Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A model of the joint determination of labor force participation and fertility decisions of married women Mazany, Robin Leigh 1982

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1982_A1 M39.pdf [ 7.88MB ]
Metadata
JSON: 831-1.0095608.json
JSON-LD: 831-1.0095608-ld.json
RDF/XML (Pretty): 831-1.0095608-rdf.xml
RDF/JSON: 831-1.0095608-rdf.json
Turtle: 831-1.0095608-turtle.txt
N-Triples: 831-1.0095608-rdf-ntriples.txt
Original Record: 831-1.0095608-source.json
Full Text
831-1.0095608-fulltext.txt
Citation
831-1.0095608.ris

Full Text

A MODEL OF THE JOINT DETERMINATION OF LABOR FORCE PARTICIPATION AND FERTILITY DECISIONS OF MARRIED WOMEN by ROBIN LEIGH MAZANY B.S.F.S., Georgetown U n i v e r s i t y , 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Economics) We ac c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Ja n u a r y 1982 © Robin L e i g h Mazany, 1 982 I n 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 t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 s t u d y . I f u r t h e r a gree 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d 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 f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Economics The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date December 30, 1981 DE-6 (2/79) i i A b s t r a c t In t h i s d i s s e r t a t i o n , we examine the economic d e t e r m i n a n t s of the l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y b e h a v i o r of m a r r i e d women u s i n g a s i m u l t a n e o u s e q u a t i o n s model. The model t a k e s i n t o account both the t r u n c a t i o n of hours of work at z e r o and the dichotomous n a t u r e of the d e c i s i o n of whether t o have a c h i l d . We e s t i m a t e the model u s i n g maximum l i k e l i h o o d methods. The d a t a used a r e from the M i c h i g a n P a n e l Survey of Income Dynamics. We f i n d the r e s u l t s from the s i m u l t a n e o u s model t o be q u i t e d i f f e r e n t from the r e s u l t s o b t a i n e d u s i n g s i n g l e e q u a t i o n methods. Our r e s u l t s thus suggest a s i m u l t a n e o u s framework i s the most a p p r o p r i a t e f o r the e x a m i n a t i o n of l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y d e c i s i o n s . We run s i m u l a t i o n s on two d i f f e r e n t t a x p o l i c i e s which might be e x p e c t e d t o a f f e c t both the l a b o r s u p p l y d e c i s i o n and the f e r t i l i t y d e c i s i o n . We f i n d the t a x p o l i c i e s do not have a l a r g e e f f e c t on e i t h e r d e c i s i o n , a l t h o u g h the e f f e c t on hours worked i s s t r o n g e r than the e f f e c t on f e r t i l i t y . We a l s o f i n d the e f f e c t s of the p o l i c i e s t o be s t r o n g e r i f t h e r e are a l r e a d y two c h i l d r e n i n the f a m i l y . i i i Table of C o n t e n t s A b s t r a c t i i L i s t of T a b l e s v i Acknowledgements v i i i Chapter One: I n t r o d u c t i o n 1 Chapter Two: T h e o r i e s of l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y d e c i s i o n s 4 I n t r o d u c t i o n 4 T h e o r i e s of Labor s u p p l y 5 I n t r o d u c t i o n 5 Labor s u p p l y as an e x t e n s i o n of consumer demand t h e o r y . 7 Labor f o r c e p a r t i c i p a t i o n and hours of work 10 Dynamic models of l a b o r s u p p l y 15 Economic t h e o r i e s of f e r t i l i t y . 18, I n t r o d u c t i o n 18 L e i b e n s t e i n and E a s t e r l i n . . 19 The h o u s e h o l d p r o d u c t i o n t h e o r y of f e r t i l i t y 21 The j o i n t e s t i m a t i o n of f e r t i l i t y and 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 s 28 Household p r o d u c t i o n models 28 S e q u e n t i a l models. 29 Summary 32 Chapter- Three: A T h e o r e t i c a l model of l a b o r s u p p l y and f e r t i l i t y d e c i s i o n s 34 I n t r o d u c t i o n 34 A t h e o r e t i c a l model of l a b o r s u p p l y and. f e r t i l i t y 36 An e c o n o m e t r i c model of l a b o r s u p p l y and f e r t i l i t y 44 S p e c i f i c a t i o n of the model 44 i v D e r i v a t i o n of the l i k e l i h o o d f u n c t i o n 47 I d e n t i f i c a t i o n of the model 51 Summary 54 Chapter Four: E s t i m a t i o n of the model I : A comparison of d i f f e r e n t e s t i m a t i o n t e c h n i q u e s 56 I n t r o d u c t i o n 56 The d a t a 57 S p e c i f i c a t i o n of the e q u a t i o n s 65 The wage e q u a t i o n 65 The hours e q u a t i o n 66 The a d d i t i o n a l c h i l d e q u a t i o n 68 A comparison of e s t i m a t e s 74 Maximum l i k e l i h o o d e s t i m a t i o n 74 A l t e r n a t i v e methods of e s t i m a t i o n 77 The wage e q u a t i o n 79 The hours e q u a t i o n 82 The a d d i t i o n a l c h i l d e q u a t i o n 85 C o n s i s t e n t e s t i m a t e s 88 Summary 92 Chapter F i v e : E s t i m a t i o n of the model I I : P o l i c y i m p l i c a t i o n s 94 I n t r o d u c t i o n 94 G e n e r a l p o l i c y i m p l i c a t i o n s 95 C h i l d c a r e t a x p o l i c i e s 105 1975 and 1976 c h i l d c a r e t a x p o l i c i e s 105 E f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d 109 E f f e c t on the hours worked 128 Summary 144 Chapter S i x : C o n c l u s i o n 148 Summary 148 F u t u r e r e s e a r c h 150 In c o n c l u s i o n 1 52 R e f e r e n c e s 154 Appendix t o Chapter Three 163 Appendix t o Chapter Four 170 v i L i s t of T a b l e s T a b l e 4.1a: D e s c r i p t i v e s t a t i s t i c s f o r the complete sample.. 62 Table 4.1b: D e s c r i p t i v e s t a t i s t i c s f o r workers 63 T a b l e 4.1c: D e s c r i p t i v e s t a t i s t i c s f o r non-workers 64 T a b l e 4.2: C h i - s q u a r e t e s t of independence of s u b - p o p u l a t i o n s . . 73 Table 4.3: L i k e l i h o o d r a t i o t e s t s on the c o v a r i a n c e s of the d i s t u r b a n c e terms 76 T a b l e 4.4: Wage e q u a t i o n e s t i m a t e s 81 T a b l e 4.5: Hours e q u a t i o n e s t i m a t e s . .84 T a b l e 4.6: A d d i t i o n a l c h i l d e q u a t i o n e s t i m a t e s 87 T a b l e 4.7: C o n s i s t e n t e s t i m a t e s 91 T a b l e 5.1: Maximum l i k e l i h o o d e s t i m a t e s 98 T a b l e 5.2: E l a s t i c i t i e s and p r o b a b i l i t i e s 103 T a b l e 5.3: P r o b a b i l i t y of h a v i n g a c h i l d - ML e s t i m a t e s : Base p o l i c y 116 T a b l e 5.4: P r o b a b i l i t y of h a v i n g a c h i l d - ML e s t i m a t e s : I n c r e a s e d p e r s o n a l d e d u c t i o n s 118 T a b l e 5.5: P r o b a b i l i t y of h a v i n g a c h i l d - ML e s t i m a t e s : Refundable tax c r e d i t 120 T a b l e 5.6: P r o b a b i l i t y of h a v i n g a c h i l d - OLS e s t i m a t e s : Base p o l i c y 122 T a b l e 5.7: P r o b a b i l i t y of h a v i n g a c h i l d - OLS e s t i m a t e s : I n c r e a s e d p e r s o n a l d e d u c t i o n s 124 T a b l e 5.8: P r o b a b i l i t y of h a v i n g a c h i l d - OLS e s t i m a t e s : Refundable tax c r e d i t 126 T a b l e 5.9: Hours of work (thousands) - ML e s t i m a t e s : Base p o l i c y 132 T a b l e 5.10: Hours of work (thousands) - ML e s t i m a t e s : I n c r e a s e d p e r s o n a l d e d u c t i o n s 134 T a b l e 5.11: Hours of work (thousands) - ML e s t i m a t e s : Refundable t a x c r e d i t . 136 v i i Table 5.12: Hours of work (thousands) - OLS e s t i m a t e s : Base p o l i c y 138 T a b l e 5.13: Hours of work (thousands) - OLS e s t i m a t e s : I n c r e a s e d p e r s o n a l d e d u c t i o n s 140 Ta b l e 5.14: Hours of work (thousands) - OLS e s t i m a t e s : Refundable t a x c r e d i t 142 v i i i Ac knowledqements I would l i k e t o thank E r n s t Berndt and C r a i g R i d d e l l , who s t a r t e d me on t h i s r e s e a r c h . S p e c i a l thanks go t o A l a n Woodland, f o r g u i d i n g me through the i n t r i c a c i e s of j o i n t d e n s i t y f u n c t i o n s ; T e r r y Wales, f o r h i s h e l p through the a c t u a l e s t i m a t i o n ; and P e t e r C h i n l o y , f o r o v e r s e e i n g i t a l l . S p e c i a l thanks a l s o go t o Mike P a t t e r s o n of the Computing C e n t r e , f o r h i s h e l p w i t h the use of the n o n l i n e a r monitor s o f t w a r e . I would a l s o l i k e t o thank Dave S t a p l e t o n and Erwin D i e w e r t f o r t h e i r h e l p f u l comments. F i n a l l y , but not l e a s t , I would l i k e t o thank my c o l l e a g u e s , D a v i d Ryan, f o r h i s h e l p w i t h the programming .in t h i s d i s s e r t a t i o n ; and F e l i c e M a r t i n e l l o , f o r h i s encouragement, support., and h i s many h e l p f u l s u g g e s t i o n s . 1 Chapter One I n t r o d u c t i o n T h i s d i s s e r t a t i o n examines the l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y b e h a v i o r of m a r r i e d women i n the U n i t e d S t a t e s . E c o n o m i s t s , as w e l l as o t h e r s o c i a l s c i e n t i s t s , have been i n t e r e s t e d i n e x p l a i n i n g b oth the obser v e d d e c l i n e i n b i r t h r a t e s and the observ e d i n c r e a s e i n m a r r i e d women's l a b o r f o r c e p a r t i c i p a t i o n over the l a s t twenty y e a r s . Economists have been p a r t i c u l a r l y p u z z l e d by the two phenomena because economic t h e o r y p r e d i c t s t h a t r i s i n g r e a l incomes, which have been e x p e r i e n c e d , s h o u l d have caused a r i s e i n the b i r t h r a t e and a f a l l i n m a r r i e d women's l a b o r f o r c e p a r t i c i p a t i o n , a l l o t h e r t h i n g s h e l d c o n s t a n t . Both phenomena have i m p l i c a t i o n s f o r the growth of the economy. Lower b i r t h r a t e s imply a s m a l l e r f u t u r e l a b o r f o r c e . They a l s o imply a change i n the a g e - d i s t r i b u t i o n of a c o u n t r y , w i t h consequences f o r the demand f o r d i f f e r e n t s e r v i c e s , such as e d u c a t i o n or old-age s e c u r i t y programs. Lower b i r t h r a t e s a l s o imply l e s s p r e s s u r e on e x i s t i n g r e s o u r c e s , such as l a n d . An i n c r e a s e i n the l a b o r 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 l s o has i m p l i c a t i o n s f o r the economy. I t i m p l i e s a change i n the c o m p o s i t i o n of the work f o r c e , an i n c r e a s e i n the s i z e of the work f o r c e , and an i n c r e a s e i n the income of h o u s e h o l d s . I n c r e a s e d l a b o r 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 w i l l a l s o have an e f f e c t on b i r t h r a t e s i f wo r k i n g and r a i s i n g a c h i l d a r e competing uses of a woman's t i m e . 2 H i s t o r i c a l l y , the f e r t i l i t y d e c i s i o n and 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 of m a r r i e d women have been t r e a t e d s e p a r a t e l y . However, i t has been a r g u e d 1 t h a t the two d e c i s i o n s s h o u l d be a n a l y z e d w i t h i n a common framework. In t h i s d i s s e r t a t i o n , we d e v e l o p a model of the j o i n t d e t e r m i n a t i o n of the f e r t i l i t y and 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 s of m a r r i e d women. By t r e a t i n g the two d e c i s i o n s j o i n t l y , we can get a b e t t e r i d e a of how each d e c i s i o n a f f e c t s the o t h e r . T h i s knowledge of the i n t e r a c t i o n between the two d e c i s i o n s may h e l p p o l i c y m a k e r s b e t t e r u n d e r s t a n d the f u l l e f f e c t s of p o l i c i e s d i r e c t e d towards e i t h e r d e c i s i o n . The e m p i r i c a l model used f o r e s t i m a t i o n i s an e x t e n s i o n of a g e n e r a l e c o n o m e t r i c model proposed by Heckman [1978b]. We e s t i m a t e the model u s i n g maximum l i k e l i h o o d e s t i m a t i o n . A l t h o u g h the maximum l i k e l i h o o d e s t i m a t i o n i s e x p e n s i v e , i t a l l o w s us t o c a p t u r e n o n - l i n e a r i t i e s i n the system which o t h e r e s t i m a t i n g t e c h n i q u e s do n o t . Indeed, we found, i n comparing d i f f e r e n t e s t i m a t i n g t e c h n i q u e s , t h a t s i n g l e e q u a t i o n e s t i m a t i n g t e c h n i q u e s o f t e n gave s i g n i f i c a n t l y d i f f e r e n t r e s u l t s from the maximum l i k e l i h o o d e s t i m a t e s . We a l s o r e j e c t e d the h y p o t h e s i s t h a t t h e r e were no r e l a t i o n s h i p s among the t h r e e e q u a t i o n s i n our model. Hence, we can c o n c l u d e t h a t use of s i n g l e e q u a t i o n t e c h n i q u e s i s i n a p p r o p r i a t e f o r t h i s model and may y i e l d m i s l e a d i n g p o l i c y i m p l i c a t i o n s . 1 See, f o r example, the papers and comments i n the supplement t o the J o u r n a l of P o l i t i c a l Economy, M a r c h / A p r i l , 1973. 3 In Chapter Two, we p r o v i d e an o v e r v i e w of the economic l i t e r a t u r e on f e r t i l i t y and the l a b o r 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. Chapter Two t h u s p r o v i d e s the background f o r our model, which i s d e r i v e d i n Chapter Three. Chapter Three a l s o c o n t a i n s a d i s c u s s i o n of the l i k e l i h o o d f u n c t i o n of the model. An appendix t o Chapter Three p r o v i d e s a d e t a i l e d d e r i v a t i o n of the l i k e l i h o o d f u n c t i o n . Chapter Four c o n t a i n s the maximum l i k e l i h o o d e s t i m a t e s of the model. A comparison of the maximum l i k e l i h o o d e s t i m a t e s w i t h e s t i m a t e s o b t a i n e d from a l t e r n a t i v e e s t i m a t i n g t e c h n i q u e s i s p r o v i d e d . Chapter F i v e c o n t a i n s a d i s c u s s i o n of the p o l i c y i m p l i c a t i o n s of the model, b o t h i n g e n e r a l and w i t h r e g a r d to two a l t e r n a t i v e tax p o l i c i e s . The f i n a l c h a p t e r p r e s e n t s a summary of our r e s u l t s and su g g e s t s some p o s s i b l e d i r e c t i o n s f o r f u t u r e r e s e a r c h . 4 Chapter Two T h e o r i e s of l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y d e c i s i o n s A. I n t r o d u c t i o n In t h i s c h a p t e r , we d i s c u s s economic t h e o r i e s of 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 s of m a r r i e d women and economic t h e o r i e s of f e r t i l i t y d e c i s i o n s . In g e n e r a l , the two d e c i s i o n s have been t r e a t e d s e p a r a t e l y . R e c e n t l y , however, t h e o r i e s have been d e v e l o p e d which examine l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y d e c i s i o n s w i t h i n a common framework. The c h a p t e r i s o r g a n i z e d as f o l l o w s . The next s e c t i o n c o n t a i n s a d i s c u s s i o n of economic t h e o r i e s of the l a b o r 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. In the t h i r d s e c t i o n , we examine economic t h e o r i e s of f e r t i l i t y . In the l a s t s e c t i o n , we loo k a t t h e o r i e s which t r e a t l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y d e c i s i o n s as j o i n t l y d e t e r m i n e d w i t h i n the framework of the t h e o r y . 5 B. T h e o r i e s of l a b o r s u p p l y B.1 I n t r o d u c t i o n 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 s of m a r r i e d women have a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n from e c o n o m i s t s i n the l a s t twenty y e a r s . M a r r i e d women a r e o f t e n secondary workers i n the f a m i l y . As a r e s u l t , they a re more l i k e l y t o r e a c t t o s m a l l changes i n economic v a r i a b l e s , such as wage r a t e s , than a re m a r r i e d men or s i n g l e p e r s o n s . T h i s i s because m a r r i e d men and s i n g l e p e r s o n s a re u s u a l l y p r i m a r y income e a r n e r s , and hence w i l l c o n t i n u e t o work even i f t h e r e a re s m a l l changes i n the wage or o t h e r economic v a r i a b l e s . As a r e s u l t , p o l i c y measures, such as minimum wage p o l i c i e s and t a x changes, a r e exp e c t e d t o a f f e c t the m a r r i e d woman worker more than the o t h e r two groups of w o r k e r s . For t h i s r e a s o n , e c o n o m i s t s a re i n t e r e s t e d i n how m a r r i e d women's l a b o r f o r c e p a r t i c i p a t i o n and l a b o r s u p p l y r e a c t t o changes i n these t y p e s of v a r i a b l e s . I n t e r e s t i n 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 s of m a r r i e d women was a l s o s t i m u l a t e d by the a p p a r e n t l y c o n t r a d i c t o r y e v i d e n c e of t i m e - s e r i e s and c r o s s - s e c t i o n a l d a t a . T i m e - s e r i e s d a t a showed a r i s e i n the p a r t i c i p a t i o n r a t e of m a r r i e d women a t the same time t h a t r e a l f a m i l y incomes were r i s i n g . These data seemed t o imply a p o s i t i v e e f f e c t of income on m a r r i e d women's l a b o r f o r c e p a r t i c i p a t i o n , c o n t r a d i c t i n g s t a n d a r d economic t h e o r y . 1 On the o t h e r hand, c r o s s - s e c t i o n a l 1 The u s u a l assumption i n demand t h e o r y i s t h a t l e i s u r e i s a normal good. Hence, i f no n - l a b o r income i n c r e a s e s , the demand f o r l e i s u r e s h o u l d i n c r e a s e . 6 dat a i n d i c a t e d a p o s i t i v e income e l a s t i c i t y of demand f o r l e i s u r e , as would be e x p e c t e d from t h e o r y . M i n c e r [1962] was the f i r s t t o r e s o l v e t h i s apparent c o n t r a d i c t i o n i n the two t y p e s of d a t a . Economic t h e o r y p r e d i c t s a r i s e i n f a m i l y income w i l l d e c r e a s e the l a b o r s u p p l y of the w i f e , a l l o t h e r t h i n g s h e l d c o n s t a n t . But a l l o t h e r t h i n g s were not b e i n g h e l d c o n s t a n t i n the r e a l w o r l d . Women's r e a l wages were a l s o r i s i n g . I t was here t h a t M i n c e r p r o v i d e d h i s i n s i g h t . The c h o i c e s f o r the m a r r i e d woman (or f o r anyone, f o r t h a t m a t t e r ) , he argued, were not s i m p l y l e i s u r e and market work. There was a t h i r d c h o i c e : home work. As t h e i r r e a l wage i n c r e a s e d , m a r r i e d women s u b s t i t u t e d market work f o r home work, r a t h e r than f o r l e i s u r e . By i n c l u d i n g home work as a t h i r d c h o i c e , M i n c e r showed t h a t the t i m e - s e r i e s d a t a d i d not n e c e s s a r i l y imply a n e g a t i v e income demand f o r l e i s u r e . R a t h e r , b o t h l e i s u r e and market work were b e i n g i n c r e a s e d at the expense of home work. 7 B.2 Labor s u p p l y as an e x t e n s i o n of consumer demand t h e o r y M i n c e r ' s model i s based on a l i f e t i m e u t i l i t y m a x i m i z a t i o n model. T h i s approach i s f o l l o w e d i n numerous o t h e r s t u d i e s , i n c l u d i n g C a i n [1966], L e u t h o l d [ 1 9 6 8 ] , R o s t e r s [1969], A s h e n f e l t e r and Heckman [1974 ], Abbott and A s h e n f e l t e r [1976], and Lau, L i n , and Y o t o p o u l o s [1978]. The b a s i c model u n d e r l y i n g these papers (sometimes i m p l i c i t l y ) i s d e s c r i b e d as f o l l o w s . 2 The consuming u n i t i s taken t o be a m a r r i e d c o u p l e who maximize l i f e t i m e u t i l i t y , s u b j e c t t o l i f e t i m e income c o n s t r a i n t s . That i s , the c o u p l e maximizes (1) U = U(X,L_,L ) t in. where X i s a H i c k s i a n composite good and L^ i s l i f e t i m e l e i s u r e , f = w i f e , m=husband, s u b j e c t t o (2a) A + EW. (T. -L. ) > PX i 1 1 1 (2b) 0 < L. < T. where A i s the f l o w of n o n - l a b o r income; i s the market wage r a t e , i=f,m; P i s the p r i c e ( i n d e x ) of X; and , i=f,m, i s the maximum amount of l e i s u r e which can be consumed i n the l i f e t i m e . The v a l i d i t y of assuming the e x i s t e n c e of a f a m i l y u t i l i t y f u n c t i o n depends upon whether we can ensure c o n s i s t e n t 2 See Heckman, et a l [1979] f o r a more complete d e s c r i p t i o n of the model and i t s i m p l i c a t i o n s . 8 p r e f e r e n c e s w i t h i n the f a m i l y . I f we f o l l o w Gorman [1 9 5 3 ] , we must assume t h a t the income d i s t r i b u t i o n w i t h i n the f a m i l y does not a f f e c t i n d i v i d u a l demands. T h i s assumption seems u n l i k e l y i n a f a m i l y c o n t e x t . For i n s t a n c e , i t might be supposed t h a t the demands of an a d o l e s c e n t i n the f a m i l y would change i f he or she r e c e i v e d a l a r g e r share of f a m i l y income. F u r t h e r , t h i s a ssumption i m p l i e s c e r t a i n r e s t r i c t i o n s on the f u n c t i o n a l form of the demand f u n c t i o n s and hence of the u t i l i t y f u n c t i o n , which may be deemed u n d e s i r a b l e by the r e s e a r c h e r . A l t e r n a t i v e l y , we can assume t h a t f a m i l y income i s always o p t i m a l l y d i s t r i b u t e d among f a m i l y members such t h a t a " f a m i l y w e l f a r e f u n c t i o n " i s maximized (Samuelson [ 1 9 5 6 ] ) . T h i s a ssumption i m p l i e s the u t i l i t y of one f a m i l y member i s independent of the l e v e l of u t i l i t y of any o t h e r f a m i l y member and t h a t t h e r e i s no j o i n t consumption. A g a i n , both c o n d i t i o n s a r e l i k e l y t o be v i o l a t e d i n a f a m i l y s i t u a t i o n . In g e n e r a l , the problem of c o n s i s t e n t p r e f e r e n c e s among f a m i l y members i s i g n o r e d . W h i l e n o t i n g the problem, we w i l l f o l l o w the l i t e r a t u r e i n assuming t h a t the f a m i l y a c t s as i f i t were m a x i m i z i n g a u t i l i t y f u n c t i o n such as e q u a t i o n (1) and t h a t t h e s e p r e f e r e n c e s are c o n s i s t e n t ( W i l l i s [1973]; Wales and Woodland [ 1 9 7 6 ] ) . D e f i n e f u l l income, the maximum amount of income the c o u p l e c o u l d e arn p l u s a s s e t income, as I = A + EW.T.. M a x i m i z a t i o n of i 1 1 e q u a t i o n (1) s u b j e c t t o e q u a t i o n (2) y i e l d s the demand e q u a t i o n s f o r X and L. , which are f u n c t i o n s of P, Wf, W , and I . 9 (4a) L. = L.(P,W^,W ,1) i=f,m l l f m (4b) X = X(P,W_,W ,1) f m Labor s u p p l y i s then c a l c u l a t e d as a r e s i d u a l . (5) H- = T.-L. = H. (P,W^,W ,1) i = f,m i i i l f m An i n t e r i o r s o l u t i o n f o r PL i s a c h i e v e d i f L^ i s not e q u a l t o e i t h e r T. or z e r o , U -X.W. =0, i=f,m, and U„ -XP=0, where X i s I L. I X I the L a g r a n g i a n m u l t i p l i e r . Two assumptions u n d e r l i e the model. The f i r s t i s t h a t non-market time i s p e r f e c t l y s u b s t i t u t a b l e a c r o s s time p e r i o d s . T h i s a ssumption i m p l i e s t h a t , except f o r " t r a n s i t o r y " f a c t o r s a f f e c t i n g o p p o r t u n i t y c o s t s at a g i v e n p o i n t i n t i m e , the t i m i n g of a woman's l a b o r f o r c e p a r t i c i p a t i o n i s random. The second assumption i s t h a t the l a b o r s u p p l y d e c i s i o n i s ge n e r a t e d by an i n t e r i o r s o l u t i o n . T h i s i m p l i e s everyone works sometime d u r i n g her l i f e t i m e . Hence, the l i f e t i m e l a b o r s u p p l y of any woman can be measured by the p r o b a b i l i t y of her b e i n g i n the l a b o r f o r c e a t a g i v e n p e r i o d of ti m e , o r , f o r a group of women, by the l a b o r f o r c e p a r t i c i p a t i o n r a t e of t h a t group, (Mincer [ 1 9 6 2 ] ) . Under these two as s u m p t i o n s , p l u s the a d d i t i o n a l assumption t h a t t h e r e i s no r a t i o n i n g of hours of work, r e g r e s s i o n s run w i t h a l a b o r f o r c e p a r t i c i p a t i o n measure or a l a b o r s u p p l y measure on P, W, W , and A w i l l y i e l d e s t i m a t e s of both the f m J H i c k s - S l u t s k y income e f f e c t and the s u b s t i t u t i o n e f f e c t (Heckman [ 1 9 7 8 a ] ) . However, i f t h e r e a r e some women who do not work a t a l l d u r i n g t h e i r l i f e t i m e s , e s t i m a t i o n of a p a r t i c i p a t i o n 10 e q u a t i o n no l o n g e r y i e l d s the income and s u b s t i t u t i o n e f f e c t s . B.3 Labor f o r c e p a r t i c i p a t i o n and hours of work I f some women i n the sample do not work, s e v e r a l d i f f i c u l t i e s a r i s e i f the whole sample i s t o be used f o r e s t i m a t i o n . F i r s t , t h e r e i s no wage r a t e i n f o r m a t i o n a v a i l a b l e f o r women who do not work. Second, t h e r e i s a sample s e l e c t i v i t y problem i n the sense t h a t the woman chooses t o work, and hence, chooses whether she w i l l have p o s i t i v e hours of work. 3 A woman w i l l be more l i k e l y t o p a r t i c i p a t e , and hence t o have an obs e r v e d wage, i f her ( p o t e n t i a l ) wage i s h i g h . T h i s s e l f -s e l e c t i o n w i l l cause b i a s e s i n the e s t i m a t e s of hours worked i f o r d i n a r y l e a s t squares i s used. To d e a l w i t h t h e s e problems, and w i t h the e s t i m a t i o n problem t h a t hours of work must be non-n e g a t i v e , new models were d e v e l o p e d . These models e s t i m a t e the parameters of the l a b o r s u p p l y and l a b o r f o r c e p a r t i c i p a t i o n f u n c t i o n s j o i n t l y . ' S u c h models a re found i n Heckman [1974,1976], Hanoch [1976], Cogan [1977], B u r t l e s s and Hausman [19 7 8 ] , Hausman [1979], and Wales and Woodland [1980]. In g e n e r a l , these models have been used t o look a t 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 s a t a g i v e n p o i n t of time r a t h e r than over the whole l i f e c y c l e . 3 S e l f - s e l e c t i v i t y may not be a problem f o r a group such as men, 30-45 y e a r s of age, s i n c e t h e i r p a r t i c i p a t i o n r a t e i s c l o s e t o 100 p e r c e n t . However, f o r m a r r i e d women, the group of i n t e r e s t h e r e , t h e r e i s a s u b s t a n t i a l number who do not work a t a g i v e n p o i n t i n ti m e . For example, i n the 1976 i n t e r v i e w year of the M i c h i g a n P a n e l Survey of Income Dynamics, 53.8% of the m a r r i e d women d i d not work. 11 The b a s i c u n d e r l y i n g model i s a b i n a r y c h o i c e model of the s o r t d i s c u s s e d by Heckman [1976, 1978a, 1979], Domenc.ich and McFadden [1 9 7 5 ] , McFadden [1974, 1976], and o t h e r s . The model i s d e r i v e d from the m a x i m i z a t i o n problem g i v e n by e q u a t i o n s (1) and ( 2 ) . Assume, f o r the moment, t h a t the husband's hours of work (and, hence, hours of l e i s u r e ) and wage r a t e a r e g i v e n . We r e w r i t e the problem as (1') max U*(X,L f) s u b j e c t t o (2a') A + E + W.H. > PX m f f (2b') 0 < L_ < T £ r f where E i s the husband's e a r n i n g s , d e f i n e d as W H , which a re m mm assumed t o be g i v e n , and the v a r i a b l e s a r e now measured f o r a g i v e n p o i n t i n t i m e , r a t h e r than f o r the whole l i f e c y c l e . The s o l u t i o n of the m a x i m i z a t i o n problem y i e l d s a system of eq u a t i o n s f o r X and the Lagrange m u l t i p l i e r s i n terms of P, E^+A, W fH f, and H f (Heckman [ 1 9 7 4 ] ) . These e q u a t i o n s can be s o l v e d f o r the shadow wage r a t e of the w i f e a t z e r o hours of work. The shadow wage r a t e w i l l a l s o be a f u n c t i o n of P, +A, W fH f, and H f. The w i f e i s ex p e c t e d t o work i f and o n l y i f the market wage r a t e i s g r e a t e r than the shadow wage r a t e at z e r o hours of work. I f we s e t the market wage r a t e e q u a l t o the shadow wage r a t e f u n c t i o n , we can s o l v e f o r a reduced form e q u a t i o n of the 1 2 e q u i l i b r i u m hours of work. I f we assume l i n e a r i t y , the l a b o r supply e q u a t i o n can be e x p r e s s e d i n g e n e r a l terms as i f X.b + u. > 0 l I o t h e r w i s e where i s the number of hours worked by i n d i v i d u a l i , X^ i s a v e c t o r of exogenous v a r i a b l e s , i n c l u d i n g the wage r a t e , b i s a v e c t o r of p a r a m e t e r s , and the u^ a r e d i s t u r b a n c e terms which a r e assumed t o be d i s t r i b u t e d as independent normal v a r i a b l e s , w i t h mean z e r o and c o n s t a n t v a r i a n c e , c2 . In o r d e r t o e s t i m a t e e q u a t i o n (6) over a sample which i n c l u d e s non-working women, i t i s n e c e s s a r y t o make the wage r a t e endogenous. T h i s a l l o w s us t o t a k e i n t o account i n f o r m a t i o n on non-workers when e s t i m a t i n g the l a b o r supply f u n c t i o n . ' A g a i n , i f we assume the f u n c t i o n s a r e l i n e a r i n t h e i r arguments, the model c o n s i s t s of the f o l l o w i n g e q u a t i o n s : i f X,. b, + W. g + u. > 0 o t h e r w i s e where X.. i s the set of exogenous v a r i a b l e s , j=1, 2; and u. and (6) H. = I X. b + u. (7) H ± = j X 1 ; Lb, + W ± g + u. (8) W± = X 2 i b 2 + e ± (9) Wi i s o b s e r v e d i f f PL > 0 " For a survey of the v a r i o u s e s t i m a t o r s proposed t o d e a l w i t h the problems r a i s e d when a sample c o n t a i n s non-working women, see Wales and Woodland [1980]. 1 3 ar e assumed t o have a j o i n t normal d i s t r i b u t i o n w i t h mean z e r o and v a r i a n c e - c o v a r i a n c e m a t r i x (10) E = tf 1 2 6 \ 2 6 2 1 a 2 2 The l i k e l i h o o d f u n c t i o n a s s o c i a t e d w i t h t h i s model has two p a r t s . 5 The f i r s t p a r t i s c a l c u l a t e d over the subsample of workers and t a k e s i n t o account the t r u n c a t i o n of the hours worked v a r i a b l e . The second p a r t i s c a l c u l a t e d over the subsample of non-workers and r e f l e c t s the f a c t t h a t wage r a t e s are not ob s e r v e d f o r the s e i n d i v i d u a l s . M N (11) L(H,W,b,g;E) = n n(u.,e.;E) n F(d.) i = l 1 1 i=M+l i where n(u.,e.,E) i s the b i v a r i a t e normal d e n s i t y f u n c t i o n f o r u x i i and e. ; d. = -(X,.b, + X 2. b 2 g ) / ( g 2 t f 2 2 + 2 g f f l 2 + tfl2)1/2; F i s X 1 1 1 the c u m u l a t i v e s t a n d a r d normal d e n s i t y f u n c t i o n , 6 and the sample i s o r d e r e d such t h a t workers comprise the f i r s t M o b s e r v a t i o n s . The main importance of t h i s model i s t h a t i t a l l o w s the use of i n f o r m a t i o n on non-workers. Thus, i t combines both the p a r t i c i p a t i o n d e c i s i o n (working v s . not working) and the l a b o r 5 For a d e r i v a t i o n of the l i k e l i h o o d f u n c t i o n , see Wales and Woodland [1980]. oo , 6 F(d- ) = / Z 1 n(u-: ,e- ;s)du. de. . For l i n e a r hours and wage X — oo —co - " - I X j. e q u a t i o n s , F(d^) = / i g ( u j _ )du^, where g(u^) i s the m a r g i n a l d i s t r i b u t i o n of Uj_ . 1 4 s u p p l y d e c i s i o n (how many hours t o work) i n t o a common framework. T h i s can perhaps be more c l e a r l y seen i f , f o l l o w i n g Wales and Woodland [1980], we r e w r i t e the l i k e l i h o o d f u n c t i o n i n the f o l l o w i n g manner. ( 1 2 ) L(H,W,b,g;I) = M , u. - e. {. n .[5 f ( 1 g g s — - )/ ( l - F ( r i ) ) ] } i = l JC cc •{ n t- f ( — ) ( 1 ~ F(^r±> )]> i = l 0 c a c 1 - F(d.) M N' { ,n ( l - F(d.)) n F(d.) }. i = l i=M+i where e < r n 2 ) 1 / 2 ] / ( f 2 i s the s t a n d a r d d e v i a t i o n -(X,. b, + W. q)/a ; f i s the s t a n d a r d 1 ,1 c f o r u^ g i v e n e^; r ^ normal d e n s i t y f u n c t i o n ; and the o t h e r v a r i a b l e s a r e d e f i n e d as i n e q u a t i o n ( 1 1 ) . The f i r s t term g i v e s the l i k e l i h o o d f u n c t i o n f o r hou r s , g i v e n hours a r e p o s i t i v e and g i v e n the wage r a t e . The second term i s the l i k e l i h o o d f u n c t i o n f o r the wage r a t e , g i v e n hours of work a r e p o s i t i v e . These two p a r t s a r e e v a l u a t e d over the subsample of worke r s . The t h i r d term i s the l i k e l i h o o d f u n c t i o n f o r 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 . I t c a l c u l a t e s the p r o b a b i l i t y t h a t a g i v e n i n d i v i d u a l w i l l be found, i n the l a b o r f o r c e . Thus, the model d e r i v e d here s o l v e s some of the e s t i m a t i o n 1 5 problems of the e a r l i e r l i f e c y c l e m odels. 7 One d i f f i c u l t y w i t h t h e s e models, however, i s t h a t they i g n o r e l i f e c y c l e c o n s i d e r a t i o n s , c o n c e n t r a t i n g on a s i n g l e p o i n t i n t i m e . In an attempt t o i n c l u d e l i f e c y c l e c o n s i d e r a t i o n s i n the a n a l y s i s , w h i l e d e a l i n g w i t h the problems which a r i s e i f some of the women do not work, dynamic models of l a b o r s u p p l y have been proposed. B.4 Dynamic models of l a b o r s u p p l y Dynamic models of l a b o r s u p p l y have been deve l o p e d i n o r d e r to c a p t u r e i n t e r p e r i o d r e l a t i o n s h i p s among the v a r i a b l e s and t o d e r i v e time paths f o r v a r i a b l e s i n which the r e s e a r c h e r i s i n t e r e s t e d . Dynamic models e x p l o r i n g 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 are found i n Ghez and Becker [1975], Heckman and W i l l i s [1977], Smith [1977], and Heckman and MaCurdy [1980]. The essence of the dynamic models can be seen i n the model de v e l o p e d by Heckman and MaCurdy [1980]. Assume i n d i v i d u a l s o p e r a t e under p e r f e c t c e r t a i n t y w i t h a known, f i n i t e l i f e s p a n , T. U t i l i t y a t age t i s g i v e n by a s t r i c t l y concave u t i l i t y f u n c t i o n , U ( X ( t ) , L ( t ) ) , where X ( t ) and L ( t ) are the i n s t a n t a n e o u s consumption of goods and l e i s u r e , r e s p e c t i v e l y , a t time t . L ( t ) i s assumed t o l i e i n the u n i t i n t e r v a l : 0 < L ( t ) < 1. L ( t ) can be i n t e r p r e t e d as the p r o p o r t i o n of t o t a l time i n p e r i o d t spent consuming l e i s u r e . Wages are taken t o be 7 T h i s b a s i c sample s e l e c t i v i t y model has been extended t o take i n t o account n o n p r o p o r t i o n a l t a x e s , i n s t i t u t i o n a l r e s t r i c t i o n s on hours worked, and f i x e d c o s t s of w o r k i n g . For a summary of these e x t e n s i o n s , see Heckman et a l [1979]. 