Patent Quality and Incentives at the Patent Office∗ Florian Schuett† October 2012

Abstract Patent examination is a problem of moral hazard followed by adverse selection: examiners must have incentives to exert effort, but also to truthfully reveal the evidence they find. I develop a theoretical model to study the design of incentives for examiners. The model can explain the puzzling compensation scheme in use at the U.S. patent office, where examiners are essentially rewarded for granting patents, as well as variation in compensation schemes and patent quality across patent offices. It also has implications for the retention of examiners and for administrative patent review. Keywords: innovation, patent examination, soft information, intrinsic motivation, delegated expertise JEL classification numbers: D73, D82, L50, M52, O31, O38 ∗ This paper is based on the first chapter of my PhD dissertation at the Toulouse School of Economics. An earlier version of this and a companion paper was circulated under the title “Inventors and Impostors: An Economic Analysis of Patent Examination.” I am indebted to my advisor Jean Tirole for his support and guidance. I am also grateful to David Martimort (the co-editor) and two anonymous referees, whose detailed comments have greatly improved the paper. In addition, I thank C´edric Argenton, Jan Boone, Jing-Yuan Chiou, Pascal Courty, Eric van Damme, Vincenzo Denicol` o, Guido Friebel, Elisabetta Iossa, Paolo Pin, Fran¸cois Salani´e, Mark Schankerman, Christoph Schottm¨ uller, Paul Seabright, Emanuele Tarantino, participants at the EVPAT summer school in Bologna, the “Knowledge for Growth” conference in Toulouse, the congress of the European Economic Association in Milan, the EARIE conference in Toulouse, the EPIP conference in Bern, the ASSET conference in Florence, as well as seminar participants at the Toulouse School of Economics, the European University Institute, the University of Alicante, Ludwig-Maximilians University Munich, the Centre for European Economic Research (ZEW), the University of Vienna, the Australian National University, and Tilburg University for helpful comments and suggestions. All errors are mine. Financial support from the 7th European Community Framework Programme through a Marie Curie Intra-European Fellowship (grant number PIEF-GA-2010-275032) is gratefully acknowledged. † Tilburg University (TILEC and CentER). Postal address: Tilburg University, Department of Economics, PO Box 90153, 5000 LE Tilburg, Netherlands. Email: [email protected].

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Introduction

Patent examiners at the U.S. Patent and Trademark Office (USPTO) receive a bonus that depends on the number of applications processed. But because a rejection is more timeconsuming than a grant, the bonus introduces a bias towards granting patents.1 Such a compensation scheme is puzzling. Apart from biasing the grant decision, it does not seem to give examiners good incentives to exert effort. Rejecting an application requires the examiner to come up with evidence that the claimed invention already exists or would have been obvious to someone skilled in the art. Granting a patent is much less demanding: the examiner can simply report not having found such evidence. If anything, shouldn’t we expect examiners to be rewarded for rejecting applications? Concerns about patent quality have recently given rise to intense policy debate.2 Observers bemoan that the USPTO is granting more questionable patents than other national patent offices, in particular the European Patent Office (EPO). Given that the EPO does not use any performance-based compensation (its examiners are paid a fixed wage only), the USPTO’s compensation scheme is a natural candidate for criticism and has often been cited as one of the causes of the quality issues (Jaffe and Lerner, 2004; Merges, 1999; Lemley and Shapiro, 2005). My analysis, however, suggests that the scheme may be a symptom rather than a cause of the problem. In this paper I develop a theoretical model of patent examination to study the design of incentives for examiners. I argue that examination can be described as a problem of moral hazard followed by adverse selection: the examiner must be given incentives to exert effort (looking for evidence to reject, within the prior art), but must also be given incentives to truthfully reveal the evidence he finds (or lack thereof). I show that the model can explain the puzzling compensation scheme in use at the USPTO, as well as variation in compensation schemes and patent quality across patent offices. To see why it can sometimes be optimal to reward a patent examiner for granting, consider the following situation. Suppose the examiner has discretion over the decision to accept or reject an application and wants to avoid mistakes. Ignore for a moment the decision how much effort to provide and focus on the grant decision. If the examiner believes that a large proportion of applications is bad, he will have little confidence that an application is good when a shallow search for evidence turns up nothing. Absent monetary incentives, his desire 1

For details of the compensation scheme in use at the USPTO, see Section 6. Both the Federal Trade Commission (2003) and the National Academy of Sciences (2004) authored influential reports voicing concerns about poor patent quality. Although in September 2011 the U.S. Congress passed a patent reform bill known as the “America Invents Act,” the quality issues are far from being resolved; see, e.g., Financial Times, “US reform: Overhaul fails to address system’s weaknesses,” June 14, 2012, available at http://www.ft.com/intl/cms/s/0/c9aeab12-b3bf-11e1-8b03-00144feabdc0.html. 2

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to avoid mistakes will then lead him to reject the application despite the lack of evidence. Inducing him to truthfully reveal the result of his search requires rewarding him for grants. In this situation, the adverse-selection problem is in conflict with the moral-hazard problem: the examiner may have to be rewarded for granting patents even though effort is positively correlated with producing evidence for rejection. Incentives are directed primarily towards inducing truthful revelation rather than inducing effort. Note however that they succeed in inducing at least some effort. In fact, ensuring truthful revelation is a prerequisite for effort provision: if the examiner anticipated not truthfully revealing the result of his search, there would be no reason for him to exert effort searching in the first place. This argument rests on two premises. First, for the examiner to have discretion over the grant decision, the signal that an application is bad must be soft information, i.e., unverifiable by the principal and third parties. This makes sense because of the technical complexity of patent applications, the vagueness of patentability criteria, and because there is little information on the quality of an examiner’s decisions in the short run. While more information becomes available in the long run (e.g., through court decisions on patent validity), this information is difficult to include in a contract. Second, examiners must have a desire to avoid mistakes that is unrelated to short-term monetary compensation. Such a desire might stem from long-term implicit incentives within the organization (promotion, dismissal, etc.), but also from recognition by peers or a concern for social welfare. With a slight abuse of language, I will refer to the desire to avoid mistakes as intrinsic motivation. In the model presented below, the government delegates patent examination to an examiner motivated by both extrinsic rewards (i.e., monetary transfers) and intrinsic rewards (i.e., utility gains from making correct decisions). The examiner must expend effort to obtain a signal about an applicant. If the applicant’s claimed invention is not truly new, the examiner can come up with a signal demonstrating the lack of novelty; I assume that the signal is soft information. Firms react to how rigorously they expect the examiner to screen applications. They submit fewer bad applications when they anticipate greater effort, which results in an endogenous application pool. This feature links intrinsic motivation to the proportion of good applications (via the examiner’s equilibrium effort), which is crucial for the model to be able to explain variation in compensation schemes. My analysis of the government’s choice of incentives and of the equilibrium of the examination game yields three main results. Proposition 1 shows that in the absence of intrinsic motivation, no examination effort can be sustained in equilibrium. The amount of effort that is implementable increases with the examiner’s intrinsic motivation. Proposition 2 demonstrates that for low levels of intrinsic motivation, the optimal incentive scheme rewards the examiner for granting patents, provided the proportion of good applications is sufficiently small when 2

applicants expect zero effort. Proposition 3 establishes a complementarity between intrinsic and extrinsic rewards: provided the proportion of good applications is sufficiently large at the first-best effort level, for high levels of intrinsic motivation it eventually becomes optimal to reward the examiner for rejecting, which feeds back positively into effort provision. Patent quality can be defined as the posterior probability that a patent issued by the patent office actually satisfies the criteria for patentability. The model then predicts that intrinsic motivation should be positively related to patent quality and negatively related to the use of short-term monetary rewards for granting patents. This prediction allows me to establish a link between observable organizational features of patent offices and observable outcomes of opposition and litigation involving patents issued by those offices. I argue that intrinsic motivation, when interpreted as implicit incentives within the patent office, is likely to be positively related to how long the examiner expects to stay at the patent office and to how timely information about the quality of his decisions becomes available. A comparison of the USPTO with the EPO shows important differences in examiner turnover and the availability of information on decision quality. The USPTO has problems retaining examiners and lacks an administrative review procedure comparable to the EPO’s opposition system that can provide timely information on quality. Thus, the model predicts that patents issued by the USPTO should be of lower quality than EPO patents, and that U.S. examiners should be more likely to be rewarded for granting through short-term compensation. In line with these predictions, examiners at the EPO receive a fixed wage, unlike their U.S. counterparts. And the available evidence from opposition and litigation, as well as the perception in the patent community, tend to confirm that problems with patent quality are more acute in the U.S. The basic model from which I derive my main results relies on reduced forms of applicant behavior, welfare, and implicit incentives within the patent office. To provide microfoundations and examine the robustness of my results, I extend the model in two ways. First, I introduce internal and external review of the examiner’s decisions, making the signal partially verifiable and allowing me to formally model implicit incentives. Proposition 4 shows that as long as transfers contingent on internal review cannot be unbounded, the essence of the previous results remains intact. Second, I allow the government to charge application fees, which are likely to be an important determinant of the quality of applications in practice. Proposition 5 derives modified conditions for the previous results to carry over, and Proposition 6 relates these to properties of the distribution of the returns to innovation. I argue that the required distributional assumptions are empirically plausible. The paper contributes to the literature on the optimal design of the patent system (see, e.g., Gilbert and Shapiro, 1990; Denicol`o, 1996; Cornelli and Schankerman, 1999; Scotchmer, 1999; Hopenhayn and Mitchell, 2001; Hopenhayn et al., 2006). More specifically, it is related 3

