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LIPRO 2.0: An application of a dynamic demographic projection model to household structure in the Netherlands. NIDI

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LIPRO 2.0: AN APPLICATION OF A DYNAMIC DEMOGRAPHIC
PROJECTION MODEL TO HOUSEHOLD STRUCTURE IN
THE NETHERLANDS
Evert van Imhoff
Nico Keilman
PREFACE
This book originated from the research project "The impact of changing house-
hold structure on future social security expenditure in the Netherlands", carried
out for the Netherlands Ministry of Social Affairs and Employment by the
Netherlands Interdisciplinary Demographic Institute (NIDI) during the years
1988-1990. A major component of this research project consisted of the devel-
opment of a multidimensional household projection model. The model has been
implemented in the computer program LIPRO 2.0. Although originally written
as a program for making household projections, LIPRO 2.0 can in fact be used
for a wide range of multidimensional demographic computations.
Realizing that the proof of the pudding is in the eating, we have included
several chapters on an illustrative application of the model to household structu-
re and social security in the Netherlands. However, the emphasis in this book
is on the methodological and computational aspects of the LIPRO model.
Almost half of the book is devoted to a detailed description of the operation of
the LIPRO computer program. The program itself is on the diskette included
in the back cover.
The LIPRO program (or rather: set of programs) has been written in Borland’s
Turbo Pascal 5.0. It can be run on MS/DOS personal computers or compatibles.
A mathematical co-processor, a hard disk, and a memory of 640 kb are required.
If the mathematical co-processor is absent, a recompiled version of the program
can be supplied.
Although the use of the LIPRO programs is free of charge (except for the price
of this book), and indeed warmly encouraged, we would like to urge anyone
intending to publish results obtained with LIPRO to include a proper reference
to this manual. In addition, neither the authors nor the NIDI can accept any
responsibility for damage as a result of errors in the software or its documenta-
tion, let alone as a result of improper use of the software. We welcome any
(written) suggestions for improvement of the program or the manual, but as yet
there are no guarantees that the funds will be available to produce regular
updates of the LIPRO package.
Several people have made valuable contributions during the production of the
program and the book. In the course of intensive collaboration with demogra-
phers at the Central School of Planning and Statistics in Warsaw, Irena
Kotowska discovered many bugs in a previous version of the program and also
made many useful suggestions for improvement. At the NIDI, Frans Willekens
urged us to give high priority to user-friendliness in programming; his
enthusiasm in promoting demographic software has been a great stimulus.
Suzanne Wolf did most of the calculations for the household projections. She
made numerous suggestions for improving the software and prepared all the
figures included in this book. Joan Vrind contributed to the final layout of the
manuscript, and Angie Pleit-Kuiper edited our non-native use of the English
language.
Part of the text of this book was written after the second author had left the
NIDI to join the Norwegian Central Bureau of Statistics. The facilities provided
by this organization, allowing us to finish this book, are gratefully acknowl-
edged.
Evert van Imhoff
Nico Keilman
The Hague / Oslo, October 1991
TABLE OF CONTENTS
PART I : INTRODUCTION ........................... 1
1. Aims and scope .................................... 3
1.1. Introduction ................................. 3
1.2. Problem formulation ........................... 4
1.3. LIPRO .................................... 5
1.4. Outline of this book ........................... 6
2. Household models: a survey ........................... 9
2.1. Concepts ................................... 9
2.2. Definitions ................................ 10
2.3. A typology of household models ................. 11
2.4. Static household models ....................... 12
2.5. Dynamic household models ..................... 13
2.6. Comparison ................................ 15
PART II : THEORETICAL ISSUES ..................... 17
3. A characterization of multidimensional projection models ...... 19
4. The exponential and the linear model .................... 23
4.1. Preliminaries ............................... 24
4.2. Formulation of the exponential model .............. 26
4.2.1. Formulas for age groupsx>1 ............ 26
4.2.2. Formulas for the youngest age group ........ 27
4.2.3. Formulas for a singular intensity matrix ...... 28
4.3. Formulation of the linear model .................. 29
4.4. Computing rates from events .................... 31
4.5. On the computation of exp[X] and inv[X] ........... 32
Appendix: Calculating sojourn times for a Markov process with
uniform entries and intensity matrix of any rank ............ 33
5. The consistency algorithm ............................ 43
5.1. Notation .................................. 44
5.2. Formulation of the consistency problem ............ 45
5.3. Solution to the consistency problem for a specific
class of objective functions ..................... 45
5.3.1. A specific class of objective functions ....... 45
5.3.2. Interpretation of the parameter p ........... 46
5.3.3. Comparison with the unidimensional
harmonic mean method ................. 47
5.3.4. Relationship between age-specific and
aggregate adjustment factors .............. 47
5.4. From adjusted events to adjusted rates ............. 48
6. Some issues in multidimensional life table analysis .......... 51
6.1. Interpretation of life tables in LIPRO .............. 51
6.2. Determining the radix of the life table ............. 51
6.3. Handling the highest age group .................. 53
6.4. Calculating mean ages ........................ 54
6.5. Fertility indicators ........................... 54
6.6. Experience tables ............................ 55
PART III : APPLICATION ........................... 57
7. The specification of the state space in the household model ..... 59
7.1. General considerations ........................ 59
7.2. The specification of household positions ............ 61
7.3. Household events ............................ 62
7.4. Consistency relations ......................... 68
8. From data to input parameters ......................... 73
8.1. Introduction ................................ 73
8.2. The Housing Demand Survey of 1985/1986
(WBO 1985/1986) ........................... 73
8.3. The initial population ......................... 74
8.4. Estimation of jump intensities ................... 74
8.4.1. Estimation of transition probabilities ........ 74
8.4.2. From transition probabilities to jump
intensities ........................... 76
8.4.3. Adjusting the intensities to achieve
internal and external consistency ........... 79
9. Demographic scenarios .............................. 81
9.1. On the term "scenario" ........................ 81
9.2. Jump intensities and the multidimensional life table .... 81
9.3. Five demographic scenarios ..................... 82
9.3.1. Constant Scenario ..................... 83
9.3.2. Realistic Scenario ..................... 83
9.3.3. Swedish Scenario ..................... 84
9.3.4. Fertility Scenario ..................... 84
9.3.5. Mortality Scenario .................... 84
9.4. Quantification of the scenarios ................... 84
10. Household projections: results ......................... 87
10.1. The Realistic Scenario ........................ 87
10.2. A comparison of the five scenarios ................ 92
10.3. The effect of a particular model specification ....... 100
10.4. Comparison with the official NCBS population forecast 101
11. Household projections and social security ................ 105
11.1. Demography and social security ................. 106
11.1.1. General ........................... 106
11.1.2. Previous studies for the Netherlands ....... 107
11.1.3. International comparative studies ......... 112
11.1.4. Conclusions ........................ 114
11.2. Method ................................ 115
11.3. Illustrative social security projections ............. 118
11.3.1. Old age pensions .................... 119
11.3.2. Survivor pensions and social welfare ....... 122
11.3.3. The usefulness of including household
structures in social security projections ..... 122
12. Summary and concluding remarks ..................... 127
PART IV : LIPRO USERS GUIDE .................... 133
13. Introduction and overview ........................... 135
13.1. Introduction ............................... 135
13.2. Hardware requirements ....................... 136
13.3. Installation ............................... 137
13.4. Menus .................................. 138
13.5. Command bars ............................. 138
13.6. Edit screens ............................... 138
13.7. Directory screens ........................... 139
13.8. Browse .................................. 141
13.9. Editing ASCII files .......................... 141
13.10. Output files ............................... 143
13.11. Turbo Pascal error codes ...................... 143
14. Getting started ................................ 145
14.1. The MAIN menu ........................... 145
14.2. Basic input files ............................ 147
14.2.1. The definition file .................... 147
14.2.2. The parameter file .................... 148
14.3. The DEFINITIONS menu ..................... 150
14.4. The STATE SPACE menu .................... 151
14.4.1. Ranges and default values .............. 153
14.4.2. Example .......................... 154
14.5. The PARAMETERS menu .................... 154
14.5.1. Ranges and default values .............. 155
14.5.2. Example .......................... 155
14.6. State labels and variable indicators ............... 156
15. Preparing data ................................ 159
15.1. The EDIT DATA menu ...................... 160
15.1.1. Edit mode ......................... 162
15.2. The CONVERT menu ....................... 163
15.2.1. Example .......................... 167
15.3. The RATES menu .......................... 168
16. Implementation of the consistency algorithm .............. 171
16.1. The input file for the consistency algorithm ......... 172
16.2. Formulas for consistency relations ............... 173
16.3. "Passive" consistency relations .................. 174
16.4. Formulas for endogenous constants .............. 175
16.5. Endogenous constants varying over time ........... 176
16.6. Comments in the input file .................... 177
16.7. A complete example ......................... 177
17. Setting scenarios ................................ 181
17.1. Overview of the SCENARIOS command .......... 182
17.2. The input file for the SCENARIOS command ....... 183
17.2.1. Assignment formulas for rates ........... 185
17.2.2. Assignment formulas for endogenous
constants .......................... 186
17.3. Example ................................. 186
17.4. The SCENARIOS menu ...................... 189
18. Projection ...................................... 191
19. Analysis ....................................... 193
19.1. The ANALYSIS menu ....................... 194
19.2. The TABLES command ...................... 194
19.2.1. Input files for the TABLES command ...... 195
19.2.2. Default values for parameters in TABLES
input files ......................... 197
19.2.3. The TABLES menu .................. 197
19.2.4. Example .......................... 199
19.3. The AGGREGATE command .................. 199
19.3.1. The AGGREGATE input file ............ 199
19.3.2. Default values for parameters
in AGGREGATE input files ............. 202
19.3.3. The AGGREGATE menu .............. 203
19.4. The LIFE TABLE ANALYSIS command .......... 204
19.4.1. The RATES command ................. 205
19.4.2. The DISTRIBUTION OF BIRTHS
command .......................... 205
19.4.3. The EXPERIENCE TABLES command .... 206
19.4.4. Example .......................... 207
19.5. The TRANSITION PROBABILITIES command ..... 208
19.5.1. The TRANSITION PROBABILITIES
input file .......................... 208
19.5.2. Default values for parameters in
TRANSITION PROBABILITIES input files . 210
19.5.3. The TRANSITION PROBABILITIES menu . 211
19.6. Exporting LIPRO results ...................... 211
20. Miscellaneous program features ....................... 213
20.1. The UTILITIES menu ....................... 213
20.2. The SCREEN COLOURS command .............. 214
20.3. The EXECUTE A PROGRAM command .......... 215
20.4. The MS/DOS INTERNAL COMMAND command . . . 215
20.4.1. Example .......................... 216
20.5. The BIN68 program ......................... 216
21. Linking user profiles ............................... 219
21.1. The SOCPROF data files ..................... 220
21.2. Calling the SOCPROF program ................. 222
21.3. The DEFINITIONS menu ..................... 223
21.4. The SHOW PROFILES menu .................. 224
21.5. The EDIT PROFILE menu .................... 225
21.5.1. The EDIT TABLE command ............ 226
21.6. The TIME SERIES menu ..................... 227
21.7. The FULL TABLES menu .................... 229
References .......................................... 231
Author index ......................................... 237
Subject index to parts I, II and III .......................... 239
Subject index to LIPRO Users Guide ....................... 243
LIST OF TABLES
3.1. Classification of events ........................... 21
7.1. Events matrix of the household model ................. 63
8.1. The population in private households according to age, sex,
and household position, the Netherlands, December 31, 1985 . . 75
8.2. Assumptions on multiple events ..................... 78
8.3. Constraints for external consistency, 1986-1990 .......... 80
9.1. Key indicators in the five scenarios ................... 86
10.1. Results of the Realistic Scenario ..................... 88
10.2. Results of the various scenarios, 2035 and 2050 .......... 93
10.3. Population by sex and household position, 1985 and 2035
(Constant Scenario), three model specifications .......... 101
10.4. Comparison of LIPRO projection with NCBS forecast ..... 102
11.1. Social security expenditures in the Netherlands,
1975-1988 ................................... 106
11.2. Expenditures for and number of recipients of demographic
schemes, the Netherlands, 1975-1988 ................. 108
11.3. Projected expenditures for AOW, AWW, and ABW,
the Netherlands ................................ 111
11.4. Average annual growth rate in public expenditure
on pensions in OECD countries .................... 113
11.5. Public expenditures for old age pensions in selected
industrialized countries (index 1980=100) .............. 114
11.6. Comparing social security data from different sources ..... 117
11.7. Projections for old age state pensions (AOW) - Realistic
Scenario ..................................... 120
11.8. AOW expenditures under various scenarios ............ 121
11.9. Projections for the sum of survivor pensions (AWW) and
welfare (ABW) - Realistic Scenario .................. 123
11.10. AWW+ABW expenditures under various scenarios ....... 124
11.11. Four methods to project social security expenditures,
Realistic Scenario .............................. 125
19.1. Part of the output file for the TABLES command ........ 201
21.1. Output of the TIME SERIES command ............... 229
LIST OF FIGURES
10.1. The population by private household position, the
Netherlands, 1985-2050 (Realistic Scenario) ............. 90
10.2. Private households by type, the Netherlands, 1985-2050
(Realistic Scenario) .............................. 90
10.3. Age pyramid for some household positions, 1985 ......... 91
10.4. Age pyramid for some household positions, 2035
(Realistic Scenario) .............................. 91
10.5. Population by household position, 1985 and 2050,
Realistic Scenario versus Swedish Scenario ............. 98
10.6. Households by household type, 1985 and 2050,
Realistic Scenario versus Swedish Scenario ............. 98
10.7. Age pyramid for some household positions, 2035
(Fertility Scenario) .............................. 99
10.8. Age pyramid for some household positions, 2035
(Mortality Scenario) .............................. 99
11.1. Interrelations between demographic developments and
social security ................................. 108
14.1. The MAIN menu ............................... 145
14.2. The STATE SPACE menu ........................ 151
14.3. The DIMENSIONS command ...................... 152
14.4. The INTERNAL POSITIONS command .............. 153
14.5. The edit screen of the PARAMETERS menu ........... 156
15.1. The EDIT DATA menu .......................... 161
15.2. The edit mode of the EDIT DATA menu .............. 162
15.3. The CONVERT menu ........................... 165
15.4. Example of the BROWSE command ................. 167
15.5. Example of the EDIT INPUT FILE command .......... 168
17.1. Three types of scenario setting ..................... 184
17.2. The SCENARIOS menu .......................... 189
19.1. The TABLES menu ............................. 200
19.2. The EDIT INPUT FILE command .................. 200
19.3. Example 1 of the EXPERIENCE TABLES command ..... 209
19.4. Example 2 of the EXPERIENCE TABLES command ..... 209
21.1. The MAIN menu of the SOCPROF program ........... 222
21.2. The EDIT PARAMETERS command ................ 224
21.3. The SHOW PROFILES menu ...................... 224
21.4. The EDIT TABLE command of the EDIT PROFILES menu 226
21.5. The PROFILE command of the TIME SERIES menu ..... 228
PART I
INTRODUCTION
1. AIMS AND SCOPE
1.1 Introduction
Household projection models developed in demography over the past few
decades are predominantly of the headship rate type. This modelling strategy
rests on the principle of comparative statics. The development of households
by number and type is described by the development of number and type of
their heads, being household or family "markers" (Brass, 1983), on the basis
of analysing trends in proportions of individuals who occupy the position of
household or family head within some broader defined population categories.
Analysis as well as projection rely on data referring to the situation at specific
points in time. Thus, the headship rate method describes the results of dynamic
processes between the time points in terms of changing headship rates, these
dynamic processes themselves remaining a black box. Like the labour force
participation rate in labour force studies, the headship rate is not a rate in the
demographic ("occurrence-exposure") sense, and its analytical use reflects a
focus on changes in stocks, rather than a focus on flows.
In spite of this disadvantage, the headship rate method is often used, and it owes
much of its popularity to two factors. The method is easy to apply and its data
demands are modest. Also, the method has been refined gradually, for instance,
to take into account aspects of household members (Linke, 1988). However, the
static nature of the headship rate method and its extensions have led to a
number of attempts to develop dynamic household models, as it is the dynamic
processes of household formation and dissolution that really cause changes in
the cross-sectionally observed stock numbers of households, and not the other
way around.
Early attempts to construct dynamic household models were largely of an ad
hoc nature, as we will see in the model review in chapter 2. Data availability
and certain practical solutions to modelling problems were largely responsible
for this ad hoc situation. But in recent years, better data on household dynamics
became available for some countries, and, at the same time, a coherent and very
general methodology for the modelling and projection of dynamic demographic
aspects of a population was developed: multidimensional demography, mainly
due to Rogers, Ledent, and Willekens. It is the purpose of this book to describe
4
the construction of a very general dynamic household model, based on the
insights of multidimensional demography, and to apply that model to trace the
present and the future household situation in the Netherlands. What are the
consequences of increasing rates of divorce for the number of one-parent
families? How many additional one-person households would there be if young
adults were to leave the parental home at a lower age? These and similar
questions may be answered by using a dynamic household model, and by
running that model on the basis of various "scenarios" regarding its exogenous
variables. A static model would provide little insight in these matters.
Changes in household structures may have profound consequences for several
other aspects of society. For example, consumption, housing, labour force
participation, commuting, and tax revenues are affected by household dynamics.
In the present study, the consequences for social security expenditures are
investigated. Some of these expenditures are very sensitive to demographic
changes, including shifts in household structures. The application of the
household model to social security enables the investigation of the possible bias
in studies into the link between demography and social security which do not
take account of household dynamics.
1.2 Problem formulation
This book reports the findings of the research project "The impact of changing
household structures on future social security expenditure in the Netherlands",
which was carried out at the Netherlands Interdisciplinary Demographic Institute
(NIDI) with financial support from the Netherlands Ministry of Social Affairs
and Employment. The problem formulation of this project is as follows.
What is the impact of demographic developments, in particular shifts in
the age structure and household composition, on future social security
expenditure in the Netherlands?
The problem was investigated in a number of steps:
1. the construction of a dynamic model for the projection of the population
broken down by age, sex, and household position;
2. the collection of data on the quantitative aspects of household formation and
dissolution;
3. the collection of data for a number of demographically relevant types of
social security expenditures, broken down by age, sex, and household
position ("social security user profiles");
4. the formulation of various scenarios for possible demographic futures of the
Netherlands’ population;
5
5. the computation of a projection of the population of the Netherlands by age,
sex, and household position;
6. the projection of social security expenditures selected under item 3 on the
basis of the future demographic estimates computed under item 5.
Many of the sections and chapters which follow are based on a research report
written in Dutch (Van Imhoff and Keilman, 1990a). However, in the present
book the emphasis is more on methodology, and less on application and
numerical results, than in the Dutch report. For instance, in the research report
three types of social security were investigated: public old age pensions, social
welfare, and child allowance. The latter is not discussed in the present book,
because it is insensitive to changes in household structure (although the size and
the age structure of the population does have an impact on child allowance).
Moreover, the present book contains an extensive presentation of the method-
ology that was used to answer the research questions, and a major part of the
book contains a user guide for the computer program which was developed
within the project.
1.3 LIPRO
Within the project, a flexible model was developed for multidimensional
demographic projection, called LIPRO ("LIfestyle PROjections"). The LIPRO
model contains a number of methodological innovations. In part II of this book
we show how existing constant intensities models have been extended to include
entries (for instance, due to immigration or childbearing). Moreover, LIPRO
contains a very general algorithm to deal effectively with the problems that arise
because the behaviour of various individuals belonging to the same household
is interrelated. In traditional demography this is known as the two-sex problem:
the number of males who marry in a certain period should equal the number
of marrying females. In LIPRO the problem has been reformulated in a much
more general manner as a so-called consistency problem. The solution we
propose can handle the relation between two or more adults, and that between
children and parents, under processes of household formation and household
dissolution.
The computer program which was written in the context of the project is called
LIPRO 2.0 - it is included on floppy disk in this book. A harddisk and a
memory of 640 Kb are required to run LIPRO 2.0. In addition, the enclosed
standard version of LIPRO requires the presence of a mathematical co-
processor. Other versions than the standard one are also available (see chapter
13).
6
1.4 Outline of this book
This book consists of four major parts: Introduction, Theoretical issues, Applica-
tion, and LIPRO user’s guide. Chapter 2 concludes the introductory part of the
book. In that chapter existing household projection models are reviewed, and
we formulate major developments that can be traced in the literature on house-
hold modelling. Issues concerning the concept and the definition of a household
are also addressed.
Part II contains a comprehensive treatment of the theoretical issues relevant to
understanding how the demographic projection program LIPRO works. A
general characterization of multidimensional projection models is given in
chapter 3. Chapter 4 contains a description of the exponential (constant
intensities) model which has its roots in statistics and the theory of Markov
processes, and the linear model which was developed within demography.
LIPRO 2.0 has an option to choose between these two versions of the multi-
dimensional projection model. The consistency problem is formulated in very
general terms in chapter 5, and we present an algorithm to solve it. Chapter 6
discusses some issues in multidimensional life table analysis. Multidimensional
demographic models require a multitude of input parameters. Life tables provide
the possibility to calculate summary indicators for these parameters, which in
turn may be used effectively when scenarios for the input parameters of the
projection model are formulated.
In part III we apply the general model sketched in part II to the future
household dynamics in the Netherlands, and we trace its consequences for some
social security expenditures. First, in chapter 7 a specification is given of the
various household positions an individual may occupy during his or her life.
Although the emphasis in the current book is on the individual person and the
household events this person experiences, the specification of household positi-
ons also enables us to identify types of households which correspond with
individual household positions. A considerable part of chapter 7 is devoted to
the identification of household events and consistency relations. The estimation
of the input parameters on the basis of data from various sources is treated in
detail in chapter 8. The larger part of the input rates was estimated using
retrospective information on current and past household status of the members
of some 47,000 private households in 1985 in the Netherlands. But in spite of
the detailed and massive nature of this survey, some approximating assumptions
had to be made. Therefore, the input parameters were constrained in such a way
that LIPRO produced demographic results which correspond to official demo-
graphic statistics for the years 1986-1990. Chapter 9 presents five demographic
scenarios which were used for the projections. A demographic scenario describes
a possible trajectory of future demographic development. Results of the house-
hold projections are given in chapter 10. Here it is demonstrated how the five
7
scenarios differ with respect to projected household composition of the populati-
on of the Netherlands until the year 2050. An enormous growth in the number
of individuals living in a one-person household is persistent in all scenarios.
