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Government's research policy and economic growth: Capital, knowledge and economic
327
Government’s research policy and economic growth: Capital, knowledge and economic structure * Wei-Bin
Zhang
Institute for Futures Studies, Stockholm, Final version
received
February
Sweden
1992
This study proposes a growth model to examine dynamic processes of capital and knowledge accumulation and price structure. The economy consists of three - agriculture, industry and service - production sectors and one knowledge creator - the university. There are four - land, capital, labor force and knowledge - factors of economic production. The university is financially supported by the government whose income comes from tax from the production sectors. Capital accumulation is due to savings by the households and knowledge growth is due to creativity of the three production sectors and the university. We guarantee the existence of long-run equilibria and provide stability conditions. We also examine effects of changes in research policy, creativity of economic activities and scientists, the territorial size, and savings rate upon long-run economic development and the price structure.
1. Introduction There is an increasing number of publications on relations between knowledge accumulation and economic development in the recent theoretical economic literature (e.g. [l-11]). These new approaches have provided much more appropriate answers to questions, such as why Japan and the Asian four “tigers” have succeeded in economic development and why China is much delayed in the industrialization process, than the economic growth models without endogenous
Correspondence to: Wei-Bin Zhang, Institute for Futures Studies, Hagagatan 23B, Box 6799, S-113 85 Stockholm, Sweden. * I am grateful to two anonymous referees for important comments. Research Policy 22 (1993) 327-336 North-Holland
0048-7333/93/$06.00
0 1993 - Elsevier
Science
Publishers
knowledge, such as the Solow-Swan model, the Leontief-von Neumann-Sraffa multisector models. This study tries to give another contribution to the literature by examining relations of a special public sector-research institutions and universities-to economic growth. Within my limited knowledge, there is no approach to deal with the problem in the same way as that proposed in this study. We treat knowledge stock as “public good”. Knowledge has the character of public good in the sense that utilization of knowledge by any economic sector will not affect that by any other sectors. There is certain knowledge, such as new innovation by a private company, which may not be accessible to the public, though new theories in mathematics, theoretical physics, economics, philosophy and the like are accessible to the public almost as soon as they are discovered. In a sense, the “share” of “private” knowledge to the total knowledge stock of a society is usually very small. Moreover, “private knowledge” will become public through imitation and other processes in the long term. Surely, in order to understand dynamic competitive processes of individual firms, it is necessary to be aware of how “private knowledge” is innovated and what effects the innovation has upon market shares and other aspects of the firms. Since we are concerned with historically long-run economic development, it is probably acceptable to neglect such microeconomic aspects of innovation. It must be emphasized that knowledge has characteristics different from those of other public goods such as transportation and communication infrastructures. For instance, knowledge tends to be further increased
B.V. All rights reserved
328
W’ei-Bin Zhang / Gormmnent ‘s research policy
during utilization processes. There are no congestion effects in knowledge utilization. The government in a competitive economy can affect economic development in different ways. Taxing private agents, government may provide supplies of public goods such as transportation and communication infrastructures and easy accessibilities to education and health care. These “material aspects” of government’s intervention in economic life have been well recognized in the economic literature, in particular, in public economics. There is another important economic issue which needs to be further examined. It is significant to investigate the effects of government’s knowledge creators, such as universities and research institutions, upon economic development. As national labor and capital resources are limited, investment in knowledge creation by the government may either benefit or decrease national economic growth. This study tries to shed some light on issues about the relationship between government’s inte~ention in R&D activities and the economic structure. The paper is organized as follows. Section 2 defines the basic model. Section 3 guarantees existence of long-run equilibria and provides stability conditions. Section 4 examines effects of government’s research policy upon economic development. Section 5 analyses how an improvement in creativity of economic activities or scientists can affect long-run capital and knowledge accumulation and the price structure. Section 6 examines effects of the territorial size upon the price structure. Section 7 examines how the savings rate may in a complicated way affect the economic growth, knowledge accumulation and the price structure. Section 8 concludes the study. In the Appendix, we provide the proof of existence of equilibria and stability in section 3.
2. The basic model
institution of pure mathematics may be established not to increase productivity of some special industry unit, but to increase knowledge stock of the society. The product of the mathematical institution - mathematical theorems - may find applications in various production sectors and thus increase national productivity in the long term. We classify production activities into three classes - agriculture, industry and service. The agricultural sector produces goods such as corn, rice and vegetables. The industrial sector produces commodities. The service sector provides services such as restaurants, shopping, hotels, ship/bus/train/air systems, information, health care, and education. The industrial product can either be consumed or saved to increase capital, though the product of the agricultural sector and the service sector can only be consumed. The product of the service sector cannot be saved in the sense that services are consumed in the process of their production. We assume that the production factors are constantly in full employment. Prices are measured in terms of the agricultural good. Let the price of the agricultural good be unity during the study period and denote pi(t) and p,(t), respectively, the ratios of the prices of the industrial commodity and the services over that of the agricultural good at time t. As we assume that the labor force, N, land, L, and capital, K, are homogeneous, the wage rate, w(r), the land rent, R(t), and the interest rate, r(t), are identical in the whole economy.
2.1. The agricultures sector
Agricultural production is a process of combining land, labor force, capital and knowledge. The production function of the agricultural sector is specified as F, = .Z”K,“NfL;,
We assume that the government may financially support knowledge creators, such as universities and research institutions, to increase knowledge for national deveIopment. For simplicity, we call government’s knowledge creators university. It should be remarked that the university may not be production oriented. For instance, a research
Ly,P,7 r 0,
CY+p-l-7=1, a 2 0,
(2.1)
in which 2 is knowledge stock of the economy, K, machines utilized by the agriculture sector, N, the number of farmers, and L, the land devoted to the agriculture production. Maximizing the
Wei-Bin Zhang / GoL~ernment‘7 research policy
profit by the agriculture ing conditions r = aTF.JK,,
sector
yields
the follow-
w = PTFJN,,
329
force employed by the industrial sector, respectively. The marginal conditions are given by r = mpiTFi/Ki,
R = rTF,/L,,
(2.2)
in which T = 1 - tax rate upon the product. In this study, we assume that the government gets its income to support the university only from the producers. There is no tax upon the properties and wage incomes. Moreover, the tax rate is identical among the three production sectors. It can be easily seen that we can relax this assumption by introducing heterogeneous tax rates upon product and taxes upon the properties and the wage incomes within the framework of this study. Let C, be the consumption of agricultural good of the population. Since we assume that agricultural good cannot be saved, we have c;, = 6,
(2.3)
2.2. The service sector
w = qp,TF,/N,
Let Y(t) denote the gross income the agricultural good. We have Y(t)
d+e=l,
F \ =ZhKdN” S 5.7
(2.4)
in which K, and N, are the capital and the labor force employed by the service sector, respectively. The profit-maximization yields
(2.9)
Denote ~(0
dK/dt
(2.10)
in which 6, is the given depreciation of rate of capital. Denote Ci the consumption of the industrial good by the population. Since the industrial product consists of investment and consumption, we have (2.11)
Ci 2 0,
in which Ci is the consumption trial good.
level of the indus-
2.4. The consumption of the households For simplicity of analysis, we assume that there is a “standard consumer” whose utility function stands for the population. We assume that the utility function takes on the following form u=
w = ep,TF,/N,
Let C, be the consumption population, we have
of
d, e > 0,
b>O,
r = &,T&/K,,
in terms
=rK+wN+RL
Ci = Fi - sY/p, , The production (in a strict sence, capacity) of the service sector is through combining knowledge, capital and the labor force. The production function of the service sector is defined by
(2.8)
(2.5)
of services
Cs = F,
of the
(2.6)
(~~/‘N)U(Ci/N)“(C~/N)x(~~/N)y,
u, c, x, y > 0 The total consumption budget is given by: (1 s)Y. Accordingly, the consumer problem is defined by min (C,/N)U(Ci/N)“(C,/N)X(L,/N)y
2.3. The industrial sector
subject
to: C, +piCi +p,C,
+ RL, = (1 - s)Y. (2.12)
We specify tion as follows F, = Z”KyNy,
the
agricultural
production
funcThe unique
m+q=l,
m,q>O,c>O, (2.7)
in which
Ki and Ni are the capital
and the labor
optimal
is given by
ci = US,Y/Pi,
c, = us,Y, c, = Xq-/Ps inwhich
solution
7
s,=(l-s)/u,
L, = yq,Y/R,
and u,=u+u+x+y.
