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A life cycle model of labor supply Katsaitis, Odysseus 1983

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A LIFE CYCLE MODEL OF LABOR SUPPLY by ODYSSEUS KATSAITIS B.A., U n i v e r s i t y of Athens, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of ECONOMICS We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1983 (c) Odysseus K a t s a i t i s , 1983 In p r e s e n t i n g 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 o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a 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 r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s •for f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of ^ C Q ^ A ' O ^ N c C  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 i i A LIFE CYCLE MODEL OF LABOR SUPPLY Research Supervisor: P r o f e s s o r T. Wales ABSTRACT This t h e s i s focusses on three areas i n the theory of i n t e r -temporal u t i l i t y maximization. F i r s t , I i n t e g r a t e the theory of labor supply and human c a p i t a l accumulation. I formulate a model of i n t e r t e m p o r a l u t i l i t y maximization i n which time i s a l l o c a t e d between l e i s u r e , schooling and work. I t i s assumed that the wage ra t e i s a f u n c t i o n of years of school i n g and experience which, i n t u r n , i s a f u n c t i o n of the t o t a l number of hours that the i n d i v i d u a l has worked so f a r . Second, I develop a new technique which allows us to estimate f u n c t i o n a l r e l a t i o n s h i p s d erived from optimal c o n t r o l problems f o r which no a n a l y t i c s o l u t i o n e x i s t s . T h i r d , I estimate the proposed model f o r two d i f f e r e n t data s e t s . F l e x i b l e f u n c t i o n a l forms are employed f o r e s t i m a t i o n purposes and every e f f o r t i s made so that the e m p i r i c a l model approximates as c l o s e l y as p o s s i b l e the t h e o r e t i c a l one. i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v ACKNOWLEDGEMENT v i CHAPTER.I INTRODUCTION 1 Footnotes to Chapter I 3 CHAPTER I I ON MODELLING HUMAN CAPITAL A. I n t r o d u c t i o n 4 B. The Investment Model 7 C. The Experience-Schooling Model 8 D. The D i r e c t Approach 10 E. Concluding Remarks 11 Footnotes to Chapter I I 14 CHAPTER I I I ON MODELLING THE SUPPLY OF LABOR: THE STATIC MODEL A. Introductory Remarks 15 B. The Conventional N e o c l a s s i c a l Model of 15 Labor Supply C. Becker's Model 17 D. Estimating the Supply of Labor I 19 Analytic/Econometric Problems E. E s t i m a t i n g the Supply of Labor I I 26 E c o n o m e t r i c / S t a t i s t i c a l Problems Footnotes to Chapter I I I 29 CHAPTER IV ON MODELLING THE SUPPLY OF LABOR: THE DYNAMIC MODEL A. Introductory Remarks 30 B. The Basic Model 30 C. Accumulation of Human C a p i t a l and the 33 Supply of Labor i v D. Occupational Choice and Human C a p i t a l 34 Accumulation E. An E m p i r i c a l l y Oriented Model 36 F. Concluding Remarks 38 Footnotes to Chapter IV 46 CHAPTER V THE PROPOSED MODEL A. D e s c r i p t i o n / P r o p e r t i e s of the Model 47 B. A Tr a c t a b l e Model 63 Footnotes to Chapter V 73 CHAPTER VI ESTIMATES A. Data 74 B. Results 80 Footnotes to Chapter VI 97 CHAPTER V I I CONCLUSIONS A. Introductory Remarks 99 B. Re s u l t s 99 C. P o s s i b l e Extensions of the Model/Directions 102 f o r Related Research Footnotes to Chapter V I I 108 BIBLIOGRAPHY 109 LIST OF TABLES VI.1 Number of Observations per Cohort. 1967 Sample 82 VI.2 Number of Observations per Cohort. 1975 Sample 84 VI.3 U t i l i t y Function Parameter Estimates 87 VI.4 Test of Curvature P r o p e r t i e s of U t i l i t y Functions 88 VI.5 Estimates of the D i f f e r e n c e 90 (Rate of I n t e r e s t Minus Rate of Time Preference) VI.6 E l a s t i c i t i e s of Labor Supply w i t h Respect to the 91 Wage Rate VI.7 Estimates of the De p r e c i a t i o n Rate of Experience 93 VI.8 Wage Function Parameter Estimates 95 v i ACKNOWLEDGEMENT I wish to express my a p p r e c i a t i o n to the members of my d i s s e r t a t i o n committee whom without t h e i r guidance and encouragement t h i s d i s s e r t a t i o n would not have been completed. I e s p e c i a l l y am indebted to Pr o f e s s o r s T. Wales, J . Kesselman and K. Nagatani whose thorough readings of many d r a f t s of my d i s s e r t a t i o n and t h e i r c r i t i c i s m s and suggestions c o n t r i b u t e d s i g n i f i c a n t l y to the improvement of t h i s study. I am a l s o g r a t e f u l to Pr o f e s s o r s R. Evans, H. Neary, C. R i d e l l and T. Lewis who read the manuscript and made val u a b l e comments. I would a l s o l i k e to express my g r a t i t u d e to Pr o f e s s o r s C. A r c h i b a l d , E. Diewert and C. Blackorby f o r f i r s t i n t r o d u c i n g me to the r i g o r o u s study of economics, and to Profe s s o r s J . Cragg and A. Woodland from whose l e c t u r e s I f i r s t learned about econometric modelling. CHAPTER I INTRODUCTION This research project presents a model of consumption, labor supply and human c a p i t a l accumulation based on the postulates of inter-temporal u t i l i t y maximizing behavior. It generalizes most of the models in human c a p i t a l accumulation which have been derived i n the t r a d i t i o n of Arrow's [1961J, Becker's [1967J, and Ben-Porath's [1967J models. Furthermore, i t contains as a spe c i a l case the standard l i f e cycle models of consumption.* The model i s tested using U.S. data. Over the past decade there have been quite a few attempts to estimate labor supply and/or human c a p i t a l accumulation functions. Most of those studies have been hampered by the following shortcomings: i ) The estimated functions have not been derived e x p l i c i t l y from a u t i l i t y maximization problem, i i ) Labor supply and human c a p i t a l accumulation functions have been estimated independently of each other, i i i ) The opportunity cost of time and the price of human c a p i t a l have been mis-specified by ignoring the shadow value of experience, i v ) Some of the individuals who have been included i n the sample might not have been i n equilibrium because of r e s t r i c t i o n s on the number of hours of work and/or 2. because they might have had a second job. In this research project, every e f f o r t has been made to r e c t i f y those shortcomings and take into account the p e c u l i a r i t i e s of the labor market. i ) The empirical model i s an isomorphic mapping of the th e o r e t i c a l one. i i ) F l e x i b l e functional forms 2 have been used for the u t i l i t y function and the wage function. Thus, the structure of the empirical model i s almost as r i c h as the structure of the th e o r e t i c a l one. i i i ) The employed data set s a t i s f i e s most of the r e s t r i c -tions imposed by the t h e o r e t i c a l model. This study i s i n seven chapters. Chapters II, III and IV consider c r i t i c a l l y some of the issues related to the modelling and estimation of labor supply and human c a p i t a l accumulation functions. Chapter V presents a l i f e cycle model of human ca p i t a l accumulation, consumption and labor supply. The properties of the t r a j e c t o r i e s of the control and state variables are explored as well as the impact on them of certain exogenous variables. Furthermore, a new technique i s proposed which allows estimation of functions derived from optimal control problems for which an analytic solution does not exist. In the la s t two chapters, the estimates of the model are presented, as well as the conclusions and directions for related research. Footnotes to Chapter I 1. See, e.g, Modigliani and Brumberg [1954J. 2. F l e x i b l e functional forms are capable of providing l o c a l second-order approximation to any function. Chapter I I 4. QN MODELLING HUMAN CAPITAL A. I n t r o d u c t i o n The major problem f a c e d by any s t u d e n t of human c a p i t a l i s the d e f i n i t i o n and measurement of human c a p i t a l . From an econom i s t ' s p o i n t of view human c a p i t a l can be d e f i n e d as any a b i l i t y , c a p a c i t y or s k i l l p o s s e s s e d by an i n d i v i d u a l w h i ch commands a n o n n e g a t i v e p r i c e . I w i l l d e f i n e by the term "human c a p i t a l " a l l m a r k e t a b l e s k i l l s , a b i l i t i e s or c a p a c i t i e s of an i n d i v i d u a l . Assuming t h a t a composite i n d e x of human c a p i t a l can be c o n s t r u c t e d , human c a p i t a l might be m o d e l l e d as f o l l o w s : 1 ( I I . 1 ) Hj = g ( S j , L j , OJTj ," MAj , P j , u-j) where f o r the j t h i n d i v i d u a l H j : human c a p i t a l , s j : y e a r s of s c h o o l i n g , L j : e x p e r i e n c e , O J T J : o n - t h e - j o b t r a i n i n g , MAJ: i n n a t e mental a b i l i t i e s , P j : p e r s o n a l i t y c h a r a c t e r i s t i c s , Uj : a l l o t h e r v a r i a b l e s which might a f f e c t the a c c u m u l a t i o n of human c a p i t a l . A p p e a l i n g to the m a r g i n a l p r o d u c t i v i t y t h e o r y and assuming t h a t markets are both s t a t i c and p e r f e c t l y c o m p e t i t i v e , we d e r i v e the f o l l o w i n g r e l a t i o n s h i p : 5. (II.2) W j = MPj = f(H j ; CI) where W j : the wage rate, M P J : the marginal product for an "hour" of input, CI: a composite index of a l l other inputs used i n the production process. A few caveats are i n order here: i ) Except for some primitive technologies the wage rate depends on the r e l a t i v e prices of a l l other inputs and the technology. Equation (II.2) i s , b a s i c a l l y , the inverse demand function for labor. Hence, most attempts to estimate such a function w i l l result i n biased estimates because of the s p e c i f i c a t i o n error. i i ) By d e f i n i t i o n a s i g n i f i c a n t number of the variables that determine human c a p i t a l are controllable by the i n d i v i d u a l . Furthermore, investment i n some of those factors depends on the real wage, ceteris paribus. For example, investment i n formal education and/or on-the-job t r a i n i n g depend on the expected wage rate in general. Ignoring those relationships w i l l r e s u l t i n biased estimates of the parameters of the wage equation because of the simultaneity bias. i i i ) Equation (II.2) cannot capture certain non-pecuniary benefits associated with s p e c i f i c occupations, e.g., prestige. Furthermore, equation (II.2) cannot capture possible payments i n kind, e.g., paid and/or extended vacations, pension plans, etc. If i t is 6. assumed that firms offer fringe benefits, observed wages w i l l under-estimate the marginal product of the worker, ceteris paribus. From an empirical point of view the most severe problem w i l l be simultaneity bias, mentioned i n ( i i ) . The reason i s that, i f the markets are "reasonably" perfectly competitive, they w i l l be characterized by perfect information and free mobility of resources. Then the rate of return on human c a p i t a l , i . e . , the wage rate, should be the same for a l l i n d u s t r i e s , thus the s p e c i f i c a t i o n bias w i l l be minimal. It should be emphasized that the aforementioned result is not v a l i d for intertemporal comparisons. Abundant evidence exists that the structure of the economy i s changing over time which results in changes i n re l a t i v e prices. If this i s the case, i . e . , i f r e l a t i v e prices are changing over time, the s p e c i f i c a t i o n bias w i l l be minimized by using cross section rather than time series data. F i n a l l y , the impact of the errors i n variables and/or s p e c i f i c a t i o n error discussed in ( i i i ) can be reduced i f : i ) The observed wage rate i s adjusted to include payments i n kind, and i i ) Equation (II.2) i s embedded i n a u t i l i t y maximization framework which can accommodate the non-pecuniary rewards of human c a p i t a l and employment. Such a model i s presented i n Chapter V. In the rest of this chapter I w i l l present a select i v e survey/ taxonomy of previous attempts to measure human c a p i t a l . 7. T h i s e x e r c i s e w i l l a l l o w us to put i n t o the pr o p e r c o n t e x t the model t h a t I w i l l p r e s e n t and d i s c u s s l a t e r on. A l l s t u d i e s t h a t w i l l be r e v i e w e d are based on the p r e v i o u s l y d e s c r i b e d model, t h a t i s , e q u a t i o n s ( I I . 1 ) and ( I I . 2 ) . However, they d i f f e r s i g n i f i c a n t l y i n the i n f o r m a t i o n t h a t i s r e q u i r e d to i d e n t i f y the d e t e r m i n a n t s of human c a p i t a l . Moreover, the f o c u s as w e l l as the p o s s i b l e p o l i c y i m p l i c a t i o n s and a p p l i c a t i o n s of those s t u d i e s v a r y c o n s i d e r a b l y . B. The Investment Model Becker [1964J s u g g e s t e d a s u r p r i s i n g l y s i m p l e t e c h n i q u e w h i c h a l l o w s us to e s t i m a t e s i m u l t a n e o u s l y the i n t e r n a l r a t e of r e t u r n on human c a p i t a l , the c o s t s of i n v e s t i n g i n human c a p i t a l i n each p e r i o d and the t o t a l c o s t of i n v e s t i n g i n human c a p i t a l . The o n l y i n f o r m a t i o n r e q u i r e d i s net e a r n i n g s . The s u g g e s t e d methodology might be d e s c r i b e d as f o l l o w s . C o n s i d e r two a c t i v i t i e s ; the f i r s t one, say E, has no i n v e s t m e n t and p r o v i d e s the net e a r n i n g s s t r e a m Eg, Ei ,... E n . The second a c t i v i t y , say E, r e q u i r e s i n v e s t m e n t f o r at l e a s t one p e r i o d and p r o v i d e s the net e a r n i n g s stream Eg, E^....E n. The amount i n v e s t e d , c o s t , d u r i n g the j t n p e r i o d i s d e f i n e d as: ( I I . 3) Cj = Ej - E j + r k Z = Q C k where r i s the r a t e of r e t u r n , CQ = E Q - Eg, C^ = E j -E i + TCQ e t c . 8. Total undiscounted costs can be estimated as: CO (11.4) c = ac-j j = o J But the present value of net earnings should be the same for both a c t i v i t i e s , say, (11.5) I Ej = E Ej j= 0 j= 0 ( l + r ) J + 1 ( l + r ) J + 1 The i n t e r n a l rate of return can be determined u t i l i z i n g equation (II.5). Investment costs and the investment period can be calculated recursively using equation ( I I . 3 ) , F i n a l l y , t o t a l costs can be estimated using equation ( I I . 4 ) , The model can be, somehow, generalized by modelling e x p l i c i t l y the maximization process. Ben-Porath [1967J, Becker [1967J, Rosen [1976J have s p e c i f i e d and/or estimated models where i t i s assumed that the i n d i v i d u a l maximizes the present value of earnings subject to an earnings function which i s assumed to be a function of human c a p i t a l . It turns out that the underlying methodology as well as the informational requirements are i d e n t i c a l to the technique suggested by Becker [1964J, so th i s "sophisticated" version of the investment model w i l l not be discussed here. C. The Experience-Schooling Model Even a cursory study of age-earnings p r o f i l e s indicates that earnings increase with years of schooling and age/experience. The natural vehicle for studying earnings as a function of 9. schooling and age/experience i s Becker's [1964J model. Suppose that Ej = 0 for every j . Define gross earnings as: (11.6) GEj = Ej + Cj Given the d e f i n i t i o n of gross earnings and the assumption that Ej= 0 2, (II.3) can be written as: j - l (11.7) GEj_i + r C j - i = rZ = GEj. k = l Solving (II.7) recursively and noting that during the years when the i n d i v i d u a l i s a f u l l - t i m e student investment costs are equal, by d e f i n i t i o n , to gross earnings, we can rewrite equation (11.7) as: (II. 8) fciGEj = fciGEg + r s s + r p I C_ t=0 GE t where r s : rate of return to school investment r p : rate of return to post-school investment, (11.8) i s not empirically tractable since data about the cost of the investment are not readily available. Assuming that investment in human c a p i t a l follows a s p e c i f i c path over the l i f e t i m e of the i n d i v i d u a l C_ can be i m p l i c i t l y defined GE t as a function of time, e.g., (11.9) k t = k 0 - k£t or k t = koe'Pt T where T i s t o t a l period of net investment and k t = C GE t Transforming (II.8) into a continuous function and substituting (II.9) we derive two di f f e r e n t s p e c i f i c a t i o n s for the gross e a r n i n g s f u n c t i o n . Q u i t e a few a u t h o r s have e s t i m a t e d d i f f e r e n t " a p p r o x i m a t i o n s " to the gr o s s e a r n i n g s f u n c t i o n or they have t r i e d to extend i t e i t h e r by i n t r o d u c i n g more e x p l a n a t o r y v a r i a b l e s or by c o n s i d e r i n g o t h e r f u n c t i o n a l f o r m s . 