Chapter 10 also discusses the effects of alternative model specifications (expo-
nential versus linear model, with and without consistency) for the demographic
results. Finally, the link between social security and demographic developments
is taken up in chapter 11. It contains a review of the literature on the interrelati-
on between social security and demography, a discussion of the estimation of
social security user profiles for public old age pensions and social welfare, and
a brief presentation of the illustrative social security projections. Our findings
suggest a much steeper increase in future expenditures for old age pensions in
the Netherlands than previous studies did, mainly because we find larger
numbers of elderly persons living alone (who receive a higher pension than
elderly persons living with a partner). Chapter 11 also discusses the usefulness
of incorporating household position in the demographic model, in addition to
age and sex. Chapter 12 contains a summary of parts I to III of the book, as
well as some questions that still require research.
Part IV of the book constitutes a complete user’s guide to LIPRO 2.0. Although
it can be read independently from parts I to III, the reader is assumed to have
a basic understanding of the theoretical issues presented in part II. The various
program features are discussed using examples from the application of the
LIPRO model presented in part III.
The introductory chapter 13 includes a discussion of hardware requirements,
program installation, the use of menus, issuing commands, the LIPRO editor,
and the various types of program output. Chapter 14 ("Getting started") introdu-
ces the reader to LIPRO’s main menu. In chapter 15 it is discussed how demo-
graphic input data (intensities, occurrence-exposure rates, initial population) are
entered into the program. The implementation of the consistency algorithm and
the formulation of scenarios is taken up in chapters 16 and 17, while chapter
18 explains how to carry out a demographic projection with LIPRO. Chapters
19 and 20 contain a discussion of various issues, including the presentation of
projection results, the analysis of multidimensional life tables, and the exporting
of data for use by other programs. Finally, chapter 21 introduces the reader to
a program which links results of demographic projections to a user profile.
Social security user profiles are used as an example here, but other applications
(e.g. consumption or tax revenues) are possible too.
8
2. HOUSEHOLD MODELS:
A SURVEY
2.1 Concepts
The household is one of the many operationalizations of the concept of a living
arrangement. Other examples are marital status and family type. The trichotomy
marital status, family type, and household type runs parallel to Ryder’s
distinction between the conjugal dimension, the cosanguineal dimension and the
co-residence dimension of family demography (see Ryder, 1985, 1987). In this
order, these alternative operationalizations describe living arrangements ranging
from a less to a more complex type of structure. First, the conjugal and the
marital status perspective explore the formation and dissolution of marital
unions. Second, the cosanguineal and the family relationship explore links
between parents and children. Kinship studies can be placed within this
framework too. Finally, we consider the household, that is a group of
individuals, familial and non-familial, which is at least identified by a
co-residence criterion (and possibly by other criteria as well, cf. below). A more
formal definition will be given later in this chapter.
Households are the most complex type of primary units, embracing all the
aspects of the less complex definitions of the above classification. A married
couple with their children and living as an isolated co-resident group can be
viewed as one specific household type. Other household types, for instance,
isolated co-resident conjugal units (childless couples), may also be included in
one or more of the three operationalizations of living arrangements.
In a certain sense, kin and household are each other’s opposite. A household
is a co-resident group regardless of cosanguineal or affinal ties; by kin we refer
to a group of relatives regardless of their residence (see De Vos and Palloni,
1989, p. 175). The latter authors note that the household and the family had
been often equated, until the extent of cohabitation was sufficient to force many
researchers to make a distinction.
10
2.2 Definitions
The United Nations recommend three possibilities for the definition of a house-
hold in population and housing censuses.
- The "housekeeping-unit" concept: a private household is either (i) a one-
person household, that is, a person who lives alone in a separate housing unit
or who occupies, as a lodger, a separate room of a housing unit but does not
join with any of the other occupants of the housing unit to form part of a
multi-person household; or (ii) a multi-person household, that is, a group of
two or more persons who combine to occupy a whole or part of a housing
unit and who provide themselves with food and possibly other necessities
of life.
- The "household-dwelling" concept: a private household is the aggregate
number of persons occupying a housing unit. This is equivalent to the co-
resident group described above.
- Institutional households and other communal relationships: an institutional
household is comprised of groups of persons living together, who usually
share their meals and are bound by a common objective and are generally
subject to common rules, for example, groups of persons living together in
dormitories of schools and universities, hospitals, old age homes and other
welfare institutions, religious institutions, prisons, military camps, and so on.
The housekeeping-unit definition of a household is applied in this book, the
main reason being the fact that our household data, provided by the Netherlands
Central Bureau of Statistics (see chapter 8), follow this convention. As the
household-dwelling definition of a household does not involve any housekeeping
criteria, the latter definition is less restrictive than the housekeeping-unit definiti-
on. Institutional households will not be considered in this book.
An inventory carried out by the Economic Commission for Europe in the early
1980s indicated that about two-thirds of the ECE countries employ the house-
keeping definition in their data collection. However, none of the definitions
given above is without problems. For instance, difficulties arise for persons with
more than one dwelling (Linke, 1988; Schwarz, 1988), for persons who are not
related to the family (subtenants, service personnel), and so on. Schmid (1988)
argues that the scoring of non-related persons present in the household has a
considerable effect on the enumeration of people in households.
Household models not only provide insight into the development of numbers
of households of various types, but also of numbers of individuals classified by
household position. It is very important to make a proper distinction between
these two levels of aggregation. Individuals who belong to the same household
may occupy different household positions. For example, in the LIPRO model
11
(which is to be discussed in chapter 7), one of the household types is "married
couple with children and possibly with other co-resident adults" (household type
MAR+ in section 7.2). A person who belongs to this household may occupy
either of the following three household positions: (i) married adult living with
spouse; (ii) child living with both parents; (iii) non-family related adult. LIPRO
deals primarily with individuals and the household events they experience.
Numbers of households are derived from projected numbers of persons in the
various household positions.
2.3 A typology of household models
Given the research questions to be addressed in this book, a model for the
projection of the population broken down by age, sex, and household position
is required for the demographic part of this study. Models of this kind may be
classified according to two dimensions: the static/dynamic dichotomy on the one
hand, and a dimension related to the link between demographic variables and
non-demographic variables on the other hand. In the latter dimension we
distinguish between purely demographic models, and models which include both
demographic and non-demographic (most often socioeconomic) variables. This
classification is broader than that of Bongaarts (1983, p. 32), who considers
purely demographic models only.
In the research project described in this book the focus is on purely
demographic dynamic household models. The advantage of dynamic household
models relative to static models will be discussed in sections 2.4 and 2.5. The
emphasis on purely demographic models does not imply that we regard the
impact of economic, social, psychological, legal, and cultural processes on
household structures to be of secondary importance. However, within household
demography, little is known of formal demographic household events occurring
to individuals in changing household structures. Therefore, we think that
structural issues in household modelling have to be resolved before any
substantive relationships can be adequately studied. Formal relationships be-
tween demographic entities have to be analysed as a prelude to the examination
of causal issues. A thorough analysis of patterns, disaggregated by demographic
and household characteristics, may itself bring forth explanatory variables which
should be considered. The issue of modelling explanatory factors in the context
of household projection models will be taken up in chapter 12.
Examples of household models containing non-demographic variables which
"explain" household variables are the Cornell model in the USA (Caldwell et
al., 1979), the IMPACT model in Australia (Sams and Williams, 1982), the
model developed at the Policy Studies Institute in Great Britain (Ermisch, 1983),
the model of the Sonderforschungsbereich 3 in Frankfurt, Germany (Galler,
1988), the UPDATE model for households in small areas in England (Duley
12
et al., 1988), and the NEDYMAS model constructed by Nelissen for the Nether-
lands (Nelissen and Vossen, 1989). These models are much less developed in
terms of household structures than the LIPRO model to be described in the
following chapters.
The review contained in sections 2.4 and 2.5 is restricted to operational
household models. Marital status models and family models will not be
discussed here. Family models were constructed by Bongaarts (1981, 1987) for
general applications, by Rallu (1985) for France, and by Kuijsten (1986, 1988)
for the Netherlands. A review of marital status projection models was given by
Keilman (1985b). A number of family and household models have been propo-
sed in the literature, but no operational version of them exists, to the best of our
knowledge. Examples are the models of Webber (1983), Muhsam (1985), Ledent
et al. (1986), and Murphy (1986).
2.4 Static household models
Headship rate models are among the oldest models to project households. They
are the typical representatives of the class of static, purely demographic, house-
hold models. The idea is to extrapolate proportions of household heads in
population categories defined by a certain combination of age, sex, and possibly
marital status. An independent projection of the population by age and sex (and
marital status) facilitates a projection of the future number of households, broken
down by demographic characteristics of the head of the household.
The prototype of this method was published in 1938 by the United States
National Resources Planning Committee. It became internationally accepted after
the 1950 round of censuses. Kono (1987) and Linke (1988) discuss the headship
rate model in detail, including its extensions, such as the household membership
rate model. The household composition model proposed by Akkerman (1980)
adds the age of the household members as an extra dimension to the headship
rate model. (However, it should be noted that Akkerman’s model is not widely
used, possibly due to numerical problems that may arise. See, for instance,
Keilman and Van Dam, 1987, p. 26).
The headship rate method for the projection of households has several charac-
teristic features: it is a simple and practical method for which the necessary data
are frequently available. Projections can be updated easily. However, its static
character is often judged a disadvantage: the model is unable to deal with the
dynamics of household formation and dissolution. The headship rate method
reflects a focus on changes in household structures at subsequent points in time
(comparative statics), whereas truly dynamic models simulate household events
over a certain period. Like the labour force participation rate in labour market
studies, the headship rate is not a rate in the demographic ("occurrence-
13
exposure") sense. It is a proportion, while dynamic models are constructed on
the basis of transition probabilities or intensities (see chapter 4).
2.5 Dynamic household models
Dynamic household models deal, in one way or another, with processes of
household formation and household dissolution. They can be used to answer
questions such as: "If children would leave the parental home two years later
than is the case presently, how would this influence household structures?" or
"How would a 25 percent increase in divorce rates affect the number of one-
parent families?". Traditional static models say little about such matters.
The history of dynamic household models is much younger than that of the
headship rate models, as the first dynamic household models were developed
after the mid-1970s. Three dynamic household models will be discussed here
briefly to illustrate the most important recent developments in household
modelling: the model developed by Möller for the Federal Republic of Germany;
the Swedish model constructed by Hårsman, Snickars, Holmberg, and others;
and finally the LIPRO model built at the NIDI, which will be presented in detail
in chapters 7 to 12. The following draws largely on an earlier review (Keilman,
1988).
Möller (1979, 1982) describes a household projection model which was
developed at the "Institut für Angewandte Systemforschung und Prognose" (ISP)
in Hanover, Federal Republic of Germany. The model was applied in a study
of future consumption patterns in the FRG. It starts from results of a population
projection model which simulates future population structures by age and sex.
Then the model further breaks down the population into dependent children,
married adults, and unmarried adults. Using an assumption on headship rates,
the number of households is calculated on the basis of male adults and unmar-
ried female adults. This means that adult males and unmarried adult females are
always considered as the head of a household. Given the number of households,
the model finally determines their distribution by number of children present
using parity-specific fertility curves by age of the mother, and "home-leaving"
curves by age of the child.
The dynamic character of the ISP model is very limited as it relies on only one
household event: leaving home by young adults. Other household features are
introduced by means of traditional ratio and headship rate methods.
During the late 1970s and early 1980s, a dynamic household model was con-
structed by Hårsman, Snickars, Holmberg and others in Sweden. To date,
several versions and applications of the model exist and the original ideas have
been updated many times. We shall discuss the model version described in a
comprehensive report by Dellgran et al. (1984), as well as in Holmberg (1987).
14
Bugge (1984) applies the method to Norwegian data and Zelle (1982) uses a
comparable approach for Austria. Hårsman and Snickars (1983) provide a useful
summary of the model.
The Swedish model follows individuals, classified by household status, over a
discrete time interval. Its key instrument is a matrix of probabilities describing
transitions in household statuses, these transitions being experienced by individu-
als between the beginning and the end of the time interval. Household status
is defined as household size (1, 2, , 5+) and whether or not a household
contains dependent children. This yields a total of nine household types.
The model first adjusts an originally observed transition matrix to a number of
external and internal constraints. After the adjustment of the transition matrix,
the population is projected forward in time over one interval. This procedure
is repeated for the whole projection period.
An example of the constraints which the transition matrix has to satisfy is that
at each point in time the number of dependent children in two-person
households must be equal to the number of adults in that category. Furthermore,
the adjusted transition matrix should resemble the original matrix as closely as
possible. This is achieved by an optimization routine.
To facilitate a comparison of household models, the most important aspects of
the LIPRO model are summarized here. More detailed descriptions are given
in later chapters.
Many of the efforts to construct LIPRO ran parallel to the development of other
dynamic household models in the Netherlands. Hooimeijer and Linde (1988)
provide a useful summary. In the LIPRO application which is presented in
chapter 7, the population in private households is broken down according to age,
sex, and household position. For the latter characteristic, 11 positions are used:
three for children, four for persons who live with a partner, one for persons who
live alone (one-person households), one for heads of a one-parent family, and
two for other household positions (see section 7.2). These 11 household positi-
ons identify 69 possible household events that individuals may experience as
they move from one household position to another. Besides household events,
the model describes birth, death, emigration, and immigration (section 7.3). An
event is expressed in LIPRO in terms of an occurrence-exposure rate (for each
relevant combination of age and sex), representing the intensity with which the
event occurs to an individual.
An important part of LIPRO is the so-called consistency algorithm (section 7.4).
The purpose of this algorithm is to guarantee consistency in the numbers of
events which members of the same household experience. For instance, the
number of males who marry during a certain period must equal the number of
females who marry, and similarly for new consensual unions. And when a
married man with children dies, both his wife and his children must be moved
to the position one-parent family (head and child, respectively).
15
2.6 Comparison
A number of major developments can be traced in the literature on household
modelling.
From static to dynamic models
More and more emphasis is given to the development of dynamic household
models. Changes in household structures cannot be studied adequately with
static models. Models of the latter type are widely used at present, particularly
because of their simplicity. But innovations in household modelling take place
in the area of dynamic models.
From households to individuals
The interest is shifted from a description of numbers of households (of various
types) to a description of (events occurring to) individuals (broken down by
household position). Headship rate models represent the first tradition, the ISP
model takes an intermediate position, while the Swedish model and LIPRO
emphasize the individual. Related to this development is that the notion of "head
of household" gradually loses significance.
More states and more events
Within the class of dynamic household models, an increasing number of house-
hold positions and household events (or transitions) can be noted. The only
household event in the ISP model is leaving the parental home. The Swedish
model has nine household positions and it deals implicitly with 41 events.
LIPRO’s state space contains 11 positions and the model describes 69 events.
Growing data requirements
Data requirements increase with growing model complexity. Headship rate
models only require proportions of head of households within population classes
(stock data). At the other end of the spectrum we find LIPRO, for which, in the
present application, information is necessary on the 69 types of events which
individuals may experience (flow data). The Swedish model and the ISP model
have lower data demands than LIPRO. The high demands of modern dynamic
household models are an important factor in the slow development and the
relatively scarce application of these models.
PART II
THEORETICAL
ISSUES
3. A CHARACTERIZATION
OF MULTIDIMENSIONAL
PROJECTION MODELS
Dynamic demographic projection models describe the development over time
of a population. A population consists of a number of individuals (or alternative
units of analysis), broken down by certain demographic characteristics (e.g. age,
sex, marital status, geographic location, household status, and the like). This
multidimensional breakdown of the population defines a state space. A vector
in the state space is called a state vector; its elements consist of numbers of
individuals at one point in time, broken down by demographic characteristics.
Theory and applications of multidimensional models (sometimes called multi-
state models) have appeared in the demographic literature since the mid-1970s.
Most applications have focussed on multiregional models (e.g. Rogers, 1975;
Willekens and Drewe, 1984; Keilman, 1985a; Rogers and Willekens, 1986;
Ledent and Rees, 1986) and marital status models (e.g. Schoen and Nelson,
1974; Krishnamoorthy, 1979; Schoen and Land, 1979; Willekens et al., 1982;
Gill and Keilman, 1990). Other applications include working life tables and
fertility by parity. For additional references, see Gill and Keilman (1990,
p. 124). The application to household dynamics is relatively new (cf. chapter
2). Most of the papers referred to above concern life tables rather than
projection models. In the present book, the focus is on projection models (cf.
the literature review in chapter 2). Hoem and Funck Jensen (1982) give a
comprehensive review of the general multidimensional life table and projection
model, as well as its Markov process formulation.
The research that gave rise to the construction of the LIPRO program belongs
to the research project "The impact of changing living arrangements on social
security expenditures in the Netherlands" which has been carried out at the NIDI
with financial support from the Netherlands Ministry of Social Affairs and
Employment. In this project a multidimensional household projection model was
developed; the household projections generated by this model are subsequently
used to trace future expenditures on social security.
20
The development over time of the population can be described in terms of
events: immediate jumps from one cell in the state vector to another. Examples
of events are: marriage, divorce, leaving the parental home, internal migration.
It is possible for an individual to experience several events within one single
projection interval; of course, the probability of multiple events increases with
the length of the projection interval.
The population under consideration is not closed: some individuals leave the
population (death, emigration), others enter the population (birth, immigration).
Such jumps into, or out of, the population are also termed events. In order to
distinguish this latter type of events from the type discussed in the previous
paragraph, jumps across the boundaries of the population will be termed external
events as opposed to internal events. External events comprise exits and entries.
Exits can be subdivided according to destination, entries according to origin.
A different classification of events is by endogenous and exogenous events. An
endogenous event is an event that is "explained" within the demographic model
itself. In purely demographic models, the occurrence of events is explained by,
first, the number of individuals occupying a certain state during a certain
interval of time, and, second, the probability that a given individual will expe-
rience some event. Consequently, events are endogenous whenever the number
of events is dependent on the distribution of individuals within the population
over the various characteristics. All internal events and all exits are endogenous.
Entries are in part endogenous (births), in part exogenous (immigration).
These various types of events have been illustrated in Table 3.1.
In LIPRO, the state space is defined by the number of categories in each
dimension of the state space, as well as by the labels attached to these
categories. These variables are stored together in the definition file. The follo-
wing variables are involved:
NSEX The number of sexes. Its possible values are 1 (no distinction
between males and females) and 2.
NAGE The number of age groups
NWAGE The width of the age groups
NPOS The number of internal positions (or states, for short)
NOUT The number of destinations for exits (external positions). Its two
possible values are 1 (mortality) and 2 (mortality and emigration).
NIN The number of origins for exogenous entries. Its two possible values
are 0 (no immigration) and 1 (immigration).
LSEX The labels for the sexes. If NSEX=2, the first sex corresponds to
females, the second to males.
LAGE The labels for the age groups. Group 1 corresponds to the age group
born during the projection interval (endogenous entries). Group 2
refers to the youngest age group present at the beginning of the
projection (or observation) interval.
21
LPOS The labels for the internal positions
LOUT The labels for the destinations for exits (external positions)
LIN The labels for the origins for exogenous entrants
Table 3.1. Classification of events
position after the event
internal positions external positions
#1 #2 #3 .. etc. dead rest world
p
o
s
i
t
i
o
n
b
e
f
o
r
e
e
v
e
n
t
internal
positions #1
#2
#3
.
etc.
internal events exits
external
positions
not yet
born endogenous entries irrelevant
rest of
the world exogenous entries irrelevant
4. THE EXPONENTIAL AND
THE LINEAR MODEL1
As stated in chapter 3, in purely demographic models, the occurrence of events
is explained by, first, the number of individuals occupying a certain cell in the
state vector during a certain interval of time, and, second, the probability that
a given individual will experience some event.
The relationship between the number of individuals in a certain cell, on the one
hand, and the number of events experienced by these individuals, on the other,
can be described by means of a so-called Markov equation. Here one implicitly
assumes that the probability that an individual in position i will, within an
infinitesimal interval of time, experience a direct jump into position j, is equal
to a constant times the length of the infinitesimal time interval. Formally:
lim
dt0
Pr[I(t dt) jI(t) i]
dt mij(t)
Here mij(t) is a constant dependent on time t, and on i and j (the position before
and after the event, respectively). This constant is termed the instantaneous
intensity or rate for the jump from i to j. When mij(t)=m
ij for all t within the
observation interval, the resulting model is known as the exponential model or
the constant intensities model (Gill and Keilman, 1990).
The data that underlie the calculations for one single projection interval are the
constant intensities mij, the distribution of the population across positions at the
start of the projection interval, and the numbers of the exogenous entries. In
section 4.2 we will formulate the full exponential model in matrix notation. This
formulation generalizes earlier work by Gill (1986) on Markov models for
closed populations to include the case of open populations.
The exponential model gives rise to quite complex expressions, requiring
iterative evaluation techniques. For computational simplicity, it is often assumed
1Some of the material covered in this chapter can also be found in Van
Imhoff (1990a).
24
that all events are uniformly distributed over the projection interval. This
so-called linear integration hypothesis (Hoem and Funck Jensen, 1982) allows
the projection to be carried out in one single computation step. The correspon-
ding model is known as the linear model.
Although its computational simplicity is obviously a great advantage, the
drawbacks of the linear model should not be underestimated. First, the assump-
tion of uniformly distributed events can only be justified on the grounds of
computational convenience; it is not based on any statistical theory, like the
theory of Markov processes underlies the exponential model. Second, and even
worse, in some cases the linear model may lead to impossible results, viz.
negative numbers of individuals in some cells of the cross-classification table
(Gill and Keilman, 1990). The exponential model does not have this unpleasant
property.
The organization of this chapter is as follows. Section 4.1 gives some intro-
ductory remarks and spells out the notation. In section 4.2 the exponential model
is derived, and the linear model is formulated in section 4.3. In section 4.4 we
briefly indicate how the reverse calculations are performed, i.e. computing rates
from events, rather than events from rates. The final section contains some
technical information on the way in which the computer program performs
special matrix operations.
4.1 Preliminaries
Throughout this chapter the following symbols will be used:
Ithe identity matrix.
ιarow vector consisting of only ones.
T operator for transposition of a vector or matrix.
