the trade depends only on ε(x) and buyers will try

to buy technology from sellers who offer the lowest ε(x) and is indifferent to the actual value of

the project, x. Hence, competition in the technology market should determine a unique market

premium and this will be independent of x, i.e., ε(x) = ε. We also focus our analysis on a steady

state where the distribution of individual firm types, and hence ε does not change over time.

These assumptions may be unrealistic if there is only a limited number of firms in the market or 6 Alternatively, we may assume that transaction costs are incurred after asymmetric information problem be resolved and two parties sign a contract for trading the technology.

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because technology changes over time. However, we believe it is acceptable for our purpose of

illustrating how R&D management policies differ among firms with different downstream assets.

Figure 1 graphically presents our three stages and the decisions options for each stage.

The following four state variables describe our system.

I2,t = 1 if the firm has a project in stage II in period t. 0 otherwise.

I3,t = -1 if the firm has no product in the market in period t. 0 if the firm has only an old product in the market in period t.

1 if the firm has a new product in the market in period t.

xt-1: true market value of the project in stage II revealed to the firm in period t. The subscript denotes that the project starts at the beginning of period t-1.

zt: posterior expected value of the product in stage I after observing the signal st at the

end of period t. The subscript denotes that the project starts at the beginning of period t.

Note that (I2,t, I3,t) indicate how urgent it is to develop or launch a product to avoid

incurring the adjustment cost. We call the firm with (I2,t, I3,t) “firm (I2,t, I3,t)”. For example, firm

(0, -1) and firm (1, -1) do not have a product in the market and thus face no imminent threat of

incurring an adjustment cost. This example corresponds to the case of a firm that has not yet

made co-specialized investments in downstream activities. In contrast, firm (0, 0) and firm (1, 0)

will incur the adjustment cost unless it launches a new product by either commercializing its

current project (feasible only for firm (1, 0)) or buying technology from another firm in the

current period. Finally, firms (0, 1) and (1, 1) have a new product in the market, which gives

them two periods with introducing a new product before incurring the adjustment costs.

Let V(zt, xt-1, I2,t, I3,t : ε) be the firm’s value function evaluated at the end of period t and

parameterized by the market premium ε. We often omit subscript t and parameter ε and use

shorter notation V(z, x, i, j) or Vi, j (z, x). Note that Vi, j does not depend on x when i = 0 because

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there is no project in the second stage, thus we omit x (i.e. V0, j(z)). Our model’s three decision

variables are:

e1,t = 1 if the firm continues a project to stage II at the end of period t. 0 otherwise.

e2,t = 1 if the firm launches a product to the market at the end of period t.

0 otherwise.

ft = 1 if the firm sells its product to another firm at the end of period t. -1 if the firm buys a product from another firm at the end of period t.

0 otherwise.

The options of launching a product (e2,t = 1) to the market includes: (1) launches own

product (ft = 0); (2) buys technology from the market (ft = -1). Or the firm may not launch a

new product in the market (e2,t = 0) with the options: (3) terminate the project in the second

stage, if there is one (ft = 0); (4) sell the project in the second stage in the technology market (ft =

1). Hence, we have e2,t + ft = {0, 1}. A dynamic linkage exists between the firm’s current

decisions of (e1,t , e2,t , ft ) and its future profits. For example, a start-up’s decision of launching

a product in the market implies that it may face in the future the possibility of incurring c. State

variables for the firm are (zt, xt-1, I2,t, I3,t), which can be treated as the strength of the pipeline.

Intuitively the strength of the pipeline will affect decisions (e1,t , e2,t , ft ). Suppose the firm has a

project in stage II and a product in the market, the impact of (zt, xt-1) on decisions (e1,t , e2,t , ft )

can be different from a firm without a project in stage II or without a product in the market. We

formally model how these decisions are made by using the dynamic programming approach.

First, the firm has to pay d if e1,t = 1 (i.e., advances a project into stage II), and c if I3,t = 0

and e2,t = 0 (i.e., sells an old product in the current period and does not launch new product in the

market). Then it earns xt-1 + ε – τ if ft = 1 (i.e., sells technology in the technology market), xt-1 if

e2,t = 1 and ft = 0 (i.e., launches own project in the market), and – ε if ft = -1 (i.e., launches a new

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product in the market by buying from the technology market). Therefore, the static profit

function can be written as follows:

3, 2,1, 0 2, 1 1 11 (1 ) 1 1t t tt t I t t e f tde c e x f

tfπ ε τ= − + == − − − + + − =

t

where 1* is an indicator function, i.e., 1* = 1 when * is true but 0 otherwise.

Let δ be the discount rate. The Bellman equation takes the following form:

3, 2,

1, 2,

1

1 2, 3, 1, 0 2, 1 1 1, ,

, | 1 2, 1 3, 1

( , , , ; ) max [ 1 (1 ) 1 1

( , , , )]

t t tt t t

t t t

t t t t t I t t e f t fe e f

z x z t t t t

V z x I I de c e x f

E V z x I I

ε ε τ

δ+

− = − +

+ + +

= − − − + + −

+

= =

)t

(1)

The transition of states from period t to t+1 is the following: If e1,t = 1, the firm will have a

project in stage II in the next period, and vice versa if e1,t = 0; hence, I2,t+1 = e1,t. If e2,t = 1, the

firm will have a new product in the market in the next period (i.e., I3,t+1 = 1); if I3,t = 1 and e2,t =

0, the firm will have an old product in the market next period (i.e., I3,t+1 = 0); if I3,t = 0 or -1, and

e2,t = 0, the firm will have no product selling in the market next period (i.e., I3,t+1 = -1). Hence,

3,3, 1 2, 0 2,1 (1tt t II e+ ≤= − − e x. Also, as we discussed above, , and 1 ( )tz f+ ∼ ( | )t tx f x z∼ .

Equation (1) simply states that the expected value of the firm this period is the maximum

of the sum of the payoff it obtains this period and the discounted expected value of the firm in

the next period conditional on all state variables. This dynamic programming approach

transforms the optimization problem into one of computing the value function V(⋅). Under some

general conditions that are all satisfied here, Blackwell’s theorem shows that the value function

can be identified uniquely and there are some well-known and quite effective algorithms for

computing the function V(⋅).7 Furthermore, the solutions (e1,t , e2,t , ft), t = 1, 2, …, to the Bellman

equation specify the optimal project selection rules for the firm. The firm should only advance a

project and/or introduce a new product, through buying from the technology market or by

7 For the conditions required for Blackwell’s theorem, see Stokey and Lucas (1989) for example.

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launching own product from stage II, when e1,t = 1 and/or e2,t =1 are the solutions to the Bellman

equation in (1).

Key to our analysis is the dynamic interdependency among project selection decisions

which, as we will explain, arises when transaction costs are positive: the firm’s decisions depend

on the strength of its R&D pipeline, which is a function of 1 2, 3,( , , , )t t t tz x I I− , and such decisions

are time-varying as state variables change over time. Notice that we do not have discrete types

as are continuous measures. The major purpose of this paper is to provide a rigorous

analysis of how the strength of the R&D pipeline will affect the decisions of heterogeneous firms

under different product and technology market conditions.

1( , )t tz x −

1. Some Preliminaries

Before proceeding with our analysis, we set up some model preliminaries that will be

useful to simplify our proofs. Let Vij denote the value function in (1) where the focal firm has I2t

= i and I3t = j. Given two types of I2t and three types of I3t, we have altogether six types of firms.

Suppose that the firm has a project in stage II, the value of which is x, and the firm is choosing

between selling the project in the technology market and abandoning it. It is straightforward that

the firm will sell only if the revenue it obtains is positive, i.e., 0x ε τ+ − ≥ . Suppose that the

firm has a project in stage II and has decided to launch a new product in the market, and the firm

is choosing between launching its own project with value x and buying another project in the

technology market, where it has to pay the premium of ε. Again, it is straightforward that the

firm will launch its own project only if 0x ε+ ≥ . Define | | max{ ,0}X X+ ≡ and

. We can break down the value function of the six types of firms

from (1) as the following:

1( | ) [ ( , ) | ]t t tE V z E V z x z+≡ t

15

01 00 10 01 11( ) max{ ( | ); ( | ); ( | ); ( | )}V z E V z d E V z E V z d E V zδ δ ε δ ε δ= − + − + − − + ,

11 00 10

01 11

( , ) max{| | ( | ); | | ( | );| | ( | ); | | ( | )},

V z x x E V z d x E V zx E V z d x E V z

ε τ δ ε τ δε ε δ ε ε δ

+ +

+ +

= + − + − + + − ++ − + − + + − +

00 0 1 1 1 01

11

( ) max{ ( | ); ( | ); ( | ); ( | )},V z c E V z d c E V z E V z

d E V z δ δ ε δ

ε δ− −= − + − − + − +

− − +

10 0 1 1 1

01 11

( , ) max{ | | ( | ); c | | ( | ) | | ( | ); | | ( | )},

V z x c x E V z d x E V zx E V z d x E V z

;ε τ δ ε τ δε ε δ ε ε δ

+ − + −

+ +

= − + + − + − − + + − ++ − + − + + − +

0 1 0 1 1 1 01 11( ) max{ ( | ); ( | ); ( | ); ( | )},V z E V z d E V z E V z d E V zδ δ ε δ ε δ− − −= − + − + − − +

1 1 0 1 1 1

01 11

( , ) max{| | ( | ); | | ( | );| | ( | ); | | ( | )}.



− + − +

+ +

= + − + − + + − ++ − + − + + − +

−

10

Next, define

1 00( ) max{ ( | ); ( | )}V z E V z d E V zδ δ= − + ,

2 01( ) max{ ( | ); ( | )}V z E V z d E V z11δ δ= − + ,

3 0 1( ) max{ ( | ); ( | )}V z E V z d E V z1 1δ δ− −= − + .