3 U n f o r t u n a t e l y most of those a n a l y s e s are r a t h e r ad hoc. Whatever the l i m i t a t i o n s of the e s t i m a t e d models we can s t i l l d e r i v e some l e s s o n s from them. i ) The hypotheses t h a t the r a t e s of r e t u r n t o s c h o o l i n g and e x p e r i e n c e are not c o n s t a n t cannot be r e j e c t e d by the d a t a , i i ) R e t u r n s to e d u c a t i o n v a r y w i t h the type of e d u c a t i o n . i i i ) B e s i d e s e d u c a t i o n and e x p e r i e n c e o t h e r f a c t o r s may i n f l u e n c e the l e v e l of e a r n i n g s . D. The D i r e c t Approach A few a u t h o r s have attempted to measure d i r e c t l y t h e m a r g i n a l p r o d u c t i v i t y of wo r k e r s of d i f f e r e n t e d u c a t i o n a l backgrounds, t h u s , they have att e m p t e d to measure d i r e c t l y the m a r g i n a l p r o d u c t i v i t y of human c a p i t a l . The two s t u d i e s t h a t are of some i n t e r e s t f o r our p u r p o s e s , are the ones by Berg [1969J and L a y a r d et a l [ 1971J. Berg examined the performance of bl u e and w h i t e c o l l a r workers w i t h d i f f e r e n t e d u c a t i o n a l backgrounds. The i n t e r p r e t a t i o n of h i s r e s u l t s r e q u i r e s extreme c a u t i o n s i n c e h i s a n a l y s i s does not go beyond a s i m p l e p r e s e n t a t i o n of the d a t a and some d e s c r i p t i v e s t a t i s t i c s . The study by L a y a r d et a l i s a much more a m b i t i o u s one. However, the methodology that they employed, as well as the v a l i d i t y of their conclusions, are quite questionable, as pointed out by Blaug [1972J. E. Concluding Remarks The picture that emerges from the review of the l i t e r a t u r e on human c a p i t a l indicates that the theory, almost twenty years old, has not yet matured. Contrary to Blaug [ 1976J, who asked: "what refutations have been encountered i n the 'protective b e l t ' of the program" I believe that the question should be whether the protective belt of the research program allows us to assess i t s degree of corroboration. The research program on human ca p i t a l i s so poorly or ingeniously designed - depending upon one's point of view - that one can hardly refute i t s predictions, i ) The model proposed by Becker [1964J i s a perfect example of an untestable model for at least two reasons. F i r s t , there i s a serious d e f i n i t i o n a l / measurement problem. According to Becker [1964; 38J " ' r e a l ' earnings are the sum of monetary earnings and monetary equivalent of psychic earnings." It i s is obvious that there i s no possible way of measuring re a l earnings i n the Beckerian framework, unless we embed the problem i n a u t i l i t y maximization framework and include i n the u t i l i t y function such variables as human c a p i t a l . Second, the fundamental assumption of "earnings 12. maximization" i s a maintained but not testable hypothesis. i i ) The "second generation" of the investment i n human c a p i t a l models - what I ca l l e d "The Sophisticated Investment Model" - present, mutatis mutandis, the same d i f f i c u l t i e s and i n f l e x i b i l i t i e s . The testable prediction of the model that investment i n human ca p i t a l i s declining throughout the l i f e - c y c l e i s f u l l y compatible with other maintained hypotheses, e.g., u t i l i t y maximization, i i i ) The apparent success of the extended earnings function model i l l u s t r a t e s , quite f o r c e f u l l y , the naivete of the schooling and/or schooling and experience models, There i s no doubt today that genetic pre-disposition and the family environment do affect earnings. However, this should not be interpreted as an unqualified acceptance of the extended earnings function. To the best of my knowledge none of the estimated earnings functions i s based on a well defined theory of learning and/or education. Thus, the researchers cannot claim that by estimating those earnings functions they are assessing the degree of corroboration of a well defined theory. A l l they can claim i s that the conjecture that certain variables affect the earnings of an i n d i v i d u a l cannot be refuted by the data. i v ) Suppose t h a t the e s t i m a t e d e a r n i n g s f u n c t i o n s were models of a t h e o r y r a t h e r than ad hoc c o n s t r u c t i o n s . The a s s u m p t i o n t h a t a number of s o c i o - e c o n o m i c v a r i a b l e s d etermine the e a r n i n g s of an i n d i v i d u a l s u g g e s t s t h a t the pr o p e r way to model such a s i t u a t i o n i s v i a a s i m u l t a n e o u s e q u a t i o n model ( s e e G r i l i c h e s [ 1 9 7 6 ] ) . Hence, the e a r n i n g s f u n c t i o n i s reduced form e q u a t i o n . But we know from e l e m e n t a r y e c o n o m e t r i c s t h a t e s t i m a t i n g o n l y one of the e q u a t i o n s of a s e t of s i m u l t a n e o u s e q u a t i o n s w i l l r e s u l t i n b i a s e d c o e f f i c i e n t s (see M o r g e n s t e r n [ 1 9 7 3 ] ) . The p r e v i o u s a n a l y s i s s u g g e s t s t h a t the economics of e d u c a t i o n i s i n bad shape. H o p e f u l l y , r e s e a r c h i n the f u t u r e w i l l attempt to i n v e s t i g a t e i n a d i r e c t way the f o l l o w i n g two t h e o r e t i c a l i s s u e s : i ) The c o n s t r u c t i o n of o p e r a t i o n a l l y u s e f u l t h e o r i e s which can d e s c r i b e and a n a l y z e the a c c u m u l a t i o n of knowledge - human c a p i t a l - as w e l l as the r e l a t i o n s h i p between knowledge and perfor m a n c e , i i ) The b e h a v i o r of the f a m i l y u n i t towards the a c c u m u l a t i o n of human c a p i t a l . F o o t n o t e s to Chapter I I See Taubman and Wales [1974]. See Becker and C h i s w i c k [ 1 9 6 6 ] , M i n c e r [1974J. A number of o t h e r v a r i a b l e s have been i n t r o d u c e d i n the e a r n i n g f u n c t i o n , e,g., c o q n i t i v e a b i l i t i e s / i n t e l l i g e n c e / p e r s o n a l i t y t r a i t s ( G r i l i c h e s and Mason [ 1 9 7 3 ] ) , G r i l i c h e s [1976J, Duncan et a l [1972J, Taubman and Wales [ 1 9 7 4 ] , House [1976J, Wise [1975]; q u a l i t y of s c h o o l s (Wales [1973], Taubman and Wales [1974]; o c c u p a t i o n a l s t a t u s ( S t o l z e n b e r g [1975b], F a r l e y [1977 J , Goodman [ 1 9 7 9 ] ) ; f a m i l y b a c k g r o u n d / r a c e / s o c i a l c l a s s ( H a r r i s o n [1972a], [ 1 9 7 2 b ] , Weiss [1970], Bowles [1973], Bowles and G i n t i s [1975J, F l a n a g a n [1974], Featherman and Hauser [1976], S t o l z e n b e r g [1975aJ, F a r l e y [1977], Duncan et a l [ 1 9 7 7 ] ) ; r e l i g i o u s p r e f e r e n c e ( G o c k e l [1969J, G r e e l e y [ 1 9 7 6 ] ) ; m i g r a t i o n ( C h i s w i c k [1977] [ 1 9 7 8 ] ) ; c o m p l e t i o n of l e v e l s of s c h o o l i n g ( L a y a r d and P s a c h a r o p o u l o s [1974], Taubman and Wales [ 1 9 7 4 ] , A l b r e c h t [ 1 974], Wise [1975 J , Goodman [1 9 7 9 J ) ; sex ( M i n c e r and P o l a c h e k [ 1 9 7 4 ] ) . Chapter III ON MODELLING THE SUPPLY OF LABOR: THE STATIC MODEL A. Introductory Remarks This chapter considers c r i t i c a l l y some of the empirical and th e o r e t i c a l studies concerned with the supply of labor. The subsequent discussion i s lim i t e d , i n the sense that I w i l l consider only quantitative aspects of labor supply. More s p e c i f i c a l l y , I w i l l focus on issues related to the construction and empirical implementation of operationally useful models of the supply of labor. U n t i l recently, the majority of economists working within the neoc l a s s i c a l paradigm have f a i l e d to recognize the d i s t i n c t i v e features of the demand for l e i s u r e - i . e . , the supply of labor - vis a*" vis the demand for other commodities, Consequently, most of the techniques that I w i l l present and discuss below are f a i r l y recent and deserve some attention. B. The Conventional Neoclassical Model of Labor Supply According to the neo c l a s s i c a l model i t i s assumed that the consumer-worker maximizes a nondecreasing, quasiconcave continuous u t i l i t y function subject to an income constraint. The model i s summarized below: (III.3) max u = U ( x ^ , X 2 , x^,£) w. r. t. x l 2. 0,. . . xj! _> 0 h > % > 0 s u b j e c t to N I + w(h- A) > Z P i X i i = l where u: the u t i l i t y f u n c t i o n , x^ : the q u a n t i t y of the i t n commodity ( r e n t a l ) , p^: the p r i c e ( r e n t a l p r i c e ) of the i t n commodity ( r e n t a l ) , I: hours of l e i s u r e , h: t o t a l number of hours a v a i l a b l e , I : n o n - l a b o r income. The c o m p a r a t i v e s t a t i c s of the model are r e a d i l y a v a i l a b l e and w i l l not be c o n s i d e r e d h e r e . I t s u f f i c e s to remind o u r s e l v e s t h a t a change of the wage r a t e may r e s u l t i n an i n c r e a s e or d ecrease of the i n d i v i d u a l ' s hours of work. ( I t seems t h a t Robbins [1930J was the f i r s t to d e r i v e t h a t r e s u l t . ) Hence, the o n l y r o u t e t h a t i s a v a i l a b l e i n o r d e r to e v a l u a t e the impact of a change of the wage r a t e on the s u p p l y of l a b o r i s by computing the e l a s t i c i t y of l a b o r s u p p l y . The i m p o r t a n c e of r e l i a b l e e s t i m a t e s of the p r e f e r e n c e o r d e r i n g of the consumer-worker f o r the p o l i c y maker cannot be o v e r s t a t e d . The major advantage of the c o n v e n t i o n a l n e o c l a s s i c a l model l i e s i n i t s s i m p l i c i t y . I t can be e a s i l y e s t i m a t e d and i t a l l o w s us to i n v e s t i g a t e a number of t h e o r e t i c a l and e m p i r i c a l i s s u e s , some of w h ich are i m p o r t a n t f o r purposes of economic p o l i c y . However, the major advantage of the model - i t s s i m p l i c i t y - may be c o n s i d e r e d i t s major d i s a d v a n t a g e . Because of i t s 17. s i m p l i c i t y one might be tempted to estimate the model without paying attention to some " t e c h n i c a l i t i e s " or"minor points". As w i l l be shown in sections (D) and (E) of this chapter, the price that we have to pay i n ignoring those "minor points" can be quite high. C. Becker's Model The f i n e s t p a r t i t i o n of the variable "time" that the con-ventional neoclassical model of labor supply can handle i s " l e i s u r e " and "hours of work". Such a p a r t i t i o n i s u n r e a l i s t i c and inadequate for analytic purposes. A c t i v i t i e s , such as, preparing a meal, cleaning the house, etc., cannot be c l a s s i f i e d as l e i s u r e . On the other hand one cannot "consume" a book, the only way to derive u t i l i t y from a book is by reading i t , or even by looking at i t , or by glancing at i t ! Contrary to the t r a d i t i o n a l n e o - c l a s s i c a l models which cannot handle those situ a t i o n s , Becker's [1965J model can. In his pathbreaking paper he suggested that the u t i l i t y function of the consumer, or consumer-worker, be defined over a c t i v i t i e s . The a c t i v i t i e s are produced by both commodities and time. Hence the demand for a c t i v i t i e s depends on the price (rental price) of commodities (rentals) and the opportunity cost of time. Becker's model can be described as follows: (III.2) max u — U ( Z i » Z 2, • • • Z[^) w.r.t. x 1 _> 0, x M > 0 T j > 0, T N > 0 Z j, Z 2 • • • ZJJ h > I > 0 subject to (III.3) Z ± = f± (X ±, T ±) M (III . 4) I + w(h- A) > l p i X i i = l N ( I I I . 5) Z T ± + (h- 4) - H i = l where: u: the u t i l i t y function, Z±: the i t h a c t i v i t y p-j^ : the price (rental price) of the i 1 - * 1 commodity ( r e n t a l ) , X j ^ : the i t n commodity ( r e n t a l ) , f x : l i n e a r homogeneous production function, I: non-labor income, (h-l): the supply of labor, w: the wage rate, H: t o t a l number of "hours" available per time period, T^: the input of the individual's own time. A l t h o u g h B e c k e r ' s work p r o v i d e s us w i t h an i m p o r t a n t g e n e r a l i z a t i o n of the n e o c l a s s i c a l model, i t s h o u l d not be c o n s i d e r e d as the u l t i m a t e model of l a b o r s u p p l y . As D i e w e r t [ 1971J shows, the model can be extended to i n c l u d e o c c u p a t i o n a l c h o i c e of the i n d i v i d u a l as w e l l as the l a b o r f o r c e p a r t i c i p a t i o n d e c i s i o n . A l t h o u g h the model can i n c l u d e a s i g n i f i c a n t number of l a b o u r s u p p l y i s s u e s i t s a t t r a c t i v e n e s s i s d i m i n i s h e d when one con- t e m p l a t e s e s t i m a t i n g i t . The i n f o r m a t i o n a l r e q u i r e m e n t s f o r e s t i m a t i o n purposes are e x c e s s i v e . Wales and Woodland [1977J have e s t i m a t e d a s i m p l e v e r s i o n of B e c k e r ' s model. An i m p o r t a n t c o n c l u s i o n of t h i s s t u d y was t h a t , f o r the d a t a s e t employed a v a r i a n t of B e c k e r ' s model performed b e t t e r than the t r a d i t i o n a l model. D. E s t i m a t i n g the Supply of Labor I  A n a l y t i c / E c o n o m e t r i c Problems In t h i s p a r a g r a p h I w i l l c o n s i d e r a number of a n a l y t i c and e c o n o m e t r i c problems which c o m p l i c a t e the s p e c i f i c a t i o n and e s t i m a t i o n of the l a b o r s u p p l y model. I t s h o u l d be emphasized t h a t most of the t o p i c s t h a t w i l l be p r e s e n t e d i n t h i s p a r a g r a p h are m u t u a l l y i n t e r d e p e n d e n t . Moreover, the taxonomy i s by no means an e x h a u s t i v e one. 1/. F u n c t i o n a l form. Except f o r a few a u t h o r s ( e . g . , Wales and Woodland [1976J, [1977J, Lau et a l [1978]) who have employed f l e x i b l e f u n c t i o n a l forms f o r e s t i m a t i o n p u r p o s e s , most a u t h o r s have used q u i t e r e s t r i c t i v e and/or ad hoc f u n c t i o n a l forms. A maintained thesis of this study i s that the use of f l e x i b l e functional forms i s imperative for labor supply models. The reason i s that f l e x i b l e functional forms may be interpreted as second order approximations to an arb i t r a r y u t i l i t y function at a point. 2/. Taxes. The introduction of taxes i n the consumer-worker maximization model - either the t r a d i t i o n a l n e oclassical model or Becker's - does not create any d i f f i c u l t i e s , per se. What does create problems i s the structure of the tax function. Given the progressivity of the income tax, the concavity of the budget constraint i s not vio l a t e d . However, for most tax systems the after tax budget w i l l be a segmented l i n e a r function. T r a d i t i o n a l econometric techniques cannot handle segmented l i n e a r functions, hence an approximation of the budget constraint i s necessary. Fortunately, as Diewert [1971J has shown, the tax function can be l i n e a r i z e d around each observation, therefore t r a d i t i o n a l econometric techniques can be employed. This technique suffers from the following shortcomings: i ) It introduces an approximation error i n the estimation process. i i ) It ignores the endogeneity of the net wage rate. An algorithm - due to Wales and Woodland [1979J - w i l l be presented below which eliminates the approximation error and takes into consideration the endogeneity of the net wage rate. 3/. Non-labor income. Non-labor income might be disaggregated into two components. i ) C a p i t a l / P r o p e r t y income which i n c l u d e s d i v i d e n d s , i n t e r e s t p a i d on bonds and s a v i n g s a c c o u n t s , r e n t s , income from l i f e i n s u r a n c e p o l i c i e s , e t c . i i ) T r a n s f e r s which i n c l u d e f a m i l y and c h i l d b e n e f i t s , n e g a t i v e income t a x schemes and wage s u b s i d i e s p a i d d i r e c t l y to the worker. A l t h o u g h from a p u r e l y t e c h n i c a l p o i n t of view ( i ) and ( i i ) can be e a s i l y i n t r o d u c e d i n t o the budget c o n s t r a i n t of the consumer, from a t h e o r e t i c a l p o i n t of view t h e i r v e r y e x i s t e n c e i l l u s t r a t e s the i n a b i l i t y of the o r d i n a r y s t a t i c model to handle such s i t u a t i o n s . C l e a r l y , the presence of c a p i t a l income v i o l a t e s the assumption of s t a t i c o p t i m i z a t i o n b e h a v i o r , w h i l e t r a n s f e r income r e s u l t s i n an endogenously d e t e r m i n e d budget c o n s t r a i n t . 4/. C o s t s of w o r k i n g . There are two k i n d s of expenses a s s o c i a t e d w i t h w o r k i n g : i ) Money c o s t s , w h i c h i n c l u d e expenses f o r t o o l s , u n i f o r m s , t r a v e l t o work, e t c . and i i ) Time c o s t s , w h ich i n c l u d e commuting t i m e , e t c . One can s a f e l y i g n o r e ( i ) s i n c e f o r most workers such expenses are s m a l l . Time c o s t s cannot be accommodated by the n e o c l a s s i c a l model. One has to use B e c k e r ' s model i n o r d e r t o s t u d y and/or e s t i m a t e such c o s t s . 2 5/. E n d o g e n e i t y of wages. We have a l r e a d y d i s c u s s e d a number of reasons f o r t r e a t i n g the wage r a t e as an endogenous v a r i a b l e , e.g., t a x e s , t r a n s f e r s , c o s t s of w o r k i n g . Even i f we had been l i v i n g i n a f r i c t i o n l e s s w o r l d w i t h o u t t a x e s and t r a n s f e r s , the wage rate would have been endogenous because of the influence of other variables. Consider the following factors which determine, up to a certain point, the wage rate, the supply of labor and the occupational choice of the i n d i v i d u a l , i ) Education, i i ) Experience, i i i ) The "agreeableness of the job", iv) P r o b a b i l i t y of unemployment, v) "Probability of success", v i ) "The trust to be reposed i n the worker", v i i ) "The cost of learning the trade", v i i i ) The relationship between productivity - hence, the wage rate - and the number of hours of work per day, and ix) Pecuniary and non-pecuniary rewards of the job. The impact of the f i r s t two variables on the supply of labor w i l l be discussed i n the next two chapters. The next f i v e , i . e . , ( i i i ) - ( v i i ) have been taken from Adam Smith's The Wealth of Nations [1961; 112-113J. P r a c t i c a l l y none of those can be included i n the t r a d i t i o n a l s t a t i c model of labor supply which ignores uncertainty and possible non-pecuniary rewards of the occupation chosen by the i n d i v i d u a l . Now l e t us be more s p e c i f i c about the non-pecuniary rewards of the job and the r e l a t i o n s h i p between the wage rate and hours of work. 5A. Non-pecuniary rewards. For pedagogical purposes we may distinguish two d i f f e r e n t kinds of non-pecuniary rewards of a job: i ) Positive (negative) c h a r a c t e r i s t i c s associated with a s p e c i f i c occupation which increase (decrease) the u t i l i t y of the consumer-worker. Some of those c h a r a c t e r i s t i c s might be available through certain commodities, but the consumer-worker would have to pay (be paid) for them. i i ) The consumption of certain goods provided at the place of work. As Lancaster [ 1971J has shown the n e o c l a s s i c a l model i s a s p e c i a l case of the c h a r a c t e r i s t i c s model. Hence, the f i r s t category of the non-pecuniary rewards of the job encompasses the second category. From an empirical point of view, the important question is how those attributes or c h a r a c t e r i s t i c s of an occupation can be introduced into a model of labor supply. Three possible answers to the previous question have been suggested i n the l i t e r a t u r e . i ) Tinbergen [1956J assumes that the u t i l i t y function of the consumer-workers is defined over income and the c h a r a c t e r i s t i c s / a t t r i b u t e s of the occupation, i i ) Lancaster [1971J assumes that the u t i l i t y function i s defined over c h a r a c t e r i s t i c s . The consumer maximizes his/her u t i l i t y subject to the trans-formation function which describes the r e l a t i o n s h i p between c h a r a c t e r i s t i c s and commodities - including time - and his/her budget constraint. i i i ) B e c k e r ' s "Theory of the A l l o c a t i o n of Time" p r o v i d e s a l e s s a m b i t i o u s b u t , p o s s i b l y , e m p i r i c a l l y t r a c t a b l e model, compared to the T i n b e r g e n or L a n c a s t e r models. D e f i n e the u t i l i t y f u n c t i o n over a c t i v i t i e s and hours of work at d i f f e r e n t o c c u p a t i o n s (see Die w e r t [ 1 9 7 1 J ) . The budget c o n s t r a i n t i s , of c o u r s e , d e f i n e d c o n f o r m a b l y w i t h the u t i l i t y f u n c t i o n . G i v e n the Wales and Woodland [1977J r e s u l t s , the s p e c i f i c a t i o n of the commodities - t i m e / a c t i v i t i e s t e c h n o l o g y s h o u l d not pose any e s t i m a t i o n problem. Some c o m p l i c a t i o n s may a r i s e due to the e x i s t e n c e of c o r n e r s o l u t i o n s and the en d o g e n e i t y of the wage r a t e 5B. Wages and hours of work. Wages might depend on the hours of work f o r the f o l l o w i n g two r e a s o n s : i ) The p r o d u c t i v i t y of the worker i s a concave f u n c t i o n of the hours of work. I n i t i a l l y the p r o d u c t i v i t y i s low because i t t a k e s some time f o r the worker t o "warm up", w h i l e by the end of the w o r k i n g day the p r o d u c t i v i t y might d e c l i n e because the worker i s t i r e d ( s e e , e.g., B a r z e l [ 1 9 7 3 ] ) . Note t h a t i f the f i r m s are not a b l e to o f f e r v a r i a b l e wage s c h e d u l e s they may impose c o n s t r a i n t s on the number of hours of work per day. i i ) C e r t a i n f i x e d c o s t s of employment p a i d by the f i r m might r e s u l t i n a v a r i a b l e wage s c h e d u l e (see Rosen [ 1 9 7 6 ] ) . Both ( i ) and ( i i ) can be e a s i l y handled by the ne o c l a s s i c a l model. The researcher has to v e r i f y whether the supply of labor i s a free variable, i , e . , whether the i n d i v i d u a l faces any r e s t r i c t i o n s on the number of work hours. 