Mathematical Supplements

We now discuss under which conditions intrinsic growth rates and stable age distributions do exist, and whether they depend on the initial popu-lation vectorn^f₀ or only on the matrixF.

Existence of a Stable Population

1.We begin with the first question, whether one can construct an intrinsic growth rate and a stable age distribution for some given matrixF. This depends on the coefficients ofF. As introduced in the previous section,F has the following structure:⁴

F =

4Matrices having this structure are often called Leslie matrices to remind of P. H.

Leslie who has first provided an extensive discussion with demographic applications, see Leslie (1945).

One can be sure thatF ≥0, meaning that all coefficients ofF are non-negative. One can also safely assume that 0< δ_τ^f<1, forτ = 1, . . . , τm−1, and consequently all entries in the subdiagonal ofFare greater than zero.

But a question concerns the birth ratesβ_τ^∗. Since the reproductive period of women is limited and, in general,τb< τm, we can assume thatβ_τ^∗_b >0 but need to observe thatβ_τ^∗= 0 forτ > τb, implying thatFhas less than full rank.

2.We can proceed, however, in two steps. In a first step we consider only the firstτb rows and and columns ofF, that is, the matrix

This is now a non-negative matrix which has full rank.⁵ Furthermore, ˜F is an irreducible matrix.⁶ This allows to apply a famous mathematical theorem by G. Frobenius.⁷ The theorem guarantees that ˜F has at least one real positive eigenvalue, say λ^∗, also called a dominant eigenvector of ˜F, with a corresponding eigenvector, say v^∗ = (v₁^∗, . . . , v_τ^∗_b)⁰, whose coefficients are all real and positive. So we can write the equation

F v˜ ^∗ = λ^∗v^∗ (17.3.1)

A further implication of the theorem that will be used below in the discus-sion of our second question is that all eigenvalues of ˜F have an absolute value (modulus) which is less than, or equal to,λ^∗.

3.We can now derive a stable age distribution and an intrinsic growth rate. The intrinsic growth rate can be simply defined byρ^∗_f :=λ^∗−1. The derivation of the stable age distribution is in two steps. In a first step we define components of a vectorn^f,∗ by

n^f,∗_τ :=

( v_τ^∗ forτ = 1, . . . , τb 1−δ_τ−1^f

λ^∗ v^∗_τ−1 forτ =τb+ 1, . . . , τm 5This is seen by the determinant of ˜Fwhich is

det( ˜F) =±σfβ_τ^∗_b

τ_b−1

Y

τ=1

(1−δ_τ^f) 6= 0 The sign depends on whetherτbis even or odd.

6By this is meant that, for any two indicesi andj (1≤i < j ≤τb), one can find further indices, sayk1, . . . , km, such that aik1ak1k2· · ·ak_mj > 0.

7We refer to Gantmacher (1971, ch. xxiii).

From equation (17.3.1) and the structure ofFit then follows that F n^f,∗ = λ^∗n^f,∗ = (1 +ρ^∗_f)n^f,∗ (17.3.2) showing that the age distribution which is represented by n^f,∗ will not change when multiplied byF; all components ofn^f,∗will grow, or shrink, with the same rate,ρ^∗_f. Therefore, to get the stable age distribution one only has to transformn^f,∗ into proper proportions:

n^f,p_τ := n^f,∗_τ X^τm

j=1

n^f,∗_j

4.To illustrate the argument we use the example of the previous section.

In this example the matrix ˜Fis given by F˜ =

Calculating eigenvalues and eigenvectors can be done with the following TDAscript:⁸

One finds that the dominant eigenvalue isλ^∗= 1.0573 and the correspond-ing eigenvector is

v^∗ = (0.7405,0.5603,0.3710)⁰

The eigenvalue provides the intrinsic growth rate, ρ^∗_f = 0.0573, which is identical with the value found in the previous section. The eigenvector can be used to calculate the components ofn^f,p:

n^f,∗₁ = 0.7405, n^f,∗₂ = 0.5603, n^f,∗₃ = 0.3710, and n^f,∗₄ = 0.6

1.05730.3710 = 0.2105

8More detailed explanations of the practical calculations will be given in Section 17.5.1.

Of course, equation (17.3.2) does not change ifn^f,∗is multiplied by an ar-bitrary scalar value. So we can rescalen^f,∗to get a frequency distribution with components adding to unity. The result is

n^f,p = (0.39,0.30,0.20,0.11)⁰

and equals the age distribution found in the previous section.

