AA Quest for Knowledge ∗Johannes Schneider † Christoph Wolf ‡ March 2021
Abstract
Which discoveries improve decision making? Which question does a re-searcher pursue? How likely will a researcher discover an answer? We proposea model in which the answers to research questions are correlated. Revealingan answer endogenously improves conjectures on unanswered questions. Adecision maker uses conjectures to address problems. We derive the benefits ofa discovery on decision making. A researcher maximizes the benefits of discov-ery net the cost of research. We characterize the researcher’s optimal choiceof research question and research effort. Benefits and cost of research cruciallydepend on the structure of pre-existing knowledge. We find that discoveriesare rare but advance knowledge beyond the frontier if existing knowledge isdense. Otherwise, discoveries are more likely and improve conjectures insidethe frontier. ∗ We thank Marco Ottaviani, Antonio Cabrales, and William Fuchs for guidance on the project.We thank Steven Callander for fruitful discussions at early stages. We are grateful to MarcoCelentani, Philipp Denter, Florian Ederer, Alex Frug, Toomas Hinosaar, Nenad Kos, Ignacio Ortuño,Nicola Pavoni, Harry Pei, Konrad Stahl, Armin Schmutzler, Carlo Schwarz, Ludo Visschers andaudiences at Pompeu Fabra, THEMA Cergy, uc3m, the Economics of Science Workshop at Bocconi,and MaCCI Annual Meeting 2020 for helpful comments.Johannes Schneider gratefully acknowledges financial support from Agencia Estatal de Investigación,grant PID2019-111095RB-I00, Ministerio Economía y Competitividad, grant ECO2017-87769-P,and Comunidad de Madrid, grant MAD-ECON-POL-CM (H2019/HUM-5891). † Carlos III de Madrid; E-mail: [email protected] ‡ Bocconi University, Department of Economics and IGIER; E-mail: [email protected] a r X i v : . [ ec on . T H ] M a r Evolution] comes through asking the right questions, because the answerpre-exists. But it’s the questions that we have to define and discover. . . . Youdon’t invent the answer. You reveal the answer. Jonas Salk.
In his letter to Franklin D. Roosevelt (
Science, The Endless Frontier ), VannevarBush (1945) pleads with the president to preserve the freedom of inquiry by federallyfunding basic research—the “pacemaker of technological progress.” That letterpaved the way for the creation of the National Science Foundation (NSF) in 1950.The NSF today, like the vast majority of governments and scientific institutions, cherishes scientific freedom and allows academic researchers to select research projectsindependently. For example, the research ministers of the European Union emphasizethat scientific freedom and access to knowledge are essential to progress in the BonnDeclaration (2020). With the power of scientific freedom comes the responsibility for “asking theright questions” that Jonas Salk advocates in the epigraph. However, what is theright question? In his guide for young researchers Varian (2016) suggests to become“a Wizard of Ahs”—to pursue questions that have “lots of implications.” Thus werefine: Which question should a researcher pursue to produce the most implications?A question to advance our knowledge to new territories outside the current frontier?A question to explore the details around existing knowledge? Or a question to bridgethe gap between established findings?In this paper, we show that each strategy can outrank the others. The rankingdepends on existing knowledge—that is, the set of questions to which the answers areknown. In particular, the ranking depends on the density of existing knowledge—thatis, the distance between questions in that set.Consider the following setting. A decision maker faces a problem. In herresponse to the problem, the decision maker builds on the information that existingknowledge provides. This information stems from two sources. First, existingknowledge provides the answers to particular questions. Second, existing knowledgeprovides conjectures on questions to which the answers are still undiscovered. Thereason for the second source follows from answers being correlated: the precision ofthe conjecture depends on the question’s location relative to existing knowledge. We conceptualize the correlation by assuming that answers to questions follow therealization of a Brownian path. Figure 1 visualizes that idea. Questions are onthe horizontal axis and the gray line depicts the answers to all questions. Dots ( )represent existing knowledge. Due to the Brownian assumption, the answer to each See, for example, OHCR (1966), Art 15. “Scientific research benefits the people and society through the advancement of knowledge.Freedom of scientific research is a necessary condition for researchers to produce, share and transferknowledge as a public good for the well-being of society.” An example of such spillovers is the COVID-19 vaccine development by Moderna which “tookall of one weekend” only. The speed was a direct consequence of researchers discovering how toreplicate the spikes of another Corona virus in research on MERS. Details are discussed for instancein This American Life (2020). r Questions A n s w e r s x l x r Questions A n s w e r Figure 1:
Existing Knowledge and Conjectures. question is normally distributed. The mean and the variance depend on existingknowledge. The solid black lines in Figure 1 represent the mean, the dashed linesprovide the band of the 95-percent prediction interval. A new discovery createsadditional information. Moving from the left panel of Figure 1 to the right panelillustrates how the discovery of an additional point improves conjectures. The benefitof the discovery measures the improvement of the decision maker’s response due tothese improved conjectures.We find the following relation between the density of the existing knowledgeand the benefit of discovery. If the existing knowledge is sparse—that is, any twoquestions in existing knowledge are far apart—it is best to narrow the gap betweentwo adjacent questions asymmetrically. A researcher should pursue a discovery ona question closer to one of the two answered questions. If the existing knowledgeis dense—that is, any two adjacent questions in existing knowledge are close—it isbest to advance knowledge beyond but not too far from the frontier. A researchershould pursue a discovery on a question that does not lie between any answeredquestions. If the existing knowledge is neither sparse nor dense—that is, there aretwo adjacent questions of intermediate distance—it is best to bridge the gap betweenthese questions. A researcher should pursue a discovery on the question equidistantto the two answered questions.The intuition follows from how knowledge improves decision making throughmore precise conjectures. Consider first the left panel of Figure 1. Only the answerto question x r is known. Now take an unanswered question x . The closer x is to x r the more precise is the conjecture. Now consider the right panel. To the left of x l and to the right of x r the same argument applies. If x lies between x r and x l , theknowledge of both x r and x l shape the conjecture about x .Suppose that initially x r is known, that is, we are in the left panel. Now, the The 95-percent prediction intervals depend on existing knowledge and describe the followingrelation: For each question, the answer lies with a probability of 95 percent between the respectivedashed lines given existing knowledge. For now, we focus only on the benefit of discovery. We introduce the cost below. x l , that is, we move to the right panel. This discovery improvesthe precision of the conjecture about any question x between x r and x l . The further x l is from x r the more questions have improved conjectures. However, the further x l is from x r the less precise is the conjecture about any question close to x r . Thatrelation defines a tradeoff: A discovery far apart from existing knowledge improvesdecision making. Many conjectures become more precise, albeit only mildly so.A discovery close to existing knowledge improves decision making as well. Someconjectures become very precise, albeit not many conjectures improve at all. Whenpursuing a question to advance knowledge beyond the frontier, a researcher resolvesthis tradeoff by selecting a question of intermediate distance.Now, suppose that the answer to both x l and x r was already known. Assumethe researcher chooses to pursue a third question, x m , that lies between x r and x l .Here, the researcher balances the tradeoff from above on two sides simultaneously.If she discovers the answer to x m , she replaces the initial area of questions spannedby x r and x l by two new areas: one area spanned by x l and x m , another spannedby x m and x r . In sum, the lengths of the two new areas ( x l to x m and x m to x r )equal the length of the initial area ( x l to x r ). From the discussion of advancingknowledge beyond the frontier, we know that ideally the researcher creates twoareas of intermediate length. However, the length of the initial area constrainsthe researcher. The following optimal strategies emerge: If the initial area is shortenough, the researcher should bridge the gap and pursue a question x m equidistantto both x r and x l . Otherwise, the researcher should narrow the gap: Pursue aquestion closer to one of the area’s bounds and thereby focus on creating one areaof intermediate length.Overall, the largest benefit comes from bridging the gap between distant, yet nottoo far apart pieces of knowledge. Advancing knowledge beyond the frontier trumpspursuing a question inside an existing area only if all available areas are short.In reality, research comes at a cost. Naturally, the cost influence the researcher’schoices. It takes time and effort to search for an answer. The more effort a researcherinvests the higher the likelihood of a discovery. We find that this likelihood varieswith existing knowledge because existing knowledge determines conjectures. Themore precise the conjecture about a question the more likely it is to discover theanswer with any given level of effort. This relationship influences the researcher’s risk:Given effort, increasing the distance to existing knowledge reduces the likelihood ofdiscovery. Mitigating that additional risk requires costly effort. Thus, introducingcost adds another dimension to the problem: the researcher balances risk and reward.The qualitative statements on the optimal choice of question remain, evenincluding the effort choice. In addition, we can address research productivity: howlikely is discovery? We find the following relation between productivity and existingknowledge. Consider a researcher who chooses to bridge a gap in existing knowledge.If the area is short, the research productivity is low. The benefit of a discovery islow and hence the researcher refrains from exerting much effort. At first, researchproductivity increases in the area length. Both the benefit of a discovery and the costincrease, yet the former dominates the latter. However, if the area is large, researchproductivity decreases in the area length. The increase in the cost dominates.If we combine the analysis of benefit and cost we obtain a clear picture: If4vailable, the most productive and valuable research bridges the gap between distant,yet not too far apart pieces of knowledge. If such gaps do not exist, the researchershould aim at narrowing a large gap asymmetrically by pursuing a question closerto one of the bounds. Finally, if existing knowledge is dense, the researcher shouldtake the risk to advance the knowledge frontier.We provide a microfounded framework to study decisions in research and theirconsequences. We make the following basic assumptions: (i) the set of potentialresearch questions is large and given by the real line. (ii) The answer to one questionis informative about the answers to other questions. We assume that answers arepredetermined by the realization of a Brownian motion. (iii) A decision maker hasaccess to the public good knowledge . Applying this knowledge, the decision makerchooses whether and how to pro-actively address a problem or to “do nothing,” e.g.,continue with business as usual. (iv) The value of knowledge corresponds to thequality of decision making. Finally, (v) we conceptualize research as choosing aquestion and an interval of possible answers that the researcher samples. If theanswer lies in the sampled interval, discovery is successful. The larger the sampleinterval, the larger the cost to the researcher.We derive both an endogenous benefit-of-discovery function and an endogenouscost-of-research function based on those assumptions. Our microfoundation impliesa set of non-standard properties of both functions. Despite these peculiarities, theresearcher’s optimization problem remains tractable. Due to these peculiarities,however, we obtain rich and intuitive results regarding the researcher’s choices.Therefore, we believe that our model is widely applicable.To illustrate our model’s applicability beyond the researcher’s problem, weembed it into a setting of science funding. A funder who respects the researcher’sscientific freedom can spend a fixed budget both on reducing the researcher’s costand on rewards for successful research. We construct the funder’s feasible set ofimplementable choices. We show that—from the funder’s perspective—novelty andoutput can be both complements or substitutes depending on the effective priceratio between cost reductions and rewards. In the analysis, we combine standardtools from consumer theory with our previous results.Overall, we make three contributions. First, we propose a modeling framework tostudy the societal value of research and a researcher’s choice of question and effort.Crucially, we link all of these to existing knowledge. Second, we characterize theresearcher’s decision for existing knowledge. We show that—if existing knowledgepermits—the most productive and valuable research bridges the gap between distant,yet not too far apart pieces of knowledge. Third, we derive the combinations ofresearch output and novelty of discovery that a budget-constrained funder caninduce. While we derive these functions with an academic researcher in mind, we believe they arealso interesting in other settings. For instance, they may be relevant when firms buy innovativestart-ups simply to obtain knowledge and to improve conjectures relevant for their own questions.Our model may be helpful to differentiate among the different effects in the ongoing debate aboutkiller acquisitions (see Letina, Schmutzler, and Seibel, 2020; Cunningham, Ederer, and Ma, 2021). The benefit of a discovery is neither single-peaked in distance to existing knowledge norglobally concave in area length; the cost are concave in distance at any point. .1 Related Literature Ample empirical literature in the science of science has documented the importanceof novelty and output for progress in science. Fortunato et al. (2018) provide anextensive summary of it. The importance of (accessible) pre-existing knowledge forresearch purposes is documented, for example, in Iaria, Schwarz, and Waldinger(2018). That literature studies the connection between benefits of discoveries,cost of search, and existing knowledge. Yet its findings are mainly based on(quasi)experiments; a formal model connecting these aspects is missing. We aimat closing this gap. The formal model we provide is simple yet flexible enoughto address several issues. The model is based on few (in the baseline model, two)parameters, which makes it identifiable and testable.Several existing theoretical models in the science of science consider particularaspects of the scientific process we have in mind. Aghion, Dewatripont, andStein (2008), for example, consider a setting in which progress has a predefinedstep-by-step sequential structure. To advance to the next question, a particularprior question has to be answered. Our model offers greater flexibility in that itposits that any question can—in principle—be addressed at any time. However,the benefits from a discovery and the effort needed for the discovery depend onprevious work. Bramoullé and Saint-Paul (2010) model the decision of a researcherto deepen knowledge in a given area or to advance the knowledge frontier. The maindriver in their model is the assumption that as an area gets increasingly crowded,the reputation a researcher gains from new developments in that area declines. We offer a decision-based microfoundation that provides a measure of uncertainty,in line with Frankel and Kamenica (2019). It reaches a similar conclusion: as theopportunities in the area become increasingly narrow, the informational content ofan additional finding decreases, and hence its value does too. However, unlike inBramoullé and Saint-Paul (2010), the researcher in our model has more discretion,as she chooses—in addition to the area—the degree of novelty and the level ofresearch intensity which directly determines the probability of success. Both choicesare continuous, and shrinking the research area may even be beneficial if it leads tobetter conjectures by closing the gap between existing pieces of knowledge. Whileour model is static, a simple dynamic extension could reproduce the core models of We want to stress that our notion of knowledge is orthogonal to that in the literature onepistemic game theory. Brandenburger (1992) provides an overview. Different to that literatureknowledge is always fully transparent and there is no strategic interaction. However, it is possibleto embed it in strategic settings to address alternative questions. There is a literature orthogonal to ours that views science as establishing links in a networkbetween known answers (e.g., Rzhetsky et al., 2015). Our model is complementary, we considerresearch as the search for answers where the links in the network are known. To (ab)use Newton’s metaphor: Any researcher can build a ladder to see farther, but theeffort required depends on the existing giants’ shoulders. Related ideas appear in Scotchmer (1991),Aghion et al. (2001), and Bessen and Maskin (2009). Similar to Bramoullé and Saint-Paul (2010) innovation fully translates to a public good inour setting. That differentiates us from most models of R&D competition. Yet, similarly to, forexample, Letina (2016), Letina, Schmutzler, and Seibel (2020), and Hopenhayn and Squintani(2021), we assume that progress corresponds to successful search in an ocean of possibilities. Unlikein those approaches, benefit and cost depend on the question’s relation to existing knowledge inour setting. The closest paper to ours in the literature on innovation is Prendergast (2019),which is complementary to ours. He, too, studies a model of innovation in which thecorrelation between questions is determined by a Brownian motion. He focuses on anagency problem in a single exogenously given research area. While we abstract fromagency concerns, the results in our mirofounded model come from the researcher’schoice between several distinct research areas and expanding knowledge beyond thefrontier. In addition, we provide an endogenous relation between the effort investedand the probability of a discovery. While neither of the two models nests the other,there is a special case of our model that corresponds to a special case of his. Wediscuss the relationship in greater detail in Section 7.Technically, we build on the literature that uses Brownian models to analyzeR&D choices, following Callander (2011). Most of that literature assumes thatthe payoffs are determined by a specific target of the stochastic process’s realization(for example, Callander, 2011) or the weighted sum of all realizations (for example,Bardhi, 2019). We differ from it in that we posit that the value of a discoveryis determined by the reduction in the variance of conjectures. Importantly, theobjective of our model is not to compare the expected realizations to a safe outcome.Instead, we care about the precision of conjectures about all realizations. Callanderand Clark (2017) is the closest work to ours within this literature. We discuss itsrelation to our model in detail in Section 7.
We build a parsimonious model of knowledge that captures the following real-worldaspects:(i.) Knowledge informs decision making.(ii.) Knowing the answer to certain questions spills over onto conjectures aboutother questions.(iii.) The set of questions available is infinite.(iv.) The impact of answering one question on another one depends on how closethe two questions are.
Questions and answers.
We represent the universe of questions by the real line, R . A specific question is an element x ∈ R . Each question x has precisely one answer, Other recent work in that area includes Akerlof and Michaillat (2018), Andrews and Kasy(2019), and Frankel and Kasy (2020). Andrews and Kasy (2019) and Frankel and Kasy (2020)study the (potentially) distorting affects of the publication process—a friction we abstract from.Akerlof and Michaillat (2018) study how false paradigms survive due to homophily. Our focusis different: we differentiate not quality of research but selection of topics. It is straightforwardto extend our analysis to research-specific preferences by introducing additional parameters; forexample, homophily in the research field. See also Callander and Hummel (2014), Garfagnini and Strulovici (2016), Callander and Clark(2017), Callander, Lambert, and Matouschek (2018), Bardhi (2019), and Callander and Matouschek(2019) and references therein. ( x ) ∈ R . A question-answer pair ( x, y ( x )) is thus a point in the two-dimensionalEuclidean space. The answer y ( x ) to question x is determined by the realization of a randomvariable, Y ( x ) : R → R . We provide more structure for Y ( x ) below. Truth and knowledge.