6/. Measurement of the supply of labor. We decided to refer to this issue l a s t because of i t s important effect on the previously mentioned t h e o r e t i c a l and empirical considerations. Both the neoclassical model and Becker's model are defined over time. Usually, time i s measured in hours, while the model refers to calendar years. These measurements are employed for p r a c t i c a l purposes - i . e . , a v a i l a b i l i t y of data - rather than for th e o r e t i c a l ones. Such an approach i s correct i f and only i f hours, weeks and months are perfect substitutes. However, as Hanoch [1980J points out, this i s not necessarily the case. For example, f i f t y hours of l e i s u r e spread over the year are not equivalent - for most individuals - to one non-working week. Hence, the u t i l i t y function perhaps should be defined over hours, days, weeks, and months. Assuming that the opportunity cost of time is fixed, one can always aggregate a l l types of l e i s u r e into one. The aggregated model can be used for forecasting purposes provided that there are no costs of working, the hourly wage rate i s fi x e d , and there are no r e s t r i c t i o n s on the hours of work. If any of those conditions i s vi o l a t e d , then the opportunity costs of time cannot be treated as a constant, since Hick's aggregation theorem i s not applicable. 26. E. Estimating the Supply of Labor II Econometric/Statistical Problems 1/. Sample s e l e c t i v i t y . Because of lack of data, or t h e o r e t i c a l considerations, economists estimate labor supply functions using a sample of families or i n d i v i d u a l s that s a t i s f y certain c r i t e r i a . For example suppose that the researcher imposes the sample selection rule that the individual's income i s less than a s p e c i f i c upper l i m i t . This sample selection rule can be expressed as: Y ± = I ± + w± (h-J^) < Y i = 1, N where Y^: t o t a l income of the i t n i n d i v i d u a l , 1^: non-labor income, w^  : the wage rate, (h-J^): hours of work, Y: The upper l i m i t of income for the selected sample. It can be shown that i f the sample is censored (see Amemiya [1973J, and Wales and Woodland [1981J), estimation of the labor supply function by ordinary least squares w i l l r e s u l t , i n general, i n biased and inconsistent estimates. A number of estimation techniques have been proposed which y i e l d unbiased and/or consistent estimates. The only drawback of these algorithms is that they are highly non-linear even in the case of l i n e a r supply functions. 2/. E n d o g e n e i t y of wages. As we have a l r e a d y seen, the i n t r o d u c t i o n of t a x e s i n the budget c r e a t e s two problems: i ) the observed net wage r a t e and the income of the consumer-worker are endogenous v a r i a b l e s , i i ) The u t i l i t y m a x i m i z i n g p o i n t may be l o c a t e d on a segment of the budget c o n s t r a i n t o t h e r t h a n the o bserved one. Wales and Woodland [1979] proposed an i n g e n i o u s a l q o r i t h m f o r o b t a i n i n g u n b i a s e d and c o n s i s t e n t e s t i m a t e s of the parameters of the u t i l i t y f u n c t i o n s o l v i n g at the same time f o r the u t i l i t y m a x i m i z i n g p o i n t . The b a s i c s t e p s of the a l g o r i t h m are the f o l l o w i n g ones: i ) F or each i n d i v i d u a l and f o r every s e t of parameters of the u t i l i t y f u n c t i o n s o l v e the o p t i m i z a t i o n problem of the i n d i v i d u a l , hence c a l c u l a t e Ij= £(Wj>P» Mj) f o r every segment of the budget, where X. i s l e i s u r e , W the wage r a t e , p the p r i c e of the composite good, and M i s the f u l l income, i i ) Check i f £j i s l o c a t e d on segment j . i i i ) I f i t i s , then Hi i s the o p t i m a l one, say li -D e f i n e e* = ( S ^ - W^  )M* where S i s the 1 s h a r e . i v ) I f i t i s n o t , check a l l c o r n e r s o l u t i o n s . P i c k up the one c o r r e s p o n d i n g to the h i g h e s t i n d i f f e r e n c e c u r v e . Define e* = (S £ - W^C)MC where Mc = a pYc + u)*c v) Evaluate E(e*) 2 for any set of parameters H and minimize i t The experimental evidence presented by Wales and Woodland proves, beyond any doubt, the supe r i o r i t y of their algorithm. Unfortunately, the attractiveness of the technique is diminished because of the computational burden involved i n the implementation of the alqorithm. (A simi l a r algorithm has been proposed by Burtless and Hausman [1978]). Footnotes to Chapter III 29. 1. See Diewert [1974J, Lau [1974J 2. The neo c l a s s i c a l model can accommodate time costs i f they are treated as exogenous variables (see Cogan [1981J). Chapter IV ON MODELLING THE SUPPLY OF LABOR: THE DYNAMIC MODEL A. I n t r o d u c t o r y Remarks The models of human c a p i t a l a c c u m u l a t i o n and l a b o r s u p p l y p r e s e n t e d i n the p r e v i o u s two c h a p t e r s are und o u b t e d l y l i m i t e d i n scope. N e i t h e r i s human c a p i t a l accumulated i n a vacuum nor i s the s u p p l y of l a b o r u n a f f e c t e d by the s t o c k of human c a p i t a l . Moreover the s i m p l i f y i n g a s s u m p t i o n of i n s t a n t a n e o u s u t i l i t y m a x i m i z a t i o n i s un n e c e s s a r y as o p t i m a l c o n t r o l t h e o r y can be a p p l i e d . In t h i s c h a p t e r I w i l l p r e s e n t a number of models t h a t s t u d y the a c c u m u l a t i o n of human c a p i t a l w i t h i n the framework of i n t e r - t e m p o r a l u t i l i t y m a x i m i z a t i o n . I t w i l l be shown t h a t t h e o r e t i c a l l y o r i e n t e d models can be t e s t e d w i t h a p p r o x i m a t i o n s ; i t w i l l a l s o be shown t h a t e m p i r i c a l l y o r i e n t e d ones can be as g e n e r a l , c o n s i s t e n t and r i g o r o u s as the t h e o r e t i c a l l y o r i e n t e d ones. I w i l l demonstrate t h a t one i m p o r t a n t a s p e c t of the c o n s t r u c t i o n of a model i s the s p e c i f i c a t i o n of i t . Even a minor m o d i f i c a t i o n or rearrangement of the v a r i a b l e s might be the c r u c i a l p o i n t which a l l o w s us to t r a n s f o r m a not e m p i r i c a l l y t r a c t a b l e model i n t o an e s t i m a b l e one. B. The B a s i c Model A c c o r d i n g to the b a s i c model i t i s assumed t h a t t h e consumer worker t r i e s to s o l v e the f o l l o w i n g m a x i m i z a t i o n problem: T (IV.1) max J e ' P t U(x(t) £(t))dt 0 w.r.t. x ( t ) , A(t) subject to: (IV.2) I = r l + w(t)(h-*(t)) - p ( t ) x ( t ) (IV.3) K O ) = I where p ( t ) : the price of the consumption good, x ( t ) : the composite good, £(t): hours of l e i s u r e per period, h ( t ) : t o t a l number of hours available per period p: the rate of time preference, r: the interest rate, 1(0): i n i t i a l wealth, T: terminal time. Assuming that an i n t e r i o r maximum ex i s t s , the f i r s t order conditions for the maximization problem are the following: where X(t) i s the costate variable associated with the state variable I ( t ) ; i t has the usual i n t e r p r e t a t i o n as the marginal u t i l i t y of wealth i n the sense that i t measures the contribution of an additional unit of I ( t ) to the u t i l i t y sum from t onwards. (IV.4) 3U(t) = X(t)e( P" r)tp(t) 8x(t) 3U(t) = X(t)e( p - r ) t w ( t ) 3£(t) In order to f a c i l i t a t e the subsequent analysis i t i s convenient to derive the dynamic demand functions for l e i s u r e and consumption by inverting equation (IV.4) The dynamic demand functions may be written as: (IV.5) x(t) = x( X ( t ) e ( p - r ) c p ( t ) , X(t)e<p- r) cw(t)) (IV.6) £(t) = ii X(t)e( P - r ) t p ( t ) , X(t)e< p- r)tw(t)) A few caveats are i n order here: i ) The demand for le i s u r e and for the composite good, during any period, depends on the pre v a i l i n g prices i n this period - the wage rate, the price of the composite good, and X(t). Thus X(t) is the only l i n k between the past and the future, i i ) X(t) depends on l i f e - c y c l e wages, i n i t i a l assets, the interest rate and the rate of time preference. While X(0) i s the marginal u t i l i t y of wealth at the start of l i f e - c y c l e , such marginal u t i l i t y does vary over the l i f e - c y c l e , because p(t) and w(t) most ce r t a i n l y w i l l . Contrary to the s p i r i t of the permanent income, theory, our focus i s on the pattern of variations i n consumption and le i s u r e of the in d i v i d u a l over his l i f e - c y c l e , i i i ) Provided that an estimate of the trajectory of X(t) can be'obtained for every period including 0 and T one can estimate the system of dynamic demand functions, equations (IV.5) and (IV.6). 33. C. Accumulation of Human Cap i t a l and the Supply of Labor A major drawback of the basic model i s the assumption of an exogenous wage rate. There i s s u f f i c i e n t evidence supporting the hypothesis that the wage rate depends on the human c a p i t a l of the worker. Heckman [1976J presented an empirically tractable model of human c a p i t a l accumulation. He formulated the maximization problem of the consumer as follows: T (IV.7) max J e - f t u ( x ( t), A ( t ) H ( t ) ) d t 0 w.r.t. £(t), x ( t ) , s ( t ) subject to: (IV.8) H(t) = f ( b s ( t ) H ( t ) , D(t)) - aH(t) (IV.9) I ( t ) = a r l ( t ) + aw(t)(h-Jt(t)-s(t)) - p D ( t ) D ( t ) -p( t ) x ( t ) H(0) = H 0 K O ) = I 0 where x ( t ) : the composite good, £(t): hours of l e i s u r e per period, H(t): human c a p i t a l , s ( t ) : time devoted to human c a p i t a l accumulation, D(t): the input of market goods, pn(t): the price of direct educational expenses, a: an exponential rate of depreciation of human c a p i t a l , p ( t ) : the price of the composite good, I ( t ) : the wealth (debt) of the i n d i v i d u a l , r: the interest rate, b: a productivity parameter 6: the rate of time preference, h ( t ) : t o t a l number of hours available per period, w(t): the wage rate per unit of human c a p i t a l , say, w(t) = RH(t) where R can be normalized to unity for every t, and 1-a: the proportional income tax rate Heckman examined certain properties of the model and s p e c i f i e d the earnings function of the i n d i v i d u a l as (IV. 10) E(t) = [w(t)(h-s(t)J - p D ( t ) D ( t ) - w(tH(t) For estimation purposes i t was assumed that the human c a p i t a l production function and the u t i l i t y function were Cobb-Douglas. The values of the estimated parameters, using U.S. census data, seem to be quite pl a u s i b l e . However, the asymptotic t s t a t i s t i c s were, except for two parameters, less than 0.5. Furthermore, as Heckman shows, a simple quadratic model performs almost as well as (IV.10). Ghez and Becker [1975J proposed an intertemporal model of labor supply/accumulation of human c a p i t a l which extends and generalizes Becker's [1965J "Theory of the A l l o c a t i o n of Time". Their model is much more general than any model which has been presented so f a r . Unfortunately, the estimation of the model i s next to impossible because of the excessive information requirements. D. Occupational Choice and Human Cap i t a l Accumulation It i s widely recognized that the two major components of human c a p i t a l are formal education and experience. In p r i n c i p l e , the s p e c i f i c a t i o n of accumulated work experience as " t o t a l number of hours that the i n d i v i d u a l has worked so f a r " i s erroneous. There i s no information available about on-the-job tr a i n i n g and the c h a r a c t e r i s t i c s of di f f e r e n t occupations so that we cannot introduce them i n an empirically tractable model of labor supply. Nevertheless i t i s i n t e r e s t i n g to see how the conclusions of the t h e o r e t i c a l model are affected by these variables. Blinder and Weiss [1976J were the f i r s t to introduce on-the-job t r a i n i n g e x p l i c i t l y i n the model. Furthermore they paid attention to periods of s p e c i a l i z a t i o n , such as f u l l - t i m e schooling and retirement. They were able to derive a number of concrete conclusions by imposing rather severe r e s t r i c t i o n s on the preferences of the consumer worker and the human c a p i t a l generating function. I believe that their model i s worth studying because i t i l l u s t r a t e s the l i m i t s of th e o r e t i c a l models. Even a cursory study of the i r model indicates that a tractable version of i t is not possible. Three of the variables of the model are unobservable: human c a p i t a l , i t s shadow price and the occupational index. The ambiguity about the f i r s t two can be resolved by normalizing one of them and getting a measure of the second one. Estimation of the t h i r d variable i s not fea s i b l e since detailed information i s not available. Now l e t us consider the model proposed by Blinder and Weiss. The maximization problem of the consumer i s sp e c i f i e d as follows: T max Je" PtUCc, i)dt + B[A(T) J 0 w.r.t. x ( t ) , h ( t ) , c ( t ) s u b j e c t t o : A = rA + g(x)hK - c K = (axh - 6)K h ( t ) + Sit) = 1 h ( t ) > 0 0 < x ( t ) < 1 where c: the composite good, £(t): the f r a c t i o n of time devoted to l e i s u r e , A ( t ) : the w e a l t h of the i n d i v i d u a l , h ( t ) : the f r a c t i o n of time devoted to market a c t i v i t y ( i n c l u d i n g work and e d u c a t i o n ) , x ( t ) : the o c c u p a t i o n a l i n d e x , and g ( x ) : the wage r a t e . The main l e s s o n t h a t we can d e r i v e from the model i s t h a t a t h e o r e t i c a l l y o r i e n t e d model, whatever i t s degree of o r i g i n a l i t y and s o p h i s t i c a t i o n , might not be e m p i r i c a l l y t r a c t a b l e . However the l a s t statement does not i m p l y t h a t we s h o u l d s a c r i f i c e the c o n s i s t e n c y and c o n c r e t e n e s s of the model s i m p l y to make i t t r a c t a b l e . I t i m p l i e s r a t h e r t h a t the r e s e a r c h e r has t o f i n d a compromise between the two o b j e c t i v e s of g e n e r a l i t y and t r a c t a b i l i t y . E. An E m p i r i c a l l y O r i e n t e d Model Heckman and MaCurdy [1980J p r e s e n t e d an e m p i r i c a l l y t r a c t a b l e model of l a b o r s u p p l y . ^ The t h e o r e t i c a l model proposed by Heckman and MaCurdy i s a v e r y s i m p l e , almost n a i v e , one, but a l l v a r i a b l e s and parameters of i can be e a s i l y computed and estimated. The econometric technique implemented by Heckman and MaCurdy was to treat X(0) - the shadow price of wealth during the f i r s t period - as a fixed effect variable, that i s , they estimated simultaneously the u t i l i t y function, the wage equation and X(0). Heckman and MaCurdy assumed that the i n d i v i d u a l t r i e s to solve the following optimal control problem: ' T max Z 1 [ A ( t ) U ( t ) ) a + B ( t ) ( c ( t ) ) T J t=0 (l+p) t w.r. to lit), c(t) subject to T 1(0) = E 1 [c(t) - w(t)(h-£(t)J t=0 ( l + r ) t A(t) = exp {Z(t)$} tow(t) = X ( t ) g where A ( t ) , B(t): "age s p e c i f i c modifiers of "tastes" or "household production"", Z ( t ) : "a vector of measured determinants of l e i s u r e choices", X ( t ) : a vector of exogenous variables, £(t): hours of l e i s u r e , c ( t ) : consumption, w(t): the wage rate. It can be shown that the demand function for l e i s u r e i s : A*(t) = 1 ( I H J ) A ( t ) o ( l + r ) X(0)w(t] i f the a * i n d i v i d u a l works £*(t) = h otherwise Heckman and MaCurdy estimated the demand function for l e i s u r e and the wage equation recognizing that the d i s t r i b u t i o n of the error term is truncated because of sample s e l e c t i v i t y . They estimated i n two stages. In the f i r s t stage they maximized the l i k e l i h o o d with respect to the parameters of the model. In the second stage they regressed the "fixed e f f e c t " v a r i a b l e — t h e shadow price of wealth - on some exogenous variables. The s p e c i f i c a t i o n and estimation of this model raises a number of issues. i ) Given their estimation procedure - i . e . , fixed effect X(0) - the estimated parameters are inconsistent, i i ) The s p e c i f i c a t i o n of the l i k e l i h o o d function and the estimation procedure are hampered by the omission of the demand function for the composite good. Therefore, estimates of B(t) - the tastes modifiers for consumption - have not been obtained, i i i ) It i s assumed that human c a p i t a l i s an exogenous variable. Therefore, the model is mispecified. iv) It is not clear i f this technique can be employed for optimal control problems with more than one equations of motion. (If human c a p i t a l i s endogenously determined at least two equations of motion are needed.) F. Concluding Remarks What general conclusions and directions for related research can we draw from the body of work discussed i n this chapter? The only unambiguous conclusion is that no " d e f i n i t e " or "generally acceptable" model has been presented so f a r . This section is designed to j u s t i f y the l a s t statement and highlight some of the d i f f i c u l t i e s faced by the researcher attempting to model a l i f e - c y c l e model of labor supply. It w i l l also be shown that s p e c i a l i z e d models, ones that focus on s p e c i f i c facets of the labor market can be s u f f i c i e n t l y general and rigorous. The most important d i f f i c u l t i e s that we face i n l i f e - c y c l e models of labor-supply - i n addition to the ones discussed i n Chapter III - can be summarized as follows: 1/. Longitudinal data versus synthetic cohorts. As a rule of thumb we can c l a s s i f y the available models into two classes. 