Then we can further reduce the value functions above into the following,

01 1 2( ) max{ ( ); ( )}V z V z V zε= − + , (2)

11 1 2( , ) max{| | ( ); | | ( )}V z x x V z x V zε τ ε ε+ += + − + + − + , (3)

00 3 2( ) max{ ( ); ( )}V z c V z V zε= − + − + , (4)

10 3 2( , ) max{ | | ( ); | | ( )}V z x c x V z x V zε τ ε ε+ += − + + − + + − + , (5)

0 1 3 2( ) max{ ( ); ( )}V z V z V zε− = − + , (6)

1 1 3 2( , ) max{| | ( ); | | ( )}V z x x V z x V zε τ ε ε− + += + − + + − + . (7)

Based on the above preliminaries, we have the following results which are useful for later

analysis:

16

Lemma 1. Compare the values of different types of firms, we have and 00 01 0 1( ) ( ) ( )V z V z V z−≤ ≤ 10 11 1 1( , ) ( , ) ( , )V z x V z x V z x−≤ ≤ . In addition,

. All inequalities are strict for some sets of (z, x) with positive measure, implying that, for any ,

10 00 11 01 1 1 0 1( , ) ( ) ( , ) ( ) ( , ) ( )V z x V z V z x V z V z x V z− −− ≥ − ≥ −

tz

10 1 00 1 11 1 01 1 1 1 1 0 1 1[ ( , ) ( ) | ] [ ( , ) ( ) | ] [ ( , ) ( ) |t t t t t t t t t t t tE V z x V z z E V z x V z z E V z x V z z+ + + + − + − +− > − > − ]

Proof in the appendix.

2. Zero Adjustment Costs

Let {e1* (zt, xt-1, I2,t, I3,t), e2

* (zt, xt-1, I2,t, I3,t), f* (zt, xt-1, I2,t, I3,t)} be the optimal strategy in

period t. We first consider a baseline case in which the adjustment cost c is zero. In this case,

firms will have the same valuations for the same project, which implies that there will be no

trade in the technology market.

Proposition 2. When c = 0, no firm trades in the technology market (i.e. f*= 0). Let x = 0 and z be such that

0 ( | )xf x z dx dδ

+∞=∫ . Then, e1

* = 1 (= 0) if z > z (z

< z ) and e2* = 1 (= 0) if x > x (x < x ) for all types of firms.


Proposition 2 shows that absence of adjustment cost eliminates the interdependency

among projects and that the firm evaluates each project separately. Given our assumption of

homogeneous R&D capability, all firms value the same project in the same way, which makes

transaction costs associated with the technology market immaterial. Therefore, a product should

be introduced to the market as long as its expected value is positive and the project should be

advanced to development as long as its option value exceeds the cost of development, which is

the independent project evaluation prescription found in Roberts and Weitzman (1981) and,

more generally, in the project finance literature (e.g., see Brealey and Myers 2002).

3. Positive Adjustment and Transaction Costs

17

Now we present our main findings for a more general case, where c > 0 and τ > 0, in the

following propositions. In this case dynamic interdependencies among projects in the pipeline

emerge. Also, the market price for technology affects the optimal R&D selection policies.

Proposition 3.1. Consider a firm that does not have a project in stage II. There is a threshold 0ˆ ( )jz ε such that the firm will advance the project from stage I to II (i.e., e1

* (z, x, 0, j) = 1) only if z ≥ 0,ˆ ( )jz ε , for j = 1, 0 and –1. Furthermore, 01 00 0 1ˆ ˆ ˆ( ) ( ) ( )z z zε ε ε−≤ ≤ . For a sufficiently high ε, 01 00 0 1ˆ ˆ ˆ( ) ( ) ( )z z zε ε − ε< < while, for a sufficiently low ε,

01 00 0 1ˆ ˆ ˆ( ) ( ) ( )z z zε ε −= = ε (See Figure 2). Proof in the appendix.

The above proposition describes how the optimal threshold, z , differs for different state

conditions when no project is in development. As displayed in Figure 2, the thresholds depend

on ε, the market premium for purchasing technology. Note that the (0, 1) firm has a new product

in the market but its R&D pipeline is empty (it has no project currently in stage II) and thus it is

eager to fill the pipeline by lowering the threshold for early-stage projects knowing that the

current product in the market expires in the next period. The higher is ε, the lower threshold

will be because the high cost of buying technology leaves advancing own project into the later

stage as the only means of avoiding to incur the adjustment cost. In contrast, the higher the

market premium for a (0, -1) type firm the higher will be the threshold

01z

0 1z − because it will be

more costly to avoid adjustment costs once it invests in downstream assets. One seemingly

counter-intuitive result is that 00 01ˆ ˆ( ) ( )z zε ε> when ε is high enough. Although the (0, 0) type

firm has the most urgent need for a new product, it will give up launching a new product when

ε becomes too high and its early-stage product is not strong enough.

18

In the preceding analysis, no project was in stage II. We now ask how a project in stage

II affects the decision to advance a project from stage I. Two propositions follow from this

question:

Proposition 3.2. Consider the firms which have a project in stage II. There is a threshold 1 ( , )jz xε such that they will advance the project from stage I to II (i.e., e1

* (z, x, 1, j) = 1) only if 1,ˆ ( , )jz z xε≥ , for j = 1, 0 and –1. Furthermore, comparing all firms, 01 11 10 00 1 1 0 1ˆ ˆ ˆ ˆ ˆ ˆ( ) ( , ) ( , ) ( ) ( , ) ( )z z x z x z z x zε ε ε ε ε− −≤ ≤ ≤ ≤ ≤ ε

x

. Proof in the appendix.

The above proposition is similar to Proposition 3.1. With similar reasoning, the (1, 1)

type firm has the most urgent need to advance the project in stage I, while the (1, -1) type firm

has the least urgent need. As 01 11ˆ ˆ( ) ( , )z zε ε≤ for any x, it implies that having a project in stage

II may reduce the urgency to advance the project in stage I. The strict inequality holds within a

range of the market premium, in which the price is too high for the (0, 1) type firm to buy but too

low for the (1, 1) type firm to sell a project in the technology market. The relationship is the

opposite if the firm has no need to fill in its pipeline (for the (1, 0) type firm, this is the case if

the market premium is too high and the value of the project in stage II too low), hence we have

1 0ˆ ˆ( , ) ( )jz x z jε ε≤ , j = 0 or -1.

A similar reasoning brings us to an interesting result in the following proposition.

Proposition 3.3. Consider the firm that has a project in stage II. 11ˆ ( , )z xε is non-decreasing in x, and

0,1 1,1 ' 1,1 1,1 ' 1,1ˆ ˆ ˆ ˆ ˆ( ) ( , ') | ( , ) ( , ') | ( , )x x tz z x z x z x zε εε ε ε ε ε<− > −= ≤ ≤ = τ ε− ;

10ˆ ( , )z xε is non-increasing in x and 1,0 1,0 ' 1,0 1,0 ' 0,0ˆ ˆ ˆ ˆ ˆ( , ) ( , ') | ( , ) ( , ') | ( )x t xz z x z x z x zε εε τ ε ε ε ε ε> − <−− = ≤ ≤ = ;

1 1ˆ ( , )z xε− is non-increasing in x and

1, 1 1, 1 ' 1, 1 1, 1 ' 0, 1ˆ ˆ ˆ ˆ ˆ( , ) ( , ') | ( , ) ( , ') | ( )x t xz z x z x z x zε εε τ ε ε ε ε ε− − > − − − <−− = ≤ ≤ = − .


19

The above proposition implies policy rules for balancing the R&D pipeline portfolio.

The (1, 1) type firm is more likely to raise the threshold for a project in stage I as the prospect for

a project in stage II improves. Projects in stages I and II are substitutes, in the sense that the firm

is in less need for a new product in the next period once the product in stage II is

commercialized. In contrast, projects in stages I and II are complements for the (1, 0) and (1, -1)

type firms. Once the product in stage II is launched, the firm will pursue more aggressively to

advance the project in stage I to lower the likelihood of incurring the adjustment costs in future.

Figures 2 and 3 display the thresholds of z for the (1, 1), (1, 0) and (1, -1) type firms, assuming

that x ε≤ − and x τ ε≥ − with fixed ε, respectively. As x rises from – ε to τ − ε, 1 ( , )jz xε

continuously shifts up on the ε-dimension.

The following propositions provide results for the optimum decisions of trading in the

technology market. Note that firms that have a project in stage II may choose to launch or

abandon the project, and sell or buy project in the technology market, while firms that do not can

only choose buying or not in the technology market.

Proposition 3.4. Consider the firms that do not have project in stage II. There are thresholds of market premium 0 1ˆ ( )zε − < 01ˆ ( )zε < 00ˆ ( )zε such that the (0, j) firm will buy in the technology market (i.e. *( , ,0, ) 1f z x j = − ) only if ε < 0ˆ ( )j zε , for j = 0, 1, or -1.


Proposition 3.4 discusses the optimal technology market buy-strategy for the types of

firms that do not have any projects in stage II. All these firms should buy technology if the

premium is sufficiently low; but the (0, 0) type that has an old product in the downstream market

is most likely to buy and the (0, -1) type that does not have co-specialized investments is least

20

likely to buy. The thresholds also depend on the prospects of their projects in stage I (i.e. z) as

0ˆ jε is a function of z. The functions of 0ˆ jε are illustrated in Figures 4 to 6.

The next proposition looks at the decision making by the (1, j) firms. It allows us to

study the decisions of whether to launch a new product in the market and whether to buy or sell a

project in the technology market, and analyze how the strength of the R&D pipeline and the

market premium ε affect those decisions. Note that 0ˆ jε in later propositions is the same as

defined in Proposition 3.4.

Proposition 3.5. Consider the firms who have a project in stage II. There are thresholds of market premium 1 1ˆ ( )zε − < 11ˆ ( )zε < 10ˆ ( )zε , where 1 ( )j zε > 0ˆ ( )j zε , and thresholds for the value of the product in stage II, 1 1 0ˆ ˆ ˆ( , ) ( , ) ( , )x z x z x zε ε ε− ≥ ≥ , where ˆ ( , )jx zε ε τ− ≤ ≤ −ε , such that the following properties hold:

i) When 1 ( )j zε ε> , the (1, j) firm never launches a new product (i.e.

