# Oxford University Press 1999
Oxford Economic Papers 51 (1999), 453±475
453
Capital accumulation, learning, and
endogenous growth
By Charles van Marrewijk
Erasmus University Rotterdam, Dept. of Applied Economics, H8±10, P.O. Box
1738, 3000 DR Rotterdam, The Netherlands; email: [email protected]
I develop a growth model with three sectors of production (®nal goods, intermediate
goods, and R&D), each with a different technology using three inputs (labor, capital,
and knowledge), and combining elements from three strands of the literature (neoclassical growth, and AK and R&D endogenous growth). The paper suggests not only
that capital accumulation and innovation are complementary processes, neither of
which would take place in the long run without the other, but also that this process
can be self-sustaining either through Hicks-neutral knowledge spillovers without
capital accumulation, or through Harrod-neutral knowledge spillovers with capital
accumulation.
1. Introduction
There are two types of endogenous growth models: AK models and R&D models,
see Jones (1995). The AK-style growth models of Romer (1986, 1987), Lucas
(1988), and Rebelo (1991) focus on capital accumulation, in the sense that there
is constant returns to a suf®ciently broad concept of capital, as the means to
generate endogenous growth. The R&D based models of Romer (1990), Grossman
and Helpman (1991a, 1991b), and Aghion and Howitt (1992) focus on technological progress, resulting from the search for innovation undertaken by pro®tmaximizing individuals, to raise productivity and generate endogenous growth. A
monetary version of this model is in Marquis and Reffett (1995). The model
developed in this paper combines the key elements of the two types of endogenous
growth models (capital accumulation and innovation) with a key element of the
neoclassical growth model, namely Harrod-neutral technological change, to generate growth (Ramsey, 1928; Solow, 1956; Burmeister and Dobell, 1970, ch. 3).
Aghion and Howitt (1997a, 1997b) also investigate the need to combine capital
accumulation and innovation to generate economic growth, rather than relying on
innovation or technological change alone. They term the latter the `consensus view'
(see Aghion and Howitt, 1997a, p. 1): `The consensus seems to be that capital
accumulation plays at best a passive, supporting role in the long run, affecting
only the level of output, not its rate of growth.' They argue (1997a, pp. 2±3) that
the consensus view is wrong because it is based on the rather special and unrealistic
assumption made in previous models with innovation and capital accumulation
that capital is not used in the R&D technology that produces innovations. At the
454
capital accumulation
very least, one would argue, capital is used in the R&D sector to minimize production costs, where we can think of relatively simple items such as buildings, personal
computers, or of®ce equipment, or we can think of vastly more sophisticated (and
expensive) items, such as laboratory equipment, main frame computers, scanning
tunneling microscopes, and super-colliders.1 The model below investigates the role
of capital accumulation in the R&D process to conclude that: `capital accumulation
and innovation are complementary processes. . . For without innovation, diminishing returns would choke off net investment, and without net investment the
rising cost of capital would choke off innovation' (Aghion and Howitt, 1997b, p.
99). My approach differs from Aghion and Howitt's in at least three ways. First, I
investigate expanding product variety (horizontal differentiation) rather than quality improvements (vertical differentiation). Second, knowledge spillovers (learning)
are associated with the extent to which human beings are involved in the production process, as captured by Harrod-neutral technical change in the neoclassical
growth model. Third, I allow for different production technologies in the R&D
sector and in the ®nal goods sector, rather than adopting the approach of identical
technologies in these two sectors used thus far in the literature, see Rivera-Batiz and
Romer (1991), Barro and Sala-i-Martin (1995, chs 6 and 7), and Aghion and
Howitt (1997a, 1997b).
2. Rival inputs, accumulation and economic growth:
a discussion
I distinguish between three different types of inputs: (i) rival and non-accumulable
inputs, (ii) rival and accumulable inputs, and (iii) non-rival and accumulable
inputs. This classi®cation is based on the dual distinction between rival and
non-rival inputs and between accumulable and non-accumulable inputs, see
Romer (1990, pp. s73±8) for a discussion of these and related concepts. The
fourth logical possibility, that of a non-rival and non-accumulable input, seems
less interesting, and would be trivial to include.
We take labor, denoted by L, as an example of a rival and non-accumulable
input. A rival input is an input that, if used in one sector, cannot simultaneously be
used in another sector. Although a secretary can work in the ®nal goods sector one
day and in the R&D sector on another day, he cannot be working in both sectors
simultaneously. Although the number of laborers can, of course, vary over time it is
a non-accumulable input (bounded from above). In the sequel, the number of
laborers L is constant.
We take capital, denoted by K, as an example of a rival and accumulable input. A
desk cannot simultaneously be used in the ®nal goods sector and in the intermediate goods sector, while the economic value of the capital stock is in principle
unbounded.
..........................................................................................................................................................................
1
Capital could also be an input to R&D as waiting Austrian style, i.e. as a wages-advanced capital cost.
charles van marrewijk 455
Finally, we take knowledge, denoted by N, as an example of a non-rival and
accumulable input. Clearly, the idea of double-entry bookkeeping can be used
simultaneously in all sectors of the economy, and a shortage of new ideas does
not appear to be imminent.
In the sequel, knowledge will be measured by the increase in the number of
intermediate goods. I want to avoid possible confusion associated with the use of
the term `capital'. Romer (1990), for example, uses the term `capital stock' for a
mixture of the number of different types of intermediate goods (or capital goods;
that is, a non-rival and accumulable input) and the number of each type of intermediate good produced (a rival and accumulable input). Moreover, he distinguishes between different types of labor, unskilled and skilled, where the latter is
termed `human capital' (a rival and non-accumulable input). The term capital then
refers simultaneously to both rival and non-rival inputs and to accumulable and
non-accumulable inputs. To avoid potential confusion I use the terms `labor',
`capital', and `knowledge' as explained above. Alternatively, the reader may
prefer to think of `human capital' as an example of a rival and non-accumulable
input, `physical capital' as an example of a rival and accumulable input, and
`knowledge capital' as an example of a non-rival and accumulable input.
