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Mathematical and Computer Modelling
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Human capital externality and chaotic
equilibrium dynamics
Hung-Ju Chen a & Ming-Chia Li b a
Department of Economics , National Taiwan University , Taipei, Taiwan
b
Department of Applied Mathematics , National Chiao Tung University , Hsinchu, Taiwan
Published online: 10 Oct 2008.
To cite this article: Hung-Ju Chen & Ming-Chia Li (2008) Human capital externality and chaotic
equilibrium dynamics, Mathematical and Computer Modelling of Dynamical Systems: Methods, Tools and Applications in Engineering and Related Sciences, 14:6, 571-586
To link to this article: http://dx.doi.org/10.1080/13873950802308901
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Human capital externality and chaotic equilibrium dynamics
Hung-Ju Chena* and Ming-Chia Lib
a
Department of Economics, National Taiwan University, Taipei, Taiwan;bDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan
(Received 23 May 2007; final version received 16 June 2008)
This study develops a two-period overlapping generations model in which adults undertake educational investment decisions on behalf of young agents. In addition to educational investment, we argue that the accumulation of human capital is also dependent upon the externality from average human capital within the economy. In a departure from the previous literature in this area, we assume that there is a reduction in the overall productivity of human capital accumulation brought about by human capital externality, and show that complicated dynamics will emerge under this circumstance. In addition to displaying the chaotic dynamics in the sense of Li and Yorke, we also verify the existence of Devaney’s chaos and Smale’s chaos.
Keywords: chaotic dynamics; externality; human capital accumulation
1. Introduction
There has been increasing interest over recent years surrounding the study of chaotic behaviour during the overall process of economic development. Utilizing a standard neoclassical model with capital accumulation in order to investigate the possibility of complicated dynamics, Day [1,2] showed that chaotic trajectories would emerge under certain conditions on savings and productivity, and indeed, chaotic motion is an important element in the study of economic development because it suggests that future economic performance cannot be predicted from a prior developmental pattern. As such, the tiniest of differences between two initial conditions will result in very different trajectories.
Following Day’s consideration of a model with a negative capital externality in [1,2],1 Boldrin et al. [3] subsequently went on to develop a two-sector endogenous growth model with positive capital externality which demonstrated that chaotic equilibrium will exist within such an environment. Adopting the model in [3] as an example, Mitra [4] provided a sufficient condition for topological chaos which is applicable to endogenous models when the Li–Yorke criterion in [5] is not satisfied.
Although the growth literature relating to chaotic dynamics has tended to focus on models with capital accumulation, Lucas [6] and Becker et al. [7] argued that human capital also plays an important role in economic growth. Our aim in this article is therefore to develop a model with human capital accumulation within which complex behaviour will be an inherent factor. We develop a two-period overlapping generations (OLG) model in
*Corresponding author. Email: [email protected] Vol. 14, No. 6, December 2008, 571–586
ISSN 1387-3954 print/ISSN 1744-5051 online Ó 2008 Taylor & Francis
DOI: 10.1080/13873950802308901 http://www.informaworld.com
which adults make educational investment decisions on behalf of young agents, and argue that in addition to educational investment, the accumulation of human capital is also dependent upon the externality from average human capital within the economy. The average human capital here refers to the common knowledge and information that exists within the economy. Notice that we do not distinguish between terminologies relating to knowledge, information and human capital in this article; however, Dasgupta and David [8] noted that knowledge is the product of research, and that information is the codification of such knowledge.
Galor and Tsiddon [9] set up an accumulation function of human capital with (local) home environment and (global) technological externalities in order to study the linkage between economic growth and income distribution. Their work captured local externalities in the form of educational investment and parental human capital, whereas the global externality was represented by the average human capital within the economy. Positive global externality was assumed by introducing a non-decreasing function of average human capital. Such a positive externality, from average human capital to human capital accumulation, was also used by De la Croix and Doepke [10,11] in order to study the role of differential fertility during the overall process of economic growth.2
Although a high level of average human capital can contribute to the overall accumulation of human capital, we argue that it can also cause a reduction in the level of productivity of human capital accumulation. Stephan [12, Subsection 9A] argued that in discovery, excessive knowledge is a bad thing because it ‘encumbers’ researchers. Moreover, when there is an increase in the average human capital, people need to spend more time on examining and digesting the knowledge or information they receive. Since the invention of the internet, people nowadays can get information and knowledge easily and even instantly. However, too much information and knowledge may cause the problem of ‘information pollution’.3Also, the wrong information and knowledge may be widespread through the internet and will lower the individual’s human capital if one accepts the wrong information or knowledge.4Hence, in contrast to the previous literature on endogenous growth theory, we assume that with a rise in average human capital, there will be a corresponding increase in the overall degradation of human capital accumulation productivity.