1 6 p a r a m e t r i c and the p r i c e of goods i s n o r m a l i z e d t o u n i t y . C a p i t a l markets are assumed t o be p e r f e c t . Assume the i n d i v i d u a l s t a r t s w i t h a s s e t s A ( 0 ) , t h a t the i n t e r e s t r a t e i s r , and t h a t the r a t e of time p r e f e r e n c e i s p. Then, the i n d i v i d u a l ' s problem i s t o T (13) max / e p t U [ X ( t ) , L ( t ) ] d t 0 s u b j e c t t o T (14) 0 = A(0) + / e [ W ( t ) ( l - L ( t ) ) - X ( t ) ] d t 0 Demand f u n c t i o n s f o r X ( t ) and L ( t ) can be d e r i v e d as f u n c t i o n s of the d i s c o u n t e d wage r a t e a t time t and the m a r g i n a l u t i l i t y of w e a l t h i n p e r i o d z e r o . The l a b o r s u p p l y f u n c t i o n a t time t i s then 1 - L ( t ) . Two measures of l i f e t i m e l a b o r s u p p l y are d e r i v e d . F i r s t , l i f e t i m e l a b o r s u p p l y i s measured by the t o t a l number of hours spent w o r k i n g over the l i f e c y c l e . T h i s measure i s g i v e n by T (15) H = / h ( t ) d t 0 where h ( t ) = 1 - L ( t ) . Second, l i f e t i m e l a b o r s u p p l y can be measured as the t o t a l number of " p e r i o d s " worked. 8 T h i s measure of l i f e t i m e l a b o r s u p p l y i s g i v e n by 8 In e m p i r i c a l work, the model was r e f o r m u l a t e d i n terms of a d i s c r e t e time framework. 1 7 T (16) S = / d ( t ) d t 0 where d ( t ) = 1 f o r each p e r i o d the i n d i v i d u a l works. Heckman and MaCurdy argue t h a t i n the one p e r i o d l i f e t i m e models, these two measures of l i f e t i m e l a b o r s u p p l y , H and S, are i d e n t i c a l . However, i n the dynamic c o n t e x t , they measure two d i s t i n c t c o n c e p t s . One measures " p a r t i c i p a t i o n " ( S ) ; the o t h e r measures " l a b o r s u p p l y " (H). Heckman and MaCurdy assume t h a t the m a r g i n a l u t i l i t y of w e a l t h a t p e r i o d z e r o , denoted X ( 0 ) , i s f i x e d . T h i s assumption g r e a t l y s i m p l i f i e s e s t i m a t i o n by e l i m i n a t i n g the need f o r i n f o r m a t i o n on wage r a t e s and o t h e r v a r i a b l e s over the whole l i f e c y c l e . C u r r e n t v a l u e s become s u f f i c i e n t f o r e s t i m a t i n g l i f e c y c l e d e c i s i o n s . However, such an assumption d e s t r o y s some of the v a l u e of u s i n g a dynamic model by o b s c u r i n g the i n t e r p e r i o d r e l a t i o n s h i p s among v a r i a b l e s . The advantage of dynamic models i s t h a t they p e r m i t the i n v e s t i g a t i o n of the t i m i n g o f . d e c i s i o n s and i n t e r p e r i o d s u b s t i t u t i o n s among v a r i a b l e s . One of t h e i r drawbacks, however, i s the l a r g e amount of data needed f o r e s t i m a t i o n p u r p o s e s . Attempts t o s o l v e the d a t a problem, such as Heckman and MaCurdy's c o n s t a n t m a r g i n a l u t i l i t y of w e a l t h demand f u n c t i o n s or Ghez and B ecker's s y n t h e t i c c o h o r t s , o f t e n reduce the a b i l i t y of the e s t i m a t e d parameters t o d e s c r i b e the time p a t h s of v a r i a b l e s i n which the r e s e a r c h e r i s i n t e r e s t e d . W i t h the a v a i l a b i l i t y of l o n g i t u d i n a l d a t a , such as the N a t i o n a l L o n g i t u d i n a l Survey (NLS) and the M i c h i g a n P a n e l Suvey of Income Dynamics (PSID) and i m p r o v i n g computer f a c i l i t i e s , i t i s e x p e c t e d t h i s d i f f i c u l t y u l t i m a t e l y w i l l be overcome. 18 C. Economic t h e o r i e s of f e r t i l i t y C.1 I n t r o d u c t i o n In d e v e l o p e d c o u n t r i e s , and i n c r e a s i n g l y i n l e s s d e v e l o p e d c o u n t r i e s , the main source of p o p u l a t i o n growth has been the b i r t h r a t e r a t h e r than d e c r e a s e s i n the death and/or m i g r a t i o n r a t e s ( E a s t e r l i n [ 1968], F u l o p [ 1 9 7 7 ] ) . A l t h o u g h most economic models take p o p u l a t i o n (both the l e v e l and the r a t e of growth) as exogenous, over time p o p u l a t i o n w i l l i n t e r a c t w i t h economic v a r i a b l e s , w i t h i m p l i c a t i o n s f o r the growth of the economy and f o r government p o l i c i e s . I t was t o study these i m p l i c a t i o n s t h a t e c onomists began t o c o n s t r u c t t h e o r i e s which f o c u s s e d on the economic a s p e c t s of p o p u l a t i o n growth; i n p a r t i c u l a r , on the economic a s p e c t s of f e r t i l i t y . One p o i n t common to most economic t h e o r i e s of f e r t i l i t y i s t h a t they p r e d i c t , o t h e r t h i n g s b e i n g e q u a l , a p o s i t i v e r e l a t i o n s h i p between income 9 and f e r t i l i t y . Y e t , c r o s s - s e c t i o n a l and t i m e - s e r i e s d a t a show a n e g a t i v e r e l a t i o n s h i p between the t w o . 1 0 Hence, the t h e o r i e s have had t o e x p l a i n why income might have a n e g a t i v e r e l a t i o n s h i p w i t h f e r t i l i t y . 9 One has be be c a r e f u l when u s i n g the term "income" t o d i s t i n g u i s h between wage income and n o n - l a b o r income, as the former w i l l have a p r i c e e f f e c t as w e l l as an income e f f e c t . The e a r l y t h e o r i s t s d i d not always make the d i s t i n c t i o n . T h i s p o i n t was r a i s e d by M i n c e r [1963]. As used i n t h i s p a r a g r a p h , "income" r e f e r s t o n o n - l a b o r income, but i t can a l s o i n c l u d e the income e f f e c t of wage income. 1 0 An e x c e p t i o n i s t i m e - s e r i e s d a t a over the b u s i n e s s c y c l e , which show a p o s i t i v e r e l a t i o n s h i p between f e r t i l i t y and income. See Simon [1969] . 19 C.2 L e i b e n s t e i n and E a s t e r l i n The f i r s t modern a n a l y s i s of the economics of f e r t i l i t y 1 1 may be c o n s i d e r e d t o be the a n a l y s i s by L e i b e n s t e i n [1957]. In the c o n t e x t of a l e s s d e v e l o p e d c o u n t r y , he t h e o r i z e d t h a t a c h i l d c o n t r i b u t e d t h r e e t y p e s of u t i l i t y : ( 1).consumption u t i l i t y , (2) work (income) u t i l i t y , and (3) ( o l d age) s e c u r i t y u t i l i t y . As a c o u n t r y d e v e l o p e d , c h i l d r e n would no l o n g e r be a l a r g e source of work or s e c u r i t y u t i l i t y . As incomes r o s e , t h e r e would be l e s s need t o have c h i l d r e n work and l e s s need f o r t h e i r s u p p o r t i n o l d age ( e s p e c i a l l y w i t h the r i s e of government or p r i v a t e p e n s i o n programs). S i m i l a r l y , as the a v a i l a b i l i t y of e d u c a t i o n s p r e a d , c h i l d r e n would no l o n g e r be a v a i l a b l e f o r work as they would be i n s c h o o l . Hence, L e i b e n s t e i n argued, because of t h e s e d e c l i n e s i n the u t i l i t i e s of c h i l d r e n , t h e r e would be a d e c r e a s e i n f e r t i l i t y r a t e s due t o the economic development of a c o u n t r y . L a t e r , L e i b e n s t e i n [1974, 1975a, 1975b] p r e s e n t e d a new microeconomic t h e o r y of f e r t i l i t y i n which c h i l d r e n were d e f i n e d as a "commitment good" s u b j e c t t o i n c r e a s i n g m a r g i n a l u t i l i t y up t o some l e v e l , and d e c r e a s i n g m a r g i n a l u t i l i t y beyond t h a t l e v e l . He combined t h i s d e f i n i t i o n w i t h the s o c i o l o g i c a l view t h a t the p o p u l a t i o n i s s t r a t i f i e d i n t o s t a t u s groups. For each s t a t u s l e v e l , t h e r e i s a c e r t a i n l e v e l of consumption which i s mandatory i f the f a m i l y i s t o m a i n t a i n i t s s t a t u s . S i n c e s t a t u s i s r e l a t i v e , as the g e n e r a l l e v e l of income r i s e s , the f a m i l y 1 1 M a l t h u s [1798], of c o u r s e , may be c o n s i d e r e d the f i r s t e conomist to t r e a t p o p u l a t i o n growth as endogenous. 20 w i l l have t o i n c r e a s e i t s p r o p o r t i o n a t e e x p e n d i t u r e on " s t a t u s goods" i n o r d e r t o m a i n t a i n i t s r e l a t i v e s t a t u s . As a r e s u l t , i t w i l l spend p r o p o r t i o n a t e l y l e s s on "commitment goods", i n c l u d i n g c h i l d r e n . At the same t i m e , w i t h i n s t a t u s groups, t h e r e w i l l be a p o s i t i v e r e l a t i o n s h i p between income and f e r t i l i t y s i n c e , a l t h o u g h the r i c h e r f a m i l y w i l l have the same u t i l i t y c o s t s ( e . g . , e x p e n d i t u r e s on " s t a t u s g o o d s " ) , i t w i l l have l e s s of a budget c o n s t r a i n t . L e i b e n s t e i n ' s t h e o r y o f f e r s one way of e x p l a i n i n g the s o c i a l as w e l l as economic f a c t o r s which may i n f l u e n c e f e r t i l i t y b e h a v i o r . However, t h e r e a re s e v e r a l d i f f i c u l t i e s w i t h h i s approach. F i r s t , the t h e o r y c o n c e n t r a t e s on income e f f e c t s , n e g l e c t i n g p r i c e e f f e c t s which a l s o have been argued as b e i n g i m p o r t a n t i n e x p l a i n i n g f e r t i l i t y b e h a v i o r . F u r t h e r , the i n t e r a c t i o n between the s o c i a l and economic v a r i a b l e s i s not c l e a r l y e x p l a i n e d . F i n a l l y , the t h e o r y has not y e t been f o r m a l i z e d i n such as way as t o y i e l d e m p i r i c a l l y t e s t a b l e h y p o t h e s e s . The second t h e o r y , proposed by E a s t e r l i n [1968, 1969], a l s o combines s o c i o l o g i c a l w i t h economic f a c t o r s i n e x p l a i n i n g f e r t i l i t y . E a s t e r l i n e x p l a i n s f e r t i l i t y i n terms of " r e l a t i v e i n t e r g e n e r a t i o n income". The consumption l e v e l e x p e r i e n c e d i n the p a r e n t s ' home i s assumed t o shape the c h i l d r e n ' s p r e f e r e n c e s f o r m a t e r i a l goods. When the c h i l d r e n become a d u l t s , the consumption l e v e l t o which they a s p i r e c o r r e s p o n d s t o the a c t u a l l e v e l of consumption t h e i r p a r e n t s had when they were l i v i n g at home. I f c h i l d r e n have more income than t h e i r p a r e n t s had, they w i l l consume more of e v e r y t h i n g , i n c l u d i n g c h i l d r e n . Hence, the 21 post-war baby boom i s e x p l a i n e d , a c c o r d i n g t o t h i s t h e o r y , by the f a c t t h a t the p a r e n t s of the 1940's and 1950's had more income than t h e i r p a r e n t s , who l i v e d d u r i n g the D e p r e s s i o n . Both L e i b e n s t e i n ' s and E a s t e r l i n ' s t h e o r i e s c o n c e n t r a t e on v a r i a t i o n s i n f e r t i l i t y over t i m e . Both emphasize the r o l e of t a s t e s i n d e t e r m i n i n g the demand f o r c h i l d r e n . E a s t e r l i n ' s r e l a t i v e i n t e r g e n e r a t i o n income h y p o t h e s i s i s an attempt t o e x p l a i n f e r t i l i t y b e h a v i o r by e x p l a i n i n g i n t e r g e n e r a t i o n a l d i f f e r e n c e s i n t a s t e . L e i b e n s t e i n a t t e m p t s t o e x p l a i n d i f f e r e n c e s i n t a s t e s by r e f e r e n c e t o " s o c i a l i n f l u e n c e groups". T a s t e s , however, are d i f f i c u l t t o measure. C.3 The hou s e h o l d p r o d u c t i o n t h e o r y of f e r t i l i t y The h o u s e h o l d p r o d u c t i o n t h e o r y has i t s r o o t s i n s e v e r a l p a p e r s , p a r t i c u l a r l y Becker [1960], M i n c e r [ 1 9 6 3 ] , and Becker [ 1 9 6 5 ] . 1 2 In h i s 1960 paper, Becker argued t h a t c h i l d r e n can be mo d e l l e d as a type of d u r a b l e good, y i e l d i n g consumer s a t i s f a c t i o n w h i l e e n t a i l i n g i n v e s t m e n t - l i k e e x p e n d i t u r e s . The u t i l i t y of a c h i l d was assumed t o depend on i t s q u a l i t y , where q u a l i t y r e f e r s t o a t t r i b u t e s which can be c r e a t e d through the a p p l i c a t i o n of time and market goods. The c o s t of the c h i l d was assumed t o be l i m i t e d t o d i r e c t c o s t s . 1 3 F e r t i l i t y was then 1 2 For t h i s r e a s o n , the t h e o r y we term "household p r o d u c t i o n " i s sometimes r e f e r r e d t o as the "Chicago-Columbia" t h e o r y of f e r t i l i t y . 1 3 C r i t i c i s m s have been made of these a s s u m p t i o n s . See, f o r i n s t a n c e , the d i s c u s s i o n s i n Duesenberry [ i 9 6 0 ] , Okun [1960], B l a k e [1968], Namboodiri [1972], and Sanderson [ n . d . ] . 22 d e t e r m i n e d by income, c o s t s , " u n c e r t a i n t y of p r o d u c t i o n " , and t a s t e s (as r e p r e s e n t e d by the u t i l i t y f u n c t i o n ) . The importance of Bec k e r ' s t h e o r y l i e s i n i t s a n a l y s i s of the demand f o r c h i l d r e n w i t h i n a u t i l i t y m a x i m i z a t i o n framework. I t i s t h i s a s p e c t of h i s a n a l y s i s , t o g e t h e r w i t h the two o t h e r s t r a n d s d e s c r i b e d n e x t , which form the b a s i s of the c u r r e n t h o u s e h o l d p r o d u c t i o n t h e o r y of f e r t i l i t y . The second s t r a n d of the t h e o r y i s found i n M i n c e r [1963]. M i n c e r argued t h a t the r e l e v a n t c o s t of a c h i l d was not so much the d i r e c t c o s t as the o p p o r t u n i t y c o s t of the mother i n terms of foregone e a r n i n g s . I t i s t h i s i d e a of the o p p o r t u n i t y c o s t of c h i l d r e a r i n g which has dominated the economic l i t e r a t u r e on f e r t i l i t y . 1 " M i n c e r a l s o p o i n t e d out t h a t the e f f e c t of income on f e r t i l i t y w i l l depend on the source of the change i n income. A change i n wage income w i l l have an o f f s e t t i n g p r i c e e f f e c t , whereas a change i n no n - l a b o r income w i l l have a pure income e f f e c t on f e r t i l i t y . The argument t h a t foregone e a r n i n g s a r e the o p p o r t u n i t y c o s t , or shadow p r i c e , of h a v i n g a c h i l d was c r i t i c i z e d by L e i b e n s t e i n [ 1 9 7 4 ] , L e i b e n s t e i n argued t h a t the o p p o r t u n i t y c o s t of the mother was not any foregone wage income, but the c o s t of s u b s t i t u t e c h i l d c a r e . However, L e i b e n s t e i n ' s argument i s not c o m p l e t e l y s a t i s f a c t o r y . I f the w i f e i s not working b e f o r e she has the c h i l d , her shadow p r i c e of time must have been h i g h e r than the market wage r a t e , a c c o r d i n g t o s t a n d a r d l a b o r s u p p l y t h e o r y . 1 " See, f o r example, Gronau [1973a, 1 973b, 1977]. 23 Hence, i t i s p o s s i b l e t h a t , u s i n g the shadow p r i c e of the w i f e ' s time as a measure of her o p p o r t u n i t y c o s t , t h a t the o p p o r t u n i t y c o s t of h a v i n g a c h i l d i s g r e a t e r f o r the non-working mother than the w o r k i n g mother ( K e e l e y [ 1 9 7 5 ] ) . However, u s i n g L e i b e n s t e i n ' s c r i t e r i o n , the woman who t a k e s c a r e of the c h i l d h e r s e l f and has no c h i l d c a r e c o s t s , has a z e r o o p p o r t u n i t y c o s t of h a v i n g a c h i l d , r e g a r d l e s s of foregone e a r n i n g s or her shadow p r i c e of t i m e . S u r e l y , however, t o have a c h i l d i s not c o s t l e s s . Perhaps a b e t t e r way of t r e a t i n g the q u e s t i o n of o p p o r t u n i t y c o s t i s the f o l l o w i n g . C l e a r l y , i f a w i f e g i v e s up working t o have a c h i l d , she g i v e s up the e a r n i n g s she would have o t h e r w i s e earned. However, i f she c o n t i n u e d working a f t e r h a v i n g the c h i l d and h i r e d c h i l d c a r e , her net r e t u r n from, working i s lowered by the amount p a i d out f o r c h i l d c a r e . S i m i l a r l y , a w i f e who d i d not work b e f o r e h a v i n g the c h i l d has some shadow p r i c e of t i m e . I f she has a c h i l d , her time must be d i v e r t e d from her p r e v i o u s a c t i v i t i e s t o r a i s i n g the c h i l d . Thus, we can t h i n k of the o p p o r t u n i t y c o s t of h a v i n g a c h i l d as e i t h e r the wage r a t e or shadow p r i c e of time (depending on whether the w i f e i s w o r k i n g ) , a d j u s t e d f o r any c h i l d c a r e c o s t s . In t h i s way, we take i n t o account both foregone e a r n i n g s (or foregone time which c o u l d have been spent on o t h e r a c t i v i t i e s ) and the c o s t of s u b s t i t u t e s f o r the mother's t i m e . The t h i r d and major s t r a n d u n d e r l y i n g the h o usehold p r o d u c t i o n t h e o r y of f e r t i l i t y comes from Becker [1965]. Becker p r e s e n t e d an a l t e r n a t i v e view of consumer demand t h e o r y . Consumers a r e assumed t o demand market goods not f o r t h e m s e l v e s , but because the market goods, t o g e t h e r w i t h t i m e , can be 24 t r a n s f o r m e d i n t o "commodities" by the household v i a a household p r o d u c t i o n f u n c t i o n . The h o u s e h o l d thus f a c e s c o n s t r a i n t s on budget and t i m e . The model of the h o u s e h o l d then has the f o l l o w i n g components: (1) a u t i l i t y f u n c t i o n w i t h "commodities" r a t h e r than "goods" as i t s arguments; (2) household p r o d u c t i o n f u n c t i o n s by which time and market goods are t r a n s f o r m e d i n t o c o mmodities; and (3) c o n s t r a i n t s on i t s income (and hence amount of market goods i t can buy) and on i t s time (which can be thought of as b e i n g a l l o c a t e d t o market work, home p r o d u c t i o n , and " l e i s u r e " ) . The household p r o d u c t i o n model may be b r i e f l y d e s c r i b e d as f o l l o w s . 1 5 The household i s assumed t o maximize a l i f e t i m e u t i l i t y f u n c t i o n (17) U=U(C,S) where S i s an aggregate commodity of n o n - c h i l d s e r v i c e s produced by the h o u s e h o l d ; and C, c h i l d s e r v i c e s , i s d e f i n e d as average q u a l i t y , Q, t i m e s the number of c h i l d r e n , N. The use of " q u a l i t y " i n the u t i l i t y f u n c t i o n d e s e r v e s some d i s c u s s i o n . Q u a l i t y , as d e f i n e d h e r e , i s a c t u a l l y average r e a l e x p e n d i t u r e per c h i l d . I t i s not what we would r e f e r t o as i n n a t e q u a l i t i e s , such as t a l e n t or p e r s o n a l t r a i t s . Thus, i t i s 1 5 T h i s model u n d e r l i e s most of the papers g i v e n a t a c o n f e r e n c e on f e r t i l i t y i n 1972 and l a t e r p u b l i s h e d as a Supplement t o the M a r c h / A p r i l 1973 J o u r n a l of P o l i t i c a l Economy . The v e r s i o n d e s c r i b e d here i s taken from Ben-Porath [1973], A more complete d e s c r i p t i o n of the model i s found i n W i l l i s [1973] and S c h u l t z [ 1 976]. Note t h a t t h i s model i s analogous t o the l i f e t i m e l a b o r s u p p l y models. 25 not c l e a r t h a t average r e a l e x p e n d i t u r e per c h i l d s h o u l d be c o n s i d e r e d a good measure of the " q u a l i t y " of the c h i l d . 1 6 Hence, the use of the term " q u a l i t y " f o r average r e a l e x p e n d i t u r e per c h i l d i s somewhat m i s l e a d i n g (Sanderson [ n . d . ] ) . F u r t h e r , i n o r d e r t o d e r i v e t e s t a b l e hypotheses from the model, r e s t r i c t i o n s must be p l a c e d on Q. 1 7 Average r e a l e x p e n d i t u r e e i t h e r can be assumed t o be f i x e d or e x p e n d i t u r e per c h i l d can be assumed t o be e q u a l f o r each c h i l d . Such a s s u m p t i o n s , however, d e s t r o y any v a l u e of i n c l u d i n g Q i n the u t i l i t y f u n c t i o n i n the f i r s t p l a c e . In t h i s e x p o s i t i o n of the model, we assume Q i s f i x e d . T h i s assumption i m p l i e s the u t i l i t y f u n c t i o n can be w r i t t e n as (17') U=U(N,S) N and S are produced by the h o u s e h o l d a c c o r d i n g t o i t s h o u s e h o l d p r o d u c t i o n f u n c t i o n s . These p r o d u c t i o n f u n c t i o n s are assumed t o be l i n e a r l y homogeneous and no j o i n t p r o d u c t i o n i s assumed. These p r o d u c t i o n f u n c t i o n s may be w r i t t e n as (18a) N = f(T_ ,T ,X ) fn mn n (18b) S = g(T_ ,T ,X ) fs ms s where T.. i s the t o t a l . t i m e i n p u t of i n d i v i d u a l i , i=f,m i n t o 1 6 For i n s t a n c e , i f p a r e n t s buy candy f o r t h e i r c h i l d r e n , t h a t i s an e x p e n d i t u r e on them. But can t h a t e x p e n d i t u r e r e a l l y be s a i d t o improve the q u a l i t y of the c h i l d r e n ? 1 7 The r e s t r i c t i o n s on Q a r e n e c e s s a r y , but not s u f f i c i e n t . See below. 26 the p r o d u c t i o n of commodity j , j=n,s, and X_. i s the t o t a l market good i n p u t i n t o the p r o d u c t i o n of commodity j , j=n,s. The r e s o u r c e c o n s t r a i n t s a r e ( 19a) T. + T. + H. = T. i = f ,m i n i s l l (19b) A + H W_ + H W = P(X +X ) f f m m n s where H. i s the number of hours worked by i n d i v i d u a l i ; A i s I n o n - l a b o r income; T. i s the t o t a l amount of time a v a i l a b l e t o l i n d i v i d u a l i ; Wj_ i s the market wage r a t e of i n d i v i d u a l i ; and P i s the p r i c e ( i n d e x ) of the H i c k s i a n composite good, X=X n+X s. M a x i m i z a t i o n of e q u a t i o n (17) s u b j e c t t o e q u a t i o n s (18) and (19) y i e l d s the demand f u n c t i o n s (20a) N = N(P,Wf,Wm,A) (20b) S = S(P,W f,W m,A) DeSerpa [1975] showed t h a t the model y i e l d s unambiguous p r e d i c t i o n s o n l y i n the two-good, t w o - a c t i v i t y c a s e , w i t h time and goods used i n f i x e d p r o p o r t i o n s ( i . e . , L e o n t i e f household p r o d u c t i o n f u n c t i o n s ) . T h i s r e s u l t a r i s e s from the s p e c i f i c a t i o n of c h i l d s e r v i c e s as the pro d u c t of the number of c h i l d r e n and average r e a l e x p e n d i t u r e per c h i l d . The use of t h i s v a r i a b l e i n the u t i l i t y f u n c t i o n means we do not have the s t a n d a r d model of demand t h e o r y . 1 8 Hence, the c o m p a r a t i v e s t a t i c s of demand t h e o r y do not c a r r y t h r o u g h . In the more g e n e r a l case, c e r t a i n 1 8 See Sanderson [n.d.] f o r a comparison of the hou s e h o l d p r o d u c t i o n t h e o r y w i t h the s t a n d a r d t h e o r y of consumer demand. 27 a ssumptions must be made r e g a r d i n g the r e l a t i v e i n t e n s i t y of time i n the p r o d u c t i o n of N and S i n o r d e r t o d e r i v e t e s t a b l e hypotheses r e g a r d i n g the s i g n s of the c o e f f i c i e n t s . The u s u a l assumptions are t h a t N i s more i n t e n s i v e i n the use of the w i f e ' s time than S and t h a t N i s not i n t e n s i v e i n the use of the husband's time ( C a r l i n e r e t a l [ 1 9 8 0 ] ) . These r e s u l t s b r i n g i n t o q u e s t i o n whether the household p r o d u c t i o n model o f f e r s any improvement over c o n v e n t i o n a l demand t h e o r y . P o l l a k and Wachter [1975] demonstrated t h a t u n l e s s the h o u s e h o l d p r o d u c t i o n f u n c t i o n s a r e l i n e a r l y homogeneous and have no j o i n t p r o d u c t i o n , the shadow p r i c e s of the h o u s e h o l d commodities w i l l depend on the commodity bundle consumed by the h o u s e h o l d as w e l l as on i t s t e c h n o l o g y and the p r i c e of market goods. Yet i n a f a m i l y , i t would seem more l i k e l y t h a t j o i n t p r o d u c t i o n and non-constant r e t u r n s t o s c a l e a r e more the r u l e than the e x c e p t i o n . T h i s r e s u l t i m p l i e s t h a t the u s e f u l n e s s of the shadow p r i c e s f o r a n a l y z i n g the demand f o r commodities i s reduced. I t a l s o c a s t s f u r t h e r doubt on the a b i l i t y of the h o u s e h o l d p r o d u c t i o n models t o o f f e r i n s i g h t s beyond those o f f e r e d by c o n v e n t i o n a l demand t h e o r y . However, the household p r o d u c t i o n models d i d p o i n t out the importance of t r e a t i n g time as a r e s o u r c e , and hence as a c o n s t r a i n t , something which neo-c l a s s i c a l demand t h e o r y tended t o i g n o r e . 28 D. The j o i n t e s t i m a t i o n of f e r t i l i t y and 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 s D.1 Household p r o d u c t i o n models In the l a s t s e c t i o n , we o u t l i n e d the household p r o d u c t i o n t h e o r y as i t has been a p p l i e d t o f e r t i l i t y d e c i s i o n s . The model can e a s i l y be extended t o i n c o r p o r a t e l a b o r s u p p l y d e c i s i o n s as w e l l as f e r t i l i t y d e c i s i o n s . A n a l y s e s u s i n g t h i s extended model i n c l u d e W i l l i s [1973], C a i n and Dooley [1976], McCabe and Rosenzweig [1976], F l e i s c h e r and Rhodes [1979], and C a r l i n e r e t a l [ 1 9 8 0 ] . The model i s a s i m p l e e x t e n s i o n of the model d e s c r i b e d by e q u a t i o n s ( l 7 ' ) - ( 2 0 ) of the p r e v i o u s s e c t i o n . In a d d i t i o n t o commodity demand f u n c t i o n s , f a c t o r demand f u n c t i o n s may a l s o be d e r i v e d . Given the c o n s t r a i n t t h a t H_ = T - T - T , the f f fn f s w i f e ' s l a b o r s u p p l y may be w r i t t e n a s : (17) H. = H.(P,W_,W ,A) r r t m The problems d i s c u s s e d i n the l a s t s e c t i o n a l s o a p p l y t o t h i s extended model. Because the household p r o d u c t i o n model does not o f f e r many unambiguous p r e d i c t i o n s , many d i f f e r e n t e m p i r i c a l e s t i m a t e s can be argued t o be c o m p a t i b l e w i t h the model w i t h o u t any c l e a r way of c h o o s i n g among them. Hence, i t i s d i f f i c u l t t o e v a l u a t e e m p i r i c a l work i n l i g h t of the model o n l y , w i t h o u t u s i n g o t h e r a p r i o r i n o t i o n s r e g a r d i n g the i n t e r r e l a t i o n s h i p s among the v a r i a b l e s . 29 D.2 S e q u e n t i a l models A b a s i c d i f f i c u l t y w i t h the household p r o d u c t i o n approach (though c e r t a i n l y not e x c l u s i v e t o i t ) i s t h a t i t s p e c i f i e s a o n e - p e r i o d s t a t i c model t o r e p r e s e n t l i f e t i m e d e c i s i o n s . U n l e s s goods, l e i s u r e , and c h i l d r e n a re p e r f e c t s u b s t i t u t e s i n t e r t e m p o r a l l y , a o n e - p e r i o d model i s not a n • a p p r o p r i a t e framework w i t h i n which t o a n a l y z e i n t e r t e m p o r a l d e c i s i o n - m a k i n g . I n s t e a d , the s e q u e n t i a l and dynamic a s p e c t s of d e c i s i o n - m a k i n g need t o be taken i n t o account e x p l i c i t l y . F u r t h e r d i f f i c u l t i e s a r i s e when m o d e l l i n g the demand f o r c h i l d r e n . T o b i n [1973] d e s c r i b e s some of the problems: They come i n d i s c r e t e i n t e g r a l lumps; they cannot be bought or s o l d i n the u s e d - c h i l d market or scrapped at w i l l ; the r e n t a l market i s h i g h l y i m p e r f e c t ; d e l i v e r y time i s more than n o r m a l l y u n c e r t a i n ; t h e i r q u a l i t i e s a re v e r y u n c e r t a i n ex ante and ex post c o n t r o l of q u a l i t y i s q u i t e l i m i t e d ; t h e i r own r e q u i r e m e n t s and t a s t e s a l t e r the h o u sehold u t i l i t y f u n c t i o n , and so on. Thus, t o t r e a t c h i l d r e n e x a c t l y l i k e consumer d u r a b l e s w i t h o u t t a k i n g i n t o account these s p e c i a l c h a r a c t e r i s t i c s i s l i k e l y t o cause problems i n the e s t i m a t i o n of the demand f o r c h i l d r e n . A f i n a l c o n s i d e r a t i o n i s t h a t the a d d i t i o n of a n o t h e r c h i l d and the t i m i n g of such a d d i t i o n s may be a more i n t e r e s t i n g q u e s t i o n than t o t a l f a m i l y s i z e . 1 9 I f , as Freedman [1 9 6 3 ] , B l a k e [1968], and o t h e r s have argued, t h e r e a re s o c i e t a l norms and 1 9 T h e o r e t i c a l and e m p i r i c a l e v i d e n c e i n f a v o r of t h i s s u g g e s t i o n i s found, f o r example, i n Ben-Porath and Welch [1972], Namboodiri [1972], and Simon [1975a, 1975b]. 30 p e r s o n a l p r e f e r e n c e s p r e s c r i b i n g minimum f a m i l y s i z e , 2 0 then c h i l d r e n of the f i r s t few p a r i t i e s may not be s u b s t i t u t a b l e w i t h o t h e r goods, w h i l e those of l a t e r p a r i t i e s may be so. There a r e o t h e r reasons f o r u s i n g each a d d i t i o n a l b i r t h as the dependent v a r i a b l e r a t h e r than t o t a l number of c h i l d r e n born. F i r s t , u n l e s s the sample i s l i m i t e d t o women o v e r , say, 4 5 y e a r s of age, completed f a m i l y s i z e i s u s u a l l y not known. Second, each b i r t h o c c u r s under i t s own s e t of c i r c u m s t a n c e s and i n f l u e n c e s f u t u r e c i r c u m s t a n c e s , i n c l u d i n g those f a c t o r s d e t e r m i n i n g the p r o b a b i l i t y and t i m i n g of f u t u r e b i r t h s . There a re a l s o reasons f o r t r e a t i n g the l i f e t i m e l a b o r f o r c e p a r t i c i p a t i o n (and hence, l a b o r s u p p l y ) d e c i s i o n s as s e q u e n t i a l . A woman may not p a r t i c i p a t e i n the l a b o r f o r c e f o r a l l of her p o t e n t i a l w o r k i n g l i f e . She may e n t e r and l e a v e the l a b o r f o r c e s e v e r a l t i m e s . The reasons f o r e n t e r i n g and l e a v i n g may v a r y a t each p o i n t i n ti m e . Even i f she works f o r a l l of her l i f e , she may change the amount she works, c h a n g i n g from p a r t -time t o f u l l - t i m e or v i c e v e r s a . Hence, f o r thes e r e a s o n s , and g i v e n the s e q u e n t i a l n a t u r e of d e c i s i o n - m a k i n g i n g e n e r a l , i t would seem t h a t the l i f e t i m e 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 and the f e r t i l i t y d e c i s i o n s h o u l d be examined w i t h i n the c o n t e x t of a s e q u e n t i a l , dynamic model. However, one d i f f i c u l t y w i t h a dynamic s p e c i f i c a t i o n i s the dat a r e q u i r e d . I t i s not p o s s i b l e t o o b t a i n d a t a f o r the whole l i f e t i m e . A l t h o u g h l o n g i t u d i n a l data s e t s l i k e the NLS and PSID c o n t a i n more than one year of data on i n d i v i d u a l s , they o n l y c o v e r p a r t of the l i f e c y c l e . One way of s o l v i n g the da t a 2 0 For example, p r e f e r e n c e s a g a i n s t h a v i n g an o n l y c h i l d . 31 r e q u i r e m e n t i s t o f o l l o w the p r o c e d u r e of Heckman and MaCurdy [1980] t o d e r i v e m a r g i n a l - u t i l i t y - o f - i n c o m e - c o n s t a n t demand f u n c t i o n s . T h i s procedure was used by Hotz [1980]. Hotz s p e c i f i e d an i n t e r t e m p o r a l a d d i t i v e u t i l i t y f u n c t i o n w i t h the number of c h i l d r e n p r e s e n t a t the b e g i n n i n g of p e r i o d t , the amount of l e i s u r e consumed by the w i f e i n p e r i o d t , and the amount of market goods consumed by the household i n p e r i o d t as the arguments. E q u a t i o n s of motion f o r the number of c h i l d r e n , time spent on c h i l d c a r e by the w i f e , " p o t e n t i a l w e a l t h " ( i . e . , f u l l income), the w i f e ' s wage r a t e , and the husband's e a r n i n g s a r e d e r i v e d . M a x i m i z a t i o n of the u t i l i t y f u n c t i o n s u b j e c t t o the e q u a t i o n s of motion y i e l d e d the demand f u n c t i o n f o r a c h i l d i n p e r i o d t> the demand f u n c t i o n f o r the w i f e ' s l e i s u r e i n p e r i o d t , and a f u n c t i o n f o r the w i f e ' s wage. By s p e c i f y i n g the f u n c t i o n s i n terms of the c o n s t a n t m a r g i n a l u t i l i t y of income, Hotz was a b l e t o use one year of d a t a t o e s t i m a t e h i s model. S i n c e some of the c o e f f i c i e n t s e s t i m a t e d were c o e f f i c i e n t s i n the e q u a t i o n s of m o t i o n , p r e d i c t i o n s c o u l d be made about the way d i f f e r e n t v a r i a b l e s would move over t i m e , even though o n l y one year of d a t a was used. Other models which take a s e q u e n t i a l approach t o l i f e t i m e l a b o r s u p p l y and f e r t i l i t y d e c i s i o n s i n c l u d e N e r l o v e and S c h u l t z [1970], Cohen and S t a f f o r d [1974 ], and Rosenzweig [1976 ]. N e r l o v e and S c h u l t z [1970] use d a t a from two c o n s e c u t i v e censuses i n P u e r t o R i c o t o e s t i m a t e t h e i r model. Cohen and S t a f f o r d use s i m u l a t i o n t e c h n i q u e s t o e s t i m a t e t h e i r model, w h i l e Rosenzweig uses o n l y one year of d a t a . Thus, some work 32 remains i n o r d e r t o e s t i m a t e i n t e r r e l a t i o n s h i p s between v a r i a b l e s over t i m e , u s i n g more than one year of d a t a . E. Summary The t h e o r e t i c a l c o n s i d e r a t i o n s d i s c u s s e d i n t h i s c h a p t e r suggest t h a t a j o i n t model of l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y s h o u l d be a s e q u e n t i a l model such t h a t p a s t f e r t i l i t y and 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 s i n f l u e n c e c u r r e n t d e c i s i o n s . The s e q u e n t i a l n a t u r e of the model s u g g e s t s t h a t the dependent v a r i a b l e i n the f e r t i l i t y e q u a t i o n s h o u l d be the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d r a t h e r than the a c t u a l number of c h i l d r e n . Two advantages of t h i s s p e c i f i c a t i o n a re t h a t i t a l l o w s the r e s e a r c h e r t o handle the problem of the d i s c r e t e n a t u r e of c h i l d r e n and t h a t i t a l l o w s one t o d e a l w i t h f a m i l i e s which are not y e t complete. However, the d a t a r e q u i r e m e n t s of such a model are f a i r l y e x t e n s i v e and e s t i m a t i o n d i f f i c u l t . In the next c h a p t e r , we p r e s e n t a model which l o o k s at f e r t i l i t y and 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 s a t a s i n g l e p o i n t i n t i m e . That i s , r a t h e r than c o n c e n t r a t i n g on l i f e t i m e l a b o r s u p p l y and f e r t i l i t y d e c i s i o n s , we w i s h t o look a t s h o r t - r u n l a b o r s u p p l y and f e r t i l i t y d e c i s i o n s . In o r d e r t o do so, we look a t c o n d i t i o n a l d e c i s i o n s . We ta k e the pa s t as g i v e n r a t h e r than s p e c i f y i n g i n t e r r e l a t i o n s h i p s between pa s t and p r e s e n t (and p r e s e n t and f u t u r e ) v a r i a b l e s . Thus, c o n d i t i o n a l on what has happened i n the p a s t , a c o u p l e makes d e c i s i o n s f o r the c u r r e n t p e r i o d on whether they w i l l have another c h i l d and on whether the w i f e w i l l work,. 