Social Experimentation with
Interdependent and Expanding Technologies
Umberto Garfagnini

Bruno Strulovici

ITAM School of BusinessNorthwestern University

June 30, 2015

Abstract
How do successive, forward-looking agents experiment with interdependent and endogenous
technologies? In this paper, trying a radically new technology not only is informative of thevalue of similar technologies, but also reduces the cost of experimenting with them, in effectexpanding the space of affordable technologies. Successful radical experimentation has mixed effects: it improves the immediate outlook for further experimentation but decreases the value andthe marginal value of experimentation in a longer term, resulting in less ambitious ‘incremental’
experimentation and in a reduced size of radical experimentation. Incremental experimentation
lowers the option value of similar technologies, which may spur a new wave of radical experimentation. However, experimentation eventually stagnates for all parameters of the model.
JEL Codes: C73, D83, O3

1

Introduction

From the ubiquitous use of transistors and integrated circuits in the electronics industry to the
exploitation of chromatography and DNA sequencing in medical research, contemporary firms may
be seen as building on the innovation of their predecessors: discoveries that took place decades or
years ago are integrated and combined by engineers and scientists to serve as a stepping stone for
new innovation (Baumol, Litan, and Schramm 2007). This process has a number of features best
described as a sequence of experimentations by successive agents. Learning occurs through twochannels: the transmission of information across generations of agents, and the experimentationthat each agent performs during his lifespan.
This type of experimentation cannot be framed as a standard multi-armed bandit problem.
Firstly, the “arms,” or technologies, have interdependent values. For example, the emergence
We are grateful to Alessandro Bonatti, Steve Callander, Xavier Durand, Michal Fabinger, Michael Mandler,
Philip Marx, Kiminori Matsuyama, Joel Mokyr, Francisco Ruiz-Aliseda, Simon Wilkie, Huanxing Yang, and seminar participants at the Spring 2011 Midwest Theory Conference, Banco de México, and the 2012 North American
Winter Meeting of the Econometric Society. Strulovici gratefully acknowledges financial support from the NSF under Grant No. 1151410 and the Alfred P. Sloan Foundation. Email addresses: umberto.garfagnini@itam.mx andb-strulovici@northwestern.edu.

1

of a new drug offers a test not only for the value of that drug, but also for the value of drugs
based on similar chemical compounds or on the same treatment strategy, as illustrated by thetreatment of various cancers.1 Secondly, technologies that seemed prohibitively costly to explore
decades ago are now within reach. The domain of affordable technologies is thus expanding overtime, and this happens through particular innovations which are “radical,” i.e., going beyond themere combination of previous innovations. Radical innovations require substantive investments
which lay the foundation for subsequent “incremental” improvements.2 Finally, to analyze thelong-run dynamics and potential stagnation of experimentation, a satisfactory model must allow
technological and payoff sets which cannot be exhausted in finite time, which means that these setsmust be unbounded.
Formulating a tractable model of interdependent and unbounded technologies and payoffs which
captures an option value of experimentation is challenging. In particular, it is well-known thatthe standard bandit model increases substantially in complexity once arms become correlated.
The classic index result of Gittins (1979) no longer holds. As we will see, experimentation withinterdependent technologies has the implication that experimentation with one technology affects
the option value of experimenting with other related technologies.
This paper provides a novel framework which addresses these issues. At each period, a new
forward-looking firm, or agent, is born and lives for two periods.3 The agent observes the technologies chosen by previous agents and their outcome. In our model, technologies lie on the positive
real half-line and their associated (but unknown) outcomes are drawn according to the path of a
Brownian motion. Technologies are thus interdependent: by trying a technology z between two
known technologies x and y, an agent reduces the variance and updates the expected value ofall technologies between x and y. This feature of the model allows us to capture “incremental”
experimentation as a convex combination of explored technologies.
The agent can alternatively choose to perform radical experimentation, such as developing anentirely new treatment to cure a disease, or a new procedure to produce a good, such as nuclear
power for commercial use in the sixties and seventies.4 This possibility is represented by the1

For example, Novartis initially developed Afinitor to treat renal cell carcinoma. Due to Afinitor’s success,scientists at Novartis believed the drug could be used to treat other types of cancer. They began testing for breast
cancer and later received approval. See “The CEO of Novartis: On Growing After a Patent Cliff ” in the Harvard
Business Review (December 2012).
2
A natural example is the birth of the internet, which may be traced back to the “radical” thinking of J.C.R.
Licklider, a scientist at the Advanced Research Projects Agency, who first conceived the idea of a network of computerswhich led to the creation of ARPANET. See Ruttan (2006, p. 116).
3
The option value of experimentation, which drives incentives to experiment, would not arise with myopic or
one-period agents. We consider agents living for more than two periods in Section 7.
4
Worldwide nuclear capacity grew from 1 gigawatt to 100 gigawatts in these two decades. The number of active
reactors grew similarly during that period. See the annual publication of “Nuclear Power Reactors in the World ”

2choice of technologies which lie beyond the rightmost known technology x. These technologies
have a variance which increases linearly in the distance from x thus generating a higher option
value of experimentation compared to incremental experimentation. Such radical experimentation,
however, entails a specific cost of fundamental research, which increases with the novelty, or size,of experimentation.
The main research questions addressed here concern the type and size of experimentation as it
unfolds over time: Does successful experimentation encourage further radical experimentation, or
does it spur a wave of less ambitious, incremental experimentation? How does the size of radical
experimentation depend on past outcomes? Does radical experimentation persist in the long-run?
Or does it converge to a well-defined technological standard, despite the unboundedness of thetechnological and payoff spaces?
These questions may be tractably studied thanks to the Brownian model, which captures acontinuum of correlated (deterministic) bandits with unbounded payoffs. The Markovian structureof Brownian motion allows us to establish a value for each interval between previously tried technologies, instead of the classic approach of assigning a value for each individual technology. We
develop new techniques to establish properties of this value and use them to characterize the shortand long run dynamics of experimentation.
Successful past experimentation turns out to have two opposite effects on subsequent experimentation. Firstly, past successes reduce not only the value of radical experimentation, but also
its marginal value. Intuitively, if past technologies provide high utility, there is little immediateneed to experiment further. More precisely, it becomes less likely that these technologies will besurpassed by radical experimentation. Likewise, the added benefit from performing more radical
experimentation decreases with past successes, as those successes reduce the probability that theoption value created by this experimentation will be exercised. Formally, the value of radical experimentation is submodular in experimentation size and the value of past technologies. Secondly,
however, a failed radical experimentation reduces the prospects of building on the corresponding
technology. Suppose that in a given field what was the most promising and groundbreaking approach turns out to be a disappointment. Such a failure can create much pessimism about further
new research in that area and result in less ambitious or ’fundamental’ research.
The combination of these two effects creates a complex dynamic of experimentation: Highly
successful radical experimentation immediately begets further radical experimentation. However,should that new attempt be less successful, the initial success becomes a drag on radical experimentation: it reduces the likelihood of further radical experimentation, and the size of such experimentation.
published by the International Atomic Energy Agency.

3

The dynamics of experimentation may be more fully described as follows. When incremental
experimentation takes place, its informational option value decreases. This makes radical experimentation more attractive and may spur a succession of radical experimentations. As newtechnologies become available, however, the option value of incremental experimentation gets replenished, making it more attractive, and radical experimentation gives again way to incremental
experimentation, generating a cycle of sort between the two types of experimentation.
Eventually, experimentation stagnates with probability one, for all specifications of the model.
The intuition may be summarized as follows: Suppose that recent radical experimentation has
been extremely disappointing relative to known technologies, which must eventually happen forany path. Then, because such radical experimentation serves as the basis of further radical experimentation, agents prefer to stay within the confines of incremental experimentation. Therefore, aninformational cascade of sort arises, in which no agent accepts to bear the cost of radical experimentation, and the outlook for further radical experimentation is henceforth frozen at a negative
level. Thereafter, only incremental experimentation occurs, and technological knowledge converges
to a well-defined limit. Establishing this convergence is particularly challenging because both theset of technologies and the range of their payoffs are unbounded – as they should be in any model
investigating whether radical experimentation is sustainable in the long run.
This work draws on several strands of literatures, concerned with experimentation, social learning, and innovation. The literature on multi-armed bandits and strategic experimentation typicallyconsiders a fixed set of independent arms. Here, in contrast, technologies are modeled as a continuum of “arms” with interdependent payoffs and varying costs. Moreover, the cost of experimenting
with a given technology depends on the technologies that have been tried in the past. These features
are new, to our knowledge, in the experimentation literature.5
Our baseline model focuses on a succession of agents living for two periods. This assumption
is made for several reasons. Firstly, it makes the model tractable. It is well-known, and is easy
to show, that when arms are correlated the optimal experimentation policy does not follow a
Gittins-like index policy. An example of this is provided in Appendix E. Secondly, the structureof the model brings it closer to the social learning literature which, since Banerjee (1992) andBikhchandani, Hirshleifer, and Welch (1992), has been an important source of inspiration to study
learning dynamics, and has served as the main paradigm to analyze knowledge accumulation by
successive agents from the adoption of new technologies to cultural change. These models examine
the asymptotic efficiency and fragility of social learning under restrictive assumptions (myopic
agents, exogenous signals, stationary and limited choice sets), however, which make them ill-suited5

Callander (2011) and Jovanovic and Rob (1990) also consider Brownian uncertainty, but their agents are myopic.

The relation to those papers is detailed below.

4

to study important knowledge accumulation processes, such as technological innovation.6 It also
makes social learning models unsuitable to study whether radical experimentation is self-sustainable
in the long run or requires external intervention, a question that has received longstanding interest
from economists and policymakers alike.7
Our model builds on Callander (2011), who uses Brownian motion to model the uncertainty ofpolicy values. In that model, however, agents are myopic and their goal is to find a zero of someBrownian path. In particular, there is no uncertainty about the value of the optimal policy. The
combination of forward-looking agents and uncertainty in the value of the optimal policy createsarguably richer dynamics. Experimentation cycles and stagnation, absent from Callander (2011),
are central features of our analysis.
Jovanovic and Rob (1990) consider a countable infinity of research dimensions, each modeled as
an independently distributed standard Brownian path over a fixed interval. Interpreting each new
dimension as a separate technology, it is unclear why the outcomes of these technologies should
be completely independent of one another. In reality, even radical breakthroughs build on existing
technologies. It is also counterintuitive, in that model, that a breakthrough cannot be followed up
by a further push in the same direction. Lining up technologies on a single dimension allows us
to investigate the consequences of path dependence, which is an important empirical phenomenon
in the history of innovation. The independent dimensions assumption also makes impossible tostudy how the size of radical experimentation is affected by past successes, a key aspect of ouranalysis. The mechanisms underlying experimentation (or innovation) cycles are also different: In
Jovanovic and Rob, agents continue to pursue radical experimentation after disappointing outcomesfrom previous radical experimentation, owing to the independence of Brownian processes driving
each new technology. Here, by contrast, successful radical experimentation begets more radical
experimentation because path dependence affects agent’s beliefs.
Both Callander (2011) and Jovanovic and Rob (1990) restrict attention to myopic agents. The
option value of experimentation, at the heart of the experimentation literature and of the present
paper, is thus absent from those works. Jovanovic and Rob circumvent this issue by allowing theagent to learn, at some fixed cost, the value of a new “technique,” and then decide whether to try
the new technique or use an old one. By disentangling information acquisition from technologicalchoice, that model restores some learning from otherwise myopic agents. In many applications,
technological choices require irreversible investments so that search and experimentation cannot bethus disentangled.
Finally, our model provides a stark prediction for long-run experimentation, with the emergence
67

Mueller-Frank and Pai (2013) study social learning when agents’ signals are costly and endogenously acquired.
See, e.g., Griliches (1992), Hall (1996), Gordon (2012). Recent press coverage includes Slywotsky (2009), Zakaria(2011). Policymakers’ interest was highlighted by President Obama’s State of the Union Address in 2011.

5

of a technological standard after finitely many cycles of radical and incremental experimentation.
The analysis establishes that experimentation and beliefs converge to a well-defined limit despite
the fact that both the technological and payoff spaces are non compact. Showing this result ischallenging. One technical contribution of the paper is to extend to unbounded sets (both forthe domain of technologies and the range of their payoffs) the techniques developed by Easley
and Kiefer (1988) to show the existence of a well-defined limit for agents’ beliefs resulting fromexperimentation.
After describing the baseline model in Section 2, we analyze incentives for radical experimentation and introduce the value of radical experimentation in Section 3 and discuss incremental
experimentation and the challenges presented by the correlation across technology payoffs in Section 4. Section 5 investigates the dynamics of social experimentation, including experimentation
cycles and long-run stagnation. The social inefficiency of this stagnation is discussed in Section 6.
Section 7 discusses how the insights developed in the baseline model can be generalized to agentsliving for more than two periods. Section 8 compares our mechanism of stagnation with those thathave been presented in the literature on growth and discusses ways in which the baseline model
could be modified to break our stagnation result. All the proofs are in the Appendix, which also
contains an extension to the case in which the cost of radical experimentation depends on theoutcome of past experimentation, and a discussion of patent policy.

2

A Model of Social Experimentation

An agent is born in each period t ∈ N = {0, 1, . . . , }, who lives for two periods, “young” and “old.”
The agent is risk neutral and chooses at each period a technology x in the technological space[0, ∞). The payoff B(x) of technology x is initially unknown, except at the origin where B(0) = 0.8
A young agent inherits from the contemporary old agent the knowledge of all technologies andpayoffs that have previously been tried. This information transmission is assumed to be costlessand non-strategic and is the only interaction across those agents.9 The history Ht at time t consistsof all technology-payoff pairs that have been experienced in the past. We let B̄t denote the highest
payoff among explored technologies, x̂t denote the rightmost explored technology, i.e., the current
frontier of the explored domain, with associated payoff B̂t ≡ B(x̂t ), and Gt ≡ B̄t − B̂t ≥ 0 denote thedifference, or gap, between the payoffs of the best explored technology and the frontier technology.
8

The function B(·) could represent utils instead of payoffs, as long as agents are expected-utility maximizers. One

could also consider the larger domain R. Radical experimentation would then have “left” and “right” components.
9
The older generation is not affected by the choice of the younger one and has therefore no incentive to manipulate
the transmission. It could benefit from selling the information. We discuss patents in Appendix F. In Niehaus (2011),
intergenerational information transmission is costly, and altruistic older generations weight this cost of transmission
with the potential benefit for the younger generation.

6

The payoff function B is distributed as a Brownian motion with drift κ and volatility σ > 0.10
The analysis assumes that κ = 0, which avoids complications with old-agent experimentation.11
The payoff of a technology x > x̂t has a normal distribution with mean B̂t and variance σ 2 (x − x̂t ).
Technologies to the right of x̂t thus have the same expected payoff, and a variance that increases
with their distance from x̂t .
A technology x lying in some bounded interval [xl , xr ] with known endpoint payoffs B(xl ) andB(xr ) has a normally distributed payoff with meanB(xl ) +B(xr ) − B(xl )(x − xl ),xr − xl(1)

and variance(x − xl )(xr − x) 2
σ .
(2)
xr − xl
The expected payoff of a technology increases linearly from the endpoint technology with the worst
payoff to the one with the highest payoff. The variance, instead, increases as one moves awayfrom either endpoint, and is maximized at the midpoint of the interval. Owing to the Markovian
structure of Brownian motion, observing the payoff of a technology in any given interval affects
only the distribution of technologies lying in that interval.
Incremental vs. Radical Experimentation
The literature on growth and technological change has taken numerous approaches to define
incremental and radical experimentation, which include informational and payoff components. The
spatial structure of the model provides a natural way to dissociate those concepts. Incremental
experimentation refers, in period t, to a technological choice x ∈ [0, x̂t ]. This type of experimentation
can be interpreted as a convex combination of, or marginal changes to old technologies. It involves
technologies that are less risky to develop but also with more limited potential for success. Radical
experimentation refers to a technology lying beyond the frontier x̂t . Radical experimentation isunbounded in its possibilities and it is characterized by higher variance which could lead to very
successful technologies as well as complete failures.
Costs of Experimentation. Radical experimentation incurs a cost that depends on how far
that technology lies from the current frontier. This cost represents the large initial investmentsthat characterize fundamental research, together with its high uncertainty. In summary, the costof experimentation is equal to c(x − x̂), where c is twice continuously differentiable, increasing,
10

More precisely, uncertainty is modeled with a filtered probability space (Ω, F, P) satisfying the usual regularity

conditions (see, e.g., Karatzas and Shreve 1991) and whose outcomes are identified with the paths of a Brownian
motion.
11
The case of an arbitrary drift is analyzed in the online appendix and does not overturn the stagnation theorem
obtained with a zero drift.

7

convex, and such that i) c(y) = 0 for y ≤ 0 and ii) either c0 (0) > 0 or c00 (0) > 0.12 Incremental
experimentation is assumed to be cheaper than radical experimentation and for simplicity weassume that it is costless. This assumption is, among others, justified by Romer (1990, p. 72):
“Once the cost of creating a new set of instructions has been incurred, the instructions can be usedover and over again at no additional cost. Developing new and better technologies is equivalent
to incurring a fixed cost. This property is taken to be the defining characteristic of technology.”
The qualitative results of the paper are robust to the introduction of a positive cost, provided
that either i) exploitation of old technologies incurs the same cost, or ii) the cost of incremental
experimentation vanishes as experimentation becomes arbitrarily close to known technologies.
This cost asymmetry is required to distinguish between incremental and radical experimentation, which is an important feature of the process of innovation in reality. The formulation also
captures the idea that radical experimentation creates a positive payoff externality on future generations, in addition to an informational externality, because all technologies between the old andnew frontiers become cheaper to explore after the initial investment has been made.
Each agent maximizes his total expected payoff, discounting his second period payoff by a factorδ ∈ (0, 1]. An old agent has no value for information and thus always chooses the best exploredtechnology (with zero drift, the expected payoff of any radical experimentation is equal to the payoffat the frontier). A young agent solves the optimization problem
U (Ht ) = sup EHt B(x) − c(x − x̂t ) + δ max{B(x), B̄t } .

