Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 836537, 6 pages
http://dx.doi.org/10.1155/2013/836537
Research Article
Nonlinear Dynamics in the Solow Model with Bounded
Population Growth and Time-to-Build Technology
Luca Guerrini1
and Mauro Sodini2
1
Department of Management, Polytechnic University of Marche, Piazza Martelli 8, 60121 Ancona, Italy
2
Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi 10, 56124 Pisa, Italy
Correspondence should be addressed to Luca Guerrini; luca.guerrini@univpm.it
Received 5 September 2013; Accepted 15 October 2013
Academic Editor: Carlo Bianca
Copyright © 2013 L. Guerrini and M. Sodini. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
The aim of this paper is to show that the Solow model equipped by realistic assumptions on technology and population dynamics is
capable of explaining a well-known stylised fact of growth, that is, the presence of persistent oscillations of demoeconomic variables.
In particular, our analysis shows that the coexistence of delays between (i) new investment and production and (ii) the birth date
and the recruitment in the labour force is a source of cyclical behaviour in capital accumulation.
1. Introduction
Following a long tradition, the theory of economic growth
is nevertheless mostly concerned with constant population
growth rates. There are only a few theoretical papers that consider
more complex and realistic assumption on population
dynamics. For example, Guerrini [1] generalizes the result of
existence and global stability of the (stationary) equilibrium
in the Solow model by assuming a bounded, but not necessarily
constant, population growth rate; Fanti and Manfredi [2]
show in a Solow-type model that the existence of distributed
delays (age structure) that influence the labour force and a
Malthusian mechanism that relates fertility and wage rates
are sources of fluctuations in growth paths; and Fabi˜ao and
Borges [3] study the effects of fluctuating population size on
capital accumulation. Other cases of nontrivial population
dynamics may be observed in models where agents allocate
their resources between childbearing and consumption (e.g.,
[4–6]).
The idea of the present paper is to study a Solowtype
model in which one of the main characteristics is
the introduction of reproductive mechanism that prevents
unbounded population growth in the long run. In particular,
we assume that (i) population dynamics is described by a
delayed logistic equation [7], which presents a constant carrying
capacity and (ii) a time-to-build technology characterizes
the productive sector. Namely, we introduce two different
sources of time-delays: with regard to the first assumption,
it implies that there exists a lag between the birth date and
the recruitment in the labour force; with regard to the second
one, it means that production occurs with a delay while new
capital is installed (investment gestation lag). Both processes
affect the capital accumulation path by means of production
function.
It is interesting to notice that the theoretical question
whether lags in productive activity induce cycles is a topic
that dates back to early economists such as Jevons [8] and
Kalecki [9], and the introduction of new mathematical tools
for the study of functional differential equations (e.g., the
influential book of Hassard et al. [10]) has revitalized this
issue. Nonetheless, the large part of the literature is focused
on investment lags (e.g., [11–14]), and the role of other
kinds of lags in economic growth models or the linkages
between population dynamics and lags in productive sector
has received little attention. Exceptions are the papers of Fanti
and Manfredi [2] described before; Bianca and Guerrini [15],
who embed the problem of population dynamics in a network
structure; Fanti et al. [16] who study problems related to
demographic transition and the emergence of low-fertility
traps in a model in which a system of differential equations
with distributed delays is introduced to describe the age
distribution of the population; Guerrini and Sodini [17] who
2 Abstract and Applied Analysis
analyze the role of the assumption of a positive or a negative
constant population growth rate in a model with a time-tobuild
technology; and Ballestra et al. [18] who consider a
Kaleckian type model of business cycle where a negative effect
of the capital stock increases disproportionally as the capital
stock builds up.
In the present paper, two alternative hypotheses on
capital depreciation are explored. In the first hypothesis, it
is assumed that the productive capital (i.e., the capital that
actually is used in the productive process) depreciates at a
constant rate (e.g., [11]). In the second one, it the case in which
depreciation affects directly the investments even if they have
not yet created productive capital is explored. The second case
represents a plausible assumption if we consider capital in
a broad sense that includes intangible capitals such as new
inventions, prototypes, and patents. It is worth noting that,
differently by Hongliang and Wenzao [19], business cycles
phenomena may be observed even in the second case.