33 and i f so, how many h o u r s . A l t h o u g h we a v o i d the dynamics of the d e c i s i o n p r o c e s s , we hope t h a t t h i s model w i l l p r o v i d e some i n s i g h t s i n t o the f a c t o r s i n f l u e n c i n g f e r t i l i t y and 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 s a t a g i v e n p o i n t i n t i m e , f o r g i v e n p a s t e x p e r i e n c e s . In t h i s way, we hope t o a v o i d some of the problems which a r i s e from assuming a o n e - p e r i o d l i f e t i m e framework, w h i l e a l s o a v o i d i n g the data r e q u i r e m e n t s and e s t i m a t i o n c o s t s of the dynamic models. 34 Chapter Three A t h e o r e t i c a l model of l a b o r s u p p l y and f e r t i l i t y d e c i s i o n s A. I n t r o d u c t i o n In Chapter Two, v a r i o u s economic t h e o r i e s which have been de v e l o p e d t o e x p l a i n l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y d e c i s i o n s of m a r r i e d women were d i s c u s s e d . In t h i s c h a p t e r , we d e v e l o p a o n e - p e r i o d model i n which the household t a k e s i n f o r m a t i o n from e a r l i e r p e r i o d s as g i v e n when making i t s d e c i s i o n s about the c u r r e n t p e r i o d . We are i n t e r e s t e d i n u s i n g the model t o examine i n t e r r e l a t i o n s h i p s between f e r t i l i t y d e c i s i o n s and l a b o r s u p p l y d e c i s i o n s i n the s h o r t r u n . For t h i s r e a s o n , we d e r i v e demand f u n c t i o n s c o n d i t i o n a l on the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y , r a t h e r than c o n v e n t i o n a l demand f u n c t i o n s , from the u t i l i t y m a x i m i z a t i o n problem. These c o n d i t i o n a l demand f u n c t i o n s a l l o w us t o examine the d i r e c t i n f l u e n c e of f e r t i l i t y b e h a v i o r on l a b o r s u p p l y and of l a b o r s u p p l y d e c i s i o n s on f e r t i l i t y . U n l i k e o n e - p e r i o d l i f e t i m e models or dynamic models, we do not examine the l o n g - r u n f e r t i l i t y and l a b o r s u p p l y b e h a v i o r of the m a r r i e d woman. Of c o u r s e , i f the c o u p l e has a c h i l d i n the c u r r e n t p e r i o d , i t can be c o n s i d e r e d t o be a l o n g - r u n d e c i s i o n i n the sense t h a t the d e c i s i o n i s i r r e v e r s i b l e (once the c h i l d i s b o r n , and assuming the c h i l d i s not g i v e n up f o r a d o p t i o n ) and has r e p e r c u s s i o n s f o r f u t u r e d e c i s i o n s . What we mean by the s h o r t - r u n f e r t i l i t y d e c i s i o n , t h e n , i s the d e c i s i o n t o have a 35 c h i l d i n the c u r r e n t p e r i o d , r a t h e r than the " l o n g - r u n " d e c i s i o n t o have a . c e r t a i n number of c h i l d r e n over the whole l i f e c y c l e . Thus, we ta k e p a s t d e c i s i o n s as exogenous d e t e r m i n a n t s of c u r r e n t b e h a v i o r , but do not model how c u r r e n t d e c i s i o n s might a f f e c t f u t u r e b e h a v i o r . The c o u p l e i s assumed t o view the f u t u r e m y o p i c a l l y . They do not p e r c e i v e w i t h p e r f e c t f o r e s i g h t how f u t u r e d e c i s i o n s w i l l be a f f e c t e d by d e c i s i o n s t h i s p e r i o d . The c h a p t e r i s o r g a n i z e d as f o l l o w s . In the next s e c t i o n , we d e r i v e the model. In the t h i r d s e c t i o n , we d e r i v e an e s t i m a b l e v e r s i o n of the t h e o r e t i c a l model and i t s l i k e l i h o o d f u n c t i o n . We a l s o d i s c u s s i d e n t i f i c a t i o n of the the model. An appendix t o t h i s c h a p t e r p r o v i d e s a more d e t a i l e d d e r i v a t i o n of the l i k e l i h o o d f u n c t i o n . 36 B. A t h e o r e t i c a l model of l a b o r s u p p l y and f e r t i l i t y In o r d e r t o c o n c e n t r a t e on the i n t e r r e l a t i o n s h i p s between a m a r r i e d woman's f e r t i l i t y and 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 s i n the s h o r t run, we make c e r t a i n s i m p l i f y i n g a s s u m p t i o n s : A1. The husband's l a b o r s u p p l y and wage r a t e (and hence h i s l a b o r income) a r e t r e a t e d as exogenous. 1 A2. The woman i s assumed t o f a c e a p e r f e c t l y e l a s t i c demand c u r v e f o r her l a b o r s e r v i c e s . A3. The w i f e i s assumed t o be i n l a b o r market e q u i l i b r i u m . That i s , a l l unemployment i s assumed t o be v o l u n t a r y and i s t r e a t e d as " n o n - p a r t i c i p a t i o n " i n the l a b o r f o r c e . 2 A4. There i s no u n c e r t a i n t y or randomness i n c o n c e p t i o n or the p r e v e n t i o n of c o n c e p t i o n . A5. A l l o t h e r home p r o d u c t i o n i s assumed to be s e p a r a b l e from c h i l d r e n and hours worked. A6. The c o u p l e i s assumed t o know exogenous c u r r e n t p e r i o d v a r i a b l e s , ( e . g . , n o n - l a b o r income), w i t h c e r t a i n t y . Each c o u p l e i s assumed t o maximize a w e l l - b e h a v e d , t w i c e -d i f f e r e n t i a b l e , q u a s i - c o n c a v e u t i l i t y f u n c t i o n 1 One way of v i e w i n g t h i s assumption i s t o assume the husband i s r a t i o n e d i n h i s observed hours of work and t h a t the r a t i o n i s b i n d i n g ( A s h e n f e l t e r [ 1 9 8 0 ] ) . A l t e r n a t i v e l y , we can t h i n k of the model as r e c u r s i v e , where the husband's hours of work are d e t e r m i n e d f i r s t (Rosen [ 1 9 7 6 ] , Nakamura et a l [ 1 9 7 9 ] ) . 2 Another way of s t a t i n g t h i s assumption i s i f the w i f e p a r t i c i p a t e s i n the l a b o r f o r c e , she i s assumed t o be employed. Unemployment means she v o l u n t a r i l y d e c i d e s not t o p a r t i c i p a t e i n the l a b o r f o r c e . A s i m i l a r assumption i s made by Heckman and W i l l i s [ 1 9 7 7 ] . 37 • ( 1 ) U. (L . ,X. ,N.. ;Z. ) i i l i . l where L.. i s the number of hours of l e i s u r e consumed by the i t h w i f e i n the c u r r e n t p e r i o d ; fcL i s the number of c h i l d r e n i n the f a m i l y ; 3 X^ i s a H i c k s i a n composite of of market goods consumed by the i t h h o u s e h o l d ; and Zi i s a v e c t o r of exogenous c h a r a c t e r i s t i c s of the i t h h o u s e h o l d . M a x i m i z a t i o n of e q u a t i o n (1) s u b j e c t t o the a p p r o p r i a t e c o n s t r a i n t s would y i e l d o r d i n a r y demand f u n c t i o n s f o r L^, X^, and N.. However, we are not i n t e r e s t e d i n the demand f o r t o t a l I number of c h i l d r e n ( N ^ ). R a t h e r , we are i n t e r e s t e d i n the demand f o r an a d d i t i o n a l c h i l d i n the c u r r e n t p e r i o d . For t h i s r e a s o n , we r e s t r u c t u r e the model i n the f o l l o w i n g way. L e t R take on two v a l u e s . I f the c o u p l e d e c i d e s not t o have a n o t h e r c h i l d , N. = N 0.. That i s , N 0. i s the number of I i i c h i l d r e n c u r r e n t l y i n the f a m i l y . I f the c o u p l e d e c i d e s t o have another c h i l d , N. t a k e s on the v a l u e N,.= N0.+ 1." G i v e n N, the I ' i ° i • ' c o u p l e maximizes e q u a t i o n (1) s u b j e c t t o 3 We s p e c i f y u t i l i t y as a f u n c t i o n of the number of c h i l d r e n o n l y . W h i l e a more r e a l i s t i c s p e c i f i c a t i o n might i n c l u d e the " q u a l i t y " of the c h i l d r e n , t h e r e a r e d i f f i c u l t i e s i n u s i n g " q u a l i t y " as a v a r i a b l e i n the u t i l i t y f u n c t i o n . See the d i s c u s s i o n on t h i s s u b j e c t i n Chapter Two. A l t e r n a t i v e s p e c i f i c a t i o n s might i n c l u d e the age c o m p o s i t i o n of the c h i l d r e n and o t h e r f a c t o r s which might be e x p e c t e d t o i n f l u e n c e the u t i l i t y r e c e i v e d from c h i l d r e n . A n a l y t i c c o n v e n i e n c e d i c t a t e s the use of s i m p l y the number of c h i l d r e n as an argument of the u t i l i t y f u n c t i o n . " I t i s assumed t h e r e are no m u l t i p l e b i r t h s . 38 (2a) W.L. + X. < W.T.(N.) + A. 1 1 l i i i l (2b) 0 < L. < T.(N.) l i i where the p r i c e of X. has been n o r m a l i z e d t o u n i t y ; 5 A. i s • I . i f a m i l y income, d e f i n e d as a f t e r - t a x n o n - l a b o r income p l u s the husband's a f t e r - t a x l a b o r income i n the i t h h o u s e h o l d ; W. i s the I i t h w i f e ' s a f t e r - t a x wage r a t e (assumed t o be endogenous); 6 T. (N. ) i s the t o t a l amount of time a v a i l a b l e t o the i t h w i f e i i a f t e r c h i l d c a r e time has been d e d u c t e d . 7 I f we s u b s t i t u t e the r e s u l t i n g demand f u n c t i o n s i n t o e q u a t i o n ( 1 ) , we d e r i v e the i n d i r e c t u t i l i t y f u n c t i o n , c o n d i t i o n a l on the number of c h i l d r e n the c o u p l e has i n the c u r r e n t p e r i o d (3) V(W,I;N,Z) = max (U(L,X;N,Z) : WL + X < A + WT(N), L,X 0 < L < T(N)} where the i s u b s c r i p t has been dropped f o r n o t a t i o n a l s i m p l i c i t y . I i s f u l l income, d e f i n e d as 5 Everyone i s assumed t o f a c e the same p r i c e f o r X i n a g i v e n time p e r i o d , but wages are assumed t o v a r y over i n d i v i d u a l s . 6 That i s , i f w i s the b e f o r e - t a x wage r a t e , W=w(l-t), where t i s the m a r g i n a l tax r a t e on f u l l income e x c l u d i n g the w i f e ' s l a b o r income. That i s , t i s the r a t e a t which e x t r a income ( i n t h i s c a s e , the w i f e ' s e a r n i n g s ) would be t a x e d . 7 We deduct c h i l d c a r e from the t o t a l amount of time a v a i l a b l e t o the w i f e because we assume t h a t c h i l d c a r e "must" be u n d e r t a k e n . Hence, i f a c h i l d i s p r e s e n t , the t o t a l amount of d i s c r e t i o n a r y time a v a i l a b l e t o the w i f e d e c r e a s e s . A g a i n , we c o u l d have a more r e a l i s t i c s p e c i f i c a t i o n by t a k i n g i n t o account the a g e - d i s t r i b u t i o n of the c h i l d r e n , e t c . For s i m p l i c i t y , we do n o t . See Hotz [1980] f o r a s p e c i f i c a t i o n of time spent on c h i l d c a r e as a f u n c t i o n of the age d i s t r i b u t i o n of the c h i l d r e n . 39 (4) I = I(N) = A + WT(N) A p p l i c a t i o n of Roy's I d e n t i t y t o e q u a t i o n (3) y i e l d s the uncompensated demand c u r v e s f o r l e i s u r e and goods, c o n d i t i o n a l on whether an a d d i t i o n a l c h i l d i s chosen. where I. i s the f u l l income a s s o c i a t e d w i t h h a v i n g N, c h i l d r e n , 3 ? j = 0,1. A s i m i l a r d e r i v a t i o n y i e l d s the demand f o r X. The demand f o r hours of work by the w i f e i s then s i m p l y d e r i v e d from H = T(N) - L. We assume the w i f e has some p o s i t i v e amount of l e i s u r e . 8 However, t h e r e i s a p o s i t i v e p r o b a b i l i t y t h a t l e i s u r e w i l l e q u a l t o t a l d i s p o s a b l e t i m e . That i s , t h e r e i s a p r o b a b i l i t y t h a t hours worked i n the c u r r e n t p e r i o d w i l l be z e r o . In t h i s c a s e , we can no l o n g e r use the market wage r a t e as an argument i n the demand f o r l e i s u r e f u n c t i o n as i t i s not observed f o r non-w o r k e r s . I n s t e a d , we must s p e c i f y the i n d i r e c t u t i l i t y f u n c t i o n i n terms of the shadow wage r a t e . E q u a t i o n (5) can then be r e w r i t t e n as 8 A l t e r n a t i v e l y , we assume t h a t the w i f e never works a l l of the time a v a i l a b l e . -3V(W,I^;N ,Z)/3W j = 0 , 1 3V(W,I.;N ,Z)/3I 3 D (6) H = H(W,I.;N.,Z) j = 0 , 1 40 -9V(W*,1 . ;N.,Z)/3W* (5') L(W*,Lj ;N, ,Z) = 3 —3 j = 0,1 •3V(W*,I .;N .,Z)/3I 3 3 where W* i s the shadow wage r a t e . Roy's I d e n t i t y s t i l l h o l d s f o r the c o r n e r s o l u t i o n i n the sense t h a t t h e r e w i l l e x i s t some shadow wage r a t e a t which the woman w i l l choose z e r o hours of work. 9 When hours of work are p o s i t i v e , the woman w i l l a d j u s t her hours of work u n t i l the shadow wage r a t e e q u a l s the market wage r a t e . When hours of work are z e r o , the shadow wage, or p r i c e of the woman's ti m e , w i l l be g r e a t e r than the market wage r a t e . E q u a t i o n (6) can then be d e r i v e d i n terms of the shadow wage r a t e r a t h e r than the market wage r a t e from e q u a t i o n ( 5 ' ) . ( 6 ' ) H = H (W* , I . ; N . , Z ) j=0,1 where W*=W i f hours of work are p o s i t i v e . I f the w i f e works, the shadow wage r a t e w i l l be a f u n c t i o n of the w i f e ' s hours of work, f a m i l y income, and the w i f e ' s e a r n i n g s , 1 0 c o n d i t i o n a l on the exogenous c h a r a c t e r i s t i c s of the f a m i l y and the number of c h i l d r e n . I f she does not work, the shadow wage r a t e w i l l depend o n l y upon f a m i l y income, 9 For an e x p o s i t i o n of the c o r n e r s o l u t i o n problem u s i n g the d i r e c t u t i l i t y f u n c t i o n , see Nakamura e t a l [19 7 9 ] , 1 0 Heckman [1974] d e r i v e s the shadow wage r a t e as a f u n c t i o n of the w i f e ' s e a r n i n g s i n the appendix t o the paper. In h i s s p e c i f i c a t i o n i n the t e x t , however, he drops the term. T h i s m o d i f i e d s p e c i f i c a t i o n assumes t h a t t h e r e i s no income e f f e c t from the w i f e ' s wage, i m p l y i n g t h a t t h e r e can be no backward-bending l a b o r s u p p l y c u r v e . See Nakamura and Nakamura [ 198' 1 3 f o r a f u r t h e r d i s c u s s i o n of t h i s p o i n t . 41 c o n d i t i o n a l on the exogenous c h a r a c t e r i s t i c s and the number of c h i l d r e n . 1 1 We can e x p r e s s these r e l a t i o n s h i p s as The c o u p l e then makes the d e c i s i o n whether t o have an a d d i t i o n a l c h i l d based on the f o l l o w i n g d e c i s i o n r u l e . (8) max {V(W*,I 0;N 0,Z), V(W*,I,;N,,Z)} where the i n d i r e c t u t i l i t y f u n c t i o n from e q u a t i o n ( 3 ) i s now e x p r e s s e d i n terms of the shadow wage r a t e t o . t a k e i n t o account women who do not work. V(W*,I 0;N 0,Z) i s the u t i l i t y d e r i v e d from not h a v i n g an a d d i t i o n a l c h i l d , and V(W*,I,;N,,Z) i s the u t i l i t y the c o u p l e r e c e i v e s i f they do have an a d d i t i o n a l c h i l d . A l t e r n a t i v e l y , we can c h a r a c t e r i z e the d e c i s i o n t o have an a d d i t i o n a l c h i l d i n a s l i g h t l y d i f f e r e n t way. D e f i n e a v a r i a b l e , C*, such t h a t (9) C* = V(W*,I,;N,,Z) - V(W*,I 0;N 0,Z) where C* may be thought of as measuring the d e s i r e f o r c h i l d r e n . 1 1 The shadow wage r a t e w i l l a l s o depend on the p r i c e of market goods. R e c a l l , however, t h a t we have n o r m a l i z e d the p r i c e of X to u n i t y , and t h a t i t i s assumed t o be the same f o r a l l h o u s e h o l d s . Hence, the p r i c e of X does not appear e x p l i c i t l y i n the shadow wage r a t e f u n c t i o n . (7) i f H > 0 i f H = 0 j = 0, 1 42 The c o u p l e w i l l have a c h i l d i n the c u r r e n t p e r i o d i f C* i s g r e a t e r than z e r o and w i l l not have a c h i l d o t h e r w i s e . I t i s e a s i l y seen t h a t C* may be e x p r e s s e d as a f u n c t i o n of f u l l income, the number of c h i l d r e n c u r r e n t l y i n the f a m i l y , exogenous c h a r a c t e r i s t i c s of the f a m i l y , and the shadow wage r a t e . 1 2 E q u a t i o n s (6') and (8) (or (9)) a r e then s o l v e d s i m u l t a n e o u s l y t o y i e l d a s o l u t i o n f o r the d e c i s i o n t o work and the d e c i s i o n t o have another c h i l d i n the c u r r e n t p e r i o d . I f we s u b s t i t u t e e q u a t i o n (7) i n t o e q u a t i o n s (6') and (9) f o r the shadow wage r a t e , we d e r i v e the f o l l o w i n g e q u a t i o n s 1 3 (1Oa) H = H(A,W;Nj,Z) (10b) C* = C*(A,W,H;N_.,Z) j=0,1 We a l s o assume the o f f e r e d wage r a t e , W, i s endogenous t o our model and s p e c i f y (10c) W = W(Z*) where Z* i s a v e c t o r of exogenous d e t e r m i n a n t s of the o f f e r e d wage. 1 2 Note t h a t t h e r e i s no e x p l i c i t demand f u n c t i o n f o r c h i l d r e n i n t h e s e s p e c i f i c a t i o n s . T h i s r e s u l t a r i s e s from the d i s c r e t e n a t u r e of the demand f o r an a d d i t i o n a l c h i l d . For a somewhat d i f f e r e n t approach t o m o d e l l i n g the demand f o r d i s c r e t e goods, u s i n g the e x p e n d i t u r e f u n c t i o n , see S m a l l and Rosen [1981]. 1 3 S i n c e I=A+WT(N) and W* i s a f u n c t i o n of A, W, and N, we can w r i t e H and C* as f u n c t i o n of A, W, and N. 43 The system of e q u a t i o n s thus d e r i v e d may be summed up as f o l l o w s . The wage r a t e i s c o m p l e t e l y determined by v a r i a b l e s exogenous t o the system. Hours of work (or l e i s u r e ) a re then d e t e r m i n e d as a f u n c t i o n of the wage r a t e , the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y , whether the c o u p l e has another c h i l d t h i s p e r i o d , and o t h e r v a r i a b l e s . Whether they have another c h i l d t h i s p e r i o d i s i n t u r n d e t e r m i n e d by the wage r a t e , hours worked, and the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y , as w e l l as o t h e r v a r i a b l e s , such as f u l l income and s ocio-demographic c h a r a c t e r i s t i c s of the f a m i l y . W i t h t h i s s p e c i f i c a t i o n , t h e n , we can examine whether t h e r e are any s i g n i f i c a n t d i r e c t e f f e c t s of each d e c i s i o n on the o t h e r , as w e l l as i n d i r e c t e f f e c t s through income. The u s u a l r e s t r i c t i o n s p l a c e d on the demand f u n c t i o n by t h e o r y h o l d i n the case of the c o n d i t i o n a l demand f u n c t i o n s f o r L (H) and X ( P o l l a k [ 1 9 6 9 ] ) . That i s , the demand f u n c t i o n s are homogeneous of degree z e r o i n p r i c e and income, and the S l u t s k y m a t r i x of s u b s t i t u t i o n e f f e c t s i s s y m m e t r i c a l and n e g a t i v e semi-d e f i n i t e . Note t h a t these r e s t r i c t i o n s h o l d c o n d i t i o n a l on the c h o i c e t o have an a d d i t i o n a l c h i l d . S i n c e no e x p l i c i t demand f u n c t i o n e x i s t s f o r an a d d i t i o n a l c h i l d , we cannot t a l k about c o n v e n t i o n a l demand r e s t r i c t i o n s i n t h i s c o n t e x t . Only r e v e a l e d p r e f e r e n c e c o n s i d e r a t i o n s , i n the sense t h a t the c o u p l e r e v e a l s t h e i r p r e f e r e n c e f o r an a d d i t i o n a l c h i l d by h a v i n g one, are a p p l i c a b l e . I f we had assumed a s p e c i f i c f u n c t i o n a l form f o r the u t i l i t y f u n c t i o n , some f u r t h e r r e s t r i c t i o n s might have been p l a c e d on the parameters of the demand f u n c t i o n s . However, 44 r a t h e r than s p e c i f i y i n g a p a r t i c u l a r f u n c t i o n a l form f o r the u t i l i t y f u n c t i o n , we l e a v e i t i n g e n e r a l f o r m . 1 " I n s t e a d , we work w i t h a l i n e a r a p p r o x i m a t i o n t o the demand f u n c t i o n s . These l i n e a r demand f u n c t i o n s may be thought of as T a y l o r s e r i e s a p p r o x i m a t i o n s of a more c o m p l i c a t e d f u n c t i o n a l form, where the h i g h e r o r d e r terms have been s e t e q u a l t o z e r o . Note t h a t by s p e c i f y i n g the demand f u n c t i o n s as l i n e a r , t h e r e i s no l o n g e r a s t r a i g h t f o r w a r d c o n n e c t i o n between t h e parameters of the demand f u n c t i o n and the parameters of the u t i l i t y f u n c t i o n . 1 5 On the o t h e r hand, t h e r e i s a g a i n i n s i m p l i c i t y , which i s imp o r t a n t when e s t i m a t i n g t h i s model. C. An eco n o m e t r i c model of l a b o r s u p p l y and f e r t i l i t y C.1 S p e c i f i c a t i o n of the model In o r d e r t o make the model o p e r a t i o n a l , we need t o s p e c i f y a d i s t u r b a n c e s t r u c t u r e . We a l s o want t o s p e c i f y the e q u a t i o n s such t h a t they t a k e . i n t o account the s p e c i a l n a t u r e of the v a r i a b l e s ; t h a t i s , t h a t hours are bounded from below by z e r o , t h a t wages a r e not obser v e d f o r non-workers, and t h a t the d e c i s i o n . t o have an a d d i t i o n a l c h i l d i s a dichotomous v a r i a b l e . A l s o , wages are assumed t o be endogenous, so a wage e q u a t i o n 1' We d i d attempt t o work out the problem i n terms of a q u a d r a t i c u t i l i t y f u n c t i o n . The r e s u l t i n g demand f u n c t i o n s were h i g h l y n o n - l i n e a r and not amenable t o e s t i m a t i o n . 1 5 However, Hausman [ 1 9 8 1 ] has d e r i v e d c o n d i t i o n s under which a l i n e a r demand f u n c t i o n can be i n t e g r a t e d up t o a w e l l - b e h a v e d e x p e n d i t u r e f u n c t i o n and, hence, u t i l i t y f u n c t i o n . 45 must a l s o be s p e c i f i e d . We s p e c i f y an a d d i t i v e d i s t u r b a n c e s t r u c t u r e . T h i s s p e c i f i c a t i o n i m p l i e s t h a t the o n l y randomness i n p r e f e r e n c e s among i n d i v i d u a l s o c c u r s i n the i n t e r c e p t term. A l t e r n a t i v e l y , a d d i t i v e d i s t u r b a n c e s imply the o n l y d i s t u r b a n c e s a r e those of o p t i m i z a t i o n (on the p a r t of the i n d i v i d u a l ) or of measurement (on the p a r t of the o b s e r v e r ) . To d e a l w i t h the problems posed by the na t u r e of the v a r i a b l e s , we use the t e c h n i q u e s of Heckman [1974, 1976] and Cox [19 7 0 ] , The complete model may be w r i t t e n a s 1 6 (11a) H*• = X,b, + a,W + d,N + k,AC + u, (11b) C* = X 2 b 2 + a2W + g 2H + d 2N + k 2AC + u 2 (11c) W = X 3b 3•+ u 3 (1 Id) H = (H* i f H* > 0 0^ o t h e r w i s e (11e) AC= (1 i f C* > 0 0 o t h e r w i s e ( 1 1 f ) W i s observed i_f_f H > 0 where X^, j = 1,2,3 are v e c t o r s of exogenous v a r i a b l e s ; N i s now i n t e r p r e t e d as the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y ; AC i s a dichotomous v a r i a b l e which t a k e s on the v a l u e of 1 i f the c o u p l e has a c h i l d i n the c u r r e n t p e r i o d , and the v a l u e of z e r o , o t h e r w i s e ; W i s the o f f e r e d wage; a, b, d, g and k are 1 6 T h i s model i s s i m i l a r t o the one d e s c r i b e d by Heckman [1978b]. 46 p a r a m e t e r s ; and U j , j = 1 ,2,3 are the d i s t u r b a n c e terms, which are assumed t o be d i s t r i b u t e d n o r m a l l y w i t h mean z e r o and v a r i a n c e -c o v a r i a n c e m a t r i x , I . H* and C* a r e l a t e n t v a r i a b l e s which d e f i n e H and AC, r e s p e c t i v e l y . H* can be thought of as a c o n t i n u o u s v a r i a b l e r e p r e s e n t i n g the d e s i r e t o work. The d e s i r e t o work can be' e x p r e s s e d as a f u n c t i o n the wage r a t e and o t h e r v a r i a b l e s as i n e q u a t i o n (11 a) (Heckman [ 1 9 7 4 ] ) . However, we o n l y observe the n o n - n e g a t i v e range of H*. T h i s t r u n c a t i o n i s s p e c i f i e d by e q u a t i o n (11d). C* i s d e f i n e d i n e q u a t i o n (9) above. I t r e p r e s e n t s the d e s i r e t o have an a d d i t i o n a l c h i l d i n the c u r r e n t p e r i o d . When the d e s i r e t o have an a d d i t i o n a l c h i l d c r o s s e s a c e r t a i n t h r e s h o l d ( i n t h i s c a s e , z e r o ) , we observe a c h i l d i s born, and AC becomes one. E q u a t i o n (11b) needs some f u r t h e r e x p l a n a t i o n . I t s t a t e s t h a t the d e s i r e f o r c h i l d r e n depends on whether a c h i l d i s born i n the c u r r e n t p e r i o d , w h i l e whether a c h i l d i s born depends on the d e s i r e f o r c h i l d r e n p a s s i n g some t h r e s h o l d . I f the d e s i r e t o have a c h i l d passes a c e r t a i n t h r e s h o l d ( i n t h i s c a s e , z e r o ) , the c o u p l e has a c h i l d . The f a c t t h a t they have a c h i l d i s then assumed t o a f f e c t t h e i r d e s i r e by amount k 2. As Heckman [1978b] p r o v e s , f o r the model t o e x i s t , AC must not appear i n the reduced form e q u a t i o n f o r C*. That i s , the reduced form c o e f f i c i e n t must eq u a l z e r o . However, i n the s t r u c t u r a l e q u a t i o n we can have t h i s s i m u l t a n e i t y . 47 C.2 D e r i v a t i o n of the 1 i k e l i h o o d f u n c t i o n The l i k e l i h o o d f u n c t i o n of the model d e s c r i b e d i n s e c t i o n B.2 i s more f u l l y e x p l a i n e d i n the appendix t o t h i s c h a p t e r . T h i s s e c t i o n g i v e s a b r i e f e x p o s i t i o n of i t s d e r i v a t i o n . I t i s u s e f u l t o w r i t e the l i k e l i h o o d f u n c t i o n i n terms of the reduced form of e q u a t i o n s (11). To d e r i v e the reduced form e q u a t i o n s , we need to c o n s i d e r two c a s e s : H > 0 and H = 0. For H > 0, the reduced form of e q u a t i o n s (11a) and (11b) i s 1 7 X,b, + X 3 b 3 a , + d,N + k,AC + u, + u 3 a , X 2 b 2 + X 3 b 3 ( a , g 2 + a 2 ) + X,b,q2 + N(d,g 2+d 2) + U i g 2 + u 2 + u 3 ( a , g 2 + a 2 ) The o t h e r e q u a t i o n s of the model remain the same, except now the H* and C* i n e q u a t i o n s (11d) and (11e) , r e s p e c t i v e l y , a re the reduced form, r a t h e r than the s t r u c t u r a l , v a r i a b l e s . For H = 0, the reduced form f o r H* i s the same as (12a). For C*, however, the H term drops o u t , and C* becomes (12b') C* = X 2 b 2 + X 3 b 3 a 2 + d 2N + u 2 + u 3 a 2 The o t h e r e q u a t i o n s are a g a i n the same, w i t h a r e d e f i n i t i o n of H* and C*, except W i s now not o b s e r v e d . We can r e w r i t e (12) i n compact n o t a t i o n . (12a) H* = (12b) C* = 1 7 R e c a l l t h a t the r e s t r i c t i o n k-g 2+k 2 = 0 has been imposed. 48 (13a) H* = Z,p, + v, ( 13b) C* = Z 2 p 2 + v 2. (13b') C* = Z 2*p 2*+ v 2 * where the d e f i n i t i o n s of the v e c t o r s , Z j , Z j * , P j , P j * , V j , V j * , j=1,2, a r e d e s c r i b e d i n the appendix t o t h i s c h a p t e r . The reduced form of the wage e q u a t i o n i s s i m p l y the s t r u c t u r a l e q u a t i o n . We can r e w r i t e the s t r u c t u r a l wage e q u a t i o n as (1 3c) ;w = z 3 p 3 + v 3 where Z 3 p 3 = X 3 b 3 and v 3 = u 3 . To d e r i v e the l i k e l i h o o d f u n c t i o n , we need t o s p e c i f y f o u r cases or s t a t e s of the w o r l d : S I : H>0, AC=1 S2: H>0, AC=0 S3: H=0, AC=1 S4: H=0, AC=0 The l i k e l i h o o d f u n c t i o n i s then composed of fo u r p a r t s , c o r r e s p o n d i n g t o each of the f o u r c a s e s . SI: H>0, AC=1 With H* g r e a t e r than z e r o , both hours and wages are obs e r v e d . However, C* i s not ob s e r v e d . We know o n l y i t s s i g n . In t h i s c a s e , we know C* i s g r e a t e r than z e r o because AC i s observe d t o be equ a l t o one. That i s , we observe a c h i l d was 49 born i n the c u r r e n t p e r i o d . L e t the t r i v a r i a t e normal d i s t r i b u t i o n of the reduced form d i s t u r b a n c e s be d e f i n e d as n ( v 1 , v 2 , v 3 ) . Then the d e n s i t y f u n c t i o n f o r a woman w i t h p o s i t i v e hours of work, who has a c h i l d i n the c u r r e n t p e r i o d i s 00 (14a) P1 = / n ( v , , v 2 , v 3 ) d v 2 - z 2 P 2 That i s , s i n c e we o n l y know C* i s p o s i t i v e but not i t s a c t u a l v a l u e , we i n t e g r a t e over t h a t p a r t of the domain of v 2 f o r which C* i s p o s i t i v e . S2: H>0, AC=0 T h i s case i s j u s t the r e v e r s e of S1. We now i n t e g r a t e over the domain of v 2 f o r which C* i s n e g a t i v e (and hence, AC=0). -Z„P 2 (14b) P2 = 7 n ( v , , v 2 , v 3 ) d v 2 — CO S3: H=0, AC=1 The p a r t s of the l i k e l i h o o d f u n c t i o n f o r S3 and S4 a r e more c o m p l i c a t e d than P1 and P2 because now H=0 and W i s not obs e r v e d . Hence, we must i n t e g r a t e over a l l t h r e e d i s t u r b a n c e terms. —z p °° 00 (14c) P3 = } 1 / / n ( v , , v 2 * , v 3 ) d v 3 d v 2 * d v ! — 00 — Z * D * —00 2^2 S i n c e the wage r a t e i s not ob s e r v e d , we i n t e g r a t e v 3 over i t s 50 e n t i r e range. In a sense, we " i n t e g r a t e o u t " the wage r a t e . v, and v 2 * a r e i n t e g r a t e d over the r e l e v a n t ranges needed t o ensure ( i n t h i s case) H* < 0 and C* > 0. S4: H=0, AC=0 P4 i s s i m i l a r t o P3; o n l y the l i m i t s of i n t e g r a t i o n f o r v 2 * c h a n g e . 1 8 - Z l P l -Z*p* -(I4d) P4 = / . / / n ( v , , v 2 * , v 3 ) d v 3 d v 2 * d v 1 — 00 —CO —CO Each P j , j = 1,...,4, i s c a l c u l a t e d f o r ev e r y i n d i v i d u a l i n the r e l e v a n t j t h c a t e g o r y . The l i k e l i h o o d i s then the pr o d u c t of a l l the P j ' s . (15) L = n PI n P2 n P3 n P4 SI S2 S3 S4 where n means the p r o d u c t over a l l women i n the j t h c a t e g o r y . C.3 Ident i f i c a t i o n of the model The model d e s c r i b e d by e q u a t i o n s (11) has two p a r t s t o i t , a p r o b i t p a r t , c h a r a c t e r i z e d by e q u a t i o n s (11b) and ( l i e ) , and a p a r t analogous t o the model i n Heckman [ 1974]. T h i s p a r t i s d e s c r i b e d by the r e m a i n i n g f o u r e q u a t i o n s . I t c o n s i s t s of t o b i t 1 8 P3 and P4 can be somewhat s i m p l i f i e d . See the appendix t o t h i s c h a p t e r . 51 e q u a t i o n s f o r hours worked ( e q u a t i o n s (11a) and ( I 1 d ) ) and the wage e q u a t i o n s ( e q u a t i o n s (11c) and ( I 1 f ) . The t o b i t s p e c i f i c a t i o n ensures e s t i m a t e d hours a r e g r e a t e r than z e r o (Tobin [ 1 9 5 8 ] ) . 1 9 The p r o b i t model p e r m i t s e s t i m a t i o n of the reduced form c o e f f i c i e n t s o n l y up t o a f a c t o r of p r o p o r t i o n a l i t y . Hence, i t w i l l not be p o s s i b l e t o r e c o v e r unique e s t i m a t e s of the s t r u c t u r a l parameters of e q u a t i o n (11b) or of the v a r i a n c e -c o v a r i a n c e m a t r i x . The reduced form c o e f f i c i e n t s f o r the hours and wage e q u a t i o n s , however, can be i d e n t i f i e d . Thus, the s t r u c t u r a l parameters of the hours and wage e q u a t i o n s can be i d e n t i f i e d . To see t h i s , r e w r i t e the reduced form of e q u a t i o n s (11) as (16a) H* = X , p n + X 3 p 1 3 + Np 1 I ( + A C p 1 5 + v, (16b) C* = X , p 2 1 ' + X 2 p 2 2 ' + X 3 p 2 3 ' + Np 2«' + v 2 (16c) W = X 3 p 3 3 + v 3 1 9 Wages, of c o u r s e , a l s o cannot be n e g a t i v e . T h i s problem i s s o l v e d by u s i n g the n a t u r a l l o g a r i t h m of the wage r a t e i n e s t i m a t i o n . 52 where P i 1 : = b i P i 3 : = b 3 a , P i U : = d, P i 5 : = k i p 2 t 1 = b^q2/u2 p 2 ! 2 = b 2 / u 2 p 2 I 3 = b 3 ( a , g 2 + a 2 ) / u 2 P2 ! 4 = ( d , g 2 + d 2 ) / u 2 P 3 3 : = b 3 where u 2 = u 2 2 1 / 2 i s the s t a n d a r d d e v i a t i o n of v 2 . I t i s easy t o see we can i d e n t i f y the f o l l o w i n g s t r u c t u r a l p a r a m e t e r s : a i = P l 3 j / p 3 3 j g 2 ' = p 2 1 j / P i 1 j b 2 ' = p 2 2 ' d1 = Pi« k1 = P i s b 3 = P 3 3 where p 1 3 j and p 3 3 j r e f e r t o the c o e f f i c i e n t s on e v e r y v a r i a b l e j found i n X 3 but not X,; p 2 1 j and p , i j r e f e r t o the c o e f f i c i e n t s on every v a r i a b l e j found i n X 1 f but not X 2 or X 3; and g 2 ' = g 2 / o 2 , b 2 ' = b 2 / o 2 r and so f o r t h . W i th these c o e f f i c i e n t s , we can c a l c u l a t e 53 a 2 ' = ( P 2 3 V P 3 3 ) " 9 2 ' a , d 2 ' = p 2 u' " g 2 ' d ! S i m i l a r l y , i t w i l l be p o s s i b l e t o e s t i m a t e o n l y p a r t of the v a r i a n c e - c o v a r i a n c e m a t r i x . We cannot i d e n t i f y u 2 2 , t n e v a r i a n c e of C*, s i n c e C* i s not o b s e r v e d . We can i d e n t i f y u 1 1 r u 1 3 , and o 3 3 s i n c e they are not concerned w i t h the a d d i t i o n a l c h i l d e q u a t i o n . We a l s o can i d e n t i f y u,,, ff13i and tf33, because u 1 w o 1 3 , and o 3 3 are f u n c t i o n s of o n l y these elements of the s t r u c t u r a l v a r i a n c e - c o v a r i a n c e m a t r i x . (See the appendix.) However, we can e s t i m a t e the o t h e r terms o n l y up t o a f a c t o r of p r o p o r t i o n a l i t y : u 1 2'= «i 2/u 2 and o 2 3 ' = u 2 3 / u 2 (Heckman [ 1 9 7 8 b ] ) . T h i s r e s u l t a l s o a p p l i e s f o r the c o r r e s p o n d i n g s t r u c t u r a l c o v a r i a n c e s . Hence, we need t o f i x u 2 2 i n o r d e r t o be a b l e t o i d e n t i f y the m o d e l . 2 0 In g e n e r a l , adherence t o the s t a n d a r d rank or o r d e r c o n d i t i o n s and a n o r m a l i z a t i o n of o 2 2 w i l l a l l o w i d e n t i f i c a t i o n of the system. 2 0 Note t h a t t h i s r equirement i s the s t a n d a r d one f o r i d e n t i f i c a t i o n of a p r o b i t model. See, f o r example, Maddala [1 9 7 7 ] . 