(3)x∈[0,∞)

3

Radical Experimentation

Incentives for radical experimentation are captured by the value of radical experimentation,
V R (Ht ), which represents the highest lifetime payoff a ‘young’ agent can get when his first-period
choice is restricted to radical experimentation, in excess of the payoff guaranteed by exploitation, B̄t . With this restriction on his first-period choice, a young agent’s value function is equal
to (1 + δ) B̄t + V R (Ht ) .
Consider a history Ht , with maximum explored payoff B̄t , frontier payoff B̂t , and gap Gt =B̄t − B̂t . Proposition 1 provides a useful representation for the value of radical experimentation.
Proposition 1 (Value of Radical Experimentation) (i) There exists a function F ∞ : R+ →R+ such that
V R (Ht ) = F ∞ (Gt ) − Gt .
12

(4)

In an extension (Appendix D), c is allowed to also depend on B̄t , thus capturing the idea that current technologies

affect the cost of further radical experimentation.

8

(ii) The value of radical experimentation is decreasing in the gap Gt . (iii) If V R (Ht ) is negative insome period t, radical experimentation is abandoned forever after.
Proposition 1 shows that the value of radical experimentation depends on history only throughthe gap Gt : distinct histories inducing the same gap yield identical incentives for radical experimentation. In particular, the payoff level of the best explored technology does not matter per se.
Accordingly, the value of radical experimentation at time t will be denoted more simply by V R (Gt ).
The incentives for radical experimentation are driven by the probability of finding a new technology whose payoff exceeds B̄t . Since the payoff distribution of new technologies is pinned down by
the payoff at the frontier, B̂t , a higher gap reduces the probability of this event. The monotonicity
property (ii) implies the stagnation result contained in (iii): By definition, the value of radical
experimentation is negative if and only if exploiting the best known technology strictly dominatesradical experimentation. Therefore, if the value of radical experimentation becomes negative, it
can never be positive again, because B̂t is frozen, while B̄t can only increase. The proposition thus
has consequences for the sustainability (or lack thereof) of radical experimentation in the long run,which are exploited in later sections.
The next result characterizes the optimal size of radical experimentation, and shows that themarginal value and the size of radical experimentation are both decreasing in the highest known
payoff, B̄t .
Let xR (Ht ) denote the technology chosen when radical experimentation takes place and ytR (Ht ) =xR (Ht ) − x̂t denote the optimal size of radical experimentation.13 Also let φ(·) denote the density
function of the standard normal distribution.
Proposition 2 (Radical Experimentation) Suppose that radical experimentation is optimal.
Then, (i) the optimal size ytR solves


Gt
δσ
=
c0 (y).
√ φ√| {z }2 y
σ y|{z} M arginal Cost

(5)

M arginal Benef it(ii) ytR is strictly increasing in σ and δ. (iii) If Gt = 0, then ytR is indeed strictly positive. (iv) If
Gt > 0 and ytR > 0, then ytR is strictly decreasing in Gt . (v) Finally, there exists a cutoff G̃ > 0
above which V R (G) < 0.
To get some intuition for Proposition 2, recall that radical experimentations have an expected
payoff of B̂t , regardless of their size. A larger size provides a higher variance, and thus an increased
probability of exceeding B̄t . With a zero gap, the marginal benefit of radical experimentation
13

There may exist several optima, although such case happens with zero probability. In such knife-edge cases, thecomparative statics in the proposition still apply in the sense of the strong set order of lattice theory.

9Marginal Benefit of Radical Experimentation

G=0

G=1

G=2

G=5

0
Size of Radical Experimentation

Figure 1: Marginal benefit of radical experimentation for a discount factor δ = 0.8 and volatility
σ = 3.
(left-hand side of (5)) is then arbitrarily large close to the frontier (see Figure 1). As the size of√radical experimentation increases, the volatility of the payoff, σ x − x̂t , increases at a decreasing
rate, and the marginal benefit of radical experimentation converges to zero. Since the marginal
cost is increasing, the optimal size of radical experimentation is therefore positive and well-defined.
The situation is quite different when the gap is positive, because the marginal benefit of radical
experimentation now converges to zero for radical experimentations close to the frontier. Close to
the frontier, a radical experimentation is accompanied by an inadequately low increase in volatility,and virtually no impact on the expected payoff of the agent when he becomes old. The “outside”
option, B̄t , is thus strictly preferred to small radical experimentations. The marginal benefit ofradical experimentation is single peaked: it is initially pushed up by the increase in the probabilityt,of discovering a payoff above the current outside option B̄t , which is given by 1 − Φ σ√Gx−x̂twhere Φ denotes the distribution function of a standard normal distribution. When the size ofG2radical experimentation reaches σ2t , the marginal benefit starts to decrease, as the probability ofobtaining a payoff greater than B̄t converges to 21 . Figure 1 illustrates the marginal benefit ofradical experimentation for different sizes of the gap.
Because volatility is the agent’s only hope of improving the best technology, a higher volatility
has a positive effect on the incentives to innovate. Similarly, a higher discount factor increases the
incentives to innovate. By contrast, an increase in the gap reduces the marginal benefit of radical
experimentation, because it reduces the probability of surpassing the current outside option. To
maintain that probability, an agent must increase the size of radical experimentation, so as to10increase payoff volatility.
Finally, if radical experimentation is disappointing enough, so that the gap exceeds some threshold G̃, then the value of radical experimentation is negative. This, combined with Proposition 1,
implies that radical experimentation is abandoned forever.

4

Incremental Experimentation

Analyzing incremental experimentation presents a particular challenge, owing to the correlation
across payoff technologies. Experimenting with a new technology reveals information not only aboutthat technology, but also about nearby technologies. Our strategy is to partition the technology
space into ‘units’, separated by previously tried technologies. The partition consists of the finitely
many bounded intervals whose endpoints have been explored and whose interior points have not, andof the unbounded interval [x̂t , ∞). A unit is defined by any such interval, along with the values of its
endpoint payoffs. The distribution of payoffs within a given unit is described by a Brownian bridge:
it is the distribution of Brownian motion on some interval with known end values.14 In particular,
it is conditionally independent of the payoff of all observed technology-payoff pairs outside of theunit. Thus, units have the appealing property that experimentation with a technology lying insidea unit changes only the payoff distribution of technologies that belong to that same unit.
The optimization problem of a young agent can be decomposed in two steps: First, choose
one of the finitely many units corresponding to history Ht . Second, choose which technology topick within that unit. The advantage of this decomposition is that the value of each unit may
be characterized according to a simple index, described by Proposition 3. The index determines
the value of experimentation of the unit, which forms a key block in the analysis of the interplay
between radical and incremental experimentation performed in the next section.
For any bounded unit u, let B L (B H ) denote the smaller (larger) of the two endpoint payoffs,
L the width of the underlying interval, and D = B H − B L ≥ 0 the difference between endpoint
payoffs. These variables clearly depend on the unit they are attached to, but the reference to u isomitted when there is no ambiguity. The unbounded unit at time t is denoted u∞ , again omitting
the reference to t when there is no ambiguity, which has already been analyzed in the previous
section. Figure 2 represents an experimentation history up to t = 3.
Proposition 3 To each bounded unit u corresponds a value of experimentation, V (u, B̄t ), with thefollowing properties:(i) It is optimal for the young agent to choose a technology within the unit (bounded or unbounded)that has the highest value of experimentation.
14

See Billingsley (1968, p. 64) for an introduction to Brownian bridges.

11Expected Payoff
B̄3

x
x1

0

x0
u′

u

x2 = x̂3u′′u∞Technology Space

Figure 2: History of units after three periods.
(ii) There exists a function F : R2+ → [1, ∞) such that the value of experimentation of a bounded
unit can be written as√L

V (u, B̄t ) = (B − B̄t ) + D F

L B̄t − B L
,
D
D

!
.

(6)

(iii) If B H < B̄t , then V (u, B̄t ) is increasing in L, B L and B H .
Moreover, it is strictly optimal, within a bounded unit, to choose a technology closer to the endpointwith the higher payoff.15
The value of experimentation of a unit u is similar to the value of radical experimentation. The
only difference is that a young agent’s choice in the first period is now restricted to a technology
within u, instead of [x̂t , +∞). With this restriction, a young agent’s value function may be written
as (1 + δ) B̄t + V (u, B̄t ) .
Some units may have a negative value of experimentation. Choosing from those units is always
suboptimal: receiving the best explored payoff B̄t in both periods dominates such choices. The
next proposition shows a stronger result: the value of experimentation of a unit is nonincreasing inB̄t . Because B̄t is nondecreasing over time, this implies that once a unit gets a negative value ofexperimentation, it is abandoned forever.
Proposition 4 (Value Monotonicity) For any bounded unit u, V (u, B̄t ) is strictly decreasingin B̄t . If, at any time, V (u, B̄t ) < 0, no technology in u is ever chosen after time t.
15

As a consequence, it is optimal to choose the midpoint of a unit whose endpoints have the same payoff.

121This monotonicity property is shown in two steps. Consider, first, a unit u that does not containthe best explored technology, and suppose that the payoff of that technology is increased fromB̄t to B̄t0 > B̄t . The increase has no effect on the payoff distribution of technologies inside u,
but it reduces the probability that any technology in u beats the best explored technology. This,intuitively, reduces the value of experimenting in that unit, implying that V (u, B̄t0 ) < V (u, B̄t ).
Now, consider a unit u whose endpoints include the best explored technology. For such a unit, ahigher value of B̄t increases, linearly, the expected payoff of all technologies inside the unit. The
variance of payoffs within that unit is unaffected (this is a standard property of Brownian bridge),
however, so exploitation with the best explored technology is relatively more appealing than before
the increase, which again reduces the value of experimentation.16
A common issue in experimentation and social learning models is whether, and with what
probability, agents converge in finite time to some specific action. The next result shows that
exploitation – choosing a known technology – is strictly suboptimal, which implies that a new unit
is created at each period and, hence, that the partitioning in units of the technology space becomes
strictly finer over time.
Proposition 5 (Exploitation) If u contains the best explored technology, then V (u, B̄t ) > 0.
This proposition implies that choosing B̄t for both periods is strictly dominated by choosing anothertechnology included in a unit u whose endpoint payoffs include B̄t . Intuitively, a slight departurefrom the best explored technology reduces the expected payoff of the agent, but it also creates anoption value that dominates that reduction, because volatility increments of Brownian motion (of√order dt) dominate expectation increments (of order dt). By the same argument, exploitation
would remain suboptimal even if incremental experimentation were costly, as long as the costof incremental experimentation goes smoothly to zero as one gets closer to known technologies.
This property is convenient for the analysis: it implies that one does not need to separate pure
exploitation from experimentation. In substance, however, one should note that there is very little
difference, in this model, between pure exploitation and a very small amount of experimentation:
the payoffs and amount of information are very similar in both cases, with very high probability.
To distinguish between incremental and radical experimentations, define the value of incremental experimentation, V I (Ht ), as the maximum value of experimentation over all bounded16

A starker intuition for this result can be obtained by appealing to the theory of large deviations (see, Dembo andZeitouni 1998). As B̄t gets arbitrarily large, the payoff distribution inside the unit u looks closer to a straight line,
joining the low-payoff extremity xl to the best technology xr with payoff B̄t : (B(x) − B(xl ))/(B̄t − B(xl )) →B̄t →∞B L + (B̄t − B L )(x − xl )/(xr − xl ) a.s., where B L is the payoff at xl . As B̄t gets arbitrarily large, therefore, theprobability that any given technology x in u surpasses B̄t , converges to zero. Exploitation of xr remains suboptimal
for all values of B̄t , however, as guaranteed by Proposition 5.

13

units. Given the concept of unit, the value of radical experimentation can be alternatively definedas the value of experimentation of the unbounded unit: V R (Ht ) = V (u∞ (Ht ), B̄t ).
From Proposition 3, an agent prefers radical over incremental experimentation if and only if
the value of radical experimentation exceeds the value of incremental experimentation.

5

The Dynamics of Social Experimentation

In this model, social experimentation takes the form of experimentation cycles, characterized inthis section, in which radical experimentation is followed by incremental experimentation. Radical
experimentation generates a positive externality on all future generations by expanding the frontierof explored technologies and, hence, the set of incremental experimentations.
The set [0, x̂t ] of technologies that are convex combinations of previously explored ones will
be referred to as the active domain. While incremental experimentation refines knowledge about
technologies in the active domain, it is characterized by more predictable outcomes and lacks upside
potential. As the learning value of incremental experimentation goes down, radical experimentation
becomes attractive again, provided that the value of radical experimentation remains positive. Following a highly successful radical experimentation, the best explored technology lies at the frontier.
In that case, further radical experimentation is equally likely to outperform or underperform thepayoff of the current frontier, and a wave of radical experimentations can occur. This and otherresults are formalized in Proposition 6.
Proposition 6 (Radical versus Incremental Experimentation) Consider a history Ht for
which V R (Gt ) > 0. Then, experimentation has the following properties:
1. The probability that radical experimentation occurs at time t or at some future date is strictly
positive, regardless of the value of incremental experimentation.
2. Suppose that Gt = 0.17 If radical experimentation takes place at time t, there exist (historyIdependent) cutoffs bRt and bt such that, letting B̂t+1 and ut denote the payoff resulting fromthat radical experimentation and, respectively, the newly created unit (based on the interval
between old and new boundaries),i) radical experimentation is optimal at time t + 1 if and only if B̂t+1 > bRt , andii) if incremental experimentation is optimal at time t+1, it takes place in the newly createdunit ut if and only if B̂t+1 > bIt .
17

In the Appendix, we analyze both the case in which Gt = 0 and Gt > 0. Here, however, we restrict attention toa zero gap for expositional clarity.

14

Thus, highly successful radical experimentation begets more radical experimentation, while
moderately successful experimentation is followed by incremental experimentation.18 To appreciate
the subtlety of i), one may observe that a high payoff increases both the value of incremental
experimentation in the new unit and the value of radical experimentation. Why does the latter
dominate the former? Intuitively, the conditional payoff distribution on the newly created unit
is, for very high payoff realizations, roughly a straight line, with very low variance.19 This meansthat technologies in the new unit have a lower expectation than the frontier technology, and a
low variance. By contrast, radical experimentation has the same expectation as the value of thefrontier technology, and a variance that is independent of that level, which makes it more attractive.
Whether incremental experimentation occurs within the newly created unit, however, depends onwhether or not the outcome of radical experimentation performed at time t, B̂t+1 , exceeds thecritical threshold bIt . Otherwise, incremental experimentation, when optimal, will take place in theold domain.
The previous results may be summarized as follows: From Proposition 5, we know that exploitation is never optimal. Thus, any wave of incremental experimentations reduces the widthof units in the active domain. This, all else equal, reduces the value of experimentation in theactive domain, by Proposition 3. The value of incremental experimentation may then decrease to
the point of triggering a new round of radical experimentation, which continues until it leads toa disappointing payoff, triggering a new wave of incremental experimentations to explore further
the units created within the new frontier, giving rise to alternating cycles between radical andincremental experimentation.
Because the technological domain is unbounded, it is a priori possible that agents indefinitely
expand the boundary of the active domain as time goes by. Such qualitative property of themodel would correspond to unlimited radical experimentation, and to a never-ending successionof radical and incremental experimentation cycles. By contrast, without radical experimentation,technology and knowledge converge to finite levels. Our next result shows that unlimited radical
experimentation never occurs. This result persists even when radical experimentation entails anarbitrarily positive drift, as long as the optimal policy is well-defined.20
Theorem 1 Radical experimentation ends in finite time with probability one. After radical experimentation has ended, the value of incremental experimentation converges to zero almost surely.
Thus, only incremental experimentation persists along the equilibrium path. As the value of
18
19

These qualitative features of the experimentation process clearly remain valid with a positive gap.
This result is well-known in the literature on large deviations, see e.g., Dembo and Zeitouni (1998). See also

Footnote 16.
20
This result is shown in Section B of the online appendix.

15

additional experimentation goes to zero, we observe the emergence of a technological standard in
the limit. This is because all limiting technologies (when more than one limit technology exists)must be payoff-equivalent almost surely.21
A key step for understanding and proving Theorem 1 is to establish comparative statics for the
incentives of radical experimentation, which we performed in Section 3. The rest of the argument
applies ideas from the experimentation literature to analyze our spatial model of technologicalspace.
Proposition 2 hints at the reason why radical experimentation cannot be sustained in the longrun: for high gaps, the optimal size of radical experimentation drops, which reduces volatility
and, hence, the value of radical experimentation, just when volatility is most needed to makeradical experimentation attractive. That proposition alone, however, does not imply that radical
experimentation terminates in finite time, because the technologies observed in equilibrium arechosen endogenously. To prove stagnation, a key step is to show that, if the value of radical
experimentation were positive at all times, the frontier would keep expanding in steps that arebounded below away from zero.22 Intuitively, the size of radical experimentation is decreasing in
the gap and we know from Proposition 2 that a high enough gap ends radical experimentation
forever. For radical experimentation to continue, therefore, the size of the radical experimentationmust be bounded below. However, radical experimentation will eventually hit a region of thetechnology space where the frontier payoff is so disappointing that the resulting gap exceeds thecritical cutoff G̃ identified in Proposition 2.
Figure 3 shows a simulated path for the technology payoff function B, along with the equilibrium
experimentation dynamics for two cost specifications. The figure highlights several interesting
features of our model. Experimentation cycles arise endogenously along the equilibrium path.
Both the length and the number of cycles is path dependent. In particular, it is not necessarily
the case that a lower cost of radical experimentation leads to more radical experimentation. The
figure shows that radical experimentation may indeed end sooner when experimentation is cheaper.
Cheaper experimentation leads to a larger size of radical experimentation when the gap is zerobut this larger size may lead an agent to explore an unattractive part of the technology space
which is exactly what happens in the simulation. In general, a wave of radical experimentations
can go on for several periods before radical experimentation ends forever. Eventually, the search
for a better technology begins to cluster around a technology discovered by a previous generation,
21

This result bears some resemblance to the informational cascades analyzed in the social learning literature,especially according to the definition provided by Lee (1993) for the case of a continuum of actions. As we alreadymentioned in the introduction, the mechanism is different here, because the amount of information is endogenously
acquired by each generation.
22
See Lemma 2.