). In this case *2 ( , ,1, ) 0e z x j = ˆ ( , )jx zε τ ε= − . The firm will not buy in the

technology market, and it will sell in the technology market (i.e. ) only if *( , ,1, ) 1f z x j = ˆ ( , )jx x zε> ;

ii) When 0ˆ ˆ( ) ( )j 1 jz zε ε ε≤ ≤ , the (1, j) firm never buys or sells in the technology

market (i.e. *2 ( , ,1, ) 0f z x j = ), for j = 0 and -1. The firm will commercialize its

stage II project (i.e. *2 ( , ,1, ) 1e z x j = ) only if ˆ ( , )jx x zε> . The function 1( , )x zε

is increasing in z and 0ˆ ( , )x zε and 1ˆ ( , )x zε− are decreasing in z; iii) When 0ˆ ( )j zε ε< , the (1, j) firm always launches a new product (i.e.

). In this case *2 ( , ,1, ) 1e z x j = ˆ ( , )jx zε ε= − . The firm will buy a product in the

technology market (i.e. *( , ,1, ) 1f z x j = − ) only if ˆ ( , )jx x zε< . (See Figures 7-12.)


Before discussing Proposition 3.5, let us present an alternative way to characterize

1 ( )j zε and ˆ ( , )jx zε in the next corollary.

Corollary 3.6.

21

i) The (1, j) firm will never buy in the technology market when 1 ( )j zε ε> , and will never sell in the technology market when 0ˆ ( )j zε ε< for any z.

ii) The firm will either launch in the market or sell in the technology market, i.e., , if * *

2 ( , ,1, ) ( , ,1, ) 1e z x j f z x j+ = ˆ ( , )jx x zε> and will either terminate the project in

stage II or buy in the technology market, i.e., * *2 ( , ,1, ) ( , ,1, ) 0e z x j f z x j+ = , if

ˆ ( , )jx x zε< for j = 1, 0 and -1 and any ε and z.

Proposition 3.5 and Corollary 3.6 discuss the optimal strategy in buying and selling a

project in the technology market. There exist thresholds for ε. Firms should sell their projects

(with x high enough) if the premium is sufficiently high. The threshold for the decision to sell is

the highest for the (1, 0) type firm (i.e. least likely to sell) and the lowest for the (1, -1) type firm

(i.e. most likely to sell). Similar to Proposition 3.4., these firms should buy a technology project

(with x low enough) if the premium is sufficiently low. This threshold for the (1, j) type firm is

identical to that for the (0, j) type firm.

The inequalities in Proposition 3.5 1 1 0ˆ ˆ ˆ( , ) ( , ) ( , )x z x z x zε ε ε− ≥ ≥ and

1 1ˆ ( )zε − < 11ˆ ( )zε < 10ˆ ( )zε imply that the (1, 0) type firm is most likely to commercialize its own

project and least likely to sell it while the (1, -1) type firm is least likely to commercialize its

own project and most likely to sell it. The reason is clear from our earlier discussion: the (1, 0)

type firm has the most urgent need to introduce a new product to avoid incurring adjustment

costs while the (1,-1) type firm has the least urgent need to introduce a new product because it

has no downstream co-specialized assets to maintain.

For an intermediate level of ε higher than 01ε but lower than 11ε , (ii) in Proposition 3.5

says that 1( , )x zε is increasing in z, suggesting that the (1,1) type firm will increase the threshold

for commercialization if there is the prospect of a successful product being introduced in the

22

future (high z). This outcome arises because such a firm can choose to terminate an unprofitable

product without the fear of incurring the adjustment cost. Therefore, as we have argued earlier,

projects in the pipeline are substitutes for the (1, 1) firm (see Figure 10). In contrast, the

prospect of a successful product being introduced in the future (high z) decreases the threshold

for commercialization in current period for (1, -1) and (1, 0) type firms. Both types will launch

instead of selling its final-stage project if it has a strong early stage project because the likelihood

of incurring the adjustment cost in future is relatively low (see Figures 11 and 12).

4. Two Extreme Cases

In the above propositions, we show that project selection rules are different among

different types of firms when transaction costs are non-negligible. In the next two subsections,

we evaluate the two extreme cases: when τ = 0 and when τ is too large for the technology

market to exist. These two cases demonstrate that managing R&D pipeline strategically is

important because of dynamic interdependencies when τ is high but the dynamic

interdependencies disappear as τ goes to zero.

4.1. No Transaction Cost Case

Proposition 4.1. If c > 0 but τ = 0, all firms will have the identical R&D selection policies. That is, * *

2( , , , ) ( , , , ) 1e z x i j f z x i j+ = only if x > - ε and only if *1 ( , , , ) 1e z x i j =

ˆ( | )xf x z dx dε

δ+∞

−>∫ . Let ε∗ be the equilibrium market premium. Then, there are

thresholds for premium, 0 1> 1ε ε ε−> such that *0 > 1ε ε ε−> always hold and

1) potential buyers are the (1,0) and (0,0) firms and potential sellers are the (1,1) and (1,-1) firms when *

0 1>ε ε ε> ; and 2) potential buyers are the (1,0), (0,0), (1,1) and (0,1) firms and potential sellers are

the (1,-1) firms when *1 1>ε ε ε−> .

Proof in the Appendix.

23

Here, we assume that the market is efficient enough so that the transaction costs are

negligible. In such setting, the access to the market for final-stage technology eliminates any

differences in R&D selection rule for early-stage projects. Compared with the case of zero

adjustment cost in Proposition 2, the values of thresholds x and z are different unless 0ε = .

Parts 1) and 2) in the proposition suggest that the (1, 0) and (0, 0) type firms always become

buyers if their own projects end up with x < - ε while the (1, -1) type firms are always sellers if

their own projects turn out to be good enough, namely x > - ε. Depending on the distribution of

R&D investment outcome, the (1, 1) type firms could become buyers or sellers of technology. In

the equilibrium, the number of buyers has to match the number of sellers and therefore, there are

upper limits and lower limits to the equilibrium market premium. This result specifies that the

range is between 1ε− and 0ε .

4.2. High Transaction Cost Case

Proposition 4.1 specifies the range for the equilibrium market premium is *0 1>ε ε ε−> .

As the transaction cost τ rises, the feasible range for *ε gets narrower and eventually the

technology market will collapse for a sufficiently high τ. The following proposition provides a

sufficient condition for the technology market to not exist. When the transaction cost becomes

too large, no price will become acceptable for buyer or seller.

Proposition 4.2. The technology market does not exist when

11 1 1 01 0 1max ( | ) ( | )z

E V V z E V V z cτ δ δ− −≥ − − − + , where

11 1 1 01 0 1max ( | ) ( | )z

E V V z E V V zδ δ− −− − − is positive.


24

Although we do not elaborate, the dynamic interdependencies between the decisions on

R&D selection and product introduction (i.e. e1* and e2

*) are greatest when the technology

market does not exist.8 In this case, the range where e1 and e2 are complements or substitutes is

not limited to a potentially narrow range of xε τ ε− < < − .

To summarize , firms in this case have different optimal project selection policies in the

management of their R&D pipelines, depending on whether they have made co-specialized

investments or not, and on the strength of their pipelines. If they already have a product in the

market and are facing the danger of incurring adjustment costs, they will try to fill up their R&D

pipeline by lowering the threshold values. Hence, we will find that, on average, the incumbents

push projects into later R&D stages or introduce products into the market with lower values than

start-ups. In a general case where transaction costs are not negligible, such dynamic

dependencies exist even with a technology market where incumbents may buy projects from

other firms. Thus, in the situation in which a technology market operates but with positive

transaction costs, these interdependencies are greater as transaction costs in the technology

market become higher.

Discussion and Conclusion

The most important implications of our model are the strategic implications that

downstream co-specialized activities, the strength of R&D pipelines, and transaction costs in

technology markets have on R&D project selection rules. Strategically managing R&D pipelines

differs from the management of independent R&D projects whenever transaction costs and

adjustment costs are greater than zero. The consideration of positive transaction costs in the

8 The detailed analyses for the case without technology market can be provided upon request.

25

technology market and state-contingent adjustment costs changes project selection incentives and

leads to selection thresholds that are state contingent and different from the standard prescription

in which R&D projects are evaluated independently. Our model provides a general formulation

of the issue in the sense that the standard prescription is a special case of our model in which the

adjustment costs are zero.

Our model offers several counterintuitive finding. First, our model indicates that firms

with identical downstream assets may choose substantially different project selection rules

depending on the strength of their R&D project portfolio. Second, firms with different

downstream assets may choose identical project selection rules if they face no transaction costs

in the technology market. Third, our model informs when firms sell projects and when they buy

projects. In some instances, firms with downstream assets may choose to sell projects, which is

an empirical regularity observed in the pharmaceutical industry (i.e., large pharmaceutical firms

sometimes sell their projects) but not explained in the literature. Fourth, our analysis indicates

that R&D projects are sometimes complements and other times substitutes, which represent a set

of predictions that are not initially obvious. The above analyses and insights show that firms

have to consider the fully specified option value of a project, one that includes the

interdependencies described above, when making decisions of whether or not to advance a

project into development or to commercialize it in the market. Fifth, we are able to specify the

upper and lower bounds for the market equilibrium premium in the technology market, as well as

the level of transaction costs over which the technology market will no longer exist.

Finally, our model provides an interesting implication for firms’ preference for risk. We

can show that the value function (Eq. (1)) is convex in the (z,x) space for a fixed ε and has an

“option”-like structure, if the distribution of (z,x) is well-behaved. The result can be easily

26

understood recalling that firms fully benefit from a high value of (z,x) but the loss from a low

value of (z,x) is limited to adjustment costs. This implies that all types of firms should be risk-

loving. However, the marginal rate of substitution between mean returns and variance of returns

for each type of firm should vary as long as the firms face transactions costs in the technology

market. Now, suppose that firms can choose in the research stage from the set of multiple

potential projects, each of which has a different expected value and level of uncertainty. Our

conjecture is that established firms are less willing to trade-off mean returns for variance of

returns than new firms. This intuition comes from comparing options with a low exercise price

to those with a high exercise price. The lower exercise price can be interpreted as the project

selection threshold of established firms and the higher one can be interpreted as that of new

firms. Assuming a normal distribution, we formally confirmed that options with a higher

exercise price gain more value in percentage terms from increased variance than those with a

lower exercise price, if these exercise prices are not too far above the mean of the distribution.

Intuitively, if you are out of money, your option value is very sensitive to the change in risk

while, if you are in the money, it is less so because the option value is high reflecting the gain

from exercising the option at the current price. The implication in our context is that young

firms with no co-specialized assets will tend to pursue high-value high-uncertainty projects while

established firms with significant co-specialized assets will prefer relatively moderate-value low-

uncertainty projects, which is consistent with our casual empiricism.

