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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Experimentation, Patents, and Innovation
Daron Acemogiu
Kostas Bimpikis
Asuman Ozdaglar
Working Paper 08-1
October 8, 2008
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E52-251
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Experimentation, Patents, and Innovation*
Daron Acemoglu^
Asuman
Kostas Bimpikis*
Ozdaglar^
October 2008.
Abstract
This paper studies a simple model of experimentation and innovation. Our analysis suggests that
may improve the allocation of resources by encouraging rapid experimentation and efficient ex
patents
Each firm receives a private signal on the success probability
and decides when and which project to implement. A successful innovation can be copied by other firms. Symmetric equilibria (where actions do not depend
on the identity of the firm) always involve delayed and staggered experimentation, whereas the optimal
allocation never involves delays and may involve simultaneous rather than staggered experimentation.
post transfer of knowledge across firms.
of one of
The
many
potential research projects
social cost of insufficient experimentation can
can implement the socially optimal allocation
be arbitrarily
large.
(in all equilibria).
Appropriately-designed patents
In contrast to patents, subsidies to
experimentation, research, or innovation cannot typically achieve this objective.
signal quality differs across firms, the equilibrium
may
We
may experiment after those with weaker signals. We show that
environment patents again encourage experimentation and reduce delays.
stronger signals
Keywords:
JEL
show that when
in this
more general
delay, experimentation, innovation, patents, research.
Classification: 031, D83, D92.
'We thank
for
also
involve a nonmonotonicity, whereby players with
participants at the September 2009 workshop of the Toulouse Network on Information Technology
comments. Acemoglu
also gratefully
acknowledges financial support from the Toulouse Network on Information
Technology.
^Dept. of Economics, Massachusetts Institute of Technology
^Operations Research Center, Massachusetts Institute of Technology
'Dept. of Electrical Engineering and
Computer
Science, Massachusetts Institute of Technology
Introduction
1
Most modern
societies provide intellectual property rights protection to innovators using a
The main argument
system.
creating ex post
monopoly
in favor of patents
rents (e.g., Arrow,
1988, Klemperer, 1990, Gilbert
is
patent
that they encourage ex ante innovation by
1962, Kitch,
Reinganum, 1981, Tirole,
1977,
and Shapiro, 1990, Romer, 1990, Grossman and Helpman, 1991,
Aghion and Howitt, 1992, Scotchmer, 1999, Gallini and Scotchmer, 2002). In
suggest an alternative (and complementary) social benefit to patents.
certain circumstances, patents encourage experimentation
We
this paper,
show
that,
we
under
by potential innovators while
still
allowing socially beneficial transmission of knowledge across firms.
We
there
construct a stylized model of experimentation and innovation.
is
and can decide
innovation
is
signal
on which of these projects
is
more
likely to lead to
to experiment with any of these projects at any point in time.
own
novator).
the
probability of success, but in the process capture
The returns from the implementation
number
of firms implementing
it.
We
A
successful
The symmetric
some
in order to increase
of the rents of this first in-
of a successful innovation are nonincreasing in
provide an explicit characterization of the (subgame
perfect or perfect Bayesian) equilibria of this
time).
an innova-
publicly observed and can be copied by any of the other potential innovators (for
example, other firms can build on the knowledge revealed by the innovation
their
game
a large number of potential projects and TV symmetric potential innovators (firms).
Each firm receives a private
tion
In our baseline
dynamic game (both
in discrete
and continuous
equilibrium always features delayed and staggered experimentation. In
particular, experimentation does not take place immediately
and involves one firm experiment-
ing before others (and the latter firms free-riding on the former's experimentation). In contrast,
the optimal allocation never involves delays and
experimentation.
The
nificant efficiency loss:
may
require simultaneous rather than staggered
insufficient equilibrium incentives for
experimentation
may
create a sig-
the ratio of social surplus generated by the equilibrium relative to the
optimal allocation can be arbitrarily small.
We
next show that a simple patent system, where a copying firm has to make a prespecified
payment
to the innovator, can
implement the optimal
When
allocation.
involves simultaneous experimentation, the patent system
makes
free-riding prohibitively costly
and implements the optimal allocation as the unique equilibrium.
involves staggered experimentation, the patent system plays a
post transmission of knowledge but
The patent system can achieve
this
still
the optimal allocation
When
the optimal allocation
more subtle
role.
It
permits ex
increases experimentation incentives to avoid delays.
because
it
generates "conditional" transfers.
An
innovator
receives a patent
payment only when copied by other
firms. Consequently, patents
encourage one
firm to experiment earher than others, thus achieving rapid experimentation without sacrificing
Moreover, we show that patents can achieve this outcome in
useful transfer of knowledge.
The
equilibria.
fact that patents are particularly well
designed to play this role
is
also
highhghted
by our result that while an appropriately-designed patent implements the optimal allocation
all equilibria,
all
subsidies to experimentation, research, or innovation cannot achieve the
in
same
objective.
In our baseline model, both the optimal allocation
and the symmetric equilibrium involve
sequential experimentation. Inefficiency results from lack of sufficient experimentation or
delays.
The
structure of equilibria
potential innovators differs
and
is
richer
when the strength (quahty)
is
of the signals received by
also private information. In this case, those with sufficiently
is
strong signals will prefer not to copy successful innovations.
of signals
from
With two
firms or
when
the support
such that no firm will have sufficiently strong signals, the "symmetric" equilibrium
of this extended
game (where
do not depend on the identity of the player)
strategies
satisfies
a monotonictty property, so that firms with stronger signals never act after firms with weaker
signals.
This ensures an
knowledge.
pattern of experimentation and efficient ex post transfer of
efficient
However, when there are more than two firms and
possible, then the equilibrium
may
violate monotonicity.
sufficiently strong signals are
Interestingly,
when
this
is
the case
patents are again potentially useful (though they cannot restore monotonicity without preventing
(some) ex post transfer of knowledge).
In addition to the literature on patents mentioned above, a
to our paper.
of
common
Keller,
First, ours is a
work
is
in this
area
(e.g.,
Bolton and Harris, 1999, 2000, and
Cripps, 2005). These papers characterize equilibria of multi-agent two-armed
bandit problems and show that there
equilibrium
of other works are related
simple model of (social) experimentation and shares a number
features with recent
Rady and
number
may be
model and can be characterized
particularly simple in our
payoff-relevant uncertainty
discussed above, there
is
is
is
The
structure of the
explicitly
because
all
revealed after a single successful experimentation. In addition, as
insufficient
a simple form: either there
or experimentation
is
insufficient experimentation.
experimentation in our model as
free-riding
delayed.
We
also
well,
though
this also takes
by some firms reducing the amount of experimentation
show that patent systems can increase experimentation
incentives and implement the optimal allocation.
Second, the structure of equilibria with symmetric firms
of attrition
games
Cannings, 1989).
(e.g.,
War
Maynard Smith,
of attrition
is
reminiscent to equilibria in war
1974, Hendricks, Weiss and Wilson, 1988, Haigh
games have been used
in the
and
study of market exit by Fudenberg
and Tirole (1986) and Bulow and Klemperer (1999), research tournaments by Taylor (1995), and
by Bulow and Klemperer (1994). In our symmetric model, as
in auctions
of attrition, players choose the stochastic timing of their actions in such a
other players indifferent and willing to mix over the timing of their
of equilibria
and the optimal allocation
is
different,
involve either simultaneous experimentation by
to those resulting in
arises
from
asymmetric
all
The
equilibria.
Finally, the monotonicity property
when
and
is
also related to
actions.
in
equilibria.
Fudenberg and
result
not hold in our model
when
rest of the
paper
there are
is
model with two symmetric
in this
model and provide
similar
on the clustering of
model with endogenous timing,
which agents with stronger signals act earlier
differs
from these
highlighted by the result that this monotonicity property does
sufficiently strong signals so that
The
is
Bulow and Klem-
Tirole, 1986,
than those with weaker signals, though the specifics of our model and analysis
is
may
model
novel beneficial role of patents in our
actions in herding models. In the context of a standard herding
previous contributions. This
The structure
players or staggered experimentation similar
Gul and Lundholm's (1995)
Gul and Lundholm construct an equilibrium
make
as to
the quality of signals differs across agents
to results in generalized wars of attrition (e.g.,
perer, 1994, 1999)
way
however, and the optimal allocation
implement such asymmetric
their ability to
own
symmetric wars
in
some
more than two firms and the support
firms prefer not to copy successful experimentations.
organized as follows.
firms.
We
is
In Section 2
we
start
with a discrete-time
characterize both asymmetric and symmetric equilibria
explicit solutions
our continuous-time analysis, which
of signals includes
more
when period length tends
tractable.
We
to zero.
This motivates
obtain the continuous-time model as
the limit of the discrete-time model and provide explicit characterization of equilibria in Section
3.
Section 4 extends these results to a setup with an arbitrary
number
of firms.
Section 5
characterizes the optimal allocation and shows that the efficiency gap between the symmetric
equilibrium and the optimal allocation can be arbitrarily large.
also
The
analysis in this section
demonstrates that an appropriately-designed patent system can implement the optimal
Section 6 extends the model to an environment in which signal
allocation (in aU equilibria).
quality differs across firms.
It
establishes the monotonicity property of equilibria
when
either
there are only two firms or the support of signal distributions does not include sufficiently strong
signals.
Section 7 shows
how
this
monotonicity property no longer holds when there are more
than two firms and sufficiently strong signals are possible.
allocation
and the
role of patents in this
Section 8 discusses the optimal
extended model. Section 9 concludes.
Two Symmetric
2
In this section,
we
start
with a discrete-time model with two symmetric firms.
n,
-
Environment
2.1
The economy
follows
consists of
two research
firms,
each maximizing the present discounted value of
Let us denote the time interval between two consecutive periods by
profits.
we
will take
A
A
>
0.
In
what
to be small.
Each firm can implement ("experiment with") one
represent the set of projects
V?
Time
Firms: Discrete
of
byAI = {1,...,M}. Each
M potential innovation projects.
Let us
firm receives a private ("positive") signal
G A^ indicating the success potential of one of the projects. The unconditional probability
that a project will be successful (when implemented)
signal the success probability of a project
signals
about
different projects.^
When
observed.
At each
experimentation
The
is
p >
We
0.
small (~
0).
Conditional on the positive
assume that the two firms always receive
success or failure of experimentation by a firm
successful,
is
is
we
refer to this as
an "innovation".
can choose one of three possible actions:
instant, a firm
(1)
is
,
publicly
,
experiment with a
project (in particular, with the project on which the firm has received a positive signal); (2)
copy a successful project;
(3) wait.
Experimentation and copying are
irreversible, so that
a firm
cannot then switch to implement a different project. In the context of research, this captures
the fact that
project
commitment
of intellectual
and financial resources
to a specific research line or
necessary for success. Copying of a successful project can be interpreted more broadly
is
as using the information revealed by successful innovation or experimentation, so
need to correspond to the second firm replicating the exact same innovation
Payoffs depend on the success of the project and whether the project
interval of length
is
TTiA
7r2A
>
>
0.
0.^
Until
In contrast,
The
future at the
A, the payoff to a firm that
if
is
common
rate r
>
is
(or product).'^
copied. During an
is
implemented by both firms, each receives
normalized to zero. Both firms discount the
(so that the discount factor per period
we introduce heterogeneity
does not
the only one implementing a successful project
a successful project
payoff to an unsuccessful project
is
it
in success probabilities,
is e^'"'^).
we maintain the
following assump-
'This means that signals are not independent.
since
M
is
large, small) probability that the
If signals were independent, there would be a positive (but
two firms receive signals about the same project. Assuming that this
event has zero probability simplifies notation.
'In line with this interpretation, we could also allow copying to be successful with some probability u £ (p, 1].
This added generality does not affect any of the main economic insights and we omit it.
It will be evident from the analysis below that all of our results can be straightforwardly generalized to the
case where an innovator receives payoff 772"^'' A when copied, whereas the copier receives ttJ'^'""'' A. Since this has
no major effect on the main economic insights and just adds notation, we do not pursue this generalization.
tion.''
Assumption
1
Let US also define the present discounted value of profits as
n^^-^for
and
=
j
l,2,
for future reference, define
^.|.
Clearly,
£ {p,l) in view of Assumption
P
Now we
t
= kA
for a firm
1.
are in a position to define strategies in this game. Let a history
some
for
(1)
is
integer k) be denoted by
a mapping from
The
/i'.
set of histories
is
up
to time
denoted by W'.
A
& A4, and the history of the game up to time
its signal, ip
(where
t
strategy
/i',
t,
to
the probability of experimentation at a given time interval and the distribution over projects.