Diag[v] operator for the formation of a diagonal matrix with the elements
of vector von the diagonal.
x index for age group. The index takes values from 1 to NAGE.
x=1 refers to the age group of individuals born during the projec-
tion interval.
s index for sex. 1=female, 2=male.
t calendar time at the start of the projection interval.
h the length of the projection interval, assumed to be equal to the
width of the age groups.
NPOS the number of additional demographic positions considered (e.g.
regions, household positions, marital statuses, etc.).
NOUT the number of destinations for exits. In most applications NOUT
will be equal to 2 (death, emigration).
25
NIN the number of origins for exogenous entries. In most applications
NIN will be equal to 1 (immigration).
(s,x,t) x=1 NAGE, s=1,NSEX. A row vector with the number of
individuals of sex s and age x at time t, ordered by position. The
vector has dimension (1 by NPOS).
Mi(s,x,t) x=1 NAGE, s=1,NSEX. A (NPOS by NPOS) matrix with
intensities for internal events. The element (a,b) is the intensity
of the jump (event) from position a to position b. The diagonal
elements of the matrix are equal to 0 (to retain one’s original
position is not an event).
Me(s,x,t) x=1 NAGE, s=1,NSEX. A (NPOS by NOUT) matrix with
intensities for exits.
Mb(s,x,t) x=1 NAGE, s=1,NSEX. A (NPOS by NPOS) matrix with
birth intensities (endogenous entries). The element (a,b) is the
intensity of the event that a woman in position a gives birth to a
child of sex s that will enter position b.
M(s,x,t) x=1 NAGE, s=1,NSEX. The (NPOS by NPOS) matrix
M(s,x,t) is a transformation of the matrices Mi(s,x,t) and
Me(s,x,t), defined by
M(s,x,t) = Mi(s,x,t) - Diag [ Mi(s,x,t) ιT+Me(s,x,t) ιT]
The matrix M(s,x,t) differs from matrix Mi(s,x,t) in that its diago-
nal does not contain zeros, but the negative of the intensity of all
events (both internal events and exits) that result in a jump out
of the corresponding position.
O(s,x,t;h) x=1 NAGE, s=1,NSEX. A (NIN by NPOS) matrix with
numbers of exogenous entries, ordered by origin and by position
of destination. The age group x refers to the age that the entrants
had at time t, not age at time of entry.
In order to be able to formulate the differential equations describing the
development of the population over time, an assumption is needed on the
distribution over time of the events of entry during the projection interval. We
will assume here that this distribution is uniform.
For the exogenous entries (immigration), this assumption can be justified by
pointing out that the number of potential immigrants is very large compared to
the intensity of immigration: thus the event "immigration" has a negligible effect
on the number of individuals exposed to the risk of immigration. Hence, the
assumption of a uniform distribution for immigration is, in practice, equivalent
to the assumption of a constant immigration intensity.
For the number of births, the assumption of a uniform distribution can hardly
be justified on theoretical grounds. Given the solution to the differential
equations for the female population of childbearing age, an exact distribution
26
of births can be obtained, at least in principle (all on the maintained assumption
of the Markov specification). However, these exact solutions are mathematically
unmanageable. Therefore, the uniformity assumption may be considered an
approximation of the exact distribution of births.
Finally, note that the model is formulated in terms of row vectors. The
traditional notation in demographic texts based on column vectors (e.g. Keyfitz,
1968) would lead to unnecessarily complicated expressions (Gill and Keilman,
1990).
4.2 Formulation of the exponential model
4.2.1. Formulas for age groupsx>1
For age groups x, x>1, the combination of the Markov assumption and the
assumption of uniformly distributed immigration gives rise to the following
differential equation:
(s,x+τ,t+τ) = (s,x+τ,t+τ)M(s,x,t) + (1/h) ιO(s,x,t;h) (4.1)
d
dτ
This is a non-homogeneous matrix differential equation. The homogeneous part
is a straightforward transformation of the well-known Kolmogorov forward
differential equation, multiplying both sides of the equation by (s,x,t). The
non-homogeneous part is added in order to account for exogenous entries.
The general solution to the homogeneous part of the equation (4.1) is:
(s,x+τ,t+τ)=CeM(s,x,t) τ(4.2)
The exponential of a square matrix is defined in terms of its Taylor power
series. Cis a vector of integration constants.
A particular solution to the non-homogeneous differential equation is the
following constant solution:
(s,x+τ,t+τ) = - (1/h) ιO(s,x,t;h) M-1(s,x,t) (4.3)
For the moment we will assume that the inverse of Mexists; in section 4.2.3
we will return to this issue.
From the initial conditions that (s,x+τ,t+τ) for τ=0 should equal (s,x,t), the
values of the constants of integration Cfollow, giving the following general
solution:
(s,x+τ,t+τ) = (s,x,t) eM(s,x,t) τ+
27
+ (1/h) ιO(s,x,t;h) M-1(s,x,t) { eM(s,x,t) τ-I} (4.4)
so that, in particular
(s,x+h,t+h) = (s,x,t) eM(s,x,t) h +
+ (1/h) ιO(s,x,t;h) M-1(s,x,t) { eM(s,x,t) h -I} (4.5)
The vector of person years (or sojourn times) spent by all individuals in the
various positions is defined by:
L(s,x,t;h) = (s,x+τ,t+τ)dτ(4.6)
h
0
Substitution of (4.4) into (4.6) and using (4.5) yields:
L(s,x,t;h) = { (s,x+h,t+h) - (s,x,t) - ιO(s,x,t;h) } M-1(s,x,t) (4.7)
The number of endogenous events can now be computed from the vector of
person years and the intensity matrices. For the numbers of internal events,
exits, and births within the projection interval we have, respectively:
Ni(s,x,t;h) = Diag[L(s,x,t;h)] Mi(s,x,t) (4.8)
Ne(s,x,t;h) = Diag[L(s,x,t;h)] Me(s,x,t) (4.9)
Nb(s,x,t;h) = Diag[L(1,x,t;h)] Mb(s,x,t) (4.10)
Note that the latter equation uses person years for females only (s=1).
For the highest, open-ended age group, the population at the end of the
projection interval is obtained by combining the survivors of age groups
(NAGE-1) and NAGE.
4.2.2. Formulas for the youngest age group
For age group x=1, the number of entries during the projection interval equals
the sum of the total number of births and the number of immigrants born
between the start of the projection interval and the moment of immigration. The
total number of births, classified by position of the mother and position of the
baby, can be obtained from (4.10):
B(s,t;h) = Nb(s,x,t;h) (4.11)
NAGE
x 2
28
This leads to the following differential equation in matrix form:
(s,1+τ,t+τ) = (s,1+τ,t+τ)M(s,1,t) +
d
dτ
+ (1/h) ιO(s,1,t;h) + (1/h) ιB(s,t;h) (4.12)
The general solution can be found in a way analogous to solving (4.1), the
initial conditions now being that (s,1+τ,t+τ) for τ=0 should equal zero. The
solution is:
(s,1+τ,t+τ) = { (1/h) ιO(s,1,t;h) + (1/h) ιB(s,t;h) }
M-1(s,1,t) { eM(s,1,t) τ-I} (4.13)
so that
(s,1+h,t+h) = (s,2,t+h) = { (1/h) ιO(s,1,t;h) + (1/h) ιB(s,t;h) }
M-1(s,1,t) { eM(s,1,t) h -I} (4.14)
The vector of person years spent by all newly-born individuals in the various
positions is defined by:
L(s,1,t;h) = (s,1+τ,t+τ)dτ(4.15)
h
0
Substitution of (4.13) into (4.15) and using (4.14) yields:
L(s,1,t;h) = { (s,2,t+h) - ιO(s,1,t;h) - ιB(s,t;h) } M-1(s,1,t) (4.16)
The number of endogenous events can now be computed from the vector of
person years and the intensity matrices, as above (cf. (4.8)-(4.9)).
To conclude, the computation scheme for the newly-born individuals is
equivalent to that of the older age groups, provided that:
- the initial population (s,1,t) is set equal to zero;
- the number of births ιB(s,t;h) (endogenous entries) is added to the number
of immigrants ιO(s,1,t;h) (exogenous entries).
29
4.2.3. Formulas for a singular intensity matrix
Until now, we have assumed that the intensity matrix M is non-singular, so that
its inverse exists. This assumption cannot be maintained if there are so-called
absorbing states or absorbing subsets of states. An absorbing state is a state
from which no jumps are possible, i.e. the corresponding row of the M matrix
is zero. One might be inclined to think that such a situation is impossible, since
at least the event "death" could be experienced from any internal position.
(Remember that the state of being "dead" is not considered as an internal state
here, since exits due to mortality are explicitly permitted). However, if the data
survey from which the intensities are derived does not contain observations on
some positions for some age groups, then such zero rows occur for empirical
reasons. Thus, empirically (though not theoretically), states may be absorbing
for some age groups.
For this reason, LIPRO contains a procedure that performs the calculations for
the projection in the special case of a singular intensity matrix. Gill (1986)
offers a comprehensive computation scheme for an intensity matrix Mof any
rank. However, his formulas are only valid for closed Markov models, i.e. they
cannot be applied to the general case which includes events of entry and exit.
The LIPRO procedure generalizes Gill’s solution to include the case of
uniformly distributed entries into the population.
Initially, the computer program proceeds under the assumption of a non-singular
intensity matrix. The generalized Gill procedure is invoked only if the ordinary
procedure breaks down as a result of the singularity of M. In that case, a
message to this respect will be displayed on the screen.
In a nutshell, the Gill procedure consists of two major steps. First, the state
space is compressed by collapsing all states that belong to some absorbing
subset into one single state. When the vector of sojourn times for this absorbing
macro-state has been computed, the sojourn times for the individual component
states are calculated by working backwards. The details of the calculation
scheme, as well as the formal proof of its validity, are given in Appendix 4A.
4.3 Formulation of the linear model
In the linear model, the concept of an occurrence-exposure rate (o-e rate) plays
a key role. The o-e rate for an immediate jump (event) from state i to state j
is defined as the ratio of the number of jumps from i to j within a certain
interval of time, and the corresponding sojourn time spent in state i during that
interval. An o-e rate (notation mij(s,x,t;h)) is the discrete-time analogue of the
continuous-time jump intensity mij(s,x,t) used in the exponential model. Hence
it will be obvious that o-e rates may be arranged in matrices with formats
similar to their exponential counterparts. Within the context of the exponential
model, the o-e rate mij(s,x,τ;h) is equal to the intensity mij(s,x,τ) for all τ
between t and t+h.
30
We start with the relationship between events, sojourn times (for the moment
still unknown) and occurrence-exposure rates. This relationship is given in
equations (4.8)-(4.10).
Analogous to the o-e matrix M, we can define the matrix Nof immediate jumps
between positions:
N(s,x,t;h) = Ni(s,x,t;h) - Diag [ Ni(s,x,t;h) ιT+Ne(s,x,t;h) ιT] (4.17)
The diagonal of Ncontains the negative of the number of jumps out of the
corresponding internal position. It follows that
N(s,x,t;h) = Diag[L(s,x,t;h)] M(s,x,t;h) (4.18)
By definition we have, from the so-called accounting equation:
(s,x+h,t+h) = (s,x,t) + ιO(s,x,t;h) + ιN(s,x,t;h) (4.19)
For the youngest age group, the initial population (s,x,t) is zero and the vector
of immigrants ιO(s,x,t;h) includes the number of births.
By assumption, events and entries are uniformly distributed over the projection
interval. Then the vector of person years equals:
L(s,x,t;h) = ½h { (s,x,t) + (s,x+h,t+h) } (4.20)
Substitution of (4.19) and (4.18) into (4.20) yields:
L(s,x,t;h) = h (s,x,t) + ½h ιO(s,x,t;h) + ½h L(s,x,t;h) M(s,x,t;h)
(4.21)
From (4.21) it follows that:
L(s,x,t;h)={h (s,x,t) + ½h ιO(s,x,t;h) } { I- ½h M(s,x,t;h) }-1
(4.22)
Combining (4.19), (4.18) and (4.22) yields (s,x+h,t+h). It is easily seen that
this computation scheme is equivalent to solving (4.18)-(4.20) directly for
(s,x+h,t+h), leading to the well-known formula:
(s,x+h,t+h) = (s,x,t) { I+ ½h M(s,x,t;h) } { I- ½h M(s,x,t;h) }-1 +
+ιO(s,x,t;h) { I- ½h M(s,x,t;h) }-1 (4.23)
31
These expressions hold for both absorbing and non-absorbing states.
4.4 Computing rates from events
Sections 4.2 and 4.3 dealt with the computation of events and person years,
given the initial population and the intensities or the o-e rates.
There are two stages at which the reverse computation has to be made, i.e.
determining the rates and the person years, given observations on events and
initial population. These stages are:
a. estimation of rates from empirical observations on events;
b. computation of adjusted rates, given adjusted events. This stage occurs if
LIPRO’s consistency algorithm (see chapter 5) is invoked and adjusted rates
are desired.
In the case of the linear model, the reverse calculation can be made in one step.
Using (4.17), (4.19), and (4.20), the vector of person years L(s,x,t;h) can be
easily calculated from events and initial population. The matrices of rates then
follow immediately from equations (4.8) to (4.10).
In the case of the exponential model, an iterative computation scheme has to
be adopted. This scheme is due to Gill and Keilman (1990). The steps in this
computation scheme are the following:
1. take an initial guess for the person years L.
In LIPRO, the initial guess usually is the solution corresponding to the linear
model, calculated from (4.17) to (4.20). However, in stage b above
(calculation of adjusted rates from adjusted events), the user may specify
that the initial guess should be taken from the vector of person years corres-
ponding to the exponential model given the unadjusted rates. Experience has
shown that the "linear model initial guess" is generally faster, except in cases
where the differences between unadjusted and adjusted events are very small.
2. calculate the rates Mfrom the events Nand the current guess for the person
years L, by solving (4.8) to (4.10).
3. calculate an updated guess for Lby applying the exponential model to the
rates obtained in step 2.
4. calculate the norm of the change in vector L.
5. repeat steps 2-4 until convergence of the vector L.
In LIPRO, iterations stop when either the Euclidian norm calculated under
step 4 becomes less than a user-supplied convergence criterion, or steps 2-4
have been repeated a certain maximum number of times. In the latter case,
LIPRO will print a message if no convergence was achieved.
32
4.5 On the computation of exp[X] and inv[X]
The exponent exp[X] of a square matrix Xis computed by Taylor series
expansion:
exp[X]=I+X+ (1/2!) X2+ (1/3!) X3+
This infinite power series is truncated after a finite number of steps. More
specifically, the iterations halt if after n steps the norm of the last term in the
power series is less than a convergence criterion CRIT supplied by the user, i.e.
Norm [ (1/n)! Xn] < CRIT
The computation of the inverse matrix inv[X]orX-1 of a square matrix Xis
a special case of the more general problem of solving a system of linear equati-
ons:
aX=b
with matrix Xand vector bknown, and vector aunknown. LIPRO contains a
subroutine that solves this linear system by eliminating subsequent rows of the
matrix X(Gaussian elimination). Before actually eliminating a row, the
subroutine checks whether the pivotal element can be increased in absolute value
by interchange of columns (permutation of the matrix X, in order to minimize
rounding errors). If the pivotal element is equal to zero, then the matrix Xis
singular and its inverse is undefined. Because of rounding errors, the pivotal
element will hardly ever be exactly zero. Therefore, a parameter SMALL is
passed to the subroutine. The matrix Xis considered singular whenever a pivotal
element is less than SMALL in absolute value during the elimination process.
APPENDIX:
Calculating sojourn times
for a Markov process
with uniform entries and
intensity matrix of any rank
This Appendix gives the details of the generalized Gill procedure, referred to
in section 4.2.3, for calculating sojourn times in the exponential model when
the intensity matrix M is singular.
Consider a constant intensity Markov process for an open population. Given are
data on the following variables:
(0) a (1 by n) vector with the initial population, classified by initial state,
where n is the number of states;
Man (n by n) matrix with intensities;
ba (1 by n) vector with the number of immigrants per unit of time, classi-
fied by the state into which they immigrate. Immigrations (or births) are
assumed to be uniformly distributed over time;
h the length of the projection interval.
The population develops over time according to:
(τ)= (τ)M+b(4A.1)
d
dτ
It is assumed that the set of all n states is absorbing. In other words, individuals
do not leave the population. This assumption sounds more restrictive then it
really is. For instance, for events like "death" or "emigration", one can
(temporarily) define internal states as "dead" or "emigrated" and rearrange the
34
intensity matrix Min such a way that the absorption property is achieved. Then,
since the population is closed to exits, the intensity matrix Msatisfies:
MιT= 0 (4A.2)
where ιis a row vector consisting of ones only, and T denotes transposition.
(4A.1) and (4A.2) together define a Markov process with constant intensity
matrix and uniformly distributed entries, i.e. the exponential model of section
4.2. The cumbersome expression "constant intensity Markov process with
uniformly distributed entries" will be abbreviated to "CIMUE process"
throughout this appendix.
From (4A.1) and (4A.2) it follows that the total population (aggregated over all
states), denoted by N, is given by:
N(τ)=N(0)+τbιT(4A.3)
Integration of (4A.1) yields:
(τ) = (0) exp[Mτ]+b{Iτ+(Mτ2/2!) + (M2τ3/3!) + } (4A.4)
where the exponent of a square matrix is defined in terms of its Taylor power
series; the fact that (4A.4) indeed solves (4A.1) can be checked by simple
differentiation. In particular:
(h) = (0) exp[Mh]+b{Ih+(Mh2/2!) + (M2h3/3!) + } (4A.5)
The vector of sojourn times until time t is defined as:
L(t) = (τ)dτ(4A.6)
t
0
From (4A.4) to (4A.6) we obtain:
L(h) M= (0) exp[Mh] - (0) + b{(Mh2/2!) + (M2h3/3!) + } =
= (h) - (0) - bh (4A.7)
And from (4A.6) and (4A.3):
L(h) ιT=(τ)ιTdτ=N(τ)dτ= N(0) h + ½ h2bιT(4A.8)
h
0
h
0
35
The problem is how to calculate the vector of sojourn times L(h). If the rank
of Mis equal to (n-1), then (4A.7) and (4A.8) together constitute a system of
(n+1) linear equations of rank n in n unknowns, which can be solved. However,
Mmay well have rank less than (n-1). In fact, the difference between Dim[M]
(=n) and Rank[M] is equal to the number of disjoint absorbing subsets of states
which may range from 1 (only the full set of states 1 n is absorbing) to n
(each state is absorbing). Thus, in the general case we have
0Rank[M]n-1 (4A.9)
The problem of calculating L(h) for the general case of Mhaving any rank is
solved in Appendix 3 to the Gill (1986) paper. This appendix generalizes Gill’s
solution to include the case of uniformly distributed entries into the population
(the special case of a population that is closed with respect to both exits and
entries is obtained by putting b=0 in (4A.1)). The method outlined below is
more general than the solution discussed by Van Imhoff (1990a) which requires
that each absorbing subset of states consists of one state only.
The algorithm proposed here uses two theorems which will be presented first.
Subsequently, the appendix explains the various steps in the calculation as they
are carried out by the LIPRO computer program.
Theorem 1
Consider the CIMUE process defined by (4A.1) and (4A.2). Assume, without
loss of generality, that the n states have been ordered in such a way that the first
r states constitute an absorbing subset of states, 1rn. An (n by r) transformati-
on matrix P*exists that collapses the first r states into one absorbing state. Let
*(0) = (0) P*(4A.10)
*(τ)=(τ)P*(4A.11)
b*=bP
*(4A.12)
M*=(P*)TMP
*(4A.13)
Then the process
*(τ)= *(τ)M*+b*(4A.14)
d
dτ
is also a CIMUE process.
36
Proof of theorem 1
If the first r states of the original process constitute an absorbing subset, then
the matrix Mcan be written as:
(4A.15)
M
M11 0
M21 M22
The matrix P*has the following form:
(4A.16)
P
ιT0
0I
The matrix M*then equals:
(4A.17)
M
00
M21 ιTM22
Now from (4A.11) and (4A.1):
*(τ)= (τ)P*=(τ)MP
*+bP
*=
d
dτ
d
dτ
=(τ)P*(P*)TMP
*+b*=*(τ)M*+b*
which establishes (4A.14).
Theorem 2 (generalization of Gill, 1986)
For any CIMUE process, let L(τ) denote the vector of sojourn times until time
τ. Let Z(t) be defined by:
Z(t) = L(τ)dτ(4A.18)
t
0
Then:
Z(t) M=L(t) - t (0) - ½ t2b(4A.19)
Z(t) ιT= ½ t2N(0) + (1/3) t3bιT(4A.20)
37
Proof of theorem 2
(4A.19) follows from (4A.18), (4A.6) and (4A.7). (4A.20) follows from (4A.18)
and (4A.8).
With these two theorems proven, an outline of the calculations can be provided.
Step 1
Determine the (n by n) access matrix Adefined by:
Aij with i=/j = 1 state i has access to state j
0 otherwise
Aii =1state i has access to any other state
0 otherwise (i.e. state i is absorbing)
State i is said to have access to state j if either Mij>0 (direct access) or state i
has access to some third state k which has access to state j (indirect access).
This suggests the following iterative procedure for the construction of the access
matrix A:
1. set A=0.
2. for all i,j=1 n, set Aij=1 whenever Mij=/ 0 (direct access).
3. set IFLAG=0.
4. for each i=1 n, consider all rows j=/ i for which Aij=1. For all k=1 n,
if Ajk=1 and Aik=0, then set Aik=1 and IFLAG=1 (new indirect access
found).
5. if IFLAG=1, return to step 3.
Step 2
Determine the number of disjoint absorbing subsets of states, as well as the
elements of these subsets and the states not belonging to any subset. For each
state i=1 n, there exist three possibilities:
1. Aii=0. Then state i is absorbing and belongs to an absorbing subset
containing state i only.
2. Aii=1 and for all states j=/i to which state i has access (Aij=1), the rows Aj
are the same as Ai. In other words: all states to which i has access also
have access to state i (Aji=1) and, in addition, do not have access to any
state to which state i does not have access (there is no k for which Ajk=1
and Aik=0). Then state i and all the states to which i has access together
constitute a communicating absorbing subset of states.
