Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/statedependentinOOacem
DEWEY
HB3i
.M415
3M
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
STATE-DEPENDENT INTELLECTUAL
PROPERTY RIGHTS POLICY
Daron Acemoglu
Ufuk Akcigit
Working Paper 06-34
December 11, 2006
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=951 788
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JAN
C 8 2007
LIBRARIES
State-Dependent Intellectual Property Rights Policy"
Daron Acemoglu
Ufuk Akcigit
MIT
MIT
December 2006
Abstract
What form
economic growth?
Should technological followers be able to license the products of technological leaders? Should
of intellectual property rights (IPR) policy contributes to
a company with a large technological lead receive the same IPR protection as a company
with a more limited lead? We develop a general equilibrium framework to investigate these
The economy
many
and firms engaged in cumulative (step-bystep) innovation. IPR policy regulates whether followers in an industry can copy the technology
of the leader and also how much they have to pay to license past innovations. With full
patent protection, followers can catch up to the leader in their industry either by making the
same innovation(s) themselves or by making some pre-specified payments to the technological
questions.
consists of
industries
leaders.
We prove the existence of a steady-state
We
equilibrium and characterize some of
then quantitatively investigate the implications of different types of
equilibrium growth rate.
growth rate
in the
The two major
IPR
its
properties.
policy on the
results of this exercise are as follows.
First, the
standard models used in the (growth) literature can be improved significantly
by introducing a simple form
of licensing. Second,
we show that
full
patent protection
is
not
optimal from the viewpoint of maximizing the growth rate of the economy and that the growth-
maximizing policy involves state-dependent IPR protection, providing greater protection to
technological leaders that are further ahead than those that are close to their followers. This
form of the growth-maximizing policy is a result of the "trickle-down" effect, which implies that
providing greater protection to firms that are further ahead of their followers than a certain
threshold increases the R&D incentives also for all technological leaders that are less advanced
than this threshold.
Keywords:
competition, economic growth, endogenous growth, industry structure, inno-
vation, intellectual property rights, licensing, patents, research
JEL
and development, trickle-down.
classification: 031, 034, 041, L16.
*We thank
Science Days,
conference and seminar participants at the Canadian Institute of Advanced Research, European
MIT, National Bureau
of
Economic Research, Toulouse Information Technology Network, and
Philippe Aghion and particularly Suzanne Scotchmer for useful comments. Financial support from the Toulouse
Information Technology Network
is
gratefully acknowledged.
1
Introduction
How
should the intellectual property rights of a company be protected? Should a firm with
a large technological lead receive the same intellectual property rights (IPR) protection as a
company with a more
limited technological lead? These questions are central to
sions of patent policy.
A
recent ruling of the
Microsoft to share secret information about
York Times, December
22, 2004).
There
is
European Commission,
its
many
discus-
example, has required
for
products with other software companies (New
a similar debate about whether Apple should make
iPod's code available to competitors that are producing complementary products. Central to
the debate about Microsoft and iPod
the fact that they have a substantial technological
is
lead over their rivals. This implies that the analysis of
emic questions requires a framework
By
policy.
state-dependent
IPR
many
for the analysis of state-dependent
policy,
we mean a
policy that
patent/IPR protection
makes the extent
or intellectual property rights protection conditional on the technology
firms in the industry. Existing
first
of patent
gap between
different
work has investigated the optimal length and breadth of patents
assuming an IPR policy that does not allow
a
and acad-
of the relevant policy
for licensing
and
is
uniform. In this paper,
we make
attempt to develop a framework that incorporates both licensing of existing patents and
state-dependent
Our
IPR
policies.
basic framework builds on and extends the step-by-step innovation models of Aghion,
Harris and Vickers (1997) and Aghion, Harris, Howitt and Vickers (2001), where a
(typically two) firms
number
of
R&D
in
engage in price competition within an industry and undertake
order to improve the quality of their product.
The technology gap between
the firms determines
the extent of the monopoly power of the leader, and hence the price markups and profits.
The purpose
of
R&D
by the follower
Schumpeterian models of innovation,
Grossman and Helpman,
is
e.g.,
(e.g.,
up and surpass the leader
standard
(as in
Reinganum, 1981, 1985, Aghion and Howitt, 1992,
1991), while the purpose of
competition of the follower and increase
general
to catch
its
R&D
markup and
by the leader
profits.
As
is
to escape the
in racing-type
models
in
Harris and Vickers, 1985, 1987, Budd, Harris and Vickers, 1993), a large gap
between the leader and the follower discourages
R&D
by both.
Consequently, overall
R&D
and technological progress are greater when the technology gap between the leader and the
follower
is
relatively small.
1
One may expect
that
full
patent protection
may be suboptimal
in
a world of step-by-step competition; by stochastically or deterministically allowing the follower
Aghion, Bloom, Blundell, Griffith and Howitt (2005) provide empirical evidence that there is greater R&D
where there is a smaller technological gap between firms. See O'Donoghue, Scotchmer and
in British industries
Thisse (1998) for a discussion of
how patent
life
may come
to
an end because of related innovations.
to use the innovations of the technological leader, the likelihood of relatively small gap between
and
leaders
may
less
and thus the amount of R&D may be
followers,
further conjecture that state-dependent
IPR
policy
raised.
may
also
2
Based on
this intuition,
one
be useful and should provide
protection to firms that are technologically more advanced relative to their competitors.
There are two problems with
this intuition. First, existing
growth models (including those
mentioned above) do not allow licensing of the leading-edge technology and the presence of
licensing changes the trade-offs underlying the above intuition.
derived from models with uniform
of state-dependent
To
IPR
policy on
IPR
policy,
R&D
3
Second, the above intuition
without a systematic study of the implications
incentives.
we construct a general equilibrium model with
investigate these issues systematically,
step-by-step innovation, potential licensing of patents and state-dependent
model economy, each firm can climb the technology ladder via three
by "catch-up R&D," that
leader;
(ii)
by
"frontier
R&D
is,
R&D,"
that
leader for a pre-specified license
technological leader.
but
of,
is
is,
policy. In our
different
methods:
(i)
building on the patented innovations of the technological
and
feature
essential for understanding
The combination
IPR
investments applied to a variant of the technology of the
fee;
The second
is
(iii)
is
as a result of the expiration of the patent of the
not present in any growth model that we are aware
how IPR
policy works in practice.
of these policies allows a variety of different policy regimes.
The
first is full
patent protection with no licensing, which corresponds to the environment assumed in existing
growth models
(e.g.,
Aghion, Harris, Howitt and Vickers, 2001) and provides
full (indefinite)
patent protection to technological leaders, but does not allow any licensing agreements
the license fees to infinity).
The second
is
full patent protection with licensing,
to
regime as
pay a patent
"full
The
third regime
2
3
than the
We
is
from the use of the leading-
uniform imperfect patent protection, which deviates
from the previous two benchmarks by allowing either expiration of patents and/or
fees that are less
fee.
patent protection" because patents never expire and followers have
fee equal to the gain in net present value resulting
edge technology. 4
sets
which allows
technological followers to build on the leading-edge technology in return for a license
refer to this
(it
full
benefit to the follower.
The
license
adjective uniform indicates that
is conjectured, for example, in Aghion, Harris, Howitt and Vickers (2001).
See Scotchmer (2005) for the importance of incorporating these types of licensing agreements into models
This
of innovation.
4
The
alternative
would be
for the followers to
pay a
fee
equal to the
damage
to the technological leader,
and further innovation. In practice,
licensing fees or patent infringement fees reflect both the benefits to the firm using the knowledge and the
damage to the original inventor (see, e.g., Scotchmer, 2005). In our analysis, we allow the license fees to be set
at any level, thus incorporating both possibilities.
resulting from the loss of the leader's
monopoly power
after their licensing
in this policy
environment
industries are treated identically irrespective of the technology
all
gap between the leader and the
The
follower.
state- dependent imperfect patent protection,
and most interesting policy regime
final
which deviates from
patent protection as a
full
function of the technology gap between the leader and the follower in the industry
allows technologically
more advanced
more or
firms to receive
policy regimes captures a different conceptualization of
Note however that the
right.
We
first
last
regime
is
IPR
less protection)
policy and
is
is
.
Each
(i.e.,
it
of these
interesting in its
own
general enough to nest the other three.
prove the existence of a stationary (steady-state) equilibrium under any of these
policy regimes and characterize a
number
example, we prove that with uniform
IPR
For
of features of the equilibrium analytically.
R&D
policy,
investments decline
when
the gap
between the leader and the follower increases.
We then
turn to a quantitative investigation of growth-maximizing
by providing a number of simulations
different forms of
1.
IPR
policies.
5
Our
of the equilibrium for plausible
(
"optimal"
)
IPR
policy,
parameter values and
for
quantitative investigation leads to two major results:
Allowing for licensing of patents increases the equilibrium growth rate of the economy
significantly.
Intuitively,
without such licensing, a large part of the
R&D
effort
goes to
R&D does not contribute to the growth rate of the economy.
Licensing implies that R&D by all firms — not just the leaders — contributes to growth
and also increases the R&D incentives of followers. In our benchmark parameterization,
duplication, and followers'
allowing for licensing increases the steady-state equilibrium growth rate of the economy
from 1.86% to 2.58% per annum.
2.
Growth-maximizing IPR policy
centive effect of relaxing
IPR
is
state dependent.
protection on
R&D,
In particular, because of the disin-
uniform
IPR
policy (either by
ma-
nipulating license fees or the duration of patents) has minimal effect on the equilibrium
growth
tion
rate.
In contrast, state-dependent
and growth
state-dependent
in the
IPR
economy. For example,
"
policy can significantly increase innova-
in
our baseline parameterization, optimal
policy increases the growth rate to 2.96% relative to the growth
rate of 2.63% under uniform
of
IPR
IPR
policy.
More important than
growth-maximizing state-dependent IPR policy
is its
form.
the quantitative effect
We
find in all cases that
Throughout, we simplify terminology by using the terms "optimal" and "growth-maximizing" interchangeis well known that welfare-maximizing and growth-maximizing policies need not coincide, and in fact,
in models with quality competition, the equilibrium may involve excessive innovation and growth (e.g., Aghion
and Howitt, 1992). Nevertheless, when the discount rate is small, welfare and growth-maximizing policies will
be close, which justifies our use of the term "optimal" in this context.
ably. It
optimal state-dependent
IPR
policy provides greater protection to technological leaders
that are further ahead than those that have only a small lead relative to their followers.
The reason
for this result
is
the trickle-down of
R&D
incentives. In particular,
particular state for the leader (say being n* steps ahead of the follower)
perform
this increases the incentives to
ahead, but for
all
leaders with a lead of size
can be powerful enough to increase
nological leaders relative to the
the
IPR
R&D
is
very profitable,
not only for leaders that are n*
n < n* — 1.
when a
—
1
steps
Notably, the trickle-down effect
—rather than reduce—the R&D investments by tech-
benchmark with
full
patent protection. Thus, reducing
protection of leaders that are few steps ahead while offering a high degree of
protection to firms that are sufficiently advanced relative to their competitors increases
R&D
incentives. Consequently, in contrast to existing models, imperfect state-dependent
IPR can
increase
R&D
investments relative to
full
protection. Moreover, contrary to the
IPR
conjecture above, the trickle-down effect implies that the optimal state-dependent
policy should provide greater protection to firms that have a larger technological lead.
Overall, our analysis illustrates the different effects of
IPR
policy
on the equilibrium growth
rate of an economy. It shows that allowing for licensing can increase growth significantly
that optimal state-dependent
relative to
uniform
policies could
IPR
IPR
policy.
policy can also increase the growth rate of an
and
economy
Consequently, the benefits of considering a rich set of
IPR
be substantial in practice. Our analysis also suggests that the structure of optimal
IPR may depend on
the equilibrium industry structure (which determines the technology gap
between leaders and followers as a function of the underlying technology of the industry).
more
IPR
detailed analysis of the relationship between industry structure
policy
is
Our paper
atures.
The
an area
is
and the optimal form of
for future research.
a contribution both to the
industrial organization
IPR
and growth
protection and the endogenous growth
literatures
though monopoly power
1981, Tirole, 1988,
liter-
emphasize that ex-post monopoly
rents and thus patents are central to generate the ex-ante investments in
progress, even
A
also creates distortions (e.g.,
R&D and technological
Arrow, 1962, Reinganum,
Romer, 1990, Grossman and Helpman, 1991, Aghion and Howitt, 1992,
Green and Scotchmer, 1995, Scotchmer, 1999,
Gallini
and Scotchmer, 2002, O'Donoghue and
Zweimuller, 2004). 6
Much
of the literature discusses the trade-off between these two forces to determine the
optimal length and breadth of patents. For example, Klemperer (1990) and Gilbert and Shapiro
J
Boldrin and Levine (2001, 2004) or
Quah
(2003) argue that patent systems are not necessary for innovation.
.
in order to provide
(1990)
show that optimal patents should have a long duration
R&D,
but a narrow breadth so as to limit monopoly distortions.
A
number
inducement to
of other papers, for
example, Gallini (1992) and Gallini and Scotchmer (2002), reach opposite conclusions. Another
literature, including the seminal
branch of the
interesting papers
(2001), adopts a
paper by Scotchmer (1999) and the recent
by Llobet, Hopenhayn and Mitchell (2001) and Hopenhayn and Mitchell
mechanism design approach
to the determination of the optimal patent
intellectual property rights protection system.
and
For example, Scotchmer (1999) derives the
patent renewal system as an optimal mechanism in an environment where the cost and value
of different projects are unobserved
and the main problem
is
to decide which projects should
go ahead. To the best of our knowledge, no other paper in the literature has considered state-
dependent patent or IPR
policy,
which
is
the focus of the current paper.
As a
first
attempt,
we
only look at state-dependent patent length and license fees (though similar ideas can be applied
to an investigation of the gains from
Our paper
is
most
making the breadth
closely related to
and extends the
of patent
awards state-dependent)
results of Aghion, Harris
and Vickers
(1997) and Aghion, Harris, Howitt and Vickers (2001) on endogenous growth with step-by step
innovation.
'
Although our model builds on these papers,
of significant ways.
we
First,
specified license fee for building
show that such
is
from them
they both compete for scarce labor.
8
IPR
firms.
We
on the equilibrium growth rate of the economy.
IPR
policy that can be state dependent. Third,
a general equilibrium interaction between production and
general model (with or without
a number
in
on the leading-edge technology developed by other
licensing has significant effects
economy there
also differs
allow licensing agreements whereby followers can pay a pre-
Second, our economy incorporates a general
in our
it
R&D,
since
we provide a number
of analytical results for the
policy), while Aghion, Harris,
Howitt and Vickers (2001)
Finally,
focus on the case where innovations are either "drastic" (so that the leader never undertakes
R&D)
or very small.
They
also
do not prove the existence of a steady state or characterize the
properties of the equilibrium in this class of economies.
Finally, our results are also related to the literature
Fudenberg, Gilbert,
Stiglitz
on tournaments and
example,
and Tirole (1983), Harris and Vickers (1985, 1987), Choi (1991),
Budd, Harris and Vickers (1993), Taylor (1995), Fullerton and McAfee
7
races, for
(1999),
Baye and Hoppe
Segal and Whinston (2005) analyze the impact of anti-trust policy on economic growth in a related model
of step-by-step innovation.
8
This general equilibrium aspect is introduced to be able to close the model economy without unrealistic
assumptions and makes our economy more comparable to other growth models (Aghion, Harris, Howit and
Vickers, 2001, assume a perfectly elastic supply of labor). We show that the presence of general equilibrium
interactions does not significantly complicate the analysis and it is still possible to characterize the steady-state
equilibrium.
and Moscarini and Squintani
(2003).
(2004). This literature considers the impact of
or exogenous prizes on effort in tournaments, races or
state-dependent
IPR
The
In terms of this literature,
policy can be thought of as state-dependent handicapping of different
players (where the state variable
To the best
R&D contests.
endogenous
is
the gap between the two players in a dynamic tournament).
of our knowledge, these types of schemes have not been considered in this literature.
rest of the
paper
organized as follows.
is
Section 2 presents the basic environment.
IPR
Section 3 characterizes the equilibrium under uniform
results to the case in
which IPR policy
is
state dependent. Section 5 quantitatively evaluates
the implications of various different types of
IPR
policy regimes
characterizes the growth-maximizing state-dependent
the Appendix contains the proofs of
Section 4 extends these
policy.
IPR
on economic growth and
policies. Section 6 concludes, while
the results stated in the text.
all
Model
2
We now
The
describe the basic environment.
different policy regimes
is
characterization of the equilibrium under the
presented in the next section.
Preferences and Technology
2.1
Consider the following continuous time economy with a unique
populated by a continuum of
supply
individuals, each with 1 unit of labor
1
inelastically. Preferences at
time
t
are given
good.
final
The economy
is
endowment, which they
by
/oo
E
where Ej denotes expectations
at date
t.
