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Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
Competing Engines of Growth:
Innovation and Standardization
Daron Acemoglu
Gino Gancia
Fabrizio Zilibotti
Working Paper 0-7
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April 23,
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Social Science Research
Competing Engines of Growth: Innovation and
Standardization*
Daron Acemoglu
Gino Gancia
Fabrizio Zilibotti
MIT
CREi and UPF
University of Zurich
April 2010
Abstract
We
study a dynamic general equilibrium model where innovation takes the
form of the introduction new goods, whose production requires
followed by a costly process of standardization, whereby these
Innovation
is
new goods
are adapted to be produced using unskilled labor.
highlights a
of growth
skilled workers.
number
of novel results. First, standardization
and a potential
barrier to
it.
As a
result,
Our framework
is
growth
both an engine
in
an inverse U-
shaped function of the standardization rate (and of competition). Second, we
characterize the growth and welfare maximizing speed of standardization.
show how optimal IPR
policies affecting the cost of standardization vary
We
with
the skill-endowment, the elasticity of substitution between goods and other
parameters. Third,
dardization
of our
may
model
we show
lead to multiple equilibria. Finally,
for the skill-premium
North-South trade to
JEL
that the interplay between innovation and stan-
the implications
illustrate novel reasons for linking
intellectual property rights protection.
classification: F43,
Keywords:
and we
we study
031, 033, 034.
growth, technology adoption, competition policy, intellectual
property rights.
*We thank seminar participants at the SED Annual Meeting (Boston, 2008), Bank of Italy,
CERGE-EI, University of Alicante and the REDg-Dynamic General Equilibrium
the Kiel Institute,
Macroeconomics Workshop (Madrid, 2008) for comments. Gino Gancia acknowledges financial support from the Barcelona GSE, the Government of Catalonia and the ERC Grant GOPG 240989.
Fabrizio Zilibotti acknowledges financial support from the ERC Advanced Grant IPCDP-229883.
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/competingenginesOOacem
1
The
new
diffusion of
technologies
and process innovations.
are often
in the
New
complex and require
economy
Introduction
often coupled with standardization of product
is
technologies,
when
first
conceived and implemented,
skilled personnel to operate.
At
this stage, their
limited both by the patents of the innovator and the skills that
is
these technologies require. Their widespread adoption and use
tasks involved in these
new
technologies to
first
necessitates the
become more routine and standardized,
How-
ultimately enabling their cheaper production using lower-cost unskilled labor.
ever,
such standardization not only expands output but also implies that the rents
accruing to innovators will
is
use
come
to an end. Therefore, the process of standardization
both an engine of economic growth and a potential discouragement to innovation.
In this paper,
The
we study
this interplay
between innovation and standardization.
history of computing illustrates the salient patterns of this interplay.
The
use of silicon chips combined with binary operations were the big breakthroughs,
starting the
ICT
revolution. During the
first
30 years of their existence, computers
could only be used and produced by highly skilled workers. Only a lengthy process
made computers and
of standardization
silicon chips
more widely
available
and more
systematically integrated into the production processes, to such a degree that today
computers and computer-assisted technologies are used
with workers of very different
skill levels.
at every stage of production
At the same time that the
simplification of
manufacturing processes allowed mass production of electronic devices and low
competition
among ICT
and then more broadly
In our model,
duced only by
firms intensified enormously,
skilled workers.
are invented via costly
This innovation process
is
R&D
first
be pro-
whereby the previously new goods are adapted to be produced
standardization will be undertaken by newcomers, which
By
and can
followed by a costly process
using unskilled labor. 1 Free entry into standardization makes
bent producers.
leaders
at a global scale.
new products
of standardization,
among few industry
first
prices,,
shifting
alleviates the pressure
some technologies
on scarce
This view has a clear antecedent
it
may
a competing process;
then displace incum-
to low-skill workers, standardization
high-skill workers, thereby raising aggregate
in
demand
Nelson and Phelps (1966), which we discuss further below.
See also Autor, Levy and Murnane (2003) on the comparative advantage of unskilled workers in
routine, or in our language "standardized," tasks. We can also interpret innovation as product
innovation and standardization as process innovation.
innovation
(e.g.,
Cohen and Klepper, 1996)
in process innovation
is
are smaller
Evidence that firms engaging in product
skill intensive than firms engaging
and more
consistent with our assumptions.
and fostering incentives
for further innovation. Yet, the anticipation of
also reduces the potential profits
from new products, discouraging innovation. This
implies that while standardization
an engine of economic growth,
slowing
Our
down
it
—and the technology adoption that
it
brings
—
is
can also act as a barrier to growth by potentially
innovation.
model
baseline framework provides a simple
Under some
standardization
we
relatively mild assumptions,
ance growth path that
is
for the analysis of this interplay.
establish the existence of a unique bal-
saddle-path stable.
We
show that equilibrium growth
is
an inverse U-shaped function of the "extent of competition" captured by the cost of
When
standardization.
cause
new products use
production and
growth
skilled
profitability.
workers
On
is
very costly, growth
for a long while
the other hand,
and
relatively slow be-
is
this reduces their scale of
when standardization
is
very cheap,
again relatively slow, this time because innovators enjoy ex post profits only
is
This inverse U-shaped relationship between competition and growth
for a short while.
is
standardization
consistent with the empirical findings in
Aghion
the theoretical channel highlighted in Aghion et
al.
et al.
(2005),
and complements
(2001, 2005), which
is
driven by
the interplay of their "escape competition" mechanism and the standard effects of
monopoly
profits
on innovation.
In our model, the laissez-faire equilibrium
in
many models
of
is
endogenous technology, there
inefficient for
two reasons.
First, as
an appropriability problem: both
is
innovating and standardizing firms are able to appropriate only a fraction of the gain
in
consumer surplus created by
low.
Second, there
is
their investment
and
makes the growth
this
a new form of "business stealing"
effect,
standardization decisions reduce the rents of innovators.
laissez-faire equilibrium
levels of
is
inefficient
and that growth
The
whereby the costly
possibility that the
maximized by intermediate
competition implies that welfare and growth maximizing policies are not
necessarily those that provide
to innovators.
maximal
intellectual property rights (IPR) protection
Under the assumption that a government can
the cost of standardization by regulating
and welfare maximizing combinations
most
is
2
rate too
of
of the literature, the optimal policy
static cost of
IPR
IPR and
is
we
competition
markups and
characterize growth
policies.
Contrary to
not the result of a trade-off between the
monopoly power and dynamic
2
protection,
affect
gains.
Rather, in our model an excess
Another form of business stealing, studied extensively in Schumpeterian models of vertical in(e.g., Aghion and Howitt 1992), is when a monopoly is destroyed by new firms introducing
a "better" version of an existing products. We suggest that standardization is also an important
novation
source of business stealing.
may harm growth by
of property right protection
increasing the overload on skilled
workers, which are in short supply.
When
IPR
the discount rate
policy involves lower protection
when markups
labor supply
fact that
for
is
when
more
profitable
tion.
We
also
we
small,
is
new products
find that
when
are higher
R&D
growth and welfare maximizing
new products)
costs (for
and when the
are lower,
ratio of skilled to unskilled
greater.
The
there
a large supply of unskilled labor, standardization becomes
is
comparative static result
latter
is
a consequence of the
and thus innovators require greater protection against standardiza-
show that when competition policy
as well as
IPR
policy can be used,
the optimal combination of policies involves no limits on monopoly pricing for
products, increased competition for standardized products and lower
than otherwise.
this
may
Intuitively, lower
IPR
IPR
new
protection
protection minimizes wasteful entry costs, but
lead to excessive standardization and
weak
incentives to innovate.
To max-
imize growth or welfare, this latter effect needs to be counteracted by lower markups
for standardized products.
countries
may
We
also
innovate and standardize. However,
IPR
policy,
Finally,
tions
it
show that trade
if
increased trade openness
coupled by optimal
we show that under
different
parameter configurations or different assump-
on competition between innovators and standardizers, a new type of multiplicity
economy
When too much of the
resources of the
are devoted to standardization, expected returns from innovation are lower
this limits innovative activity.
rates
is
always increases welfare and growth.
of equilibria (of balanced growth paths) arises.
and
liberalization in less-developed
create negative effects on growth by changing the relative incentives to
Expectation of lower innovation reduces interest
and encourages further standardization.
Consequently, there exist equilibria
with different levels (paths) of innovation and standardization.
this multiplicity does not rely
and emphasized
whereby the
with
expectations.
initial
Our paper
is
noteworthy that
on technological complementarities (previously studied
in the literature),
equilibria,"
It is
and has much more of the
change
flavor of "self-fulfilling
in order to
support equilibria consistent
related to several different literatures.
In addition to the endoge-
relative prices
nous growth and innovation literatures
(e.g.,
Aghion and Howitt, 1992, Grossman
and Helpman, 1991, Romer, 1990, Segerstrom, Anant and Dinopoulos, 1990, Stokey,
1991), there are
now
several
ogy adoption. These can be
complementary frameworks
classified into three groups.
for
the analysis of technol-
The
first
includes models
based on Nelson and Phelps's (1966) important approach, with slow diffusion of technologies across countries (and across firms), often related to the
human
capital of
the workers employed by the technology adopting firms.
This framework
porated into different types of endogenous growth models, for example,
(2000),
Acemoglu, Aghion and
among
in
incor-
Howitt
and Acemoglu (2009, Chapter
more microeconomic foundations
Several papers provide
include,
Zilibotti (2006),
is
for
slow diffusion.
18).
These
and Lach (1989), Jovanovic and Nyarko (1996), Jo-
others, Jovanovic
vanovic (2009) and Galor and Tsiddon (1997), which model either the role of learning
or
human
capital in the diffusion of technologies.
The second group
includes pa-
pers emphasizing barriers to technology adoption. Parente and Prescott (1994)
is
a
well-known example. Acemoglu (2005) discusses the political economy foundations of
why some
The
final
societies
may
choose to erect entry barriers against technology adoption.
group includes models
in
which diffusion of technology
is
slowed
down
or
prevented because of the inappropriateness of technologies invented in one part of
the world to other countries (see,
Stiglitz, 1969,
e.g.,
Acemoglu and
Basu and Weil, 1998 and David, 1975). Gancia and
propose a unified framework
for
Our paper
is
to, all
is
different from,
three groups of papers.
also related to
Krugman's (1979) model
technology diffusion, whereby the South adopts
in turn,
Zilibotti (2009)
studying technology diffusion in models of endoge-
nous technical change. Our approach emphasizing standardization
though complementary
Atkinson and
Zilibotti, 2001,
of North-South trade
new products with
a delay.
was inspired by Vernon's (1966) model of the product cycle and
and
Krugman,
his
approach
has been further extended by Grossman and Helpmarf (1991) and Helpman (1993). 3
Our approach
make
differs
from
all
different use of skilled
these models because innovation
and unskilled workers and because we focus on a closed
economy general equilibrium setup rather than the
ically
advanced and backward countries
our alternative set of assumptions
is
and standardization
is
as in these papers.
that, differently
an inverse-U function of standardization.
paper characterizes the optimal
IPR
interactions between technolog-
policy
A new
implication of
from previous models, growth
More importantly, none
and how
it
above
varies with skill abundance.
Grossman and Lai (2004) and Boldrin and Levine (2005) study the
governments have to protect intellectual property
of the
in a trading
incentives that
economy. Their frame-
work, however, abstracts from the technology adoption choice and from the role of
skill
which are central to our analysis.