Let Y denote ®nal output, let t denote time, let x_ denote the change of variable x
over time, that is x_ ˆ dx=dt, and let x~ be the growth rate of variable x over time,
_
that is x~ ˆ x=x,
for any variable x. We can now write general reduced form
equations for ®nal output and the accumulation of knowledge (see also RiveraBatiz and Romer, 1991)
…1†
Y ˆ F…Ky ; Ly ; N†
N_ ˆ R…Kn ; Ln ; N†
…2†
Equation (1) indicates that the production of ®nal goods rises from increases in
the rival inputs, capital Ky and labor Ly . Final output also rises if the stock of
knowledge N rises. Since knowledge is a non-rival input it does not require a subindex, that is it can be used simultaneously in the ®nal goods sector and in the R&D
sector. Equation (2) indicates that the production of knowledge rises from
increases in the rival inputs, capital Kn and labor Ln and from a rise in the nonrival input knowledge. In the discussion in the remainder of this section we investigate constant growth rates of knowledge and ®nal output for a given division of
non-accumulable labor over the two sectors (Ly and Ln constant). Using Kaldor's
(1961, pp. 178±9) stylized fact we restrict the rate of capital accumulation to equal
~ Notation-wise, we let " j be the input
the rate of output growth, that is Y~ ˆ K.
i
j
elasticity of factor i for production function j, that is "i ˆ …@j…:†=@i† …i=j† for i ˆ l,
k, n and j ˆ F, R.
2.1 Neoclassical growth model
The neoclassical growth model (summarized in Burmeister and Dobell, 1970, ch. 3)
uses Harrod-neutral technical change, with a given growth rate, and constant
returns to scale in the ®nal output sector, where all capital and labor is employed,
456
capital accumulation
for the economy to grow. In the notation of eqs (1) and (2) this can be written as
Y ˆ F…K; NL† with F…K; NL† ˆ F…K; NL† and N_ ˆ gN, where g is a constant,
exogenously given rate of technical change. Thus, the exogenous growth rate of the
stock of knowledge …N~ ˆ g† in combination with the above rule and notation
determines the growth rate of ®nal output, Y~ ˆ …"Fl =1 ÿ "Fk †g ˆ g, where the last
equality follows from "Fl ‡ "Fk ˆ 1.
2.2 AK endogenous growth model
An AK-type endogenous growth model assumes constant returns to a broad measure of accumulable capital (with or without spillovers) as the driving force behind
economic growth. In the notation of eqs (1) and (2) this can be written as
Y ˆ F…K; L† with F…K; L† ˆ KF…1; L† and N_ ˆ 0. This production function
immediately implies that ®nal output grows in proportion to the capital stock,
~ The growth rate itself is determined by confronting savings and investment
Y~ ˆ K.
incentives.
2.3 R&D endogenous growth model
An R&D-type endogenous growth model without capital assumes constant returns
to knowledge in the R&D sector and human effort into that sector to be the driving
force behind economic growth. In the notation of eqs (1) and (2) this can be
written as Y ˆ F…Ly ; N† and N_ ˆ R…Ln ; N† with R…Ln ; N† ˆ NR…Ln ; 1†. Thus, the
growth rate of the stock of knowledge is endogenous, N~ ˆ R…Ln ; 1†, and determined by confronting revenue and cost in the R&D sector. The growth rate of ®nal
output is then determined by the production function, Y~ ˆ "Fn R…Ln ; 1†:
This R&D type model can be easily extended to include capital accumulation
into the ®nal goods sector only, that is Y ˆ F…K; Ly ; N† and N_ ˆ R…Ln ; N† with
R…Ln ; N† ˆ NR…Ln ; 1†. Since there is no feedback between capital accumulation and
the rate of accumulation of the stock of knowledge the growth rate of ®nal output
is then simply Y~ ˆ "Fn =…1 ÿ "Fk †R…Ln ; 1†; see e.g. Grossman and Helpman (1991a,
p. 121).
2.4 Capital accumulation in the R&D sector
If capital, labor, and knowledge are used simultaneously to produce new knowledge
and to produce ®nal output this links the growth rates of eqs (1) and (2). This link
implies that we have to impose restrictions on the production functions F and R for
consistency. A way out of this problem is to effectively investigate a one-sector
model by having the same production functions F and R for ®nal output and the
accumulation of knowledge, see e.g. Rivera-Batiz and Romer (1991), Barro and
Sala-i-Martin (1995, ch. 6, 7), or Aghion and Howitt (1997a, 1997b). The approach
adopted in this paper, however, allows for different production functions and
different capital intensities in the ®nal goods sector and in the R&D sector by
combining the key elements of the neoclassical growth model, the AK endogenous
growth model and the R&D endogenous growth model as outlined above. To this
charles van marrewijk 457
end we use Harrod-neutral knowledge spillovers and constant returns to scale in
the R&D sector of the economy, and constant returns to accumulable factors in the
®nal output sector of the economy. In the notation of eqs (1) and (2) this can be
written as Y ˆ F…Ky ; Ly ; N† with F…Ky ; Ly ; N† ˆ F…Ky ; Ly ; N†, that is constant
returns to accumulable factors for ®nal output, and N_ ˆ R…Kn ; NLn † with
R…Kn ; NLn † ˆ R…Kn ; NLn †, that is constant returns to R&D with Harrod-neutral knowledge spillovers. Differentiation for constant growth rates and a stable
~ and N~ ˆ "Rk K~n ‡ "Rl N:
~ For
distribution of the labor force gives: Y~ ˆ "Fk K~y ‡ "Fn N,
F
F
R
~
~
~
~
Y ˆ Ky ˆ Kn ˆ N this can be consistent in view of "k ‡ "n ˆ 1 and "k ‡ "Rl ˆ 1:
2.5 Overview
I proceed in two steps. The ®rst step of the paper is taken in Section 3, which
incorporates capital in all sectors of the economy with Hicks-neutral knowledge
spillovers, but without capital accumulation. Section 4 discusses some of the possibilities and complications of this model using three examples. Since there is no
capital accumulation in Sections 3 and 4 of the paper the term capital as used in
these sections is not strictly in accordance with the terminology at the beginning of
this section. The whole purpose of these sections, however, is to demonstrate the
possibilities and complications of allowing for such capital accumulation. The
second step of the paper is taken in Section 5, which amends the model of Section
3 to allow for capital accumulation by using Harrod-neutral knowledge spillovers
in the R&D and intermediate goods sectors. We illustrate and discuss the amended
model in Section 6 by using a Cobb±Douglas technology. Section 7 concludes. In
setting up the model structure we follow Grossman and Helpman (1991a, ch. 5.1).