to a small number of recent papers investigating patent examination. Langinier and Marcoul (2009) and Atal and Bar (2010) study inventors’ incentives to search for and disclose relevant prior art to the patent office. R´egibeau and Rockett (2010) examine the optimal duration of patent examination as a function of the importance of an innovation. They find that, controlling for the position in the innovation cycle, more important innovations should be examined faster, a prediction which is borne out by evidence from a sample of U.S. patents. Caillaud and Duchˆene (2011) focus on the “overload problem” facing the patent office: when flooded with large numbers of applications, the average quality of examination declines, leading to a vicious circle by encouraging even more bad applications. All of these papers consider a benevolent patent office maximizing social welfare. Therefore, they are unable to make predictions about examiner compensation. The paper is also related to the literature on delegated expertise (Demski and Sappington, 1987; Gromb and Martimort, 2007). In this literature, a principal contracts with an expert to produce an unverifiable signal that is informative about the optimal project to undertake. Transfers to the expert can condition on the reported signal and the outcome of the chosen project. My model differs from this literature in several respects. My information gathering technology confounds the case where the expert has exerted low effort with the case where the project is good – in both cases, the signal is empty. This technology fits the specificities of patent examination and is crucial for the result that it may be optimal to reward the examiner for reporting no signal. I also assume (in the basic model) that transfers cannot be conditioned on outcome information. Instead, outcome information enters the expert’s utility through what I refer to as intrinsic motivation. Moreover, unlike in the literature on delegated expertise, the proportion of good projects (i.e., applications) is endogenous. The remainder of the paper is organized as follows. Section 2 presents the basic model. Section 3 studies the government’s choice of incentives and derives the equilibrium of the examination game. Section 4 considers internal and external review of the examiner’s decisions and endogenizes intrinsic motivation by modeling implicit incentives. Section 5 introduces application fees and provides micro-foundations for applicant behavior. Section 6 identifies empirical proxies for the strength of implicit incentives and provides evidence on patent quality and examiner compensation in Europe and the U.S. Section 7 summarizes the results of the model and comments briefly on policy implications. Proofs are relegated to the Appendix.

2

A simple model of patent examination

Consider the following setup. There are three types of players: a benevolent planner, a patent examiner, and firms. Firms file patent applications which can be either good (G), i.e., true

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inventions, or bad (B), i.e., non-inventions which already exist or would have been obvious to someone skilled in the art. Granting a firm patent protection generates a welfare gain of V > 0 if the firm’s application is good and a welfare loss of L > 0 if its application is bad. For both types of applications, the welfare effect of refusing patent protection is normalized to zero. To receive patent protection, an application must first pass an examination, which is administered by the examiner. The examiner. The examiner does not observe the type of the application but believes that a proportion p is good and a proportion 1 − p is bad. He conducts a prior-art search that allows him to receive a signal σ about the application. The distribution of the signal depends on the type of the application and on the examiner’s effort, which is unobservable. If the application is good (G), the examiner never obtains any signal (σ = ∅). If the application is bad (B), he obtains a perfectly informative signal σ = B with probability e, and no signal with probability 1−e, where e ∈ [0, 1] is the effort that he puts into patent examination. Such an asymmetry in the information gathering technology is inherent in patent examination: it is conceptually impossible to find evidence that something is new, i.e., that it has never been done before. One can only search the stock of existing knowledge for evidence that something (similar) has already been done. Since the stock of knowledge is large, the search can never cover its entirety (or only at prohibitive costs). Accordingly, in practice examiners have to search for prior art, i.e., previously published literature (patents, scientific articles, etc.) proving that the claimed invention was already known or obvious (so that it does not satisfy the patentability standards of novelty and non-obviousness).3 I make the following assumption on the nature of the signal: Assumption 1 (Soft information). Patent examination produces soft information: the signal σ = B is unverifiable by the planner or third parties. Patent applications are inherently technical and have increased in complexity over time. Moreover, patentability criteria such as the non-obviousness standard are somewhat vague, ill-defined concepts. As noted by Jaffe and Lerner (2004, p. 172), “there is an essentially irreducible aspect of judgment in determining if an invention is truly new. After all, even young Albert Einstein faced challenges while assessing applications (...) in the Swiss Patent Office.” Because of ambiguity in patentability criteria and the technical complexity of applications, patent office management cannot easily verify an examiner’s signal. The assumption of soft 3

This is reflected in the patent statutes and their interpretation by the courts. For example, title 35 of the U.S. code specifies in §102 that “A person shall be entitled to a patent unless” the invention was previously known. The courts have interpreted this language as requiring the patent office to accept an application unless it can demonstrate that the claimed invention does not meet the patentability criteria (FTC, 2003, Ch. 5, p. 8).

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information thus arguably represents a good approximation of reality. In Section 4, I relax this assumption by introducing an internal review making the signal partially verifiable. The examiner has utility U = t + αy − γ(e), where t is the monetary transfer he receives from the planner, y is an intrinsic reward, α ≥ 0 is a parameter measuring the strength of intrinsic rewards, and γ(e) is the cost of effort (increasing and convex with γ(0) = γ 0 (0) = 0 and γ 0 (1) = ∞). I assume that the examiner is protected by limited liability (i.e., transfers must be non-negative). The value taken by the intrinsic reward y depends on the type of application and the approval decision, as indicated in Table 1.

Decision Grant Rejection

Application Good Bad yG 0 0 yB

Table 1: Intrinsic rewards Assumption 2 (Intrinsic motivation). Intrinsic rewards satisfy yG > 0 and yB > 0. According to Assumption 2, the examiner derives an intrinsic reward from accepting good applications and from rejecting bad ones.4 This reward structure formalizes the idea that the examiner cares about making the right decision. Several interpretations are possible, one of which, pursued further in Section 4, is that some information about the quality of an examiner’s decisions may transpire over time. Although this information cannot be contracted on, it can be used in subjective performance evaluation and thus be brought to bear on promotion and dismissal decisions, which are part of the organization’s implicit incentives. The information may also be learnt by the examiner’s peers, whose esteem he may value. Alternatively, the examiner may have genuine intrinsic motivation, i.e., he may care about the impact of his decisions on others (in this context, particularly technology users). The planner. The planner’s objective is to maximize social welfare. For the moment, her only instrument is choosing an incentive scheme for the examiner; in Section 5, I consider application fees as an additional instrument. The incentive scheme consists of a transfer t ≥ 0 and a grant rule x ∈ {0, 1}, specifying whether a patent is granted (x = 1) or not (x = 0), conditional on the examiner’s report.5 I assume that the shadow cost of public funds is zero. 4

The fact that the top-right and lower-left fields are set to zero is a normalization. All that matters for the examiner’s decision is the comparison between the intrinsic rewards of granting and rejecting a given type of application. Note that the expected intrinsic reward also depends on the examiner’s posterior belief that an application is good given the result of his prior-art search. 5 Thus, I restrict attention to deterministic grant rules. A justification for this restriction is that probabilistic grant rules (0 < x < 1 for some report) are subject to commitment problems: to avoid deadweight loss, the patent office may be tempted to refuse patent protection even to good applicants. The patent office can more easily commit to a deterministic grant rule because deviations from the rule are then readily detected.

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Firms. The patent office is likely to receive fewer bad applications when applicants anticipate more rigorous examination. In Section 5, I derive the behavior of potential applicants from an explicit model of filing strategies. For now I will adopt a reduced-form approach that relies on assumptions about how the number of good and bad applications depends on the examiner’s effort. Assumption 3 (Endogenous application pool). The number of good applications is NG > 0 for all e. The number of bad applications is determined by a continuously differentiable function NB (e) ≥ 0 satisfying NB (0) > 0 and NB0 ≤ 0. In words, the number of good applications is strictly positive and does not depend on e. The number of bad applications is strictly positive in the absence of examination and decreases with e.6 I will interpret the function NB (e) as the applicants’ best-response function. Section 5 explains why this interpretation makes sense. A direct consequence of Assumption 3 is that the quality of the pool from which applications are drawn is endogenous. The proportion of good applications is given by p(e) = NG /(NG + NB (e)), with p0 ≥ 0. So far, nothing in the model ensures that having a patent system is worthwhile. If bad applications are common and the cost of bad patents is large, society may prefer to abolish the patent system altogether (which the planner can accomplish by setting x = 0 irrespective of the examiner’s report). For expositional convenience, I assume in what follows that the patent system is socially desirable, including for low levels of examination effort: Assumption 4 (Desirability of the patent system). NG V ≥ NB (0)L. Essentially, this assumption says that a registration system, in which applicants simply register patents without examination, is welfare superior to not having a patent system at all.7 Timing. The timing of the game is as follows (see Figure 1). At the beginning of the game, the planner selects an incentive scheme. Then, firms file applications, and the examiner decides how much examination effort to provide. Finally, signals are drawn, acceptance and rejection decisions are made, and payoffs are realized. The important assumption here is that the examiner cannot commit to a level of examination effort e before potential applicants For similar arguments, see Khalil (1997) in the context of auditing, and Nocke and Whinston (2011) in the context of merger review. 6 In principle, the number of applications should also depend on the grant rule x. I have chosen to neglect this dependence here to simplify the exposition. Clearly, if the planner never grants a patent to any applicant (i.e., if x = 0 regardless of the examiner’s report), the number of applications should be zero. If the planner grants patents to all applicants (x = 1 regardless of the report), the number of bad applications should be NB (0) and the number of good applications NG . 7 Note that, as documented by Lerner (2005), many countries use (or have in the past used) a registration system, leaving the determination of validity entirely to the courts.

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Planner chooses incentive scheme.

Firms file patent applications.

Patent examiner chooses e.

T =0

T =1

T =2

Signal σ ∈ {B, ∅} realized. Acceptance/rejection. Payoffs realized. time T =3

Figure 1: Timing of the game make filing decisions. This implies that the examiner does not take into account the effect of his effort on the proportion of good and bad applications.

