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Stochastic models of changes in population distribution among categories Gerchak, Yigal 1980

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STOCHASTIC MODELS OF CHANGES IN POPULATION DISTRIBUTION AMONG CATEGORIES by YIGAL GERCHAK B.A., T e l - A v i v U n i v e r s i t y , 19 70 M.Sc, T e l - A v i v U n i v e r s i t y , 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF (Fa c u l t y of Commerce and Business A d m i n i s t r a t i o n ) We accept t h i s t h e s i s as conforming DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1980 Y i g a l Gerchak, 1980 In p resent ing t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t ha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and study. I f u r t h e r agree that permiss ion f o r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n permi s s ion . The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 Supervisor^: Dr. Shelby L. Brumelle i i ABSTRACT There are very many processes i n the n a t u r a l and s o c i a l sciences which can be represented as a.set of flows of objects or people between categories of some kind. The Markov chain model has been used i n the study of many of them. The basic form of the Markov chain model i s , however, r a r e l y adequate to describe s o c i a l , occupational and geographical mobility processes. We s h a l l therefore discuss a number of generalizations designed to introduce greater realism. In Chapter I we formulate and investigate a general model which r e s u l t s from relax i n g the assumptions of sojourn-time's memorylessness and independence of o r i g i n and d e s t i n a t i o n states, and of population homogeneity. The model (a mixture of semi-Markov processes) i s then used i n two ways. F i r s t , i t provides a framework i n which various s p e c i a l cases (which correspond to models which were used by s o c i a l s c i e n t i s t s ) can be a n a l y t i c a l l y compared. We pay p a r t i c u l a r a t t e n t i o n to comparisons of rate of m o b i l i t y i n r e l a t e d versions of various models and to compatability of popular parametric forms with observed mobility patterns. Second, any r e s u l t obtained for the general model can be s p e c i a l i z e d for the various cases and subcases. In Chapter II we formulate a system-model allowing i n t e r a c t i o n among in d i v i d u a l s (components), which has been motivated by Conlisk. We define processes on t h i s model and analyze t h e i r properties. A major e f f o r t i s then devoted to e s t a b l i s h i n g that when the population s i z e becomes large, t h i s rather complex stochastic model can be approximated by a s i n g l e deterministic recursion due to Conlisk (1976). Nevertheless, we draw attention to c e r t a i n aspects ( p a r t i c u l a r l y steady-state behavior) in which the approximation may f a i l . i i i In Chapter I II we address ourselves to the issue of measurement of (what we r e f e r to as) s o c i a l inheritance i n intergenerational mob il it y processes. We d i s t i n g u i s h between various aspects and concepts of s o c i a l inheritance and o u t l i n e the implications that c e r t a i n " s o c i a l values" may have on constructing a measure (or index). In the mathematical discussion which follows c e r t a i n mechanisms for generating " f a m i l i e s " of measures are indicated, and the properties of some p a r t i c u l a r combinations are investigated. i v TABLE OF CONTENTS Page ABSTRACT i i ACKNOWLEDGEMENT v i INTRODUCTION 1 CHAPTER I ANALYTICAL COMPARISON OF MOBILITY MODELS IN A HETEROGENEOUS SEMI-MARKOV CONTEXT • 8 1.1 A General Model and Some of i t s P r o p e r t i e s 10 1.2 S p e c i a l Cases 13 1.3 A Comparison of Popu l a t i o n Heterogeneity Induced through T r a n s i t i o n Rates and through P r o b a b i l i t y T r a n s i t i o n Functions 18 1.4 Rate of M o b i l i t y i n the Various Models 19 1.4.1 I n t e r - T r a n s i t i o n Times and Number of T r a n s i t i o n s per Unit Time 20 1.4.2 Diagonal Elements and Eigenvalues 22 1.5 C o m p a t i b i l i t y of Parametric F a m i l i e s of D i s t r i b u t i o n s w i t h Observed P r o p e r t i e s of Career P a t t e r n s 26 1.5.1 Decreasing Rate of I n e r t i a Accumulation . . . . 27 1.5.2 Heterogeneity of Subpopulations Selected by t h e i r Durations 28 CHAPTER I I A STOCHASTIC MODEL ALLOWING INTERACTION AMONG INDIVIDUALS 32 2.1 The F i n i t e P o p u l a t i o n Model and i t s P r o p e r t i e s . . . . 35 2.1.1 Formulation 35 2.1.2 Example 37 2.1.3 P r o p e r t i e s 38 2.1.4 Conditions f o r I r r e d u c i b i l i t y and A p e r i o d i c i t y 41 2.1.5 Some Computational Aspects 44 V Page 2.2 Approximation of the P r o f i l e Process 45 2.2.1 Approximation over F i n i t e Time Horizon . . . . 47 2.2.2 Approximation of Steady State 52 2.3 Weak Convergence of Sequence of Markov Chains . . . . 56 2.4 Concluding Remarks 62 CHAPTER I I I THE MEASUREMENT OF SOCIAL INHERITANCE 63 3.1 P r e l i m i n a r y D i s c u s s i o n 66 3.1.1 Basic Requirements 66 3.1.2 Concepts of S o c i a l I nheritance 67 3.2 Mathematical, Model Based D i s c u s s i o n 69 3.2.1 D e f i n i t i o n s 69 3.2.2 Mathematical Statement of D e s i r a b l e P r o p e r t i e s 71 3.2.3 Measuring the Non-Constancy of the Operator. . 73 3.2.4 Some S p e c i a l Cases 76 3.2.5 From Non-Constancy to S o c i a l I n h e r i t a n c e -Introducing Period-Consistency 84 3.3 Conclusion 88 BIBLIOGRAPHY 89 v i ~ ACKNOWLEDGEMENT I wish to thank my thesis supervisor, Professor Shelby L. Brumelle, for his invaluable help i n t h i s research. His knowledge'and rigorousness as well as patience and f r i e n d l i n e s s were a r e a l asset to me, and I learned a l o t from him. I am also g r a t e f u l to my committee members, Professors K. R. MacCrimmon, I. Vertinsky and J . Walsh f o r t h e i r help and encouragement. This i s an opportunity to thank the following people f o r helping me through the Ph.D. program. To Professors M. A v r i e l and W. T. Ziemba f or "bringing" me to U.B.C.; to Ms. B. Wynne-Edwards and Professor K. R. MacCrimmon for helping me i n obtaining f i n a n c i a l assistance; and to the U.B.C. awards-office f o r granting i t ; to Professor L. G. Mitten f o r en-abling me to get teaching experience and to Ms. Colleen Colclough and Ms. Suzan Altimas f o r excellent typing of t h i s t h e s i s . I would also l i k e t o thank my fellow doctoral students, and e s p e c i a l l y E. Choo, J. Kallberg and R. Solanki f o r t h e i r constant help and f r i e n d l i n e s s . F i n a l l y , I owe a s p e c i a l debt of thanks to my family, without whom a l l t h i s would have been impossible. - 1 -INTRODUCTION Since the e a r l y f i f t i e s , researchers i n v a r i o u s - d i s c i p l i n e s have attempted to devise models to describe the dynamics of s o c i a l systems. The f i r s t models described changes i n v o t i n g behavior (Anderson 1954), s o c i a l m o b i l i t y ( P r a i s 1955) and occupational m o b i l i t y (Blumen et a l . 1955). These models were s t o c h a s t i c i n nature, and used Markov chains as t h e i r main t o o l . Markov chains were used l a t e r to model other types of s o c i a l systems, such as geographical m o b i l i t y [ f o r a recent survey of models used i n v a r i o u s m o b i l i t y processes see Stewman (1976)], d i f f u s i o n of i n n o v a t i o n s , educational systems, and flow of people among d i f f e r e n t s t a t e s of h e a l t h [see Bartholomew (1973) f o r a general survey]. They were a l s o used to model buying behavior (brand s w i t c h i n g ; e.g. Massy et a l . 1970), i n t e r n a l labour markets (e.g. Vroom and MacCrimmon 1968) and h e a l t h d e l i v e r y systems (e.g. Meredith 1973)/""^ In the area of i n t e r g e n e r a t i o n a l s o c i a l m o b i l i t y , i n a d d i t i o n to the i n t e r e s t i n d e v i s i n g d e s c r i p t i v e models, there was an accompanying i n t e r e s t i n measuring m o b i l i t y of c e r t a i n s o c i e t i e s . The f i r s t attempt to devise an index (measure) of m o b i l i t y was made by P r a i s (1955a), and was f o l l o w e d , among others, by Matras (1960), Yasudo (1964), Goodman (1969), Bartholomew (1973, pp. 23-7), Boudon (1973), Pullum (1975) and Sommers and C o n l i s k (1979). Recently Shorrocks (1978) -appfoaclied t h i s i s s u e more.: s y s t e m a t i c a l l y , .setting (*) S i m i l a r work was done i n the b i o s c i e n c e s f o r nonhuman populations (e.g. Thompson and V e r t i n s k y 1975). (**) The term " s o c i a l systems" to be used here should be broadly i n t e r p r e t e d to i n c l u d e a l l the above systems. - 2 -some d e s i r e d p r o p e r t i e s of such i n d i c e s as axioms and making inferences from them. The above mentioned measures were suggested f o r a Markov chain model, and are a c t u a l l y f u n c t i o n s defined on p r o b a b i l i t y t r a n s i -t i o n matrices ("mobility t a b l e s " ) . More r e c e n t l y , however, researchers found i t necessary to g e n e r a l i z e and extend the b a s i c model. In order to understand what motivated t h i s trend, one should r e a l i z e t h a t , i n gene r a l , a new model of a c e r t a i n phenomenon u s u a l l y emerges as a r e s u l t of one of three causes. The f i r s t , and most n a t u r a l to those who view models as devices f o r f i t t i n g data, i s poor f i t (or inadequate p r e d i c t i o n s ) < o f the e x i s t -i n g model. In the area of i n t r a g e n e r a t i o n a l m o b i l i t y , researchers observed an e m p i r i c a l r e g u l a r i t y i n the form of " c l u s t e r i n g on the main diagonal", which standard Markov chain models were not able to account f o r . This prompted some researchers to r e l a x the assumption of "popula-. t i o n homogeneity" ( i m p l i c i t i n e a r l y models), r e s u l t i n g i n the "Mover-Stayer Model" and i t s subsequent extensions (Blumen..ettally. 1955, McFarland 1970, Spilerman 1972a, 1 9 7 2 b ) . ^ Another r a t i o n a l e f o r extending e x i s t i n g models occurs when some be h a v i o r a l p a t t e r n s , which were unknown before, are discovered. T y p i c a l examples, i n the area of geographical m i g r a t i o n processes, i n c l u d e the r e a l i z a t i o n that duration-.in one's current l o c a t i o n reduces the chances, of l e a v i n g i t ("Axiom of Cumulative I n e r t i a " ; McGinnis 1968, Huff and C l a r k 1978), the demonstration that the d e s t i n a t i o n "one moves to- i s not inde-pendent of the d u r a t i o n ' i n one's current l o c a t i o n (Ginsberg 1978a) and (*) S i m i l a r extensions were proposed a l s o i n buying-behavior l i t e r a t u r e (Jones 1973, Givon and Horsky 1978). - 3 -observation of d e c l i n e .of m o b i l i t y r a t e w i t h age (Mayer 1972). An attempt to accomodate such e f f e c t s gave r i s e to the " C o r n e l l M o b i l i t y Model" (McGinnis 1968, Henry et a l . 1971) and, more g e n e r a l l y , to a semi-(*) Markov model (Ginsberg 1971). A somewhat r e l a t e d avenue of research i n v o l v e s hypothesizing some l o g i c a l patterns and using a model to t e s t them. An example i s the work by Wise (1975) on the e f f e c t s of academic achievement on graduates' careers. In order to understand the t h i r d c l a s s of r a t i o n a l e s f o r extend-i n g models, i t i s important to r e a l i z e that "...not a l l mathematical models are intended to f i t e m p i r i c a l data; not i n f r e q u e n t l y mathematical models are developed to work out i m p l i c a t i o n s of p o s t u l a t e s . . . " (McFarland 1974, p. 883). In f a c t , t h i s l a t t e r r a t i o n a l e f o r modelling becomes c e n t r a l when we move from the f i e l d of Sociology (where the above c i t a t i o n appeared) to the p o l i c y sciences, where one f r e q u e n t l y wants to assess f u t u r e impacts of p o l i c i e s which might be implemented. With p a r t i c u l a r a t t e n t i o n to the r a t i o n a l e f o r modelling d i s -cussed i n the above paragraph., and i n order to come up w i t h a compre-hensive model which captures more features of r e a l systems, C o n l i s k (1976) argued i n favor of dropping the assumption of i n d i v i d u a l s "moving" independently of each other, which was i m p l i c i t i n a l l above mentioned models. C o n l i s k formulated a model which allows f o r i n t e r a c t i o n s among i n d i v i d u a l s , and Smallwood and C o n l i s k (1978) made an i n t e r e s t i n g use of some of i t s v e r s i o n s to model adaptive behavior of poorly informed consumers. (*) This has been made p o s s i b l e by the development of the theory of semi-Markov processe°s i n the e a r l y s i x t i e s and by using c e r t a i n notions from r e l i a b i l i t y theory which were developed about the same time. While Markov and semi-Markov models could make d i r e c t use of a v a i l -able s t o c h a s t i c process theory, once pop u l a t i o n heterogeneity or i n t e r -a c t i o n between i n d i v i d u a l s i s introduced, the r e s u l t i n g models have no immediate counterparts i n c l a s s i c a l theory, so some s p e c i f i c a n a l y s i s i s r e q u i r e d . A p a r t i c u l a r problem i s that " T r a d i t i o n a l Markov chain theory p e r t a i n s to a s i n g l e object moving from s t a t e to s t a t e ; but i n a p p l i c a t i o n s to s o c i a l m o b i l i t y and other s o c i a l systems, one considers an e n t i r e p o p u l a t i o n , each person moving p r o b a b i l i s t i c a l l y from s t a t e to s t a t e " (McFarland 1970, p. 463). As long as the processes which i n d i v i d u a l s i n the p o p u l a t i o n f o l l o w were assumed to be ( s t o c h a s t i c a l l y ) the same and independent of each other, the law of l a r g e numbers j u s t i f i e s approximating the p r o p o r t i o n of the p o p u l a t i o n (assumed large) i n a given s t a t e by an i n d i v i d u a l ' s / p r o b a b i l i t y of being i n that s t a t e . Without these assumptions, however, one should be very c a r e f u l when for m u l a t i n g models and attempting to i n f e r p o p u l a t i o n - l e v e l (macroscopic) q u a n t i t i e s from i n d i v i d u a l - l e v e l (microscopic) ones and v i c e versa. This work addresses shortcomings of e x i s t i n g l i t e r a t u r e on three l e v e l s : 1. By a c a r e f u l , f o r m u l a t i o n and a n a l y s i s of models. Since r e l a x i n g the assumption that the i n d i v i d u a l s behave independently of each other seems to be the most reward-i n g i n terms of the spectrum of systems i t may enable us to model, and s i n c e i t seems to be the most d i f f i c u l t step towards a comprehensive system-model, we devote a major e f f o r t to such a model. 2. By p r o v i d i n g a l o g i c a l h i e r a r c h y f o r v a r i o u s models (with - 5 -p a r t i c u l a r a t t e n t i o n to those w i t h heterogeneous populations) w i t h i n the theory of s t o c h a s t i c processes. 3. - By t e s t i n g a n a l y t i c a l l y whether some models indeed have the des i r e d p r o p e r t i e s , and whether they always a l t e r p r e d i c -t i o n s i n a manner which i s c o n s i s t e n t w i t h the way they were motivated. 4. By d i f f e r e n t i a t i n g among v a r i o u s a s p e c t s . o f ' s o c i a l m o b i l i t y , s i n g l i n g out, f o r purposes of measurement, a pure s o c i a l -i n h e r i t a n c e aspect, and approaching i t s y s t e m a t i c a l l y on both s u b s t a n t i v e and mathematical l e v e l s . In Chapter I we formulate and i n v e s t i g a t e a general model which r e s u l t s from r e l a x i n g the assumptions of sojourn-time's memorylessness and independence of o r i g i n and d e s t i n a t i o n states, and of pop u l a t i o n homogeneity. The model (a mixture of semi-Markov processes) i s then used i n two ways. F i r s t , i t provides a framework i n which various s p e c i a l cases (which correspond to models which were used by s o c i a l s c i e n t i s t s ) can be a n a l y t i c a l l y compared. We pay p a r t i c u l a r a t t e n t i o n to comparisons of r a t e of m o b i l i t y i n r e l a t e d v e r s i o n s of various models and to c o m p a t a b i l i t y of popular parametric forms w i t h observed m o b i l i t y p a t t e r n s . Second, any r e s u l t obtained f o r the general model can be s p e c i a l i z e d f o r the various cases and subcases. In Chapter I I we formulate a system-model a l l o w i n g i n t e r a c t i o n among i n d i v i d u a l s (components), which has been motivated by Co n l i s k . We def i n e processes on t h i s model and analyze t h e i r p r o p e r t i e s . A major e f f o r t i s then devoted to e s t a b l i s h i n g that when the popu l a t i o n s i z e becomes l a r g e , t h i s r a t h e r complex s t o c h a s t i c model can be approximated - 6 -by a s i n g l e d e t e r m i n i s t i c r e c u r s i o n due to C o n l i s k (1976). Nevertheless, we draw a t t e n t i o n to c e r t a i n aspects ( p a r t i c u l a r l y steady-state behavior) i n which the approximation may f a i l . In Chapter I I I we r e t u r n to the i s s u e of measurement of (what we r e f e r to as) s o c i a l i n h e r i t a n c e i n i n t e r g e n e r a t i o n a l m o b i l i t y processes. We d i s t i n g u i s h between various aspects and concepts of s o c i a l i n h e r i -tance and o u t l i n e the i m p l i c a t i o n s that c e r t a i n " s o c i a l v a l u e s " may have on c o n s t r u c t i n g a measure (or index). In the mathematical d i s c u s s i o n which f o l l o w s , c e r t a i n mechanisms f o r generating " f a m i l i e s " of measures are i n d i c a t e d , and the p r o p e r t i e s of some p a r t i c u l a r combinations are i n v e s t i g a t e d . Although the models i n Chapters I and I I were motivated on sub-s t a n t i v e grounds, some of the observations and r e s u l t s there may be viewed as con t e x t - f r e e and, h o p e f u l l y , perhaps of an independent proba-b i l i s t i c i n t e r e s t . Chapter I I I , on the other hand, i s geared, essenti^-a l l y , to i n t e r g e n e r a t i o n a l s o c i a l m o b i l i t y issues only. Despite the common sources and r a t i o n a l e s of the v a r i o u s models and problems addressed by t h i s work (which were o u t l i n e d i n t h i s i n t r o -duction) , the t e c h n i c a l aspects of the various chapters (and s e c t i o n s ) vary s i g n i f i c a n t l y . This i s one of the reasons why the three chapters are s e l f - c o n t a i n e d and w i t h l i t t l e c r oss-references. The main mathemat-i c a l " t o o l s " used i n t h i s work are: - Chapter I : Markov and semi-Markov processes (Sections 1.1 -1.3; Ross 1970, C i n l a r 1975); s t o c h a s t i c dominance (sub-s e c t i o n 1. 