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Economic development with accumulation of physicalcapital and human capitalWEI-BIN ZHANG aa Institute for Future Studies , Hagagatan 23B, Box 6799, Stockholm, S-l13 85, SwedenPublished online: 27 Apr 2007.

To cite this article: WEI-BIN ZHANG (1993) Economic development with accumulation of physical capital and human capital,International Journal of Systems Science, 24:1, 65-77, DOI: 10.1080/00207729308949472

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INT. J. SYSTEMS SCI., 1993, VOL 24, NO. I, 65-77

Economic development with accumulation of physical capitaland human capital

WEI-BIN ZHANGt

The purpose of this study is to develop an economic growth model to explaindynamic interactions among physical capital accumulation, population adjust­ment, and human capital improvement. We assume that human capital can beincreased by education and learning by doing. Population growth may be affectedby knowledgeand economicconditions. We examine the effects of changeson someparameters by traditional comparative static analysis under presumed stability. Wealso guarantee the existence of permanent oscillations when the system loses itsstability due to shifts in the population adjustment speed parameter.

1. IntroductionAbout 200 years ago Malthus put forward a theory of the relationship between

the population and economic development that still survives today. In his Essay onthe Principle of Population which was first published in 1798, Malthus postulated,upon the concept of diminishing returns, a universal tendency for the population of acountry, unless checked by dwindling food supplies, to grow at a geometric rate,doubling every 30-40 years. At the same time, because of diminishing returns to thefixed factor, land, food supplies could only expand roughly at an arithmetical rate.Since the growth in food supplies could not keep pace with the burgeoning popu­lation, per capita incomes (defined in an agrarian society simply as per capita foodproduction) would have a tendency to fall so low as to lead to a stable populationbarely existing at or slightly above the subsistence level. Malthus therefore con­tended that the only way to avoid this condition of chronic low levels of living or'absolute poverty' was for people to engage in 'moral restrain' and limit the numbersof their progeny.

In fact, 'An Essay on the Principle of Population' was written by Malthus indirect response to the kind of perfectibilitarianism which can be seen in Condorcetand Godwin. Malthus himself describes the Essay as 'remarks on the speculations ofMr. Godwin, M. Condorcet, and other writers.' Godwin had dealt expressly with thepossibility of overpopulation, but insisted that the progress of reason would reducehuman desire for fertility, adding, 'myriads of centuries of still increasing populationmay pass away, and the earth be yet found sufficient for the support of its habitants.'Although this point of view has been realized in developed nations, it is invalid forunderdeveloped nations such as China.

It should be noted that after the publication of the first edition, the Essay wassoon recognized as a powerful book. Even Godwin congratulated Malthus for thisbrilliant thesis. However, Malthus did not change Godwin who still believed in thepower of human reason to supply a moral check upon the progress of fertility.Godwin wrote to Malthus to argue that Malthus failed to give proper due toanother check upon the increasing population which operated very powerfully and

Received 18 June 1991.t Institute for Future Studies, Hagagatan 23B, S-113 85, Box 6799, Stockholm, Sweden.

0020-7721/93 $10.00 to 1993 Taylor &. Francis Ltd.

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66 Wei-Bin Zhang

extensively. Godwin identified this positive check as 'that sentiment, whether virtue,prudence, or pride, which continually restrains the universality and frequent repe­tition of the marriage contract. ... The more men are raised above poverty and a lifeof expedients, the more decency will prevail in their conduct and sobriety in theirsentiments. '

This point of view had a strong influence on Malthus. In the second and allsubsequent editions of the Essay, Malthus gives full recognition to the efficiencyof positive checks. Malthus moved over a period of years from pessimism rootedin biology to an optimism that sprang from conservation to a sociological­progressive perspective. In the later editions of the Essay on Population he says: 'Inmost countries, among the lower classes of people, there appears to be somethinglike a standard of wretchedness, a point below which they will not continue to marryand propagate their species ... The principle circumstances which contribute to raise(this standard) are liberty, security of property, the diffusion of knowledge, and ataste for the comforts of life. Those which contribute principally to lower it aredespotism and ignorance.'