The
exp (-p(s-t)) log C{s)ds,
t
at time
t,
p
>
the discount rate and
is
is
Y (t)
be the total production of the
closed and the final
C (t) = Y (t).
good
is
,
is
good
final
The standard Euler equation from
this equation defines g
the interest rate at date
t.
and the discount rate
at time
used only for consumption
.
^*)3
where
C (t)
is
consumption
logarithmic preferences in (1) facilitate the analysis, since they imply a simple
relationship between the interest rate, growth rate
Let
(1)
(t)
C(t)
cw
as the
=
(1)
Y(t)
yW
t.
(i.e.,
We
there
(see (2) below).
assume that the economy
is
no investment), so that
then implies that
= r (t)_p
'
'
growth rate of consumption and thus output, and r
(2)
(t)
.
The
final
Y
good
produced using a continuum
is
1
of intermediate goods according to the
Cobb-Douglas production function
\nY(t)= f
Jo
where y
the output of jth intermediate at time
assume that there
is
free entry into the final
have the following demand
Each intermediate
infinitely-lived firms.
compete a
firms
at time
j
6
[0, 1]
We
Y (t)
y^ =
p-{jjy
la Bertrand.
where k(j,t)
is
comes
two
in
difference
(j, t)
We
.
These assumptions,
final
good producer
Firm
i
=
1
(4)
different varieties, each
produced by one of two
varieties are perfect substitutes
and these
or 2 in industry j has the following technology
=
qi
{j,t)h(j,t)
(5)
the employment level of the firm and qi(j,t)
The only
t.
imply that each
by p
^e[0,l).
assume that these two
9
(3),
sector.
t
for intermediates
y(j,t)
time
Throughout, we take the price of the
t.
good production
together with the Cobb-Douglas production function
will
(3)
good as the numeraire and denote the price of intermediate j
final
also
(j, t) is
\ny(j,t)dj,
between two firms
is
their technology,
is
its level of
which
will
technology at
be determined
endogenously.
Each consumer
in the
economy holds a balanced
sequently, the objective function of each firm
The production function
producing intermediate j
is
for intermediate
for firm
i
to
portfolio of the shares of
maximize expected
goods,
and industry j
(5),
at time
all firms.
Con-
profits.
implies that the marginal cost of
t is
MGiU,t)-=^r
where
w (t)
When
is
the wage rate in the economy at time
this causes
(6)
t.
no confusion, we denote the technological leader in each industry by
i
and the follower by —i, so that we have:
Qi {j, t)
>
q-i
(j, t)
Alternatively, we can assume that these two varieties are imperfect substitutes,
output of intermediate j is given by
y
with a
>
1.
(j, t)
=
[<pyi ti, t)
"
+
(i
-
<p)
yz
(j, t)
for
example, so that the
•
This generalization has no effect on any of our qualitative results, but increases notation.
therefore prefer to focus on the case where the two varieties are perfect substitutes.
We
Bertrand competition between the two firms implies that
all
intermediates will be supplied by
10
the leader at the "limit" price:
w
&(3,t)=
Equation
(4)
(t)
);.y
n
(7)
then implies the following demand for intermediates:
v(M-.*ffiY®.
R&D
(8)
and IPR Policy
2.2
Technology,
R&D
by the leader or the follower stochastically leads to innovation.
the leader innovates,
The
follower,
its
>
technology improves by a factor A
on the other hand, can undertake
R&D
nology or to improve over the frontier technology. 11
The
We
assume that when
1.
to catch
up with the
first possibility,
frontier tech-
catch- up
R&D,
be thought of as discovering an alternative way of performing the same task as the
We
leading-edge technology (applied to the follower's variant of the product).
because this innovation
R&D
efforts,
it
is
for the follower's variant of the
can
.current
assume that
product and results from
its
own
does not constitute infringement of the patent of the leader, and the follower
does not have to make any payments to the technological leader in the industry. Therefore,
the follower chooses the
first possibility, it will
leader (as applied to
own
patent
fee.
its
The
have to retrace the steps of the technological
variant of the product), but in return,
For follower firm —i in industry j at time
a -i U,
alternative, frontier
leading-edge technology.
R&D,
If this
t)
=
t,
we denote
R&D
it
will
not have to pay a
this type of
R&D
by
0.
involves followers building
type of
if
on and improving the current
succeeds, the follower will have improved the
leading-edge technology using the patented knowledge of the technological leader, and thus
have to pay a pre-specified patent
will
10
If
fee to the leader.
12
We
the leader were to charge a higher price, then the market would be captured by the follower earning
positive profits.
A
lower price can always be increased while making sure that
prefer the intermediate supplied
by the leader
i
always gains by increasing
A
it
third possibility
is
first
making the
for the follower to
is
at
good producers
even
demand curve
if
infinite,
is
still
the latter were
the leader
price given in (7) the unique equilibrium price.
climb the technology ladder step- by-step, meaning that, for example,
some technology rung
discover technology n-i 3
"slow catch-up" in
—i
and the follower itself is at n- j (t) < n l} (t) — 1,
have investigated this type of environment with
a previous version of the paper. Since the general results are similar, we do not discuss this
the current leader
must
its price,
all final
rather than that by the follower
supplied at marginal cost. Since the monopoly price with the unit elastic
when
specify the patent fees below.
(t)
+
1,
n;, (t)
et cetera.
%
We
variation to save space.
12
Clearly, the follower will never license the technology of the leader for production purposes, since this
lead to Bertrand competition and zero ex post profits for both parties.
would
This strategy
denoted by
is
=
a-i(jyt)
Throughout, we allow a_;
G
(j, t)
[0, 1]
for
!•
mathematical convenience, thus a should be
R&D by the follower.
R&D by the follower,
inter-
preted as the probability of frontier
R&D
by the
leader, catch-up
and
frontier
R&D
by the follower
have different costs and success probabilities. Nevertheless, we simplify the analysis by
may
assuming that
all
In particular, in
three types of
cases,
all
R&D
have the same costs and the same probability of success.
we assume
that innovations follow a controlled Poisson process, with
R&D
investments. Each firm (in every industry) has access to
the arrival rate determined by
the following
R&D
technology:
XiO,t) = F{hi(j,t)),
where
x, (j, t)
hired by firm
is
i
the flow rate of innovation at time
in industry j to
work
in the
R&D
(9)
and
t
hi
process at
t.
(j, t)
is
assume that
<
F'(O)
<
(0)
(0,
labor
oo implies that there
w
is
(t),
is
R&D
the cost for
and the assumptions on
satisfies
G'
(•)
>
0,
x*
is
=
(•)
>
for all
no Inada condition when
is
(j, t)
At +
0,
F"
h >
r
hi (j, t)
o (At).
()
h.
—
0.
<
0,
The
The
an upper bound on the flow rate of
not essential but simplifies the proofs). Recalling that the wage rate for
is
therefore
G(i
G"
(•)
F
>
(j\t))
i
w (t) G (xj
0,
G"
(0)
>
(j,i))
where
= F-\( ? .0",*)),
immediately imply that
G
is
and lim x _ s G'
x
is
oo) such that F' (h)
assumption, on the other hand, ensures that there
innovation (which
is
twice continuously differentiable and satisfies F'
is
oo and that there exists h G
assumption that F'
last
F
of workers
This specification implies that
within a time interval of At, the probability of innovation for this firm
We
number
the
(10)
twice continuously differentiable and
(x)
=
= F(h)
oo,
where
(11)
the maximal flow rate of innovation (with h defined above).
We
next describe the evolution of technologies within each industry. Suppose that leader
in industry j at
time
t
i
has a technology level of
qi (j,t)
and that the follower — i's technology
at time
q- i (j,t)
=
t
=
^,
(12)
n
\ -<^,
(13)
X
n
is
where mj
>
(t)
n-ij
and
(t)
the follower in industry
If
j.
n-ij
riij (i),
(t)
We refer to rij
j.
£ Z+ denote the technology rungs
= riij
(t)
—n^ij
(t)
(t)
of the leader
as the technology gap in industry
the leader undertakes an innovation within a time interval of At, then
increases to
qi (j, t
+
At)
A nyt+1 and technology gap
—
more innovations within the
probability of two or
rises to
interval
and
+
rij (t
At
=
At)
its
technology
rij (t)
+
1
(the
be o(At), where o(At)
will
represents terms that satisfy lim& t ^o o (At) /At).
When
At, then
the follower
it
catches
successful in catch-up
is
up with the
and
leader
R&D
R&D
technology, so
and pays the
license fee
(i.e.,
a_j
—
X ^,
In contrast,
0.
(j, t)
=
1),
then
if
At)
=
rijt+At
=
(j,t
and the technology gap variable becomes
is
successful
surpasses the leading-edge
it
+
X
1
n
(
^ +1
an d from
this point
and —i are swapped, since the previous follower now becomes the
In addition to catching
R&D,
the follower
we have
q. l
i
within the interval
0)
n
+ At) =
and the technology gap variable becomes njt +At
in frontier
=
a_, (j,t)
technology improves to
its
q^(j,t
(i.e.,
followers
onwards, the labels
leader).
up with or surpassing the technology
frontier with their
can also copy the technology frontier because IPR policy
is
own
such that some
patents expire. In particular, we assume that patents expire at some policy-determined Poisson
rate
q- l
77,
(j,t
and
+
after expiration followers
At)
n
=
\ »K
This description makes
the patent;
it
license fees.
(ii)
can costlessly copy the frontier technology, jumping to
13
clear that there are
In our
two aspects to IPR
7]
for all
t
>
r\
(n)
=
r\
n
<
00
technological leader that
is full
13
the length of
two functions:
:
N
->
R+
0,
C(t)
Here
(i)
most general policy regime, we allow both of these to be
state dependent, so they are represented by the following
and
policy:
is
is
:
N
-»
R+ U {+00}
.
the flow rate at which the patent protection
n
steps ahead of the follower.
When
7?
n
=
is
0, this
removed from a
implies that there
protection at technology gap n, in the sense that patent protection will never be removed.
Alternative modeling assumptions on
IPR
policy,
such as a fixed patent length of
T>
innovation, are not tractable, since they introduce a form of delayed-differential equations.
10
from the time of
In contrast,
gap n
r\
—»
n
oo implies that patent protection
reached. Similarly, £ (n,
is
=
t)
Cn
{t)
removed immediately once technology
is
denotes the patent fee that a follower has to pay in
order to build upon the innovation of the technological leader,
industry
£
(£)
is
= {C
n
steps.
x (t)
,
(2
14
(*)
i
Our formulation imposes that
••}
license fees should not
is
This
a function of time.
remain constant. Below, we
when the technology gap
={77^ 772,'...}
77
in the
time-invariant, while
is
natural, since in a growing economy,
is
£ grows at the same rate
will require that
as aggregate output in the economy.
When
£n
(t)
—
0,
there
technology at zero cost.
is
prohibitively costly.
protection
is
imperfect.
15
is
no protection because followers can license the leading-edge
In contrast,
when
Note however that
(n
(t)
=
n
<
00 does not necessarily imply that patent
C,
what
In particular, in
follows
is
and-neck
being at a technology gap of 0) as
regime as uniform
IPR
intellectual property
protection
law treats
all
if
both
firms
also
n >
n.
assume that there
Given
this specification,
exists
77
"full
and £
will interpret
(i.e., rj
uni
protection".
(t)
and industries
gap between the leader and the follower
We
we
a situation in which
equal to the net extra gain from surpassing the leader rather than being neck-
the license fee
(i.e.,
00, licensing the leading-edge technology
16
We
also refer to a policy
are constant functions,
meaning that
identically irrespective of the technology
= {77,
77, ...}
some n < 00 such that
we can now write the law
r)
of
n
and C um
=
"q
n
(t)
and ( n
=
{(
(t)
(t)
=
,
(n
£
(t)
(t)
,
...}).
for all
motion of the technology gap
in
14
Throughout, we assume that £ is a policy choice and firms cannot contract around it. An alternative
approach would be to allow firms to bargain over the level of license fees. In this case, it is plausible to presume
that the legally-specified infringement penalties or license fees will affect the equilibrium in the bargaining game,
so the effect of policies we investigate would still be present. We do not allow bargaining between firms over
the license fees in order to simplify the analysis.
11
Throughout, we interpret f„
(t)
=
as £n
(t)
'= e with e
J.
0,
so that followers continue not to license the
new technology without innovation (recall the comment in footnote 12).
16
In other words, we interpret "full protection" to correspond to a situation
in which £n (t) = V\ (t) — Vo (t),
one step ahead of its rival and Vo is the value of a firm
that is neck-and-neck with its rival (naturally, £ n (t) > Vi (t) — Vo (t) would also correspond to full protection,
since in this case no follower would choose to license). Alternatively, full protection could be interpreted as
corresponding to the case in which the follower pays a license fee equal to the loss of profits that it causes for
the technology leader (see Scotchmer, 2005). In our model, this would correspond to £n (t) = Vo (t) — V-i (t),
where V-i is the net present value of a firm that is one step behind the technology leader. This expression
applies, since without licensing the follower would have made its own innovation, in which case the leader would
have fallen to the state of a neck-and-neck firm rather than being surpassed by one step. In all equilibria we
compute below, we find that the second amount is significantly less than the first, thus our interpretation of full
protection licensing fee is large enough to cover both possibilities. In any case, what value of £ is designated
as "full protection" does not have any bearing on our formal analysis, since we characterize the equilibrium for
any £ and then find the growth-maximizing policy sequence.
where Vi
refers to the net present value of
a firm that
is
11
industry j as follows:
nj
n-j (t
+ At)
=
(t)
+
<
nj
(t)
with probability
((1
with probability
i
-<*-*
_ fx
+ a-i
At
Xi (j, t)
with probability
1
(jY*))
^
.
t)
x_,
(j, t)
x -i -Oi*)
+ x _.
( j,
:
^
t)
t)At
+ Vn
+
j (i))
^
+ o (At)
At+ °(At)
Ai _
N
(
Ai ))
(14)
Here o (At) again represents second-order terms,
in particular, the probabilities of
one innovations within an interval of length At. The terms
rates of innovation
is
is
by the leader and the
follower; a_i
up with a leader or surpass
trying to catch
and
rj
allowed to copy the technology of a leader that
a-i (j,t)
—
leader
is
successful (flow rate Xj (j,t)) or
2.3
Profits
We
of
it;
(j, t)
n
./
is
t
\
€
is
Xj (j,t)
[0, 1]
if
the follower
rij (t)
is
expenditures and license
fees). Profits of
leader
w(t)
i
steps ahead.
rij (t)
to
Intuitively,
rij (t)
+
rij (t)
= n^
(£)— n_jj
(t) is
the
1 if either
(i.e.,
the profits exclusive
in industry j at
time
t
are
Y(t)
qi(j,t)J Piti't)
l-A-M^F^)
where
when
successful (flow rate x_j (j,t)).
w(t) \
q-i(j,t)
the flow
denotes whether the follower
next write the instantaneous "operating" profits for the leader
R&D
(j,t) are
the flow rate at which the follower
the technology gap in industry j increases from
1,
and X-i
more than
(
the technology gap in industry j at time
t.
The first
line
15 )
simply
uses the definition of operating profits as price minus marginal cost times quantity sold.
The
= w (t) /q-i
(j, t)
second
line uses the fact that the equilibrium limit price of firm
as given
and
is
(13).
by
(7),
The
and the
is
p
final equality uses the definitions of qi (j,t)
7;
(j, t)
and
q-i (J,t)
from (12)
expression in (15) also implies that there will be zero profits in an industry that
neck-and-neck,
profits, since
i
i.e.,
in industries
with
rij (t)
=
0.
Also
clearly, followers
always make zero
they have no sales.
The Cobb-Douglas aggregate production function
profits (15), since
it
implies that profits only
and aggregate output. This
in (3)
is
responsible for the form of the
depend on the technology gap of the industry
will simplify the analysis
below by making the technology gap
each industry the only industry-specific payoff-relevant state variable.
12
in
.
The
objective function of each firm
profits (operating profits
this,
each firm
output
(t)]
to
maximize the net present discounted value of net
expenditures and plus or minus patent
take the sequence of interest rates,
will
[Y
levels,
R&D
minus
is
sequence of wages, [w
t>0 the
(t)]
,
and
policies as given.
2.4
Equilibrium
t
>
[r(i)]
,
t>0
,
R&D
the
fees).
In doing
the sequence of aggregate
decisions of
all
other firms
= {/i n (£)}^Lo denote the distribution of industries over different technology gaps, with
(£) denotes the fraction of industries in which the firms are
S^Lo^n (*) = 1- For example,
Let
fi (t)
/j,
neck-and-neck at time
t.
