Entry and Growth in a Perfectly Competitive Vintage Model
Peter Funk
University of Cologney
22nd June 2007
For helpful suggestions and comments I am grateful in particular to Raouf Boucekkine, Kerstin Burghaus, Giacomo
Corneo, Martin Hellwig, Ulrich Kamecke, an associate editor and two anonymous referees.
y
Staatswissenschaftliches Seminar, Universität zu Köln, Albertus-Magnus-Platz, 50923 Köln, Germany.
+49/221/4704496, Fax: +49/221/4705143, Email: [email protected]
1
Tel.:
Abstract
A perfectly competitive vintage-knowledge model of Schumpeterian growth is introduced to study
the relation between growth, technology-lifetime, entry, and productivity-dispersion. The incentive
to innovate is generated by the productivity-dispersion (latent in traditional vintage models) between
new and old plants, rather than by monopoly rents. The model has a unique steady-state REE
with endogenous growth. The endogenous extent of entry constitutes a bu¤er, dampening the e¤ect
of research-e¢ ciency and completely neutralizing the e¤ect of population size or population growth
rates on per-capita income levels and growth rates. Variations of research-e¢ ciency lead to a negative relation between growth and vintage-lifetime and a non-monotonic relation between growth and
productivity-dispersion.
Keywords: Endogenous Growth, Productivity-Dispersion, Vintage-Model, Competition
JEL: D40, D90, O30, O40
2
1
Introduction
This paper introduces a vintage-knowledge model of perfect competition and endogenous growth to
study the interaction between productivity growth, active lifetime of technology generations, productivitydispersion, and entry (or rejuvenation).
Real economic growth, driven by technological progress, is characterized by the gradual replacement of old methods of production by new methods. Known techniques and commodities neither
remain in use perpetually nor are they “scrapped” instantaneously as soon as new and better techniques (or products) are introduced. Related to this …rst observation is the following second: not
all plants and …rms producing close substitutes are equally productive. In fact, in their review of
empirical studies using longitudinal microdata on productivity, Bartelsman and Doms [5] summarize
evidence about wide gaps in productivity between best and worst …rms and plants. They remark that
“of the basic …ndings related to productivity and productivity growth, perhaps the most signi…cant
is the degree of heterogeneity across establishments and …rms in productivity in nearly all industries
examined.” Extensive productivity-dispersion (between …rms producing close substitutes) naturally
entails a certain dispersion of (short-run) pro…tability. On the one hand, the fact that relatively ine¢ cient …rms or plants remain active allows more e¢ cient …rms to realize short-run rents which are
the higher, the larger their productivity-distance to the least e¢ cient active competitor. On the other
hand the productivity-dispersion shows that more e¢ cient …rms are either not willing or not able to
serve their whole market.
The present paper captures this pattern of technical change, gradual replacement, and productivitydispersion as complementary consequences of only few key assumptions. Progress is modelled as a
process of creative destruction in which the rate at which new technologies are introduced as well as the
active lifetime of technologies and the resulting productivity-dispersion are determined endogenously:
Many small …rms produce a single consumption commodity. Due to previous investment in research and adoption (henceforth “R&A”) directed towards adopting and possibly improving existing
knowledge as well as embodying this knowledge into installed capacity, each incumbent …rm has a
predetermined capacity and predetermined labor productivity, which (at equilibrium) only depends
on the …rm’s technology vintage. Relatively young technologies are more productive than older technologies. Only su¢ ciently productive (and hence su¢ ciently young) technologies will be active in
each period. How many vintages are active, depends on actual labor employment in the …nal-good
production and is determined by the condition that the least productive and currently oldest active
…rm just breaks even. The age-dispersion of active technologies naturally generates a productivity3
dispersion among active technologies, which in turn translates into positive rents for any (but the
marginal) …rm not yet driven into obsolescence. The anticipation of these pro…ts motivates costly
R&A-activity aimed at reaching or surpassing the current frontier of applied knowledge. The extent
of entry attracted by the prospect of future pro…ts and the amount of R&A by individual entrants
in each period shapes the future distribution of …nal-good …rms. While this R&A-activity builds on
all incumbent knowledge and thus on the fruits of previous R&A, each individual future …nal-good
…rm has nevertheless to perform his own R&A to embody old as well as possibly new or improved
knowledge into its future capacity. Decreasing returns to scale with small e¢ cient scales of individual
…rms (given R&A) prevent fresh innovations from serving the whole market and allow established
technologies to remain active for a while. At the same time these decreasing returns to scale render
strategic considerations or monopolistic features inessential to generate rents and an incentive to innovate. The prospect of (quasi-)rents necessary to …nance today’s innovation are thus generated as a
natural by-product of the productivity-dispersion inherent in traditional vintage models, rather than
by the local monopoly-power assumed in most endogenous growth models. In other words, there will
be perfect competition on all levels of the model.
The …rst aim of this paper is to show that the productivity-dispersion already present in older
vintage models does indeed provide the necessary incentive to innovate and to build a genuine model of
endogenous technical change: at the unique steady state rational expectations equilibrium the growth
rate of productivity and per capita income is strictly positive.
The second aim is to highlight the relation between productivity growth, technology lifetime and
productivity-dispersion lying at the heart of the present approach, but absent from more standard
growth models. For instance, an increase of R&A-e¢ ciency raises productivity growth and reduces
technology lifetime.
Thus an empirical dispersion of R&A-e¢ ciency induces a negative relation
between productivity (and per capita income) growth on the one hand and the active lifetime of
technology vintages on the other hand.1 In contrast, there is no such clear cut relation between the
productivity-dispersion and growth. Depending on the exact speci…cation of the R&A production
1
This prediction is in accord with [27], [29], [50]. For instance, Hsieh [29] observes that the lifetime of airplanes in
di¤erent countries – varying between 5 and 35 years – is much longer in countries with slow productivity growth and
provides some preliminary evidence for a negative correlation between active technology lifetime and productivity growth
rates. Williamson [50] quotes evidence that on Rhode Island a cotton mill installed in 1813 and embodying the then up
to date technology remained active for at most 14 years, while similar equipment had a much longer active lifetime in
England at the same time.
4
function the relation is either non-monotonic or positive.
The third aim of this paper is to study the role of the endogenous rate of entry or rejuvenation for
comparative statics, in particular where it a¤ects standard comparative statics results. For instance,
increasing the R&A-e¢ ciency will attract more entry and duplication, dampening the positive growth
e¤ect the R&A-e¢ ciency would otherwise have. More strikingly, the extent of entry constitutes a
perfect bu¤er, completely neutralizing the e¤ect of population size on the steady state growth rate
or the steady state level of per capita income. This may help explain why, following [43], neither
population size nor population growth rate are “signi…cantly related to growth”. An increase of the
population growth rate has no permanent e¤ect on the per-capita income growth rate, a negative
e¤ect on the level of per-capita income, and a positive e¤ect on the R&A-intensity and its share in
GDP. The present approach may thus help reconcile the observed constancy of per-capita growth
with the observed surge of resources spent on research (in contrast to the Romer-type endogenous
growth models) without giving up the possibility of perpetual growth even when these resources and
population no longer grow (in contrast to the Jones-type semi-endogenous growth models). Finally,
policy can in principle raise the long-run per-capita growth rate by subsidizing R&A-activity, although
this e¤ect is insigni…cantly weak in plausible numerical examples.
Relation to previous vintage models Many of the leading growth theorists of the 1960s have
described growth within perfectly competitive vintage models. They were interested in the relation
between growth rates and the endogenous active lifetime of technology generations (see for instance
[41], [48], [47]). Exogenous technical progress raises wages, reducing the short-run pro…ts that can be
realized by incumbent technologies of older vintages until they are scrapped. A similar “in‡ationary
wage scheme”determines the active lifetime of technologies and the age structure of active technologies
in the present paper, except that the rate of technical progress in‡ating wages is endogenous too.
Although the simultaneous activity of machines and techniques of di¤erent ages is an obvious fact,
only the latest technology vintage is active in most post-1960 growth models with creative destruction.2
2
The reason for this narrowing of scope lies in the technical di¢ culties of vintage models. The present paper avoids
most of these di¢ culties by restricting itself to steady state equilibria. Motivated by the omnipresence of heterogeneous
vintage structures in reality and the inconsistencies of non-vintage growth theory with aggregate data (see [16]) research
in vintage models has recently been resumed (see for instance [4], [6], [11], [13], [15], [29]). A common issue of this
literature is the possibility of endogenous ‡uctuations (at non-steady state equilibria), caused by the replacement echoes
of investment. [14] show that these ‡uctuations can help explain the observed negative short-term relationship between
investment and growth. [12] highlight the importance of incorporating the negative obsolescence e¤ect and the positive
modernization e¤ect of embodied technical progress on the incentive to invest and to innovate.
5
Notable exceptions are [4], [14], or [29]. In contrast to these previous models with endogenous growth
and endogenous and overlapping (…nite) active vintage lifetime, the present framework remains in a
world of perfect competition with small …rms. This adds an endogenous variable (number of entering
…rms). In exchange I exogenously …x another variable (capacity or physical capital), which is an
important endogenous variable in most vintage capital models.
Relation to previous perfectly competitive models with endogenous technical change
The idea of a formal model of endogenous growth with perfect competition can already be found in
Shell [46]. While Shell’s model is partial equilibrium, [22] and [28] analyze general equilibrium models
with perfect competition and endogenous growth (and exogenously …xed vintage lifetime).3
Perfectly competitive models of endogenous technical change share the assumption that –given its
technological knowledge –an innovating …rm cannot replicate productive capacity without experiencing decreasing returns to scales. This may be interpreted as a short-cut to capture the fact that any
idea is initially known only to a …nite number of people or is initially embodied only in …nite amount
of capital. A …rm “owning” an idea embodied in its human and physical capital cannot instantly
double its output by doubling the number of workers and equipment. Transferring the knowledge
to new workers will take time and may in addition bind the already knowing workers away from
production. This observation is made explicit in the theory of Boldrin and Levine [8], [9], [10]. In
contrast, the present model boldly assumes the limited returns to scale of the individual …rm given
its R&A-investment. An alternative way to justify this assumption is to interpret the initial investment in technology not as an investment that creates (and implements) new knowledge but rather as
investment that merely implements or adopts existing abstract knowledge to a particular production
3
Concerning the nature of competition and endogenous growth the present model is a representative consumer
continuous-time version with rational expectations of the discrete time OLG model in [22]. In contrast to the present
paper, new knowledge can be imitated free of cost after an exogenously …xed number of periods. Similarly, in [28] new
knowledge can already be freely copied after one period. In [28] each …rm chooses the e¢ cient scale of inputs (‘capacity’)
of the next period’s small …nal good technology generated by the innovation as well as the productivity at these scales.
