Malthus’ (1798) population hypothesis is inconsistent with the demographic transition and the concurrent massive expansion of incomes observed among industrialised countries. This study shows that eliminating the income-effect on the demand for children from the Malthusian model makes it harmonise well with industrial development.demographic transition; fertility; Malthus

Jacob L. Weisdorf

English

Research Papers in Economics

Discussion Papers 
Department of Economics 
University of Copenhagen 
 
 
 
No. 07-03 
 
 
Malthus Revisited: 
Fertility Decision Making based 
on Quasi-Linear Preferences 
 
 
Jacob L. Weisdorf 
 
 
 
 
 
 
 
Studiestræde 6, DK-1455 Copenhagen K., Denmark 
Tel. +45 35 32 30 82 - Fax +45 35 32 30 00 
http://www.econ.ku.dk 
 
ISSN: 1601-2461 (online) 
 
 Malthus Revisited:
Fertility Decision Making based on Quasi-Linear Preferences∗
Jacob L. Weisdorf
University of Copenhagen
January 22, 2007
Abstract
Malthus’ (1798) population hypothesis is inconsistent with the demographic transition and
the concurrent massive expansion of incomes observed among industrialised countries. This
study shows that eliminating the income-eﬀect on the demand for children from the Malthusian
model makes it harmonise well with industrial development.
JEL classiﬁcation codes: J13, N30, O10
Keywords: Demographic Transition, Fertility, Malthus
∗I gratefully acknowledge the feedback from seminar participants at University of Copenhagen and at the First
Summer School of the Marie Curie Research Training Network ‘Unifying the European Experience,’ especially Tommy
E. Murphy. Contact: jacob.weisdorf@econ.ku.dk.
11I n t r o d u c t i o n
Malthus, in his Essay on the Principle of Population,i d e n t i ﬁes England as a preventive check
society, meaning a society in which fertility respond to changes in economic conditions (Malthus,
1798). If income levels decrease and the price of provisions rises, then it becomes harder to rear a
family; this, Malthus argues, results in fewer marriages, leading ultimately to lower birth rates.
In England, a u-shaped relationship between income and birth rates has been observed (Figure
1). Ironically, the initial positive relationship between income and birth rates, for which Malthus is
a spokesman, breaks down shortly after the Principle is published. Subsequently, higher incomes
accompany lower birth rates, an episode today recognised as the demographic transition.
25
30
35
40
45
50
1675 1695 1715 1735 1755 1775 1795 1815 1835 1855
Figure 1
Crude Birth Rates and Real Wages, England 1675-1865 
Sources : Clark (2001); Wrigley and Schofield (1981)
60
65
70
75
80
85
90
95
100
Crude Birth Rate Real Wages
An incomplete list of studies that investigate the long-run relationship between income and
birth rates from a theoretical viewpoint includes Boucekkine et al. (2002), Galor and Weil (2000),
Lucas (2002), Strulik (2003) and Tamura (1996). The common story told is that human capital
accumulation is key: on the one hand, it leads to income increase; on the other, it induces parents
to trade oﬀ child-quantity for child-quality.
The current study diﬀers from the related literature from two perspectives: First, human capital
accumulation is not required to generate a demographic transition. Second, the present analysis is
kept entirely within the boundaries of a Malthusian economy. Using a simply analytical framework,
the current study shows that eliminating the income-eﬀect on the demand for children from the
Malthusian model makes it well in tune with industrial development.
More speciﬁcally, using so-called quasi-linear (‘zero-income eﬀect’) preferences, the (indirect)
eﬀect on birth rates of income changes (which enter through the price of provisions) turns out
to be sector-dependent: agricultural sector income growth leads to higher birth rates, industrial
sector income growth to lower birth rates. According to the theory presented below, therefore,
England’s u-shaped relationship between income and birth rates was caused by initial acceleration
in agricultural productivity (an ‘agricultural revolution’), succeeded by acceleration in industrial
productivity (an ‘industrial revolution’).
22T h e M o d e l
Consider a two-sector, two-goods, closed economy. An agricultural sector produces foods, an in-
dustrial sector produces manufactured goods. The labour force is divided endogenously between
the two sectors. Unless explicitly stated, all variables are considered in period t.
Output Food production is subject to constant returns to land and labour.1 Land, measured
by X,i sﬁxed and set to unity. Total food output is thus
YA = ΩALα