3. Otherwise state i does not belong to an absorbing subset of states.
38
Let r be the resulting number of disjoint absorbing subsets and s the number
of states not belonging to any absorbing subset. Construct the allocation vector
B defined by:
Bi=k,krstate i belongs to subset k
k, k>r state i is the (k-r)-th state not belonging to any subset
(r<kr+s)
Step 3
Construct the n by (r+s) transformation matrix P*defined by:
P*
ij =1Bi=j
0 otherwise
The matrix P*transforms the original n-state CIMUE process (to be referred
to as the O-process) into an (r+s)-state CIMUE process (to be referred to as the
*-process) by collapsing each of the disjoint absorbing subsets of states into a
single absorbing state. In addition, the states are reordered in such a way that
the absorbing states come first.
By virtue of theorem 1, the *-process is a CIMUE process. Therefore, all
expressions derived for the O-process also hold true for the *-process, after
transformation of the process variables by means of the matrix P*.
Step 4
Construct the (r+s) by (1+s) transformation matrix P** defined by:
P**
ij =1( j=1 ir)( j>1 i-r=j-1 )
0 otherwise
The matrix P** transforms the (r+s)-state *-process into a (1+s)-state **-process
by collapsing the r absorbing states of the *-process into a single absorbing
state.
Step 5
Calculate the final population for the O-process according to (4A.5):
(h) = (0) exp[Mh]+b{Ih+(Mh2/2!) + (M2h3/3!) + }
Construct the initial population, final population, vector of immigration density,
and intensity matrix for the *-process and the **-process:
39
*(0) = (0) P*
*(h) = (h) P*
b*=bP
*
M*=(P*)TMP
*
**(0) = *(0) P**
**(h) = *(h) P**
b** =b*P**
M** =(P**)TM*P**
Step 6
Solve L**(h) from the following system of equations:
L**(h) M** =**(h) - **(0)-hb**
L**(h) ιT= N(0) h + ½ h2bιT
This is a system of (2+s) equations in (1+s) unknowns. The system is solvable
because, by construction, M** is a Markov matrix with Rank[M**]=
Dim[M**] - 1 = s, or, equivalently, because the **-process contains one and only
one absorbing subset of states, i.e. state 1. Compare the theorem in Appendix
1 to Gill (1986).
Step 7
Applying theorem 2 to the **-process, we have the following system of
equations:
Z**(h) M** =L**(h)-h **(0) - ½ h2b**
Z**(h) ιT= ½ h2N(0) + (1/3) h3bιT
Again, with L**(h) known from step 6, this system of rank (1+s) can be solved
for the (1+s) elements of Z**(h).
Step 8
Applying the first part of theorem 2 to the *-process, we have the following
system of (r+s) equations:
L*(h)=h *(0) + ½ h2b*+Z*(h) M*
The vector Z*(h) is unknown. However, by construction, its last s elements are
equal to the last s elements of Z**(h) which are known from step 7.
Furthermore, the first r rows of M*are equal to zero so that we can write:
40
Z*(h) M*=[0Z
**(h) ] M*
where the row vector 0has length r-1.
Now the system becomes
L*(h)=h *(0) + ½ h2b*+[0Z
**(h) ] M*
which can be solved for L*(h) in a straightforward way.
Step 9
From the transformation
L*(h) = L(h) P*
it can be deduced:
- for each absorbing subset i, i=1 r: the total sojourn time spent in the
states belonging to this subset, which equals L*
i(h).
- for each state j not belonging to an absorbing subset (Bj=r+i, with
i=1 s): the sojourn time spent in this state, Lj(h), which equals L*
r+i(h).
At the present stage, the s elements of the vector L(h) corresponding to states
not belonging to an absorbing subset are known. These elements are stored into
the (1 by s) vector L0(h).
For each of the absorbing subsets i, i=1 r, we can now proceed as follows:
1. denote the vector of unknown sojourn times for the nistates belonging to
subset i by the (1 by ni) vector Li(h). Construct the corresponding vectors
of initial population i(0), final population i(h), and immigration density
bi.
2. construct the (niby ni) matrix that is obtained by deleting all rows and
colums from matrix Mthat do not correspond to states in subset i. Denote
this matrix by Mii.
3. construct the (s by ni) matrix that is obtained by deleting all columns of
matrix Mthat do not correspond to states in subset i, and by deleting all
rows that do not correspond to any of the the s states not belonging to any
absorbing subset. Denote this matrix by M0i.
4. from the definition of absorbing subset i, the equations in (4A.7) that
correspond to the states belonging to subset i can be written as:
Li(h) Mii +L0(h) M0i =i(h) - i(0)-hbi(4A.21)
41
(4A.21) is a system of niequations of rank (ni-1) in the niunknown elements
of Li(h). An additional independent equation can be obtained from the trans-
formation equation referred to above:
Li(h) ιT=L
*
i(h) (4A.22)
From (4A.21) and (4A.22), Li(h) can be solved.
Going through steps 1-4 for each of the absorbing subsets completes the compu-
tation of L(h).
5. THE CONSISTENCY
ALGORITHM
The central endogenous variables in dynamic demographic projection models
are the numbers of events: immediate jumps from one cell in the state vector
to another. Except for trivial breakdowns of individuals (e.g. by age and sex
only), these demographic models give rise to problems of consistency. Consis-
tency can be defined as a situation in which the endogenous variables satisfy
certain constraints (Keilman, 1985a).
Such constraints may arise from various sources. Some constraints stem from
the nature of the cross-classification chosen. For instance, if individuals are
classified by sex and marital status, then a natural constraint on the number of
events would be that the total number of males experiencing a jump into the
married state should equal the number of females experiencing such a jump;
this is the well-known two-sex problem in nuptiality models (Keilman, 1985b).
Other constraints may occur because of interrelationships between different
models. For instance, numbers of events computed from regional models may
be required to add up to the corresponding numbers in the national population
forecast. Keilman (1985a) terms these two types of consistency as internal and
external consistency, respectively.
The existence of constraints on the endogenous variables gives rise to a
consistency problem as soon as the variables fail to meet the constraints. In the
approach followed here, solving the consistency problem amounts to adjusting
the initially calculated numbers of events in such a way that the constraints are
satisfied. The procedure by which these adjustments are realized will be termed
aconsistency algorithm.
Generally speaking, inconsistencies between numbers of events can be said to
be caused by inadequate modelling. For example, the two-sex problem typically
arises because of the use of sex-specific nuptiality rates, disregarding the
interaction between the sexes in the marriage market (Pollard, 1977). This is
not to say that there are no good reasons for such inadequate modelling. Our
knowledge of the determinants of the outcome of the bargaining process in the
marriage market is so limited that using sex-specific rates is probably the best
we can do.
Even more complex interactions between individuals occur in household projec-
tion models. Here, household formation, household dissolution, and household
44
change are the result of interactions between two or more individuals. There are
good reasons why household projection models should take the individual as
the unit of analysis, instead of the household itself (McMillan and Herriot, 1985;
Keilman and Keyfitz, 1988). But the other side of the coin is that, precisely
because of the "wrong" unit of analysis, the consistency problem appears in full
force. This is not to say that modelling households instead of individuals would
altogether remove problems of consistency, but they would certainly be smaller.
This chapter presents a very general characterization of the consistency problem,
as well as a slightly less general algorithm to solve it. We will restrict ourselves
to the main features of the consistency algorithm; proofs and additional technical
details will be given elsewhere (Van Imhoff, 1992).
Section 5.1 spells out the notation used throughout this chapter. Section 5.2
gives the general formulation of the consistency problem. Section 5.3
investigates the properties of the solution to the consistency problem for the
special (and analytically convenient) case of linear first-order conditions. Section
5.4 discusses the relationship between adjusted numbers of events and the under-
lying model parameters (jump intensities or occurrence-exposure rates).
5.1 Notation
The following notation will be used throughout this chapter:
Ithe identity matrix, with ones on the diagonal and all off-diagonal ele-
ments equal to zero;
ιarow vector consisting of ones only;
T operator for transposition of a vector or matrix;
xoperator for the formation of a diagonal matrix with the elements of
vector xon the diagonal;
fxif f(x) is a scalar function of vector x, then fxdenotes the row vector of
partial derivatives [f1fn];
Na(1byK)row vector with projected numbers of events;
na(1byK)row vector with adjusted numbers of events.
The vector nshould satisfy the consistency conditions (constraints):
nA=c
with ca non-negative (1 by C) vector and Aa (K by C) matrix with full rank.
Of course, CK. The full-rank condition boils down to requiring that no
superfluous consistency conditions (i.e. dependent on other consistency
conditions) are included in the matrix A.
45
5.2 Formulation of the consistency problem
A consistency algorithm finds a vector of adjusted numbers of events n
satisfying the consistency conditions and being "optimal" according to some
criterion yet to be specified. The projection model computes these numbers of
events given values for the model parameters Θ. Consequently the vector Ncan
be written as a function of these model parameters: N=G(Θ). Similarly, the
vector of adjusted numbers of events nis a function of the adjusted model
parameters θ:n=G(θ). Generally speaking, the optimality criterion corresponds
to some measure of closeness of θwith respect to the original vector Θ.
Thus a consistency algorithm solves the following optimization problem:
find θto minimize F(Θ,θ) subject to n(θ)A=c(5.1)
The Lagrangean for this constrained optimization problem is given by:
L(θ,λ)=F(Θ,θ)+{n(θ)A-c}λT(5.2)
where λis a (1 by C) vector of Lagrange multipliers.
First order conditions are:
Fθ(Θ,θ)+λATnθ=0(5.3)
n(θ)A-c=0(5.4)
The solution vectors θand λcan be deduced from (5.3) and (5.4). In general,
the system of equations (5.3)-(5.4) is nonlinear, so that an iterative procedure
is required.
5.3 Solution to the consistency problem for a specific class
of objective functions
5.3.1. A specific class of objective functions
Let us specify the objective function F( ) as follows:
F(N,n;p) = Norm2[(n-N)N-p ]=(n-N)N-2p (n-N)T(5.5)
For this particular specification, the optimization problem becomes linear in the
unknown parameters. This attractive property is achieved by combining two
things: the choice of the adjusted events vector nas control variable; and the
quadratic specification of F( ).
46
Of course, the solution vector nremains a function of the adjusted model
parameters θ. If these model parameters are intensities or o-e rates, then this
function gives an invertible correspondence between θand n, so that one could
also write: θ=G
-1(n). Thus once the adjusted numbers of events have been
found, the underlying adjusted model parameters follow immediately. We will
return to this issue in section 5.4.
It can be shown (Van Imhoff, 1992) that the solution to the problem of
minimizing F( ) in (5.5) subject to (5.4) is given by:
n=N{I-A(ATN2pA)-1 ATN2p }+c(ATN2pA)-1 ATN2p (5.6)
The first term on the right-hand side of (5.6) can be interpreted as an averaging
term, reshuffling numbers of events in order to meet conditions of internal
consistency. The second term involves a complete shift of the events vector,
reflecting the pressure imposed by the conditions of external consistency.
Since we have assumed Ato have full rank, the matrix ATN2pAin (5.6) is
invertible whenever either Ndoes not contain too many zeros or p=0. For p=/0
this can be achieved by leaving any zero element of Nout of the optimization,
putting the corresponding element of nequal to zero as well.
5.3.2. Interpretation of the parameter p
Minimization of the objective function (5.5) is equivalent to weighted least
squares optimization, where each element of the adjustment vector (n-N)is
weighted by some inverse power of the corresponding element of the vector N
of numbers of events from which the adjustment starts. For example, if p=0,
F( ) reduces to the square of the Euclidian distance between the vectors of
projected and of adjusted numbers of events, respectively.
Now minimizing (the square of) Euclidian distances is, at least in special cases,
equivalent to taking arithmetic averages. That is, for p=0 one would expect our
consistency algorithm to yield adjusted numbers of events that in some way
correspond to arithmetic averages of the originally projected numbers of events.
More generally, it can be shown that our class of objective functions (5.5)
corresponds to a certain class of averages of the elements of the vector N,
characterized by the parameter p. This correspondence can be illustrated by
considering the case in which the matrix Ais such that all elements of nare
restricted to be equal, i.e. n=mιwhere m is some average of the elements of
Nand ιdenotes a vector with all elements equal to one.
If we substitute mιfor nand minimize F(N,n) as a function of the single
variable m, we obtain the following general expression:
(5.7)
mK
i 1 N1 2p
i
K
i 1 N2p
i
47
For p=0, (5.7) reduces to , which is the arithmetic mean of them Ni/K
elements of the vector N. For p=1, we obtain , whichm N 1
i/ N 2
i
corresponds to the minimization of the relative distance between Nand mι,
rather than the absolute distance under p=0. Finally, for p=½we have mK/ N1
i
, which is the harmonic mean. The latter expression corresponds to the harmonic
mean solution to the two-sex problem in nuptiality models (Keilman, 1985b).
5.3.3. Comparison with the unidimensional harmonic mean method
If the consistency matrix Ais blockwise diagonal, then the consistency problem
can be split up in independent subproblems. This is the case, for instance, if
each nienters at most one consistency relation. Then for each of the constraints,
the adjusted (consistent) number of events can be found independently of the
other constraints.
However, the more detailed the classification of events, the larger the number
of consistency relations, and also the higher the probability that numbers of
events are restricted in multiple ways. For these multidimensional consistency
problems a matrix formulation becomes unavoidable. In fact, the procedure
adopted by Van Dam and Keilman (1987) in order to circumvent the problem
of cells entering several consistency relations boils down to artificially putting
some elements of Aequal to zero, thus dividing the consistency problem into
several independent unidimensional sub-problems. One of the innovations made
possible by the present matrix formulation is that such rather arbitrary
assumptions are no longer required.
5.3.4. Relationship between age-specific and aggregate adjustment factors
In multidimensional demography, it will generally be the case that the
consistency conditions read in terms of events aggregated over all age groups
only. This is so because these models typically trace the age of the individual
only, not the ages of other persons with whom the individual under consider-
ation interacts.
Now the question arises whether there exists a simple relationship between the
adjustments of the age-specific numbers of events, on the one hand, and the
adjustments of the aggregated numbers of events, on the other hand. If such a
simple relationship is found, then the consistency problem could be greatly
simplified by dropping the age dimension from the variables involved in the
optimization problem.
It can be shown (Van Imhoff, 1992) that, for objective functions of type (5.5),
the age-specific adjustments in numbers of events can be written in terms of
the adjustments in aggregate numbers of events if and only if either p=0 or p=½.
In other words, our simple relationship exists only for the (generalized) arithme-
tic mean and for the (generalized) harmonic mean specification of the consisten-
cy problem. This result gives strong reasons in favour of one of these specifica-
48
tions, as the solution to the consistency problem can be greatly simplified by
taking the short-cut via consistent aggregate numbers of events. Indeed, the
implementation of the consistency algorithm in LIPRO is such that only aggre-
gate consistency relations can be imposed.
For the arithmetic (p=0) mean we have:
nx=Nx+(nΣ-NΣ)/NAGE (5.8)
where the subscript x refers to events for age group x and the subscript Σto
events aggregated over all age groups. Thus, for the arithmetic mean, the age-
specific adjustments are equal in absolute terms.
For the harmonic mean (p=½), we have:
nx={ι+(nΣ-NΣ)NΣ
-1 }Nx(5.9)
Thus, for the harmonic mean, the age-specific adjustments are equal in relative
terms. This result justifies the proportional adjustment method of Van Dam and
Keilman (1987).
5.4 From adjusted events to adjusted rates
A multidimensional projection model specifies jump intensities (or occurrence-
exposure rates for models based on the linear integration hypothesis) for all
possible transitions. Once adjusted numbers of events have been found by the
consistency algorithm, the corresponding adjusted rates can be computed.
Adjustments in the numbers of events affect the numbers of person-years lived
within each cell of the state vector. Therefore, all rates change, even although
the numbers of only some of the events are adjusted.
This is of course theoretically unsatisfactory. The rates are to some extent
behavioural parameters, reflecting an individual’s tendency to experience a
certain event (cf. Schoen, 1981). If consistency is required for aggregate behavi-
our, then one would like to adjust only those behavioural parameters which are
directly related to the initial inconsistencies, leaving the other rates exactly as
they were. On the other hand, if only some of the rates are adjusted, then the
correspondence between adjusted numbers of events and adjusted rates is lost.
The previous discussion suggests an iterative approach which is capable of
leaving the rates intact which are not directly related to the consistency relations,
without losing the correspondence between rates and numbers of events. This
iterative scheme is optional in the LIPRO program. The procedure consists of
the following steps:
1. compute rates corresponding to adjusted numbers of events;
49
2. replace the unrelated rates by their initial values;
3. compute numbers of events for rates obtained under step 2;
4. replace the constrained numbers of events by their adjusted values as calcu-
lated by the constrained optimization procedure;
5. repeat steps 1-4 until convergence has been reached.
In LIPRO, iterations stop when either the Euclidian norm of the change in
the vector of person years becomes less than a user-supplied convergence
criterion, or steps 1-4 have been repeated a certain maximum number of
times. In the latter case, LIPRO will print a message if no convergence was
achieved.
It should be pointed out that, for the linear version of the projection model, an
alternative one-step computation scheme exists (Keilman, 1985a, pp. 1483-
1484).
6. SOME ISSUES IN
MULTIDIMENSIONAL
LIFE TABLE ANALYSIS
The LIPRO program offers several options for the construction and analysis of
multidimensional life tables. This chapter discusses several technical issues
concerning the way in which these life tables have been implemented in LIPRO.
6.1 Interpretation of life tables in LIPRO
As LIPRO is a program for demographic projection, its demographic data are
of the period-cohort type. On the other hand, life table analysis in the strict
sense involves data of the age-cohort (or simply cohort) type. That is, a life
table analyses events experienced by an imaginary group of people born at a
particular instant, while within the context of LIPRO, events are experienced
by a group of people born within a particular time span. Therefore, the life
tables produced by the program are not life tables in the strict sense of the word.
Rather, they should be interpreted as a summary of the events experienced by
an imaginary cohort over its life cycle.
Starting point for the life table analysis is a set of rates (either intensities for
the exponential model, or occurrence-exposure rates for the linear model). In
creating the life tables, LIPRO sets the rates for international migration equal
to zero. That is, the population is assumed to be closed.
6.2 Determining the radix of the life table
In single-dimensional life tables, the radix of the table is a single number
(usually: 100,000) indicating the size of the group of people to which the life
table refers. In multidimensional life tables, the radix of the table is a vector
of numbers, each element corresponding to the number of people starting their
life in a particular state. The sum of the elements of the vector equals the size
of the imaginary cohort. In LIPRO, this sum equals 100,000.
52
As long as the state space is such that all individuals are born into the same
state i, the radix is a vector consisting of the element 100,000 in cell i and zeros
elsewhere. For example, in a marital status model, all individuals start their life
in the "single" state. If the states are labelled "single", "married", "widowed",
and "divorced", then the radix equals: [ 100,000000].
However, it is not always the case that the radix is so easily determined. There
are many specifications of the state space which lead to a situation in which
children can be born into more than one state. In regional models, for example,
children are born into the state occupied by their mother at the time of birth.
And in the NIDI household model, the position of the child is dependent on the
position of the mother: children of married women are born into the state "child
with married parents", children of cohabiting women are born into the state
"child with cohabiting parents", and so on.
We are therefore confronted with the problem how to determine the distribution
of our imaginary cohort across states of birth. It is clear that each distribution
leads to a different life table (except in the not very interesting case in which
all rates are the same across states of origin, implying that the classification
chosen is demographically irrelevant).
If we start from a given radix, with a particular distribution across states of
birth, we can compute a corresponding life table. The female members of the
imaginary cohort will bear children during the course of their life cycle. The
distribution of these children across states of birth will in general differ from
the distribution within the radix from which the life table for their mothers was
calculated.
The only case in which the distribution of the children across states of births
is exactly equal to the distribution of their mothers across the states into which
they themselves were born, is the case of a stable population. A stable
population is defined as a population that remains constant over time in all
respects except size. In particular, in a stable population the distribution of
newly-born children across states of birth remains constant.
The problem of determining the stable distribution of births across states can
be solved as follows. Assume that there are K possible states into which chil-
dren can be born. In regional models, K equals the number of regions. In the
household model to be presented in chapter 7, K equals 4 (being the states
CMAR, CUNM, C1PA, and OTHR; cf. section 7.2). The index ik, k=1 K,
gives the index of the state corresponding to the k-th state of birth; in the
household model, we have i1=1, i2=2, i3=3, i4=11. For each of these states ik,
k=1 K, we construct a life table based on a unitary female radix rkwith
100,000 in position ikand zeros elsewhere. From this life table, we can calculate
the vector of female births bk. The vector bkrepresents the number of girls,
classified by the state into which they are born, born out of mothers who
themselves were born into state ik.
We now have K radices rkand K corresponding birth vectors bk. The next
problem is to determine the stable radix s, such that the vector of female births
53
cgenerated by a female radix sis a multiple of s. The stable radix can be
written as a weighted sum of the K unitary radices:
s=wR= 100,000 w(6.1)
where Ris a (K by K) matrix, the rows of which are given by the unitary
radices rk, k=1 K, and wis a row vector of weights. The vector of births
cgenerated by sis also a weighted sum of the K birth vectors:
c=wB (6.2)
where Bis the (K by K) matrix, the rows of which are given by the birth
vectors bk, k=1 K. We require cto be a multiple of s:
c=fs(6.3)
with f a scalar. From (6.1)-(6.3) we obtain:
wB= f 100,000 w=λw(6.4)
From (6.4), we see that the unknown vector wis an eigenvector of the birth
matrix Band λits corresponding eigenvalue. From the mathematical theory of
stable population, it follows that the appropriate eigenvector is the one corres-
ponding to the dominant eigenvalue, since this is the distribution to which any
initial population will eventually converge. The dominant eigenvalue λ*is also
known as the net reproduction factor in the stable population.
6.3 Handling the highest age group
For the highest age group, which is open ended, the demographic rates
determine the number of events that will be experienced during a period, the
length of which equals the length of the projection interval (h). However, after
such a time span, a number of the individuals present at the start of the interval
will still be alive. For the calculation of the life table, these surviving individuals
should be exposed to another interval of demographic risk. This process should
be repeated until all members of the imaginary cohort have died.
