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Working Papers / Documents de travail
WP 2023 - Nr 26
Speed of Convergence in a Malthusian World:
Weak or Strong Homeostasis?
Arnaud Deseau
Speed of Convergence in a Malthusian World:
Weak or Strong Homeostasis?∗
Arnaud Deseau†
November 27, 2023
Abstract
The Malthusian trap is a well recognized source of stagnation in per capita income prior
to industrialization. However, previous studies have found mixed evidence about its exact
strength. This article contributes to this ongoing debate, by estimating the speed of conver-
gence for a wide range of economies and a large part of the Malthusian era. I build a simple
Malthusian growth model and derive the speed of convergence to the steady state. A cali-
bration exercise for the English Malthusian economy reveals a relatively weak Malthusian
trap, or weak homeostasis, with a half-life of 112 years. I then use β-convergence regres-
sions and historical panel data on per capita income and population to empirically estimate
the speed of convergence for a large set of countries. I find consistent evidence of weak
homeostasis, with the mode of half-lives around 120 years. The weak homeostasis pattern is
stable from the 11th to the 18th century. However, I highlight significant differences in
the strength of the Malthusian trap, with some economies converging significantly faster
or slower than others.
Keywords: Convergence, Homeostasis, Malthusian trap, Preventive checks, Marriage,
Fertility, Malthusian model, Beta-convergence
JEL Codes: J1, N1, N3, O1, O47
∗I am grateful to Hugues Annoye, Thomas Baudin, Pierre de Callataÿ, Greg Clark, David de la Croix, Frédéric
Docquier, Oded Galor, Marc Goñi, Andreas Irmen, Nils-Petter Lagerlöf, Hélène Latzer, Anastasia Litina, Jakob
Madsen, Fabio Mariani, Stelios Michalopoulos, Luca Pensieroso and David Weil for their valuable comments and
suggestions. I also thank participants at the 2018 University of Luxembourg CREA Workshop on Culture and
Comparative Development; the 2017 Université Saint-Louis – Bruxelles CEREC Workshop on Macroeconomics
and Growth; and seminars at Brown University and the Catholic University of Louvain. I acknowledge financial
support from the French government under the “France 2030” investment plan managed by the French National
Research Agency Grant ANR-17-EURE-0020, and by the Excellence Initiative of Aix-Marseille University -
A*MIDEX.
†Aix-Marseille University, CNRS, AMSE, Marseille, France. 5 Boulevard Maurice Bourdet CS 50498 F-13205
Marseille cedex 1, France. E-mail: arnaud.deseau@univ-amu.fr).
In four centuries [1300-1700], the [French] population only increased by 2 million persons in
all! And some say less! [...] Thus, an extraordinary ecological equilibrium is revealed. Of course, it
did not exclude possibly prodigious, but always temporary, upheavals and negative fluctuations in its
time like those experienced by animal population.
Emmanuel Le Roy Ladurie (1977), Motionless History.
1 Introduction
One of the most central prediction of the Malthusian theory is that standards of living were
stagnant before the onset of industrialization. Stagnation however does not literally mean con-
stant, or flat, per capita income. In fact, any shock striking a Malthusian economy generates
fluctuations in the standards of living, namely temporary or non-sustained economic growth.
Indeed, a simple Malthusian model predicts that a positive shock on the technolo level –
say the introduction of better cultivation techniques – increases income per capita in the short
run only; in the long run, population increases and the economy returns to its initial level of
income per capita. This is the so-called “Malthusian trap” mechanism, that has been recognized
as one of the major obstacles to achieve sustained economic growth during millennia (Kremer,
1993;Galor and Weil,2000;Hansen and Prescott,2002;Clark,2007;Ashraf and Galor,2011;
Galor,2011).
While the existence of the Malthusian trap is widely established empirically, previous litera-
ture has found mixed evidence about its exact strength. A first group of studies, mainly focusing
on England, finds evidence of a weak Malthusian trap, known as weak homeostasis1(Lee,1993;
Lee and Anderson,2002;Crafts and Mills,2009;Fernihough,2013;Bouscasse et al.,2023).
In these studies, the half-life of adjustment to shocks is typically about one century, and can be
as long as four centuries. On the other hand, Madsen et al. (2019) find the first evidence of a
1Homeostasis comes from the Greek homoios “similar” and stasis “steady”, meaning “staying the same”. In
demography, it refers to a population equilibrium maintained by density-dependent checks (Lee,1987).
1
strong and widespread Malthusian trap, or strong homeostasis, with estimated half-lives between
one and three decades.
In this article, I reinvestigate the question of the strength of the Malthusian trap by ex-
amining the speed of convergence of Malthusian economies – i.e. how quickly they tend to
return to their steady state following a shock. I argue that the speed of convergence in Malthu-
sian times should be relatively slow, and thus reflects weak homeostasis. The main reason is
that the Malthusian trap involves demographic fluctuations which, by definition, are long and
take generations to unfold. As argued by Malthus (1798) himself, the channels through which
population adjusts to the amount of resources per capita are the age at marriage, fertility and
mortality (the so-called preventive and positive checks). This is confirmed by empirical stud-
ies finding a significant but small response of demographic variables to changes in per capita
incomes in pre-industrial times.2
To investigate this conjecture, I first build an overlapping-generations Malthusian growth
model including both preventive and positive checks as means of population adjustment. In
particular, agents first choose to marry (or not), influencing the extensive margin of fertility,
and then choose the number of children within marriage, influencing the intensive margin of
fertility. Both choices depend on income per capita, in a Malthusian fashion. I show that the
speed of convergence of a Malthusian economy to its steady state depends on four parameters:
the land share of output and the elasticities of fertility, marriage and survival with respect to
income per capita. I calibrate the model for England and show that, under plausible param-
eter values, the speed of convergence indicates weak homeostasis, with a half-life of 112 years.
Alternative calibration scenarios using the 10th percentile and 90th percentile of the long-run
elasticities estimated in the literature for England also indicate weak homeostasis, with half-lives
between 64 and 230 years. I also provide a quantitative analysis, showing that weak homeostasis
is consistent with the centuries long reaction of the English Malthusian economy to the Black
2Empirical estimates of these elasticities can be found in Lagerlöf (2015) and Klemp and Møller (2016), among
others. See Section 3.1 and Section Bof the Appendix for further details.
2
Death.
Second, I employ β-convergence regressions à la Barro and Sala-i Martin (1992) to pro-
vide empirical estimates of the speed of convergence for a large set of economies and a large
part of the Malthusian period. I first use the data compiled in the latest update of the Mad-
dison Project, which offers the most comprehensive GDP per capita data available to study
Malthusian economies. I also run the same regressions using historical population levels from
McEvedy et al. (1978). To gain in precision, and explore the temporal and spatial heterogeneity
of the speed of convergence in a more comprehensive way than previous studies, I employ two
additional datasets that possess a much higher cross-sectional and time dimension than the two
aforementioned sources. The first one, coming from Lagerlöf (2019), gives simulated GDP per
capita series based on the same empirical moments as the Maddison Project data. The second
dataset, coming from Reba et al. (2016), compiles the historical urban population series origi-
nally produced by Chandler (1987) and Modelski (2003). In all cases, I find consistent evidence
of a relatively weak homeostasis, with the mode of the estimated half-lives around 120 years. I
find evidence of a stable pattern of weak homeostasis throughout much of the Malthusian era,
from the 11th century to the end of the 18th century. On the other hand, I find significant
differences in the speed of convergence between countries, with some Malthusian economies
converging significantly slower or faster than others. In particular, some economies are found
compatible with strong homeostasis, in the same magnitudes as found by Madsen et al. (2019).
There is one main concern in my empirical analysis: weak Malthusian dynamics may be
the result of a serious omitted variable bias in the β-convergence regressions. I employ several
strategies to mitigate that concern. My empirical analysis includes country and time fixed ef-
fects, which respectively account for unobserved time-invariant characteristics at the country
level (e.g. geography), and common trends (e.g. technolo diffusion) that could simultane-
ously affect growth and initial development levels. In addition, I also include a time-varying
control variable (Statehist), developed by Borcan et al. (2018), capturing state presence during
3
the Malthusian period. In principle, this variable captures general institutional changes likely
to affect the steady-state position of Malthusian economies, thus reducing the omitted variable
bias. Finally, to address remaining endogeneity concerns, I employ an instrumental variable
approach (GMM), which uses the lagged values of the endogenous regressors as instruments.
Typically, I find that GMM estimates confirm the weak homeostasis pattern of my fixed-effects
regressions.
This article contributes to the growing literature examining the existence and strength of the
Malthusian trap (Lee and Anderson,2002;Nicolini,2007;Crafts and Mills,2009;Kelly and
Gráda,2012;Fernihough,2013;Møller and Sharp,2014;Lagerlöf,2015;Madsen et al.,2019;
Cummins,2020;Jensen et al.,2021;Attar,2023). Typically, the literature finds evidence of
the existence of Malthusian dynamics in a particular country, albeit small in magnitude. For
instance, Crafts and Mills (2009) study Malthusian dynamics in England (1540-1870) using
structural modelling, and conclude that there is a very weak Malthusian trap. Similarly, Fer-
nihough (2013) finds evidence of weak homeostasis in Northern Italy (1650-1881), using VAR
methods. I contribute to this literature in two main respects. First, rather than focusing on a
specific country, this article is the first, to my knowledge, to provide evidence of weak home-
ostasis across a wide range of Malthusian economies and for a large part of the Malthusian era.
