Applied Mathematical Sciences, Vol. 1, 2007, no. 40, 1991 - 2000
Closed Form Solutions to a Generalization
of the Solow Growth Model
Juan Gabriel Brida1
School of Economics and Management
Free University of Bolzano, Italy
JuanGabriel.Brida@unibz.it
Erick José Limas Maldonado
Universidad Autónoma de Ciudad Juárez
Posgrado en Ciencias Económicas; Mexico
limas.erick@gmail.com
Abstract
The Solow growth model assumes that labor force grows exponentially. This is not a realistic assumption because, exponential growth
implies that population increases to infinity as time tends to infinity.
In this paper we propose replacing the exponential population growth
with a simple and more realistic equation - the Von Bertalanffy model.
This model utilizes three hypotheses about human population growth:
(1) when population size is small, growth is exponential; (2) population
is bounded; and (3) the rate of population growth decreases to zero as
time tends toward infinity. After making this substitution, the generalized Solow model is then solved in closed form, demonstrating that the
intrinsic rate of population growth does not influence the long-run equilibrium level of capital per worker. We also study the revised model’s
stability, comparing it with that of the classical model.
Mathematics Subject Classification: 91B62
Keywords: Solow growth model, population growth
1
Introduction
This paper describes a generalization of the Solow economic growth model
that allows population growth rate to diminish over time. The original Solow
1
Corresponding author. Tel.: +39 0471 013492, Fax: +39 0471 013 009
1992
J. G. Brida and E. J. Limas Maldonado
growth model assumes that the labor force L grows at a constant rate n > 0. In
continuous time, the population growth rate used in the original Solow model
is
n=
∂L
L̇
= ∂t
L
L
(1)
for any initial level L0 , at time t the level of the labor force is
L(t) = L0 ent
(2)
This assumption is realistic only for small values of the labor force because,
with unlimited exponential growth, as time tends toward infinity, so too does
the labor force. Several studies support the hypothesis that the world’s population growth rate is deceasing and tends toward zero.2 Natural resources are
limited, implying that food shortages, unemployment, and pollution eventually
limit population growth. This limit is usually called the carrying capacity of
the environment (denoted herein as L∞ ) and forms a numerical upper bound
on the population size.
Therefore, as described by Maynard Smith [9], a more realistic model of
the growth of the labor force L(t) must exhibit the following properties:
1. when population is small enough in proportion to environmental carrying
capacity L∞ , then L grows at a constant rate n > 0.
2. when population is large enough in proportion to environmental carrying
capacity L∞ , the economic resources become scarcer, reducing the rate
of population growth.
3. population growth rate decreases to 0 over time.
In this paper we assume that the labour force L(t) exhibits all these properties. In particular, we introduce the von Bertalanffy equation (von Bertalanffy
(1938)).3 This model is widely used in population studies and data analysis4 and is one of the simplest realistic models of population dynamics that
incorporate all the properties previously introduced.
2
See, for example, Day (1996)
See Mingari Scarpello et al. (2003) for a similar generalization, which uses the logistic
model of population growth. In this paper the authors focus upon the solutions to the
equation of motion in a closed form by using the special functions.
4
Cloern et al. (1978) apply the von Bertalanffy equation to predict body growth for
Macoma balthica. Jurado-Molina et al. (1992) conduct a study in which the von Bertalanffy
equation is applied to Mugil curema to calculate its weight growth. Xiao (2000) uses the
von Bertalanffy model to calculate the parameters from a set of tagging data concerning
times at liberty, lengths at release, and lengths at recapture of a Lates calcarifer. Anislao,
3
Generalization of the Solow growth model
1993
The von Bertalanffy function is the solution of the initial value problem:
L̇ = r (L∞ − L)
(3)
L(0) = L0
where L∞ is a theorical maximum asymptote size of the labor force (carrying
capacity), L0 is the labor force at time t0 and r is a constant which determines
the speed at which the labor force reaches the asymptote. This equation shows
that the growth rate is a decreasing linear function of the population size. In
addition, the equation verifies four relevant facts:
1. The growth rate is greatest when the population size is smallest;
2. The growth rate decrease to zero as the population size approaches the
carrying capacity;
3. If the population size is greater than the carrying capacity, the growth
rate is negative; and
4. There is a constant upper limit on population size (L∞ ).
This equation has only one steady state L∞ and the solution is given by:
L(t) = L∞ − (L∞ − L0 )e−rt
(4)
Note that, with the von Bertalanffy model, the growth rate becomes:
n(t) =
r (L∞ − L)
r (L∞ − L0)
L̇(t)
=
=
−rt
L(t)
L∞ − [L∞ − L0 ] e
(L0 − L∞ ) + L∞ ert
(5)
which decreases monotonically to 0 as t tends to infinity.
