Discussion Paper No. 994

REWARDING MEDIOCRITY?
OPTIMAL REGULATION
OF R&D MARKETSWITH REPUTATION CONCERNS

Chia-Hui Chen
Junichiro Ishida

March 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Rewarding Mediocrity? Optimal Regulation of
R&D Markets with Reputation Concerns
Chia-Hui Chen∗ and Junichiro Ishida†
March 23, 2017

Abstract
In this paper, we consider a dynamic signaling model of an R&D market in which a
researcher can choose either a safe project (exploitation) or a risky project (exploration) at
each instance. We argue that there are substantial efficiency gains from rewarding minor
innovations above their social value and further that it is indeed superior to rewarding
major innovations directly, even when those minor innovations are intrinsically valuelessin themselves. When only major innovations are rewarded, the R&D market eventually
shuts down due to a version of the lemons problem. Rewarding minor innovations is
actually conducive to major innovations as it induces self-sorting among researchers,
which is essential in providing time and resources necessary for more productive ones
to take riskier but more ambitious approaches. This result draws clear contrast to thestatic counterpart where such a scheme can never be optimal. Our model also exhibits
reputation dynamics which capture a pervasive view in academia that “no publications
are better than a few mediocre publications” at an early stage of one’s career.

JEL Classification Number: D82, D83
Keywords: dynamic signaling, career concerns, exploration, exploitation, deadline.
∗†

Institute of Economic Reserach, Kyoto University. Email: chchen@kier.kyoto-u.ac.jp
Institute of Social and Economic Research, Osaka University. Email: jishida@iser.osaka-u.ac.jp

1

Introduction

Since the seminal work of Price (1965), it has been well recognized that the distributionof citations is heavily skewed to the right, with a substantial fraction of scientific papers
receiving no citations at all. Even many decades later, the situation has remained roughly
the same. More recently, based on the citation data from Web of Science, Wallace et al. (2009)
document that although “uncitedness” has declined somewhat in recent years, roughly 60%
of papers published in 1996 are cited less than 5 times in the 10-year window.1 On the righttail of the distribution, Redner (1998) documents that in the samples of papers published injournals catalogued by the Institute of Scientific Information (ISI), only 64 and 2103 out of
783,339 papers (0.008% and 0.268%!) are cited more than 1000 and 200 times, respectively,
and shows that the distribution of citations N (x) has a power law form x−α for a large x with
α close to 3.2 Granted that the number of citations is only one way to measure the social
value of a scientific work, there is little doubt that this is one of those “superstar industries”
where output is heavily concentrated among a few top individuals, and most of the effort isbound to fail in the end.
The outlook is quite different, however, once we turn to how researchers are compensatedfor their achievements: the distribution of reward, broadly defined, is typically much more
gradual and compressed than that of scientific impact.3 Although there are presumably manyreasons for why this is so, this compensation practice raises a concern from the efficiency pointof view if we take into account the inherent nature of knowledge creation. As widely acceptedin the innovation literature, different types of innovation typically require fundamentallydifferent approaches and methods from the outset, where more radical and path-breaking
innovations (hereafter, major innovations) often necessitate the exploration of new ideas whilemore incremental and obscure ones (minor innovations) build on the exploitation of old ideas.
Rewarding minor innovations disproportionately may then distort researchers’ attention awayfrom exploration to exploitation, thereby systematically reducing the frequency of majorinnovations that can advance our scientific knowledge. At a glance, it seems to make more
sense to reward major innovations more heavily in order to encourage researchers to take1
A similar pattern is observed in economics as well. For instance, Hamermesh (2015) finds that the
distribution of citations is highly right-skewed even among those paper published in “top 5” journals.
2
Redner (1998) also shows that the distribution has a stretched exponential form exp[−(x/x0 )β ] withβ ≈ 0.44 for a small x.
3
In a seminal analysis of the economics of superstars, Rosen (1981) argues that “in certain kinds of economic activity there is concentration of output among a few individuals, marked skewness in the associateddistributions of income and very large reward at the top.” The contrast is perhaps clear with other superstar industries such as sports and entertainment where top performers earn enormous reward while the vast
majority can barely survive.

1

riskier but more ambitious approaches.
In light of this concern, this paper aims to explore welfare consequences of a compensation practice under which reward is not proportionally aligned with scientific impact. We
argue that although the above reasoning sounds intuitive and even convincing, this “reward
compression” can be welfare-enhancing under certain conditions, for it is actually conducive
to major innovations. Two features of our setup characterize the R&D market of our focus
and play a crucial role in deriving this conclusion.
• Reputation matters, especially at an early stage of a researcher’s career: good reputation helps attract external resources such as research grants and productive work
environments, constituting a crucial factor of production for R&D activities.
• There are different approaches to innovation with different degrees of risk involved: togenerate a major innovation, one must try untested ideas and methods which necessarily
entail more uncertainty and are most likely to fail.
The former gives rise to reputation concerns on the part of researchers, while the latter defines
the information structure of our setting. We show that when these two features are combined
together, there arise substantial efficiency gains from rewarding minor innovations above theirsocial value, even when they are intrinsically valueless and any attempt to generate them issocially wasteful.
More specifically, we analyze a dynamic signaling model in which an agent (researcher)undertakes R&D activities over time. The agent’s productivity, which is either high or low, is
his private information. To capture the route through which reputation affects productivity,
we assume that the agent needs an upfront payment, or an investment flow, from the market
to undertake R&D activities, as often assumed in the experimentation literature (Bergemannand Hege, 1998, 2005; Hörner and Samuelson, 2013). The market forms a belief by observing asequence of innovation outcomes and provides an investment flow based on its belief. At eachpoint in time, the agent chooses one of the three possible actions: safe project (exploitation),risky project (exploration), and no effort (shirking). The return to the safe project is small
but predictable in that the agent can come up with a minor innovation for sure by working
on it. In contrast, the return to the risky project is unpredictable and sporadic: if theagent chooses the risky project, he generates a major innovation only with some (possibly
very small) probability while he fails and ends up with no visible output with the remaining
probability. Moreover, we assume that the risky project is ability-intensive in that its successprobability depends on the agent’s ability type, and in particular that only the high type cansucceed in it.
2A snapshot of our argument goes as follows. The resource constraint faced by the agent
implies that he must establish reputation to secure the requisite investment flow from themarket. This process of reputation building becomes tricky when only output, but not input,
is publicly observable because the risky project is most likely to fail and produce no visible
output in the end, thereby making it very difficult to distinguish those who explore fromthose who slack off. This fact gives rise to a defining feature of our model where the hightype may have no feasible means to fully separate from the low type at his own will, nomatter how much cost he is willing to incur: if the high type settles on the safe project, thelow type can surely pool with the high type by following the same path; if the high typetakes on the risky project, the low type can still easily mimic (at an observational level) withhigh probability by simply slacking off. The bottom line is that the risky project has a small
informational value because of its unpredictable and sporadic nature, which in turn gives riseto the signaling role of the safe project.
In a competitive environment where there is no direct return to minor innovations, thelow type simply has no choice but to slack off. Given this, if the high type is to choose therisky project, the two types are observationally indistinguishable, which necessarily lowers
the market belief about the agent’s productivity. Moreover, as the high type achieves a majorinnovation sporadically over time and “exit” from the market, the market belief, conditional
on no major innovation having occurred, monotonically declines and eventually reaches a level
where supporting the risky project can no longer be justified, at which point the R&D market
shuts down entirely. In an extreme case where the market holds a relatively pessimistic belief
to begin with, the R&D market fails in a stark manner, and no R&D activities, either safeor risky, can take place at all.
This incentive structure changes rather drastically once minor innovations are rewarded
via “subsidies” so that the low type has an incentive to take on the safe project with somepositive probability.4 When the subsidy is lucrative enough, the low type occasionally chooses
the safe project and voluntarily separates from the high type by achieving a minor innovation.
This self-sorting behavior is crucial, as it keeps the market belief high and within some tightbounds, which degenerate to a single point in the continuous-time limit. We derive thesocially optimal subsidy scheme and show that the incidence of major innovations is enhanced
substantially by rewarding minor innovations. Moreover, we also analyze a scheme which4

It should be noted that we use the term “subsidy” purely for expositional purposes and very broadly:
it intends to capture any gains that accrue from each innovation above its intrinsic value. It includes not
only financial supports and grants from public entities but also other more intangible benefits such as social
recognition and reputation. Some of them are chosen consciously as policy variables whereas others should
more appropriately be regarded as exogenous. Our main aim here is hence to investigate the consequences ofartificially creating value for minor innovations regardless of its origin.