(2.13)
330
Wei-Bin Zhang / Goc;emmentS research policy
2.5. Knowledge growth We propose growth dZ/d
t = T, F,/Z’
the following + ri F,/Z”
+ rpZgK;N;
possible
knowledge
+ rs F,/Z”
- 6,Z,
(2.14)
in which K, and N, are capital and scientists employed by the university, respectively, 6, (2 0) is the depreciation rate of knowledge, and ~(0 2 E I 11, 7r(O I7r I 0, a(0 I V 2 l), 6, 7, 2, 7, (2 O), 7i (2 O), rS (2 O), rP (2 0) are parameters. We further require, O
=a&‘Fi(n)
dnj”+o,,
in which a,, a2 and a3 are positive parameters, and 0 I a2 I 1. The above equation implies that effects of industrial activities upon knowledge accumulation have properties of decreasing scale. It is not difficult to see that a, and a2 can be interpreted as measurements of efficiency of “learning by doing” by the industrial sector. Taking the derivatives of the equation yields: dZ/dt = riFi/ZT, in which ri = u1u2, 7~ = 1 - a2. If the industrial sector is very effective in creating knowledge, then r is very small; and vice versa. We can similarly interpret the terms r,F,/Z’ and T,F,/Z”. It should be remarked that in (2.14), we omit the possibility of R&D by private companies. From the framework of this study, one can see that it is not impossible to introduce this actual important source of knowledge growth. For instance, we may divide K, and Ni into Ki, and Ki2, Ni, and Niz, respectively, where Ni, and Ki, are labor force and capital employed by the industry to produce commodities and Ni2 and Ki, are labor force and capital employed by the industry to do R&D activities. The industry decides the employment structure by maximizing the long-run profit. This obviously involves many other complicated issues of modeling. For simplicity of analysis, we limit ourselves to the narrow case of knowledge growth. We call the term, T~Z~K~N~, creativity of the university. A similar functional form for knowl-
edge creation by scientists is suggested by Zhang [9]. The term, rpZgKGN:, implies that creativity of the university is positively related to the number of scientists and capital employed. On the one hand, as knowledge stock is increased, the university may more effectiveIy utilize traditiona knowledge to discover new theorems. On the other hand, a large stock of knowledge may make discovery of new knowledge difficult. This implies that the parameter, g, may be either positive or negative, depending upon the characteristics of knowledge creation of the university. According to the above interpretations, we may call r,Fa/Zc, riFi/ZT, T~F,/Z~, and T~Z~K;N~~, respectively, the creativity of the agricultural sector, the industrial sector, the service sector, and the university. It should be emphasized that a sector may be very creative in one economy, while it may be not creative at all in another. For instance, creativity of the agricultural sector may be reasonably ignored in many developed economies. But we cannot omit it in analysing, for instance, current economic development in China. Since we assume that the financial resource of the university is from government’s taxes upon the product, we have (1 - T)( F, +piFi +p,F,)
= wN, + rK,
(2.15)
We now have to design a way to determine N, and K,. We assume that the government decides the number of scientists and the capital stock of the university in the following way N, = nN,
K, = kK,
0
k < 1,
(2.16)
in which n and k are the policy variables fixed by the government. We assume that n and k are exogenously given. We are interested in the effects of the policy parameters n and k upon the economic system. In this study, we assume that the tax rate is endogenously given in the sense that the government will tax the producers only to satisfy (2.15). That is, (1 - T) is equal to (wN, + rlK,)/( F, + pi Fi + ps F,) at any point in time. 2.6. Full employment of the labor force, capital and land The assumption that the labor force, capital and land are always fully employed yields the
331
W&Bin Zhang / Government’s research policy
following
fixed, the tax rate is constant during the study period. We thus can write the dynamics of capital and knowledge in the following form:
equations L,+L,=L,
N,+N,+N,=(1-n)N, K,+K,+K,=(l-k)K,
(2.17)
in which N and L are fixed. We have thus built the model which explains both distribution of the labor force, capital and land and accumulation of capital and knowledge. It should be remarked that the assumptions that population, land, and savings rate are exogenously given can be, in principle, relaxed within the framework of this model. There are 25 endogenous variables, K, K,, Ki,
y + UTT),
L,
= YL/(
Y + UTT)
K, = uh,K,,
9
= 1 + nu,/s,
N, =xes,N,
+ ku,/s,,
(2.21)
+ r,F,/Z”
+ r,,ZgKZ - S,Z,
(2.22)
70 = rp(nN)“k”, and F,, Fi and of K and Z given by:
F, are
FH= A,ZbKd, (2.23)
where A, = (au~,)“(u&N)~L;, A,=
(a!~s,)~(xe.s~N)~, (2.24)
Ai = (mv,s,)m(qv,s,N)q In what dynamic
follows, system.
3. Equilibria
we analyse
and stability
the behavior
sF,/(vs,+s) r,F,/Z’
as a solution
+ r,F,/Z”+
rOZRKz = S,Z (3.1)
From the first equation,
we have
K = AZc’q,
(3.2)
in which A = (sAi/S,(vs, + s))“~ > 0. Substituting (3.2) into the second equation in (3.1) yields 3 QJaZa-‘+&4--l + 7iAiAmZ-.rr+c/q-1
au + mu + d-x + S/Q),
+ 7 A s
+
Adzb-o+dc/q-1 s
TOAZZg+CZ/q-l
_
6,=()
(3.3)
S2=(1-n)/(/3u+w,+-),
l/(U
-t v +s/s,
From (2.211, we ratios n and k, government tax and the taste). satisfies: 0 < T
+x)
see that the government fixes the the tax rate is determined (by the policy, the production structure From (2.211, one sees that T < 1. As all the parameters are
of the
=S,K,
+ riFi/ZK
v~=v+s/so,
z.40 =
of the
conditions
An equilibrium is defined following equations
H(Z)
in which s1 = (1 - k)/(
+ riFi/ZK
Ki = mv,s,K,
(2.20) l/T
= r,F,/Z’
-S,K,
Fi = A,Z’K”,
(2.19) Ni = qU,S, N,
dZ/dt
F, = A,Z”K*,
(2.18) K, = aus,K,
=sF,/(vs,+s)
in which functions
KS, K,, Na, Ni, Ns, Np’ La, L,, Z, Fa, Fi, Fs, C,, C,, C,, pi, ps, r, R, w, 1 - T, Y. It can easily be checked that the system consists of 25 independent equations (2.1)~(2.10), and (2.13)~(2.17). Hence, under appropriate conditions the system may have solutions. We now show that the dynamics of our system can be written in terms of K and 2. That is, our dynamic analysis will be reduced to that of a two-dimensional differential equations system. As shown in Appendix Al, we can write L, and L, as functions of L, N,, Ni and N, as functions of N, K,, Ki and K, as functions of K, and T as function of the research policy, respectively, as follows L, = urTL/(
dK/dt
Appendix
A2 proves
the following
proposition.
3.1. Proposition (3.1) If a+cc-u/q
b+cd/q< is a unique
332
W&Bin Zhang / Go~lernment S research policy
If a+m/q>l+t, c/q>l+r, h+cd/q> 1 + CT, and g + cz/q > 1, then there is a unique equilibrium. The system is unstable. In each of the cases (l&(14) defined in appendix A2, the system has either no equilibrium or two equilibria. When the system has two equilibria, the one with higher values of K and Z is unstable and the other one is stable.
To interpret the proposition, we examine the stable case: a + ca/q < 1 + E, c/q < 1 + V, b + cd/q < 1 + CT, g + cz/q < 1. Since a, b and c are the marginal production of knowledge in the agricultural, service and the industrial sectors, respectively, we may conclude that an increase in the marginal production of knowledge tends to destabilize the system. On the other hand, as E, (T and rr are the measurements of decreasing scale effects in knowledge accumulation by the agricultural, service and the industrial sectors, respectively, we may conclude that an increase in the decreasing scale effects tends to stabilize the system. In other words, an increase in creativity of the economic activities tends to destabilize the economic system. Since (Y, d and m (= 1 - q) are the marginal production of capital in the agricultural, the service and the industrial sector, respectively, we can say that if the marginal production of capital is increased, the system tends to become unstable. In g + cz/q < 1, the parameter g describes the effects of knowledge stock upon knowledge creation and z measures the effects of capital in helping scientists to discover new knowledge. Accordingly, this condition can be interpreted as that when creativity of the university is not very high, then the system tends to be stabilized. Surely, whether the system is stable or unstable depends upon the combined conditions of the economic structure and the characteristics of knowledge accumulation. In what follows, we carry out comparative static analyses. As the analyses are meaningful only under presumed uniqueness and stability of the equilibrium, we assume a + ecu/q < 1 + E, c/q < 1 + 7~, b + cd/q < 1 + u, and g + cz/q < 1. We will retain this assumption in the remainder of this study. It must be emphasized that the effects of changes in the parameters obtained below may be different from those when this stability assumption is relaxed.
