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Pareto.uab.es

Research and the Approval Process An agent sequentially collects information to obtain a principal's approval, such as a pharmaceutical company seeking FDA approval to introduce a new drug. To capture suchenvironments, we study strategic versions of the optimal stopping time problem …rst proposedby Wald (1945). Our ‡exible model allows us to consider di¤erent types of rules and commit-ments by the principal as well as strategic withholding of information by the agent. We shedlight on current regulation and proposed reforms of the drug approval process. The modelalso captures situations such as a …rm seeking antitrust approval to merge with a competitor,a manager proposing a project to the …rm's headquarters or an author submitting a paper toan editor.
Keywords: Research, organization, approval, regulation.
JEL Classi…cation: D83 (Search; Learning; Information and Knowledge; Communication;Belief), M38 (Government Policy and Regulation).
Ottaviani acknowledges …nancial support from the European Research Council through ERC Advanced Grant 295835. We thank Umberto Sagliaschi for excellent research assitance.
ySciences Po and CEPRzBocconi and CEPR Pharmaceutical companies run costly clinical trials on new drugs to demonstrate the safety and e¤ectiveness necessary to obtain regulatory approval. Similarly, a …rm intending to take over a competitor searches for evidence of synergies to convince the antitrust authority to approve the transaction. A manager collects information to push the …rm headquarters to validate his project. And an author submitting a paper conducts research and robustness checks to convince the editor to accept. In turn, the regulator, the headquarters or the editor can can also conduct additional independent research on these issues.
In all these situations an agent searches sequentially for evidence to convince a principal, with a priori di¤erent preferences, of the desirability of an activity with uncertain private and social payo¤s; the principal, in turn, also conducts some research. How does the organization of the approval process a¤ect the information that gets produced and the quality of the …nal decision? How does the possibility for the agent to withhold information a¤ect the process? Consider our leading application to the drug approval process. After identifying a promising compound, pharmaceutical companies conduct an extensive and well de…ned series of clinical trials to obtain the approval of the regulator in charge of drug safety.1 Pharmaceutical research is conducted sequentially, so that at each point in time one of three decisions is made: continue research by acquiring additional information, abandon the project altogether, or ask for approval for introducing the drug to market.2 This corresponds to the current organization of the drug approval process, in particular in the US. This process has evolved greatly over time and new issues are currently emerging (as described in detail in section 2). The historical evolution of legislation on drug approval has tended to strengthen the powers of the FDA to mandate research and created opportunities for the agency to commit to standards and to a precise process of approval. In this paper, we will study the bene…ts from these di¤erent types of commitment. Currently the attention has turned to both the issue of withholding of information, following some heavily publicized scandals,3 and 1 In the US the Food and Drugs Administration (FDA) regulates the approval of new drugs. In the European Union, pharmaceutical companies can choose between applying to a EU-wide authority, the European MedicinesAgency (EMA), or to one of the national authorities, such as the Medicines and Healthcare products RegulatoryAgency (MHRA) in the UK or the Agenzia Italiana del Farmaco (AIFA) in Italy.
2 Obtaining additional evidence can mean conducting additional clinical trials or adding patients to a trial. As reported in Nundy and Gulhati (2005), increasingly Western drug companies conduct trials in India to decreasecosts and bene…t from easier regulatory approval. Moreover, several scandals involve illegal trials not approved bythe Indian authorities.
3 Scandals include the allegation that for several years Merck withheld evidence on adverse e¤ects of its block- buster drug Vioxx. There has been a recent push to impose stronger disclosure requirements. For example, the question of post-approval regulation. Our theoretical analysis will shed light on these new concerns and on the necessary regulatory steps.
Our model captures these types of environments in a simple, tractable and ‡exible way. A choice needs to be made between rejection and approval. Rejection yields a zero payo¤ while the bene…ts from approval depend on a binary state of nature; they are positive if the state is high and negative if it is low. Research can be conducted to obtain additional information about the state. It is costly on two accounts: there is a direct …nancial cost and, in addition, research delays decision making with an associated opportunity cost. The arrival of new information is conveniently modeled in continuous time as a Wiener process with a drift that depends on the If the same player were in full control of both research and approval, the model would boil down to a version of the classic single-agent optimal stopping problem that has been widely analyzed in the statistical decision theory literature on sequential analysis pioneered by Wald (1945). The well-known solution involves two threshold values (or standards) for the belief, such that it is optimal to abandon the project when the belief that the state is high is su¢ ciently low (below the rejection standard) and such that it is optimal to adopt the project when the belief is su¢ ciently high (above the approval standard). When the belief lies within these thresholds, it is optimal to continue researching— this is Wald's celebrated sequential probability ratio test.
The payo¤s of agent and principal are typically misaligned. For instance, because the phar- maceutical …rm does not internalize all externalities, it typically gets a higher payo¤ than the regulator in the bad state. In most of the paper we thus focus on situations where, for the stand-alone Wald Problem, the agent searches more than the principal at the lower end, when the state is bad, and less at the upper end, when the state is good. It is the case most people have in mind for drug approval: pharmaceutical companies are eager to stop researching, obtain approval, and adopt the drug earlier than the regulator when information is good and more reluctant to abandon the project when the state is bad.
In practice, the research and approval processes are not single agent problems; intuitively the agent controls the research decision, at least initially, while the principal has the hold on the approval decision. The baseline situation we thus study is one where the agent chooses the lower benchmark of search and the principal simultaneously chooses the upper benchmark (the Nash medical journals and regulatory authorities have pushed for early registration of trials and for disclosure of theresults obtained in the trials.
Equilibrium).4 In equilibrium, we show that the principal will compromise at a lower standard for approval compared to the principal's non-strategic standard. Intuitively, given that the agent now chooses when to abandon research, the principal's value of information is reduced— thus the principal becomes more eager to approve.
Two properties of this Nash equilibrium solution stand out. First, the principal can obtain a negative payo¤ in equilibrium. Indeed he controls the upper benchmark, and can force immediate approval but cannot force rejection. If the agent has su¢ ciently misaligned preferences and searches too much at the lower end, the principal at the upper benchmark gets a negative expected bene…t. The goal of research in such situations is that it will provide su¢ ciently bad news that the lower benchmark will be reached and the agent can be convinced to abandon. Second, more research is conducted in the Nash equilibrium solution than in either stand-alone problems.
The principal thus has an interest to commit to a course of action that discourages research by the agent at the lower end. We …rst consider commitments ex ante to an approval standard, what we call the Stackelberg outcome. The principal chooses his preferred point on the agent's best response curve. Depending on the value of the initial belief, di¤erent types of commitments are optimal. When the belief is low, a high blocking commitment that discourages research by the agent is preferred: this allows the principal to avoid the negative payo¤ that sometimes characterizes the nash outcome. For intermediate values of the belief, the principal will commit to an interior commitment, allowing for some research by the agent. Finally, when the initial belief is very favorable, the principal will commit to immediate approval.
In practice, this type of commitment is not easy to achieve and furthermore, as suggested by the above discussion, cannot be uniform and independent of the current state of knowledge. In fact, as described in section 2, this is not the approach chosen for drug regulation. Rather, there is a commitment to a well de…ned sequential procedure for approval. We thus study a model where the interaction occurs in two stages as follows: 1. In the …rst stage the agent conducts research, and then decides when to transfer the decision- making power to the principal.
2. Once that happen, in the second stage, the principal conducts research and eventually decides whether to approve.
4 This is in fact the outcome of a game where the principal can mandate research: in each period the agent chooses between three actions, research, submit or wait and if the agent chooses submit, the principal choosesbetween research, approve and wait.
Clearly, the solution in the second stage corresponds to the non-strategic solution for the principal.
Expecting that outcome, in the …rst stage the agent has less incentive to undertake research at the bottom because of the extra research conducted at the top in comparison to the baseline Nash equilibrium. Our analysis shows that this type of commitment is not necessarily optimal from the point of view of the principal. Whereas in the Nash Equilibrium outcome, the issue was excessive research by the agent at the bottom, the issue is here one of excessive research at the top. This suggests that regulation of drug approval should be reoriented towards a commitment Recent regulation of the drug approval process has focused on the issue of withholding of information. As recent scandals suggest, the agent can withhold some of the evidence, but this is nevertheless costly; Merck for instance in the Vioxx case has paid over 4.8 billion dollars for settling complaints. We enrich the model and suppose that the agent can report any belief, but is then subject to an expected penalty when the state turns out to be low. The expected penalty increases in the distance between the actual state of knowledge and the report (the case is harder to defend in court when the lie is big).
Even though the agent will always lie in equilibrium, by exploiting the knowledge of the bounds of the agent's research interval, the principal is able to perfectly invert the information and not be deceived. In fact, we demonstrate that the principal will actually bene…t from the agent's ability to costly misreport information. The reason is that the cost of lying decreases the value of information for the agent, leading to a reduction of research at the lower end, something that is bene…cial for the principal. We show that it is optimal for the principal to choose a penalty for misrepresentation that is not in…nite so that the agent carries out some costly misrepresentation in equilibrium.