3

directly subsidizes major innovations and make a case for why rewarding minor innovations
is superior to rewarding major innovations directly. This result draws clear contrast to thestatic counterpart where such a scheme can never be optimal.
Although the main contention of this paper is largely normative, our model also exhibits
reputation dynamics that capture a norm peculiar to the academic profession. In academia,
there is a pervasive belief that “no publications are better than a few mediocre publications,”
especially at an early stage of one’s career. The presence of such a norm is an indicationthat researchers do indeed pursue different approaches and self-sort themselves into different
categories. In our setup, under the optimally chosen subsidy, the high type chooses the risky
project while the low type randomizes between the safe project and no effort. Note that,
as we have emphasized earlier, the choice of the safe project results in an outcome thatis informationally distinct and hence fully reveals the agent’s type, which gives rise to theunique reputation dynamics of our model: along the equilibrium path, the reputation of anagent who has achieved a minor innovation is actually lower than that of an agent who has
achieved nothing at all.5
Related literature: The paper that is most closely related to ours in spirit is Manso (2010)
who analyzes a two-period moral-hazard model where the agent decides not only how much
effort to supply (work or shirk) but also how much risk to take (explore or exploit) in eachperiod. In this extended setup, he argues that it is essential to tolerate, or even reward,
early failure in order to encourage exploratory activities.6 Our analysis shares the same
motivation and builds on a similar (trinary-choice) framework that incorporates a choice
between exploration and exploitation. The main difference is that Manso (2010) assumes
only one type of success where exploration in his context is about finding the true successprobability. He also builds on a bandit problem with a risky arm whose payoff is unknown
to the agent whereas ours is cast in the framework of a signaling model in which the agent
is privately informed about his innate productivity. Given these differences, our focus isplaced more on the principal’s learning and ways in which to regulate this dynamic learning
process. In addition, we also explicitly consider an optimal stopping problem in a infinitehorizon model – in particular the continuous-time limit of the equilibrium – which allows us
to illustrate the optimal extent of tolerance for early failure in further detail.
Our analysis also contributes to the diverse literature on dynamic signaing. In our model,
5This phenomenon is to be distinguished from the case of counter-signaling where very high types refrainfrom signaling. In models of counter-signaling, it is typically the case that only medium types engage in costly
signaling while high and low types pool on no signaling. See Feltovich et al. (2002), Araujo et al. (2007) and
Chung and Eso (2013) for various examples of counter-signaling.
6
Also, see Ederer and Manso (2013) who provide experimental evidence that the combination of tolerance
for early failure and reward for long-term success is quite effective in motivating innovation.

4

low-type agents reveal their productivity gradually by achieving minor innovations, and theresultant dynamics are hence governed by their indifference condition. In this regard, thepaper is closely related to a strand of literature which examines the consequences of dynamic
signaling via exit decisions (of firms/sellers) in competitive environments (Hörner, 2002; BarIsaac, 2003; Daley and Green, 2012; Gul and Pesendorfer, 2012; Atkeson et al., 2015).7 The
paper is in particular closely related to Atkeson et al. (2015) who consider an environment
where firms make unobservable investments upon entry and consumers gradually learn theirtypes over time. With the slow diffusion of information, a mild version of the lemons problem
persists where low-type firms enter the market and pool with high-type firms. They then showthat in a competitive market environment with spot prices, entry regulation can complement
the role of reputation and improve welfare. Two differences from our work are worth noting
here. First, their regulatory scheme aims at correcting the inefficiency at the entry level
whereas our scheme works through the regulation of exit behavior, reflecting the fact thatentry regulation is typically not viable in industries of our interest. Second, in their model,
firms choose whether to continue or not once they decide their types upon entry. In contrast,
there is no explicit choice of exit in our model: low-type agents effectively “exit” from themarket whenever they adopt a strategy which fully reveals their type, but what choice ofaction constitutes the exit option, if any, depends also on the high type’s strategy.
From the market’s point of view, the problem we analyze can be seen as a variant of
the canonical two-armed bandit problem with one safe arm (terminating the project) and
one risky arm (continuing the project). There are now increasingly many works, often called
strategic experimentation, where a group of individuals, rather than a single individual, facea bandit problem (Bolton and Harris, 1999; Bergemann and Välimäki, 2000; Keller et al.,2005; Klein and Rady, 2011; Bonatti and Hörner, 2011). In a broad sense, our model also
falls into this strand of literature in that the experimentation process involves decisions ofmultiple individuals. A key departure is that we consider an environment where experiments
are “intermediated” by an informed party, i.e., the agent, who may or may not behave as themarket wishes.8
7

Along this line, Murto and Välimäki (2011) analyze a setting where players are ex ante symmetricallyinformed but gradually learn their types and exit when they become sufficiently pessimistic. Chen and Ishida(2015b) provide a model of hierarchical experimentation where the less confident principal gradually terminates
the project over time.
8
Chen and Ishida (2015a) consider an optimal stopping problem in a similar framework where the informed
agent chooses either to work or to shirk while the principal learns his type through a sequence of outcomes. The
major difference from this work is that the agent in the current setup faces not only a choice between working
and shirking but also that between exploration and exploitation. In a different vein, Guo (2016) analyzes
a model of “delegated experimentation” in which a principal delegates experimentation to an (uninformed)
agent and solves for the optimal delegation rule.

5

2

Model

Environment: Consider an infinite-horizon dynamic signaling model in which an agent(e.g., researcher/entreprenuer) seeks to obtain external resources from a market of potential
employers (e.g., university/investor) to undertake R&D activities. The agent’s ability is either
high (a = h) or low (a = l) where the prior probability that the agent is of the high typeis p0 ∈ (0, 1). The ability type is the agent’s private information which cannot be observed
directly by the principal.
Timing: For most part, we describe and analyze the model in discrete time to clarify its
timing structure although we will eventually focus on the continuous-time limit for clarity
and tractability. More precisely, the agent and the market take actions at times 0, ∆, 2∆, ...,
where we generally assume that ∆ is some arbitrarily small number. The timing of events ineach period proceeds as below:
1. The market offers a flow of funding, which we refer to as the investment flow, based on
its belief about the agent’s ability.
2. The agent chooses which project to work on from a set of feasible alternatives.
3. The outcome is realized, depending on the agent’s ability and project choice.
4. The market belief is updated given the realized outcome.
Investment flows: At the beginning of each period, the market provides the investment
flow upfront to the agent. Let wt denote the investment flow (per unit of time) offered attime t, which the agent can either invest in his project or divert for his private benefits. The
investment flow is necessary for the agent to undertake costly R&D activities: to implementa project, the investment flow must be at least as large as its implementation cost (to be
discussed below). We assume that the market is perfectly competitive and simply offers wt
which equals the agent’s expected productivity for the period [t, t + ∆). As such, throughout
the analysis, we simply take the market as a nonstrategic offer process which computes theagent’s expected productivity and matches the investment flow.
Project choices: Upon receiving wt , the agent privately chooses which project to work on
from a set of alternatives X := {R, S, N }. There are three possible choices for the agent:risky project R (exploration of new ideas), safe project S (exploitation of existing ideas) andno effort N (shirking). Let xt ∈ X denote the agent’s project choice at time t. Different
project choices lead to different outcomes in that the risky project may potentially generate
major innovation while the safe project can constantly generate minor innovation.
6

Innovation outcomes: Given the agent’s ability and project choice, the innovation outcome
yt ∈ Y := {R, S, ∅} is realized and publicly observed. There are three potential outcomes,
where yt = R denotes major innovation, yt = S minor innovation, and yt = ∅ failure (noinnovation). Two properties characterize the underlying R&D process of our setting. First,
the risky project is highly ability-intensive in that only the high type can successfully complete
it. Second, the risky project succeeds abruptly at random points in time while the safe project
constantly yields some output with no uncertainty. Formally, we capture these properties by
the following specification of production technologies:• If xt = R, the agent succeeds and generates one unit of major innovation (yt = R) withprobability ga ∆ and fails (yt = ∅) with probability 1 − g a ∆ where gh = g > g l = 0.9• If xt = S, the agent succeeds with probability one and generates ∆ units of minor
innovation (yt = S) regardless of the ability type.
• If xt = N , the agent fails (yt = ∅) with probability one regardless of the ability type.
xt \a
R

l
yt = ∅ with prob. oneShyt = R with prob. g∆
yt = ∅ with prob. 1 − g∆yt = S with prob. one