4. 1; Brumelle and Vickson 1975); s p e c t r a l represen-t a t i o n s (subsection 1.4.2; C i n l a r 1975, Appendix); r e l i a b i l i t y - 7 -theory ( p a r t i c u l a r l y i n s e c t i o n 1.5; Barlow and Proschan 1975). - Chapter I I : Products of f i n i t e non-negative square matrices (subsection 2.1.4; Hajnal 1958); mathematical p r o b a b i l i t y (throughout: Breiman 1968); weak convergence of p r o b a b i l i t y measures ( s e c t i o n 2.2 and, p a r t i c u l a r l y 2.3; B i l l i n g s l e y 1968). - Chapter I I I : Linear Transformations (subsection 3.2.1; Halmos 1958); Orders (subsection 3.2.2; Krantz et a l . 1971); metrics and norms (subsection 3.2.3; Royden 1963); ergodic c o e f f i c i e n t , r a t e of convergence of Markov chains (subsection 3.2.4; Isaacson and Madsen 1976); i n f o r m a t i o n theory (subsection 3.2.4; Khin c h i n 1957); r a t e of convergence notions (subsection 2.3.5; Ortega and Rheinboldt 1970). Due to t h i s r a t h e r " l o c a l " use of v a r i o u s concepts, we have chosen to define concepts and quote r e s u l t s only when we need them. Nevertheless, we attempted to make the work v i r t u a l l y s e l f - c o n t a i n e d . - 8 -CHAPTER I ANALYTICAL COMPARISON OF MOBILITY MODELS IN A HETEROGENEOUS SEMI-MARKOV CONTEXT When Markov chain models f a i l e d to describe c e r t a i n aspects of human m o b i l i t y , and when e m p i r i c a l evidence showed that t h e i r assumptions were not -compatible w i t h human behavior i n c e r t a i n types of s o c i a l systems, researchers suggested two major d i r e c t i o n s of extending them. One was the Mover-Stayer Model and i t s subsequent extensions (Blumen, Kogan and McCarthy 1955, McFarland 1970, Spilerman 1972b), which introduced popu-l a t i o n heterogeneity. The other d i r e c t i o n was semi-Markov models (McGinnis 1968, Ginsberg 1971), which allowed the d i s t r i b u t i o n of time between moves to depend on the o r i g i n and d e s t i n a t i o n s t a t e s . A semi-Markov model has a l s o been used to model movement of personnel through a h i e r a r c h i c a l o r g a n i z a t i o n ( G r i n o l d and M a r s h a l l 1977, Se c t i o n 4.4). Since researchers were t y p i c a l l y i n t e r e s t e d i n e x p l a i n i n g p a r t i -c u l a r phenomena and modeling s p e c i f i c systems, they u s u a l l y formulated r a t h e r s p e c i a l i z e d models (e.g. "cumulative i n e r t i a " of d u r a t i o n l e n g t h - a p a r t i c u l a r form of a semi-Markov process). Even when they r e l a t e d models to each other, the b a s i s of comparison was t h e i r r e l a t i v e success i n f i t t i n g a given set of data. Very l i t t l e was5 done towards arranging the va r i o u s models i n some l o g i c a l order, e s t a b l i s h i n g r e l a -(**) t i o n s among them, and comparing t h e i r p r e d i c t i o n s a n a l y t i c a l l y . (*) Some exceptions are the works of Ginsberg (1971), Singer and Spilerman (1974, 1976) and Schinnar and Stewman (1978). (**) One exception, i n the area of personnel p r e d i c t i o n models, was a t h e o r e t i c a l comparison of a c r o s s - s e c t i o n a l (Markov) model and a l o n g i t u d i n a l (cohort) model by M a r s h a l l (1973) [see a l s o G r i n o l d and M a r s h a l l (1977, S e c t i o n 4.5)]. - 9 -In t h i s chapter, f o l l o w i n g Singer and Spilerman (1974), we formulate a general model (a mixture of semi-Markov processes) which i n c l u d e s a l l of the models mentioned above as s p e c i a l cases. We use t h i s model i n two ways. F i r s t , i t provides a framework i n which the va r i o u s s p e c i a l cases can be compared. Second, any r e s u l t obtained f o r the general model w i l l hold f o r the va r i o u s s p e c i a l cases ( p o s s i b l y assuming some s p e c i a l forms). This i s done i n Sections 1 through 3. As was pointed out i n t h e ! \ l i t e r a t u r e , (e.g. Singer and Spilerman 1974) there are two equivalent ways of i n t e r p r e t i n g mixtures of s t o c h a s t i c pro-cesses. The popu l a t i o n may be considered to c o n s i s t of subpopulations (*) which f o l l o w d i s t i n c t processes. A l t e r n a t i v e l y , each i n d i v i d u a l may be considered to "draw" the process that he w i l l f o l l o w from some proba-b i l i t y d i s t r i b u t i o n over processes (or parameters). The wording of our formulations w i l l f o l l o w the l a t t e r i n t e r p r e t a t i o n . One property to which we pay p a r t i c u l a r a t t e n t i o n i s the r a t e of m o b i l i t y (Section 4 ) . Many of the above extensions of the simple Markov model were motivated by the e m p i r i c a l l y observed f a c t that the simple model overestimated some measures of m o b i l i t y . We show that p a r t i c u l a r extensions of continuous-time Markov chains i n the d i r e c t i o n of "cumula-t i v e i n e r t i a " d u r a t i o n times (McGinnis 1968) and i n the d i r e c t i o n of Mover-Stayer models (Spilerman 1972b) r e s u l t i n s t o c h a s t i c a l l y longer durations and i n s t o c h a s t i c a l l y fewer t r a n s i t i o n s i n any time i n t e r v a l (subsection 4.1). Another comparison concerns the "extended mover-stayer model w i t h (*) See also L a z a r s f e l d and Henry (1968). - 10 -rate heterogeneity" (Spilerman 1972b). Bartholomew (1973, pp. 48-49) wondered whether an attempt to model such a system as a discrete-time Markov chain always underestimates the proportion of i n d i v i d u a l s who remain i n t h e i r i n i t i a l state. A counterexample shows that t h i s i s not always true, but we give some s u f f i c i e n t conditions f o r i t s v a l i d i t y (subsection 4.2). We also check whether some s p e c i f i c parametric forms of the models, which were suggested i n the l i t e r a t u r e , have properties which March and March (1977, p. 380) consider desirable for models of career patterns (Section 5). 1.1 A General Model and Some of I t s Properties Suppose that K i s a f i n i t e set of categories, e.g. K regions K = {1,2, K}. Let X(t) be the category of a given i n d i v i d u a l at time t. The stochastic process {X(t):t>0} i s defined as follows. (i) The given i n d i v i d u a l chooses a parameter Z from a set . (*) of parameters A , according to a p r o b a b i l i t y measure u. i . e . , f o r any event A C A , y(A) = Pr[Z e A]. If A = we s h a l l denote the corresponding d i s t r i b u t i o n function by G. ( i i ) Given Z = z we assume that {X z(t):t>0} i s a semi-Markov (**) process. In order to characterize the semi-Markov (*) A can be interpreted as the population. Individuals can be i d e n t i -f i e d with z G A . (**) Note that the unconditional process {X(t):t>0} w i l l : i n general not be semi-Markov. - 11 -process ( C i n l a r 1975) one needs to s p e c i f y the two q u a n t i t i e s B(t) and Q(t) defined next, z z a) B(t) i s the d i s t r i b u t i o n of i n i t i a l c o n d i t i o n s at time 0. z More p r e c i s e l y , i t i s a matrix-valued f u n c t i o n such that B..(t) i s the -joint ( c o n d i t i o n a l of Z = z) d i s t r i b u t i o n z i ] of i n i t i a l category i , the category j to which the f i r s t t r a n s i t i o n i s made and the time u n t i l t h i s t r a n s i t i o n . Let v ( i ) be the (marginal) p r o b a b i l i t y of X(0) = i (given z Z = z) as s o c i a t e d w i t h t h i s j o i n t d i s t r i b u t i o n ; i . e . , K v ( i ) =[Pr X(0)=i|Z=z]= £ :B..(°°). For every i , l e t j = l Z 1 3 r be the c o n d i t i o n a l p r o b a b i l i t y measure of Z given X(0) = i ; i . e . , r (A) = Pr[ZeA|x(0)=i]. I t i s obtained from v z ( i ) and u v i a Bayes' Formula. b) Q..(t) i s the c o n d i t i o n a l p r o b a b i l i t y , given that Z = z z ^1 and that t r a n s i t i o n i n t o category i has been made at time that the next t r a n s i t i o n w i l l be i n t o s t a t e j and w i l l occur before time s + t . Since t h i s p r o b a b i l i t y does not depend on s, the process {X z(t):t>0} i s time-homogeneous. For every semi-Markov process {X z(t):t>0} d e f i n e : P . = Q..(°°) = p r o b a b i l i t y that the s t a t e which w i l l be occupied a f t e r i i s j . F..(t ) = Q . . ( t ) / P.. = p r o b a b i l i t y t h a t , given that z 1J Z 1] z 1] r J ° the process occupies s t a t e i at time s and l a t e r moves to s t a t e j , t h i s t r a n s i t i o n w i l l take place before time s + t . (I f P.. = 0 then F..(t) i s a r b i t r a r y , z i ] z i ] - 12 -K u. = f td( J Q..(t)) = mean time between t r a n s i t i o n s i n z 1 o A z 13 s t a t e i . N(t) = number of t r a n s i t i o n s i n ( 0 , t ] . z z P ± j ( t ) = P r [ X z ( t ) = j | X z ( 0 ) = 1]. In g e n e r a l , any f u n c t i o n defined on the s t o c h a s t i c processes { X z ( t ) : t > 0 } , w i l l be indexed by z. The same f u n c t i o n defined on {X(t):t>0} w i l l be denoted by the same symbol, but without the index. For example, P..(t) = P r [ X ( t ) = j|X(0) = i ] = / P r [ X ( t ) = j |x(0) = i , Z = z] r . ( d z ) . •U A 1 We w i l l , i n t h i s case, use the matrix n o t a t i o n P ( t ) = E( zP(t:)) f o r the above i n t e g r a l s . The behavior of the process {X(t):t>0} can be deduced from the behavior of the processes { X z ( t ) : t > 0 } . Of p a r t i c u l a r i n t e r e s t are r e s u l t s about the l i m i t i n g behavior of the process {X(t):t>0}, e.g. about l i m P ( t ) , which i s equal to l i m E( P ( t ) ) . Since each z p i - ( t ) 1 S bounded (between zero and one), i t f o l l o w s from Lebesgue's Dominated Convergence Theorem that l i m E„P(t) = E l i m P ( t ) . However, si n c e the processes t->°° f*°° {X z(t):t>_0} are semi-Markov, Theorem 5.16 i n Ross (1970, p. 104) show that i f P i s i r r e d u c i b l e and a p e r i o d i c w i t h steady s t a t e p r o b a b i l i t i e s z TT ., then Z X TT . * U . l i m P..(t) = -rr- 2 ? - J — f o r every k . (1.1) ^ I TT . • y. .^L Z X Z X x=l - 13 -Hence Z ^ i * Z P i l i m P . (t) = E l i m „P, . (t) = E v J ^ f o r every k. ^ ki z. ki K I z*± • z y ± d.2) 1=1 Consider a s p e c i a l case i n which f o r every Z = z, u. = u f o r every i . z 1 z Then the above equation reduces to l i m P, . (t) = E^TT. = E l i m CP™), . 4.^, kj Z i m Z k i f o r every k. (The l i m i t i n g behavior of the s p e c i a l case y = 1 f o r every z and y was analyzed by Morrison . i et a l . 1971.) I f , i n a d d i t i o n , T^T = TT f o r every z, then l i m P, . (t) = TT. . (1.3) k i i Some of the s p e c i a l i z e d models discussed i n the next subsection have these p r o p e r t i e s . 1.2 S p e c i a l Cases In t h i s s e c t i o n we i d e n t i f y some important s p e c i a l cases of the general model which has been used to model m o b i l i t y . We assume through-out t h i s s e c t i o n that a t r a n s i t i o n has j u s t occured at time 0, so that B. . (t) = v (t) • Q . . ( t ) . z 13 z v z ^ i j The models are -divided i n t o two main c l a s s e s . Those i n c l a s s A correspond to homogeneous populations and those i n c l a s s B correspond to heterogenepus populations. A. In the models numbered 1 through 4 below, we s h a l l assume that zQ and that vz do not depend on z, so that zQ = Q and V z = v f o r each z 6 A. The process {X(t):t>0} then reduces to a semi-- 14 -(*) , Markov process^ ' (Ginsberg 1971, 1978a,.1978b, G r i n o l d and M a r s h a l l 1977, Section 4.4). 1) F i i ( t ) = { l t > 1 f ° r 6 V e r y ± , i This i s the case where X ( t ) i s a d i s c r e t e - t i m e Markov chain -the " c l a s s i c a l " model (see, f o r example, Blumen et a l . 1955). - A . t 2) F _ ( t ) = 1 - e 1 , t >_ 0 f o r every i , j . T his i s the case of a continuous-time Markov chain (see, f o r example, Coleman 1964, Tuma, Hannan and Groenveld 1979). A s p e c i a l subcase i s (2s): X^ = X f o r every i . {N(t):t>0} then becomes a Poisson process. 3) The F i j ' s a r e "decreasing f a i l u r e r a t e " (DFR) d i s t r i b u t i o n s ; i . e . F „ (x + t ) / F (t) i s i n c r e a s i n g i n t > 0 f o r each x >_ 0. This i s equivalent to the "Axiom of Cumulative I n e r t i a " . a) F . . ( t ) i s a r i t h m e t i c , i . e . F . . ( t + ) ^ F..(t~) only i f t i s an i n t e g e r . This i s the " C o r n e l l M o b i l i t y Model" (McGinnis 1968, Henry et a l . 1971). b) F (t) = F ( t ) f o r every i and j (which reduces N(t) to a renewal process), and F ( t ) i s a mixture of exponential -Yt d i s t r i b u t i o n s ; i . e . F ( t ) = E ( l - e ), t >_ 0, where the random v a r i a b l e Y has d i s t r i b u t i o n f u n c t i o n L. Bartholomew (1973, p. 54) argues that "The long-run behavior of such a system [Class A] w i l l depend only on the t r a n s i t i o n matrix [of the embedded Markov c h a i n ] . . . " . As can be seen from equation (1.1), t h i s i s only true i f = y f o r every i . - 15 -For a proof that a mixture of exponential d i s t r i b u t i o n s i s a DFR d i s t r i b u t u i o n see Barlow and Proschan (1975, p. 103, Theorem 4.7(a)). This p o i n t was a l s o mentioned by Ginsberg (1971, pp. 253-4). 4) F (t) = F ^ ( t ) f o r every i , j , where the F^'s are a r i t h m e t i c and F.(°°) < 1 ( i . e . the F.'s are d e f e c t i v e d i s t r i b u t i o n s ) , l l This case corresponds to Mayer's "Absorbing State Model" (Stewman 1976, pp. 218-9). B. In the f o l l o w i n g cases z Q ( t ) and v z ( i ) do depend on z, correspond-ing to models with heterogeneous populations. The processes {X^t) :tH)} are d i s c r e t e - t i m e Markov chains. The process {X(t):t_>0} corresponds to an Extended Mover-Stayer Model that p o s t u l a t e s p o p u l a t i o n heterogeneity w i t h respect to t r a n s i t i o n p r o b a b i l i t i e s (McFarland 1970, Morrison et a l . 1971, Spilerman 1972a, Bartholomew 1973, pp. 34-7., Singer and Spilerman 1974, pp. 375-5, (*) Example 2). A "promotion" model due to Wise (1975) i s , e s s e n t i a l l y , a s p e c i a l subcase w i t h P.. = p and z i x *z P ,, = 1 - p f o r every i . z i , i + l *z b) A continuous-time v e r s i o n of 5a: F..(t ) = F ( t ) = 1-e z i j t >_ 0 f o r every z, i , j . For t h i s model we get: 5) a) t < 1 t > 1 f o r every z, i , j . (*) For a r e l a t e d model see L a z a r s f e l d and Henry (1968, S e c t i o n 9.3). - 16 -z P r = e A t C P - I ) . z and thus P ( t ) = Ee At( zP-I) (1.4) 6) P = P f o r every z, A = R and F..(t) = 1 - e Z + Z 1 T - z t t >•' 0 f o r every i , j . This case corresponds to an Extended Mover-Stayer Model t h a t postu-l a t e s p o p u l a t i o n heterogeneity w i t h respect to t r a n s i t i o n r a t e . Such models were considered by Spilerman (1972b), Bartholomew (1973, pp. 46-54), and Singer and Spilerman (1974, pp. 375-9, Example 3), although they a l s o assumed that v ( i ) = v ( i ) f o r every z. For t h i s model we have Note that t h i s model s a t i s f i e s the assumptions under which (1.3) was obtained, so i t s l i m i t i n g behavior c o i n c i d e s w i t h that of i t s embedded Markov chain. This was.also proved s p e c i f i c a l l y f o r Model 6 by Spilerman (1972b, Appendix A). [For a r e l a t e d r e s u l t see Bartholomew (1973, p. 52)]. A f t e r addressing mover-stayer models l i k e Model 6, Bartholomew (1973, p. 54) concludes by saying ".. . the general semi-Markov model [ i . e . C l a s s A] ... in c l u d e s them a l l as s p e c i a l cases". This c l a s s i f i c a t i o n was repeated by Singer and Spilerman (1974, p. 377) and Stewman (1976, Table 3 ) , and i t seems that they i d e n t i f i e d zt(P-I) and thus P ( t ) = Ee Zt(P-I) (1.5) - 17 -Model 6 with Model 3b (with corresponding parameters). These models are, however, d i s t i n c t . In Model 6, Z i s chosen at time 0 and i t s r e a l i z a t i o n i s adhered to throughout, w h i l e under Model 3b, Y i s rechosen a f t e r each t r a n s i t i o n . Under Model 6 "... the length of stay between two successive moves ... are dependent because people w i t h h i g h p r o p e n s i t i e s to move are l i k e l y to have two short i n v e r v a l s and people w i t h low p r o p e n s i t i e s to move are l i k e l y to have long ones" (Ginsberg 1971, p. 254). The time i n t e r v a l s between t r a n s i t i o n s are independent of each other i n Model 3b, as w e l l as i n any semi-Markov process f o r which F.. = F f o r every (**) 1 > J • K "V 7) A = R , P = P f o r each zeA, and F..