Alternative Endogenous Innovation Frameworks with Endogenous
Population Growth
By
Norman Sedgley*
Department of Economics
Sellinger School of Business and Management
Loyola College in Maryland
Baltimore, MD 21210
and
Bruce Elmslie
Department of Economics
University of New Hampshire
McConnell Hall
Durham, NH 03824
January 16, 2006
Abstract: Most growth theorists agree that understanding the economics of innovation
and technological change is central to understanding why some countries are richer
and/or grow faster than other countries. Over the past two decades the models explaining
innovation as the endogenous outcome of economic activity have evolved greatly and
become very complex. This increasing complexity is, in large part, due to a desire to
eliminate scale effects from the first generation models of endogenous innovation and
make the models more consistent with the lack of scale effects reported in the empirical
literature. In this paper we ask what can be learned about first generation growth models
and the alternative models that eliminate scale effects by carefully considering the
endogenous demographics. Our analysis leads us to some important conclusions about
the comparative equilibrium outcomes of the various approaches once a falling
population growth rate consistent with a demographic transition is confronted in a
standard endogenous innovation setting with an endogenous savings rate.
Keywords: Endogenous Innovation, Demographic Transition, Scale Effects
JEL classification O31, O41
*Corresponding Author. Telephone 410-617-2848. Email [email protected]
I. Introduction
Nearly two decades ago Paul Romer (1986, 1990) revived the literature on
economic growth by challenging economists to reconsider the economics of ideas and
technological change. The new literature, built on advances in economic modeling
pioneered by Dixit and Stiglitz (1977), answered this challenge quickly and grew into the
endogenous innovation literature.
Early models, referred to here as first generation models, followed two different
paths but provided very similar implications. One approach endorsed by Grossman and
Helpman (1991) looked at increasing product variety as an index of complexity in the
economy. With this horizontal innovation along the extensive margin more complexity
implies a higher level of technology, thus the number of products an economy offers
provides an index of the level of technology. Products are introduced by profit seeking
entrepreneurs. Another approach was taken by economists following Aghion and Howitt
(1992, 1998). Inspired by the writings Joseph Schumpeter they modeled a similar
innovation process; but the process was one of creative destruction. Thus innovation was
modeled along the vertical dimension or intensive margin. Though slightly different in
nature these two first generation approaches to endogenous innovation where very similar
in spirit and substance.
Two similarities are of paramount importance here. First neither approach
considered population growth as endogenous. In fact population growth was typically
assumed to be equal to zero. Consequently a true steady state equilibrium growth rate
could be examined in a reasonably tractable setting. Also, and related to the first point,
the two approaches shared the prediction of scale effects. These models, in their simplest
1
form, suggested a scale effect; ceteris paribus a large economy innovates and grows at a
faster rate than a smaller economy. Early on, scale effects where viewed as a unique and
testable implication of the new literature on economic growth. Though Kremer (1993)
reported some support for scale effects over very long time periods, this unique
implication was generally accepted to be at odds with the evidence (Grossman and
Helpman, 1994; Jones, 1995a and 1995b).
The critique of first generation growth models, based on the observation that a
growing population implies a counterfactual accelerating growth rate due to scale effects,
is most closely associated with the work of Charles Jones (1995a and 1995b). The Jones
critique led to a concerted effort to eliminate scale effects from the first generation
models. Basically two approaches to eliminating scale effects have been taken (Jones,
1999) and they are reviewed below. Developing and assessing these two approaches has
been the focus of research in the endogenous innovation literature since circa 1995.
Economists working in other areas of economic growth theory have continued to
do theoretical and empirical work focusing on transitional dynamics and the relationship
between endogenous demographics and economic growth (Becker and Barro, 1988;
Becker et al, 1990; Dahan and Tsiddon, 1998; Brezis, 2001; McDermott, 2002; and Galor
and Weil, 2000). These economists have tackled questions relating to the tendency of
fertility and population growth rates to fall as an economy develops. In this paper we ask
what the micro foundations of the demographic transition can teach us about the various
approaches to modeling endogenous innovation. The results are, at first glance,
surprising. Once the demographic transition is considered it is shown that the Jones
model does not fully address the scale effects issue. The focus on the functional form of
2
the production function for new ideas has received too much weight in the theoretical
debate concerning scale effects. When more than one factor of production is used in the
production function for new ideas the often cited relationship between the growth of
knowledge and population growth is reversed, with a negative correlation between the
growth of knowledge and population growth. In the steady state only a second
generation model along the lines of those proposed by Aghion and Howitt (1992, 1998) is
consistent with population growth other than zero. Section two of this paper reviews the
various approaches to modeling endogenous innovation. Section three extends the
models presented in section two along lines that allow for the microeconomic foundations
of the demographic transition. Section four concludes.
II Related theories and background.
During the meetings of the American Economics Association in 1999 Charles
Jones presented a paper that neatly highlights the differences between three classes of
endogenous innovation models. The approach taken by Jones outlined the linkages
between first generation endogenous innovation models and their descendants. In this
section we follow a similar approach. Our approach differs, however, in two important
ways. First we use a more general production function that allows for more accurate
comparisons amongst growth models once the model is updated to account for
endogenous population growth1. Secondly, a more general functional form for the
production function of new ideas is used. It turns out to be equivalent to Jones’s
functional form when used in the context of a reduced form growth model with one factor
1
This appears to be an improvement over the functional form used by Jones. In his paper the growth of
output per capita is the sum of two additive terms, but only one of the terms is related to growth in
technology. The implication is that per capita growth in the steady state can depend on some factors other
than technological change. This odd result leads to a model with Jones’s (1995a and 1995b) implications
asymptotically. Our formulation avoids these peculiarities.
3
of production. It is more general once multiple inputs are allowed to enter the production
function. The general functional forms for the production of output and ideas are
presented in equations 1 and 2 respectively.
1
(1) Y  z1
Q
Q 1  Az 2,i di .
0