For this reason, LIPRO lets h→∞ for the highest, open-ended age group. If for
this age group the initial population is denoted by , the events matrix by N,
the intensity matrix by Mand the vector of person years by L, then:
final population = 0=+ιN=+ι(LM)= +LM
from which
54
L=- M-1
6.4 Calculating mean ages
For some of the calculations performed by LIPRO, the mean age at which a
particular event is experienced has to be calculated. Within a particular age
group, the mean age could in principle be different across events: in general,
the higher the intensity of experiencing a particular event, the lower the mean
age at which it occurs. The exact expressions for calculating these mean ages
are given by Van Imhoff (1990b).
Since such sophisticated calculations would be very time-consuming (especially
in the exponential model), LIPRO uses an alternative calculation as an
approximation. The implicit assumption behind this formula is that for a particu-
lar age group, all events on average occur at the same mean age. The formula
is given by:
ae=a
0+L/(P
0+P
h)
where:
ae= average age at time of event
a0= average age at start of interval ( = midpoint of age group )
L = number of person years lived during interval
P0= number of survivors at start of interval
Ph= number of survivors at end of interval
6.5 Fertility indicators
LIPRO optionally calculates six fertility indicators for each state as well as for
the aggregate across states. These indicators are the following:
1. Total fertility rate = sum of fertility rates across age groups
2. Average number of children = average number of births per woman, taking
inter-household changes and mortality into account
3. Gross reproduction rate = sum of female fertility rates across age groups
4. Net reproduction rate = total number of female births per woman
5. Average length of generation = average age of mother at birth of daughters
6. Annual growth rate = annual growth rate of female population in stable
population = ln [net reproduction rate] / (average length of generation)
55
6.6 Experience tables
In some applications, it is useful to analyse the occurrence of a particular
demographic event, or group of events, over the lifetime of the average
individual. Examples include the analysis of the proportion ever-married as a
function of age, or the probability of experiencing at least one dissolution of
a relationship.
For this type of life table analysis, LIPRO offers the opportunity to construct
so-called experience tables. An experience table is a life table, restricted to those
members of the life table population who have experienced at least one event
of a particular type earlier in their life. Examples include "ever been married",
"ever experienced the loss of a spouse", "ever lived in a one-parent family".
An experience table corresponds to a particular events set. The events set
specifies the type of events, the experience of which changes the status of an
individual from "never experienced" to "at least experienced once". An events
set can include:
- internal events
- births into a particular state
The experience table is constructed like an ordinary life table: starting from an
initial radix, the age-specific rates are applied to the surviving population. The
steps in the construction of an experience table are the following:
- the state space is extended with a second external state, namely "at least
experienced once". This state is comparable to the external state "dead" in
that it is absorbing: an individual once in the state "at least experienced
once" can never leave it.
- the rates for internal events are redirected in accordance with the speci-
fication of the events set: for each event in the set, the rate is added to the
initially zero rate for exit into the state "at least experienced once", at the
same time setting the rate for the original event equal to zero.
- an "inexperience table" is constructed starting from the initial radix. If the
events set specifies births into one or more particular states, the correspon-
ding element of the radix is set equal to zero. This "inexperience table" is
a life table for the individuals with the status "never experienced".
- finally, the "inexperience table" is substracted from the original life table to
yield the experience table.
PART III
APPLICATION
7. THE SPECIFICATION OF
THE STATE SPACE IN
THE HOUSEHOLD MODEL
7.1 General considerations
The general multidimensional projection model described in chapters 3 to 6 was
applied to a study into the impact of population dynamics on future social
security expenditures in the Netherlands. In this study, the term population
dynamics is interpreted as changes in household structure (household formation
and dissolution) and age structure of the population. Regarding social security,
the emphasis is on those social security schemes that are particularly sensitive
to the living arrangement and/or age of the recipient. The consequences of
population dynamics for social security expenditure are traced by linking the
results of macrosimulations produced by the LIPRO household projection model,
to fixed user profiles for social security schemes. The technique of employing
user profiles is dealt with in more detail in chapter 11. The current chapter and
chapters 8 to 10 will focus on the demographic dimension of the problem.
A meaningful classification of individuals should be such that the resulting
categories are relatively homogeneous, with respect to both demographic
behaviour and the use of social security schemes.
Classification by age and sex is obviously necessary for demographic reasons:
mortality, fertility, and household formation and dissolution (e.g. marriage,
tendency to start one-person households, tendency to become head of one-parent
family) are to a large extent determined by age and sex. Formally, the
distinction between the sexes is of less importance for the use of social security,
as social security legislation is approaching full equality of men and women in
the Netherlands. In practice, however, the distinction is highly relevant. This
is because the economic position, which is a major determinant of the eligibility
for social security benefits, of men is generally very different from that of
women. For instance, one-parent families with labour income less frequently
apply for social welfare than one-parent families without labour income. Labour
force participation rates for women are lower than for men (in the Netherlands
60
they are much lower). Consequently, we expect average social security benefits
for female single parents to be higher than for male single parents.
The choice of a classification of living conditions is the most problematic. On
the one hand, one would like a very detailed classification in order to make the
categories as homogeneous as possible. In particular, household size should be
included among the classification criteria (important for child allowance and
social welfare). On the other hand, there are technical limitations to the number
of categories that can be distinguished. These restrictions stem from the finite
capacity of computers, but especially from the limited availability of the neces-
sary input data.
Although the actual choice of classification may vary with the special character-
istics of the social security system in the country under consideration, a minimal
classification of household positions would be the following:
1. dependent child
2. one-person household
3. partner (either married or cohabiting)
4. head of one-parent family
5. other
The categories "one-person household" and "head of one-parent family" can be
subdivided according to the event that caused the particular household position
(exit from existing household, divorce, death of spouse). The type of event
leading to the present living condition will determine the degree to which
individuals apply for various social security schemes. For the sum of all social
security schemes, such a sub-classification would be less important.
An important consideration in the choice of household positions is that the
classification should facilitate the transformation of a projection of individuals
into a projection of households. The more detailed the classification of
individuals is, the more accurate the distribution of households can be. This is
particularly relevant for the scoring of children into households of various types.
Distinction of household positions "partner" and "head of one-parent family",
according to the number of children present in the household, would be desira-
ble; however, a complete breakdown would lead to insurmountable data and
computing problems. As a first approximation, a distinction could be made
between families with and without children.
Within the category "partner", a further distinction could be made between
married couples and consensual unions. In the Netherlands, the distinction has
by now almost completely disappeared from social security legislation. From
a purely demographic point of view, the distinction is theoretically interesting.
With respect to international comparability, the distinction can hardly be missed.
For social security, an important argument in favour of the distinction is that
the registration of consensual unions is much less complete than the registration
of marriages, implying differences between the two categories in both the degree
61
to which couples apply for social security allowances and the degree to which
the executive authorities will honour these applications.
7.2 The specification of household positions
After weighing all the arguments set out in the previous section, it was decided
to use a classification which contains 11 household positions. We feel that this
classification offers a reasonable compromise between the conflicting objectives
of completeness and feasibility. An individual in a private household may
occupy, at a certain point in time, one of the following household positions.
1. CMAR child in family with married parents
2. CUNM child in family with cohabiting parents
3. C1PA child in one-parent family
4. SING single (one-person household)
5. MAR0 married, living with spouse, but without children
6. MAR+ married, living with spouse and with one or more children
7. UNM0 cohabiting, no children present
8. UNM+ cohabiting with one or more children
9. H1PA head of one-parent family
10. NFRA non family-related adult (i.e. an adult living with family types 5
to 9)
11. OTHR other (multi-family households; multiple single adults living
together)
These 11 household positions together define 7 household types:
1. SING a one-person household
2. MAR0 a married couple without children, but possibly with other adults
3. MAR+ a married couple with one or more children, and possibly with
other adults
4. UNM0 a couple living in a consensual union without children, but
possibly with other adults
5. UNM+ a couple living in a consensual union with one or more children,
and possibly with other adults
6. 1PAF a one-parent family, possibly with other adults (but no partner to
the single parent!)
7. OTHR multi-family households, or multiple single adults living together
without unions.
No upper age limit was used with respect to the definition of "child". In practi-
ce, a situation could be encountered in which an adult child, even at an advan-
ced age, belongs to the same household as his or her parent(s). For instance,
62
a household consisting of an elderly mother and her co-residing daughter aged
50 is labeled here as a "one-parent family", although in reality the mother might
have joined her daughter to form one household (and thus the household would
be "other"). The solution we chose was based on the data which were available
for this project: the Housing Demand Survey of 1985/1986 (see chapter 8)
contains information regarding the structure of the household of each respon-
dent, but no clear guidelines were given to the interviewer as to how to define
a child. Thus the notion of child used here is not always the same as that used
in social security regulations. However, because the number of households
containing "old children" is probably small, the bias is likely to be limited.
(Moreover, in the empirical part of this study, intensities for jumps from
CMAR, CUNM or C1PA to SING for age groups 50-54 and over were set at
an arbitrarily high value, so as to avoid a large number of older "children", see
section 8.4.2).
Numbers of households of various types may be easily inferred from numbers
of persons in the 11 household positions. Thus a household projection in terms
of individuals may be translated into one in terms of households:
1. the number of households of type SING equals the number of persons in
household position SING;
2. the number of households of type MAR0 equals half the number of persons
in household position MAR0;
3. the number of households of type MAR+ equals half the number of persons
in household position MAR+;
4. the number of households of type UNM0 equals half the number of persons
in household position UNM0;
5. the number of households of type UNM+ equals half the number of persons
in household position UNM+;
6. the number of households of type 1PAF equals the number of persons in
household position H1PA;
7. the number of households of type OTHR equals the number of persons in
household position OTHR divided by the average number of persons in
OTHR households. This average household size was 2.82 persons in 1985.
It is assumed that the average size of OTHR households remains unchanged
throughout the projection period.
7.3 Household events
Given the classification of household positions (the definition of the state space),
a matrix of household events can be identified. The events matrix given in Table
7.1 is based on the following assumptions:
63
1. spouses who divorce or separate no longer co-reside;
2. a return to one of the positions for a co-residing child (CMAR, CUNM,
C1PA) is only possible from the position SING (or OTHR);
3. adults can only leave the household through the (intermediate) position of
SING.
Given these assumptions, it is possible to identify the demographic event which
causes the internal or external event. These events are listed below (impossible
events are denoted with an asterisk). Items on the main diagonal in Table 7.1
need some clarification. Most of them are "non-events" and these are omitted.
Table 7.1. Events matrix of the household model
from: to: 1234567891011dead emigr
1. CMAR * * *
2. CUNM * *
3. C1PA * *
4. SING *
5. MAR0 * * * * ****
6. MAR+ * * * * * * *
7. UNM0 * * * * * * *
8. UNM+ * * * * * *
9. H1PA * * *****
10. NFRA * * * ******
11. OTHR *
birth by child’s position after birth:
mother’s state
before birth:
1. CMAR ********** * *
2. CUNM ********** * *
3. C1PA ********** * *
4. SING * * ******** * *
5. MAR0 ********** * *
6. MAR+ ********** * *
7. UNM0 * ********* * *
8. UNM+ * ********* * *
9. H1PA * * ******** * *
10. NFRA ********** * *
11. OTHR ********** * *
immigration **
* = impossible event.
= possible event.
However, for some "aggregate" positions, such as UNM+ (consensual union
with 1 or more children), MAR+ (with marriage partner and 1 or more chil-
dren), or H1PA, the arrival of an additional child (due to birth or return to
64
parental home), or the exit of a child (due to home-leaving, death or emigration
- when sufficiently many children stay behind) causes the adults to remain in
the same household position. On the other hand, not every person who remains
in the position UNM+, MAR+, or H1PA, experiences the arrival or exit of a
child. In fact, most of them will not. Thus for these "aggregate" positions the
one-to-one correspondence between an event, on the one hand, and a pair of
positions, on the other, does not hold. This bears some implications for the
consistency between (numbers of) events of adults and those of children; see
below.
From To Demographic event
CMAR CMAR * (no event)
CMAR CUNM * (assumption 1)
CMAR C1PA a) divorce or separation of parents
b) death of parent
CMAR SING leaving the parental home to start one-person household
CMAR MAR0 marriage
CMAR MAR+ marriage with a lone parent
CMAR UNM0 start of a consensual union
CMAR UNM+ start of a consensual union with a lone parent
CMAR H1PA *
CMAR NFRA entrance into an existing family
CMAR OTHR a) entrance into or formation of OTHR household
b) child, while living with both parents, gets own child
c) second family moves in
CUNM CMAR marriage of cohabiting couple with child(ren)
CUNM CUNM *
CUNM C1PA a) divorce or separation of parents
b) death of parent
CUNM SING leaving the parental home to start a one-person household
CUNM MAR0 marriage
CUNM MAR+ marriage with lone parent
CUNM UNM0 start of a consensual union
CUNM UNM+ start of a consensual union with a lone parent
CUNM H1PA *
CUNM NFRA entrance into an existing family
CUNM OTHR a) entrance into or formation of OTHR household
b) child, while living with cohabiting adults, gets own child
c) second family moves in
C1PA CMAR marriage of lone parent
C1PA CUNM lone parent starts consensual union
65
C1PA C1PA *
C1PA SING a) leaving the parental home to start a one-person household
b) death of a lone parent with one co-residing child
C1PA MAR0 marriage
C1PA MAR+ marriage with lone parent
C1PA UNM0 start of a consensual union
C1PA UNM+ start of a consensual union with a lone parent
C1PA H1PA *
C1PA NFRA entrance into an existing family
C1PA OTHR a) entrance into or formation of OTHR household
b) child, while living with one parent, gets own child
c) second family moves in
d) lone parent of two or more co-residing children dies
SING CMAR return to married parents
SING CUNM return to cohabiting parents
SING C1PA return to single parent
SING SING *
SING MAR0 marriage
SING MAR+ marriage with lone parent
SING UNM0 start of a consensual union
SING UNM+ start of a consensual union with a lone parent
SING H1PA a) return of child to lone parent
b) birth
SING NFRA entrance into an existing family
SING OTHR entrance into or formation of non-family household
MAR0 CMAR * (assumption 2)
MAR0 CUNM * (assumption 2)
MAR0 C1PA * (assumption 2)
MAR0 SING a) divorce or separation
b) death of partner
MAR0 MAR0 *
MAR0 MAR+ a) birth
b) return of child to married parents
MAR0 UNM0 * (assumption 1)
MAR0 UNM+ * (assumption 3)
MAR0 H1PA * (assumption 3)
MAR0 NFRA * (assumption 3)
MAR0 OTHR entrance into or formation of non-family household
66
MAR+ CMAR * (assumption 2)
MAR+ CUNM * (assumption 2)
MAR+ C1PA * (assumption 2)
MAR+ SING divorce or separation
MAR+ MAR0 last child leaves the parental household
MAR+ MAR+ *
MAR+ UNM0 * (assumption 3)
MAR+ UNM+ * (assumption 3)
MAR+ H1PA a) divorce or separation
b) death of partner
MAR+ NFRA * (assumption 3)
MAR+ OTHR a) entrance into or formation of non-family household
b) co-residing daughter gets a child
UNM0 CMAR * (assumption 2)
UNM0 CUNM * (assumption 2)
UNM0 C1PA * (assumption 2)
UNM0 SING a) divorce or separation
b) death of partner
UNM0 MAR0 marriage
UNM0 MAR+ * (assumption 3)
UNM0 UNM0 *
UNM0 UNM+ a) birth
b) return of child to cohabiting parents
UNM0 H1PA * (assumption 3)
UNM0 NFRA * (assumption 3)
UNM0 OTHR entrance into or formation of non-family household
UNM+ CMAR * (assumption 2)
UNM+ CUNM * (assumption 2)
UNM+ C1PA * (assumption 2)
UNM+ SING divorce or separation
UNM+ MAR0 * (assumption 3)
UNM+ MAR+ marriage
UNM+ UNM0 last child leaves parental household
UNM+ UNM+ *
UNM+ H1PA a) divorce or separation
b) death of partner
UNM+ NFRA * (assumption 3)
UNM+ OTHR a) entrance into or formation of non-family household
b) co-residing daughter gets a child
H1PA CMAR * (assumption 2)
H1PA CUNM * (assumption 2)
67
H1PA C1PA * (assumption 2)
H1PA SING last child leaves parent
H1PA MAR0 * (assumption 3)
H1PA MAR+ marriage
H1PA UNM0 * (assumption 3)
H1PA UNM+ start of a consensual union
H1PA H1PA *
H1PA NFRA * (assumption 3)
H1PA OTHR a) entrance into or formation of non-family household
b) co-residing daughter gets a child
NFRA CMAR * (assumption 2)
NFRA CUNM * (assumption 2)
NFRA C1PA * (assumption 2)
NFRA SING exit from family and start of one-person household
NFRA MAR0 * (assumption 3)
NFRA MAR+ * (assumption 3)
NFRA UNM0 * (assumption 3)
NFRA UNM+ * (assumption 3)
NFRA H1PA * (assumption 3)
NFRA NFRA *
NFRA OTHR a) second family moves in
b) co-residing daughter gets a child
OTHR CMAR a) married parents leave multi-family household
b) return to married parents from OTHR household
OTHR CUNM a) cohabiting parents leave multi-family household
b) return to cohabiting parents from OTHR household
OTHR C1PA a) lone parent leaves multi-family household
b) return to lone parent from OTHR household
OTHR SING exit from OTHR household and start of one-person
household
OTHR MAR0 a) marriage
b) married couple without children leaves multi-family
household
OTHR MAR+ a) marriage with lone parent
b) married couple with child(ren) leaves multi-family
household
OTHR UNM0 a) start of consensual union
b) cohabiting couple without children leaves OTHR
household
c) two co-residing adults start a consensual union
68
OTHR UNM+ a) start of consensual union with lone parent
b) cohabiting couple with child(ren) leaves OTHR
household
c) two co-residing adults of whom at least one is a lone
parent start a consensual union
OTHR H1PA one-parent family leaves multi-family household
OTHR NFRA family with co-residing adult leaves multi-family household
OTHR OTHR *
7.4 Consistency relations
Consistency relations were formulated on the basis of the household events
which were identified in the previous section. As discussed in chapter 5,
consistency relations describe constraints which the projected numbers of events
have to satisfy.
The events matrix for the LIPRO household model, depicted in Table 7.1,
contains 69 internal events, 22 exits, and 22 entries. For two sexes, this amounts
to a total of 226 events. Formulation of consistency relations between these 226
events led to 37 restrictions in terms of 127 variables. Four assumptions turned
out to be necessary in addition to the three assumptions listed in section 7.3:
4. divorced partners do not continue to live together;
5. adoption can be disregarded for the entry of a first child into the household;
6. the formation and dissolution of homosexual consensual unions can be
disregarded as far as the two-sex requirement for cohabitation is concerned;
7. only complete households can migrate.
The 37 consistency relations are listed below. All relations are in terms of
numbers of events. Each type of event is described using the following notation:
T(S,ORIG,DEST)
in which:
T stands for type of event. T may be:
I internal event
X external event (exit)
B birth (endogenous entry)
N immigration (exogenous entry)
S stands for the sex experiencing the event. S may be:
F female
M male
M+F both sexes
69
ORIG stands for the household position, or range of household positions,
from which the event takes place. In the case of births, ORIG
indicates the position of the mother prior to the moment of birth.
DEST stands for the household position, or range of household positions, to
which the event leads. In the case of births, DEST indicates the
household position that the newly-born child occupies.
Constraints for households of type MAR0
household dissolution:
1) I(F,MAR0,SING) + X(F,MAR0,DEAD) = I(M,MAR0,SING) +
X(M,MAR0,DEAD)
birth of child, or return of child to parents:
2) I(M,MAR0,MAR+) = I(F,MAR0,MAR+)
formation of OTHR households:
3) I(M,MAR0,OTHR) = I(F,MAR0,OTHR)
emigration:
4) X(M,MAR0,REST) = X(F,MAR0,REST)
immigration:
5) N(M,REST,MAR0) = N(F,REST,MAR0)
Constraints for households of type MAR+
marriage dissolution:
6) I(M,MAR+,SING) + I(M,MAR+,H1PA) + X(M,MAR+,DEAD) =
I(F,MAR+,SING) + I(F,MAR+,H1PA) + X(F,MAR+,DEAD)
7) I(M,MAR+,SING) + I(M,MAR+,H1PA) + X(M,MAR+,DEAD) =
I(M+F,CMAR,C1PA) / [Mean number of children per married couple]
exit of last child:
8) I(M,MAR+,MAR0) = I(F,MAR+,MAR0)
formation of OTHR households:
9) I(M,MAR+,OTHR) = I(F,MAR+,OTHR)
emigration:
10) X(M,MAR+,REST) = X(F,MAR+,REST)
11) X(M,MAR+,REST) = X(M+F,CMAR,REST) / [M.n.o.c.p.m.c.]
immigration:
12) N(M,REST,MAR+) = N(F,REST,MAR+)
13) N(M,REST,MAR+) = N(M+F,REST,CMAR) / [M.n.o.c.p.m.c.]
Constraints for households of type UNM0
marriage:
14) I(M,UNM0,MAR0) = I(F,UNM0,MAR0)
70
household dissolution:
15) I(F,UNM0,SING) + X(F,UNM0,DEAD) =
I(M,UNM0,SING) + X(M,UNM0,DEAD)
birth of child, or return of child to parents:
16) I(M,UNM0,UNM+) = I(F,UNM0,UNM+)
formation of OTHR households:
17) I(M,UNM0,OTHR) = I(F,UNM0,OTHR)
emigration:
18) X(M,UNM0,REST) = X(F,UNM0,REST)
immigration:
19) N(M,REST,UNM0) = N(F,REST,UNM0)
Constraints for households of type UNM+
marriage:
20) I(M,UNM+,MAR+) = I(F,UNM+,MAR+)
21) I(M,UNM+,MAR+) =
I(M+F,CUNM,CMAR) / [Mean number of children per cohabiting couple]
marriage dissolution:
22) I(M,UNM+,SING) + I(M,UNM+,H1PA) + X(M,UNM+,DEAD) =
I(F,UNM+,SING) + I(F,UNM+,H1PA) + X(F,UNM+,DEAD)
23) I(M,UNM+,SING) + I(M,UNM+,H1PA) + X(M,UNM+,DEAD) =
I(M+F,CUNM,C1PA) / [M.n.o.c.p.c.c.]
exit of last child:
24) I(M,UNM+,UNM0) = I(F,UNM+,UNM0)
formation of OTHR household:
25) I(M,UNM+,OTHR) = I(F,UNM+,OTHR)
emigration:
26) X(M,UNM+,REST) = X(F,UNM+,REST)
27) X(M,UNM+,REST) = X(M+F,CUNM,REST) / [M.n.o.c.p.c.c.]
immigration:
28) N(M,REST,UNM+) = N(F,REST,UNM+)
29) N(M,REST,UNM+) = N(M+F,REST,CUNM) / [M.n.o.c.p.c.c.]