(t) = 1 - e f o r each + z Z 1 ] t 0 and i , j eK where z = ( z n , z 0,...,z ). This i s a g e n e r a l i z a -1 JL K t i o n of Model 6 (Singer and Spilerman 1974, pp. 380-5, Example 4) i n which K parameters (z., , z 0,. .. , z v ) are chosen at time 0; z. i s the t r a n s i t i o n r a t e i n category i . For each zeA, l e t z be the diagonal matrix z = (*) To put i t i n other words, the r e a l i z a t i o n of the f i r s t d u r a t i o n time T^ provides some in f o r m a t i o n about Z, which i n t u r n i n f l u e n c e the p o s t e r i o r d i s t r i b u t i o n of T 2. More on t h i s i n subsection 1.5.2 (**) Though not f o r the general Class A as claimed by Ginsberg (1971, 1978a, 1978b). - 18 -t z ( P - I ) Then P ( t ) = e , and {X ( t ) : t > 0} i s a Markov process w i t h z z generator ^k = z ( P - I ) . Thus P ( t ) = E e t Z ( P I } . (1.6) 1.3 A Comparison of P o p u l a t i o n Heterogeneity Induced through T r a n s i t i o n Rates and through P r o b a b i l i t y T r a n s i t i o n Functions  In t h i s s e c t i o n we show that Models 6 and 7, which introduce popula-t i o n heterogeneity through the t r a n s i t i o n r a t e are, subject to a bound on the t r a n s i t i o n r a t e , a s p e c i a l case of Model 5b, which has constant t r a n s i t i o n r a t e A, but heterogeneous p r o b a b i l i t y t r a n s i t i o n m a t r i c e s . Consider Model 7, and suppose there e x i s t s a number X such that (*) Z. £ A w i t h p r o b a b i l i t y 1 f o r every i . For each z, define where ^k i s the generator defined i n Model 7. Then by Theorem 8.4.31 i n C i n l a r (1975), a v e r s i o n of Model 5b wi t h p r o b a b i l i t y t r a n s i t i o n matrices { P:z S A} and r a t e A w i l l have e x a c t l y the same t r a n s i t i o n z J f u n c t i o n P ( t ) as does the v e r s i o n of Model 7 w i t h which we s t a r t e d , z So we conclude that extensions of the b a s i c Models 1 and 2 i n the (*) I f the d i s t r i b u t i o n G of Z does not s a t i s f y t h i s c o n d i t i o n , we can p i c k some (lar g e ) A and truncate G to o b t a i n ( G(z)/G(A,...,A) i f 0 <_ z. <_ X f o r every i , 1 otherwise. Since the d i s t r i b u t i o n s of times between t r a n s i t i o n s w i l l s t i l l be mixtures of exponentials, they w i l l r e t a i n the DFR property. Hence the c o n d i t i o n i s not too r e s t r i c t i v e . - 19 -d i r e c t i o n of p o p u l a t i o n heterogeneity through the t r a n s i t i o n r a t e are, e s s e n t i a l l y , s p e c i a l cases of the extension that p o s t u l a t e s p o p u l a t i o n heterogeneity through t r a n s i t i o n p r o b a b i l i t i e s . P o p u l a t i o n heterogeneity i s expressed through one parameter i n Model 6 and K parameters i n Model 7, which are the " r a t e - d i f f e r e n c e " cases, w h i l e i t i s expressed through K parameters i n Model 5, which p o s t u l a t e s " t r a n s i t i o n p r o b a b i l i t y (*) d i f f e r e n c e s " . 1.4 Rate of M o b i l i t y i n the Various Models The r a t e aspect of ( i n t r a g e n e r a t i o n a l ) m o b i l i t y which was not accounted f o r p r o p e r l y by Markov chain models was the tendency to remain i n the same category. In f a c t , the other models were introduced s p e c i f i -c a l l y i n order to decrease the " r a t e of m o b i l i t y " p r e d i c t e d by Markov chains. In order to be able to assess t h e i r success i n a c h i e v i n g t h i s g o a l , we have to focus on q u a n t i t i e s which are r e l a t e d to the " r a t e of m o b i l i t y " i n the system. Three q u a n t i t i e s come to mind: a) durations ( i n t e r - t r a n s i t i o n times); b) number of t r a n s i t i o n s i n a given time i n v e r v a l ; and c) diagonal elements of P ( t ) matrices. (*); Yet another k i n d of p o p u l a t i o n heterogeneity (through the order of the Markov chain) was p o s t u l a t e d i n the brand-choice l i t e r a t u r e (Jones 1973, Givon and Horsky 1978). Consumers are c l a s s i f i e d i n t o three c a t e g o r i e s : those of the f i r s t type choose a brand indepen-d e n t l y of t h e i r previous choice (a "zero order" Markov c h a i n ) : the choice of the consumers of the second type i s a f f e c t e d only by t h e i r most recent choice (a " f i r s t order" Markov c h a i n ) , w h i l e the choices of the r e s t are a f f e c t e d by a l l t h e i r previous choices ( l i n e a r l e a r n i n g - an " i n f i n i t e order" Markov c h a i n ) . - 20 -Since the f i r s t q u a n t i t y i s most meaningful i n models which evolve i n continuous-time, we use i t to compare Model 2s to Models 3b and 6. Models 3b and 6 may be viewed as two a l t e r n a t i v e methods of i n t r o d u c i n g an a d d i t i o n a l s t o c h a s t i c component to Model 2s. A comparison between these models i n terms of the second q u a n t i t y can then be deduced using some monotonicity p r o p e r t i e s . H i s t o r i c a l l y , however, the d i r e c t m o t i v a t i o n f o r i n t r o d u c i n g the more complex models was that Model 1 underestimated the t h i r d q u a n t i t y . I t i s thus of i n t e r e s t to check whether i g n o r i n g p o p u l a t i o n heterogeneity ( i . e . using Model 1 when the system i s a c t u a l l y Class B type) r e s u l t s i n systematic u n d e r p r e d i c t i o n of diagonal elements. 1.4.1 I n t e r - T r a n s i t i o n Times and Number of T r a n s i t i o n s Per U n i t Time In t h i s s e c t i o n we compare Models 2s, 3b, and 6. Assume that the three models have the same p r o b a b i l i t y t r a n s i t i o n m a t r i x P. R e c a l l that i n Model 2s there i s a constant t r a n s i t i o n rate" A; i n Model 3b the t r a n s i t i o n r a t e Y i s chosen anew at each t r a n s i t i o n from a d i s t r i b u t i o n L; and i n Model 6 the t r a n s i t i o n r a t e Z i s chosen i n i t i a l l y from a d i s -t r i b u t i o n G. Assume f o r the purposes of comparing the three models that G = L (we w i l l use G to denote the common d i s t r i b u t i o n , even i n the context of Model 3b), and that EZ = A (= EY). Let T^3h\ T ^ and (2s) T be random i n t e r - t r a n s i t i o n s times f o r these models. Then w h i l e P r ( t ( 3 b ) > t ) = P r ( T ( 6 ) > t) = Ee Z t , t > 0, (1.7) Pr(T< 2 s> > t) = e " X t = e " t E Z , t > 0. (1.8) - 21 -— x t Since e i s a convex f u n c t i o n i n x, we have by Jensen's i n e q u a l i t y that E e " Z t > e - ^ . (1.9) Consequently P r ( T ( 3 b ) > t ) = p r ( T ( 6 ) > t ) ^ p r ( T ( 2 s ) > t ) , t >_ 0. (1.10) When two random v a r i a b l e s X and Y have the same d i s t r i b u t i o n we w r i t e X S = t Y and when Pr(X >^  x) >_ Pr (Y > x) f o r every x we w r i t e X ^ :Y (see, f o r example, Brumelle and Vickson 1975). Using t h i s n o t a t i o n , (1.10) can u •«.•. m(3b) s t ^ (6) s t m ( 2 s ) be r e w r i t t e n as T = T > T . Consider now the number of t r a n s i t i o n s i n the. time i n t e r v a l (0,t]' (**) pr e d i c t e d by these three models. Let N 2 s ^ b e t b e n u m b e r o f t r a n s i -t i o n s i n (0,t] f o r Model 2s. Since ( t ) : t > 0} i s a Poisson process, i t s renewal f u n c t i o n m„ (t) = EN„ (t) equals Xt. { N , ( t ) : t > 0} i s Is Zs z 6 also a Poisson process, so m^(t) = z t ; un c o n d i t i o n i n g we get z O nig(t) = X t . Hence Models 2s and 6 (with corresponding parameters) pre-(*) F of Model 3b i s DFR with mean E T ( 3 b ) = E ( E ( T ( 3 b ) | Y ) ) = E ( i ) = E(|) . Hence, i n combination w i t h the upper-bound on the s u r v i v a l d i s t r i -b u t i o n provided by Barlow and Proschan (1975, p. 116, Theorem 6.10), we get -tEZ „ ,t3» , r e - t / E ( l / Z ) f o r t < E ( l / Z ) e < Pr(T^ > t) l { E ( 1 / z ) / e t f o r t , E ( 1 / z ) . (**) I f P.. ^ 0 f o r some i , some of the t r a n s i t i o n s w i l l not i n v o l v e a i i r e a l category change. But sin c e P i s the same f o r a l l three models, the p r o p o r t i o n of t r a n s i t i o n of t h i s type w i l l be the same i n a l l of them, so we can use the t o t a l number of t r a n s i t i o n s as means of comparison. For any semi-Markov process, there e x i s t s some semi-Markov process wit h P. . = 0_for.every i which has the same d i s t r i b u t i o n of- sample paths. Moreover, a Markov process r e t a i n s i t s Markovian property under such transformation. - 22 -d i e t the same expected number of t r a n s i t i o n s . Let us now compare N 2 s ^ w :*- t n N 3 b ^ * B ^ d e f i n i t i o n N (t) = sup {n: £ T . ( 2 s ) < t} Z i = l 1 and N„,(t) = sup {n: £ T . ( 3 b ) < t } . JO . . 1 — i = l The N ( t ) ' s are thus decreasing f u n c t i o n s of the corresponding T_/s. Since by (1.10) T ^ 3 ^ S£ T^^ S^ , i t f o l l o w s from the independence of the T. 1s that l N 3 b ( t ) S< N 2 s ( t ) , t > 0. (1.11) Hence m 3 b ( t ) £ m & ( t ) = m 2 g ( t ) , t _> 0. So we conclude t h a t , compared to Model 2s, Model 3b reduces the r a t e of m o b i l i t y as measured by both durations and expected number of t r a n s i -t i o n s per u n i t time, w h i l e Model 6 does so w i t h respect to durations only. 1.4.2 Diagonal Elements and Eigenvalues I t has been already shown by a counterexample (Bartholomew 1973, p. 37) that Model 1 does not n e c e s s a r i l y underpredict the diagonal elements of the P ( t ) ' s of a process that a c t u a l l y evolves according to '(*) Model 5a. For a process that evolves according to (the more s p e c i -(*) Bartholomew provides a s u f f i c i e n t c o n d i t i o n ( r e v e r s i b i l i t y of the processes {X (t):t>_0}, f o r each z) under which u n d e r p r e d i c t i o n w i l l occur. - 23 -a l i z e d ) Model 6, previous examples (Spilerman 1972b, Bartholomew 1973, pp. 48-9) d i d e x h i b i t u n d e r p r e d i c t i o n of diagonal elements by Model 1, but no general proof i s a v a i l a b l e . We s h a l l prove that i f the eigenvalues of P are r e a l , the sum of the diagonal elements generated by Model 6 w i l l indeed be l a r g e r than the one generated by Model 1. In other words, under t h i s hypothesis, the expected t o t a l number of i n d i v i d u a l s who at any f u t u r e time are i n the s t a t e they s t a r t e d from i s l a r g e r i n Model 6 than would have been pre-d i c t e d by a Markov chain model. We s h a l l r e s t r i c t ourselves to the case i n which ^ z ( i ) = v ( i ) f o r every z. Theorem 1.1 In Model 6, i f a l l the eigenvalues of P are r e a l , then t r a c e P(k) > t r a c e P ( l ) k k = 1,2,3, ... (1.12) Proof: F i r s t , note that Z(V-I) k t r a c e P(k) = t r a c e E[e ^ ; ] and that . t r a c e P ( l ) k = t r a c e [ E e Z ( P _ I ) ] k . I f the eigenvaluesr'of P are r e a l , so are those of P-I. Denote the zk(P—I) eigenvalues of P-I by A , A . The eigenvalues of e are then 1 K ( C i n l a r 1975, Appendix) e z k ^ l , e z k^K, and i t can be shown^ ^ that (*) Suppose that f o r every x, [ .A i s an eigenvalue of the generator A, i . e . there e x i s t s a v e c t o r V such that ( A)V = V( A). Then t a k i n g X X X expectations of both sides (with respect to the d i s t r i b u t i o n of X) we get (E A)V = V E ( r A ) . Hence E( A) i s an eigenvalue of E( A). - 24 -, „ Zk(P-I) _ ZKAi ^ ZkA K c . ^ . , _ those of Ee are Ee , . . . , Ee K . Since a matrix s t r a c e equals the sum of I t s eigenvalues, we have tr a c e P(k) = I E ( e Z A j ) k j = l and t r a c e P ( l ) k = f ( E e Z X J ) k . j = l z A i k Let g(z) = (e J ) . Then f o r k > 1, g i s a convex f u n c t i o n of z. Hence by Jensen's i n e q u a l i t y Eg(Z) > gE(Z); i . e. ZA ZA. E(.e' J) l ( E e J ) j = l , ...,K. (1-13) Summing over j completes the proof. Remarks: 1) Since 0 i s always an eigenvalue of ( P - I ) , and since the sum of the eigenvalues, being equal to the t r a c e , i s always r e a l , the other eigenvalue i n the two-categories case i s a l s o r e a l . Hency f o r two-categories systems the a s s e r t i o n of the Theorem always holds. 2) I n e q u a l i t i e s (1.13) are stronger than the a s s e r t i o n of the Theorem. They imply that each eigenvalue of a k-step t r a n s -it i t i o n m a t r i x i s l a r g e r than the corresponding one of P ( l ) . - 25 -We now provide an example f o r which the r e l a t i o n (1.12) does not hold. This example, of course, has complex eigenvalues and shows the n e c e s s i t y of some r e s t r i c t i o n on P such as the assumption of r e a l eigenvalues i n the Theorem. Example Let P = .5 .0 .5 .3 . 7 .0 .0 .4 .6 and l e t w i t h p r o b a b i l i t y w i t h p r o b a b i l y .91 i t y . l j We now wish to c a l c u l a t e the t r a c e s of the matrices P(k) of Model 6 and of the corresponding Model 1. Since we s h a l l do so by summing eigenvalues, i t should be noted that complex eigenvalues appear i n con-jugate p a i r s . Let A = a + b i be an eigenvalue of ( P - I ) , and l e t A be i t s conjugate. Now, Akz , Akz kz(a+bi) , k z ( a - b i ) e + e = e v + e _ g a k z ^ Q g b k ^ + i g - j ^ b k : ^ 9.1cZ + e [cos(-bkz) + i s i n ( b k z ) ] = 2e a' C Zcosbkz. A l s o , (Ee ) + (Ee ) = 2Re(Ee ) o 7 1, 2Re{E[e (cosbZ+isinbZ)]} In our example the eigenvalue of (P-I) are 0, -.6 + .3317i and -.6 - .3317i. Hence t r a c e P(k) = 1 + 2E(e cos.3317kZ) and t r a c e P ( l ) k = 1 + 2Re{E[e~* 6 2(cos.3317Z+isin.3317Z)]} k. Using d o u b l e - p r e c i s i o n , we obtained the f o l l o w i n g values: k trac e P(k) tr a c e P ( l ) k 1 .. 1.9.33120 1.933120 2 1.427165 1.383567 3 1.161984 1.130630 4 1.039448 1.028467 5 0.992150 0.994745 6 0.979972 0.988163 7 0.981579 0.990234 8 0.986916 0.993771 9 0.991969 0.996566 10 0.995607 0.998313 We see that there are some cases (e.g. k=6) where trac e P(k) < t r a c e P ( l ) . T h i s , of course, i m p l i e s that at l e a s t one diagonal element of P(6) i s smaller than the corresponding one i n P ( l ) . So some r e s t r i c -t i o n s on P have to be imposed i f (1.12) i s to hold. } 1.5 C o m p a t i b i l i t y of Parametric F a m i l i e s of D i s t r i b u t i o n s w i t h Observed P r o p e r t i e s of Career Patterns  In t h i s s e c t i o n we mention s e v e r a l observations made by March and March (1977) about career patterns. We then check whether some s p e c i f i c parametric forms of the models, which were suggested i n the l i t e r a t u r e , are compatible w i t h these p r o p e r t i e s . (*) I t i s a l s o evident that i t i s not s u f f i c i e n t (as conjectured by Blumen et a l . 1955) that the diagonal elements of P w i l l be l a r g e . - 27 -1.5.1 Decreasing Rate of I n e r t i a Accumulation Recent m o b i l i t y s t u d i e s i n d i c a t e d t h a t , i n a d d i t i o n to the "cumula-t i v e - i n e r t i a " phonomenon, the d i s t r i b u t i o n s of time between t r a n s i t i o n s e x h i b i t some a d d i t i o n a l p r o p e r t i e s . Tuma (1976) and March and March (1977) argue that "...although s e l e c t i o n , r e t e n t i o n , adaptation and d e p l e t i o n processes c h a r a c t e r i s t i c a l l y r e s u l t i n changes i n average d u r a b i l i t y , the r a t e of change d e c l i n e s over the d u r a t i o n of the match" (March and March 1977, p. 380). Translated to our vocabulary, t h i s means that although i n e r t i a i s being accumulated, the r a t e of accumulation i s decreasing. Assuming ( f o r s i m p l i c i t y ) that F i s twice d i f f e r e n t i a b l e , we can express i t as: 3 F(t+x) It [ F ( t ) •* - ° f ° r e v e r y x ( D F R ) 2 < 0 f o r e v e r y x> < 1- 1 4) 3 t 2 F ( t ) where F = 1 - F. Denote by T a random time between successive t r a n s i t i o n s , and l e t u = ET. Then i t i s known (Barlow and Proschan 1975) that DFR i m p l i e s a property c a l l e d "New Worse than Used i n Expectation", which can be s t a t e d as E(T - t|.T > t ) > u . A popular DFR d i s t r i b u t i o n i n the m o b i l i t y l i t e r a t u r e ( S i l c o c k 1954, Spilerman 1972b) i s a mixture of exponential d i s t r i b u t i o n s w i t h gamma mixing d i s t r i b u t i o n . However, Morrison (1978) proved that f o r such d i s t r i b u t i o n E(T - 11T > t) = at + b a > 0. - 28 -Consequently, -— E ( T — t | T > t ) = a > 0 dt s2 —j E(T - t|T > t) = 0 f o r every t . at Hence t h i s d i s t r i b u t i o n does not s a t i s f y (1.14). The Weibul d i s t r i b u t i o n F ( t ) = 1 - e ^ A ; t > 0 A > 0 wi t h 0 < a _< 1, and the Gompertz d i s t r i b u t i o n F ( t ) - ! _ e " A ( e B t - l ) / 3 t > Q w i t h 3 < ; 0 , have been a l s o suggested f o r modelling m o b i l i t y (Ginsberg 1978a). These d i s t r i b u t i o n s are DFR, but i n both cases as t -> °° the "hazard r a t e " approaches zero (Ginsberg 1971, p. 253, Barlow and Proschan 1975, p. 73). So (1.14) w i l l not be s a t i s f i e d here e i t h e r . A DFR d i s t r i b u t i o n which does s a t i s f y (1.14) i s a gamma, which has den s i t y , sCt-1 -At _, . A(At ) e „ , „ £( f c> = r ( a ) t > 0 , A > 0 , i f 0 < a <_ 1 (Ginsberg 1971, p. 253, Barlow and Proschan 1975, pp. 73-5). 1.5.2 Heterogeneity of Subpopulations Selected by Their Durations R e f e r r i n g to career patterns March and March (1977, p. 380) c l a i m that s e l e c t i o n and r e t e n t i o n r u l e s commonly used by employers "... reduce the variance [and thus] reduce over the dur a t i o n of the matches the heterogeneity of populations". In our n o t a t i o n , t h i s statement becomes - 29 -Var (Z|T > t) i s decreasing i n t v ', where T denotes a sojourn time. Not every j o i n t d i s t r i b u t i o n of Z and T w i l l have t h i s property, how-ever. Suppose that a l a r g e m a j o r i t y of the pop u l a t i o n have short sojourn times w i t h low v a r i a b i l i t y , making the o v e r a l l variance of Z s m a l l . Now, i f a r e a l i z a t i o n of sojourn time turns out to be long, the i n d i v i d u a l i s l i k e l y to belong to the l e s s mobile m i n o r i t y , which may have a high v a r i a b i l i t y among i t s members. In the p a r t i c u l a r case of Model 6 Pr(T > t ) = E e " Z t ; so we o b t a i n that -Zt /_,_ -Zt \ Hence af- Var(Z|T > t ) = 1 — [ 3 E e ~ Z T E Z e " Z t E Z 2 e " Z t ? t ( E e _ Z T ) 3 - ( E e - Z t ) 2 EzV Z t - 2 ( E Z e - Z t ) 3 ] . A r e l a t e d property i s Var (Z | T 1 = t ) <_ Var Z f o r every t _> 0, where T]_ i s the sojourn time i n the i n i t i a l category. These kinds of questions, comparing p r o p e r t i e s of p o s t e r i o r d i s t r i b u t i o n s to those of p r i o r ones, are common i n Bayesian a n a l y s i s , and the answers depend on the j o i n t d i s t r i b u t i o n of Z and T. Upon r e p l a c i n g v a r i a n c e by entropy as a measure of d i s p e r s i o n , on the other hand, s i m i l a r p r o p e r t i e s become v a l i d f o r any two random v a r i a b l e s ( c f . Khinchin 1957, pp. 2-9). - 30 -For the d e s i r e d property to h o l d , the d i s t r i b u t i o n G of Z has to be such that 3 Ee" Z t EZe" Z t E Z 2 e " Z t - ( E e ^ ) 2 EzV Z t - 2 ( E Z e - Z t ) 3 < 0. Consider, f o r example, the d i s t r i b u t i o n ^ _ f X w i t h p r o b a b i l i t y 3 ^ M w i t h p r o b a b i l i t y l-f3 , where X has an exponential d i s t r i b u t i o n w i t h r a t e 1 and M i s a constant. Here we get (t+i) 4 and hence _ 9 V a r ( z i T > t ) ^ e " 2 M t 3 ( l - B ) ( t + l ) [ M 3 ( t + l ) 3 - 3 M 2 ( t + l ) 2 + 3 M ( t + l ) - l ] [ ( l - g ) ( t + l ) - g e M t ]  3 t ( t + l ) 6 ( E e " Z t ) 3 T h U S ^ V a r ( Z | T > t ) < 0 i f and only i f t +. 