Population dynamics in a spatial Solow model
with a convex-concave production function
Vincenzo Capasso∗ , Ralf Engbers† , Davide La Torre‡
Abstract
In this paper the classical Solow model is extended, by considering
spatial dependence of the physical capital and population dynamics, and
by introducing a nonconcave production function. The physical capital
and population evolution equations are governed by semilinear parabolic
differential equations which describe their evolution over time and space.
The convergence to a steady state according to different hypotheses on
the production function is discussed. The analysis is focused on an Sshaped production function, which allows the existence of saddle points
and poverty traps. The evolution of this system over time, and its convergence to the steady state is described mainly through numerical simulations. Key words: The Solow model, Economic geography, Convexconcave production function, Poverty traps.
1
Introduction: the Solow model with a convexconcave production function
The Solow model [21], introduced in 1956, represents one of the milestones of
endogenous growth literature; despite its relative simplicity, it provides a first
dynamic model that is still used in today’s macroeconomic theory. Solow’s
purpose was to develop a model to describe the dynamics of the growth process
and the long-run evolution of the economy, ignoring short-run fluctuations.
For many economic growth models based on inter-temporal allocation the
hypothesis of a concave production function has played a crucial role. The production function is the most important part of an economic model. It specifies
the maximum output for all possible combinations of input factors and therefore determines the way the economic model evolves in time. The Cobb-Douglas
production function (see Figure 1) is by far the most used production function
for describing situations with substitutional input factors although there are of
course alternatives. Nonetheless, even if a Cobb-Douglas production function
∗ Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences
(ADAMSS) and Department of Mathematics, University of Milan, Via C. Saldini 50, 20133
Milan, Italy ([email protected])
† Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte
Mathematik, Einsteinstr. 62, D 48149 Münster, Germany ([email protected])
‡ Department of Economics, Business and Statistics, University of Milan, Via Conservatorio
7, 20122 Milan, Italy ([email protected])
§ The original publication is available at www.springerlink.com.
1
is not imposed, usually a production function f is assumed to be non-negative,
increasing and concave and also fulfill the so called Inada conditions (see [1]).
From an economic point of view, the Inada conditions say that it is possible
to gain infinitely high returns by investing only a small amount of money. This
obviously can not be realistic. Before getting returns it is necessary to create
prerequisites, by investing a certain amount of money. After establishing a
basic structure for production, one might get only small returns until reaching
a threshold where the returns will increase greatly to the point where the law
of diminishing returns takes effect. In literature this fact is known as poverty
traps (see [18]). In other words, we should expect that there is a critical level of
physical capital having the property that if the initial value of physical capital
is smaller than such a level, then the dynamic of physical capital will descend
to the zero level, thus vanishing any possibility of economic growth.
What happens to the Solow model if we do not assume Cobb-Douglas production functions or, more general, a nonconcave production function? A small
amount of money may have an effect in the short-run but this effect will tend
to zero in the long-run, if there are no more investments. Thus it makes sense
to assume that only an amount of money bigger than some threshold will lead
to returns.
A first model with nonconcave production function was introduced by Clark
[8] and Skiba [19]. Recently several contributions have focused on the existence
and implications of critical levels. In this paper we consider a parametric class
of nonconcave production functions
F (K, L) =
α1 K p L1−p
1 + α2 K p L−p
(1)
where K and L denote capital and labor respectively and all involved parameters are nonnegative. Note that, by setting α1 = 1, α2 = 0 and p = α, we
end up with the same equation as in the Cobb-Douglas case, thus (1) can be
understood as an extension of the Cobb-Douglas production function. It does
not satisfy some of the classical Inada conditions, thus allowing a larger variety
of dynamics and include the possibility of poverty traps. In particular, this production function shows an S-shaped behavior for p ≥ 2, and it can be classified
in the class of nonconcave or convex-concave production functions. Obviously
it fulfills lim(K,L)→(0,0) F (K, L) = 0 with a smooth junction in the area of the
threshold.
2
Spatially structured Solow model with population dynamics
At this point we introduce a spatial component to the model. Following some
other papers in literature (see, for instance, [2, 3, 9, 12]), we are assuming a
continuous space structure, as a mathematical representation of the assumption
that in modern economies all locations have access to goods. Thus, K(x, t)
denotes the capital stock held by the representative household located at x,
at date t, in a bounded domain Ω ⊂ Rn , n = 1, 2, t ≥ 0. We assume that
population (raw labor) coincides with the available number of workers and it is
2
driven by the following PDE

