An issue often discussed in relation to agricultural development is the effect on agricultural labour productivity of more intensive land-use. Introducing aspects of seasonality into a stylized Malthusian model, we unify two diverging views by showing that labour productivity may go up or down with agricultural intensification, depending on whether technological progress emerges in relation to cultivation or harvesting activities. Our result rests on evidence reported by Boserup (1965) and others, which suggests that harvest seasons in traditional agriculture are characterized by severe labour shortage.agricultural intensification ; Boserup ; labour surplus ; Malthus ; seasonality

Jacob L. Weisdorf

Paul Sharp

English

Research Papers in Economics

   
 
 
 
 
  WORKING PAPER N° 2008 - 03 
 
 
 
A Malthusian model for all seasons 
 
 
 
 
 
 
Paul Sharp 
 
Jacob Weisdorf 
 
 
  JEL Codes: J22, N13, O10 
  Keywords: agricultural intensification, Boserup, labour 
surplus, Malthus, seasonality 
   
 
 
PARIS-JOURDAN SCIENCES ECONOMIQUES 
LABORATOIRE D’ECONOMIE APPLIQUÉE - INRA 
 
48, BD JOURDAN – E.N.S. – 75014 PARIS 
TÉL. : 33(0) 1 43 13 63 00   –   FAX : 33 (0) 1 43 13 63 10 
www.pse.ens.fr 
 
CENTRE NATIONAL DE LA  RECHERCHE SCIENTIFIQUE – ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES 
ÉCOLE NATIONALE DES PONTS ET CHAUSSÉES – ÉCOLE NORMALE SUPÉRIEURE 
 
