Modelling Kinship with LISP - A Two-Sex Model of Kin-Counts

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

Modelling Kinship with USP

a two-sex

model o f Icin-counts

J. Bartlema

L.

Winkelbauer

November 1986 WP-86-69

Working Rzpers are interim r e p o r t s on

work of t h e International Institute f o r Applied Systems Analysis and have received only

lim-

ited review.

Views

or opinions expressed herein do not neces- sarily r e p r e s e n t those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS

2361

Laxenburg, Austria

Foreword

The Population P r o g r a m

at

IIASA deals with various a s p e c t s of population aging phenomena in developed countries. The c r u c i a l problem r e l a t e d to aging i s how to provide s u p p o r t f o r t h e increasing p r o p o r t i o n of t h e elderly. The measure and way of t h i s s u p p o r t depends on t h e kinship p a t t e r n f o r a p a r t i c u l a r population.

The p a p e r develops t h e a p p r o a c h

to

modeling t h e kinship. The r e s u l t s of modeling show t h a t t h e a p p r o a c h c a n b e successfully implemented

to

t h e analysis of t h e family dynamics.

Anatoli Yashin Deputy Leader Population Program

Acknowledgements

W e would like t o e x p r e s s o u r indebtedness

to

Michael Stoto of Harvard University and Doug Wolf of IIASA, who commented upon a n e a r l i e r d r a f t of this paper. W e benefitted from conversations with

a

number of s c h o l a r s o v e r a relatively long period of time, during which t h e r e s u l t s contained in t h i s p a p e r gradually took shape. J o r g e Somoza of CELADE and Hein Moors of NIDI made comments which increased o u r understanding of c e r t a i n a s p e c t s of kinship modelling. T.Pullum provided t h e computer program with which t h e r e s u l t s of t h e GKP m o d e l

were

calculated. The computer division of IIASA w a s v e r y collaborative in helping

to

worm o u r way through t h e suc- cession of bottlenecks w e encountered. In t h i s context a special word of thanks

to

P.Pronay and D.Legault is in it's place.

This p a p e r w a s written

at

IIASA

as

p a r t of a r e s e a r c h projeot c u r r e n t l y being undertaken by t h e Catholic Universtity of Tilburg, The Netherlands, on t h e use of Colleotive Provisions by t h e elderly. The pro- ject i s supervised by prof .G.A.B.Frinking. Funds f o r t h e s t a y in Austria were supplied by t h e subfaculty of social cultural sciences of Tilburg University and by t h e IIASA Foundation in t h e Netherlands.

Jan Bartlema i s a r e s e a r c h associate

at

t h e S b b f a c u l t y of Social Culturcrl Sciences of t h e C a t h o l t c U n t v e r s t t y of n l b u r g , t h e Netherlands.

Lothar Winkelbauer i s r e s e a r c h assistant

at

t h e Advanced Computer A p p l i c a t i o n s p r o j e c t of IIASA.

CONTENTS

1. Introduction

2. Characteristics of LISP 3. Kinship Models

4. The GKP Model: an elaboration

5. Translating Aggregate Measures into Kin Counts

Appendices

1 : An illustrative analysis of stepfamily relations 2 : A note on the sister's riddle

3 : LISP source code f o r assignation of fathers to children 4 : References

Yodelling Kinship with

LISP

a two-sex m o d e l of kin-comb

J. Bartlema and L. Winkelbauer

1.

Introduction

I t i s f r e q u e n t in family sociology and c u l t u r a l anthropology t o conceive of kinship s t r u c t u r e s as a socio-cultural s u p e r s t r u c t u r e on a biological basis

The concept of t h e family w e shall adopt in t h i s aontext is derived from R. Adams' t h e o r e t i c a l discussion (1971) according

to

which t h e r e are t w o distinct dyadic r e l a t i o n s which may b e considered as t h e elementary atoms from which all human kinship s t r u c t u r e s are constructed: t h e mother-child and wife-husband dyads. From a demographic point of view t h i s implies t h a t , given prevailing levels of mortality, t h e p r o c e s s e s of f e r t i l i t y and nuptiality a r e of key importance t o understand and model kinship. While ohildbearing c a n b e considered t o b e essentially a biological f a c t , m a r r i a g e i s a c u l t u r a l phenomenon, a l b e i t o n e with t h e function of regulating a biological f a c t . In t h e

t e r m s

of Firth ''Kinship i s fundamentally a r e i n t e r p r e t a t i o n in social t e r m s of t h e f a c t s of p r o c r e a t i o n and regularized s e x union." (1948).