Second, I am able to characterize, for the first time, the full distribution of convergence speed
during the Malthusian period. I show that most countries are characterized by weak homeostasis
of around a century, while highlighting significantly stronger or weaker Malthusian traps for
some countries. In particular, I find that the Spanish Malthusian trap is close to strong home-
ostasis, with a half-life of less than 50 years; whereas in other countries, such as England, the
half-life is of the order of a century or more. The article closest to mine is Madsen et al. (2019),
which find evidence of widespread strong homeostasis in a panel of 17 countries (900-1870).
The main difference between the two articles lies in the approach to the data and the estimation
method. Whereas Madsen et al. (2019) rely on data interpolated from heterogeneous historical
4
sources and use a SUR model, I employ the data as they appear in their original sources and
use the standard techniques developed in the empirical growth literature to estimate the speed
of convergence, such as fixed-effects models and GMM.
This article also adds to the literature studying Malthusian dynamics in an overlapping-
generations frameworks. The existing overlapping-generations Malthusian frameworks con-
sider the intensive margin of fertility as the only channel through which population adjusts
(Ashraf and Galor,2011;Lagerlöf,2019). I build on these previous models by incorporating,
for the first time, marriage as an explicit channel through which the population adjusts, as orig-
inally argued by Malthus (1798). The marriage channel allows me to incorporate the extensive
margin of fertility, since unmarried people typically had no children in the Malthusian era, and
therefore allows me to model richer population dynamics.
Finally, this article relates to the literature deriving the speed of convergence in growth
models. Working in continuous time, Irmen (2004) and Szulga (2012) find that the speed of
convergence of a Malthusian economy depends on the land share of output and the elasticities of
the birth rate and death rate to income per capita. I contribute to this literature by showing that
the elasticity of the marriage rate to income per capita also matters to characterize the speed of
convergence. In a modern context, this article relates also to the seminal work of Barro (1991)
and Barro and Sala-i Martin (1992).
The rest of this article is organized as follows. Section 2presents my Malthusian growth
model. Section 3presents my calibration exercise, discussing the parameters I use and presenting
my simulations. Section 4derives the speed of convergence implied by my model and discuss
it in relation to the literature. Section 5describes my empirical strate and the data I use
to estimate the speed of convergence. Section 6presents and discusses my empirical results.
Section 7concludes.
5
2 Theoretical Framework
In this section, I present the core elements of the Malthusian model I use to study the dynamics
of GDP per capita and population in the Malthusian era. I consider an overlapping-generations
economy with time modelled as discrete and going from zero to infinity, and where agents live
two periods. In the first period of their life, they are inactive children entirely supported by
their parents; they make no decisions. In the second period of their life, they work, earn an
income and make decisions about consumption, marriage and fertility.
I deviate from textbook Malthusian models by modelling explicitly marriage, celibacy and
childlessness decisions. In brief, that means that I am considering both the extensive margin
of fertility, i.e. whether or not an individual marries and can have children, and the intensive
margin of fertility, i.e. variations in individual’s number of surviving children within marriage.
These two elements are crucial as they directly affect the response of fertility to income per
capita, and therefore the speed with which a Malthusian economy returns to its steady state
after a shock. Both are consistent with empirical studies showing the importance of the so
called preventive checks, advocated by Malthus (1798) himself, in affecting fertility. Indeed,
Cinnirella et al. (2017) show that real wages affect negatively birth spacing within marriage and
the time of marriage and first child in England for the period 1540-1850. Cummins (2020)
finds similar results with a negative effect of living standards on the age at first marriage in
France between 1650 and 1820. de la Croix et al. (2019) show that singleness and childlessness
are key elements to take into account when estimating reproductive success in pre-industrial
times. Therefore, modelling both the extensive and intensive margins of fertility appears crucial
to a rigorous analysis of population dynamics during the Malthusian era.
I model childlessness and celibacy together, leaving the possibility to procreate only to
married agents. This is fully consistent with historical studies showing very low illegitimate
birth rates in pre-industrial Europe (Hajnal,1965;Segalen and Fine,1988;Wrigley et al.,
6
1989). Marriage offers the opportunity for agents to gain utility from another source than just
pure consumption.3On the other hand, the disutility of marriage is represented by a search
cost that agents need to pay in order to match with a partner.4Agents are assumed to be
heterogeneous in their search cost, which is exogenously given. At the beginning of their adult
life, agents draw a search cost λiwith λi∼ U (1, b)and bbeing the maximum of the uniform
distribution. Agents maximize their utility and therefore a marriage occurs only if the utility of
being married is superior to the utility of being single. Within marriage, I let the agent’s fertility
depend on his income per capita, according to the standard Malthusian theory and empirical
evidence (Cinnirella et al.,2017;de la Croix et al.,2019;Cummins,2020).
Preferences and Budget Constraints.— The utility of a married agent iof generation tis
defined à la Baudin et al. (2015):
UM
i,t =ln ct+γln (nt+ν)−ln λi,(1)
where ctdenotes consumption, γ > 0is a child preference parameter, ntis the number of
surviving children, ν > 0allows for childlessness as the individual utility remains defined when
nt= 0, and λiis the utility cost of marriage.
It follows that the utility of an unmarried agent of generation tis given by:
US
i,t =ln ct+γln (ν).(2)
Agents allocate their income between consumption and child rearing such that we have the
following budget constraint:
ct=yt−f(nt),(3)
3This means that parents only care about the quantity of surviving children, as in a standard Malthusian model.
4Alternatively, one can think the cost as representing a dowry that agents need to pay in order to marry.
7
where ytis agent’s income, and f(nt)is the cost of having ntchildren in terms of goods.
A convenient functional form for f(·)capturing both the idea of childlessness (f(0) = 0)
and allowing for different types of returns to scale in the production of children is the following
one:
f(nt) = q(nt+ν)1/δ−q ν1/δ,(4)
with q > 0being the unit cost of a child, and δ > 0a parameter influencing the degree of
return to scale in child production.
Fertility.— Maximizing (1) subject to (3), I obtain the optimal fertility behaviour of a
married agent of generation t:
nt=κ·yt+q ν1/δδ−ν≡nt(yt),(5)
where κ=q
γδ +q−δ. Thus, in accordance with Malthusian theory, the number of sur-
viving children within marriage depends positively on income per capita (∂nt/∂yt>0).
Marriage.— An agent is indifferent between being married and single if utility is the same
in both situations. I define λas the draw from the search cost distribution that makes an agent
indifferent between being married and single. The condition for an agent to be married is:
λi< λ with λi∼ U (1, b). I can therefore compute the probability for an agent of generation t
to be married as:
pt=P(λi< λ) = λ(yt)−1
b−1≡pt(yt),(6)
where bis the maximum of a uniform distribution and the threshold draw λdepends on an
individual’s income.5Since I work at the generation level, ptis also equivalent to the marriage
rate in that Malthusian economy. In the rest of the article, I will use ptas the marriage rate.
5The full expression of λis available in Section Aof the Appendix
8
Thus, in line with the idea of Malthus (1798), an increase in income lowers the age of marriage,
resulting in a higher marriage rate at the generation level in the model (∂pt/∂yt>0).
Production.— Total output in period tis given by:
Yt= (AtT)αL1−α
t,(7)
where Atis a land-augmenting technolo factor, Tis total land area, Ltis the size of the
labour force that is equivalent to the adult population in my analysis and α∈(0,1) is the land
share of output.
I assume that workers are self-employed and earn an income equal to the output per worker
in t. Using (7) and normalizing land area to unity (T= 1), we obtain:
yt=At
Ltα
.(8)
Following Lagerlöf (2019), I consider sustained but constant growth in land productivity.
The technological level in period tis given by:
At=A0(1 + g)t,(9)
where A0is the initial technological level and gis an exogenously given and constant rate
of technological progress.
Mortality.—Malthus (1798) and the Malthusian theory assert that population adjusts via
the so called positive and preventive checks. My model includes the two types of Malthusian
population adjustment: (i) preventive checks, as both the decision to marry and the number of
kids within marriage result from agents’ optimization, and (ii) positive checks as I model the
survival rate of adult agents as directly depending on their income in the following way:
st=s yϕ
t,(10)
9
where sis a parameter calibrated to target an initial survival rate and ϕis the elasticity of
the survival rate to income per capita. Thus, in accordance with the Malthusian theory, adult’s
survival is increasing along income since s > 0and ϕ > 0.