Our ®nal model differs from theirs in (i) having general production technologies
for the intermediate goods sector and the R&D sector, (ii) allowing for the use of
capital and capital accumulation in all three sectors of the economy (in particular
for R&D), and (iii) introducing Harrod-neutral knowledge spillovers in the R&D
and intermediate goods sector in combination with constant returns to accumulable factors in the ®nal goods sector to enable balanced endogenous growth.
3. An R&D model without capital accumulation
I incorporate capital in an R&D based model of endogenous growth by investigating a closed economy with three sectors of production: intermediate goods, ®nal
goods, and R&D. All sectors of the economy use capital and labor as productive
inputs, while the ®nal goods sector also uses intermediate goods. Total demand for
capital and labor by these three sectors of the economy must, in equilibrium, be
equal to total supply, which is perfectly inelastic at exogenous levels K and L,
respectively. The possibility of accumulation of physical capital will be discussed
in Section 5. Many small ®rms manufacture ®nal goods subject to constant returns
to scale. We aggregate these to the industry level and use the following technology;
see Benhabib and Jovanovich (1991) or Mankiw et al. (1992)
458
capital accumulation
Y ˆ Ay Kyk Dd Ly l ;
with k ‡ d ‡ l ˆ 1
… N
1=
‰x… j†Š
Dˆ
0
…3†
…4†
Where Ay is a constant, Ky denotes the capital stock allocated to the ®nal goods
sector, D represents an index of intermediate goods, and Ly denotes employment of
labor in the ®nal goods industry. Equation (4) gives the index of intermediate
goods, making use of the by now well-known expanding variety speci®cation,
where x… j† represents the input of component j in the production of ®nal goods,
N represents the number of varieties and 1=1 ÿ " is the elasticity of substitution between any two varieties. Let w1 be the wage rate, wk the rental rate on capital,
and pd the minimum cost to manufacturers of obtaining one unit of the quantity
index D. The price index pd , therefore, re¯ects both the prices of the underlying
intermediates and the state of technology. The ®nal goods market is perfectly
competitive, so the price py equals the production costs, which (through choice
of units Ay ) is given by
py ˆ pd d wkk wll
…5†
The producers of intermediates face a constant elasticity of demand 1=1 ÿ and,
in a monopolistically competitive environment, charge a constant mark-up 1=
over marginal production cost given by the unit cost function cx …wk ; wl †. That is,
intermediates are produced under constant returns to scale taking input prices as
given. The optimal pricing rule for producers of intermediates, who all charge the
same price, is
c …w ; w †
…6†
px ˆ x k l
Producers of ®nal goods, therefore, use all available varieties in equal quantities and
the index of intermediate inputs is given by
…7†
D ˆ Ad X
where X ˆ Nx denotes the aggregate volume of intermediate output and
Ad ˆ N 1ÿ= ˆ N 1="ÿ1 represents the productivity index. From pd D ˆ px X then
follows
p
…8†
pd ˆ x
Ad
The producer of a particular variety makes instantaneous pro®ts, …t† say,
because the price charged is higher than the production cost. Let …t† denote
the value of a ®rm
…z
…1
eÿ‰R…†ÿR…t†Š …†d;
where R…z† ˆ r…s†ds
…9†
…t† ˆ
t
0
The term R…t† represents the discount factor from time t to time 0, while r…s† is the
instantaneous interest rate at time s. Equation (9) states that …t† equals the present
discounted value of future pro®ts. Differentiating eq. (9) with respect to time t
gives the no-arbitrage condition
charles van marrewijk 459
~ ˆ r ÿ
v
…10†
New varieties of intermediates can be developed under constant returns to scale
given the general level of knowledge, which is treated as a partially public good. We
make this general knowledge proportional to the number of varieties N available,
which means that each R&D project contributes to the general stock of knowledge
independently of the aggregate amount of R&D that has been undertaken in the
past. The private cost of developing a new variety is pn ˆ cn …wk =N; wl =N †, where N
is treated as exogenous by the producers of new varieties. Thus, increases in the
general stock of knowledge reduce the private cost of developing new varieties.
These knowledge spillovers in the R&D sector are Hicks-neutral. Free entry into the
development of new varieties ensures that the costs of developing new varieties
cannot fall below the value of a new variety
w w c …w ; w †
with equality if N~ > 0
…11†
pn ˆ cn k ; l ˆ n k l v
N N
N
The representative household maximizes intertemporal utility over an in®nite horizon. Preferences take the simple form
…1
C…†1ÿ ÿ 1
d
…12†
eÿ…ÿt†
Ut ˆ
1ÿ
t
Maximization of (12) subject to an intertemporal budget constraint that allows
consumers to borrow or lend freely at the instantaneous interest rate r
…1
…1
eÿ‰R…†ÿR…t†Š py …†C…†d eÿ‰R…†ÿR…t†Š wl …†d ‡ W…t†
…13†
t
t
where W…t† is initial wealth at time t, requires spending E ˆ py C to evolve
according to
1
…14†
E~ ˆ …r ÿ †
To pin down the price level at any time we are free to set the time path of one
nominal variable and to measure prices at any moment of time against the chosen
numeÂraire. This has, of course, no effect on the evolution of real variables, such as
production or relative prices. It is most convenient to follow Grossman and Helpman and normalize prices so that nominal spending remains constant
E…t† ˆ 1; 8t;
hence r…t† ˆ ; 8t
…15†
Let V py Y=N be the ratio of aggregate output to the aggregate value of the
0
stock market, let ! wl =wk be the wage-rental ratio, and let ci;z
…:† be the derivative
of function ci ; i ˆ x; n; with respect to argument z ˆ wk ; wl ; then:
Proposition 1 The positive balanced growth rate of the economy with Hicksneutral knowledge spillovers in the R&D sector and without capital accumulation
is given in eqs (16)±(18)
460
capital accumulation
…1 ÿ †d V ˆ N~ ‡ 0
…1; !†N~ ‡ fL1 …!†V ˆ L;
cn;w
l
0
…1; !†N~ ‡ fK1 …!†V ˆ K;
cn;w
k
Proof
2
fL1 …!† 6l ‡
4
3 1
7 cn
;1
5
!