Finally, our
emphasis on the role of
and unskilled workers
skilled workers in the
in the production of standardized
production of new goods
goods makes our paper also
3
Similar themes are also explored in Bonfiglioli and Gancia (2008), Antras (2005), Dinopoulos
and Segerstrom (2007, 2009), Lai (1998), Yang and Maskus (2001).
and wage
related to the literature on technological change
Acemoglu
others,
Moav
Galor and
and Violante (2002),
(1998, 2003), Aghion, Howitt
(2000),
Greenwood and Yorukoglu
Rios-Rull and Violante (2000).
The approach
inequality; see,
in
and
(1997),
Galor and
predictions for
wage inequality similar to
Caselli (1999),
Krusell, Ohanian,
Moav
larly related, since their notion of ability-biased technological
among
(2000)
is
particu-
change also generates
though the economic mechanism and
ours,
other implications are very different.
The
paper
rest of the
is
organized as follows. Section 2 builds a dynamic model of
endogenous growth through innovation and standardization.
It
provides conditions for
the existence, uniqueness and stability of a dynamic equilibrium with balanced growth
and derives an inverse-U relationship between the competition from standardized
products and growth. Section 3 presents the welfare analysis. After studying the
best allocation,
it
characterizes growth
policies as function of parameters.
first
and welfare maximizing IPR and competition
As an application
of these results,
we
discuss
how
trade liberalization in less developed countries affects innovation, standardization
and the optimal
policies.
Section 4 shows
how
a modified version of the model
may
generate multiple equilibria and poverty traps. Section 5 concludes.
2
2.1
A Model of Growth through Innovation and Standardization
Preferences
The economy
C
sumption
t
of agents:
is
populated by infinitely-lived households who derive
and supply labor
inelastically.
Households are composed by two types
high-skill workers, with aggregate supply
aggregate supply L.
The
H, and
utility function of the representative
U=
/
e~
pt
from con-
utility
log
low-skill workers,
household
with
is:
C dt,
t
Jo
where p >
is
the discount rate.
plan to maximize
utility,
The
representative household sets a consumption
subject to an intertemporal budget constraint and a No-
Ponzi game condition. The consumption plan
^=r
where r
t
is
t
satisfies
the standard Euler equation:
-p,
the interest rate. Time-indexes are henceforth omitted
no confusion.
(1)
when
this causes
Technology and Market Structure
2.2
Aggregate output, Y,
in the
economy. As
is
a
CES
Romer
in
function denned over a measure
(1990), the
measure of goods
A
A
of goods available
captures the level of
technological knowledge that grows endogenously through innovation. However,
assume
upon introduction, new goods involve complex technologies that can
that,
only be operated by skilled workers.
production process
is
simplified
remain unaltered so that
is
defined
ized)
Ah
and the good can then be produced by unskilled
all
/ fA
\
$=i
ZU V
di
/ [ Al
«=*
= z (A
)
the measure of hi-tech goods,
is
goods and
The term Z = A
A = AH + AL
c
~l
is
.
>
e
1 is
«=i
x L,i
Al
is
l'
di
To
productivity,
Yj
(2),
A»
jo
final
is
the relative
,
(2)
the elasticity of substitution between goods.
is
linear in tech-
good production function consistent with balanced
R&D
in
see why, note that with this formulation
(Ax),
\ «^
«=i
xfr dij
the measure of low-tech (standard-
growth (without introducing additional externalities
see below).
+
a normalizing factor that ensures that output
nology and thus makes the
From
output symmetrically. Thus,
varieties contribute to final
as:
Y=
where
After a costly process of standardization, the
Despite this change in the production process, good characteristics
workers too.
Y
we
equal to
demand
A — as
for
AK
in
models.
any two goods
/
\
technology as we will
when
X{
—
x,
aggregate
4
i,j
E
A
is:
~ 1/e
'"
'
We
choose
one unit of
Y
Y
»
(3)
.
to be the numeraire, implying that the
minimum
cost of purchasing
must be equal to one:
Each hi-tech good
one unit of
*
is
produced by a monopolist with a technology that requires
skilled labor per unit of output.
Each low-tech good
is
produced by a
4
The canonical endogenous growth models that do not feature the Z term and allow for a ^ 2
Grossman and Helpman, 1991) ensure balanced growth by imposing an externality in the
innovation possibilities frontier (R&D technology). Having the externality in the production good
(e.g.,
function instead of the
R&D
technology
is
no
less general
and
simplifies our analysis.
monopolist with a technology that requires one unit of labor per unit of output. Thus,
the marginal cost
is
wH
equal to the wage of skilled workers,
for hi-tech firms
,
and
the wage of unskilled workers, wl, for low-tech firms. Since high-skill worker can be
employed by both high- and low-tech
When
firms,
then
wh >
wl-
standardization occurs, there are two potential producers (a high- and a
The competition between
low-tech one) for the same variety.
described by a sequential entry-and-exit game. In stage
and produce a standardized version of the intermediate
(i)
i.e.,
when a
low-tech firm becomes a monopolist.
compete d
Bertand (stage
la
market
hi-tech firm leaves the
(hi)).
If
We
(ii)
Then,
variety.
Exit
in stage
(ii),
assumed to be
is
cannot go back to
it
and the
the incumbent does not exit, the two firms
assume that
produce (and thus pay the cost of producing) at
this assumption, stage
it
is
a low-tech firm can enter
the incumbent decides whether to exit or fight the entrant.
irreversible,
these producers
firms entering stage
all
least £
>
(iii)
must
units of output (without
would be vacuous, as incumbents would have a "weakly
dominant" strategy of staying
in
and producing x
—
in stage
(iii)).
Regardless of the behavior of other producers or other prices in this economy, a
subgame-perfect equilibrium of this game must have the following features: standardization in sector j will
Wh >
wl-
If
be followed by the
exit of the high-skill
the incumbent did not exit, competition in stage
of the market being captured
by the low-tech firm due to
incumbent would make a
on the
as long as the skill
loss
premium
is
£
>
units that
positive, firms
its
it is
incumbent whenever
(iii)
would
result in all
cost advantage
forced to produce. Thus,
contemplating standardization can
nore any competition from incumbents. However,
if
wH < w L
incumbents would
entrants and can dominate the market. Anticipating this, standardization
itable in this case
is
and
will
not take place. Finally, in the case where
a potential multiplicity of equilibria, where the incumbent
fighting
and
exiting. In
what
follows,
tie-breaking rule that in this case the
in Section 4).
We
we
is
fights
ig-
fight
not prof-
is
wh —
wl, there
indifferent
will ignore this multiplicity
incumbent
summarize the main
and the
between
and adopt the
(we modify this assumption
results of this discussion in the following
Proposition.
Proposition
1 In
scribed above, there
any subgame-perfect equilibrium of the entry-and-exit game deis
only one active producer in equilibrium.
Whenever
all hi-tech
firms facing the entry of a low-tech competitor exit the market.
w h < wl
hi-tech
incumbents would
fight entry,
and no standardization
wh > wl
Whenever
occurs.
=
we
In the rest of the paper,
equilibrium there
markup
focus on the limit
=
wH
1
(
and
pL
down the
labor market clearing pin
=
and x #
-7-Ax,
H
recall that
is
the
number
fraction of revenues:
^
PhH
nH =
this point,
it
is
is,
endowment
of skilled workers.
is
wL
1
j
>
0.
5
Since in
(5)
.
scale of production of each firm:
= T~>
Ah
(
6)
employed by hi-tech firms and
pricing implies that profits are a constant
.
and
*l
=
PlL
7al
(7)
n
useful to define the following variables:
H/L. That
n
(
of skilled workers
Markup
the remaining labor force.
At
=
J
xl
is
—
over the marginal cost:
Symmetry and
L
as £
only one active producer, the price of each good will always be a
is
Ph
where
economy
= An/ A
the fraction of hi-tech goods over the total and h
Then, using demand
(3)
and
(6),,
is
and
ft.
=
the relative
we can
solve for
relative prices as:
^v
\xlJ
ȣ
1/e
= ( h^-^y
(
Pl
\
n
1A
(8)
J
and
fc^
n
(9)
.
)
Intuitively, the skill
skill (h
= H/L)
skilled workers.
premium wh/w l depends
and positively on the
Note that
wh = wl
relative
For simplicity, we restrict attention to
of Proposition
always remain in the interval
this interval, over
5
which
1,
number
if
s
demanding
h+1
initial states of
we
start
technology such that
from n > n mm
n € [n mm ,l]. We can
skilled
of hi-tech firms
at:
n min
As an implication
negatively on the relative supply of
,
n > n min
.
the equilibrium will
therefore restrict attention to
workers never seek employment in low-tech firms.
The focus on the limit economy is for simplicity.
and assume away stage (ii). Although conceptually
We could
alternative
similar, this case
is
model the game
less tractable.
differently,
Using
(7)
and
(8) yields relative profits:
^
1-1/6
l_-n
/
(10)
This equation shows that the relative profitability of hi-tech firms, tth/^l,
ing in the relative supply of
decreasing in the relative
effect
is
that a larger
H/L, because
skill,
number
number
is
increas-
of a standard market size effect
The reason
of hi-tech firm, Ajj/Al.
and
for the latter
of firms of a given type implies stiffer competition for
labor and a lower equilibrium firm scale.
Next, to solve for the level of profits,
we
first
symmetry
use
1/(6-1)
l-e
_l
P
(l-n) (
-Ph
A
+n
1
A
and
;n)
and
(12)
1/(6-1)
l-e
PL
into (4) to obtain:
Ph_
—n+n
Pl
Using these together with
(8) into (7) yields:
i
E-]
H
Tiff
i
e
M
L
KL
I
-
formalizes
Lemma
1
e
>
2.
is
remain constant. Moreover, the following
of the profit functions:
mm
Then, for n € [n
dn
Moreover, ni
:i-n)$=i
1
some important properties
Assume
,
n
that, for a given n, profits per firm
lemma
n<-- 1
i
n
- -
(
e
Note
- -
<
and
,
l]
:
——
9n
>
0.
(13)
a convex function of n.
Proof. See the Appendix.
The condition
e
>
2
is
sufficient
—though not necessary—
for the effect of
tition for labor to be strong enough to guarantee that an increase in the
compe-
number
of
hi-tech (low-tech) firms reduces the absolute profit of hi-tech (low-tech) firms. In the
9
we assume
rest of the paper,
that the restriction on
e in
Lemma
1 is satisfied.
6
Standardized Goods, Production and Profits
2.3
Substituting (6) into
(2),
the equilibrium level of aggregate output can be expressed
as:
Y = A (l-n)<L— +n*H—
showing that output
is
elasticity function of
H and L.
linear in the overall level of technology, A,
dY
dn
From
is
we have
(14),
maximized when the
is
sion:
by
shifting
is
L
(
"
\
(15)
it
= xH
and
— n) —
(1
fraction of hi-tech products
some technologies
a constant-
e
maximized when nj
important in that
is
U-»J
of skilled workers in the population, so that x L
goods. Equation (14)
£-1
l
\n)
and
that
£-1
r
A -W\
e — 1
which implies that aggregate output
production
(14)
,
is
h. Intuitively,
equal to the fraction
prices are equalized across
highlights the value of technology diffu-
to low-skill workers, standardization "alleviates"
the pressure on scarce high-skill workers, thereby raising aggregate demand.
shows that the
effect of
It
also
standardization on production, for given A, disappears as
goods become more substitutable (high
e).
smoothing consumption across goods (xl
=
In the limit as
xh) so that
Y
e
—
>
oo, there
is
no gain to
only depends on aggregate
productivity A.
Finally, to better
it
is
effect of
technology diffusion on innovation,
also useful to express profits as a function of Y.