Truth is the collection of all question-answer pairs. It isthe realization of all random variables Y ( x ) over the entire domain, R . Knowledge isthe collection of known question-answer pairs. We denote it by F k = { ( x i , y ( x i )) } ki =1 .For notational convenience, we assume that F k is ordered such that x i < x i +1 .The key assumption of our model concerns the truth-generating process Y ( x ).We assume that Y ( x ) follows a standard Brownian motion defined over the entirereal line. This assumption captures the notion that the answer to question x issimilar to the answer to a close-by question x . As the distance to the originalquestion increases, there remains a correlation. Yet the noise increases the largerthe distance to the closest piece of knowledge. Knowledge F k determines a partition of the real line consisting of k + 1 elements X k := { ( −∞ , x ) , [ x , x ) , · · · , [ x k − , x k ) , [ x k , ∞ ) } . We refer to each element of the partition as an area and define it by the index ofits lower bound. Thus, question x ∈ [ x i , x i +1 ) is in area i of length X i := x i +1 − x i .For x < x we say x is in area 0 of length X = ∞ , and for x > x k it is in area k oflength X k = ∞ . Conjectures.
Based on the commonly known truth-generating process, Y ( x ),and the existing knowledge, F k , it is straightforward to form a conjecture —thatis, to compute the distribution of the answer —for each question x . We denotethe conjecture about question x by the conditional distribution function G x ( y |F k ),which is defined over the answer domain, R . Conjectures about questions to whichthe answer is known are trivial. The conjecture G x i ( y |F k ) = y ≥ y ( x i ) is a right-continuous step function jumping to 1 at y = y ( x i ) if ( x i , y ( x i )) ∈ F k . The conjecturefor a yet-to-be-discovered y ( x ), G x ( y |F k ), is a well-defined cumulative distributionfunction. Because Y ( x ) is determined by a Brownian motion, any G x ( y |· ) follows anormal distribution with mean µ F k ( x ) and variance σ F k ( x ). Figure 2 depicts the Our assumption implies that the relation between two questions can be obtained in a singledimension. While projecting all available questions across all disciplines onto a line might implya sizeable loss, that loss is smaller if we think of our universe of questions as being within onespecific discipline . As in Callander (2011), the realized truth, Y , is a random draw from the space of all possiblepaths, Y , generated by a standard Brownian motion going through some initial knowledge point,( x , y ( x )). While the process has fully realized at the beginning of time, knowledge is the filtrationknown to the observer, F k . We choose a standard Brownian path with zero drift and varianceof one for convenience only. Our model extends naturally to other (Gaussian) processes. Wewant to emphasize that the x dimension should not be confused with a sequential structure offinding answers. To the contrary, the entire process Y ( X ) has realized at the beginning and anyquestion-answer pair ( x, y ( x )) / ∈ F k is both discoverable and yet to be discovered. We use the Euclidean distance on the x dimension, | x − x | , throughout when we refer todistance. . y G x ( y | F ) d = 0 d = 1 d = 4 d = 16
38 40 42 44 4600 . . y g x ( y | F ) Figure 2:
Distributions of answers for different distances, d , to knowledge when F = (0 , x = 0, d = 1 depicts the distribution of answers toquestions x = − x = 1, d = 4 the distribution of answers to questions x = − x = 4, and so on. All answers have the same mean (42), but the variance and thus theprecision of the conjecture differs with the various distances, d , to the only known point, F . For d = 0, the answer is known and G ( y |F ) is a step function. Questions withlonger distance have larger variances. The left panel depicts the respective distributionfunctions; the right panel depicts the densities. distributions for different distances to the existing piece of knowledge, assuming itis F = (0 , µ F k and σ F k follow immediately from the properties of the Brownian motion.We have to differentiate between settings in which we are on a Brownian bridge( x ∈ [ x , x k ]) and those in which we are outside the current frontier ( x / ∈ [ x , x k ]). Property 1 (Expected Value) . Given knowledge F k , the answer to question x hasthe following expectations: µ F k ( x ) = y ( x ) if x < x y ( x i ) + x − x i X i ( y ( x i +1 ) − y ( x i )) if x ∈ [ x i , x i +1 ) , i ∈ { , ..., k − } y ( x k ) if x ≥ x k Property 2 (Variance) . Given knowledge F k , the answer to question x has thefollowing variance: σ F k ( x ) = x − x if x < x x i +1 − x )( x − x i ) X i if x ∈ [ x i , x i +1 ) , i ∈ { , ..., k − } x − x k if x ≥ x k Decision making.
For each question x ∈ R the decision maker takes an action a ∈ R ∪ ∅ . The symbol ∅ represents the act of “doing nothing”, or continuing withthe status quo action. Any number a ∈ R represents a proactive decision regardingquestion x .We assume that the expected payoff of selecting a = ∅ is finite, safe (that is,independent of the true answer y ), and question-invariant. We denote it by − q . The9ayoff of addressing x with some action a = ∅ is represented by a quadratic lossaround the answer, y ( x ), to question x . u ( a ; x ) = − ( a − y ( x )) if a = ∅ − q if a = ∅ The decision maker has access to F k only.The purpose of the safe option a = ∅ is the following: If a question is far fromexisting knowledge, the expected payoff for any pro-active address a ∈ R becomesarbitrarily negative because of a lack of knowledge. The safe option guarantees thatthe expected payoff is well defined nonetheless. Graphical example.
Before moving to the analysis, we present a short graphicalexample that highlights our main model features and fosters intuition for theremainder of the paper. Suppose the following snapshot of the realization of theBrownian path constitutes the truth on [ − , -2 -1 0 1 24042444648 x y ( x ) Figure 3:
The color of the truth is gray
The next two graphs depict the situations in which the answer to a single questionis known, F = { (0 , } , and in which two answers, F = { ( − . , . , (0 , } ,are known.In the situation represented in the left panel of Figure 4, under F , only theanswer to question 0, which is 42, is known. We represent that knowledge by a dot( ). Given the martingale property of a Brownian motion, the current conjecture isthat the answer to all other questions is normally distributed with mean 42. Werepresent the mean of the conjecture by the solid lines. However, the farther aquestion is from 0, the less precise is the conjecture (see Figure 2). We depict thelevel of precision by the dashed 95-percent prediction interval. For each question x ,the truth lies, with a probability of 95 percent, between the two dashed lines giventhe knowledge F .In the right panel of Figure 4, in addition to F , the answer to question x = − . .
6, is known. The additional knowledge changes the conjectures for Making q stochastic and dependent on x is straightforward and does not alter our arguments.The only substantial assumption we make is that an action with a finite expected payoff exists. x y ( x ) -2 -1 0 1 24042444648 x y ( x ) Figure 4:
Conjectures and their precision under F (left) and F (right) The red dots represent known question-answer pairs. The solid lines represent the expectedanswer to each question x given the existing knowledge. The dashed line represents the95-percent prediction interval—that is, the interval in which the answer to question x lies,with a probability of 95 percent, given F k . questions in the negative domain compared to the left panel. The conjecture aboutquestions between − . − . . − .
2. Moreover,uncertainty decreases for all questions in the negative domain, and the predictionbands become narrower. The positive domain is unchanged because of the martingaleproperty of Brownian motion.Now, consider moving to knowledge F = { ( − . , . , (0 , , (1 . , . } (leftpanel of Figure 5) and then to F = { ( − . , . , (0 , , (0 . , . , (1 . , . } (right panel of Figure 5). -2 -1 0 1 24042444648 x y ( x ) -2 -1 0 1 24042444648 x y ( x ) Figure 5: Conjectures and their precision under F (left) and F (right) Moving from F to F , the change is similar to that from F to F , but thistime in the positive domain. All conjectures in the positive domain become moreprecise, but the negative domain is unaffected. Further, a Brownian bridge betweenthe known points (0 ,
42) and (1 . , .
8) arises.Moving from F to F , knowledge of an answer to a question that lies between twoalready-answered questions is added. That implies that conjectures about answers11o questions between 0 and 1 . . < . . F . Moreover, the expected answers are decreasingin x from 0 to 0 . . . Discovery occurs if a new question-answer pair is found and added to the existingknowledge ( F k ). In this section, we derive a formulation that measures the benefitsof such additions for the decision maker. Knowledge is important for decision making. The decision maker responds to allquestions simultaneously. For each individual question x , the decision maker usesher conjecture about the answer to x to evaluate her options. Suppose the decisionmaker takes a particular action a = ∅ . Her expected utility, given F k , is determinedas follows: U ( a = ∅ ; x |F k ) = − Z ( a − y ) dG x ( y |F k )Whenever the decision maker addresses the question, she selects a ∗ := arg max a ∈ R U ( a = ∅ ; x |F k ) = µ F k ( y )and her expected payoff is U ( µ F k ( x ); x |F k ) = − Z ( µ F k ( x ) − y ) dG x ( y |F k ) = − σ F k ( x ) . Addressing the question is only optimal if σ F k ( x ) ≤ q . Otherwise, the decisionmaker exercises the safe option a = ∅ with payoff − q . We denote the optimalaction to question by a ∗ ( x ) := arg max a ∈{ R ∪ ∅ } U ( a ; x |F k ). The value of knowing F k . If y ( x ) is known, the decision maker addresses x optimally. She selects a = y ( x ) and obtains u ( y ( x ); x ) = 0. All other questionsare addressed through the imperfect a ∗ ( x ) ∈ { µ F ( x ) , ∅ } . The resulting expectedutility to the decision maker is U ( a ∗ ; x |F k ) = max {− q, − σ F k ( x ) } . The total value Alternatively, she could face a random draw of a subset, [ x, x ], of the set of all questions R . Ifthat interval is large enough all results remain unchanged. Regarding the status quo, we have in mind long-standing policies to which the expected payoffis finite. The question the decision maker asks is whether she should revise her policies giventhe existing knowledge. Take the discussion about how to respond to climate change: Since theKyoto Protocol, decision makers have reevaluated policies based on the evolution of knowledgeby constantly trying to decide whether to continue with business as usual or to change policy indifferent areas (for example, transportation, energy, protection of nature). x y ( x ) − − . . qx σ ( x ) qσ v ( F ) Figure 6:
The value of knowing F The left panel is identical to the left panel in Figure 4. The right panel shows the varianceof the conjectures about questions. The expected payoff from taking an action equal to themean of the question, a = µ ( x ), is the area above the variance but below q . We assume thevalue of a = ∅ is − q = −
1. The variance σ ( x ) = d ( x ). For d ≤ a = µ ( x ) is preferredto a = ∅ . The net payoff from a = µ ( x ) relative to a = ∅ is q − d ( x ). The total value of F is the shaded area. of knowing F k , v ( F k ), captures how much the decision maker gains compared tonot knowing anything and thereby responding with the safe option a = ∅ regardlessof the problem. We use the (normalized) utility q − U ( a ∗ ( x ); x |F k ) q to describe how much the decision maker benefits from the knowledge F k in herresponse to x . To keep the analysis focused, we abstract from any exogenousprioritization. The decision maker cares about the improvement in her actions inresponse to any question in the same way, that is, there is no exogenous ranking ofquestions. Thus, the value of knowing F k is v ( F k ) := Z max ( q − σ F k ( x ) q , ) dx. The right panel of Figure 6, using our previous example, provides a graphicalrepresentation of the value of knowing F . The upper-right panel of Figure 7represents the value of knowing F , the lower-right panel that of knowing F . The benefits of discovery.
The benefits of a discovery describe how much thevalue of knowledge improves due to the discovery. Formally, adding an additionalpoint ( x, y ( x )) provides the benefit: V := v ( F k ∪ ( x, y ( x ))) − v ( F k ) It is straightforward to (numerically) incorporate a weighting function for questions; however,it comes at the cost of clarity. x y ( x ) -2 -1 0 1 2011.52 qx σ ( x ) qσ vV (1 . q ; ∞ ) -2 -1 0 1 24042444648 x y ( x ) -2 -1 0 1 2011.52 qx σ ( x ) qσ vV (0 . q ; 1 . q ) Figure 7:
The benefits of discovery
Upper panels: Benefit of a knowledge-expanding discovery.
The left panel isidentical to the right panel in Figure 4. Outside the frontier, x / ∈ [ − . , σ ( x ) = d ( x ). Inside, it is σ ( x ) = d ( x )( X − d ( x )) /X , where X = 1 . − . , d ( x ) because of inference from both knowledge points. As in Figure 6, the benefit of F is the area (shaded in gray) below q but above σ ( x ). The net benefit of discovering the answer to question x = 1 . Lower panels: Benefit of knowledge-deepening discovery.
The left panel isidentical to the right panel in Figure 5. The right panel shows the value of knowledgeand the benefit of discovery when research deepens knowledge. The dark-gray areais the net benefit of discovery of point ( x, y ( x )) = (0 . , .
8) over existing knowledge F = { ( − . , . , (0 , , (1 . , . } . As in the previous figures, the total value of F isthe area above the variance and below q . V depends on the choice of x . We distinguish two scenarios: expanding knowledge and deepening knowledge. We say a discovery expands knowledge if it ison a question x / ∈ [ x , x k ] outside the current frontier. We say a discovery deepensknowledge if it is on a question x ∈ [ x , x k ] inside the current frontier.We first derive the benefit-of-discovery function. We then state two corollariesthat characterize properties of the benefit of discovery. The two main ingredientsto determine this benefit are the distance to knowledge , which we formally definebelow, and the research area . It turns out that the length X is a sufficient statisticfor the research area. Recall that X = ∞ for areas outside the current frontier. Definition 1.
The distance of a question x to knowledge F k is the smallest Euclideandistance to a question to which the answer is known: d ( x ) := min ξ ∈{ x ,x ,...x k } | x − ξ | We now state the benefits of discovery.
Proposition 1.
Take a discovery ( x, y ( x )) with distance d = d ( x ) and in a researcharea of length X . The benefit of discovery, V ( d, X ) is V ( d ; ∞ ) = − d q + d + d> q ( d − q ) / √ d q , if knowledge is expanded, and V ( d ; X ) = 16 q (cid:18) dX − d + d> q √ d ( d − q ) / − X> q √ X ( X − q ) / + X − d> q √ X − d ( X − d − q ) / (cid:19) if knowledge is deepened. The terms in V ( d ; · ) without an indicator function measure the direct changein variance and hence the effect on decision making conditional on a proactiveaction a = ∅ . The terms with an indicator function, , become active whenever thecorresponding area contains questions with conjectures sufficiently imprecise (seee.g., Figure 8, right panel). The imprecise conjectures induce the decision maker torefrain from a proactive action. Instead the decision maker chooses the safe option ∅ which limits losses to − q . The terms with an indicator function that enter positivelycorrespond to newly created areas. The term with an indicator function that entersnegatively corresponds to an old replaced area.Figure 7 illustrates the benefits of discovery for expanding knowledge (upperpanels) and deepening knowledge (lower panels). The right panel of Figure 9 onpage 18 illustrates the functions for different area lengths X . To gain intuition, it isuseful to discuss expanding knowledge and deepening knowledge separately. Expanding knowledge is the process of discovering an answer outside the currentdomain of knowledge ([ x , x k ]). We focus on discovering the answer to x < x , which15 qx σ ( x ) qσ vV (3 q ; ∞ ) -5 -4 -3 -2 -1 0011.52 qx σ ( x ) qσ vV (5 q ; ∞ ) Figure 8:
Benefit-maximizing (left) and too large (right) distance of x given F Given that the value of doing nothing is − q = −
1, the benefit-maximizing distance whenexpanding knowledge is d = 3. The left panel depicts the benefit-maximizing choice, given F , when expanding to the negative domain using d = 3 q and thus x = −
3; the right panelshows the effect of a choice that is too far away ( x = − , d = 5 q ). The gain in knowledge V ( d ; ∞ ) is the dark-shaded area. It is larger in the left panel than in the right panel. is the case when moving from Figure 6 to the upper row of Figure 7. Case x > x k isanalogous. The benefit of expanding knowledge comes from the new research area itcreates. A discovery of y ( x ) with x < x pushes the boundary to the left and createsa new research area [ x, x ). The benefit of that discovery is precisely the value ofthat area (the dark-shaded area in the upper row of Figure 7). The value of adding an area depends on (i) the amount of questions in thatarea, and (ii) the degree of improvement in decision making compared to the default a = ∅ . The latter depends on the precision of the conjectures. The precision ofthe conjecture to each question in [ x, x ) is determined by its distance to bothbounds of the area. Increasing either of these distances decreases the precision ofthe conjecture. Consider two areas [ x, x ) and [ x , x ) with x < x and a question z ∈ [ x, x ) with d ( z ) = x − z . The conjectures about z depend on whether x or x is discovered. Since x is further apart from z than x , the conjecture becomesis precise, σ F k ∪ [ x ,x ) ( x ) > σ F k ∪ [ x,x ) ( z ). A less precise conjecture implies a lessimproved decision. Thus, increasing the area length implies a decrease in conjecturesfor questions of a given distance d .Increasing the area length of the newly created area has two opposing effects onthe value of discovery. It increases the amount of questions to which the conjectureimproves—an increase in the benefits of discovery. It decreases the precision of allquestions in the area—a decrease in the benefits of discovery. The upper right panelof Figure 7 illustrates the benefits of discovery of creating a (too) short area, theleft panel of Figure 8 illustrates the largest attainable benefits of discovery, and To be precise, the conjectures about questions to the left of the old frontier are replaced byconjectures inside the new research area and conjectures to the left of the new frontier become alsomore precise. However, as can be seen in the upper right panel of Figure 7, the variance reductionto the left of the frontier is always of equal value. Hence the benefits are as if there was only a newarea added. d ( x ) = 3 q ) where allconjectures have a variance strictly smaller than q . Corollary 1.