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Designing incentives for the examiner

In this section, I study the optimal design of incentives for the examiner. The planner faces a problem of moral hazard followed by adverse selection: the examiner’s effort determines the distribution of “types” (in this case, the distribution of signals). We can work backwards from the adverse-selection stage and invoke the revelation principle, according to which a direct revelation mechanism is without loss of generality. The planner offers a menu of contracts (tσ˜ , xσ˜ ), where σ ˜ ∈ {B, ∅} is the signal reported by the examiner. That is, the planner asks the examiner to report his signal σ. If he reports B, the planner pays tB and grants a patent according to the rule xB . If he reports ∅, the planner pays t∅ and grants a patent according to the rule x∅ . The following lemma streamlines the exposition. Lemma 1 (Grant rules). Under Assumption 4, restricting attention to grant rules such that xB = 0 and x∅ = 1 entails no loss of generality. According to Lemma 1, we can restrict attention to grant rules whereby the applicant is granted a patent if and only if the examiner reports σ = ∅. This result is based on three observations. As shown in the proof, a necessary condition for incentive compatibility is that an application for which the examiner finds signal B has a weakly lower probability of acceptance than one for which he finds no signal (xB ≤ x∅ ). This eliminates the counterintuitive rule xB = 1−x∅ = 1. Assumption 4 implies that abolishing the patent system (xB = x∅ = 0) cannot be optimal either. Finally, setting xB = x∅ = 1 amounts to a registration system, which the planner can replicate by setting xB = 1 − x∅ = 0 and inducing zero effort.8 Because by Lemma 1 there is a one-to-one relationship between the examiner’s report and the decision to grant or reject, the decision can be interpreted as being his. In what follows, I will therefore use the terms “reporting B” and “rejecting” as well as the terms “reporting ∅” and “granting” interchangeably. 8

Recall that e = 0 implies σ = ∅ with probability 1.

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Consider the case where the examiner has exerted equilibrium effort e∗ > 0 and come up with signal σ = B. For him to prefer to report B, it must be the case that tB + αyB ≥ t∅ .

(1)

Given signal B, he knows with certainty that the application is bad, but he only enjoys the intrinsic reward from rejection, αyB , if he reports B. If, on the other hand, the examiner obtains no signal (σ = ∅), he will prefer to report ∅ provided t∅ + αˆ pyG ≥ tB + α(1 − pˆ)yB ,

(2)

where pˆ ≡ Pr[G|∅] is the examiner’s posterior belief that the application is good given that he has found no evidence to the contrary. His expected intrinsic reward from granting is αˆ pyG , while that from rejecting is α(1 − pˆ)yB . Turning to the moral-hazard stage, suppose the examiner anticipates truthfully revealing the signal he finds. He then chooses e to maximize   p[t∅ + αyG ] + (1 − p) e[tB + αyB ] + (1 − e)t∅ − γ(e).

(3)

With probability p, the application is good, so that he cannot find any grounds for rejection. The transfer he receives is t∅ , and the intrinsic reward is αyG . With probability 1 − p, the application is bad, for which he finds evidence with probability e. He is paid tB and enjoys an intrinsic reward of αyB . With probability 1 − e, the examiner finds no evidence. He receives a transfer of t∅ and no intrinsic reward. Differentiating with respect to e leads to the first-order condition (1 − p)[tB − t∅ + αyB ] = γ 0 (e).

(4)

This equation defines the examiner’s best-response function, determining his effort as a function of the proportion of good applications. It follows from (4) that, for a given p, effort is increasing in tB −t∅ . Moreover, positive examination effort is only sustainable if the examiner expects there to be some bad applications (p < 1). A final set of constraints comes from the possibility that the examiner deviates from both the equilibrium effort and truthful reporting. Two cases are relevant: always reporting ∅, and always reporting B.9 In both cases, choosing e = 0 is optimal (if the examiner anticipates that his report will not depend on his signal, there is no point in exerting effort). To rule out such double deviations, the equilibrium utility with truthful reporting must be larger than 9

A third strategy, which would consist in always reporting the opposite of the signal found, leads to an optimal effort of zero under condition (1), and therefore reduces to the strategy of always reporting B.

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the utility with zero effort and either report (∅ or B). Letting U ∗ denote the examiner’s equilibrium utility, we must have t∅ + αpyG ≤ U ∗ tB + α(1 − p)yB ≤ U ∗ ,

(5) (6)

  with U ∗ = p[t∅ + αyG ] + (1 − p) e∗ [tB + αyB ] + (1 − e∗ )t∅ − γ(e∗ ). Social welfare when the examiner reports truthfully and xB = 1 − x∅ = 0 is given by W (e) ≡ NG V − NB (e)(1 − e)L − (NG + NB (e))γ(e). The three terms represent the social value of innovation, the social cost of bad patents, and the cost of patent examination. An increase in e reduces the number of bad applications submitted (NB ) and the share of bad applications which obtain patent protection (1 − e). At the same time, it increases the examination cost per application (γ). The planner solves10 max

tB ≥0,t∅ ≥0

W (e∗ )

subject to

(1), (2), (5), (6), and

γ 0 (e∗ ) = (1 − p∗ )[tB − t∅ + αyB ] NG . p∗ = NG + NB (e∗ )

(7) (8)

Equations (7) and (8) state that (e, p) must be an equilibrium of the examination game, i.e., examination effort e and the proportion of good applications p, summarizing applicant behavior, must be best responses to each other. As the following lemma shows, equilibrium exists and is unique. Lemma 2 (Existence and uniqueness). Suppose that xB = 1−x∅ = 0 and that (tB , t∅ ) satisfy (1), (2), (5), and (6). Then, there exists a unique equilibrium (p∗ , e∗ ) of the examination game characterized by (7) and (8). Constraints (1), (2), (5), and (6), together with the equilibrium conditions (7) and (8), determine which levels of e the planner can implement through an appropriate choice of transfers. The next lemma provides a simpler representation of the planner’s problem. Lemma 3. The planner’s problem is equivalent to max W (e) e≥0

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subject to

g(e) ≤ α(yG + yB ),

I have omitted the examiner’s individual-rationality constraint,   p[t∅ + αyG ] + (1 − p) e[tB + αyB ] + (1 − e)t∅ − γ(e) ≥ u,

(9)

(IR)

where u is the examiner’s outside opportunity. Since public funds are assumed to be costless and the examiner risk-neutral, this constraint does not play any role. The solution of the relaxed problem will determine tB − t∅ ; the absolute levels can be obtained by combining (IR) with the limited liability constraints.

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where

    NG NB (e) 0 [(1 − e)γ (e) + γ(e)] + 1 + γ 0 (e). g(e) ≡ 1 + NG NB (e)

(10)

According to Lemma 3, it is as if the planner could choose e directly, but is limited by the constraint g(e) ≤ α(yG + yB ), which implicitly defines the set of implementable effort levels. The key element in the proof of this result is that constraints (5) and (6) impose a lower and an upper bound on transfers, given by − αyB +

γ(e) γ(e) ≤ tB − t∅ ≤ α[ˆ pyG − (1 − pˆ)yB ] − . e(1 − p) p + (1 − p)(1 − e)

(11)

Intuitively, soft information gives the examiner discretion over the signal he reports. If we pay him too much for reporting B, he will always report B. If we pay him too much for reporting ∅, he will always report ∅. In both cases, it is not worthwhile for him to exert effort because he knows his report will not depend on his signal. To be willing to exert effort, he must anticipate truthfully revealing the signal he finds and obtaining a sufficiently large equilibrium utility. Monetary incentives can only induce additional effort to the extent that they do not make it too tempting to deviate. In particular, the upper bound imposed by (6) limits the power of incentives. Evaluating tB −t∅ at its upper bound given by (11), the examiner’s best response function can be used to determine the maximum level of effort that can be implemented for a given p. This maximum effort is implicitly defined by γ 0 (e)[1 − e(1 − p)] + (1 − p)γ(e) = p(1 − p)α(yG + yB ).

(12)

This result is illustrated in Figure 2, where the blue curve corresponds to equation (12). Its inverted-U shape has an intuitive explanation. If p = 0 or p = 1, the examiner knows in advance whether he is facing a good or bad application. There is no point in exerting effort to acquire information that is redundant; thus, only e = 0 is implementable. Noticing that an equilibrium needs to be a best response for applicants as well, and that the applicants’ best response depends on e, the maximum amount of effort that can be implemented – denoted e¯ in Figure 2 – is found at the intersection of the blue curve with the p(e) curve. The e(p) curve in the figure corresponds to the examiner’s best-response function, implicitly defined by (4), evaluated at some feasible pair of transfers; it is shown for illustrative purposes. The maximum effort that the planner can implement depends on α. The constraints caused by soft information mean that extrinsic rewards can only play a limited role in inducing effort, thus assigning a crucial role to intrinsic motivation, as the following proposition shows. Proposition 1 (Importance of intrinsic motivation). If α = 0, no examination effort can be sustained in equilibrium. An increase in α weakly increases welfare and strictly increases the maximum implementable effort. 11

p

6 1 (e, p) such that (12) holds



Q k Q p(e)





e(p) p∗

0

e∗

-e

e¯

Figure 2: The maximum implementable effort Some amount of intrinsic motivation is essential for effort provision. If α = 0, no effort is implementable, e = 0.11 An increase in intrinsic motivation relaxes the constraint on e, which makes it possible to induce greater examination effort. An examiner who cares more about making the right decision can be induced to exert more effort, whatever the proportion of good and bad applications (as long as 0 < p < 1). The following proposition relates to the optimal compensation scheme for moderate levels of intrinsic motivation. Let eα denote the solution to the planner’s problem, (9). Proposition 2 (Rewarding grants). Suppose yB /(yB + yG ) > p(0).

(13)

Then, for α small but strictly positive, eα > 0 and tB < t∅ . Proposition 2 says that, when intrinsic motivation is low and yB /(yB + yG ) > p(0), the compensation scheme rewards the examiner for granting. Were he not compensated for granting by means of a monetary transfer, the examiner would reject all applications. The intuition is that in an equilibrium in which effort is low and the proportion of bad applications relatively large, the best the examiner can do to avoid mistakes is reject everything. If the planner wants to make the examiner truthfully reveal his signal, she must reward him for granting. Moreover, unless the examiner anticipates being truthful, he will not exert any effort. Welfare is increasing in e at e = 0 as W 0 (0) = (NB (0)−NB0 (0))L−(NG +NB (0))γ(0)− NB0 (0)γ 0 (0) > 0 because NB (0) > 0 and NB0 ≤ 0 by Assumption 3 and γ(0) = γ 0 (0) = 0. Thus, the planner wants to induce e > 0. By rewarding the examiner for granting, she gets 11 This result would also hold if I allowed for random grant rules. It does not hold in the presence of internal review, however; see Section 4.

