THREE ESSAYS IN INDUSTRIAL ORGANIZATION
By
Kyoungbo Sim

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics – Doctor of Philosophy
2017

ABSTRACT
THREE ESSAYS IN INDUSTRIAL ORGANIZATION
By
Kyoungbo Sim
Chapter 1: Optimal Use of Patents and Trade Secrets for Complex Innovations
In this paper, the optimal protection of “complex" innovations that involve multiple complementary
components is considered. The innovator can protect each component through either patent or
trade secret protection, so that the complete innovation may be protected through a mix of patents
and trade secrets. The innovator faces potential entrants who might acquire the components either
by costly imitation or through licensing. It is shown that: (i) secrecy is optimal when the patent
length is relatively short; otherwise, a combined use of patents and trade secrets is optimal. (ii)
The innovator is (weakly) over-rewarded compared to an inventor with a simple innovation (with
identical profitability and imitation cost). (iii) A “Strict Utility Requirement" policy (that precludes
the use of a patents-secrets mix) enhances allocative efficiency ex-post but stifles R&D incentives
ex-ante.

Chapter 2: Welfare Effects of Certification under Latent Adverse Selection (with Anthony
Creane and Thomas D. Jeitschko)
Asymmetric information is a classic example of market failure that undermines the efficiency associated with perfectly competitive market outcomes: the “lemons" market. Credible certification,
that substantiates unobservable characteristics of products that consumers value, is often considered a potential solution to such market failure. This paper examines welfare effects of certification
in markets in which there is asymmetric information, but without an adverse selection problem. We
analyze the market equilibrium when the certification technology becomes available and contrast
this with the equilibrium without certification. We find that despite an improvement in allocative
efficiency, overall welfare may decrease due to the possibility of certification when such certifi-

cation is either costly or inaccurate. In fact, most of these results are not derived from the direct
welfare cost of certification, but rather from certification’s effect on the market(s).

Chapter 3: A Role of Non-practicing Entity (NPE) when Patent Ownership is Fragmented
This paper presents a mechanism by which a non-practicing entity (NPE) enhances returns to individual inventors. Assuming an environment in which: (i) a product, manufactured by a practicing
entity (PE) must adopt all complementary innovations patented and owned by individual inventors;
and (ii) a threat of litigation by an individual inventor is not credible so that patents are sold through
a sequential auction, the following two games are studied in turn. First, I consider the game played
by two PEs in which they compete to acquire the patents, and then negotiate licensing fees based
on their patent portfolio in the shadow of litigation. Next, the same game played by a PE and a
NPE. I highlight the two key findings: (i) in the former game, the two PEs buy all patents at 0
prices, and thus the aggregate payoff to the individual inventors is also 0 (for almost all parameter values); and (ii) in the latter game, on the contrary, if total number of patents is sufficiently
large, the existence of the NPE induces the PE to buy some patents at a positive price and thereby
increases the aggregate payoff to the individual inventors.

ACKNOWLEDGEMENTS

I would like first to acknowledge the chair of my dissertation committee, Jay Pil Choi for his
invaluable advice, guidance, encouragement and mentorship. Without his help, I would not have
been able to finish this dissertation and grow as an economist. I would also like to thank the
other dissertation committee members, Thomas D. Jeitschko, Arijit Mukherjee, and Aleksandr
Yankelevich. They have consistently provided me with thoughtful insights and suggestions.
It was also a pleasure for me to meet many great friends in my graduate career. I especially
thank Hyeongjun Bang, Yung Shin Jang, Yeonjei Jung, Hyunsoo Kim, Jaesoo Kim, Soo Jin Kim,
Kyong Hyun Koo, Mark Tremblay and Nikolay Ushakov for their useful comments on my papers.
I am grateful for the financial and administrative support I received from the Graduate School
and the Department of Economics at Michigan State University. My sincere thanks go to Todd
Elder, Steven Haider, Margaret Lynch, and Lori Jean Nichols.
Last, but not least, I cannot thank my family enough. I am eternally indebted to my parents,
brother and parents-in-law for the love, prayers and moral support they have given to me throughout
my academic journey. I have saved my last acknowledgement for my daughter Haeun, who makes
me happy every day, and my wife Eunhye, who not only has sacrificed everything so that I could
focus on my study but also has been my best friend over the years. I could not have completed this
journey without them by my side. I thank them with all my heart.

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TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
CHAPTER 1

OPTIMAL USE OF PATENTS AND TRADE SECRETS FOR COMPLEX INNOVATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Optimal Use of Patents and Trade Secrets . . . . . . . . . . . . . . . . . . . .
1.4.1 Expected Payoffs in Each Protection Mode . . . . . . . . . . . . . . .
1.4.1.1 Patenting Both Components . . . . . . . . . . . . . . . . . .
1.4.1.2 Relying only on Secrecy . . . . . . . . . . . . . . . . . . . .
1.4.1.3 Mixing Patents and Secrecy . . . . . . . . . . . . . . . . . .
1.4.2 Optimal Protection Decision and Complex vs. Simple Innovations . . .
1.5 Strict Utility Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Ex-post Welfare Implication of SUR . . . . . . . . . . . . . . . . . . .
1.5.2 Ex-ante Welfare Implication of SUR . . . . . . . . . . . . . . . . . . .
1.6 Extensions and Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 N Component Innovation . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.2 Probabilistic Patents . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 2

2.1
2.2
2.3

2.4

WELFARE EFFECTS OF CERTIFICATION UNDER LATENT ADVERSE SELECTION (WITH ANTHONY CREANE AND THOMAS D.
JEITSCHKO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Base Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Two Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3.1 No Information (No Certification) Equilibrium . . . . . . . .
2.3.3.2 Full Information Equilibrium . . . . . . . . . . . . . . . . .
2.3.3.3 Welfare Comparison . . . . . . . . . . . . . . . . . . . . . .
Certification Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Certification Environment . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Price Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Certification Equilibrium and Welfare Analysis . . . . . . . . . . . . .
2.4.3.1 Analysis when only 1-market Equilibria Exist . . . . . . . . .

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2.4.3.2 Analysis when 1-market and 2-market Equilibria Coexist . . . . . 70
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
CHAPTER 3 A ROLE OF NPE WHEN PATENT OWNERSHIP IS FRAGMENTED
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Equilibrium Analysis: PE vs. PE . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Licensing in the Shadow of Litigation . . . . . . . . . . . . . . . . .
3.4.2 Patent Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Equilibrium Analysis: PE vs. NPE . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Licensing in the Shadow of Litigation . . . . . . . . . . . . . . . . .
3.5.2 Patent Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

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LIST OF TABLES

Table 3.1: Minimum Threshold Number of Patents . . . . . . . . . . . . . . . . . . . . . . 100

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LIST OF FIGURES

Figure 1.1: Payoff Comparison between Complex Innovator and Simple Innovator . . . . . 20
Figure 1.2: Ex-post Welfare with and without SUR . . . . . . . . . . . . . . . . . . . . . . 23
Figure 1.3: Example of Ex-ante Welfare with and without SUR . . . . . . . . . . . . . . . 25
Figure 1.4: Equilibrium Protection Constellations . . . . . . . . . . . . . . . . . . . . . . 31
Figure 2.1: No Information Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Figure 2.2: Full Information Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 2.3: Welfare Comparison between NI and FI Benchmarks . . . . . . . . . . . . . . 57
Figure 2.4: Certification Equilibrium Constellations . . . . . . . . . . . . . . . . . . . . . 68
Figure 2.5: Welfare Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Figure 2.6: Welfare Comparisons under Cost-less, but Imperfect Certification when γ0 = .5

71

Figure 2.7: Price Equilibrium Outcomes under Costly, but Perfect Certification when
γ0 = .5 and c¯ = .1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Figure 2.8: Welfare Comparisons under Costly, but Perfect Certification when γ0 = .5 . . . 74
Figure 2.9: Certification Equilibrium Constellations . . . . . . . . . . . . . . . . . . . . . 79
Figure 3.1: PE1 vs. PE2 Game for the Case of N = 7 and K = 3 . . . . . . . . . . . . . . . 93
Figure 3.2: PE vs. NPE Game for the Case of N = 7 and K = 3 . . . . . . . . . . . . . . . 98

viii

CHAPTER 1
OPTIMAL USE OF PATENTS AND TRADE SECRETS FOR COMPLEX INNOVATIONS

1.1

Introduction

Firms protect their innovations not only by patents but also by secrecy.1 These two protection
modes help the firms appropriate returns from their innovations through different mechanisms.
Patents grant patents holders a legal right to exclude others from using protected innovations for
a fixed period of time but require disclosure of all information. In contrast, trade secrets do not
have a fixed time period and do not require any disclosure of information but they are vulnerable
to independent reverse engineering.2
In the intellectual property rights literature, initially, patents received far more attention than
trade secrets. Subsequently, several authors have highlighted trade secrets as an important appropriability method (e.g., Levin et al., 1987; and Cohen et al., 2000), and analyzed the secrecy
mechanism as a viable alternative to patents (e.g., Friedman et al., 1991). However, most of this
work does not consider a potential for a combined use of patents and trade secrets, the focus of this
paper.3
To fill this gap in the literature, this paper considers a situation in which an innovator has
a “complex” innovation that involves multiple complementary components. The innovator can
protect each component either by a patent or by secrecy, i.e., the innovation may be protected
through a mixed use of patents and trade secrets. In such a setting we ask the following questions.
Under which condition, does the innovator opt for the mix use? Would the innovator with the
1 More

generally, Hall et al. (2014) point out that there are two different types of intellectual
property rights (IPRs): formal IPRs including patents, trademarks, designs and copyright, and
informal IPRs including secrecy, confidentiality agreements and lead time advantage. In this paper,
I focus only on patents and secrecy.
2 Coca-Cola’s secret formula, which has been locked away in a vault for more than 90 years, is
a good example of well guarded trade secrets.
3 Hall et al. (2014) offer a comprehensive survey on firms’ choice between formal and informal
IPRs including both theoretical and empirical work.

1

complex innovation earn more than an innovator with a simple innovation when the mix use of
patents and trade secrets is feasible? Is restricting her choice set to the binary one, either patenting
or relying on secrecy, socially desirable?
Substantial anecdotal evidence suggests that patents and trade secrets are combined in many
settings. Premarin, which has been sold by Wyeth (merged with Pfizer in 2009), has been the most
popular hormone-therapy drug for menopausal women since 1942. Wyeth filed two patents to
protect the innovation in the early 1940s. However, the company effectively extended the duration
for its exclusive position in the industry beyond the length of the patents. The reason for this
success was that no rivals could uncover closely secured trade secrets, more specifically the socalled Brandon Process: a secret chemical process extracting conjugated estrogen from pregnant
mare urine (Jorda, 2007; and Orly, 2013).
Jorda (2007) also attributes a success story of GE in the business of industrial (or synthetic)
diamonds to the use of patents in conjunction with trade secrets. GE patented a part of its technology for making diamonds, and therefore the disclosed information was publicly available to its
rival companies after the expiration of patents. However, as GE also closely kept the remaining
part secret, it could maintain a dominant position in that market. Pan (2010) reports another example; an Illinois based energy company, Coskata, applied for patents to protect technology on
a bio-reactor while keeping the identity of the micro-organism fed into the bio-reactor secret. In
addition, there are many other examples showing a similar strategy pattern in the food industry
(see the introduction in Belleflamme and Bloch, 2013) and in the organic chemical industry in the
19th century (Arora, 1997). In fact, the web pages of many patent attorneys and law firms advise
their customers to use this type of strategy. 4
This paper first examines the best protection mode for an innovator facing potential imitators
after the discovery of a complex innovation. To this end, it is important to figure out the inventor’s
ability to appropriate returns from the innovation under each protection mode: the all-by-patent
4 For

example, visit http://www.peloquinlaw.com/combining_protection.html. (retrieved on
8/4/2016)

2

mode, the all-by-secrecy mode and the optimal patent-secret mix. Instead of imposing ad hoc
assumptions in that regard, my framework extends Henry and Ponce (2011)’s work in the context
of complex innovations. In this model, entry times of the imitators under each protection modes are
endogenously determined in equilibrium. More specifically, the length of patents (T ) and the rates
of entry by the imitators (λ ), which are driven by imitation costs (κ), are key variables governing
the entry times of the imitators under patent and trade secret protection respectively.
The first key finding of the paper is that the all-by-secrecy is optimal when the patent length
is relatively short, otherwise, the optimal combined use of patents and trade secrets is optimal. To
understand this result, it should be first noted that the all-by-patent is dominated by any patentsecret mix because the inventor can effectively extend the duration of monopoly beyond T by
mixing the two protection mechanisms; due to the perfect complementarity of the components,
patented components make entry by the imitators infeasible during the patent period, and the remaining components kept by secrecy delay entry after T . Moreover, the optimal patent-secret
mix requires patenting just the component with the minimal imitation cost because the prolonged
monopoly duration is determined by the aggregate imitation costs of all components kept under
secrecy. Therefore, the optimal protection mode has a binary feature: depending on T , choosing
either the all-by-secrecy or the optimal patent-secret mix; the optimal patent-secret mix is more
advantageous than the all-by-secrecy in the sense that the mix takes advantage of the patent system, but it is also less advantageous in the sense that it requires the inventor to disclose a part of
the innovation, thereby increasing the rates of entry.
Next, this paper compares the optimal decision and the payoff to the innovator with the complex innovation (the “complex innovator” hereafter) with that to another inventor with a “simple”
innovation comprised of a single component (the “simple innovator” hereafter). Ceteris paribus
(i.e., with identical profitability and imitation costs), the complex innovator is more inclined to
patent because she can take advantage of the patent system with only a partial disclosure of the
whole innovation. Moreover, the complex innovator is (weakly) over-rewarded compared to the
simple innovator. When T is relatively short, so that both inventors rely only on secrecy, they earn

3

the same expected payoff regardless of whether the innovation can be fragmented or not, but if
T falls into the region in which the complex innovator opts for the optimal patent-secret mix, the
complex innovator earns strictly more than the simple innovator does.
In addition to the positive analysis of ex-post innovation problem, this paper further derives
ex-ante and ex-post policy implications of imposing a “Strict Utility Requirement” (SUR). For exante welfare analysis, we also take into account ex-ante investment incentives of the innovator in
the R&D stage. Under the SUR, the patent office turns down all patent applications that discloses
only a part of innovation in compliance, and thus the inventor cannot use a patent-secret mix. The
complex inventor’s payoff schedule converts into one that is exactly same as that of the simple
inventor’s. Whereas the SUR always enhances ex-post welfare, the ex-ante welfare implication is
ambiguous. This is because the availability of the combined use of patents and trade secrets might
boost the inventor’s ex-ante investment incentive.

1.2

Related Literature

There is a large body of literature viewing patents and trade secrets as mutually exclusive alternatives in a variety of contexts. Anton and Yao (2004) study the innovator’s choice between patenting
and relaying on secrecy, focusing on signaling (or strategic disclosure) aspects. Kultti et al. (2007)
and Kwon (2012) consider a similar problem in a patent race model in which two or more innovators compete to develop the same innovation simultaneously, and they make some normative
arguments regarding socially desirable patent policies. Denicolo and Franzoni (2004) focus on
whether granting a prior user right to the first inventor (i.e., the subsequent innovator is able to
patent and exclude the first inventor when the first inventor resorted to trade secrets) increases
social welfare or not. Erkal (2005) and Bhattacharya and Guriev (2006) consider a cumulative
innovation framework. More specifically, Erkal (2005) considers two consecutive R&D races between two firms in which the winner of the first race has an option to protect it either by a patent or
by secrecy, and compares several patent policy regimes. Bhattacharya and Guriev (2006) build a

4

model in which the upstream inventor has two potential licensees in the downstream market. They
show which of the two protection modes is more efficient from the inventor’s and the social planner’s point of view. All of these papers offer great insights in understanding distinctions between
patents and trade secrets as substitutes for each other, but neglect the possibility of the combined
use of those two mechanisms.
Although there is a nascent strand of literature exploring the patent-secrecy mix mechanisms,
research papers on this topic are still sparse. Lee (2015) envisions secrecy as a mechanism for
entry deterrence, but patents as a more nuanced instrument of transferring entrants’ surplus to the
inventor or deterring entry. He shows that the mix use of protection can arise in equilibrium when
the two mechanisms perform different functions. Unlike his model, I show similar results that
the two mechanisms could be complements with the opposite assumption that knowledge covered
by trade secrets can be transferred or licensed to the potential imitators while patents only deter
entry. The two papers most closely related to my own approach are Cugno and Ottoz (2006) and
Belleflamme and Bloch (2013). Both models start with a situation in which the inventor has come
up with a complex innovation, and show that the mix of patents and trade secrets may provide
optimal protection. They, however, consider an innovation which can be fragmented continuously.
Cugno and Ottoz (2006) consider a framework in which it is a prerequisite for an imitator’s entry,
given a patent-secrecy portfolio, to not only uncover a part protected by secrecy but also find a
way to invent around the other part protected by patents. Belleflamme and Bloch (2013) study a
broader set of competitive configurations (e.g., an imitator may be able to get some surplus even
with access to only a small portion of the innovation) as well as protection modes including dynamic patenting. Albeit my paper considers complex innovations with finite number of perfectly
complementary components, it is based on a micro-foundation which determines the strength of
each protection mode endogenously, and thus provides a detailed mechanism by which the evolution of the imitation game and inventor’s profit is fully described. Moreover, my framework also
explicitly considers the ex-ante investment incentive of the inventor, which is absent in their works.
The remainder of the paper is organized as follows. In section 2, the base model and as-

5

sumptions are introduced. Section 3 provides the main analysis including the calculation of the
inventor’s expected payoffs under each protection mode, to derive the inventor’s optimal decision
and to compare her payoff to the one of the simple innovator. In Section 4, policy implications
regarding SUR are drawn. Section 5 contains two possible extensions and checks the robustness
of the main results, and the final section presents the conclusion.

1.3

The Model

This section introduces the basic setup of the model, which builds upon the framework in Henry
and Ponce (2011). Consider an infinite horizon model in which ∆ represents the length of each time
period. More specifically, we consider the continuous time framework which could be understood
as the limit of a discrete time framework as ∆ approaches to 0.
There are three players: an innovator, denoted by i, and two potential imitators, indexed by
b ∈ { j, k}. All parties are assumed to be risk neutral agents maximizing their (expected) discounted
payoffs, and have the same per-period discount factor, δ := e−r∆ , where r > 0 represents a discount
rate.
The innovator already has a “complex” innovation at time period t = 0 allowing her to introduce a new product to the market. Here, the complex innovation is comprised of two perfectly
complementary components of knowledge: x and y, which differ by the extent of difficulties in
reverse engineering.5 In the beginning, the innovator can protect each component by a patent or by
secrecy; there are four possible choices: protecting all components by a patent, all by trade secrets,
x by secrecy and y by a patent, and y by secrecy and x by a patent. Note that the choice set of
innovation protection incorporates some implicit assumptions about patent examination process. I
assume that the novelty requirement is strict in the sense that the patent office treats an invention
already on sale as a prior art and thus rejects a patent application for that.6 Consequently, the
5 We

relax this assumption and consider N-component model in section 5.
the America Invents Act (AIA) was enacted in 2011, some have argued that invention
claims on a “secret” sale is no longer considered as a prior art, based upon a new interpretation
6 After

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inventor cannot rely on a dynamic protection method combining patents and trade secrets, protecting x by a patent and y by secrecy initially, for example, and then at some point later trying to get
another patent for y. However, I assume the weak utility requirement, which allows the inventor to
get a patent even when hiding part of the invention and disclosing only the other part.
An imitator can “enter” the market and start to earn profits in the market only with access
to both knowledge components. There are two different ways for the imitators to “adopt” (or
learn) the knowledge components; they can either buy a component from the innovator through a
contract or imitate the component through a reverse engineering process. The imitation costs of x
and y are given by κx and κy respectively. Without loss of generality, I assume that κy ≥ κx > 0.
Put differently, the component y is more difficult or costly to imitate.
Regardless of whether a firm is the innovator or an imitator, the per-period profit is the same for
all selling in the market and is determined only by the number of firms in the market; πn represents
a single firm’s per-period profit with n firms in the market. Then, the present value of stream of
all profit flows per firm with n firms in the market is equal to Πn ≡ πn /r. All parameter values,
including profits given the number of firms (πn ), imitation costs (κx , κy and κ) and the discount rate
(r), are common knowledge. Throughout the paper, I maintain the following assumption stating
that joint industry profits are dissipated by more entrants.
Assumption 1: Π1 > 2Π2 > 3Π3 .
Furthermore, to make the analysis non-trivial, I assume that the sum of imitation costs is not
so high that both imitators have an incentive to enter the market.
Assumption 2: Π3 > κ := κx + κy , where κy ≥ κx > 0.
The game unfolds as follows under each protection mode chosen by the innovator. First, when
the inventor protects both components by the patent, it confers a temporal monopoly to the inventor
for a fixed term T in exchange for disclosing information about the innovation. As a result, the
of an amended clause about the prior art. There have been considerable number of debates supporting or opposing the view. Here, I accept the old meaning. For more details, see for example,
http://patentlyo.com/patent/2012/10/did-the-aia-eliminate-secret-prior-art.html.

7

imitators become aware of how the innovation works without incurring any costs, but have to wait
to enter the market until the patent expires. In summary, the imitators choose their entry times after
T without incurring any costs in all-by-patent scenario.
In the second scenario of all-by-secrecy protection, as soon as an imitator uncovers how to
“adopt” the two components through licensing or imitating, he is able to enter the market. Henry
and Ponce (2011) suggest two possible types of contracts: transferable and nontransferable. Under
a transferable contract, unlike a non transferable one, an imitator can resell the knowledge to the
remaining imitator. Since Henry and Ponce (2011) show that offering transferable contracts is
optimal and the only way for the inventor to earn a stream of profits similar to the one obtained
through patent protection, I will focus only on transferable contracts.7 Transferable contracts
specify the price levels offered by the seller (the inventor) to the buyers (the imitators) before the
first purchase takes place and by the sellers (the inventor and the first buyer) to the remaining buyer
after that moment for each knowledge component.
A more detailed description of the game under all-by-secrecy mode is given below. In any
time period t from the beginning until the first knowledge adoption occurs for one of the two
components, the game unfolds as follows: (i) The innovator i announces prices for x and y, pxt
ib
yt

and pib , to buyer b ∈ { j, k}. (ii) Given the offered prices, the imitators simultaneously choose to
learn each component by accepting the contract (ctib ), to learn each component by imitating (mtb )
or to wait (wtb ). Thus, the action set of an imitator b at time t is given by Atb = {ctib , mtb , wtb } ×
{ctib , mtb , wtb } in which the first and second elements represent his action for component x and y
respectively. Suppose now that the first knowledge adoption arises by one of the two imitators.
Without loss of generality, say imitator j learned the component x. In that case, there would then
be two sellers, i and j, for component x while there is still only one seller, i, for component y.
7 It

is not difficult to find a technology transfer agreement in which a licensor
grant a licensee a right to sublicense or resell the licensed technology (a transferable contract).
Such technology transfer agreements can be found in the archive
of the U.S. Securities and Exchange Commission webpage.
(For example, see
https://www.sec.gov/Archives/edgar/data/1223112/000119312510190208/dex1074.htm.
Retrieved: Oct/12/2016.)

8

From this period on, until somebody acquires another component, the game proceeds as follows in
xt
each period: (i) Two sellers of x offer prices, pxt
ik and p jk , to k, and the sole seller i for component
yt

yt

y announces prices, pi j and pik . (ii) Given the price offers, the imitator j chooses atj ∈ Atj =
{cti j , mtj , wtj } for the remaining component y, while, at the same time, the imitator k chooses atk ∈
Atk = {ctik , ctjk , mtk , wtk } × {ctik , mtk , wtk }. Note that now k has three different options for getting
component x: by buying from i (ctik ), by buying from j (ctjk ) and by imitating (mtk ). The game
goes on in a similar manner until everybody possesses both knowledge components and enters the
market.
At time period t, every agent observes and recalls the history up to t − ∆. The equilibrium
concept, which will be used for the analysis, is symmetric Markov perfect equilibria (MPE) in
which each agent’s strategy is optimal given the state variable and other agents’ strategy. The state
variable is denoted by θ t := (θ xt , θ yt ) at time t in which θ xt (θ yt ) indicates which imitator has
obtained the component x (y) in addition to the inventor, i.e., θ xt , θ yt ∈ {{i}, {i, j}, {i, k}, {i, j, k}}.
The last case to be considered is that of the patent-secret mix. For expositional simplicity,
assume that x is protected by a patent and y by secrecy. Then, the imitators can use x freely after T ,
while they still need to adopt y for entry. Then, the imitators will adopt y in a similar way as was
described in the all-by-secrecy mode. But now the state variable θ t should also specify whether
the current time period t is before or after the patent expires. If t ≥ T , the imitators start to sell as
soon as they adopt y. If t < T , however, the remaining duration until patent expiration is different
at each t, which results in different payoff structures given the same entry time after T . Thus, under
this scenario I derive a symmetric MPE of the game in which the state variable is θ t := (τ, θ yt ).
Here, the first element indicates whether it is before or after the expiration of the patent. If it is
before T , then, furthermore, the first element specifies at which period the players are currently.
(i.e., τ = t if t < T and τ = T+ otherwise).