Thus the time
t
strategy can be written as
:M xH' ^
CT^
where
[0, 1]
ing) at
time
[0, 1]
X
A(A4),
denotes the probabihty of implementing a project (either experimenting or copyt
and
A (A4)
denotes the
set of probability distributions over
the set of projects,
corresponding to the choice of project when the firm implements a project. The latter piece of
generality
is
largely unnecessary (and will be omitted), since there will never be mixing over
projects (a firm will either copy a successful project or experiment with the project for which
it
has received a positive signal). Here a'
a'
((/5,
/i')
—
tation or copying of a successful project.
firm has experimented
and
we
a
s*
both
a''
(0,
) corresponds to waiting at time
up
to time
[(fi,h^^
and
i,
{0, 1}
m' £ A^ denoting which project
this choice
cr* (^ip , a''
,
m^
,
was
successful.
it
structure of equilibria without this assumption
is
trivial as
i
=
1,2 by
has chosen in that case,
slight
strategies.
abuse of notation
Although
of incomplete information, the only source of asymmetric information
of the project about which a firm has received a positive signal
The
t
and
denoting whether the other
With a
to denote time
s^'j
t
which could be experimen-
t,
Let us also denote the strategy of firm
can be summarized by two events a' G
e {0,1} denoting whether
will use
game
t
=
implementing project j at time
(l,j) corresponds to
History up to time
[ip, /i')
is
this is
the identity
and no action other than the
our analysis
in
Section 5 shows.
implementation of a project reveals information about
focus on
subgame
subgame
perfect equilibrium (or simply equilibrium)
this.
In light of this,
it
sufiicieat to
is
perfect equilibria (rather than perfect Bayesian equilibria) in this
the best response to 0"_, in
histories h^
all
E H^
for
a strategy profile
is
=
i
such that
(cti, CT2)
A
game.
<7j is
1,2.
Asymmetric Equilibria
2.2
Even though firms are symmetric
symmetric and asymmetric
(in
equilibria.
terms of their payoffs and information), there can be
Our main
interest
strategies are independent of the identity of the player.
is
with symmetric equilibria, where
Nevertheless,
with asymmetric equilibria. These equilibria are somewhat
it
is
convenient to start
because, as we will see,
less natural,
they involve one of the players never experimenting until the other one does.
In an asymmetric equilibrium, one of the firms, say
which
and
received a signal. Firm 2 copies firm
it
tries its
own
1
immediately attempts the project for
1,
in the next
time period
if
the latter
is
successful
project otherwise. In terms of the notation above, this asymmetric equilibrium
would involve
for
t
=
0,
A,2A,.... In words, this means that firm
has not experimented yet until
Firm
signal, j.
2,
t)
chooses to experiment immediately
1
and experiments with the project on which
J,
=
o-\m\s^)
I
(1,/)
if
a'
(l,j)
if
a'
=
=
m'
=f
1, 771'
=/
5'
=
1,
s*
=
0,
by these strategies
is
1,
ifa'
(0,-)
=
t
0,A,2A,....
The
experiments until firm
1
crucial feature highlighted
(J2 is
Given
(72,
follows
in the
A
time
t,
since
during the
t
=
0).
first
its
and
and
0,
experimentation
t
r~^ log
(where
t
straightforward to verify
it is
(/3
+
= kA
will
+
1
—
for k
p)-
What about
G N).
If
firm
1
cti?
now
e-'-(*+^)n2),
be successful with probability
period following the success (equivalent to e~'''/TiA
Then according
that firm 2 never
expected payoff
l/[(]=p(e-^'7riA
at
< A* =
suppose that the game has reached time
will receive
=
proof of Proposition 2 below,
a best response to aj provided that
(7i, it
its
does.
Using the same analysis as
that
has received
on the other hand, uses the strategy
^2(^2=
for
it
(if it
to
(T2,
p, yielding a profit of tt]
when discounted
firm 2 will copy the successful project
the present discounted value e~''*'^^^n2 from then on.
If,
and firm
at this point, firm
1
1
A
to time
will receive
chooses not
game proceeds
to experiment, then the
(aj"*"^, a^2
)' it
to time
is
not profitable. This discussion establishes the following proposition
(proof in the text). Throughout the paper,
to denote the firm
^
the experimentation of firm
More
its
one firm,
i
=
A* = r~^
log
+
(/3
1
—
Then
p).
1,2, tries its project with probability
tries its project unless
i,
observes the outcomes of
it
firm,
~
i
copies
= j,a\mls^) =
1,2.
ifa'=\,ml^j' ands' =
if a^ = I, ml = j' and s' =
(1,/)
(
(l,j)
l
it if
successful
,,
l,
0,
ifa'^0,
(0,-)
,:,,.,
'
.
.'
•
.•
,
;
'
,_
Symmetric Equilibria
Asymmetric
i, is
i
project otherwise.
[
2.3
A<
holds and that
never
i,
.
i=
we use the notation ~
formally, the two equilibria involve strategies of the form:
al^{<fi^,
for
there are two firms,
Following experim,entation by
i.
own
~
'
,
1
equilibria. In each,
immediately and the other, firm
and experiments with
when
i.
asymmetric
there exist two
1
i'
Suppose that Assumption
1
A, and according to the strategy profile
will receive payoff equal to
Therefore this deviation
Proposition
+
i
equilibria explicitly condition
on the identity of the
treated differently than the firm with label
particular,
can be verified easily that firm
it
of Proposition
equilibria rely
1
than firm
i.
~
~
i
i.
firm:
one of the
firms,
with label
This has important payoff consequences. In
has
strictly greater payoffs in the
equilibrium
In addition, as already noted in the previous section, asymmetric
on the understanding by both firms that one of them
will not
experiment until the
other one does. In this light, symmetric equilibria, where strategies are not conditioned on firms'
"labels,"
and firms obtain the same equilibrium payoffs are more natural. In
we study such symmetric
equilibria.
We
this subsection,
focus on the case where the time interval
A
is
strictly
positive but small.
As defined above a
firm's strategy
of implementing a project.
a given time
••
Our
is
t
either
or
We
I.
first result
shows that
a
mapping from
refer to a strategy as
That
a^
..
is
is,
if
the experimentation probability at
a pure strategy takes the form
-.MxHf
for small
pure
information set to the probability
its
^
{0,1} X A4.
,
A, there are no pure-strategy symmetric
equilibria.
that
Assumption
Then
there exist
Proposition 2 Suppose
recall that
=
P
Il2/Tii).
holds
1
and
A<
that
A* =
r
no symmetric pure-strategy
Then
(1, j).
some time
io
and history
(with
h'°
to
consider a deviation by firm
+ A,
The
copying firm
2's
project
payoff to this strategy
V
V [to
cr'
|
,
a*]
> V
=
/c
+
1
-
p) (ujhere
This imphes that
exists.
G N) such that
a^"
((/?
=
j,
h^)
=
is
e-'-'°pU,.
after history
and experimenting with
successful,
if
its
/i'°,
waiting until date
own
project otherwise.
is
-
k',a*]
[fo
[to
\
e-'-('°+^)
(^^2
+
(pRa
+
at time to
1
and there
cF*,a*]
g-r((o+A)
or
for
which involves,
to a',
1
since firm 2 experiments with probability
Clearly,
= kA
following this history the payoff to both firms
V\to\(T*,a*]
Now
to
(/?
equilibria.
Proof. Suppose, to obtain a contradiction, that such an equilibrium a*
there exists
log
^
(1
is
(1
- p)pn:)
and
is
successful with probabihty p.
a profitable deviation
if
- p)pni) > e-"'vni,
if
A<
Here A*
>
since,
profitable deviation
Proposition 2
for the
A* ~r-Mog(/3 + l-p).
from Assumption 1,0
= U2/^\ >
This establishes the existence of a
P-
and proves the proposition.
is intuitive.
Asymmetric
equilibria involve
one of the firms always waiting
other one to experiment and receiving higher payoff. Intuitively, Proposition 2 implies
that in symmetric equihbria both firms would like to be in the position of the firm receiving
higher payoffs and thus delaying their
the other firm.
in order to benefit
from that of
These incentives imply that no (symmetric) equilibrium can have immediate
experimentation with probabihty
by either
1
Proposition 2 also implies that
over,
own experimentation
all
firm.
symmetric equilibria must involve mixed
strategies.
any candidate equilibrium strategy must involve copying of a successful project
Assumption
we can
1
in
Moreview of
and immediate experimentation when the other firm has experimented. Therefore,
restrict attention to
a'{^
time
t
strategies of the form
= j,a',m\s') =
{
a'
{!,]')
if
(l,j)
if a'
(g(t)A,j)
=
=
1,
1,
m' = / and
m' = / and
ifa'
=
0.
s*
s*
=
=
1,
0,
(2)
for
=
i
A, 2A,
0,
where q{t)A
...,
is
the probabihty of experimenting at time
experimentation by either firm up to time
(all
t
<
=
q{t)
i.e.,
We
consider a symmetric mixed-strategy equilibrium a* and suppose that the
game
(and period length
t
t
Let v^
without experimentation.
i
when
~
firm
plays
i
ct*
[t
A] and Ve
\
and firm
i
A]
[t
=
Ve
history h^ (with no experimentation
suffices to characterize
up
to
t), i.e.,
i
equation
holds for
(3)
i's
Similarly, its payoff
where firm
case firm
1
—
i
i
receives
from waiting
copies
if
no
v.^
[f -1-
[i
i
A
+A
is
-
'
payoff'
today and with probability gA, firm
the experimentation
A] from the left-hand side of
-
(1
A
A—
[t
+A
'Here we use
A] I
r
v, since
and taking the
-^
+
V
it
lim^
M
q A-^o \
+
(1
then receives
(5)
~
i
A\A]y
if
receives
—
Vu,
p)pll\.
[i
+A
|
-
(5)
experiments, in which
and experiments with
successful
is
- gA) e-^^) -{v^[t +
(1
Dividing both sides by
lim ..
i
I
A]
(4)
"
;
does not experiment, and firm
I
it
qA).
otherwise, with expected continuation return pll2
Hu,
Therefore,
payoffs from experimenting:
v^[t\A]=e-'^{qA{pU2 + {l-p)pUi) + {l-qA)v^,\t +
~
/l^
has also experimented during the same time interval (probability qA), and
otherwise (probability
firm
such
all
any
after
and receives continuation value Hi
successful with probability p
is
(3)
symmetric equilibria involve mixing
a* such that (3) holds. First, consider firm
since in this case firm
112
we need
,
I
all
t
chooses to wait or to experiment
Ve[t\A]=qApUi + {l-qA)pU2,
i
\
A]
[i
I
of Proposition 3 below shows that
A] denote the time
[t
A).^ For a mixed-strategy equilibrium to exist,
is
v^
~
0.
>
that firms use a constant probability of experimentation over time,
continuation payoffs to firm
firm
—
A
more
has reached time
The proof
no
(Proposition 3 relaxes this assumption and establishes uniqueness
q for all
generally).
oo
1.
unique symmetric equilibrium as
explicit characterization of the
we assume
In the text,
conditional
such histories are identical, hence we write q[t)
instead of q{h^)). Clearly, feasibility requires that q{t)A
Next we derive an
t
With
its
own
probability
project
1
— gA,
Adding and subtracting
A].
and rearranging, we obtain
A\A]-
limit as
v^
[t
\
A^
+ A A]^
= e'^^qA
(pHs
+
(1
.
,.
- p)pUi)
yields
v^[t
\
\
A] \
A
denotes the value discounted back to
A])
.•
J
i
=
0.
^
'
^
''"
+q
^^^ +
(1
- p)pU.]
.
(6)
From equation
all
(thus for
/z*
that
Uu,
[i
+A
(4),
all i),
=
Aj
we
see that Ve
we have
Uu,
v-u, [t
|
\t
A] does not depend on
\
=
A]
Ve
\t
\
A]. Therefore, the second
[i
I
lim
where
recall,
this is
(1),
that
=
A]
|
Ug
+A
[t
|
A], implying
(6)
must be
A
Du;
+
A
q(0
=
fi
Al
—>
=
we obtain
in (4),
lim Ug
A]
fi
=
,,
.
pn2.
I
I
=
(5
<?*
=
j^P
for
alH,
(7)'
/
112/111.
constant for
is
and shows that
all t
indeed the unique symmetric equilibrium.
that
symmetric equilibrium. In
{t)
Proof.
'H'
+A
term on the left-hand side of
proposition relaxes the assumption that q{t)
Proposition 3 Suppose
with q
[i
(6), this yields
from
The next
Wu-
I
equal to zero. Moreover, taking the hmit as
Combined with
A] and
Since equation (3) holds for
t.
=
Assumption
1
A
holds and
this equilibrium, both
Then
^- 0.
there exists a
unique
firms use the mixed strategy a as given in (2)
q* as in (7).
We first
show that any symmetric equilibrium must involve mixing
along which there has been no experimentation.
establishes that after any such history
We next show
that there
is
V* (discounted back
,
T
equilibrium there must exist some time
and innovation.
Now
mLxed-strategy equilibrium, V* [T
to time
|
A]
V* > pH2, yielding a contradiction and
>
t
=
t
0.
T
there
A]
=
T>
is
0.
1.
First note that
V* >
pll2,
positive probability
and
this
establishing the desired result.
establishes that there cannot exist any time interval {T,T'), with T'
=
is
T >
0.
By
therefore, in
any
suppose that
e~^'^pll2,
e~^^pll2. However, for
€
This implies that in any
to obtain a contradiction,
\
t
0), satisfies
=
such that after time
the argument preceding the proposition, limA-.o Ve \T
—
/i*
there cannot be experimentation with probability
h'',
by experimenting at time
since each firm can guarantee this
any history
proof of Proposition 2
in the
positive probability of experimentation at time
the equilibrium-path value to a firm,
of experimentation
The argument
after
than
strictly less
The same argument
>
also
T, along which there
is
no mixing.
Hence, along any history
/i'
where there has not been an experimentation, both firms must
be indifferent between waiting and experimenting. This implies that
q{t)A denote the probability of experimentation at time
time
•
^
t is
given by an expression similar to equation
Ve[t\A]=qit)ApUi
10
+
t.
Firm
i's
(3)
must hold
for all
Let
t.
payoff for experimenting at
(4),
{\-q{t)A)pU2.
'
(8)
We
next show that the probability of experimentation q{t) in a symmetric equilibrium
a continuous function of
=
q{t+) (where q{i+)
Ve \t+
not continuous at some
t
>
then follows from (8) and Assumption
1
that Vg
Suppose that
t.
lim4|fg(i)),
A]. This implies that firm
it
q{t) is
>
q{t)
we have
q{t+),
probability
1
function of
t.
A
But
t.
<
/^]
this
at time
Ve\t\ A]
> Ve\t+
A], implying that firm
\
i
all
will
such
experiment -with
again yielding a contradiction. This establishes that q(t)
t,
Similarly,
t.
is
a continiaous
derivation similar to that preceding the proposition then shows that equation (6) holds
when
q
is
replaced by q{t). In particular,
hm..[t +
A|A]--^limf --'^ + ^'^J---'^l^] U^4-bn2 + (l-p)pn,].
-
.
:
Since v^
[t
|
=
A]
v^
\t
|
A] and
u^,
[t
+A
|
A]
=
Wg
[i
+A
^
lim
"
1^
+^
^] ~
I
(
A—O
^"'
,
,.
\^\^] \
A
V
^^^(Ve[t + A\A]-V^\t\A]
^
A-^0
/
A
V
lim (g(£
".
.
"',
,:•..-;-'•
(9)
.
A] we can write the second term on
|
the left-hand side of equation (9) as
.'
{t\
I
contradicts the fact that the symmetric equilibrium must involve mixing at
if
<
If (?(F)
0.
has an incentive to delay experimentation at time
i
is
.
\
=
where the second equality follows from equation
^
0,
(8)
+ A)-q(i))Ap(ni-n2)
.
.
•
•
.,::.:.-
;,
/.;
and the third equality holds by the continuity
of the experimentation probabihty ^(f). Substituting for
Um
A^O
in equation (9)
and solving
Vw\t
+ A\A]=
for q{t) yields q{t)
lim Vg
A—
=
[i
|
Al
=
pn2,
q* as in (7), completing the proof.
A
is
which Proposition 3 shows
is
Proposition 3 characterizes the unique mixed strategy equilibrium when period length
small.
The
analysis
is
in discrete time.
The
limit
where
A
—^
0,
well behaved, should also correspond to the (symmetric) equilibrium of the
directly in continuous time.
We
establish this in the next section
with the continuous-time model, which
3
Two Symmetric
The model
is
and we work
is
slightly
more
up
tractable.
Time
identical to that introduced in the previous section, except that time
directly with flow rates of experimentation. Strategies
U
set
and subsequently work directly
both more economical and
Firms: Continuous
same model
and
is
continuous
equilibria are defined
similarly
and Assumption
1
still
This implies that in any equilibrium a firm that has
applies.
not experimented yet will copy a successful innovation. Using this observation,
strategies by a function A
R+
:
-^
R+
experimentation at each date until there
probability
time
t
1
at
time
corresponds to A
t
corresponds to A
[t)
=
is
(i)
The
+00.
distribution of "stopping time," which
R+ = R+ U
(where
=
while experimenting with probability
0,
at
1
some
flow rate of experimentation A induces a stochastic
we denote by
other plaj'er not having experimented until then.
A
=
at
any time
t
G 1R+ conditional
on.
pure strategy simply specifies r £ M.+
example, the strategy of experimenting immediately
represented by r
The stopping time r designates the
r.
happen
is
{+00}) specifying the flow rate of
experimentation by one of the players. Waiting -with
probability distribution that experimentation will
other firm's experimentation
we can represent
=
r
is
0,
.
the
For
whereas that of waiting for the
+00. The t notation
is
convenient to use for
the next two propositions, while in characterizing the structure of equihbria
we need
to use A
(thus justifying the introduction of both notations).''
Proposition 4 Suppose Assumption
each equilibrium, ti
=
and t^i
1
= +00
Then
holds.
for
=
i
there exist two asym.m,etric equilibria.
1,2.
Proof. The proof follows from the same argument as
Our next
result
In
in subsection 2.2
and
is
omitted.
the parallel of Proposition 2 and shows that there are no symmetric
is
pure-strategy equilibria in the continuous-time model.
Proposition 5 Suppose Assumption
Then
1 holds.
there exist no
symmetric pure-strategy equi-
libria.
Proof. Suppose, to obtain a contradiction, that a symmetric pure-strategy equilibrium
Then
r*
=
t
€
K+
for
i
=
1, 2,
yielding payoff
1/(r*,T*)
to
Now
both players.
a time interval
e
time interval. As
consider a deviation
and copying a
e
—
»
0, this
V
exists.
t'
>
=
t
e-''Vni
for
one of the
successful innovation
if
there
firms,
is
which involves waiting
for
such an innovation during this
strategy gives the deviating firm payoff equal to
(r',
T*)
= hm e-"('+^)
[pHs
+
(1
- p)pUi]
'Here we can be more formal and follow Simon and Stinchcombe's (1989) formal model of continuous-time
games with jumps. This amounts to defining an extended strategy space, where stopping times are defined for all
t € R+ and also for t+ for any t € R+. In other words, strategies will be piecewise continuous and right continuous
functions of time, so that a jump immediately following some time t G R-|- is well defined. Throughout, we do
allow such jumps, but do not introduce the additional notation, since this is not necessary for any of the main
economic insights or proofs.
12
Assumption
As
V (r', r*) > V {r*, r*),
implies that
1
in the discrete-time
model, we
establishing the result.
next show that there exists a (unique) symmetric
will
mixed-strategy equihbrium. In what follows, instead of working with the stopping time
more convenient
at time
and
be used
Lemma
work
that synmietric equilibria
in the characterization of
The support of mixed strategy
1
Proof. The proof comprises three
First,
we show that
=
t
=
>
inf{i: X{t)
>
0}
all
t
G R+
equilibria is
^+-
steps.
belongs to the support of mixing time (so that there
Then with
0.
must involve mixing on
mixed-strategy equilibria.
interval with zero probability of experimentation).
ti
it is
directly with A(t), which designates the flow rate of experimentation
The next lemma shows
t.
will
to
r,
the
particular because experimenting after
t\
Suppose, to obtain a contradiction, that
same argument
is
in the
no time
is
as in the proof of Proposition
3
(in
support of the mixed-strategy equilibrium),
equilibrium payoffs must be
Vi
Now
for
consider deviation where firm
any
ti
>
i
=
e-''^'pU2.
chooses A (0)
time r (induced by A)
(i)
firms once the
since
ii
=
=
is
T <
V
4-oo
and
=
let ti
game reaches time
=
inf{(: A(t)
time; this
(r
=
t),
+oo})
V {t),
or
is diff'erent
infji;
A(i)
than
ti
=
it
implies that there exists
t
£ [0,T]
This implies that the payoff to both
-|-oo}.
without experimentation (which has positive probability
ti
is
ti)
=
e-^^'pU2
.
.
denotes present discounted value as a function of experimentation
V
consider a deviation by firm
has reached
'
-
oo such that the support of the stopping
within [0,r]. Suppose not, then
V{T =
(where
This has payoff
4-oo.
yielding a contradiction.
0,
Second, we show that there does not exist
such that A
=
\t],
i
which referred to the value
to strategy r',
at
time
which involves waiting
and copying a successful project by firm
~
i
(if
there
is
t
in subsection 2.2).
for e
>
after the
Now
game
such a success). This has
payoff
V{t')
since firm
1
~
i
imphes that
is still
V (r')
A (ii)
is
=
-|-oo
=
and
e-'-^'+'>
will thus
strictly greater
than
\pn2
+
{I
- p)pU
experiment with probability
V [t =
13
1
at
ij) for e sufficiently small.
ti.
.
Assumption
Finally,
exist
the
we show
and
t-[
first
>
^2
that A
*i
>
(t)
for all
=
such that A(i)
part, the payoff
experimentation until
t.
Again suppose,
for
t
t
to obtain a contradiction, that
Then, with the same argument as
(ti,i2)-
from the candidate equilibrium strategy r to firm
i
conditional
ii is
,
=
V'(r)
However, deviating and choosing
t'
=
in
on no
.
e-^*^pn2.
yields
ti
V{t' =
there
ti)
=e-"*>pn2 >ViT).
This contradiction completes the proof of the lemma.
Lemma
1
implies that in
experimentation).
is
all
Using
symmetric equilibria there
will
be mixing at
all
times (until there
unique symmetric
this observation, Proposition 6 characterizes a
equilibrium. Let us illustrate the reasoning here by assuming that firms use a constant flow rate
of experimentation (the proof of Proposition 6 relaxes this assumption). In particular, suppose
~
that firm
i
innovates at the flow rate A for
choosing t
(i.e.,
=
t)
for firm
V{t)= (
i
€ K+. Then the value of innovating at time
all t
t
is
\pll2
+
(1
This expression uses the fact that when firm
~
i
Xe-^'e-''
- p)pni] dz
+ e-^'e-''Vn2.
(1"0)
Jo
of
its
experimenting at the flow rate
experimentation has an exponential distribution, with density
in (10)
t
is
is
the expected discounted value from the experimentation of firm
~
the
the timing
first
between
i
term
and
(again taking into account that following an experimentation, a successful innovation will be
copied, with continuation value ^112
firm
~
i
does not experiment until
t,
+
(1
The second term
—p)'plli).
is
the probability that
which, given the exponential distribution,
multiphed by the expected discounted value to firm
t
Then
Ae~'^*.
A,
i
when
it is
the
first to
is
equal to
e""^',
experiment at time
(given by e~'^^pY[2)-
Lemma
V'
[t)
1
implies that
must be equal
y'(()
for all
t.
V [t)
to zero for
must be constant
all
t
in
in
t
for all
£ 1R+. Therefore,
t
its
derivative
any symmetric equilibrium implying that
= Ae-^'e-'->n2 + (l-p)pn,]-(r +
A)e-^*e-'-Vn2
=
0,
^
(11)
This equation has a unique solution:
'
A*
=
-^
-p
for all
t.
.
(12)
1
The next proposition shows that
this result also holds
experimentation rates.
14
when both
firms can use time-varying
Proposition 6 Suppose Assumption
Then
holds.
1
there exists a unique
symmetric equilibrium.
This equilibrium involves both firms using a constant flow rate of experimentation X* as given by
~
Firm
(12).
if firm
~
i,
firm
i
immediately copies a successful innovation by firm
~
i
and experim ents
experiments unsuccessfully.
i
~
Proof. Suppose that firm
experiments at the flow rate A
i
= f
m{t)
{t)
at
time
G R+. Let us define
t
X{z)dz.
(13)
Jo
Then the
equivalent of (10)
V{t)= I
is
\ [z)
\pU2
e-"^(--)e-'--'
+ {l-p) pUi]
+
dz
e'"''-'^e-''pU2.
(14)
Jo
Here J^^ X{z)e~^'-~'dz
tween times
ij
and
the probabihty that firm
is
^2,
and
experimented before time
experimentation of firm
this event).
e""*'''
i
1
Thus the
t.
~
=
first
t
term
(given
by
Lemma
in
t,
the probability that
~
has not
i
(discounted and multiplied by the probability of
~
again the probability that firm
is
when
i
it is
i
the
does not experiment until
first
to experiment at
t
time
e~'''pn2).
1
implies that
V (i)
must be constant
this implies that its derivative
V'{t)
.
is
the expected discoimted value from the
is
multiplied by the expected discounted value to firm
t
(using strategy A) will experiment be-
i
Jq X{z) e^^^^-'^dz
and
between
The second term
—
~
=
=
,
V
(t)
A(t)e-"^(*)e-'"*[pn2
for all
Moreover, note that m{t)
is
in
for all
t
must be equal
t
G
E-|-.
Since
to zero for all
+ (l-p)pni]-
(r
+
t.
V (t)
differentiable
m'(0)e-'"(')e-''Vn2
..=
and m'(t)
=
different! able
Therefore,
.:-,
t.
is
.,
^
equation
X{t). Therefore, this
,
is
equiva-
lent to
,
_,
The unique
A(0bn2 +
solution to (15)
is
(1
-p)pni] =
(r
+
(12), establishing the
A(i))pn2 for allf.
.
'
(15)^
uniqueness of the symmetric equilibrium
without restricting strategies to constant flow rates.
This proposition contains exactly the same economics as Proposition 3 derived in the previous
section.
The
flow rate of innovation A* in (12)
is
also clearly identical to g* in (7). This establishes
formally that the limit of the discrete-time model and the continuous-time model give the same
economic and mathematical answers. In what
since
it is
more tractable and
follows,
necessitates less notation.
15
we
will use
the continuous-time model,
Multiple Firms
4
Let us
now suppose
that there are A^ firms, each of which receives a positive signal about one
The
probability that the project that has received a positive signal will succeed
of the projects.
p and each firm
is still
,
from a project that
is
receives a signal about a diff'erent project. Let tt^ denote the flow payoff
implemented by n other firms and define
—
TT
Once
again,
The
/?