∂L

 ∂t (x, t) = ∆L(x, t) + L(x, t)gL (x), (x, t) ∈ Ω × [0, +∞),
∂L
(x, t) ∈ ∂Ω × [0, +∞),
∂n = 0,


L(x, 0) = L0 (x),
x ∈ Ω,
(2)
where gL is the population growth rate.
Furthermore, it appears evident to consider net flows of capital to a given
location or space interval to describe the motion of capital. We normalize the
saving capacity to one for simplicity, because we are more interested in the way
the production function and population dynamics affect economic growth. Thus
the budget constraint can be written as
∂K
(x, t) = ∆K(x, t) + F K(x, t), L(x, t) − δK(x, t)
(3)
∂t
for all (x, t) ∈ Ω × [0, +∞), where F is the production function, ∆ represents
the Laplacian operator and δ the physical capital depreciation. In addition to
(3) we assume that the initial capital distribution, K(x, 0), is known and that
there is no capital flow through the boundary ∂Ω, i.e. the directional derivative
∂K
∂n is equal to zero.
By combining all these equations, the model can be rewritten in a compact
form as
 ∂K
(x, t) = ∆K(x, t) + F K(x, t), L(K, t) − δK(x, t),

∂t


 ∂L (x, t) = ∆L(x, t) + L(x, t)g (x),
L
∂t
∂L
 ∂K
=
0
=
0,

∂n

 ∂n
K(x, 0) = K0 (x), L(x, 0) = L0 (x),
(x, t) ∈ Ω × [0, +∞),
(x, t) ∈ Ω × [0, +∞),
(4)
(x, t) ∈ ∂Ω × [0, +∞),
x ∈ Ω.
It is easy to verify that if a Cobb-Douglas production function is assumed,
then one can easily prove the existence of a nontrivial global stable equilibrium.
This is not true anymore if we assume nonconcave production functions; in fact,
under this hypothesis, classical results in literature concerning the existence
of nontrivial equilibria cannot be applied. The analysis of the S-shaped case
is much more complicated due to lack of concavity of the relevant evolution
operator. In particular, global uniqueness of a nontrivial steady state solution
is lost and the analysis of a saddle point behavior is required. Some analytical
results about local stability of steady states can be found in [4, 6, 7, 10, 13, 14,
20]. Here we shall limit ourselves to show the behavior of the system through
numerical simulations and we then discuss the obtained results.
3
Numerical simulations
The main goal of this section is to describe the long run behavior of the model