1) ones for which data sets are available and the researchers have implementation problems; 2) ones for which no data sets are available and have implementation problems. During the l a s t decade economists have been able to estimate labor supply functions using either longitudinal data or synthetic cohorts. The simulation of the l i f e - c y c l e p r o f i l e of an i n d i v i d u a l using synthetic cohorts suffers from the following drawbacks (see Smith [1977J, Heckman and MaCurdy [1980]): i ) If expectations about the future are biased then the v a r i a t i o n between the age c e l l s w i l l not correspond to the v a r i a t i o n over the l i f e - c y c l e . i i ) Because of increases i n r e a l wage rates, during the recent years, the expected wealth of younger cohorts is greater than that of older cohorts. i i i ) It i s assumed that a l l individuals belonging to a s p e c i f i c age c e l l have the same l e v e l of u t i l i t y and the same amount of wealth. It i s obvious that studies based on longitudinal data w i l l not be affected by the formation of expectations - i n the sense of ( i ) - but they w i l l be affected by the change of the r e a l wage rate, unless complete p r o f i l e s are available. So the only comparative disadvantages of synthetic cohorts are related to the formation of expectations and the l e v e l of u t i l i t y and wealth. I f i n d the previous treatment of expectations s l i g h t l y schizophrenic. Except for one model a l l other models assume either i m p l i c i t l y or e x p l i c i t l y that the consumer is able to foresee the future. Given this assumption there is no place i n the model for formation of expectations. The discussion would have been relevant i f uncertainty and/or i n a b i l i t y to get a perfect forecast of the future had been included, e x p l i c i t l y , i n the model. The question that should be asked is not which approach allows for variations i n expectations but rather i f the model is equipped to handle expectations at a l l . Mutatis  mutandis, the same is true about real wages. Therefore, from a technical point of view, the question is whether X(0), which is a function of "expectations", i s corr e c t l y s p e c i f i e d . I w i l l return to this point in (2) below. An advantage of l o n g i t u d i n a l data is that the r e s u l t i n g estimates are more e f f i c i e n t because the estimation procedure u t i l i z e s a ceteris paribus, larger and richer sample. That advantage of longitudinal data should be c a r e f u l l y contrasted w i t h a s e r i o u s d i s a d v a n t a g e , t h a t c e r t a i n v a r i a b l e s , such as e x p e r i e n c e , are s u b j e c t to measurement e r r o r . The o n l y unambiguous s h o r t c o m i n g of s y n t h e t i c c o h o r t s i s the assumption t h a t a l l i n d i v i d u a l s b e l o n g i n g to a p a r t i c u l a r age c e l l are at the same l e v e l of u t i l i t y and possess the same amount of w e a l t h . T h i s can be viewed as some s o r t of s p e c i f i c a t i o n e r r o r . The impact of the a f o r e m e n t i o n e d s h o r t c o m i n g can be sub-s t a n t i a l l y reduced by s t r a t i f y i n g the sample a c c o r d i n g to more than one c h a r a c t e r i s t i c . For example, a r e a s o n a b l e s t r a t i f i c a -t i o n i s by age and y e a r s of e d u c a t i o n , or age and i n i t i a l w e a l t h or age, e d u c a t i o n and i n i t i a l w e a l t h . F u r t h e r m o r e , the u t i l i t y f u n c t i o n can be d e f i n e d over consumption, l e i s u r e and y e a r s of e d u c a t i o n . On a p r i o r i grounds, one e x p e c t s t h a t as the p a r t i t i o n becomes f i n e r the d i f f e r e n c e s i n the l e v e l of u t i l i t y and the amount of w e a l t h of the i n d i v i d u a l s w i t h i n each c e l l w i l l d e c r e a s e . T h i s proposed t e c h n i q u e i s e s p e c i a l l y a t t r a c t i v e i n view of the f a c t t h a t measurement of the w e a l t h of the i n d i v i d u a l w i t h l o n g i t u d i n a l d a ta s e t s i s not p o s s i b l e y e t . We can draw the f o l l o w i n g l e s s o n from the p r e v i o u s d i s c u s s i o n : N e i t h e r of the two t e c h n i q u e s has a d e f i n i t e advantage over the o t h e r . E r r o r s i n v a r i a b l e s are more p r o b a b l e w i t h l o n g i t u d i n a l d a t a s e t s , w h i l e p a n e l data r e s u l t i n some s o r t of m i s s p e c i f i c a t i o n . The c h o i c e depends on the p r e f e r e n c e s of the r e s e a r c h e r , the s u b j e c t i v e e v a l u a t i o n of the advantages and d i s a d v a n t a g e s of each t e c h n i q u e . 2/. E s t i m a t i n g the shadow v a l u e of w e a l t h . The demand f u n c t i o n s f o r l e i s u r e and the composite good which c o r r e s p o n d to the simplest model - which I c a l l e d the Basic Model - can be written as: x(t) = f [ Xe< P^ r) tw(t), Xe( p - r ) t p ( t ) J £(t) = g[ Xe( p - r ) t w ( t ) > Xe(p- r) cp(t)J The main d i f f i c u l t y in estimating those equations is the lack of observations for X(0). In p r i n c i p l e , data for X(0), or any X(t) are not required; i t s u f f i c e s to solve the optimal control problem and express X(t) as a function of the other variables and parameters of the model. It is quite s u r p r i s i n g that no researcher has pursued this straightforward approach. Although the r e s u l t i n g d i f f e r e n t i a l equations may be so "messy" that an analytic solution i s impossible, i t i s up to the researcher to specify the model in such a way that an a n a l y t i c solution of the two boundary problems can be obtained. Without any loss of generality we can c l a s s i f y a l l techniques for estimating the foregoing intertemporal equations as follows: i ) The f i r s t approach implemented by Ghez and Becker [1974J manages to get around the problem by estimating approximations of the two demand equations, so that X(t) is subsumed into the parameters of the estimated functions. This technique can be employed either with cross section or longitudinal data. With some luck, this approach allows us to recover, at most, Hicks- Slutsky income and s u b s t i t u t i o n e f f e c t s , i i ) The second approach suggested by Mincer [ 1962 J i s to decompose the current income and the wage rate into a permanent and a transitory component. The underlying assumption i s that the response of the demand for l e i s u r e with respect to those two components d i f f e r s . As Heckman and MaCurdy (1980J point out, neither the t h e o r e t i c a l rationale for this decomposition nor the empirically oriented d e f i n i t i o n s of these two components are well defined. i i i ) The t h i r d approach due to MaCurdy [1978J i s to proceed by treating X(0) as a fixed effects variable. As we have seen, this approach results i n inconsistent estimates, i v ) The fourth approach implemented i n this research project i s to solve e x p l i c i t l y for the X(0). Now l e t us focus on the proposed technique. Notice that for most l i f e - c y c l e models the equation of motion of X(t) i s very simple: X(t) = -r X(t) This d i f f e r e n t i a l equation can be e a s i l y solved i f either X(0) or X(T) are known. Define the bequest function as: B[I(T); cJ where B[ J: the bequest function, I(T): wealth at time T, i . e . , terminal time, £: a vector of c h a r a c t e r i s t i c s of the i n d i v i d u a l and/or variables evaluated at time T. It i s well known that X(T) i s given by: w T x = 3 B [ I ( T ) ; g]  K K ' 3 I ( T ) Hence, the equation of motion of X(t) can be wri t ten as: X(t) = X ( T ) e - r ( t - T ) For est imation purposes i t suf f ices to specify a func t iona l form for the bequest f u n c t i o n , 2 subst i tute the la s t equation into the system of demand equations and estimate simultaneously the parameters of the u t i l i t y funct ion and the bequest func t ion . The suggested technique is quite useful because i t allows us to estimate intertemporal demand functions without imposing r e s t r i c t i v e approximations of the u t i l i t y funct ion or employing techniques which resu l t in incons is tent estimates. Furthermore, i t i s useful for s imulat ing the impact of pension plans, taxes, and the l i k e , on the t r a j e c t o r i e s of the costate var iables of the i n d i v i d u a l . The same technique can be employed, mutatis mutandis, for any costate v a r i a b l e , provided that the associated state var iab le is an argument of the bequest funct ion . If i t i s not , one may s t i l l approximate the t ra jec tory of the costate var iab le by u t i l i z i n g the t r a n s v e r s a l i t y condit ions; such a case i s presented in the next chapter. 3/. Model l ing human c a p i t a l . I have already demonstrated that human c a p i t a l and i t s shadow pr ice are not observable. The only observable var iab le is the wage ra te , which is assumed to be a funct ion of the stock of human c a p i t a l of the i n d i v i d u a l . In order to s impl i fy the subsequent analys is l e t us suppose that human c a p i t a l is a funct ion of years of schooling and 45. experience. The human c a p i t a l accumulation/earnings part of the model can be written as follows: • • • H(t) = f ( H ( t ) , s,L) w(t) = h(H(t)) where H(t): human c a p i t a l , s: years of schooling, L: experience, and w: the wage rate Footnotes to Chapter IV MaCurdy [1978J [1981] presented a similar model. A l t e r n a t i v e l y , T can be defined as date of retirement and B[I(T)J as the u t i l i t y y i e l d of net wealth at retirement. Chapter V THE PROPOSED MODEL A. Description/Properties of the Model This paragraph presents a l i f e - c y c l e model of labor supply that i s almost empirically tractable. The model generalizes most of the models which have been proposed so far, save for the one due to Becker and Ghez. The maximization problem of the consumer i s written as follows: T (V.l) max /e~ 6 tU[x,(h-t(s)-t(w)),sJdt + B[I(T)J to w.r.t. x, t ( s ) , t(w) subject to (V.2) s = t(s) (V.3) L = t(w)- yL (V.4) 1 = r l + f[w(L,s)t(w)J - px - C[t(s)J s(0) = "s" » o (V.5) L(0) = L » 0 1(0) =1 where U: the u t i l i t y function, 6: the rate of time preference, B: the u t i l i t y y i e l d of net wealth at retirement (UYONWAR), tg: i n i t i a l time, T: date of retirement, x: the composite good, t(w): hours of work per period, t ( s ) : hours of schooling per period, h-t(s)-t(w): hours of l e i s u r e per period, s: "years" of schooling, L: experience, I: the wealth of the i n d i v i d u a l , r: the inte r e s t rate, w: the wage rate p: the price of the composite good, C: cost of schooling, y: the depreciation rate of experience ft J: after tax earnings The following assumptions are made with regard to the optimization problem. i ) The u t i l i t y function and the UYONWAR function are defined over the pos i t i v e orthant and are continuous, increasing, strongly concave functions, are f i n i t e for a l l f i n i t e values of their arguments and s a t i s f y the usual Inada conditions, e.g.: Mm U x = °°, V[h-t(s)-t(w) J e.t.c. x -»o i i ) The after tax earnings function i s a p o s i t i v e , continuous, strongly concave function,., i i i ) C apital income i s non-taxable, or r i s the" after tax V return on c a p i t a l income. iv) The cost of schooling function is a s t r i c t l y p o s i t i v e , continuous, strongly convex function. Before proceeding to study the properties and the predictions of the model, i t w i l l be necessary to examine the maintained hypotheses. i ) The r eqularity conditions imposed on the u t i l i t y function are s l i g h t l y more r e s t r i c t i v e than required in order to generate convex indifference curves. Let us consider each condition: - The continuity assumption is imposted in order to simplify the calculus and, i n any case, i s quite harmless. The Inada conditions exclude the p o s s i b i l i t y of corner solutions; i t is assumed that the subsistence l e v e l of consumption is s t r i c t l y greater than zero and the i n d i v i d u a l has to take some rest during each period. - The strong concavity of the u t i l i t y function i s imposed i n order to ensure the existence of a unique intertemporal plan. Furthermore, i f the Hamiltonian is strongly concave, then the necessary conditions are also s u f f i c i e n t for an optimum, a result which considerably s i m p l i f i e s manipulation of the model. i i ) A l l except one assumption for the after-tax-earnings function are imposed in order to f a c i l i t a t e the calculus and, i n any case, are quite harmless. The only assumption that must be j u s t i f i e d i s strong concavity. Abundant evidence supports the hypothesis of a concave earnings function, (see, e.g., [Mincer, 1974J). This well-established property has been strengthened in order to ensure the strong concavity of the Hamiltonian. i i i ) The assumption that c a p i t a l income i s non-taxable is imposed in order to simplify the notation as well as the handling of the model. The assump-tion also f a c i l i t a t e s the modelling of the tax func-t i o n , a l b e i t at some loss of generality. Three things should be noted. F i r s t , for most working in d i v i d u a l s , c a p i t a l income i s pretty low. Second, a s i g n i f i c a n t part of c a p i t a l income is non-tax-able. Third, i n the empirical implementation of the model, every e f f o r t has been made to approximate as accurately as possible the actual amount of taxes paid by the i n d i v i d u a l , iv) It was decided to present the model in continuous form so that the notation could be kept at manageable l e v e l s . A discrete version of the model w i l l be examined in the second part of this chapter. (v) The u t i l i t y function i s defined over consumption, le i s u r e and education. There are two reasons for doing so. F i r s t , abundant evidence supports the hypothesis that education l e v e l affects the u t i l i t y function.^ Second, there i s enough evidence suggesting that non-pecuniary factors affect the demand for schooling and occupational choice. 2 The Hamiltonian of the optimal control problem can be written as e - 6 t U ( ) + ot(s) + X[t(w)-yLj + u[rl + f( ) -px - C( )] The f i r s t order conditions for a maximum are: ( v . 6) e " 6 t U x = vp (v. 7) -e-«tU A = e - & U t ( s ) = o + y C t ( s ) t s > 0 (V. 8) -e- f i tU A= e - f c u t ( w ) = -A-uf'w tw > 0 (V. 9) [ e - « t u t ( s ) + o ~ i C t ( s ) J < 0 t s = 0 (V. 10) [e~ 6 t u t ( w ) + X + uf 'wj < 0 tw = 0 (V. 11) *0 = - e - fitUg - 'wst(w) a(T) = 0 (V. 12) \ = - uf 'wLt(w) + \y X(T) = 0 (V. 13) u = - ur u(T) = B' [1(T)J Very few individuals are part time students and part time workers for the entire length of the i r working l i v e s . For most in d i v i d u a l s , the f i r s t phase of their l i f e - c y c l e i s the one of f u l l time s c h o o l i n g , succeeded p o s s i b l y by p a r t time s c h o o l i n g and p a r t time w o r k i n g , w h i l e d u r i n g the t h i r d phase they are w o r k i n g f u l l t i m e . A most i n t e r e s t i n g p r e d i c t i o n of the model i s t h a t the a f o r e m e n t i o n e d s u c c e s s i o n of phases i s , i n g e n e r a l , the o p t i m a l one f o r an i n d i v i d u a l who does not f a c e any c o n s t r a i n t s , save f o r the a l r e a d y p r e s e n t e d ones. I w i l l r e t u r n to t h i s p o i n t below. For the time b e i n g , l e t us suppose t h a t t h i s i s the o p t i m a l p a t h . B e f o r e s t u d y i n g the t h r e e phases of the l i f e - c y c l e , i t i s n e c e s s a r y to deve l o p some n o t a t i o n ; the t h r e e phases w i l l be denoted by , T2, T3. 1/. F u l l time s c h o o l i n g . D u r i n g the f i r s t phase of the l i f e - c y c l e , the f i r s t o r d e r c o n d i t i o n s can be w r i t t e n a s : (V.14) e - « S t U t ( s ) + o - vC t(s) = 0 o = -e" 6 t U s < 0 (V.15) e _ l S t U t ( w ) + X + uf'w < 0 X ( t ) = Xy teTx The f i r s t o r d e r c o n d i t i o n s can a l s o be w r i t t e n a s : (V.16) . e " 6 t U % = a - yC t( s) (V.17) e - <5tu £ > X + uf'w Hence, (V. 18) a ~ yC t( s) > X + uf *w That i s - , the net d i s c o u n t e d b e n e f i t s of e d u c a t i o n are g r e a t e r than the net b e n e f i t s of w o r k i n g d u r i n g the f i r s t phase. Rearrange equations (V.16) and (V.17) as follows: \ (t) = X7 teTi The f i r s t order conditions can also be written as: (V.16) e-^U^ = a - i£ t( s) (V.17) e" 6 tU £ > X + uf 'w Hence, (V. 18) o - uC t( s) > X + uf 'w That i s , the net discounted benefits of education are greater than the net benefits of working during the f i r s t phase. Rearrange equations (V.16) and (V.17) as follows: (V.19) a > X + uf *w + yC t( s) Equation (V.19) implies that during the f i r s t phase, the shadow price of schooling is greater than the sum of foregone earnings, shadow price of experience, and marginal cost of schooling. A few caveats are in order here: i ) Equations (V.18) and (V.19) should not be interpreted as a "proof" of the succession of phases. Full-time schooling implies (V.18) and (V.19) and vice-versa. (Mutatis mutandis the previus discussion applies to (V.20) and (V.21)). i i ) A true proof must be that a p a r t i c u l a r sequence and only this sequence guarantees the continuity of shadow prices over-time. As i t w i l l be shown below such a proof appears to be impossible. However, i t w i l l be shown that for "most" individuals the aforementioned succession of phases guarantees "almost everywhere" the continuity of shadow prices over time. Obviously, this i s not a general proof. Therefore, the results that are presented below should be interpreted with caution. 2/. F u l l time working. In order to f a c i l i t a t e the subsequent analysis, I w i l l consider the t h i r d phase f i r s t and then discuss the second one. The f i r s t order conditions for the t h i r d phase can be written as: (V.20) e _ l 5 t U t ( s ) + a - \£t(s) < 0 a = - e~ S t u s - uf'w st(w) (V.21) e _ 6 t U t ( w ) + X + uf'w = 0 X = - uf 'wLt(w) + Xy Equations (V.20) and (V.21) can be written as: (V.22) e ' ^ U ^ ^ cr UC t ( s) (V.23 ) e" 5 tU % = X + uf 'w (V.24) o - u C t ( s ) i . x + "f'w t e T 3 Equation (V.23) shows that the marginal u t i l i t y of l e i s u r e is not equal to the marginal af t e r tax wage rate but i t is equal to the marginal after tax wage rate plus the shadow value of experience. Therefore, a l l s t a t i c models underestimate the price of l e i s u r e which, of course, results in inconsistent estimates. Equation (V.24) summarizes nicely the c h a r a c t e r i s t i c s of that phase; i t i s the period during which the benefits, of schooling pecuniary and non-pecuniary, are less than the benefits of working. During the th i r d phase, hours of work and earnings reach a peak and then decline. The wage rate by assumption i s a non-decreasing function of experience and w i l l reach a peak at a point around the age of retirement. Fuchs [ 196 7 J shows that this conjecture i s supported by U.S., cross-section data for a l l levels of education, except for those with post-graduate degrees. Given the shape of the wage rate function, one would expect hours of work to reach a peak before earnings reach th e i r maximum point. Assuming that the u t i l i t y function i s strongly separable i n consumption and l e i s u r e , the foregoing statement i s supported by the model. Let us consider the case of maximum hours of work: Suppose that the u t i l i t y function i s strongly separable. Define the after tax wage rate as: Notice that the la s t transformation i s absolutely harmless since we are going to investigate the properties of the trajectory of the wage rate around a point. Taking the derivative, with respect to time, of equation (V.23) we obtain: (V.25) cw where c: a constant. (V.26) - 6e- 6 t U SL-e UM t(w) = X- urcw + pc[wLt(w) + w st(s)J Noticing that t(w) and t(s) are by assumption equal to zero and u t i l i z i n g the equation of motion for A we get (V.27) -6e _ < S tU i l= - urcw + Ay Using equation (V.23), equation (V.27) can be written as (V.28) - 6[ A+ pcwj = - urcw + Ay or, (V.29) A[ fit- A] r *6 = cw u[r- SJ A/ u i s the r e l a t i v e shadow price of experience i n terms of non-human c a p i t a l . Equation (V.29) shows that hours of work w i l l reach a peak at the point where the r e l a t i v e price of experience is equal to the marginal after tax wage rate, i f 6=r and y=0. If r> 6 and y = 0 hours of work w i l l reach a peak at a point where the r e l a t i v e price of experience i s lower than the marginal after-tax wage rate. F i n a l l y , i f the rate of time preference i s greater than the interest rate, then, obviously, the previous analysis does not go through and we have to conclude that hours of work reach a peak during the second phase. Now l e t us consider the other important point of the t h i r d phase the point where earnings reach a peak. Imposing strong s e p a r a b i l i t y on the u t i l i t y function, multiplying equation (V.23) by t(w) and taking the derivative of the product with respect to time, we obtain: (V.30) - 6e~ <Stu At(w) - e" 5 tU M t ( w ) t ( w ) + e~ fitU £t(w) = • • • = Xt(w) + Xt(w) - urcwt(w) + yc[wt(w)J But, yc[wt(w)J = 0 e - < S tU£t(w) = Xt(w) + ucwt(w) Xt(w) = -ucw Lt(w) t(w) + Xyt(w) = yc[wt(w)J t(w) + Xyt(w) = Xyt(w) Substituting the previous equations into ( V . 