16

30
Path of
Brownian motion

25

20

Payoffs

Experimentation pathwith β=11
15

10

5

Experimentation pathwith β=9

0

−5
0

5

10

15
Technology Space

20

25

30

2

y
Figure 3: Dynamics of experimentation: δ = 1, c(y) = 2β
. The payoff function is the realized pathof a Brownian motion with zero drift and volatility σ = 3.
yielding the endogenous-information equivalent of an informational cascade. Society converges to a
suboptimal technology, which is largely path dependent: a single experiment at, say, x = 10 would
completely change the dynamics of the search process when radical experimentation is cheaper,
shifting experimentation towards a different part of the technological space which would uncover
better technologies.

6

Stagnation and Social Inefficiency

Our stagnation result points to a general inefficiency problem with social experimentation. Radical
experimentation generates two positive externalities on future generations, by increasing knowledge
and by expanding the active domain. If these externalities were taken into account by current
generations, wouldn’t they systematically push radical experimentation beyond the equilibrium
level characterized in earlier sections? We formalize this question by introducing an infinitely-lived
social planner who discounts payoffs with a discount factor δS < 1.
Comparing the equilibrium dynamics of experimentation with the social optimum raises several
difficulties. Firstly, it is well-known that optimal experimentation with an infinitely lived agent andcorrelated technologies cannot be characterized by the arm-specific index policies that Gittins andJones (1974) have identified for the standard multi-armed bandit problem with independent arms.23
This is intuitive: trying a new technology teaches something about surrounding technologies and23

See Appendix E for an example.

17affects the set of units.24 Secondly, any comparison between equilibrium and efficient experimentation has to be of a probabilistic nature. Indeed, it is easy to construct specific Brownian paths forwhich equilibrium experimentation will last longer than the social optimum, by assigning an implausibly low payoff to the first technology tried by the social planner and, repeatedly high payoffs
for the radical experimentations arising in equilibrium. The resulting histories are consistent withpossible Brownian paths, and clearly result in equilibrium radical experimentations lasting longer
than what is prescribed by the efficient policy. Thirdly, it is not clear how the social planner should
discount, if at all, future generations.
We circumvent all those difficulties by considering a limiting result as δS goes to 1, i.e., when
the social planner becomes infinitely patient. Let x̂F B (δS ) denote the frontier at which a social
planner with discount factor δS stops radical experimentation. Despite our inability to characterize
the social optimum notwithstanding, we can prove the following result.

Theorem 2 As δS goes to 1, Prob limδS →1 x̂F B (δS ) = +∞ = 1.
Theorem 2 implies, as a corollary, that equilibrium experimentation is with high probability
inefficiently low compared to the social optimum.

7

Beyond Two-Period Agents

Our earlier focus on two-period agents allowed us to tractably analyze experimentation with interdependent technologies and emphasize how past experimentation affects the option value of newtechnologies. This section discusses whether the insights developed in the baseline model extend toagents living for more periods. In particular, does stagnation still occur in this case? Longer-lived
agents have stronger incentives to experiment, in a sense that we will quantify, and the possible
effects of early experimentation on future periods are more complex than with two-period agents.
To maintain tractability, we mainly focus on a modification of the baseline in which agents live forthree periods and each generation is born in the last (third) period of the previous generation.
To avoid confusion, we will refer to an agent in his first period as newly born, while an agent
in his second and third period will be referred to as young or old, respectively, as in the baselinemodel. Given a discount factor δ > 0, period index T = 0, 1, 2 and history Ht , let U I (Ht ; T ) andU R (Ht ; T ) denote the value functions of an agent with 2 − T periods left in the future when histime-t choice is restricted to incremental and radical experimentation, respectively. Recalling the24

Readers may wonder whether units, which are conditionally independent, can be characterized by Gittins-likeindices. We have explored this possibility and concluded that this was highly unlikely: an index of the kind that we
obtained for two-period lived agents does not seem to exist for infinitely lived agents.

18

convention that c(y) = 0 for any y ≤ 0, we have
"
#
2−T
X
I
s
U (Ht ; T ) ≡supEHt B(xt ) +δ (B(xt+s ) − c(xt+s − x̂t+s )) ,{xt+s }2−T
s=0

s.t.

(7)

s=1

xt ∈ [0, x̂t ], xt+s ∈ R+ , s = 1, ..., 2 − T

and
"
R

U (Ht ; T ) ≡
s.t.

sup
{xt+s }2−T
s=0

EHt B(xt ) − c(xt − x̂t ) +

2−T
X

#
s

δ (B(xt+s ) − c(xt+s − x̂t+s )) .

(8)

s=1

xt ≥ x̂t , xt+s ∈ R+ , s = 1, ..., 2 − T

The values of incremental and radical experimentation are defined as in the baseline model bycomparing the agent’s restricted value function to what he would get by exploiting the best known
technology at all periods:
I

I

V (Ht ; T ) ≡ U (Ht ; T ) −

2−T
X

s

δ B̄t ,R

R

and V (Ht ; T ) ≡ U (Ht ; T ) −

s=0

2−T
X

δ s B̄t .

(9)

s=0

From this simple normalization, it follows that the agent prefers incremental over radical experimentation at history Ht if and only if V I (Ht ; T ) ≥ V R (Ht ; T ).
The cases T = 1 and T = 2 correspond to the young and old agents already analyzed in ourbaseline model. We now consider the case T = 0 of a newly born agent. The value of radical
experimentation can be written recursively as25

V R (Ht ; 0) = sup EHt B(xt ) − c(xt − x̂t ) − B̄t + δ max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}xt ≥x̂t+δ(1 + δ) max{B̄t , B(xt )} − B̄t



.

(10)

The main challenge is to characterize the marginal value of radical experimentation for a newlyborn agent, since it is this value which, compared with the marginal cost, determines the size ofradical experimentation. This marginal value turns out to be significantly more complicated withthree periods, not least because radical experimentation in the first period can now create new
incentives for radical experimentation in the second period of a 3-period agent. This effect wasabsent with two periods because the agent was already old after performing a radical experiment.
Radical experimentation by a newly born agent affects both the value of radical experimentationand the value of incremental experimentation in the agent’s second (young) period. First, it createsa new unit which becomes available for incremental experimentation when the agent is young.
Second, a radical experiment also affects the size of the gap in the following period. It is for example
25

A similar recursive expression holds for the value of incremental experimentation.

19

possible that the size of the gap when the agent is young is smaller than when the same agent was
newly born due to a successful radical experiment. Thus, the value of radical experimentation
for a young agent may be increased, compared to the first period, as an outcome of first period
experimentation.
Nonetheless, a newly born agent has a stronger incentive to perform radical experimentation
than a young agent, all else equal.26
Proposition 7 For any history Ht of tried technologies and corresponding outcomes, an agent’s
value of radical experimentation is decreasing in his age: V R (Ht ; 0) ≥ V R (Ht ; 1) ≥ V R (Ht ; 2) =−Gt .
While intuitive, this result is challenging to establish due to the various implications of radical
experimentation by a newly-born agent.
The interplay between radical and incremental experimentation is also more complex, and thevalue of radical experimentation for a newly-born agent is no longer simply determined by the gap
B̄t − B̂t . Nevertheless, an increase in the highest-known payoff, which increases the gap, still reduces
the value of radical experimentation, all else equal. Moreover, if the value of radical experimentation
becomes negative, then all future generations still discard radical experimentation.
Proposition 8 (i) The value of radical experimentation for a newly born agent is decreasing inthe size of the highest known payoff, B̄, all else equal. (ii) If the value of radical experimentationis negative at some time t, radical experimentation is abandoned by all current and future agents.
The optimal radical experiment, xR,0t (Ht ), for a newly born agent now solves the first-ordercondition
δ(1 + δ)σ√φ
2 x − x̂t

G√ t
σ x − x̂t
dEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x+δ
= c0 (x − x̂t ),
dx

(11)

which generalizes equation (5). The first term of (11) is the immediate benefit of a bolder radical
experimentation on the probability of exceeding the current best payoff B̄t , through a higher variance. This effect is stronger than in the baseline model because the benefit of the current radical
experiment extends over a longer horizon. The second term of (11) represents the intertemporal
benefit of bolder radical experimentation, and was absent from the baseline model. It captures the
positive effect that a radical experiment today generates on the experimentation opportunities of
the agent one period ahead. While the intertemporal marginal benefit boosts incentives to perform
radical experimentation for a newly born agent, its effect can be showed to be uniformly bounded
above. This observation is used to show the existence of a gap threshold above which radical
experimentation becomes suboptimal.
26

For the results presented in this section, we assume that the value functions are differentiable.

20

Proposition 9 (i) If Gt = 0, then y R,0 (Ht ) > 0. (ii) There exists a value of the gap Ĝ such that
V R (Ht ; 0) < 0 whenever Gt > Ĝ.
Just like radical experimentation, incremental experimentation becomes harder to characterizewith longer-lived agents. The two-period equation (6) does not have a simple equivalent with three
periods. Some insights developed of the baseline do extend to a longer life-span. For example,
exploitation is suboptimal for a newly born agent, just as it was suboptimal for a young agent.
Moreover, there still is a positive probability that radical experimentation occurs at or after any
history where the value of radical experimentation is positive (Proposition 6.1 in the baselinemodel).
New issues also arise. For example, successful radical experimentation by a newly born agent
replenishes the value of further radical experimentation, but the same agent, when young, may
prefer to switch to an incremental experiment in the newly created unit. In addition, the shortrun dynamics of experimentation are now affected by the life cycle of new generations: each new
generation brings in higher incentives to experiment radically due to an exogenously longer life-spancompared to the aging previous generation.
Despite this additional complexity, it is possible to prove that stagnation surely arises in thelong run.
Theorem 3 Consider the extended model with three-period-lived agents. Radical experimentationends in finite time with probability one.
In summary, the trade-off between radical and incremental experimentation is more complex thanin the case of two-period agents, but the long run dynamics look the same. While the size of the gap
needed to induce a negative option value of radical experimentation may be larger with a longer
life-span, past successful incremental experimentation or disastrous radical experimentation will
eventually extinguish any incentive to perform radical experimentation due to the path-dependenceof the process of experimentation.

Generations overlapping for more than one period
Suppose now that new generations are born when the previous generation is still young. Possible strategic effects are free riding and encouragement effects. Free riding might occur because anagent can avoid radical experimentation in the hope that some other agent will incur the cost of
expanding the current frontier. Radical experimentation pursued by one agent may, however, also
encourage other agents to pursue more radical experimentation when previous experimentation led
to the discovery of an attractive technology. Nevertheless, the long-run dynamics of experimentation would still be affected by large failures at the frontier which should cause the end of radical

21

experimentation once the gap exceeds a critical level. The short-run dynamics of social experimentation will instead be affected by free-riding and encouragement effects. While clearly beyond thescope of this paper, these effects could in principle be investigated through the way in which theyaffect the option values of radical and incremental experimentation.27
In our framework, new strategic issues would also arise, beyond free-riding and encouragement
effects. Consider the experimentation game between a young agent and a newly born agent. Suppose that the newly born agent experiments radically. From the point of view of the young agent,a smaller radical experiment is less valuable than what it would be in the absence of the otheragent’s radical experiment. This is due to two factors. First, an additional radical experiment
involves a duplication of costs. Second, a larger radical experiment has a larger option value ofexperimentation. Therefore, the young agent may prefer to experiment incrementally this period
and use the information generated by the other agent when old. Similarly, a radical experiment by
the young agent may also reduce the option value of radical experimentation for the newly born
agent even if such an agent might have performed a larger radical experiment had he been the only
active agent. Thus, a multiplicity of equilibria may follow.

8

Discussion

8.1

Experimentation Dynamics: An Example

The history of fuel efficiency illustrates both socially inefficient experimentation and the negative
relationship between past successes and incentives for radical experimentation. After the oil crisis of
the 1970s, fuel efficiency for passenger vehicles doubled within ten years, but it has stayed roughly at
the same level for nearly 20 years.28 Past successful efforts to increase fuel efficiency have channeledfurther investments toward improvements of the same technologies, instead of pushing the search
for new radical technologies. There is, however, a broad agreement that radical experimentationis required to significantly increase fuel efficiency, which is close to its theoretical upper boundunder current technologies. While the myopic benefits associated with the exploration of radical
technologies may be low compared to the presumably high development costs, the unquestionable27

With infinitely-lived agents, some of these effects have been studied by Bolton and Harris (1999), Keller, Rady,and Cripps (2005), and Keller and Rady (2010). Akcigit and Liu (2013) identifies another interesting aspect ofstrategic experimentation, where agents hide their research failures, resulting in inefficient replication of unsuccessful
research. The limitation of those models is that they focus exclusively on two-arm settings with independent arms.
While we restrict attention to forward-looking but non-strategic behavior, we complement the literature by enriching
the action space to encompass a continuum of technologies with interdependent and unbounded payoffs.
28
See the report “History of Fuel Economy: One Decade of Innovation, Two Decades of Inaction” by the PEW
Environment Group, available online at http://www.pewenvironment.org.

22large social value that such technologies could generate suggests a fitting illustration of inefficientstagnation.

8.2

Breaking Stagnation

The mechanism behind our stagnation result is simple: as radical experiments with uncertain
and unbounded payoffs expand the technological frontier, the frontier must at some point get so
unattractive relative to the best known technology that no agent wishes to experiment beyond it.
This evolution captures Gordon (2012)’s view that radical innovations eventually get stuck as future
generations become unwilling to make the necessary investments to further push the technologicalfrontier. It is the willingness of each generation to undertake bold experimentation that fades awayover time, not the possibility to innovate radically.
There are two approaches to break the stagnation result: by increasing the new generation’svalue of radical experimentation, or by decreasing the cost of radical experimentation. Consider,first, the cost reductions. A natural extension is to allow the cost of radical experimentation todepend on known technologies: incremental innovations are helpful to produce more radical ones.
In Appendix D, we analyze a setting in which the cost of radical experimentation is decreasing in
the value of the best known technology. As long as this cost remains strictly positive and convex,
however, we show that stagnation still arises with probability 1. The result persists if we add a
positive fixed cost for incremental experimentation, which could a priori help by making radical
experimentation more attractive relatively to incremental experimentation.
A more radical extension of the model would feature population growth as in Kortum (1998)
or capital accumulation, as in standard growth models (Acemoglu 2008). The cost of any given
radical experimentation would then decrease in per capita terms or become small relative to thecapital available. These features would make it more affordable for later generations to increase thesize of radical experimentation, perhaps to the point at which an unattractive technology at thefrontier becomes irrelevant relative to much bolder innovations. This approach would likely suffice
to break stagnation as long as the population or capital available grows in each period regardlessof the technological choices made by current generations.
Turning to the value of experimentation, we analyze in the online appendix (Section B), avariation of the model in which the Brownian path has an arbitrary positive drift.29 Intuitively,
this drift captures the idea that even when the frontier technology is disappointing, the outlook ofmoving beyond the frontier gets increasingly better with the size of this move. However, we show
29

We require that the marginal cost of radical experimentation exceeds this drift as the size the of experimentation

becomes arbitrarily large. Without this assumption, the optimal size of radical experimentation may be infinite andthe problem is not well defined.

23

that this channel does not work: stagnation still arises with probability one.
Our model assumes that radical experimentations are building on the current frontier. In reality,
there are often multiple directions to follow for radical experimentation, and if the frontier along
one direction is not promising, one can always experiment along another direction. With finitely
many directions, however, this extension of the model would not help: one would still reach withprobability 1 a point at which all directions look unappealing.30
Another approach is to consider a fully multidimensional technological space, instead of justmultiple one-dimensional technological spaces. A natural modeling choice here would be to use
a Brownian sheet, which is a multidimensional extension of Brownian motion (see, Revuz and
Yor 1999).31 In such a model, each previously tried technology would be informative about thepayoff distribution of the new technology. This feature is perhaps realistic, since even remotelyconnected technologies may be helpful to each other. But this would also significantly complicate
the analysis by breaking the Markov property which was instrumental in analyzing the “units”
developed in this paper.
In modeling a single decision at each period, we followed the multi-armed bandit and social
learning literatures where agents must choose a single bandit or action at each time. A slight
modification of this paradigm (but not anodyne for our stagnation result, as discussed above) is tofollow Jovanovic and Rob (1990) by allowing the agent to peek at new technologies before deciding
which one to choose. This possibility may be unrealistic when projects have large fixed costs.
For example, the only way to really learn the value of a large particle collider is to build one.
Alternatively, agents may be allowed to split their resources between multiple technologies. For
example, the current generation could exploit a known technology to guarantee enough immediate
consumption but also invest some of its resources in a more radical experiment. If utility fromconsumption is concave and the generation is sufficiently long-lived, patient, or altruistic towards
future generations, such a model would deliver more radical experimentation. However, this alone
would likely not suffice to overturn our stagnation result: agents with a fixed lifespan would always
reach a period at which the technology at the frontier is so poor as to deter them from making theradical investment.
In our model, generations do not directly interact with one another. With longer lived generations and transfers across them, patents would present another potential channel to foster radical
experimentation. If, in particular, older generations are allowed to patent the technologies devel30

Jovanovic and Rob (1990) consider a setting with infinitely many such dimensions. Even in their setting, however,stagnation would obtain if agents were not allowed to “peek” at the values of new technologies before choosing theirtechnology.
31
Brownian sheets have multiple indices. They are not equivalent to n dimensional Brownian motions, and indeed
are much more complex mathematical objects.

24oped when they were young, this may push young generations to make their own experiment, either
to avoid paying licensing fees or in the hope of receiving licensing fees in later periods. These mechanisms are discussed in Section F of the appendix, but they have well-known limits. In particular,
the perspective of future experimentation can also reduce today’s incentives to experiment, as newtechnologies quickly become obsolete. The process of creative destruction can be detrimental toradical experimentation through direct and equilibrium effects, as shown by Aghion and Howitt(1992), and also lead to stagnation.
Finally, if each generation consists of multiple agents, competition may also create incentives
for experimentation. Our model has assumed that there was a single firm experimenting in eachperiod, and that the payoff of any given technology was exogenous. In practice, however, various
firms may compete in the use of any given technology, which may crowd out profit and increases thecosts of the relevant resources. This may spur some firms to choose different technologies, assumingthat technologies are not perfect substitutes, in order to achieve horizontal differentiation.
As this section illustrates, many interesting questions remain open for successfully incorporating
the experimentation paradigm into the innovation and growth literatures. Our paper provides one
step in this direction and a framework on which to base further research on the topic.