4. Effects of the number of scientists of the university
and capital
First, we examine effects of changes in the parameter n upon the behavior of the system. Taking derivatives of (3.3) and (3.2), respectively, with respect to n yields @ dZ/dn
= -((Y +p)~,F,/(l
-n)Z’
- er,F,/(
1 - n)Z”
- T,PF,/(
1 - n) Z”
+,,(8/n-z/(l-n))zgKz, (4.1) (l/K)
dK/dn
= (c/qZ)
dZ/dn
- l/(
1 - n), (4.2)
in which CDis a positive number defined in (3.6). From (4.11, we can see that if 6/n -z/(1 - n) is negative, then the knowledge stock will be reduced as the number of scientists in the university is increased. If S/n -z/(1 -n> is positive, then an increase in the number of scientists may either increase or decrease the equilibrium level of knowledge stock, depending upon the other terms in (4.1). As the two parameters, 6 and z, represent the “marginal creativity” of the scientists and the “marginal creativity” of capital, respectively, we may interpret the sign of 6/n z/(1 -n> as follows: if the marginal creativity of scientists is appropriately larger than that of capital, then 6/n - z(1 - n) is positive. The above discussion means that if the marginal creativity of scientists is very low, then an increase in the number of scientists will certainly reduce knowledge stock of the society in the long term. When the marginal creativity of scientists is quite high and the creativity of the economic activities is very low (i.e. r,, ri and 7, are very small), then an increase of the number of scientists tends to increase knowledge stock of the society. Eq. (4.2) implies that if dZ/dn is negative, then equilibrium value of capital will be reduced as more scientists are allocated to the university. Even if dZ/dn is positive, if the knowledge stock of society is quite high and dZ/dn is relatively small, then equilibrium capital also tends to be reduced. Only when Z is relatively low and dZ/dn is quite large, equilibrium level of capital will be increased as more scientists are employed by the university.
W&Bin
Zhang
/ Gocernment’s
333
research policy
It should be remarked again that the above conclusion is dependent upon the assumed condition: @ > 0. Since any case among (1)-X14) may not satisfy this requirement, the effects of change in II may be opposite to those just obtained in any case among (1)-(14X In this sense, whether increasing the number of scientists will benefit the economy depends upon the special characteristics of the economic system under consideration in a rather complicated way. Since dT/dn = - T*u,/s,(l -n) < 0, we see that as the number of scientists is increased, the tax rate has to be increased in order to keep the government budget in balance. It is easy to check the effects upon the output of the economic activities as follows
We can interpret effects of changes in k similarly to the case of n. Summarizing the above discussions, we can generally conclude that, from a purely economic point of view, whether the government should strongly financially support sciences is much dependent upon the creativity of economic activities and purely scientific research and the effects of knowledge upon the economic activities. As, in modern economic systems, “private” economic activities such as large companies and individual artists can create new knowledge, to enlarge the university through government intervention might not benefit the economic development.
(l/Fa)
dF,/dn=(l/Z)(a-e+ac/q)
dZ/dn
5. Efficiency
(l/F,)
dFi/dn
-(a+P)/(l = (l/Z)(c/q-n)
(l/F,)
dF,/dn
-e/(1-n), = (l/Z)( b - u + de/q)
dZ/dn
Since under the presumed stability, changes in r,, rs, ri, rP have similar effects upon the variables, it is sufficient for us to carry out the analysis only with respect to one parameter, e.g. Ti. The effects of a shift in the knowledge accumulation efficiency parameter, TV, are given by
-n), dZ/dn
-P/(1 -n) (4.3) One can see that even when dZ/dn is positive, the product may be reduced as more labor force is shifted to the university. The effects are also uncertain in the sense that they may be either positive or negative, depending upon the parameter values. As dY/dn = (l/ys,) dF,/dn, dY/dn is positive only when dFJdn > 0. From r = aTFJK,, w = PTFJN,, R = TTFJL,, pi = (us0 + s)Fa/ us0 F,, pS = xF,/uI?,, one can directly get the effects of changes in n upon the interest rate, wage rate, land rent, and prices of the industrial good and agricultural good. It is not difficult to see that an increase in the number of scientists may either have positive or negative effects upon these variables, depending upon the further assumptions one may add to the values of the parameters. We have the following effects of changes in capital of the university upon the knowledge stock and the total capital of the economy @ dZ/dk=
-(a+rn/q)T,Fa/(l-k)Z’ - m(l + l/q)T,F,/(l
- k)Z”
(l/K)
dK/dk=(c/qZ)
- mz/q(
@ dZ/dri
= (cAK/qZ) = FiZ1-=
accumulation
dZ/dr,
> 0,
> 0,
(5.1)
in which @ is a positive constant, defined in (3.6). One can directly check that Na, Ni, N,, F,, Fi, F,, Y, C,, C,, Ci, w, R, are increased as the efficiency of knowledge accumulation is improved. As dr/dri=
(r/Z){a
- (1 -a)cA/q}
dZ/dri,
if the marginal production, a, of knowledge of the agricultural sector is extremely low, then an improvement in the efficiency of knowledge accumulation will reduce the interest rate. If a is appropriately large, then the interest rate will be increased. As the parameter a seems to be very low in an industrialized society, it appears reasonable to observe a decline in the interest rate as knowledge is increased. Utilizing pi = (us0 + s>F,/~~,,F,, p, = xFa/uFS, we obtain
-~,d(l+d/q)F,/(l-k)ZP + m,( l/k
dK/dri
of knowledge
1 - k))ZgKZ,
dZ/dk-m/q(l-n) (4.4)
Z dpi/dri
=pi{ a - c + ((Y - m)c&q)}
Z dp,/dri
=p,{ a - b + ((Y - d)cA/q}
dZ/dri, dZ/dri (5.2)
W&Bin Zhang / Government’s research policy
334
5.1. Pmposition (5.1)
The effects upon the prices are determined
As the efficiency of knowledge accumulation by the industrial sector is improved, the equilibrium values of capital and knowledge are increased. The output of the three sectors, the gross income, and the consumption of the output of the three sectors are increased as Ti is increased. The wage and the land rent are increased as ?i is increased. The prices of the industry good and services (in terms of agricultural good) may be either increased or decreased as the efficiency is improved. For instance, in the case of a and b being extremely small, m > CY> d, dr/dTi < 0, dp,/dr, < 0, dp,/d~i > 0.
6. The te~itorial
size
Although the territorial size is usually not changeable, it is still important to analyse effects of changes in L upon the economic system, for instance, when we compare American and Japanese economic structures. With regard to L, we have: @ d Z/d L = n,F,/LZ’, dK,‘dL
= (cK,‘qZ)
dZ,‘dL
(6.1)
With all the other conditions equal, a country with a larger territory will have more capital and knowledge. Since the agricultural sector has usually a very low efficiency in knowledge creation in an industrialized economy, the effects upon capital and knowledge accumulation are, in fact, very weak. One may also check that K,, Ki, K,, L,, L,, F,, Fi, F,, Y, C,, C,, Ci, and w are increased if L is increased. As (Z/R)
dR/dL
= (a +ac/q) -(I
-7)2/L,
(Z/Pi1
dPi/dL
=(a-c+(vm)c/q) -(l
(6.2)
one sees that, under the condition that knowledge stock is not strongly affected by the agricultural production, a country with a large territory tends to have lower land rent. An interesting case, though rarely happening, is that if dZ/dL is quite large, then dR/d L might be positive.
dZ/dL
-T)Z/L,
(Z/P,) dps/dL = (a -b + (cy - d)c/q) -(l
dZ/dL
- T)Z/L
Accordingly, under the very acceptable hypothesis that dZ/d L is extremely small, we may conclude that an economy with a large territory tends to have cheap industrial goods and services (in terms of agricultural product).