3 0 ) we obtain ( V . 3 1 ) ( 6+X)e" S tU % t(w) + e~ 6 tU ^ At(w)t(w) - yrcwt (w)- Xucwt (w) Hence, ( V . 3 2 ) ( 6 + y ) U £ + U^ £t(w) » 0 i f r > y A s u f f i c i e n t but not necessary condition for ( V . 3 2 ) to hold is t(w) to be negative, which implies that hours of work are decreasing when earnings reach a peak, As 6 becomes smaller, p r o b a b i l i t y that hours of work w i l l reach a peak increases. Another way to look at the problem is by writing ( V . 3 2 ) as: ( V . 3 3 ) ( 6+Y) + U jut(w) » Q which i l l u s t r a t e s that the point where earnings reach a peak depends on the curvature of the u t i l i t y function, or the e l a s t i c i t y of marginal u t i l i t y . 58. 3/. Part time working - Part time schooling. Very few new things can be said about the second phase of the l i f e - c y c l e . The only i n t e r e s t i n g exercise i s to determine the duration of this phase. This exercise has already been performed when we determined the terminal point of the f i r s t period and the i n i t i a l point of the t h i r d period. This i s a good point to consider the optimality of the pa r t i c u l a r succession of phases that we have studied. That i s , i t has to be shown that an i n d i v i d u a l working on a f u l l time basis w i l l not, in general, return to school. As we have seen, during the phase of f u l l time working, the optimality conditions can be written as: (V.24) a - yC t( s) < A + uf'w Suppose that the i n d i v i d u a l returns to school on a part time basis. Then the optimality conditions can be described by the following equation: a ~ l j C t ( s ) = x + Uf'w F i n a l l y , i f the i n d i v i d u a l switches to f u l l time schooling inequality (V.24) i s reversed, i . e . , 0 ~ M ct(s) ^ * + uf ' w Equivalently, a necessary but not s u f f i c i e n t condition for an i n d i v i d u a l to return to school i s that the net shadow price of schooling increases at a faster rate than the opportunity cost of working, or equivalently, the opportunity cost of working decreases at a faster rate than the net shadow price of schooling. The change of the net shadow price of schooling d u r i n g f u l l time working i s g i v e n by • • • o " ( vCt(s)) = a + r u C t ( s ) • s i n c e C t ( s ) i s independent of time. Given (V.20) a i s n e g a t i v e , hence the net shadow p r i c e of s c h o o l i n g might be i n c r e a s i n g over time because of the decrease of the shadow v a l u e of wealth. Assuming that the m a r g i n a l tax i s c on st an t, the chanqe of the o p p o r t u n i t y cost of working i s g i v e n by • • X + (pew) = -ycwLt(w) + Xy - rucw + ucwLt(w) = Ay - r yew The l a s t e q u a t i o n i m p l i e s that the o p p o r t u n i t y cost of working might be a d e c r e a s i n q f u n c t i o n of time because of the decrease of the shadow value of wealth. Now l e t us c o l l e c t the p r e v i o u s r e s u l t s . I t has been shown that the net shadow p r i c e of s c h o o l i n g might be i n c r e a s i n g and the o p p o r t u n i t y cost of working might be d e c r e a s i n g . Hence, one cannot exclude the case that the net shadow p r i c e of s c h o o l i n g w i l l o vertake, i . e . , w i l l become l a r g e r , the o p p o r t u n i t y cost of working. The q u e s t i o n that a r i s e s i s i f such an event should be viewed as an i n t e g r a l p art of the o p t i m a l t r a j e c t o r y of a " t y p i c a l " i n d i v i d u a l . The answer to the l a s t q u e s t i o n i s p r o b a b l y n e g a t i v e . The p r e v i o u s a n a l y s i s i n d i c a t e s that only i n d i v i d u a l s who have accumulated enough a s s e t s w i l l c o n s i d e r the p o s s i b i l i t y of r e t u r n i n g to s c h o o l , e i t h e r on a part time or f u l l time b a s i s . Moreover, the prime m o t i v a t i o n f o r doing so i s not an increase of their wage rate, which may or may not increase when they w i l l return to f u l l time working. (If experience depreciates over time, the wage rate w i l l d e f i n i t e l y decrease i f they become f u l l time students. On the other hand, the wage rate w i l l increase because their l e v e l of education w i l l improve). The ramifications of the statement that "the prime moti-vation for returning to school i s not an increase of the wage rate" are quite important. The aforementioned statement implies that the behavior of individuals who return to school i n order to increase their wage rate i s , i n general, suboptimal. I n t u i t i v e l y speaking, i f investment i n education i s p r o f i t a b l e , then the in d i v i d u a l i s better off by investing i n i t as early as possible. There are at least two reasons for a consumer to return to school i n order to improve his/her earnings capacity. The f i r s t one i s the p o s s i b i l i t y of unforeseeable events which might a l t e r d r a s t i c a l l y the intertemporal plans of an otherwise r a t i o n a l i n d i v i d u a l . The second one i s the existence of various constraints which might hamper the implementation of an optimal plan during the early stages of the l i f e cycle. Imperfect c a p i t a l markets, time constraints, lack of part time jobs and a host of other variables may force the consumer/worker to enter the labor force although such a strategy can be suboptimal. We w i l l return to this point in the next chapter. We now turn to the comparative dynamics of the model. Even i n this s i m p l i f i e d model, the influence of certain exogenous variables, l i k e taxes or the inte r e s t rate, on the t r a j e c t o r i e s of the state and control variables is rather moot. F i r s t , an increase of the interest rate reduces the discounted wealth of the i n d i v i d u a l who, as a consequence of the increased shadow price of wealth, w i l l decrease his/her demand for l e i s u r e . Contrariwise, an increase of the interest rate reduces the price of future consumption, hence the age consumption p r o f i l e w i l l be pushed upwards. A sim i l a r problem arises i n s t a t i c models of labor supply where nothing can be said about the r e l a t i v e importance of the income and substitution e f f e c t s . In a dynamic model, the wealth effect should also be taken into consideration; this makes things more complicated. The i n a b i l i t y of the model to provide us with unambiguous q u a l i t a t i v e results underscores the necessity of empirical work. Second, the l i f e - c y c l e response to a v a r i a t i o n of the marginal tax is ambiguous. Ignoring, temporarily, wealth, we can see that an increase of the marginal tax rate i s equivalent to a decrease of the wage rate. By introducing wealth, the picture becomes foggier since the wealth may decrease while i t s shadow price w i l l increase. An increase of the shadow price of wealth represents an increase of a l l prices, including l e i s u r e ; thus the f i n a l outcome cannot be determined. The impact of taxes is quite profound during the f i r s t phase. Here an increase of the marginal tax reduces the p o t e n t i a l wage rate and hence encourages heavier investment in formal education, provided that the increase of the shadow value of wealth does not outweigh the decrease of the p o t e n t i a l wage rate. The purpose of this paragraph was to present a p o t e n t i a l l y empirically tractable l i f e - c y c l e model of labor supply. Therefore my task i s to show how the model can be estimated given the available information. Before doing so, i t i s necessary to indicate possible generalizations of the model. i ) It appears as a straightforward exercise to define the model over a family rather than an i n d i v i d u a l . It s u f f i c e s to include the hours of l e i s u r e of the second member (or members) of the family into the u t i l i t y function and modify appropriately the equations of motion. Unfortunately, such a model w i l l be limited i n scope and realism. Nowadays storks do not bring children, which implies that family formation i s an endogenous variable. The introduction of family formation i n the model w i l l complicate i t and i t i s not clear i f the f i n a l product w i l l be a testable one. i i ) By disaggregating the composite good, we can generate a system of intertemporal demand functions. As a matter of f a c t , this i s the only technique which w i l l generate a system of intertemporal demand functions consistent with neoclassical economics. i i i ) The wealth can be disaggregated into i t s components, such as pension plans, savings, bonds, etc. The estimation of such a model should be quite useful for policy purposes since i t provides us with a system of demand functions for assets. iv) F i n a l l y , other possible extensions may include i n t r o -duction of the costs of vocational t r a i n i n g , cost of equipment necessary for the job, time required for commuting, etc. B: A Tractable Model Given the available data sets, information for the f i r s t two phases of the l i f e - c y c l e i s not available. The best that we can do is to estimate the parameters of the model using information from the t h i r d phase only. The estimates w i l l be e f f i c i e n t given the available information. The estimate of only a part of the l i f e - c y c l e can be j u s t i f i e d by appealing to the well celebrated p r i n c i p l e of optimality. Bellman [1957J expresses the p r i n c i p l e as An optimal policy has the property that, whatever the i n i t i a l state and decision, etc., the remaining decisions must constitute an optimal policy with regard to the state r e s u l t i n g from the f i r s t decision The p r i n c i p l e of optimality can be described as follows. Con-sider a consumer/worker who has solved the optimal control problem presented at the beginning of the chapter. Therefore, he/she has derived the optimal t r a j e c t o r i e s for the control and state variables for the entire time horizon, i . e . , ( t Q , T ) . Suppose that the optimal policy i s such that the i n d i v i d u a l switches to f u l l time schooling at time t]_, t± > t Q . Denote the accumulated stocks of wealth, experience and education at time t j ^ as I* ( t ^ ) , L* (t j ^ ) and s* (t^) respectively. The p r i n c i p l e of optimality states that the remaining part of the tr a j e c t o r i e s of the said optimal control problem - the t r a j e c t o r i e s for the time i n t e r v a l (tj_,T) - must be optimal with respect to the i n i t i a l conditions I* ( t ^ ) , L* ( t ^ ) , and s * ( t ^ ) . Equivalently, the t r a j e c t o r i e s for the time i n t e r v a l (tj_,T) may be viewed as i f they had been derived from the aforementioned optimal control problem defined over the period (t]_,T) and with i n i t i a l conditions I * ( t i ) , L*(t^) and s*(t±). Therefore, the optimal t r a j e c t o r i e s for an i n d i v i d u a l who optimizes over the time i n t e r v a l (tj_,T) and starts with i n i t i a l stocks of I * ( t i ) , L*(t^) and s*(t^) w i l l be i d e n t i c a l to the t r a j e c t o r i e s of an i n d i v i d u a l who has accumulated I * ( t i ) , L*(t^) and s*(t^) at t]_. (It i s , of course, assumed that both in d i v i d u a l s are of the same age and the two optimal control problems are, save for i n i t i a l t i m e , i d e n t i c a l ) . In the context of the model, the followinq two points should be treated with some caution: i ) The u t i l i t y function during the t h i r d phase i s variable with respect to education. Hence, i t should be written as U(x,£.;s) rather than as U(x, I,s). In any case, since the demand for schooling i s by assumption equal to zero during the t h i r d phase, i t i s not possible to recover a l l the parameters of an ordinary u t i l i t y function using data from the t h i r d phase. i i ) I n i t i a l stocks should be evaluated at the i n i t i a l time of the t h i r d period. The maximization problem of the consumer can be written as follows: T (V.34) maximize J e ~ 6 , :U(x, l;s)dt + e " r T I ( T ) w.r.t. x, £ subject to (V.35) L = (h-4) - yL (V.36) I = r l + f [w(L,s)(h- Jt) J - px (V.37) s ( t ) = s(t!) » 0 te [ t ! , T j (V.38) I ( t L ) = I 0 (V.39) L ( t i ) = L 0 » 0 The f i r s t order conditions for the optimal control problem are the following: (V.40) e _ < s t u x = yp (V.41) e ~ 6 t u A = yf' w + X (V.42) *u = - yr w ( t ) = e _ r T (V.43) X = - uE'w L(h-£) + yL, X(T) = 0 Quite c l e a r l y , the previous set of equations cannot be estimated d i r e c t l y because they are in continuous form. There are two ways to get around this d i f f i c u l t y . The f i r s t approach would be some sort of discrete time approximation, e.g., x = x t - x t_jL. The second approach would be to set up the model in discrete time. The former technique creates some problems of i t s own, such as introducing an approximation error i n the estimation process. The l a t t e r technique not only i s free of the aforementioned shortcoming but i t also has the extra advantage of presenting the model in a more " r e a l i s t i c " form. The optimal control of an i n d i v i d u a l who has completed his academic endeavours can be written, i n discrete time form, as follows: T - l (V.44) max Z _JL U ( x ( t ) , * ( t ) ; s ( t i ) ) + 1 . I <T) t ^ K l + j ) 1 U+r)'1' w. r. t. x, SL subject to (V.45) L(t+1) = [h-*(t)J + ( l - y ) L ( t ) (V.46) Kt+l) = ( l + r ) I ( t ) + f [w(L,s)(h-A)J - px (V.47) I ( t l ) = I Q (V.48) L ( t x ) = L Q 0 The f i r s t order conditions are the following: (V.49) ( l + 6 ) _ t U x ( t ) = y(t+l)p (V.50) ( l + 6 ) - t U A ( t ) = y(t+l)f'w(t) + X(t+1) (V.51) y(t) = u(t+l)(l+r) y(T) = 1 (l+r)T (V.52) X(t) = (l-y)X(t+l) + y( t+1) f 1 wL (t) [ h- £( t ) J , X(T)=0 The model i s not empirically tractable yet because X(t) i s an unobservable quantity. There are two ways to solve this p a r t i c u l a r problem. The f i r s t one i s to solve the optimal control problem, hence express X(t) as a function of the para-meters of the problem and time. The second one is to approximate the path of X(t). The former approach can be implemented i f and only i f one is w i l l i n g to impose very r e s t r i c t i v e assumptions on the u t i l i t y function and the wage function so that an analytic solution to the optimal control problem can be derived. There-fore, this technique w i l l not be pursued i n this research project. Before considering the second technique, i t i s necessary to dispose of some t e c h n i c a l i t i e s . F i r s t , notice that an analytic solution for u(t) is available, i . e , : (V.53) 1 u(t) = ( l + r ) c Second, after some experimentation, i t was clear that the result s were sensitive to the value of the interest rate. Moreover, information about the exact int e r e s t rate that every consumer faced i s not available. So i t was decided to estimate the difference between the interest rate and the rate of time preference. To that e f f e c t , the following approximation i s s u f f i c i e n t : (V.54) ( l + r ) t + 1 a ( l + r - 6 ) t (l+6) c 68. Using equations (V.53) and (V.54) equations (V.49) (V.50) and (V.52) can be written as (V.55) (1+r-6) tU x(t) = p (V.56) (1+r- 6) tU A ( t ) = f'w(t) + X(t+1) (V.57) X(t) = ( l - Y ) X ( t + l ) + f'w L[h-Jl(t)J where X(t) = X ( t ) ( l + r ) t + 1 that i s , X(t) i s the r e l a t i v e price of experience. Consider the following approximation for X(t): X(t) = (l-y)X(t+l) + f 'wL [h- £(t) J = ( l - y ) X ( t + l ) + c t Awt [ h-A ( t ) J AL, (V.58) =(1-Y) X (t+1) + [ c t + l w t + l - c tw TJ [h-£(t)J X(T) = 0 where c t is equal to one minus the marginal tax Inspection of the la s t set of equations reveals that the path of X(t) can be computed recursively for every i n d i v i d u a l i n the sample. -For estimation, the following f l e x i b l e functional form for the variable u t i l i t y function w i l l be employed. (V.59) U = 4 ( a 0 i x 1 / 4 s + b 0 1 x 1 / 4 s 1 / 2 + a 0 2 A 1 / 4 s + b 0 2 ^ 1 / 4 s + l / 2 a 1 1 x 1 / 2 s + l / 2 b 1 1 x 1 / 2 s 1 / 2 + a 22 I 1 / 2 s + l / 2 b 2 2 £ 1 / 2 s 1 / 2 + a 1 2 x 1 / 4 * 1 / 4 s + b 1 2 x 1 / 4 £ 1 / 4 s 1 / 2 ) One can e a s i l y show that i f a l l the parameters of the u t i l i t y function are nonnegative and at least one s t r i c t l y p o sitive then for fixed s, U is nonnegative, nondecreasing and quasiconcave i n x and £. For fixed x and Si, U i s nonneqative, nondecreasing and quasiconcave i n s. The inverse demand functions (V.55) and V.56) corresponding to (V.59) are: (V.60) ( l + r - 6 ) H a 0 i x - 3 / 4 s + b 0 i x _ 3 / 4 s 1 / 2 + a 1 1 x ~ 1 / 2 s + b 1 1 x " 1 / 2 s 1 / 2 + a 1 2 x " 3 / 4 £ 1 / 4 s + b 1 2 x - 3 / 4 £ 1 / 4 s 1 / 2 J = p (V.61) (1+r- 6)' [ a 0 2 J T 3 / 4 s + b 0 2 r " 3 / 4 s 1 / 2 + + a 2 2 J T 1 / 2 s + b 2 2 JT 1 / 2 s 1 / 2 + a 1 2 x 1 / 4 J T 3 / 4 s + b 1 2 x 1 / 4 r 3 / 4 s 1 / 2 J = f'w(t) + X(t+1) Now l e t us consider the models which w i l l be estimated. Model 1 i s defined by equations (V.60) and (V.61) and X(t) i s computed via (V.58). The stochastic structure of the model i s sp e c i f i e d as follows: Define by e x, e^ the disturbance term corresponding to equations (V.60) and (V.61) respectively. Denote by e i t the 2 X 1 vector of disturbances, i . e . , (e x,e£), for i n d i v i d u a l i of time t. It is assumed that e i t are independently and normally d i s t r i b u t e d and E ( e i t ) = 0, E ( e i t e i t ) = Z, E ( e i t e ' i t + s ) = 0 for s * 0, E ( e i t e ' j t + S ) = 0. The previous set of assumptions implies that the vector of disturb-ances i s both, s e r i a l l y and contemporaneously independently d i s -tributed, while for i n d i v i d u a l i at time t the disturbances in the two equations are j o i n t l y d i s t r i b u t e d . Maximum l i k e l i h o o d estimates of the parameters of the u t i l i t y function can be derived by maximizing the logarithm of the concentrated l i k e l i h o o d function K (V.62) L = -NK(Jtn2n+1) - K f e i l s l + Z £n(abs|j k|) T~ T~ ' ' k=l 1 1 where N is the number of equations, K the number of observations, S i s the 2 x 2 sample covariance matrix of disturbances, and abs | J k | i s the absolute value of the determinant of the 2 x 2 matrix of the derivatives of the error term with respect to the endogenous variables of the model, that i s x, I. It is assumed that the disturbances are due to unexpected (unforseeable) events which forced the i n d i v i d u a l away from the optimal path. Two points should be noted: i ) The