Ph — A
A
understand the
Using
(2)-(4) to substitute
i-2
<'
(Y/xh)
into (7), profits of a hi-tech firm can be written as:
similar expression holds for
demand, Y. Thus,
ization raises Y,
it
ttl.
Notice that profits are proportional to aggregate
as long as faster technology diffusion (lower n)
also tends to increase profits.
On
through standard-
the contrary, an increase in
6
An elasticity of substitution between products greater than 2 is consistent with most empirical
evidence in this area. See, for example, Broda and Weinstain (2006).
10
n
> n mm
reduces the instantaneous profit rate of hi-tech firms
d%H n
dn
txh
1
dY n
e
dn
e
—
1
Y
<
0.
(17)
Innovation and Standardization
2.4
We model
ization,
both innovation,
i.e.,
i.e.,
the introduction of a
new
We
follow the "lab-equipment" approach
these activities in terms of output, Y. In particular,
hi-tech
hi-tech
and standard-
the process that turns an existing hi-tech product into a low-tech variety,
as costly activities.
new
hi-tech good,
good requires
good
costs
fx
we assume
that introducing a
units of the numeraire, while standardizing an existing
H
units of
\x L
and define the costs of
Y We may
.
think of
fi L
as capturing the technical
cost of simplifying the production process plus any policy induced costs
new
regulations restricting the access to
we
Next,
define
V# and Vl
due to
IPR
technologies.
as the net present discounted value of a firm producing
a hi-tech and a low-tech good, respectively. These are given by the discounted value of
the expected profit stream earned by each type of firm and must satisfy the following
Hamilton- Jacobi-Bellman equations:
rVL =
rVH =
where
m
is
tt
h
+ Vh -
m, a
mVH
hi-tech firm
is
is
,
endogenous and depends on the
These equations say that the instantaneous
from running a firm plus any capital gain or
from lending the market value of the firm
rate
(18)
the arrival rate of standardization, which
intensity of investment in standardization.
profit
+ VL
nL
losses
must be equal to the return
at the risk- free rate, r.
Note
that, at a flow
replaced by a low-tech producer and the value
Vh
is lost.
Free-entry in turn implies that the value of innovation and standardization can
be no greater than their respective
costs:
VH < H
fi
If
its
VH < fiH
cost
(Vl
<
and there
Ml)'
will
tri
and
vl <
Ml-
en the value of innovation (standardization)
be no investment in that
11
activity.
is
lower than
Dynamic Equilibrium
2.5
A
dynamic equilibrium
is
a time path for (C,
maximize the discounted value
by
free entry in innovation
We
will
now show
for
consumption
This
is
equation
is
determined
is
for prices
is
consistent
consistent with household
that a dynamic equilibrium can be represented as a
C
X
first differential
pA such that monopolists
and standardization, the time path
solution to two differential equations. Let us
The
r,
of profits, the evolution of technology
with market clearing and the time path
maximization.
A, n,
Xi,
=A
y
;
first define:
Y
=A>
A
9
=A"
the law of motion of the fraction of hi-tech goods, n.
is
the state variable of the system. Given that hi-tech goods are replaced by a
low-tech goods at the endogenous rate m, the flow of newly standardized products
Al — tuAh- Prom
The second
and the
this
definition
differential equation
—
n
=
(A
—
— Al)/A we
obtain:
h
=
is
the law of motion of x- Differentiating
(1
n) g
is
ran.
(19)
x and using
the consumption Euler equation (1) yields:
Z = r -p-g
(20)
X
Next, to solve for
consumption
g,
we use the aggregate resource
equal to production minus investment in innovation,
is
standardization,
/i
L A^. Noting that
X
A/ A =
(20) gives the following
g and
= y~ Vh9 ~
Substituting for g from this equation
(i.e.,
g
is
(y
—
ii L
two equation dynamical system
=
n
A L /A = mn, we
r
-p
\
n
J
fi
V„
H
—
V
a function of n (see equation (14)).
12
In particular,
/j,
h A, and
in
can thus write:
^L mn
—
x
Note that y
constraint.
mn —
x) I ^h) in ^° (19) an d
in the (n, x) space:
(21)
/W
Finally, r
and
m
can be found
1
n from the Hamilton-Jacobi-Bellman equations.
as functions of
there
is
is
positive imitation
(m >
then free-entry implies
0),
=
constant, Vl must be constant too, Vi
>
(g
two
0),
then
V# =
0.
Likewise,
0.
VL —
there
if
First,
is
fi
L
.
note that,
Given that
fi
if
L
positive innovation
Next, equations (18) can be solved for the interest rate in the
cases:
r
=
ttl
m>
if
(23)
Ml
tth
=
r
m
g
if
>
0.
(24)
Mff
We
summarize these findings
Proposition 2
A dynamic
in the following proposition.
equilibrium
is
characterized by
(i)
autonomous system
the
of differential equations (21)-(22) in the (n,x) space where
y
=
y(n)
r
=
r i(n)
= (l-n)'LV + n «#V
\
(^(n)
= max <
{
if
r>
if
r ^
?/(n)-x
and
are given by (12),
anrf
7T/y
and
(Hi) the transversality condition
2.6
A
ttl (n)
* Lin)
^
1
(n)
n
/%
Ml
(
m = m (n) =
n H (n)
,
(ii)
nH
^ —m
a pair of initial conditions, no and Aq,
lim^oo exp
(
— fQ
r s ds) f
'
Vidi
=
0.
Balanced Growth Path
Balanced Growth Path (BGP)
hence, the
skill
premium and the
is
a dynamic equilibrium such that h
interest rate are at a steady-state level.
= rh = and,
An "interior"
BGP where, in addition, m > and g > 0. Equation (19) implies that
an interior BGP must feature m ss — g (1 — n) jn = (r — p) (1 — n) jn. To find the
associated BGP interest rate, we use the free-entry conditions for standardization and
BGP
is
a
innovation. Using (12), the following equation determines the interest rate consistent
with
m
>
0:
Tl (n)
= Ik
(25)
Ml
i
i
L
-
(l-n)*=*
n
13
BGP
Next, the free-entry condition for hi-tech firms, conditional on the
BGP:
tion rate, determines the interest rate consistent with the
— -m
=
r%(n)
n— + (l-n)p
=
ss
standardiza-
M//
(26)
/Ah
i
i
H
£-1
n*-
(n,r), the
=
BGP
value of
and standardization,
0)
n can be found
BGP, both innovation and
along a
Proposition 3 An
r L (n
BGP
ss
ss
and
)
BGP
interior
rjy (n
interest rate
—
p) (1
AL Y
,
and
—n
C
ss
)
all
/n
Due
r tf(n).
ss
,
grow
respectively. In the space
words,
in other
as their crossing point:
a dynamic equilibrium such that n
is
We
— n ss
where
)
ss
= r™(n s °),
in n.
8
—
ttl
(n
where n^
set of
ss
is
same
at the
The
is
/ pL
)
rate,
ss
g
is
non-monotonic
intuition for the
=
r
ss
—
little.
(first
among
when n
is
This tends to reduce
—
n) jn) and this brings
1
shows the
7
It
0,
BGP
the equilibrium must
is
on the (dashed)
and
A
ttjj
(n)
and
When n
many
down and
+
1)
is
this
aggregate
tasks to perform,
even further.
When n
high as well (since,
n.
r L (n) curve.
recall,
Note
An
that, as long as
interior
BGP
must
can also be verified straightforwardly that the allocation corresponding to this crossing point
satisfies the transversality condition.
8
77,
the return to innovation.
relationship between r
lie
Ah,
Finally,
as follows.
higher than hj (h
m = g (1
>
is
hi-tech firms brings tth
high, but the flow rate of standardization
Figure
=
and convex. Provided that p
increasing and then decreasing) and
non monotonicity
down
m
is
increasing
is
ix
J.
to study the properties of
H
low,
ss
n — n
rate
ss
the
,
ss
p.
are low, because skilled workers have too
while unskilled workers too
Given n ss
(26), respectively.
and the standardization
BGP, we need
lowers the return to innovation. Moreover,
Y
,
(27)
as in (12) [evaluated at
high, competition for skilled workers
productivity and
and
are given by (25)
to the shape of itl, r L (n)
not too high, r^ (n)
concave
ss
r
is
To characterize the
S
m
p.
satisfies
and
is
n)
in the following proposition:
r L (n°°)
is
—
standardization must be equally profitable.'
summarize the preceding discussion
(r
(1
curves r jf (n) and r^ (n) can be interpreted as the (instantaneous) return from
innovation (conditional on h
n
+
s
The
ss
1
n
/% e
formal argument can be found in the proof of Proposition
14
4.
.
0.06
t
05--
0.04-
0.03"
0.020.4
0.3
0.6
0.5
0.7
0,9
1.0
n
Figure
also
lie
on the
of a
BGP, and
Assumption
r#
(solid)
the intersection.
The
This condition
(n) curve.
BGP
< p < H/
:
is
Solid
=
r^
(n),
=
Dashed
r L (n)
BGP
Thus, the interior
value of
standard:
is
it
guarantees that innovation
is
—A__.
Assumption
2 ensures that rff (n
case in which standardization
by
and uniqueness
sufficiently profitable
Assumption 2
H < ^ LTT7
identified
(/%e)
and that the transversality condition
[i
is
interior.
to sustain endogenous growth
;
n
following assumptions guarantee the existence
that such
1
1:
is
mm
>
)
r L (n
min
)
,
is
satisfied.
ruling out the uninteresting
always more profitable than innovation when n
expected to stay constant, and guarantees that the
state the existence and uniqueness of the
BGP
Proposition 4 Suppose Assumptions 1-2
hold.
librium.
15
in
BGP
is
interior
and unique.
is
We
a formal proposition.
Then
there exist a unique
BGP
equi-
Proof. See the Appendix.
Proposition 4 establishes the existence and uniqueness of a
denoted by (n
ss
,x
ss
The next
)-
goal
dynamics
is
BGP.
to this
equilibrium,
and uniqueness
to prove the (local) existence
is
dynamic equilibrium converging
of a
BGP
Unfortunately, the analysis of
complicated by several factors. First, the dynamic system (21)-(22)
highly nonlinear. Second,
it
may
is
exhibit discontinuities in the standardization rate
ss
ss
(and thus in the interest rate) along the equilibrium path. Intuitively, at (n ,x )
there
is
is
both innovation and standardization (otherwise we could not have n
relatively easy to prove that, similar to
models of directed change
(e.g.,
=
0). It
Acemoglu
2002 and Acemoglu and Zilibotti 2001), there exists a dynamic equilibrium converging
to the
BGP
left,
reached.
in
ss
).
This implies that when the economy approaches the
,
BGP we
have
of hi-tech firms
m
>
Since throughout there
0.
must remain constant
at
Vh
is
is
= Affi
Consequently, there must be an exactly offsetting
BGP
or only standardization
BGP from
the standardization rate and the interest rate both jump once the BGP is
ss
there is no standardization, thus, m — 0, while
In particular, when n < n
(when n > n
the
n < n ss )
featuring either only innovation (when
innovation and thus the value
there can be no
jump
reached and the standardization rate, m, jumps.
However,
dynamic
it
jump
in interest rate r
in r
+ m.
when the
9
turns out to be more difficult to prove that there exist no other
equilibria.
In particular,
we must
rule out the existence of equilibrium
trajectories (solutions to (21)-(22)) converging to (n
ss
ss
,x
)
with both innovation and
standardization out of BGP. Numerical analysis suggests that no such trajectory exists
as long as
Assumptions
the condition that
1
and
2 are satisfied. In particular, the
m = tth (n) /f -H~ n L {n) /Ml
J
both innovation and standardization)
we can only prove
this analytically
impose the following parameter
Proposition 5 Suppose
that
(i- e
-i
system (21)-(22) under
under the condition that there
globally unstable around (n
is
ss
,
x
ss
)-
is
However,
under additional conditions. In particular, we must
restriction:
10
Assumptions
1
and 2 and (28)
hold.
Then
there ex-
9
Note that the discontinuous behavior of the standardization rate and interest does not imply
any jump in the asset values, Vh and/or Vl- Rather, the rate of change of these asset values may
jump
10
locally.
This restriction ensures that n mm
strategy (see Appendix). For example,
e
= 3,
n min = h/{l + h)>
=
h/
when
(1
e
+ h) be not too small, which is key in the proof
= 2, it requires n min = hj (1 + h) > 0.28 and when
0.21.
16
ists
>
p
such
particular, if
ss
n to < n
],
that,
n to
is
<
for p
BGP
BGP
the interior
p,
in the neighborhood of
its
is
value,
n
t
[t
,
ss
,
and n to > n
BGP
then there exists a unique path converging to the
and h T <
> t we have r &
t\, m T > 0, g T =
(i.e.,
and h T > 0], and the economy attains the BGP at
ss
ss
ss
m T — m and g T = g ).
nT = n
finite
In
locally saddle-path stable.
,
t
[resp.,
such that for some
[resp.,
for
ss
all
mT =
>
r
t,
0,
gT
>
we have
,
,
Proof. See the Appendix.
Growth and Standardization: an Inverse-U Relationship
2.7
How does
the cost of standardization,
this question
is
policies such as
is
,
affect the
BGP growth rate,
ss
g
?
Answering
important from both a normative and a positive perspective. First,
IPR
protection are likely to have an impact on the profitability of
Therefore, knowing the relationship between standardization
and
a key step for policy evaluation. Second, the difficulty to standardize
may
standardization.
growth
pL
vary across technologies and over time.
The
cost of standardization affects
amounts to
cost of standardization
the intersection,
form n = n mm
rate depends in turn
but not
(n),
rr,
Thus, increasing the
shifting the tl (n) curve in Figure 1
(low
to
pL )
—
n
>
1
(high
on the relationship between g
g°°(n)
rjf (n).
ss
p L ). The
and
effect
and therefore
on the growth
n:
= r%(n)-p = n(^M-p
This expression highlights the trade-off between innovation and standardization: a
high standardization rate (and thus a low n) increases the instantaneous profit rate
ir
H
(n),
but lowers the expected profit duration. Taking the derivative and using (17)
yields:
dg ss
_
(n)
7i
dn
H
pH
KHJn)
ep H
From
8Y (n) /dn =
dY (n) /dn = — oo.
(15),
lim,,-,!
dg ss
at
^
\
|
dY{n)
dn
n
Y (n)
n min For n > n min we have 8Y
.
,
(n)
Thus:
(n)
dn
=
n
n dn H (n)
dn
pH
(n)
=
n=nmi „
H+L
PH e
2
dg
—
dn
ss
p
17
and
lim
n-i
(n)
= — oo.
/dn <
with
Provided that p
< ^-§, g ss
(n)
the wage of unskilled workers
them
technologies to
is
n.
Intuitively, at
=
n
1
zero and hence the marginal value of transferring
is
terms of higher aggregate demand and thus also profits)
(in
Instead, at
infinite.
an inverse-U function of
is
=
n
n min
Y
aggregate output
,
is
maximized and marginal
changes in n have second order effects on aggregate production. Moreover, given that
future profits are discounted, the impact of prolonging the expected profit stream
(high n) on innovation vanishes
n* 6 (n
min
1)
,
if
p
is
_
ejjjj
condition p
< ^-§
BGP
n
is
is
nH
satisfied
is
and which guarantees g >
scarce. It
p
<
jpjj,
growth
maximized at
is
that solves:
P
The
When
high.
0)
and
dY(n*)
n*
dn*
Y(n*Y
(n*)
whenever Assumption
e
<
+
1
1/h,
i.e., if
1
{
'
(which we imposed above
skilled workers are sufficiently
when p and p H are sufficiently low. Now recalling that in
function of fiL we have the following result (proof in the
also satisfied
an increasing
,
text):
Proposition 6 Let g 33 be the BGP growth rate and assume p < ^-^
an inverse U-shaped function of the cost of standardization.
Figure
1
Then, g 3S
is
provides a geometric intuition. Starting from a very high n L such that
rff (n) is in its
decreasing portion, a decrease in
along the schedule
rfj (n)
.
This yields a lower n
in this region, a decrease in
T"H (n)
.
schedule
is
pL
\x
ss
L
moves the equilibrium to the
and thus higher growth. Therefore,
increases growth. However, after the
passed, further decreases in p L reduce
Proposition 6 also has interesting implications for the
in this model, the skill-premium
is
left
maximum
of the
n and growth.
skill
premium. Recall that,
the market value of being able to operate
technologies and produce hi-tech goods. For this reason,
tion of hi-tech firms (see equation (9)). Since growth
is
it is
new
increasing in the frac-
an inverse-U function of
n,
the model also predicts a inverse U-shaped relationship between growth and wage
inequality, as
shown
in Figure
sign that standardization
is
2.
Intuitively, a very high skill-premium could
so costly to slow
down growth. A very low
skill
be a
premium,
however, might be a sign of too fast standardization and thus weak incentives to
innovate.
18
0.021
0.020
"
"
^^^^
""""
^\
/
0.019-
^\
/
^
0.018-
0.017-
0016-
1
1
1
h-
1
1
i
12
1.0
1.4
-H
1
1.6
1
1
1
1
1
2.2
2.0
1.8
1
1
1
1
1
1
1
2 6
2.4
2.8
,
,
3,0
wh/wl
Figure
3
Growth and the
2:
Skill
Premium
Welfare Analysis and Optimal Policies
We now turn to the normative analysis. We start by characterizing the Pareto optimal
allocation for a given
/j,
L
,
representing the technical cost of standardization.
This
allows us to identify the inefficiencies that are present in the decentralized equilibrium.
Next, we focus on the constrained
with a limited
efficient allocation
level of competition.
optimal
3.1
we
allow the government to increase
IPR
regulations and to influence the
set of instruments. In particular,
the cost of standardization above
we
Finally,
fi
L through
that a government could achieve
briefly discuss
how North-South
trade affects the
policies.
Pareto Optimal Allocation
The Pareto optimal
allocation
is
the one chosen by a social planner seeking to max-
imize the utility of the representative agent, subject to the production function (14)
and
for given costs of innovation,
Hamiltonian
for
the problem
fj,
H and
,
standardization,
is:
U = In (Y -IH - IL + Si?
19
+SL
\i
L
.
The
current value
where
and 1^ are investment
Ijj
control variables are IH and
variables £ H
n
and £ L
,
in innovation
and standardization,
II, while the state variables are
Prom the
respectively.
solves:
first
A
The
respectively.
and Al, with co-state
order conditions, the Pareto optimal
—
— 9Y
dA~^~dA~ ]r
1
L
That
is,
}
(
L
the planner equates the marginal rate of technical substitution between hi-
The Euler equation
tech and low-tech products to their relative development costs.
for the planner
is:
C _dYJ_
C~
By comparing
P
'
is
inefficient for
To
a standard appropriability problem whereby firms only appropriate
is
isolate this inefficiency, consider the simplest case
no standardization and
innovation
is
H
Y = AH,
appropriability effect also applies
is
too
stealing externality:
benefit
e
=
temporary.
is
7r#
=
while the private return
Second, there
see that the
two reasons.
R&D
a fraction of the value of innovation/standardization so that
low.
we can
these results to those in the previous section,
decentralized equilibrium
First, there
dAfj,H
much
H/e.
is
when L >
0,
is
too
so that there
is
In this case, the social return from
= H/e < H. The same
only r
form of
0.
standardization relative to innovation due to a business
the social value of innovation
A
L =
investment
is
permanent while the private
particularly simple case to highlight this inefficiency
is
when
2 so that (30) simplifies to:
n
_
1-n
h
fVL + V H
Vh
V
^
In the decentralized equilibrium, instead, the condition r L (n)
HClearly,
To
n
is
On
h
m + rJ
/j,
r"
(n) yields:
h
too low in the decentralized economy.
correct the
needed.
=
=
first inefficiency,
subsidies to innovation (and standardization) are
the other hand, the business stealing externality can be corrected by
introducing a licensing policy requiring low-tech firms to compensate the losses they
impose on hi-tech
pay a one-time
firms.
In particular, suppose that firms that standardize
licensing fee
c
/^ to the
must
original inventor. In this case, the free-entry
20
conditions together with the Hamilton-Jacobi-Bellman equations (18) become:
VH
=
ith
Clearly, the business stealing effect
firms
is
m (VH - I2f) =
nH
~
removed when
compensate the hi-tech produces
=
fi
fi
when low-tech
H We summarize
.
allocation can be decentralized using a subsidy to
Constrained Efficiency: Optimal
Proposition 7 shows
is,
,
11
innovation and a license fee imposed on firms standardizing
3.2
H that
for the entire capital loss
these results in the proposition (proof in the text).
Proposition 7 The Pareto optimal
c
fi'l
.
fi
how the Pareto optimal
new
products.
L
allocation can be decentralized. However,
the subsidies to innovation require lump-sum taxes and in addition, the government
would need to
set
up and operate a system
these might be difficult.
efficient policy,
12
where we
limit the instruments of the
regulations restricting the access to
policy be in this case.
11
In practice, both of
Motivated by this reasoning, we now analyze a constrained
we assume that the government can only
13
of licensing fees.
More
new
In particular,
affect the standardization cost
technologies,
precisely,
government.
we
through
IPR
and ask what would the optimal
find the (constant)
L that maximizes
\x*
due to monopoly pricing. This is because in our model
and unskilled labor). Thus, markups do not
distort the allocation. In a more general model, subsidies to production would also be needed correct
Note that there
is
no
static distortion
firms only use inelastically supplied factors (skilled
all
for the static inefficiency.
12
and
Some
reasons emphasized in the literature
monopoly
why
licensing
may fail include asymmetric
information
Bessen and Maskin, 2006). See also Chari et al. (2009) on the
difficulties of using market signals to determine the value of existing innovations.
13
We do not consider patent policies explicitly for a number of reasons. Patents are often perceived
as offering relatively weak protection of IPR, less than lead time and learning-curve advantage in
preventing duplication. Patents disclose information, the application process is often lengthy and
cannot prevent competitors from "inventing around" patents. Overall, Levin et al. (1987) found that
patents increase imitation costs by 7-15%. This support our approach of modeling IPR protection
bilateral
(e.g.,
as an additional cost.
21
BGP
utility:
OO
max pW
= pi ln(- +ln(A
e
~ pt
9t
)
J
dt
(31)
o
+—
= ln&"
The optimal
rate.
turn, g (p, L = n [~kh (n) I Ph ~ p]
= y( n ~ 9 (Pl) Ip-h + Ml U — n )L
ss
In
)
ss
and x
problem
)
(31) does not
>
0,
results.
consumption
level
and
its
growth
evaluated at the n (p L ) that solves (27)
evaluated at the same
In general,
77.
cases.
Optimal/ Growth Maximizing Policy: p
3. 2. 1
—
of the
have a closed-form solution. Nonetheless, we can make progress
by considering two polar
As p
sum
policy maximizes a weighted
•
the optimal policy
Manipulating the
maximize g ss For
to
is
—
.
*
this case,
we have simple
order condition (29), the optimal n*
first
analytic
is
implicitly
is
increasing
defined by:
Note that the
in n,
LHS
decreasing in n, from infinity to zero, while the
is
ranging from minus infinity to 1/e. Thus, the solution n*
is
RHS
always interior and
unique. Using the implicit function theorem yields:
9_f±
oe n*
=
t^l
>0
—
^A =
and
ah
1
e
because, from (32), en*
—
fraction of hi-tech goods
+
e
is
1
=
e
(n*)
7
(1
(1
_n
*
n*
— n*)~r
h^
increasing in the relative
_ e + n e)>0
.
)(1
>
0.
skill-
That
is,
(33)
the optimal
endowment and
in
the
elasticity of substitution across products.
Once we have
n*,
standardization, W77i
=
g (1
—
n) jn
we
=
we can use the
^tt to solve
-,
indifference condition
for the p*L that
between innovation and
implements
When
p
—
>
and
obtain:
^ = ^I^fi^/,^
= ^I
n
n
p
7r
The
n*.
L
/ijr
)
\
L
n
(34)
equilibrium fraction of hi-tech goods, n, depends on relative profits (tth/Vl),
22
R&D
relative
costs (hh/^l)' an d the standardization risk faced by hi-tech firms
(captured by the factor 1/n). Note that a decline in the relative
to a
more than proportional
L^L-h)
n because tth/^l
fall in
and
falls
skill
m
rises.
from (32) we can find the optimal standardization cost
£
„!-«,(»•- +
l
This expression has the advantage that
i)"'
=
r
supply, h, leads
Substituting
as:
-f^.
(35,
only depends on h through n* and can be
it
used to study the determinants of the optimal policy. Differentiating (35) and using
(33),
we obtain the
following proposition (proof in the text).
Proposition 8 Consider
when
the case p
the cost of standardization
pL
—
BGP
0.
>
satisfies (35)
and growth are maximized
welfare
and n*
the solution to (32).
is
The
following comparative static results hold:
=
dfij h
-en*(l-n*) <
~dh^l
The
results
1
—1
>
- 1
d/i*L
e
n*e
de
fi*
L
e
summarized
0.
Changes
in this proposition are intuitive.
in the cost of
innovation should be followed by equal changes in the cost of standardization, so as to
keep the optimal n constant.
A
decline in the relative supply of skilled workers
makes
technology diffusion relatively more important. However, the incentive to standardize
increases so
much (both because
the equilibrium
more
costly.
calls for
m
change
in
instantaneous profits and because
increases too) that the optimal policy
IPR
protection. Finally, given that
makes the
e
>
the other polar case.
To
2,
a lower elasticity of
more important
lA
.
Optimal Policy: high p
To understand how the optimal
14
make standardization
diffusion of technologies to low-skill workers
for aggregate productivity and reduces the optimal IPR, p*L
3.2.2
to
is
Thus, somewhat surprisingly, a higher abundance of unskilled worker
stronger
substitution
of the
policy changes with the discount rate,
In particular, assume that p
see this, recall that en*
-e+
1
>
0.
Then,
23
e
>
— ^—. As we
->
2 implies (en*
-
1) / (e
-
we
consider
will see, this is
1)
>
0.
the highest p compatible with positive growth. In this case, Section 2.7 shows that
g
is
maximized
^^
=
jth
at the corner
=
and 3
^^
be close to zero yields x
-p
=
n min
=
->
(since p ->
sa
which
V-,
discounting the optimal policy
is
the
+
h/(h
also
is
same
^7).
maximized
as the
»
n™
we have
Next, the result that 3 must
at
n mm
Thus, with high
.
one that maximizes static output
(and consumption) only. Reaching this point requires setting p*L
in this
—
Moreover, for n
1).
= pH
.
Note that,
extreme scenario, the optimal policy becomes independent of h and other
parameters. Comparing the policy p*L
= pH
to (35) shows, not surprisingly, that high
discounting implies a lower optimal protection of IPR.
Other Competition Policies
3.3
In practice, several other competition policies, besides licensing fees and intellectual
property rights, are used in order to affect the profitability of standardization.
now
briefly discuss the implications of
can directly
can
set exx
>
markups
affect
e
and ex >
When markups
e
BGP
sectors. In particular,
it
in the pricing equations (5).
and
\\-n)
,r
.t±(5)*
e
H \n
(36)
J
'
Hl an d the above expressions, it is immediate to see that the
n only depends on the product p L e L This result highlights that competition
r L (n)
policy (ex)
ttl/
.
and IPR protection (p L ) are
(high ex) for low-tech firms, there
can
Suppose that the government
and the low-tech
in the hi-tech
-£L (JL-f
eL
=
policies.
vary across firms, profits (16) become:
,L
From
such
We
contrary, g
(n)
is less
mark-ups
entry in the L-sector. Yet, the government
by reducing p L so that it becomes easier to standardize. On the
does not depend on e L so that n* is as before. Given that intervening
offset this effect
ss
substitutes. Intuitively, with lower
,
,
on p L or ex is equally effective to implement a desired
on the relative costs of the two policies.
Now when we
also
have p
—
>
0,
/*
Then, under the assumptions that
n*, the optimal
mix depends
equation (35) becomes:
-?
ex
ex.
^
- r^r^— +
1
e
n*e
can be changed at no
24
cost,
it is
easy to see that
the optimal policy
is:
£h
=
a
=
(-L
=
^r
I
M
where
IPR
tion;
/j.™'
n
>
is
the
minimum
protection). Intuitively, full
jj,*
l
among
=
pt
l_ e + n
*
€
"technical" cost of standardization
monopoly among
(i.e.,
with no
hi-tech firms ensures high innova-
Lmin minimizes the resources spent on standardization; high competition
low-tech firms yields the optimal n*.
If
L should be adjusted
cannot be achieved, then
3.4
min
\i*
the desired level of competition
upward
accordingly.
ex,
15
North-South Trade and IPR Policy
We now
ask
affects the
how
trade opening in countries with a large supply of unskilled workers
optimal
settled debate
IPR
policy.
on whether trade
This question
IPR policies, which serve
economies.
We
interesting because there
is
an un-
liberalization in less developed countries should
accompanied by tighter IPR protection,
less strict
is
as implied
be
by the TRIPs Agreement, or by
to encourage technology diffusion to less advanced
can investigate this question using our model.
Consider an integrated world economy (the North), described by the model in
Section
2.
For simplicity,
let
us also assume that there
is
a single large developing
country endowed with unskilled workers only (the South). Without trade, we assume
that Northern technologies are copied at no cost by competitive firms in the South.
However, this form of technology transfer
is
introduced in the South, labor productivity there
is
no innovation
when a low-tech good
imperfect:
is
only a fraction
ip
6
(0, 1].
is
There
in the South.
Now imagine that the
South opens
its
economy
We assume that economic
to trade.
integration allows Northern firms to produce in the South. In the
new
integrated equi-
librium factor prices are equalized (or else firms would relocate to the country where
15
Another way to highlight the same
result
is
that policy does not affect markups, but rather
patent duration in the low-tech sector. In the model considered so
the low-tech sector. Suppose, however, that patent duration
the good
is
far,
finite
patent length
is
infinite in
and, once the patent expires,
Here, the key trade-off
is produced by unskilled workers under competitive conditions.
between the cost of standardization and the duration of the subsequent monopoly position in
the low-tech sector. The gist of the argument is that the best combination is, in a sense, low
IPR everywhere (low ji L and short patents). However, it has to be carefully tailored, since the
cost-relative-to-duration must be pinned down so as to get the right n.
is
25
labor
is
cheaper) and Southern firms are replaced by their Northern counterpart. This
stems from the fact that Northern firms are more productive and can capture
result
the entire market by charging a price equal to or lower than the marginal cost of
the Southern imitators, pi
< Wl/p-
However,
>
if ip
—
(1
1/e),
Northern firms must
compress their markup to keep Southern imitators out.
In sum, the effect of trade opening in the South
world endowment of
in the
L and
margins of low-tech firms (higher
rate and the optimal
7Tl (see
What
ex,).
are the implications for the
The change
policy?
1
may
up
either shift
up because the greater supply
the profitability of hi-tech products.
be higher or lower.
maximum
L/H
As a
result, in the
Despite this ambiguity,
is
how
increases the optimal level of
fall
on
effects
among
If
IPR
if
new BGP, n ss and g ss may
policy,
shift of the r
/j,
s
^
fi L
,
is
correctly adjusted.
is
(n) curve, implying that
l should be changed. As already seen, a higher
IPR
protection,
/j.*
l
.
On
the other hand, higher
low-tech firms, e^, calls for a reduction in
The
the liberalizing country
is
large
and
pressure posed by imitators on low-tech firms
should be followed by a tightening of
IPR
fi L
,
to
compensate
for
net effect depends on which force
inefficient (low (p), the
competitive
weak, while the threat to hi-tech
is
due to the increased incentives to standardize,
4
(n) curve, instead, always
easy to see that trade opening
it is
in profit margins (see equation (37)).
dominates.
firms,
have opposite
£/,
profit
growth
attainable r must be higher.
crucial question, then,
competition
the
BGP
goods increases the price and thus
of low-tech
This follows immediately from the upward
The
L and
in
down. The r#
or
necessarily growth (and welfare) enhancing
the
markup and
possibly a reduction in the
equation (36)) and hence on the return from standardization, so the r L (n)
curve in Figure
shifts
IPR
isomorphic to an increase
is
policies.
is
high. In this case, integration
10
Extension: Multiple Equilibria and Poverty Traps
We have so far
a BGP n stays
sumption that,
assumed
that, at
in the range (n
at
wh =
wH = wL
min
,
1).
This
,
standardization stops implying that in
is
an immediate consequence of the
wi, incumbent hi-tech firms
fight (see
Proposition
1)
as-
so that
low-tech firms do not find entry profitable. Under this assumption and Assumption
2,
Proposition 4 established the uniqueness of a
BGP
equilibrium.
However, either
16
These are the policies that a world planner would choose starting from the optimum. Yet,
governments of individual countries face different incentives, because an increase in l leads to a
higher skill premium and redistributes income towards skill-abundant countries. This conflict of
interests between the North and the South is studied, among others, by Grossman and Lai (2004).
/j,
26
when we adopt
the alternative tie-breaking rule
facing the entry of a low-tech competitor exit
—whereby
at
wh = wl
—or relax Assumption
2,
hi-tech firms
the model
generate multiple equilibria and potential poverty traps. In this section,
we
may
briefly
discuss this possibility.
For brevity, we focus on the case where Assumption 2
tion between hi-tech firms and entrants at
wh = w L
is
still
holds but the competi-
resolved according to the polar
opposite tie-breaking rule. Even under this alternative tie-breaking rule,
that
But
wh > wl
is
low
now standardization may continue even
we
step in the analysis of this case,
first
wH = wL
levels of n. Recall that
constant at
wh = wl
and some high
a profit rate of
=
ttjj
at
n
skill
For n
.
< n mm
,
the
.
premium
skill
.
This yields h
determined
h, is
= n/(l — n)
In other words, for sufficiently low n,
~r^-
n < n min
at
workers are employed in low-tech firms.
wh = w L
(9) after setting
=
irjj
have
characterize the static equilibrium
= n mm
In this case, the allocation of labor between the two type of firms,
endogenously by equation
still
since skilled worker can always take unskilled jobs.
