Why does the rate of population growth decline in the face of economic growth? We show that growing product variety may induce a permanent reduction in the demand for children and a continuous rise in income and consumption.demographic transition; demand for children; product variety; Malthus

Jacob Weisdorf

Ricardo Andrés Guzmán

English

Research Papers in Economics

Discussion Papers 
Department of Economics 
University of Copenhagen 
 
 
 
 
 
 
 
 
Øster Farimagsgade 5, Building 26, DK-1353 Copenhagen K., Denmark 
Tel.: +45 35 32 30 01 – Fax: +45 35 32 30 00 
http://www.econ.ku.dk 
 
 
ISSN: 1601-2461 (online) 
 
 
No. 09-25 
 
 
 
Product Variety and the Demographic Transition 
 
 
 
Ricardo Andrés Guzmán, Jacob Louis Weisdorf 
 
 
 
 
 
 
 
 
 
 
  
  
 
  
 Product variety and the demographic transition
Ricardo Andrés Guzmán∗ Jacob Louis Weisdorf†
(Forthcoming in Economics Letters)
December 8, 2009
Abstract
Why does the rate of population growth decline in the face of economic growth? We show
that growing product variety may induce a permanent reduction in the demand for children
and a continuous rise in income and consumption.
Keywords: Demographic transition; demand for children; product variety; Malthus.
JEL classiﬁcation: J13; N30; O10.
1I n t r o d u c t i o n
Starting in the late 19th century, net reproduction rates in western Europe dropped from an average
of three surviving children per woman to just below two children in the early 21st century (Maddison,
2001). This is known as the demographic transition. Yet, over the same period, income per capita
has increased ninefold (ibid.). If we believe that children are normal goods, then the fall in the
demand for children must be explained by negative price eﬀects that overrode the positive income
eﬀect. While more expensive children are certainly part of the explanation (e.g., Bergstrom, 2007;
Galor, 2005; Galor and Weil, 1999, 2000; de la Croix and Licandro, 2009), we show that a continuous
increase in the consumption goods variety may also depress the demand for children and speed up
the growth of income and consumption. Two conditions are needed for this: children and other
consumption goods must be normal goods, and they must be substitutes for each other.
∗Facultad de Gobierno and Facultad de Economía y Negocios, Universidad del Desarrollo. Address: Av. Plaza
700, San Carlos de Apoquindo, Las Condes, Santiago, Chile. E-mail: rnguzman@uc.cl.
†Department of Economic History, Lund University, and Department of Economics, University of Copenhagen.
Address: Öster Farimagsgade 5, DK-1353, Copenhagen, Denmark. Phone: +45 35323033. Fax: +45 35323000.
E-Mail: jacob.weisdorf@econ.ku.dk. (Corresponding author)
12 The model
2.1 Setup
Consider a small, open economy. Time is continuous, indexed by  ≥ 0. The number of adults
alive in time  is ()  0.A d u l t sl i v ef o ro n ep e r i o d :t h o s ea l i v ei nt i m e will be dead in time
 + ∆,w h e r e∆ ∈ R+ is any positive real number. When the adults die, they are replaced by their
children. All adults are identical.
At y p i c a lt i m e -  adult maximizes a CES utility function:
()=
³
()
−1
 +( 1− )()
−1

´ 
−1
 (1)
where () is his consumption of a composite good, and ()  0 is the number of his children.
Parameter  ∈ (01) is the weight of children on the utility, and 1 is the elasticity of substitution
between consumption and children. Because 1, consumption and children are gross substitutes.
The composite encompasses () ∈ R+ diﬀerent consumption goods in time :
()=
ÃZ ()
0
()
−1
 d
! 
−1
 (2)
where () is the adult’s consumption of good  ∈ [0()].W ec a l l() the variety. Parameter
1 represents the elasticity of substitution between the diﬀerent types of goods. The fact that
1 implies that adults will want to diversify consumption.
Each time- adult earns a nominal wage ()  0. The typical adult faces the following budget
constraint:
() ≥ ()()+()()
where () denotes the price of the composite in time ,a n d()  0 denotes the price of a child.
The economy is small and open, so all prices are exogenous.
Standard calculations yield the following Marshallian demands for consumption and children:
()=

()−()

()1− +( 1− )
 ()1− (3)
()=
(1 − )
 ()−()

()1− +( 1− )
 ()1− (4)
Since all goods cost the same, they will be consumed on equal amounts:
(0)=(), for all  ∈ [0()] (5)
2It follows that the total expenditure in the composite is
()()=
Z ()
0
()d = (0)() (6)
where  is the price of each individual good type. Using equations (2) and (5), we obtain the
following:
()=(0)()