Nyt = ∅ with prob. one

yt = ∅ with prob. one

yt = S with prob. one

Table 1: innovation outcomes
Implementation costs: The cost of implementing a project varies over time, depending on
the past innovation outcomes. We in particular assume that only the first success is costly toachieve – an assumption that is made to capture that there is a long-term gain from achieving
an innovation.10 More precisely, let cxt denote the cost of implementing x (per unit of time) attime t. We initially assume that cx0 = cx where cR > cS ≥ cN = 0. Once the agent succeeds in
x at time t, however, the game enters the post-innovation stage in the next period where theagent incurs no implementation cost (as long as he works on the same project), i.e., cxs = 0
for all s > t. As noted above, wt ≥ cxt is required to implement x at time t.
9
We use g∆ to approximate 1 − e−g∆ for an arbitrarily small ∆ throughout the analysis. The same
approximation applies to the discount factor which we introduce below.
10
For a more practical interpretation, in the case of R&D intensive firms, the post-innovation stage corresponds to the phase where an innovating firm appropriates rent from its newly invented product. In academia,
the post-innovation stage reflects the idea that first success is always the most difficult one to achieve, e.g.,
because one must identify a right topic and a right approach to the topic.

7

Payoffs: Each innovation carries an intrinsic benefit which we refer to as the social value.
Let η y denote the (per unit) social value of innovation outcome y where η R > η S ≥ η ∅ ≥ 0.
Here, we consider an environment where the market value of an innovation may differ fromits social value because an innovation may be “subsidized,” as we discuss later. The market
value is given by the sum of its social value η y and a subsidy byt ≥ 0 and is hence denoted by
πty = η y + byt . Given byt , the principal’s payoff in [t, t + ∆) isR
if yt = R,
πt − wt ∆PS
ut = (πt − wt )∆ if yt = S,


−wt ∆
if yt = ∅,whereas the agent’s payoff isxuA
t = (wt − ct )∆.

Note that the agent does not derive any direct benefit from producing an innovation perse; his benefit comes indirectly as a success boosts his reputation and raises the subsequent
investment flows. Both players discount future payoffs by a common discount factor 1 − r∆.
Simplifying assumptions: Throughout the analysis, we make two assumptions to highlight
our main contention.
Assumption 1 η R = v > η S = 0.
Assumption 2 cR = c > cS = 0.
Assumption 1 states that minor innovations are intrinsically valueless, which we assume
strictly to emphasize that our main conclusion (that it is optimal to subsidize minor innovations) holds even when the safe project is socially inefficient. Assumption 2 is also innocuous
as the level of cS becomes totally irrelevant in the continuous-time limit: all of our subsequent
results hold for any cS ∈ (0, cR ) as ∆ → 0 (see Appendix B for more detail). In the interestof reducing the number of parameters and simplifying exposition, therefore, we analyze thecase where cS = 0, so that we can summarize the cost structure entirely by a single variable
cR = c.

3

A static benchmark

We begin our analysis with a static version of the model to illustrate the nature of our setting
as well as to provide a preliminary discussion for our subsequent analysis. In this example,
we suppose that the agent simply receives a post-innovation rent π y at the end of the game8

when the outcome is y. For now, the post-innovation rents are taken as exogenous while they
will be endogenized in the following sections when we analyze the dynamic model.
The incentive compatibility constraints are all straightforward in this static setting withno dynamic reputation concerns. First, it is clear that the low type never chooses R, giventhat he can never succeed in it. The low type thus chooses S if π S > 0 and N otherwise. In
contrast, the high type chooses R if
gπ R − c ≥ max{π S , 0} = π S .
Assume gv > c, so that R is a socially efficient choice. Since minor innovations are intrinsicallyvalueless, the first-best allocation is such that the high type chooses R while the low typechooses N . Below, we examine when this allocation can be implemented.
First, consider the case with no subsidies, i.e., π y = η y for all y ∈ Y . Since π S = 0
by assumption, no one has an incentive to choose S. This means that the low type always
chooses N while the high type chooses R if it is feasible, i.e., w ≥ c, under the maintained
assumption. If the low type chooses N and the high type R, perfect competition forces themarket to offer
w = p0 gv.
If p0 is large enough to satisfy
p0 gv ≥ c,the first-best allocation can be supported as an equilibrium outcome. If this condition fails
to hold, on the other hand, the high type is unable to choose R, and a version of the lemonsproblem surfaces. The only equilibrium in this case is such that the market offers w = 0 andboth types have no choice but to choose N .
Given the scarcity of major innovations, it is reasonable to assume that p0 is relatively
small and hence c > p0 gv, in which case no valuable innovations would be generated under
perfect competition. Even in this case, the equilibrium allocation can be substantially improved by appropriately subsidizing innovations. Since the ultimate goal here is to induce
major innovations, it is perhaps natural to consider a scheme that subsidizes b for each majorinnovation, i.e., π R = v + b. This scheme indeed works and realizes the first-best projectchoice if b is set large enough to satisfy
p0 g(v + b) ≥ c.
Note that this scheme has no detrimental incentive effect as it keeps all the incentive conditions intact for both types.
9Although it appears a bit awkward, one may also consider a scheme that subsidizes minor
innovations to raise the investment flow above the requisite level c. Now suppose that b is
subsidized for each minor innovation, i.e., π S = η S + b = b. Under this scheme, it is now
strictly optimal for the low type to choose S, and the investment flow is obtained as
w = p0 gv + (1 − p0 )b.
The high type then chooses R if
gv − c ≥ b and w = p0 gv + (1 − p0 )b ≥ c,provided that there exists b that can satisfy both conditions.
Does it ever make sense to subsidize minor innovations so as to induce major innovations?
In this static benchmark, the answer is a resounding no because that would distort bothtypes’ incentives. First, under the scheme that subsidizes minor innovations, the low typeis necessarily induced to work on the inefficient safe project (although this efficiency loss is
assumed to be arbitrarily small throughout the analysis).11 Second, this scheme also has a
distortionary incentive effect for the high type: it makes the safe project more attractive,
giving the high type a stronger incentive to choose S instead of R. In particular, if
gv > (2 − p0 )c,
there is no subsidy level b that can induce the high type to choose R.
This argument suggests that there is no efficiency rationale to subsidize minor innovationsin this static environment, as any compensation practice which rewards minor innovations
above their social value can never be welfare-enhancing. As we will argue below, however, thisconclusion is reversed once dynamic incentives are explicitly taken into account: a schemethat subsidizes minor innovations exhibits several nice properties while minimizing the distortionary incentive effect for the high type; in fact, the distortion disappears altogether inthe continuous-time limit.

4

On the failure of the R&D market

We now examine a fully dynamic version of the model where the agent needs to be incentivized
continuously over time. In this section, we analyze the model with no subsidies, i.e., πty = η y ,y ∈ Y , for all t, to show once again that the R&D market can break down rather starkly
under some feasible conditions. The underlying reasoning is basically the same as in thestatic case but is now augmented with Bayesian belief updating.
11

Throughout the analysis, we assume that only the first success is costly to achieve. Since the agent isguaranteed to succeed in S at a first try, the total cost of working on S is cS ∆. As such, the cost converges
to zero in the continuous-time limit for any cS .

10

4.1

Equilibrium investment flows

In the dynamic setting, the market observes a sequence of outcomes and continuously updates
its belief at each instance. Formally, let ht := (y0 , y∆ , ..., yt−∆ ) ∈ Ht denote the history ofS
the game at time t with h0 = ∅, and H := ∞n=0 Hn∆ the set of all histories. We denote by
pt = p(ht ) the market belief at time t, which is defined as the probability that the agent is of
the high type conditional on ht .
Given the current belief, the market offers the investment flow wt for the period [t, t + ∆).
Under perfect competition, the market is forced to offer the amount that equals the agent’s
expected productivity. Letting σta (x) be the behavior strategy of type a, the equilibrium
investment flow is obtained as
wt = pt σth (R)gπtR + σth (S)πtS + (1 − pt ) σtl (R)gπtR + σtl (S)πtS .

taking πtR and πtS as given.