[0, 1]
in contrast to Proposition 1
As a
for
any n €
for
we
it
is
and
as
if
pn = Pl, and so are profits.
and r^ (n) schedules over the entire
workers were perfect substitutes, prices are equalized
To
we draw the tl
find the steady states,
domain n €
(n)
Figure 3 shows the determination of n ss for two possible
[0, 1].
schedules, corresponding to different values of
part of both schedules
The
equal to one.
line)
and
r L in)
a straight
is
interior
(dashed
BGPs
line, as
n
states" such that
two
g
p >
i.e.,
1LL
incentive to standardize,
n
=
=
m
> ^i^ —
premium
the
1,
first
constant and
is
are again the intersections between the
—
and
=
r^
(n) (solid
r
now
p.
there might exist "corner steady-
A
corner steady state can arise in
=
n
0, there is no incentive to innovate nor to
and p > 1La- — ^±^: (ii) at n = 0, firms have an
p < M±L^ but there are no goods to standardize, since
at
(i)
i.e.,
discouraged by the expectation that
is
goods would trigger a high standardization
that to
skill
to Figure
= ^±^
Moreover, innovation
0.
Compared
.
(n)
line) schedules.
different circumstances:
standardize,
L
there the
In addition to balanced growth equilibria,
=
\i
rj,
p whenever n
>
0.
rate.
new
hi-tech
Formally, innovating firms expect
This conjecture does not violate the resource
constraint since the absolute investment in standardization would be infinitesimal
when n
in
=
even though the standardization rate were high. The uninteresting case
which r L (n) lays above r^ (n)
As shown
in Figure 3,
for all
n
is still
ruled out by Assumption
depending on the standardization
cost,
2.
there are two
regimes:
High p L
:
For
\x
L
> (H +
L) / (pe) (lower r L (n) schedule in Figure
27
3),
there
is
a
0.08
0.07
0.06"
0.05
0.04"
0.03
002
Figure
3:
Solid
=
r#
(n),
Dashed
=
r^ (n)
unique steady state (BGP) corresponding to the unique crossing point of the
r L (n)
Low
fi L
:
and r#
For
fi L
(n) schedules.
< (H +
L) / (pe) (upper tl (n) schedule) there are two interior
a corner steady state.
3.
The two
can be seen in Figure
interior steady states
In this case, a corner steady state also exists, since rjf (0)
(H +
L) J
The reason
(/J,
L e)
.
Hence, standardization
for the potential multiplicity
decisions by firms. If firms expect
n
is
is
and
profitable at
n
=
=
p
<
r L (0)
=
0.
a complementarity between investment
to be high in the
BGP, they
also anticipate a
low
standardization rate, m, and this encourages further innovation. Greater innovation
in turn increases the
demand
than consumption) and
for resources
(i.e.,
raises the interest rate.
the
A
demand
for "investment" rather
greater interest rate reduces the
value of standardization more than the value of innovation), confirming the expectation of a low
m. In
contrast,
when a
large fraction of the resources of the
are devoted to standardization, expected returns from innovation decline
economy
and
this
limits innovative. Expectation of lower innovation reduces the interest rates, leading
to reverse reasoning
—
i.e.,
encouraging standardization (more than innovation) and
28
.
17
Note that this complementarity was also
confirming the expectation of a high m.
present in the model analyzed in the previous sections. Yet,
multiplicity because
Assumption
correspond to levels
mm which was
of n below n
baseline tie-breaking rule.
unskilled labor
is
strictly
ruled out by Proposition
Thus, the fact that standardization
is
1
under our
profitable only
if
cheaper than skilled labor prevents the economy from falling
where innovation
to low-growth traps
did not give rise to
2 guarantees that the other candidate steady states
,
18
it
is
discouraged by the expectation of a very fast
standardization rate.
We
summarize the characterization
of the set of steady-state equilibria in the
following proposition (proof in the text).
Proposition 9 Suppose
1.
If
2.
If
Hi > (H
i±
L
+
that
Assumptions
1
and 2
L) J (pe) there exists a unique
< (H +
hold.
BGP
Then:
which
BGP
L) J (pe) there exist two interior
is interior.
equilibria
and a corner
steady state.
It is
fi
also
noteworthy that the non-monotonic relationship between the g ss and
L and the policy analysis derived in the previous sections
interior
may
BGP. The main
novelty, however,
is
now apply
to the higher
that too low a cost of standardization
lead to multiple steady states, with the equilibrium determined by self-fulfilling
expectations, and stagnation.
Concluding Remarks
5
New
technologies often diffuse as a result of costly adoption and standardization de-
cisions.
for
Such standardization
also creates cheaper
example, substituting cheaper unskilled labor for the more expensive skilled labor
necessary for the production of
17
is
ways of producing new products,
To
new complex
see the role of the interest rate, consider a
products.
This process endogenously
more general formulation
of preferences
where 9
the inverse of the intertemporal elasticity of substitution. In this case, the Euler equation takes
the form
C/C —
(r
— p) /6.
Using
this,
ss
=
r H (n)
equation (26) becomes:
On
nB
+
1
it a
-nti H
(
1
—n
\nd +
1
Note that, as 8 —
the rjf (n) curve becomes flat and the BGP is necessarily unique.
18
When Assumption 2 is relaxed, it is possible that the ri (n) and rj/ (n) schedules cross twice
over the range n e [n mln l]
»
,
29
generates competition to original innovators. In this paper,
tions of this costly process of standardization, emphasizing
of
growth and
its
studied the implica-
both
its role
as
an engine
potential negative effects on innovation (because of the "business
stealing" effect that
Our
we
it
creates).
analysis has delivered a
number
of
new
First, the tension
results.
between
innovation and standardization generates an inverse U-shaped relationship between
competition and growth. Second, while technology diffusion
it
can also have destabilizing
equilibria (multiple
growth paths).
and IPR policy and how
it
be protected more when
are in short supply
Finally,
we
characterized the optimal competition
depends on endowments and other parameters, such as the
We
found that innovation rents should
skilled workers are perceived as scarcer, that
and when the
showed that these
also
Standardization can open the door to multiple
of substitution between products.
elasticity
We
effects.
potentially beneficial,
is
elasticity of substitutions
results provide
new
is,
when they
between goods
high.
is
reasons for linking North-South trade
to intellectual property rights protection.
It is also
tition
worth noting that a key feature of our analysis
is
the potential compe-
between standardized products and the original hi-tech products.
that this
is
takes place
a good approximation to a large
by
different firms
(and often
number
in the
of cases in
form of
Nevertheless, the alternative, in which standardization
innovator,
(2010),
is
another relevant benchmark.
we study a model
of offshoring,
We
believe
which standardization
slightly different products).
is
carried out by the original
In our follow-up work,
Acemoglu
where offshoring can be viewed
process of standardization carried out by the original innovator to
et al.
as a costly
make goods
pro-
ducible in less developed countries with cheaper labor.
Our model
particular,
it
yields a
number
of novel predictions that
suggests that competition and
IPR
can be taken to the data. In
policy should have an impact on skill
premia. Furthermore, data on product and process innovation might be used to test
the existence of a trade-off between innovation and standardization at the industry
level.
These seem interesting directions
for future
30
work.
Appendix
6
Proof of Lemma
6.1
Recall
i\
which
is
H
=
f^ = &&.
e
>
2,
,
To
(11)
it is
&& <
immediate to see that
establish the properties of
m;
,
^=
lim „_i
has no
^^
From
true in equilibrium.
5n
For
1
i=i
1
fc
if
p H > pL
,
note that:
t^l,
-n
Convexity of iii follows immediately because the function
oo.
critical point.
Notes on Figures
6.2
The benchmark economy used
p
=
to
draw
=
=
2;
pB
0.02; r
=
0.02;
0.02;
e
all figures
uL
=
m = 0.02;
n
22.7,
has the following parameter values:
59.1;
H=
L =
1;
3
implying in steady state:
s
=
Proof of Proposition
6.3
A BGP
must be a
Thus any
1.
must be a zero
where n
ss
of the
must be
satisfies r L (n
tence of a unique interior
such that r L (n
that at (n
We
ss
5S
,x
)
= r ^ (n
ss
),
first
Moreover
tion 2)
rem
r|f
and \im n ^i
(n
min
(r L
We denote such
(see again Proposition 3).
)
by showing that there
that there
is
is
in
first
note that
view of Proposi-
r L (n
ss
)
is
a zero by (n
We
ss
,
x
min
€ (n
a unique corresponding value of
is
ss
it
ss
),
prove the exis-
a unique value n
x
ss
,
l)
and
satisfied.
ss
step by establishing that r# (n )
U-shaped function whereas
tion.
(n
sa
the transversality condition
prove the
1
We
interior as defined in Proposition 3, or equivalently,
rj^
BGP
s
)
ss
=
)
1.5.
boundaries n = n mm and n =
dynamical system (21)-(22).
ss
—=
dynamical system (21)-(22).
rest point of the
BGP
0.5;
4
there cannot be a rest point at the
tion
=
is
a continuous inverse
a continuous, increasing and convex func-
min
)
(n
ss
> r L (n ) (which follows immediately from Assump— rj^ (n ss )) = oo. Then, the intermediate value theo-
)
BGP,
establishes the existence of such a
31
while the shape of the two func-
,
tions implies uniqueness.
h
>
Let
=
(f>(x)
x
Standard algebra establishes that
0.
cave function, such that lim I _o
4>
maximum in the unit
+ (1 — n ss p. Since r^ (n ss
(n
it
)
•
if/ (/%e)
)
where
e
>
is
= — oo.
0' (x)
and
2
Thus,
(x)
<f>
=
Next, note that r^ (n ss )
a linear transformation of
U-shaped concave function, with a unique
also a continuous inverse
is
lim^^
interval.
)
h~r
£
l)
a continuous inverse U-shaped con-
is
°° ar>d
has a unique interior
ss
—
x (i
(x)
—
(z)
+
(n
ss
)
interior
maximum in the unit interval. Consider now r L (n ss Since r L (n ss = ix^ (n ss / p, L
Lemma 1 establishes that 77, (n ss is increasing and convex, with lim^s^ n, (n ss = 00
ss
3
ss
— r^
(implying that lim n M_i (r^, (n
{n )) = 00).
)
.
)
)
)
,
)
)
Next, straightforward algebra immediately implies that, conditional on n
m = m (n
ss
and z
)
=
z (n
ss
there exists a unique value
)
the dynamical system (21)-(22). Finally, since in
Assumptions
1-2),
the transversality condition
\
BGP
is
r
= X ss
=p+
= n ss
,
that yields a zero of
>
g
satisfied in the
g (from (1)
and
unique candidate
BGP.
Proof of Proposition
6.4
5
Recall that dynamic equilibria are given by solutions to dynamical system (21)-(22)
with boundary conditions given by the
By
condition.
the
same argument
initial
We
will
— n mm
and
either converge to the unique (interior)
show
neighborhood of n
(the
and that there cannot be
BGP), there
n —
>
BGP
from any
in this proof that starting
ss
=
n
no and the transversality
as in the proof of Proposition 4, there cannot
any dynamic equilibrium path where n
must thus
condition
1
.
(n
Any dynamic
ss
,x
ss
)
equilibrium
or involves cycles.
condition n
initial
be
=
no in the
unique path converging to (n ss \ ss )
exists a
,
cycles, thus establishing local saddle-path stability of the
dynamic equilibrium.
Because there are two sources of technical change (innovation and standardization),
we
(which
first
may
CASE
this case,
distinguish between three possible types of potential
converge to the
1:
VH =
BGP,
p H and
(n
3S
VL <
,
x
dynamic
equilibria
ss
))-
p L (=»
m =
and g
=
{y (n)
- *)//%)•
In
from Proposition 3 the dynamics are governed by the following system of
ordinary differential equations:
(n)
X
X
=
h
= (1-n)
717/
y{n)- X
- P-
(38)
P-H
y{n)
-x
Ph
32
CASE
VH < p H
2:
and
VL = pL
m=
(=*>
(y (n)
-
=
x) / ("Ml) and 5
0).