system of demand equations should be, i n general, exactly i d e n t i f i e d . Notice that without any loss of generality we have dropped the demand for "savings" or "wealth" function. The dynamic model should be c a r e f u l l y contrasted to the s t a t i c one. In the s t a t i c model one demand equation i s redundant, because given the leisure/labor supply choice of the i n d i v i d u a l and the demand for n-1 commodities, the demand for the la s t one can be computed as a re s i d u a l . In the dynamic model, the savings (or debt payment) decision can be treated as the residual, i i ) For the model that I have presented, normalization of the parameters of the u t i l i t y function i s not required. Such a normalization has already been imposed throuqh the UYONWAR function which has been defined as Model II A maintained hypothesis of the proposed model is that the wage rate i s an endogenous variable; as a result estimates of the parameters derived from the previous model may be biased because the wage rate was assumed to be exogenous. The model is extended by estimating simultaneously the inverted demand functions and the wage equation: al(T) + b and i t has been normalized to: K T ) that i s , a was set to be equal to unity and b equal to zero. (V.60) ( l + r - 6 ) t U x = p (V.61) ( l + r - 6 ) t U £ = f'w(t) + X(t+1) (V.63) w = ag + a^L + a 2 S + a3s^/^L^/2 where X(t) i s estimated via equation (V.61) The s t o c h a s t i c s t r u c t u r e of the model i s s p e c i f i e d as f o l l o w s : Denote by e x , e ^ , e^ the d i s t u r b a n c e s c o r r e s p o n d i n g to the t h r e e e q u a t i o n s . I assume t h a t the (3x1) v e c t o r of d i s t u r b a n c e s e = ( e x , e^, ey) i s n o r m a l l y and b o t h contemporaneously and s e r i a l l y i n d e p e n d e n t l y d i s t r i b u t e d . Maximum l i k e l i h o o d e s t i m a t e s of the parameters of the u t i l i t y f u n c t i o n and the wage f u n c t i o n can be o b t a i n e d by m a x i m i z i n g the l o g a r i t h m of the c o n c e n t r a t e d l i k e l i h o o d f u n c t i o n w i t h r e s p e c t t o those parameters K L = -NK( & i 2 r r r l ) - K toils I + Z t o i ( a b s | j k ) T~ 1 k=l ' where N i s the number of e q u a t i o n s , K the number of o b s e r -v a t i o n s , S i s the 3 x 3 sample c o v a r i a n c e m a t r i x of d i s t u r b a n c e s , and abs |^k| i s t n e a b s o l u t e v a l u e of the d e t e r m i n a n t of the 3 x 3 m a t r i x of the d e r i v a t i v e s of the e r r o r term w i t h r e s p e c t t o the endogenous v a r i a b l e s of the model, t h a t i s , x t , £ t , w t . Footnotes to Chapter V See Wales and Woodland [1976J. A number of authors have used education l e v e l as an explanatory variable i n labor supply models [see Masters and Garfinkel [1974], Ham [1982 J). Hence, they have shown that education l e v e l affects d i r e c t l y the consumption/leisure choice of the i n d i v i d u a l . See Freeman [1971], Lucas [1977]. Chapter VI ESTIMATES A. Data The model i s estimated f or 1967 and 1975. The two samples are drawn from the University of Michigan Research Center's "A Panel Study of Income Dynamics". Following the methodology of Ghez and Becker [ 1975 J I use synthetic cohorts to approximate the p r o f i l e of a t y p i c a l i n d i v i d u a l . That i s , the data are cross-tabulated according to age and years of schooling; the representative i n d i v i d u a l for every cohort i s simply the "average" i n d i v i d u a l . Thus the number of observations i s equal to (years of schooling)x(time periods). The two samples are constructed as follows: i ) I include i n the samples only male, married, not self-employed i n d i v i d u a l s , 28-50 years old, whose education ranges from grade six up to college graduation. This r e s t r i c t i o n was imposed i n order to make the sample as homogeneous as possible, i i ) I exclude in d i v i d u a l s who receive any welfare payments, or any bonuses, commissions or extra income from overtime work, since such payments may affect the l a b o r / l e i s u r e choice of the i n d i v i d u a l . i i i ) I exclude in d i v i d u a l s whose wives were working, as well as those who hold a second job or were attending school. Quite c l e a r l y the behavior of such individuals cannot be captured by the model which was presented in Chapter V. iv) I also exclude households for which the taxable income of a l l dependents - other than the wife -i s qreater than $1000 (1967) or $1612 for 1975 1, since the labor supply response of the head of the household may be affected i f the contribution of the dependents to the family budget is substantial. v) F i n a l l y , I exclude individuals who were not s a t i s f i e d with the number of hours spent on the job. This r e s t r i c t i o n was imposed in order to ensure that the maximization process of a l l individuals included i n the sample corresponds as closely as possible to the t h e o r e t i c a l model. If the i n d i v i d u a l is not s a t i s f i e d with the number of hours spent on the job, that implies that i n s t i t u t i o n a l constraints are binding. Hence the proposed model i s not d i r e c t l y applicable. Before discussing the construction of the variables, i t i s important to understand the l i m i t a t i o n s of the model. i ) Sample s e l e c t i v i t y i s a problem in my samples. Unfortunately, given the large number of r e s t r i c t i o n s , i t i s next to impossible to incorporate those constraints i n the l i k e l i h o o d function. Hence,some caution i s required i n the i n t e r p r e t a t i o n of the r e s u l t s . 76. i i ) Individuals belonging to minority groups, e.g., blacks, are not excluded from the samples for two reasons. F i r s t , there i s no reason to believe that t h e i r preferences are d i f f e r e n t . Second, there i s evidence supporting the hypothesis that differences of the wage rate between whites and blacks are minimal, i f any, provided that one controls for education. 2 The variables of the model are constructed as follows: i ) Consumption. Given the r e s t r i c t i o n s on the samples, t o t a l income i s equal to labor income plus interest payments, dividends and rents. An index of annual consumption i s constructed as t o t a l income plus the renta l value of the house - i f the household owned one - plus the rental value of the car( s ) , minus income taxes, minus property taxes, minus annual mortgage payments, minus annual payments on car debts. Information about the rental value of the car(s) and annual payments on car debts was not available for the 1975 sample. i i ) Experience. The index of experience i s defined as follows: L (t+1) = (1-y)L(t)+[h-Jt(t) J , L(0) is given. Thus the index i s constructed simultaneously with the estimation of the model. Notice that there is a one-to-one map-ping between the d e f i n i t i o n of experience i n the theo-r e t i c a l model and the index employed i n my empirical 77. work. The only problem i n constructing the index i s that a measure of experience up to age 28 must be provided. Hours of experience up to age 28 can be approximated by (hours of work per year)x(years of work). Years of work can be estimated as 21 minus years of schooling, Hours of work per year can be approximated as 45x40. Notice that I assumed that the in d i v i d u a l was working only 45 weeks per year, hence this ad hoc measure might underestimate the actual experience of in d i v i d u a l s with few years of schooling. However, one might claim that most employed young adults and/or adolescents are casual workers, so that th e i r experience should be weighted by a lower weight, i i i ) The rest of the variables, such as years of schooling, gross wage rate and hours of work are taken d i r e c t l y from the data s e t . 3 A few caveats are in order here: i ) Since there i s no information on l o c a l and state taxes, 1 use as a proxy for the marginal tax rate the federal marginal income tax rate. The taxable income i s calculated as t o t a l income minus the standard deduction minus t o t a l exemptions. 4 The after tax marginal wage rate is the gross wage rate multiplied by one minus the marginal tax rate, i i ) Exact information about savings i s not available. The households were asked i f their savings "were greater 78. than , but less than,..." or " i f savings amount to as much as months' income'". For most households reported savings were less than two months' income, a substantial part of which should be transactions balances. Moreover, there is no information about contractual savings. One might be tempted to approximate t o t a l savings as c a p i t a l income divided by the interest rate. But this technique suffers from two drawbacks. F i r s t , i t ignores contractual savings and assets with accruing forms of income. Second, the actual rate of return on the savings is unknown. As a re s u l t , I had to assume that savings were equal to zero. The lack of information about savings is the most serious deficiency of the employed data set. i i i ) The rental value of automobiles for 1967 was computed as the price of automobiles times t h i r t y per cent. According to the U.S.A. Tax Guide, the allowable depreciation rate for automobiles is between t h i r t y and forty per cent dependinq on the nature of the business. Since those depreciation rates are too large for p r i v a t e l y owned automobiles, a depreciation rate of twenty-five per cent was employed. The rental price was calculated as twenty-five per cent plus f i v e per cent, the p r e v a i l i n g interest rate during the period. This approach leaves much to be desired because the 79. depreciation rate of automobiles i s , a choice variable for the household, i v ) The rental value of houses i s calculated at the value of the house times the int e r e s t rate, less c a p i t a l appreciation, plus depreciation, plus property taxes.^ It should be kept i n mind that the depreciation rate of houses is also an endogenous variable of the household, v) An index of the shadow value of experience i s constructed using equation (V.58), i . e . , X(t) « ( l - y ) X (t+1) +[Ct+l"t+l ~ c t w t J [h-A(t) J h-£(t) - T L t X(T) = 0 The following assumptions are employed in the construction of this index. (a) It is assumed that the shadow value of experience is zero by the age of 60 [X(60)=0] (Note that the index of the shadow value of experience is constructed using a larger data set -28-60 years old individuals - than the one used for estimation purposes - 28-50 years old.) (b) Whenever cohorts are missing the wage rate and hours of work are approximated by i n t e r p o l a t i o n so that the index of the shadow value of experience and the index of experience can be computed using the approximations. For example, i f the i t h cohort i s missing, the wage rate is approximated as the arithmetic mean of the wage rates of the i-1 and i+1 cohorts. v i ) The index of consumption and the after tax wage rate for the 1975 sample are expressed i n 1967 prices. They are deflated by the consumer price index - so that the two samples would be as comparable as possible. The numbers of observations per cohort for the two years are presented i n Tables VI.1 and Table VI.2. B. Results Before discussing the empirical results of the model, i t i s necessary to consider a few technical points. i ) As Heckman and MaCurdy [1980J point out, the estimates of labor supply models are not invariant with respect to the choice " t o t a l number of hours available per year" Most authors define " t o t a l number of hours per year" as equal to 8760, i . e . , 24 x 365. Given the d e f i n i t i o n of l e i s u r e as the number of hours i n a year minus hours of work, we overestimate hours of l e i s u r e by including such a c t i v i t i e s as sleep as part of l e i s u r e . I assume that approximately ten hours per day are required for sleep, commuting, preparation of meals and other home productions so that l e i s u r e i s defined as 5000 minus hours of work, i i ) The index of education is constructed as follows: (a) 1967 sample 81. 7. 6-8 grades 10. 9-11 grades 12. 12 grades (completed high school) 13. 12 grades plus non-academic t r a i n i n g 14. college, no degree 16. college, bachelor degree (A.B., B.S., etc.,) (b) 1975 sample 7. 6-8 grades 10. 9-11 grades 12. 12 grades 14. 13-15 grades (including individuals who have completed high school and have some non-academic training) 16. 16 grades Notice that the 1975 sample has been aggregated into 5 groups rather than 6. This i s due to the fact that the 1975 sample i s much smaller than the 1967 sample (see Tables VI.1 and VI.2). i i i ) For estimation purposes, the index of consumption and hours of l e i s u r e are divided by 10,000. The index of education i s divided by 2. Time-t-is measured i n increments of .5.^ 82. TABLE VI.1 NUMBER OF OBSERVATIONS PER COHORT - 1967 SAMPLE AGE EDUCATION 6-8 9-11 12 13 14-15 16 Grades Grades Grades Grades Grades Grades 28 2 - - 1 1 1 29 1 1 1 1 30 - 1 2 1 2 1 31 1 - 1 2 1 1 32 2 1 1 - 1 -33 3 1 1 - 1 -34 2 3 3 1 - 2 35 3 2 2 1 1 1 36 1 1 - - - 1 37 3 - 1 - 2 2 38 1 - 3 1 1 -39 1 4 2 - 1 1 40 1 1 4 - - 3 41 3 3 2 2 - 1 42 3 2 1 - 1 1 43 2 1 - 1 - 2 44 2 2 - - 2 -45 1 1 1 - 2 2 46 1 47 - - 2 1 2 1 48 1 2 - 1 - -49 1 2 1 - 1 -TABLE VI.1 (cont'd.) AGE 6-8 9-11 12 13 14-15 16 Grades Grades Grades Grades Grades Grades 50 2 3 - - 1 1 51 2 2 2 1 - -52 1 3 - - - 1 53 - 1 - - 1 -54 1 1 - - 1 -55 2 1 - - 1 1 56 3 1 1 - 1 -57 1 58 1 3 1 - - -59 1 1 3 - 2 -60 1 2 - - 1 -50 46 35 14 27 23 84. TABLE VI.2 NUMBER OF OBSERVATIONS PER COHORT - 1975 SAMPLE AGE EDUCATION 6-8 9-11 12 13-15 16 Grades Grades Grades Grades Grades 28 1 - 5 2 3 29 1 1 2 1 1 30 - - 3 1 -31 1 2 3 2 -32 - 3 3 1 -33 - 2 2 1 -34 - 2 1 2 2 35 - 1 4 - 1 36 2 2 1 - 1 37 - - 3 1 1 38 1 3 39 - 1 1 - -40 - 2 1 1 -41 1 42 - - 2 - -43 1 1 1 1 3 44 1 - - - 1 45 1 - 2 2 1 46 47 1 2 - 1 2 48 - 2 - 1 -49 - 2 TABLE VI.2 (cont'd.) AGE 6-8 9-11 12 13-15 16 Grades Grades Grades Grades Grades 50 2 2 1 - -51 - - - - 1 52 - 1 2 - 1 53 3 2 2 2 -54 2 - 2 - 1 55 - - 1 - -56 1 1 1 57 - - 2 - -58 - - - 1 2 59 1 1 1 - -60 1 - 1 - -21 33 47 20 21 86. v) The l i k e l i h o o d functions were maximized using a quasi-Newton algorithm due to Fletcher [1972J i n order to locate the approximate maximum point and the Berndt et^ al_ [1974J algorithm to obtain the point exactly. A l l f i r s t p a r t i a l s of the l i k e l i h o o d functions were computed a n a l y t i c a l l y , save for the one corresponding to y which was computed numerically. The convergence of the algorithm was quite time consuming for two reasons. F i r s t , the Jacobian had to be evaluated for every observation and function evaluation. Second, the indices for X and L had to be reevaluated at every i t e r a t i o n . Now l e t us consider the r e s u l t s . 1. U t i l i t y function. Inspection of the parameters of the estimated u t i l i t y functions (see Table VI.3) indicates that the functions cannot s a t i s f y globally the properties of monotonicity and quasiconcavity. However, monotonicity was s a t i s f i e d everywhere, while quasiconcavity was s a t i s f i e d for most sample points, (see Table VI.4). This result indicates that the assumption of u t i l i t y maximization cannot be rejected by the data. Quasiconcavity was tested by checking the following Hessian uxx u x l Uxl " l l TABLE VI.3 '01 01 11 12 '12 '02 02 Ml '11 '22 UTILITY FUNCTION PARAMETER ESTIMATES 1967 SAMPLE 1975 SAMPLE Model I Model I I Model I Model 0.127 -0.007 -0.014 0.026 (1.4) (1.6) (2.1) (1.9) -0.102 0.145 0.136 -0.298 (0.9) (1.1) (0.9) (1.3) 0.061 -0.171 0.278 0.008 (2.3) (2.4) (2.8) (2.9) -0.210 -0.198 0.035 -0.149 (3.9) (3.9) (3.6) (4.9) 0.214 0.561 0.124 0.233 (1.8) (2.2) (2.2) (1.5) 0.640 0.349 0.417 0.562 (1.8) (1.5) (1.3) (1.6) -1.103 -1.924 -1.633 -1.326 (3.1) (2.7) (2.7) (4.1) -0.275 0.413 -0.182 -0.491 (0.9) (0.9) (1.3) (1.2) 0.597 0.442 0.469 0.762 (2.4) (1.9) (1.9) (1.7) 1.369 2.051 1.691 1.776 (3.7) (4.6) (5.8) (4.2) *Asymptotic " t " s t a t i s t i c s i n parentheses. TABLE VI.4 TEST OF THE CURVATUEE PROPERTIES OF UTILITY FUNCTIONS* 1967 SAMPLE 1975 SAMPLE Model I Model I I Model I Model I I 87/95 88/95 56/64 57/64 ^ F r a c t i o n of sample points f o r which the u t i l i t y f u n c t i o n i s concave. 89. The function s a t i s f i e d quasiconcavity l o c a l l y i f U X X and U i i were negative and the determinant of the matrix p o s i t i v e . 2. Rate of time preference (see Table VI.5) Although the rate of time preference was not estimated, i t was shown that the difference (interest rate minues rate of time preference) i s p o s i t i v e . This supports some of the t h e o r e t i c a l predictions of the model which required the rate of time preference to be smaller than the int e r e s t rate. Furthermore, i t is reasonable to expect the interest rate to be at least equal or larger than the rate of time preference because of market forces and transaction costs. The estimated rates of time preference are consistent with results obtained by Rosen [1976J and Zabalza [1979J. Their estimates were between 0.064 and 0.0875. Given the p r e v a i l i n g interest rates during 1967 and 1975 - 5% and 8.5% respectively - my estimates are, p r a c t i c a l l y , of the same magnitude. 