p
1−p

 ∂K (x, t) = ∆K(x, t) + α1 K(x, t) L(x, t)
− δK(x, t),

∂t
p

1 + α2 K(x, t) L(x, t)−p


∂L
(x, t) = ∆L(x, t) + L(x, t)gL (x),
∂t

∂K

 ∂n = 0 ∂L
= 0,

∂n


K(x, 0) = K0 (x), L(x, 0) = L0 (x),
3
(x, t) ∈ Ω × [0, +∞),
(x, t) ∈ Ω × [0, +∞),
(x, t) ∈ ∂Ω × [0, +∞),
x ∈ Ω,
(5)
Figure 1: Cobb-Douglas type production Figure 2: Production function according
function
to (1)
when an S-shaped production function is assumed. Due to its complexity, we
shall limit ourselves to show this analysis through numerical simulations and
then discuss the obtained results.
For our numerical results we use (1) with α1 = α2 = 0.0005 and p = 4 (see
Figure 2).
Our production function is in some sense flattened when the amount of labor
is larger than a certain amount of capital and this will allow the possibility of
poverty traps. We examine the behavior of our spatial Solow model for different
combinations of values. In particular we will look at some threshold situations
where the solution depends on even small changes of the parameters. Concerning
the instantaneous depreciation rate of physical capital (δ = 0.05) we follow the
baseline specification of Mulligan and Sala-i-Martin ([15], p. 761).
We provide two numerical simulations with, respectively, gL (x) = 0.0144 and
x2
gL (x) = 0.0144e− 2 . The parameter gL , giving the change of population size
over time, represents the so-called growth rate coefficient. In the first numerical
simulation, we attribute to this parameter the value of 0.0144, which is the
average growth rate of the labor-force in the U.S. private business sector over
the period 1948-1997 ([11], Table 1, p.73). Such a value is not distant from
the estimate of 0.019 obtained for the population of Great Britain from 1801
to 1971 ([16], p.360). In the second numerical simulation we assume spatial
heterogeneity with gL (x) = 0.0144e−
3.1
x2
2
.
Numerical results and discussion
In this part of this section we assume to have homogeneous labor growth with
a constant gL (x) = 0.0144.
Figure 3 shows the long run behavior of L over space and time. Depending
on the initial capital, the behavior of the solution over space and time can vary.
We first assume that the initial capital is shaped as a piecewise linear function
as shown in Figure 4. Figure 5(a) shows that the solution tends to the trivial
stationary solution when there is not enough money. On the other hand, with
more initial capital available, the solution will grow showing a long run behavior
similar to that occurring for labor (see Figure 5(b)).
x2
Let us now consider the heterogenous case by assuming gL (x) = 0.0144e− 2 .
Since we run these numerical simulations when x ∈ [0, 1], we rescaled the func-
4
30
25
20
k
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 4: Initial distribution of physical
Figure 3: Labor growth gL (x) = 0.0144 capital
(a) Homogeneous labor growth – tending to zero, because there is not
enough money to sustain labor growth
(b) Homogeneous labor growth –
will be increasing according to labor
growth because there is enough money
initially available
Figure 5: Comparing simulations for homogeneous labor growth and different initial
capital
tion gL (x) to obtain a bell-shaped function in this interval. So gL (x) actually
−
(x−0.5)2
looks like gL (x) = 0.0144e 2×0.22 . Figure 6 shows the behavior of L(x, t) over
time and space.
The simulations we obtain for the case of heterogeneous labor growth are
provided in Figure 7. Figure 7(a) describes the situation in which we assume
an high initial capital level. For those spatial locations in which there is enough
initial capital to sustain labor growth, the model exhibits a long-run behavior
similar to that occurring for L(x, t), i.e. the locations with the highest labor
growth will show the highest economic growth as well. The other locations, in
which a low level of initial capital is assumed, will remain stuck in the poverty
trap: there is no available money to escape from there (only small amounts
of money are flowing into that region due to diffusion). The case in which a
medium initial capital level is assumed is shown in Figure 7(b). Under this initial
condition, only the locations (on the left part of the interval [0,1]) which show
an initial high capital and low labor growth are able to sustain economic growth.
The locations in the center of the interval, which show an high population growth
rate but high labor growth, are stuck in the poverty trap.
To better illustrate the effect of the heterogeneous labor growth, we now run
two numerical simulations with constant initial capital levels. In Figure 8(a)
5
−
Figure 6: Labor growth according to gL (x) = 0.0144e
(a) High initial capital - will tend to a
solution similar to L(x, t)
(x−0.5)2
2·0.22
(b) Medium initial capital - high labor
growth is a disadvantage at first
Figure 7: Comparing simulations for homogeneous labor growth and constant labor
- high initial capital
we assume k1,0 (x) = 5.3 while in Figure 8(b) we assume k2,0 (x) = 5.5. In both
cases, at the very beginning of the simulation, the level of capital in the center
of the interval (which shows an high population growth rate) will quickly tend
to zero while the remaining parts of the interval show economic growth. The
situations radically changes in the long-run: in the first case there is not enough
flow of capital to sustain the economy and, as a consequence, the level of capital
will decay to zero. However, in the second case, the spatial flow of capital is
enough to sustain economic growth and this will affect the level of capital in the
center of the interval which will escape the poverty trap.
4
Conclusion
By comparing the dynamics of this model with the one presented in [5], where an
higher technological level was always better than a lower one, here higher values
of L are not an advantage in all situations. The economic motivations of these
numerical evidences are quite easy to interpret; according to Romer’s definition
of ideas [17], the use of ideas by one person does not diminish others’use and
therefore ideas are non-rival goods. If the level of technology increases this is a
benefit for the whole economy. On the other hand, an high level of population
together with a low/medium level of capital are not enough to sustain economic
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(a) Constant initial capital - all capital
decays
(b) Constant initial capital - center escapes poverty trap
Figure 8: Comparing simulations for homogeneous labor growth and constant labor
- high initial capital
growth; in some situations the flow of capital can help to escape the poverty
traps, but in other cases this does not happen and the economy collapses.
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