By …xing e¢ cient scales, the present paper simpli…es on an important issue, which is explicitly modelled in [28].
While in the present paper (as in [22], [46]) there are many small innovators in each period, there is a single small
innovator per period, whose achievements can be copied free of cost after one period in [22], [23], [24], [25]. [24] also
relates the perfectly competitive endogenous growth models to the literature of induced change of the 60s ([18],[33],
[44]). In contrast to [23], [24], [25], [28], [46], [49], the active lifetime of individual technology vintages is determined
endogenously in the present model.
6
process. This interpretation relates the present model to the literature on technology adoption (see for
instance [37], [38], [39]). When assuming the pure adoption interpretation, the externality of today’s
R&A-activity on future R&A-e¢ ciency (which makes possible persistent growth) translates into an
externality of today’s adoption activity on the public stock of abstract knowledge.
The remaining sections are organized as follows. Section 2 introduces the vintage model of perfect
competition and of endogenous growth and de…nes (steady state) equilibrium. Section 3 derives essential equilibrium conditions and shows existence and uniqueness of steady state equilibrium. Section 4
studies the comparative statics. Section 5 concludes.
2
A vintage model of perfect competition and endogenous growth
The model has only two levels of activity at the production-side of the economy:
1. the production of a homogeneous …nal-consumption good by many small …rms, each of which is
characterized by a level of “knowledge-capital” determining its labor productivity and
2. the production and installation of new knowledge-capital by many small R&A-labs (entrants or
established …rms) who can later use their knowledge-capital as …nal-good producers. Building
on (economy-wide) past knowledge each individual …rm has to produce and install his “own”
knowledge-capital. The quality of the knowledge-capital (future labor productivity in …nal-good
production) depends on the individual R&A-intensity in a deterministic way.
Consumers are modelled as in the Ramsey-Cass-Koopmans model and equilibrium will be rational
expectations equilibrium, con…ned to steady-state equilibria.
2.1
Final-good production
At period t 2 <+ …nal-good producer j of type
2 ( 1; t] is characterized by his capacity, …xing
the e¢ cient-scale amount of labor, and by the number A of output units he can produce with one
unit of labor. A is ’s knowledge-capital arising from past R&A investment which will be described
below. His type will later be identi…ed with the time he …rst entered the market or last up-dated his
knowledge-capital, which I call his vintage. In the basic version of the model I normalize his capacity
to one unit of labor (if he decides to be active).4 At t the index of the most recent technology is
4
= t.
Note that the crucial assumption allowing a sound micro-foundation of the perfect competition assumed here is that
given past decisions, individual …rms have e¢ cient input scales that are small relative to the total supply of these inputs
7
The total number (mass) of small …nal-good producers of type
at t is completely described by the range (A
f
g
2( 1;t]
1 ; At ]
is
. The …nal-good sector
of known technologies at t and the distribution
of incumbents over this range. I denote by (t) the vintage of the oldest technology still
in use at t, which needs of course to be determined endogenously. Tt := t
(t) then is the active
lifetime of this technology.
There is perfect competition in the …nal-good market and the labor-market among all existing
…rms of all types at t; in the sense that each …rm takes output prices (normalized to one) and wages
wY t as given. The short-run pro…t of a …rm with productivity A at t therefore is A
wY t if it is
active and 0 if it remains inactive. Short-run pro…t-maximization implies that all incumbent …rms
with A
wY t > 0 are active at t, incumbent …rms with A
wY t = 0 are indi¤erent between being
active or inactive, while all other …rms are inactive. Given wages wY t , the oldest active vintage (t)
is thus determined by the condition wY t = A
(t)
younger …rms’production:5
Yt =
Z
and the aggregate output at t is the integral over all
t
A d .
(1)
d .
(2)
(t)
Total labor employed in the …nal-good sector is
LY t =
Z
t
(t)
Equivalently, given LY t (2) determines the oldest active vintage (t), wages wY t = A
(t) ,
and thus,
using (1), aggregate production at t. The resulting aggregate short-run production function (Yt as a
function of LY ) has the standard concave form (see Figure 1), since reducing employment LY t drives
out of business the least productive …rms …rst. If many …rms have recently entered (Tt = t
(t)
small) with technologies almost as good as the currently best, the short-run aggregate production
function (as a function of labor input) is nearly linear on a fairly large domain (Figure 1A). If, in
(see [21]). This is also the essential assumption behind the absence of scale-e¤ects. To simplify the exposition I normalize
this e¢ cient scale input to one unit of labor. The simplest way to guarantee that an active individual producer will in
fact employ exactly one unit of labor is to assume that an individual technology
produces no output with less than
one unit of labor and produces A units of output with at least one unit of labor. Note that while this extreme form of
decreasing returns to scale simpli…es the exposition, it is neither necessary for the perfect competition assumed here (see
[36]) nor for the absence of scale e¤ects. Nevertheless, by …xing the amount of physical capital embodying technological
knowledge, I simplify on an important issue, which is explicitly modelled in all mentioned capital-vintage models and
also in [28].
5
I am already assuming here, that
of , and that all
…rms of vintage
is a continuous function of , that A is a non-decreasing continuous function
have chosen the same technology A (as will later be shown to be optimal).
8
At L
At L
slope: Aθ ( t )
At LYt
b g
At LYt
Yt LY
slope: Aθ (t )
Yt
b g
Yt LY
Yt
agg. Profits/wt
LYt
LY
aggregate Profits/wt
A: low dispersion, high output, small profits
LYt
LY
B: strong dispersion, low output, large profits
Figure 1: Aggregate short-run production function
contrast, relatively few …rms know technologies almost as good as the currently best, then the shortrun aggregate production function exhibits rather strong decreasing marginal returns (Figure 1B).
A microeconomic foundation of the aggregate short-run production function …tting into the present
setting is given in [36] in a Cournot-Walras framework. A similar foundation can be given in the
Bertrand-Walras setting of [21].
2.2
Research and adoption
The incentive for R&A The productivity-dispersion inherent in the vintage-structure of the …nalgood sector generates an incentive to innovate in the absence of any monopolistic element. A …rm
j that installs technology Aj at t can look forward to earning the short-run pro…ts Aj
future periods
wY
in
> t provided that Aj > wY . Thus, as in other endogenous growth models, the
future short-run pro…ts an entrant6 anticipates provide the necessary incentive for his R&A. In most
monopolistically competitive endogenous growth models these pro…ts arise due to a constant mark-up
of prices over marginal costs. Typically this mark-up is …xed exogenously either by the elasticity of
substitution between the innovator’s brand and other brands or –closer to the present setting –by the
6
Note that in the present framework incumbent …rms have the same R&A possibilities as newcomers. Since further-
more no strategic issues are involved, established …rms and newcomers have the same incentive to innovate. Therefore,
when I write “entrant” it may as well be an established …rm, which – once its old technology is no longer pro…table –
renews its technology. This is important as [19] …nds that …rm age and technology vintage are generally uncorrelated in
a wide range of industries.
Also note that in the present model, if there is no population growth the constant process of technological rejuvenation
at steady state equilibrium may happen without entrance of any new …rm.
9
…xed quality-adjusted productivity distance of the best to the second best …rm (limit-pricing in case of
a non-drastic vertical process or product innovation). As has been explained in the previous section,
in the present framework, the limit-…rm is not the second best …rm but rather the endogenously
determined oldest active incumbent. As a consequence
1. the mark-up 1
A
( ) =Aj
productivity-distance A
at
for any given incumbent j is determined endogenously by the
( ) =Aj
between j and the oldest active incumbent as well as by the
active technology lifetime T =
( ) which determines who is the oldest active incumbent
( ),
2. the mark-up for any given incumbent j is not constant, because the productivity-distance
between j and the currently oldest active incumbent shrinks in each period (until j becomes the
oldest current …rm himself before being driven out of business).
Return to Figure 1: In Figure 1A, productivity dispersion At =A
A
(t) =At
(t)
and hence the mark-up 1
of the price (normalized to 1) on the most e¢ cient …rm’s marginal cost (wY t =At = A
(t) =At )
is small. The current incentive for new …rms to enter is weak. If in contrast, in Figure 1B, productivity
dispersion as well as mark-ups are large. The current incentive to enter the market with new knowledgecapital is strong.
The cost of R&A
So far I have talked about the incentive to install new knowledge-capital. I
now explain what resources are required. On the basis of already known technologies, R&A-lab j can
develop or adopt and install existing or improved technologies. The extent of the improvement over
known technologies depends on the basis of knowledge accessible to research …rm j and on the amount
hj of labor employed by j. An R&A-lab employing hj units of labor at t comes up (with certainty)
with a technology of productivity
Z
Aj = f (ahj )
A d ,
1< <t
10
(3)
where A is the productivity of the leading technology at
and a > 0 is an e¢ ciency parameter.7
Thus, slightly deviating from the language used in much of the new growth literature, …rms do not
‘create a number A_ t of new designs’or products, but rather implement known or new technologies for
producing the …nal-good output parametrized by their productivity Aj . It will be shown later that at
equilibrium all R&A-labs of vintage t choose the same number of researchers hj = ht and therefore
R
generate the same technology Aj = At = f (aht ) <t A d . Hence at a steady state equilibrium (with
constant ht = h)
A_ t = f (ah)At ,
(4)
which is equivalent to the more standard expression for research production functions at steady state:
if ht = h is constant, then the growth rate of labor productivity g = A_ t =At = f (ah) is constant as
well. The scale elasticity of At in knowledge-capital production is therefore one, as in …rst generation
endogenous growth models.
