Download Malthus Revisited: Fertility Decision Making based on Quasi

Malthus Revisited: Fertility Decision Making
based on Quasi-Linear Preferences∗
Jacob L. Weisdorf
University of Copenhagen
FIRST DRAFT
September 7, 2006
Abstract
The traditional interpretation of Malthus’ population theory makes it
incompatible with the demographic transition and the massive expansion
in income per capita which most industrialised countries have experienced.
This study proposes a new interpretation of Malthus’ ‘preventive check’
hypothesis which makes his theory perfectly consistent with the development path of an industrialised economy. Putting together a simple analytical framework based on quasi-linear (‘zero income-effect’) preferences,
this modified version of the Malthusian model predicts that a u-shaped
relationship between income and fertility observed in historical England
would have been caused by initial acceleration in agricultural productivity growth (i.e., an ‘agricultural revolution’), succeeded by acceleration in
industrial productivity growth (i.e., an ‘industrial revolution’).
1
Introduction
Malthus, in his Essay on the Principle of Population, identifies England as a
‘preventive check’ society, meaning a society in which fertility rates respond
to changes in economic conditions (Malthus, 1798). That is, if income levels
decrease and the price of provisions rises, then it becomes harder to rear a
family; this, Malthus reasons, results in fewer people getting married and fewer
early marriages taking place, leading ultimately to lower birth rates.
In England, a u-shaped relationship between income and birth rates has
been observed, whereby income growth leads initially to higher birth rates, and
later to lower birth rates (see Figure 1). Ironically, the positive relationship
between income and birth rates, i.e., the first half of the u-shaped relationship,
for which Malthus is a spokesman, breaks down shortly after the Principle is
∗ I am greatful for the comments and suggestions made by the participants at the First
Summer School of the Marie Curie Research Training Network ‘Unifying the European Experience: Historical Lessons of Pan-European Development,’ especially Tommy E. Murphy.
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made public: the preventive check mechanism, seemingly, ceases to operate.
Subsequently, increasing income levels are accompanied by a reduction in birth
rates, an episode today recognised to as the demographic transition.
[Figure 1 about here]
Using a simple analytical framework, this study provides a theory of the
u-shaped relationship between income and fertility, which, aside from a technicality, is perfectly consistent with Malthus’ preventive check hypothesis.
More specifically, the current theory relies on a special type of preference
function, namely, so-called quasi-linear (or zero income-effect) preferences. Such
preferences imply, in line with Malthus’ theory, that peoples’ demand for children responds inversely to changes the price of provisions; however, unlike what
Malthus imagines, the income-effect on the demand for children is absent.
The use of a quasi-linear preference function brings out an interesting result.
Namely, that the effect on fertility of income variation–which according to the
current theory affects people’s birth rates through the price of provisions–turns
out to be sector-dependent: agricultural sector income growth leads to higher
birth rates, whereas industrial sector income growth leads to lower birth rates.
According to the current work, therefore, the u-shaped relationship between
income and birth rates outlined above is caused by initial acceleration in agricultural productivity growth (i.e., an ‘agricultural revolution’), succeeded by
acceleration in industrial productivity growth (i.e., an ‘industrial revolution’).
1.1
Related Literature
It is normally accepted that one of the most striking shortcomings of Malthus’
theory about the principle of population is its lack of power to forecast not
only the massive expansion in income per capita but also an event such as the
demographic transition, both of which nearly all industrialised country have
experienced.
Not surprisingly, therefore, nearly all studies that try to rationalise the ushaped relationship between income and birth rates attribute the effects of
industrial development, especially that of technical progress, to the impetus
behind the demographic transition.
An incomplete list of papers that investigates the long-run relationship between income and fertility from a theoretical viewpoint includes Becker et al.
(1990), Doepke (2004), Galor (2005), Galor and Weil (2000), Hansen and Prescott
(2002), Jones (2001), Kalemli-Ozcan (2002), Lagerlöf (2003), Lucas (2002),
O’Rourke et al. (2005), Strulik (2004), Tamura (2006) and Weisdorf (2004).
The argument most commonly forwarded to motivate the demographic transition is that parents face a trade-off between child quantity and child quality.
That is, given a fixed amount of resources, parents can choose between raising
many children of a low quality (little education or human capital) and having
only a few children of a high quality (much education or human capital).