The paper is organised as follows. In Section 2, the models
are formally introduced. Section 3 is focused on the study of
the first model specification. In Section 4, the second model is
analyzed. In Section 5, the global properties of the models are
investigated via numerical simulations. Section 5 concludes.
2. The Models
2.1. First Specification. We introduce a Solow [20] model with
the following characteristics. We assume that at time 𝑡 identical
and competitive firms produce a homogeneous good,
𝑌, by combining capital and labour 𝐾 and 𝐿, respectively,
through the constant returns to scale Cobb-Douglas technology.
New investments display a delay of period 𝜏 before
they can be used for production (time-to-build technology).
Thus, the evolution of capital accumulation is governed by the
following equation:
̇𝐾 = 𝑠𝐴𝐾 𝛼
𝑑 𝐿1−𝛼
− 𝛿𝐾 𝑑, (1)
where 𝐴 is a positive parameter representing (exogenous)
technological progress, 𝛼 ∈ (0, 1) measures the productivity
of capital, 𝑠 ∈ (0, 1) is the exogenous constant saving rate,
𝐾 𝑑 := 𝐾(𝑡 − 𝜏) is the state value of 𝐾 at time 𝑡 − 𝜏, and 𝛿
is the constant rate of productive capital depreciation (in this
specification, obsolescence only affects capital if and only if
this last is used in a productive process).
We assume that population of workers evolves according
to the delayed logistic equation introduced by Hutchinson
(see Arino et al. [21] for an alternative formulation):
̇𝐿 = 𝐿 (𝑎 − 𝑏𝐿 𝑑) , (2)
where 𝐷 = 𝑎/𝑏 is the constant carrying capacity and
𝐿 𝑑 := 𝐿(𝑡 − 𝜏) is the state value of 𝐿 at time 𝑡 − 𝜏.
Notice that the presence of the delayed variable 𝐿 𝑑 captures
the lag between the birth date and the recruitment in the
labour force while 𝐷 defines a limit to population growth.
Other specifications (see [16]) that use delayed differential
equations are mostly concerned with the description agespecific
fertility and mortality rates.
Setting 𝑘 = 𝐾/𝐿, we have
̇𝑘 =
̇𝐾
𝐿
−
̇𝐿
𝐿
𝑘 =
𝑠 (𝐴𝐾 𝛼
𝑑 𝐿1−𝛼
) − 𝛿𝐾 𝑑
𝐿
−
̇𝐿
𝐿
𝑘
= 𝑠𝐴(𝐿−1
𝐿 𝑑)
𝛼
𝑘 𝛼
𝑑 − 𝛿 (𝐿−1
𝐿 𝑑) 𝑘 𝑑 −
̇𝐿
𝐿
𝑘.
(3)
Hence, the model is described by
̇𝑘 = [𝑠𝐴(𝐿−1
𝐿 𝑑)
𝛼
] 𝑘 𝛼
𝑑 − [𝛿 (𝐿−1
𝐿 𝑑)] 𝑘 𝑑 − (𝑎 − 𝑏𝐿 𝑑) 𝑘,
̇𝐿 = 𝐿 (𝑎 − 𝑏𝐿 𝑑) .
(4)
2.2. Second Specification. The second model differs from the
first one in the fact that obsolescence affects capital regardless
of its utilization in the productive process. Thus, (4) is
replaced by the following:
̇𝐾 = 𝑠𝐴𝐾 𝛼
𝑑 𝐿1−𝛼
− 𝛿𝐾. (5)
Setting 𝑘 = 𝐾/𝐿, we obtain
̇𝑘 =
̇𝐾
𝐿
−
̇𝐿
𝐿
𝑘 =
𝑠 (𝐴𝐾 𝛼
𝑑 𝐿1−𝛼
) − 𝛿𝐾
𝐿
−
̇𝐿
𝐿
𝑘
= 𝑠𝐴(𝐿−1
𝐿 𝑑)
𝛼
𝑘 𝛼
𝑑 − 𝛿𝑘 −
̇𝐿
𝐿
𝑘.
(6)
As a result, the model is now represented by
̇𝑘 = [𝑠𝐴(𝐿−1
𝐿 𝑑)
𝛼
] 𝑘 𝛼
𝑑 − [𝛿 + (𝑎 − 𝑏𝐿 𝑑)] 𝑘,
̇𝐿 = 𝐿 (𝑎 − 𝑏𝐿 𝑑) .