AppendicesA

Proofs of propositions from Sections 3 and 4

Proof of Proposition 1
Given an arbitrary history H,32 the expected utility from a radical experimentation x > x̂ is
E[B(x) − c(x − x̂) + δ max{B(x), B̄}] = (1 + δ)B̂ − c(x − x̂)+ δE max{B(x) − B̂, B̄ − B̂}= (1 + δ)B̂ − c(x − x̂) + δE max{k(x), G},
where k(x) = B(x) − B̂. Recall that k(x) ∼ N (0, σ 2 (x − x̂)). Therefore,




√
G
G
√
√
+ GΦ
.
E max{k(x), G} = σ x − x̂φ
σ x − x̂σ x − x̂

(12)

Taking the supremum over x ≥ x̂, we obtain





√
G
G
√
√
+ δGΦ
sup (1 + δ)B̂ − c(x − x̂) + δσ x − x̂φ
σ x − x̂σ x − x̂
x≥x̂






1
G
G
√
= (1 + δ) B̂ + sup−c(y) + δσ yφ
+
δGΦ
,
√
√
σ y
σ y
y≥0 1 + δ
where y ≡ x − x̂. Defining





1
G
G
√
F (G) =
sup −c(y) + δσ yφ
+ δGΦ
,
√
√
1 + δ y≥0
σ y
σ y
∞

32

We drop the time subscripts from now on to simplify the exposition.

25

(13)

h
i
and V R (H) = B̂ + F ∞ (G) − B̄ = F ∞ (G) − G yields the formula claimed by Proposition 1.
Next, we need to evaluate the effect of an increase in the size of the gap, G, on the value of radical
experimentation V R (G) = F ∞ (G) − G.33 For this and other comparative statics results, we will apply
differential techniques to some value functions. These techniques can be applied because the relevant functions are always left and right differentiable, a result which follows from Corollary 4 of Milgrom and Segal(2002).34 All the comparative statics arguments to follow can be applied to the right derivative (or, in caseof a decrease of the parameter, to the left derivative). For expositional simplicity, we drop reference to theside of the derivative.
R∞
If there exists an interior solution y ∗ > 0 to (13) when G > 0, then ∂V∂G(G) = dF dG(G) − 1 =
δ
1+δ Φ

G
√
σ y∗

R

δ
δ
−1 < 0. If there is no interior solution, then F ∞ (G) = 1+δ
G which gives ∂V∂G(G) = 1+δ
−1 < 0.
R

When G = 0, V R (G) = F ∞ (0) and ∂V∂G(G) = 0.
Finally, if V R (Gt ) < 0, the previous comparative statics, monotonicity of the sequence {B̄t }, and thefact that B̂t is unaffected by incremental experimentation, imply that radical experimentation will never be
undertaken for any t0 > t, as it is always dominated by exploitation of the best explored technology.

Proof of Proposition 2
If G = 0, the expected utility from radical experimentation of size y > 0 is equal to√E B(x̂ + y) − c(y) + δ max{B(x̂ + y), B̄} = (1 + δ)B̂ − c(y) + δσ yφ(0),from (13). The first-order condition yields Equation (5) with G = 0. The right-hand side of (5) is increasing
in y, while the left-hand side is strictly decreasing. The left-hand side is also unbounded around 0, andconverges to 0 as y → +∞. Therefore, there always exists a solution to Equation (5) when G = 0, and it isunique. The second-order condition
δσφ(0)− 3/2 − c00 (y) < 0,4yis satisfied, guaranteeing that the first-order condition characterizes maxima.
If G > 0, the expected utility from radical experimentation of size y > 0 is




G
G
√
(1 + δ)B̂ − c(y) + δσ yφ
+
δGΦ
.
√
√
σ y
σ y


δG
G
√
The first-order condition is again given by (5). Differentiating (5) with respect to G, we obtain − 2σy
3/2 φ
σ y ,
which is strictly negative. Since the objective function in (13) is submodular in (y, G), the Strict Monotonicity Theorem 1 of Edlin and Shannon (1998) implies that the optimal size of radical experimentation isdecreasing in G. Similarly, the objective function is supermodular in (y, δ) and (y, σ).
We now show the 
existence of a cutoff G̃ above which radical experimentation has a negative value. LetGδσA(y, G) ≡ 2√y φ σ√y denote the marginal benefit of radical experimentation, given a gap G and a size yof radical experimentation.35
Lemma 1 For any G > 0,
1. limy→+∞ A(y, G) = limG→+∞ A(y, G) = limy→0 A(y, G) = limy→0 ∂A(y,G)
= 0;
∂y
33

As explained in the main text, we adopt the convention of representing the value of radical experimentation

directly through its dependence on the gap.
34
The value function may fail to be differentiable at parameter values for which there exist multiple maximizers.
At such values, however, the value function is still left and right differentiable, with the left (right) derivative being
evaluated at the maximizer that minimizes (maximizes) the derivative of the objective with respect to the parameter.
See Milgrom and Segal (2002).
35
A(y, G) corresponds to the left-hand side of equation (5).

26

2. ∂A(y,G)
< 0.
∂G
2

3. A(y, G) is strictly quasi-concave in y, and maximized at G
σ2 .
Proof. The first two limits in 1) and the sign of the derivative in 2) directly follow from the properties
1
1/2
of φ(·). The limit limy→0 A(y, G) is computed using the fact that limy→0 √1y e− y = limz→∞ zez = 0. One
shows similarly that limy→0 ∂A(y,G)= 0. Strict quasiconcavity of A in y comes from the fact that
∂y
 2
 

G
∂A(y, G)
δσ
G
= 3/2
−1 φ
,
√
∂y
σ2 y
σ y
4y
2

2

G

which is nonnegative below G
σ 2 and negative above. This also shows that σ 2 maximizes A(·, G).
To establish the existence of the threshold G̃ > 0, we start by showing that y ∗ = 0 is the unique maximizerof (13), whenever G is high enough. If c0 (0) > 0, Lemma 1 implies that limy→0 A(y,G) = 
0 < c0 (0) for

any G > 0. Also by Lemma 1, ∂A(y,G)< 0, limG→+∞ A(y, G) = 0, and A(y, G) ≤ A∂GG2
σ2 , G

2= δσ
2G φ(1).

This, combined with the properties of c, implies the existence of a large enough threshold G̃ such thatc0 (y) > A(y, G) for all G > G̃ and y > 0. A similar argument applies if c0 (0) = 0 and c00 (0) > 0, because
δG= 0, from Lemma 1. Substituting y ∗ = 0 into (13) yields F ∞ (G) = 1+δ
. Therefore, thelimy→0 ∂A(y,G)∂yG< 0, for G > 0.
value of radical experimentation is equal to V R (G) = − 1+δ
Proof of Proposition 3
Given that the distribution of payoffs within each unit is conditionally independent from other units, fix
an arbitrary bounded unit u. Suppose, without loss of generality, that B H = B(xr ) > B(xl ) = B L (the caseof an equality is obtained by taking the limit as, say, B L is increased to B H ). Then, for any x ∈ [xl , xr ],
BH − BL(x − xl )(xr − x) 2
B(x) ∼ N B L +(x − xl ),σ ,xr − xl
xr − xl
where N (κ, σ 2 ) denotes the distribution of a Gaussian variable with mean κ and variance σ 2 . Lettingla(x) = xx−x
,
r −xl

B(x) − B L ∼ N D a(x), a(x)(1 − a(x))Lσ 2 .
We also define k(x) = B(x) − B L and B̄ 0 = B̄ − B L to obtain an explicit formula for the expected payoff:!
"!#B̄ 0 − Da(x)B̄ 0 − Da(x)00pp+ Da(x) 1 − ΦE max{k(x), B̄ } = B̄ Φ
σ a(x)(1 − a(x))Lσ a(x)(1 − a(x))L!
p
B̄ 0 − Da(x)p+ σ a(x)(1 − a(x))Lφ,
σ a(x)(1 − a(x))Lwhere Φ and φ are the CDF and pdf of the standard normal distribution. This leads to
Eu B(x) + δ max{B(x), B̄} = (1 + δ)B L + Da(x) + δE max{k(x), B̄ 0 }
(
!!
B̄ 0
− a(x)LD√ p= (1 + δ)B + D a(x) 1 + δ − δΦσ DL a(x)(1 − a(x))
!
√
B̄ 0
p
−
a(x)
L
√ pD
+ δσ a(x)(1 − a(x))φ
Dσ DL a(x)(1 − a(x))!)
B̄ 0− a(x)
B̄ 0
D
√ p
+δ Φ
.
D
σ L a(x)(1 − a(x))
D

27

(14)

√

0

Define q1 ≡ DL , q2 ≡ B̄D , and1F̄ (a, q1 , q2 ) =1+δ

(a 1 + δ − δΦq −ap2
σq1 a(1 − a)+δq2 Φ

q −ap2
σq1 a(1 − a)!!

!
p+ δσ a(1 − a)q1 φ

q −ap2
σq1 a(1 − a)

!)
.

(15)

√
As the common argument of Φ and φ depends only on a(x), L/D and B̄ 0 /D, we can write
(
!)
√
0


L
B̄
Eu B(x) + δ max{B(x), B̄} = (1 + δ) B L + D F̄ a(x),
.
,
D D

(16)

Maximizing the expected payoff over x ∈ [xl , xr ] yields"

U (u; Ht ) ≡ max Eu B(x) + δ max{B(x), B̄} = (1 + δ) B L + D Fx∈[xl ,xr ]√L B̄ 0
,
D D

!#
,

(17)

where
F (q1 , q2 ) = max F̄ (a, q1 , q2 ) .
(18)
a∈[0,1]ih√L− B̄ yields the characterization of the value of experimentation
Letting V (u, B̄) = B L + D F DL , B̄−B
D
for a bounded unit.
The value of experimentation of each unit, whether bounded or unbounded, corresponds to a normalization of the agent’s value function when the domain of choice is restricted to that particular unit. Therefore,
the agent optimally chooses a technology within the unit with the highest value of experimentation.
Next, notice that(!
!)
1q2 − aδσq1 1 − 2a∂ F̄ (a, q1 , q2 )q2 − appp=
1 + δ − δΦ
+
φ
∂a
1+δ
2
σq1 a(1 − a)a(1 − a)σq1 a(1 − a)
≥

1
> 0,
1+δ

for any a ≤ 12 , because the sum of the first three terms in curly brackets is weakly greater than 1 by theproperty of Φ and the last term is positive if, and only if, a ≤ 21 . Thus, any a∗ ∈ arg maxa∈[0,1] F̄ (a, q1 , q2 )lmust be greater than 21 . As a(x) = xx−x
, the corresponding optimal technology from a unit with intervalr −xlxr +xl∗[xl , xr ], x (u, B̄), must lie in
2 , xr whenever B(xl ) < B(xr ).
We suppose once again that the value function (18) is differentiable. The Envelope Theorem implies, forany bounded unit u, that36!
r∂V (u, B̄)δ σ a∗ (1 − a∗ )B̄ 0 − Da∗
p
=
φ
> 0,
(19)
∂L
1+δ 2
L
σ a∗ (1 − a∗ )L(u,B̄)(u,B̄)
where a∗ ∈ arg maxa∈[0,1] F̄ (a, q1 , q2 ). Similarly, for B H 6= B̄, ∂V∂B
> 0 and ∂V∂B
> 0, as can be seenH
Lfrom equations (23) and (24) in the proof of Proposition 6.
Proof of Proposition 4
For any bounded unit u with B H < B̄ we have37!
dF (q1 , q2 )1 ∂F (q1 , q2 )δ
q2 − a∗
p
=
=
Φ
,
(20)
D
∂q2
(1 + δ)DdB̄
σq1 a∗ (1 − a∗ )36

An argument similar to the proof of Proposition 5, which holds independently of the present comparative statics,

shows that any optimum x∗ is always in the interior of a unit.
37
Again, we omit dependence on side-derivatives. See Footnote 34 and the discussion surrounding it.

28

√

L

∗
where, as before, q1 ≡ DL , q2 ≡ B̄−B
D , and a ∈ arg maxa∈[0,1] F̄ (a, q1 , q2 ). This implies that!
∂V (u, B̄)δB̄ − B L − Da∗p=Φ− 1 < 0.
1+δ∂ B̄σ a∗ (1 − a∗ )L √LIf, instead, u contains the best explored technology, then V (u, B̄) = B L − B̄ + (B̄ − B L )F B̄−B
L,1 .
 √
 √


√
B̄)
L
L ∂F
L
This yields ∂V∂(u,= F B̄−B− B̄−B, 1 − 1, which is strictly negative from (25).
L,1
L ∂q
B̄
B̄−B L
1
Exploitation of the best explored technology is always feasible for a young agent, and yields a payoff of(1 + δ)B̄t . Consider a unit u such that V (u, B̄t ) < 0. Then,V (u, B̄t ) < 0 ⇐⇒U (u; Ht )< B̄t ⇐⇒ U (u; Ht ) < (1 + δ)B̄t ,1+δ

where the second equivalence follows from (17). Therefore, choosing the best explored technology dominateschoosing any technology in u. Since B̄t is nondecreasing in t, once the value of experimentation of a unit isnegative, it remains negative.
Proof of Proposition 5
Suppose that B H = B̄. Differentiating F̄ (a, q1 , 1) with respect to a yields
"
!#
!
r
r
1
δσq1 1 − 2a
1−a
1
1−a
∂ F̄ (a, q1 , 1)p
= 1 + δ − δΦ+,φ
(1 + δ)
∂a
σq1
a
2
σq1
a
a(1 − a)δ
which tends to −∞ as a goes to 1. Also, F̄ (0, q1 , 1) ≡ lima→0 F̄ (a, q1 , 1) = 1+δ
< 1 = lima→1 F̄ (a, q1 , 1) ≡F̄ (1, q1 , 1). We have thus shown that any solution to (18) lies in (0,1) whenever q1 > 0 and q2 = 1. Consider
any history Ht and unit u with B̄ as one of its endpoint payoffs. Then, D = B̄ − B L and
!
!
√
√
L
L
L
L
L
L
, 1 > B − B̄ + (B̄ − B )F̄ 1,,1
V (u, B̄) = B − B̄ + (B̄ − B )F
B̄ − B L
B̄ − B L= B L − B̄ + (B̄ − B L ) = 0,
which proves that there is always at least one bounded unit with strictly positive value of experimentation.
Thus, exploitation is strictly suboptimal.
Proof of Proposition 6
If V R (Gt ) > V I (Ht ), then radical experimentation occurs with probability 1 at time t. Thus, consider ahistory Ht such that V I (Ht ) > V R (Gt ) > 0. By continuity, there exists ε > 0 such that the value of radical
experimentation remains positive if the current gap is increased to Gt + ε. Let B(ε) denote the set of pathsof B on [0, x̂t ] which are compatible with Ht and are bounded above by B̄t + ε. B(ε) occurs with positive
probability. By construction, the sequence {B̄t0 }t0 >t stays below B̄t + ε for any path in B(ε), which implies
that the value of incremental experimentation converges to zero over time, by Theorem 1. Thus, the valueof incremental experimentation must fall below the value of radical experimentation, which is uniformly
bounded away from zero for any path in B(ε).
Next, suppose that radical experimentation occurs at an arbitrary history Ht . Let B̂t+1 denote thecorresponding payoff of radical experimentation, and let ut denote the newly created bounded unit. For
the rest of this proof, it turns out to be more convenient to work with the auxiliary indexes Q(u, B̄) =V (u, B̄) + B̄ = B L + D F (q1 , q2 ) and Q(u∞ , B̄) = F ∞ (G) + B̂ which are a simple normalization of thevalue function of a young agent when his first period choice is restricted to a unit u. Thus, the young agent
experiments with a technology within the unit with the highest (auxiliary) index Q(u, B̄). In particular, we
can define the indexes for incremental and radical experimentation by
QI (Ht ) = max Q(u, B̄)u∈P(Ht )and29

QR (Ht ) = Q(u∞ , B̄),(21)

for any history Ht , where P(Ht ) is the collection of bounded units induced by a history Ht . After radical
experimentation has taken place, the new index for incremental experimentation at the start of time t + 1 is












I
Q (Ht+1 ) = max
max Q(u, max{B̄t , B̂t+1 }), Q(ut , max{B̄t , B̂t+1 }) .
u∈P(Ht )




{z
}


|

Q̄(max{B̄t ,B̂t+1 })We first consider how Q̄(max{B̄t , B̂t+1 }) varies with B̂t+1 . If B̂t+1 ≤ B̄t , Q̄(·) is unaffected by B̂t+1 . If
B̂t+1 > B̄t , consider any unit u ∈ P(Ht ). Then,38
!#
! 
"
√
√

∂
L B̂t+1 − B L∂FL B̂t+1 − B Lδ∂Q(u, B̂t+1 )L=
B + DF,=,∈ 0,, (22)
D
D
∂q2
D
D
1+δ
∂ B̂t+1∂ B̂t+1
by (20). Next, we consider the index of the new unit ut . To this end, recall that B̂t is the payoff associatedwith the left endpoint of the newly created unit ut . If B̂t < B̂t+1 ≤ B̄t , then!#"√LB̄t − B̂t∂
∂Q(ut , B̄t )B̂t + (B̂t+1 − B̂t )F,=B̂t+1 − B̂t B̂t+1 − B̂t∂ B̂t+1∂ B̂t+1"!#a∗q2 − a∗p=
1 + δ − δΦ
> 0,
(23)
1+δ
σq1 a∗ (1 − a∗ )where a∗ ∈ arg maxa∈[0,1] F̄ (a, q1 , q2 ). If, instead, B̂t+1 < min{B̄t , B̂t }, then"!#√∂Q(ut , B̄t )∂L
B̄t − B̂t+1
B̂t+1 + (B̂t − B̂t+1 )F=,∂ B̂t+1∂ B̂t+1B̂t − B̂t+1 B̂t − B̂t+1
"
!# 

1
δ
q2 − a∗∗p= (1 − a ) 1 −
Φ
∈ 0,
.
1+δ
1+δ
σq1 a∗ (1 − a∗ )(24)

The upper bound follows from the fact that a∗ ∈ 21 , 1 , then
"
!#


δ
1
1
q2 − a∗δ 12+δ
∗p(1 − a ) 1 −
Φ
≤
<
,
1−
=
∗
∗
1+δ
2
1+δ2
4(1 + δ)1+δ
σq1 a (1 − a )for any δ ∈ [0, 1]. Finally, suppose that B̂t+1 > B̄t (≥ B̂t ), then we have
"
!# 
r

∂Q(ut , B̂t+1 )1
1
1 − a∗
δ∗∗=a (1 + δ) + δ(1 − a )Φ
∈
,
1
.
1+δ
σq1
a∗
1+δ
∂ B̂t+1(25)

The upper bound follows from a∗ < 1, since q2 = 1 in this case, while the lower bound follows from a∗ ≥ 12 ,by Proposition 3.
Below B̄t , Q̄ is flat as a function of B̂t+1 , while the index of the new unit strictly increases in B̂t+1 , byδ(23) and (24). Above B̄t , Q̄ grows at most by 1+δ
following an increase in B̂t+1 , while the derivative ofδ
the index of the new unit is at least 1+δ as shown by (25). If Q̄(B̄t ) > Q(ut , B̄t ) B̂t+1 =B̄t , the index of the
new unit is strictly below Q̄(B̄t ) for any B̂t+1 < B̄t . Since, above B̄t , the lowest slope of the index of the
new unit is strictly higher than the largest slope of Q̄(·), the two indexes intersect exactly once. If insteadQ̄(B̄t ) < Q(ut , B̄t ) B̂ =B̄ , the two indexes necessarily cross only once at some B̂t+1 < B̄t , but never above
t+1

38

t

Again dropping dependence on side-derivatives, see Footnote 34.