Our model, in addition to understanding the dynamics of project selection rules, may be

useful in explaining several phenomena. For instance, it may provide at least one explanation for

the enigma of R&D productivity, in which R&D productivity apparently declines with firm size

(for example, see Cohen and Klepper 1996). So long as the investment in co-specialized

27

downstream assets is positively correlated with firm size, larger firms are more willing to accept

projects with less chance of blockbuster success for more secured profit in research and

development activities than small firms, which, given the exit of unsuccessful small firms, could

lead to the observed pattern. Our model also resonates with Williamson’s (1985) arguments that

specific investments have substantial implications for innovative behavior and firm performance.

Unlike Williamson’s discussion, however, we provide a formal model of how downstream co-

specialized assets affect investment decisions.

The model also has implications for research on joint ventures (JV) and mergers and

acquisitions (M&A). Take the pharmaceutical industry. By now, there is a large literature on

joint ventures and alliances, much of which is developed around the context of the

pharmaceutical industry, that indicates these structures arise to reduce transaction costs. This

research provides prima facie evidence that even the pharmaceutical industry, which is often

characterized as the prototypical technology market, faces positive transaction cost, which is a

necessary condition for the conclusions of our model. Also note that our model indicates that the

value of a project is state-contingent. This implies that a project in the development stage is

likely, on average, to have more value for an established pharmaceutical firm than for a biotech

venture, for example. Hence, a biotech firm should sell its successful projects to large

pharmaceutical firms in the form of a JV or M&A rather than launching the product in the

market on its own.

So, in the pharmaceutical industry, we anticipate that those established pharmaceutical

firms that end up with few candidate drugs in their R&D pipeline will be more likely to form

joint ventures with biotechnology firms whose drug discoveries are well matched with the

incumbent’s co-specialized distribution channels. Relatedly, we anticipate that mergers and

28

acquisitions may be made for similar reasons, which imply both weakness in a firm’s pipeline

and the match between development pipelines and distribution channels are relevant factors.

Gans and Sterns (2000) is the first research that explicitly models the entrant’s option of

licensing the innovation in the R&D race and analyzes how the incentive to innovate is shaped

by this option. Our approach to this issue would be to evaluate the interactions between the

R&D pipeline management and the decisions on such options as JV and M&A before appealing

to the assumption of asymmetric market power between the incumbent and the entrant. Future

research should model such options to evaluate how adjustment costs affect not only the choice

of JV and M&A but also the price of such transactions, which are likely to be especially costly

for the buyer when its desire to avoid incurring adjustment costs is great.

Another area of potential research involves the number of research projects initiated and

which project among multiple potential candidates should be chosen to advance to later stages.

Our model does not explicitly consider multiple projects in each stage, which is an important

element of managing R&D portfolios. A more complete understanding of portfolio management

could be derived from a model in which multiple projects per stage are considered. Such a

model presents a substantial opportunity for future research.

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Cooper, R. G., and S. J. Edgett (2001). “Portfolio Management for New Products: Picking the Winners.” Product Development Institute, Ancaster, Ontario, Canada.

Gans, J. S., and S. Stern (2000). “Incumbency and R&D Incentives: Licensing the Gale of Creative Destruction,” Journal of Economics and Management Strategy, 9(4): 485-511.

Gee, R. E. (1971). “A Survey of Current Project Selection Practices,” Research Management, 14(5): 38-45.

Kamien, M. I. and N. L. Schwartz (1978). “Self-financing of and R&D Project,” American Economic Review, 68: 252-261.

Merton, R. C. (1973). “Theory of Rational Option Pricing,” Bell Journal of Economics, 4(1): 141-83.

Pharmaceutical Business News. (1992). “Zeneca’s drug pipeline critical for ICI’s future.” Reinganum, J. F. (1983). “Uncertain Innovation and the Persistence of Monopoly,” American

Economic Review, 73(4): 741-48. Reinganum, J. F. (1989). “The Timing of Innovation: Research, Development, and Diffusion,”

in Handbook of Industrial Organization. Volume 1, 849-908, R. Schmalensee and R. D. Willig eds., New York: Elsevier Science.

Roberts, K. and M. L. Weitzman (1981). “Funding Criteria for Research, Development, and Exploration Projects,” Econometrica, 49(5): 1261-1288.

Schumpeter, J. A. (1942). Capitalism, Socialism, and Democracy. New York: Harper and Row. Schwartz, W. S. and L. Trigeorgis (2001). Real Options and Investment under Uncertainty.

Cambridge, MA: The MIT Press. Stokey, N. L. and R. E. Lucas (1989). Recursive Methods in Economic Dynamics.

Massachusetts: Harvard University Press. Tanouye, E. and R. Langreth (1997). “Time's Up: With Patents Expiring On Big Prescriptions,

Drug Industry Quakes -- Top Firms Gird for Onslaught From Generics, Scramble To Develop New Products --- Bet on Fewer Blockbusters,” Wall Street Journal, (August 12): A1.

Wheelwright, S. C. and K. B. Clark (1992). Revolutionizing Product Development. New York: The Free Press.

Williamson, O. W. (1985). The Economics Institutions of Capitalism: Firms, Markets, and Relational Contracting. New York: Free Press.

30

Stage I:Research

cost is sunk and omitted

Stage II:Development

cost is d > 0

Stage III:Commercialization

adjustment cost is c

xt is expected value of project

st = xt + εt

E[εt] = 0

zt=E[xt|st]

xt is fully revealed

Decide to launch, terminate, or participate in the technology market

Decide to continue or terminate the project

profit of xt in first year,

product lasts two periods

Figure 1: Modeling Assumptions with Technology Market and Adjustment Costs

If participating in the technology market, buy or sell projectτ = τΒ + τS

31

1,1 0,1

1

( - | )or ( )

E V V z dz z

δ =

1,0 0,0

0

( - | )or ( )

E V V z dz z

δ = 1, 1 0, 1

1

( - | )or ( )

E V V z dz z

δ − −

−

=

0,1 0, 1

0 1

( - | )or ( )

E V V zz

δ εε

−

−

=

1,1 1, 1

0 1

( - | )or ( )

E V V zz

δ εε

−

−

=

1,1 1,0

01

( - | )or ( )

E V V zz

δ εε

=

0,1 0,0

01

( - | )or ( )

E V V zz

δ εε

=

0,1 0, 1

00

( - | )or ( )

E V V z cz

δ εε

− + =

1,1 1, 1

00

( - | )or ( )

E V V z cz

δ εε− + =

0 1

1 1

ˆ ( )ˆ ( , )zz x

εε

−

−

=00

10

ˆ ( )ˆ ( , )zz x

εε

=01

11

ˆ ( )ˆ ( , )zz x

εε

=

ε

z

Figure 2: and when x < - ε01ˆ ( )z ε 00ˆ ( )z ε 0 1ˆ ( )z ε−01ˆ ( )z ε 00ˆ ( )z ε 0 1ˆ ( )z ε− 11ˆ ( , )z xε 10ˆ ( , )z xε 1 1ˆ ( , )z xε−11ˆ ( , )z xε 10ˆ ( , )z xε 1 1ˆ ( , )z xε−

1,1 0,1

1

( - | )or ( )

E V V z dz z

δ =

1,0 0,0

0

( - | )or ( )

E V V z dz z

δ = 1, 1 0, 1

1

( - | )or ( )

E V V z dz z

δ − −

−

=

0,1 0, 1

1 1

( - | )or ( )

E V V z tz

δ εε

−

−

+ =

1,1 1, 1

1 1

( - | )or ( )

E V V z tz

δ εε

−

−

+ =

1,1 1,0

11

( - | )or ( )

E V V z tz

δ εε

+ =

0,1 0,0

11

( - | )or ( )

E V V z tz

δ εε

+ =

0,1 0, 1

10

( - | )or ( )

E V V z c tz

δ εε

− + + =

1,1 1, 1

10

( - | )or ( )

E V V z c tz

δ εε

− + + =

1 1ˆ ( , )z xε−10ˆ ( , )z xε11ˆ ( , )z xε

ε

z

Figure 3: when x > t - ε11ˆ ( , )z xε 10ˆ ( , )z xε 1 1ˆ ( , )z xε−11ˆ ( , )z xε 10ˆ ( , )z xε 1 1ˆ ( , )z xε−

32

01ˆ ( )zε

ε

z

Figure 4: The (0,1) firm and The (1,1) firm with x <

1

2

1 (develop)1 (launch)

1 (buy)

eef

=== −

1

2

0 (do not develop)1 (launch)

1 (buy)

eef

=== −

1

2

1 (develop)0 (do not launch)0 (do not trade)

eef

===

1

2

0 (do not develop)

0 (do not launch)

0 (do not trade)

e

e

f

=

=

=

01ˆ ( )z ε

1,1 1,0

01

( - | )or ( )

E V V zz

δ εε

=

0,1 0,0

01

( - | )or ( )

E V V zz

δ εε

=

1 ( , )x zε

00ˆ ( )zε

ε

z

Figure 5: The (0,0) firm and The (1,0) firm with x <

1

2


1 (buy)

eef

=== −

1

2


1 (buy)

eef

=== −

1

2


eef

===

1

2

0 (do not develop)0 (do not launch)0 (do not trade)

eef

===

0,1 0, 1

00

( - | )or ( )

E V V z cz

δ εε

− + =

1,1 1, 1

00

( - | )or ( )

E V V z cz

δ εε− + =

00ˆ ( )z ε

0ˆ ( , )x zε

33

0 1ˆ ( )zε −

ε

z

Figure 6: The (0,-1) firm and The (1,-1) firm with x <

1

2


1 (buy)

eef

=== −

1

2


1 (buy)

eef

=== −

1

2


eef

===

1

2


eef

===

0,1 0, 1

0 1

( - | )or ( )

E V V zz

δ εε

−

−

=

1,1 1, 1

0 1

( - | )or ( )

E V V zz

δ εε

−

−

=

0 1ˆ ( )z ε−

1ˆ ( , )x zε−

11ˆ ( )zε

ε

z

Figure 7: The (1,1) firm with x >

1

2

1 (develop)0 (do not launch)1 (sell)

eef

===

1

2

1 (develop)1 (launch)0 (do not trade)

eef

===

1

2

0 (do not develop)1 (launch)0 (do not trade)

eef

===

1

2

0 (do not develop)