Schneider (1965) gives a n inventarization and c r i t i q u e of a number of defin- itions of t h i s n a t u r e . The modal applied h e r e adopts s e p a r a t e technical instruments

to

model t h e biological and t h e cultural dimension.

The objective of modelling kinship within a n applied-demographic con- t e x t i s

to

produce a r e p l i o a which makes b e s t use of t h e e ~ i l t k b l e informa- tion and works o u t t h e implications of t h i s input under a n a p p r o p r i a t e set of assumptions with a useful d e g r e e of detail. The output we want t o g e n e r a t e i s of a global n a t u r e . W e would like t o give a g e n e r a l idea of t h e consequences f o r kinship s t r u c t u r e s in society of t h e developments in f e r t i l i t y and

mortality t h a t have b e e n taking place o v e r t h e c o u r s e of t h e c e n t u r y . A second topic of i n t e r e s t i s t h e e f f e c t t h a t a n a l t e r a t i o n of t h e nuptiality s t r u c t u r e from monogamy

to

s e r i a l monogamy would have upon kinship net- works.

Modelling institutions i s becoming more problematic t h a n i t was in t h e r e c e n t p a s t due to t h e f a c t t h a t western societies are undergoing a p r o c e s s of de-institutionalization. The normative, role-defining power of o u r institu- tions with r e s p e c t

to

f o r example t h e formation a n d dissolution of unions, or t h e e n t r a n c e a n d e x i t from t h e workforce, t h e educational system and so f o r t h i s decreasing. In t h e p r o c e s s a l l kinds of hybrid v a r i a n t s of t h e solid institutions of t h e post-war

era

are being g e n e r a t e d . This c o n f r o n t s t h e r e g i s t r a t i o n systems of industrial s o c i e t i e s with novel conceptual ambigui- ties. The social scientist's o b j e c t of study i s becoming increasingly difficult

to

classify into well-defined d i s c r e t e c a t e g o r i e s . I t makes s e n s e in such a situation

to

look f o r tools t h a t

can

g r a s p s o f t material. A programming language t h a t i s p a r t i c u l a r l y suited

to

manipulate symbols r a t h e r t h a n numbers might t h e r e f o r e b e helpful

to

complement existing mathematical a n d s t a t i s t i c a l p r o c e d u r e s in modelling institutions.

I t i s a l s o reasonable, upon experiencing a growing s e n s e of indeter- minacy

to

fall back upon t h e things we d o know by biological necessity:

*

people are b o r n a n d t h e r e f o r e have f a t h e r s a n d mothers, some people e n t e r a f i r s t r e p r o d u c t i v e union ,

people die.

W e r e s t r i c t t h e input of t h e model

to

f e r t i l i t y rates by a g e of p a r e n t , survival-rates by

sex

a n d two-sex first-marriage matrices

,

a l l in 5-year a g e groups.

The model designed

to

transform t h e input we h a v e into t h e o u t p u t w e d e s i r e consists of t w o distinct phases. F i r s t t h e numerical r e l a t i o n s between kin of d i f f e r e n t 5-year a g e g r o u p s are calculated in a two-sex s t a b l e popula- tion. T h e r e a f t e r t h e s e a g g r e g a t e measures

are

t r a n s l a t e d into a hypotheti- c a l population in which e a c h individual i s identified, with h i s or h e r network of n u c l e a r kin. The f i r s t p h a s e of t h e model u s e s s t a n d a r d biomathematical p r o c e d u r e s , while t h e second applies LISP. The f i r s t p h a s e i s macro- analytic, while t h e second u s e s s t o c h a s t i c p r o c e d u r e s . The r e s u l t i s a model with t r a i t s of macro-

as

well as micro-models.

The emphasis in t h i s p a p e r i s upon methodological issues. W e

are

mainly discussing t h e m e r i t s of a model. and not t h e implications of shrink- ing kinship s u p p o r t networks f o r t h e elderly. After a brief introduction into LISP and t h e field of kinship modelling t h e Goodman. Keyfitz, Pullum a p p r o a c h i s summarized a n d a n application discussed. T h e r e a f t e r a simuh- tion p r o c e d u r e i s described. In a n annex a n illustrative application i s p r e s e n t e d , giving a n impression of t h e e f f e c t of a n a l t e r a t i o n from

a

strictly monogamous mating system t o one in which individual lifecycles may contain

t w o

successive reproductive unions.