To simplify exposition I assume in the main text that the R&A-function has a constant8 elasticity
7
This necessary e¤ort may be very moderate if the …rm contents itself with a rather old-fashioned technology and –
depending on the shape of f ( ) –may be substantial if it wants to come up with an up-to-date or even leading technology:
By choosing a su¢ ciently small hj , entrant j merely adopts knowledge that has already been used for a while by other
…rms. In contrast, by choosing a su¢ ciently large hj …rm j may raise its own productivity signi…cantly (depending on
the form of f ) beyond the best currently used technique. At steady state equilibrium both extremes will not occur: at
each moment, R&A-labs implement marginal improvements upon the best currently used technologies (see Eq.4).
8
The R&A-function is expressed as a function f (ah) of e¢ ciency units ah (rather than of h) to ease immediate
extension of all results concerning R&A-e¢ ciency a to more general R&A-functions. In fact, it would be disturbing if the
results depended too strongly on the speci…c example used in the main text. While the simple example R&A-function
facilitates concentration on the essentials of the analysis, it seems restrictive and implausible to assume that the elasticity
of the R&A-function does not decline with rising h and that correspondingly the momentaneous productivity growth
rate f (ah) resulting from research of an individual research lab is in principle unbounded g := limh!1 f (ah) = 1: In my
view it would be more realistic to assume (or at least not to exclude) that the momentaneous productivity growth rate is
bounded (g < 1), so that for already high research intensity the elasticity of f declines with increasing h. Furthermore
one may want to allow for increasing returns to research (f 00 > 0) for small h.
All propositions (and lemmas) of the paper hold for the following much wider class of R&A-functions f (e
h), where f
is de…ned for e¢ ciency units e
h = ah, so that both the interpretation of a as well as all results concerning a retain their
validity in the general setting: f (e
h) is increasing in e
h (f 0 (e
h) =
Its elasticity
(g) :=
(there exists a g m in
e
e
df (h)
h
e f (h)
e
dh
for g = f (e
h) is non-increasing
0 such that
e
df (h)
e >
dh
( d dg(g)
(g) < 1 for all e
h with f (e
h) > g m in ).
0) with limh!0
f (e
h) = 0 and limh!1
f 0 (e
h) = 0.
e
e
0) and less than unity for su¢ ciently large e
h
In the example of the main text (Eq. (5)) (g) = , g m in = 0 and g = 1. An examples with g < 1 is f (e
h) =
with , > 0 (Example 2), where either
is positive but declining (d (g)=dg =
1 (Example 2a) or
g (1
)=
> 1 (Example 2b). In Example 2 (g) =
). Another admissible example is f (e
h) = 1
11
e
e
h
(1
e
h
e
1+h
g 1= )
(Example 3). In
with 0 <
< 1:
f (ah) = (ah)
(5)
Once more, note that each R&A-lab turns into a single …nal-good …rm only. A …rm which considers
to enter the …nal-good market or to upgrade its technology has to put in some e¤ ort of its own (or
to have it done by a down-stream R&A-lab), even if competitors are developing similar techniques.
This re‡ects the rival part of knowledge which takes into account that to some extent ideas are
always embodied in human or physical capital. Nevertheless knowledge keeps its partially public good
character: when trying to enter the market or upgrade its knowledge a …nal-good producer (or its
downstream R&A-lab) can build on all knowledge which has been introduced by previous innovators
R
(the term <t A d in (4)).9
The objective of R&A-labs Each individual R&A-lab succeeds with its envisaged production of
knowledge-capital. Given knowledge-capital Aj R&A-lab j turns into a …nal-good producer using the
new knowledge capital to produce Aj units of output with one unit of labor. R&A-lab j of vintage
t thus chooses the number of researchers hj to maximize the excess of this present value over the
research cost wAt hj :
max
hj
Z
t+T (hj )
[Aj
wY ]e
r(
t)
d
wAt hj ;
t
where T (hj ) is the length of time the …rm plans to be active on the …nal-good market and where I
have already assumed that interest rates are constant.
There is free entry to R&A. This will make sure that at equilibrium the revenues generated by
one R&A-lab just cover wages wAt hj : Since each R&A-lab turns into exactly one …nal-good …rm, the
number
t
of R&A-labs at t determines the extent of entry by new (or rejuvenated) …rms into the
…nal-good market. At equilibrium all R&A-labs at t will choose the same knowledge-capital At , so
that
t
determines the number of future …nal-good competitors with common knowledge-capital At .
If there are
sector is
t ht
t
R&A-labs at t, each employing ht workers, then total labor demand of the research
= LAt . Thus, while
of entering …rms. Together
t
t
measures the extent of entry, ht determines the productivity
and ht will shape the curvature of future …nal-good technologies and
both examples g = 1. Furthermore, in Example 2b f 00 > 0 for small h < 0. The example exhibits increasing returns to
research for 0 < f (e
h) < g m in = [1
(1=
)] and decreasing returns for f (e
h) > g m in > 0 although the elasticity
everywhere strictly decreasing.
R
9
In the pure adoption interpretation of the model
(g) is
A d is the “stock of existing knowledge” and the activity hj
R
is pure adoption activity (rather than as research and adoption if hj is large).
A d grows because of a positive
<t
<t
externality from the adoption of the currently most advanced active technology to the stock of existing knowledge.
12
hence the future intensity of competition and of pro…ts. An increase of total R&A-activity LAt does
not automatically enhance productivity growth of the leading …nal-good technology. It may merely
increase the extent of entry.
2.3
Households
The household sector is modeled as in the standard Ramsey-Cass-Koopmans model. The representative household has the dynastic utility function
Z
1
e
t
u (ct ) Lt dt with u (ct ) = ln ct
t=0
where ct = Ct =Lt is the per capita consumption of the household at t and Lt the number of household
members.
Total population and labor-supply Lt grows at an exogenous rate n :=
dLt =dt
Lt
workers LY t are employed in the …nal-good sector at wage wY t , while LAt = Lt
0. Of the Lt
LY t are employed
in the research sector at wage wAt . All workers are identical and can freely choose in which sector
to work. The allocation of labor to the two sectors and the employment ratio z := LAt =LY t will be
determined endogenously such as to equalize the two wages (Section 4.4 brie‡y analyses what happens
when the employment ratio z exogenously …xed).
Households own the existing …nal-good …rms and can acquire shares of new …nal-good …rms on
the equity market from successful R&A-labs. At each instant they receive the short-run pro…ts of
…nal-good …rms corresponding to their shares (Households also own R&A-labs. However, as there is
free entry to R&A, shares in R&A-labs will pay no dividend and will have price zero).
Final-good producer assets are priced at their fundamental value, i.e. at the present value of
the corresponding ‡ow of future pro…ts evaluated at market interest rates. Under these conditions,
a necessary condition of utility maximization given the sequence of interest rates rt is the standard
Euler-equation c_t =ct = rt
2.4
.
Steady State Equilibrium
At each instant of time …nal-good …rms maximize instantaneous pro…ts, taking factor prices (wY t ;
wAt ; and rt ) and the output price (normalized to 1) as given. Each active R&A-lab j chooses its
research input hj such as to maximize the di¤erence between the value of the …nal-good …rms’shares
they generate and the cost of R&A. There is an in…nite number of potential R&A-labs, which are
indi¤erent between becoming active or not, when pro…ts are zero (Free Entry to R&A). Households
13
maximize dynastic utility within their …nancial means, taking as given initial asset holdings and the
correctly foreseen sequence of wages and interest rates. A perfectly competitive equilibrium, given
an initial distribution of …rms over known A
(0) ,
A0 , f
g
2( (0);0] ,
(wY t ; wAt ; and rt ) and of quantities (ht , LAt , LY t , Yt , Ct , At ,
is de…ned as a sequence of prices
t,
Tt ) such that under the listed
assumptions the markets for labor, …nal-goods, and shares clear at any instant.
A steady state equilibrium is an equilibrium at which rt , ht , lAt := LAt =Lt ; lY t := LY t =Lt ;
t
=
t =Lt ,
and Tt are constant and wY t ; wAt ; yt := Yt =Lt ; ct := Ct =Lt ; and At grow exponentially at
the same constant rate g.
Mechanical steady-state relations The remaining of this section shows that the constancy of two
variables, say g and lA , mechanically determines constant values of h, lY = 1 lA , , T , Yt =(At Lt ),
R
and wY t =At . As has already been remarked, at steady state A_ t = f (ah) <t A_ d = f (ah)At and
thus g := A_ t =At = f (ah). Thus the R&A-function f de…nes a one to one relation between h and the
productivity growth rate g, the number of workers needed to generate a given g is h(g) := f
1 (g)=a.
Of course h(g) is constant if g is.
Now consider employment in the R&A-sector. Each of
researchers, so that LAt =
t
t
individual R&A labs employs h(g)
h(g), or in per-capita terms
lA =
h(g).
(6)
Given lA and h(g) this determines the per-capita number of R&A-labs:
= lA =h(g). Next consider
employment in the …nal-good sector. Each active …nal-output …rm employs one unit of labor.
Thus per-capita employment by …rms of vintage
2 [ (t); t] is
=
=L , the per-capita number of
…rms that have entered at . Therefore per-capita employment at t in manufacturing is lY = LY t =Lt =
Rt
Rt
Rt
R
L
n(t ) d or, with I(n; T ) = T e n d ,10
(t) Lt d =
(t) L Lt d =
t Tt e
0
1
lA = lY =
I(n; T ).
We have thus shown that given constant g and lA ,
10
Throughout the paper we will deal with present values
(7)
and T are constant as well.