AX1−α ≡ ΩALα
A,α ∈ (0,1),X≡ 1. (1)
where ΩA measures total factor productivity in agriculture and LA is the number of farmers. The
net rate of agricultural productivity growth between any two periods is denoted γA. Without
property rights over land, the land rent is zero,2 and a farmer’s income is
wA =
YA
LA
=
ΩA
L1−α
A
. (2)
Turning to industrial production, this is subject to constant returns to labour. Total industrial
output is thus YM = ΩMLM,w h e r eΩM measures industrial labour productivity (M for manufac-
turing) and LM the number of manufacturers. The net rate of growth of industrial productivity
between any two periods is denoted γM, and a manufacturer’s income is
wM =
YM
LM
= ΩM. (3)
Labour Force Dynamics Consider an overlapping-generations economy where people live
for up to two periods: childhood and possibly adulthood. Income-generating activities take place
only during adulthood. Development in the size of the adult population, i.e., the labour force, is
aﬀected by birth rates and death rates. That is, at the end of each period, the adult generation
dies out and is replaced by a new generation consisting of children surviving childhood. The
number of surviving children per adult equals the adult’s (endogenous) birth rate b multiplied by
the survival probability. With d ∈ (0,1) denoting the (exogenous) risk of dying before adulthood,
1 − d measures survival probability. With identical agents, the number of surviving children per
adult is n =( 1− d)b. Evolution in the size of the labour force from one period to the next is
therefore Lt+1 = ntLt =( 1− d)btLt, and the net rate of growth of the labour force between any
two periods is thus
γL ≡ (1 − d)bt − 1. (4)
Labour Market Equilibrium The total labour force L consists of farmers and manufactur-
ers, i.e., L = LA +LM.L e tp ≡ pA/pM denote the relative price of foods and manufactured goods.
Labour market equilibrium then requires that p adjusts, so that
pwA = wM. (5)
1Throughout, we suppress the use of capital in production, an assumption commonly used in the related literature
(see, e.g., Galor and Weil, 2000).
2In the related literature, this is also not an unusual assumption (e.g., Galor and Weil, 2000).
3Food Market Equilibrium Suppose that, over the course of a lifetime, an individual con-
sumes a ﬁx e dq u a n t i t yo ff o o d s( o rc a l o r i e s )m e a s u r e db yη. For simplicity, food is demanded only
during childhood and some of it stored for adulthood.3 Setting η ≡ 1, total food demand equals
total food supply, given by (1), when bL = ΩALα
A.
Preferences People, in a Darwinian sense, derive utility from the number of surviving oﬀ-
spring n and from the number of manufactured goods consumed m. The utility function of a
representative adult is of a quasi-linear type whereby u(n,m)=l nn + m. This implies that the
price-eﬀect on the demand for children is negative whereas the income-eﬀect is zero (see further
below).
The Budget Constraint Historically, the costs of bringing up children are largely equal to
the costs of the foods that they consume.4 The total costs of bringing up b children, measured
in terms of manufactured goods, are thus pb. The budget constraint of a representative adult is
therefore w = pb + m,w h e r e ,f o l l o w i n g( 5 ) ,w ≡ pwA = wM.
Utility Maximisation The solution to the utility maximisation problem: maxb u(n,m) sub-
ject to w = pb + m, is that an adult’s demand for children is b∗ =1 /p and the demand for
manufactured goods m∗ = w − 1.
General Equilibrium Conditions Below, we ﬁrst explore the static general equilibrium
conditions. Then, we study the general equilibrium conditions along a balanced growth path.
Combining the food and labour market equilibrium conditions with the solution to the utility
maximisation problem and the wage rates, the static general equilibrium birth rate becomes
bSGE =
ΩA
Ωα
ML1−α (6)
while the manufactured goods consumption level becomes
mSGE = ΩM − 1. (7)
Equation (6) says that birth rates (i) increase with agricultural productivity growth, but de-
crease (ii) with growth in the size of the labour force and (iii) with industrial productivity growth.
Unlike the third eﬀect, the ﬁrst and the second eﬀects are well-know elements in the traditional
Malthusian framework: Starting from a constant population, an upward shift in agricultural pro-
ductivity raises agricultural sector income. Through its negative eﬀect on the price of provisions,
this leads to higher birth rates and therefore to population growth. Due to diminishing returns to
labour in agriculture, population growth causes agricultural income reduction. This reduces birth
