JOURNAL OF ECONOMIC DEVELOPMENT 103
Volume 28, Number 2, December 2003
THE SOLOW’S MODEL WITH ENDOGENOUS POPULATION:
A NEOCLASSICAL GROWTH CYCLE MODEL
LUCIANO FANTI AND PIERO MANFREDI
*
Dipartimento di Scienze Economiche
and
Dipartimento di Statistica e Matematica Applicata all’Economia
It is shown here that the Solow (1956) neo-classical growth paradigm not only explains
the “first” stylised fact of economic growth, namely the existence of a globally stable state of
balanced growth, but, once endowed with a demographically founded formulation of the
labour supply, is also capable to endogenously explain a second main stylised fact of growth,
i.e., the generation of globally stable oscillations around the path of balanced growth.
Keywords: Solow’s Balanced Growth Model, Endogenous Population, Neoclassical
Growth-Cycle Model
JEL classification: E3, J0
1. INTRODUCTION
The centrality of the neo-classical growth model of Solow (1956) for economic
theory is witnessed by the current persistency of new contributions stimulated by his
work (for instance Bajo-Rubio (2000)). The aim of the present paper is to investigate the
effects of endogenous labour supply within Solow’s model.
This endogenisation is based on a coupling of an age structure argument (absent in
the original Solow’s work) plus a Malthusian relation between fertility and income,
which was well recognised by Solow himself. A realistic formulation of labour supply
must take into account past demographic behaviours in that the new entries into the
labour force at time t are the outcome of fertility behaviours of past generations. This
requires the introduction of the age structure prevailing in the population. This fact was
well known to the “classical” economists: “the supply of labourers in the market can
neither be speedily increased when wage rise, nor speedily diminished when they fall.
*
The authors thank an anonymous referee of this journal for several useful comments that greatly
improved the quality of the paper. Usual disclaimers apply.
LUCIANO FANTI AND PIERO MANFREDI104
When wages rise a period of eighteen or twenty years must elapse before the stimulus,
given the principle of population, can be felt in the market” (McCulloch (1854, p. 34)).
The previous McCulloch’s sentence contains, according to the current literature in
population economics, a terse definition of Malthusian cycles. Malthusian cycles are a
firm theoretical point in population economics. They are the consequence : “...of the lags
between the response of fertility to current labour market conditions and the time when
the resulting births actually enter the labour force” (Lee (1997, p. 1097)). The existence
of such cycles was lucidly predicted, at least theoretically, by Malthus in several loci in
his famous Essay (Malthus (1992)).
Long run oscillations of population and of the economy are an important, though
sometimes neglected (for instance Barro and Sala i Martin (1995), disregard it), stylised
fact of economics. They are well documented in the recent empirical study by Lee and
Anderson (2002) (see in particular their figs. 1, 2, 3 reporting UK historical time series
of the death and birth rates, and of the wage rate). They state: “Both series, especially
the birth rate, show broad swings over hundreds of years, and short term fluctuations are
substantial, especially prior to 1750” (ibidem, p. 201). Another fundamental example is
represented by the so called post-transitional fertility fluctuations, such as the babybooms
observed in most of the Western World, sometimes explained by the Easterlin’s
effect (Easterlin (1961)). We finally feel that if Lowest Low Fertility, as recently
observed in the Western World (Kohler et al. (2002)), should finally come to an end and
this is considered more or less a necessary event by many scholars (Bongaarts
(1999)) - this would almost necessarily give rise to a new fertility wave.
In this paper the age structure of the population, which governs Malthusian cycles, is
embedded in our Solow-type model by resorting, in a manner very close to the classical
economists thought, to time-lags.1
We show that persistent oscillations may occur in the
Solow’s model when the rate of change of the labour supply is correctly assumed to
depend (even in the simplest manner) on past demographic behaviours. Thus we can
argue that the neoclassical growth paradigm, e.g., the Solow’s model, is capable not only
to explain the stylised fact of balanced growth, but, once endowed with a correctly
demographically founded formulation of the labour supply, also to endogenously explain
a second main stylised fact of economic growth, namely the appearance of steady
oscillations around the “average” path of balanced growth.
In order to position the present paper in the current literature on economic growth,
we note that our model is a Solovian model of balanced growth with endogenous
population. In this sense it may be considered an “endogenous” growth model, in that it
1
The idea that time-delays represent a simple and clever strategy to embed age structure within complex
models, dates back to Vito Volterra famous 1926 paper on interactions between biological species. For a
modern presentation see McDonald (1978). This strategy is of course an approximation of the true underlying
age structure model, based on the assumption of “stability” of the age composition, i.e., the constancy of the
ratios between different cohorts of the underlying population.
