In this paper, we solve a Solow model in continuous time and space. We prove the existence of a solution to the problem and its convergence to a stationary solution. The simulation of various scenarios in the last section of the paper illustrates the convergence issueSolow Model; Growth; Geographic economics

Benteng, ZOU

Carmen, CAMACHO

English

Research Papers in Economics

The spatial Solow model¤
Carmen Camacho
IRES
Benteng Zou
IRES
26th September 2003
Abstract
In this paper, we solve a Solow model in continuous time and space. We prove the
existence of a solution to the problem and its convergence to a stationary solution.
The simulation of various scenarios in the last section of the paper illustrates the
convergence issue.
Keywords:Solow model, Growth, Geographic economics.
JEL classiﬁcation:R1; O0.
1 Introduction
The inclusion of the geographical space in economic analysis has regained relevance in the
recent years. The emergence of a new economic geography discipline is indeed one of the
major events in the economic literature of the last decade (see Krugman, 1991 and 1993,
Fujita, Krugman and Venables, 1999, and Fujita and Thisse, 2002). Departing from the
early regional science contributions which are typically based on simple ﬂow equations
(eg. Beckman, 1952), the new economic geography models use a general equilibrium
framework with a reﬁned speciﬁcation of local and global market structures, and some
precise assumptions on the mobility of production factors.
Two main characteristics of the new economic geography contributions are: (i) the discrete
space structure, and (ii) the absence of capital accumulation. Since capital accumulation
¤We acknowledge the ﬁnancial support of the Belgian French speaking community (Grant ARC 99/04-
235 “Growth and incentive design”) and of the Belgian Federal Government (Grant PAI P5/10, “Equilib-
rium theory and optimization for public policy and industry regulation”). Address for correspondance:
Carmen Camacho, Universit´ e catholique de Louvain, place Montesquieu 3, B-1348 Louvain-la-Neuve,
Belgium. camacho@ires.ucl.ac.be
1is not allowed, the new economic geography models are losing a relevant determinant
of migrations, and more importantly, an engine of growth. It seems however clear that
many economic geography problems (eg. uneven regional development) have a preeminent
growth component, and vice versa. Thus, there is an urgent need to unify in some way
the new economic geography and the growth theory, or at least to develop some junction
models.
This paper constitutes a ﬁrst step following exactly this line of research. We study the
Solow model with space. Space is continuous and inﬁnite, and capital accumulation are
space dependent. In line with Mossay (2003), we shall allow for both dispersion and
convergence forces. The convergence force is the well known neoclassical mechanism
according to which poor regions attract capital because of decreasing returns to this
factor. Dispersion mechanisms are linked to space heterogeneity, given by region speciﬁc
technology and/or saving rate. If a region produces using a more advanced technology, it
attracts capital from less advanced regions, despite decreasing returns to scale. The same
result holds for a region that saves, and therefore invests, at a larger rate.
Neoclassical economic theory predicts that regions will converge in the long run under
perfect competition. However, this is not so. An argument that has been put forward
is that technological transfers between regions is far from perfect. The lack of transferee
expertise and poor training in the technology importing region, together with Govern-
ment barriers may impede an eﬀective technological transfer (see Niosi, Hanel and Fiset
(1995)). Boucekkine, Martinez and Saglam (2003) point out the role of capital goods
technological embodiment in technology adoption decisions. A developing country may
not adopt the most sophisticated technique since it implies replacing existing capital and
lose their technology-speciﬁc skills. The spatial Solow model allows to study the link
between technology transfers and development. Indeed, it is ﬂexible enough to study the
existence of technological poles with partial transfer to neighboring regions, as well as
more complicated patterns of knowledge diﬀusion across space and time.
The paper is organized as follows. Section 2 presents the spatial Solow problem. Section 3
is devoted to prove the existence of solutions, providing explicit solutions for the Ak case,
and their convergence to a steady state. Section 4 presents diﬀerent scenarios that bring
out the relevance of initial conditions and of space dependent technology and savings.