Although interactions among human capital, the population, and economicconditions have been emphasized since Malthus, effects of human capital upon thepopulation growth are not introduced into growth models. There are some economicgrowth models which have treated the population as an endogenous variable (Swan1956, Niehans 1963, Pitchford 1974, Becker 1976, 1981, 1988, Cigno 1981, 1986,Cigno and Zhang 1988, Zhang 1990 a, b), but no model has explicitly introducedknowledge as a checking force for population growth. We need to further investigatehow knowledge may affect the population growth, not to mention economicdevelopment.

This study is organized as follows: §2 develops the basic model which describesdynamic interactions between capital, population and human quality; §3 guaranteesthe existence of a unique long-run equilibrium and provides its stability conditions;§4 describes our comparative statics analysis with respect to some parameters underpresumed stability; §5 illustrates the complexity of the behaviour of the dynamicsystem by proving the existence of limit cycles with the Hopf bifurcation theorem; §6concludes the study. The theorem of the existence of limit cycles in §5 is proved inthe Appendix.

2. Economic growth with human quality improvementWe consider a standard one-sector neoclassical growth economy (e.g. Zhang

1990b). On product is produced by a combination of capital and a qualified labourforce. Product is identical over time in the entire economy. The product can either beconsumed by the labour or invested in the nation that produces it. Let the capital ofthe economy at time t be denoted by K(t). 'Quality' of capital is identical over thestudy period. Hence, we neglect the aspects of how knowledge can affect the designof machines (Johansen 1959).

We denote the population by N(t). Both the population and human capital interms of productivity or knowledge are changeable. To describe the human capital ofthe population, we introduce the concept of a qualification index, z(t), of the popu­lation of the nation (see Samuelson 1965, Drandakis and Phelps 1966, Romer 1986,Lucas 1988, Zhang 1990a, b, 1991, 1992). The quality index is an aggregatedmeasurement of population quality and knowledge. It is determined by average

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Economic developmenl with capital accumulation 67

education level, knowledge, working skills, and other aspects of human quality of thepopulation of the nation under consideration. We assume that Z(I) ~ 0 for 1 ~ O.z(l) = 0 implies that at time I, there are no productive activities. Using the concept ofa qualification index, we define the qualified labour force of the economy by z(I)N(I).The qualified labour force is determined by the combination of the population andits qualification index.

Assume that the production is given by

Y = F(K, z(I)N(I)) (2.1)

We consider a special case where F takes on the Cobb-Douglas form:F= vK'(zN)'-", where v> 0, I> Q > O. In a sense, the parameter v describesthe efficiency of the institutions and organizations of the economy, while the termsK"(zN)'-" descibe the 'designed' capacity of the production (see Zhang 1990b).

According to the definitions, we have the capital accumulation equation

dK/dl = sF - oK (2.2)

where 0 is the depreciation rate of capital and s is the savings rate.Before we introduce the dynamics of human quality, we look at (2.2), which

describes economic growth without international interactions and without the possi­bility of human quality improvement. In the case where human quality is constant,the system is identical with the standard one-sector neoclassical growth model if weallow the population to grow at an endogenously fixed growth rate. It is well knownthat this system has a unique equilibrium. Moreover, the equilibrium is globallystable (see Zhang 1990b). We thus can conclude that if there are no possiblechanges in the population and human quality, the economy is globally stable.

Before we suggest a possible dynamics for population growth, we discuss someextensions of the Malthusian growth model. It is well known that, in the 'Malthusiangrowth model', the population grows at a constant rate with no limitations on itsresources. That is, dN/ dr = aN. Such a population growth may be valid for a shorttime, but it clearly cannot go on forever. There are limitations of natural resourceswhich prevent the population from limitlessly growing. The logistic growth model,which is defined by dN/dl = aN(I - N/ K), takes account of the checking effectsof natural resources upon population growth. However, for human society,resources are not given, but are 'endogenous variables'. To analyse how productionaffects population growth, Haavelmo (1954) suggests the following model:dN/dl = aN(l - bN/ Y), where Y is the production of the society. Haavelmofurther assumes that Y = Nf3, 0 < (3 < I. This population model is valid for anargicultural economy where no capital accumulation is allowed.