THE SOLOW’S MODEL WITH ENDOGENOUS POPULATION 105
endogenously explains the rate of total income growth.2
Additionally, it obviously is a
“descriptive” model, rather than a “fully optimized” one as those surveyed in Tamura
(2000). Further, our model has the following features: 1) it is a growth cycle model; 2) it
endogeneises the age structure of the population in a macroeconomic setting. The two
latter features are not shared by most available growth models, whether they belong to
the Solovian tradition or to the “new growth theory”.
The present paper is organised as follows. In section two a basic Solow-type model
with endogenous population depending on current fertility is introduced, and its
properties are studied. In section three we introduce a more general model embedding a
time-lag in the reaction of the rate of change of the labour supply to past income, and its
properties are investigated by means of local stability analysis plus Hopf bifurcation.
Section four completes the analysis of the model by investigating its global properties
via numerical simulations and discussing the economic meaning of its main results.
The “core” results are summarised in the conclusions.
2. A BASIC SOLOW’S MODEL WITH ENDOGENOUS POPULATION
The original Solow growth model (Solow (1956)) is defined by the ordinary
differential equation (ODE):
knksfk s−)(=& , (1)
where k = K/L denotes the capital-labour ratio, )(kfy = the production per unit of
labour, s the saving rate )10( << s , 0>sn the rate of change of the supply of labour
LL/& , which was initially assumed exogenous by Solow. As in the Solow 1956 paper,
we have assumed 0=δ where δ is the rate of capital depreciation. When a CobbDouglas
production function is chosen (ibidem, example 2, p. 76), the model collapses
into the ODE:
knskk s−= α& , (2)
where 10 <<α . Contrary to most subsequent developments, where the supply of
labour was treated as exogenously determined, Solow also tried, still in his 1956 paper,
to endogenise it. He wrote (ibidem, p. 90) the rate of change of the supply of labour as a
function of the current level of per-capita income: ))(( kfnn ss = , which becomes in the
Cobb-Douglas case
2
We note that conversely our model does not allow a positive rate of growth of the per-capita income
growth, which is a major task of the “new growth theory”.
LUCIANO FANTI AND PIERO MANFREDI106
)( α
knn ss = . (3)
He also gave some qualitative analysis of the effects of a general non monotonic sn
function.3
As in this paper we are essentially interested in the effects of forces of
“fundamental” nature, in what follows we assume, for simplicity, that the function sn
be linear and increasing, i.e., we consider α
nkns = where 0>n is a constant
parameter,4
tuning the reaction of the rate of change of the labour supply to changes in
per-capita income. We therefore have the model:
αα +1
−= nkskk& . (4)
It is easy to see that Equation (4) always admits, as in the basic Solow’s model, the zero
equilibrium )( 0E and a unique positive equilibrium )( 1E which is globally asymptotically
stable (the previous conclusions are preserved if we assume )(⋅n to be a general
increasing function, saturating or not). In particular the capital-labour ratio at 1E is
nskk /1 == .
It is worth noting that the long term growth rate of total income )(Y , in model (4)
is:
ααα
snnkn
y
y
Y
Y
s
−
==+= 1
1
&&
. (5)
3
For sake of precision, we notice that Solow considered a qualitatively complicated form of the fertility
function, generating three equilibria, and also predicted, in a pioneering way, the possibility of endogenous
self-sustained growth allowed by the endogeneity of labour supply. This idea will be better formalised many
years later, for instance by Becker, Murphy and Tamura (1990).
4
According to a purely demographic interpretation of labour supply, the positive relationship between
labour supply and wage adopted here may be interpreted as a Malthusian relation between fertility and the
wage, rather than a “well behaved” labour supply function. We did not explicitly considered here incomerelated
mortality but the extension is straightforward.
Moreover, referring to Malthus’ original ideas, the rate of change of labor supply could also be specified
as ))((/ °−= ykfnLL& , where °y is subsistence income and 0>n . In this specification, the growth of the
labor supply would be negative when per capita income is below the subsistence level, which is a realistic
fact. But also this specification (as well as any non linear saturating specifications) leads to the same results
presented here under the simpler specification adopted above.