=
and
sections
its
proof
is
is
is
that
holds, so that
1
/?
>
p.
established using similar arguments to those in the previous two
omitted.
Proposition 7 Suppose
exist
and Assumption
112/111 as specified in (1)
following proposition
^"
•
Assumption
no symmetric pure-strategy
1
holds and that there are TV
Moreover
equilibria.
>
Then there
2 firms.
the support of the mixed- strategy equilibria
R+.
straightforward to
It is also
example, when
N—
to copy
1
As
Tlj\j/Y[i
if
>
p,
it
show that there
is
an equihbriura
this firm is successful. If
in the previous
exist
it is
asymmetric pure-strategy
experiment and the remaining
for firm 1 to
let
us
and so
unsuccessful, then firm 2 experiments
two sections, symmetric equilibria are of greater
the structure of symmetric equilibria,
on.
To characterize
interest.
suppose that
first
For
equilibria.
.
'
n„ = n2
for all
,
''
n > 2
.
and also to simplify the discussion, focus on symmetric equilibria with constant fiow
In particular, let the rate of experimentation
subgame
N — n,
to
starting at time
it)
there are
n >
2 firms be A„.
with n firms that have not yet experimented (and
=
r*
it
chooses to experiment with probability
\n[n-
Jto
1) e-^"("-i)(^-''>)e-^(^-'«)
[pDj
+
(1
1
at time to
-
p) v^-i] dz
-\-
+
1
rates.
Consider a
all
experiments have been unsuccessful). Then the continuation value of firm
onwards) when
vn
io
when
(16)
i
previous,
(from time
is
e-^-^^-^^'e-^'pUi,
.
(17)
where fn-i
is
the
maximum
value that the firm can obtain
when
there are
have not yet experimented (where we again use v since this expression
value from time
of the
n—
happens,
1
to
onwards). Intuitively, A„ (n
—
1) e~'^"("~'')(-~'°) is
successful with probability p
and
will
16
be copied by
all
1
firms that
refers to the continuation
the density at which one
other firms mixing at the rate A^ will experiment at time z G
it is
n —
(io, to
-\-
1).
When
this
other firms, and each will
.
receive a value of e~^^'^~^°'Il2 (discounted back to to)- If
the niunber of remaining firms
time
until
t,
event
this
firm
is
-
1
equilibrium, Vn
derivative
its
i
and
1,
this gives a value of Vn-i-
A„ (n
-
1) e-^"("-^)(--*°)(iz
needs to be independent of
must be equal
i
=
and moreover,
Vn
its
which
is
— p),
no firm experiments
If
The
probability of
it is
clearly differentiable in t.
So
to zero. This implies
Proposition 7 imphes that there has to be mixing in
thus
1
e-^"("-i)^ As usual, in a mixed strategy
A„(n-l)[pn2 + (l-p)^;„-ij = (A„(n-
Intuitively,
unsuccessful (probability
chooses to experiment at this point and receives e~'^^pll2.
//j""^*
(i)
n —
is
it is is
=
mixing implies that the firm
is
indifferent
+
r)pn2.
all histories,
n >
pYl2 for all
l)
(18)
thus
2.
(19)
between experimentation and waiting, and
continuation payoff must be the same as the payoff from experimenting immediately,
pll2-
Combining
(18)
and
(19) yields
=
^"
71
(1
w"' nrr
-p)(ni)ni
(20)
'
This derivation implies that in the economy with TV firms, each firm starts mixing at the
flow rate A^v- Following an unsuccessful experimentation, they increase their flow rate of exper-
imentation to A/v_i, and so on.
The
this, let
derivation leading
up
'
'
'-/
';'.,-'''i'-
'
to (20) easily generalizes
'-
''
"
1
to:
'
"'
To demonstrate
relax (16).
.
2
n„ >
value of experimenting at time
of (17):
/
Vn {t)=
'*'
when we
us relax (16) and instead strengthen Assumption
Assumption
The
;
r
Xn (n -
t
plf] for all n.
(starting with
n firms)
.
.
1)
e-^n("-i)(~'-to)e-r(.-to) [pn„
+
now
given by a generalization
.;
;
,
is
(1
-
p) i;„_i] dz
'•
-
+
-
e-^-^^-i^'e-'-'pn^.
Jto
Again differentiating
this expression
with respect to
t
and
gives the equivalent indifference condition to (18) as
.'
-
for
.
n
=
2, ...,N.
setting the derivative equal to zero
,
A„(n-l)[pn„ + (l-p)Dn-i] = (A„(n-l) + r)pn„.
In addition,
we
still
have
Vn
=pn„
.
for all
17
n >
.
(21)
'
.
2.
Combining
this
with (21), we obtain
=
-^"
and
7^
(1
=
us adopt the convention that Aj
let
forn
Nr""im
-p){ni)n„_i
=
2,..„7V,
(22)
+oo.
This derivation estabUshes the following proposition.
Proposition 8 Suppose
that
librium. In this equilibrium,
Assumption 2
when
there are
holds.
=
n
1,2,
Then
there exists a unique
..., A'^
firms that have not yet experimented,
A
each experiments at the constant flow rate An as given by (22).
mediately copied by
all
followed by
all
firms
is
An
remaining firms.
An
probability of further experimentation
may
is
at the
decline.
Whether
fast Ifn decreases in n.
5
Patents and Optimal Allocations
The
analysis so far has estabUshed that both in discrete
mixed
strategies, potential delays,
that with probability
1,
one of the firms
will
>
3
flow rate A„_i
that after an unsuccessful experimentation, the
how
libria involve
successful innovation is im-
unsuccessful experimentation starting with n
remaining firms experimenting
interesting feature of Proposition 8
symmetric equi-
and
this
is
the case or not depends on
and continuous time, symmetric equi-
also staggered experimentation
(meaning
experiment before others). Asymmetric equilibria
avoid delays, but also feature staggered experimentation. Moreover, they are less natural, be-
cause they involve one of the firms never acting (experimenting) until the other one does and
also because they give potentially very different payoffs to different firms.
first
We
establish the inefficiency of (symmetric) equilibria.
5.1
It is
N
>
2 firms,
we
we
then suggest that an appropriately-
designed patent system can implement optimal allocations. While
hold for
In this section,
all
of the results in the section
focus on the case with two firms to simplify notation.
Welfare
straightforward to see that symmetric equilibria are Pareto suboptimal. Suppose that there
exists a social planner that
can decide the experimentation time
that the social planner would like to maximize the
two firms.
sum
for
each firm.
Suppose also
of the present discounted values of the
Clearly, in practice an optimal allocation (and thus the objective fimction of the
social planner)
may
also take into account the implications of these innovations
However, we have not so
far specified
how consumer
18
welfare
is
on consumers.
affected by the rephcation of
new
successful innovations versus
mal allocations from the viewpoint of
cessful innovation
firms.
we focus on opti-
This would also be the optimal allocation taking
when consumer
consimier welfare into account
Therefore, in what follows,
innovations.
surpluses from a
implemented by two firms are proportional to
new innovation and from a
Jl]
and
we
we
2112, respectively. If
would
take the differential consumer surpluses created by these innovations into account, this
only affect the thresholds provided below, and for completeness,
suc-
also indicate
what these
alternative thresholds would be (see, in particular, footnote 7).
The
1.
adopt one of two strategies:
social planner could
Staggered experimentation: this would involve having one of the firms experiment at
successful, then the other firm
if it is
i
=
0;
would copy the innovation, and otherwise the other
firm would experiment immediately. Denote the surplus generated by this strategy by SC.
2.
Simultaneous experimentation: this would involve having both firms experiment immediately at
It is clear
time as
A—
'
Intuitively,
>
i
=
0.
Denote the surplus generated by
this strategy
by
52'
that no other strategy could be optimal for the planner. Moreover, both in discrete
and
continuous time,
in
--
5f and
S2 have simple expressions. In particular,
S^ = 2pU2 + {l-p)pUi.
'
one of the firms experiments
and
first
^•'
^
1
—
p,
the
first
firm
(23)
successful with probability p.
is
happens, the other firm copies a successful innovation, with total payoff
plementary probabihty,
'•^
2Il2.
When
With the com-
unsuccessful, and the second firm experiments
is
independently, with expected payoff pIli. Both in continuous time and in discrete time as
these payoffs occur immediately after the
The
alternative
to have
is
first
this
A—
>
0,
experimentation and thus are not discounted.
both firms experiment immediately, which generates expected
surplus
S^ =
The comparison
optimal when 2p <
of
1
5f and
+ p.
the firms experimenting
in the
S2
when
2/3
>
1
+ p,
If
is
stated
text).'''
that
2/3>l-fp,
'
is
the optimal allocation involves one of
and the second firm copying successful innovations. This
next proposition (proof in the
Proposition 9 Suppose
(24)
implies that simultaneous experimentation by both firms
In contrast,
first,
2pU,.
(25)
consumer surpluses from a new innovation and from the two firms implementing the same project were,
19
then the optimal allocation involves staggered experimentation, that
firm and copying of successful innovations.
If (25) does not hold,
involves immediate experimentation by both firms.
and immediate experimentation
tation
now compare
Let us
When
2/3
is,
experimentation hy one
then the optimal allocation
= l+p,
both staggered experimen-
are socially optimal.
this to the equihbria characterized so far. Clearly,
asymmetric equilibria
and thus generate surplus 5j
are identical to the first strategy of the planner
(recall
subsection
In contrast, the (unique) symmetric equilibrium generates social surplus
2.2.).
/oo
=
S^
2A*e-(2^'+'")'[2pn2
+ (l-p)pni]di
(26)
Jo
= w^^[^-p^2 + {i-p)pni],
za + r
where A*
the (constant) equilibrium flow rate of experimentation given by (12).
is
of (26) applies because the time of
experimentation corresponds to the
first
first
The
first line
realization of
one of two random variables, both with an exponential distribution with parameter A* and time
discounted at the rate
is
surplus
is
r.
If
the
experimentation
first
equal to 2112, and otherwise (with probability
with expected payoff pIli.
The second
A
A
-^
(since (12)
is
in a simple
manner. Let S^
is
second
always
= max {5f
S^
is
inefficient.
,
5.f }
<
Clearly, 5
let
us
first
1,
is
s
the more inefficient
so that the equilibrium
is
s
(n2
+ C2)/(ni +Ci),
=
=
it is
2p(n2
+
C2)
kHi, then
S[
.
Therefore, the
Moreover, this inefficiency can be quantified
measure of
ratio of equilibrium social surplus
inefflciency:
inefficient as stated above.
is
More
specifically,
delayed experimentation.
+ (i-p)p(ni+Ci)
2p(n, +c,).
then clear that condition (25) would be replaced by 27 > 1 + p and
As noted in the beginning of this section, if C2 = Kn2 and
the rest of the analysis would remain unchanged.
=
than
and 2C2, then we would have
5r
sf
Ci
(strictly) less
the equilibrium.
always
is
always
suppose that (25) holds. Then, the source of inefficiency
respectively, Ci
Denoting 7
the second firm also experiments,
line of (26) applies in the discrete-time
and consider the
to the social surplus in the optimal allocation as a
Naturally, the lower
— p),
p,
identical to (7)).
straightforward comparison shows that
unique symmetric equilibrium
which has probability
successful,
obtained by solving the integral and substituting for
line is
(23). It is also straightforward to verify that the
model with
1
is
this condition
would be identical to
(25).
20
In this case,
2A*
_
2/3
2A*+r ~ 2^ + 1-p'
where the
(1
+p) /2
last
(its
equahty simply uses
lower
bound given
It is
(12).
(25)). In that case,
_
In addition, as p
1
minimized, for given
is
p, as /3
=
we have
+p
can be as low as 1/2.
0, s
].
clear that s
Next consider the case where
(25) does not hold.
Then
•-'2
2 A*
2A*
where the
applies
that
>
p),
equilibrium
<
1
+p,
and thus
is
/?
because
'-
=
2pUi
P,
it
,..,
.
can be arbitrarily small as long as p
in this case s | 0. In
is
Since this expression
(1).
small (to satisfy the constraint
both cases, the source of inefficiency of the symmetric
generates insufficient incentives for experimentation. In the
first
case
delayed experimentation, and in the second, as lack of experimentation by
this exhibits itself as
one of the
+
+ (i-p)pn
again uses (12) and the definition of p from
last equality
when
2pn2
r
,...,,,
firms.
This discussion establishes (proof in the
text).**
•
.
,
,
...
:.
.
.
,
Proposition 10
when
2.
1.
Asymm.etric equilibria are Pareto optimal and maxim.ize social surplus
(25) holds, but fail to
The unique symmetric equilibrium
surplus.
When
When
is
social surplus
when
(25) does not hold.
always Pareto suboptimal and never maximizes social
(25) holds, this equilibrium involves delayed experimentation,
(25) does not hold, there
3.
maximize
is
and when
insufficient experimentation.
(25) holds, the relative surplus in the equilibrium compared to the surplus in the
optimal allocation,
5,
can be as small as 1/2.
When
(25) does not hold, the symmetric
equilibrium can be arbitrarily inefficient. In particular, s
|
as
p
J.
and
/3
J,
0.