54 D. Summary In t h i s c h a p t e r we d e r i v e d a model t o e x p l a i n the demand f o r hours of work by the w i f e , c o n d i t i o n a l on whether another c h i l d i s born, and the demand f o r an a d d i t i o n a l c h i l d , c o n d i t i o n a l on hours worked by the w i f e , w i t h i n a s i m u l t a n e o u s e q u a t i o n s framework. We use c o n d i t i o n a l demand f u n c t i o n s because we w i s h t o t e s t whether t h e r e a re any d i r e c t e f f e c t s , as w e l l as i n d i r e c t e f f e c t s , of each c h o i c e on the o t h e r . The model i s a s i n g l e p e r i o d model. I t l o o k s at o n l y one p e r i o d i n a m u l t i - p e r i o d l i f e t i m e . A more r e a l i s t i c model would have a dynamic s p e c i f i c a t i o n of the demands f o r hours of work and f o r an a d d i t i o n a l c h i l d . However, d a t a and c o m p u t a t i o n a l r e q u i r e m e n t s l e d us t o choose the s i m p l e r s p e c i f i c a t i o n . In an e f f o r t , t o c o n c e n t r a t e on the e s s e n t i a l a s p e c t s of the problem, we have made c e r t a i n s i m p l i f y i n g a s s u m p t i o n s . These a s s u m p t i o n s , such as t a k i n g the husband's hours of work and wage r a t e as g i v e n and assuming p e r f e c t f e r t i l i t y c o n t r o l , l e s s e n the r e a l i s m of the model. However, the l i k e l i h o o d f u n c t i o n d e r i v e d f o r the model i s c o m p l i c a t e d , even w i t h o n l y t h r e e endogenous v a r i a b l e s . Any more would make e s t i m a t i o n i n t r a c t a b l e . A l s o , i n the case of f e r t i l i t y c o n t r o l , l a c k of data p r e v e n t s us from t a k i n g i t i n t o a c c o u n t . We have a l s o s i m p l i f i e d the model by assuming l i n e a r demand f u n c t i o n s . W h i l e under c e r t a i n c o n d i t i o n s i t i s p o s s i b l e t o i n t e g r a t e a l i n e a r demand e q u a t i o n back t o a w e l l - b e h a v e d u t i l i t y f u n c t i o n (Hausman [ 1 9 8 1 ] ) , the l i n e a r s p e c i f i c a t i o n may not c o r r e c t l y p o r t r a y the. d e c i s i o n - m a k i n g of the c o u p l e . D e s p i t e these s i m p l i f i c a t i o n s , we f e e l the model w i l l be 55 u s e f u l f o r i n s i g h t s i n t o the f a c t o r s a f f e c t i n g a c o u p l e ' s d e c i s i o n s on f e r t i l i t y and the w i f e ' s l a b o r s u p p l y a t a g i v e n p o i n t i n t i m e . Thus, a l t h o u g h we may not be a b l e t o say a n y t h i n g about l i f e t i m e d e c i s i o n s , we can s t i l l make p r e d i c t i o n s about d e c i s i o n s made r e g a r d i n g a g i v e n time p e r i o d . 56 Chapter Four E s t i m a t i o n of the model I : A comparison of d i f f e r e n t e s t i m a t i o n t e c h n i q u e s A. I n t r o d u c t i o n In Chapter Three, we deve l o p e d a model t o e x p l a i n the i n t e r r e l a t i o n s h i p s between f e r t i l i t y and l a b o r s u p p l y d e c i s i o n s by m a r r i e d women. Our model d i f f e r s from e a r l i e r models by s p e c i f y i n g d i r e c t e f f e c t s of t h e . f e r t i l i t y d e c i s i o n on the l a b o r s u p p l y d e c i s i o n , and v i c e v e r s a , as w e l l as s p e c i f y i n g i n d i r e c t e f f e c t s t h r o u g h income and o t h e r v a r i a b l e s . 1 In t h i s c h a p t e r , we p r e s e n t e m p i r i c a l e s t i m a t e s of our model. In the next s e c t i o n , we d i s c u s s the data used f o r e s t i m a t i o n . In the t h i r d s e c t i o n , we d e s c r i b e the s p e c i f i c a t i o n of the e q u a t i o n s . In the f o u r t h s e c t i o n , we d i s c u s s the maximum l i k e l i h o o d e s t i m a t i o n p r o c e d u r e and compare the maximum l i k e l i h o o d e s t i m a t e s w i t h e s t i m a t e s o b t a i n e d by a l t e r n a t i v e methods of e s t i m a t i o n . Such a comparison o f f e r s some i n d i c a t i o n of b i a s e s a r i s i n g from t r e a t i n g the l a b o r s u p p l y and f e r t i l i t y d e c i s i o n s as u n r e l a t e d t o one a n o t h e r . However, a Monte C a r l o s t u d y , where the t r u e parameters a r e known, would be n e c e s s a r y 1 M i n c e r [1963] argues t h a t f e r t i l i t y and hours of work a r e not c a u s a l l y r e l a t e d . R a t h e r , they a re determined s i m u l t a n e o u s l y by. the same economic v a r i a b l e s . Our s p e c i f i c a t i o n of the model o f f e r s one t e s t of t h i s h y p o t h e s i s . 57 f o r a d e f i n i t i v e statement r e g a r d i n g any b i a s . In the l a s t s e c t i o n , we p r e s e n t a summary of our r e s u l t s . B. The data The d a t a used f o r e s t i m a t i n g the model come from the M i c h i g a n P a n e l Survey of Income Dynamics. The survey i s l o n g i t u d i n a l w i t h d a t a on over 6,000 f a m i l i e s 2 f o r an e l e v e n -year p e r i o d , 1968-1978. In 1976, q u e s t i o n s were asked of wives s e p a r a t e l y from those asked of the head of the h o u s e h o l d . These q u e s t i o n s p r o v i d e d g r e a t e r d e t a i l on m a r r i e d women's working h a b i t s and f e r t i l i t y p a t t e r n s than had h e r e t o f o r e been a v a i l a b l e . For t h i s r e a s o n , the 1976 i n t e r v i e w was chosen as the base i n t e r v i e w . The 1976 i n t e r v i e w i n c l u d e s d a t a on both 1975 v a r i a b l e s ( e . g . , income, hours of work) and 1976 v a r i a b l e s ( e . g . , age of youngest c h i l d , number c u r r e n t l y i n the f a m i l y ) . Hence, 1975 became our base y e a r , w i t h the 1976 i n t e r v i e w as the source of the 1975 d a t a . The 1976 i n t e r v i e w d a t a were supplemented by d a t a from the 1975 i n t e r v i e w where needed. We made our p e r i o d of o b s e r v a t i o n t o be two y e a r s , t o take i n t o account the nine-month g e s t a t i o n p e r i o d . We e n v i s a g e d the c o u p l e on January 1, 1975, making t h e i r d e c i s i o n s r e g a r d i n g f e r t i l i t y and the w i f e ' s l a b o r s u p p l y f o r the " c u r r e n t p e r i o d " , 2 Because of " s p l i t - o f f s " , i . e . , sons or dau g h t e r s l e a v i n g t o form t h e i r own h o u s e h o l d s , or d i v o r c e , or d e a t h , and because of a t t r i t i o n , the number of f a m i l i e s may v a r y from year t o y e a r . 58 where the " c u r r e n t p e r i o d " i s assumed t o run from January 1, 1975 t o December 31, 1976. Hence, we used d a t a f o r 1975 and 1976 from the 1975-1977 i n t e r v i e w i n g y e a r s . We r e q u i r e d our sample t o have c e r t a i n c h a r a c t e r i s t i c s . F i r s t , we r e q u i r e d the w i f e t o be l e s s than 50 y e a r s o l d a t the time of the 1975 i n t e r v i e w . T h i s c u t - o f f age was a r b i t r a r i l y chosen t o take i n t o account both b i o l o g i c a l and p e r s o n a l c o n s t r a i n t s a g a i n s t h a v i n g another c h i l d . We f e e l the b i a s from e x c l u d i n g women over 50 y e a r s of age who d e s i r e d t o have a n o t h e r c h i l d and who were p h y s i c a l l y a b l e t o do so was s m a l l . 3 Second, we r e q u i r e d the woman t o be m a r r i e d w i t h spouse p r e s e n t f o r a t l e a s t the y e a r s 1974-1976. T h i s c o n s t r a i n t was implemented t o e x c l u d e c o u p l e s who might have been m a r r i e d i n 1974 and d i v o r c e d i n 1975, or v i c e v e r s a . I f we had i n c l u d e d d i v o r c e d (or widowed) p e o p l e , such a s i t u a t i o n would not a l l o w us t o examine the economic d e t e r m i n a n t s of f e r t i l i t y b e h a v i o r . That i s , the f e r t i l i t y b e h a v i o r of a d i v o r c e d or widowed woman would (most l i k e l y ) be de t e r m i n e d s o l e l y by her m a r i t a l s t a t u s and not by any of the o t h e r v a r i a b l e s we wished t o i n v e s t i g a t e . T h i r d , our t h e o r e t i c a l model r e q u i r e s the w i f e t o be i n l a b o r market e q u i l i b r i u m . We were a b l e t o se g r e g a t e the sample a l o n g these l i n e s because i n the 1976 i n t e r v i e w the wives were asked whether they wanted t o work more h o u r s , but c o u l d n o t , and whether they wanted t o work fewer h o u r s , but c o u l d n o t . Wives 3 Given a l l of the r e q u i r e m e n t s on the data d i s c u s s e d below, but a l l o w i n g the women t o be any age, no woman over f i f t y y e a r s of age r e p o r t e d h a v i n g a c h i l d i n the r e l e v a n t p e r i o d . 59 a n s w e r i n g yes t o e i t h e r q u e s t i o n were e l i m i n a t e d . " Wives who r e p o r t e d themselves t o be unemployed were a l s o e l i m i n a t e d . 5 F i n a l l y , we e l i m i n a t e d m u l t i - f a m i l y h o u s e h o l d s , because d i s e n t a n g l i n g the c h i l d r e n was too d i f f i c u l t . O b s e r v a t i o n s were a l s o d e l e t e d i f t h e r e were m i s s i n g d a t a on c r u c i a l v a r i a b l e s such as hours worked and the wage r a t e . O b s e r v a t i o n s were a l s o d e l e t e d i f a p o s i t i v e wage was r e p o r t e d w i t h z e r o hours of work or i f p o s i t i v e hours were r e p o r t e d , w i t h a z e r o wage r a t e or l a b o r income. 6 F i n a l l y , because of h i g h e s t i m a t i o n c o s t s , a o n e - q u a r t e r random sample was t a k e n . The r e s u l t was a sample of 417 o b s e r v a t i o n s . Out of t h i s sample, 290 women worked and 49 had a c h i l d i n the r e l e v a n t p e r i o d . Of those who had a c h i l d , 31 worked. The c o u p l e was assumed t o have d e c i d e d t o have a c h i l d i n the p e r i o d i f a c h i l d was born sometime between September 1975 and August 1976. U n f o r t u n a t e l y , the month of b i r t h was a v a i l a b l e o n l y from the 1976 i n t e r v i e w , and hence was not a v a i l a b l e f o r a c h i l d born a f t e r the mother's i n t e r v i e w d a t e . Hence, a c o u p l e was assumed t o have d e c i d e d t o have a c h i l d i f a c h i l d was born 4 Note t h a t t h i s r e s t r i c t i o n w i l l l e a d t o sample s e l e c t i v i t y b i a s . However, our model i s not d e s i g n e d t o d e a l w i t h women who are not i n l a b o r market e q u i l i b r i u m . 5 E l i m i n a t i o n of i n v o l u n t a r i l y unemployed women means the sample w i l l u n d e r e s t i m a t e women i n the l a b o r f o r c e . I n v o l u n t a r y unemployment can be thought of as l a b o r f o r c e p a r t i c i p a t i o n w i t h l a b o r . s u p p l y e q u a l t o z e r o . V o l u n t a r y unemployment, on the o t h e r hand, i m p l i e s n o n - p a r t i c i p a t i o n . However, o n l y 1.8% of the t o t a l PSID sample r e p o r t e d b e i n g unemployed or t e m p o r a r i l y l a i d o f f d u r i n g the p e r i o d i n q u e s t i o n . Hence, the e f f e c t of e l i m i n a t i n g these women from the subsample s h o u l d be n e g l i g i b l e . 6 Note t h a t t h i s r e s t r i c t i o n w i l l e l i m i n a t e any un p a i d f a m i l y workers i n the sample 60 i n September 1975 or a f t e r , or i f a c h i l d was born i n 1976. W h i l e t h i s assumption p o t e n t i a l l y b i a s e s upwards the number of b i r t h s by i n c l u d i n g p o s s i b l e b i r t h s a f t e r August 1976, i t was f e l t t h i s b i a s would be s m a l l . 7 Hours of work and income were c a l c u l a t e d over the two-year p e r i o d . Income and wages were a d j u s t e d f o r f e d e r a l t a x e s , but not s t a t e t a x e s . 8 The adjustment f o r t a x e s r e f l e c t s the assumption t h a t people r e a c t t o changes i n the income they a c t u a l l y r e c e i v e , t h a t i s , income net of t a x e s , r a t h e r than g r o s s income. A f t e r - t a x f a m i l y income was c a l c u l a t e d as the husband's l a b o r e a r n i n g s , p l u s a l l a s s e t e a r n i n g s of the c o u p l e , minus f e d e r a l t a x e s p a i d . The a f t e r - t a x wage r a t e was c a l c u l a t e d by m u l t i p l y i n g the w i f e ' s r e p o r t e d wage r a t e by one minus the m a r g i n a l t a x r a t e . T h i s c a l c u l a t i o n assumes the w i f e i s the secondary worker and t h a t the c o u p l e f i l e s a j o i n t r e t u r n . 9 Thus, the t a x r a t e r e l e v a n t f o r c a l c u l a t i n g the w i f e ' s a f t e r - t a x wage r a t e i s what the tax r a t e would be w i t h the w i f e ' s e a r n i n g s added t o f a m i l y income, or the m a r g i n a l t a x r a t e . I n f o r m a t i o n on 7 However, t h e r e i s no way of measuring how many women d i d have a c h i l d between September and December 1976. 8 No d a t a on s t a t e t a x e s were a v a i l a b l e . 9 D u r i n g the 1975 and 1976 t a x y e a r s , t a x e s p a y a b l e by a m a r r i e d c o u p l e were lower i f they f i l e d , a j o i n t r e t u r n than i f they f i l e d s e p a r a t e r e t u r n s . Hence, the assumption of j o i n t r e t u r n s i s not an u n r e a s o n a b l e one. 61 the m a r g i n a l t a x r a t e i s r e p o r t e d i n the PSID d a t a . 1 0 F a m i l y c h a r a c t e r i s t i c s , such as number of c h i l d r e n i n the f a m i l y or number of y e a r s worked s i n c e 18 were taken from the 1975 i n t e r v i e w . The v a r i a b l e s used i n our a n a l y s i s are found i n the a p p e n d i x . D e s c r i p t i v e s t a t i s t i c s of the data are found i n Table 4.1. T a b l e 4.1a c o n t a i n s s t a t i s t i c s f o r the whole sample. T a b l e 4.1b and T a b l e 4.1c c o n t a i n s t a t i s t i c s f o r workers and non-w o r k e r s , r e s p e c t i v e l y . 1 0 I f we wished t o implement the i d e a s d i s c u s s e d i n Chapter Two, S e c t i o n C.3, r e g a r d i n g the o p p o r t u n i t y c o s t of h a v i n g a c h i l d , we c o u l d s p e c i f y the net wage as W'=W(l-t)-c, where t i s the m a r g i n a l t a x r a t e and c i s the m a r g i n a l c o s t of c h i l d c a r e i n c u r r e d w h i l e the w i f e worked. A l t h o u g h d a t a on c h i l d c a r e , c o s t s a r e a v a i l a b l e i n the PSID, t h e r e are many m i s s i n g o b s e r v a t i o n s . However, i t would be i n t e r e s t i n g t o pursue t h i s s p e c i f i c a t i o n as i t r e f l e c t s the i d e a t h a t i f m a r g i n a l c h i l d c a r e c o s t s are l a r g e r than l a b o r market e a r n i n g s , the w i f e ' s net r e t u r n from market work i s n e g a t i v e . Hence, the w i f e would work o n l y i f she earned more than she p a i d out f o r c h i l d c a r e (assuming no p o s i t i v e u t i l i t y from market work). 62 T a b l e 4.1a D e s c r i p t i v e s t a t i s t i c s f o r the complete sample (N = 417) Var i a b l e Mean Wi f e ' s hours 1.5562 of work ('000) Log of w i f e ' s h o u r l y wage Years worked s i n c e 18 Number of months i n c u r r e n t p o s i t i o n Number of y e a r s of , s c h o o l i n g Number of c h i l d r e n l e s s than 6 y e a r s Number of c h i l d r e n Age of youngest c h i l d Age of husband Age of w i f e Age of w i f e at m a r r i a g e F a m i l y income 3 ('0000) Gap c 0.56930 7.7602 18.470 11.940 0.64029 1 .9424 5.2086 35.441 32.590 19.669 2.3760 -1.3118 S t a n d a r d d e v i a t i o n 1.5393 0.62377 6.4989 29.121 2.2551 0.80569 1.6631 4.9819 9.5536 8.7027 3.4237 1.3242 2.2625 V a r i a n c e M i n. Max 2.3694 42.236 848.02 5.0853 0.64914 2.7659 24.819 91.271 75.738 1 1 . 722 1 .7536 0.0 5.452 0.38908 -1.386 2.93 0.0 29.0 0.0 98.0 2.0 17.0 0.0 4.0 0.0 10.0 0.0 17.0 19.0 63.0 17.0 49.0 12.0 45.0 0.1047 13.001 5.1189 -10.0 4.0 63 Table 4.1b D e s c r i p t i v e s t a t i s t i c s f o r workers (N = 290) St a n d a r d Var i a b l e Mean d e v i a t i o n W i f e ' s hours 2.2377 1.3713 of work ('000) a Log of w i f e ' s 0.81862 0.59604 h o u r l y wage Years worked 9.1793 6.7107 s i n c e 18 Number of 26.559 31.704 months i n c u r r e n t p o s i t i o n Number of 12.200 2.2510 y e a r s of b s c h o o l i n g Number of 0.52759 0.71629 c h i l d r e n l e s s than 6 y e a r s Number of 1 .6897 1 .4434 c h i l d r e n Age of 5.2724 5.1158 youngest c h i l d Age of husband 35.262 9.6018 Age of w i f e 32.410 8.7752 Age of w i f e 19.690 3.2847 at m a r r i a g e F a m i l y income a 2.2687 1.2679 ( ' 0000) V a r i a n c e M i n. Max. 1.8804 Gap c.. -1.0448 2.0465 45.033 1005. 1 5.0671 0.51308 2.0833 26.171 92.194 77.004 10.789 1.6076 4. 1 883. 0.105 5.452 0.35526 -1.386 2.93 0.0 29.0 0.0 98.0 4.0 17.0 0.0 0.0 4.0 7.0 0.0 17.0 19.0 59.0 17.0" 49.0 12.0 45.0 0.1047 13.001 -7.0 3.0 64 T a b l e 4.1c D e s c r i p t i v e s t a t i s t i c s f o r non-workers (N = 127) St a n d a r d Var i a b l e Mean d e v i a t i o n Var i a n c e Min. Max. Years worked s i n c e 18 4.5197 4.5754 20.934 0.0 20.0 Number of y e a r s of b s c h o o l i n g 11 .346 2.1580 4.6568 2.0 17.0 Number of c h i l d r e n l e s s than 6 y e a r s 0.89764 0.93295 0.87039 0.0 3.0 Number of c h i l d r e n 2.5197 1.9674 3.8706 0.0 10.0 Age of youngest c h i l d 5.0630 4.6781 21.885 0.0 17.0 Age of husband 35.850 9.4676 89.636 19.0 63.0 Age of w i f e 33.000 8.5551 73.190 .18.0 49.0 Age of w i f e at m a r r i a g e 19.622 3.7351 13.951 12.0 44.0 F a m i l y income a COOOO) 2.6212 1.4196 2.0153 0.390 10.131 Gap c -1 .921 3 2.5992 6.7557 -10.0 4.0 a H o u r s of work year p e r i o d . and f a m i l y income are measured over the two-Years of s c h o o l i n g over 17 were r e p o r t e d as 17. GAP i s d e f i n e d as the d e s i r e d number of c h i l d r e n minus the a c t u a l number of c h i l d r e n . 65 C. Spec i f i c a t i o n of the equat i o n s C.1 The wage e q u a t i o n In o r d e r t o implement e m p i r i c a l l y the model d e v e l o p e d i n Chapter Three, we need t o s p e c i f y the exogenous v a r i a b l e s d e t e r m i n i n g the wage r a t e , hours worked, and the d e s i r e t o have an a d d i t i o n a l c h i l d . The wage e q u a t i o n i n the model i s s p e c i f i e d as a f u n c t i o n of exogenous v a r i a b l e s o n l y . 1 1 We s p e c i f y the wage e q u a t i o n as (1) LNW = b 3 0 + b 3 1 * Y R l 8 + b 3 2*MONPOS + b 3 3*EDUC + b 3„*MONPOS 2 + b 3 5*EDUC 2 The n a t u r a l l o g a r i t h m of the wage i s chosen as the dependent v a r i a b l e t o ensure the e s t i m a t e d wage i s p o s i t i v e . 1 2 The s p e c i f i c a t i o n of the wage e q u a t i o n i s d e r i v e d as a f u n c t i o n of p r o x i e s f o r human c a p i t a l : the number of y e a r s worked s i n c e 18 (YR18); the number of months worked i n c u r r e n t p o s i t i o n (MONPOS); and the number of y e a r s of s c h o o l i n g (EDUC). The squared terms r e f l e c t n o n - l i n e a r i t i e s i n the e f f e c t of human c a p i t a l p r o x i e s on the wage. P r e v i o u s work ( e . g . , M i n c e r [ 1 9 7 4 ] , Lee [ 1 9 7 8 ] , Schmidt [1978]) s u g g e s t s t h a t the c o e f f i c i e n t s on the human c a p i t a l v a r i a b l e s s h o u l d be p o s i t i v e , w h i l e those on the squared terms s h o u l d be n e g a t i v e . T h i s i m p l i e s the wage f u n c t i o n i n c r e a s e s w i t h a d d i t i o n s t o human c a p i t a l , but a t a 1 1 See, however, Rosen [1976], who argues t h a t the wage a l s o depends on hours of work. While h i s assumption may be more r e a l i s t i c , i t would add t o the c o m p l e x i t y of our model. 1 2 See a l s o the argument i n Min c e r [1974]. 66 d e c r e a s i n g r a t e above a g i v e n l e v e l . However, the s e a p r i o r i e x p e c t a t i o n s about the s i g n s of the c o e f f i c i e n t s may not h o l d t r u e , e s p e c i a l l y i n the case of m a r r i e d women, because of d i s c o n t i n u i t i e s i n work e x p e r i e n c e . A l t h o u g h some d a t a were a v a i l a b l e on e x i t s and e n t r i e s from and t o the work f o r c e , t h e r e were not enough d a t a t o p r o v i d e a complete p i c t u r e of a woman's work h i s t o r y . 1 3 C.2 The hours e q u a t i o n The hours e q u a t i o n , as g i v e n i n Chapter Three, i s s p e c i f i e d as a f u n c t i o n of the ( n a t u r a l l o g a r i t h m of the) wage r a t e ; AC, a dichotomous v a r i a b l e r e f l e c t i n g whether the c o u p l e has a c h i l d i n the c u r r e n t p e r i o d ; the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y ; and exogenous v a r i a b l e s r e f l e c t i n g c e r t a i n c h a r a c t e r i s t i c s of the f a m i l y . From Chapter Three, we know t h a t one of the exogenous v a r i a b l e s i s f a m i l y income, d e f i n e d as a f t e r - t a x a s s e t income p l u s the husband's a f t e r - t a x l a b o r income. We s p e c i f y the hours e q u a t i o n , t a k i n g i n t o account the above c o n s i d e r a t i o n s , as (2) H* = b 1 0 + b,,*LNW + b 1 2*AC + b 1 3 * Y R l 8 + b,,*CHILD + b,5*FAMINC 1 3 For a study of women's e a r n i n g s . w h i c h t a k e s i n t o account the e f f e c t of d i s c o n t i n u i t i e s i n a woman's work h i s t o r y , see Min c e r and Polachek [1974]. They found the c o e f f i c i e n t on the e x p e r i e n c e squared term t o be n e g a t i v e . 67 where LNW, AC, and YR18 have been d e f i n e d above; CHILD i s the number, of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y ; and FAMINC i s f a m i l y income. The dependent v a r i a b l e i s hours of work over the two-year p e r i o d . The wage r a t e w i l l have an ambiguous e f f e c t on hours worked because t h e r e i s both an income and s u b s t i t u t i o n e f f e c t , which work i n o p p o s i t e d i r e c t i o n s . The income e f f e c t s t a t e s t h a t as the wage goes up, income goes up. I f l e i s u r e i s a normal good, i t s h o u l d i n c r e a s e and hours of work d e c r e a s e , as income i n c r e a s e s . On the o t h e r hand, as the wage i n c r e a s e s , the p r i c e of l e i s u r e a l s o i n c r e a s e s , i m p l y i n g a s u b s t i t u t i o n away from l e i s u r e . In g e n e r a l , the s u b s t i t u t i o n e f f e c t has been found t o dominate f o r m a r r i e d women (Heckman et a l [ 1 9 7 9 ] ) . AC and CHILD r e p r e s e n t the d i r e c t e f f e c t of c h i l d r e n on hours worked. The e f f e c t of a newborn c h i l d i n the f a m i l y i s assumed t o have a d i f f e r e n t e f f e c t from h a v i n g one more c h i l d of an o l d e r age. A more complete s p e c i f i c a t i o n , however, would take i n t o account the age d i s t r i b u t i o n of the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y . 1 " The e f f e c t of the number of c h i l d r e n a l r e a d y i n the f a m i l y i s ambiguous. The g r e a t e r the number of c h i l d r e n (and the younger the c h i l d r e n ) , the more d i f f i c u l t i t w i l l be f o r the w i f e t o f i n d a s u b s t i t u t e f o r her home t i m e . On the o t h e r hand, the l a r g e r the number of c h i l d r e n , the g r e a t e r the f i n a n c i a l burden on the f a m i l y . Hence, i t may be n e c e s s a r y f o r the w i f e t o work i n or d e r t o c o n t r i b u t e t o f a m i l y income. The e f f e c t of an a d d i t i o n a l , newborn, c h i l d i s expected 1 ' See, f o r example, Nakamura et a l [1979], 6 8 t o have a n e g a t i v e e f f e c t on hours worked, a l l o t h e r t h i n g s h e l d c o n s t a n t . YR18 r e f l e c t s a t a s t e v a r i a b l e . I f the woman has a s t r o n g l a b o r f o r c e a ttachment, r e f l e c t e d by many y e a r s of work, she i s more l i k e l y t o work i n the c u r r e n t p e r i o d , a l l o t h e r t h i n g s h e l d c o n s t a n t . F a m i l y income w i l l be i n v e r s e l y r e l a t e d t o hours of work i f l e i s u r e i s a normal good. C.3 The a d d i t i o n a l c h i l d equat i o n The d e s i r e f o r c h i l d r e n e q u a t i o n has been s p e c i f i e d as a f u n c t i o n of the wage r a t e , a c t u a l hours of work, the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y , whether another c h i l d i s bor n , and exogenous v a r i a b l e s . A g a i n , the t h e o r y i n Chapter Three s u g g e s t s t h a t one of these exogenous v a r i a b l e s s h o u l d be f a m i l y income. We s p e c i f y the c h i l d e q u a t i o n as (3) C* = b 2 0 + b 2 1*LNW + b 2 2*CHILD + b 2 3 * H + b 2 „*AGECH + b 2 5*AGEWMAR + b 2 6*FAMINC + b 2 7*GAP + b 2 8*AC where LNW, CHILD, FAMINC, and AC have been d e f i n e d above; H i s a c t u a l hours of work; AGECH i s the age of the youngest c h i l d i n the f a m i l y a t the time of the d e c i s i o n t o have a n o t h e r c h i l d ; AGEWMAR i s the age of the w i f e a t m a r r i a g e ; and GAP i s the d i f f e r e n c e between the d e s i r e d number of c h i l d r e n and the number of c h i l d r e n a l r e a d y i n the f a m i l y . The wage r a t e r e p r e s e n t s the o p p o r t u n i t y c o s t of the w i f e ' s t i m e . The h i g h e r the wage, the l e s s l i k e l y she w i l l be t o have a c h i l d i f she has to g i v e up market work t o do so. On the o t h e r 69 hand, the wage w i l l a l s o have an income e f f e c t . The income e f f e c t i s e x p e c t e d , t h e o r e t i c a l l y , t o be p o s i t i v e i f c h i l d r e n a r e a normal good. However, the observed r e l a t i o n s h i p between income and the number of c h i l d r e n i n a f a m i l y i s n e g a t i v e . Thus, what the a c t u a l s i g n of the income e f f e c t w i l l be i s a m b i g u o u s . 1 5 The number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y w i l l have an ambiguous e f f e c t . A p o s i t i v e e f f e c t c o u l d r e f l e c t a " t a s t e " f o r c h i l d r e n , w h i l e a n e g a t i v e e f f e c t c o u l d r e f l e c t the d e c i s i o n of the c o u p l e not t o have a l a r g e r f a m i l y . The v a r i a b l e , GAP, i s an attempt t o i s o l a t e these e f f e c t s more c l e a r l y by measuring whether the c o u p l e has reached t h e i r d e s i r e d number of c h i l d r e n . I t i s e x p e c t e d t o have a p o s i t i v e e f f e c t on the d e s i r e f o r an a d d i t i o n a l c h i l d . Hours of work are e x p e c t e d t o have a n e g a t i v e e f f e c t on the d e s i r e f o r c h i l d r e n , on the assumption t h a t r a i s i n g a c h i l d (at l e a s t f o r the f i r s t f i v e y e a r s ) competes w i t h market work f o r the woman's t i m e . 1 6 E m p i r i c a l l y , f a m i l y income has been found t o be i n v e r s e l y c o r r e l a t e d w i t h the number of c h i l d r e n . T h i s r e s u l t does not n e c e s s a r i l y imply c h i l d r e n a r e an i n f e r i o r good. A r i s e i n income may change the p r i c e of the w i f e ' s time such t h a t t h e r e i s a s u b s t i t u t i o n away from t i m e - i n t e n s i v e a c t i v i t i e s , , such as 1 5 See the d i s c u s s i o n of the e f f e c t of FAMINC below. 1 6 See, however, McCabe and Rosenzweig [1976], They found t h a t , i n the extended f a m i l y c o n t e x t and l e s s i n s t i t u t i o n a l i z e d work p l a c e s i n l e s s - d e v e l o p e d c o u n t r i e s , h a v i n g a c h i l d d i d not p r e s e n t a c o n f l i c t i n g use of time w i t h market work. 70 r a i s i n g c h i l d r e n ( M i c h a e l [ 1 9 7 3 ] ) . 1 7 AGECH i s i n c l u d e d i n an attempt t o get some i n d i c a t i o n of c h i l d s p a c i n g d e s i r e s . The ex p e c t e d s i g n of the c o e f f i c i e n t i s ambiguous, however. A p o s i t i v e c o e f f i c i e n t on AGECH would i n d i c a t e a d e s i r e f o r some s p a c i n g between c h i l d r e n ; the younger the youngest c h i l d , the l e s s l i k e l y the c o u p l e would be t o have a c h i l d i n the c u r r e n t p e r i o d . A n e g a t i v e c o e f f i c i e n t would i n d i c a t e a d e s i r e t o have c h i l d r e n c l o s e t o g e t h e r , perhaps t o m i n i m i z e the amount of time when c h i l d r e n under s c h o o l age would be a t home. 1 8 The age of w i f e at m a r r i a g e a t t e m p t s t o measure b i o l o g i c a l c o n s t r a i n t s as w e l l as perhaps t a s t e s . A g a i n , the e x p e c t e d s i g n of the c o e f f i c i e n t i s ambiguous. I f a w i f e i s o l d e r a t m a r r i a g e , she may be i n c l i n e d t o have c h i l d r e n soon and c l o s e t o g e t h e r i n o r d e r t o have the d e s i r e d number of c h i l d r e n b e f o r e f e c u n d i t y d e c r e a s e s . A l t e r n a t i v e l y , a h i g h e r age a t m a r r i a g e may r e f l e c t a g r e a t e r commitment t o c a r e e r s by the s e women and hence r e f l e c t t a s t e s a g a i n s t c h i l d r e n . 1 9 A l s o a h i g h e r age a t m a r r i a g e c o u l d 1 7 Simon [1969] a l s o argues the e f f e c t of income on f e r t i l i t y w i l l be ambiguous because a change i n income g i v e s r i s e t o i n d i r e c t , as w e l l as d i r e c t , e f f e c t s . Some a u t h o r s ( e . g . , Becker [ 1 9 6 0 ] , de Tray [1973]) have found a p o s i t i v e e f f e c t of income on f e r t i l i t y . O t hers ( e . g . , W i l l i s [ 1 9 7 3 ] , Hotz [1980]) have found t h a t income has a n e g a t i v e e f f e c t , w h i l e s t i l l o t h e r s ( e . g . , M i c h a e l [1973], Khan and S i r a g e l d i n [1977]) have found i t to have an i n s i g n i f i c a n t e f f e c t on f e r t i l i t y . Hence, the e f f e c t of income on f e r t i l i t y i s s t i l l an open q u e s t i o n . 1 8 AGECH does not e n t i r e l y measure s p a c i n g s i n c e we have not c o n t r o l l e d f o r the age of the w i f e . 1 9 See Ross [1974] f o r some e v i d e n c e s u p p o r t i n g t h i s s u g g e s t i o n . 71 mean lower f e c u n d i t y and hence a b i l i t y t o have a c h i l d . I t w i l l be noted both r a c e and r e l i g i o n have been e x c l u d e d as e x p l a n a t o r y v a r i a b l e s a l t h o u g h both have been found t o be s i g n i f i c a n t i n p r e v i o u s w o r k . 2 0 P r e l i m i n a r y i n v e s t i g a t i o n i n d i c a t e d t h a t they were not s i g n i f i c a n t i n e x p l a i n i n g f e r t i l i t y or l a b o r s u p p l y b e h a v i o r of our sample. To f u r t h e r t e s t these r e s u l t s , a c h i - s q u a r e t e s t was run on the p r o p o r t i o n s of C a t h o l i c s i n each of the f o u r c a t e g o r i e s a g a i n s t the p r o p o r t i o n s of n o n - C a t h o l i c s i n each c a t e g o r y . The c h i - s q u a r e t e s t was a l s o run on the p r o p o r t i o n s of b l a c k s found i n each c a t e g o r y a g a i n s t the p r o p o r t i o n s on n o n - b l a c k s i n each group. The t e s t s l e d t o the r e j e c t i o n of the h y p o t h e s i s t h a t the two groups had s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s i n f e r t i l i t y and l a b o r f o r c e p a r t i c i p a t i o n b e h a v i o r from the r e s t of the sample. The r e s u l t s a r e summarized i n T a b l e 4.2. There a r e s e v e r a l p o s s i b l e reasons f o r t h e s e r e s u l t s . F i r s t , w i t h r e g a r d t o r e l i g i o n , we a r e l o o k i n g a t the p r o b a b i l i t y of h a v i n g another c h i l d i n a g i v e n time p e r i o d r a t h e r than a t the t o t a l number of c h i l d r e n . The l a t t e r v a r i a b l e u s u a l l y has been the dependent v a r i a b l e i n p r e v i o u s r e s e a r c h . Hence, C a t h o l i c s c o u l d have s i m i l a r p r o b a b i l i t i e s of h a v i n g a c h i l d a t a g i v e n p o i n t i n t i m e , c e t e r i s p a r i b u s , as the r e s t of the sample, but have l a r g e r f a m i l i e s i n t o t a l . Second, the use of c o n t r a c e p t i v e s by C a t h o l i c s i n the U n i t e d S t a t e s i s p o s s i b l y 2 0 See, f o r example, C a i n and W e i n i n g e r [1970], Nakamura et a l [ 1979] , and Hotz [1980]. 72 no l o n g e r s i g n i f i c a n t l y d i f f e r e n t from t h e i r use i n the r e s t of the p o p u l a t i o n . The r e s u l t s w i t h r e g a r d t o b l a c k s a re a b i t more d i f f i c u l t t o e x p l a i n . P o s s i b l y our r e s t r i c t i o n t h a t the c o u p l e be m a r r i e d f o r a g i v e n p e r i o d has e x c l u d e d a number of women who had c h i l d r e n born out of wedlock. 73 T a b l e 4.2 C h i - s q u a r e t e s t of independence of s u b - p o p u l a t i o n s T o t a l sample (N=417) Category S1 S2 S3 S4 Number 31 259 18 1 09 P e r c e n t 7% 62% 4% 26% C a t e g o r y S1 S2 S3 S4 C a t h o l i c (N=97) Number P e r c e n t X 2 ( 3 ) = .9839 9 60 3 25 9% 62% 3% 26% N o n - C a t h o l i c (N=320) Number P e r c e n t 22 199 1 5 84 7% 62.5% 5% 25.5% C a t e g o r y S i S2 S3 S4 B l a c k (N=80) Number P e r c e n t 5 48 4 23 6% 60% 5% 29% Non-Black (N=337) Number P e r c e n t 26 21 1 1 4 86 8% 62.5% 4% 24.5% X 2 ( 3 ) = .546 where S1 i s the c a t e g o r y of workers who have a c h i l d t h i s p e r i o d ; S2, workers who do not have a c h i l d t h i s p e r i o d ; S3, non-workers who have a c h i l d t h i s p e r i o d ; and S4, non-workers-who do not have a c h i l d t h i s p e r i o d . 74 D. A comparison of e s t i m a t e s D.1 Maximum l i k e l i h o o d e s t i m a t i o n E q u a t i o n s ( l ) - ( 3 ) of the p r e c e d i n g s e c t i o n were the o n l y s p e c i f i c a t i o n of the model e s t i m a t e d by the ML method. 2 1 The t h r e e e q u a t i o n s were e s t i m a t e d u s i n g a program by F l e t c h e r [ 1 9 7 2 ] . 2 2 The l i k e l i h o o d f u n c t i o n was too c o m p l i c a t e d t o a l l o w the c a l c u l a t i o n of a n a l y t i c d e r i v a t i v e s . N u m e r i c a l d e r i v a t i v e s were used i n s t e a d . Hence, c a r e s h o u l d be taken when t e s t i n g hypotheses as the t - s t a t i s t i c s w i l l not n e c e s s a r i l y be as a c c u r a t e as i f they were c a l c u l a t e d from a n a l y t i c a l d e r i v a t i v e s . There were some problems w i t h convergence i n our model. In an attempt t o a l l e v i a t e t h i s problem, we made use of some p r i o r i n f o r m a t i o n about the v a r i a n c e - c o v a r i a n c e m a t r i x . Hotz [1980] found the c o v a r i a n c e between the d i s t u r b a n c e terms of h i s c h i l d e q u a t i o n and wage e q u a t i o n not t o be s i g n i f i c a n t l y d i f f e r e n t from z e r o . Hence, we imposed the r e s t r i c t i o n t h a t c 2 3 r the c o v a r i a n c e of the d i s t u r b a n c e s terms of the a d d i t i o n a l c h i l d e q u a t i o n and the wage e q u a t i o n , be z e r o . T h i s r e s t r i c t i o n e n a b l e d us t o rea c h convergence w i t h o u t f u r t h e r problems. 2 1 Cost c o n s i d e r a t i o n s r u l e d out u s i n g maximum l i k e l i h o o d e s t i m a t i o n on o t h e r s p e c i f i c a t i o n s . Cost c o n s i d e r a t i o n s a l s o l i m i t e d the number of v a r i a b l e s which c o u l d be used i n t h i s s p e c i f i c a t i o n . 