In this
from Proposition 3 the dynamics are governed by the following system of
case, again
ordinary differential equations:
-
y
ttt (n)
= -Z±-±-p
X
Ml
n
-x
'y{n)
= —
(39)
Ml
CASE
(2/ (
Vb = Pn
3:
n ) — p L mn — x)
system of ordinary
anc
^
*
In
/Mff)-
=
=
(=> rn
I^l
this case, the
n €
[0,y(n)j,
Therefore, (n
.
ss
ss
,x
is
)
above or below n
ss
)
and
will
the economy will converge in
a switch to
economy
if
if
ss
n < n
dynamics
if
to (n
ss
,
Lemma
x
1
or
CASE
show that
any point with
CASE
(n, x) 7^ {n
CASE
time to (n ss ,x ss )
3 at that point,
BGP
and
establishing several
ss
,
ss
ss
,
will
,
)
(it is
x
it
is
and
not
3 cannot
ss
and then a jump
and since (n
stay at (n
Lemmas.
ss
,
ss
First,
,
x
x
ss
)
ss
)
)-
Instead,
is
would imply n <
0).
Second,
immediate that,
if
n > n
ss
,
in
m
and
r
a zero of (40), the
thereafter.
Lemma
ss
2 establishes that,
,v
Lemma
there exists a unique trajectory converging to (n
(39)
ss
,
/i#
2 (depending on whether
there exists no trajectory converging to (n
(39), since these
n > n
dynamics
first
at
will
but
(40),
ss
there exists a unique trajectory converging to (n S3 ,x )
,
£
be unique. Then under the equilibrium dynamics,
finite
have reached the
prove by
n < n
that,
will
CASE
we
Nevertheless,
=
This implies that V#
0.
a zero of the dynamical system
the equilibrium will be given by either
We
mW
V
and g >
dynamic equilibrium behavior
will create
=
(40)
BGP m >
a zero either of (38) or of (39).
is
/^ L and 5
[n min ,l].
Recall that in the
n
L (n)
three cases, the differential equations are defined over the region x
all
describe
ix
Ph
Pl
mh
= pL
//% —
(n)
dynamics are governed by the following
p
X
Vl
h
differential equations:
- =
In
tt
(it is
ss
)
immediate
following the
3 establishes that,
ss
,
x
ss
)
following the
there exists no trajectory converging
following the dynamics given by (38), since these imply
h >
4 provides a complete characterization of equilibrium dynamics
0).
Third,
when the
transition involves either only innovation or only standardization, followed by a jump
33
in either the innovation of standardization rate as the
but continuous changes in asset values.
economy reaches (n ss ,x ss ),
Lemma
Fourth,
5 establishes that (under
the sufficient conditions of the Proposition) there exists no trajectory converging
to (n ss ,x
ss
dynamics
either
following the dynamics (40).
)
in the
CASE
1
Lemma
Finally,
6 rules out transitional
BGP in which there is a jump from CASE 3 to
between CASE 1 and CASE 2. These lemmas together
neighborhood of the
CASE
or
2 or
establish local saddle-path stability.
Lemma
< n ss
2 Suppose n
Then, there exists a unique trajectory attaining {n ss
.
CASE
in finite time following the dynamics of
monotonic convergence
shown
tions (38)
x
1]
over the region where
(i- e ->
particular, in the interior of the region [n min
h >
and x |
<= X
% X n
i
)
>
,
n 6
x
1]
and x 6 [0)2/ ( n )D- n
S3
ss
y(n)], and hence at (n ,x ),
[n min ,l]
[0,
= Hp +
where X in)
from Assumption
last inequality follows
the system of differential equa-
This system has no zero over the feasible region
in Figure 4.
[0,J/(n)]
[i
initial level of
the control variable,
In particular, since (n ss ,x ss )
^-
- nH
y (n)
x
)
is
<
(n)
Although whether x ("
1.
general ambiguous, the phase diagram shows that there
unique
ss
)
nSS
< x( )
finite
initial
and
is
and
(the length of the transition)
(no, T", x'o)
with
is
T ^T'
n
i
,
value problem with (n^,
From the standard
solution. Fixing the initial condition
ensures that this solution
1S
time to (n ss x ss )
Xo-
n
Xt)
and
result of existence
uniqueness of solutions for systems of ordinary differential equations, this
T
The
.
not a zero of the system (38), the determination of
is
being the boundary (terminal) condition.
for
y (n)
a unique trajectory (and a
converging in
the converging trajectory can be expressed as an
problem has a unique
)
0).
Consider the phase diagram depicting
Proof.
["min,
n (n >
in
ss
x
This trajectory features
(38).
1,
,
initial
value
yields a unique solution
The monotonicity
of the dynamics of
n
there does not exist two solutions (no, T, Xo)
unique,
i.e.,
and Xo
7^ Xo-
This argument also proves that convergence
attained in finite time.
Lemma
3 Suppose no
> n ss
in finite time following the
monotonic convergence
Proof. The proof
is
(n
ss
Then, there exists a unique trajectory attaining (n
.
dynamics of
<
similar to that of
no zero over the feasible region
and X >
0.
The
and x >
latter follows
[n min
,
1]
CASE
2,
(39).
,
x
ss
)
This trajectory features
0).
Lemma
x [0,y
2.
(n)].
The dynamical system again has
In particular, for n
from the observation that x
34
^
<=>
Hl
> n ss n >
,
("•)
/«l
%
Pi
C/A
Figure
Saddle Path, n
4:
< n sa
and n L (n ss ) / p L > p, implying that n L (n) / p L > p for all n > n ss
The phase diagram in Figure 5 shows that there is a unique trajectory converging in
where
finite
<
l (n)
tt'
.
time to (n ss x ss )
This trajectory features monotonic dynamics.
,
The previous two Lemmas
Lemma
If no
4 There
< n
ss
the
,
together imply our key characterization result.
economy converges
VH = p H
reaches (n
ss
ss
,x
)
,
,
If Uq
> n
the
ss
)
=
r'
the
(n
economy converges
differential equations (39), with
VH <
convergence,
p H)
economy reaches (n
VL =
ss
,
and g
= pL
=
(y (n)
—
x
)
i
)
+ m (n ss
)
.
\
X)
characteristics.
following the system
)
Throughout
n.
/p H
Thereafter,
in finite time to (n
ss
monotonic convergence
pL
ss
ss
,
m =
there
increase in innovation such that y (n
and Vl
,
ss
.
When
the
this
con-
economy
a discrete increase in standardization offset by a fall in the
is
interest rate such that r (n
ss
m=
Vl < Pi,
there
in finite time to (n
ss
monotonic convergence in
of differential equations (38), with
vergence,
dynamics with the following
exists equilibrium
is
ss
)
(y (n)
—
ss
,x
in
)
Vh = Ph
=
^l-
following the system of
n and
\) / {np L )
anc^ ^l
>
\.
Throughout
and g
=
0.
this
When
a discrete fall in standardization offset by an
—x
remains constant. Thereafter, Vh
= Ph
.
Proof. The proof follows from Lemmas 2 and 3 combined with the following obser35
C/A
1.0
n
Figure
vations.
Suppose we
start with
5:
< n ss
n
> n ss
Saddle Path, n
,
then the dynamic equilibrium
the system of differential equations (38), so from
Lemma
2 until
we have
T,
m
At T, we reach {n ,x ) and
jumps from zero to its steady state, m
offset by an equal jump down in r implying that Vjj does not change (i.e.,
ss
at
Vh = n H ). Moreover,
Note that there
m
ss
is
at T,
VL
no discontinuity
attains
its
steady state
(BGP)
given by
is
ss
.
0.
This
is
remains
it
value,
m—
Vl
change in
in the asset value Vl, since the
=
r
fi
L
.
and
are perfectly anticipated, causing a continuous change in the value Vl before the
actual change occurs to reach Vl
that there
Thus
is
=
^l exactly
no arbitrage opportunity
at this point, the
in
at
T
buying and
(the continuity of Vl ensures
selling shares of L-sector firms).
dynamics switch to those given by the system of
equations (40) with both innovation and standardization. Since (n
ss
ss
(40), the economy stays at (n ,x ) thereafter.
transversality condition follows by the
The
ss
fact that this
same argument
,
x
ss
is
)
path
differential
a zero of
satisfies
the
as in the proof of Proposition
4.
Next, suppose that we start with n
by the system of
in
.
differential equations (39)
T, investment in standardization and
and investment
> n ss Then
mr
the dynamic equilibrium
and from
fall
Lemma
3, until
T, g
=
discretely (the latter declining to
innovation and g jumps up (the latter increasing to g
36
is
ss
).
given
0.
m
At
ss
There
),
is
no change
investment and thus neither
m jumps down, Vh attains
As
at T.
in overall
Vh
As a
is
by
result,
system of
(n
ss
s$
,x
path
this
We
m
T
exactly at
\i
(note
perfectly anticipated
is
with both innovation and standardization. Since
ss
/SS
a zero of (40), the economy stays at (n
\ ) thereafter.
The
,
satisfies
that
fact
the transversality condition again follows by the same argument..
next show that transitional dynamics converging to (n ss x ss ) cannot feature
,
Vh =
both
VH — H
change
Vj,
again at T, the dynamics switch to those given by the
differential equations (40)
is
)
stead state value,
continuous at T, as the change in
that the path of
investors).
its
nor consumption nor
r,
fJ.
H an d Vl — Hl
since the system (40)
unstable in the neighborhood of
is
(n",x").
Lemma
5 Suppose n
^ n ss
.
Then, under the (sufficient) conditions of the Proposi-
no trajectory converging
tion, there exists
to (n
ss
,x
ss
)
following the dynamical system
(40).
Proof. The proof, which
Lemmas
Vh <
fJ-H
is
long,
presented in the next subsection.
is
2-5 establish that in the
or ^i
<
neighborhood of (n ss ,x ss )
^L' implying that there
different regimes while
n
t
^
n
ss
,
and thus
either
either only innovation or only standard-
is
However, the results established so
ization.
we must have
,
far
do not
cycles.
between
rule out "switches"
Moreover, with such switches, the
equilibrium might also be indeterminate, with multiple paths starting from some
tial
n converging to the BGP.
Lemma 6
that in the neighborhood of the
BGP,
rules out all of these possibilities
neighborhood of
Lemma
we
by showing
there cannot be a switch from the dynamics
given by any one of (38), (39) and (40) to one of the other two.
convenience, in this lemma,
ini-
write Vn,t
=
[J-h
*°
mean
(For notational
= /%
that Vh,v
for
in a
t'
t).
6 Consider an equilibrium trajectory in the neighborhood of the
BGP,
(n
3S
,
\
ss
Then, there cannot be a switch from any one of (38), (39) and (40) to one of the other
two,
i.e., if
at
t
in the neighborhood of (n
then an equilibrium cannot involve Vn,t
ss
,
x
ss
)>
we have
< /% and/or
Vi,t
Vn,t Q
<
—
/•*#
Hl for
t
and
>
fi
](o
,
=
I^l-
if we have
to!
= n H o,nd Vi,t Q < f^L> then an equilibrium cannot involve Vn,t <
Vff,t
= L then an
Vlj = Ml for t > to; and if we have VHtto < H and Vl
cannot involve Vn,t = /% and/or Vi < \x L for t > to.
fJ-
Vj,,to
P-h
and/or
equilibrium
<t
We will
VL,ta = Uli
Proof.
prove that
and
^ en
t
>
to.
The other
if
in the
neighborhood of (n ss
an equilibrium cannot involve
cases are analogous.