9

1.4

Optimal Use of Patents and Trade Secrets

This section explores how the innovator would optimally protect her complex innovation by using
patents and trade secrets. To this end, I solve the innovator’s profit maximizing problem depending
on modes of protection: patenting both components, relying only on secrecy, and mixing patents
and trade secrets. A comparison of profits is provided next, followed by an illustration of how the
payoff of the complex innovator relates to that of the simple innovator.

1.4.1
1.4.1.1

Expected Payoffs in Each Protection Mode
Patenting Both Components

I study the case in which all components are protected by the patent first. Here, notice that we
only consider ironclad patents which grant the inventor an exclusive right to use the innovation
until T . In addition, the imitators do not need to pay to use the innovation (i.e., both components)
once the patent expires. Consequently, at time T , it is optimal for the imitators to enter the market
immediately and simultaneously. Hence, the discounted expected payoff of the inventor under
protecting all by patent mode is given by

ViP = (1 − e−rT )Π1 + e−rT Π3 − f
in which f represents a patent application fee.

1.4.1.2

Relying only on Secrecy

How much can the innovator appropriate from her innovation when she relies only on trade secrets?
Working backward, I consider cases after the first adoption of a knowledge component takes place.
Before doing that, for expositional simplicity, let us define a subgame, in which there is one seller
for a component, as a monopoly game with respect to that component. Similarly, define a subgame, in which there are two sellers for a component, as a competitive game with respect to that

10

component.8 Thus, for example, if there are two sellers for one component and there is only the
original innovator for the other component, then it is called a monopoly game with respect to the
former and a competitive game with respect to the latter.
Lemma 1 (i) In the unique MPE of a competitive game with respect to both components or with
respect to only one last component, the component(s) with two sellers is (are) sold to the remaining
buyer(s) at a zero price(s) immediately; and (ii) In an MPE of a game which is competitive with
respect to one component and monopolistic with respect to the other, the component with two
sellers is sold to the remaining buyer at a zero price, and the MPE is unique up to equilibrium
payoff regardless of the timing of its sale.
Whenever there is competition in selling a knowledge component, it is sold to the remaining
buyer at 0 price, and it can be assumed that the timing of the sale, in general, is immediate. The
intuition behind result (i) is straightforward. For expositional ease, suppose θ = ({i, j}, {i, j}).
Since price offers are stationary in MPE and entry is profitable for k, entry by him is immediate if
he buys from one of the sellers or imitates both components. Anticipating that, i and j compete
in prices to sell the two components, and thus equilibrium prices are driven to 0. Now, take a step
back and consider a subgame in result (ii). For expositional ease, say j has just adopted x, i.e.,
θ = ({i, j}, {i}). Note that since x and y are perfect complements, the stream of market profits for
each firm depends entirely on the timing of the first adoption of y after which the entry by both
imitators follows promptly. Then, it is never optimal for k to buy x at some positive prices or to
imitate x now taking price offers and imitators’ adoption decision for y as given. This is because
k can wait instead until the first adoption of y arises and buy it at 0 price. Therefore, k buys x
from one of the sellers at 0 price in equilibrium, which allows one to assume that a subgame with
θ = ({i, j}, {i}) proceeds to a subgame with θ = ({i, j, k}, {i}) instantly. The remaining analysis
for the subgame directly follows from the main results of Henry and Ponce (2011) in which they
consider a single component innovation. Even if, for simplicity, the following lemma characterizes
8 Recall

that as only transferable contracts are considered, the first adopter becomes an additional seller for the corresponding component.

11

a unique MPE for only the case in which the first adoption has occurred for x, it is applicable in a
similar way to the other case as well.9
Lemma 2 (Henry and Ponce (2011)) In a unique symmetric MPE in a monopolistic game
with respect to y only (i.e., θ = ({i, j, k}, {i})), (i) it is optimal for the inventor to induce the
y

imitators to mix adopting and waiting for the remaining component y by setting pib = κy for b ∈
{ j, k}; (ii) entry times of each imitator follow the exponential distribution with hazard rate, λ y|x =
r (Π3 − κy )/κy ; and (iii) the expected equilibrium payoffs of the inventor and of the imitators are
y|x

Vi

y|x

= µ y|x Π1 + (1 − µ y|x )(Π3 + κy ), where µ y|x := r/(r + 2λ y|x ) ∈ (0, 1), and Vb = Π3 − κy

respectively.
Given the occurrence of the first adoption of x, the discounted expected payoff of the inventor
consists of three parts; she will earn monopoly profits for a random duration until the first adoption for y arises (µ y|x Π1 ), a licensing revenue from selling y when the first adoption for y takes
place ((1 − µ y|x )κy ), and triopoly profits thereafter ((1 − µ y|x )Π3 ) in which y|x denotes a variable
associated with a monopolistic game with respect to y given x was adopted by both imitators. A
delay in the imitators’ entry can be attributed to the equilibrium outcome of balancing the benefits
and costs of immediate entry as next explained. From Lemma 1 and result (i) in Lemma 2, we
know that the first and second adopter pay κy and 0 to learn component y respectively while they
end up obtaining the same market profits because of almost simultaneous entry. Thus, on the one
hand, an imitator has an incentive to wait to save the cost of knowledge acquisition. On the other
hand, delaying entry entails costs as well: they need to sacrifice market profits which would have
been earned by immediate entry. Hence, in symmetric equilibrium, the imitators mix waiting and
buying y from the inventor in a way that such benefits and costs are offset exactly. A speed of entry
is described by the hazard rate of entry, λ y|x , and µ y|x represents the natural duration of monopoly
9 For

θ = ({i}, {i, j}) or ({i}, {i, k}), instead the inventor sets pxib = κx for b ∈ { j, k}; hazard
x|y
rate is λ x|y = r (Π3 − κx )/κx ; the expected payoffs of the inventor and the imitators are Vi =
x|y

µ x|y Π1 + (1 − µ x|y )(Π3 + κx ), where µ y|x := r/(r + 2λ y|x ) and Vb

12

= Π 3 − κx .

under secrecy protection.10 Simple comparative statics shows that the higher imitation costs of y
(κy ), the later the entry by the imitators.
Now consider the game starting at t = 0, in which the inventor is the only seller of both knowledge components, x and y. Unlike the single innovation component case, the inventor can induce
several different sequences of adoption by manipulating price offers, which will result in different
speeds of entry and timings of knowledge sale.
Lemma 3 In the monopoly game with respect to both components, the inventor never charges
yt

pxt
ib > κx and pib > κy for b ∈ { j, k}.
Lemma 3 states that it is never optimal for the inventor to charge such high prices for both
components, letting the imitators incur imitation costs rather than buying from her. Once one rules
out the price structure in Lemma 3, there remain three candidates for the symmetric equilibrium
yt

yt

yt

xt
xt
price structures: (1) pxt
ib ≤ κx and pib > κy , (2) pib > κx and pib ≤ κy and (3) pib ≤ κx and pib ≤ κy

for b ∈ { j, k}. The price structure (1) induces a step-by-step adoption sequence (denoted by x → y)
in which the imitators adopt x first and then y next (i.e., mixing cw and ww in the current stage).
Similarly, (2) induces the imitators to adopt y first and then x next (denoted by y → x). Lastly,
(3) induces an all-or-nothing adoption sequence (denoted by xy) in which the imitators adopt x
and y together (i.e., mixing cc and ww). To see why, suppose we have the price structure (1).
The imitators are aware that if a first adoption occurs for x, they promptly enter a subgame with
y

θ = ({i, j, k}, {i}) in which pib = κy and they end up getting Π3 − κy . Therefore, the imitators are
willing to adopt only x for now.11
It turns out that inducing the imitators to follow the sequence xy is optimal for the inventor.
10 In some other papers, e.g., Denicolo and Franzoni (2004) and Belleflamme and Bloch (2013),
µ y|x /r is instead defined as the discounting adjusted duration of monopoly under secrecy protec-

tion.
11 One might ask why they do not mix cm and ww instead. In fact, the imitators are indifferent
to the choice between mixing cm and ww and mixing cw and ww because we are assuming the
continuous time framework. I assume the tie-break rule here is to choose knowledge sale rather
than imitation.

13

The proposition below summarizes the results.12
Proposition 1 In a unique symmetric MPE when the inventor only relies on secrecy, (i) it is
optimal for the inventor to induce the imitators to mix adopting and waiting for both components by
y

setting pxib = κx and pib = κy for b ∈ { j, k}; (ii) entry times of each imitator follow the exponential
distribution with hazard rate, λ xy = r (Π3 −κ )/κ; and (iii) the expected equilibrium payoffs of the
inventor and of the imitators are Vi = µ xy Π1 + (1 − µ xy )(Π3 + κ ), where µ xy := r/ (r + 2λ xy ) ∈

(0, 1), and Vb = Π3 − κ respectively.
In the appendix, I show that the optimal payoffs of the inventor under sequence xy and y → x
are given by, respectively,
xy

Vi = µ xy Π1 + (1 − µ xy )(Π3 + κ )
where µ xy and λ xy are as in result (ii) in Proposition 1, and
y→x

Vi

= µ y Π1 + (1 − µ y ) κy + µ x|y Π1 + (1 − µ x|y )(Π3 + κx )

where µ y = r/(r + 2λ y ), λ y = r (Π3 − κ )/κy , µ x|y = r/(r + 2λ x|y ) and λ x|y = r (Π3 − κx )/κx .
Here, y stands for variables associated with an adoption game for y given optimally chosen price
structure (2), conditional on not having any adoption for both, and x|y for variables associated with
y→x

an adoption game for x, given that the adoption of y occurs. Subtracting Vi

xy

from Vi shows the

trade-offs between the two mechanisms.
xy

y→x

Vi −Vi

=

µ xy − µ y − (1 − µ y ) µ x|y (Π1 − Π3 )
−( µ xy − µ y )κy − µ xy − µ y − (1 − µ y ) µ x|y κx

One can readily see that the expected time until entry is longer under xy than under y → x, i.e.,
µ xy − µ y − (1 − µ y ) µ x|y > 0. This is because, roughly speaking, the imitators are more aggressive
12 There

exists a trivial asymmetric MPE as well. However, in the literature, papers touching
on a variant of wars of attrition games (or timing games more generally) with the continuous time
framework like mine mostly focus only on symmetric equilibrium. For example, see Choi (1998),
Fudenberg and Tirole (1991) p.117~130, footnote 7 in Henry and Ponce (2011) p.966 and the cited
papers therein.

14

in entering in an y → x sequence, in which they may have a chance to split the licensing fees,
than in xy. Thus, the advantage of xy relative to y → x results from a longer duration of monopoly
(the first term on RHS). However, there is a disadvantage associated with xy: the inventor collects
revenues from knowledge sales later (the second and third terms on RHS). Given assumptions 1
and 2, though, the advantages outweigh the disadvantage, and hence the sequence xy is optimal.
Let ViS denote the maximized payoff of the inventor under protecting all-by-secrecy mode.
Then,
ViS = µ xy Π1 + (1 − µ xy )(Π3 + κ ).
It is worth noting that the profit flow of the innovator under secrecy-only protection resembles the
one under patent-only protection in that both mechanisms yield a monopoly profit for some time
and a triopoly profit after the simultaneous entry by the imitators.

1.4.1.3

Mixing Patents and Secrecy

The final category to analyze is the case where the innovator patents one component and keeps the
other secret. The analysis is analogous in either of the two possible variations in this category, i.e.,
x patented and y kept secret, or y patented and x kept secret. For expositional simplicity, let us
examine the former case of the two, in which case the imitators cannot use x until time T but can
do so freely afterwards. Thus, due to the perfect complementarity of x and y, the imitators’ entry
during the patent term is not feasible. However, the timing of entry after T is closely related with
the timing of adoption of component y.
Let us consider a subgame starting at T . There are two scenarios to investigate: (i) a competitive
game with respect to y, (i.e., the first adoption of y has occurred before T ), θ = (T+ , {i, j}) or

(T+ , {i, k}), and (ii) a monopoly game with respect to y, θ = (T+ , {i}). Notice that these subgames
have a stationary property as true of the games analyzed using the secrecy mechanism, and thus the
analysis therein is directly applicable. From Lemma 1 and 2, we can infer that under scenario (i), at
time T , the discounted expected payoffs of the inventor and the imitators are the same as discounted

15

sum of triopoly profits (i.e., Vi |θ =(T ,{i, j}) = Vb |θ =(T ,{i, j}) = Π3 ) since the entry by the second
+
+
imitator is immediate. In contrast, under scenario (ii), the adopting game for y begins from this
period on, and thus, Vi |θ =(T ,{i}) = (1 − µ y|x )Π1 + µ y|x (Π3 + κy ) and Vb |θ =(T ,{i}) = Π3 − κy
+
+
as in Lemma 2.
With the results above in mind, we will derive the maximum payoff of the inventor. Observe
first that eliminating the chance of the first adoption of y during the patent term strictly increases
the inventor’s payoff as it extends the duration of monopoly even beyond T . Second, the inventor
yt

is able to make the imitators to preclude the imitators from adopting y before T by setting pib
relatively high for b ∈ { j, k} and t < T . To understand this point, conditional on not having the
first adoption, at time t < T an imitator j would get e−r(T −t ) (Π3 − κy ) in a worst-case scenario
in which no adoption takes place until time T either by himself or by the rival imitator, k. Instead,
yt

if j decides to obtain y now, his payoff would be e−r(T −t ) Π3 − min{pi j , κy } which is strictly less
than his minimum possible payoff unless the inventor offers quite low price for y. Put differently,
yt

if pib > e−r(T −t ) κy for all t < T , adopting y earlier than T is strictly dominated by waiting for the
imitators.
The discussion above is summarized in the following proposition.
Proposition 2 Suppose the inventor protects x by patenting and y by relying on secrecy. In any
symmetric MPE, (i) it is optimal for the inventor to induce the imitators not to adopt y until the
y

patent expires (t < T ) by setting high enough prices and to set pib = κy for b ∈ { j, k} thereafter
(t ≥ T ); (ii) entry times of each imitator follow the exponential distribution with hazard rate,
λ y|x = r (Π3 − κy )/κy starting from T ; and (iii) the expected equilibrium payoffs of the inventor
and of the imitators are Vi = (1 − e−rT )Π1 + e−rT {(1 − µ y|x )Π1 + µ y|x (Π3 + κy )} − f , where
µ y|x := r/(r + 2λ y|x ) ∈ (0, 1), and Vb = e−rT (Π3 − κy ) respectively.
Proposition 2 states that under the patent-secret mix the monopoly period is extended beyond
the term of the patent in the expected sense, depending on the imitation cost of the component
kept by secrecy. It is also certain that the component protected by secrecy is not adopted by the

16

imitators until T for sure as the adopting game for the component begins after T .
It remains to examine which mixed use is more profitable: either patenting x and keeping y
secret as in the previous discussion or patenting y and keeping x secret. As shown in Corollary 1,
the inventor (weakly) prefers the former over the latter because the more costly the component kept
secret, the longer duration of monopoly beyond T while the term of patent is invariant regardless
of which component to patent.
Corollary 1 When the inventor uses a patent-secret mix, (i) it is optimal for her to keep the
component y, which is the more costly to imitate, secret if κy > κx ; and (ii) she is indifferent
between protecting x by a patent and y by secrecy and protecting y by a patent and x by secrecy if
κy = κx .
If we denote the maximum payoff of the inventor under the optimal mixed protection mode by
ViM ,
ViM = (1 − e−rT )Π1 + e−rT µ y|x Π1 + (1 − µ y|x )(Π3 + κy ) − f .
This result is comparable to that found by Anton and Yao (2004). They show that an innovator
may patent small innovations but keep secret big innovations with the view that patents and trade
secrets are mutually exclusive substitutes. Though my model also suggests that the inventor may
optimally protect the more important y (more costly to imitate) by secrecy and x with a patent, such
behavior results from the combined use of those two protection mechanisms here. In addition, the
reasons for doing that in the two models are different. Here, keeping y secret is to generate a
longer duration of monopoly, whereas, in their framework, where the strategic disclosure works
as a signaling devise, the inventor faced with limited legal protection is reluctant to disclose big
innovations as they are valuable enough to make the imitator want to imitate.

17

1.4.2

Optimal Protection Decision and Complex vs. Simple Innovations

In this section, I derive one of the key findings of this study by comparing the inventor’s payoffs.
If the patent filing fee is too high, the inventor would always avoid patenting. In order to exclude
those trivial cases, we impose the following assumption.
Assumption 3: f < Π1 −ViS .
First, note that
ViM −ViP = e−rT µ y|x (Π1 − Π3 ) + (1 − µ y|x )κy > 0
for all T > 0. There are two different elements which make using a patent and a trade secret
together more appealing than using a patent alone. After the patent expires, not only can the
inventor be a monopolist for some more random periods (the first term in the brace), but she also
can collect revenues from the sale of y (the second term in the brace). Consequently, the innovator
never protects both components by a patent. Then, by comparing ViS and ViM , we obtain the
following proposition.
Proposition 3 (i) There exists a unique patent length T

Com

such that the inventor protects the

complex innovation through the optimal patent-secrecy mix (as described in Corollary 1) if the
patent length is relatively long (T ≥ T

Com

holding all other variables constant, T

) and otherwise only by secrecy (T < T

Com

Com

); and (ii)

decreases as the composition of the imitation costs

become more asymmetric, i.e., κy increases and κx decreases by the same amount.
There are two facets of optimal protection worth noting. First, the inventor’s optimal protection
has a binary feature: either all-by-secrecy or patenting x and keeping y secret. Second, ceteris
paribus the inventor becomes more aggressive in patenting (i.e., T

Com

gets smaller) under a more

asymmetric structure of the imitation costs. The intuition behind this result is as follows. ViS
depends not on the composition of the imitation costs but on the aggregate imitation costs (see
Proposition 1), and thus it is invariant to such cost changes. However, ViM increases as κy increases
and κx decreases by the same amount because term T is given the same as before while the expected

18

duration for monopoly beyond T will be lengthened. As a result, the threshold patent length at
which ViS and ViM are equal gets shorter.
What if the discovery rate of secrecy (λ ) is exogenously given regardless of what portion of an
innovation is kept secret? When λ is exogenous and constant, as long as at least one component has
been protected by secrecy, keeping either x or y secret generates the same λ . Likewise, patenting
either x or y makes the inventor a monopolist until T . Given that x and y are perfect complements,
to take advantage of both mechanisms, the inventor would always choose to combine both of them
in use. In addition, determining which component to patent and which to keep secret is never a
concern for the inventor because she is indifferent to the choice between the two possible mixing
methods. Lastly, an imitator may uncover the inventor’s trade secrets even during the patent term,
which is in sharp contrast to the prediction of this paper that under a patent-secret mix entry never
takes place until the patent expires.
Now, let us compare the payoffs of inventors having the complex innovation (the “complex
innovator” hereafter) and a simple innovation (the “simple innovator” hereafter) respectively. The
simple innovation consists of a single and not separable component of which imitation cost is
κ. We also assume that the identical profit structure for the simple innovation. Note that, the
available protection for the simple innovator is binary: either protecting all-by-patenting or allby-secrecy. Then, it is easy to see that the simple innovator’s equilibrium payoff is ViSim (T ) =
max{ViS ,ViP (T )} while the complex innovator’s is ViCom (T ) = max{ViS ,ViM (T )} in which ViP ,
ViS and ViM are defined as in section 3.1.
Proposition 4 There exists a unique patent length T

Sim

such that the inventor protects the

simple innovation only by a patent if the patent length is relatively long (T ≥ T
secrecy otherwise (T < T

Sim

). Moreover, T

Com

<T

Sim

Sim

) and only by

.

As shown in figure 1, both the complex innovator (red dashed line) and the simple innovator
(blue line) have the same payoff schedules until T

Com

. If T is intermediate i.e., T ∈ [T

Com

,T

Sim

),

the payoff of the complex innovator starts to grow, while the simple innovator’s is still fixed. When

19

Figure 1.1: Payoff Comparison between Complex Innovator and Simple Innovator

Note: Vertical and horizontal axes represent the innovator’s payoffs and length of patent, respectively. The two dashed lines near the horizontal axis show the optimal protection modes for the
simple innovator (upper blue) and for the complex innovator (lower red).

T exceeds T

Sim

, the payoff of the simple innovator also begins to increase but never catches up to

the complex innovator’s. From the optimal protection choice depicted in the bottom, we also find
that the complex innovator is more inclined to apply for a patent.
One important observation is that without the patent system the inventor would get the same
payoffs as the simple inventor would. To understand this, recall that the equilibrium payoff of the
inventor under secrecy protection does not depend on the composition of imitation costs but only
on the aggregate costs (Proposition 1). This implies that it is not per se the existence of a number
of perfectly complementary components which may make the complex innovator more profitable.
Coexistence of the patent system and the secrecy mechanism is crucial to have that result.

20

1.5

Strict Utility Requirement

Up to this point, only the positive analysis has been conducted. In particular, it was shown that the
inventor with a complex innovation is (weakly) over-rewarded. However, if the patent examination
office imposes a strict utility requirement (SUR), under which the innovator needs to disclose both
components to get a patent, it can transform the inventor’s payoff schedule into that of another
investor having a simple innovation. This section addresses policy implications of SURs whether
they are socially desirable or not.
There are numerous research studies investigating optimal patent policy regarding various policy variables (e.g., the patent scope or the term of patent). However, there is no clear consensus on
this topic, and results heavily depend on underlying environments and assumptions.13 For the sake
of a thought experiment, we only consider the length of patent as the policy instrument of interest.
We then consider a simple model in which the social planner decides whether to impose an SUR
or not after the discovery (ex-post welfare implication) and before the discovery (ex-ante welfare
implication) of the complex innovation in that order. For simplicity, throughout this section we
assume the two components are symmetric in terms of imitation costs. (i.e., κx = κy = κ/2)

1.5.1

Ex-post Welfare Implication of SUR

In order to first calculate ex-post welfare after the discovery of the innovation with and without
SUR, given the patent length, T , we denote the ex-post welfare functions under each protection
mode, P, M and S, by W P (T ), W M (T ) and W S as follows

W P (T ) = (1 − e−rT )W1 + e−rT W3
W M (T ) = (1 − e−rT )W1 + e−rT µ 1W1 + (1 − µ 1 )W3
W S = µ 2W1 + (1 − µ 2 )W3
13 For

example, Gilbert and Shapiro (1990) and Gallini (1992) show sharply contrasting results.

21

where µ 1 and µ 2 are the natural duration of monopoly generated by keeping 1 component and 2
components by secrecy, respectively.14 As is usually assumed in the literature, W1 and W3 represent
social welfare, defined as the sum of industry profits and consumer surplus under monopoly and
triopoly market structure respectively, and social welfare is higher under a triopoly situation than
under a monopoly one, i.e., W1 < W3 .
As the inventor’s optimal decision is opting for S if T ≤ T

Sim

and P otherwise under SUR,

ex-post welfare with SUR is

W SUR

(T ) =




W S

if T ≤ T



W P (T )

otherwise

Sim

.

Similarly, ex-post welfare without SUR is given by



W S
NO
W (T ) =


W M (T )

if T ≤ T

Com

.
otherwise

The schedules of the two welfare functions can be compared by first identifying their characteristics. Observe first that W M is always less than W P because the patent-secret mix always
entails a longer duration for a monopoly. Next, while W S does not vary by the patent length T ,
W M and W P decrease as T increases. Last, and most interestingly, there is a sudden drop in the
two welfare functions. More specifically, W SUR suddenly decreases at the threshold, T

Sim

. The

reason for this discontinuity is that while the optimal payoff of the inventor remains the same in the
two protection modes at T

Sim

, the expected monopoly duration is shorter under secrecy protection.

To see this, it is useful to compare the natural duration of the secret, µ 2 and the natural patent
length, (1 − e−rT

Sim

) where T

Sim

= (1/r ) ln[(Π1 − Π3 )/{(1 − µ 2 )(Π1 − Π3 − κ ) − f }] by the

symmetric imitation cost structure and the definition of T

(1 − e−rT

Sim

) − µ2 =

Sim 15
.