2 2 The program uses the v a r i a b l e m e t r i c method f o r c a l c u l a t i n g a minimum. L e t x ( k ) be an a p p r o x i m a t i o n t o the s o l u t i o n f o r x at the k t h i t e r a t i o n . Then x ( k + l ) i s chosen t o be x ( k ) + a p ( k ) , where p(k) = - H ( k ) g ( k ) . H(k) i s an a p p r o x i m a t i o n t o the i n v e r s e of the H e s s i a n m a t r i x and g(k) i s the g r a d i e n t v e c t o r a t x ( k ) . The v a l u e of a i s chosen t o m i n i m i z e the v a l u e of the f u n c t i o n , e v a l u a t e d at x ( k ) + a p ( k ) , w i t h r e s p e c t t o a. 75 T a b l e 4.3 p r e s e n t s the l i k e l i h o o d r a t i o t e s t s f o r two hypo t h e s e s . The f i r s t h y p o t h e s i s i s t h a t the t h r e e c o v a r i a n c e s of the d i s t u r b a n c e terms a r e z e r o . T h i s h y p o t h e s i s i m p l i e s t h a t t h e r e i s no r e l a t i o n s h i p among the t h r e e d i s t u r b a n c e terms. I f t h i s h y p o t h e s i s i s t r u e , i t means we can e s t i m a t e the model u s i n g s i n g l e e q u a t i o n e s t i m a t i o n t e c h n i q u e s w i t h o u t f e a r of s i m u l t a n e o u s e q u a t i o n s b i a s , l o s s of e f f i c i e n c y or sample s e l e c t i v i t y b i a s . The second h y p o t h e s i s a l l o w s the c o v a r i a n c e between the d i s t u r b a n c e term i n the hours e q u a t i o n s and the d i s t u r b a n c e term i n the wage e q u a t i o n t o be non-zero, but assumes the o t h e r c o v a r i a n c e s a re e q u a l t o z e r o . T h i s h y p o t h e s i s i m p l i e s t h e r e i s no r e l a t i o n s h i p between the d i s t u r b a n c e terms of the hours or wage e q u a t i o n s and the a d d i t i o n a l c h i l d e q u a t i o n . Both hypotheses a r e r e j e c t e d a t the 99% l e v e l of c o n f i d e n c e . T h i s r e s u l t i n d i c a t e s t h a t i t i s c o r r e c t t o t r e a t the wage r a t e , hours worked, and the c h o i c e of whether t o have ano t h e r c h i l d as j o i n t l y d e t e r m i n e d . The r e j e c t i o n of the two hypotheses a l s o suggests t h a t s i n g l e e q u a t i o n methods, which i m p l i c i t l y assume z e r o c o v a r i a n c e s , can l e a d t o i n e f f i c i e n t and p o s s i b l y b i a s e d r e s u l t s . 2 3 2 3 See the d i s c u s s i o n i n s e c t i o n s D.2-D.5 below. 76 Table 4.3 L i k e l i h o o d r a t i o t e s t s on the c o v a r i a n c e s of the d i s t u r b a n c e terms 1 . Ho: <r! 2 = <s 1 3 •= 0 a' k H1 : * , 2_ * 0 , * , 3' * 0 L o g - 1 i k e l i h o o d of r e s t r i c t e d model: -1000.091 L o g - l i k e l i h o o d of u n r e s t r i c t e d model: -939.485 L i k e l i h o o d r a t i o s t a t i s t i c : 121.212 Degrees of freedom: 2 C r i t i c a l X 2 v a l u e a t 1% l e v e l of s i g n i f i c a n c e : 9.210 2. Ho: tf,2 = 0 H1 : cr, 2 * 0 L o g - l i k e l i h o o d of r e s t r i c t e d model: -949.520 L o g - l i k e l i h o o d of u n r e s t r i c t e d model: -939.485 L i k e l i h o o d r a t i o s t a t i s t i c : 20.070 Degrees of freedom: 1 C r i t i c a l X 2 v a l u e a t 1% l e v e l of s i g n i f i c a n c e : 6.635 tf23 i s a l r e a d y f i x e d at z e r o . See the d i s c u s s i o n i n the t e x t . k t f 1 2 i s the c o v a r i a n c e f o r the hours and a d d i t i o n a l c h i l d e q u a t i o n s . «r T 3 i s the c o v a r i a n c e f o r the hours and wage e q u a t i o n . 77 D.2 A l t e r n a t i v e methods of e s t i m a t i o n We compared f o u r a l t e r n a t i v e methods of e s t i m a t i o n w i t h maximum l i k e l i h o o d (ML) e s t i m a t i o n . Each a l t e r n a t i v e method has i m p l i c a t i o n s f o r the s t r u c t u r e of the v a r i a n c e - c o v a r i a n c e m a t r i x of the m o d e l . 2 " F i r s t , we a p p l i e d o r d i n a r y l e a s t squares (OLS) t o the t h r e e e q u a t i o n s s e p a r a t e l y . T h i s method i s the most n a i v e method. Not o n l y does i t i g n o r e any s i m u l t a n e i t y among the v a r i a b l e s , but i t a l s o i g n o r e s the t r u n c a t i o n of ob s e r v e d hours of work a t z e r o , any sample s e l e c t i v i t y , and the dichotomous n a t u r e of the d e c i s i o n t o have an a d d i t i o n a l c h i l d . I f we a p p l y OLS t o each of the t h r e e e q u a t i o n s , we i m p l i c i t l y assume t h a t the c o v a r i a n c e s between the d i s t u r b a n c e terms of the t h r e e e q u a t i o n s a r e z e r o . I f the c o v a r i a n c e s were i n f a c t z e r o , a p p l y i n g OLS t o the e q u a t i o n s would be e q u i v a l e n t t o u s i n g ML e s t i m a t i o n w i t h o u t t a k i n g i n t o account the t r u n c a t i o n of hours worked or the dichotomous n a t u r e of the d e c i s i o n t o have an o t h e r c h i l d . However, as noted i n s e c t i o n D.I, the assumption of z e r o c o v a r i a n c e s was r e j e c t e d f o r t h i s model. In the second method, we a p p l i e d t o b i t a n a l y s i s t o the hours e q u a t i o n and p r o b i t a n a l y s i s t o the c h i l d e q u a t i o n . These methods a l l o w account t o be taken of the t r u n c a t i o n of the hours 2 * In our d i s c u s s i o n , we assume the maximum l i k e l i h o o d e s t i m a t e s a r e the " c o r r e c t " e s t i m a t e s . T h i s assumption r e l i e s on the assumption of a t r i v a r i a t e normal d i s t r i b u t i o n of the d i s t u r b a n c e terms. I f the d i s t u r b a n c e s a r e not n o r m a l l y d i s t r i b u t e d , or i f t h e r e i s o t h e r s p e c i f i c a t i o n e r r o r , our m a i n t a i n e d h y p o t h e s i s need not be c o r r e c t (see Cragg [ 1 9 6 8 ] ) . We d i d not t e s t the v a l i d i t y of the n o r m a l i t y a s s u m p t i o n , however. 78 e q u a t i o n and of the dichotomous n a t u r e of AC, but s i m u l t a n e o u s e q u a t i o n s b i a s and sample s e l e c t i v i t y a r e s t i l l not acc o u n t e d f o r . 2 5 A g a i n , t h i s method i m p l i e s t h a t t h e r e i s no r e l a t i o n s h i p among the d i s t u r b a n c e s of the t h r e e e q u a t i o n s . I f the c o v a r i a n c e s were i n f a c t z e r o , the l i k e l i h o o d f u n c t i o n d e r i v e d i n Chapter Three would reduce t o the s e s i n g l e e q u a t i o n methods a p p l i e d s e p a r a t e l y t o the a p p r o p r i a t e e q u a t i o n s . The t h i r d method was t o a p p l y the sample s e l e c t i v i t y model to the hours and wage e q u a t i o n s . 2 6 T h i s method t a k e s i n t o account sample s e l e c t i v i t y and end o g e n e i t y of the wage r a t e and hours worked. However, i t i g n o r e s the endogeneity of the d e c i s i o n t o have another c h i l d . T h i s method, t o g e t h e r w i t h the p r o b i t e s t i m a t e of the a d d i t i o n a l c h i l d e q u a t i o n , assumes the second h y p o t h e s i s t e s t e d i n s e c t i o n D.1. That i s , a non-zero c o v a r i a n c e i s assumed t o e x i s t between the d i s t u r b a n c e terms of the hours and wage e q u a t i o n s , but the c o v a r i a n c e s between the d i s t u r b a n c e terms of thes e two e q u a t i o n s and the a d d i t i o n a l c h i l d e q u a t i o n a r e assumed t o be z e r o . A g a i n , we note t h a t t h i s h y p o t h e s i s was r e j e c t e d f o r our model. The f o u r t h method, a c o n s i s t e n t , but not e f f i c i e n t , e s t i m a t o r , i s d i s c u s s e d s e p a r a t e l y below. 2 5 T o b i t e s t i m a t i o n a l l o w account t o be taken of the d e c i s i o n whether t o work as w e l l as the number of hours of work. However, i t does not d e a l w i t h the problem t h a t wage r a t e s a re not obs e r v e d f o r non-workers. 2 6 By the sample s e l e c t i v i t y method, we mean the method which t a k e s i n t o account the t r u n c a t i o n of hours worked a t z e r o and the f a c t t h a t the wage r a t e i s not observed f o r non-workers. See Chapter Two, s e c t i o n B.3 above. 79 D.3. The wage equat i o n The d i f f e r e n t e s t i m a t o r s f o r the wage e q u a t i o n a r e found i n Table 4.4. We compare t h r e e e s t i m a t o r s : OLS, sample s e l e c t i v i t y ( S S ) , and ML. The a l t e r n a t i v e c o n s i s t e n t e s t i m a t o r w i l l be d i s c u s s e d i n s e c t i o n D.6. Of the t h r e e e q u a t i o n s , the wage e q u a t i o n has the l e a s t d i f f e r e n c e s i n the parameters e s t i m a t e d by d i f f e r e n t t e c h n i q u e s . T h i s r e s u l t s h o u l d not be s u r p r i s i n g s i n c e the wage e q u a t i o n i s de t e r m i n e d s o l e l y by v a r i a b l e s exogenous t o the system. Hence, any problems w i t h 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 be l e s s l i k e l y t o o c c u r . However, t h e r e s t i l l may be problems w i t h sample s e l e c t i v i t y . In g e n e r a l , the sample s e l e c t i v i t y e s t i m a t e s a r e c l o s e r t o the ML e s t i m a t e s than t o the OLS e s t i m a t e s . Both OLS and the sample s e l e c t i v i t y method u n d e r e s t i m a t e the c o e f f i c i e n t on the number of y e a r s worked s i n c e 18, r e l a t i v e t o the ML e s t i m a t e . However, the sample s e l e c t i v i t y c o e f f i c i e n t i s almost i d e n t i c a l t o the ML e s t i m a t e . The c o e f f i c i e n t on the number of months i n c u r r e n t p o s i t i o n i s u n d e r e s t i m a t e d by both OLS and the sample s e l e c t i v i t y method, r e l a t i v e t o the ML e s t i m a t e . The sample s e l e c t i v i t y e s t i m a t e i s c l o s e s t t o the ML e s t i m a t e . The l a r g e s t d i f f e r e n c e i n c o e f f i c i e n t s comes on number of y e a r s of s c h o o l i n g . A l l t h r e e e s t i m a t i n g t e c h n i q u e s f i n d number of y e a r s of s c h o o l i n g t o be i n s i g n i f i c a n t . The sample s e l e c t i v i t y method a l s o f i n d s EDUC 2 t o be i n s i g n i f i c a n t . The OLS e s t i m a t e of the EDUC 2 c o e f f i c i e n t i s s i g n i f i c a n t a t the 5% l e v e l of s i g n i f i c a n c e , w h i l e the ML e s t i m a t e i s s i g n i f i c a n t a t the 10% 80 l e v e l . The c o e f f i c i e n t s on MONPOS2 a r e u n d e r e s t i m a t e d ( i n a b s o l u t e v a l u e ) by OLS and the sample s e l e c t i v i t y s e l e c t i v i t y method. L i k e the c o e f f i c i e n t s on MONPOS, the sample s e l e c t i v i t y e s t i m a t e i s the c l o s e s t t o the ML e s t i m a t e . These r e s u l t s i n d i c a t e the sample s e l e c t i v i t y e s t i m a t i n g t e c h n i q u e p r o v i d e s a b e t t e r a p p r o x i m a t i o n t o the ML e s t i m a t e s than OLS. However, the wage e q u a t i o n p l a y s an i m p o r t a n t r o l e i n the e s t i m a t i o n of the hours e q u a t i o n and a d d i t i o n a l c h i l d e q u a t i o n . Hence, how i t i s e s t i m a t e d may be i m p o r t a n t , even i f the a l t e r n a t i v e parameter e s t i m a t e s a re c l o s e t o the ML e s t i m a t e s . T a b l e 4.4 V a r i a b l e C o n s t a n t YR 1 8 MONPOS EDUC MONPOS 2EDUC 2 s 2 Wage e q u a t i ^ b OLS 0.655 ( 1 .2355) 0.0121 (2.3476) 0.0122 (3.5044) -0. 1 003 (-1.1671) -0.000107 (-2.8662) 0.00736 (2.1180) 0.2940 e s t i m a t e s 3 SS 0.452 (2.3214) 0.0131 (2.5338) 0.0186 (3.9152) -0.0166 (-0.5151) -0.000166 (-3.7746) 0.00138 (0.9279) 0.3156 ML 0.497 ( 1 .8328) 0.01 36 (2.6305) 0.0217 (6.5030) -0.0507 (-1.1377) -0.000195 (-6.0999) 0.00347 (1.7853) 0.31 30 a T - s t a t i s t i c s a r e i n p a r e n t h e s e s . C r i t i c a l v a l u e s f o r a two-t a i l e d t - t e s t a r e 1.96 f o r the 5% l e v e l of s i g n i f i c a n c e and 1.645 f o r the 10% l e v e l of s i g n i f i c a n c e . OLS was run on o n l y the subsample of worker s . 82 D.4 The hours e q u a t i o n T a b l e 4.5 c o n t a i n s f o u r d i f f e r e n t e s t i m a t e s of the hours e q u a t i o n . Both of the s i n g l e e q u a t i o n methods, OLS and t o b i t a n a l y s i s , u n d e r e s t i m a t e the c o e f f i c i e n t on the n a t u r a l l o g a r i t h m of the wage r a t e by a l a r g e amount r e l a t i v e t o the ML e s t i m a t e . OLS p r e d i c t s the e f f e c t of LNW on a n n u a l hours t o be more than seven t i m e s s m a l l e r than the ML e s t i m a t e . The t o b i t c o e f f i c i e n t i s over t w i c e as s m a l l . The sample s e l e c t i v i t y c o e f f i c i e n t i s c l o s e s t t o the ML e s t i m a t e . However, i t o v e r e s t i m a t e s the e f f e c t of LNW on hours worked. A l l t h r e e a l t e r n a t i v e e s t i m a t i o n t e c h n i q u e s g r o s s l y u n d e r e s t i m a t e the impact of an a d d i t i o n a l c h i l d on hours worked, r e l a t i v e t o the ML e s t i m a t e . The sample s e l e c t i v i t y method p r e d i c t s the s m a l l e s t d e c l i n e i n hours worked by the w i f e i f she has another c h i l d : o n l y 618 hours over a two-year p e r i o d . T h i s i s a p p r o x i m a t e l y 15.5 weeks (assuming a 40-hour work week), or a l i t t l e l e s s than f o u r months. T h i s r e s u l t ' s e e m s c o n s i s t e n t w i t h the average l e n g t h of time a l l o w e d by many companies f o r m a t e r n i t y l e a v e . ML e s t i m a t i o n , on the o t h e r hand, p r e d i c t s the w i f e w i l l s t o p work f o r a t l e a s t a y e a r a f t e r h a v i n g a c h i l d . The number of y e a r s worked s i n c e 18 r e c e i v e s mixed r e s u l t s . The s i n g l e e q u a t i o n methods f i n d i t t o be s i g n i f i c a n t . However, both the sample s e l e c t i v i t y method and the ML methods f i n d the number of y e a r s worked s i n c e 18 t o be i n s i g n i f i c a n t i n e x p l a i n i n g hours worked by the w i f e . T h i s r e s u l t s u g g e s t s t h a t y e a r s worked s i n c e 18 may o n l y a f f e c t hours worked i n d i r e c t l y , t h r o ugh the wage r a t e , r a t h e r than d i r e c t l y . The f o u r methods of e s t i m a t i o n y i e l d s i m i l a r e s t i m a t e s f o r 83 the e f f e c t of the number of c h i l d r e n a l r e a d y i n the f a m i l y on the number of hours worked. The t h r e e a l t e r n a t i v e t e c h n i q u e s u n d e r e s t i m a t e the e f f e c t , r e l a t i v e t o the ML e s t i m a t e , w i t h the OLS c o e f f i c i e n t b e i n g the f u r t h e s t away from the ML e s t i m a t e . The e s t i m a t e d c o e f f i c i e n t s on f a m i l y income a r e a l s o s i m i l a r . The sample s e l e c t i v i t y model u n d e r e s t i m a t e s the c o e f f i c i e n t , r e l a t i v e t o the ML e s t i m a t e . The s i n g l e e q u a t i o n methods o v e r e s t i m a t e the c o e f f i c i e n t , w i t h the OLS c o e f f i c i e n t b e i n g the c l o s e s t of the t h r e e e s t i m a t e s t o the ML e s t i m a t e . In g e n e r a l , the sample s e l e c t i v i t y method a g a i n appears t o produce c o e f f i c i e n t s which are c l o s e s t t o the ML e s t i m a t e s . T h i s r e s u l t i s not s u r p r i s i n g s i n c e the sample s e l e c t i v i t y model can be c o n s i d e r e d a s p e c i a l case of the t h r e e - e q u a t i o n model. The l a r g e s t d i f f e r e n c e s i n c o e f f i c i e n t s among the f o u r methods o c c u r s i n the c o e f f i c i e n t s on LNW and AC. Each of these v a r i a b l e s a r e d e t e r m i n e d endogenously. Thus, i t seems t h a t the s i m u l t a n e o u s e q u a t i o n s b i a s i n the c o e f f i c i e n t s induced by i g n o r i n g the e n d o g e n e i t y of the wage r a t e and the d e c i s i o n t o have another c h i l d i s s i g n i f i c a n t . Table 4.5 Hours e q u a t i o n e s t i m a t e s Var i a b l e OLS T o b i t SS ML Con s t a n t 2.407 (11.072) 1.012 (4.1896) ( -1.634 -1.6303) 0. 130 (0.2539) LNW 0.461 (3.7579) 1 .545 , ( 1 1 .204) 5.572 (3.8937) 3.684 (5.8600) AC -0.684 (-2.8914) -0.818 (-3.2271 ) ( -0.618 -2.7135) -2.669 (-8.9092) YR1 8 0.0446 (4.0775) 0.0665 (5.2570) ( -0.018 -0.4645) 0.00534 (0.2437) CHILD -0.169 (-3.3132) -0.216 (-4.1261) ( -0.209 -5.8637) -0.267 (-4.9650) FAMINC -0.264 (-4.6819) -0.304 (-4.8870) ( -0.195 -3.2475) -0.260 (-3.9896) S 2 1.4525 1.4830 10.5524 5.1261 a T - s t a t i s t i c s a re i n p a r e n t h e s e s . C r i t i c a l v a l u e s f o r a two-t a i l e d t - t e s t are 1.96 f o r the 5% l e v e l of s i g n i f i c a n c e and 1.645 f o r the 10% l e v e l of s i g n i f i c a n c e . OLS was run on the subsample of workers o n l y . 85 D.5 The a d d i t i o n a l c h i l d equat i o n T a b l e 4.6 c o n t a i n s the d i f f e r e n t e s t i m a t e s of the a d d i t i o n a l c h i l d e q u a t i o n . In the OLS and p r o b i t e q u a t i o n s , AC i s the dependent v a r i a b l e . In the ML e q u a t i o n , the u n o b s e r v a b l e C* i s the dependent v a r i a b l e . A l l t h r e e methods e s t i m a t e a p o s i t i v e c o e f f i c i e n t f o r the wage r a t e . Both the OLS and p r o b i t c o e f f i c i e n t s a r e s i g n i f i c a n t a t the 10% l e v e l of s i g n i f i c a n c e . The ML e s t i m a t e i s s i g n i f i c a n t a t the 1% l e v e l . Both s i n g l e e q u a t i o n methods u n d e r e s t i m a t e the. c o e f f i c i e n t , r e l a t i v e t o the ML c o e f f i c i e n t . ML e s t i m a t i o n y i e l d s a c o e f f i c i e n t over t w i c e as l a r g e as the p r o b i t c o e f f i c i e n t and over t w e l v e t i m e s as l a r g e as the OLS c o e f f i c i e n t . T h i s r e s u l t s u g gests t h a t s i n g l e e q u a t i o n e s t i m a t i o n t e c h n i q u e s w i l l u n d e r e s t i m a t e the e f f e c t s of the wage r a t e on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d . The two a l t e r n a t i v e methods p r e d i c t a p o s i t i v e e f f e c t of the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d . The maximum l i k e l i h o o d e s t i m a t e , on the o t h e r hand, i s i n s i g n i f i c a n t . T h i s r e s u l t s u g g e s t s t h a t the e f f e c t of the number of c h i l d r e n c u r r e n t l y i n the f a m i l y has o n l y i n d i r e c t e f f e c t s , t h r ough hours worked, on the p r o b a b i l i t y of h a v i n g another c h i l d . W i t h r e g a r d t o hours of work, a l l t h r e e methods e s t i m a t e a n e g a t i v e c o e f f i c i e n t . The OLS e s t i m a t e i s the s m a l l e s t c o e f f i c i e n t ( i n a b s o l u t e v a l u e ) . The v a l u e of the ML e s t i m a t e f a l l s a l most e x a c t l y i n between the v a l u e s of the OLS and p r o b i t c o e f f i c i e n t s . The p r o b i t e s t i m a t e of the e f f e c t of the age of the 86 youngest c h i l d on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d i s l a r g e r ( i n a b s o l u t e v a l u e ) than the ML e s t i m a t e . OLS, on the o t h e r hand, tends t o u n d e r e s t i m a t e the e f f e c t of the age of the youngest c h i l d on the p r o b a b i l i t y of h a v i n g another c h i l d . A l l t h r e e methods p r e d i c t a n e g a t i v e e f f e c t . A l l t h r e e methods e s t i m a t e a p o s i t i v e e f f e c t f o r the age of the w i f e a t mar r i a g e on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d . The p r o b i t i s c l o s e r t o the ML e s t i m a t e than i s the OLS c o e f f i c i e n t . F a m i l y income i s i n s i g n i f i c a n t , a c c o r d i n g to a l l t h r e e e s t i m a t i n g t e c h n i q u e s . T h i s r e s u l t i s somewhat s u r p r i s i n g s i n c e f a m i l y income i s suggested by t h e o r y t o be an imp o r t a n t e x p l a n a t o r y v a r i a b l e . GAP, the d i f f e r e n c e between the d e s i r e d number of c h i l d r e n and the a c t u a l number of c h i l d r e n , i s p r e d i c t e d t o have a p o s i t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d by a l l t h r e e methods. P r o b i t a n a l y s i s o v e r e s t i m a t e s the e f f e c t , w h i l e OLS u n d e r e s t i m a t e s i t , r e l a t i v e t o the ML e s t i m a t e . In g e n e r a l , the p r o b i t c o e f f i c i e n t s a re the c l o s e s t of the two a l t e r n a t i v e e s t i m a t i n g t e c h n i q u e s t o the ML e s t i m a t e s . T h i s r e s u l t s u g g e s t s t h a t the l i n e a r p r o b a b i l i t y model may not g i v e e s t i m a t e s as c l o s e t o the ML e s t i m a t e s as the p r o b i t model. However, the p r o b i t method d i d not g i v e c o e f f i c i e n t s s u f f i c i e n t l y c l o s e t o the ML r e s u l t s f o r i t t o be c o n s i d e r e d a c o m p l e t e l y s a t i s f a c t o r y a l t e r n a t i v e e s t i m a t i n g t e c h n i q u e . 87 T a b l e 4.6 A d d i t i o n a l c h i l d e q u a t i o n e s t i m a t e s 3 V a r i a b l e OLS P r o b i t ML C o n s t a n t -0.0111 (-0.11874) ( -1.7 97 -3.34.12) ( -0.947 -3.3525) LNW 0.0469 (1 .6580) 0.280 ( 1 .6630) 0.633 (4.4180) CHILD 0.0555 (3.3156) 0.203 (2.1958) ( -0.0592 -0.9757) H -0;0377 (-3.1313) ( -0.243 -2.9058) ( -0.309 -6.4794) AGECH -0.00831 (-2.6359) ( -0.0909 -2.9754) ( -0.0515 -39.1369) AGEWMAR 0.00954 (2.2064) 0.0484 (2.0306) 0.0300 (3.4088) FAMINC 0.000604 (0.051553) ( -0.00728 -0.088479) ( -0.0347 -0.6734) GAP 0.0710 (5.6802) 0.275 (4.5134) 0. 107 (2.9986) AC b — — -0.825 T - s t a t i s t i c s are i n p a r e n t h e s e s . C r i t i c a l v a l u e s f o r a two-t a i l e d t - t e s t a re 1.96 f o r the 5% l e v e l of s i g n i f i c a n c e and 1.645 f o r the 10% l e v e l of s i g n i f i c a n c e . b F o r OLS and p r o b i t e s t i m a t i o n , AC i s the dependent v a r i a b l e , r a t h e r than C*. The ML c o e f f i c i e n t on AC i s c o n s t r a i n e d t o e q u a l minus the c o e f f i c i e n t on H t i m e s the hours e q u a t i o n ' s c o e f f i c i e n t on AC. See Chapter Three, s e c t i o n C.1. 88 D.6 C o n s i s t e n t e s t i m a t e s Maximum l i k e l i h o o d e s t i m a t i o n proved t o be e x p e n s i v e . C o n s i s t e n t e s t i m a t e s were d e r i v e d f o l l o w i n g the t e c h n i q u e s d e s c r i b e d i n Wales and Woodland [1980] and Heckman [1978b]. The c o n s i s t e n t e s t i m a t e s are not a s y m p t o t i c a l l y e f f i c i e n t . F u r t h e r , the u s u a l formulae f o r computing t - s t a t i s t i c s a r e not a p p r o p r i a t e . 2 7 Hence, l i t t l e can be s a i d about the s i g n i f i c a n c e of the c o n s i s t e n t e s t i m a t e s . However, i t i s of some i n t e r e s t t o compare the e s t i m a t e d v a l u e s of the c o n s i s t e n t c o e f f i c i e n t s w i t h those d e r i v e d from maximum l i k e l i h o o d e s t i m a t i o n . The c o n s i s t e n t parameter v a l u e s a re p r e s e n t e d i n Ta b l e 4.7 The c o n s i s t e n t e s t i m a t e s t a k e i n t o account two e s t i m a t i o n problems. F i r s t , C* i s not ob s e r v e d . Second, t h e r e may be sample s e l e c t i o n b i a s . These two problems are taken i n t o account i n the f o l l o w i n g way. W r i t e the reduced form e q u a t i o n s i n g e n e r a l n o t a t i o n as (4a) H = Z,p, + V T (4b) C* = Z 2 p 2 + v 2 (4c) W = Z 3 p 3 + v 3 The p o p u l a t i o n r e g r e s s i o n f u n c t i o n s , c o n d i t i o n a l on H > 0, can be w r i t t e n as (Johnson and Kotz [1972]) 2 7 C a l c u l a t i o n of the t - s t a t i s t i c s would i n v o l v e not o n l y c o r r e c t i n g the v a r i a n c e - c o v a r i a n c e m a t r i x of the parameters f o r sample s e l e c t i v i t y . (See, f o r example, the formulae i n Heckman [1976].) In a d d i t i o n , adjustment would have t o be made f o r the f a c t t h a t C* i s not obs e r v e d . 89 (5a) H = Z,p, + R(d) + v,* (5b) C* = Z 2 p 2 + R(d) + v 2 * (5c) W = Z 3 p 3 + R(d) + v 3 * where R(d) = f ( d ) / ( 1 - F ( d ) ) ; d = - Z , p , / ( u , , ) 1 / 2 ; f i s the s t a n d a r d normal d e n s i t y f u n c t i o n ; F i s the s t a n d a r d normal d i s t r i b u t i o n f u n c t i o n ; and v^*, i = 1 ,2,3, a re the c o n d i t i o n a l reduced form d i s t u r b a n c e s . 2 8 S i n c e d i s not known, i t i s n e c e s s a r y t o e s t i m a t e i t . A c o n s i s t e n t e s t i m a t e of d can be o b t a i n e d by a p p l y i n g p r o b i t a n a l y s i s t o the problem of whether the w i f e works. Once an e s t i m a t e of d, and hence R ( d ) , i s o b t a i n e d , OLS can be a p p l i e d t o e q u a t i o n (5c) f o r the subsample of worke r s . T h i s g i v e s us both c o n s i s t e n t e s t i m a t e s of the wage e q u a t i o n and an e s t i m a t e d wage r a t e . P r o b i t a n a l y s i s i s a p p l i e d t o e q u a t i o n (4b) t o o b t a i n an e s t i m a t e of the p r o b a b i l i t y of h a v i n g a c h i l d . Both the e s t i m a t e d wage and the e s t i m a t e d p r o b a b i l i t y of h a v i n g a c h i l d a r e s u b s t i t u t e d i n t o e q u a t i o n ( 5 a ) . E q u a t i o n (5a) i s then e s t i m a t e d u s i n g OLS, which y i e l d s c o n s i s t e n t e s t i m a t e s of the hours e q u a t i o n p a r a m e t e r s . In o r d e r t o e s t i m a t e e q u a t i o n ( 5 b ) , i t i s n e c e s s a r y t o r e w r i t e i t w i t h hours as the dependent v a r i a b l e . T h i s n o r m a l i z a t i o n i s n e c e s s a r y s i n c e C* i s never o b s e r v e d (Heckman [ 1 9 7 8 b ] ) . The e s t i m a t e d wage r a t e , the p r o b a b i l i t y of h a v i n g a c h i l d and the ex p e c t e d v a l u e of C* are s u b s t i t u t e d f o r the See Wales and Woodland [1980]. 90 r e s p e c t i v e endogenous v a r i a b l e s . The e q u a t i o n i s then e s t i m a t e d u s i n g OLS. The r e s t r i c t i o n t h a t k,g 2 = - k 2 can be imposed by f o r m u l a t i n g j o i n t l e a s t square e s t i m a t o r s f o r e q u a t i o n s (5a) and ( 5 b ) . In the hours e q u a t i o n , o n l y the c o e f f i c i e n t FAMINC i s of the same magnitude as t h a t from ML e s t i m a t i o n . In p a r t i c u l a r , the wage c o e f f i c i e n t i s lower and the AC c o e f f i c i e n t h i g h e r ( i n a b s o l u t e v a l u e ) w i t h the c o n s i s t e n t e s t i m a t i o n t e c h n i q u e . The c o n s i s t e n t e s t i m a t i o n t e c h n i q u e p r e d i c t s p o s i t i v e e f f e c t s f o r CHILD and YR18 on hours worked, w h i l e the ML c o e f f i c i e n t f o r the former i s n e g a t i v e . In the a d d i t i o n a l c h i l d e q u a t i o n , the c o n s i s t e n t c o e f f i c i e n t on the wage r a t e i s lower than t h a t o b t a i n e d by ML e s t i m a t i o n . The c o n s i s t e n t e s t i m a t e s a r e of the same magnitude as the ML c o e f f i c i e n t s on hours worked, age of youngest c h i l d , and the gap between d e s i r e d and a c t u a l number of c h i l d r e n . The c o n s i s t e n t e s t i m a t e p r e d i c t s a p o s i t i v e e f f e c t f o r f a m i l y income and c h i l d r e n a l r e a d y i n the f a m i l y on the p r o b a b i l i t y of h a v i n g a nother c h i l d . The ML c o e f f i c i e n t s f o r these v a r i a b l e s have a n e g a t i v e s i g n . I t a l s o p r e d i c t s a n e g a t i v e e f f e c t of the age of the w i f e a t m a r r i a g e , whereas ML p r e d i c t s a p o s i t i v e e f f e c t . The wage e q u a t i o n r e s u l t s are s i m i l a r a c r o s s the two e s t i m a t i n g t e c h n i q u e s . T h i s r e s u l t s h o u l d not be s u r p r i s i n g g i v e n the r e s u l t s i n s e c t i o n D.3 above. More emphasis i s g i v e n t o EDUC and l e s s t o MONPOS by the c o n s i s t e n t e s t i m a t i o n t e c h n i q u e s than by ML e s t i m a t i o n . I f we compare the c o n s i s t e n t e s t i m a t e s w i t h the a l t e r n a t i v e e s t i m a t e s found i n T a b l e s 4.4-4.6, we f i n d t h a t the c o n s i s t e n t 91 e s t i m a t e s do not do unambiguously b e t t e r , r e l a t i v e t o the ML e s t i m a t e s , than the o t h e r e s t i m a t i o n methods. In some c a s e s , ( f o r example, the c o e f f i c i e n t s on CHILD i n the hours e q u a t i o n ) , the c o n s i s t e n t e s t i m a t e s a re f u r t h e r , from the ML e s t i m a t e s than the o t h e r a l t e r n a t i v e e s t i m a t e s . In o t h e r c a s e s , they a r e c l o s e r , ( f o r example, the c o e f f i c i e n t on GAP i n the a d d i t i o n a l c h i l d equat i o n ) . T a b le 4.7 C o n s i s t e n t e s t i m a t e s Wage e q u a t i o n Hours e q u a t i o n C h i l d e q u a t i o n C o n s t a n t 0.84118 Constant 2.5523 Con s t a n t -1.20452 YR18 0.010114 LNW 0.55239 LNW 0.17849 MONPOS 0.0079818 AC -4.6696 CHILD 0.16368 EDUC -0.10812 YR18 0.027969 H -0.79891 MONPOS2 -0.000069164 CHILD 0.0046112 AGECH -0.02448 EDUC 2 0.0074851 FAMINC -0.16061 AGEWMAR -0.00329 FAMINC 0.28129 GAP 0.11562 AC -3.7306 92 E. Summary In t h i s c h a p t e r , we d i s c u s s e d the maximum l i k e l i h o o d e s t i m a t e s of the model d e r i v e d i n Chapter Three. We t e s t e d two hypo t h e s e s . The f i r s t h y p o t h e s i s i s t h a t the c o v a r i a n c e s among the d i s t u r b a n c e terms of the t h r e e e q u a t i o n s a r e a l l z e r o . The second h y p o t h e s i s i s t h a t the c o v a r i a n c e s between the d i s t u r b a n c e terms of the a d d i t i o n a l c h i l d e q u a t i o n and the hours and wage e q u a t i o n s , r e s p e c t i v e l y , a re z e r o . Both hypotheses were r e j e c t e d by the d a t a . These r e s u l t s i m p ly t h a t i t i s c o r r e c t t o model the d e c i s i o n t o have another c h i l d j o i n t l y w i t h 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 . We compared the maximum l i k e l i h o o d e s t i m a t e s of the model w i t h e s t i m a t e s o b t a i n e d from s e v e r a l a l t e r n a t i v e e s t i m a t i n g t e c h n i q u e s . D i f f e r e n t t e c h n i q u e s were based on d i f f e r e n t a s sumptions r e g a r d i n g the s t r u c t u r e of the v a r i a n c e - c o v a r i a n c e m a t r i x of the model. For example, a p p l i c a t i o n of OLS t o a l l t h r e e e q u a t i o n s , or of OLS t o the wage e q u a t i o n and p r o b i t a n a l y s i s and t o b i t a n a l y s i s t o the a d d i t i o n a l c h i l d and hours e q u a t i o n s , r e s p e c t i v e l y , i m p l i e d the h y p o t h e s i s t h a t a l l of the c o v a r i a n c e s of the model a r e z e r o . As t h i s h y p o t h e s i s was r e j e c t e d by the d a t a , i t i s not s u r p r i s i n g t h a t the OLS c o e f f i c i e n t s performed the worst i n e s t i m a t i n g the t h r e e e q u a t i o n s . S i m i l a r l y , a p p l i c a t i o n of the sample s e l e c t i v i t y model t o the hours and wage e q u a t i o n s , t o g e t h e r w i t h p r o b i t a n a l y s i s of the a d d i t i o n a l c h i l d e q u a t i o n , i m p l i e d t h a t o n l y the c o v a r i a n c e between the d i s t u r b a n c e terms of the hours and wage e q u a t i o n s was non-zero. T h i s h y p o t h e s i s was a l s o r e j e c t e d by the d a t a . 93 However, these c o e f f i c i e n t s were c l o s e r , over a l l , t o the ML e s t i m a t e s than were the s i n g l e e q u a t i o n c o e f f i c i e n t s . We a l s o d e r i v e d c o n s i s t e n t e s t i m a t e s of the model. The c o n s i s t e n t e s t i m a t i o n t e c h n i q u e d i d not do w e l l i n e s t i m a t i n g the wage e q u a t i o n , r e l a t i v e t o the o t h e r a l t e r n a t i v e methods. Of the t h r e e ' e q u a t i o n s , the c o n s i s t e n t e s t i m a t e s comes c l o s e s t t o the ML e s t i m a t e s i n the a d d i t i o n a l c h i l d e q u a t i o n . As we were unable t o d e r i v e the s t a n d a r d e r r o r s of the s e e s t i m a t e s , i t i s not p o s s i b l e t o c o m p l e t e l y determine the r e l a t i v e m e r i t of the s e e s t i m a t e s , r e l a t i v e t o the o t h e r a l t e r n a t i v e e s t i m a t e s . The i m p l i c a t i o n of these r e s u l t s i s t h a t the sample s e l e c t i v i t y method, t o g e t h e r w i t h p r o b i t a n a l y s i s appears t o be the best a l t e r n a t i v e t o ML e s t i m a t i o n . However, the s e two methods t o g e t h e r imply t h e r e i s no r e l a t i o n s h i p between the a d d i t i o n a l c h i l d e q u a t i o n and the o t h e r two e q u a t i o n s . T h i s h y p o t h e s i s was r e j e c t e d e a r l i e r . Hence, t h e r e may be some b i a s i n the c o e f f i c i e n t s u s i n g the s i m p l e r e s t i m a t i o n t e c h n i q u e s . A Monte C a r l o s t u d y , however, i s n e c e s s a r y f o r a more p r e c i s e statement of any p o t e n t i a l b i a s . However, as the maximum l i k e l i h o o d e s t i m a t i o n was e x p e n s i v e , t h i s l e s s c o s t l y method might be p r e f e r r e d . 94 Chapter F i v e E s t i m a t i o n of the model I I : P o l i c y I m p l i c a t i o n s A. I n t r o d u c t i o n In the l a s t c h a p t e r , we d e s c r i b e d the maximum l i k e l i h o o d e s t i m a t e s of our model. In t h i s c h a p t e r , we loo k a t the p o l i c y i m p l i c a t i o n s of those e s t i m a t e s . The c h a p t e r i s o r g a n i z e d i n the f o l l o w i n g way. In the next s e c t i o n , we d i s c u s s the g e n e r a l p o l i c y i m p l i c a t i o n s of our par a m e t e r s . We c a l c u l a t e the e l a s t i c i t i e s of hours and wages w i t h r e s p e c t t o the e x p l a n a t o r y v a r i a b l e s . We a l s o c a l c u l a t e the e f f e c t of the e x p l a n a t o r y v a r i a b l e s on the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d . In the t h i r d s e c t i o n , we compare the e f f e c t s of two d i f f e r e n t c h i l d c a r e t a x p o l i c i e s on the l a b o r s u p p l y and f e r t i l i t y b e h a v i o r . We show t h a t u s i n g the maximum l i k e l i h o o d e s t i m a t e s f o r the s i m u l a t i o n g i v e s q u i t e d i f f e r e n t p o l i c y i m p l i c a t i o n than i f s i n g l e e q u a t i o n e s t i m a t e s a re used. In the l a s t s e c t i o n , we summarize our r e s u l t s . 95 B. G e n e r a l p o l i c y i m p l i c a t i o n s T a b l e 5.1 c o n t a i n s the 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 t h r e e e q u a t i o n s . I t a l s o c o n t a i n s the e s t i m a t e s of the v a r i a n c e - c o v a r i a n c e m a t r i x o f _ t h e d i s t u r b a n c e terms. The s i g n s on YR18, EDUC, and EDUC 2 i n the wage e q u a t i o n a r e as e x p e c t e d . However, EDUC i s i n s i g n i f i c a n t , w h i l e EDUC 2 i s s i g n i f i c a n t o n l y a t the 10% l e v e l . MONPOS has a p o s i t i v e c o e f f i c i e n t , w h i l e MONPOS2 has a n e g a t i v e c o e f f i c i e n t . One e x p l a n a t i o n i s t h a t o n l y a few months i n a p o s i t i o n may r e p r e s e n t a pro m o t i o n . Hence, the i n c r e a s e i n the wage r a t e from the o l d p o s i t i o n t o the new p o s i t i o n may be h i g h e r than the wage r a t e i n c r e a s e s which occur as the woman c o n t i n u e s i n the same p o s i t i o n . In the hours e q u a t i o n , a l l of the v a r i a b l e s a r e s i g n i f i c a n t a t the 5% l e v e l except the number of y e a r s worked s i n c e 18. The wage r a t e has a l a r g e p o s i t i v e e f f e c t on hours worked, w h i l e the pre s e n c e of a new c h i l d has a l a r g e n e g a t i v e e f f e c t . The number of c h i l d r e n i n the f a m i l y has a n e g a t i v e e f f e c t on hours worked. 