37
ss
,
,\
),
we have
VH<t = H
[i
and
Vn,t
Vj,,t
— Hh
< Hl
f° r
,
Suppose to obtain a contradiction that
where Vi
=
\i
< Hl
First, Vi,t'
VLi t< =
have
Case
T
h by
(i.e.,
for all
t'
>
we write
2:
=
Vl,t
>
the case and denote the last instance
We
0.
and second, there
t,
Vl# <
P>l f° r alii'
equilibrium path will converge to the
Case
f° r £
is
need to distinguish two cases.
>
exists T"
t,
such that we again
fi£.
the fact that
1:
< Pl
Vl,t+e
this
t
contradicts the hypothesis that the
BGP.
Vl,t as follows:
exp
/
>
-
(
r{nv )dv
/
(n T ) dr
ir L
J
=
exp
/
-
I
n L (n T ) dr
r{n v )dv
/
+ exp
j
—
dr
r (n T )
/
J
where the equality exploits the
exp
/
f
-
r{nv )du
/
i\
by hypothesis
fact that
have, again by hypothesis, that Vl,t
— Md
L (n T ) dr
=
Vl,t'
—
Ml- Moreover,
.
all
r 6
r G
for all
we
also
I
- exp
1
-
I
/
r (n T )
dr
/i
L
.
(41)
But then from
Lemma 5, this implies fir >
Lemma 1, 7T£,(n T > 7T£,(n ss for
— M#
<
instability result in
[T, T"].
Moreover, since Vh, t
[T, T'].
,
J
Suppose next that n^ > n ss By the
and thus n T > n
L
which implies
J
S3
/z
J
)
)
an<^ Vl,t
Mli
we a^ so nave that
for all
r6[r,n
r (nr )
,
!E*Jfe)
> 2£fit) >
f£&3
Mh
Ml
Ml
where the second inequality again follows from
nT > n
ss
exp
/
for all r
(
-
/
G
[T,T'].
r(n v )dv
Lemma
=
1
r (n -)
in
view of the fact that
But then,
7r
L (n r ) dr
<
/
exp
I
—
7
r(nv )dv
J
<
f
1
- exp I
/
j
r{n T )dT
[T, T'\.
This inequality contradicts
thus
nT < n
ss
for all r
G
[T,T'}.
<
r (n
(n ss )
j
ss
)
L
,
for all r
G
(41).
Suppose instead that n T < n ss By the
.
fact that r (n r )
TTx,
J
/i
J
where the second inequality follows from the
dr
instability result in
Lemma
Moreover, by the same reasoning
38
5,
nT <
for all r
G
and
[T,T'],
ss
^h^t) > ^H{n ) and
\'h,t
=
expf-/
/
<
since Vl, t
m
Ml;
(
n r)
=
0-
+ m{nv )) dv
{r{n v )
Therefore,
ir
J
exp
-
I
r(n u )dv
/
7r# (n T )
H (nT ) dr
-
+ exp
dr
r(n T )dT
/
\x
J
But
since for
r 6
all
first
term
in (42)
contradicts Vh,t
Lemmas
—
is
in
/i
H
view of
=
—/%
strictly greater
>
-
Ms
than
(
1
— exp — ft
r (n T ) dr
[
fi
J
2-6 establish the results of the Proposition. In particular,
or
5,
ruled out by
VL <
or
fi L
.
BGP, then Lemmas
H and thus
6.
Lemmas
5
ss
t
to switch to a regime where Vjj
If
we have Vh < Hh
or
<
fi
< Ml n
Vi,
'
H
or
VL <
fi L ,
which
2-4 imply that there exists a unique path converging to the
We
Proof of Lemma
BGP.
5
take a linear approximation of the dynamical system (40) around (n
£ ~ Fx
ss
(x
,n
ss
)-( X
-X
ss
)
+ Fn(x ss ,n ss )-(n-n ss
)
A.
2 ~ G x X ss n ss )
,
(
•
(
ss
X- X ) +
Gn
ss
(
X
,
n ss )
•
(n
-
n»°)
where subscripts denote partial derivatives and
x
p,
F{x,n)
_
= r{n)-p
G( X ,n)
a
/
v
y
(
n ~ Mjg (n)n-x
)
,
(l^)yJ^A- m {n)(l + il-n)^
\
n J
nH
V
mm
is
the neighborhood of the
This completes the proof of the Proposition.
6.5
and
^ n in the neighborhood of the BGP we must have
Vh — Hh an d Vl = Ml; either we diverge from the BGP
n
If
we have
Lemma
J
m
M#-
6 imply that starting at
VH <
(42)
.
[T, T'\
r (n T )
the
H
Mh
ss
,
x
ss
'
)
implying that:
Fxix
,
= -L >0,G x
n)
(
x ,n )
=
-(^f)±<
=
Solving for the schedules such that, respectively, \
X(n)lx=o
X
(n)
U=o
and n
=
=
y(ji)-ti L m(n)n-fj, H {r(n)-p)
=
V in)
-
// L
m (n) n - u H m (n)
.
yields:
n
,1-n
with slopes:
,,
Fn
,,
Suppose there were
converging to (n
ss
,
would be negative.
x
(
x ",n")
-j^^)
*<"^=° =
ss
We show that
= Fx
A2
1
this
ss
(
is
impossible and that under the sufficient con-
both eigenvalues must be
X
,n
ss
by Ai and
+ Gn
)
ss
ix
,n
A2.
positive. Let the
We know
two eigenvalues
that
ss
)
The following inequality holds
Fn
Fx
Hence, A x
both innovation and standardization
= Fx (xss ,n°°)-Gn (xa3 ,nsa )-Fn (x aa ,n ss )-Gx (xa3 ,n aa ).
Ai-A 2
Claim
.
)
Then, either one or both eigenvalues of the linearized system
)
of the linearized system be denoted
+
G„(x",~")
=„„„
0x(x
trajectories featuring
ditions of the Proposition
Ai
X-(n)U=.
«""
•
A2
ss
ix
,n
ss
)
ss
x \n
<
s
(
)
G n X ss ,n ss
G x ix ss ,n°°)'
(
)
>
We need to show that x' {n ss x =o < x' {n ss |n=o- Define A(n) = x in) x =o —
max
>
X n |n=o = pH m n {t~h) ~ Ph r n — P)- We know that m (n) = for n > n
n ss Thus, at n = n max we have:
Proof.
)
)
i
(
)
(
(
)
\
\
)
.
A (n max = -pH
)
(r (n)
-
p)
= -u H
(
V
by Assumption
1.
Next, recall that at n mm
40
^
= -~^ we
Ph
- p) <
1
have y (n)
=
H + L, m (n) =
H+L
f-i-
- ±)
A
and
r (n)
3±k. Thus:
H+L
min v
,
=
e
by Assumption
x{
X
nSS
Moreover, we know that n ss
2.
Since
>X n
(
we know that A r A2 >
0,
has two positive eigenvalues and
be written
in
W
Gn
U= = -
(n)
e [tiith
+
where
A (n) =
?t# (n)
n
(1
The
condition Xi
{x
ss
+ A2 >
can
,n™)
— n) t\l
us
and
(n)]
is
first
dA n
is
m (n)
tt«(")
T L (n)
f*H
VL
=
+ "Ml
1-n'/*a
T^hKn,
+
1
dn \ith
For p
r -*
0, in
T^r^.
c(l-n) ^ H
Thus:
the
'
(n)
BGP
we have
9A n
9n
>
maximum
nL
{n)
e
(n)
n2
^H
H
(1
sufficient condition for x'
(
n)
/Lijj
n)
|n=o
e-1
e
{n)
+
/j„
n]
> n min
for
n £
—
0.
>
Using
(nji*
|n=0-
A (n)
•
that the factor
corresponds to
(it
[n min ,n*]
is
1
e
1
(1-n)
41
\^h
(n)
-
pH
1
e-
€ (I
then
—
(n)
H
+ €-- +
e
1
kl
2
and #dn (-M
Vir (n)!)
71
he
n
x
We know
- n)
(l
£
>
into
[th («)
at n*
Thus,
^n
=
e-1
Ml
—
be upward sloping
a sufficient
0:
nH
n
fig
consider the case where p
n
^^
=
that the locus n
g characterized in Section 2.7).
>
(43)
l-nj
(
be satisfied
BGP. Let
1
1
condition for x' («)i„=o
A
.
FX (X
Gx Xss ,n**)
>
inverted U-shaped, with a
is
maximum
dn
x =o
S
=y(n)-m (n)
X ( n ) k=o
the
\
therefore unstable.
s
Gx
a neighborhood of the
=
=
< X ( n ) U=o for all n > n ss m
establishes that the system
Ax + A 2 >
and x ( n )
showing that
is
lx=o
Then, by the intermediate value theorem,
.
ss
(x
sufficient condition for (43) to
y (n)
ss
Thus, x (^ ss )
unique.
is
as:
X
A
n
n < n
|n=o f°r all
)
Ml/
VMff
)\n=o for a unique value of
(n) \x=o
/# + L
V ^l
1\,
I
f
z/fi
^
—n
J
pL
=
d
dn
+-
*l [n
\tt h (n)
— ^tt^t-
Mw
Ml
t
ir
H (n) (l-n)n
1
J
1
—n
Finally, note that in the
^\
BGP
= &-n
= ^r?i =
-> **
:
(±=%h)
Thus,
n.
the sufficient condition become:
1
e-1
1
- +
e --
—,
n
Since this expression
is
(1
e
fl-n,\
+
c
—
(
h
n
n)
we only need
increasing in n,
n
)
>0
to verify that
positive at
it is
£-1
Timin-
At n m m we have (^p/i)
=
'
-
e-1
= h/ (1 +
n >
Finally, for
n
(
From equations
n'H (n)
(16)
7r# (n)
Substituting
_ T«
7r ff
and
(n)}~T
~
[y
0.
(44)
yields (28) in the text.
/i)
(
n
)
,
(n)
—
1
&4
n
<9nA
1
n
_ f
,
\1 ~ n / ^hX ( n ) k=o
(17):
1_ l[y(n)]^
n
e
e — 1
_
+n>
n)
e (1
n*, rewrite the necessary condition as:
n ) U=o
x( n )|n=o
x'
—e-1
—
1
+e--+
e
Substituting n min
and the condition becomes:
1
(1
—
/
H\^
l[y(n)]^ /
\
n
e
/
e
H~
—
1
\1
I \ ^'
— n/
n) 7
L~ +
1) i
tt^ (n)
(I-")"
1
<
+
n
*H{n)
+ ( _^)i h'-r"
n'<-
into this expression,
1
ne
we have
1
1
e (e
-
\
This implies that
using the fact that in
+(T^r)
BGP ^44 = ^n,
Thus, a sufficient condition for x' ( n )
£i +
f
f-l+
" ^
t
~1
e(l-n)
l-^ + n.+
The LHS
at n*,
it
is
increasing in n, the
will also
be satisfied
be verified that condition (45)
ficient to
f
—
~ir lV
^§^jb; > 777^
1
^
1— n
|n=o
>
+^a
^
sufficient for x' (n) |n=o
we obtain that
1
Now
0.
1
-
^+n+
e.
i
1
1+fe)
decreasing. Thus,
if
always satisfied
when
i
fc
(A5)
^
this condition
for higher values of n. Since n*
is
>
i s:
e(«-l)
is
A (n) =
__J
Mt.
e
RHS
is
'
>
is
satisfied
1/2 (see (32)),
it
can
sum,
is
suf-
(44) holds. In
(44)
prove that the dynamical system with both innovation and standardization
42
is
locally unstable in the limit
for
p < p
for
some p >
where p
—
0.
By
continuity, the
same
result applies
sufficiently small.
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