As initially suggested, our model is rich enough to cover other cases of research and approval than our main application to the pharmaceutical market. We derive some further results in this case, varying some assumptions to better …t the applications. We …rst depart from the assumption that the agent cares about the cost of the principal's research (i.e. has to pay for research) and revisit the two stage commitment game. We show that even though this introduces a free riding incentive, the agent still has incentive to do research to move the principal away from his rejection threshold which is ine¢ ciently high from the agent's point of view.
5 There is some discussion of relevant standards in the literature. For instance, Ocana and Tannock (2011) argue that, even though the FDA has tended to accept any trial showing statistically signi…cant results, they shouldbecome stricter and request "clinically important di¤erences", i.e., statistical di¤erences large enough to make itworth running the risk of introducing a new drug.
Related Literature.
The original problem of sequential research, examining the tradeo¤ be- tween the cost of an extra signal and the bene…t of a more informed decision, was introduced by Wald (1945, 1947) and Wald and Wolfowitz (1948).6 The ensuing applied probability literature of this non-strategic problem has a large impact on the actual design of clinical trials. Closely building on Wald's decision-theoretic foundational framework, we focus on the strategic issues that arise when the decisions to collect information and to make the …nal decision are made by two di¤erent players.
Our paper thus relates to the literature on strategic experimentation (see Bolton and Harris 1999) and especially to Strulovici (2010), who highlights how the loss of control of decision making (determined through voting in his model) reduces the incentives to acquire information and thus induces a status quo bias; see also Fernandez and Rodrik (1991). Our model is closest to Gul and Pesendorfer (2012), Lizzeri and Yariv (2011), and Chan and Suen (2012) who consider strategic settings in which public information arrives over time to voters. In Gul and Pesendorfer's (2012) model information is provided by the party that leads, whereas in Lizzeri and Yariv (2011) and Chan and Suen (2012) voters decide collectively themselves when to stop acquiring public information and reach a decision. In their setting information is revealed publicly to all voters, while we focus on the sequential interaction between an agent who collects private information and then reports (or possibly misreports) it to a principal who makes the approval decision. We also analyze the commitment solution in which the principal moves …rst by setting the approval standard, and then extend the model to analyze approval in multiple stages.
For our baseline analysis we constrain reporting of the belief (corresponding to the …nal re- sults) to be truthful at the moment of application, for example because misrepresentation is in…nitely costly as in the disclosure models of Grossman (1981) and Milgrom (1981). We also consider the possibility of costly misreporting. While Kartik, Ottaviani and Squintani (2007) and Kartik (2009) characterize the amount of equilibrium costly lying in static models of strategic communication, in our dynamic model we show that ex post lying costs reduce the ex ante incen- tives for information collection. See also Shavell (1994), Henry (2009), and Dahma, González, and Porteiro (2009) for strategic analysis of partial disclosure of research results. In Henry (2009), pharmaceutical …rms are worse o¤ when their research e¤orts are not observed by the regulator as they are forced to do additional tests to convince him they are not hiding any evidence. Our 6 Moscarini and Smith (2001) recently advanced this literature on non-strategic sequential analysis by analyzing a continuous-time model in which the decision maker can the vary number of experiments in each period. Ourformulation is also in continuous time, but we focus on the simpler case with one experiment per period.
setup is di¤erent in the sense that information is not veri…able: in fact the possibility of hiding information here reduces research because of the cost of lying. This turns out to be bene…cial for the regulator who wants to limit research at the lower end.
Finally, we do not allow our principal to use monetary transfers, in line with the literature on mechanism design without transferable utility; see HolmstrÃ{m (1977) and Alonso and Ma- touschek (2008), Armstrong and Vickers (2010), and Taylor and Yildirim (2011). This approach delivers a number of important insights on the functioning of approval processes that we observe in a number of practical settings where, by and large, transfers are actually not used. A com- plementary literature analyzes the problem of optimal incentive provision for innovation, search, and experimentation where transfers are allowed; recent papers in this area are Manso (2011), Lewis and Ottaviani (2008), Lewis (2012), Gerardi and Maestri (2012), HÃ{rner and Samuelson (2012), and Halac, Kartik, and Liu (2012).
Drug Approval Process We present a brief overview of the drug approval process, past and present, in the US. This exercise will guide our theoretical analysis: we will both analyze the possible e¤ects of past regulations and consider potential consequences of current regulatory e¤orts.
The evolution of the legislation on drug approval was a series of reactions to resounding scandals. Prior to 1938, the role of the US Food and Drug Administration (FDA) was mostly limited to preventing misleading statements on drug labeling. In 1937, a drug company developed a liquid preparation that was not tested prior to marketing and contained a poisonous solvent.
The drug killed over a 100 people. In reaction, the 1938 Food, Drug and Cosmetic Act was swiftly passed. The main feature of this law is that it required that research results be submitted to obtain approval for the drugs, although the FDA had little power to mandate further research if the initial evidence was unsatisfactory.
It was an important step: it introduced the New Drug Application (NDA) necessary to obtain approval, a procedure that still exists today.7 However, the power of the FDA still remained limited. For instance it had only 60 days to examine the evidence and there was no speci…cation of the rules for testing.
A new scandal in 1962 highlighted the need for regulation of the process of testing. A hypnotic 7 The NDA had to include "all clinical investigations, a full list of the drug's components and copies of both the packaging and labelling of the new drug" known as thalidiomide was discovered in Europe to lead to birth defects. It was not allowed the US, but several thousands of samples had been sent to US doctors who gave them to patients without mentioning it was experimental, leading to a number of cases of a¤ected babies. In reaction, the 1962 Drug Amendments introduced the process of drug testing as we know it. The main features of the amendments can be summarized as follows: It put in place a system of pre-clinical testing noti…cation so that regulators could judge whether it was safe to start testing on humans It gave more power to the regulator to mandate research. As explained by Junod (2008), "FDA was given the authority to set standards for every stage of drug testing from labo- ratory to clinic".
However the law did not set very strict legal standards for approval. The law required that there be "substantial evidence"that the drug be e¤ective. As pointed out in Junod (2008), alternative stronger language such as "preponderance of evidence"or "evidence beyond any reasonable doubt" could have been used.
The current phase of regulation is another example of a reaction to a scandal, this time involving misreporting of evidence by …rms, in particular information on side e¤ects of drugs. A case in particular, the allegation that for several years Merck withheld evidence on adverse e¤ects of its blockbuster drug Vioxx, has led to a recent push to impose stronger disclosure requirements.
The FDA Modernization Act of 1997 created the clinical trial registry ClinicalTrials.gov. The FDA Amendments Act of 2007 expanded the types of clinical trials needed to be registered and the amount of details that should be included. Some legislators are trying to push for further Model and Best Responses To capture most of the features of the interactions between an agent and a principal, such as a pharma …rm and the FDA, we consider the following model. A choice needs to be made between two alternatives, adoption A or rejection R. The bene…ts derived from these alternatives depend 8 For example, medical journals and regulatory authorities have pushed for early registration of trials and for disclosure of the results obtained in the trials.
on the state of the world ! that can be either high H or low L. The payo¤ for player i in state j if the choice is k is given by vi . We assume that the payo¤ from rejection is zero for all players, regardless of the state. Furthermore we assume that, for any player i, accepting a good project provides positive payo¤s while accepting a bad one provides a negative one: vi For convenience, we use the following log-likelihood parametrization of beliefs so that the probability that the state is high is given by e = (1 + e ). All players share a common prior 0. Given the restrictions we imposed on the payo¤s, if player i is forced to make a decision , there exists a threshold value (or standard) of the belief , such that A is chosen if > and R is chosen if < . That value solves Research can be conducted to learn the value of the state. The arrival of new information is modeled as a Wiener process d . The drift is determined by the state. Speci…cally, the process has positive drift and variance 2 if the state is H or drift and variance 2 if the state is L. Accumulating information over a period of time dt costs cidt, where the cost of collecting information can vary across individuals.
Finally, payo¤s are discounted, so that if an alternative is chosen at date t it is discounted at rate ri. There are therefore two costs associated with searching for more information: the direct …nancial cost and the opportunity cost associated to the delay in the accrual of the decision Suppose player i undertakes research until time t. The accumulated information at date t is t. The log-likelihood ratio of observing in the two states is given by where h is the density of a standard normal distribution. According to Bayes' rule, the log posterior probability ratio is equal to the sum of the log prior probability ratio and the log- likelihood ratio. Thus, the posterior belief at time t is given by where d 0 is a Wiener process with drift 2 2= 2 if the state is H and 2 2= 2 if the state is L and instantaneous variance 4 2= 2.
When the same player i makes both the search and approval decisions, for a belief that is , there may be a bene…t of searching for more information to make a more informed decision. This is a standard stopping time problem: there exists two values of , s and S (s < s the player stops researching and rejects; < S the player conducts research; > S the player stops researching and approves.
It is immediate to characterize the utility function of the player when u( ) = e rdtE[u( + d 0)] Following Stokey (2009, Chapter 5), starting in the intermediate region, we let T be the …rst time the belief hits either s or S. The direct monetary cost of searching is given by c e rT . Once we de…ne, as in Stokey (2009): 2 (s; S) is given by The …rst line corresponds to the case where the state is high and the upper benchmark S is reached …rst. The second line is the case where the state is low but the upper benchmark is reached …rst, and so on.