0
d cx;w
…1; !†
l
1
;1
cx
!
2
3
1
0
1; 7
d cx;wk
6
!
4
5
cn …1; !†
fK1 …!† k ‡
cx …1; !†
…16†
…17†
…18†
See Appendix 1.1.
~ V and ! along a balanced
Equations (16)±(18) together determine the values of N,
growth path. Equation (16) gives combinations of V and N~ for which the ratio of
output to the aggregate value of the stock market does not change …V~ ˆ 0†, while
eqs (17) and (18) represent factor market clearing conditions for labor and capital,
respectively.
The situation is illustrated in Fig. 1. Both the labor market clearing condition
and the capital market clearing condition can be depicted as straight lines in
~ V †-space given any particular value of the wage-rental ratio, !0 say.
…N;
Both market clearing conditions also intersect the upward sloping relationship V~ ˆ 0 given by eq. (16). If this point of intersection does not coincide,
as is the case for the L0 L0 curve and the K0 K0 curve in Fig. 1, intersecting
at points A0 and B0 respectively, then this particular value of the wage-rental
ratio !0 is not consistent with balanced growth. Now an increase in the wagerental ratio from !0 to !1 shifts the LL curve outward to L1 L1 and the KK
curve inward to K1 K1 such that the two points of intersection with the
V~ ˆ 0 curve coincide, that is A1 ˆ B1 . Clearly, the wage-rental ratio ! can
adjust instantaneously to ensure that the economy immediately jumps to the
balanced growth path.
0
…1; !† and
To be somewhat more precise, it is clear from the properties of cn;w
l
0
cn;wk …1; !† that an increase in ! will shift the point of intersection of the LL
curve with the horizontal axis outward, while the point of intersection of the
KK curve with the horizontal axis will shift inward. The results with respect to
the vertical axis are somewhat more cumbersome. However, using the f L1 …!†
and f K1 …!† functions and standard properties of the cost function it follows
that an elasticity of substitution between capital and labor in the production of
intermediate goods of at least 1 is a suf®cient condition for an inward shifting
KK curve and an outward shifting LL curve if ! rises. The balanced growth rate
of innovation for any given value of the capital stock is therefore unique if
the elasticity of substitution between capital and labor in the production of
intermediate goods is larger than or equal to unity. In general, however, multiple
solutions to (16)±(18) may exist, or none at all (for example if l < …1 ÿ †
in Section 4.1 below).
charles van marrewijk 461
Fig. 1. Endogenous growth without capital accumulation and with Hicksneutral knowledge spillovers in the R&D sector
4. Examples without capital accumulation
Below we brie¯y discuss three examples of the R&D based endogenous growth
model without capital accumulation developed in Section 3. In particular, these
examples give us information on the possibilities and dif®culties of extending
the model to include capital accumulation. Throughout this section we assume
Cobb±Douglas technology in the intermediate goods sector, that is
cx …wk ; wl † ˆ …wk †k …wl †l ; for 0 k 1 and k ‡ l ˆ 1.
4.1 Only labor or only capital as an innovative force
4.1.1 Labor as an innovative force The ®rst example, in which we assume that only
labor is required to produce new varieties and intermediate goods, is taken from
Grossman and Helpman (1991a, ch. 3). Suppose that pn ˆ cn …wk ; wl †=N ˆ wl =N:
This reduces the factor market clearing eqs (17) and (18) to
N~ ‡ bl V ˆ L;
bk V ˆ K=!;
bl l ‡ l d
…19†
bk k ‡ k d
…20†
462
capital accumulation
Fig. 2. Hicks-neutral knowledge spillovers in the R&D sector, where labor is the
only input
The situation is illustrated in Fig. 2. The labor market clearing condition (19) is
independent of the wage-rental ratio and together with eq. (16) determines the
balanced growth rate of innovation N~l say
N~l ˆ l L ÿ …1 ÿ l †;
0 < l …1 ÿ †d
<1
…1 ÿ †d ‡ bl
…21†
Note that a positive rate of innovation requires l L > …1 ÿ l †, that is the LL
curve must intersect the vertical axis above the V~ ˆ 0 curve. We see from eq. (21)
that the rate of innovation N~l is independent of the capital stock. Equation (20)
implies that the KK curve becomes a horizontal line, see Fig. 2. The wage-rental
ratio ! in eq. (20) simply adjusts to ensure that the capital market clears, that is
such that the horizontal KK curve intersects the LL curve and the V~ ˆ 0 curve. The
labor market plays a decisive role in determining the rate of innovation, while the
capital market simply plays an accommodative role. What happens if we allow for
capital accumulation in this example, that is let K increase? It simply results in an
offsetting rise in the wage-rental ratio (see Grossman and Helpman, 1991a, ch. 5.1,
for further details).
charles van marrewijk 463
4.1.2 Capital as an innovative force Investigating the mirror image of example
4.1.1 in which capital, rather than labor, is the only input in the R&D sector, that is
pn ˆ cn …wk ; wl †=N ˆ wk =N reduces the factor market clearing eqs (17) and (18) to
bl V ˆ L!
…19 0 †
N~ ‡ bk V ˆ K
…20 0 †
This time the capital market clearing condition …20 0 † is independent of the wagerental ratio and together with eq. (16) determines the balanced growth rate of
innovation, N~k say
N~k ˆ k K ÿ …1 ÿ k †;
0 < k …1 ÿ †d
<1
…1 ÿ †d ‡ bk
…21 0 †
The wage-rental ratio simply adjusts to clear the labor market. A one-time increase
in the capital stock increases the growth rate of the economy.
4.2 Both capital and labor as an innovative force
The third example assumes that both capital and labor are necessary to develop
new varieties and produce intermediate goods, that is pn ˆ cn …wk ; wl †=N ˆ
…wk †k …wl †l =N; with 0 < k < 1 and k ‡ l ˆ 1. This reduces the factor market
clearing eqs (17) and (18) to
…22†
l N~ ‡ bl V ˆ L!k
k N~ ‡ bk V ˆ K=!l
…23†
Using eqs (16), (22), and (23) one may calculate that the balanced rate of
innovation, N~ say, is the solution to
1
1 ÿ k k 1
1 ÿ l l
ÿ l N~ ‡
ÿ k N~ ‡
…24†
Ll K k ˆ
k
l
k
l
The left-hand-side of eq. (24) is constant, while the right-hand-side of eq. (24) is
~ Thus a positive balanced growth rate of innovation,
monotonically rising in N.
which depends in a rather complicated way on the various parameters in the
economy, exists and is unique if and only if Ll K k > ‰1 ÿ k =k Šk ‰1 ÿ l =l Šl :
The situation is illustrated in Fig. 3. Note from eqs (22) and (23) that an increase
in the wage-rental ratio leads to a parallel outward shift of the LL curve and a
parallel inward shift of the KK curve. Both curves are in principle unbounded. The
~ V -space for a given value of !, as
LL curve is steeper than the KK curve in N;
drawn in Fig. 3, if and only if l =k > bl =bk ; which, for the sake of argument, I now
assume. This holds, for example, if the ®nal goods sector is most capital intensive,
the R&D sector is most labor intensive and capital intensity in the intermediate
goods sector is intermediate, that is l =k > l =k > l =k : It follows from the
intersection of eqs (22) and (23) that N~ and V are both positive if and only if
l K
b K
>!> l
k L
bk L
…25†
464
capital accumulation
Fig. 3. Cobb-Douglas example with Hicks-neutral knowledge spillovers in the
R&D sector, which uses both capital and labor; one-time capital stock increase
The ®rst inequality in eq. (25) follows from the condition that V be positive and
the second inequality from the condition that N~ be positive.
Figure 3 illustrates what happens with a one-time increase in the capital
stock K. First, at the initial equilibrium wage-rental ratio (at E0 ), the KK
curve shifts out from K0 K0 to K1 K1 This parallel outward shift is incompatible
with a balanced growth equilibrium since the L0 L0 curve intersects the V~ ˆ 0
curve at a different point than the K1 K1 curve. Equilibrium is restored at E1
through a rise in the wage-rental ratio which causes a parallel inward shift of
the K1 K1 curve to K2 K2 and a parallel outward shift of the L0 L0 curve to L2 L2 .
An increase in the capital stock therefore raises the balanced growth rate of
~ the wage-rental ratio ! and the value of output relative to the
innovation N,
stock market value V . There are two effects: a higher capital stock reduces the
price of capital and then (i) research costs fall, thus stimulating innovation, and
(ii) the costs of producing intermediaries falls, implying that more of each good is
produced, so that the value of innovation rises, which leads to more innovation for
given research costs. Similar reasoning shows that a rise in the labor force also
increases N~ and V , but lowers !, while a rise in the rate of time preference raises !
~
and V , but lowers N:
charles van marrewijk 465
Fig. 4. Response of the rate of innovation to changes in the capital share in
R&D; Cobb-Douglas example; par 1 ˆ parameter set 1; par 2 ˆ parameter set 2
(see Appendix 1.2)
Clearly, eqs (22) and (23) reduce to eqs (19) and (20) if l ˆ 1 ÿ k ˆ 1, while
they reduce to equations (19 0 ) and (20 0 ) if l ˆ 1 ÿ k ˆ 0, therefore
lim N~ ˆ z Z ÿ …1 ÿ z † ˆ N~z ;
z !1
for z ˆ K; L
…26†
This is illustrated for two different parameter sets in Fig. 4, showing the
impact on the rate of innovation as the capital share in R&D varies between
0 and 1. As indicated in eq. (26) the endpoints are given by the pure labor
and pure capital rates of innovation …N~l f or k ˆ 0 and N~k f or k ˆ 1;
respectively). As illustrated in Fig. 4, the transition from one extreme to another
as the capital share varies can be either monotone (parameter set 1) or nonmonotone (parameter set 2), that is the Cobb±Douglas structure is richer than
either of the two extreme special cases; see Appendix 1.2 on the values for
parameter sets 1 and 2.
466
capital accumulation
5. Capital in an R&D model with accumulation
The model of Section 3 incorporates capital in all three sectors of an R&D-based
endogenous growth model, but (i) does not allow for the accumulation of capital,
and (ii) does not explain where the capital stock is coming from. We now modify
the model of Section 3 to address these issues.
5.1 Capital accumulation
The example in Section 4.2, where both capital and labor are necessary to produce
new varieties, clearly illustrates the problem of capital accumulation in the model of
Section 3. A one-time increase in the capital stock raises the growth rate of the
economy, see Fig. 3. Therefore, if we allow for a continuous increase in the capital
stock the growth rate will rise inde®nitely. The only exception to this rule was given
in the ®rst example of Section 4. If labor is the only input in the R&D sector, capital
accumulation simply leads to a rise in the wage-rental ratio, without affecting the
rate of innovation (see Fig. 2).
The model of Section 3 can be easily modi®ed to allow for accumulation of the
(non-depreciating) capital stock and simultaneously explain where the capital stock
is coming from if we assume, as is customary, that ®nal goods can be either
consumed or added to the capital stock. As explained above this requires other
changes in the model to avoid ever increasing growth rates. Recall, from eq. (15),
that C ˆ E=py ˆ 1=py : Since ®nal goods can be either consumed or invested
market clearing in the ®nal goods sector is given by
dK 1
‡ ˆY
…27†
dt py
The price of a unit of the capital good must be equal to the present discounted
value of future rental payments, that is
…1
eÿ‰R…†ÿR…t†Š wk …†d
…28†
py …t† ˆ
t
Differentiating eq. (28) with respect to time t, while using the normalization (15),
gives the no-arbitrage condition
wk
‡ p~y ˆ …29†
py
that is, the rate of return on capital goods plus the capital gains must be equal to
the instantaneous interest rate.