Population Dynamics.— The size of the population of the next generation t+ 1 is given by:
Lt+1 =ntptstLt.(11)
Income per capita Dynamics.— Forwarding (8) to period t+ 1 and using (8), (9) and (11),
I obtain a first-order difference equation giving the income per capita of the next generation:
yt+1 =1 + g
nt(yt)pt(yt)st(yt)α
·yt≡ψ(yt).(12)
Steady State.— The steady state of the economy is defined by a situation in which:
y∗≡1 + g
n(y∗)p(y∗)s(y∗)α
= 1 .(13)
At the steady state, the rate of population growth equals the rate of technological progress,
such that income per capita remains constant period after period.6
3 Quantitative Analysis
In this section, I simulate the reaction of the English Malthusian economy to the Black Death
in order to illustrate the convergence process of a Malthusian economy after a shock. I start
by discussing the identification of the parameters that I use to calibrate the English Malthusian
economy. I then discuss the simulation results of the calibration exercise and compare them
with existing data.
6Section Aof the Appendix shows that ψ(yt)has a unique and locally stable steady state y∗>0, provided that
ytis not too low.
10
3.1 Identification of the Parameters and Initial Conditions
In order to simulate the evolution of a Malthusian economy and study its speed of convergence,
I first set the value of some parameters a priori, while some others are set to match some
target following an exact identification procedure. I focus on England as the literature already
provides a rich array of parameter values for that economy during the Malthusian period. Table
1summarizes and explains my calibration strate.
Table 1: Benchmark Parameter Values
Parameter Value Interpretation and comments
t25 Number of years per generation. Fixed a priori
γ1Preference for children. Fixed a priori
q1Unitary cost of a child. Fixed a priori
δ0.074 Gives preventive checks-income per capita elasticity of 0.21. Fixed a priori
ϕ0.1Gives positive checks-income per capita elasticity of 0.1. Fixed a priori
α0.5Land share of output. Fixed a priori
g0.023 Rate of technological progress per generation. Fixed a priori
s0.178 Minimum of the survival rate. To match s∗= 0.71
ν0.662 Child quantity preference parameter. To match n∗= 1.62
b3.48 Maximum of the search cost distribution. To match p∗= 0.89
Notes: See text for more details on the sources.
First, the length of a period or generation tis fixed at 25 years, meaning that an agent
is living at most 50 years in my model.7This is in line with life expectancy figures in pre-
industrial England as reported by Wrigley et al. (1997). Life expectancy at the age of 20 was as
high as 33-34 years on the period 1550-1799. Conditional on their survival until the age of 20,
Malthusian agents have therefore good chances to reach the age of 50. This is also in line with
the evidence on the so-called European Marriage Pattern (EMP) from Hajnal (1965). Indeed,
the EMP is characterized by a late age of first marriage for women (between age of 24 and 26)
and low illegitimacy birth rates. In my setting, agents marry and procreate only in the second
period of their life, that is to say between age of 25 and 50 as indicated by the EMP.
Next, I normalize γand q, respectively the agent’s preference for children and the cost of
7de la Croix and Gobbi (2017) make a similar assumption in a modern context with developing economies.
11
raising a child, to one.
Elasticity parameters δand ϕare particularly important in my setting, as they directly affect
the speed of convergence (see Section 4). Since I am working at the generation level, I consider
these parameters as representing respectively the long-run elasticity of the preventive checks
(fertility and marriage) and the long-run elasticity of the positive checks (survival) to income
per capita.8The empirical literature testing the Malthusian model in England provides various
estimates of these long-run elasticities based on wage, Crude Birth Rate (CBR), Crude Marriage
Rate (CMR) and Crude Death Rate (CDR) time-series (Lee,1981;Lee and Anderson,2002;
Nicolini,2007;Crafts and Mills,2009;Klemp,2012;Møller and Sharp,2014). I set δ= 0.074
and ϕ= 0.1in my benchmark specification to match the median of the long-run elasticities
provided by the aforementioned literature. This corresponds to a long-run elasticity of 0.21
for the preventive checks and 0.1 for the positive checks. Table B-1 in the Appendix provides
a complete list of studies, elasticity values, and details the method used to calibrate δand ϕ.
Setting δ < 1means that my model consider decreasing returns to scale in the production
of children, while most standard Malthusian models assume constant returns to scale (δ=
1).9As pointed out by Lagerlöf (2019), we may interpret decreasing returns to scale in the
production of children as stemming from an implicit production function for child survival
featuring two inputs: parental time devoted to each child and each child’s food intake. More
children automatically yields less time per child, leading to an increase in the per-child amount
of the consumption good necessary to ensure the survival of each child. Furthermore, the
aforementioned empirical literature consistently finds values well below unity for the long-term
elasticities of the preventive and positive checks. For instance, using exogenous cross-county
variations in Swedish harvest between 1816 and 1856, Lagerlöf (2015) finds long-run elasticities
of fertility, marriage and mortality of 0.1, 0.16 and -0.09, respectively.
The land share of output αfor England is set at 0.5, corresponding to its estimated long-run
8The long-run elasticity is the sum of elasticities at various time lags.
9See, for instance, Ashraf and Galor (2011).
12
value for the Malthusian period (Federico et al.,2020).
In standard Malthusian models with constant technological progress, total population at
the steady state is not constant. In fact, (13) shows that population grows at the same pace as
technolo; this is a necessary condition to keep income per capita constant at the steady state.
Consequently, gis calibrated using 25-years average population growth using Broadberry et al.
(2015) data for the period 1270-1675.
Consider next the three remaining parameters, s,νand bthat are calibrated to match re-
spectively the steady-state survival rate for adults (s∗), agent’s steady-state fertility (n∗) and the
steady-state marriage rate (p∗) following an exact identification procedure. The first target s∗is
set to 0.71 as in Wrigley (1968). This corresponds to the survival rate of population of 25 years
old until the age of 50 for the period 1538-1624 in England. The second target p∗is set to 0.89,
which corresponds to a percentage of never married women of 11% as reported by Dennison
and Ogilvie (2014) for England. This figure is the average of the percentage of never married
women for England across 45 historical studies and is also very close to the value reported in
the seminal study of Wrigley et al. (1989). Knowing the two first targets, the third target n∗is
given by the steady-state condition in (13). To find the value of these three remaining param-
eters, I also set the steady-state level of income per capita y∗to an arbitrarily high initial level,
by adjusting the initial level of technolo A0.
3.2 Simulation Results
This section shows the overall ability of my model to reproduce Malthusian dynamics and
match some of the long-run dynamics of the English economy after the Black Death. To do
so, I simulate a Black Death alike shock killing 60% of the population at t= 5. This is in line
with Benedictow et al. (2004), who finds an overall mortality of 62.5% for England.
Figure 1shows the evolution of income per capita (yt), fertility (nt), the marriage rate
(pt) and the survival rate (st) under my benchmark parametrization and under two alterna-
13
tive specifications, across 20 generations. The two alternative specifications are identical to
the benchmark, with the exception of the long-run elasticity values used to calibrate δand ϕ.
Whereas I calibrate the benchmark using the median of the long-run elasticity values found in
the literature for the preventive and positive checks, I calibrate the two alternative specifications
using the 10th and 90th percentiles of the long-run elasticities (see Table B-1 for an overview
of the long-run elasticity values I consider).
Standard Malthusian theory predicts that an exogenous negative shock on the population
level (or Black Death) increases income per capita in the short run only.10 After the shock,
population increases and the economy gradually converges back to its steady state such that,
at the long-run, the income per capita is constant. This is, by construction, what I observe in
my model. Figure 1shows that, right after the plague onset, the surviving agents enjoy indeed
a temporarily higher level of income per capita. These better material conditions mean that
agents have better chances to survive, they marry more and are able to raise more surviving
children inside marriage. This translates into faster population growth, which in turn triggers
the convergence process of income per capita to its steady state.
The top left panel of Figure 1also display the half-life of convergence for the benchmark
and the two alternative specifications. In the three cases, the elasticities imply long adjustments
to shocks, indicating weak homeostasis. The half-life is about 112 years (4.47 generations ×25
years) in the benchmark scenario, 64 years (2.55 generations) in the 90th percentile scenario
and 230 years (9.21 generations) in the 10th percentile scenario. It implies that any shock
striking the Malthusian English economy is persistent across several generations. It takes, at
least, 2.5 generations to fill half of the gap with respect to the steady-state.
As a complementary and illustrative exercise, Figure 2evaluates the ability of the model to
replicate the dynamic of income per capita after the Black Death, using English historical GDP
per capita data from Broadberry et al. (2015). To do so, I first extract the cyclical component
10Jedwab et al. (2022) find evidence that the Black Death was indeed a plausibly exogenous shock to the Euro-
pean economy.