(7)
It is simple to verify that systems (4) and (7) have the same
(nontrivial) equilibria. By setting ̇𝑘 = 0 and 𝑘 𝑑 = 𝑘 = 𝑘∗, ̇𝐿 =
0, and 𝐿 𝑑 = 𝐿 = 𝐿∗ for all 𝑡, we find that there exists a unique
nontrivial equilibrium (𝑘∗, 𝐿∗), where
𝑘∗ = (
𝑠𝐴
𝛿
)
1/(1−𝛼)
, 𝐿∗ =
𝑎
𝑏
. (8)
Notice that because the population dynamics is bounded
from above and the level of technological progress is fixed,
in the present work, differently from Guerrini and Sodini
[17], a balanced growth path is not a feasible solution and
(𝑘∗, 𝐿∗) identifies the stationary solution of the model, for
which 𝐾 = 𝑘∗ 𝐿∗.
3. Local Analysis of System (4)
In order to study the local properties of the system around
nontrivial equilibrium, we linearize system (4) at (𝑘∗, 𝐿∗).
This gives
[
̇𝑘
̇𝐿
] = [
0 𝛿𝐿−1
∗ (−𝛼𝐿−1
∗ + 𝑘∗)
0 0
] [
𝑘 − 𝑘∗
𝐿 − 𝐿∗
]
+ [
(𝛼 − 1) 𝛿 (𝛼 − 1) 𝛿𝐿−1
∗ 𝑘∗ + 𝑏𝑘∗
0 −𝑏𝐿∗
] [
𝑘 𝑑 − 𝑘∗
𝐿 𝑑 − 𝐿∗
] .
(9)
Abstract and Applied Analysis 3
The characteristic equation can be derived from
det {[
0 𝛿𝐿−1
∗ 𝑘∗ (−𝛼𝐿−1
∗ + 1)
0 0
] − 𝜆𝐼
+𝑒−𝜆𝜏
[
(𝛼 − 1) 𝛿 (𝛼 − 1) 𝛿𝐿−1
∗ 𝑘∗ + 𝑏𝑘∗
0 −𝑏𝐿∗
]} = 0,
(10)
where 𝐼 is the identity matrix. A direct calculation shows that
the above equation can be rewritten as
𝑃 (𝜆, 𝜏) = 𝑃1 (𝜆, 𝜏) ⋅ 𝑃2 (𝜆, 𝜏) = 0, (11)
where
𝑃1 (𝜆, 𝜏) = 𝜆 + (1 − 𝛼) 𝛿𝑒−𝜆𝜏
, 𝑃2 (𝜆, 𝜏) = 𝜆 + 𝑎𝑒−𝜆𝜏
.
(12)
By putting 𝜏 = 0 in (11), we obtain [𝜆 + (1 − 𝛼)𝛿](𝜆 + 𝑎) =
0. We derive that the two roots of this equation are negative.
Hence, the equilibrium (𝑘∗, 𝐿∗) is stable in case there is no
delay. As 𝜏 increases, the stability of the equilibrium point will
change when (11) under consideration has zero or a pair of
purely imaginary eigenvalues. The former occurs when 𝜆 = 0,
and this is not possible. The latter deals with the existence of
a root 𝜆 = 𝑖𝜔 for (11). This means we need to investigate when
𝑃1(𝑖𝜔, 𝜏) = 0 and/or 𝑃2(𝑖𝜔, 𝜏) = 0. Without loss of generality,
since the complex roots of (11) appear as complex conjugate
pairs, we assume that 𝜔 > 0.
Lemma 1. Equation 𝑃1(𝜆, 𝜏) = 0 (resp., 𝑃2(𝜆, 𝜏) = 0) with
𝜏 = 𝜏(1)
𝑗 (resp., 𝜏 = 𝜏(2)
𝑗 ), 𝑗 ∈ N0
= N ∪ {0}, has a pair of purely
imaginary roots 𝜆 = ±𝑖𝜔1 (resp., 𝜆 = ±𝑖𝜔2), where
𝜔1 = (1 − 𝛼) 𝛿 (resp. 𝜔2 = 𝑎) ,
𝜏(1)
𝑗 =
1
𝜔1
(
𝜋
2
+ 2𝑗𝜋) (resp. 𝜏(2)
𝑗 =
1
𝜔2
(
𝜋
2
+ 2𝑗𝜋)) .
(13)
Proof. Consider 𝑃1(𝑖𝜔, 𝜏) = 0; that is, 𝑖𝜔 + (1 − 𝛼)𝛿𝑒−𝑖𝜔𝜏
=
0. Separating real and imaginary parts of this equation, we
obtain
𝜔 = (1 − 𝛼) 𝛿 sin 𝜔𝜏, 0 = (1 − 𝛼) 𝛿 cos 𝜔𝜏. (14)
From cos 𝜔𝜏 = 0 in (14), and 𝜔 and (1 − 𝛼)𝛿 being both
positive, it follows that 𝜔𝜏 = 𝜋/2 + 2𝑗𝜋 (𝑗 = 0, 1, 2, . . .) and
𝜔 = (1 − 𝛼)𝛿. We also obtain a sequence of the critical values
𝜏(1)
𝑗 of 𝜏. Next, suppose 𝑃2(𝑖𝜔, 𝜏) = 0; that is, 𝑖𝜔 + 𝑎𝑒−𝑖𝜔𝜏
= 0.
Then
𝜔 = 𝑎 sin 𝜔𝜏, 0 = 𝑎 cos 𝜔𝜏, (15)
which lead to 𝜔 = 𝑎 and the existence of 𝜏(2)
𝑗 . This concludes
the proof.
We easily obtain the following result.
Proposition 2. (1) If 𝑎 ̸= (1− 𝛼)𝛿, then characteristic equation
(11) has a pair of purely imaginary roots 𝜆 = ±𝑖𝜔 𝑚 at 𝜏 = 𝜏(𝑚)
𝑗 ,
𝑚 = 1, 2, 𝑗 ∈ N0
.
(2) If 𝑎 = (1 − 𝛼)𝛿, then characteristic equation (11) has a
multiple root with the multiplicity of two.
Remark 3. When 𝑎 = (1 − 𝛼)𝛿, the system (4) is a
degenerated case and it is difficult to determine the crossing
direction of the characteristic roots through the imaginary
axis. Moreover, the criteria of stability switches fail to analyze
the stability switches of system (4).
Henceforth, we assume 𝑎 ̸= (1 − 𝛼)𝛿, so that (11) has no
repeated roots.
Lemma 4. For 𝜏 = 𝜏 𝑚
𝑗 , 𝑗 ∈ N0
, ±𝑖𝜔 𝑚, 𝑚 = 1, 2, are simple
roots of (11) with
𝑑 (Re 𝜆)
𝑑𝜏
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝜆=𝑖𝜔 𝑚
> 0. (16)
Proof. Both 𝜆 = 𝑖𝜔1 and 𝜆 = 𝑖𝜔2 are simple. If we suppose,
by contradiction, that, for example, 𝜆 = 𝑖𝜔1 is a repeated
root, then 𝑃(𝑖𝜔1, 𝜏) = 0 and 𝑑𝑃(𝑖𝜔1, 𝜏)/𝑑𝜆 = 0 lead to
a contradiction. Let 𝜆(𝜏) = 𝜇(𝜏) + 𝑖𝜔(𝜏) be the branch of
characteristic roots of (11) such that 𝜇(𝜏(1)
𝑗 ) = 0 and 𝜔(𝜏(1)
𝑗 ) =
𝜔1 (𝑗 = 0, 1, 2, . . .). Taking the derivative of 𝜆 with respect to
𝜏 in (11), it follows that
(
𝑑𝜆
𝑑𝜏
)
−1 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝜆=𝑖𝜔1
= −
1
𝜆2
−
𝜏
𝜆
. (17)
Then, we get
sign {
𝑑 (Re 𝜆)
𝑑𝜏
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝜆=𝑖𝜔1
}
= sign {Re (
𝑑𝜆
𝑑𝜏
)
−1 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝜆=𝑖𝜔1
}
= sign {(
𝑑𝜆
𝑑𝜏
)
−1 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝜆=𝑖𝜔1
} = sign {
1
𝜔2
1
} .
(18)
Similarly for 𝜆 = 𝑖𝜔2, the statement holds.
Therefore, a pair of pure imaginary roots will cross the
imaginary axis from left to right as 𝜏 increases. Furthermore,
this also yields to the existence of a Hopf bifurcation at the
equilibrium when 𝜏 = 𝜏 𝑚
𝑗 . Using this last lemma and the
previous analysis about the existence of purely imaginary
roots of (11), we can state the following theorem.
Theorem 5. Let 𝜏(𝑚)
𝑗 , 𝑚 = 1, 2, 𝑗 ∈ N0
, be defined as in (13),
and let 𝜏0 = min{𝜏(1)
0 , 𝜏(2)
0 }, where 𝜏(1)
0 = 𝜋/[(1 − 𝛼)]𝛿, 𝜏(2)
0 =
𝜋/(2𝑎).
(1) The equilibrium point (𝑘∗, 𝐿∗) of (4) is locally asymptotically
stable for 𝜏 ∈ [0, 𝜏0) and unstable for 𝜏 > 𝜏0.
(2) System (4) undergoes a Hopf bifurcation at the equilibrium
(𝑘∗, 𝐿∗) when 𝜏 = 𝜏(𝑚)
𝑗 , 𝑚 = 1, 2, 𝑗 ∈ N0
.
4 Abstract and Applied Analysis
4. Local Analysis of System (7)
In this section, we consider the model described by system
(7). It is known that the modification of the Zak model
[11] with the assumption that capital obsolescence affects
the present capital makes the stationary equilibrium globally
asymptotically stable for any value of time-delay (no Hopf
bifurcation may occur) (see [19]). We will see that the
introduction of population dynamics drastically changes the
results.
As in the previous model, in order to study the local
dynamics, we linearize system (7) at (𝑘∗, 𝐿∗). We have
[
̇𝑘
̇𝐿
] = [
−𝛿 −𝛼𝛿𝐿−2
∗ 𝑘∗
0 0
] [
𝑘 − 𝑘∗
𝐿 − 𝐿∗
]
+ [
𝛼𝛿 𝛼𝛿𝐿−1
∗ 𝑘∗ + 𝑏𝑘∗
0 −𝑏𝐿∗
] [
𝑘 𝑑 − 𝑘∗
𝐿 𝑑 − 𝐿∗
] .
(19)
The associated characteristic equation of system (19) is
det {[
−𝛿 −𝛼𝛿𝐿−2
∗ 𝑘∗
0 0
] − 𝜆𝐼
+𝑒−𝜆𝜏
[
𝛼𝛿 𝛼𝛿𝐿−1
∗ 𝑘∗ + 𝑏𝑘∗
0 −𝑏𝐿∗
]} = 0
(20)
and takes the form
𝑄 (𝜆, 𝜏) = 𝑄1 (𝜆, 𝜏) ⋅ 𝑄2 (𝜆, 𝜏) = 0, (21)
where
𝑄1 (𝜆, 𝜏) = 𝜆 + 𝛿 − 𝛼𝛿𝑒−𝜆𝜏
, 𝑄2 (𝜆, 𝜏) = 𝜆 + 𝑎𝑒−𝜆𝜏
.
(22)
When 𝜏 = 0, all roots of (21) are negative, so that the system
is stable. We will let 𝜏 vary and investigate possible stability
switches and bifurcations. For (𝑘∗, 𝐿∗) to become unstable,
characteristic roots have to cross the imaginary axis to the
right when 𝜏 increases. Let 𝜆 = 𝑖𝜔 (𝜔 > 0) be a purely
imaginary root of (21). Then, it satisfies 𝑄1(𝑖𝜔, 𝜏) = 0 and/or
𝑄2(𝑖𝜔, 𝜏) = 0. Let 𝑄1(𝑖𝜔, 𝜏) = 0. This can be rewritten as the
following two equations:
𝜔 = −𝛼𝛿 sin 𝜔𝜏, 𝛿 = 𝛼𝛿 cos 𝜔𝜏, (23)
which imply
𝜔2
= (𝛼2
− 1) 𝛿2
< 0. (24)
Therefore, 𝑄1(𝜆, 𝜏) has no purely imaginary roots. Noticing
that 𝑄2(𝜆, 𝜏) = 𝑃2(𝜆, 𝜏), with 𝑃2(𝜆, 𝜏) defined as in (12),
it follows that 𝑄2(𝜆, 𝜏) has a unique pair of simple purely
imaginary roots ±𝑖𝜔2 at a sequence of critical values 𝜏(2)
𝑗 ,
where 𝜔2 and 𝜏(2)
𝑗 are defined by (13). Let 𝜆(𝜏) = 𝜇(𝜏) + 𝑖𝜔(𝜏)
denote the roots of (21) near 𝜏 = 𝜏(2)
𝑗 satisfying 𝜇(𝜏(2)
𝑗 ) = 0
and 𝜔(𝜏(2)
𝑗 ) = 𝜔2 (𝑗 ∈ N0
). Then, the following transversality
condition 𝑑[Re 𝜆(𝜏(2)
𝑗 )]/𝑑𝜏 > 0 holds and so the root crosses
the imaginary axis from left to right (at 𝜏 = 𝜏(2)
𝑗 ) as 𝜏
increases. Since (𝑘∗, 𝐿∗) is locally asymptotically stable when
𝜏 = 0, then it becomes unstable at the smallest value of 𝜏 for
which an imaginary root exists. Thus, we can get the following
results.