(2) A   ( N / Q)  ,   1 .
Y is output, z1 and z 2 are inputs. A is the level of technology and Q represents the
number of sectors supplying the input z 2 . Equation 2 is a standard generalized
production function for new ideas. The “dot” notation signifies a derivative with respect
to time.  is a research productivity parameter, and N represents the total resources,
measured in units of output, that are devoted to the R&D effort.  is a parameter that
describes the nature of returns to scale in the production of new ideas. It can take
positive or negative values.
The review begins with a look at the various alternative endogenous innovation
growth models. Following Jones (1999) allow for one factor of production, labor.
Setting z 2,i  LY ,i and   1 the production function becomes:
Q
(3) Y  C  N   ALY ,i di  AQL .
0
In equation 3 L  QLY ,i is the total labor force taken to be equal to the population and C
is consumption. It is assumed, as is the case in this class of models, that the demand for
consumables is symmetric and LY ,i  LY , j for all i and j . Allow population to grow at
an exogenous rate, g L . The model is closed by assuming that a constant fraction of
4
resources, s, are used in the production of new ideas, N  sY and C  (1  s)Y . The
number of sectors, Q , evolves according to the equation:
(4) Q  L ,   0 .
Equations 2, 3, and 4 imply:
s  L (1  )
(5) g A  g y  
.
A1
In this paper, unless otherwise noted, g denotes growth rates and lower case
letters denote variables measured in per capita terms. This generalized equation is used
to highlight the implications of first generation endogenous growth models and their
descendants. The simplest model of endogenous growth that we consider presently is a
first generation growth model with only one sector,   0 and   1 . These are the
restrictions on the production functions necessary to highlight the implications of the first
generation models. Equation 5 becomes:
(6) g y  sL .
A lot of controversy surrounds equation 6. The two notable features are that
policy, proxied by s, the fraction of resources devoted to R&D, impacts the growth rate.
Furthermore, the scale effect is evident. The larger is the population the higher is the rate
of growth. It is the lack of evidence surrounding this scale effect that has led to the
evolution of these models into far more complex formulations of the same processes.
The evolution has followed two distinct paths. One path seeks to eliminate scale effects
but retain the prediction that policy impacts the growth rate of per capita output. These
models have been pursued by Aghion and Howitt (1998), Dinopoulos and Thompson
5
(1998), Peretto (1998) and Young (1998). We refer to this class of models as second
generation Aghion and Howitt2 models.
The implications of second generation AH models are found by assuming that
research along the extensive margin has no impact on productivity growth. By allowing
Q to grow proportionally with population the resources for research are spread out over
more and more sectors as population increases. This neutralizes the scale effect as long
as the growth in sectors is exactly equal to the rate of population growth. In other words
set   1 in equation 4. Continue to assume   1 and equation 5 leads to a new equation
for per capita growth.
(7) g y  s
It is clear that the scale effect is eliminated but policy implications are not 3.
Another approach to eliminating the scale effect has its focus on the functional
form of equation 5. By forcing  <1 increasing or decreasing returns can be imposed on
the production of new ideas. This class of models has been proposed by Jones (1995b),
Kortum (1997), Segerstrom (1998), and Hall and Jones (1999). We refer to these models
as second-generation Jones models. Allow   0 once again. Given that the steady
state is defined as a state with a constant rate of growth in technology, differentiation of
equation 5 with respect to time leads directly to equation 8.
(8) g y 
g L
.
(1   )
2
We use this terminology for brevity only. We do not intend to discount the contributions of the other
authors in this field.
3
Note that the growth rate in this model need not depend on the rate of population growth as Jones (1999)