4.2

Continuation equilibria with no reputation concerns

The amount of information revealed by each innovation outcome depends on the strategies
of both types. In some cases, an innovation outcome can fully reveal the agent’s type, i.e.,either pt = 0 or pt = 1, after which the agent no longer has any reputation concerns. With
no reputation concerns, the agent’s problem becomes essentially static, and the continuation
equilibria in these contingencies can be straightforwardly characterized as below.
• If pt = 0, it is (at least weakly) optimal for the low type to choose S. We supposethat the low type always chooses S thereafter. Let VtS denote the continuation payoffin this contingency which emerges in equilibrium when the low type achieves a minor
innovation.
• If pt = 1, it is strictly optimal for the high type to choose R. We suppose that the hightype always chooses R thereafter. Let VtR denote the continuation payoff in this contingency which emerges in equilibrium when the high type achieves a major innovation.

4.3

Project choice with reputation concerns

When pt ∈ (0, 1), the agent must take into account how his action affects his reputation.
Define pyt+∆ (p), or simply pyt+∆ , as the next-period belief when pt = p ∈ (0, 1) and yt = y.
We also denote by Uta (p) the continuation payoff for type a as a function of the current

11belief.12 Then, given pt and wt ≥ c, if the agent chooses R in period t, the expected payoff
is given byaaR(wt − c)∆ + (1 − r∆) (1 − g a ∆)Ut+∆(p∅t+∆ ) + g ∆Vt .

(1)

Similarly, if the agent chooses S and N , the expected payoffs areawt ∆ + (1 − r∆)Ut+∆(pSt+∆ ),(2)awt ∆ + (1 − r∆)Ut+∆(p∅t+∆ ),(3)

andrespectively.
Observe that since gl = 0, N strictly dominates R for the low type, so that he would
never choose R under any circumstance. Similarly, from (1) and (3), N strictly dominates R
for the high type if
h
(p∅
Ut+∆
t+∆ ) >

(1 − r∆)gVtR − c.
(1 − r∆)g(4)

2This condition always holds in the limit if c > gVtR ≥ grv , in which case there is no feasible
way to let the high type take on R.13 Since this is clearly uninteresting for the purpose of
our study, we assume that the value of a major innovation is high enough for the high typeto choose R when the continuation payoff is sufficiently small.
2

Assumption 3 g rv ≥ c.

4.4

Equilibrium with no subsidies

The essential aspect of the current setting is that the agent faces a resource constraint andneeds to obtain external resources to carry out R&D activities. Because of this feature, we
may have an equilibrium in which the principal offers wt = 0 and the agent has no choice but
to choose xt = N , which we call trivial in what follows. Note that the trivial equilibrium is aform of pooling equilibrium which by itself provides no additional information to the market.
As long as the trivial equilibrium prevails, therefore, the market belief stays constant overtime, which bounds the investment flow from above. In particular, when the initial prior p0
is not high enough to begin with, the investment flow is perpetually kept below the requisite
level and only the trivial equilibrium can emerge as a consequence.
12
In principle, the continuation payoff may depend not only on the current belief but also on the entire
history of the game. In what follows, however, we only look at cases where the continuation payoff depends
only on the current belief for a given t.
13
When pt = 1, the high type always chooses R and hence wt = gv when no subsidies are provided. Thisimplies that VtR ≥ gv
r

12

Proposition 1 If c > pt gv, the only continuation equilibrium is the repetition of the trivial
equilibrium, i.e., ws = 0 and both types choose xs = N for all s ≥ t.
Proof: When only major innovations generate value as in this setup, the equilibrium
investment flow wt cannot exceed pt gv. This implies that c > pt gv ≥ w0 , so that theinvestment flow cannot be high enough to support R. Given this, the agent can choose
either S or N , but in either case, his expected productivity is zero by assumption. The
principal thus offers wt = 0, and the agent has no choice but to choose N as a consequence.
Since no additional information is revealed in the trivial equilibrium, we have pt+∆ = pt .
As this argument applies to all the subsequent periods, the only possible equilibrium is therepetition of the trivial equilibrium where wt = s and both types choose xs = N for all s ≥ t.
The R&D market failure is most severe when the initial prior belief is not high enoughto support R from the beginning, i.e., c > p0 gv. Even when the condition does not hold attime 0, however, the problem can still persist. Suppose that p0 gv ≥ c and it is also optimalfor the high type to choose R at time 0. Given this, the belief in the next period is given byp∆ =(1 − g∆)p0,
1 − p0 + (1 − g∆)p0which is evidently lower than p0 . As can be seen from this, the belief monotonically declines
over time as long as the high type chooses R (while the low type chooses N ), and any R&D
activities are bound to disappear sooner or later.
This simple argument suggests that the R&D market can break down rather starkly due
to a version of the lemons problem when only major innovations are valuable and hence
rewarded. One way to solve this dilemma is then to artificially create value for innovations,be it major or minor, via “subsidies” so as to raise the agent’s expected market value. In
the next section, we in particular argue that a simple subsidy scheme – namely the one that
exclusively subsidizes minor innovations – can alter the equilibrium allocation and correct
this market failure to a considerable extent. To emphasize this point, we focus on a situation
where the initial prior is not high enough to support R at the outset, so that the R&D market
would break down in the most severe manner without any intervention.
Assumption 4 gv > c > p0 gv.

5Subsidizing minor innovations

We are now ready to evaluate the consequences of rewarding minor innovations above theirsocial value in the environment described above. To make our contention, however, we take13

an approach from the reverse direction: instead of directly evaluating the welfare impactof rewarding minor innovations, we first identify a subsidy scheme that can implement thesecond-best allocation, and then show that one of those schemes is a simple scheme thatsubsidizes minor innovations at a constant rate.14 From this result, we argue that the reward
compression, which appears to overcompensate minor innovations, is not so harmful after all
and can even be welfare-improving under some plausible conditions.

5.1

The social Planner’s problem

We consider a social planner who can commit to a subsidy scheme at the outset of the game.
There is a transfer (or taxation) cost in that the social planner incurs a cost θ > 0 for eachdollar transferred. The social welfare in this context is then defined as the discounted sumof values generated from R (hereafter, simply the value of innovation) net of the discounted
sum of transfer costs (the transfer cost). A feasible subsidy scheme is a pair (bS , bR ) whereby : H → R+ maps a current history to a real-valued transfer for each y = S, R.
The social planner’s objective is to device a subsidy scheme which maximizes the social
welfare. To this end, it is important to note that although the whole set of histories can bequite large, some of the histories are actually irrelevant for the purpose of our study and canbe excluded from consideration without loss of generality. This fact allows us to reduce thedomain and simplify the problem to a large extent. Below, we first comment on the relevant
domain of the social planner’s problem which we consider in the subsequent analysis.
First, it is often convenient to divide our game into two distinct stages – before and after
a major innovation is achieved – as the problems in these stages are quite different and in factindependent from each other. As we have seen above, once a major innovation is achieved,
the agent’s type is fully revealed, and the high type chooses R indefinitely even without
any subsidies. This indicates that there is no feasible way to improve the social welfare via
subsidies once the game enters into this post-innovation stage. For this reason, when deriving
an optimal scheme, we suppose that no subsidies are provided once a major innovation is15
achieved and hence VtR = V R := gv
r for all t.
14Note that we do not consider direct revelation mechanisms because our goal here is not to identify a classof optimal mechanisms per se, but rather to evaluate the efficiency properties of a particular compensation
practice. This draws contrast to Halac et al. (2016) who analyze optimal contracts for experimentation withhidden information about the agent’s ability type.
15
Of course, there is a possibility that rewarding subsequent major innovations further enhances welfare
because a higher post-innovation rent V R can provide a stronger incentive for the high type to choose R(for ht ∈ H t ), thereby allowing the social planner to extend the contract duration. However, given that
our aim is to make an argument for why it makes sense to reward minor innovations, this possibility is an
issue of independent interest: any subsidy in the post-innovation stage can affect the equilibrium allocation
only through its effect on V R ; as such, V R is a sufficient statistic of whatever happens in this stage. Since
our argument holds for any V R that satisfies the maintained assumptions, if it is ever optimal to provide

14

Second, since the expected productivity of an agent who has achieved no major innovation
gradually declines over time, it must become optimal at some point not to support R any
more. This implies that we can focus on schemes which terminate and provide no subsidies
after some finite time.16 Let T denote such a termination date (which we also interchangeablyrefer to as the contract duration) after which no subsidies are offered, i.e., the agent mustachieve a major innovation until time T or otherwise he will receive a continuation payoff ofzero after that.
Given these restrictions, we can now define
H t := {ht : ys 6= R for all s < t}.
as the set of histories in which no major innovation has yet been achieved, and
T

T

H :=

∆
[

H n∆ .

n=0

as the set of all such histories up to time T . Then, the above restrictions are formallyTequivalent to bS (h) = bR (h) = 0 for all h ∈/ H , and hence H

Tconstitutes the relevant

domain of our problem. We say that a subsidy scheme is optimal if it implements the secondbest allocation, i.e., the one that maximizes the social welfare among all feasible schemes on
the relevant domain.