In this study, we extend Haavelmo's population growth model in the followingway:

dN/dl = aN(1 - b(z)N/F(K,zN)) (2.3)

where b(z) > O. This model means that an increase in production makes it possiblefor society to support more people to materially survive, but human quality makesanother contribution to checking the population growth rate. If the function, b(z), isconstant, the term, bN/ F(K, zN), can be similarly interpreted as in the Haavelmopopulation growth model. But we assume that b(z) is dependent upon humanquality. We consider that human quality has a significant checking force uponpopulation growth. As discussed in the previous section, this aspect of humanquality on population growth has been well recognized for a long time.

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68 Wei-Bin Zhang

We have to specify some properties of the function b(z). Relationships betweenknowledge (human quality) and population growth rate are very complicated(Becker 1975,1981). At an individual level, an increase in human quality may resulteither in an increase or a decrease in the number of children. It is not difficult to seethat the relationship between human quality and population is rather uncertain at anaggregated level. But it seems safe to say that improved human quality tends toreduce population growth rate. For instance, wide spread education tends to resultin reduced family sizes in developing nations. In this study, for simplicity of analysiswe assume that b(z) takes on the following functional form: b(z) = cz", where c and/3 are positive constants. From

dN/dt = aN[1 - cz f3N/vKO(zN)I-oj

= aN[1 - cz f3/v(K/Nt(z)l-oj

we see that, in a sense, the assumed /3 < I implies that the 'checking force' of humanquality upon the population growth is weaker than that of the material condition(since /3 < a + (I - a) = I). The condition (3 > I implies that the checking force ofhuman quality on population growth is stronger than that of the output. As shown later,some results of comparative statics analysis are very sensitive to whether /3 > I or not.

We now discuss some possible dynamics of human quality change. Surely, thereare many ways to affect human quality. We have at least four main aspects oflearning processes to take into account. They are: division of labour, imitation,learning by doing (such as by producing, trading, consuming) and learning bytraining (such as eduction). Our main problem is how we can mathematicallyhandle these factors in the economic growth model. In what follows, we suggest apossibility of dynamics of human quality:

dz/dt = H( Y, N, z) - rz (2.4)

where r is the depreciation rate of the human quality. We suggest H has the followingproperties. First, for a fixed z, an increase in the output tends to increase H. That is,the more output is produced, the higher the human quality tends to become. Surely,the increased product should have both positive and negative contributions tohuman quality improvement. Taking account of effects of division of labour, invest­ment in education, learning by doing and imitation, it is generally acceptable toassume that an increase in the output tends to improve human quality. There aresome factors that cause improvement in human quality to slow down as the popu­lation becomes richer. Such negative factors may be not so strong as to dominateother positive factors in most cases. Second, H has the property that as z increases, Hshould at least not increase. This simply means that human quality becomes moreand more difficult to improve. For simplicity, we suggest H has the following form:

H=pF/(1 +hz) (2.5)

where p and h are positive constants. This functional form satisfies the propertiesrequired above.

We have thus described the whole dynamic system which consists of the follow­ing equations:

dK/dt = svKfi(zN)I-o - s«, }dN/dt = aN[1 - cz f3N/vKO(zN)I-0j,

dz/dt = pvK o(ZN)I-fi /(1 + hz) - rz

(2.6)

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Economic development with capital accumulation 69

(3.1)

3. Existence of equilibrium and stability conditionsIn this section, we first examine whether the system has a long-run equilibrium

and then find the stability conditions of the equilibrium.A positive equilibrium is determined as a solution of the following equations:

svK"(zN)I-" = 6K, I = czfJN/vK"(zN)I-" }

pvK"(zN) 1-" /(1 + hz) = rz

It is not difficult to check that a unique equilibrium is given by

K' = srz'(1 + hz')/p6, N' = 6K '/csz'fJ}

z' = (c/v)I/(I-fJ)(6/sv)W(3.2)

(3.5)

where w = a/(I - a)(1 - (3). In the remainder of the study, we assume that f3 is notequal to unity.