suggests.
6
The implications are twofold. First, scale effects are replaced with a prediction that the
rate of growth of per capita output is positively correlated with the population growth
rate. Furthermore, policy variables do not impact the rate of growth. It can easily be
shown that policy variables and scale impact the long run level of per capita output.
The question of policy and scale impacts in levels vs. growth rates is now a
dominant theme in the endogenous innovation literature. This debate, however, has its
roots in the original empirical rejection of scale effects. Each of the three classes of
models studied to this point make unique predictions about the growth process.
The main question asked in the rest of this paper is which model presented
provides the easiest basis for extending endogenous innovation to explain the broad facts
of growth and demographics? We close this section by emphasizing that theoretical and
empirical work in the area of endogenous innovation almost always ignores transitional
dynamics and endogenous population growth (Kremer 1993; Jones 1995a, 1995b; Todo
and Miyamoto, 2002). Of those studies that do consider transitional dynamics the
relationship between the theory and the empirical model is not well specified (Hall and
Jones, 1999; Jones, 2002). Each study that controls for transitional dynamics, however,
finds that transition is an important part of the data generating process.
III. Models of endogenous innovation accounting for the demographic
transition and transitional dynamics.
The general endogenous innovation framework extended to include the
demographic transition and, therefore, transitional dynamics has the tendency to become
complicated very quickly. It is our desire to show that adding endogenous demographics
7
can be done in a relatively simple and well accepted framework. The analysis that
follows borrows from Becker and Barro (1988) and Barro and Sala-i-Martin (1999) in
assuming a linear cost relationship in raising children. Their analysis suggests that the
fertility rate is related to the ratio of consumption to the capital stock. An increase in the
level of consumption increases child quality and the demand for children. The increase
in the capital stock increases the marginal product of labor and the opportunity cost of
having children.
Another recent approach taken by Galor and Weil (2000) looks at a
quantity/quality tradeoff in reproductive choice. Their model is based on the idea that
greater future technological change will cause families to shift from quantity to quality
since the reward to human capital investment is expected to rise. This approach allows
the actual transition from rising population growth rates to falling population growth rates
to be modeled. The purpose of this paper is not to predict the point of transition from
increasing population growth to decreasing population growth. The reason for our choice
of modeling strategy is simplicity. The Becker and Barro model, familiar to and accepted
by most economists, allows for a focus on the relationship between capital accumulation,
innovation, demographics, and the scale effects implications of the model. Extending the
model and pinning down the point of transition in the population growth rate is an
important extension that we leave to future research.
It is necessary to include capital in the model. This, it seems to us, is the most
natural way of introducing transitional dynamics. We follow the literature in assuming
that final output is produced with intermediate goods and labor, intermediate goods are
produced with capital alone, and more advanced intermediate goods are more capital
8
intensive in production. We generalize the model presented in section two to allow for
optimal choice, endogenous population growth and capital accumulation.
Production
Our departure point is equations 1 & 2. Allow   1, z1  L and z 2,i  xi , where
x i is now an intermediate good and i indexes the brand of intermediate. I is gross
investment in capital, K . The production functions for output and intermediates are:
1
(9) Y  C  I  N  L Q
 1
Q
Ax
i

i
di ,
0
(10) xi  x 
Ki
Ai
Equation 10 demonstrates a fundamental complimentarity between capital accumulation
and technology. We agree strongly with the position that the process of capital
accumulation and the process of technological change are best viewed as complementary.
As in the model presented in Section 2 we allow Q  L . The total capital stock is:
Q
Q
0
0
(11) K   Ai xi di  x  Ai di .
Q
We define the average level of productivity to be A   Ai di / Q and the leading edge
0
technology to be AMAX . Define the productivity adjusted capital stock per sector as
Q
(12) kˆ  K / AQ  x  Ai di / AQ  xi  x
0
This equation together with equation 9 shows that the production technology is Cobb
Douglas.
9
1
(13) Y  L Q
( 1)
Q
K
 A ( AQ )

i
di  K  ( AL)1
0
The model follows the existing literature in the specification of the production
function for new ideas but allows for any value of  . A Poisson process is specified, with
an arrival rate proportional to the level of resources per sector devoted to R&D. Equation
2 becomes:


A A MAX

(14) g A  

A AMAX
AMAX


N
  N 
  
  ,
AMAX  L 
Q
where  is the Poisson parameter and  is a research productivity parameter. According
to equation 4 the arrival of horizontal innovations, which do not enhance productivity, is


Q
L
 .
Q
L
Utility Maximization
The model of utility maximization follows the continuous time version of Barro’s
model developed fully in Barro and Sala-i-Martin (1999). Parents derive utility from
consumption, total family size, and the number of children. The problem is to maximize
the present value of dynastic utility. The intertemporal utility function is:

(15) U   e  t ( ln L  ln c   ln( b  d ))dt ,
0
where   0 is the discount rate, c is per capita consumption, L represents family size, b
is the birth rate and d is the mortality rate.   0 and   0 are elasticities.
Utility is maximized subject to the following constraints:

(16) k  w    rk  (b d )k  Bbk  c  n ,
10

(17) L  (b  d ) L ,
Where w is the wage rate;  is per capita firm profits; Bbk is a linear cost function for
rearing children, such that the present value of expenditures per child is directly
proportional to the capital to labor ratio; r is the interest rate, and n is the resources
devoted to R&D per capita.
This type of dynamic optimization problem is familiar to economists. The
problem has two choice variables, c and b. The state variables are k and L . Denote the
Hamiltonian as H. Define the multipliers for each constraint respectively as  and  .
The first order conditions can be summarized as:
a.
H
H
 0 , b.
0,
c
b

c.   

H
H
, d.   
,
k
L
First order conditions a and c provide the Euler equation.
(18) g c  r  (   (b  d )  Bb ) .
Equations a, b, d, and g L  b  d defines the birth rate at each point in time, equal to:
(P) b  d 
 (c / k )
.
 (1  B)   (c / k )
Equations 18 and P hold at all times, including any time period of transition to the
steady state. Equation 18 is the familiar Euler equation where (b  d ) increases the
discount rate; this is due to the diminishing marginal utility of children. Bb is subtracted
from r because child rearing costs, Bbk , increase with k . As k rises the return to
capital falls.
11
As equation P shows, the birth rate varies directly with the death rate, and
inversely with the cost of child rearing, B . The relationship between c / k and b is
direct. c represents an income effect (a higher c improves child quality and raises b )
and k represents a substitution effect (a higher k increases the opportunity cost of child
rearing and lowers b .)
Profit Maximization and Innovation
Intermediate goods are supplied to the producers of final goods under the
conditions of imperfect competition. Differentiation of equation 9 with respect to x
yields the derived demand for an intermediate good. Profits, therefore, are:

(19)   Ai L(1 )(1 ) xi  (r   ) Ai xi ,
where  is the rate of capital depreciation. The first order condition from maximizing
equation 19 together with equation 12 implies that the demand for intermediates is
symmetric (a fact expressed in equations 10 and 12 and verified here) and:
(20) r     2 (kˆ / L(1  ) )  1 .
Resources flowing into innovation are determined by equating the marginal
benefits of increasing R&D resources with the marginal costs of increasing R&D
resources. Allow V to represent the present value of innovating. Equations 10, 19, and
20 allow the present value of profits to be expressed as:
(21) V 
AMAX L(1 )(1  ) (1   )( kˆ)
r
 N

.
 
AMAX  L 
12
N / L 
The marginal benefit of increasing
 
AMAX
by a unit is V and the marginal cost is
AMAX . Setting marginal benefits equal to marginal costs gives a familiar research
arbitrage equation that depends, in part, on the parameters  and  .
(22) 1 
L(1 )(1  ) (1   )( kˆ)
r
 N

.
 
AMAX  L 
Equilibrium
Equations 9 through 22 completely define the dynamics of the model economy,
both during the transitional period and during the steady state. We follow Barro and
Sala-i-Martin (1999) and express the model in terms of the average product of capital,
~
Z  k  1 and the ratio c / k .
Proceed by deriving two unique relationships between Z and g A . The first
relationship needs hold only in the steady state. In the Steady state g A  g c . Using the
Euler equation this implies r  g A    (b  d )  Bb . Combine this equation with
~
equation 20 and note that kˆ  k L / Q :
(Z) g A   2 Z  (     d )  (1  B)b .
13
Figure 1
Z
gA
A
gA *
R
Z*
Figure
c/k
Z
2
P
(c/k)*
A
d
b
This Z equation is plotted in figure 1 as an upward sloping schedule since the partial
derivative,
g A
  2 , is greater than zero. The upward slope is interpreted as the impact
Z
14
of a change in g A on the Z. An increase in g A increases the growth rate of consumption
via the Euler equation. This implies an increase in the rate of interest and an increase in
the rental cost of capital. The demand for capital falls. As capital falls the average
product of capital increases. Fertility is a shift factor for Z. A fall in fertility, b, shifts the
schedule in figure 1 to the left. This is the impact of a fall in fertility on innovation. A
fall in b implies fewer resources devoted to child rearing and more resources for
innovation. Clearly b must be constant and the Z schedule stationary in a steady state
equilibrium.
The second relationship between Z and g A is derived from the research arbitrage
equation, an equation that holds even during the transition to the steady state. Equation
(1 )(1  )
22 is rewritten as r  g A /    (1   ) L

~
kˆ . Since k   Z  1 equation 20 can be
used to express the arbitrage equation as:
1 
(R) g A   (1   ) L Z

 1
  2 Z  
The R equation is plotted as a downward sloping line in Figure 1 because the partial
1
g A


 (1   ) LZ  1   2 , is less than zero. The intuition is, again,
derivative,
Z
 1
straight forward. A higher Z implies a lower capital stock. This directly raises the rental
rate of capital and diminishes the present value of an innovation (see equation 21.) With
the marginal benefits of raising R&D resources falling relative to the marginal costs of
raising R&D resources, resources to R&D are lowered as is g A . The scale effect appears
in this equation. An increase in L shifts the R curve up and to the right. This scale effect
is independent of the value of  because it enters the model from the demand side rather
15
than the supply side. It reflects the fact that a larger population leads to a shift to the right
in the derived demand for intermediates. This enhances the profits from innovation and
increases the steady state rate of growth. This scale effect is independent of the
parameter  .
The model is now expressed in the form of the three equations Z, R, and P.
(Z) g A   2 Z  (     d )  (1  B)b

(R) g A   (1   ) L1  Z  1   2 Z  
(P) b  d 
 (c / k )
 (1  B)   (c / k )
Next we compare various formulations of endogenous innovation models using these
three equations.
First allow   0 and   1 , producing a first generation growth model. The
equations are :
(Z - 1st Gen) g A   2 Z  (     d )  (1  B)b

(R – 1st Gen) g A   (1   ) LZ  1   2 Z  
(P – 1st Gen) b  d 
 (c / k )
 (1  B)   (c / k )
The equilibrium steady state for this specification is represented by point A in Figures 1
and 2. The traditional scale effect shows up in the model as L in the R equation. In
equilibrium a larger economy will experience a faster growth rate, ceteris paribus. It
should be pointed out that the size of the population is now endogenous with its growth
rate determined at each point in time by equation P, as graphed in Figure 2. A steady
state in a first generation model requires zero population growth, as noted by Jones
16
(1995a, 1995b, 1999). Point A in Figure 2 is drawn such that b=d, which holds
asymptotically as c/k falls.
Next consider the second-generation Jones model. We stated that too much is
made of the parameter  in the production function for new ideas, equation 2. If   0
and   1 then the steady state growth rate is still related to g L as suggested by equation
8. The reader can verify that changing the production function for knowledge in this way
has no impact on the R and Z and P schedules as outlined immediately above. A steady
state continues to require a constant L, pinning the population growth rate in the steady
state to zero. This is true because a demand side scale effect remains and is in no way
addressed by specifying a particular value for  less than unity.
In the simple one sector model scale is entirely a supply side phenomenon. It is
easy to classify an economy as large or small based on population size. The problem
becomes more ambiguous as more factors are added to the production function for ideas.
In the model outlined here the scale effect originates on the demand side; in the demand
for intermediates as expressed in the derived demand for intermediate goods. More
people imply a greater demand for output, a greater demand for intermediates, more
profits for successful innovators, and a higher rate of innovation. Changing the
parameter  in equation 2 does not change this chain of events in any way. Thus the
second generation Jones model is only consistent with a steady state growth rate of zero.
This intuition can be mathematically verified by log differentiation of equation 14
yielding g A  g N . In the steady state g N  g A  (b  d ) . Since (b-d) must be equal to
zero and   1 it must be true that g A  0 .
17
If a second generation Aghion and Howitt model is formulated by setting   1
and   1 then the equations are:
(Z-AH) g A   2 Z  (     d )  (1  B)b
(R-AH) g A   (1   )Z
(P-AH) b  d 