5.2

The high type’s optimal strategy

For any ht ∈ H t , given wt ≥ c, the high type chooses R over N only ifhh∅(wt − c)∆ + (1 − r∆) g∆V R + (1 − g∆)Ut+∆(p∅t+∆ ) ≥ wt ∆ + (1 − r∆)Ut+∆ (pt+∆ ),which is simplified toh(1 − r∆)g V R − Ut+∆(p∅t+∆ ) ≥ c.

By rearranging and taking the limit, we obtaincgvh− ≥ Ut+∆(p∅t+∆ ),
rg(5)

additional incentives by subsidizing subsequent major innovations to raise V R , the social planner can do sowhile maintaining the elements of the optimal scheme we will discuss below. Alternatively, our analysis canbe seen as analyzing a situation where V R is already optimally set, possibly by subsidies to subsequent majorinnovations.
16
Although it is in principle possible to design an optimal scheme which provides subsidies indefinitely overtime, we can always rewrite this into a scheme which terminates at some termination date since only thingthat matters is the continuation payoff. It is hence without loss of generality to focus on this class of schemes.

15where we often refer to the left-hand side of (5) as the static gain from working on R. The
key to this dynamic incentive structure is that a higher continuation payoff diminishes thecurrent incentive to take on R, implying that wt should be kept as small as possible whilesatisfying the resource constraint when necessary.
Suppose for now that there is some exogenously given termination date T and moreoverthat the high type always chooses R and wt = c for any history. Under these conditions, thecontinuation payoff at any time t < T converges to
Z T −t 2
1

1 
g v −(r+g)s
1 − e−(r+g)(T −t) .
eds = gv−r
r r+g0
From this, (5) can be written as

1

c
1 
gv
1 − e−(r+g)(T −t) .
− > gv
−
r
g
r r+g

(6)

Among other things, the high type’s current incentive to work on R depends crucially on
2g v
T − t, i.e., the remaining time the agent has until the termination date. If c > r+g, we candefine T such thatfrom which we obtain

1

c
1 
gv
1 − e−(r+g)T ,− = gv
−
r
g
r r+g
T =

(7)

g3 v
1
,
ln
r + g r (r + g)c − g 2 v

Since the high type’s incentive to work on R is maximized when wt = c as long as he fails, thisis the upperbound of the contract duration in which the high type can be induced to work2g v
on R for all t ∈ [0, T ]. If r+g≥ c, on the other hand, (6) can be satisfied for any remaining

time, and we let T = ∞ in this case. The upper bound plays a crucial role in shaping the
optimal scheme because there exists no equilibrium in which the high type chooses R for allt ∈ [0, T ] if T is set to be larger than T . The optimal design of the subsidy scheme thus
eventually comes down to how much to subsidize each innovation and for how long.

5.3

Optimal indirect scheme

With a termination date and time discounting, we can show that the second-best allocation
should have effort spread out evenly over the entire contract duration. In fact, in the secondbest allocation to be implemented, the high type chooses R with some constant probabilityk ∈ (0, 1] for all t ∈ [0, T ]; the implemented equilibrium is hence characterized thoroughly by apair (k, T ). For clarity, we say that an equilibrium is full-funding if k = 1 and partial-funding
if k < 1.
16

To illustrate how to pin down the optimal contract duration, consider a full-funding
equilibrium. In this equilibrium, conditional on no major innovation having occurred, the
−gt

e
17
(limit) probability that the agent is of the high type is given by pt := 1−pp00+p−gt for t ≤ T .
0e

If the contract duration is extended for one more period at time T , the agent generates amajor innovation with probability pT g∆, and the expected value generated is pT g∆(v + (1 −r∆)V R ).18 In the meanwhile, the market needs to pay c∆ to the agent, so that the amount ofsubsidy needed is c∆ − pT gv. This cost and benefit must be balanced at the optimal contract
duration, i.e.,
pT g v + (1 − r∆)V R = θ(c − pT gv).

whose continuous-time limit is given byg
p0 e−gT
gv 1 + θ += θc.
r 1 − p0 + p0 e−gT

(8)

Let Tb denote the solution to (8) which is given by

!
g
)
−
θc
gv(1
+
θ
+
1
p
0
r
Tb :=
+ ln
ln
,
g
1 − p0θcwhich is the break-even point under the presumption that the high type always chooses R.
Note that if θ is too large, it is clearly optimal to provide zero subsidies, and no social
surplus is generated under the maintained assumptions.19 As this only results in a trivial
solution, we assume that θ is sufficiently small, so that Tb > 0 and the socially optimal
allocation involves strictly positive subsidies.
θcAssumption 5 p0 gv > 1+θ+g .
v

For expositional purposes, we say that a subsidy scheme is history-independent if by (h) =Tby for all h ∈ H . The next proposition is the main result of this paper, which states thatthe second-best allocation can be implemented by a simple history-independent scheme that
exclusively subsidizes minor innovations at a constant rate.
Proposition 2 Let k̃ be a solution to

R Tk̃b k̃g2 v −(r+k̃g)tcdt = gv0r e
r − g . Then, the following

history-independent scheme is optimal: for all h ∈ H

T∗,bR (h) = 0, bS (h) = k∗ c,(9)17Note that pt is to be distinguished from the belief pt since H t contains a history where the agent has
achieved a minor innovation, in which case pt = 0.
18
After the termination date, only the agent who has achieved a major innovation can continue to work,and the continuation value is precisely V R .
19
As we will see below, if Assumption 5 fails to hold, it is optimal to set T = 0. Given no subsidies, theagent has no choice but to choose N for all t as we have seen above.

17

where• if T ≥ Tb, k∗ = 1 and T ∗ = Tb;b• if Tb > T , k∗ = k̃ and T ∗ = Tk̃ .

In the implemented equilibrium, the high type chooses R with probability k∗ for all t ∈ [0, T ∗ ].
Proof: See Appendix A.
In words, the optimal scheme subsidizes k∗ c for each minor innovation until time T ∗ , giventhat no major innovation has been achieved. For expositional purposes, we refer to this as anindirect scheme as it attempts to induce major innovations indirectly by subsidizing minor
innovations. The optimal indirect scheme implements a full-funding equilibrium in which thehigh type is induced to choose R for all t ∈ [0, T ∗ ] if T > Tb, i.e.,!
gv(1 + θ + gr ) − θc
1
p0
1
g3 v
≥
ln
.
(10)
ln
+ ln
r + g r (r + g)c − g2 v
g
1 − p0θc

A key determinant of k∗ is the marginal transfer cost θ. Under the maintained assumptions,
there exists some θ such that (10) holds for θ ≥ θ.
It is worth pointing out that the equilibrium implemented by this optimal scheme exhibits
unique reputation dynamics: on the equilibrium path, the market belief declines to zero assoon as the agent succeeds in S whereas it is strictly positive when he has achieved nothing.
That is, throughout the contract duration, the agent actually loses reputation by achievinga minor innovation, which may capture a pervasive view in academia that “no publications
are better than a few mediocre publications” at an early stage of one’s career. This rather
peculiar property of the reputation dynamics is a reflection of the fact that exploration of new
ideas is inherently time-consuming and requires persistent effort that pays off only randomly
over time. Most notably on this point, March (1991) notes that “returns from exploration
are systematically less certain, more remote in time, and organizationally more distant fromthe locus of action and adaptation,” and “the search for new ideas, markets, or relations
has less certain outcomes, longer time horizons, and more diffuse effects than does further
development of existing ones.” The time-consuming and uncertain nature of exploratory
activities is a crucial driving force which determines the structure of innovations, both in ourmodel and in reality.