Throughout, we focus on Markov Perfect Equilibria (MPE), where
strategies are only functions of the payoff-relevant state variables.
the dependence on industry
leader that
n
is
us denote the
by fn(*)
=
(
thus
j,
refer to
R&D
a_„ and x_„
steps ahead and by
list
we
of decisions by the leader
x n{t),Pi{j,t),yi{j,t)) and £_„
for
17
decisions by
This allows us to drop
xn
a follower that
is
for the technological
n steps behind. Let
and follower with technology gap n
(t)
=
{a- n
,X- n {t))P
(t)
indicate the whole sequence of decisions at every state, £
(£)
=
at time
Throughout, £
{£„ {t)}'^= _ 00
We
t
will
define an
allocation as follows:
Definition 1 (Allocation) Let
cation
is
(rj,
[C(*)]t>o)
^e
a sequence of decisions for a leader that
a sequence of
R&D
decisions for a follower that
sequence of wage rates [w
(t)]
t>0
,
is
^ e ^^
is
n =
n
—
P°^ C V sequences.
0,1,.. .,oo step ahead,
l,...,oo step behind,
and a sequence of industry
Then an
allo-
[£ n (t)]
t>0
[£-7iW]t>o
j
,
a
distributions over technology gaps
[MOW
For given
IPR
sequences
r\
payoff- relevant state variables,
and
[£ (i).]t>o>
^PE strategies,
which are only a function of the
can be represented as follows
x
a
:
:
Zx
R2_-> R+,
Z_\
{0} x R2.-»
[0, 1]
17
MPE is a natural equilibrium concept in this context, since it does not allow for implicit collusive agreements
between the follower and the leader. While such collusive agreements may be likely when there are only two
firms in the industry, in most industries there are many more firms and also many potential entrants, making
collusion more difficult. Throughout, we assume that there are only two firms to keep the model tractable.
1
The price and output decisions, Pi{j,t) and yi{j,t), depend not only on the technology gap, aggregate
output and the wage rate, but also on the exact technology rung of the leader, riij (t). With a slight abuse of
notation, throughout we suppress this dependence, since their product p; (j,t)yi {j,t) and the resulting profits
for the firm, (15), are independent of n^ (t), and only the technology gap, n 3 (£), matters for profits, R&D,
aggregate output and economic growth.
13
R&D
The
first
it is
a leader in an industry) with an
function represents the
when
decision of a firm (both
n
it is
a follower and when
step technology gap (n e Z), and the aggregate level of
output and the wage ((Y,w) £ K+). The second function represents the decision of a follower
a function of the
(as
R&D
output and the wage in the economy) of whether to direct
level of
its
to reaching or surpassing the leading-edge technology (or
more
precisely,
the probability with which the follower will choose to undertake
R&D
to surpass the leading
edge technology)
.
it
represents
Consequently, we have the following definition of equilibrium (below
we
will
define a steady-state equilibrium in terms of a solution to maximization problems):
Definition 2 (Equilibrium) Given an IPR policy sequence
by a sequence
fect Equilibrium, is given
and
[y* (j,t)]
[£*
t>0 implied by
taking aggregate output [Y*
the
R&D policies
given by (3); and
(t)]
,x*
wages [w*
,
of other firms [a*
(iv) the labor
[a* (t)
i.e.,
t>0
w*
,
(i)
,x*
(t)]
(t)]
(t)]
t>0
market clears
,
{t)]
t>0
such that
R&D policies
(i)
Markov Per\pl {j, t)] t>0
[a* (t)
x*
,
{t)]
t>0
maximizes the expected
profits of firms
government policy
[C(i)] t >
(77,
as given; (Hi) aggregate output [Y*
at all times given the
Since only the technological leader produces, labor
(t)
=
n can be expressed
)
(i)]
wage sequence [w*
an d
t>0
(t)]
is
t>0
.
demand
X
=
n
(t)
^
neZ +
for
w(t)
In addition, there
industries.
R&D
Using
is
(9)
demand
for labor
and the
coming
for
definition of the
K
G
R&D
(16)
from both followers and leaders
function,
Note that
refers to the
(t))
+ G(x- n (t))
if
neN
if
n=
its
X- n
(t))
of a follower in an industry with a technology gap of n.
(t)
refers to the
optimal choice of a_ n
(t)
R&D
6
effort of
[0, 1]).
The
for
R&D
coming from the two
a follower that
is
n steps
Moreover, this expression takes
into account that in an industry with neck-and-neck competition,
be twice the demand
(17)
,
2G{x
demand
in this expression,
behind (conditional on
G(x n
in all
we can express industry demands
(t)={
{
<
with technology gap
.
labor as
f
where n
in industry j
as
In (t)
for
Y*
(8); (ii)
t>0
t>0
,
[C(0]t>o)> a
The Labor Market
2.5
nj
(t)
and
t>0 satisfy (7)
{t)]
are best responses to themselves,
[£* (t)
(77,
i.e.,
with n
=
0,
there will
firms.
labor market clearing condition can then be expressed
as:
1
!>£>„(*) u (t) X- + G(x n (t)) + G(x- n (t))
n=0
14
(18)
and
is
u> (t)
>
0,
with complementary slackness, where
the labor share at time
supply
is
equal to
1,
then the wage rate,
this will never
The
t.
and the demand cannot exceed
w (£),
be the case
this
and thus the labor share, w
in equilibrium).
production (the terms with
for
labor market clearing condition, (18), uses the fact that total
neck-and-neck industries (2G (xq
from leaders and followers
in the
co
demand
If
falls
short of
1,
have to be equal to zero (though
(t),
The right-hand
side of (18) consists of the
demand
R&D workers from the
for R&D workers coming
denominator), the demand for
when n =
(t))
amount.
in other industries
and the demand
0)
(G
(x n
+ G (x- n
(£))
(£))
when n >
0).
Defining the index of aggregate quality in this economy by the aggregate of the qualities
in the different industries,
i.e.,
lnQ(t)= [
\nq(j,t)dj,
(20)
Jo
19
the equilibrium wage can be written as:
w{t)
^\
t
(21)
Steady State and the Value Functions
2.6
Let us
now
focus on steady-state (Markov Perfect) equilibria, where the distribution of in-
dustries n(t)
=
{fi n
(£)}^L
*s
stationary,
economy, are constant over time.
characterize a
by
= Q{t)\-^=° n
definition,
rate.
number
we have
of
its
will establish the existence of
properties. If the
Y (t) —
The two equations
We
Y^e 9
*
1
defined in (19) and g, the growth rate of the
oj (t)
economy
in steady state at
w (t) = woe 9 where
u (t) — oj* for all t > 0.
and
also imply that
is
such an equilibrium and
',
the parameters are such that the steady-state growth rate g*
is
g*
is
time
t
=
0,
then
the steady-state growth
Throughout, we assume that
positive but not large
enough
to violate the transversality conditions. This implies that net present values of each firm at
all
point in time will be finite and enable us to write the maximization problem of a leader that
is
n >
steps ahead recursively.
First note that given
leader that
is
n
an optimal policy x
steps ahead at time
exp
199 T
Note that
(16).
Thus we
t
for
a firm, the net present discounted value of a
can be written
(-r- (s
-
t)) [II (s)
as:
+Z(s)-w (s) G (x (a))] ds
Y (t) = JQ \nq(j,t) {j,t)dj = L lniji (j,t) + In ^JA nj
have In Y (t) = J [In qt (j, t) + In Y (t) — In w (t) — rij In A] dj.
In
and writing exp / n 3
I
;
In Xdj
=
A
_ ^"=° nM " (t)
we obtain (21).
,
15
dj,
where the second equality uses
Rearranging and canceling terms,
where
the operating profit at time
IT (s) is
paid) by a firm which
s
>
the leader and
is
All variables are stochastic
t.
s
>
t,
Z (s)
the patent license fees received (or
is
w(s)G(x(s)) denotes the
R&D
expenditure at time
and depend on the evolution of the technology gap within
the industry.
Next taking the equilibrium
wage
rate,
w*
(t),
R&D policy of other firms,
as given, this value can be written as:
x*_ n (t)
and
and equilibrium
a*_ n (t),
20
Vn (t) = max {[nn (i) - w*(t)G(x n (t))] At +o (At)
(22)
x n (t)
{x n
+
-exp(-r(t
+
(r)
part of this expression
interval of length At.
Vn +i
(t)
and Vo
(t)
ahead and a firm
(l
-
Vn+1
At
a*_ n (t)) x*_
n (t)
(t
+
At)
+ o (At)) V
The second
is
(l
{a*_ n (t) x*_ n (t)
- xn
(t)
At -
rj
n At
the flow profits minus
part
is
At +
-
o (At)) (T/_a
+
At)
+ At) + CJ
At-o (At)) Vn (t + At)
x*_ n (t)
R&D
(t
expenditures during a time
the continuation value after this interval has elapsed.
are defined as net present discounted values for a leader that
an industry that
in
{t
>
+
first
+
o {At))
At) At)
+
The
n At
At +
{t)
is
neck-and-neck
(i.e.,
n
=
is
The second
0).
n
+
1
steps
part of the
expression uses the fact that in a short time interval At, the probability of innovation by the
leader
is
first line
xn
(t)
At
+ o (At),
where o (At) again denotes second-order terms. This explains the
of the continuation value. For the remainder of the continuation value, note that the
probability that the follower will catch up with the leader
in particular,
undertaking
if a*_ n (t)
R&D
1,
this eventually will never
(l
a*_ n (t)
=
1.
in the third line, conditional
—
a*_ n (t)) x*_ n (t)
fee (n
.
There are two differences between the second and third
on success by the
follower, a leader
Finally, the last line applies
when no
R&D
20
Vn (t)
from both
sides, divide
Clearly, this value function could be written for
set the
R&D
main body
line,
lines;
moves to the position
(ii)
patents continue to be enforced, so that the technology gap remains at
Next, subtract
+ o (At);
This explains the third
of a follower rather than a neck-and-neck firm (V-\ instead of Vq);
dependent patent
At
happen, since the follower would be
not to catch up but to surpass the leader.
which applies when
(i)
=
is
it
receives the state-
effort is successful
n
and
steps.
everything by At, and take the limit as
At
—
»
0.
any arbitrary sequence of R&D policies of other firms. We
and ol„ (t), to reduce notation in the
policies of other firms to their equilibrium values, x*_ n (t)
of the paper.
16
.
This yields:
r(t)Vn (t)-Vn
=
(t)
maxIp^-^WG^^+XnWlK+iW-KW]
(23)
x n (t)
where V^
(i)
.
((1
+
(a*_„ (t) x*_ n (t)
a*_ n (i)) x*_ n (t)
denotes the derivative of
present value of a firm that
neZ +
-
+
is
n
Vn (t)
what
n ) [V
+
[C„
- Vn
(t)
T/ (*)
-
V-i
will also
grow
[*)]
(t)]},
In steady state, the net
at a constant rate g* for all
Let us then define the normalized values as
n €
=
YV
(24)
Z, which will be independent of time in steady state,
follows
we assume that
will ensure the existence of
(t)
—
vn
.
Similarly, in
Cn(*)
=
tn
vn
i.e.,
up by GDP, so that
license fees are also scaled
f.
which
r,
with respect to time.
Vn{t)
for all
+ Vn )
Vn (i),
steps ahead,
+
Y
^
-
a (stationary) steady-state equilibrium (the use of £ n
(t)
for
the original patent fees and C n f° r the scaled-up fees should not cause any confusion).
Using (24) and the fact that from
value function (23) can be written
pvn
= max{ (l +
where
x*_ n is
[(1
-
(2), r (t)
+
rj
the equilibrium value of
steady-state labor share (while x n
is
n]
+
(x n )
[up
R&D
now
xn
+ p,
= max {-oj*G (x
\v n+1
- v^ +
the recursive form of the steady-state
-
vn
]
a*_ n x*_ n [y-i
explicitly chosen to
)
(25)
,
by a follower that
Similarly the value for neck-and-neck firms
pvQ
(t)
g
as:
A" n ) - u*G
a*_ n ) x*_ n
—
is
-v n + C n ]}
n
for
steps behind,
neN,
and
ui* is
the
maximize vn ).
is
+x
[v\
- v + x*
[v-\
]
-
v
XQ
]}
(26)
,
while the values for followers are given by
pv- n
=
max {-ui*G (x- n +
)
C — n )0> — n
+a_„x_ n
which takes into account that
it
will
become the new
if
[vi
-
V- n
-
[(1
Cn
]
- a_ n x- n + rjn
+ xn
)
[V- n -l
the follower decides to build
leader, but will
-
[v
- v- n
U~n]} for
(27)
]
71 <E
N,
upon the leading-edge technology
have to pay the patent
17
]
fee Q n
.
,
For neck-and-neck firms and followers, there are no instantaneous
in (26)
and
and
(27). In the
in the latter case,
former case this
because neck-and-neck firms
is
because followers have no
emphasize that, because of growth, the
profits,
sales.
which
is
reflected
marginal cost,
sell at
These normalized value functions
effective discount rate
is
r(t)
— g (t) =
p rather than
r(t).
The maximization problems
R&D
rium
policies, (o*,x*),
immediately imply that any steady-state equilib-
in (25)- (26)
must
satisfy:
=
€
1
if
vi
[0, 1]
if
vi
=
if Vl
-
(n
>
Cn
=
vo
Cn
<
Vq
vq
(28)
and
<
= m ax {G'-^
[Vn+1
Vn]
u:
1
-
a*_ n
),o\
[vq
)
-
(29)
v- n
]
+ a*_ n
[vi
- v^ n -
(„]
,0
ur
where
G~
(•) is
differentiable
[Vl ~
l±
= max^g- 1
x*
Vq]
UJ
,o|,
,0
\
G'~ l
strictly concave,
incremental value of moving to the next step
by the normalized wage
rate, u*, is less.
be equal to
is
which
The response
economic force in
leaders that are
n +
1
1
here relaxing
IPR
greater
max
form of lower
and when the
>
0,
whenever the
R&D,
these
as
R&D
measured
levels
can
operator.
—
vn
,
is
the key
For example, a policy that reduces the patent protection of
r/
effect of relaxing
IPR
policy.
n+1 or reducing Cn+i)
—
v n and x*n
Q
may encourage
directly contributing to aggregate growth.
x* +1
This corresponds to the
effect.
This positive incentive
.
further frontier
R&D
by the
followers,
Second and perhaps somewhat more paradoxically,
lower protection for technological leaders that are
— v n+ \ and
being
equation (30) shows, weaker patent protection in the
First, as
license fees (lower
.
wm make
In contrast to existing models, however,
policy can also create a positive incentive
thus increasing v n+ 2
cost of
to the increments in Values, v n+ \
steps ahead (by increasing
has two components.
G is twice continuously
rates, the rc*'s, will increase
steps ahead less profitable, thus reduce v n+ \
standard disincentive
effect
,
and since
also that since G' (0)
taken care of by the
this model.
n +
is
Note
of innovation rates, x*
G function,
continuously differentiable and strictly increasing.
is
These equations therefore imply that innovation
zero,
(31)
,
the inverse of the derivative of the
and
(30)
We
n+1
steps ahead will tend to reduce v n+ \
will see that this latter effect plays
18
an important
role in the
may
IPR
form of optimal state-dependent
also create a beneficial composition effect; this
decreasing sequence, which implies that x*_j
4 and 5 below).
Weaker patent protection
more
industries into the neck-and-neck state
R&D
in the economy.
21
The optimal
determined by the interplay of these three
Given the equilibrium
across states
/j.*
R&D
ir*
for
n >
1
—
protection
v n }^Lo
*s
a
(see Propositions
the form of shorter patent lengths) will shift
and potentially increase the equilibrium
and structure
level
IPR
because, typically, {v n+ i
is
higher than
is
(in
Finally, relaxing
policy.
of
IPR
policy in this
economy
level of
be
will
forces.
decisions (a*, x*), the steady-state distribution of industries
has to satisfy the following accounting identities:
«+l + Z-n-l + Vn+l) Mn+l = <Mn
for
n e
N
(
'
32 )
oo
+ llj + T]^
{x\
= 2x^0 + Yl a -n x -nVn,
Ml
(33)
.
n=l
oo
2sSM0
=
E
( i
1
- a ~n) X ~n +
In)
<
(34)
n.=l
The
first
expression equates exit from state
more step ahead or the
follower catching
n + 1 (which
up
surpassing the leader) to entry into the state
for
n making one more
(which takes the form of a leader from the state
equation, (33), performs the
this state
same accounting
comes from innovation by
1,
innovation)
.
The second
taking into account that entry into
two firms that are competing neck-and-neck.
either of the
innovation by a follower in any industry with
The labor market
for state
with entry into this state, which comes from
from state
Finally, equation (34) equates exit
takes the form of the leader going one
n>
1.
clearing condition in steady state can then be written as
J_ + G{x
1>EX
A
n=0
k
T
*
n)
+G
(
x *_ n )
and u* >
(35)
0,
with complementary slackness.
The next
proposition characterizes the steady-state growth rate.
results in the paper, the proof of this proposition
Proposition
(/x*,
1
is
As with
all
the other
provided in the Appendix.