14
Figure 1: Responses of the English Malthusian Economy to a Black Death
Median p10
p90
100 120 140 160
GDP per capita
0 5 10 15 20
Generations
100 102 104 106 108
Surviving children per adult
0 5 10 15 20
Generations
100 102 104 106 108
Marriage rate
0 5 10 15 20
Generations
100 105 110 115
Survival rate
0 5 10 15 20
Generations
Notes: This figure plots the response of per capita income (top-left panel), fertility (top-right panel), marriage
(bottom-left panel) and survival (bottom-right panel) to a Black Death alike shock, killing 60% of the population
at t= 5. The solid line indicates the benchmark scenario, using the median of the long-run elasticities of the
preventive and the positive checks provided by the literature to calibrate the model (see Section Bof the Appendix
for more details). The longdashed and doted line indicates an alternative calibration, using the 10th percentile of
the long-run elasticities. The dashed line indicates another alternative calibration, using the 90th percentile of the
long-run elasticities. Vertical lines in the top-left panel indicate half-lives of the shock.
in the data using an Hodrick–Prescott filter.11 This is necessary step, as my model analyses
the dynamic of convergence to a unique and fixed steady state. On the contrary, fluctuations
in the data might reflect changes in the position of the Malthusian steady state, as well as the
transition to a fixed steady state. As argued by North and Thomas (1973) and Acemoglu and
Robinson (2012), the Black Death might have affected the steady state of the English economy
11I set the smoothing parameter to 100 given that I use yearly data.
15
itself, through institutional changes.12
Figure 2shows that a simple calibrated Malthusian model is able to generates a path for
GDP per capita similar to the cyclical component of the data in the years following the Black
Death for the English economy. This result is remarkable because the path predicted by the
model is governed only by the initial demographic shock and the long-run elasticities provided
by the empirical literature.
As my model has no stochastic components, deviations from the predicted trajectory reflect
subsequent shocks hitting the Malthusian economy. The important point is that the overall
trend remains within the limits of the three scenarios, all of which reflect a relativity weak
Malthusian trap.
12For example, institutional changes allowing for an increase in the rate of technological progress g, would
modify the position of the Malthusian steady state.
16
Figure 2: GDP per capita Dynamic after the Black Death: Simulated Paths vs. Data
80 100 120 140 160
Cyclical component of GDP per capita (1300s=100)
1300 1350 1400 1450 1500 1550 1600 1650
10th percentile
Median
90th percentile
Data
Notes: This figure plots the cyclical component of GDP per capita from Broadberry et al. (2015) (solid line),
and the simulated post-Black Death GDP per capita paths from the benchmark calibration using the median of
the long-run elasticities (dashed and short dotted line) and two alternative calibrations using the 10th percentile
(dashed and long dotted line) and the 90th percentile (dashed line) of the long-run elasticities. Data are normalized
on the period 1300-1325, the last period before the occurrence of the Black Death in England (1348).
4 The Speed of Convergence in a Malthusian World
In this section, I start by deriving the speed of convergence of a Malthusian economy to its
steady state, as implied by my model. Next, I use the derived formula and parameter values
found in the literature to calculate the speed of convergence for various Malthusian economies,
and compare it with the literature.
The speed at which GDP per capita converges to its steady state in a Malthusian economy
17
is given by:
β∗=α(ϵnt+ϵpt+ϵst),(14)
where ϵnt,ϵptand ϵstare the elasticities of fertility, marriage and survival with respect to
income per capita. Section Cof the Appendix provides further details on the derivation of the
speed of convergence. The speed of convergence is therefore determined by the product of the
land share of output αand the sum of the elasticities representing the preventive checks ϵntand
ϵpt, and the positive checks ϵst. Similar results are found by Irmen (2004) and Szulga (2012) in
continuous time.
In Table 2, I compare the half-life obtained from my calibration of the English Malthusian
economy with the speed of convergence found in the literature for other Malthusian and de-
veloping economies. In particular, I use equation (14) and long-run elasticity values provided
by Galloway (1988), Lagerlöf (2015), Klemp and Møller (2016) and Pfister and Fertig (2020)
to calculate the speed of convergence implied by my model for Denmark, Norway, Sweden,
Germany and the median European Malthusian economy. I also report the half-life directly
estimated by other studies for comparison purposes.
Despite the differences in period and context, the half-lives obtained for England appear to
be in line with much of the literature. In particular, my benchmark result is very close to the
half-life estimated by Fernihough (2013) for Northern Italy (112 years), or calculated using
Galloway’s (1988) long-run elasticity values for the median European Malthusian economy
(115 years). My benchmark falls also close to a half-life of one century as found by de la
Croix and Gobbi (2017) for Sub-Saharan Africa, or calculated using Lagerlöf’s (2015) long-
run elasticity values for Sweden. However, my estimations appear to be substantially higher
than the half-lives found by Madsen et al. (2019) in a panel of 17 Malthusian economies.
18
Table 2: Speed of Convergence in my Calibrations and in the Literature
Country Period Authors Half-Life (years) Comments
England Present study 112 Benchmark specification, calibrated using the median
of the long-run elasticities reported in Table B-1 (δ=
0.074;ϕ= 0.1;α= 0.5)
England Present study 230 Alternative specification, calibrated using the 10th
percentile of the long-run elasticities reported in Ta-
ble B-1 (δ= 0.045,ϕ= 0.06 and α= 0.5)
England Present study 64 Alternative specification, calibrated using the 90th
percentile of the long-run elasticities reported in Ta-
ble B-1 (δ= 0.09,ϕ= 0.23 and α= 0.5)
Sub-Saharan Africa 1990 de la Croix and Gobbi (2022) 198 Table 3, population regression
Europe 1540-1870 Galloway (1988) 115 Using (14), α= 0.5and reported elasticities
Northern Italy 1650-1881 Fernihough (2013) 112 Table 2, VAR estimates
Developing countries 1990 de la Croix and Gobbi (2017) 100 Table 7, population regression
Sweden 1816-1870 Lagerlöf (2015) 100 Using (14), α= 0.5and reported elasticities
Norway 1775-1853 Klemp and Møller (2016) 91 Using (14), α= 0.5and reported elasticities
Denmark 1821-1890 Klemp and Møller (2016) 84 Using (14), α= 0.5and reported elasticities
Germany 1730-1830 Pfister and Fertig (2020) 58 Using (14), α= 0.5and reported elasticities
Sweden 1775-1873 Klemp and Møller (2016) 48 Using (14), α= 0.5and reported elasticities
17 countries 1470-1870 Madsen et al. (2019) 29 Table 2, income regression
17 countries 1470-1870 Madsen et al. (2019) 12 Table 3, population regression
17 countries 1470-1870 Madsen et al. (2019) 11 Table 1, wage regression
19
5 Empirical Framework
In this section, I first present the data I use to empirically estimate the speed of convergence for
a wide range of Malthusian economies. Then, I detail my main estimating equation and discuss
potential threats to my identification strate.
5.1 Data
In the empirical analysis that follows, I use two main types of datasets: (i) panel data on GDP
per capita (historical or simulated) and (ii) panel data on historical population levels (total or
urban population). The historical GDP per capita series come from the Maddison Project
Database (Bolt and Van Zanden,2020). Building on the pioneering work of Maddison (2003),
the Maddison Project provides standardized historical GDP per capita series spanning several
centuries. These series are regularly updated and enriched by researchers in the field of historical
national accounting. However, as discussed in more detail in the next section, the uncertainty
associated with past economic fluctuations is one of the concerns associated with the use of these
sources. To limit measurement error issues, I focus on the period 1000-1800, and consider only
countries with good data availability – i.e. countries for which GDP per capita data are available
annually or every ten years before 1800. Following these two criteria, I consider a panel of
twelve countries, including core (e.g. Italy, England, China) and more peripheric (e.g. Mexico,
Poland, Sweden) Malthusian economies.
To complete my analysis, I also use simulated GDP per capita series from Lagerlöf (2019).
Lagerlöf (2019) shows that a Malthusian model with stochastic and accelerating growth in land
productivity is able to match the moments of historical GDP per capita series presented in
Fouquet and Broadberry (2015). Simulations are available for 1,000 model economies and 501
years, making it very useful to circumvent the lack of GDP per capita data inherent to the pre-
industrial period. From an econometric point of view, it corresponds to an ideal setting where
20
both the cross-sectional and the time dimensions are large, limiting the bias of the different
estimators on the speed of convergence.
For historical population series, I first use McEvedy et al.’s (1978) data. Population figures
from this source have been widely used to answer various questions in the comparative develop-
ment literature, with most of the contributions exploiting cross-country variations over a few
years (Acemoglu et al.,2001;Nunn,2008;Nunn and Qian,2011;Ashraf and Galor,2011,
2013).13 My aim, on the other hand, is to exploit population changes within a country, and so
I have coded McEvedy et al.’s (1978) data in their panel dimension. Although widely used in
the literature, these data are also highly criticized, mainly for measurement error issues (Guin-
nane,2021). To mitigate this problem, I use only a specific time frame and set of countries.
First, I consider only the period between the years 1000 and 1750, which avoids the sizeable
uncertainty surrounding population figures at the end of the Roman Empire and the beginning
of the Middle Ages. Second, within that selected period, I keep only countries for which popu-
lation figures are reported with maximum frequency – i.e. every century before 1600 and every
half-century after 1600. Following these two criteria, I consider a panel of eighteen countries
from this source for my empirical analysis.
