Capital-Induced Labor Migration in a Spatial Solow Model

Capital-Induced Labor Migration in a
Spatial Solow Model
João Plı́nio Juchem Neto∗, PPGMAp/UFRGS, jpjuchem@gmail.com
Julio Cesar Ruiz Claeyssen, PPGMAp/UFRGS, jcrclaeyssen@gmail.com
Abstract
In this work we propose a generalization of the Spatial Solow Model for economic growth, considering that workers move from regions with lower density of
capital to regions with higher density of capital, and that the labor force grows
following a logistic law. Then we show, through a linear stability analysis, that
the introduction of this capital-induced labor migration in the model is a necessary
condition for the model to reach an unstable regime, in which it can generate rich
spatio-temporal dynamics. Moreover, we present numerical simulations in one spatial dimension showing that, depending on the intensity of this migration and on
the size of the economy, this model can develop four kinds of behaviors, making the
economy endogenously: (i) converge to a homogeneous steady-state; (ii) converge
to a non-homogeneous steady-state; (iii) develop periodic spatio-temporal cycles;
(iv) and develop irregular and aperiodic spatio-temporal cycles.
JEL Classification: R12, O40.
Keywords: Spatial Solow Model, Regional Economics, Industrial Agglomeration.
Thematic Area: (3) Regional and Urban Economics.
Resumo
Neste trabalho propomos uma generalização para o Modelo de Solow Espacial de crescimento econômico, considerando que os trabalhadores se movem de
regiões com baixa densidade de capital para regiões com alta densidade de capital,
e que o crescimento orgânico da força de trabalho segue a lei logı́stica. Através de
uma análise de estabilidade linear, mostramos que a introdução desta migração de
mão-de-obra induzida pelo capital é uma condição necessária para o modelo entrar
em um regime instável, onde pode gerar uma rica variedade de dinâmica espaçotemporais. Além disso, apresentamos simulações numéricas em uma dimensão espacial mostrando que, dependendo da intensidade desta migração e do tamanho
da economia, este modelo pode gerar endogenamente quatro tipos de comportamentos, fazendo a economia: (i) convergir para um estado estacionário homogêneo;
(ii) convergir para um estado estacionário não-homogêneo; (iii) desenvolver ciclos
espaço-temporais periódicos; (iv) e desenvolver ciclos espaço-temporais irregulares
e aperiódicos.
Classificação JEL: R12, O40.
Palavras-Chave: Modelo de Solow Espacial, Economia Regional, Aglomeração Industrial.
Área Temática: (3) Economia Regional e Urbana.
∗
The author acknowledges financial support from Petrobras, under the program PFRH-PB 16.
1
1
Introduction
The standard neoclassical economic growth model was proposed by Solow and Swan in
the 50’s [28, 29]. In this model, the saving rate and the malthusian labor growth are
exogeneously given, and as in most economic growth models found in the literature, it
is spatially homogeneous, that is, it not take into account migration of capital and labor
through space. In the 70’s, Isard and Liossatos proposed a very general model, which
takes into account such migration in a continuous space-time framework [15, 16, 17],
and whose dynamics is then used in space-time dynamical optimization problems. As
we will see in Section 2.2 of this paper, the particular case of the Isard-Liossatos Model
that we are interested in can generate negative solutions, what is unrealistic. In the 90’s
there was a new interest in the regional sciences with the advent of the new economic
geography [19, 12], with the development of continuous space models in circle domains.
In these models, the authors use a general equilibrium framework to analyse the interaction and aggregation of spatially distributed two-factor economies: in one model the
interaction between an agricultural sector (perfect competition) with an industrial sector
(monopolistic competition); and in other, the interaction between two industrial sectors
in a monopolistic competition framework [12]. One limitation of these models is that
they lack capital accumulation [6, 3], viewing agglomeration as a result of the interplay
between transportation costs and increasing returns, and considering, in some of them, a
moving industrial labor force [12, 3].
In the 2000’s, Camacho and Zou [6] proposed a continuous Spatial Solow Model in an
unidimensional unbounded domain, considering only capital as a production factor, and
explicitly modelling capital accumulation. In fact, this is a direct spatial generalization
of the standard Solow-Swan Model, besides of being a special case of the Isard-Liossatos
Model – the reaction-diffusion case [15, 16, 17]. As an improvement, Brito [4] considered
this same model, also in an unbounded domain, but introducing an exogenous labor force
following a malthusian growth. In these two models there is agglomeration only with the
introduction of some exogenous asymmetry. When considering an endogenously saving
rate in such a model, Brito also showed that agglomeration may be endogenously generated [3]. Then, Cappasso et al. [7] considered the model proposed by Camacho and
Zou in a bounded domain, but using an S-shaped production function, which implied
the existence poverty traps. Recently, Capasso et al. [8] also introduced an enxogenous
space-dependent malthusian labor force in their model. In this work, we will follow this
more recent literature, introducing a capital-induced labor migration in the Spatial Solow
Model, and showing that such a modification in the model can gererate, endogenously,
different types of spatio-temporal dynamics. One of these dynamics, in fact, is the convergence of the economy to a stable non-homogeneous steady-state, showing capital end
labor agglomeration.
In this work we consider, on one hand, that both capital and labor diffuse from regions
with higher density to regions with lower densities of capital and labor, respectively. This
behavior is in agreement with the neoclassical principle of diminishing marginal return
of the factors and with the models considered in the literature up to now. On the other
hand, and this is the main contribution of this work, it considers that the labor force
moves to regions with higher density of capital – where workers can find more factories to
work, for example – with an intensity proportional to the labor present at location x (this
2
proportionality guarantee the non-negativity of solutions, fixing the problem of negative
solutions in the Isard-Liossatos Model – at least in the particular case considered here).