Let the steady-state distribution of industries and
R&D
decisions be given by
a*, x*), then the steady-state growth rate is
g*
=
\n\
2 V0 X
+ Yl &
(
X«
+ a -n X -n)
(36)
71=1
21
Weaker patent protection also creates a beneficial "level effect" by affecting equilibrium prices (as shown in
equation (7) above) and by reallocating some of the workers engaged in "duplicative" R&D to production.
19
This proposition
by three
that the steady-state growth rate of the economy
is
determined
factors:
1.
R&D
2.
The
3.
Whether
decisions of industries at different levels of technology gap, x*
distribution of industries across different technology gaps,
followers are undertaking
=
frontier, a*
IPR
clarifies
R&D
to catch
up with the
fj,*
= {^^j^L.^.
=
{f^^^Lo-
frontier or to surpass the
{a*}"!.^.
policy affects these three margins in different directions as illustrated by the discussion
above.
We now
define a steady-state equilibrium in a
more convenient form, which
and derive some of the properties of the equilibrium.
establish existence
Definition 3 (Steady-State Equilibrium) Given an IPR policy
librium
is
a tuple
(32), (33)
and
be used to
will
v,
(fx*,
£), a steady-state equi-
(77,
a*, x*, u*, g*) such that the distribution of industries ll* satisfy
(34), the values
v=
{v n }n=-oo satisfv ffi), (%6) and (2 V> th ^
R& D
decisions
a* and x* are given by (28), (29), (30) and (31), the steady-state labor share u* satisfies (35)
and the steady-state growth
We
is
given by (36).
next provide a characterization of the steady-state equilibrium, starting
case in which there
is
uniform
IPR
first
with the
policy.
Uniform IPR Policy
3
Let us
r)
rate g*
<
first
IPR policy, whereby
um
and £ um In this
which we denote by r]
assume that the only available policy option
=
00 and Cn
C
(27) implies that the
<
n E N,
°° f°r an
problem
is
is
uniform
.
identical for
all
v- n
followers, so that
=
nn
=
case,
V-\ for n G N.
Consequently, (27) can be replaced with the following simpler equation:
pv-i
= max {-u*G{x-i) +
X— ,a_
1
[(l-a-i)x-i+ii\[vo-v-i] + a-iX-i[vi-v-i-£]},
(37)
1
implying optimal
R&D
decisions for
xU = max JG'-
all
followers of the form
m&X ([t
=
'°
1
(
V ~ l]
J
Vl
-
V~ l
~
Let us denote the sequence of value functions under uniform
C])
)
IPR
,
0]
(38)
.
as {v n }'^L_ 1
.
with a number of results that characterize the form of a steady-state equilibrium in
omy.
20
We
start
this econ-
Proposition 2 Consider
a uniform
v, a!_ l5 x*, u>*, g*)
<(^z*,
bounded and
The next
is
n>
(fi*-,
(r)
un1
uni
£
,
and suppose
^
and {vn }^=0 forms a
vq
shows that we can
a
finite
^
and a corresponding steady-state
restrict attention to
policy
(r)
un%
c\
,
um
N
Then, there exists n* £
v, a!_ 1; x*, u*, g*\.
sequence of values:
such that x*
=
for
all
n*.
The next
= {x^}^=1
proposition shows that x*
a decreasing sequence, which implies that
is
technological leaders that are further ahead undertake less
further
R&D
Proposition 4 Consider a uniform IPR
librium with uniform IPR, x^ +1
-"A
R&D
and
final
R&D.
markup
comparison
<
R&D
um and um
£
rj
N
x*n for alln £
between the
is
policy
(recall (15)).
Then
.
in
and moreover, £* +1
any steady-state
<
£*
if
x*n
>
The next proposition shows that
neck-and-neck industries are "most
R&D
Proposition 5 Consider a uniform IPR
Moreover,
The
if
G'~ l ((l
- A" 1 )
condition G'~ l ((l
/ {p
Xq
/ (p
in the steady-state equilibrium (see
+
>
x\ and x^
policy
r)
and C
>
rj))
Remark
um and um
£
>
A
steady state with with
there are no profits, thus
it
ui*
<
would then imply that there
is
1, x*_j
below and
1
1 is
Then Xq
.
>
x*_
>
x\ and Xq
>
Lemma
in the
1
(//*,
(36) together
no economic growth,
=
1 (so
with
i.e.,
jig
v, a*_ 1} x*,
g*
=
R&D
u>*
,
g*) with
when
uj*
=
1,
firms are neck-and-neck
that
all
=
and x*n
1
.
Appendix).
economically more interesting, since
(j,q
x*_ 1
v
here ensures that there will be positive
must be the case that
and there are no markups). Equation
=
x\, so that without licensing,
then x*
0,
Next, we prove the existence of a steady-state equilibrium
1.
0.
intensive".
+ r])) >
— A -1 )
>
equi-
neck and-and-neck firms, Xq, and the
levels of
investments of followers and leaders in industries with a technology gap of n
x\.
<
Intuitively, the benefits of
are decreasing in the technology gap, since greater values of the technology gap
translate into smaller increases in the equilibrium
ui*
<
Then, v-\
that
sequence converging to some positive value v^.
Proposition 3 Consider a uniform IPR
equilibrium
policy
a steady-state equilibrium.
strictly increasing
result
IPR
—
for all
n £ Z+
0.
Establishing the existence of a steady-state equilibrium in this economy
is
made compli-
cated by the fact that the equilibrium allocation cannot be represented as a solution to a
maximization problem. Instead, as emphasized by Definition
taking the
R&D
3,
each firm maximizes
its
value
decisions of other firms as given; thus an equilibrium corresponds to a set of
21
R&D
and a labor share (wage
decisions that are best responses to themselves
clears the labor market.
We
rate) lu* that
establish the existence of a steady-state equilibrium in a
number
of steps.
First, note that
each x n belongs to a compact interval
Now
of innovation defined in (11) above.
(Markovian) steady-state strategies for
all
that given
while
its
some z
optimal
we denote
=
{Co,
R&D
R&D
result characterizes this
[z],
xn
(z)
Xn
e
letters
and
refer to their
policy
(r)
um
£
,
policies of all other firms are given by z
R&D
uniquely determined,
policies
convex-valued for each z G
A-\
[z]
C
what
follows,
elements by lowercase
[z].
um
=
^ and
suppose that the labor share
Then
(<D,a_i,x).
mization problem of an individual firm leads to a unique value function
and optimal
is
problem and shows
choices are given by a convex-valued correspondence. In
Proposition 6 Consider a uniform IPR
and the
the maximal flow rate
and a sequence (a_i,x) of
a_i,x), the value function of an individual firm
a_i (z) G A-\
letters, e.g.,
[0, 1]
is
other firms in the economy, and consider the dynamic
and correspondences by uppercase
sets
where x
u G
a labor share
fix
Our next
optimization problem of a single firm.
[0, x],
and
[0,1]
X
[z]
:
{
Z and upper hemi- continuous
— 1} U Z+
v
=1
in z (where
[z]
[0,
v
the
[z]
:
x]
=
dynamic
opti-
{-1}UZ + — R +
»
are compact
and
{v n [z]}^__ 1 and
X [] = {*«[«]}"_!;}.
Now
z
6, this
xn
[z]
E
let
is
us start with an arbitrary z
mapped
Xn
[z].
into optimal
From
R&D
=
{Co,
R&D decision sets
we
policies (d_i,x),
equations (32), (33) and (34).
Then we can
Z = [0,
a_i, x) 6
A-i
calculate
/x
X
[0,
x]°°.
From Proposition
[z],
where a_i 6 A-\
[a_i,x]
= {p, n [a-i,x],}°L
and
[z]
x
1]
[z]
and
using
rewrite the labor market clearing condition (35)
as
,,im
OjAl
yT
+ G (x n u + G {x- n
)
)
(39)
where due to uniform IPR,
<!>
[z]
=
((p
(z)
,
A^i
[z]
,
X
x*_ n
=
[z]),
which maps Z into
x*_ 1 for all
n >
$:
That $ maps Z
into itself follows since z G
Z
Z
0.
Next, define the mapping (correspondence)
itself,
that
is,
=J Z.
(40)
consists of a_i
and the image of z under $ consists of a_i G
[0, 1]
and x G
clearly in [0,1] (since the right-hand side is nonnegative
22
G
[0, 1],
x G
[0,x]°°,
[0,
x]°°
and w G
and moreover,
and bounded above by
1).
[0, 1],
(39)
is
Finally,
from Proposition
A-\
6,
[z]
upper hemi-continuous in
and
z,
Proposition 7 Consider
G"
_1
)
IPR
Then a
0.
>
>
then g*
0,
Remark
policy
a trivial equilibrium in which x*
which imply that Xq
= T,
jjLq
—
_1
((l
for all
>
x*n for
all
n^O).
of output equal to
—
x*_ x
x*_i
<
-1
A
/ (p
)
n G Z+,
>
>
n > n*
out
/j,q
—
=
>
77
is
oj*
is
=
—
A
if
-1
3,
(since there is
either n
in the
that G'"
((l
no innovation
-A
)
in industries
all
>
C)
/ (p
)
A
0.
compete a
+ 77)) >
followers,
industries
22
To
R&D
/j,*
why
see
implies v-i
=
or
is
no
5,
0,
Bertrand
la
and a labor share
on the other hand,
either
i.e.,
>
77
or
sufficient condition to ensure that
(32)
is
that
pn
=
for all
with technology gap greater than n*). Thus
jump
a*_ x
=
1), this
Markov
chain.
Moreover,
to the neck-and-neck state
Markov chain
is
(when
irreducible (and
This
is
stated
next proposition.
policy
(r)
uni
uni
,
£
^
and a steady-state equilibrium
Then, there exists a unique steady-state distribution of
decisions (a*_ ly x*).
.
this condition
and
is
sufficient
(25) implies vi
>
(l
l
1
to positive growth
is
suppose that 7? =
and also that zLj =0. Then (37) immediately
-1
A ) /p. Moreover, from (38) and the fact that u>* < 1, we have
—
-1
> G'- (ui - v-i - C) > G'~ ((1 - A
hypothesis that ilj = 0, and implies xlj >
x-i
>
22
combined with
industries
technology gap of one (when
Proposition 8 Consider a uniform IPR
sequence of
g*)
relaxed, then there exists
aperiodic), thus converges to a unique steady-state distribution of industries.
and proved
to*,
then this equilibrium would also
0,
-1
also strictly positive.
immediate consequence of Proposition
0) or to the
al l5 x*,
that
addition the steady-state equilibrium
If in
1.
-
is
1
because after an innovation by a follower,
=
suppose
and from Propositions 4 and
1
the law of motion of an industry can be represented by a finite
a!_!
establish:
an equilibrium in which there
i.e.,
if
((l
/p
77
and
^
cost, leading to zero aggregate profits
and thus
1
then the growth rate
when
An
+ 7?)) >
some probability of catch- up or innovation by the
0,
UTU
In addition,
1.
Lemma
Moreover,
The assumption that G'~ l
1.
sufficient to rule
involves
£
,
so that in every industry two firms with equal costs
and charge price equal to marginal
is
un%
steady-state equilibrium (/n*, v,
innovation and thus no growth (this follows from
involve
(r)
we can
0.
assumption that G'
1 If the
each z 6 Z, and also
for
continuous. Using this construction,
uniform
a
+ 77)) >
/ (p
<p is
compact and convex-valued
are
[z]
Moreover, in any steady-state equilibrium u*
exists.
x*_ t
—A
((l
-1
Xn
and
)
0.
/p'-t). Therefore, G''
The reason why
77
>
related to the composition effect discussed above.
23
1
((l
- A"
can, under
1
)
/p - C)
>
contradicts the
some circumstances, contribute
.
IPR
State-Dependent
4
We now
IPR
Policy
extend the results from the previous section to the environment with state-dependent
The main
policy.
from the previous section generalize, but the argument
results
is
slightly
modified.
It is
IPR
no longer necessarily
the leader.
that the sequence of values {^nj^L-oo
is
increasing, since
be very sharply increasing, making a particular state very unattractive
policies could
23
the. case
For this reason, we do not have the equivalent of Proposition
can be established that only a
finite
number
for
Nevertheless,
2.
it
of states will have positive weight in the steady-
state distribution:
Proposition 9 Consider
(fi*,
u*
v, a*_i, x*,
such that
jj,*
n
—
for
<?*)
,
all
policy
(77,
Then
a steady-state equilibrium.
is
n>
IPR
state-dependent
the
and
0,
suppose
that
G
there exists a state n*
N
n*
Because of the reasons highlighted in footnote 23, Proposition 4 may not necessarily hold
with state-dependent IPR. Nevertheless, the proofs of these propositions make
as long as
(77,
£)
not too far from uniform IPR, the conclusions in these propositions will
is
continue to hold. In fact, our numerical results with optimal state-dependent
IPR
the conclusions of Proposition 4 (but interestingly not those of Proposition
5;
Section
clear that
it
always verify
see Figure 5 in
5).
Our next
result
is
a generalization of Proposition
maximization problem
is
R&D
which shows that each individual firm's
well-behaved with state-dependent IPR.
Proposition 10 Consider
share and the
6,
the state- dependent
IPR
policy
(77,
policies of all other firms are given by z
£)
=
and suppose
(<I>,
Then
a, x).
optimization problem of an individual firm leads to a unique value function
A
convex-valued for each zgZ
A = {A n [z]};Loo ^d X
and optimal
R&D
policies
[z]
Z^\{0}
=4 [0,1] and
X
and upper hemi- continuous
[z]
[z]
:
= {Xn
[*]}« _,
[z]
:
Z
=J
[0,
in z (where
that the labor
x]
v
[z]
v
the
[z]
:
dynamic
Z — R+
are compact
=
{v n [z]}^L_ 1
r)
n
—
an ^ Vn+i ~* °°> which would imply that v„+i
24
,
and establishes the existence of a steady-
state equilibrium with positive growth.
we could have
and
).
Finally, the next result generalizes Proposition 7
'For example,
>
—
v„
is
negative.
Proposition 11 Consider
G'~ l ((l
>
)
+ Tyj)) >
/ (p
IPR
state- dependent
the
Then
0.
>
then g*
0,
policy
(77,
a steady-state equilibrium
Moreover, in any steady-state equilibrium
exists.
x*_ x
-A
-1
ui*
<
1.
and
£)
v, a*_ 1
(/j,*,
In addition,
if
suppose
,
x*,
that
g*)
oj*,
either n^
>
or
0.
Although the analysis so
and characterized some of
dependent IPR policy
far has established the existence of a steady-state equilibrium
its
properties,
it
is
not possible to determine the optimal state-
analytically. For this reason, in Section 5,
we undertake a quantitative
and structure of optimal state-dependent IPR policy using plausible
investigation of the form
parameter values.
A
Optimal IPR Policy:
5
In this section,
we
investigate the implications of various different types of
R&D
the equilibrium growth and
Our purpose
librium.
Quantitative Investigation
policy under plausible parameter values.
values
we
5.1
Methodology
We
As we
take the annual discount rate as 5%,
R&D
given by
(9).
function. For example,
1
The
The
.
its
implications for optimal
will see, the structure of
optimal
relatively invariant to the range of
IPR
pol-
parameter
is
i.e.,
p year
—
0.05
and without
loss of generality,
we
theoretical analysis considered a general production function
empirical literature typically assumes a Cobb-Douglas production
Kortum
(1993) considers a function of the form
Innovation
where Bq
model and
consider.
normalize labor supply to
for
on
not to provide a detailed calibration of the model economy, but to
is
and the innovation gains from such policy are
icy
policies
rates using numerical computations of the steady-state equi-
highlight the broad quantitative characteristics of the
IPR
IPR
a constant and exp (nt)
(t)
is
=
Bq exp
(nt)
(R&D
a trend term, which
inputs) 7
(41)
,
may depend on
general technological
trends, a drift in technological opportunities, or changes in general equilibrium prices (such as
wages of researchers
fact that
to
R&D
etc.).
most empirical work estimates a
inputs.
of this
form
is
not only
its simplicity,
but also the
single elasticity for the response of innovation rates
Consequently, they essentially only give information about the parameter
in terms of equation (41).
concave.
The advantage
For example,
A
low value of 7 implies that the
Kortum
R&D
production function
is
7
more
(1993) reports that estimates of 7 vary between 0.1 and 0.6
25
,
Pakes and Griliches, 1980, or Hall, Hausman and Griliches, 1988). For these reasons,
(see also
throughout, we adopt a
R&D
production function similar to (41):
x
where
J3,
where
w
the exp
7 >
term).
(tit)
and can be
24
=
checks for 7
number
we
=
and 7
0.1
0.6.
benchmark value
The
0-05 also pins
We
that
if
down
7
g*
i.e.,
as the
=
and then report
0.35,
~
licensing.
sensitivity
chosen so as to ensure
benchmark economy
This growth rate together with
=
0.069 from equation
the expected duration of time between any two consecutive innovations
benchmark
value,
1
and check the robustness of the
=
(2).