To complement my analysis with historical population series, I am also using data from Reba
et al. (2016), who compiled and geocoded urban population figures from Chandler (1987) and
Modelski (2003). In particular, the database provides population level for cities worldwide
from 3700 BC to 2000 AD. I apply the same procedure as for the other datasets, namely I first
select urban population levels during the period 1000-1800.14 Next, I focus on cities with a
good data availability – i.e. cities for which a population figure is available for at least seven
half-centuries (out of the seventeen potentially available) between the years 1000 and 1800.15
13For example, Ashraf and Galor (2011) use McEvedy et al. (1978) data as dependent variable, and exploit its
cross-sectional variation in year 1, 1000 and 1500.
14When both Chandler (1987) and Modelski (2003) data are available for the same city and year, I take the
average between the two figures. This was the case for 20 cities, only for year 1000.
15That threshold corresponds to the median of data availability.
21
5.2 Empirical Strate
To empirically assess the speed of convergence of Malthusian economies, I rely on a standard
β-convergence model. Such models have been extensively used in the growth literature to
quantify the speed at which modern economies converge to their steady state (Barro,1991;
Barro and Sala-i Martin,1992;Islam,1995;Caselli et al.,1996;Barro,2015). More recently,
this framework has also been used in the Malthusian context by Madsen et al. (2019).
My main specification is the following dynamic panel:
ln(yi,t)−ln(yi,t−τ)
τ=βln(yi,t−τ) + γ′Xi,t+δt+αi+εi,t ,(15)
where i= 1, ..., N indicates my unit of analysis which can be either a country or a city and
t= 1, ..., T corresponds to a given year. The left-hand side of equation (15) corresponds to the
growth rate of my variable of interest y, which can be either GDP per capita or population
levels, depending on the specification. The parameter τindicates the number of years between
two available data points, so that the dependent variable is always the average annual growth
rate of ybetween period t−τand t.
My coefficient of interest is β, which gives the average annual speed at which Malthusian
economies converge to their steady state. Obtaining unbiased estimates of the speed of conver-
gence is challenging in many ways. First, endogeneity is a concern, as past levels of economic
development and current economic growth may be jointly determined by omitted factors. To
mitigate that issue, equation (15) includes fixed effects αithat control for time-invariant deter-
minants of economic development, such as geography, climate and, to some extent, culture.
While partially solving the problem of omitted variables, country fixed effects are them-
selves recognized as a source of upward bias in the measurement of convergence speed in dy-
namic panels, known as the Hurwicz-Nickell bias (Hurwicz,1950;Nickell,1981). This is a
potential problem, as it would constitute a systematic bias against weak homeostasis in my anal-
22
ysis. However, as highlighted by Barro (2015), the Hurwicz-Nickell bias tends towards zero
when the overall sample length in years tends towards infinity. This means that the risk of a
sizeable Hurwicz-Nickell bias is strongly mitigated in my analysis by the length of the overall
sample, which spans several centuries.16
To address the endogeneity issue arising from time-varying omitted factors, the vector Xi,t
includes Statehist and its squared level as control variables (Borcan et al.,2018). Statehist is an
index retracing state development every half-century from 3500 BC until today. I use it to proxy
broad institutional changes that can affect the steady-state position of Malthusian economies. In
equation (15), I also include time fixed effects δtto control for global changes in the steady-state
determinants, such as the spread of new technologies or global climatic changes.17
To further address the endogeneity concerns, I provide results using an instrumental variable
approach. In particular, I use the Arellano and Bond (1991) and the Blundell and Bond (1998)
GMM estimators (hereafter referred to as AB and BB, respectively). These estimators have long
been used in the context of growth regressions, either to estimate the speed of convergence of
modern economies or to measure the effect of steady-state determinants.18 Their advantage over
the fixed effects estimator is the ability to instrument endogenous regressors, while controlling
for country and time fixed effects.19 However, one recognized potential issue using AB is the
weakness of its instruments, which is known to bias βestimates towards their fixed effects
counterparts. BB is more robust to that issue, but requires a stationarity assumption to deliver
consistent results, which is found to not necessary hold in practice (Hauk and Wacziarg,2009).
Given the merits and drawbacks of each method, throughout the article I systematically present
16On the contrary, Barro (2015) finds that the Hurwicz-Nickell bias on the speed of convergence coefficient is
sizeable in the modern growth context, where the analysis runs typically over 50 years.
17For instance, my analysis spans from the 11th to the 19th century, the period during which certain global
climatic events, such as the Medieval Warm Period or the Little Ice Age, occurred. Time fixed effects can control
for these events, provided that they affected a large part of the sample.
18This procedure was first used by Caselli et al. (1996) in the growth context to address both the Hurwicz-
Nickell bias and the endogeneity of regressors.
19In particular, the AB estimator takes the first-difference of the regression equation and uses the lagged levels of
the endogenous variables as instruments. The BB estimator complements AB, also using the lagged first differences
of the endogenous variables as instruments for their levels.
23
estimates based on both estimators for comparison purposes.
Another source of concern is the measurement error of the lagged dependent variable. In
presence of classical measurement error, i.e. random errors in the measurement of an explana-
tory variable, βwill suffer from an attenuation bias, increasing the estimated speed of conver-
gence. To limit this possibility, I implement several strategies. First, as detailed in Section 5.1, I
systematically avoid using the most uncertain data on population or GDP per capita, excluding
figures prior to the year 1000. Indeed, as pointed recently by Guinnane (2021), we simply “do
not know the population” going that far back in the past where standardized and systematic
censuses were not operated. Population and output measures between the years 1000 and 1800
also contain a sizeable part of uncertainty. However, local censuses, parish registers or proxy
variables such as urbanization are increasingly available on that period, reducing measurement
error. I also only consider countries or cities with the best, or at least above median, data
coverage in each source. Second, I follow the usual practice in the empirical macroeconomic
literature and calculate 50-year averages of the explanatory variables when the data is available
at a lower frequency.20 This allows me to avoid spurious changes and focus on long-term dy-
namics. Third, AB and BB estimators would also mitigate this source of bias, as instrumental
variables can in principle deal with classical measurement error.
Nevertheless, there remains the possibility of non-classical measurement error, such as sys-
tematic and persistent differences over time in the measure of explanatory variables between
countries. If this type of measurement error is highly persistent over time, it will be treated by
the country fixed effects.21 To account for less persistent measurement error across countries, I
also systematically run fixed effects regressions with year-interacted lagged dependent variables.
In this case, any varying differences in measurement correlated with initial population or initial
GDP per capita levels will be taken into account. Typically, I find that this approach do not
20This means that I take a minimal τof 50 years in equation (15).
21Similarly, time fixed effects can deal with measurement errors that vary over time and are common to the
countries in the sample, such as the gradual improvement in population figures as we approach year 1800.
24
differ significantly from the results of my baseline fixed effects regressions.
In my estimation strate, I consider only the “within country” class of estimators (fixed
effects, AB and BB), while growth regressions have also been estimated using the between or a
random effects estimator. Monte Carlo simulations on β-convergence regressions in the context
of modern growth have found mixed evidence about the ability of the two classes of estimators
to accurately estimate the speed of convergence. Hauk (2017) finds that the speed of conver-
gence is best estimated with the within-country class of estimators when endogeneity bias on
the steady-state determinants is the main concern. On the contrary, Hauk and Wacziarg (2009)
find that the speed of convergence is best estimated using the between or random effects esti-
mator when regressor measurement error is the dominant issue. In the present case, I consider
the endogeneity bias to be the most serious threat and therefore use the within-country type of
estimator for two main reasons. First, there are very few control variables available for a large
sample and a long period of analysis in the Malthusian context, opening the possibility of a sub-
stantial endogeneity bias stemming from omitted variables. Second, measurement error is dealt
to a certain extent by the various strategies described in the previous two paragraphs. Further-
more, Hauk and Wacziarg (2009) show that the within-county estimators imply a higher speed
of convergence. This means that measurement error will ultimately constitute a bias against
the weak homeostasis hypothesis. In my results, I show consistent evidence of weak homeostasis,
suggesting that measurement error is indeed a second-order concern.
6 Results
In this section, I present my empirical estimates of the speed of convergence for various Malthu-
sian economies. I start by presenting my results using historical and simulated per capita income
data. Then, I present my results using historical data on total and urban population.
25
6.1 Speed of Convergence using GDP per capita Data
In Table 3, I report the estimations of specification (15) using OLS and fixed effects. The
dependant variable is the average annual growth rate of GDP per capita calculated from Mad-
dison Project’s data (Bolt and Van Zanden,2020).22 I first present the relationship between
the dependent variable and the initial level of GDP per capita, controlling for time fixed effects
(columns 1 and 4). Then, I add country fixed effects (column 2 and 5). Finally, I add Statehist
and its squared level as control variables (columns 3 and 6).