The three eigenvalues are given by the following equation

e3 + a le2 + a2e + a3 = 0 (3.3)

where R = (I - a)r+ rz/(I + hz), al = R +aa/N + 6 - a6,

a2 = (1- a)[arf3 + ara - ar - aa6 - ar6+ 6R + aa6/N + aaR/(1 - a)N]}(3.4)

a3 = -a6r(1 - a)[1 - a 2- f3 + R + a 2

/ N - R/N]

The necessary and sufficient conditions for stability are known as the Routh­Hurwitz criterion: (i) al > 0; and (ii) ala2 - a3 > O. As the expressions of a,(i = 1,2,3) are too complicated, we cannot make a general conclusion aboutwhether the equilibrium is stable.

For convenience of examining the stability conditions, we specify the values ofsome parameters. We take: s = 0'2, v = I, a = f3 = 0,5, 6 = r = 0·1. Hence, theremaining parameters are a, c, p, and h, which are practically much more difficultto identify. With the specified parameters, we have:

a\ = 1/IO+c2/10(4+hc2

) + 40ap/(4 + hc2) }

a2 = 4ap(4 + hc2+ c2)/(4 + hc2)2 + c2/200(4 + hc2) + 1/200 - 12a/200

a3 = -(a/200)[3/10 + c2/IO(4 + hc2) + (64p + 8phc2)/(4 + hc2)2]

Since for a positive a, a3 is always negative, we see that the system is unstable. Wemay consider the case of a = O. As a = 0 implies that the population is not adaptedto the changeable conditions, we see that N is constant during the study period. Wemay fix N(t) by N(t) = 6K'/csz'fJ, where z' and K' are equilibrium values of K(t)and z(t) for a = O. It is easy to show that the reduced two-dimensional system isstable for a = O. A meaningful question is what will happen to the system if theparameter a takes on a sufficiently small positive value. From (3.5), we know that assoon as the adjustment parameter a is not equal to zero, the system loses its stability.It should be noted that from bifurcation theory, we know that the loss of stabilitydoes not imply that the system is destroyed in the long run. A new equilibrium (ormultiple ones) may be bifurcated from the old equilibrium (a = 0).

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70 Wei-Bin Zhang

(4.1 )

Ifwe replace the values of 0< and {3 by a = (3 = 3/4 and use v, r, 6 and s with thesame values as in the previous example, a, are given by

al = 1/20 + [2c4 + 175(163)ap]/20{l6 + hc4)

a2 = [(5 - 7a)/100 + (c4 + 75(163)ap)/50(16 + hc4) + 3(163)apc4/(l6 + hc4)2]/8

a3 = a[3/8 - {c4 + 430(162)p}/10(16 + hc4) - 8c4(162)p/(16 + hc4)2]/400

In this case, a, is positive, but a2 and a3 may be either positive or negative. It can beseen that for an appropriately large a and a small p, a2 may be negative. Similarly, itis possible for a3 to be positive, or zero, or negative.

From these two examples, we may generally conclude that the system may beeither stable or unstable, dependent upon the parameter values. As these conditionsare too complicated to economically interpret their meaning, it is reasonable for usto investigate what will happen to the system if some assumptions on the parametervalues are accepted. In what follows, we examine the behaviour of the system underpresumed stability and instability, respectively.

4. Comparative statics analyses under presumed stabilityFrom bifurcation theory, we know that if the system is subjected to instability,

then small shifts in parameters may result in structural changes of the systembehaviour. Hence, comparative analysis may become very complicated in the caseof instabilities. For simplicity of analysis, in this section we assume that the system isalways stable, i.e. the Routh-Hurwitz criterion is satisfied.

4.1. Effects of the savings rateFirst, we investigate the effects of changes in the savings rate on economic

growth. From (3.2), we see that shifts in the equilibrium are given by

dK/ds = rC[(1 - w)s-W+ hC( I - 2w)s-2wj/p6 )

dN/ds = (r/pc)[(l - (3)z-fJ + h] dz/ds

dz/ds = -wCs-w-

1

where C = (c/v)I/(I-fJ)(6/vf > 0, and we omit '*' on K, Nand z. We see that anincrease in the savings rate will definitely decrease human quality and population inthe long term, but the effects on the population are uncertain. If I < w, i.e.I + 0<{3 < 20< + {3, then the population will be reduced as the savings rate isincreased. If I > 2w, i.e. I + 0<{3 > 30< + {3, then the population will be increased asthe savings rate is increased. But if w < 1 < 2w, the effects are much more difficult todetermine. If (I - w) > hC(2w - 1)s-W, then dK/ds > 0; otherwise, dK/ds < O. Ifweassume that {3 > I, then these conclusions tend to be the opposite. Hence, how thehuman quality enters into population growth dynamics has a significant role indetermining the effects of a shift in the savings rate.