7. The savings rate This study makes a very strict assumption about the savings behavior of the households. In reality, saving behavior will adapt to a new social and economic environment in the long term. Although we do not treat the savings rate as an endogenous variable in this study, it is important to examine the effects of an exogenous shift in the savings rate upon the economic system. Effects of changes in s upon Z and K are given by a, dZ/ds = -ra(LYSi f /3szq/N)u,F,/Z’ + ~gyr,F,/( -(ms,
y + mT)Z”
+qs*/N-s,/(~~,s,+S))Ul~if;li/Za
- (ds, f es,q/Wu,r,WZ” +T,ZgK”q{l/s+
(r:-ul)/uI(uso+s)
-m/s&(1 4/%Q( (l/K)
- k)(l
-s)
1 - a)( I - s)}/z,
dK/ds
= (c/qZ)
dZ/dL
by
+{[u/s(l
d.Z/ds -s)
+u,Js,/(Us,+s)
- (ms, -+qs,q/N)u,}q,
(7.1)
in which T, = dT/ds = - T’qnu,u,/(l - ~>~(l n) - T2ku,u,/(l - s>‘(l - k) < 0. From (7.11, we see that the effects of changes in the savings rate are affected by different forces. An increase in the savings rate may either expand or reduce
We&Bin Zhang / Goc;ernment’s research policy
capital and knowledge. As the conditions are so complicated to interpret, we can get a definite conclusion about the effects of shifts in the savings rate upon capital and knowledge only when we further specify the values of the parameters. The effects upon the other variables are also very complicated, and we will not carry out a further analysis. It should be remarked that although our model is developed within the framework of neoclassical growth theory (e.g. the Solow-Swan model, Uzawa’s two sector model) in the sense that full employment of the product factor is assumed, the effects of the savings behavior upon the system are much more complicated than those forecast by the models just mentioned. Obviously, the complexity is due to the introduction of endogenous knowledge into the system. Whether an increase in propensity to save will stimulate economic development or not depends upon complicated interactions among the taste structure, production functions and characteristics of knowledge accumulation. 8. Concluding
remarks
We have answered certain questions about interactional relations between knowledge and capital accumulation. The problems are analysed under some strict assumptions. Some of these assumptions can be relaxed in our approach. For instance, it is also possible to introduce an endogenous savings rate in our analysis. Contribution to knowledge accumulation by private sectors is much more complicated than assumed in this study. It should be remarked that we can solve the equations because we specify production and knowledge accumulation functions in the Cobb-Douglas formula. If these functions are taken on more general functional forms, it might be quite difficult to get definite analytical conclusions. It is clear that if we allow international interactions for trades and for exchanges of ideas, the system will become extremely complicated. Appendix Al. The dynamics Z
in the term of K and
We now show (2.18j42.21) given in section 2. From R = rTF,/L,, F, = us,Y, L, = ys,Y/R, we have: La/L, = urT/y, in which T is an en-
335
dogenous variable. The ratio of the land employed by the agriculture sector and the land devoted to the housing is positively related to the marginal utility of the agricultural product and the marginal product of the land, but negatively related to the marginal utility of housing and the tax rate. From L = L, + L,, we have (2.18). As the tax rate T is, as will be shown, constant, one sees that L, and L, are constant during the study period. From (2.21, (2.5) and (2.81, one has: Ki/K,
= ( mc/aq)
NJN,,
K/K,
= (me/qd)
N/N,
From
(2.11)
one has pi = (us0 + Utilizing these relations and (2.2), (2.5) and (2.8), we get
s)Fa/us,Fi, K/K,
and
(Al)
(2.131,
p, = xFJuF,.
= m( cso + s) /aus,,
KS/K, = dx/cuu
(A21 Equations (Al) and (A2) imply that the ratios of the labor force or the capital employed by the three sectors are determined by the marginal productivity and the marginal utilities of the factors. Accordingly, if the savings rates, u, v, x and y are not affected by knowledge and wealth, then the ratios will be kept constant during the study period. In fact, since the savings rate, s, and in particular, the “life-style parameters” u, L’, x and y are much affected by economic conditions and varieties of goods and services, it is possible to examine how the ratios of the labor force or the capital employed by the three sectors are affected by economic development. In this study, we shall not be involved with the complexity of endogenous changes in s, u, v, x and y. Substituting (A2) into K, + Ki + K, = (1 - k)K yields (2.19). From (A2) and (2.19), and N, + N, + h$ = nN, one has (2.20). Utilizing 1 - T = (wNP = rK,)/(Fa + piFi + p,F,) in which F, +piFi +p,F, = F,(l + (us0 + s)/uso +x/u), wN, = wnN = nwN,/u@, rK, = rkK = krK,/aus,
= nTF,/us,, = kTF,/us,,
(A3)
we have (2.21).
Appendix A2. Proving Proposition
3.1.
First, we find conditions such that H(Z) = 0 has positive solutions. Let’x, = a + at/q - E - 1,
336
We&Bin Zhang / Gor,ernment S research policy
x,=c/q-r-1 and x,=b+dc/q-a-1, xq = g + cz/g - 1. We exclude the case of x1 =x2 = x2 =x4 = 0. One can directly check that if x, r 0, x2 2 0, x3 2 0, x4 2 0 (i.e., H(0) < , H(w) > 0 and H’>OforZ>O)orx,~0,x,~O,x,~1,x,~1 (i.e., H(0) > 0, H(m) < 0 and H’ < 0 for Z > O), then the system has a unique positive equilibrium. If (1) x,pO, x2 < 0, xg < 0, x4 < 0, (2) x,pO, x2 > 0, x3 < 0, x4 -=c 0, (3) “$0, x2 -C 0, x3 > 0, x4 < 0, (4) x1 < 0, x2 < 0, x3 > 0, x4 < 0, (5) x1 < 0, x2 > 0, x3 < 0, x4 < 0, (6) x1 < 0, x2 > 0, x3 > 0, x4 < 0, (7) x,pO, x2 < 0, x3 < 0, x4 > 0, (8) x,pO, x2>o, x,o, (9) x,/30, x,o, x4 > 0, (10) xr < 0, x2 < 0, xj > 0, x4 > 0, (11) x, < 0, x2 > 0, x3 < 0, x4 > 0, (12) x1 < 0, x2 > 0, lx, > 0, lx, > 0, (13) x1 < 0, x2 < 0, x3 < 0, lx, > 0, (14) lx, > 0, lx, > 0, lx, > 0, x4 < 0, then the system has either two equilibria or none. We just prove case (1). The other cases can be similarly checked. Since a + at/q - E > 1, c/q n0 and H(w) > 0. This implies that H(Z) = 0 has either no solution or multiple solutions. Since ZH’( Z) =xlr,A,AaZX~
+x2riAiAmZX*
+ x3r,A,AdZX”
+ X4T0AZZX4,
(Ad)
we see that H’(Z) may be either positive or negative, depending upon the parameter values. If H(Z) = 0 has more than two solutions, there are at least two values of Z such that H’(Z) = 0. Since d(ZH’)/dZ > 0 strictly holds for Z > 0, it is impossible for H’(Z) = 0 to have more than one solution. Accordingly, H(Z) = 0 has either no solution or two solutions. A necessary and sufficient condition for the existence of two equilibria is that there exists a value of Z* such that H(Z’) < 0 and H’(Z *) = 0. The proofs of cases Q-(14) can be similarly carried out. The Jacobian at an equilibrium is given by
(A3 in which a, = r,F,/Z’
+ mriFi/ZT
+ zr,,ZgKZ>
0,
+ dr,F,/Z”
a,=(l+E-a)r,Fa/Zt+(l+r-c)riFi/ZT +(l
+a-b)r,F,/Z”+
(1 -g)rOZ”KZ
> 0
The two eigenvalues,
(‘46) 4r and &,
are determined
by +2 + (Sq + a2/Z)4
+ a,Sq/Z
- Sca,/Z
= 0 (A?
Accordingly, if a,q < ca,, the system is unstable; if a,q > ca,, the system is stable. The condition, a,q > ca, can be rewritten as @ = -x,r,F,/ZE
- x2riFi/Z”
- x3r,&/Zv
-x4r,ZgK’>0 We thus have proved
(A8) Proposition
(3.1).
References [l] ..&E. Andersson and W.B. Zhang, Endogenous Technological Changes and Economic Growth, in: Manas Chatteji and Robert Kuenne (Editors), Dynamics and Conflict in Regional Structural Change (The Macmillan Press, London, 1990). [2] G.S. Becker, K.M. Murphy and R. Tamura, Human Capital, Fertility and Economic Growth, Journal ofPolitical Economy 98 (5) (1990) 12-37. [3] G.M. Grossman and E. Helpman, Quality Ladders in the Theory of Growth, Review of Economic Studies 58 (1991) 43-61. [4] R.E. Lucas, On The Mechanics of Economic Development, Journal of Monetary Economics 22 (1988) 3-42. [5] P.M. Romer, Increasing Returns and Long-Run Growth, Journal of Political Economy 94 (1986) 1002-1037. [6] P.M. Romer, Endogenous Technological Change, Journal of Political Economy 98 (5) (1990) 71-102. [7] G.W. Stadler, Business Cycle Models with Endogenous Technology, The American Economic Review 80 (1990) 163-778. [8] N.L. Stokey, The Volume and Composition of Trade Between Rich and Poor Countries, Review of Economic Studies 58 (1991) 63-80. [9] W.B. Zhang, Economic Growth and Technological Change, International Journal of Systems Science 21 (1990) 1933-1949. [lo] W.B. Zhang, Regional Dynamics with Creativity and Knowledge Diffusion, The Annals of Regional Science 59 (1991) 179-191. [ll] W.B. Zhang, Economic Development with Creativity and Knowledge Diffusion, Socio-Spatial Dynamics 2 (1991) 19-30.