Table 3: Speed of Convergence using GDP per capita Data from the Maddison Project
Sample Used: Full Europe
OLS FE FE OLS FE FE
(1) (2) (3) (4) (5) (6)
log(GDPpc) -0.0006 -0.0057** -0.0057*** 0.0000 -0.0046** -0.0046**
(0.001) (0.002) (0.002) (0.001) (0.001) (0.001)
Time FE Yes Yes Yes Yes Yes Yes
Country FE No Yes Yes No Yes Yes
Statehist No No Yes No No Yes
Observations 85 85 85 69 69 69
adj. R-sq -0.01 0.16 0.14 -0.05 0.11 0.08
Half-Life 1197 122 121 -18766 150 152
Half-Life 95% C.I. [-434,252] [422,71] [391,72] [-356,370] [587,86] [663,86]
Notes: This table presents OLS estimates of the speed of convergence using GDP per capita data from the Maddison Project at the country
level. Columns 1-3 present results obtained from the full sample of countries considered from the Maddison Project data, and columns 4-6
show results obtained by focusing on European countries. For each sample, I first display the relationship controlling for time fixed effects in
column 1, then include country fixed effects and finally add Statehist as control. Standard errors clustered at the country level are in parentheses.
* p<0.1, ** p<0.05, *** p<0.01.
Starting with the most parsimonious specification, with only time fixed effects as controls,
column 1 reveals that the lagged dependent variable coefficient is not statistically different from
zero. This is not really surprising as the omitted variable bias is substantial in this case, driv-
ing the lagged dependent coefficient toward zero. Moreover, as my theoretical model suggests,
22In this case, GMM estimates are not reported due to the lack of observation units. Indeed, as Roodman
(2009) advises, a useful rule of thumb to avoid weak instrument problems in GMM estimations is to keep the
total number of instruments below the number of observation units. This is not possible with the current sample
from the Maddison Project, as we have eleven countries and fifteen instruments in the most parsimonious case,
resulting in unitary Hansen test p-values.
26
Malthusian economies should display conditional convergence rather than absolute conver-
gence, as the steady-state position of each economy depends on its characteristics.23
Adding country fixed effects, column 2 reveals a negative and significant relationship be-
tween GDP per capita growth and the initial level of GDP per capita, indicating conditional
convergence of Malthusian economies. The estimated coefficient implies a half-life of 122 years
(ln(2)/0.0057), with a 95% confidence interval giving half-lives between 422 years and 71 years.
Therefore, the most comprehensive and up-to-date historical GDP per capita series are consis-
tent with weak homeostasis of Malthusian economies, as it takes at least several generations to
absorb half of a shock. Compared to other studies, the results in column 2 are close to Fer-
nihough (2013), who find a half-life of 112 years for Northern Italy (1650-1881) using VAR
methods. However, this result is in great contrast with Madsen et al. (2019), who find a half-
life of 29 years for income per capita and conclude in favor of strong homeostasis of Malthusian
economies.24
One possible reason for the distortion of the estimated convergence speed in favor of weak
homeostasis is the presence of a severe omitted variable bias. In particular, column 2 does not
control for time-varying determinants of GDP per capita growth at the country level, as it
includes only time and country fixed effects. To limit that concern, column 3 adds Statehist
and its squared level as controls. The speed of convergence is almost unaffected, as the reported
half-life is now slightly higher at 121 years.
As a robustness check, columns 4 to 6 replicate the analysis, restricting the sample to Euro-
pean countries, giving similar results. In particular, column 6 indicates an even slower speed of
convergence on average, with a half-life of 152 years, confirming the weak homeostasis pattern
found in the previous columns. However, I find no significant differences in the estimated
23From the steady-state condition in (13), it is clear that two economies, with for instance different rates of
technological progress g, will not converge to the same steady state.
24Note that my article has several methodological differences with respect to Madsen et al. (2019). First, they
rely on interpolated data coming from heterogeneous sources for GDP per capita and population data, while I
take the data as given from each source. Second, they use seemingly unrelated regression (SUR) models, a random
effects family estimator, while I use within-country estimators (LSDV, AB and BB).
27
speed of convergence between the two country samples.
Figure 3displays the fixed effects estimations of columns 3 and 6, adding an interaction term
between time fixed effects and the initial level of GDP per capita. This allows me to examine
the heterogeneity of the speed of convergence through time, and to check the possible influence
of non-classical measurement errors. Overall, the point estimates are negative and statistically
different from zero at the 5% level. Whether considering the full or the European sample of
countries, the vast majority of the estimated coefficients are not statistically different from a
half-life of 115 years, as found for Europe using the long-run elasticities of Galloway (1988) in
Section 4. This indicates a clear and stable pattern of weak homeostasis during a large part of the
Malthusian period. On the contrary, strong homeostasis, as represented by the highest half-life
(about 30 years) found in Madsen et al. (2019), is always rejected at the 5% level.
Turning to the heterogeneity of the speed of convergence by country, Figure 4displays
point estimates of the fixed effects estimations of column 3 and 6, adding an interaction term
between the country fixed effects and the initial level of GDP per capita. Figure 4reveals mixed
results as some countries are found compatible with weak homeostasis (e.g. the Netherlands),
and some other countries rather lean towards strong homeostasis (e.g. Poland). Some countries,
like France or Spain, are even found to be compatible with both types of homeostasis. However,
precision of estimates is clearly an issue in that specification. Indeed, as shown in Figure 4,
confidence intervals are generally large.
In addition to the above results, Table 4reports OLS, fixed effects and GMM estimates of
specification (15), where the dependent variable is the average annual growth rate of GDP per
capita calculated using Lagerlöf’s (2019) simulated data. In particular, the simulated GDP per
capita series are produced from a Malthusian model with stochastic and accelerating growth
in land productivity. Under plausible parameter values, Lagerlöf (2019) shows that the model
is able to accurately reproduce the empirical moments of the historical GDP per capita series
presented in Fouquet and Broadberry (2015) for several European economies between 1300 and
28
Figure 3: Speed of Convergence per period using the Maddison Project Data
Half-life=115 years
Galloway (1988)
Half-life=30 years
Madsen et al. (2019)
-.02 0 .02 .04
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
Half-life=115 years
Galloway (1988)
Half-life=30 years
Madsen et al. (2019)
-.02 0 .02 .04
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
Notes: This figure plots estimates of the speed of convergence using GDP per capita data from the Maddison
Project. It corresponds to the FE estimations in column 3, Table 3 (left panel) and in column 6, Table 3 (right
panel), adding year-interacted lagged GDP per capita levels as controls. 95% confidence intervals are reported.
Figure 4: Speed of Convergence per country using the Maddison Project Data
Half-life=115 years
Galloway (1988)
Half-life=30 years
Madsen et al. (2019)
-.04 -.03 -.02 -.01 0 .01
China
France
Italy
Mexico
Netherlands
Peru
Poland
Portugal
Spain
Sweden
United Kingdom
Half-life=115 years
Galloway (1988)
Half-life=30 years
Madsen et al. (2019)
-.04 -.03 -.02 -.01 0 .01
France
Italy
Netherlands
Poland
Portugal
Spain
Sweden
United Kingdom
Notes: This figure plots estimates of the speed of convergence using GDP per capita data from the Maddison
Project. It corresponds to the FE estimations in column 3, Table 3 (left panel) and in column 6, Table 3 (right
panel), adding country-interacted lagged GDP per capita levels as controls. 95% confidence intervals are reported.
29
1800. The original series presented in Fouquet and Broadberry (2015) are still part of the latest
Maddison Project database for some countries (e.g. Holland and Italy), or are updated versions
using the same methodolo (e.g. England and Sweden). Consequently, the main advantage
of using this simulated series to estimate convergence speed is to gain in precision, since the
simulated data correspond to the same moments while possessing a much greater temporal and
cross-sectional dimension.
Table 4: Speed of Convergence using simulated GDP per capita Data from Lagerlöf (2019)
OLS FE GMM-AB GMM-BB
(1) (2) (3) (4)
log(GDPpc) -0.0019*** -0.0052*** -0.0063*** -0.0047***
(0.000) (0.000) (0.002) (0.001)
Time FE Yes Yes Yes Yes
Country FE No Yes Yes Yes
Observations 10000 10000 9000 10000
adj. R-sq 0.09 0.18 . .
AR(7) 0.17 0.18
Hansen 0.22 0.23
Diff. Hansen . 0.21
Instruments 13 15
Half-Life 363 133 110 146
Half-Life 95% C.I. [403,330] [141,126] [212,75] [250,103]
Notes: This table presents OLS and GMM estimates of the speed of convergence using simulated GDP per capita data from Lagerlöf (2019)
at the country level. Column 1 controls for time fixed effects, and the subsequent columns add country fixed effects. The GMM estimations
in columns 3 and 4 use the seventh and further lagged levels of GDP per capita as instruments. I use a collapsed matrix of instruments and
report the number of instruments. The AR(7) row reports the p-value of a test for the absence of seventh-order correlation in the residuals.
Standard errors clustered at the country level are in parentheses. * p<0.1, ** p<0.05, *** p<0.01.
As expected, the speed of convergence is now estimated with much more precision. The
fixed effects estimation in column 2 shows a half-life of 133 years, with a 95% confidence
interval giving half-lives between 141 and 126 years. These results lie within the wide confidence
intervals of the previous results in Table 3using Maddison Project’s data. The Hurwicz-Nickell
bias is very unlikely to affect the estimates, as this is a setting where the time dimension is very
large (T= 500).