Numerous works have concentrated on the study of the complicated dynamics that are present in OLG models. Based upon the assumption that children inherit their consumption tastes from their parents, De la Croix [16] showed that this would generate endogenous oscillations. Nishimura and Shimomura [17] extended the model in [16] to a trade model and went on to argue that such child–parent externalities, along with international trade, will together generate chaos. Chen and Li [18] and Chen et al. [19] showed that for OLG models, expectation formation is an important determinant to the occurrence of chaos. Medio and Negroni [20] examined the complicated dynamics that occurred in a two-dimensional OLG model with production function, whereas Yokoo [21] subsequently proposed a two-dimensional OLG model with government debt.
When analysing the one-dimensional dynamical system, the Li–Yorke criterion has been the general focus of the literature studying chaotic motion in economic models. Day [1] and Boldrin et al. [3] examined Li–Yorke chaos in growth models. Complex dynamics can also easily arise in a monetary economy. Auray et al. [22] also used the Li–Yorke criterion to examine the presence of chaotic motion in a ‘cash-in-advance’ model with habit persistence based upon catching up with the Joneses literature. A further contribution of this article is that, in addition to demonstrating chaotic dynamics in the sense of Li and Yorke, we also use the first-order nonlinear difference equation in human
H.-J. Chen and M.-C. Li
capital, generalized by our model, to verify the presence of Devaney’s chaos and Smale’s chaos.
The remainder of this article is organized as follows. The next section presents a simple model with human capital accumulation. We then derive the law of motion of human capital in an economy occupied by homogeneous agents. In Section 3, we show that with human capital externality, the chaotic equilibrium dynamics in the sense of Li and Yorke will present. Besides the Li–Yorke chaos, we also illustrate the presence of Devaney’s and Smale’s chaos in Section 4. A numerical example is also given in this section. The final section provides the conclusions drawn from this study.
2. The model
We adopt an infinite-horizon, discrete time OLG model within which agents live for two periods, corresponding to childhood (young agents) and adulthood (old agents). Each adult gives birth to a single child, there is no population growth, and we normalize the population size to one. Earnings for an adult are equal to his/her human capital, ht.
2.1. Schools
We assume that parents make educational investment decisions qtfor their children,5that
is they decide how much tuition they want to pay. Moreover, we make the following ‘assumptions of schools’ (AS):
(AS1) For any level of educational expenditure chosen by a parent, there always exists a school charging the tuition at the same amount to accept his/her child.
(AS2) A school will charge the same tuition fees for all types of students. (AS3) Schools earn zero profit.
(AS4) School quality is measured by its expenditure per student.
(AS1) implies that every young agent has a school to attend. (AS2) indicates that there is no price discrimination amongst students within a school; therefore, private schools can be perfectly segregated by their tuition fees.6(AS3) along with (AS4) illustrate that school quality can be represented by its tuition.
2.2. Human capital accumulation function
What makes this article different from the previous literature of human capital is the accumulation function of the human capital. The key to the model is that there is a negative externality of the average human capital when forming human capital accumulation. The justification given for this is the remark of Stephan [12, p. 1220] in discovery:
Fourth, there is anecdotal evidence that ‘too’ much knowledge can be a bad thing in discovery in the sense that it ‘encumbers’ the researcher. There is the suggestion, for example, that exceptional research may at times be done by the young because the young ‘know’ less than their elders and hence are less encumbered in their choice of problems and in the way they approach a questions.
Besides, a negative externality of the average human capital may arise when some of knowledge/information is wrong. One good example is the problem of the information
quality on the internet. The invention of the internet provides another way for agents to exchange information and to accumulate human capital. However, the discovery of the internet makes users overwhelmed with information. When there is too much information, agents need to spend lots of time on digesting it, not to mention that some of the information might be wrong and the wrong information would lower the human capital accumulation. During his interview by the BBC [14], Jakob Nielsen mentioned that:
the entire ideology of information technology for the last 50 years has been that more information is better, that mass producing information is better. But the net is now so much a machine with all the answers instantly, it has mutated into ‘procrastination apparatus’ which spews information without much prioritisation.
He defined ‘information pollution as information overload taken to the extreme’ and likened it to pollution in the physical environment. He also argued that ‘information pollution can become an impediment to your ability to get your work done’. This indicates that information pollution may reduce working productivity.
We then incorporate the ideas of Stephan [12] and Nielsen [14] when forming the human capital accumulation function. One can think that there is a saturation level of the average human capital representing the situation in which there is so much knowledge/ information within the economy that people cannot distinguish between right and wrong information/knowledge and hence an increase in the average human capital contributes nothing to (or even reduces) the accumulation of human capital.
Following the literature, we assume that human capital is accumulated according to a Cobb–Douglas learning technology:7
htþ1¼ AqtZHtdðm HtÞb; ð2:1Þ
where A 4 0 is the productivity of human capital accumulation, qt is the educational
investment, Htis the average human capital for the society, m 4 0 is the saturation level of
Ht. The parameters Z, d, b2 [0,1] are the corresponding elasticity of qt, Htand (m 7 Ht)
to future human capital.
Equation (2.1) is a variation of the traditional human capital accumulation function by including the externality term. As we have explained in the introduction, there is a saturation level of the average human capital Htin economy, which is represented by m.