THE SOLOW’S MODEL WITH ENDOGENOUS POPULATION 107
Thus the long term growth rate of total output depends on the joint action of population
(as summarised through the Malthusian parameter n) and saving. The interest for this
result lies in the relevance of its message, and in the simplicity with which it is obtained:
contrary to Solow’s original result, where, paradoxically, saving determines the
accumulation but does not play any role for long term growth of total income, saving
matters for growth. This is widely reasonable, as it is only through saving, e.g., new
investments, that population change may promote economic growth.
Clearly, the assumption that the rate of change of the supply of labour is a function
of current income, as in (4), might perhaps reflect an underlying participation effect, but
is hardly defendable as determined by true population change. We argue, however, that
the simple model (4) is useful to cast the investigation of the consequences of fully
endogenous population dynamics, as will be done in the next section by resorting to
time-lags.
3. AGE STRUCTURE AND LABOUR SUPPLY:
THE DEMOGRAPHIC DELAY
3.1. Motivations
To take into account of the overall demographic mechanism of age structure in
economic models would lead to a larger dimension and thus more analytical complexity.
Since the mathematical literature (Mac Donald (1978)) has shown that time-delays often
represent simple approximations of age structure mechanisms, in this paper we follow
this route. The intuitive idea here is that the current rate of change of the supply of
labour is related to past fertility, and thus to past levels of the wage, following a
prescribed pattern of delay. There are two main alternatives: fixed delays and distributed
delays. The former is better suited when there is no variability in the process of
transmission of the past into the future: for instance when we assume that all individuals
are recruited in the labour force at the same fixed age. Conversely when recruitment may
occur at different ages, i.e., with different delays (for instance because the time needed
to complete formal education is heterogeneous within the population), distributed delays
appear more suitable.5
The introduction of a distributed delay in the population term in
(4) leads to the following integro-differential (IDE) equation:
( )( ) ( ) kdtGknskk
t






−−= ∫
∞−
ττταα& , (6)
5
See Invernizzi and Medio (1991) for an economically oriented discussion of distributed delays.
LUCIANO FANTI AND PIERO MANFREDI108
where the term ( ) tkn <)( ττα
, captures past (rather than current), income-related
fertility, and )( τ−tG is the corresponding delaying kernel. As well known (MacDonald
(1978)), by assuming that the G kernel belongs to the so-called Erlangian family,6
IDEs as (6) may be reduced to higher order systems of ordinary differential equations.
3.2. The Model: Equilibrium and Dynamical Analysis
In economics the most frequent type of delaying kernel is based on the notion of
exponentially fading memory (obtained for 1=r in the Erlangian family). However an
exponentially fading memory, which implies to give maximal weight to the “more
recent” past, cannot be considered a reasonable representation of the demographic
mechanism. Rather the new entries in the labour supply at each moment t of time are
due to individuals born in a rather narrow interval of time in the past. We argue that
“humped” memories, rather than exponential ones, give a satisfactory representation of
the demographic process (i.e., past fertility plus delayed transition into the labour force).
We therefore choose as the delaying kernel the second member of the Erlangian family
)2( =r :
( ) ( ) ,0;0,2
,2 >>== −
ββ β
β xxexErlangxG x
(7)
because it is the simplest type of “humped” Erlangian density (having a mean delay
β/2=T ). This implies that the maximal weight is given to some appropriate time
interval in the past, which heuristically well corresponds to the idea of demographic
delay.7
Under (7) the IDE (6) may be reduced to an ODE system by the linear chain
trick. This is obtained, in our case, by introducing 2=r auxiliary variables, the first
one of which is the integral term in (6) where )(,2 xErlangG β= , and the second one is
obtained by replacing, in the integral term, )(,2 xErlang β with )(,1 xErlang β . We
obtain, after the further simplifying change of variable Zk =α
, the 3-dimensional ODE
system:
6
A density function )(xf is erlangian with parameters ),( βr when it has the form:
( )
( )
.0;,...2,1;0,
!1
,; 1
>=>
−
= −−
β
β
β β
rxex
r
rxf xr
r
7
The use of a “more humped” distribution in the Erlangian family (as erlangian densities of higher order,
i.e., =r 3,4,5...), which could perhaps more realistically capture the phenomenon of the delayed entry into
the labour force, simply confirms the more important results of this paper (e.g., the onset of persistent
oscillations as shown in next section). We did not investigate analytically the properties of higher order
systems with higher order kernels because they become more and more analytically cumbersome without
offering any extra “qualitative” insight from the economic point of view.
THE SOLOW’S MODEL WITH ENDOGENOUS POPULATION 109
( )
( ).
,
,
1
RZR
XRX
nXsZZZ
−=
−=








−=
−
β
β
α α
α
&
&
&
(8)
In a nutshell the linear trick procedure implies r additional linear differential equations.
In particular every additional equation represents a continuous time adaptive mechanism,
as evident from (8). For technical details on the linear chain trick see Mac Donald
(1978), Fanti and Manfredi (1998).
Model (8) has the zero equilibrium 0E and the positive equilibrium
),,( 1111 RXZE = where:
α






===
n
s
RXZ 111 . (9)
Thus, besides the change of variable Zk =α
, model (8) preserves the equilibria of its
unlagged counterpart (4), so that Equation (5) holds with its interesting economic
interpretation.
The local stability analysis of the positive equilibrium 1E gives the third order
characteristic polynomial: 32
2
1
3
)( aXaXaXXP +++= with coefficients:
( ) ( ) ,,21 1
2
3
2
121 =+−12=,−1+= nZanZanZa ββαβαβ
which are all strictly positive. Therefore, by the Routh-Hurwitz stability test, 1E will
be locally stable provided the condition 03212 >−=∆ aaa holds. This leads to the
stability condition:
( ) ( )( ) 0>−12+5−4+2=)
2
11
2
nZnZf αβαββ( . (10)
The discussion of (10) shows that: i) for 88.02 =<αα no loss of stability is possible;
ii) for 2αα > , )(βf always has two strictly positive real roots, 1β , 2β (with
LUCIANO FANTI AND PIERO MANFREDI110
21 ββ < ), implying that losses of stability may occur. In particular both 1β , 2β values
represent Hopf bifurcation values of the β parameter.8
They are given by:
( ) ( )( )αααβ α
α
8945
4
1
2,1 −±−=
−
s
n
. (11)
The latter formula provides a nice relation between the average demographic delay (we
recall that the mean delay is β/2=T and the economic parameters.
We may summarise our main findings by the following:
Proposition 1: When the profit share α is below a prescribed threshold value
( 2αα < ), system (8) replies the behaviour of the original Solow’s model, with
convergence to the unique and globally stable equilibrium 1E . When 2αα > , system
(8) continues to converge to the globally stable equilibrium 1E only when β is
sufficiently large or sufficiently small, i.e., for 2ββ > , or 1ββ < . In the whole window
21 βββ << the 1E equilibrium is locally unstable. At the points 1ββ = , 2ββ = ,
Hopf bifurcations occur.
In words: an endogenous labour supply according to the demographic mechanism
adopted here is able to destabilise the equilibrium of the Solow’s model and to generate
persistent oscillations9
via Hopf bifurcations.
4. SIMULATIVE EVIDENCE AND WORKING OF THE SYSTEM
The fact to know that a Hopf bifurcation exists nothing says about the stability
properties of the involved periodic orbits, e.g., it does not say whether the bifurcation is
supercritical or subcritical (that is, whether the periodic orbit is locally stable or
unstable). Unfortunately the investigation of stability of periodic orbits emerged via
8
The formal proof requires to show that (Marsden and McCracken, (1976)) : i) purely imaginary
eigenvalues exist for the linearised system at 1ββ = or 2ββ = due to a “continuous” movement of a pair
of complex eigenvalues; ii) the crossing of the imaginary axis by the involved complex pair occurs with
nonzero speed. The proof, omitted here for brevity, is available on request.
9
Of course this is not the only avenue through which to obtain population induced persistent oscillations
in a neoclassical growth model: for instance, as suggested by a referee of this Journal, this could be obtained
by a two-period overlapping generation model - rather than the Solovian one developed in our paper including
a model of adaptive expectation on fertility determination.
THE SOLOW’S MODEL WITH ENDOGENOUS POPULATION 111
Hopf bifurcation at dimensions greater than the second is a hard task (Marsden and
McCracken (1976)). Moreover the predictions of the Hopf theorem are “local”: they
nothing say about global behaviours.
We therefore resorted to numerical simulation to clarify the stability nature of the
Hopf bifurcations occurred at the points 1ββ = , or 2ββ = and more generally to
investigate the global properties of our model.
Let us now summarise our main dynamical findings and illustrate the working of the
model. There is a region, defined by 88.02 =<αα , in which the traditional behaviour
of Solow’s model is confirmed: the economy converges to a long term globally stable
steady state. Conversely, for very large )( 2ααα > the economy may be destabilised by
the action of the demographic delay. More in detail, as long as β is very large (in
relative terms), i.e., for 2ββ > (we recall that this means “very” small mean
demographic delays), the 1E equilibrium preserves its stability. But as β is decreased
(this happens for increasing mean delays) stability may be lost. This happens for 2ββ =
where a first Hopf bifurcation occurs and 1E exchanges its stability with a stable limit
cycle. Finally, by further decreasing β , a further bifurcation occurs at 1ββ = where
the local stability of the 1E equilibrium is restored. Therefore for very large mean
delays the Solow’s behaviour is restored once again. Table 1 reports a synoptic view of
the process of phase transition of the positive equilibrium when 2αα > .10
Table 1. Windows of the delay parameter β and the corresponding behaviour of
the model as summarised by the nature of its unique positive equilibrium 1E .
β ),0( 1β ),[ 21 ββ ),[ 2 +∞β
1E is : stable (node or focus) unstable (surrounded by a
stable limit cycle)
stable (node or focus)
It is worth to remark the following further facts suggested by simulation: i) both the
points 1ββ = , 2ββ = , generate supercritical bifurcations (i.e., locally stable oscillations);
ii) the whole window 21 βββ << is a region of stable oscillations, iii) all the
10
We notice that, although persistent oscillations seem to require a rather large value of the elasticity of
capital α (even higher than those estimated by using a broad definition of capital stock including the
human capital), this is just a feature of the low-order kernel ( r = 2) considered here for purposes of analytical
simplicity. Indeed by resorting to higher order distributions of the delaying kernel (e.g., erlangian densities of
order r = 3,4,5...), which are more realistic, we obtained, via simulation, more realistic “critical” values of
the elasticity of capital.
LUCIANO FANTI AND PIERO MANFREDI112
properties of the model seem to hold globally: when the 1E equilibrium is either
locally stable, or surrounded by locally stable limit cycles, the system always seems
globally stable.
Thus our results are consistent with two important stylized facts: 1) the existence of
global stability of the economy, thus confirming Solow’s result; 2) the presence of
(stably) fluctuating behaviours around the state of balanced growth of the economy. The
latter fact greatly enriches Solow’s result.
Figures 1a and 1b report a two-dimensional view of the emerging cycles respectively
at the largest bifurcation point 064.02 =β , and at the smaller one 0056.01 =β (other
parameter values: 92.0=α , 3.0=s , 01.0=n ).
Figure 1a. A Stable Limit Cycle appeared at 064.02 == ββ
THE SOLOW’S MODEL WITH ENDOGENOUS POPULATION 113
Figure 1b. A Stable Limit Cycle appeared at 0056.01 == ββ
The economic interpretation of our findings is straightforward: when returns are “not
too decreasing” (i.e., 2αα > ), an increase in the age of entrance in the labour force may
destabilise the equilibrium of the Solow’s model, and trigger persistent oscillations.
These latter appear to be bounded, so that the global stability of the economy is
preserved. These fluctuations appear to be the outcome of the demographic memory
operating through the age structure delay. The fact that periodic behaviours may persist
also for very long mean delays seems to suggest the existence of possible
“supergenerational” echoes, not necessarily related only to past fertility but to other long
lasting delayed effects.
5. CONCLUSIONS
In this paper we have shown that the Solow’s model, once extended to take account
for: i) the existence of a delay in the process of recruitment in the labour force, due to
the age structure of the population, ii) the existence of a Malthusian relation between
fertility and wage, is capable of generating stably fluctuating growth paths. This fact
shows that the neoclassical growth paradigm not only explains the stylised fact of
balanced growth, but, once endowed with a correctly demographically founded
LUCIANO FANTI AND PIERO MANFREDI114
formulation of the labour supply, becomes capable to endogenously explain the other
main stylised fact of economic growth, namely the generation of globally stable
oscillations around a path of balanced growth. An interesting consequence of the
presence of endogenous population is that, contrary to Solow’s original result, where, as
well known, saving determines the accumulation but does not play any role for long
term growth, saving matters for (long term) growth. The main message is thus that it is
only through savings, (and therefore investments), that population growth may
eventually promote economic growth.
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Mailing Address: Dipartimento di Scienze Economiche and Dipartimento di Statistica e
Matematica Applicata all’Economia, Via Ridolfi 10, 56124 Pisa, ITALY. Tel: 0039-(0)50-
2216317, Fax: 0039-(0)50-2216375. E-mail: manfredi@ec.unipi.it
Manuscript received December, 2002; final revision received July, 2003.