The benefit of expanding knowledge is single peaked in d . The benefit-maximizing distance in the expanding area is d ( ∞ ) = 3 q . The benefit of optimallyexpanding knowledge is V (3 q ; ∞ ) = q . Deepening knowledge is the process of discovering an answer y ( x ) to a question x inan area i with two bounds, x i and x i +1 . The answers y ( x i ) and y ( x i +1 )) are known.An illustration is the lower panel of Figure 7. The difference to expanding knowledgeis that instead of creating a new area, deepening knowledge replaces an old area([ x i , x i +1 )) by two new areas [ x i , x ) and [ x, x i +1 ).The benefit of a discovery depends on the combination of improved decisionmaking in either of the areas. We know from Corollary 1 that the largest benefitscome from an area of length 3 q . Thus, if the old area i had length X i = 6 q discoveringthe midpoint provides the largest benefits. However, if X i = 6 q at least on of thetwo areas has to have a length different from 3 q and thus provide smaller benefits.If X i = 6 q two forces are at play. First, there is a benefit of replacing the oldarea with two symmetric new areas each half the length of the old. Increasing thelength of an area comes at the cost of decreasing precision in conjectures. However,the larger the area the greater that loss in precision. The reason is that spilloversfrom the boundaries decline. As a result, it is beneficial to (marginally) decreasethe length of the larger area at the expense of an increase the length of the smallerarea. Inspection of the lower right panel of Figure 7 provides a graphical intuition.Second, benefits decline if an area length is far from 3 q . Maintaining symmetrywould imply that the length of both newly created areas is far from 3 q if the lengthof the old X i is far from 6 q .If the old X i was small, that tradeoff is solved optimally in favor of symmetry.It is better to balance spillovers even if each area falls short of having length 3 q . Ifhowever, the old X i was large, the tradeoff is solved in favor of having one high-valuearea. It is better to suffer from low spillovers onto questions in the middle of thelarger area than to create two areas with imprecise conjectures each. As a resultthe smaller area is close to yet above length 3 q , the other area is much larger. Asthe length of area grows large, X i → ∞ , the impact of the larger area becomes lessand less important and the length of the smaller area converges to 3 q . A cutoff˜ X ∈ [6 q, q ] exists such that it is most beneficial to create two symmetric areas ifand only if X i < ˜ X Finally, one might ask which initial area length X i provides the largest benefitof being transformed into 2 new areas. As explained above, two areas of length 3 q provide the largest value. However, we have to take into account that the two newareas also replace an old area. The larger the old area the lower the value it initiallyprovided. Thus, there is a benefit to replace large areas. Yet, the larger the old area The results of this and the next corollary follow directly from the analysis of V ( · ; · ) derivedin Proposition 1. However, their derivations are not entirely straightforward—hence, we providethem in detail in the appendix. ˆ X ˇ X ˜ X V ∞ Area Length X B e n e fi t s V q q q q qV ( d ; 3 q ) V ( d ; 6 q ) V ( d ; 10 q ) V ( d ; ∞ )Distance d B e n e fi t s V Figure 9:
The benefit of discovery
Left panel: Benefit of discovery as a function of the area length.
Thegraph plots the benefit of discovery in each interval when choosing the optimal distance d ( X ) therein (dashed line) and the benefit of choosing the optimal distance d ( ∞ ) on anexpanding interval. Deepening is preferred to expanding if X > ˆ X ≈ . q . The maximumbenefit is at ˇ X ≈ . q ; d ( X ) < X/ X > ˜ X . Right panel: Benefit of discovery given X as a function of distance d . Expanding knowledge (solid line) and deepening knowledge (nonsolid lines) for arealengths X ∈ { q, q, q } . The benefits are nonmonotone in X . Note:
Plots for deepening knowledge end at the maximum distance in each area, d = X/ (beyond 6 q ) the lower the benefit the two new areas provide. As a result the arealength whose replacement provides the largest benefits, ˇ X ≈ . q , is above 6 q . Expanding vs deepening knowledge.
A natural question is to ask when are the benefitsof expanding knowledge larger than those of deepening knowledge. On the one hand,creating new areas has the benefit that no knowledge is replaced as all old areasremain. On the other hand, deepening knowledge has the benefit of creating twonew areas with relatively precise conjectures. If an area is small, the benefits ofreplacing that area by two areas with more precise conjectures is small. Conjecturesand hence decisions do not improve much. If an area is large before knowledge isdeepened, it does not contribute much to the value of knowledge. In particular,if the area contains questions to which conjectures are so imprecise that decisionremain at a = ∅ deepening may be beneficial. We find that indeed deepening isonly beneficial if the initial area contains such questions with imprecise conjecturesand are hence larger than 4 q . We determine a cutoff ˆ X ≈ . q such that deepeningis more beneficial than expanding if X i > ˆ X .Below, we summarize all findings in a corollary to Proposition 1. Figure 9provides the graphical illustration. 18 orollary 2. There are three cutoff area lengths, q < ˆ X < q < ˇ X < ˜ X < q ,such that the following holds: • If F k is such that X i < ˆ X for all intervals i , then expanding knowledgemaximizes the benefit of discovery. Otherwise, deepening knowledge maximizesthe benefit of discovery. • The benefit of discovery is largest for area length ˆ X . It is increasing in arealength for X < ˆ X and decreasing in area length for X > ˆ X . • The benefit-maximizing distance to knowledge (given area X ), d ( X ) , is in-creasing for X < ˜ X and decreasing for X > ˜ X . If X < ˜ X , the midpointmaximizes the benefit, d ( X ) = X/ . Otherwise d ( X ) < X/ . • As X → ∞ , d ( X ) → d ( ∞ ) and V ( d ; X ) → V ( d ; ∞ ) uniformly. So far, we have only considered the benefits of a discovery. However, not all researchleads to discovery and there are costs of research. Searching for an answer requirestime and effort. We assume—in line with our assumption that research is a discoveryprocess—that having selected a question, the researcher samples an interval of herchoice on the real line for the answer. If the answer is within that interval, theresearcher has found the answer; otherwise, the researcher has not found the answer.We make the standard assumption that the marginal cost of search are increasingin the interval length covered. Formally, we assume that if a researcher samples aninterval [ a, b ] on the real line to see whether the answer y ( x ) to question x lies inthat interval, she incurs a cost proportional to ( b − a ) . We now characterize the (endogenous) cost of research incurred by a researcherwho aims at obtaining the answer to question x with a particular likelihood ρ . Forany ρ the researcher selects the shortest interval [ a, b ] that contains an answer withprobability ρ . There are three steps leading to our characterization: First, thatshortest interval is centered around the expected answer. Second, the length ofthe interval is proportional to the standard deviation and thus a function of thedistance to existing knowledge d ( x ; F k ) and the length of the research area, X .Third, fixing distance d and X , the length of the interval is proportional to theinverse Gaussian error function, erf − ( ρ ). Combining the three steps, we obtain asimple, reduced-form cost function that is separable in d and ρ . We begin with adefinition. Definition 2 (Prediction Interval) . The prediction interval α ( x, ρ ) is the smallestinterval [ a, b ] ⊆ R such that the answer to question x lies within [ a, b ] with aprobability of at least ρ . The values of the cutoffs ˆ X , ˇ X are analytically yet implicitly computed in Appendix B.5.We obtain ˆ X ≈ . q, ˇ X ≈ . q . We are not able to analytically compute ˜ X in a meaningfulway, but we can bound its value by ˇ X from below and by 8 q from above. The quadratic formulation is for convenience only. It allows us to describe closed-formsolutions. What matters for our results is that the cost is increasing and convex in the length ofthe interval sampled. See Proposition 2. x y ( x ) x y ( x ) Figure 10:
Cost of research and interference
The left panel corresponds to the right panel in Figure 5. The right panel is a close-upof the left panel. On the right, the two dotted lines represent the 95-percent predictionintervals for the answers to questions x = 1 and x = 1 . F = { ( − . , . , (0 , , (0 . , . , (1 . , . } . Both questions have the same distanceto F ( d ( x ) = 0 . x = 1 is shorterbecause the variance is smaller, in turn because researching x = 1 implies deepeningknowledge. Research on question x = 1 . The dashed line in Figure 10 plots the 95-percent prediction interval for eachquestion x as a function of existing knowledge. We now characterize the predic-tion interval analytically. The cost function is a corollary to the following simpleproposition regarding normal distributions. Proposition 2.
Suppose α ( x, ρ ) is the prediction interval for probability ρ andquestion x , where the answer y ( x ) is normally distributed with mean µ and standarddeviation σ . Then any prediction interval has the following two features:1. The interval is centered around µ .2. The length of the prediction interval is / erf − ( ρ ) σ , where erf − is theinverse of the Gaussian error function. Figure 10 illustrates how the length of the prediction interval depends on thelocation of the question. Two questions with the same distance to existing knowledge(that is, distance to question x = 0 .
6) have different 95-percent prediction intervalsdepending on whether the research deepens knowledge or expands it. That differencetranslates to a difference in the cost function. The cost-of-research function (seeFigure 11) follows from a simple corollary to Proposition 2.
Corollary 3.
For given knowledge F k , fix a probability ρ and a question x . Theminimal cost of obtaining an answer to question x with probability ρ is proportionalto c ( ρ, x ; F k ) = 8( erf − ( ρ )) σ F k ( x )Corollary 3 yields an intuitive characterization of the cost of discovering ananswer to question x given initial knowledge F k with probability ρ . Because theinverse error function is increasing and convex, the cost is increasing and convex20 q q q q qd c o s t c ( d ; 3 q ) c ( d ; 6 q ) c ( d ; 10 q ) c ( d ; ∞ ) . . . . ρ c o s t c ( d ; 3 q ) c ( d ; 6 q ) c ( d ; 10 q ) c ( d ; ∞ ) Figure 11:
Cost of research as a function of distance to knowledge (left) and probability ofsuccess (right)
The cost become smaller for a given ( d, ρ ) as interval length X decreases. Moreover,the cost are linear in distance when expanding knowledge but concave when deepeningknowledge. The cost function is always convex in ρ . The left panel holds ρ = 95% fixed;the right panel holds d = 3 q/ X < q end at the maximum possible distance ( d = X/ y -axis areomitted, as costs are only proportional to the depicted functions (see Corollary 3). in the probability of finding an answer. Finding an answer with certainty impliesinfinite cost, as there is always a chance that the answer is outside the sampledinterval.Moreover, the more imprecise the conjecture about question x , the higher thecost of discovering an answer with a certain probability ( ρ ). Hence, if a more distantquestion is to be answered, the ceteris paribus cost of doing so is higher.Finally, when comparing two questions with the same distance but differentresearch areas, the costs are different. To illustrate this, consider a question that, ifanswered, would expand knowledge (for example, some x e = x − d ) and a questionthat, if answered, would deepen knowledge (for example, some x d = x + d = x − d ).Distance d is the same for both questions. However, the conjecture is more precisein the deepening interval, as information from two known answers can be used toform that conjecture. As a consequence, for the same distance ( d ) and probability( ρ ), the cost is lower in the deepening interval. In this section, we introduce a researcher to our model of knowledge and characterizeher optimal choice.
Consider a researcher that can search for a discovery. Her expected payoff iscomposed of the benefits she provides to the decision maker if she finds an answer21that is, the function from Section 3) and her own cost of research (that is, the costfunction from Section 4). She is unconstrained in her choice of the research questionand her effort, but whenever she fails to obtain an answer, the benefit of her researchis zero. The closest real-world analogue to our model researcher is perhaps a tenureduniversity professor. She is free to select a research question and the intensity withwhich to work on it. The quality of her research is mainly evaluated through peerreview and depends on the (expected) impact of her research on decision making.The more the researcher invests in discovery, the more resources she needs in suchforms as research assistants or time taken away from other projects. To capture therelative weighting of cost and benefits, we introduce a cost parameter, η > x andresearch interval [ a, b ] imply a distance ( d ), an expected probability of finding ananswer ( ρ ), and an area length ( X ). These three variables are sufficient to describethe researcher’s payoff: u R ( d, ρ ; X ) := ρV ( d ; X ) − ηc ( ρ, d ; X )For the above formulation and for what follows, we abuse notation slightly tofacilitate understanding. We (re)define the cost c ( ρ, d, X ) := c ( ρ, x ; F k ) /
8, where d ( x ) = d and X is the length of the area containing x . In addition, to easecomputation, we divide the function from Corollary 3 by 8 and capture the constantwithin η . We now characterize the optimal choice of a researcher with access to knowledge F k .The researcher’s problem can be formulated as follows:max X ∈{ X ,...,X k } max d ∈ [0 ,X/ ,ρ ∈ [0 , ρV ( d ; X ) − ηc ( ρ, d ; X ) | {z } =: U R ( X ) If there is no cost ( η = 0), we can use the insights from Section 3 to obtain theresearcher’s optimum. For any X , the researcher selects ρ = 1. For X < ˜ X sheselects d = X/ d ∈ (3 q, X/
2) for ˜ X < X < ∞ . If X = ∞ , the researcherchooses d = 3 q . She prefers to expand knowledge if and only if X i ≤ ˆ X for anyarea X i < ∞ defined by F k (see Corollary 2). A rationale for discarding nonfindings comes from moral hazard concerns: science is complex,and it is impossible to distinguish the absence of a finding from the absence of proper search. Ourmodel can easily account for the possibility of publishing nonfindings; unsurprisingly, these increasethe value of knowledge as well. The difficulty of publishing the absence of evidence, however,has been long recognized in the literature. See, for example, Sterling (1959). In principle, it isrelatively straightforward to compute updated answer distributions based on null results in oursetting. Including this in our researcher model, however, is beyond the scope of this paper. X X q X r Xd X d (a) Distance d p X X q X r Xρ Xρ (b) Probability ρ p X X q X r XU R XU R (c) Payoff U R Figure 12:
Outcomes of the researcher’s choices in areas of different length
The graphs indicate optimal choices conditional on area length, X . We compare themwith optimal choices on the expanding area X = ∞ (the horizontal line in each graph).On the x -axis we indicate the cutoffs ˆ X, ˇ X, ˙ X, and ˜ X from Proposition 3.The solid graph plots the optimal choice conditional on X being the best available area.For small areas ( X < ˆ X ), the researcher prefers to expand knowledge. For areas of length X > ˆ X , deepening knowledge is preferred to expanding knowledge. If the area has length X < ˜ X , the researcher selects the largest distance possible in area X —that is, d = X/ X > ˜ X , it is optimal to select a distance d < X/ X < ˙ X ), ρ ( X ) increases with X . For large areas ( X > ˙ X ), ρ ( X ) decreases in X (apart from a discontinuous jump at ˜ X ). The researcher’s payoffincreases in area length for X < ˇ X and decreases for X > ˇ X . The order of the cutoffs isindependent of the value of the cost parameter, η . We now characterize the researcher’s optimal decision when η >
0, that is, whenconsidering the cost of research as well. Let d ( X ) and ρ ( X ) be the researcher’schoices conditional on an area of length X , and let U R ( X ) be the associated payoff.Analogously, d ∞ , ρ ∞ , and U ∞ R are the respective objects for expanding research.Figure 12 illustrates the following proposition.23 roposition 3. Fix η ∈ (0 , ∞ ) . On the expanding area, we obtain optimal choices d ∞ ∈ (0 , q ) and ρ ∞ ∈ (0 , , which deliver utility U ∞ R ∈ (0 , ∞ ) . There are cutoffs, ˆ X ≤ ˙ X ≤ ˇ X ≤ ˜ X , such that the following holds: The optimal distance d ( X )• increases in X if X < ˜ X with d ( X ) = X/ and • declines on average on the interval of area lengths ( ˜ X, ∞ ) . The optimal probability ρ ( X )• increases in X if X < ˙ X , • decreases in X if X ∈ ( ˙ X, ˜ X ) , • discontinuously increases at ˜ X , and • declines on average on the interval of area lengths ( ˜ X, ∞ ) . The researcher’s payoff, U R ( X ),• is smaller than U ∞ R if and only if X < ˆ X , • increases in X if X < ˇ X , and • decreases in X if X > ˇ X .Moreover, as X → ∞ , d ( X ) → d ∞ , ρ ( X ) → ρ ∞ , and U R ( X ) → U ∞ R from above. Introducing cost, η >
0, distorts the researcher’s decision from the choice thatmaximizes the benefits of a discovery. There are two additional effects: First, ceterisparibus the cost of research increase in the variance of the conjectures—a force thatmitigates the researcher’s incentives to aim for higher d . Second, cost increase fasterin d when expanding knowledge because the variance increases faster—a force thatmitigates the researcher’s incentive to aim for expanding knowledge. Yet, in termsof the choice of question we obtain the same qualitative pattern.Figure 12 sketches the researcher’s expected value of conducting research inareas of different lengths and the corresponding probability of finding an answer.The optimal within-area distance is d = X/ X ∈ [0 , ˜ X ] and d < X/ X > ( ˜ X, ∞ ). The maximum payoff is achieved for research in area ˇ X . Expandingknowledge is the optimal strategy if all available interval lengths X < ˆ X . As thearea length X → ∞ , the value of conducting research on the area converges to thatof expanding research, X = ∞ . If several X are available, the researcher choosesthe one with the largest value. We discuss the effect of η on the cutoff values below.Yet, the cost of research introduce a second, interlinked dimension: the re-searcher’s likelihood of finding an answer.It is instructive to begin with the expanding-knowledge case, X = ∞ . Theresearcher simultaneously chooses a probability of finding an answer, ρ , and adistance to current knowledge, d . Her expected benefit is ρV ( d ; ∞ ), and her cost is ηerf − ( ρ ) d . By setting ρ = 0—that is, by not looking for an answer at all—theresearcher can guarantee herself a payoff of u R = 0. The marginal cost of increasingthe probability, starting at ρ = 0, are 0 and non-increasing. However, the marginalbenefit of increasing ρ is V ( d ; ∞ ) which is positive if d >
0. Thus, a higher utilitythan u R = 0 is attainable by looking for the answer to a question with ( d, ρ ) > ρ →
1, the researcher has to look for an answer on the entire Numerically, d ( X ) and ρ ( X ) are decreasing throughout on ( ˜ X, ∞ ). However, we provide noformal proof. d > ρ is interior.By Proposition 1 we know that the benefit of discovery is increasing at first butdecreasing once d > q . The cost of research, in turn, is monotone in d . The moredistant the question, the more imprecise the conjecture and hence the more effortrequired to answer the question with any given probability ρ . The optimal distanceis smaller than absent cost (0 < d ∞ < q ). The probability of finding a solutionis strictly positive but not 1 (0 < ρ ∞ < < U ∞ R < V (3 q ; ∞ ).Now consider a researcher that deepens knowledge. The basic logic is identi-cal. However, deepening knowledge is ceteris paribus less costly than expandingknowledge. For any given ( ρ, d ) the variance is lower the smaller X i and, hence, thecheaper is the search. Regarding the researcher’s choice of d the logic follows thediscussion of Proposition 1.The optimal ρ depends on the size of the area too. Consider a small area X i . Thescope for improvement is small as conjectures are already precise. Thus, investinginto discovery has a small expected payoff. Albeit cost are small, the researcherdoes not invest much into the search for an answer— ρ is small. Now consider alarge area. The benefits of deepening researcher are larger than in the small area.However—because conjectures are imprecise—the cost are larger too. The researcherdoes not invest much into a discovery— ρ is small. In an area of intermediate length,the benefits of a discovery are (relatively) high, yet conjectures are sufficiently preciseand limit the cost of research. The return on investment is largest and ρ is high.We obtain a similar pattern for ρ as that for d : while ρ is low for small and largeareas, it is higher for intermediate X i .Moreover, the researcher only trades off d ( X ) against ρ ( X ) if X is of intermediatelength. If X is small, an increase in X increases the benefits of research. Cost aresmall, and the researcher has an incentive to increase d ( X ) and ρ ( X ). As X becomeslarger, the marginal increase in the benefits of research declines, yet the marginalcost of research increase both for d ( X ) and ρ ( X ). Eventually the researcher faces atradeoff: should she lower ρ ( X ) to maintain d ( X ) = X/
2? It turns out that thisis optimal. While the researcher wants to remain at a boundary in her choice of d ( X ) she mitigates the increased cost by lowering ρ ( X ). As X increases further, theresearcher eventually also lowers d ( X ). After a discrete jump upwards at ˜ X of ρ ( X )due to the jump downwards of d ( X ), the two start to co-move again. Both decline.The researcher’s overall most preferred area length, ˇ X , is in a region in which atrade-off between ρ ( X ) and d ( X ) exists. The researcher would like to enter a larger X to increase the benefits of research, she would also prefer a smaller X to reducethe cost of finding an answer. Thus, d ( X ) is increasing and ρ ( X ) decreasing at thepoint at which U R ( X ) is maximal.Note that we have only characterized the researcher’s decision conditional on anarea length X and compared these values so far. An explicit analytical characteriza-tion of the researcher’s choice thus depends on the existing knowledge and thus allavailable X ’s. Solving for such an optimum is straightforward but not instructive. The extensive margin—that is, which X to choose—depends on the precise A computer program to numerically calculate the optima for different parameter values isavailable from the authors. X, ˙ X, ˇ X , and ˜ X . We summarize the comparative statics ofthese cutoffs in the next corollary. Corollary 4.