1 > Mt * t+1 + e So f o r a given M, the s i g n of the d e r i v a t i v e may depend on t . However, the commonly used mixture of exponentials w i t h gamma mixing d i s t r i b u t i o n has the d e s i r e d property. Let the den s i t y of Z be g ( z ) = za"l e - z / r ( a ) z >_ 0 , a > 0 Then E Z m e " Z t = r(a-Hn)/r(a)(t+l) a 4 i n, - 31 -so Var(Z T > t) = r(g)r(g+2) - [r.;(g+i)] [r(a)] 2 ( t + i ) 2 (t+1)' which i s c l e a r l y decreasing i n t. - 32 -CHAPTER I I A STOCHASTIC MODEL ALLOWING INTERACTION AMONG INDIVIDUALS Many systems, a r i s i n g i n a v a r i e t y of contexts, c o n s i s t of a number of i n d i v i d u a l s moving among various categories or s t a t e s . T y p i c a l l y , the i n d i v i d u a l s i n such systems do not move independently of each other, but i n s t e a d i n t e r a c t i n some way. Several authors have pointed out the need f o r e x p l i c i t l y modeling such i n t e r a c t i o n s . S o c i a l m o b i l i t y (Matras 1967) and promotion chances i n an o r g a n i z a t i o n (White 1970) are a f f e c t e d by the existence of o p p o r t u n i t i e s , which are created (among other reasons) by other people's movements. P o p u l a r i t y a f f e c t s p o l i t i c a l (*) a f f i l i a t i o n (Holley and L i g g e t t 1975), consumers' brand s w i t c h i n g (Smallwood 1975, Smallwood and C o n l i s k 1979), and modal choice (Krishnan and Beckman 1979). The e f f e c t of crowding on i n t e r n a l m i g r a t i o n was modeled by Cordey-Hayes and Gleave (1974). Models of epidemics '.and d i f f u s i o n of rumours (e.g. Bartholomew 1973, Ch. 9, 10) i n c o r p o r a t e e f f e c t s of human contacts. C o n l i s k (1976) introduced a g e n e r a l i z a t i o n of Markov chains i n which an i n d i v i d u a l ' s next category depends on h i s current category and on the d i s t r i b u t i o n of the pop u l a t i o n among the c a t e g o r i e s . Models of t h i s type, which combine "push" flows w i t h " p u l l " flows (Bartholomew 1973, p. 26) were a l s o suggested by Matras (1967) and Smallwood (1975) and are used by Smallwood and C o n l i s k (1979). This type of model appears to be appropriate f o r systems such as those mentioned i n the f i r s t paragraph. (*) This model o r i g i n a t e d from s t a t i s t i c a l mechanics (see a l s o S p i t z e r 1970). - 33 -C o n l i s k ' s model assumes that the p r o b a b i l i t y Q of moving from category i to category j i s a f u n c t i o n of how the other i n d i v i d u a l s are d i s t r i b u t e d among the categories. Thus i n s t e a d of having one p r o b a b i l i t y t r a n s i t i o n m a t r i x , t h i s model has a p r o b a b i l i t y t r a n s i t i o n matrix Q(y) f o r each v e c t o r y ( c a l l e d a p r o f i l e ) , whose components are f r a c t i o n s of the p o p u l a t i o n i n each category. However, C o n l i s k does not analyze the above model, which we c a l l the f i n i t e p opulation model. In f a c t , he does not de f i n e i t unambigously. We thus s t a r t s e c t i o n 1 of t h i s chapter by c a r e f u l l y d e f i n i n g a f i n i t e p o p u l a t i o n model. In s t r u c t u r i n g our model, we have attempted to be c o n s i s t e n t w i t h the m o t i v a t i o n and examples which are so n i c e l y developed i n C o n l i s k (1976) and Smallwood and C o n l i s k (1979). We then i n v e s t i g a t e some of the model's p r o p e r t i e s and computational aspects. C o n l i s k argues that f o r a l a r g e p o p u l a t i o n the p r o f i l e s at time t and time t+1 (denoted by row ve c t o r s y f c and y t +^> r e s p e c t i v e l y ) should be r e l a t e d by y t + l = Y t Q ( Y t } ' ( 2 . 1 ) ( * } In t h i s model, which we c a l l the i n f i n i t e p o p u l a t i o n model, the process y 0 ' y l ' y2'"*" ^ S d e t e r m i n i s t i c o n c e Y Q i s s p e c i f i e d ( i n the f i n i t e popu-l a t i o n model the p r o f i l e s w i l l be t r u l y s t o c h a s t i c ) . C o n l i s k (1976, p. 158) s t a t e s that " [ y t ] i s s t o c h a s t i c [ i n the f i n i t e p o p u l a t i o n model] and the equation [(2.1)] must be viewed as approximate; but, f o r a l a r g e p o p u l a t i o n the approximation e r r o r i s n e g l i g i b l e " . The i n f i n i t e popula-(*) Matras (1967) a l s o suggests < t h i s r e l a t i o n . I f the f u n c t i o n a l Q happens to be one-to-one, (2.1) becomes a s p e c i a l case of a demo-graphic model due to Cohen (1976). - 34 -t i o n model (2.1) i s a l s o used i n Smallwood and C o n l i s k (1978) w i t h a s i m i l a r comment about the approximation. However, n e i t h e r paper sub-s t a n t i a t e s the c l a i m that the approximation e r r o r i s n e g l i g i b l e . Now, sinc e the i n f i n i t e p o p u l a t i o n model has obvious computational advantages over the corresponding f i n i t e p o p u l a t i o n one, the l a t e r s e c t i o n s of t h i s chapter are devoted to examining the v a l i d i t y of (2.1) as an approximation of the f i n i t e p o p u l a t i o n model. In Sec t i o n 2, we i n v e s t i g a t e the degree to which the i n f i n i t e popula-t i o n model approximates p r o f i l e s i n the f i n i t e p o p u l a t i o n model. Loosely speaking, a r e a l valued f u n c t i o n (with some c o n t i n u i t y assumptions) defined on the f i n i t e p o p u l a t i o n p r o f i l e process converges to the same f u n c t i o n defined on the i n f i n i t e p o p u l a t i o n process. Over a f i n i t e h o r i -zon, the c o n t i n u i t y assumptions on the f u n c t i o n s are not very r e s t r i c t i v e . However, over an i n f i n i t e h o r i z o n they are bothersome. In p a r t i c u l a r , one must be very c a r e f u l about making inferences about the e q u i l i b r i u m or steady s t a t e behavior of the f i n i t e p o p u l a t i o n model from the c o r r e s -ponding behavior of the i n f i n i t e p o p u l a t i o n model. For example, the la c k of a g l o b a l l y s t a b l e f i x e d point f o r the map y _ >yQ(y) i n the i n f i n i t e p o p u l a t i o n model does not imply that the f i n i t e p o pulation p r o f i l e pro-cess l a c k s asymptotic s t a t i o n a r i t y ( f o r d e f i n i t i o n s , see subsection 2.2.2). Put d i f f e r e n t l y , i n general l i m l i m Pr(Y NeB) ^ l i m l i m P r ( Y N e B ) , N where Y i s the p r o f i l e at time t i n a model with population s i z e N, and B i s some set. However, we do show that i f the map y-yyQ(y) i s g l o b a l l y s t a b l e then the i n f i n i t e p o p u l a t i o n model does approximate the f i n i t e - 35 -population model even over i n f i n i t e time h o r i z o n . S e c t i o n 2.3 develops the mathematical theory needed f o r the d i s c u s s i o n i n S e c t i o n 2.2. The b a s i c r e s u l t i s that i f the p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s of a sequence of Markov processes converge weakly and uniformly to some p r o b a b i l i t y t r a n s i t i o n f u n c t i o n p, then the sequence of Markov processes converges weakly to a Markov process w i t h p r o b a b i l i t y t r a n s i -t i o n f u n c t i o n p. Although the theory developed i n S e c t i o n 2.3 i s used i n S e c t i o n 2.2, i t i s placed l a t e r so as not to i n t e r u p t the d i s c u s s i o n of the model. 2.1 The F i n i t e P o p u l a t i o n Model and i t s P r o p e r t i e s 2.1.1 Formulation Let K = {1,2,...,K} be a set of c a t e g o r i e s (e.g. s o c i a l or occupa-t i o n a l c l a s s e s , geographical r e g i o n s , brands, e t c . ) . Let S = K { y = (.y^>Y2' • • >y^} : ly± = I a n c* y. ^ 0 ( j = l , 2 , . .. ,K) }. An element of i = l ^ yeS i s c a l l e d a p o p u l a t i o n p r o f i l e and y i s the f r a c t i o n of the popula-t i o n i n category j . Let N be the p o p u l a t i o n s i z e . For a p a r t i c u l a r p o p u l a t i o n s i z e , only c e r t a i n p r o f i l e s can occur and these are i n c l u d e d i n S^, which i s the subset of S c o n s i s t i n g of p r o f i l e s y such that each (*) component of Ny i s i n t e g e r . We w i l l now define a s t o c h a s t i c process (X^: t=Q,l,2,...} f o r each (*) The number of p r o f i l e s i n i s given by r (N + K -"1"\ - 36 -of the N . i n d i v i d u a l s , which are named i=l,2,...,N. The random v a r i a b l e i s the category of i n d i v i d u a l i at time p e r i o d t . The p r o f i l e at N time t i s defined to be the random vector Y whose k-th component i s N (*)' Y t W'-N^V-k] ( 2- 2 ) 1=1 t fo r each category k. The f i r s t assumption i s that the f u t u r e of the i n d i v i d u a l s depends only on t h e i r current categories and not on how they a r r i v e at t h e i r c a t e g o r i e s . More f o r m a l l y , we assume that the s t o c h a s t i c process {(X^,X 2,...,X^):t=0,l,2,...} i s a vec t o r valued Markov chain; that i s Pr [X^ + 1=k l 5 X 2 + 1=k 2,. . . . X ^ - k J (X^,X 2,. . . ,X^) , 11=0,1,2,... ,t] (2.3) = P r [ X t + l = k l ' x 2 + l = k 2 ' - - - ' X t + l = k K l X t ' X t - ' - ' X t ] (**) f o r each t=0,l,2,... and ca t e g o r i e s k^ei\. The second assumption i s that each i n d i v i d u a l ' s d e c i s i o n as to h i s next category depends only on h i s current s t a t e and the current p r o f i l e , and i s taken independently of the other i n d i v i d u a l s ' d e c i s i o n s . This assumption means that the model does not e x p l i c i t l y a l l o w f o r l e a d e r s h i p of i n f l u e n t i a l i n d i v i d u a l s , since any i n d i v i d u a l ' s d e c i s i o n does not depend on the p a r t i c u l a r category of any other p a r t i c u l a r i n d i v i d u a l , but only on the d i s t r i b u t i o n of the other i n d i v i d u a l s among the va r i o u s (*')' 1^ i s the i n d i c a t o r f u n c t i o n of the event A; i . e . 1^ = 1 i f A occurs, and I., = 0 i f A does not occur. ' A (**) For a general d i s c u s s i o n of vector-valued Markov processes see Moyal (1962). - 37 -c a t e g o r i e s . More f o r m a l l y , define Q., (y) to be the p r o b a b i l i t y that 3 k an i n d i v i d u a l moves from category j to category k i f the current p r o f i l e i s y. That i s , f o r each i=l,2,...,N and ye-S^, Q j k(y):= P r [ X ^ + 1 = k|xj=j, Y t=y] f o r t=0,l,2,... (2.4) Then the assumption can be formulated as P r [ x t + r k i ' x^ 1-k 2.,„.^ + 1-k N| Xi- J 1, xt2=J2,...,x^=JN] N = 11 Q. . (y) f o r t=0,l,2,.. .and f o r any c a t e g o r i e s j . and k., X=l J X X (**) where y = Yfc i s defined i n terms of Xfc by (2.1). So f a r the f o r m u l a t i o n of the model. A continuous time v e r s i o n of the s p e c i a l case K=2 was s t u d i e d by Holland and Leinhardt (1977) i n the context of s o c i a l networks. 2.1.2 Example (Smallwood and C o n l i s k 1979) Consider a product whose q u a l i t y can be t e s t e d only by using i t , and not by simple i n s p e c t i o n or p r i o r i n f o r m a t i o n . The product l a s t s one p e r i o d (e.g. automobile insurance p o l i c y ) , and each of the N consumers buys one u n i t of the product each period. There are K' e q u a l l y - p r i c e d brands, the q u a l i t y of which i s defined s o l e l y i n terms of t h e i r "break-(*) However, the presence of leaders can be accommodated i n t h i s model by creating an extra set of categories for each leader. (**) The s p e c i a l case, i n which Q i s a constant function of y ( i . e . no i n t e r a c t i o n among i n d i v i d u a l ) , corresponds to the " c l a s s i c a l " model i n which the i n d i v i d u a l s follow independent and i d e n t i c a l l y d i s t r i -buted Markov chains. - 38 -down" p r o b a b i l i t i e s b 1,...,b . Let us assume that f o r the product of i K: i n t e r e s t breakdown merely means moderately u n s a t i s f a c t o r y product p e r f o r -mance. I f the product does not breakdown, the consumer repurchases the same brand. I f i t does breakdown, he chooses h i s next p e r i o d brand randomly among a l l brands, with p r o b a b i l i t i e s p r o p o r t i o n a l to the current N a market shares [Y ( i ) ] i=l,...,R, where a i s a non-negative parameter which may be i n t e r p r e t e d as the degree of confidence i n market p o p u l a r i t y i m p l i c i t i n consumer behavior. Hence we have a s p e c i a l case of the f i n i t e p o p u l a t i o n model i n which Q has the f o l l o w i n g form 2.1.3 P r o p e r t i e s I. The d i s t r i b u t i o n of the s t o c h a s t i c process { ( X ^ , X 2 , . . . , X ^ ) : t=0,l,2,...} i s defined r e c u r s i v e l y by (2.3) and (2.5) once the d i s t r i b u -1 2 N t i o n of (XQ , X Q , . . . , X Q ) i s s p e c i f i e d . The parameters of t h i s time homo-geneous, v e c t o r valued Markov chain are the po p u l a t i o n s i z e N, the matrix 1 2 N valued f u n c t i o n Q ( * ) > and the d i s t r i b u t i o n of ( X ^ , X Q , . . . , X Q ) . The pro-N f i l e process {Y : t = 0 , l , 2,......} i s defined by (2.2) and i s a l s o a time homogeneous v e c t o r valued Markov chain. I t f o l l o w s from (2.5) that i f Q i s continuous i n y (an assumption which we s h a l l make throughout) a small change i n Q w i l l not cause a la r g e change i n the p o p u l a t i o n - l e v e l t r a n s i t i o n p r o b a b i l i t i e s . More - 39 -. s p e c i f i c a l l y , (2.5) i m p l i e s that i f sup I Q..(y) - Q',,(y) I < <5 i,jeK 1 3 1 3 yeS then sup J l " ^1 * " zK P r r X t + l 1 = k l " - - X t + l = k N v 1-- v N-- 1 t J l ' t JN " P r [ X t i l = k l > ' • • ' X t + l = k N X t 1 = j l " ' * ' X t N = V < « N . Since, i n p a r t i c u l a r , steady-state p r o b a b i l i t i e s vary continuously w i t h (*) t r a n s i t i o n p r o b a b i l i t i e s , the above i m p l i e s that the model's e q u i l i -brium w i l l be " s t a b l e " under s m a l l p e r t u r b a t i o n s of Q. Due to the model's symmetry i n i n d i v i d u a l s , the f o l l o w i n g statements are equivalent f o r given N and Q(*): a. {(X^, X 2,...,X^):t=0,l,2,...} i s i r r e d u c i b l e and a p e r i o d i c . N b. {Y :t=0,l,2,...} i s i r r e d u c i b l e and a p e r i o d i c . c. There e x i s t s t such that Pr(X 1=k1x^=1,Y^=y) > 0 t ' 0 0 f o r every yeS^ and c a t e g o r i e s j and k. These three statements w i l l be used interchangeably i n the next sub-s e c t i o n , where s u f f i c i e n t c o n d i t i o n s f o r the above w i l l be given. (*) The steady-state p r o b a b i l i t i e s are a s o l u t i o n of a system of l i n e a r equations, which i s known to be continuous i n the c o e f f i c i e n t s . - 40 -I I . One would l i k e to be able to i n t e r p r e t Y^(k) as the p r o b a b i l i t y that an i n d i v i d u a l i s i n category k; i . e . , Y^(k) = Pf(xJ = k|Y^). (2.6) This i s c l e a r l y v a l i d i f i n d i v i d u a l i i s chosen a t random (with the i n d i v i d u a l s e q u a l l y l i k e l y to be chosen) at time t . E q u i v a l e n t l y , one can number the i n d i v i d u a l s at time 0 at random (with each of the N! permu-t a t i o n s e q u a l l y l i k e l y ) a f t e r they have been assigned to t h e i r i n i t i a l c a t e g o r i e s . Then (2.6) holds f o r each i = 1,2,...,N. R e l a t i o n (2.6) has the f o l l o w i n g i n t u i t i v e i n t e r p r e t a t i o n . In decid -i n g , say, on our p o l i t i c a l a f f i l i a t i o n at time t , our choice w i l l be N s t o c h a s t i c a l l y the same (given the population p r o f i l e Y ) i f we choose N p a r ty k w i t h p r o b a b i l i t y Y (k) or i f we s e l e c t a person at random and swi t c h to the party which he c u r r e n t l y supports. Taking the expectation of each s i d e of (2.6) gives EY^(k) = Pr(xj=k). (2.7) We s h a l l make use of t h i s e q u a l i t y l a t e r on. I I I . Although QC') has been defined on a l l of S, some of i t s values w i l l have no impact on the behavior of the f i n i t e p o p u l a t i o n model (*) v regardless of population s i z e . In p a r t i c u l a r , i f f o r some keK we l e t A^={yeS:y(k)=0} then f o r any yeA^ the values of Q^.(y) w i l l have no e f f e c t on the model's behavior. Hence any c o n d i t i o n imposed on such values i s not at a l l r e s t r i c t i v e . This f a c t should be kept i n mind when (*) They w i l l have no impact on the i n f i n i t e p o p u l a t i o n model e i t h e r . - 41 -co n s i d e r i n g the c o n d i t i o n s imposed on Q(') i n the next subsection. 2.1.4 Conditions f o r I r r e d u c i b i l i t y and A p e r i o d i c i t y Without some r e s t r i c t i o n s , the p r o p e r t i e s of the above mentioned pro-cesses can be s i g n i f i c a n t l y dependent on the po p u l a t i o n s i z e N. We s h a l l 1 2 N thus look f o r co n d i t i o n s under which the process { ( X t , X f c,...,X t):t=0,1,2,... } i s i r r e d u c i b l e and a p e r i o d i c f o r every N. We s h a l l w r i t e A > 0 (where A and 0 are matrices of the same order, the l a t t e r of which c o n s i s t s of zeroes only) i f a „ i s ( s t r i c t l y ) p o s i -t i v e f o r every i and j . The b a s i c r e s u l t i s the f o l l o w i n g : Theorem 2.1 I f there e x i s t s an i n t e g e r n such that f o r any y meS m = 0,1,...,n n 1 2 N n Q(y ) > 0 , then {(X,., X^,... , X J : t=0,1,2, } i s i r r e d u c i b l e and - m f t t m=0 a p e r i o d i c f o r any popu l a t i o n s i z e N. Proof: The c o n d i t i o n n II Q(y ) > 0 f o r any y eS m=0,l,...,n _ m m m=0 i s equivalent to K K n T . . . J n Q. . (y ) > 0 f o r any y eS m = 0,1,...,n f o r i =1 l =1 m=0 m, m+1 1 n a n y V W ^ ' - 42 -which i m p l i e s that f o r any y ES m=0,l,...,n and any i ^ j i ^ ^ e K there e x i s t i , , . . . , i .,eK such that 1 n-1 n I Q . . (y ) > 0. m=(J m, m+1 (2.8) 1 2 N Now, sin c e { ( X t , Xt»...,X ): t=0,l,2,..,} i s a Markov ch a i n , f o r any . 1 . N cate g o r i e s i n , . . . , i and l N m ii+1 " \+l n+l ~ xn+l | X0 " V ' * , X 0 ~ XQ ' n T, ,TT 1 . 1 ,/ N . N n Pr(X =iv,, ,-..,x =1., m+1 m+1 m+1 m+1 = I I , .1 N 1 N- m=0 " T 7" 7. l < i ...,i*<K l < i ^ , . . . , i <K x l = i l . m m x =i ) m m m m By the model's assumptions the product i n the l a s t expression equals n N ' i N n IT. Q.