It is easy to understand why the effects of changes in the savings rate on equilib­rium capital level are so complicated. It seems necessary to explain how it happensthat the population and its quality are definitely reduced by an increase in the savings rate.

We assume that at the initial point of time, the dynamic system (2.6) is at astationary state. Assume that some factors cause the population to increase the rateof the product. This will first cause the product F to increase, which will obviously

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Economic development with capital accumulation 71

result in increases in capital, human quality and the population at the initial stage ofthe disturbance. However, after a certain period of time, the 'negative' checkingforces on the population (z13N) in the population dynamic equation and the decreas­ing effects in learning (1/ (1 + hz)) will function to cut down the rate of change in thepopulation growth and human quality improvement. Hence, after a certain period oftime both the population and its quality may tend to decrease. Hence, the actualdynamics are due to different forces upon the system. At the new equilibrium, the popu­lation and its quality are reduced due to the net results of conflicting dynamic forces.

It is also significant to look at how the output is changed as the parameter s isshifted. It can be easily shown that dF/ds < 0 and d(F/ N)/ds < o. Hence, the totaloutput and the output per capita are reduced as the savings rate is increased.

From the analysis above, one can see that increases in the savings rate tend to beundesirable for society in the long term if the checking effects of human quality areweak (i.e. (3 is smaller than unity) and the system is stabilized.

In the Solow-Swan one-sector growth model an increase in the savings rate tendsto increase capital per capita (and hence the total output). We see that this conclu­sion may be invalid in a model where the population adapts to the system's actualconditions.

4.2. Effects of shifts in the depreciation rates

A shift in the depreciation rate of capital will cause the equilibrium to move in thefollowing way:

dK/d6 = (w - I - hz)K/6(1 + hz)

dN/d6 = (sw/pcsz 13)[1 - (3 + h(2 - (3)z] dz/d6

dz/d6 = wz/6 (4.2)

Hence, if the checking force of human quality is relatively weak (i.e. (3 < I), anincrease in the depreciation rate of capital will cause the population and its qualityto be increased in the long term, and vice versa. However, the effects on capitalaccumulation are dependent on the sign of (w - I - hz). If it is positive, then theequilibrium value of capital tends to increase as a result of faster depreciation ofcapital; if (w - I - hz) is negative, the equilibrium value of capital tends to decrease.Hence, a fast depreciation rate does not necessarily imply a reduced equilibrium levelof capital.

It is not difficult to see that the effect on the total output is given bydF/d6 = wK/(1 + hz). Hence, if the checking force of human quality is relativelyweak, fast capital depreciation makes the equilibrium output increase, irrespective ofthe direction of the effects on the capital accumulation.

4.3. Increased efficiency of institutions

We have interpreted the parameter v as a description of the efficiency of insti­tutions. It is important to examine its effects on the equilibrium.

We havedK/dv = (K/z) dz/dv + (hsrz/p6) dz/dv }

dN/dv = {r[1- (3 + (2 - (3)hzllz 13pc} dz/dv

dz/dv = -z/v(1 - (3) - wz]»

(4.3)

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72 Wei-Bin Zhang

It seems very important to note that increased efficiency of institutions does notnecessarily mean improvement in the society as a whole in our model. If the check­ing force of human quality is relatively weak in comparison with that of the produc­tion, then increased efficiency of institutions will reduce the population, its qualityand capital.

On the other hand, if the checking force of human quality on the populationgrowth is stronger than that of the output, then increased efficiency of institutionswill improve human quality and increase capital in long-term equilibrium. However,effects upon the equilibrium population are uncertain, dependent on the sign of1 - (3 + (2 - (3)hz. If it is positive, the population will be increased; while if it isnegative, the population will be reduced.

5. Complexity of instabilities: an illustration using the existence of economic cyclesIn § 3, we showed that the economic system may be either stable or unstable,

depending on the parameter values. The previous section analysed the effects ofchanged parameters upon the long-term behaviour of the system under presumedstability. It is natural to ask what will happen to the system if it is unstable.