2
The model
The original Solow model assumes that:
1. There is an aggregate production function Y = F (K, L), which is assumed to satisfy a series of technical conditions, such as:
Auró, and González (2002) present a work in which the speed of growth of Cyprinus carpio
was estimated with data including length and weight of Cyprinus carpio, a work in which
they employ the von Bertalanffy equation. Finally, De Graaf and Prein (2005) compare
three approaches to the multivariate analysis of Oreochromis niloticus growth, based upon
the von Bertalanffy equation.
1994
J. G. Brida and E. J. Limas Maldonado
(a) it is increasing in both factors;
(b) it shows decreasing marginal returns for each factor;
(c) it displays constant returns to scale; and
(d) it satisfies the Inada conditions.
Among all the possible production functions satisfying these properties, we
shall assume the Cobb-Douglas function, as it is the most often cited in the
literature:
F (K, L) = K αL1−α ; 0 < α < 1
(6)
2. The change in capital stock equal the gross investment I = sF (K, L)
minus the capital depretiaton δK. It is
K̇ = sF (K, L) − δK
(7)
3. There exists a law of motion for the stock of capital per worker. In
continuous time, this law is:
k̇ = sk α − (δ + n) k
(8)
4. The population grows at rate n, which equals:
L(t) = L0 ent
(9)
In this paper the last assumption is set aside. In its place we assume
that the labor force follows a von Bertalanffy model:
L̇ = r (L∞ − L)
(10)
L(0) = L0
In growth theory it is convenient to express all the variables of interest in
per capita terms. Thus, we shall use small letters to denote the variables
is the capital per worker we have
in per worker terms. If k = K
L
K̇ L̇
k̇
=
−
k
K L
(11)
1995
Generalization of the Solow growth model
and then
sF (K, L) − δK
r (L∞ − L0 )
k̇
=
−
= sk α−1 − δ − n(t)
rt
k
K
(L0 − L∞ ) + L∞ e
(12)
From this, we obtain the equation of motion for the stock of capital per worker
for the modified Solow growth model, which uses the von Bertalanffy labor
growth mode. This describes how capital per worker varies over time:
k̇ = sk α − (δ + n(t)) k
(13)
r(L∞ −L0 )
where n(t) = (L0 −L
In the next section, we solve the differential
rt .
∞ )+L∞ e
equation (13) and analyze the stability of the model obtaining the asymptotic
value of the variables.
3
Closed form solutions and stability analysis
It has recently been pointed out that the classical Solow growth model with
Cobb-Douglas technology has closed form solutions. See, for example, the recent textbook of Barro and Sala-i-Martin [2] and the references [7] and [11]. In
a recent paper, Mingari Scarpello et al. [10] showed that this result can be extended when the population growths following the logistic law. In this section,
we will show that equation 13 can also be solved in closed form. Equation
(13) is a Bernoulli type equation that can be transformed by the change of
variables
u = k 1−α
(14)
u̇ + (1 − α) (δ + n(t))u = (1 − α) s.
(15)
into the linear equation
In order to find the solutions to this linear differential equation, one must
remember that, given the continuous functions a(t) and b(t), the solution of a
linear differential equation:
ẋ + a(t)x = b(t)
with the initial condition x0 = x(0) is
t
A(t)
−A(τ )
x(t) = e
b(τ )e
dτ
x0 +
(16)
(17)
0
where
A(t) = −
t
a(τ )dτ.