£ ,. £ \ [y(X ,. . . ,X )] , which i n tu r n equals m=0 1=1 m m+1 N n I N II n Q.£ .£. [y(X ,...,X ) ] . But by (2.8), f o r each £ we can choose . , n I i m m £=1 m=0 m, m+1 n i ^ , . . . , i £ such that IT Q.£, . £ [yCX 1,. . . ,X N)] > 0 i n _ l l .. m m m=0 m m+1 For such { i ^ , . . . , 1 ^ } ^ . 1 n x = ± N n I N n n Q.£, .£ [y(x ,... ,x )] > o £=1 m=0 m m+1 Hence P r ( x i + 1 = i n + J , , X N ' =i N ' Xn+1 V l X ^ i 1 xo V ' xN=iN) 'xo V - 43 -. , . r . .1 . N J . 1 . N i s s t r i c t l y p o s t i v e f o r any categ o r i e s 1Q , . . . , 1Q and ,:Ln+]_»• • • » x n+]_ • The Markov chain { (X^, X 2,.. . ,X^) :t=0,l,2,... :.} i s thus i r r e d u c i b l e and a p e r i o d i c f o r every N . _ Remark: In the n o n - i n t e r a c t i v e case [Q(y) = Q for.every yeS] the Theorem reduces to s t a t i n g that i f the i n d i v i d u a l - l e v e l Markov chain {X^:t=0,1,2,...} i s i r r e d u c i b l e and a p e r i o d i c then so i s the popul a t i o n l e v e l chain {(X^,...,X^):t=0,l,2,...}, which i s what one would expect i n t u i t i v e l y . What we would l i k e now i s to f i n d some v e r i f i a b l e c o n d i t i o n s on Q(') under which the Theorem's hypothesis w i l l be s a t i s f i e d . Define the p a t t e r n of Q to be a matrix P such that P.. = i n f Q..(y). Thus P.. > 0 i f and only i f Q..(y) > 0 f o r every yeS. 1 J yeS ± 3 1 J 1 : 1 We say that P i s r e g u l a r i f P n > 0 f o r some n. The theory of products of f i n i t e non-negative square matrices (Hajnal 1958) i m p l i e s that i f Q has a r e g u l a r p a t t e r n then there e x i s t s n such that n n Q(y ) >0 f o r any y eS, m=0,l,...,n. _ m m m=0 Thus i f Q has a r e g u l a r p a t t e r n then the process {(X^,X 2,...,X^):t=0,l,2,...} i s i r r e d u c i b l e and a p e r i o d i c f o r any popula-t i o n s i z e N . The f o l l o w i n g s p e c i a l case of the Theorem i s p a r t i c u l a r l y u s e f u l . C o r o l l a r y 2.1 N - i N I f Q(y) > 0 f o r every yeS then f o r any N and t PrC * + 1=y|Y t=y) > 0 f o r every y,y E S ^ . - 44 -2.1.5 Some Computational Aspects The f i n i t e p o p u l a t i o n model can be simulated i n a s t r a i g h t f o r w a r d manner: at each p e r i o d t the random v a r i a b l e s X^, i = l , . . . , N w i l l be randomly generated according to the p r o b a b i l i t y mass f u n c t i o n Q...(y) i N where J = x t _ ^ a n c* y = Y t - l * w ^ i e n t h e p o p u l a t i o n s i z e becomes very l a r g e , however, the above procedure becomes r a t h e r c o s t l y . Instead, we can make use of the f o l l o w i n g observation. N N Given Y =y the random v a r i a b l e N Y t + - ^ can be w r i t t e n as a sum of K independent multinomialy d i s t r i b u t e d random v a r i a b l e s M^, k=l,2,...,K N with parameters (Ny(k); (y) ,. . . sQ^Cy) ) • Hence the v a r i a b l e s NY ^ can be generated by generating the multinomials M^, k=l,2,...,K. Now, there are two ways of generating a multinomial. F i r s t , by combining a l l the categories (and parameters) but one we can generate one of i t s components by generating a binomial v a r i a b l e . Then, updating the p o p u l a t i o n s i z e and the parameters, s i n g l e out another category from the remaining ones and continue i n that f a s h i o n (Bishop, Feinberg and Holland 1975, S e c t i o n 13.4). Second, there are r o u t i n e s which generate multinomials d i r e c t l y . The computational saving i s due to the w e l l known f a c t t h a t , even f o r moderately l a r g e p o p u l a t i o n s , binomials can be adequately a p p r o x i -mated by normal d i s t r i b u t i o n s . Moreover, multinomial d i s t r i b u t i o n s can be d i r e c t l y approximated by the m u l t i v a r i a t e normal d i s t r i b u t i o n (Bishop et a l . 1975, pp. 469-70). Hence the f o l l o w i n g computational scheme seems reasonable. I f N Ny(k) i s s m a l l , generate M by a s s i g n i n g each i n d i v i d u a l s e p a r a t e l y ( v i a K. Q k > ( y ) ) . I f i t i s l a r g e , generate M^ by a m u l t i v a r i a t e - n o r m a l approxi-- 45 -mation. Thus the computational e f f o r t r e q u i r e d f o r the s i m u l a t i o n i s p r o p o r t i o n a l to the number of categories and, e s s e n t i a l l y , does not depend on the population s i z e . 2.2 Approximation of the P r o f i l e Process R e c a l l from the i n t r o d u c t i o n that C o n l i s k (1976) introduced the d e t e r m i n i s t i c r e c u r s i o n Yt+^ = y t Q ( y t ) r e l a t i n g the p r o f i l e vectors at times t and t+1, and suggested i t as a model f o r the e v o l u t i o n of popu-l a t i o n p r o f i l e s f o r l a r g e populations. He a l s o assumed that each component of Q(y), say Q „ ( y ) , i s a continuous f u n c t i o n of y. We, too, make t h i s assumption f o r both the f i n i t e and i n f i n i t e p o p u l a t i o n models. This s e c t i o n i n v e s t i g a t e s the degree to which C o n l i s k 1 s i n f i n i t e p o p u l a t i o n model approximates the f i n i t e p o p u l a t i o n model. Theorem 2.2 i m p l i e s that the approximation i s "good" over one p e r i o d , and Theorem 2.3 i m p l i e s that the approximation i s "good" over any f i n i t e h o r i z o n . Theorem 2.4 i m p l i e s that approximation of steady s t a t e behavior i s "good" i f the i n f i n i t e p o p u l a t i o n model has a g l o b a l l y s t a b l e f i x e d p o i n t . Denote the one step p r o b a b i l i t y t r a n s i t i o n f u n c t i o n of the f i n i t e -p o p u l a t i o n p r o f i l e process by p N ( y , B ) : = P r [ Y ^ + 1 eB|Y^=y] f o r y e ^ a n d BeB(S), (*) I t should be noted that i n the " c l a s s i c a l " n o n - i n t e r a c t i v e model (constant Q) t h i s type of r e c u r s i o n i s f r e q u e n t l y used to approximate the s t o c h a s t i c p r o f i l e process. (**) A continous time v e r s i o n of the r e c u r s i o n was considered i n C o n l i s k (1978b). - 46 -where B(S) i s the Borel a - f i e l d of 5. I t s t-step p r o b a b i l i t y t r a n s i t i o n function i s s i m i l a r l y denoted by N r N r N p t(y,B): = P r L Y ^ ^ eB|Y =y] , and i t s i n i t i a l p r o b a b i l i t y measure by P N ( B ) : = Pr[Y^eB] for Be8(S). The recursion ( 1 . 1 ) determines the i n f i n i t e population process {Y t:t= 0,l , 2 ,...} once Y^ i s s p e c i f i e d . Let Y^ have p r o b a b i l i t y measure u defined on B(S); i . e . y(B) = Pr[Y QeB], BeB(S). Note that conditioned on YQ = y, the p r o f i l e process i s deterministic with Y^ +^ = Y F CQ(Y ). It i s convenient to introduce some a d d i t i o n a l notation f o r the i n f i n i t e population model. Let F( y ) : = F^ "*"^  (y) :=yQ(y) and l e t F ( t + 1 ) ( y ) : = F ( F ( t ) ( y ) ) for yeS. Let p.(y,B) : = I [ F ( ) e B ] an<* P t(y>B): = I j p ( t ) ^ y ) e j3] f o r Y £^' B e B ( S ) . Note that p i s the (determin-i s t i c ) p r o b a b i l i t y t r a n s i t i o n function of the Markov chain Y^jY^jY^,... . Since Conlisk's model allows p r o f i l e s to take any value i n 5, the compari-N son between our model and Conlisk's i s f a c i l i t a t e d by extending p to a p r o b a b i l i t y t r a n s i t i o n function on a l l of S. Define the extension N N N p by p (y,B): = p (y,B) where y i s the point i n nearest to y. Any fi x e d rule can be used to break t i e s ; e.g. i f several points i n S^ are equally close to y, then choose the l e x i c o g r a p h i c a l l y smallest. This extension i s only f o r technical convenience. Note that i t does not N N a l t e r the d e f i n i t i o n of the f i n i t e population model and p (y,B) = p (y,B) for a l l ye-S^ j- For typographical convenience, the '"*" w i l l be suppressed N and p (y,B) w i l l always r e f e r to the extended p r o b a b i l i t y t r a n s i t i o n function. - 47 -The noti o n of convergence used i n the paper i s summarized here. Suppose 1/ i s any metric space (e.g. any of the spaces , t = l , 2,. .. ,°°) N and Z and Z are random elements t a k i n g values i n 1/ w i t h r e s p e c t i v e N N p r o b a b i l i t y measures y and y. I f Ef (Z )->-Ef(Z) as N-*» f o r each r e a l N bounded continuous f on \J, then we say that Z converges i n distrib u t i o n . ; . N p N N w to Z(Z —*Z) or that y converges weakly to y (y —>y) ( B i l l i n g s l e y 1968). N R e c a l l that p and p are, r e s p e c t i v e l y , p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s N f o r the Markov chains {Y , t=0,l,2,...} and {Y t:t=0,1,2,...} rep r e s e n t i n g the f i n i t e and i n f i n i t e p o p u l a t i o n p r o f i l e processes. We say that p^—5p uniformly on S i f N f ( y , x ) p (y,dx) > f(y,x)p(y,dx) 2 uniformly i n y f o r each r e a l f u n c t i o n f continuous on S 2.2.1 Approximation over F i n i t e Time Horizon N w Theorem 2.2 p —»-p uniforml y on S. Proof: During some t r a n s i t i o n , say the t - t h , l e t Z be the number of i n d i v i d u a l s moving from category i to category j . Given that Y^ = y = (y^,y2» • • • ,y R) w e have that Z^,...,ZK_. are independent f o r each j , and that Z „ has a binomial d i s t r i b u t i o n c h a r a c t e r i z e d by EZ. . = Ny.Q..(y), and Var Z.. = Ny.Q. . (y) (1-Q. . (y)) <Ny./4. N 1 ^ Since Y ( j ) = =• I Z ., i t f o l l o w s that - 48 -E [ Y t + l ( j ) ' Y t = y ] = y Q ' j ( y ) and (2.9) V a r [ Y ^ + 1 ( j ) | Y ^ = y] < 1/4N f o r j=l,2,...,K, ( * ) where Q.j(y) i s the j - t h column of Q(y) For yeS, define |y| = sup|y.| . The f u n c t i o n y-*-|y| i s a norm on S i 1 and any f u n c t i o n continuous with respect to Euclidean norm (e.g. F(y)=yQ(y)) w i l l be continuous w i t h respect to t h i s norm. Since F i s continuous and 5 i s compact, F i s uniformly continuous. Thus there e x i s t s a f u n c t i o n 6 ( G ) such t h a t |F(y) - F ( y ) | < G/2 whenever |y-y| < 6 ( G ) . Note that f o r any yeS, the dis t a n c e |y-y| between y and i t s nearest neighbor y i n cannot exceed 1/N. Given G > 0, choose N^ l a r g e enough so that 1/NQ < 6 ( G ) . Then f o r each N > N^ and yeS with nearest neighbor ye-S^, we have : |y-y| <6(G) ,'.and i t f o l l o w s that P r [ l Y t + l " F ( y ) l > G l Y t = y ] P r U ^ + 1 " F(y)|> G |y^ = y] < P r [ l Y t + l " F ( y ) l + l F ( y ) " F ( y ) l > G l Y t = y ] < P r [ ' Y t + l ' F ( y ) I" 6 / 2 l Y t = y ] (*) The argument so f a r i s s i m i l a r to Smallwood and C o n l i s k (1979, Footnote 11). - 49 -K K By (2.9), Chebychev's i n e q u a l i t y , and the i n e q u a l i t y P r ( U A . ) < £ P r ( A . ) , 1=1 1 1=1 1 P r [ l Y t + l " F ^ \ > e / 2 \ Y t = y]<K/e 2N. Consequently, f o r each yeS (2.10) Pr[|Y - F ( y ) | > e|Yfc = y] + 0 uniformly i n y as N-x». Let feC(S) and yeS be a r b i t r a r y , but f i x e d throughout the remainder of the proof. Since f i s continuous and S i s compact, f i s uniformly continuous; hence, given G>0 there e x i s t s &(G) such that xeA(G): = [xeS: | f ( x ) - f(yQ(y))|< G] whenever |x - yQ(y)|< 6(e). So (2.10) can be r e w r i t t e n as P r t Y ^ A(e)-b£ = y J < In order to show that f ( x ) p N ( y , d x ) + f ( y Q ( y ) ) = f ( x ) p ( y , d x ) , we w r i t e • • as + • S A(e) A(e) c F i r s t note that N f ( x ) p (y,dx) - f(x)p(y,dx) A(G): A(G) N f(x)p"(y,dx) - f(yQ(y)) A(G) < G P N ( A (G),y) < G - 50 -Next, r f ( x ) P N ( y , . d x ) - f ( x ) p ( y j d x ) A ( G ) « A ( G ) ' N f ( x ) p (y.,dx) A ( G ) < (supf) • P N ( A ( e ) c , y ) 6(e)2N supf, where supf: = sup(f(y):yeS) i s f i n i t e s i n c e f i s continuous and S compact. N w Consequently, p (y, • )->-p (y, •) and the convergence i s uniform i n y. Since Q(.*) i s continuous, f(y>v)p(y,dv) = f(y,yQ(y))eC(5) whenever 2 N w feC(S ). This observation, together w i t h the f a c t that p -»-p uniformly on 5 allow us to apply Theorems 2.5 and 2.6 from s e c t i o n 2.3. To i l l u s -t r a t e the usefulness of these theorems i n the present context, some simple consequences are given i n the next theorem. Theorem 2.3 N V ( i ) I f Y + y and i f f i s a bounded, measurable, r e a l f u n c t i o n on 00 S which i s continuous (with respect to the product topology) at ( y , F ( y ) , F ( 2 ) ( y ) , . . . ) , then Ef (Y^.Y^Y^,. .. )->f (y, F (y) , F ( 2 ) (y) ,. . .) as N-*». N w ( i i ) —»-p uniformly on S f o r each t . N V ( i i i ) I f Y Q — y y ( i . e . y i s the random v a r i a b l e which only takes the value yeS), then f o r each G>0 and each t , Pr Max n=0,l,2,. ,t Y N - F ( n ) ( y ) n -KL as N-x=° - 5 1 -N V Remark: I f • -> y, then i t f o l l o w s from ( 2 . 7 ) and the above Theorem that f o r l a r g e N, the i n d i v i d u a l - l e v e l P r ( X^ = k) approximately (t) (*) equals the k-th coordinate of F ( y ) . Proof ( i ) This i s j u s t a restatement of Theorem 2.6 i n the present context. oo ( i i ) Theorem 2.5 i m p l i e s that f o r feC(S ) E [ f ( Y ^ , Y ^ , . . . , Y ^ ) | Y^=y]. - f (y,F (y) ,F ( 2 ) (y) ,. . . , F ( t ) ( y ) ) . CO uniformly i n y. Let i r t be the map ( y 0 , y 1 > y 2 , . .. )->-yt on S . Then f o r 00 each g e C ( S ) , g(u (• ))eC(S ), and so E [ g ( Y ^ ) | v j j = y] * g ( F ( t ) ( y ) ) uniformly i n y. N W r. But t h i s i s equivalent to p ->> p^ _ uniformly on 5. N V ( i i i ) Theorem 2.5 and YQ —»• y imply that ( Y » , Y ! ; , . . . Y » , Z ww,if l l W , . j f t l W ) . Since the l a t t e r q u a n t i t y i s a constant, convergence i n d i s t r i b u t i o n i s equivalent to convergence i n p r o b a b i l i t y ( B i l l i n g s l e y 1 9 6 8 , pages 2 4 , 2 5 , and Theorem 5.1 C o r o l l a r y 2 ) , which proves ( i i i ) . (*) In an e a r l y paper (Gerchak 1 9 7 8 ) , t h i s property was proved by f i r s t showing that the v a r i a b l e s and X^_ are a s y m p t o t i c a l l y (when N-*») independent. A r e s u l t s i m i l a r to part ( i i i ) of Theorem 2.3 then followed. - 52 -Note that part ( i ) of Theorem 2.3 does not i n c l u d e steady s t a t e r e s u l t s , because of the c o n t i n u i t y requirement. E.g. i f we define 1 v ^T,(YmY-, J o ) • • •) J=lim T T T / I r m a s the f r a c t i o n of time the t r a -il U 1 z t + l . _ ty.eiJJ t-K>° i=0 '••'I j e c t o r y of p r o f i l e vectors (y_,y.,y 0,..•) i s i n the set B, then f^ i s not U 1 i. D oo continuous on S . 2.2.2 Approximation of Steady State Before we i n v e s t i g a t e the r e l a t i o n between the steady s t a t e behavior of the f i n i t e and i n f i n i t e p o pulation models some re l e v a n t concepts from the theory of Markov chains (e.g. Breiman 1968) are summarized and the s p e c i a l meaning they obtain i n the ( d e t e r m i n i s t i c ) i n f i n i t e popula-t i o n model i s i n d i c a t e d . Steady s t a t e behavior of Markov chains i s c h a r a c t e r i z e d by the equivalence c l a s s e s of communicating s t a t e s and t h e i r corresponding s t a t i o n a r y measures. A measure n of a Markov chain with p r o b a b i l i t y t r a n s i t i o n f u n c t i o n q i s s t a t i o n a r y i f TT(B) = |q(v,B )Tr(dv) f o r each Bo r e l set B. For the Markov chain {Y^,Y ^,.,. .}, IT i s thus s t a t i o n a r y i f TT(B) = I r w x „iTr(dy) f o r each Be8(S). The Markov chains J [F(y)eBJ { Y Q , Y ^ , . . . } are f i n i t e s t a t e Markov chains. For each yeS, TT(y,*) defined by t Tr(y,B) = E f l i m ^ £ V e B ] l*Jfy] t-x» n=0 i s a s t a t i o n a r y p r o b a b i l i t y measure. There i s a unique s t a t i o n a r y measure i f and only i f the chain i s i r r e d u c i b l e . I f i t i s al s o a p e r i o d i c , N N then TT defined by IT(B) = limPr[Y teB] e x i s t s f o r any d i s t r i b u t i o n of Y^ - 53 -and i s the unique s t a t i o n a r y measure, and the chain i s s a i d to be as y m p t o t i c a l l y s t a t i o n a r y . The Markov chain {Y^,Y ,Y^,...} has the d e t e r m i n i s t i c p r o b a b i l i t y t r a n s i t i o n f u n c t i o n p(y,B) = I r T W x „ n and, [F(y) £B] given YQ = y, Y t = F ^ (y) • Again, Tr(y,«) defined by Tr(y,B) = 1 v l i m I I|.p(n). . e x i s t s f o r each yeS and BeB(S) and i s a t-x» n=0 s t a t i o n a r y p r o b a b i l i t y measure. Define to be the measure on 8(S) such that e (B) = f 1 1 I 0 o i f yeB y C o otherwise. Then y i s a f i x e d point of F if.land only i f ± s a s t a t i o n a r y measure f o r { Y Q J Y ^ J Y ^ , . . . } . A f i x e d p o i n t of F e x i s t s by Brouwer's f i x e d p o i n t theorem, si n c e S i s convex and compact and F i s continuous. The f u n c t i o n F i s s a i d to be g l o b a l l y s t a b l e i f there e x i s t s a y*eS such that (t) * l i m F (y) = y f o r each yeS. I t i s s t r a i g h t forward to check that F t-x» has a g l o b a l l y s t a b l e f i x e d p o i n t i f and only i f there e x i s t s an unique s t a t i o n a r y measure y of {YQ , Y ^ , Y2,...} and i n t h i s case y=e^*. N N Theorem 2.4 Suppose Y ^ has p r o b a b i l i t y measure y and Y ^ p r o b a b i l i t y measure y. ( I ) I f y xs a s t a t i o n a r y measure f o r {Y^,Y^,Y^,. ..} and y —s-y, the y i s a s t a t i o n a r y measure f o r { Y Q , F ( Y Q ) , F ^ 2 ^ (YQ),...}. (2) ( i i ) I f y i s the unique s t a t i o n a r y measure of { Y ^ , F ( Y Q ) , F V ' ( Y ) , . . . } (or e q u i v a l e n t l y , y=£i* where y* i s a g l o b a l l y s t a b l e f i x e d p o i n t of F ) , N N N and i f f o r each N, y i s a s t a t i o n a r y measure of {Y Q , Y ^ , . . . } , then Nw -j. , . y —>-y =Gy*> In t h i s case, - 54 -Pr[Max | Y N - y* | < € ] —* 1 as N-*», and n=0,l,2,...t n N N * * * Ef C Y Q . Y ^ . . . ) f (y ,y ,y ,...) as N-~> oo f o r each bounded, measurable, r e a l f u n c t i o n f on S continuous at * & * (y »y ,y , • • • ) • Remark: I f F i s g l o b a l l y s t a b l e w i t h f i x e d point y , i t f o l l o w s from (2.7) and the above Theorem that the i n d i v i d u a l - l e v e l Pr(X^=k) i s approximately equal to the k-th coordinate of y f o r l a r g e N and l a r g e t . Proof N ( i ) By hypothesis, y (B) N N p (y,B)y (dy). Also by hypothesis, y^-*y, and by Theorem 3.1, p^-^p uniforml y on 5. Hence by Lemma 2.2 [ i n t e r p r e t q N ( y , 0 = y N(«) and q(y,«) =• y (.