## Modelling the Long-Term Dynamics of Population Structures. the Reference Age Structure

**Gustavo De Santis**, *University of Florence*

**Giambattista Salinari**, *University of Sassari*

In this paper, we present and test ahypothesis that we believe is a novelty in demography: the age structure of anypopulation in any period tends towards a specific shape, here called “referenceage structure”, which is the age structure of the current stationary population.We define both of them in Section 1. In Section 2, we show that this tendencycannot be proved mathematically in all circumstances. Indeed, the reverse istrue: there are theoretical situations in which this tendency will *not*emerge. However, these situations are so peculiar that the probability thatthey materialize is close to zero. In “normal” circumstances, the forces thatcause this convergence operate, albeit only weakly (see next point).

Next, we turn to the empirical analysis,exploiting the UN-DESA database. In Section 3, we show that when the twostructures do not coincide, the current age structure tends to its reference counterpart.This convergence is slow, and is usually obscured by the other forces at play, startingfrom the demographic transition. However, in the long run it proves effective,and it can be detected statistically using an ECM (Error Correction Model), thetwo relevant parameters of which turn out to be significant and with theexpected sign (see below for the details).

In Section 4, we show that the current andthe reference age structures are indeed normally close to each other, except inspecific (time-limited and easily identifiable) periods. Indeed, the deviationsof the current from the reference age structure follow a rather regular scheme:they are relatively high for countries at the beginning, or in the midst, oftheir demographic transition; they decline markedly for countries towards theend of their demographic transition, and they are almost negligible for more“mature” countries. Other specific exceptions (comparatively large differencesbetween the reference and the actual age structure) are noted: they emerge whenrelatively small countries adopt very strong and selective migration policies,attracting a large influx of young adults. These effects, however, are typicallyshort-lived.

In Section 5, we briefly discuss some possibleuses of the notion of reference age structure. On a theoretical plan, it throwsnew light on old concepts and discussions: how to measure the demographic bonus(or malus, for that matter), how to assess the relative importance of mortality,fertility, and migration in the process of population ageing, etc. Theexistence of a reference age structure may also have practical consequences: asit drives the future evolution of the population, it may prove helpful in thedesign of long-lasting societal compacts, starting from pension systems. DETAILS: DYNAMIC AND STATIC ANALYSIS

Let c_{x,t} be the age structure ofthe population at time t, or current age structure. It is defined as therelative share of population aged x, P_{x}, to the total population P:c_{x}=P_{x}/P. Similarly, let k_{x,t} be the agestructure of the stationary population, calculated on the cross sectional lifetable at year t. If L_{x} are the person-years lived at age x, and T_{0}is their sum, or total number of person-years lived at all ages current, theratios k_{x}=L_{x}/T_{0} form the age structure of thestationary population, which we also call “reference” age structure in thispaper. Our hypothesis is that the c_{x,t} tend to move towards theirreference counterpart k_{x,t}. To prove this, we run the following ErrorCorrection Model (ECM) on UN data

1) D*c _{x,t}* = b

_{0}+ b

_{1}D

*k*+ b

_{x,t}_{2}(

*k*-

_{x,t-1}*c*) + e

_{x,t-1}

_{x,t}where D*c _{x,t}* =

*c*-

_{x,t}*c*, D

_{x,t-1}*k*=

_{x,t}*k*-

_{x,t}*k*, and e

_{x,t-1}_{x,t}is the error term. Our main object of interest is the long-termcoefficient β

_{2}, which we expect to be significantly greaterthan zero. If this is true, the

*c*

_{x}_{,t}seriestend to converge in the long run on the

*k*series, for thereason shown in Figure 1, which represents the actual and the reference agestructure at time

_{x,t}*t*-1 of a hypothetical population.

As for the short-term dynamics representedby β_{1}, we expect the reference and the current age structure tomove in opposite directions and thus β_{1} to be negative. Forinstance, improved survival translates into an older reference and a youngeractual age structure during the initial part of the demographic transition.

Our empirical results are neat (Table 1):regardless of the world area examined, in the past 60 years, the parameters thatmeasure this convergence were always positive and virtually always verysignificant.

We also measured the distance between theactual and the reference age structure with the index of dissimilarity D, whichproved small (at most, only about a fourth of its theoretical maximum) and witha tendency to decline over time (especially in the future, according to the UNforecast, medium variant). This corroborates our convergence hypothesis, and reinforcesits practical utility.

Presented in Session 1233: Posters