From bifurcation theory, we know that loss of stability does not necessarily meanthe destruction of a dynamic system. Rather complicated behaviour such as multipleequilibria, oscillations and even chaos may result from instabilities. Unfortunately,there are no general methods for analysing the complexity of non-linear dynamicsystems. The methods for analysing non-linear phenomena depend on the character­istics of the system under consideration. As our system is potentially unstable withcomplicated non-linear terms, we cannot know all the possibilities of its behaviour.In what follows, we only provide an example oflong-term time-dependent behaviourby identifying the existence of limit cycles under certain conditions.

As traditional comparative statics only deals with the movement of time­independent equilibrium when parameters are changed, the effect of shifting a policyparameter on a variable is not changeable with time. We now know that suchconclusions are invalid if the system is located near a critical point (Zhang 1991).For instance, if the jacobian of the dynamic economic system evaluated in anequilibrium point has a pair of purely imaginary eigenvalues, then a small changein the parameter may result in time-dependent limit cycles (Hopf bifurcation). Thevalues of the variables on the cycle may be either greater or smaller than the values ofthe variables in the equilibrium, which are dependent on the phase of motion of thedynamic system.

To guarantee the existence of limit cycles, we first make the followingassumption.

Assumption 5.1

There are appropriate (economically meaningful) values of the parameters suchthat aj > 0 and ala2 = a3'

In what follows, we explain that this assumption is valid under certain con­ditions. As al is always positive, the conditions of the assumption are satisfied if,for instance, for r = 6, a = (3, the following equations hold:

a2 > 0, i.e. 6R + aa[6 + R/{l - a)l/N > ar(1 + 6 - a) (5.la)

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Economic development with capital accumulation

a3 > 0, i.e. I + R < (R - ci)/N + {3 + ci

al02 = a3, i.e. RR(8 + ao/(I - o)N) + a202(8 + R)/NN

= (20 - 8)oa82 + (I - 0)(1 - 8)8R + oa8(2a - {3 - I)/N

+(8+202/(I-o»a8R/N

73

(5.1 b)

(5.1 c)

Omitting all the terms associated with the parameter a and using the definition of R,we can rewrite (5.1 c) approximately in the following form: rz/(I + hz) =(I - 0)(1 - 28) where (c/v)I/(I-fJ)(8/sv)w. We see that it is reasonable to assumethe existence of appropriate values of the parameters such that (5.1 c) is valid.Hence, if the parameter a is sufficiently small, then (5.1 a) and (5.1 c) can be satisfiedby choosing appropriate parameter values.

To show the validity of (5.1 b), i.e. 1 + R + 02/N < 0

2 + {3 + R/N, we note thatR = (I - o)r + rz/(I + hz), N = (I + hz)/p(s/8)8, where 8 = 1/(1 - 0), v = I. Wesee that it is possible to make N appropriately large by choosing p large, and Rappropriately small by choosing r very small. In this case, the condition may besatisfied if 0

2 + {3 > I.The above discussion for the validity of (5.2) is based on the specified conditions,

o = {3, 8 = r and v = 1. The following results are not dependent on thesespecifications.

We now show that under Assumption 5.1, the jacobian has a pair of purelyimaginary eigenvalues, and the three eigenvalues are given by

8 8 ± . 1/ 2 ±'81 = -aI, 2,3 = 102 = I 0 (5.2)

where a; are defined in (3.4). This can be easily shown by comparing the parametersof the two sides of the equations 8 3 + al 8 2+ a28 + a3 = (8 + b, )(82+ b2) = O.

In the following bifurcation analysis, we select the population adjustment speed aas a bifurcation parameter. We denote by ao, the value of a that satisfies (5.2). Asmall shift in a from ao is expressed by 0', i.e.

O'=a-ao (5.3)

The eigenvalues are continuous functions of a. We denote by 8(0'), the eigenvaluethat equals i80 at 0' = 0 (i.e. a = ao). In the Appendix, 8 0 (0) is explicitly calculated.

Assumption 5.2

Assume that the real part of 8 u(0) is not equal to zero at a = ao.

The Appendix shows that this assumption is quite acceptable. We now presentour main result.