When Htis much less than m, we have that higher average human capital is beneficial for
the accumulation of human capital by the factor Hd
t. By contrast, when Htis close to m,
the benefit no longer exists. The factor (m 7 Ht)b represents the impact of reduction
in the productivity of human capital accumulation caused by excessive information. As Ht approaches m, this factor approaches zero. Thus, as Ht becomes larger the human
capital htþ1 accumulates at a lower rate. If Ht is large, the human capital declines. We
restrict all factors devoted to the accumulation of human capital to exhibit diminishing or constant returns to scale.
2.3. The maximization problem for households
We assume parents care about their consumption ct and their children’s human capital
htþ1. All agents have the same utility function over their life cycle, which is:
ln ctþ o ln htþ1; ð2:2Þ
where o 4 0 represents preference towards human capital. H.-J. Chen and M.-C. Li
Parents need to decide how to allocate their income between consumption and educational investment for their children. Hence, the budget constraint for adults is
ctþ qt ¼ ht: ð2:3Þ
2.4. Equilibrium
Given h1, an equilibrium comprises sequences of individual human capital stock {ht}t1,
the average human capital stock {Ht}t1, and individual decisions {ct,qt}t1such that:
(1) the household maximization problem is solved by {ct,qt}t1, maximizing the utility
function subject to Equations (2.1) and (2.3); and
(2) young agents will accumulate human capital following Equation (2.1).
2.5. The law of motion of human capital
It is easy to see that for the maximization problem, the optimal choice of educational investment that parents choose for their children is
qt¼
Zo
1þ Zoht: ð2:4Þ Equation (2.4) shows that investment in education increases along with any increase in parental human capital and is a linear function of ht. By substituting Equation (2.4) within
the human capital accumulation function of Equation (2.1), we have the law of motion of human capital: htþ1¼ A Zo 1þ Zo Z hZtH d tðm HtÞ b : ð2:5Þ
Equation (2.5) indicates that the human capital accumulation function is an increasing function of parental human capital, all other things being held constant. Under an economy occupied with homogeneous agents, the law of motion of human capital becomes
Htþ1¼ A
Zo 1þ Zo Z
HZþdt ðm HtÞb: ð2:6Þ
Equation (2.6) shows that the economy can be represented by a one-dimensional dynamical system in human capital.
3. Chaotic dynamics: Li–Yorke chaos
In this section, we study the dynamical behaviour of the average human capital based on the law of motion of human capital (2.6). Without loss of generality, we may assume that
m¼ 1. Let l ¼ A1þZoZo Z, a¼ Z þ d , and x ¼ Ht, then the model (2.6) turns into the
family of functions fl,a,b: [0,1] ! R defined by
fl;a;bðxÞ ¼ lxað1 xÞb; ð3:1Þ
where l 4 0, 0 5 a 2 and 0 5 b 1 are real parameters. For simplicity, we write f¼ fl,a,b, denote the identity function by f0, and inductively define fn¼ f fn71 for
positive integer n.
Figure 1 indicates that the dynamic behaviour of Equation (3.1) with a¼ 2 and b ¼ 1 varies from simple dynamics to chaotic dynamics as l increases.
Before proving the existence of chaotic dynamics, we give elementary analysis on the model (3.1). By calculating the derivative f0, we find that the maximum of f occurs at the
critical point a
aþb. Moreover, f is strictly increasing on
0; a
aþb
and strictly decreasing on a aþb;1 . Hence, f a aþb
1 if and only if f ([0,1]) \ [0,1]. By computing the second derivative f00, one has that f0 is strictly increasing on [0,a] for some 0 5 a 5 1 and is
strictly decreasing on [a,1] if aþ b – 1 4 0, and f0 is strictly decreasing on [0,1] otherwise. By the intermediate value theorem, f a
aþb
> a
aþb implies that there exist
0 < p<aþba < p <1 such that
fðpÞ ¼ fðpÞ ¼ p: ð3:2Þ
3.1. Li–Yorke chaos
Following the article of Li and Yorke [5], we define the Li–Yorke chaos.
Definition 3.1: Let h : I! I be a map, where I is an interval. We say that h exhibits Li–Yorke chaoson I if:
Figure 1. The bifurcation diagram of fl,2,1(x) in l.
H.-J. Chen and M.-C. Li
(1) h has periodic points of all periods; here by a periodic point p of period n, we mean that fn(p) ¼ p and fi(p) 6¼ p for 0 5 i 5 n;
(2) there exists an uncountable set S \ I such that
(i) if x, x, y2 S with x 6¼ y then
lim sup
n!1
jhnðxÞ hnðyÞj > 0 and lim inf
n!1 jh
n
ðxÞ hnðyÞj ¼ 0;
(ii) if x2 S and y 2 I is periodic then
lim sup
n!1
jhnðxÞ hnðyÞj > 0:
We recall some related theorems. The Li–Yorke Theorem in [5, Theorem 1] says that any continuous map on an interval with a periodic point of period three exhibits Li–Yorke chaos. A periodic point of period one is also called a fixed point. The existence of a fixed point is guaranteed by the well-known fixed point theorem: if I is a closed interval, h : I! R is a continuous function, and h(I) [ I, then f has a fixed point in I.