2. C h a r a c t e r i d i c a of

LISP

If w e define kinship modelling f o r t h e purpose

at

hand as

"

t h e genera- tion of f o r m a l representations of numerical relations between kin

",

then i t is c l e a r t h a t mathematics has traditionally produced t h e

tools to

do i t with.

From t h e classical t h e o r i e s of branching processes,

to

t h e

m o s t

r e c e n t s b chastic micro-models: all a r e mathematical. Meaningful models cannot res- t r i c t themselves however

to

t h e construction of trees and networks of kin- ship but must move into t h e direation of representing t h e dynamics of family formation and dissolution in terms of cultural developments, psychological processes, and group-dynamios. What w e would like

to

model are t h e forces which make t h e oomponents of o u r model -persons- behave as they do. Some of these f o r c e s a r e e x t e r n a l t o these persons and have

to

do with t h e socio-economic s t r u c t u r e s within which they a r e embedded. Others a r e internal: psychological processes, and o t h e r s again have t o do with t h e interaction between t h e external and t h e internal: oultural and socio- psychological variables.

There are numerous verbal theories and empirical studies on such issues, but attempts

to

construct models which work o u t in a formal fashion what t h e implications of c e r t a i n qualitative postulates would b e on a hypothetical population, such models d o not yet exist. They can b e made, but r e q u i r e t h e use of instruments which oan handle symbols as well as numbers.

The conviction t h a t i t might b e useful

to

think in such a direction lies behind t h e decision

to

use a computer language a p p r o p r i a t e f o r this kind of task.

To those of us who are not familiar with t h e approach and who might be wil- ling

to

consider t h e possiblity

to

complement o u r quantitative results with formalized qualitative thought, w e propose

a

brief digression into t h e struc- t u r e of LISP. The r e a d e r interested in kinship modelling 'tout sec' may skip t h e following paragraph. Before presenting t h e introduction

to

t h e pro- gramming language,

w e

hastily add t h a t t h e application of LISP presented in this p a p e r i s of t h e

m o s t

primitive nature.It is a dealaration of intention.

Artificial Intelligence has been described

as

t h e

art

of making t h e computer d o things t h a t would r e q u i r e intelligence if performed by human beings. In 1958

John

M c C a r t h y c r e a t e d t h e programming language LISP (LISt Processing)

to

give t h e pioneers of t h e Artifical Intelligence commun- ity a tool which allows

to

process symbols (i.e. qualitative terms) in addition

to

n u m e r i c a l c a l c u l a t i o n s (i.e. processing quantitative terms) which

are

t h e c e n t r a l aim of conventional programming Lahguages such

as

FOR- TRAN, PASCAL, PL/I or COBOL.

Although of t h e programming languages still

in

use, only FORTRAN is older than LISP one could have said t h a t until very recently LISP

w a s

t h e only A1 language used by A1 programmers.

INTERLISP-D VAX LISP

(Xerox) (DEC

⁾

COMMON LISP

PruuLI SP LM LISP

(VnIX) (LMI)

\ /

MacLI SP

USP 1.5

LISP ( 1 9 9 )

Figure 1: The f a m i l y tree of LISP

McCarthy d e s c r i b e s LISP

as

follows: (McCarthy in B a r r a n d Felgen- baum, 1982b)

1.

Computing w i t h symbolic expressions rather t h a n numbers; t h a t

is,

bit patterns

in

a computer's memory and registers c a n stand for a r b i t r a r y symbok, not j u s t those of ar%thmettc.

2.

List processing, t h a t i s , representing data

as

linked-list structures

in

the machine a n d a s multiLevel l i s t s o n paper.

3.

CvntroL s t r u c t u r e based o n the computation of functions to form more complex functions.

4.

Recursion

^as

a w a y to descdbe processes and problems.

5 .

Representation of LISP programs i n t e r n a l l y

^as

Linked Lists and ezternally

^as

muLtiLevel l i s t s , t h a t i s ,

in

the same f i r m

^as

d l data are represented.

6 . h e

function WAL, w r i t t e n

in

L W itself, serves

^as

a n interpreter for LISPand

^as

a f i r m a L d e n t t i o n of t h e Language.

will

f i g u r e

2: h e

basic LlSP data s t u c t u r e

T h e r e i s n o essentlal d i f f e r e n c e between d a t a a n d programs, h e n c e LISP p r o g r a m s c a n u s e o t h e r LISP p r o g r a m s as d a t a . LISP i s highly r e c u r s i v e , and d a t a a n d programs are r e p r e s e n t e d as nested lists. I t d o e s not always make f o r easy-to-read syntax, but i t allows f o r elegant solutions

to

oomplex problems t h a t are difficult

to

solve in t h e v a r i o u s conventional programming languages.