Then, it is easy to see that

(1 − µ 2 )κ + f
> 0.
Π1 − Π3

= r/(r + 2λ 1 ), λ 1 = r (Π3 − κ/2)/(κ/2), µ 2 = r/(r + 2λ 2 ), λ 2 = r (Π3 − κ )/κ.
15 See the proof of Proposition 4.
14 µ 1

22

Figure 1.2: Ex-post Welfare with and without SUR

Note: Vertical and horizontal axes represent respectively social welfare and length of patent
The intuition is simple: under secrecy protection as the inventor has another channel of appropriating her innovation, i.e., selling the knowledge components, a shorter monopoly is required for
her to achieve the expected payoff she would have earned under patent protection, with the threshold T

Sim

. This implies when the inventor is indifferent between the secrecy and patent protection

modes, ex-post welfare is higher under secrecy protection, i.e., W S > W P (T
can be applied to show W S > W M (T

Com

Sim

). A similar logic

).

From the discussion above, it is clear that W SUR and W NO must be drawn as depicted in figure
2. The figure shows that W NO is equal or less than W SUR for all T , implying that availability of a
patent-secret mix decreases ex-post welfare, and thus, imposing SUR is advisable.
Proposition 5 Imposing the strict utility requirement is socially desirable ex-post (strictly desirable for T > T

Com

).

23

1.5.2

Ex-ante Welfare Implication of SUR

This section turns from the above examination of ex-post welfare implications of SUR to the exante ones, taking into account the inventor’s investment incentives. To this end, the first question
is to determine which T will be chosen by the social planner facing an innovator when only the
patent protection mode is available. Then, such T is a solution of the following problem:
max W P (T ) = IˆP · W P − D(IˆP )
T

subject to
IˆP = arg max I ·ViP − D(I )
I

where I is the probability that the innovation process turns out to be successful; D(I ) is a cost
function of such R&D process given I; IˆP is the optimal investment decision of the inventor given
T . Furthermore, let us assume that the objective function subject to the optimal investment decision
is strictly quasi-concave in T , and so the solution exists uniquely. There is a trade-off of giving
more incentive for the inventor to invest by increasing T , which increases the probability that the
innovation project is successful but also decreases ex-post social welfare by extending the duration
for which the inventor is a monopolist. Thus, if we denote the solution of the above problem
by T P∗ ∈ (0, ∞), it is set by the social planner in such a way that those two marginal effects are
balanced. We can consider the same problem with mixed protection. Let T M∗ ∈ (0, ∞) be the
solution of such social planner’s problem.
Then, the ex-ante social welfare functions with and without SUR are defined again by the
optimal decision rules of the innovator:

W SUR

(T ) =

and
W NO

(T ) =




W S

if T ≤ T



W P (T )

otherwise

Sim




W S

if T ≤ T



W M (T )

otherwise

Com

.

24

Figure 1.3: Example of Ex-ante Welfare with and without SUR

(a) Case (I): κ = .04

(b) Case (II): κ = .085

Note: Vertical and horizontal axes represent social welfare and length of patent, respectively.
W SUR and W NO are the ex-ante welfare functions given T with and without SUR respectively.

Even if the socially optimal length of patent is well defined for P and M, a complete analysis for
all underlying primitives is somewhat challenging. First, given T , optimally chosen investment
levels under P, M and S are all different. Moreover, due to the discontinuity nature of the ex-post
welfare functions introduced in the previous subsection, the ex-ante welfare functions are also not
continuous and even, at times, not monotone. Therefore, in what follows, I will demonstrate, by
offering an example, that policy implications are sensitive to parameter values of the model and
ambiguous in general.
Example: Consider the following two cases which only differ by the imitation costs: for case
(I), κ = .04 and for case (II), κ = .085. Except for that difference, the two cases share the same
modeling assumptions: D(I ) = I 2 , Π1 = .4, Π3 = .1, W1 = .5, W3 = .8, r = .1 and f = 0. Figure
3 shows welfare functions for the two cases. Note that the policy implication to be drawn is that
imposing SUR is socially desirable whenever W SUR > W NO . First, consider case (I) in which the
strength of secrecy mechanism (relatively low κ), by which the lower bound of ex-ante investment
incentives is determined, is relatively low. In this situation, T

25

Com

and T

Sim

are less than T M∗ and

T P∗ , respectively, and this implies the secrecy only protection is not giving enough investment
incentives to the innovator. Hence, in this case, for some relatively intermediate T , allowing the
inventor to choose M may increase the social welfare. However, the policy implication is not
monotone in T . Roughly speaking, imposing SUR increases social welfare over either the lower
or the upper tail range of T but decreases it over the intermediate range of T . In contrast, in case
(II), the strength of secrecy is so high that even without the patent system, the inventor is willing to
invest aggressively. Therefore, adding M option to the innovator’s choice set always lowers social
welfare. This example illustrates that the policy implication of SUR is, in general, ambiguous.

1.6

Extensions and Robustness

This section considers two extensions to check the robustness of the main findings. The complex innovation is comprised of N (> 2) components rather than 2 components is first discussed,
followed by an exploration of the case in which patent rights are probabilistic.

1.6.1

N Component Innovation

We have assumed so far, for simplicity, that the complex innovation has just two complementary
components. Here, suppose we have a complex innovation which is comprised of N different
complementary components. The imitation costs of each component are given by κ1 ≤ . . . ≤ κn ≤
. . . ≤ κN for n ∈ {1, 2, . . . , N}, and let κ ≡ ∑N
n=1 κn < Π3 denote the aggregate imitation cost. I
show that the main results we had for the model with 2 components carry over to this situation.
When the inventor applies for a patent containing all N components, her payoff remains the
same as before:
ViP = (1 − e−rT )Π1 + e−rT Π3 − f .
Regarding the secrecy only mode and the patent-secret mix, equilibria in each scenario also shows
a pattern that is similar to the 2 component model. We formally characterize the equilibria in the
next two lemmas.

26

Lemma 4 In the N component model, when the inventor relies only on secrecy, the unique
symmetric MPE is characterized as follows: (i) it is optimal for the inventor to induce the imitators
to mix adopting and waiting for all N components by setting pnib = κn for all n ∈ {1, 2, . . . , N} for
b ∈ { j, k}; (ii) entry times of each imitator follow the exponential distribution with hazard rate,
λ = r (Π3 − κ ) /κ; and (iii) the expected equilibrium payoff of the inventor is ViS = µΠ1 + (1 −
µ )(Π3 + κ ) where µ = 1/(r + 2λ ).
Lemma 5 In the N component model, (i) the optimal patent-secret mix is patenting only component 1, which is the least costly to imitate, and keeping all the other components by secrecy;
(ii) the inventor induces the imitators not to adopt all components except component 1 until the
patent expires (t < T ) by setting high enough prices and sets pnib = κn for all n ∈ {2, 3, . . . , N}
for b ∈ { j, k} thereafter (t ≥ T ); (ii) entry times of each imitator follow the exponential distribution with hazard rate, λ −1 = r (Π3 −κ−1 )/κ−1 starting from T ; and (iii) the expected equilibrium
payoff of the inventor is ViM = (1−e−rT )Π1 + e−rT {µ −1 Π1 + (1− µ −1 )(Π3 + κ−1 )}− f where
µ −1 = 1/(r + 2λ −1 ) and κ−1 = ∑N
n=2 κn .
ViS is the same expression as in the 2 component model, and ViM is also similar to the counterpart of the 2 component model. Hence, we reach to the same conclusion that the optimal protection
mode is either S or the optimal M because P is always dominated by the optimal M, and thus the
complex innovator weakly earns more than the simple innovator.
The N component model also offers some additional insights through a simple comparative
static analysis. Assuming symmetric imitation costs, as the innovation becomes more complex
(larger N), both the expected duration of prolonged monopoly (higher λ −1 ) and the inventor’s expected payoff (higher ViM ) increase in the optimal patent-secrecy mix, and, as a result, the threshold
equating ViS and ViM (T

Com

) gets shorter.

Proposition 6 In the N component model, (i) there exists a unique patent length T

Com

such that

the inventor protects the complex innovation through the optimal patent-secret mix (as described
in Lemma 4) if the patent length is relatively long (T ≥ T

27

Com

) and only by secrecy otherwise

(T < T

Com

); and (ii) when all components are symmetric in terms of imitation cost, i.e., κ1 =

. . . = κn = . . . = κN , T

1.6.2

Com

shortens as the innovation becomes more complex (i.e., as N rises).

Probabilistic Patents

How much of the previous results would be changed when we assume that patents are probabilistic
rather than ironclad? Put differently, what if filing for a patent no longer absolutely assures the
inventor of T -period monopoly position, but only provides some probability? Here, assuming
patents are probabilistic, we determine under which conditions the combined use of patents and
secrecy turns out to be optimal, at least for some values of patent length, and thus the complex
innovator is still (weakly) over-rewarded.
To formalize the idea, consider the following simplified version of a post-grant review in the
U.S. patent system or an opposition proceeding in the European patent system.16 Right after a
patent is granted to the inventor, the imitators have a chance to challenge it at the patent office.
For simplicity, assume that the challenge process takes place instantly without any delay at 0 cost,
which implies it is optimal for the imitators to challenge every patent granted. Let us further assume
that f = 0 and that the two components are symmetric in terms of imitation cost (κx = κy = κ/2)
and the probability that each component will be upheld as valid through the trial proceeding when
the components are challenged. Denote by α the probability that at least one component survives
the challenge and is upheld as valid when the inventor has chosen P. Similarly, define ρα as
the probability that a component is upheld as valid when the inventor has chosen M where ρ is
a measure of the inverse degree of complementarity between the two components in obtaining a
valid patent and takes a value in [1/2, 1]. 17
16 United States Patent and Trademark Office states “Post grant review is a trial proceeding con-

ducted at the Board to review the patentability of one or more claims in a patent on any ground that
could be raised under §282(b)(2) or (3). Post grant review process begins with a third party filing
a petition on or prior to the date that is 9 months after the grant of the patent or issuance of a reissue patent.” (For more details, visit https://www.uspto.gov/patents-application-process/appealingpatent-decisions/trials/post-grant-review. Last retrieved on Aug/19/2016.)
17 To see this, note that the joint distribution of trial results for the two components is as follows

28

Then, the equilibrium payoff of the inventor under S remains as in the basic model, i.e.,
ViS = µΠ1 + (1 − µ )(Π3 + κ ), whereas the equilibrium payoffs under the other two protection
modes change. When the inventor opts for P, she earns monopoly profit until T and triopoly profit
afterwards only with probability α, and with the complementary probability, 1 − α, the imitators
can enter the market from the beginning. Hence,
ViP = α (1 − e−rT )Π1 + e−rT Π3 + (1 − α )Π3 .
When the inventor chooses M, she becomes a monopolist until T and then the adopting game for
the remaining component by the imitators follows right after T if the patent is upheld as valid (with
probability ρα). Otherwise (with the complementary probability 1 − ρα), the same adopting game
starts from the beginning. Thus,

ViM = ρα (1 − e−rT )Π1 + e−rT V 1 + (1 − ρα )V 1 ,
where V 1 = {µ 1 Π1 + (1 − µ 1 )(Π3 + κ/2)}, µ 1 = r/(r + 2λ 1 ) and λ 1 = r (Π3 −
κ/2)/(κ/2).18
since x and y are symmetric in terms of validity:
Component x \ y Valid Not valid
Valid
A
B
Not valid
B
C
Then, α = A + 2B and ρα = A + B, and thus, ρ = (A + B)/(A + 2B) which takes a value
between 1/2 and 1. For example, depending on whether the trial results of x and y are perfectly
positively correlated (B = 0), independent
(B = ρα (1 − ρα )) or perfectively negatively correlated
√
(B = α/2), ρ equals to 1, (1 − 1 − α )/α, or 1/2 respectively.
18 Alternatively, we can assume that patent rights are probabilistic in the sense that patent examiners at the patent office reject a claim (component), which is included in a patent application and
does not meet one of the patentability requirements, with a certain probability. (In the U.S. patent
law, the patentability requirements are patentable subject matter, novelty, non-obviousness and utility). If rejected claims are disclosed to the public, the equilibrium payoffs will be the same as those
described in the post-grant review scenario. If rejected claims are not disclosed to the public, and
thus if the imitators still needs to incur imitation costs or to buy from the inventor in order to adopt
those components, the equilibrium payoffs with P and M are modified as follows respectively:
ViP = (2ρ − 1)α{(1 − e−rT )Π1 + e−rT Π3 } + 2(1 − ρ )α{(1 − e−rT )Π1 + e−rT V 1 } + (1 − α )ViS
and ViM = ρα{(1 − e−rT )Π1 + e−rT V 1 } + (1 − ρα )ViS . The qualitative results do not, however,
change under this alternative scenario.

29

In order to find the optimal protection mode depending on the patent length, first observe that
if T = 0, for all α and ρ,
ViS > ViM |T =0 > ViP |T =0 .
This tells us that when T is 0, it is better to protect as many components as possible by secrecy
so that the inventor can have a longer random monopoly period regardless of α and ρ. This is
somewhat obvious: recall that we had the same ranking between the payoffs when T = 0 even
in the basic model, and patenting, moreover, now became less attractive due to the probabilistic
nature of patent rights. By the continuity of payoff functions, when patent duration is relatively
short, S would be still optimal compared to M or P. However, as T increases, the ranking between
the payoffs may be reversed since, for all α and ρ,
∂V S
∂V M
∂ViP
= αre−rT (Π1 − Π3 ) > i = ραre−rT (Π1 −V 1 ) > i = 0
∂T
∂T
∂T
where the first inequality follows from the fact that ρ ≤ 1 and V 1 > Π3 . In fact, the full optimal
schedule of protection mode over T ∈ [0, ∞) crucially depends upon the ranking of ViP , ViM and
ViS in the limit (i.e., T → ∞). Thus, to fully characterize equilibrium constellation, let us define the
following thresholds: α ∗ , ρˆ (α ) and ρ˜ (α ) such that respectively
ViS ⇐⇒ α

α ∗,

ViS ⇐⇒ ρ

ρˆ (α ),

lim ViP (α ) ⇐⇒ ρ

ρ˜ (α ).

lim ViP (α )
T →∞
lim ViM (α, ρ )
T →∞
lim ViM (α, ρ )

T →∞

T →∞

Then, the equilibrium protection constellations are as shown in Proposition 7 and illustrated in
figure 4.
Proposition 7 In the probabilistic patents model, given (α, ρ ) ∈ (0, 1] × [1/2, 1], the optimal
protection modes chosen by the inventor are as follows:
(i) if α ≤ α ∗ and ρ ≤ ρˆ (α ), S always;
(ii) if α > α ∗ and ρ ≤ ρ ∗ , S if T is small and P otherwise;

30

Figure 1.4: Equilibrium Protection Constellations

(iii) if α > α ∗ and ρ ∗ < ρ < ρ˜ (α ), S if T is small, M if T is intermediate and P otherwise;
(iv) if ρ˜ (α ) ≤ ρ and ρˆ (α ) < ρ, S if T is small and M otherwise
V S −Π

V S −V 1

i
where α ∗ = Πi −Π3 ∈ (0, 1), ρˆ (α ) =
, ρ˜ (α ) =
α (Π1 −V 1 )
1
3
ρ˜ (α ∗ ) = ρˆ (α ∗ ) ∈ (1/2, 1).

α (Π1 −Π3 )−(V 1 −Π3 )
α (Π1 −V 1 )

and ρ ∗ =

Proposition 7 confirms that in the probabilistic patents model the combined use of patents and
trade secrets (for some T ) arises in equilibrium as before insofar as the probability of getting a valid
patent with the two components is relatively high (i.e., relatively high α), and the gap between such
probabilities with the two components and with a component is not that high (i.e., relatively high
ρ). That corresponds to the area (iii) and (iv) in figure 4, and in this region it is also robust that
the complex innovator is weakly over-rewarded.
It is noteworthy that now there is a pair (α, ρ ) under which P dominates M for all T (area (ii)).
Recall that in the main analysis, either M or P ensured obtaining a patent, and, as a result, P was
always inferior to M as it only made the inventor give up a chance to prolong the monopoly period
beyond T by disclosing all information. That is no longer the case since P has an advantage over M
in the sense that it has a higher probability of acquiring a patent, and the degree of that advantage
depends on ρ. More specifically, for each α > α ∗ , the optimal pattern of the inventor’s strategy
changes, as ρ increases, from the one in (ii) of Proposition 7 to the one in (iii) and finally to the
one in (iv). Another interesting point is that some dubious innovations, which would not have been

31

patented at all if they were a simple innovation, might be patented if ρ is high (corresponding to
the subarea of (iv) such that α < α ∗ ).

1.7

Conclusion

This paper studies the optimal use of patents and trade secrets by an inventor who has a complex
innovation. I introduce the mechanism of how the patent-secret mix can extend the duration of
monopoly beyond the length of patent, and thus dominate the patent protection. For that reason,
the inventor’s optimal choice has a binary feature: either opting for the optimal patent-secret mix
or secrecy only. The optimality of patent-secret mix is carried over to the case of probabilistic
patent rights as well on a broad set of parameters.
As one may easily expect, with a richer set of protection modes, including the mixed one, the
inventor who has the complex innovation is over-rewarded compared to another inventor who has
a simple innovation. However, the policy implication of imposing SUR considering the ex-ante
investment incentives is ambiguous and sensitive to underlying parameters. Therefore, unfortunately, this paper informs us that designing optimal intellectual property rights policies calls for
a case-by-case scrutiny with the patent-secret mix as the inventor’s protection strategy. In this regard, it would be also beneficial to conduct a similar analysis to illuminate how and when firms
optimally use a patent-secret mix in different innovation frameworks, such as, a cumulative innovation setting, a patent portfolio setting and a complementary innovation setting associated with
patent pooling.

32

APPENDIX

33

Appendix for Chapter 1

Proof of Lemma 1
Result (i): I explore all possible competitive games with respect to a knowledge component in
turn. First, consider a competitive game with respect to both components. There are two possible
scenarios: (1) an imitator is a seller for both components besides i, i.e., θ = ({i, j} , {i, j}) or

({i, k} , {i, k}); (2) each imitator is a seller for different components in addition to i, i.e., θ ∈
({i, j}, {i, k}) or ({i, k}, {i, j}). Assume scenario (1) first and say j the seller without loss of
generality, i.e., θ = ({i, j} , {i, j}). I will show that both of components are sold immediately at 0
prices in equilibrium. The proof for this claim consists of the following three steps, and these can
be applied to the rest of three cases in a similar manner.
Step 1: Given price offers, k’s best respond is


y


csk csk if pxs ≤ κx and ps ≤ κy







c m if px ≤ κ and py > κ
sk k




mk csk







mk mk

x

s

s

y

y

if pxs > κx and ps ≤ κy
otherwise.

where s denote the seller who offers the lower price. This is straightforward from the fact (1) that
including wk for any components is never a best response because the immediate entry, which is
possible only when k adopts x and y, is profitable for k and (2) that it is optimal for k to pick the
cheapest way of adopting a component.
Step 2: In any MPE, all components are sold to k through contracts offered by one of the two
y
sellers. Suppose not, which implies either pxs > κx or ps > κy . For simplicity, say pxs > κx . Then,

it is profitable for either i or j to set px = κx at which k buys x rather than imitating because it
increases her or his payoff by κx without changing the timing of entry by k at all since it takes
place anyway in that period.

34

y

y

Step 3: In a unique MPE, pxik = pxjk = 0 and pik = p jk = 0. Again, suppose it were not the
case. If both sellers offer a pair of positive prices for a component, at least one of the two has an
incentive to set a price just below the rival seller’s and collect more revenue without affecting the
present value of each firm’s profit stream form the market. Similarly, if one of them offers a strictly
positive price, but the other offers 0, then the latter has an incentive to increase her price. Thus, in
any competitive game with respect to both components, there is a unique MPE where all remaining
buyers obtains knowledge components still not in their possession at 0 price immediately.
Next, consider another class of competitive games in which there are two sellers for a component, and the other was obtained by all firms. For simplicity, assume that all firms learned x. Then,
there are two possible cases to consider, θ = ({i, j.k} , {i, j}) or ({i, j.k} , {i, k}). The analysis is
analogous to the previous one but simpler than that because it is now sufficient to consider only
one remaining component to be sold.
Result (ii): Consider a game that is competitive with respect to a component but monopolistic
with respect to the other. I show that the remaining buyer of the former component purchases it
at a 0 price in an MPE. For simplicity, say x was adopted by j at time t, i.e., θ = ({i, j} , {i}).
First, note that until the first adoption takes place for y as well, i is the monopolist in the market
because of perfect complementarity of x and y. If k chooses to wait for x, he starts to earn triopoly
profit once the first adoption for y takes place since he can get x for free almost immediately
(from result (i)). Conversely, if k chooses either buy or imitate at time t, he enters a subgame of
θ = ({i, j, k} , {i}) right away in which he ends up earning the same profit stream with imitators’
actions for y remaining the same as in the previous case. As so far as the minimum price of x
offered to k is positive, imitating or buying x at time t is dominated by waiting, in any MPE of this
subgame, x should be sold to k at 0 price and the equilibrium payoff is independent of the timing
of sale of x.

35

Proof of Lemma 2
Together with Lemma 1 in this paper, see Lemma 2 and Proposition 2 in Henry and Ponce (2011)
except calculating the buyers’ payoffs in equilibrium. Note that
y|x

Vb

=

λ y|x (Π3 − κy ) + λ y|x Π3
r + 2λ y|x

.
y|x

Once λ y|x in Lemma 2 is plugged into the above equation, it follows that Vb

= Π 3 − κy .

Proof of Lemma 3
y
Suppose there is an MPE in which the innovator offers pxib > κx and pib > κy for b ∈ { j, k}. Then,

in any equilibrium it must be the case that the first adoption of a component by an imitator19 is associated with imitation by himself rather than buying knowledge through a contract. Given that if
the seller offers a price equal to its imitation cost for that component, the only thing changes would
be the mode of adoption of it from imitating to buying with all other aspects of the equilibrium
unchanged. Then, clearly by doing this the seller will be strictly better off because now she additionally obtains revenue from selling knowledge. This contradicts the supposition that pxib > κx
y

and pib > κy for b ∈ { j, k} are MPE price offers.

Proof of Proposition 1
Step 1: First, let us calculate the expected payoffs to the innovators and imitators given different
y

price combinations. If pxib ≤ κx and pib > κy , mxb is never optimal because at least b can resort to
cxib to adopt x allowing b to save imitation cost, keeping all other aspects of the game unchanged.
y

y

By the same token, cib is never optimal as b can instead choose mb . Then, a candidate for an
y

y

y

y

equilibrium is among cxib wb , wxb wb , cxib mb and wxb mb . However, if the equilibrium should be assoy

y

y

ciated either with cxib mb or wxb mb , i always has a profitable deviation of adjusting pib = κy , which
19 Depending on an equilibrium strategy profile, a component and an imitator need to be modified

as components and the imitators respectively.

36

increases i’s revenue from selling y without changing other aspects of the game. Since that is a
y

y

clear contradiction, it must be the case that b chooses either cxib wb or wxb wb if this is an equilibrium.
This implies that knowledge adoption takes place for x first and for y later in turn in equilibrium.
(x → y)
Let Fb : [0.∞) × [0, κx ] → [0, 1] refer to the distribution function of entry times of b. Taking F−b
as given and conditional on no adoption for x and y until time t, the expected benefit of waiting
infinitesimal dt measure of time more (comparing to purchasing x now) is given by f−b pxib dt/(1 −
F−b ) since b can get x freely instead of paying pxib if −b enters during this time window. However,
there are also costs associated with waiting. b needs to sacrifice some surplus which he would
otherwise have been able to earn, r (Π3 − κy − pxib )dt. 20 In equilibrium the above two terms
should be equal, which yields
r (Π3 − κy − pxib )
f−b
.
=
1 − F−b
pxib
Note that λ−b := f−b /(1−F−b ) = r (Π3 −κy − pxib )/pxib is stationary, and thus adoption strategies
of the imitators follow an exponential distributions with the hazard rates.
The timing of the first adoption for x is a random variable which follows exponential distribution with the hazard rate of λ j + λk since it is the minimum of adoption times of the two imitators
which are independently and exponentially distributed with hazard rates of λ j and λk respectively
given px = ( pxij , pxik ) ∈ [0, κx ]2 . By applying the technique used by Henry and Ponce (2011) along
with Lemma 2 21 , if we write down the expected payoff of the inventor,
y|x
λ j pxij + λk pxik
(λ j + λk )Vi
rΠ1
+
+
.
r + λ j + λk
r + λ j + λk
r + λ j + λk

x→y x
Vi
(p ) =

x→y

One can easily check that Vi

( px ) is strictly increasing in both price levels. In order to see

this, observe
x→y

λj
∂Vi
1 ∂λj
y|x
= − 2 x rΠ1 − rVi − (r + λ j ) pxik + λ j pxij
x
∂ pi j
φ
φ ∂ pi j
20 Recall

from Lemma 1 and 3 that once the first adoption takes place for x, they enter the
subsequent subgame immediately in which imitators’ expected payoffs in equilibrium is Π3 − κy .
21 See the proof of Proposition 2 in Henry and Ponce (2011) for more details

37

where φ := r + λ j + λk . Since ∂ λ j /∂ pxij < 0, it suffices to check the term inside of the squared
y|x

bracket is positive. Plugging in the expression for Vi

from Lemma 2 and r (Π3 − κy − pxik ) for

λ j pxik and collecting terms,
y|x

rΠ1 − rVi

=

− (r + λ j ) pxik + λ j pxij

r (2Π1 −4Π3 −κy )(Π3 −κy )
(2Π3 −κy )

+ λ j pxij

,
x→y

and the above term is positive as 2Π1 − 4Π3 − κy > 0 by assumption 1 and 2. Hence Vi

is

y
maximized when pxib = κx and pib > κy for b ∈ { j, k}.