Therefore, in this model, that for simplicity we shall consider in one spatial dimension,
the spatial distribution of labor determines the distribution of capital, and vice-versa.
The model introduced here consists of a system of two coupled reaction-diffusion-advection
partial differential equations, which is similar with models that arise in the study of
chemotaxis motion in the field of Biomathematics. It is known that these models may
lead to the formation of spatial patterns [18, 32, 21, 11], and show more complex spatiotemporal behaviors [25, 24, 1]. In our model we will consider a Cobb-Douglas production
function, an exponential depreciation of the capital, and a labor force following the Verlust’s logistic law. This later law was already considered in the standard non-spatial
Solow model [9, 20, 13]. After we discuss the model, we carry out a standard linear
stability analysis, and look for conditions that may lead to a richer spatio-temporal behavior. References that have already used this technique in the economics literature are,
for example: Brock and Xepapadeas [5], in in the study of stable pattern-formation in a
coupled economic-ecological system described by a reactive-diffusive model; and Fujita,
Krugman and Venables [12] in the study of industrial agglomeration in the models of the
new economic geography commented above.
The structure of this paper is as follows: in section two we introduce the capital-induced
labor migration in the Spatial Solow Model; in section three we derive conditions that
make the model presents linear instability, regime in which we may observe non-trivial
dynamics; and in section four we present some numerical results, checking the linear
stability results and ilustrating the kinds of spatio-temporal dynamics produced by the
model. Finally, we close with conclusions and perspectives for future research.
2
Derivation of the Model
The model considered in this work describes the spatio-temporal evolution of capital
and labor densities in a continuum of local economies represented as a compact interval
Ω = [0, l], so l > 0 is the size of the economy. At each point x ∈ Ω, there is a density
of capital K(t, x) ≥ 0 and a density of labor L(t, x) ≥ 0, factors that will be used to
produce an aggregate good through a given production function f (K, L).
2.1
Equation describing the evolution of capital
The equation giving the balance of capital in the economy is derived from the following
conservation law:
∫
∫ b
∂ b
K(t, x)dx =
h(K, L)dx + U (t, a) − U (t, b)
(1)
∂t a
a
where the term in the left hand side is the rate of ∫change of the total amount of capital
b
in a finite segment (a, b) of the whole economy Ω; a h(K, L)dx is the net rate of capital
produced in (a, b); and U (t, a)−U (t, b) is the net rate of capital flowing into (a, b). Here we
consider h(K, L) = f (K, L) − δK, where f (K, L) is a production function, and δK is the
3
depreciation of capital. Using the fundamental theorem of calculus, and assuming that
the flux of capital U (t, x) is continuously differentiable, we can rewrite U (t, a) − U (t, b)
∫b
as − a ∂U
(t, x)dx. Then, considering K(t, x) continuously differentiable, (1) becomes:
∂x
∫ b(
a
∂K
∂U
−h+
∂t
∂x
)
dx = 0.
(2)
By assuming that the produced capital h is continuous in Ω, the integrand turns out a
continuous function. Since the integral must be zero for any finite segment (a, b) of Ω, it
follows that the integrand must be identically zero in Ω. Thus we obtain the continuity
equation for the capital:
∂K
∂U
=h−
.
(3)
∂t
∂x
For the flux of capital U (t, x), we consider that:
U (t, x) = −dK
∂K
∂x
(4)
where dK ≥ 0 is the capital diffusion coefficient. The dynamics given by (4) considers
that the capital moves from regions with high density of capital to regions with low
density of capital (what obeys the neoclassical principle of diminishing marginal return of
capital). Then, differentiating (4) with respect to x, and substituting into (3), we obtain
the reaction-diffusion partial differential equation (pde) governing the spatio-temporal
evolution of the capital density:
∂K
∂ 2K
= h + dK 2 .
∂t
∂x
2.2
(5)
Equation describing the evolution of labor force
For the balance of the labor force in the economy, the derivation is very similar to the
one presented previously. The conservation equation in this case is given by:
∫
∫ b
∂ b
L(t, x)dx =
g(L)dx + V (t, a) − V (t, b)
(6)
∂t a
a
where V (t, a) − V (t, b) is the net rate of workers flowing into a finite segment (a, b) of Ω
at time t, and g(L) gives the natural growing rate of the labor force. Considering L, V
and g smooth enough, we arrive on:
∂V
∂L
=g−
∂t
∂x
(7)
where the labor flux V (t, x) is given by:
V (t, x) = −dL
∂L
∂K
+ χL L
.
∂x
∂x
(8)
Here, the term −dL ∂L
makes the labor force to move from regions of high density of
∂x
workers to regions with low density of workers (diminishing marginal return of labor); and
χL L ∂K
, the capital-induced labor migration term, causes workers to move into regions
∂x
4
with higher density of capital. Then, combining (8) with (7), we obtain the reactiondiffusion-advection pde governing the evolution of the labor force:
(
)
∂L
∂ 2L
∂
∂K
= g + dL 2 − χL
L
(9)
∂t
∂x
∂x
∂x
where dL ≥ 0 is the labor diffusion coefficient, and χL ≥ 0 is the capital-induced labor
migration coefficient.
Remark 1 - Instead of using the term χL L ∂K
in (8), Isard and Liossatos [17] proposed
∂x
∂K
the use of the term χL ∂x without further analysis. The problem is that the solution
for labor density can assume negative values in this case, even for nonnegative initial
conditions, what is unrealistic. The modification presented here solves this problem, but
at the cost of increasing the nonlinearity of the problem. That the solutions will always
be nonnegative follows easily, with minor changes, from the proof presented in Edelstein
[10] □
2.3
Initial and Bondary Conditions
To complete the model, we need to assign convenient initial and boundary conditions.
For the initial conditions we will consider nonnegative initial densities of capital and
labor K(0, x) = K0 (x) and L(0, x) = L0 (x), respectively. For the boundary conditions,
we will assume that the economy Ω is an autarchy, i.e., there is no flux of capital or labor
with the exterior. Mathematically, this means that we have homogeneous Neumann (or
no-flux) boundary conditions for K and L:
∂K ∂L = 0 and
= 0.
∂x x∈∂Ω
∂x x∈∂Ω
Then, considering Sections 2.1 and 2.2, we can write down the general formulation of our
model:

∂K
∂ 2K



=
h
+
d
, for x ∈ (0, l), t > 0
K

2

∂t
∂x

(
)

2


 ∂L = g + d ∂ L − χ ∂ L ∂K for x ∈ (0, l), t > 0
L
L
∂t
∂x2
∂x
∂x
(10)



K(t, x) = K0 (x), L(t, x) = L0 (x), for x ∈ Ω = [0, l], t = 0





∂L
∂K


= 0,
= 0, for x ∈ ∂Ω = 0, l, t > 0
∂x
∂x
In this work we will consider that the natural growing rate of the labor force follows the
logistic law:
g(L) = aL − bL2
(11)
where a ≥ 0 is the population growing rate, and ab ≥ 0 is the labor maximum capacity
of each local economy. The production function will be given by a Cobb-Douglas [2]
f (K, L) = sAK ϕ L1−ϕ , where s ∈ [0, 1] is the constant saving rate, A > 0 is the constant
total factor productivity, and ϕ ∈ (0, 1) is the output elasticity of capital. In this way,
we have that h is given by:
h(K, L) = sAK ϕ L1−ϕ − δK,
5
(12)
being δ > 0 the depreciation rate of capital. Note that if we consider spatial homogeneous
initial conditions, there will be no spatial migration of capital and labor in the economy,
and the solution of (10) will be the same as in the non-spatial model considered in [20];
and if, besides of that, we set b = 0 in the logistic law, we have malthusian growth,
and therefore recover the classical Solow-Swan model [28, 29]. On the othe hand, if we
consider an unbounded domain Ω = R and a = b = dL = χL = 0, we recover the Spatial
Solow Model considered by Camacho and Zou in [6]; if Ω = R and b = χL = 0, then we
have the model considered by Brito in [4].
The model (10) is similar with models that arise in the study of chemotaxis motion in
Biomathematics, which may lead to the formation of spatial patterns [18, 32, 21, 11], and
more complex spatio-temporal dynamics [25, 24, 1].
As far as existence of solutions goes, Senba and Suzuki [26] proved, in chapter 7 of their
book, local existence (in time) of classical solutions for the problem (10). In [22], [23] and
[14] was established global existence for some particular cases of h and g. If χL = 0, we
have a reaction-diffusion system, for which existence conditions can be found in Smoller
[27] and Zeidler [31].
3
Linear Stability Analysis
The goal of this section is to carry out a linear stability analisys of the model, about the
spatially homogeneous equilibrium point of coexistence, following the analysis presented
in [32] and [24]. Firstly, it is convenient to consider the system (10) in its adimensional
form. Introducing the new variables:
√
K
L
a
∗
∗
∗
∗
K =
, L =
, t = at, x =
x,
K∞
L∞
dK
substituting in (10), and dropping out the asterisks in order to mantain notation simplicity, we get the system in adimensional form:

∂ 2K
∂K



=
βh
+
, for x ∈ (0, l), t > 0


∂t
∂x2

(
)

2


 ∂L = g + d ∂ L − χ ∂ L ∂K for x ∈ (0, l), t > 0
∂t
∂x2
∂x
∂x
(13)



K(t, x) = K0 (x), L(t, x) = L0 (x), for x ∈ Ω = [0, l], t = 0





∂L
∂K


= 0,
= 0, for x ∈ ∂Ω = 0, l, t > 0
∂x
∂x
where β = aδ , d =
dL
,
dK
χ=
a χL
b dK
1
( sA ) 1−ϕ
δ
, and, from now on, we redefine h and g as being:
h(K, L) = K ϕ L1−ϕ − K
g(L) = L − L2 .
6
(14)
Defining the vector U = (K, L)T , we can write the system (13) in vector form:

∂ 2U
∂U


=
f
(U)
+
D
+ w (U)


∂x2
 ∂t
U(0, x) = U0 (x) on Ω




 ∂U(t, x) = 0 on ∂Ω
∂x
where:
D=
(
1 0
0 d
)
(
, f (U) =
βh(K, L)
g(L)
)
(
and w(U) =
0(
∂
−χ ∂x
L ∂K
∂x
)
(15)
)
.
(16)
The spatially homogeneous equilibrium points of equation (15) are the solutions of
f (U) = 0. It turns out that:
U∞ = (K∞ , L∞ )T = (1, 1)T
(17)
is the unique spatially homogeneous equilibrium point of coexistence between capital and
labor for this equation. Defining the spatially heterogeneous small perturbation:
(
)T
u = U − U∞ = uK , uL ,
we can linearize (15) about U∞ . Using Taylor Theorem and dropping the higher order
terms, we obtain:
∂u
∂ 2u
+ Dχ 2 = Au
(18)
∂t
∂x
where:
(
) ( ∂h ∂h ) a11 a12
∂K
∂L
A=
=
∂g
∂g
a21 a22
U∞
∂K
∂L
is the Jacobian matrix of f evaluated at U∞ , and:
(
)
1 0
Dχ = −
.
−χ d
By applying the spatial Fourier transform in (18), we obtain the time dependent first
order system:
∂ũ
− k 2 Dχ ũ = Aũ
(19)
∂t
where ũ(t, k) is the Fourier transform of u(t, x). Thus the perturbation can be represented
as a sum of spatial harmonic waves e−ikx ũ(t, k):
∫ ∞
1
e−ikx ũ(t, k)dk
(20)
u(t, x) =
2π −∞
where ũ(t, k) is the solution of (19), that is:
)
∂ũ (
= A − k 2 Dχ ũ.
∂t
(21)
We can solve this later system by using the Euler method that involves the eigenvalues
of the matrix (A − k 2 Dχ ) for each wave number k. Seeking exponential solutions eσt v of
(21), we obtain the eigenvalue problem:
)
(
(22)
σv = A − k 2 Dχ v
7
or, equivalently, we need to find non zero solutions v of the system:
(
)
Mv = σI − k 2 Dχ − A v = 0.
In terms of values, we have the homogeneous linear system:
(
)(
) ( )
(σ − a11 + q)
−a12
C
0
=
−(a21 + χq) (σ − a22 + dq)
D
0
(23)
2
2
where
( q =)k , and σ(k ) is the growing rate of the mode of wavenumber k and amplitude
C
v=
.
D
The linear system (23) admits nontrivial solutions if and only if M is singular, that is:
P (σ) = det M = (σ − a11 + q)(σ − a22 + dq) − a12 (a21 + χq) = 0.
(24)
This later characteristic equation can be conveniently written as:
P (σ) = σ 2 + z(q)σ + w(q) = 0
(25)
z(σ) = (1 + d)q − tr A
w(σ) = dq 2 − (a11 d + a22 + a12 χ)q + det A.
(26)
where:
By linear superposition, we have that for any set of constants c1 and c2 :
ũ(t, k) = c1 eσ1 t v1 + c1 eσ2 t v2
(27)
is a solution of (21). Here v1 and v2 are the corresponding eigenvectors to the eigenvalues
σ1 and σ2 . When the roots σ1 and σ2 of (25) are distinct, the constants can be determined
in terms of the transformed initial value ũ(0, k).
Definition - The equilibrium point U∞ is said linearly stable whenever Re{σ} < 0,
that is, when the modes go to zero as t → ∞.
Proposition 1 - Conditions for Linear Stability
i. If d = 0 and χ = 0, then U∞ is linearly stable if and only if:
tr A − q < 0 and det A − a22 q > 0.
ii. If d ̸= 0 and χ = 0, then U∞ is linearly stable if and only if:
(1 + d)q − tr A > 0 and dq 2 − (a11 d + a22 )q + det A > 0.
iii. If d ̸= 0 and χ ̸= 0, then U∞ is linearly stable if and only if:
(1 + d)q − tr A > 0 and
dq 2 − (a11 d + a22 + a12 χ)q + det A > 0.
8
Proof: We will show only (iii), since (i) and (ii) follows directly if we take: (i) the capitalinduced migration and diffusion coefficients equal zero; and (ii) only the capital-induced
labor migration coefficient equals zero. From (25) we have that:
σ1 =
)
)
√
√
1(
1(
−z + z 2 − 4w and σ2 =
−z − z 2 − 4w .
2
2
Note that the real part of both roots σ1 and σ2 must be negative in order to U∞ to be
stable, and this is true if z > 0 and w > 0. Thus (iii) follows ■
Now, considering the Cobb-Douglas production function and the logistic growth, so we
