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A MODEL OF GROWTH THROUGH
CREATIVE DESTRUCTION
Philippe Aghion
Peter Hewitt
No.
527
May 1989
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
A MODEL OF GROWTH THROUGH
CREATIVE DESTRUCTION
Philippe Aghion
Peter Howitt
No.
527
May 1989
J/fT
>n*OtCQ
,r
A Model of Growth through Creative Destruction
by
Philippe Aghion* and Peter Howitt**
May, 1989
*
Department of Economics, M.I.T.
** Department of Economics, University of Western Ontario
The
authors wish to acknowledge the helpful
comments and
criticisms of
Roland Benabou,
Olivier Blanchard, Patrick Bolton, Mathias Dewatripont, Dick Eckaus, Zvi Griliches, Rebecca
Henderson, Louis Phaneuf, and Patrick Rey. The second author also wishes to acknowledge
helpful conversations with Louis Corriveau.
INTRODUCTION
1.
This paper presents a model of endogenous stochastic growth, based on Schumpeter's
idea of creative destruction.
starting with
is
Solow (1957),
attributable to
We
begin from the belief, supported by
that a large proportion of
improvement
in
from the
We
1988)
in
activities
economic growth
in
empirical studies
developed countries
technology rather than the accumulation of capital. The
paper models technological progress as occurring
result
many
in the
form of innovations, which
in turn
of research firms.
depart from existing models of endogenous growth (Romer, 1986, 1988; and Lucas,
two fundamental
respects.
we emphasize
First,
the fact that technological progress
creates losses as well as gains, by rendering obsolete old skills, goods, markets, and
manufacturing processes. This has both posidve and normative implications for growth. In
positive terms, the prospect of a high level of research in the future can deter research today
by threatening the
fruits
of that research with rapid obsolescence. In normative terms,
obsolescence creates a negative externality from innovations, and hence a tendency for
laissez-faire to generate too
much
Obsolescence does not
models have only positive
generate too
little
model innovations
than existing Ones.
fit
growth.
well into existing models of endogenous growth. Those
externalities, in the
growth. The closest in
spirit to
consist of the invention of
Once invented
because of the lengthening of the
form of technology
new
good remains
a
list
our model
is
spillovers,
that
of
Romer
and thus tend
to
(1988). In that
intermediate goods, neither better nor worse
in
production forever. Growth takes place
of available intermediate goods. This model does a good
job of capturing the division—of—labor aspect of growth. But adding obsolescence, by
allowing old goods to be displaced by the introduction of
this
goods,
may
eliminate growth in
kind of model (see Deneckere and Judd, 1986 1 ).
Our second departure
the
new
is
that
we view
view taken by many economic historians
the growth process as discontinuous.
We
share
that individual technological breakthroughs
have
aggregate effects, that the uncertainty of innovations does not average out across industries.
1
This view
is
supported by recent empirical evidence of authors such as Nelson and Plosser
Mankiw
(1982) and Campbell and
includes a substantial
(1987) that the trend component of an economy's
random element. 2
Existing models of endogenous growth do not produce a
exogenous technology shocks are added
We
to the
model, as
is
random
trend, unless
done by King and Rebelo (1988).
take the view that technology shocks should not be regarded as exogenous in an analysis
that seeks to explain the
Instead,
in
GNP
GNP
we suppose
are
economic decisions underlying
the accumulation of
that they are entirely the result of such decisions.
knowledge.
Statistical innovations
produced by economic innovations, the distribution of which
is
determined by the
equilibrium amount of research. This makes the distribution of technology shocks endogenous
to the
model. 3
In
making both of
these departures
our inspiration from Schumpeter (1942,
from recent models of endogenous growth we take
p. 83, his
emphasis):
The fundamental impulse that sets and keeps the capitalist engine in motion
comes from the new consumers' goods, the new methods of production or
transportation, the new markets,... [This process] incessantly revolutionizes the
economic structure from within incessantly destroying the old one, incessantly
,
creating a new one. This process of Creative Destruction
about capitalism.
In Schumpeter's view, capitalist growth
is
inherently uncertain.
is
the essential fact
Fundamental
breakthroughs are the essence of the process, and they affect the entire economy. However,
the uncertainty
upon
is
endogenous
the level of research,
"the prizes offered
"economic
by
life itself
to the
which
in turn
flexible
Although
enough
depends upon the the monopoly rents
that constitute
capitalist society to the successful innovator." (1942, p. 102)
changes
its
own
The present paper develops
elements.
system, because the probability of a breakthrough depends
it is
data,
by
a simple
quite special in
to serve as a prototype
fits
and
model
starts."
(1934,
Thus,
p. 62).
that articulates these basic
some dimensions we believe
it is
Schumpeterian
also simple and
model of growth through creative destruction. 4
THE BASIC MODEL
2.
a.
Assumptions
There are four classes of txadeable objects: land, labor, a consumption good, and a
There
continuum of
continuum of intermediate goods
i
infinitely— lived individuals, with
mass N, each endowed with
6 [0,1].
is
also a
a
identical
one— unit flow of
labor,
and
each with identical intertemporally additive preferences defined over lifetime consumption,
>
with the constant rate of time preference
r
constant marginal utility of consumption
at
in the
economy. There
is
no
disutility
0.
Except
in section 5
each date; thus
from supplying
below,
There
a
unique rate of interest
r is also the
labor.
we assume
also a fixed supply
is
H of
land.
The consumption good
Since
constant returns.
H
is
is
produced using land and the intermediate goods, subject
fixed
we can
express the production function
to
as:
1
y=\
(2.1)
where F' >
input
i,
and
0,
F" <
(F[x(i)]/c(i)}di
0,
y
is
the flow output of
consumption good,
x(i) the
flow of intermediate
parameter indicating, for given factor prices, the unit cost of producing the
c(i) a
consumption good using the intermediate input
Each intermediate good
i
is
i.
produced using labor alone, according
to the linear
technology:
x(i)
(2.2)
where
L(i) is the
=
L(i)
flow of labor used
Labor has an alternative use
research,
X,
i.
to producing intermediate goods.
It
can also be used in
which produces a random sequence of innovations. The Poisson
innovations in the
and
in intermediate sector
economy
at
any instant
is
Xn, where n
is
arrival rate of
the flow of labor
a constant parameter given by the technology of research.
There
is
used
in research
no memory
in this
technology, since the arrival rate depends only upon the current flow of input to research, not
upon past research. 5
Time
continuous, and indexed by x >
is
starting with the
st
t+1
innovation (or with x =
t
The length of each
.
research in interval
t,
its
Each innovation
interval
The symbol
in the case
random.
is
0.
mind such
in
=
=
0)
0,1...
denotes the interval
and ending
flow of labor n
just before the
is
applied to
length will be exponentially distributed with parameter Xn
new
consists of the invention of a
in
line
.
of intermediate goods, whose
We
producing consumption goods.
"input" innovations 6 as the steam engine, the airplane, and the computer,
whose use made possible new methods of production
An
t
If the constant
use as inputs allows more efficient methods to be used
have
of
t
mining, transportation, and banking.
in
innovation need, not, however, be as revolutionary as these examples, but might consist
instead of a
new
generation of intermediate goods, similar to the old ones.
Specifically, use of the
in (2.1)
by the factor y
new
e (0,1).
most modern intermediate good
during interval
t
c
(2.3)
is
the
(i)
=
t
where c~
is
same
is
assume away
all
always produced
goods reduces the cost parameters
lags in the diffusion of technology. 7
in
c(i)
The
each sector, and the unit—cost parameter
for all sectors; thus:
= CqY
c
We
line of intermediate
1
Vie
[0,1],
t
=
0,1,...,
£
the initial value given by history.
(Of course,
it is
always possible
consumption good using an old technology, with a correspondingly old
to
produce the
line of intermediate
goods.) 8
A
sector,
successful innovator obtains a continuum of patents, one for each intermediate
each one granting the holder the exclusive right to produce the newly invented
intermediate good in that sector.
from retaining
a firm that
that right in a positive
becomes
assume, however, that anti— trust laws prohibit anyone
measure of
sectors.
So
the innovator sells each patent to
the local monopolist in that sector until the next innovation occurs.
assume perfect competition
patents.)
We
in all
(We
markets other than those for the intermediate goods and for
The innovator
offers each patent for sale at a price equal to the expected present value
of the monopoly rents accruing to the patent.
The buyer pays
that price as a
lump sum
in
exchange for unconstrained use of the patent. Thus we implicitly rule out royalties and other
The
contingent contracts between the innovator and the local monopolist.
restriction will
parties
be discussed
would want
rationalized either
in (d)
below where
it is
argued
The
a contract with negative royalties.
by noting
that
force of this
without the restriction the two
might therefore be
restriction
by referring
that negative royalties are not in fact observed, or
to
costs of monitoring the output of the intermediate good.
No
The
idea behind this assumption
before
its
new
patent covers the use of a
is
invention, to be patented.
intermediate good in the consumption— good sector.
that potential uses of the
The assumption might
new good
are too obvious,
also be rationalized
even
by the
observation that important innovations like the ones mentioned above have had widespread
uses too numerous and diffuse for anyone to monopolize them.
b.
The Intermediate Monopolist's Decision Problem
The
local monopolist's objective
of profits over the current interval.
is
When
to
maximize
the expected present value of the flow
the interval ends so
do the
rents.
uncertainty in the decision problem arises because the length of the interval
Because no one operates
amount of research
Because of
simply
to
this
at
in a positive
is
measure of sectors the monopolist takes
random.
as
given the
each time, and hence also takes as given the length of the interval.
and because
maximize
The only
the cost of the patent
the flow of operating profits
In equilibrium all prices
symmetry, the same quantity x
i.e.:
K
sunk, the monopolist's strategy will be
at
each
instant.
and quantities are constant throughout the
will be
interval.