Naturally, if we take into account consumer surpluses as discussed in footnote 7 and it is not the case that
C2 = Kn2 and C\ = Kill, then equilibrium social surplus relative to social surplus in the optimal allocation can
be even lower because of the misalignment between firm profits and consumer surpluses resulting from different
types of successful research projects.
21
Patents
5.2
The previous subsection
established the inefficiency of the symmetric equihbrium resulting
delayed and insufficient experimentation. In this subsection,
or ameliorate this problem.
Our main argument
we
discuss
from
how patents can solve
that a patent system provides incentives for
is
greater experimentation or for experimentation without delay.
We model
a simple patent system, whereby a patent
is
granted to any firm that undertakes a
successful innovation. If a firm copies a patented innovation,
holder of the patent.
next subsection.
achieve
tv,'o
discuss the relationship
between
this pa3rment
appropriately-designed patent system
objectives simultaneously. First,
beneficial for the
is
An
We
it
has to make a payment
it
(i.e.,
and licensing
it
can allow firms to copy others when
fees
in the
it is
it
can
77)
socially
when
it
can provide compensation to innovators, so that incentives to free-
on others are weakened. In particular, when staggered experimentation
ride
to the
the appropriate level of
knowledge created by innovations to spread to others (and prevent
not beneficial). Second,
77
is
optimal, a patent
system can simultaneously provide incentives to one firm to innovate early and to the other firm
to copy
When
an existing innovation.
rj
is
chosen appropriately, the patent system provides
However, more
incentives for the ex post transfer of knowledge.
innovation because an innovation that
innovation and paying the patent
is
fee.
crucially,
also encourages
it
copied becomes more profitable than copying another
The key
here
is
that the incentives provided by the
patent system are "conditional" on whether the other firm has experimented or not, and thus
induce an "asymmetric" response from the two firms.
profitable
when
This makes innovation relatively more
the other firm copies and less profitable
when
the other firm innovates.
incentive structure encourages one of the firms to be the innovator precisely
is
these asymmetric incentives imply that,
symmetric equilibrium no longer
other firm
it is
to
when the other firm
Consequently, the resulting equihbria resemble asymmetric equilibria.
copying.
is
exists.
when the patent system
It is less
is
This
Moreover,
designed appropriately, a
profitable for a firm to innovate
when
the
also innovating, because innovation no longer brings patent revenues. Conversely,
not profitable for a firm to wait
when the
other firm waits, because there
is
no innovation
copy in that case.
Our main
result in this subsection formahzes these ideas.
proposition and then provide most of the proof, which
Proposition 11 Consider
the
is
We state this result
intuitive, in the text.
model with two firms. Suppose that Assumption
22
in the following
1
holds.
Then:
1.
When
(25) holds, a patent system with
^^(i-p)n.
n2-pni
2
(which
is
feasible in view of (25)),
gered experimentation, in
and
the other
implements the optimal allocation, which involves stag-
all equilibria.
That
in all equilibria one firm experiments first,
is,
one copies a successful innovation and experiments immediately following
an unsuccessful experimentation.
2.
When
(25) does not hold, then the optimal allocation, which involves simultaneous exper-
imentation,
is
implemented as the unique equilibrium by a patent system with
77
That
> U2-PII1.
there exists a unique equilibrium in which both firms immediately experiment.
is,
Let us start with the
first
Observe that since
claim in Proposition 11.
77
<
112
— pHi,
the
equilibrium involves copying of a successful innovation by a firm that has not acted yet. However,
now has an
incentives for delaying to copy are weaker because copying
innovation has an additional benefit
T>
innovate at some date
to firm
when
i
it
77
'
It is
i
is
Suppose that firm
imitating.
has not done so until then).
chooses experimentation and waiting are
wait
-.
_:.
the other firm
(provided that firm
experiment now
•'
if
clear that for
any
T>
0,
= p{U2 +
=
is
rj
>
show that
^
"^
all
'
-
is
Waiting
+ (i-p)pni).
a strict best response, since
.
is
waiting
so.
Then
is
optimal.
To
the payoff to each
whether they experiment or wait, would be
wait
,
is
the payoffs
,
,
not an equilibrium. Suppose they did
experiment now
.,
will
implement the optimal allocation, we also need to show that both firms
experimenting immediately
firm, as a function of
i
•
'
So experimenting immediately against a firm that
equilibria
~
.
p{U2+rj)>p{U2-Tl) + {l'P)P^i
since
Then
and
r],
'.
r})
e-^^(p(n2-7])
experimenting
additional cost
=
pITi
•
'
•
= p(n2 -
77)
+
(1
~p)pni.
"
a strict best response since
.
p(n2-?7) + (i-p)pni >pUi
23
,
•
which holds
view of the fact that
in
<
i]
II2
- pH].
This argument makes
patents induce an equihbrium structure without delay: waiting
other firm
when the
other firm
waiting. However, to establish this formally,
is
Lemma
When
2
equilibria.
This
is
done
in the
to prove that tliere
next lemma.
equation (25) holds, there does not exist any equilibrium with mixing.
Proof. Let us write the expected present discovmted value of experimenting at time
i
when
~
firm
except that
experiments at the flow rate X{t) as in (14)
i
we now take patent payments
successful innovation
V{t)= f
is still
A(2)e-"^(-')e-''~Xn2-??)
all
t
m
in
(t) is
the support of the mixed-strategy equilibrium.
must
77
>
(i)
b (n2 if
r?)
+
(1
-
=
p) pDi]
is
The argument
(r
+
- pHi. Then
it is
in
in the
proof of Proposition
A
(t))
P (n2
+ 77)
the proposition. Suppose that (25)
not profitable
for
{t)
G IR+
The preceding
discussion and Proposition 11
is
making copying
is
all
all
parties
is
(or "free-riding") unprofitable.
equihbria.
It is
which coincides
9.
show how an appropriately-designed patent
for experimentation.
socially beneficial, a patent
socially beneficial, the patent
not satisfied and
to experiment immediately,
system can be useful by providing stronger incentives
experimentation by
is
a firm to copy a successful innovation. Therefore,
with the optimal allocation characterized in Proposition
delays in
This expression must be constant for
6.
(25) holds this equation cannot be satisfied for any A
both firms have a unique optimal strategy which
knowledge
copying a
Therefore, there does not exist any equilibrium with mixing.
t).
112
(25) so that
....
Let us next turn to the second claim
let
the proof of Proposition 6
satisfy
can be verified easily that
any
firm
+ (l-p)pni]dz + e-"We-^'p(n2+??),
given by (13) in the proof of Proposition
A
(for
for
t
:
6 establishes that A (t)
It
in
and use equation
into account
profitable. This expression
-'0
where
optimal
(strictly)
is
we need
intuitive that
optimal when the
(strictly)
experimenting immediately and experimenting immediately
is
mixed strategy
are no
is
it
On
simultaneous
system can easily achieve
the other hand,
system can instead ensure
important to emphasize that,
When
when ex post
this
by
transfer of
this while also preventing
in the latter case, the
patent system
provides such incentives selectively, so that only one of the firms engages in experimentation
and the other firm potentially benefits from the innovation of the
first
patents, simple subsidies to research could not achieve this objective. This
firm.
is
In contrast to
stated in the next
proposition and highlights the particular utility of a patent system in this environment.
24
Proposition 12 Suppose equation (25)
w>
There exists no
that experiments.
w>
Consider a direct subsidy
holds.
such that
all equilibria
given to a firm
with subsidies correspond to the
optimal allocation.
Proof. This
is
straightforward to see.
and
firms experiment immediately
If
w>
<
if u;
112
II2
-pHi
,
there exists an equilibrium in which
both
- pHi, the symmetric mixed-strategy equilibrium
with delayed experimentation survives.
It is clear
that the
same argument
applies to subsidies to successful innovation or any
com-
bination of subsidies to innovation and experimentation.
Patents and License Fees
5.3
The
and
analysis in the previous subsection
in return
it
has to
make some
assumed that a firm can copy a successful innovation
pre-specified
payment
patents often provide exclusive rights to the innovator,
With
or discovery to other firms.
to the original innovator. In practice
77
who
this interpretation, the
then allowed to license
is
payment
rj
its
product
would need to be negotiated
between the innovator and the (potential) copying firm rather than determined in advance.
While licensing
is
an important aspect of the patent system
the theoretical insights
To
we would
illustrate this, let us
like to
exclusive rights to the innovator but
system.
How
In particular, suppose that
and the innovator brings a lawsuit,
will receive
damages equal to k
environment,
let
us interpret
77
(111
—
the court system fimctions
will
112),
where k >
is
We
max
(this has
no
effect
{pHi;
112
"
/^^ (fl]
is
—
Without
of 112
+
/"^(IIi
—
112)
if
112
~
/"^(111
(0, 1)
can be represented by a take-
2112.
If
112)
>
pill
where the max operator takes care
112)},
25
and
this subsection). If
they disagree, then the outside
may be
to experiment
licensing, the innovator will receive
—
and the innovator
between the potential copying firm
on the conclusions of
of the fact that the best alternative for the "copying" firm
explicit licensing agreement.
also part of the patent
ignore legal fees. Given this legal
this negotiation
the two firms agree to licensing, their joint surplus
option of the copying firm
0.
as a license fee negotiated
by the innovator
is
succeed with probability p £
and the innovator. For simphcity, suppose that
it-or-leave-it offer
developing a different but highly
a firm copies a successful innovation without licensing
if
it
is
the second firm copies a successful innovation, the court
if
system needs to determine damages.
not essential for
Suppose further that the patent system gives
innovation.
first
is
it
emphasize.
suppose that the copying firm
substitutable product to the
in practice,
IIi
otherwise.
if
there
is
no
an expected return
This implies that the
negotiated licensing
fee,
as a function of the parameters of the legal system, will be
pk{Ui-U2)
(
U2-PU1
ri{p,K)=l
<n2-p/^(ni-n2),
ifpDi
if
[00
pDi >n2--/9K(ni -02) and2n2 > Hi,
otherwise,
where 00 denotes a prohibitively expensive licensing
Clearly,
by choosing p and
does not hold and
is
k,
can be ensured that
it
between
(1
-
ni/2 and
p)
112
77
—
such that no copying takes place.
fee,
{p, k) is
greater than 112
when
pIli
it
—
pfli
when
holds. This illustrates
(25)
how
an
appropriately-designed legal enforcement system can ensure that equilibrium licensing fees play
exactly the
same
role as the pre-specified patent fees did in Proposition 11.
Model with Heterogeneous Information
6
In this
and the next two
now
Instead,
identical precision.
to
sections,
we
relax the assumption that
indicated project
we assume
g{p) over
its
support
realizations for others
We
continue to define
equilibria
c
[a, 6]
and
that p
We
distribution function G(p).
continue
But the information
single project.
also
is
it
by p
(or
drawn from a
G
assume that
The
[0,1].
by
p; for firm
i).
Throughout
this
and the
distribution represented by the cumulative
has strictly positive and continuous density
realization of
p
for
each firm
is
independent of the
private inform.ation.
is
/3
We
parameterize signal quality by the probability with which the
and denote
successful
is
next two sections,
We
differs.
firms receive signals with
signal quality differs stochastically across firms.
assume that each firm receives a positive signal about a
content of these signals
all
=
We
112/111 as in (1).
where strategies do not depend on firm
also focus
identity.
on the equivalent of symmetric
Asymmetric
equilibria are discussed
briefly in Section 8.
In this section,
we
will establish a
monotonicity property showing that
in equilibrium firms
with higher p (stronger signals) experiment earher (no later) than firms with lower
turns out to be true under two scenarios:
firms but the upper support,
b,
of
G
is
when
there are two firms or
6.1
The
This result
there are multiple
not "too large". In the next section, we show
monotonicity result does not generalize to an environment
b>
when
p.
in
how
this
which there are multiple firms and
i3.
Two Firms
following
an important
lemma
is
straightforward and follows fiom the definition of
role in the analysis that follows (proof omitted).
26
/?
in (1). It will play
Lemma
3 Suppose that firm
ment with
As a
its
first
has support
own
~
<
project. If pi
/?,
firm
step towards characterizing
C
[a, 6]
Under
[0,^].
>
has innovated successfully. If pi
i
prefers to experi-
i
prefers to copy the successful project.
i
equilibrium with two firms,
ttie
assumption, we
this
P, firm
will
us suppose that p
let
prove that there exists a symmetric
equilibrium represented by a strictly decreasing function t{p) with t(6)
=
which maps signals
The
to time of experimentation provided that the other player has not yet experimented.
following proposition formalizes this idea and
Proposition 13 Suppose
is
proved by using a
that the support of.G is [a,h\
C
lemmas.
series of
Define
[0,/?].
b
T{p)
Then
=
\
rp
[log G{h) (1
-
-
6)
log G{p) (1
-
p)
+ /
log
G{z)d
(27)
Jp
I
symmetric equilibrium takes the following form:
the unique
each firm copies a successful innovation and immediately experiments
1.
if the
other firm
experiments unsuccessfully;
2.
firm
i
with signal quality pi experiments at tim.e t [p^) given by (27) unless firm
experimented before time T
{pi)
•
.
Proof. The proof uses the following lemmas.
Lemma
e
>
4 r(p) cannot
such that T{p)
=
be locally constant.
for
t
^
•.
,
-
let pi
time
is
t
e P. Firm
'
.
That
is,
there exists no interval
P=
\p,p+
e]
=
for all
p E P.