Best Response Analysis We start by characterizing the best responses of the research problem. Speci…cally, for a given value of the lower standard s (resp. upper standard S) we characterize the optimal choice of the upper standard S = BR(s) (resp. lower benchmark s = br(S)). This best response analysis allows for a better understanding of our problem and will serve as a building block for the next sections. We start by characterizing br(S). For the moment we drop the subscript i.
Proposition 1 For a given upper benchmark S: 1. The best response br(S) is independent of the current belief and is such that br(S) = S 2. The best response br(S) is decreasing in vHA, vLA and increasing in c.
3. The length of the research interval l(S) = S br(S) is increasing in S and converges to a …nite value l when S converges to +1, where l is solution to vHA = (eg(l) The …rst result states that there is dynamic consistency in the sense that the best response is independent of the current belief. It also states that br(S) = S for values S < . This result is natural, since when S < , approval at S gives a negative value and the player can guarantee himself a zero payo¤, when S, by choosing s = S and imposing rejection.
, the …rst-order condition characterizing the best response (derived in the appendix) bene…t of gaining more information …nancial cost of research 1(s; S) > 0 and where: is the expected bene…t from approval when the belief is At the lower benchmark s, the tradeo¤ expressed by (3) is clear. There are typically two costs associated with research: …rst the direct …nancial cost, proportional to c=r and second the cost of delaying the decision. At the lower benchmark of research rejection yields a zero payo¤.
Thus there is no cost of delay and the only cost is the …nancial one. This expected cost has to be equal to the expected value of information which is proportional to VA(S), i.e., the value if the upper standard is reached. Overall this gives condition 3. The comparative statics then naturally follow. Increasing the cost c naturally decreases research. On the other hand increasing vHA or vLA has the e¤ect of increasing the value of information without a¤ecting the cost and thus decreases the lower benchmark.
Interestingly, result 1.3 indicates that the length of the research interval, measured by S br(S) (an indirect measure of the quantity of research), is increasing in S. The intuition is the following: for a given length l between s and S, the expected bene…t VA(S) is higher for large values of S and the expected cost of moving from s to S is lower since there is a higher probability that the drift will be positive. Thus, if for a certain value of S, the optimal choice is a length br(S) of the research interval, for higher values of S, the interval will be larger. This property is visible in Figure 1 where we plot both br(S) and BR(S). As indicated in result 1.3, at the limit, when S goes to in…nity, the value of l(S) converges to l such that vHA = (eg(l) This limit value depends only on c=r and vHA: when S goes to in…nity, the player is sure the state is good and at the lower benchmark br(S) he is indi¤erent between stopping immediately and getting a zero payo¤ or incurring the cost of research and obtaining vHA when the upper benchmark is reached.
At the upper benchmark, the tradeo¤ is more intricate since the cost of research now has the two components mentioned above: the direct cost and the cost of delaying the decision. Overall Proposition 2 For a given lower benchmark s 1. the best response BR(s) is independent of the current belief 2. the best response BR(s) is decreasing in vHA, vLA and c.
3. the length of the research interval l(s) = BR(s) s is decreasing in s and converges to a …nite value l when s converges to 1, where l is solution to vLA = (e g(l) + 1) cr The …rst order condition characterizing the best response to a given value of s can be expressed in the following way (where 2(s; S) > 0 and bene…t of information cost / bene…t of delaying decision …nancial cost of research For the interpretation of these conditions we distinguish between the case where VA(S) (i.e S > ) and the case VA(S) < 0. Consider the …rst case, that occurs when s is not too low.
At the upper benchmark S the cost of research is composed of the direct …nancial cost and of the cost of delaying the decision, which is proportional to VA(S). Information has value since it can lead to avoiding the negative payo¤ vLA if the state is in fact low. Condition (4) re‡ects this tradeo¤ between cost and value of information. In the second case, when s is very low, the tradeo¤ is di¤erent: for these values, it will be too costly in terms of expected cost of research, to choose a value of S > p. At the upper benchmark S, the player will thus incur a loss VA(S) < 0.
Research then has value to try since it can allow to reach the lower benchmark where a zero payo¤ can be obtained. Thus, in these cases, at the upper benchmark S the loss VA(S) has to be equal to the expected cost of research needed to reach the lower benchmark.
Finally, result 2.3 indicates that the length of the research interval decreases with s. As stated above, when s is small, the purpose of research at the upper benchmark S is to avoid incurring the loss VA(S) by performing research to reach s and get a zero payo¤. When s is small, the loss is particularly large and furthermore, the expected time cost to reach s will be smaller, since there are more chances that the drift is negative. This property is represented in Figure 1. At the limit, l converges to a value l that depends only on vLA and c=r. The intuition is similar to that of 2.3. The player is sure that the state is bad and at BR(s), he is indi¤erent between getting the sure loss vLA and searching in the hope of reaching the lower benchmark, with a cost proportional to c=r.
These best responses are a natural building block for the rest of our argument. If the same player was making both the research and approval decisions, his optimal choice (s ; S ) would be characterized by the intersection of the best response curves. However in practice these decisions are typically made by di¤erent agents and these strategic interactions are the focus of our paper.
To clarify the exposition of the rest of the paper, we add more structure on how the preferences of agent and principal are misaligned. We will focus on the leading case where the agent does not bear the entire social cost of a wrongful adoption, so that the payo¤s of adoption in the low > vp . For instance, in the application to drug approval the key concern is that the pharmaceutical company does not fully compensate patients who su¤er from taking an unsafe drug because of the di¢ culty in identifying these individuals and the company's ability to shelter from liability (the judgement proofness problem). According to the previous comparative statics, this implies that the agent prefers to stop earlier at the upper end but to conduct more research at the lower end.
The comparison in the high state is less obvious. It seems reasonable to think that va i.e., that the submitter has more at stake than society at large. For instance an author that has a paper accepted gets the full private bene…ts from that decision, but does not take into account the negative externality he imposes on other authors. In the case of a private …rm conducting research, this can be less obvious. Indeed, it is often thought that a …rm cannot capture the full social bene…t from an innovation; see for instance Bloom, Schankerman, and Van Reenen (2012).
Of course the factor mentioned above, that goes in the other direction, is still present: the …rm that innovates, in the case of non radical innovations, takes some pro…ts away from the current market leader, an e¤ect typically not internalized. Our results for the leading case hold provided the externality associated to adoption in the high state is either negative or positive but lower than the negative externality associated to adoption in the low state: va Given this type of con‡ict in preference, the bliss point of the principal (sp; Sp) and of the agent (sa; Sa) are such that the agent conducts more search at the lower end sa < sp and less search at the upper end Sa < Sp as represented in Figure 2. One property is notable and will 9 There is an additional factor speci…c to the pharmaceutical industry that can push vaHA above vp . Given drug users typically do not pay directly but are reimbursed, pharmaceutical companies might be able to obtainprivate bene…ts higher than social bene…ts.
prove useful in the rest of the paper: the bliss point corresponds to the maximum of the upper best response and the minimum of the lower best response.
Approval Regulation In practice, in all the applications we have in mind, the same player does not control the full research process and at the same time make the approval decision. As described in section 2, in the case of drug regulation, separating these roles was precisely the purpose of the 1938 law, that introduced an approval requirement before marketing.
A natural way of thinking of the e¤ects of such a relatively weak regulation is that the …rm conducts research and, at some point submits the evidence to the principal who, based on the evidence, makes a decision to approve or wait. The 1938 law did not give the principal power to mandate further research. Thus, in a subgame perfect equilibrium, the principal approves any drug when the evidence is above p (i.e. any evidence that gives the principal a positive expected bene…t). With small con‡icts of interest (va ), the research interval is then the …rst best of the agent, (sa; Sa). With intermediate con‡ict of interest s.t. Ba (ba ( p)) < p, the agent searches in (ba ( p) ; p). While with large con‡ict of interest such that p > S de…ned as the lowest S above p such that S = ba (S), agent does not do any research.
The 1962 Amendments gave the further power to the principal to mandate research. We examine in Section 4.1, how this extra power a¤ects the outcome of the game. We then show in Section 4.2 that committing ex ante to an approval standard can improve the payo¤ of the designer. This was not the path chosen by the lawmakers, who instead chose to organize the regulation as a sequential process that we examine in Section 4.3. Throughout this section we will maintain the assumption that the regulator cannot misreport the information he obtains, an assumption we relax in Section 5.