Government’s research policy and economic growth: Capital, knowledge and economic structure * Wei-Bin
Zhang
Institute for Futures Studies, Stockholm, Final version
received
February
Sweden
1992
This study proposes a growth model to examine dynamic processes of capital and knowledge accumulation and price structure. The economy consists of three - agriculture, industry and service - production sectors and one knowledge creator - the university. There are four - land, capital, labor force and knowledge - factors of economic production. The university is financially supported by the government whose income comes from tax from the production sectors. Capital accumulation is due to savings by the households and knowledge growth is due to creativity of the three production sectors and the university. We guarantee the existence of long-run equilibria and provide stability conditions. We also examine effects of changes in research policy, creativity of economic activities and scientists, the territorial size, and savings rate upon long-run economic development and the price structure.
1. Introduction There is an increasing number of publications on relations between knowledge accumulation and economic development in the recent theoretical economic literature (e.g. [l-11]). These new approaches have provided much more appropriate answers to questions, such as why Japan and the Asian four “tigers” have succeeded in economic development and why China is much delayed in the industrialization process, than the economic growth models without endogenous
Correspondence to: Wei-Bin Zhang, Institute for Futures Studies, Hagagatan 23B, Box 6799, S-113 85 Stockholm, Sweden. * I am grateful to two anonymous referees for important comments. Research Policy 22 (1993) 327-336 North-Holland
0048-7333/93/$06.00
0 1993 - Elsevier
Science
Publishers
knowledge, such as the Solow-Swan model, the Leontief-von Neumann-Sraffa multisector models. This study tries to give another contribution to the literature by examining relations of a special public sector-research institutions and universities-to economic growth. Within my limited knowledge, there is no approach to deal with the problem in the same way as that proposed in this study. We treat knowledge stock as “public good”. Knowledge has the character of public good in the sense that utilization of knowledge by any economic sector will not affect that by any other sectors. There is certain knowledge, such as new innovation by a private company, which may not be accessible to the public, though new theories in mathematics, theoretical physics, economics, philosophy and the like are accessible to the public almost as soon as they are discovered. In a sense, the “share” of “private” knowledge to the total knowledge stock of a society is usually very small. Moreover, “private knowledge” will become public through imitation and other processes in the long term. Surely, in order to understand dynamic competitive processes of individual firms, it is necessary to be aware of how “private knowledge” is innovated and what effects the innovation has upon market shares and other aspects of the firms. Since we are concerned with historically long-run economic development, it is probably acceptable to neglect such microeconomic aspects of innovation. It must be emphasized that knowledge has characteristics different from those of other public goods such as transportation and communication infrastructures. For instance, knowledge tends to be further increased
B.V. All rights reserved
328
W’ei-Bin Zhang / Gormmnent ‘s research policy
during utilization processes. There are no congestion effects in knowledge utilization. The government in a competitive economy can affect economic development in different ways. Taxing private agents, government may provide supplies of public goods such as transportation and communication infrastructures and easy accessibilities to education and health care. These “material aspects” of government’s intervention in economic life have been well recognized in the economic literature, in particular, in public economics. There is another important economic issue which needs to be further examined. It is significant to investigate the effects of government’s knowledge creators, such as universities and research institutions, upon economic development. As national labor and capital resources are limited, investment in knowledge creation by the government may either benefit or decrease national economic growth. This study tries to shed some light on issues about the relationship between government’s inte~ention in R&D activities and the economic structure. The paper is organized as follows. Section 2 defines the basic model. Section 3 guarantees existence of long-run equilibria and provides stability conditions. Section 4 examines effects of government’s research policy upon economic development. Section 5 analyses how an improvement in creativity of economic activities or scientists can affect long-run capital and knowledge accumulation and the price structure. Section 6 examines effects of the territorial size upon the price structure. Section 7 examines how the savings rate may in a complicated way affect the economic growth, knowledge accumulation and the price structure. Section 8 concludes the study. In the Appendix, we provide the proof of existence of equilibria and stability in section 3.
2. The basic model
institution of pure mathematics may be established not to increase productivity of some special industry unit, but to increase knowledge stock of the society. The product of the mathematical institution - mathematical theorems - may find applications in various production sectors and thus increase national productivity in the long term. We classify production activities into three classes - agriculture, industry and service. The agricultural sector produces goods such as corn, rice and vegetables. The industrial sector produces commodities. The service sector provides services such as restaurants, shopping, hotels, ship/bus/train/air systems, information, health care, and education. The industrial product can either be consumed or saved to increase capital, though the product of the agricultural sector and the service sector can only be consumed. The product of the service sector cannot be saved in the sense that services are consumed in the process of their production. We assume that the production factors are constantly in full employment. Prices are measured in terms of the agricultural good. Let the price of the agricultural good be unity during the study period and denote pi(t) and p,(t), respectively, the ratios of the prices of the industrial commodity and the services over that of the agricultural good at time t. As we assume that the labor force, N, land, L, and capital, K, are homogeneous, the wage rate, w(r), the land rent, R(t), and the interest rate, r(t), are identical in the whole economy.
2.1. The agricultures sector
Agricultural production is a process of combining land, labor force, capital and knowledge. The production function of the agricultural sector is specified as F, = .Z”K,“NfL;,
We assume that the government may financially support knowledge creators, such as universities and research institutions, to increase knowledge for national deveIopment. For simplicity, we call government’s knowledge creators university. It should be remarked that the university may not be production oriented. For instance, a research
Ly,P,7 r 0,
CY+p-l-7=1, a 2 0,
(2.1)
in which 2 is knowledge stock of the economy, K, machines utilized by the agriculture sector, N, the number of farmers, and L, the land devoted to the agriculture production. Maximizing the
Wei-Bin Zhang / GoL~ernment‘7 research policy
profit by the agriculture ing conditions r = aTF.JK,,
sector
yields
the follow-
w = PTFJN,,
329
force employed by the industrial sector, respectively. The marginal conditions are given by r = mpiTFi/Ki,
R = rTF,/L,,
(2.2)
in which T = 1 - tax rate upon the product. In this study, we assume that the government gets its income to support the university only from the producers. There is no tax upon the properties and wage incomes. Moreover, the tax rate is identical among the three production sectors. It can be easily seen that we can relax this assumption by introducing heterogeneous tax rates upon product and taxes upon the properties and the wage incomes within the framework of this study. Let C, be the consumption of agricultural good of the population. Since we assume that agricultural good cannot be saved, we have c;, = 6,
(2.3)
2.2. The service sector
w = qp,TF,/N,
Let Y(t) denote the gross income the agricultural good. We have Y(t)
d+e=l,
F \ =ZhKdN” S 5.7
(2.4)
in which K, and N, are the capital and the labor force employed by the service sector, respectively. The profit-maximization yields
(2.9)
Denote ~(0
dK/dt
(2.10)
in which 6, is the given depreciation of rate of capital. Denote Ci the consumption of the industrial good by the population. Since the industrial product consists of investment and consumption, we have (2.11)
Ci 2 0,
in which Ci is the consumption trial good.
level of the indus-
2.4. The consumption of the households For simplicity of analysis, we assume that there is a “standard consumer” whose utility function stands for the population. We assume that the utility function takes on the following form u=
w = ep,TF,/N,
Let C, be the consumption population, we have
of
d, e > 0,
b>O,
r = &,T&/K,,
in terms
=rK+wN+RL
Ci = Fi - sY/p, , The production (in a strict sence, capacity) of the service sector is through combining knowledge, capital and the labor force. The production function of the service sector is defined by
(2.8)
(2.5)
of services
Cs = F,
of the
(2.6)
(~~/‘N)U(Ci/N)“(C~/N)x(~~/N)y,
u, c, x, y > 0 The total consumption budget is given by: (1 s)Y. Accordingly, the consumer problem is defined by min (C,/N)U(Ci/N)“(C,/N)X(L,/N)y
2.3. The industrial sector
subject
to: C, +piCi +p,C,
+ RL, = (1 - s)Y. (2.12)
We specify tion as follows F, = Z”KyNy,
the
agricultural
production
funcThe unique
m+q=l,
m,q>O,c>O, (2.7)
in which
Ki and Ni are the capital
and the labor
optimal
is given by
ci = US,Y/Pi,
c, = us,Y, c, = Xq-/Ps inwhich
solution
7
s,=(l-s)/u,
L, = yq,Y/R,
and u,=u+u+x+y.