By using the theorems mentioned above, we establish the existence of Li–Yorke chaos for our model for the case when the maximum of f is equal to one.
Theorem 3.2: Let f¼ fl,a,b be given by Equation (3.1). If f
a aþb
¼ 1, then f has periodic orbits of all periods and exhibits Li–Yorke chaos on[0,1].
Proof: Let I1¼ 0;aþba and I2¼ a aþb;1 . Because f (0)¼ 0, f (1) ¼ 0, and f a aþb ¼ 1, f(I1) [ I1[ I2and f(I2) [ I1[ I2. Because f(I1) [ I1, there is a closed subinterval A1of
I1such that f (A1)¼ I1. Because f (I2) [ I1 [ A1, there is a closed subinterval A2of I2such
that f (A2)¼ A1. Again, because f (I1) [ I2 [ A2, there is a closed subinterval A3of I1such
that f (A3)¼ A2. Hence f3(A3)¼ f2(A2)¼ f (A1)¼ I1 [ A3. Because f3 is continuous,
the fixed point theorem implies that f3has a fixed point, namely z, in A3. Then f(z) 2 A2
and f3(z)¼ z. Because the common point of I1 and I2 is aþba and f3
a
aþb
¼ 0, z 6¼ a aþb.
Therefore, z is a periodic point of period three for f. By the Li–Yorke Theorem, f exhibits
Li–Yorke chaos. ¤
Before the maximum faþba attains the number one, we can have that f2 exhibits Li–Yorke chaos. Although similar results can be found in [4, Proposition 2.3], our methodology is very different from his. We use the method of interval covering to prove the existence of Li–Yorke chaos for f2.
Theorem 3.3: Let f¼ fl,a,b be given by Equation (3.1), where l, a, b satisfy f
a
aþb
>aþba , and let p_be given by Equation(3.2). If f2
a
aþb
p, then f has periodic orbits of all even
periods on[0,1] and f2exhibits Li–Yorke chaos on [0,1].
Proof: Let I1¼ p;aþba and I2¼ a aþb; p . Because f2(p7)¼ f2(p)¼ p and f2 a aþb p,
the continuity of f2 implies that f2(I1) [ I1[ I2 and f2(I2) [ I1[ I2. By the same
argument as in the proof of Theorem 3.2, f6 has a fixed point, namely z, in I1,
and f2(z)2 I2. Because f2
a
aþb
p, z6¼aþba . Therefore, the point z is a periodic
point of period three for f2and of period six for f. By the Li–Yorke Theorem, f2exhibits
Li–Yorke chaos. ¤
One can see from Figure 2 that the model fl,a,bin Equation (3.1), with l¼ 6.5, a ¼ 2
and b¼ 1, satisfies the conditions of Theorem 3.3.
If the dynamics of Equation (2.6) exhibits Li–Yorke chaos, then irregular cycles will emerge with the development of economics. Hence, unlike [1] and [3], which concentrated on the possibility of endogenous fluctuations in an economy with capital accumulation, our result enriches this line of studying by showing that it is also likely to obtain Li–Yorke chaos in an economy with human capital accumulation.
4. Other types of chaos
Besides verifying the possibility of the presence of chaos in the sense of Li and Yorke when there is negative externality of human capital, in this section we show that this nonlinear first-order difference equation in human capital can also exhibit Devaney’s and Smale’s chaos under certain conditions. Although most theoretical studies of complex dynamics in economic models tend to focus on the examination of Li–Yorke chaos due to the mathematical convenience, we provide the other two alternative considerations of chaos. A numerical example is given at the end of the section.
4.1. Devaney’s chaos
In his popular textbook, Devaney [29] gives the following definition for chaos.
Figure 2. The graph of f6.5,2,1(x).
H.-J. Chen and M.-C. Li
Definition 4.1: Let h : I! I be a map, where I is a closed interval. We say that h exhibits Devaney’s chaos on I if the following conditions are satisfied:
(1) the set of periodic points is dense in I;
(2) the map h is topologically transitive, i.e. for any given pair of nonempty open sets Uand V in I, there is a positive integer n such that fn(U)\ V 6¼ ø; and
(3) the map h has sensitive dependence on initial conditions, i.e. there exists a 4 0 such that for any x2 I and any E 4 0, there are y 2 I and n 2 N such thatjx – yj 5 E and j hn(x) – hn(y)j 4 a.
We also need the following definition. For a C3map h : I! I, where I is an interval, the Schwarzian derivative of h is defined by
ShðxÞ ¼ h000ðxÞ h0ðxÞ 3 2 h00ðxÞ h0ðxÞ 2
for x2 I with h0(x)6¼ 0. By using the chain rule, one has that Sh5 0 implies Sh2 <0.
Thus, we have the following property that
if Sh<0; then Shn<0 for all n 1: ð4:1Þ
Moreover, Sh5 0 implies that h0cannot have a positive local minimum or a negative local
maximum. Indeed, if c is a critical point of h0, then h000ðcÞ=h0ðcÞ ¼ S
hðcÞ < 0 and hence
h000(c) and h0(c) have opposite signs. Therefore, by continuity of h0, we have that if h0 6¼ 0 and Sh5 0 on [a, b] then for any x2 (a,b),
either h0ðxÞ > min fh0ðaÞ; h0ðbÞg > 0 or h0ðxÞ < max fh0ðaÞ; h0ðbÞg < 0: ð4:2Þ
Return to our study on the model f¼ fl,a,b in Equation (3.1). Assume a 1 and
f a aþb
¼ 1. Then there are p7 5 p as defined in Equation (3.2). Because f0 is strictly
decreasing on [0,1], the mean value theorem implies that 0 and p are the only fixed points of f. Because f2(p7)¼ f2(p)¼ p and f2
a
aþb
¼ 0 < p, the intermediate value theorem
implies that there exist p75 ‘15 r15 p such that
f2ð‘1Þ ¼ f2ðr1Þ ¼ p: ð4:3Þ
In fact, such a pair ‘1and r1satisfying Equation (4.3) is unique due to the monotonicity of
fonp;aþba and a aþb;1 .
For the case when the maximum attains one, we have shown that the model (3.1) exhibits the Li–Yorke chaos in Theorem 3.2. Furthermore, Devaney’s chaos may exist.
Theorem 4.2: Let f¼ fl,a,bbe given by Equation (3.1) with a 1 and f
a
aþb
¼ 1 and let p7 5 ‘15 r15 p be as in Equations (3.2) and (4.3). If min{j f0(p7)j, j f0(p)j} 4 1,
maxr1aþba ;aþba ‘1
< pand the Schwarzian derivative Sf(x) 5 0 for x2 [0, 1], then f
exhibits Devaney’s chaos on[0,1].
Proof: Let J¼ ½p; pn
a
aþb
. For x2 J, define t(x) ¼ min{n 2 N : fn(x)2 [p7,p]}. Then
t(x) is well defined. Indeed, let x2 J then f(x) 2 [p, 1) and so f2(x)2 (0, p]. Because f(y) 4 y for all y2 (0, p7), there exists a positive integer n such that fn(x)2 [p7, p].
First, we claim jðftðxÞÞ0ðxÞj > 1 for all x 2 J: ð4:4Þ For n 1, let In ¼ x2 a aþb; p :tðxÞ ¼ n and ^In¼ x2p;aþba :tðxÞ ¼ ng. Then J¼ [1
n¼1ðIn[ ^InÞ, I1¼ {p}, Iˆ1¼ {p7}, andj f0(x)j 4 1 for x 2 I1[ Iˆ1. Consider n 2.
The continuity of f implies that In¼ [rn72, rn71) and Iˆn¼ (‘n72, ‘n71] for some
p7 ¼ ‘0 ‘n725 ‘n715 rn725 rn71 r0¼ p. It is easy to check that fn(rn72)¼ p7,
fn(rn71)¼ p, and fnmaps
a
aþb; rn2
and [rn72, rn71] homeomorphically onto [0, p7] and
[p7, p], respectively. By the mean value theorem applied to fnon
a
aþb; rn2
and [rn72,
rn71], respectively, one gets that there exist yn2
a aþb; rn2 and zn2 (rn72, rn71) such that ð fnÞ0 ðynÞ ¼rn2paþba and ðfnÞ0 ðznÞ ¼rn1pprn2. Because a aþb; rn2 a aþb; r1 , r1aþba < p and (rn72, rn71)Ü (p7,p), we haveð fnÞ0ðynÞ r1paþba
>1 and (fn)0(zn) 4 1.
Because Sf5 0, by (4.1) we have Sfn<0 and hence by Equation (4.2) applied to fn, we
obtain (fn)0(rn72) 4 min{( fn)0(yn), ( fn)0(zn)} 4 1. Because r0¼ p we have (f2)0(r0) 4 1.
Thus, inductively we have that for n 3,
ðfnÞ0ðrn1Þ ¼ f0ðfn1ðrn1ÞÞ ðfn1Þ0ðrn1Þ ¼ f0ðpÞ ðfn1Þ0ðrn1Þ > 1:
By Equation (4.2) again, we get that (fn)0(x) 4 min {(fn)0(rn72), (f n
)0(rn71)} 4 1 for all
x2 In¼ [rn72, rn71). By using the same argument as above, we have that (f n
)0(x) 5 –1 for all x2 ^In¼ ð‘n2; ‘n1. The desired claim follows.