12

him to exert positive (albeit low) effort. Condition (13) says that rejecting bad applications needs to give the examiner a sufficiently high intrinsic reward, relative to allowing good ones. The ratio of intrinsic rewards must be large compared to the proportion of good applications in the absence of examination. The next proposition considers what happens as intrinsic motivation becomes larger. Let eo

denote the unconstrained maximizer of the planner’s objective, i.e., eo ≡ arg max W (e). e

Proposition 3 (Complementarity). Suppose yB /(yB + yG ) < p(eo ).

(14)

Then, there exists α ˆ ≥ 0 such that tB > t∅ for all α > α ˆ. The model yields a complementarity between intrinsic and extrinsic rewards: higher intrinsic motivation increases the examiner’s effort not only by itself, but also by allowing the planner to use monetary incentives more effectively. Assuming yB /(yB + yG ) < p(eo ), for sufficiently large values of α it is possible to reward the examiner for rejecting applications without impeding truthful revelation. When the proportion of good applications is large and the examiner has more confidence in the result of his prior-art search, he is no longer tempted to reject everything. Rewarding rejection then has a positive feedback effect on effort. As shown in the proof of Proposition 3, there exists a threshold level of p above which rewarding rejection is optimal. The threshold is bounded above by yB /(yB + yG ). Condition (14) ensures that the first-best effort yields a p exceeding the threshold. Then, there exists α ¯ such that a level of effort eˆ with p(ˆ e) > yB /(yB + yG ) is implementable, and for levels of intrinsic motivation exceeding some value α ˆ<α ¯ , the planner pays the examiner for rejecting. For exogenous variation in α to be able to explain a difference in the compensation schemes used by two patent offices operating under otherwise identical conditions, conditions (13) and (14) must be satisfied simultaneously, holding all other parameters constant. If this is the case, the following corollary applies. Corollary. If p(0) < yB /(yB + yG ) < p(eo ), then tB < t∅ for α sufficiently small and tB > t∅ for α sufficiently large. Note that if p were exogenous and constant, only one of the conditions (13) and (14) could hold (so that either Proposition 2 or 3 would apply). The assumption of an endogenous quality pool (Assumption 3) is crucial for the corollary, which relies on p(0) < p(eo ). Patent quality in this model can be measured by the posterior probability that an issued patent is good, pˆ ≡ p/[p + (1 − p)(1 − e)]. As pˆ is increasing in both e and p, patent quality increases as α goes from 0 to α ¯. 13

Hard information. It is instructive to compare the results in this section to what happens when, contrary to Assumption 1, the signal σ = B is hard information. The planner then faces a simple moral hazard problem for which standard results apply. She should pay the examiner for coming up with a verifiable signal that the application is bad. The intuition is that σ = B is informative about the examiner’s effort, whereas σ = ∅ is not. Risk neutrality and costless public funds mean that the planner can achieve the first best (or an arbitrarily close approximation thereof), irrespective of the examiner’s intrinsic motivation.12 Even under the favorable assumptions of risk neutrality and costless funds, soft information constrains what the planner can achieve and lends importance to intrinsic motivation. The next section examines the intermediate case in which the signal is partially verifiable.

4

Internal and external review of the examiner’s decision

The analysis in the preceding sections assumes that the examiner’s signal is unverifiable for the planner, so that transfers cannot be contingent on the quality of the examiner’s decisions. In this section, I extend the model by allowing for the possibility of internal and external review, both of which generate information about whether an application handled by the examiner is good or bad. Internal review takes place within the patent office while external review comes in the form of court decisions on the validity of issued patents. My modeling emphasizes one particular interpretation of what I have termed intrinsic motivation in the previous analysis, namely, the implicit incentives within the patent office. I assume that the outcome of the internal review can be contracted on, so that transfers to the examiner can depend on it. The outcome of the external review cannot be contracted on, but the planner can decide not to rehire the examiner if the courts invalidate a patent he had granted. This setup allows me to formally link the parameter α from the basic model to the availability of an alternative employment opportunity for the examiner, which determines the effectiveness of implicit incentives. Throughout this section I ignore the effects of internal and external review on innovation and bad patents. I am only interested in the role these two types of review play in designing the incentive scheme for the examiner. Assume there are two periods. In each period, the timing is as depicted in Figure 1, except for stage T = 3, which is modified as follows. After the acceptance/rejection decision, but prior to transfers being paid to the examiner, an internal review occurs with probability η. If the application handled by the examiner was bad, the internal review discovers this 12

The only potential impediment to achieving the first best in this model is the following. Ex ante the planner may want to implement an allocation such that all bad applications are deterred, in which case the first best would entail p = 1. But ex post, if p = 1 there are no bad applications, so the examiner has no incentive to search for σ = B no matter how powerful the incentive scheme is. In that case, the planner can achieve an allocation that is close to but not equal to the first best.

14

with probability 1. The examiner is paid trB if the review shows that he correctly rejected the application, tr∅ if the review shows that he correctly accepted the application, and zero otherwise. With probability 1 − η, there is no internal review; the examiner is paid tB for rejecting and t∅ for accepting the application. Between the two periods, the following additional events take place: 1. With probability 1 − α, an alternative employment opportunity for the examiner comes along, which would yield him a second-period payoff of U . With probability α, there is no such opportunity (his outside option is zero). The examiner decides whether to leave the patent office. If he stays, the alternative employment opportunity disappears. 2. If the examiner accepted the patent application he handled in period 1, the patent is litigated. If the patent is bad, it is invalidated by the courts. 3. The planner decides whether to rehire the examiner for the second period. There is no time discounting. The important assumption here is that the parties cannot sign a long-term contract. Several of the other simplifying assumptions could easily be relaxed. Although internal review is assumed to be random and its probability of occurrence (η) exogenous, this can be interpreted as reflecting the cost of conducting the review (in a richer setting, η could be a choice variable for the planner). Both internal and external review are perfect, in the sense that no mistakes are made in assessing the patentability of an invention. More generally, it would suffice that a review reaches the correct conclusion more than half the time. Similarly, the assumption that court review always occurs when a patent is granted (and never when it is rejected) is not essential for the results below. Finally, the incentive scheme is assumed to reward the examiner when the internal review confirms his decisions, and pay the lowest possible transfer (zero) if the review reveals mistakes. This is clearly optimal and thus simplifies the exposition. Assumption 5. The transfer paid to the examiner if internal review confirms an acceptance decision is bounded from above: tr∅ ≤ t¯, with t¯ > 0. An upper bound on tr∅ can be justified, for instance, by the threat of collusion between the examiner and the supervisor who carries out the internal review (Tirole, 1986). Suppose that when the application is bad the internal review generates a verifiable signal B and otherwise generates ∅. (Thus, the supervisor can suppress the signal B but cannot fabricate B if he finds ∅.) Then, a large transfer tr∅ makes it tempting for the supervisor and examiner to collude in case the examiner reports ∅ and the supervisor discovers B.13 There might also be other reasons for an upper bound on transfers. Bureaucracies such as the patent office face 13 For a model in which the principal optimally reduces the power of the agent’s incentives to reduce the stake of collusion, see Laffont and Tirole (1991).

15

budgetary constraints which might render extremely large transfers infeasible. Alternatively, large transfers handed out “at random” might create resentment among other examiners, thereby damaging morale. This is a minimal assumption to keep the analysis interesting. Similar results could be derived if all transfers (not only tr∅ ) were capped. Even in the presence of internal review, moral hazard under limited liability implies that if the examiner stays at the patent office in period 2, he obtains a strictly positive information rent U2∗ > 0. The examiner would thus like to be rehired (at least if no alternative employment opportunity comes along). The planner can use this to provide implicit incentives by committing not to rehire the examiner when either the internal or the external review reveals ¯ > U ∗ , i.e., the alternative employment opportunity that he made a mistake. Assume that U 2

is more lucrative than the examiner’s rent from staying at the patent office. This means that α is a measure of the effectiveness of implicit incentives and plays the same role as intrinsic motivation in the previous analysis. Proposition 4. When η > 0, the planner can induce strictly positive equilibrium effort even for α = 0. Suppose the planner wants to minimize the examiner’s rent subject to inducing some level e > 0 (with p(e) < 1). Under Assumption 5, there exists ηˆ > 0 such that tB < t∅ for η < ηˆ and tB = t∅ = 0 for η ≥ ηˆ. The threshold ηˆ decreases with α. Proposition 4 contains three results. First, internal review allows the planner to induce effort provision through explicit incentives even when implicit incentives are absent. Second, assuming there is an upper bound on tr∅ , implementing a given e > 0 may still require rewarding the examiner for granting if the frequency of internal review is sufficiently low. As η increases, the need to reward granting diminishes, and for η greater than a threshold ηˆ, transfers should condition exclusively on the outcome of internal review. Third, an increase in the effectiveness of implicit incentives makes it less likely that the examiner has to be rewarded for granting. This shows that, although internal review softens Proposition 1, the flavor of the previous results is preserved as long as internal review does not occur too frequently (perhaps because conducting review with high frequency would be too costly).14

5

Application fees and applicant behavior

In this section, I enrich the basic model in two ways. First, I develop an explicit model of applicant behavior, endogenizing the function NB . Second, I introduce application fees as an additional instrument for the planner. This is an important robustness check. If the planner were to charge prohibitive application fees (deterring all bad applications) when only low 14

If trB were also subject to an upper bound, one could obtain a result where tB > t∅ is part of an optimal incentive scheme (as in Proposition 3) and where the maximum implementable effort increases with α.