Also, by
chosen by each monopolist. This quantity will also be
the total output of intermediate goods in period
of labor in manufacturing;
is
t,
which by
(2.2) equals the total
employment
1
1
=
x
1
We
=
x. (i) di
J
l
L
J
can then re—express (2.1)
F(x
(i)
di
•
l
as:
)
-^t
yt =
The
inverse
demand curve
facing a monopolist charging the price p
is
the marginal product
schedule:
p t = F'(x t)/c t
(2.4)
Thus
w
the decision
as given.
is to
The necessary
w
c
(2.5)
problem
.
t
Assume
)
+ x F"(x
t
t
when F comes from
a
CES
—w
]x
,
taking c and
is
is
downward—sloping:
V x >
when F" <
This condition holds automatically
)/c
).
t
2F"(x) + x F"(x) <
:
maximize [F'(x
marginal—revenue schedule
that the monopolist's
Assumption A.l
to
first—order condition
= F'(x
t
choose x so as
0;
production function.
we show
It
in
Appendix
1
that
it
also holds
follows from (2.5) and A.l that there
is
a
unique solution to the decision problem:
w
x = x(c
(2.6)
f
t
where x
is strictly
)
t
decreasing. Thus the
demand
for labor in the intermediate industry
is
a
decreasing function of the cost (in terms of the consumption good) of producing one efficiency
unit of an intermediate good.
Likewise
(2.7)
we can
express the monopolist's price and flow of profits
p t = p(c t w t )/c t iF'(x(c twt))/c t
,
and
(2.8)
=
7C
5r(c
t
t
=
w t)/c
t
w -cw
[(P(c
)
)
t
t
t
w )]
= -[x(c
£
t
w )]/c
x(c
t
t
w ))
F"(x(c
t
t
t
t
as:
where p
is strictly
%
increasing and
is strictly
decreasing.
For future use, we also assume:
Assumption A.2
We
=
x(0)
:
0.
have not allowed the local monopolist's decision problem
potential competition
from
were potentially binding;
example
=
x(°°)
<*>,
the holder of the previous patent.
i.e. if
the innovation
were non-drastic
Tirole, 1988, ch.10) then the current patent
who would
of the previous patent,
This
be constrained by
because
if
the constraint
in the usual sense (see, for
would be of
face no such competition.
is
to
greatest value to the holder
Thus our model
predicts that
non—drastic innovations would be implemented by incumbent producers, whereas
innovations would generally be implemented by
new
firms.
This
is in
drastic
accordance with several
empirical studies to the effect that incumbent firms implement less fundamental innovations
new
than do
(See, for example, Scherer, 1980.)
entrants.
Whether or not
the innovation
is
drastic
is
determined by whether or not a competitive
producer of the consumption good would incur a loss buying from the previous monopolists,
a price equal to the unit cost
C(p^,w
)
>
w
.
Thus
the condition for a drastic innovation
at
is
1
^c"
,
t
where C(
p
is
)
is
the unit—cost function uniquely associated with the production function F,
The
the equilibrium price of land.
latter is
determined by the conditions for competitive
equilibrium in the markets for land and the consumption good:
C(p^,p
)
=
1
c"
t
where p
is
The
the equilibrium market price of
special case of a
F(x
(2.9)
)
all
Cobb—Douglas
It is
= x^
straightforward to verify that in this case:
(2.10)
x^Cc^a2 17^"
)
1
^
other intermediate goods, given by (2.7).
production
t
and
is
defined by:
(2.11)
P^Wj/a,
(2.12)
7C
=
- a)/a]w x
[(1
t
and innovations
and only
will be drastic if
y<a a
(2.13)
,
t
t
if:
.
Research
c.
There are no contemporaneous spillovers
employing
in the research sector; that
z will experience innovations with a Poisson arrival rate Xz,
—
be independent of other firms' research employment n = n
research firm after the
innovation.
V
i .
w
z.
Thus
takes
It
is
The firm
innovation.
w
Xz.
its
max
0=
max
XzV
> 0}
[z
The Kuhn-Tucker condition
w
(2.16)
>
The value
whose length
(2.17)
v
'
X
z
> o)
z
is
XV
V
is
, ,
V
V.
+ X(n
t+i
on
W
Then
,.
t+i
W
-w
=
is
the
(W
Xn
this
event at
,
,
t+i
0, so (2.14)
z
>
0,
).
Its
flow of labor cost
- W.) - w.z
t
t
can be expressed
as:
z
C
with at least one equality (waloe).
the expected present value of the constant flow
:
is
any firm makes an
—W
.
t
is:
V=—\+-V-.
An
r
be the value of a
Bellman equation: 10
exponendally distributed with parameter Xn
t
W
arrivals will
from successful innovation
— W when
.
+ z)(W.
c
and these
The density of
, .
the expected payoff per unit time
free entry into research.
(2.15)
will receive
and n as given. Thus we have
{
Let
a research firm
be the value of patents from the
expected rate of capital gain
t
Assume
Thus
it
will realize a capital gain
rW, =
(2.14)
V
let
firm makes an innovation
If the
the current instant
Xz
innovation, and
t
z. 9
is,
K over an
interval
is
that there is an important intertemporal spillover in this
Note
reduces costs forever.
by
c
the
same
that will be
model.
An
innovation
allows each subsequent innovation to reduce the unit—cost parameter
It
and with the same probability An
fraction y,
lower by the fraction y than
it
,
but from a starting value c
,
would otherwise have been. The producer of an
innovation will capture (some of) the rents from that cost reduction, but only during the next
interval.
After that the rents will be captured by other innovators, building upon the basis of
the present innovation, but without
compensating the present innovator. 11 This intertemporal
spillover will play a role in the welfare analysis of section 4 below.
The model
is
an act of creation aimed
at
capturing
monopoly
rents that motivated the previous creation.
the
denominator of
V
in (2.17).
the current monopolists,
d.
Wage
(2.18)
"!'
-
'.
w
But
it
also destroys the
monopoly
Creative destruction accounts for the term An
in
research this period will reduce the expected tenure of
and hence reduce the expected present value of
their
flow of rents. ]2 / /
aggregate research employment, (2.16) implies:
is
>
Combining
W
+1
,
n > 0, waloe.
(2.8), (2.17), (2.18)
x + n =
(2.19)
More
rents.
Determination
Because n
.
Each innovation
thus embodies Schumpeter's idea of "creative destruction".
t
N;t =
and the equilibrium condition:
0,l,....
t
yields the condition:
^gj#p^
w
-
(2.20)
.--.-..
^r^Wmk
--•-.^
::
",
:;
:
>
r
"^tVi,
+ An
t+i
=
ft c t+i
w t+i )/c t+i
,
(rA) +
n
-
x
(c
t+1
w t+1 )
<M
N
x <
waloe.
,
;
r
Condition (2.20) describes the supply of labor to the intermediate sector.
unless that sector absorbs
in research,
all
the economy's labor, the
wage
It
says that
will equal labor's opportunity cost
XV~[7.- This condition, together with the labor-demand schedule
(2.6), jointly
10
w
determine
and x
in
terms of
w
,
and x
,,
shown
as
Figure
in Figure
1.
1
clear
It is
now why
a successful innovator
Such a contract would induce each
royalties.
This would have two effects on
below
maximal value given by
the
reducing n
,
Thus the
would be
as given
(2.8).
to offer a contract with negative
local monopolist to
by
(2.17).
Second,
it
seller
of the
second—order small by
a
line
t
creative destruction during interval
creative destruction
royalties,
,
would want
First, it
would
produce more than x(c
would reduce
also reduce the
w
).
the numerator
denominator by
through (2.19). In the neighborhood of the zero—royalty solution analyzed above
the former effect
not.
V
x
N
X.=X(Wt .ct )
on
their
because each one
would
of patents would want negative royalties so as to discourage
t.
The
local monopolists
own by producing more
is
the envelope theorem, but the latter
too small to affect n
.
than x(c
would not attempt
w
)
in the
to
discourage
absence of negative
11
PERFECT FORESIGHT EQUILIBRIA
Using (2.6) and the
fact that
x
is
(PFE)
an invertible function,
we can
Multiplying both sides of (2.20) by c and using the identity: c
= yc
,
w
write
we
=
x
(x )/c
.
obtain:
_1
h* t )Z Trk+ H
x
(3.1)
(x
rc(x
,
which we can re-express by
= ^(x
x
(3.2)
t
t+1
l
))/ Y
l\ +]
<N
x
-.
;
waloe
t
the following first—order difference equation:
= min(G(x
)
t+1
),
N)
where: 13
1
(x
tc(x
G(x
t+1
= x
)
We define
t+1
N -
r/X +
))/y
x
t+1
a perfect foresight equilibrium (PFE) as a sequence {x
satisfying (3.2) for all
t
>
endogenous variables (w
In
0.
,
x
,
y
PFE everyone
,
%
,
n
,
V
)
Lemma
1
V
,
with certainty. 14 However, the length of each
—
i
»
G(x)
is
decreasing in
random. The
x.
is
as follows:
A
foreseen increase in x
hand
it
it
raises the flow of
wage
w
,
will raise the reward
monopoly
rents
tt.(x
X
(
x
t+ i))
reduces the amount of research and hence the amount of creative
destruction next interval (the effect in the denominator).
equilibrium
is
simplified by the following:
next innovator: on the one hand
the other
N]
follows from the fact that both x and k are decreasing functions. The
1
interpretation
to the
and on
is
The mappine x
:
Lemma
economic
PFE
in [0,
can predict the future evolution of the
interval (in real rime %), as well as the identity of each innovator,
characterization of
},-.
this
period
(if
The
increase in
X
V
,
will raise the
x < N), which in turn will induce the intermediate
monopolists to reduce their demand for labor
this period,
x
.
12
Because
(3.2)
is
forward—looking, history does not determine a unique value of x
n
PFE. Typically, there will be a continuum of PFE indexed by the
there will only be a finite
asymptotically.
Proposition
1
More
periodic trajectories to which any
—
converge asymptotically
are, or
with zero growth
—a
"real"
—
"no— growth
a
2—cycle,
growth
(n
=
SE
(NGT)
remains constant. Growth
will be innovations if
We
and only
define real
2—cycle
a real
if
2—cvcle
x° = ^(x
(3.3)
Then
= ^(x).
1
),
x
1
labor
In
SE
the
economy experiences balanced
<
N
and zero
positive if x
is
allocated to research.
= ^(x°), x
to
unique SE.
is
as a pair (x
corresponds
a
.
employment between manufacturing
in the sense that the allocation of
N — x)
PFE
.
or
trap"
as a steady state x
growth
:
.