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1A Malthusian Model for all Seasons∗
Paul Sharp†and Jacob Weisdorf‡
February 5, 2008
Abstract
An issue often discussed in relation to agricultural development is the eﬀect on
agricultural labour productivity of more intensive land-use. Introducing aspects of
seasonality into a stylized Malthusian model, we unify two diverging views by showing
that labour productivity may go up or down with agricultural intensiﬁcation, depend-
ing on whether technological progress emerges in relation to cultivation or harvesting
activities. Our result rests on evidence reported by Boserup (1965) and others, which
suggests that harvest seasons in traditional agriculture are characterized by severe
labour shortage.
JEL classiﬁcation codes: J22, N13, O10
Keywords: Agricultural Intensiﬁcation, Boserup, Labour Surplus, Malthus, Sea-
sonality
∗We gratefully acknowledge the feedback from seminar and conference participants at the University of
Copenhagen, the European University Institute, the Paris School of Economics, the Seventh Conference of
the European Economic History Society, the Globaleuronet Summer School ‘Why (not) Europe’, and the
2nd Sound Economic History Workshop.
†University of Copenhagen, Department of Economics, Studiestraede 6, DK-1455 Copenhagen, Denmark;
phone: (+45) 35 32 30 37; fax: (+45) 35 32 30 00; e-mail: paul.sharp@econ.ku.dk.
‡Paris School of Economics and University of Copenhagen, Department of Economics, Studiestraede
6, DK-1455 Copenhagen, Denmark; phone: (+45) 35 32 30 33; fax: (+45) 35 32 30 00; e-mail: ja-
cob.weisdorf@econ.ku.dk. Corresponding author.
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11 Introduction
The so-called ‘Malthusian’ model is frequently used by development economists, growth
economists and economic historians to capture development in a traditional agricultural so-
ciety. An issue often discussed in relation to agricultural development–but about which the
Malthusian model is silent–is the eﬀect on agricultural labour productivity of more inten-
sive land-use. According to Hunt (2000), scholars divide into two camps. On the one hand,
Boserup (1965) makes a strong case that agricultural intensiﬁcation raises labour costs per
unit of food produced. In her view, therefore, agricultural labour productivity is negatively
correlated with land productivity, a notion referred to by Hunt as the ‘decline thesis’. Other
scholars, however, have pointed to the nearly fourfold increase of England’s population level
between 1500 and 1800 and compared it to the halving of the share of England’s population
employed in agriculture over the same period as convincing evidence that agricultural labour
productivity is positively correlated with land productivity. Following Hunt, this position
h a sb e c o m ek n o w na st h e‘ r i s et h e s i s ’ .
In this paper, we demonstrate that the ‘rise thesis’ and the ‘decline thesis’ can both be
understood within the context of a single framework through an acceptance of the importance
of seasonality in agriculture. This we demonstrate by adding two salient features about
traditional agriculture to a stylized Malthusian model. The ﬁrst feature involves making a
distinction between two types of seasonal activities in agriculture: cultivation and harvesting.
T h es e c o n df e a t u r eb u i l d so ne v i d e n c er e p o r t e db y Boserup (1965) and others, which suggests
that land productivity in traditional agriculture is limited, not by how much food can be
grown, but by how much labour is available in the harvest season. In our seasonality-
augmented version of the Malthusian model, the sign of the correlation between land and
labour productivity depends simply on the relative rate of growth of technological progress
in cultivation and harvesting activities.
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12T h e m o d e l
Following Clark (2007), a deﬁning characteristic of a traditional agricultural society is that
technological progress over the long run permits a larger population but does not aﬀect
income per capita which remains at the level of subsistence. The stylized Malthusian model
captures this through the use of two components: (i) a positive relationship between income
and population growth, and (ii) diminishing returns to labour in production. Our modiﬁed
version of the Malthusian model builds on both components. The former is described in
detail in the following; the latter is described further below.
Consider a one-sector, one-good agricultural economy. Economic activities extend over
inﬁnite (discrete) time. Unless explicitly noted, all variables are considered in period t.I n
period t, the population consists of Nt identical individuals. Consistent with the stylized
Malthusian model, the reproductive success of an individual is ‘checked’ by the size of its
income. Symbolically, the number of surviving children of an individual is given by the func-
tional relationship n = n(w),w h e r ew is income per individual and where n(·) is continuous
and monotonic, with n(0) = 0 and n(∞) > 1. This implies that a ‘subsistence’ income
level exists, deﬁned as the income level at which an individual is able to raise exactly one
surviving oﬀspring. Since all individuals are identical, change in the size of the labour force
between any two periods is given by Nt+1 = ntNt.
To introduce aspects of seasonality, suppose that goods are generated according to a
two-step procedure. This involves ﬁrst the production of intermediate goods (think: crops
cultivated for the purpose of harvesting), then the production of ﬁnal goods (think: harvested
crops). Thus, there are two types of seasonal activities in the model: cultivation (seeding,
weeding, harrowing, hoeing etc) and harvesting.
Suppose that the production of output is subject to constant returns to land and labour.
Each individual is endowed with one unit of labour, which, as will be explained below, is
divided endogenously (but, for the individual, parametrically) between output-generating
activities and unemployment. Consistent with the stylized Malthusian model, land, mea-
sured by X,i si nﬁxed supply, and is normalized to one (X ≡ 1). For simplicity, there is no
property rights over land, and each period the land mass is divided equally between the indi-
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1viduals. In symbolic terms, the land mass per individual is thus given by x = X/N ≡ 1/N.
Let subscript C and H denote Cultivation (intermediate goods) and Harvesting (ﬁnal
goods) activities, respectively. Output per individual in season i ∈ {C,H} can thus be
written as
yi = Ai (γiei)
α (x)
1−α ≡ Ai (γiei)
α
µ
1
N
¶1−α
,α ∈ (0,1),i∈ {C,H}, (1)
where γi ∈ (0,1) is the fraction of the year for season i ∈ {C,H} (implicitly, therefore,
γC + γH ≤ 1); ei ∈ (0,1) is the labour input of an individual in season i ∈ {C,H};a n dAi
is the level of technology (total factor productivity) in season i ∈ {C,H}. It follows that
income per individual is equal to the individual’s ﬁnal output, i.e. w = yH.
Time-budget studies done in traditional agricultural societies suggest that harvest seasons
are normally characterized by severe labour shortage (hence the term peak season), and that
the harvest season labour supply, therefore, sets the upper limit to agricultural output (see