Constraints for households of type H1PA
marriage:
30) I(M+F,H1PA,MAR+) =
I(M+F,C1PA,CMAR) / [Mean number of children per lone parent]
cohabitation:
31) I(M+F,H1PA,UNM+) = I(M+F,C1PA,CUNM) / [M.n.o.c.p.l.p.]
emigration:
32) X(M+F,H1PA,REST) = X(M+F,C1PA,REST) / [M.n.o.c.p.l.p.]
71
immigration:
33) N(M+F,REST,H1PA) = N(M+F,REST,C1PA) / [M.n.o.c.p.l.p.]
Constraints for the formation of marriages and consensual unions
formation of marriages of type MAR0:
34) I(M,CMAR..SING,MAR0) + I(M,UNM0,MAR0) +
I(M,OTHR,MAR0) = I(F,CMAR..SING,MAR0) +
I(F,UNM0,MAR0) + I(F,OTHR,MAR0)
formation of marriages of type MAR+:
35) I(M,CMAR..SING,MAR+) + I(M,UNM+..H1PA,MAR+) +
I(M,OTHR,MAR+) = I(F,CMAR..SING,MAR+) +
I(F,UNM+..H1PA,MAR+) + I(F,OTHR,MAR+)
formation of consensual unions of type UNM0:
36) I(M,CMAR..SING,UNM0) + I(M,OTHR,UNM0) =
I(F,CMAR..SING,UNM0) + I(F,OTHR,UNM0)
formation of consensual unions of type UNM+:
37) I(M,CMAR..SING,UNM+) + I(M,H1PA,UNM+) +
I(M,OTHR,UNM+) = I(F,CMAR..SING,UNM+) +
I(F,H1PA,UNM+) + I(F,OTHR,UNM+)
Note that a constraint for household type MAR+ similar to constraint number
2 for type MAR0 cannot be formulated because MAR+ is an aggregate position.
Hence in the present LIPRO application there is no guarantee that the number
of males who are originally in position MAR+, and who experience the arrival
of an additional child in the household, equals the corresponding number of
females. The same holds for males and females in aggregate positions UNM+,
H1PA, and OTHR.
To illustrate the flexibility of the LIPRO computer program, it should be pointed
out that the form in which the consistency relations are entered as input to the
program is exactly the same as that of the expressions listed here (see chapter
16).
8. FROM DATA TO INPUT
PARAMETERS
8.1 Introduction
A projection of future household positions requires two types of input data:
1. an initial population at the start of the projection interval (in our case
December 31st, 1985);
2. data on jump intensities, or, alternatively, data on jumps and exposed
population from which jump intensities can be estimated.
Our main source of demographic data is the so-called "Woningbehoeften-
onderzoek 1985/1986" ("Housing Demand Survey 1985/1986") or simply
WBO 1985/1986. A short description of the data set is given in section 8.2. The
next two sections describe how the initial population (8.3) and the jump intensi-
ties (8.4) were determined.
8.2 The Housing Demand Survey of 1985/1986 (WBO 1985/1986)
The Housing Demand Surveys ("Woningbehoeftenonderzoeken" or WBO’s) are
conducted by the Netherlands Central Bureau of Statistics (NCBS) at four year
intervals. The 1985/1986 edition started from a sample of 72,071 addresses. The
field work took place during the last few months of 1985 and the first few
months of 1986. Because of non-response, institutional households, and other
factors, the WBO 1985/1986 contains detailed information on 46,730 house-
holds. The data include the household situation of the respondents at the time
the survey was taken, and their household situation one year earlier. A slight
drawback of the WBO 1985/1986 is that the questionnaire focuses on private
households: only a few basic questions are included for persons living in
institutions.
The WBO 1985/1986 gives us the household position of all individuals in the
sampled private households at the survey date. This information was used to
construct the initial population for the simulation, together with data on the
74
distribution by age, sex, and marital status of the WBO respondents living in
institutional households (see section 8.3). The WBO data were corrected so as
to correspond to the observed population structure by age, sex, and marital
status as per December 31, 1985.
Information on jumps between the 11 household positions can be obtained from
variables indicating the household position of each person one year earlier, to
be reconstructed from a small number of "retrospective" questions included in
the questionnaire. Unfortunately, this "retrospective" information is incomplete,
requiring the use of simplifying assumptions and approximation methods.
In all computations employing WBO 1985/1986 data, we used weight factors
provided by the NCBS in order to achieve national representativeness.
8.3 The initial population
The starting point of our projections is the situation as per December 31, 1985.
From the WBO 1985/1986, the number of persons in each of the 11 household
positions can be calculated, by age and sex. Since these WBO 1985/1986 figures
are subject to sampling error, they have been adjusted to bring them in line with
the official NCBS population statistics for December 31, 1985. These statistics
give the population according to age, sex, and marital status.
First, the population statistics were adjusted to eliminate the population living
in institutions, using estimated age, sex, and marital status specific numbers
living in institutions as given by Faessen and Nollen-Dijcks (1989). Next, the
age- and sex-specific numbers from the WBO 1985/1986 were adjusted propor-
tionally over the 11 household positions, equalizing the sum of the numbers in
positions MAR0 and MAR+ to the numbers in marital status "married", and
equalizing the sum of the numbers in the other 9 household positions to the sum
of the numbers in marital states "never married", "widowed" and "divorced".
This procedure resulted in the population in private households as of December
31, 1985, according to age, sex, and household position. Table 8.1 summarizes
this information.
8.4 Estimation of jump intensities
8.4.1. Estimation of transition probabilities
Because the parameters of the household model are jump intensities, whereas
the WBO 1985/1986 provides information on (most household) transitions, an
algorithm was devised to construct intensities from transition data. The first step
was the determination of transition probabilities. If, for each individual in the
sample, his/her household position at some previous point in time were known,
then transition probabilities could be calculated from the simple age- and
75
Table 8.1. The population in private households according to age, sex,
and household position, the Netherlands, December 31, 1985
Females Males House-
0-19 20-64 65+ total 0-19 20-64 65+ total holds
in thousands
cmar 1705 237 0 1942 1792 474 0 2266 -
cunm 29 2 0 31 36 6 0 42 -
c1pa 178 49 0 227 193 114 0 307 -
sing 13 456 411 880 8 502 98 609 1489
mar0 4 865 357 1226 0 778 445 1224 1225
mar+ 3 2145 34 2182 0 2122 67 2189 2186
unm0 8 197 13 217 1 218 12 230 224
unm+ 0 41 0 42 0 44 1 45 44
h1pa 1 235 29 265 0 39 7 46 311
nfra 0 13 18 31 1 18 10 29 -
othr 20 66 47 133 20 80 17 117 89
total 1963 4306 909 7177 2052 4395 656 7104 5567
Source: Data constructed on the basis of WBO 1985/1986.
sex-specific cross-tables of past versus present household positions. (It should
be noted that transition probabilities thus computed are net of emigration and
mortality.) Unfortunately, the variables in the WBO 1985/1986 do not allow an
exact reconstruction of past household positions, at least not in terms of our 11-
cell classification. Therefore, an approximation method was devised.
For each individual, the WBO 1985/1986 gives the following relevant variables
on the household situation one year before the survey date:
- did the individual live in the same household?
- relation to head of household (RELTOHEAD) in which the individual lived
at the time (whether or not this was the same household as the present one),
coded as follows:
1. head of one-person household
2. head of multi-person household
3. married spouse of head
4. unmarried partner of head
5. (step)child of head and/or partner
6. other adult
7. other child
8. not yet born (i.e. born during last year)
9. living abroad (i.e. immigrated during last year);
- marital status;
- the number of individuals who left the household during the past year.
76
The households in the sample fall into one of the following three categories.
1. Households without entries and without exits.
For these households, reconstruction of the state of its members one year
ago is unnecessary. The only point to bear in mind is the possibility of a
transition from an unmarried to a married couple.
2. Households with entries.
For the non-entrants (i.e. the members of the household who were already
present one year earlier), the household position can be found by "sub-
tracting" the entrants from the present household composition.
For the entrants, the previous household position has been approximated by
considering their previous value on the variable RELTOHEAD and assuming
that the age- and sex-specific cross-tabulation of RELTOHEAD versus our
11-cell classification did not change between 1984 and 1985.
Immigrants were not taken into account, as the WBO 1985/1986 data are
not reliable in this respect.
For those individuals born during the year prior to the survey date, we tried
to reconstruct the household position of the mother at the moment of child-
birth. If necessary, we used simplifying assumptions. Once the household
position of the mother has been determined, the household position into
which the child is born follows automatically. If this household position at
birth is crossed with the household position at the survey date, the transition
matrix for age group 0 can easily be calculated.
3. Households with exits.
Here too, an approximation method had to be used since the household
position of the person(s) who left, and consequently the household position
of the remaining household members before the departure, is unknown.
For the stayers, their previous values on the variables RELTOHEAD, marital
status, and household size were considered, assuming an unchanging age-
and sex-specific distribution across the 11 household positions within each
combination of RELTOHEAD, marital status, and household size.
The departed persons do not need separate treatment. If they moved into
another household, they can be assumed to be included in the entrants
discussed under item 2. If they left the population (through death or
emigration), the corresponding jump intensity is estimated from different
sources.
8.4.2. From transition probabilities to jump intensities
The computations thus far have yielded single-year transition matrices for
internal events, for each age/sex group. What we need are intensities, being the
fundamental parameters of the exponential multidimensional projection model
(Van Imhoff, 1990a). The mathematical relationship between a transition matrix
Tand an intensity matrix Mis given by:
77
T= exp[Mh] (8.1)
where h is the length of the observation interval (here 1 year, except for age
group 0 where h is approximately equal to ½). Then
M= log[T] / h (8.2)
where the log of a matrix is defined in terms of its Taylor power series. It may
happen that the latter power series does not converge. In that case the empirical
transition matrix Tis said to be non-embeddable, i.e. inconsistent with the
assumptions of the exponential model (Singer and Spilerman, 1976). In our
application, only four out of 176 transition matrices turned out to be non-
embeddable. By putting the probabilities for some very improbable transitions
equal to zero, embeddability could be achieved for these four cases. Impossible
transition intensities (i.e. those denoted by an asterisk in Table 7.1) were
subsequently put equal to zero.
However, even for embeddable transition matrices, application of expression
8.2 leads to very unreliable estimates of the intensities. This is caused by the
fact that the logarithm of a matrix is very sensitive to small changes in one of
its elements. Since the empirical transition matrices Tare subject to a high
degree of sampling error, the resulting intensities exhibit a very irregular and
unrealistic pattern when plotted as a function of age.
Therefore it was decided to follow a different approach. This approach rests on
the assumption that an observed transition can be identified with an event. That
is, it is assumed that each individual experiences one event at most during the
observation period. Since the observation period is rather short (1 year), this
assumption appears to be quite reasonable. The only exceptions were made for
transitions that are impossible events according to the events matrix of Table
7.1. For example, if a woman was observed to be in position UNM0 one year
before being observed to be in state MAR+, it has been assumed that she
experienced two events, namely first the event from UNM0 to MAR0 and then
the event from MAR0 to MAR+. A full list of these assumptions on multiple
events in the case of "impossible transitions" is given in Table 8.2.
From the events matrices constructed in this way, intensity matrices can be
estimated using the moment estimator developed by Gill (1986). The computer
program for estimating intensities from events matrices of any rank is described
by Van Imhoff (1989).
The intensity matrices obtained in this way refer to internal events only. In order
to estimate household position-specific mortality and emigration intensities, we
used marital status as an approximation. From the NCBS population statistics
1981-1985, marital status-specific exit intensities were estimated using the
method of Gill and Keilman (1990). These intensities were subsequently trans-
formed into intensities by household position, using the age-
Table 8.2. Assumptions on multiple events
78
Observed transition Assumed events
cmar cunm cmar c1pa cunm
mar0 cmar mar0 sing cmar
mar0 cunm mar0 sing cunm
mar0 c1pa mar0 sing c1pa
mar0 unm0 mar0 sing unm0
mar0 unm+ mar0 sing unm+
mar0 h1pa mar0 mar+ h1pa
mar0 nfra mar0 sing nfra
mar+ cmar mar+ sing cmar
mar+ cunm mar+ sing cunm
mar+ c1pa mar+ sing c1pa
mar+ unm0 mar+ sing unm0
mar+ unm+ mar+ h1pa unm+
mar+ nfra mar+ sing nfra
unm0 cmar unm0 sing cmar
unm0 cunm unm0 sing cunm
unm0 c1pa unm0 sing c1pa
unm0 mar+ unm0 mar0 mar+
unm0 h1pa unm0 unm+ h1pa
unm0 nfra unm0 sing nfra
unm+ cmar unm+ sing cmar
unm+ cunm unm+ sing cunm
unm+ c1pa unm+ sing c1pa
unm+ mar0 unm+ sing mar0
unm+ nfra unm+ sing nfra
h1pa cmar h1pa sing cmar
h1pa cunm h1pa sing cunm
h1pa c1pa h1pa sing c1pa
h1pa mar0 h1pa sing mar0
h1pa unm0 h1pa sing unm0
h1pa nfra h1pa sing nfra
nfra cmar nfra sing cmar
nfra cunm nfra sing cunm
nfra c1pa nfra sing c1pa
nfra mar0 nfra sing mar0
nfra mar+ nfra sing mar+
nfra unm0 nfra sing unm0
nfra unm+ nfra sing unm+
nfra h1pa nfra sing h1pa
and sex-specific marital status distribution for each household position as
weights. A similar approximation method was used to produce estimates of the
79
immigrant population by household position from immigration statistics
1981-1985 by marital status.
Since the number of estimated intensities is very large compared with the
number of observations, the resulting estimates are subject to large random
variations. In order to reduce this variation, the one-year/single age group
intensities were transformed into five-year/five-year age group intensities. A
secondary advantage of this transformation is that it reduces the number of
computations for a given projection by a factor of 25. The transformation
involves a weighted average of the single-year intensities, using the average
population (over the year) in each household position as weights, and taking into
account the fact that a five-year age group over a period of five years involves
9 different one-year age groups. This procedure was applied to internal intensi-
ties, to the exit and fertility intensities, and to the numbers of immigrants, and
resulted in 38 sets of intensity matrices (i.e. 2 sexes and 19 age groups, 18
ranging from 0-4 to 85+ and one for the age group born during the five-year
period).
Finally, intensities for jumps from CMAR, CUNM or C1PA to SING for age
groups 50-54 and over were made equal to one. No observations are available
for these events (and thus intensities cannot be computed). The procedure
sketched here guarantees that an ever-growing group of older "children" during
the projections is avoided.
8.4.3. Adjusting the intensities to achieve internal and external consistency
The five-year intensities were used to make a household projection over a single
projection interval, i.e. the five-year period 1986-1990. Not surprisingly, the
projected numbers of events failed to satisfy the conditions for internal
consistency. In addition, the results on vital events in several respects diverged
from the official numbers of the NCBS, i.e. the sum of observed numbers for
the years 1986-1987, and the corresponding numbers in the national population
forecast for the years 1988-1990. Using the consistency algorithm, the numbers
of events projected by LIPRO were adjusted to yield internal consistency, as
well as external consistency with the official numbers on seven vital events:
- number of births;
- number of deaths;
- number of marriages;
- number of marriage dissolutions;
- number of male entries into widowhood;
- number of female entries into widowhood;
- net international migration.
The precise constraints for these events are listed in Table 8.3. From the
internally and externally consistent numbers of events, the intensities were
80
reconstructed using Gill’s algorithm. It is this adjusted set of jump intensities
that constitutes the basis of the projections to be discussed in chapter 10.
Table 8.3. Constraints for external consistency, 1986-1990
{1. Number of live births}
938200 = B(M+F,cmar..c1pa,othr) + B(M+F,sing,c1pa) +
B(M+F,mar0..mar+,cmar) + B(M+F,unm0..unm+,cunm) +
B(M+F,h1pa,c1pa) + B(M+F,nfra..othr,othr);
{2. Number of deaths}
610600 = X(M+F,cmar..othr,dead);
{3. Number of marriages}
446700 = I(M,cmar..sing,mar0..mar+) + I(M,unm0,mar0) +
I(M,unm+,mar+) + I(M,h1pa,mar+) + I(M,othr,mar0..mar+);
{4. Number of marriage dissolutions}
427100 = I(M,mar0..mar+,sing) + X(M,mar0..mar+,dead) +
I(M,mar+,h1pa);
{5. Number of new widows}
208500 = X(M,mar0..mar+,dead);
{6. Number of new widowers}
80000 = X(F,mar0..mar+,dead);
{7. Net immigration}
179100 = N(M+F,rest,cmar..othr) - X(M+F,cmar..othr,rest);
9. DEMOGRAPHIC
SCENARIOS
9.1 On the term "scenario"
In daily life, a scenario gives a more or less detailed description of the way the
future evolution of a process is perceived. A film scenario contains the details
of what will result in a movie. A journalist who wants to reveal certain dubious
aspects of a politician’s behaviour imagines how various parties will react to
his reports, and his publication strategy is based upon these perceptions. Both
the film director and the journalist construct a scenario.
A scientific scenario involves a coherent description of the way the researcher
perceives the future. Thus a scenario is an instrument used in futures studies.
A scenario may be of a quantitative or of a qualitative nature, depending on the
process involved. Envisioning the geo-political constellation in the 21st century
requires a qualitative scenario. Demographers who explore future population
trends formulate quantitative scenarios.
The examples given here illustrate that a scenario differs from a forecast. A
forecast is an unconditional statement of the most likely future trends,
formulated on the basis of insights into current processes. A scenario is
conditional: it develops the consequences of the assumptions that are made.
More specifically, the scenarios reported here are a quantification of assumed
demographic patterns regarding household events and vital events, and they
indicate how household structures will develop in the future if these assumptions
are borne out. The scenarios consist of a set of values for jump intensities (and
numbers of immigrants), being the basic parameters (exogenous variables) of
the household projection model presented in chapter 7. These values are formu-
lated for the entire projection period.
9.2 Jump intensities and the multidimensional life table
The value and the implications of a given set of age-specific jump intensities
are difficult to assess, because time series for most of these intensities are
lacking. Hence it is impossible to analyse these parameters for the past, and
82
extrapolations cannot be made (except for such simple events as marriage,
marital birth, and death).
Therefore, multidimensional life table analyses were used as an intermediate step
in the construction of household scenarios. In a life table analysis, the life
course is explored for a fictitious cohort by assuming that the members of this
cohort follow a given set of age-specific jump intensities over their life span.
The life table is a procedure for calculating a large number of summary
indicators, such as life expectancy, the mean number of children, the average
number of years spent in married life, etc.
The life table offers the opportunity to judge a given set of jump intensities on
the basis of corresponding summary indicators which can be easily interpreted.
For instance, assume that a scenario has to be designed with increased propen-
sities (relative to some observed pattern) to start a consensual union, and at the
same time with decreased marriage propensities. The problem then is by how
much the consensual union propensities should be increased, given the fact that
the scenario should result in reasonable values for all summary indicators. A
50 percent growth in consensual union propensities resulting in proportions ever
married below 30 percent would clearly be unrealistic.
The following life table indicators were used to assess the various scenarios
(values as of 1985 are given in parentheses):
1. Mortality
life expectancy at birth, males (70.9)
life expectancy at birth, females (78.1)
2. Fertility
total fertility rate (1.391)
proportion of births outside wedlock (8%)
3. Marriage
proportion ever-married, males (72.2%)
proportion ever-married, females (79.1%)
9.3 Five demographic scenarios
Five demographic scenarios were explored. In this section they will be described
verbally, whereas section 9.4 shows how they were quantified. Model parame-
ters for which an explicit trend has been extrapolated until a certain year in the
1This value has been calculated on the basis of WBO 1985/1986 information.
Three factors explain why it differs slightly from the TFR calculated on the
basis of vital statistics (1.51 in 1985, see NCBS, 1990, p. 21): the WBO
estimate is subject to sampling error; it is obtained from surviving women
only; and it has been calculated on the basis of 5-year age groups.
83
future (e.g. 2010 for mortality in four scenarios) are kept at a constant level
from that year until the year 2050, the end of the projection period.
9.3.1. Constant Scenario
In the Constant Scenario, it is assumed that all intensities (and numbers of
immigrants) are constant during the entire projection period, except for possible
adjustments due to consistency relations. Thus the Constant Scenario illustrates
how the population in private households in the Netherlands would evolve, if
household dynamics as observed in the year 1985 would also apply to the
future.
9.3.2. Realistic Scenario
The Realistic Scenario fits in, as much as possible, with the demographic
expectations contained in the official population forecast of the Netherlands
Central Bureau of Statistics (the medium variant of the 1989-based forecast, see
Cruijsen, 1990). The Realistic Scenario does not include extreme trends, as
demographic developments are extrapolated smoothly into the future. The
following "reasonable" assumptions were made:
1. Mortality:
A gradual further increase in life expectancy until the year 2010. The
increase for males is somewhat stronger than that for females. Thus the
difference in life expectancies between the sexes will diminish slightly.
2. Fertility:
A moderate growth of period fertility rates until the year 2025, to a level
which is still much lower than replacement level. At the same time a steep
increase in the proportion of extra-marital births.
3. Household formation:
A decrease in the proportion ever-married by about 8 percentage points (to
the year 2025). The accompanying drop in first marriage intensities is
completely compensated for by higher intensities to start a consensual union.
Thus a modest substitution of formal marriages by consensual unions is
anticipated, which is also expressed by a firm increase in the fertility of
cohabiting couples. Moreover, remarriage propensities for lone parents are
slightly raised.
4. Household dissolution:
A rise until the year 2025 of divorce propensities (which are low compared
to other countries), both for marriages and consensual unions.
5. International migration:
A modest decline of immigration numbers and of emigration propensities
by 10 percent until the year 2000.