Furthermore there always exists
define
However
PFE could converge
one of the following
to
a stationary equilibrium (SE) with positive
— SE
We
value x^.
precisely:
PFE
All
:
number of
initial
in
,
1
in
x
)
such
* x° and
if
(x)
and research
x = N, because there
that:
N
t
{x°, x
1
}.
which manufacturing employment
oscillates
between two different values with each succeeding innovation. High manufacturing
employment
in
odd
intervals raises the
reward
to research during
even
intervals,
depresses manufacturing employment during even intervals, through the
and hence
mechanism discussed
above. Likewise, low manufacturing employment in even intervals raises manufacturing
employment
A
but
N
e
in
odd
intervals.
"no—growth"
{x
,
x
}.
trap
is
As shown
a pair (x
in
,
x
)
such that the
Figure 2 below,
NGT
first
exists
three conditions of (3.3) hold,
when G(N) <
G
(N) = x
.
13
Figure 2
**
Even though
NGT defines
stop after period
The
From
1.
interpretation of
=
#0
an infinite sequence (x
then on the
NGT is
economy
that the prospect
~,
}
will
the oscillation of the
perform as
if in
SE
economy
will
with zero growth.
of low manufacturing employment in even
periods so depresses the incentive to research in odd periods that research stops.
As we
shall
can happen even in an economy that possesses a posidve balanced— growth
see, this
equilibrium.
Proposition
chaotic PFE.
It
1
rules out complicated periodic trajectories such as
follows immediately from
follows a monotonic path in
[0,
N].
Lemma
1,
which implies
k—cycles
that x
Therefore the sequence converges.
By
in
with k >
3,
or
odd periods
continuity, the
">
.
.
.
limit point is a fixed point of the
2—cycle
(either real or
intersection xe [0,N]
second—iterate map
NGT). Since ^(x)
is
3f ", which
non—increasing
corresponds to either
in x, there is a
between the graph of J?and the 45 —line;
in other
SE
unique
words, there
is
a
unique SE. 15
We
conclude
this section
with a brief discussion of each type of asymptotic PFE.
or a
14
a.
Stationary Equilibrium (SE)
In the
Cobb-Douglas
case,
-
.
1—a
x
a
x = G(x) =
positive growth
is
determined by:
-|1/(<X-1)
a
y
_r/X +
i.e.
SE with
N -
X
by the simple equation:
—1-oc
a
Y = -
(3.4)
rfX
For growth
to
be positive,
Y
(3.5)
+
•
X
N
-
it is
x
necessary and sufficient to have:
<^.1^.N
In particular, for a fixed value of
Y, X, r,
N, a necessary and sufficient condition for positive
*
growth
is
To
that the
parameter
be sufficiently small:
a < a = XN/(kN +
interpret this result note that
a
is
1—a
is
the Lerner (1934)
intermediate sector.
(price
a
Specifically,
minus marginal cost divided by
intermediate monopolist, and
1—a
sector accruing to the monopolist,
is
yr)
<
\.
an inverse measure of monopoly power
(1—a)
price),
is
in
each
measure of monopoly power
the elasticity of
demand faced by an
the fraction of equilibrium revenue in an intermediate
—
n
-
*
t
.
Thus,
if
monopoly power
is
too
weak (a > a
)
then the flow of monopoly rents from the next innovation would not be enough to induce
positive research
aimed
at capturing those rents
creative destruction in the next interval.
b.
"No—growth" Trap
As we have
seen,
NGT
will exist
iff:
even
if
they could be retained forever, with no
15
G(N)<x C
(3.6)
=
G
!
(N)
^c\J-\
N6J
6(H)
/
X
t
+ 2.
Figure 3
In particular
NGT
will
always exist when the interest
smaller r (or r/X), the easier
are
more
c.
A
dj?
-j
it is
likely to exist in an
Real
to satisfy (3.5):
economy
rate r
is
in other words,
that possesses a positive
sufficient condition for a real 2-cycle to exist
<
1,
"no— growth
Note
that the
trap" equilibria
balanced— growth equilibrium!
2—cycle
is that:
2
(x)
sufficiently small. 16
where x
is
the unique SE. 17
(Figure 4 below).
G(N) < x
(i.e.
(3.6))
and
16
= &C*lJ)
k' 5
s"l.fc
^ aA
Figure 4
Proposition 2
When R =
:
r/X. is
sufficiently small
and the elasticity of demand for intermediate
-
input
(V)
18
)
is
sufficiently close to
The assumption
that
R
is
1.
there exists a real
small guarantees that
7(1
^'(X)=(T1-1)
(3.7)
1
(See Appendix
2).
-
2—cycle.
G(N) < x
.
(See (b) above). Also:
Tl)
1
-I
XT]
Tl
Tl
This expression approaches zero as
r\
approaches
1.
Proposition 2 then
follows from the fact that:
jf-(x) = >'(x)
In the
^'(^(x)) = (^'(x))
•
Cobb—Douglas
case
we
2
have:
1
(3.8)
we know
Then, from Proposition
2,
G(N) < x (rA
We
small).
that there exists a real
2—cycle whenever a
can summarize the conclusions obtained
in the
is
small and
Cobb—Douglas
case
17
by the following picture. Again a can be interpreted as an inverse measure of
monopoly power
the degree of
in the intermediate industry.
-X.
£
Pofitvc
L^
*
\
1
\
W.I«nUi
ZCTt
'.II
c#)
°<
[r*<»~uJ
growth
it
NH1
.
r l^WT)
uuri
(
Figure 5
The reason
in the
model of
for cycles in this
Shleifer (1986). 19
model
is
similar, but not identical, to the reason for cycles
In the Shleifer
model innovations
are exogenous,
and there
are multiple equilibrium strategies for implementing them, because of an aggregate—demand
externality.
That externality
intersectoral form.
As
(3.1)
is
present in this model in an intertemporal rather than
makes
clear, higher
flow of monopoly rents next period. As
manufacturing output next period raises the
we have
seen,
it is
this effect,
of creative destruction, that makes x depend inversely upon x
,
,
together with the effect
and which therefore makes
high manufacturing employment in odd intervals together with low manufacturing employment
in
even intervals a possible equilibrium configuration.
There are several questions that would need
empirical relevance of these
2—cycles. One
is
to
be addressed before judging the likely
their observability
under
realistic expectational
assumptions; equilibria that are stable under perfect foresight (as in Figure 4 above) are often
unstable under learning (Grandmont and Laroque, 1986).
Another
is
the size of the resulting
output fluctuations; modern economies typically allocate no more than 2 or 3 percent of their
18
We
labor force to research.
issues,
and even
if the
Learning raises
regard these questions as open.
movement of
it
could produce large proportional
fluctuations in research, and hence in the growth rate of output.
we
unresolved
labor between manufacturing and research would cause
small proportional fluctuations in the level of output
questions further
many
Rather than pursue these
paper on balanced growth equilibria.
shall focus the rest of the
Accordingly, the main interest of the above analysis
is
not to propose a theory of cycles but to
describe the logic of the model of creative destruction.
BALANCED GROWTH
4.
Time—Series
a.
One of
Properties of Output
the benefits of endogenizing technical
average rate of growth
(AGR)
of the economy.
change
is
we can endogenize
that
Rather than have
AGR depend
exogenous population growth and/or exogenous technical progress as
growth model,
we have made
it
depend upon various
benefit
is
that
what appear
determine
in the
AGR.
no growth,
in
we can endogenize
We
model
factors that affect the incentive to
In this sense
(1986), Lucas (1988) and others.
the variability of the
growth
rate
we
do
are
Another
(VGR). The variance of
technology shocks depends upon the same economic factors that
In the extreme case
VGR
shall
as
Romer
upon
in the neoclassical
research and the fruitfulness of research in reducing production costs.
following the seminal contributions of
the
where the incentive
to
do research
is
so small as to result
will also be zero.
now proceed under
the assumption that the
economy
is
always
in a situation
of positive balanced growth. Thus: 20
x = x
(4.1)
X/y
r
To
(n
see
how
= N — x),
l
•
tc(x
(x))
+ X(N
- x)
a change in the parameters
it
= G(x) (more
r,
X, y,
N
precisely G(x,
r,
X, y, N))
affects the equilibrium level of research
suffices to determine the sign of the partial derivatives J&',
J&,
<-%^,
<%\r,
c^
19
where
<^fis
defined by:
<#(x,
X, y,
r,
N) = G(x)
The following
Lemma
-
is
x.
an immediate consequence of
Lemma
1
and the
fact that
jt
and x are decreasing:
Lemma
We
2
^ = G^ -
:
1
<
0; <#.
>
0;
^
<
0; <%.
'
>
0;
^
+ Jj^ <
0.
then obtain:
Proposition 3
:
The amount of research performed
in a positive
balanced growth equilibrium
will:
fa)
decrease with the rate of interest
(b) increase with the arrival
(c)
parameter X
This proposition
research,
speed
at
is
(i.e.
decrease with
r
means an increase
an increase
in the arrival rate
(the effect
an increase
in the size of
increasing the size of next interval's
fi,
in the required rate
effect will be to reduce the total investment in
X
will
an increase in
total labor
R&
of return on
D.
have both the positive effect of raising the
on the numerator of
increasing creadve destruction (effect on the denominator).
(d)
and
intuitive:
which research pays
(c)
y),
endowment N.
an increase in the interest rate
whose
(b)
,
increase with the size of each innovation
(d) increase with the total labor
(a)
r.
4.1),
The
and the negative effect of
first effect
dominates. 21
each innovation (decrease in y) also increases research by
monopoly
rents relative to today's research costs.
endowment
N
increases research by increasing, for a given
the size of the market that can be "monopolized" by a successful innovator.
20
We now
complete our comparative
of monopoly power
good
in the intermediate industry.
measure of monopoly power was
(inverse)
proceed
in a similar
way and
F(x) = (f(x))
(4.2)
< a <
where
We
Lemma
3
:
statics analysis
The
1.
In the
the
by studying the effect of the degree
Cobb— Douglas
case,
we saw
that a
we can
parameter a. In the general case
define:
a
,
elasticity
r\
again depends positively on a. 22
can then show:
d% >
0,
where <#(x,
ct)
= G(x, a)
—
x.
Proof:
!