All cutoff values are non-increasing in η . While ˙ X is constant for η > , the cutoffs ˆ X, ˇ X, and ˜ X are strictly decreasing and thus strictly smaller thantheir counterparts from Corollary 2. In this section, we illustrate our model through an application. Consider a fundinginstitution (the funder) with two instruments. Ex ante cost reduction for theresearcher (by, for example, providing grants to reduce the researcher’s cost) and expost rewards (by, for example, handing out prizes for seminal contributions). Inaddition, assume that the funder has to respect scientific freedom and can spend herbudget on any combination of cost reductions and rewards. Which levels of novelty d and productivity ρ can the funder implement.In the main text we present the stylized model to address this question in ourframework and it’s main result: a characterization of the feasible set in the ( d, ρ )-space. In appendix C, we provide the full analysis of the mechanism combining ourresults from above with standard arguments from consumer theory. Setup.
Consider the following stylized model. Knowledge consists of a singlequestion-answer pair, F = ( x , y ( x )). The researcher’s cost parameter is η .The funder has a fixed budget K to invest in the researcher and has two fundingtechnologies: ex ante cost reductions, h , and ex post rewards, ζ . Marginal cost ofboth are constant, and the cost ratio κ is such that the funder’s budget constraintis given as follows: K = ζ + κh A cost reduction of h implies that the researcher faces cost parameter η ≡ η − h .An ex post reward gives the researcher additional utility of ζ if she finds an answer.We assume rewards only come for “seminal contributions,” that is, contributionsthat are sufficiently novel and thus difficult to obtain. We proxy that relation by afunction f ( σ F k ) : R → [0 ,
1] that determines the probability of a reward. To keepthe analysis simple, we assume a piecewise linear relationship f ( σ ) = σ s if σ < s s >
0. The parameter s determines the minimum level of difficulty thatguarantees a reward. To shorten notation define ˜ c ( ρ ) := ( erf − ( ρ )) . Regarding the comparative statics for the intensive margin, d ( X ) and ρ ( X ), we conjecturethat both are decreasing in η as well. However, so far we have been unable to provide a formalproof for the case X > ˜ X . There is a large literature debating different forms of science funding. For recent contributions,see Price (2019) and Azoulay and Li (2020) and references therein. The crucial assumption here is that f is a bounded function, which is true whenever f isindeed a probability. If f instead was unbounded, the researcher would naturally (for any ζ > . . . . . . . . . . . . . ρ d . . . . . . . . . . . . . ρ d Figure 13:
Feasible set for different reward standards
The shaded area shows the implementable ( ρ, d ) combinations of a funder for a givenbudget K . All points on a solid line require the same amount of funding, K . In bothpanels, the funder has a budget of K = 10, the price ratio is κ = 6, and the baseline costfactor is η = 2. The status quo parameter has value q = 1. In the left panel, the rewardtechnology is parameterized with s = 40; in the right panel, it is s = 200. In our stylized model the researcher’s problem becomes the following:max d,ρ ρ (cid:18) V ( d ; ∞ ) + f ( σ ( d ; ∞ )) ζ (cid:19) − η ˜ c ( ρ ) σ ( d ; X ) . The feasible set.
In reality, funding may have many different objectives and canrange from maximizing the externalities of research to answering particular questions.A demand-side discussion of research is beyond the scope of this paper. Instead, wecharacterize the feasible set of choices ( d, ρ ) that a funder can induce with a givenbudget. Computing this set provides a useful tool to analyze the optimal fundingscheme given a particular preference relation over ( d, ρ ) bundles. As in a standardconsumer problem, it can be readily applied, with this set being the analogue of abudget set. Let ˜ c ρ ( ρ ) := ∂ ˜ c/∂ρ ( ρ ) . Proposition 4.
The set of implementable ( d, ρ ) -combinations for a given cost ratio κ and a budget K is described by the ( d, ρ ) implementation frontier defined over [ ρ, ρ ] ,which are the endogenous upper and lower bounds of ρ . These bounds are determinedby the extreme funding schemes ( ζ = 0 , η = η − K/κ ) and ( ζ = K, η = η ) . Theresearch-possibility frontier between those polar points is as follows: d ( ρ ; K ) = 6 q ( K + s − κη ) ρ ˜ c ρ ( ρ ) − ˜ c ( ρ )2 sρ ˜ c ρ ( ρ ) − s ˜ c ( ρ ) − κρ (1) d ( ρ ; K ) can be increasing or decreasing, depending on whether d and ρ behave assubstitutes or complements at the ( ζ, η ) mix inducing the current level of ( d, ρ ) . The proof of Proposition 4 is, together with the rest of the formal discussion ofthis application, in appendix C. The main takeaway from Proposition 4 is that the choose to select d = ∞ . The specific form, in turn, is chosen only for ease of computation. ρ , implies an increasein novelty, d , as the two are complements from the funder’s perspective. If theslope is negative, an increase in output implies a decrease in novelty, as the two aresubstitutes from the funder’s perspective.We illustrate the feasible set for two levels of parameter s in Figure 13. In theleft panel ( s = 40), novelty and output are substitutes as long as ρ < ˜ ρ ≈ s increases to 200, that threshold decreases to ˜ ρ ≈ d (conditional on discovery) would choose to implement a riskychoice by the researcher ( ρ ≈ s = 40. The same funder would choose a muchsafer strategy when s = 200 ( ρ ≈ Novelty and output become complements because an increase in ρ has twoeffects: (i) it increases the marginal benefit of distance, V d ( d, X ) + ζ/sσ d ( d, X ), byincreasing the probability that the researcher finds an answer, and (ii) it increasesthe marginal cost of distance, η ˜ c ( ρ ) σ d . The uncertainty of the conjecture aboutquestions, σ ( d, X ), is increasing in distance. Moreover, for any distance, the intervalthat has to be covered to find an answer with probability ρ is increasing in thisprobability. For certain parameter constellations, it may be that, for example, anincrease in ζ increases the weight placed on the marginal-benefit effect relatively morethan the resulting increase in η increases the marginal-cost effect. Straightforwardly,the larger s , the smaller the effect on the marginal benefit. In this section, we relate our model and our findings to the two closest models inthe literature, those in Callander and Clark (2017) and Prendergast (2019).Callander and Clark (2017) consider judicial decision making. Judges are inter-ested in learning the realization of the entire path of a Brownian motion. As in ourmodel, a decision maker (a lower court in their framework) adjusts decisions to theunknown state of the world. The decision maker’s knowledge is the realization of theBrownian process at values that a higher court reveals. Aside from the application,the main differences between our model and that in Callander and Clark (2017) arethe following. First, we assume that the researcher selects, in addition to the researchquestion, her intensity of research. The latter affects both the cost of research andthe probability of finding an answer. Second, we aim to learn where exactly theanswer lies, in contrast to the “above or below the threshold” problem that Callanderand Clark (2017) analyze. Unlike in their model, the value of a discovery in ourframework does not lie in how the expectations relate to the threshold but in howthey relate to other known points: as the conjectures become more precise andfinding an answer comparatively easy, the benefits of research shrink too. Third, inCallander and Clark (2017), it is eventually optimal to stop discovery. In our model,science never stops. Although the cost of asking a specific question decreases when We assume in this exercise that s is exogenous to the funder. Under most funding schemes,the winners of a prize are determined by a jury of peers rather than by the funder itself. Thus, thethe standard may not be under the funder’s control. X (so that expandingknowledge is not beneficial) but smaller than ˜ X (so that selecting the midpoint isalways optimal). The novelty depending probability-of-success function he assumesin the extensions matches the features we derive for our endogenous probability ofsuccess. Thus, we can construct a special case in which model predictions coincide.Our focus is on the microfoundation of the functions while his is on agencyconcerns in a reduced-form model. Yet, in terms of modeling approaches our modelof knowledge differs at least in two crucial dimensions. First, we assume that itis possible to expand research beyond the frontier and show that it can also beoptimal to do so. Prendergast (2019) instead assumes that research always takesplace between two existing findings. Second, our decision maker has an outsideoption that limits her expected losses if conjectures are too imprecise. Existence ofan outside option implies that there are bounds on the benefits of novelty. Oncenewly created areas become too large, conjectures in that area become too imprecisewhich mitigates the value of the area. Therefore, and different from Prendergast(2019), we obtain a nonmonotonicity in the value of novelty.Finally, Prendergast (2019) assumes an exogenous cost function. While ourendogenous cost function has the basic properties he assumes (increasing andconcave), we provide additional structure by microfounding the model as a searchprocess. Moreover, if cost follow from the search for an answer, they endogenouslydepend on the area size: the smaller the area, the smaller the variance and hencethe smaller the cost.
We propose a tractable model of knowledge and show how to embed it in a frameworkthat allows us to study which questions researchers aim to answer and to what extentthey invest in finding the answer. The main features of our model are that (i) findingthe answer to one question spills over onto the conjectures about other questions,(ii) questions in close proximity to existing knowledge are easier to answer thanquestions that are far away from existing knowledge, (iii) the benefit of discoverydepend on its effect on decision making, and (iv) researchers are motivated by thebenefit their findings provide for decision making, but bear a cost of searching foran answer.Using these four elements, we set up a model to derive an endogenous benefits-of-discovery function and an endogenous cost-of-research function and to characterizethe researcher’s optimal selection of research question and likelihood of finding an That result also formalizes the argument in Price (2020) that focusing on providing incentivesthat maximize novelty can backfire, as knowledge may be scattered to disconnected islands. Inour model, research that is too far away from the current body of knowledge has little positivespillovers on society’s conjectures about unanswered questions. From observed decisions, such asthose documented in Rzhetsky et al. (2015), we could identify the distribution ofthese cost parameters. Our model would then allow us to compute the counterfactualoutcomes of the various mitigation strategies proposed by Rzhetsky et al. (2015).One such strategy is to incentivize the publication of null results, which weassume to be noncommunicable (an assumption motivated by observed reality).Within our model, knowing that the answer is not in a certain interval has a clearvalue that is straightforward to compute. In addition, because answers to questionsare correlated, the knowledge provided by null results also generates a computablespillover onto the conjectures about other questions. Thus, if the researcher cancredibly claim to have searched in an interval without finding the answer, publishingnull results should be part of the benefit function and inform subsequent researchers’selection and search strategies.Another dimension present in the analysis of Rzhetsky et al. (2015) is to considerresearch as establishing links between questions while the implicit assumption inour model is that links are ex-ante known and given by a line network. A naturaldirection for future research is to combine our search for answers with the task ofdiscovering the network that links questions.Our paper starts by emphasizing the role of scientific freedom. Preserving thatfreedom remains a challenging task for science-funding institutions when designinga funding architecture that provides researchers the support to engage in researchactivities that might not be undertaken otherwise. The NSF emphasizes that it aimsat funding high-risk/high-reward research to advance the knowledge frontier. Whilethe question of optimal market design is beyond the scope of this paper, we hope thatour modeling framework will serve as a stepping stone toward developing a structuralmodel to evaluate funding incentives and to provide meaningful counterfactuals thatcan inform decision makers about how to provide incentives optimally. One could also have researchers arrive with an intuition or stroke of genius—that is, a privatesignal about the location of a specific answer. We thank Alex Frug for this suggestion. ppendix A Notation and Properties of erf − Notation:
We use argument subscripts to denote the partial derivatives with respectto the argument. We omit function argument whenever it is convenient. we use thenotation d f ( x,y )dx to indicate the total derivative ( f x + f y y x ). Properties of erf − . From Appendix B.5 onwards we rely heavily on the proper-ties of the inverse error function. It is instructive to keep in mind that˜ c ( ρ ) := erf − ( ρ ) is convex and increasing on [0 ,
1) with ˜ c (0) = 0 and lim ρ → ˜ c ( ρ ) = ∞ . Thederivative ˜ c ρ ( ρ ) = √ πe ˜ c ( ρ ) erf − ( ρ )is increasing and convex with the same limits.Further, we make use of the fact that for ρ ∈ (0 ,
1) ˜ c ( ρ ) has a convex andincreasing elasticity bounded below by 2 and unbounded above. Its derivative ˜ c ρ ( ρ )has an increasing elasticity bounded below by 1 and unbounded above. We want toemphasize that these properties are not special to our quadratic cost assumption.To the contrary, erf − ( x ) k for any k ≥ ρ ˜ c ρρ ( ρ )˜ c ρ ( ρ ) ∈ (1 , ∞ ) and increasing, ρ ˜ c ρ ( ρ ) − ˜ c ( ρ ) ∈ (0 , ∞ ) and increasing,˜ c ( ρ ) ρ ˜ c ρ ( ρ ) ∈ (0 , .
5) and decreasing.
B Proofs
At various points we make use of inequality relations the proof of which we relegateto supplementary appendix D. In each of these cases proving the inequalities is donevia algebra that produces little additional insight. Due to this limit and the researcher’s ability to choose ρ = 1, we augment the support of thecost function to include ρ = 1 with ˜ c (1) = ∞ . However, the optimal ρ is always strictly interiorunless the cost parameter η is chosen to be zero in which case we assume that η ˜ c ( ρ = 1) = 0. .1 Proof of Proposition 1 Proof.