...}
,
C,
=
{00, 00,
tection and licensing,
More
specifically, (42)
results to A
i.e.,
is
about 3 years
= 1.05
A = 1.2
26
We
=
1.01
all
of the parameters
1.05.
year and 12 years, respectively). This completes
model except the IPR
= {0, 0,
24
is
0.019, in the
the annual interest rate as r year
take A
and
policy.
As noted above, we begin with the
(77
numerical
choose the value of A using a reasoning similar to Stokey (1995). Equation (36) implies
(expected duration of
of the
...}).
(77
We
full
patent protection regime without licensing,
then compare this to an economy with
= {0, 0, ...} ,£ =
{CiCj
•••})>
where (
=
v\
—
full
vq.
patent pro-
27
After
(t)
=
Bui
(t)
"
(R&D
some
level h.
'
and h
large,
it
= min {Bh? + eh; Bh 1 + eh}
makes no difference to our simulation results.
benchmark parameterization with full protection without
In particular, in our
licensing,
24%
of industries
are in the neck-and-neck state. This implies that improvements in the technological capability of the
R&D
make
For example, the following generalization of (42),
x
for e small
this,
1
expenditure)'
thus would be equivalent to (41) as long as the growth of w (i) can be approximated by constant rate.
25
For example, we could add a small linear term to the production function for R&D, (42), and also
flat after
i.e.,
can be alternatively written as
Innovation
26
far
Following the estimates reported in
an industry, then a growth rate of about 1.9% would require A
in
captured by
results, since in all
other parameter in (42), B,
which features indefinitely-enforced patents and no
=
25
of
it is
[x/B]~i w,
boundary conditions we imposed so
satisfy the
of states are reached.
start with a
G (x) —
(thus in terms of the above function,
an annual growth rate of approximately 1.9%,
Pyear
(42)
do so without affecting any of the
easily modified to
(1993),
economy
Equation (42) does not
exercises only a finite
Kortum
Bh"<
In terms of our previous notation, equation (42) implies that
0.
the wage rate in the
is
=
76%
R&D
economy
both the leaders and
the followers in 24% of the industries. Therefore, the growth equation implies that we have g ~ In A x 1.24 x x,
where x denotes the average frequency of innovation in a given industry. A major innovation on average every
three years implies a value of A ~ 1.05.
27
For the interpretation of full patent protection as C = vi — vo, recall the discussion in footnote 16. Note also
is
driven by the
efforts of the leaders in
of the industries
26
and the
efforts of
>
we move
IPR
to the optimal state-dependent
optimal value of each
late the
state-dependent
IPR
technology gap of n
IPR
=
r\
Since
policy.
and ( n we
n
5 or more. In other words, the
policy
(Vl, Cl)
—
n —
^^
none
s-
little effect
IPR
and make an
growth rate g*
,
and then check
Depending on whether there
process
56789
flexibility (e.g., allowing
relies
process
and
is
is
then repeated
we
(
£, are
demand
excess
IPR
take the
We now
and
or £ 5
7] e
iteration.
R&D policies,
is
r)y.
On
We
—
vo
and £
conditional on w*,
}^
=
After convergence,
_ OQ
,
is
state-
{a n }nL-<x>
we compute the
market from equation
in the labor
is
(18).
varied and the whole
IPR policy
we
find the
is
computed.
IPR
policy sequences,
growth-maximizing value
then move the next component,
for
example,
determined, we go back to optimize over
present our simulation results.
IPR
rj-y
T]
2
-
for
Once
again,
and
results
We
start with the
parameter values A
=
and
1.05
change when we vary these parameters.
Protection Without Licensing
benchmark with
that at a license fee of
v1
77
repeated recursively until convergence.
start with the
in this case
policies
policies.
at a time, until
772 is
and then show how the
Full
{x*
or supply of labor, w*
IPR
IPR
on whether there
full
protection and no licensing, which
existing literature has considered so far (e.g., Aghion, Harris, Howitt
=
and
775
Then
uj*.
case, the optimal (growth-maximizing)
computed one element
procedure
5.2
market clearing
for
the growth-maximizing value of
0.35,
first
or { un}^l_i depending
of industries, {/^}^Lo-
for different
that component, for example,
C
10 11
on uniformization and value function
repeated until the entire steady-state equilibrium for a given
is
In the state-dependent
We
">
results.
of values {^nl^L-oo
and the steady-state distribution
=
(V5> Cs)
,
4
dependent IPR policy or not), and we derive the optimal
7
form of
industries that have a
all
C4)
,-^,
guess for the equilibrium labor share
initial
we generate a sequence
this
>-um
£"
,
details of the uniformization technique are described in the proof of Proposition 6.
as given
77
('74.
3
value function iteration, see Judd (1999). In particular,
The
um
policy matrix takes the form:
^s^
2
The numerical methodology we pursue
The
r]
computationally impossible to calcu-
C2) 073> C3)
,—^~,
(172.
1
on our
policy
limit our investigation to a particular
,
checked and verified that allowing for further
C 6 to differ) has
it is
whereby the same V and C applies to
policy,
Technology gap:
We
IPR
to a comparison of the optimal (growth-maximizing) uniform
£,
followers are indifferent between a
we always suppose
— e for e 0.
that they choose o
=
J.
27
1.
=
and a
Thus
=
1,
and
in
is
the case that the
and Vickers, 2001). In
computing the equilibrium
alternatively one might wish to think that
terms of our model, this corresponds to
=
(28) implies that a*_ n
for all n.
benchmark has an annual growth
The
value function for this
We
r]
=
n
B
co for
terms of
in
benchmark case
shown
is
n >
0,
in
Figure
which
R&D
(with the solid
1
for firms that are
no state-dependent IPR
also
shown
neck-and-neck
and
efforts for leaders
R&D
followers in this
policy, all followers
The
undertake the same
benchmark
a leader declines as the
level of
at the technology
(i.e.,
line).
no state dependence in the IPR
technology gap increases. Moreover, consistent with Propositions 4 and
is
(42), so that the
consistent with Proposition 4,
is
level for all followers (since there is
(again with the solid lines). This figure shows that the
R&D
Equation
all n.
rate of 1.86%.
Figure 2 shows the level of
policy).
—
n and £n
choose the parameter
value function has decreasing differences for
and features a constant
for all
5,
the highest level of
—
gap of n
0).
R&D
level of
Since there
is
which
is
effort,
in the figure.
Figure 3 shows the distribution of industries according to technology gaps (again the solid
benchmark
line refers to the
n
=
1,
but there
The mode
case).
also a concentration of industries at technology
is
them
implies that innovations by the followers take
The
B
above,
column of Table
first
of the distribution
1
The
the bottom row.
is
reasonable.
28
the workforce
Full
5.3
We
In the
GDP.
Instead,
next turn to
1
— u*
shows that
=
technology
C
is
—
vi
~
a*_ n
=
equal to 0.0186, which
is
recorded at
1
and
2.
The
IPR
is
all
this
in this
lo*
u> is
should not be thought
5%
in
GDP, which
is
benchmark parameterization 3.2%
of
corresponds to
also consistent
protection with licensing.
v o for
steady-state value of
with
US
of
data.
29
With Licensing
protection with licensing as corresponding to
and Cn
0,
benchmark parameterization,
Protection
full
because
measures the share of pure monopoly profits
working as researchers, which
IPR
is
the only factor of production in the economy,
Finally, the table also
is
0,
table also shows the gap between x*Q and x*_ l (0.029 versus 0.018), the
of as the labor share in
value added.
=
gap n
gap of
benchmark simulation. As noted
also reports the results for this
frequencies of industries with technology gaps of
0.952. Since labor
at the technology
to the "neck-and-neck" state.
chosen such that the annual growth rate
is
is
n
r]
n
=
As
specified above,
for all
n
(so that the license fee for
we think
of
full
IPR
(so that patents never expire)
making use
of a leading-edge
equal to the net present discounted value gap between being a one step ahead
Bureau of Economic Analysis (2004) reports that the ratio of before-tax profits to GDP in the US economy
was 7% and the after-tax ratio was 5%.
29
According to National Science Foundation (2006), the ratio of scientists and engineers in the US workforce
in 2001 is about 4%.
28
in 2001
28
and a neck-and-neck
leader
effort levels
and distribution of industries
is
pay the
no longer a spike
license
R&D
is
effort at
similar to that in the
n
=
0,
however, since
column
(in fact, as
Table
2 of
1
shows
considerably higher than in the benchmark case. In particular,
The
resulting growth rate
is
growth comes not from the increase
now
of the followers
Table
2.58% instead of 1.86%
in the
R&D
shows that
1
economy without
now
is
no
licensing.
firms always prefer to
level of
xl x
in the
effort overall,
this considerably higher
is
R&D
now
growth rate
by followers
is
0.021 rather than
but from the fact that the
economy
equation (36)). In
(recall
be no
benchmark. The boost to
also advances the technological frontier of the
and build on the leading-edge technology
license
Since there
.
in equilibrium there will
More importantly, the
industries in the neck-and-neck state).
0.018.
lines)
and jump ahead of the leading-edge technology. This makes the neck-and-neck
no longer special
state
in
dashed
for this case (with the
state-dependent policy the general pattern
There
R&D
Figures 1-3 show the corresponding value functions,
firm).
R&D
since they can
fact,
column
2 of
achieved with a lower fraction of
is
the workforce, only 2.6%, working in the research sector.
This
which
result,
is
robust across different parameterizations of the model,
is
the
first
important implication of our analysis. Relative to existing models of step-by-step innovation,
such as Aghion, Harris, Howitt and Vickers (2001), which do not allow for the possibility of'
licensing, here the
R&D
effort
by followers can directly contribute to economic growth and
this
increases the equilibrium growth rate of the economy.
Optimal Uniform IPR Protection
5.4
We
next turn to optimal (growth-maximizing)
that
r)
life
=
rj
and Cn
Column
rate.
to
n
=
C f° r au
3 of Table
1
n and look
>
IPR
policy with licensing.
for values of
77
That
is,
and £ that maximize the growth
shows that the growth-maximizing values of
77
and £ are both equal
the benchmark parameterization. This corresponds to zero license fees and indefinite
in
of patents, so that followers can never copy the leading-edge technology without
they can always advance one step ahead of the leader
efforts
(without paying any license
The
resulting value function,
when they
The
figures
R&D
effort levels
and column 3 of Table
1
are successful in their
IPR
R&D
and industry distributions according to
lines).
— vq
declines significantly), but encourages
by the followers, since when successful they do not have to pay the
optimal uniform
but
show that the growth-maximizing IPR policy discour-
ages leaders (which can be seen from the fact that v\
effort
R&D,
fees).
technology gaps are shown in Figures 4-6 (with the solid
R&D
we impose
increases the growth rate
29
by only a
little,
license fee.
The
however. While the growth
economy with
rate of the
increase
full
IPR
protection with licensing was 2.58%,
it is
now 2.63%. This
associated with a very modest rise in the share of the labor force working in research
is
(from 2.6% to 2.7%).
Optimal State-Dependent IPR Without Licensing
5.5
We
next turn to our second major question; whether state-dependent
difference relative the uniform IPR.
the comparison
is
benchmark
to the
and look
for the
shown
column 4 of Table
in
Two
column
We first
now 2.03%
column
1).
we
In particular,
set ( n
—
oo for
all
n
The
results are
growth rate increases noticeably
relative to
that maximizes the growth rate.
...,775}
1.
features are worth noting.
1; it is
a significant
investigate this question without licensing (so that
case in
combination of {77^
IPR makes
First, the
instead of 1.86%. Nevertheless, this increase
IPR
to the benefits of licensing. Therefore, state-dependent
is
quite small relative
policy with no licensing
is
not a
substitute for licensing.
Second, we see an interesting pattern (which
investigations).
The optimal state-dependent
policy
technological leaders that are further ahead.
involves
=
rj^
0.054,
7?
=
2
0.005,
and
=
773
77
4
in fact quite general in all of
is
{77 l7
...,
T75}
In particular,
=
t]
5
=
0.
our quantitative
provides greater protection to
we
find that the optimal policy
This corresponds to very
little
patent
protection for firms that are one step ahead of the followers, which can be seen by comparing
77 !
=
0.054 to
followers are
x*_ .j
=
0.010.
more than
This comparison implies that firms that are one step behind
five times as likely to
catch up with the technological leader because
of the expiration of the patent of the leader as they are likely to catch
successful
R&D. Then,
there
is
leaders that are two steps ahead,
and
?7 2
This pattern of greater protection
is
it is
economy by bringing more
little
R&D)
is full
or
more
is
IPR
policy should try to boost the growth
into neck-and-neck competition. This composition effect
as follows:
force, the trickle-down effect.
by providing secure patent protection
steps ahead of their rivals, optimal state-dependent
secure protection of their
own
ahead may
industries with large technology gaps (where leaders
leaders that are one and two steps ahead as well
less
protection and patents never expire.
for technological leaders that are further
dominated by another, more powerful
trickle-down effect
own
l/10th of 77^ Perhaps even more remarkably, after
go against a naive intuition that state-dependent
engage in
their
a steep increase in the protection provided to technological
a technology gap of three or more steps, there
rate of the
up because of
—despite the
intellectual property.
30
IPR
The
is
present, but
intuition for the
to firms that are three
increases the
R&D
fact that these firms
In fact, this
is
effort of
now
face
precisely because of
low protection
for technological leaders that are
one or two steps ahead combined with the
promise of much greater protection once they reach a technology gap of three steps or more.
Mechanically, high levels of
^
advanced firms increases vn
for
and
r\
n >
2
IPR
reduce v\ and V2, while high
Consequently,
3.
that are one or two steps ahead to undertake
R&D
it
becomes more
protection for more
beneficial for leaders
to reach the higher level of
IPR
protection.
R&D for leaders under state-dependent IPR contrasts with uniform
IPR, which always reduces R&D by all technological leaders. The possibility that imperfect
state-dependent IPR protection can increase R&D incentives is a novel feature of our approach,
This pattern of increased
IPR
since previous models have not considered state-dependent
Optimal State-Dependent IPR With Licensing
5.6
we turn
Finally,
to the
most general policy choice within the context of our approach and
investigate which combinations of
{r7 l5 ...,
?? 5
and {Cii—jCs} maximize the growth rate and
}
the contribution of the optimal state-dependent
shown
of this exercise are
uniform
to
policies.
IPR
in
column
5 of
column
policy with licensing in
maximize the aggregate growth
Table
rate.
1.
3,
IPR
policies to
Now the
economic growth. The results
comparison should be to the optimal
where uniform IPR
The value
functions,
policies
R&D
economy
distribution over different levels of technology gaps in this
r\
efforts
and C were chosen
and the industry
shown
are
in Figures 4-6
(with the dashed lines).
We
see in
column
5 that
patents never expire.
growth-maximizing IPR policy involves
Nevertheless,
IPR
paying any license
From
fees.
is
=
0,
of the
economy
policy,
2.63%
rate of the
is
in
full
there on, £ increases to £ 2
column
economy
3.
is
for all n, so that
is
not
In
full.
one step ahead of their own knowledge without
patent protection (since v\
2.96%, which
=
which implies that followers can
=
significantly higher
— vq =
which
0.82,
patent protection. After three or more steps, however, we have £ 3
which are very close to
n
protection for technological leaders
particular, the growth- maximizing policy involves £i
build on the leading-edge technology that
rj
=
1.96).
is still less
1.94
The
and C4
—
than
Cs
—
full
1-95,
resulting growth rate
than the growth rate under uniform IPR
This shows that state-dependent policies can increase the growth
significantly.
State-dependent policies again achieve this superior growth performance by exploiting the
trickle-down effect, which
we already saw
in the case
without licensing. In particular, £n
is
an increasing sequence, which implies that technological leaders that are further ahead receive
greater protection.
boosting the
R&D
As
in the previous subsection, this pattern of
effort of technological leaders
31
IPR
is
used as a way of
that are one or two steps ahead of their
rivals (see
Figure 5)
Since these leaders receive
.
increase both their profits
and
their
IPR
little
protection and understand that they can
protection by undertaking further innovations, they
have relatively strong innovation incentives and undertake high
the fraction of the labor force working in
in Table 1)
R&D
and again
illustrates
how
R&D to 3.9%
(which
is
levels of
R&D.
greater than
IPR
imperfect state-dependent
This increases
the other cases
all
protection can increase
incentives relative to full protection.
The growth
ers that are
rate of the
economy
also receives a boost
one step behind the leaders,
from the
R&D
effort of the follow-
R&D
thanks to licensing, followers'
since,
directly
contributes to the advancing the technological frontier of the economy. Figure 5 shows that
followers that are one step behind the frontier
case with growth-maximizing uniform
this pattern of
R&D
efforts
is
IPR
now have
R&D
a higher
(which involved (n
that the growth-maximizing
—
IPR
effort
than even
for all n).