RT
0
e
x
d of a unit-output stream over the interval [0; T ],
where the constant discount rate x will be the productivity growth rate, the population growth rate, the interest rate,
or the utility discount rate. In the following I use the notation
8
Z T
<
I(x; T ) :=
e x d =
:
0
14
1 e xT
x
if x 6= 0
T if x = 0
Notation
Aggregate production becomes Yt =
e
Lt At 1
(g+n)T
g+n
Rt
(t)
A d =
t At
Rt
(t)
t
A
At d
= Lt At
Rt
(t) e
(g+n)(t
)d
and grows at the rate g +n of the number Lt At of labor e¢ ciency units. Consumption
and output per labor e¢ ciency unit is
Yt
= I(g + n; T )
At Lt
(8)
Thus, as in the Solow-model, given g, T , and , the population size Lt of the economy has no impact
on the level of per-capita steady-state incomes, while the population growth rate n has a negative
e¤ect on these incomes. Per capita output yt = Yt =Lt and consumption ct = yt grow at the rate g.
At competitive equilibrium …nal-good sector employees are paid at their marginal productivity,11
thus wY t = A
(t) ,
or
wY t
=e
At
gT
,
which also is the inverse of the steady-state productivity-dispersion At =A
mark-up over the youngest active …rm’s marginal cost is 1
3
e
(t)
= egT : The price
gT .
Existence and Uniqueness of Steady State Equilibrium
In this section I show that the present perfectly competitive vintage growth model has a unique steadystate rational expectation equilibrium. The main tools are two functions relating the (endogenous)
technology lifetime T and the (endogenous) productivity growth rate g: The …rst of these functions
results from the individual R&A-lab’s optimal choice of its R&A intensity hj and from a “free entry
to research” condition: Equalizing the return per researcher from opening a new R&A-lab (“return
at the extensive margin”) and the return per researcher from expanding existing R&A-labs (“return
at the intensive margin") results in a decreasing relation between g and T (see Figure 2). Equalizing
research wages and …nal-good sector wages results in the second equilibrium relation between g and
T , which will be strictly increasing (see Figure 3). Together, the decreasing …rst and the increasing
second relation between g and T determine unique general equilibrium values g and T (see Figure
4) independently from the labor market conditions (6) and (7), which will be used to determine the
lA ) and the extent of entry
employment ratio z = lA =(1
11
.
The short-run relation between LY t and Yt exhibits the usual properties of a short-run neoclassical production
function (see Figure 1): short-run aggregate production is a strictly concave and increasing function of labor. Assuming
constant past
and g this function is Yt (LY ) = Lt At 1
e (g+n)T (LY t )
,
g+n
where T (LY t ) is de…ned by inverting (7). Thus
g+n
Yt (LY ) =
Y = Lt )] n
Lt At 1 [1 (nLg+n
of labor therefore is
becomes
dYt (LY t )
dLY
dYt (LY t )
dLY
= At e
gT
=
=A
for LY
Lt At e
L=n and Y (LY ) = LAt 1
(g+n)T dT (LY )
.
dLY
Inserting
(t) .
15
dT (LY )
dLY
e gLY = L
g
=
df
if n = 0. The marginal product
ln[1 (n= )(LY =Lt )]=ng
dLY
=
1
enT
Lt
this
=
3.1
3.1.1
Optimal individual R&A-intensity and Free Entry
Optimal R&A-intensity of the individual R&A-lab
The market value of an R&A-lab is the present value of the expected ‡ow of quasi-pro…ts generated
by the corresponding …nal-good producer. Thus the individual R&A-lab j at t chooses hj to maximize
the excess of this present value over the research cost wAt hj :
max
fhj :T (hj ) 0g
Z
(hj ) = max
hj
t+T (hj )
[Aj
wY ]e
r(
t)
d
wAt hj ;
t
where T (hj ) is the length of time a …rm using Aj would be active in the goods market and where
I have already assumed that interest rates are constant. Note that for su¢ ciently small hj , say for
0 < hj < hmin
t , the productivity Aj realized by research hj is smaller than that of the least e¢ cient
presently active incumbent technology: Aj < A
(t) ,
where (t) is the vintage of the oldest active …rm
at t. An intensity hj , with 0 < hj < hmin
will never be chosen, since hiring researchers to come up
t
with already outdated techniques is not pro…table (for hj > 0 and T (hj ) = 0 pro…ts (hj ) <
wAt hj
are negative). Perfect competition on the …nal-good market leads to marginal product real wages
wY = A
( ),
where ( ) is the vintage of the oldest active …rm at . Thus the R&A-lab solves
max
in
hj hm
t
Z
t+T (hj )
[Aj
A
( ) ]e
r(
t)
d
wAt hj .
(9)
t
The …rst order condition requires that the marginal present value of the expected ‡ow of quasi-pro…ts
(return at the intensive margin of R&A) equals the research wage:
Z
t
t+T (hj )
dAj
e
dh
r(
t)
dT (hj )
[Aj
dh
d +
A
(t+T (hj )) ]e
rT (hj )
= wAt .
Appendix 6.1 shows that the …rst order condition is su¢ cient if all other steady state equilibrium
conditions are satis…ed. Since by de…nition of ( ) and T (hj ), (t + T (hj )) = t, the second term is
zero such that the condition shortens to
dAj
dh
Z
T (hj )
e
r
d = wAt .
(10)
0
Since the condition is identical for all R&A-labs of any given vintage t0 , R&A-labs of vintage t0
will all choose the same ht0 and Tt0 = T (ht0 ) at equilibrium. Consider a steady state equilibrium
candidate with constant h, T and g = f (ah). R&A-lab j of vintage t, considering the pro…tability
of a deviation, faces the research function (3). If I denote the “latest” technology before j’s decision
R
by At := lim !t A = A eg(t ) for
< t, (3) becomes Aj = f (ahj ) 1< <t At e g(t ) d =
<
16
f (ahj )At
R0
g
1e
d = f (ahj )At
1
g
f (ahj )
12
f (ah) At .
=
Thus
dAj
af 0 (ahj )
=
At
dh
f (ah)
so that, with hj = ht , (10) can be rewritten as
Z Tt
af 0 (aht )
e
f (ah)
0
r
d =
If there is a steady state equilibrium with constant h, T , g,
wAt
At
=
(11)
wAt
.
At
wAt
At
(12)
(12) has to hold at ht = h, Tt = T and
wAt
At
(hence At = At , which also follows from the constancy of g). Again using the notation
RT
xT
0T
I(x; T ) := 0 e x d = 1 ex
(with 1 e0
= T ), the marginal revenue condition for h …nally
becomes
I(r; T ) =
where
3.1.2
wAt
h,
At
(13)
ah
= f 0 (ah) f (ah)
is the elasticity of the R&A production function.13
Free entry to research and zero pro…t for research-…rms
The present value of future quasi-pro…ts expected by one innovator at t employing hj hmin
workers in
t
R t+T (hj )
research (assuming constant interest rate r) is t
[Aj A ( ) ]e r( t) d
wAt hj where T (hj ) 0
is the length of time the …rm plans to be active on the good market. At equilibrium free entry to
research requires these pro…ts to be zero, the return from opening a new R&A-lab (return at the
extensive margin of R&A) must equal the cost of the R&A-lab:
Z t+T (hj )
(Aj A ( ) )e r( t) d = wAt hj
t
or
Z
t+T (hj )
A( )
)e
Aj
(1
t
r(
t)
d =
wAt
hj
Aj
All R&A-labs j of vintage t choose the same hj = ht and Aj = At . At steady state with constant
RT
ht = h, T (h) = T , and g = f (ah), again using the notation I(x; T ) := 0 e x d , the condition
becomes (see the Appendix 6.2 for the straightforward calculation)
I(r; T )
e
gT
I(r
g; T ) =
wAt
h
At
(14)
12
Note that the deviation of a single R&A-lab of generation t or even of the complete vintage t from a given path
R
fA g has no in‡uence on the basis of knowledge <t0 A d in (3) for generation t0 6= t. Thus a deviation of vintage t
does not e¤ect the decisions of other R&A-labs of any vintage.
13
More generally, for the class of R&A-functions of Footnote 8 the constant
evaluated at g = f (ah).
17
is simply replaced by the function
(g)
Conditions (14) and (13) do not depend on the size of the economy (L) nor on its growth rate
(n) because neither the technology of the individual …nal-good producer nor the industrial wage
wY t = e
3.1.3
gT A
t
depend on L or n.
Equal return to R&A at the intensive and the extensive margin
Comparison of the optimal individual R&A-condition (13) and the free-entry condition (14) shows:
I(r; T )
e
gT
I(r
g; T ) = I(r; T )
.
(15)
Inserting the Euler-equation yields
I(g + ; T )
e
gT
I( ; T ) = I(g + ; T )
.
(16)
Lemma 1 Equal return at the two margins: For every T > 0 equilibrium condition (16) has a unique
and strictly decreasing solution
g eqr (T ).
(17)
g eqr (T ) does not depend on either n, a, or z and is increasing in .14
g
g eqr (T )
−
T
Figure 2: Decreasing relation between T and g: equal return to R&A at the intensive and the extensive
margin.
Proof. See Appendix 6.3.
Eq.(15) re‡ects the fact that at equilibrium the return from opening a new R&A-lab (“return at
the extensive margin”) has to match the return of investing the same amount of additional R&A in
existing R&A-labs (“return at the intensive margin”). The resulting relation g eqr (T ) is decreasing
14
Note that the Lemma 1 is valid for all R&A-functions of the general class in Footnote 8, even if
The function takes values on the full range of f : limT !0 g
eqr
(T ) = g and limT !1 g
eqr
(T ) = g
speci…c R&A-function f (Figure 2 depicts the main example with g min = 0 and g = 1).
18
min
(g) depends on g.
, which depend on the
because the “return at the extensive margin” of R&A (LHS of (15)) increases more than the “return
at the intensive margin" (RHS of (15)) with both g and T , such that in order to keep equal the returns
at the two margins when T is raised, g has to fall. At the heart of this result lies the fact that an
increase of both g and T raises the productivity dispersion egT and thus the price mark-up 1
e
gT .
This raises the return at the extensive margin (opening a new R&A-lab) but leaves una¤ected the
return at the intensive margin because the latter does not depend on the price mark-up (A
( )
in (9)
does not depend on hj and therefore does not a¤ect (10) or (13)).