0 (do not launch)

1 (sell)

e

e

f

=

=

=

11ˆ ( , )z xε

1,1 1,0

11

( - | )or ( )

E V V z tz

δ εε

+ =

0,1 0,0

11

( - | )or ( )

E V V z tz

δ εε

+ =

1 ( , )x zε

34

10ˆ ( )zε

ε

z

Figure 8: The (1,0) firm with x >

1

2


eef

===

1

2


eef

===

1

2


eef

===

1

2

0 (do not develop)0 (do not launch)1 (sell)

eef

===

10ˆ ( , )z xε

0,1 0, 1

10

( - | )or ( )

E V V z c tz

δ εε

− + + =

1,1 1, 1

10

( - | )or ( )

E V V z c tz

δ εε

− + + =

0ˆ ( , )x zε

1 1ˆ ( )zε −

1,1 0,1( - | )E V V z dδ =1, 1 0, 1( - | )E V V z dδ − − =ε

z

Figure 9: The (1,-1) firm with x >

1

2


eef

===

1

2


eef

===

1

2


eef

===

1

2


eef

===

0,1 0, 1

1 1

( - | )or ( )

E V V z tz

δ εε

−

−

+ =

1,1 1, 1

1 1

( - | )or ( )

E V V z tz

δ εε

−

−

+ =

1 1ˆ ( , )z xε−

1ˆ ( , )x zε−

35

36

1

2


eef

===

z

x

t-ε

Figure 10: The (1,1) firm with

1

2


1 (buy)

eef

=== −

1

2


eef

===

1

2


eef

===

1x

-ε

11z

01 11ˆ ˆ( ) ( )z zε ε ε≤ ≤

z

x

t-ε

Figure 11: The (1,0) firm when

1

2


eef

===

1

2


eef

===

1

2


1 (buy)

eef

=== −

1

2


eef

===

0x

10z

-ε

00 10ˆ ˆ( ) ( )z zε ε ε≤ ≤

z

x

t-ε

1

2


eef

===

1

2


eef

===

1

2


1 (buy)

eef

=== −

1

2


eef

===

0x

0z

1 1z −

-ε

Figure 12: The (1,-1) firm when 0 1 1 1ˆ ˆ( ) ( )z zε ε ε− −≤ ≤

37

Appendix

Proof of Lemma 1 By comparing (4) with (6), and (5) with (7), it is immediate that , and . Hence, it can be derived that,

0 1 00 0 1( ) ( ) ( )V z c V z V z− −− ≤ ≤

1 1 10 1 1( , ) ( , ) ( , )V z x c V z x V z x− − ≤ ≤ −

3

2

3 1( ) ( ) ( )V z c V z V z− ≤ ≤ . (A.8) Then, by definition we have the first result:

00 01 0 1( ) ( ) ( )V z V z V z−≤ ≤ and

10 11 1 1( , ) ( , ) ( , )V z x V z x V z x−≤ ≤ . From the above inequalities and by definition, . (A.9) 1 2 3( ) ( ) ( )V z V z V z≤ ≤Next, define

1 1{ | ( ) ( )}Z z R V z V zε≡ ∈ ≥ − + ,

0 3{ | ( ) ( )}2Z z R c V z V zε≡ ∈ − + ≥ − + ,

1 3{ | ( ) (2 )}Z z R V z V zε− ≡ ∈ ≥ − + where R is the set of all real numbers. From (A.8),

0 1 1Z Z Z−⊂ ⊂ (A.10) Now,

10 00 2 3 0

0

( , ) ( ) max{| | ,| | ( ) ( ) } for | | for

V z x V z x x V z V z c z Zx z Z

ε τ ε εε

+ +

+

− = + − + − + − + ∈= + ∉

11 01 2 1 1

1

( , ) ( ) max{| | ,| | ( ) ( )} for | | for

V z x V z x x V z V z z Zx z Z

ε τ ε εε

+ +

+

− = + − + − + − ∈= + ∉

1 1 0 1 2 3 1

1

( , ) ( ) max{| | ,| | ( ) ( )} for | | for

V z x V z x x V z V z z Zx z Z

ε τ ε εε

− − + +

+ −

− = + − + − + − ∈= + ∉

−

From (A.8) and (A.10), we can derive the second result: 10 00 11 01 1 1 0 1( , ) ( ) ( , ) ( ) ( , ) ( )V z x V z V z x V z V z x V z− −− ≥ − ≥ −

where the inequalities are strict for some sets of (z,x) with positive measure given that f is continuous and f(x) > 0 for all x. Proof of Proposition 2

Suppose c = 0. By definition, 00 0 1( ) ( )V z V z−= and 10 1 1( , ) ( , )V z x V z x−= . Then, from the definition of ’s, . This leads to iV 1 3( ) ( )V z V z= 1 2 3( ) ( ) ( )V z V z V z= = by (A.9). Now, from definition, 0 ( ) max{0; } ( )jV z V z1ε= − + for all (0,j) firm and the firm should buy a

technology only if ε ≤ 0. At the same time, 1 1( , ) max{| | ; | | } ( )jV z x x x V zε τ ε ε+ += + − + − + . When

ε > τ > 0, the firm should sell its technology only if x τ ε≥ − . When τ > ε > 0, the firm never trades in the technology market and commercializes its product only if x ≥ 0. When ε < 0, the firm should buy a technology only if x ε≤ − . Therefore, there exists no market premium for which there are both buyers and sellers. Hence ft = 0 for all t. This result simplifies the value functions as follows: 0 00( ) max{ ( | ); ( | )}jV z E V z d E V z10δ δ= − + ,

38

1 00 10 00

0

( , ) max{ ( | ); ( | ); ( | ); ( | )}

| | ( )j

j

V z x E V z d E V z x E V z d x E V z

x V z10δ δ δ δ

+

= − + + − + +

= +

They mean that the project in stage II should be commercialized if and only if x > 0 and the project in stage I should be advanced to stage II only if

00 10( | ) ( |E V z d E V z)δ δ≤ − + for all types . Furthermore, the right-hand side of this inequality is increasing in z while the left-hand side does

not depend on z thus there is a critical threshold such that the product should be developed if and only

if . is determined by . This concludes the proof.

z

ˆz z> z 10 000

( | ) ( |d E V V z xf x z dδ δ+∞

= − = ∫ ) x

Proof of Proposition 3.1 First, we solve for 0,1ˆ ( )z ε . Recall

01 00 10 01 11( ) max{ ( | ); ( | ); ( | ); ( | )}V z E V z d E V z E V z d E V zδ δ ε δ ε δ= − + − + − − + . Define 0( )z ε , 1( )z ε and 01( )z ε by

00 0 10 0( | ) ( |E V z d E V z )δ δ= − +)

, (A.11)

01 1 11 1( | ) ( |E V z d E V zδ δ= − +

)

(A.12) and

10 01 01 01( | ) ( |d E V z E V zδ ε δ− + = − +)

. (A.13) Note that 1( |jE V z is increasing in z but 0( |j )E V z is independent of z. (A.11)-(A.12) and

Lemma 1 imply that 11 01 1 10 00 0 11 01 0( | ) ( | ) ( |d E V V z E V V z E V V z )δ δ δ= − = − > − . Hence,

1 0( ) ( )z zε ε> . Because depends on the steady state market premium ε, , and also depend

on ε. With some algebra, ijV 0z 1z 01z

01 10( | )E V V zδ ε− − is decreasing in ε. Therefore, 01( )z ε is decreasing in ε. Next, define 01ε and 01( )zε by

00 01 01( | ) ( |E V z E V z)δ ε δ= − + (A.14) and

10 01 11( | ) ( |E V z E V z)δ ε δ= − + (A.15) where 01ε does not depend on z because the project is terminated in the next period while 01( )zε does.

With some tedious algebra, you can show that 01 00( | )E V V zδ ε− − and 11 10( | )E V V zδ ε− − are decreasing in ε. From Lemma 1, 01 00 11 10( , ) ( ) ( , ) ( )V z x V z V z x V z− ≥ − implying 01ε > 01( )zε .

Now define 0 0

0,1 01 01 01

1 0

( ) when ˆ ( ) ( ) when ( )

( ) when ( ).

zz z z

z z

1

1

ε ε εε ε ε ε

ε ε

≥⎧⎪= <⎨⎪ ≤⎩

εε

<

01 01( )zε ε ε< < means 01 00 11 10( | ) (E V V z E V V z| )δ ε δ− > > − , which then is equivalent to

10 00 10 01 11 01( | ) ( | ) (E V V z E V V z E V V z| )δ δ ε δ− > − + > − . With (A.11)-(A.13), these inequalities imply

1 01 0( ) ( ) ( )z z zε ε ε> > . Now, we are ready to show that *

1 1e = (= 0) if 0,1ˆ ( )z z ε> ( 0,1ˆ ( )z z ε< ).