The value of knowing F k is Z max ( q − σ F k ( x ) q , ) d x. No matter which point of knowledge ( x, y ( x )) is added to F k , the value of knowledgeoutside the frontier is identical for both F k and F k ∪ ( x, y ( x )). Area lengths X = X k = ∞ do not depend on F k and neither does the variance for a question x < x or x > x k with a given distance d to F k . The conjectures about all questionsoutside [ x , x k ] deliver a total value of2 Z q q − xq d x = q, which is independent of F k . Moreover, answering question ˆ x ∈ [ x i , x i +1 ] with( x i , y ( x i )) , ( x i +1 , y ( x i +1 ) ∈ F k only affects questions in the area [ x i , x i +1 ], i.e., G ( x |F k ) = G ( x |F k ∪ (ˆ x, y (ˆ x )) ∀ x / ∈ ( x i , x i +1 ).To simplify notation, let us consider the points in terms of distance to the lowerbound of the area with X , d ≡ x − x i .The value of the area [ x i , x i + ] is (with abuse of notation) v ( X ) = Z X max q − d ( X − d ) X q , d d. Note that whenever X ≤ q , d ( X − d ) X ≤ q . Hence, we can directly compute the valueof any area with length X ≤ q as v ( X ) = − X q + X. Whenever
X > q , value is only generated on a subset of points in the area. Asthe variance is a symmetric quadratic function with X/ X/ q . The pointswith variance equal to q are given by d , = X ± √ X √ X − q . Hence, the valueof an area with X > q is (due to symmetry) v ( X ) = 2 Z d q − d ( X − d ) X q d d = − X q + X + X − q q √ X q X − q. Hence, if a knowledge point on the boundary is added, a new area is created and32o area is replaced. The value created is thus V ( d ) = v ( d ) = − d q + d + , if d ≤ q d − q q √ d √ d − q, if d > q. If a knowledge point is added inside an area with length X with distance d tothe closest existing knowledge, it generates two new areas with length d and X − d that replace the old area with length X . The total value of the two intervals new is v ( d ) + v ( X − d ) = − d q + d + , if d ≤ q d − q q √ d √ d − q, if d > q +( − ( X − d ) q ) + X − d + , if X − d ≤ q X − d − q q √ X − d √ X − d − q, if X − d > q . The benefit of discovery is then V ( d ; X ) = v ( d ) + v ( X − d ) − v ( X ) which isprecisley the expression from the proposition. B.2 Proof of Corollary 1
Proof.
The optimality of d = 3 q follows directly from the first-order condition for d ≤ q which is ∂V ( d ; ∞| d ≥ q ) ∂d = − d q + 1 = 0and the observation that the benefit is decreasing in d for d > q which can be seenfrom the derivative with respect to d which is ∂V ( d ; ∞| d > q ) ∂d = − d q + 1 + s d − qd d − q q < . The inequality follows from Lemma 23 in appendix D.
B.3 Proof of Corollary 2
We proceed in a series of lemmata. We outline the mapping to Corollary 2 first• The first part of the first bullet point follows from Lemma 1 and Corollary 1.The second part follows from Lemma 2 to 6 and Lemma 8.• The second bullet point follows from Lemma 2, 6 and 8• The third bullet point follows from Lemma 2 to 7.• The fourth bullet point follows from Lemma 5.• Lemma 9 proves the order of the cutoffs.• Lemma 10 provides an approximation to ˆ X .Throughout, we refer to the distance d that maximizes V ( d ; X ) as d ( X ). Werelegate the algebra behind some inequalities to the supplementary appendix (see33eferences below). Proof.
Lemma 1. d ( X ) = X/ if X ≤ q .Proof.
1. Assume X ≤ q. The benefits of discovery are V ( d ; X | X ≤ q ) = 13 q ( Xd − d )which is increasing in d and hence maximized at d = X/
2. Moreover, V ( X/ X ) = X / (12 q ) which is increasing in X .
2. Assume X ∈ (4 q, q ](i) d ≥ X − q implies (since d ≤ q ) V ( d ; X | d ≥ X − q, X ∈ (4 q, q ])) = 16 q (cid:16) dX − d − √ X ( X − q ) / (cid:17) which is the same as in the first case up to constant −√ X ( X − q ) / . Thus theoptimal d conditional on d ≥ X − q is d = X/ d ≤ X − q the benefit becomes V ( d ; X | d ≤ X − q, X ∈ (4 q, q ])) =16 q (cid:16) dX − d + √ X − d ( X − d − q ) / − √ X ( X − q ) / (cid:17) , with derivative V d = 13 q X − d − ( X − d − q ) s X − d − qX − d . which is positive for d ≤ X − q, X ∈ (4 q, q ] by Lemma 24 from appendix D.Hence, V d ( d ; X | d ≤ X − q, X ∈ [4 q, q ]) > d and X in the considereddomain. Thus, d = X − q maximizes V ( d ; X | d ≤ X − q, X ∈ (4 q, q ])) and hence d = X/ V ( d ; X | X ∈ (4 q, q ]). Lemma 2. If X > q then d ( X ) = X/ .Proof. Take d = 4 q < X/
2. That implies V ( d ; X | X > q ) = 16 (cid:18) Xq − q − √ X ( X − q ) / + q ( X − q )( X − q ) / (cid:19) . By comparison V ( X/ X | X > q ) = 16 X − √ X ( X − q ) / + 12 √ X ( X − q ) / ! V ( d ; X |· ) − V ( X/ X |· )= 16 q (cid:18)q X − q ( X − q ) / − √ X X − q ) / − ( X − q ) (cid:19) = 16 ( X − q ) / (cid:18) q X − q − √ X − q ( X − q ) (cid:19) , which is positive if 4( X − q ) > X − q ⇔ X > q and holds by assumption. Lemma 3. d ( X ) < X/ ⇒ dV ( d ( X ); X ) dX < .Proof. By the envelope theorem, dV ( d ( X ); X ) dX = V X ( d ( X ); X ) . This derivative is negative for X ≥ q and for all d ∈ [0 , X − q ] by Lemma 25in appendix D. If X ≥ q that claim is sufficient. By Lemma 1 we know that X ≥ q whenever d ( X ) = X/
2. In 2 . ( i ) in the proof of Lemma 25, page 61, weshow that V d > d ∈ [ X − q, X/
2) if X ≤ q . Hence if d ( X ) = X/
2, then d ( X ) ≤ X − q and the inequality proved in Lemma 25 proves the lemma. Lemma 4. d ( X ) < X/ for some X ∈ [6 q, q ] ⇒ d ( X ) < X/ for all X > X , X ∈ [6 q, q ] .Proof. We prove the claim by showing that V ( d ( X ); X ) for d ( X | X < < X/ V ( X/ X ) from below at any intersection point. Thus, there is at most oneswitch from d ( X ) = X/ d ( X ) < X/ V ( d ; X ) is a continuously differentiable function in X and d . Thus any interior (lo-cal) optimum d ( X ) is continuous as well and so are V ( d ( X ); X ) and V ( X/ X ). Wenow show that if V ( d ( X ); X ) = V ( X/ X ) for some d ( X ) < X/ X ∈ [6 q, q ],then d V ( d ( X ); X ) / d X > d V ( X/ X ) / d X . Note that d V ( d ( X ) , X ) / d X < V ( X/ , X )is decreasing and must be such that d V ( X/ , X ) / d X < d V ( d ( X ) , X ) / d X . Weprove that this is the only potential intersection in Lemma 26 in appendix D wherewe show that d V ( X/ , X ) / (d X ) < V ( d ( X ) , X ) / (d X ) > Lemma 5. As X → ∞ , d ( X ) → d ∞ .Proof. Take any sequence of increasing X n with lim n →∞ X n = ∞ . For any δ ( d ) , ∃ n such that V n ( d ; X ) − V ( d ; ∞ ) < δ ( d ). Hence V ( d ; X ) converges uniformly to V ( d ; ∞ )and the result follows. Lemma 6. If d ( X ) ≤ X/ then V ( d ( X ); X ) > V ( d ∞ , ∞ ) . roof. By Lemma 1, we know that d ( X ) < X ⇒ X > q . But given any X < ∞ we obtain V (3 q ; X ) > V (3 q ; ∞ ). Lemma 7. V ( d ( X ); X ) is continuous in X .Proof. V ( d ( X ); X ) = max d V ( d ; X ) with V ( d ; X ) continuous in both d ∈ [0 , X/ X . Hence, the max is continuous as well. Lemma 8. V ( d ( X ); X ) is single peaked with an interior peak. ˇ X ≈ . q Proof.
Follows from continuity of V ( X/ X ) (by Lemma 7) and Lemma 1 to 5.The peak can be computed. It is the solution to XX − q = 2 √ X − q √ X . (2)Replacing X ≡ mq the above reduces to mm − > s ( m − m For m >
4, the LHS decreases in m while the RHS increases in m for m > m ≈ . Lemma 9. q < ˆ X < q < ˇ X < ˜ X .Proof. q < ˆ X < q by Lemma 1 and 6; ˇ X > q by Lemma 8; 8 q > ˜ X > ˇ X byclaims Lemma 2, 4 and 8 Lemma 10. ˆ X ≈ . q .Proof. For X ∈ [4 q, q ] all we need to consider is d ( X ) = X/ V ( X/ X ) = X q − √ X ( X − q ) / q with V (3 q ; ∞ ) = q . Replacing X ≡ ‘q and simplifying we the two intersect if q ‘ − √ ‘ ‘ − − / ! = 0which has a unique solution for ‘ < ‘ ≈ . .4 Proof of Proposition 2 Proof.
As the normal distribution is symmetric around the mean with a densitydecreasing in both directions, it follows directly that the smallest interval thatcontains the realization with a particular likelihood is centered around the mean.Take an interval of length
Z < ∞ that is symmetric around the mean µ andassume a total mass of ρ is inside the interval. Then (1 − ρ ) / z l of theinterval has (by symmetry of the interval) distance µ − Z/ z l ) = 1 / erf z l − µσ √ !! = 1 / erf − Z/ σ √ !! . Solving (using symmetry of erf )1 / (cid:18) − erf (cid:18) Zσ / (cid:19)(cid:19) = 1 − ρ erf (cid:18) Zσ / (cid:19) = ρ ⇔ Z = 2 / erf − ( ρ ) σ. B.5 Proof of Proposition 3
We proceed in a series of lemmata. We outline the our strategy below• Lemma 11 proves existence of an interior optimum.• Lemma 12 characterizes the choices hen expanding research.• Lemma 13 and 14 show that the value of a boundary solution is strictlyincreasing on [0 , q ] and has a unique maximum on (4 q, ˇ X ].• Lemma 15 proves existence and uniqueness of cutoff ˜ X .• Lemma 16 proves the jump upwards in ρ at ˜ X .• Lemma 17, shows that the researcher’s value is single-peaked in X .• Lemma 18 to 20 derived the order of the cutoffs.We make frequent use of properties of ˜ c ( ρ ) = erf − ( ρ ) as defined in appendix A. Proof.
Lemma 11.
There is a non-trivial optimal choice with ∞ > d > , > ρ > onany interval with positive length, X ∈ (0 , ∞ ) .Proof. Recall that the researcher can always guarantee a non-negative payoff bychoosing either d = 0 or ρ = 0. Hence, the researcher’s value is bounded from below, U R ( X ) ≡ max d,ρ u R ( d, ρ ; X ) ≥
0. Next, note that u R ( ρ = 0 , d > ε ; X ) = 0 for somesmall ε > ∂u R ( ρ =0 ,d>ε ; X ) ∂ρ = V ( ε, X ) > X there is a maximum with d > , ρ > V ( d, X ) ≤ M < ∞ and lim ρ → ˜ c ( ρ ) = ∞ . Therefore the optimal ρ <
1. Finally, V ( d, ∞ ) is decreasing in d for d large enough while the cost η ˜ c ( ρ ) σ ( d, ∞ ) is increasing in d . Hence, theoptimal distance is bounded d ≤ D < ∞ . Lemma 12.
On the expanding interval, X = ∞ , the optimal choice is characterizedby the first-order conditions (FOCs). The FOCs are sufficient and the optimal d ∞ ∈ (2 q, q ) . The researcher’s value is strictly positive U R ( X = ∞ ) > .Proof. Fix any ρ ≥
0. Since σ ( d ; ∞ ) is increasing it is immediate that the re-searcher’s utility is non-increasing in d if V ( d ; ∞ ) decreases in d . Thus, it issufficient to restrict attention to d ≤ q .By Lemma 11, the researcher’s optimal choice is interior and, hence, characterizedby the first-order conditions. That the value is positive follows immediately fromthe choice being strictly interior and X > ρV dd − ηcσ dd = − ρ q < σ dd = 0 and the second is given by thedeterminant of the Hessian which is positive − ρV dd η ˜ c ρρ ( ρ ) σ ( d ; ∞ ) − ( V d − η ˜ c ρ ( ρ ) σ d ( d ; ∞ )) = ρ q η ˜ c ρρ ( ρ ) d − − d q + 1 − η ˜ c ρ ( ρ ) ! (3)= ρ ˜ c ρρ ( ρ )˜ c ρ ( ρ ) V ( d ; ∞ )3 q − − d q + 1 − V ( d ; ∞ ) σ ( d ; ∞ ) ! where the last equality follows from substituting using the first-order conditions ρV d ( d ; ∞ ) − η ˜ c ( ρ ) σ d ( d ; ∞ ) = 0 (4) V ( d ; ∞ ) − η ˜ c ρ ( ρ ) σ ( d ; ∞ ) = 0; (5)in particular, ησ ( d ; ∞ ) = V ( d ; ∞ )˜ c ρ ( ρ ) , η ˜ c ρ ( ρ ) = Vσ . Rearranging the last term ofequation (3) and substituting for V under the assumption that d ≤ q . ρ ˜ c ρρ ( ρ )˜ c ρ ( ρ ) > − d q + 1 − − d q + dd q − d q + d ⇔ ρ ˜ c ρρ ( ρ )˜ c ρ ( ρ ) > d (6 q − d ) . where the inequality follows by the properties of ˜ c ( ρ ) from appendix A implying LHS ≥ RHS ∈ [0 ,
1] for d ≤ q . Distance 3 q is an upperbound for every critical d because the d -first-order condition yields d ∞ = 3 q − η ˜ c ( ρ ) ρ ! < q. (6)38eplacing η via equation (5) and solving for d we obtain d ∞ = 3 q − ˜ c ( ρ )2˜ c ρ ( ρ ) ρ − ˜ c ( ρ ) ! ∈ (2 q, q )where the lower bound follows from the properties of the error function. Lemma 13.
Fix d = X/ and a assume that an interior optimum exists. Then U R ( X | d = X/ is maximal only if the total differential d V ( d = X/ X )d X ≥ .Proof. Under the assumption d = X/ U R ( X ) is defined and continuously differ-entiable for all X ∈ [0 , ∞ ) despite the indicator functions. Because X = 0 implies U R ( X = 0) = 0 and Lemma 11 holds, there is an interior X at which U R ( X ).Then, if U R ( X ) is maximal for some interior and differentiable X it needs tosatisfy ∂U R ∂X = 0By assumption we have d ( ˇ X ) = X/ ρ holds. At an interior and differentiable X we need that ρ d V ( d = X/ X )d X = η c ( ρ ) . The right hand side is non-negative, which implies the desired result. Lemma 14.
The value of the deepening boundary solution d = X peaks at ˇ X ∈ (4 q, ˇ X ] .Proof. Note that U R ( d ≡ X/ X ) > X ∈ [0 , q ] as in this case U R ( d ≡ X/ , X ) = ρ X q − η ˜ c ( ρ ) X and, hence, U R ( d ≡ X/ , X ) = ρ X q − η ˜ c ( ρ ) . Usingoptimality of ρ via the FOC X q = η ˜ c ρ ( ρ ) ⇒ X q = η ˜ c ρ ( ρ )2which yields U R ( X ) = ρ η ˜ c ρ ( ρ )2 − η ˜ c ( ρ ) 14= ˜ c ρ ( ρ )4 ρη − ˜ c ( ρ ) ρ ˜ c ρ ( ρ ) ! > c ( ρ ).Moreover, U R ( X ) is strictly concave on [4 q, q ] as V ( d = X/ , X ) is concave on The inequality is weak as for η = 0, ρ ( X ) = 1 and U R ( X ) = V ( X ). σ XX ( d = X/ , X ) = 0 implying U R ( X ) = ρ dd V ( d = X/ X )d X d X < . For
X > ˇ X , d V ( d = X/ X )d X < X . By Lemma 13, it follows thatthe maximizing X ∈ (4 q, ˇ X ] Lemma 15.