Section 5 concludes.
2 The model
In contrast to the standard Solow model, the law of motion of capital does not rely
entirely on the saving capacity of the economy under consideration: the net ﬂows of
capital to a given location or space interval should also be accounted for. Suppose that
households locate along the real line. The technology at work in location x is simply
y(x;t) = A(x;t)f(k(x;t)), where A(x;t)(¸ 0) stands for total factor productivity at
2location x and date t, and f(¢) is the production function, which satisﬁes the following
assumptions:
(A1) f(¢) is non-negative, increasing and concave;
(A2) f(¢) veriﬁes the Inada conditions, that is,
f(0) = 0; lim
k!0
f
0(k) = +1; lim
k!+1
f
0(k) = 0:
Moreover we assume that the production function is the same whatever is the location.
A(x;t) could be another heterogeneity factor. However, we will assume it is time inde-
pendent in the crucial parts of this paper, and hence, this heterogeneity could be omitted
from now on. The budget constraint of household x 2 R is
@k(x;t)
@t
= s(x;t)A(x;t)f (k(x;t)) ¡ ±k(x;t) + ¿(x;t); (1)
where s(¸ 0) is the savings rate and ¿ is the household’s trade balance. Since the economy
is closed:
Z
R
µ
@k(x;t)
@t
¡ s(x;t)A(x;t)f (k(x;t)) + ±k(x;t) ¡ ¿(x;t)
¶
dx = 0:
And if regions are considered as closed economies, then for any given region R ½ R:
Z
R
µ
@k(x;t)
@t
¡ s(x;t)A(x;t)f (k(x;t)) + ±k(x;t) ¡ ¿(x;t)
¶
dx = 0:
Capital ﬂows searching regions with high marginal productivity. So that capital move-
ments tend to eliminate geographical diﬀerences. Applying the fundamental theorem of
calculus to region R, then, the trade balance is equal to:
Z
R
¿(x;t)dx =
Z
R
@2k
@x2(x;t)dx:
Therefore, the budget constraint can be written as:
@k(x;t)
@t
¡
@2k(x;t)
@x2 = s(x;t)A(x;t)f (k(x;t)) ¡ ±k(x;t): (2)
The initial distribution of capital, k(x;0), is assumed to be known and C0. Moreover, we
assume that, if the location is far away from the origin, there is no capital ﬂow, that is
lim
x!§1
@k
@x
= 0:
We can write the problem as:
3(P)
8
> > > <
> > > :
@k
@t
(x;t) ¡
@2k
@x2(x;t) = s(x;t)A(x;t)f (k(x;t)) ¡ ±k(x;t);
k(x;0) = k0(x) > 0; x 2 R;
lim
x!1
@k
@x
= lim
x!¡1
@k
@x
= 0:
in R £ (0;1). k0 is deﬁned in L1
+(R), where L1
+(R) = fy 2 L1(R)jy(x) ¸ 0 for almost
every x 2 Rg.
3 Mathematical results
3.1 Existence
The literature on Partial Diﬀerential Equations provides us with an existence theorem for
problem P:
Theorem 1 If s, A are continuous and if f veriﬁes (A1) and (A2), there exists a unique
global continuous nonnegative solution to problem P.
Proof: If (x;t) 2 R£(0;T), where T < 1, it is well known there exists a unique bounded
solution to problem P (see Ladyzhenskaja, Solonnikov and Ural’ceva (1968)). Following
Hofbauer and Simon (2001), we obtain the global existence and uniqueness of the solution
to P in R £ (0;1).