4

w

I

* w

*

learn LISP NIL

*

We

1

&

T h e r e are only a few basic LISP functions; a l l o t h e r LISP functions are defined in t e r m s of t h e s e basic functions. This means t h a t one c a n easily create new higher-level functions. Hence, one c a n create

a

LISP o p e r a t i n g system and t h e n work up

to

whatever h i g h e r level one wishes

to

g o

to.

Because of t h i s g r e a t flexibility, LISP h a s n e v e r been stand- ardized in t h e way t h a t languages such as FORTRAN and BASIC have.

Instead, a core of basic functions h a s been w e d

to

c r e a t e a wide v a r i e t y of LISP d i a l e c t s (see Figure 1).

LISP i s unique among programming languages in s t o r i n g i t s pro- grams as s t r u c t u r e d data. The basic d a t a s t r u c t u r e s in LISP are t h e

atom,

any d a t a o b j e c t t h a t cannot b e f u r t h e r broken down, and t h e C t Y B node.

Each

atom

h a s

an

associated p r o p e r t y list t h a t oontains informa- tion about t h e atom, including i t s name, i t s value, and a n y o t h e r pro- p e r t i e s t h e programmer may d e s i r e .

A COAL5 node is a d a t a s t r u c t u r e t h a t consists of

t w o

fields, each of which contains a pointer

to

a n o t h e r LISP d a t a object. CONS nodes c a n b e linked t o g e t h e r

to

form d a t a s t r u c t u r e s of any d e s i r e d size or

com-

plexity (Figure 2). To change or extend a d a t a s t r u c t u r e in a LISP list, f o r example, o n e need only

to

change a pointer

at

a CONS node.

Elements of l i s t s need not b e a d j a c e n t in memory

-

i t i s all done with pointers. This not only means t h a t LISP i s modular, i t also means t h a t i t manages s t o r a g e s p a c e v e r y efficiently and f r e e s t h e progmm-

m e r to

create complex and flexible programs.

Conventional programming languages normally consist of sequen- t i a l statements and associated subroutines. LISP consists of a g r o u p of modules, e a c h of which specializes in performing

a

p a r t i c u l a r h s k . This makes i t e a s y f o r programmers

to

subdivide t h e i r e f f o r t s i n t o numerous modules, e a c h of which c a n b e handled independently.

F o r t h i s r e a s o n LISP h a s been used f o r many A1 p r o j e c t s in t h e following fields: Knowledge-based Systems ( E x p e r t Systems), Natural Language Understanding Systems, Computer Vision, Robotics, Gaming Programs, Learning Systems. I t i s used h e r e

to

program t h e assignation of r e l a t i v e s

to

e a c h o t h e r . The dialect of LISP

w e

used i s Franz Lisp.

3. KINSHIP YODELS

Kinship modelling h a s been a topic of i n t e r e s t f o r demographers since t h e discipline emerged from t h e context of mathematics, and t h e behavioral sciences. In a n implicit way notions derived from kinship s t r u c t u r e s are used when r e f e r r i n g

to

t h e n e t reproduction r a t e , t h e total f e r t i l i t y

rate

and so f o r t h . Recently, the subject of modelling kin- ship and household s t r u c t u r e s h a s received renewed attention, c r e a t - ing a body of l i t e r a t u r e , a common t h e o r e t i c a l p e r s p e c t i v e and a

set

of methods. In s h o r t a distinct field of inquiry c a n said

to

b e originating with t h e study of kinship and household s t r u c t u r e s as i t s objective.

No attempt

to

review t h e field will b e given h e r e . (See Keyfitz, 1984; Bongaarts,lg82; d e Vos and Palloni, 1984 ). For o u r purposes however a useful distinction between t h e t y p e s of models used i s between t h o s e based on macro-analytic expressions and those based on micro-simulation procedures. To t h e f i r s t family belong t h e Goodman, Keyfitz ,Pullurn (1974), Krishnamoorty (1979), Le B r a s (1973), Madan (1986) models and s o f o r t h , while t h e second class contains such models as t h e e a r l y Hyrenius models (Hyrenius and Adolfsen, 1964; Hyrenius, Adolfsen, Holmberg 1966, Holmberg, 1968), t h e Universtity of North Carolina POPSIM model t h e Le Bras (1984) model, and t h e Wolf (1986) m o d e l based on s t o c h a s t i c simulation procedures. The analytic models have t h e advantage t h a t t h e mechanics of constructing kin-groups are easily accessed, through mathematical expressions on a n a g g r e g a t e level, while with a stochastic p r o c e d u r e kin-structures r e s u l t