The researcher’s optimal choice of distance is on the midpoint of theinterval, d = X , for X ≤ ˜ X and interior, d < X , for X > ˜ X with ˜ X > ˇ X . Itconverges from above to d ∞ , lim X →∞ d ( X ) = d ∞ . Any optimal distance choice satisfies d ≤ q .Proof. Note first that the choice d = X always constitutes a local maximum asthe marginal cost of distance is zero at this point, ∂σ ( d,X ) ∂d = 1 − dX , and, for anychoice of d , there is a unique ρ that solves the first-order condition with respect to ρ because the first-order condition with respect to ρ for any d , V ( d,X ) σ ( d,X ) = η ˜ c ρ ( ρ ), has acontinuous, strictly increasing, unbounded right-hand side that starts at ˜ c ρ (0) = 0and a constant left-hand side. Hence, the boundary solution with d = X is always acandidate solution.We first show that for X ≤ q , the optimal choice will always be the boundarysolution with d = X/ ρV d ( d, X ) − η ˜ c ( ρ ) σ d ( d, X ) = 0 V ( d, X ) − η ˜ c ρ ( ρ ) σ ( d, X ) = 0 . Replacing η from the second ( ρ ’s) first-order condition in the first ( d ’s) first-ordercondition, we obtain V d ( d,X ) σ d ( d,X ) V ( d,X ) σ ( d,X ) = ˜ c ( ρ ) ρ ˜ c ρ ( ρ ) . It follows from the properties of ˜ c ( ρ ) that the RHS ∈ [0 , /
2] and decreasing. Thus,whenever the
LHS > / ρ , the boundary choice d = X will be optimal. For X ≤ q V d σ d Vσ = X − d ) X − dX dX − d ) d ( X − d ) X = 1 . Hence, for small intervals, the boundary choice is indeed optimal.Next, we show that for
X > q , the boundary solution is suboptimal to someinterior solution. Note first that the variance of the question on the boundary is Note that we totally differentiate the value twice and all ρ ( X ) and ρ ( X ) terms drop outby optimality of ρ by applying the first-order condition directly and total differentiation of thefirst-order condition. σ = d ( X − d ) X is increasing in d . Hence,if the benefit of research V is larger for an interior question than for the boundaryquestion, the researcher can obtain a higher payoff by choosing an interior questionwith the same ρ as for the boundary question: the cost will be lower, the successprobability the same and the benefit upon success higher. The benefit of findingan answer on the boundary of an interval with X > q is always smaller than forsome interior distance by Lemma 2 from the proof of Corollary 2. Hence, an interiorchoice is optimal for X > q .For X ∈ (4 q, q ) and X − d < q , V d ( d,X ) σ d ( d,X ) V ( d,X ) σ ( d,X ) = 2 d ( X − d ) − d + 2 dX − √ X ( X − q ) / which is decreasing in d with limitlim d → X/ d ( X − d ) − d + 2 dX − √ X ( X − q ) / = X / X / − √ X ( X − q ) / which, in turn, is increasing in X and 1 for X = 4 q . Hence, any interior solutionmust be such that X − d > q as otherwise, the first-order condition with respect to d is always positive.Thus, we know that (i) on intervals with X < q , the researcher’s distance choiceon the deepening interval will be a boundary solution, (ii) on intervals with X > q the researcher’s distance choice will be interior, (iii) on intervals with X ∈ [4 q, q ]the researcher’s distance choice may be interior or on the boundary, and (iv) anyinterior choice has to satisfy X − d > q and d < q . It remains to show that the values, U R of d = X/ d ( X ) < X/ d ( X )solving the first-order condition of d ( X ) and ρ i ( d i , X ) chosen optimally cross onlyonce. We use three observations to show this.1. First, at the area length X for which U R ( d ; X ) = U R ( d ; X ), the payoff at theboundary must be decreasing faster than the payoff in the interior as the firstswitch is from the boundary solution to the interior solution by continuousdifferentiability of all terms and the observation from above that d ( X ) = X/ X < q .2. Second, on the interval [4 q, q ] the payoff of the boundary solution has astrictly lower second derivative with respect to X for all X than the interiorsolution. Hence, the two values can cross at most once on this interval.3. Third, the value of the boundary solution is bounded from above by the valueof the interior solution for all X ≥ q .The first observation is immediate.For the second observation follows from totally differentiating U R for the twotypes of local maxima. Using envelope conditions we obtain that the payoff isconcave in the boundary solution and convex in the interior solution which impliesthe second observation. Define ϕ ( X ) := max ρ u ( d = X/ , ρ, X ) for the boundary, we From Lemma 2, 4 and 6 any interior choice that maximizes V (ignoring cost) satisfies X − d > q and d < q . ϕ ( X ) is concave. In Lemma 28 in appendix Dwe in turn show that max ρ,d u ( d, ρ, X ) is convex in X provided that the maximizer d ( X ) < X/ X → ∞ , V ( d, X ) converges to V ( d, ∞ )and σ ( d, X ) to σ ( d, ∞ ) and the researcher’s optimization on the deepening intervalconverges to the optimization on the expanding interval which has a unique andinterior maximum at ( d ∞ , ρ ∞ ). In particular, if such an interior optimum exists, theenvelope condition implies that U R ( X ) = ρV X ( d, X ) − η ˜ c ( ρ ) σ X ( d, X ) < V X ( d, X ) < X > q and X − d > q and σ X ( d, X ) > X . Because thepayoff is continuous in X it follows that ˜ X > ˇ X where ˜ X denotes the first intervallength such that the interior value with d < X/ d = X/ Lemma 16. lim X & ˜ X ρ ( X ) > lim X % ˜ X ρ ( X ) . Proof.
At ˜ X there are two solutions to d ( X ), a boundary solution d b = X/ d i < X/
2. Each comes with an associate ρ d , ρ i respectively. Weshow that ρ i ( X ) > ρ d ( X ) which proves the claim as ρ ( X ) is continuous in X if d ( X )is continuous in X which is true by the first-order conditions.By the definition of ˜ X and continuity of U R we have that U R ( d b ; ˜ X ) = U R ( d i ; ˜ X ) ρ b V ( d b ; ˜ X ) − η ˜ c ( ρ b ) σ ( d b ; ˜ X ) = ρ i V ( d i ; ˜ X ) − η ˜ c ( ρ i ) σ ( d i ; ˜ X ) . (7)Taking the first-order conditions w.r.t. ρ we get V ( d b ; ˜ X ) = η ˜ c ρ ( ρ b ) σ ( d b ; ˜ X ) V ( d i ; ˜ X ) = η ˜ c ρ ( ρ i ) σ ( d i ; ˜ X )Replacing V ( · ) accordingly in equation (7) and dividing by η we obtain σ ( d b ; ˜ X ) ρ b ˜ c ρ ( ρ b ) − ˜ c ( ρ b ) ρ b ! = σ ( d i ; ˜ X ) ρ i ˜ c ρ ( ρ i ) − ˜ c ( ρ i ) ρ i ! Or equivalently σ ( d i ; ˜ X ) σ ( d b ; ˜ X ) = ρ b (cid:16) ˜ c ρ ( ρ b ) − ˜ c ( ρ b ) ρ b (cid:17) ρ i (cid:16) ˜ c ρ ( ρ i ) − ˜ c ( ρ i ) ρ i (cid:17) . We use & ( % ) to describe the one-sided limit "from above" ("from below"). d b > d i ⇒ σ ( d b ; ˜ X ) > σ ( d i ; ˜ X ) and ρ (cid:16) ˜ c ρ ( ρ ) − ˜ c ( ρ ) ρ (cid:17) increasing in ρ by theproperties of the error function it follows that ρ i > ρ b which proves the claim. Lemma 17.
The researcher’s value U R ( X ) is single-peaked in X .Proof. Follows from the value being decreasing for the interior solution and thesingle-peakedness of the boundary value (increasing for
X < q and concave on X ∈ [4 q, q ]). Lemma 18.
Suppose d = X/ is optimal for a range [ X, X ] . Then the optimal ρ ( X ) is single peaked in that range. It is highest at ˙ X = ( π ) √ Proof.
By Lemma 13 we know that d V ( d = X/ X )d X ≥ X > ˆ X .Moreover, recall σ ( d = X/ X ) = X/
4. The first-order condition with respect to ρ becomes V ( X/ X ) X = η c ρ ( ρ ) , With V ( X/ X ) X = X q − X> q ( X − q ) / √ X q . The latter is continuous and concave. Since ˜ c ( ρ ) is an increasing, twice continu-ously differentiable and convex function, ρ increases in X if and only if V ( X/ X ) /X increases in X . By concavity of V ( X/ X ) /X that implies single peakedness.Thus, ˙ X is independent of η and given by ˙ X = ( π ) √ ≈ . q. Lemma 19. lim X & ˆ X ρ ( X ) > ρ ∞ and ˆ X decreases in η .Proof. We begin to show that the first claim holds if ˆ
X < q , then we show thesecond claim which together with the observation that ˆ X < q is sufficient to provethe lemma.At ˆ X we have U R ( ˆ X ) = U R ( ∞ ) ρ ( ˆ X ) V ( ˆ X/
2; ˆ X ) − η ˜ c ( ρ ( ˆ X )) ˆ X ρ ∞ V ( d ∞ ; ∞ ) − η ˜ c ( ρ ∞ ) d ∞ . (8)where the fact that d ( ˆ X ) = ˆ X/ V ( d ∞ ; ∞ ) = η ˜ c ρ ( ρ ∞ ) d ∞ (FOC ρ ∞ ) V ( ˆ X/
2; ˆ X/
2) = η ˜ c ρ ( ρ ( ˆ X )) ˆ X ρ ˆ X ) Step 1: ρ ∞ < ρ ( ˆ X ) if ˆ X < q . Using (FOC ρ ∞ ) and (FOC ρ ˆ X ) we obtain thatby the properties of the error function ρ ( ˆ X ) > ρ ∞ if and only if43 V ( ˆ X/
2; ˆ X/ X > V ( d ∞ ; ∞ ) d ∞ . Case 1: ˆ X > q . Substituting for the V ( · )’s the above becomes ˆ X q − q ( ˆ X − q ) / q ˆ X > − d ∞ q ⇔ d ∞ + 2 ˆ X − X − q ) / q ˆ X | {z } < ( ˆ X − q ) > q A sufficient condition for the above to hold is thus that d ∞ − X + 10 q > d ∞ > q by Lemma 12 we obtain that sufficient condition for ρ ( ˆ X ) > ρ ∞ is that ˆ X < q . Case 2: ˆ X ∈ (2 q, q ] . Performing the same steps only assuming that ˆ X ∈ [2 q, q ]we ˆ X q > − d ∞ q ⇔ X > q − d ∞ > q which implies the desired result. Case 3: ˆ X < q We show that case 3 never occurs, that is ˆ
X > q . To do sowe compare U R ( d = 2 q ; ∞ ) with U R ( d = 1 q ; X = 2 q ) and show that the former isalways larger. Hence X = 2 q < ˆ X for any η .For X = d = 2 q we have that ˆ X q = 1 − d q , and thus ρ ( X = 2 q ) = ρ ( d ; ∞ ) = ρ (cf. case 2). Moreover we have that V (1 q ; 2 q ) = q/ V (2 q ; ∞ ) = 4 / q, and (FOC ρ X ) implies 4 V (1 q ; 2 q ) / q = 2 / η ˜ c ρ ( ρ )Since ˜ c ρ ( ρ ) > ˜ c ( ρ ) /ρ for any ρ > η ˜ c ( ρ ) /ρ < / Since ˆ X ≤ ˇ X ≤ q that case is irrelevant. U R ( d = 2 q ; ∞ ) − U R ( X = 2 q ) ρ q − η ˜ c ( ρ ) − ρ q η ˜ c ( ρ ) q q (cid:18) ρ − η ˜ c ( ρ ) (cid:19) , which is positive whenever η ˜ c ( ρ ) /ρ < / U R ( d = 2 q ; ∞ ) > U R ( X = 2 q ) and therefore ˆ X < q . Step 2: If ρ ∞ < ρ ( ˆ X ) then ˆ X decreases in η . Using (FOC ρ ∞ ) and (FOC ρ ˆ X ) to replace the V ( · )’s in equation (8) and dividingby η we obtain d ∞ ( ρ ∞ ˜ c ρ ( ρ ∞ ) − ˜ c ( ρ ∞ )) = ˜ X/ (cid:16) ρ ( ˆ X )˜ c ρ ( ρ ( ˆ X )) − ˜ c ( ρ ( ˆ X )) (cid:17) from which we get ˆ X/ d ∞ ( ρ ∞ ˜ c ρ ( ρ ∞ ) − ˜ c ( ρ ∞ )) (cid:16) ρ ( ˆ X )˜ c ρ ( ρ ( ˆ X )) − ˜ c ( ρ ( ˆ X )) (cid:17) . Now we use the envelope theorem to calculate ∂U R ( ˆ X ) − U R ( ∞ )) ∂η = ˜ c ( ρ ( ˆ X )) ˆ X − ˜ c ( ρ ∞ ) d ∞ . Replacing for ˆ X implies that the RHS is positive if and only if(˜ c ( ρ ∞ )) − ˜ c ( ρ ( ˆ X )) ρ ∞ ˜ c ρ ( ρ ∞ ) − ˜ c ( ρ ∞ ) ρ ( ˆ X )˜ c ρ ( ρ ( ˆ X )) − ˜ c ( ρ ( ˆ X )) > . Using that ρ ˜ c ρ ( ρ ) > ˜ c ( ρ ) by the properties of the error function and factoringout he denominator of the first term, the above holds if and only if˜ c ( ρ ∞ ) ρ ∞ ˜ c ρ ( ρ ( ˆ X )) − ˜ c ( ρ ( ˆ X ) ρ ∞ ˜ c ρ ( ρ ∞ ) > ρ ( ˆ X )˜ c ρ ( ρ ( ˆ X ))˜ c ( ρ ( ˆ X )) > ρ ∞ ˜ c ρ ( ρ ∞ )˜ c ( ρ ∞ )which holds if and only if ρ ( ˆ X ) > ρ ∞ by the properties of the error function. Thus,ˆ X decreases if ρ ( ˆ X ) > ρ ∞ . Conclusion:
Since ˆ X ∈ [2 q, q ], ⇒ ρ ∞ < ρ ( ˆ X ) ⇒ ˆ X is decreasing in η . Lemma 20. ˆ X ≤ ˙ X < ˇ X ≤ ˜ X Proof.
Step 1: ˇ X > ˙ X . By the envelope theorem we need for X = ˇ X∂U R ( ˇ X ) ∂X = ρ d V ( d = ˇ X/
2; ˇ X )d X − η c ( ρ ) = 0 . (9)45he FOC for ρ implies V ˇ X = η c ρ ( ρ )Now assume for a contradiction that ρ ( ˇ X ) is increasing, then by Claim 11 V ( · ) / ˇ X must be increasing which holds if and only ifd V ( d = ˇ X/
2; ˇ X )d X ˇ X > V ( d = ˇ X/
2; ˇ X ) . But then we obtain the following contradiction to U R ( ˇ X ) being maximald V ( d = ˇ X/
2; ˇ X )d X > V ( d = ˇ X/
2; ˇ X )ˇ X = η c ρ ( ρ ) > η c ( ρ ) ρ . The first inequality follow because V ( d = ˇ X/
2; ˇ X ) / ˇ X must be increasing, the equal-ity follows by equation (9). The last inequality is a consequence of the properties of erf . By Lemma 18, ρ ( X ) is single peaked which proves the claim. Step 2: Ordering.
By Lemma 19 we know that ˆ
X < ˆ X . Thus ˆ X < ˙ X ⇒ ˆ X < ˙ X . Moreover, ˜ X > ˇ X by Lemma 15 which concludes the proof. B.6 Proof of Corollary 4
Proof.
The last claim follows from Lemma 18, and the first bullet point of the thirdclaim follows from Lemma 19.
Lemma 21. ∂ ˜ X/∂η < .Proof. To see hat ˜ X is decreasing in η we use the same arguments as those in step2 of the proof of equation (8). WE provide the steps again for completeness.Recall our notations ( ρ b , d b ) for the boundary solution at ˜ X and ( ρ i , d i ) for theinterior solution at ˜ X . Using the first order conditions with respect to ρ we obtain ησ ( d i ; ˜ X ) (cid:16) ρ b ˜ c ρ ( ρ b ) − ˜ c ( ρ b ) (cid:17) = ( ˜ X/ η (cid:16) ρ i )˜ c ρ ( ρ i )) − ˜ c ( ρ i )) (cid:17) from which we get ˜ X/ σ ( d i ; ˜ X ) (cid:16) ρ b ˜ c ρ ( ρ b ) − ˜ c ( ρ b ) (cid:17) ( ρ i )˜ c ρ ( ρ i )) − ˜ c ( ρ i ))) . Now we use the envelope theorem to calculate ∂U R ( d b ; ˜ X ) − U R ( d i ; ˜ X ) ∂η = ˜ c ( ρ ( ˆ X )) ˜ X − ˜ c ( ρ ∞ ) σ ( d i ; ˜ X ) . Replacing for ˜ X/ ˜ c ( ρ i ) (cid:17) − ˜ c ( ρ b ) ρ i ˜ c ρ ( ρ i ) − ˜ c ( ρ i ) ρ b ˜ c ρ ( ρ b ) − ˜ c ( ρ b ) > . Using that ρ ˜ c ρ ( ρ ) > ˜ c ( ρ ) by the properties of the error function and factoringout he denominator of the first term, the above holds if and only if˜ c ( ρ i ) ρ i ˜ c ρ ( ρ b ) − ˜ c ( ρ b ρ i ˜ c ρ ( ρ i ) > ρ b ˜ c ρ ( ρ b )˜ c ( ρ b ) > ρ i ˜ c ρ ( ρ i )˜ c ( ρ i )which holds if and only if ρ b > ρ ∞ by the properties of the error function. Thus, ˆ X decreases if ρ b > ρ i . Lemma 22. ∂ ˇ X/∂η < .Proof. Because ˇ X is a maximizer of U R ( X ) we know that ∂U R ( ˇ X ) ∂X = ρ ( ˇ X ) d V ( ˇ X/
2; ˇ X )d X − η ˜ c ( ρ ) d σ ( ˇ X/
2; ˇ X )d X = 0 . (10)Taking derivatives with respect to η∂U R ( ˇ X ) ∂X∂η = ∂ ˇ X∂η ∂U R ( ˇ X ) ∂X∂X + ∂ρ ( X ) ∂η ∂U R ( ˇ X ) ∂X∂ρ − ˜ c ( ρ ( ˇ X )) d σ ( ˇ X/
2; ˇ X )d X = 0 , or equivalently ∂ ˇ X∂η ∂U R ( ˇ X ) ∂X∂X = − ∂ρ ( ˇ X ) ∂η ∂U R ( ˇ X ) ∂X∂ρ + ˜ c ( ρ ( ˇ X )) d σ ( ˇ X/
2; ˇ X )d X . (11)(FOC ρ X ) implies that for a given XF OC ρ ( η, ρ ) := V ( ˇ X/
2; ˇ X ) /σ ( ˇ X/
2; ˇ X ) − η ˜ c ρ ( ρ ( ˇ X )) = 0which in turn implies via the IFT that ∂ρ ( X ) ∂η = − ∂F OC ρ ∂η∂F OC ρ ∂ρ = − ˜ c ρ ( ρ ( X ))˜ c ρρ ( ρ ( X )) η . Observe that ∂U R ( ˇ X ) ∂X∂ρ = d V ( ˇ X/
2; ˇ X )d ˇ X − η ˜ c ρ ( ρ ( ˇ X ) d σ ( ˇ X/
2; ˇ X )d X , and after replacing d V ( ˇ X/
2; ˇ X )d X using equation (10) ∂U R ( ˇ X ) ∂X∂ρ = η ˜ c ( ρ ( ˇ X )) ρ ( ˇ X ) − ˜ c ρ ( ρ ( ˇ X )) ! d σ ( ˇ X/
2; ˇ X )d X . ∂ ˇ X∂η ∂U R ( ˇ X ) ∂X∂X = ˜ c ( ρ ( ˇ X )) ρ ( ˇ X ) − ˜ c ρ ( ρ ( ˇ X )) ! ˜ c ρ ( ρ ( ˇ X ))˜ c ρρ ( ρ ( ˇ X )) + ˜ c ( ρ ( ˇ X )) ! d σ ( ˇ X/
2; ˇ X )d X .