39

1) When 01ε ε≥ , from (A.14) and (A.15), 00 01( | ) ( |E V z E V z)δ ε δ≥ − + and

10 11( | ) ( |E V z E V z)δ ε δ> − + where the second inequality is derived from 01( )zε ε> , and

0 ( )z z ε> is equivalent to 00 10( | ) ( |E V z d E V z)δ δ< − + from (A.11). Then, from definition,

and , and *1 1e = * *

2 0e f= = 01 10( ) ( | )V z d E V zδ= − + . 2) When 01 01( )zε ε ε< < , from (A.14) and (A.15), 00 01( | ) ( |E V z E V z)δ ε δ< − + and

10 11( | ) ( |E V z E V z)δ ε δ> − + , and 01( )z z ε> indicates 10 01( | ) ( |d E V z E V z)δ ε δ− + > − +

from (A.13). Then, and *1 1e = * *

2 0e f= = , and 01 10( ) ( | )V z d E V zδ= − + . 3) When 01( )zε ε≤ , 10 11( | ) ( |E V z E V z)δ ε δ≤ − + , and 1( )z z ε> implies

00 10( | ) ( |E V z d E V z)δ δ< − + and 01 11( | ) ( |E V z d E V z)δ δ< − + from (A.11) and (A.12)

where the second one is obtained from 0 ( )z z ε> . Then, * *1 2 1e e= = and , and * 1f = −

01 11( ) ( | )V z d E V zε δ= − − + .9 A similar argument holds for 0,1ˆ ( )z z ε< . To sum up, for any ε,

00 01 0,101

10 11 0,1

ˆmax{ ( | ); ( | )} when ( )( )

ˆmax{ ( | ); ( | )} when ( )E V z E V z z z

V zd E V z d E V z z z

δ ε δ εδ ε δ ε

− + <⎧= ⎨ − + − − + >⎩

which indicates that the (0,1) firm advances its project from stage I to stage II only if 0,1ˆwhen ( )z z ε>

and does not do so if 0,1ˆ ( )z z ε< . Similarly, for 0,0ˆ ( )z ε and 0, 1ˆ ( )z ε− , define 1( )z ε− , 00( )z ε and 0 1( )z ε− by

0 1 1 1 1 1( | ) ( |E V z d E V z )δ δ− − − −= − +)

, (A.16)

0 1 00 11 00( | ) ( |c E V z d E V zδ ε δ−− + = − − +

)

(A.17) and

0 1 0 1 11 0 1( | ) ( |E V z d E V zδ ε δ− − −= − − + . (A.18) Also, define 00( )zε , 00ε , 0 1( )zε − and 0 1ε − by

1 1 00 11( | ) ( |c E V z E V z)δ ε δ−− + = − + (A.19)

0 1 00 01( | ) ( |c E V z E V z)δ ε δ−− + = − + . (A.20) 1 1 0 1 11( | ) ( |E V z E V z)δ ε δ− −= − + (A.21) and

0 1 0 1 01( | ) ( |E V z E V z)δ ε δ− −= − + . (A.22) where 00ε and 0 1ε − do not depend on z while 00( )zε and 0 1( )zε − do. Again, Lemma 1 implies 1 1 0( ) ( ) ( )z z zε ε− > > ε and 00 00 ( )zε ε ε< < is equivalent to

1 00 1( ) ( ) ( )z z zε ε− > > ε and 0 1 0 1( )zε ε ε− −< < is equivalent to 1 0 1 1( ) ( ) ( )z z zε ε ε− −> > .

9 Strictly speaking, when 01( )zε ε= , the firm is indifferent between * * *

1 2( , , ) (1,1, 1)e e f = − and

. * * *1 2( , , ) (1,0,0)e e f =

40

Define 1 0

0,0 00 00 00

1 0

( ) when ( )ˆ ( ) ( ) when ( )

( ) when

z zz z

z

ε εε ε ε ε ε

0

0

zε

ε ε ε

− ≥⎧⎪= <⎨⎪ ≤⎩

<

1

1

zε

and

1 0

0, 1 0 1 0 1 0 1

1 0

( ) when ( )ˆ ( ) ( ) when ( )

( ) when

z zz z

z

ε εε ε ε ε ε

ε ε ε

− −

− − − −

−

≥⎧⎪= <⎨⎪ ≤⎩

< .

Again, the (0,0) firm and (0,-1) firm should advance their projects from stage I to stage II if and only if 0,0ˆwhen ( )z z ε> and 0, 1ˆwhen ( )z z ε−> , respectively. Namely, we can easily show the optimal strategy for j = 0 and -1 as follows:

1) When 0 jε ε≥ , and , *1 1e = * *

2 0e f= =

2) When 01 01( )zε ε ε< < , and * *1 2 1e e= = * 1f = − ,

3) When 01( )zε ε≤ , and * *1 2 1e e= = * 1f = − .

From Lemma 1, implying 11 1 1 01 0 1( , ) ( ) ( , ) ( )V z x V z V z x V z− −− ≥ − 0 1( )zε − ≥ 0 1( )zε − and

00( )zε ≥ 00( )zε . Furthermore,

00 3 2

3 2

( ) max{ ( ); ( )}max{ ( ); ( )} ( )

V z c c V z V z cV z V z V z0 1

δ δ εδ ε δ −

+ = − + − + +> − + =

which implies 00( )zε > 01( )zε . And,

1 1 3 2

3 2

10

( , ) max{| | ( ); | | ( )}max{ | | ( ); | | ( )}

( , )

V z x x V z x V zc x V z x V z

V z x

δ δ ε τ ε εδ ε τ ε εδ

− + +

+ +

= + − + + − +> − + + − + + − +=

which implies that 01( )zε > 0 1( )zε − . Now, we have shown

1 1 0( ) ( ) ( )z z zε ε− > > ε

z

(A.23) and

00 00 01 01 0 1 0 1( ) ( ) ( )z zε ε ε ε ε ε− −> > > > > , (A.24) which then imply 0,1 0,0 0, 1ˆ ˆ ˆ( ) ( ) ( )z z zε ε ε−≤ ≤ .

Figure 1 illustrates our result and clearly shows that, for a sufficiently high ε, 01 00 0 1ˆ ˆ ˆ( ) ( ) ( )z z zε ε −< < ε and, for a sufficiently low ε, 01 00 0 1ˆ ˆ ˆ( ) ( ) ( )z z zε ε ε−= = .

Proof of Proposition 3.2 and Proposition 3.2 We prove the results for three separate ranges for x: x < - ε; x > - ε + τ ; and - ε ≤ x ≤ - ε + τ . When x < - ε: No firm with the project sells or launches the product. Hence, 1 0( , ) ( )jV z x V zj= for all z and x <

- ε and thus, for 1, 0,ˆ ˆ( , ) ( )jz x z jε ε= , e1* = 1 (= 0) only if 1,ˆ ( , )jz z xε> ( 1,ˆ ( , )jz z xε< ).

When x > - ε + τ:

41

Any firm with the project sells or launches the product. Define 11ε , 11( )zε , 10 ( )zε , 10ε , 1 1( )zε − and 1 1ε − by

00 11 01( | ) ( |E V z E V z)τ δ ε δ− + = − + (A.25)

10 11 11( | ) ( |E V z E V z)τ δ ε δ− + = − + (A.26)

1 1 10 11( | ) ( |c E V z E V z)τ δ ε δ−− − + = − + (A.27)

0 1 10 01( | ) ( |c E V z E V z)τ δ ε δ−− − + = − + (A.28) 1 1 1 1 11( | ) ( |E V z E V z)τ δ ε δ− −− + = − + (A.29) and

0 1 1 1 01( | ) ( |E V z E V z)τ δ ε δ− −− + = − +Note that these equations are almost identical to (A.14), (A.15), (A.19)-(A.22) except that the left-hand sides contain – t leading to

. (A.30)

1 0( ) ( )j jz zε ε> and 1 0( ) ( )j jz zε ε> . (A.31) From Lemma 1, you can show

10 10 11 11 1 1 1 1( ) ( ) ( )z z zε ε ε ε ε ε− −> > > > > . (A.32) Also, define 11( )z ε , 10 ( )z ε , and 1 1( )z ε− by

10 11 01 11( | ) ( |d E V z E V z )τ δ ε δ− − + = − +)

(A.33)

0 1 10 11 10( | ) ( |c E V z d E V zτ δ ε δ−− − + = − − +

)

(A.34) and

0 1 1 1 11 1 1( | ) ( |E V z d E V zτ δ ε δ− − −− + = − − +x

. (A.35) Define new thresholds 1,ˆ ( , )jz ε for x > - ε + τ as follows.

0 1

1,1 11 11 11

1 1

( ) when ˆ ( , ) ( ) when ( )

( ) when ( )

zz x z z

z z

1

1

ε ε εε ε ε ε

ε ε

≥⎧⎪= <⎨⎪ ≤⎩

εε<

0

0

ε

1 1

1,0 10 10 10

1 1

( ) when ( )ˆ ( , ) ( ) when ( ) ( )

( ) when

z zz x z z z

z

ε εε ε ε ε ε

ε ε ε

− ≥⎧⎪= <⎨⎪ ≤⎩

<

1

1

and

1 1

1, 1 1 1 1 1 1 1

1 1

( ) when ˆ ( , ) ( ) when ( )

( ) when ( )

zz x z z

z z

ε ε εε ε ε ε

ε ε

− −

− − −

−

≥⎧⎪= <⎨⎪ ≤⎩

εε

−< .

It is easy to show that e1* = 1 (= 0) only if 1,ˆ ( , )jz z xε> ( 1,ˆ ( , )jz z xε< ) as is done for

Proposition 3.1. From (A.31), (A.32) and (A.23), it is straightforward to show 01 11 10 00 1 1 0 1ˆ ˆ ˆ ˆ ˆ ˆ( ) ( , ) ( , ) ( ) ( , ) ( )z z x z x z z x zε ε ε ε ε− −≤ ≤ ≤ ≤ ≤ ε for x > - ε + t. To find it correct, compare

Figure 1 with Figure 2. When - ε ≤ x ≤ - ε + τ:

Firms never buy or sell technology and the value functions are simplified to 11 00 10 01 11( , ) max{ ( | ); ( | ); ( | ); ( | )}V z x E V z d E V z x E V z d x E V zδ δ δ δ= − + + − + +

42

10 0 1 1 1 01 11( , ) max{ ( | ); c ( | ); ( | ); ( | )}V z x c E V z d E V z x E V z d x E V zδ δ δ δ− −= − + − − + + − + +

1 1 0 1 1 1 01 11( , ) max{ ( | ); ( | ); ( | ); ( | )}.V z x E V z d E V z x E V z d x E V zδ δ δ δ− − −= − + + − + +

Define 11( , )z xε , 10 ( , )z xε and 1 1( , )z xε− as follows:

(A.36) 10 11 01 11( | ) ( |d E V z x E V zδ δ− + = + ))

)x

0 1 10 11 10( | ) ( |c E V z d x E V zδ δ−− + = − + + (A.37) and

0 1 1 1 11 1 1( | ) ( |E V z d x E V zδ δ− − −= − + + . (A.38) Then, define thresholds 1,ˆ ( , )jz ε for - ε ≤ x ≤ - ε + t as follows:

1,1 0 1 11ˆ ( , ) max{ ( ), min{ ( ), ( , )}}z x z z z xε ε ε= ε

ε

ε−

, (A.39)

1,0 1 1 10ˆ ( , ) max{ ( ), min{ ( ), ( , )}}z x z z z xε ε ε−= , (A.40) and

1, 1 1 1 1 1ˆ ( , ) max{ ( ), min{ ( ), ( , )}}z x z z z xε ε ε− −= . (A.41)

Then, suppose 11ˆ ( , )z z xε> . By definition (A.39), either 1( )z z ε> , which is equivalent to

01 11( | ) ( |E V z d E V z)δ δ< − + , or 11( , )z z xε> , which is equivalent to

10 01( | ) ( |d E V z x E V z)δ δ− + > + . (A.39) also implies 0 ( )z z ε> , which is equivalent to

00 10( | ) ( |E V z d E V z)δ δ< − + . Hence, e1* = 1 and

11 10 11( , ) max{ ( | ); ( | )}V z x d E V z x E V zδ δ= − + + . Similarly, if 11ˆ ( , )z z xε< , (A.39) indicates

11( , )z z xε< , which is equivalent to 10 01( | ) ( |d E V z x E V z)δ δ− + < + , or 0 ( )z z ε< , which is equivalent to 00 10( | ) ( |E V z d E V z)δ δ> − + . The definition also implies 1( )z z ε< , which is equivalent to 01 11( | ) ( |E V z d E V z)δ δ> − + . Therefore, e1

* = 0 and

11 00 01( , ) max{ ( | ); ( | )}V z x E V z x E V zδ δ= + . Similarly, we can show that e1* = 1 if z >

1,ˆ ( , )jz xε and e1* = 0 if z < 1,ˆ ( , )jz xε , for j = 0 and –1, too.

By definition, 11( , )z xε is increasing in x, and 10 ( , )z xε and 1 1( , )z xε− are decreasing in x. You can easily verify that 11 01( , ) ( )z zε ε ε− = , 11 11( , ) ( )z t zε ε ε− = ,

10 00( , ) ( )z zε ε ε− = , 10 10( , ) ( )z t zε ε ε− = , 1 1 0 1( , ) ( )z zε ε ε− −− = and 1 1 1 1( , ) ( )z t zε ε ε− −− = by comparing (A.36)-(A.38) with (A.13), (A.17), (A.18), (A.33)-(A.35).

Therefore, 1,ˆ ( , )jz xε is continuous in the entire region and 11ˆ ( , )z xε is non-

decreasing in x, and 0,1 1,1 ' 1,1 1,1 ' 1,1ˆ ˆ ˆ ˆ ˆ( ) ( , ') | ( , ) ( , ') | ( , )x xz z x z x z x zε τ εε ε ε ε ε<− > −= ≤ ≤ = τ ε− ;

10ˆ ( , )z xε is non-increasing in x and 1,0 1,0 ' 1,0 1,0 ' 0,0ˆ ˆ ˆ ˆ ˆ( , ) ( , ') | ( , ) ( , ') | ( )x xz t z x z x z x zτ ε εε ε ε ε ε> − <−− = ≤ ≤ = ε

x;

1 1ˆ ( , )z ε− is non-increasing in x and

43

1, 1 1, 1 ' 1, 1 1, 1 ' 0, 1ˆ ˆ ˆ ˆ ˆ( , ) ( , ') | ( , ) ( , ') | ( )xz z x z x z x zτ ε εxε τ ε ε ε ε ε− − > − − − <−− = ≤ ≤ = − . This concludes the proof of Proposition 3.3.

To complete the proof of Proposition 3.2, we need to show 01 11 10 00 1 1 0 1ˆ ˆ ˆ ˆ ˆ ˆ( ) ( , ) ( , ) ( ) ( , ) ( )z z x z x z z x zε ε ε ε ε− −≤ ≤ ≤ ≤ ≤ ε for any (ε, x). Because we have already

shown 01 11 10 00 1 1 0 1ˆ ˆ ˆ ˆ ˆ ˆ( ) ( , ) ( , ) ( ) ( , ) ( )z z x z x z z x zε ε ε ε ε− −≤ ≤ ≤ ≤ ≤ ε

ε

for x > - ε + τ and x < - ε,

01 11 11 11 10 10

10 00 1 1 1 1 1 1 0 1

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( , ) | ( , ) ( , ) | ( , ) | ( , )ˆ ˆ ˆ ˆ ˆ ˆ( , ) | ( ) ( , ) | ( , ) ( , ) | ( )

x x x

x x x

z z x z x z x z x z xz x z z x z x z x z

ε τ ε τ ε

ε τ ε

ε ε ε ε ε εε ε ε ε ε

<− > − > −

<− − > − − − <− −

= ≤ ≤ ≤ ≤≤ = ≤ ≤ ≤ = ε

This concludes the proof of Proposition 3.2.

Proof of Proposition 3.4 Define 0ˆ ( )j zε as follows:

01 0

10,1 01 0 1

01 1

if ˆ ( ) ( ) ( ) if

if

z zz z z z z

z z

εε

ε

−

<⎧⎪= ≤⎨⎪ >⎩

z≤

00 11

0,0 00 1 1

00 1

if ˆ ( ) ( ) ( ) if

if

z zz z z z z z

z z

εε

ε

−−

−

<⎧⎪= ≤⎨⎪ >⎩

≤

−≤

and 0 1 1

10, 1 0 1 1 1

0 1 1

if ˆ ( ) ( ) ( ) if

if

z zz z z z z z

z z

εε

ε

−−

− −

− −

<⎧⎪= ≤⎨⎪ >⎩

where 0 ( )j zε ’s and 0 ( )j zε ’s are defined by (A.14), (A.15), (A.19)-(A.22) and is the inverse function of

10( )jz −

0 ( )jz ε which is defined by (A.13), (A.17) and (A.18). It is straightforward

from the definition to show that *( , ,0, ) 0f z x j = when ε > 0ˆ ( )j zε and when ε <

*( , ,0, ) 1f z x j = −

0ˆ ( )j zε where j = 0,1,-1 (See Figures 3-5). We will only show the proof for j = 1. The proofs for j = 0, -1 are similar.

Recall the value function for the (0,1) type is 01 00 10 01 11( ) max{ ( | ); ( | ); ( | ); ( | )}V z E V z d E V z E V z d E V zδ δ ε δ ε δ= − + − + − − + .

First, 01ε ε> and are equivalent to 0z z< 00 01( | ) ( |E V z E V z)δ ε δ> − + and

00 10( | ) ( |E V z d E V z)δ δ> − + , respectively. The second inequality also induces

01 11( | ) ( |E V z d E V z)ε δ ε δ− + > − − + by Lemma 1. Therefore, * 0f = and 01 00( ) ( | )V z E V zδ= . If

01ε ε< instead, 00 01( | ) ( |E V z E V z)δ ε δ< − + , thus * 1f = − and 01 01( ) ( | )V z E V zε δ= − + . Next, is equivalent to 1

01( ) ( )zε −> z 01( )z z ε> because is decreasing in ε. Then, 01z 01( )z z ε> and

0 ( ) ( )z z z1ε ε≤ ≤ mean, by definition, 10 01( | ) ( |d E V z E V z)δ ε δ− + > − + ,

00 01( | ) ( |E V z E V z)δ ε δ≤ − + and 01 11( | ) ( |E V z d E V z)δ δ≥ − + . Therefore, and * 0f =

01 10( ) ( | )V z d E V zδ= − + . If

44

101( ) ( )zε −< z for 0 1( ) ( )z z zε ε≤ ≤ , 10 01( | ) ( |d E V z E V z)δ ε δ− + < − + , and and * 1f = −

01 01( ) ( | )V z E V zε δ= − + . Finally, when 01ε ε> and , 1z z> 10 11( | ) ( |E V z E V z)δ ε δ> − + and

01 11( | ) ( |E V z d E V z)δ δ< − + . The second inequality implies 00 10( | ) ( |E V z d E V z)δ δ< − + by

Lemma 1. Hence, and * 0f = 01 10( ) ( | )V z d E V zδ= − + . If 01ε ε< , it is immediate that * 1f = − and 01 11( ) ( | )V z d E V zε δ= − − + . Because 01( )z ε is decreasing in ε and 00 ( )z ε and 0 1( )z ε− are increasing in ε, 0,1ˆ ( )zε is nonincreasing in z and 0,0ˆ ( )zε and 0, 1ˆ ( )zε − are nondecreasing in z. From (A.24),

00 00 01 01 0 1 0 1( ) ( ) ( )z z zε ε ε ε ε ε−> > > > > − leading to 0 1ˆ ( )zε − < 01ˆ ( )zε < 00ˆ ( )zε . Proof of Proposition 3.5

Define 1,ˆ ( )j zε as follows (See Figures 7-12):

, 11 0

11,1 11 0 1

11 1

if ˆ ( ) ( ) ( ) if

if

z zz z z z z

z z

εε

ε

−

<⎧⎪= ≤⎨⎪ >⎩

z≤

10 11

1,0 10 1 1

10 1

if ˆ ( ) ( ) ( ) if

if

z zz z z z z z

z z

εε

ε

−−

−

<⎧⎪= ≤⎨⎪ >⎩

≤

−≤

and 1 1 1

11, 1 1 1 1 1

1 1 1

if ˆ ( ) ( ) ( ) if

if

z zz z z z z z

z z

εε

ε

−−

− −

− −

<⎧⎪= ≤⎨⎪ >⎩

where 1 ( )j zε ’s and 1 ( )j zε ’s are defined by (A.25)-(A.30) and 11( )jz − is the inverse function of

1 ( )jz ε which is defined by (A.33)-(A.35). We only prove the proposition for j = 1. The proofs for j = 0 and -1 are similar. The

value function for the (1,1) type is

11 00 10

01 11

( , ) max{| | ( | ); | | ( | );| | ( | ); | | ( | )}.



+ +

+ +

= + − + − + + − ++ − + − + + − +

i) When 1,1ˆ ( )zε ε> :

Suppose 11ε ε> and . Then, by definition, 0z z< 00 01( | ) ( |E V z E V z)τ δ ε δ− + > − + and

00 10( | ) ( |E V z d E V z)δ δ> − + . By Lemma 1, the second inequality also implies

01 11( | ) ( |E V z d E V z)δ δ> − + . Therefore, 11 00( ) | | ( | )V z x E V zε τ δ+= + − + , and and *2 0e = * 1f =

if 0x ε τ+ − > and if * *2 0e f= = 0x ε τ+ − < .