•)], y s a t i s f i e s y(B) = p(y,B)y(dy) and i s thus s t a t i o n a r y . N ( i i ) Since S i s compact, {y } i s r e l a t i v e l y compact and hence any N subsequence of y contains a f u r t h e r subsequence which converges weakly N' ( B i l l i n g s l e y 1968, Prokorov's Theorem). Suppose the subsequence y N' converges weakly to IT. Then sin c e each y i s a s t a t i o n a r y measure, i t (2) f o l l o w s from ( i ) that II i s a s t a t i o n a r y measure f o r { Y Q , F ( Y Q ) , F V ( Y ^ ) , . . . } . By hypothesis, y i s the unique s t a t i o n a r y measure; hence II = y. Thus N each subsequence of {y } contains a f u r t h e r subsequence w i t h converges to y. By B i l l i n g s l e y (1968, Theorem 2.3), t h i s i m p l i e s that yN-^ >-y. The l a s t statement holds by Theorem 2.3, sin c e y = and y = F ( y ). - 55 -In l i g h t of the previous Theorem, i t i s tempting to conjecture that for a s u f f i c i e n t l y large population, i f the i n i t i a l p r o f i l e i s y, then the p r o f i l e a f t e r a long enough period of time t ( i . e . "at steady state") should be near F^1"^ (y) with high p r o b a b i l i t y . More p r e c i s e l y , t h i s conjecture i s that l i m Pr.[|Y^ - F ^ (y) | < G | Y^ = y] should be close t-x» to 1 for N s u f f i c i e n t l y large. Of course, Theorem 2.4 shows that the conjecture i s true i f F i s g l o b a l l y stable. The following example shows that the above conjecture i s not true i n general. In the example, the f i n i t e population models are asymptoti-N i N N c a l l y stationary f o r each N, so that l i m Pr[Y teB|Yp = y] = u (B) exi s t s t-»°° for each Be8(5) and does not depend on y. However, lim F ^ ^ X y ) does t-*» depend on y since F has three f i x e d points i n the example. Consequently, the conjecture cannot be true i n general. For the example, suppose that there are two categories (K=2), and that Q i s defined by Q n ( y ) 1/4 + ( 7/8 ) y ; L 0 < y x < 5/7 7/8 5/7<Vl<l, 255 y l and Q 2 2 ( y ) = ^ f f - where Q 1 2 (y) = 1 - Q n ( y ) , Q 2 1 ( y ) = 1 - Q 2 2 ( y ) , and y = (y^, 1-y^). The function F (y) = yQ(y) has three f i x e d points A AA AAA A AA y , y , and y , where y 1 = 0.015751, y = 0.661332, and AAA A AAA AA y^ = 0.752568. The fi x e d points y and y are stable; y i s unstable. Since Q(y) > 0 for each y , i t follows that P r [ Y t + 1 = y|Y = y] > 0 for N each y , y e S N a n ( i f ° r each N. Hence the processes {Y : t = 0,1,2,.. . } - 56 -are a s y m p t o t i c a l l y s t a t i o n a r y f o r each N, although F has s e v e r a l f i x e d p o i n t s . N e i t h e r i s uniqueness of f i x e d point s u f f i c i e n t . C o n l i s k (1976, Appendix I , Case 3) provides a two-categories example f o r which the behavior of the i n f i n i t e p o p u l a t i o n model i s c h a r a c t e r i z e d by a unique but unstable f i x e d p o i n t and a s t a b l e l i m i t c y c l e (see graph on p. 178 there ) . However f o r that example, again, Q(y) > 0 f o r each y, so the f i n i t e population processes are a s y m p t o t i c a l l y s t a t i o n a r y . Hence the con-j e c t u r e f a i l s even here. 2.3 Weak Convergence of Sequences of Markov Chains In t h i s s e c t i o n we show that i f the p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s p^(',y) -^-.p(',y) uniformly i n y and i f the i n i t i a l d i s t r i b u t i o n s N w N N y —>- y, then the Markov chains corresponding to p and y converge weakly to the Markov chain corresponding to p and y. The Markov chains considered i n t h i s s e c t i o n are general and are not r e s t r i c t e d to the p r o f i l e pro-cesses defined i n the previous s e c t i o n s . The key r e s u l t i s Lemma 2.2, which demonstrates that uniform weak convergence i s preserved when p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s are com-posed. Theorem 2.5 then a p p l i e s t h i s r e s u l t to show that the Markov chains, considered over a f i n i t e number of peri o d s , converge weakly. A standard r e s u l t from weak convergence then allows us to conclude that the e n t i r e processes converge weakly. Let <R be the r e a l l i n e and (R^  the space of k-dimensional r e a l v e c t o r s . For any me t r i c space V, l e t B(l/) be the B o r e l sets of V; i . e . - 57 -8(1/) i s the smallest a - f i e l d c o n t a i n i n g a l l of the open sets i n I/. Let C(l/) be the set of r e a l - v a l u e d , continuous, bounded fu n c t i o n s on (/. I f U i s a l s o a met r i c space, define the map (u,V)-*p(u,V) f o r ueU and Ve8(l0 to be a p r o b a b i l i t y t r a n s i t i o n f u n c t i o n from (J to 1/ i f f o r each ueU, p(u,«) i s a p r o b a b i l i t y measure on B(l/), and i f f o r each Ve8(l/), p(«,V) i s a measurable f u n c t i o n on (J. I f v and (N=0,l,2,.. .) are p r o b a b i l i t y measures defined on 8(1/), w then vN—»- v i f and only i f f ( x ) v N ( d x ) f ( x ) v ( d x ) f o r each f e C ( l / ) . * ( B i l l i n g s l e y 1968) I f p and p N are p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s from U to 1/ then p^ p uniformly on U i f and only i f r r f ( u , v ) p N ( u , dv) —>• f (u,v)p (u,dv) uniformly on U as N-*» f o r each feC(Uxl/). The next lemma i s used i n the proof of Theorem 2.5 and f o l l o w s immediately from the d e f i n i t i o n of c o n t i n u i t y . Lemma 2.1 I f jf (u,v)p(u,dv)eC(U) whenever feC(Uxl/) then V f(s,u,v)p(u,dv) EC(SxU) whenever f eC(SxUxl/). 1/ Lemma 2.2 Suppose that 5, U, and 1/ are separable metric spaces, that q and q^(N=0,1,2,...) are p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s from S to 0 , and that p and p^(N=0,l,2,...) are p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s from U to V. * We sometimes omit the domain of i n t e g r a t i o n when i t i s the whole space. - 58 -I f Q j j ^ q uniformly on S, PN^ ~*P uniformly on U, and Jf(u,v)p(u,dv)eC(U) whenever feC(Uxl/), then v ^ v uniformly on S, where v and v N (N=0,l,2,...) are p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s from S to UxV and are defined by v N(s,UxV): = p N(u,V)q N(s,du) and v(s,UxV): = p(u,V)q(s,du) U Note that and v are a c t u a l l y p r o b a b i l i t y t r a n s i t i o n f u n c t i o n s from S to UxV. Since U and V are separable, any measure defined on {UxV:UeB(U) ,\. VeB(l/)} has an unique extension to B(Uxl/) ( B i l l i n g s l e y 1968, page 225). By Theorem I I I . 2 . 1 (Neveu 1965) v ( - , W ) and v N ( - , W ) are measurable f o r each WEUXU. Proof We need to show that f(s,w)v N(s,dw) Uxt/ If(s,w)v(s,dw) uniformly i n seS Ux(/: f o r each feC(SxUxl/) . By F u b i n i ' s Theorem and the d e f i n i t i o n s of and v given i n our Theorem, t h i s may be r e w r i t t e n as f (_s,u,v)p N(u,dv)q N(s,du)- f (s,u,v)p(u,dv)q(s,du) U V uniformly i n seS f o r each f C(Sx(Jxl/), - 59 -Applying the t r i a n g l e i n e q u a l i t y . f (s,u,v) p N(u,dv)q N(s,du) IT V f (s,u,v)p(u,dv)q(s,du) U. 1/ f( s , u , v ) p N ( u , d v ) q N ( s , d u ) U 1/ U 1/ f (s,u,v)p(u,dv)q N(s,du) + • r U V f(s,u,v)p(u,dv)q N(s,du) U 1/ f(s,u,v)p(u,dv)q(s,du) Suppose we are given e > 0. Since, by hypothesis f o r seS we have • f(s,u,v)p^(u,dv)—>• f(s,u,v)p(u,dv) uniformly on U, 1/ 1/ i t f o l l o w s that the f i r s t term on the r i g h t hand s i d e of the above i n -e q u a l i t y i s l e s s than e/2 f o r N s u f f i c i e n t l y l a r g e . A l s o by hypothesis, w f (s,u,v)p(u,dv) i s continuous and bounded on Sxtl and q^ —>-q uniformly on S; V ,. ' - , ........ ,- - • . -hence' the second term i s l e s s "than G/2 f o r N * s u f f i c i e n t l y ' l a r g e . Thus the-desired l i m i t " has been e s t a b l i s h e d . g Assume that S i s a P o l i s h space, i . e . , a complete, separable, m e t r i c space (e.g. 5 Let {Y ,Y^,Y2»...} be a time homogeneous Markov chain w i t h s t a t e space S, w i t h p r o b a b i l i t y t r a n s i t i o n f u n c t i o n p from S to 5, and with the i n i t i a l c o n d i t i o n s given by the p r o b a b i l i t y measure p(.B) = Pr [ Y Q £ B ] , BeBCS). Given that Y. = y, denote the j o i n t d i s t r i b u t i o n of Y , „ Y _ , . . . Y ^ . 0 1 2' t by v t(y,B):=Pr[(Y 1,Y 2,Y 3.,,,Y t)eB|Y ( ) = y] f o r yeS, BeB(5 t). Using Theorem V . l . l from Neveu (1965), v i s a p r o b a b i l i t y t r a n s i t i o n f u n c t i o n - 60 -from 5 to s'. The fu n c t i o n s v are also c h a r a c t e r i z e d by the r e c u r s i v e r e l a t i o n V t + l ( y ' B l x B ) = v f c(x,B)p(y,dx) (2.11) f o r B CS and BOS 1". Unconditioning on Y^, de f i n e the j o i n t d i s t r i b u t i o n f u n c t i o n v t(B):= Pr[.(Y 0,Y 1,...,Y t)eB] f o r B e E ( S t + 1 ) . The measures v are c h a r a c t e r i z e d by the r e l a t i o n v t ( B 1 x B ) = v (y,B)y(dy) f o r BjCS and BCS (2.12) B N Now, i n a d d i t i o n , consider a f a m i l y of Markov chains {Y^jY^jY^,...} f o r N = 1,2,..., each defined on the same s t a t e space S. For each pro-N N N ~N cess, define p , u , v^, and v as above. (Since the p r o b a b i l i t y space N a c t u a l l y changes as we change N, we should a l s o use Pr. However, the p a r t i c u l a r p r o b a b i l i t y space w i l l be c l e a r from the context and t h i s s u p e r s c r i p t w i l l be omitted.) Of course, (2.11) and.(2.12) apply w i t h the s u p e r s c r i p t e d measures. N w Theorem 2.5 I f p -*p uniforml y on S and i f f(u,v)p(u,dv)eC(5) when-ever f e C ( S 2 ) and ueS, then f o r each t , ~~*v£uniformly on S as N-*°°. I f , N w ~N w~ i n a d d i t i o n , y —>-y, then f o r each t ^ t ~^ v t a s or e q u i v a l e n t l y N N N N V ( Y Q , Y ^ , Y ^ ,. . . »Y^.) (YQ,Y . •. - 61 -Proof We w i l l prove by induction that (i) f(u,v) v (u,dv)e C(S) whenver f e C ( S t + 1 ) , and N (ii) v :v uniformly on S as N-*». Since = p, the above statements are true by hypothesis when t = 1. Suppose statements (i) and (i i ) hold for some t. Then f f(y,x,x)v t(x,dx) p(y,dx) ^t+1 xe-S X E 5 f(y,x)v t + 1(y,dx) 2 follows from (2.11). The inner integral belongs to C(S ) by Lemma 2.1 and induction hypothesis ( i ) . Consequently, the outer integral belongs N to C(S). Now apply Lemma 2.2 to show that v t + 2 ' characterized by v t + 1 ( y ) B l x B ) = *1 v^(x,B) p^(y,dx), converges weakly and uniformly on S to v -j_ characterized by (2.11). This completes the proof of (i) and ( i i ) . The second conclusion of the Theorem follows by applying Lemma 2.2 -N to the characterization of v and vfc given by (2.12). The hypotheses N w of Lemma 2.2 are satisfied by ( i ) , ( i i ) , and the assumption that u —>-u uniformly on S. Theorem 2.6 Assume that the hypotheses of Theorem 2.5 hold, including N w N N N V U —*y. Then (Y Q,Y 1,Y 2,. ..) —>(YQ Y^Y^...). And for any bounded, oo measurable, real valued function f defined on S with discontinuity set Df such that v(D f) = 0, Ef(Y^,Y^,Y^,...) ->Ef(Y 0,Y 1,Y 2,...). (2.13) - 62 -Proof The Theorem f o l l o w s from Theorem 2.5 and the d i s c u s s i o n at the beginning of Section 5 and page 19 i n B i l l i n g s l e y . (Although B i l l i n g s l e y assumes S C fi , only completeness i s used i n h i s argument.) I f feC(S°°), V then (2.13) f o l l o w s from the d e f i n i t i o n of Theorem 5.2 i n B i l l i n g s l e y shows that the c o n t i n u i t y of f can be r e l a x e d to the s t a t e d c o n d i t i o n s . • 2.4 Concluding Remarks Although a major part of t h i s chapter was concerned with the approx-imation of the f i n i t e p o p u l a t i o n model by the i n f i n i t e p o p u l a t i o n one, the p o s s i b i l i t y of using the f i n i t e p o p u l a t i o n model d i r e c t l y should not be ignored. As we pointed out, i t can be simulated without d i f f i -c u l t y . A l s o , f a i r l y general and e a s i l y checked c o n d i t i o n s f o r asymptotic s t a t i o n a r i t y were given, whereas comparable r e s u l t s f o r g l o b a l s t a b i l i t y (of the i n f i n i t e p o p u l a t i o n model) have been obtained ( C o n l i s k 1976, 1978a, Smallwood and C o n l i s k 1979) only f o r s p e c i a l cases. Nevertheless, f o r c e r t a i n parametric f a m i l i e s of the i n f i n i t e popu-l a t i o n model i t i s sometimes f e a s i b l e (Smallwood and C o n l i s k 1979) to discover the dependence of the model's e q u i l i b r i u m on some underlying parameters. This i s c e r t a i n l y an advantage of the i n f i n i t e p o p u l a t i o n model which, coupled w i t h i t s computational s i m p l i c i t y , j u s t i f i e s the e f f o r t we have put i n t o proving i t s v a l i d i t y . - 63 -CHAPTER I I I THE MEASUREMENT OF SOCIAL INHERITANCE A major aim of s o c i a l s c i e n t i s t s , who engaged i n modelling i n t e r -g e n e r a t i o n a l m o b i l i t y processes, was oft e n e i t h e r to study the i n t e r -n a t i o n a l d i f f e r e n c e s i n the r a t e s of m o b i l i t y , or to analyze m o b i l i t y trends over time. One aspect, of p a r t i c u l a r i n t e r e s t to s o c i o l o g i s t s , was the dependence of a person's s o c i a l ( o c c u p a t i o n a l ; income) c l a s s on that of h i s f a t h e r . This gave r i s e to what i s f r e q u e n t l y r e f e r r e d to as "measurement of m o b i l i t y " and l e d to the formal problem of b u i l d i n g an acceptable index of m o b i l i t y . For h i s t o r i c a l account and survey see Boudon (1973). The term " r a t e of m o b i l i t y " i s , however, q u i t e ambiguous and may be given e n t i r e l y d i f f e r e n t i n t e r p r e t a t i o n s even w i t h i n the context of a s i n g l e phenomenon l i k e , say, occupational m o b i l i t y . A common i n t e r p r e t a -t i o n i s that of the net r e d i s t r i b u t i o n of the working f o r c e by f u n c t i o n a l c a t e g o r i e s - i n d u s t r i e s and occupations (sometimes r e f e r r e d to as " s t r u c t u r a l m o b i l i t y " ) . This aspect of occu p a t i o n a l m o b i l i t y i s of par-t i c u l a r i n t e r e s t i n the context of development economics, si n c e economic growth i s known to be accompanied by such net r e d i s t r i b u t i o n (see Smelser and L i p s e t 1966, e s p e c i a l l y the c o n t r i b u t i o n by Duncan). S o c i a l s c i e n t i s t s w i t h primary i n t e r e s t i n issues of equi t y , on the other hand, focus more on patter n s of gross m o b i l i t y . They are i n t e r e s t e d i n assessing the d e v i a t i o n of a given s o c i e t y from some s o c i a l i d e a l , such as the one i n which son's c l a s s does not depend on that of previous generations i n h i s f a m i l y l i n e . Shorrocks (1978) r e f e r s to t h i s aspect as the " p r e d i c t a b i l i t y " of s o c i e t y - the extent to which - 64 -f u t u r e p o s i t i o n s are d i c t a t e d by the current place i n the d i s t r i b u t i o n . (Sometimes r e f e r r e d to a l s o as " c i r c u l a t i o n " m o b i l i t y ) . Since the p r e d i c t a b i l i t y ( d e v i a t i o n from s o c i a l i d e a l ) aspect of s o c i a l m o b i l i t y seems most i n t e r e s t i n g and c h a l l e n g i n g as far' as measure-ment i s concerned, we s h a l l focus our a t t e n t i o n on i t . We make i t e x p l i c i t by r e f e r r i n g to the problem s p e c i f i c a l l y as measurement of " s o c i a l i n h e r i -tance", ' r a t h e r than "measurement of m o b i l i t y " . Since high degree of s o c i a l i n h e r i t a n c e corresponds to what was p r e v i o u s l y l a b e l l e d as low (*) S t i l l another aspect of m o b i l i t y , of p a r t i c u l a r importance i n m i g r a t i o n and other types of i n t r a g e n e r a t i o n a l m o b i l i t y , i s the " p h y s i c a l " r a t e at which i n d i v i d u a l s change l o c a t i o n s (e.g. number of residence changes per u n i t time; see Long (1970) and Section 1.4). Although "...as more movement i s observed i t would be normal to expect the c l a s s occupied i n the f u t u r e to become l e s s dependent on the present p o s i t i o n . In g e n e r a l , t h e r e f o r e , they Jthe r a t e of movement and the n o n - p r e d i c t a b i l i t y of s o c i e t y ] should be i n harmony" (Shorrocks 1978, p. 1016), these two aspects of m o b i l i t y are not p e r f e c t l y c o r r e l a t e d , and should hot be confused. {See a l s o Sommers and C o n l i s k (1979, pp. 254-255)]. (**) Pullum (1975) used the term "occupational i n h e r i t a n c e " . Another p o s s i b l e term i s "measurement of e q u a l i t y of s o c i a l opportunity". (***) This w i l l a l s o be c o n s i s t e n t w i t h an argument made by Duncan (1966) according to which a v a i l a b l e m o b i l i t y t a b l e s provide o n l y " i n h e r i -tance", and not " m o b i l i t y " , data. - 65 -degree of mobility, our ordering w i l l reverse the more common one. v '' v An underlying assumption i n the following discussion i s that the processes of i n t e r e s t are time-homogeneous, which seems to have some empirical support as f a r as intergenerational m o b i l i t y i s concerned (Bartholomew 1973). However, even i f the process i s not r e a l l y time-homogeneous, i t might be i n t e r e s t i n g to ask what are the long term implications of the current trend. The degree of s o c i a l inheritance i n two s o c i e t i e s can be compared on the basis of data pertaining to .. ... s i m i l a r dates, perhaps combining "generation-specific" indices to an (ft**) o v e r a l l one. In Section 1 we discuss, in non-mathematical terms, the proper-t i e s one would l i k e such measures to have (they are r e l a t e d to the ones mentioned by Shorrocks 1978). Various concepts through which s o c i a l inheritance i s manifested are then discussed. The second section then formulates a (Markovian) model as a map-ping over the u n i t simplex. It then states desirable properties and inheritance concepts mathematically and discusses t h e i r implications. Various concept-dependent ways to measure "non-constancy" of the operator are suggested, and i t i s shown that many known measures, as well as new ones, can be obtained i n t h i s way. Some sp e c i a l cases are analyzed i n d e t a i l . A method of introducing "period consistency" (Shorrocks 1978) i s then discussed. (*) Though i t w i l l be consistent with Goodman's (1969) notion of "status persistence" and with Sommers and Conlisk's (1979) "Immobility". (ftft) j 7 o r a n i n t e r e s t i n g discussion of measures of ( s t a t i c ) income ine q u a l i t y see Kolm (1976). (ftft*) see also Shorrocks (1978, p. 1021). - 66 -3.1 P r e l i m i n a r y D i s c u s s i o n As we s h a l l soon see, even zeroing i n on t h i s s i n g l e aspect of m o b i l -i t y does not provide s u f f i c i e n t guidance f o r c o n s t r u c t i n g measures. But already at t h i s stage we can l a y down some general requirements from such a measure. For the moment, they w i l l be stat e d i n non-mathematical form; i n S ection 2, we s h a l l r e s t a t e them mathematically. An und e r l y i n g assumption i s , however, that s o c i a l m o b i l i t y over generations i s Markovian. The " s t a t e s " may, however, be taken to be e i t h e r s o c i a l c l a s s e s or d i s t r i -butions over c l a s s e s ( " p r o f i l e s " ) so the set-up i s general enough to in c l u d e Markovian models of previous chapters. 