84
9.3.3. Swedish Scenario
In the Swedish Scenario, the consequences for the Netherlands are traced of a
household formation and dissolution pattern which tends towards the pattern
observed in Sweden in 1985. When compared with the Realistic Scenario, the
Swedish Scenario is characterized by:
- relatively little importance attached to formal marriage and, at the same time,
much more emphasis on consensual unions;
- a rather high instability of affective relationships; and
- a relatively high fertility.
9.3.4. Fertility Scenario
The Fertility Scenario is identical to the Realistic Scenario, with the exception
of fertility assumptions. In the Fertility Scenario, the Total Fertility Rate
increases to replacement level (2.1 children per woman) over a period of 40
years. Given current fertility conditions in the Netherlands, this is a rather
extreme assumption.
9.3.5. Mortality Scenario
The Mortality Scenario is an alternative to the Realistic Scenario as well.
Compared with the latter scenario, an additional rise in life expectancy is
assumed: 1 year extra for females, 3 years extra for males.
9.4 Quantification of the scenarios
The upper panel of Table 9.1 presents the target values of various summary
indicators in the scenarios.
In multidimensional models such as LIPRO, many variables show strong
interrelations. For instance, a reduction in marriage propensities affects not only
proportions married, but through resulting changes in the household structure
of the population, also fertility (fertility intensities for married persons are higher
than those for unmarried persons) and mortality (unmarried persons have lower
survival probabilities than married persons). Therefore, it is impossible to
determine unequivocally how intensities should be adjusted in order to obtain
a given value of a certain summary indicator. This explains why a "trial-and-
error" approach was followed when determining appropriate values for the
intensities.
All intensities for a given event were adjusted proportionally, irrespective of age.
This implies an unchanged age pattern for the event in question. The lower
panel of Table 9.1 shows proportional adjustment factors which resulted in
target values for summary indicators as presented in the upper part of the table.
Note the strong adjustment for the fertility of cohabiting women without chil-
dren. The reason is the low level of fertility for this group in the Netherlands
85
in the mid-1980s. As formal marriages will gradually be substituted by consen-
sual unions, cohabiting couples will tend to show a childbearing behaviour
closer to married couples than at present. Even a modest degree of substitution
requires rather strong adjustments in fertility intensities of cohabiting persons.
The adjustment factors shown in Table 9.1 apply to the projection intervals from
the year 2025 onwards, and in some cases even earlier. Adjustment factors for
intervals between 1986-1990 and the target year develop according to a straight
line, starting from 1 and ending at the values shown in the table.
86
Table 9.1. Key indicators in the five scenarios1
Constant Realistic Swedish Fertility Mortality
(1985)
Target values for the year 2025,
life table indicators:
Life expectancy at birth:
males (in 2010) 70.9 75.0 75.4 R 78.0
females (in 2010) 78.1 81.5 81.7 R 82.5
Total fertility rate 1.39 1.65 1.90 2.10 R
Proportion births outside wedlock 8% 33% 50% R R
Proportion ever-married:
males 72% 64% 67% R R
females 79% 70% 67% R R
Target values for the year 2025,
intensities (relative to the 1985 value):
Consensual unions * * * *
Remarriage of lone parents +33% +50% R R
Divorce of formal marriages +50% - R R
Divorce of married couples with children - +100%
Separation of cohabiting partners +50% - R R
Emigration (in 2000) -10% R R R
Immigration (in 2000) -10% R R R
Adjustment factors for intensities:
Mortality:
males (in 2010) -31% -34% R -45%
females (in 2010) -27% -28% R -33%
Marriage:
lone parents +33% +50% R R
other * -36% -40% R R
Divorce and separation:
married couples without children +50% +10% R R
married couples with children +50% +100% R R
cohabiting partners +50% +10% R R
Fertility:
married women +20% constant +55% R
single women +80% +300% +133% R
cohabiting women without children +440% +340% +600% R
cohabiting women with children +80% +300% +133% R
lone parents +20% +150% +55% R
other women +20% +100% +55% R
International migration:
emigration (in 2000) -10% R R R
immigration (absolute numbers) (in 2000) -10% R R R
1 R: Realistic Scenario value
* A reduction of marriage intensities is accompanied by a rise in corre-
sponding intensities to form a consensual union, to the extent that the sum
of these two intensities (i.e. the intensity to start a partner relation) remains
unchanged.
10. HOUSEHOLD
PROJECTIONS:
RESULTS
10.1 The Realistic Scenario
Table 10.1 presents results of the Realistic Scenario. Multidimensional models
produce thousands of numbers, when used for projections of the population
according to age, sex, and additional characteristics for a number of years in
the future. Thus, to avoid "hay-stacking" effects in the output, we selected only
a few outcomes. The initial population is shown, as well as that for the years
2000, 2015 (when the youngest members of the post-World War II cohorts reach
retirement age), 2035 (when aging is at its maximum), and 2050.
The development of the total population by household position is illustrated in
Figure 10.1. Total population size reaches its maximum in 2025 (16.4 million),
after which a gradual decline sets in. Developments of the population by
household position are characterized by:
- a decrease in the number of children;
- a strong decline in the number of couples with children;
- an increase in the number of lone parents which is modest in the absolute
sense, but much stronger in the relative sense;
- a diminishing average household size;
- an enormous growth in the number of individuals living in a one-person
household.
The tremendous increase in the number of persons living alone is the most
striking result of the application of the LIPRO model to household projections
in the Netherlands, irrespective of the scenario chosen. Regarding this particular
household trend we can speak of a forecast, since there is little uncertainty left:
a variety of input scenario’s persistently resulted in an upward trend regarding
persons living alone.
88
Table 10.1. Results of the Realistic Scenario
Females Males
31 Dec. House-
1985 0-19 20-64 65+ total 0-19 20-64 65+ total holds
in thousands
cmar 1705 237 0 1942 1792 474 0 2266 -
cunm 29 2 0 31 36 6 0 42 -
c1pa 178 49 0 227 193 114 0 307 -
sing 13 456 411 880 8 502 98 609 1489
mar0 4 865 357 1226 0 778 445 1224 1225
mar+ 3 2145 34 2182 0 2122 67 2189 2186
unm0 8 197 13 217 1 218 12 230 224
unm+ 0 41 0 42 0 44 1 45 44
h1pa 1 235 29 265 0 39 7 46 311
nfra 0 13 18 31 1 18 10 29 -
othr 20 66 47 133 20 80 17 117 89
total 1963 4306 909 7177 2052 4395 656 7104 5567
Females Males
31 Dec. House-
2000 0-19 20-64 65+ total 0-19 20-64 65+ total holds
in thousands
cmar 1522 312 0 1833 1578 579 0 2157 -
cunm 49 3 0 52 63 8 0 71 -
c1pa 222 140 0 361 227 292 0 520 -
sing 0 677 615 1292 0 777 190 967 2259
mar0 1 1010 480 1490 0 903 585 1488 1489
mar+ 1 1948 45 1994 0 1951 50 2001 1997
unm0 0 178 5 183 0 187 9 196 190
unm+ 0 60 2 61 0 62 2 64 63
h1pa 0 408 38 445 0 92 14 106 551
nfra 0 23 14 37 0 26 6 32 -
othr 22 78 32 132 25 90 9 124 91
total 1816 4836 1230 7882 1893 4967 867 7727 6641
Females Males
31 Dec. House-
2015 0-19 20-64 65+ total 0-19 20-64 65+ total holds
in thousands
cmar 1332 319 0 1650 1377 578 0 1955 -
cunm 82 4 0 86 97 9 0 106 -
c1pa 232 190 0 422 241 378 0 620 -
sing 0 927 841 1768 0 960 332 1293 3061
mar0 1 1039 649 1689 0 893 793 1686 1687
mar+ 1 1610 58 1669 0 1607 69 1676 1672
unm0 0 195 6 202 0 197 17 214 208
unm+ 0 97 4 101 0 98 5 104 102
h1pa 0 429 62 491 0 134 25 159 650
nfra 0 26 16 42 0 43 11 54 -
othr 23 80 35 138 27 101 12 140 98
total 1670 4916 1670 8257 1742 5000 1265 8007 7480
89
Table 10.1. Results of the Realistic Scenario (end)
Females Males
31 Dec. House-
2035 0-19 20-64 65+ total 0-19 20-64 65+ total holds
in thousands
cmar 1213 255 0 1468 1244 480 0 1724 -
cunm 155 5 0 160 173 10 0 183 -
c1pa 284 195 0 479 305 392 0 697 -
sing 0 974 1324 2298 0 1003 585 1588 3886
mar0 1 790 750 1541 0 638 900 1538 1539
mar+ 1 1309 55 1365 0 1304 68 1372 1368
unm0 0 204 10 214 0 194 33 227 220
unm+ 0 150 9 159 0 153 9 162 161
h1pa 0 408 71 479 0 142 35 178 657
nfra 0 27 26 53 0 46 25 71 -
othr 22 74 52 147 26 101 24 151 106
total 1675 4390 2297 8362 1748 4463 1680 7891 7938
Females Males
31Dec. House-
2050 0-19 20-64 65+ total 0-19 20-64 65+ total holds
in thousands
cmar 1145 253 0 1398 1171 465 0 1636 -
cunm 164 7 0 171 183 13 0 196 -
c1pa 288 204 0 492 311 414 0 725 -
sing 0 1000 1310 2311 0 992 569 1562 3872
mar0 1 739 615 1355 0 610 742 1353 1354
mar+ 1 1224 42 1267 0 1221 53 1274 1271
unm0 0 211 9 220 0 195 38 233 226
unm+ 0 151 11 162 0 156 9 164 163
h1pa 0 402 60 462 0 151 32 183 644
nfra 0 27 31 58 0 46 26 72 -
othr 21 72 51 144 25 100 24 148 104
total 1619 4289 2131 8039 1690 4364 1493 7546 7634
This trend is even stronger when the number of households is considered,
instead of the number of persons by household position. Figure 10.2 shows the
development in the number of households of various types for the Realistic
Scenario. The increase in the proportion of one-person households is dramatic:
from 27 percent in 1985 to no less than 51 percent in 2050.
The rise in the number of persons living alone goes hand in hand, to a large
extent, with the general aging of the population. In 1985, 34 percent of the
persons living alone was aged 65 or over; in 2035 and in 2050 the share is 49
percent. Also note the diminishing share of the traditional family in Figure 10.2,
i.e. the married couple with one or more children. Changes in the age structure
only partially explain this: elderly couples are more frequently in the "empty
nest" phase than younger couples. However, changes in household formation
patterns are more important: more couples remain childless, less persons marry,
and more marriages are dissolved at a relatively early stage.
90
Figure 10.1. The population in private households by household position,
the Netherlands, 1985-2050 (Realistic Scenario)
Figure 10.2. Private households by type, the Netherlands, 1985-2050
(Realistic Scenario)
91
Figure 10.3. Age pyramid for some household positions, 1985
Figure 10.4. Age pyramid for some household positions, 2035
(Realistic Scenario)
92
The age pyramids in Figures 10.3 and 10.4 further illustrate the age-specific
developments in household structure between 1985 and 2035. The strong growth
of the number of elderly persons living alone comes out very clearly. Improved
longevity is responsible for the rising numbers of elderly couples (both married
and in consensual union) without children. Unmarried cohabitation will slowly
become more popular among the elderly in the Realistic Scenario, but the
numbers involved will remain low, as Table 10.1 indicates. Stated differently,
the mean age of cohabiting persons will rise in the first half of the 21st century,
especially among males.
10.2 A comparison of the five scenarios
Main results of the five scenarios for the years 2035 and 2050 are presented in
Table 10.2 and in Figures 10.5 to 10.8.
A comparison between the Realistic Scenario and the Constant Scenario reveals
a large number of substantial differences. The size of the total population in
private households, in particular, differs strongly between these two scenarios:
in 2050, it is almost 2 million persons lower in the Constant Scenario. Also note
the relatively high share, in the Realistic Scenario, of one-person households:
in spite of the higher population size in this scenario, the share is 51 percent
in 2050, as compared to 45 percent in the Constant scenario.
The remaining scenarios were designed as variants of the Realistic Scenario,
and thus they will be compared with the latter scenario. The Constant Scenario,
with its rather unrealistic assumption of constant demographic rates (note, in
particular, the extremely low fertility as of 1985), is much less suitable for
purposes of comparison, and hence will not be used as a benchmark.
Although the Swedish Scenario results in a somewhat larger share of one-parent
families than the Realistic Scenario does, the differences between the two
scenarios are surprisingly small, see for instance Figures 10.5 and 10.6. The
limited impact for household structures in the Swedish Scenario of relatively
high rates for starting a consensual union and the high share of births taking
place outside wedlock is explained largely by the high marriage propensities of
cohabiting couples with children (UNM+) in the Netherlands. In spite of the
large influx into the UNM+ position, this position is rather unstable and a
substantial part of the influx moves on quickly to the position MAR+. This
makes the scenario less Swedish than it perhaps could have been.
Of course, the Fertility Scenario results in a larger total population; compare,
for instance, Figure 10.7 and Figure 10.4. However, the consequences for
household structures are very small indeed. The most important effect of
increased fertility is a higher average household size for households with
children; yet the share of such households in all households remains more or
93
Table 10.2. Results of the various scenarios, 2035 and 2050
Constant Scenario
Females Males
31 Dec. House-
2035 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 83 6 0 10 83 11 0 24 -
cunm 3 0 0 1 3 0 0 1 -
c1pa 13 4 0 5 12 8 0 7 -
sing 0 19 59 25 0 18 31 17 45
mar0 0 24 31 21 0 19 58 22 23
mar+ 0 34 2 19 0 33 4 21 21
unm0 0 4 1 2 0 4 2 3 3
unm+ 0 1 0 1 0 1 1 1 1
h1pa 0 7 4 5 0 2 2 2 7
nfra 0 1 1 1 0 1 1 1 -
othr 1 2 2 2 1 1 1 2 1
in thousands
100% 1433 4317 1939 7689 1492 4346 1345 7183 7117
Constant Scenario
Females Males
31 Dec. House-
2050 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 84 6 0 19 83 11 0 24 -
cunm 3 0 0 1 3 0 0 1 -
c1pa 12 4 0 4 12 8 0 7 -
sing 0 19 62 26 0 18 32 17 45
mar0 0 24 28 20 0 20 56 22 22
mar+ 0 33 2 19 0 33 4 21 21
unm0 0 4 1 2 0 4 2 3 3
unm+ 0 1 0 1 0 1 1 1 1
h1pa 0 7 3 5 0 2 2 2 7
nfra 0 1 1 1 0 1 1 1 -
othr 1 2 2 2 1 2 1 2 1
in thousands
100% 1319 4077 1746 7142 1373 4100 1170 6644 6612
94
Table 10.2. Results of the various scenarios, 2035 and 2050 (continued)
Realistic Scenario
Females Males
31 Dec. House-
2035 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 72 6 0 18 71 11 0 22 -
cunm 9 0 0 2 10 0 0 2 -
c1pa 17 4 0 6 17 9 0 9 -
sing 0 22 58 27 0 22 35 20 49
mar0 0 18 33 18 0 14 54 20 19
mar+ 0 30 2 16 0 29 4 17 17
unm0 0 5 0 3 0 4 2 3 3
unm+ 0 3 0 2 0 3 1 2 2
h1pa 0 9 3 6 0 3 2 2 8
nfra 0 1 1 1 0 1 1 1 -
othr 1 2 2 2 1 2 1 2 1
in thousands
100% 1675 4390 2297 8362 1748 4463 1680 7891 7938
Realistic Scenario
Females Males
31 Dec. House-
2050 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 70 6 0 17 69 11 0 22 -
cunm 10 0 0 2 11 0 0 3 -
c1pa 18 5 0 6 18 9 0 10 -
sing 0 23 62 29 0 23 38 21 51
mar0 0 17 29 17 0 14 50 18 18
mar+ 0 29 2 16 0 28 4 17 17
unm0 0 5 0 3 0 4 3 3 3
unm+ 0 4 1 2 0 4 1 2 2
h1pa 0 10 3 6 0 3 2 2 8
nfra 0 1 1 1 0 1 2 1 -
othr 1 2 2 2 1 2 2 2 1
in thousands
100% 1619 4289 2131 8039 1690 4364 1493 7546 7634
95
Table 10.2. Results of the various scenarios, 2035 and 2050 (continued)
Swedish Scenario
Females Males
31 Dec. House-
2035 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 61 5 0 15 59 10 0 19 -
cunm 15 0 0 3 15 0 0 4 -
c1pa 23 5 0 7 24 10 0 11 -
sing 0 20 57 25 0 23 36 20 48
mar0 0 19 33 19 0 15 53 20 20
mar+ 0 27 2 15 0 27 3 15 16
unm0 0 6 1 3 0 5 3 3 3
unm+ 0 4 1 2 0 4 1 2 2
h1pa 0 12 3 7 0 3 2 2 10
nfra 0 1 1 1 0 1 1 1 -
othr 2 2 2 2 2 2 1 2 1
in thousands
100% 1812 4402 2295 8509 1892 4484 1748 8124 8011
Swedish Scenario
Females Males
31 Dec. House-
2050 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 58 5 0 15 56 10 0 19 -
cunm 16 0 0 4 17 1 0 4 -
c1pa 24 5 0 8 25 11 0 12 -
sing 0 21 61 26 0 23 39 20 49
mar0 0 18 29 17 0 15 48 18 18
mar+ 0 26 2 14 0 25 3 15 15
unm0 0 6 1 3 0 5 3 4 4
unm+ 0 4 1 2 0 4 1 2 2
h1pa 0 12 3 7 0 3 2 2 10
nfra 0 1 1 1 0 1 2 1 -
othr 2 2 3 2 2 2 2 2 1
in thousands
100% 1842 4377 2128 8347 1923 4468 1565 7951 7772
96
Table 10.2. Results of the various scenarios, 2035 and 2050 (continued)
Fertility Scenario
Females Males
31 Dec. House-
2035 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 71 6 0 20 70 11 0 24 -
cunm 10 0 0 2 11 0 0 3 -
c1pa 17 5 0 7 18 9 0 10 -
sing 0 21 57 25 0 22 35 19 48
mar0 0 16 33 16 0 13 53 17 18
mar+ 0 31 3 16 0 30 4 17 18
unm0 0 5 0 2 0 4 2 3 3
unm+ 0 4 0 2 0 4 1 2 2
h1pa 0 10 3 6 0 3 2 2 9
nfra 0 1 1 1 0 1 1 1 -
othr 1 2 2 2 2 2 1 2 1
in thousands
100% 2148 4580 2298 9025 2242 4663 1675 8581 8087
Fertility Scenario
Females Males
31 Dec. House-
2050 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 69 7 0 21 68 12 0 25 -
cunm 11 0 0 3 12 0 0 3 -
c1pa 18 5 0 7 19 10 0 11 -
sing 0 22 61 26 0 22 38 19 49
mar0 0 15 29 14 0 12 59 15 16
mar+ 0 30 2 16 0 29 4 17 18
unm0 0 5 0 3 0 4 3 3 3
unm+ 0 4 1 2 0 4 1 2 3
h1pa 0 10 3 6 0 3 2 2 9
nfra 0 1 1 1 0 1 2 1 -
othr 1 2 2 2 1 2 2 2 1
in thousands
100% 2256 4763 2129 9148 2355 4862 1485 8702 8055
97
Table 10.2. Results of the various scenarios, 2035 and 2050 (end)
Mortality Scenario
Females Males
31 Dec. House-
2035 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 73 6 0 17 71 11 0 21 -
cunm 9 0 0 2 10 0 0 2 -
c1pa 17 4 0 6 17 9 0 8 -
sing 0 22 55 27 0 23 35 21 49
mar0 0 18 36 20 0 14 54 20 20
mar+ 0 30 2 16 0 30 4 17 17
unm0 0 5 0 3 0 4 2 3 3
unm+ 0 3 0 2 0 4 1 2 2
h1pa 0 9 3 6 0 3 2 2 8
nfra 0 1 1 1 0 1 1 1 -
othr 1 2 2 2 1 2 1 2 1
in thousands
100% 1682 4403 2399 8484 1757 4509 1917 8183 8173
Mortality Scenario
Females Males
31 Dec. House-
2050 0-19 20-64 65+ total 0-19 20-64 65+ total holds
percentages
cmar 71 6 0 17 69 11 0 21 -
cunm 10 0 0 2 10 0 0 3 -
c1pa 18 5 0 6 18 9 0 9 -
sing 0 23 58 28 0 23 38 21 50
mar0 0 18 32 18 0 14 49 19 19
mar+ 0 29 2 16 0 28 3 16 16
unm0 0 5 0 3 0 4 3 3 3
unm+ 0 4 1 2 0 4 1 2 2
h1pa 0 9 3 6 0 3 2 2 8
nfra 0 1 1 1 0 1 2 1 -
othr 1 2 2 2 1 2 2 2 1
in thousands
100% 1628 4303 2257 8188 1700 4415 1759 7874 7917
98
Figure 10.5. Population by household position, 1985 and 2050,
Realistic Scenario versus Swedish Scenario
Figure 10.6. Households by household type, 1985 and 2050,
Realistic Scenario versus Swedish Scenario
99
Figure 10.7. Age pyramid for some household positions, 2035
(Fertility Scenario)
Figure 10.8. Age pyramid for some household positions, 2035
(Mortality Scenario)
100
less unchanged. The projection period is too short for the increased fertility to
have any impact on the size and the composition of the elderly population. The
share of this sub-group in the total population will diminish somewhat: for tha
population aged 65 and over the figure is 20 percent in the Fertility Scenario
in 2050, compared to 23 percent in the Realistic Scenario.
The Mortality Scenario results in even stronger aging effects than the Realistic
Scenario. The elderly comprise 25 percent of the total population in 2050.
Household structures are mainly affected for women: numbers of women living
alone (mostly widows) are somewhat lower, and those for women living with
their spouse a bit higher; compare Figure 10.8 with Figure 10.4.
10.3 The effect of a particular model specification
The LIPRO model for the projection of households contains a number of
innovations, compared to other demographic household projection models. Two
important aspects of LIPRO are, firstly, its focus on household events for
individuals, and secondly, the use of constant intensities (the exponential model)
during the unit projection interval. As a consequence of the first aspect, it was
necessary to formulate consistency relations for various household events to
satisfy. Thus the occurrence of "impossible" numbers of household events was
avoided as much as possible. The use of the exponential model is attractive from
a theoretical perspective, but it requires much more computer time than the
linear model.