30

B̄t . We ignore the case Q̄(B̄t ) = Q(ut , B̄t ) B̂t+1 =B̄t , which occurs with zero probability. The intersection
is denoted by bIt and it is such that incremental experimentation would occur, when optimal, in the newly
created unit if and only if B̂t+1 > bIt .
We now analyze how the index of radical experimentation varies with B̂t+1 , which again denotes the
new frontier payoff. With a slight abuse of notation, theindex of radical experimentation
at time t + 1 isR∞∞given by Qt+1 (B̂t+1 ) = B̂t+1 + F (Gt+1 ) = B̂t+1 + F
max{B̄t − B̂t+1 , 0} as a function of the frontier
∂QR

t+1
∞
payoff B̂t+1 . For any B̂t+1 ≥ B̄t , QR
= 1.
t+1 (B̂t+1 ) = B̂t+1 + F (0) and ∂ B̂
t+1

Suppose instead that B̂t+1 < B̄t . From Proposition 2, there exists a threshold G̃ above which the
optimal size of radical experimentation is 0. Define B̃ by B̄t − B̃ = G̃. Then, for any B̂t+1 ≤ B̃,B̂t+1 +δ B̄t
QR, because the time-t + 1 generation would prefer the frontier technology to any
t+1 (B̂t+1 ) =
1+δ
∂QR

1
radical experimentation. Thus, in this case ∂ B̂t+1 = 1+δ
.

t+1


∂QR
δ
Φ B̄tσ−√B̂yt+1
For B̂t+1 ∈ (B̃, B̄t ), the maximizer of (13) may be interior. In that case, ∂ B̂t+1 = 1 − 1+δ
∈
∗
t+1


1
∗
1+δ , 1 , where y is the optimal size of radical experimentation following history Ht+1 . Thus, the derivative1is always at least 1+δ
.
We need to consider two cases.
Case 1 : Gt = 0. Let bIt denote the cutoff such that incremental experimentation occurs in the new unit if,
and only if, B̂t+1 ≥ bIt , whose existence we just proved. If bIt ≥ B̄t , the index of incremental experimentation
is flat for any B̂t+1 < B̄t , and
 

δ

I
from (22) if B̄t ≤ B̂t+1 < bIt ,0,∂Qt+1 1+δ ∈δ
∂ B̂t+1 , 1 from (25) if B̂t+1 ≥ bIt ,1+δ

which follows from our previous analysis. Since the slope of the index of radical experimentation is always1
δI
at least 1+δ
≥ 1+δ
below bIt and 1 above bIt (as bIt ≥ B̄t ), it follows that QR
t+1 and Qt+1 as functions of B̂t+1
R
cross exactly once. Let bt denote such intersection.
Similarly, if bIt < B̄t , the index for radical experimentation is unchanged, but
if B̂t+1 < bIt , = 0∂QIt+1  ∈ 0, 1
from (24) if bIt ≤ B̂t+1 < B̄t , 1+δ ∂ B̂t+1 δ ∈, 1 from (25) if B̂t+1 ≥ B̄t .
1+δ

A direct comparison of the slopes of the indexes shows that there exists exactly one intersection.
Case 2 : Gt > 0. If bIt ≥ B̄t , the analysis is the same as for Case 1. Thus, there exists a uniqueintersection bRt .
I
If bt < B̄t , the slope of the index of incremental experimentation over the range [max{bIt , Bl }, B̄t ) is
given by (23), which cannot be compared with the slope of QR
t+1 in an unambiguous way. Thus, we cannot
exclude the possibility of multiple intersections between the two indexes.
B

Proof of Theorem 1

Step 1: We first show that radical experimentation ends in finite time almost surely. From Proposition 2,
there exists a threshold G̃ > 0 such that radical experimentation ends after any history such that Gt ≥ G̃.
Therefore, it suffices to show that this threshold is reached almost surely.
Consider an experimentation path along which radical experimentation happens infinitely often, andR
let {ϕ(t)} denote the sequence of times at which radical experimentation occurs. In particular, yϕ(t)=R
xϕ(t) − x̂ϕ(t) > 0 for all t.

31

Lemma 2 {x̂ϕ(t) } is unbounded a.s.
R
Proof. Suppose that x̂ϕ(t) , which is increasing in t, converges to some finite limit x̃. This implies that yϕ(t)Rconverges to 0. From Proposition 2, yϕ(t)is decreasing in the gap. Therefore, there must exist a subsequence{ψ(t)} of {ϕ(t)} such that {Gψ(t) = B̄ψ(t) − B̂ψ(t) } is increasing. Since that sequence is bounded above by G̃,it must converge to some strictly positive limit ρ, and B̄ψ(t) converges to the limit ρ + B(x̃). For sufficiently
high t, the expected payoff from radical experimentation is approximately equal to B(x̃) + δ(ρ + B(x̃)), while
the payoff from exploitation is approximately (1 + δ)(ρ + B(x̃)). Since ρ + B(x̃) > B(x̃) by construction, an
agent will eventually strictly prefer to exploit the best explored technology yielding a payoff close to ρ + B(x̃)
over the radical experimentation corresponding to times in {ϕ(t)}, a contradiction.

From Proposition 2, ȳ = y R (0) is an upper bound on the optimal size of radical experimentation for any
size of the gap. Therefore, |x̂ϕ(t+1) − x̂ϕ(t) | ≤ ȳ. For any s > 0 and path B, let


0
00
0
As (−G̃) = sup x − x : 00max 0 {B(x )} < −G̃, and x < x < s .
x ∈[x,x ]Lemma 3 As (−G̃) > ȳ almost surely as s → +∞.
Proof. By the recurrence property of Brownian motion, there exists a.s. an x̃ > 0 such that B(x̃) < −G̃
and, hence, some s > 0 such that As (−G̃) > 0. The result then follows from the scaling property of Brownian
motion.

Lemma 3 means that if radical experimentation goes far enough, with each leap size bounded aboveby ȳ, the frontier is bound to “fall” into a region where its payoff is less than −G̃. Because B̄ is always
nonnegative, the gap B̄ − B̂ after such history will exceed G̃, prompting radical experimentation to stop.
Combined with Lemma 2, this guarantees that radical experimentation must stop in finite time, almost
surely.
Step 2: Belief convergence.
After radical experimentation stops, all experimentation takes place in a compact domain. The payoff
distribution over that domain is characterized by finitely many Brownian bridges, whose endpoints correspond to previously explored technologies. We now establish that the beliefs resulting from the subsequent
experimentation converge to a well-defined limit. Let K = [0, x̂] denote the domain of experimentation afterradical experimentation has stopped, and µ0 denote the distribution of B on K, given the history leading up
to the end of radical experimentation. For notational simplicity, we will reset to 0 the time at which radical
experimentation has stopped.
Let Θ denote the space of continuous functions on K starting at 0. At any time t the belief µt is a
probability distribution over Θ: µt ∈ ∆(Θ). Some arguments that we need to use hold only for compact
spaces. Because of this, we will sometimes need to replace the paths B by some bounded counterpart. For Λ >1, we will consider any transformation T (·, Λ) of R such that i) T (·, Λ) is continuous and strictly increasing,ii) T (x, Λ) = x for all x such that |x| < Λ − 1, and iii) limx→−∞ T (x, Λ) = −Λ and limx→+∞ T (x, Λ) = +Λ.
Such function is easily built, and is bounded by Λ.
For any Brownian path B, the transformed path bΛ : x 7→ T (B(x), Λ) is continuous and bounded by Λ,
and is homeomorphic to B. In particular, bΛ and B are observationally equivalent. Any belief µ about B
has a corresponding belief µΛ about bΛ and vice versa. Let Θ(Λ) denote the subset of Θ whose elements arebounded in absolute value by Λ, and ∆(Λ) denote the set of distributions over Θ(Λ).
Given a sequence {xt }t≥0 of technology choices, let {µΛt } denote the sequence of beliefs in ∆(Λ) about theΛunderlying transformed path b , obtained through Bayesian updating. It is well-known that this sequence isa martingale and converges to some limit µΛ . This result follows from the Martingale Convergence Theorem,
and is proved similarly to Theorem 4 in Easley and Kiefer (1988). Translating this result in terms of B, thisshows that the sequence {µt } of beliefs about the path B also converge to some limit µ.

32

For any history H leading to the belief µ, let Z Λ (µ) = sup{bΛ (xt ) : xt contained in H}. As is easily
checked, Z Λ (µ) is independent of the particular history leading up to the limiting belief µ, and continuousin µ. We can similarly let Z(µ) = max{B(xt )} = T −1 (Z Λ (µ), Λ)39 where, for each Λ, T −1 (·, Λ) denotes theinverse of T (·, Λ).
Step 3: Technology Convergence and Vanishing Value of Incremental Experimentation.
The next step is to characterize the limit to which technologies converge. For any (z1 , z2 ) ∈ R × R+ ,let r(z1 , z2 ) = z1 + δ max{z1 , z2 } denote the payoff of an agent if the payoff of his chosen technology when
young is z1 and the best explored payoff until then was z2 . Given a technology x, payoff z, and belief µ, letZu(x, µ, z) =r(B(x), z) dµ(B),ΘanduΛ (x, µ, z) =Zr(T (B(x), Λ), z) dµ(B).
ΘUsing the distribution µΛ implied on Θ(Λ) by µ, we haveuΛ (x, µ, z) = v(x, µΛ , z),where we define v, for any µ̃ ∈ ∆(Λ), byZr(bΛ (x), z) dµ̃(bΛ ).

v(x, µ̃, z) =Θ(Λ)

We will use the following lemma, which is proved at the end of this Appendix (Section B.1):
Lemma 4 v(x, µ, z) is continuous over K × ∆(Λ) × [−Λ, Λ].
Given a belief µ with corresponding maximum explored payoff z, a young agent solves the maximization
problem:U (µ, z) = max u(x, µ, z).
x∈K∗
The equilibrium technological path, denoted {xt } is such that, for each t, x∗t maximizes u(x, µt , zt ).

We
now derive properties for the long-run technologies arising in equilibrium. Given a sequence of technologies{xt }∞
t=0 , let M({xt }) be the set of its limit points.
Proposition 10 For any history H, limiting belief µ, and x ∈ M({x∗t }), x ∈ argmaxx0 ∈K u(x0 , µ, Z(µ))
and B(x) = Z(µ).
Proof. Let {x∗tk } denote a subsequence converging to x. By construction,u(x∗tk , µtk , Z(µtk )) ≥ u(x0 , µtk , Z(µtk )),(26)

for any x0 ∈ K. Because Lemma 4 applies only to bounded payoffs, we cannot directly take the limit in theprevious inequality. Instead we will approximate it by its equivalent when the payoffs are bounded by Λ forΛ arbitrarily large. Let Ω(Λ) = {B ∈ Θ : maxx∈K |B(x)| > Λ − 1}. We have for any x, µ̂, zZZΛ|u(x, µ̂, z) − u (x, µ̂, z)| ≤|r(B(x), z) − r(T (B(x), Λ), z)|dµ̂(B) ≤ 2|r(B(x), z)|dµ̂(B).
(27)
Θ

Ω(Λ)

We now show that the right-hand side converges to zero as Λ goes to infinity, uniformly on the domain
K × ∪t {µt } × [0, Z(µ)]. For all x ∈ K and z ∈ [0, Z(µ)], |r(B(x), z)| ≤ (1 + δ)(Z(µ) + maxx∈K |B(x)|).
Therefore, the right-hand side of (27) is bounded above by40Z
2(1 + δ)Z(µ) + max B(x) − min B(x) dµ̂(B).
Ω(Λ)
39
40

x∈K

x∈K

This maximum is also well defined, because B is continuous on the compact domain K.
The inequality relies on the fact that, since B(0) = 0, maxx∈K B(x) ≥ 0 and minx∈K B(x) ≤ 0.

33

We will show that



Z
sup
µt :t≥0Z(µ) + max B(x) − min B(x) dµt (B),
x∈K

Ω(Λ)

x∈K

converges to zero as Λ goes to infinity. For this, it suffices to show the convergence for


Z
sup
max B(x) dµt (B),µt :t≥0Ω(Λ)

x∈K

since the other two terms can be treated similarly.41 We will establish a stronger result, whose proof is in
Appendix B.2. Let P(K) denote the set of all finite partitions of K and, for each Π ∈ P(K) and Z̄ ≥ 0, letµZ̄Π denote the probability measure over Θ corresponding to Brownian bridges with endpoints at consecutive
elements of Π and endpoint values identically equal to Z̄.
Lemma 5 For any constant Z̄ ≥ 0,
(
lim

Λ→+∞

(Z
sup

Π∈P(K)

Ω(Λ)

))


max B(x) dµZ̄
= 0.
Π (B)
x∈K

For each µt , maxx∈K B(x) is clearly dominated, in the sense of first-order stochastic dominance, by the same
maximum under the distribution µΠ , whose partition corresponds to the units of µt , and whose endpoints
are equal to Z(µ), which is greater than Z(µt ).42 Applying Lemma 5 to Z̄ = Z(µ) thus proves the desired
uniform convergence.
This implies that there exists, for any ε > 0, a positive threshold Λ(ε) such that |uΛ (x, µt , z) −u(x, µt , z)| < ε for all (x, t, z) ∈ K × N × [0, Z(µ)] and Λ > Λ(ε). Therefore, (26) implies that, for asequence converging to x, we haveuΛ (x∗tk , µtk , Z(µtk ))≥ u(x∗tk , µtk , Z(µtk )) − ε≥ u(x0 , µtk , Z(µtk )) − ε≥ uΛ (x0 , µtk , Z(µtk )) − 2ε.

Taking the limit as Λ goes to infinity, and using Lemma 4, we obtain thatu(x, µ, Z(µ)) ≥ u(x0 , µ, Z(µ)) − 2ε.
Since ε was arbitrary, this proves that u(x, µ, Z(µ)) ≥ u(x0 , µ, Z(µ)). Proposition 5 also implies thatu(xtk , µtk , Z(µtk )) > (1 + δ)Z(µtk ), which shows that U (µ, Z(µ)) = u(x, µ, Z(µ)) ≥ (1 + δ)Z(µ). Moreover, B(x) ≤ Z(µ), since Z(µ) = supt {B(xt )}, B is continuous, and x is a limit point of {xt }, andu(x, µ, Z(µ)) = B(x) + δZ(µ) ≤ (1 + δ)Z(µ), where the equality holds because B(x) is known given µ.
Therefore, U (µ, Z(µ)) = (1 + δ)Z(µ). In particular, B(x) = Z(µ).

Proposition 10 and its proof also show that the value of incremental experimentation converges to zero
over time: for any x, u(x, µt , Z(µt )) − (1 + δ)Z(µt ) becomes nonpositive.
41

The last term obtains by symmetry, the first term with the constant Z(µ) can in fact be incorporated in Z̄ inthe argument following Lemma 5.
42
Indeed, µΠ is obtained from µt by raising to the level Z(µ) the payoff of each technology explored by time t.