5.4

Discussion: how and why the indirect scheme works

Proposition 2 states that the second-best allocation can be implemented by a simple stationary scheme where the subsidy rate is held constant for t ∈ [0, T ]. The reason why this simple
18

scheme works is because it induces self-sorting among low-type agents to sustain the market
belief constant over time. The key mechanism at work is that the low type is held indifferent
between S and N throughout the contract duration: if the high type chooses R, the low
type can separate from the high type by choosing S, in which case he chooses S and receivesbS (h) = c until time T ; if he instead chooses N , he partially pools with the high type andreceives wt = c. On the equilibrium path, therefore, the low type randomizes between S and
N to keep the expected productivity invariably at c for any history. This means that at any
t ∈ [0, T ∗ ), we must havept gv + (1 − pt )σtl (S)c = c ⇔ σtl (S) =c − pt gv,(1 − pt )ctaking pt as given. The belief in the next period is then obtained aspt+∆ =(1 − g∆)cpt (1 − g∆)=,
pt (gv−c)g(v− c∆)(1 − pt ) (1−pt )c + pt (1 − g∆)cwhich is constant over time and converges to gv
as ∆ → 0.

An important virtue of the proposed scheme is that it has no distortionary incentive effectfor the high type, which draws clear contrast with the static counterpart where rewarding
minor innovations necessarily distorts the high type’s incentives. The key difference is thatthe agent needs to be incentivized continuously over time in the dynamic setting. In thesecond-best allocation, the investment flow must equal c whenever the high type is supposed
to choose R, implying that the agent can secure at least a payoff of c in this contingency.
As is common in this type of dynamic setting, the presence of this dynamic rent inevitably
weakens the current incentive to work hard because even if the agent fails today, he can still
have another chance tomorrow.
The proposed scheme is designed to exploit this property, using this dynamic rent to
induce the low type to occasionally choose S and “exit” from the market. To see this,observe that the high type chooses R over S ifhSh(wt − c)∆ + (1 − r∆) g∆V R + (1 − g∆)Ut+∆(p∅t+∆ ) ≥ wt ∆ + (1 − r∆)Ut+∆ (pt+∆ ).

h (p∅ ) ≥ U h (pSGiven that (5) holds, this condition is necessarily satisfied if Ut+∆t+∆ t+∆ ), whicht+∆suggests that rewarding minor innovations is entirely harmless as long as the continuation
payoff of choosing S does not exceed that of choosing N . Now suppose that the high typedeviates and chooses S. In this contingency, the market falsely updates its belief to pt = 0,
but still offers wt = c under the proposed scheme, thereby giving him exactly the sameh (p∅ ) = U h (pScontinuation payoff after the deviation, i.e., Ut+∆t+∆ t+∆ ). As such, the incentivet+∆

to choose S is no stronger than the incentive to choose N , and there is hence no incentive
distortion from subsidizing minor innovations.
19

5.5

Subsidizing major innovations?

In the current setting, there actually exist other subsidy schemes that can equivalently implement the second-best allocation. In fact, it is possible to construct an optimal scheme that
exclusively subsidies major innovations as in the static benchmark; for expositional purposes,
we call this a direct scheme as it subsidizes major innovations directly to induce major innovations. Here, we briefly discuss this possibility and build on this argument to make a casefor why rewarding minor innovations is superior to rewarding major innovations directly.
TConsider an alternative scheme in which bS (h) = 0 for all h ∈ H . In this hypothetical environment where only major innovations are valuable and hence rewarded, the low
type clearly has no incentive to choose S. Since no minor innovations are observed on theequilibrium path under this scheme, we defineT

H t := {ht : ys = ∅ for all s < t} ⊂ H t , H T :=∆[H n∆ .

n=0

Given some history ht ∈ H t and the corresponding belief pt = p(ht ), the investment flow attime t is then obtained aswt = pt g v + bR (ht ) ,which must be at least as large as c to induce R from the high type. Since the low type chooses
N for all t, the market belief pt must decline monotonically over time. This immediately
suggests that no history-independent scheme that exclusively subsidizes major innovations
can replicate the second-best allocation, as it necessarily implies wt > c for t ∈ [0, T ) if it is
to induce R for all t ∈ [0, T ]. To keep the investment flow at c, the subsidy rate must thenincrease over time to offset a decrease in the market belief.
Proposition 3 The following direct scheme is optimal:(Sb (h) = 0 for h ∈ H

T∗R, b (h) =∗cp(h)g − v

for h ∈ H T ,0for h ∈/ HT ,∗

where T ∗ is determined as in Proposition 2.
Proof: Under the proposed scheme, the low type earns zero payoff regardless of his action
and hence chooses N with probability one. The expected productivity of the agent is theninvariably c for any given pt . Now consider the same offer process as in Proposition 2 which
offers wt = c with probability k and wt = 0 with the remaining probability. If k and T
are set appropriately, the high type then chooses R whenever wt = c. Moreover, since the20amount of subsidy is exactly the same as under the optimal indirect scheme, this indeed
implements the second-best allocation.
The optimal direct scheme generates major innovations at the same rate while requiring
the same amount of subsidy as does the optimal indirect scheme. Both of these schemes
hence implement the second-best allocation and are equivalent in terms of social welfare. We
argue, however, that there are reasons to believe that the indirect scheme is superior to thedirect scheme in a practical sense. Below, we raise three major reasons for why it makes more
sense to reward minor innovations to induce major innovations.
Simplicity: A major advantage of the indirect scheme is that it is history-independent andhence substantially easier to implement. In contrast, in the optimal direct scheme, the subsidy
rate depends on the market belief and must increase over time at a rate which depends onthe minute details of the underlying environment. We argue that this virtue of simplicity isparticularly important when we attempt to regulate a large and diverse community, such as
academia, where it is prohibitively costly to tailor an elaborated scheme for each individual
and the class of enforceable schemes can be quite limited.
Manipulability: The more elaborated an incentive scheme becomes, the easier it is to
manipulate the scheme, as illustrated most notably by Holmström and Milgrom (1987). This
reasoning also applies to our direct scheme: the subsidy rate that monotonically increases
over time can give rise to an incentive to game the system if outside parties cannot observe
when a major innovation is achieved. To see this possibility in more detail, suppose that themarket (which should now be interpreted as an individual employer) is now also a strategic
player, and further that the arrival of a major innovation is observed only privately by themarket and the agent but not by the social planner. Upon the arrival of a major innovation,
the market can then choose when to “publicize” it to maximize the joint payoff of the market
and the agent. Suppose that a major innovation is achieved at some time t. If it is disclosed
R
immediately, the joint continuation payoff is bR
t + v + V ; if disclosed at the next instanceR
t + ∆, it is (1 − r∆)(bRt+∆ + v + V ). It is then optimal for the principal to wait ifRbRt+∆ − btR≥ r(bRt+∆ + v + V ).
∆Letting ∆ → 0, we obtainc(1 − p0 )cRḃR≥ r(bRt = − 2 ṗt =t + v + V ),p0 e−gt
pt gwhich holds if r is small relative to g. Note that no such incentive arises under the indirectscheme since the value of each major innovation stays constant over time.
21Externalities: We have thus far assumed that minor innovations are intrinsically valuelessto make our conclusion more emphatic. In all likelihood, however, it is a little too extreme
to assume that those minor innovations offer absolutely zero value to society.
particularly true when they yield some external effects.

This is

For instance, it is reasonable

to argue that even when minor innovations per se are useless, they may still raise theoverall productivity due to knowledge spillovers: after all, a situation where a sizable groupof researchers work hard at generating something new is more conducive to innovations
than to a situation where the vast majority simply slack off and do nothing. If minor
innovations provide positive externalities even slightly, the indirect scheme becomes strictlysuperior to the direct scheme that can induce effort only from a limited fraction of researchers.
Before we conclude, it should be emphasized that the aim of our analysis is to provide astylized description of when and under what conditions rewarding minor innovations entails
less incentive distortion, and we make some strong assumptions to make this point clearer.
Our theoretical results should thus be regarded as more of a theoretical benchmark, forthey hold only under a set of stringent conditions. In particular, when we consider a more
realistic case with cS > 0, the assumption that the total cost of implementing S, as captured
by cS ∆, vanishes to zero in the limit plays a crucial role for our indirect scheme to be(approximately) optimal.20 Needless to say, this is a very strong assumption which is madestrictly for expositional clarity rather than for realism.
In contrast, when cS ∆ is bounded away from zero, the indirect scheme entails an additional social cost as it induces the low type to waste resources on the inefficient safe project.
As we have argued, however, our argument still holds true in a qualitative sense even inthis case: rewarding minor innovations induces self-sorting among low-type agents, thereby
sustaining the market belief within some tight bounds, while its distortionary incentive effort
becomes less attenuated when the agent needs to be incentivized continuously over time. In
this more realistic scenario, this efficiency loss must be traded off against the aforementioned
benefits; obviously, the benefits dominate the cost when cS ∆ is sufficiently small.