Now we use Inada conditions to prove that the solution is nonnegative. Deﬁne k(x;t) =
e¡±tv(x;t), then v(x;t) satisﬁes the following problem:
(M)
8
<
:
@v
@t
(x;t) ¡
@2v
@x2(x;t) = s(x;t)A(x;t)f
¡
e
¡±tv(x;t)
¢
;
v(x;0) = k0(x) > 0; x 2 R;
By the ﬁrst part of this proof, we know that there exists a unique solution v(x;t) to prob-
lem (M). Using a comparison theorem, we assert that the above solution v is nonnegative,
provided that (A1) and (A2) hold. So does k(x;t). ¤
The following theorem gives an explicit solution for the Ak model.
Theorem 2 Suppose that the production function, f(k(x;t)) = k(x;t). If A and s are
constants, then the solution to problem P is given by
k(x;t) = e
(sA¡±)t
Z
R
Γ0(x ¡ y;t)k0(y)dy; (3)
4where
Γ0(x;t) =
8
<
:
1
(4¼t)
1
2
e
¡ x2
4t ; t > 0;
0; t < 0:
If A = A(x;t), s = s(x;t), the solution to problem (P) is given by
k(x;t) =
Z
R
Γ(x ¡ »;t)k0(»)d»;
where Γ is deﬁned as
Γ(x ¡ »;t ¡ ¿) = Γ0(x ¡ »;t ¡ ¿) +
Z t
¿
Z
R
Γ0(x ¡ ´;t ¡ ¾)Φ(´ ¡ »;¾ ¡ ¿)d´d¾;
and Φ satisﬁes
Φ(´ ¡ »;¾ ¡ ¿) =
1 X
º=1
(LΓ0)º(´ ¡ »;¾ ¡ ¿):
The operator L is recursively deﬁned, and it is given by
(LΓ0)1 = LΓ0 = (s(x;t)A(x;t) ¡ ±)Γ0(x;t);
(LΓ0)º+1(´ ¡ »;¾ ¡ ¿) =
Z ¾
¿
Z
R
((LΓ0)(´ ¡ y;¾ ¡ s))(LΓ0)º(y ¡ »;s ¡ ¿)dyds:
Proof: See Ladyzenskaja, Solonnikov and Ural’ceva (1968) and Friedman (1983).
Remark As proved by Ladyzenskaja, Solonnikov and Ural’ceva (1968), if k0(x) does not
increase too rapidly for jxj ! +1 (for example, not faster than ex2), then the integral in
(3) converges. Hence we can get the same order of growth rate as in the standard Solow
model.
Theorem 2 allows to clearly study the long run behavior of k(x;t) when A(x;t) = A and
s(x;t) = s, where s and A are constants. For if sA · ±, then from (3) one can check that
lim
t!1
k(x;t) = 0:
If an economy does not save at least to compensate for depreciation, then it will decay
until no capital is left.
If, on the contrary, sA > ±, we obtain that:
lim
t!1
k(x;t) = 1:
This implies that, as in the 1-dimensional case, the spatial Ak model does not have a
steady state.
53.2 Steady State and Convergence
We deﬁne a steady state solution to (1) by the standard conditions
@k(x;t)
@t = 0, A(x;t) =
A(x) and s(x;t) = s(x):
(PS)
8
> <
> :
@2k(x)
@x2 + s(x)A(x)f (k(x)) ¡ ±k(x) = 0;
lim
x!1
@k
@x
= lim
x!¡1
@k
@x
= 0;
We can reduce the problem into an ordinary diﬀerential equation 2-dimensional system:
@k(x)
@x
= w(x);
@w(x)
@x
=
@2k(x)
@x2 = ¡s(x)A(x)f(k(x)) + ±k(x):
then a solution to this system is given by,
k(x) =
Z x
¡1
w(z)dz;
w(x) =
Z x
¡1
(¡s(x)A(x)f(k(z)) + ±k(z)dz:
Any solution (k(x);w(x))x2R must also verify that,
lim
x!1
@k
@x
(x) = lim
x!¡1
@k
@x
(x) = 0; (4)
this implies that, limx!§1 w(x) = 0;. If k veriﬁes that
¡s(x)A(x)f(k(x)) + ±k(x) = 0; 8x (5)
then, the boundary conditions are veriﬁed and k is a particular solution of PS. Unfortu-
nately, the stationary solutions are not unique.