Second, we claim that for every x2 [0,1] whose orbit does not go through a aþb, there
exists a positive integer nxsuch that
jðfnxÞ0ðxÞj > 1: ð4:5Þ
For x2 J, claim (4.5) follows Equation (4.4) by taking nx¼ t(x). Next, we consider
x2 [0,p7]. Because Sf5 0 and f0 4 0 on [0, p7], by Equation (4.2) we have that
f0(x) min {f0 (0), f0 (p
7)} 4 1 for all x2 [0,p7]. Thus, Equation (4.5) holds for
x2 [0,p7] by taking nx¼ 1. Finally, consider x 2 [p,1]. Then f (x) 2 [0,p] and so the
above result implies that jðfmxÞ0ðfðxÞÞj > 1 for some integer m
x 1. Because Sf5 0
and f05 0 on [p, 1], by Equation (4.2) we have f0(x) max{f0(p), f0(1)} 5 –1.
Thus jðfmxþ1Þ0ðxÞj ¼ jðfmxÞ0ðfðxÞÞ f0ðxÞj > 1. Therefore, Equation (4.5) holds by taking
nx ¼ mxþ 1. We have finished the proof of the claim.
Third, we claim that for any nonempty open set U \ [0,1], there exists a positive integer
nsuch that
fnðUÞ ½0; 1: ð4:6Þ Let U be an interval in [0,1]. Because f (x) 4 x for x2 (0,p7] and f ([p,1)) \ (0,p], there are
a positive integer n and a subinterval U0 \ U such that fn(U0) \ J. For convenience, we
denote R(x)¼ ft (x) (x) for x 2 J. The claim (4.4) says that R expands the lengths of
H.-J. Chen and M.-C. Li
intervals in J and hence there exists an integer k 4 0 and a subinterval V0 \ fn(U0) such
that Rk (V0) contains a discontinuity point of R. Thus, there exists m 4 0 such that
p2 fm(V0). Now it remains to prove that fmþ ‘ (V0)¼ [0,1] for some ‘ 4 0. Because
f maps [p, 1] homeomorphically onto [0, p], there exists a unique d2 [p,1] such that fðdÞ ¼ a aþb. Then f mþ2^‘ðV 0Þ a aþb; d
for some ^‘ >0. Indeed, becausej (f2)0(y)j ¼ j
( ft(x))0(x)j 4 1 for x2 [r1,p] and f2ðxÞ ¼ fðfðxÞÞ fðfðr1ÞÞ ¼ p<aþba < x for
x2 a aþb; r1
, we have that f2(x) 5 x for all x2 a aþb; p
and hence f2(x) 4 x for all x2 (p,d ]. Thus, there exists ^‘ >0 such that fmþ2^‘ðV
0Þ a aþb; d . Because f2aþba ; d¼ f a aþb;1 ¼ ½0; 1, fmþ2^‘þ2ðV 0Þ f2 a aþb; d
¼ ½0; 1. The proof of the desired claim is complete.
Finally, we are in position to obtain the three properties of Devaney’s chaos. Let U be any nonempty open interval in [0,1]. Then there exist a nonempty open interval V and a closed interval W such that V \ W \ U. By claim (4.6), there exists a positive integer n
such that fn(V) \ [0,1] and hence fn(W) [ W. By the fixed point theorem, fnhas a fixed point in W. Therefore, f has a periodic point in W and so in U. We have proved that the set of periodic points is dense in [0,1]. The claim (4.6) immediately implies that f is topologically transitive. For sensitive dependence of f, we take Z¼1
4. Let x2 [0,1] and
E40 be arbitrary. Take U to be the interval x; xþE 2 or xE 2; x provided it is well defined. By claim (4.6), we have fn(U) [ [0,1]. Thus, there exists y 2 U such that
jfnðxÞ fnðyÞj >1
4¼ Z. The proof of the theorem is complete. ¤
Now we consider the case when a 4 1 and faþba >aþba . Let p75 p be as in
Equation (3.2). Then f0(0)¼ 0 and hence there exists a unique point, namely q, in (0, p7)
such that
fðqÞ ¼ q: ð4:7Þ
Similar to Equation (4.3), we have that if f2 a aþb
q, then there exist p75 ‘15 r15 p
such that
f2ð‘1Þ ¼ f2ðr1Þ ¼ p: ð4:8Þ
Moreover, if f2 a aþb
< q, the intermediate value theorem implies that there exist p7 5 ‘15 q75 qþ5 r15 p such that
f2ðqÞ ¼ f2ðqþÞ ¼ q: ð4:9Þ
Under the condition f2aþba ¼ q, Theorem 3.3 says that f2exhibits Li–Yorke chaos.
In fact, the existence of Devaney’s chaos is also possible.
Theorem 4.3: Let f¼ fl,a,bbe given by Equation(3.1) with a 4 1 and f2
a aþb
¼ q, where q is in Equation (4.7), and let ‘1, r1 as in Equation (4.8). If min {j f0(q)j, j f0(p) j} 4 1,
maxr1aþba ;aþba ‘1
< p q and the Schwarzian derivative Sf(x) 5 0 for
x2q; f a aþb
, then f exhibits Devaney’s chaos onq; f a aþb
.
The proof is similar to the one for Theorem 4.2. We omit it here.
If Devaney’s chaotic motion presents, then the initial condition will be an important determinant to the future development for the economy. This result contradicts the traditional Solow growth model because two economies which only differ from each other
at initial conditions will behave very differently not only in the short run but also in the long run.