3.2
Equal wages for R&A and …nal-good production
Since all workers are equally skilled to perform any task in the economy, equilibrium wages in both
sectors are identical:
wAt
wY t
=
=e
At
At
Inserting the wage equality condition wAt =At = e
gT ,
gT
equal wages
the inverse R&A-function h = h(g), and the
Euler equation into the two R&A equilibrium conditions (13) and (14) yields two equations in two
endogenous variables g and T alone:
I(g + ; T ) egT = h(g)=
optimal h
I(g + ; T ) egT = I( ; T ) + h(g)
free entry to R&D
or, equalizing the two right hand sides:
h(g) =
I( ; T ).
1
(18)
This directly de…nes a simple strictly increasing relation between g and T : g eqw (T ) :=f a 1
I( ; T )
(see Figure 3).15
Lemma 2 Equal wages in the two sectors: Equalizing wages de…nes a positive relation between growth
and technology lifetime. For every T > 0 the equilibrium condition wY = wA has a unique and strictly
15
Note that the lemma is expressed in a way valid for the wider class of R&A-functions de…ned in Footnote 8. In
general the elasticity
(g) of f may vary with g, so that the RHS of (18) depends on g. Since
(g) is non-increasing,
eqw
it is easily seen that (18) nevertheless de…nes a strictly increasing function g (T ): Solving (18) for T ; with h(g) =
h
i
f 1 (g)
1 "(g) f 1 (g)
yields
T
=
ln
1
= which is uniquely de…ned for all g 2 (g m in ; g m a x ), where g m a x is given by
a
"(g)
a
f
1
m ax
"(g
)
(g m a x ) 1 "(g
= a= . Obviously, this T is increasing in g and thus has a well de…ned increasing inverse g e q w (T );
m ax )
which is increasing in a and takes values from limT !1 g e q w (T ) = g min to limT !0 g e q w (T ) = g m a x < g:
19
increasing solution,
g eqw (T ):
g eqw (T ) is increasing in a, decreasing in
(19)
and does not depend on n or z.
g
g max =
a α
ρ 1−α
a
A
g eqw (T | a)
+
+
T
Figure 3: Increasing relation between T and g: equal wages in the two sectors.
The positive relation between g and T induced by the global research allocation corresponds to
the negative relation between competition and growth in more standard endogenous growth models
(with the exogenous intensity of competition usually captured by a constant mark-up of prices over
marginal costs): As we have seen, an increase of T (given g) raises productivity-dispersion egT , markup 1
e
gT
and future pro…ts, which raises the incentives to enter and to increase h and thus g: Note
that the equality of the returns to R&A at the extensive and intensive margin has also been used in
the derivation of 19.
3.3
General Equilibrium
Using the steady state R&A-function h(g) := f
1 (g)=a
and the standard Euler-equation g = r
, the
optimal-research condition (13) together with the free-entry condition (14) have led to the decreasing
function g eqr (T ) in Section 3.1 (17). Section 3.2 has translated the equal wage condition into an
increasing function g eqw (T ) of T: The positive relation g eqw (T ) between g and T from Lemma 2
together with the negative relation g eqr (T ) from Lemma 1 immediately yield the following proposition:
Proposition 3 The economy has a unique steady state equilibrium with constant knowledge vintage
lifetime T and strictly positive productivity and per-capita income growth rate g = g eqw (T ).
The equilibrium values g and T have been determined independently of the resource constraints
lA =
h(g) (6) and 1
lA = lY =
I(n; T ) (7). Given g and T
20
these conditions determine
g
g max =
a α
ρ 1−α
a
A
g eqw (T | a)
+
+
g * (a)
+
g eqr (T )
−
T * (a )
T
−
Figure 4: Steady-state equilibrium productivity growth and vintage lifetime
z = lA =(1
lA ) = h(g )=I(n; T ), lA = 1=[1 + 1=z ] and
= lA =h(g ). Total research employment
is LAt = Lt =[1 + 1=z ], total manufacturing employment is LY t = Lt =[1 + z ], and the research-share
in GDP is
4
wAt LAt
Yt +wAt LAt
=
e gT LAt
Lt I(g+n;T )+e
gT L
At
=
e gT (lA = )
I(g+n;T )+e gT (lA = )
=
e g T h(g )
.
I(g +n;T )+e g T h(g )
Comparative Statics
4.1
The e¤ect of R&A-e¢ ciency and of R&A-subsidies
Positive growth e¤ect of R&A-e¢ ciency dampened by endogenous extent of entry A rise
of the R&A-e¢ ciency a raises the incentive to invest into R&A, lifting upward the function g eqw (T )
of Section 3.2, without directly a¤ecting the relative incentives to enlarge individual R&A-intensity
and to create more R&A-labs thus letting the function g eqr (T ) of Section 3.1 unchanged (see Figures
5 and 4). As a consequence
Proposition 4 A rise of the R&A-e¢ ciency a raises the steady state equilibrium growth rate g
and reduces the technology lifetime T .
The impact of a on lA is very weak. This is best seen by considering an economy with n =
may serve as a useful benchmark: With n =
(ratio (6) to (7)) and h(g) =
z =
1
1
the steady state resource constraint z = h(g)=I(n; T )
I( ; T ) (de…ning the increasing function g eqw (T )) directly determines
. The elasticity of the R&A-function thus directly …xes the global R&A-intensity lA =
More generally z =
1
I( ;T )
I(n;T )
is mainly determined by the R&A-elasticity
realistically small values such that
16
which
I( ;T )
I(n;T )
as long as
!
and n take
is close to 1.16 Since T falls with rising a, it follows directly
Also note that for local comparative statics, these observations do not hinge on the example R&A-function f (ah) =
(ah)
used in the main text. Even if
(g) depends on g (as in most examples in the class de…ned in Footnote 8)
21
from the resource condition lY = 1
lA =
I(n; T ) (7) that the extent of entry
rises with rising a
(keeping in mind that lA barely changes). Thus a surge in R&A-e¢ ciency a attracts more entry
the expense of individual R&A-intensity h = lA =
at
, dampening the positive growth e¤ect of a (i.e.
the e¤ect of a on g = f (alA = ) given lA = ).
R&A-subsidy has positive but insigni…cant growth e¤ect To study the general equilibrium
impact of a policy aimed at raising productivity growth, assume the government pays a subsidy on
research wages …nanced by a lump-sum tax: for each output unit a R&A-lab pays to its workers, the
government adds a subsidy of s 2 [0; 1] output units. The wage paid by a research …rm then is wAt =
worker , where w worker is the wage received by the worker. Workers must be indi¤erent between
s)wAt
At
(1
worker = w
the two sectors, so that wAt
Y t or wAt = (1
s)wY t . The relation wY t =
dYt (LY t )
dLY
is neither a¤ected by the lump-sum tax nor by the research subsidy, so that wAt =At = (1
= At e
s)e
gT
gT .
With the new interpretation of wAt as net wage paid by research …rms condition (13) and (14) and
thus condition (15) still apply. Inserting wAt =At = (1
s)e
gT
and h(g) = f
1 (g)=a
into (13) and
(14) now yields
(g) I(r; T ) = e
I(r; T )
e
gT
I(r
g; T ) = e
gT
1
s
a
gT
1
s
a
f
1
(g)
optimal h
f
1
(g)
free entry to R&D
As one can see, increasing the R&A-subsidy s has qualitatively the same e¤ect on g and T as increasing
the R&A-e¢ ciency a. Since the Euler-equation g = r
too remains una¤ected by the lump-sum
tax, we have shown:
Proposition 5 Increasing the subsidy on research wages (…nanced by a lump-sum tax) by 1%,
raises the steady state growth rate and reduces the active lifetime of technologies in the
same way as an increase of the R&A-e¢ ciency parameter a by s=(1
s)%.
Thus, the present model is a model of endogenous growth not only in the sense that growth occurs
due to pro…t-seeking R&A, but also in the sense that the growth rate can be a¤ected by public policy.
However, the e¤ect of s on g is insigni…cantly weak in numerical examples that lead to plausible values
of g and T .17 Thus, while qualitatively growth is endogenous also in the sense that its rate can be
z =
1
I( ;T )
I(n;T )
remains valid, such that, whenever the examples are calibrated in a way that generates any given value
for lA , the elasticity
17
=
lA
I( ;T )
I(n;T )
I( ;T
)
lA I(n;T )
1
is close to lA for all these examples if
and n are small.
Suppose the subsidy initially is 10% of research wages. Increasing the subsidy by 1% acts as if a were raised by
about 0:1% (
s=(1
s)%). If h does not react drastically g = f (ah ) will rise by merely 0:1
22
%: If for instance
a¤ected by public policy, quantitatively the e¤ect of subsidies is more in line with empirical studies,
which typically fail to …nd a signi…cant growth e¤ect of public policy (see for instance [20]).
4.2
The e¤ect of population size and population growth rate
We have seen that the e¤ect of an increase of R&A-e¢ ciency a on the growth rate g = f (aLAt =
t)
is
dampened by the fact that it attracts more entry. The dampening e¤ect of endogenous entry is much
more accentuated in case of an increase of the size of the economy:
Proposition 6 An increase of the total labor force L0 leads to a proportional increase in the extent
of entry without a¤ ecting per-capita entry, the R&A-intensity lA , per-capita income growth g , or the
steady-state level (Yt =At Lt ) . An increase of the population growth rate n ...
... has no in‡uence on g and T
... raises the employment ratio z =
h(g )
I(n;T )
... raises the per-capita intensity of entry
and the global R&A-intensity lA =
1
1+1=z
= lA =h(g )
... reduces per labor-e¢ ciency unit income level (Yt =At Lt )
... raises the R&A-share in total income
wAt LAt
Yt +wAt LAt
=
e g T h(g )
I(g +n;T )+e g T h(g)
The independence of g and T from L and n immediately follows from the fact that neither of
the R&A equilibrium conditions (g eqr (T ) = g eqw (T ), wAt =At = e
gT )
depend on L or n. The reason
for this independence are …rst that the e¢ cient scale size of the individual …nal-good producer does
not depend on the size of the market and second that installed knowledge is …rm speci…c so that each
individual …nal-good producer has to perform its own additional R&A to reach or improve upon the
current knowledge frontier. An increase of total resources thus increases proportionally the steady
state extent of entry
t
without a¤ecting the growth rate or per capita incomes. In other words,
= 0:1 (remember that this more or less corresponds to lA = 0:1) even doubling the subsidy to 20% would raise g by
merely 0:1 0:1 100% = 1%, or (for g = 0:02) by some insigni…cant 0:02 percentage points.