We investigated the sensitivity of projection results for alternative model
assumptions: consistency vs. no consistency, and the exponential vs. the linear
model. Table 10.3 shows projection results for the Constant Scenario in the year
2035. When the consistency algorithm is switched off in the program, important
differences arise for married persons and for children in one-parent families;
compare columns 3-6. After 50 years, one observes a shortage of 50,000
married males without children, and a surplus of about the same amount among
married males with children. Because birth rates and death rates differ by
household position, total population size is also affected when consistency
constraints are left out: it is 150,000 persons less in 2035.
The use of the linear model also leads to changes in the results, albeit smaller
than in the previous case (compare columns 3 and 4 with columns 7 and 8). In
particular, differences for single persons and for children can be observed, and
hence also for the total population.
101
Table 10.3. Population by sex and household position, 1985 and 2035
(Constant Scenario), three model specifications
2035
1985 Exponential, Exponential, Linear,
consistency no consistency consistency
FM FM FM FM
in thousands
cmar 1942 2266 1456 1723 1446 1709 1416 1691
cunm 31 42 40 54 35 51 39 53
c1pa 227 307 348 518 324 471 333 516
sing 880 609 1954 1218 1932 1189 2011 1287
mar0 1226 1224 1608 1606 1575 1626 1627 1625
mar+ 2182 2189 1496 1504 1528 1477 1480 1489
unm0 217 230 186 198 161 221 188 200
unm+ 42 45 46 49 45 52 46 49
h1pa 265 46 385 121 405 112 385 123
nfra 31 29 42 58 42 57 40 59
othr 133 117 129 133 135 127 130 131
total 7177 7104 7689 7183 7629 7091 7695 7223
10.4 Comparison with the official NCBS population forecast
In constructing the Realistic Scenario, we have tried to adhere, as much as
possible, to the demographic expectations contained in the official population
forecast of the Netherlands Central Bureau of Statistics (1990). Although quite
a number of differences exist between the LIPRO model and the NCBS
projection model, both projections should produce at least roughly comparable
results.
Table 10.4 compares the most important results of the NCBS projection
(medium variant, projection 1990; see De Beer, 1990) with those of the Realistic
Scenario. As far as the younger age groups are concerned, the results are
reasonably close; there are some minor differences, notably for the age group
15-39, that could well be attributed to differences in the time path of the
assumed demographic trends, and perhaps also to differences in the age pattern
of international migration.
102
Table 10.4. Comparison of LIPRO projection with NCBS forecast
(in thousands)
NCBS forecast 1990, medium variant, population per January 1
1991 1995 2000 2005 2010 2020
0-14 years 2735 2834 2952 2958 2809 2504
15-39 years 6007 5906 5678 5402 5181 5148
40-64 years 4331 4638 5065 5556 5944 5801
65-79 years 1497 1559 1650 1694 1825 2397
80 years and over 438 484 515 582 618 666
total 15008 15420 15860 16192 16377 16517
LIPRO, Realistic Scenario, population per December 31
1990 1995 2000 2005 2010 2020
0-14 years 2719 2763 2779 2691 2565 2471
15-39 years 5977 5811 5565 5315 5081 4942
40-64 years 4317 4712 5161 5611 5945 5715
65-79 years 1455 1549 1629 1700 1853 2466
80 years and over 307 387 456 563 639 790
total 14774 15223 15589 15880 16082 16385
For the older age groups, the differences between the two projections are
significantly larger. The numbers in the LIPRO projection are initially below
those of the NCBS forecast, which can be explained by the fact that the LIPRO
calculations do not take into account the population in collective households.
However, aging is more rapid in the LIPRO projections. The most probable
explanation for these differences lies in the assumed mortality trends. Although
the Realistic Scenario was set in such a way that the increase in life expectancy
exactly follows the life expectancy in the NCBS forecast, the corresponding
projected number of deaths is much smaller in the LIPRO projection as
compared to the NCBS forecast.
This rather peculiar phenomenon can be explained from the fact that LIPRO
and the NCBS have a different way of handling mortality differentials between
household positions or marital states. Although the NCBS model recognizes
mortality probabilities that differ across marital states, the model calculates both
life expectancy and the projected number of deaths without taking the composi-
tion of the population by marital state into account; it is only in the second
phase of the projection process that the projected aggregate number of deaths
103
is subdivided into numbers of deaths by marital status. In the LIPRO model,
on the other hand, deaths are immediately attributed to the relevant household
positions. The result is that, with constant mortality rates, changes in the marital
composition of the population do not affect the life expectancy in the NCBS
model, while changes in household structue do affect the life expectancy in the
LIPRO model.
Two groups of hypotheses for explaining differences in mortality rates across
living arrangements (marital status or household position) can be distinguished
(e.g., Beets and Prins, 1985, p. 53). The protection hypothesis states that the
living arrangement explains mortality; single persons die earlier than married
persons because the former have a less healthy life style. The selection
hypothesis argues exactly the opposite: persons with a high mortality risk have
a lower probability of marriage than persons with a normal mortality risk.
From the point of view of the selection hypothesis, changes in the household
structure of the population should not be allowed to influence life expectancy,
while it should be allowed to have an effect according to the protection
hypothesis. Thus, one might conclude that in the NCBS model mortality is
projected from the selection approach, while in the LIPRO model it is projected
from the protection approach.
11. HOUSEHOLD
PROJECTIONS AND
SOCIAL SECURITY
Household projections such as those presented in the previous chapter may be
used for several policy purposes, such as the planning of housing facilities, for
tracing future consumption patterns, etc. In the present study, the projections
served to assess the impact of changes in age structure and household
composition on social security expenditures in the Netherlands. Hence the social
security projections were not an end in itself, and therefore they are by and large
of an illustrative nature.
It should be stressed that social security in the sense used here involves much
more than just old age pensions. Welfare, survivor pensions, unemployment
benefits, and other state subsidies to maintain a certain minimum income level
are also included in the notion of social security. However, not all types of
expenditures will be considered here. The focus will be on those expenditures
that are particularly sensitive to household developments.
Section 11.1 presents a general discussion of the interrelations between social
security and demography. Earlier studies applicable to the Netherlands are most
relevant here, but a few international comparative studies are also reviewed
briefly. Section 11.2 shows how LIPRO has been extended with a social security
cost module. This module consists of a number of so-called social security user
profiles, one for each type of benefit. These user profiles constitute a straight-
forward generalization of the age profiles which were employed earlier in
studies into the impact of aging on social security expenditures (e.g. Holzmann,
1987; IMF, 1987). Section 11.3 contains the results of the illustrative social
security projections.
106
11.1 Demography and social security
11.1.1. General
Social security expenditures in the Netherlands amounted to 115 billion Dutch
guilders in 1988; see Table 11.1. About 93 billion guilders were spent on social
insurance. Within the latter group of regulations, the Ministry of Social Affairs
and Employment of the Netherlands distinguishes the so-called demographic
regulations, which constitute nearly one-third of all social security expenditures.
These demographic regulations involve old age state pensions ("Algemene
Ouderdomswet" or AOW), early retirement schemes ("Vervroegde Uittreding"
or VUT), child allowance ("Algemene Kinderbijslagwet" or AKW), and survivor
pensions ("Algemene Weduwen- en Wezenwet" or AWW). Section 11.2 presents
a more extensive review of the Dutch social security system.
A certain social security scheme falls within the group of demographic
regulations if its eligibility criterion is a demographic one. For instance, age (for
old age state pension/AOW, and for child allowance/AKW), number and age
of the children (AKW), age, marital status, and number of children under
Table 11.1. Social security expenditures in the Netherlands,
1975-1988
1975 1980 1985 1988
in DFl billion
Total social security*52.6 89.8 110.6 114.7
Including
- social insurance 41.7 73.4 86.2 92.8
- social benefits 7.2 11.5 19.6 16.7
Share of total expenditures in
net national income 26% 30% 29% 29%
*Excluding supplementary private pension insurance and regulations.
Source: Financiële Nota Sociale Zekerheid 1990, p. 152.
107
18 (for survivor pension/AWW) are relevant criteria for many of the demo-
graphic regulations. Non-demographic criteria, such as health and household
income or individual income, are applied for eligibility assessment of the
remaining regulations. The most important scheme in the latter group is welfare
("Algemene Bijstandswet" or ABW), which aims at ensuring a minimum income
level.
In the present study, we selected three social security schemes: AOW, AWW,
and ABW. The latter scheme was chosen because, in practice, a large part of
the expenditures are supplied to lone mothers with insufficient income. On the
other hand, we made projections for child allowance (AKW) but these will not
be presented here, because the impact of the household structure of the
population on AKW expenditures is very limited (Van Imhoff and Keilman,
1990a).
A number of authors have examined the link between demography and social
security. Van den Bosch (1987) gives a systematic review, and he also discusses
how these two issues are related to economic developments. Macro-demographic
aspects such as population size and population structure are highly relevant here.
However, micro-demographic aspects should also be considered: individual
demographic behaviour, e.g. divorce, number of children, death, etc. Van den
Bosch distinguishes two components with respect to social security:
- expenditures, incomes, and distributional aspects;
- structure and models of social security.
Consequently, the relation between demography and social security may be
studied at two levels, and in two directions; see Figure 11.1. The present study
analyses the relationship denoted by the upper arrow: the impact of macro-
demographic developments on social security expenditures.
11.1.2. Previous studies for the Netherlands
Table 11.2 shows the development of expenditures for the social security
schemes of AOW, AWW, AKW, and ABW. The increase from DFl 21 billion
in 1975 to the present level of DFl 38 billion is due to a number of factors:
demographic factors (e.g. growth and aging of the population, changes in
household structure), changes in eligibility criteria, variations in benefit levels,
and inflation. Between 1975 and 1988, the population size rose from 13.6
million to nearly 14.7 million persons. The resulting average annual population
growth rate of 0.6 percent is much lower than the average annual growth rate
of 4.7 percent for expenditures of the four schemes presented in Table 11.2 for
the same period. Hence demographic factors other than population size, as well
as non-demographic factors have accounted for the larger share of the growth
in expenditures.
108
Figure 11.1. Interrelations between demographic developments and
social security
DEMOGRAPHIC DEVELOPMENTS SOCIAL SECURITY
* population size
* population structure * expenditure and
income
* distribution
aspects
↑↑
* individual
demographic behaviour * structure and
models of
social security
Based on: Van den Bosch (1987, p. 246).
Table 11.2. Expenditures for and number of recipients of demographic
schemes, the Netherlands, 1975-1988
(see text for abbreviations)
Expenditures (in DFl billion) Recipients (* 1000)
1975 1980 1988 1975 1980 1988
AOW 11.7 19.4 24.9 1159 1280 1893
AWW 1.6 2.5 2.9 162 168 195
AKW 4.4 7.1 5.9 5186 1) 4865 1) 3585 1)
ABW 2) 3.1 4.5 4.0 229 162 216
1) Estimated on the basis of number of households by number of eligible
children.
2) Senior citizens homes and "Rijksgroepsregeling Werkloze Werknemers/-
RWW" excluded.
Source: Financiële Nota Sociale Zekerheid 1990, pp. 8-10, pp. 152-155.
109
International comparative studies, and studies pertaining to the Netherlands (both
to be discussed below) have revealed that, generally speaking, the demographic
component in the expenditures growth is less important than the non-demograp-
hic component. Increases in real benefits and inflation contribute much more
than changes in demographic structure or population growth. These findings
must be considered in conjunction with the rise of the welfare state in the
industrialized world since the 1960s. A number of functions which has previous-
ly been performed by the family and other kin, were increasingly taken over
by other institutions, in particular the state. Old age pensions and welfare are
the most important examples here.
The approach which is generally taken to assess the impact of demographic and
other components on social security expenditures is as follows. The population
is divided into a number of relevant sub-groups, for instance according to age,
or age and marital status. Two factors are determined for each sub-group:
coverage, i.e. the proportion of persons relying on social security within the sub-
group, and (average) level of benefits. Using standardization techniques, the
contribution can be determined of the demographic component (size as well as
demographic structure), the coverage component and the benefits level compo-
nent (in nominal terms or in real terms) to overall expenditures.
Nelissen and Vossen (1984) analysed the impact of demographic factors on
various social security expenditures in the Netherlands between World War II
and 1982, and they considered the consequences of future population trends on
these expenditures. The method used in the present project (see section 11.2)
extends the approach employed by Nelissen and Vossen into two directions: the
variety of future demographic trends is larger, and the household concept is used
instead of marital status.
In chapters 9 and 10 we formulated a number of demographic scenario’s and
computed their implication for the future age and household structure of the
population. Nelissen and Vossen limited themselves to the population forecast
of the Netherlands Central Bureau of Statistics (NCBS), which constitutes a
probable (and hence relatively narrow, in terms of the span of possible futures)
trajectory of the population.
In the population forecast, the NCBS categorizes the population by age, sex,
and marital status. In the present project we go one step further: the individual’s
household position is used instead of his or her marital status, because the
former notion is more relevant regarding eligibility for many social security
allowances than the latter notion is (see section 7.1). In their suggestions for
further research, Nelissen and Vossen made this point as well (see also Van den
Bosch, 1987, p. 251). However, the construction of an adequate household
model was beyond the scope of their project.
The findings of Nelissen and Vossen for the period until 1982 can be
summarized as follows. Expenditures for AOW, AWW, and AKW rose by 67
110
to 85 percent between 1975 and 1982. Non-demographic factors accounted for
a growth of 60 to 63 percent, and inflation (43 percent) was much more impor-
tant than real growth (12 to 16 percent). Demographic effects (growth in
population size, changes in structure by age and marital status) resulted in a
growth of 13 and 15 percent for AOW and AKW, respectively, and only 4
percent for AWW. For AOW and AWW expenditures, these demographic
effects were almost entirely due to changes in population size (of the elderly
and of widows, respectively). Changes in marital status structure of the
population lead to a 5% decrease in AWW expenditures, whereas they had a
negligible impact on AOW. These findings illustrate that for AOW, AWW, and
AKW, demographic factors are much less important than non-demographic
factors. Inflation was the major cause of the rise in expenditures between 1975
and 1982.
Because of a lack of data, Nelissen and Vossen could only analyse ABW
benefits supplied to divorced women. The findings differ strongly from those
for AOW, AWW, and AKW. The increase in the number of ABW benefits (81
percent in the period 1975-1982) was much sharper than that for the other three
schemes. This is entirely due to demographic factors, in particular, growing
numbers of divorced women. Other factors had a negative impact (minus 7
percent).
The relatively modest role of demographic factors in trends in social security
developments was also assessed by Goudriaan et al. (1984). These authors
investigated expenditures for AOW, AWW, ABW (supplied to divorcees), as
well as a number of other schemes for the period 1970-1981. They found a rise
in real expenditures for these schemes of 94 percent. On the basis of
demographic factors alone (i.e. population growth, and changes in population
structure according to age and marital status), the growth would have been only
16 percent.
Two important non-demographic causes for the growth in social security
expenditures are:
1. more extensive coverage;
2. a real growth of benefit levels.
Goudriaan et al. concluded that the first cause was the major one behind the
growth in expenditures for disability schemes. A real growth of benefit levels
is related to economic growth, and leads to an increase in the spending power
of households which solely rely on social security. In fact, a real benefit growth
should be interpreted the same way as inflation.
On the basis of their analysis of observed trends, Nelissen and Vossen projected
trends in future social security expenditures as far as they are related to
demographic factors. Concerning non-demographic factors, these authors
111
assumed no changes in the social security system - in particular this means that
real benefit levels and coverage rates for each population sub-group are kept
constant. Table 11.3 is due to Nelissen and Vossen (1984, pp. 145-146), who
based their calculations on the NCBS population forecast of 1980. Most striking
are the strong rise in AOW expenditures (in particular after 2010, when the
post-World War II baby boom reaches retirement age) and ABW expenditures
for divorced women (until approximately the year 2010).
These findings agree rather well with those from other studies. Van den Bosch
(1987, p. 258) reviewed three projections for social security expenditures for
the period 1985-2020. Although these studies apply to different sets of schemes
(they represent between 48 and 56 percent of all social security expenditures),
the results are basically the same: between 1985 and 2000 a rise in expenditures
of 15-19 percent can be noted, and over the period 2000-2020 the growth is 10-
15 percent. Finally, we mention the conclusions of a committee, installed by
the Netherlands Ministry of Social Affairs and Employment, that reported in
1987 on financial issues related to future old age pensions ("Commissie Finan-
ciering Oudedagsvoorziening"). For social security (AOW, AWW, AKW, ABW
for divorcees, and two disability schemes), the committee noted an increase in
expenditures of 23-30 percent for the period 1985-2030, depending on future
demographic developments. The rise would be particularly strong
Table 11.3. Projected expenditures for AOW, AWW, and ABW, the
Netherlands (see text for abbreviations)
AOW AWW ABW*
in DFl billion
1982 21.4 2.6 -
index (1982 = 100)
1985 104 105 120
1990 113 102 152
2010 139 117 196
2030 197 97 158
* Only for divorced women under 65.
Source: Nelissen and Vossen (1984, pp. 145-146).
112
between 1985 and 2010: 18-21 percent. For AOW alone, the growth in
expenditures would be around 104 percent until 2030 (42 percent until the year
2010). Note how close the latter results are to those of Nelissen and Vossen (see
Table 11.3).
11.1.3. International comparative studies
International comparative analyses into the impact of demographic factors on
future social security expenditures were recently carried out by the OECD and
the IMF; see for instance Holzmann (1987) and Heller et al. (1986). A major
difference between these studies and the Dutch studies reviewed above is that
in the former, demographic effects are restricted to shifts in the age structure,
while in the latter studies the population was broken down by age and marital
status.
Holzmann (1987) investigated public expenditures on pensions in the twenty
OECD countries during the period 1960-1984. Table 11.4 contains a selection
of his main findings. It includes three countries which take an extreme position
with respect to demographic effects on expenditure growth: Ireland, Austria and
Japan. The latter country’s population is aging relatively quickly. Hence in both
sub-periods it shows the strongest demographic effects on expenditure growth:
about twice as much (4.8 and 3.4 percent per annum) as the average OECD
growth (2.3 and 1.6 percent). Irish and, to a certain extent, also Austrian
populations are aging much slower than the OECD countries as a whole. Thus
for the periods 1960-1975 and 1975-1984, respectively, these countries have the
weakest demographic component. In the Netherlands, the demographic effects
are close to the OECD average (2.4 and 1.7 percent). Table 11.4 clearly shows
that the modest contribution of purely demographic effects to public expenditu-
res for pensions in the Netherlands in the past can also be found in other OECD
countries. In the OECD as a whole, the contribution of demographic factors to
real expenditures growth is roughly one-third (2.3/8.4 and 1.6/4.7). For the
Netherlands, the ratio is 21 percent for the period 1960-1975, but for the
subsequent period the demographic component constitutes almost half (47
percent) of the total real growth.
Regarding future trends in (real) public expenditures to old age pensions,
Holzmann assumed that average life expectancies at birth in Member countries
would increase by approximately two years until 2050. This led him to conclude
that expenditures in the OECD would rise by 35 percent in the period 1985-
2010, and by 87 percent during the years 1985-2030. These results agree rather
well with those of Nelissen and Vossen for AOW in the Netherlands; see Table
11.3.
A strong future rise in public expenditures for old age pensions can also be
noted from the study carried out by staff of the IMF. Table 11.5 summarizes
the main findings for a number of industrialized countries. The calculations are
made on the basis of a benchmark scenario, in which birth rates and mortality
rates, as observed at the end of the 1970s, were kept constant.
113
Table 11.4. Average annual growth rate in public expenditure on
pensions in OECD countries
Nominal Real growth
growth Total Components
Demographic Coverage Expenditure
per recipient
percentage points
Unweighted average for all OECD countries
1960-19751) 14.6 8.4 2.3 1.6 4.6
1975-1984 14.7 4.7 1.6 1.5 2.2
Selected OECD countries:
1960-1975
Netherlands 17.4 11.2 2.4 0.1 8.5
Ireland 16.6 8.4 0.8 1.3 6.2
Japan 21.4 13.0 4.8 4.4 2.6
1975-1984
Netherlands 9.2 3.6 1.7 0.9 1.0
Austria 9.3 3.9 -0.5 2.3 2.1
Japan 12.6 11.1 3.4 2.2 5.1
1) Excluding Greece, Portugal, and Spain.
Source: Holzmann (1987, p. 420).
114
Table 11.5. Public expenditures for old age pensions in selected
industrialized countries (index 1980=100)
2000 2010 2025
Canada 154 189 317
France 175 230 325
Federal Republic of Germany 180 218 300
Italy 198 285 498
United Kingdom 159 199 290
United States 145 178 306
Source: Heller et al. (1986, p. 31).
The growth in expenditures in the six selected countries is stronger than that
in the Netherlands: expenditures rise by between 80 (the US) and 185 (Italy)
percent during the period 1980-2010, and by no less than 190 (the UK) to 400
(Italy) percent during 1980-2025. The relatively low growth rates for the Nether-
lands are caused by the rather high birth rates that could be observed in that
country until the mid-1960s. Compared with the six other countries, this leads
to a low proportion of the population aged 65 or over for the first few decades
of the 21st century.
11.1.4. Conclusions
Three general conclusions become immanent on the basis of the existing
literature.
1. The impact of demographic factors on most social security expenditures has
been rather limited in the last few decades. At least as important (if not more
important) were non-demographic factors such as inflation and the level of
the benefits. For expenditures as a proportion of net national income, the
impact of demographic factors is, of course, much stronger.
2. On the basis of demographic factors alone (and assuming no changes in the
social security system), a sharp increase can be expected in public social
security expenditures for the next few decades, in particular for old age
pensions.
3. Concerning methodology, the approach taken in the current project is a
further refinement of earlier approaches. A number of Dutch studies dealt
with the impact of age, sex, and marital status on expenditures. In the
115
current project, this approach is extended to include household position of
the individuals concerned, instead of marital status. Demographic character-
istics other than age and sex are rarely used in international studies.
11.2 Method1
If one is prepared to make certain assumptions about the way in which social
security expenditures are distributed across the population, then demographic
projections can be used to yield projections of social security outlays. Such
assumptions would concern participation rates and average benefit levels for
each cell in the demographic cross-classification table. For both variables we
use the term user profile.
An example may clarify this approach. Let us assume that the number of female
heads of a one-parent family in the age group 40-44 in the Netherlands will
increase from 30,000 now to 50,000 in 1995. A user profile indicates that 80