2
Probably the most prominent example is Galor and Weil (2000). According to their theory, people divided up their time between income-generating
activities, on the one hand, and child rearing, on the other.
Each person requires a minimum consumption level in order to subsist. At
relatively low income levels, therefore, people have to devote a relatively large
fraction of their time to income-generating activities, to meet their subsistence
requirements.
As a result, productivity (i.e., technology) growth enables people to reduce
time spent on income-generating activities and increase time allocated to the
upbringing of children. At relatively low income levels, therefore, productivity
growth increases a person’s income potential, and therefore its birth rate, which
is how the authors explain the first half of the u-shaped relationship described
above.
Higher birth rates leads to a larger population. Population size constitutes a
positive scale-effect on the growth rate of new technology. This creates a virtues
circle whereby technology growth leads to higher birth rates, which leads to
population increase, which accelerates technology growth.
Eventually, however, technology growth becomes so significant that keeping
up with technological progress demands schooling. This, according to the authors, is why parents ultimately trade off child quantity for child quality, and
is what triggers the demographic transition, i.e., accounts for the second half of
the u-shaped relationship outlined above.
The current study differs from the existing literature from two perspectives.
First, a new interpretation of Malthus’ preventive check hypothesis is proposed,
which is perfectly consistent with the u-shaped relationship between income and
fertility described above. That is, we modify the Malthusian model by removing
the income-effect on the demand for children while holding on to the price-effect.
Second, a trade-off between child quantity and child quality is not the engine
behind the demographic transition in this paper. Instead, the fertility decline
in this model appears as a result of changes in the price of provision. More
specifically, the drop in birth rates results from the fact that industrial (i.e., nonfood) productivity growth drives up the price of foods relative to manufactured
goods. The increase in the price of foods, which makes it more expensive to rear
a family, shifts demand away from children, causing ultimately fertility decline.
In the following, a simple analytic framework is provided, which explains in
detail the paper’s theory.
2
The Model
Consider a two-sector, closed economy. An agricultural sector produces foods;
an industrial sector produces manufactured goods. As will become apparent
below, the economy’s labour force is divided endogenously between the two
sectors, so that some people are employed as farmers, others as manufacturers.
3
2.1
The Agricultural Sector
Suppose that food production is subject to constant returns to land and labour.1
We assume that the mass of land, measured by X, is in fixed supply, and that
the amount is set to unity. The total food output can thus be written as
1−α
YA = ΩA Lα
≡ ΩA Lα
AX
A , α ∈ (0, 1) , X ≡ 1.
(1)
The variable ΩA measures the agricultural sector’s total factor productivity
(subscript A for agriculture). The variable LA is agricultural labour input, i.e.,
the number of farmers in the economy. Unless explicitly stated, all variables in
the model are considered in time-period t.
For simplicity, there are no property rights over land, which means that the
land rent is zero.2 A farmers, therefore, gets the average agricultural output,
meaning that a farmer’s income is
wA =
YA
ΩA
= 1−α .
LA
LA
(WA)
To be used in the following, we define the net rate of growth of agricultural
productivity between any two periods as
γA ≡
2.2
ΩA,t+1 − ΩA,t
.
ΩA,t
(2)
The Industrial Sector
Suppose that manufactured goods production is subject to constant returns to
labour. Total industrial output can thus be written as
YM = ΩM LM ,
where the variable ΩM measures industrial labour productivity (subscript M for
manufacturing). The variable LM is industrial labour input, i.e., the number of
manufacturers in the economy.
It follows that a manufacturer’s income is
wM =
YM
= ΩM .
LM
(WM)
To be used below, we define the net rate of growth of industrial productivity
between any two periods as
γM ≡
ΩM,t+1 − ΩM,t
.
ΩM,t
(3)
1 Throughout, we suppress the use of capital in production, an assumption commonly used
in the related literature (see, e.g., Galor and Weil, 2000).
2 In the related literature, this is also not an unusual assumption (e.g., Galor and Weil,
2000).
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2.3
The Labour Market Equilibrium Condition
The economy’s total labour force, measured by the variable L, consists of farmers
and manufacturers. That is,
L = LA + LM .