Now of course h will typically not remain constant when s is raised. In fact lA = h as well as
the benchmark economy with
determine z =
1
1 s 1
= n the two equilibrium conditions h(g) = z I(n; T ) and (1
or lA =
1 s+ s
a = (0:02)
leading to g
0:02 and T
1
I( ; T ) now
). However the e¤ect of s on g remains insigni…cant in numerical examples
that lead to plausible values of g and T . Consider an economy with
10
s)h(g) =
may change (In
= 0:1, n =
= 0:02; s = 0 (thus lA = 0:1) and
10. An increase of s from 0 to 0:05 raises g insigni…cantly to g
0:0201 and
even the growth e¤ect of a juicy 50% research subsidy (s = 0 raised to s = 0:5, half of all R&A-expenses …nanced by the
subsidy!) is extremely weak, raising g to only g
0:021: In other numerical examples these e¤ects are similarly weak.
23
the extent of entry constitutes a perfect bu¤er, completely neutralizing the e¤ect of mere size on the
steady state growth rate or the steady state level of per capita income.18
The employment ratio z =
lA
lY
=
h(g )
I(n;T )
=
h(g )
I(n;T )
is raised by an increase of n because I(n; T )
falls (Given T , an increase of n reduces the number of old active …rms relative to the number of new
…rms
t
= Lt ). Note that if the extent of entry
were …xed, an increase of z and hence of the total
R&A-intensity lA would raise the growth rate g = f (alA = ). Instead, the positive e¤ect of raising lA
on g is completely absorbed by a proportional rise of the rate of entry
. The property that the R&A-
share in total income is increased when n is raised without a¤ecting per-capita income-growth may
help reconcile two stylized facts within an endogenous growth model that can generate perpetually
balanced growth without depending on perpetual population growth: the increasing aggregate research
intensity observed in recent decades with the more or less constant productivity and per-capita income
growth rates.
4.3
The relation between productivity-growth and technology lifetime
Negative relation between productivity growth and technology lifetime We have seen
that a variation of a or s a¤ects g and T in opposite directions: any variation of a or s (given the
other parameters) induces a negative relation between growth g and technology lifetime T . These
comparative statics results are thus consistent with the negative empirical relation between growth
and technology lifetime reported in [29] and [50] if one assumes that observed di¤erences in (g; T ) are
due to di¤erences of a or n:
Corollary 7 Negative relation between growth and technology lifetime. A variation of a and
s induces a negative relation between g and T:
18
This conclusion distinguishes the present model –to di¤erent degrees –from monopolistically competitive endogenous
growth models: The absence of any bu¤er between the growth rate and the total amount of research in the …rst generation
of endogenous growth models ([2], [26], [42]) is responsible for the growth e¤ect of the sheer size of an economy in these
models. In order to reconcile the observed increase of resources allocated to research with the observed constant growth
rates, the semi-endogenous growth models of ([30], [31], [32], [34], [45]) dismiss the assumption that a constant amount
of labour devoted to innovating activity can generate constant growth rates. This also eliminates the possibility of
persistent productivity growth without persistent population growth.
The e¤ect of size on the steady state growth rate has also been eliminated in the two-dimensional ‘generation 98’
endogenous growth models [17], [40], [51], or Chapter 12 of [3]. Jones [31] has already emphasized that each of these
models involves scale e¤ects: an increase of L increases its steady state per-capita income levels and an increase of n
raises per-capita income growth.
24
Productivity growth and productivity-dispersion The productivity-dispersion egT = At =At
T
depends positively on both g and T . As we have just recalled, g and T are a¤ected in opposite directions
by variations of a and s. The net e¤ect depends on which of the two e¤ects dominates and one may
expect that such variations will only moderately in‡uence the productivity-dispersion egT . In fact,
in numerical examples, variations of any of the exogenous parameters, even when strongly altering g
and T have only very weak e¤ect on egT . If two economies have very di¤erent levels of g and T , their
productivity adjusted wage wY t =At = e
gT ,
as well as their R&A-share in total GDP
wAt LAt
Yt +wAt LAt
will
nevertheless be very similar if the di¤erence in productivity growth mainly results from di¤erent a’s.19
Furthermore, the relation between g and egT is non-monotone.20
Productivity growth and competition Although there is perfect competition in the sense that
all …rms act as price-takers, the knowledge asymmetry between small incumbent …rms makes it meaningful to talk about more intense or less intense (perfect) competition. The intensity of competition is
high (low) when there are many (few) incumbent …rms close to the current frontier of knowledge. This
corresponds to a colloquial usage of the term21 and is also captured by standard measures of competition based on price-costs margins. To …x attention I follow [1] to measure the intensity of competition
19
In the above example with parameters
= 0:02; s = 0 (thus lA = 0:1) and a = (0:02)10 (leading
= 0:1, n =
to g = 0:0200 and T = 10:024) the productivity-dispersion is about eg
potential productivity as wage (wY t =At = e
the productivity dispersion At =At
T
g T
=e
g T
T
= 1:22 and a worker receives 81:8% of his
= 0:818). While g rises fast with increasing research-e¢ ciency a,
and wages wY t =At = e
g T
remain nearly constant (doubling a almost
doubles g to 0:038, while wY t =At slightly falls to 0:816). While g rises fast with increasing a, T
falls to a similar
extent.
20
For the R&A-function of the main text (f (ah) = (ah) ) or for Example 2a of Footnote 8 (f (ah) =
with
< 1), variations of a induce a U-shaped relation between g and e
f (ah) = (ah) ,
= 0:02, wages per e¢ ciency unit e
= 0:1, n =
g T
g T
(ah)
1+(ah)
. In the above example economy with
take a minimum (with respect to variations of
a) at 81:29%. This minimum is however attained at nonsensically high values of g ; which are possible because g = 1
in Example 1 (see Footnote 8). In Example 2a with
is g = 1:95% and T
= 0:1,
= 1,
= 10:04 , while wages per e¢ ciency unit are e
= 0:02 and a = (0:02)10 productivity growth
g T
0.510 raises g from 0:1% to 8%, reduces T from 34 to 2 and …rst reduces e
g = 2:8%) and than raises e
21
g T
= 82:15%. Increasing a from 0:00510 to
g T
from 83% to 82.14% (at a = 0:0310 ,
back to 83%.
In an exam at graduate school, a beauty or music contest one says that competition is intense when there are many
almost equally learned, beautiful or skilled contenders even if no individual contender takes into account potential e¤ects
of his action on others’action.
25
P
pro…tj 22
1
j active outputj ,
# active …rms
Rt
A wY t
d , or at steady
A
t (t)
d
by a weighted sum of individual Lerner-indices ([Price-MC]/Price) 1
which in the present atomless setting becomes Compt := 1
Rt
t
state Compt =
e
gT I(n
g;T )
.
I(n;T )
1
(t)
This measure for the intensity of competition is indeed closely linked to
the (inverse of the) productivity-dispersion egT between the most productive and the least productive
active …rm, which also is the leading …rms’ mark-up of their price (= 1) over their marginal cost
R1
gT
)
= I(gT; 1) = 0 e (gT ) d . The above
(= e gT ). For n = 0 for instance, Compt = e I(T g;T ) = I(g;T
T
non-monotone relation between g and e
g T
thus directly translates into a similar non-monotone
relation between competition and growth.23
4.4
Fixed employment ratio
So far I have assumed that the two types of labor are perfect substitutes. The allocation of labor to
the two sectors was determined endogenously such as to equalize the two wages. In reality the skills
needed in the two sectors –even in the long-run –may be less than perfect substitutes. To check the
robustness of the above comparative static results with respect to the degree of substitutability, this
section brie‡y considers the opposite polar case in which households supply two distinct supplies of
labor, one that can only be used in …nal-good production and one that can only be used in R&A. The
supplies LAt and LY t = Lt
LAt are …xed exogenously, so that the employment ratio z = LAt =LY t is
…xed. Since wY t and wAt need no longer be identical, condition (19) de…ning the increasing function
g eqw (T ) is no longer relevant. Instead, for exogenously …xed z, the labor-market conditions (6) and
(7) de…ne an alternative strictly increasing equilibrium relation between T and g: dividing (6) by (7)
yields the ratio of researchers to industrial labor z := lA =lY = h(g)=I(n; T ), or h(g) = z I(n; T ).
Recalling the de…nition h(g) := f
1 (g)=a
of h this immediately de…nes the steady-state relation
g Mecha (T jz ; a ; n) := f (a z I(n; T )) between productivity growth g and technology lifetime T which
+ + +
must of course hold at steady-state-REE (see Figure 5):24 The increasing function g Mecha (T jz; a; n)
22
The simpler measure 1
to ta l p ro …t
to ta l o u tp u t
would yield similar steady state dependencies of competition on growth and
technology lifetime.
23
This shape of the relation between g and Comp hinges on the example R&A-function used in the main text. For
f (h) = (ah) and also for Example 2a (f (ah) =
(ah)
1+(ah)
with
< 1) Comp is non-monotone function of a (and
thus of g) (as in the cross-sectional study [1]). However Comp is a strictly increasing function of a for Example 2b
(
> 1) (as in the cross-sectional studies [35], [7]).
24
Equivalently, for 0 < g < f (z=n) this can be written as T (g jz ; n) =
+
+
ln[1 nh(g)=z]
n
if n 6= 0 and T (g jz ) =
+
h(g)
z
if
n = 0. Once more consider an economy at steady state with constant population (n = 0): the constant active lifetime
of each knowledge-vintage equals the number of employees per research lab multiplied by the ratio of total employment
in R&D to employment in manufacturing.
26
and the decreasing function g eqr (T ) (17 from Section 3.1) have a unique intersection. The economy
with exogenously …xed employment-ratio z thus has a unique steady-state equilibrium with constant
knowledge vintage lifetime T (z) and strictly positive productivity and per-capita income growth rate
g(z). The in‡uence of z, n, a on g(z) and T (z) can be read directly from Figure 5). An increase of
g
 az 
f 
 n
z, a
A
g Mecha (T | z, n, a )
n
g * ( z , n, a )
+ + − +
A
+ − +
g eqr (T )
−
T
*
T ( z , n, a)
− + −
Figure 5: Steady-state equilibrium with …xed global R&D-intensity
a, or z raises g(z) and reduces T (z), while an increase of the population growth rate n reduces g(z)
and raises T (z).25 Thus, the negative relation between productivity-growth and technology lifetime
discussed in Section 4.3 is strengthened in an economy in which the two kinds of labor are less perfect
substitutes: a variation of any of the parameters a, s, z, or n (given the other parameters) induces a
negative relation between g and T .
5
Conclusions
This paper shows that the productivity-dispersion between technology generations provides the necessary incentive for pro…table R&A and endogenous growth even in the absence of monopoly rents: the
perfectly competitive vintage knowledge model has a unique steady state REE with strictly positive
endogenous productivity growth.
The joint determination of growth, technology lifetime, and entry generates a negative relation between growth and technology lifetime and a non-monotonic relation between growth and
productivity-dispersion or competition. An increase of the R&A-e¢ ciency attracts additional entry
and duplication, reducing the positive growth e¤ect. The endogenous extent of entry constitutes a
25
Recall that in an economy with perfectly substitutable labor, n has no e¤ect on g or T . In that economy, the e¤ect
of n also depends on how z reacts to a change of n. The negative direct e¤ect of n on g (given z) was exactly neutralized
by a positive indirect e¤ect on g which acted through an increase of z.
27
bu¤er, dampening the e¤ect of R&A-e¢ ciency and completely neutralizing the e¤ect of population
size or population growth rates on per-capita income levels and growth rates. The model can reconcile
the observed rise of global R&A-expenditures with the constancy of income growth rates without giving up the possibility of persistent endogenous growth. Considering the inconclusive current empirical
evidence, it seems unwarranted to exclusively bid on models in which continuous per capita income
growth depends on continuous population growth or/and on models that are ‘inextricably tied to scale
e¤ects’. This paper shows that from a theoretical point of view this specialization is not necessary:
endogenous growth without scale e¤ects is possible and does not require population growth.
6
6.1
Appendix
Appendix. Optimal Individual R&A-intensity
I show su¢ ciency and necessity of the …rst order condition for the general class of R&A-function
de…ned in Footnote 8.
R t+T (hj )
[Aj
hmin
t
t
Necessary …rst order condition The R&A-lab solves maxhj
wAt hj . The …rst derivative is
0
Z
(hj ) =
t+T (hj )
t
dAj
dh
=
Z
dAj
e
dh
t+T (hj )
e
r(
t)
r(
t)
d +
d
dT (hj )
[Aj
dh
A
(t+T (hj )) ]e
r(
A
t)
( ) ]e
r(
t) d
wAt
wAt ,
t
since by de…nition of ( ) and T (hj ) the term Aj
A
(t+T (hj ))
= 0 for any choice of Aj by j at t:
Technology Aj ceases to be active (by de…nition at t + T (hj ) ) exactly when the wages w will be
determined by Aj .
Su¢ ciency of the …rst order condition I show su¢ ciency of the …rst order condition by already
using the fact that at steady state equilibrium all (other) …rms are determining the constant steady
state variables h, g, and T . The lifetime T (hj ) of …rm j (deviating at t) is determined by the condition
that j becomes inactive when its technology Aj matches the least e¢ cient non-deviating incumbents’
technology, i.e. by the condition Aj = At+T (hj )
is f (ahj ) = geg(T (hj )
T)
T
= eg(T (hj )
T )A .
t
or
T (hj ) = T +
ln(f (ahj )=g)
.
g
28
With Aj =At = f (ahj )=f (ah) this
Note that this condition also determines hmin
= f
t
ited by the upper bound limhj !1 T (hj ) = T +
1 (ge gT )=a.
Furthermore j’s lifetime is lim-
ln(1=g)
.
g
Given that all other …rms choose their
R t+T (hj )
steady state variables …rm j’s expected pro…ts are (hj ) = t
[Aj A ( ) ]e r( t) d
wAt hj =
ln(f (ahj )=g)
R t+T (hj ) f (ahj ) r( t)
f (ah ) R T +
g
e r d
At e gT
At t
eg[( t) T ] e r( t) ]d
wAt hj = At g j 0
[ g e
r g
j )=g)
R T + ln(f (ah
(r g)T (f (ah )=g)
g
f (ahj ) 1 e rT (f (ahj )=g) r=g
g
wAt
j
(r g) d
gT 1 e
e
w
h
=
A
e
t
At j
0
g
r
r g
At hj =
i
h
r g
g r=g
wAt
1
1
gT = A [c f (ah ) + c f (ah ) c3
rT f (ah )
g
c4 hj c5 ] ;
At rg
f (ahj ) + r(r
t 1
j
2
j
j
At hj
r ge
g) e
where c1 = 1=rg; c2 = g r=g =r(r
and c5 = [1=(r
gT
g)] e
= [1= ] e
for the individual R&A-lab j.
[c1 f (ahj ) + c2 f (ahj )
c3
c1
(c3 +1)
c2 c3 f
gT
rT
= g r=g =r
j)
3
= (r
c5 ]. First consider the function
0. The …rst derivative of
is strictly increasing from
g)=g =
=g; c4 = wAt =At
hmin
to maximize e (hj ) :=
t
is
0 (h
j)
(hj ) := c1 f (ahj ) + c2 f (ahj )
:= c1
c2 c3 f
1 (when hj ! 0) to c1
0 (h
for hj > h1 , since f 0 > 0. Furthermore limhj!1
0 (h
rT ; c
are strictly positive numbers, which are exogenously given
hj ! 1)26 and is zero for a unique hj , say h1 . Thus
derivative
e
The R&A-lab j thus chooses hj
c4 hj
on the complete domain hj
g) e
0 (h
j)
j)
c3 1
c2 c3 g
0 (h
< 0 for hj < h1 ,
1)
c3
af 0 . The term
c3 1
> 0 (when
0 (h
= 0, and
j)
>0
= 0 since limhj!1 f 0 = 0: Hence, the …rst
has a unique global maximum at some hj > h1 , say h2 with
00 (h
2)
= 0. Since c4 > 0
1. 25
0
0
0. 5
1
1. 5
2
hj
-1. 25
-2. 5
-3. 75
Figure 6:
0 (h
the function
and nowhere if
j)
0 (h
0 (h
j)
c4 therefore takes zero value exactly twice if
1)
0 (h
1)
< c4 . We can exclude the latter case because e 0 (hj ) =
has at least one solution hj = h. The case that
(hj )
0 (h
> c4 , once if
0 (h
j)
1)
= c4
c4 = 0
c4 hj has only one interior extremum
is also excluded since e would be everywhere strictly decreasing except at h1 which contradicts
e (hmin
t ) =
(wAt =At )hmin
< 0 = e (hj = h) since T > 0; thus h > hmin
t
t . Thus
(hj )
two local extrema. The …rst (which lies between h1 and h2 ) must be a local minimum of
26
Otherwise
0
(hj ) would be negative for all h which contradicts
steady state all innovators satisfy the …rst order condition.
29
0
c4 hj has
(hj )
c4 hj
(hj ) = c4 at hj = h; since by construction at
and hence of e (hj ) and the second (which is larger than h2 ) must be a local maximum. Because
f (ahj )
c3
(hj ) and e (hj ). Furthermore
tends to in…nity when hj tends to zero, the same holds for
limhj !1 (hj ) = limhj !1 e (hj ) =
1 because limhj !1 f (ahj ) = 0 and c4 > 0. Therefore e (hj ) is
strictly positive for small hj , falls to attain a local minimum, then rises to reach a local maximum and
…nally falls with increasing hj .
0.5
0.25
0
0
0.5
1
1.5
2
hj
-0.25
-0.5
hmin > 0, e (hj ) is the individual innovator’s pro…t function at steady state
Figure 7: for hj
equilibrium
e min
Remember that e (hj ) is not relevant for j’s maximization for hj < hmin
t . We know that (ht ) =
(wAt =At )hmin
< 0. Therefore at its local minimum e must be negative too. Furthermore we know
t
that at steady state equilibrium (hj ) = (h) = 0 so that the …rst order condition used to determine
the steady state must in fact correspond to the local maximum of e (hj ) which is a global maximum
on the allowed domain (hmin
t ; 1).
6.2
Appendix. The Free Entry Condition
R t+T
The free entree condition is t (1 eg[
R t+T r( t)
R t+T (g r)(
wAt
e
d
e
At h or t
t
wAt
At h
6.3
or I(r; T )
e
gT I(r
g; T ) =
( ) t] )e r(
t) gT d
=
t) d
=
wAt
At h
or
wAt
At h.
R t+T r( t)
R t+T g[ ( ) t] r( t)
wAt
e
d
e
d
At h or t
t
R t+T r( t)
R
t+T
e
d e gT t
e (r g)( t) d =
t
Appendix. Proof of Lemma 1
To prove Lemma 1 I …rst show the following Lemma 8. Note that the equilibrium condition (15) can
be rewritten as D := [1
(g)] egT
I(r g;T )
I(r;T )
= 0:
Lemma 8 At D = 0; the function D(g; r; T; ) := (1
…rst three variables and decreasing in
.
30
)egT
I(r g;T )
I(r;T )
is strictly increasing in the
=
Proof: 1. Both integrals I(r
g; T ) and I(r; T ) decrease with the interest rate r. However I(r
decreases faster than I(r; T ), as the ‘discount rate’r
(for g > 0). Therefore
g; T )
g is always smaller than the ‘discount rate’r
g; T )=I(r; T ) and hence D are increasing in r, whenever g > 0:
h
i
1
1
T
gT
(r
g)T
2. @D(r; g; T; )=@g > 0 at D = 0 if (1
)e
T > r g I(r g; T ) r g e
I(r;T )
1
r I(r; T )
T
rT I(r g;T ) at D = 0. Substituting (1
re
[I(r;T )]2
i
h
(r g)T
I(r g;T )
by I(r;T ) yields T > r 1 g r T g eI(r g;T )
inequality
i
h
rT
Te
1 e
1
r
I(r
rT
, or T >
h
T
1
r g
(r g)T
e
positive for r > 0, since T <
g > r, since then T >
e(r g)T
g; T ) =
r
1
r g
g < r I show that already T >
T = I(0; T ) > I(r
erT
1 e
=
1
r g
1
1
i
=
RT
0
h
erT
T
(r g)T
e
1
, or T
g)T
(<)re(r
rT
i
and dividing both sides of the
h
i
T e (r g)T
, or T > r 1 g
(r g)T
1 e
. The second term in parentheses is always
The …rst term in parentheses is negative if
d . In this case the desired inequality follows. If
, in which case the inequality is also shown. In fact
e
(r g)T
e
1
>
)egT g >
1
hr
e
g,
or T (1 +
(r g)T
I(r;T )
I(r g;T )
I(r;T )
1
(r g)T
)>
1 i
e rT I(r g;T )
I(r;T ) I(r;T )
e
h
1
r g.
at D = 0. Substii
e rT
g;T )
I(r;T ) or g >
(r g)T
g;T )
)egT by I(r
and dividing both sides by
yields g > eI(r
i I(r;T )
r
. Multiplying both sides with (e(r g)T 1)(erT 1); this becomes ge(r g)T (erT 1) >
erT 1
1
R
R
g)T (egT 1) or erT 1 > (<) egT 1 if r g > (<)0. This is equivalent to T er d > (<) T eg d
r
g
0
0
tuting (1
h
r g
I(r g;T )
I(r;T )
T e
r I(r;T )
1
r
(r g)T
3. @D(r; g; T; )=@T > 0 at D = 0 if (1
e(r
1
er d .
g)
e(r
(r g)T
r g
T
1
r
RT
0
i
)egT by
h
if r > (<)g, which is always satis…ed.
Proof of Lemma 1:27 If g min > 0 and g < g min , than D < 0 for any T
no solution in T . Furthermore, by l’Hôpital’s rule, limT !0 = [1
while limT !1 = [1
(g)]eg1
(g)]
1=
0, so that D = 0 has
(g) < 0 for any g,
(g + )= = 1 for any g > g min . Thus D = 0 has a solution in T for
any g 2 (g min ; g). Lemma 8 guarantees uniqueness and monotony with respect to g of this solution
T . Inverting yields the desired g eqr (T ). The signs of all derivatives also follow from Lemma 8. It is
straightforward to check that limT !0 g eqr (T ) = g and limT !1 g eqr (T ) = g min .
References
[1] P. Aghion, N. Bloom, R. Blundell, R. Griffith, and P. Howitt. Competition and
innovation: An inverted-u relationship. Quarterly Journal of Economics 120, forthcoming (May
2005).
[2] P. Aghion and P. Howitt. A model of growth through creative destruction. Econometrica
60, 332–351 (1992).
27
Note that I prove the proposition for the general class of R&A-function de…ned in Footnote 8, where
on g.
31
may depend
[3] P. Aghion and P. Howitt. “Endogenous Growth Theory”. The MIT Press, Cambridge,
Massachusetts, London, England (1998).
[4] P. Bardhan and R. Priale. Endogenous growth theory in a vintage capital model. UC
Berkeley, CIDER WO C96-069 (1996).
[5] E. J. Bartelsman and M. Doms. Understanding productivity: Lessons from longitudinal
microdata. Journal of Economic Literature XXXVIII, 569–594 (2000).
[6] J. Benhabib and A. Rustichini. Vinatge capital, investment, and growth. Journal of Economic
Theory 55, 323–339 (1991).
[7] R. Blundell, R. Griffith, and J. V. Reenen. Market share, market value and innovation
in a panel of British manufacturing …rms. Review of Economic Studies 66, 529–554 (1999).
[8] M. Boldrin and D. K. Levine.
Perfectly competitive innovation.
Available at
http://www.dklevine.com (1999).
[9] M. Boldrin and D. K. Levine. Factor saving innovation. Journal of Economic Theory 105,
18–41 (2002).
[10] M. Boldrin and D. K. Levine. The case against intellectual monopoly. International Economic
Review 45, 327–350 (2004).
[11] R. Boucekkine, F. del Rio, and O. Licandro. Endogneous vs exogenously driven ‡uctuations in vintage capital models. Journal of Economic Theory 88, 161–187 (1999).
[12] R. Boucekkine, F. del Rio, and O. Licandro. Obsolescence and modernization in the
growth process. Journal of Development Economics 77, 153–171 (2005).
[13] R. Boucekkine, M. Germain, O. Licandro, and A. Magnus. Creative destruction, investment volatility and the average age of capital. Journal of Economic Growth 3, 361–384 (1998).
[14] R. Boucekkine, O. Licandro, L. Puch, and F. del Rio. Vintage capital and the dynamics
of the AK model. Journal of Economic Theory 120, 39–72 (2005).
[15] R. Caballero and M. Hammour. On the timing and e¢ ciency of creative destruction. Quarterly Journal of Economics 111, 805–851 (1996).
32
[16] T. Cooley, J. Greenwood, and M. Yorukoglu. The replacement problem. Institute of
Empirical Macroeconomics DP Nr. 95 (1994).
[17] E. Dinopoulos and P. Thompson. Schumpeterian growth without scale e¤ects. Journal of
Economic Growth 3, 313–335 (1998).
[18] E. M. Drandakis and E. S. Phelps. A model of induced invention, growth, and distribution.
Economic Journal LXXVI, 823–840 (1966).
[19] T. Dunne. Patterns of technology usage in U.S. manufacturing plants. Rand Journals of Economics 25, 488–499 (1994).
[20] W. Easterly and S. Rebelo. Fiscal policy and economic growth: An empirical investigation.
Journal of Monetary Economics 32, 417–58 (1993).
[21] P. Funk. Bertrand and Walras equilibria in large economies. Journal of Economic Theory 67,
436–466 (1995).
[22] P. Funk. Temporary equilibrium, endogenous growth and the direction of change. SFB 300
Bonn DP A- 506 (1996).
[23] P. Funk. Satiation and underdevelopment. Journal of Development Economics 57, 319–341
(1998).
[24] P. Funk. Induced innovation revisited. Economica 69, 155–71 (2002).
[25] P. Funk and T. Vogel. Endogenous skill bias. Journal of Economic Dynamics and Control
28, 2155–2193 (2004).
[26] G. M. Grossman and E. Helpman. Quality ladders in the theory of growth. Review of
Economic Studies pp. 43–61 (1991).
[27] H. J. Habakkuk. “American and British Technology in the Nineteenth Century”. Harvard
Univ. Press, Cambridge, M.A. (1962).
[28] M. Hellwig and A. Irmen. Endogenous technical change in a competitive economy. Journal
of Economic Theory 101, 1–39 (2001).
[29] C. Hsieh. Endogenous growth and obsolescence. Journal of Development Economics 66, 153–171
(2001).
33
[30] C. I. Jones. R & D-based models of economic growth. Journal of Political Economy 103,
759–784 (1995).
[31] C. I. Jones. Growth: With or without scale e¤ects? American Economic Association Papers
and Proceedings 89, 139–144 (1999).
[32] C. I. Jones. Sources of U.S. economic growth in a world of ideas. American Economic Review
92, 220–239 (2002).
[33] C. Kennedy. Induced bias in innovation and the theory of distribution. Economic Journal 74,
803–819 (1964).
[34] S. S. Kortum. Research, patenting, and technological change. Econometrica 65, 1389–1419
(1997).
[35] S. Nickel. Competition and corporate performance. Journal of Political Economy 104, 724–746
(1996).
[36] W. Novshek and H. Sonnenschein. Small e¢ cient scale as a foundation for Walrasian equilibrium. Journal of Economic Theory 22, 243–256 (1980).
[37] S. L. Parente. Technology adoption, learning-by-doing, and economic adoption. Journal of
Economic Theory 63, 346–369 (1994).
[38] S. L. Parente. A model of technology adoption and growth. Economic Theory 6, 405–420
(1995).
[39] S. L. Parente and E. C. Prescott. Barriers to technology adoption and development. Journal
of Political Economy 102(2), 298–321 (1994).
[40] P. Peretto. Technological change and population growth. Journal of Economic Growth 3,
283–311 (1998).
[41] E. Phelps. Substitution, …xed proportions, growth and distribution. International Economic
Review (1963).
[42] P. M. Romer. Endogenous technological change. Journal of Political Economy 98(5, part II),
S71–S102 (1990).
34
[43] X. Sala-i-Martin, G. Doppelhofer, and R. Miller. Determinants of long-term growth:
A bayesian averaging of classical estimates (BACE) approach. American Economic Review 94,
813–835 (2004).
[44] P. A. Samuelson. A theory of induced innovation along Kennedy-Weizsäcker lines. Review of
Economics and Statistics XLVII(4), 343–56 (1965).
[45] P. S. Segerstrom. Endogenous growth without scale e¤ects. American Economic Review 88,
1290–1310 (1998).
[46] K. Shell. Inventive activity, industrial organization and economic growth. In J. Mirrlees and
N. Stern, editors, “Models of Economic Growth”, pp. 77–100. Wiley, New York (1973).
[47] R. Solow. Substitution and …xed proportions in the theory of capital. Review of Economic
Studies 24, 207–218 (1962).
[48] R. Solow, J. Tobin, C. von Weizsäcker, and M. Yaari. Neoclassical growth with …xed
factor proportions. Review of Economic Studies 33, 79–115 (1966).
[49] K. Wälde. Endogenous growth cycles. International Economic Reveiw 46, 867–894 (2005).
[50] J. Williamson. Optimal replacement of capital goods: The early New England and British
textile …rm. Journal of Political Economy 79, 1320–1334 (1971).
[51] A. Young. Growth without scale e¤ects. Journal of Political Economy 106, 41–63 (1998).
35