Theorem 6. Let 𝜏(2)
𝑗 , 𝑗 ∈ N0
, be defined as in (13), where 𝜏(2)
0 =
𝜋/(2𝑎).
(1) The equilibrium point (𝑘∗, 𝐿∗) of (7) is locally asymptotically
stable for 𝜏 ∈ [0, 𝜏(2)
0 ) and unstable for 𝜏 > 𝜏(2)
0 .
(2) System (7) undergoes a Hopf bifurcation at the equilibrium
(𝑘∗, 𝐿∗) when 𝜏 = 𝜏(2)
𝑗 for 𝑗 ∈ N0
.
Remark 7. Notice that bifurcation value of both the models is
the same if 𝛼 > 𝛿/(2 + 𝛿).
5. Numerical Simulations
In order to understand the dynamical properties of the
model, we consider a numerical specification and let 𝜏 vary.
In particular, we set
𝐴 = 1; 𝛿 = 0.09; 𝛼 = 0.43;
𝑎 = 0.6; 𝑏 = 0.4; 𝑠 = 0.8.
(25)
For this parameter specification (𝑘∗
, 𝐿∗
) = (1.5, 46.2) and
from Theorem 5, we have 𝜏1 < 𝜏0. Thus, the bifurcation values
for the two models are the same and we have verified by
means of several numerical experiments that the associated
dynamics look very similar. Therefore, we focus on the
simulations for model 1. Figure 1 shows the evolution of long
run dynamics when 𝜏 is varied. It is interesting to note
that after bifurcation, differently from Guerrini and Sodini
[17], both the state variables oscillate permanently along the
unstable equilibrium. If 𝜏 is increased, the enlargement of the
limit cycle induces larger and larger oscillations. However, by
means of several numerical experiments, it seems that only
nonchaotic dynamics arise.
6. Conclusions
In this paper, we have analyzed the dynamics in Solow-type
model with investment lags and nonstationary but bounded
population. We have shown that the interaction between
the population dynamics and time-to-build technology may
be an engine of cyclical behaviour of capital accumulation.
Formally, by applying the techniques developed by Hassard
et al. [10] for delayed differential equations, we have proved
that a Hopf bifurcation occurs once lag 𝜏 passes through
a critical value and a family of periodic orbits bifurcates
from the stationary equilibrium. Numerical simulations confirm
the existence of oscillating dynamics in the demoeconomic
variables. More complex structures of time-delays and
microeconomic foundation of the model are left for future
research.
Abstract and Applied Analysis 5
46.9
46.8
46.7
46.6
46.5
46.4
46.3
46.2
46.1
0 50 100 150 200 250
t
k(t)
(a)
1.515
1.51
1.505
1.5
1.495
1.49
L(t)
0 50 100 150 200 250
t
(b)
500
450
400
350
300
250
200
150
100
50
0
0 50 100 150 200 250
t
k(t)
(c)
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0 50 100 150 200 250
t
L(t)
(d)
Figure 1: (a) Time evolution of size capital per worker (𝜏 = 2.3); (b) time evolution of population (𝜏 = 2.3); (c) time evolution of capital per
worker (𝜏 = 3.3); and (d) time evolution of population (𝜏 = 3.3).
Acknowledgment
The authors would like to thank the referees for the careful
review and the valuable comments. Mauro Sodini acknowledges
that this work has been performed within the activity
of the PRIN Project “Local Interactions and Global Dynamics
in Economics and Finance: Models and Tools,” MIUR, Italy.
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