16

levels of examination effort are implementable, this would undermine the previous arguments relying on the proportion of bad applications being large. Assume firms can file applications by drawing on an exogenously given stock of n technologies. Some of the technologies have already been developed, while others have not. I will refer to the latter as innovative technologies and to the former as known technologies. Innovative technologies allow firms to file good patent applications. Known technologies only allow them to file bad applications. The number of innovative technologies is nG > 0, while the number of known technologies is nB > 0, with nG + nB = n. Technologies are indexed by a profitability parameter π drawn from a continuously differentiable cumulative distribution function F with density f on [π, π], where 0 ≤ π < π. For a firm to appropriate π, the technology has to be protected by a patent. Applying for a patent requires payment of an application fee φ ≥ 0, set by the planner.15 An innovative technology π, once developed and patented, yields the patent holder an expected payoff of π−φ. A known technology that is patented yields the patent holder a payoff of βπ − φ. The parameter β ∈ (0, 1) reflects the difference in the probability of enforcement between good and bad patents. The idea is that good patents (i.e., patents on innovative technologies) are more likely to be enforced by the courts than bad patents (patents on known technologies). Innovative technologies that firms decide to develop create social benefits; bad patents create social costs. The social benefit from an innovative technology π is given by v(π) ≥ 0, with strict inequality for π > π. A patent on a known technology π causes a social cost of `(π) > 0, which might include deadweight loss and litigation costs. Notice first that when the planner has the application fee as an additional instrument, a registration system in which the fee is optimally chosen weakly dominates not having a patent system at all. The planner can always choose φ so large that no bad applications are submitted; it suffices to set φ ≥ βπ. Thus, the equivalent of Assumption 4 will be satisfied, and the result of Lemma 1 continues to apply: it is without loss of generality to restrict attention to grant rules x∅ = 1 and xB = 0. I now relate the proportion of good and bad applications to e and φ and derive the welfare function. Firms develop innovative technologies if and only if π − φ ≥ 0. This defines ˆG (φ) a threshold π ˆG (φ) = min{π, max{π, φ}} such that innovative technologies with π ≥ π are developed (and patented). Firms apply for patents on known technologies if and only if (1 − e)βπ − φ ≥ 0. This defines a threshold π ˆB (e, φ) = min{π, max{π, φ/((1 − e)β)}} such that known technologies with π ≥ π ˆB (e, φ) are submitted to the patent office. Firms 15 The model can be generalized to allow for an alternative appropriation mechanism for innovative technologies (e.g., secrecy). It can also be augmented by an explicit model of R&D investment a ` la Cornelli and Schankerman (1999). Neither of these variations change the qualitative results. For details, see an earlier working paper version (Schuett, 2011).

17

correctly anticipate the examiner’s equilibrium effort; they self-screen and submit fewer bad applications the more often they expect to be caught. It follows that the number of good applications is NG (φ) ≡ nG [1 − F (ˆ πG (φ))] and the number of bad applications is NB (e, φ) ≡ nB [1 − F (ˆ πB (e, φ))]. The social welfare function is given by Z π Z π v(π)dF (π) − nB (1 − e) `(π)dF (π) − [NG (φ) + NB (e, φ)] γ(e). W (e, φ) ≡ nG π ˆG (φ)

π ˆB (e,φ)

(15) As in the simpler model of Section 2, welfare is the sum of three terms: the social value of innovation, the social cost of bad patents, and the cost of examination. The planner’s problem becomes max

φ≥0,tB ≥0,t∅ ≥0

W (e∗ , φ)

subject to

(1), (2), (5), (6), and

γ 0 (e∗ ) = (1 − p∗ )[tB − t∅ + αyB ] NG (φ) . p∗ = NG (φ) + NB (e∗ , φ)

(16) (17)

In the remainder of this section, I revisit the results of Section 3 within the more general framework presented here. Proposition 1 continues to hold; it is easily verified that its proof does not depend on the presence of application fees or the shape of the welfare function. To extend Propositions 2 and 3, I impose the following assumption: Assumption 6. The parameters satisfy π/π < β and v(π) = 0. The density function satisfies f (π) > 0 for all π ∈ [π, π) and f (π) = 0. The assumption that π/π < β means that the planner faces a tradeoff in setting the application fee when e = 0: she cannot deter all bad applications without also deterring some good ones. The assumption that f (π) = 0 formalizes the idea that there are few technologies at the top of the distribution, so that the welfare cost of decreasing the fee slightly below the level where all bad applications are deterred is second order whereas the benefit from inducing more firms to develop innovative technologies is first order. Assumption 6 thus ensures that the optimal application fee when e = 0 is interior. In other words, the presence of a tradeoff between good and bad applications makes it plausible that the planner will not charge prohibitive application fees even when she can only implement low levels of examination effort. Let (eα , φα ) denote the solution to the planner’s problem for a given α, and let (eo , φo ) denote the unconstrained maximizer of W (e, φ). The following analogue to Propositions 2 and 3 can then be established. Proposition 5. Suppose the planner can charge an application fee φ ≥ 0. (i) If yB /(yB + yG ) > p(0, φ0 ), then for α small but strictly positive, eα > 0 and tB < t∅ . 18

(ii) If yB /(yB + yG ) < p(eo , φo ), then there exists α ˆ such that tB > t∅ for all α > α ˆ. Proposition 5 shows that the results from the basic model carry over to a setting in which the planner can charge firms an application fee. The conditions for the incentive scheme to reward the examiner for granting or rejecting now have to take into account that the planner adjusts the fee to the level of intrinsic motivation α. Claim (i) is based on the proportion of good applications that would be filed in an optimal registration system, in which there is no examination (e = 0) and the fee is set optimally (φ = φ0 ). Claim (ii) is based on the proportion of good applications filed in the hypothetical situation where the planner is not constrained by agency problems and can choose examination effort directly. For both Claims (i) and (ii) to be potentially applicable at the same time, it must be the case that p(0, φ0 ) < p(eo , φo ). Exogenous variation in α can then explain differences in compensation schemes and patent quality. Unlike in the basic model, here the proportion of good applications depends on both e and φ; p being increasing in e therefore no longer suffices for the inequality to hold. Sufficient conditions are given in the next proposition. Proposition 6. The proportion of good applications satisfies p(0, φ0 ) < p(eo , φo ) if either of the following holds: (i) π follows the Pareto distribution; (ii) the distribution of π has an increasing hazard rate, and φo ≥ φ0 . As α increases, the planner tends to implement greater effort, but may increase or decrease φ. Moreover, while p unambiguously increases with e, it may increase or decrease with φ. In general, therefore, the net effect of increasing α on p is ambiguous. Proposition 6 identifies two particular cases in which the effect is clear. If π follows the Pareto distribution, a change in φ leads to the same percentage change in the number of both good and bad applications, so that p does not vary with φ. If the distribution of φ has an increasing hazard rate, p increases with φ. Thus, if φo ≥ φ0 , the change in the fee reinforces the effect of increased effort. Both of these cases are interesting. Empirical studies usually find that the observed distribution of returns to innovation is best fit by either the Pareto or the lognormal distribution (Pakes and Schankerman, 1984; Schankerman and Pakes, 1986; Harhoff et al., 2003; Silverberg and Verspagen, 2007). Although the hazard rate associated with the lognormal is nonmonotonic (it is first increasing and then decreasing), many other distributions with singlepeaked densities are characterized by increasing hazard rates; see Bagnoli and Bergstrom (2005). Note also that the condition φo ≥ φ0 is sufficient but not necessary. More generally, when the hazard rate is increasing, p(0, φ0 ) < p(eo , φo ) as long as φo is not too small.

19

6

Intrinsic motivation, patent quality, and examiner compensation in the U.S. and Europe

According to the theory, higher intrinsic motivation should be associated with higher patent quality and make it less likely that examiners are rewarded for granting. This section identifies determinants of intrinsic motivation and provides empirical evidence on patent quality and examiner compensation in Europe and the U.S. consistent with the theoretical predictions. Determinants of examiner motivation. Section 4 suggests that implicit incentives within the patent office can play the role of what is termed intrinsic motivation in the basic model. If the examiner cares about correct decisions in part because they affect his future with the patent office, a case can be made that how much he cares depends on how long he expects to stay at the patent office. He is likely to care more if he expects to stay long-term because, in the long run, more information about the quality of his decision-making becomes available. For the same reason, the strength of implicit incentives depends on the precise meaning of “long run.” That is, how timely does information about the examiner’s decisions become available? If such information only becomes available after the examiner has left the patent office, it cannot be used within the implicit incentive system. On both of those dimensions, the U.S. and European patent offices differ considerably. At the EPO, examiners usually stay for a long time, whereas at the USPTO, examiners often leave after short periods of time, making long-term incentives largely irrelevant. Friebel et al. (2006) report that 25 percent of EPO examiners had been at the office for more than fifteen years, compared with only 10.2 percent of USPTO examiners. The USPTO’s problems in hiring and retaining examiners have been extensively documented; see, e.g., GAO (2007), finding that more than 1600 examiners left the USPTO from 2002 to 2006 (in 2006, the USPTO employed a total of about 4800 examiners), and that 70 percent of those that left had been at the agency for less than five years. The EPO also has the edge in terms of timely information about decision quality, thanks to its widely-used opposition system. Opposition allows private parties to mount a challenge against questionable patents through the patent office itself. It can be triggered within nine months after a patent grant. The opposition procedure produces faster results than judicial review through the court system. Although the USPTO has a similar procedure called re-examination, it is rarely used. According to Graham et al. (2002), during the 1981-1998 period, only 0.2 percent of patents were reexamined in the U.S., whereas 8.3 percent were opposed in Europe. Patent quality. Even though claims about how badly the USPTO performs compared to other national patent offices are legion and anecdotal evidence about questionable patents abounds, there is surprisingly little systematic empirical evidence about patent quality. The

20

available evidence comes from U.S. court decisions and the European opposition procedure. Although the numbers cannot easily be compared and should be interpreted with caution, they nevertheless lend some support to the perception that U.S. patents are of lower quality than European ones. Allison and Lemley (1998) study the population of all 299 final validity decisions in district courts and the Court of Appeals for the Federal Circuit (CAFC) between 1989 and 1996. They find that 46 percent of patents were held invalid. Similarly, in a study of 182 validity decisions by the CAFC between 1997 and 2000, Cockburn et al. (2002) report that 50 percent of patents where found to be invalid. The European opposition procedure is studied by Graham et al. (2002) who use a sample of 2021 opposed patents that were granted between 1980 and 1997. They find that opposition resulted in revocation of the patent in 35.1 percent of the cases. Thus, the probability that a litigated U.S. patent survives a validity challenge appears to be considerably lower than the probability that a European patent survives opposition.16 Examiner compensation. The compensation scheme for examiners at the USPTO has both fixed and variable components. The variable components include a bonus that rewards examiners who exceed a target number of “counts” (GAO, 2007). A count is awarded for each “first office action” and each “disposal.” The first office action is an official letter notifying applicants about the patentability of their invention; disposal occurs either when the examiner allows the application, or when the applicant abandons the application, files a request for continued examination (RCE), or files an appeal to which the examiner responds.17 This means that the fastest way for an examiner to obtain two counts is to dispose of an application through a first-action allowance. Disposing of an application through an abandonment or RCE usually requires working through a series of responses and amendments by the applicant and issuing a second office action, none of which earns the examiner any counts.18 As a result, it is more time-consuming to earn the second count through a rejection than through a grant.19 As others have noted, the count system thus essentially rewards examiners for granting patents (Merges, 1999; Jaffe and Lerner, 2004; Lemley and Shapiro, 2005). 16

Obviously, both litigated and opposed patents are subject to selection bias. There is reason to believe, however, that selection bias would reinforce the picture painted by these figures. Litigation is usually initiated by the patent holder, seeking to assert its patent against an alleged infringer. Economic theory suggests that only those disputes for which the patent holder’s probability of winning is relatively large will be litigated to trial (Meurer, 1989; Chiou, 2008). In contrast, opposition is initiated by competitors of the patent holder. It seems plausible that challengers will select patents whose probability of being valid is relatively low. 17 See USPTO, Manual of Patent Examining Procedure, section 1705, available at http://www.uspto.gov/ web/offices/pac/mpep/documents/1700 1705.htm. 18 Recently proposed changes to the count system (see http://www.uspto.gov/web/offices/ac/ahrpa/opa/ documents/briefing for corps-final draft-093009-external-jrb.pdf) leave the essence of the system unchanged. 19 Note that the difference in the necessary amount of time stems from the administrative part of examination (i.e., writing office actions and responding to the applicant), rather than from the search part. For any given level of search effort, a rejection takes longer.

21

At the EPO, examiners receive only a fixed wage (Friebel et al., 2006). There are no explicit monetary incentives tied to performance (productivity or other), although the implicit incentive system includes a regular performance evaluation that is used for promotion decisions. Performance is evaluated on four dimensions: productivity, quality, attitude, and aptitude. The productivity measure is based on the number of actions completed.20

7

Conclusion

I present a model of patent examination in which a benevolent planner delegates patent examination to an examiner, who receives applications filed by firms. Examination is modeled as a problem of moral hazard followed by adverse selection: the examiner must be induced to provide effort but also to reveal the signal he finds, the assumption being that the signal is soft information (unverifiable by third parties, including the planner). I also assume that the examiner has a desire to make the right decisions, which I term intrinsic motivation. Finally, I model the proportion of good applications as endogenously determined by the effort that firms expect the examiner to provide. I show that soft information severely constrains the design of incentives, so that intrinsic motivation becomes a crucial determinant of the equilibrium outcome. When intrinsic motivation is low, monetary incentives may be reduced to the role of ensuring truthful revelation, leading to a seemingly paradoxical compensation scheme that rewards examiners for granting. Yet this scheme succeeds in inducing the examiner to provide effort: if the examiner anticipated not being truthful, he would optimally choose zero effort. As intrinsic motivation increases, extrinsic (monetary) incentives can be used more effectively, reinforcing the provision of effort. The proportion of bad applications falls, resulting in higher patent quality. I argue that the modeling assumptions I use (most notably soft information and intrinsic motivation) provide a reasonable description of how patent examination works in practice. Examining patents requires assessing complex scientific evidence. Moreover, there is little short-term information about the quality of the examiner’s decisions; such information only becomes available after a delay and is difficult to contract on. It may, however, be used in the organization’s promotion and dismissal decisions, which provide long-term implicit incentives. These implicit incentives tend to create a desire to make correct decisions on the examiner’s part, consistent with how I define intrinsic motivation. Examiners are likely to care more about making correct decisions the longer they expect to stay at the patent 20

EPO management recently modified the way productivity is measured; refusals now count twice as much as grants or withdrawals. The change was pointed out to the author by EPO controller Ciaran McGinley. In correspondence with the author, McGinley justifies the move by the fact that “experience has shown that refusals take twice as long as other finalisation processes (grants or withdrawals).”

22

office, and the quicker information about their decisions becomes available. A comparison of examiner turnover and procedures for administrative review at the European Patent Office (EPO) and the U.S. Patent and Trademark Office (USPTO) reveals that EPO examiners generally have longer tenure than their US counterparts, and that the administrative review procedure at the EPO is much more widely used than the one at the USPTO. In the light of these considerations, which suggest that intrinsic motivation, as defined in this paper, is higher at the EPO than at the USPTO, the model can explain why U.S. examiners are essentially rewarded for granting patents, but also why European examiners do not face a similar compensation scheme and instead receive a fixed wage. In addition, its predictions are consistent with the fact that the quality of patents issued is generally perceived to be lower in the U.S. than in Europe. The main policy implications concern examiner retention and administrative patent review. Retaining examiners for more than a few years allows long-term incentives to become effective, and a functioning system of administrative review makes information on the quality of examiners’ decisions available in a more timely manner. Moreover, short-term incentives can be improved by implementing a random internal review of examiners’ decisions and conditioning bonus payments on the outcome of the review. The analysis suggests that retaining examiners and creating an administrative review are desirable for reasons beyond those typically mentioned in the patent-reform debate, which has focused on the fact that more experienced examiners perform better work and that private parties may be better informed about prior art than examiners. Rather, the argument here is that both measures improve examiners’ incentives to make correct decisions and increase the scope for reinforcing effort provision through short-term compensation.

Appendix: Proofs Proof of Lemma 1. With general grant rules (xB , x∅ ) ∈ {0, 1}2 , the examiner’s incentivecompatibility constraints are tB + (1 − xB )αyB ≥ t∅ + (1 − x∅ )αyB t∅ + α[ˆ px∅ yG + (1 − pˆ)(1 − x∅ )yB ] ≥ tB + α[ˆ pxB yG + (1 − pˆ)(1 − xB )yB ],

(A.18) (A.19)

where pˆ ≡ Pr[G|∅]. Adding up (A.18) and (A.19) and simplifying yields xB ≤ x∅ . Thus, incentive compatibility rules out xB = 1 − x∅ = 1. If xB = x∅ = 0 or xB = x∅ = 1, it is clearly optimal for the planner to induce e = 0. Setting xB = x∅ = 0 leads to zero welfare. Setting xB = x∅ = 1 is equivalent to setting xB = 1 − x∅ = 0 and inducing e = 0 (because γ(0) = 0). Expected welfare then is NG V + NB (0)L ≥ 0, where the inequality is due to Assumption 4. 23

Proof of Lemma 2. The examiner’s strategy set is the unit interval, [0, 1], which is a nonempty, convex and compact subset of R. His payoff function is continuous in (e, p) and concave in e (because γ 00 > 0). Because NB (e) is continuous by Assumption 3, the firms’ best response, as summarized by the function p(e), is continuous in e and also takes values on [0, 1]. By the existence theorem for Nash equilibria in infinite games with continuous payoffs (see, e.g., Theorem 1.2 in Fudenberg and Tirole (1991)), equilibrium exists. By equation (4), e is monotonically decreasing in p: if tB − t∅ + αyB ≤ 0, the examiner’s best response is e = 0 for any p; otherwise e is strictly decreasing in p. By Assumption 3, NB0 ≤ 0, implying that p is monotonically increasing in e. Therefore, the equilibrium is unique. Proof of Lemma 3. Rewriting (2) and (6) as tB − t∅ ≤ α[ˆ pyG − (1 − pˆ)yB ] (tB − t∅ )[p + (1 − p)(1 − e)] ≤ α[pyG − (1 − p)(1 − e)yB ] − γ(e),

(A.20) (A.21)

respectively, and using pˆ = p/[p + (1 − p)(1 − e)] so that (A.21) becomes tB − t∅ ≤ α[ˆ pyG − (1 − pˆ)yB ] −

γ(e) , p + (1 − p)(1 − e)

(A.22)

we see that (A.22) implies (A.20) (and thus that (6) implies (2)). Similarly, rewriting (1) and (5) respectively as tB − t∅ ≥ −αyB tB − t∅ ≥ −αyB +

(A.23) γ(e) , e(1 − p)

(A.24)

it becomes apparent that (A.24) implies (A.23) (and thus that (5) implies (1)). Therefore, the relevant constraints are (A.24) and (A.21), while (A.23) and (A.20) can be neglected. Combining (A.24) and (A.21) yields the inequality −αyB +

γ(e) γ(e) ≤ α[ˆ pyG − (1 − pˆ)yB ] − e(1 − p) 1 − e(1 − p) ⇔

γ(e) ≤ p(1 − p)α(yG + yB ). e

(A.25)

Inequality (A.25) describes the set of values of e for which there exist some transfers simultaneously satisfying both constraints. Denote this set Ω. Not every element in Ω is implementable as an equilibrium. Since effort is increasing in tB − t∅ , the maximum implementable effort, for a given p, is obtained by substituting for tB − t∅ in (4) using the right-hand side of (A.22), yielding γ 0 (e) = (1 − p)ˆ p[yG + yB − γ(e)/p]. 24

(A.26)

Denote the set defined by this upper bound Ω0 . What needs to be shown is that Ω0 is a subset of Ω. Rewrite (A.26) as γ 0 (e)[1 − e(1 − p)] + (1 − p)γ(e) = p(1 − p)α(yG + yB ). The left-hand side is nondecreasing in e as

∂ 0 ∂e [γ (e)[1

(A.27)

− e(1 − p)] + (1 − p)γ(e)] = γ 00 (e)[1 −

e(1 − p)] ≥ 0, where the inequality follows from the convexity of γ. A necessary and sufficient condition for Ω0 to describe a subset of Ω thus is γ 0 (e)[1 − e(1 − p)] + (1 − p)γ(e) ≥ γ(e)/e

⇔

γ 0 (e) ≥ γ(e)/e.