 1
  2 Z  
 (c / k )
 (1  B)   (c / k )
The scale effect in the resource constraint is effectively eliminated. The population
growth rate must be constant in the steady state, but there is no requirement that L be
constant. Thus the equilibrium in this model as represented in Figures 3 and 4 show
equilibrium at a positive population growth rate.
Figure 3
Z
gA
A
gA *
R
Z*
Z
18
Figure
c/k
4
P
A
(c/k)*
d
b
b
The transitional dynamics in this case are straightforward. Barro and Sala-iMartin (1999) show that, according to equation P, an economy with a falling population
growth rate is converging from below, where the growth in consumption is lower than
growth in the capital stock. In other words the capital to labor ratio during the transition
is below the steady state capital to labor ratio. Due to the diminishing marginal returns to
capital in the production function the average product of capital, Z, must be falling over
time as the c / k ratio falls. The fall in the c / k ratio and the demographic transition are
demonstrated in figure 4. The R schedule is now fixed and still must be satisfied at
every point in time. Recall that we must satisfy the Z equation only in the steady state.
These facts define the R schedule as a stable transitional path to the steady state
consistent with convergence from below.
19
The results are interesting. Rather than a positive relationship between the
population growth rate and innovation the model suggests just the opposite. Innovation
increases in spite of a falling population growth rate. The reason population growth is
not needed is the complimentarity between the capital accumulation process and
technological change. During the transition the growth rate output per capita responds
according to:
(22) g y  g k  (1   ) g A .
Contrary to Jones’ (1995b) main argument, it is immediately clear that the stationary
growth rate of per capita income is consistent with a transition to the steady state and a
falling population growth rate since the growth rate of capital is falling due to
convergence from below. Decreases in g K are offset by increases in g A during the
transition. The problem may be related to the reduced form approach to modeling growth
taken by Jones. The microeconomic foundations provided by the consumer’s choice and
research arbitrage leads to a scale effect through derived demand for intermediates. The
reduced form approach assumes that scale effects originate from the supply side in the
sense that more people represent more potential ideas and faster growth. The scale
effects in a model with microeconomic foundations can be considerably more complex,
originating from the extent of the market’s impact on the profitability of innovating.
V Conclusion
There is little doubt that understanding of the economics of endogenous
innovation is central to understanding why some countries are rich and some are poor.
Over the past two decades, however, the models explaining innovation as the endogenous
outcome of economic activity have become far more complex in an effort to eliminate
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scale effects from the first generation models of endogenous innovation and make the
models more consistent with the apparent lack of scale effects reported in the empirical
literature.
This paper asks what can be learned about first generation growth models, and
their decedents that eliminate scale effects, by carefully considering a falling endogenous
population growth rate as seen in the data for all developed economies. Our recognition
of the demographic transition provides useful guidance in incorporating transitional
dynamics into innovation based growth models. Our analysis leads us to some important
conclusions about the nature of scale effects when consumer optimization, capital
accumulation and fertility choice are included in the model. We find that the original
models of endogenous innovation with scale effects in growth rates are consistent with a
steady state population growth rate of zero. This is not a surprise. More interesting is the
fact that specifying diminishing returns to R&D in production function for new ideas by
setting   1 in equation 5 does not eliminate the scale effect in a multi factor model with
an endogenous savings rate. In fact the only plausible steady state is one where
population growth and the growth rate of per capita output are zero. Too much attention
is paid to the value of  in the production function for knowledge, since important scale
effects can enter the model from the demand side rather than from the supply side.
The 2nd generation Aghion and Howitt approach provides the most tractable
model with transitional dynamics allowing for positive steady state population growth.
The implications are consistent with a non-trending growth rate (typically thought to
imply steady state equilibrium) and a falling population growth rate which implies
transitional dynamics. Thus, a model where policy variables impact the long run growth
21
rate of per capita output, not just the long run level of per capita output, appears to be the
most logical model considering the nature of scale effects and demographics in an
endogenous growth model with an endogenous savings rate.
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