34

B.1

Proof of Lemma 4

Let Θ̃ = Θ(Λ) (Λ is fixed throughout, so there is no ambiguity about the underlying space). Let {(xn , µn , zn )}be a sequence from K × ∆(Θ̃) × [−Λ, Λ] which converges to (x, µ, z) ∈ K × ∆(Θ̃) × [−Λ, Λ]. Then,Z
Zr(bΛ (xn ), zn ) dµn −|v(xn , µn , zn )−v(x, µ, z)| =r(bΛ (x), z) dµ
Θ̃
Θ̃
Z
Z
Λ
Λ
[r(bΛ (xn ), z) − r(bΛ (x), z)] dµ[r(b (xn ), zn ) − r(b (xn ), z)] dµn +
≤
Θ̃
Θ̃
Z
Z
Z
Λ
Λ
r(bΛ (x), z) dµn −r(bΛ (xn ), z) dµ[r(b (xn ), z) − r(b (x), z)] dµn +
+
Θ̃
Θ̃
Θ̃
Z
Λ
Λ
≤ δ|zn − z| + 2|r(b (xn ), z) − r(b (x), z)| dµΘ̃ZZZr(bΛ (x), z) dµ .
r(bΛ (x), z) dµn −|r(bΛ (xn ), z) − r(bΛ (x), z)| dµn +
+
Θ̃

Θ̃

Θ̃

The last term converges to zero by weak convergence of the beliefs. We focus on the second termZZ|r(bΛ (xn ), z) − r(bΛ (x), z)| dµ ≤ (1 + δ)|bΛ (xn ) − bΛ (x)| dµ,Θ̃Θ̃which converges to zero by the Bounded Convergence theorem. Next,ZZΛΛ|r(b (xn ), z) − r(b (x), z)| dµn ≤ (1 + δ)|bΛ (xn ) − bΛ (x)| dµn .
Θ̃Θ̃K is compact and every bΛ ∈ Θ̃ is continuous, hence also uniformly continuous. Fix ε > 0 and let

 

1
1
Λ
Λ
Λ
A
, ε = b ∈ Θ̃ : ∃ λ > s.t. |x − y| < λ =⇒ |b (x) − b (y)| < ε .
j
j
Λ
Λ
Λ
By the previous observations, it also follows
 that for any b ∈ Θ̃, there exists j = j(b ) such that b ∈S∞
A j10 , ε , ∀ j 0 > j. Thus, Θ̃ = j=1 A 1j , ε .
n oNext, since A 1j , εconverges to Θ̃, it follows that for any ϑ > 0, there exists J > 0 such that µ A 1j , ε> 1 − ϑ2 , ∀ j > J. Fix j̃ > J, by weak convergence of beliefs, there exists N > 0 such that
 
c 
 
c 
µn A 1j̃ , ε− µ A 1j̃ , ε< ϑ2 , for any n > N .

Since xn → x, there exists N 0 > N such that |xn − x| < 1j̃ , for any n > N 0 . Finally, we obtain, forn > N 0,
Z
Z
Z
Λ
Λ
Λ
Λ
|b (xn ) − b (x)| dµn =   |b (xn ) − b (x)| dµn +  c |bΛ (xn ) − bΛ (x)| dµn
Θ̃

A

1
,ε
j̃

A

1
,ε
j̃

 
c 
1
≤ sup
|b(xn ) − b(x)| + 2Λµn A
,ε


j̃
b∈A 1j̃ ,ε

 
c 
 
c 
c   
1
1
1≤ ε + 2Λ µn A
,ε
−µ A
,ε
+ µ A
,ε
j̃
j̃
j̃
≤ ε + 2Λϑ.
Since ε and ϑ were arbitrary, this completes the proof.

35



B.2

Proof of Lemma 5

Letting Ω+ (Λ) = {B : maxx∈K B(x) > Λ − 1} and Ω− (Λ) = {B : minx∈K B(x) < −(Λ − 1)}, we haveΩ(Λ) = Ω+ (Λ)∪Ω− (Λ). Since Z̄ ≥ 0, maxx∈K B(x) is nonnegative for all B in the support of any µΠ ∈ P(K).
Therefore,Z
Z
Z
max B(x)dµΠ (B) ≤max B(x)dµΠ (B) +max B(x)dµΠ (B).
Ω(Λ) x∈KΩ+ (Λ) x∈KΩ− (Λ) x∈K

We will prove that the first term (the harder one) converges to zero as Λ → ∞, uniformly in Π. The second
term can be treated similarly.
Put in the language of probability theory, we need to show that, for each Z̄ ≥ 0, the family of randomvariables {XΠ = maxx∈K B(x) : B ∼ µΠ }Π∈P(K) is uniformly integrable.43 To show uniform integrability,
it suffices to prove that there exists p > 1 such that:44p
sup E [XΠ
] < ∞.

Π∈P(K)

Without loss of generality, we set Z̄ = 0 (other cases follow by translation) and K = [0, 1] (other cases follow
by the scaling property of Brownian motion). For each Π, we haveY
P r(XΠ ≤ Λ) =P r(Xi ≤ Λ),
πi ∈Π

where {πi }i describes the units of the partition Π, Xi is the maximum of B over πi , and we are using the factthat the variables {Xi }i are independently distributed. Moreover, each Xi is the maximum of a Brownian
bridge with width ηi (the width of π) and endpoints equal to 0. This implies that45
2

2

P r(Xi ≤ Λ) = 1 − e−2Λ /ηi .
Therefore, we can compute the density of XΠ , and obtain, for p = 2,


Z

X Y
2
2
2
2 4Λ
2

E[XΠ
]=
Λ2
1 − e−2Λ /ηj  e−2Λ /ηi 2 dΛ.
ηi
R+
i
j6=i

For each i, the product with respect to j is bounded by 1, implying thatX Z Λ3 e−2Λ2 /ηi2
2
E[XΠ
]≤4
dΛ.
ηi2
R+
i
Making, for each i, the change of variable ui = Λ/ηi , we obtain
XZ
2
2
E[XΠ ] ≤ 4
ηi u3i e−2ui dui .
i

Since, for any partition Π,R+Pi ηi = 1, we conclude that2
sup E[XΠ] ≤ C,
Π∈P(K)

2

where C = 4 R+ u3 e−2u du < ∞.
R

43

A family {Xi }i∈I is uniformly integrable if limΛ→+∞ {supi {E[|Xi | : |Xi | > Λ]}} = 0.
See, e.g., Durrett (1996), Exercise 4.5.1., p. 260.
45
See, e.g., Durrett (1996), Exercise 7.8.1., p. 433. The formula given there is for a Brownian bridge with width
44

equal to 1. The general formula obtains by the scaling property of Brownian motion, which is easily shown to be
inherited by the Brownian bridge: letting {Bta }t∈[0,a] denote a Brownian bridge with endpoints 0 and a and endpointvalues equal to 0, Bta has the law of Bt − at Ba , where B is the standard Brownian motion.

36

C

Proof of Theorem 2

We first need to define the values of incremental and radical experimentation for the social planner. Givena discount factor δS and an arbitrary history Ht , define the following auxiliary functions:46
"
#
+∞
X
p−t
F B,I
U
(Ht ; δS ) ≡sup EHt B(xt ) +δS (B(xp ) − c(xp − x̂p )) ,(28){xp }+∞
p=t

s.t.

p=t+1

xt ∈ [0, x̂t ], xp ∈ R+ , p = t + 1, ....

and
"
U

F B,R

(Ht ; δS ) ≡supEHt B(xt ) − c(xt − x̂t ) +{xp }+∞
p=t

s.t.

+∞
X

#
δSp−t (B(xp ) − c(xp − x̂p ))

.

(29)

p=t+1

xt ≥ x̂t , xp ∈ R+ , p = t + 1, ....

F B,I

U
(Ht ; δS ) is the value function of a social planner with discount factor δS when his time-t choice
is restricted to technologies that lie within the frontier associated with an arbitrary history Ht . Similarly,
U F B,R (Ht ; δS ) represents the value function when the time-t choice must be greater than or equal to thefrontier. As for the baseline model, if the social planner finds optimal to exploit the best explored technology
at an arbitrary time t, then he will stick to that technology forever. This is because no learning occurs from
exploitation so that the social planner would start t + 1 with exactly the same beliefs that made exploitation
optimal in the previous period. Thus, despite the higher complexity of the social planner’s problem, we candefine the values of incremental and radical experimentation in a similar way to the baseline model up tonormalization:B̄tB̄t, and V F B,R (Ht ; δS ) ≡ U F B,R (Ht ; δS ) −.
(30)
V F B,I (Ht ; δS ) ≡ U F B,I (Ht ; δS ) −1 − δS1 − δSAs the values of incremental and radical experimentation are simply an affine transformation of the valuefunctions of the social planner’s maximization problem (restricting the domains of his first choice within
and outside the frontier, respectively), it also follows that the social planner prefers incremental over radical
experimentation at a history Ht if, and only if, V F B,I (Ht ; δS ) ≥ V F B,R (Ht ; δS ).
We start the analysis with a technical result.
Lemma 6 There exists a sequence {G̃(δS )} with G̃(δS ) → +∞ as δS → 1 such that the value of radical
experimentation for a social planner with discount factor δS is positive for any gap G ≤ G̃(δS ).
Proof. Let G and x̂ denote the gap and the frontier associated with an arbitrary history H. Define a 1-steppolicy as a policy such that the social planner experiments only once and then exploits forever after. Given ahistory H, let V 1,R (G; δS ) denote the value of radical experimentation when the social planner is restricted tousing only a 1-step policy at any history H with associated gap G.47 Clearly, V F B,R (H; δS ) ≥ V 1,R (G; δS ) asa 1-step policy is a feasible policy for the social planner. The restriction to 1-step policies has the advantagethat we can explicitly write down the associated maximization problem as
(



)
B̂
δS
G
δS
G
B̄
√
1,R
V
(G; δS ) = sup− c(y) +
σ yφ
+
GΦ
−
√
√
1 − δS1 − δSσ y
1 − δSσ y
1 − δS
y≥0





1
G
G
√
=
sup −G − (1 − δS )c(y) + δS σ yφ
+ δS GΦ
,(31)√√
1 − δS y≥0
σ y
σ y
46
47

Recall the convention that c(y) = 0 for any y ≤ 0.
When restricted to 1-step policies, the value of radical experimentation depends on the history only through thegap as in the baseline model. This is not the case for general policies which can instead depend on the history in a
non-trivial way.

37

which is equivalent to the maximization problem for an agent in our baseline model with a discount factor
δS
equal to 1−δ. Note that, for δS = 1, the maximization problem reduces to,48
S





G
G
√
sup −G + σ yφ
+ GΦ
= +∞.
(32)
√
√
σ y
σ y
y≥0
Thus, V 1,R (G; 1) = +∞, for any G ≥ 0. This implies that there exists a δ̄ ∈ (0, 1) such that, for any δS ≥ δ̄,
we can find a positive number G̃(δS ) such that V 1,R (G̃(δS ); δS ) > 0. By Proposition 1, it also follows thatV F B,R (H; δS ) ≥ V 1,R (G; δS ) > 0, for any G ≤ G̃(δS ).
Suppose by way of contradiction that there is no sequence {G̃(δS )} with δS > δ̄ as defined in the proof
of Lemma 6 such that the sequence diverges. Take any such sequence. Then, there must exist a Ḡ > 0 suchthat G̃(δS ) < Ḡ along that sequence.49 This implies that there also exists G0 > Ḡ and a subsequence suchthat V 1,R (G0 ; δS ) < 0 for any δS along the subsequence. However, evaluating (32) at G = G0 implies that
V 1,R (G0 ; δS ) will eventually be positive as δS → 1. We thus reached a contradiction.
Fix a path of Brownian motion. Let x̂F B (δS ) denote the frontier at which a social planner with discountfactor δS will stop radical experimentation. Suppose by way of contradiction that lim inf δS →1 x̂F B (δS ) = x̆ <+∞. Take a subsequence x̂F B (δS,n ) such that x̂F B (δS,n ) ↑ x̆ and G̃(δS,n ) → +∞ as n → +∞.50 For x̆ < +∞to be the case, it must be that the sup norm of the path of Brownian motion over the interval [0, x̂F B (δS,n )],defined as supx∈[0,x̂F B (δS,n )] |B(x)|, exceeds

G̃(δS,n )2

for any n. Suppose to the contrary that there exists n̄

G̃(δS,n̄ )such that supx∈[0,x̂F B (δS,n̄ )] |B(x)| ≤. By Lemma 6, the value of radical experimentation for a social
2
planner with discount factor δS,n̄ would be bounded away from zero over [0, x̂F B (δS,n̄ )], regardless of thehistory, because the associated gap can never exceed G̃(δS,n̄ ) over that interval. As the social planner keeps
experimenting within the fixed interval [0, x̂F B (δS,n̄ )], he will eventually explore every unit that still generates
a positive value of experimentation.51 This is because the frontier stays the same during experimentationand the payoff function over [0, x̂F B (δS,n̄ )] is bounded with probability 1 by the almost sure continuity ofBrownian motion, as in the baseline model. Therefore, the value of incremental experimentation for the social
planner must eventually converge to zero and, consequently, below the value of radical experimentation. Thismeans that the social planner will also eventually find radical experimentation optimal, which leads to aG̃(δS,n )for every n along thecontradiction. Hence, it must be the case that supx∈[0,x̂F B (δS,n )] |B(x)| >2
subsequence that we are considering.
Finally, recall that G̃(δS,n ) → +∞ as n → +∞ (or δS,n → 1). This implies that the path of Brownian
motion must have an infinite sup norm over the interval [0, x̆] but this cannot be the case because the a.s.
continuity of Brownian motion implies that supx∈[0,x̆] |B(x)| < +∞ with probability 1. We thus reached thedesired contradiction. Finally, we can conclude that lim inf δS →1 x̂F B (δS ) = +∞ with probability 1.
D

Cost Externalities

In the baseline model, the path of experimentation affects incentives for radical experimentation through i)
the expected value of new technologies, and ii) the opportunity cost of forgoing incremental experimentation.
It is perhaps more realistic to allow radical experimentation to be directly affected by past technologies, if
those technologies are helpful as inputs to produce radical experimentations. Our model may be extended
along this direction, by assuming that the cost of radical experimentation is decreasing in the best available
48

σThe derivative of the objective function with respect to y is 2√
φ
y

49


σ

G
√
y



> 0.

If necessary, we can restrict attention to a subsequence.
Such a subsequence exists by Lemma 6.
51
Even if we have not explicitly defined the value of experimentation of a unit, such a concept can be easily defined50

following (28) and (30).

38technology: the cost is given by a function c(x − x̂, αB̄) for x > x̂ with α ≥ 0 (α = 0 corresponds to thebenchmark model). We assume that, for each w, the function c(·, w) satisfies the same assumptions as in
the baseline model.
Let y R (Gt , αB̄t ) denote the optimal size of radical experimentation at a history Ht with associated gap
Gt , and current best payoff B̄t . The following result is established similarly to Proposition 2 and is statedwithout proof.52
Proposition 11 (Comparative statics) Suppose that α > 0 and that c is submodular. Then, y R (G, αB̄)is increasing in B̄ and in α.
Monotonicity with respect to B̄ of the optimal size of experimentation does not necessarily imply thatradical experimentation itself is fostered by a higher B̄, even if we also assume that the cost c(y, αB̄)is decreasing in its second component. Indeed, when the gap is positive, a higher value of B̄ reduces
the cost of radical experimentation, but also reduces the value of radical experimentation, as shown by
Proposition 1. When the gap is zero, however (following successful radical experimentation), an increase in
the best available technology always stimulates radical experimentation. Therefore, this link between cost
and the best technology should result in longer waves of radical experimentation.
Monotonicity of experimentation size with respect to the gap, which was established by Proposition 2,
may fail in the presence of the cost externality studied here. Indeed, the reduction in the marginal benefitof radical experimentation following a larger gap might be more than compensated by a decrease in the costand marginal cost of radical experimentation. Without the cost externality, an increase in the gap reduces
the marginal benefit while leaving the marginal cost unaffected. In that case, we already know that there isa threshold for the gap above which an agent would always set the size of radical experimentation to zero.
The next result shows that stagnation still occurs as long as the marginal cost of radical experimentation
is bounded below away from zero as the best available technology becomes arbitrarily large. We now assume
that c is decreasing in its second component. Let c̄(y) = limB̄→+∞ c(y, αB̄) denote the lower envelope of thecost functions {c(·, w)}.53
Theorem 4 (Stagnation) Fix α > 0, and suppose that c̄(·) is increasing in a right neighborhood of y = 0.
Then, radical experimentation ends in finite time with probability one.
Proof. Since c̄(·) is increasing in a right neighborhood of y = 0, the properties of each cost function c(·, αB̄)then guarantee that c̄(·) is increasing everywhere. Replicating the proof of Proposition 2, one can show theexistence of a threshold G̃ > 0 above which the marginal benefit of radical experimentation is strictly lessthan the marginal cost, at any y > 0, for any B̄ > 0. Thus, y R (G, αB̄) = 0 for any G > G̃, regardless of theabsolute level of B̄.
For fixed B̄, y R (0, αB̄) is still an upper bound on the size of radical experimentation for any G > 0,
because the incentives to perform radical experimentation are maximal with a zero gap. Moreover, y R (0, αB̄)
increases in B̄, because the cost function is submodular in (y, αB̄). However, the marginal cost is bounded
away from zero by the properties of the lower envelope c̄(·), which implies that there exists 0 < Λ < +∞such that limB̄→+∞ y R (0, αB̄) < Λ. Finally, we can now repeat the same proof as in Step 1 of Theorem 1.
Even if the long-run dynamics is the same with and without intergenerational cost externalities, theshort-run pattern of experimentation might be significantly different in the two scenarios.
52

The agent’s objective function is submodular in (y, α). When there are multiple maximizers, Proposition 11

holds in the sense of the strong set order (see footnote 13).
53
The function c̄(·) is well-defined because, for each y ≥ 0, the sequence {c(y, w)} is decreasing in w and nonnegative.

39

E

Suboptimality of the Gittins Index Policy with Correlated Arms:
an Example

Consider a model with two arms, 1 and 2, and state space S = {s1 , s2 , s3 }.54 In state s1 arm 1 returns
10 and arm 2 returns 5. Pulling arm 1 in state s1 sends the world into state s2 , in which arm 1 continues
returning 10 and arm 2 returns 0. Pulling arm 2 in state s1 sends the world into state s3 , in which the payofffrom arm 1 doubles to 20 and arm 2 continues to pay out 5. States s2 and s3 are absorbing states which
are never left. Since arm 1 returns 10 forever when pulled in state s1 , it has a Gittins index of 10, and the10
optimal payoff from pulling arm 1 first is 1−δ
. Similarly, arm 2 has a Gittins index of 5. However, by pulling20δ
arm 2 first and arm 1 from then on, the agent can obtain a payoff of 5 + 1−δ, which is strictly better thanthe prescribed policy for δ sufficiently close to one.