Sfa
(x))
x
-
T](x,a)
(x)
1
This expression decreases with a. The
We
lemma
then follows from (4.1).
then obtain the following intuitive result:
Proposition 4
:
An
increase in
monopoly power
(reduction in a) increases the
amount of
research in a balanced growth equilibrium.
We now
derive an explicit expression for both the average growth rate
variability of the
output
(i.e.
growth
yt =
(VGR)
in a
the-
balanced growth equilibrium. The volume of real
consumption goods) during interval
the flow of
ia i\
(4.3)
rate
(AGR) and
t
is:
J^
F(x)
whirh imnlies:
_1
(4-4)
Thus
y t+1
=y
•
yt
.
the time path of the log of real output b\ y(x)
—where
x
is
real time
—
will be a
random
21
step— function starting
at in
y« = mF(H,x) —
(n c~, with the size
of each step equal to the
constant —in y > 0, and with the time between each step {A,,A~,...} a sequence of
variables exponentially distributed with parameter Xn.
iid.
This statement, together with
(4.1),
fully specifies the stochastic process driving output, as a function of the parameters of the
model.
Not surprisingly,
this stochastic
made
at discrete points in
(4.5)
m
where
e(x) is -in
above discussion
y(x+l) =
time
m
process
+
y(x)
e(x); x
=
Suppose observations were
(4.4):
0,1,...
y times the number of innovations between x and x+1.
that
\
1
\ '—,
.
is
...
in y(x+l)
=
follows from the
as:
-lnlhy+
in y(x)
It
a sequence of iid. variables distributed Poisson with
parameter Xn. Thus (4.5) can be rewritten
(4.6)
nonstationary.
Then from
unit apart.
1
is
e(x); x
=
0,1,...
with
e(x) iid., E(e(x))
(4.7)
where
e(x)
=
e(x)
+ Xn
in y.
=
0,
From
var e(x) = Xn(in y)
(4.6)
and
(4.7), the discrete
output follow a random walk with constant positive
AGR
(4.8)
Combining
changes on
AGR
= -Xn
in y,
VGR
= Xn(in
(4.8) with Propositions 3
and
VGR.
2
drift.
lower
AGR. The
it.
.
and 4 we can now sign the impact of parameter
Increases in the arrival parameter, the size of innovations, the size
all raise
AGR.
Increases in the rate of
These parameter changes have the same qualitative effect on
effects are intuitive
with our earlier finding
is
also follows that:
2
y)
of labor endowment and the degree of monopoly power
interest
It
sequence of observations on log
needed before growth
for the growth process.
(in the
is
and straightforward. The
Cobb—Douglas
effect of
VGR
as
on
monopoly power, combined
case) that a minimal degree of
monopoly power
even possible, underline the importance of imperfect competition
22
The
Schumpeterian theme: the tradeoff between current
effects also highlight another
output y = F(x)/c and growth.
Any parameter change
(other than
N)
that raises
reduces y (taking c as predetermined), by drawing labor out of manufacturing.
of
a
that effect is
amplified by the attendant
consumption—good
sector.)
However, the
a static efficiency loss of monopoly.
shift
loss of current output
economy
and research. Whether the economy picks
a
also
(In the case
of the production function in the
Our assumption of
imperfect competition from putting the
AGR
inside
is
not, as
inelastic labor
its
Schumpeter argued,
supply prevents
current frontier of manufacturing
good point on
examined
that frontier will be
in
the next subsection.
The
has noted
positive effect of
N
on
AGR
in a similar context, that larger
footnote 20 above implies that, in the
more
than double the growth rate.
international, since
We
has the unfortunate implication, which
we have
economies should grow
Cobb—Douglas
Romer (1988)
In fact, (4.1
faster.
case, a doubling of population should
(These comparisons must be intertemporal rather than
not dealt with the question of transboundary technology flows.)
accept the obvious implication that this class of models has
little to
say, without
considerable modification, about the relationship between population size and growth
The
variability of output in this
model should be distinguished from
existing real business cycle models, for the following reasons.
has been endogenized here.
An
innovation in the
statistical
that
respond
Second, output never
to
its
rate.
variability in
of shocks
First, the variability
sense in real output
an innovation in the economic sense, the distribution of which depends as
economic decisions
') in
is
we have
the effect of
seen upon
parameter changes.
falls in this
model, contrary to what
is
observed in recessions, and
contrary to what happens in existing real business cycle models. This
innovations are increments in knowledge.
knowledge but a smaller than average
A
is
because
all
negative e(x) indicates not a decrease in
increase.
We
accept the implication that the random
fluctuations of output around the balanced growth path of our
model
tell
a seriously
—
23
incomplete story of the business cycle. Instead they are best thought of as portraying the
(Output can
stochastic trend of output.
however,
fall,
We now
compare the
laissez-faire
AGR
chosen by a social planner whose objective was
c
derived above with the
to
maximize
the
AGR
that
would be
expected present value of
of consumption y(t). The maximized value of the planner's objective function,
historically given,
is
2—cycle.)
Welfare
b.
utility
in a
is
PF(xJ
1
,
determined by the following Bellman equation:
max
rZ =
(4.9)
Z when
+ X(N-x
C
(x <N)
(Z
•
)
t
-Z
t+1
)
t
t
t
The
first—order condition
F'(x
(4.10)
It is
easily
checked
t
Z
-Z
F(x )/c
*-*
+ X(N
- x )(1
[$$
- x*]
r
(4.12)
Y=
.
+
r
Thus
the social
XN
—
*
*
)
=
Xn.
( 1
optimum
same stochastic process
—x
-
zp
y
)
solves:
Hi-y)
X(N
is:
is:
*
to the
problem
)
=
'
where x
this
t
.
=-V
1
t+1
that the solution to (4.9)
*
Zt
(4.11)
= ^(Z
)/c
t
an interior solution for
instead of
-
— yl
y)
a balanced
is
growth path, with real output growing according
as before, but with innovations arriving at the rate
X(N —
x)
=
A.n.
Accordingly laissez-faire produces an
AGR
more
*
(less) than
opumal
if
Which way
x < (>) x
these inequalities go can be checked by comparing (4.1) and (4.12).
former can be rewritten
y
.
as:
7u(x~
(x))
—^~
:
i
(4.13)
Y=
r
+ X(N -
x)
The
24
which, from the definition of % and
X(l-y)(£ - 1)
f^
y=
(4.14)
where
*-
w
=
C
—
wc
'
/
*"
can be rewritten
x,
+ X y
x
x
£
^-
\
is
between price and marginal cost
the constant ratio
The
intermediate industry on the balanced growth path.
reward
as:
to research, deflated
by next period's
real
monopolistic
in the
right side of (4.14) is the private
wage. The right side of (4.12)
the
is
corresponding social reward, deflated by the shadow cost of research next period.
There are three differences between (4.12) and (4.14): The
presence of the term
— XNy
in the
denominator of
A
from
it
his innovation for
comes from
Z
an innovation,
V
X>X
,
.
is
the
private innovator will capture rent
one interval only, whereas the social rent continues forever. Formally
the presence of the term
value function
difference
This corresponds to the
(4.12).
intertemporal spillover effect discussed in section 2(c).
first
XZ
,
in the definition (4.9)
whereas no such term' appears
of the social planner's
in the definition (2.17)
of the private value of
This effect leads the laissez-faire economy to underinvest in research:
i.e.
.2
The second difference
is
X y *- x
the presence of the term
in the
numerator of
(4.14).
This corresponds to a "business— stealing" effect. The private innovator does not internalize
the loss of surplus to the monopolist
presence of
— X(N —
whose
x )Z in the social planner's
subtraction appears in the
third difference
the numerator of (4.14).
maximand
It
comes formally from
in (4.9),
is
i.e.
it
<x
the
whereas no corresponding
problem (2.15) solved by the private research
private firms to overinvest in research:
The
rents he destroys.
This effect leads
firm.
.
the presence of the term
d
(-t-
—
-
F(x*)
l)x instead of [p ') *p
x
This corresponds to the monopolistic distortion effect.
It
— x*
•
]
in
arises
because the flow of returns whose capitalized value the social planner attempts to maximize
total output,
whereas the pnvate value
can work in either direction. In the
V
is liie
capitalized value of profits,
Cobb—Douglas
case
it is
7t
.
is
This distortion
non-existent, since in that case
—x=
F/F'
to
)x
(
=
—
(-2-
l)x.
one another: K = (l-ct)y
,
(In the
Cobb—Douglas
case output and profits are proportional
so the effect amounts to nothing
more than
a multiplicative
constant in the social—planner's decision— problem.)
spillover effect will dominate
The intertemporal
(i.e.
y
social reward
when
—
—
small)
is
there
is
reward
in this case the private
and laissez-faire
innovations are not too large
dominate, leading to an
AGR
(i.e.
y
is
the size of innovations
compared
to research will be small
AGR
will generate an
much monopoly power (a
when
less than optimal. 24
close to zero in the
Cobb—Douglas
On
is
large
to the
the other
big
hand
case) and
not too small), the business—stealing effect will
under laissez-faire which exceeds the optimal level of average
growth. 25 (This case cannot arise in Romer's (1988) model where there
is
no destruction of
existing monopolistic activities). 26
c.
Introducing Learning by Doing
We now
introduce a second source of growth besides innovations:
accumulation of learning— by-doing (lbd)
formalize lbd
is
assume
to
intermediate industry can
that in the time interval
still
instant,
t,
and
x
is
the
amount of labor devoted
natural
way
to
its
intermediate goods through
value of the unit cost parameter
to
at
time x
manufacturing intermediate goods
in
at that
we assume:
| = -gx
(4.15)
where g >
in the
if
if c (x) is the
A
between two successive innovations, the
improve upon the quality of
manufacturing them: more formally,
interval
in the intermediate industry.
namely, the
0.
Thus, learning—by—doing
consumption good sector
in the
at the rate
innovation occurs, productivity will
jump
intermediate industry will increase productivity
g x between two innovations.
as before
by the factor y
When
a
new
.
Following A. Young (1928) and more recently Arrow (1962), Romer (1986), and
Dasgupta—Stiglitz (1988), we assume
that the returns
from learning by doing are shared by
all
26
In particular the intermediate firms
firms in the economy.
their lbd,
which also
over into the research sector. Because of
spills
intermediate firm will choose
its
optimal level of intermediate output
lbd on the other intermediate producers; that
In
in (4.15).