In equilibrium, the relative price of the two types of goods, foods and manufactured goods, adjusts, so that workers receive the same income regardless of
their choice of occupation.
Let p denote the relative price of the two goods, or, more specifically, the
number of units of manufactured goods that it takes to obtain a unit of food.
In other words, p measures agricultural terms of trade.
The labour market equilibrium condition (LEC) then requires that agricultural terms of trade adjust, so that
p=
2.4
wM
.
wA
(LEC)
Population and Labour Force Dynamics
Consider an overlapping-generations economy in which people live for a maximum of two periods: childhood and possibly adulthood. During childhood,
there are no productive or income-generating activities. Such activities take
place only during adulthood. The income that adults generate through productive activities, as will become apparent below, is divided endogenously between
children and consumption goods.
The development in the size of the adult population, which, by construction,
equals the size of the labour force, is affected by birth rates and death rates.
That is, at the end of each period, the total adult generation leaves the economy
(i.e., die out). It is then replaced by a new generation of adults, consisting of
the children who survive childhood.
The number of children surviving childhood, in turn, is equal to the former
period’s birth rate, denoted b, multiplied by the probability of outliving childhood. If we let d ∈ (0, 1) denote the risk of dying before adulthood, then 1 − d
reflects the chances making it until adulthood.
It is assumed that the death risk d cannot be affected by any actions taken by
individuals. Throughout, therefore, the death risk is considered as exogenous.
Individuals are, however, capable of affecting their birth rate, the size of which
will be determined endogenously further below.
With all individuals being identical, the number of children of a representative individual surviving until adulthood can thus be written as
n = (1 − d) b.
The evolution in the size of the labour force from one period to the next is
therefore
Lt+1 = nt Lt = (1 − d) bt Lt
(4)
5
It follows that the net rate of growth of the labour force between any two periods
can be written as
Lt+1 − Lt
γL ≡
= (1 − d) bt − 1.
(5)
Lt
2.5
The Food Market Equilibrium Condition
Suppose that an individual, over the course of a lifetime, consumes a fixed
quantity of foods–i.e., a certain amount of calories–measured by η. To simplify
matters, a person’s lifetime food consumption is demanded during childhood and
some of it then stored for adulthood.3
Without loss of generality, the food demanded by an individual can be set
to unity, i.e., η ≡ 1. Therefore, the total amount of foodstuff demanded in the
economy in any given period equals the birth rate multiplied by the size of the
labour force. That is, the total food demand is bL.
By setting total food demand equal to total food supply–the latter given
by equation (1)–it follows that the food market equilibrium condition (FEC)
is fulfilled when
bL = ΩA Lα
(FEC)
A.
2.6
Preferences
In line with the Beckerian approach (see Becker, 1991), we assume that children
provide utility alongside other goods. Furthermore, as was explained in the
introduction section above, we want to capture the idea proposed by Malthus
(1798) that people’s demand for children respond to changes in the price of
provisions.
Suppose, in accordance with these views, that the utility function of a representative individual is of a quasi-linear type,4 whereby
u (n, m) = ln n + m.
It follows that people derive utility from the number of their children making
it to adulthood, measured by n, and from the amount of manufactured goods
consumed, measured by m.
2.7
Utility Maximisation
Suppose that the costs of bringing up children consist of the costs of provisions,
i.e., the costs of the foods that the children consume.5 Since the total amount of
3 If the individual does not survive until adulthood, then it is assumed that foods left over
are either wasted or consumed by the child’s siblings. It would not affect the qualitative
nature of the results, if, instead, the individual’s food demand where to be divided over two
periods. However, such a construction would severely complicate matters.
4 The quasi-linear preference type is also known as zero income-effect preferences.
5 A construction by which children, in addition to foods, require a certain amount of parents’
time or a certain number of manufactured goods will not change the quantitative nature of
the results. Such a construction, however, severely complicate the analysis.
6
foods demanded by an individual is set to unity (i.e., η ≡ 1), and since p is the
price of a unit of food, it follows that the total costs of bringing up b children,
measured in terms of manufactured goods, equals pb.
Normalising the price of manufactured goods to one, it follows that the
budget constraint of a representative individual can be written as
w = pb + m,
where, according to the labour market equilibrium condition equation (LEC),
w ≡ pwA = wM .