The e¤ect of granting the principal the power to mandate research (as in the 1962 Amendments) in her interaction with an agent who initiated the research process, is naturally captured by the following baseline model. In each period t, agent and principal move sequentially. First, the agent chooses between three actions research Ra, submit Sa or wait/withdraw Wa. Second, if the agent submits Sa, the principal chooses between research Rp, approve Ap or wait Wp. Research is the period's outcome if either the agents chooses research Ra or the agent chooses submit Sa and the principal chooses research Rp (in that sense the principal can mandate research). Approval A is the period's outcome if the agent chooses Sa and the principal Ap. Finally withdrawal W is the period's outcome if the agent chooses Wa or the agent chooses to submit Sa and the principal We assume that the cost of research enters symmetrically in the agent's and principal's utilities regardless of who conducts the research. In the case of the FDA, the principal can mandate research but integrates this research cost in her welfare function, since these costs re‡ect the costs to patients. We show in the appendix that the outcome of all Markov Perfect Equilibria of the game above, with as the state variable, correspond to what we call the Nash equilibrium solution and denote (sN ; SN ). This equilibrium is at the intersection of the best response curve of the agent to the upper benchmark bra(S) and the best response curve of the principal to the lower one BRp(s) . In other words, in this setting, the principal controls the upper standard S while the agent controls the lower standard s. We have: Proposition 3 There exists a unique Nash equilibrium such that: 1. the principal conducts less search at the upper end and the agent less search at the lower end: Sa < SN < Sp and sp > sN > sa; 2. the length of the research interval is larger than for the stand-alone problems: SN The strategic interaction between the agent and the principal of course a¤ects the research decision compared to the non strategic benchmark. These changes can be decomposed in two e¤ects. First, there is an e¤ect on the extensive margin: the range of values of research is conducted changes. Second, there is an e¤ect on the intensive margin: the length of the research interval is a¤ected.
Result 3.1 above refers to the extensive margin. Compared to the principal's …rst best, in the Nash solution, more research is conducted at the lower end and less at the upper end. The logic of the result is clear. Since both the agent and the principal now control only one benchmark, the value of information is decreased: the principal conducts less research at the upper end and the agent less research at the lower end then in their respective stand-alone problems. These ideas are illustrated in Figure 1. The solid lines correspond to the best response S to a given s and the dotted ones to the best response s to a given S. The equilibrium (sN ; SN ) is at the Figure 1: Best replies, the Nash equilibrium, the commitment solution, and comparison with theprincipal's and the agent's unconstrained solutions.
intersection of the lower best response curve bra(S) of the agent and the upper best response curve of the principal BRp(s). The …gure illustrates the fact that SN < Sp and sN > sa. Indeed, as the lower benchmark s moves away from sp, the principal's best response decreases (Sp being the maximal value), since the value of information is decreasing.
Result 3.2 above refers to the intensive margin. Surprisingly, introducing strategic interactions increases the intensity of research (the length of the research interval is increased). This runs contrary to the classical intuition that tends to …nd the opposite e¤ect (for instance Strulovici 2010). To understand this result, consider the agent's problem. The principal conducts more research at the upper end than the agent would like him to do (SN > Sa). Thus, when the agent considers the choice of the lower benchmark, expressed in equation (3), his incentives to search are higher: if he reaches the upper benchmark, he gets a higher value (VA(SN ) > VA(Sa)) and moreover he reaches it faster in expectation since the belief that the state is H is higher for larger values of S. These two e¤ects (underlying the result of Proposition 1.3) imply that the search interval is larger SN We now discuss the payo¤ of the principal and the agent in the Nash Equilibrium solution.
These values are plotted as a function of in Figure 1. The key message is that the utility of the principal can be negative at the Nash equilibrium solution. Consider a belief sN < 1 0 A similar logic leads to the result SN principal was picking both benchmarks alone, he would not be able to obtain a positive utility and he would thus choose sp > , in other words immediate rejection, to guarantee himself a zero payo¤. In the Nash equilibrium solution, this is not an option since the agent controls the lower payo¤ and actually sets it below Proposition 4 At the unique Nash equilibrium: the agent gets a positive payo¤ for all values of the principal gets a negative payo¤ for 2 [sN ; brp(SN )], In practice, it seems natural that the principal would try to achieve a higher payo¤ by committing ex ante to a certain behavior. The most natural form of commitment, that we consider in the next section, would be to commit ex ante to a certain standard of approval.
As highlighted in section 2, this was not the approach chosen in the 1962 Amendments, who chose rather weak legal language in terms of standards. The legislator chose rather to commit to perform the evaluation in a prede…ned number of rounds, something we consider in section 4.3.
Commitment: Stackelberg Solution We study in this section the case where the principal has the ability to commit to an approval standard that depends only on the current state of knowledge (and not on the path or time taken to get there). Clearly, if the principal could commit to an approval rule that could be conditioned on the entire path, the principal would be able to obtain the unconstrained optimal solution sp; Sp . Such commitment, however, might be di¢ cult to achieve in practice so we consider a simpler and more realistic commitment to approval rules that depend only on the current state of knowledge with the following cuto¤ form: approve if and only if is the type of commitment, although weak, that was introduced in the 1962 law with the terms We now characterize the path of the commitment solution; see Figure 4.2. Dynamic con- sistency no longer applies: the optimal choice of commitment by the principal depends on the as described in the following result. We use the notation bra for the upper inverse function constructed by inverting bra for S > Sa.
Proposition 5 In the Stackelberg equilibrium with commitment by the principal, there exist be- liefs 2 (sp; bp) such that, if the con‡ict of interest is not too large, i.e SN > bp: 1. If the initial belief < , the principal chooses a blocking commitment inducing no research: sa, any commitment above is part of an equilibrium: Sc( ) 2 ( ; +1); < , any commitment above bra ( ) is part of an equilibrium: Sc( ) 2 (bra ( ); +1).
2 ( ; SN ), the principal chooses an interior commitment Sc( ) decreasing in .
There is a discontinuity in commitment at : Sc( ) < bra ( ).
> SN , the principal chooses an approval commitment Sc( ) If the con‡ict of interest is large (SN bp, the principal chooses a blocking commitment and if > bp, he chooses immediate approval.
For low values of the initial belief , the optimal commitment is what we call a blocking commitment: the principal commits to an upper benchmark that induces the agent to do no sa, the initial belief is so low that even the agent would not want to do any research, regardless of the commitment. For slightly higher, the commitment to be blocking, i.e induce the agent to do no research, needs to be above bra ( ) (by de…nition of bra(S)). Note that this minimum blocking commitment is an increasing function of When the initial belief , starts to be su¢ ciently favorable, blocking research by the agent by committing to rejection becomes too costly and there is a preferable interior commitment.
This happens at belief , which is the belief at which the zero iso-utility curve of the principal is tangent to the lower best response curve of the agent: if the principal chose his preferred point on the best response curve of the agent bra ( ), he would get a zero utility and he is thus indi¤erent. If the belief is above that value, an interior commitment is strictly preferable. At this point there is a discontinuity in the commitment: there is a discrete downwards jump from a blocking commitment to the optimal interior commitment.
Proposition 5, indicates that the value is above sp. Intuitively, for beliefs lower than sp, the principal cannot obtain a positive utility even when in full control; a fortiori, the lowest at which the principal can only obtain a zero utility when the agent controls the lower standard must be above sp. [In addition, at = brp(SN ), the principal obtains a zero payo¤ at the Nash equilibrium solution and could do strictly better by committing to a di¤erent point on the agent's lower best response curve. The belief for which the principal is indi¤erent between interior commitment and the blocking commitment (yielding zero payo¤) thus has to occur earlier.] The fact that interior commitment is strictly above SN for 2 ( ; SN ) re‡ects the tradeo¤ between two e¤ects: A second-order negative direct e¤ ect: Holding …xed the agent's strategy s, an excessive amount of research is induced at the upper end, which induces a loss for the principal. This loss is clearly second order by the envelope theorem because we start from the principal's optimal choice of S holding …xed the agent's choice of s.
A …rst-order positive strategic e¤ ect: The agent's strategic response of the increase in S is to increase s given that the agent's best reply is upward sloping in the relevant range— strategies are strategic complements in the terminology of Bulow, Geanakoplos, and Klem- perer (1985). Intuitively, the increased loss of control at the upper end further reduces the agent's value of information at the lower end. Given that the agent's choice of s at the lower end was originally lower than the principal would have liked, this increase in s bene…ts the principal. This is …rst-order e¤ect because the envelope theorem does not apply given that the agent, not the principal, chooses s.
As we show, an increase in increases the direct e¤ect and reduces the strategic e¤ect. Thus, as the optimal interior commitment decreases in = SN the strategic e¤ect becomes zero, and commitment has e negative value for > SN . When the initial belief the optimal choice for the principal is to chose immediate approval, what we call an approval commitment. This occurs for beliefs below Sp (the point where the principal would chose immediate approval if in full control), since the principal cannot control the upper benchmark.
In fact, as indicated in Proposition 5, for beliefs above SN immediate approval is optimal.
In terms of welfare, it is clear that the principal can always do weakly better by committing.
In fact, the following proposition indicates that she does strictly better whenever the initial belief is in (sa; SN ). This result naturally follows from the previous discussion. First, the blocking com- mitment is chosen for beliefs where the Nash equilibrium gives negative utility to the principal, so that commitment is strictly preferable. Second, when an interior commitment is chosen, the principal's optimal commitment is strictly above SN indicating that a better commitment exists.
Proposition 6 In the Stackelberg equilibrium, the principal's payo¤ is: weakly higher than the payo¤ in the Nash equilibrium for all values of strictly higher for 2 (sa; SN ).
The …nal essential consideration is the e¤ect of commitment on both the extensive and in- tensive margins of research discussed in the previous sections.
Proposition 7 In the Stackelberg equilibrium: The extensive margin of research is decreased compared to the outcome in the stand-alone principal solution: ( ; SN ) The intensive margin of research is increased compared to both the Nash Equilibrium and the stand-alone principal outcomes: for 2 ( ; SN ), Sc bra(Sc) > max(SN sN ); Sp sp).
Proposition 7 suggests that, in the Stackelberg solution, there are less instances where some research is conducted, but for the values where this is the case, more research will be performed.
There are two instances where no research is performed in the Stackelberg solution: for low values , the principal chooses a blocking commitment and for high values he induces immediate approval. The …rst result is then due to the fact that the principal, since he does not control the lower benchmark, wants to prevent research for more values of . The second result echoes the result on the extensive margin for the Nash outcome: for beliefs such that research is conducted, more research will be performed.
Commitment: Sequential Research In practice a commitment is not always easy to achieve. In fact, as stated earlier, the 1962 law explicitly chose not to commit to a strict standard. As suggested in the previous section, one of the main reason could be that the optimal level of commitment is speci…c to the baseline state of knowledge, which can vary across types of drugs, and thus there is no uniform standard that can be applied. Of course, the close interaction between the …rm and the regulator could allow for individualized commitments, but even those are not easy to credibly make. A di¤erent way of committing is in the way research is organized. Often the communication between the agent and the principal is organized in a number of rounds. This was the approach chosen by the 1962 law that organized the interaction between the …rm and the regulator in a well de…ned series of clinical trials.
We consider in this section a model where the interaction is organized in two rounds of inde…nite length. First the agent conducts research and at some point decides to transfer this information to the principal. The principal then decides how much additional research to perform before making the approval decision.11 We maintain the assumption that the agent cares about the research cost of the principal. This assumption is sensible in the application to the FDA that can mandate research. It is less so for other applications and we consider those in section 7. We denote (sseq; Sseq) the choice of the agent, where seq stands for sequential.
When the agent submits the information to the principal, the principal performs research and makes the approval decision as in the stand-alone case since there will be no more interaction with the agent: the principal will then perform research if 2 (sp; Sp). Thus the agent will never want submit before Sp is reached because the agent bears the full research cost regardless of who performs the research but would lose from submitting to the principal before Sp is reached because then principal would carry too little research at the lower end in the eyes of the agent.
Furthermore, at the lower end the agent chooses the best response to Sp: sseq = bra(Sp). These results are summarized in the following proposition: Proposition 8 In the sequential problem: 1. In equilibrium the agent conducts research whenever is in (bra(Sp); Sp), abandons for lower values and submits the evidence for higher ones 1 1 In most applications, there could be additional round but we will focus on the one round case without loss of generality of the message.
2 (sN ; ), the principal obtains a higher payo¤ in the sequential than in the Nash 2 ( ; Sp), the principal obtains a higher payo¤ in the Nash equlibrium than in the 4. The length of the research interval is larger in the sequential than in the Nash or in the stand-alone problems The Nash equilibrium, the Stackelberg commitment solution considered in section 4.2 and the sequential research procedure correspond each to a di¤erent point on the best response curve of the agent, bra(S). Clearly, the Stackelberg point results in highest expected payo¤ for the principal. Results 8.2 and 8.3, indicate that the comparison between the Nash and sequential outcomes is potentially ambiguous. This suggests that the type of commitment put in place through the 1962 law was not necessarily welfare enhancing, in particular in cases where the initial belief is quite favorable as in 8.3.
Does the principal prefer the Nash equilibrium outcome or the sequential research outcome? Compared to the Nash equilibrium, the sequential procedure results in more research at the top and less research at the bottom. The principal is not necessarily better o¤, as illustrated in Figure 2. On the one hand, the principal bene…ts from reduction of s and increase in S along p's BRp(s); the movement toward North-East along BRp (s) from (sN ; SN ) to (bra (Sp) ; BRp (bra (Sp))) in- creases the principal's expected payo¤. On the other hand, the principal loses for additional increase in S; the upward movement from (bra (Sp) ; BRp (bra (Sp))) to (bra (Sp) ; Sp) results in a reduction in the principal's expected payo¤. The dashed indi¤erence curve corresponds to a setting in which the principal prefers the sequential solution to the Nash equilibrium; the opposite ranking holds with the continuous indi¤erence curve. As can be seen graphically, a su¢ cient condition for the principal to prefer the sequential solution to be Nash equilibrium is that SC > Sp = Sseq; otherwise the ranking is ambiguous.
In the region where the blocking commitment is the optimal Stackelberg commitment ( (sN ; )), the principal prefers sequential commitment to the Nash outcome because sequential commitment results in the same outcome as the Stackelberg commitment in some cases and limits the amount of research performed in others. When instead 2 ( ; Sp), the optimal commitment is to approve immediately and this is also the outcome with the Nash solution, whereas additional research is performed under the sequential commitment thus leading to a lower payo¤ for the Figure 2: Welfare comparison between the Nash solution and sequential research.
Misrepresentation of Information The new phase of regulation has started focusing on the regulation of the disclosure of clinical trial results. The alleged withholding of negative results by pharmaceutical companies in the recent cases of Vioxx (an anti-in‡ammatory drug proven to increase the risk of cardiovascular events) or Paxil (an anti-depressant that could increase the suicide rates among children) generated major uproar and large demands for compensation.
Withholding information has a potential cost for the …rm itself: in the case of Vioxx, Merck paid over 4.85 billion dollars for settling individual complaints from patients. In 2011, it agreed to plead guilty and pay 950 million to the federal government to settle the criminal and civil charges …led against it. These costs seem to be an increasing function of the size of the misrep- resentation. It is because Merck was shown to have withheld evidence that the penalties were of that magnitude. A larger lie makes it easier for the plainti¤s to win their case in court.
To capture such situations, we enrich the model and assume that the agent who collects the evidence can misreport it at no cost.12 However, if the state turns out to be low the agent expects a …ne F . The probability of obtaining this sentence depends on the size of the misreporting. If the agent has collected evidence showing that the state is and conveys information probability of being convicted is given by P ( 0) and we denote the overall expected sentence 0) where C (i.e., P ) is increasing and C(0) = C0(0) = 0.
1 2 In our model, evidence comes in in…nitesimal amounts so that the agent can for any state, always hide a su¢ cient quantity of negative results to be able to present veri…able evidence consistent with that state.
We concentrate on the commitment case as in Section 4.2 where the principal can commit to an approval standard, that we denote SM (M stands for withholding). The rule is thus: "approve if and only if the reported evidence is above SM ". In this setting, the agent is in full control of both extremities of the search interval, but the agent expects a lower value when deciding to < SM provided that the state turns our to be low.
Two limit cases are of special interest. When F = +1 we are back to the commitment case of section 4.2 where truthful reporting was assumed, (sC; SC). When F = 0, the agent's stand-alone solution of Section 3 results, (sa; Sa). However nothing refrains the principal from choosing intermediate values of F . He now has two instruments available that are indeed used in practice: the …ne F imposed for misreporting and the standard for acceptance SM .
In the context where F takes intermediate values, the cost of lying has two e¤ects on research: The cost of lying weakly decreases the payo¤ from approval and thus a¤ects research at The cost of lying creates an additional incentive to conduct more research at the upper end, to accumulate further positive evidence in order to decrease the expected …ne.
We show in the following results that this modi…ed research problem has a unique solution (sM ; SM ), which are the lower and upper benchmark of the search interval chosen by the agent when the principal sets a standard for acceptance SM . In other words, there is full information revelation in equilibrium. The agent will lie but the principal will know exactly the state at which search was stopped. Sequential search is key for this result— if, instead, the agent was just observing the state and reporting, di¤erent types would pool on the same report given the binary decision made by the principal.
Proposition 9 With private information and misreporting, there is a unique solution (sM ; SM ) such that the agent conducts research if and only if 2 (sM ; SM ) and such that, if SM > Sa, lying occurs in equilibrium, i.e the agent chooses an upper benchmark of search strictly lower than the standard for acceptance: SM < SM .
Proposition 9 indicates that there will always be lying in equilibrium. The intuition is that the marginal cost of lying when the report is close to the truth is zero while the added value of getting more information is strictly negative since the standard set by the principal is greater than the optimal upper benchmark of the agent (SM > Sa). Thus the agent prefers to stop a bit earlier and incur the expected lying cost.
We now examine the modi…ed best responses in this case. For the lower benchmark, the best reply to S is characterized by a condition identical to condition (3), bene…t of gaining more information …nancial cost of research except that the value VA upon stopping is given by Clearly, for given values of S less than SM the agent will search less at the lower end than he would in the absence of misreporting since his value in case the state turns out to be bad will be lower due to the expected penalty. It thus decreases the value of information which is proportional to VA.
For the best response to s, there is an additional e¤ect: searching more provides evidence that directly decreases the expected cost in case the state is in fact low. The …rst order condition bene…t of information value of evidence cost of delaying decision …nancial cost of research At the upper benchmark three factors push the agent to do more research than in the stand-alone The value of information (proportional to S)) increases since there is a higher cost to avoid in case the state is shown to be low.
The cost of delaying the decision, proportional to VA is smaller.
There is a value of collecting evidence (independent of the value of information), propor- S), this extra evidence making a court case less likely.
We can now use this best response analysis to compare the welfare of the principal in this case compared to the commitment case studied in Section 4.2. Note that the commitment case with no misreporting is equivalent to this model with an in…nite penalty. The question we therefore address is the following: Should penalties be in fact limited? In Section 4.2 we characterized the optimal commitment of the principal SC in the case of no misreporting, i.e., the principal's preferred point on the agent's best response bra(S). In what follows, we focus on the case of interior commitment, i.e according to Proposition 5, the case where the initial belief 2 ( ; ).13 The …rst thing to note is that in the case of limited penalties, the principal can always choose a large value of SM > SC such that the agent chooses in equilibrium the upper benchmark of search to be SC. Indeed, for any value of s the best response BRa(s) is an increasing function of SM . So SM can be chosen so as to induce the agent to do the same amount of research at the upper end as in the commitment case analyzed in Section 4.2. However in this case with misreporting, the agent will also search less at the lower end for the reasons outlined above. If this extra research is not excessive, this will be bene…cial for the principal: Proposition 10 For 2 ( ; ), the principal can always …nd a combination of instruments (F; SM ) with F < +1 that results in a strictly higher expected payo¤ than in the commitment case without misreporting.
Proposition 10 indicates that it is optimal in equilibrium to allow some misreporting. Mis- reporting induces the agent to conduct less excessive research at the lower end. The intuition is presented in Figure 9.
1 3 In the case where without misreporting the optimal commitment is a blocking commitment, this can also be obtained with misreporting.
Post-Approval Regulation After the approval of the drug by the regulator, the drug is prescribed to patients and further information is thus accumulated on a large scale. If very bad information is accumulated, the principal can decide to order a withdrawal of the drug. However, this information accumulation strongly depends on the monitoring e¤ort of the principal.14 In practice, it appears that the monitoring e¤ort of the FDA in the United States is rather weak: for instance there is no central recording system of all adverse e¤ects. The FDA intervenes only in the case where litigation starts for serious injuries due to the drug.
We thus extend our model to this setting and it further demonstrates the ‡exibility and applicability of our setup. We assume that while the drug is on the market, the regulator obtains a ‡ow bene…t w and the …rm a ‡ow pro…t . However, if the state is in fact low, at some point in time very serious adverse e¤ects will occur. We assume that the players then gets a negative payo¤ Pi (where i 2 a; p) whose arrival time is distributed according to an exponential distribution of parameter . Furthermore we assume that Pi is large enough so that if the regulator is sure the state is bad, he will immediately withdraw (condition formally stated below).
Initially, before the start of the game, the principal chooses how closely he will monitor the drug post approval. Speci…cally, we assume he chooses a costless monitoring e¤ort e.15 Given a choice of e, post approval, there is a probability (e) that principal obtains further information.
In this case he chooses a benchmark of withdrawal W . With probability 1 information is collected.
The timing of the game is therefore: 1. The principal chooses e at no cost 2. The agent and the principal simultaneously choose s and S (Nash equilibrium as in Section 3. If the decision is to approve, with probability (e) further information is obtained and the principal chooses the withdrawal benchmark W Stage 2 is unchanged compared to the Nash Equilibrium resolution of section 4.1. However, the addition of stage 3 puts structure on the payo¤s in the di¤erent states (vLA and vHA): 1 4 It also depends on the amount of sales, something that could also give rise to a di¤erent extension of our 1 5 In the case of the FDA, this corresponds to setting up a central system to collect information on adverse If there is no additional information accumulated post approval (probability 1 agent and the principal have no decision to make, they just collect bene…ts and potentially su¤er from penalties. If the state is high, the principal obtains w and the agent state is low, they collect these bene…ts until the adverse e¤ect arises: the principal obtains p) and the agent obtains r If additional information is accumulated (probability (e)), the principal has to choose W the withdrawal benchmark. In the bad state, bene…ts will be obtained up to the point where either the belief reaches the withdrawal benchmark W or the adverse shock occurs leading to withdrawal. In the good state, bene…ts will be continuously obtained unless the belief reaches the withdrawal benchmark W Given the choice of W we denote fL(tj ) (resp. fH(tj )) the distribution of the …rst time the belief reaches the withdrawal benchmark W < S for a belief . The distribution fL(tj ) is an inverse Gaussian of parameters ( Conditional on the low state, the expected bene…t of the principal for a choice of W at belief fL(T j ; W )dT dt e ( +r)tP [T > t When the state is high, this is If the state is low, there are two countervailing e¤ects of an increase in W : it decreases the chance of getting the penalty but increases the expected ‡ow of bene…ts w obtained. If the penalty is large enough, the …rst e¤ect dominates and vLA increases in W . If the state is high, an increase in W unambiguously decreases the expected payo¤ vHA. Thus, if the belief becomes su¢ ciently negative, the regulator has an incentive to withdraw: Proposition 11 In the three stage game with post approval, there exists a value P such that, if 1 6 Distribution of the …rst time a Brownian motion hits a particular value the principal imposes a withdrawal benchmark W > 1 and achieves a strictly higher payo¤ than when no information is obtained.
the agent's lower best response and the principal's upper best responses are decreasing in e.
When the penalty is high, choosing partial monitoring, i.e. picking a low value of e, is a way for the principal to decrease the value obtained when the state is bad vLA and thus to commit to perform more research at the upper end. It also forces the agent to do less at the lower end.
This could explain the behavior of the FDA that chooses not to have a systematic review of post approval evidence.
Other Applications: Research with Private Costs Our model can be extended to analyze a wide range of situations involving an agent initiating research and a principal making approval decisions. As discussed in the introduction, most relevant are the cases of an author trying to convince an editor, …rms wanting to merge trying to in‡uence the antitrust authority or a manager trying to push his project at the headquarters.
This section extends the model to address these applications.
Throughout the previous sections, we maintained the assumption that both the principal and the agent cared about the full cost of the research, even if the other party was performing it. It was natural in the case of drug approval: the …rms conduct the research (since the regulator can mandate it) and thus care about the cost and the regulator also cares as a social planner. In the applications discussed above, instead, the agent does not typically bear the cost of the research undertaken by the principal.
Of course, it is always possible for the principal to charge the agent for the cost of his research.
For instance the competition authority could charge a fee proportional to the number of sta¤ members it puts on a particular case, or the …rm headquarters could ask the manager to pay a fee for examination of the case. However, it does not seem to be the approach taken in most applications. We examine in this section, under what conditions it is optimal for the principal to charge the agent for the additional research.
We place ourselves in the context of the sequential two stage game considered in section 4.3.
A di¤erent interpretation of the setting considered in that section is that it corresponds to the case where the agent has to pay for the principal's research cost. We are now going to derive the other polar case.
The …rst thing to note is that, regardless of whether the agent pays or not for his research e¤ort, once he transfers the information, the behavior of the principal will be identical: after the agent's report is received, if the report of the agent is in the interval (sp; Sp), the principal will conduct additional research using these stopping times (sp and Sp).17 Sseq) the agent's search interval when he does not pay for the principal's research ((sseq; Sseq) is the analogous interval in the case where he does pay, derived in section 4.3), we see that from the principal's point of view, the welfare bene…ts from charging for the principal's research, will only depend on the di¤erence between b sseq and sseq since at the upper end the …nal outcome will be identical (search will be conducted until Sp). We obtain the following result that shows that it is not necessarily optimal for the agent to charge for his research: Proposition 12 It is optimal for the principal to charge for his research if and only if In the case where the agent pays for the principal's research (case of section 4.3), the agent searches in the interval (br(Sp); Sp). Indeed, he knows that if he stops searching before Sp, the principal will keep searching and he will run the risk that bad information arrives and the principal abandons the search ine¢ ciently early since they do not agree on the lower benchmark of search. Since he pays in any case for the research, he has no incentive to stop before Sp.
As a reminder, in this case, the lower best response to S is characterized by: bene…t of gaining more information …nancial cost of search In the case where he does not pay, a tradeo¤ emerges: the agent might want to stop earlier to save on research costs even though he runs the risk that the principal stops early. When he stops at a belief S < Sp, the agent knows that the principal will conduct additional search and thus uncertainty remains. If the state is high, rather than expecting a payo¤ of vHA, he p(S; H )vHA that we denote vHS and if the state is low, p(S; L)vLA replaces vLA that we denote vHS, where p(S; I ) < 1 characterizes the expected time for the principal to reach his upper benchmark when the state is I.
1 7 Given the result of Proposition 2, the boundaries of the principal's search interval will be independent of the report of the agent.
The …rst-order condition characterizing the best reply is then bene…t of gaining more information …nancial cost of search So it is the same condition as in the case with the fee except for the value of information (VH rather than VA). The key to result in Proposition 12 is that VH can be lower (resp. higher than) vLA is small (resp. large) compared to vHA. The intuition is clear: the fact that the principal conducts more search has a negative impact if the state is high (delay good decision) and a positive one if the state is low (potentially avoid bad decision). If vLA is large compared to vHA, the positive impact outweighs the negative one and the agent would in fact search more.18 For the sake of concreteness, most of the paper focused on illustrating how our model sheds light on current and considered regulation of the drug approval process. Similar considerations are relevant for approval regulation in other areas, from competition and consumer policy to …nancial regulation. These applications are particularly relevant given the recent trend toward increased consumer protection and the move toward using the ex ante approval approach when regulating systemically important …nancial institutions.
Given its tractability, the model can be extended to address a number of other applications.
In the case of managers proposing projects to the headquarters, it would be natural to consider competing agents. Considering an agent facing a sequence of principals would be a natural extension. For example, in the case of authors submitting papers, upon rejection in one journal, authors can submit to another outlet. In some applications, the cost of research might di¤er across players. It is worth revisiting the question of delegation of decision rights by the principal to the agent within this model.
1 8 Note that reasoning on the lower best response functions is su¢ cient since, given the smooth pasting condition, the equilibrium in both the case with and without the fee will correspond to the maximum of the best reply.
We can rewrite the utility is the research area as where, for a given process with mean 2 and optimal stopping times s and S, standard results as in Stokey (2009) give . We can derive useful relations among these values using the fact R1 + R2 = 1. We have e(S s) e(R2 1)(S s) = e S+s+R2(S s) e S+s+R1(S s) = es S eR2(S s) eR1(S s)=e We now determine the best response to a given upper benchmark S. The …rst order condition w.r.t. s is Substituting the expressions for @ =@s R2) e R1(S s) e R2(S s) R2eR2(S s) + R1eR1(S s) We can rewrite the …rst-order condition as This establishes the …rst part of 1.1: the best response is independent of the current belief fact, br(S) is implicitly de…ned by It is useful to introduce the following variables, Dividing the previous expression by eS s we can rewrite the br(S) as , we obtain as in the main text We show below that 1(s; S) > 0. We have eg > 1 (eg is increasing in S s and is equal to 1 for s = 0), so the sign is that of egeS s R1e R2(S s) + e S eR2(S s) R1eR1(S s) + eR2(S s) 1(s; S) > 0.
Comparative Statics for br (S) We use to derive the comparative statics, the following expression for br(S) Comparative Statics with Respect to vHA Taking derivatives of this …rst order condition, we have Comparative Statics with Respect to vLA As above, we have Using the same arguments as above, we have Comparative Statics with Respect to c Follow immediately from the fact that 1(s; S) > 0.
br(S) increasing in S br(S). Equation (13) can be rewritten as Taking derivatives with respect to S, we have Following the same arguments as above we can show @g > 0 and @g0 < 0.
At l = 0 (i.e s = S), eg = 1 so that @l > 0. Equation 16 then implies that @g0 < 0 is always increasing and asymptotically approaches the value l solution to vHA = (eg(l) 1) c , as indicated in result 3.
We use the notation ( ; H) and for ( ; H). Using the expression for the utility derived in the proof above, the …rst order condition with respect to S is given by: We now derive the expressions for the partial derivatives of and with respect to S. We R1e R1(S s) + R2e R2(S s) From the de…nition of we have Given that R1 + R2 = 1, collecting eS we obtain @ = (R2 R1)eS e R2s R1 After adding and subtracting R2 to the …rst exponential in the numerator and, similarly, after adding and subtracting R1 to the second exponential in the numerator, this is equal to @ = (R2 R1)eS e R2s R1 +R2 R2 Substituting the de…nition of ( ; L) = eR2(S s) eR1(S s) ( ; H), we conclude that Exploiting the following relation eR1(S s) = e(1 R1)(S s)e(1 R2)(S s) = e(S s) e R1(S s) rewrite the condition above in a more compact way as ( R2e R2(S s) R1e R1(S s) ( R2e R2(S s) R1e R1(S s) We can now express the best reply as We can rewrite the implicit equation de…ning BR(s) as (s; S)VA + (s; S) c = (s; S) = ef es S(1 + eS) > 0 (s; S) = ef0 > 0: (s; S), we use the fact that Note that ef > 1 and we can show that ef e S e s + ef0e s is of the same sign as Comparative Statics To derive the comparative statics on BR(s), we use the following characterization of the best Comparative Statics with Respect to vHA Taking derivatives of (19) R1e R1(S s) + R2e R2(S s) if VA(S) > 0, we clearly have If VA(S) < 0, we can rewrite the expression Given that @f < 1, we also obtain the result.
Comparative Statics with Respect to vLA Thus the same arguments allow us to establish that Comparative Statics with Respect to c Follow immediately from the fact (s; S) > 0.
s decreasing in s We denote l = BR(s) s. Equation (19) can be rewritten as Taking derivatives with respect to s, we have This can be written as Following the same arguments as for the proof of the comparative statics with respect to vHA, we have that the denominator is positive. Since ef > 1, the numerator is negative and we obtain We now establish the result by taking the limit of the above …rst order condition. We obtain (ef es S + ef0) vLA + We can show that ef es S + ef0 = eg, so that indeed, l is solution to vLA = (e g(l) + 1) cr Foundation for the Nash Equilibrium Solution At each instant t, agent and principal move sequentially: First, the agent chooses research Ra, submit Sa or wait/withdraw Wa Second, if the agent submits Sa, the principal chooses research Rp, approve Ap or wait Wp R results if Ra or (Sa; Rp); A results if (Sa; Ap), W results if Wa or (Sa; Wp) We solve for the Markov Perfect Equilibria where the state variable is given by the current in- formation . We show that the unique MPE outcome is: R for < sN and A for This unique outcome can supported by multiple equilibrium strategies.
We prove the result in a number of steps: bri(S) reaches its minimum at S and BR i(s) reaches its maximum at si According to the smooth pasting condition, given that rejection always yields a zero value, Taking derivatives with respect to S, taking into account that s = br(S) yields Furthermore, given that @ui (S ) = 0, we have The best response function br(S) reaches a minimum for S = S .
Taking derivatives with respect to s yields The best response function BR(s) reaches a maximum for s = s .
If bra and BRp cross it is for values such that BRp(s) is increasing in s and bra(S) is decreasing in S.
In Proposition 1 and 2, we showed that bra(S) brp(S) and BRp(s) BRa(s). According to step 1 and 2, BRp crosses brp at (sp; Sp) where BRp is maximum. Given that bra(S) and that brp is single peaked, it has to be the case that bra and BRp cross for values such that BRp(s) is increasing in s. The same logic applies to show that bra(S) is increasing in S when bra and BRp cross.
@BR(s) < 1.
We derived in the proof of Proposition 2, that BR(s) is de…ned by s, and taking derivatives of the implicit equation above, we have This can be rewritten as Furthermore we have Given that ef > 1 the right hand side is positive. Furthermore, we have that along BR(s) that > 0. Therefore, since we established that @f > 0, we conclude that @br 1(s) > 1 for values such that br(S) is decreasing in S.
br(S) is de…ned by the following implicit function We have shown that for s > s , br(S) is an increasing function. On this interval, br 1(s) is a well de…ned function. Taking derivatives of the implicit equation above, we have Grouping terms we have We showed in the proof of Proposition 1 that: e Seg0 + e S ege s < 0. Given that vLA < 0, to have the equation above satis…ed, since we are deriving a property of the curve over which @S > 0, it has to be that We also showed in the proof of Proposition 1 that @g0 < 0 and @g > 0. So the above inequality implies that @S > 1.
Step 5: Deriving the results.
Step 3 implies that if a crossing between bra(S) and BRp(s) occurs, it will be for values of (s; S) such that the properties of step 4 and 5 apply. Furthermore these properties imply both that the curves do cross and that the crossing is unique: when the curves cross, @br 1(s) < 1 and @BR(s) < 1, so the curves cannot cross again.
Result 1. then follows from step 1.
Finally from Proposition 1.3, we know S br(S) is increasing in S, so since SN < Sa and sN = bra(SN ), we have SN sa. Similarly, using Proposition 2.3, we have SN sp. This establishes result 3.
Result 1: We have SN > Sa > a. So, if SN , the agent gets a strictly positive payo¤ < SN he can guarantee himself a zero payo¤ by choosing s = . Thus the agent's payo¤ is positive for all values of Result 2: Given a value of the upper benchmark SN , the principal gets a zero payo¤ when he chooses the lower benchmark = brp(SN ) since it implies rejection. In the Nash equilibrium solution, since sN = bra(SN ) < brp(SN ), he gets a strictly negative payo¤. The result is true 2 [sN ; brp(SN )] since even if the principal was choosing alone, he could only guarantee a zero payo¤. However, when sN > , the principal, by de…nition of s, gets a zero We …rst examine the case of small con‡ict of interest: sa, the agent never experiments, since even when he controls both bench- marks, he gets a negative payo¤ when experimenting. Thus, in this case, as indicated in Propo- sition 5.1, any commitment Sc( ) > will be equivalent for the principal and will give the zero rejection payo¤. Furthermore, choosing a commitment below leads to immediate approval and a negative payo¤. We conclude on the …rst result: if sa then Sc( ) 2 ( ; +1).
> sa, the principal has the choice between an interior commitment that leads the agent to perform research and a blocking commitment, such that the agent performs no research: even though the principal cannot force rejection because the lower threshold s is controlled by the agent, the principal can induce the agent to reject immediately by committing to a su¢ ciently high adoption threshold S such that bra (S) = s . The lowest possible such blocking commitment Sc = bra ( ), where bra is the upper inverse function constructed by inverting bra for S > Sa. Clearly, any S strictly above bra ( ) is equally blocking and results in exactly the same outcome and gives the principal a zero payo¤.
Consider the other option of an interior commitment. We …rst show that if, for a starting , an interior commitment gives a strictly positive value (i.e is strictly preferred to a blocking commitment), then it will also be the case for 0 > . Consider any point (bra (S); S) (point on the agent's best response curve). The utility is given by ) + R1eR1(S ) < 0: On the contrary, we have We need to show that the second e¤ect dominates to prove that @u(s; S; ) > 0 and thus if u(s; S; ) > 0, then for 0 > , u(s; S; 0) > 0. Therefore, there exists a value of the initial belief such that the interior commitment is preferred to the blocking commitment if and We conclude the proof of result 1 by showing that 2 (sp; p). For (s; S) gives the principal a negative or zero payo¤, so naturally > sp. For equilibrium outcome (sN ; SN ) gives the principal a positive payo¤ since he could always choose S = p and guarantee a zero payo¤. So we have Result 2. According to result 1, if is above and close to , an interior commitment is optimal. We …rst show that the interior commitment is decreasing in where @up and @up are derived in the proof of Propositions 1 and 2 and ds corresponds to the lower best response of the agent to a choice of S. The …rst term is the non strategic direct e¤ect of S on the principal's utility. The second is the strategic e¤ect of S working through the indirect impact of S on the agent's choice.
We …rst examine how the direct non strategic e¤ect reacts to an increase in Using the formulas derived in Proposition 2, we have that @2up is proportional to Both terms are positive so that For the strategic e¤ect, we use the fact that the agent's best response to S is independent , so that ds is independent of . Using the formulas derived in Proposition 1, we have that @2up is proportional to The …rst term is positive while the second is negative. Thus we have Overall we need to show that the strategic e¤ect dominates and we have In a second step we show that We showed in the proof of result 2 that @2up < 0 and ds > 0.
We have established in the proof of Proposition 2 In the proof of Proposition 2 when we examine the comparative statics with respect to vHA, we established that the second term is also decreasing in S. Thus we have We also have d2s , so that overall, the utility the principal is a concave function of the level of commitment S.
This establishes that up has a unique interior maximum de…ned by dup = 0. Furthermore, we established that d2up < 0, so that u p will reach a maximum for lower values of S as i.e the optimal commitment SC( ) is decreasing in Result 3. We now identify the belief where the principal will switch from an interior com- mitment to an approval commitment where the principal chooses immediate approval. Denote , the …rst belief such that the principal is indi¤erent between commitment and immediate ap- proval. Such a belief exists: indeed, the payo¤ from approval and from interior commitments are continuous and, for Sp, the optimal commitment is Sc = since even when the principal has full control, he would adopt immediately. For belief , we must have SC( ) = . Furthermore, since we showed above that the interior commitment SC( ) is decreasing in , we will also have the interior commitment is below and the principal therefore chooses immediate We conclude the proof by showing that = SN . Note that we necessarily have < SN , at the Nash equilibrium, the principal could choose immediate approval but does not do so, in other words, for those values of , a commitment SC = SN does better than immediate = SN , an interior commitment at S = SN gives the payo¤ of immediate approval.
We conclude the proof by showing that at = SN , the optimal interior commitment is indeed SC = SN . We have At (s; S) = (sN ; SN ), we have @up (s N ; SN ) = 0. Furthermore, for ) (multiple of ) is equal to zero. Thus, we have for dup (sN;SN; = SN) = 0: As we established before, the unique interior commitment is characterized by dup = 0 so that the optimal commitment is SC = SN . This concludes the proof.
This follows immediately from the results in Proposition 5. For 2 (sa; ), the optimal commitment is either a blocking commitment or an interior commitment that gives a strictly higher payo¤ than the Nash. For < sa, both yield a zero payo¤. For approval payo¤.
In Proposition 5 we showed that > sp and < Sp. This establishes the …rst result: SN . We showed in Proposition 1 that S bra(S) is increasing bra(SN ). The fact that Sc immediately from Proposition 3.
Result 1: As explained in the main text, in the second phase, the principal will search in the interval (sp; Sp). By backwards induction, the agent will thus search in the interval (bra(Sp); Sp).
Indeed, since the agent pays the cost of research even if the principal searches, stopping for decreases the utility of the agent since sp > bra(Sp) and thus with some probability, the principal would stop too early.
2 ( ; Sp), the Nash equilibrium outcome coincides with the optimal com- mitment and is strictly di¤erent from the outcome of sequential search. The principal will thus achieve a strictly higher level of utility Result 4: The length of the research interval is larger in the sequential case than in the Nash since (bra(Sp); Sp) is a point on the agent's best response curve for a higher value of S.
The result in Proposition 2.3 yields the result. It is also clearly larger than for the standalone problem since the value of S = Sp is the same but the lower benchmark s = bra(Sp) < brp(Sp).
Given an acceptance standard SM , when the agent chooses a search interval (s; S) with SM , the utility is The change compared to the benchmark case is that an additional cost is incurred if the state is low. The best response to the upper benchmark is given by Thus the best response is above the best response in the benchmark case.
We now examine the best response to the lower benchmark. The …rst order condition with respect to S is slightly modi…ed ( R2e R2(S s) R1e R1(S s) We can then rewrite the …rst order condition in a more compact way as (s; S)VA + (s; S) = We conclude the proof by showing that the upper benchmark chosen SM by the agent is such that SM < SM . Suppose this was not the case and the agent chose SM = SM . Then the best response above would be the same as the best response for the agent's stand-alone problem. However, given that SM = SM > Sa and that the agent's stand-alone upper best response reaches its maximum at a value S = Sa, SM cannot be on the agent's best response curve. We therefore reach a contradiction. As can be veri…ed grapically in Figure 5, the agent's upper best response with misreporting is strictly below the horizontal line S = SM .
We are in the case 2 ( ; ), i.e the case of an interior commitment. At S = Sc( ), by de…nition of the interior commitment, the iso-utility curve of the principal is tangent to the lower best response curve of the agent for a …ne F = O (bra(S), i.e. the best response of the agent used in all the previous sections).
For a …ne F > 0 and an approval standard SM = Sc, the best response curve of the agent bra(S; F ) is modi…ed. It is also tangent to bra(S) at S = Sc( ) since C0(0) = 0. However, locally, for F large enough it is to the right of the iso-utility curve of the agent: for a given S, s is strictly increasing in F . As a consequence, the intersection with the agent's upper best response BRa(S; F ), (sM ; SM ) belongs to a higher iso-utility curve. All these properties are represented in Figure 9.
If the state is low, the expected bene…t of the principal for a choice of W at belief by (5), is decreasing in W for su¢ ciently large values of Pp. If the state is high, (6) is increasing in W . Thus, there will be a benchmark belief W where these two e¤ects cancel each other and such that withdrawal should be chosen, whenever Given that the expected payo¤ upon approval are vHA = p(e)vHA(W ) + (1 vLA = p(e)vLA(W ) + (1 LA is increasing in e and vHA is increasing in e. For Pp su¢ ciently large, the e¤ect on vLA dominates and according to Proposition 2, the best response of the agent is decreasing in e. A similar argument can be made for the lower best response of the agent.
The utility of the agent is given by where the expected value upon stopping research depends on the uncertainty due to the principal's research process, We …rst examine the …rst order condition characterizing the lower benchmark of search s. Since p(S; H ) depends on s, the …rst order condition is identical to the base case except for the value of VA. We have VH (S) = 1(s; S) c=r using the fact that p(S; H ), we have We examine how VH compares to VA. We have that for S = Sp, VH(S) = VA(S). Thus, we look at the comparative statics of VH (S) with respect to Sp, for S 1+eS e R1(Sp sp) e R2(Sp sp) R2e R2(Sp sp) (vHA + e Sp vLA) e Sp (e R1(Sp sp) 1+eS e R1(Sp sp) e R2(Sp sp) R2e R2(Sp sp) vHA + e Sp R2e R1(Sp sp) + R1e R2(Sp sp)) vLA : Therefore we have @VH R1e R1(Sp sp) + R2e R2(Sp sp) e Sp R2e R1(Sp sp) R1e R1(Sp sp)+R2e R2(Sp sp) i < 1. We use the notation R2e R1(Sp sp) R1e R2(Sp sp) R1e R1(Sp sp) + R2e R2(Sp sp) e Sp R2e R1(Sp sp) Note that this condition is independent of S.
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