It is easy to apply the same logic for the rest of price structures to find out optimal pairs of
prices. If the innovator wants a component to be adopted by the imitators in the current state, she
charges a price equal to its imitation cost. Otherwise, a price for that component is higher than its
imitation cost.
Step 2: Candidates for symmetric MPE and corresponding maximized expected payoffs of the
y

innovator are as follows. If pxib = κx and pib > κy in the beginning (inducing an adoption sequence
of x first and y later, x → y),

x→y

Vi

= µ x Π1 + (1 − µ x ) κx + µ y|x Π1 + (1 − µ y|x )(Π3 + κy )

where µ x = r/(r + 2λ x ), λ x = r (Π3 − κ )/κx , µ y|x = r/(r + 2λ y|x ) and λ y|x = r (Π3 − κy )/κy .
y
If pxib > κx and pib = κy , (inducing an adoption sequence of y first and x later, y → x),

y→x

Vi

= µ y Π1 + (1 − µ y ) κy + µ x|y Π1 + (1 − µ x|y )(Π3 + κx )

where µ y = r/(r + 2λ y ), λ y = r (Π3 − κ )/κy , µ x|y = r/(r + 2λ x|y ) and λ x|y = r (Π3 − κx )/κx .
y
If pxib = κx and pib = κy , (inducing simultaneous adoption for both components, xy),
xy

Vi = µ xy Π1 + (1 − µ xy )(Π3 + κ )

38

where µ xy = r/(r + 2λ xy ) and λ xy = r (Π3 − κ )/κ. One can verify that an imitator’s payoff is
x→y

= Vby→x = Vbxy = Π3 − κ.22

the same in all cases as Vb

Step 3: Now we show that in the monopoly game with respect to both components, it is optimal
y∗
xy
for the inventor to induce a simultaneous adoption by setting px∗
ib = κx and pib = κy since Vi >
y→x

Vi

x→y 23
.

≥ Vi

(The last inequality is strict if and only if κx < κy .)
xy

y→x

First, let us show Vi > Vi
xy

y→x

Vi −Vi

=

y→x

. If we take Vi

xy

from Vi , we get

µ xy − µ y − (1 − µ y ) µ x|y (Π1 − Π3 − κx ) − ( µ xy − µ y )κy .

By replacing µ xy , µ y and µ x|y with the expressions in step 2, we have
xy

y→x

Vi −Vi

2κx κy (Π3 − κ )
(Π − Π3 − κx )
(2Π3 − κ − κx )(2Π3 − κx )(2Π3 − κ ) 1
(2Π3 − κ − κx )κ − (2Π3 − κ )κy
κy
−
(2Π3 − κ − κx )(2Π3 − κ )
2κx κy (Π3 − κ )(Π1 − Π3 − κx )
2(Π3 − κ )κx κy
=
−
(2Π3 − κ − κx )(2Π3 − κx )(2Π3 − κ ) (2Π3 − κ − κx )(2Π3 − κ )
2κx κy [(Π3 − κ )(Π1 − Π3 − κx ) − (Π3 − κ )(2Π3 − κx )]
=
(2Π3 − κ − κx )(2Π3 − κx )(2Π3 − κ )
2κx κy (Π3 − κ )(Π1 − 3Π3 )
>0
=
(2Π3 − κ − κx )(2Π3 − κx )(2Π3 − κ )

=

since all terms in the numerator and the denominator are positive by assumption 1 and 2.
x→y

22 In

order to derive this, plug in corresponding hazard rates into the following three: Vb
=
y→x
xy
x
x
y
y
λ {(Π3 −κ ) + (Π3 −κy )}/(r + 2λ ), Vb
= λ {(Π3 −κ ) + (Π3 −κx )}/(r + 2λ ) and Vb =
xy
xy
λ {(Π3 − κ ) + Π3 }/(r + 2λ ).
23 In fact, when px = κ and py = κ , not only the sequence xy but any other sequences can take
x
y
ib
ib
place because buyers are indifferent among the following actions: cc, cw, wc and ww. If cw or wc
y
is included in an optimal behavioral strategy of b given pxib = κx and pib = κy with strictly positive
probability, i would lower prices for each component by ε in order for b to mix only adopting
y
or waiting for both and thereby to increase her expected payoff. (pxib and pib approach to κx and
κy from below, but cannot hit that point.) With that said, if we do not specify optimal behavior
of the imitators in this case, a price equilibrium may not exist. To guarantee the existence of
equilibrium, we assume that the imitators behave in such a way that the inventor’s expected payoff
is maximized when they are indifferent among several actions. Another simple justification, which
yields a similar consequence, would be that the inventor can sell knowledge components through
a bundling.

39

y→x

Next, to show Vi
y→x

Vi

x→y

−Vi

x→y

≥ Vi

=

, write the difference between the two as follows:

(1 − µ x )(1 − µ y|x ) − (1 − µ y )(1 − µ x|y ) (Π1 − Π3 )
+(1 − µ y ) κy + (1 − µ x|y )κx − (1 − µ x ) κx + (1 − µ y|x )κy .

Substitution of µ x , µ y|x , µ y and µ x|y given in step 2 into the above equation gives, after a similar
tedious algebra as shown above,
y→x

Vi

x→y

−Vi

=

4κx κy (Π1 − 3Π3 )(Π3 − κ )(κy − κx )
,
{2(Π3 − κ ) + κx } 2(Π3 − κ ) + κy (2Π3 − κx )(2Π3 − κy )

which is 0 if κx = κy and strictly positive otherwise.
Step 1 through 3 completes the proof since we only consider symmetric MPE. However, one
xy

might wonder whether there exists an asymmetric sequence generating higher payoff than Vi .
Step 4 demonstrates that when the inventor induces asymmetric sequences, the inventor’s payoff is
xy

y→x

a weighted average of some two of the three expected payoffs: Vi , Vi

x→y

and Vi

implying it is

never optimal for her to do so.
Step 4: Here, I only offer a proof for the case in which the inventor induces j to choose x → y
y

y

(pxij = κx , pi j > κy ) and k to choose y → x (pxik > κx , pik = κy ), which gives enough description
x→y/y→x

on how to prove the remaining cases. Let Vi

denote the expected payoff of the investor

gets from this pricing strategy. Then,
x→y/y→x

Vi

=

r

y Π1 +

λx
κx + (1 − µ y|x )Π1 + µ y|x (Π3 + κy )
x
y
r+λ +λ

r +λx +λ
λy
+
κy + (1 − µ x|y )Π1 + µ x|y (Π3 + κx ) .
r +λx +λy
x→y

Rearranging terms and writing it again in terms of Vi
x→y/y→x

Vi

=

y→x

and Vi

completes the proof.

1
× rΠ1 + 2λ x κx + (1 − µ y|x )Π1 + µ y|x (Π3 + κy )
2(r + λ x + λ y )

+rΠ1 + 2λ y κy + (1 − µ x|y )Π1 + µ x|y (Π3 + κx )
=
x

r + 2λ x
r + 2λ y
x→y
y→x
V
+
Vi
i
x
y
x
y
2(r + λ + λ )
2(r + λ + λ )
y

x

y

r +2λ
r +2λ
r +2λ
r +2λ
where 2(r+
and 2(r+
∈ (0, 1), and 2(r+
+ 2(r +
= 1.
λ x +λ y )
λ x +λ y )
λ x +λ y )
λ x +λ y )

40

Proof of Corollary 1
If the inventor patents x and keeps y secret, from Proposition 2,
Vi = (1 − e−rT )Π1 + e−rT µ y|x Π1 + (1 − µ y|x )(Π3 + κy ) − f .
On the contrary to this, if the inventor chooses the opposite mix use,
Vi = (1 − e−rT )Π1 + e−rT µ x|y Π1 + (1 − µ x|y )(Π3 + κx ) − f .
If we subtract the second expression from the first, we have
e−rT ( µ y|x − µ x|y )(Π1 − Π3 ) + (1 − µ y|x )κy − (1 − µ x|y )κx

=

2 (Π1 + Π3 − 2κ )Π3 + κx κy (κy − κx )
>0
erT (2Π3 − κx )(2Π3 − κy )

by assumption 1 and 2.

Proof of Proposition 3
Result (i): In the text, we have already shown that protecting all by patent is dominated by the
optimal mix use of a patent and secrecy. Thus, we only need to show the existence of T

Com

.

Note that ViM |T =0 < ViS , limT →∞ ViM > ViS by assumption 3 and ViM is strictly increasing in T .
Therefore, there exists a unique T
T
Result (ii): Given T

Com

Com

Com

∈ (0, ∞) which solves ViM = ViS . More specifically,

(1 − µ y|x )(Π1 − Π3 − κy )
1
.
= ln
r (1 − µ xy )(Π1 − Π3 − κ ) − f

, note that as κy increases and κx decreases by the same amount, µ y|x

increases. Thus, the numerator of the log on RHS decreases while all other things remain constant,
which causes T

Com

to decrease.

Proof of Proposition 4
For the simple innovation, the innovator’s optimal payoff is max{ViP ,ViS }. Since ViP |T =0 < ViS ,
limT →∞ ViP > ViS by assumption 3 and ViP is strictly increasing in T , there exists a unique T

41

Sim

∈

(0, ∞) which solves ViP = ViS , and
T

Sim

=

1
Π1 − Π3
ln
.
xy
r (1 − µ )(Π1 − Π3 − κ ) − f

Hence, it follows that the optimal protection mode is patenting if T < T
secrets otherwise. Taking away T
T

Sim

−T

Com

Com

Sim

and relying on trade

, which was obtained in Proposition 3, from T

=

Sim

, we have

1
Π1 − Π3
.
ln
r (1 − µ y|x )(Π1 − Π3 − κy )

Then, since Π1 − Π3 > (1 − µ y|x )(Π1 − Π3 − κy ), it follows T

Sim

>T

Com

.

Proof of Lemma 4
Let us start from the case N = 3. Once we rule out the price structure in which the inventor
charges higher than imitations costs for all components by the same logic in Lemma 3, equilibrium
candidate price structures fall into one of the following three categories: trying to sell (1) one
2t
3t
component (for example, p1t
ib ≤ κ1 , pib > κ2 and pib > κ3 ), (2) two components (for example,
2t
3t
1t
2t
p1t
ib ≤ κ1 , pib ≤ κ2 and pib > κ3 ) and (3) all three components (for example, pib ≤ κ1 , pib ≤ κ2

and p3t
ib ≤ κ3 ) for b ∈ { j, k} given that no adoption has occurred. Note that (1) is analogous to the
2 component model because once an imitator adopts a component, they enter a subgame in which
the remaining two components are sold in the all-or-nothing adoption fashion as in Proposition
2. Since we showed that in Proposition 2 that the step-by-step adoption sequence in the two
component model is less profitable than the all-or-nothing adoption sequence, the inventor prefers
price structure (3) to (1). Also, with a very similar reason, (3) will be chosen over (2).
Applying this logic recursively, it is straightforward to see that for all N > 3, the inventor will
sell knowledge in all-or-nothing fashion by setting pnt
ib = κn for all n ∈ {1, . . . , N} and b ∈ { j, k}.
Hence, slight modification of Proposition 2 allows us to have Lemma 4 as stated in the body.

42

Proof of Lemma 5
Define components kept by secrecy in a patent-secret mix as NS . Then, NS should be an element
in 2{1,...,N} , where 2{1,...,N} is a power set of {1, . . . , N}, excluding two extreme protection modes:
patenting all (NS = 0)
/ and keeping all-by-secrecy (NS = {1, . . . , N}). We know that if all components in NS are not adopted by the imitators until T , the inventor’s discounted expected payoff
starting from T is given by
µ NS Π1 + (1 − µ NS )(Π3 + κNS )
where κNS = ∑n∈NS κn , µ NS = 1/(r + 2λ NS ) and where and λ NS = r (Π3 − κNS )/κNS from the
−r (T −t ) κ for all n ∈ N by
Lemma 4. Also, note that it is optimal for the inventor to choose pnt
n
S
ib > e

the same logic in Proposition 2. Then, the optimal patent-secret mix is a solution to the following
problem:
max

NS ∈2{1,...,N} \{{1,...,N},0}
/

(1 − e−rT )Π1 + e−rT µ NS Π1 + (1 − µ NS )(Π3 + κNS ) − f .

It is easy to see that the solution NS∗ maximizes κNS , and so NS∗ = {2, . . . , N}, i.e., it is optimal
patenting only component 1 least costly to imitate and keeping all the other components as trade
secrets.

Proof of Proposition 6
As the proof for the point (i) is exactly the same as the one in Proposition 3, here only the proof
for the point (ii) is provided. When κ1 = κ2 = . . . = κN = κ/N,
T

Com

=

1 (1 − µ −1 )(Π1 − Π3 − (1 − N1 )κ )
ln
.
r
(1 − µ )(Π1 − Π3 − κ ) − f

Com
Only the numerator in the log is a function of N. Since µ −1 increases in N, T
gets shorter as

N increases.

43

Proof of Proposition 7
We will first show (α ∗ , ρ ∗ ) is in the interior of (0, 1] × [0, 1/2], which implies each area is not an
empty set, and then investigate equilibrium protection mode in each area in turn.
Step 1: We show that (α ∗ , ρ ∗ ) is in the interior of [0, 1] × [1/2, 1]. Since α ∗ = (ViS −
Π3 )/(Π1 − Π3 ) ∈ (0, 1) is obvious, it is sufficient to show that ρ ∗ = ρ˜ (α ∗ ) = ρˆ (α ∗ ) =
(ViS −V 1 )(Π1 −Π3 )
∈ (1/2, 1). First, ρ ∗ < 1 since (ViS − Π3 )(Π1 −V 1 ) − (ViS −V 1 )(Π1 − Π3 ) =
(ViS −Π3 )(Π1 −V 1 )
(Π1 −ViS )(V 1 − Π3 ) > 0. Next, observe ρ ∗ > 1/2 is equivalent to

2(ViS −V 1 )(Π1 − Π3 ) − (ViS − Π3 )(Π1 −V 1 ) > 0.
After some algebra, one can easily verify that the above inequality is equivalent to the following
condition:

(Π1 − Π3 )(ViS −V 1 ) − (Π1 −ViS )(V 1 − Π3 ) > 0.
Since Π1 − Π3 > Π1 −ViS , it remains to show ViS −V 1 > V 1 − Π3 . Now to conclude, note that
ViS −V 1 − (V 1 − Π3 ) =

( µ 2 − 2µ 1 )(Π1 − Π3 ) − ( µ 2 − µ 1 )κ =
(Π1 − 3Π3 )κ 2
(4Π3 − κ )(2Π3 − κ )

> 0.

where the second equality follows from replacing µ 2 and µ 1 with the expressions given in the
body and rearranging terms.
Step 2: Patterns of optimal protection in each area are as follows.
Area (i): For all T , ViS > ViP and ViS > ViM given α ≤ α ∗ and ρ < ρˆ (α ). Hence, S is trivially
optimal for the inventor for all T .
Area (iv): Given ρ˜ (α ) ≤ ρ ≤ 1, ViM > ViP for all T given α, which implies P is never optimal.
Also, since we consider the area ρˆ (α ) < ρ ≤ 1 where ViS > ViM |T =0 but ViS < limT →∞ ViM given
α, there exists a threshold T

M

such that ViS = ViM |T =T¯ M by the Intermediate Value Theorem.

Moreover, since ViM is monotonically increasing in T the threshold is unique. Thus, the optimal
protection mode is S for T ≤ T

M

M

and M for T > T .

44

Area (ii) and (iii): First, consider the following subarea where ρ < ρˆ (α ) and α > α ∗ . Here,
applying a similar logic used in a proof for area (iv), we can show that the optimal protection mode
P

P

P

is S for T ≤ T and M for T > T where T is such that .ViS = ViP |T =T¯ P . Next, consider the rest
of the area. Here, for any (α, ρ ) limT →∞ ViP > limT →∞ ViM > ViS , and recall that ViS > ViM |T =0 >
ViP |T =0 . Therefore, by continuity of payoff functions we know that S is optimal when T is small
and P is optimal when T is large. One remaining question is whether there are some intermediate
P

T values under which M is optimal. The previous statement is true if and only if T > T
T

P

and T

M

M

where

are defined as above. That is because in that case as T increases ViM surpasses ViS
P

first while ViP is still less than ViS even if it is overtaken by ViP eventually. To see T > T

M

is

equivalent to ρ > ρ ∗ , observe the following:
P

T >T

M

α (Π1 −Π3 )
ρα (Π1 −V 1 )
1 ln
>
r
α (Π1 −Π3 )−(ViS −Π3 )
ρα (Π1 −V 1 )−(ViS −V 1 )
α (Π1 −Π3 )−(ViS −Π3 )
ρα (Π1 −V 1 )−(ViS −V 1 )
<
α (Π1 −Π3 )
ρα (Π1 −V 1 )
1
S
(V −V )(Π1 −Π3 )
ρ > iS
= ρ∗
(Vi −Π3 )(Π1 −V 1 )

⇐⇒ 1r ln
⇐⇒
⇐⇒

.

In sum, in area (iii) the optimal protection mode is S if T is small, M if T is intermediate and P
otherwise whereas S if T is small and P otherwise in area (ii).

45

CHAPTER 2
WELFARE EFFECTS OF CERTIFICATION UNDER LATENT ADVERSE SELECTION
(WITH ANTHONY CREANE AND THOMAS D. JEITSCHKO)

2.1

Introduction

Asymmetric information is a classic example of market failure that undermines the efficiency associated with perfectly competitive market outcomes—the “lemons” market (Akerlof, 1970). This
issue has recently regained more detailed scrutiny with the rise of “organic,” “fair trade,” “local,”
“sustainable” etc. products. In markets for such products, the concept of product quality in consumer preferences has extended to include the process of production and distribution. When such
properties are claimed by firms, along with specific corporate philosophies or policies, it is very
hard or impossible for consumers to ascertain the quality or the process of production of those
products even after their consumption experience in many cases. Due to the aforementioned credence good nature,1 quality assurance mechanisms, such as offering warranties (e.g., see Spence,
1977) and building up a reputation (e.g., see Klein and Leffler, 1981; and Shapiro, 1983), are not
expected to function well to address the information problem in those markets.2 Accordingly,
credible (third-party) certification is often considered the only potential solution for overcoming
informational asymmetries. In addition, while “quality” in the traditional, lemon’s sense (vertical
differentiation) may be hard to measure, consumers’ valuation of the production process, as in
the examples above, makes certification more easily quantifiable. Although the degree to which
such claims are backed up by formal certification or regulated by law varies, various certification
schemes are in use in many marketplaces.
In spite of the seeming attractiveness of credible certification, it is still important to substantiate
whether introducing certification increases or decreases social welfare through a formal analysis,
1 For

more detailed definition and discussion of credence goods see, for example, Darby and
Karni (1973) and Dulleck and Kerschbamer (2006).
2 See Bonroy and Constantatos (2015) and Dranove and Jin (2010) for detailed discussion on
this point.

46

especially considering its popularity. To this end, in this paper we consider markets in which
goods have unobservable characteristics that consumers value. We analyze the market equilibrium
when a technology is available that credibly verifies the relevant attributes of products and contrast
this with the equilibrium without certification. We assume that the certification technology could
be costly and imperfect. One effect the certification technology has is to potentially create two
markets (market segmentation)—one with certified products and the other without—while without
certification there is only one pooled market. As a result, one has to consider both the firm’s
decision as to whether to seek certification as well as the consumer’s choice as to which market to
patronize.
We find that certification may be welfare-reducing when the certification technology is costly
or imperfect. This implies that even when the certification technology is perfectly accurate—and
therefore is able to resolve informational asymmetries in the market place—overall welfare may
decrease due to the possibility of certification when such certification is costly. Similarly, even
when the certification technology is costless, welfare may decrease when such certification is quite
imperfect.
The underlying reason for this is that while the certification process admits better information
and therefore increases efficiency by reallocating high quality goods to consumers with relatively
higher valuations, it also brings a negative impact on the average quality of non-certified goods.
The latter effect can push some high quality producers out of the non-certified market or the entire
group (a market collapse). In contrast, absent any certification, the increased presence of products
with desirable characteristics provide a sufficiently strong positive externality to sustain an equilibrium that entails a larger number of high-quality products. When the certification technology
is relatively costly or imperfect, only a small portion of high type producers are present in in the
certified market in equilibrium, and this may still cause the non-certified market to collapse, which
results in a quite low welfare level. Most of these results are not a result from the direct welfare
cost of certification, but rather from certification’s effect on the market(s).

47

2.2

Related Literature

This paper contributes to the literature on certification (or labeling), or more broadly, quality disclosure. There are two streams of the certification literature. The first strand focuses on the role
of strategic certification intermediaries. Some studies aim to investigate how competition between
certifiers affects market outcomes including optimal pricing schemes, the amount of information
transmitted to consumers or quality provision (e.g., Albano and Lizzeri, 2001; Fischer and Lyon,
2014; Hvide, 2009; and Lizzeri, 1999). Some other papers (e.g., Benabou and Laroque, 1992)
explore whether a reputation concern can mitigate the incentive problems of certification intermediaries. Some of these papers have something in common with ours in that they show a potential
source of inefficiency associated with certification. However, while they emphasize the role of
strategic certifiers as a source of inefficiency, we are more interested in sellers’ incentives for getting certified and their incidence on markets taking certification environments as given.
The other strand of this literature focuses on a seller’s incentive for quality disclosure. Since
the “unraveling result”3 was presented by Grossman (1981) and Milgrom (1981), many subsequent
studies have explored the subject of when and why unraveling fails to hold (e.g., see De and Nabar,
1991; Guo and Zhao, 2009; Grubb, 2011; Hotz and Xiao, 2013; Shavell, 1994; and Viscusi, 1978).
Closely related with such work, several authors also studied whether mandatory disclosure laws
enhance or hinder efficiency when voluntary disclosure fails to occur (e.g., see Bar-Isaac et al.,
2012; Board, 2009; Gavazza and Lizzeri, 2007; Harbaugh et al., 2011; Jovanovic, 1982; and
Matthews and Postlewaite, 1985). Though these papers share several features with us, the most
important difference in ours is that we compare the situations with and without certification, rather
than comparing a situation in which sellers voluntarily choose to get certified with another situation
in which sellers must get certified to sell.4
3 The

unraveling result refers to the situation in which each seller voluntarily discloses her
quality to consumers for differentiating herself from other sellers with a lower quality if quality
disclosure is credible and costless.
4 For more examples, see cited papers in Dranove and Jin (2010).

48

There are only a relatively small number of studies that focus on welfare effects of certification.
The authors of these studies point that welfare may decrease with the availability of certification
mostly in the context of Eco-labeling, as we do in this paper. Among these studies, Baltzer (2012),
Bonroy and Constantatos (2008) and Zago and Pick (2004) differ from ours in that the underlying
mechanisms that derive welfare decreasing results in their work are other than exacerbated adverse
selection problem by certification. The most closely related papers to this one are Mason and
Sterbenz, 1994 and Mason, 2011. Taking into account sellers’ incentive for opting for certification,
the above two papers show that certification can aggravate adverse selection, and thus certification
maybe welfare reducing. However, the main drivers of the exacerbated adverse selection problem
in their approaches and our approach are different: unlike they are more focused on the mimicking
incentives of low type sellers, we are more interested when certification causes the collapse of the
non-certified market. More importantly, we assume a downward sloping demand curve in order to
seriously consider allocative efficiency, associated with certification, which is an important factor
when examining welfare effects of certification.
The remainder of this paper proceeds as follows. Section 2 describes the base model. To
explain the value of information, in section 3, we compare two benchmarks: full information
and no information case. We derive a certification equilibrium and conduct a welfare analysis in
Section 4. Finally, Section 5 concludes. All omitted proofs are in the appendix.

2.3

The Base Model

We consider markets in which there is asymmetric information between consumers and firms about
the product quality.

2.3.1

Consumers

In order to introduce certification (which implies a segmentation of the market and hence sorting
of demand according to distinct quality/grades) later in the model, a richer modeling structure for

49

demand is needed. Following Mussa and Rosen (1978), we adopt the vertical differentiation model
of gradations g with a continuum of consumer types each of whom has a unit demand. Consumers
are of total mass 1 and distributed on [θ , θ¯ ) according to the strictly increasing cdf, F (θ ) . Then,
a type-θ consumer’s net utility, when paying price p, is given by
U (g, p|θ ) = θ g − p,
while a consumer gets 0 when buying nothing. All consumers are price takers.

2.3.2

Firms

On the supply side suppose there is a mass (measure) n of firms which are also all price takers. A
firm produces (or serves) either high (g)
¯ or low (g) quality product (or service). Since we do not
focus on a moral hazard problem but on an adverse selection problem, we assume these quality
levels are exogenouly given. A proportion of high grade producers is indexed by γ0 ∈ (0, 1), and
γ0 n firms can produce a unit of high quality at cost of c.
¯ The rest of producers, (1 − γ0 )n, can
produce a unit of low quality at cost of c. Each seller knows the true quality of her own product,
but consumers do not. In order to make the analysis of the lower grade market simpler when a
segmentation of the market arises (under full information or with certification), let us normalize
g = c = 0.

2.3.3

Two Benchmarks

In the rest of this section, we study the equilibrium configurations when there is no certification
technology (No Information/No Certification) and when the quality of product is observable to
consumers (Full Information) in turn as two benchmark cases before delving into equilibrium
configurations with certification. By comparing welfare under the two benchmarks, we will explain
why certification can potentially be valuable to the society.

50

2.3.3.1

No Information (No Certification) Equilibrium

Without any quality certification the market is subject to the law of one price, therefore only overall
market demand for different compositions of quality on offer is required. Consider the supply
side first. No high quality goods are offered if the price is below their production costs. More
specifically,

qS ( p) =





κ (1 − γ0 )n, κ ∈ [0, 1]







(1 − γ )n
0




(1 − γ0 + κγ0 )n, κ ∈ [0, 1]







n

if p = 0,
if p ∈ (0, c¯),
(2.1)
if p = c,
¯
if p > c.
¯

Since consumers rationally anticipate the supply schedules, the demand structure is given by



1 − F ( p/γ g¯) if p ≥ c,
¯
D
q ( p) =


0
if p < c,
¯
in which γ is beliefs of consumers, and so γ g¯ represents ex-ante average quality in the market. To
make the analysis non-trivial, we take the following assumption.
Assumption 1: 0 < c¯ < γ0 g¯θ¯
Assumption 1 specifies cost structure under which adverse selection need not necessarily happen. The last inequality implies that at least some trade of high quality given the prior belief is
efficient. The potential for adverse selection comes from the first inequality, which implies that all
high producers are driven out if consumers believe that there are no high type in the market. This
is because in that case even for the highest type the consumer valuation would be lower than c.
¯
Since we are interested in inquiring about if certification can decrease social welfare, we further
assume that the number of firms is relatively small so that there is asymmetric information in the
market but not necessarily an adverse selection problem.
Assumption 2: n ≤ 1 − F (c/γ
¯ 0 g¯)

51

Figure 2.1: No Information Equilibrium

Note: The demand curves need not necessarily be linear.
Under Assumption 2, in any equilibrium in which a positive quantity is traded (at an equilibrium price greater than c),
¯ rational expectation implies that consumer beliefs must be the same as
prior beliefs, i.e., γ = γ0 , and all n firms serve consumers.5 As seen in Figure 1, although there
exists a trivial one in which no transaction occurs (pNI = 0 and qNI = 0) as well in general, we
select the equilibrium with the higher quantity and the higher level of welfare.
Thus, in equilibrium
qNI = n;

pNI = γ0 gH
¯ (1 − n)

NI = pNI − c;
πH
¯

πLNI = pNI

where H (τ ) ≡ F −1 (τ ) a quantile function defined on [0, 1).6
5 In

principle, γ could differ from γ0 if n is sufficiently large because there may be some high
quality producers staying out of the market in equilibrium when the market price is equal to c.
¯ See
Creane and Jeitschko (2016) for more on this point.
6 qNI = n, or θ NI = H (1 − n).

52

2.3.3.2

Full Information Equilibrium

Under full information assumption, firms will be sorted into two distinct groups. And thus, we
have to examine the two separate markets. In the low grade market (market L), it is optimal for low
grade producers to produce as long as the market price (pL ) is greater than their production cost
c = 0. All low quality firms are indifferent between producing or not when pL = 0. Therefore, the
supply correspondence in market L is given by,



κ (1 − γ0 )n, κ ∈ [0, 1]
S
qL ( pL |n) =


(1 − γ0 )n

if pL = 0,
if pL > 0.

However, it is optimal for any type of consumer, θ , not to purchase in this market if pL is positive
because U g, pL |θ = −pL < 0 for all consumer types. This implies the demand correspondence
in market L is given by,
qD
L ( pL |n) =




∈ [0, 1]

if pL = 0,



0

if pL > 0.

In principle, the equilibrium quantity can vary though the unique equilibrium price is 0 (pL = 0).
FI
In all such situations, we assume there is no transaction by setting qFI
L = 0 and pL = 0. Note that

the normalization assumption in market L (g = c = 0) results in the normalization of surplus in
that market as well. 7
Now consider the high grade market (market H). The supply schedule can be derived in a
7 This

do not harm generality much for total welfare analysis. However, when it comes to consumer surplus, this simplifying assumption makes a difference. With the normalization assumption, the surplus of the marginal consumer in market H is 0. In contrast, if we assume gθ¯ > c, so
that if some lower quality products can be served in an equilibrium, the marginal consumer of the
higher grade market should get some surplus since now she has an outside option of buying from
the lower grade market. Therefore, consumer surplus from the high grade market may decrease
with the normalization in market L.

53

Figure 2.2: Full Information Equilibrium

(a) High Grade Goods Market

(b) Low Grade Goods Market

Note: The demand functions need not necessarily be linear.
similar way as we did for market L:

qSH ( pH |n) =





0





if pH < c,
¯

κγ0 n, κ ∈ [0, 1]






γ0 n

if pH = c,
¯
if pH > c.
¯

In order to derive demand, note that a type-θ consumer would purchase given pH if and only if
U (g,
¯ pH |θ ) = gθ
¯ − pH ≥ 0.
If we define the marginal consumer who is indifferent between buying or not as θHFI ≡ pH /g,
¯
consumers with θ ≥ θHFI buy high grade goods, while the remainder (θ < θHFI ) do not. The
demand schedule is thus given by

qD
H ( pH ) =




1 − F ( pH /g¯)

if pH ∈ [0, g¯θ¯ ],



0

if pH > g¯θ¯ .

54

Assumption 2 implies that under full information all high types serve in the market,8 and thus the
equilibrium outcome can be summarized as follows,
qFI
H = γ0 n;

qFI
L = 0;

pFI
¯ (1 − γ0 n) > c;
¯
H = gH

pFI
L = 0;

FI = pFI − c;
πH
¯
H

πLFI = 0,

FI and π FI stand for equilibrium profits of high and low quality firms under full informawhere πH
L

tion respectively.9

2.3.3.3

Welfare Comparison

Having the equilibrium configurations in the two benchmarks, we show that information increases
social welfare. We take social welfare as the sum of firm profits and consumer surplus. Since
price paid by consumers is just transferred to firms, total welfare under no information and full
information can be written as gross consumer benefits net of production costs:
W NI =
W FI =

θ¯
θ NI
θ¯
FI
θH

γ0 gθ
¯ dF (θ ) − γ0 nc,
¯
gθ
¯ dF (θ ) − γ0 nc.
¯

Proposition 1 (Information Increases Welfare) Welfare is strictly greater under full information
than under no information regardless of a proportion of high grade producers, i.e.,
W FI > W NI , ∀γ0 ∈ (0, 1).

8 Mathematically,

this is because Assumption 2 implies n < 1 − F (c/γ
¯ 0 g¯) <
=
gH
¯
(
1
−
γ
n
)
≥
c.
¯
(1/γ0 ) [1 − F (c/
¯ g¯)] and thus, n ≤ (1/γ0 ) [1 − F (c/
¯ g¯)] ⇐⇒ pFI
0
H
9 qFI = γ n, or θ FI = H (1 − γ n).
0
0
H

55

Proof: Subtracting W NI from W FI and rewriting it shows a trade-off between the two situations as
follows:
W FI −W NI

θ¯

=
=

gθ
¯ dF (θ ) −

FI
θH
θ¯

FI
θH

θ¯
θ NI

gθ
¯ dF (θ ) −

γ0 gθ
¯ dF (θ )

θ¯
FI
θH

FI
θH
γ0 gθ
¯ dF (θ )
θ NI

γ0 gθ
¯ dF (θ ) −

Gains from reallocating

Losses from not served

high quality to θ ≥ θHFI

consumers (θ NI < θ < θHFI )

.

θ

Now we show that the above expression is always greater than 0. Note that θ 2 θ dF (θ ) =
1
θ

τ

θ2 F (θ2 ) − θ1 F (θ1 ) − θ 2 F (θ )dθ = τ12 H (τ )dτ for any arbitrary θ1 < θ2 and corresponding
1
τ1 < τ2 . The first equality follows from integration by parts and the second from the definition of
H (·). This implies,
W FI −W NI = g¯ (1 − γ0 )

1
FI
τH

H (θ )dθ − γ0

FI
τH
H (θ )dθ
τ NI

,
τ

FI ) and θ NI = H (τ NI ). Moreover, since (τ − τ )H (τ ) < 2 H (τ )dτ < (τ −
where θHFI = H (τH
2
1
1
2
τ1

τ1 )H (τ2 ) for any increasing H (·)10 , we have
FI )θ FI − γ (τ FI − τ NI )θ FI
W FI −W NI > g¯ (1 − γ0 )(1 − τH
0 H
H
H

= g¯ (1 − γ0 )γ0 nθHFI − γ0 (1 − γ0 )nθHFI
= 0.

Proposition 1 states that restoring full information improves welfare. The intuition behind this
result is straightforward. Even though the market is not subject to adverse selection under no
information, the market still suffers from an allocation inefficiency associated with asymmetric
information. This is because those with relatively higher valuations may obtain a low quality
10 Since F (·)

is strictly increasing, so is H (·).

56

Figure 2.3: Welfare Comparison between NI and FI Benchmarks

Note: The demand function needs not necessarily be linear.
product and vice versa. When we move from the no information case to the full information case,
while the aggregate production costs remain the same, allocative efficiency is restored because now
high quality goods are allocated only to the group of consumers with relatively higher valuations.
To understand this point further, refer to Figure 3. The measure of γ0 n consumers above θHFI
gain at least g¯(1 − γ0 )θHFI each from the reallocation of goods, while the measure of (1 − γ0 )n
consumers between θ NI and θHFI lose at most γ0 gθ
¯ HFI . Hence, welfare under full information is
always greater.

2.4

Certification Equilibrium

The previous section shows that information increases social welfare. One immediate implication
of the result is that perfect and cost-less certification, which coincides with the full information
benchmark, is welfare enhancing. However, what if the certification technology is costly or imperfect? Are there any cases in which welfare with certification is even lower than that under no

57

information? In order to answer these questions, we derive a price equilibrium given certification
decisions of firms and then characterize a certification equilibrium. The welfare analysis follows
next and shows that certification may actually decrease social welfare.
For simplicity and tractability of the model, we employ further assumptions. Throughout this
section, let us assume that consumers are uniformly distributed on the line segment [0, 1], i.e.,
F (θ ) = θ
on [0, 1], and H (τ ) = τ. Also, for simplicity we normalize g¯ = 1. To reduce the equilibrium
constellations to consider, we also restrict our interests to the case in which Assumption 2 binds
and denote such measure of producers by n.
¯ With the uniform distribution assumption, note that
n¯ = 1 − c/γ
¯ 0.

2.4.1

Certification Environment

We consider a certification market in which all certifiers are homogeneous in their certification
costs and accuracy of certification test. More specifically, each certifier needs to incur the same
fixed cost of certification, z ≥ 0. We further assume that the certification market is perfectly competitive so that the equilibrium price for certification is always equal to z.
Certification technology may be imperfect in the following sense. Let g denote the grade
discovered by the certification test. Then,
Pr {g = g|g
¯ = g}
¯ = x;

Pr g = g|g
¯ = g = 0;

Pr g = g|g = g¯ = 1 − x;

Pr g = g|g = g = 1,

where 0 < x ≤ 1. So, the test is informative in the sense that the test perfectly weeds low grades
out and identifies a high type with some precision. Put differently, the certification test commits
a type-I error with probability 1 − x but not a type-II error. However, x can be lower than 1/2.
Certification is imperfect unless x = 1.
If the test reveals that a seller is of high type (g˜ = g),
¯ she gets certified (Crt). Otherwise
(g˜ = 0), the seller can only sell in the pool of sellers who did not apply for certification and applied

58

but failed to get certification (NC). Put differently, consumers can only condition on whether the
good is certified or not certified. Finally, the sellers cannot shop around for certification, and so
the first certification result sticks. This assumption precludes the possibility that the sellers use an
imperfect and cost-less certification repeatedly in such a way that full information outcome may
be replicated.
We consider the following two-period model when certification is an available option for sellers:
Period 1 (Certification Decision): In the first period each seller makes a decision whether to opt
for certification test or not.
Period 2 (Walrasian Price Formation): In the second period, with there being asymmetric information partially resolved through the certification (given sellers’ certification decisions and outcomes) process, trade takes place in which Walrasian mechanism forms prices in both markets:
certified (pCrt ) and non-certified (pNC ) market. Depending on offered prices in each market, high
and low grade producers decide whether to produce at production costs of c¯ and 0 respectively
(produce to demand) or to shut down, and consumers choose to buy in the Crt market or in the NC
market or not to buy.
In fact, firms need to go through a certification process before producing in many cases. For
example, to get the USDA Organic label, a farm must develop and implement an organic system
plan first and then hire a USDA-accredited certifying agent to get inspected before producing. Note
that in this case the certification cost, z, can include all expenses for adopting the organic system
as well as an agency fee.11

2.4.2

Price Equilibrium

Given the assumptions on certification environment, there is no chance that the low type is identified as the high type by the test. Thus, without loss of generality, we can assume that the low
11 See

https://www.ams.usda.gov/services/organic-certification/becoming-certified (last retrieved on May 1, 2017).

59

type never seeks certification, and the price equilibrium constellations depend only on the high
producers’ certification decisions. Let α denote the proportion of high type sellers who choose
to opt for certification. Then, there are three cases to consider: a pooling equilibrium (α = 0), a
semi-separating equilibrium (α ∈ (0, 1)) and a fully separating equilibrium (α = 1). In a pooling
equilibrium in which no high type seeks certification (α = 0), the resulting outcomes coincide with
the no information benchmarks, and there is nothing to be analyzed further. In a semi-separating
equilibrium or a fully separating one, certification environment gives rise to the following two interesting features. First, third party certification results in market segmentation. Second, if a large
portion of high grade firms congregate in one of the two markets, this may give rise to the collapse
of the other market due to the intensified adverse selection even if there is no adverse selection
problem in the beginning when all firms are pooled together in one market.
In order to derive a price equilibrium for each α ∈ (0, 1], write down the supply schedules
in each market which are similar to (2.1) with the modification of population of high and low
grade producers. In the Crt market there are αxγ0 n¯ measure of high producers deciding whether to
produce or not. Similarly, in the NC market, the rest of producers, the pool of (1 − αx)γ0 n¯ measure
of high types and (1 − γ0 )n¯ of low types, make the same decisions. And thus, supply schedules in
the two markets are given by

qCrt/S ( pCrt ) =

qNC/S ( pNC ) =




0





if pCrt ∈ [0, c¯),

ραxγ0 n,
¯ ρ ∈ [0, 1]






αxγ0 n¯

if pCrt = c,
¯
if pCrt > c;
¯





κ (1 − γ0 )n,
¯ κ ∈ [0, 1]







(1 − γ )n¯
0




¯ κ ∈ [0, 1]
[(1 − γ0 ) + κ (1 − αx) γ0 ] n,







[(1 − γ0 ) + (1 − αx) γ0 ] n¯

60

(2.2)

if pNC = 0,
if pNC ∈ (0, c¯),
(2.3)
if pNC

= c,
¯

if pNC > c.
¯

Since consumers rationally expect the supply schedules, the posterior belief for the Crt market is
given by
γ Crt ≡ γ (g|Crt
¯
)=

αxργ0
=1
αxργ0

as long as pCrt ≥ c¯ and γ (1|Crt ) = 0 otherwise. Note that in the NC market low grade producers
are willing to sell at all positive prices while high types would only do so when pNC ≥ c,
¯ which
implies in any equilibrium associated with some transaction and with shutting down by some firms,
the exiting firms must be of high type. From this argument we know that,
γ NC ≡ γ (g|NC
¯
)=

(1 − αx) γ0
(1 − αx) κγ0
≤
(1 − αx) κγ0 + (1 − γ0 ) (1 − αx) γ0 + (1 − γ0 )

in which κ is 1 for pNC > c,
¯ any number in [0, 1] for pNC = c¯ and 0 for pNC < c.
¯ Thus, γ Crt = 1 >
γ0 ≥ γ NC for all α ∈ (0, 1], which basically means that certification allows high quality producers
to differentiate themselves from low quality producers with some probability while lowering the
average quality in the NC market .
Given these beliefs and two price levels - pCrt and pNC , consumers are divided into three
separate groups depending on the following decision rule: a type-θ consumer



buys in market i if γ i θ − pi ≥ max{γ j θ − p j , 0},


does not buy

if 0 ≥ max{γ Crt θ

− pCrt , γ NC θ

(2.4)

− pNC },

where i, j ∈ {Crt, NC} and i = j.
After defining a price equilibrium formally, we will derive some preliminary results.
Definition 1 A price equilibrium in the second period, given high types’ certification decision (α)
and certification precision (x), is a quadruple of ( pCrt , γ Crt , pNC , γ NC ) such that
1. Walrasian Market Clearing in Both Markets: given pCrt and pNC , the quantities supplied,
determined by (2.2) and (2.3), equal to the quantities demanded, derived by (3.4) and beliefs:
γ Crt and γ NC , in each market.

61

2. Consistent Beliefs: there exists ρ (and κ ) ∈ [0, 1] describing high grade firms’ production
decisions in (2.2) (and (2.3)) which implies equilibrium quantity in the certified market (and
non-certified market) and γ Crt (and γ NC ).

Note that there could be two potential types of price equilibria: with only one active market
and the collapse of the other, and with two active markets. Let’s denote the former one as “1market” equilibrium and the latter as “2-market.” To verify the equilibrium characteristics and its
uniqueness, first we derive some conditions that equilibrium prices should meet.
Lemma 1 In any price equilibrium in which some high quality goods are traded,
1. the equilibrium price in the Crt market is always greater than that in the NC market (i.e.,
pCrt > pNC );
2. if some high quality producers shut down, the equilibrium price for that market must be
equal to c.
¯

There are several important implications of Lemma 1. First, since pCrt is higher than pNC in
equilibrium (and γ Crt > γ NC ), there is no 1-market equilibrium in which transaction occurs only
in the NC market. This implies that 1-market and 2-market equilibria have to result in the price
structures pCrt ≥ c¯ > pNC = 0 and pCrt > pNC ≥ c¯ respectively.
Second, there exists a marginal consumer who is indifferent between buying in Crt market and
NC market. The cutoff consumer’s type θ Crt is derived from the following condition:
γ Crt θ Crt − pCrt = γ NC θ Crt − pNC .

(2.5)

Without considering the possibility of not buying, all consumers with θ ≥ θ Crt are affiliated with
the Crt market while the remainder belongs to the NC market.
Third, all high types with a certification should serve (ρ = 1) in both 1-market and 2-market
equilibria. To see this, recall that the equilibrium price under no information with n¯ is equal to

62

c¯ implying the margin is 0 for the high grade producers. Now, there are only αxγ0 n¯ number of
producers in the Crt market, and they serve consumers with relatively high valuations from the
top (i.e., θ ∈ [θ Crt , 1]). Also, note that each consumer’s willingness to pay is also higher than that
under no information because γ Crt > γ0 . Therefore, it must be the case, pCrt > c¯ and ρ = 1.
Fourth, in any 2-market equilibria, pNC = c¯ and κ < 1. In no information benchmark, θ NI type consumer’s willingness to pay, γ0 θ NI , was equal to c.
¯ Now since γ NC < γ0 , his willingness to
pay is lower than c,
¯ and so some high types have to shut down. It is optimal that some high types
choose to sell and the others do not only when those two actions yield the same payoffs of 0 to
them, which implies pNC = c.
¯
If both type of equilibria coexist, even though the previous claim implies that given α, x and
n,
¯ social welfare from the Crt market is invariant with the equilibrium quantity equal to αxγ0 n,
¯
we may have multiple price equilibria due to the nature of the NC market. Hence, equilibrium
selection is necessary to make a prediction about optimal certification decisions. To be consistent
with the no information benchmark and as we want to show welfare under certification might be
lower than under no information, we select the most favorable equilibrium in terms of welfare.
Note that if there exist both 1-market and 2-market equilibrium, the welfare maximizing one is the
2-market equilibrium. Otherwise, the welfare maximizing equilibrium is the 1-market one. The
lemma below characterizes such price equilibria.
Lemma 2 The price equilibrium constellations are as follows.
1. If the production cost of high quality good is relatively high (i.e., c¯ ∈ [γo2 , γ0 )), the NC market
always collapses (κ = 0), so a 1-market equilibrium is the unique welfare maximizing price
equilibrium for all α ∈ (0, 1] and x ∈ (0, 1]; and
2. if the production cost of high quality good is relatively low (i.e., c¯ ∈ (0, γo2 )), there exists a
threshold ξ such that
a) for αx ∈ (ξ , 1], the NC market always collapses, so a 1-market equilibrium is the

63

unique welfare maximizing price equilibrium; and
b) for αx ∈ (0, ξ ], a 2-market equilibrium with κ (α, x) ∈ (0, 1) is the unique welfare
maximizing price equilibrium where κ (α, x) is the larger root solving pNC = c.
¯

The lemma states conditions under which we have 1-market and 2-market equilibria. In order to grasp a deeper understanding, first we explain two different negative effects of high type’s
departure, from the pool of entire producers to the Crt market, on equilibrium price in the NC
market. Not only it causes the average quality in the NC market (γ NC ) to fall (rotation in the NC
market demand curve), but also it skims consumers with highest valuations from the pool (left shift
of the NC market demand curve). However, that does not necessarily result in the collapse of the
NC market if some portion of high type shut down. With a downward sloping demand curve, this
exit by high grade producers has two countervailing effects on equilibrium price down in the NC
market: pNC might increase due to the decrease in quantity supplied (move along the NC market
demand curve) but decrease due to the further drop in the average quality in the NC market (further rotation in the NC market demand curve). If the former effect dominates the latter, 2-market
equilibria may emerge. Therefore, as stated in Lemma 2 price equilibrium configurations crucially
depend on the magnitude of c.
¯ 2-market equilibria do not emerge at all when c¯ is relatively large
(or n¯ relatively small) because, roughly speaking, the smaller n,
¯ the less potential for such positive
(quantity) effects on pNC .
Given any certification decisions (characterized by α), the corresponding welfare-maximizing
price equilibrium is unique and well defined. Corollary 1 formally summarizes price equilibrium
outcomes.
Corollary 1 The welfare maximizing price equilibrium outcomes are as follows (given α and x).

64

1. When only 1-market equilibria exist,
γ Crt = 1;

γ NC = 0;

θ Crt = 1 − αxγ0 n;
¯

θ NC = θ Crt ;

pCrt = θ Crt ;

pNC = 0;

Crt = pCrt − c;
πH
¯

NC = π NC = 0.
πH
L

2. When 1-market and 2-market equilibria coexist,
γ Crt = 1;

γ NC = γ NC (κ (α, x)) ;

θ Crt = 1 − αxγ0 n;
¯

θ NC = 1 − [1 − (1 − αx) {1 − κ (α, x)}] n;
¯

pCrt = (1 − γ NC )θ Crt − pNC ;

pNC = γ NC θ NC = c;
¯

Crt = pCrt − c;
πH
¯

NC = 0;
πH

πLNC = pNC ;
where κ (α, x) is defined as in Lemma 2, and θ Crt and θ NC represent the marginal consumer
in the Crt and NC market respectively.

2.4.3

Certification Equilibrium and Welfare Analysis

Assuming all aspects and results of the model up to now are common knowledge to the agents, now
we explore optimal decisions of high types on certification and conduct welfare analysis. As stated
earlier, we are interested in checking whether availability of certification is socially beneficial or
not. In order to highlight the welfare decreasing result without involving 2-market equilibria, first
we consider the case 1 in Lemma 2 where the collapse of the NC market arises whenever some
high types get certification. Next, we confirm that similar results also hold for the other cases in
which 2-market equilibria may emerge.