A representative individual’s utility maximisation problem is thus
max u (n, m) = ln n + m
b,m
st. w
= pb + m,
The solution to the maximisation problem (SMP) implies that, in optimum
(signified with an asterisk), the number of children demanded by an individual,
i.e., its birth rate, is
1
b∗ =
(SMP)
p
Note that the income-elasticity on the demand for children is zero while the
price-elasticity is negative; that is, children are an ordinary good.
It also follows from the solution to the maximisation problem that the number of manufactured goods demanded in optimum is
m∗ = w − 1.
2.8
The Static General Equilibrium Condition
The aim in the following is to sort out the factors that determine the general
equilibrium birth rate as well as the number of manufactured goods consumed
by each individual.
Combining the solution to the optimisation problem (SMP), the labour
market equilibrium condition (LEC) and the food market equilibrium condition (FEC) with the income of farmers (WA) and the income of manufacturers
(WM), it follows that, in a static general equilibrium (SGE), the birth rate is
bGE =
ΩA
α
ΩM L1−α
(b-SGE)
while manufactured goods consumption level is
mGE = ΩM − 1.
(m-SGE)
The static general equilibrium birth rate, i.e., equation (b-SGE), reveals
the response of fertility to changes in economic conditions. It tells us that
birth rates increase with agricultural productivity growth, but decrease with
industrial productivity growth and with growth in the size of the labour force.
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On the one hand, therefore, the result in equation (b-SGE) is well in line
with the traditional interpretation of Malthus’ theory. That is, starting from a
constant population, an upward shift in agricultural productivity raises agricultural sector income. Through its negative effect on the price of provisions, this
leads to fertility increase, and thus to population growth. Due to diminishing
returns to labour in agriculture, however, population growth causes agricultural
income reduction. This brings down fertility, until population growth eventually
peters out.
On the other hand, the results of equation (b-SGE) provide an insight that
contrasts with the traditional interpretation of Malthus’ theory. That is, starting
from a constant population, an upward shift in industrial productivity raises
industrial sector income. Though its positive effect on the price of provisions,
this creates a decline in birth rates, thus causing a negative rate of growth of
population. Via diminishing returns in agriculture, a smaller population fosters
a higher agricultural income, which increases birth rates, until the population
growth rate is back to zero.
The key to understanding this result is that the effect on fertility of productivity change enters through the price of provisions. When, for example,
industrial sector productivity grows, manufacturers receive a higher income.
By the labour market equilibrium condition, an adjustment in the price of provisions, i.e., in agricultural terms of trade, takes place, so that people obtain
the same income regardless of their choice of occupation. In the case of growing
industrial productivity, for example, food goods need to become more expensive
relative to manufactured goods in order for farmers to obtain the same income
as manufacturer. As a result of higher food prices, and because of a negative
price-elasticity on the demand for children, birth rates ultimately drop.6
Turning instead to the general equilibrium manufactured goods consumption
level, i.e., equation (m-SGE), this shows that the number of manufactured goods
produced and consumed depends solely on the level of industrial productivity
ΩM . That is, because industrial output is not adversely affected by the population level, the economy’s standard of living is subject to limitless increase,
restricted only by industrial productivity growth.
2.9
Dynamics and Stability
Equation (b-SGE) gives us the static equilibrium birth rate in the economy in
period t. To see how the static equilibrium birth rate evolves over time, we now
characterize the equilibrium along a steady-state balanced growth path. Then,
we explore the dynamics and the stability properties of this path.
6 If instead we had allowed for a postive income-effect on the demand, for example, if
preferences had been of a Cobb-Douglas type instead which normally employed the related
literature, i.e., if
u (n, m) = β ln n + (1 − β) ln m,
then the static general equilibrium birth rate would have been bGE = β α ΩA /L1−α . Using
a Cobb-Douglas type preference function, therefore, the effect of changes in industrial sector
productivity drops out. As a result, the Malthusian model becomes incapable of capturing
the effect of development in the industrial sector.
8
A balanced growth path is a path along which all variables grow at constant
geometric rates (possibly zero). We will look for a balanced growth path in
which the birth rate b is constant.