3. E l a s t i c i t y of labor supply (see Table VI.6). In evaluating these e l a s t i c i t i e s , caution i s i n order. These e l a s t i c i t i e s should not be compared with e l a s t i c i t i e s derived from s t a t i c Marshallian functions. The ones which are presented here can be interpreted as short-run or impact effect e l a s t i c i t i e s . They are calculated as follows: d(h-l) cw , . ( l + r - 6 ) - T U X X cw ————————— , — 9 ...... ._. ... dw (h-1) U X x u l l ~ U x 1 2 (h-1) TABLE VI.5  ESTIMATES OF THE DIFFERENCE (RATE OF INTEREST MINUS RATE OF TIME PREFERENCE) 1967 SAMPLE 1975 SAMPLE Model I Model I I Model I Model I I 0.0159 0.0172 0.0128 0.0156 (2.4) (1.9) (1.9) (1.6) ^Asymptotic " t " s t a t i s t i c s i n parentheses. TABLE VI.6 ELASTICITIES OF LABOR SUPPLY WITH RESPECT TO THE WAGE RATE* 1967 Sample Education 6-8 grades 9-11 grades 12 grades 12 grades plus non-academic t r a i n i n g College no degree College bachelor's degree Model I 0.16 0.16 0.10 0.11 0.12 0.09 Model I I 0.06 0.09 0.05 0.08 0.07 0.04 1975 Sample Education Model I Model I I 6-8 grades 0.31 0.25 9-11 grades 0.25 0.18 12 grades 0.23 0.20 13-15 grades 0.18 0.14 16 grades 0.21 0.18 ^Aggregate e l a s t i c i t i e s were c a l c u l a t e d by weighting cohort e l a s t i c i t i e s by hours of labo r supply and number of i n d i v i d u a l s per cohort. where an estimate of the derivative i s computed by employing the i m p l i c i t function theorem. The computed e l a s t i c i t i e s measure the response of an i n d i v i d u a l given a temporary change of the net wage rate. As Nagatani (1978) has shown, this response should not be confused with steady-state comparative dynamics. It can be e a s i l y shown that the "wealth effect constant" e l a s t i c i t y of l e i s u r e i s always nonpositive i f the i n d i v i d u a l i s a u t i l i t y maximizer. Moreover, this r esult can be j u s t i f i e d i n terms of t r a d i t i o n a l economic theory (see Layard and Walters [1978, p. 138J). The sign of subs t i t u t i o n effects i s always negative. Notice that i f we allow for wealth effects - t e c h n i c a l l y by allowing X(t) to vary - the sign of the e l a s t i c i t y w i l l be ambiguous. A l l e l a s t i c i t i e s presented i n Table VI.6 have the correct sign and are reasonably low. It should be noted that the e l a s t i c i t i e s reported by Smith [1977J and Ghez and Becker [1975J - who employed a completely d i f f e r e n t l i f e - c y c l e model - are of the same magnitude. Moreover, e l a s t i c i t i e s of labor supply computed via s t a t i c models (see Wales and Woodland [1976J [1977J) are quite consistent with the results of this paper. Depreciation rate of experience (see Table VI.7). The estimated depreciation rates of experience are quite consistent with the observed decline of wages around the TABLE VI.7 ESTIMATES OF THE DEPRECIATION RATE OF EXPERIENCE* 1967 Sample Model I Model I I 1975 Sample Model I Model I 0.045 (0.9) 0.037 (1.3) 0.039 (0.9) 0.047 (0.8) ^Asymptotic " t " s t a t i s t i c s i n parentheses. l a t e f o r t i e s or e a r l y f i f t i e s . Moreover, my e s t i m a t e s are q u i t e s i m i l a r t o the ones r e p o r t e d by Heckman [1976J and Rosen [1976J. However, the i n t e r p r e t a t i o n of the e s t i m a t e s r e q u i r e s some c a u t i o n . F i r s t , the s t a n d a r d e r r o r s of the e s t i m a t e s are q u i t e l a r g e , as shown by the s m a l l t a b u l a t e d a s y m p t o t i c " t " s t a t i s t i c s . Second, the e s t i m a t e s are v e r y s e n s i t i v e w i t h r e s p e c t to the s p e c i f i c a t i o n of the model and the employed d a t a s e t . Wage f u n c t i o n (see T a b l e VI.8) The e s t i m a t e d wage f u n c t i o n s are c o n s i s t e n t w i t h t h e assumptions of the t h e o r e t i c a l model i n the sense t h a t the wage r a t e i s an i n c r e a s i n g f u n c t i o n of e d u c a t i o n and e x p e r i e n c e . However, the r e s u l t s s h o u l d be i n t e r p r e t e d w i t h extreme c a u t i o n f o r the f o l l o w i n g r e a s o n s : (a) G i v e n the l a r g e number of r e s t r i c t i o n s on my samples, sample s e l e c t i v i t y i s a problem. Hence, the e s t i m a t e s may be i n c o n s i s t e n t . (b) In view of the c o m p l e x i t y of the l i k e l i h o o d f u n c t i o n , i t was not p o s s i b l e to i n t r o d u c e i n the wage f u n c t i o n s c e r t a i n v a r i a b l e s w h ich may a f f e c t the wage r a t e , such as q u a l i t y of s c h o o l i n g . ( c ) I t i s w e l l known t h a t the r e a l wage r a t e i s an i n c r e a s i n g f u n c t i o n of time because of economic growth. The l a s t statement s u g g e s t s t h a t time s h o u l d be i n t r o d u c e d as an e x p l a n a t o r y v a r i a b l e i n the wage f u n c t i o n . U n f o r t u n a t e l y , " t i m e " and e x p e r i e n c e a r e 95. TABLE VI.8 WAGE FUNCTION PARAMETER ESTIMATES*1'b 1967 Sample 1975 Sample a l a2 a 3 a o -0.002 -0.0009 (1.1) (0.6) 0.442 0.507 (6.1) (6.4) 0.013 0.021 (1.6) (2.3) 1.051 1.376 (2.6) (1.8) a The wage f u n c t i o n i s defined as w = a^ + a^L + a^s + a ^ s 2 L 2 b Asymptotic " t " s t a t i s t i c s i n parentheses highly correlated, so i t was not possible to introduce time in the model. 97. Footnotes to Chapter VI 1. The consumer price index for 1975 was 161.2 (1967 = 100) see, Economic Report of the President, U.S.G.P.O. Washington, D.C, 1979, p. 240. 2. See Ham [1977J [1982 J and Masters [1975]. There is a lot of contrary evidence (see Stolzenberg [1975aJ and Lazear [1979]). However, this evidence should be interpreted with caution because of possible misspecificationsof the employed models. Ham's analysis suggests that race acts as a proxy for underemployment. Therefore, by excluding indi v i d u a l s who were not s a t i s f i e d with the number of hours spent on the job we eliminate the spurious c o r r e l a t i o n between race and labor supply. 3. The following variables have been taken from the 1967 (1975) data sets: 5(18): House value, 8(22): Annual Mortgage Payments, 6(25): Property taxes, missing (24): Whether Mortgage Payments include property taxes, 22(missing): Annual payments on car debts, 47(32): Head's annual hours working for money, 53(44): Wife's annual hours working for money, 74(22): Head's income from wages, 76(86): Total taxable income of head and wife, 79(106): Taxable income of others i n family unit, 114(15): Family composition, 252(75): Bonus, overtime, Commissions, 117(136): Age of Head, 119(137): Sex of Head, 196(158): Employment status, 197(159): Occupation Code, 198(161): Whether work for s e l f , someone else or both, 227(218): Whether extra job, 228(219): Occupation extra job, 313(384): Education, 230(224): Whether could have worked more, 231(227): Whether wanted more work, 232(228): Whether could have worked le s s , 233(229): Whether wanted less work, 256(736): Bkt. ADC, AFDC - Head and wife, 257(737): Bkt. other welfare - Head and wife, 258(738): Bkt. Social Security - Head and wife, 259(739): Bkt. Other retirement, 260(740): Unemployment, 261(741): Alimony, 145(missing): value of a l l cars owned. Due to lack of data, I was not able to take into account the exclusion of half of long-term gains. The rental value of houses for 1967 is constructed as f o l -lows : Long-term interest rate: 5% (See Review, October 1981, Federal Reserve Bank of St. Louis), Capital appreciation: 3.3% (See 1970 HUD S t a t i s t i c a l Yearbook, U.S. Department of Housing and Urban Development, U.S.G.P.O. Washington, D.C, 1971, p. 313), Property taxes and depreciation: 4.8% (See Shelton [1968 J). During the early and mid-seventies house prices exhibited substantial short-term f l u c t u a t i o n s . As a r e s u l t , the e s t i -mated rental value is sen s i t i v e to alternative assumptions about the appreciation rate. Gillingham [1980J suggested to apply a 15-year weighted average of the appreciation rate to house values. His analysis indicates that the rental values of houses increased by 30 to 60 per cent between 1966/1967 and 1974/1975. So, a rental value of 9.5% is employed for the 1975 data set. The results are not sen s i t i v e to these rescalings. CHAPTER VII CONCLUSIONS A. Introductory Remarks What general conclusions and directions for related research can we draw from this research? What is the usefulness and significance of the proposed model as a piece of empirical research? In order to be able to answer the l a t t e r question the former one should be answered f i r s t . In the next section I present the main results of this study, those which may be viewed either as o r i g i n a l ones or as improvements of previously established r e s u l t s . In the l a s t section, I present possible extensions of the model and directions for related research. B. Results 1. Estimation of systems of intertemporal functions. The well known aphorism, "in the long run we are a l l dead" i s hardly an acceptable excuse for ignoring time as a parameter of the u t i l i t y maximization problem. After a l l , the human being i s distinguished by his a b i l i t y to plan over a long time horizon. Assuming that inter-temporal u t i l i t y maximization is the appropriate framework for modelling the behavior of the consumer and assuming that a l l prices are given exogenously, the s p e c i f i c a t i o n of the model i s a straightforward exercise. However, certain issues have to be confronted by estimating the derived demand functions. As we have seen, the demand functions depend on current prices and the shadow value of wealth. This can be expressed, for every instant of time, as the appropriately discounted shadow value of wealth evaluated at either the i n i t i a l time or at the terminal time. The main drawback to using the i n i t i a l time i s that i t r e s t r i c t s d r a s t i c a l l y the number of procedures which can be employed for estimation purposes. Indeed, the only workable estimation procedure i s to treat the shadow value of wealth as a fixed effects variable, a procedure which i n general yields inconsistent estimates. Note that we are forced to use this technique because an estimate of the shadow value of wealth evaluated at the i n i t i a l time cannot, i n general, be derived.^ The technique which was suggested and implemented i n this research provides a clear solution to the problem of estimating intertemporal demand functions. It was shown that the shadow value of wealth can be expressed as a function of the bequest function of the i n d i v i d u a l . Given a functional form for the bequest function and information about terminal wealth, 2 one can estimate simultaneously the parameters of the instantaneous u t i l i t y function and the bequest function. Furthermore, i f one i s w i l l i n g to assume a line a r bequest function that accompanies r i s k neutral behavior, then the shadow value of wealth depends on only the parameter of the bequest function. In general, there i s no reason to believe that a l l prices are exogenously given to the consumer. Consequently, empirical work based on the assumption that a l l prices are exogenous w i l l r e s u l t i n b i a s e d e s t i m a t e s as a consequence of the m i s s p e c i f i c a t i o n of the model. A l t h o u g h the s p e c i f i c a t i o n of a model t r e a t i n g the wage r a t e as an endogenous v a r i a b l e does not c r e a t e any d i f f i c u l t i e s , p e r  se , the e s t i m a t i o n of the c o r r e s p o n d i n g c o s t a t e v a r i a b l e s does r a i s e some d i f f i c u l t i e s . I t was shown t h a t i t i s p o s s i b l e to approximate the time path of the c o s t a t e v a r i a b l e s u s i n g i n f o r m a t i o n from the sample. Even i f the e m p i r i c a l r e s u l t s of t h i s paper were not s a t i s f a c t o r y , the u s e f u l n e s s of the proposed t e c h n i q u e s would not be a f f e c t e d . T h e i r u s e f u l n e s s l i e s i n the f a c t t h a t they t r a n s f o r m a p r a c t i c a l l y u n t r a c t a b l e model i n t o one t h a t can be r e a d i l y e s t i m a t e d . S p e c i f i c a t i o n of the model. The proposed model g e n e r a l i z e s most of the models which have been s u g g e s t e d so f a r , except f o r those d e v e l o p e d i n the t r a d i t i o n of the "Theory of the A l l o c a t i o n of Time". The model i n t e g r a t e s the two approaches of m o d e l l i n g human c a p i t a l , the one due to B e n - P o r a t h and the one due to Arrow. The human c a p i t a l a c c u m u l a t i o n p a r t of the model i s q u i t e s i m i l a r to the one s u g g e s t e d by M i n c e r L1974J, y e t , the u n d e r l y i n g methodoloqy of M i n c e r ' s approach i s s u b s t a n t i a l l y d i f f e r e n t . M i n c e r ' s work i s based on the assumption of income m a x i m i z a t i o n w h i l e i n the p r e s e n t paper human c a p i t a l a c c u m u l a t i o n was embedded i n t o the f a r more g e n e r a l framework of u t i l i t y m a x i m i z a t i o n . The u t i l i t y m a x i m i z a t i o n framework not o n l y i n c l u d e s income m a x i m i z a t i o n as a s p e c i a l case but a l s o a l l o w s us t o i n t e g r a t e the model w i t h i n the paradigm of mainstream n e o c l a s s i c a l economics. The major advantage of the proposed model l i e s i n i t s s p e c i f i c a t i o n . A l l v a r i a b l e s and e q u a t i o n s are d i r e c t l y o b s e r v a b l e and measurable , w hich d e p a r t s from the t r a d i t i o n of the l i t e r a t u r e to p r e s e n t such models i n terms of u n o b s e r v a b l e q u a n t i t i e s , such as human c a p i t a l . M oreover, the r e l a t i o n s h i p between the t h e o r e t i c a l model and i t s s t o c h a s t i c s t r u c t u r e i s c l e a r and the l a t t e r can be j u s t i f i e d and r a t i o n a l i z e d i n terms of the f o r m e r . F i n a l l y , i f the t h e o r e t i c a l model i s i d e n t i c a l to the e s t i m a t e d one, the t h e o r e t i c a l one p r o v i d e s us w i t h the r e q u i r e d r e s t r i c t i o n s f o r the e m p i r i c a l one. In our c a s e , the c l o s e s i m i l a r i t y between the two models p r o v i d e d us w i t h at l e a s t two u s e f u l " t e c h n i q u e s " f o r e s t i m a t i o n purposes - the use of a c o n d i t i o n a l u t i l i t y f u n c t i o n and the a l g o r i t h m f o r a p p r o x i m a t i n g the shadow v a l u e of e x p e r i e n c e . P o s s i b l e E x t e n s i o n s of the M o d e l / D i r e c t i o n s f o r R e l a t e d  R e s e a r c h . P o s s i b l e e x t e n s i o n s of the model can be c l a s s i f i e d i n t o two c a t e g o r i e s . " P e d e s t r i a n " ones s i m p l y g e n e r a l i z e the model by a p p e a l i n g to " b r u t e f o r c e " , and o t h e r e x t e n s i o n s may r e q u i r e r e c o n s i d e r a t i o n of the m a i n t a i n e d h y p o t h e s e s . We have a l r e a d y d i s c u s s e d p o s s i b l e e x t e n s i o n s which belong to the f i r s t c l a s s and we w i l l o n l y summarize them h e r e . i ) Household u t i l i t y f u n c t i o n , i i ) D i s a g g r e g a t i o n of the composite qood. i i i ) D i s a g g r e g a t i o n of the w e a l t h of the i n d i v i d u a l ( f a m i l y ) , i v ) C o r r e c t i o n f o r sample s e l e c t i v i t y , v) I n t r o d u c t i o n of o t h e r c h a r a c t e r i s t i c s , b e s i d e s e d u c a t i o n , i n the u t i l i t y f u n c t i o n , v i ) E n d o g e n e i t y of t a x e s . Now l e t us c o n s i d e r some e x t e n s i o n s of the model w h i c h appear to be p r o m i s i n g and i n t e r e s t i n g . I t seems s t r a n g e t h a t such an " e x p e n s i v e t o y " as a l i f e - c y c l e model cannot produce a " t r i v i a l " s t a t i s t i c such as the e l a s t i c i t y of the s u p p l y of l a b o r . The o n l y way to get e s t i m a t e s of the e l a s t i c i t y of the s u p p l y of l a b o r i s by s i m u l a t i n g the f u l l model. Note t h a t s i m u l a t i o n does not mean t o s u b s t i t u t e i n the model a c o u p l e of numbers and s o l v e f o r the r e s u l t i n q e s t i m a t e of the e l a s t i c i t y of the s u p p l y of l a b o r . We have to s o l v e the complete o p t i m a l c o n t r o l problem u s i n g the same time s e r i e s f o r wages, except f o r one o b s e r v a t i o n , and compare the e s t i m a t e d responses to the observed ones. U s i n g t h i s t e c h n i q u e y i e l d s an e s t i m a t e of the i n s t a n t a n e o u s e l a s t i c i t y , t h a t i s , the s h o r t run impact e f f e c t , as w e l l as a measure of the impact e f f e c t over t h e l i f e c y c l e . A l t h o u g h t h i s e x e r c i s e i s not beyond the c a p a c i t i e s of a modern h i g h speed computer, the c o m p u t a t i o n a l c o s t w i l l be s i g n i f i c a n t . Be t h a t as i t may, I t h i n k i t would be unwise to i n v e s t i n t h i s k i n d of s t u d y f o r the time b e i n g , s i n c e we do not know yet what i s the best approach to model the s u p p l y of l a b o r over the l i f e c y c l e . The r e s u l t s t h a t Heckman and MaCurdy d e r i v e d as w e l l as the r e s u l t s of t h i s model are r e a s o n a b l e , q u i t e c o n v e n t i o n a l and p r o m i s i n g f o r f u r t h e r r e s e a r c h . N e v e r t h e l e s s , more r e s e a r c h i s r e q u i r e d i n o r d e r t o get a more complete p i c t u r e . G i v e n these r e s u l t s , the e x t r a e f f o r t , and money r e q u i r e d f o r e s t i m a t i n g more c o m p l i c a t e d models can be j u s t i f i e d . I t i s w e l l known t h a t q u i t e a few consumers f a c e a number of time and c r e d i t c o n s t r a i n t s which may a f f e c t t h e i r o p t i m a l p a t h s . For example, as I have shown i n the p r e v i o u s c h a p t e r , a r a t i o n a l i n d i v i d u a l who r e t u r n s to s c h o o l most p r o b a b l y does so because h i s / h e r c h o i c e s e t was c o n s t r a i n e d when he/she was younger. T h e r e f o r e , i t would be i n t e r e s t i n g to i n t r o d u c e e x p l i c i t l y t h o s e c o n s t r a i n t s i n the o p t i m i z a t i o n problem. T e c h n i c a l l y s p e a k i n g , the s a i d g e n e r a l i z a t i o n of the model w i l l n o t , i n g e n e r a l , c r e a t e any i n s u r m o u n t a b l e d i f f i c u l t i e s . U n f o r t u n a t e l y , such an e x t e n s i o n of the model seems to be q u i t e d i f f i c u l t i f not i m p o s s i b l e . The r e a s o n i s the l a c k of d a t a . One can o n l y hope t h a t s t a t i s t i c a l a g e n c i e s w i l l u n d e r s t a n d the i m p o r t a n c e and u s e f u l n e s s of such s t u d i e s and w i l l g e n e r a t e a p p r o p r i a t e d a t a s e t s . 3 The economics of uncertainty and replanning seem to be other f r u i t f u l areas of research. The maintained hypothesis of perfect knowledge of the future is limited i n scope and realism. A most reasonable conjecture is the one suggested by Nagatani [1972J that the consumer adjusts every "year" his/her plan according to his/her expected wealth. There are two possible ways to model such a s i t u a t i o n . Note that the demand for l e i s u r e and the composite good depends on current prices and the shadow value of wealth, which, i n turn, depends on the parameter of the bequest function and the terminal wealth. Therefore, i t s u f f i c e s to substitute the value of expected wealth into the system of equations. Since information about expected wealth i s not available, i t must be estimated as a fixed effects v a r iable. Furthermore, i f the prices that w i l l p r e v a i l during a period are not precisely known at the beginning of the period, we may assume that the consumer maximizes his/her u t i l i t y subject to a vector of planning prices which w i l l be functions of the prices that prevailed i n the previous period. The equations for the planning prices can be estimated simultaneously with the demand functions. My only qualm about these models i s that they are ad_ hoc. The formulation of expectations about wealth and prices i s treated as a black box. Even i f the estimates are acceptable from a s t a t i s t i c a l point of view, we cannot assess whether that i s due to the robustness of the model or, simply, the time trend of the employed data set. If the planning equations are generated within the model - for example, the wage rate - one cannot raise any objections to their usefulness as ana l y t i c and empirical tools. The model that I proposed can easi l y handle the above mentioned technique for estimating the value of expected wealth. It su f f i c e s to define the bequest function as, say, a quadratic one, hence the shadow value of wealth w i l l be a line a r function of wealth. Expected wealth can be estimated as a fixed effects variable. The second approach to uncertainty i s to attempt to model e x p l i c i t l y the formation of expectations. The advantage of t h i s approach i s that i t subsumes a number of other problems that may arise because of uncertainty, such as unemployment and the unknown returns to investment. Unfortunately, this area of research i s s t i l l i n i t s infancy. We conclude by examining two c r u c i a l question; ( i ) are there any l i m i t a t i o n s to the model, and ( i i ) i f so, w i l l the empirical model i d e n t i f y such situations? The answers to both questions are affirmative. ( i ) If any of the assumptions of the model i s vi o l a t e d , then the model i s not d i r e c t l y a p p l i c a b l e . 