65

2.4.3.1

Analysis when only 1-market Equilibria Exist

Throughout this subsection, assume that c¯ ∈ [γ02 , γ0 ) is relatively large so that even an arbitrarily
small exit of high quality triggers the collapse of the NC market. In period 1, each high type
will decide whether to get certified or not given a pair of a certification cost and a test precision,

(z, x) ∈ Ω ≡ R+ × (0, 1]. From the two benchmark cases and Corollary 1, given all other agents’
strategies (given α), an individual high type’s payoff of opting for certification is given by
EΠH (Cert|α ) = x {(1 − αxγ0 n¯ ) − c}
¯ −z
Crt only with probability x and 0 with the complementary probability, but pays
because she gets πH

z regardless of the certification outcome. In contrast, the payoff of not getting certified is always
NC = 0.
given by EΠH (No Cert|α ) = 0 regardless of α because in either case she ends up getting πH

The optimal decision rule for a high type is derived by comparing the payoffs with and without
certification. However, since in our model each agent is atomic, technically it is hard for us to
imagine how each individual’s deviation affects the equilibrium prices. To this end, we define the
certification equilibrium as follows which is particularly useful for the analysis associated with
2-market equilibria:
Definition 2 Given (z, x), α ∗ characterizes a certification equilibrium if there is no profitable εdeviation among the high grade producers. A profitable ε-deviation means that there exists an
arbitrarily small positive number ε ≈ 0 such that EΠH (Cert|α ∗ ) > EΠH (No Cert|α ∗ − ε ) and
EΠH (No Cert|α ∗ ) > EΠH (Cert|α ∗ + ε ).
Since here EΠH (Cert|α ) and EΠH (No Cert|α ) are monotone and constant with respect to
α respectively, we have three different certification equilibrium configurations depending on the
costs of certification fixing the certification precision. Given x, when the certification cost is relatively small, a fully separating equilibrium arises (α ∗ = 1). When the certification cost is relatively
high, a pooling equilibrium arises (α ∗ = 0). When the certification cost is intermediate, a semiseparating equilibrium arises with α ∗ ∈ (0, 1) in which all high quality producers are indifferent

66

between getting certified or not, i.e., EΠH (Cert|α ∗ ) = EΠH (No Cert|α ∗ ). The preceding arguments are summarized in the following proposition.
Proposition 2 Suppose only 1-market equilibria exist (i.e., c¯ ∈ [γ02 , γ0 )). Then, given x ∈ (0, 1], in
equilibrium all high types apply for certification (α ∗ = 1) when z is relatively small, only some of
them ( α ∗ ∈ (0, 1)) when z is intermediate and no high types (α ∗ = 0) when z is relatively large.
Specifically,

α∗ =





1





if z ∈ [0, z˜(x)] ,

1 1 − z + c¯
xγ
x

0 n¯





0

if z ∈ (z˜(x), zˆ(x)) ,
if z ∈ [zˆ(x), ∞) ,

where z˜(x) ≡ x {(1 − xγ0 n¯ ) − c}
¯ and zˆ(x) ≡ x (1 − c¯).
As each (z, x) ∈ Ω has a unique certification equilibrium, we can define three mutually exclusive subsets of Ω in which fully separating, semi-separating, and pooling equilibria arise respectively:
ΩFS ≡ {(z, x) ∈ Ω|z ≤ z˜(x) and x ∈ (0, 1]} ,
ΩSS ≡ {(z, x) ∈ Ω|˜z(x) < z < zˆ(x) and x ∈ (0, 1]} ,
ΩP ≡ {(z, x) ∈ Ω|ˆz(x) ≤ z and x ∈ (0, 1]} .
Figure 4 illustrates the three subsets of Ω on the z − x plane. A simple comparative statics analysis
suggests that more high quality producers seek certification, ceteris paribus, as z decreases on
parameter spaces in which a semi-separating equilibrium emerges (i.e., ∀ (z, x) ∈ ΩSS ).
For (z, x) in ΩFS or ΩSS , the welfare function defined as the sum of consumer surplus and
firms’ expected profits net of certification costs can be written as
W Crt (z, x) =

1
θ Crt

θ dθ − α ∗ xγ0 n¯ c¯ − α ∗ γ0 nz
¯

where θ Crt = 1 − α ∗ xγ0 n¯ and α ∗ as defined in Proposition 2. Otherwise, i.e., (z, x) ∈ ΩP ,
W Crt (z, x) = W NI . Depending on the parameter values, we have three qualitatively different re-

67

Figure 2.4: Certification Equilibrium Constellations

Note: Vertical and horizontal axes represent accuracy of the certification technology and the certification cost respectively.
sults. If certification environments are similar enough to the full information benchmark (put
differently, if z is low, and x is high), then welfare with certification is greater than welfare under no information (W Crt (z, x) > W NI ). In this situation, certification is profitable for all high type
producers. Even though certification causes the NC market collapse and accrues certification costs,
allocative efficiency enhanced by certification outweighs such loss. In the opposite extreme cases,
if the cost of certification is too large or the certification technology is too inaccurate, nobody has
an incentive to opt for certification. Thus, welfare with certification coincides with welfare under
no information (W Crt (z, x) = W NI ).
The most interesting case is the intermediate area in between the above two extremes in which
welfare with certification is lower than welfare under no information (W Crt (z, x) < W NI ). In order
to formally state these results, let us define ΩNI ≡ (z, x) ∈ ΩFS |W Crt (z, x) = W NI
contour set of welfare level W NI as ΩNI + ≡ (z, x) ∈ ΩFS |W Crt (z, x) > W NI .

68

and upper

Figure 2.5: Welfare Comparisons

Note: Vertical and horizontal axes represent accuracy of the certification technology and the certification cost respectively. z∗ |x=1 represents a value such that W Crt (z∗ , 1) = W NI . In a similar
fashion, x∗ |z=0 represent a value such that W Crt (0, x∗ ) = W NI .
Proposition 3 Suppose only 1-market equilibria exist (i.e., c¯ ∈ [γ02 , γ0 )). Then,




> W NI if (z, x) ∈ ΩNI + ,




W Crt (z, x) = W NI if (z, x) ∈ ΩNI ∪ ΩP ,






< W NI otherwise,
where ΩNI + is a convex proper subset of ΩFS which is located around the point (0, 1) as shown in
Figure 5.
Figure 5 shows a graphical illustration of Proposition 3. It is worth noting that it is not per
se certification cost which makes W Crt lower than W NI . As we discussed in the full information
benchmark, certification brings welfare gain from reallocating high quality to consumers whose
value for the good is relatively high. However, if only a small measure of high grade producers
are selected into the Crt market, such welfare gain with certification will be negligible. Moreover,
given that we consider a parameter space on which the NC market always collapses with certification, welfare loss will be huge because most of high type producers now shut down. In other

69

words, the main driving force for a welfare-decreasing result is adverse selection caused by a negative impact of certification on the NC market. To see this point clearly, consider the vertical axis in
Figure 5. If the certification technology is relatively inaccurate (x < x∗ |z=0 ), we see W Crt < W NI
even though the certification cost is 0.

2.4.3.2

Analysis when 1-market and 2-market Equilibria Coexist

We now turn our attention to the other case c¯ ∈ (0, γ02 ) so that 2-market equilibria may exist. For
simplicity, we investigate only the following two special cases in turn: “a cost-less (z = 0), but
imperfect (x ∈ (0, 1]) test” and “a costly (z ∈ [0, ∞)), but perfect (x = 1) one”.
Cost-less (z = 0), but Imperfect (x ∈ (0, 1]) Test
Crt > 0 and π NC = 0 for all α > 0 (i.e.,
In this case, from Corollary 1 one can easily see that πH
H

regardless of whether there exists a 2-market equilibrium or not). Then, since certification is free
here, expected payoffs of high types are always higher with certification, i.e., EΠH (Cert|α ) =
Crt > EΠ (No Cert|α ) = 0 for all α > 0, which implies the unique certification equilibrium is
xπH
H

a fully separating one (α ∗ = 1).
What remains unknown for welfare calculation is to check when we would have a 2-market
equilibria. From Lemma 2, 2-market equilibria are expected for only small x values. Let κ − and
κ + the smaller and larger solutions solving for
pNC (κ|α ∗ = 1, x · · · ) = c¯
respectively. Then, if real roots of the above equation exist, then according to our equilibrium
selection criteria, we choose κ (α ∗ = 1, x) = κ + . Hence, welfare in this case is given by
W Crt

1

(x ) =

θ Crt

θ dθ − xγ0 n¯ c¯ +

θ Crt
θ NC

Welfare from Crt market

γ NC θ dθ − (1 − x)(1 − κ + )γ0 n¯ c¯,
Welfare from NC market

where all corresponding variables are defined as in Lemma 2-2 with κ (α ∗ = 1, x) = κ + . If we

70

Figure 2.6: Welfare Comparisons under Cost-less, but Imperfect Certification when γ0 = .5

(a) c¯ = .02

(b) c¯ = .05

(c) c¯ = .15

(d) c¯ = .25

Note: Vertical and horizontal axes represent welfare levels and the certification accuracy respectively.
have just 1-market equilibria, welfare is given by
W Crt (x) =

1
θ Crt

θ dθ − xγ0 n¯ c,
¯

where all corresponding variables are defined as in Lemma 2-1.
We demonstrate welfare comparisons of a 2-market equilibrium through simulations when γ0 =
.5 for various values of production cost c¯ in Figure 6. In all 3 panels except c¯ = .25(= γ02 ), we
have cutoffs of test precision determining before which 2-market equilibria emerge and otherwise
1-market equilibria. Put differently, if test is more precise than the cutoffs, certification results in

71

the collapse of the NC market in equilibrium. The simulation shows that W Crt could be lower than
W NI here too as long as production costs are not too small.
Costly (z ∈ [0, ∞)), but Perfect (x = 1) Test
Now we study the case of costly but perfect certification. More specifically, consider the case
where γ0 = .5 and c¯ = .1. Then, depending on certification decisions, whole trajectories of equilibrium prices and profit levels of high grade producers are as shown in Figure 7. As seen in the
case of cost-less and imperfect certification, 2-market equilibria prevail if only small number of
high types choose to get certified (relatively small α). This causes discontinuities in the price and
profit functions.
Given the price equilibrium outcomes, optimal certification decisions depend on the sizes of
Crt − z and EΠ (No Cert|α ) = 0. Note that if the certification cost is relatively
EΠH (Cert|α ) = πH
H

small (z ≤ z˜), a fully separating equilibrium is obtained (refer to the right panel of Figure 7).
Similarly, if the certification cost is relatively large (z ≥ zˆ), a pooling equilibrium results. Otherwise
Crt (α ∗ ) =
(˜z < z < zˆ), a certification equilibrium candidate must satisfy the following condition: πH

z. Note that for values of z slightly above z˜, there might be two equilibrium candidates (α) making
high types indifferent between opting for certification and not doing so. However, the certification
equilibrium associated with the smaller candidate of the two is not ε-stable because high types
currently not opting for certification expect the equilibrium price and their profits to increase if
a small measure of them deviate. Therefore, the certification equilibrium is characterized by α ∗
Crt (α ∗ ) = z and ∂ π Crt /∂ α|
such that πH
α =α ∗ < 0.
H

One thing to note is that here the highest certification cost supporting a semi-separating equilibrium, zˆ, is lower than the one we would have obtained if we did not consider 2-market equilibria
(which would have been given as zˆ = 1 − c¯ = .9 as in Proposition 2). Whenever 2-market equilibrium exist, the price in Crt market should be set in such a way leaving the exact same amount
of surplus to the marginal type consumer (θ Crt ) which he would get when purchasing in the NC
market instead. For this reason, when 2-market equilibria exist, the upper bound supporting a

72

Figure 2.7: Price Equilibrium Outcomes under Costly, but Perfect Certification when γ0 = .5 and
c¯ = .1

(a) Equilibrium Prices in Crt and NC Markets

(b) Equilibrium Profits of High Types

Note: Vertical and horizontal axes represent price and profit levels and measure of high grade
producers opting for certification respectively.
semi-separating certification equilibrium gets smaller.
Welfare calculations again depend on the type of equilibrium. If we have 2-market equilibria,
W Crt (z) =

1
θ Crt

θ dθ − α ∗ γ0 n¯ c¯ +

θ Crt
θ NC

γ NC θ dθ − (1 − α ∗ )(1 − κ + )γ0 n¯ c¯ −

Welfare from Crt market

Welfare from NC market

α ∗ γ0 nz
¯

,

Certification Costs

where all corresponding variables are defined as in Lemma 2-2 with κ (α ∗ , x = 1) = κ + and
Proposition 2. If we have just 1-market equilibria,
W Crt (z) =

1
θ Crt

θ dθ − α ∗ γ0 n¯ (c¯ + z),

where all corresponding variables are defined as in Lemma 2-1 and Proposition 2. Figure 8 summarizes simulation results regarding welfare levels for various values of production cost. It confirms
once again that welfare could be lower with certification. In all cases, as the certification cost, z,
increases from 0 to a certain threshold, welfare with certification monotonically decreases from
the full information level to the one lower than the no information level. Beyond the threshold, it
becomes the same as the no information level.

73

Figure 2.8: Welfare Comparisons under Costly, but Perfect Certification when γ0 = .5

(a) c¯ = .05

(b) c¯ = .10

(c) c¯ = .20

(d) c¯ = .25

Note: Vertical and horizontal axes represent welfare levels and certification costs respectively.
2.5

Conclusion

It is well-known that markets with asymmetric information about quality or other vertically differentiating goods characteristics suffer from allocative inefficiencies. Overcoming these asymmetries can lead to welfare increases. However, this understanding overlooks the close relationships
between markets. Allowing for certification may create the co-existence of both a certified and
a non-certified market. Both consumers and firms subsequently self-select into one market, the
other market, or opting out. Total welfare is a function of these self-selections, and thus welfare

74

implications cannot be considered in isolation. If certification is imperfect or costly, then positive externalities of high quality in non-certified markets can be destroyed without a complete
offsetting through added welfare in the certified markets. This implies subtle policy-implications
when evaluating the introduction of government certification programs and regulations of industry
self-governance with respect to certification programs.
We close this paper by discussing one feasible extension that may offer additional interesting
insights. It would be interesting to add a first stage to the game in which firms make their entry
decisions. This will allow us to see long-run consequences of certification depending on different
certification environments, which have received little interest in most of existing studies. We leave
this idea for future research to consider.

75

APPENDIX

76

Appendix for Chapter 2

Proof of Proposition 1
Proof of proposition 1 is in the text of the paper.

Proof of Lemma 1
Let us show pCrt > pNC first. Suppose there exist a price equilibrium where at least some high
quality goods are traded and pCrt ≤ pNC . Then, since γ Crt ≥ γ NC ,
γ Crt θ − pCrt > γ NC θ − pNC ∀θ ∈ [0, 1].
Then, all consumers θ ≥ PCrt /γ Crt wants to buy in Crt. Having some equilibrium output of
high quality goods, it must be the case that the cutoff type should be in the interior of support
of consumer types i.e., PCrt /γ Crt ∈(0, θ¯ ). Otherwise, there is no demand for both markets, and
such an equilibrium does not exist. Thus, it is sufficient to only consider the following range of
pCrt ∈ (0, γ Crt ): If 0 < pCrt < c,
¯ there is no supply in Crt market even though there is demand,
so such prices are not market clearing. If c¯ ≤ pCrt < γ Crt , there are always some low quality
producers willing to sell in NC since pNC ≥ pCrt ≥ c¯ > 0 while there is no demand at all for that
market. Hence, these prices are not market clearing either, which implies pCrt > pNC must be true.
The second argument is obvious from the binary feature of firm’s optimal decision rule.

Proof of Lemma 2
Given n¯ (or c)
¯ and α ∈ (0, 1), for a 2-market equilibrium to exist, the following conditions must
hold:
1. Existence of marginal types (θ Crt ) between buying in Crt market and buying in NC market:
γ Crt θ Crt − pCrt = γ NC θ Crt − pNC ;

77

2. Existence of marginal types (θ NC ) between buying in NC market and not buying:
γ NC θ NC − pNC = 0;
3. Market clearing condition in Crt market
qCrt/D ≡ 1 − θ Crt = αγ0 n¯ ≡ qCrt/S ;
4. Market clearing condition in NC market
qNC/D ≡ θ Crt − θ NC = [(1 − αx) κγ0 + (1 − γ0 )] n¯ ≡ qNC/S ;
(1−αx)κγ

(1−αx)γ

0
0
5. Consistent beliefs: γ Crt = 1, γ NC = (1−αx)κγ +(1−γ
≤ (1−αx)γ +(1−γ
where κ ∈ [0, 1].
0
0)
0
0)

6. From high grade firm’s indifferent condition between selling in NC and being inactive and
the definition of n,
¯ for all κ ∈ [0, 1]
pNC = c.
¯
From the condition 3, θ Crt = 1 − αγ0 n.
¯ Plugging this in the condition 4, we end up having θ NC =
{1 − [1 − (1 − αx)(1 − κ )γ0 ] n}.
¯ Since pCrt can be adjusted according to the condition 1 once
pNC is determined, the existence of a 2-market equilibrium pins down to find a proper κ satisfying
pNC = c.
¯ More specifically, from the condition 2, 5 and 6, if there exists κ ∈ [0, 1] which solves
the following equation,
pNC (κ|α, x · · · ) ≡

(1 − αx) γ0 κ
{1 − [1 − (1 − αx)(1 − κ )γ0 ] n}
¯ = c,
¯
(1 − αx) γ0 κ + (1 − γ0 )

a 2-market equilibrium exists. Before solving the equation, let us show that welfare is maximized
with the largest value κ (α, x · · · ) satisfying the above equation. First, note that welfare generated
from the Crt market will be the same no matter what kind of equilibrium we have because qCrt
and γ Crt are invariant. What only changes in the Crt market is pCrt which is a net transfer from
consumers to producers over different kinds of equilibria. Therefore, whenever a 2-market equilibrium exists, a 1-market equilibrium is not a welfare maximizing one. Then, it remains to pick up

78

Figure 2.9: Certification Equilibrium Constellations

Note: We assume that γ0 = .5, c¯ = .1 and x = 1. Vertical and horizontal axes represent pNC and κ
respectively.
a 2-market equilibrium resulting in the highest welfare level, which is obviously the largest value
κ (α, x · · · ) such that pNC = c.
¯
The first and second derivative of pNC (κ|α, x · · · ) with respect to κ are respectively,



 γ0 (1 − xα ) × 1 − γ0 − γ 2 (1 − xα )2 κ 2
0
∂ pNC
∂κ
∂ 2 pNC
∂ κ2

=









+2γ0 (1 − γ0 )(1 − xα )κ + (1 − γ0 ) (1 − γ0 (1 − xα ))} n¯ ] 

[1 − γ0 + γ0 (1 − xα )κ ]2
2γ02 (1 − γ0 )(1 − γ0 xα n¯ )(1 − xα )2
< 0.
= −
[1 − γ0 + γ0 (1 − xα )κ ]3

Note that pNC (κ|α, · · · ) has a similar shape with a concave quadratic curve passing through the
origin around which the slope of the curve is positive. Furthermore, as α increases from 0 to 1, this
curve gets closer to the horizontal axis fixing κ and has a (weakly) larger argmax of the function
(i.e., ∂ pNC /∂ α < 0 given κ for all κ ∈ [0, 1] and ∂ (arg maxκ pNC )/∂ α ≥ 0). Even though, Figure
9 is drawn for the case where γ0 = .5, c¯ = .1 (so n¯ = .8) and x = 1, the shape of pNC is similar for
all other parameter values. Also, we will have the same properties when it comes to x rather than
α because pNC (κ|α, x · · · ) is symmetric with respect to α and x.

79

Note that pNC (κ = 1, α = 0) = c¯ always by construction. So, if
∂ pNC
∂κ

≥ 0,
α =0,κ =1

2-market equilibria do not exist. Now observe that
sign

∂ pNC
∂κ

= sign

1 − γ0 − γ02 + 2γ0 (1 − γ0 ) + (1 − γ0 )2 n¯

α =0,κ =1

= sign (1 − γ0 − n¯ ) .
NC
Therefore, ∂ ∂p κ

α =0,κ =1

≥ 0 ⇐⇒ 1 − γ0 − n¯ ≥ 0 which is equivalent to c¯ ≥ γ02 .

Even though we may have some 2-market equilibria if c¯ < γ02 , that is the case only for small
α. Finally, whenever a 2-market equilibria exists, the welfare maximizing κ (α ) is the larger root
of the equation pNC (κ|α, · · · ) = c.
¯ Now it completes the proof to define ξ as a positive number
making pNC (κ|αx = ξ , · · · ) = c¯ have only one solution.

Proof of Proposition 2
Obvious from the discussion in the text and thus omitted.

Proof of Proposition 3
We will walk through a sequence of steps leading to the desired results.
Step 1: W Crt (z, x) is strictly decreasing in z and strictly increasing in x on ΩFS since
∂W Crt (z, x) /∂ z = −xγ0 n¯ < 0 and
∂W Crt (z, x) /∂ x = (1 − xγ0 n¯ )γ0 n¯ − γ0 n¯ c¯

= {(1 − xγ0 ) − c¯(1 − x)} γ0 n¯
> (1 − c¯) (1 − x) γ0 n¯
>0

80

where the first inequality follows from γ0 < 1, and the second inequality follows from Assumption
1.
Step 2: If x is too small, W Crt (z, x) < W NI regardless of z on ΩFS . To see this, fix z = 0.
Since W Crt (0, 1) = W FI > limx→0 W Crt (0, x) = 0 and
such that W Crt (0, x)

W NI if and only if x

∂W Crt (z,x)
∂x

> 0, there exists a unique x∗

x∗ by the Intermediate Value Theorem. Then, for all

(z, x) ∈ ΩFS such that x < x∗ , W Crt (z, x) < W NI since W Crt (z, x) strictly decreases in z.
Step 3: For each x > x∗ on ΩFS , there exists z∗ (x) < z˜(x) such that W Crt (z, x)
only if z

W NI if and

z∗ (x). First, let us show that limz→˜z(x) W Crt (z, x) < W NI .
lim W Crt (z, x) −W NI = (1/2) 1 − (1 − xγ0 n¯ )2 − xγ0 n¯ c¯ − γ0 n˜
¯ z(x )

z→˜z(x)

− (1/2)γ0 1 − (1 − n¯ )2 − γ0 n¯ c¯

= γ0 n¯ (1/2) x2 γ0 n¯ − (2 − n¯ ) + c¯
< γ0 n¯ {(1/2) [n¯ − (2 − n¯ )] + c}
¯

= −γ0 n¯ (1/γ0 − 1) c¯
<0
where the second equality follows from arranging terms; third equality follows from arranging
terms after plugging in 1 − (c/γ
¯ 0 ) for n;
¯ the first inequality follows from x2 γ0 < 1. Along with
this results and the fact that W Crt (0, x) > W NI for all x > x∗ , we proved the point again by the
Intermediate Value Theorem.
Step 4: From Step 2 and 3, ΩNI + = (z, x) ∈ ΩFS |z < z∗ (x) and x > x∗ and ΩNI +

ΩFS .

Step 5: For all (z, x) ∈ ΩSS , W Crt (z, x) < W NI . The welfare was defined as the sum of consumer surplus. Compared to the fully separating case, now less amount of consumers are served
in equilibrium given x while consumer valuation for the good remains the same as before so that
consumer surplus decreases. In addition to this, the aggregate expected profits are lower in semiseparating equilibrium because by definition producers earn 0 now while their profits are positive in fully separating equilibria. In sum, W Crt (z, x) is lower than W Crt (z˜(x), x), which implies

81

W Crt (z, x) < W NI for all z ∈ (z˜(x), zˆ(x)) given x.