(2.13)
330
Wei-Bin Zhang / Goc;emmentS research policy
2.5. Knowledge growth We propose growth dZ/d
t = T, F,/Z’
the following + ri F,/Z”
+ rpZgK;N;
possible
knowledge
+ rs F,/Z”
- 6,Z,
(2.14)
in which K, and N, are capital and scientists employed by the university, respectively, 6, (2 0) is the depreciation rate of knowledge, and ~(0 2 E I 11, 7r(O I7r I 0, a(0 I V 2 l), 6, 7, 2, 7, (2 O), 7i (2 O), rS (2 O), rP (2 0) are parameters. We further require, O
=a&‘Fi(n)
dnj”+o,,
in which a,, a2 and a3 are positive parameters, and 0 I a2 I 1. The above equation implies that effects of industrial activities upon knowledge accumulation have properties of decreasing scale. It is not difficult to see that a, and a2 can be interpreted as measurements of efficiency of “learning by doing” by the industrial sector. Taking the derivatives of the equation yields: dZ/dt = riFi/ZT, in which ri = u1u2, 7~ = 1 - a2. If the industrial sector is very effective in creating knowledge, then r is very small; and vice versa. We can similarly interpret the terms r,F,/Z’ and T,F,/Z”. It should be remarked that in (2.14), we omit the possibility of R&D by private companies. From the framework of this study, one can see that it is not impossible to introduce this actual important source of knowledge growth. For instance, we may divide K, and Ni into Ki, and Ki2, Ni, and Niz, respectively, where Ni, and Ki, are labor force and capital employed by the industry to produce commodities and Ni2 and Ki, are labor force and capital employed by the industry to do R&D activities. The industry decides the employment structure by maximizing the long-run profit. This obviously involves many other complicated issues of modeling. For simplicity of analysis, we limit ourselves to the narrow case of knowledge growth. We call the term, T~Z~K~N~, creativity of the university. A similar functional form for knowl-
edge creation by scientists is suggested by Zhang [9]. The term, rpZgKGN:, implies that creativity of the university is positively related to the number of scientists and capital employed. On the one hand, as knowledge stock is increased, the university may more effectiveIy utilize traditiona knowledge to discover new theorems. On the other hand, a large stock of knowledge may make discovery of new knowledge difficult. This implies that the parameter, g, may be either positive or negative, depending upon the characteristics of knowledge creation of the university. According to the above interpretations, we may call r,Fa/Zc, riFi/ZT, T~F,/Z~, and T~Z~K;N~~, respectively, the creativity of the agricultural sector, the industrial sector, the service sector, and the university. It should be emphasized that a sector may be very creative in one economy, while it may be not creative at all in another. For instance, creativity of the agricultural sector may be reasonably ignored in many developed economies. But we cannot omit it in analysing, for instance, current economic development in China. Since we assume that the financial resource of the university is from government’s taxes upon the product, we have (1 - T)( F, +piFi +p,F,)
= wN, + rK,
(2.15)
We now have to design a way to determine N, and K,. We assume that the government decides the number of scientists and the capital stock of the university in the following way N, = nN,
K, = kK,
0
k < 1,
(2.16)
in which n and k are the policy variables fixed by the government. We assume that n and k are exogenously given. We are interested in the effects of the policy parameters n and k upon the economic system. In this study, we assume that the tax rate is endogenously given in the sense that the government will tax the producers only to satisfy (2.15). That is, (1 - T) is equal to (wN, + rlK,)/( F, + pi Fi + ps F,) at any point in time. 2.6. Full employment of the labor force, capital and land The assumption that the labor force, capital and land are always fully employed yields the
331
W&Bin Zhang / Government’s research policy
following
fixed, the tax rate is constant during the study period. We thus can write the dynamics of capital and knowledge in the following form:
equations L,+L,=L,
N,+N,+N,=(1-n)N, K,+K,+K,=(l-k)K,
(2.17)
in which N and L are fixed. We have thus built the model which explains both distribution of the labor force, capital and land and accumulation of capital and knowledge. It should be remarked that the assumptions that population, land, and savings rate are exogenously given can be, in principle, relaxed within the framework of this model. There are 25 endogenous variables, K, K,, Ki,
y + UTT),
L,
= YL/(
Y + UTT)
K, = uh,K,,
9
= 1 + nu,/s,
N, =xes,N,
+ ku,/s,,
(2.21)
+ r,F,/Z”
+ r,,ZgKZ - S,Z,
(2.22)
70 = rp(nN)“k”, and F,, Fi and of K and Z given by:
F, are
FH= A,ZbKd, (2.23)
where A, = (au~,)“(u&N)~L;, A,=
(a!~s,)~(xe.s~N)~, (2.24)
Ai = (mv,s,)m(qv,s,N)q In what dynamic
follows, system.
3. Equilibria
we analyse
and stability
the behavior
sF,/(vs,+s) r,F,/Z’
as a solution
+ r,F,/Z”+
rOZRKz = S,Z (3.1)
From the first equation,
we have
K = AZc’q,
(3.2)
in which A = (sAi/S,(vs, + s))“~ > 0. Substituting (3.2) into the second equation in (3.1) yields 3 QJaZa-‘+&4--l + 7iAiAmZ-.rr+c/q-1
au + mu + d-x + S/Q),
+ 7 A s
+
Adzb-o+dc/q-1 s
TOAZZg+CZ/q-l
_
6,=()
(3.3)
S2=(1-n)/(/3u+w,+-),
l/(U
-t v +s/s,
From (2.211, we ratios n and k, government tax and the taste). satisfies: 0 < T
+x)
see that the government fixes the the tax rate is determined (by the policy, the production structure From (2.211, one sees that T < 1. As all the parameters are
of the
=S,K,
+ riFi/ZK
v~=v+s/so,
z.40 =
of the
conditions
An equilibrium is defined following equations
H(Z)
in which s1 = (1 - k)/(
+ riFi/ZK
Ki = mv,s,K,
(2.20) l/T
= r,F,/Z’
-S,K,
Fi = A,Z’K”,
(2.19) Ni = qU,S, N,
dZ/dt
F, = A,Z”K*,
(2.18) K, = aus,K,
=sF,/(vs,+s)
in which functions
KS, K,, Na, Ni, Ns, Np’ La, L,, Z, Fa, Fi, Fs, C,, C,, C,, pi, ps, r, R, w, 1 - T, Y. It can easily be checked that the system consists of 25 independent equations (2.1)~(2.10), and (2.13)~(2.17). Hence, under appropriate conditions the system may have solutions. We now show that the dynamics of our system can be written in terms of K and 2. That is, our dynamic analysis will be reduced to that of a two-dimensional differential equations system. As shown in Appendix Al, we can write L, and L, as functions of L, N,, Ni and N, as functions of N, K,, Ki and K, as functions of K, and T as function of the research policy, respectively, as follows L, = urTL/(
dK/dt
Appendix
A2 proves
the following
proposition.
3.1. Proposition (3.1) If a+cc-u/q
b+cd/q< is a unique
332
W&Bin Zhang / Go~lernment S research policy
If a+m/q>l+t, c/q>l+r, h+cd/q> 1 + CT, and g + cz/q > 1, then there is a unique equilibrium. The system is unstable. In each of the cases (l&(14) defined in appendix A2, the system has either no equilibrium or two equilibria. When the system has two equilibria, the one with higher values of K and Z is unstable and the other one is stable.
To interpret the proposition, we examine the stable case: a + ca/q < 1 + E, c/q < 1 + V, b + cd/q < 1 + CT, g + cz/q < 1. Since a, b and c are the marginal production of knowledge in the agricultural, service and the industrial sectors, respectively, we may conclude that an increase in the marginal production of knowledge tends to destabilize the system. On the other hand, as E, (T and rr are the measurements of decreasing scale effects in knowledge accumulation by the agricultural, service and the industrial sectors, respectively, we may conclude that an increase in the decreasing scale effects tends to stabilize the system. In other words, an increase in creativity of the economic activities tends to destabilize the economic system. Since (Y, d and m (= 1 - q) are the marginal production of capital in the agricultural, the service and the industrial sector, respectively, we can say that if the marginal production of capital is increased, the system tends to become unstable. In g + cz/q < 1, the parameter g describes the effects of knowledge stock upon knowledge creation and z measures the effects of capital in helping scientists to discover new knowledge. Accordingly, this condition can be interpreted as that when creativity of the university is not very high, then the system tends to be stabilized. Surely, whether the system is stable or unstable depends upon the combined conditions of the economic structure and the characteristics of knowledge accumulation. In what follows, we carry out comparative static analyses. As the analyses are meaningful only under presumed uniqueness and stability of the equilibrium, we assume a + ecu/q < 1 + E, c/q < 1 + 7~, b + cd/q < 1 + u, and g + cz/q < 1. We will retain this assumption in the remainder of this study. It must be emphasized that the effects of changes in the parameters obtained below may be different from those when this stability assumption is relaxed.