Columns 3 and 4 present AB and BB GMM results. In both cases, the estimated speed of
30
convergence is highly significant and consistent with weak homeostasis. In particular, I find that
both GMM estimates are not statistically different from the speed of convergence estimated by
the fixed effects model in column 2. This may seem worrying, as it is generally considered to
be a a sign of the weak instrument problem in the literature. However, as mentioned above,
the fixed effect estimation of column 2 takes place in an ideal setting where its main source of
bias – i.e. the Hurwicz-Nickell bias – is expected to be small. Under these conditions, it is
plausible that GMM and fixed effects estimations give similar results.
The classical GMM post-estimation tests give also clear signs that the moment conditions are
globally satisfied. In particular, I reject the null hypothesis of seventh-order serial correlation
in the residuals (AR(7) test), meaning that using the seventh (and greater) lag of GDP per
capita as instruments does not violate the exclusion restriction. Second, I reject both the null
hypothesis of the Hansen test and the difference in Hansen test for all GMM instruments,
indicating that the set of used instruments are plausibly exogenous. Overall, I consider that the
GMM estimates provide converging evidence of weak homeostasis.
Figure 5investigates the time heterogeneity of the speed of convergence. All the coeffi-
cients are statistically different from zero and very precisely estimated, thanks to the large time
and sample size. The speed of convergence is fairly stable over time. Half of the estimated
coefficients cannot reject a half-life of 115 years at the 5% level, as found for Europe using
the long-run elasticities of Galloway (1988). In addition, all the remaining coefficients show a
slower speed of convergence, again indicating a weak homeostasis of Malthusian economies.
The large cross-sectional dimension of Lagerlöf’s (2019) data allows me to study the range
of plausible half-lives in Malthusian economies with greater consistency than with Maddison
Project’s data. To do so, I perform the fixed effects estimation in column 2, adding an interac-
tion term between the country fixed effects and the initial level of GDP per capita to estimate
the speed of convergence for each Malthusian economy. Figure 6displays the kernel density of
31
Figure 5: Speed of Convergence per period using Lagerlöf (2019) Data
Half-life=115 years
Galloway (1988)
Half-life=30 years
Madsen et al. (2019)
0-.005-.01-.015-.02
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
Notes: This figure plots estimates of the speed of convergence by period using simulated GDP per capita data from
Lagerlöf (2019). It corresponds to the FE estimation in column 2, Table 4, adding year-interacted lagged GDP
per capita levels as controls. 95% confidence intervals are reported.
Figure 6: Speed of Convergence per country using Lagerlöf (2019) Data
Half-life=30 years
Madsen et al. (2019)
Half-life=115 years
Galloway (1988)
020 40 60 80
Density
-.04 -.02 0 .02
Speed of Convergence - β coefficients
kernel = epanechnikov, bandwidth = 0.0013
Notes: This figure plots the kernel density of the estimated speed of convergence by country using simulated
GDP per capita data from Lagerlöf (2019). It corresponds to the LSDV estimation in column 2, Table 4, adding
country-interacted lagged GDP per capita levels as controls.
32
the estimated speed for the 1000 simulated Malthusian economies in Lagerlöf (2019).25 Con-
sistent with the previous country-level evidence and my results, it appears that the mode of
the distribution is very close to a half-life of 115 years, as found for Europe using the long-run
elasticities of Galloway (1988). As a result, most pre-industrial economies were in a moderate
Malthusian trap or weak homeostasis. Interestingly, some Malthusian economies appear to have
lived under a strong Malthusian trap, with half-lives of 30 years or less, as found by Madsen
et al. (2019).
6.2 Speed of Convergence using Population Data
In Section Cof the Appendix, I show that the speed of convergence of population to its steady
state in my Malthusian model is the same as for GDP per capita. Therefore, in this section, I
use the same β-convergence models and population data to provide additional estimates of the
speed of convergence during Malthusian times.
In Table 5, I present my results based on OLS, fixed effects and GMM estimations of
equation (15). The dependent variable is the average annual population growth rate, calculated
from McEvedy et al. (1978) population figures. I first present the relationship between the
dependent variable and the initial population levels, controlling for time fixed effects (column
1). Then, I add country fixed effects (column 2) and Statehist and its squared level as control
variables (column 3). Finally, I perform AB and BB GMM estimations (columns 4 and 5).
Controlling for time and country fixed effects, column 2 reveals a negative and highly signif-
icant relationship between population growth and its initial level. The implied half-life is about
147 years, which is in line with my previous results using historical GDP per capita series (see
Table 3, column 3, and Table 4, column 2). The 95% confidence interval indicates half-lives
between 224 and 109 years, which stays clearly in the range of weak homeostasis.
25Figure D-1 in the Appendix delivers the point estimates along with their 95% confidence intervals for the 200
first simulated economies in Lagerlöf (2019).
33
Table 5: Speed of Convergence using Population Data from McEvedy et al. (1978)
OLS FE FE GMM-AB GMM-BB
(1) (2) (3) (4) (5)
log(Population) -0.000*** -0.005*** -0.006*** -0.009*** -0.004*
(0.000) (0.001) (0.001) (0.003) (0.002)
Time FE Yes Yes Yes Yes Yes
Country FE No Yes Yes Yes Yes
Statehist No No Yes Yes Yes
Observations 180 180 180 162 180
adj. R-sq 0.48 0.60 0.61 . .
AR(2) 0.69 0.36
Hansen 0.94 0.99
Diff. Hansen . 0.87
Instruments 18 22
Half-Life 4414 147 125 73 167
Half-Life 95% C.I. [12873,2663] [224,109] [231,86] [215,44] [-1176,78]
Notes: This table presents OLS and GMM estimates of the speed of convergence using population data from McEvedy et al. (1978) at the
country level. Column 1 controls for time fixed effects, the subsequent columns add country fixed effects and Statehist. The GMM estimations
in columns 4 and 5 use the second to fourth lagged levels of population as instruments. Statehist is and its squared level are treated as endogenous
and instrumented with the same set of lags as population. I use a collapsed matrix of instruments and report the number of instruments. The
AR(2) row reports the p-value of a test for the absence of second-order correlation in the residuals. Standard errors clustered at the country
level are in parentheses. * p<0.1, ** p<0.05, *** p<0.01.
Dealing further with the omitted variable issue, column 3 adds Statehist and its squared level
as controls. Convergence tends to be faster on average, with a half-life of 125 years. However,
I do not find significant differences in the speed of convergence between columns 2 and 3.
Columns 4 and 5 use GMM estimation procedures. Starting with the AB estimation, col-
umn 4 shows a faster average speed of convergence than the fixed effects results, with a half-life
of 73 years. This is potentially problematic, as it could reflect the influence of weak instru-
ments. Keeping the same set of instruments, the BB estimation indicates weak homeostasis, but
is only weakly significant. The difficulties associated with using GMM in this case stem from
the fact that the cross-sectional dimension is small using McEvedy et al.’s (1978) data relative to
the number of instruments used. This is the well-known problem of “too many instruments”,
highlighted by Roodman (2009).26 Under these conditions, my preferred specification is the
26Symptomatic of this problem, column 4 and 5 of Table 5reveal Hansen’s test p-value very close to one. This
34
fixed effects model with controls in column 3, assuming the Hurwicz-Nickell bias is small. As
mentioned in Section 5.2, this is all the more plausible given that the time dimension spans over
several centuries in this Malthusian context.
Figure 7displays the point estimates for the fixed effects estimation in column 3, adding year-
interacted initial population levels. All estimated coefficients are statistically different from zero
and consistent with a half-life of 115 years, as found for Europe using the long-run elasticities
of Galloway (1988). The point estimates are fairly stable in terms of magnitude, and within a
range compatible with weak homeostasis, confirming my previous results using GDP per capita
data.
Figure 8investigates the cross-country heterogeneity of the speed of convergence. Confi-
dence intervals are narrower than for the Maddison Project data, highlighting significant differ-
ences in the speed of convergence between Malthusian economies. The strongest Malthusian
trap is in Spain, with an half-life of 48 years and a 95% confidence interval between 40 and
61 years. On the other side of the spectrum, the weakest Malthusian trap is in Japan, with an
half-life of 118 years. The estimated half-life of the English Malthusian economy is 85 years,
with a 95% confidence interval between 71 and 106 years. This figure is lower than the half-life
obtained from the calibration of my benchmark Malthusian model in Section 3.2 (112 years).
However, the order of magnitude remains similar, as the estimated half-life lies between the two
alternative calibration scenarios, which give half-lifes of 64 and 230 years. These significant dif-
ferences among Malthusian economies suggest that a common shock could persist substantially
longer in England than in Spain, which is closer to strong homeostasis. Despite these significant
differences, the overall pattern remains compatible with a relatively weak homeostasis, since it
takes at least several generations to absorb half of a shock.
In Table 6, I present my results based on OLS, fixed effects and GMM estimations of
equation (15). The dependent variable is the average annual urban population growth rate,
is due to the fact that the number of countries in the sample (18 in this case) is very close to or less than the
number of instruments, even when considering a parsimonious instrumentation.