6

Conclusion

This paper illustrates how an R&D market can fail due to a version of the lemons problem
and how optimally R&D activities can be subsidized to overcome this problem. We argue that20

Under the assumption that only the first success is costly, the total cost the agent needs to incur is cS ∆even if he works on S for all t. Even when we suppose cS > 0, the optimal indirect scheme we identified canapproximately implement the second-best allocation as ∆ → 0. See Appendix B for more detail.

22

there are substantial efficiency gains from rewarding minor innovations above their market
value, and further that it is superior to rewarding major innovations directly even whenminor innovations per se are intrinsically valueless. Rewarding minor innovations induces selfsorting among less productive agents which is essential in ensuring more time and resources
for more productive ones to try out riskier but potentially more valuable approaches. Our
argument implies that the current compensation practice, which appears to overcompensate
minor innovations, is not necessarily welfare-reducing and can even be conducive to majorinnovations when dynamic incentives are explicitly taken into account.
As a final note, we consider a pure signaling context where the agent knows his own
type precisely throughout the analysis. A possible extension of this setting is therefore toincorporate the agent’s learning about his own type. Although the extension clearly involves
technical issues of independent interest, the same economic principle should apply in thiscase as well: less confident agents eventually choose the safe project and drop out when the
expected return of doing so is sufficiently high; to induce this self-sorting, minor innovations
must be rewarded sufficiently to compensate for the loss of reputation. The essential elementof our story is hence that the agent has some private information, regardless of whether his
information is perfect or not. Nevertheless, since both elements of learning and signaling must
be present in reality, it is of some interest to consider the possibility of imperfect informationin this type of environment.

Appendix A: proof of Proposition 2
To characterize the optimal scheme, we first take the contract duration T as given and, foreach T , identify subsidy schemes that maximize the value of innovation, which we refer toas surplus-maximizing for clarity. Among all those surplus-maximizing schemes, we thenidentify the one that entails the minimum transfer cost. Since there is at least one such
scheme for each T , this gives us a set of candidate schemes across all T . Identifying this set is
an important intermediate step because the optimal scheme by definition must exist within
this class.
First, if T > T , it is possible to construct a full-funding equilibrium. To do this, theT

expected productivity of the agent must be c for any history in H . Assumption 4 (c > p0 gv)
then implies that the low type must be induced to separate from the high type by choosing S
with some probability. Now consider a scheme with k = 1. Under this scheme, the low type
always receives c until time T regardless of his strategy: even if he chooses S and separates
from the high type, he continues to receive wt = bSt = c. The low type is therefore indifferent
between N and S for the entire contract duration under the proposed scheme, and wt = c23can be sustained by having the low type randomize between N and S. More precisely, wemust havept gv + (1 − pt )σtl (S)c = c ⇔ σtl (S) =c − pt gv,(1 − pt )c2g v≥ c, the high type prefers R over N as wetaking pt as given. Given that wt = c and r+ghS
have already seen. It is also easy to see that uht+∆ (p∅t+∆ ) > ut+∆ (pt+∆ ), suggesting that it isT

optimal for the high type to work on R as long as bS (h) = c for h ∈ H .
The situation becomes more complicated when the contract duration exceeds the upperbound T , in which case it is no longer feasible to implement a full-funding equilibrium: evenif wt ≥ c for all t ∈ [0, T ], the high type has no incentive to choose R early on when thecontract is still far from termination. Since the social planner wants to induce R earlier thanlater due to time discounting, a solution in this case is to provide only partial funding inorder to keep the continuation payoff low enough. More precisely, it is essential to keep theccontinuation payoff weakly below gv
r − g throughout the contract duration. The following

statement derives the optimal scheme when the termination date exceeds the upperbound,which implements a partial-funding equilibrium where the high type is induced to work on
R only for some fraction of time.
Lemma 1 Let k̂T be a solution to

R T k̂T g2 v −(r+k̂ g)tc
Tdt = gvr e
r − g . Then, for a given T > T ,0

the following scheme minimizes the transfer cost among all surplus-maximizing schemes: forT

all h ∈ H ,bR (h) = 0, bS (h) = k̂T c.

(11)

Proof: The proposed scheme implements an equilibrium in which the high type chooses
R with probability k̂T over the whole contract duration. For the sake of the argument, we
think of the market as a non-strategic offer process which gives wt = c with probability k̂T
and wt = 0 with the remaining probability.21 Now suppose that the high type chooses R
whenever wt ≥ c. Given the offer process, his continuation payoff in the limit is then givenby
Uth (pt ) =Z T −t0k̂T g2 v −(r+k̂T g)s
gv
c
e
ds <
− ,
r
r
g

21
We can think of this as an equilibrium outcome of some strategic bidding process when we consider acompetitive market of potential employers. To see this, note that the current investment flow has no incentive
effect in this setting. The only difference it makes is when the investment flow is so low that the resource
constraint binds, in which case some employer may induce the high type to choose R by offering wt ≥ c.
However, the expected productivity would be c in this equilibrium, so that employers are indifferent between
offering wt = c and wt = 0 and hence have no strict incentive to deviate.

24

for any t > 0. Therefore, by (5), the high type will choose R over N whenever wt ≥ c. Notealso that if the high type ever deviates by choosing S, he will simply receive k̂T c (by workingon S), so the continuation payoff converges to the level when he repeatedly chooses N until
period T . As this yields a strictly lower payoff, the high type chooses R whenever wt ≥ c andN otherwise.
The low type’s optimal strategy is effectively the same with a minor modification. Again,
the low type is indifferent between pooling with the high type and separating from the hightype (by choosing S): in any contingency, the expected per-period payoff for the low type isk̂T c. The only difference is that we now have some time intervals with wt = 0, in which casethe low type chooses N to pool with the high type. On the other hand, when wt = c, thelow type adopts a similar mixed strategy to keep the expected productivity of the agent at
c, i.e.,pt gv + (1 − pt )σtl (S)k̂T c = c ⇔ σtl (S) =c∆ − pt gv(1 − pt )k̂T c,taking pt as given. This shows that the proposed subsidy scheme, along with the offer process,
implements a partial-funding equilibrium in which the high type chooses R with probabilityk̂T .
Given this, we now show that the implemented partial-funding equilibrium is surplusmaximizing for a given T > T . To see this, note that since the high type appropriatesgvga constant fraction v+rgv = r+gof the value of innovation, the problem is equivalent tormaximizing the high type’s expected payoff in period 0. Given T > T , once the continuationcpayoff goes above gv
r − g , the high type would not choose R and no additional surplus couldcbe generated. This means that even if the continuation payoff goes above gv
r − g , it does

not contribute to increasing the value of innovation; as such, the continuation payoff must
R T k̂T g2 v −(r+k̂ g)tcc
Tdt = gv
be bounded by gv
r − g . Since 0r er − g by definition, the implemented

equilibrium maximizes the value of innovation.

Finally, we need to show that the proposed scheme minimizes the transfer cost among all
surplus-maximizing schemes. To see this, consider an alternative equilibrium in which thehigh type chooses R with probability κt at time t as long as ht ∈ H t . It follow from the above
argument that any surplus-maximizing scheme must satisfy
Z T
g2 v −(r+κt g)t
gv
c
κt
e
dt =
−
r
r
g
0
Z T
Z T
κt e−(r+κt g)t dt =⇒k̂T e−(r+k̂T g)t dt.
0

0

25

Since e−κt t < e−k̂T t if and only if κt > k̂T , we have
Z T
Z T
−rt
κt e dt ≥k̂T e−rt dt.