Romer
knowledge, hence
optimum.
Let
in
(x),
which x
Likewise
V
7U
w
(x)
(x)
over interval
follows
(x)
= x
=
grow
and
economy—wide
is
average x
less than the social
however, the spillover of lbd can generate the opposite
too
to a research firm
We
As
(x).
(x)
grows
before,
at the
V
of making the
t
innovation
at
time
x.
confine attention to balanced growth equilibria,
Then x = x(c (x)w
(constant).
Tt(cw)/c
result; i.e. that
fast. 27
tt(x) analogously.
Since K
t.
taking as given the
each
x(i) as before, relying for
an average growth rate under laissez-faire which
denote the value
(x)
is,
this externality,
(1986), this type of externality leads private firms to underinvest in
In this paper,
private economies
Define x
to
experience a complete spillover of
(x)) as
(x) is the
before, so c (x)w (x)
= cw
(constant).
expected present value of k evaluated
constant exponential rate gx over this interval,
it
that:
V
(4.16)
(x)
Vx)
=
+ An - gx
r
Therefore:
£(x
V
(4.17)
(x)
Thus
(x))/c (x)
f
=
r
The
-1
+ XN
+ g)x
first—order condition for positive research to
the level of output x in a
(4.18)
- (X
y
~
rc(*
r
+
(
XN
(non—degenerate) SE
*))/x
(x)
- (?t+g)x
is
occur
is,
given by:
as before,
w
(x)
= XV
,
(x).
27
This equation
is
identical to (4.1) except for the term
—gx
in the
denominator.
previous analysis of stationary states was a special case, with g =
The
AGR
under laissez-faire
)
number of innovations
the
is
that
lim
oo
E(-)
X
°°
occurred until date
innovations follow a Poisson process with arrival rate X
X-t
0.
= limi(gxt-E(t)my)
X-t
T-t eo
t
the
given by:
is
AGR =Umi- E(my(x)-my
(where
Thus
h,
we
x.)
Given our assumption
that
have:
= ln.
Hence:
(4.
AGR =
19)
g(N
- n) - Xn&ry.
In words, the average growth rate
component due
to Ibd
process (— Xnmy).
On
(g(N
—
n))
now
derives from two components:
and a random component corresponding
the other hand,
VGR is
additional growth generated through lbd
VGR =
(4.20)
By
directions:
Xfi(foy)
is
deterministic:
2
example, an increase
equilibrium, thereby reducing
to the innovation
given by the same formula as before since the
contrast with Section 4(a) above, parameters
for
a deterministic
in r will
VGR. The
may
affect
AGR
and
VGR in opposite
reduce the amount of research n performed in
effect
on
AGR is
ambiguous: the random part will
28
be reduced as before but the deterministic part will increase together with the amount of
N — h.
manufacturing labor x =
An
effect
on
AGR
In particular
increase in the speed of learning by doing
AGR
of research
ri.
(equal to
28
When
effect will increase
Ag(N -
n)).
It
AGR. When
(i.e. in
g) will
will also indirectly affect
the initial value of g
is
the initial g
> —Xfiry.
will increase if g
have
AGR
a positive direct
by increasing the
sufficiently small (— Xlny
—
g >
0), the indirect
large however, a further increase in g
is
have a perverse effect on growth by inducing too much research
at the
level
may
expense of learning by
doing.
The tendency
following welfare analysis. Let
satisfies the
Z
(x)
rZ(c
(x))
max
=
^
F(x)
(x<N)
additional term
according
= Z(c
(x))
research
is
brought out more clearly by the
be the social planner's value function.
It
following Bellman equation, analogous to (4.9) above:
(4.21)
The
much
for lbd to induce too
—Z'(c
(x))gx c (x)
+ X(N-x) [Z(y
is
-1
x
c (t))
-
Z(c
t
the social return
(T))]
- Z'(c
t
(x))gxc (T)
t
t
from learning by doing
at
time
x,
to (4.15).
It is
straightforward to verify that
if
research
is
positive then the solution to (4.21)
is:
yF'(x*)
1
=
Z(cfx))
VL
(4.22)
t
where x
"-c^xT|>(l -y)
solves the equation:
(X(1_y)
-
gyXpWO-
*
x
}
y:
(4.23)
r
The comparison between
level x
SY
at the social
as before,
i.e.
-y -i;
L
+ XN(1
)
the equilibrium level x under laissez-faire
optimum
(i.e.
and the corresponding
between (4.18) and (4.23)) involves the same three
the intertemporal spillover effect, the business stealing effect
monopolistic distortion. But there
is
now
and the
an additional source of distortion due to the
introduction of learning by doing as a second factor lbd; terms in (— g) appear in the
effects
29
denominator of (4.18) and
in the
numerator of (4.23):
research under laissez-faire, but reduces
"business—stealing effect" mentioned
The economics of
the
reward to research
innovation.
But
it
this result
activities,
it
words introducing lbd increases
in other
optimum! This
in the social
effect reinforces the
earlier.
can be summarized as follows:
learning by doing raises
by raising the net present value of rents from a successful
does not raise the marginal profitability of hiring manufacturing workers,
since the benefits of lbd are external to the firm (our spillover assumption).
a laissez-faire
economy
will respond to an increase in g
by investing more
This explains
in
why
research at the
expense of manufacturing. With the social planner things will go the other way around:
internalizing the effect of lbd in his decisions, the social planner will respond to an increase in
g by putting
To
more workers
see that learning by doing can
the following example.
Start with an
*
is
into manufacturing, thus fewer workers into research.
almost equal to x
make
a laissez-faire
economy without
economy grow
too fast, consider
learning by doing, where x
= x(g =
0)
*
=x
(g
^ (AGR* - AGR) =
=
0). 29
j|g(g
Then:
+ X
(x*(g)
in y)
-
x(g))
*
=
As we have
x*-x
+
(g
+
dx
dx
seen, -t-
-r-|
\mY)-(g--g)
^
n >
0.
Therefore,
introducing learning by doing will generate too
if
x
(0) is sufficiently close to x(0),
much growth
at
the margin:
AGR*(dg) < AGR(dg).
5.
CONSUMPTION - SMOOTHING: MAKING INNOVATIONS LARGER CAN
DETER RESEARCH
In this section
we
relax the assumption of constant marginal
utility.
Instead, the
instantaneous utility function has a constant elasticity of marginal utility equal to
a>
0.
Thus
30
o"
=
u(c)
1—
1
c
t
.
The pure
rate
of time preferences
elasticity of intertemporal substitution is
case of
a=
With a >
0.
consumers
0,
l/o".
will
is still
The preceding
want
to
>
the constant r
0.
Thus
the
analysis dealt with the special
smooth consumption over time, which
will
affect the equilibrium.
We
confine attention to situations of
are identical, so
have equal consumption
all will
innovation will reduce the marginal
u ' (y
,
/N)/u ' (y 7N)
of substitution equal
=y
to
consumption good just
.
w
=
t
where
V
.
is
Each shareholder of a research firm
have a marginal
Accordingly, a research firm will discount the payoff
after.
,
and
its
rate
and the
to a
marginal condition for positive research will be
XyCT V t+1
That value continues to be determined by
The
will thus
just before an innovation
good of
the market value in terms of the consumption
(2.17). 30
From
(5.1)
and
the
t
st
+
1
innovation.
(2.17):
Vi
*
,„,
equal to y/N. Thus each
in equilibrium,
between the consumption good
y
All households
of consumption by the factor
utility
successful innovation by the factor y
(5.1)
non—degenerate balanced growth.
l
local monopolist's choice
problem
will be the
same
as before.
Thus
in a
balanced growth
equilibrium (5.2) implies:
1
(5.3)
y
Equation
(5.3)
^
1
7
= X ftf*" (*))/*"* (*>
r + X(N - x)
determines
x,
and hence also the level of research
ri
=
N — x.
difference between (5.3) and the previous equation (4.13)
is
The average growth
determined by
comparative
rate
and
statics effects
its
on
n,
variability continue to be
AGR
and
VGR remain
the presence of
The only
C on
(4.8).
unchanged except
the left side.
Thus
all
the
for the effect of the
size of innovations.
Specifically, an increase in the size of innovations (decrease in y)
research, and hence increases the average growth rate and
its
still
increases
variability, if the elasticity of
31
intertemporal substitution exceeds unity (a <
of (5.3), which causes a decrease
left side
substitution
is less
than unity (a >
reduction in research.
payoff
raises the
If the elasticity
marginal
effect
utility
if
the elasticity of intertemporal
AGR
measured
and
new
VGR.
possibility
in units of
is
consumption good. But
of consumption after the innovation relative to before.
when a >
negatively sloped savings schedule that occurs
overlapping generations model with
An
all
1
in the
endowments accruing
The
effect
standard
to the
because
reduces the
it
research by reducing the rate at which marginal utility
The welfare
maximized value of
its
falls after
analysis of balanced— growth equilibria
the social planner's objective function
max [j-^—[F(x )/c
rZ =
l
<N)
(x
~ a
l
1-CT
.]
is
Z
easily verified that if research
z* =
z^/y
1
^=
FCx*)"
is
a_1
where x* solves:
1
^(1-Y
(5 5)
^)
F(x*)
*
F'(x
y l ~° r
x
)
a-1
+ 7iN(l - y
)
*
increases the
variability,
determined by the Bellman
+ X(N
- x.)(Z
.
t+l
-Z)}.
positive then the solution to (5.4)
F'(x*)A(y
o~)
an innovation.
t
t
It is
similar to the
almost the same as before. The
is
t
l
the latter
raises the utility— rate of return to
equation:
(5.4)
is
1
young.
research, and hence also increases the average growth rate and
of (5.3). Intuitively,
also reduces the
two—period
increase in the intertemporal elasticity of substitution (decrease in
left side
it
decrease in y
When a >
amount of
it
A
straightforward.
so strong as to reduce the utility rate of return to research.
is
the
sufficiently close to zero this could lead to such a
is
interpretation of this
to research as
But
in x.
y decreases
then an increase in the size of innovations leads to a
1),
large reduction in research as to reduce
The economic
In this case the decrease in
1).