3.1.1 B a s i c Requirements I f i n e q u i t i e s due to d i f f e r e n c e s i n o r i g i n e n t i r e l y disappear w i t h i n few generations, the degree of s o c i a l i n h e r i t a n c e i s as low as i t can (*) be. A l l other types of s o c i e t i e s e x h i b i t some degree of i n h e r i t a n c e . The slower the " r a t e " at which i n e q u i t i e s due to o r i g i n disappear, the higher should the s o c i e t y rank on a s o c i a l i n h e r i t a n c e s c a l e . At the extreme, we f i n d s o c i e t i e s where any person's c l a s s i s always i d e n t i c a l to that of h i s f a t h e r - a p e r f e c t caste system. Those should be assigned the highest value of the measure. Since one may wish to compare s o c i a l i n h e r i t a n c e i n s o c i e t i e s f o r which the l e n g t h of time i n t e r v a l f o r which data are a v a i l a b l e might be d i f f e r e n t (say, one generation vs. two), measures should be r e q u i r e d to (-*) Within t h i s c l a s s , we may rank s o c i e t i e s according to the number of generations i t takes. - 67 -be "Period C o n s i s t e n t " (Shorrocks 1978). This i s to say that a s o c i e t y whose degree of s o c i a l i n h e r i t a n c e i s ranked higher than another's on the b a s i s of one g e n e r a t i o n a l i n h e r i t a n c e data, should be so ranked on the b a s i s of data corresponding to any number of generations. 3.1.2 Concepts of S o c i a l I n h e r i t a n c e The aspect of s o c i a l m o b i l i t y ( i n h e r i t a n c e ) which we wish to measure has many f a c e t s which are not n e c e s s a r i l y p e r f e c t l y c o r r e l a t e d w i t h each other. We s h a l l now d i s c u s s some f a c e t s of s o c i a l i n h e r i t a n c e and t h e i r i m p l i c a t i o n s f o r purposes of measurement. As we s a i d , a measure of s o c i a l i n h e r i t a n c e should i n d i c a t e the r a t e at which i n e q u i t i e s , due to d i f f e r e n c e s i n o r i g i n c l a s s e s , disappear over generations. Now, what i f f o r a c e r t a i n s o c i e t y , r e g a r d l e s s of how many generations pass, some " b a s i c " i n e q u i t i e s due to o r i g i n s t i l l pre-v a i l (while others, perhaps, disappear)? Loosely speaking, i n such cases one could lump s o c i a l c l a s s e s together and end up w i t h a p e r f e c t caste system over the lumped c l a s s e s . Hence we f e e l that there i s a j u s t i f i c a t i o n to a s s i g n a l l such processes (and not only those which correspond to p e r f e c t caste systems over the o r i g i n a l c l a s s e s ) the highest value of the measure. The above poi n t g i v e s r i s e , however, to a general fundamental i s s u e , which has ( s u r p r i s i n g l y ) achieved l i t t l e a t t e n t i o n i n the m o b i l i t y measure-ment l i t e r a t u r e . Given any concept of s o c i a l i n h e r i t a n c e , a measure can focus on a f a c e t . which e x h i b i t s " h i g h e s t " s o c i a l i n h e r i t a n c e , or i t can "average" a l l the f a c e t s (circumstances). For example, s o c i a l c r i t i c s - 68 -f r e q u e n t l y point out that the r a t e of r e d u c t i o n i n some s p e c i f i c i n e q u i t i e s i n a given s o c i e t y (say, a gap i n access to high education between c e r t a i n groups) i s too slow, and d i s r e g a r d ( p o s s i b l e ) f a s t r e d u c t i o n s i n e q u i t i e s simultaneously t a k i n g place i n another part (or aspect) of s o c i e t y . S p e c i f i c a l l y , the above d i s t i n c t i o n i s p a r t i c u l a r l y r e l e v a n t i n the treatment of : a. " D e s t i n a t i o n " c l a s s e s (or p r o f i l e s ) . The chances of j o i n i n g c e r t a i n s o c i a l c l a s s e s may depend on o r i g i n c l a s s more than chances to j o i n other c l a s s e s do. The measure w i l l then be an "average" weighted over a l l d e s t i n a t i o n c l a s s e s , and the weights w i l l "express" the user's s o c i a l p r i o r i t i e s . At one extreme, a l l weight w i l l be assigned to the most origin-dependent d e s t i n a t i o n , and on the other a l l d e s t i n a t i o n s w i l l be weighted e q u a l l y . b. " O r i g i n " c l a s s e s (or p r o f i l e s ) . Using " d i f f e r e n c e s " between d i s t r i b u t i o n s over d e s t i n a -t i o n c l a s s e s f o r each p a i r of o r i g i n c l a s s e s as a b a s i c t o o l , the measure w i l l be a weighted average over a l l these p a i r s . At one extreme, a l l weight w i l l be assigned to the p a i r of o r i g i n s which generates the l a r g e s t d i f f e r e n c e . At the other extreme, a l l p a i r s w i l l be weighted e q u a l l y . Even i f i n e q u i t i e s due to d i f f e r e n c e i n o r i g i n v i r t u a l l y disappear a f t e r a s u f f i c i e n t number of generations, t h i s may take longer f o r the o f f s p r i n g s of some o r i g i n -c l a s s e s (or, more g e n e r a l l y , f o r some i n i t i a l d i s t r i b u -- 69 -t i o n s ) than f o r others. Again, one can focus on. e i t h e r the "slowest" o r i g i n - e f f e c t to disappear, or compute an average over a l l o r i g i n s . One m a n i f e s t a t i o n of the r a t e at which i n e q u i t i e s due to d i f f e r e n c e i n o r i g i n - c l a s s disappear are the d i f f e r e n c e s between the d i s t r i b u t i o n over c l a s s e s of i n d i v i d u a l s whose f a t h e r s belonged to d i f f e r e n t - c l a s s e s ( i . e . the amount of " s o c i a l scrambling" that takes place from one generation to the n e x t ) . The r e l a t i o n of such two successive generations' d i s t r i b u t i o n s to each other a f t e r many generations had passed may serve as a ba s i s f o r a measure of s o c i a l i n h e r i t a n c e . We s h a l l now tu r n to a mathematical d i s c u s s i o n . 3.2 Mathematical D i s c u s s i o n 3.2.1 D e f i n i t i o n s Let K = { 1 , 2 , ... ,K} be. a set. of categ o r i e s ( s o c i a l c l a s s e s or d i s -t r i b u t i o n s over c l a s s e s ) . Let P^" be a c o n d i t i o n a l p r o b a b i l i t y mass f u n c t i o n over K, so that P i s , say, the p r o b a b i l i t y that a son of c l a s s - i - f a t h e r belongs to c l a s s j . Let P be the set of p r o b a b i l i t y mass fu n c t i o n s over K, i . e . i f p£p then p=(p ,. . . ,p ) such that p . = proba-J_ K i b i l i t y that a given i n d i v i d u a l belongs to c l a s s i (or that the p r o f i l e i s of type i ) . P can be modelled by a K - l dimensional simplex S, which i s a convex C K „ r r , _ R . The p o i n t s {V ,...,V } are a r b i t r a r y p o i n t s i n R except f o r the r e s t r i c t i o n that - 70 -they be l i n e a r l y independent. The v e r t i c e s V.. ,V„ correspond to the p r o b a b i l i t y mass f u n c t i o n s (or p r o f i l e s ) (1,0,...,0) (0,...,0,1) r e s p e c t i v e l y . The p r o b a b i l i t y d i s t r i b u t i o n p=(p 1 }...p -) corresponds to 1 K K the point ^ p.V, . i = l The c o n d i t i o n a l p r o b a b i l i t y mass f u n c t i o n s (P..:i=l,2,...,K) can be c h a r a c t e r i z e d as a map T from S. to i t s e l f , w i t h the property that K K K T( I P.V.) = I p.T(V.) f o r each J p.V.eS. ; There i s a unique 1=1 1 1 1=1 1 1 1=1 1 1 extension of T from S to a l i n e a r map on Br'. Let T ^ n + 1 ^ ( X ) = T ( T ^ ( X ) ) , where T ^ ( X ) = T(X). I f {V ,...,VR} happens to be used as the b a s i s of R , then the matrix r e p r e s e n t a t i o n of T i s P whose i - t h row i s the v e c t o r (**) K P i A , and Tp = pP. ' I f a d i f f e r e n t b a s i s { b ^ , . . . ^ ^ i n R i s chosen, then there e x i s t s an i n v e r t i b l e matrix A such that i f x=T p.V. then x = I(pA a n ( i t n e matrix r e p r e s e n t a t i o n of T i s APA (see, f o r example, Halmos 1958). T w i l l be considered a "constant" map i f there e x i s t s p* such that Tp = p* f o r every peS. I t i s p o s s i b l e that although T i t s e l f i s not con-s t a n t , there e x i s t s n such that T ^ p = p* f o r every peS (Brosh and Gerchak 1978). T i s an " i d e n t i t y " map (denoted by I) i f TV ± = V f o r every i e (*) We s h a l l sometimes a b r e v i a t e T(X) by TX. (.**) This r e p r e s e n t a t i o n i s the common one i n the l i t e r a t u r e since i t describes the Markovian nature of the process. K' K (***) For any peS } a 1 > . . . ,a K. such t h a t p = ^ N o w » 1 P = !( I a.V.) = i = l i = l 1 1 K K / a.IV. = y a.V. = p. Hence the above i s enough to c h a r a c t e r i z e an . - . I l . . . i i ° 1=1 1=1 i d e n t i t y map over a l l S. - 71 -3.2.2 Mathematical Statement of D e s i r a b l e P r o p e r t i e s Let us f i x {V^:i=l,...,K}, use i t as a b a s i s , and consider the aspect of s o c i a l i n h e r i t a n c e . What we a c t u a l l y want i s to Impose a simple order on maps of the kind defined above. A simple order i s a r e l a t i o n which i s connected, t r a n s i t i v e and antisymmetric ( f o r d e t a i l s , see Krantz, Luce, Suppes and Tversky 1971). Now, the number of K-state r a t i o n a l -valued p r o b a b i l i t y t r a n s i t i o n s matrices i s countable and t h i s set i s dense i n the set of a l l K-state p r o b a b i l i t y t r a n s i t i o n m a trices. Hence Theorem 2 i n Krantz et a l . (1971, p. 40) i m p l i e s that f o r any simple order over those t r a n s i t i o n matrices there e x i s t s an isomorphism i n t o the r e a l l i n e . Hence we s h a l l r e s t r i c t ourselves to r e a l - v a l u e d f u n c t i o n s (measures). Moreover, s i n c e i n the previous s e c t i o n we have i d e n t i f i e d processes f o r which we wish the measure to a t t a i n i t s maximum and m i n i -mum, we can r e s t r i c t o u r s e l v e s to any cl o s e d i n t e r v a l , and i n p a r t i c u l a r to the common choice [0,1]. Denote f u n c t i o n s which map the T's to [0,1] by M. Some of the general p r o p e r t i e s r e q u i r e d above can now be stated as f o l l o w s , a. M(T) = 0 i f and only i f there e x i s t s an i n t e g e r n such that i s a constant map. The " i f " part i m p l i e s Shorrocks 1 " P e r f e c t M o b i l i t y " c o n d i t i o n [ " i f T i s constant M(T) = 0"]; h i s "strong" v e r s i o n , however, assigns the extreme value only i f n=l. b) M(T) = 1 i f and only i f l i m T ^ p i s not constant i n p. This n-*"° i m p l i e s Shorrocks' "Immobility" c o n d i t i o n (M(I)=1) since the i d e n t i t y map has t h i s property. His "stro n g " v e r s i o n , however, assigns the extreme value to the measure only i f T i s an i d e n t i t y map. - 72 -Thus the remaining task i s to rank non-constant i r r e d u c i b l e -aperiodic Markov chains by the degree of s o c i a l inheritance they e x h i b i t . The "Period Consistency" condition can be stated as follows: n 'n (*) i f M(T) ^M(T') then M(T ) >_ M(T ) for every n. J Shorrocks, who assumed that i n d i v i d u a l s follow independent and i d e n t i c a l l y d i s t r i b u t e d Markov chains ( i . e . the categories i n his model were the usual s o c i a l classes) introduced the following p a r t i a l order over the p r o b a b i l i t y t r a n s i t i o n matrices, which he c a l l e d "Monotonicity": i % P P' i f for every i ^ i P.. < P.. and f o r some j ^ i P.. < P... J x i - x i J r XJ XJ However, since constant maps were previously assigned the lowest value of the measure, then i n order to achieve consistency between the two some further r e s t r i c t i o n of the p a r t i a l order i s necessary. Indeed, Shorrocks r e s t r i c t e d i t to P's which have a "quasi-maximal diagonal": there e x i s t s p o s i t i v e numbers y^,...,y R such that V^P^ — ^k^ik e v e r Y ^ a r u * ^' The above p a r t i a l order w i l l be too strong i f one wishes to focus on the extremes of society. In p a r t i c u l a r , i f what we wish to express i n the measure i s the " l a r g e s t " inequity i n society, "improvements" i n other parts (facets) of society should not a f f e c t the measure. This may be achieved by weakening the notion of Monotonicity to "Weak Mono-t o n i c i t y " : P ^ P 1 i f for every j ^ i P — P i j • Although i t w i l l be nice i f the t o t a l orders we s h a l l come up with w i l l be consistent with the above p a r t i a l order (and some w i l l indeed :(*) Suppose that M(T) ^ M(T') =>M(Tn) >_ M(T' n) for every T,T' and n _> 1, and assume that f o r some T,T' and n* M(T n*) > M(T' n*). Suppose now that for these T and T 1 M(T) < M(T'). Then, by the f i r s t assumption, M(T n) £ M(T' n) for every n, and i n p a r t i c u l a r f or n*. Contradiction. Hence "Period Consistency", the way i t was defined, implies the seemingly stronger property " i f M(T^) ^M(T'k) for some integer k'then M(T n) >. M(T' n) f o r every integer n". - 73 -be), i t i s not that c r u c i a l given our approach as i t has been w i t h Shorrocks'. We s h a l l e s s e n t i a l l y a r r i v e at measures i n a c o n s t r u c t i v e manner, using v a r i o u s f a c e t s of s o c i a l i n h e r i t a n c e as bases, w h i l e Shorrocks had only "Monotonicity" to help him evaluate the s u i t a b i l i t y of a r b i t r a r y f u n c t i o n s as measures. As pointed out by Sommers and C o n l i s k (1979, p. 254) ".. . i m m o b i l i t y might be thought of as the slowness w i t h which the s t a t e p r o b a b i l i t i e s of a Markov chain 'escape' the e f f e c t s of i n i t i a l c o n d i t i o n s on route to t h e i r e q u i l i b r i u m values. The slower t h i s convergence, the more s t r o n g l y the parent's s t a t e i n f l u e n c e s the l i f e chances of the c h i l d , g r a n d c h i l d , and so on,. That i s , the 'mathematical' problem of measuring the slowness of a Markov chain's convergence to e q u i l i b r i u m i s c l o s e l y a k i n to the immobility measurement problem". I t w i l l be n i c e , then, i f our measures w i l l be r e -l a t e d to t h i s r a t e of convergence. 3.2.3 Measuring the Non-Constancy of the Operator Measuring second-generation i n e q u i t i e s due to d i f f e r e n c e s i n o r i g i n n a t u r a l l y i n v o l v e s measuring the non-constancy of the operator T. I f we (*) " a l l o w " the process to s t a r t from any d i s t r i b u t i o n , i t amounts to eval u -a t i n g d i f f e r e n c e s among a l l {T(p>; pes}. I f we r e s t r i c t our i n t e r e s t to v e r t e x - o r i g i n s only, we evaluate differencesamong only {T(V.): i = l , . . . , K } . (n) * I f l i m T (p) = p f o r every peS, we may choose to measure the di s t a n c e s between (T(p): peS jp=V±: i = l , . . . , K ] } and pT» (*) Say, v a r y i n g a random mechanism f o r choosing an i n d i v i d u a l . - 74 -We now wish to o p e r a t i o n a l i z e the n o t i o n of d i f f e r e n c e s / d i s t a n c e s between d i s t r i b u t i o n s . Let p and q be any p r o b a b i l i t y mass f u n c t i o n s over K, and l e t d(p,q) be a " d i s t a n c e - f u n c t i o n " . Such a r e a l - v a l u e d f u n c t i o n i s r e f e r r e d to as met r i c (Royden 1962) i f f o r every p,q and r on K: i . d(p,q) > 0; i i . d(p,q) = 0 i f and only i f p=q; i i i . d(p,q) = d(q,p); and i v . d(p,q) _< d(p,r) + d(r,q) . I f c o n d i t i o n ( i i ) i s relaxed, to read d(p,p)=0, the,.;,f u n c t i o n , d Is .ca l l e d a pseu-dometric. A most common way to construct m e t r i c s (pseudometrics) i s by d e f i n i n g a norm over d i f f e r e n c e s of d i s t r i b u t i o n s . A norm i s a non-negative r e a l -valued f u n c t i o n II . II such that 1. Ilxll = 0 i f any only i f x = o; 2. Ilx+yll < Ilxll + ll y l l ; and 3. Haxll = [a' Ilxll . I f c o n d i t i o n (1) i s r e l a x e d to read II Oil = 0 the f u n c t i o n i s c a l l e d a pseu-donorm. A n a t u r a l step i n the d i r e c t i o n of c o n s t r u c t i n g measures of non-constancy i s thus to s e l e c t a f u n c t i o n g and a norm 11.11, and consider expressions l i k e llg(Tp)-g(Tq) II. The most obvious f u n c t i o n to consider i s j u s t the i d e n t i t y f u n c t i o n , i . e . to focus on H Tp - Tq II. The d i f f e r e n c e s T(p) - T(q) are K-dimensional, so a n a t u r a l f a m i l y of norms to consider w i t h such g i s - 75 -K l / a L = (I !x.| ) , For a=l we obtain the summation ("averaging") a i = l ' 1 K norm £ |x. | (to be denoted by II ",) . For a = 0 0 we get the supremum 1=1. X norm sup |x.| (to be denoted by II H°°). i Another n a t u r a l d i r e c t i o n to proceed i s to t r y and come up w i t h a f u n c t i o n g which by i t s e l f maps from the boundary of the K-dimensional u n i t simplex to the r e a l l i n e ( a l l L -type norms then reduce to the a absolute-value norm). A well-known such"function which turned :out .to be use-f u l i n many contexts, i s the entropy of the d i s t r i b u t i o n (e.g. Khinchin 1957). For any p r o b a b i l i t y mass f u n c t i o n p i t i s defined as r 1 » (*)' i i H =/, P- l o§ / • The pseudometric |H(Tp)-H(Tq) [ i s then an " a l t e r -P i 1 ^ i ( * * ) II n a t i v e " to the pseudometric » T(p) - T(q)H as a " b a s i s " f o r a measure of non-constancy. Let us now t r y to combine these pseudometrics w i t h the n o t i o n s of s o c i a l ' i n h e r i t a n c e discussed 'in subsection 3.1.2 and i n the beginning of t h i s subsection. A measure of non-constancy w i l l be a choice of combination of: a. nature of o r i g i n s ( v e r t i c e s vs. d i s t r i b u t i o n s ) ; b. d i s t a n c e generated by p a i r s of o r i g i n s vs. d i s t a n c e to steady s t a t e ; c. pseudometric over d e s t i n a t i o n s (see above); d. : norm over o r i g i n s . (.