as

t h e less t r a n s p a r e n t outoome of random assignation p r o c e d u r e s on a n indi- vidual basis. On t h e o t h e r hand t h e d e g r e e of detail provided by t h e micro-approach i s s u p e r i o r , since all c h a r a c t e r i s t i c s of a real popula- tion t h a t w e might b e i n t e r e s t e d in c a n b e simulated. F o r example, where i t i s f r e q u e n t

to

have expressions f o r expected values and possi- bly distributions of kin by a g e under r a t h e r simple assumptions in t h e analytic models, more e l a b o r a t e oorrelations between t h e oomponent v a r i a b l e s c a n b e introduced in t h e stochastic models. I t i s possible t o simulate r e l a t i o n s f o r which no analytio expressions c a n b e formulated.

I t h a s generally been less problematic t o e n c o r p o r a t e t h e interac- tion of fertility a n d mortality in kinship models than t h a t of nuptiality.

This i s due

to

a number of f a c t o r s :

while j e r t i l i t y c a n be readily studied

as

a renewable event, n u p t i a l i t y i s u s u a l l y conceptualized

as

a process of entrance a n d e z i t w i t h respect to the i n s t i t u t i o n of marriage Leading to state-time models (eg. Wow ,2986; WilLekens, ShaA, Ramachan- d r a n ,

11882.)

. As a resuLt the compLezCty of the models

i s

increased.

Data requirements p r the encorporation of n u p t i a l i t y are o n e n diJ7tcult to meet, specially if the model specfJ%cation u s e s a g e - s p e M c t r a n s i t i o n rates from never-married to married states, from married to divorced states, from marriage to widowhood a n d vice versa,

as

weLl

as

mortality rates specfJ%c for each civil-status.

m i L e the m a r g i n for ambiguity

as

to the d f l n i t i o n of a live- b i r t h i s small, t h i s c a n not be said of marriage. The d i s t i n c t i o n between registered marrtage and consensual u n i o n s

has

Long been recognized

as

one of degree and not of k i n d (see for exam- ple Van de Walle,

lQ68).

Measuring the age- s p e m c occurrence of matrimony

is

becoming a problem not o n l y

in

countrtes w i t h d f l d e n t statistics, b u t also

in

the so-called injbrmation sodetLes of the west.

The

problem i s not t h a t there are n o data;

the problem i s t h a t the v a l i d i t y of the ir&wmation we have o n

membership of i n s t i t u t i o n s

in

o u r societies i s becoming ques-

tionable ( h r t l e m a a n d Vossen,

11884).

Table 1. E x p e c t e d Numbers of kin in t h r e e simulated t h e o r e t i c a l populations, approximating t h e Netherlands 1939, 1984 a n d t h e CBS middle v a r i a n t f o r e c a s t f o r 2030.

Grand

Grand Grand

Age Daughters Daughters Daughters Mothers S i s t e r s N i e c e s Aunts Cousins

While there c a n be n o doubt that biomathemattcs has developed adequate tools to s t u d y t h e btological aspects of demographic behavior,

this

c a n n o t be said

with

s u c h c o m d e n c e w i t h respect to t h e c u l t u r a l component of demographic variables.

For these reasons i t can b e justified

to

develop a model which attempts t o accomplish t h e following:

Do

as m u c h as possible

with

a n a l y t i c expressions u t t l t z i n g reliable a n d valid in$ormatton w i t h respect to f e r t i l t t y , mor-

tality

a n d entrance i n t o

first

reproductive u n i o n . The interac- t i o n between m o r t d i t y a n d

Jkrtiltty

determines t h e n u m b e r s

of

people

in

btological d y a d i c k t n relattons tn

a

g i v e n popula- tion.

Use

a

tried btomathematicd approach for t h e biological basis a n d explore t h e p o s ~ b t l t t i e s of employing

a

s d e r i n s t r u m e n t to model t h e subtler material of c u l t u r e .

Work o u t t h e consequences o n k i n s h i p networks of

an

altera- t i o n of m a t i n g principles

u s i n g as

simple

an

i n p u t

as

posstble to achieve

this

end. We do not k n o w how n u p t t a l i t y variables

will

develop, a n d yet would l t k e to h a v e

an

t m p r e s s i o n of w h a t t h e sects would be of c e r t a i n possible courses of aggregate behavior. I n o u r appltcation for ezample

an

a l t e r a t i o n from

a

s y s t e m of s t r i c t monogamy w i t h n o remarriage to one w i t h serial monogamy

will

be simulated a n d t h e meets u p o n t h e d e n s i t y of t h e k i n networks will be studied.