4 ( i i ) By construction, the empirical model i s , p r a c t i c a l l y , an isomorphic mapping of the t h e o r e t i c a l one. Consequently, i f any of the assumptions of the t h e o r e t i c a l model are v i o l a t e d , the empirical model w i l l detect them. In other words, the possible li m i t a t i o n s of the t h e o r e t i c a l moddel is not a serious issue for the simple reason that a l l i t s assumptions are refutable. Thus, i f the t h e o r e t i c a l model can not pass the test that w i l l provide supportive empirical evidence, this should prompt the researcher to consider alternative structures that might not otherwise have been considered. A caveat i s i n order here. It i s , of course, assumed that the researcher w i l l not employ defensive strategies i n testing for evading refutations of hypotheses. Footnotes to Chapter VII Such an estimate can be derived i f an analytic solution of the optimal control problem is available. As we have already seen, terminal time can be interpreted as the date of retirement and the bequest function as the u t i l i t y y i e l d of net wealth at retirement. Such data sets w i l l include information about savings, contractual savings, assets with accruing forms of income and possible credit constraints. As with any other model, the present model has some possible l i m i t a t i o n s , such as: uncertainty about future prices, wages and employment opportunities, myopic expectations (Strotz [1956]), unanticipated changes of the u t i l i t y function and so on. However, as we have already discussed, the model can in p r i n c i p l e be extended to handle such s i t u a t i o n s . BIBLIOGRAPHY A l b r e c h t , J.W. [1974], "The Use of E d u c a t i o n a l I n f o r m a t i o n by Employers". Paper p r e s e n t e d at the W i n t e r , 1974, E c o n o m e t r i c S o c i e t y M e e t i n g s . Amemiya, T. [1973J, " R e g r e s s i o n A n a l y s i s when the Dependent V a r i a b l e i s T r u n c a t e d Normal", E c o n o m e t r i c a , 41, 997-1016. Arrow, K . J . [1962], "The Economic I m p l i c a t i o n s of L e a r n i n g by D o i n g " , Review of Economic S t u d i e s , 29, 155-173. A s h e n f e l t e r , 0. & J . J . Heckman [1974], "The E s t i m a t i o n of Income and S u b s t i t u t i o n E f f e c t s i n a Model of F a m i l y Labor S u p p l y " , E c o n o m e t r i c a , 42, 73-85. B a r z e l , Y. [ 1973], "The D e t e r m i n a t i o n of D a i l y Hours and Wages", Q u a r t e r l y J o u r n a l of Economics, 87, 220-238. B a r t h , P.S. [1967], "A C r o s s - S e c t i o n a l A n a l y s i s of Labor F o r c e P a r t i c i p a t i o n Rates i n M i c h i g a n " , I n d u s t r i a l and  Labor R e l a t i o n s Review, 20, 234-249. B e c k e r , G.S. [1964], Human C a p i t a l , New York: Columbia U n i v e r -s i t y P r e s s . B e c k e r , G.S. [1965], " A Theory of the A l l o c a t i o n of Time", Economic J o u r n a l , 75, 493-517. B e c k e r , G.S. [1967], Human C a p i t a l and the P e r s o n a l D i s t r i b u -t i o n of Income: An A n a l y i c a l Approach, Ann A r b o r , M i c h i g a n : I n s i t i t u t e of P u b l i o c A d m i n i s t r a t i o n . B e c k e r , G.S. & B. C h i s w i c k [1966J, " E d u c a t i o n and the D i s t r i b u -t i o n of E a r n i n g s " , American Economic Review, 56, 358-369. B e c k e r , G.S. & G.H. Lewis [1973J, "On the I n t e r a c t i o n between Q u a n t i t y and Q u a l i t y of C h i l d r e n " , J o u r n a l of  P o l i t i c a l Economy, 82, S279-S288. B e c k e r , G.S. & N. Tomes [1967J, " C h i l d Endowments and the Q u a n t i t y and Q u a l i t y of C h i l d r e n " , J o u r n a l of  P o l i t i c a l Economy, 84, 5143-5162. B e l l m a n , R. [1957], Dynamic Programming, P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s . B e n - P o r a t h , Y. [ 1 9 6 7 ] , "The P r o d u c t i o n of Human C a p i t a l and the L i f e C y c l e of E a r n i n g s " , J o u r n a l of P o l i t i c a l  Economy, 75, 352-365. Berg, I. L1969 J, Education and Jobs: The Great T r a i n i n g  Robbery, Boston: Beacon. Berndt, E . K . , B . H . H a l l , R . E . H a l l & T . A . Hausman, [1974J, "Estimation and Inference in Nonlinear S t r u c t u r a l Models", Annals of Economic and S o c i a l Measurement, 4, 653-665. Blaug, M. [ 19 72 J , "The C o r r e l a t i o n Between Education and Earnings , What Does i t S ign i fy?" , Higher Educat ion, 1, 53-76. Blaug, M. [1976J, "Human C a p i t a l Theory: A S l i g h t l y Jaundiced Survey", Journal of Economic L i t e r a t u r e , 24, 827-855. B l i n d e r , A . S . & Y. Weiss [1976J, "Human C a p i t a l and Labor Supply", Journal of P o l i t i c a l Economy, 84, 449-472. Boskin, M.S. 11973J, "The Economics of Labor Supply", in Income  Maintenance and Labor Supply, G.G. Cain & H.W. Watts (eds.) Chicago: Markham. Boskin, J . M . [1979J, "The Ef fec t s of Government Taxes and Expenditures on Female Labor", American Economic  Review, 64, 251-256. Bowen, W.G. & A . T . Finegar [1969J, The Economics of Labor Force  P a r t i c i p a t i o n , Pr inceton: Princeton Univers i ty Press . Bowles, S. [1973J, "Understanding Unequal Economic Opportunity", American Economic Review, 63, 346-356. Bowles, S. & H. G i n t i s [1975J, "The Problem with Human C a p i t a l Theory - A Marxian C r i t i q u e " , American Economic Review, 65, 74-82. B u r t l e s s , G. & J . A . Hausman [1978J, "The Effect of Taxation on Labor Supply: Evaluat ion of the Gary Income Maintenance Experiment", Journal of P o l i t i c a l Economy, 86, 1103-1130. Chiswick, B.R. [1977], "Sons of Immigrants: Are they at an Earnings Disadvantage?", American Economic Review, Papers and Proceedings, 67, 376-380. Chiswick, B.R. [1978], "The Ef fec t of Americanizat ion on the Earnings of Foreign-born Men", Journal of P o l i t i c a l  Economy, 86, 909-921. Cogan, J . [1980], Labor Supply with Time and Money Costs of P a r t i c i p a t i o n " i n Female Labor Supply: Theory and  Est imat ion , J . P . Smith ( ed . ) , Princeton: Princeton Univers i ty Press . Cogan, J . [1981], " F i x e d C o s t s and Labour S u p p l y " , E c o n o m e t r i c a , 49, 945-963. D i e w e r t , W.E. [1971J, "Choice on Labour Markets and the Theory of the A l l o c a t i o n of Time", Ottawa: Department of Manpower and I m m i g r a t i o n . D i e w e r t , W.E. [1974], " A p p l i c a t i o n s of D u a l i t y Theory", i n F r o n t i e r s of Q u a n t i t a t i v e Economics, Volume I I , M.D. I n t r i l i g a t o r and D.A. K e n d r i c k , Amsterdam: N o r t h H o l l a n d P u b l i s h i n g Company. Duncan, O.D. & O.L. Featherman & B. Duncan [1972], S o c i o e c o n o m i c  Background and Achievement, London: Seminar P r e s s . E a s t e r l i n , R.A. [ 1 9 7 5 ] , "The Economics and S o c i o l o g y of F e r t i l i t y : A S y n t h e s i s " , i n H i s t o r i c a l S t u d i e s of  Changing F e r t i l i t y , C. T i l l y ( e d . ) , P r i n c e t o n . P r i n c e t o n U n i v e r s i t y P r e s s . F a r l e y , F. [1977J, "Trends i n R a c i a l I n e q u a l i t i e s : Have the Gains of the 1960's D i s a p p e a r e d i n the 1970's?" A m e r i c a n S o c i o l o g i c a l Review, 42, 189-208. Featherman, D.S. & R.M. Hauser [1976], " S e x u a l I n e q u a l i t i e s and Socioeconomic Achievement i n the U.S., 1962-1973, American S o c i o l o g i c a l Review, 41, 462-483. F l a n a g a n . R.J. [1974], "Labor F o r c e , E x p e r i e n c e , J o b - T u r n o v e r , and R a c i a l Wage D i f f e r e n t i a l s " , Review of Economics  and S t a t i s t i c s , 21, 308-312. F l e t c h e r , R. [1972J, " F o r t r a n S u b r o u t i n e s f o r M i n i m i z a t i o n by Quasi-Newton Methods", Report R712S AERE, H a r w e l l , E n g l a n d . Freeman, R. [1971J, The Market f o r C o l l e g e - T r a i n e d Manpower, Cambridge: H a r v a r d U n i v e r s i t y P r e s s . Ghez, G. & G. Becker [1975], "The A l l o c a t i o n of Time and Goods  over the L i f e C y c l e " , New York: Columbia U n i v e r s i t y P r e s s . G i l l i n g h a m , R. [1980], " E s t i m a t i n g the User Cost of Owner-Occupied H o u s i n g " , Monthly Labour Review, 103, 31-35. G o c k e l , G.L. [1969], "Income and R e l i g i o u s A f f i l i a t i o n : A R e g r e s s i o n A n a l y s i s " , A m erican J o u r n a l of S o c i o l o g y , 74, 632-646. G o l d b e r g e r , A.S. [1975], " L i n e a r R e g r e s s i o n i n T r u n c a t e d Samples", M a n u s c r i p t , S o c i a l Systems R e s e a r c h I n s t i t u t e , U n i v e r s i t y of W i s c o n s i n . Goodman, J.D. [1979J, "The Economic R e t u r n s of E d u c a t i o n : An Assessment of A l t e r n a t i v e Models", S o c i a l S c i e n c e  Q u a r t e r l y , 60, 269-283. G r e e l e y , A.M. [1976J, E t h n i c i t y , Denomination and I n e q u a l i t y , Sage R e s e a r c h Papers i n the S o c i a l S c i e n c e s , S e r i e s 90-029, B e v e r l y H i l l s : Sage P u b l i c a t i o n s . G r i l i c h e s , Z. [1977], " E s t i m a t i n g the Returns to S c h o o l i n q : Some E c o n o m e t r i c Problems", E c o n o m e t r i c a , 45, 1-22. G r i l i c h e s , Z. & W.M, Mason (197 2 J , " E d u c a t i o n , Income and A b i l i t y " , J o u r n a l of P o l i t i c a l Economy, 80, 574-5103. G r i l i c h e s , Z. & W.M. Mason [1972], " E d u c a t i o n , Income, and A b i l i t y " , i n Investment i n E d u c a t i o n : The E q u i t y - E f f i c i e n c y Quandary, T.W. S c h u l t z ( e d . ) , C h i c a g o : Chicago U n i v e r s i t y P r e s s . H a l l , E.R. [1973], "Wages, Income and Hours of Work i n the U.S. Labor F o r c e " , i n Income Maintenance and Labour  S u p p l y , G.G. C a i n and H.W. Watts (eds:) Chicago Markham. Ham, J.C. [1977], " R a t i o n i n g and the Supply of Labor: An E c o n o m e t r i c Approach", Working Paper, P r i n c e t o n U n i v e r s i t y . Ham, J.C. [ 1 9 8 2 j , " E s t i m a t i o n of a Labour Supply Model w i t h C e n s o r i n g Due to Unemployment and Underemployment", Review of Economic S t u d i e s , 49, 335-354. Hanoch, G. [1980], "Hours and Weeks i n the Theory of Labor S u p p l y " , i'n Female Labor Supply: Theory and  E s t i m a t i o n , J.P. S m i t h , (ed.) P r i n c e t o n , P r i n c e t o n U n i v e r s i t y P r e s s . H a r r i s o n , B. [1972aJ, " E d u c a t i o n and Underemployment i n the Urban G h e t t o " , A m e r i c a n Economic Review, 62, 796-812. H a r r i s o n , B, [1972bJ, E d u c a t i o n , T r a i n i n g and the Urban G h e t t o , B a l t i m o r e : Johns Hopkins U n i v e r s i t y P r e s s . House, J.C. [1975], " A b i l i t y and S c h o o l i n g as De t e r m i n a n t s of L i f e t i m e E a r n i n g s , or I f You're So Smart, Why A r e n ' t You R i c h ? " , i n E d u c a t i o n , Income and Human B e h a v i o r , F.T. J u s t e r ( e d . ) , New York: M c G r a w - H i l l . Heckman, J.J. [1974], " L i f e C y c l e Consumption and Labor Supply: An E x p l a n a t i o n of The R e l a t i o n s h i p between Income and Consumption over the L i f e C y c l e " , American Economic  Review, 64, 188-194. Heckman,J.J. [ 1 9 7 6 j , "A L i f e - C y c l e Model of E a r n i n g s , L e a r n i n g and Consumption", J o u r n a l of P o l i t i c a l Economy, 84 , S11-S44. Heckman, J . J . & T. MaCurdy [1980 J, " A L i f e C y c l e Model of Female Labor S u p p l y " , Review of Economic S t u d i e s , 47, 42-74. L a n c a s t e r , K.J [ 1971J, Consumer Demand: A New Approach, New York: Columbia U n n i v e r s i t y P r e s s . L a y a r d , P.R.G., J.D. Sargan, M.E. Argan, and D.J. Jones [1971], Q u a l i f i e d Manpower and Economic Per f o r m a n c e , London: Pen g u i n P r e s s . L a y a r d , P.R.G., & G. P s a c h a r o p o u l o s [1974], "The S c r e e n i n g H y p o t h e s i s and the R e t u r n s to E d u c a t i o n " , J o u r n a l  of P o l i t i c a l Economy, 82, 98S-98. L a y a r d , P.R.G. & A.A. W a l t e r s [1978], M i c r o e c o n o m i c Theory, New York: M c G r a w - H i l l Book Co. Lau, L . J . [1973], " E c o n o m e t r i c s of M o n o t o n i c i t y , C o n v e x i t y , and Q u a s i c o n v e x i t y " , Working Paper, S t a n f o r d U n i v e r s i t y . Lau, L . J . [1974J, "Comments" i n F r o n t i e r s of Q u a n t i t a t i v e  Economics, Volume I I , M.D. I n t r i l i g a t o r and D.A. K e n d r i c k , (eds.) Amsterdam, N o r t h - H o l l a n d P u b l i s h i n g Company. Lau, L . J . , W.L. L i n and P.A. Y o t o p o u l o s [1978], "The L i n e a r L o g a r i t h m i c E x p e n d i t u r e System: An A p p l i c a t i o n t o Consumption L e i s u r e C h o i c e " , E c o n o m e t r i c a , 46, 843-868. L a z e a r , E. [1979], "The N a r r o w i n g of B l a c k - W h i t e Wage D i f f e r e n t i a l s i s I l l u s o r y , " American Economic Review, 69, 553-564. L e i b e n s t e i n , M. [1974J, "An I n t e r p r e t a t i o n of the Economic Theory of F e r t i l i t y : P r o m i s i n g P a t h or B l i n d A l l e y " , J o u r n a l of Economic L i t e r a t u r e , 12, 457-479. L e u t h o l d , J.H. [1968], "An E m p i r i c a l Study of Formula Income T r a n s f e r s and the Work D e c i s i o n of the P o o r " , J o u r n a l  of Human R e s o u r c e s , 3, 312-323. L u c a s , R.E.B. [1977], "Hedonic Wage E q u a t i o n s and P s y c h i c Wages i n the R e t u r n s to S c h o o l i n g , " American Economic Review, 67, 549-558. MaCurdy, T. [1978], Two E s s a y s on the L i f e C y c l e , u n p u b l i s h e d , Ph.D. t h e s i s , U n i v e r s i t y of C h i c a g o . MaCurdy, T. [1981J, "An E m p i r i c a l Model of Labour Supply i n a L i f e - C y c l e S e t t i n g " , J o u r n a l of P o l i t i c a l Economy, 89, 1059-1085. M a s t e r s , S. [1975J, B l a c k - W h i t e Income D i f f e r e n t i a l s : E m p i r i c a l  S t u d i e s and P o l i c y I m p l i c a t i o n s , New York: Academic P r e s s . M a s t e r s , S. & I . G a r f i n k e l L19 7 4 j , E s t i m a t i n g the Labour Supply E f f e c t s of Income Maintenance A l t e r n a t i v e s , New York: k Academic P r e s s . May, D.J, & D.M. Heer [1968J, "Son S u r v i v o r s h i p M o t i v a t i o n and F a m i l y L i f e i n I n d i a : A Computer S i m u l a t i o n " , P o p u l a t i o n S t u d i e s , 22, 199-210. M i c h a e l , R.T. [1974J, " E d u c a t i o n and the D e r i v e d Demand f o r C h i l d r e n " , i n Economics of the F a m i l y : M a r r i a g e , C h i l d r e n and Human C a p i t a l , T.W. S c h u l t z ( e d . ) , C h i cago U n i v e r s i t y of Chicago P r e s s . M i n c e r , J . [1962J, "Labor F o r c e P a r t i c i p a t i o n of M a r r i e d Women: A Study of Labor S u p p l y " , i n A s p e c t s of Labor  Economics, N a t i o n a l Bureau of Economic R e s e a r c h , P r i n c e t o n , P r i n c e t o n U n i v e r s i t y P r e s s . M i n c e r , J . [1974J, S c h o o l i n g , E x p e r i e n c e and E a r n i n g s , New York, Columbia U n i v e r s i t y P r e s s . M i n c e r , J . & J . P o l a c h e k [1974], " F a m i l y Investment i n Human C a p i t a l : E a r n i n g s of Women", The Economics of the  F a m i l y , T.W. S c h u l t z ( e d . ) , C h i c a g o : U n i v e r s i t y of Chicago P r e s s . M o d i g l i a n i , F. and R. Brumberg [1954J, " U t i l i t y A n a l y s i s and the Consumption F u n c t i o n : An I n t e r p r e t a t i o n of C r o s s S e c t i o n D a t a " , i n Po s t K e y n e s i a n Economics, K. K u r i h a r a ( e d . ) , New B r u n s w i c k , N.J.: Ru t g e r s U n i v e r s i t y P r e s s . M o r g e n s t e r n , R.D. [1973], " D i r e c t and I n d i r e c t E f f e c t s on E a r n i n g s of S c h o o l i n g and Socio-Economic Background", Review of Economics and S t a t i s t i c s , 55, 22S-233. N a g a t a n i , K. [1972J, " L i f e - C y c l e S a v i n g : Theory and F a c t " , American Economic Review, 62, 344-3S3. N a g a t a n i , K. [1978], "Toward a G l o b a l A n a l y s i s of Macrodynamics", D i s c u s s i o n Paper, U n i v e r s i t y of B r i t i s h Columbia. R o b b i n s , L. [1930], "On the E l a s t i c i t y of Demand f o r Income i n Terms of E f f o r t , Economica, 10, 123-129. Rosen, S. [ 1976 J, "Taxes i n a Labor Supply Model w i t h J o i n t Wage - Hours D e t e r m i n a t i o n " , E c o n o m e t r i c a , 46, 485-507. Rosen, S. [1976J, "A Theory of L i f e E a r n i n g s " , J o u r n a l of  P o l i t i c a l Economy, 86, 545-568. S h e l t o n , J . [1968J, "The Cost of R e n t i n g Versus Owning a Home", Land Economics, 44, 59-72. S m i t h , A. [1961J, The Wea l t h of N a t i o n s , London: Methuen. S m i t h , J.P. [1977J, " F a m i l y Labor Supply over the L i f e C y c l e " , E x p l o r a t i o n s i n Economic R e s e a r c h , 4, 205-276. S t o l z e n b e r g , R. [1975aJ, " B l a c k / W h i t e D i f f e r e n c e s i n O c c u p a t i o n , E d u c a t i o n and Wages", American J o u r n a l of S o c i o l o g y , 81, 299-323. S t o l z e n b e r g , R. [1975bJ, " O c c u p a t i o n s , Labor Markets and the P r o c e s s of Wage A t t a i n m e n t " , American S o c i o l o g i c a l  Review, 11, 447-461. S t r o t z , R. [1956J, "Myopia and I n c o n s i s t e n c y i n Dynamic U t i l i t y M a x i m i z a t i o n , " Review of Economic S t u d i e s , 23, 165-180. Taubman, P.J. and T. Wales [1974], H i g h e r E d u c a t i o n and E a r n i n g s , New York: M c G r a w - H i l l Book Co. T i n b e r g e n , J . [1956], "On the Theory of Income D i s t r i b u t i o n " , W e l t w i r t s c h a f l i c h e r A r c h i v , 77, 155-173. Wales, T. [1973], " E s t i m a t i o n of a Labor Supply Curve f o r S e l f - E m p l o y e d B u s i n e s s P r o p r i e t o r s " , I n t e r n a t i o n a l  Economic Review, 14, 69-80. Wales, T.J. and A.D. Woodland [1976], " E s t i m a t e s of Household U t i l i t y F u n c t i o n s and Labor Supply Response", I n t e r -n a t i o n a l Economic Review, 17, 397-410. Wales, T.J. & A.D. Woodland [1977J, " E s t i m a t i o n of the A l l o c a t i o n of Time f o r Work, L e i s u r e and Housework", E c o n o m e t r i c a , 45, 115-132. Wales, T.J. & A.D. Woodland 11979], "Labour Supply and P r o g r e s s i v e Taxes", Review of Economic S t u d i e s , 46, 83-95. Wales, T.J. & A.D. Woodland [1981], " Sample S e l e c t i v i t y and the E s t i m a t i o n of Labour Supply F u n c t i o n s " , I n t e r n a t i o n a l  Economic Review, 21, 437-468. 116. Weiss, R.D. [1970J, "The Effe c t of Education on the Earnings of Blacks and Whites", Review of Economics and S t a t i s t i c s , 52, 150-159. Weiss, Y. [1972J, " Learning by Doing and Occupational S p e c i a l i -zation", Economic Journal, 82, 1293-1315. Wise, D.A. [ 19 7 5 J, "Academic Achievement and Job Performance", American Economic Review, 65, 350-366. Zabalza, A, [1979J, "A Note on the Estimation of Subjective Rates of Discount from Labour Supply Functions" Economica, 46, 197-202. 

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