This inequality is satisfied for all e since by assumption γ is an increasing and convex function with γ(0) = 0; hence its marginal is always above its average. Replacing p = NG /(NG +NB (e)) in (A.27), I conclude that Ω0 is the set of values of e satisfying   NB (e) NB (e)NG eNB (e) + γ(e) ≤ α(yG + yB ). γ 0 (e) 1 − NG + NB (e) NG + NB (e) (NG + NB (e))2 Rearranging yields (10). Proof of Proposition 1. If α = 0, the planner’s program is max W (e) e≥0

subject to g(e) ≤ 0.

In equation (10), defining g, the terms 1 + NB (e)/NG and 1 + NG /NB (e) are always strictly positive, and by assumption γ(e), γ 0 (e) > 0 for all e > 0. Therefore, e = 0 is the unique solution satisfying g(e) ≤ 0. Let eα denote the solution to program (9) for a given α, and let µα denote the Lagrange multiplier associated with the constraint g(e) ≤ α(yB + yG ). By the envelope theorem, dW (eα ) = µα (yB + yG ) ≥ 0. dα For the claim that the maximum implementable effort increases with α, it further needs to be shown that g 0 (¯ e) > 0, where e¯ ≡ max{e|g(e) = α(yB + yG )}. Letting G(e, q) ≡ (1 + q)[(1 − e)γ 0 (e)+γ(e)]+(1+1/q)γ 0 (e) and q ≡ NB (e)/NG , we have dg/de = ∂G/∂e+(∂G/∂q)(dq/de). Furthermore, ∂G = (1 + q)(1 − e)γ 00 + (1 + 1/q)γ 00 > 0, ∂e

(A.28)

where the inequality follows from the convexity of γ, and dq/de = NB0 (e)/NG ≤ 0 by Assumption 3. Being a ratio, q can take values on [0, ∞). We have   ∂G 1 0 = (1 − e)γ + 1 − 2 γ, ∂q q 25

(A.29)

which is positive as q → ∞ and negative as q → 0. For ∂G/∂q ≤ 0, g 0 > 0 follows immediately. For ∂G/∂q > 0, note that e¯ can alternatively be defined as the largest value of e that solves the system {G(e, q) = α(yB +yG ), q = NB (e)/NG }. Equations (A.28) and (A.29) imply that the implicit function defined by G(e, q) = α(yB + yG ) giving e as a function of q is first increasing and then decreasing in q. Let ρ denote its inverse defined on the decreasing part (where ∂G/∂q > 0), implying ρ0 (e) = −

∂G/∂e . ∂G/∂q

By the definition of e¯, it must be the case that ρ0 (¯ e) < NB0 (¯ e)/NG . Thus, for ∂G/∂q > 0, g 0 (¯ e) = (∂G/∂q) (dq(¯ e)/de − ρ0 (¯ e)) > 0. Proof of Proposition 2. Denote ∆t the upper bound on transfers. From (11), we have ∆t ≡ α[ˆ pyG − (1 − pˆ)yB ] − γ(e)ˆ p/p. Suppose α = 0. By Proposition 1, we then have e = e0 = 0, so tB − t∅ ≤ ∆t = 0 and p = p(0). Compute   d∆t dˆ p pˆ de 0 γ(e) dˆ p dp = α[yG + yB ] + pˆyG − (1 − pˆ)yB − γ (e) − 2 p− pˆ . dα dα p dα p dα dα Noting that pˆ = p for e = 0 and γ(0) = γ 0 (0) = 0, evaluating this expression at α = 0 yields d∆t = p(0)yG − (1 − p(0))yB < 0. dα α=0 We have W 0 (0) = (NB (0) − NB0 (0))L − (NG + NB (0))γ(0) − NB0 (0)γ 0 (0) > 0 because NB (0) > 0 and NB0 ≤ 0 by Assumption 3 and γ(0) = γ 0 (0) = 0. Thus, the constraint g(e) ≤ α(yG + yB ) must be binding at α = 0 and, by continuity, in its vicinity. It follows that deα yB + yG = . dα e=0 g 0 (0) As shown in the proof of Proposition 1, g 0 (¯ e) > 0, and e¯ = 0 for α = 0. Hence, for small positive values of α, we have eα > 0 and tB − t∅ < 0. Proof of Proposition 3. I start by establishing that 0 < eo < 1. Differentiating W yields W 0 (e) = (NB (e) − NB0 (e))L − (NG + NB (e))γ(e) − NB0 (e)γ 0 (e). By Assumption 3 and because γ(0) = γ 0 (0) = 0, we have W 0 (0) > 0. Moreover, because γ 0 (1) = ∞, we have W 0 (1) < 0, so 0 < eo < 1. It follows that γ(eo ) and γ 0 (eo ) are finite. Next, I show that under condition (14) there exists eˆ < eo such that g(ˆ e) is finite, p(ˆ e) > yB /(yG + yB ), W 0 (ˆ e) > 0, and W (e) < W (ˆ e) for all e < eˆ. Note that given 0 < eo < 1 and NG > 0, g(ˆ e) is finite if and only if NB (ˆ e) > 0. There are two cases. If NB (eo ) > 0, existence of eˆ is trivial: because NB is continuous by Assumption 3, implying continuity of p and W , we can always find ε > 0 such that eˆ = eo − ε satisfies the above conditions. If NB (eo ) = 0, existence of eˆ follows from NB (0) > 0 and the continuity of NB , p, and W . 26

Finiteness of g(ˆ e) implies that there exists α ¯ > 0 such that g(ˆ e) = α ¯ (yB + yG ). Because W 0 (ˆ e) > 0 and W (e) < W (ˆ e) for all e < eˆ, the constraint g(e) ≤ α(yB + yG ) must be binding at α = α ¯ , and eα ≥ eˆ for all α ≥ α ¯ . By construction, when the constraint is binding, tB − t∅ = ∆t. Let us find the locus in (α, p) space such that ∆t = 0, that is, α[ˆ pyG − (1 − pˆ)yB ] = γ(e)ˆ p/p. Using the definition of pˆ, and solving for p, we obtain p=

γ(e)/α + (1 − e)yB . yG + (1 − e)yB

(A.30)

From (4), we have, for tB − t∅ = 0, γ 0 (e) = (1 − p)αyB . By the convexity of γ, therefore γ(e) ≤ e(1 − p)αyB , which we can use to obtain an upper bound on p in (A.30), given by p ≤ yB /(yG + yB ). Note that this upper bound is independent of e and α. For any p > yB /(yG + yB ), the associated ∆t is positive. By definition of eˆ, p(ˆ e) > yB /(yB + yG ). Hence, ∆t > 0 for α = α ¯ , proving that there must exist α ˆ ≥ 0 as claimed. Proof of Proposition 4. The examiner’s utility is the sum of his period-1 and period-2 payoffs, ¯ with probability 1 − α. If the U = U1 + U2 , where U2 = U ∗ with probability α and U2 = U 2

examiner exerts effort e and truthfully reveals his signal, his expected utility is   U = p ηtr∅ + (1 − η)t∅ + αU2∗ + (1 − p) [e [ηtrB + (1 − η)tB + αU2∗ ] + (1 − e)(1 − η)t∅ ] ¯ . (A.31) − γ(e) + (1 − α)U The incentive-compatibility constraints are ηtrB + (1 − η)tB + αU2∗ ≥ (1 − η)t∅

(A.32)

(1 − η)tB + (1 − pˆ) [ηtrB + αU2∗ ] ≤ (1 − η)t∅ + pˆ ηtr∅ + αU2 . 

 ∗

(A.33)

The first-order condition determining effort provision is (1 − p) [ηtrB + (1 − η)(tB − t∅ ) + αU2∗ ] = γ 0 (e).

(A.34)

The double-deviation constraints are   ¯ ≤ U∗ p ηtr∅ + (1 − η)t∅ + αU2∗ + (1 − p)(1 − η)t∅ + (1 − α)U ¯ ≤ U ∗; p(1 − η) [tB + αU2∗ ] + (1 − p) [ηtrB + (1 − η)tB + αU2∗ ] + (1 − α)U

(A.35) (A.36)

once again, these imply the incentive compatibility constraints, (A.32) and (A.33). I first prove the claim that e∗ > 0 is possible when η > 0 and α = 0. Rearranging (A.35) yields (1 − p)e [ηtrB + (1 − η)(tB − t∅ ) + αU2∗ ] ≥ γ(e). By (A.34), γ 0 (e) ≤ (1 − p) [ηtrB + (1 − η)(tB − t∅ ) + αU2∗ ] for all e ≤ e∗ . Hence, Z e∗ ∗ γ(e ) = γ 0 (e)de ≤ (1 − p)e [ηtrB + (1 − η)(tB − t∅ ) + αU2∗ ] , 0

27

so (A.35) is satisfied for all e∗ < 1 regardless of α. Rearranging (A.36) yields 
 p η tr∅ + αU2∗ + (1 − η)(t∅ − tB ) ≥ γ(e∗ ) + (1 − p)(1 − e∗ ) [ηtrB + (1 − η)(tB − t∅ ) + αU2∗ ] . (A.37) Choose some 0 < e < 1 such that p(e) < 1 and set tB = t∅ = 0. By (A.34), when α = 0, e∗ = e requires trB = γ 0 (e)/[η(1 − p(e))]. Because by assumption η > 0, e < 1, and p(e) < 1, such a trB exists and is finite. Satisfying (A.37) then simply requires setting tr∅ ≥ Γ(e)/η, where Γ(e) ≡ [γ(e) + (1 − e)γ 0 (e)]/p(e). Next, I prove existence of a threshold ηˆ as claimed in the proposition. In a first step, I ignore the constraint tr∅ ≤ t¯ of Assumption 5. Noting that, as shown above, (A.35) is satisfied for any e∗ < 1, the planner solves the following program: minr

tB ,t∅ ,tB ,tr∅

  p ηtr∅ + (1 − η)t∅ + αU2∗ + (1 − p) [e [ηtrB + (1 − η)tB + αU2∗ ] + (1 − e)(1 − η)t∅ ]

subject to tσ˜ ≥ 0, trσ˜ ≥ 0, σ ˜ = B, ∅, (A.34) and (A.37). The latter constraint being binding at the optimum, this program is equivalent to min

tB ≥0,t∅ ≥0

(1 − η) [p(tB + αU2∗ ) + (1 − p)t∅ ] + γ 0 (e),

the solution of which is tB = t∅ = 0. The transfers trB and tr∅ can then be determined as a function of e using (A.34) and (A.37), yielding trB = [γ 0 (e)/(1 − p(e)) − αU2∗ ]/η and tr∅ = Γ(e)/η − αU2∗ . Now impose Assumption 5. For any finite t¯ and e > 0, we have limη→0 Γ(e)/η − αU2∗ > t¯, so that the solution derived ignoring the constraint tr∅ ≤ t¯ is not feasible for small η. Satisfying (A.37) then requires setting t∅ − tB = [(γ(e) + (1 − e)γ 0 (e))/p(e) − η(t¯ + αU2∗ )]/(1 − η) > 0, establishing existence of ηˆ ∈ (0, 1]. To prove that ηˆ decreases with α, note first that if t¯ ≤ Γ(e) − U2∗ then ηˆ = 1 for all α so dˆ η /dα = 0. Suppose instead t¯ > Γ(e) − U2∗ . We then have

 ηˆ =

1 Γ(e)/(t¯ + αU2∗ )

for α < (Γ(e) − t¯)/U2∗ for α ≥ (Γ(e) − t¯)/U2∗ ,

so dˆ η /dα ≤ 0 as claimed. Proof of Proposition 5. I start by extending Lemmata 2 and 3 to the setup of Section 5. Lemma 2 can be extended as long as the examination game is well defined, which requires min{ˆ πG , π ˆB } < π. Noting that β < 1 implies π ˆG ≤ π ˆB , the examination game is well defined if and only if φ < π. The examiner’s strategy set and payoff function are unchanged. Because F is continuous, the firms’ strategies, summarized by π ˆG and π ˆB , lead to a function p that is continuous in e and takes values on [0, 1], establishing existence of equilibrium. As for 28

uniqueness, the examiner’s best-response function is unchanged, so it suffices to show that p is monotonically increasing in e for any φ. If βπ ≤ φ < π, p = 1 regardless of e. If φ < βπ, we have p=

nG [1 − F (φ)] 
 nG [1 − F (φ)] + nB 1 − F φ/[β(1 − e)]

for e < 1 − φ/(βπ),

and p = 1 otherwise. Thus,
NG (φ)nB φf φ/[β(1 − e)] ∂p =
2 > 0 ∂e β(1 − e)2 NG (φ) + NB (e, φ) for e < 1 − φ/(βπ), and ∂p/∂e = 0 otherwise, implying that the equilibrium is unique. It is straightforward to translate Lemma 3 to the case with application fees. Applying the reasoning used in the proof of Lemma 3 shows that the planner’s problem is equivalent to max

e≥0,φ≥0

W (e, φ)

subject to g(e, φ) ≤ α(yG + yB ),

(A.38)

where     NB (e, φ) NG (φ) 0 g(e, φ) ≡ 1 + [(1 − e)γ (e) + γ(e)] + 1 + γ 0 (e). NG (φ) NB (e, φ)

(A.39)

For later reference, note that the partial derivatives of W with respect to e and φ are Z π    γ(e) φ ∂W = nB `(ˆ πB ) + f (ˆ πB ) − (NB + NG )γ 0 (e) (A.40) `(π)dF (π) + ∂e β(1 − e) 1−e π ˆB ∂W f (ˆ πB ) = −nG [v (ˆ πG ) − γ(e)] f (ˆ πG ) + nB [(1 − e)` (ˆ πB ) + γ(e)] . (A.41) ∂φ β(1 − e) The rest of the proof proceeds as follows. First, I show that the application fee absent examination (and thus, by Proposition 1, when α = 0) is interior. This allows me to apply the implicit function theorem at (e, φ) = (0, φ0 ) to establish claim (i). Second, I show that the unconstrained maximizer of problem (A.38), (eo , φo ), is such that π ˆB (eo , φo ) < π, implying that g(eo , φo ) is finite. This allows me to establish claim (ii). Claim (i). Because π ˆG = π for any φ ∈ [0, π], the planner will always choose φ ≥ π. Evaluating ∂W/∂φ at (e, φ) = (0, π) yields ∂W (0, π)/∂φ = −nG v(π)f (π)+nB ` (π/β) f (π/β) /β > 0, where the inequality follows from the assumptions that v(π) = 0, `(π) > 0 for all π, f (π) > 0 for π < π, and π/β < π (Assumption 6). Evaluating ∂W/∂φ at (e, φ) = (0, βπ) yields ∂W (0, βπ)/∂φ = −nG v (βπ) f (βπ) + nB `(π)f (π) /β < 0, where the inequality follows from the assumptions that f (βπ) > 0 and f (π) = 0. Thus, welfare is strictly increasing at φ = π and strictly decreasing at φ = βπ, implying that the optimal application fee absent examination is interior: letting φ0 ≡ arg maxφ W (0, φ), we have π < φ0 < βπ. It follows from Proposition 1 that (0, φ0 ) is the solution to (A.38) when α = 0. By continuity, the optimal fee must be interior in the vicinity of α = 0 as well. Moreover, the 29

constraint g(e, φ) ≤ α(yG + yB ) must be binding at α = 0: noting that π ˆB (0, φ) = φ/β, we have ∂W (0, φ0 ) = nB ∂e

"Z

# φ0 `(π)dF (π) + [`(φ0 /β) + γ(0)] f (φ0 /β) − (NB + NG )γ 0 (0) > 0, β φ0 /β π

where the inequality again follows from Assumption 6 and γ 0 (0) = 0. Thus, in the vicinity of α = 0, (eα , φα ) solves the first-order conditions for an interior solution of program (A.38), given by ∂W ∂g −µ =0 ∂e ∂e ∂W ∂g −µ =0 ∂φ ∂φ α(yB + yG ) − g(e, φ) = 0, where µ denotes the Lagrange multiplier. By the implicit function theorem, ∂eα = ∂α 2

∂ g µ ∂e∂α

2

  h 2 i  2 ∂2g ∂2g ∂2g ∂g ∂ W ∂ W (y + y ) − µ − µ − µ ∂e∂φ (yB + yG ) ∂φ − ∂g + ∂e∂φ B G 2 2 ∂e ∂φ∂α ∂φ ∂φ .   2      2  2 2 2 2 2 2 ∂ g ∂ g ∂g ∂ g ∂g ∂ W ∂g ∂ W ∂ W 2 ∂g − µ − µ − − µ − 2 2 2 2 ∂e ∂φ ∂e∂φ ∂e∂φ ∂φ ∂e ∂e ∂e ∂φ ∂φ



∂g ∂φ

(A.42) Clearly, ∂ 2 g/∂e∂α = ∂ 2 g/∂φ∂α = 0. Letting q(e, φ) ≡ NB (e, φ)/NG (φ), at (0, φ0 ),    ∂g(0, φ0 ) ∂q 1 0 = γ(0) + γ (0) 1 − = 0, ∂φ ∂φ (q(0, φ0 ))2 which follows from γ 0 (0) = γ(0) = 0 and the fact that NB (0, φ0 ) = nB [1 − F (ˆ πB (0, φ0 ))] > 0 because φ0 < βπ, implying q(0, φ0 ) > 0. By the same argument, ∂ 2 g/∂φ2 = 0 at (0, φ0 ) as well. Moreover, ∂g(0, φ0 )/∂e = (1 + q(0, φ0 ))γ 00 (0) > 0. After simplifying, we obtain ∂eα yB + yG = > 0. ∂α (e,φ)=(0,φ0 ) ∂g(0, φ0 )/∂e Hence, for small positive values of α, we have eα > 0. The argument in the proof of Proposition 2 can then be applied to establish that if p(0, φ0 ) < yB /(yB +yG ), then for small α, tB −t∅ < 0. Claim (ii). Clearly, ∂W (0, φ0 )/∂e > 0 and γ 0 (1) = ∞ imply 0 < eo < 1. I claim that ˆB (eo , φo ) = π π ˆB (eo , φo ) < π, which is equivalent to φo < (1−eo )βπ. Suppose otherwise, i.e., π or equivalently φo = (1 − eo )βπ. From (A.40), we have ∂W (e, (1 − e)βπ)/∂e ≤ 0 for any e, implying eo = 0, a contradiction. It follows from 0 < eo < 1 and π ˆG < π ˆB < π that g(eo , φo ) is finite. Thus, there exists α ¯ such that g(eo , φo ) = α ¯ (yG + yB ). The argument used in the proof of Proposition 3 can then be applied to establish that if p(eo , φo ) > yB /(yG + yB ), then tB > t∅ for sufficiently large α. 30

Proof of Proposition 6. Letting q = NB /NG , we can write p = 1/(1 + q). Thus, ∂p/∂e = −∂q/∂e (1 + q)−2 and ∂p/∂φ = −∂q/∂φ (1 + q)−2 . We have   φ nB φ f 2 β(1−e) ∂q β(1−e) =− < 0, ∂e nG [1 − F (φ)] implying ∂p/∂e > 0. The strict inequality follows from (eo , φo ) being interior, as established in the proof of Proposition 5. Furthermore,   h  i φ φ nB nG − f [1 − F (φ)] + n n f (φ) 1 − F G B β(1−e) β(1−e) β(1−e) ∂q = . 2 2 ∂φ nG [1 − F (φ)] Thus, ∂q/∂φ ≤ 0 if and only if 1 h β(1 − e)



φ β(1 − e)

 ≥ h (φ) ,

where h(π) ≡ f (π)/(1 − F (π)) is the hazard rate. Under condition (ii), h0 ≥ 0. Since, moreover, β(1 − e) < 1, we then have ∂p/∂φ ≥ 0. Hence, eo > 0 and φo ≥ φ0 imply p(0, φ0 ) < p(eo , φo ). Under condition (i), F (π) = 1 − (π/π)a with a > 0, π > 0, and π = ∞. The associated hazard rate is h(π) = a/π. It follows that ∂q/∂φ = ∂p/∂φ = 0. Hence, eo > 0 implies p(0, φ0 ) < p(eo , φo ).

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