35
Figure 7: Speed of Convergence per period using McEvedy et al. (1978) Data
Half-life=115 years
Galloway (1988)
Half-life=30 years
Madsen et al. (2019)
.0060-.006-.012-.018-.024
1000
110 0
1200
1300
1400
1500
1600
1650
1700
1750
Notes: This figure plots estimates of the speed of convergence by period using population data from McEvedy
et al. (1978). It corresponds to the FE estimation in column 3, Table 5, adding year-interacted lagged population
levels as controls. 95% confidence intervals are reported.
Figure 8: Speed of Convergence per country using McEvedy et al. (1978) Data
Half-life=112 years
Benchmark - England
Half-life=64 years
90th percentile - England
Half-life=30 years
Madsen et al. (2019)
.0060-.006-.012-.018-.024
Austria
Bel. and Lux.
China Proper
Czechoslovakia
England & Wales
France
Germany
Hungary
Italy
Japan
Korea
Pak. India & Bang.
Poland
Portugal
Romania
Russia in Europe
Scandinavia
Spain
Notes:This figure plots estimates of the speed of convergence by country using population data from McEvedy
et al. (1978). It corresponds to the FE estimation in column 3, Table 5, adding country-interacted lagged total
population levels as controls. 95% confidence intervals are reported.
36
calculated using Reba et al. (2016) data. Using urban population data to estimate the speed of
convergence is interesting because the frequency of observations and the sample size are higher
than for the country-level population data of McEvedy et al. (1978), which increases precision.
I perform city-level estimations in columns 1-4 and estimations with urban population data
aggregated at the country level in columns 5-9. In each case, I first present the relationship
between the dependent variable and the initial population level, controlling for time fixed effects
(columns 1 and 5). Then, I add respectively city and country fixed effects (columns 2 and 6).
When possible, I add Statehist and its squared level as control variables (column 7). Finally, I
provide GMM estimation results (columns 3, 4, 8 and 9).
Starting with the city-level estimations, column 2 reveals a negative and highly significant
relationship between urban population growth and the initial level of urban population, con-
ditional on time and city fixed effects. The corresponding half-life is 95 years, with a 95%
confidence interval indicating half-lives between 155 and 68 years.
The GMM estimates in columns 3 and 4 confirm the results of the fixed effects estimation,
with half-lives of 97 and 104 years respectively, and similar confidence intervals. Both GMM
estimates reject the presence of second-order correlation in the residuals (AR(2) test), demon-
strating the validity of the set of instruments used. It is worth noticing that the AB estimation
fails to satisfy the Hansen test at the usual confidence levels. Reassuringly, using the same set
of instruments, the Hansen test of overindentifying restrictions and the difference in Hansen
test indicate that the moment conditions are satisfied in the BB estimation in column 4.
Turning to the country-level estimations, column 6 reveals a negative and highly significant
relationship between urban population growth and its initial level, conditional on time and
country fixed effects. The half-life is almost identical to the previous fixed effects estimate using
city-level data in column 2, but is now estimated with greater precision.
Aggregating at the country level, I am now able to control further for time varying determi-
nants of population growth. Column 7 adds Statehist and its squared level as control variables.
37
Table 6: Speed of Convergence using Urban Population Data from Reba et al. (2016)
Observational Unit: City Country
OLS FE GMM-AB GMM-BB OLS FE FE GMM-AB GMM-BB
(1) (2) (3) (4) (5) (6) (7) (8) (9)
log(Population) -0.003*** -0.007*** -0.007*** -0.007*** -0.004*** -0.007*** -0.008*** -0.009** -0.006**
(0.001) (0.001) (0.002) (0.001) (0.001) (0.001) (0.001) (0.004) (0.002)
Time FE Yes Yes Yes Yes Yes Yes Yes Yes Yes
City FE No Yes Yes Yes No No No No No
Country FE No No No No No Yes Yes Yes Yes
Statehist No No No No No No Yes Yes Yes
Observations 1706 1706 1239 1706 509 509 509 411 509
adj. R-sq 0.08 0.22 . . 0.16 0.24 0.26 . .
AR(2) 0.49 0.50
AR(3) 0.10 0.10
Hansen 0.07 0.13 0.51 0.46
Diff. Hansen 0.92 0.37
Instruments 30 32 24 28
Half-Life 258 95 97 104 192 94 84 80 109
Half-Life 95% C.I. [500,174] [155,68] [212,63] [171,75] [270,149] [133,73] [118,64] [481,44] [480,62]
Notes: This table presents OLS and GMM estimates of the speed of convergence using urban population data from Reba et al. (2016) at the city and country level. Columns 1-4 present results
using city-level data and columns 5-9 show results using urban population aggregated at the country level. For each sample, I first display the relationship controlling for time fixed effects, then
include city or country fixed effects and finally add Statehist and its squared level as controls (this was not possible for the city-level estimates). The GMM estimations in columns 3 and 4 use the
second and further lagged levels of urban population at the city level as instruments. Columns 8 and 9 use the third to fifth lagged levels of urban population aggregated at the country level as
instruments. Statehist and its squared level are treated as endogenous and instrumented with the same set of lags as urban population in columns 8 and 9. I use a collapsed matrix of instruments
and report the number of instruments. The AR(2) row reports the p-value of a test for the absence of second-order correlation in the residuals for the city-level GMM estimations and the
AR(3) row reports the p-value of a test for the absence of third-order correlation in the residuals for the country-level GMM estimations. Standard errors clustered at the city level in columns
1-4 and at the country level in columns 5-9 are in parentheses. * p<0.1, ** p<0.05, *** p<0.01.
38
The estimated speed of convergence is now faster with a half-life of 84 years, but remains con-
sistent with weak homeostasis. In particular, the 95% confidence interval indicates half-lives
between 118 and 64 years.
Columns 8 and 9 present AB and BB GMM estimates of the speed of convergence at the
country level. Both estimates confirm the weak homeostasis pattern found in the previous col-
umn, with half-lives estimated at 80 and 109 years, respectively. In both cases, the GMM
estimates appear to be much less precise than the fixed effect estimate in column 7, as the
confidence intervals for the half-lives are now larger, while remaining compatible with weak
homeostasis.
Overall, my results using historical urban population data clearly confirm the weak home-
ostasis pattern found in the previous sections. Most of the estimated half-lives in Table 6are
close to one century, and the smallest half-life found is 80 years.
Figure 9explores the time heterogeneity of the speed of convergence, both for the city-level
and country-level estimations. In both cases, a stable pattern of weak homeostasis over time is
confirmed. This is particularly striking for the city-level data, where all the point estimates
starting from the year 1250 onwards cannot reject a half-life of 115 years at the 5% level.
Figure 10 plots the kernel density of the estimated speed of convergence for a sample of 185
cities.27 It reveals a pattern similar to my previous findings using Lagerlöf’s (2019) simulated
data, with the mode of the distribution very close to a half-life of 115 years. Moreover, the
distribution is also more concentrated around that value than my previous estimates (Figure
6), giving additional support to the widespread of weak homeostasis across Malthusian societies.
Finally, as in Figure 6, a strong Malthusian trap cannot be rejected for some cities.
Figure 11 shows the heterogeneity of the speed of convergence at the country level, using
urban population data. Alongside the previous estimates using total population data (Figure 4)
and the above results at the city level (Figure 10), Figure 11 highlights significant differences
27Figure D-2 of the Appendix shows the point estimates along with their 95% confidence intervals for the 185
cities in the sample.
39
Figure 9: Speed of Convergence per period using Reba et al. (2016) Data
Half-life=
115 years
(Galloway,
1988)
Half-life=
30 years
(Madsen et al.,
2019)
.0060-.006-.012-.018-.024
1050
110 0
115 0
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
Half-life=
115 years
(Galloway,
1988)
Half-life=
30 years
(Madsen et al.,
2019)
.0060-.006-.012-.018-.024
1050
110 0
115 0
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
Notes: This figure plots estimates of the speed of convergence by period using urban population data from Reba
et al. (2016). It corresponds to the FE estimations in column 2, Table 6 (left panel) and in column 7, Table 6 (right
panel), adding year-interacted lagged urban population levels as controls. 95% confidence intervals reported.
Figure 10: Speed of Convergence per city using Reba et al. (2016) Data
Half-life=30 years
Madsen et al. (2019)
Half-life=115 years
Galloway (1988)
020 40 60 80
Density
-.1 -.05 0 .05
Speed of Convergence - β coefficients
kernel = epanechnikov, bandwidth = 0.0017
Notes: This figure plots the kernel density of the estimated speed of convergence by city using urban population
data from Reba et al. (2016). It corresponds to the FE estimation in column 2, Table 6, adding city-interacted
lagged urban population levels as controls.
40
in the strength of the Malthusian trap across countries. Some Malthusian economies, such
as Algeria, Germany, Peru and Thailand are found under a strong Malthusian trap regime,
with half-lives close to 30 years. The results also confirm that Spain has one of the strongest
Malthusian trap, with an half-life of 31 years, and a 95% confidence interval between 28 and 35
years. This result is even stronger than the previous estimate based on total population figures
of McEvedy et al. (1978).
On the other hand, countries such as Denmark, Israel and Portugal are found with the
weakest estimated Malthusian trap. For instance, the estimated half-life for Denmark is 322
years. Contrary to the previous findings