The reason
more than once step behind
followers that are
Overall, the results
consider the
show that state-dependent IPR
economy without
R&D
level of
lower than in the economy with uniform IPR.
is
can increase the equilibrium
policies
growth rate of the economy substantially and that the trickle-down
when we
for
and the trickle-down
policy
have increased the value of being a technological leader. In contrast, the
effect
in the
licensing,
but also
in the
effect
is
powerful, not only
presence of licensing.
Robustness
5.7
Tables 2-5 show the robustness of the patterns documented in Figures 1-6 and in Table
1.
In particular, each of these tables changes one of the two parameters A and 7 (increasing or
reducing A to 1.2 or 1.01, and increasing or reducing 7 to 0.6 or 0.1) and show the results
corresponding to each one of the
we change the parameter
case,
economy with
full
IPR
B
regimes and discussed so
five different policy
in
all
cases
we
protection without licensing
see a significant increase in the
smallest increase
R&D,
is
when 7
=
0.6,
In each
equation (42) so that the growth rate of the benchmark
is
the
same
as in Table
Notably, the qualitative, and even the quantitative, patterns in Table
In
far.
1
1,
1.86%.
are relatively robust.
growth rate when we allow
licensing.
The
presumably because with limited diminishing returns to
incentives were already sufficiently strong without licensing.
the growth rate increases only from 1.86% to 1.98%. In
increases the growth rate to above 2.5%, which
is
all
As a
result, in this case,
other cases, allowing for licensing
a very large increase relative to the baseline
of 1.86%.
Moreover, in
further,
all
cases,
moving
to state-dependent
IPR
policy increases the growth rate
though the extent of the increase varies depending on parameters. The most modest
32
when 7 =
gain comes
because in this case the economy with
0.1,
already achieves a very high growth rate of 2.78%, and thus there
an increase in growth rate from state-dependent IPR. In
growth rate
is
is
the fact that in
shaped by the trickle-down
is
considered, there
ahead, but
IPR
is little
or
only
little
room
left for
other cases, the increase in the
In
effect.
all
all
cases,
growth-maximizing state-dependent
of the various parameterizations
no protection provided
we have
to technological leaders that are one step
protection grows as the technology gap increases. This
implied by the trickle-down
We
is
quite substantial.
Perhaps, more noteworthy
IPR
all
protection and licensing
full
is
the typical pattern
effect.
therefore conclude that both the substantial benefits of licensing in models of tech-
and the benefits of state-dependent
nological competition
policies,
mostly coming from the
trickle-down effect, are robust across different specifications.
Conclusions
6
In this paper,
we developed a general equilibrium framework
to investigate the impact of the
extent and form of intellectual property rights (IPR) policy on economic growth.
major questions we focused on are whether
leading-edge technology in return of a license
rate
licensing,
fee,
The two
which allows followers to build on the
has a major impact on the equilibrium growth
and whether the same degree of patent protection should be given to companies that are
further ahead of their competitors as those that are technologically close to their competitors.
In our
model economy, firms engage
in
cumulative (step-by-step) innovation. Leaders can
innovate in order to widen the technology gap between themselves and the followers, which
enables
them
to charge higher markups.
On
the other hand, followers innovate to catch up with
or surpass the technological leaders in their industry. Followers can advance in three different
ways.
First, the
patent of the technological leader
industry to copy the leading-edge technology.
R&D
to improve
its
own
when
successful,
In the model economy,
it
will
be
far followers,
expire, allowing the follower in the
Second, each follower can undertake catch-up
variant of the product to catch
follower can undertake frontier
in this case,
may
and
R&D,
it
IPR
will
up with the
Third, each
building on and improving the leading-edge technology;
have to pay a license
fee to
the technological leader.
policy regulates whether licensing
also the length of the patents.
steady-state equilibrium and proved
leader.
its
We
existence under general
is
possible
and how costly
characterized the form of the
IPR policies. We
then used this
framework to investigate the form of "optimal" (growth-maximizing) IPR policy quantitatively.
33
The major
1.
A move
findings of this quantitative exercise are as follows:
from an IPR policy without licensing to one that allows
on the equilibrium growth
icant effect
for licensing has a signif-
For the benchmark parameterization of our
rate.
model, licensing increases the growth rate from 1.86% to 2.58% per annum. This substantial increase in
2.
State-dependent
rate. In
growth rate
IPR
is
robust to a large range of variation in the parameters.
also leads to a significant
improvement
in the equilibrium
growth
our benchmark parameterization, the growth rate of the economy increases from
2.57% under the growth-maximizing uniform IPR policy to 2.93% under state-dependent
IPR
Perhaps more interesting than
policy.
growth rate
intuition,
is
we
the form of the optimal state-dependent
find that the
impact on the equilibrium
this substantial
IPR
policy.
Contrary to a naive
growth-maximizing IPR policy provides greater protection to
firms that are further ahead of their rivals than those that are technologically close to
their competitors.
The
effect.
Underlying
this
form of the optimal IPR policy
the trickle-down
trickle-down effect implies that providing greater protection to sufficiently
advanced technological leaders not only increases their
R&D
is
efforts of all technological leaders that are less
the reward to innovation
now
R&D
but also raises the
efforts,
advanced than
this level,
because
includes the greater protection that they will receive once
they reach this higher level of technology. Our results suggest that the trickle-down
is
powerful both with and without licensing, and
insensitive to the exact
The
policy
industry).
move
move towards optimal state-dependent
increase innovation and economic growth.
IPR
may depend on
form and magnitude are
relatively
parameter values used in the quantitative investigation.
analysis in this paper suggests that a
particular, a
its
effect
The
to a richer
menu
of
policies with licensing
results also
IPR
may
policies,
in
significantly
show that the form
of optimal
the industry structure (and the technology of catch-up within the
Nevertheless, these conclusions are based on a quantitative evaluation of a rather
simple model.
It
would be interesting to investigate the robustness of these
results to different
models of industry dynamics and study whether the relationship between the form of optimal
IPR
policy and industry structure suggested by our analysis also applies
when
variation in
industry structure has other sources (for example, differences in the extent of fixed costs).
most important area
IPR
policy, using
also structural
for future
work
is
a detailed empirical investigation of the form of optimal
both better estimates of the
models where the
The
effects of
IPR
effect of different policies
34
policy on innovation rates
and
on equilibrium can be evaluated.
,
,
Appendix: Proofs
Proof of Proposition
Equations (19) and (21) imply
1.
Q(i)A-£-° n/i " (1)
u (t)
«>(t)
Y(t)
(t)
'
Since
=
u> (t)
w* and {/^}^L
Y (t)
are constant in steady state,
grows at the same rate as Q(t).
Therefore,
,
=
q
Now
+ At)-
In <?(£
,.
lim
.
At-o
At
At (i) in the fraction /i* of the industries with
£*At + o(At); (ii) in the same industries, the
note the following: during an interval of length
technology gap
n >
the leaders innovate at a rate
1
Q(i)
In
followers innovate at the rate a*_ n x*_ n At+o {At);
(iii) in the fraction Pq of the industries with technology
both firms innovate, so that the total innovation rate is 2x$ At + o (At)); and (iv) each
innovation increase productivity by a factor A. Combining these observations, we have
gap of n
=
0,
\nQ(t
Subtracting
+
=
At)
\nQ
(t),
ln<2(£)
+lnA
dividing by
2p* x* At
&< A
+ J2
At and taking
the limit
*
+
At
9 (At)
—
+ JT a*_ n x*_ n At +
o (At)
gives (36).
>
Proof of Proposition 2. Let (a_i, {x n }^°__ 1 } be the R&D decisions of the firm and {v n }'^'__ l
be the sequence of values, taking the decisions of other firms and the industry distributions, {xn}™=-i>
{/OnL-i' w * anc^ 5' as gi ven By choosing x n = for all n > —1, the firm guarantees v n > for all
-
n > — 1. Moreover,
that {w n }^= _ 1
since flow profit satisfy
Suppose, first,
Proof of v\ > vq
problem in equilibrium (26) implies vq
:
(25) implies that
v\
>
v
when
since
1 for all
x*_ x
=
0, V\
>
i>i
<
^o,
=
v\
=
(l
— A -1
n > — 1, i>„ < 1/p
n > — 1.
n > — 1,
establishing
then (31) implies Xq = 0, and by the symmetry of the
As a result, from (30) we obtain xlj = 0. Equation
0.
)
/ (p
+ rj) >
v\
—
77/
(p
C,
<
<
0,
yielding a contradiction
and proving that
Suppose, to obtain a contradiction, that u_i > vq.
— 0. This implies i>_i = 771J0/ (p + 77), which contradicts v-i
Y
Thus we must have vi — £ > vq, which implies that a*_ x = 1. Imposing a*_ x
vq
'
vq, (30) yields x*_
+ 77) <
1.
the value function of a neck-and-neck firm can be written
pv
=
max{-u*G(x
Xq
>
ma,x {-gj*
XQ
>
-u*G
=
pV-l,
which contradicts the hypothesis that
Proof of v n
(25)
< vn+ i
:
)
+
+ x*_
(xlj)
tj_i
+x
)
max{-w*G(x
>
)
G (x + x
XQ
and
for all
1/p] for all
[0,
.
Proof of v-i
If
nn <
a bounded sequence, with v n £
is
>
vq
l
x
\vi
=
Vq
1,
as:
{vi-v ]+x °[v^i
[vi
-
v
[vi
-
v-\
-
-
C]
-
>
V-l
}}
-v
(43)
}}
,
£]}
+
7?
,
[v
-
V-i]
and establishes the claim.
Suppose, to obtain a contradiction, that v n
>
v n +\.
-
f„]
Now
(29) implies x*n
=
0,
becomes
pv n
=
(1
- A" n ) + xl a
[a*_!
(u-i
+
C)
35
+
(1
-
all) ^o
+
V bo
-
t>n]
(44)
.
+
Also from (25), the value for state n
pvn+i
>
(1
1 satisfies
- A""- 1 ) + x*_ x
[a'_
(vj-i
1
+
C)
+
(1
-
ali) ^o
-
+
^n+i]
- vn+ i]
[vo
t?
(45)
Combining the two previous expressions, we obtain
- A" n ) + as!.!
~
- \- n + xl
[a*
(1
l
>
_n_1
1
< A~ n
j
+ C) + (l - all) v - v n + [v - vn
(v-i + C) + (1 - all) ^o - «h+i] + V N - «*ti]
(u_!
[at!
1
r\
]
]
•
this implies u n < n n +i, contradicting the hypothesis that v n > vn +i, and establishing the desired result, v n < v n +i- Consequently, {v n }^__ 1 is nondecreasing and {vn}^L * s
(strictly) increasing. Since a nondecreasing sequence in a compact set must converge, {v n }'^'__
con1
Since A
verges to
its limit
has a nonnegative
,
point, v^o, which
initial value.
Proof of Proposition
must be
strictly positive, since {v n }^L
strictly increasing
is
and
This completes the proof.
The
3.
maximization of the value function
first-order condition of the
(25) implies:
G'
—
>
(x-n)
and x n >
0,
with complementary slackness. G' (0) is strictly positive by assumption. If (v n +\ — v n ) /ui* < G' (0),
then x n = 0. Proposition 2 implies that {v n }'^__ l is a convergent and thus a Cauchy sequence, which
implies that there exists 3n* € N such that v n+ i — v n < u>*G' (0) for all n > n*
u
.
Proof of Proposition
From equation
4.
5 n+ i
is
sufficient to establish that
=
x n+l < x*n
v n+x
(29),
-v n <v n - v n _i =
6n
(46)
.
Let us write:
pv n
= max{(l -
A~
n
)
- u>*G(x n +
)
x* ,-[v n +i ~'Vn]
+^-i
[all (.V-i
+
+
(l
-
a*_
x )
v
]
+tjv
}
,
(47)
where p
=
p
+
x*_ x
+
r\.
Since x n+1
,
x* and x*n _ Y are maximizers of the value functions v n+ i, v n and
v n -!, (47) implies:
- ™" 1
pu n+ l
=
1
-A
pv n
>
1
- A" n - u'G
pv n
>
1
pVn-x
=
1
- W*G
«
+1 )
+ <+i Kn-2 -
Un+l]
+
X*-\ [«-l (^-1
+
'+ (l
~
a -l) ^o]
+
TO),
(48)
Now
+ x*n+l [v n+ - Vn] + x*_! [all («-i + C) + (l - all) vQ + tjvq,
n
- A" - uj*G «_x) + <_i K+i - v n + x*_i [al (v_i + C) + (l - a*-i) v + TO),
- A" n+1 - LJ*G «_ + X*n _! [Vn - Vn-i] + itj. [alj {V-i + C) + (1 - ali) Wb] + TO(x* +1 )
i
]
]
]
x
x )
taking differences with pv n and using the definitions of
n
-
-1
<5
n 's,
we obtain
-
pSn+i
<
A~
P5n
>
A-^^l-A-^+^.i^n+l-^)-
(l
A
)
+
x n+1
(<5
n+2
S n +i)
Therefore,
(P
+ <-i) (<Wi -
5n )
< -k n + xn+1
{5 n+2
where
kn
= (A-l) 2 A- n_1 >0.
36
- S n+ i)
,
(49)
.
Now
—
to obtain a contradiction, suppose that <5„+i
5n
>
0.
.
From
(49), this implies
<5
n+ 2 — 6 n+ i >
Repeating this argument successively, we have that if 5 n + \ — 8 n > 0,
However, we know from Proposition 2 that {v n }^L Q is strictly
then 5 n+ i — 5 n >
for all n > n'
increasing and converges to a constant v x This implies that 5 n
0, which contradicts the hypothesis
that <5 n+ i — S n >
for all n > n' > 0, and establishes that x* +1 < x^- To see that the inequality is
strict when x* > 0, it suffices to note that we have already established (46), i.e., 6 n+ i — 5 n < 0, thus if
equation (29) has a positive solution, then we necessarily have x* +1 < x*n
since k n
is
strictly positive.
i
'
.
.
J.
.
>
Proof of Proposition
al x =0, and
(26)
5.
Proof of Xq
can be written as
=
pv
We
have vq
>
from Proposition
2.
-lj*G
x*_ 1
(xq)
Suppose
+ Xq
>
i>o
Suppose
:
>
>
-v
V\
This inequality combined with
Inequality (51)
strictly
We
now
a"_
1
=
and
(31)
0,
Then
0.
V-i+vi-2vq
+vi-
[v-i
that C
first
2v
>
>
[v
~
>
and
>
(38) }delds Xq
>
=
0.
G ()
is
Suppose next that vq
x"L±.
x*_
x
x*_
is
Moreover, since
.
x
=0.
an optimal
policy, so that
-uj*G(xI)+Xq[vi-vo]+Xo[v-i-vo\
-uj*G{xq)+x*q [vi-V-i -C} + v[vo -v-i}.
Subtracting the second expression from the
P
(28) implies
V-i.
.
pv-i
Then
(51)
Vq-
given by (31), (50) then implies Xq =
and thus
next show that when £ < v\ — vo, Xq > x'L 1 In this case, a*_ 1 = 1
=
vq.
(50)
is
pv
-
)
(50) implies Xq
holds as a weak inequality and implies that Xq
convex and Xq
v\
U_i]
<
we obtain
first,
+
X*Q [V-i
C
-
Vq]
+
{X*
+ Tj)
[V-l
~
Vq]
,
and therefore
-v-i] <
[vq
+C-t'o]-
[v-i
< vo, and therefore V-i + C > vq. Next observe that with a*_ l = 1,
>
imply
that
x*_
if and only if v\ — Vq > V\ — V-\ — Q, or equivalently if and only if
Xq
(38)
x
>
we
vq.
Thus
have
established
that Xq > x*_ x both when £ > v\ — vq and when C < vi — vq.
C
Proposition 2 implies that v_\
(31)
u_i
and
+
Lemma
Lemma
1
1
and proof:
We
next
first
prove a lemma, which will be used in the rest of the Appendix.
1
Suppose that G'^ 1 ((l - A^ )
/ (p
+
Proof of Lemma: Suppose, to obtain a
x*_ 1 = 0- Then (25) implies
-q))
>
0,
>
then Xq
contradiction, that Xq
then implies that
pv\
Equation (26) together with Xq
=
>
gives vq
v1
Combined with
1
=
0,
A
+
[vq
j]
-
vi]
and hence
-v >
1-A"
P + V
1
this inequality, (31) implies
naxfG'- 1
A
1
(
maxJG'-M
37
.
r
-
.
{p
i
+ rf)
A
A"
-
P+V
)
n
.n
and v >
=
0.
0.
The
first
part of the proof
,
.