e.g. Boserup 1965; Jones 1964). Following this approach, individuals in the model employ
their entire labour resources in the harvesting season (i.e. eH =1 ). Assuming absence of
uncertainty, the number of intermediate goods produced equals the number of ﬁnal goods
that can be harvested, i.e. yC = yH. It then follows that the labour input per individual
needed in the cultivation season (for intermediate goods production) is given by
eC =
γH
γC
µ
AH
AC
¶ 1
α
.
Notice the season-speciﬁce ﬀects of technological progress on the demand for labour in
cultivation activities. Technological progress that occurs in relation to harvesting activities
is labour-demanding (when more ﬁnal output can be generated in the harvest season, it
takes more labour input in the cultivation season to produce more intermediate goods).
Technological progress that occurs in relation to cultivation activities, by contrast, is labour-
saving (it needs less labour input to generate the number of intermediate goods needed in
the harvest season).
With reference to seminal work by Lewis (1954), the labour resources not used for output-
generating activities, i.e. γc (1 − ec), can be thought of as surplus labour or disguised unem-
ployment.
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13 Stability and Comparative Statics
We want to analyze the correlation between land and labour productivity in a steady state.
In a steady state, all variables grow at a constant rate (possibly zero). Speciﬁcally, this
means that the steady state population growth rate is constant. A constant rate of growth
of population implies that income per individual must be constant over time. For tractability
of the analysis, we parameterize the Malthusian relationship between income and population
growth, so that n(w)=w. Then, using (1), it follows that the gross rate of growth in income
per individual is given by
wt+1
wt
=
AH (γH)
α
³
1
ntNt
´1−α
AH (γH)
α
³
1
Nt
´1−α =
µ
1
wt
¶1−α
.
Hence, in steady state (marked with an asterisk), income per individual is w∗ =1 .N o t i c e
that stability of the steady state requires that ∂wt+1/∂wt |wt=w∗ < |1|, a condition that is
fulﬁlled when α ∈ (0,1), which is equivalent to having diminishing returns to labour in
production.
By use of the parameterization above, having w∗ =1means that each individual is able
to raise exactly one surviving oﬀspring, i.e. n∗ =1 . In steady state, therefore, there is no
population growth, so the population level is constant (Nt+1 = Nt). Thus, by construction,
w∗ =1is the subsistence income. Such a steady state is normally referred to as a ‘Malthusian’
equilibrium.
In the following, we calculate land and labour productivity in steady state. Using (1), it
follows that land productivity–deﬁned as ﬁnal output per unit of land–in steady state is
D
∗ ≡
yH
x
=( ( γH)
α AH)
1
1−α . (2)
Labour productivity–deﬁned as ﬁnal output per unit of land divided by labour input per
unit of land–in steady state is
E
∗ ≡
yH/x
(1/x) ·
P
i={C,H} γiei
=
1
γH(1 + (
AH
AL)1/α)
(3)
Notice once again the season-speciﬁce ﬀects of technological progress. Starting in a steady
state, a positive shock to harvesting technology means that individuals can produce more
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1ﬁnal output, and, therefore, need to cultivate more intermediate goods. In the absence of
technological progress in cultivation activities, individuals thus have to put in more labour
in the cultivation season. Altogether, therefore, the annual labour input of an individual
goes up in response to technological progress in harvesting activities.
At the same time, though, because each individual also generates more ﬁnal output,
at least to begin with, income per individual increases. This leads to growth in the size
of the population. However, more people means less land per individual, which ultimately
drives down income per individual until population growth is brought to a halt. Eventually,
therefore, the economy returns to a steady state with a higher population level, but where
income per individual is back at the level of subsistence, a result identical to that of the
stylized Malthusian model (see Clark 2007).
However, the introduction of seasonal aspects allows us to take the analysis one step
further. Since output per individual over the long run is unaﬀected whereas individual labour
input has gone up, a positive shock to harvesting technology in the current model increases
land productivity but reduces labour productivity. According to the present framework,
therefore, if technological progress occurs only in relation to harvesting activities, then the
‘decline thesis’ holds true.
Meanwhile, we still need to analyze the eﬀects on land and labour productivity of techno-
logical progress in relation to cultivation activities. It follows from (2) that a positive shock
to cultivation technology has no eﬀect on land productivity. However, technological progress
in cultivation means that the amount of intermediate goods needed for the harvesting season
can now be generated using less labour. Obtaining the same output using less labour input,
labour productivity, therefore, has gone up as a result of technological progress in cultivation.
The latter result helps shed light on how land and labour productivity can be positively
correlated. To see this, let gi = 0 denote the net rate of growth of technology in season
i ∈ {C,H}. It then follows from (2) and (3) that, if gH >g C ≥ 0, then the ‘decline thesis’
applies. By contrast, if gC >g H > 0, then the ‘rise thesis’ applies. In words: if the labour-
demanding eﬀect of technological progress in harvesting is dominated by the labour-saving
eﬀect of technological progress in cultivation, then land and labour productivity are positively
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1correlated, and the ‘rise thesis’ is valid. If, by contrast, the labour-saving eﬀect is dominated
by the labour-demanding eﬀect, then land and labour productivity are negatively correlated,
and the ‘decline thesis’ is valid.
4C o n c l u s i o n
This paper introduces aspects of seasonality in agriculture to a stylized Malthusian model.
The analysis shows that labour productivity in agriculture may go up or down with agri-
cultural intensiﬁcation, depending on the relative rate of growth of technological progress
in harvesting and cultivation activities, respectively. The theory should be testable through
an analysis of changes in relative seasonal wage rates following season-speciﬁc technological
progress in traditional agriculture.
References
[1] Boserup, E. (1965), The Conditions of Agricultural Growth, Earthscan, London.
[2] Clark, G. (2007), A Farewell to Alms: A Brief Economic History of the World,P r i n c e t o n
University Press, Princeton.
[3] Hunt, R.C. (2000), ‘Labor Productivity and Agricultural Development: Boserup Revis-
ited’, Human Ecology 28, 251-77.
[4] Jones, E.L. (1964), Seasons and Prices: The Role of the Weather in English Agricultural
History, London: George Allen & Unwin.
[5] Lewis, W.A. (1954), ‘Economic development with unlimited supplies of labour’,
Manchester School of Economic and Social Studies 22, 139-191.
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A Malthusian model for all seasons?

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