have g and h given by (14), we have that the Jacobian A becomes:
(
) (
)
a11 a12
−β(1 − ϕ) β(1 − ϕ)
A=
=
,
(28)
a21 a22
0
−1
what implies tr A = −1 − β(1 − ϕ) < 0 and det A = β(1 − ϕ) > 0, since ϕ ∈ (0, 1) and
β > 0. Therefore (i) is always satisfied. Besides of that, (ii) is always valid, since:
tr A < 0 ⇒ (1 + d)q − tr A > 0
(29)
dq 2 − (a11 d + a22 )q + det A = dq 2 + (β(1 − ϕ)d + 1)q + det A > 0
(30)
and:
where d, q = k 2 ≥ 0.
Proposition 2 - If there is no capital-induced labor migration, i.e., if χ = 0, then U∞
is linearly stable.
Proof - Note that, in this case:
w(q) = dq 2 + (β(1 − ϕ)d + 1)q + β(1 − ϕ)
is non-negative since β, q = k 2 ≥ 0 and 0 < ϕ < 1 ■
Remark 2 - Following Turing [30], we say that (13) presents diffusive instability if: (i)
U∞ is linearly stable in the absence of all difusive and reactive terms; and (ii) U∞ is
linearly unstable in the presence of the reactive and difusive terms only. We can easily
show that (13) satisfies (i) but, by Proposition 2, not (ii). Then, our model does not show
diffusive instability. Therefore, it is necessary the presence of capital-induced labor migration (χ > 0) to have linear instability, regime in which the economy may develop more
complex spatio-temporal dynamics, instead of converging to a homogeneous steady-state
□
Proposition 3 - The equilibrium point U∞ is linearly unstable if the following dispersion relations is satisfied:
w(q) = dq 2 + [1 + β(1 − ϕ)(d − χ)] q + β(1 − ϕ) ≤ 0.
(31)
Proof - Since (29) is always true, the only way the system can show linear instability is
9
if the second inequality in (iii) is not satisfied, that is, if we have the following dispersion
relation:
w(q) ≤ 0 ⇔
dq 2 − (a11 d + a22 + a12 χ)q + det A ≤ 0.
Now, it is enough to consider (28) to get the result ■
Corollary 1 - Consider d > 0. Then, the equilibrium point U∞ is linearly unstable if
the following inequality is satisfied:
[√
χ≥
]2
√
1
+ d =: χc .
β(1 − ϕ)
(32)
Proof - Since d > 0, the parabola w(q) given by (31) will have a global minimum at:
q∗ = −
1
[1 + β(1 − ϕ)(d − χ)]
2d
and then, to have instability, we need w(q ∗ ) ≤ 0, what implies (32) ■
Remark 3 - Rewriting (32) in terms of the dimensional variables:
(
χL ≥ b
δϕ
sA
1 (√
) 1−ϕ
√
δdL
+
a
dK
1−ϕ
)2
=: χ̃c ,
(33)
we can see that a bigger diffusion coefficient, dK or dL , and a bigger capital depreciation
rate, δ, implies in a bigger critical capital-induced labor migration coefficient χ̃c ; a smaller
a
labor maximum capacity for the economy (keeping a fixed) and a smaller technological
b
factor, A, also implies in a bigger χ̃c . On the other hand, a bigger labor growth rate a
and a bigger saving rate s imply in a smaller χ̃c □
Remark 4 - The previous linear stability analysis is valid for unbounded domains. If we
restrict these results to the bounded region Ω = [0, l], and consider homogeneous no-flux
boundary conditions, we get that the eigenvalues – values of k that generate nontrivial
solutions – are the discrete wavenumbers given by:
k=
nπ
n2 π 2
⇒ q = k 2 = 2 , n = 0, 1, 2, . . .
l
l
(34)
and then, the perturbation u(t, x) given in in (20) can be written as cosine Fourier series:
( nπx )
∑ ( Cn )
σ (k2 )t
cos
e
(35)
u(t, x) =
Dn
l
where the constants Cn e Dn depend on the initial conditions. If (32) is satisfied, we have
that σ(k 2 ) is positive only in an interval of wavenumbers:
√ √
k 2 = q ∈ (q1 , q2 ) ⇔ k ∈ (k1 , k2 ) = ( q1 , q2 ).
10
Here q1 and q2 are the positive roots of the parabola (31), and depends on χ. Since the
wavenumbers are discrete, we have that1 :
⌈
⌉
⌊
⌋
k1 (χ)l
k2 (χ)l
n1 =
and n2 =
.
(36)
π
π
From this we can see that, in addition to χ satisfy (32), the domain size, l, must be large
enough to guarantee the existence of natural numbers n ∈ [n1 , n2 ], and therefore the
existence of unstable modes. We call this smaller size, such that exist unstable modes,
of lc = lc (χ). The number of unstable modes, as a function of χ and l, is given by:
{ ⌊
⌋ ⌈
⌉
}
k2 (χ)l
k1 (χ)l
max{0, n2 − n1 + 1} = max 0,
−
+1 .
(37)
π
π
Therefore, (35) can be approximated as the sum of the growing modes only:
)
n2 (
( nπx )
∑
Cn
σ (k2 )t
u(t, x) ≈
e
cos
□
Dn
l
(38)
n=n1
4
Numerical Simulations
In this section we present some numerical approximations for the solutions of the model,
which were obtained using a standard explicit finite-difference scheme. In this way we
verify the linear stability analysis results presented in the previous section, and ilustrate
the kinds of spatio-temporal dynamics that this model can generate. We consider that
the capital and labor diffusion coefficients are equal, what implies d = 1, that β = 2.5,
what follows if we consider the capital depreciation rate 2.5 bigger than the labor growth
rate (δ = 2.5a), and that ϕ = 0.5. With these parameters, from (32) we have that
χc = 95 + √45 ≈ 3.59. So, if χ < χc , U∞ is linearly stable, and the economy will converge
to a spatially homogeneous steady state. Otherwise, if χ > χc , U∞ is linearly unstable,
and we should expect more complex dynamical behaviors.
In Figure 1 (a) we can see the unstable discrete modes as a function of the capital-induced
labor migration coefficient, χ, when l = 12 – the set of unstable modes for a given χ consists of all natural numbers between n1 and n2 (n1 and n2 are given by (36)). Note that
we only have unstable modes for χ > χc . In (b) we can see that the number of unstable
modes increases with the value of χ (this graph was obtained using (37)). In Figure
(c)-(d) we have the unstable discrete modes as a function of the economy’s size, l, for
χ = 5 and χ = 10. In (c) we have the interval of unstable modes, and in (d) the number
of unstable modes. As we can see, the number of unstable modes grows linearly with the
economy’s size (d), but sublinearly with the capital-induced labor migration coefficient
(b). This follows directly from (31), (36) and (37).
In Figure 2 we present the spatio-temporal evolution of an initial small random perturbation of the labor spatially homogeneous equilibrium L∞ = 1, for some representative
values of χ, when l = 12 (the evolution for the capital density is qualitatively similar). As
Note that ⌊x⌋ is the largest integer not greater than x, and ⌈x⌉ is the smallest integer not smaller
than x.
1
11
(a)
(b)
20
20
n
18
n
16
2
18
16
Number of Unstables Modes
1
n1, n2
14
12
10
8
6
4
12
10
8
6
4
2
2
0
14
0
5
10
15
20
0
25
0
5
10
(c)
25
60
80
100
(d)
90
chi=5
chi=10
Number of Unstable Modes
90
80
70
n1, n2
20
100
100
60
50
40
30
20
10
0
15
chi
chi
80
70
60
50
40
30
20
10
0
20
40
60
80
0
100
l
0
20
40
l
Figure 1: (a)-(b) Unstable discrete modes as a function of the capital-induced labor
migration coefficient, χ, for an economy of size l = 12. (c)-(d) Unstable discrete modes
as a function of the economy’s size, l, for χ = 5 and χ = 10.
we can see, if χ = 3 < χc , the labor distribution converges to its spatially homogeneous
equilibrium L∞ = 1. If χ > χc , we can observe three kinds of spatio-temporal behaviors:
the labor force (and the capital) converges to a new non-homogeneous stable equilibrium
(χ = 5, 7); the economy shows periodic spatio-temporal cycles (χ = 10); and the economy develops aperiodic spatio-temporal cycles (χ = 17.5), what suggests the existence
of a chaotic behavior [24]. Note that, in this figure (and in the following), white means
a labor density bigger than one, gray a density of one, and black a density smaller than
one. A greater χ implies in more unstable modes, and therefore we have a more complex
spatio temporal behavior.
In Figures 3 and 4 we show the spatio-temporal evolution of an initial non-homogeneous
small random perturbation of the labor spatially homogeneous equilibrium L∞ = 1, for
some representative values of l, when χ = 5 and χ = 10, respectively. In both cases
the initial distribution of capital is given by its homogeneous equilibrium K0 (x) ≡ 1.
When χ = 5 (Figure 3), lc = 1.65, and then, for l ≥ lc the economy converges to a
new non-homogeneous steady-state, with the formation of labor and capital clusters. If