- Dc|-°.
is
32
Comparison of
(5.5) with (5.3) reveals the
So, as before, laissez-faire
ENDOGENOUS
6.
may produce
SIZE
same
three differences as before
either too
much
or too
little
between x and x
.
research.
OF INNOVATIONS: INNOVATIONS ARE TOO SMALL
UNDER LAISSEZ-FAIRE
This section generalizes the analysis of balanced growth developed in section 4 by
allowing research firms to choose not only the frequency but also the size of innovations.
show
that the equilibrium size of innovations will be
except the technology of research firms. Thus
analyzed remain unchanged. There
is,
independent of everything in the model
comparative— statics effects previously
all the
however, a change
in the welfare analysis.
Specifically, under laissez-faire innovations will be too small.
make
the intertemporal— spillover effect in tending to
the
Rather than assume a linear research technology
This
new
economy grow
at the
effect will reinforce
too slowly.
individual firm level,
helpful to assume an infinitely elastic supply of identical research firms with
curves.
Assume
that to experience innovations with a cost reduction factor
firm must hire v(z,y) units of labor, with v, >
v~ <
0,
0.
with a constant aggregate level of research employment
z and
y within each research
number of
v(z,y)/z
is
firm,
and with c
w
x =
N — n,
(2.5)
and
n,
y
it is
U— shaped
cost
at the rate
with z and y equal to the constants
for
all
t.
Since n/v(z,y)
demand
for
manufacturing labor,
given by:
c
w
=
c
w
v
V
t+1
=F
st
'
(x)
+ x F"(x),
innovation
^ (CW)/C
t+1
= r+(A/k)n
is:
= k(cw)/cw
r+(
k)n
U
A
w t+l = Aw
t+1
is
the
= (Vk)n, where k =
a constant.
is
Xz a
look for a stationary equilibrium
firms, the aggregate arrival rate of innovations is [n/v(z,y)]?t.z
the value of the t+1
r*n
(6J)
We
= cw = constant
In a stationary equilibrium, the intermediate
We
i.e.
3
1
1
33
If a
makes
firm choosing y
w t+i
/z: ->\
(6.2)
c
t
During interval
and
(6.2),
A
and takes
t,
will
make
,cw>~=wy
(—
)y
c
+
1
wage
rate rise to:
.
t
w
and
the
t
the research firm takes as given that
max - w
(6.3)
it
~—
cw =
=
innovation,
the (t+1)
as given.
v(z,y)
Thus
+ A,zAy~
its
V
.
decision problem
will be
determined by
(6.1)
is:
w
(z.y)
Assume
an interior solution.
that (6.3) has
must equal
From
zero.
(6.4)
Vj = vz
(6.5)
v
this
By
free entry, the
maximized expression
in (6.3)
and the first—order conditions:
_1
_1
From
average cost
2
= -vy
(6.3) the output of research
proportional
is
K(z,y)
=y
It
proportional to z/y. Therefore a research firm's
to:
v(z,y)/z.
Our assumption of U-shaped
Assumption A.
is
cost
is:
K(z,y)
:
is strictly
convex.
follows from (A. 3) that (6.4) and (6.5) define a unique equilibrium combination (z,y);
specifically, the
combination
(z,y)
(6.6)
=
arg
that
min
minimizes average
K(z,y).
Since none of the parameters (r,X,N) enter
of them. Similarly, the size of each innovation
power. Thus
all
cost:
comparative— statics
results
is
(6.6),
it
follows that y and z are independent
independent of the degree of monopoly
go through
as before except for those describing
—
34
which
the effects of varying the size of innovations,
are
no longer relevant since
the size
is
endogenous.
To conduct
function Z(c
rZ(c)= max
(6.7)
1
where
m
The
(
{z,y,m}
number of
the
is
F(N ~ mv ( z '7»
+ Xmz(Z( Tc
c
1
—F'
easily
c"
checked
7r
Z(c
1
^
)
mv~ + Xzm
c
\
)
,
l
,y )/z
,
Vj
= vz*" 1
(6.9)
v
=
Our main
=
n
(6.8)
2
)
that the solution to (6.7)
= J-l-7,
Z(c
y
m
1
-<l-Y*r vY*~
))
=
=
is:
-n*)
F(N
*-,
-4
=
l
v(z
are:
=
))
Z'(yc
r
= v(z
l
t
+ A.m(Z(7C )-Z(c
1
where k
problem
+ Xz(Z(yc )-Z(c
m V][
c"
- Z(c .)))
firms. 31
t
- F'
)
l
t
first—order conditions for this
-F' vc;
It is
maximized objective
given by the Bellman equation:
is
)
the welfare analysis, note that the social planner's
,y
)
(k/k )(y
,
-l)n
w/c C
and:
1
result is that innovations are too small in equilibrium:
*
Proposition 5
Proof
:
(6.10)
Y> Y
:
Define a s y
(z
,y
)
v(z*,Y*)/z*(l-y*)
= arg min{K(z,Y) +
ay}.
>
0.
From
A.3, (6.8) and (6.9):
35
From
(6.6)
and
(6.10):
K(z,y) +
From
ay > K(z
these inequalities:
This result
K(zV).
<
K(z,y)
ay > ay
,y
)
+ ay
.
Since a > 0, therefore y > y
.
.
a
another manifestation of the business— stealing effect. The smaller the
is
innovation the larger the losses through obsolescence relative to the gains of an innovation.
The
private firm ignores this cost of raising
By
free entry the
_
y=
(6.11)
__i
rc(g
maximand
i
(x))/*
krA + N -
in (6.3) equals zero.
W
„
to (6.7) yields:
F'(x*)
(6.12)
x *
(1-Y*)
y =
F'(x
)
.
_1
k*rA +
(l-y*
)
Comparison of (6.11) with (6.12) reveal
amount of
fact that
too
k
much
7.
research.
* k
or too
This implies:
x
where k = v(z,y)/z. The solution
*
y.
N
the
same
In addition, the fact that
three distortions as before
y> y
will also generate a distortion.
As
on the equilibrium
will tend to create too little research.
before, the total level of research
may
The
be
little.
RANDOM ARRIVAL PARAMETER: CREATIVE DESTRUCTION CAN DESTROY
GROWTH
In this section
illustrates the force
world
to deter
value of
X
in
innovation a
the arrival parameter
state
,..,X
new X
X
to vary
randomly. This generalization
of creative destruction by allowing an increase in
research in other states. In fact,
one
Let {X.
we allow
}
is
becomes
be the
infinite, the
finite set
we
in
some
states
of the
present an example where in the limit as the
average growth rate
of possible values of
drawn, according
X
to the transition
X.
falls to zero.
At the moment of any
matrix A, and
all
research firms learn
36
Transition into a high—X state could represent a fundamental breakthrough that
this value.
wave of innovations, whereas
leads to a Schumpeterian
transition to a
low
state
could represent
the exhaustion of a line of research.
stochastic equivalent of balanced growth equilibrium
The
manufacturing employment depends only on the
the value of
making
the
an equilibrium in which
of the world, not on time. Let V./c be
state
innovation arid moving into state
t
is
j.
In any state
i,
the marginal
m
expected return to research
in interval
t
is
S
X1
:
positive research occurs in state
i.
If
_
a-.
j
V./c
U
J
research occurs in
This will equal the wage
t+ r
*
all states,
the V-'s
must
satisfy the
Bellman equations:
vY =
(7.1)
Sfl.jIa.jVj/y)
[
j
The
AGR equals —f /h
n.
=N-x(X.
t
l
v
y,
- ^[N-x^Ia-.V./yXIV.
J
J
where
j
f is the
J
;
i=l,..,m
J
asymptotic frequency of innovations. Define
Za.. V./y). Then:
y
J
m
f= I
(7.2)
.
where
q- is
i=l
X. n. q.
i_i. q »
the asymptotic fraction of time spent in state
Our example has
m=
2 and
a..
q1 =
l_ q2 =
;)
1/2 V...
__7_
X~n~
(7.3)
=
It is
if
i.
easily verified that in this case:
37
From
(7.2)
and
(7.3):
=
(? 4)
-
'
To complete
and
r
=
the
N = X,
2 ^1^-9
n
Vl
^2
When
X~ =
1,
=
Cobb—Douglas
Using (2.10) and
1.
=
1/3.
f
=
—2n—n~
[\6X (W
2
1
=
jj
°°,
is
W
Thus
0.
= x
)
and suppose
a =y=
1/2,
(2.12), (7.1) can be rewritten as:
2
}V
1
)\-
2
is
the solution
is
1
-\2 {l
V. = V~ =
V, =
1/4,
2
-[4L,(V 1+ V )r )V
2
2
«
7
V?
/8,
=
which implies n, = n~ =
1/3,
and
which-implies n, =
n^ =
1,
3/4,
raising the productivity of research in
be discouraged in the other
growth
l+
case (F(x)
^(l-^Vj+V^]
the solution to (7.5)
When Xj =
f
state,
one
state
0,
and
can cause research
through the threat of creative destruction, to such an extent
eliminated.
CONCLUSION
8.
We
have presented a model of economic growth based on Schumpeter's process of
creative destruction.
results
Growth
results exclusively
from competition among research firms
consists of a
new
line
from technological progress, which
that generate innovations.
in turn
Each innovation
of intermediate goods that can be used to produce final output more
efficiently than before.
that
9
example, take the
V2 =
that
n
fV 1 = [16(V 1+ V 2 )]
(7.5)
to
+
i
Research firms are motivated by the prospect of monopoly rents
can be captured when a successful innovation
is
patented.
But those
rents in turn will be
destroyed by the next innovation, which will render obsolete the existing line of intermediate
goods.
38
The model possesses
a unique
balanced— growth equilibrium,
constant allocation of labor between research and manufacturing.
of
GNP
growth
follows a
rate,
random walk with
and the variance of
drift.
The
the increments
in
a
economy's average
the variability of the
these parameters of growth are endogenous to the model, and
is
In that equilibrium the log
size of the drift is the
is
which there
growth
depend upon the
rate.
Both
rate of time
preference and elasticity of intertemporal substitution of the representative household, and
upon
the size
and likelihood of innovations resulting from research.