Next, is equivalent to 111( ) ( )zε −> z 11( )z z ε> because is decreasing in ε. Then, 11z

11( )z z ε> and 0 ( ) ( )z z z1ε ε≤ ≤ mean, by definition, 10 01( | ) ( |d E V z E V z)τ δ ε δ− − + > − + ,

00 01( | ) ( |E V z E V z)δ ε δ≤ − + and 01 11( | ) ( |E V z d E V z)δ δ≥ − + leading to

45

11 10( ) | | ( | )V z d x E V zε τ δ+= − + + − + . Therefore, *2 0e = and * 1f = if 0x ε τ+ − > and

if * *2 0e f= = 0x ε τ+ − < . Finally, when 11ε ε> and , 1z z> 10 11( | ) ( |E V z E V z)τ δ ε δ− + > − + and


Lemma 1. Hence, 11 10( ) | | ( | )V z d x E V zε τ δ+= − + + − + , and *2 0e = and if * 1f = 0x ε τ+ − >

and if * *2 0e f= = 0x ε τ+ − < .

To summarize, 1( , )x zε τ ε= − satisfies the property of ˆ jx for j = 1 and 1,1ˆ ( )zε ε> specified in Proposition 3.5.

ii) When 0,1 1,1ˆ ˆ( ) ( )z zε ε ε≤ ≤ :

First, 01 11ε ε ε≤ ≤ and induce 0z z< 00 01( | ) ( |E V z E V z)δ ε δ≥ − + ,

00 01( | ) ( | )E V z E V zτ δ ε δ− + ≤ − + 00 10( | ) ( | )E V z d E V zδ δ> − +)

, and

01 11( | ) ( |E V z d E V zε δ ε δ− + > − − + . The last two inequalities induce

11 00 01( , ) max{| | ( | ); | | ( | )}V z x x E V z x E V zε τ δ ε ε δ+ += + − + + − + and the first two inequalities imply that, for 1 00 01ˆ ( | )x E V V zδ≡ − 1x, ε τ ε− ≤ ≤ −

01

and

1 00 00 1 01 1ˆ ˆ ˆ| | ( | ) ( | ) ( | ) | | ( | )x E V z E V z x E V z x E V zε τ δ δ δ ε ε δ+ ++ − + = = + = + − + . Therefore, and if *

2 1e = * 0f = 1x x> and if * *2 0e f= = 1x x< .

Second, and 1 101 11( ) ( ) ( ) ( )z z zε− −≤ ≤ z 10z z z≤ ≤

d E V z E V z imply

10 01( | ) ( | )δ ε δ− + ≥ − + 10 01( | ) ( | )d E V z E V z, τ δ ε δ− − + ≤ − +)

,

00 01( | ) ( |E V z E V zδ ε δ≤ − + and 01 11( | ) ( |E V z d E V z)δ δ≥ − + . Again, the last two inequalities suggest 11 10 01( , ) max{ | | ( | ); | | ( | )}V z x d x E V z x E V zε τ δ ε ε δ+ += − + + − + + − + and the first two inequalities imply that, for 1 10 01 )ˆ ( |x d E V V zδ≡ − + − 1x, ε τ ε− ≤ ≤ −

01

and

1 10 10 1 01 1ˆ ˆ ˆ| | ( | ) ( | ) ( | ) | | ( | )d x E V z d E V z x E V z x E V zε τ δ δ δ ε ε δ+ +− + + − + = − + = + = + − + . Therefore, and if *

2 1e = * 0f = 1x x> and * *2 0e f= = if 1x x< .

Finally, when 01 11ε ε ε≤ ≤ and , 1z z> 10 11( | ) ( |E V z E V z)δ ε δ≥ − + ,

10 11( | ) ( | )E V z E V zτ δ ε δ− + ≤ − + 01 11( | ) ( | )E V z d E V z, δ δ< − +)

and

00 10( | ) ( |E V z d E V zδ δ< − + . The last two inequalities induce

11 10 11( , ) max{ | | ( | ); - | | ( | )}V z x d x E V z d x E V zε τ δ ε ε δ+ += − + + − + + + − + and the first two inequalities imply that, for 1 10 11ˆ ( | )x E V V zδ≡ − 1x, ε τ ε− ≤ ≤ −

11 | ) and

1 10 10 1

1 11

ˆ ˆ| | ( | ) ( | ) (ˆ| | ( | ).

d x E V z d E V z d x E V zd x E V z

ε τ δ δ δε ε δ

+

+

− + + − + = − + = − + += − + + − +

.

Therefore, and if *2 1e = * 0f = 1x x> and * *

2 0e f= = if 1x x< . To sum up, when 0,1 1,1ˆ ˆ( ) ( )z zε ε ε≤ ≤ ,

46

00 01 0

1 10 01

10 11 1

( | ) for ˆ ( , ) ( | ) for

( | ) for

E V V z z z

0 1x z d E V V z z zE V V z z z

δε δ

δ

− <⎧⎪≡ − + − ≤ ≤⎨⎪ − >⎩

z

satisfies the property of ˆ jx for j = 1 and 0,1 1,1ˆ ˆ( ) ( )z zε ε ε≤ ≤ specified in Proposition 3.5.

iii) When 0,1ˆ ( )zε ε< : This part is similar to the proof for Proposition 3.4. First, 01ε ε< and are

equivalent to 0z z<

00 01( | ) ( |E V z E V z)δ ε δ< − + and 00 10( | ) ( |E V z d E V z)δ δ> − + , respectively. The second inequality also induces 01 11( | ) ( |E V z d E V z)ε δ ε δ− + > − − + by Lemma 1. Therefore,

11 01( ) | | ( | )V z x E V zε ε δ+= + − + . Thus, *2 1e = and * 0f = if 0x ε+ > and and *

2 1e = * 1f = − if 0x ε+ < . Next, by definition, and 1

01( ) ( )zε −< z 10 ( ) ( )z z zε ε≤ ≤ are equivalent to

10 01( | ) ( | )d E V z E V zδ ε δ− + < − + 00 01( | ) ( | )E V z E V z, δ ε δ≤ − +

) and

01 11( | ) ( |E V z d E V zδ δ≥ − + . Therefore, 11 01( ) | | ( | )V z x E V zε ε δ+= + − + . This indicates *2 1e =

and if * 0f = 0x ε+ > and and *2 1e = * 1f = − if 0x ε+ < .

Finally, when 01ε ε< and , 1z z> 10 11( | ) ( |E V z E V z)δ ε δ< − + and


Lemma 1. Hence, 11 11( ) | | ( | )V z d x E V zε ε δ+= − + + − + . Again, *2 1e = and if * 0f = 0x ε+ >

and and if *2 1e = * 1f = − 0x ε+ < .

In sum, 1( , )x zε ε= − satisfies the property of ˆ jx for j = 1 and 0,1ˆ ( )zε ε< specified in Proposition 3.5.

This concludes the proof for j = 1 (See Figures 7,8). . From (A.32), it is straight-forward to show 1 1 0ˆ ˆ ˆ( , ) ( , ) ( , )x z x z x zε ε ε− ≥ ≥ .

Proof of Proposition 4.1 Substitute in τ = 0 into,(3), (5) and (7) and obtain

11 01( , ) | | ( )V z x x V zε += + + ,

10 00( , ) | | ( )V z x x V zε += + + and

1 1 0 1( , ) | | ( )V z x x V zε− += + + − . Hence, all firms with a project in the second stage (i.e. the (1,1), (1,0) and (1,-1) firms) should launch the product if and only if x > - ε. Furthermore, since 11 01( , ) ( )V z x V z− = 10 00( , ) ( )V z x V z− =

, 1 1 0 1( , ) ( )V z x V z− −− 0( )z ε = 1( )z ε = 1( )z ε− from (A.11), (A.12) and (A.16), and

01 01 11 11( ) ( )z zε ε ε ε= = = , 00 00 10 10( ) ( )z zε ε ε ε= = = , 0 1 0 1 1 1 1 1( ) ( )z zε ε ε ε− − − −= = = from (A.14), (A.15), (A.19)-(A.22) and (A.25)-(A.30). Therefore,

0,1ˆ ( )z ε = 0,0ˆ ( )z ε = 0, 1 1,1 1,0 1, 1ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )z z z z zε ε ε ε− = = = ≡−

) x

by definition where is defined by

.

z

1 0 ˆ ˆ( | ) ( |j jd E V V z xf x z dε

δ δ+∞

−

= − = ∫

47

Finally, define 1 01 01 11 11( ) ( )z zε ε ε ε ε≡ = = = , 0 00 00 10 10( ) ( )z zε ε ε ε ε≡ = = = ,

1 0 1 0 1 1 1 1( ) ( )z z 1ε ε ε ε− − − −≡ = = = ε − . Then, 0 1> 1ε ε ε−> from (A.24). Because demand and supply have to balance, Proposition 3.4 and 3.5 imply that *

0 > 1ε ε ε−> always hold and: 1) potential buyers are the (1,0) and (0,0) firms and potential sellers are the (1,1) and (1,-1)

firms when *0 1>ε ε ε> ; and

2) potential buyers are the (1,0), (0,0), (1,1) and (0,1) firms and potential sellers are the (1,-1) firms when *

1 1>ε ε ε−> . Proof of Proposition 4.2

Proposition 3.4 and 3.5 say that the (1,j) firms sell their products in technology markets only when 1,ˆ ( )j zε ε> and the (0,j) and (1,j) firms buy products in technology markets only when

0,ˆ ( )j zε ε< . Therefore, if 1, 0,, ,ˆ ˆmin ( ) max ( )jz j z j

z j zε ε≥ , there exists no market premium which

attracts both buyers and sellers. From the definition of ,ˆ ( )i j zε and (A.24) and (A.32), this condition is equivalent to 1, 1 0,0min ( ) max ( ')

z zz zε ε− ≥ , and to

01 0 1 11 1 1min ( | ) max ( | ))z z

E V V z c E V V zτ δ δ− −+ − ≥ + − . Because 01 0 1(E V V z− | )− does not depends on

z, the condition can be modified to 11 1 1 01 0 1max ( | ) ( | )z

E V V z E V V z c cτ δ δ− −≥ − − − + > .

48

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