The age-specific growth function of an observed population and the reproductive value function based on the population´s current vital rates determine the intrinsic rate of growth implied by those vital rates through the simple relationship given in equation (1). That equation establishes the analytical significance of age-specific growth, and leads to relationships that quantify a population´s approach to stability and that specify the extraordinarily close connection between reproductive values and population momentum.age-specific growth, intrinsic r, population momentum, reproductive value, stable equivalent, stable population

Robert Schoen

English

Research Papers in Economics

Demographic Research a free, expedited, online journal
of peer-reviewed research and commentary
in the population sciences published by the
Max Planck Institute for Demographic Research
Konrad-Zuse Str. 1, D-18057 Rostock ¢ GERMANY
www.demographic-research.org
DEMOGRAPHIC RESEARCH
VOLUME 24, ARTICLE 33, PAGES 825-830
PUBLISHED 10 JUNE 2011
http://www.demographic-research.org/Volumes/Vol24/33/
DOI: 10.4054/DemRes.2011.24.33
Formal Relationship 15
Age-speciﬁc growth, reproductive values,
and intrinsic r
Robert Schoen
c ° 2011 Robert Schoen.
This open-access work is published under the terms of the Creative
Commons Attribution NonCommercial License 2.0 Germany, which permits
use, reproduction & distribution in any medium for non-commercial
purposes, provided the original author(s) and source are given credit.
See http://creativecommons.org/licenses/by-nc/2.0/de/Table of Contents
1 Introduction 826
2 Proof 827
3 A brief history 827
4 Related results 828
5 Acknowledgements 829
References 830Demographic Research: Volume 24, Article 33
Formal Relationship
Age-speciﬁc growth, reproductive values, and intrinsic r
Robert Schoen1
Abstract
The age-speciﬁc growth function of an observed population and the reproductive value
function based on the population’s current vital rates determine the intrinsic rate of growth
implied by those vital rates through the simple relationship given in equation (1). That
equation establishes the analytical signiﬁcance of age-speciﬁc growth, and leads to rela-
tionships that quantify a population’s approach to stability and that specify the extraordi-
narily close connection between reproductive values and population momentum.
1 Population Research Institute, Pennsylvania State University, University Park, PA 16802 USA.
Mailing address: 3145 Turk Blvd, #103, San Francisco, CA 94118 USA. E-mail: rschoen309@att.net.
http://www.demographic-research.org 825Schoen: Age-speciﬁc growth, reproductive values, and intrinsic r
1. Introduction
A simple and elegant relationship links the intrinsic rate of natural increase (Lotka’s r)
with age-speciﬁc growth and age-speciﬁc reproductive value functions. Speciﬁcally
r =
Z
r(x;t)q(x;t)dx (1)
where the integral ranges over all ages x; the age-speciﬁc growth function, r(x;t) is
deﬁned as
r(x;t) =
£@N(x;t)
@t
¤
N(x;t)
(2)
where @ indicates partial differentiation and N(x;t) is the population age x at time t; and
reproductive fraction q(x;t) is given by
q(x;t) =
N(x;t)v(x)
Q(t)
(3)
As deﬁned below, Q(t) is the stable equivalent number of births at time t. The repro-
ductive function v(x) represents the contribution from age x to the number of births in
the stable population that is speciﬁed by the given vital rates underlying v(x). Mathemat-
ically
v(x) =
1 R
x
e¡rap(a)f(a)da
[A¤e¡rxp(x)]
(4)
where p(a) is the probability of surviving from birth to age a under the given mortality
rates, f(a) is the fertility rate at age a, and A¤ is the mean age at childbearing in the
stable population (cf. Schoen 2006, 30). The product v(x) ¢ A¤ is a reproductive value
that can be interpreted as the "present value" of future children born to a person age x, i.e.
the future number of children that a person age x will have discounted back to age x at
an interest rate equal to intrinsic r. Division by the stable mean age at childbearing, A¤,
expresses that reproductive value on an annual basis.
The stable equivalent number of births at time t, Q(t), can be thought of as the number
of births in the time t stable population that, growing exponentially at rate r, becomes
identical in size and composition to the stable population that ultimately arises from the
observed time t population, i.e. the N(x;t), as it grows under the given vital rates. That
stable equivalent number of births can be written as
Q(t) =
Z
N(x;t)v(x)dx (5)
826 http://www.demographic-research.orgDemographic Research: Volume 24, Article 33
where the integral spans all ages x. The function q(x;t) can thus be interpreted as the
fraction of the time t stable equivalent number of births that is contributed by persons age
x.
Equation (1) indicates that if vital rates are constant for even an instant at time t,
then the intrinsic growth rate (r) implied by those vital rates is a weighted average of
the observed population’s age-speciﬁc growth rates, where the weights are each age’s
fractional contribution to the time t stable equivalent number of births. Although both
the r(x;t) and the q(x;t) are mixtures of vital rates and the number of persons in the
observed time t population, their accumulated product, r, is completely independent of
the N(x;t).
2. Proof
The proof of equation (1) follows from considering how the stable equivalent number of
births, Q(t), grows over time. Since the vital rates are assumed to be constant, we must
have
dQ(t)
dt
= rQ(t) (6)
because, at all ages, the stable equivalent grows exponentially at the intrinsic rate implied
by the constant vital rates. At the same time, equation (5) indicates that
dQ(t)
dt
=
d
£R
N(x;t)v(x)dx
¤
dt
=
Z
r(x;t)N(x;t)v(x)dx (7)
where the second equality follows from equation (2). Equating equations (6) and (7) and
dividing by Q(t) yields
r =
Z
r(x;t)
N(x;t)v(x)
Q(t)
dx (8)
which, by equation (3), is the desired result.
3. A brief history
The concept of the reproductive value goes back to the great statistician R.A. Fisher.
In the 1930s, Fisher interpreted demography’s stable population characteristic equation
http://www.demographic-research.org 827Schoen: Age-speciﬁc growth, reproductive values, and intrinsic r
as describing how individuals repaid the "loan" of a life at birth by their reproduction
over the life course. The age x reproductive value was simply the amount still owed
at age x (cf. Schoen 2006, 30). The later development of the discrete stable model by
Leslie (1945) and others showed that reproductive value functions were the elements of
the dominant left eigenvector of a population projection (Leslie) matrix. Along with
Lotka’s r (the dominant eigenvalue of a Leslie matrix), and the stable population age
composition (the dominant right eigenvector of a Leslie matrix), the reproductive value
emerged as a fundamental demographic quantity.
The concept of age-dependent growth emerged in work by Bennett and Horiuchi
(1981), which was generalized by Preston and Coale (1982). The motivation was to
develop new demographic relationships that could be used to estimate a broad range of
population measures. Kim (1986) extended the "variable-r" approach to discrete rela-
tionships, but pointed out that the variable-r relationships were essentially tautologies. If
data were available to calculate the age-dependent growth rates, those data could be used
directly, without introducing r(x;t) values. In some applications, however, the variable-r
approach has been shown to be analytically useful (cf. Preston, Heuveline, and Guillot
2001, 184-189).
Preston (1986) pointed out that in any observed population there was an age at which
the sum of the age-speciﬁc growth rates up to that age equaled the intrinsic growth rate
implied by the population’s current vital rates. Preston (1986) approximated that age
by Lotka’s generation length, T, though Wachter (1988) later showed that T was not a
reliable estimate. Schoen and Kim (1991) sought analytical linkages between the r(x;t)
function and conventional demographic variables. That work derived equation (1) and
explored the relationships discussed in the next section.
4. Related results
Three noteworthy analytical relationships follow from equation (1). First, the change in
the q(x;t) function over time depends directly on the difference between r(x;t) and r, as