< 1.
where the second inequality follows from the fact that ui*
The assumption that
-1
then implies Xq > 0, thus leading to a contradiction and establishing
G'~ ((l — A ) / (p + r/)) >
l
>
that x^
>
G'"
when G'~ l
-
0M
>
then implies vq
£
x*_!
((l
A
-1
1
((l
when
>
together with Xq
and C > 0: Given Lemma 1,
+ w)) >
) / (p
- A" ) / {p + r))) >
implies that Xq > 0. Then the first part of the proof implies that
> vi — vo, Xq > x'L 1 Next suppose that < C < v i — vo- Then the same argument as above
that Xq > x*_ 1 if and only if vi — vq > vi — V-i — £, or equivalently if and only if v^i + £ > vq.
Proof of Xq
1
G ()
Strict convexity of
0.
.
implies
Suppose
this
thus v-x
=
Then from the first part of the proof, we have that Xq = x*_ x = 0, and
which implies u_i +
> vq and thus Xq > x*_ 1 This yields a contradiction and
not the case.
is
=
vq
0,
(,
>
completes the proof that Xq
when
x*L x
Proof of Xq > x\ To prove that Xq
proof of Proposition 4:
>
:
.
>
£
G/_1
and
— A" 1 )
((l
+
/ (p
>
77))
x\, let us write the value functions v 2
0.
and
t>i
,
vq as in (48)
in the
Now
4,
1- \~ 2 -u*G{x*2 )+X2lv3 -v 2 }+xt
=
pv 2
~pv x
>
pv x
>
pv
=
[a*_
1
+ (l-a*_ )vo}+r)v
+ (l -a*-i)v Q +V v o,
{v-i+C,)
1
1
- X" -oj*G{x 2 + x*2
1
\v 2 -v{\ + x*_ 1 [alj (v-i +
1- A" -uj*G{x* )+x*q[v 2 -v }+x*_ [al^-i + C) +
-LJ*G(xq) + Xq [Vl - Vq] +nVQ + X*_ Vo + Xq [V-i - Vq]
1
)
]
1
1
1
,
(l
- alj
v
]
+
nvo,
.
x
taking differences with pv n and using the definitions of
<5
n 's as in (46) in the proof of Proposition
we obtain
p5 2
ph
pSi
p5i
<
X-
1
(I
-
- X-1
1
)
+
+ xl(5 3 -
52 )
-
+ xl
>
(1
>
>
(l-X- )+x
-1
(l - A
) +Xq
A
)
xq {5 2
1
'(5 2
(6 2
b~i)
-5
-
1 )
5i)
(52)
x*_ 1 a*-i(+
+
-
(x*_
—
of the proof has established that x*_ x
—
xo\ [v-i
—
Vq]
>
0,
and the
1
tu.i
combining
+
(p
—
last inequality
with the
this inequality
1
<
Xq
pSi> (1-
Now
a*_
first
-
x* ) (6 2
<5i)
+
(u-i
l
+
Next note that Proposition 2 implies that
[x*_±
.[al
l
+
C)
(1
-v
v
]
-x^[v-i-.v
],
{x*-ia*-i-xZ)[v-i-vo\,
-
Xq) [u_i
<
vq
v
]
Moreover, the
0.
Combining
0.
this
with
first
part of the
a*_ x
<
1
first
part
establishes that
then implies
A" 1
+x*
)
{5 2
- <5i).
inequality of (52),
< -
- a*_ x )
- A"
(1
2
1
)
we obtain
+ x2
{S 3
-
62 )
(53)
.
Proposition 2 has established 62 > S3, so that the right-hand side is strictly negative, therefore, we
must have 5 2 — 5\ < 0, which implies that Xq > x\ and completes the proof.
Proof of Proposition
problem of a representative
pv n =
max
Fix z
6.
=
(uj, {.T n
}^
=
_1
,
{a n }~= _ 00
),
and consider the optimization
firm, written recursively as:
{ (1
- A~ n ) - uG {x n ) +
xn
[v n+1
-
vn
)
x„e[o,x]
+
x-i (a_i
Pvq
[v-i
-
vn
+
C]
+
(1
= max {-loG (xq) +x
- a_i)
[vi
[v
-
+
v[ v o
[v-i
-v
v n ])
-v ]+ x
-vj)
for
n 6 N
}}
io£|0,i]
pv-i
=
max
{—u>G(xo)
+ x-.i(a-i[vi—v-i-Q + (l-a-i)[vo-v-i])
i_i€[0,x], a_i£[0,l]
+t][vq -v-i]}.
38
We now
transform this dynamic optimization problem into a form that can be represented as a
mapping using the method of "uniformization" (see, for example, Ross, 1996, Chapter
contraction
5).
Let I
=
({x n }~ _ 1 ,{a n } n= _ c
and
be n' starting with state n when the firm
and the
R&D
policy of other firms
=
policy, (x- n ,a- n )
(i_i,a_i)
(l-Q-l)z_i+7)
X-1+I7
p-lo €l€
po.ikk,
Pn,-1 (£
—
Pn,0
£J
Xn+ i_,+ n
£
,
{a n } n _
X_!+7)
(l-q-i)x_i+7;
x n +£_i+77
I
Pn,n+1
Uniformization involves adding fictitious transitions from a state into
itself,
=
£
I
Hx-i+t;
which do not change the
value of the program, but allow us to represent the optimization problem as a contraction.
purpose, define the transition rates
tp n
These transition rates are
(*
+ X-i+T] for n e
+ for n = — 1
for n =
x-i
2x n
t)
I
For this
as
xn
4'n
IPR
Xo + Xq
£
I
({x n )^__ l
transition probabilities can
P-a,iUI£ =
=j^
=
Using the fact that, because of uniform
be written as:
£.
n £ N, these
po,-i«K
I
I) be the probability that the next state will
I
in question chooses policies £
given by
is
for all
Pn,n> («
finite since ip n
I
£
£
|
{1,2,
...}
r\
<
)
=
ip
+
2x
<
rj
and
00 for
all
n,
where x
is
the
maximal
by assumption).
Now following equation (5.8.3) in Ross (1996), we can use these transition rates and define the new
transition probabilities (including the fictitious transitions from a state to itself) as:
flow rate of innovation defined in (11) in the text (both x
fw
are finite
77
V'nQsie)
Pn,n' (t
I)
I
1
if
n
7^ n'
if
n
=
1
n'
This yields equivalent transition probabilities
P_ 1 ,_ 1 (^IC) =
P0.-1 (€ \l)
Pn,0 (€
1
g)
=
=
1-^
2§bj
-a
(1
and also defines an
^
P-1,0 («
1
P~n,n
(« ||)
1+ "
effective discount factor
1
/3
given by
+ V"
p
the per period return function (profit net of
n
f
T
l-
J
+
+
77
+
R&D
'
,
,
1
(
?"\
I
1
tj
'
o_-is_i
2x+r,
i)
=
^
£
£
—
I
J
Tv n
,
I Il n
for all
(x„)
+ P^2p n ,n'
n£Z.
39
Pn,-1
(fi
I
1)
=
77
u"^
1
(54)
otherwise
Using these transformations, the dynamic optimization problem can be written
max
2J+r)
expenditures) be
P+2S+7,
—
(€
Pn,n+1
2x
25
ip
=
po,i
^fe^
-
(e
„
P-1,1 (s
a£,-
=
p
let
-B
Po.o(«l€)=l-^
1+T;
/3
Also
(1
=
e)
Un
\
I)
*V
>
,
for all
as:
n G
Z,
(55)
5-ii-i
22+7)
v={v n }^L_
where
V = {v
functions
=
z
(<I>,
:
and the second
1
{^1} U Z + — M + }
»
defines
line
into
the operator T,
T
itself.
a_i, {in}^L_i), P ossesses a unique fixed point v*
clearly
is
=
a
mapping from the space
contraction,
Q
{v*l } ^= _ l
thus,
for
of
given
Stokey, Lucas and Prescott,
(e.g.,
1989).
Moreover, x n G [0,5], a^\ G [0, 1], and v n for each n = -1, 0, 1, ... given by the right-hand side of
is continuous in a n and x n (a n applying only for n = —1), so Berge's Maximum Theorem (Aliprantis
and Border, 1999, Theorem 16.31, p. 539) implies that the set of maximizers (Al x {X^}'^'__ 1 ) exists,
(55)
,
is
nonempty and compact-valued
for
each z and
is
upper hemi-continuous
in z
= (w,a_i, {x n }^L_-
i
Moreover, concavity of v n in a n and x n for each
convex- valued for each z, completing the proof.
n = —1, 0,
1,
...
implies that (A*_ 1;
Z4 Z constructed in
7.
We first show that the mapping $:
-1
_1
and then establish that when G'
((l - A
+ ??)) >
) / (p
Proof of Proposition
fixed point,
to a steady state with u>*
First,
in the
it
<
{X*}^L_ 1 \
is
).
also
(40) has a
this fixed point corresponds
1.
3> maps Z into itself. We next show that Z is compact
a subset of a locally convex Hausdorff space. The first part follows
has already been established that
product topology and
is
Z can be written as the Cartesian product of compact subsets, Z = [0, 1] x [0, 1] x
Then by Tychonoff's Theorem (e.g., Aliprantis and Border, 1999, Theorem 2.57, p. 52;
Kelley, 1955, p. 143), Z is compact in the product topology. Moreover, Z is clearly nonempty and also
convex, since for any z, z' £ Z and A G [0, 1], we have Az+ (1 — A) z' € Z. Finally, since Z is a product
from the
n^L-i
fact that
[OjS].
on the
of intervals
Lemma
1999,
Next,
n G
{
if is
real line,
5.54, p.
it is
a subset of a locally convex Hausdorff space (see Aliprantis and Border,
192).
a continuous function from
— 1} U Z+
Z
into
are upper hemi-continuous in
z.
[0, 1]
and from Proposition
Consequently,
<&
=
(p
[z]
,
6, A_i
A-i [z]
(z)
,
X
and
Xn (z)
[z])
has closed
for
X
graph in z in the product topology. Moreover, each one of <p{z), A-i (z) and n (z) for n = —1,0, ... is
nonempty, compact and convex-valued. Therefore, the image of the mapping $ is nonempty, compact
and convex-valued for each z G Z. The Kakutani-Fan-Glicksberg Fixed Point Theorem implies that if
the function $ maps a convex, compact and nonempty subset of a locally convex Hausdorff space into
itself and has closed graph and is nonempty, compact and convex- valued z, then it possesses a fixed
point z* G $(z*) (see Aliprantis and Border, 1999, Theorem 16.50 and Corollary 16.51, pp. 549-550).
This establishes the existence of a fixed point z* of $.
To complete the proof, we need to show that the fixed point, z*, corresponds to a steady state
equilibrium. First, since a n (ui* a*_ l {i* ,}^L_ 1 ) = a*-\ and x n (ui*, alj, {x*n }'^'= _ 1 ) = x*n for n G
,
,
^' _ ) constitutes an R&D policy vector
= 1
by steady-state equilibrium (Definition 3). Next, we need to
prove that the implied labor share ui* leads to labor market clearing. This follows from the fact that the
fixed point involves w* < 1, since in this case (39) will have an interior solution, ensuring labor market
clearing. Suppose, to obtain a contradiction, that u>* = 1. Then, as noted in the text, we must have
for n € { — 1} U Z + However, Lemma 1 implies
Hq = 1. From (32), (33) and (34), this implies x*n =
1
that this is not possible when G'~ l ((l — A" ) / (p + r])) > 0. Consequently, (39) cannot be satisfied
at ui* = 1, implying that w* < 1. When u* < 1, the labor market clearing condition (35) is satisfied
at ui* as an equality, so ui* is an equilibrium given {x* }'^L_ 1 and thus z* = (ui*, a*_ lt {a;* }^__ 1 ) is a
{
— 1} U Z + we have
,
that
is
that given a labor share of
best response to
itself,
c
ui*
,
(a!_j,
{x*
l
}
as required
.
l
,
steady-state equilibrium as desired.
Finally,
g*
>
0.
if
rj
>
0,
from Lemma 1, equation (36) implies
then (34) implies that p,*, > 0. Since Xq >
if x*_
> 0, then g* > follows from (36), completing the proof.
l
Alternatively,
and a;* > for
Proof of Proposition 8. We will establish that there exists n* such that £*. =
n > n* which will imply that the states n < n* are transient and can be ignored, and that {Pn) n =o
forms a finite and irreducible Markov chain over the states n = 0, 1, ..., n*.
is strictly increasing. Then it follows
First, recall that Proposition 2 has -established that {v n } n€ z
for all n >
from the proof of Proposition 3 that there exists a state n** G N such that a;** =
all
,
40
,
n S N:u n+ i — v n < oj*G' (0)}. Such an n* exists, since the set
{ ,...,n--} {
-1
and nonempty because of the assumption that G'~ l ((l — A ) / (p + 77)) > 0.
Then by construction x* > for all n < n* and x*. = as desired. Now denoting the probability of
T
T
for
being in state n starting in state n after r periods by P (n, n), we have that lim T _>00 P (n, n) =
finite
Markov
chain
over
the
states
n = 0, 1, ...,n*
all n > n* and for all n. Thus we can focus on the
Now
n**.
*s
{Mn)n=o
n=
finite
is
and {Mn}"=o
^s
the limiting (invariant) distribution of this
un iq ue ly
0, 1, ...,n*
in Stokey,
= min ne
n*
let
{0, ...,n**}
—
so that
1,
Markov
chain.
Given
a*_
x
and {x*}™__ 1
Markov chain is irreducible (since x* >
communicate with n = or n = 1). Therefore, by Theorem
defined. Moreover, the underlying
all
states
Lucas and Prescott (1989,
Proof of Proposition
Then it
such that x* =
for all n >
p. 62)
there exists a unique stationary distribution
,
for
11.2
{aC}^1
.
There are two cases to consider. First, suppose that {v n } n£Z is
from the proof of Proposition 3 that there exists a state n* g N
n* and as in the proof of Proposition 8, states n > n* are transient (i.e.,
for all n>n*.
for all n > n* and for all n), so /a* =
lim T _ 00 P T (n, n) =
>
Second, in contrast to the first case, suppose that there exists some n** 6 Z + such that
v n —+\. Then, let n* = min n6 {o,...,n"} { n £ N fn+i — v n < ui*G' (0)}, which is again well defined.
for all states with n < n*, and
Then, optimal R&D decision (29) immediately implies that x* >
T
for all n > n* and for all n,
since x* = 0, all states n > n* are transient and lmv^oo P (n, h) =
strictly increasing.
9.
follows
,
iv
:
.
completing the proof.
Proof of Proposition
The proof
10.
follows closely that of Proposition
6.
In particular, again
using uniformization, the maximization problem of an individual firm can be written as a contraction
mapping
tp
=
max n
2x
{r)
similar to (55) there.
+ max n
{rj
n}
<
n } is finite, since
The
00 (this
each
rj
finiteness of the transition probabilities follows, since
is
n 6
a consequence of the fact that x defined in (11) is
and by assumption, there exists n < 00 such that
K+
Z — M+
Theorem 16.31, p.
This contraction mapping uniquely determines the value function v
Maximum Theorem
ip n
[z]
:
>
(£
|
£j
<
finite
and
=
r\^).
r\
n
.
and Border, 1999,
539) again implies
that each of A n (z) for n € Z_\ {0} and
n (z) for n G Z is upper hemi-continuous in z = (w, a, 5c),
and moreover, since v n for n € Z is concave in a n and x n the maximizers of v [z], A = {^4- n }^Li and
X = {^n}^^^) are nonempty, compact and convex- valued.
Berge's
(Aliprantis
X
,
Proof of Proposition
KKLoo
11.
The proof
follows
that
of
Proposition
7
closely.
Fix
(*,
{«n}i-oo>. and define Z = [0, 1] x IT=-i [0, 1] x ir=-oo [0,x]. Again by Tychonoff's Theorem, Z is compact in the product topology.
Then consider the mapping $: Z =3 Z
are defined in Proposition 10.
constructed as $ = (v?, A, X), where ^ is given by (39) and
and
Clearly $ maps Z into itself. Moreover, as in the proof of Proposition 7, Z is nonempty, convex, and
a subset of a locally convex Hausdorff space. The proof of Proposition 10 then implies that $ has
closed graph in the product topology and is nonempty, compact and convex-valued in z. Consequently,
the Kakutani-Fan-Glicksberg Fixed Point Theorem again applies and implies that $ has a fixed point
z* € $ (z*). The argument that the fixed point z* corresponds to a steady-state equilibrium is identical
to that in Proposition 7, and follows from the fact that within argument identical to that of Lemma 1,
-1
implies that Xq > 0. The result that u* < 1 then follows immediately.
G'" 1 ((l — A ) / (p + ??i)) >
z
=
,
A
Finally, as in the proof of Proposition 7, either
rj
1
>
41
or x!_ x
>
X
is
sufficient for g*
>
0.