(12)

0

0

RT
In the equilibrium constructed above with k̂T , the low type’s expected payoff is 0 k̂T ce−rt dt;
RTin the alternative equilibrium, the low type’s expected payoff is at least 0 κt ce−rt dt, whichRT
is larger than 0 k̂T ce−rt dt by (12). In contrast, these two equilibria yields the same valueof innovation and hence the high type’s payoff, meaning that the transfer cost is higher in
the alternative equilibrium. Since this holds for any nonconstant sequence of κt , this provesthat the proposed scheme minimizes the transfer cost among all surplus-maximizing scheme.
Given the set of candidate schemes, we now simply need to find the optimal contract
duration T . The argument is essentially the same as above: the social planner sets T at apoint where the marginal cost equals the marginal benefit. It follows from (8) that in the
full-funding equilibrium, the belief reaches this point at time Tb. Clearly, if T ≥ Tb, it is

optimal to set k∗ = 1 and T ∗ = Tb. If not, the high type cannot be induced to work on R for
all t ∈ [0, Tb], and it is hence optimal to set k∗ < 1. Note that if the high type chooses R withb

probability k, the belief then reaches this point at time Tk . The optimal scheme is thus givenbby k∗ = k̃ and T ∗ = Tk̃ where k̃ solves
Z Tb
k̃

0

1
b
T
gv
k̃g 2 v −(r+k̃g)t1 c
1 − e−(r+g̃) k̃ =edt = gv−− .
rr r + k̃grg

Appendix B: the case with cS > 0
In this appendix, we argue that it is innocuous to assume cS = 0 by showing that as ∆ → 0,
there exists a sequence of equilibria converging to the equilibrium in Proposition 2 for any
cS > 0. For the appendix, we focus on the case where T ≥ Tb, so that it is optimal to

implement a full-funding equilibrium. The argument can be easily extended to the case of apartial-funding equilibrium.

For some given ∆ > 0, we first show how to select bS (h) = b to implement a full-funding
equilibrium. Given cS > 0, the low type (who has never chosen S before) has no incentive
to choose S. Suppose that the last period in which the low type chooses S with positive26

probability is T − n∆. In this case, we must havenX
(1 − r∆)i (b − wT −(n−i)∆ ) = cS .

(13)

i=1

In a period earlier, the low type also chooses S with positive probability, which means thatnX
(1 − r∆)i (b − wT −(n+1−i)∆ ) = cS .

(14)

i=1

It follows from these thatnb − wT −n∆ =

r∆ S
r∆ X
(1 − r∆)i (b − wT −(n−i)∆ ) =c .
1 − r∆
1 − r∆
i=1

Similarly,b − wt =r∆ Sc ,1 − r∆

for t = ∆, 2∆, · · · , T − (n + 1)∆.
r∆ S
c for t =This means that the agent’s expected productivity must be b − 1−r∆∆, 2∆, ..., T − n∆. We can then derive a belief p such that
pgv + (1 − p)gb = b −

r∆ S
c .
1 − r∆(15)

The left-hand side is the expected productivity when the low type chooses S with probabilityg∆ in every period (until T − n∆). Given this strategy, the belief stays constant over time
for t ∈ [∆, T − n∆]. At t = 0, as p0 is given and in general p0 < p, the low type needs tor∆ Schoose S with a higher probability to satisfy p∆ = p (and hence w0 > b − 1−r∆c . On theother hand, from period T − (n − 1)∆ on, the low type only chooses N . The belief thus pt
goes down until period T , and so does
wt = pt gv.

(16)

To induce R from the high type, it is necessary to have wt ≥ c for all t ∈ [0, T ]. Since
the belief declines monotonically for t ∈ [T − (n − 1)∆, T ], we let wT = c. Note that fort ∈ [T − (n − 1)∆, T ], the belief follows
pt+∆ =

pt (1 − g∆).
1 − pt g∆

This completely characterizes the dynamic paths of pt and wt , from which we can find b andn to satisfy (13) and (15), which we denote b(∆) and n(∆) respectively.

27

We now show that the sequence of equilibria constructed above converges to the equilibrium in Proposition 2 as ∆ → 0, i.e., lim∆→0 b(∆) = b and lim∆→0 n(∆)∆ = 0. Supposeon the contrary that lim∆→0 n(∆)∆ > 0. Then, since wt < b is strictly decreasing fort ∈ [T − n(∆)∆, T ], the left-hand side of (14) is strictly positive in the limit while the righthand side converges to zero. As this is a contradiction, lim∆→0 n(∆) = 0. It then follows
from this that given wT = c, lim∆→0 wT −n∆ = c, and both wt and b converge to c by (15).

References
Araujo, A., Gottlieb, D., and Moreira, H., 2007, A Model of Mixed Signals with Applications
to Countersignaling, RAND Journal of Economics, 38, 1020-43.
Atkeson, A., Hellwig, C., and Ordonez, G., 2015, Optimal Regulation in the Presence of
Reputation Concerns, Quarterly Journal of Economics, 130, 415-64.
Bar-Isaac, H., 2003, Reputation and Survival: Learning in a Dynamic Signalling Model,
Review of Economic Studies, 70, 231-51.
Bergemann, D. and Hege, U., 1998, Venture Capital Financing, Learning, and Moral Hazard,
Journal of Banking and Finance, 22, 703-35.
Bergemann, D. and Hege, U., 2005, The Financing of Innovation: Learning and Stopping,
RAND Journal of Economics, 36, 719-52.
Bergemann, D. and Välimäki, J., 2000, Experimentation in Markets, Review of Economic
Studies, 67, 213-34.
Bolton, P. and Harris, C., 1999, Strategic Experimentation, Econometrica, 67, 349-74.
Bonatti, A. and Hörner, J., 2011, Collaborating, American Economic Review, 101, 632-63.
Chen, C.-H. and Ishida, J., 2015a, Dynamic Performance Evaluation with Deadlines: The
Role of Commitment, revised version of ISER Discussion Paper No. 919.
Chen, C.-H. and Ishida, J., 2015b, Hierarchical Experimentation, ISER Discussion Paper No.
949.
Chung, K.-S. and Eso, P., 2013, Persuasion and Learning by Countersignaling, Economics
Letters, 121, 487-91.
Daley, B. and Green, B., 2012, Waiting for News in the Market for Lemmons, Econometrica,
80, 1433-1504.

28

Ederer, F. and Manso, G., 2013, Is Pay for Performance Detrimental to Innovation? Management Science, 59, 1496-1513.
Feltovich, N., Harbaugh, R., and To, T., 2002, Too Cool for School? Signaling and Countersignaling, RAND Journal of Economics, 33, 630-49.
Gul, F. and Pesendorfer, W., 2012, The War of Information, Review of Economic Studies,79, 707-34.
Guo, Y., 2016, Dynamic Delegation of Experimentation, American Economic Review, 106,
1969-2008.
Halac, M., Kartik, N., and Liu, Q., 2016, Optimal Contracts for Experimentation, Reiew of
Economic Studies, 83, 1040-91.
Hamermesh, D.S., 2015, Citations in Economics: Measurement, Uses and Impacts, NBER
Working Paper 21754.
Holmström, B., and Milrom, P., 1987, Aggregation and Linearity in the Provision of Intertemporal Incentives, Econometrica, 55, 303-28.
Hörner, J., 2002, Reputation and Competition, American Economic Review, 92, 644-63.
Hörner, J. and Samuelson, L., 2013, Incentives for Experimenting Agents, RAND Journal of
Economics, 44, 632-63.
Keller G., Rady, S., and Cripps, M., 2005, Strategic Experimentation with Exponential
Bandits, Econometrica, 73, 39-68.
Klein, N.A. and Rady, S., 2011, Negatively Correlated Bandits, Review of Economic Studies,78, 693-732.
March, J.G., 1991, Exploration and Exploitation in Organizational Learning, Organizational
Science, 2, 71-87.
Manso, G., 2011, Motivating Innovation, Journal of Finance, 66, 1823-69.
Murto, P. and Välimäki, J., 2011, Learning and Information Aggregation in an Exit Game,
Review of Economic Studies, 78, 1426-61.
Price, D.J. de Solla, 1965, Networks of Scientific Papers, Science, 149, 510-15.
Redner, S., 1998, How Popular is Your Paper? An Empirical Analysis of the Citation Distribution, European Physical Journal B, 4, 131-34.
Wallace, M.L., Lariviere, V., and Gingras, Y., Modeling a Century of Citation Distributions,
Journal of Informetrics, 3, 296-303.
29