dq(x;t)
dt
= q(x;t)[r(x;t) ¡ r]: (9)
Equation (9) is reminiscent of the central relationship in Guillot (2009), a previous con-
tribution to the Formal Relationships Special Collection.
Second, equation (1) leads to a "Fundamental Principle" of population dynamics that
sees every observed population as always moving toward the stable population implied
by its current vital rates. Let us deﬁne the stable equivalent population at age x and time
828 http://www.demographic-research.orgDemographic Research: Volume 24, Article 33
t, S(x;t), by
S(x;t) = Q(t)e¡rx p(x) (10)
and age-speciﬁc momentum at time t, ­(x;t), as
­(x;t) =
S(x;t)
N(x;t)
(11)
Here, the term "momentum" is used to indicate the growth associated with any transition
to stability, not just with the transition to a zero-growth population. The extent to which
an observed population moves toward stability is then given by the q-weighted covariance
between the r(x;t) and the natural logarithm of the ­(x;t) (cf. Schoen 2006, 71).
Third, the discrete form of equations (1) and (2) leads to an intimate relationship
between reproductive contributions and population momentum (Schoen and Kim 1992).
Let Nijt be the number of persons in component i and age group j of the population
at time t, where component 1 is the dominant component of the spectral decomposition
of the Leslie matrix. The symbol Njt represents the observed number of persons, in all
components, in age group j at time t. Age-component-speciﬁc momentum, ­ijt, is then
­ijt =
Nijt
Njt
(12)
Analogously, the age-component-speciﬁc reproductive contribution function, qijt, can be
written
qijt =
Njt vij
Qit
(13)
If matrix ­t has ­ijt as its element in row i and column j, and matrix qt has qijt as its
(i;j)th element, then
­t = q
¡1
t (14)
or the inverse of the age-component-speciﬁc reproductive contribution matrix gives the
age-component-speciﬁc population momentum matrix.
5. Acknowledgements
The many contributions to this work by the late Young J. Kim are gratefully acknowl-
edged.
http://www.demographic-research.org 829Schoen: Age-speciﬁc growth, reproductive values, and intrinsic r
References
Bennett, N. and Horiuchi, S. (1981). Estimating the completeness of death registration in
a closed population. Population Index 47(2): 207–221.
Guillot, M. (2009). The effect of changes in fertility on the age distribution of stable pop-
ulations. Demographic Research 20(24): 595–598. doi:10.4054/DemRes.2009.20.24.
Kim, Y.J. (1986). Examination of the generalized age distribution. Demography 23(3):
451–461. doi:10.2307/2061441.
Leslie, P. (1945). On the use of matrices in certain population mathematics. Biometrika
33(3): 183–212.
Preston, S. (1986). The relation between actual and intrinsic growth rates. Population
Studies 40(3): 343–351. doi:10.1080/0032472031000142286.
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of population dynamics. Demography 28(3): 455–466. doi:10.2307/2061467.
Schoen, R. and Kim, Y.J. (1992). Covariances, roots, and the dynamics of age-speciﬁc
growth. Population Index 58(1): 4–17.
Wachter, K. (1988). Age group growth rates and population momentum. Population
Studies 42(3): 487–494. doi:10.1080/0032472031000143606.
830 http://www.demographic-research.org

Age group growth rates and population momentum.

Age structure, growth, attrition, and accession: A new synthesis.

Covariances, roots, and the dynamics of age-speciﬁc growth.

Dynamic Population Models.

Estimating the completeness of death registration in a closed population.

Examination of the generalized age distribution.

M.(2001). Demography: Measuringandmodeling population processes.

Movement toward stability as a fundamental principle of population dynamics.

On the use of matrices in certain population mathematics.

The effect of changes in fertility on the age distribution of stable populations. Demographic Research

The relation between actual and intrinsic growth rates.

Age-specific growth, reproductive values, and intrinsic r

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Age-specific growth, reproductive values, and intrinsic r

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