Theorem 3 If the production function f veriﬁes (A1) and (A2), then the nonnegative
solution k(t;x;k0) to problem P converges to a stationary solution as t ! 1.
Proof: The proof requires some minor changes to the proof provided in Bandle, Pozio
and Tesei (1987) for a similar problem.
63.3 Dynamic simulations
We illustrate in this section the behavior of solutions to P under diﬀerent scenarios. In
particular, we simulate the spatial Solow model with a Cobb-Douglas production function.
We consider various cases depending on initial conditions and on whether A and s are
constant or space-dependent.
Example 1: We shall consider in this ﬁrst example that, initially, all households are
equally endowed with one unit of physical capital. There are no geographical diﬀerences,
so that they save at a rate s(x;t) = 0:2 and the technological coeﬃcient A(x;t) = 10. The
capital share in the production production, ®, equals 1=3 and physical capital depreciation
± = 0:05.
-4
-2
0
2
4 0
5
10
15
20
25
0
25
50
75
100
-4
-2
0
2
4
Figure 1: Simulation results when space is homogenous.
Simulated capital reproduces a neoclassical growth path (see ﬁgure 1). Marginal produc-
tivity of capital is the same along the real line, so that investors are indiﬀerent among all
locations. Since there is no source of heterogeneity, all points produce and grow at the
same rates.
Example 2: We introduce heterogeneity at the initial endowment of capital to study
whether diﬀerences across regions may persist in the long run. We assume that k(x;0) =
e¡x2t. The rest of parameters take the same values as in example 1. Figure 2 shows that
after some iterations, initial diﬀerences are smoothed out and that, in the long run, all
points in space will be equally rich.
Example 3: In this example, A(x;t) = e¡x2, that is the central region uses a more advanced
technology. The central locations produce using a more advanced technology, and since
there is no technology transfer they remain leaders forever. The ﬁrst graph in ﬁgure 3
shows the growth path when the initial condition is spatially homogenous, k(x;0) = 1. In
the second graph, k(x;0) = e¡x2, which adds a further source of heterogeneity. Results
show that whichever the initial condition, any diﬀerence in technology which is not subject
to modiﬁcation through time (i.e. if there are no technological spill-overs from the center
to the periphery), leads to a non homogenous steady state.
7-4
-2
0
2
4 0
1
2
3
4
5
0
5
10
15
-4
-2
0
2
4
-4
-2
0
2
4 0
5
10
15
20
25
0
25
50
75
100
-4
-2
0
2
4
Figure 2: Simulation results for diﬀerent time horizons. On the left t 2 [0;5] and on the right
t 2 [0;25].
-4
-2
0
2
4 0
20
40
60
80
100
0.6
0.8
1
-4
-2
0
2
4
-4
-2
0
2
4 0
20
40
60
80
100
0
0.25
0.5
0.75
-4
-2
0
2
4
Figure 3: Left: k(x;0) = 1, 8x. Right: k(x;0) = e¡x2
, 8x
4 Conclusion
In this paper, we solve a Solow model in continuous time and space. We prove at the
same time, the existence of a solution to the problem and the convergence to a stationary
solution. Results coincide with the non-spatial neoclassical intuition. We obtain that
in the Ak case, the model does not have a steady state; furthermore, with a standard
neoclassical production function, this steady state exists and we prove convergence. If
space is homogenous, i.e. if all locations produce using the same technology and they save
at the same rate, then at the steady state, all locations have the same level of physical
capital. This is true whatever the initial condition. However, if spatial heterogeneity is
introduced at the level of the technology or savings rate, regional diﬀerences persist.
Further research in this ﬁeld should lead to the generalization of our results. A natural
8continuation is the extension to the Ramsey model, in which we are already working.
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The Spatial Solow Model

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