l = 1 < lc , the economy converges to a homogeneous steady-state. Note that the number
of clusters increases with the size of the economy, since in this case more modes are
unstable. On the other hand, when χ = 10 (Figure 4), what implies in lc = 0.99, we
have more unstable modes for a given l ≥ lc than in the case where χ = 5 (see Figure 1
12
(d)). So we observe a richer variety of behaviors: (i) for l = 1, 2, 4, 10, 15, the economy
converges to a non-homogeneous steady-state; (ii) for l = 7, 9, 12, the economy develops
periodic spatio-temporal cycles; and for l = 14, 19, 22, 24, the economy develops irregular
and aperiodic spatio-temporal cycles. In Figure 5 we ilustrate the case (i) for l = 15.
Here we show the initial (t = 0) and final (t = 400) spatial profiles for the evolution of
initial gaussian distributions of capital and labor. As we can see, the economy converges
to a non-homogeneous equilibrium, with the formation of locally coincident clusters of
capital and labor.
Figure 2: Spatio-temporal behavior of the labor force L(t, x) as a function of χ, for
an economy of size l = 12. Labor initial condition is a random perturbation of the
equilibrium L ≡ 1, and capital initial condition is given by K ≡ 1. White means a labor
density bigger than one, and black a density smaller than one.
In Figure 6 we show the temporal evolution of capital and labor for some local economies
for the case (i) – l = 15, and for the economy’s aggregate capital, KT , and labor LT ,
which is shown in the last column. In the last line we have the phase diagrams K vs. L,
where we can check the convergence to a stable equilibrium. In Figure 7 we do the same
for the case (ii), when l = 9. In the phase diagrams we can see the periodic temporal
cycles. Finally, in Figure 8, we have case (iii), for l = 23. Again, in phase diagrams we
can see irregular cycles in some local economies (x = 9.20 and x = 11.50), and for the
aggregate economy. This messy phase diagrams suggest the existence of chaotic behavior
in the system [24, 25].
13
Figure 3: Spatio-temporal behavior of the labor force L(t, x) as a function of l, for χ = 5.
Labor initial condition is a random perturbation of the equilibrium L ≡ 1, and capital
initial condition is given by K ≡ 1. White means a labor density bigger than one, and
black a density smaller than one.
Figure 4: Spatio-temporal behavior of the labor force L(t, x) as a function of l, for χ = 10.
14
2
0.7
t=0
t=400
1.8
0.6
1.6
1.4
0.5
1.2
K
L
0.4
1
0.3
0.8
0.6
0.2
0.4
0.1
0.2
0
0
1
2
3
4
5
6
7
8
0
9 10 11 12 13 14 15
0
5
x
10
15
x
Figure 5: Formation of stable clusters of capital and labor when l = 15 and χ = 10,
considering initial gaussian distributions.
0.5
0
400
1.5
0
200
t
1
0
200
t
0.6
0.4
0
400
0
200
t
2
L(t,0)
0.3
0.2
0
200
t
LT(t)
20
0
200
t
0
400
1.5
60
1
40
0.5
0
400
0
200
t
400
0
200
t
400
20
0
200
t
0.4
60
0.2
0.1
0.1
L(t,3.75)
0
0
0.5
L(t,6.0)
0
40
20
0.2
0
0
400
0.3
0.6
K(t,6.0)
K(t,3.75)
0
0
0.8
0.1
0
0
400
0.4
0
400
0.4
0.5
200
t
0.2
0.5
1
0
40
0.6
0.2
0
2
1
0.8
K(t,6.0)
K(t,3.75)
0.5
60
0
400
0.8
1
3
1
KT(t)
200
t
1.5
KT(t)
0
80
0.5
1
0
4
K(t,7.5)
2
1
L(t,6.0)
L(t,3.75)
L(t,0)
3
2
L(t,7.5)
1.5
4
K(t,7.5)
5
0
0.05
L(t,7.5)
0.1
0
0
50
LT(t)
Figure 6: Time evolution of some local economies for the case where the economy converges to a non-homogeneous steady-state (χ = 10, l = 15). In the last columns we have
the time evolution for the aggregate economy, and in the last column the corresponding
phase diagrams.
15
0
1.5
0
200
t
400
0.8
0.4
0.2
0
200
t
0
400
1.4
K(t,2.25)
1.2
1
0.8
0.6
0.4
0
T
L (t)
0
200
t
0
400
1.5
0
200
t
0.4
400
0
200
t
400
0
200
t
0
400
50
0.6
0.6
0.6
45
0.4
0.4
0.2
0.2
0.2
0
0
0
0
1
L(t,2.25)
2
200
t
400
40
50
LT(t)
60
20
0
0.8
5
0
40
0.5
0.8
L(t,0)
400
1
0.8
0.4
200
t
60
0.6
0
0
80
0.2
K(t,3.60)
0
400
1
K(t,3.60)
K(t,2.25)
0.5
200
t
0.8
0.6
1
0
T
400
20
0
K (t)
200
t
0.5
0
40
1
0
1
L(t,3.60)
2
T
0
0.5
2
K (t)
1
80
60
1
K(t,4.50)
2
1
3
L(t,4.50)
2
1.5
K(t,4.50)
3
0
2
1.5
L(t,2.25)
4
L(t,3.60)
5
40
35
0
0.5
L(t,4.50)