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C Science Funding
In this part, we disccus the model of science funding from Section 6 in greater detail.
Setup.
The setting is the same as in Section 6. For convenience we repeat thediscussion here. We take the simplest possible setup within our modeling framework.Suppose there is only one known question-answer pair, F = ( x , y ( x )). In sucha case, the researcher—absent any funding incentives—solves the following specialcase of the problem in Section 5:max d,ρ ρV ( d ; ∞ ) − η ˜ c ( ρ ) σ ( d ; ∞ ) , with ˜ c ( ρ ) = erf − ( ρ ) .Now assume that the funder has a fixed budget K to invest in the researcherand has two funding technologies: ex ante cost reductions, h , and ex post rewards, ζ . Marginal cost of both are constant, and the cost ratio κ is such that the funder’sbudget constraint is given as follows: K = ζ + κh From the researcher’s perspective, a cost reduction of h implies that she faces costparameter η ≡ η − h . An ex post reward gives the researcher additional utility of ζ if she finds an answer. Since research prizes are typically competitive, we assumethat the researcher obtains the reward more likely if her contribution is seminal;otherwise the amount goes to some other (unmodeled) recipient. The probability ofobtaining the prize positively correlates with the difficulty of solving the problemand is bounded. For the sake of concreteness, we assume the correlation is linear invariance: f ( σ ) = σ s if σ < s s > We first consider two polar objectives of the funder: maximizing output andmaximizing novelty. Then we determine the set of ( ρ, d ) combinations the fundercan achieve. At the end, we consider what happens as prior knowledge expands. Economically, this assumption means that either the cost structure of the two instrumentsis linear everywhere or it can be interpreted as implicitly assuming that the funder’s budget, K ,is sufficiently small such that a linear cost structure is a decent approximation of the fundingopportunities. The crucial assumption here is that f is a bounded function, which is true whenever f isindeed a probability. If f instead was unbounded, the researcher would naturally (for any ζ > d = ∞ . The specific form, in turn, is chosen only for ease of computation. . . . . . . . η ζ . . . . . . η ζ Figure 14:
Iso- ρ and iso- d curves with budget lines Dashed lines represent the budget lines for κ — K < K for the lower budget and
K > K for the higher budget.The left panel shows the iso- ρ curves. An output-maximizing funder follows the followinginvestment rule: for K < K , invest all of the budget in increasing ζ , z = K, h = 0; for thelarger budget (
K > K ), invest all of the budget in decreasing η , z = 0 , h = K/κ .The right panel shows the same budget lines but with iso- d curves for three levels of d . Thesecurves are concave and flatter than the iso- ρ curves. Consequently, a novelty-maximizing funder can invest entirely in increasing ζ in situations in which an output-maximizingfunder invests entirely in decreasing η . Note: the x -axis moves in reverse order to plot decreases in the cost-function parametercompared to the base level η = 2. After observing the funding choices, the researcher’s problem becomes the fol-lowing: max d,ρ ρ ( V ( d ; ∞ ) + f ( σ ( d ; ∞ )) ζ ) − η ˜ c ( ρ ) σ ( d ; X )With our framework, we can straightforwardly apply basic consumer theory tostudy the funder’s choices. Example . The funder’s objective is to maximize the re-searcher’s output: max h,ζ ρ Proposition 5.
For an output-maximizing funder, it is optimal to focus exclusivelyon either of the two funding options. Moreover, there is a cutoff budget K such thatthe funder invests exclusively in ex post rewards if K < K and exclusively in ex antecost reductions if
K > K .Proof.
For the proof we characterise the “marginal rate of substitution” for ρ betweenthe two funding options. That is, the (negative) of the ratio of the marginal effectsof a an increase in η and ζ on ρ , M RS ρζη := − ∂ρ∂η∂ρ∂ζ .
52h MRS defines the slope of the funder’s indifference curves. Thus we can use hesimple economics from consumer theory to determine the funder’s optimal decision.Throughout, observe that since X = ∞ we have that σ d ( d ; ∞ ) = 1 and σ dd ( d ; ∞ ) = 0. As we restrict attention to small K (or large s ) we implicitlyassume that d ( ζ, η ) < q and hence V d = 1 − d/ (3 q ) and V dd = − / (3 q ).In Lemma 29 in appendix D on page 64 we show that M RS ρζη = 2˜ c ρ ( ρ ) − ˜ c ( ρ ) /ρ. The marginal rate of substitution is a positive, increasing and convex function of ρ . The properties of the iso- ρ curves follow immediately. As it only depends on ρ , itis immediate that along an iso- ρ curve M RS ρ is constant.To maximize output, the funder chooses the highest iso- ρ curve on the budgetline. Note that the steepest iso- ρ curve is the one with the highest ρ . Note thathigher iso- ρ levels occur for higher ζ and lower η . Hence, it is sufficient to checkwhether the slope at one of the corner solutions is steeper or flatter than the costratio of η and ζ . Consider a corner solution in which all investment goes into ζ andthe corresponding level of ζ . If M RS ρ < κ , then this is the optimal funding scheme.If M RS ρ > κ , then the optimal funding scheme invests only into ζ . Clearly, as thebudget increases, the highest level of ρ that can be implemented increases. Thus, the M RS ρ at the optimal funding scheme increases and a switch from a corner solutionwith only ζ to one with only η might occur.The intuition behind Proposition 5 builds on the properties of iso- ρ curves in( ζ, η ) space as depicted in Figure 14. Like the consumer in a consumer problem, thefunder chooses ζ and h (or equivalently η ≡ η − h ), which together imply a certain ρ . An iso- ρ curve depicts all choices ( ζ, η ) that induce the same ρ . The slope of theiso- ρ curves is described by the marginal rate of substitution (MRS) between ζ and η : M RS ρζη ( ρ ) = s (2˜ c ρ ( ρ ) − ˜ c ( ρ ) /ρ )From the MRS and the properties of the inverse error function it is apparent thatthe slope of the iso- ρ curves is (i) linear with (ii) an increasing slope in ρ . For a givenrelative price ( κ ), the funder’s solution is therefore generically a corner solution.Moreover, the slope determines whether it should focus on ex ante incentives or expost incentives. As the slope changes with the output level ρ , so may the optimalfunding scheme. If that ρ level is large, the iso- ρ curve is steep and ex ante costreductions are optimal. If the ρ level is small, the iso- ρ curve is flat and ex postrewards are optimal. The highest level of ρ , and thus the type of the corner solution,depends on the available budget ( K ).Intuitively, at low levels of ρ , the cost effect—which is convex—is not yet dominantand increasing the benefit of a discovery is more effective in incentivizing output.However, as ρ increases, the cost effect becomes more pronounced and weakening itwith decreases in η becomes more effective instead. Example . The funder’s objective is to maximize the novelty Note that with ˜ c ρ , we denote the derivative of ˜ c with respect to ρ to simplify notation.
53f the researcher’s question of choice: max h,ζ d Proposition 6.
For a novelty-maximizing funder, it is optimal to focus on either ofthe two funding options. Moreover, there is a threshold s < . such that whenever s > s and an output-maximizing funder finds it optimal to focus on ex post rewards,so does a novelty-maximizing funder. The reverse, however, does not hold.Proof. Using the implicit function theorem results from the proof of Lemma 29above, we obtain for the marginal rate of substitution between ζ and η on theexpanding interval by substituting first-oder conditions M RS dζη = ˜ c ρ ˜ c/ρ − ˜ c ρ + ˜ c ˜ c ρ ˜ c ρρ ˜ c/ρ − ˜ c ρ + ρ ˜ c ρρ . This is a positive, increasing and convex function in ρ . Moreover, we obtain that M RS dζη > M RS ρζη as the difference reduces to˜ c ρ − ˜ c/ρ + ˜ c ρρ ρ ˜ c ρ − ˜ c ˜ c/ρ − ˜ c ρ + ρ ˜ c ρρ > c ρ > c/ρ and the denominator being positive.Hence, moving along an iso- d curve ρ is decreasing and therefore the M RS d isdecreasing along the iso- d curve which is concave. Thus, a corner solution maximizes d on a given budget line.Whenever M RS ρ < κ , it follows from M RS ρ > M RS d that M RS d < κ . Hence,if the corner solution for ρ uses only ex post rewards so does the corner solution for d . The reverse is not true as Figure 14 illustrates.The iso- d curves are concave and flatter than the iso- ρ curve crossing at eachpoint. Their slope is M RS dζη ( ρ ) = ˜ c ρ ( ρ ) ˜ c ( ρ ) /ρ − ˜ c ρ ( ρ ) + ˜ c ( ρ )˜ c ρ ( ρ ) ˜ c ρρ ( ρ ) ˜ c ( ρ ) /ρ − ˜ c ρ ( ρ ) + ρ ˜ c ρρ ( ρ ) , which can be shown to be smaller than M RS ρζη for any s > . Thus, if at anypoint it is optimal for the output-maximizing funder to invest more in increasing ζ ,the novelty-maximizing funder would follow the same strategy. The reverse need notbe true. The right panel of Figure 14 depicts the iso- d curves alongside the samebudget lines as those in the left panel. Thus, the two funder types are observationallyequivalent for small budgets, but it is possible to tell them apart by observing theirbehavior once the budget grows larger. Note that the pictures in Figure 14 are plotted on a reverse x -axis. eneral funding objectives. In reality, funding may have many different ob-jectives and can range from maximizing the externalities of research to answeringparticular questions. A demand-side discussion of research is beyond the scope ofthis paper. Instead, we characterize the feasible set of choices ( d, ρ ) that a fundercan induce with a given budget. Computing this set provides a useful tool to analyzethe optimal funding scheme given a particular preference relation over ( d, ρ ) bundles.As in a standard consumer problem, it can be readily applied, with this set beingthe analogue of a budget set. We restate (and prove) the Proposition 4 from themain text
Proposition (Identical to Proposition 4 from page 27) . The set of implementable ( d, ρ ) -combinations for a given cost ratio κ and a budget K is described by the ( d, ρ ) implementation frontier defined over [ ρ, ρ ] , which are the endogenous upper andlower bounds of ρ . These bounds are determined by the extreme funding schemes ( ζ = 0 , η = η − K/κ ) and ( ζ = K, η = η ) . The research-possibility frontier betweenthose polar points is as follows: d ( ρ ; K ) = 6 q ( K + s − κη ) ρ ˜ c ρ ( ρ ) − ˜ c ( ρ )2 sρ ˜ c ρ ( ρ ) − s ˜ c ( ρ ) − κρ (12) d ( ρ ; K ) can be increasing or decreasing, depending on whether d and ρ behave assubstitutes or complements at the ( ζ, η ) mix inducing the current level of ( d, ρ ) .Proof. Deriving the Research Possibility Frontier:
Using the two first orderconditions of the researcher and solving for ζ and η we obtain η = d q ρρ ˜ c ρ − ˜ cζ = d q − d q ˜ cρ ˜ c ρ − ˜ c ! s. (13)Using the calculated M RS ρζη and
M RS dζη we observe that any ( ρ, d ) can at most beimplemented through one ( ζ, η ) combination because each iso- ρ curve crosses eachiso- d curve at most once: both slopes are positive and the slope of the iso- ρ curvesis steeper throughout than the slope of the iso- d curves.Given budget K ζ ∈ [0 , K ], and η = η − h ∈ [ˇ η, η ] where ˇ η = η − K/κ .Moreover the budget line is K = κh + ζ .The polar solutions induce ( ζ = 0 , ˇ η ) is a direct application of Proposition 3.More generally, we can plug conditions (13) into the the budget line and rearrangeto obtain d ( ρ ) = 6 q ( K + s − κη ) ρ ˜ c ρ ( ρ ) − ˜ c ( ρ )2 sρ ˜ c ρ ( ρ ) − s ˜ c ( ρ ) − κρ . (14)our research possibility frontier. Deriving the Boundary ρ ’s on the Research Possibility Frontier The firstterm in brackets, K + s − κη can be both positive or negative depending on thechosen parameters. Observe, however, that the minimum attainable cost factor55 η = η − K/κ . Thus we can rewrite K + s − kη = s (1 − ˇ ηκ ) . Replacing ˇ η = 1 / (2˜ c ρ ( ˇ ρ ) − ˜ c ( ˇ ρ ) / ˇ ρ which is the solution of equation (5) from the proofof Lemma 12 performing the same steps as when solving for equation (6) (which ispossible because ζ = 0 in that case) that implies K + s − κη = s − κs (2˜ c ρ ( ˇ ρ ) − ˜ c ( ˇ ρ ) / ˇ ρ ) | {z } = MRS ρζη (ˇ ρ ) = s M RS ρζη ( ˇ ρ ) − κM RS ρζη ( ˇ ρ ) ! . Thus, K + s > κη ⇔ M RS ρζη ( ˇ ρ ) > κ. Now consider the last term, ( ρ ˜ c ρ ( ρ ) − ˜ c ( ρ )) / (2 sρ ˜ c ρ ( ρ ) − s ˜ c ( ρ ) − κρ ). It is positiveif and only if 2 sρ ˜ c ρ ( ρ ) − s ˜ c ( ρ ) > κρκ < s ρ ˜ c ρ ( ρ ) − ˜ c ( ρ ) ρ !| {z } = MRS ρζη . If M RS ρζη ( ˇ ρ ) > κ then any iso- ρ curve that crosses the budgetline must satisfy ρ > ˇ ρ and thus M RS ρζη ( rho ) > κ by Proposition 5. Moreover, the largest implementable ρ on the research possibility frontier comes from the polar case ζ = K, η = η .Similarily if M RS ρζη ( ˇ ρ ) < κ then all iso- ρ curves that cross the budgetline mustsatisfy ρ < ˇ ρ and the lowest implementable ρ on the research possibility frontiercomes from the polar case ζ = K, η = η Deriving the slope of the Research Possibility Frontier:
Let n ( ρ ) be thenumerator of the last term and dn ( ρ ) the denominator. Then, the last term isincreasing in ρ if and only if n ( ρ ) dn ( ρ ) > n ( ρ ) dn ( ρ )or equivalently using that n ( ρ ) = ρ ˜ c ρρ ( ρ ) > dn ( ρ ) = s (2 ρ ˜ c ρρ + ˜ c ρ ) − κ if andonly if κs < ˜ c ρ ( ρ )˜ c ( ρ ) + ρ ˜ c ( ρ )˜ c ρρ ( ρ ) − ρ (˜ c ρ ( ρ )) ˜ c ρρ ( ρ ) ρ − ρ ˜ c ρ ( ρ ) + ˜ c ( ρ ) | {z } = MRS dζη ( ρ ) . Thus d ( ρ ) is increasing if and only if ( M RS ρζη ( ˇ ρ ) − κ )( M RS dηζ ( ρ ) − κ/s ) > . While the expression in Equation (12) looks cumbersome at first sight, Proposi-tion 5 and Proposition 6 give guidance on its interpretation. Any change in ρ can56 . . . . . . . . . . . . . ρ d . . . . . . . . . . . . . ρ d Figure 15:
Feasible set for different reward standards
The shaded area shows the implementable ( ρ, d ) combinations of a funder for a givenbudget K . All points on a solid line require the same amount of funding, K . In bothpanels, the funder has a budget of K = 10, the price ratio is κ = 6, and the baseline costfactor is η = 2. The status quo parameter has value q = 1. In the left panel, the rewardtechnology is parameterized with s = 40; in the right panel, it is s = 200. only be implemented through a movement along the ( ζ, η ) budget line. Whether thefunding scheme has to increase rewards and reduce grants to increase the successprobabilities is determined by the MRS of ρ between ζ and η relative to the costratio κ . In the proof of Proposition 4, we show that K − κη + s has the same signas s c ρ ( ˇ ρ ) − ˜ c ( ˇ ρ )ˇ ρ !| {z } = MRS ρζη (ˇ ρ ) − κ, where ˇ ρ is the researcher’s choice of ρ given the polar case ( ζ = 0 , η = η − K/κ ).Moreover, the denominator in Equation (12) is equivalent to ρ ( M RS ρζη ( ρ ) − κ ).Thus, an equivalent formulation of equation (12) is d ( ρ ) = M RS ρζη ( ˇ ρ ) − κM RS ρζη ( ˇ ρ ) 1 M RS ρζη ( ρ ) − κ ˜ c ρ ( ρ ) − ˜ c ( ρ ) ρ ! M, where M > ρ curves that cross the budget line have to be in the same relationto the budget line’s slope, κ . Thus, if M RS ρζη ( ˇ ρ ) > κ then M RS ρζη ( ρ ) > κ andvice versa. For the special case M RS ρζη ( ˇ ρ ) = κ , the research-possibility frontier is avertical line because in that case, the slope of the budget line, κ , is precisely thatof the M RS ρζη ( ρ ), which implies that moving along the budget line is the same asmoving along the iso- ρ curve for all ( ζ, η ) combinations. It is possible to implementvarious levels of distance, d , while ρ remains constant.The slope of the research-possibility frontier can be positive or negative. If theslope is positive, changing the allocation of funds to increase output, ρ , implies anincrease in novelty, d , as the two are complements from the funder’s perspective. If57he slope is negative, an increase in output implies a decrease in novelty, as the twoare substitutes from the funder’s perspective.The slope of the research frontier is positive and finite if and only if K + s − κη | {z } = s MRSρζη (ˇ ρ ) − κMRSρζη (ˇ ρ ) ˜ c ρ ( ρ )˜ c ( ρ ) + ρ ˜ c ( ρ )˜ c ρρ ( ρ ) − ρ (˜ c ρ ( ρ )) ˜ c ρρ ( ρ ) ρ − ρ ˜ c ρ ( ρ ) + ˜ c ( ρ ) | {z } = MRS dζη ( ρ ) − κ/s > . Note that both
M RS dζη ( ρ ) and M RS ρζη ( ρ ) are increasing in ρ by the properties ofthe inverse error function.To illustrate, consider the two scenarios from Figure 15. Given the parameters, K + s > κη for both levels of s . In the left panel ( s = 40), novelty and output aresubstitutes as long as ρ < ˜ ρ ≈ M RS dζη crosses the threshold κ/s = 0 .