As Schumpeter argued,
the degree of market
power available
an innovation will also be an important determinant of growth.
degree of market power, and shown that
rate
and
its
it
We
to
someone implementing
have parametrized the
has a positive effect upon both the average growth
variability.
The average growth
rate
and
its
variability are also affected
by the extent of
learning—by—doing in the manufacturing sector of the economy. The greater the coefficient of
learning— by—doing, the more research will be undertaken. This will raise the average rate of
growth attributable
to research, but
doing, because the only
way
it
may
for society to
decrease the growth attributable to learning—by-
engage
in
more research
is to
take labor out of
manufacturing.
The average growth
two counteracting
rate
distortions.
may
The
be more or less than socially optimal, because there are
first is a
technology spillover.
A
successful innovation
produces knowledge that other researchers can use without compensation
The counteracting
distortion
internalize the loss to others
is
a "business— stealing" effect.
due
that innovations are too small
to
The
to the innovator.
private research firm does not
obsolescence resulting from his innovations.
We
also
show
under laissez-faire, again because of the business—stealing
effect.
Learning by doing introduces a further distortion because
manufacturing firm. This distortion tends
much
research.
Whether
it
produces too
to
produce too
much
little
it is
external to the individual
manufacturing, and hence too
or too litde growth depends upon the relative
39
efficacy of learning by doing and research as sources of growth.
Under some circumstances
with balanced growth.
finite
it
One
is
a
model possesses
the
"no— growth
trap" in
time because of the (rational) expectation that
would be followed by
a flurry of research activity.
equilibria in addition to the unique one
which the economy stops growing
if
one more innovation were
This expectation makes
it
to take place
so likely that
the rents accruing to the producer of the next innovation will be destroyed quickly that
to create the innovation,
worth trying
real
2—cycle,
in
and research
which manufacturing employment
cycles, as well as those arising
stops.
Another possible equilibrium
oscillates
in
it is
not
is a
between two values. These
from random imitation or a random
arrival
parameter of
innovations, have the interesting implication of a negative correlation between the cyclical
components of productivity
(y
c/N) and
real
wages (w c
),
because y
cVN = F(x(w
c ))/N.
t
Relatively high real wages arise from a relatively high incentive to research, and discourage
the output of goods.
Two
the
final results are
model has a
sufficiently
size of innovations
rate of
is
a
one
worth mentioning here.
low intertemporal
representative household in
First, if the
elasticity of substitution then
can actually reduce the amount of research, and hence reduce the average
growth in the economy. Second, when the Poisson arrival
random function of
state
an increase in the
the
amount of
rate of innovations
research, an increase in the likelihood of innovations in
of the world will result in extra creative destruction in that
state; this
can discourage
research in other states to such an extent that the economy's average growth rate
We
size of
conclude by nodng direcdons for future research.
innovadon eventually
to fall, to
It
would be useful
allow for the possibility that technology
bounded. Also to include a richer intersectoral structure
to study
diffusion and the normative effects of intersectoral spillover.
falls.
to
is
allow the
ulrimately
both the positive dynamics of
The model would
also gain
40
richness and realism
if capital
technical change, or
Rand
D
were introduced, either physical or human capital embodying
capital that affects the arrival rate of innovations. 32
unemployment by introducing search
structure of
demand
demand
externality,
externalities into the labor market, and
for intermediate goods so as to allow for a
might also generate multiple equilibria and allow us
feasible because of the simplicity
changing the
contemporaneous aggregate
reciprocal interaction between technical change and the business cycle.
seem
Allowing
to
study the
All these extensions
and transparency of the basic model.
41
APPENDIX
Assumption A.l
Let F(x) = (x p +
We
HP
1/p
)
,
have:
!/
F' = l/p(xP + HP) P-
Hence:
is
1
satisfied in the
< p <
where
V"
1
1
= xP-
CES
case .
1.
•
F" =
(p
-
1)xP- 2 (xP + hP) 1 ^" 1 + xP" 1
=
(p
-
l)(xP
(xP +
•
HP) 1
(1/p
-
2
2
2
1/
+ hP) P- [xP" xP + xP- RP
^
1)( X
1
P + nP) 1
- x 2 P- 2
^^
2
1
]
p- 2 xP-2 hP<o
= ( P -D(xP + hP) 1/
and
1Hp
•
F" =
-
(p
l)(l/p
-
2)(xP +
HPj^P^pxP-^P- 2
2
3
1/
+ (p-1)(xP+hP) P- ( P -2)xP-
=
(p
-
l)(xP
=
(p
-
l) X
3
!/
+ HP) P- [(1
-
2p)x
2 P"3
+
P- 3 (xP + HP) 17 P-3 [(p - 2)HP -
(p
(1
- 2)x 2 P~3
+
3
+ HPxP" (p -
p)xP].
Therefore:
2F" + xF" =
(x p
1/p_3 hP(p
+ hP)
2
+ xP- [(p-2)HP-(l +
=
(p
<0
-
1)
•
hp
•
(xP +
np
l/
)
-
l)[2xP~2 (x p +
Hp
)
p)xP]]
P-3 [J-2 ][xP(i -
p)
-f
p hP]
2)]
42
APPENDIX
2
Derivation of (3.7)
We
have:
N-
r/X +
Using
(2.5), (2.8),
and the
or' /-N
J? (x)
= x(wc), we
fact that x
X'
=
+
rfk
[
- x(wc)
1/y
(5ic
+
t/X
wc
N
get:
TI^WC)/!
WC]'—
N
x
L
x
'
(wc)
[— x(wc) + ywc x'(wc)]
X(WC)
—
~,
'WC
7t(wc)
=
.
where
„x
=
ri
i)(-yn;
—
wc x (wc)
^
'
~,
'wc
By
_
(r,
,
,m x
(—
YH'wc
v
-
—
1N
1)'
'
N
c
-
1),
.
x(wc)N
the first—order condition (2.5):
F '^Xl-^y)=wcDifferentiating this w.r.t. (wc),
we can
'wc
1
-
-
reexpress
xtV
1/T|
Hence:
Y(l-
^'(X) = (T1-1)
.1
-
ri)
XT]'
1/T|
7t
r\
as:
(X
?7X +
'(wc) + ywc x'(wc))
- x(wc)
7t(wc)
=
+
(x))/y
N -x
43
References
Aghion, Philippe, and Peter Howitt, "Growth and Cycles through Creative Destruction,"
Unpublished, University of Western Ontario, 1988.
Arrow, Kenneth
J.,
"The Economic Implications of Learning by Doing," Rev. Econ. Stud. 29
155-73.
(June 1962):
Azariadis, Costas, "Self— Fulfilling Prophecies," Journal of
Economic Theory 25
(December 1981): 380-96.
Campbell,
J.
and N.G. Mankiw, "Are Output Fluctuations Transitory?" Quart.
(November
Cochrane, John H.,
Econ. 102
857-880.
1987):
"How Big
(October 1988):
J.
is
the
Random Walk
in
GNP?" Journal
of Political
Economy 96
893-920.
Corriveau, Louis, "Entrepreneurs, Growth, and Cycles," Unpublished, University of
Western Ontario, 1988.
Dasgupta, Partha, and Joseph
and Trade
Deneckere,
Policies,
Raymond
J.,
Stiglitz,
"Learning by Doing, Market Structure, and Industrial
Oxford Economic Papers 40 (June 1988): 246—68.
and Kenneth L. Judd, "Cyclical and Chaotic Behavior
Dynamic Equilibrium Model, with
in a
Implications for Fiscal Policy," unpublished,
Northwestern University, 1986.
Grandmont, Jean—Michel, and Guy Laroque,
"Stability of Cycles
Journal of Economic Theory 40 (October 1986):
and Expectations,"
138—151.
Griliches, Zvi, "Issues in Assessing the Contribution of Research
and Development
Productivity Growth," Bell Journal of Economics 10 (Spring, 1979):
Griliches, Zvi, "Patents:
Hausman,
Jerry A.,
Bronwyn
909-38.
92-116.
Recent Trends and Puzzles," unpublished, Harvard University, 1989.
Hall, and Zvi Griliches, "Econometric
with an Application to the Patents
1984):
in
—R
and
D
Models
for
Count Data
Relationship," Econometrica 52 (July
44
King, Robert G., and Sergio T. Rebelo, "Business Cycles with Endogenous Growth,"
unpublished. University of Rochester, 1988.
Lerner,
Abba
P.,
"The Concept of Monopoly and the Measurement of Monopoly Power,"
Review of Economic Studies
Lucas, Robert E.
Jr.,
"On
the
1
157—175.
(June 1934):
Mechanics of Economic Development," Journal of
Monetary Economics 22
(July 1988):
3^2.
Nelson, Charles, and Charles Plosser, "Trends and
Random Walks
Journal of Monetary Economics 10 (Sept. 1982):
Pakes, Ariel, "Patents as Options:
Some
in
Macroeconomics,"
139—62.
Estimates of the Value of Holding European Patent
Stocks," Econometrica 54 (July, 1986):
755-84.
Reinganum, Jennifer, "Innovation and Industry Evolution," Quarterly Journal of Economics
100 (February 1985):
,
Jennifer,
81-99.
"The Timing of Innovation: Research, Development and Diffusion,"
unpublished, University of Iowa, 1987.
Romer, Paul M., "Increasing Returns and Long— Run Growth," Journal of Political
Economy 94 (October
,
1986):
1002-1037.
"Endogenous Technical Change," unpublished, University of Chicago, 1988.
Scherer, F.M., Industrial
Market Structure and Economic Performance. 2nd
ed. Chicago:
Rand McNally, 1980.
,
Innovation and Growth: Schumpeterian Perspectives. Cambridge,
MA: MIT
Press, 1984.
Schumpeter, Joseph A., The Theory of Economic Development.
New
York: Oxford University
Press, 1934.
,
Capitalism, Socialism
and Democracy.
New
York:
Harper and Brothers, 1942.
Shleifer, Andrei,
"Implementation Cycles," Journal of Political Economy 94 (December 1986):
1163-1190.
45
Solow, Robert M., "Technical Change and the Aggregate Production Function, Review of
Economics and
Tirole, Jean,
Statistics
39 (August, 1957): 312—20.
The Theory of Industrial Organization. Cambridge,
MA:
M.I.T. Press, 1988.
Woodford, Michael, "Stationary Sunspot Equilibria," unpublished, University of Chicago,
1986.
Young, Allyn, "Increasing Returns and Economic Progress," Economic Journal 38 (Dec.
1928):
527-42.
.
46
FOOTNOTES
In the Deneckere— Judd model a constant proportion of existing intermediate goods
disappear each period. The equilibrium of the model exhibits no growth.
!
2 More
recent
work by Cochrane (1988) shows
that this
evidence
is
not,
however,
conclusive.
3
Thus policy analysis based on
critique because
it
treats the variability
King— Rebelo model would be subject to the Lucas
of growth as invariant to the policy determinants of
the
growth.
4 Mention should also be made of the preliminary work of Corriveau (1988), who is
developing a similar model of growth and cycles in a discrete time setting. The discreteness
of time In Corriveau's model introduces complications from the possibility of simultaneous
innovations, which we avoid by our assumption of continuous time. In Corriveau's framework
all innovations are non—drastic, and it is not possible to parameterize the degree of monopoly
power in the economy as we have done.
5
Some consequences
of introducing
memory
are discussed in section 8 below.
6 Scherer (1984) combines process— and input—oriented R and D into a measure of
"used" R and D, which he distinguishes from pure product R and D. He estimates that during
the period 1973—1978 in U.S. industry the social rate of return to "used" R and D lay between
71% and 104%, whereas the return to pure product R and D was insignificant.
7 Gradual
diffusion could be introduced by allowing the cost parameter after each
innovation to follow a predetermined but gradual path asympotically approaching the limit c
and then
to
jump
to c
upon
the next innovation
and follow a gradual path approaching c
,
,
This would produce a cycle in research within each interval, as the gradual fall in costs
induces manufacturing firms to hire more and more workers out of research until the next
innovation occurs.
8
Note that
this description
of technology implies increasing returns in the production of
which are land, labor (as embodied in the
intermediate goods) and knowledge (as embodied in the c's). If land and labor alone were
doubled, then the flow of intermediate input would also double, and hence, by the assumption
of constant returns, final output would just double. Since there are constant returns in these
two factors holding the third constant, there are increasing returns in all three.
The technological specification of the economy follows that of Griliches (1979). There
exist empirical studies casting doubt on our assumptions that there are constant returns in
research in that a doubling of research will double the rate of innovations (Griliches, 1989),
that there is no variability in the size of innovations (Pakes, 1986), and that there is no
variability in the Poisson arrival rate of innovations (Hausman, Hall, and Griliches, 1984).
The first two assumptions can be relaxed with only notarional difficulties. The third is relaxed
in section 7 below.
Finally, we could modify the intermediate technology (2.2) to allow a specialized fixed
factor in addition to labor. We could interpret the specialized factor as unskilled labor, and,
interpret the labor that can be used in either manufacturing or research as skilled labor, thus
adding plausibility to the model. The work of Romer (1988) suggests that no substantive
difficulties would be created.
the consumption good, the ultimate inputs of
The alternative assumption that research firms have identical U—shaped functions with
a small efficient scale produces identical results. This alternative is employed in section 6
9
below.
10
Bellman equations
like (2.14) are standard in the
patent—race literature (see Tirole,
47
1988, ch. 10). Several more are introduced below. This footnote presents an alternative
Let T, T', and T" be, respectively, the time of the next innovation, the time at
which the other firms would experience their first innovation if they held employment constant
derivation.
and the time
forever,
at n
employment constant
at
which
firm would experience
this
Under
at z forever.
the
its first
innovation
no— spillover assumption, T' and T"
if
it
held
its
are
independent random variables, expoentially distributed with respective parameters An and Xz,
T = min
and
(T", T").
+
with parameter X(n
follows
It
z)
and
already mentioned,
T
that the probability that the next
on the date of
firm, conditional
that, as
that innovation,
is
exponentially distributed
innovation will be
Prob {T = T' |T)
is
made by
the constant z/(n
+
this
z).
'
Thus, conditional on
',
-rT
W (T)
the value of the research firm
V/
t
t+ 1+
Prob{T=T'|T}V
is:
—
t+1
'
o
e
w.zdx
t
Thus:
W (T) = e -rT W t+1 +
By
+ z))V
(z/(nt
t
-rT N
-1 n
(1-e
)
t+1
w
z
t
definition:
W
1
= Max
{z >0}
-rT
") =
Since E(e
EW
f
(T).
l
X(n +
z)/(r
+
+
A.(n
z)):
f
+ z)W
X(n
W
t
Max
=
1
[z
> 0}
t
r.
+1
+ ?L(n
+ XzV
.+„
i+{
- w
z
t
z)
t
The
last
equation
n It
is
is
equivalent to (2.14).
straightforward to relax the
assumpuon of complete
spillover by allowing the
th
t
st
innovator to have an advantage in finding the t+1
innovation; to have an arrival parameter
X' > X. We have worked out this analysis under the assumpuon of U—shaped cost with small
efficient scale (see footnote 9), with no important change in results.12Our analysis of research is derived from the patent—
race literature surveyed by
Reinganum (1987) and Tirole (1988, ch. 10). Within that literature the paper closest to ours is
Reinganum (1985), which emphasizes the affinity to creative destruction. Most other papers
do not have the sequence of races necessary for creative destruction. Our paper goes beyond
that literature by embedding the sequence of races in a general—equilibrium setting. Thus
(a) the flow of profits Jt
resulting from an innovation will depend upon the amount of
,
research during the next interval, (b) our welfare criterion is the expected lifetime utility of the
representative consumer, (c) we characterize the time—series properties of GNP, and (d) in
section 5 we find that the size of innovations affects the rate of discount applied to research.
In addition, the analysis in secdon 6 of the endogenous determination of the size of
48
innovations, and the analysis of cycles in section
patent—race
13
G
3,
go beyond anything we have found
in the
literature.
is
well defined on (0,N), by A. 2.
The methods of Azariadis (1981) or Woodford (1986) could also be used to construct
rational—expectations sunspot equilibria, at least in some cases.
14
This distinguishes our model from the growth
(1986), which may have no steady state.
15
Romer
model with
externalities
developed
in
n
16
When
\~,
—
7t(x
r=0, x
is
c
(x ))/y
defined by:
1,
l
= N;
N
whereas
l
G(N) = x
By
5t(x
(N)Vj =
x(+») =
< x, by
(T
A.2.
continuity, (3.6) will hold for r (or r/X) positive but small.
17
occur
A
2—cycle also exists if G(N) > x and ^(x) > 1. However,
Cobb—Douglas case (see Aghion and Howitt, 1988).
real
in the
18T1(X)
|
|
this
can never
F'OQ
H
xF'(x)
19 Likewise,
•
the reason for cycles
is
similar to that underlying the chaotic fluctuations in
model of Deneckere and Judd (1986). They model innovations as Romer (1988) does, with
no creative destruction. Their dynamics are backward looking, because the incentive to
innovate depends upon the existing number of products, which depends upon past innovative
activity, whereas their assumption that monopoly rents are not destroyed by future innovations
makes the incentive independent of future innovative activity. Their cycles arise because the
only rest—point of their backward—looking dynamics is unstable.
the
20 In the
Cobb—Douglas
X
(4.1'):
Y
r
21 In the
22r|(x,a)
1-q
a
+ X(N -
case, (4.1) can be expressed by:
1-a
a
x)
(N -
t/X
+ n
extension of the model considered in section 7 the second effect
= -F'(x)/xF"(x) =
f(x)
CZ (x)
(l^x)x[f'(x)]
23
h)
may
dominate.
.
-xf(x)f'(x)
Two
additional kinds of spillover can easily be included. First, researchers could
from the flow of others' research, so that an individual firm's arrival rate would be a
constant returns function X(z,n) of its own and others' research. Second, there could be an
exogenous Poisson arrival rate \i of imitations that costlessly circumvent the patent laws and
benefit
clone the existing line of intermediate goods. Both would have the effect of lowering AGR.
Also as we showed in Aghion and Howitt (1988), the inclusion of fi would introduce another
source of cycles in the economy, since each imitation would make the intermediate industry
perfectly competitive, which would raise manufacturing employment, until the next innovation
a
49
arrives.
24 More formally, as
approaches a
^In
will
—XN
-—
1—
—
— > rAN,
then n
falls to the
.
*
-rrr,
approaches zero, whereas x
x
strictly positive limit.
the
when y^- >
lower limit
.
y
Cobb-Douglas
case, if
>
0, for all
y (see 3.5) whereas n* =
XN/ctr.
26 There seems to be a presumption among empirical students of R and D that spillovers
dominate business stealing in a welfare comparison. See, for example, Griliches (1979, p.
99).
27 The
formal source of these different results is that in our model research is a separate
from the production of the goods embodying knowledge. The activity whose doing
activity
creates external learning
is
Doing
the production.
it
more intensively requires
less research to
be undertaken. Romer follows Arrow (1962) in assuming that the two activities are
inseparably bundled, thereby eliminating this tradeoff.
28 This
follows immediately from equation (4.18), using the fact that both
decreasing function.
the Cobb-Douelas
intnecobb
Douglas
2 9Tn
casecase.
(1
T?
)N
(i_ a )
ayv/X ~ ( y)N "
1
*T
"T
+ £y " (l-y)(l—oc)
~
tc
and x are
^
'
30 Someone
who gave up a marginal unit of consumption to buy shares in a local patent
would receive a constant dividend of tcTV until T, the date of the next innovation. The
expected
utility
gain of the transaction would be:
U'(C)7C/V
T
Equating
3
this to the
expected
utility cost
u'(c
)
yields (2.17)...
_
_
.
..
We assume that no non—negativity constraints bind.
We have examined the consequences of a limited kind of R and D capital by
allowing for the arrival parameter X to increase randomly in the middle of an interval, and then
to remain high until the next innovation, so that the probability of an innovation depends not
just upon the flow of research but also upon the length of time over which the innovation has
been sought. The analysis is formally like that of section 7 above, and most of the
comparative—statics results of section 4 go through.
32
5S&U
Date Due
6
J
1991
NOV 15
mi
.
.
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3 1 awt
•
MAR.
24
FEB
1996
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Lib-26-67
MIT LIBRARIES DUPL
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