30
30
1
Figure 7: Time evolution of some local economies for the case where the economy develops
periodic cycles (χ = 10, l = 9). In the last columns we have the time evolution for the
aggregate economy, and in the last column the corresponding phase diagrams.
2
0
4
3
3
2
1
0
200
t
0
400
1.5
2
1
0
200
t
0
400
1.5
60
40
LT(t)
4
3
L(t,11.50)
L(t,0)
L(t,5.75)
4
4
L(t,9.20)
6
2
20
1
0
200
t
0
400
1
0
200
t
0
400
1.5
0.5
0.4
1
KT(t)
0.6
0.5
0.2
0
0
200
t
0
400
1.4
0
200
t
0
400
400
0
200
t
400
40
LT(t)
60
40
K(t,11.50)
0.5
K(t,9.20)
1
200
t
50
0.8
1
0
30
20
10
0
200
t
0
400
1.5
0
200
t
0
400
1
45
1
0.8
1
0.5
0.6
0.4
0.6
0.4
5
L(t,0)
10
0
0
2
L(t,5.75)
4
0
0.4
0
2
L(t,9.20)
0
35
30
0.2
0.2
0
40
0.6
KT(t)
0.8
0.8
K(t,11.50)
1
K(t,9.20)
K(t,5.75)
1.2
0
1
L(t,11.50)
2
25
20
Figure 8: Time evolution of some local economies for the case where the economy develops
irregular and aperiodic cycles (χ = 10, l = 23). In the last columns we have the time
evolution for the aggregate economy, and in the last column the corresponding phase
diagrams.
16
5
Conclusions
In this work we proposed the introduction of a capital-induced labor migration in the
Spatial Solow Model found in the literature, and showed that the inclusion of this feature
in the model gives rise to a rich variety of spatio-temporal dynamic for the capital and
labor densities in a bounded autarchic economy, when a Cobb-Douglas production function, and a logistic growth for the labor force are considered. In addition, we noted that
the model proposed here is similar to chemotaxis models found in the Biomathematics
literature, what suggests that a deeper interaction between these two fields may prove to
be fruitful.
Using a conservation of capital and labor argument, we first derived the equations of
the model in one spatial dimension, and then, for simplicity, considered its adimensional
form. Analysing the linear stability of its unique spatially homogeneous equilibrium of
coexistence between capital and labor, we derived conditions for which this equilibrium
is unstable. In particular, we derived a critical value for the capital-induced labor migration coefficient, χc , such that: if χ < χc , the homogeneous equilibrium is stable, and the
economy always converge to a homogenous steady-state; and if χ > χc , the equilibrium
is unstable, and the economy shows a more complex behavior. Besides of that, we found
a critical size for the economy, lc , depending on χ, such that we have stability if l < lc
and instability if l > lc .
When the economy is in its unstable regime, i.e., when χ > χc and l > lc (χ), we identified
three kinds of spatio-temporal behaviors, with the distributions of capital and labor: (i)
converging to a spatially non-homogeneous steady-state; (ii) developing periodic spatiotemporal cycles; and (iii) developing irregular and aperiodic spatio-temporal cycles, what
suggests that the model can show a chaotic behavior.
In terms of the dimensional parameters of the model, we have instability if χL > χ̃c , and
χ̃c depends on the economic parameters in the following way: a bigger diffusion coefficient, dK or dL , and a bigger capital depreciation rate, δ, imply in a bigger χc ; a smaller
a
labor maximum capacity of the economy (keeping a fixed) and a smaller technological
b
factor, A, also imply in a bigger χ̃c . On the other hand, a bigger labor growth rate a
and a bigger saving rate s imply in a smaller χ̃c . A bigger χ̃c means that it is necessary
a more intense capital-induced labor migration in order to destabilize the homogeneous
steady-state.
The stability results presented here can be easily generalized for the case of a two dimensional economy. Future research topics may include numerical simulations in two
dimensions, the inclusion of a labor-induced capital migration in the model, the use of
such a model as the basic dynamics in optimal space-time development problems, and
show the well-posedness of such a class of models.
17
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