V
(t
I
since with probability
t.
In this case, firm
probability,
it is
where C
Since 112
=
= E [p
>
i,
Pr
The
t)
—
payoff after the
G (p)
successful,
p £ P]
PiHi,
is
112.
~
i
(1
-
has p £
G (p + e) + G (p)) Hs]
P
and thus
not copied and receives
Now
IIi.
we have
'
-
,
>
also experiments at
With
consider the deviation t(Pj)
'
time
the complementary
= t+6
for 5
>
and
is
+ e)-G{p)){CU2 + {l-C)p^n^) + (l-G{p +
is
strictly positive density,
firm
t
game has reached (without experimentation)
+ e)-G (p)) ni +
payoff to this
e-''[{G{p
|
\{G [p
G{p +
when
t)
copied and receives
arbitrarily small.
va{t\p,)
=
P^)
with
p ^ P.
all
(time
z's
has
i
r
Proof. Suppose, to obtain a contradiction, that the equilibrium involves r(p)
Then,
~
e)
+
G{pJ))p^n2],
the expected probability of success of a firm with type in the set P.
(^Il2
+
G {p+
— OPi^i > Pi^i- Moreover, by the assumption that G has
— G {p) > 0. Thus for 5 sufficiently small, the deviation is
(1
e)
profitable. This contradiction establishes the
lemma.
.27
'
Lemma
5 r(p)
continuous in
is
Proof. Suppose t
lim£|o T {p
6
>
r
>
+
>
e)
discontinuous at p.
(p) is
T (p-)
;
[a,b].
=
+
e
;';,
for e
i;!^
'
loss of generality that r
firms with signal
< t (p+) -
'
'
Assume without
Then
lime|o '''(?+£)•
can experiment at time t (p-)
;,,;,
= p+ 6
p
,:'
{p+)
for sufficiently
r (p-) and increase their payoff since
"'"
6 t(p)
monotone on
is strictly
[a,b].
''
~
expected profit when
Pi
=
=
it
>
q2
chooses to experiment at time
e-'^^PHp^2
^(P)
{P-
and
q
V{q,t)= f
where Phejore
qi
=
such that r(gi)
r(ij2)
=
the equilibrium strategy characterized by r(p) and consider firm
follovv^s
i
v
;
Proof. Suppose, to obtain a contradiction, that there exist
Suppose that
-
+ {l-p)qni)dG(p) +
*} ^"'^ -^oj/er
=
>
iP- '"(p)
=
small
0.
Lemma
>
t.
i's
This can be written as
e-''qU2
dG{p),
[
Notice that V(g,t)
^}-
t.
is
(28)
linear in q.
For t{p) to characterize a symmetric equilibrium strategy and given our assumption that
=
r(gi)
T{q2)
=
we have
f,
V{qi,f)>V{q,,t') andV{q2,f)>V{q2,t')
for all
eR+.
t'
Now
=
take q
that for any
t 7^
+
aqi
f,
T {aqi
the
+
(1
middle
—
Q:)q2)
this contradicts
The
that T
b
three
is
i
—
Q-')'?2
{1
-
a)q2,t)
inecjuality
=
f for
Lemma
a G
invertible,
with signal
for
some a
<E
(0, 1).
By
the linearity of
V {q, t),
this implies
q.
=
aV{qut) + {l'a)V{q2,t)
<
aV{qi,T(qi))
=
V{aqi + {l-a)q2,f),
exploits
[0, 1].
+ [l-a)V{q2,T{q2))
This
(29).
string
of
inequalities
Therefore, r must be constant between
implies
and
qy
92-
that
But
'
4,
establishing the current
lemmas together
establish that t
with inverse t"^
— e could experiment
firm
(1
we have
V {aqi +
where
(29)
earlier
[t).
is
lemma.
•
continuous and strictly monotone. This implies
Moreover, r
and increase
•
its payoff.
[b)
=
Now
0,
since otherwise a firm v/ith signal
consider the maximization problem of
This can be written as an optimization problem where the firm in question
28
chooses the threshold signal p
particular, this mciximization
ma^ f
P^la.b]
where the
first
i's
t
^
rather than choosing the time of experimentation t
(t)
In
.
problem can be written as
+ {l-p^^)gn^)dG{p^^) +
e-'^^P-'^ip^^U2
e-'^^P^G{p)qU2,
(30)
Jp
term
the second term
=
is
~
the firm
when p^j <
the expected return
is
when
the expected return
p,
has signal quality p^i G
i
so that firm
~
i
and
[p, b]
copy from
will necessarily
successful innovation.
Next, suppose that r
Then the
(we will show below that r must be differentiable).
differentiable
is
objective function (30)
also differentiable
is
and the
first-order optimality condition
can be written (after a slight rearrangement) as
9{p)
rr'lp)
l_P^(l_p)^-i
i —
G{p)
9
In a symmetric equilibrium, the function r [p)
corresponds to p
=
Therefore,
g.
"
'
'
when
must be a best response
differentiable, r (p)
itself,
which here
a solution to
is
= -|^(i-p)r'-
^^'(p)
to
:
-;'
.
(31)
Integrating this expression, then using integration by parts and the boundary condition r(6)
we obtain
the unique solution (when t (p)
differentiable) as
is
\ogG{b){l~b)-\ogG{p){l-p)+
r/3
6 imply that t (p)
T (p)
i.e.,
is
it
proof,
Then
to establish that this
also differentiable. Recall that a
strictly
is
f
\ogG{z)dz
the unique solution.
monotone. The result follows
monotone function
is
\
.
Lemmas
if
+
e).
some
Then
p.
101,
Theorem
sufficiently small
e
3.23).
>
(p)
5
and
differentiable almost everywhere,
Take p to be a point of
such that r
,
we prove that
can can have at most a countable number of points of non-differentiabihty
there exists
on {p,p
we need
must be continuous and
example, FoUand, 1999,
0,
rb
1
To complete the
=
is
it
for
non-differentiability.
differentiable on (p
(31) holds on both of these intervals. Integrating
(see,
—
e,
p)
over these intervals,
and
we
obtain
I
t{p)
= T{p-e)-—
rp
r{p)
=
T ip)
-
fP
g(z)
{l-z)-—-—dz
Jp-^e
^ j\l
-
forpe (p-e,p), and
G{z)
z)
^^dz
29
,:
for
pe(p,p+e),
Now
taking the limit
T (p+)
= T(p—
But then t
is
{p) is
differentiable
A
—
6
<
on both
>
intervals,
we have
either
given by (27) and
is
/3.
(ii)
must apply.
t
(p)
example
to Proposition 13
G
obtained when
is
is
uniform over
[a,
b]
In that case
interesting feature of
= —rip -\ogp-b +
symmetric equilibria
close to 0, experimentation
is
(ii)
thus differentiable. This argument establishes that
\ogb] for
all
rp
that this
t (jo— ); or
everywhere and proves the uniqueness of equilibrium.
t{p)
An
^
t {p+)
(i)
of these two possibilities contradicts continuity, so
first
particularly simple
< a <
for
The
).
£
may
be delayed
in this case
is
p €
[a, b].
evident from (32): for a arbitrarily
long time.
for arbitrarily
(32)
It
can be verified from (27)
a general feature (for types arbitrarily close to to the lower support
a,
—
log
G {p)
is
arbitrarily large).
when both
Proposition 13 characterizes the unique equilibrium
plete characterization of eciuilibrium
is
provided
cation of this equihbrium characterization
mentioned
of
in the Introduction
asymmetric
is
the next proposition.
in
An
P.
The com-
important impli-
the monotonicity property of symmetric equilibria
(we will see in Section 8 that this
typically not the property
is
Firms with higher signal quality never experiment
equilibria).
<
firms have p
after firms
with
lower signal quality.
Proposition 14 Let
t{p)
=
rPG
Then
G
the support of
C
be [a,b]
[0, 1]
and define
b
logG(6)(l-6)-logG(p)(l-p)+
=
min{/3,6} and
logGi = )dz
/
(33)
(6)
the unique sym.metric equilibrium involves:
ifp>.8
^^P^
I
That
ts,
firms with p
>
/3
f{p}
'ifpe[a,p)
experiment immediately and firms with p G
time t (p) unless there has been an experimentation at
at
t
<
t[p), then a firm with p G [a,^) copies
experiments immediately
if it
first
step of the proof
firms with p £
[a,/?)-
is
The proof
of experimenting at time
t
for
if
<
f
(p).
If there is
the previous attempt
experiment
at
experimentation
was successful and
was unsuccessful.
Proof. The proof consists of three
The
it
t
[a, j3)
steps.
to
show that firms with p >
is
by a single-crossing argument.
a firm with p g
[a,/3) is
30
/3
will
always experiment before
First, recall that the value
given by (28). Defining PafterA3+
~
iP-
t{p)
>
and p > P} and
t
firm with q e
~
PafteTA/S-
iP-
''"(P)
>
* ^'^'^
^
P
value of experimenting for a
f^}' ^'^^
can be rewritten as
[a, /?)
ip^PL,before
+ /
dG{v)
^ /
V9
tiG(p)
/?
(34)
which exploits the
fact that
when p £
when p e
Pbefore °^
PlfterAi3+' there will
be no copying,
and when p G PafterAB-^ ^^^ innovation (which takes place again with probability
copied, for a payoff of II2
q) will
be
= PUi
=
Next, turning to firms with p
>
q'
P, first recall that these firms prefer not to
Lemma
prior successful experimentation (from
Therefore, their corresponding value
3).
copy
can be
written as
V{q',
=
t)
-
q'n2
e— (P)dG(p) + e-^'
/
-
dG{p)
/
+ /
dG{p)
(35)
Note
also that
when
the experimentation time
and the second expression
integral gives us the cost of such a change
Now
bracketed term) gives the gain.
meaning that
crossing property,
>
for q'
than
>
1//3
and
greater for q G
there exists
q'
>
r (p)
r
>
is,
T (p)
T
and thus
than
for q'
=0
strictly greater
>
p.
such that t [p)
for all
p >
From
<T
To show
P.
0.
T
for all
The
[a,/3),
PafterhB-
=
for all
final step is to
First
p >
to
(e"''*
<
t'
<
t'
t,
the
first
times the square
always strictly more valuable
is
i
by the expression
integral in (34)
than
p >
for all
all
/?
and t
firms with
this, first
> T
(p)
p > P
note that
all
are
(35)
a convex combination
is
is
always strictly
argument
for all
will
and
in (34)
so that the cost
1//3,
this strict single-crossing
p G
it
[a,
follows that
p).
experiment immediately,
terms in (35) are multiplied by
any firm with p > P must be identical. Moreover, since
'^
identical for
Therefore, the unique optimal strategy for
[a.,P).
exactly.
p G
first
is
P, so the optimal set of solutions for
>
a reduction to
t
step of the proof establishes that
Therefore, r (p)
p e
1//3,
[a, /?)
some
The next
that
any
First, the gains, given
[a,/3).
t
the comparison of (34) to (35) establishes the single-
Second, the term in parenthesis in the
identical.
of 1/q
E
for q
at
reduced, say from
is
all t
all
6
p >
[0,
/3
T],
is
and
i
>
is
costly because
to experiment immediately.
/?.
combine the equilibrium behavior of firms with p > P with those of
suppose that
Next suppose that
b
>
b
<
p.
Then
p, so that
the characterization in Proposition 13 applies
some
firms might have signals p
>
p.
step of the proof has estabhshed that these firms will experiment immediately.
firms with p e [a,P) will copy a successful innovation at time
unsuccessful experimentation at
t
=
0. If
there
is
31
i
=
or experiment
no experimentation
at
t
=
0,
The previous
Subsequently,
if
there
is
an
then equilibrium
behavior (of firms with p £
now P and
by
G {p).
(a,/?)) is
the relevant distribution
given by Proposition 13 except that the upper support
is
G (p)
conditional on p G
[a, /?),
thus
all
is
terms are divided
This completes the proof of the proposition.
Multiple Firms
6.2
The previous subsection established
that
when there
are
two
firms, the equilibrium satisfies the
monotonicity property whereby firms with stronger signals always experiment earher than those
The same
with weaker signals.
long as the support of
G {p)
result generalizes to the case
where there are
N
does not include "very strong" signals. In particular,
>
2 firms as
let
~
/?N
=
Monotonicity then requires that the support of
see that monotonicity no longer holds
Proposition 15 Suppose
where
(3pf is
m
defined
when
that there are
(36).
Then
N
^.
G
[a,b]
>
(36)
is [a, b]
^
C
In the next section,
[0, /3yv]-
we
will
[0,/3yv]-
2 firms
there exists a unique
and
the support of
G
is
[a,b]
C
[0,/3j^],
symmetric equilibrium. This equilibrium
takes the following form:
•
a firm with signal p experiments at time t^ (p)
mentations
•
all
before,
firms copy immediately
Proof. The proof
Lemmas
is
for each.
Given T2
(p),
7
there
is
N—n
unsuccessful experi-
continuous for n
=
0, 1,
...,
N—
2.
a successful innovation.
similar to that of Proposition 13. It involves establishing the equivalents of
Then we
we can then
are identical to the
there has been
(p) is strictly decreasing and.
if
game
4-6 in the continuation
experiments
13.
where r„
if
arguments
in
which there have been n
solve for T2 (p) using an identical
0, 1,...,A''
—
2 unsuccessful
argument to that of Proposition
n
solve recursively for r^ (p) for each
in
=
=
3,
...,
TV
—
2.