4. Effects of the number of scientists of the university
and capital
First, we examine effects of changes in the parameter n upon the behavior of the system. Taking derivatives of (3.3) and (3.2), respectively, with respect to n yields @ dZ/dn
= -((Y +p)~,F,/(l
-n)Z’
- er,F,/(
1 - n)Z”
- T,PF,/(
1 - n) Z”
+,,(8/n-z/(l-n))zgKz, (4.1) (l/K)
dK/dn
= (c/qZ)
dZ/dn
- l/(
1 - n), (4.2)
in which CDis a positive number defined in (3.6). From (4.11, we can see that if 6/n -z/(1 - n) is negative, then the knowledge stock will be reduced as the number of scientists in the university is increased. If S/n -z/(1 -n> is positive, then an increase in the number of scientists may either increase or decrease the equilibrium level of knowledge stock, depending upon the other terms in (4.1). As the two parameters, 6 and z, represent the “marginal creativity” of the scientists and the “marginal creativity” of capital, respectively, we may interpret the sign of 6/n z/(1 -n> as follows: if the marginal creativity of scientists is appropriately larger than that of capital, then 6/n - z(1 - n) is positive. The above discussion means that if the marginal creativity of scientists is very low, then an increase in the number of scientists will certainly reduce knowledge stock of the society in the long term. When the marginal creativity of scientists is quite high and the creativity of the economic activities is very low (i.e. r,, ri and 7, are very small), then an increase of the number of scientists tends to increase knowledge stock of the society. Eq. (4.2) implies that if dZ/dn is negative, then equilibrium value of capital will be reduced as more scientists are allocated to the university. Even if dZ/dn is positive, if the knowledge stock of society is quite high and dZ/dn is relatively small, then equilibrium capital also tends to be reduced. Only when Z is relatively low and dZ/dn is quite large, equilibrium level of capital will be increased as more scientists are employed by the university.
W&Bin
Zhang
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research policy
It should be remarked again that the above conclusion is dependent upon the assumed condition: @ > 0. Since any case among (1)-X14) may not satisfy this requirement, the effects of change in II may be opposite to those just obtained in any case among (1)-(14X In this sense, whether increasing the number of scientists will benefit the economy depends upon the special characteristics of the economic system under consideration in a rather complicated way. Since dT/dn = - T*u,/s,(l -n) < 0, we see that as the number of scientists is increased, the tax rate has to be increased in order to keep the government budget in balance. It is easy to check the effects upon the output of the economic activities as follows
We can interpret effects of changes in k similarly to the case of n. Summarizing the above discussions, we can generally conclude that, from a purely economic point of view, whether the government should strongly financially support sciences is much dependent upon the creativity of economic activities and purely scientific research and the effects of knowledge upon the economic activities. As, in modern economic systems, “private” economic activities such as large companies and individual artists can create new knowledge, to enlarge the university through government intervention might not benefit the economic development.
(l/Fa)
dF,/dn=(l/Z)(a-e+ac/q)
dZ/dn
5. Efficiency
(l/F,)
dFi/dn
-(a+P)/(l = (l/Z)(c/q-n)
(l/F,)
dF,/dn
-e/(1-n), = (l/Z)( b - u + de/q)
dZ/dn
Since under the presumed stability, changes in r,, rs, ri, rP have similar effects upon the variables, it is sufficient for us to carry out the analysis only with respect to one parameter, e.g. Ti. The effects of a shift in the knowledge accumulation efficiency parameter, TV, are given by
-n), dZ/dn
-P/(1 -n) (4.3) One can see that even when dZ/dn is positive, the product may be reduced as more labor force is shifted to the university. The effects are also uncertain in the sense that they may be either positive or negative, depending upon the parameter values. As dY/dn = (l/ys,) dF,/dn, dY/dn is positive only when dFJdn > 0. From r = aTFJK,, w = PTFJN,, R = TTFJL,, pi = (us0 + s)Fa/ us0 F,, pS = xF,/uI?,, one can directly get the effects of changes in n upon the interest rate, wage rate, land rent, and prices of the industrial good and agricultural good. It is not difficult to see that an increase in the number of scientists may either have positive or negative effects upon these variables, depending upon the further assumptions one may add to the values of the parameters. We have the following effects of changes in capital of the university upon the knowledge stock and the total capital of the economy @ dZ/dk=
-(a+rn/q)T,Fa/(l-k)Z’ - m(l + l/q)T,F,/(l
- k)Z”
(l/K)
dK/dk=(c/qZ)
- mz/q(
@ dZ/dri
= (cAK/qZ) = FiZ1-=
accumulation
dZ/dr,
> 0,
> 0,
(5.1)
in which @ is a positive constant, defined in (3.6). One can directly check that Na, Ni, N,, F,, Fi, F,, Y, C,, C,, Ci, w, R, are increased as the efficiency of knowledge accumulation is improved. As dr/dri=
(r/Z){a
- (1 -a)cA/q}
dZ/dri,
if the marginal production, a, of knowledge of the agricultural sector is extremely low, then an improvement in the efficiency of knowledge accumulation will reduce the interest rate. If a is appropriately large, then the interest rate will be increased. As the parameter a seems to be very low in an industrialized society, it appears reasonable to observe a decline in the interest rate as knowledge is increased. Utilizing pi = (us0 + s>F,/~~,,F,, p, = xFa/uFS, we obtain
-~,d(l+d/q)F,/(l-k)ZP + m,( l/k
dK/dri
of knowledge
1 - k))ZgKZ,
dZ/dk-m/q(l-n) (4.4)
Z dpi/dri
=pi{ a - c + ((Y - m)c&q)}
Z dp,/dri
=p,{ a - b + ((Y - d)cA/q}
dZ/dri, dZ/dri (5.2)
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5.1. Pmposition (5.1)
The effects upon the prices are determined
As the efficiency of knowledge accumulation by the industrial sector is improved, the equilibrium values of capital and knowledge are increased. The output of the three sectors, the gross income, and the consumption of the output of the three sectors are increased as Ti is increased. The wage and the land rent are increased as ?i is increased. The prices of the industry good and services (in terms of agricultural good) may be either increased or decreased as the efficiency is improved. For instance, in the case of a and b being extremely small, m > CY> d, dr/dTi < 0, dp,/dr, < 0, dp,/d~i > 0.
6. The te~itorial
size
Although the territorial size is usually not changeable, it is still important to analyse effects of changes in L upon the economic system, for instance, when we compare American and Japanese economic structures. With regard to L, we have: @ d Z/d L = n,F,/LZ’, dK,‘dL
= (cK,‘qZ)
dZ,‘dL
(6.1)
With all the other conditions equal, a country with a larger territory will have more capital and knowledge. Since the agricultural sector has usually a very low efficiency in knowledge creation in an industrialized economy, the effects upon capital and knowledge accumulation are, in fact, very weak. One may also check that K,, Ki, K,, L,, L,, F,, Fi, F,, Y, C,, C,, Ci, and w are increased if L is increased. As (Z/R)
dR/dL
= (a +ac/q) -(I
-7)2/L,
(Z/Pi1
dPi/dL
=(a-c+(vm)c/q) -(l
(6.2)
one sees that, under the condition that knowledge stock is not strongly affected by the agricultural production, a country with a large territory tends to have lower land rent. An interesting case, though rarely happening, is that if dZ/dL is quite large, then dR/d L might be positive.
dZ/dL
-T)Z/L,
(Z/P,) dps/dL = (a -b + (cy - d)c/q) -(l
dZ/dL
- T)Z/L
Accordingly, under the very acceptable hypothesis that dZ/d L is extremely small, we may conclude that an economy with a large territory tends to have cheap industrial goods and services (in terms of agricultural product).
7. The savings rate This study makes a very strict assumption about the savings behavior of the households. In reality, saving behavior will adapt to a new social and economic environment in the long term. Although we do not treat the savings rate as an endogenous variable in this study, it is important to examine the effects of an exogenous shift in the savings rate upon the economic system. Effects of changes in s upon Z and K are given by a, dZ/ds = -ra(LYSi f /3szq/N)u,F,/Z’ + ~gyr,F,/( -(ms,
y + mT)Z”
+qs*/N-s,/(~~,s,+S))Ul~if;li/Za
- (ds, f es,q/Wu,r,WZ” +T,ZgK”q{l/s+
(r:-ul)/uI(uso+s)
-m/s&(1 4/%Q( (l/K)
- k)(l
-s)
1 - a)( I - s)}/z,
dK/ds
= (c/qZ)
dZ/dL
by
+{[u/s(l
d.Z/ds -s)
+u,Js,/(Us,+s)
- (ms, -+qs,q/N)u,}q,
(7.1)
in which T, = dT/ds = - T’qnu,u,/(l - ~>~(l n) - T2ku,u,/(l - s>‘(l - k) < 0. From (7.11, we see that the effects of changes in the savings rate are affected by different forces. An increase in the savings rate may either expand or reduce
We&Bin Zhang / Goc;ernment’s research policy
capital and knowledge. As the conditions are so complicated to interpret, we can get a definite conclusion about the effects of shifts in the savings rate upon capital and knowledge only when we further specify the values of the parameters. The effects upon the other variables are also very complicated, and we will not carry out a further analysis. It should be remarked that although our model is developed within the framework of neoclassical growth theory (e.g. the Solow-Swan model, Uzawa’s two sector model) in the sense that full employment of the product factor is assumed, the effects of the savings behavior upon the system are much more complicated than those forecast by the models just mentioned. Obviously, the complexity is due to the introduction of endogenous knowledge into the system. Whether an increase in propensity to save will stimulate economic development or not depends upon complicated interactions among the taste structure, production functions and characteristics of knowledge accumulation. 8. Concluding
remarks
We have answered certain questions about interactional relations between knowledge and capital accumulation. The problems are analysed under some strict assumptions. Some of these assumptions can be relaxed in our approach. For instance, it is also possible to introduce an endogenous savings rate in our analysis. Contribution to knowledge accumulation by private sectors is much more complicated than assumed in this study. It should be remarked that we can solve the equations because we specify production and knowledge accumulation functions in the Cobb-Douglas formula. If these functions are taken on more general functional forms, it might be quite difficult to get definite analytical conclusions. It is clear that if we allow international interactions for trades and for exchanges of ideas, the system will become extremely complicated. Appendix Al. The dynamics Z
in the term of K and
We now show (2.18j42.21) given in section 2. From R = rTF,/L,, F, = us,Y, L, = ys,Y/R, we have: La/L, = urT/y, in which T is an en-
335
dogenous variable. The ratio of the land employed by the agriculture sector and the land devoted to the housing is positively related to the marginal utility of the agricultural product and the marginal product of the land, but negatively related to the marginal utility of housing and the tax rate. From L = L, + L,, we have (2.18). As the tax rate T is, as will be shown, constant, one sees that L, and L, are constant during the study period. From (2.21, (2.5) and (2.81, one has: Ki/K,
= ( mc/aq)
NJN,,
K/K,
= (me/qd)
N/N,
From
(2.11)
one has pi = (us0 + Utilizing these relations and (2.2), (2.5) and (2.8), we get
s)Fa/us,Fi, K/K,
and
(Al)
(2.131,
p, = xFJuF,.
= m( cso + s) /aus,,
KS/K, = dx/cuu
(A21 Equations (Al) and (A2) imply that the ratios of the labor force or the capital employed by the three sectors are determined by the marginal productivity and the marginal utilities of the factors. Accordingly, if the savings rates, u, v, x and y are not affected by knowledge and wealth, then the ratios will be kept constant during the study period. In fact, since the savings rate, s, and in particular, the “life-style parameters” u, L’, x and y are much affected by economic conditions and varieties of goods and services, it is possible to examine how the ratios of the labor force or the capital employed by the three sectors are affected by economic development. In this study, we shall not be involved with the complexity of endogenous changes in s, u, v, x and y. Substituting (A2) into K, + Ki + K, = (1 - k)K yields (2.19). From (A2) and (2.19), and N, + N, + h$ = nN, one has (2.20). Utilizing 1 - T = (wNP = rK,)/(Fa + piFi + p,F,) in which F, +piFi +p,F, = F,(l + (us0 + s)/uso +x/u), wN, = wnN = nwN,/u@, rK, = rkK = krK,/aus,
= nTF,/us,, = kTF,/us,,
(A3)
we have (2.21).
Appendix A2. Proving Proposition
3.1.
First, we find conditions such that H(Z) = 0 has positive solutions. Let’x, = a + at/q - E - 1,
336
We&Bin Zhang / Gor,ernment S research policy
x,=c/q-r-1 and x,=b+dc/q-a-1, xq = g + cz/g - 1. We exclude the case of x1 =x2 = x2 =x4 = 0. One can directly check that if x, r 0, x2 2 0, x3 2 0, x4 2 0 (i.e., H(0) < , H(w) > 0 and H’>OforZ>O)orx,~0,x,~O,x,~1,x,~1 (i.e., H(0) > 0, H(m) < 0 and H’ < 0 for Z > O), then the system has a unique positive equilibrium. If (1) x,pO, x2 < 0, xg < 0, x4 < 0, (2) x,pO, x2 > 0, x3 < 0, x4 -=c 0, (3) “$0, x2 -C 0, x3 > 0, x4 < 0, (4) x1 < 0, x2 < 0, x3 > 0, x4 < 0, (5) x1 < 0, x2 > 0, x3 < 0, x4 < 0, (6) x1 < 0, x2 > 0, x3 > 0, x4 < 0, (7) x,pO, x2 < 0, x3 < 0, x4 > 0, (8) x,pO, x2>o, x,
+x2riAiAmZX*
+ x3r,A,AdZX”
+ X4T0AZZX4,
(Ad)
we see that H’(Z) may be either positive or negative, depending upon the parameter values. If H(Z) = 0 has more than two solutions, there are at least two values of Z such that H’(Z) = 0. Since d(ZH’)/dZ > 0 strictly holds for Z > 0, it is impossible for H’(Z) = 0 to have more than one solution. Accordingly, H(Z) = 0 has either no solution or two solutions. A necessary and sufficient condition for the existence of two equilibria is that there exists a value of Z* such that H(Z’) < 0 and H’(Z *) = 0. The proofs of cases Q-(14) can be similarly carried out. The Jacobian at an equilibrium is given by
(A3 in which a, = r,F,/Z’
+ mriFi/ZT
+ zr,,ZgKZ>
0,
+ dr,F,/Z”
a,=(l+E-a)r,Fa/Zt+(l+r-c)riFi/ZT +(l
+a-b)r,F,/Z”+
(1 -g)rOZ”KZ
> 0
The two eigenvalues,
(‘46) 4r and &,
are determined
by +2 + (Sq + a2/Z)4
+ a,Sq/Z
- Sca,/Z
= 0 (A?
Accordingly, if a,q < ca,, the system is unstable; if a,q > ca,, the system is stable. The condition, a,q > ca, can be rewritten as @ = -x,r,F,/ZE
- x2riFi/Z”
- x3r,&/Zv
-x4r,ZgK’>0 We thus have proved
(A8) Proposition
(3.1).
References [l] ..&E. Andersson and W.B. Zhang, Endogenous Technological Changes and Economic Growth, in: Manas Chatteji and Robert Kuenne (Editors), Dynamics and Conflict in Regional Structural Change (The Macmillan Press, London, 1990). [2] G.S. Becker, K.M. Murphy and R. Tamura, Human Capital, Fertility and Economic Growth, Journal ofPolitical Economy 98 (5) (1990) 12-37. [3] G.M. Grossman and E. Helpman, Quality Ladders in the Theory of Growth, Review of Economic Studies 58 (1991) 43-61. [4] R.E. Lucas, On The Mechanics of Economic Development, Journal of Monetary Economics 22 (1988) 3-42. [5] P.M. Romer, Increasing Returns and Long-Run Growth, Journal of Political Economy 94 (1986) 1002-1037. [6] P.M. Romer, Endogenous Technological Change, Journal of Political Economy 98 (5) (1990) 71-102. [7] G.W. Stadler, Business Cycle Models with Endogenous Technology, The American Economic Review 80 (1990) 163-778. [8] N.L. Stokey, The Volume and Composition of Trade Between Rich and Poor Countries, Review of Economic Studies 58 (1991) 63-80. [9] W.B. Zhang, Economic Growth and Technological Change, International Journal of Systems Science 21 (1990) 1933-1949. [lo] W.B. Zhang, Regional Dynamics with Creativity and Knowledge Diffusion, The Annals of Regional Science 59 (1991) 179-191. [ll] W.B. Zhang, Economic Development with Creativity and Knowledge Diffusion, Socio-Spatial Dynamics 2 (1991) 19-30.