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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 USER’S 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 User’s 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

dt→0

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

0≤Rank[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, 1≤r≤n. 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 =1⇔state 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,k≤r⇔state i belongs to subset k

k, k>r ⇔state i is the (k-r)-th state not belonging to any subset

(r<k≤r+s)

Step 3

Construct the n by (r+s) transformation matrix P*defined by:

P*

ij =1⇔Bi=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 ∧i≤r)∨( 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, C≤K. 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