rates until population growth eventually peters out.
The third eﬀect, however, which appears as a result of the quasi-linear preference function,
provides a new facet to the Malthusian framework: Starting from a constant population, an upward
shift in industrial productivity raises industrial sector income. Though its positive eﬀect on the
3It will not aﬀect the qualitative nature of the results, if, instead, individual food demand is divided over two
periods. Such a construction, however, severely complicates matters.
4A construction whereby children, in addition to foods, require a certain amount of parents’ time, or a given
number of manufactured goods, will not change the quantitative nature of the results but severely complicates the
analysis.
4price of provisions, this causes birth rates to decline and therefore leads to negative population
growth. Via diminishing returns in agriculture, a smaller population fosters higher agricultural
income, which increases birth rates until the population growth rate is back to zero. As will
become apparent below, the third eﬀect plays a key role for the demographic transition.
Turning to the second static general equilibrium condition, equation (7), this shows that the
number of manufactured goods produced and consumed depends solely on the level of industrial
sector productivity. Unlike the traditional Malthusian model, consumption levels per capita in this
modiﬁed population model are subject to sustained growth, restricted only by industrial produc-
tivity growth.
Dynamics and Stability To see how the static equilibriums evolve over time, we now char-
acterize their steady-state balanced growth paths. Then, we explore the dynamics and stability
properties of the two paths.
A balanced growth path is a path along which all variables grow at constant geometric rates
(possibly zero). The left-hand-side of (6) is constant by deﬁnition, so the right-hand-side must
also be constant. Using the growth rates of productivity and labour deﬁned above, the equilibrium
birth rate remains ﬁxed over time when γA = αγM +( 1− α)γL. Inserting (4), the steady state
birth rate therefore becomes
bst.st. =
1
1 − d
µ
1+
γA − αγM
1 − α
¶
.
Abstracting from variation in the (exogenous) death risk, changes in the steady state birth rate
will be caused by a shift in the ratio of agricultural to industrial productivity growth. Deﬁning
γ ≡ γA/γM,a ni n c r e a s ei nγ increases the birth rate, and vice versa. Note that the eﬀect of
income variations on γ is sector-dependent: agricultural sector income growth leads to an increase
in γ, whereas industrial sector income growth leads to a reduction in γ. Accordingly, controlling
for changes in the death rate, England’s u-shaped relationship between income and birth rates
(Figure 1) would have been caused by initial acceleration in agricultural productivity growth (γ
up), succeeded by acceleration in industrial productivity growth (γ down). In other words, as has
also been observed in England (e.g., Overton, 1996), an ‘agricultural revolution’ is followed by an
‘industrial revolution.’
Finally, the steady state rate of growth of manufactured goods consumption is γm = γM,w h e r e
γM is the net rate of growth of industrial productivity and γm denotes the net rate of growth
of manufactured goods consumption (or economic growth) between any two periods. Evidently,
sustained economic growth is perfectly consistent with the current version of the Malthusian model.
3 Testable Implications
Testing the current theory means testing if ratio of agricultural to industrial productivity growth,
i.e., γ, moves in the expected directions, keeping in mind that the theory considers a closed economy,
and that, during industrialisation, countries typically become open up to foreign trade.
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Agricultural Revolution in England: The Transformation of the Agrarian Economy 1500-1850,

From decay to growth: A demographic transition to economic growth,”

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Population, Technology and Growth: From Malthusian Stagnation to tpwhe Demographic Transition and

The Industrial Revolution: Past and Future, Cambridge:

The Population History of England 1541-1871,L o n -don,

Vintage Human Capital, Demographic Trends, and Endogenous Growth,”

Malthus Revisited: Fertility Decision Making based on Quasi-Linear Preferences

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Malthus Revisited: Fertility Decision Making based on Quasi-Linear Preferences

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