15. As s increases to 200, that threshold decreases to κ/s = 0 .
03 and˜ ρ ≈ ρ ≈ s = 40, the same funder would choosea much safer strategy when s = 200 ( ρ ≈ Distance and success probabilities become complements because an increasein ρ has two effects: (i) it increases the marginal benefit of distance, V d ( d, X ) + ζ/sσ d ( d, X ), by increasing the probability that the researcher finds an answer, and(ii) it increases the marginal cost of distance, η ˜ c ( ρ ) σ d . The uncertainty of theconjecture about questions, σ ( d, X ), is increasing in distance. Moreover, for anydistance, the interval that has to be covered to find an answer with probability ρ isincreasing in this probability. For certain parameter constellations, it may be that,for example, an increase in ζ increases the weight placed on the marginal-benefiteffect relatively more than the resulting increase in η increases the marginal-costeffect. Straightforwardly, the larger s , the smaller the effect on the marginal benefit. Extensive margin.
By assumption, the statements in Proposition 5, Proposi-tion 6, and Proposition 4 refer to the intensive margin of ρ and d within the(unbounded) area outside the knowledge frontier. As we have seen in Section 5, sucha focus covers only part of the researcher’s decision. We conclude this applicationby illustrating how the funding architecture affects the extensive margin. To thatend, we consider F such that X = ˆ X . In that case, the researcher is indifferentbetween expanding knowledge and deepening knowledge. She chooses the midpointˆ X/
2, given the funding architecture ( η = η , ζ = 0). We can straightforwardlycompute the MRS between rewards and grants that keeps the researcher indifferentbetween the two intervals. Comparing this to the relative prices of rewards andgrants, 1 /κ , determines how a funding institution can incentivize a shift towardexpanding knowledge by shifting more funds either toward rewards or toward grants. We assume in this exercise that s is a parameter to the funder. Under most funding schemes,the winners of a prize are determined by a jury of peers rather than the funder itself. Thus, thestandards may not be entirely under the funder’s control. M RS ˆ Xζη = ˜ c ( ρ ∞ ) /ρ ∞ s ˜ cρ ( ρ ∞ )˜ c ( ρ ∞ ) /ρ ∞ − ˜ cρ ( ρ ˆ X )˜ c ( ρ ˆ X ) /ρ ˆ X − − ˜ c ρ ( ρ ∞ ) − ˜ c ( ρ ∞ ) /ρ ∞ ˜ c ρ ( ρ ˆ X ) − ˜ c ( ρ ˆ X ) /ρ ˆ X − > . Derivation of
M RS ˆ Xζη . Recall that at ˆ X , the researcher is indifferent betweenexpanding and deepening at d = ˆ X/ ρ ∞ ( V ( d ∞ ) + ζs d ∞ ) − η ˜ c ( ρ ∞ ) d ∞ = ρ ˆ X ( V ( ˆ X/ , ˆ X ) + ζs ˆ X − η ˜ c ( ρ ˆ X ) ˆ X ρ -FOC, we obtain ρ ∞ (˜ c ρ ( ρ ∞ ) − ˜ c ( ρ ∞ ) /ρ ∞ ) d ∞ = ρ ˆ X (cid:16) ˜ c ρ ( ρ ˆ X ) − ˜ c ( ρ ˆ X ) /ρ ˆ X (cid:17) ˆ X X d ∞ ρ ∞ ˜ c ρ ( ρ ∞ ) − ˜ c ( ρ ∞ ) ρ ˆ X ˜ c ρ ( ρ ˆ X ) − ˜ c ( ρ ˆ X ) . From the envelope theorem it follows that dU R ( ζ, η ; X ) dζ = ρ X σ ( X ) sdU R ( ζ, η ; X ) dη = − ˜ c ( ρ X ) σ ( X ) . Then, the marginal rate of substitution defined as
M RS ˆ Xζη = − ddη ( U R ( ˆ X ) − U R ( ∞ )) ddζ ( U R ( ˆ X − U R ( ∞ ))is M RS ˆ Xζη = ˜ c ( ρ ˆ X ) ˆ X − ˜ c ( ρ ∞ ) d ∞ ρ ˆ X ˆ X s − ρ ∞ d ∞ s and the expression from the text follows by replacing ˆ X from above and factoring d ∞ and s out. D Omitted Proofs
Here we provide the steps that we have omitted in the proofs because they involvecumbersome algebraic manipulation with little economic or mathematical insight.
Lemma 23. ∂V ( d ; ∞| d> q ) ∂d < . roof. ∂V ( d ; ∞| d > q ) ∂d = − d q + 1 + s d − qd d − q q Letting τ := d/q the statement is negative if3 − τ s τ − τ τ − < τ and converges to 0 as τ → ∞ . Lemma 24. q (cid:16) X − d − ( X − d − q ) q X − d − qX − d (cid:17) > if d ∈ [0 , X − q ] .Proof. We show that the derivative V d is a convex function which is positive at itsminimum on [0 , X − q ] and hence throughout on that domain.The relevant derivatives to consider are V d = 13 q X − d − ( X − d − q ) s X − d − qX − d .V dd = 13 q − √ X − d − q ( X − d ) / (( X − d − q )( X − d ) + ( X − d − q )2 q ) ! .V ddd = 4 q ( X − d ) / ( X − d − q ) / > . where V ddd > X − d ) > X − d − q ) >
0. Itfollows that, V d is strictly convex over the relevant range. The maximal distance inthis range, d = X − q , V d | d = X − q = q − X q > d = 0 or at someinterior d such that V dd = 0. Suppose the minimum is at d = 0, then V d | d =0 = q (cid:18) X − ( X − q ) q X − qX (cid:19) > X − qX < V d attains an interior minimum. In thiscase, V dd = 0 must hold at the minimum and hence q X − d − q ( X − d ) / = ( X − d − q )( X − d ) + ( X − d − q )2 q . The first derivative can be rewritten as V d = 13 q X − d − √ X − d − q ( X − d ) / ( X − d − q )( X − d − q )( X − d ) ! and plugging in for the minimum condition we obtain V d | V dd =0 = 13 q X − d − X − d − q )( X − d − q )( X − d )( X − d − q )( X − d ) + ( X − d − q )2 q ! = 13 q ( X − d )(( X − d − q )( X − d ) + ( X − d − q )2 q ) − X − d − q )( X − d − q )( X − d )( X − d − q )( X − d ) + ( X − d − q )2 q .
60s the denominator and q are both positive, the sign of V d at its minimum isdetermined by the sign of its numerator only. Note that the numerator is increasingin d because its derivative is 2( X − q )( X − d − q ) >
0. Thus, the numerator of thederivative of V d evaluated at the interior minimum d such that V dd = 0 is greaterthan − X ( X − qX + 10 q ) = − X (( X − q ) − q ) > . Lemma 25. V X ( d ( X ); X ) < if X ≥ q and d ∈ [0 , X − q ] .Proof. Observe that for any X ≥ q and d ≤ X − qV Xd = 124 q − s X − dX − d − q − (5( X − d ) + 4 q ) √ X − d − q ( X − d ) / ! . Denote a := X − d , this is an increasing function in a as dV Xd da = 4 q a / ( a − q ) / > . Hence, the highest value of V Xd is attained for a → ∞ andlim a →∞ q − s aa − q | {z } → − a √ a − qa / | {z } → +4 q √ a − qa / | {z } → = 0 . It follows that the V Xd converges to zero from below implying that V Xd <
0. Thus, V X ( d ( X ) , X ) < V X ( d = 0 , X ) and we obtain V X ( d, X | d ≤ q, X − d ≥ q )= 13 q d + ( X − d − q ) s X − d − qX − d − ( X − q ) s X − qX < V ( d = 0 , X | d ≤ q, X − d ≥ q )= 13 q ( X − q ) s X − qX − ( X − q ) s X − qX = 0as desired. Lemma 26. d V ( X/ , X ) / (d X ) < and d V ( d ( X ) , X ) / (d X ) > . roof. Considering the boundary solution we obtain d V ( X/ , X ) dX = − X − qX − q qX / √ X − q + 16 qd V ( X/ , X ) dX = 4 q X / ( X − q ) / > d V ( X/ ,X ) dX ≤ d V (4 q, q ) dX with d V (4 q, q ) dX = − q − q − q q / q / q / + 16 q = − q √ q + 16 q = 8 − / √ q < . Next, consider the value of any interior solution and apply the envelope andimplicit function theorem to obtain dV ( d ( X ) , X ) dX = V X + d ( X ) V d |{z} =0 by optimality of d = V X d V ( d ( X ) , X ) dX = V XX + d ( X ) ( V Xd + V dd d ( X )) | {z } =0 by IFT on FOC + d ( X ) V d |{z} =0 by optimality = V XX ( d ( X ) , X ) . Observing that V XXd ( d, X | d ≤ q, X − d ≥ q ) = 24 q ( X − d ) / ( X − d − q ) / > V XX ( d ( X ) , X ) = 124 q s X − dX − d − q − s XX − q ! + 6 s X − d − qX − d − s X − qX + (cid:18) X − qX (cid:19) / − X − d − qX − d ! / ≥ V XX ( d = 0 , X )= 0implying that d V ( d ( X ) , X ) / (d) ≥ Lemma 27.
Assume X ∈ [4 q, q ] , then d U R ( d = X/ X ) / (d X ) < .Proof. Take the case of the boundary solution: we are analyzing a one-dimensionaloptimization problem with respect to ρ . Denote the objective f ( ρ ; X ) and theoptimal value by ϕ ( X ) = max ρ f ( ρ ; X ). Then, the optimal ρ solves f ρ = 0. We62btain ϕ ( X ) = f ρ |{z} =0 by optimality ρ ( X ) + f X ϕ ( X ) = f ρ |{z} =0 by optimality ρ ( X ) + ( f ρρ ρ ( X ) + f Xρ ) | {z } =0 by total differentiation of FOC ρ ( X ) + f XX = f XX = ρ ( X ) V XX ( X/ X ) < . Hence, the optimal value at the boundary solution is strictly concave as σ XX ( X/ X ) =0 and V XX < Lemma 28.
Let d i < X/ be an local maximum of u r ( rho, d, X ) . If d i ( X ) existson X ∈ [4 q, q ] , then d U R ( d = d i ( X ); X ) / (d X ) > .Proof. The implicit function theorem yields for d ( X ) and ρ ( X ) d ( X ) ρ ( X ) ! = − f dd f ρρ − f ρd f dX f ρρ − f ρX f dρ f ρX f dd − f dX f dρ ! . Note that − f dd f ρρ − f ρd < − det ( H ) and the determinant of the secondprincipal minor being positive is a necessary second order condition for a localmaximum given that the first ( f ρρ ) is negative.Denote the objective f ( ρ, d ; X ) and the optimal value by ϕ ( X ) = max ρ,d f ( d, ρ ; X ).Then, the optimal ( d, ρ ) solves f ρ = 0 and f d = 0. Differentiating the value of theresearcher twice with respect to X yields ϕ ( X ) = f ρ |{z} =0 by optimality ρ ( X ) + f d |{z} =0 by optimality d ( X ) + f X ϕ ( X ) = f ρ |{z} =0 by optimality ρ ( X )+ d ( X ) ( f dX + f dd d ( X ) + f dρ ρ ( X )) | {z } =0 by total differentiation of foc wrt d + ρ ( X ) ( f ρX + f ρd d ( X ) + f ρρ ρ ( X )) | {z } =0 by total differentiation of foc wrt ρ + f dX d ( X ) + f ρX ρ ( X ) + f XX = f dX d ( X ) + f ρX ρ ( X ) + f XX . Observe first that f XX > f XX = ρV XX ( d ; X ) − η ˜ c ( ρ ) σ XX ( d ; X ) and V XX > σ XX ( d ; X ) = − d X . Next, we show f dX d ( X )+ f ρX ρ ( X ) > f ρρ f dd > f ρd . f dX d ( X ) + f ρX ρ ( X ) = − f dX f dX f ρρ − f ρX f dρ f dd f ρρ − f ρd ! − f ρX f ρX f dd − f dX f dρ f dd f ρρ − f ρd ! .
63s we only need the sign of this expression we can ignore the positive denominatorto verify − f dX ( f dX f ρρ − f ρX f dρ ) − f ρX ( f ρX f dd − f dX f dρ ) > f dX f ρρ + f ρX f dd − f dX f ρX f dρ < f dX f ρX f ρρ f dρ + f ρX f dX f dd f ρd > . where we used the signs of the terms that follow because f ρρ = − η ˜ c ρρ ( ρ ) σ ( d ; X ) < f ρX = V X − η ˜ c ρ ( ρ ) σ X ( d ; X ) < V X − η ˜ c ( ρ ) ρ σ X ( d ; X ) < f dρ = V d − η ˜ c ρ ( ρ ) σ d ( d ; X ) < V d − η ˜ c ( ρ ) ρ σ d ( d ; X ) = 0 f dX = ρV dX − η ˜ c ( ρ ) σ dX < f ρρ f dd − f ρd >
0, we can replace f ρρ f dρ with f dρ f dd as f ρρ f dρ > f dρ f dd yielding2 < f dX f ρX f dρ f dd + f ρX f dX f dd f ρd which is true as the right-hand side can be written as g ( a ) = a + a with a = f dX f ρX f dρ f dd > g ( a ) is a strictly convex function for a > a = 1 with g ( a = 1) = 2. Lemma 29.
M RS ρζη = 2˜ c ρ ( ρ ) − ˜ c ( ρ ) /ρ .Proof. For any ( η, ζ ) the system of first-order conditions for a non-boundary choiceis given by V d ( d, ∞ ) + ζσ d ( d, ∞ ) /s = η ˜ c ( ρ ) /ρV ( d, ∞ ) + ζσ d ( d, ∞ ) /sd = η ˜ c ρ ( ρ )For an interior optimal choice of ( d, ρ ), we obtain using σ ( d, X ) = d, σ d ( d, X ) = 164nd σ dd ( d, X ) = 0 dddηdddζdρdηdρdζ = − det ( H ) d (˜ c ρ ( V d + ζ/s − η ˜ c ρ ) + η ˜ c ˜ c ρρ ) − d ( V d + ζ/s − η ˜ c ρ + ρη ˜ c ρρ ) − ρσ ˜ c ρ V dd + ˜ c ( V d + ζ/s − η ˜ c ρ ) − ρ/s ( V d + ζ/s − η ˜ c ρ − dV dd ) . where det ( H ) is the determinant of the Hessian matrix of the objective functionwhich is given by − ησ ˜ c ρρ ρV dd − ( V d + ζ/s − η ˜ c ρ ) > . Note that the determinant of the Hessian matrix for a local maximum is positive asthe Hessian is negative semidefinite and the first principal minor − η ˜ c ρρ σ < It follows that the sign of the derivatives are determined only by the negative ofthe sign of the respective terms in the matrix. Using the first-order conditions torewrite these equations yields dddη = − dηdet ( H ) ˜ c ρ ˜ cρ − ˜ c ρ ! + ˜ c ˜ c ρρ ! < dddζ = dηdet ( H ) ˜ cρ − ˜ c ρ + ρ ˜ c ρρ ! > dρdη = − ρηdet ( H )(2˜ c ρ − ˜ c/ρ ) (˜ c ρ − ˜ c/ρ ) < σ V dd = − d q and from the first-order conditions we knowthat d q = 2 η (˜ c ρ − ˜ c/ρ ). The properties of ˜ c imply that ˜ c ρ > ˜ c/ρ . Finally, dρdζ = ρη/sdet ( H ) (˜ c ρ − ˜ c/ρ ) > We suppress arguments of the functions for readability. In our case, one can actually show that this has to hold given that d < ∞ . Plugging in fromthe first-order conditions yields η (˜ c ρ − ˜ c/ρ )2( ρ ˜ c ρρ − ˜ c ρ + ˜ c/ρ ) > c . dddη < dddζ > dρdη < dρdζ > . Moreover, define we obtain for the marginal rate of substitution between ζ and η on the expanding interval − dρdηdρdζ = M RS ρζη = s (2˜ c ρ − ˜ c/ρc/ρ