82

CHAPTER 3
A ROLE OF NPE WHEN PATENT OWNERSHIP IS FRAGMENTED

3.1

Introduction

A non-practicing entity (NPE)1 is a company or an individual who amasses a few or large number
of patents only so as to monetize its patents through a licensing negotiation or litigation, without
exerting any efforts to commercialize, develop and sell a product adopting technologies covered
by its patents. According to the RPX report, the proportion of NPE lawsuits had rapidly increased
from 35 percent to 66 percent from 2010 to 2012 and remained between 66–70 percent since then
until 2015.2
This remarkable rise of NPE lawsuits, seen in recent years, have aroused considerable controversy on roles of NPEs. Some scholars expressed their concern for negative impacts of a NPE
on incentives of practicing entities (PEs) to develop a new product by underscoring opportunistic
patent acquisition and assertion strategies employed by NPEs (e.g., see Chien, 2010; and Lemley
and Melamed, 2013). On the contrary, a common defense of NPEs is that NPEs offer small individual inventors a better means of enforcing their patents and thereby increase the payoff to the
individual innovators (e.g., see Hagiu and Yoffie, 2013; and Risch, 2012). Even though a number
of studies started to analyze strategic behavior of NPEs and its incidence on innovation market
outcomes, to the best of my knowledge, there is still no research on how NPEs can increase the
payoff of the individual innovators and why that can be an important issue.
In order to address the above question, this paper considers an environment in which a product,
manufactured by a PE must adopt all complementary innovations patented and owned by individual
inventors; and a threat of litigation by an individual inventor is not credible so that patents are sold
1 Sometimes,

it is also called a “patent troll” or a patent assertion entity (PAE).
“2015 Report, NPE Litigation, Patent Marketplace, and NPE Cost” by RPX (last retrieved on May 1, 2017 from https://www.rpxcorp.com/wp-content/uploads/sites/2/2016/07/RPX2015-Report-072616.FinalZ.pdf).
2 Source:

83

through a sequential auction. The following two games are studied in turn. First, I consider the
game played by two PEs in which they compete to acquire the patents, and then negotiate licensing
fees based on their patent portfolio in the shadow of litigation. Next, the same game played by a
PE and a NPE is analyzed.
In the former game, it turns out equilibrium outcomes significantly depend on whether there
exists a state of “patent acquisition stalemate (PAS).” A PAS state refers to an intermediate patent
allocation in which none of the two PEs litigation threat becomes credible eventually regardless of
the results of the remaining patent auctions from that point on. It is shown that whenever a PAS
state exists, the two PEs buy all patents at 0 prices, and thus the aggregate payoff to the individual
inventors is also 0 in equilibrium. This implies that in fact patent ownership fragmentation can
lead to sharp decrease in the individual innovators’ payoff.
In sharp contrast to the above results, in the latter game, if total number of patents is sufficiently large, the existence of the NPE induces the PE to buy some patents at a positive price and
thereby increases the aggregate payoff to the individual inventors. Therefore, this implies that
fragmented patent ownership can significantly lower the payoff to the individual innovators, and
in this situation the NPE can play an important role of helping such small innovators get paid for
their innovations.

3.2

Related Literature

This paper contributes to a burgeoning literature on roles of NPEs. As in this model, there are
some papers that touch on strategic patent acquisitions by PEs and NPEs (Choi and Gerlach, 2017;
Cosandier et al., 2014; and Scott Morton and Shapiro, 2014). In a different vein, there are some
other papers that study how NPEs (or PEs) strategically assert their patents in order to maximize
their licensing revenues from an established patent portfolio (Choi and Gerlach, 2015; Hovenkamp,
2013; Lemley and Melamed, 2013; and Scott Morton and Shapiro, 2014).
Though the previous studies offer keen insights regarding various effects of NPEs on innovation

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market outcomes, this paper is different from the previous studies in either of the following two
aspects: First, this paper is the only one attempting to study the mechanism through which NPEs
can increase the payoff of the small individual innovators. Another distinction of this paper is
patent acquisitions with endogenous valuations. While a patent acquisition takes place at once, if
there is any, in most of previous studies, patent acquisitions take place sequentially in this paper.
As a result, a value of an ex-ante identical patent can vary depending on in which stage of the
sequential auction, which is characterized by each party’s intermediate patent portfolio, the patent
is sold.
This paper proceeds in the following way. Section 2 presents the model. In Section 3 and 4, I
analyze the game played by two PEs and that by a PE and NPE respectively. Section 5 concludes.

3.3

Model

While I consider the two games played between two PEs and between a PE and a NPE3 in turn, for
simplicity, here I delineate the model as for only the former game. Necessary model modifications
are elaborated later (right before the latter game is to be analyzed.)
There are two firms (PEs), 1 and 2 , who produce a complex technology product adopting
N > 2 complementary functionalities. The functionalities are complementary in the sense that the
product is not marketable without incorporating those features. All N technologies, which enable
respective functionalities, are protected by patents owned by N different individual inventors. Thus,
both of PEs take a risk of infringing N patents by selling the product unless they get a right to use
them through a license contract with patent owners or buy those patents. The duopoly profit is
denoted by π.4
3 Usually,

a NPE refers to all entities that own patents but do not produce any goods using
technologies covered by their patents. In this sense, individual inventors are also sometimes called
as NPEs. However, I make a distinction between individual inventors and the NPE as a patent
aggregator throughout the paper.
4 The product market competition is abstracted by assuming that each PE earns π.

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In the beginning, each individual innovator and each PE engage in licensing in the shadow
of litigation. Let α (∈ (0, 1]) denote the probability that a patent is upheld valid in court. This
probability is assumed to be the same and independent across all N patents. Define R(> 0) as a
per patent damage award for infringement and L(> 0) as a litigation cost. The damage award per
a patent is also the same for all patents, and all agents incur the same litigation cost. If a litigation
threat of an individual inventor is credible against a PE,5 i.e., αR ≥ L, they negotiate on a licensing
fee through a Nash bargaining with equal bargaining power. They are settled out of court in order
to avoid litigation costs and agree on a licensing fee of αR.6 In this situation, there is nothing to
further analyze, and the aggregate payoff to individual innovators is 2NαR and the payoff to each
firm π − NαR. By assuming that the product market is lucrative enough to cover licensing fees and
the litigation cost i.e., π − NαR − L > 0, I abstract from analyzing the firms’ entry or exit decision.
If the threat of litigation by an individual innovator is not credible, i.e., αR < L, each innovator
is not able to collect licensing revenue. However, even in this scenario, if the bundle of all patents
gave them a credible litigation threat (i.e., NαR ≥ L), the individual innovators would have formed
a patent pool (or a licensing coalition) to profitably license their patents. If this is possible, there
is no room for a third party NPE to play a role of increasing returns to the individual innovators.
I rule out this scenario by making the following assumption to study the role of the NPE when
cooperation is not an available option for the individual inventors.7
Assumption 1: (Information Structure) (i) An individual innovator is aware of what kind of
product is going to be released by PEs and how much licensing revenues (αR) will be determined in
5 In

this paper, a credible litigation threat is always in regard to a single defendant because
“Leahy-Smith America Invents Act,” which was enacted in 2011, precludes a plaintiff to file a
patent suit against multiple defendants,
6 Assuming equal bargaining power, in a Nash bargaining between an individual inventor and
a PE, the individual inventor’s payoff is αR − L + (1/2)2L = αR while the PE’s is −αR − L +
(1/2)2L = −αR.
7 Note that the assumption that I need in this paper needs not be asymmetric information between the individual innovators and PEs. It is sufficient to assume that forming a licensing coalition
is not feasible for some reason.

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court in expected sense, but does not know which other (N − 1)-patents are going to be potentially
infringed by the product and who are the owners of those patents ex-ante. (ii) On the contrary, two
PEs and a NPE know all that information.
In fact, this is not an unrealistic assumption because many NPEs are founded by former lawyers
of a PE or patent attorneys.8 Given Assumption 1, which implies that ex-ante coordination among
individual inventors is impossible, the individual innovators put their patents up for a sale through
an auction, and PEs engage in the patent acquisition game. Finally, given a patent allocation, a
licensing fee between PEs are negotiated and transferred in the shadow of patent litigation.
Put up for sale, N patents are sold through a sequence of second-price auctions. Thus, the
setup for the patent acquisition game introduced here is similar to the model used in Gale and
Stegeman (2001). Since all patents are symmetric in terms of expected licensing fee ex-ante so
that the order of sequence does not have any effect on equilibrium characteristics.9 Any stage
of the game is described by its current state of patent allocation, which is denoted by (n1 , n2 )
where n1 and n2 are the number of patents possessed by firm 1 and 2 respectively. Let F =
{(n1 , n2 ) ∈ I2+ |n1 + n2 = N} and A = {(n1 , n2 ) ∈ I2+ |n1 + n2 < N} denote a collection of all
final and intermediate patent allocations respectively where I+ = {0, 1, 2, ...}. Firm i’s bidding
strategy is a function bi : A → R+ . Let bi (n1 , n2 ) be the firm i’s optimal bidding in (n1 , n2 ) ∈ A .
Once a final patent allocation is determined through the patent acquisition game, firms negotiate
about licensing fees in the shadow of litigation. For an arbitrary final allocation (i.e., (n1 , n2 ) ∈ F ),
let Vi (n1 , n2 ) denote the net licensing revenue of firm i in the shadow of litigation. Then, the payoff
8 For example, Intellectual Ventures, a well-known NPE, was founded by Nathan Myhrvold and

Edward Jung, both of whom once held important positions at Microsoft, among many others (e.g.,
see Ewing and Feldman, 2012).
9 This does not necessarily mean that equilibrium prices are all alike on the equilibrium path due
to the endogenous valuation nature of this game. As well documented in auctions with endogenous
valuations literature (e.g. Krishna 1993, 1999), patents, that are the same ex-ante, could have
different values to different buyers depending on a patent allocation.

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of firm i is
Πi (n1 , n2 ) = π + Vi (n1 , n2 ) − (sum of payments for obtaining ni patents),
where i = {1, 2}, and the aggregate payoff of individual innovators is the sum of auction revenues.
The equilibrium concept is Subgame Perfect Nash equilibrium (SPNE).

3.4

Equilibrium Analysis: PE vs. PE

In this section, I solve the game between two PEs by backward induction confining our attention to
the case in which the threat of litigation by an individual innovator is not credible (i.e., αR < L), but
the sum of individual patent’s licensing revenue is greater than the licensing cost (i.e., NαR ≥ L).
The game between a PE and a NPE is analyzed in the next section.

3.4.1

Licensing in the Shadow of Litigation

In terms of calculating equilibrium licensing fees, the model shares many things in common with
Lemus and Temnyalov (forthcoming). Since the validity of a patent (α) is determined independently across all N patents, the probability that firm i’s product infringes on x number of patents in
court out of firm j’s patent portfolio of size n j follows the following binomial distribution:


 nj  x
n −x

 α (1 − α ) j
x
and vise versa. Thus, the expected damage awards that firm i needs to pay to firm j is n j αR once
litigation takes place. However, because firm i can counter-litigate without incurring additional
costs, she also collects ni αR from firm j in expected sense. Taking this into consideration, one of
the two will initiate litigation only when the net transfer is greater or equal to the litigation cost,
i.e., |ni − n j |αR ≥ L. If this is the case, as similarly discussed in section 2, a symmetric Nash
bargaining results in an out-of-court settlement in which the net payment of firm i is (ni − n j )αR.

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In all other cases in which no PE has any incentives to initiate litigation, there are no licensing
payments.
The following lemma, which is basically a reminiscence of Proposition 1 in Lemus and Temnyalov (forthcoming), summarizes the above arguments and formally characterizes Vi (n1 , n2 ) in
equilibrium for an arbitrary final allocation.
Lemma 1 PEs always cross-license their patents through an out-of-court settlement (no litigation)
whenever one of the two has a credible litigation threat, and the net licensing revenue of firm i is
given by



(ni − n j )αR if |n1 − n2 |αR ≥ L,
Vi (n1 , n2 ) =


0
otherwise.

(3.1)

For notational simplicity, define a minimum gab in numbers of patents owned by the two PEs,
which makes one of the two PE’s litigation threat credible, K as follows:
Definition 1: K is a positive natural number such that (K − 1)αR < L ≤ KαR.
Also, let us divide the set of final allocations (F ) into two mutually exclusive subsets: a set of
allocations with a credible litigation threat,
L ≡ {(n1 , n2 )|(n1 , n2 ) ∈ F , |n1 − n2 | ≥ K}
and without a credible litigation threat, F \L .

3.4.2

Patent Acquisition

With the equilibrium characterization in the licensing stage in mind, now let us turn to the subgame
perfect Nash equilibrium of the patent acquisition game. It should be noted that this game can be
considered as auctions with endogenous valuations first studied by Krishna (1993, 1999). Put
differently, even though all patents have identical expected damage awards ex-ante, when they are

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auctioned sequentially, a value of such an ex-ante identical patent can vary depending on a patent
allocation in which it is sold.
As similarly defined for final allocations, for an arbitrary intermediate patent allocation (i.e.,

(n1 , n2 ) ∈ A ), let Vi (n1 , n2 ) denote the net continuation valuation of firm i in the subgame starting
from that patent allocation. Thus, on an equilibrium path, Vi (n1 , n2 ) represents the sum of a net
licensing fee which will be determined by a final allocation and payments for obtaining additional
patents in firm i’s patent portfolio from that time on.
Before solving the game, equilibrium selection and its implications are briefly discussed.10
Equilibrium selection is needed because the second-price auction under complete information has
multiple equilibria including seemingly quite unreasonable ones. Following the approach taken
in many previous studies in the literature, I select the equilibrium in which each bidder bids the
marginal increase in valuation of winning the patent given a patent allocation, (n1 , n2 ), i.e.,11
b1 (n1 , n2 ) = V1 (n1 + 1, n2 ) −V1 (n1 , n2 + 1),

(3.2)

b2 (n1 , n2 ) = V2 (n1 , n2 + 1) −V2 (n1 + 1, n2 ).

(3.3)

Given the above optimal bidding strategies, if b1 (n1 , n2 ) > b2 (n1 , n2 ), firm 1 wins the patent
at the payment of firm 2’s bid, and the game proceeds to (n1 + 1, n2 ). Therefore,
V1 (n1 , n2 ) = V1 (n1 + 1, n2 ) − b2 (n1 , n2 ),

(3.4)

V2 (n1 , n2 ) = V2 (n1 + 1, n2 ).

(3.5)

Similarly, if b2 (n1 , n2 ) > b1 (n1 , n2 ), firm 2 wins the patent, and the game proceeds to (n1 , n2 + 1),
and
V1 (n1 , n2 ) = V1 (n1 , n2 + 1),

(3.6)

V2 (n1 , n2 ) = V2 (n1 , n2 + 1) − b1 (n1 , n2 ).

(3.7)

10 Most

of these preliminary results follow directly from Gale and Stegeman (2001).
more detailed explanations, see, for example, Dudey (1992), Gale and Stegeman (2001)
and Krishna (1993, 1999). The above equilibrium selection is usually justified by applying iterated
elimination of weakly dominated strategies in the literature.
11 For

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When bids are tied, i.e., if b1 (n1 , n2 ) = b2 (n1 , n2 ), it is immediate to see that (3.4) and (3.6) coincide with (3.5) and (3.7) respectively by using (3.2)~(3.3). This implies that equilibrium payoffs
are uniquely determined even when the game involves multiple equilibria due to some ties. An
equilibrium price is given by min{b1 (n1 , n2 ), b2 (n1 , n2 )}.
One important remark is that on an equilibrium path, the game must proceed towards the state
with the higher total valuation due to the optimal bidding rules ((3.2) and (3.3)). To see this, define
the total valuation function, V (n1 , n2 ) := V1 (n1 , n2 ) + V2 (n1 , n2 ). Then, firm 1 wins the auction
at (n1 , n2 ) if and only if V (n1 + 1, n2 ) ≥ V (n1 , n2 + 1) while firm 2 wins the auction if and only
if V (n1 + 1, n2 ) ≤ V (n1 , n2 + 1). Following Gale and Stegeman (2001), I call this property of the
equilibrium path the “principle of maximizing local valuation (MLV ).”
Given the above preliminary results, one can easily analyze the last stages in which the N-th
auction takes place (i.e., at allocations (n1 , n2 ) ∈ A such that n1 + n2 = N − 1). There are three
cases to consider: (i) regardless of the result of this auction, the game moves on to a state with
credible litigation threat; (ii) regardless of the result of this auction, the game moves on to a state
without credible litigation threat; (iii) depending on the result of this auction, the game moves on
to either a state with credible litigation threat or a state without it.
For expositional ease, assume that firm 1 has more patents in its patent portfolio (n1 ≥ n2 ).
In case (i), i.e, (n1 + 1, n2 ), (n1 , n2 + 1) ∈ L , by (3.1), (3.2) and (3.3), both firms bid 2αR in
equilibrium, i.e., bi (n1 , n2 ) = 2αR for i = {1, 2}. Then, since n1 + n2 = N − 1, (3.4) - (3.7) imply
the payoff to the individual innovator selling her patent increases by this much, i.e., Vi (n1 , n2 ) =

(2ni − N )αR for i = {1, 2}. In case (ii), (n1 + 1, n2 ), (n1 , n2 + 1) ∈ F \L . Hence, no firm has
an incentive to obtain an additional patent at a positive price, which gives rise to bi (n1 , n2 ) = 0
and Vi (n1 , n2 ) = 0. Lastly, in case (iii), the result of the last auction determines whether firm 1
ends up having a credible litigation threat and collecting a licensing fee from firm 2. Thus, firms
bid most aggressively here compared to the previous two cases. To see this, note that bi (n1 , n2 ) =

[(n1 + 1) − n2 ] αR because (n1 + 1, n2 ) ∈ L but (n1 , n2 + 1) ∈ F \L . Also, since only the cases
in which K ≥ 2 are considered, bi (n1 , n2 ) ≥ 2αR, which implies V1 (n1 , n2 ) = 0 and V2 (n1 , n2 ) =

91

− [(n1 + 1) − n2 ] αR ≤ −2αR. I call this case a “critical patent allocation.”
Definition 2: Suppose n1 ≥ n2 without loss of generality. Then, (nˆ 1 , nˆ 2 ) is a “critical patent
allocation” if and only if nˆ 1 + nˆ 2 = N − 1, and (nˆ 1 + 1, nˆ 2 ) ∈ L but (nˆ 1 , nˆ 2 + 1) ∈ F \L .
By taking a similar approach recursively, I can characterize equilibria of subgames starting
from patent allocations that are relatively close to a final allocation in the following lemma.
Lemma 2
1. (Active Patent Acquisition) In any subgames starting from (n1 , n2 ) ∈ A such that |ni −
n j | − (N − ni − n j ) ≥ K, bi (n1 , n2 ) = 2αR for all remaining patents, and thus Vi (n1 , n2 ) =

(2ni − N )αR in equilibrium where i = {1, 2}.
2. (Patent Acquisition Stalemate (PAS)) In any subgames starting from (n1 , n2 ) ∈ A such that

(N − ni − n j ) + |ni − n j | < K, all remaining patents are sold at 0 prices in equilibrium, i.e.,
bi (n1 , n2 ) = 0, and thus Vi (n1 , n2 ) = 0 where i = {1, 2}.
3. A critical patent allocation is never reached on the equilibrium path if a state of PAS exists.

First, if a patent allocation is fairly asymmetric, firms bid 2αR (result 1). In this case, the litigation threat will be eventually credible, and thus winning an additional patent has both offensive
(by increasing ni by 1) and defensive values (by not letting n j increase by 1). In contrast, if a
patent allocation is relatively symmetric, all remaining patents are auctioned off at 0 prices (result
2). Here firms have no incentive to acquire additional patents at positive prices because the patent
acquisition game will end in a final state without a credible threat of litigation anyway. Lastly,
result 3 explains the property of the critical patent allocations. If a PAS allocation exists, the last
patent determines whether the leading firm will be eventually able to earn a licensing fee (greater
than 2αR) or not, and thus both firms bid most aggressively at the critical patent allocations. The
principle of MLV implies that anticipating this, firms would try to avoid reaching the critical patent
allocations in the previous stage.

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Figure 3.1: PE1 vs. PE2 Game for the Case of N = 7 and K = 3

Note: Numbers in dashed line rectangles indicate a final patent allocation (n1 , n2 ) ∈ F . Numbers
in and below rectangles indicate a patent allocation (n1 , n2 ) ∈ A and continuation payoffs of
players, divided by αR, at that allocation respectively. Numbers above or below arrows or dashed
lines indicate firm 1’s bid and firm 2’s, divided by αR, respectively. An arrow shows to which
direction the game proceeds at a patent allocation in equilibrium. Finally, blue, red, and yellow
rectangles represent a state of active patent acquisition, a critical patent allocation, and a state of
PAS respectively.
Example 1 Consider the case of N = 7 and K = 3. As shown in Figure 1, a set of states of active
patent acquisition is {(5, 0), (5, 1), (6, 0), (0, 5), (1, 5), (0, 6)} at each of which b1 (·) = b2 (·) =
2αR; a set of states of PAS {(3, 3)} at which b1 (·) = b2 (·) = 0. A set of critical patent allocations
{(4, 2), (2, 4)}.
The results in Lemma 2 are not only useful for deriving the equilibrium of the entire game
(as will be formulated in Proposition 1 shortly after) but also have an interesting implication in
themselves. The aggregate payoff of individual innovators crucially depends on the initial patent
allocation. To see this point, instead of assuming that all N patents are owned by the individual
innovators, suppose that firm 1 and 2 have some patents in the beginning. Then, when the initial
patent allocation is an active patent acquisition state, individual innovators sell their patents at the

93

price of 2αR. On the other hand, when the initial allocation falls in the PAS region, individual
innovators earn nothing.
Since Lemma 2 studies only for the subgames starting from patent allocations relatively close
to the final allocations, a further analysis regarding all preceding subgames is necessary in order to
find the equilibrium of the entire game. In the appendix, I show that only two qualitatively different
type of equilibria can emerge depending on whether a state of PAS exists or not.
Proposition 1
1. If a state of PAS does not exist (K = 2 and N is even), each patent is sold at 2αR in any
equilibrium, and thus the aggregate payoff to the individual inventors is the same as they
would have earned through individual licensing contracts (2NαR).
2. If a state of PAS exists (K ≥ 3, or K = 2 and N is odd), the last auction (N-th patent) takes
place at one of the PAS states, and each patent is sold at 0 in any equilibrium. Therefore, the
aggregate payoff to the individual inventors is 0 .

Proposition 1 tells us that when a PAS state exists (corresponding to most cases), surprisingly,
PEs acquire all patents at 0 prices despite the competition between the two firms. Put differently,
the existence of a PAS state enables PEs to pay nothing to use patented technologies and settle down
a final allocation without a credible litigation threat. Loosely speaking, this is because whenever
a PAS state exists, in an earlier stage of the game the losing firm is able to catch up in number
of patents by bidding more aggressively in any equilibrium. To better understand this point, see
Figure 1 which illustrates the analysis for the case of N = 7 and K = 3. For example, the firm 2,
who is losing at (2, 0) or (2, 1), is able to win the patent by paying 0 and thereby avoid the game
moving on closer to a state of active patent acquisition. Anticipating this, there is no incentive for
any firm bids positive amount to be the leader in preceding stages.
Another implication of Proposition 1 is that the aggregate payoff of individual innovators decreases dramatically when patent ownership is fragmented whenever a PAS state exists. To see

94

this, suppose all patents are owned by a single innovator instead. In sharp contrast to the above
results, when the entire patents are sold all together at once, that is there is no chance that the firms
can acquire a portion of patents, this all-or-nothing feature of the auction leads each firm to bid
2NαR . Thus, the innovator can earn 2NαR either through an auction or a licensing negotiation.

3.5

Equilibrium Analysis: PE vs. NPE

In this section, I solve for the equilibrium of the game played between a PE and a NPE and contrast
this with that of the game between the two PEs. As in the previous section, taking final patent
allocations as given, I study the licensing problem first, and then the patent acquisition problem
later.

3.5.1

Licensing in the Shadow of Litigation

A few modifications to the previous model are needed. Let (n, m) denote a patent allocation in
which n and m represent the number of patents possessed by the PE and the NPE respectively.
Since the NPE does not produce a product, the only source of profit for the NPE is licensing fees.
Thus, NPE’s payoff function is given by
Πnpe = Vnpe (n, m) − (sum of payments for obtaining m patents),
where Vnpe (n, m) regarding a final and an intermediate patent allocations is defined as in the previous model.
The main difference between the two games is that the NPE is not subject to counter-litigation
because the NPE produces nothing and thus will not infringe any patents of PE. This implies the
one-way nature of litigation between them: only the NPE can have a credible litigation threat
when m ≥ K. Thus, now the set of patent allocations with a credible litigation threat is L ≡
{(n, m)|(n, m) ∈ F , m ≥ K}.
It is straightforward to show Lemma 3, the counterpart to Lemma 1 through an analogous
analysis.

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Lemma 3 A PE and a NPE always settle out-of-court (no litigation) whenever a litigation threat
of a NPE is credible, and the net licensing revenues of the PE to the NPE are given by



−mαR if m ≥ K,
V1 (n, m) =


0
otherwise;



mαR if m ≥ K,
Vnpe (n, m) =


0
otherwise.