Along a balanced growth path the left-hand-side of equation (b-SGE) is
constant by definition, so the right-hand-side must also be constant. Using the
growth rates defined in equations (2) and (3), the equilibrium birth rate remains
fixed over time when
γ A = αγ M + (1 − α) γ L ,
where γ A and γ M are the rates of growth of productivity in the agricultural and
industrial sector, respectively, and where γ L is the rate of growth of population.
Inserting the growth rate of population as was defined in equation (5), it
follows that, along a balanced growth path, the steady state birth rate is
¶
µ
1
1
st.st.
=
b
1+
(γ − αγ M ) .
1−d
1−α A
As expected, a reduction in death rates in steady state leads to lower fertility.
More importantly, abstracting from changes in the death risk, which we have
taken to be exogenous, variation in the demand for children, along a balanced
growth path, is caused by a shift in the ratio of agricultural productivity growth
to industrial productivity growth. If we define this ratio as
γ
(6)
γ≡ A,
γM
then it follows that an increase in γ will increase fertility whereas a reduction
in γ has the opposite effect.
Accordingly, the u-shaped relationship between income and birth rates described above would be caused by initial acceleration in agricultural productivity
growth, i.e., an increase in γ, succeeded by acceleration in industrial productivity growth, i.e., a reduction in γ. In other words, the current theory predicts that
an ‘agricultural revolution’ where to be succeeded by an ‘industrial revolution,’
in order to create the u-shaped relationship illustrated in Figure 1.
Note that the steady state birth rate is not necessarily equal to one. That
is, the steady state population level is not necessarily constant. Rather, the
size of the population will increase or decrease along a balanced growth path,
depending on the relative rate of growth of productivity in the agricultural and
the industrial sector. Of course, in the absence of any productivity growth, as
Malthus predicted, the steady state population level would be constant.
Turning next to the steady state manufactured goods consumption level, the
right-hand-side of equation (m-SGE) is constant only when the level of industrial
productivity remains fixed. This implies that the growth rate of manufactured
goods consumption, in steady state, is
γm = γM
where γ M is the rate of growth of productivity in the industrial sector, and where
γ m is defined as the net rate of growth of manufactured goods consumption–or,
in this model, economic growth–between any two periods.
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In the traditional interpretation of the Malthusian population hypothesis,
productivity growth, via higher income, will lead to population increase. And
population increase ultimately “eats up” all the extra income that agricultural
productivity growth brings about. Clearly, this feature is well captured by the
current model, at least as long as the increase in productivity stems from the
agricultural sector.
Industrial productivity growth, however, have a different effect. Not only
does it cause lower birth rates, and, therefore, slower population growth. It
also leads to a higher industrial income, which is not “eaten up” by a larger
population because it does not boost up popolation growth.
Finally, since industrial productivity growth brings about a drop in fertility,
this will limit the demand for agricultural labour. All other things being equal,
this means that workers, as a result of industrial productivity growth alone, can
be transferred from agricultural to industry, another important feature of the
process of industrialisation.
3
Conclusion
In England, a u-shaped relationship between income and birth rates has been
observed, whereby income growth leads first to higher birth rates, and then
later to lower birth rates. The latter part of the u-shape, often referred to as
the demographic transition, is normally taken to go against Malthus’ theory
about the principle of population.
Using a simple analytical framework, this study provides a new interpretation of the Malthus’ population hypothesis, which is perfectly consistent with
the demographic transition and with a steady growth in income per capita. It
is demonstrated that if people have so-called quasi-linear preferences, i.e., if the
demand for children varies negatively with the price of foods but is unaffected
by changes the individual’s income, then the effect of productivity growth on
fertility ends up being sector-dependent: agricultural productivity growth increases fertility whereas industrial productivity growth reduces fertility. The
theory predicts that the u-shaped relationship between income and birth rates
observed in England would result from an ‘agricultural revolution,’ which is
succeeded an ‘industrial revolution.’
In principle, this modified version of Malthus population model easily put
to the test. It can be done in at least two ways. Either by testing if birth rates
respond inversely to changes in agricultural terms of trade. Or by testing if the
γ ratio from equation (6) moves in the direction dictated by the model.
An impediment to testing the model’s results, however, arises from the fact
that this is a model of a closed economy. The fact that, during the process
of industrialisation most countries become open to foreign trade and exchange,
breaks down the labour market equilibrium condition.
10
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Figure 1
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