The
details
Proposition 13 and are omitted.
Nonmonotonicity
Equilibria characterized in the previous section involve inefficient delay as in the model with
firms with symmetric signals.
Nevertheless, there
with stronger signals experiment
of
G
satisfies [a,b]
(f.
a monotonicity property, whereby firms
than firms with weaker signals.
earlier
show that monotonicity does not always
and the support
is
apply.
[0,/3],
In particular,
when
In this section,
there are
N
>
we
2 firms
any symmetric equilibrium necessarily involves
nonmonotonicity with positive probability.
32
The main
result of this section
is
provided in the next proposition, which estabUshes that
monotonicity no longer holds in this case.
Proposition 16 Suppose
N
that there are
>
and
3 firms
ofG
the support
satisfies [a,
^
b]
[0,/3].
Then:
There does not
1.
t
2.
=
exist a
symmetric equilibrium in which
In any symmetric equilibrium, there
is
positive probability that a firm with p
experiment earlier than a firm with
>
p.
N
=
p'
and that
3,
Consider firm
i
with pi
that both firms have p
i
>
Let Xo
j3.
t>e
>
experiments at time
^(Pi,0)
Intuitively,
when none
=
t
=
when
,
112.
i
|
p >
/3].
>
/?
0.
and X2 be the probability
Since pi
>
/3,
by hypothesis,
,
,
p >
when
P,
successful, the firm
both of the other two firms have p >
receives Hi.
t
=
i
X2Pi^l-
When
experiments at time
at
is
n2
ni
of the other two firms have
successful, firm
this other firm also
= E [p
expected payoff
+ X-iPi
XoPi^2
immediately, receiving payoff
copying, so
Its
0.
will
/3
the probability that none of the other two firms have
Let us also define C
p.
>
> P experiment
firms with p
all
Xi be the probability that one of the other two firms has p
/?.
at
=113. Suppose, to obtain a contradiction,
112
that there exists a symmetric equilibrium where
firm
experiment
f3
0.
Proof. (Part 1) Suppose that
P >
>
firms with p
all
=
When
and
is
there
/3,
one of the other two firms has p >
successful with probability ^
copied
is
= E [p
is
P,
|
p
no
then
>
/?]
In that case, in a symmetric equilibrium the third firm copies each one of the two successful
innovations with probability 1/2.
with p
> P
Now
is
unsuccessful,
With
the complementary probability,
and the third firm necessarily copies firm
consider the deviation to wait a short interval
e
>
1
—
C>
the other firm
i.
before innovation. This will have
payoff
.
.,
.
I
iimV(pi,e)
=
xoP^n2
>
v{p,,o).
+
xiPr(cni
+
-C)n2) +
(i
X27'.:ni
£lO
The
first line
,
,,
•
,
,
of the previous expression follows since, with this deviation,
other firm with p
follows since flj
>
>
/?,
112,
the third firm necessarily will copy the
first
when
innovator.
The
establishing that there cannot be an equilibrium in which
33
there
all
is
one
inequality
firms with
p > P experiment
in
which
N
>
3
at time
and n„s
(Part 2) Part
1
=
i
0.
This argument generalizes, with a httle modification, to cases
differ.
imphes that any symmetric equihbrium must involve mixing over the time
>
of experimentation by firms with p
proof of Proposition 14,
that
maximize
their
all
firms with p
> P
the same argument following from (35) in the
will
have the same
set of
experimentation times
expected payoffs, so must use the same mixed strategies
equilibrium. Therefore, there
a positive probability that any one of
is
establishing the claim in Part
The next
By
p.
them
in
will
any symmetric
experiment
first,
2.
when
proposition characterizes the form of the mixed strategy equilibrium
the
monotonicity property no longer holds. To simplify the exposition, we focus on an economy in
N=3
which
G
and
uniform on
is
(relatively straightforwardly)
to any distribution G,
The
[0, 1].
N
extended to
though
this
is
characterization result in this proposition can be
>
3.
We
also conjecture that
it
can be extended
less trivial.
Proposition 17 Consider an economy with
N
=^
3 firms, 112
= ^3 ind G
uniform over
[0, 1].
Then, the following characterizes the unique symmetric equilibrium.
•
Firms with p > P experim.ent
mented
until
t.
flow rate ^{t) as long as no other firm has experi-
They experim.ent immediately following another (successful or unsuccessful)
experiment. There exists
T<
oo such that
That
> P
will
all
is,
firms with p
Firms with p < P
a flow rate A2
(t)
have necessarily experimented within the interval
[0,
T]
immediately copy a successful innovation and experiment according
to
(or equivalently, limt^T
•
at the
£,
{i)
= +00/
.
follounng an unsuccessful experimentation and at the rate A3
has been no experimentation until time T.
Proof. From Proposition
16, firms
subset of IR+. Let us define
time
t
has p^,
Now
and
>
p.
ij,{t)
with p
that
consider the problem of firm
this firm
> P must mix
experiments at time
t,
i
its
G
with
is
over experimentation times on
uniform over
pi
if
there
-
as the probability that firm
The assumption
(t)
>
/?.
If
~
i
that has not experimented until
[0, 1]
implies that /i(0)
34
+
=
1-/3.
there has yet been no experimentation
payoff (discounted to time
V{p,,t)=p,e-^'[Tl,p.{ty~
some
t
=
U2{l-m%
0)
is
since
/.t
With the complementary
probability,
its
experimentation by some small amount dt
V{p^,t
+ dt) =
where p
^(i).
=
> P and
the probability with which both other firms have p
is
(()
innovation will be copied.
>
p^e-''^^+'^*^\2p^{t)ij{t)dm^
E[p|p
>
j3]
and we use the
0,
then
-
its
payoff
+
(1
+
2^{t)ii{t){l
Alternately,
>
U2{1 -
ii{t
^
dtf)]
+ n2(l -
/i(t
+
dt))]
+ dtf +
p)dt[YlipL{t -V dt)
with p
fact that other firms
l3
experiment at the rate
—
In a mixed-strategy equilibrium, these two expressions must be equal (as dt
these equal and rearranging,
In addition, the evolution of beliefs
£,
(i)
-^
ji
(t)
ri^\t)
let
0).
Setting
=
(37)
nT^n^-
given the uniform distribution and flow rate of experi-
can be obtained as
-..-"-:•
'^
/.(f)
—-,-——i
=
^^i_.;, ^^,
e-/of(-)'^-(l_/j)+/3
Now
>
we obtain
^^^ + 2mKm - Kt))\p + Kt)] mentation at
delays
if it
is:
2({t)ij{t)dt)lUifi{t
-
will thus not copy.
;.,..,:,.,,-,,
,.
(38)
.
us define
"
'-
/(i)=e--^«(^)'^(^)(l-/]).
'
-
.
'
(39)
.'
Using
(39), (38)
can be rewritten as
"
.
\
,
fit)
lj{t)
f{t)+P'
which
in
turn implies
fit)
l-l^it)-
Moreover
(39) also implies that
nt)
-
m'(o
'^'^-m- ii-.itmt)
'
Substituting these into (37),
we obtain the
:
'
-
;.
.^''^
j
following differential equation for the evolution of
Kt)-
.'_
,
2ii{t)ii'{t)
+
2at)Kt){l -
fi{t))\p
+
/lit)]
+
(Z?-^
-
ri.?{t)
=
Further substituting ^(t) from (40), we obtain
a'{t)
=
-,
(1
35
-
1) /i^(0)-
l_
^.
,
This differential equation satisfies the Lipschitz condition and therefore
it lias
a unique solution,
which takes the form
arctan
m(0 =
rJ^-T
vr'^i
with boundary condition
firms with p
^(i)
=
\f^
t'^"
>
P,
ciC2(l
£,
+
(t), is
=
fi{0)
1
-t
Pir'-i)
/^.(O)
(
+
v^/3"^
-
1
VP^^i
1-/3. Given this solution, the flow rate of experimentation for
obtained from (40) as
tan(ci(-C2t
1
+ cg))^)
+
ci
tan(ci(-C2f
+
1
+
C3))
tan(ci(-C2i
+
C3))J
where
arctan
ci
It
=
^
\Ij3
-
1,
= - ^_,
c2
—, and
(M(o)v'r^
C3
J
can then be verified that
lim f
T =
where
C3/C2.
It
it)
can also be verified that
=
00,
for all
(
G
[0,T],
where firms with p
> 8
are
experimenting at positive flow rates, firms with p < P strictly prefer to wait. The equilibrium
behavior of these firms after an unsuccessful experimentation or after time
T
is
reached
is
given by an analysis analogous to Proposition 14 and again involves mixing. Combining these
observations gives the form of the equilibrium described in the proposition.
Notice also that Proposition 17 implies that nonmonotonicity in this case affects the allocation of resources mainly by inducing delay (only firms with
[0,T]),
p > P act during the time interval
because following a single vmsuccessful experimentation in
[0,
T], other firms
with p > p
experiment immediately and these firms would have preferred not to copy a successful innovation.
8
Patents, Heterogeneity, and Nonmonotonicity
In this section,
we
discuss the optimal allocation with heterogeneity,
its
comparison to equilib-
rium, and the role of patents in the economy with heterogeneity.
8.1
Welfare
Consider a social planner that
is
interested in maximizing total surplus (as in Section
5).
What
the social planner can achieve will depend on her information and on the set of instruments
that she has access to. For example,
firm,
then she can achieve a
much
if
the social planner observes the signal quality, p, for each
better allocation than the ecjuilibrium characterized above.
36
However,
it is
more
plausible to limit the social planner to the
same information structure.
same equilibrium
that case, the social planner will have to choose either the
allocation as in
the symmetric equilibria characterized in the previous two sections, or she will
asymmetric equilibrium, where one of the firms
p
(this
carmot be conditioned on p since p
More
G
of
is
specifically, let us focus
[a, 5]
c
[0,/?].
is
is
instructed to experiment
first
In
implement an
regardless of
its
private information).''*
on the economy with two firms and suppose that the support
In this case, without eliciting information about the realization of firm
types, the p's, the planner has three strategies.
1.
Staggered asymmetric experimentation: in this case, the social planner would instruct one
of the firms to experiment immediately
and then have the other firm copy
successful innovation. Since the social planner does not
experimenting firm randomly.
2.
We
know
the
p's,
if
there
is
a
she has to pick the
denote the social surplus generated by this strategy by
Staggered equilibrium experimentation: alternatively, the social planner could
let
the firms
play the symmetric equilibrium of the previous two sections, whereby a firm of type p will
experiment at time t
firm.
We
(p) unless there
denote the social surplus generated by this strategy by 5^, since this
as the equilibrium outcome. ^'^
3.
has previously been an experimentation by the other
';:
Simultaneous experimentation: in
to experiment immediately.
We
.
;:!-,;.
this case, the social
m
''
is
the
same
.,:..';.'.''-,'
/:
planner would instruct both firms
denote the social surplus generated by this strategy by
S?
'2
The
from these
social surpluses
different strategies are given as follows.
In the case of
staggered asymmetric experimentation, we have
Pi2n2
5r
Yet another alternative
solution to a
is
+ (l-pi)
/
p2dG{p2)\Tl^
dGipi).
to specify exactly the instruments available to the planner
mechanism design problem by the planner. However,
if
and characterize the
these instruments allow messages and include
payments conditional on messages, the planner can easily elicit the necessary information from the firms.
'Without eliciting information about firm types and without using additional instruments, the social planner
cannot implement another monotone staggered experimentation allocation. For example, she could announce
that if there is no innovation until some time i > 0, one of the firms will be randomly forced to experiment. But
such schemes will not preserve monotonicity, since at time t, it may be the firm with lower p that may be picked
for experimentation. In the next subsection, we discuss how she can implement better allocations using patent
'
payments.
37
In contrast, the expected surplus
,.:'
be written as
j
g-rr(max{pi,p2})
max
,.',
•
.
,
SRs
+
(1
/ min{pi,p2}n
(pi po })
,
in the
equihbrium as specified
the firm with the stronger signal (higher p) will experiment
max
firm experiments
the
If
firm
first
fails
max
—
1
G
in
Proposition
{pi,P2}, in which case
ma:x {pi,P2}), then the second
and succeeds with probability min {pi,p2}. Since both
drawn independently from G, we integrate over
The
(probability
dG(pi)dG(p2)
so there will be delay until
first,
{pi,p2}- At that point, this firm will succeed with probability
the second firm will copy.
is
- max
by observing that
';'"
-.:_..::
'
{pi ps)
Intuitively, this expression follows
13,
from the unique (mixed-strategy) symmetric equihbrium can
and p2
pi
are
randomly
twice to find the expected surplus.
surplus from simultaneous experimentation, on the other hand, takes a simple form and
given by
rb
S^ =
2Yl,
/
vdG{p),
Ja
since in this case each firm
is
and generates payoff
successful
Il\
with probability p distributed
with distribution function G.