F

Patents and Radical Experimentation

One way of improving inefficiently short radical experimentation is to use patents. In our model, patents
would naturally consist of transfers between consecutive generations and would be defined by exploiting
the spatial nature of the technological space. In particular, one could assume that, whenever an agent
undertakes radical experimentation, the set of technologies that become available at no cost, due to thisradical experimentation, are patented to this agent.
Such a patent protection system can affect incentives for radical experimentation through two channels:
1. Receiving Royalty Fees Radical experimentation now brings a positive probability that the payoffof radical experimentation will be in some intermediate range at which further radical experimentation issuboptimal for the next generation, but incremental experimentation takes place in the new unit, as shownby Proposition 6. When this happens, a small royalty fee just paid by the new generation increases the exante incentives for radical experimentation. Perhaps counter-intuitively, though, even in the case of a flat
royalty (let alone a more complex royalty structure), the level of that royalty affects the marginal value ofradical experimentation and, therefore, the optimal size of radical experimentation. Moreover, this value
may be non-monotonic in the patent level.55
2. Avoiding Royalty Fees Precisely because an incoming generation has to pay a cost to the previous
generation in order to innovate in a newly created unit, this reduces the value of incremental experimentation
for that generation, relative to radical experimentation: the new generation is “pushed” towards further
radical experimentation.
These incentives are substitutes of each other: the more the new generation avoids royalty fees (say,because they are fixed at high levels), and the lower the patent incentives for the old generation, and viceversa. However, regardless of which effect dominates, royalty fees, structured in this way, would foster radical
experimentation.
54

This example was proposed by Philip Marx for an assignment in Strulovici’s graduate course on dynamic methods(2014).
55
This may be explained as follows: a bolder radical experimentation has a higher variance and, for given mean,a higher probability of reaching very high outcomes. When this happens, the next generation has a strong incentive
to use the technological domain created by this radical experimentation, which generates royalty fees for the oldgeneration. A higher royalty fee may therefore increase the expected marginal value of radical experimentation,
spurring radical experimentation and increasing the size of radical experimentation. However, as the royalty fee gets
arbitrarily large, the new generation prefers to forgo this opportunity and to innovate in older units, causing thenon-monotonicity.

40References
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41

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42

Online Appendix: Social Experimentation
Umberto Garfagnini

Bruno Strulovici

ITAM School of BusinessNorthwestern University

June 29, 2015

Section A of this online Appendix contains the omitted proofs of the results in Section 7 ofthe paper about three-period agents. Appendix B extends the baseline model to the case inwhich the Brownian motion has a positive drift.
We assume that differentiability holds whenever derivatives are taken. We refer to the proofof Proposition 1 and Footnote 33 in the paper for possible ways to deal with non-differentiability.

A

Three-Period Agents

We first need to formally define the values of incremental and radical experimentation for an
arbitrary agent. Given a discount factor δ and an arbitrary history Ht , define the following
auxiliary functions, for any T = 0, 1, 2:1"
U I (Ht ; T ) ≡EHt B(xt ) +sup{xt+s }2−T
s=0

2−T
X

#
δ s (B(xt+s ) − c(xt+s − x̂t+s )) ,

(1)

s=1

xt ∈ [0, x̂t ], xt+s ∈ R+ , s = 1, ..., 2 − T

s.t.
and
"
U R (Ht ; T ) ≡supEHt B(xt ) − c(xt − x̂t ) +{xt+s }2−T
s=0

s.t.

2−T
X

#
δ s (B(xt+s ) − c(xt+s − x̂t+s )) .

(2)

s=1

xt ≥ x̂t , xt+s ∈ R+ , s = 1, ..., 2 − T

We can define the values of incremental and radical experimentation in a similar way to the1

Recall the convention that c(y) = 0 for any y ≤ 0.

1

baseline model up to normalization:
V I (Ht ; T ) ≡ U I (Ht ; T ) −2−TXδ s B̄t ,and V R (Ht ; T ) ≡ U R (Ht ; T ) −

s=0

2−T
X

δ s B̄t .

(3)

s=0

We focus on the problem of a newly born agent. We can write the value of radical experimentation
recursively as
V R (Ht ; 0) = sup EHt B(xt ) − c(xt − x̂t ) − B̄t + δ max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}xt ≥x̂t+δ(1 + δ) max{B̄t , B(xt )} − B̄t



.

(4)

A similar recursive expression holds for the value of incremental experimentation.
Lemma A.1 characterizes an important property of the intermediate expected value of exper
imentation, EHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x , for an arbitrary radical experiment x, which
will be useful later on.
Lemma A.1. Fix an arbitrary history Ht with associated gap Gt > 0, and a technology x ≥ x̂t .
Then,


dEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x
1+δ
,
(5)
<
2
dB̄t
where B̄t denotes the highest known payoff at time t.
Proof. Let x ≥ x̂t , then
EHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x
Z


I
R
2
=
max{V (Ht ∪ {(x, B)} ; 1), V (Ht ∪ {(x, B)} ; 1)} ψ B; B̂t , σ (x − x̂t ) dB,

(6)

R




where ψ B; B̂t , σ (x − x̂t ) is the pdf of a normal distribution with mean B̂t and varianceσ 2 (x − x̂t ). Notice that V R (Ht ∪ {(x, B)} ; 1) = (1 + δ)V R (max{B̄t − B, 0}), by definition of thevalue of radical experimentation in the baseline model. Also,2

 II
V I (Ht ∪ {(x, B)} ; 1) = (1 + δ) max Vold
(Ht ∪ {(x, B)}), Vnew
(Ht ∪ {(x, B)}) .

(7)

I
Vold
(Ht ∪ {(x, B)}) is the value of incremental experimentation among the old units, which isaffected by the radical experiment at time t through the possible change in the size of the highestI
known payoff. Vnew
(Ht ∪{(x, B)}) is the value of incremental experimentation of the unit created2

by the radical experiment x, of length x − x̂t , with associated left endpoint outcome B̂t and rightendpoint outcome B.
Suppose that the gap is positive at time t, that is, Gt = B̄t − B̂t > 0, so that the technology
with the highest known payoff lies strictly within the frontier x̂t at time t. We want to consider
the effect of an increase in the highest known payoff at time t, B̄t . Differentiating with respect
to B̄t , we obtain that
dEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|xdB̄t
 
Z +∞ 
d max{V I (Ht ∪ {(x, B)} ; 1), V R (Ht ∪ {(x, B)} ; 1)}2=ψ B; B̂t , σ (x − x̂t ) dB.
dB̄t
−∞

(8)

First, notice that an increase in B̄t weakly reduces V R (Ht ∪ {(x, B)} ; 1) for any fixed B due
to the monotonicity of the value of radical experimentation in the baseline model established in
Proposition 1. Second, let us consider the effect on the value of incremental experimentation.
Consider values of B such that B ≤ B̄t , then the value of incremental experimentation of theold units is also decreasing in the highest known payoff from Proposition 4. Similarly, the valueof incremental experimentation of the new unit is decreasing in B̄t given that we assumed thatB̄t ≥ B. Therefore, 
d max{V I (Ht ∪ {(x, B)} ; 1), V R (Ht ∪ {(x, B)} ; 1)}
ψ B; B̂t , σ 2 (x − x̂t ) dB < 0. (9)
dB̄t
−∞

Z B̄t

Next, suppose that B > B̄t . Now, an increase in B̄t increases the value of experimentationof the two old units which share the technology with the highest known payoff at time t as an
endpoint by Proposition 3 because B̄t+1 = B > B̄t . The value of experimentation of the newunit is instead unaffected given that B > B̄t renders it independent from small increases in B̄t .
The same holds for the value of radical experimentation because B > B̄t implies that Gt+1 = 0

3and thus V R (Ht ∪ {(x, B)} ; 1) = (1 + δ)V R (0). Therefore, 
Z +∞ 
d max{V I (Ht ∪ {(x, B)} ; 1), V R (Ht ∪ {(x, B)} ; 1)}
ψ B; B̂t , σ 2 (x − x̂t ) dB
dB̄t
B̄t
Z +∞

I
dV (Ht ∪ {(x, B)} ; 1) ≤ψ B; B̂t , σ 2 (x − x̂t ) dB
dB̄t
B̄t
Z +∞

dV I (Ht ∪ {(x, B)}) 
2(1 + δ)=ψ B; B̂t , σ (x − x̂t ) dB
dB̄t
B̄t
Z +∞


(1 + δ) ψ B; B̂t , σ 2 (x − x̂t ) dB
≤
B̄t



2
= (1 + δ) 1 − Φ B̄t ; B̂t , σ (x − x̂t )< (1 + δ) 1 − Φ B̂t ; B̂t , σ 2 (x − x̂t )
=

1+δ
.
2

(10)

The first inequality follows from the previous discussion. The second inequality instead follows
from equation (23) in the paper, because 1 is the upper bound on the derivative of the valueof experimentation of the units that contain the previous highest known payoff, B̄t , but do
not contain the current highest payoff, as B̄t < B = B̄t+1 , with respect to an increase in B̄t .
Finally, the third inequality follows from the fact that B̄t > B̂t and the symmetry of the normal
distribution. Putting (9) and (10) together yields the result.
We then show that the value of radical experimentation is decreasing in an agent’s remaining
life-span.
Proposition A.1. For any arbitrary history Ht , the value of radical experimentation is (weakly)
decreasing with age, that is, V R (Ht ; 0) ≥ V R (Ht ; 1) ≥ V R (Ht ; 2) = −Gt .
Proof. The observation that V R (Ht ; 1) ≥ V R (Ht ; 2) = −Gt follows from the analysis of thebaseline model. Next, we are going to show that V R (Ht ; 0) ≥ V R (Ht ; 1), that is, for a given
history, the value of radical experimentation is weakly higher for a newly born agent than for ayoung agent. Note that
V R (Ht ; 0) = sup −Gt − c(x − x̂t ) + δEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x
x≥x̂t





√
Gt
Gt
√
√
+ GΦ
,
+δ(1 + δ) −Gt + σ x − x̂φ
σ x − x̂σ x − x̂

4

(11)







√
Gt
Gt
√
√
V (Ht ; 1) = sup −c(x − x̂t ) + δσ x − x̂φ+ δGt Φ− (1 + δ)Gt .
σ x − x̂σ x − x̂
x≥x̂t
(12)
Then,
R

V R (Ht ; 0) − V R (Ht ; 1)
= sup −δ 2 Gt − c(x − x̂t ) + δEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x
x≥x̂t





√
Gt
Gt
√
√
+δ(1 + δ) σ x − x̂φ+ Gt Φ
σ x − x̂σ x − x̂





√
Gt
Gt
√
√
+ δGt Φ.
− sup −c(x − x̂t ) + δσ x − x̂φ
σ x − x̂σ x − x̂
x≥x̂t

(13)

Let x∗,1 = x∗,1 (Gt ) denote the optimal technology for the maximization problem associated withthe value function V R (Ht ; 1), which depends only on Gt from our baseline analysis. Then,
V R (Ht ; 0) − V R (Ht ; 1)
≥ −δ 2 Gt − c(x∗,1 − x̂t ) + δEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x∗,1





√
Gt
Gt
∗,1
√
√
+ δ(1 + δ) σ x − x̂φ+ Gt Φ
σ x∗,1 − x̂
σ x∗,1 − x̂





√
Gt
Gt
∗,1
∗,1
√
√
+ δGt Φ− −c(x − x̂t ) + δσ x − x̂φ
σ x∗,1 − x̂
σ x∗,1 − x̂= δEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x∗,1






√
Gt
Gt
2
∗,1
√
√
+ δ σ x − x̂φ+ Gt Φ
− Gt .
σ x∗,1 − x̂
σ x∗,1 − x̂

(14)Let y ∗,1 (G) = x∗,1 (G) − x̂ and define the functionp
F(G) = σ y ∗,1 (G)φNotice that F(0) = σ

p

!

G
σ

p
y ∗,1 (G)

+ GΦ

!

G
σ

p
y ∗,1 (G)

− G.

(15)

y ∗,1 (0)φ(0) > 0, andGσF 0 (G) = p
φ
2 y ∗,1 (G)σpy ∗,1 (G)!

dy ∗,1 (G)
+Φ
dG

5

G
σ

p
y ∗,1 (G)!
− 1 < 0,

(16)

∗,1

because dy dG(G) < 0 by Proposition 2 in the main text. Therefore,Gp
inf F(G) = lim F(G) = lim σ y ∗,1 (G)φ

G≥0

G↑+∞

σ

G↑+∞

!

p
y ∗,1 (G)G

− lim G 1 − Φ

σ

G↑+∞

p
y ∗,1 (G)

!!
.

(17)
The first limit in (17) equals zero due to the properties of the standard normal pdf and the factthat y ∗,1 (G) ∈ [0, y ∗,1 (0)] for any G ≥ 0.
By applying L’Hôpital’s rule for an arbitrary y > 0, we obtain thatlim G 1 − Φ

G→+∞

G
√

φ


= lim

σ y

G→+∞



G
√

σ y

G−2


= 0.

(18)

As y ∗,1 (G) is bounded and monotonically decreasing in G, it also follows that0≤G 1−Φ

!!

Gσ≤G 1−Φ

py ∗,1 (G)Gσpy ∗,1 (0)

!!
.

(19)

Taking limits gives that inf G≥0 F(G) = 0. This shows that V R (Ht ; 0) − V R (Ht ; 1) ≥ 0 given that

EHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x∗,1 ≥ EHt V I (Ht+1 ; 1)|x∗,1 ≥ 0,(20)

by Proposition 5. This completes the proof.
We can now show the following result which generalizes the idea that a larger gap reduces thevalue of radical experimentation even for a longer life span. As in the baseline model, we also
show that a negative value of radical experimentation leads to the abandonment of that type ofexperimentation.
Proposition A.2 (Value of Radical Experimentation). (i) The value of radical experimentation for a newly born agent, V R (Ht ; 0), is decreasing in the size of the highest known payoff,B̄t , all else equal. (ii) If the value of radical experimentation is negative at some time t, radical
experimentation is abandoned forever after.
Proof. (i) Differentiating V R (Ht ; 0) with respect to B̄t , while keeping everything else fixed, andapplying the envelope theorem gives− 1 − δ(1 + δ) 1 − Φ



G
√ t
σ x∗,0 − x̂dEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x∗,0
+δ
,
dB̄t

6

(21)

where x∗,0 is a maximizer. Lemma A.1 implies that (21) is negative. Therefore, it follows thatthe value of radical experimentation is decreasing in the size of the highest known payoff.
(ii) Fix a history Ht . Recall that V R (Ht ; 1) = (1 + δ)V R (Gt ) where Gt is the gap inducedby history Ht . The monotonicity of V R (G) with respect to the size of the gap G, which follows
from Proposition 1 in the main text, implies that, for any given technology x lying inside thecurrent frontier and any associated outcome B(x), V R (Ht ; 1) ≥ V R (Ht+1 ; 1) where Ht+1 =Ht ∪ {(x, B(x))} because max{B̄t , B(x)} ≥ B̄t .
Next, notice that V R (Ht ; 0) < 0 implies that the newly born agent at time t will choose either
exploitation or incremental experimentation in his first period. Let x∗t denote the technology thus
chosen and let B(x∗t ) denote the associated outcome. Then, the observation made in the previous
paragraph and Proposition A.1 imply that V R (Ht+1 ; 1) ≤ V R (Ht ; 1) ≤ V R (Ht ; 0) < 0, where
Ht+1 = Ht ∪{(x∗t , B(x∗t ))}. Therefore, the newly born agent will also find radical experimentation
suboptimal when young.
Finally, let us consider the incentives to perform radical experimentation of the subsequent
newly born generation at time t + 2. We know from our previous analysis that no radical
experimentation has taken place at times t and t + 1. Clearly, if exploitation has taken placeat both t and t + 1, then V R (Ht+2 ; 0) = V R (Ht ; 0) < 0. However, incremental experimentation
might have taken place at either or even both times. As incremental experimentation weakly
increases the highest known payoff, the incentives to perform radical experimentation can only
decrease.
Next, let xR,0t (Ht ) denote the technology which maximizes the value of radical experimentation for a newly born agent.
Proposition A.3. If Gt = 0, then xR,0t (Ht ) > x̂t .
Proof. The first order condition of the maximization problem of a newly born agent pursuing
radical experimentation is given byδ(1 + δ)σ√φ
2 x − x̂t

G√ t
σ x − x̂t
dEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x+δ
= c0 (x − x̂t ).
dx

7

(22)

Assuming differentiability once again, we also have that
dEHt max{V I (Ht+1 ; 1), V R (Ht+1 ; 1)}|x
=
dx


Z

d max{V I (Ht ∪ {(x, B)} ; 1), V R (Ht ∪ {(x, B)} ; 1)} 
2=ψ B; B̂t , σ (x − x̂t ) dB
dx
R


2
Z
dψ B; B̂t , σ (x − x̂t )+ max{V I (Ht ∪ {(x, B)} ; 1), V R (Ht ∪ {(x, B)} ; 1)}
dB.
(23)
dx
R
Fix B ∈ R and let us consider the effect of a small increase in the size of the radical experiment
on V R (Ht ∪ {(x, B)} ; 1). Such an increase has no effect on the value of radical experimentation
starting from t + 1 because the new value of radical experimentation depends only on the sizeRof the gap which, fixing B, is independent from x. Thus, dV (Ht ∪{(x,B)};1)= 0. Next, considerdx
the effect of a similar increase on the value of experimentation of the old units, such an effect isalso zero because the radical experiment at time t affects the value of experimentation of the oldunits at time t + 1 only through the outcome of the experiment, not the size of the experiment.
dV I (H ∪{(x,B)};1)= 0. However, the size of the radical experiment at time t has a
Therefore, old t dx
direct effect on the value of experimentation of the newly created unit because of the increase inI∪{(x,B)};1)> 0. This shows that theits length. From the baseline model, we know that dVnew (Htdx
first integral in (23) is non-negative.
Next,