0
(18)
1996
J. G. Brida and E. J. Limas Maldonado
Observe that
|x0 − x1| eA(t)
(19)
gives the difference between two different solutions with initial conditions x0
and x1 . Then, a solution of the linear differential equation (16) is stable if and
only if the function A(t) is bounded from above in [0, +∞). If, besides, it is
lim A(t) = −∞
(20)
t→+∞
then the solutions are globally asymptotically stable. Observe that the solutions of equation (16) have an horizontal asymptote if there exists the limit:
lim
b(t)
= x∞
(21)
lim x(t) = x∞
(22)
t→+∞ a(t)
and, in this case we have:
t→+∞
In fact, by the L’Hopital’s rule,
t
b(t)
A(t)
−A(τ )
= x∞
x0 +
lim x(t) = lim e
b(τ )e
dτ = lim
t→+∞
t→+∞
t→+∞ a(t)
0
(23)
Now we can employ these observations to solve the equation (15). In this case,
the continuous functions a(t) and b(t) are:
r (L∞ − L0 )
(24)
a(t) = (1 − α) (δ + n(t)) = (1 − α) δ +
(L0 − L∞ ) + L∞ ert
b (t) = (1 − α) s
and then we have that
t
A(t) = (1 − α) δt + 0
= (1 − α) δt − rt + ln
eA(t) = e
=e
and
t
(1−α) δt−rt+ln
(1−α)(δ−r)t
r(L∞ −L0 )
dτ
(L0 −L∞ )+L∞ erτ
(L0 −L∞ )+L∞ ert
,
L0
(L0 −L∞ )+L∞ ert
L0
(L0 −L∞ )+L∞ ert
L0
1−α
α−1
t
rτ
−A(τ )
(α−1)(δ−r)τ (L0 −L∞ )+L∞ e
dτ
b(τ
)e
dτ
=
(1
−
α)
se
L0
0
0
α−1
(1−α)s t (α−1)(δ−r)τ
= Lα−1 0 e
((L0 − L∞ ) + L∞ erτ )
dτ.
0
(25)
(26)
(27)
1997
Generalization of the Solow growth model
Then, to obtain the closed form solution of equation (15) we have to compute
the integral
t
e(α−1)(δ−r)τ ((L0 − L∞ ) + L∞ erτ )α−1 dτ.
(28)
I (t) =
0
This can be done in terms of the hypergeometric function 2 F1 obtaining that.
L −L α−1
∞
0
L∞
(α−1)(δ−r)t
ert L∞
∞
−
e
A
(29)
I (t) = (α−1)(δ−r) A L0L−L
2
L0 −L∞
∞
where
δ
δ
A(Z) =2 F1 1 − α, 1 − α + , 2 − α + , Z .
r
r
In the paper [10] there is a detailed description of the evaluation of integral
I (t) and an Appendix with a short outline on the hypergeometric function.
Then we have that the solutions to equation (15) are given by
u(t) = e
(1−α)(δ−r)t
(L0 − L∞ ) + L∞ ert
L0
1−α
k01−α +
(1 − α) s
I (t)
L0α−1
(30)
where k0 is the initial value verifying k0 = k (0) and I (t) is given by equation
(29). This is the closed form solution of equation (15). From this and the
1
change of variables k = u 1−α we can obtain the closed form solution of equation
(13), representing the generalized Solow model.
Note that A(t) is bounded from above in [0, +∞) and tends to −∞ as
t → +∞. Thus, the solutions of equation (15) are globally asymptotically
stable. Finally, we have that:
b(t)
(1 − α) s
= lim
r(L∞ −L0 )
t→+∞ a(t)
t→+∞
(1 − α) δ + (L0 −L
rt
∞ )+L∞ e
lim
=
s
δ
(31)
and then all the solutions of (15) have the horizontal asymptote at level sδ as
1
t → +∞. The change of variables k = u 1−α transforming solutions of equation
(15) into solutions of (13) is continuous. This implies that equation (13) is
globally asymptotically stable, and, as time approaches to infinity, capital per
1
worker k tends to the long run limit value δs 1−α . This equilibrium value
1
k̂ = δs 1−α is not a steady state, since it is not a solution of equation (13).
Nevertheless, it is the long run value of the per worker level of capital k.
Observe that the intrinsic rate of population growth n(t) does not influence
the long-run equilibrium per worker level of capita k. For any initial condition,
1998
J. G. Brida and E. J. Limas Maldonado
capital per worker converges to the value k̂. This is true since the model is
asymptotically stable.