In this case, there
metric equihbrium
social
is
may
no longer any guarantee that
max {^f
,
> S^
^2^}
Therefore, the sym-
.
generate a higher expected surplus (relative to allocations in which the
planner does not have additional instruments). To illustrate
example, where p has a uniform distribution over
us consider a specific
this, let
In this case, staggered asymmetric
[0,/?].
experimentation gives
re
?f
=
/
r0
/
+
2pin2
(1
- pi)p2nidp2dpi = Do
1
~
(
2
+
+
3
^/3
4^
whereas simultaneous experimentation gives
fB
S^ =
2pY[idp=mi =n2.
/
Jq
Comparing simultaneous experimentation and staggered asymmetric experimentation, we can
5f > S2 whenever
with common signals,
> 2/3 and 5f <
whenever
<
2/3, showing that, as
conclude that
/?
in the case
either simultaneous or staggered experimentation might be
optimal. Next,
[0,j3],
we can
also
compare these surpluses
S2
to
S^
.
Since p
(5
is
uniformly distributed in
(27) implies that
t{p)
=
-7;
b-
logP -13
As a consequence, max{pi,p2} has a Beta(2,l)
distributed Beta(l,2).
Then evaluating
the
+
log/?]
.
distribution (over
[0,/3])
while min{pi,p2}
expression for 5^, we find that
38
when
<
/3
<
is
2/3,
> S^
5.2"
/3
<
,
~
/3*
so simultaneous experimentation gives the highest social surplus.
5f > 5^,
0.895,
social surplus.
asymmetric experimentation gives the highest
so that staggered
when
Finally,
<
j3*
gives higher social surplus than
<
/?
1,
When 2/3 <
5^ > Sf >
S2
so the symmetric equilibrium
,
both staggered asymmetric experimentation and simultaneous
experimentation.
Finally,
it is
a particular value
p,
we can repeat the same argument
symmetric equilibrium can be arbitrarily
G to be highly concentrated
by choosing
also straightforward to see that
as in subsection 5.1
around
and show that the
the optimal allocation.
inefficient relative to
Equilibrium with Patents
8.2
Equilibria with patents are also richer in the presence of heterogeneity. Let us again focus on
the case in which there are two firms.
one discussed
Suppose that there
a patent system identical to the
is
whereby a firm that copies a successful innovation pays
in subsection 5.2,
r]
to the
innovator. Let us define
It is clear,
with a reasoning similar to
payment
the patent system specifies a
equilibria with patents.
6
>
=
of
77.
3,
that only firms with p
The next proposition
,
p^ will copy
when
characterizes the structure of
;
that there are two firms
and the patent system
Let p^ be given by (^l), the support of
for copying.
<
'
,
Proposition 18 Suppose
77
Lemma
G
he
[a, 6]
specifies a
C
[0,1],
payment
and define
min{6,p''} and
logG(6)(ni-2r7-6ni)~logG(p)(ni-277-pni)+ni f \ogG{z)dz
Jp
f^[p)
r{Il2
+
r/)G (6)
.
,
Then
the unique
(v)
^P'
-
ifp>p^
f (p)
firms with p
>
•
tHp)
for any
rj'
>
rj
.'
ifpG[a,p^)
'at t
<
f^ {p).
'
rj
tends to reduce delay. In particular:
such that b
strict inequality
•:''--'
p^ experiment imm.ediately and firms with p G {a,?^) experiment at time
unless there has been an experimentation
Moreover, a higher
'^
'
^
I
is,
(42)
symmetric equilibrium involves:
t"^
That
_
<
whenever t^
p^ and b
(p)
>
<
p^
,
0;
39
we have
t'I
[p)
<
t^ {p) for
all
p £
[a, b],
with
for any
•
T)
such that
rj'
starting at
t]
=
>
b
for
rj'
p^
there exists p* (r/)
,
p*
p E
all
,p^
{rj')
Proof. The proof mimics that of Proposition
problem of firm
i,
max f
with signal
=
p^
+
r?)
will receive
the unique equihbrium
?;.
=
b
m
h
f (p)
that
show that
b
=
b).
t''
is
its
=
(p)
firm
is
Repeating the same argument as
in
77
,
if
implies that
t''
r'^'
(p)
strictly decreasing for all
=
r]
>
any
for
since log G(6)
>
log
>
b
p'^
.
>
>
We
rj.
f (p)
is
=
p^
and
effect of
r]
first line is
dr''''
on
p^
so that
decreasing in
is
show
therefore only need to
differentiable,
b.
{p) / drj (in the
sufficient to
it is
(02
+
^^^
?7'
Hi -
+
VgW)
'
'°^
27?'
again strictly negative and so
p and tends to
p*
Ip* [f]')
[rj'),
,p^
it is
),
as
is
the
p approaches p^ (from
no larger than the
T^ (p)
is
become
In particular, whenever
first
again decreasing
important role of patents
creases, T (p) tends to
p''
<
<0,
^
'
^"
5(P"
,
)
-f
in
in
77,
in
is
(43)
(p)
77',
this
The second
line.
term
is
(77')
decreasing
such that
This establishes that
line.
is
highlighted by this result.
is less
for
this does not necessarily apply for very low p's,
but
When
r;
in-
delay and thus "time runs faster".
reduced by an increase
40
77')
completing the proof.
experimentation
t (p)
G (p)
expression in the second
the second
"steeper" so that there
b,
^°S
Therefore, there exists p*
(42)).
term
-
G{p^')Ui
first
neighborhood of
In particular:
expression in the second line could be positive, however. For given
b,
p^
\a,p^).
1
-p"' Hi
G(p'?')nir(n2+7/)G(p'0
>
<
p^ and b
'
In that case b
g{p^')
p^
<
0.
'"''
An
will
it
Proposition 14 establishes that
+ ^^{logG{b)-logG{p))
terms because of the
dr/
p &
,
rj)
f (p)
G(p) and
dT'''(p)
>
v)
negative. This follows by differentiating (42) (with
is
f^{p)
+
(0-2
will include additional
p
+
{U2
innovator, then
first
p £
for
77'
[a,p^). Since
1
Next, suppose that
for
(p) q
In particular,
di]
in
0.
.
f (p)
=
(p)
p £
derivative with respect to
dT'^(p)
The
the
of the proposition, first suppose that b
both cases. Recall also that
so that
rj,
and
i
To prove the second part
>
(30) to
- p^,) qU,)clG (p^J + e^'P^G
(1
decreasing in
(p) is
with the only difference that the maximization
given by (42).
is
such that t^
)
with strict inequality whenever t^ {p)
,
14,
which takes into account that copying has cost
be copied and
[0,p''
now modified from
g, is
-
e-''^(P-) (p^, (Ha
G
is still
in
patent payments.
When
true for high p's. Overall, this
common
result implies that as in the case with
when
incentives and reduce delay. In the limit,
patents tend to increase experimentation
p's,
becomes
77
arbitrarily large, the equilibrium in-
volves simultaneous experimentation. Nevertheless, as discussed in subsection 8.1, simultaneous
experimentation
may
not be optimal in this case.
Alternatively (and differently from Proposition 18), a patent system can also be chosen such
that the socially beneficial ex post transfer of knowledge takes place.
In particular, suppose
that there has been an innovation and the second firm has probabihty of success equal to p. In
this case, social surplus
second firm
with
jD
<
there
if
copying, and
is
is
it
+ pYli
equal to Hi
if
the
forced to experiment. This implies that to maximize ex post social welfare, firms
is
2/3
equal to 2112
is
—
should be allowed to copy, whereas firms with p
1
experiment. Clearly, from (41) choosing
18, this will typically lead to
r]
=
Y\.i
—
achieves
TI2
>
2/3
this.
—
1
should be induced to
Naturally, from Proposition
This argument
an equilibrium with staggered experimentation.
establishes the following proposition (proof in the text).
Proposition 19 ^ patent
system, with
experimentation behavior for
The juxtaposition
all
p £
7/
[a, 6],
of Propositions 18
=
IIi
-
112
induces the socially efficient copying and
but typically induces delayed experimentation.
and 19 implies that when
signal quality
is
heterogeneous
and private information, the patent system can ensure either rapid experimentation
socially beneficial ex post transfer of
will
knowledge (and experimentation by the right types), but
not typically be able to achieve both objectives simultaneously.
.
,
Patents and Nonmonotonicity
8.3
Our
or the
shows that patent systems cannot prevent the nonmonotonicity identified
last result
in the
previous section.
Proposition 20 Consider
[0, /3],
the
economy with
N
>
3 firms
and
the support of
G
given by
so that the symmetric equilibrium without patents involves nonmonotonicity.
any patent
there exists
i],
there exists
an equilibrium
no patent system such that
either with
no copying or nonmonotonicity.
all equilibria will
[a, b]
Then for
That
suppose that
r;
<
'
knowledge.
Proof. Clearly, any patent with
111
proof of Proposition
—
112.
16.
Then
77
let
>
Ifj
—
is,
avoid nonm^onotonicity while ensuring
'
efficient transfer of
^
112 will
us adopt the
prevent
same
efficient transfer of
definitions of C,
Xc
knowledge.
So
Xi ^-nd X2 ^^ i^ ^^^
Suppose, to obtain a contradiction, that there exists an equilibrium in
41
which
all
does so
p >
firms with
/3
experiment immediately. Then the
payoflF to firm Pi
>
/3
when
it
^^
is
=
V{p^,0)
X0P^{^2
+
'2r,)
+
C(^
XlPl
+
^)+(i-0(n. + .)
+
X2P'^-i
which takes into account the patent payments from the other firms into account.
Now
consider a deviation for firm
that involves waiting for a short interval
i
e
>
before
innovation. This will have payoff
limV^(p„e)
since
< Hi —
t]
+
=
X'oP.(^2
>
V{p,,0),
> P and
delay by firms with p
XlP,(C^l
+ (l-C)(^2 +
r/))+X2P^^l
,
Therefore, whenever
112.
+
277)
ij
< Hi —
112,
the symmetric equilibrium will involve
thus potential nonmonotonicity.
Conclusion
9
This paper studied a simple model of experimentation and innovation.
private signal on the success probability of one of
when and which
We
show
project to implement.
both
that,
A
many
potential research projects and decides
successful innovation can be copied by other firms.
and continuous time, symmetric
in discrete
Each firm receives a
where actions do
equilibria,
not depend on the identity of the firm, necessarily involve delayed and staggered experimentation.
Wlien the
signal quality
the
is
same
for all players, the
(pure-strategy symmetric equilibria do not exist).
equilibrium
is
When
equilibrium
is
in
mixed
strategies
signal quality differs across firms, the
represented by a function r (p) which specifies the time at which a firm with signal
quality p experiments.
As
environment with
in the
common
signal quality, the equilibrium
may
involve arbitrarily long delays.
We
large.
also
show that the
The optimal
social cost of insufficient experimentation incentives can
allocation
may
be arbitrarily
require simultaneous rather than staggered experimentation.
In this case, the efficiency gap between the optimal allocation and the equilibrium can be arbitrarily large. Instead,
equilibrium
is
when
inefficient
the optimal allocation also calls for staggered experimentation, the
because of delays.
in the ecjuilibrium to that in the
One
of the
main arguments
We
show that
in this case the ratio of social surplus
optimal allocation can be as low as 1/2.
of the paper
is
that appropriately-designed patent systems en-
courage experimentation and reduce delays without preventing
edge across firms. Consequently, when signal quality
is
the
efficient ex post transfer of
same
for all firms,
designed patent system can ensure that the optimal allocation results in
42
all
knowl-
an appropriately-
equilibria. Patents
are particularly well-suited to providing the correct incentives
when the optimal
allocation also
requires staggered experimentation. In this case, patents can simultaneously encourage
one
of
the firms to play the role of a leader in experimentation, while providing incentives to others
Technically, appropriately-designed patents destroy
to copy successful innovations.
equilibria,
which are the natural equihbria
in the
absence of patents but
may
symmetric
involve a high
degree of inefficiency. That patents are an attractive instrument in this environment can also be
seen from our result that, while patents can implement the optimal allocation, there exists no
simple subsidy (to experimentation, research, or innovation) that can achieve the same policy
objective.
When
signal quality differs across firms, an additional dimension of efficiency
in
which firms with different signals experiment.
or
when
there
is
We
is
the sequence
show that when there are only two firms
an arbitrary number of firms but the support of the signal quality distribution
does not include "very strong" signals, the equilibrium has the following monotonicity property:
firms with stronger signals always experiment earlier (no later) than firms with weaker signals.
However,
this result
no longer holds when there are more than two firms and "very strong"
nals are possible. In this case,
of nonmonotonicity.
we show
that the equihbrium necessarily involves
it is
some amount
Patents are again useful in encouraging experimentation in this case and
reduce delays. Nevertheless, interestingly,
monotonicity,
sig-
when
the equilibrium without patents involves non-
impossible for patents to both ensure efficient ex post transfer of knowledge
and restore monotonicity.
We
believe that the role of patents in encouraging socially beneficial experimentation
more general than the simple model used
ignored the consumer side.
It is
in this paper. In particular,
possible that
new
throughout the paper we
innovations create benefits to consumers that
are disproportionately greater than the use of existing successful innovations (as
relative profitabihties of the
are even greater
innovations.
lines versus
The
same
and patents can
is
activities). In this case, the social benefits of
compared
to the
experimentation
also be useful in preventing copying of previous successful
investigation of the welfare
and policy consequences
experimenting with new, untried research lines
area.
43
is
of pursuing successful
an interesting and underresearched
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