5.It would suffice to calculate the dominant eigenvalue of ˜Fbecause the corresponding eigenvector, and consequently the stable age distribution, can be derived from the death rates. Let the dominant eigenvalue,λ^∗, be given. Since the corresponding eigenvector,v^∗, is determined only up to an arbitrary multiplicative factor, we can setv^∗₁ = 1. All further elements ofv^∗ can be calculated recursively with the formula

v^∗_τ = 1−δ_τ−1^f

λ^∗ v^∗_τ−1 (forτ = 2, . . . , τm)

The argument also shows that, if λ^∗ = 1, the stable age distribution de-pends only on the death rates, not on the birth rates. But, of course,λ^∗ also depends on birth rates.

Convergence to a Stable Age Distribution

6.We now turn to the second question, whether, beginning with an ar-bitrary initial female populationn^f₀, the sequence n^f_t =F^tn^f₀ finally con-verges to an equilibrium defined by the intrinsic growth rate,ρ^∗_f, and the stable age distribution, n^f,p.⁹ As will be shown, the answer is positive under quite general conditions. To develop the argument, we first consider the sub-matrix ˜F which consists of the first τb rows and columns of F.

Correspondingly, we refer to the firstτbelements ofn^f_t by the vectorn^f,a_t . Since ˜Fis an upper block-diagonal matrix, it follows that

n^f,a_t = ˜F^tn^f,a₀ (17.3.3)

We now show that, given an additional assumption to be explained be-low, n^f,a_t converges to a vector which is proportional to v^∗, that is, the eigenvector corresponding to the dominant eigenvalue of ˜F.

7.This requires to refer to all eigenvalues of ˜Fwhich will be denoted by λj, with corresponding eigenvectors vj, for j = 1, . . . , τb. One of these eigenvalues, sayλj^∗=λ^∗, is the dominant one and has the corresponding eigenvectorv_j∗=v^∗. So we can write the equations

Fv˜ _j = λjv_j (forj = 1, . . . , τb)

9It will be assumed that there is at least one woman of an age under, or equal to,τb.

which, by defining Λ := diag (λ1, . . . , λτb) and V := (v1, . . . ,v_τb), may also be written as a single matrix equation

FV˜ = VΛ

As mentioned above, ˜F has full rank and its eigenvectors are therefore linear independent. This implies that V is an invertible matrix and we may write ˜F=VΛV⁻¹, from which it follows that

F˜^t = VΛ^tV⁻¹ This then allows to write

n^f,a_t = ˜F^tn^f,a₀ = VΛ^tV⁻¹n^f,a₀ = VΛ^tu

where, for the last equation, we have used the abbreviationu:=V⁻¹n^f,a₀ . In a next step this equation can be written in the following way:

n^f,a_t = (v1, . . . ,v_τb)

which shows thatn^f,a_t is a weighted mean of the eigenvectors of ˜F. Finally, dividing byλ^t_j∗, we get

8.This equation can be used to think about the convergence problem.

From the theorem of Frobenius we already know thatλj^∗ ≥ |λj| for all j= 1, . . . , τb. We now introduce a further assumption, to be discussed be-low, thatλj^∗ > |λj| for allj6=j^∗. Given this assumption, it follows that the second term on the right-hand side of equation (17.3.4) will converge to zero and this, in turn, implies the convergence

1 λ^t_j∗

n^f,a_t −→ uj^∗v_j∗

This shows that, for sufficiently larget, n^f,a_t+1 ≈ λj^∗n^f,a_t

andn^f,a_t will be approximately proportional to the eigenvectorv^∗. More-over, also the remaining components of n^f_t will converge to a stable age distribution. This is seen from the fact that these remaining components only depend on the growth of the female population at age τb and the

death rates at ages greater than, or equal to,τb. Therefore, if eventually the number of women at ageτb grows, or shrinks, with a constant (intrin-sic) rate, this will propagate to all higher ages. The stable age distribution for all ages may then be calculated as shown in the first part of this section.

9.It remains to discuss the assumption that the dominant eigenvalue of ˜F is greater, in magnitude, than all other eigenvalues. This is not necessarily the case. For example, the matrix

F˜ :=

0 1 0.8 0

has two real eigenvalues, 0.8944 and -0.8944, having the same magnitude.

In this example, as shown by equation (17.3.4),n^f,a_t will not converge to a unique stable age distribution but oscillate between two different distribu-tions. Such cases are, however, exceptional. A sufficient condition for the existence of a dominant eigenvalue which is greater, in magnitude, than all other eigenvalues is that there are at least two successive ages with a positive birth rate.¹⁰ Therefore, cyclical solutions will only occur if one uses a highly aggregated Leslie matrix; for instance, a matrix that only distinguishes three age groups, belowτa, betweenτa andτb, and aboveτb. If one distinguishes at least two age groups in the reproductive period one can safely assume the existence of a stable age distribution.

Im Dokument Concepts, Data, and Methods (Seite 130-133)