Theorem 5.1Assume that Assumptions 5.1 and 5.2 hold. Then, the system (2.6) has limit

cycles around the equilibrium (K', N', z'). The bifurcated cycles have period

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74 Wei-Bin Zhang

(5.5)

(5.4)

21r/M(x), and can be approximately expressed as

K(I, x) = K* + 2x{ YI\ cos (MI) - Y12 sin (MIn + O(x2) }

N(I, x) = N* + 2x{ Y21 cos (MI) - Y22 sin (MIn + O(x2)

z(l, x) = z" + 2x{ Y31 cos (MI) - Y32 sin (MIn + O(x2)

where x is the expansion amplitude parameter defined in the Appendix, Yij are realnumbers explicitly determined in the Appendix, and

<7(€) = a - ao = X2<72/2 + O(x4) }

M(x) = 8 0 + x2M2I2 + O(x4)

where <72 and M2 are constants given in the Appendix. When Re (8u ) > 0, if <72 > 0,the cycle is supercritically stable, while if <72 < 0, the cycle is unstable. WhenRe (8u ) < 0, if <72 is negative, the cycle is subcritically stable, while if <72 is posi­tive, the cycle is unstable.

The theorem is proved in the Appendix.In Theorem 5.1, we omit the higher order approximations since the expressions

are too complicated. Supercritical bifurcations mean that if the bifurcation par­ameter a is increased, the system is stabilized, while if it is decreased, the systembecomes unstable and bifurcations take place.

We now examine the behaviour of the system when the parameter a is shiftedfrom ao. For convenience, we limit our discussion to the case of supercritical bifur­cations. We look at the system (2.6). When the population adjustment speed a is lessthan ao, the system is stable. Hence, the discussion of the previous section is valid aslong as shifts in other parameters do not destabilize the system. When a is increasedsuch that a = ao, the system becomes 'neutral'. Its stability is sensitive to a smallchange in the parameter. As the parameter a is further increased, i.e. a > aQ, thesystem loses its stability. According to the Hopf bifurcation theorem, limit cyclesappear around the equilibrium. It is not a stationary state but permanent oscillationsthat characterize the dynamic behaviour of the system.

The existence of limit cycles is due to the fact that there are different 'competitiveforces' acting on each variable of the system. For instance, there are both negativeand positive 'checking forces' on the population growth as human quality isimproved and output is increased. This is similarly true for the dynamics of humanquality.

6. Concluding remarksWe have developed an economic growth model consisting of the dynamics of

capital, population and human quality. The results from the comparative staticsanalysis under presumed stability revealed that some conclusions from the standardneoclassical one-sector growth model (e.g. the Solow-Swan model) may be theopposite to the results from our model, dependent on the 'checking force' of humanquality on the population growth rate. This implies that it is rather dangerous toneglect population problems in dynamic economic analysis if the growth of thepopulation is dependent on the actual situations in an economic system. In asense, our model is important for developing economies, in. which populationgrowth is much affected by its economic structure and human quality.

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Economic development with capital accumulation 75

Although our model may be an appropriate description of certain economicphenomena, we cannot understand all possible behaviour of the system because itmay be unstable. From modern mathematics, we know that the behaviour ofa three­dimensional non-linear unstable dynamic system may be so complicated that one canhardly forecast its behaviour. In this study, we illustrate the complexity of dynamicbehaviour by guaranteeing the existence of limit cycles.

AppendixProof of Theorem 5.1

We apply the Hopfbifurcation theorem (Marsden and McCracken 1976) and thebifurcation method of Iooss and Joseph (1980) to prove the theorem. We firstidentify the conditions for Assumption 5.2 and then construct the limit cycles givenin the theorem.

As we are only concerned with the stability of equilibria and the local behaviourof the system, it is very convenient to express the system in the local form near theequilibrium. We introduce

v, = K - K', V2 = N - N', V 3 = z - z" (A I)

where (K, N, z) satisfies (2.6), and V (= [VI V2 V3IT ) are sufficiently small. Sub­stituting (A I) into (2.6) yields

dV/dt = JV + P(V, V) + O(IVI 3) (A2)

where A is the jacobian evaluated at equilibrium and P( V, V) is the quadratic term.J is given by

[

(0: - l)sF/K (I - o:)sF/N

J= o:aN/K o:a

o:rz/ K rz( I - 0:) / N

(I - o:)sF/z ]aN(1 - 0: - (3)/z

-o:r - hrz/(I + hz)

(A 3)

P( V, V) is not explicitly given since we rarely use them in the remainder of the study.Define the eigenvector Yand the adjoint eigenvector y' with the eigenvalue i80

by

JY= i8oY,

(Y, Y') = 0,

JT y ' = -i8oY'}

(Y,Y)=I(A4)

where (.,.) is the product operator in C3. The following vectors satisfy (A4):

Y, = Yll + iY12 = K/ao:zN[1 + (1- 0: - (3)z/{i8 + ZW'Y2 = Y2, + Y22 = l/z[i8 - ao: - ar(l - 0: - (3)(1 - 0:)/{i8 + ZW'Y3 = Y3, + Y32 = rz[o:Y1/ K + (I - 0:) Yz/NJ/{i8 + Z)

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76 Wei-Bin Zhang

Yt = [Y1 + Y2{((1 - a)sF - i8K)/a - rzG }/aN + G]

Yi = [((I - a)sF - i8K)/a - rzG] YUaN

Yj = GYt, Z = or + hrz/(I + hz)

G = [(I - a)sF + (i8 + aa)((1 - a)sF + i8K)/aa]/[rz(i8 + aos]« - rz(1 - a)]

(A 5)

which are evaluated at (K', N', z", ao). In (A 5), obviously, Yij are real numbers.Thus, we can easily calculate Y and Y', although their expressions are verycomplicated.

We now identify the condition for loss of stability. According to Iooss andJoseph (1980), we know that

in which

[

0 0 0 ]Jq(O) = aN/K a N(I - a - (3)/z

o 0 0

(A6)

(A 7)

It is easy to calculate 8 q(0) according to (A 5) and (A 6). Moreover, one can see thatAssumption 5.2 is acceptable.

We have shown that if Assumptions 5.1 and 5.2 are satisfied, according to theHopf bifurcation theorem, for a small a there are limit cycles around the equilibrium(K', N', z'). In what follows, we construct the cycles given in the theorem.

We introduce two linear operators Jo and Join Pz" space

Jo = -80 d/dm + J, Jo= 8 0 d/dm + JT (A 8)

where m = 8 01. According to Iooss and Joseph, the bifurcated cycle can be con­structed in a series of powers of amplitude x defined by x = [V, exp (im) Y'], where[.,.J is the product operator in Ph. The solution is constructed in the form of thefollowing series:

[v(m,x)] [vn(m)]

a(x) = fxn/n! anM(x) ,=1 M;

where an and M; are parameters to be determined and

V(m, x) = V(m + 27r), V(m,O) = 0 }

m = M(X)I, M(O) = 8 0 , <r(0) = 0

(A 9)

(A 10)

As x is very small, it is sufficient to determine the coefficients with respect to the loworders of x. According to Iooss and Joseph (1980), we have

a2n-1 =M2n_ 1 =0, n= 1,2, ... (A 11)

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Economic development with capital accumulation 77

(A 12)

Vi = Y exp (-iMt) + Y exp (iMt)

[

YII cos (Mt) - Yl2 sin (Mt)]

= 2 Y21 cos (Mt) - Y22 sin (Mt)

Y31 cos (MI) - Y32 sin (MI)

From (A 9) and (A 12), we obtain the expression of the cycles in the theorem. V 2 aredetermined as a solution of the following differential equation: JoV 2 = - P( V I, V I),in which V I are known functions of time. Moreover, a2 and M2 are determined by

ia2 - eu M2= [P( Vi, V I), Y' exp (iMt)J

= PI +iP2

from which we obtain

a2 = -P,/Re (eu), M2 = a2 Irn (eu) + P2

To identify the stability conditions, introduce a real number

Q(x) = -x2a2 Re (eu ) + O(x4)

(A 13)

(A 14)

(A 15)

Then, according to the factorization theorem ofIooss and Joseph (1980), we knowthat if Q(x) is negative, the bifurcated cycle is stable; while if Q(x) is positive, thenthe cycle is unstable. Thus all the results of the theorem have been proved. 0

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