1 T h i s r e s u l t s u g gests t h e r e e x i s t s some t r a d e - o f f between h a v i n g c h i l d r e n and w o r k i n g . F a m i l y income a l s o has a n e g a t i v e e f f e c t on hours worked, as ex p e c t e d from t h e o r y ( g i v e n 1 Nakamura and Nakamura [1981] and Rosenzweig [1976] f i n d the number of c h i l d r e n i n the f a m i l y t o have a n e g a t i v e e f f e c t on hours worked. 96 l e i s u r e i s a normal g o o d ) . 2 In the a d d i t i o n a l c h i l d e q u a t i o n , the wage r a t e has a p o s i t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d . 3 These r e s u l t s suggest t h a t the presence of an a d d i t i o n a l c h i l d r a i s e s the w i f e ' s shadow wage r a t e . That i s , i f the w i f e i s w o r k i n g , she w i l l have another c h i l d o n l y i f her market wage r a t e i s h i g h enough t o o f f s e t the i n c r e a s e i n her shadow wage r a t e caused by h a v i n g the c h i l d . " Hours of work have a n e g a t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d , as does the age of the youngest c h i l d . These r e s u l t s a g a i n suggest a t r a d e - o f f between w o r k i n g o u t s i d e the home and h a v i n g c h i l d r e n . The age of the w i f e a t m a r r i a g e has a s m a l l p o s i t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d . T h i s r e s u l t i m p l i e s o l d e r wives a t m a r r i a g e may f e e l the e f f e c t of the b i o l o g i c a l c o n s t r a i n t on h a v i n g c h i l d r e n . GAP, the d i f f e r e n c e between the d e s i r e d number of c h i l d r e n and the a c t u a l number of c h i l d r e n , has the ex p e c t e d p o s i t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d . 2 The n e g a t i v e r e l a t i o n s h i p between f a m i l y income and hours worked i s s t a n d a r d i n the l i t e r a t u r e . See, f o r example, Mincer [ 1 9 6 2 ] , C a i n [1966], C a r l i n e r ' e t a l [ 1 9 8 0 ] , Hotz [ 1 9 8 0 ] , and Nakamura and Nakamura [1981]. Note, however, t h a t the husband's e a r n i n g s , r a t h e r than f a m i l y income, i s o f t e n used as the income var i a b l e . 3 Not a l l p r e v i o u s r e s e a r c h uses the wage r a t e as an e x p l a n a t o r y v a r i a b l e d e s p i t e i t s t h e o r e t i c a l i m p o r t a n c e . The u s u a l reason i s l a c k of d a t a . Years of s c h o o l i n g of the w i f e i s o f t e n used as a proxy f o r the wage r a t e . (See, f o r example, W i l l i s [ 1 9 7 3 ] , Heckman and W i l l i s [1977].) McCabe and. Rosenzweig [1976] f i n d a p o s i t i v e e f f e c t of a p r e d i c t e d wage v a r i a b l e on the number of c h i l d r e n ever born. " See the d i s c u s s i o n i n Chapter Four on the o p p o r t u n i t y c o s t of h a v i n g a c h i l d . 97 The number of c h i l d r e n a l r e a d y i n the f a m i l y i s i n s i g n i f i c a n t i n e x p l a i n i n g the p r o b a b i l i t y of h a v i n g another c h i l d . 5 F a m i l y income i s a l s o i n s i g n i f i c a n t . 6 The v a r i a n c e - c o v a r i a n c e m a t r i x of the d i s t u r b a n c e terms i s a l s o g i v e n i n T a b l e 5.1. G i v e n t h a t the c o v a r i a n c e of the d i s t u r b a n c e terms of the a d d i t i o n a l c h i l d and wage e q u a t i o n s i s f i x e d a t z e r o , the e s t i m a t e d c o r r e l a t i o n c o e f f i c i e n t s between the d i s t u r b a n c e terms of the hours and a d d i t i o n a l c h i l d e q u a t i o n s , and the hours and wage e q u a t i o n s , are .63251 and -.74050, r e s p e c t i v e l y . As d i s c u s s e d above, hours have a n e g a t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d , and the presence of an a d d i t i o n a l c h i l d has a n e g a t i v e e f f e c t on hours worked. However, the p o s i t i v e c o r r e l a t i o n c o e f f i c i e n t between the d i s t u r b a n c e terms of the two e q u a t i o n s s u g g e s t s t h a t t h e r e i s some p o s i t i v e r e l a t i o n s h i p between the u n e x p l a i n e d p o r t i o n of the v a r i a n c e s of the two dependent v a r i a b l e s . 5 P r e v i o u s r e s e a r c h used the number of c h i l d r e n i n the f a m i l y as the dependent v a r i a b l e . However, Khan and S i r a g e l d i n [1977] use p r o b i t a n a l y s i s on data from P a k i s t a n . They a l s o found a n e g a t i v e r e l a t i o n s h i p between the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y and the p r o b a b i l i t y of h a v i n g another c h i l d . Hotz [1980] found the v a r i a b l e t o be i n s i g n i f i c a n t i n h i s model. 6 Most s t u d i e s use the husband's e a r n i n g s as the income v a r i a b l e , r a t h e r than f a m i l y income, which i n c l u d e s a s s e t s . Ben-P o r a t h [1973] and Khan and S i r a g e l d i n [1977] f i n d the husband's e a r n i n g s t o be i n s i g n i f i c a n t i n most c a s e s . Hotz [1980] f i n d s the husband's e a r n i n g s t o have a n e g a t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g a c h i l d . Table 5.1 Maximum l i k e l i h o o d e s t i m a t e s Wage e q u a t i o n Var i a b l e C o n s t a n t YR1 8 MONPOS EDUC MONPOS 2EDUC 2 Coef f i c i e n t 0.49689 0.013568 0.021669 -0.050721 -0.00019531 0.0034671 Asymptot i c t - s t a t i s t i c 1 .8328 2.6305 6.5030 -1.1377 -6.0999 1.7853 Hours e q u a t i o n Var i a b l e C o n s t ant LNW AC YR1 8 CHILD FAMINC Coef f i c i e n t 0.13042 3.68365 -2.66855 0.0053365 -0.26718 -0.26014 Asymptot i c t - s t a t i s t i c 0.2539 5.8600 -8.9092 0.2437 -4.9650 -3.9896 Table 5.1 ( c o n t . ) 3. A d d i t i o n a l c h i l d e q u a t i o n Asymptot i c V a r i a b l e Coef f i c i e n t t - s t a t i s t C o n s t ant -0.94683 -3.3525 LNW 0.66319 4.4180 CHILD -0.059186 -0.9757 H -0.30924 -6.4794 AGECH -0.051493 -39.1369 AGEWMAR 0.029954 3.4088 FAMINC -0.034700 -0.6734 GAP 0.10745 2.9887 AC -0.82521 a 4. A s y m p t o t i c v a r i a n c e - c o v a r i a n c e m a t r i x of the d i s t u r b a n c e s H 5.12609 C 1.43207 1.00000 b W -0.93802 0.0 C 0.31302 The c o e f f i c i e n t on AC i s r e s t r i c t e d t o e q u a l minus the c o e f f i c i e n t on H times the c o e f f i c i e n t on AC i n the hours e q u a t i o n . S i n c e AC was e s t i m a t e d as the p r o d u c t of the s e two c o e f f i c i e n t s , no n u m e r i c a l d e r i v a t i v e c o u l d be c a l c u l a t e d . . b . The v a r i a n c e of the C* e q u a t i o n must be n o r m a l i z e d f o r i d e n t i f i c a t i o n . c A s d i s c u s s e d i n Chapter Four, the c o v a r i a n c e between the d i s t u r b a n c e s of the a d d i t i o n a l c h i l d e q u a t i o n and the wage e q u a t i o n was f i x e d a t z e r o . 100 S i m i l a r l y , the r e g r e s s i o n r e s u l t s imply a p o s i t i v e r e l a t i o n s h i p between wages and hours worked, w h i l e the n e g a t i v e c o r r e l a t i o n c o e f f i c i e n t between the two d i s t u r b a n c e terms suggests a n e g a t i v e r e l a t i o n s h i p between the u n e x p l a i n e d p o r t i o n of the v a r i a n c e s . T a b l e 5.2 p r e s e n t s the e s t i m a t e d e l a s t i c i t i e s of the wage r a t e and hours of work w i t h r e s p e c t t o the e x p l a n a t o r y v a r i a b l e s . The e f f e c t of a change i n the e x p l a n a t o r y v a r i a b l e s on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d i s a l s o g i v e n . The e l a s t i c i t i e s i n T a b l e 5.2 suggest t h a t months i n c u r r e n t p o s i t i o n has the l a r g e s t e f f e c t of the two s i g n i f i c a n t v a r i a b l e s on the wage r a t e . One more year i n the c u r r e n t p o s i t i o n (a 64% i n c r e a s e a t the means), w i l l i n c r e a s e the wage r a t e by 19%, or 33 c e n t s per hour a t the means. The number of y e a r s worked s i n c e 18 has the s m a l l e s t e f f e c t on the wage r a t e . 7 The e l a s t i c i t y of hours of work over the two-year p e r i o d w i t h r e s p e c t t o the wage r a t e i s 1.52. That i s , a 10% i n c r e a s e i n the ( a f t e r - t a x ) wage r a t e w i l l r e s u l t i n a 15.2% i n c r e a s e i n hours worked a t the mean, or 238 hours over two y e a r s . The h i g h p o s i t i v e wage e l a s t i c i t y s u g g e s t s m a r r i e d women a r e s e n s i t i v e t o t h e i r a f t e r - t a x wage r a t e . T h i s r e s u l t i s c o n s i s t e n t w i t h s e v e r a l o t h e r s t u d i e s , which found a h i g h , p o s i t i v e wage 7 Note t h a t the e l a s t i c i t i e s a r e c a l c u l a t e d w i t h r e s p e c t t o the wage r a t e i t s e l f , not the n a t u r a l l o g a r i t h m of the wage r a t e . T h i s i s a l s o t r u e i n the hours e l a s t i c i t i e s ; the a c t u a l wage r a t e i s used i n the c a l c u l a t i o n s . The change i n the p r o b a b i l i t y of h a v i n g another c h i l d due to a change i n the wage r a t e , however, i s e x p r e s s e d i n terms of the l o g a r i t h m r a t h e r than the a c t u a l l e v e l . 101 e l a s t i c i t y . 8 We can a l s o c a l c u l a t e the e f f e c t on hours worked of a change i n the m a r g i n a l ta x r a t e i t s e l f . For example, a 10% i n c r e a s e i n the m a r g i n a l t a x r a t e r e s u l t s i n a 4% d e c r e a s e i n hours worked a t the means (62 hours over two y e a r s ) . T h i s low r e s u l t o c c u r s because a 10% i n c r e a s e i n the m a r g i n a l tax r a t e l o w e r s the a f t e r - t a x wage r a t e by o n l y 2.6% a t the means. However, t h e r e i s a l a r g e d e c r e a s e i n hours worked i f the m a r g i n a l tax r a t e i n c r e a s e s by t e n p e r c e n t a g e p o i n t s , a 48% i n c r e a s e i n the m a r g i n a l ta x r a t e a t the means. I f the m a r g i n a l tax r a t e i n c r e a s e s by ten p e r c e n t a g e p o i n t s , hours worked w i l l d e c r e a s e by 19.2% a t the means (299 hours over two y e a r s ) . The e f f e c t of h a v i n g an a d d i t i o n a l c h i l d on hours worked i s l a r g e . An a d d i t i o n a l c h i l d w i l l cause the w i f e t o d e c r e a s e her hours of market work by 2669 hours a t the means over a two-year p e r i o d . That i s , the model p r e d i c t s the w i f e w i l l withdraw from the l a b o r f o r c e f o r s l i g h t l y over one year (out of a two-year p e r i o d ) when she has a c h i l d . 9 T h i s r e s u l t may seem h i g h i n view of i n s t i t u t i o n a l c o n s t r a i n t s on m a t e r n i t y l e a v e and hours of work. However, our model assumes the w i f e has the f l e x i b i l i t y t o change her hours of work as she w i s h e s . The number of c h i l d r e n a l r e a d y i n the f a m i l y and f a m i l y income have the e x p e c ted n e g a t i v e s i g n s . I f t h e r e i s one more c h i l d a l r e a d y i n t h e f a m i l y (a 50% i n c r e a s e a t the means), the 8 See, f o r example, Rosen [1 9 7 6 ] , who found a wage e l a s t i c i t y of 2.3. Heckman [1977] f i n d s a wage e l a s t i c i t y of 4.5. Nakamura and Nakamura [1981 ], on the o t h e r hand, f i n d m o s t l y s m a l l ( i n a b s o l u t e v a l u e ) , n e g a t i v e wage e l a s t i c i t i e s . 9 A l t e r n a t i v e l y , i t p r e d i c t s she w i l l s w i t c h from f u l l - t i m e work to p a r t - t i m e work. 102 w i f e w i l l work 10.7% fewer hours (167 hours a t the means). An i n c r e a s e of 10% i n f a m i l y income a t the means w i l l d e c r e ase the w i f e ' s hours of work by 2.5%. C l e a r l y , t h e n , the w i f e ' s hours of work a r e most r e s p o n s i v e t o her a f t e r - t a x wage r a t e and whether she has an o t h e r c h i l d . The p r o b a b i l i t y of h a v i n g a c h i l d a t the means i s 11.9%. T h i s e stimated, p r o b a b i l i t y almost e x a c t l y e q u a l s the a c t u a l p r o p o r t i o n of women i n the sample who have a c h i l d ( 1 1 . 8 % ) . The p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d i s almost i d e n t i c a l f o r the subsamples of workers and non-workers. One i n t e r e s t i n g r e s u l t i s t h a t the a f t e r - t a x wage r a t e has a p o s i t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d . An i n c r e a s e of one d o l l a r i n the a f t e r - t a x wage r a t e , a t the means, w i l l i n c r e a s e the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d by 7%. Hours of work has a n e g a t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d , as e x p e c t e d . An i n c r e a s e i n hours worked a t the means, from 1556 t o 4000 ( f u l l - t i m e f o r two y e a r s ) i m p l i e s a d e c r e a s e of 9% i n the p r o b a b i l i t y of h a v i n g a c h i l d . The age of the youngest c h i l d has a n e g a t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d . An i n c r e a s e i n the age of the youngest c h i l d by one year w i l l d e c r e ase the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d by s l i g h t l y l e s s than 1% a t the means. These r e s u l t s suggest t h a t any t a x p o l i c y w i l l have an e f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d o n l y through i t s e f f e c t on the a f t e r - t a x wage r a t e and through i t s e f f e c t on hours worked. A t a x p o l i c y w i l l i n f l u e n c e hours worked both Table 5.2 E l a s t i c i t i e s and p r o b a b i l i t i e s 1. Wage e q u a t i o n 1 03 Var i a b l e YR1 8 MONPOS EDUC A l l women  E l a s t i c i t y .1172 .2973 .4264 Workers  E l a s t i c i t y . 1 1 96 .2881 .3970 2. Hours e q u a t i o n 0 Var i a b l e WAGE YR1 8 CHILD FAMINC A l l women E l a s t i c i t y 1.5174 0.01706 -.2138 -.2546 Workers  E l a s t i c i t y 1.3091 0.01741 -. 1604 -.2097 The e l a s t i c i t i e s a r e c a l c u l a t e d w i t h r e s p e c t t o the a c t u a l wage r a t e . T a b l e 5.2 (cont.) 3. A d d i t i o n a l c h i l d e q u a t i o n Var i a b l e LNW CHILD H AGECH AGEWMAR FAMINC GAP A l l women E f f e c t on p r o b a b i 1 i t y .'. 1 256 -.01174 -.06135 -.01022 .005943 -.006885 .02132 Workers E f f e c t on p r o b a b i l i t y . 1 257 -.01175 -.06140 -.01022 .005947 -.006889 .02133 Non-workers E f f e c t on p r o b a b i 1 i t y .1254 -.01 172 -.061 26 -.01020 .005934 -.006874 .02129 Prob(AC=l) f o r t o t a l sample = .1186 Prob(AC=l) f o r workers = .1187 Prob(AC=l) f o r non-workers = .1184 105 through changes i n a f t e r - t a x f a m i l y income and i n the a f t e r - t a x wage r a t e . In the next s e c t i o n , we examine the e f f e c t s of two d i f f e r e n t t a x p o l i c i e s on hours worked and the p r o b a b i l i t y of ha v i n g another c h i l d . C. C h i Id c a r e t a x p o l i c i e s C.1 1975 and 1976 c h i l d c a r e tax p o l i c i e s In the 1975 t a x y e a r , a c o u p l e c o u l d c l a i m the f o l l o w i n g c h i l d c a r e expenses as a d e d u c t i o n . I f the c h i l d c a r e took p l a c e i n the c o u p l e ' s home, the maximum d e d u c t i o n was $400 per month. I f the c h i l d c a r e was o u t s i d e the home, the maximum d e d u c t i o n was $200 per month f o r one c h i l d , $300 f o r two c h i l d r e n , and $400 f o r t h r e e or more c h i l d r e n . T h i s d e d u c t i o n c o u l d be c l a i m e d by the c o u p l e o n l y i f they f i l e d a j o i n t r e t u r n and o n l y i f they i t e m i z e d d e d u c t i o n s . F u r t h e r , the c h i l d c a r e had t o be i n c u r r e d t o p e r m i t the t a x p a y e r t o be " g a i n f u l l y employed" or t o seek such g a i n f u l employment. That i s , b a b y s i t t i n g c o s t s f o r an ev e n i n g out were not d e d u c t i b l e . 1 0 In 1976, the law was changed. C h i l d c a r e expenses became a tax c r e d i t r a t h e r than a d e d u c t i o n . A c o u p l e c o u l d c l a i m 20% of c h i l d c a r e expenses as a tax c r e d i t . The maximum amount of c h i l d c a r e expenses t o which the c r e d i t c o u l d be a p p l i e d was $2000 f o r one c h i l d and $4000 f o r two or more c h i l d r e n . Hence, the maximum c r e d i t f o r one c h i l d was $400 (20% of $2000) and $800 f o r two or 1 0 U.S. Master Tax Guide, 1976, paragraph 1039. 106 more c h i l d r e n . A g a i n , the c o u p l e had t o f i l e a j o i n t r e t u r n and the c h i l d c a r e expenses had t o be i n c u r r e d t o p e r m i t g a i n f u l employment. However, the c o u p l e no l o n g e r had t o i t e m i z e d e d u c t i o n s t o r e c e i v e the t a x c r e d i t . 1 1 The d i f f e r e n c e between h a v i n g c h i l d c a r e expenses as a d e d u c t i o n (which l o w e r s t a x a b l e income) and as a c r e d i t (which lowers the amount of t a x p a y a b l e f o r a g i v e n income) can be seen i n the f o l l o w i n g example. Suppose a c o u p l e has one c h i l d and a j o i n t income of $20,000 a f t e r a l l d e d u c t i o n s except the c h i l d c a r e d e d u c t i o n . Assume the c o u p l e can c l a i m the f u l l amount of c h i l d c a r e expenses p o s s i b l e . Then, under the 1975 law, the $20,000 t a x a b l e income i s reduced t o $17,600. The t a x on t h i s income would be $3648. Under the 1976 law, t a x a b l e income remains $20,000, but the t a x p a y a b l e i s reduced by $400 t o $ 3 9 2 0 . 1 2 In t h i s c a s e , the c o u p l e i s b e t t e r o f f , i n terms of t a x l i a b i l i t y , under the 1975 law. However, at lower income l e v e l s , the c o u p l e would be b e t t e r o f f under the 1976 law. For example, i f income were $10,000, a f t e r a l l d e d u c t i o n s except c h i l d c a r e , the c o u p l e would have p a i d $1732 i n t a x e s i n 1975, but o n l y $1215 i n 1976. In t h i s section., we examine the e f f e c t of two d i f f e r e n t tax p o l i c i e s 1 3 on hours worked and on the p r o b a b i l i t y of h a v i n g 1 1 U.S. Master Tax Guide, 1977, par a g r a p h 175D. 1 2 There was no change i n the m a r g i n a l t a x r a t e between the two y e a r s . 1 3 The two p o l i c i e s d e s c r i b e d here a r e d e s i g n e d more t o i l l u s t r a t e the i m p l i c a t i o n s of our model than t o o f f e r e x a c t p o l i c y recommendations. 1 07 another c h i l d . As a base p o l i c y f o r comparison, we ran a s i m u l a t i o n assuming the c o u p l e c l a i m e d o n l y the $750 exemption per dependent and d i d not i t e m i z e d e d u c t i o n s . A g a i n s t t h i s base p o l i c y , we compared the f o l l o w i n g two p o l i c i e s . The f i r s t p o l i c y assumes t h e r e are no d e d u c t i o n s or c r e d i t s f o r c h i l d c a r e expenses. However, i t assumes the p e r s o n a l exemption d e d u c t i o n s are i n c r e a s e d from $750 t o $1000 per d e p e n d e n t . 1 4 The second p o l i c y assumes p e r s o n a l exemptions remain at $750 per dependent, but a l l o w s a r e f u n d a b l e tax c r e d i t f o r c h i l d c a r e e x p e n d i t u r e s . 1 5 The maximum t a x c r e d i t a l l o w e d i s assumed to be 25% of $2000 f o r one c h i l d , of $4000 f o r two c h i l d r e n , and of $6000 f o r t h r e e or more c h i l d r e n . We assume the s e expenses can be c l a i m e d as a t a x c r e d i t o n l y i f they a r e i n c u r r e d t o p e r m i t g a i n f u l employment. We a l s o assume the t a x p a y e r i s a b l e to c l a i m the maximum amount p e r m i s s i b l e . The r e s u l t of the two p o l i c i e s i s not o n l y t o change a f t e r -tax f a m i l y income, but a l s o t o change the a f t e r - t a x wage r a t e , i f the m a r g i n a l tax r a t e a l s o changes as a r e s u l t of the p o l i c y . Hence, hours worked w i l l be a f f e c t e d by both income and wage changes, as w e l l as by the induced change i n the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d . S i m i l a r l y , the p r o b a b i l i t y of h a v i n g 1' As w i t h the base p o l i c y , t h i s d e d u c t i o n a p p l i e s t o the t a x p a y e r and spouse, as w e l l as t o c h i l d r e n . 1 5 A r e f u n d a b l e tax c r e d i t means t h a t i f , when the c r e d i t i s a p p l i e d t o t a x e s owing, the r e s u l t i s a n e g a t i v e number, the t a x p a y e r can c l a i m t h i s amount as a r e f u n d . A n o n - r e f u n d a b l e tax c r e d i t would mean t h a t t a x e s would go t o z e r o , and t h e r e would be no r e f u n d . In our s i m u l a t i o n s , a r e f u n d o c c u r s o n l y f o r f a m i l i e s w i t h an income o f • $ 10,000/year and t h r e e or more c h i l d r e n . 108 another c h i l d w i l l not o n l y be a f f e c t e d by the change i n income and wages, but a l s o by the induced change i n hours worked. We make two s e t s of comparisons w i t h our s i m u l a t i o n s . F i r s t , we compare the e f f e c t of the two p o l i c i e s on hours worked and the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d w i t h the p r e d i c t e d hours and p r o b a b i l i t i e s o b t a i n e d under the base p o l i c y . Sec/ond, we compare the s i m u l a t i o n s of the t h r e e p o l i c i e s (base p o l i c y , an i n c r e a s e i n p e r s o n a l exemptions, and a c h i l d c a r e t a x c r e d i t ) run u s i n g the maximum l i k e l i h o o d e s t i m a t e s w i t h s i m u l a t i o n s run u s i n g the o r d i n a r y l e a s t squares e s t i m a t e s . We f i n d t h e r e a re l a r g e d i f f e r e n c e s between the s i m u l a t i o n s run u s i n g ML e s i m t a t e s and those run u s i n g OLS e s t i m a t e s . In our s i m u l a t i o n s , we make the f o l l o w i n g a s s u m p t i o n s . F i r s t , we assume the w i f e i s w o r k i n g . We make t h i s assumption s i n c e , under U.S. t a x law, c h i l d c a r e d e d u c t i o n s or t a x c r e d i t s have been a l l o w e d o n l y i f the c h i l d c a r e i s n e c e s s a r y t o m a i n t a i n or t o o b t a i n employment. Second, we assume t h a t the w i f e has n i n e y e a r s of work e x p e r i e n c e , t h a t the age of the youngest c h i l d i s f i v e y e a r s o l d , t h a t the age of the w i f e a t m a r r i a g e i s 20 y e a r s o l d , and t h a t the c o u p l e d e s i r e s one more c h i l d than they a l r e a d y have. We examine the e f f e c t s of the p o l i c i e s f o r f a m i l i e s w i t h d i f f e r e n t incomes, wages, and number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y . 109 C.2 E f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d T a b l e s 5.3-5.8 show the e f f e c t of the t h r e e p o l i c i e s on the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d . T a b l e s 5.3-5.5 were c a l c u l a t e d u s i n g the ML e s t i m a t e s . T a b l e 5.6-5.8 were c a l c u l a t e d u s i n g the OLS e s t i m a t e s . We c a l c u l a t e the p r o b a b i l i t i e s of h a v i n g a n o t h e r c h i l d f o r f a m i l i e s w i t h z e r o t o f o u r c h i l d r e n , 1 6 f o r b e f o r e - t a x wage r a t e s of $3.00/hour t o $9.00/hour, and f o r b e f o r e - t a x f a m i l y incomes of $l 0 , 0 0 0 / y e a r t o $25,000/year. For g i v e n income l e v e l s , wage r a t e s , and number of c h i l d r e n , t h e r e i s l i t t l e d i f f e r e n c e i n the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d under any of the t h r e e p o l i c i e s . However, t h e r e a r e d i f f e r e n c e s among d i f f e r e n t l e v e l s of wage r a t e s , incomes, and number of c h i l d r e n . L e t us f i r s t l o o k a t T a b l e s 5.3-5.5. D e s p i t e the f a c t t h a t the wage r a t e has a p o s i t i v e e f f e c t on the p r o b a b i l i t y of h a v i n g a nother c h i l d , the p r o b a b i l i t y d e c r e a s e s as the wage r a t e i n c r e a s e s . T h i s r e s u l t h o l d s , r e g a r d l e s s of the number of c h i l d r e n or income l e v e l s . The reason f o r t h i s r e s u l t i s t h a t as the wage r a t e i n c r e a s e s , hours of work a l s o i n c r e a s e . In t h i s c a s e , the n e g a t i v e e f f e c t of hours worked on the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d outweighs the p o s i t i v e wage e f f e c t . As income i n c r e a s e s , the p r o b a b i l i t y of h a v i n g a c h i l d i n c r e a s e s , f o r g i v e n wage r a t e s and number of c h i l d r e n . For example, w i t h t h r e e c h i l d r e n and a b e f o r e - t a x wage r a t e of 1 6 Note t h a t w i t h z e r o c h i l d r e n , t h e r e i s no t a x c r e d i t . Hence, the r e s u l t s f o r f a m i l i e s w i t h z e r o c h i l d r e n a r e the same f o r the tax c r e d i t p o l i c y as f o r the base p o l i c y . 110 $5.00, the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d i n c r e a s e s from 11.9% t o 15.7% under the base p o l i c y , as income r i s e s from $10,000 per year t o $25,000 per y e a r ; from 11.7% t o 15.7% under the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y ; and from 11.2 t o 15.5% under the t a x c r e d i t p o l i c y . We a l s o see an i n c r e a s e i n the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d as the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y i n c r e a s e s , f o r g i v e n wage r a t e s and income l e v e l s . T h i s r e s u l t r e f l e c t s the f a c t t h a t as the number of c h i l d r e n i n c r e a s e s , d e d u c t i o n s (and c r e d i t s ) i n c r e a s e , c h a n g i n g both a f t e r - t a x income and the a f t e r - t a x wage r a t e ( t h r o u g h changes i n the m a r g i n a l t a x r a t e ) . The i n c r e a s e i n p r o b a b i l i t y as the number of c h i l d r e n a l r e a d y p r e s e n t i n the f a m i l y i n c r e a s e s i s l a r g e s t f o r the h i g h income l e v e l s . For example, a t a $7.00 b e f o r e - t a x wage r a t e , the i n c r e a s e i n the p r o b a b i l i t y of h a v i n g another c h i l d a t a y e a r l y income of $10,000 i s .9% under the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y . At a y e a r l y income of $25,000, the p r o b a b i l i t y i s 1.4%. T h i s g e n e r a l r e s u l t i s a l s o t r u e f o r the o t h e r two p o l i c i e s . A l t h o u g h the d i f f e r e n c e s i n p r o b a b i l i t i e s f o r g i v e n wage r a t e , income, and number of c h i l d r e n a r e s m a l l , t h e r e i s a g e n e r a l t r e n d of f a l l i n g p r o b a b i l i t i e s . That i s , p r o b a b i l i t i e s under the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y a re s m a l l e r than those under the base p o l i c y , and those under the t a x c r e d i t p o l i c y a r e s m a l l e r than those under the o t h e r two p o l i c i e s . T h i s r e s u l t s u g g e s t s t h a t the n e g a t i v e hours e f f e c t dominates the p o s i t i v e wage e f f e c t . The e f f e c t of moving from the base p o l i c y t o the t a x c r e d i t p o l i c y , v i a the i n c r e a s e d p e r s o n a l d e d u c t i o n s 111 p o l i c y , i s t o decrease the m a r g i n a l t a x r a t e f o r a g i v e n income l e v e l and number of c h i l d r e n , as w e l l as t o decrease the a b s o l u t e amount of t a x e s p a i d . Hence, both a f t e r - t a x wage r a t e s a n d . a f t e r - t a x incomes i n c r e a s e . However, as w i l l be seen i n the next s e c t i o n , hours worked a l s o i n c r e a s e . The hours worked e f f e c t dominates the e f f e c t of the i n c r e a s e i n a f t e r - t a x wage r a t e s . 1 7 I f we compare the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y w i t h the base p o l i c y , we f i n d the f o l l o w i n g r e s u l t s . At a low income l e v e l ($10,000 per y e a r ) , the i n c r e a s e i n p e r s o n a l d e d u c t i o n s has i t s l a r g e s t e f f e c t when t h e r e a r e t h r e e or more c h i l d r e n , r e g a r d l e s s of the wage r a t e . For example, w i t h a $3.00 b e f o r e -t a x wage r a t e and one c h i l d , the p e r c e n t a g e decrease i n the p r o b a b i l i t y of h a v i n g another c h i l d i s o n l y .6%, whereas w i t h f o u r c h i l d r e n , i t i s 1.6%. At an income l e v e l of $15,000 per y e a r , t h e r e i s a r e l a t i v e l y l a r g e e f f e c t f o r f a m i l i e s w i t h no c h i l d r e n and w i t h two c h i l d r e n , r e g a r d l e s s of the wage r a t e . The lo w e s t e f f e c t ( i n p e r c e n t a g e terms) i s f o r f a m i l i e s w i t h one c h i l d . With t h r e e or more c h i l d r e n , the e f f e c t s a re r e l a t i v e l y l a r g e r a t the h i g h e r wage r a t e s ($7.00 and $9.00). At an income l e v e l of $20,000 per y e a r , the l a r g e s t e f f e c t of the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y o c c u r s f o r two or 1 7 A l t h o u g h a f t e r - t a x f a m i l y income was found not t o be s i g n i f i c a n t i n e x p l a i n i n g the p r o b a b i l i t y of h a v i n g another c h i l d (see S e c t i o n B ) , i t does have a n e g a t i v e c o e f f i c i e n t . Hence, we see i n our s i m u l a t i o n s t h a t a r i s e i n a f t e r - t a x f a m i l y income due t o a p o l i c y change a l s o works t o decrease the p r o b a b i l i t y of h a v i n g another c h i l d . 1 1 2 more c h i l d r e n , r e g a r d l e s s of the wage r a t e . For a y e a r l y income of $25,000, t h e r e i s not a r e l a t i v e l y l a r g e e f f e c t u n t i l f o u r c h i l d r e n . These r e s u l t s suggest the p o l i c y has i t s l a r g e s t e f f e c t on e x t r a - m a r g i n a l c h i l d r e n . That i s , c o u p l e s may have two c h i l d r e n r e g a r d l e s s of income l e v e l s or o t h e r economic v a r i a b l e s , but the t h i r d and subsequent c h i l d i s more l i k e l y t o be i n f l u e n c e d by economic f a c t o r s . 1 8 W i t h the tax c r e d i t p o l i c y , we see somewhat s t r o n g e r e f f e c t s . For a g i v e n income l e v e l , the per c e n t a g e d e c r e a s e i n the p r o b a b i l i t y of h a v i n g a c h i l d r i s e s w i t h the number of c h i l d r e n , f o r a g i v e n wage r a t e , w i t h some l e v e l i n g at f o u r c h i l d r e n . T h i s l e v e l i n g a t f o u r c h i l d r e n i s t o be ex p e c t e d s i n c e the t a x c r e d i t p o l i c y has an i n c r e a s e i n the tax c r e d i t o n l y up t o the t h i r d c h i l d . That i s , h a v i n g more than t h r e e c h i l d r e n does not y i e l d any e x t r a t a x b e n e f i t s . For example, w i t h a y e a r l y income of $15,000 and a b e f o r e - t a x wage r a t e of $7.00, the p e r c e n t a g e decrease i n the p r o b a b i l i t y of h a v i n g another c h i l d ranges from 1.6% w i t h one c h i l d t o 3.5% w i t h f o u r c h i l d r e n . 1 9 At f o u r c h i l d r e n , t h e r e i s a s m a l l d e c rease i n the p r o b a b i l i t y from t h r e e c h i l d r e n (3.7% f o r t h r e e c h i l d r e n t o 3.5% f o r f o u r ) . The p e r c e n t a g e d e c r e a s e i n the p r o b a b i l i t y of h a v i n g 1 8 See Simon [1969] f o r f u r t h e r e v i d e n c e i n su p p o r t of t h i s h y p o t h e s i s . 1 9 R e c a l l t h a t the the t a x c r e d i t p o l i c y , the p r o b a b i l i t i e s f o r f a m i l i e s w i t h z e r o c h i l d r e n w i l l be i d e n t i c a l t o the p r o b a b i l i t i e s f o r f a m i l i e s w i t h z e r o c h i l d r e n under the base p o l i c y . 1 13 a n o t h e r c h i l d a l s o r i s e s w i t h the wage r a t e , g i v e n the number of c h i l d r e n . For example, w i t h a y e a r l y income of $15,000 and two c h i l d r e n , the p e rcentage d e c r e a s e ranges from 2.6% a t a b e f o r e -tax wage r a t e of $3.00 t o 3.3% a t a wage r a t e of $9.00. F u r t h e r , the p e r c e n t a g e d e c r e a s e i n the p r o b a b i l i t y of h a v i n g a c h i l d , r e l a t i v e t o the base p o l i c y , d e c l i n e s as income i n c r e a s e s , f o r g i v e n wage r a t e s and number of c h i l d r e n . For example, w i t h a y e a r l y income of $10,000 and a $9.00 b e f o r e - t a x wage r a t e , the p e rcentage d e c r e a s e i n the p r o b a b i l i t y of h a v i n g a nother c h i l d , r e l a t i v e t o the base p o l i c y , ranges from 1.7% (one c h i l d ) t o 6.3% ( f o u r c h i l d r e n ) . At a y e a r l y income of $25,000, the range i s from .8% t o 2%. Thus, i n g e n e r a l , the two a l t e r n a t i v e p o l i c i e s tend t o d e c r e a s e the p r o b a b i l i t y of h a v i n g another c h i l d , r e l a t i v e t o the base p o l i c y . F u r t h e r , the two a l t e r n a t i v e p o l i c i e s tend t o have a l a r g e r e f f e c t when t h e r e are two or more c h i l d r e n , r e g a r d l e s s of the wage r a t e or income l e v e l . I f we compare T a b l e s 5.3-5.5 w i t h T a b l e s 5.6-5.8, we f i n d t h e r e a r e s i g n i f i c a n t d i f f e r e n c e s between the s i m u l a t i o n r e s u l t s o b t a i n e d u s i n g the maximum l i k e l i h o o d e s t i m a t e s and those o b t a i n e d u s i n g the o r d i n a r y l e a s t squares e s t i m a t e s . F i r s t , the e s t i m a t e d p r o b a b i l i t y of h a v i n g another c h i l d i s s i g n i f i c a n t l y h i g h e r at a l l wage r a t e s , income l e v e l s , and number of c h i l d r e n w i t h the OLS e s t i m a t e s than w i t h the ML e s t i m a t e s . The ML e s t i m a t e d p r o b a b i l i t i e s range from 6.4% t o 23.1%, whereas the OLS p r o b a b i l i t i e s range from 56.6% t o 67.6%. C l e a r l y , the p o l i c y c o n c l u s i o n s reached would be q u i t e d i f f e r e n t depending on whether the ML e s t i m a t e s or the OLS e s t i m a t e s were 1 1 4 used. Second, t h e r e i s even l e s s d i f f e r e n c e among the t h r e e p o l i c i e s a c c o r d i n g t o the OLS e s t i m a t e s than t h e r e i s w i t h the ML e s t i m a t e s . For example, a t a $10,000 y e a r l y income l e v e l , t h e r e i s no d i f f e r e n c e , a c c o r d i n g t o the OLS e s t i m a t e s , between the base p o l i c y and the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y , e x c ept i n a few ca s e s a t the h i g h e r wage r a t e s ($7.00 and $9.00). Even t h e n , most of those d i f f e r e n c e s a r e on the o r d e r of .1% t o .2%. T h i r d , the g e n e r a l t r e n d s of the OLS e s t i m a t e s d i f f e r from tho s e of the ML e s t i m a t e s . The OLS e s t i m a t e s show the p r o b a b i l i t i e s of h a v i n g another c h i l d i n c r e a s i n g as we move from the base p o l i c y t o the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y and then t o the t a x c r e d i t p o l i c y . The ML e s t i m a t e s show the p r o b a b i l i t i e s d e c r e a s i n g . The OLS e s t i m a t e s a l s o d i f f e r i n t h e i r p r e d i c t i o n s when the wage r a t e i n c r e a s e s , f o r g i v e n income l e v e l s and number of c h i l d r e n . The ML e s t i m a t e s p r e d i c t a dec r e a s e i n the p r o b a b i l i t y of h a v i n g another c h i l d as the wage r a t e i n c r e a s e s . The OLS e s t i m a t e s p r e d i c t an i n c r e a s e . T h i s r e s u l t most l i k e l y a r i s e s because the OLS e s t i m a t e s do not take i n t o account the e f f e c t an i n c r e a s e i n the wage r a t e has on hours worked. Hence, the OLS e s t i m a t e s c a p t u r e o n l y p a r t of the e f f e c t of the change i n the wage r a t e on the p r o b a b i l i t y of h a v i n g another c h i l d . W i t h r e g a r d t o an i n c r e a s e i n the number of c h i l d r e n and an i n c r e a s e i n f a m i l y income, the OLS e s t i m a t e s p r e d i c t an i n c r e a s e i n the p r o b a b i l i t y of h a v i n g another c h i l d . These r e s u l t s a re c o n s i s t e n t w i t h the ML r e s u l t s . 1 1 5 Thus, use of OLS can l e a d t o p o l i c y i m p l i c a t i o n s which are q u i t e d i f f e r e n t from those o b t a i n e d from u s i n g ML. I f we a c c e p t the h y p o t h e s i s t h a t the c o r r e c t model of l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y b e h a v i o r i s one which t r e a t s the two d e c i s i o n s " j o i n t l y , then these s i m u l a t i o n r e s u l t s p o i n t out t h a t the use of s i n g l e e q u a t i o n methods can y i e l d m i s l e a d i n g p o l i c y c o n c l u s i o n s . T a b l e 5.3 P r o b a b i l i t y of h a v i n g a c h i l d - ML e s t i m a t e s Base p o l i c y 1. B e f o r e - tax f a m i l y income: 10,000/year B e f o r e - tax wage 3. 00 5.00 7.00 9. 00 0 0. 1 68 0.111 0.0820 0. 0644 Number 1. 0. 1 70 0.113 0.0833 0. 0654 of 2 0. .1 74 0.115 0.0855 0. 0673 c h i l d r e n 3 0. 1 79 0.119 0.0886 0. 0699 4 0. 1 84 0. 123 0.0917 0. 0725 2. B e f o r e - tax f a m i l y income: 15,000/year 0 0. 183 0.122 0.0911 0. 072 0 Number 1 o. 185 0.124 0.0927 0. 0733 of 2 0. 191 0.128 0.0960 0. 0761 c h i l d r e n 3 0. 1 94 0.131 0.0985 0. 0782 4 0. 200 0. 1 36 0. 1 02 0. 081 1 1 1 7 Tab l e 5.3 ( c o n t . ) 3. B e f o r e - tax f a m i l y income: 20,000/year B e f o r e - t a x wage 3. 00 5.00 7. 00 9. 00 0 0. 198 0. 134 0. 101 0. 0800 Number 1 0. 203 0. 138 0. 1 04 0. 0829 of 2 0. 209 0.143 0. 108 0. 0859 c h i l d r e n 3 0. 214 0. 147 0. 1 1 1 0. 0889 4 0. 220 0.151 0. 115 0. 0920 4. B e f o r e - tax f a m i l y income: 25,000/year 0 0. 21 1 0.144 0. 1 09 0. 0868 Number 1 0. 216 0. 148 0. 1 1 2 0. 0899 of 2 0. 222 0.153 0. 1 1 6 0. 0931 c h i l d r e n 3 0. 228 0.157 0. 1 20 0. 0963 4 0. 234 0. 162 0. 1 24 0. 0997 T a b l e 5.4 P r o b a b i l i t y of h a v i n g a c h i l d - ML e s t i m a t e s I n c r e a s e d p e r s o n a l d e d u c t i o n s 1. B e f o r e - tax f a m i l y income: 10,000/year B e f o r e - t a x wage 3. 00 5.00 7.00 9. 00 0 0. 167 0.110 0.0813 0. 0638 Number 1 0. 169 0.112 0.0826 0. 0649 of 2 0. 1 73 0.115 0.0849 . 0. 0668 c h i l d r e n .3 0. 176 0.117 0.0871 0. 0687 4 0. 181 0.121 0.0902 0. 0712 2. B e f o r e - t a x f a m i l y income: 15,000/year 0 0. 180 0. 120 0.0896 0. 0707 Number 1 0. 184 0.123 0.0920 0. 0727 of 2 0. 188 0. 126 0.0944 0. 0748 c h i l d r e n 3 0. 1 92 0. 1 30 0.0969 0. 0769 4 0. 198 0.134 0.101 0. 0799 1 19 T a b l e 5.4 ( c o n t . ) 3. Before--tax f a m i l y income: 20,000/year B e f o r e - t a x wage 3. 00 5.00 7.00 9. 00 0. 0. 1 97 0. 1 33 0.0998 0. 0793 Number 1 0. 202 0.137 0. 1 03 0. 0822 of 2 0. 206 0. 1 40 0. 106 0. 0844 c h i l d r e n 3 0. 212 0. 1 45 0.110 0. 0875 4 0. 213 0. 1 46 0.110 0. 0880 4. B e f o r e - t a x f a m i l y income: 25,000/year 0 0. 209 0.143 0. 108 0. 0859 Number 1 °-215 0. 1 47 0.112 0. 0892 of 2 0. 221 0. 1 52 0.115 0. 0925 c h i l d r e n 3 0. 227 0. 1 57 0.119 0. 0959 4 0. 231 0. 160 0. 1 22 0. 0984 T a b l e 5.5 P r o b a b i l i t y of h a v i n g a c h i l d - ML e s t i m a t e s Refundable t a x c r e d i t 1. B e f o r e -t a x f a m i l y income: 10,000/y ear B e f o r e - t a x wage 3. 00 5.00 7.00 9. 00 0 0. 1 68 0.111 0.082 0. 0644 Number 1 0. 1 68 0.111 0.0819 0. 0643 of 2 0. 168 0.111 0.0821 0. 0645 c h i l d r e n 3 0. 1 70 0.112 0.0832 0. 0654 4 0. 1 75 0.116 0.0862 0. 0679 2. B e f o r e - t a x f a m i l y income: 15,000/y ear 0 0. 183 0. 1 22 0.0911 0. 0720 Number 1 0. 183 ' 0. 122 0.0912 0. 0721 of 2 0. 186 0.125 0.0930 0. 0736 c h i l d r e n 3 0. 189 0. 1 27 0.0949 0. 0752 4 • 0. 1 94 0.131 0.0984 0. 0781 Table 5.5 ( c o n t . ) 3. B e f o r e - t a x f a m i l y income: 20,000/year B e f o r e - t a x wage 3. 00 5.00 7.00 9. 00 0 0. 198 0.134 0.101 0. 0800 Number 1 0. 200 0. 1 36 0. 1 02 0. 0814 of 2 0. 205 0. 1 39 0. 1 05 0. 0837 c h i l d r e n 3 0. 209 0. 1 43 0. 108 0. 0862 4 0. 215 0. 147 0.112 0. 0893 4. B e f o r e - t a x f a m i l y income: 25, 000/year 0 0. 21 1 0. 1 44 0. 109 0. 0868 Number 1 0. 215 0. 1 47 0.112 0. 0892 of 2 0. 219 0.151 0.115 0. 0918 c h i l d r e n 3 0. 224 0. 1 55 0.118 0. 0944 4 0. 230 0. 1 59 0. 122 0. 0977 Table 5.6 P r o b a b i l i t y of h a v i n g a c h i l d - OLS e s t i m a t e s Base p o l i c y 1. B e f o r e -t a x f a m i l y income: 10,000/y ear B e f o r e - t a x wage •3. 00 5.00 7.00 9. 00 0 0. 560 0.566 0.570 0. 573 Number 1 0. 585 0.591 0.594 0. 597 of 2 0. 609 0.615 0.619 0. 621 c h i l d r e n 3 0. 633 0.639 0.642 0. 645 4 0. 656 0.662 0.665 0. 668 2. B e f o r e - tax f a m i l y income: 15,000/y ear 0 0. 562 0.568 0. 572 0. 575 Number 1 0. 587 0.593 0.597 0. 600 of 2 0. 612 0.617 0.621 0. 624 c h i l d r e n 3 0. 636 0.641 0.645 0. 648 4 0. 659 0.664 0.668 0. 671 1 23 Tab l e 5.6 ( c o n t . ) 3. Before--tax f a m i l y income: 20,000/year B e f o r e - t a x wage 3. 00 5.00 7.00 9. 00 0 0. 565 0.571 0.575 o. 578 Number 1 0. 589 0.595 0.599 0. 602 of 2 0. 614 0.619 0.623 0. 626 c h i l d r e n 3 0. 637 0.643 0.647 0. 649 4 0. 660 0.666 0.670 0. 672 4. B e f o r e - t a x f a m i l y i ncome: 25,000/year 0 0. 567 0.573 0.577 0. 580 Number 1 0. 592 0.598 0.602 0. 604 of 2 0. 616 0.622 0.626 0. 628 c h i l d r e n 3 0. 640 0.645 0.649 0. 652 4 0. 663 0.668 0.672 0. 675 T a b l e 5.7 P r o b a b i l i t y of h a v i n g a c h i l d - OLS e s t i m a t e s I n c r e a s e d p e r s o n a l d e d u c t i o n s 1. B e f o r e - tax f a m i l y income: 10,000/y ear B e f o r e - t a x wage 3. 00 5.00 7.00 9 .00 0 0. 560 0.566 0.570 0 .573 Number 1 0.585 0.591 0.595 0 .598 of 2 0. 609 0.615 0.619 0 .622 c h i l d r e n 3 0. 633 0.639 0.643 0 .645 4 0. 656 0.662 0.666 0 .668 2. B e f o r e - tax f a m i l y i ncome: 15,000/y ear 0 0. 563 0.569 0.573 0 .576 Number 1 0. 588 0.594 0.597 0 .600 of 2 0. 612 0.618 0.622 0 .625 c h i l d r e n 3 0. 636 0.642 0.645 0 .648 4 0. 659 0.665 0.668 0 .671 1 25 Table 5.7 (cont. ) 3. Before-tax family income: 20,000/year Before-tax wage 3. 00 5.00 7.00 9 .00 0 0. 565 0.571 0.575 0 .578 Number 1 0. 590 0.595 0.599 0 .602 of 2 0. 614 0.620 0.624 0 .626 children 3 0. 638 0.643 0.647 0 .650 4 0. 662 0.668 0.671 0 .674 4. Before- tax family income: 25,000/year 0 0. 568 0.573 0.577 0 .580 Number 1 0. 592 0.598 0.602 0 .605 of 2 0. 616 0.622 0.626 0 .629 children 3 0. 640 0.646 0.650 0 .652 4 0. 664 0.669 0.673 0 .675 Table 5.8 P r o b a b i l i t y of h a v i n g a c h i l d - OLS e s t i m a t e s Refundable t a x c r e d i t 1. B e f o r e - tax f a m i l y income: 10,000/year. B e f o r e - t a x wage 3 .00 5.00 7.00 9 .00 0 0 .560 0.566 0.570 0 .573 Number 1 0 .585 0.591 0.5959 0 .598 of 2 0 .610 0.616 0.620 0 .622 c h i l d r e n 3 0 .634 0.640 0.644 0 .647 4 0 .658 0.663 0.667 0 .669 2. B e f o r e - t a x f a m i l y income: 15,000/year 0 0 .562 0.568 0.572 0 .575 Number 1 0 .588 0.594 0.598 0 .601 of 2 0 .613 0.618 0.622 0 .625 c h i l d r e n 3 0 .637 0.642 0.646 0 .649 4 0 .660 0.666 0.669 0 .672 1 27 Table 5.8 ( c o n t . ) 3. B e f o r e - t a x f a m i l y income: 20,000/year B e f o r e - t a x wage 3 .00 5.00 7.00 9. 00 0 0 .565 0.571 0. 575 0. 578 Number 1 0 .590 0.596 0.600 0. 602 of 2 0 .614 0.620 0.624 0. 627 c h i l d r e n 3 0 .638 0.644 0.648 0. 650 4 0 .662 0.667 0.671 0. 673 4. B e f o r e - t a x f a m i l y income: 25,000/year 0 0 .567 0.5.73 0.577 0. 580 Number 1 0 .592 0.598 0.602 0. 605 of 2 0 .617 0.622 0.626 0. 629 c h i l d r e n 3 0 .641 0.646 0.650 0. 653 4 0 .664 0.669 0.673 0. 676 1 28 C.3 E f f e c t on hours worked T a b l e s 5.9-5.14 show the e f f e c t of the t h r e e p o l i c i e s on the hours worked by the w i f e . 2 0 T a b l e s 5.9-5.11 were o b t a i n e d u s i n g the ML e s t i m a t e s . T a b l e s 5.12-5.14 were o b t a i n e d u s i n g the OLS e s t i m a t e s . We see a somewhat l a r g e r d i f f e r e n c e among the t h r e e p o l i c i e s i n t h e i r e f f e c t s on hours worked than we d i d i n t h e i r e f f e c t s on the p r o b a b i l i t y of h a v i n g another c h i l d . We can d i s c e r n the f o l l o w i n g t r e n d s under a l l t h r e e p o l i c i e s . F i r s t , as the wage r a t e i n c r e a s e s , hours of work i n c r e a s e , 2 1 a t a l l l e v e l s of income and number of c h i l d r e n . As we saw i n s e c t i o n B, the wage r a t e has a l a r g e e f f e c t on hours worked. Second, f o r g i v e n income l e v e l s and wage r a t e s , as the number of c h i l d r e n i n c r e a s e s , the number of hours worked d e c r e a s e s . T h i s r e s u l t r e f l e c t s our e a r l i e r f i n d i n g t h a t t h e r e i s a t r a d e - o f f between market work and t a k i n g c a r e of 2 0 S i n c e the c h i l d c a r e t a x d e d u c t i o n s or c r e d i t s a r e o n l y a v a i l a b l e i f the c h i l d c a r e i s n e c e s s a r y t o p e r m i t g a i n f u l employment, we have assumed the w i f e i s employed i n our s i m u l a t i o n s . 2 1 The p r e d i c t e d hours of work are f o r two y e a r s of work. At the h i g h e r wage r a t e s , some of the p r e d i c t e d hours of work are u n r e a l i s t i c . T h i s r e s u l t i s due t o two t h i n g s . F i r s t , we have assumed i n our model t h a t the w i f e can work as many hours as she wishes a t a g i v e n wage r a t e . Hence, we have not imposed any c o n s t r a i n t on hours worked ( o t h e r than t h a t they be p o s i t i v e ) . Second, t h e r e were few high-wage women i n our sample. Hence, the c o e f f i c i e n t on the wage r a t e w i l l r e f l e c t t h i s " b i a s " i n the sample. 1 29 c h i l d r e n . 2 2 F i n a l l y , f o r g i v e n wage r a t e s and number of c h i l d r e n , as f a m i l y income i n c r e a s e s , hours of work d e c r e a s e . T h i s r e s u l t i s c o n s i s t e n t w i t h our e a r l i e r f i n d i n g s and w i t h most of the l i t e r a t u r e on l a b o r s u p p l y . I f we compare a c r o s s the t h r e e p o l i c i e s , we f i n d the two a l t e r n a t i v e p o l i c i e s tend to i n c r e a s e hours worked, r e l a t i v e t o the base p o l i c y , f o r g i v e n wage r a t e s , number of c h i l d r e n , and income l e v e l s . The tax c r e d i t p o l i c y has the l a r g e s t e f f e c t of the two a l t e r n a t i v e p o l i c i e s . T h i s r e s u l t i s not s u r p r i s i n g s i n c e the t a x c r e d i t p o l i c y has a l a r g e r e f f e c t on the m a r g i n a l tax r a t e and t o t a l t a x e s p a i d than does the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y . For both a l t e r n a t i v e p o l i c i e s , the e f f e c t s a r e s t r o n g e s t a t a low wage r a t e and s e v e r a l c h i l d r e n . For example, under the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y , w i t h a y e a r l y income of $10,000 and a b e f o r e - t a x wage r a t e of $3.00, hours worked i n c r e a s e by 1.4%, r e l a t i v e t o the base p o l i c y , when t h e r e a r e no c h i l d r e n , and by 3.8% when t h e r e a r e f o u r c h i l d r e n . When the wage r a t e i s $9.00, the p e r c e n t a g e s a r e .5% and 1.2%, r e s p e c t i v e l y . S i m i l a r l y , f o r the tax c r e d i t p o l i c y , at the same income, hours worked i n c r e a s e from 2.7% (one c h i l d ) t o 14.2% ( f o u r c h i l d r e n ) at $3.00, r e l a t i v e t o the base p o l i c y . At a b e f o r e - t a x wage r a t e of $9.00, these p e r c e n t a g e s a r e 1% and 4.3%, r e s p e c t i v e l y . 2 2 The one e x c e p t i o n t o t h i s r e s u l t i s the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y when f a m i l y income i s $25,000 per y e a r . In t h i s c a s e , the percentage i n c r e a s e i n hours worked drops s l i g h t l y , r e l a t i v e t o the base p o l i c y , from z e r o c h i l d r e n t o one c h i l d , remains a t t h i s new l e v e l f o r two and t h r e e c h i l d r e n , then i n c r e a s e s s h a r p l y when t h e r e a re f o u r c h i l d r e n . 1 30 These r e s u l t s suggest women a r e more s e n s i t i v e t o changes i n t h e i r a f t e r - t a x wage r a t e a t lower wage r a t e s than they a r e at h i g h e r wage r a t e s . A l s o , the more c h i l d r e n they have, the more s e n s i t i v e they are t o changes i n t h e i r a f t e r - t a x wage r a t e . T h i s r e s u l t h o l d s f o r a l l wage r a t e s , but i s s t r o n g e s t a t the lower wage r a t e s . For example, w i t h an income of $20,000 per year and t h r e e c h i l d r e n , hours of work i n c r e a s e by 6% as a r e s u l t of the i n c r e a s e d p e r s o n a l d e d u c t i o n s p o l i c y when the wage r a t e i s $3.00, but o n l y 1.3% when i t i s $9.00. For the t a x c r e d i t p o l i c y , the p e r c e n t a g e s a r e 9.5% and 2.5%, r e s p e c t i v e l y . In g e n e r a l , the t a x c r e d i t p o l i c y has the l a r g e s t e f f e c t of the two a l t e r n a t i v e p o l i c i e s . Both p o l i c i e s tend t o i n c r e a s e hours worked, s u g g e s t i n g a s t r o n g wage e f f e c t . Both p o l i c i e s a l s o tend t o have t h e i r g r e a t e s t e f f e c t a t low wages and a l a r g e number of c h i l d r e n , i r r e s p e c t i v e of income l e v e l s . I f we compare the ML r e s u l t s i n T a b l e s 5.9-5.11 w i t h the r e s u l t s o b t a i n e d u s i n g the OLS e s t i m a t e s , found i n T a b l e s 5.12-5.14, we a g a i n f i n d l a r g e d i f f e r e n c e s between the two r e s u l t s . F i r s t , as i n the case of the p r o b a b i l i t i e s of h a v i n g another c h i l d , t h e r e i s o n l y a s m a l l d i f f e r e n c e among the t h r e e p o l i c i e s , a c c o r d i n g t o the OLS r e s u l t s . P e rcentage changes range between z e r o and 1.3%. Second, the OLS r e s u l t s p r e d i c t a d e c r e a s e i n hours worked, r e l a t i v e t o the base p o l i c y , i f e i t h e r of the two a l t e r n a t i v e p o l i c i e s a r e implemented. The ML r e s u l t s show an i n c r e a s e . T h i r d , the OLS r e s u l t s do not show a l a r g e d i f f e r e n c e i n hours worked as the wage r a t e i n c r e a s e s , f o r g i v e n income l e v e l s and number of c h i l d r e n . For example, under the t a x c r e d i t 131 p o l i c y , w i t h an income l e v e l of $15,000 and one c h i l d , the OLS e s t i m a t e s p r e d i c t hours worked w i l l r i s e from 1988 hours t o 2486 hours as the wage r a t e i n c r e a s e s from $3.00 t o $9.00. The ML e s t i m a t e s p r e d i c t an i n c r e a s e from 2350 hours t o 6692 h o u r s . The OLS e s t i m a t e s show a s i m i l a r p a t t e r n t o the ML e s t i m a t e s o n l y i n t h a t they p r e d i c t a d e c r e a s e i n hours worked as income r i s e s and an i n c r e a s e i n hours worked as the wage r a t e i n c r e a s e s . Thus, we a g a i n f i n d t h a t the OLS e s t i m a t e s y i e l d q u i t e d i f f e r e n t p o l i c y i m p l i c a t i o n s than do the ML e s t i m a t e s . Table 5.9 Hours of work (thousands) - ML e s t i m a t e s Base p o l i c y 1 . Before--tax f a m i l y income: 10,000/year B e f o r e - tax wage 3. 00 5.00 7.00 9. 00 0 2. 847 4.880 6. 197 7. 1 70 Number 1 2. 686 4.721 6.038 .7. 012 of 2 2. 479 4.516 5.835 6. 810 c h i l d r e n 3 2. 228 4.269 5.590 6. 565 4 1 . 981 4.024 5.347 6. 324 2. B e f o r e - t a x f a m i l y income: 15,000/year 0 2. 440 4.483 5.805 6. 782 Number 1 2. 273 4.317 5.641 6. 618 of 2 2. 022 4.069 5.395 6. 374 c h i l d r e n 3 1 . 815 3.865 5. 1 92 6. 1 72 4 1 . 562 3.615 4.945 5. 926 1 33 Tab l e 5.9 ( c o n t . ) 3. B e f o r e - t a x f a m i l y income: 20,000/year B e f o r e - t a x wage 3 .00 5.00 7.00 9. 00 0 2 .040 4.093 5.421 6. 402 Number 1 1 .794 3.849 5. 1 79 6. 1 62 of 2 1 .546 3.605 4.937 5. 921 c h i l d r e n 3 1 .298 3.360 4.694 5. 679 4 1 .050 3.115 4.451 5. 438 4. B e f o r e -•tax f a m i l y income: 25,000/year 0 1 .734 3.793 5. 1 27 6. 1 1 1 Number 1 1 .486 3. 549 4.884 5. 870 of 2 1 .238 3.303 4.641 5. 628 c h i l d r e n 3 0 .988 3.057 4.397 5. 386 4 0 .738 2.810 4. 1 52 5. 1 42 T a b l e 5.10 Hours of work (thousands) - ML e s t i m a t e s I n c r e a s e d p e r s o n a l d e d u c t i o n s 1 . Before' -tax f a m i l y income: 10,000/year B e f o r e - t a x wage 3. 00 5.00 7.00 9. 00 0 2. 886 4.919 6.235 7. 207 Number 1 2. 722 4.756 6.073 7. 046 of 2 2. 512 4.549 5.867 6. 841 c h i l d r e n 3 2. 304 4. 343 5.663 6. 638 4 2. 057 4.099 5.421 6. 397 2. B e f o r e - t a x f a m i l y income: 15,000/year 0 2. 518 4.559 5.881 6. 857 Number 1 2. 309 4.353 5.676 6. 654 of 2 2. 1 00 4. 1 46 5.471 6. 449 c h i l d r e n 3 1 . 889 3.938 5.264 6. 243 4 1 . 633 3.685 5.013 5. 994 1 35 T a b l e 5.10 ( c o n t . ) 3. B e f o r e - t a x f a m i l y i n c o m e : 2 0 , 0 0 0 / y e a r B e f o r e - t a x wage 3 .00 5.00 7.00 9. 00 0 2 .083 4.1 34 5.462 6. 442 Number 1 1 .831 3.886 5.216 6. 198 of 2 1 .628 3.685 5.016 5. 999 c h i l d r e n 3 1 .376 3.436 4.770 5. 754 4 1 .264 3.325 4.659 5. 644 4. B e f o r e -•tax f a m i l y i n c o m e : 2 5 , 0 0 0 / y e a r 0 1 .782 3.840 5.173 6. 1 57 Number 1 1 .523 3. 585 4.920 5. 906 o f 2 1 .269 3.335 4.672 5. 659 c h i l d r e n 3 1 .015 3.083 4.422 5. 41 1 4 0 .807 2.878 4.219 5. 208 Table 5.11 Hours of work (thousands) - ML e s t i m a t e s Refundable t a x c r e d i t 1. B e f o r e - tax f a m i l y income: 10,000/year B e f o r e - t a x wage 3. 00 5.00 7.00 9. 00 0 2. 847 4.880 6. 1 97 7. 1 70 Number 1 2. 759 4.792 6.109 7. 082 of 2 2. 661 4.694 6.011 6. 984 c h i l d r e n 3 2. 514 4.549 5.866 6. 839 4 2. 263 4.301 5.621 6. 595 2. B e f o r e - tax f a m i l y income: 15,000/year 0 2. 440 4.483 5.805 6. 782 Number 1 2. 350 4.393 5.715 6. 692 of 2 2. 1 72 4.217 5.541 6. 518 c h i l d r e n 3 • 1 . 990 4.037 5.362 6. 340 4 1 . 735 3.785 5.112 6. 092 1 37 Table 5.11 ( c o n t . ) 3. Before--tax f a m i l y income: 20,000/year B e f o r e - t a x wage 3 .00 5.00 7.00 9. 00 0 2 .040 4.093 5.421 6. 402 Number 1 1 .878 3.932 5.261 6. 243 of 2 1 .663 3.719 5.050 6. 033 c h i l d r e n 3 1 .445 3.504 4.837 5. 821 4 1 . 1 95 3.257 4.592 5. 578 4. B e f o r e - t a x f a m i l y i ncome: 25,000/year 0 1 .734 3.793 5. 1 27 6. 1 1 1 Number 1 1 .523 3.585 4.919 5. 905 of 2 1 .309 3.373 4.710 5. 696 c h i l d r e n 3 1 .093 3. 1 60 4.498 5. 486 4 0 .841 2.911 4.251 5. 241 T a b l e 5.12 Hours of work (thousands) - OLS e s t i m a t e s Base p o l i c y 1. B e f o r e - tax f a m i l y income: 10,000/year B e f o r e - tax wage 3 .00 5.00 7.00 9. 00 0 2 .406 2.638 2.790 2. 904 Number 1 2 .224 2.456 2.608 2. 722 of 2 2 .038 2.270 2.422 2. 536 c h i l d r e n 3 1 .848 2.080 2.233 2. 346 4 1 .662 1 .894 2.047 2. 161 2. B e f o r e - tax f a m i l y income: 15,000/year . 0 2 . 1 84 2.416 2.568 2. 682 Number 1 1 .992 2.223 2.375 2. 489 of 2 1 .799 2.031 2. 1 83 2. 297 c h i l d r e n 3 1 .612 1 .844 1 .996 2. 1 1 0 4 1 .421 1 .652 1 .805 1 . 9 1 9 1 39 Tab l e 5.12 ( c o n t . ) 3. B e f o r e - tax f a m i l y income: 20,000/year B e f o r e - tax wage 3 .00 5.00 7.00 9 .00 0 1 .978 2.210 2.362 2 .476 Number 1 1 .787 2.019 2.171 2 .285 of 2 1 .596 1 .828 1.980 2 .094 c h i l d r e n 3 1 .406 1 .637 1 .790 1 .904 4 1 .216 1 .448 1 . 600 1 .714 4. B e f o r e - tax f a m i l y i ncome: 25,000/year 0 1 .783 2.014 2. 1 67 2 .280 Number 1 1 .590 1 .822 1 .974 2 .088 of 2 1 .398 1 .629 1 .782 1 .896 c h i l d r e n 3 1 .206 1 .437 1 . 590 1 .704 4 1 .014 1 .245 1 .398 1 .512 T a b l e 5.13 Hours of work (thousands) - OLS e s t i m a t e s I n c r e a s e d p e r s o n a l d e d u c t i o n s 1. B e f o r e - t a x f a m i l y income: 3.00 0 2.406 Number 1 2.222 of 2 2.034 c h i l d r e n 3 1.849 4 1.664 2. B e f o r e - t a x f a m i l y income: 0 2.179 Number 1 1.988 of 2 1.798 c h i l d r e n 3 1.609 4 1.415 10,000/year B e f o r e - t a x wage 5.00 7 .00 9 .00 2.637 2 .790 2 .904 2.453 2 .606 2 .720 2.266 2 .418 2 .532 2.081 2 .233 2 .347 1 .896 2 .048 2 . 1 62 15,000/year 2.411 2 .563 2 .677 2.220 2 .372 2 .486 2.030 2 . 182 2 .296 1 .840 1 .993 2 . 1 07 1 .647 1 .799 1 .913 T a b l e 5.13 ( c o n t . ) 3. B e f o r e - t a x f a m i l y income: 20,000/y ear B e f o r e - t a x wage 3.00 5.00 7.00 9 .00 0 1 .977 2.208 2.360 2 .474 Number 1 1.782 2.013 2. 1 66 2 .279 of 2 1 .592 1 .824 1 .976 2 .090 c h i l d r e n .3 1.399 1 .631 1.784 1 .898 4 1 . 187 1.419 1 .571 1 .685 4. B e f o r e - t a x f a m i l y income: 25,000/y ear 0 1 .785 2.016 2. 1 69 2 .282 Number 1 1.583 1.815 1 . 967 2 .081 of 2 1 .386 1.618 1 .770 1 .884 c h i l d r e n 3 1 . 190 1 .421 1 .574 1 .688 4 0.998 1 .230 1 .382 1 .497 T a b l e 5.14 H o u r s o f work ( t h o u s a n d s ) - OLS e s t i m a t e s R e f u n d a b l e t a x c r e d i t 1. B e f o r e -t a x f a m i l y i n c o m e : 1 0 , 0 0 0 / y e a r B e f o r e - t a x wage 3 .00 5.00 7.00 9 .00 0 2 .406 2.638 2.790 2 .904 Number 1 2 .221 2.452 2.605 2 .719 o f 2 2 .035 2.267 2.419 2 .533 c h i l d r e n 3 1 .845 2.077 2.230 2 .344 4 1 .659 1 .891 2.043 2 . 1 57 2. B e f o r e -•tax f a m i l y i n c o m e : 1 5 , 0 0 0 / y e a r 0 2 . 184 2.416 2.568 2 .682 Numbe r 1 1 .988 2.220 2.372 2 .486 o f 2 1 .793 2.0.24 2. 1 77 2 .291 c h i l d r e n 3 1 .597 1 .828 1 .981 2 .095 4 1 .405 1 .637 1 .789 1 .903 143 T a b l e 5.14 ( c o n t . ) 3. B e f o r e - tax f a m i l y income: 20,000/year B e f o r e - t a x wage 3. 00 5.00 7 .00 9 .00 0 978 2.210 2 .362 2 .476 Number 1 1. 785 2.016 2 . 1 69 2 .283 of 2 1. 586 1.817 1 .970 2 .084 c h i l d r e n 3 1. 387 1.619 1 .771 1 .885 4 U 1 97 1.429 1 .581 . 1 .695 4. B e f o r e - tax f a m i l y income: 25,000/year 0 1 . 783 2.014 2 . 167 2 .280 Number 1 1 . 582 1.814 1 .966 2 .080 of 2 1 . 382 1.614 1 .766 . 1 .880 c h i l d r e n 3 1 . 182 1.414 1 .566 1 .680 4 0. 990 1.222 1 .374 1 .488 1 44 D. Summary In t h i s c h a p t e r , we l o o k e d a t the p o l i c y i m p l i c a t i o n s of the maximum l i k e l i h o o d e s t i m a t e s of our model. We found t h a t the a f t e r - t a x wage r a t e has a l a r g e , p o s i t i v e e f f e c t on hours worked, w h i l e the number of c h i l d r e n , the a d d i t i o n of another c h i l d , and f a m i l y income a l l have n e g a t i v e e f f e c t s on hours worked. The p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d i s p o s i t i v e l y a f f e c t e d by the wage r a t e and the d e s i r e to have more c h i l d r e n . Hours worked and the age of the youngest c h i l d have a n e g a t i v e e f f e c t . We compared the e f f e c t s of two t a x p o l i c i e s on hours worked and on the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d a g a i n s t a base p o l i c y . The base p o l i c y assumed t h a t the .couple o n l y c l a i m e d a $750 d e d u c t i o n per dependent. The f i r s t a l t e r n a t i v e p o l i c y assumed.there were no c h i l d c a r e d e d u c t i o n s or c r e d i t s , but i n c r e a s e d the amount of the d e d u c t i o n per dependent t o $1000. The second a l t e r n a t i v e p o l i c y i n c r e a s e d the c h i l d c a r e tax c r e d i t from i t s 1976 l e v e l , w h i l e k e e p i n g p e r s o n a l d e d u c t i o n s at t h e i r o r i g i n a l l e v e l of $750. Both p o l i c i e s caused the p r o b a b i l i t y of h a v i n g another c h i l d t o d e c r ease by a s m a l l amount. These r e s u l t s suggest both f e r t i l i t y d e c i s i o n s and l a b o r s u p p l y d e c i s i o n s w i l l not be s t r o n g l y a f f e c t e d by income tax changes. E i t h e r l a r g e changes i n the t a x s t r u c t u r e need t o occur or more d i r e c t p o l i c i e s need t o be implemented i f the government wishes t o a f f e c t e i t h e r d e c i s i o n . We found, w i t h r e g a r d t o the p r o b a b i l i t y of h a v i n g another 1 45 c h i l d , t h a t t h e r e was o n l y a s m a l l d i f f e r e n c e among the t h r e e p o l i c i e s . In g e n e r a l , we found t h a t as the wage r a t e i n c r e a s e d , under a l l t h r e e p o l i c i e s , the p r o b a b i l i t y of h a v i n g another c h i l d d e c r e a s e d , f o r g i v e n income l e v e l s and number of c h i l d r e n . T h i s r e s u l t s uggested t h a t the n e g a t i v e e f f e c t of an i n c r e a s e i n hours worked, induced by the i n c r e a s e i n wage r a t e s , outweighed the p o s i t i v e e f f e c t of the i n c r e a s e i n the wage r a t e i t s e l f on the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d . We a l s o found t h a t as the number of c h i l d r e n i n c r e a s e d , f o r g i v e n wage r a t e s and income l e v e l s , the p r o b a b i l i t y of h a v i n g a c h i l d i n c r e a s e d under a l l t h r e e p o l i c i e s . We found t h a t the two a l t e r n a t i v e p o l i c i e s had lower p r o b a b i l i t i e s of h a v i n g another c h i l d , f o r g i v e n l e v e l s of income, wage r a t e s , and number of c h i l d r e n . T h i s r e s u l t a g a i n s u g g ested the induced hours worked e f f e c t outweighed any p o s i t i v e income or wage e f f e c t s . We found t h a t the two a l t e r n a t i v e p o l i c i e s had t h e i r l a r g e s t e f f e c t s when t h e r e were two or more c h i l d r e n a l r e a d y i n the f a m i l y . At the h i g h e s t income l e v e l ($25,000), the e f f e c t s were not l a r g e u n t i l the f o u r t h c h i l d . When we compared the ML r e s u l t s w i t h the OLS r e s u l t s , we found s u b s t a n t i a l d i f f e r e n c e s both i n the magnitude of the p r e d i c t e d p r o b a b i l i t i e s and i n the d i r e c t i o n of change of the p r o b a b i l i t y under the two a l t e r n a t i v e p o l i c i e s . In g e n e r a l , the OLS r e s u l t s o v e r p r e d i c t e d the p r o b a b i l i t i e s of h a v i n g another c h i l d , r e l a t i v e t o the ML r e s u l t s . F u r t h e r , the OLS r e s u l t s i n d i c a t e d almost no e f f e c t of the two a l t e r n a t i v e p o l i c i e s on the p r o b a b i l i t y of h a v i n g another c h i l d , r e l a t i v e t o the base 1 46 p o l i c y . A l t h o u g h the ML r e s u l t s i n d i c a t e d o n l y s m a l l e f f e c t s , t hese e f f e c t s were s t r o n g e r than the O L S . e f f e c t s . F u r t h e r , where t h e r e were d i f f e r e n c e s among the t h r e e p o l i c i e s , the OLS r e s u l t s p r e d i c t e d the two a l t e r n a t i v e p o l i c i e s would i n c r e a s e the p r o b a b i l i t y of h a v i n g another c h i l d , r e l a t i v e t o the base p o l i c y . The ML r e s u l t s i n d i c a t e d the p r o b a b i l i t i e s would d e c r e a s e . W i t h r e g a r d t o hours worked, the e f f e c t s were s t r o n g e r . Under a l l t h r e e p o l i c i e s , we found t h a t hours worked rose w i t h the wage r a t e , f o r g i v e n income l e v e l s and number of c h i l d r e n . There was an i n v e r s e r e l a t i o n s h i p between hours worked and number of c h i l d r e n , f o r g i v e n wage r a t e s and income l e v e l s . Hours worked a l s o d e c r e a s e d as income r o s e , f o r g i v e n wage r a t e s and number of c h i l d r e n . These r e s u l t s a r e c o n s i s t e n t w i t h our f i n d i n g s i n S e c t i o n B. We found the two a l t e r n a t i v e p o l i c i e s i n c r e a s e d hours worked, r e l a t i v e t o the base p o l i c y . T h i s r e s u l t s uggested t h a t the e f f e c t of the p o l i c y on the wage r a t e outweighed i t s e f f e c t on income l e v e l s . We found both a l t e r n a t i v e p o l i c i e s t o have t h e i r l a r g e s t e f f e c t s a t low wage r a t e s and s e v e r a l c h i l d r e n . The t a x c r e d i t p o l i c y had the l a r g e s t e f f e c t on m a r g i n a l tax r a t e s and the t o t a l t a x b i l l . Hence, i t had the l a r g e s t e f f e c t on hours worked. A g a i n , we found d i f f e r e n c e s between the ML r e s u l t s and the OLS r e s u l t s . In g e n e r a l , the OLS r e s u l t s tended t o u n d e r p r e d i c t hours worked r e l a t i v e t o the ML r e s u l t s . L i k e the p r o b a b i l i t y r e s u l t s , the OLS e s t i m a t e s found l i t t l e d i f f e r e n c e among the 1 4 7 t h r e e p o l i c i e s i n terms of t h e i r e f f e c t s on hours worked. F u r t h e r , the OLS r e s u l t s p r e d i c t e d a d e c r e a s e i n hours worked, r e l a t i v e t o the base p o l i c y , as a r e s u l t of the two a l t e r n a t i v e p o l i c i e s . The ML r e s u l t s p r e d i c t e d an i n c r e a s e . The c o n c l u s i o n s we can draw from these r e s u l t s a r e the f o l l o w i n g . Tax p o l i c i e s of the type d e s c r i b e d i n t h i s c h a p t e r w i l l t e n d t o have t h e i r l a r g e s t e f f e c t s on women w i t h low wage r a t e s and s e v e r a l c h i l d r e n . F u r t h e r , a t l e a s t w i t h our sample, hours worked w i l l have the s t r o n g e s t e f f e c t on the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d , o u t w e i g h i n g the p o s i t i v e wage e f f e c t . F i n a l l y , OLS e s t i m a t e s g i v e q u i t e d i f f e r e n t p r e d i c t i o n s from the ML e s t i m a t e s . To the e x t e n t we a c c e p t the h y p o t h e s i s t h a t f e r t i l i t y and 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 s a r e j o i n t l y d e t e r m i n e d , s i n g l e e q u a t i o n methods w i l l not c a p t u r e the i n t e r a c t i o n s of the two d e c i s i o n s . Hence, p r e d i c t i o n s based on s i n g l e e q u a t i o n methods w i l l most l i k e l y g i v e d i f f e r e n t p o l i c y c o n c l u s i o n s from those based on s i m u l t a n e o u s e q u a t i o n s methods. 1 48 Chapter S i x C o n c l u s i o n A. Summary The purpose of t h i s d i s s e r t a t i o n was t o examine the l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y d e c i s i o n s of m a r r i e d women w i t h i n a common framework. We d e v e l o p e d a model which i s an e x t e n s i o n of the g e n e r a l model p r e s e n t e d i n Heckman [1978b], Our model d i f f e r s from e a r l i e r models by l o o k i n g a t the d e c i s i o n t o h a v i n g another c h i l d , r a t h e r than t r y i n g to d etermine the t o t a l number of c h i l d r e n a c o u p l e w i l l have. I t a l s o d i f f e r s from o t h e r models by m o d e l l i n g d i r e c t e f f e c t s as w e l l as i n d i r e c t e f f e c t s of each d e c i s i o n on the o t h e r . We e s t i m a t e d t h i s model by maximum l i k e l i h o o d t e c h n i q u e s . We used d a t a from the M i c h i g a n P a n e l Survey of Income Dynamics. We found t h e r e were s i g n i f i c a n t d i r e c t as w e l l as i n d i r e c t e f f e c t s of the f e r t i l i t y d e c i s i o n on 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 s and v i c e v e r s a . U n l i k e o t h e r s t u d i e s , we found t h a t r a c e and r e l i g i o n d i d not p l a y a s i g n i f i c a n t r o l e i n e x p l a i n i n g e i t h e r d e c i s i o n . W i th r e g a r d t o l a b o r s u p p l y , we found t h a t h a v i n g a c h i l d reduced the amount of hours s u p p l i e d t o the market over a two-year p e r i o d . F a m i l y income a l s o had a n e g a t i v e e f f e c t on hours s u p p l i e d , as p r e d i c t e d by t h e o r y . The p r o b a b i l i t y of h a v i n g an a d d i t i o n a l c h i l d was p o s i t i v e l y i n f l u e n c e d by the w i f e ' s wage r a t e . Hours worked 1 49 n e g a t i v e l y a f f e c t e d the p r o b a b i l i t y of h a v i n g a n o t h e r c h i l d . T h i s r e s u l t s u g gests t h e r e a r e c o n f l i c t i n g e f f e c t s from working on the p r o b a b i l i t y of h a v i n g another c h i l d . The wage earned from w o r k i n g has a p o s i t i v e e f f e c t , w h i l e the hours worked has a n e g a t i v e e f f e c t . Thus, t h e r e i s a c o n f l i c t between the income c o n s t r a i n t ( i n c r e a s e d by e a r n i n g a wage) and the time c o n s t r a i n t ( d e c r e a s e d by hours spent a t work). The i m p l i c a t i o n i s t h a t the two ob s e r v e d phenomena of d e c r e a s e d b i r t h r a t e s and i n c r e a s e d l a b o r f o r c e p a r t i c i p a t i o n by m a r r i e d women are c a u s a l l y r e l a t e d . These r e s u l t s suggest t h a t as more m a r r i e d women j o i n the l a b o r f o r c e , we s h o u l d see a c o n t i n u a t i o n of low b i r t h r a t e s and perhaps even an a d d i t i o n a l d e c l i n e . We compared the ML e s t i m a t e s of the model w i t h s i n g l e e q u a t i o n methods and one method which e s t i m a t e d the hours and wage e q u a t i o n s j o i n t l y . In g e n e r a l , the s i n g l e e q u a t i o n methods d e v i a t e d s i g n i f i c a n t l y from the ML e s t i m a t e s . The sample s e l e c t i v i t y model gave the c l o s e s t e s t i m a t e s t o the ML e s t i m a t e s f o r the hours and wage e q u a t i o n s . These r e s u l t s suggest t h e r e w i l l be a t r a d e - o f f between the c o s t of maximum l i k e l i h o o d e s t i m a t i o n and the p o t e n t i a l b i a s e s i n c o e f f i c i e n t s from u s i n g s i m p l e r methods. We a l s o d e r i v e d c o n s i s t e n t e s t i m a t e s f o r the model. In g e n e r a l , the c o n s i s t e n t e s t i m a t e s d i d not pe r f o r m as w e l l as the sample s e l e c t i v i t y model d i d f o r the hours and wage e q u a t i o n s , nor as w e l l as the p r o b i t model f o r the a d d i t i o n a l c h i l d e q u a t i o n . We compared the e f f e c t of two a l t e r n a t i v e t a x p o l i c i e s on the p r o b a b i l i t y of h a v i n g a c h i l d and on hours worked by the 1 50 w i f e . The two a l t e r n a t i v e p o l i c i e s had o n l y a s m a l l e f f e c t on the p r o b a b i l i t y of h a v i n g another c h i l d . T h e i r e f f e c t was g r e a t e s t on f a m i l i e s where t h e r e were s e v e r a l c h i l d r e n , r e g a r d l e s s of income l e v e l s or wage r a t e s . We found the two a l t e r n a t i v e t a x p o l i c i e s d e c r e a s e d the p r o b a b i l i t y of h a v i n g a nother c h i l d , r e l a t i v e t o the base p o l i c y . The e f f e c t on hours worked was l a r g e r . Here, the e f f e c t s were l a r g e s t f o r f a m i l i e s w i t h s e v e r a l c h i l d r e n and low wage r a t e s . Hours worked tended t o i n c r e a s e as a r e s u l t of the two a l t e r n a t i v e p o l i c i e s . The r e s u l t t h a t the e f f e c t s of the two a l t e r n a t i v e p o l i c i e s on the p r o b a b i l i t y of h a v i n g another c h i l d and on hours worked were not s u b s t a n t i a l s u g g e s t s t h a t i f the government wishes t o a f f e c t e i t h e r d e c i s i o n , a more d i r e c t p o l i c y i s needed. The r e s u l t s a l s o suggest t h a t changes i n t a x e s implemented f o r o t h e r p o l i c y reasons w i l l not have l a r g e r e p e r c u s s i o n s on e i t h e r f e r t i l i t y or the m a r r i e d woman's l a b o r s u p p l y . B. F u t u r e r e s e a r c h There are s e v e r a l e x t e n s i o n s which can be made t o the r e s e a r c h undertaken i n t h i s d i s s e r t a t i o n . On the t h e o r e t i c a l e c o n o m e t r i c f r o n t , the prop e r formulae f o r c a l c u l a t i n g the s t a n d a r d e r r o r s of the c o n s i s t e n t e s t i m a t o r s need t o be dev e l o p e d . On the a p p l i e d e c o n o m e t r i c f r o n t , d i f f e r e n t s p e c i f i c a t i o n s of the model need t o be t e s t e d . W i th r e g a r d t o the t h e o r e t i c a l model, s e v e r a l d i r e c t i o n s are i n d i c a t e d . F i r s t , our e m p i r i c a l model d i d not c o n t a i n a l l of 151 the i m p l i c a t i o n s of t h e o r y i t might have. A model c o u l d be devel o p e d which more t h o r o u g h l y r e f l e c t s the t h e o r e t i c a l r e s t r i c t i o n s of consumer demand t h e o r y than does our model. Second, Hausman's [1981] t e c h n i q u e of d e r i v i n g u t i l i t y f u n c t i o n s from l i n e a r demand f u n c t i o n s c o u l d be a p p l i e d t o the model t o see i f c o u p l e s who d e c i d e t o have a c h i l d t r u l y get more u t i l i t y from h a v i n g the c h i l d than from not h a v i n g i t (and v i c e v e r s a ) . T h i r d , our model assumes the woman can work as many hours as she d e s i r e s . C l e a r l y , t h i s assumption does not h o l d i n the r e a l w o r l d . The model c o u l d be m o d i f i e d t o ta k e i n t o account i n s t i t u t i o n a l r e s t r i c t i o n s on hours worked. F i n a l l y , our model l o o k s a t o n l y one p e r i o d i n a c o u p l e ' s l i f e t i m e . I t would be u s e f u l t o extend the model a l o n g the l i n e s of Hotz [1980] i n o r d e r t o account e x p l i c i t l y f o r dynamic i n t e r a c t i o n s among the v a r i a b l e s . An even more i n t e r e s t i n g e x t e n s i o n would be t o c o n s t r u c t a model so as t o be a b l e t o take advantage of a l l e l e v e n y e a r s of the PSID d a t a . 