 "
# 
2
ψ
B;
B̂
,
σ
(x
−
x̂
)
dψ B; B̂t , σ 2 (x − x̂t )2tt(B − B̂t )= −1 + 2.
(24)dx
σ (x − x̂t )2(x − x̂t )which is also positive for radical experiments close to the frontier. Therefore, when Gt = 0, thesize of radical experimentation is strictly positive given that the first term in (22) is unboundedclose to the frontier.
Proposition A.4. There exists Ȳ > 0 such that xR,0t (Ht ) − x̂t < Ȳ for any history Ht .
Proof. Suppose by way of contradiction that the claim is false. Take a monotone sequence {Ȳk }with limk→+∞ Ȳk = +∞. Then, for any k, there must exist a history Hk such that ykR,0 ≡xR,0 (Hk ) − x̂k > Ȳk , and without loss of generality we can also assume that the sequence {ykR,0 }is monotone increasing. From the first-order condition (22), it follows that the left-hand side
converges to +∞ along the sequence {ykR,0 } by the strict convexity of the cost function, while8the first term on the right-hand side converges to zero. Therefore, it must also be the case that
dEHk max{V I (Hk,+1 ; 1), V R (Hk,+1 ; 1)}|x
lim
k→+∞
dx

= +∞,

(25)

x=xR,0 (Hk )where Hk,+1 ≡ Hk ∪ (xR,0 (Hk ), B(xR,0 (Hk ))).
IFirst, suppose that max{V I (Hk,+1 ; 1), V R (Hk,+1 ; 1)} = Vold
(Hk,+1 ; 1) which is the value ofI (HdEHk [Voldk,+1 ;1)|x]
incremental experimentation over the old units. Notice thatis necdx
x=xR,0 (Hk )essarily uniformly bounded because the radical experiment affects the value of incremental experimentation over the old units only through the outcome at the new frontier, not through thesize of the experiment. Thus, to complete the proof, it suffices to show that whenever experimentation is either radical or incremental over the new unit, the derivative of the expectation isstill bounded above which would deliver the desired contradiction.
We can write the derivative of the expectation asI
dEHt max{Vnew(Hk,+1 ; 1), V R (Hk,+1 ; 1)}|xR,0 (Hk )
=
dx
h
n
o
n
o
i
Z d max{V I (Hk ∪ (xR,0 , B) ; 1), V R (Hk ∪ (xR,0 , B) ; 1)} 

new
k
k
=
ψ B; B̂k , σ 2 ykR,0 dB
dx
R
(26)


Z
n
o
n
o
dψ B; B̂k , σ 2 ykR,0R,0I
R+ max{Vnew(Hk ∪ (xR,0dB. (27)
k , B) ; 1), V (Hk ∪ (xk , B) ; 1)}dx
R

9Let us focus on the integral (26),
h
n
o
n
o
i
Z d max{V I (Hk ∪ (xR,0 , B) ; 1), V R (Hk ∪ (xR,0 , B) ; 1)} 

new
k
k
2 R,0
dB
ψ B; B̂k , σ ykdxRon
Z dV I (Hk ∪ (xR,0 , B) ) 

new
k
≤
ψ B; B̂k , σ 2 ykR,0 dB
dx
R
o
n
I
Z

dVnew
(Hk ∪ (xkR,0 , B) ; 1) 
= (1 + δ)ψ B; B̂k , σ 2 ykR,0 dB
dx
ZR


δσφ(0)q≤ψ B; B̂k , σ 2 ykR,0 dB
R 4 y R,0k
δσφ(0).
= q
4 ykR,0(28)

The first inequality follows from the discussion in the proof of Proposition A.3, while the second
inequality follows from (19) in the main text. Therefore, as k goes to +∞, the limit superior of(26) is bounded above by 0.
dψ (B;B̂k ,σ 2 ykR,0 )> 0,Next, consider
theintegral
(27).
Let
A
⊆
R
denote
the
set
over
which
k
dx


q
that is, Ak = B ∈ R : |B − B̂k | > σ ykR,0 . Then, we can write the integral in (27) as


2 R,0
n
o
n
o
dψ B; B̂k , σ yk
R,0
I
R
dB
max{Vnew(Hk ∪ (xR,0k , B) ; 1), V (Hk ∪ (xk , B) ; 1)}
dx
R


R,0
2
Z
n
o
n
o
dψ B; B̂k , σ yk
R,0I
R
max{Vnew=(Hk ∪ (xR,0
,
B)
;
1),
V
(H
∪
(x
,
B)
;
1)}
dB
k
k
k
dx
R\Ak
|
{z
}
≤0


Z
n
o
n
o
dψ B; B̂k , σ 2 ykR,0R,0R
I
dB. (29)+
max{Vnew(Hk ∪ (xR,0k , B) ; 1), V (Hk ∪ (xk , B) ; 1)}
dx
Ak

Z

10

Then,
oonndψ B; B̂k , σ 2 ykR,0R,0R
I
max{Vnew(Hk ∪ (xR,0dB
k , B) ; 1), V (Hk ∪ (xk , B) ; 1)}
dx
Ak


Z
o
o
n
n
ψ B; B̂k , σ 2 ykR,0
R,0
R
I
=−
max{Vnew(Hk ∪ (xR,0dB
k , B) ; 1), V (Hk ∪ (xk , B) ; 1)}
2ykR,0
Ak
{z
}
|
≤0


Z
n
o
n
o
ψ B; B̂k , σ 2 ykR,0
R,0
I
R
2
+
max{Vnew(Hk ∪ (xR,0dB
k , B) ; 1), V (Hk ∪ (xk , B) ; 1)}(B − B̂k )2σ 2 (ykR,0 )2
Ak


Z
n
o
n
o
ψ B; B̂k , σ 2 ykR,0
R,0
I
R
2
≤2
max{Vnew(Hk ∪ (xR,0dB,
k , B) ; 1), V (Hk ∪ (xk , B) ; 1)}(B − B̂k )+2σ 2 (ykR,0 )2
Ak
(30)

Z

where A+
k =




q
R,0
B ∈ R : B > B̂k + σ yk and B − B̂k > 0 . The last inequality follows fromthe observation that the maximum of the two values of experimentation is larger when B > B̂k ,for any k along the sequence.
+,IN
Let us further partition A+
and A+,Rsuch thatk into the two subsets Ak
k


2 R,0
n
o
n
o
ψ B; B̂k , σ yk
R,0
I
R
2
dB
max{Vnew(Hk ∪ (xR,0
,
B)
;
1),
V
(H
∪
(x
,
B)
;
1)}(B
−
B̂
)
k
k
k
k
2σ 2 (ykR,0 )2
A+
k


2 R,0
Z
n
o
ψ B; B̂k , σ yk
I
2
Vnew
=
(Hk ∪ (xR,0
,
B)
;
1)(B
−
B̂
)
dB
k
k
2σ 2 (ykR,0 )2
A+,IN
k


2 R,0
Z
n
o
ψ B; B̂k , σ yk
2
dB.
(31)
+
V R (Hk ∪ (xR,0
,
B)
;
1)(B
−
B̂
)
k
k
2σ 2 (ykR,0 )2
A+,R
k

Z

11

,
Next, notice that, for any B ∈ A+,IN
k
I
(Hk ∪
Vnew

n
o
R,0
(xk , B) ; 1)






(B̄k − B̂k ) − (B − B̂k )a q= −(1 + δ)(B̄k − B̂k ) + max (B − B̂k )a 1 + δ − δΦ 
a∈[0,1] σ a(1 − a)ykR,0qp(B̄k − B̂k ) − (B − B̂k )a q
+ δσ a(1 − a) ykR,0 φ 
σ a(1 − a)ykR,0(B̄k − B̂k ) − (B − B̂k )a q+δ(B̄k − B̂k )Φ 
,
(32)

R,0
σ a(1 − a)yk
from (14) in the main text. Over the set A+,IN, an upper bound for the value of incrementalkexperimentation of the new unit is provided by a similar unit in which the payoffs at both
endpoints are all equal to the largest known payoff, that is, B L = B H = B̄k if B ≤ B̄k andB L = B H = B if B > B̄k . It can be shown from the analysis of our baseline model thatthe optimal technology in a bounded unit with equal payoffs at the endpoints is given by the
mid-point technology. Putting all these observations together shows that, for any B ∈ A+,IN
,
k
I
Vnew
(Hk ∪

q
n
o
σδ
R,0
ykR,0 φ(0).
(xk , B) ; 1) ≤2

(33)

The value of radical experimentation for a young agent is instead uniformly bounded by (1 +12δ)V R (0) regardless of the history. Thus,


2 R,0
n
o
ψ B; B̂k , σ yk
I
2
dB
Vnew
(Hk ∪ (xR,0
,
B)
;
1)(B
−
B̂
)
k
k
2σ 2 (ykR,0 )2
A+,IN
k


2 R,0
Z
n
o
ψ B; B̂k , σ yk
2
V R (Hk ∪ (xR,0
,
B)
;
1)(B
−
B̂
)
+
dB
k
k
2σ 2 (ykR,0 )2
A+,R
k


2 R,0
Z
q
ψ B; B̂k , σ yk
σδdB≤ykR,0 φ(0)(B − B̂k )22
2σ 2 (ykR,0 )2
A+,IN
k


Z
ψ B; B̂k , σ 2 ykR,0+(1 + δ)V R (0)(B − B̂k )2dB+,R2σ 2 (ykR,0 )2
Ak
#Z
"


V R (0)
δφ(0)
2
2 R,0
+
(1
+
δ)
(B
−
B̂
)
ψ
B;
B̂
,
σ
y
dB
≤
k
k
k
4σ(ykR,0 )3/2
2σ 2 (ykR,0 )2 R"#
V R (0)δφ(0)=+ (1 + δ)
σ 2 ykR,0R,0 3/2
R,0 2
24σ(yk )2σ (yk )Z

δσφ(0)V R (0)= q+ (1 + δ) R,0 ,2yk
4 ykR,0(34)

where the second inequality follows from the increase in the range of integration and the nonnegativity of both integrands, while the first equality follows from the definition of variance andthe normal distribution. The expression in (34) converges to zero as k increases.
We have thus shown that the limit superior in (25) is bounded above, which gives a contradiction.
Proposition A.5. There exists a value of the gap Ĝ such that V R (Ht ; 0) < 0 whenever Gt > Ĝ.
Proof. Let xR,0 (H) denote the optimal size of radical innovation for a newly born agent at anarbitrary history H. Taking the limit with respect to the size of the highest known payoff B̄

13

givesnoR,0lim sup V (H; 0) ≤ lim sup −c(xt (H) − x̂)
R

B̄→+∞

(35)

B̄→+∞




+ lim sup −G(B̄) + δEH max{V I (H+1 ; 1), V R (H+1 ; 1)}|xR,0 (H)
B̄→+∞
!!)
(
G(B̄)
p
+ lim sup −δ(1 + δ)G(B̄) 1 − Φ
σ xR,0 (H) − x̂
B̄→+∞
!
p
G(
B̄)
p
+ lim sup δ(1 + δ)σ xR,0 (H) − x̂φ,
σ xR,0 (H) − x̂B̄→+∞
where G(B̄) = B̄ − B̂. By Proposition A.4, xR,0 (H) − x̂ ∈ 0, Ȳ , for any arbitrary history
H. Therefore, the first limit in (35) is nonnegative. The second limit is −∞ given that the
expectation grows in B̄ at a rate bounded above uniformly by 1+δ
, by Lemma A.1, while the2third and fourth limits are both zero. Thus, there exists a large enough gap Ĝ such that thevalue of radical experimentation V R (H; 0) is negative for any size of the gap above Ĝ.
Putting together all the results obtained so far allows us to replicate the same steps as in the
proof of Theorem 1 in the main text. We can thus state the following result.
Theorem A.1. Consider the extended model with three-period-lived agents. Radical experimentation ends in finite time with probability one.

B

Positive Drift: Optimistic Beliefs and Stagnation

Theorem 1 in the main text assumed that the drift κ which determines the expected payoff of
new technologies, was equal to zero. What happens if agents are more optimistic about κ? Could
that be enough to sustain radical experimentation in the long run?
For this problem to have a well-defined solution, we must assume that the incremental costof radical experimentation eventually exceeds the drift κ. Otherwise, young agents may want tochoose an infinite amount of radical experimentation. Precisely, we assume that limy→+∞ c0 (y) >κ(1 + δ). For technical reasons, we also assume that c00 (0) > 0.
The scenario of a positive drift creates an incentive for old agents to innovate as well, as
captured by the objective function
U O,R (Ht ) =max EHt [B(x) − c(x − x̂t )] = B̂t + κ(x − x̂t ) − c(x − x̂t ).

x∈[x̂t ,+∞)

14

(36)

This slightly complicates the short-run equilibrium behavior of experimentation, because agentsnow learn from both old and young generations. This does not, however, change the insight ofTheorem 1.
Theorem B.1. Radical experimentation stops in finite time, almost surely.
This extension of Theorem 1 is not completely straightforward, and may be explained asfollows. The proof can be found at the end of the section.
The size of radical experimentation by the old generation, whenever it takes place, is characterized by the first-order conditionκ = c0 (x − x̂t ).
(37)
Let y O,R denote the solution to (37) and ξ = κy O,R − c(y O,R ) > 0. An old agent prefers radical
experimentation over exploitation if and only if
B̂t + κy O,R − c(y O,R ) ≥ B̄t , or, equivalently, if Gt ≤ ξ.

(38)

When choosing the size of his experimentation, a young agent considers the effect of his
action today on his incentives tomorrow. This effect can be quantified in a perceived reductionof the gap from Gt to Gt − ξ due to the possibility of performing radical experimentation in
the second period, which results in higher incentives to perform radical experimentation today.
However as the gap increases, ξ becomes negligible and eventually the relative benefit of radical
experimentation over exploitation falls short of the explicit cost of experimentation. Thus the
young agent will eventually opt for incremental experimentation, which is still strictly better
than exploitation.
Proof of Theorem B.1. We start by showing the existence of a threshold G̃ above whichy R (G) = 0.2 Suppose, first, that κ ≤ c0 (0), so that an old agent chooses the technology with thehighest known payoff (as observed in the main text). The expected utility of a young agent fromchoosing technology x > x̂ is






G − κyG − κy√+ σ yφB̂ + κy − c(y) + δ (B̂ + κy) 1 − Φ
√
√
σ y
σ y


G − κy+(G + B̂) Φ
√
σ y





G − κyG − κy√= (1 + δ)[B̂ + κy] − c(y) + δ (G − κy)Φ+ σ yφ
,
√
√
σ y
σ y
2

(39)
(40)
(41)

We express the dependence of the optimal size of radical innovation on the gap G rather than the history H
from expositional clarity.

15

where G = B̄ − B̂. The first-order condition is





δσ
G − κyG − κy+ √ φ= c0 (y).
κ 1 + δ − δΦ
√
√
σ y
2 y
σ y

(42)

The left-hand side of (42) approaches κ as G increases and, for fixed G, the left-hand side
converges to κ(1 + δ) as y → +∞, and to κ as y → 0. Also, the left-hand side is bounded abovebyδσ 2 φ(1)κ(1 + δ) +,2G
which converges to κ(1 + δ), as G increases. Thus, limy→+∞ c0 (y) > κ(1 + δ) (and c00 (0) > 0, if
κ = c0 (0)) implies that there exists G̃ > 0 such that G > G̃ implies y R (G) = 0.
Differentiating (42) with respect to G yieldsδφG− √ κ+< 0,
2σ yy
for all G, y > 0. This implies that y R (0) ≥ y R (G) for all G > 0. Repeating the argument used
to prove Theorem 1, we conclude that radical experimentation ends in finite time a.s.
Suppose now that κ > c0 (0). An old agent experiments radically with a size equal to y O,R > 0
if and only if G ≤ ξ, as shown in the main text. We assume without loss of generality thatG > ξ, so that an old agent does not experiment today and then the expected utility today of
any x > x̂ for a young agent is simply
E[B(x) − c(x − x̂)] + δ{E[B(x) + ξ|B(x) ≥ B̄ − ξ] Prob(B(x) ≥ B̄ − ξ) + B̄ Prob(B(x) < B̄ − ξ)}





G − ξ − κyG − ξ − κy√+ ξ + (G − ξ − κy)Φ.
= (1 + δ)(B̂ + κy) − c(y) + δ σ yφ
√
√
σ y
σ y
The first-order condition is





δσ
G − ξ − κyG − ξ − κy
κ 1 + δ − δΦ+ √ φ
= c0 (y).
√
√
σ y
2 y
σ y

(43)

The right-hand side is always greater than or equal to κ. Since G − ξ > 0, the right-handside converges to κ as y → 0, and to κ(1 + δ) as y → +∞. Since also c0 (0) < κ, there must
exist an interior solution to the first-order equation (43). When limy→+∞ c0 (y) > κ(1 + δ), thesolution is unique for high values of G because the left-hand side of (43) approaches κ pointwise
as G becomes arbitrarily large. As G increases, y R (G) approaches y O,R : the optimal size ofexperimentation for a young agent converges to the optimal size for an old agent. This follows
16

from the fact that the first-order condition becomes, dropping negligible terms, κ ≈ c0 (y). The
maximized expected utility is approximately equal to"(1 + δ)(B̂ + ξ) + δ κy O,Rp+σ y O,R φ

1−Φ

G − ξ − κy O,Rp
σ y O,RG − ξ − κy O,Rp
σ y O,R!
+ (G − ξ)Φ

!!

G − ξ − κy O,Rp
σ y O,R!#≈ B̂ + ξ + δ B̄.
Since we assumed that G > ξ, it follows that B̂ + ξ + δ B̄ < (1 + δ)B̄. The only candidate forradical experimentation gives an expected payoff which is lower than what the young agent could
get by simply exploiting. Thus, there exists G̃ > 0 such that the young agent prefers exploitation
for any gap greater than G̃.
In order to replicate the steps used to prove Theorem 1, we still need to show that there isan upper bound on the equilibrium size of experimentation. When G > ξ, the right-hand sideof (43) is strictly decreasing in G. Thus, the unique positive solution of the first-order condition
when G = ξ provides the desired upper bound. When G ≤ ξ, the old agent is experimenting
with a fixed size y O,R (independent of the gap). Replicating the argument for κ = 0, one may
show that, on the range G ∈ [0, ξ], the value and size of radical experimentation are decreasing in
G, providing an upper bound on the size of radical experimentation for a young agent, uniform
over all histories.

17