References
Aghion, Philippe, Nick Bloom, Richard Blundell, Rachel Griffith and Peter
Howitt (2005) "Competition and
Innovation: an Inverted-U Relationship" Quarterly Journal
of Economics, 120, pp. 701-728.
Aghion, Philippe, Christopher Harris, Peter Howitt and John Vickers (2001)
"Competition, Imitation and Growth with Step-by-Step Innovation" Review of Economic Studies, 68,
pp. 467-492
.
Aghion, Philippe, Christopher Harris and John Vickers (1997) "Step-by-step
Competition, Innovation and Growth:
An Example" European Economic
Review, 41, pp. 771-
782.
Aghion, Philippe and Peter Howitt (1992) "A Model
of
Growth Through Creative
Destruction" Econometrica, 110, 323-351.
Aliprantis,
sis:
Charalambos D. and Kim C. Border (1999)
New
Hitchhiker's Guide, Springer- Ver lag,
Infinite
Dimensional Analy-
York.
Arrow, Kenneth (1962) ''Economic Welfare and
the Allocation of Resources for In-
vention" in R.R. Nelson, ed. Universities-National Bureau of Economic Research Conference
Series;
The Rate and Direction of Economic
ton University Press,
New
Economic and
Social Factors, Prince-
York.
Baye, Michael R., and Heidrun C.
seeking, innovation,
Activities:
Hoppe
(2003) "The strategic equivalence of rent-
and patent-race games," Games and Economic Behavior,
44, 217-226.
Boldrin, Michele and David K. Levine (2001) "Perfectly Competitive Innovation,"
Univesity of Minnesota and
UCLA
mimeo.
Boldrin, Michele and David K. Levine (2004) "Innovation and the Size
ket," Univesity of
Minnesota and
UCLA
of the
mimeo.
Budd, Christopher, Christopher Harris and John Vickers (1993) "A Model
Evolution of Duopoly: Does the
Review of Economic Studies,
Asymmetry Between Firms Tend
and
Product
of the
to Increase or Decrease?"
60, 540-573.
Bureau of Economic Analysis (2004)
come
Mar-
Accounts"
Bureau
of
"Annual Revision of the National
Economic
Analysis
News.
BEA
In-
04-31.
www.bea.gov/bea/rels.htm
Choi, Jay P. (1991) "Dynamic
Rand Journal
R&D
Competition under Hazard Rate Uncertainty,"
of Economics, 22, 596-610.
Fudenberg, Drew, Richard Gilbert, Joseph
42
Stiglitz
and Jean Tirole (1983)
"Pre-
emption, leapfrogging and competition in patent races," European Economic Review 22, 3-31.
Fullerton,
Robert and Preston McAfee (1999) "Auctioning
entry into tournaments,"
Journal of Political Economy, 107, 573-605.
Gallini,
Nancy T.
(1992) "Patent Policy and Costly Imitation"
RAND
Journal of
Economics, 23, 52-63.
Gallini,
it
Nancy
T. and Suzanne Scotchmer (2002) "Intellectual Property;
the Best Incentive Mechanism?" Innovation Policy and the Economy,
Gilbert, Richard,
2,
When
is
51-78.
and Carl Shapiro (1990) "Optimal Patent Length and
Breadth,"
Journal of Industrial Economics, 21(1), 106-112.
Green, Jerry and Suzanne Scotchmer (1995) "On the Division
of Profit in Sequential
Innovation," Journal of Industrial Economics, 26(1), 20-33.
Grossman, Gene and Elhanan Helpman (1990) Innovation and Growth
Economy, Cambridge, MA,
Hall,
Is
Bronwyn
There a Lag?"
MIT
H., Jerry
International
Harris, Christopher
in the Global
Press.
Hausman and
Zvi Griliches (1988) "Patents and
Economic Review,
R&D:
27, pp. 265-83.
and John Vickers (1985)
"Perfect Equilibrium in a
Model
of
Race" Review of Economic Studies, 52, 193-209.
Harris, Christopher and
Economic
John Vickers (1987)
"Racing with Uncertainty" Review of
Studies, 54, 1-21.
Hopenhayn, Hugo and Matthew Mitchell (2001)
Breadth"
RAND
Journal of Economics, 32, 152-166.
Judd, Kenneth (1999) Numerical Methods
Kelley,
"Innovation, Variety and Patent
John
in Economics,
MIT
L. (1955) General Topology, Springer- Verlag,
Klemperer, Paul (1990) "How Broad Should the Scope
New
Press, Cambridge.
York.
of Patent Protection Be?" Jour-
nal of Industrial Economics, 21, 113-130.
Kortum, Samuel
(1993). "Equilibrium
American Economic Review,
Llobet, Gerard,
Sequential Innovators:
R&D
and the Patent-R&D Ratio: US Evidence"
83, pp. 450-457.
Hugo Hopenhayn and Matthew
Prizes, Patents
Mitchell (2001) "Rewarding
and Buyouts," Federal Reserve Bank of Minneapolis
Research Department Staff Report 273, mimeo.
Moscarini, Giuseppe and Francesco Squintani (2004) "Competitive Experimentation with Private Information," Cowles Foundation Discussion Paper, No. 1489.
National Science Foundation (2006)
"Industrial
R&D
Employment
in the
United
States and in U.S. Multinational Corporations" Division of Science Resource Statistics.
43
NSF
05-302. Arlington,
VA: National Science Foundation, www.bsf.gov/sbe/srs/stats.htm
O'Donoghue, Ted, Suzanne Scotchmer and Jacques Frangois Thisse (1998)
"Patent Breadth, Patent Life and the Pace of Technological Progress" Journal of Economics
and Management
Strategy,
7,
1-32.
O'Donoghue, Ted and Joseph Zweimuller (2004)
Growth" Journal of Economic Growth,
9,
"Patents in a Model of Endogenous
81-123.
Pakes, Ariel and Zvi Griliches (1980) "Patents and
Look." Economic Letters,
at the
Firm
Level:
A
First
pp. 377-381.
5,
Quah, Danny (2004)
R&D
"24/7 Competitive Innovation,"
Reinganum, Jennifer (1981) "Dynamic Games
LSE Mimeo.
of Innovation," Journal of
Economic
Theory, 25, 21-24.
Reinganum, Jennifer (1985)
"Innovation and Industry Evolution," Quarterly Journal
of Economics, 100, 81-100.
Romer, Paul M. (1990) "Endogenous
omy,
Technological Change" Journal of Political Econ-
98, S71-S102.
Ross, Sheldon (1996) Stochastic Processes, John Wiley
Scotchmer, Suzanne (1999) "On the Optimality
&
Sons, Inc.
of the Patent System,"
The Rand
Journal of Economics, 30, 181-196.
Scotchmer, Suzanne (2005) Innovations and
Incentives,
MIT Press,
Cambridge, Massa-
chusetts.
Segal, Ilya
tional
Bureau
of
and Michael Whinston (2005)
Economic Research Working Paper.
Taylor, Curtis (1995) "Digging
American Economic Review,
Tirole,
"Anti-Trust in Innovative Industries," Na-
golden carrots: an analysis of research tournaments,"
for
85, 872-890.
Jean (1988) The Theory
of Industrial Organization,
MIT
Press, Cambridge,
Massachusetts.
Stokey,
Nancy (1995) "R&D and Economic Growth" Review
of
Economic
Studies, 62,
pp. 469-489.
Stokey,
Nancy and Robert Lucas with
Economic Dynamics, Harvard University
Press.
44
E. Prescott (1989) Recursive Methods in
Table
Full
IPR
1.
Full
Protection
Benchmark Results
IPR
Optimal
Uniform
Optimal
State
State
IPR
Dependent
Dependent
Protection
without
with
licensing
licensing
Optimal
with
without
with
licensing
licensing
licensing
1.05
1.05
0.35
0.35
A
1.05
1.05
1.05
7
0.35
0.35
0.35
Vi
0.054
V2
0.005
Vz
Vi
V5
CO
3.52
CO
C2
CO
3.52
CO
0.82
C3
CO
3.52
CO
1.94
C4
CO
3.52
CO
1.95
Cs
CO
3.52
CO
1.95
-
2.70
3.52
1.71
1.59
1.96
zil
x
0.018
0.021
0.023
0.010
0.026
0.029
0.033
0.023
0.022
0.024
Mo
0.238
Ml
0.326
0.481
0.506
0.195
0.446
M2
0.189
0.253
0.253
0.135
0.232
W*
0.952
0.931
0.936
0.961
0.937
0.032
0.026
0.027
0.028
0.039
0.0186
0.0258
0.0263
0.0203
0.0296
V\
'
Ci
Do
0.440
Researcher
ratio
9*
Note: This table gives the results of the benchmark numerical computations with p
7
=
0.35 under
five different
the difference in the values V\
R&D
IPR
—
policy regimes.
vq\ the
rate of neck-and-neck competitors,
fraction of industries at a technology
force working in research;
state-dependent
IPR
xj$;
gap of n
and the growth
policies
R&D
It
=
0.05,
A
=
1.05,
reports the steady-state equilibrium values of
rate of a follower that
is
one step behind,
X*_±; the
fraction of industries in neck-and-neck competition,
—
1;
the value of "labor share,"
rate, g*. It also reports the
with or without licensing. See text for
45
U)*;
^ij-J;
the ratio of the labor
growth-maximizing uniform and
details.
Table
Full
IPR
Optimal
State
State
IPR
Dependent
Dependent
with
without
with
licensing
licensing
licensing
Protection
without
with
licensing
licensing
1.01
Optimal
Uniform
IPR
Full
Protection
A=
2.
Optimal
A
1.01
1.01
1.01
1.01
1.01
7
0.35
0.35
0.35
0.35
0.35
m
0.272
%
0.017
V3
V4
%
Ci
oo
0.19
C2
CO
0.19
CO
0.04
C3
CO
0.19
CO
0.10
C4
CO
0.19
CO
0.10
Cs
CO
0.19
CO
0.10
-v
*U
CO
0.14
0.19
0.09
0.08
0.10
0.090
0.106
0.107
0.053
0.126
x
0.139
0.162
0.107
0.103
0.115
Mo
0.245
Mi
0.330
0.496
M2
0.186
CJ*
0.991
v1
0.452
0.501
0.191
0.455
0.251
0.251
0.137
0.232
0.987
0.987
0.993
0.988
0.008
0.007
0.007
0.007
0.010
0.0186
0.0257
0.0257
0.0203
0.0293
Researcher
ratio
5*
Note: This table gives the results of the benchmark numerical computations with p
7
=
0.35 under
five different
the difference in the values V\
R&D
IPR
—
policy regimes.
vq; the
R&D
It
=
0.05,
A
—
1.01,
reports the steady-state equilibrium values of
rate of a follower that
is
one step behind,
x*_^\ the
rate of neck-and-neck competitors, Xq\ fraction of industries in neck-and-neck competition,
fraction of industries at a technology
force working in research;
state-dependent
IPR
gap of
and the growth
policies
n
=
1;
rate, g*
.
the value of "labor share,"
It also
the ratio of the labor
reports the growth-maximizing uniform and
with or without licensing. See text for
46
ui*\
//j-j;
details.
Table
Full
IPR
3.
A=
IPR
Full
Optimal
Uniform
Optimal
State
State
IPR
Dependent
Dependent
with
without
with
Protection
Protection
1.2
Optimal
without
with
licensing
licensing
licensing
licensing
licensing
A
1.20
1.20
1.20
1.20
1.20
7
0.35
0.35
0.35
0.35
0.35
Vi
0.014
V2
0.002
m
0.001
V4
V5
Ci
CO
24.85
CO
C2
oo
24.85
CO
5.69
C3
CO
24.85
CO
12.25
C4
CO
24.85
CO
14.75
C5
oo
24.85
CO
14.98
-V
xU
20.11
24.85
13.69
12.40
14.99
0.005
0.006
0.007
0.003
0.008
*
0.008
0.010
0.007
0.006
0.007
Mo
0.221
Ml
0.318
0.452
0.523
0.206
0.445
fJ-*2
0.199
0.260
0.261
0.142
0.236
U*
0.806
0.743
0.783
0.847
0.760
0.068
0.056
0.070
0.058
0.090
0.0186
0.0261
0.0284
0.0200
0.0306
Vi
x
0.418
Researcher
ratio
9*
Note: This table gives the results of the benchmark numerical computations with p
7
=
0.35 under five different
the difference in the values v\
R&D
IPR
—
policy regimes.
vq; the
R&D
It
—
0.05,
A
—
1.2,
reports the steady-state equilibrium values of
rate of a follower that
is
one step behind,
x*_i\ the
rate of neck-and-neck competitors, Xq\ fraction of industries in neck-and-neck competition, /ig;
fraction of industries at a technology
force working in research;
state-dependent
IPR
gap of
and the growth
policies
n
=
1;
rate, g*
.
the value of "labor share,"
It also
the ratio of the labor
reports the growth-maximizing uniform and
with or without licensing. See text for
47
uj*;
details.
Table
Full
IPR
Full
4.
IPR
Protection
Protection
without
with
licensing
licensing
7=
0.1
Optimal
Uniform
Optimal
State
State
IPR
Dependent
Dependent
with
without
with
licensing
licensing
licensing
Optimal
A
1.05
1.05
1.05
1.05
1.05
7
0.1
0.1
0.1
0.1
0.1
0.291
^71
V2
V3
V4
V5
V\
Cl
00
3.11
00
C2
00
3.11
00
0.77
C3
00
3.11
00
1.90
C4
00
3.11
00
1.90
C5
00
3.11
00
1.90
2.20
3.11
1.64
0.48
1.90
.
-Vq
0.022
0.023
0.024
0.016
0.025
x
0.024
0.026
0.024
0.021
0.024
Mo
0.314
Ml
0.333
0.497
0.501
0.096
0.492
M2
0.172
0.251
0.251
0.058
0.248
W*
0.944
0.916
0.917
0.981
0.917
0.008
0.007
0.008
0.003
0.010
0.0186
0.0278
0.0280
0.0220
0.0286
»i'i
0.777
Researcher
ratio
9*
Note: This table gives the results of the benchmark numerical computations with p
7
=
0.1 under five different
difference in the values v\
IPR
—
—
0.05,
force working in research;
state-dependent
IPR
vq; the
R&D
gap of
n =
and the growth
policies
=
1.05,
policy regimes. It reports the steady-state equilibrium values of the
rate of a follower that
is
one step behind,
x*_^\ the
rate of neck-and-neck competitors, Xq\ fraction of industries in neck-and-neck competition,
of industries at a technology
A
1;
the value of "labor share,"
rate, g*. It also reports the
with or without licensing. See text for
48
to*;
/1q;
R&D
fraction
the ratio of the labor
growth-maximizing uniform and
details..
Table
Full
IPR
Full
Protection
7=
5.
IPR
Protection
without
with
licensing
licensing
0.6
Optimal
Uniform
Optimal
State
State
IPR
Dependent
Dependent
Optimal
with
without
with
licensing
licensing
licensing
A
1.05
1.05
1.05
1.05
1.05
7
0.6
0.6
0.6
0.6
0.6
Vi
0.070
0.007
V2
0.019
Vz
0.007
V4
0.003
V5
Ci
00
5.24
5.24
00
C2
00
5.24
5.24
00
Cs
00
5.24
5.24
00
1.69
C4
00
5.24
5.24
00
2.13
Cs
00
5.24
5.24
00
2.35
-
0.59
4.38
5.24
5.24
2.25
s-1
0.011
0.015
0.015
0.002
0.027
x*
0.053
0.070
0.070
0.020
0.021
Mo
0.092
0.327
0.037
Mi
0.257
0.413
0.413
0.094
0.242
M2
0.193
0.252
0.252
0.074
0.128
w*
0.942
0.937
0.937
0.9320
0.9307
0.073
0.044
0.044
0.087
0.010
0.0186
0.0198
0.0198
0.0230
0.0303
Vl
v
.
2.35
Researcher
ratio
5*
Note: This table gives the results of the benchmark numerical computations with p
7
—
0.6 under five different
difference in the values V\
—
IPR
=
0.05,
vq\ the
of industries at a technology gap of
state-dependent
IPR
R&D
n
—
and the growth
policies
=
1.05,
policy regimes. It reports the steady-state equilibrium values of the
rate of a follower that
is
one step behind,
x*_^\ the
rate of neck-and-neck competitors, Xq\ fraction of industries in neck-and-neck competition,
force working in research;
A
1;
the value of "labor share,"
rate, g*
.
It also
fraction
the ratio of the labor
reports the growth-maximizing uniform and
with or without licensing. See text for
49
oj*;
/j,q\
R&D
details.
R&D
Efforts
Optimal Uniform IPR
Optimal State-Dependent IPR
Technology Gap
Figure
5.
R&D
Efforts.
Industry Shares
0.5
Optimal Uniform IPR
Optimal State-Dependent IPR
Technology Gap
Figure
600k
6.
Industry Shares.
52
36