5.2 Knowledge spillovers
The model of Section 3 has a constant capital±labor ratio in the R&D sector along a
balanced growth path, which requires a constant wage-rental ratio. If the capital
stock accumulates over time Kn =Ln cannot be constant if Kn accumulates and Ln is
bounded above by the available number of laborers. If we maintain the essential
feature of knowledge spillovers in the R&D sector and the growth path is to leave
the capital/output ratio in this sector unaltered, we know from neoclassical growth
charles van marrewijk 467
models that factor-augmenting technical change should be Harrod-neutral, that is
eq. (11) changes to
w
pn ˆ cn wk ; l ;
with equality if N~ > 0
…30†
N
The Appendix shows that this implies that for any given value of !=N both the
demand for labor and the effective capital-labor ratios Kn =NLn and Ky =NLy in the
R&D sector and in the ®nal goods sector remain constant. That is, we go from a
constant value of capital per worker in these sectors to a constant value of capital
relative to the stock of knowledge per worker. Indeed a constant value of Ln and of
Kn =NLn is exactly what is required to keep the rate of innovation N~ constant.
Consequently, a constant ratio of !=N is an obvious candidate for a balanced
growth path. This implies that the demand for labor and the effective capitallabor ratio Kx =NLx in the intermediate goods industry must be constant for a
given value of !=N as well. The latter, in turn, requires the intermediate goods
industry to bene®t from labor related knowledge spillovers, just as in the R&D
sector. That is, eq. (6) must change to
w
cx wk ; l
N
…31†
px ˆ
5.3 Endogenous growth with capital accumulation
With these changes to the model of Section 3 we get:
Proposition 2 The positive balanced growth rate of the R&D model of this
section, with capital accumulation and Harrod-neutral knowledge spillovers in
both the R&D sector and the intermediate goods sector, is given in eqs
(32)±(34), provided condition (35) holds. On this path we have:
_
~k ; w
~l ˆ 0 and K=Y
~ ˆ ÿp~y ˆ ÿw
is constant
N~ ˆ K~ ˆ Y~ ˆ !
…32†
…1 ÿ †d V ˆ N~ ‡ !
!
0
N~ ‡ fL2
V ˆL
…33†
cn;wl 1;
N
N
with
2
! 3
0
!
c
! d x;wl 1; N 7
N
6
! 5cn
;1
4 l ‡
fL2
N
N c 1;
!
x
N
!
0
~ ‡ fK2 ! V ˆ K
…34†
N
1;
cn;w
k
N
N
N
with
2
! 3
0
!
! N d cx;w
1;
k
6
7
N
! 5
;1
4k ‡
c
fK2
N
N n !
cx 1;
N
d
…35†
1 ˆ k ‡
…" ÿ 1†
468
capital accumulation
Fig. 5. Endogenous growth with capital accumulation and Harrord-neutral
knowledge spillovers in the R&D sector and the intermediate goods sector
Proof On eqs (32)±(34) see Appendix 1.3; on eq. (35) and the last sentence see
below.
~ V and !=N given L
Eqs (32)±(34) determine the balanced growth values of N;
and the initial value of K=N, which will be maintained on the growth path. The
~ V )±space in Fig. 5. Equation (32) is an upward
situation is illustrated in (N;
sloping straight line, while eqs (33) and (34) are downward sloping straight lines
for any given value of !=N. A balanced growth equilibrium exists when the three
curves intersect at the same point, which can be achieved by a suitable choice of
!=N. The model in this section with capital accumulation is therefore quite similar
to the model of Section 3 without capital accumulation, with !=N in this section
playing the role of ! in Section 3 and K=N in this section playing the role of K in
Section 3.
There is one more important point to make. We have allowed both capital K and
the number of intermediates N to accumulate. Balanced growth in general only
exists if there is constant returns to accumulable factors and this model is no
exception. Note that the curves in Fig. 5 do not shift for any value of !=N if
and only if the capital stock grows at the same rate as the rate of innovation,
~ Using eq. (27) and the fact that V is constant, this implies
that is if K~ ˆ N:
charles van marrewijk 469
~ From (3) we get Y~ ˆ k K~y ‡ d D:
~ Combining this with K~y ˆ K~ ˆ N~
that Y~ ˆ K:
~
~
and D ˆ 1=…" ÿ 1†N shows that consistency implies that eq. (35) should hold, that
is there should be constant returns to accumulable factors.
5.4 Discussion
The balanced growth equilibrium of Proposition 2 incorporates a number of
desirable properties identi®ed in the literature.
As emphasized by Romer (1994, pp. 12±13) production takes place by many
®rms in a decentralized market economy. Knowledge is a non-rival good, which in
conjunction with a replication argument for physical activities implies increasing
returns to scale in production. Technological advance is not exogenous but requires
deliberate effort and investment in R&D activities. These investments pay off
because ®rms have market power and earn monopoly rents on their discoveries.
As emphasized by Kaldor (1961, pp. 178±9) the investment/output ratio and the
capital/output ratio are constant. At the same time the capital/labor ratio and the
output/labor ratio are rising. So each worker is using more and more capital over
time and producing more and more output. Consequently, the real wage is rising.
On the other hand, the rate of interest, and the relative shares of capital and labor
in national income remain constant.
There are some additional characteristics. For example, the knowledge/output
ratio and knowledge/capital ratio are constant. This implies that the knowledge/
labor ratio is rising. So each worker is also using more and more non-rival knowledge over time. There are different production functions for the ®nal goods sector,
the intermediate goods sector and the R&D sector. The latter two bene®t from
Harrod-neutral knowledge spillovers. Capital accumulation and the rate of innovation jointly keep the economy growing. Given the structure of the model, capital
accumulation and innovation are complementary processes, that is neither would
take place in the long run without the other.
6. An example with capital accumulation
As in the example of Section 4.2 we assume that production in the R&D sector and
the intermediate goods sector is Cobb±Douglas, this time with Harrod-neutral
knowledge spillovers, that is pn ˆ cn …wk ; wl =N † ˆ …wk †k …wl =N †l and
cx …wk ; wl =N † ˆ …wk †k …wl =N †l ; with 0 < k ; dk < 1; k ‡ l ˆ k ‡ l ˆ 1, and
subject to condition (35). The factor market clearing eqs (33) and (34) then
reduce to
! k
…36†
l N~ ‡ bl V ˆ L
N
K ! ÿl
…37†
k N~ ‡ bk V ˆ
N N
where the parameters bl and bk above and the parameters l and k below are
de®ned in Section 4. Using eqs (32), (36), and (37) one may calculate that the
~ say, to
balanced growth rate is the solution, N
470
capital accumulation
L l
k K
1
1 ÿ k k 1
1 ÿ l l
ˆ
ÿ l N~ ‡
ÿ k N~ ‡
k
l
N
k
l
…38†
This result is similar to the example in Section 4.2 with K=N, the stock of capital
relative to the stock of knowledge, replacing K, the stock of capital.