4.2. Smale’s chaos
Based on the pioneering article by Smale [30] in dynamical systems, one can define Smale’s chaos as follows (see also [31]).
Definition 4.4: Let h : R! R be a map and L be a subset of R. We say that h exhibits Smale’s chaos on L if there exist an integer N 4 1 and a function j : L! SN, where
SN¼ fs0s1s2 j si¼ 1; 2; , or N for all i 0}, such that j is continuous and one to one
from L onto SN, its inverse j71is continuous, and for any x 2 L,
jðhðxÞÞ ¼ sðjðxÞÞ;
where s is the shift map on SNdefined by
sðs0s1s2 Þ ¼ s1s2s3 :
For the case when f2 a aþb
< q, our model can exhibit Smale’s chaos.
Theorem 4.5: Let f¼ fl,a,b be given by Equation(3.1) with a 4 1 and f2
a aþb
< q, where q is in Equation (4.7), and let p75 ‘15 q75 qþ5 r1 as in Equations (4.8) and
(4.9). Let
L¼ fx 2 ½q; fðqþÞ : fnðxÞ 2 ½q; fðqþÞ for all n 0g:
If one of the following holds:
(H1) min{j f0(q7)j, j f0(qþ)j} 4 1;
(H2) max{r1 – qþ, q7 – ‘1} 5 p7 – q, f0(p) 5 –1 and the Schwarzian derivative
Sf(x) 5 0 for x2 [q, f(qþ)],
then:
(1) the set L is invariant under f and is a Cantor set (i.e. closed, bounded, totally connected, and perfect);
(2) the map f has periodic points of all periods in L; (3) the map f exhibits Smale’s chaos on L; and
(4) every orbit with an initial point in [0,1]\L converges to the origin.
Proof: Consider the case when (H1) holds. For item (1), first we show that the invariance f(L)¼ L. It follows immediately from the definition of L that f (L) \ L. We prove L \ f (L) by contraction. Let x 2 L. Because f ([q, q7])¼ [q, f(qþ)], there exists y 2 [q,q7]
such that f (y)¼ x. Suppose y 62 L. Then there is m 1 such that fm(y) 62 [q, f(qþ)] and so
fm71(x)¼ fm71(f(y))¼ fm(y)62 L. This contradicts the fact that f (L) \ L. Second, we show that L is compact. Because L \ [0,1], L is bounded. Let J0¼ [0,1] and inductively
H.-J. Chen and M.-C. Li
define Jn¼ {x 2 [0,1] : f(x) 2 Jn71} for n 1. By the definition of L, we have
L¼ \1n¼0Jn. Because f is continuous, Jnis closed for all n 0. Hence, L is closed and
so is compact.
Next, we claim that there exists l 4 1 such that for any x 2 L \ [q, f(qþ)],
jf0ðxÞj l: ð4:10Þ
Because 0, q, p are fixed points for f, the mean value theorem implies that there are 0 5 a 5 q 5 b 5 p such that f0(a) ¼ f0(b) ¼ 1. Because f00has at most one root on (0,1),
(H1) implies that f0(x) 4 1 for all x2 [q,q7] and f0(x) 5 –1 for all x2 [qþ, f (qþ)]. The
desired claim follows from the continuity of f0.
Now we prove that L is totally disconnected. Suppose, on the contrary, that L [ [y,z]
for some y 5 z. Because L is invariant for f, fn([y,z]) \ fn(L) \ L \ [q, f (qþ)] \ [0,1]. This
leads to a contradiction. Indeed, the mean value theorem and the claim (4.10) together imply that
jfnðyÞ fnðzÞj ¼ jðfnÞ0
ðxÞjjy zj lnjy zj ! 1 as n ! 1:
To prove that L is perfect, first notice that Jnconsists of 2ndisjoint closed intervals. We
order the 2n components of Jn from left to right on the real line and denote the ith
component by Jn,i. Also notice that the endpoints of each Jn,i are contained in L.
Let x2 L and for n 0, let Jn,i(n,x) be the component of Jn that contains x. Then
Jnþ1, i(nþ1,x) \ Jn, i(n,x)for all n 0 and x 2 \1n¼0Jn;iðn;xÞ. Because L is totally disconnected,
the length of Jn,i(n,x)converges to 0 as n goes to ? . Therefore, there are endpoints from
Jn,i(n,x)0s arbitrarily close to x. This proves that L is perfect. We have finished the proof of
item (1).
For items (2) and (3), let I1¼ [q, q7] and I2¼ [qþ, f (qþ)]. Then f (I1) [ I1[ I2 and
f(I2) [ I1[ I2. By using the same argument as in the proof of Theorem 4.2, the result in
item (2) follows. Let s be the shift map on S2. Define j : L! S2 by j(x)¼ s0s1s2. . .,
where si¼ 1 if fi(x) 2 I1 and si¼ 2 if fi(x) 2 I2. Let x2 L, j(x) ¼ s0s1s2. . ., and
j(f (x))¼ t0t1t2. . .. Then for any i 0, fi(f(x))¼ fiþ1(x)2 ISiþ1 and f
i( f (x))
2 Iti.
Because I1\I2¼ ø, siþ1¼ ti. Thus, j( f (x))¼ s(j(x)). Based on claim (4.10), it becomes
a routine process to prove that j is continuous, one to one, and onto, and j71 is continuous. For details, refer to the proof of Theorem 5.1 in [31, Chapter II]. Hence, the statement of item (3) is true.