With t h e s e goals in mind a model

w a s

developed which

starts

by elaborating on t h e Goodman, Keyfitz, Pullum expressions. The elabora- tion consists in making t h e model age-specifla f o r both t h e participants in t h e kin-dyad under study, and in applying t h e general line of think- ing

to

a two-sex stable population. The original model, i t is understood, i s r e s t r i c t e d

to

t h e single sex case in which i t calculates average numbers of kin with r e s p e c t

to

ego by age. To model t h e consequences of altering mating principles LISP w a s used. The result i s a model which

starts

with macro-analytical expressions and produoes an output which i s similar

to

t h a t of a micro- simulation model. I t gives a hypothetical population in which each individual is identified, and of whom t h e (nuclear) kinship networks

are

specified. An illustrative application t o t h e Netherlands is given.

4. The

GKP

Model, an elaboration.

W e d e p a r t from t h e analytical expressions f o r expected numbers of kin in a female stable population, given by Goodman, Keyfitz and Pul- lum (1974). The a v e r a g e numbers of survivng daughters of women aged a

at

time

t

is simply:

h e r e t(xf r e f e r s

to

t h e lifetable s u r v i v o r s h i p function f o r t h e female population and fj(xf is the f e r t i l i t y rate

at

e x a c t a g e x f o r g i r l children of women. The i n t e g r a l goes from a, t h e beginnning of t h e r e p r o d u c t i v e p e r i o d until a t h e a g e of Ego

at

time

t.

The d o t between b r a c k e t s

at

t h e r i g h t of t h e D symbol f o r d a u g h t e r s indicates t h a t w e are integrating o v e r a l l a g e s of A l t e r . In t h e f e r t i l i t y rate t h e female symbol as a r i g h t u p p e r s u p e r s c r i p t indicates t h a t t h e g e n d e r of t h e ohild i s female while t h e l e f t s u p e r s c r i p t informs u s t h a t t h e

rates

are by a g e of mother, r a t h e r t h a n f a t h e r . In t h e s e e x p r e s s i o n s all s u r - vivorship r e f e r s

to

t h e female population as indicated.

The a v e r a g e number of surviving mothers p e r woman a g e d a

at t

i s

where x i s t h e a g e of t h e mother

at

b i r t h of Ego a n d N,,'(~) r e f e r s

to

t h e number of women of a g e x

at

^time

t-a.

^The

f a c t o r which w e e n c o u n t e r h e r e

as

well as in t h e e x p r e s s i o n s f o r siblings i s a weighting distribution f o r b i r t h of Ego. In t h i s case t h e n i t r e p r e s e n t s t h e distribution of female b i r t h s by a g e of mother. The i n t e g r a l g o e s from t h e beginning

to

t h e end of t h e r e p r o d u c t i v e period.

The a v e r a g e number of s i s t e r s of women i s given as t h e sum of t h e i r e l d e r and t h e i r younger s i s t e r s . These are calculated s e p a r a t e l y beoause a

term

f o r t h e survivorship of mothers h a s

to

b e included in t h e e x p r e s s i o n f o r younger s i s t e r s . The e x p r e s s i o n s f o r e l d e r and younger s i s t e r s are respectively:

The a g e y r e f e r s

to

mother's a g e when s h e had Ego's s i s t e r .

Although t h e Goodman, Keyfitz, Pullum a r t i c l e g o e s o n

to

e x p r e s - sions f o r

m o r e

d i s t a n t kin w e limit o u r s e l v e s in t h i s c o n t e x t

to

t h e n u c l e a r families of origin and p r o c r e a t i o n . (see Keyfitz, 1977 f o r a

more

e l a b o r a t e d e s c r i p t i o n of t h e model, and Keyfitz ,1986 f o r a r e c e n t application)

.

^{W e} note t h a t t h e GKP analytical e x p r e s s i o n s are n o t r e s t r i c t e d

to

t h e s t a b l e case. If w e h a v e c o h o r t survivorhip and t h e weighting distributions f o r b i r t h of Ego are known, e x p e c t e d numbers of kin c a n b e oalculated exactly in a closed population. The assumption of stability i s made h e r e f o r convenience, not by conceptual necessity.