Naturally, we are interested in the behavior of the economy if condition (35)
does not hold. Suppose that the right-hand-side of (35) is smaller than 1. Under
those circumstances the growth rate of the capital stock can ultimately not keep
pace with the rate of innovation. This, other things being equal, causes an inward
shift of the K=N curve until innovation comes to a complete halt. Appendix 1.4
shows that this occurs at the value, N , given by
2
31=l ‡l d ÿk ‡1=
!
2k l ‡l d =k
d
…1
ÿ
†
Lb
d
k
5
…39†
N ˆ 4
bl Beyond the cut-off point N no new varieties are invented. This in turn determines
K, Ky , Kx , Y, etc. The pro®t rate of developing new varieties is ultimately driven
down to the level of the discount rate, which implies that at that point there is no
further incentive to invest in R&D. We can now see a rather complicated process in
action in which the development of new varieties increases the productivity of
capital goods, which leads to capital accumulation. This capital accumulation, in
turn, increases the productivity and pro®tability of developing new varieties, which
leads to new inventions, which leads to further capital accumulation, etc. This
process is not self sustaining, unless there are constant returns to accumulable
factors. It is clear that the time span over which productivity growth ultimately
comes to a halt can be very long indeed. This implies that, considering the limited
data we have available, it may be a little bit too early to conclude that there is
balanced growth of per capita output.
7. Conclusions
I develop an endogenous growth model with three sectors of production (®nal
goods, intermediate goods, and R&D), using three types of inputs (labor, capital,
and knowledge), and combining elements from three strands of the literature
(neoclassical growth, AK endogenous growth, and R&D endogenous growth). Production from the, perfectly competitive, ®nal goods sector can be consumed or,
later in the paper, added to the capital stock. The monopolistically competitive
intermediate goods sector produces a range of different varieties of goods or services for the ®nal goods sector. The number of varieties N produced by the intermediate goods sector is used as a measure of knowledge. It is the task of the R&D
sector to invent new varieties, and thus increase the range of intermediate goods
produced and the extent of our knowledge.
I distinguish between three different types of input: (i) rival and non-accumulable inputs (labor), (ii) rival and accumulable inputs (capital), and (iii) non-rival
and accumulable inputs (knowledge). Alternatively, and at the expense of calling
charles van marrewijk 471
everything `capital', these can be called human-capital, physical-capital, and
knowledge-capital, respectively. The three inputs are used in all three sectors of
the economy which, unlike previous studies, all have different production
technologies.
As discussed in Section 2, the neoclassical growth model uses exogenous knowledge increases together with capital accumulation and Harrod-neutral knowledge
spillovers in the ®nal goods sector to generate balanced growth. The AK endogenous growth model uses constant returns to accumulable (capital) inputs in the ®nal
goods sector together with capital accumulation to generate balanced growth. The
R&D endogenous growth model explains the increases in knowledge by introducing an R&D sector with pro®t maximizing ®rms together with Hicks-neutral
knowledge spillovers in the R&D sector to generate balanced growth. The fact that
these spillovers are Hicks-neutral is usually obscured by having only one (rival and
non-accumulable) input in the R&D sector. The paper discusses two variants of the
model. The model in Section 3 does not allow for capital accumulation and uses
Hicks-neutral knowledge spillovers in the R&D sector to generate balanced growth.
The model in Section 5 does allow for capital accumulation and uses both capital
accumulation and Harrod-neutral knowledge spillovers in the R&D sector to generate balanced growth. As explained in Section 2, the link created in the model of
Section 5 between the growth rate of the ®nal goods sector and the R&D sector
requires constant returns to accumulable factors in the ®nal goods sector for consistency. The model of Section 5 thus combines Harrod-neutrality from the neoclassical model and constant returns to accumulable factors for ®nal output from
the AK model with innovation undertaken by pro®t maximizing individuals who
have some market power and earn monopoly rents from the R&D model.
The paper suggests that capital accumulation and innovation are complementary
processes, neither of which would take place in the long run without the other. The
development of new varieties increases the productivity of capital goods, which
leads to capital accumulation. This capital accumulation, in turn, increases the
productivity and pro®tability of developing new varieties, which leads to new
inventions, which leads to further capital accumulation, etc. The paper also suggests that this process can be self-sustaining in two alternative ways, namely (i)
through Hicks-neutral knowledge spillovers without accumulation of rival inputs
in the R&D sector (Proposition 1), or (ii) through capital accumulation and
Harrod-neutral knowledge spillovers in the R&D sector (Proposition 2). In general,
therefore, non-rival and accumulable knowledge spillovers are only needed for the
rival and non-accumulable labor inputs, and not for the rival and accumulable
capital inputs to keep the economy going.
Acknowledgements
I would like to thank four anonymous referees, Andrew Bernard, Joseph Francois, Nathalie
Igot, and workshop participants at the SOM research school, University of Groningen,
and the Tinbergen Institute, Erasmus University Rotterdam, for useful comments and
472
capital accumulation
suggestions, and Anna Bogaards-Kok for assistance. I am particularly grateful to one anonymous referee for (i) drawing my attention to Aghion and Howitt's work, (ii) pointing out
the two effects described at the end of Section 4, (iii) providing me with formula (39), and
(iv) numerous detailed and constructive comments. Moreover, I thank Peter Howitt for
sending me some of his unpublished research material. Naturally, all errors are mine.
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charles van marrewijk 473
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Appendix
1.1 Proof of Proposition 1
Applying Shephard's lemma to the model of Section 3 gives: (i) the aggregate demand for
labor, capital and intermediates by ®nal goods producers in eqs (40)±(42); (ii) the aggregate
demand for labor and capital by the producers of intermediates in eqs (43) and (44); and
(iii) the aggregate demand for labor and capital by the producers of new varieties in eqs (45)
and (46), where the equalities follow from the homogeneity of the cost functions
py Y
Ly ˆ l
…40†
wl
py Y
Ky ˆ k
…41†
wk
py Y
D ˆ d
…42†
pd
0
0
Lx ˆ cx;w
…wk ; wl †X ˆ cx;w
…1; !†X
l
l
1
0
0
Kx ˆ cx;w
;
1
X
…w
;
w
†X
ˆ
c
k
l
x;wk
k
!