Now we will contrast the long run levels of capital per worker k̂ (modified
Solow model with von Bertalanffy equation) and k̃ (original Solow model),we
can see that:
s
k̂ =
δ
1
1−α
>
s
δ+n
1
1−α
= k̃
(32)
comparing the long run levels of output per worker ŷ (modified Solow model
with von Bertalanffy equation) and ỹ (original Solow model) we have that:
s
ŷ =
δ
α
1−α
>
s
δ+n
α
1−α
= ỹ
(33)
finally, we show the long run levels of consumption per capital ĉ (modified
Solow model with von Bertalanffy equation) and c̃ (original Solow model)
s
ĉ = (1 − s)
δ
α
1−α
> (1 − s)
s
δ+n
α
1−α
= c̃
(34)
That is, if population growth follows the von Bertalanffy law, the long run
levels of the Solow model are improved.
4
Concluding Remarks
The original Solow model assumes that population growth is exponential. This
is not a realistic assumption because, with limited resources, population growth
must be bounded. In this paper we have used the von Bertalanffy equation
instead of the exponential equation to model labor growth in the Solow model.
The von Bertalanffy equation is the simplest model of population growth that
has the following characteristics: (1) population size is bounded; and (2) the
rate of population growth decreases to zero as time tends toward infinity.
These are most remarkable hypotheses concerning human population growth.
Adapting this model to the labor force, we have solved the generalized Solow
model in closed form in terms of the hypergeometric function, and we have
analyzed the stability of this model. This paper demonstrates that, using
the von Bertalanffy equation, the intrinsic rate of population growth does
not influence the long-run equilibrium levels of the per capita variables, and
Generalization of the Solow growth model
1999
that the equilibrium levels of consumption per capita, capital, and output are
improved.
Acknowledgements. A preliminary version of this paper was presented
at the XXXVIII National Congress of the Mexican Mathematical Association, Mexico City, 23-28 October 2005 and at the XV Mexican Colloquium
of Mathematical Economics and Econometrics, Tijuana 7-11 November 2005.
Financial support by the Project ”Multiregime Dynamics in Economics” - Free
University of Bolzano.
References
[1] Auró, A., V. Anislao, J., González, Evaluation of the common carp (Cyprinus carpio, var. comunnis) growth fed when ensiled pig feces, Vol. 33, Issue
2, 2002.).
[2] R. J. Barro and X. Sala-i-Martin. Economic Growth, Cambridge University Press, 2004.
[3] Bertalanffy, L. von, A quantitative theory of organic growth (Inquiries on
growth laws. II), Human Biol. 10:181-213, 1938.
[4] Cloern, J., F. Nichols, A von Bertalanffy growth model with seasonally
varying coefficient, J. Fish. REs. Board. Can. 35: 1479-1482, 1978
[5] Day, J. C., Population Projections of the United States by Age, Sex, Race,
and Hispanic Origin: 1995 to 2050, U.S. Bureau of the Census, Current
Population Reports, P25-1130, U.S. Government Printing Office, Washington D.C., 1996.
[6] De Graaf, G. and M. Prein, Fitting growth with the von Bertalanffy growth
function: a comparison of three approaches of multivariate analysis of
fish growth in aquaculture experiments, Aquaculture Research, Volume
36, Issue 1, January 2005.
[7] Irmen, A., Malthus and Solow: a note on closed-form solutions, Economic
Bulletin, 10, 1 2004.
[8] Jurado-Molina, J., D. Salas and R. Villaseñor-Talavera, Solution of the
von Bertalanffy´s weight growth differential equation (1938) by two distinct methods, Anales del Instituto del Mar y Limnologı́a, Vol. 19, 2, 1992.
[9] Maynard Smith, J., Models in Ecology, Cambridge University Press: Cambridge, 1974.
2000
J. G. Brida and E. J. Limas Maldonado
[10] Mingari Scarpello, G. and D. Ritelli, The Solow model improved through
the logistic manpower growth law, Annali Università di Ferrara -Sez VII
-Sc. Mat. (2003) 73-83.
[11] Simonovits, A., Mathematical methods in dynamical economics, MacMillan Press, (2000).
[12] Xiao, Yongshun, Use of the original von Bertalanffy growth model to describe the growth of barramundi, Lates calcarifer - Bloch, Fishery Bulletin,
Oct 2000.
Received: January 21, 2007