*),- H . =. H does not imply that p=q (consider, f o r example, any two y q permutations). (**) T(p) = T(q) does not imply that p=q. - 76 -We s h a l l now s p e c i f y and d i s c u s s some of the more i n t e r e s t i n g com-b i n a t i o n s . 3.2.4 Some S p e c i a l Cases A. For every p a i r of o r i g i n v e r t i c e s , c a l c u l a t e t h e i r images under T and t h e i r d i f f e r e n c e using the L^-norm. Take the measure to be the supremum of the above over a l l such p a i r s . We obtain the f u n c t i o n supnTV.-TV || . i , k Sup||TV. - TV || equals (twice) the " d e l t a c o e f f i c i e n t " of T, which equals one minus the "ergodic c o e f f i c i e n t " of T - a u s e f u l t o o l i n the a n a l y s i s of Markov chains (e.g. Isaacson and Madsen 1976). The ergodic c o e f f i c i e n t has been a c t u a l l y suggested, i n an e n t i r e l y d i f f e r e n t context, as a measure of the "scrambling power of a m a t r i x ... the degree to which i t approaches a matrix w i t h i d e n t i c a l rows which scrambles a l l t r a c e s of the past" (Hajnal 1958, p. 236). Since t h i s n o t i o n seems c l o s e to that of constancy, we s h a l l now " o p e r a t i o n a l i z e " the d e l t a c o e f f i c i e n t as a measure of s o c i a l i n h e r i t a n c e , and i n v e s t i g a t e i t s proper-t i e s . This w i l l be done using {V ,...,V } as a b a s i s - the I K common r e p r e s e n t a t i o n . In t h i s s e t t i n g the d e l t a - c o e f f i c i e n t becomes (Isaacson and Madsen 1976) - 77 -• K 6(P) = 1 - min £ P.. A P i , k d = l ' 1 J k j where P. . A P. . = i n f (P. ., P. .) . A delta-coefficient-based o r d i n a l which we s h a l l discuss as a measure i s the following: M(P) < M(P') i f and only i f 6(P n) < 6 ( P , n ) , where n = inf{m: 6(P m) t 6(P' m)}; i f n = » then M(P) = M(P'). (' V ) We s h a l l now state (and, for some non-standard assertions, prove) some of the properties of the above measure. a. 0 j< 6(P) •<_••! for every p r o b a b i l i t y t r a n s i t i o n matrix P [Shorrocks (1978) referred to t h i s property as "Normalization"]. b. -Jm sue h that 6(P m) < 1 i f and only i f }k s u c h that P k > 0 k ( i . e . a l l elements of P are s t r i c t l y p o s i t i v e f o r some k). Thus <5(P ) = 1 for every n i f and only i f P i s reducible. Viewed i n another way, 6(P) < 1 i f and only i f , f o r any two o r i g i n classes, there e x i s t s at l e a s t one destination c l a s s whose occupants might have come from either one of them ( i . e . i f and only i f some s o c i a l "scrambling" takes place) — an a t t r a c t i v e feature of t h i s measure. Note that the above statement "looked" at the processes backwards i n time, which i s rather natural i n (*) It i s known that i f inf{m : 6(P m) < 1} < 00 then inf{m : 6(P m) < 1} (K-l) / 5. t 2 + 1 ] , where [ ] denote the "integer part of". Hence, in order to determine t h i s infimum, only a r e l a t i v e l y small number of powers of P has to be checked. - 78 -the context of "future's independence of past", c. <5(P m) = 0 i f and only i f a l l rows of P m are equal, i . e . i f and only i f the Markov chain converges i n a f i n i t e number of periods. d. Let us now r e s t r i c t the p a r t i a l order of "weak mono-t o n i c i t y " ^ to a c l a s s P such that PeP only i f for every (ft) ft ft i P.„ 2l P J.J_ f o r every j . Given any PeP and i , j e K d e f i n e P as f o l l o w s : 1 J 13 p ft i 3 f o r i £ i f o r ft ft i =. i j £ I • j V P . * . *-£ f o r i = i i = i ft ft P . *.* + £ f o r i = i j = j i 3 ;here e > 0 i s any number such that P.*.* - £ >_ f ° r x x ft e every k ^ i , and that P.*.* + £ < P.*.* (so that P £P) 3-3 - 3 3 K Now, 1 - 6(P £) = min J P;.£ A P. . i,k j = l 13 K {min Y P.. A P, .} A {min Y P.*. AP ' . } -.•j.-ft ^ i l kj ^ i 3 ki k*i* J _ 1 k j = l But K mxn k in J P.*. L i i k j = l J A P , . = min{ y P.*.AP. . + (P.*.*-e)AP, .*+(P.*.*+e)Ap, .*} kj . L x j kj x x kx x j kj K • J. • ft 3 f i 3r3 (*) This condition i s somewhat more r e s t r i c t i v e than Shorrocks' "quasi-maximal diagonal" (for which i t i s s u f f i c i e n t ) . We chose to use i t here since i t i s more convenient to work with, as well as having an i n t u i t i v e appeal. Also, as pointed out by Shorrocks, i t i s "easy to confirm by inspection and holds f o r a large number of t r a n s i t i o n matrices [reported i n the empirical l i t e r a t u r e ] " . - 79 -= min { j. . P.*..AP, .+(P . *. *+e) AP, .*} K > min I P . A . A P . k j = l 1 1 k j Hence 1 - 6(P £) > 1 - S(P) , i . e . 6 (P £) £ <5(P) . So we proved that i f f o r every i P >_ P^'j r o r every j f i , then 6(P) £ 6(P'). This i s the "weak" v e r s i o n of "Monotonicity", which i s n a t u r a l s i n c e the d e l t a c o e f f i c i e n t focuses on the p a i r of o r i g i n s which' generates the l a r g e s t d i f f e r e n c e i n next-generation d i s t r i b u t i o n s . n * * Suppose that l i m P = P (where P has equal rows), and n-*°° l e t II All = sup I |a..| f o r any matrix A. Then ||pn-p"|| = || I-p n-P*-p n|| i i 1 J * n = II (I-P )P I I . Hence by Lemmas V.2.3 and V.2.4 i n Isaacson and Madsen (1976) llp n-P*H -<- II I-P*H 6(p n) < II I-P*ll [ 6 ( P ) ] n . We see then-; that•the'., geometric orate ofu.cbhvergence'of P n to P can be expressed in-.terms./of the delta: c o e f f i c i e n t of, P, as w e l l as i n terms of .the second l a r g e s t eigenvalue>of P (to be discussed n e x t ) . Let T be the operator which maps d i s t r i b u t i o n s over p r o f i l e s i n t o d i s t r i b u t i o n s over p r o f i l e s . Define the p r o f i l e - l e v e l d e l t a c o e f f i c i e n t 6 (T) = max|| Tp-Tp' || . I f the i n d i v i d u a l -p r o f , p.p' l e v e l processes are independent and i d e n t i c a l l y d i s t r i b u t e d N then Tp = _ I TV , where V i s the " l o c a t ion v e c t o r " of the N _ n n=l n-th i n d i v i d u a l . As before, on i n d i v i d u a l l e v e l 6. (T) = maxiiTV. - TV, ||. , and denote the p a i r of o r i g i n i n d . . ' I k 1 i i k - 80 -classes for which this maximum is attained by ( i ,k ). Since we can choose p to be a profile corresponding to V n = i n=l,...,N .[In this case• Tp = TV *] and p' to be the * one corresponding to V = k n=l,...,N, i t follows that 6 (T) > <5 . . (T). Now, by the triangle inequality pop — xnd N N II T T(V -V')ll < IITV -TV II . Hence u, n n — L . n n n=l n=l . N 6 (T) = max \\± I T(V -V ) || Pop r N _ n n P»P n=l 1 N < — max I IITV - TV I . — N n n : ( V " - ' V n = 1 : ( v - , . . . , v ^ ) Taking the maxima sequentially (individual by individual i f i r s t over (V-^ ,V ) etc.), we get N rr max N J IITV - Tv' l l = ^ N|| T V . * - TV *ll = 6. J ( T ) . L n n N I k md ( V 1 S . . . , V N ) n-1 ( v * , . . . , v ; ) Hence 6 (T) = 6. ,(T) . pop md Conclusion: In the i . i . d . case the delta coefficient assigns equal values to the profile-level operator as to the i n d i -vidual-level one. B. Consider distributions over origins and the L distance of their ° 00 images under T to the s teadyKstate distribution. Consider then the norm * of the map (T( ) - P*) (Royden 1963, p. 160), namely sup I  T ( p ) ~ p T||°° . P I P-P THoo - 81 -This f u n c t i o n i s nothing but | | , where i s the second-largest ( i n norm) eigenvalue of T. I f i s the eigenvector of T c o r r e s -ponding .to \~ then the supremum .defined above i s a t t a i n e d f o r p = P T + a b 2 , where a =. max {a:p + a b 2 £ S>, i . e . p represent s the "worst" d i r e c t i o n ( i n terms of r a t e of convergence to steady s t a t e ) . IX | has been suggested as a measure of (what we r e f e r to as) s o c i a l i n h e r i t a n c e by T h e i l (1972, Ch. 5) and Shorrocks(1978), who a l s o discussed some of i t s p r o p e r t i e s . A strong case i n favor of (*) t h i s measure was made r e c e n t l y by Sommers and C o n l i s k (1979). Among other t h i n g s , they show i t s i n t i m a t e r e l a t i o n to p a r e n t - c h i l d status c o r r e l a t i o n measure, and to the process' r a t e of r e g r e s s i o n to the mean. I f , instead of c o n s i d e r i n g the worst d i r e c t i o n , we K would average by summing over a l l eigenvector d i r e c t i o n s b^,...,b (**) we would get the sum of the eigenvalues, which ( i n the usual s e t t i n g ) i s nothing but the t r a c e of P. I t has been discussed as ( b a s i s f o r ) a measure of (what we r e f e r to as) s o c i a l i n h e r i t a n c e by Shorrocks. (*) Sommers and C o n l i s k a l s o suggested to use the second l a r g e s t eigenvalue of the m a t r i x P g = \(P+IT -1?^]!) , where IP = d i a g ( f f ) . (**) I f , instead of summing, we would have m u l t i p l i e d the eigenvalues by each other, we would have obtained the determinant of the t r a n s i t i o n matrix. I t was discussed as a measure by Bartholomew (1973) and Shorrocks (1978). This f u n c t i o n i s , however, h e a v i l y i n f l u e n c e d by the aspect of s o c i e t y w i t h l e a s t i n h e r i t a n c e , and as such i s not very i n t e r e s t i n g . Perhaps a more rewarding f u n c t i o n of t h i s nature w i l l be 1 - II (1-A.). i=2 X (***) When K=2 i t can be e a s i l y shown that|a(P)[ = | l - t r a c e P|= |X |. Thus i n t h i s s p e c i a l case the above mentioned measures c o i n -c i d e . - 82 -Consider now the pseudometric obtained by taki n g the d i f f e r e n c e i n e ntropies of two d i s t r i b u t i o n s . In p a r t i c u l a r , consider |H , . - H.*| weighted by the steady-state p r o b a b i l i t i e s u r p T X p i = l , . . . , K . E x p l i c i t l y , we are r e f e r r i n g to the measure 1=1 1 j = l J 1 J j = l 3 J The above f u n c t i o n can be shown to be the logarithm of the ftp • * expression H p . i j P i . This l a s t expression, i s , i n t u r n , nothing but the l i k e l i h o o d - r a t i o f o r t e s t i n g "H : f o r every j ; ft (ft) P.. = p. f o r every i " (Hoel 1954, Anderson and Goodman 1957). Since t h i s H q corresponds to a constant map, any s t a t i s t i c used f o r t e s t i n g i t can be a b a s i s f o r a measure of non-constancy. The expression £ TT. P.. log P.. i s u s u a l l y r e f e r r e d to i j 1 1 J 1 3 (e.g. Khinchin 1957) as the entropy of a Markov chain w i t h p r o b a b i l i t y t r a n s i t i o n matrix P and steady-state p r o b a b i l i t i e s ir. K TT.log TT. i s then the entropy of the corresponding constant j = l 2 2 map. In order to prove "strong p e r f e c t m o b i l i t y " , we have to show that t h i s pseudometric indeed a t t a i n s i t s minimum (zero) when and only when P i s the constant map. We s h a l l now prove that t h i s i s indeed the case. For general r e l a t i o n s between information measures (e.g. entropy) and l i k e l i h o o d r a t i o s , see Kul.lback (1959). - 83 -Lemma 3.1 K K The f u n c t i o n H = -T ) TT. P . . l o g P . . a t t a i n s i t s maximum i-1 j = l 1 1 J 1 J over P, subject to the c o n s t r a i n t s P . . > 0 f o r every i and i : i j -K 7 P . . - 1 = 0 f o r every i : and j = l ^ K J T T . P . . - T T . = 0 f o r every j , at P . . = TT. f o r every i and j . This maximum i s unique i f TT. > 0 f o r J 3 every j . (*) P r o o f : v ' K K The f u n c t i o n H = -/ / Tr. P.. l o g P.. i s concave and i f TT. > 0 x=l 3=1 J J f o r every i (which i s what we assume) i t i s s t r i c t l y concave ( j = i < 0 ) . Since we maximize i t subject to l i n e a r c o n s t r a i n t s , 3P . . P . . the Kuhn-Tucker c o n d i t i o n s ( Z a n g w i l l 1 9 6 9 ) are necessary and s u f f i c i e n t f o r maximum. Let the sets of corresponding m u l t i p l i e r s be { a _ } , {8.} and {Y.}. The Kuhn-Tucker c o n d i t i o n s f o r t h i s case are i J T T.(log P..+l)=a..+B. + Y. T . f o r every i and j , and i i j i j i 3 i a.. P.. = 0 f o r every i and i . For the p a r t i c u l a r c h o i c e , P.. = TT. f o r every i and i , they reduce to TT. l o g TT. + TT. = g. + TT. Y . f o r x 6 3 x x x 3 every i and j . Since t h i s system has a s o l u t i o n (8^ = n \ f o r every i , (.*) This proof may be viewed as a n a t u r a l g e n e r a l i z a t i o n of the one commonly used (e.g. T h e i l 1 9 7 2 ) i n the case of ( s t a t i c ) d i s t r i b u -t i o n s . I am indebted to E. Choo and J . K a l l b e r g f o r the idea. - 84 -= log f • f o r every j ) , our choice corresponds to a (unique) maximum. • The idea of using entropy as a "measure of d i s t a n c e " i n the s o c i a l m o b i l i t y context i s not new; i t has been used by T h e i l (1972, Ch. 5). However, the n o t i o n of d i s t a n c e that he used, /, 1- log -± > I s n o t 1 1 P i a pseudo-metric, s i n c e i t does not s a t i s f y the t r i a n g l e i n e q u a l i t y . Our n o t i o n of the "entropy excess" of a Markov map over i t s correspond-ing constant map thus seems more n a t u r a l . Nevertheless, T h e i l made a p a r t i c u l a r use of h i s n o t i o n of d i s t a n c e to o b t a i n a measure of m o b i l i t y which we wish to g e n e r a l i z e i n the l a s t subsection. 3.2.5 From Non-Constancy to S o c i a l I n h e r i t a n c e - Introducing P e r i o d -Consistency  As we already mentioned i n the previous s e c t i o n , a n a t u r a l way of co n s t r u c t i n g measures of s o c i a l i n h e r i t a n c e i s obtained by observing the r a t e at which the measures of non-constancy go to zero, i . e . the r a t e of convergence to zero of the sequence { f ( T n ) } . Now, there are two common f a m i l i e s of i n d i c a t o r s of assymptotic r a t e of convergence (see Ortega and Rheinboldt 1970). For any measure of non-constancy f ( T ) , those w i l l be the "quotient convergence f a c t o r s " f ( T n ) Q (f,T)= l i m sup — : — - me[l,°°), n + oo t f ( T n _ 1 ) ] m and the "root-convergence f a c t o r s " - 85 -. r t / r F n l / n (*) i f m = 1 fl i m sup Lf(T ) J n -> oo ] Rm( f ' T ) m I i - n l i m sup [ f ( T n ) ] / m i f m > 1 n °° What i s encouraging i s that both f a m i l i e s of measures are p e r i o d - c o n s i s -tent f o r every non-constancy measure f : Lemma 3.2 Both Q (f,T) and R (f,T) are period c o n s i s t e n t f o r every non-Hi m constancy measure f . Proof: We can assume without l o s s of g e n e r a l i t y , that m = 1. Now, consider maps T and T' such that (^(f,T) _> Q (f,T*) and R^(f ,T) _> R^f,T') Q l ( f ,T k) = l i m sup * * Vl, = 1 1 » sup i ^ L -n -* °° f [(T ) ] n + °° f (T ) r f ( T k n ) .. f ( T k n - 1 ) f ( T k n - k + l ) n V " P f C T ^ " 1 ) fc/""2) f ( T k n - k ) = 11 l i m sup A l s o , 1=0 n -> * f ( T k n 1 X) k-1 k n - i > . v 1 ^ Z ^ - \ = Q i ( f ' } i=0 n -»• 0 0 f ( T ' ) R 1 ( f , T k ) = l i m sup { f [ ( T k ) n ] } 1 / n  n -> oo l i m sup { [ f ( T k n ) ] 1 / k n } k n -> oo (*) This p a r t i c u l a r convergence f a c t o r was used by Sommers and C o n l i s k (1979) to con s t r u c t a s t a t u s - c o r r e l a t i o n measure. - 86 -{li m sup I f ( T k n ) ] 1 / k n } k n -*• 0 0 > { l i m sup [ f ( T ' k n ) ] 1 / k n } k = R ( f , T , k ) n -»- °° One co n c l u s i o n from t h i s Lemma i s that any f u n c t i o n M(T) which can be w r i t t e n i n the form Q^f.T) or R^Cfjt) f o r some f (although t h i s may not be the n a t u r a l way of d e f i n i n g or c a l c u l a t i n g i t ) i s p e r i o d -c o n s i s t e n t . In p a r t i c u l a r , T h e i l (1972, Ch. 5) showed that the second l a r g e s t eigenvalue can be obtained as Q^(f,T). Hence |X | i s period c o n s i s t e n t (Shorrocks 1978). Let M be a measure of s o c i a l i n h e r i t a n c e and l e t T and T' be maps such that M(T) > M(T') > 0, M(T' k) > 0 and M(T k) = 0 f o r some k k k ( i . e . T i s a constant map while T' i s n o t ) . Such measure M i s not period c o n s i s t e n t , and both d e l t a - c o e f f i c i e n t and entropy s u f f e r from t h i s d e f i c i e n c y . The entropy-based measure, however, e x h i b i t s a property, r e l a t e d to pe r i o d - c o n s i s t e n c y , which i s not without some appeal, and w i l l be described below. (*) T h e i l defined I to be /, TT log Y~(T) ' a P P r o x l m a t e d x t bY t n e 1 t , (Y ( i ) -TT . ) 2 quadratic approximation — £ — , and showed that i i 1 / 2 I I l i m t I 1 _ _ i = A . Shorrocks a l s o obtained | X \ by a l i m i t i n g pro-t-x» t i l I cedure of t h i s nature, using the concept of "assymptotic h a l f l i f e " . (**) This p a r t i c u l a r problem, r e l a t e d to f i n i t e - c o n v e r g e n t Markov chains, can be el i m i n a t e d by some appropriate domain r e s t r i c t i o n , but i t i s do u b t f u l whether complete period consistency w i l l be achieved. - 87 -Let P be the ( j o i n t ) p r o b a b i l i t y that a process which l k l k 2 ' - - r s t a r t s i n c l a s s i f i r s t goes to k^, then to k^ e t c . , defined over r (r) generations. Denote the entropy of t h i s d i s t r i b u t i o n by , and l e t H = J TT.H. , where H = H i s the entropy of the Markov chain as 1=1 1 1 defined p r e v i o u s l y . Then i t can be shown (Khinchin 1957) that H ( r + S ) = H ( r ) + H ( s ) f o r every r and s. - 88 -3.3 Conclusion In t h i s chapter we investigated the problem of measuring s o c i a l inheritance. Although due to the m u l t i p l i c i t y of issues involved no clear-cut procedures emerge, we believe that the systematic scheme of generating measures presented can a s s i s t s o c i a l s c i e n t i s t s i n t h i s task. Generally, once the s o c i a l inheritance aspect of s o c i a l m o b i l i t y has been singled out as the object to be measured, i t i s of c e n t r a l impor-tance to make one's " s o c i a l p r i o r i t i e s " within t h i s object e x p l i c i t . Operationalizing these concepts i s then a major step towards constructing a measure of non-constancy, though a c e r t a i n amount of f l e x i b i l i t y s t i l l remains i n the actual choice, which w i l l be made by trading-off mathe-matical properties and convenience. Measures of s o c i a l inheritance can then be constructed by observing the rate at which the measures of non-constancy, for incr e a s i n g l y long periods, converge to zero. 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