…43†
…44†
0
cn;w
…wk ; wl † dN
0
l
ˆ cn;w
…1; !†N~
…45†
l
dt
N
0
cn;wk …wk ; wl † dN
0
Kn ˆ
ˆ cn;w
…1; !†N~
…46†
k
dt
N
Total demand for labor and capital must be equal to total supply, see (47) and (48)
…47†
Ln ‡ Lx ‡ Ly ˆ L
Ln ˆ
Kn ‡ Kx ‡ Ky ˆ K
…48†
Note that eq. (49) holds if innovation takes place. Using (7), (42), (8), (6), and the homogeneity of cx (.), respectively, gives (50)
1
1
1
1
ˆ
ˆ
ˆ
…49†
1
N cn …wk ; wl † wk cn …1; !†
wl cn ; 1
!
Xˆ
d p y Y d p y Y
d py Y
d py Y
D
ˆ
ˆ
ˆ
ˆ
ˆ
pd A d
px
cx …wk ; wl † wk cx …1; !†
Ad
d py Y
1
;1
wl cx
!
…50†
From (49) we get py Y=wl ˆ V cn …1=!; 1† and py Y=wk ˆ V cn …1; !†: Substituting this information in eqs (40), (41), (43), and (44), these eqs together with (45) and (46) in the factor
market clearing, eqs (47) and (48), gives eqs (17) and (18) in the main text. Recall that the
aggregate stock of capital goods K is exogenously given, so that the only demand for ®nal
goods derives from consumption. In view of (15) market clearing is therefore given by (51)
474
capital accumulation
1
ˆY
py
…51†
Now note that pro®ts per ®rm are a fraction (1ÿ) of spending on each intermediate which,
using eq. (42), is given by d py Y=N. Thus = ˆ …1 ÿ †d py Y=N ˆ …1 ÿ †d V .
Using this, the no-arbitrage condition (10), the normalization (15) the de®nition of
V and (51) gives (52)
V~ ˆ ÿ…N~ ‡ ~† ˆ …1 ÿ †d V ÿ …N~ ‡ †
…52†
If V does not change (52) gives eq. (16) in the main text. Equations (16)±(18) in the main
~ ! and V , that is the rate of innovation, the wage-rental
text determine the variables, N;
ratio and the ratio of output to the stock market value.
1.2 Parameter space for Fig. 4
The general parameters used in Fig. 4 were ˆ 0:5; ˆ 0:12; k ˆ 0:5; k ˆ 0:4; and
K ˆ 20. Moreover, I used d ˆ 0:3 and L ˆ 10 for parameter set 1, and d ˆ 0:2 and
L ˆ 20 for parameter set 2. Obviously, k varies between 0 and 1.
1.3 Proof of Proposition 2
Applying Shephard's lemma to (30) changes eqs (45) and (46) to (53) and (54)
w
!
w @cn wk ; l dN
0
0
N
Ln ˆ cn;w
ˆ cn;…!=N†
N~
…53†
wk ; l
1;
l
@wl
dt
N
N
!
w
0
0
~
~ ˆ cn;w
Kn ˆ cn;w
wk ; l NN
1;
NN
…54†
k
k
N
N
Equation (30) also implies that eqs (49), (40), and (41) change to (55), (56), and (57),
respectively
1
1
1
ˆ
…55†
ˆ
N
wl
N
;1
wl c n
Ncn wk
!
N
py Y
N
Ly ˆ l
; 1 V
ˆ l cn
…56†
!
wl
py Y
N
Ky ˆ k
ˆ k !cn
…57†
; 1 V
wk
!
Applying Shephard's lemma to (31) changes eqs (43) and (44) to (58) and (59), respectively
!
0
pY
d cx;w
1;
l
w
! X
N y
0
0
Lx ˆ cx;w
ˆ
wk ; l X ˆ cx;w
1;
l
l
w
N N
N
Ncx wk ; l
N
!
!
N
0
0
p Y d cx;w
c
; 1 !V
d cx;w
1;
1;
l
l
N y
N n !
ˆ
ˆ
…58†
!
!
Ncx 1;
Nwk cx 1;
N
N
!
N
0
1;
c
;
1
!V d cx;w
k
wl
!
N n !
0
0
Kx ˆ cx;wk wk ;
Xˆ
X ˆ cx;wk 1;
…59†
!
N
N
cx 1;
N
charles van marrewijk 475
Substituting these demands for labor and capital in eqs (47) and (48) gives eqs (33) and (34)
in the main text.
1.4 Derivation of eq. (39)
From (36) and (37) it follows that ! ˆ bl K=bk L if N~ = 0, while ! ˆ …l =k † …K=L† if V* = 0.
Therefore, l =k K=L ! …bl =bk † …K=L†, as in the example of Section 4.2. Suppose that
the rate of innovation and capital investment are both zero. It follows from (37) that
V ˆ …K=bk N †k …L=bl †l . Moreover, from (32) V ˆ =…1 ÿ †d . Combining this information determines K as a linear function of N
1=k l =k
bl
1
K ˆ AN;
with A …60†
…1 ÿ †d
bk
L
From (55) it follows that V ˆ …!=N †k =wl , which in turn implies wl ˆ bl =L (compare
Grossman and Helpman, 1991a, p. 53). From (29) we get wk ˆ py , which, when substituted
in (5), together with (6), (8), and wl ˆ bl =L determines py as a function of N
py ˆ BN …61†
with
1ÿ
l ‡
bl ÿd d k ‡k 1=l ‡l d
B … †
; l ‡ l d
L
We now have two expressions for ! as a function of N: (i) ! ˆ …bl =bk † …K=L† ˆ …bl A=bk L†
…N†; and (ii) ! ˆ wl =wk ˆ bl =Lpy ˆ …bl =LB† …N ÿ †: Consistency requires these two
expressions for ! to coincide, which gives eq. (39) in the main text.