For item (4), let x2 ½0; 1nL. Then there exists m 0 such that fm(x) 2 [0,q) [ ( f(q þ),
1] and hence fmþ1(x)2 [0, q). Because f (y) 5 y for all y 2 (0,q) and 0 is the unique fixed point in [0,q) for f. The continuity of f implies that fn(x) tends to 0 as n goes to the infinity. Consider when hypothesis (H2) holds. By using the same argument as in the proof of Theorem 4.2, we have the result, similar to claim (4.5), that for every x2 L, there exists an integer nx 1 such that jð fnxÞ0ðxÞj > 1. Based on this, the rest of the proof is very similar
to the one given above for hypothesis (H1). We leave the details to the readers. ¤
Theorem 4.5 implies that the dynamical system (2.6) will display cycles of all periods under certain conditions, and hence Smale’s chaos is possible. From Theorems 3.2, 3.3, 4.2, 4.3 and 4.5, the chaotic motion depends crucially on parameter values. Let us consider the model (3.1) with a¼ 2 and b ¼ 1, and l varying, that is fl (x) ¼ lx2 (1 – x).
If 0 < l <27
4, then f ([0,1]) \ [0,1]. Moreover, if l 4 4, then 0 and 1þ ffiffiffiffiffiffi14
l
p
2 are fixed points
for f. Also a simple calculation implies that its Schwarzian derivative
SfðxÞ ¼
6ð1 4x þ 6x2Þ
ð2x 3x2Þ2 ;
which is negative for all x2 (0,1) except the critical point 2
3. See also Figure 1 for its
bifurcation diagram. Corresponding Theorems 3.2, 3.3, 4.2, 4.3 and 4.5, we have the following results.
Example 4.6: Let fl:½0; 1 ! R be given by fl(x) ¼ lx2(1 – x), where 4 l 6.75 is a
parameter. Then one has the following properties:
(1) If l¼ 6.75, then fl has periodic orbits of all periods and fl exhibits Li–Yorke
chaos.
(2) If 6 5l5 6.75, then fl has periodic orbits of all even periods and fl2 exhibits
Li–Yorke chaos.
(3) There exists l 6.545 such that flexhibits Devaney’s chaos.
(4) If 6.6l 6.75, then there is a Cantor subset L of [0,1] such that flhas periodic
points of all periods in L and exhibits Smale’s chaos on L, and every orbit with an initial point in [0,1]\L converges to the origin. Note that since L has Cantor structure, Figure 1 appears that for 6.6 l 5 6.75, almost all orbits converge to the origin.
5. Conclusion
In this article, we have shown the existence of chaotic behaviour in an overlapping generations model with human capital accumulation. In addition to presenting the chaos in Li and Yorke condition, we also illustrate the chaotic trajectories in the sense of Devaney and Smale. Unlike other studies, we assume that excessive knowledge/ information will reduce the productivity to accumulate human capital. Because traditional approach of human capital accumulation does not take the negative externality of human capital into account, it excludes the possibility of complex dynamics. Hence, our study highlights the important role of human capital externality and indicates that a more accurate estimation of the human capital accumulation function will be needed in the future study. Furthermore, the policy implications of our results considering the negative externality of human capital will be very different from those obtained from traditional studies.8Another work for the future study is to extend the model to a higher-dimensional dynamical system. This can be achieved by including persistent habits in the model or by changing the formation of human capital accumulation.
Acknowledgements
The first author (Chen) gratefully acknowledges financial support from the Program for Globalization Studies at the Institute for Advanced Studies in Humanities at the National Taiwan University (grant number: 95R0064-AH03-03). The second author (Li) was partially supported by an NSC grant of Taiwan (grant number: NSC 96-2115-M-009-004-MY3).
H.-J. Chen and M.-C. Li
Notes
1. Day [1] argued that an excessive amount of capital will reduce overall productivity because of the ‘pollution effect’.
2. Average human capital was used by De la Croix and Doepke [10] to represent the quality of teachers.
3. The concept of ‘information pollution’ was introduced by Nielsen [13]; see also his interview by the BBC in [14].
4. For example, Eysenbach et al. [15] demonstrated the importance of the quality of medical information on the internet because misinformation can damage one’s human capital or even life.
5. We assume that there are only private schools in the economy. For studies considering educational systems, see [23–25].
6. Caucutt [26] has developed a model where schools could engage in price discrimination amongst students.
7. See [6], [10], [11], [23], [27], [28].
8. A good discussion about implications of policy decision when chaos presents can be found in Bullard and Butler [32].
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H.-J. Chen and M.-C. Li