3.5.2

(3.8)

(3.9)

Patent Acquisition

In order to solve the subgame perfect Nash equilibrium of the patent acquisition game, I take a
similar strategy as that used in the game between the two PEs. A critical state is defined first.
Lemma 4, the counterpart of Lemma 2, follows. Finally, based on Lemma 4, I derive Proposition
2, the counterpart of Proposition 1.
Definition 3: (n,
ˆ mˆ ) is a critical state if nˆ + mˆ = N − 1, (n,
ˆ mˆ + 1) ∈ L and (nˆ + 1, mˆ ) ∈ F \L .12
Lemma 4
1. (Active Patent Acquisition) In any subgames starting from (n, m) ∈ A such that m ≥ K,
b1 (n, m) = bnpe (n, m) = αR for all remaining patents, and thus V1 (n, m) = −(N − n)αR
and Vnpe (n, m) = mαR in equilibrium.
2. (Patent Acquisition Stalemate) In any subgames given (n, m) ∈ A such that (N −n) < K, all
remaining patents are sold at 0 prices in equilibrium, i.e., bi (n, m) = 0, and thus Vi (n, m) = 0
where i = {1, npe}.
12 Note

that in the previous game, assuming nˆ 1 ≥ nˆ 2 , at the critical state, nˆ2 is an integer such
that (N − K )/2 − 1 < nˆ2 ≤ (N − K )/2. Here, (n,
ˆ mˆ ) = (N − K, K − 1). Thus, mˆ is constant when
N changes while all other numbers vary.

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3. A critical patent allocation,(N − m, m − 1), is never reached on the equilibrium path.

The results in Lemma 4 are similar to those in Lemma 2 in that in a state of active patent
acquisition, both the PE and the NPE bid positive amount (result 1); in a state of PAS, they bid 0
(result 2); and in a critical patent allocation they bid most aggressively (result 3).
However, Lemma 4 differs significantly from Lemma 2 in two respects. First, as already
pointed out, the NPE is the only one who can earn positive licensing revenue. More importantly, it
is not how much a patent allocation is symmetric but how many patents the PE has that determines
a PAS state. By the same token, it is not how much a patent allocation is asymmetric but how many
patents the NPE has that determines that determines an active patent acquisition state.
Example 2 Consider the case of N = 7 and K = 3. As shown in Figure 2, a set of states of
active patent acquisition is {(0, 3), (1, 3), (0, 4), (2, 3), (1, 4), (0, 5), (3, 3), (2, 4), (1, 5), (0, 6)} at
each of which b1 (·) = bnpe (·) = αR; a set of states of PAS {(5, 0), (6, 0), (5, 1)} at each of which
b1 (·) = bnpe (·) = 0. A set of critical patent allocations {(4, 2)}.
In order to fully characterize the equilibrium of this game, it remains to analyze what happens
at the patent allocations not studied in Lemma 4. To this end, given N and K, I only consider
allocations (n, m) ∈ A such that n ≤ N − K and m ≤ K. Assuming N large enough,13 K different
classes of allocations are studied in turn where class-i (i ∈ {1, · · · , K}) allocations are defined
as (n, K − i) where n ∈ {N − K, . . . , 0}. That is, by backward induction I study first for class1 allocations from (N − K, K − 1) to (0, K − 1), next class-2 allocations from (N − K, K − 2) to

(0, K − 2), and so on all the way up to class-K allocations. Before reporting results for general
cases, it is helpful to see the example below first.
Example 3 Consider the case of N = 7 and K = 3 in Figure 2 as an example again. Note that
an equilibrium path must be a sequence of patent allocations from the initial state (0, 0) to a final
13 See

the proof for Proposition 2 for the reason why we can assume this without loss of gener-

ality.

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Figure 3.2: PE vs. NPE Game for the Case of N = 7 and K = 3

Note: Numbers in dashed line rectangles indicate a final patent allocation (n, m) ∈ F . Numbers
in and below rectangles indicate a patent allocation (n, m) ∈ A and continuation payoffs, divided
by αR, of players at that allocation respectively. Numbers above or below arrows or dashed lines
indicate PE’s bid and NPE’s, divided by αR, respectively. An arrow shows to which direction
the game proceeds at each patent allocation in equilibrium. Finally, blue, red, and yellow rectangles represent a state of active patent acquisition, a critical patent allocation, and a state of PAS
respectively.
allocation linked by arrows. Consider class-1 allocations (i.e., (4, 2), (3, 2), . . . , (0, 2)) first. As
seen in the figure, the NPE first strictly wins a patent at the allocation of (3, 2), which is lead
to a final allocation with a credible litigation threat (i.e., (4, 3) or (3, 4) ∈ F ) in equilibrium.
Note that it takes two stages for the NPE to strictly win an auction for the first time from the
back. A similar pattern is observed for class-2 allocations (i.e., (4, 1), (3, 1), . . . , (0, 1)) as well.
While when n is relatively large ((4, 1), (3, 1), or (2, 1)), the PE wins the patent, and the game
leads to an allocation without a litigation threat, when n is relatively small, an auction is tied or
won by the NPE. At (0, 1), NPE strictly wins the auction, and the game reaches a final allocation
with a litigation threat. Since the second and the 6th patent are auctioned off at (0, 1) and (3, 2)
respectively, it takes 4 additional stages for the NPE to strictly win an auction for the first time
from the back. I denote such additional stages for class-i allocations as ∆(i). Also, it is shown that

98

in any equilibrium, the PE only pays αR to acquire the first auction; the all the rest of patents are
sold at 0; and the game reachs one of a final allocations without a litigation threat.
It turns out that ∆(i), defined as in the above example, is an important variable in characterizing the equilibrium of this game. By formally constructing a sequence {∆(i)}K−1
i=1 , I report the
subgame perfect Nash equilibrium for general N and K in the following proposition.
Proposition 2 The game ends in a final allocation without a credible litigation threat (i.e., the
auction for N-th patent always takes place at one of the PAS states) in equilibrium. Moreover,
1. if the number of patents is sufficiently large (N > ∑K−1
i=1 ∆(i)) given K, PE wins each patent
a few auctions during earlier stages (up to the (N − ∑iK−1
=1 ∆(i))-th auction) at αR, and then
the remaining patents are sold at 0. Thus, the aggregate payoff to the individual inventors is
N − ∑K−1
i=1 ∆(i) αR;
2. Otherwise, all patent is sold at 0 in any equilibrium. Therefore, the aggregate payoff to the
individual inventors is 0.
Here, ∆(i) is a sequence constructed recursively as follows: for i = 2, ∆(1) = 2, and for i ≥ 2,
∆(i) is the smallest integer ∆ satisfying the following inequality
∆>

2(K − i) + 1 + ∑i−1
s=1 ∆(s) + [Vnpe (n˜ (i − 1), K − i) /αR] ;
K −i

(3.10)

where Vnpe (n˜ (i − 1), K − (i − 1)) is a sequence constructed recursively as follows: for i = 2, (K −
1)αR, and for i ≥ 3,
Vnpe (n(i − 2), K − (i − 2)) + ∆(i − 1) [K − (i − 2)] −

i−1

∑ ∆(s) − 2(K − i) − 3.

(3.11)

s=1

Though the NPE never acquires any patent in equilibrium, the existence of the NPE induces
the PE to purchase a few patents at αR in earlier stages if the total number of patents is relatively large. This is in sharp contrast to the result obtained in the game between PEs because the

99

Table 3.1: Minimum Threshold Number of Patents
K
∑K−1
i=1 ∆ (i)

2
2

3
6

4
12

5
20

6
30

7
42

8
56

9
72

10
90

···
···

aggregate payoff to the individual innovators may be positive even with a PAS state. The main discrepancy between the results of the two games stems from the different litigation nature between
the two games. In contrast to the previous game in which PEs amass patents for both offensive and
defensive purposes, in this game, PE acquires patents for only defensive purposes while the NPE
does so for only offensive purposes.
The amount paid to the individual innovators depends on the relative magnitudes of N and
∑K−1
i=1 ∆(i). The intuition behind this result is as follows. In each class-i allocations, as illustrated
in Example 3, the PE tries to prevent the game moving towards a state of active patent acquisition
by acquiring a patent. However, this defensive patent acquisition is costly for the PE as well.
Thus, the PEs bid aggressive to win only when an allocation is relatively close to a PAS state.
Loosely speaking, ∑K−1
i=1 ∆(i) captures the upper limit to which the PE can use such strategy when
the game is analyzed up to class-(K − 1) allocations. On the other hand, when N increases, the
NPE has more opportunities to acquire a patent for offensive purposes. This is because when N
increases, the area of PAS remains the same as before while the area of active patent acquisition is
extended. Therefore, only when N is greater than ∑iK−1
=1 ∆(i), the aggregate payoff to the individual
innovators is given by the difference between the two terms. Interestingly, the minimum threshold
number of patents ∑K−1
i=1 ∆(i), which guarantees a positive payoff to the individual innovators, is
not proportional but increases more rapidly along with an increase in K. I report the values of
∑K−1
i=1 ∆(i) using (3.10) and (3.11) for a couple of different values of K in Table 1.

100

3.6

Conclusion

This paper considers a situation in which the threat of litigation by an individual innovator is not
credible, and thus individual innovators’ patents are sold through a sequential auction. The two
games are studied in turn: the game played by the two PEs in which they compete to acquire the
patents, and then negotiate licensing fees based on their patent portfolio in the shadow of litigation,
and then, the same game played by PE and NPE.
It is shown that in the former game, all patents are sold at 0 prices whenever there exists an
intermediate state of patent acquisition stalemate (PAS). This is surprising considering that if all
patents were possessed by an innovator and auctioned together, they would have been sold at a
price equal to the highest attainable licensing fees (2NαR). In contrast, in the latter game, even
with a PAS state, the existence of NPE induces PE to purchase some patents at positive prices as
long as the total number of patents are sufficiently large. Therefore, in sum, this paper suggests
that fragmented patent ownership can make it hard for the individual innovators to monetize their
patents, and in this situation NPE can play an important role of increasing the aggregate payoff to
the individual innovators without any opportunistic behaviors though such role of NPE is limited
in the sense that the amount paid to the individual innovators is less than the highest attainable
licensing fees (NαR).
Since this analysis is based on many stylized assumptions, one can think of many natural
extensions of the current framework. For example, one can check to what extent the key findings
of this paper are robust when the model allows for injunction possibilities in addition to damage
awards. Another important extension is to consider the game played by multiple PEs and a NPE.
One technical difficulty associated with such extension is that continuation payoffs of each player
in an intermediate patent allocation may not be uniquely defined because multiple equilibria with
different winners may emerge in many cases. These remain interesting venues for future research.

101

APPENDIX

102

Appendix for Chapter 3

Proof of Lemma 1
The proof directly follows from Proposition 1 in Lemus and Temnyalov (forthcoming).

Proof of Lemma 2
Assume n1 ≥ n2 without loss of generality.
Result 1: when the N-th auction takes place in this class of subgames, V1 (n1 , n2 ) =

(2n1 − N )αR and V2 (n1 , n2 ) = (2n2 − N )αR. Let us go one stage back, and consider only
patent allocations being lead to one in L . Then, in every auction for the (N − 1)-th patent,
bi (n1 , n2 ) = 2αR. By doing this recursively back to the (N − t )-th patent auction, it is easy to
see that for any (n1 , n2 ) ∈ A such that n1 + n2 = N − (t + 1), b1 (n1 , n2 ) = b2 (n1 , n2 ) = 2αR,
V1 (n1 , n2 ) = (2n1 − N )αR, V2 (n1 , n2 ) = (2n2 − N )αR and V (n1 , n2 ) = 2 [(n1 + n2 ) − N ] αR.
Result 2: when the N-th auction takes place in this class of subgames, V1 (n1 , n2 ) =
V2 (n1 , n2 ) = 0, which immediately implies b1 (n1 , n2 ) = b2 (n1 , n2 ) = 0. Then, backward induction implies result 2.
Result 3: note that V1 (nˆ 1 , nˆ 2 ) = 0 and V2 (nˆ 1 , nˆ 2 ) = − [(nˆ 1 + 1) − nˆ 2 ] αR. Also, note that there
are only two allocations which can be lead to (nˆ 1 , nˆ 2 ): (nˆ 1 , nˆ 2 − 1) and (nˆ 1 − 1, nˆ 2 ). Thus, it is
sufficient to show that (i) firm 1 always wins the auction at (nˆ 1 , nˆ 2 − 1), which is true if and only
if V (nˆ 1 + 1, nˆ 2 − 1) > V (nˆ 1 , nˆ 2 ) by MLV , and (ii) firm 2 at (nˆ 1 −, nˆ 2 ), which is true if and only if
V (nˆ 1 − 1, nˆ 2 + 1) > V (nˆ 1 , nˆ 2 ). To prove claim (i), let us compare the magnitudes of V (nˆ 1 + 1, nˆ 2 −
1) and V (nˆ 1 , nˆ 2 ). Since (nˆ 1 + 1, nˆ 2 − 1) is an allocation of ‘Active Patent Acquisition,‘
V (nˆ 1 + 1, nˆ 2 − 1) = 2(nˆ 1 + 1 + nˆ 2 − 1 − N )αR = −2αR.
However, whenever a PAS state exists, i.e., either when K ≥ 3 or when K = 2 and N is odd,
V (nˆ 1 , nˆ 2 ) = − [(nˆ 1 + 1) − nˆ 2 ] αR < −2αR,

103

which implies V (nˆ 1 + 1, nˆ 2 − 1) > V (nˆ 1 , nˆ 2 ). The proof of claim (ii) is analogous.

Proof of Proposition 1
Result 1: When K = 2 and N is even, at all final allocations but (N/2, N/2), litigation threat is
credible (i.e., F \ {(N/2, N/2} = L ). This implies that Vi (n1 , n2 ) = (ni −n j )αR for all (n1 , n2 ) ∈
F and bi (n1 , n2 ) = 2αR for all (n1 , n2 ) ∈ A such that n1 + n2 = N − 1 where i = {1, 2}. By doing
this recursively, it is easy to see that bi (n1 , n2 ) = 2αR for all (n1 , n2 ) ∈ A .
Result 2: Consider the case K ≥ 3 and N is an odd number. The proof for the other cases
(K = 2 and N is odd, or K ≥ 3 and N is even) are analogous. I walk through the following two
steps.
Step 1: First, I prove that result 2 holds when N ≥ K = 3 by induction. It is straightforward
to check that result 2 holds by backward induction when N = K = 3. Now it suffices to check
that if result 2 holds for N > 3, then so does result 2 for N + 2. To prove this, define N = N + 2
and denote all corresponding allocations associated with N by (n1 , n2 ) ∈ A ∪ F where A =

(n1 , n2 )|n1 + n2 < N + 2 and F = (n1 , n2 )|n1 + n2 = N + 2 . Note that
F = (n1 , n2 )|(n1 , n2 ) = (n1 + 1, n2 + 1), ∀ (n1 , n2 ) ∈ F ∪ (N , 0), (0, N ) .
Then, the game associated with N is equivalent to the one with N in terms of payoff structure for
all final allocations except extreme allocations: (N, 0) and (0, N ), i.e.,



(n − n )αR if |n − n |αR ≥ L
i
j
i
j
Vi (n1 , n2 ) =


0
otherwise

= Vi (n1 , n2 )
for all (n1 , n2 ) ∈ F \{(N , 0), (0, N )} where n1 = n1 + 1 and n2 = n2 + 1 since ni − n j = ni − n j .
This implies that the equilibrium consequences of the subgame including only allocations in which
both firms have at least 1 patent ((n1 , n2 ) ∈ A ∪ F such that n1 , n2 > 0) and the entire game with
N are exactly the same.

104

Suppose that result 2 holds for the case of N but not for the case of N . Then, since the game
ends in PAS given N, it must be the case, by Lemma 2-3, that there exists (n∗1 , 1) on the equilibrium
path of the subgame starting from (1, 1), which leads the firms to PAS in the end. Note that (n∗1 , 1)
must be such that n∗1 ≤ nˆ 1 − 1 , b1 (n∗1 , 1) > b2 (n∗1 , 1) and V1 (n∗1 , 1) = V2 (n∗1 , 0) = 0 where nˆ 1
is the firm 1’s patent share in the critical allocation given N . Then, since all (n1 , 1) such that
n1 ≤ n∗1 are lead to PAS, at (n∗1 , 0) it must be the case b1 (n∗1 , 0) ≥ b2 (n∗1 , 0), or V (n∗1 + 1, 0) ≥
V (n∗1 , 1) = 0. However, if the game could end in an allocation in L , some patents will be sold at
a positive price in equilibrium of the subgame starting from (n∗1 + 1, 0). If this is the case, then it
must be the case V (n∗1 + 1, 0) < 0 because the price paid weakly decreases along any equilibrium
path, by Theorem 1A in Gale and Stegeman (2001), which is a contradiction.
Step 2: It completes the proof to show that given an arbitrary odd number N, result 2 holds for
all K ∈ {3, 4, · · · , N}. By step 1, given N, result 2 must hold for the case K = 3. Now it remains to
check if result 2 holds for K > 3, then so does result 2 for K + 1. This can be shown following an
approach analogous to that used in step 1.

Proof of Lemma 3
The proof is analogous to that of Lemma 1 and omitted.

Proof of Lemma 4
The proof is analogous to that of Lemma 2 and omitted.

Proof of Proposition 2
Given K and N, I formalize the idea discussed in the text by considering and studying allocations (n, m) ∈ A such that n ≤ N − K and m ≤ K. First, I claim that assuming any arbitrary N
given K is innocuous. Similar to the PE vs. PE game, the game associated with N = N + 1 is

105

equivalent to the one with N in terms of payoff structure for all final allocations except extreme
allocations: (0, N ). Put differently, if (n , m ) ∈ A ∪ F where A = (n , m )|n + m < N + 1
and F = (n , m )|(n , m ) = (n + 1, m) ∀ (n1 , n2 ) ∈ F ∪ {(0, N )}, the game associated with
N is equivalent to the one with N in terms of payoff structure for all final allocations except the
extreme allocation: (0, N ), i.e.,


 −m αR
V1 (n , m ) =

 0


 m αR
Vnpe (n , m ) =

 0



if m ≥ K 
= V1 (n, m)


otherwise


if m ≥ K 
= Vnpe (n, m)


otherwise

for all (n , m ) ∈ F \{(0, N )} where n = n + 1 since n is independent of credibility of NPE’s
litigation threat. This implies that the equilibrium consequences of the subgame including only
allocations in which PE has at least 1 patent ((n , m ) ∈ A ∪ F such that n > 0) and the entire
game with N are exactly the same. Furthermore, regarding newly added allocations such that (0, m)
and m > K, the analysis is straightforward by result 1 in Lemma 4.
Now consider the following three cases in turn: (i) class-1 allocations, (ii) from class-2 to
class-(K − 1) allocations, and (iii) class-K allocations.14
Case (i): By Lemma 2, it is immediate to see that NPE wins the (N − 2)-th auction since
b1 (n, K − 1) = αR < bnpe (n, K − 1) = KαR, but all the other auctions are tied at b1 (n, K − 1) =
bnpe (n, K − 1). This implies that for all class-1 allocations, all patents are sold at αR, and thus,
V1 (n, K − 1) = −(N − n)αR and Vnpe (n, K − 1) = (K − 1)αR .
Case (ii): First, consider class-2 allocations: (n, K − 2) to (n, K − 2) where n = {N − K, N −
K −1, . . . 0}. When n = N −K, by Lemma 4 we know that PE wins the auction by paying 0, and the
game proceeds to a PAS state. Hence, V1 (N − K, K − 2) = Vnpe (N − K, K − 2) = 0. Now consider
the case n = N − K − 1. From the previous result and the result in case (i), NPE’s bid is (K − 1)αR
while PE’s bid is (K + 1)αR, and thus PE wins this auction. However, as n decreases, it becomes
14 Note

that when K = 2, only case (i) and (iii) are relevant.

106

too costly for PE to win the auction to avoid entering into a state of active patent acquisition. Given
N is large enough, NPE wins the auction finally at some n and thereafter the winning bid becomes
αR until we reach the case n = 0.
In a similar way, one can demonstrate that similar properties hold for all the other classes. From
applying a similar approach shown above for the class-2 allocations and using (3.2) - (3.7), one
can construct a sequence {∆(i)}K−1
i=1 , which characterizes an important equilibrium consequence
for these class of allocations. ∆(i) denotes additional stages, needed for NPE to win the auction
for the first time from the back when considering class-i allocations, on top of the total number
of such stages up to class-(i − 1) allocations, i.e., ∑i−1
s=1 ∆(s). It turns out that how much PE pays
to the individual innovators in equilibrium crucially depends on ∑K−1
i=1 ∆(i), we focus on how to
derive this sequence.
For the case of i = 1, from the result in case (i), we know that ∆(1) = 2. For all the other cases,
k ≥ 2, one can easily verify that ∆(i) is constructed recursively as follows:
(ii)-A. For a relatively large n such that N − (K − i) − ∑si−1
=1 ∆(s) ≤ n ≤ N − K, the winning bid is
always 0, and thus, V1 (n, K − i) = Vnpe (n, K − i) = 0;
(ii)-B. For a relatively small n such that 0 ≤ n < N − (K − i) − ∑is=1 ∆(s), the winning bid is
always αR, and thus, V1 (n, K − i) = −(N − n)αR and Vnpe (n, K − i) = (K − i)αR;
(iii)-C. For an intermediate n such that N − (K − i) − ∑is=1 ∆(s) ≤ n < N − (K − i) − ∑si−1
=1 ∆ (s),



< bnpe (n, K − i) if n = N − (K − i) − ∑k ∆(s),
s=1
b1 (n, K − i)


≥ bnpe (n, K − i) otherwise.
For notational simplicity let n(i) ≡ N − (K − i) − ∑is=1 ∆(s). With this definition, N − (K − i) −
∑i−1
s=1 ∆ (s) can also be simply rewritten as n(i − 1) + 1.
Now, to figure out ∆(i), let us calculate the value of bnpe (n, K −i) first in this range of n. Recall
that,

bnpe (n, K − i) = Vnpe (n, K − (i − 1)) −Vnpe (n + 1, K − i) .

107

Note that in this range of n, Vnpe (n + 1, K − i) = 0 by (ii)-A. Therefore, considering (ii)-B,



Vnpe (n, K − (i − 1)) ≤ (K − (i − 1)) αR if n = n(i − 1),
bnpe (n, K − i) =
(3.12)


(K − (i − 1)) αR
otherwise.
Then, this implies that for n ranging from n(i − 1) − 1 to n(i),

n(i−1)

V1 (n + 1, K − i) = − 

∑



bnpe (s, K − i)

(3.13)

s=n+1

since by construction PE wins or ties as long as n > n(i), and by (ii)-A V1 (n, K − i) = 0 for
n ≥ n(i − 1) + 1. On the other hand, for n ranging from n(i − 1) − 1 to n(i), by (ii)-B,
V1 (n, K − (i − 1)) = −(N − n)αR.
Therefore, here

b1 (n, K − i) = (N − n)αR − 



n(i−1)

∑

bnpe (s, K − i) .

(3.14)

s=n+1

From the discussion so far, it must be the case that ∆(i) ≡ n(i − 1) − n(i) + 1 is a smallest
positive integer ∆ such that

=

bnpe (n(i − 1) − ∆ + 1, K − i) − b1 (n(i − 1) − ∆ + 1, K − i)





 [K − (i − 1)] αR − [N − n(i − 1) + ∆ − 1] αR


 +Vnpe (n(i − 1), K − (i − 1)) + (∆ − 2) [K − (i − 1)] αR 

>0

where the equality follows from (3.12) and (3.14). This is equivalent to, by the definition of n˜ (i),
2(K − i) + 1 + ∑i−1
s=1 ∆ (s) −Vnpe (n(i − 1), K − (i − 1)) /αR .
∆>
K −i
Now it is sufficient to figure out Vnpe (n(i − 1), K − (i − 1)). If i = 2,
Vnpe (n(i − 1), K − (i − 1)) = (K − 1)αR.

108

If i ≥ 2, since NPE wins at (n˜ (i − 1), K − (i − 1)),
Vnpe (n˜ (i − 1), K − (i − 1))

= V (n(i − 1), K − (i − 2)) −V1 (n(i − 1) + 1, K − (i − 1))
= [n(i − 1) − N + K − (i − 2)] αR
+ bnpe (n(i − 2), K − (i − 1)) + (∆(i − 1) − 2) (K − (i − 2)) αR
= Vnpe (n(i − 2), K − (i − 2))
i−1

−

∑ ∆(s) + ∆(i − 1) (K − (i − 2)) − 2(K − i) − 3

s=1

where the second equality follows from (ii)-A and (3.13), and the third equality follows from
rearranging terms and (3.13).
Case (iii): Having all the results drawn in case (i) and (ii), one can investigate (n, 0) allocations in a similar way. It turns out that if n > n(K − 1), an auction is tied, or PE wins the auction
at 0 price, leading to a PAS state. Otherwise, i.e., n ≤ n(K − 1), PE wins the auction by paying αR. Hence, if N > ∑K−1
i=1 ∆(i), the aggregate payoff to the individual innovators is given by
N − ∑K−1
i=1 ∆(i) αR.

109

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110

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