1 1 However, perhaps the b i n d i n g c o n s t r a i n t here i s the computer t e c h n o l o g y (and c o s t ! ) n e c e s s a r y t o e s t i m a t e such a model. 152 C. In c o n c l u s i o n There i s an i n c r e a s i n g amount of ev i d e n c e t h a t l a b o r 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 and f e r t i l i t y a r e n e g a t i v e l y r e l a t e d i n the i n d u s t r i a l w o r l d . 2 . T h i s r e l a t i o n s h i p has s e v e r a l i m p l i c a t i o n s f o r the economic f u t u r e of the i n d u s t r i a l n a t i o n s . As more m a r r i e d women e n t e r the work f o r c e , b i r t h r a t e s w i l l c o n t i n u e a t low l e v e l s , u n l e s s c o m p a t i b i l i t y between working and r a i s i n g a c h i l d i n c r e a s e s . There a re v a r i o u s consequences of t h i s low b i r t h r a t e . S c h o o l e n r o l l m e n t w i l l c o n t i n u e t o be low, a f f e c t i n g the employment of those i n the e d u c a t i o n s e c t o r . The age d i s t r i b u t i o n of the p o p u l a t i o n w i l l s h i f t , w i t h o l d e r people f o r m i n g a l a r g e r p o r t i o n of the p o p u l a t i o n . T h i s change i n the age d i s t r i b u t i o n w i l l p l a c e s e r i o u s burdens on old-age s e c u r i t y and h e a l t h s e r v i c e s . The p o t e n t i a l f u t u r e work f o r c e w i l l be s m a l l e r than o t h e r w i s e . There w i l l be l e s s p r e s s u r e on f i x e d r e s o u r c e s , such as l a n d . Lower b i r t h r a t e s a r e n e i t h e r i n h e r e n t l y good or bad. However, they w i l l have an e f f e c t on the economy. I f t h e r e a re not t o be undue h a r d s h i p s on c e r t a i n s e c t o r s of the economy, the i m p l i c a t i o n s of a lower b i r t h r a t e s h o u l d be r e c o g n i z e d and p o l i c i e s p l a n n e d t o m i n i m i z e the d i s l o c a t i o n s . Our r e s u l t s show t h a t the government can have l i t t l e e f f e c t on f e r t i l i t y b e h a v i o r t h r o u g h t a x p o l i c i e s which a f f e c t o n l y income. More d i r e c t e f f e c t s would be needed i f the government 2 R e s e a r c h e r s have found t h i s n e g a t i v e r e l a t i o n s h i p t o be l e s s s t r o n g i n l e s s - d e v e l o p e d c o u n t r i e s . See, f o r example, McCabe and Rosenzweig [ 1 9 7 6 ] . 1 53 wished t o have a major impact on f e r t i l i t y b e h a v i o r . I n c r e a s e d l a b o r 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 w i l l a l s o have an impact on the economy. They i n c r e a s e the c u r r e n t work f o r c e . T h i s e f f e c t may be good or bad, depending on the demand f o r l a b o r i n the economy. I f the w i f e i s the secondary worker i n the f a m i l y , f a m i l y income w i l l i n c r e a s e , which may i n c r e a s e the demand f o r goods and s e r v i c e s . Not o n l y w i l l demand i n c r e a s e , but the types of goods and s e r v i c e s demanded w i l l a l s o change. For example, i t i s l i k e l y t h a t demand f o r day c a r e s e r v i c e s and convenience foods w i l l i n c r e a s e . The f u l l e f f e c t s of lower b i r t h r a t e s and i n c r e a s e d l a b o r 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 on the economy cannot be known w i t h o u t the use of a g e n e r a l e q u i l i b r i u m model. However, p a r t i a l e q u i l i b r i u m models, such as the one devel o p e d i n t h i s d i s s e r t a t i o n , can p r o v i d e some i n s i g h t i n t o the i n t e r r e l a t i o n s h i p s between l a b o r f o r c e p a r t i c i p a t i o n and f e r t i l i t y b e h a v i o r , on the one hand, and economic v a r i a b l e s , such as income and wage r a t e s , on the o t h e r . 1 54 R e f e r e n c e s A b b o t t , M. and 0. A s h e n f e l t e r [ 1 9 7 6 ] , "Labor s u p p l y , commodity demand and the a l l o c a t i o n of t i m e . " Review of Economic  S t u d i e s , 43 (October 1976), 389-411. A s h e n f e l t e r , 0. [1980]. "Unemployment as d i s e q u i l i b r i u m i n a model of l a b o r s u p p l y . " E c o n o m e t r i c a , 4 8 ( 3 ) , 547-564. A s h e n f e l t e r , 0. and 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 l a b o r s u p p l y . " Econometr i c a , 4 2 ( 1 ) , 73-85. B e c k e r , G.S. [ i 9 6 0 ] , "An economic a n a l y s i s of f e r t i l i t y . " In Demographic and Economic Change i n Developed C o u n t r i e s . U n i v e r s i t i e s - N a t i o n a l Bureau Conference S e r i e s no. 11, ( 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 ) , 1960, 209-231. - [1965]. "A t h e o r y of the a l l o c a t i o n of t i m e . " The  Economic J o u r n a l , 75(299), 493-517. B e n - P o r a t h , Y. [1973]. "Economic a n a l y s i s of f e r t i l i t y i n I s r a e l : p o i n t and c o u n t e r p o i n t . " J o u r n a l of P o l i t i c a l  Economy, 8 1 ( 2 ) , P a r t I I , S202-S233. - and F. Welch [1 9 7 2 ] , Chance, Chi Id T r a i t s and the Choice  of F a m i l y S i z e . R-1117-NIH/RF, (Santa Monica, Ca.: The Rand C o r p o r a t i o n ) . B l a k e , J . [ 1 9 6 8 ] . "Are b a b i e s consumer d u r a b l e s ? A c r i t i q u e of the economic t h e o r y of r e p r o d u c t i v e m o t i v a t i o n . " P o p u l a t i o n  S t u d i e s , 2 2 ( 1 ) , 5-25. B u r t l e s s , G. and J.A. Hausman [1978]. "The e f f e c t of t a x a t i o n on l a b o r s u p p l y : e v a l u a t i n g the Gary income maintenance e x p e r i m e n t . " J o u r n a l of P o l i t i c a l Economy, 8 6 ( 6 ) , 1103 — 1 1 30. C a i n , G. [1966]. M a r r i e d Women i n the Labor F o r c e : An Economic  A n a l y s i s . ( C hicago: The U n i v e r s i t y of Chicago P r e s s ) . ' - and M.D. Dooley [ 1 9 7 6 ] . " E s t i m a t i o n of a model of l a b o r s u p p l y , f e r t i l i t y , and wages of m a r r i e d women." J o u r n a l of  P o l i t i c a l Economy, 8 4 ( 4 ) , P a r t 2, S179-S201. 1 55 - and A. Weininger [1970]. "Economic d e t e r m i n a n t s of f e r t i l i t y : r e s u l t s from c r o s s - s e c t i o n a l aggregate d a t a . " I n s t i t u t e f o r Research on P o v e r t y , D i s c u s s i o n Paper, U n i v e r s i t y of W i s c o n s i n , December 1970. C a r l i n e r , G. , C. Robinson and N. Tomes [1980]. "Female l a b o u r s u p p l y and f e r t i l i t y i n Canada." Canadian J o u r n a l of  Economics, 13(1), 46-64. Cogan, J . [1977]. Labor Supply w i t h Time and Money C o s t s of  P a r t i c i p a t i o n . R-2044-HEW, (Santa Monica, Ca.: The Rand C o r p o r a t i o n ) . Cohen, M.S. and F.P. S t a f f o r d [ 1 9 7 4 ] . "A l i f e c y c l e model of the h o u sehold's time a l l o c a t i o n . " A n n a l s of Economics and  S o c i a l Measurement, 3 ( 3 ) , 447-461. Cox, D.R. [ 1 9 7 0 ] . The A n a l y s i s of B i n a r y Data. (London: Methuen). Cragg, J.G. [1968]. "Some e f f e c t s of i n c o r r e c t s p e c i f i c a t i o n on the s m a l l - s a m p l e p r o p e r t i e s of s e v e r a l s i m u l t a n e o u s -e q u a t i o n e s t i m a t o r s . " I n t e r n a t i o n a l Economic Review, 9 ( 1 ) , 63-86. DeSerpa, A.C. [1975]. "On the c o m p a r a t i v e s t a t i c s of time a l l o c a t i o n t h e o r y . " Canadian J o u r n a l of Economics, 8 ( 1 ) , 101-111. de T r a y , D.N. [1973]. " C h i l d q u a l i t y and the demand f o r 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, 8 1 ( 2 ) , P a r t I I , 570-579. Domencich, M. and D~. McFadden [ 1 9 7 5 ] . Urban T r a v e l Demand: A B e h a v i o r a l A n a l y s i s . (Amsterdam: N o r t h - H o l l a n d ) . Duesenberry, J.S. [ i 9 6 0 ] . "Comment." In Demographic and  Economic Change i n Developed C o u n t r i e s . U n i v e r s i t i e s -N a t i o n a l Bureau Conference S e r i e s no. 11, ( 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 ) , 231-234. E a s t e r l i n , R.A. [1968]. P o p u l a t i o n , Labor F o r c e , and Long  Swings i n Economic Growth: The American E x p e r i e n c e . N a t i o n a l Bureau of Economic R e s e a r c h , TNew York: Columbia U n i v e r s i t y P r e s s ) . 156 - [1969]. "Towards a socioeconomic t h e o r y of f e r t i l i t y : a survey of r e c e n t r e s e a r c h on economic f a c t o r s i n American f e r t i l i t y . " In Behrman, S.J., L . J . C o r s a , J r . , and R. Freedman, eds., F e r t i l i t y and F a m i l y P l a n n i n g : A World  View. (Ann A r b o r : U n i v e r s i t y of M i c h i g a n P r e s s ) , 127-156. F l e i s c h e r , B.M. and G.F. Rhodes, J r . [1979]. " F e r t i l i t y , women's wage r a t e s , and l a b o r s u p p l y . " The American  Economic Review, 6 9 ( 1 ) , 14-24. F l e t c h e r , R. [1972]. FORTRAN 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. H a r w e l l , E n g l a n d : Research Group Report 7125, T h e o r e t i c a l P h y s i c s D i v i s i o n , Atomic Energy Research E s t a b l i s h m e n t . Freedman, D.S. [1963]. "The r e l a t i o n of economic s t a t u s t o f e r t i l i t y . " The American Economic Review, 5 3 ( 3 ) , 414-26. F u l o p , M. [1977]. "A survey of the l i t e r a t u r e on the economic t h e o r y of f e r t i l i t y b e h a v i o r . " The Amer i c a n Economist, 2 1 ( 1 ) , 5-13. Ghez, G. and 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 ) . Gorman, W.M. [1953]. "Community p r e f e r e n c e f i e l d s . " E c o n o m e t r i c a , 2 1 ( 1 ) , 63-80. Gronau, R. [1973a], "The e f f e c t of c h i l d r e n on the ho u s e w i f e ' s v a l u e of t i m e . " J o u r n a l of P o l i t i c a l Economy, 8 1 ( 2 ) , P a r t I I , SI68-S199. - [1973b]. "The i n t r a - f a m i l y a l l o c a t i o n of t i m e : the v a l u e of the housewive's t i m e . " The Amer i c a n Economic Review, 63, (September 1973), 634-651. - [1 9 7 7 ] , " L e i s u r e , home p r o d u c t i o n , and work -- the t h e o r y of the a l l o c a t i o n of time r e v i s i t e d . " J o u r n a l of P o l i t i c a l  Economy, 8 5 ( 6 ) , 1099-1123. Hanoch, G. [1976]. Hours and Weeks i n the Theory of Labor Su p p l y . R-1787, (Santa Monica, Ca.: The Rand C o r p o r a t i o n ) . 1 57 Hausman, J.A. [1979]. "The e f f e c t of wages, t a x e s , and f i x e d c o s t s on women's l a b o r f o r c e p a r t i c i p a t i o n . " Working Paper No. 238, M a s s a c h u s e t t s I n s t i t u t e of Technology, May 1979. [1981]. "Exact consumer's s u r p l u s and deadweight l o s s . " The Amer i c a n Economic Review, f o r t h c o m i n g . - and D.A. Wise [1 9 7 9 ] , " A t t r i t i o n b i a s i n e x p e r i m e n t a l and p a n e l d a t a : the Gary income maintenance e x p e r i m e n t . " E c o n o m e t r i c a , 4 7 ( 2 ) , 455-473. Heckman, J . J . [1974]. "Shadow p r i c e s , market wages and l a b o r s u p p l y . " E c o n o m e t r i c a , 4 2 ( 4 ) , 679-694. - [1976]. "The common s t r u c t u r e of s t a t i s t i c a l models of t r u n c a t i o n , sample s e l e c t i o n and l i m i t e d dependent v a r i a b l e s and a s i m p l e e s t i m a t o r of such models." A n n a l s of  Economic and S o c i a l Measurement, 5 ( 4 ) , 4 75-492. - [1 9 7 7 ] , "Sample s e l e c t i o n b i a s as a s p e c i f i c a t i o n e r r o r w i t h an a p p l i c a t i o n t o the e s t i m a t i o n of l a b o r s u p p l y f u n c t i o n s . " NBER Working Paper #172, March 1977 ( r e v i s e d ) . - [1978a]. "A p a r t i a l survey of r e c e n t r e s e a r c h on the l a b o r s u p p l y of women." The Amer i c a n Economic Review, (Papers a n d . P r o c e e d i n g s ) , 6 8 ( 2 ) , 200-207. - [1978b], "Dummy endogenous v a r i a b l e s i n a s i m u l t a n e o u s e q u a t i o n s system." Econometr i c a , 4 6 ( 4 ) , 931-959. - [1 9 7 9 ] , "Sample s e l e c t i o n as a s p e c i f i c a t i o n e r r o r . " E c o n o m e t r i c a , 4 7 ( 1 ) , 153-161. -, M.R. K i l l i n g s w o r t h , and T. MaCurdy [1979]. "Recent t h e o r e t i c a l and e m p i r i c a l s t u d i e s of l a b o r s u p p l y : a p a r t i a l s u r v e y . " mimeo, 1979. - and T. MaCurdy [ 1 9 8 0 ] , "A l i f e c y c l e model of of female l a b o u r s u p p l y . " Review of Economic S t u d i e s , 47(1) - and R. W i l l i s [ 1 9 77]. "A beta l o g i s t i c model f o r the a n a l y s i s of s e q u e n t i a l l a b o r f o r c e p a r t i c i p a t i o n by m a r r i e d women." J o u r n a l of P o l i t i c a l Economy, 8 5 ( 1 ) , 27-58. 1 58 Hotz, V . J . [1980]. "A l i f e c y c l e model of f e r t i l i t y and m a r r i e d women's l a b o r s u p p l y . " C a r n e g i e - M e l l o n U n i v e r s i t y , Graduate S c h o o l of I n d u s t r i a l A d m i n i s t r a t i o n , Working Paper no. 17-80-81 . I n s t i t u t e f o r S o c i a l R e s e a r c h . A P a n e l Survey of Income Dynamics. Waves V I I I - X (1975-1977), U n i v e r s i t y of M i c h i g a n . Johnson, N.L. and S. Kotz [1972]. D i s t r i b u t i o n s i n S t a t i s t i e s :  C o n t i n u o u s M u l t i v a r i a t e D i s t r i b u t i o n s . (New York: John W i l e y and Sons, I n c . ) K e e l e y , M.C. [1975]. "A comment on 'An i n t e r p r e t a t i o n of the economic t h e o r y of f e r t i l i t y ' . " J o u r n a l of Economic L i t e r a t u r e , 1 3 ( 2 ) , 461-468. Khan, M.A. and I . S i r a g e l d i n [ 1 9 7 7 ] . "Son p r e f e r e n c e and the demand f o r a d d i t i o n a l c h i l d r e n i n P a k i s t a n . " Demography, 14 ( 4 ) , 481-495. K o s t e r s , M. [1 9 6 9 ] , " E f f e c t s of an income t a x on l a b o r s u p p l y . " In H a r b e r g e r , A.C. and M.J. B a i l e y , eds., The T a x a t i o n of  I ncome from C a p i t a l . (Washington, D.C: The B r o o k i n g s I n s t i t u t i o n ) , 301-324. Lau, L . J . , W. 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 c o n s u m p t i o n - 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 , 4 6 ( 4 ) , 843-868. Lee, Lu n g - F e i [1978]. "Unionism and wage r a t e s : a s i m u l t a n e o u s e q u a t i o n s model w i t h q u a l i t a t i v e and l i m i t e d dependent v a r i a b l e s . " I n t e r n a t i o n a l Economic Review, 1 9 ( 2 ) , 415-433. L e i b e n s t e i n , H. [1957). Economic Backwardness and Economic Growth: S t u d i e s i n the Theory of Economic Development. (New York: John WileyTT - [1 9 7 4 ] . "An i n t e r p r e t a t i o n of the econo m i c . t h e o r y of f e r t i l i t y . " J o u r n a l of Economic L i t e r a t u r e , 1 2 ( 2 ) , 457-479. - [1975a]. "The economic t h e o r y of f e r t i l i t y d e c l i n e . " The  Q u a r t e r l y J o u r n a l of Economics, 89(354), 1-31. - [1975b]. "On the economic t h e o r y of f e r t i l i t y : a r e p l y t o K e e l e y . " J o u r n a l of Economic L i t e r a t u r e , 1 3 ( 2 ) , 469-471. 159 L e u t h o l d , J.H. [19 6 8 ] , "An e m p i r i c a l study of formu l a income t r a n f e r s and the work d e c i s i o n s of the poor." J o u r n a l of  Human Resources, 3 ( 3 ) , 312-323. Maddala, G.S. [1977]. E c o n o m e t r i c s . (New York: M c G r a w - H i l l ) . M a l t h u s , T.R. [1798]. An Essay on the P r i n c i p l e of P o p u l a t i o n  and a Summary View of the P r i n c i p l e of P o p u l a t i o n . ( B a l t i m o r e : PenguinTT (Penguin e d i t i o n p u b l i s h e d 1970.) McCabe, J . L, and M.R. Rosenzweig [1976]. "Female l a b o r - f o r c e p a r t i c i p a t i o n , o c c u p a t i o n a l c h o i c e , and f e r t i l i t y i n d e v e l o p i n g c o u n t r i e s . " J o u r n a l of Development Economics, 3 ( 2 ) , 141-160. McFadden, D. [1974]. " C o n d i t i o n a l l o g i t a n a l y s i s of q u a l i t a t i v e c h o i c e b e h a v i o r . " In Zarembka, P., ed., F r o n t i e r s i n  Econometr i e s . (New York: Academic P r e s s ) , 105-142. - [1976], "Quantal c h o i c e : a s u r v e y . " A n n a l s of Economic  and S o c i a l Measurement, 5 ( 4 ) , 363-390. M i c h a e l , R.T. [1973]. " 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 . " J o u r n a l of P o l i t i c a l Economy, 8 1 ( 2 ) , P a r t I I , SI28-S164. M i n c e r , J . [1962], "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 l a b o r s u p p l y . " In 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 , " T 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 ) , 63-97. - [1963]. "Market p r i c e s , o p p o r t u n i t y c o s t s , and income e f f e c t s . " In C h r i s t , C a r l , et a l , eds., Measurement i n  Economics: S t u d i e s i n M a t h e m a t i c a l Economics and  Ec o n o m e t r i c s i n Memory of Yehuda G r u n f i e l d . ( S t a n f o r d : S t a n f o r d U n i v e r s i t y P r e s s ) , 67-82. - [1974]. 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 ) , NBER. - a n d S. Polachek [ 1 9 7 4 ] . " F a m i l y i n v e s t m e n t s i n human c a p i t a l : the e a r n i n g s of women." J o u r n a l of P o l i t i c a l  Economy, 8 2 ( 2 ) , P a r t I I , S76-S108. 160 Mood, A.M. and F.A. G r a y b i l l [1963]. I n t r o d u c t i o n t o the Theory  of S t a t i s t i e s , 2nd e d i t i o n . (New York: McGraw-Hi11). Nakamura, A. and M. Nakamura [1981 ]. "A comparison of the l a b o r f o r c e b e h a v i o r of m a r r i e d women i n the U n i t e d S t a t e s and Canada, w i t h s p e c i a l a t t e n t i o n t o the impact of income t a x e s . " E c o n o m e t r i c a , 49(2 ) , 451-490. Nakamura,M., A. Nakamura, and D. C u l l e n [1979]. "Job o p p o r t u n i t i e s , the o f f e r e d wage, and the l a b o r s u p p l y of m a r r i e d women." The Amer i c a n Economic Review, 6 9 ( 5 ) , 787-805. N a m b o o d i r i , N.K. [1972]. "Some o b s e r v a t i o n s on the economic framework f o r f e r t i l i t y a n a l y s i s . " P o p u l a t i o n S t u d i e s , 2 6 ( 2 ) , 185-206. N e r l o v e , M. and T.P. S c h u l t z [1970]. Love and L i f e Between the  Censuses: A Model of F a m i l y D e c i s i o n Making i n P u e r t o R i c o ,  1950-1960. RM-6322-AID, (Santa Monica, Ca. : The Rand C o r p o r a t i o n ) . Okun, B. [ i 9 6 0 ] , "Comment." In Demographic and Economic Change  i n Developed C o u n t r i e s . U n i v e r s i t i e s - N a t i o n a l Bureau Conference S e r i e s no. 11, ( 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 ) , 235-240. P o l l a k , R.A. [1969]. " C o n d i t i o n a l demand f u n c t i o n s and consumption t h e o r y " , The Q u a r t e r l y J o u r n a l of Economics, 8 3 ( 1 ) , 60-78. and M.L. Wachter [ 1 9 7 5 ] . "The r e l e v a n c e of the household p r o d u c t i o n f u n c t i o n and i t s i m p l i c a t i o n s f o r the a l l o c a t i o n of t i m e . " J o u r n a l of P o l i t i c a l Economy, 8 3 ( 2 ) , 255-277. Rosen, H.S. [1976]. "Taxes i n a l a b o r s u p p l y 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 , 4 4 ( 3 ) , 485-507. Rosenzweig, M.R. [1976], "Female work e x p e r i e n c e , employment s t a t u s , and b i r t h e x p e c t a t i o n s : s e q u e n t i a l d e c i s i o n - m a k i n g i n the P h i l i p p i n e s . " Demography, 13(3), 339-356. Ross, S.G. [19 7 4 ] , The Timing and Sp a c i n g of B i r t h s and Women's  Labor Fo r c e P a r t i c i p a t i o n : An Economic A n a l y s i s. Ph.D. t h e s i s " (New York: Columbia U n i v e r s i t y P r e s s ) . 161 Samuelson, P.A. [ 1 9 5 6 ] , " S o c i a l i n d i f f e r e n c e c u r v e s . " The  Q u a r t e r l y J o u r n a l of Economics, 7 0 ( 1 ) , 1-21. Sanderson, W.C. [ n . d . ] . "Economic t h e o r i e s of f e r t i l i t y : what do they e x p l a i n ? " S t a n f o r d U n i v e r s i t y , mimeo. Schmidt, P. [1978]. " E s t i m a t i o n of a s i m u l t a n e o u s e q u a t i o n s model w i t h j o i n t l y dependent c o n t i n u o u s and q u a l i t a t i v e v a r i a b l e s : the u n i o n - e a r n i n g s q u e s t i o n r e v i s t e d . " I n t e r n a t i o n a l Economic Review, 1 9 ( 2 ) , 453-465. S c h u l t z , T. P. [ 1 9 7 6 ] , "Determinants of f e r t i l i t y : a m i c r o -economic model of c h o i c e . " In C o a l e , A . J . , ed., Economic  F a c t o r s i n P o p u l a t i o n Growth. (New York: John W i l e y and Sons ) , 89-124. Simon, J.L. [ 1 9 6 9 ] . "The e f f e c t of income on f e r t i l i t y . " P o p u l a t i o n S t u d i e s , 2 3 ( 3 ) , 327-341. [1975a]. "The mixed e f f e c t s of income upon s u c c e s s i v e b i r t h s may e x p l a i n the convergence phenomenon." P o p u l a t i o n  S t u d i e s , 2 9 ( 1 ) , 109-122. [1975b]. " P u z z l e s and f u r t h e r e x p l o r a t i o n s i n the i n t e r r e l a t i o n s h i p s of s u c c e s s i v e b i r t h s w i t h husband's income, spouses' e d u c a t i o n and r a c e . " Demography, 12(2), 259-274. S m a l l , K.A. and H.S. Rosen [1981]. " A p l i e d w e l f a r e economics w i t h d i s c r e t e c h o i c e models." Econometr i c a , 4 9 ( 1 ) , 105-130. Smith, J.P. [1977]. " F a m i l y l a b o r s u p p l y 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 ( 2 ) , 205-276. T o b i n , J . [1958]. " E s t i m a t i o n of r e l a t i o n s h i p s f o r l i m i t e d dependent v a r i a b l e s . " E c o n o m e t r i c a , 2 6 ( 1 ) , 24-36. - [1973]. "Comment." J o u r n a l of P o l i t i c a l Economy, 8 1 ( 2 ) , P a r t I I , S275-S278. Wales, T.J. and A.D. Woodland [1976]. " E s t i m a t i o n of household u t i l i t y f u n c t i o n s and l a b o r s u p p l y r e s p o n s e . " I n t e r n a t i o n a l  Economic Review, 17(2), 397-410. 1 62 - [1977]. " 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 , 4 5 ( 1 ) , 115-132. - [1 9 8 0 ] . "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 l a b o u r s u p p l y f u n c t i o n s . " I n t e r n a t i o n a l Economic Review, 2 1 ( 2 ) , 437-468. H i s , R.J. [1973]. "A new approach t o the economic t h e o r y of f e r t i l i t y b e h a v i o r . " J o u r n a l of P o l i t i c a l Economy, 8 1 ( 2 ) , P a r t I I , S14-S64. Appendix t o Chapter Three In S e c t i o n B of Chapter Three, we d e r i v e d the model (A.1.a) H* = X,b, + a 1 w + d , N + k , A C + u 1 ( A . l . b ) C* = X 2 b 2 + a2W + g 2H + d 2N + k 2AC + u 2 (A.1.c) W = X 3 b 3 + u 3 (A.I.d) H = (A.1.e) AC= ( H* 0 1 0 i f H* > 0 o t h e r w i se i f C* > 0 o t h e r w i se ( A . l . f ) W i s observed i f f H > 0 The d i s t u r b a n c e terms are assumed t o be n o r m a l l y d i s t r i b u t e d w i t h mean z e r o and v a r i a n c e - c o v a r i a n c e m a t r i x I = e 1 1 c i 2 ' 1 3 62 1 "22 '2 3 '31 '32 '33 I i s assumed t o be symmetric and p o s i t i v e - d e f i n i t e . In o r d e r t o d e r i v e the l i k e l i h o o d f u n c t i o n , we need c o n s i d e r f o u r c a s e s : S I : H>0, AC=1 S2: H>0, AC=0 S3: H=0, AC=1 S4: H=0, AC=0 1 64 To c a l c u l a t e the l i k e l i h o o d f u n c t i o n , i t i s n e c e s s a r y t o d e r i v e the d e n s i t y f u n c t i o n c o r r e s p o n d i n g t o each c a s e . I t i s e a s i e r t o d e r i v e the d e n s i t y f u n c t i o n s i n terms of the reduced-form equat i o n s . In o r d e r t o d e r i v e the reduced form e q u a t i o n s , we f i r s t d e r i v e a p a r t i a l l y reduced form by s u b s t i t u t i n g (A.1.c) i n t o (A.1.a) and (A.1.b). (A.2.a) H* = X,b, + X 3 b 3 a , + d,N + k,AC + u, + u 3 a , (A.2.b) C* = X 2 b 2 + X 3 b 3 a 2 + g 2H + d 2N + k 2AC + u 2 + u 3 a 2 To o b t a i n the f u l l reduced form, we have t o c o n s i d e r the c a s e s where H > 0 and H = 0 s e p a r a t e l y . I f H > 0, H = H* and e q u a t i o n (A.2.a) can be s u b s t i t u t e d i n t o e q u a t i o n (A.2.b) t o g i v e us: (A.3.b) C* = X 2 b 2 + X 3 b 3 ( a , g 2 + a 2 ) + X,b,g 2 + N ( d 1 g 2 + d 2 ) + (k!g 2+k 2)AC + u,g 2 + u 2 + u 3 ( a , g 2 + a 2 ) As p o i n t e d out i n the t e x t , k,g 2+k 2 must e q u a l z e r o f o r the model t o make sense. Hence, the AC term drops out of e q u a t i o n (A.3.b). I f H = 0, the Hg 2 term i n e q u a t i o n (A.2.b) i s z e r o , and we are l e f t w i t h (A.3.b') C* = X 2 b 2 + X 3 b 3 a 2 + d 2N + u 2 + u 3 a 2 The reduced form of the model i s then 1 65 (A.4.a) H* = X,b, +'X 3b3a, + d,N + k,AC + u, + u 3 a , (A.4.b) C* = X 2 b 2 + X 3 b 3 ( a , g 2 + a 2 ) + Xib,q2 + N ( d 1 g 2 + d 2 ) + + u,g 2 + u 2 + u 3 ( a , g 2 + a 2 ) (A.4.b') C* = X 2 b 2 + X 3 b 3 a 2 + d 2N + u 2 + u 3 a 2 (A.4.c) W = X 3 b 3 + u 3 (A.4.d) H = [H* i f H* > 0 ( o o t h e r w i s e (A.4 . e ) AC= ( 1 i f C* > 0 v 0 o t h e r w i s e ( A . 4 . f ) W i s observed i f f H > 0 f o r S I , S2 f o r S3, S4 To s i m p l i f y n o t a t i o n , we w r i t e e q u a t i o n s (A.4.a)-(A.4.c) a s : (A.5.a) H* = Z,p, + v, (A.5.b) C* = Z 2 p 2 + v 2 (A.5.b') C* = Z 2 * p 2 * + v 2 * (A.5.c) W = Z 3 p 3 + v 3 where Z,p, = X,b, + X 3 b 3 a , + d,N + k,AC Z 2 p 2 = X 2 b 2 + X 3 b 3 ( a , g 2 + a 2 ) + X , * ) ^ + N C d ^ + d z ) Z 2 * p 2 * = x 2 b 2 + X 3 b 3 a 2 + d 2N Z 3 p 3 = X 3 b 3 v , = u , + u 3 a , v 2 = u,g 2 + u 2 + u 3 ( a 1 g 2 + a 2 ) v 2 * = u, + u 3a1 v 3 = u 3 1 66 The v ' s , by v i r t u e of t h e i r r e l a t i o n s h i p t o the u's, are a l s o . d i s t r i b u t e d n o r m a l l y w i t h mean z e r o . The v a r i a n c e -c o v a r i a n c e m a t r i x i s n = DID', where D i s a m a t r i x of c o e f f i c i e n t s such t h a t v=Du; v and u are the v e c t o r s of the reduced form and s t r u c t u r a l d i s t u r b a n c e terms, r e s p e c t i v e l y . S i n c e D w i l l v a r y depending on whether we are concerned w i t h c a s e s SI and S2, where v' = [v, v 2 v 3 ] , or w i t h c a s e s S3 and S4, where v' = [v, v 2 * v 3 ] , we must d i s t i n g u i s h two v a r i a n c e -c o v a r i a n c e m a t r i c e s . We d e f i n e Q t o be the v a r i a n c e - c o v a r i a n c e m a t r i x c o r r e s p o n d i n g t o S1 and S2, where « , , = « , , + 2a,ff, 3 + a 1 2 t f 3 3 "12 = 9 2 ' i i + ( a 1 g 2 + a 2 )a ,ff 3 3 + ( 2a , g 2+a 2) <r, 3 + « 1 2 + a^e23 " l 3 = ' l 3 + a 1 * 3 3 "22 = 9 2 2 ' i i + '22 + ( a , g 2 + a 2 ) 2 < j 3 3 + 2 g 2 * 1 2 + 2 ( a , g 2 + a 2 ) * 2 3 + 2 g 2 ( a i g 2 + a 2 ) f f , 3 "23 = 9 2 * 1 3 + '23 + (a ,g 2+a 2) <y3 3 "33 = '33 We d e f i n e n* t o be the v a r i a n c e - c o v a r i a n c e m a t r i x c o r r e s p o n d i n g t o S3 and S4, where 1 1 * = = u 1 i 1 2 * = 1 3 * = "2 2 * = 1 2 3 * = "3 3 * = '13 + a , f f 2 3 + a 2 t f , 3 + a , a 2 t f 3 3 " 1 3 '22 + 2 a 2 f f 2 3 + a 2 2 t f 3 3 '23 + a 2<r 3 3 "3 3 1 67 The l i k e l i h o o d f u n c t i o n i s the pr o d u c t of the d e n s i t y f u n c t i o n s c o r r e s p o n d i n g to each c a s e . L e t the d e n s i t y f u n c t i o n f o r v,, v 2 , v 3 be w r i t t e n as n ( v , , v 2 , v 3 ) , and l e t the d e n s i t y f u n c t i o n f o r v,, v 2 * , and v 3 be n * ( v , , v 2 * , v 3 ) , where n ( v , , v 2 , v 3 ) and n * ( v , , v 2 * , v 3 ) are t r i v a r i a t e normal d i s t r i b u t i o n s . In the case S1, H > 0, so both H and W are obser v e d and p o s i t i v e . What i s not observed i s C*. We o n l y know C* i s p o s i t i v e . Hence, the d e n s i t y f u n c t i o n c o r r e s p o n d i n g t o S1 i s oo (A.6.a) P1 = / n ( v , , v 2 , v 3 ) d v 2 - z 2 P 2 The l i m i t s of i n t e g r a t i o n a r e o b t a i n e d by n o t i n g t h a t when C*=0, v 2 = - Z 2 p 2 . The j o i n t d e n s i t y f u n c t i o n , n ( v 1 , v 2 , v 3 ) can be r e w r i t t e n as the pr o d u c t of the m a r g i n a l d i s t r i b u t i o n of v, and v 3 , and the c o n d i t i o n a l d i s t r i b u t i o n of v 2 , c o n d i t i o n a l on v, and v 3 . S i n c e the m a r g i n a l d i s t r i b u t i o n does not depend on v 2 , we can take i t o u t s i d e the i n t e g r a l s i g n . P1 becomes oo (A.6.a') P1 = g ( v 1 , v 3 ) - / h ( v 2 | v , , v 3 ) d v 2 where g ( v , , v 3 ) i s the m a r g i n a l d i s t r i b u t i o n of v, and v 3 , which i s a b i v a r i a t e normal d i s t r i b u t i o n (Mood and G r a y b i l l [ 1 9 6 3 ] ) . The i n t e g r a l of h ( v 2 | v 1 , v 3 ) i s the u n i v a r i a t e normal c u m u l a t i v e d e n s i t y f u n c t i o n (where the mean and s t a n d a r d d e v i a t i o n a re now the c o n d i t i o n a l mean and s t a n d a r d d e v i a t i o n ) . P2, the d e n s i t y f u n c t i o n c o r r e s p o n d i n g t o S2, i s the same as P1, e x c e p t , s i n c e C* i s n e g a t i v e , the l i m i t s of i n t e g r a t i o n change. 1 68 -Z 2p 2 (A.6.b) P2 = g ( v , , v 3 ) f h ( v 2 | v , , v 3 ) d v 2 P3 and P4, the d e n s i t y f u n c t i o n s c o r r e s p o n d i n g t o S3 and S4, r e s p e c t i v e l y , are more c o m p l i c a t e d . Now, H = 0 and W i s not ob s e r v e d . To e l i m i n a t e W, we i n t e g r a t e v 3 over i t s e n t i r e range (Heckman [ 1 9 7 4 ] ) . H = 0 i m p l i e s H* i s n e g a t i v e , thus we i n t e g r a t e v, over the p o r t i o n of i t s range such t h a t H* i s n e g a t i v e . The i n t e g r a t i o n over v 2 * i s the same as i n P1 and P2, except f o r the change i n v a r i a b l e . CO CO - Z 1P1 (A.6.c) P3 = / / / n * ( v , , v 2 * , v 3 ) d v 3 d v 2 * d v 1 -°° -Zfp* -°° " Z L P L "Z*p* oo (A.6.d) P4 = / / f n * ( v , , v 2 * , v 3 ) d v 3 d v 2 * d v , — CO — C O — C O P3 and P4 can be s i m p l i f i e d . L e t g ( v 1 , v 2 * ) be d e f i n e d as CO (A.7) g ( v 1 f v 2 * ) = f n * ( v 1 , v 2 * , v 3 ) d v 3 — CO where g ( v , , v 2 * ) i s the m a r g i n a l d i s t r i b u t i o n of v, and v 2 * , and i s a g a i n a b i v a r i a t e normal d i s t r i b u t i o n . P3 and P4 can then be r e w r i t t e n a s : " Z l P l oo (A.6.C') P3 = / / g ( v , , v 2 * ) d v 2 * d v , 00 - z * P * -Z p -Z*p* 1 f 1 2 f 2 (A.6.d') P4 = ' / / g ( v , , v 2 * ) d v 2 * d v , -CO —CO 1 69 That i s , P3 and P4 can be w r i t t e n as b i v a r i a t e c u m u l a t i v e normal d i s t r i b u t i o n s . The d e n s i t i e s (P1,...,P4) are c a l c u l a t e d f o r each woman i n the c o r r e s p o n d i n g c a t e g o r y . The l i k e l i h o o d f u n c t i o n i s then the product of these d e n s i t i e s , as g i v e n by e q u a t i o n (15) i n the t e x t . 1 70 Appendix t o Chapter Four The data used were taken from the M i c h i g a n P a n e l Survey of Income Dynamics t a p e . The da t a were taken from the 1975-1977 i n t e r v i e w y e a r s . The v a r i a b l e s used a r e : LNW The l o g a r i t h m of the wage r a t e as r e p o r t e d by the w i f e i n the 1976 i n t e r v i e w , a d j u s t e d by the m a r g i n a l t a x r a t e . AC A dummy v a r i a b l e e q u a l t o 1 i f the c o u p l e had a c h i l d i n the c u r r e n t p e r i o d . T h i s was deter m i n e d from date of b i r t h i n f o r m a t i o n from the 1976 i n t e r v i e w and from d a t a on whether a c h i l d was born i n 1976. (See Chapter Four, s e c t i o n B.) YR18 Number of y e a r s the w i f e worked s i n c e she was 18, from the 1976 i n t e r v i e w . CHILD Number of c h i l d r e n i n the f a m i l y , from the 1975 and 1976 i n t e r v i e w s . FAMINC Husband's l a b o r income p l u s a s s e t income a f t e r t a x e s , measured i n $10,000, f o r the two-year p e r i o d , from the 1976 and 1977 i n t e r v i e w s . MONPOS Number of months the w i f e has worked i n her c u r r e n t p o s i t i o n , from the 1976 i n t e r v i e w . EDUC Number of y e a r s of s c h o o l i n g of the w i f e , from the 1976 i n t e r v i e w . H Number of hours worked by the w i f e over the two-year p e r i o d , from the 1976 and 1977 i n t e r v i e w s . AGECH Age of youngest c h i l d , from the 1975 i n t e r v i e w AGEWMAR Age of the w i f e at m a r r i a g e , from the 1976 i n t e r v i e w . GAP D e s i r e d number of c h i l d r e n minus CHILD, from the 1976 i n t e r v i e w . 171 T=2.210 DR=0 $10.42, $10.47T $SIG 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0095608/manifest

Comment

Related Items