−1 (7)
And combining equations (6) and (7), we get the price of the composite:
()=()
− 1
−1 (8)
The economy produces goods of one type, while the remaining types of goods are imported
from abroad. Labor is immobile, and the domestic labor supply is inelastic and equal to ().T h e
nominal wage is given by
()=()()− (9)
where ()  0 is the total factor productivity (TFP) in time ,a n d ∈ (01).B e c a u s e ∈ (01),
the wage falls as population rises.
Finally, the following equation governs population dynamics:
dln()
d
= () − ¯  (10)
where ¯  is the replacement fertility rate. Equations (9) and (10) constitute the classical Malthusian
assumptions.
2.2 Equilibrium
Assume that TFP, the price of children, and variety change at constant, non-negative rates:
dln()
d
=  (11)
dln ()
d
=  (12)
dln()
d
=  (13)
where  ≥ 0.
3Combining equations (4), (8), and (9), we obtain the demand for children:
()=
(1 − )
 ()−()()−


1−
 ()
−
1−
−1 +( 1− )
 ()1−

Log-diﬀerentiating the above equation with respect to , taking limits and rearranging, we get:
lim
→∞[() − ¯ ]= − lim
→∞
dln()
d
− 
− lim
→∞
dln[
1−
 ()
−
1−
−1 +( 1− )
 ()1−]
d
 (14)
where we have used equations (10)—(13).
In the long run, the demand for children is constant:
lim
→∞()=lr (15)
lim
→∞
dln()
d
=0  (16)
where lr denotes the long-run demand for children. Inserting (15) and (16) into equation (14), we
get an expression for the long-run demand for children:
lr =¯  +
1

Ã
 −  − lim
→∞
dln[
1−
 ()
−
1−
−1 +( 1− )
 ()1−]
d
!

It is straightforward to show that
lim
→∞
dln[
1−
 ()
−
1−
−1 +( 1− )
 ()1−]
d
=
 − 1
 − 1
 (17)
Hence, the long-run demand for children can be expressed as
lr =¯  +
1

µ
 −  −
 − 1
 − 1

¶
 (18)
Once we know lr, the rates of change in the real wage () and in consumption are easily
obtained:
 =  +
 − 1
 − 1
 (19)
 = 
µ
 +
1
 − 1

¶
 (20)
4Two main results emerge from equations (18) and (19). First, consistent with previous results,
equation (18) shows that a rising cost of children (  0)w i l ld a m p e nt h ep o s i t i v ee ﬀect of
technological progress (  0) on the demand for children. Since more expensive children moderate
the growth of population, this helps to generate growth in real income per capita and consumption.
Implicitly, this is caused by diminishing returns to labor in production. Second, if children and
consumption goods are gross substitutes (i.e. if   1), then, in response to more product
variety, adults will reduce their demand for children. Since more product variety moderates the
growth of population, this, too, helps to generate growth in real income per capita.
Note that the existing literature overlooks the eﬀect of more product variety on the demand for
children because of the widespread use of Cobb-Douglas preferences. In the Cobb-Douglas case,
the elasticity of substitution between consumption goods and children equals one ( =1 ), which
eliminates the product variety from equations (18) and (19).
References
Bergstrom, T.C., 2007. Some evolutionary economics of family partnerships. American Economic
Review 97, 482—486.
De la Croix, D., Licandro, O., 2007. ‘The Child is Father of the Man:’ Implications for the Demo-
graphic Transition. CEPR Discussion Paper No 6493.
Galor O., 2005.From stagnation to growth: Uniﬁed growth theory, in: P. Aghion and S. Durlauf,
eds., Handbook of Economic Growth, Vol. 1, Elsevier/North-Holland, Amsterdam, pp. 171—293.
Galor O., Weil, D.N., 1999. From Malthusian stagnation to modern growth. American Economic
Review 89, 150—154.
Galor O., Weil, D.N., 2000. Population, technology, and growth: From Malthusian stagnation to
the demographic transition and beyond. American Economic Review 90, 806—828.
Maddison, A., 2001. The world economy: A millennial perspective. Development Centre of the
OECD, Paris.
5

From Malthusian stagnation to modern growth.

Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyond.

Some evolutionary economics of family partnerships.

stagnation to growth: Uniﬁed growth theory,

The Child is Father of the Man:’ Implications for the Demographic Transition. CEPR Discussion Paper No 6493.

The world economy: A millennial perspective.

Product Variety and the Demographic Transition

http://www.econ.ku.dk/english/research/publications/wp/dp_2009/0925.pdf/

Product Variety and the Demographic Transition

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics