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An econometric analysis of the effects of labour market rationing on household labour supply Ryan, David Leslie 1983

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AN ECONOMETRIC ANALYSIS OF THE EFFECTS OF LABOUR MARKET RATIONING ON HOUSEHOLD LABOUR SUPPLY by DAVID LESLIE RYAN B . E c , Monash Univers i ty , 1975 M . E c , Monash Un ivers i ty , 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of ECONOMICS We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1983 © David Ryan, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ECONOMICS  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 25 JULY 1983 i i ABSTRACT Unt i l recent ly , much of the labour supply l i t e ra tu re has been predicated on the assumption that a l l ind iv iduals are free to choose the amount of labour that they supply at the preva i l ing wage ra te . However, empir ical studies have shown that many indiv iduals are rationed to work more or fewer hours than they des i re . In th is thesis we formulate and estimate two models of household labour supply behaviour which take account of the fact that constraints in the labour, market prevent some (male) household heads from supplying the i r desired amount of labour. In the f i r s t model we consider the case where a male is constrained in the number of hours of labour he can supply in a working week. In the second model we decompose male le isure into le isure during working weeks and le isure during non-working weeks, and consider rat ions on both. To be consistent with the economic theory of consumer behaviour we derive these models by assuming that households, based on the i r preferences, perform an optimization procedure to determine the i r desired quant i t ies of male le isure (provided i t is not rat ioned), female le isure and goods consumption. Since we model household behaviour, by j o i n t l y estimating demand equations for rationed and non-rationed households we are able to examine the ef fects that rat ioning of the labour supply of the household head has on the labour supply of his spouse and on household goods consumption. We f ind that there is no general pattern to the way in which the marginal budget shares and labour supply e l a s t i c i t i e s d i f fe r among rationed and i i i non-rationed households. Further, the re la t ionship between the rationed and non-rationed marginal budget shares and e l a s t i c i t i e s d i f fe rs according to the demographic charac ter is t i cs of the household. We also f ind that our estimates d i f f e r from those obtained when rationed households are ei ther ignored or omitted from the sample. In view of th is resu l t , and the fact that marginal budget shares and labour supply e l a s t i c i t i e s for rationed households cannot be calculated using the misspeci f ied models, we conclude that i t is important to use the ent i re sample of rationed and non-rationed households to estimate the model in which each type of rat ioning is modelled appropr iately. iv TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS iv LIST OF TABLES v i i i LIST OF FIGURES x ACKNOWLEDGEMENT xi CHAPTER 1 INTRODUCTION 1 CHAPTER 2 MODELLING CONSTRAINED LABOUR SUPPLY BEHAVIOUR 11 2.1 Introduction 11 2.2 Modelling Labour Supply Behaviour when Rationed Agents are Not Ident i f ied 13 2.3 Models which U t i l i z e Agent-Speci f ic Rationing Information 18 CHAPTER 3 A SINGLE-RATION MODEL OF HOUSEHOLD LABOUR SUPPLY 28 3.1 Introduction 28 3.2 Model Spec i f ica t ion 32 3.2.1 Formulation of the Basic Model 33 3.2.2 Stochastic Spec i f ica t ion 34 3.2.3 Derivation of the Likel ihood Function 35 3.2.4 Sample Se lec t i v i t y Correction 39 3.3 Al ternat ive Representations of Household Preferences 40 3.3.1 Klein-Rubin or Stone-Geary U t i l i t y Function 41 3.3.2 Constant E l a s t i c i t y of Subst i tut ion (CES) U t i l i t y Function 42 3.3.3 Deaton-Muellbauer Cost Function 43 V CHAPTER 3 A SINGLE-RATION MODEL OF HOUSEHOLD LABOUR SUPPLY 3.3.4 Indirect Translog U t i l i t y Function 44 3.3.5 Direct Translog U t i l i t y Function 48 CHAPTER 4 RESULTS AND ANALYSIS FOR THE SINGLE-RATION MODEL 51 4.1 Introduction 51 4.2 Choice among Al ternat ive Representations of Household Preferences 52 4.3 Inclusion of Demographic Variables 57 4.4 Expected Effects of Including the Sample Se lec t i v i t y Correction 61 4.5 Empirical Results for the Complete Rationing Model 69 4.5.1 Parameter Estimates 70 4.5.2 E l a s t i c i t y Estimates 78 4.5.3 Comparison with Other Studies 85 4.6 Results using only Non-Rationed Households and Comparison with Estimates from the Complete Model 89 4.7 Results t reat ing A l l Households as Non-Rationed and Comparison with Estimates from the Complete Model 95 4.7.1 Comparison of Parameter and E l a s t i c i t y Estimates 95 4.7.2 Evidence on the Effects and Importance of Labour Market Rationing 100 4.8 General Evaluation of Results 108 4.9 Empirical Results for Models where Unemployed are Grouped with Underemployed 112 vi CHAPTER 5 RATIONING OF WEEKS AND HOURS OF LABOUR SUPPLY 121 5.1 Introduction 121 5.2 Model Spec i f ica t ion 123 5.2.1 Formulation of the Basic Model 124 5.2.2 Stochastic Spec i f ica t ion 127 5.2.3 Derivation of the Likel ihood Function 139 5.2.4 Sample Se lec t i v i t y Corrections 143 5.3 Equations for the Stone-Geary Spec i f ica t ion 145 5.4 Ident i f i ca t ion of Parameters 148 CHAPTER 6 RESULTS AND ANALYSIS FOR THE TWO-RATION MODEL 158 6.1 Introduction 158 6.2 Lower Triangular Decomposition of Variance-Covariance Matrices 159 6.3 Empir ical Results for the Complete Rationing Model 163 6.3.1 Parametric Rest r ic t ions 163 6.3.2 Parameter Estimates 168 6.3.3 Estimates of E l a s t i c i t i e s and Marginal Budget Shares 177 6.3.4 Comparison with Results of the Single-Ration Model 191 6.4 Results using only Non-Rationed Households and Comparison with Estimates from the Complete Model 201 6.5 Results t reat ing A l l Households as Non-Rationed and Comparison with Estimates from the Complete Model 207 6.6 Comparison with Results from Other Studies 213 CHAPTER 7 CONCLUSIONS 218 v i i BIBLIOGRAPHY 225 APPENDIX 1 DATA 231 APPENDIX 2 FORMULATION OF LABOUR SUPPLY ELASTICITIES IN THE SINGLE-RATION MODEL 237 APPENDIX 3 REGULARITY CONDITIONS FOR THE SINGLE-RATION MODEL 239 APPENDIX 4 FORMULATION OF THE MARGINAL BUDGET SHARES AND LABOUR SUPPLY ELASTICITIES FOR THE HOURS AND WEEKS MODEL 244 v i i i LIST OF TABLES Table 4.1 Parameter Estimates for the Single-Rat ion Model 72 Table 4.2 Median E l a s t i c i t i e s for the Single-Rat ion Model 82 Table 4.3 Parameter Estimates when only Non-Rationed Observations are used 91 Table 4.4 Median E l a s t i c i t i e s when only Non-Rationed Observations are used 93 Table 4.5 Parameter Estimates when A l l Observations are Treated as Non-Rationed 97 Table 4.6 Median E l a s t i c i t i e s when A l l Observations are Treated as Non-Rationed 98 Table 4.7 Parameter Estimates for the Single-Rat ion Model Under Data Rec lass i f i ca t ion 114 Table 4.8 Median E l a s t i c i t i e s for the Single-Ration Model Under Data Rec lass i f i ca t i on 115 Table 4.9 Parameter Estimates Under Data Rec lass i f i ca t i on when Only Non-Rationed Observations are used 117 Table 4.10 Median E l a s t i c i t i e s Under Data Rec lass i f i ca t ion when Only Non-Rationed Observations are used 119 Table 5.1 Relat ionship Between Rations on Weeks and Hours 128 Table 6.1 Comparison of Parameter Estimates with Dif ferent Y I Values 169 Table 6.2 Parameter Estimates for the Two-Ration Model 172 Table 6.3 Corre lat ion Estimates for the Two-Ration Model 176 Table 6.4 Marginal Budget Shares for the Two-Ration Model 179 Table 6.5 Median E l a s t i c i t i e s for the Two-Ration Model 183 ix Table 6.6 Median Cross-Rationing E l a s t i c i t i e s for the Two-Ration Model 189 Table 6.7 Comparison of Parameters in the Single-Ration and Two-Ration Models 193 Table 6.8 Comparison of Median E l a s t i c i t i e s in the Single-Ration and Two-Ration Models 198 Table 6.9 Comparison of Median Cross-Rationing E l a s t i c i t i e s in the Single-Ration and Two-Ration Models 200 Table 6.10 Parameter Estimates when only Non-Rationed Observations are used 203 Table 6.11 Median E l a s t i c i t i e s when only Non-Rationed Observations are used 205 Table 6.12 Parameter Estimates when A l l Observations are Treated as Non-Rationed 209 Table 6.13 Median E l a s t i c i t i e s when A l l Observations are Treated as Non-Rationed 211 Table Al Descr ipt ive S t a t i s t i c s for A l l Data 234 Table A2 Dis t r ibut ion of Households in the Single-Ration Model 235 Table A3 Dis t r ibu t ion of Households in the Hours and Weeks Model 236 X LIST OF FIGURES Figure 1 46 xi ACKNOWLEDGEMENT In wr i t ing th is thesis I have been most fortunate to have had Terry Wales as my research supervisor. Without his invaluable advice, suggestions and encouragement, th is d isser ta t ion would not have been completed. A number of other people have also helped with various aspects of th is t hes i s . In par t i cu la r I would l i ke to thank Craig Riddel 1, John Weymark, John Cragg, Grayham Mizon and Michael McAleer. In add i t ion , James Zidek of the Department of Mathematics provided assistance with some of the s t a t i s t i c a l problems I encountered. I also wish to express my grati tude to Alan Woodland. Although he l e f t before the model bui ld ing and empir ical work began, his ideas and encouragement were espec ia l ly important in the formative days of th is t hes i s . F i n a l l y , I owe an immeasurable debt to Donna for her support and encouragement, pa r t i cu la r l y in the d i f f i c u l t times when "things d idn ' t work". - 1 -CHAPTER ONE INTRODUCTION Although there have been many studies in the l i te ra tu re which examine labour supply decisions of indiv iduals and households under various circumstances, only recent ly have researchers taken an interest in examining the ef fects of rat ioning in the labour market on these dec is ions. While various empirical studies have revealed that labour markets are not always in equi l ibr ium, these studies have tended to concentrate on modelling th is d isequi l ib r ium at the aggregate l e v e l . In general, neither the actual mechanism for a l locat ing the aggregate d isequi l ib r ium among agents, nor the impl icat ions of th is aggregate d isequi l ib r ium for indiv idual labour supply behaviour, have been considered. In order to examine the ef fects on indiv idual behaviour of aggregate d isequ i l ib r ium in the labour market, i t is necessary to take account of the par t icu lar features of th is market. S p e c i f i c a l l y , as noted by Ashenfelter [1980] and others, while many agents in the labour market are not affected at a l l by the aggregate d isequ i l ib r ium, others are affected to varying degrees. Thus, any model which is used to examine indiv idual labour supply behaviour when the aggregate market is in d isequi l ib r ium must be s u f f i c i e n t l y f l e x i b l e to re f l ec t these i ns t i t u t i ona l features. S p e c i f i c a l l y , in such a model, i t cannot reasonably be assumed that e i ther a l l or none of the indiv idual agents are rat ioned. One of the main empirical requirements for a model which meets these spec i f i ca t ions is a data set in which i t is possible to ident i fy which - 2 -ind iv idua ls are rationed and which are in equi l ibr ium supplying the i r desired number of hours. However, most cross-sect ion data sets which are used to estimate labour supply functions do not contain rat ioning information. Consequently, studies such as those by Ashenfelter [1980] and Blundel l and Walker [1982] have examined constrained labour supply behaviour ei ther by making use of extraneous information, such as the aggregate unemployment ra te , or by assuming that a l l ind iv iduals are rat ioned. One data set which does contain rat ioning information is the Univers i ty of Michigan Survey Research Centre's "A Panel Study of Income Dynamics". Questions asked in th is survey are such that heads of households who wanted to work more or less hours than they are current ly working can be i d e n t i f i e d . Thus, households can be c l a s s i f i e d as non-rat ioned, underemployed or overemployed according to the response of the household head to the questions concerning the re la t ionsh ip between his desired and actual number of hours worked. Despite the a v a i l a b i l i t y of the Michigan data se t , r e l a t i v e l y few studies have made use of the rat ioning information i t contains. Wales and Woodland [1976], [1977] use th is information to ident i fy non-rationed or "equi l ibr ium" households. On f ind ing that the parameters of the u t i l i t y function d i f f e r s i g n i f i c a n t l y for rationed and non-rationed households, these authors r e s t r i c t further analysis to the subsample of non-rationed households. In a ser ies of recent papers, Ham [1977], [1980], [1982] argues that th is procedure w i l l resu l t in inconsistent estimates of the parameters of the labour supply funct ion. In order to obtain consistent est imates, Ham proposes and estimates various econometric models which allow the parameters of the labour supply function to be estimated for a - 3 -sample of males, some of whom are rationed in the number of hours of labour they can supply. While Ham's models represent a s ign i f i can t advance from ea r l i e r models where i t is assumed that a l l ind iv iduals are supplying the i r desired amount of labour even though the aggregate market may be in d isequ i l ib r ium, they do not provide an overal l picture of the ef fects of rat ion ing on indiv idual behaviour. S p e c i f i c a l l y , i f due to rat ioning an ind iv idual is forced to supply less labour than he des i res , his consumption decisions are also l i k e l y to be af fected. Further, i f the indiv idual whose hours are rationed in the labour market is a member of a mult ip le person household, the ef fects of his rat ioning may be such as to a l te r the labour supply behaviour of other household members as well as household consumption patterns. In order to model any of these e f fec ts , i t is c l ea r l y not su f f i c ien t simply to estimate the labour supply equation of the male household member. Rather, i t is necessary to specify and estimate a model which allows a l l these ef fects to be present. In th is thesis we formulate and estimate a l ternat ive models of household behaviour which take account of the fact that in some households, the household head is unable to supply his desired amount of labour. To construct these models we assume that households, based on the i r preferences, perform an optimization procedure to determine desired quant i t ies of male le isure (for non-rationed households only) and female le isure and the i r desired consumption of a composite commodity. Consequently, we are able to examine the ef fects on female labour supply and on household goods consumption of constraints on the labour supply of the (male) household head. While concurrent analysis by Ransom [1982] is also concerned with examining these e f fec t s , our study d i f fe rs in a number - 4 -of ways. 1 One of the most important of these dif ferences concerns the spec i f i ca t ion of the funct ional forms that are estimated. Since the demand equations we estimate are derived from the optimization procedure described above, they are consistent with the economic theory of consumer behaviour. However, l i ke the models of Ham [1977], [1982] in which labour supply functions are assumed to be l i nea r , Ransom's model is ad hoc in nature since he a r b i t r a r i l y spec i f ies l inear approximations to the f i r s t order condit ions of a u t i l i t y maximization problem. To implement the models, we adopt a spec i f i c funct ional form for household preferences and assume that i t applies to both rationed and non-rationed households. Various a l ternat ive funct ional forms are considered, including the Indirect Translog u t i l i t y function and the Stone-Geary u t i l i t y function which, for non-rationed households, gives r i se to the well known Linear Expenditure System. While some of the funct ional forms we consider are eas i l y amenable to analysis with both rationed and non-rationed households, others such as the Indirect Translog cause considerable d i f f i c u l t i e s when rationed households are examined. Consequently we are unable to obtain estimates of the parameters for several of these spec i f i ca t i ons . For those funct ional forms that can be estimated, we examine the parameter estimates as well as estimated values of marginal budget shares and labour supply e l a s t i c i t i e s for rationed and for non-rationed households. In order to evaluate the ef fects and importance of cor rec t ly modelling the behaviour of rationed ind iv idua ls , we compare our estimates to those obtained when estimation is performed using only the subsample of non-rationed households. Comparisons are also made with estimates obtained 1 See Section 3.1 for a more deta i led explanation of these d i f ferences. - 5 -in t rad i t i ona l labour supply s tudies; that i s , where a l l households are treated as though they are non-rat ioned. We begin in Chapter 2 by examining various studies which have modelled labour supply behaviour when there are constraints in the labour market. While we concentrate mainly on the cross-sect ion studies by Ham and Ransom, which are c losest in nature to those developed here, we also consider studies which do not assume knowledge of which ind iv iduals or households are subject to rat ioning const ra in ts . In addi t ion, we also examine some theoret ica l r esu l t s , based on analysis by Neary and Roberts [1980]. These resul ts are pa r t i cu la r l y useful as they enable us to derive rationed demand equations from a l ternat ive representations of household preferences. In Chapter 3 we speci fy a household model in which the labour supply of the male household head is subject to rat ioning const ra in ts . In order to f a c i l i t a t e comparisons of the labour supply behaviour of the non-rationed female in households where the male is rationed to her behaviour in households where he is non-rat ioned, and to calculate the ef fects of changes in the male rat ion level on her labour supply behaviour, we j o i n t l y estimate the rationed and non-rationed equations using the Maximum Likel ihood technique. Hence in th is chapter we derive the l i ke l ihood funct ion which comprises the product of the l ike l ihoods for the three rat ioning cases. Since the demand equations are functions of both the male and female wage ra tes , i t is necessary to estimate the model using only those households in which the female is working, and hence where her wage rate information is ava i l ab le . As shown by Wales and Woodland [1980] and others, such a procedure introduces a sample s e l e c t i v i t y bias into the resu l t s . Corrections to the l i ke l ihood function - 6 -which w i l l enable us to avoid th is bias are therefore speci f ied in th is chapter. While the model and l i ke l ihood function are derived in general terms, in order to obtain estimates i t is necessary to adopt a par t i cu la r funct ional form for household preferences. In Chapter 3 we consider several a l ternat ive forms for these preferences, and in each case derive the non-rationed demand equations. Using techniques described in Chapter 2, the corresponding equations for households in which the male is rationed are then obtained. Possible problems which may ar ise with several of these funct ional forms are also discussed. In Chapter 4 we present and analyze the resul ts obtained when th is model is estimated using a sample of 492 households drawn from Wave XI (1978) of the Univers i ty of Michigan data set . Problems encountered with some of the funct ional form representations of household preferences are discussed and decisions concerning which demographic var iables to include in the model, and the method to be used to incorporate them, are explained. Parameter and e l a s t i c i t y estimates are analyzed and compared to resu l ts obtained in other s tudies. In addi t ion, attention is paid to the ef fect on the estimates of including the sample s e l e c t i v i t y correct ion which takes account of the fact that households with non-working wives are excluded from our ana lys is . While th is correct ion has often been found to have l i t t l e effect on the parameter estimates in other labour supply s tud ies , we examine th is issue here in the context of a rat ioning model. We show that in general the d i rect ion of the sample s e l e c t i v i t y bias cannot be ana l y t i ca l l y determined in th is model. Hence in order to examine 2 Complete de ta i l s of the c r i t e r i a used to select households and the method by which they are al located to the various rat ioning c a t e g o r i e s -underemployed, overemployed and non-rationed—are contained in Appendix 1. - 7 -the nature and extent of th is b ias , we present estimates obtained both with and without the sample s e l e c t i v i t y correct ion included in the model. Chapter 4 also includes an analysis of the resul ts obtained when we estimate the model f i r s t using only the subsample of non-rationed households and then t reat ing a l l 492 households as though they are not rationed in the labour market. In order to evaluate the ef fects and importance of cor rec t ly modelling the behaviour of rationed ind iv idua ls , these resul ts are compared to those obtained using the cor rec t ly spec i f ied model. Evidence on th is issue from other studies which have examined labour supply rat ion ing is also discussed. One pa r t i cu l a r l y unappealing aspect of the parameter estimates presented in Chapter 4 is that they imply that l e i su re , pa r t i cu la r l y male l e i s u r e , is an in fe r io r good. In order to determine the reasons for th is resu l t we examine various other labour supply studies which have also found le isure to be i n f e r i o r . While there appears to be no s ingle feature common to a l l these studies which can account for th is resu l t , we show that l i n e a r i t y of the labour supply or labour earnings function is possibly one of the pr inc ipa l contr ibut ing fac to rs . A conceptual d i f f i c u l t y with the model developed in Chapter 3 concerns the treatment of ind iv iduals who claim to have been unemployed during the year for a non-zero number of weeks. In our model th is unemployment information is ignored, with the rat ioning status of the household based simply on responses of the household head to questions concerning the re la t ionsh ip between his actual and desired hours worked. An a l ternat ive approach, adopted by Ham [1977], [1980], involves grouping households in which the head is unemployed for any time during the year with those households in which the head is underemployed. One disadvantage - 8 -of th is approach is that many of the household heads who are unemployed claim not to desire any more than the i r current number of hours. However, in an attempt to improve the estimates obtained, we re-estimate the rat ioning model with unemployed and underemployed households grouped together. Despite th is data r e c l a s s i f i c a t i o n , no s ign i f i can t improvements in the parameter estimates or labour supply e l a s t i c i t i e s are observed. An a l ternat ive way to deal with the problem that many non-rationed households are unemployed for some time during the year is to view the rat ion ing problem as having greater dimensional i ty than previously considered. S p e c i f i c a l l y , as recognized by Hanoch [1980a], an indiv idual is usual ly not ind i f ferent between working 100 hours per week for 20 weeks or working 40 hours per week for 50 weeks, even though both a l ternat ives resu l t in 2000 annual hours of work. S i m i l a r l y , i t would seem l i k e l y that an indiv idual who is unemployed for some time during the year may want to work more weeks but does not want to work more hours per week during the weeks that he is working. In order to allow th is p o s s i b i l i t y , in Chapter 5 we develop a household model in which both the hours worked by the head in a working week and the weeks worked by the head in a year are subject to rat ion ing const ra in ts . Here, weeks worked by the head are assumed to be less than the desired number i f the head spent any time during the year unemployed. Thus in th is model we have 6 rat ioning categories defined by whether the head is underemployed, overemployed or non-rationed with respect to hours worked in a working week, and whether he is working his desired number, or less than his desired number of weeks in a year. In order to construct the hours and weeks model, we assume that households maximize a u t i l i t y funct ion which depends on le isure of the head during working weeks, his le isure during non-working weeks, le isure - 9 -of his spouse and household goods consumption. Maximization of th is funct ion subject to the household budget constraint y ie lds demand equations for non-rationed households. Maximization subject to the addit ional constraint that weeks worked are constrained, or that hours worked per week are constrained, or that both hours and weeks are constrained, y ie lds the remaining demand equations in the various rationed cases. Due to the nature of the avai lab le rat ion ing information and the complexity of the equations to be estimated, considerable attention is paid to the appropriate stochast ic spec i f i ca t ion for the model. Again Maximum Likel ihood methods are used, so that the demand equations for the various rat ioning cases can be estimated j o i n t l y . We therefore derive the relevant l i ke l ihood function which again includes a sample s e l e c t i v i t y correct ion to account for the fact that non-working wives are excluded from our sample. In view of the complex nature of the model speci f ied in Chapter 5, demand equations are only derived for the case where preferences are represented by the Stone-Geary u t i l i t y funct ion. Despite the s imp l i c i t y of the non-rationed equations derived from th is u t i l i t y function in the non-rationed case, equations in the various rat ion ing cases — pa r t i cu la r l y when only hours in a working week are rationed - - are quite nonl inear. Consequently i t is necessary to ca re fu l l y examine whether the st ructura l parameters and the elements of the variance-covariance matrix of the error terms are i den t i f i ed . Results for th is model, which is estimated using the same sample of 492 households, are presented and analyzed in Chapter 6. Due to the complexity of the l i ke l ihood funct ion, several problems were encountered during est imat ion. We thus begin th is chapter by describing various - 10 -s t a t i s t i c a l techniques that were used and parametric res t r i c t i ons that were imposed in order to overcome these problems. In addit ion to examining the parameter estimates, in th is chapter we also analyze the marginal budget shares and labour supply e l a s t i c i t i e s for the d i f ferent rat ion ing cases. These resul ts are then compared to those presented in Chapter 4 for the s ing le - ra t ion model. Here we f ind that as a resul t of separating male le isure into le isure during working weeks and le isure during non-working weeks, male le isure is no longer an in fe r io r good. In f ac t , male le isure during non-working weeks is i n f e r i o r , but an increase in income w i l l increase male le isure during working weeks by a greater amount than i t w i l l decrease male le isure during non-working weeks. In order to evaluate the ef fects and importance of cor rec t ly modelling the behaviour of rationed ind iv idua ls , we again re-estimate the model f i r s t using only non-rationed households and then t reat ing a l l households in the sample as though they are non-rat ioned. The parameters, marginal budget shares and labour supply e l a s t i c i t i e s are then compared to the estimates obtained using the cor rec t ly speci f ied model. In addi t ion, we also compare our resul ts to those obtained by Hanoch [1980b] who estimates an hours and weeks model assuming that a l l ind iv iduals are non-rat ioned. We conclude th is thesis in Chapter 7 by discussing the main f indings and suggesting a number of possible areas for further research. - 11 -CHAPTER TWO MODELLING CONSTRAINED LABOUR SUPPLY BEHAVIOUR 2.1 Introduction In t rad i t i ona l models of labour supply, i t is assumed that a l l workers can choose their-hours of work f ree ly and that a l l unemployment is voluntary. However, pa r t i cu la r l y in the shor t - run, the ef fect ive choice facing an ind iv idual worker is whether to work a standard day (or week or year) or not to work at a l l . In th is context, i t is possible for the indiv idual to work more hours than he would prefer , in which case the worker is said to be overemployed, to work less than he would prefer , in which case he is underemployed, or not to work at a l l , in which case he is ei ther vo lun ta r i l y or invo lun tar i l y unemployed. Recently, several studies have taken account of the fact that some ind iv iduals are rationed in the labour market. As well as the studies of Ham [1977], [1980], [1982], Ashenfelter [1980], Ashenfelter and Ham [1979], Ransom [1982] and others which e x p l i c i t l y recognize the fact that ind iv iduals do not always supply the i r desired number of hours, the aggregate d isequi l ibr ium study of Rosen and Quandt [1978] i m p l i c i t l y assumes that not a l l ind iv iduals are in equi l ibr ium where the i r actual and desired supplies of labour co inc ide. Before examining some of these studies in greater d e t a i l , i t is important to know the empirical s ign i f icance of the labour market rat ioning that these studies are seeking to exp la in . Using the Michigan Panel data for the years 1967 to 1974, Ham [1977] shows that for males aged 25-50 in 1967, the percentage who are - 12 -rat ioned varies from a low of 23% in 1967 to a high of 34% in 1970. 1 Furthermore, during the 8 year sample per iod, over 72% of the sample i s rationed at least once. Here, those who are considered to be rationed include the underemployed, overemployed and unemployed. Although questions asked in the survey allow an easy iden t i f i ca t i on of the f i r s t two of these three groups, i t is d i f f i c u l t to ident i fy unemployed workers. The operational measure Ham uses to ident i fy an indiv idual as being unemployed is whether or not that indiv idual lost one or more work days that year due to unemployment. In th is chapter we examine various studies which have modelled labour supply behaviour when there are constraints in the labour market. While several of these studies have made use of the Michigan data in which rationed households are i d e n t i f i e d , we begin our analysis by considering studies which do not assume knowledge of which indiv iduals or households are subject to rat ioning const ra in ts . In th is section we also consider some pa r t i cu la r l y useful theoret ical resu l ts which are based on analysis by Neary and Roberts [1980]. Using these resul ts we show how labour supply or le isure demand equations for the non-rationed members of a rationed household can be derived from al ternat ive representations of household preferences. Following th is ana lys is , in Section 2.3 we examine studies by Ham and Ransom which make use of the rat ion ing information contained in the Michigan data. Univers i ty of Michigan, Survey Research Centre: "A Panel Study of Income Dynamics". - 13 -2.2 Modelling Labour Supply Behaviour when Rationed Agents are Not  Ident i f ied One of the f i r s t empirical studies to deal with the fact that some ind iv iduals are constrained in the i r labour market choices, and that these constraints have repercussions on other fami ly members, is the study by Ashenfelter [1980]. Assuming a Stone-Geary u t i l i t y function which is common to both rationed and unrationed households, Ashenfelter derives commodity demand and labour supply equations f i r s t for the case where the household is unrationed, and then for the constrained case where members of the household are prevented from supplying the i r desired amount of labour. As Ashenfelter notes, a par t i cu la r advantage of the Stone-Geary formulation is that i t allows the constrained expenditure and earnings equations to be writ ten in terms of the respective unconstrained equations. To implement the model, Ashenfelter obtains aggregate earnings and commodity expenditure equations by taking a weighted sum of the rationed and unrationed funct ions, where the weights are the proportions of workers constrained and unconstrained respect ive ly . Rearranging and s impl i fy ing the resu l t ing expressions, Ashenfelter is able to derive aggregate earnings and expenditure equations which only depend on (observed) unrationed earnings and expenditure var iab les , and on the aggregate unemployment ra te . These equations are then estimated for 7 commodity groups using annual time ser ies data from 1930 to 1967. While Ashenfelter concludes from his study that i t is of interest empi r ica l ly as well as theo re t i ca l l y to t reat unemployment as a constraint on choice, he does note that the preference and aggregation structures he used were very simple. - 14 -An a l ternat ive procedure, developed by Neary and Roberts [1980], is to use dua l i ty theory and the notion of v i r tua l pr ices to l ink rationed and non-rationed demands. V i r tua l p r i ces , f i r s t defined by Rothbarth [1940-41], are the prices that would induce an unrationed household to behave in the same manner as when faced by a given vector of rat ion const ra in ts . As shown by Neary and Roberts, i t is possible to re late the propert ies of rationed demand and supply functions to the properties of unrationed demand and supply functions evaluated at these v i r tua l p r i ces . To demonstrate th is method, we consider the case where the rat ion is on the i nd i v idua l ' s consumption of l e i su re , XQ , which has pr ice p 0 and rat ion level Xo. Here the constrained cost funct ion is given by (2.1) C c (u ,p ,p 0 ,Xo ) = min x {p.X + p 0 X 0 : U(X,T0) > u} = p.X c + PfjXo, where X c is the constrained consumption of other goods which have prices p. Under cer ta in assumptions, Neary and Roberts show that v i r tua l pr ices ex is t and they character ize the i r form; in pa r t i cu la r , v i r tua l pr ices for unrationed goods are shown to be equal to the i r actual p r i ces . Denoting the v i r tua l pr ice for Xo by~p~o> th is v i r tua l price is i m p l i c i t l y defined by (2.2) C(u,p,po) = min {p.X + "poXo : U(X,X 0 ) >_ u} X,Xrj = p.X c + l>o~Xo> that i s , i t is the pr ice that resul ts in the same quant i t ies being purchased as in the constrained case. - 15 -Now combining (2.1) and ( 2 . 2 ) , we ob ta in (2 .3 ) C c ( u , p , p 0 , X 0 ) = C(u ,p ,po) + (Po - P o ) X 0 . De f i n ing expend i tu re as M when the household faces the c o n s t r a i n t Xo = Xo and p r i c e s p and p 0 , so that C c ( u , p , p 0 , X Q ) = M, then the expendi ture requ i red to a t t a i n the same u t i l i t y l e v e l , u, when the household faces p r i c e s p and po i s , from ( 2 . 3 ) , (2 .4) C ( u , p , p 0 ) = M + (po - P o ) X 0 . Hence the unrat ioned M a r s h a l l i a n demand f unc t i ons eva luated at v i r t u a l p r i c e s "po and at the expendi ture l e v e l g iven by (2.4) can be equated to the ra t i oned M a r s h a l l i a n demand f u n c t i o n s : (2 .5 ) X C ( p , p 0 , M , X o ) = XU(p ,p 0 ,M+(p 0 - p 0 ) X 0 ) . Thus with t h i s method, ins tead of ob ta i n i ng the ra t i oned demand f u n c t i o n s d i r e c t l y from the cons t ra ined pr imal problem a s , f o r example, A s h e n f e l t e r [1980] does in the method descr ibed e a r l i e r , they are obta ined from the unrat ioned demand f unc t i ons by us ing the no t ion of v i r t u a l p r i c e s . Hence, once the unconst ra ined expendi ture equa t i ons , p-jXi and PoXo are obta ined by maximizing the u t i l i t y f unc t i on sub jec t to the budget c o n s t r a i n t , the cor responding cons t ra ined expendi ture equat ions are obta ined by making the s u b s t i t u t i o n s i n d i c a t e d in equat ion (2.5) and then rea r rang ing the r e s u l t i n g e x p r e s s i o n s . S p e c i f i c a l l y , in both the p-jX^1 and poXo e q u a t i o n s , X^ i s rep laced by the cons t ra ined consumpt ion, X ^ , Xo i s rep laced by the r a t i o n l e v e l X o , a n c l the p r i c e p 0 i s rep laced by the v i r t u a l p r i c e , po • In a d d i t i o n , income M i s rep laced by M + (p"o - Po)Xo-Next , the new equat ion poXo i s rearranged to ob ta in an express ion f o r po which i s then s u b s t i t u t e d in the p-jX^ equa t i ons . Rearrangement of the - 16 -resu l t ing expressions y ie lds constrained expenditure equations which depend only on observable v a r i a b l e s . 2 An important advantage of th is procedure is that one need not s tar t with the spec i f i ca t ion of a u t i l i t y funct ion. Thus, unconstrained expenditure functions could be derived from a spec i f i ca t ion of the cost func t ion , and then the v v i r tua l pr ices method of Neary and Roberts could be applied to these expenditure equations to y ie ld expenditure equations for the constrained case. Essen t i a l l y , th is is the procedure suggested by Deaton [1981], who notes that i t is d i f f i c u l t in general to character ize rationed demands in re la t ion to unrationed demands from consideration of the d i rect u t i l i t y funct ion. To use th is method, Deaton suggests a cost function which is one of the class of cost functions defined in the context of labour supply by Muellbauer [1981], having the general form (2.6) C(u,p ,p 0 ) = u.po [ a ( p ) ] 1 " 6 + b(p) .p 0 + d(p), where a(p) and d(p) are homogeneous of degree one (HD1) while b(p) is HDO. As shown by Deaton and Muellbauer [1981], an advantage of using a cost function which has th is general form is that the resu l t ing labour earnings equation, PoXrj, w i l l be l inear both in the wage rate and in income. In add i t ion , the cost function (2.6) represents preferences which are not assumed to be weakly separable between rationed and non-rationed goods. 3 As Deaton [1981] notes, an advantage of such a spec i f i ca t ion is that the rat ion has addit ional ef fects on the marginal budget shares which are not present when preferences are assumed to be weakly separable, as is the case when they are represented by the Stone-Geary u t i l i t y function 2 For spec i f i c examples showing how th is method is appl ied, see Section 3.3 . 3 For a de f in i t i on of separable preferences see, for example, Deaton and Muellbauer [1980, p. 127]. - 17 -used by Ashenfelter [1980]. In fact one useful feature of the cost function (2 .6) , which is u t i l i z e d by Blundell and Walker [1982], is that i t allows us to test whether or not preferences are separable. S p e c i f i c a l l y , a necessary and su f f i c ien t condit ion for (2.6) to represent separable preferences is that b-j(p) = 0 for a l l i , that i s , i f the der ivat ives of b(p) with respect to each pr ice p-j are zero. 1* While Deaton suggests appl icat ions of these methods to household data where some households are free to vary the i r hours of work while others are invo lun ta r i l y unemployed or work f ixed hours, the only appl icat ions he actua l ly makes are to time ser ies data, where the rat ion is on housing. However, Blundell and Walker [1982] use the cost function (2.6) to estimate f i r s t an unrationed commodity demand and labour supply system and then a rationed system in which a l l males are assumed to be rationed at the i r observed hours of work. In th is study, Blundell and Walker use 1974 household data, and make use of information on both male and female labour supply. Due to the fact that wage rates are not observed for the many females who do not par t ic ipate in the workforce, only those observations for which female labour supply is pos i t ive are included, with the resu l t ing sample s e l e c t i v i t y (SS) bias being corrected by introducing extra var iab les , as in the method explained by Heckman [1979]. In estimating the i r model, Blundell and Walker formulate the dependent var iables as budget shares in order to remove possible problems of heteroskedast ic i ty . In addi t ion, various parameters are assumed to depend on socio-economic var iables such as the age structure and s ize of k Although Deaton [1981] does not explain the der ivat ion of th is r e s t r i c t i o n , i t is based on Theorem 4 in Goldman and Uzawa [1964,p. 392]. Blundel l and Walker [1982,p. 355-356] reduce the theorem to a simple condit ion which they show y ie lds Deaton's r es t r i c t i on for the cost funct ion (2 .6 ) . - 18 -the household. Turning to the empirical r esu l t s , p lausib le parameter estimates were obtained with both the rationed and unconstrained models, while separab i l i t y of goods from le isure was rejected in both cases. Unfortunately, due to the complex nature of the dif ferences in stochast ic spec i f i ca t ion between the two hypotheses, Blundell and Walker are unable to test whether the rationed or unrationed model is more appropriate. While the studies by Ashenfelter [1980] and Blundell and Walker [1982] attempt to model the rat ioning that is present in the labour market, they are unable to take account of cer ta in key features of the rat ioning process which Ashenfelter described. S p e c i f i c a l l y , both models f a i l to handle the fact that not a l l those who are rationed are affected to the same extent. In addi t ion, in Blundell and Walker's model i t is necessary to assume ei ther that a l l households are rationed or that a l l are non-rat ioned. In both cases these problems stem from the lack of a su i tab le data set in which rat ioned and non-rationed ind iv iduals or households can be i den t i f i ed . However, su f f i c ien t information is avai lab le in the Michigan data set to determine which households are constrained in the i r labour supply behaviour. We now examine several studies which have made use of th is data and the rat ioning information i t contains. 2.3 Models which U t i l i z e Agent-Speci f ic Rationing Information While the studies by Ham [1977], [1980], [1982] and Ransom [1982] make use of sample information that iden t i f i es which indiv iduals claim to be constrained and the d i rect ion of the i r constraint (that i s , whether they are underemployed, overemployed or unemployed), they do not require information on the extent of these const ra in ts . This is an important aspect of these studies since although cross-sect ion data may be such that - 19 -constrained ind iv iduals can be iden t i f i ed (as in the Michigan data) , i t w i l l not usual ly reveal the degree to which they are constrained. Consequently i t is not in general possible to adjust observed hours worked to obtain an ind iv idua l ' s actual labour supply. Ham [1977], [1982] i den t i f i es three main approaches which have been used to estimate indiv idual labour supply functions from data which includes ind iv iduals that are, or claim to be, rat ioned. The most common approach to the problem is to ignore the fact that some indiv iduals are constrained. In i t s simplest form, th is approach comprises the spec i f i ca t ion of a l inear equation re la t ing desired hours of work, h*, to cer ta in explanatory var iab les , X-J: (2.7) h* = X i '3 + e r This equation is then estimated by OLS for a l l households, including those whose labour supply is constrained, by equating observed hours, h ° , with h*. However using observed hours, (2.7) can be expressed as (2.8a) h°. = X . ' P + e. - (h* - h<j) , that i s , (2.8b) h!j = X i ' B + u . . Now, taking par t ia l der ivat ives with respect to the j th var iable in X-j, we obtain (2.9) ah^./BXj. = &. - 3(h*r - h ° ) / 3 X j r Thus, a problem with th is approach is that the resu l t ing least squares est imates, equal to 3h l ? /3X. i , w i l l compound the effect of a var iab le on desired labour supply with i t s ef fect on unemployment or underemployment. Further, i f , for example, the rationed are l i k e l y to have low wages or unearned income, or l i t t l e education, these explanatory - 20 -var iables w i l l be correlated with the error terms, so that inconsistent estimates w i l l be obtained. A second approach is to remove a l l the constrained indiv iduals from the sample. Apart from the fact that th is wastes information, a more serious problem is that i f the rat ioning is not random, removing the constrained indiv iduals w i l l mean that we are truncating the sample on the basis of the dependent var iab le . For example, i f unemployment is voluntary le isure and occurs because an indiv idual has a low value of h*, then we w i l l only be including an observation in the sample i f h* exceeds some threshold l e v e l . Hence, unless the estimation procedure takes account of t h i s truncation on the basis of the dependent var iab le , inconsistent estimates w i l l be obtained. Another method for dealing with the fact that some indiv iduals are rationed is to adjust hours of work for the ra t ion ing . Thus, labour supply is defined as hours worked plus unemployed hours. One problem with th is method is that operat iona l ly , unemployed hours are measured by unemployment, so that underemployment is ignored. Another problem is that i f some unemployment is l e i su re , or is involuntary but is p a r t i a l l y compensated by working more at other times during the period being examined, then th is method w i l l resu l t in the addit ion of a posi t ive error to the RHS of the labour supply equation. Further, i f th is new error term is correlated with any of the explanatory var iab les , inconsistent estimates w i l l again be obtained. In his three papers, Ham develops a l ternat ive approaches for dealing with the fact that some indiv iduals are constrained in the i r labour supply decisions while others are not constrained. In his f i r s t model, Ham [1977] notes that while actual hours are equal to desired hours for an - 21 -unconstrained i nd i v i dua l , so that the appropriate labour supply function is (2 .7) , for an unemployed or underemployed i nd i v idua l , actual hours, h ° , are less than desired hours h*, so that from (2.7) , (2.10) e i > h°. - X^e . Thus, assuming e-j to be normally d is t r ibuted with zero mean and variance a 2 , the probab i l i t y that h° <^  h* is given by Pr ( (e i /a ) > Z-j), which is equal to 1 - F(Z-j), where F(. ) is the cumulative normal d is t r ibu t ion funct ion and Z i = (h° - X^B)/© . S im i l a r l y , the probab i l i t y that h? > h*, and hence that the household is overemployed, is calculated as F(Z-j). Using f ( . ) to refer to the standard normal density funct ion, the l i ke l ihood function is therefore obtained as (2.11) L = n [1 - F(Z.) ] . n F(Z.) . n ( 1 / a ) . f ( Z . ) , U 0 NR where i e u i f the indiv idual is unemployed or underemployed, i e 0 i f the ind iv idual is overemployed, and i e NR i f the indiv idual is not rat ioned. Ham [1977] estimates the parameters of L in (2.11) using data from the Michigan Survey for 1971. Those included in his sample are males who were aged between 25 and 50 in 1967, who were not part of the 1967 Michigan non-random poverty subsample, who remained in the sample for a l l 8 years (1967-1974), who had not re t i red from the labour fo rce , and who were not both overemployed and underemployed in the same year. Apart from wage and income var iab les , Ham also includes various socio-economic var iables in the labour supply equation. In order to compare his technique with ex is t ing s tudies, Ham estimates the labour supply parameters using four a l ternat ive methods. Apart from ML on the f u l l sample, he also considers OLS applied to the f u l l sample, to the subsample of non-rationed indiv iduals and f i n a l l y , to the subsample of rationed ind iv idua ls . He found that although several of - 22 -the ML coef f i c ien ts were s imi lar to those obtained with OLS on the f u l l sample, other coe f f i c ien ts — pa r t i cu la r l y those on the demographic var iables ~ varied great ly . Although the ML estimation procedure adopted by Ham [1977] uses a l l ava i lab le information and in addit ion avoids the problem of obtaining biased estimates ~ pa r t i cu la r l y of demographic var iables - - which w i l l occur i f rat ioning is ignored, one of the problems which remains with th is method is that di f ferences in tastes (incorporated in the error term) may be correlated with some of the explanatory var iab les . To deal with th is problem, Ham [1980] generalizes th is model to the case where panel data is avai lab le by redef ining the error term to include a f ixed ef fects component. The advantage of a f ixed ef fects model is that i t controls for taste di f ferences among workers. Thus, th is avoids problems such as possib ly regarding a group of ind iv iduals as constrained just because they have an above average desire for l e i su re , and hence work fewer hours. Again i t is only necessary to ident i fy which workers suffer a spel l of unemployed hours; in pa r t i cu la r , de ta i l s concerning the number of hours they are unemployed are not required. Further, i t is not necessary to make a general assumption that a l l workers wish to work the same proportion of the i r unemployed hours. Returning to the case of a s ingle c ross -sec t ion , Ham [1982] respec i f ies the model he developed e a r l i e r in an attempt to avoid the problem of inconsistent estimates which would resu l t i f cer ta in assumptions were not s a t i s f i e d . S p e c i f i c a l l y , in Ham [1977] i t was i m p l i c i t l y assumed f i r s t that those who claim to be rationed are r e a l l y constrained, and second, that for these constrained workers actual hours are exogenous; that i s , no attempt is made to explain those hours worked. - 23 -However, since observations on those workers who claim to be rationed enter the l i ke l ihood function (2.11) in a d i f ferent way to those who are unconstrained, errors w i l l resu l t i f these workers are in fact not constrained at a l l . Further, although an explanatory equation such as (2.7) is included for desired hours, hours worked by ind iv iduals claiming to be constrained are assumed to be exogenously f i x e d , and only information concerning the d i rect ion of displacement from desired hours is used. However i f these constrained hours are not r ea l l y exogenously determined, th is procedure w i l l also resul t in inconsistent est imates. 5 Ham's method is to exclude the underemployed and unemployed from the sample (overemployed workers are ignored) when estimating labour supply behaviour, and to adjust the estimation procedure to take account of the resu l t ing sample s e l e c t i v i t y (SS) b ias . Essen t ia l l y the method used involves an extension of Heckman's [1979] SS technique to the case where two correlated select ion rules generate the sample. Here b ivar ia te probit analysis is used f i r s t to estimate separate equations for the incidence of unemployment and the incidence of underemployment. Once these equations have been estimated and the coe f f i c ien ts obtained, the labour supply equation is then estimated for the unconstrained workers only, with the resu l t ing SS bias being corrected by subst i tu t ing the coe f f i c ien ts obtained in the e a r l i e r b ivar ia te probit analysis into th is equation using a version of Heckman's [1979] method. This select ion rule (SR) approach has the advantages of being simple (at least theore t i ca l l y ) to implement, and of y ie ld ing consistent 5 Ham [1982] does not explain how or why a worker who is not working his desired hours w i l l be doing so for reasons other than an exogenously imposed const ra in t . Of course one way th is problem could occur is i f the worker is not r e a l l y constrained, although th is case is already covered by the f i r s t assumption. - 24 -estimates of the parameters in the desired labour supply equation, even i f those claiming to be rationed are not r e a l l y constrained, since in th i s case the SR method estimates the covariances between the errors in the desired hours equation and those in the equations for the incidence of unemployment and of underemployment, when these covariances are in fact r e a l l y equal to zero. However i t is not c lear that consistent estimates w i l l be obtained i f only some and not a l l the ind iv iduals claiming to be rationed are not constrained, since in th is case i t would seem that these covariances could be biased away from the i r true values. In order to determine whether previous techniques have caused s ign i f i can t b iases, Ham [1982] estimates the labour supply parameters f i r s t using the SR approach and then using 2SLS on the f u l l sample and on the censored sample. The data used is the same as in Ham [1977], although the select ion c r i t e r i on concerning overemployment is omitted as these ind iv idua ls are not considered here. Examining the estimates obtained using the SR approach, Ham f inds that the coef f i c ien ts on the two addit ional terms — included in the labour supply function to correct for the SS bias which resul ts from excluding unemployed and underemployed households ~ are j o i n t l y s ign i f i can t at the 1% l e v e l . Thus, th is indicates that the labour supply parameters are biased when estimates are obtained using only non-rationed households and no SS correct ion is included. From a pairwise comparison of the SR coef f i c ien ts and the corresponding least squares coe f f i c i en t s , s ign i f i can t di f ferences are observed to occur mainly for the coe f f i c ien ts on various demographic var iables such as education, race and number of ch i ld ren . However a spec i f i ca t ion test used to compare the set of estimates obtained with the SR method to those obtained using least squares on the f u l l sample reveals - 25 -that the overal l change in coe f f i c ien ts is s i g n i f i c a n t l y d i f ferent from zero at the 1% l e v e l . 6 In contrast to Ham's studies which model male labour supply behaviour when some ind iv iduals are rat ioned, Ransom [1982] develops a household model in which both male and female labour supply are subject to rat ioning const ra in ts . While the Michigan data provides su f f i c ien t information to iden t i f y males as underemployed, overemployed or non-rationed, no rat ioning information is avai lable on females. To overcome th is d i f f i c u l t y , Ransom assumes that a l l non-working wives are underemployed, while a l l females who are working are considered to be non-rat ioned. Ransom begins his analysis by considering the constrained u t i l i t y maximization problem that households face. However instead of der iv ing the f i r s t order condit ions and solving for the labour supply or le isure demand equations, Ransom notes (pl6) that u t i l i t y functions commonly used in empir ical work lead to funct ional forms for the f i r s t order condit ions that are very d i f f i c u l t to work wi th. To avoid problems of th is type, Ransom determines the economic var iables that would appear in the two f i r s t order condit ions ~ labour suppl ies , wage rates and household income — and spec i f ies that the f i r s t order condit ions are l inear in these va r iab les , some addit ional taste (demographic) var iables and an addi t ive random disturbance term. Thus unl ike the models of Ham examined prev ious ly , Ransom appends the disturbances to the f i r s t order condit ions rather than to the labour supply functions which can be derived from these f i r s t order condi t ions. Consequently, information concerning the rat ion ing status of the indiv idual is also used in conjunction with the f i r s t order condit ions rather than with the labour supply funct ions. 6 For a more detai led explanation of th is t es t , see Section 4 .7 .2 . - 26 -In order to derive the l i ke l ihood funct ion , which comprises the product of the l ike l ihoods for each of the s ix rat ioning cases he considers, Ransom analyzes the f i r s t order condit ions which apply for rationed and non-rationed ind iv idua ls . He notes that for a non-rationed i nd i v i dua l , the f i r s t order condit ion requires that the der ivat ive of the ind i rect u t i l i t y function with respect to that i nd iv idua l ' s labour supply, 9V/9h-j = f-j, must equal ze ro . 7 However when the labour supply of an indiv idual is rat ioned, an increase in the i r labour supply w i l l increase u t i l i t y , so that the f i r s t order condit ion becomes f-j > 0. Recal l ing that f-j is defined by Ransom to be the sum of g-j(.) and a random error term e-j, where g-j(.) is a l inear function of the economic and demographic var iab les , we see that f-j > 0 implies that e-j > -g- j( . ) , while f-j = 0 implies g-j(.) + e-j = 0. Thus, assuming the two random error terms e m and ef are j o i n t l y normally d i s t r i bu ted , with the i r density function denoted by f ( e m , e f ) , the l ike l ihood for the case where both ind iv iduals are rat ioned, is obtained by integrat ing th is density over the unobserved range for each var iab le . Hence, oo 0 0 (2.12) L l - J / f ( e m , e f ) . d e f . d e m - g m ( . ) - g f ( . ) Simi la r l y the l i ke l ihood functions when only one indiv idual is constrained w i l l involve a single i n teg ra l , while in the case where both ind iv iduals are non-rat ioned, the l i ke l ihood is simply the b ivar ia te normal densi ty . ' Here Ransom refers to the ind i rect u t i l i t y function as being the function obtained when goods consumption is substi tuted out of the d i rec t u t i l i t y function using the budget const ra in t . Thus, V = U(h ,h^,w h +w.rh^+Y), where h m and hf are the male and female v m' f m m f f ' ' 1,1 1 labour suppl ies , % and wf are the i r respect ive wage rates and Y is unearned income. - 27 -The overal l l i ke l ihood function is then obtained as the product of these various l ike l ihoods for the six rat ioning cases. Since Ransom does not exclude households in which the female is not working from his sample, no correct ion is required to take account of the sample s e l e c t i v i t y bias which, as shown by Wales and Woodland [1980], such a procedure would introduce. However, Ransom does not explain how the required wage rate information is obtained for those females who did not par t ic ipate in the workforce. In his paper Ransom presents some prel iminary resu l ts obtained by estimating his model using the Michigan data. He f inds that the main di f ference between these resu l ts and those obtained when estimation is performed on the unconstrained subsample concerns the wi fe 's wage response, although coef f i c ien ts on some of the demographic var iables also d i f f e r . He concludes that the interact ion of constraints and labour supply does not seem to be very important. - 28 -CHAPTER THREE A SINGLE-RATION MODEL OF HOUSEHOLD LABOUR SUPPLY 3.1 Introduction While the various labour supply models developed and estimated by Ham take account of the fact that some indiv iduals in the sample are rationed in the amount of labour they can supply, a number of features of these models severely l im i t the i r usefulness. F i r s t , they deal exc lus ive ly with males, and as such provide no information about the ef fects that the rat ioning of the labour supply of one member of a household has on the labour supplies of other members of the household, or on household consumption behaviour. Although the study by Blundell and Walker [1982] avoids th is l im i t a t i on , these authors assume ei ther that a l l households are non-rationed or that the male in every household is rationed at his observed hours of labour supply. Although th is l im i ta t ion could be overcome by an appropriate respec i f i ca t ion and re-est imation of Ham's models using data on households rather than ind iv idua ls , the method of estimation used in Ham [1982] imposes addit ional res t r i c t i ons on the resul ts that can be obtained. As explained in the previous chapter, th is model is estimated using an extension of Heckman's [1979] Sample Se lec t i v i t y (SS) correct ion technique in which rationed households are only used in the f i r s t stage of the estimation procedure. Here equations for the incidence of unemployment and underemployment are estimated by b ivar ia te probit analysis using these rationed households. Then the coef f i c ien ts obtained from th is analysis are - 29 -used in the second stage, where the labour supply function is estimated for the subsample of non-rationed households. Consequently, even i f household data is used, since th is method does not y ie ld separate estimates of the labour supply functions for rationed and for non-rationed households, i t is again impossible to evaluate the ef fects that rat ioning of the labour supply of one household member has on the labour supply behaviour of other (non-rationed) household members, and on household consumption behaviour. A further l im i ta t ion of Ham's [1977] and [1982] models concerns the spec i f i ca t ion of the funct ional forms which are estimated. S p e c i f i c a l l y , in these models i t is assumed that the labour supply function is l i nea r . While the main reason for choosing a l inear function is to ease the computational burden, several other funct ional forms derived from a u t i l i ty -maximiz ing or cost-minimizing framework would be less ad hoc in nature and in many cases may not unduly complicate the computations. In fact th is problem of funct ional spec i f i ca t ion also ar ises with the probi t equations in Ham's [1982] model, since with the SS correct ion technique i t is necessary to specify equations for the incidence of unemployment and underemployment. Since these equations are also speci f ied on an ad hoc bas is , the i r funct ional form is a rb i t ra ry , and Ham again simply adopts l inear spec i f i ca t i ons . The household labour supply model developed by Ransom [1982], which i s estimated using Maximum Likel ihood (ML) methods, avoids most of these l im i ta t i ons . Since the l i ke l ihood function comprises the product of the density functions for rationed and non-rationed households, Ransom is able to estimate labour supply functions for non-rationed indiv iduals both in non-rationed households and in households where the other household member - 30 -i s rat ioned. Consequently the labour supply behaviour of non-rationed ind iv iduals can be compared in the two cases. In addi t ion, Ransom calculates cross-rat ion ing ef fects which measure the ef fect on the non-rationed indiv idual of a change in the rat ion level of the constrained member of the household. While Ransom does not specify l inear labour supply equations, his model is nonetheless also ad hoc in nature. S p e c i f i c a l l y , as explained in deta i l in the previous chapter, Ransom determines the variables that would appear in the f i r s t order condit ions of the u t i l i t y maximization problem and simply spec i f ies that the f i r s t order condit ions are l inear in these var iables and a random error term. Accordingly, the f i r s t order condi t ion, and hence the st ructura l form labour supply funct ion, has the same funct ional form for a non-rationed indiv idual in a household regardless of whether or not the other member of the household is ra t i oned . 1 In add i t ion , due to the manner in which the f i r s t order condit ions are a r b i t r a r i l y spec i f i ed , the labour supply equations (and hence the corresponding le isure demand equations) derived from these f i r s t order condit ions are not homogeneous of degree zero in pr ices and income. F i n a l l y , one other important aspect of Ransom's analysis which may severely l im i t the interpretat ion of his resul ts is that he defines a female as underemployed ~ that i s , desi r ing more work ~ i f she did no work at a l l during the survey year. In contrast to the ad hoc models of Ham and Ransom, the model we construct here is e x p l i c i t l y derived from the assumption that households, 1 Note however that the reduced form labour supply functions for th is non-rationed indiv idual would d i f f e r in the two cases, since the s t ructura l form equations depend on the labour supply of the other indiv idual in the household, and th is var iable is exogenous or endogenous according to whether or not th is other indiv idual is rationed or non-rat ioned. - 31 -based on the i r preferences, perform an optimizat ion procedure to determine the i r desired quant i t ies of le isure (both male and female), and the i r desired consumption of a composite commodity. Several a l ternat ive representations of household preferences are spec i f i ed , and in each case we derive the corresponding commodity demand and le isure demand equations (which in a l l cases w i l l be homogeneous of degree zero in pr ices and income), both for the case where the household head is rationed and the case where he is not rat ioned. In order to compare the labour supply behaviour of the non-rationed female when the male is rationed to her behaviour when he is non-rat ioned, and to ca lcu la te the cross-rat ion ing e f fec t s , we j o i n t l y estimate the systems of rationed and non-rationed equations using the ML technique. However our l i ke l ihood function d i f fe rs quite s i g n i f i c a n t l y from the function maximized by Ransom. F i r s t , our l i ke l ihood only involves the product of three densi t ies ~ for households in which the male is underemployed, for households in which he is overemployed and for households in which he is non-rationed ~ since we do not consider his wife to be rationed simply because she is not working. However, since we require the wage rates of a l l females, we exclude households from the sample i f the wife is not working, and amend the l i ke l ihood function to correct for the SS bias that would otherwise resu l t . Second, our l i ke l ihood function d i f fe rs because we use the rat ioning status of the household to determine the re la t ionsh ip between actual and desired hours worked by the male, rather than whether the f i r s t order condit ion holds with equal i ty or inequal i ty . In a sense th is is a more i n t u i t i v e l y appealing approach, since when male le isure is rat ioned, the household - 32 -cannot maximize with respect to male l e i su re , and thus has no e x p l i c i t f i r s t order condit ion for th is va r iab le . The plan of the remainder of th is chapter is as fo l lows. In the next section we speci fy the household model and derive the l i ke l ihood funct ion for the case where le isure and commodity demands are obtained from a d i rect u t i l i t y funct ion. Although the Neary and Roberts [1980] procedure, explained in the previous chapter, can be used to obtain rationed demand equations regardless of the representation of household preferences from which the non-rationed demand equations are der ived, to s impl i fy the exposit ion we use a conventional constrained u t i l i t y maximization approach to derive both the rationed and non-rationed equations in the general case. However in Section 3.3 where we consider spec i f i c funct ional form representations of household preferences, including funct ional forms for the d i rect u t i l i t y funct ion, the ind i rect u t i l i t y funct ion and the cost func t ion , the Neary and Roberts method is used to derive the demand equations for rationed households. 3.2 Model Spec i f i ca t ion In th is section we derive the s ing le - ra t ion model of labour supply which is to be estimated using a sample which includes underemployed, overemployed and non-rationed households. We begin by speci fy ing the constrained u t i l i t y maximization problems in these cases and then consider the appropriate stochast ic spec i f i ca t ion for the resu l t ing demand equations. Next we derive the l i ke l ihood function which here consists of the product of the appropriate densi t ies for each rat ioning case. F i n a l l y , in Section 3.2.4 we speci fy the sample s e l e c t i v i t y correct ion which we include in the model to take account of the fact that since we require - 33 -wage information for both males and females, our sample only includes households in which the wife works outside the home. 3.2.1 Formulation of the Basic Model We assume that the household maximizes a u t i l i t y function (3.1) U = U ( L m , L f , X ) , where L m and Lf are the le isure of the male and female respect ive ly and X is a Hicksian composite good representing consumption of a l l other commodities. In i t s maximization of u t i l i t y , the household is faced with the budget constraint (3.2a) Y + (T - L j . p m + (T - - U ) .p- = p„X , v ' x m m' rm x f f r f r c o r , (3.2b) M = Y + p m T m + p f T f = PCX + P m L m + P f L f , where M is f u l l income, Y is unearned income, T m and Tf are the to ta l hours avai lab le to the male and female respec t ive ly , Pm and pf are t he i r respective wage rates and p c i s the pr ice of the consumption good, which is set equal to unity for a l l households. Performing th is constrained maximization, we obtain expressions for the desired quant i t ies of L m , Lf and X as functions of the wage rates and f u l l income: < 3 ' 3 a ) Lm" = L m(Pm'Pf ' M ) (3.3b) L* = L f ( p m , p f , M ) (3.3c) X* = X ( p m , p f , M ) . Now consider the case where the le isure of the male is rationed at r m . Denoting the quant i t ies of female le isure and of goods consumption when the male is rationed by L r and X r respect ive ly , the problem now facing the household is to maximize (3.4) U* = U * ( L r , X r ; I m ) , - 34 -subject to the budget constraint (3.5) M = p X + p i L + p-L . x ' r c r rtn m rf r Performing th is maximization we thus obtain expressions for desired female le isure and desired consumption when the male is rat ioned: (3.6a) L* = L r ( p m , P f , M , L m ) (3.6b) X* = X r ( P r n , p f , M , L m ) . 3.2.2 Stochastic Spec i f i ca t ion From the budget constraint (3.2b), we observe that there is a l inear dependence among the quant i t ies L m , Lf and X. Hence, once we determine two of these quant i t ies , the th i rd quantity can be obtained using the budget const ra in t . Consequently, i f we append error terms to the equations for the three unrationed quant i t ies , the vector of these error terms w i l l also be l i nea r l y dependent, and the resu l t ing variance-covariance matrix w i l l be s ingu lar . The solut ion to th is problem is to delete one equation pr ior to formulating the l i ke l ihood funct ion. Moreover, as shown by Barten [1969], the choice of which equation to omit is arb i t rary since we lose no information concerning the demand behaviour of the good which has i t s equation deleted from the ana lys is . Hence, since we are pr imar i ly interested in the labour supply behaviour of the household members, we drop the equation for goods consumption, (3 .3c) . S im i la r l y in the rationed case, L r and X r are also l i nea r l y dependent, as can be seen from the budget constraint (3 .5) . Thus in th is case we again drop the equation for goods consumption, (3.6b). Before appending error terms to the remaining equations, (3.3a), (3.3b) and (3.6a), in order to reduce heteroskedast ic i ty problems we f i r s t express these equations in share form by mul t ip ly ing both sides by the - 35 -relevant pr ice var iable and then d iv id ing by f u l l income. Thus the three equations to be estimated are: (3.7a) p m L*/M = p m L m (p m ,p . ,M) /M + e m (3.7b) p f L* /M = p f L f ( p m , p f , M ) / M + e f (3.7c) p f L* /M = p f L r ( p m , p f , M , L m ) / M + e r , where the disturbance terms em, ef and e r represent random errors of opt imizat ion. Although separate error structures for the rationed and unrationed systems could be spec i f i ed , and each system estimated separately using appropriate data, our aim here is to j o i n t l y estimate the parameters of the u t i l i t y function when some households are rationed and others are not constrained. Hence we assume the 3x1 vector of disturbances (e ,e.c,e ) to m T r be normally d is t r ibuted with zero mean and covariance matrix z , where r * 2 m CTmr (3.8) S = mf •? a f r a mr f r a2 r Thus the various disturbances for a par t i cu la r indiv idual are allowed to be cor re la ted, although the disturbances for d i f ferent ind iv iduals are assumed to be independent. 3.2.3 Derivation of the Likel ihood Function Since our sample consists of some households in which the male is rat ioned and some where he is not ra t ioned, in order to derive the l i ke l ihood function we need to consider each type of household separately. Begining with non-rationed households (NR), we use a theorem from Anderson [1958] to specify the appropriate probab i l i t y densi ty. This - 36 -theorem (#2.4.3, p24) states that i f a vector of var iab les , X, is normally d is t r ibuted with mean vector y and covariance matrix E, the marginal d i s t r i bu t i on of any set of components of X is mul t ivar iate normal with means, variances and covariances obtained by taking the proper components of U and £ respect ive ly . Thus for unrationed households when we only consider the two equations (3.7a) and (3.7b), the error terms on these equations, e m and e-f, are j o i n t l y normally d is t r ibuted with zero means and covariance matrix where (3.9) r o 2 a : m mf a j. a 2, mf f Hence the density for the unrationed households, g i (e m ,e - f ) , is simply a b ivar ia te normal density function with zero means and covariance matrix To determine the relevant terms in the l i ke l ihood function for households in which the male is rationed,-we need to consider two separate cases. The f i r s t case is where the male is underemployed, (U), that i s , where he is constrained to work fewer hours than he des i res , so that his actual l e i su re , L ° , exceeds his desired l e i su re , L*. Hence P m L m /M exceeds p L*/M, so from (3.7a) we see that mm (3.10) e m < Z i , where (3.11) Z l = pm(L° - L m ( . ) ) / M . To obtain the l i ke l ihood for households when the male is underemployed, we note that although the constrained le isure demand of the male is observed - 37 -in th is s i t ua t i on , the only information avai lable on e m is the inequal i ty (3.10). Thus denoting the jo in t density of males and females when the male is rationed by g2(em>er)» t o obtain the l i ke l ihood we need to integrate out e m over i t s unobserved range. Thus the density when the male is underemployed is given by Zi (3.12) g„ = / g 2 ( e m,ej dem . u m' r m — 0 0 Expressing the jo in t density g 2 as the product of a marginal and a condi t ional densi ty, we have therefore Zi (3.13) g u = h ! ( e r ) . / M e k ) dem . Now since e m and e r are j o i n t l y normally d i s t r i bu ted , the marginal density h i ( e r ) is just the normal density for a var iable with mean zero and variance a 2 . Thus, (3.14) Me r) = ( l / o r ) . f ( P f ( L ° - L r ( . ) ) / M / o r ) , where f ( . ) is the standard normal density funct ion. From Mood and Graybi l l [1974,p202], we note that i f two random var iables x and y are j o i n t l y normally d is t r ibuted with means u and u , variances a 2 and a2 and x y x y cor re la t ion p = a / ( a .a ), then the condi t ional d i s t r i bu t ion of x given xy x y y is also normal, with the spec i f i ca t i on : (3.15) (x|y) — N(y x + p ( a x / a y ) ( y - p y ) , a 2 ( l - p 2 ) ) . Thus in th is case, h 2 ( e m | e r ) is the density function for a normally d is t r ibuted var iable z with mean y_, and variance a 2 , where (3.16a) u z = P m r . ( a m / a r ) . p f ( L ° - L r ( . ) ) / M , (3.16b) a 2 = a 2 .(1 - p 2 ) , v ' z m v mr' ' (3.16c) P = a / ( a .a ) . w ' mr mr v m r' Converting th is to a standard normal densi ty, denoted f ( . ) , we have - 38 -(3.17) h 2 (z ) = ( l / a z ) . f ( ( z - M z ) / a 2 ) . Thus subst i tu t ing in (3.13) and changing the var iable of in tegrat ion, we obtain z* la (3.18) g u = Me ). / Z f(e) de . 0 0 = (l/«r).f(pf(Lj - L r ( . ) ) / M / a r ) . F ( z * / o - z ) , where (3.19) z* = Zi - y z , and F( . ) is the standard normal d i s t r i bu t ion (cumulative density) funct ion. Note also that the probab i l i t y that a male w i l l be underemployed i s , using (3.10), given by (3.20) P r (e m < Z x ) = F ( Z i / c m ) . Turning now to the overemployed (0) , who are constrained to work more than the i r desired hours and hence consume less than thei r desired amount of l e i su re , we see from (3.7a) that (3.21) L°< I* <==> pmL°/M < P m l * / M <==> e m > lx . v ' m m "mm rm m m 1 Thus, fo l lowing the same procedure as in the underemployed case and integrat ing out e m over the unobserved range defined in (3.21), the density for the overemployed case is obtained as (3.22) g 0 - ( l / o r ) . f ( p f ( L j - L r ( . ) ) / M / a r ) . F ( - z * / o z ) , with the probab i l i t y of being overemployed given by (3.23) Pr (e m > Zi) = F ( - Z i / o m ) . Note that i f e ,^ and e r are independent, so that h 2 ( e m | e r ) = h 1 ( e m ) , the cumulative d i s t r i bu t ion functions in (3.18) and (3.22) would be replaced by those in (3.20) and (3.23) respect ive ly . Now combining the three groups, the overal l l i ke l ihood function is given by - 39 -(3.24) L i = n g i ( e e f ) . n g n g 0 , NR m T U u 0 that i s , Li = n g i ( e m , e f ) . n (1/a ) . f ( p f ( L ° - L ( . ) ) /M/o ).n F ( z * / a ).n F ( - z * / a ) NR m T U + 0 r U z 0 where g i ( e m , e f ) is the jo in t normal density function with mean zero and covariance matrix given by Z' defined in (3 .9) , f ( . ) and F( . ) are the standard normal density and d is t r i bu t ion functions respect ive ly , and z* and az are defined in equations (3.19), (3.16a) and (3.16b). One important observation concerning the l i ke l ihood function (3.24) is that ofr, an element of the variance-covariance matrix £ defined in (3.8) does not appear. Consequently th is covariance is not iden t i f ied and cannot be estimated. 3.2.4 Sample S e l e c t i v i t y Correction Since we only consider households in which the female is working, due to the fact that wage rates are not avai lable for women who are not working, we need to take account of sample s e l e c t i v i t y , as demonstrated by Wales and Woodland [1980]. Here the appropriate l i ke l ihood is obtained by d iv id ing the l i ke l ihood Li in (3.24) by the probab i l i t y that the wife is working. Since the wife is never rat ioned, for households where the male is also not ra t ioned, the wife works i f l_£ < T^, that i s , i f p f L^/M < p f T f / M . From (3.7b), th is inequal i ty implies that < (p^T^/M - p^L^(.) /M), so that the probab i l i t y that the wife works is given by (3.25) P r ( e f < p f ( T f - L f ( . ) ) /M) = F ( p f ( T f - L f (.))/M/o f). I f the male is ra t ioned, the wife works i f L* < T f , that i s , i f P f L * / M < p f T f / M . Using (3 .7c) , th is inequal i ty implies - 40 -e r < (p fT^/M - p f L r ( . ) / M ) , so that the probab i l i t y that the wife works in the rationed cases is given by (3.26) P r ( e r < p f ( T f - L p ( . ) ) /M) = F ( p f ( T f - L r ( . ))/M/o r). Hence the corrected l i ke l ihood function which takes account of the sample s e l e c t i v i t y is defined as (3.27) L 2 = L i / n F ( p f ( T f - L f ( . ) ) / M / a )/ n F ( p f ( T f - L ( .)) /M/a ) NR T U+0 T T r r where Li is the or ig ina l l i ke l ihood funct ion, (3.24). 3.3 Al ternat ive Representations of Household Preferences In order to perform empirical ana lys is , i t is necessary to speci fy funct ional forms for L ( . ) , L j r ( . ) and L ( . ) . In th is section we consider m f r several a l ternat ive representations of household preferences, and in each case derive the expenditure or share equations for non-rationed households and for households in which the amount of labour which the head can supply is ra t ioned. In addi t ion, we also examine various problems which may ar ise with use of each par t i cu la r funct ional form. While i t is also important to incorporate demographic var iables in the analysis in order to allow households with d i f ferent demographic charac te r i s t i cs to react d i f f e ren t l y to spec i f i c labour force s t i m u l i , such as rat ioning of the household head, we defer consideration of th is issue un t i l the next chapter. To s impl i fy the expos i t ion, we now adopt a s l i g h t l y d i f ferent D notat ion, re fer r ing to the le isure demands L m , L f and l_r as X m , X f and X f , D and the consumption demands X and X r as X c and X c . - 41 -3.3.1 Klein-Rubin or Stone-Geary U t i l i t y Function The Klein-Rubin [1947-48] or Stone-Geary (Stone [1954], Geary [1950]) u t i l i t y function can be written as (3.28) U = £3i ln(Xi - Y i ) Maximization of (3.28) subject to the budget constraint (3.2b) y ie lds the equations of the Linear Expenditure System (LES): (3.29) p i X i = p ^ + ^ ( M - Z p.y.) , i = M,F,C, where s3 i = 1. In order to express these equations in share form, we div ide both sides by f u l l income, M. Rather than redoing the constrained maximization problem with male le isure held f ixed at i t s rationed level X^, expenditure equations for female le isure and goods consumption when the male is rationed are obtained using the method of Neary and Roberts [1980], explained in the previous chapter. Here we take the expenditure equation (3.29) for male l e i s u r e , replace Pm by the v i r tua l wage Pm> f u l l income M by M + (p" - p ) .X R and the quant i t ies X„ , X^ and X„ by the i r rationed v r m nm' m ^ m' f c J counterparts, XJ*, X^ and X^. Rearranging, we obtain an e x p l i c i t so lut ion for the v i r tua l wage, Pm: ( 3 . 3 0 ) p m . ( e m / ( ( l - e m ) ( x £ - r m ) ) ) . ( M - £ p j T j - P | n x £ ) . Now to obtain the rationed expenditure equations, we make the same subst i tu t ions in the unrationed expenditure equations (3.29) for female le isure and goods consumption as were out l ined above for the male le isure equation, and in addit ion subst i tute the expression (3.30) for the v i r tua l wage. In th is way the rationed expenditure equations are obtained as (3.31) p/.= P i T i M 3 i / ( l - 3 m ) ) . ( M - ^ p . Y j - p m x R ) , i=F,C Again we divide both sides of these equations by M to convert them to - 42 -share form. As indicated in the previous sec t ion , the share equations for goods consumption in both the rationed and non-rationed systems are dropped in est imat ion, and normally d is t r ibuted error terms are appended to the RHS of the remaining equations. 3.3.2 Constant E l a s t i c i t y of Subst i tut ion (CES) U t i l i t y Function Replacing factor quant i t ies in the CES production function (Arrow et_ al [1961]) by supernumerary consumption, (X-j - y-j), we obtain an expression for what may be termed the CES u t i l i t y function (Wales [1971]), (3.32) U = [SB. (X i - y . ) - p y l h . Maximization of th is u t i l i t y function subject to the budget constraint (3.2b) y ie lds the unrationed expenditure equations (3.33) p.X. = p.y. + (B° p^- a/Z3?pl- a).(M - Z p j Y j ) , i = M,F,C where a , the e l a s t i c i t y of subst i tu t ion is defined as a = l / ( l+p) . Using the Neary and Roberts method, expenditure equations when the male is rationed are now obtained as (3.34) p . X * = p . Y . + (B° p i ' 0 / I B ° p ^ 0 ) . ( M - Z p r - p X * ) , i = F,C i i j^m j*m J J Again, expenditure equations for goods consumption are dropped in estimation and error terms are appended to the remaining equations which are f i r s t expressed in share form by d iv id ing both sides of each equation by f u l l income, M. 3.3.3 Deaton-Muellbauer Cost Function The next spec i f i ca t ion we consider is the cost function (2.6) suggested by Deaton [1981] on the basis of the study by Muellbauer [1981]. Here we use the par t icu lar form of th is funct ion, adopted by - 43 -Blundel l and Walker [1982], in which both male and female wage rates appear separately: (3.35) C (u ,p ,p f , p m ) = u . p ^ p ^ [ a ( p ) ] 1 - 6 f - 6 " i + b f (p)p f + bm(p)pm+ d(p) where a(p) and d(p) are HD1 while bm(p) and bf(p) are HDO. Ignoring the demographic var iables considered by Blundell and Walker, these three functions are speci f ied as (3.36a) a(p) = n p£k , z*k = 1 (3.36b) b.(p) = pjjik , S3 i k = 0, i = M,F, k (3.36c) d(p) = EY k P k . However, since we only consider one composite good in th is study, these expressions s impl i fy as fo l lows: (3.37a) a(p) = p c , (3.37b) bi(p) = Yi , i = M,F, (3.37c) d(p) = Y C P C . Subst i tut ing these expressions in the cost function (3.35), we obtain (3.38) C ( u , p c , P f , p m ) = u . p ^ p > J - 6 f - 6 " + Y f P f + V m + V c . Now we obtain unrationed expenditure equations using Shepard's lemma by d i f f e ren t i a t i ng (3.38) with respect to p-j and subst i tu t ing for u using M = C(u ,p c ,p f .Pm) . Note however that when we fo l low th is procedure we obtain the same unrationed expenditure equations as were obtained using the Stone-Geary u t i l i t y function in Section 3 .3 .1 . Thus when only one consumption good is considered, the cost function used by Blundell and Walker y ie lds the LES equations (3.29) in the unrationed case and (3.31) in the rationed case, so that th is spec i f i ca t ion is observat ional ly equivalent to the Stone-Geary spec i f i ca t ion considered e a r l i e r . - 44 -3.3.4 Indirect Translog U t i l i t y Function We now consider the more general f l e x i b l e funct ional forms for the u t i l i t y function which have the property that unlike the LES and CES forms, they do not assume that the e l a s t i c i t i e s of subst i tu t ion between the commodities are constant. The f i r s t form we examine is the Indirect Translog u t i l i t y func t ion , the logarithm of which Christensen, Jorgenson and Lau [1975] define to be (3.39) In V = a 0 + So . l n p* + (1/2) E Z B ^ l n p*. ln p*. , where p* = p^/M, i = M,F,C are normalized p r i ces . Budget share equations for each commodity are obtained using a logarithmic form of Roy's i d e n t i t y : 2 (3.40) p.X./M = (31n V/ain p*) / [s (aln V/3ln p^)] . j J Applying (3.40) to the Indirect Translog (3.39), we obtain (3.41) p.X./M = ( a . + ZB- j ln P j ) / [ Z « k + E E \ j l n P j L i=M,F,C J To obtain the share equations for female le isure and goods consumption when the male is rat ioned, we again u t i l i z e the method of Neary and Roberts. However, in th is case no e x p l i c i t solut ion can be found for the v i r t ua l wage. Rather, Pm i s i m p l i c i t l y defined by (3.42) ? m X m - = [M + (p m - p m)xJ5].NUMl / DEN1 , where: N U M 1 = "m +.l V n P J + 3 mm l n ^m " ^ ^ + ( V VnK^ J i l l J DEN1 = Zak + z Z 3 k . l n P j + z p ^ l n P m - ( ^ k j ) .ln[M + (pm - p j X * ] , K j^rn - K so that < 3 ' 4 3> Pm = Pm(Pm'Pf'Pc'XS>M ' e)' 2 See, for example, Diewert [1982]. - 45 -where 9 is the vector of parameters. Since an explicit solution cannot be found for the virtual wage, we are consequently unable to obtain explicit solutions for the share equations for female leisure and goods consumption when the male is rationed. Implicitly, they are defined in the usual way by following the Neary and Roberts method. Thus, (3.44) p.X^ /M = [M + (p m- pm)X*].NUM2 / (M.DEN1), i=F,C where: NUM2 = a. + Z 8 in p. + 0. In p m - (£B,.).ln[M + (p - p m)X?], i .. IJ r j im m • i j m rm' mJ' J*m J j DEN1 is defined as in (3.42) and Pm 1 s t 0 D e substituted out using the solution obtained in (3.43). To estimate the model, we again drop the rationed and unrationed goods consumption equations, and append joint l y normally distributed errors to the remaining three share equations. We impose the symmetry condition that B-jj = Bji for all i * j and in addition, since the share equations are homogeneous of degree zero in the parameters, an arbitrary normalization is imposed by assuming that the parameters sum to unity; that i s , (3.45) Z a k + EEB k j = 1. To obtain estimates of the errors on the rationed share equations, i t is necessary f i r s t to compute a value for the virtual wage, Pm- This value is obtained from equation (3.42) using a search procedure in which, given values of the parameters and of the variables other than Pm> alternative values of ~pm are tried until one is found that satisfies the equation. However, experimental results obtained when (3.42) is evaluated for a grid of ~pm values, using the actual data points and various values of the parameters, reveal that while there are two solutions for ~pm for some observations, in other cases there are no solutions. In addition to - 46 -the theoret ica l problems th is presents, as discussed below, i t also presents problems of empirical implementation. S p e c i f i c a l l y , in order to allow the routines searching for the optimum parameter values to proceed, i t is necessary to decide which v i r tua l wage value to select when there are two so lu t ions , and at what value to set the v i r tua l wage when there is no so lu t ion . While i t is c lear that obtaining two v i r tua l wage solut ions is not consistent with a l inear le isure demand funct ion, i t is consistent with a le isure demand function which bends backwards and gives r i se to the backward bending labour supply function which has been reported in several I I Ld empir ical labour supply s tud ies. This s i tuat ion is i l l u s t ra ted in Figure 1. Here Lrj is the desired le isure demand funct ion, and the male is rat ioned at l_R hours of l e i su re . The v i r tua l wage, which is the wage at which the indiv idual just wishes to consume his rationed amount of le isure i s e i ther wi or w 2 . With a l inear le isure demand funct ion, the only v i r tua l wage solut ion would of course be w i . Examining Figure 1 in more d e t a i l , i t is c lear that with the rat ion at LJjj, the indiv idual w i l l be overemployed (and thus want more le isure See, for example, the study by Hall [1973] - 47 -than he is current ly consuming) i f w > w 2 , or w < w i . Conversely, the indiv idual w i l l be underemployed i f wx < w < w 2 . Thus i f an indiv idual i s overemployed, e i ther both v i r tua l wages should exceed his current wage, or both should l i e below i t . I n tu i t i ve l y , i t would thus seem that the v i r t ua l wage closest to his actual wage should be the one se lected. If an indiv idual is underemployed, his current wage should be bracketed by the v i r t ua l wages, so that here choosing the v i r tua l wage c losest to th is actual wage is also i n t u i t i v e l y appealing. However, much of th is i n tu i t i ve appeal is lost on empirical implementation when the current wage for some of the overemployed workers is bracketed by the i r v i r tua l wages, while both v i r tua l wages l i e above or both l i e below the actual wage for some ind iv iduals claiming to be underemployed. One possible explanation for th is resu l t l i e s with the data. Since ind iv iduals were only asked i f they were underemployed or overemployed, but not how much they were constrained or for any evidence of these const ra in ts , i t is possible that they r ea l l y were not rat ioned. While th is may be true in some cases, i t is also possible that the resu l ts obtained are not inconsistent with the above theoret ica l ana lys is : unt i l the optimum values of the parameters are obtained, there is no reason for the le isure demand functions to be well-behaved, as in Figure 1 or any other par t icu lar conf igurat ion. Due to these problems when two v i r tua l wage solut ions are obtained and s imi lar problems when there is no v i r tua l wage so lu t ion , i t i s desirable to f ind an a l ternat ive spec i f i ca t ion which, while s t i l l providing a f l e x i b l e funct ional form, w i l l permit problems of the type out l ined here, both theoret ica l and empi r i ca l , to be avoided. One such funct ional spec i f i ca t ion is the Direct Translog u t i l i t y funct ion. - 48 -3.3.5 Direct Translog U t i l i t y Function With the Direct Translog u t i l i t y funct ion, shares depend on quant i t ies consumed rather than p r i ces , and as a r esu l t , th is function allows an e x p l i c i t solut ion to be obtained for the rationed system of equations. Christensen, Oorgenson and Lau [1975] define the negative of the logarithm of the Direct Translog u t i l i t y function to be (3.46) -In U = cx0 + Zo . l n X. + ( l / Z j Z S p ^ l n X . . l n X^ . Maximization of (3.46) subject to the budget constraint (3.2b), and rearrangement of the resu l t ing expression, y ie lds the budget share equations: (3.47) p .X^M = (a. + se.jln + E E & k j 1 n Xj]> i=M,F,C Now fol lowing the method of Neary and Roberts, we obtain an e x p l i c i t so lu t ion for the v i r tua l wage Pm: < 3 - 4 8 ) Pm = ^ " P m ^ - ( a m + S f J m j l n X j ) 1 <Xm-D E N 2> • J where: DEN2 = Z o. + Z S3. . In X^ . k*m K k*m j K J J The fact that an e x p l i c i t solut ion can be found for the v i r tua l wage when using the Direct Translog but not the Indirect Translog u t i l i t y funct ion is due to the form in which prices and f u l l income enter the share equations. While these var iables occur in the i r o r ig ina l units in (3.47), they occur in logarithmic form in the share equations (3.41) derived from the Indirect Translog, making manipulations of the type required by the Neary and Roberts method extremely d i f f i c u l t . Continuing with the Neary and Roberts procedure, share equations for female le isure and consumption of other goods when the male is rationed are obtained as - 49 -(3.49) p-X^/M = (M - P m X | J ) . ( a . + U..}n X*) / ( M.DEN2), i=F,C J Again, in order to estimate the model, we drop the goods consumption equation in both the rationed and non-rationed cases. One problem with using the Direct rather than the Indirect Translog u t i l i t y function is that the endogenous var iables X-j and xB appear on both sides of the i r respective equations. Thus the Jacobian, which is the absolute value of the determinant of the matrix of der ivat ives of the disturbances on the remaining share equations with respect to the endogenous var iab les , is no longer equal to unity or a constant. Rather, i t w i l l be a function of a l l the parameters of the system of share equations. Thus, the log l i ke l ihood function we maximize here comprises the o r ig ina l log l i ke l ihood function with the Jacobian terms appended. S p e c i f i c a l l y , we now maximize (3.50) In L 3 = In L 2 + E l n (abs l j ^ i ) + z l n ( a b s j j . l ) , U+0 NR 1 1 1 where In L 2 is the logarithm of the l i ke l ihood defined in (3.27) and jB and J-j are the matrices of der ivat ives of the disturbances with respect to the endogenous var iables in the rationed and non-rationed systems of share equations, r e s p e c t i v e l y . k In estimating th is model we impose the symmetry condit ions which have been used throughout the above ana lys is , namely that 3-jj = flj-j for a l l As is the case with the Indirect Translog, the share equations in (3.47) and (3.49) are homogeneous of k Note that in order to ca lcu la te these Jacobians i t is f i r s t necessary to subst i tute in the remaining equations for the endogenous var iable which has i t s equation deleted from the ana lys is . Thus in the non-rationed case, X c is substi tuted out of equations (3.47), i=M,F, using the budget constraint (3.2b) . Hence the matrix J-j has dimension 2x2. In the rationed case we use (3.5) to subst i tute for X^ in (3.49), which is only estimated for the case where i=F. Thus, jB is a sca la r . - 50 -degree zero in a l l the parameters, so that we again impose an arb i t rary normalization by assuming that a l l the parameters sum to uni ty , as in (3.45). - si -CHAPTER FOUR RESULTS AND ANALYSIS FOR THE SINGLE-RATION MODEL 4.1 Introduction In th is chapter we present resul ts obtained using the model developed in Chapter 3. We begin by discussing various estimation problems encountered with many of the funct ional form representations of household preferences considered in the previous chapter. As a resul t of these d i f f i c u l t i e s , we proceed with analysis using the Stone-Geary s p e c i f i c a t i o n , and in Section 4.3 consider which demographic var iables to include in the model and the appropriate method to use to incorporate them within the adopted spec i f i ca t i on . Pr ior to presentation of the resul ts for th is model we examine the expected ef fects of including the Sample S e l e c t i v i t y (SS) correct ion for working wives. We show in general that the d i rec t ion of the SS bias cannot be ana l y t i ca l l y determined in a rat ion ing model of th is type. Subsequently we obtain estimates of our model both with and without th is SS correct ion included so that th is issue can be further examined. Estimates for the complete rat ioning model are presented in Section 4 . 5 . As well as the parameter estimates, various labour supply e l a s t i c i t i e s are also ca lcu la ted . Despite the d i s t i nc t i ve features of our model, an attempt is made to compare these e l a s t i c i t i e s and the estimated income ef fects with values obtained in other labour supply s tud ies. To evaluate the ef fects and importance of cor rec t l y modelling the behaviour of rationed ind iv idua ls , in the fol lowing sections we re-estimate the model f i r s t using only non-rationed households and then t reat ing a l l - 52 -households in the sample as though they are non-rationed. Evidence on th is issue from other studies which have examined labour supply ra t ion ing is discussed and i t is shown that although tests have been used in some of these studies to determine whether rat ioning has a s ign i f i can t ef fect on the parameter estimates, s imi lar tests cannot in general be used with the model estimated here. In Section 4.8 an overa l l evaluation of the empir ical resul ts is presented, while in the f i na l sec t ion , in an attempt to improve the estimates obtained, the models are re-estimated with unemployed and underemployed households grouped together. 4.2 Choice among Al ternat ive Representations of Household Preferences Of the four observat ional ly d is t inguishable funct ional forms considered in the previous chapter, resu l ts are only presented for the case where the rationed and non-rationed demands are derived from the Stone-Geary u t i l i t y funct ion. While i t is desired to compare the estimates obtained using d i f ferent representations of household preferences, a number of factors prevent such comparisons being under-taken. In the CES case, maximum l ike l ihood estimates could not be obtained due to d i f f i c u l t i e s in converging the l i ke l ihood funct ion. Essen t i a l l y the problem in th is case was that the 3i parameters of equations (3.33) and (3.34) became negative, so that B a was undefined for non-integer values of a . An attempt to overcome th is problem by redef in ing B° as and optimizing over the D- parameters also proved to be unsuccessful . With the Indirect Translog share equations, problems we ant ic ipated in the previous chapter, e i ther with two v i r tua l wages occurring or with no v i r tua l wage being obtained, proved insurmountable. Attempts at overcoming these problems by using one or other of the two - 53 -v i r t ua l wage so lu t ions , or replacing the v i r tua l wage by the actual wage i f there was no so lu t ion , did not resu l t in convergence of the ML algori thm. With both the CES and Indirect Translog forms, attempts were made to achieve convergence by excluding the offending observations. In the CES case, observations were deleted i f there were large dif ferences between estimated shares, evaluated using the parameters from the la test i t e r a t i o n , and actual shares. With the Indirect Translog form, observations were deleted i f no v i r tua l wage solut ion was obtained. Although th is procedure permitted optimizat ion to continue, convergence could s t i l l not be achieved in ei ther case, as the par t icu lar observations causing the problems varied between i t e ra t i ons . While i t is tempting the delete "suspect" observations in order to obtain a "c lean" sample, one problem with such methods is that the resu l t ing sample may exhibi t l i t t l e var ia t ion in the dependent va r iab le . As a r e s u l t , the estimated re la t ionsh ips tend to be of l i t t l e use in explaining labour supply behaviour or , as noted by Da Vanzo, De Tray and Greenberg [1976, p.319], inferences about labour supply w i l l depend on the r e l a t i v e l y few persons remaining in the sample whose hours of work d i f f e r subs tan t ia l l y from the mean. For example, in the CES case where 3i is estimated to be approximately zero (although the l i ke l ihood function is not f u l l y converged), i t can be seen from equations (3.33) and (3.34) that the resu l t ing estimate Y.. « X i . Thus in th is case delet ing observations which d i f f e r great ly from the mean w i l l be even more l i k e l y to resul t in 3^ = 0 and Y ^ = X\j. Due to these p o s s i b i l i t i e s and our lack of success with these methods so f a r , no further attempts have been made to obtain convergence by excluding observat ions. - 54 -D i f f i c u l t i e s in achieving convergence were also encountered using share equations derived from the Direct Translog u t i l i t y func t ion . In th i s case however, the reason for lack of convergence could be iden t i f i ed as being due to the estimated corre la t ion coef f i c ien t between the error on A , the male and female share equations in the unrationed case, p m f , tending towards a value of negative uni ty . Since attempts to bound th is coef f i c ien t away from th is value also proved unsuccessful in converging the l i ke l ihood funct ion, fur ther analysis was conducted to t ry to determine the factors causing th is parameter to approach negative un i ty . I n i t i a l experimentation revealed that i f the appropriate Jacobian term was omitted from the l i ke l ihood funct ion, which would only be correct i f pr ices rather than quant i t ies are endogenous, convergence could be . A achieved, with p m f ra - 0.96. Since th is outcome was observed for both the f u l l model and the model using only the subsample of non-rationed households, subsequent analysis of th is issue was conducted with the simpler non-rationed model. To further s impl i fy the ca lcu la t i ons , including the Jacobian, th is model was re-estimated both with and without the Jacobian term, after f i r s t imposing homotheticity of the Direct Translog u t i l i t y func t ion . For the share equations (3.47), th is r e s t r i c t i o n , imposed by set t ing £ 3 . . = 0 for a l l i , used in conduction with the arb i t rary normalization (3.45) means that the denominator of the rhs of these share equations s imp l i f i es to un i ty . However even in th is s imp l i f ied case, convergence again could only be achieved when the Jacobian was omitted; pmf approached negative unity when the Jacobian term was included. Our f i na l experiments with th is funct ional form consisted of taking a sub-sample of observations from the non-rationed households and re-est imating the - 55 -l i ke l ihood funct ion, Jacobian included, with p .^ held constant at a par t i cu la r value. This procedure was repeated for a grid of pm^ values in the range ( -1 , 1) and the maximized log l i ke l ihood values compared. This function was found to increase cont inual ly as p ^ decreased from 0.99 to -0 .99 , suggesting that the tendency for p ^ to approach negative unity could not be at tr ibuted to the (attempted) attainment of a local maximum when a higher (possibly global) maximum existed for some other p^^ va lue . 1 In view of the above r e s u l t s , i t is in terest ing to note that there do not appear to be any studies in the l i t e ra tu re in which ML estimates of the parameters of the Direct Translog share equations - t reat ing quant i t ies as endogenous and including the appropriate Jacobian term -have been obtained. The f i r s t authors to estimate th is form of share equations appear to be Berndt and Christensen [1973], [1974]. However, they view both prices and quant i t ies as endogenous and adopt an Instrumental Variables approach to purge the quant i t ies of the i r cor re la t ion with the addit ive disturbances. A la ter study by Christensen and Manser [1977] estimates share equations for both the Direct and Indirect Translog spec i f i ca t ions without including a Jacobian term in e i ther case. The authors argue that each of the two forms is a d i f ferent representation of preferences, since neither can be obtained from the other except when a l l second.order terms are dropped. Thus, they s ta te , while quant i t ies are endogenous and prices exogenous with the Indirect 1 Of course while the fact that no other point yielded a higher log l i ke l ihood is i nd i ca t i ve , i t does not ensure that there does not ex is t a global maximum, for some other pmf value; to be certa in we would be required to evaluate the l i ke l ihood function at an i n f i n i t e number of points for each p f value. - 56 -form, pr ices are endogenous and quant i t ies exogenous when the Direct Translog Spec i f ica t ion is used. However, since the Direct Translog share equations are derived by choosing quant i t ies to maximize a u t i l i t y function subject to a budget const ra in t , i t i s d i f f i c u l t to support t h i s view and hence the consequent omission of a Jacobian. In a recent paper, McLaren [1981] uses a ML procedure in an attempt to estimate share equations of the Direct Translog form where quant i t ies are treated as endogenous and the Jacobian is included. However, he also f a i l s to achieve convergence of the l i ke l ihood function and as a resul t uses l inear approximations to the share equations which, he observes, f a i l to converge to "sensible parameter estimates" (p. 25). As a resul t he questions the appropriateness of appending error terms to the Direct Translog share equations, arguing that they should r ea l l y be added to the reduced form equations which, however, cannot be obtained. While in general there are no clear reasons (other than computational s imp l i c i t y ) for preferr ing to append errors to reduced form rather than st ructura l form equations, i t is d i f f i c u l t to disagree with McLaren's conclusion that the Indirect Translog u t i l i t y function is more useful as a funct ional form for representing preferences when one is interested in (estimating) the resul tant systems of (unrationed) demand equations. While i t is hoped that further research w i l l p rec ise ly ident i fy the factors which prevent parameter estimates being obtained with these various funct ional forms, so that eventual ly we w i l l be able to estimate rat ioning models using these a l ternat ive spec i f i ca t i ons , such research is beyond the scope of th is d i sse r ta t i on . We thus proceed with an examination of the estimates obtained for our model of household labour supply under rat ioning using the LES formulat ion. However, before - 57 -analyzing these resul ts in d e t a i l , we f i r s t turn our attention to the important issues of determining the par t icu lar demographic var iables that are to be included in our model and the method that is to be used to incorporate them. 4.3 Inclusion of Demographic Variables As outl ined in the previous chapter, i t is desired to incorporate various demographic var iables in the estimation of the rationed and non-rationed LES demands given by equations (3.29) and (3.31). In pa r t i cu la r , based on ex is t ing studies of non-rationed demands, such as those by Wales and Woodland [1976], Hall [1973] and Da Vanzo et a l . [1976], and the work of Ham [1977], [1982], Ransom [1982] and Blundell and Walker [1982] involving labour supply in the presence of ra t i on ing , i t is expected that the presence (or absence) of young chi ldren in the household, as well as the eduction levels attained by the head and his spouse, and the i r ages, w i l l be important determinants both of household consumption and of male and female labour supply d e c i s i o n s . 2 With the LES the conventional method for incorporating such var iables is by demographic t r a n s l a t i n g , 3 in which the subsistence parameters, , are spec i f ied to be l inear functions of the required demographic va r iab les , 7 SL = 1 i • 2 Other var iables found to be important in some cases include a dummy var iable for race (Ham [1977], [1982], Ransom [1982]) and a dummy var iab le which equals unity i f the head suffers a health l im i ta t ion (Ham [1982]). These var iables are not considered here since with the method ul t imately used to allow consumption and labour supply patterns to d i f f e r across households with d i f ferent charac te r is t i cs (see below), inc lusion of fur ther demographic or socio-economic var iables would resu l t in cer ta in categories of households having an insu f f i c ien t number of observations ( in some cases no observations) for r e l i a b l e estimates to be obtained. 3 For de ta i l s of th is procedure, see Pol 1ak and Wales [1978]. - 58 -L (4.1) y. = y • + £ y^o^o » i = l . - . - . n commodities. It is in terest ing to note at th is point that in estimating the usual goods-only LES using a s ingle sample of household budget data, extraneous information is required in order to ident i fy the parameters, and that for each demographic var iable added v ia equation (4 .1) , one addit ional piece of extraneous information is required in order to ident i fy the corresponding y0. parameters.1* Essen t ia l l y th is requirement is caused by the lack of avai lable pr ice information on the d i f ferent goods when a s ingle cross-sect ion is used. However with the form of the LES estimated here, where goods demand equations are augmented by le isure demand equations, a l l parameters are iden t i f i ed even i f demographic var iables are included using (4 .1) . This resu l t occurs for two reasons. F i r s t , the pr ices of le isure - the male and female wage rates - are observed for a l l households in the sample. Second, a l l consumption goods (for which we have no pr ice information) are col lapsed into a s ingle composite commodity. One problem with th is method of incorporating demographic var iab les i s that i t resu l ts in the addit ion of 3 parameters for each demographic var iable that is included. Since the l i ke l ihood function is very non- l inear , th is represents a considerable computational burden. To ascertain which demographic var iables are s ign i f i can t determinants of consumption and labour supply behaviour, experimentation was undertaken incorporating one demographic var iable at a time in the basic model by means of (4 .1) . Here, the s ign i f icance of the par t icu lar demographic var iable being considered is evaluated by Likel ihood Ratio (LR) t es t s , in which the negative of twice the di f ference between the logarithm of the k See, for example, Ryan [1976]. - 59 -l i ke l ihood when the demographic var iable is omitted and the logarithm of the l i ke l ihood when i t is included in the form (4.1) , is asymptot ical ly d is t r ibu ted as a chi-square with 3 degrees of freedom. The resu l ts obtained with these tests reveal that while the presence of chi ldren less than 6 years old in the household, and the educational level attained by the head or spouse are separately s i g n i f i c a n t , in that the inc lusion of e i ther of these var iables resul ts in a s ign i f i can t improvement in the l i ke l ihood over the value for the basic model, the age of the head or h is spouse is not found to be s ign i f i can t using th is c r i t e r i o n . In order to include both education level and the presence of young chi ldren in the household as determinants of consumption behaviour, yet not increase the d i f f i c u l t y involved in maximizing the l i ke l ihood func t ion , the data was separated into four groups according to whether or not there were young chi ldren ( less than 6 years) in the household and whether or not ei ther the head or his spouse completed at least one year of co l lege . While the LR tests reported above suggest that the presence of young chi ldren and the education level of the head and spouse are separately s ign i f i can t when included in the model in the form (4 .1) , they do not reveal whether consumption patterns w i l l be d i f ferent for each of the four groups when a l l parameters are allowed to vary, as is the case when estimation is performed separately for each group. To test whether th is is indeed the case, maximization of the l i ke l ihood is also done separately for a l l observations in the sample, for a l l households with young chi ldren and for a l l households with no young ch i ld ren . The LR s t a t i s t i c is used to determine whether the parameter estimates for the chi ldren and no chi ldren groups are s i g n i f i c a n t l y d i f f e ren t , or whether the two groups should be combined and a s ingle set of estimates - 60 -obtained. Here, minus twice the di f ference between the log of the l i ke l ihood for a l l observations and the sum of the log l i ke l ihoods obtained when the chi ldren and no chi ldren groups are estimated separately is asymptotical ly a chi-square with 10 degrees of feedom, since in addit ion to the f i ve st ructura l parameters of the model, the f i ve i den t i f i ab le elements of the variance-covariance matrix of the res iduals are also allowed to vary between the chi ldren and no chi ldren groups. For the model the calculated chi-square values are 57.858 and 54.616 in the two cases where the SS correct ion is f i r s t omitted and then included in the l i ke l ihood funct ion. Since these values both exceed 23.21 which is the c r i t i c a l value of the chi-square s t a t i s t i c at the 1% s ign i f icance l e v e l , i t is c lear that the parameter estimates are s i g n i f i c a n t l y d i f fe rent for households with and without ch i ld ren . It is in terest ing to note that the LR s t a t i s t i c comparing the sum of the l ike l ihoods for the two groups with the value obtained when a dummy var iable for the presence of young chi ldren in the household is incorporated by way of equation (4 .1 ) , is also s i g n i f i c a n t . For the SS-uncorrected and SS-corrected groups, th is s t a t i s t i c is calculated as 29.844 and 27.870 respec t ive ly , while the c r i t i c a l chi-square at the 1% level for 7 degrees of freedom (the 2 3 parameters and the 5 elements of the variance-covariance matrix which do not vary using the dummy var iable approach) is 18.48. Thus, th is indicates that a l l the parameters vary across these groups, not just the Y.j parameters. Given that the parameters d i f f e r s i g n i f i c a n t l y for households with and without ch i l d ren , the next task involves test ing whether, for each of these two groups, the parameters d i f f e r i f the head or his spouse completed at least one year of co l lege . For each group, LR tests are - 61 -used, with the s t a t i s t i c calculated as minus twice the di f ference between the log l i ke l ihood for the whole group and the sum of the separate log l i ke l ihoods for members of the group with education and for members without education. Again there are 10 degrees of freedom, so that the c r i t i c a l chi-square value at the 1% level of s ign i f icance is 23.21. For households with ch i ld ren , the calculated s t a t i s t i c s in the SS-uncorrected and SS-corrected cases are 30.780 and 29.128 while for households without chi ldren the corresponding s t a t i s t i c s are 23.868 and 28.460. Hence consumption and labour supply patterns are s i g n i f i c a n t l y d i f ferent i f e i ther the head or spouse attended co l lege , both for households with young chi ldren and for those with no young ch i ld ren . Consequently, in the remainder of our analysis with th is model, separate ML estimates are presented for each of the four groups of observations. 4.4 Expected Ef fects of Including the Sample S e l e c t i v i t y Correct ion For each of the four groups of households we obtain two sets of estimates corresponding to the s i tuat ions where the SS cor rec t ion , which takes account of the fact that we only examine households in which the female is working, is f i r s t omitted and then included in the l i ke l ihood funct ion . While i t is c lear that omission of the SS correct ion means that we confuse meaningful s t ructura l parameters with the parameters of the funct ion which determines the probab i l i t y that an observation is included in the non-random sample 5 , in many cases the fact that the ef fect on the parameter estimates of including an SS correct ion has been found to be neg l ig ib le has been used as an argument for keeping the computations 5 Heckman [1980, p. 206] - 62 -simple by omitt ing the co r rec t i on . 6 In the context of models examining female labour supply behaviour, Heckman [1980] found that when the SS bias is corrected by adding addit ional var iables to the equations, the implied labour supply e l a s t i c i t y with respect to hourly wage rates could vary quite considerably as compared to the non-SS case. He concluded (p. 238) that SS bias is an important phenomenon in estimating labour supply functions but is not as important in estimating wage funct ions. However, Blundell and Walker [1981] f ind for the i r model examining male and female labour supply and household commodity demands that correct ing for SS bias only causes minor changes in a small number of parameter est imates. It is hoped therefore that a comparison of the SS-uncorrected and SS-corrected estimates for our model w i l l provide addit ional information on the importance of the SS cor rec t ion , pa r t i cu la r l y in the context of models which take account of rat ioning in the labour market. An addit ional reason for wishing to compare estimates with and without the SS correct ion included concerns the d i rect ion of the bias which resul ts i f the SS correct ion is omitted. Both Heckman [1980] and Cogan [1980] provide evidence showing that parameter estimates obtained from estimating female labour supply functions using a sample of working women are downward biased in absolute value i f a correct ion to take account of the sample s e l e c t i v i t y is not made. Further evidence on th is point is provided by Wales and Woodland [1980] who estimate a simple female labour supply model using experimental data. By generating samples of 1000 and 5000 observations in which approximately 70% of the sample is not working and where the true parameters of the labour supply function 6 See, for example, Ham [1982, p. 341 and fn 8] where truncation of the dependent var iable in a male labour supply equation has l i t t l e ef fect on the parameter estimates and is subsequently omitted. - 63 -are known, Wales and Woodland are able to examine the performance of various estimation methods in estimating these parameters. For a s ing le equation model of labour supply, OLS estimates (which ignore the sample se lec t i v i t y ) are found to be downward biased in absolute value, with the coe f f i c ien t on the constant term in general having the incorrect s ign . Condit ional ML methods which incorporate the SS cor rec t ion , s imi lar to those used in th is study, are found to y ie ld estimates which are very s imi la r to the true parameters. For a simultaneous two-equation model of hours and wage determination OLS estimates are again found to be downward biased in absolute value. In the context of a model of male and female labour supply where there is no ra t ion ing , the bias due to sample s e l e c t i v i t y can be i l l u s t r a t e d as fo l lows. Ignoring household subscripts for notational s imp l i c i t y and using y i and y 2 to denote male and female hours of labour supply, households are only included in the sample i f y 2 > 0, since required wage information is only avai lab le for females who are working. Assuming, for genera l i ty , that the var iables in the male and female hours equations are not necessar i ly the same, such a model may be writ ten as (4 .2) y i = M X i . & i ) + u x h 2 ( X 2 , B 2 ) + u 2 i f h 2 ( X 2 , 3 2 ) + u 2 > 0 (4 .3 ) Y2 = otherwise. 0 Now y 2 > 0 <=> u 2 > h 2 ( X 2 , B 2 ) . Hence, (4 .4) E ( y i | y 2 > 0) h i ( X i , B i ) + E ( U i | u 2 > - h 2 ( X 2 , B 2 ) ) and - 64 -(4.5) E ( y 2 | y 2 > 0) = h 2 ( X 2 , 3 2 ) + E ( u 2 | u 2 > - h 2 ( X 2 , 3 2 ) ) Assuming ui and u 2 are j o i n t l y normally d i s t r i bu ted 7 with zero means and covariance matrix £ , where £ = r 2 01 °12 2 °\2 a2 , then from Johnson and Kotz [1970, p. 81]. [1972, p. 112], (4.6) E ( u i | u 2 > - h 2 ( X 2 , 3 2 ) ) = Sll X(e) (4.7) E ( u 2 | u 2 > - h 2 ( X 2 , 3 2 ) ) = o 2 X ( 9 ) where f(e) (4.8) X ( 9 ) = [1 - F(6)] • where f ( . ) and F(. ) are the standard normal density and d is t r ibu t ion functions respec t ive ly , and where (4.9) 9 = - h 2 ( X 2,e 2 ) / o 2 . Subst i tut ing (4.6) and (4.7) in (4.4) and (4 .5) , we obtain the population regression functions for male and female hours condit ional on female hours being pos i t i ve : °12 (4.10) E(yi | y 2 > 0) = M X i . B i ) + ^ X(6) (4.11) E ( y 2 | y 2 > 0) = h 2 ( X 2 , 3 2 ) + a 2 x ( 9 ) Since the data points for a sample of workers w i l l be d is t r ibuted about these functions rather than h i ( X i . B i ) and h 2 ( X 2 , B 2 ) , estimation methods 7 Olsen [1980] demonstrates that while b ivar ia te normality is a su f f i c i en t condit ion for der iv ing the condit ional expectation in (4.11) below, i t is not a necessary condi t ion. In fact normality of u 2 and l i n e a r i t y of E (u i | u 2 ) in u 2 w i l l also y ie ld the same expression. - 65 -which ignore the sample s e l e c t i v i t y w i l l y ie ld biased estimates of 3i and 3 2 . Moreover, by d i f fe ren t ia t ing these population regression functions with respect to par t icu lar var iab les , we can ascertain the l i k e l y d i rec t ion of th is bias in various slope coe f f i c i en t s . Denoting the j t h var iable in X2 by X 2 j , we have, from (4.11), (4.12) 3 E ( y 2 | y 2 > 0) = 3 h 2 ( X 2 , 3 2 ) ^ ^ 3A(9) 3 X 2 j 3 X 2 j ^^2j Now using the fact that the der ivat ive of the d is t r ibu t ion function is the densi ty funct ion, while the der ivat ive of the normal density function f ( 9 ) with respect to i ts argument is - 8f ( e ) , i t is straightforward that 9 X t 9 ) = - ex(e) + x 2 ( 9 ) . 8 Hence subst i tu t ing in (4.12) we have 3 9 (4.13) 1 1^2 > 0) = 3 M X 2 , 3 2 ) C l + e x { 0 ) _ X 2 ( Q ) ] 3 X 2 j 3 X 2j Since the term in square brackets in (4.13) has been shown by Heckman [1980, p. 216] to l i e in the interval bounded by zero and u n i t y 9 , i t i s c lear that (4.14) 3 E ( y 2 | y 2 > 0) , 3 h 2 ( X 2 , 3 2 ) 3 X2j 3 ^ 2 j so that slope coef f i c ien ts in the female hours equations w i l l be downward biased in absolute value (but should not change sign) i f no SS correct ion is made. For the j th var iable in X i , denoted X i j , we have While the f i r s t of these statements is d e f i n i t i o n a l , the second is not as obvious. Since f(e) = (1//2H) .exp{g(9)} , where g(e) = - 9 2 / 2 , i t i s c lear that 9g(9)/86 = -9 and hence, using the chain rule of d i f f e r e n t i a t i o n , that 3f(8)/38 = -ef(e). 9 While Heckman does not actua l ly prove th is condi t ion, the term in square brackets is the variance of a standardized normal var iable which has been truncated from below, and as such is posi t ive and, as stated by Johnson and Kotz [1972, p. 112] not greater than uni ty . - 66 -(4.15) K > 0 ) = a h l ( X i , g i ) + ^ 2 . [ - 9 A ( 9 ) + X 2 ( 9 ) ] . (- 1 ) 3 h 2 ( X 2 , 3 2 ) 3 Xi j 3 X i j o"2 o 2 3X i j Now i f X i j does not appear among the var iables in X2, or i f p, the cor re la t ion between ui and U2 , is zero (so that o 1 2 = p o ^ = 0 ) , i t is c lear that the estimates in the male hours equation w i l l not be biased i f the sample s e l e c t i v i t y problem is ignored. However, since with the model devised in the previous chapter the same explanatory var iables appear in each equation, the SS bias w i l l only be zero i f p = 0. Now since the term in square brackets in (4.13) l i e s between zero and uni ty , so too does the term in square brackets in (4.15). Hence the sign of the bias in the male hours equation, (4.16) - • 3 h 2 ( X z ' g 2 ) • [- ex(9) + x 2 ( 9 ) ] , a2 3 X1j depends on the sign of the cor re la t ion coe f f i c ien t between \ix and u 2 and the sign of the corresponding slope coe f f i c ien t in the female hours equation. If both these magnitudes have the same s ign , the bias w i l l be negative while i f they have opposite signs the bias w i l l be pos i t i ve , with the SS-uncorrected ef fect overstat ing the true e f fec t . However unl ike the slope coef f i c ien ts in the female hours equation, i t is possible for the SS-corrected and SS-uncorrected estimates of the slope coef f i c ien ts in the male hours equation to be of opposite s igns . The consequences of the above analysis for estimation of the LES labour supply equations are as fo l lows . Since the der ivat ive of female 3 h f - 3 f hours with respect to income, = is downward biased in absolute 3M Wf value, i t fol lows that 3f w i l l be downward biased in absolute value i f the SS correct ion is omitted. Although is is also known that - 67 -W 1 1 + V I I T and are downward biased in absolute value in the SS-uncorrected 3wm 3 w f case, the expressions for these d e r i v a t i v e s 1 0 are functions of 3f and a l l the T i ( i = m, f , c ) , so that the d i rec t ion of the bias in these Yi parameters is unknown. 1 1 However, since i t is known that the der ivat ives are downward biased, i t is c lear that e l a s t i c i t i e s of female hours with respect to male or female wage rates should also increase in absolute value when the SS correct ion is incorporated in the model. Of course a l l the above analysis concerning the d i rect ion of the bias i f the SS correct ion is omitted is presented in the context of a t rad i t i ona l labour supply model in which rat ioning is ignored. It is thus important to consider whether these same re la t ionships concerning the ef fects of the SS correct ion hold in a model which takes account of rat ioning in the labour market. In f ac t , the rat ioning of labour supply in i t s e l f is a sample s e l e c t i v i t y problem. This is most evident in the approach adopted by Ham [1982] who uses an extension of Heckman's [1979] method, in which addit ional var iables are added to the labour supply equation to correct for the SS due to including only non-rationed ind iv iduals in the sample . 1 2 In a ML context, i t is not possible to correct for th is type of SS by d iv id ing the l i ke l ihood funct ion for non-rationed ind iv iduals by the probab i l i t y of being non-rat ioned. This is due to the fact that th is p robab i l i t y is always zero since the sum of the p robab i l i t i es of being overemployed and of being underemployed, which is the sum of the areas under a density function to the l e f t and to the 1 0 See Appendix 2 for these expressions. 1 1 It is thus somewhat surpr is ing that when incorporation of the SS correct ion in the i r model resu l ts in increases in the estimated Bf and (T-Yf) parameters, Blundell and Walker [1981, p.13] regard both changes as being in the "expected" d i rec t i on . Ham [1982] ac tua l ly ignores overemployed workers when using th is method. - 68 -r igh t of a par t i cu la r value, is always uni ty . However with the ML method adopted here th is probab i l i t y is not required since a l l data — both rationed and non-rationed ~ is used. Although the ML procedure used here is therefore not d i r ec t l y comparable to Ham's select ion rule approach in that we use a l l observations in the ent i re estimation procedure, i t is convenient to use Ham's formulation to i l l u s t r a t e that for a rat ioning model we cannot, in general , predetermine the ef fects on the estimates of the true labour supply function parameters of including an SS correct ion which takes account of the fact that only working wives are included in our sample. Since the signs of the SS bias in the male hours equation cannot in general be signed even in the t rad i t i ona l labour supply model, we concentrate here on the female hours equation. Bas i ca l l y , Ham's [1982] method 1 3 consists of using OLS to estimate the fo l lowing formulation for non-rationed ind iv idua ls : (4.17) y 2 = h 2 ( X 2 , S 2 ) + o 1 5 X 5 ( 6 5 ) + a 1 6X 6(e 6) + u 2 where X 5(e 5) and X 6 ( e 6 ) , which are essen t i a l l y b ivar ia te analogues of x(e) defined in (4 .8) , are estimates obtained from b ivar ia te probit analysis using only rationed ind iv idua ls . Now, for convenience, rewrite (4.17) as (4.18) y 2 g 2 ( X 2 , B 2 , e 5 , e 6 ) + u 2 i f y 2 > 0 0 otherwise. This equation is s imi la r in nature to (4.3) so that the previous analysis concerning the d i rec t ion of the SS bias is again appropriate. However while th is analysis indicates the d i rec t ion of bias for 3 g 2 ( - ) / 9 X 2 j , i t is not in general possible to infer from th is the ef fect on 3 h 2 ( . ) / 3 X 2 j 13 For further de ta i l s of th is method, see Chapter 2. - 69 -of ignoring the SS cor rec t ion . Of course one case where the ef fect on h 2 ( . ) can be inferred occurs when the var iable X 2 j is not contained in 6 5 or e 6 . In Ham's ana lys is , th is would imply that X 2 j would be a var iab le which affects female labour supply, but not the probab i l i t y that the female experiences unemployment or underemployment; Ham does not include any of the var iables used here ~ wage rates and income — in th is category. Hence, i t does not seem possible to form general pr ior expectations as to the d i rec t ion of the SS b ias . It is thus hoped that a comparison of the SS-corrected and SS-uncorrected estimates for the rat ion ing model estimated here w i l l provide some addit ional information concerning these issues. 4.5 Empir ical Results fo r the Complete Rationing Model We now turn to a consideration of the empirical resul ts obtained for the complete rat ioning model using the Michigan d a t a . l l + We begin by examining the parameter estimates obtained for the four groups of households using the LES formulat ion. In addit ion to comparing these parameters across groups, we also invest igate the ef fects of the SS correct ion and the extent to which the regu la r i t y condit ions are s a t i s f i e d . Due to the nature of the estimated income coe f f i c i en t s , comparisons are made with various estimates obtained in other s tud ies . Following th is ana lys is , in Section 4.5.2 e l a s t i c i t y estimates are evaluated, with par t icu lar attention being paid to the nature of the c ross- ra t ion ing e f fec t s . F i n a l l y , in Section 4 .5 .3 , we present a comparison of these e l a s t i c i t i e s with estimates obtained in various other labour supply s tudies. l l + Deta i ls of the data used in th is study are contained in Appendix 1. - 70 -4.5.1 Parameter Estimates The c o e f f i c i e n t s 1 5 in Table 4.1 have the i r usual LES in terpre ta-t i ons , with the t rans la t ion parameters, Y i , representing subsistence quant i t ies (provided they are non-negative), and the Si being the marginal budget shares in the unrationed case. Examining the Yi parameters f i r s t , we see that the subsistence quantity of le isure for the male houshold head tends to be larger for groups with no education, ranging from 7240 hours le isure (equivalent to working approximately 30 hours per week) for these groups to 6550 hours le isure (a working week of approximately 44 hours) for the group of households with education but no ch i l d ren . A s imi la r pattern is observed with the subsistence quantity of le isure for the female, Yf, which however tends to be larger for each group than the corresponding Ym magnitude. The largest value of Yf is 8220 hours of le isure (equivalent to working approximately 11 hours per week) which occurs for households which have chi ldren but no education. For households which have education but no ch i ld ren , values of female subsistence le isure of 7220 hours (working 31 hours per week) or 7040 hours (working 34 hours per week) are observed in the SS-uncorrected and SS-corrected cases respect ive ly . Note that in a l l cases, both Ym and Yf are s i g n i f i c a n t l y d i f ferent from zero at the 1% level of s i g n i f i c a n c e . 1 6 1 5 ML parameter estimates are obtained using FLETCH, a quasi-Newton routine for unconstrained optimization which uses numerical der i va t i ves . FLETCH is contained in the NLP group of nonlinear function optimization routines avai lab le at the U.B.C. Computing Centre. For d e t a i l s , see Patterson [1978]. S igni f icance tests are based on asymptotic theory under which the parameter estimates are asymptot ical ly normally d is t r ibuted with standard errors as presented in the relevant tables of r esu l t s . Hence c r i t i c a l values for 2 - t a i l tests are 1.96 (5% leve l ) and 2.576 {1% l e v e l ) . Throughout the ana lys is , except where e x p l i c i t l y speci f ied otherwise, s ign i f i cance tests are conducted at the 5% l e v e l . - 71 -Since the price of the aggregate consumption commodity has been a r b i t r a r i l y set equal to uni ty , Yc can be interpreted as the subsistence expenditure on th is commodity. From Table 4.1 we observe that Yc exh ib i ts considerable var ia t ion across the four groups, tending to be pos i t ive and s ign i f i can t for the groups with education (other than in the SS-corrected case for households which also have chi ldren) but negative and not s i g n i f i c a n t l y d i f ferent from zero for households without education. For households with both chi ldren and education, the subsistence expenditures of $7540 in the SS-uncorrected case or $4940 in the SS-corrected case compare to a mean expenditure on the consumption good by th is group of $19320, while for households with education but no chi ldren the corresponding subsistence expenditures of $20370 and $15300 compare with a mean consumption expenditure of $22243. Turning now to the marginal budget shares (mbs) in the unrationed case, we f ind that 3 m , the mbs for male l e i su re , is general ly negative and s i g n i f i c a n t , although for the group with education and no chi ldren and for the SS-corrected estimates in the groups with no education, g m is not s i g n i f i c a n t l y d i f ferent from zero. Note that th is male mbs tends to be larger in absolute value for groups with ch i l d ren , where i t l i e s in the range -0.25 to - 0 .4 . The mbs for female l e i su re , 3f , is not s i g n i f i c a n t l y d i f ferent from zero except for the group with education and no chi ldren where i t is s i g n i f i c a n t l y pos i t ive (0.09 and 0.135) and the SS-uncorrected estimate for households with both chi ldren and education, where i t is s i g n i f i c a n t l y negative (-0.145). In general 3 m is larger in absolute value than 3f except for the group with education and no ch i l d ren , where the order is reversed. - 72 -TABLE 4.1: PARAMETER ESTIMATES FOR THE SINGLE-RATION MODEL C.E C,NE NC.E NC.NE PARAMETER NO SS ss NO SS SS NO SS SS NO SS SS Ym .705* .707* .724* .722* .655* .650* .724* .720* (.022) (.020) (.026) (.027) (.059) (.006) (.013) (.013) Y f .748* .715* .822* .822* .722* .704* .742* .725* (.015) (.035) (.015) (.021) (.012) (.016) (.014) (.017) Y C .754* .494 -.276 -.321 2.037* 1.530* -1.658 -2.213 (.354) (.477) (.205) (.254) (.400) (.305) (1 .008) (1 .548) 3 m -.394* -.296* -.256* -.249 .053 .049 -.103* -.081 (.116) (.146) (.124) (.137) (.030) (.040) (.052) (.048) 0f -.007 .130 -.145* -.103 .091* .135* -.013 .010 (.059) (.082) (.056) (.069) (.025) (.031) (.015) (.013) 0 c 1.401* 1.166* 1.401* 1.352* .856* .816* 1.116* 1.072* (.140) (.188) (.168) (.185) (.040) (.048) (.063) (.049) °m .048* .048* .029* .030* .033* .033* .031* .031* (.005) (.005) (.003) (.004) (.002) (.002) (.002) (.002) °f .036* .042* .028* .034* .024* .026* .028* .031* (.004) (.005) (.003) (.008) (.001) (.002) (.002) (.002) °r .018* .020* .029* .032* .033* .036* .029* .032* (.003) (.004) (.005) (.007) (.004) (.005) (.003) (.004) Pmf -.003 .048 -.236 -.343 .040 .064 -.251* -.291* (.153) (.173) (.174) (.223) (.088) (.093) (.088) (.094) Pmr -.072 -.032 -.093 .115 .006 .0001 -.352* -.264 (.320) (.350) (.297) (.422) (.170) (.176) (.157) (.180) rmbs-f -.005 .101 -.116* -.083 .096* .142* -.012 .009 rmbs-c 1.005 .899* 1.116* 1.083* .904* .858* 1.012* .991* (.043) (.068) (.037) (.050) (.027) (.032) (.014) (.012) Code: C = Youngest c h i l d less than 6 years o l d NC = No c h i l d r e n younger than 6 years E = Head or spouse compIeted at I east one year of coI Iege NE = Neither head nor spouse attended c o l l e g e SS = Includes sample s e l e c t i v i t y c o r r e c t i o n for working wives NO SS = Does not include sample s e l e c t i v i t y c o r r e c t i o n Notes: Numbers in parentheses are asymptotic standard e r r o r s 3 C Is obtained as 1 - 3 m ~ 3 f Y j are a l l s c a l e d by 1/10000. rmbs are marginal budget shares when the male i s r a t i o n e d . * i n d i c a t e s t h a t the estimate i s s i g n i f i c a n t l y d i f f e r e n t from zero at the 5$ l e v e l . - 73 -As a resul t of these negative 3 m and 3f values, the mbs for goods consumption, 3 C , exceeds unity for a l l households other than those with education but no ch i ld ren , where i t is s i g n i f i c a n t l y less than uni ty. However in the other groups, the only cases in which the estimated value of 3 C s i g n i f i c a n t l y exceeds unity occur for households with chi ldren when the SS correct ion is omitted. While the resu l ts discussed above are contrary to our pr ior expectations in that negative values for the mbs imply that l e i su re , pa r t i cu la r l y male l e i su re , is an in fe r io r good, i t is in terest ing to note that they are consistent with the resu l ts obtained in several other labour supply s tud ies . In the majority of these studies in which le isure is found to be an in fe r io r good cross-sect ion data is used, although in the study of Rosen and Quandt [1978], an aggregate d isequi l ibr ium labour market model is estimated using time ser ies data. On viewing the labour supply decision in a l i f e - c y c l e context, Rosen and Quandt suggest that the resu l t may be due to asset income being simultaneously determined with work e f f o r t . However a rede f in i t ion of var iables to deal with th is problem is not found to change the resu l t . Ham [1977] uses a s imi la r explanation when he f inds le isure to be an i n fe r i o r good in a rat ioning model estimated using cross-sect ion data on males. He suggests that unearned income is pos i t i ve l y correlated with the error term on the labour supply equation, since ind iv iduals that work long hours possibly have a high taste for assets and hence a high level of unearned income. Da Vanzo et a l . [1976] also suggest that net worth and labour supply are j o i n t l y determined, but in a ser ies of experiments they f a i l to f ind le isure to be a normal good except in one par t i cu la r case where the sample is res t r i c ted by excluding a l l households that "pose - 74 -specia l estimation problems" (p. 314) and where imputed values are used for wages and net worth. Other studies using household data which also f ind le isure to be an in fe r io r good include Schultz [1980] and Heckman [1980], although in these cases the coe f f i c ien ts on unearned income in the labour supply equation are general ly not s i gn i f i can t . On the basis of the many studies which f ind le isure to be an in fe r io r good, Smith [1980a] estimates a simple l i f e cycle model in an attempt to better understand th is phenomenon. Not su rp r i s ing ly , he concludes that the standard pract ice of using assets to measure the ef fects of wealth on the demand for le isure is theore t i ca l l y inappropriate in that i t ignores, or f a i l s to s u f f i c i e n t l y emphasize, important l i f e cycle considerat ions. One charac te r i s t i c feature of these studies which f ind le isure to be an in fe r io r good is that they employ l inear or , in the study by Rosen and Quandt, a l inear in logarithms spec i f i ca t ion of the labour supply func t ion . Further, in th is study where we also f ind le isure to be i n f e r i o r , the labour earnings function derived from the LES formulation i s also l i nea r . Hence i t seems possible that l i n e a r i t y of the labour supply or labour earnings function may be one of the pr inc ipa l factors which induce th is resu l t . However, in contrast to the resul ts described above, in the study by Ham [1982] in which a l inear labour supply function is estimated from cross-sect ion data, and in the study by Abbott and Ashenfelter [1976], [1979] in which the LES including a l inear labour earnings equation is estimated from t ime-ser ies data, le isure is not found to be i n f e r i o r . In the study by Wales [1978] which uses the nonlinear Generalized Cobb-Douglas formulat ion, le isure is found to be i n f e r i o r , although the income e l a s t i c i t y for le isure which is very close to zero is a market value which comprises a weighted sum of both pos i t ive and - 75 -negative indiv idual e l a s t i c i t i e s . Further, when le isure is combined with commuting time (which is the s i tuat ion most comparable to our study here) the e l a s t i c i t y is pos i t i ve . While i t is c lear from the con f l i c t i ng evidence outl ined above that we cannot conclude d e f i n i t i v e l y that l i n e a r i t y of the labour supply or labour earnings function is responsible for the impl icat ion that le isure is an in fe r io r good, further supporting evidence for th is hypothesis is provided by other labour supply studies which employ less r e s t r i c t i v e funct ional forms. Ransom [1982], using essen t i a l l y the same data as in th is study, derives nonlinear labour supply equations from l inear approximations to the f i r s t order condit ions for an arb i t rary u t i l i t y funct ion and f inds le isure to be a normal good. The same resul t is obtained by Blundell and Walker [1982] who employ commodity demand and labour earnings equations derived from a cost function formulation which, unl ike the LES, does not impose weak separab i l i t y between le isure and other commodities in the u t i l i t y func t ion . Wales and Woodland [1976] who use the Generalized Cobb-Douglas and Indirect Translog formulations also f ind le isure to be a normal good. In add i t ion , in resu l ts not reported here, estimation of the model developed in the previous chapter using the Indirect Translog spec i f i ca t i on , although ignoring the rat ioning ca lcu la t ions and t reat ing a l l households as non-rat ioned, also resu l ts in pos i t ive marginal budget shares for l e i s u r e . So while i t is not c lear that our resu l t that le isure is an in fe r io r good has as i t s fundamental cause l i n e a r i t y of the labour supply or labour earnings funct ions, use of other less r e s t r i c t i v e funct ional forms seems, in general , to be an appropriate way to avoid th is theore t i ca l l y unappealing resu l t . - 76 -Comparing the resul ts in Table 4.1 when the SS correct ion is included and when i t is omitted, we are unable to detect any systematic pattern as to the way in which the 3-j or Yi parameters change, although i t appears that the predominant ef fect of the SS correct ion is to decrease the parameter est imates. However the magnitudes of these changes in the parameters when the SS correct ion is included are general ly less than the s ize of the standard error on the coe f f i c i en t s , and hence do not appear to be s i g n i f i c a n t . In th is respect these resu l ts are thus consistent with those obtained in ex is t ing s tud ies, examined previously , in which the ef fects of including the SS correct ion have been invest igated. It is in terest ing to note however that for the only household group in which 3f is s ign i f i can t in both the SS-corrected and the SS-uncorrected cases, t h i s parameter is larger in absolute value in the SS-corrected case. This resu l t is thus consistent with our previous analysis concerning the expected d i rec t ion of the SS bias in a t rad i t i ona l (non-rationing) labour supply model. For completeness, Table 4.1 also includes estimates of the iden t i f i ed elements of the variance-covariance matrix of the error terms which are obtained d i r e c t l y from the estimation procedure since the l i ke l ihood function cannot be concentrated. While the standard deviat ions of the errors on the three share equations are s i g n i f i c a n t l y d i f ferent from zero in a l l cases, the corre la t ion coe f f i c ien ts between the various errors are not s i g n i f i c a n t l y d i f ferent from zero except for households with neither chi ldren nor education, where they are s i g n i f i c a n t l y negative, ly ing in the range -0.25 to -0 .35. It is in terest ing to note that the standard deviat ions am and of are larger for households with both chi ldren and education, while ar is smallest for th is group. Note also that with the - 77 -inc lus ion of the SS cor rec t ion , o-f and a r both increase while am remains almost unchanged. The f i na l rows of Table 4.1 contain estimates of the mbs for female le isure and for goods consumption for the case where the male household head is rat ioned. In th is LES case, these rationed mbs are eas i l y calculated as 3-j/(3f + 3 C ) , i = f , c . Again, apart from households with education but no chi ldren where the rationed mbs for female le isure is s i g n i f i c a n t l y pos i t i ve , and the SS-uncorrected estimate for households with chi ldren and no education where th is rationed mbs is s i g n i f i c a n t l y negative (so that the rationed mbs for goods consumption is s i g n i f i c a n t l y greater than un i t y ) , estimated values of the rationed mbs for female le isure are not s i g n i f i c a n t l y d i f ferent from zero. In addit ion to examining the values of these estimated parameters, i t is also of interest to determine the extent to which the estimated funct ional forms sa t i s f y the propert ies of demand funct ions. S p e c i f i c a l l y , based on the concavity of the expenditure function in p r i ces , we require that the Slutsky matrix of compensated subst i tu t ion terms be symmetric and negative semi-def in i te . With the LES, symmetry is automatical ly imposed, so that i t is only necessary to check the property of negative semi-def in i teness. However, rather than checking the signs of various determinants of the Slutsky matr ix, with the LES we only need to check, for a l l observations, that a l l 3i and (Xi-Y-J ) terms are p o s i t i v e . 1 7 Hence from the resul ts in Table 4 . 1 , i t is c lear that the only case where regu la r i t y condit ions are sa t i s f i ed at a l l is for the group of households with education but no ch i ld ren . For these households, the proportion of sample points at which the regu lar i ty condit ions are 1 7 For de ta i l s of these regu la r i t y condi t ions, see Appendix 3. - 78 -s a t i s f i e d varies according to the rat ioning state of the household. In the SS-uncorrected case, these proportions are 0.28(U), 0.40(0) and 0.59(NR), so that overal l 54.5% of the sample points sa t i s f y the condi t ions. In the case where the SS correct ion is included, the corresponding proportions are la rger , being 0.44(U), 0.73(0) and 0.88(NR), so that here the overal l proportion sa t is fy ing the condit ions i s 82.6%. While i t is disappointing that these condit ions are not sa t i s f i ed for other groups, i t is encouraging that for households with education and no ch i ld ren , these to ta l proportions are at least as great as those obtained by Wales and Woodland [1976] who use the more f l e x i b l e Generalized Cobb-Douglas and Indirect Translog funct ional forms. 4.5.2 E l a s t i c i t y Estimates In Table 4.2 we present the implied labour supply e l a s t i c i t i e s corresponding to the parameter estimates in Table 4 .1 , evaluated at the 18 estimated point . Since these e l a s t i c i t i e s vary from observation to observat ion, we present median values of the e l a s t i c i t i e s for each household group. A par t icu lar advantage of using the median is that unl ike a weighted average, i t is not affected by extreme values. Thus, even i f some observations have large predict ion errors and hence extreme estimated e l a s t i c i t i e s , the median w i l l be representative of the majority of the indiv idual values for each e l a s t i c i t y . However when the number of observations is less than 5, no median is ca lcu la ted , since with so few observations even the median cannot necessar i ly be considered representa-t i ve of the indiv idual values. In add i t ion , since par t icu lar e l a s t i c i t i e s may be.negative for some ind iv iduals and pos i t ive for others, medians are 1 8 Expressions for these e l a s t i c i t i e s are contained in Appendix 2. - 79 -ca lcu lated separately for the pos i t i ve and negative values of each e l a s t i c i t y , provided there are at least f i ve observations of each type. One of the in terest ing aspects of the ML method used here to estimate the parameters is that , unl ike Ham [1982], we use data on both rationed and non-rationed households, and consequently are able to ca lcu la te separate e l a s t i c i t i e s for these two g roups . 1 9 In f ac t , e l a s t i c i t i e s for the rationed groups are calculated separately for households in which the head is underemployed (U) and overemployed (0) . In these rationed cases, since the le isure of the male (and hence his hours of labour) is f i x e d , the only avai lable e l a s t i c i t i e s are of female hours with respect to the female wage ra te , HFPF, and with respect to the male wage ra te , HFPM. In the non-rationed case (NR), in addit ion to HFPFand HFPM, we also have the corresponding e l a s t i c i t i e s for male hours of labour, HMPF and HMPM. By comparing the female e l a s t i c i t i e s for rationed and non-rationed households, we are able to evaluate the ef fect of the rat ioning of the male household head on the response of the wife to changes in e i ther her own or the male wage ra te . A suggestion as to what may be expected in th is regard is provided by Kneisner [1976] who shows that an increase in the male wage rate has considerably d i f ferent ef fects on male le isure depending on whether or not the i nd i v idua l ' s wife is working outside the home. While the s i tuat ion examined by Kneisner is not en t i re l y analogous to the circumstances of our model, in that the fact that a wife is not working does not necessar i ly imply that she is ra t ioned, i t does suggest that we could f ind that the e l a s t i c i t i e s d i f f e r markedly in the rationed and non-rationed cases. 1 9 In f a c t , Ham does use data on rationed households, but only in the f i r s t stage of his estimation procedure. Consequently, labour supply parameters cannot be estimated for these households. - 80 -In addit ion to these own and cross pr ice e l a s t i c i t i e s , estimates obtained also allow us to examine the so-ca l led "added worker hypothesis". In context of the rat ioning model estimated here, th is hypothesis states that when the rat ion on the male is increased, the female w i l l be more l i k e l y to increase her hours of work in order to make up the household income lost due to the increase in the male ra t i on . Of course working against th is is the "discouraged worker e f fec t " whereby the female regards the increased rat ion level on her husband's hours as being symptomatic of a t ightening of the employment s i tuat ion so that jobs w i l l be harder to f i n d . As a consequence she is discouraged from entering the labour force or t ry ing to increase the hours that she works. While evidence provided by Cain [1966] and Bowen and Finegan [1969] indicates that th is la t te r ef fect p reva i l s , both these studies are concerned with ef fects due to increasing levels of unemployment rather than changes in underemployment or overemployment. Hence i t is of some interest to examine the cross rat ioning ef fects in our model to determine whether a change in the rat ion level of the male increases or decreases the labour supply of the unrationed female. With the LES, the e l a s t i c i t y of female hours with respect to the level of the male rat ion is numerically equal to the e l a s t i c i t y of the w i fe ' s labour supply with respect to the male wage ra te , HFPM, in the rationed c a s e . 2 0 Hence the proportional response of the female, in terms of hours of labour suppl ied, is the same for a proportional increase in the wage rate of the rationed male as i t is for a proport ional easing of the rat ion level in the underemployed case or a proportional increase in the rat ion level in the overemployed case. 20 For de ta i l s of th is e l a s t i c i t y , see Appendix 2. - 81 -Turning f i r s t to a comparison of the values of the various e l a s t i c i t i e s for the SS-corrected and SS-uncorrected cases, we see from Table 4.2 that although these e l a s t i c i t i e s are s imi lar in s ize and sign for households with chi ldren and no education, d i f f e r ing resul ts are obtained for the other three groups of households. For the group with neither chi ldren nor education, the e l a s t i c i t i e s in the two cases have s im i la r s izes but opposite s igns, except for the e l a s t i c i t i e s with respect to male labour supply in the non-rationed case, where they have s imi la r signs and s i z e s . In fact these male labour supply e l a s t i c i t i e s tend to be of the same sign and approximately the same s ize for the two cases in each of the four household groups. E l a s t i c i t i e s with respect to female labour supply tend to have opposite signs and quite d i f ferent s izes in the group with chi ldren and education. Here the SS correct ion tends to change the sign and increase the magnitude of these e l a s t i c i t i e s . F i n a l l y , for the group with education but no chi ldren the signs appear to be general ly the same in the two cases although the s izes d i f f e r for some e l a s t i c i t i e s . However i t is quite d i f f i c u l t to reach any de f in i te conclusions for th is group due,to the mixture of pos i t ive and negative values for a l l the HFPF e l a s t i c i t i e s . In our previous analysis concerning the d i rec t ion of the SS b ias , we noted that at least in a non-rationing model of labour supply, we could expect the female e l a s t i c i t i e s to be downward biased in absolute value when the SS correct ion is omitted. While there is evidence of th is for households with both chi ldren and education (despite the sign changes), the other groups of households provide l i t t l e support for th is content ion. In addit ion to the fact that th is re la t ionsh ip was not necessar i ly expected to hold in a rat ioning model, the issue here is TABLE 4.2: MEDIAN ELASTICITIES FOR THE SINGLE RATION MODEL ELASTICITY c ,E c ,NE NC,E NC,NE NO SS ss NO SS ss NO SS SS NO SS SS U: HFPF + t -.006 .143 -.532 -.464 t -.119 .112 (8) -.136 (10) -.096 .096 HFPM + .009 -.195 .322 .272 -.147 -.219 .030 -.025 0: HFPF + -.007 .259 t t t -.089 .078 (7) -.030 (8) -.082 .078 HFPM + .018 -.401 t t -.230 -.359 .030 -.024 NR HFPF + t -.010 .304 t -.463 -.398 .063 (43) -.108(102) .164 (79) -.090 (66) -.085 .081 HFPM + .017 t -.330 .260 .214 -.210 t -.329 .022 -.018 HM=F + .126 .117 .042 .041 -.025 -.026 .037 .033 HNPM + .091 (25) -.113 (23) .029 (10) -.125 (38) -.177 -.182 .017 (6) -.050(139) .008 (26) -.024(119) -.222 -.219 Notes: t i n d i c a t e s fewer than 5 observations Numbers of households are contained in parentheses For Codes and a d d i t i o n a l notes, see Table 4.1 and t e x t . - 83 -fur ther confused by evaluation of these e l a s t i c i t i e s at the estimated point which means that in the SS-uncorrected case, they are further affected by the biased estimate of female hours. While analysis of the SS-corrected estimates reveals no general pattern in terms of the signs or re la t i ve sizes of the various e l a s t i c i t i e s which is common to a l l four household types, cer ta in re la t ionsh ips among some of these e l a s t i c i t i e s can be discerned. The e l a s t i c i t y of female labour supply with respect to the female wage ra te , HFPF, tends to be of opposite sign to the e l a s t i c i t y with respect to the male wage ra te , HFPM, except for those households with education but no chi ldren where the mixture of pos i t ive and negative e l a s t i c i t i e s tends to cloud th is r e s u l t . For educated households, both with and without ch i l d ren , the response of female hours worked is larger in magnitude for changes in the male wage rate than i t is for changes in the female wage ra te . However for non-educated households th is order is reversed, with |HFPF| > |HFPM|. Turning now to the d i f ferent types of rationed households -underemployed (U), overemployed (0) and non-rationed (NR) - we see that within each of the four groups of households, the signs of HFPF and HFPM do not d i f f e r according to the type of ra t ion ing . Further, apart from the group of households which have both chi ldren and education, the magnitudes of these e l a s t i c i t i e s tend to be s imi la r for underemployed and overemployed households. However for households with chi ldren and education, both e l a s t i c i t i e s are much larger in s ize for overemployed rather than underemployed households. Comparing the e l a s t i c i t i e s for rationed and non-rationed households, we see that for houshold groups with education, HFPF is larger in the NR case, although the non-rationed value - 84 -of HFPM is bracketed by the two rationed HFPM e l a s t i c i t i e s . However for the other household groups, both these e l a s t i c i t i e s are smaller in absolute value in the NR case. E l a s t i c i t i e s of male labour supply ( in the NR case) with respect to the male wage rate are predominantly negative for a l l groups, indicat ing that the male labour supply curve is "backward bending" at the estimated point . However an increase in the female wage rate w i l l increase the hours worked by the male except in households with education but no ch i l d ren . In teres t ing ly , HMPF is largest (0.12) for households with both chi ldren and education, which is the same group for which HFPF in the NR case takes i t s largest pos i t ive value (0.30). Hence for th is group of non-rationed households, le isure of the male and female appear to be gross complements in that with an increase in the male wage both work l e s s , while with an increase in the female wage ra te , both work more. For NR households in the group with chi ldren but no education, the two types of le isure appear to be gross subst i tutes since an increase in the female wage rate resu l ts in the female working less and the male working s l i g h t l y more, while for an increase in the male wage ra te , the opposite resul t i s observed. The HFPF e l a s t i c i t y tends to be pos i t i ve for households with both ch i ldren and education and with neither chi ldren nor education, but negative for those with chi ldren and no education. For the remaining households with education but no ch i l d ren , the sign is indeterminate. However in terms of absolute magnitudes, the largest responses of hours worked by the female to a change in her wage rate occur for households with ch i l d ren . On the other hand, the e l a s t i c i t y of female hours with respect to the male wage rate tends to be negative except for the group - 85 -with chi ldren and no education. The s ize of th is e l a s t i c i t y is s imi la r for households other than those with no chi ldren and no education where i t is very close to zero. In the non-rationed case, the HFPM response is larger for the educated groups of households. Interpret ing HFPM as the e l a s t i c i t y of female hours with respect to a change in the male rat ion l e v e l , i t appears that except for the group of households with chi ldren and no education, the added worker ef fect is predominant. Here, an easing of the rat ion for underemployed workers (that i s , an increase in the number of hours that can be worked by the male) resu l ts in a decrease in hours worked by the female, while in the overemployed case a t ightening of the ra t i on , meaning that the male is working even more hours than des i red, also causes a decrease in female hours. It is in terest ing to note that Ransom [1982] f inds th is e l a s t i c i t y to be pos i t i ve , suggesting that the discouraged worker e f fect predominates. Although Ransom does not e x p l i c i t l y ca lcu late th is e l a s t i c i t y , an estimate of 0.224, which is obtained on the basis of h is estimated parameters by evaluation at the sample means of his data for male and female hours, is of a s imi la r order of magnitude to the values of HFPM computed here. 4 . 5 . 3 . Comparison with Other Studies Several features of our analysis make comparison of resul ts with those obtained from ex is t ing labour supply studies quite d i f f i c u l t . F i r s t , our e l a s t i c i t i e s are calculated from parameter estimates obtained using a method which takes account of the fact that some indiv iduals in the sample are rat ioned. Further, since we use a l l observations at a l l stages of the estimation procedure, separate e l a s t i c i t i e s can be - 86 -calculated for rationed and non-rationed households. Second, we calcu late e l a s t i c i t i e s for both male and female labour supply, whereas many other studies tend to concentrate on one or other o f these measures. In addit ion such studies often focus on estimating labour supply responses for low income groups rather than the general popu la t i on , 2 1 or omit the relevant sample s e l e c t i v i t y c o r r e c t i o n . 2 2 Th i rd , rather than incorporate demographic var iables d i r ec t l y into our labour supply equations, separate estimation is performed, and e l a s t i c i t i e s obtained, for households with d i f ferent demographic cha rac te r i s t i cs . F i n a l l y , rather than evaluate our e l a s t i c i t i e s at sample (or subsample) averages, indiv idual e l a s t i c i t i e s are f i r s t calculated and then the median e l a s t i c i t y is computed separately for the pos i t ive and negative values o f these e l a s t i c i t i e s . Despite these d i s t i nc t i ve features of our study, i t is nonetheless o f in terest to determine whether our e l a s t i c i t y estimates are in any way s im i la r in s ize or sign to those obtained in various other s tudies. Considering f i r s t those studies which in some way take account o f ra t ion ing in the labour market, i t is c lear that the most relevant study for purposes o f e l a s t i c i t y comparisons is that o f Ransom [1982]. While Ransom does not ac tua l ly compute these e l a s t i c i t i e s , using his estimated parameters and information concerning the means o f relevant var iables over the ent i re sample, estimates of HFPM = -0.62 and HFPF = -0.19 can be obtained in the case where the male is ra t ioned, while for non-rationed households estimated values of HFPF = -0.20 and HMPM = -0.15 can be o b t a i n e d . 2 3 Other e l a s t i c i t i e s obtained in a rat ioning context by Ham include HMPM for non-rationed households of -0.31 (Ham [1977]) and -0.16 2 1 See, for example, the studies in Cain and Watts [1973]. 2 2 See, for example, Wales and Woodland [1976]. 2 3 Note that Ransom [1982] does not include a SS correct ion in his ana lys is . - 87 -(Ham [1982]), while Blundell and Walker [1982], assuming a l l households are rat ioned, estimate HFPF e l a s t i c i t i e s ly ing in the range -0.30 to 0.65, depending on the number of ch i ld ren . When th is assumption is removed, and a l l households are treated as non-rationed (due to the absence of any rat ion ing information in the data set which they use), Blundell and Walker estimate HMPM as -0 .29 , while HFPF in th is case ranges between -0.19 and 0.43. Although the values of the HMPM e l a s t i c i t i e s obtained in these various other studies are broadly s imi la r in both s ize and sign to the values obtained here, for rationed households the magnitude of HFPM derived from Ransom's parameter estimates is almost twice the s ize of the largest estimate of th is e l a t i c i t y obtained here for any of the four groups. A comparison of the HFPF e l a s t i c i t i e s is pa r t i cu la r l y d i f f i c u l t in view of the mixtures of signs and s izes we obtain for the d i f ferent groups. However i t appears from our resu l ts that when HFPF is negative, i t is much larger in absolute value than the estimates of -0.2 and -0 .3 found in these other s tud ies, although in those cases where i t is (predominantly) pos i t i ve , i t s absolute value does not great ly d i f f e r from these magnitudes. Turning now to estimates obtained from non-rationing s tud ies, Abbott and Ashenfelter [1976], [1979] obtain estimates of the male own wage e l a s t i c i t y varying between -0.07 and -0.14 using aggregate data, while Ashenfelter and Heckman [1973] obtain an estimate of -0.15 using cross-sect ion data. These authors also c i t e evidence from various other studies suggesting that HMPM l i e s in the range -0.07 to -0 .35 . C lea r l y , apart from the group of households with education and no ch i ld ren , our estimates of HMPM for non-rationed households l i e predominantly in th is - 88 -range. In the context of female labour supply funct ions, Schultz [1980] f inds HFPF to l i e in the pos i t ive range 0.3 to 2 .1 , while the value of HFPM is contained in the in terval -0.4 to -1 .6 . Heckman [1980] obtains point estimates of HFPF varying from 1.45 to 4.83 i f experience is treated as endogenous and the appropriate SS correct ion is made. While these estimates d i f f e r from those obtained here, in the non-rationed case HFPF is predominantly pos i t ive and HFPM predominantly negative except for households with chi ldren and no education. F i n a l l y , i t is of interest to compare our resu l ts with the non-rationed values obtained by Wales and Woodland [1976], [1977]. A par t i cu la r reason for undertaking th is comparison is due to the fact that these authors also separate pos i t ive and negative values of each e l a s t i c i t y , and in addi t ion, evaluate these e l a s t i c i t i e s for groups of households with par t i cu la r demographic p r o f i l e s , although these do not exact ly correspond to the p ro f i l es used here. One of the most noticeable di f ferences between the two sets of estimates is that while Wales and Woodland f ind a mixture of pos i t ive and negative values for almost a l l the i r e l a s t i c i t i e s , in th is study a mixture is found only for households with education, pa r t i cu la r l y those who also have no ch i ld ren . Looking at the resu l ts for the i r ent i re sample, Wales and Woodland (using no SS correct ion) f ind HMPF and HFPF to be predominantly pos i t i ve , HFPM to be predominantly negative and HMPM to have almost the same number of pos i t i ve and negative values. In our study, using the SS-corrected e l a s t i c i t i e s for non-rationed households, we f ind HFPF to be predominantly negative, HFPM and HMPF to be predominantly pos i t i ve except for households with education and no chi ldren and HMPM to be general ly negative. In add i t ion , the magnitudes of the e l a s t i c i t i e s in the two studies are broadly s im i l a r . - 89 -Comparing values for educated and non-educated households, Wales and Woodland [1976] f ind that the e l a s t i c i t i e s tend to be smaller for non-educated households, while HFPF which which has a s imi lar number of pos i t ive and negative values for educated households, is predominantly pos i t i ve for non-educated households. While i t is d i f f i c u l t to observe s imi la r types of re la t ionships among the e l a s t i c i t i e s in Table 4 .2 , due to the fact that we do not have a s ingle value for educated households and another value for non-educated households, we see that for households without ch i l d ren , the female hours e l a s t i c i t i e s tend to be smaller in magnitude for non-educated households, although for households with ch i l d ren , the magnitudes tend to be larger for those without education. In comparing households with young chi ldren to other households, Wales and Woodland [1976] f ind that the e l a s t i c i t i e s tend to have large pos i t ive and/or negative components for ch i l d less households, while HFPF which i s predominantly pos i t ive for households with young chi ldren becomes predominantly negative for other households. However, in terms of the e l a s t i c i t i e s we obtain for households with and without ch i l d ren , our resu l ts tend to support those found in Wales and Woodland [1977], where magnitudes of the male labour supply e l a s t i c i t i e s are in most cases smaller for households without ch i l d ren . 4.6 Results using only Non-Rationed Households and Comparison with  Estimates from the Complete Model In order to evaluate the ef fects and importance of cor rec t l y modell ing the behaviour of rationed households, in Tables 4.3 and 4 .4 , we present parameter estimates and labour supply e l a s t i c i t i e s obtained using the somewhat s imp l i f ied l i ke l ihood function which is appropriate when - 90 -estimation is performed using only those households in which the head is non-rat ioned. Thus, these are the types of estimates that would be obtained in the Wales and Woodland [1976], [1977] cases, for example, where a l l households not in "equi l ibr ium" are excluded from the sample. For these households, LR s t a t i s t i c s for determining whether or not to disaggregate data into households with chi ldren and households without c h i l d r e n 2 4 are calculated as 49.848 in the SS-uncorrected case and 48.152 in the SS-corrected case. For determining whether to disaggregate each of these two groups further into educated and non-educated households, the LR s t a t i s t i c s are 22.710 and 26.996 for the SS-uncorrected and SS-corrected cases for households with ch i l d ren , while for households without ch i ld ren , the corresponding s t a t i s t i c s are 30.180 and 33.416. Since the c r i t i c a l chi-square value at the 1% s ign i f icance level with 8 degrees of freedom is 20.09, these values are a l l s ign i f i can t so that we again estimate separately for each of the four groups. Comparing the parameter estimates in Table 4.3 with those for the complete model, as presented in Table 4 . 1 , we observe that although the coe f f i c ien ts in the two tables are general ly s im i l a r , certa in di f ferences can be detected, pa r t i cu la r l y for the group of households which have chi ldren and no education. In pa r t i cu la r , for th is group of households, subsistence quant i t ies of both male and female le isure decrease from the i r values in the f u l l model by between 740 and 800 hours for Ym and by between 450 and 540 hours for Yf. However subsistence expenditure on commodities by th is group increases in the NR only case (Table 4.3) and is now pos i t ive and, when the SS correct ion is excluded, s i g n i f i c a n t l y 2 i * Formally s ta ted, th is is a test of the nul l hypothesis that the parameters are the same for households with and without chi ldren against the a l ternat ive that they are d i f ferent for the two groups of households. - 91 -TABLE 4.3: PARAMETER ESTIMATES WHEN ONLY NON-RATIONED OBSERVATIONS ARE USED C,E c. NE NC.E NC.NE PARAMETER NO SS SS NO SS SS NO SS SS NO SS SS Y m .709* .696* .644* .648* .652* .648* .733* .730* (.023) (.023) (.014) (.012) (.006) (.007) (.013) (.013) Yf .773* .757* .777* .768* .715* .697* .742* .726* (.020) (.043) (.020) (.024) (.013) (.017) (.017) (.021) Y c .694 .878* .686* .943 1.852* 1.432* -1.111 -1.334 (.365) (.332) (.334) (.533) (.371) (.313) (.615) (.810) 0 m -.435* -.393* .063 .061 .048 .044 -.133* -.115* (.126) (.171) (.072) (.056) (.033) (.041) (.055) (.053) 0f -.055 .233 -.134* -.114 .098* .139* -.016 .010 (.075) (.185) (.054) (.067) (.026) (.031) (.022) (.022) 3c 1.490* 1.160* 1.071* 1.053* .855* .817* 1.149* 1.105* (.158) (.293) (.093) (.082) (.042) (.050) (.069) (.061) am .046* .047* .027* .027* .032* .032* .029* .030* (.005) (.005) (.003) (.003) (.002) (.002) (.002) (.002) ° f .036* .048* .027* .029* .024* .025* .028* .031* (.004) (.009) (.003) (.004) (.001) (.002) (.002) (.002) Pmf -.048 -.005 -.046 -.053 .044 .058 -.255* -.264* (.157) (.217) (.165) (.177) (.084) (.088) (.084) (.092) For notes and codes, see Table 4.1 - 92 -d i f fe rent from zero. Subsistence expenditure on goods also increases s l i g h t l y for households with chi ldren and education in the SS-corrected case, so that when only NR observations are considered, i t is s i g n i f i c a n t l y d i f ferent from zero. Turning to the marginal budget shares, changes also occur for the chi ldren and no education group when only NR observations are considered, with the mbs for male le isure now becoming pos i t i ve , although i ns i gn i f i can t , while the mbs for goods consumption decreases. In addi t ion, for households with no chi ldren and no education, 3 m is now s i g n i f i c a n t l y negative in both the SS-corrected and SS-uncorrected cases. A comparison of the labour supply e l a s t i c i t i e s when only the NR observations are used, as presented in Table 4.4 , with the NR e l a s t i c i t i e s obtained from the f u l l model, presented in Table 4 .2 , also reveals several d i f fe rences, pa r t i cu la r l y for the two groups of households which have ch i l d ren . While th is is to be expected for the group with chi ldren and no education due to changes in the estimated parameter values between Tables 4.1 and 4 .3 , the nature of these changes is somewhat su rp r i s ing . For th is group both own-wage e l a s t i c i t i e s , HFPF and HMPM, change from large negative values in the f u l l model to a mixture of smaller pos i t ive and negative values in the NR only model. Of the two cross-wage e l a s t i c i t i e s , HFPM remains pos i t ive although i t increases in magnitude, while HMPF changes sign and decreases in s i z e . For households with chi ldren and education, only one f u l l model e l a s t i c i t y changes sign when only NR observations are used, although the magnitude of several of the e l a s t i c i t i e s change quite not iceably. Both HFPF and HFPM tend to increase in s i z e , while in the SS-corrected case, HMPM changes from predominantly negative to predominantly pos i t i ve . F i n a l l y , regu lar i ty condit ions are TABLE 4.4: MEDIAN ELASTICITIES WHEN ONLY NON-RATIONED OBSERVATIONS ARE USED C,E C,NE NC,E NC.NE ELASTIC TY NO SS SS NO SS ss NO SS SS NO SS SS HFPF + t .379 (31) .136 (6) .164 (14) .081 (58) .157 (91) .060 - -.092 -1.754 (17) -.170 (31) -.092 (23) -.096 (87) -.081 (54) -.083 HFPM + .152 4.793 (10) .333 .288 t .026 - -.799 (38) -.227 -.331 -.017 HNPF + .112 .1 16 .049 .048 - -.021 -.022 -.023 -.024 HMPM + .091 (27) .121 (32) .006 (6) t .017 (10) .009 (38) - -.123 (21) -.090 (16) -.021 (31) -.030 -.035(135) -.018(107) -.230 -.225 Notes: t indicates fewer than 5 observations Numbers of households are contained in parentheses For Codes and additional notes, see Table 4.1 and text. - 94 -again only sa t i s f i ed for the group of households with education but no ch i l d ren . In the SS-uncorrected case, 70.3% of the sample points sa t i s f y these condi t ions, while with the SS-correct ion included, th is proportion increases to 91.7%. From these comparisons of the parameter estimates in Tables 4.1 and 4.3 and the e l a s t i c i t i e s in Tables 4.2 and 4.4, we see that the magnitudes, and in some cases the signs and s ign i f icance l e v e l s , of these measures change when we include rationed indiv iduals in the sample and appropriately model the i r behaviour. However even more important, the nature of these changes between the two models is unpredictable and var ies for the d i f fe rent demographic groups and for the d i f ferent parameters and e l a s t i c i t i e s . Thus, in view of these r e s u l t s , there seems l i t t l e reason to pers is t with models which are only estimated using the non-rationed households in the sample. While there are no studies with which we can d i r ec t l y compare the nature of the changes in parameter values and e l a s t i c i t i e s described above, studies by Ham [1977], [1982] f ind that in moving from estimates based only on the NR observations to those obtained when account is taken of male ra t ion ing , coe f f i c ien ts on the wages and income terms in the male labour supply equation are v i r t u a l l y unchanged. However, Ham f inds that s ign i f i can t changes do occur in the magnitude of the coef f i c ien ts on various socio-economic or demographic var iables such as education level ( in both s tud ies) , race and age ( in Ham [1977] only) and on the constant term (in Ham [1982] on ly ) . In add i t ion , for some var iab les , coe f f i c ien ts which are found not to be s ign i f i can t using only the NR observations, are s i g n i f i c a n t l y d i f ferent from zero in the f u l l model, and v ice -versa . In terms of di f ferences in wage e l a s t i c i t i e s , addit ional evidence based on - 95 -parameter estimates provided by Ransom [1982] suggests that while the male own wage e l a s t i c i t y does not vary in the two cases, taking values of -0.15 and -0 .16, the HFPF e l a s t i c i t y , evaluated at sample means, changes from -.136 using only the NR observations to -0.204 in the f u l l model. 4.7 Results Treating A l l Households as Non-Rationed and Comparison with  Estimates from the Complete Model In th is section we examine resu l ts obtained when rat ion ing information is ignored and a l l households in the sample are treated as non-rat ioned. We begin by examining the parameter and e l a s t i c i t y estimates obtained in th is case and compare them to those obtained for the complete model. Following these comparisons, we consider resu l ts from other studies which have attempted to evaluate the ef fects and importance of rat ion ing essen t i a l l y by invest igat ing the di f ferences between estimates obtained from rationed and non-rationed models. Various tests used for th is purpose by Ham [1982] are also examined, and we conclude by demonstrating that these or s imi lar tests cannot in general be applied to the models estimated here. 4.7.1 Comparison of Parameter and E l a s t i c i t y Estimates Tables 4.5 and 4.6 contain the parameter and e l a s t i c i t y estimates which are obtained i f the problem of some observations being rationed i s ignored; that i s , with a l l households in the sample treated as i f they are non-rat ioned. Here LR s t a t i s t i c s of 49.646 for the SS-uncorrected case and 49.104 for the SS-corrected case exceed the c r i t i c a l 1% chi-square value for 8 degrees of freedom of 20.09, reveal ing that consumption and labour supply behaviour d i f f e rs for households with chi ldren and - 96 -households without ch i ld ren . For households without ch i l d ren , consumption and labour supply patterns for those with education are s i g n i f i c a n t l y d i f fe rent at the \% level from those without education, the calculated LR s t a t i s t i c s being 30.570 and 31.572 in the SS-uncorrected and SS-corrected cases respect ive ly . LR s t a t i s t i c s of 19.556 and 18.826 are calculated for tes t ing whether the parameters are s i g n i f i c a n t l y d i f ferent for households with chi ldren and education and households with chi ldren but no education. While these s t a t i s t i c s are less than the 1% c r i t i c a l chi-square value, they exceed the 5% c r i t i c a l value which, for 8 degrees of freedom, is 15.51. Hence, estimates are again presented separately for the four groups of households. Comparing the parameter estimates in Table 4.1 (the f u l l model) with those in Table 4 .5 , we see that the main di f ferences occur for the group of households with no chi ldren and no education, although some di f ferences also occur for households with chi ldren and education. Considering f i r s t those households which have neither chi ldren nor education, i t is c lear that while the subsistence quantity of male l e i su re , Y m , increases, subsistence expenditure on goods, Yc» decreases in absolute value, so that while i t remains negative i t is now closer to zero but s ign i f i can t in the SS-uncorrected case. For these households changes also occur in the marginal budget shares, with g m increasing in s ize to the extent that i t is now s i g n i f i c a n t l y negative and 6 C increasing in s ize so that i t is now s i g n i f i c a n t l y greater than uni ty . Households with chi ldren and no education have larger estimated subsistence quant i t ies of both male and female le isure when a l l households are treated as non-rat ioned. As expected, di f ferences between the e l a s t i c i t i e s in Table 4.6 and those for the f u l l model in Table 4.2 also occur mainly for households - 97 -TABLE 4.5: PARAMETER ESTIMATES WHEN ALL OBSERVATIONS ARE TREATED AS NON-RATIONED C,E C.NE NC,E NC.NE PARAMFTFR r f\r\Aiwir_ i t_r\ NO SS SS NO SS SS NO SS SS NO SS SS Y m .742* .735* .738* .734* .658* .655* .754* .748* (.020) (.022) (.022) (.023) (.005) (.006) (.011) (.012) Yf .775* .761* .818* .812* .724* .702* .754* .733* (.016) (.030) (.015) (.020) (.012) (.017) (.015) (.021) Y c .268 .316 -.191 -.234 2.071* 1.492* -.374* -.438 (.321) (.377) (.180) (.237) (.424) (.308) (.179) (.230) 3m -.412* -.381* -.274* -.244* .035 .027 -.254* -.220* (.111) (.141) (.122) (.125) (.027) (.038) (.059) (.059) 3f -.069 .048 -.152* -.107 .093* .146* -.044 .006 (.051) (.090) (.060) (.071) (.027) (.034) (.030) (.035) 3c 1.481* 1.334* 1.426* 1.351* .872* .827* 1.298* 1.214* (.140) (.200) (.169) (.180) (.038) (.047) (.080) (.082) °m .048* .048* .030* .030* .032* .032* .033* .033* (.004) (.004) (.003) (.003) (.002) (.002) (.002) (.002) a f .032* .039* .029* .032* .026* .028* .029* .031* (.003) (.005) (.003) (.004) (.001) (.002) (.001) (.002) Pmf -.151 -.140 .003 -.014 .031 .044 -.197* -.191* (.126) (.169) (.136) (.153) (.076) (.080) (.074) (.082) For notes and codes, see Table 4.1 TABLE 4.6: MEDIAN ELASTICITIES WHEN ALL OBSERVATIONS ARE TREATED AS NON-RATIONED ELASTICITY C ,E c >NE NC,E NC,NE NO SS SS NO SS SS NO SS SS NO SS SS HFPF + -.138 .122 -.450 -.368 .073 (43) -.110(135) .163 (98) -.092 (80) -.138 .022 HFPM + .134 -.122 .251 .198 -.213 t -.354 .062 -.010 HMPF + .1 10 .114 .050 .049 -.015 -.014 .087 .087 HMPM + t -.156 .006 (7) -.147 (62) -.185 -.183 .009 (9) -.034(169) .005 (37) -.013(141) -.255 -.247 Notes: t indicates fewer than 5 observations Numbers of households are contained in parentheses For Codes and additional notes, see Table 4.1 and text. - 99 -with no chi ldren and no education and those with chi ldren and education. In households with neither chi ldren nor education, a l l e l a s t i c i t i e s in Table 4.6 are larger in magnitude than those in Table 4.2 except for HFPF and HFPM in the SS-corrected case, which are both smaller in magnitude. For households with both chi ldren and education, the HFPF and HFPM e l a s t i c i t i e s in Table 4.6 are larger in magnitude in the SS-uncorrected case and smaller in magnitude in the SS-corrected case than the corresponding e l a s t i c i t i e s in Table 4 .2 . The HMPF e l a s t i c i t i e s tend to be smaller for th is group in Table 4 .6 , while in th is same table HMPM has a larger proportion of negative values, which, in addi t ion, are larger in s ize than those in Table 4 .2 . In moving from the f u l l model to the model in which a l l households are treated as non-rationed, some small changes are also observed in various e l a s t i c i t i e s for the other two household groups. S p e c i f i c a l l y , the female e l a s t i c i t i e s for the group with chi ldren and no education tend to be s l i g h t l y larger in Table 4 .6 , while the male labour supply e l a s t i c i t i e s tend to be smaller in magnitude for households who have education but no ch i ld ren . F i n a l l y , when a l l households are treated as non-rat ioned, 53.9% of sample points in the SS-uncorrected case and 89.3% in the SS-corrected case sa t i s f y the regu la r i t y condit ions for these households which have education but no ch i ld ren . Again for a l l other types of households these condit ions are not sa t i s f i ed at any of the sample points . - 100 -4.7.2 Evidence on the Ef fects and Importance of Labour Market Rationing At th is point i t is pert inent to consider resul ts obtained in various other studies which in some way are concerned with the problem of rat ioning in the labour market and which attempt to evaluate the importance of th is ra t ion ing . While the studies by Ham [1977], [1982] are the most e x p l i c i t in th is regard, we f i r s t consider analysis conducted by Wales and Woodland [1976] and Blundell and Walker [1982]. While neither of these studies estimates a rat ioning model s im i la r to the one used here, Wales and Woodland [1976] f ind that when they estimate a t rad i t i ona l labour supply model f i r s t for equi l ibr ium (non-rationed) households and then separately for the rationed households, the hypothesis that both groups of households have u t i l i t y functions with the same parameters i s re jec ted, showing that the two groups d i f f e r s i g n i f i c a n t l y in the i r responses to wage va r ia t i ons . Using a l l the observations in the i r sample (possib ly including non- ident i f iab le rationed households), Blundell and Walker [1982] estimate the i r model f i r s t assuming there are no rationed households and then assuming that a l l males are rationed at the i r observed hours. Although the i r parameters d i f f e r in the two cases, with the mbs in the rationed case depending on the rat ion level of the male, and therefore varying across households, di f ferences in the parameters that can be compared do not appear to be s i g n i f i c a n t . In fact the authors ident i fy the major changes in the rationed case as being a decrease in the subsistence quantity of female le isure and an increased degree of responsiveness of female labour supply to the number of ch i l d ren , th i s l a t t e r ef fect being ref lected in di f ferences in female own wage e l a s t i c i t i e s calculated in the two cases for households with d i f fe rent numbers of ch i ld ren . - 101 -While Ham [1977] also f inds coef f i c ien ts based on a sample of non-rationed households to be s i g n i f i c a n t l y d i f ferent from those obtained from a sample composed of rationed households, for our purposes here the major importance of th is study and Ham [1982] l i e s in the i r comparison of resu l ts obtained from the f u l l rat ioning model, based on a sample in which some of the households are rat ioned, with those obtained when a l l households in the sample are treated as non-rat ioned. In both studies Ham found that for the male labour supply equation, the estimated coe f f i c ien ts obtained in these two cases were s imi lar for wage and income var iables but d i f fered on some of the included demographic or socio-economic va r iab les . S p e c i f i c a l l y , coe f f i c ien ts on age and number of chi ldren both increased in s ize and became s ign i f i can t in the f u l l model, while estimated coe f f i c ien ts for education and a dummy var iable re f lec t ing rac ia l di f ferences decreased in s ize and tended to become i n s i g n i f i c a n t . Thus, while coe f f i c ien ts on the major economic var iables af fect ing male labour supply were r e l a t i v e l y insens i t ive to the correct treatment of rationed ind i v idua ls , Ham [1977] concluded that the ef fects of certa in demographic var iables could be ser ious ly biased i f the rat ioning problem was ignored. In his 1982 study, Ham performs various tests to determine whether the sets of coe f f i ce in ts in these two cases are s i g n i f i c a n t l y d i f f e ren t . If the coe f f i c ien ts are found to be s i g n i f i c a n t l y d i f f e ren t , th is would, according to Ham, provide evidence that those ind iv iduals claiming to be rationed are r e a l l y constrained. To understand the nature of these tests i t is useful to re f l ec t on the select ion rule approach used by Ham to obtain estimates when some households are ra t ioned. As outl ined ea r l i e r in th is chapter, Ham's method consists of estimating equation (4.17) using only non-rationed ind iv iduals after f i r s t using data on the rationed - 102 -ind iv idua ls to estimate the X(.) var iables in that equation. Thus by comparing (4.17) with the usual labour supply function (4.3) estimated for a l l i nd i v idua ls , the essent ia l d i f ference with Ham's approach can be seen to be the fact that two extra var iables appear in (4.17), so that two addit ional coe f f i c ien t est imates, a 1 5 and a 1 6 are obtained. Hence one simple method for examining whether the coe f f i c ien ts d i f f e r in the two approaches is to conduct an F test on these addit ional coe f f i c ien ts to see i f they are s i g n i f i c a n t l y d i f ferent from zero. As noted by Heckman [1980, p. 215], i f these coef f i c ien ts are not found to be s i g n i f i c a n t , the disturbances af fect ing the sample select ion are independent of the distrubances af fect ing the behavioral functions of in te res t , so that the X(.) var iables may be omitted as regressors; in other words, OLS estimates obtained by estimating a simple labour supply function using only non-rationed indiv iduals w i l l be unbiased. While Ham [1982] conducts the appropriate F test and f inds the addi t ional coe f f i c ien ts to be s i g n i f i c a n t l y d i f ferent from zero, th is in i t s e l f only shows that OLS estimates of .the coef f i c ien ts of a labour supply function w i l l be biased i f use is only made of non-rationed ind iv iduals and no correct ion is made for th is f a c t . Although th is resu l t could be interpreted as indicat ing that the rationed are r ea l l y constrained, since i f they are not constrained we would expect the coe f f i c ien ts on these addit ional var iables to be i ns i gn i f i can t , i t does not e x p l i c i t l y show whether ignoring the rat ioning information and t reat ing a l l ind iv iduals as though they are non-rationed w i l l lead to s i g n i f i c a n t l y d i f ferent est imates. To examine th is issue, Ham uses another procedure based on spec i f i ca t ion tests developed by Hausman [1978]. Ham argues that the condit ions required for such tests are met in - 103 -h is model since estimates obtained using the Select ion Rule approach w i l l be consistent even i f there is no rat ioning (since 0^5 and which are being estimated would r e a l l y be zero) , although they w i l l not be asymptot ical ly e f f i c i en t in that case. Hence, denoting 3Q^<. as the estimates obtained by applying OLS to the f u l l sample and 8<.R as the estimates obtained with the Select ion Rule Approach, the quadratic form <S'[V(<$)]" 6 w i l l be asymptot ical ly d is t r ibu ted as chi-square with q degrees of freedom. Here, 6 = ( B SR" S 0LS^ ' V ^ E V ^ S R ^ _ V ^ 0 L S ^ ' w n e r e V(.) denotes the estimated variance-covariance matrix of the relevant parameters, and q is equal to the number of parameters being compared. In addit ion to f inding that the parameter estimates are s i g n i f i c a n t l y d i f ferent in the case where OLS is applied to the sample comprising a l l ind iv idua ls as compared to the case where the Select ion Rule method is used, so that those who claim to be rationed appear to be genuinely constra ined, Ham also f inds that the change in the parameter estimates between these two cases is in the expected d i r ec t i on . As noted in Chapter 2, i f some indiv iduals are r e a l l y constrained but a l l are treated as being non-rat ioned, OLS estimates w i l l compound the ef fect of a var iable on desired labour supply with i t s ef fect on unemployment or underemployment. Thus, the coe f f i c ien ts on var iables that decrease unemployment or underemployment w i l l be overestimated while coe f f i c ien ts on those var iables which increase these measures w i l l be underest imated. 2 5 Hence, from the signs of the estimated coe f f i c ien ts in the probit equations determining the probab i l i t y of unemployment and of underemployment, we can determine the expected d i rect ions of the biases i f OLS is used. Since age and education have negative coe f f i c ien ts in these equations while the race For d e t a i l s , see Ham [1982, p. 337]. - 104 -dummy var iable and number of chi ldren have pos i t ive coe f f i c i en t s , when comprison of 3QL«~ and B ^ reveals that the coef f i c ien ts of age and education are overestimated using OLS while those for number of chi ldren and race are underestimated, the resu l ts are as expected. Thus th is adds fur ther support for Ham's conclusion that those who claim to be rationed are r ea l l y constrained. While i t would be desirable to formal ly conduct tests of a s imi lar nature using the models estimated here, i t is quite straightforward to show that the par t icu lar l i ke l ihood function formulation of the rat ioning model that we use does not eas i l y lend i t s e l f to such t e s t s . Using notation and analysis from the previous chapter, and ignoring SS correct ions for ease of expos i t ion , the l i ke l ihood function for the f u l l ra t ion ing model, L R , and for the model in which a l l households are treated as non-rat ioned, LN,R, can be expressed as: ( 4 * 1 9 ) 4 = NR 9 ( V 6 f ) ' S ^ ( e " i ' e r ) d e " i ' J f 9 ( V e r ) d e m (4.20) L R = n g(e , e f ) m NR+U+0 where A = L ° - L (.), e is the error on the male le isure equation, e r m nr ' ' m ^ ' ' and ef are the errors on the female le isure equations when the male is rat ioned and unrationed respec t ive ly , and U, 0 and NR refer to underemployed, overemployed and non-rationed households. One major d i f f i c u l t y in test ing the nul l hypothesis that there is no rat ioning is the fact that these two l ike l ihoods are, in general , non-nested in the sense that there appear to be no parametric res t r i c t i ons which, when applied to LR, w i l l y ie ld LN R . This general non-nested property can - 105 -be i l l u s t ra ted in two d i f ferent ways. F i r s t , by comparing the funct ional expressions for the female le isure share when the male is ra t ioned, (3 .31) , and when there is no rat ioning (3.29), i t is c lear that not only do these expressions d i f f e r , but in addit ion there are no parametric r es t r i c t i ons whereby (3.31) can be s imp l i f ied to (3.29). Thus, even ignoring the truncation aspects of the rat ioning model and t reat ing rat ion ing and non-rationing as being two d i f ferent regimes in a simple switching regression model, i t is c lear that the l ike l ihoods are non-nested. A second way of viewing th is non-nestedness question is to concentrate on the truncation rather than the rat ioning aspects of the model. Omitting common terms from the two l ike l ihoods and wr i t ing a l l j o in t densi t ies as the product of marginal and condi t ional dens i t i es , parameter res t r i c t i ons required for L ^ R to be nested in LR are those that w i l l ensure the equal i ty : (4.21) n h(e„) -n / A g (e le )de • n f g(eJe )de = n g(ef )• n g(e |ef) U+o r U -» 1 O A 1 m U+0 U+0 1 Now to concentrate on the truncation aspect, assume for the time being that there do ex is t parameter res t r i c t i ons such that the female share equation in the case where the male is rationed s imp l i f i es to the same expression as in the unrationed case. In such a s i t ua t i on , the expressions subst i tuted for ef and e r in (4.19) and (4.20) w i l l be the same under the nul l hypothesis, so that the unconditional univar iate densi t ies on both sides of (4.21) can be cancel led. However even in th is s imp l i f ied case i t is c lear that while the remaining term on the rhs would be the product of condit ional dens i t i es , those on the lhs const i tute the - 106 -product of the areas contained in par t icu lar regions enclosed by these 26 same density funct ions. Thus, in general , LR and L^R are non-nested spec i f i ca t ions so that t e s t s , such as the Likel ihood Ratio test used e a r l i e r , are inapp l icab le . For essen t ia l l y s imi lar reasons, use of the Hausman [1978] t es t , as applied by Ham [1982], is also inappropriate with the present model formulat ion. Use of th is test would f i r s t require completion of the non- t r i v i a l task of demonstrating that the estimates obtained using the f u l l rationed model w i l l be consistent under the nul l hypothesis of no ra t ion ing . While such analysis is considerably beyond the scope of th i s study, even i f the required consistency was obtained we would s t i l l be unable to d i r e c t l y apply th is test to the parameter estimates obtained here. The reasons for th is can be seen from an examination of the parameter estimates and standard errors in Tables 4.1 and 4 .5 . Comparing values in the two tab les , i t is c lear that while the standard errors of some parameters in the f u l l rationed model exceed the corresponding magnitudes in the model where a l l households are treated as non-rationed, for other parameters the d i rec t ion of th is inequal i ty concerning re la t i ve s ize of standard errors is reversed. Consequently, we are unable to fo l low Hausman's procedure in obtaining an estimate of the variance-covariance matrix of the di f ferences in parameter estimates between the two models as the di f ference between the respective variance-covariance matrices for the two sets of est imates, since some of the variances obtained in th is way would be negative. On the basis of the above analysis i t appear that in order to test whether di f ferences between parameter estimates in the rat ioning model and 2 6 This qua l i f i ca t i on is included to cover the un l i ke ly p o s s i b i l i t y that the value of the integral is exact ly equal to the densi ty. - 107 -those in the model where a l l households are treated as non-rationed are s i g n i f i c a n t , use of some other non-nested tests is required. While many of these t es t s , based on the work of Cox [1961], [1962], have been developed in the context of l inear regression problems, only recent ly with the work of Pesaran and Deaton [1978] have these tests been extended to deal with choices between competing systems of nonlinear regression equations. However, in general the development of these tests has far surpassed the i r appl icat ions to prac t ica l empirical problems, possibly due to the fact that in dealing with the complexit ies which tend to be charac te r i s t i c of in terest ing applied problems, few of these tests are computationally t rac tab le . Although Deaton [1978] performs non-nested tests for a l ternat ive spec i f i ca t ions of nonlinear systems of demand equations, his methods are not d i r e c t l y appl icable here since the demand functions he considers are assumed to have well-behaved normally d is t r ibu ted disturbances, whereas the error d is t r ibu t ions here are truncated due to the rat ioning of male l e i su re . While Cox-type tests for comparing probit and log i t models involving d iscrete dependent var iables have been developed by Morimune [1979], appropriate tests do not appear to have been devised for Tobit-type estimation which deals with truncation of the dependent var iable at some point . An addit ional problem concerning the appl icat ion of non-nested tests to problems of the type formulated here is that many of these tests require use to be made of analyt ic expressions for the der ivat ives of the l i ke l ihood funct ion . However with a complex l i ke l i hood , such as the one estimated here, optimization is performed using numerical der ivat ives for the very reason that such analyt ic der ivat ives cannot be eas i l y obtained, or evaluated. One study which addresses th is type of problem is - 108 -Aneuryn-Evans and Deaton [1980], where numerical methods are used to evaluate expectation terms which cannot be determined a n a l y t i c a l l y . While the reason for th is complexity in the i r model is due to truncation of the error term in one of the i r two competing hypotheses, the i r test procedure is developed in the spec i f i c context of test ing l inear versus logarithmic regression models, and as such is not d i r e c t l y appl icable to the rat ioning model which we estimate here. Thus i t seems clear from the preceding analysis that in terms of test ing the rat ioning versus the non-rat ioning models using the l i ke l ihood funct ion approach, neither the required theoret ica l resu l ts nor a computationally t ractable method of implementation are current ly ava i lab le . C lear ly these remain as important issues for further reasearch. 4.8 General Evaluation of Results At th is stage i t is convenient to summarize our main f indings both in terms of the di f ference between the f u l l model and models which involve jus t the non-rationed households or which t reat a l l households as non-rat ioned, and on the basis of observed re la t ionships among the estimated parameters and e l a s t i c i t i e s in these three models. Comparing the resu l ts obtained with the complete model as presented in Tables 4.1 and 4.2 with the resul ts in the other tab les , i t is c lear that while there are di f ferences between the a l ternat ive sets of estimates, there seems to be no systematic pattern to these d i f ferences. In moving from the complete model to an incor rec t ly speci f ied model which uses only the non-rationed observations, di f ferences in the parameter estimates are seen to occur mainly for the two groups of households with ch i l d ren , pa r t i cu la r l y those which also have no education. However, i f instead of estimating the - 109 -correct complete model we treat all households as though they are non-rationed, the main differences in parameter estimates tend to occur for the groups of households which have neither children nor education and those which have both children and education. Unfortunately, our overall analysis is limited to some degree by the lack of significance of many of the parameter estimates. For households with education but no children, all parameters except 3 m are significantly different from zero in the correct model and the two alternative models. It is also interesting to note that this is the only group of households for which the regularity conditions are satisfied at any sample point. For these households the parameter estimates are very similar in all three models, so that the effects of misspecification here would appear to be minimal. Of course despite these similarities in the parameter estimates, one major disadvantage of the misspecified models is that they do not yield any information in the rationed case, such as marginal budget shares or elasticities. In terms of the effect of the sample selectivity correction for working wives, the parameter estimates obtained here seem to provide further evidence that the effect of this correction on the estimated coefficients is quite small. However, it is interesting to observe that despite what may appear to be negligible changes in the parameter estimates, the combined effect of these changes is sufficient in many cases to change the size, and in several instances the sign, of some of the implied labour supply elasticities. In relation to our earlier analysis concerning the likely effects on the estimated parameters and elasticities of including the SS correction, the results are quite ineffective in providing any general information - 110 -concern ing the d i r e c t i o n of t h i s b i a s . However, from an examinat ion of the va lues of the est imated parameters which are s i g n i f i c a n t in both the SS-co r rec ted and SS-uncor rec ted c a s e s , i t i s p o s s i b l e to make some general o b s e r v a t i o n s . The mbs fo r female l e i s u r e , 3 f , i s always l a rge r f o r households wi th educat ion but no c h i l d r e n when the SS c o r r e c t i o n i s i n c l u d e d , wh i le the mbs fo r male l e i s u r e , Bm» i s g e n e r a l l y sma l le r in abso lu te va lue when the SS c o r r e c t i o n i s inc luded f o r households with both c h i l d r e n and educat ion and those wi th ne i t he r c h i l d r e n nor e d u c a t i o n . For a l l g roups, 3 C i s g e n e r a l l y sma l le r wh i le of and ( i n the ra t i oned model) o r are always l a r g e r in the SS-co r rec ted c a s e . U n f o r t u n a t e l y , in most of these c a s e s , these d i f f e r e n c e s are sma l le r than the standard e r r o r of the c o e f f i c i e n t s , and thus do not appear to be s i g n i f i c a n t . S i m i l a r mixed evidence concern ing the e f f e c t s of the SS c o r r e c t i o n i s obta ined from the est imated e l a s t i c i t i e s . For the f u l l model and the model us ing on l y the NR o b s e r v a t i o n s , the female e l a s t i c i t i e s tend to be l a r g e r in abso lu te va lue f o r households wi th c h i l d r e n and educat ion when the SS c o r r e c t i o n i s i n c l u d e d , and there i s some weak evidence tha t t h i s a l so a p p l i e s to those wi th educat ion but no c h i l d r e n . However f o r households wi th c h i l d r e n but no e d u c a t i o n , these e l a s t i c i t i e s are sma l l e r in abso lu te va lue when the SS c o r r e c t i o n i s i n c o r p o r a t e d . For the model i n which a l l observa t ions are t rea ted as n o n - r a t i o n e d , the female e l a s t i c i t i e s are g e n e r a l l y sma l le r in abso lu te va lue in the S S - c o r r e c t e d case except f o r households which have educat ion but no c h i l d r e n . From an examinat ion of the est imated parameters in the th ree mode ls , i t i s c l e a r tha t on ly the subs is tence q u a n t i t i e s of male and female l e i s u r e , Y m and Y f r e s p e c t i v e l y , and the mbs f o r goods consumpt ion, 3 C , are always s i g n i f i c a n t . In general Yf exceeds Ym. p a r t i c u l a r l y - I l l -for households with chi ldren and no education, where yf takes i t s largest values. On the other hand, ym is smallest for households with education but no ch i ld ren . For households without education, the only s ign i f i can t mbs is the value for goods consumption, 3 C . Of the remaining households, those with chi ldren have a s ign i f i can t negative value of the mbs for male l e i su re , 3 m , so that a unit increase in income would resu l t in a decrease in male le isure of 0.3 un i ts , an increase in female le isure of 0.13 units (which, however, is not s i g n i f i c a n t l y d i f fe rent from zero) and an increase in goods consumption of 1.17 un i ts . Households with education and no chi ldren have a mbs for female le isure which is s i g n i f i c a n t l y pos i t i ve , so that a one unit increase in income for these households would resu l t in an increase in female le isure of 0.14 un i t s , in goods consumption of 0.82 un i ts , and would leave male le isure v i r t u a l l y unchanged. General resul ts which are obtained with the SS-corrected e l a s t i c i t i e s for non-rationed households in the correct model and the two misspeci f ied models can be summarized as fo l lows. The male own wage labour e l a s t i c i t y HMPM, is negative for a l l household types, and is smaller for the groups of households with education and no ch i l d ren . The corresponding cross wage e l a s t i c i t y , HMPF, which tends to be pos i t ive except for th is group of households, is larger for households with chi ldren and education. Except for households with chi ldren but without education, the female cross wage e l a s t i c i t y , HFPM is negative, taking i t s smallest value for households with neither chi ldren nor education. F i n a l l y , HFPF is predominantly pos i t i ve for households with chi ldren and education and with neither chi ldren nor education, predominantly negative for households with chi ldren and no education, and indeterminate in sign for the remaining - 112 -households which have education but no ch i l d ren . Like HFPM, th is e l a s t i c i t y also tends to be smallest for households with neither ch i ldren nor education. In the three d i f ferent models HFPF tends to be f a i r l y constant for households with education and no ch i ld ren , although i t var ies considerably for households with ch i ld ren , pa r t i cu la r l y in the model estimated using only non-rationed households. 4.9 Empir ical Results for Models where Unemployed are Grouped with  Underemployed One problem with the analysis to th is point concerns the treatment of ind iv idua ls who claim to have been unemployed during the year for a non-zero number of weeks. In c lass i f y i ng households as underemployed, overemployed or non-rat ioned, th is information has essen t i a l l y been ignored here, with decisions on the rat ioning state of the household being based on responses of the household head to questions concerning the re la t ionsh ip between desired and actual hours worked. While th is approach is common to several studies which use the Michigan data, including Wales and Woodland [1976] and Ransom [1982], an a l ternat ive procedure adopted by Ham [1977], [1980] involves grouping households in which the head is unemployed for any time during the year with those households in which the head is underemployed, even though he may not desire to work any more than his current number of hours. In order to determine the ef fect of th is r e c l a s s i f i c a t i o n on the parameter and e l a s t i c i t y est imates, both the complete model and the model using only the NR observations are re-estimated with the data c l a s s i f i e d as in Ham's s tudies. Tables 4.7 and 4.8 contain the parameter estimates and the implied labour supply e l a s t i c i t i e s for the complete model re-estimated with th is - 113 -a l ternat ive data c l a s s i f i c a t i o n . Again resu l ts are presented separately for the four groups. LR s t a t i s t i c s of 56.444 and 52.926 in the SS-uncorrected and SS-corrected cases exceed the c r i t i c a l 1% chi-square value with 10 degrees of freedom of 23.21 so that parameters are s i g n i f i c a n t l y d i f ferent for households with and without ch i ld ren . While parameters for educated and non-educated households with chi ldren are s i g n i f i c a n t l y d i f fe rent at the 1% l e v e l , the calculated LR s t a t i s t i c s being 32.40 and 34.34 in the SS-uncorrected and SS-corrected cases, fo r households without chi ldren the corresponding LR s t a t i s t i c s of 18.952 and 22.82 are only s ign i f i can t at the 5% l e v e l , exceeding the c r i t i c a l value of 18.31. Comparing the parameter estimates in Table 4.7 with those in Table 4.1 we see that the major ef fect of the r e c l a s s i f i c a t i o n is to change the s ign i f i cance of some of the parameter est imates, pa r t i cu l a r l y the mbs for male l e i s u r e , 3 m . S p e c i f i c a l l y , s ix of the parameter estimates change from being s ign i f i can t in Table 4.1 to being ins ign i f i can t in Table 4 .7 , although f i ve of these six estimates occur in the case where the SS correct ion is omitted. In add i t ion , two other parameter estimates which were not s i g n i f i c a n t l y d i f ferent from zero in Table 4 . 1 , are found to be s ign i f i can t in Table 4.7. Despite these changes in s ign i f icance l eve l s , the actual values of the parameters do not appear to change s i g n i f i c a n t l y ; in pa r t i cu la r , for households with education but no chi ldren the estimates are remarkably s imi la r in the two tab les . Examining the e l a s t i c i t i e s in Table 4 .8 , i t is c lear that the r e c l a s s i f i c a t i o n of households tends to have very l i t t l e ef fect for households without ch i l d ren . For those households with chi ldren and no education the e l a s t i c i t i e s are s imi lar to those in Table 4.2 in terms of - 114 -TABLE 4.7: PARAMETER ESTIMATES FOR THE SINGLE-RATION MODEL UNDER DATA RECLASSIFICATION C,E C,NE NC,E NC.NE PARAMETER NO SS SS NO SS SS NO SS SS NO SS SS Ym .693* .682* .720* .711* .654* .648* .721* .719* (.024) (.026) (.023) (.027) (.007) (.006) (.013) (.012) Yf .744* .695* .813* .804* .727* .711* .736* .721* (.016) (.030) (.017) (.026) (.012) (.016) (.013) (.016) Y c .575 .008 -.910 -1.470 2.215* 1.662* -2.525 -3.042 (.506) (.649) (.612) (1.571) (.446) (.327) (1 .856) (1 .918) 3m -.293* -.153 -.179 -.123 .058* .055 -.082 -.070 (.121) (.099) (.098) (.116) (.028) (.037) (.052) (.039) 3f .006 .121 -.078 -.028 .086* .129* -.007 .009 (.053) (.068) (.047) (.046) (.026) (.031) (.011) (.011) 3C 1.287* 1.032* 1.257* 1.151* .857* .817* 1.089* 1.062* (.144) (.110) (.138) (.153) (.039) (.045) (.059) (.038) CTm .047* .048* .025* .025* .032* .032* .030* .030* (.005) (.005) (.003) (.004) (.002) (.002) (.002) (.002) Of .037* .045* .028* .035* .024* .026* .028* .030* (.004) (.006) (.004) (.008) (.001) (.002) (.002) (.002) ° r .019* .019* .030* .035* .031* .033* .030* .033* (.003) (.003) (.004) (.007) (.003) (.004) (.002) (.003) Pmf .008 .060 -.234 -.322 .064 .081 -.196* -.249* (.157) (.170) (.192) (.237) (.089) (.095) (.096) (.103) Pmr -.477 -.224 -.206 .054 -.007 -.010 -.444* -.377* (.247) (.385) (.365) (.482) (.164) (.169) (.142) (.163) rmbs-f .005 .105 -.066 -.025 .091* .136* -.006 .008 rmbs-c .995* .895* 1.066* 1.025* .909* .864* 1.006* .992* (.041) (.058) (.036) (.039) (.028) (.032) (.010) (.010) For notes and codes, see Table 4.1 TABLE 4.8: MEDIAN ELASTICITIES FOR THE SINGLE RATION MODEL UNDER DATA RECLASSIFICATION ELASTICITY c ,E c ,NE NC ,E NC.NE NO SS SS NO SS SS NO SS SS NO SS SS U: HFPF + .008 t .362 -.446 -.273 t -.150 .077 (9) -.144 (17) -.066 .100 HFPM + -.013 -.303 .157 .071 -.135 -.204 .015 -.021 0: HFPF + t t t t t -.128 .026 (6) -.059 (8) -.054 .081 HFPM + t t t t -.218 -.345 .016 -.021 NR HFPF + .013 .400 (38) -4.522 (6) -.416 -.247 .074 (31) -.119(107) .130 (72) -.117 (66) -.060 .091 HFPM + -.014 3.502 (6) -.286 (38) .152 .069 -.202 t -.329 .012 -.017 HMPF + .096 .070 .032 .025 -.024 -.025 .029 .028 HMPM + .043 (12) -.101 (32) -.121 -.239 -.239 t -.061 .011 (19) -.033(119) -.235 -.238 Notes: t Indicates fewer than 5 observations Numbers of households are contained in parentheses For Codes and additional notes, see Table 4.1 and text. - 116 -s ign , although except for HMPM, the e l a s t i c i t i e s in Table 4.8 tend to be smaller in magnitude than those in Table 4 .2 . The major di f ferences between the e l a s t i c i t i e s in the two tables occur for households with both chi ldren and education, where rec lass i f y ing the observations causes a change in the signs on the e l a s t i c i t i e s in the SS-uncorrected cases, except for the male e l a s t i c i t i e s for NR households. Thus unl ike the values in Table 4 .2 , e l a s t i c i t i e s for th is group of households now have the same sign in both the SS-corrected and SS-uncorrected cases. Despite these changes in s igns, the magnitudes of the SS-uncorrected e l a s t i c i t i e s tend to be s imi la r in the two tab les . However the SS-corrected e l a s t i c i t i e s for the underemployed households in th is group are larger in magnitude in Table 4.8, while the magnitudes of the male e l a s t i c i t i e s HMPF and HMPM both decrease in the SS-corrected case when the data are r e c l a s s i f i e d . F i n a l l y , regu la r i t y condit ions are again only sa t i s f i ed for households with education and no ch i ld ren . With the data r e c l a s s i f i c a -t i o n , the proportion of sample points at which the regu la r i t y condit ions are sa t i s f i ed decreases from the previously obtained values, so that without the SS correct ion the proportions are 0.23(U), 0.21(0) and 0.46(NR), or a to ta l proportion of 41% while the proportions when the SS correct ion is included are now 0.42(11), 0.64(0) and 0.80(NR) for a to ta l proportion of 73.6%. To complete our analysis we examine the ef fects of grouping unemployed with underemployed households on the estimated parameters and e l a s t i c i t i e s obtained from the model which is estimated using only non-rationed households. Comparing the parameter estimates for th is case, as presented in Table 4 .9 , with those for the model using only NR households from the or ig ina l grouping of observat ions, as presented in Table 4 .3 , i t - 117 -TABLE 4.9: PARAMETER ESTIMATES UNDER DATA RECLASSIFICATION WHEN ONLY NON-RATIONED OBSERVATIONS ARE USED. C.E C.NE NC ,E NC ,NE PARAMETER NO SS SS NO SS SS NO SS SS NO SS SS Ym .708* .684* .624* .641* .651* .647* .731* .729* (.024) (.030) (.018) (.034) (.006) (.007) (.013) (.013) Yf .780* .754* .780* .763* .719* .696* .736* .720* (.022) (.056) (.024) (.043) (.014) (.021) (.017) (.020) Yc .472 .895* .917* 1.274 1.969* 1.448* -1.661 -1.942 (.452) (.448) (.279) (.885) (.419) (.363) (1.016) (1 .215) 3 m -.362* -.320 .1 62* .135 .050 .041 -.108* -.096* (.132) (.195) (.066) (.078) (.031) (.044) (.053) (.049) 3f -.071 .242 -.159* -.124 .095* .141* -.009 .011 (.065) (.225) (.067) (.127) (.027) (.034) (.017) (.017) 3 C 1.433* 1.078* .997* .989* .856* .818* 1.117* 1.085* (.161) (.362) (.088) (.092) (.042) (.051) (.062) (.051) O m .046* .047* .024* .024* .031* .031* .028* .028* (.005) (.005) (.003) (.003) (.002) (.002) (.002) (.002) Of .037* .050* .025* .028* .024* .026* .028* .030* (.004) (.010) (.003) (.005) (.001) (.002) (.002) (.002) Pmf -.081 .013 .011 -.029 .065 .071 -.202* -.215* (.166) (.249) (.199) (.227) (.085) (.090) (.091) (.098) For notes and codes, see Table 4.1 - 118 -i s c l e a r that on ly minor changes in these es t imates are reco rded . While the SS-co r rec ted va lue of B m i s no longer s i g n i f i c a n t l y d i f f e r e n t from zero f o r households with c h i l d r e n and e d u c a t i o n , the va lue in the SS-uncorrected case f o r households wi th c h i l d r e n and no educat ion i s now s i g n i f i c a n t . In f a c t the on ly notab le changes in parameter va lues occur f o r t h i s group of households, wi th B m i n c r e a s i n g and Ym d e c r e a s i n g . The parameter es t imates are again very s i m i l a r in the two t ab les f o r households wi th educat ion but no c h i l d r e n . R e g u l a r i t y c o n d i t i o n s which are on ly s a t i s f i e d f o r t h i s group of households are met f o r 63% of the sample po in ts in the SS-uncor rec ted case compared to 70.3% with the o r i g i n a l data c l a s s i f i c a t i o n , wh i le in the case where the S S - c o r r e c t i o n i s i n c l u d e d , the p ropor t i ons which s a t i s f y the c o n d i t i o n are the same under both data c l a s s i f i c a t i o n s at 92%. F i n a l l y , the e l a s t i c i t i e s in Tables 4 .4 and 4.10 a l so appear to be qu i t e s i m i l a r , p a r t i c u l a r l y f o r households wi th no c h i l d r e n . For those households wi th c h i l d r e n and no e d u c a t i o n , the s igns of the e l a s t i c i t i e s are the same in both cases a l though , f o r the most p a r t , they tend to be l a r g e r in Table 4.10 under the data r e c l a s s i f i c a t i o n . For households wi th c h i l d r e n and e d u c a t i o n , SS-uncor rec ted es t imates of female e l a s t i c i t i e s tend to be l a r g e r in magnitude in Table 4.10 than in Table 4 . 4 , wh i le SS-uncor rec ted es t imates of male e l a s t i c i t i e s tend to be sma l le r in magnitude in Table 4 . 1 0 . As a r e s u l t of t h i s a n a l y s i s us ing data in which unemployed households are grouped wi th underemployed househo lds , i t appears tha t t h i s s imple data r e c l a s s i f i c a t i o n i s i n s u f f i c i e n t to cause any s i g n i f i c a n t improvement in the parameter es t imates or imp l ied labour supply e l a s t i c i t i e s . Thus, in order to improve the r e s u l t s obta ined i t would TABLE 4.10: MEDIAN ELASTICITIES UNDER DATA RECLASSIFICATION WHEN ONLY NON-RATIONED OBSERVATIONS ARE USED C,E C,NE NC,E NC,NE ELASTIC TY NO SS SS NO SS SS NO SS SS NO SS SS HFPF + .306 (26) .249 (5) .188 (11) .075 (48) .172 (88) .081 - -.156 -2.093 (18) -.174 (23) -.031 (17) -.102 (90) -.086 (50) -.057 HFPM + .161 4.453 (12) .451 .348 t .015 - -.828 (32) -.224 -.356 -.019 HM°F + .088 .100 .040 .039 - -.046 -.046 -.022 -.020 HM=>M + .044 (12) .085 (28) t t .017 (6) .007 (35) - -.120 (32) -.075 (16) -.071 -.102 -.042(132) -.017(103) -.240 -.241 Notes: t indicates fewer than 5 observations Numbers of households are contained in parentheses For Codes and additional notes, see Table 4.1 and text. - 120 -appear necessary ei ther to use d i f ferent funct ional forms for both the rationed and unrationed demand equations or to adopt a d i f ferent approach to the rat ioning problem. In view of the unresolved d i f f i c u l t i e s encountered with a l ternat ive representations of household preferences as explained previously , in the fol lowing chapter we pursue the la t te r a l ternat ive by decomposing the rat ion on the male household head into two components. - 121 -CHAPTER FIVE RATIONING OF WEEKS AND HOURS OF LABOUR SUPPLY 5.1 Introduction One pa r t i cu la r l y noticeable charac te r i s t i c of the data used in th is study concerns the seemingly inconsistent responses of household heads to cer ta in questions. S p e c i f i c a l l y , while many household heads claim to be non-rationed in the hours they work, they also spent part of the year unemployed. Ham's [1977] solut ion to th is problem is to group such households with the underemployed, suggesting therefore that in these households, the head desired to work more hours. While i t is ce r ta in ly possible that questions concerning the re la t ionsh ip between desired and actual hours were simply misunderstood by the respondents, a more in t r igu ing p o s s i b i l i t y is that the problem of rat ioning has a greater dimensional i ty than previously considered. Thus, while the household head may be content with hours worked per week, he may desire to work a d i f ferent number of weeks per year than his current employment d i c ta tes . The idea of analyzing the multidimensional nature of time a l locat ion appears to have been considered f i r s t by Hanoch [1980a]. While recognizing that an indiv idual is not usual ly ind i f ferent between working 100 hours per week for 20 weeks and working 40 hours per week for 50 weeks, he also noted that the only use made of the d i s t i nc t i on between hours of work and annual weeks of work were in studies such as that by Rea [1974], where these var iab les are considered as a l ternat ive measures of the dependent var iab le in a labour supply equation. In order to allow for the fact that le isure during working weeks and le isure during non-working weeks are not - 122 -perfect subst i tu tes , Hanoch includes them as separate arguments in the u t i l i t y funct ion. Although Hanoch [1980a] develops a l ternat ive models for ind iv idua ls and for households consist ing of two people, empirical appl icat ions by Hanoch [1980b] are l imi ted to the indiv idual model which is estimated using a sample of married women. In th is chapter we construct a household model which u t i l i z e s the idea on which Hanoch's studies are based. Thus we specify a household u t i l i t y function which includes the le isure of the household head during working weeks, his le isure during non-working weeks, the le isure of his spouse and household consumption as separate arguments. As in Hanoch's models, le isure of the head during working weeks and le isure during non-working weeks are not assumed to be perfect subst i tu tes . However, in order to avoid the addit ional complexit ies involved in examining the separate ef fects of rat ioning of both types of male le isure on two types of female l e i su re , we s impl i fy our analysis by assuming that le isure of the spouse is s ing ly dimensioned. With th is new u t i l i t y function formulat ion, we now extend the model of the previous section by taking account of the fact that the household head may be rationed in the number of weeks he can work in a year as well as in the number of hours he can work in any week. Thus, in addit ion to being overemployed, underemployed or non-rationed in terms of whether desired le isure during a working week is greater than, less than, or equal to actual le isure during that week, the household head may be ei ther underemployed or non-rationed in terms of whether the number of weeks he desires to work during the year exceed, or are equal to , the actual number of weeks worked during the year ( including paid vacat ions) . Although in some cases the household head may desire to work fewer than the number of - 123 -weeks he ac tua l ly works in a year, we exclude th is case from the model since the avai lab le data does not permit these par t icu lar ind iv iduals to be i d e n t i f i e d . The operational measure used to ident i fy households as being underemployed or non-rationed in the number of weeks worked i s whether or not the household head was unemployed for any weeks during the year. Due to the complexity of the resu l t ing model, and the lack of success experienced using other funct ional forms even with the simpler s ing le - ra t ion model, we l im i t our attention here to equations which are derived from the Stone-Geary u t i l i t y func t ion . The plan of the remainder of th is chapter is as fo l lows. We begin our analysis of the two-ration model by considering the relevant u t i l i t y maximization problem under a l ternat ive forms of ra t ion ing . Next, we examine various issues concerning the appropriate stochast ic spec i f i ca t ion for the demand equations obtained from these constrained maximization problems. The l i ke l ihood function is derived and necessary sample s e l e c t i v i t y correct ions spec i f i ed . In Section 5 .3 , the spec i f i c demand functions corresponding to the Stone-Geary u t i l i t y function are obtained. Questions concerning iden t i f i ca t i on of the parameters of these equations are examined in the f i n a l sec t ion. 5.2 Model Spec i f i ca t ion In th is section we develop a model which allows us to examine household labour supply and commodity demand behaviour for a mixed sample of rationed and non-rationed households. The three forms of rat ioning that are considered include the s i tuat ions where the le isure of the male during working weeks is ra t ioned, the le isure of the male during non-working weeks is rationed and where both these forms of male le isure are - 124 -ra t ioned. We begin by formulating the u t i l i t y maximization problem and der iv ing the corresponding le isure demand and goods consumption equations for these various rat ioning s i tua t ions . Next, in Section 5 .2 .2 , attention is paid to the appropriate stochast ic spec i f i ca t ion for the model. While the f i r s t proposal considered is shown to resul t in computational complexit ies which would make estimation of the model in t rac tab le , an a l ternat ive spec i f i ca t ion involves a nonlinear constraint among the stochast ic disturbances. After determining the stochast ic s p e c i f i c a t i o n , the l i ke l ihood function is derived in Section 5 .2 .3 . Detai ls of the sample s e l e c t i v i t y cor rec t ion , required since we only include households in which the wife is working, are contained in the f i n a l sec t ion . 5.2.1 Formulation of the Basic Model We assume that the household maximizes a u t i l i t y funct ion (5.1) U = U ( L ! , L 2 , L f , X ) , where Li and L 2 are the hours of le isure of the head during working weeks and non-working weeks respect ive ly , both converted into annual hours, Lf i s the le isure of the female, and X is household consumption of a composite commodity. In i t s maximization of (5.1) the household faces the budget cons t ra in t : 1 (5.2) M = p cX + PmLi + PmL2 + p f L f , where M is f u l l income, Pm and pf are the male and female net wage rates and p c is the pr ice of the consumption good which is set equal to un i ty . Performing th is constrained maximization, we obtain expressions for the desired quant i t ies of L i , L 2 , Lf and X: 1 Throughout the ana lys is , except where i t is otherwise confusing or is necessary for the exposit ion of the argument, a l l household subscripts are deleted. - 125 -(5.3a) L? (5.3b) l_2 (5.3c) L* (5.3d) X* Although i t is convenient in the u t i l i t y function formulation to express Li and L 2 in terms of annual hours, the actual information ava i lab le for each household concerns the number of weeks which the head worked during the year, K, the number of hours which he worked in a working week, H, and the re la t ionships between observed and desired values of each of these var iab les . While we can use values of K and H to ca lcu la te Li and L 2 using the re la t ionsh ips (5.4) L i = K . (T - H) (5.5) L 2 = T . (N - K) , where T is the number of hours in a week (168) and N is the number of weeks in a year (52), the appearance of the weeks worked var iable in both these equations complicates the der ivat ion of the rationed demand equations in the case where only hours are rat ioned. This is best i l l u s t r a t e d by considering f i r s t the der ivat ion of the demand equations in the other two rat ioning s i tua t ions . When only weeks are rat ioned, i t is c lear from (5.4) and (5.5) that although L 2 is f i x e d , since K = K" and both N and T are constants, Li remains a choice var iab le . Thus rationed demand equations in th is case are obtained by maximizing (5.1) subject to (5.2) with the added constraint that L 2 = U 2 . In th is way we obtain the rationed demand equations (5.6a) L ? k = L 1 ( p m , p f , M , L 2 ) (5.6b) L* k = L k ( p m , p f , M , L 2 ) = L i ( p m , p f , M ) = L 2 ( p m , p f ,M ) = L f ( P r a . P f . M ) = X ( p m , p f , M ) . - 126 -(5.6c) X * k = X k ( p m , p f , M , L 2 ) , where the superscript ' k ' indicates that weeks are ra t i oned . 2 S i m i l a r l y , when both hours and weeks are rat ioned, the rationed demand equations (5.7a) L * h k = Lf k(p m ,p f ,M ,r i ,r 2 ) (5.7b) X * h k = X h k ( p m , p f , M , r ! , L 2 ) are obtained by performing the u t i l i t y maximization subject to the addi t ional constraints that l_i = L~i and L 2 = L"2. When only hours are rationed i t is c lear that both L± and L 2 remain as choice var iables since the unrationed quantity K appears in both (5.4) and (5 .5) . However there remains a constraint to be imposed in th is case since choosing a level of Li implies a par t icu lar value of K which in turn f i xes the level of L 2 . A l t e rna t i ve l y , choosing L 2 f i xes the level of K and hence of Hence to derive the rationed demand equations when hours are rationed at the level TT i t i s necessary to rearrange (5.4) and (5.5) to obtain the relevant rat ioning const ra in t : (5.8) H = T - (T.L i ) / (NT - L 2 ) . In th is case maximization of (5.1) subject to (5.2) and (5.8) y ie lds rationed demand equations for Lf , X, and ei ther L i or L 2 : 3 (5.9a) L* h = L5 (p m ,P f ,M,H) (5.9b) L* h = L j ( p m , p f ,M,H) (5.9c) X * h = X h ( p m , p f , M , H ) . 2 Note that these equations could be writ ten as a function of K rather than T 2 , since T 2 = T(N-K). 3 Note that due to the complexity of th is ca lcu la t ion i t is general ly not possible to e x p l i c i t l y solve for the reduced form equations presented here. S p e c i f i c a l l y , as shown in Section 5.3 using the Stone-Geary u t i l i t y func t ion , the endogenous var iable K (or L 2 ) cannot be eliminated from the rhs of the demand equations. - 127 -5.2.2 Stochastic Spec i f i ca t ion The decision as to what par t icu lar stochast ic form to assume for th is model proves to have important impl icat ions for the estimation of the rationed equations of the model. One stochast ic spec i f i ca t ion which could be assumed for th is model is analogous to the form adopted for the s ing le - ra t ion model considered e a r l i e r . This spec i f i ca t ion is based on the observation that in each of the four d i f ferent rat ioning s i tuat ions — no ra t ions , only weeks rat ioned, only hours rat ioned, both hours and weeks rationed — one equation can be dropped in estimation due to the fact that the quant i t ies are related by the budget constraint in each case. Thus j o i n t l y normally d is t r ibuted error terms would be appended to the remaining 8 demand equations. Hence i f , as before, we were to a r b i t r a r i l y el iminate the equation for goods consumption in each case, we would append errors to equations (5.3a), (5.3b) and (5.3c) in the NR case, (5.6a) and (5.6b) when weeks are rat ioned, (5.7a) when both weeks and hours are rationed and (5.9a) and (5.9b) when only hours are rat ioned. Again, when considering a par t i cu la r rat ioning s i tuat ion we would make use of Anderson's theorem that the density function of the errors on any subset of these equations w i l l also be normally d i s t r i bu ted . A d i f f i c u l t y with adopting th is par t i cu la r stochast ic spec i f i ca t ion ar ises from the nature of the avai lable rat ioning information. S p e c i f i c a l l y , while avai lab le information allows us to determine the re la t ionsh ip between observed and desired values of H, and between observed and desired values of K, th is information does not always reveal the re la t ionsh ip between observed and desired values of Li . This can be seen in Table 5.1 where the possible combinations of re la t ionships between - 128 -observed and desired values of K and H are converted into re la t ionsh ips between observed and desired values of L i and Lz.** TABLE 5.1: RELATIONSHIP BETWEEN RATIONS ON WEEKS AND HOURS Hours worked during year Weeks worked during year actual < desired actual = desired (K° < K*) (K° = K*) actual < desired (H° < H*) Case 1: L° ? L° > 2 L* Case 4: L° > L* 1 1 L° = L* 2 2 actual = desired (H° = H*) Case 2: L° < L»> L* l Case 5: L° = L* l l actual > desired (H° > H*) Case 3: L° < L! L*2 Case 6: L» < L* L» - L* Thus, for example, with Case 1 where observed weeks worked, K°, are less than desired weeks worked, K*, while the actual number of hours worked in a week, H°, is less than the number des i red, H*, we see that although L° > L*, the re la t ionsh ip between L° and L* is indeterminate due to the 2 2 1 1 in teract ion of these two e f fec ts . From th is table i t is also apparent that i f we convert the H and K var iables to L i and L 2 using (5.4) and (5 .5) , then Cases 2 and 3 w i l l be ind is t inguishable in the resu l t ing data. Throughout, the superscript ' o ' is used to indicate observed values while ' * ' denotes desired values. - 129 -Since we are unable in a l l cases to convert the rat ioning constraints to re la t ionsh ips between observed and desired values of l_i and l2, or in some cases do not wish to perform such conversions due to the resu l t ing loss of information, der ivat ion of the relevant parts of the l i ke l ihood for the various rat ioning cases is more complex than in the s ing le- ra t ion model. S p e c i f i c a l l y , without determining the re la t ionships between L° and L* and between L° and L*, i t is not possible to determine the range in which the desired var iables are unobserved, and hence the range of integrat ion of the density function in the various rat ioning s i tua t ions . In order to determine the l i ke l ihood function component for each rationed case i t is therefore necessary to transform the normal density function so that i t is defined in terms of var iables for which the range over which they are unobserved is known. Consider, for i l l u s t r a t i v e purposes, the density function for Case 1 where both hours and weeks are rationed to be less than the i r desired l e v e l s , g ( L * , L * , L * n k ) . 5 Now in Case 1 we know L° > L* so we would want to integrate with respect to L* over the range for which L* is not observed; (-°°,L°). We would also wish to integrate with respect to L* but here the unobserved range is unknown. The solut ion to th is problem is to convert the density g(L* ,L* ,L*hk) to another densi ty h ( H * , L * , L ^ h ' < ) , so that as well as integrat ing out L* as before, we could integrate with respect to H* over i t s unobserved range which is 5 Since the Jacobian from the error terms to the endogenous var iables is un i ty , the density function can be writ ten in terms of e i ther the errors or the endogenous var iab les . - 130 -(H°,») ; that i s , whenever H* > H 0 . 6 A problem with assuming that normally d is t r ibuted error terms are appended to the desired le isure equations, so that g ( L * , L * , L * n k ) is a normal density funct ion, is that the transformed density function h(H*,L*,L*h' < ) w i l l no longer be a density function for a normal d i s t r i b u t i o n . The reason for th is can be seen by rearranging (5.4) and (5.5) to obtain the equation for the desired quantity H*, (5.10) H* = T - (T . l * ) / (NT - L*) . From th is equation i t is c lear that the Jacobian for the transformation from L* to H* involves another va r iab le , L*. Hence the change in var iab le from L* to H* involves more than a simple l inear scal ing of the o r ig ina l density func t ion , so that the transformed density function w i l l not have the normality property assumed for the or ig ina l densi ty. Although th is does not present any theoret ica l problems, i t does cause d i f f i c u l t i e s in est imat ion, pa r t i cu la r l y for Cases 1 and 3, but also for Cases 2, 4 and 6. The d i f f i c u l t y is that while there are e f f i c i en t routines for evaluating s ing le and double integrals of a normal density funct ion, th is is not in general true for other funct ional forms. Since these in tegra ls have to be evaluated many times in the course of opt imizat ion, the e f f i c iency with which they can be evaluated is very important, so that the fact that e f f i c i en t integrat ion routines are not avai lable for non-normal functions makes th is approach inadvisable. 6 Note that since the rat ion is on K rather than L 2 , we could also replace L* by K* in the converted density function and integrate with respect to K*. However since the integrat ion range for L 2 is always def ined, such a subst i tu t ion is not necessary. - 131 -An a l ternat ive stochast ic spec i f i ca t ion which avoids these d i f f i c u l t i e s is based on appending error terms to equations determining H and K rather than those determining the quant i t ies Li and L 2 . To explain th is stochast ic spec i f i ca t i on , we consider the non-rationed case; analysis in the various rat ioning cases fol lows analogously. Since we plan to estimate the equations in pseudo-share form in order to minimize possible heteroskedast ic i ty problems in a manner s imi la r to that adopted with the s ing le - ra t ion model, i t is convenient to use share notation throughout. Thus the determinist ic parts of the four equations in the non-rationed case, derived from equations (5.3a) - (5.3d), (5.4) and (5.5) can be expressed as: (5.11a) s* = Pi(T-H*)/M = p x L i ( . ) / (M .K* ) (5.11b) s* = p 2T(N-K*)/M = p 2 L*/M = p 2 L 2 ( . ) / M (5.11c) s* = p f L* /M = p f L f ( . ) / M (5 . l i d ) s* = p cX*/M = p c X( . ) /M An addit ional advantage of wr i t ing the equations in th is share form is that i t i l l u s t r a t e s quite c lea r l y the re la t ionsh ip that ex is ts between the various equations in the system even under th is revised formulat ion. Here however th is re la t ionsh ip is not the simple adding up condit ion encountered previously . Rather, in th is non-rationed case, i t is c lear from the budget constraint (5.2) and the de f in i t i ona l equations (5.4) and (5 .5 ) , that (5.12) K* . s * + s* + s* + s* = 1. Now i f we assume that due to errors of opt imizat ion the observed shares d i f f e r from these desired shares by an addit ive random term, the share equations we would wish to estimate in th is non-rationed case would be those in equations (5.11a) - ( 5 . l i d ) , with a l l desired values (denoted by - 132 -aster isks) replaced by observed values and with error terms u^, u k , Uf and u c respect ive ly appended to each of the four equations. Thus a re la t ionsh ip of a s imi lar form to (5.12), except with the desired value K* replaced by the observed value K, w i l l hold among the observed shares and also among the determinist ic rhs of the share equations. Consequently, the error terms on the four equations w i l l be related by the equation (5.13) K.u. + u. + u. + u. = 0. h k f c While the re la t ionsh ip (5.13) suggests that the error terms are l i nea r l y dependent and hence that the corresponding variance-covariance matrix of these residuals is s ingu lar , the s i tuat ion here is more complex than that usual ly encountered with systems of demand equations. In the usual case, l inear dependence of the disturbance vector and the consequent s ingu la r i t y of the covariance matrix is handled by delet ing one of the equations from the system pr ior to formulating and estimating the l i ke l ihood funct ion . Furthermore, as shown by Barten [1969], the choice of which equation to drop is arb i t rary since there is no loss of information concerning the demand behaviour for the good which has i t s equation deleted. Indeed th is is the case with the s ing le- ra t ion model developed in Chapter 3, where the goods consumption equation is deleted from the ana lys is . A problem with adopting a s imi lar procedure for the two-ration model can be seen to ar ise from the fact that here the re la t ionsh ip (5.13) among the error terms involves the endogenous var iable K. Rewriting (5.11b) in terms of observable var iables and error terms ~ that i s , in the form in which i t is to be estimated ~ we have (5.14) p 2T(N-K)/M = p 2 L 2 ( . ) / M + u k . - 133 -Now rearranging to obtain an equation with K as the sole lhs var iable we obtain (5.15) K = K + Z.uk, where (5.16) K = N - L 2 ( . ) / T , (5.17) Z = -M/ (p 2 T) , and both K and Z involve only exogenous var iab les . Now subst i tu t ing (5.15) in (5.13) and rearranging y ie lds (5.18) K .u h + Z . u n u k + \ + u f + u c = 0. From th is expression i t is c lear that rather than l inear dependence, the re la t ionsh ip among the error terms is nonl inear. Such circumstances give r i se to a number of problems, one of the most important of these being that i t is no longer possible to demonstrate that the covariance matrix of the error terms is singular and hence that an arb i t rary equation can be dropped from the analysis pr ior to est imat ion. That th is d i f f i c u l t y i s caused by the appearance in (5.18) of the cross-product term u^u^ can be seen as fo l lows. Mul t ip ly ing (5.18) by u n , for example, and taking expected values we obtain (5.19) k .E(u 2) + Z .E(u 2 u k ) + E(u h u k ) + E (u h u f ) + E(u h u c ) = 0, where K and Z are assumed to be independent of the error terms. If we also assume temporari ly that a l l the error terms have zero means, then the f i r s t term in (5.19) is the variance of u n mul t ip l ied by K while the las t three terms are the covariances between u n and the other three error terms. Thus without the second term, (5.19) would indicate s ingu la r i t y of the covariance matrix since the other four terms are the elements in the f i r s t row or column of the covariance matr ix. A s imi lar re la t ionsh ip for the other rows and columns of th is matrix can be found - 134 -simply by mul t ip ly ing (5.18) by a d i f ferent error term pr ior to taking expected values. However the d i f f i c u l t y is that a l l such expressions involve a term such as E(u^u k) in (5.19). Consequently, due to the cross product term, the re la t ionsh ip (5.18) does not resu l t in s ingu la r i t y of the covariance matr ix. Rather, i t implies res t r i c t i ons which involve covariances together with higher order moments of the d i s t r i b u t i o n . Despite the fact that the covariance matrix cannot be shown to be s ingu la r , i t is c lear that we cannot simply assume that the four error terms are j o i n t l y normally d is t r ibuted with a non-singular covariance matr ix , since th is would f a i l to take account of the res t r i c t i ons implied by (5.18). On the other hand, determining the d is t r ibu t ion which the four error terms must share is a non- t r i v ia l s t a t i s t i c a l problem which is beyond the scope of th is d i sse r ta t i on . In order to proceed with the formulation and estimation of th is model i t is therefore desired to assume a d i s t r i bu t ion for these errors which is not inconsistent with the r e s t r i c t i o n s . One such d is t r i bu t ion is obtained by assuming that three of the four error terms are j o i n t l y normally d i s t r i bu ted . Under such an assumption we w i l l be acting as though the covariance matrix had been s ingu lar , in that we w i l l simply delete one equation pr ior to est imat ion. Further, since we only work with three of the four error terms, the fourth error term w i l l act simply as a residual and as such w i l l ensure that none of the res t r i c t i ons w i l l be prevented from holding. The major disadvantage of the above d is t r i bu t iona l assumption concerns the fact that since the covariance matrix is not s ingu lar , our decision as to which equation to delete is no longer a rb i t ra ry . S p e c i f i c a l l y , the estimates of the st ructura l parameters which we obtain w i l l not be invar iant to the equation which is dropped pr ior to - 135 -est imat ion. Further, we w i l l be unable to ret r ieve information on the elements of the variance-covariance matrix which are not estimated d i r e c t l y ; that i s , the variance of the error term on the omitted equation and i t s covariances with the other error terms. In order to f a c i l i t a t e comparisons with our analysis in the s ing le - ra t ion model and due to our predominant interest being in the parameters of the labour supply equations, we choose to again delete the equation for goods consumption from our ana lys is . Thus in the non-rationed case the remaining equations are (5.11a), (5.11b) and (5.11c) with random error terms appended in each case. In each of the three d i s t i nc t rat ioning cases ~ rat ion on H, rat ion on K and rat ions on both H and K — we also drop the goods consumption equation. Thus in addit ion to equations for s£, s£ and s^ in the if i / ^k L non-rationed case, we also have equations for s^ and s^ , s k and s^ and f i n a l l y for s^hk^ where, as before, the superscript ' h ' indicates that the var iable refers to the case where hours in a working week are ra t ioned, 'k ' indicates that working weeks in a year are rationed and 'hk' indicates that both hours and weeks are rat ioned. While the d is t r i bu t iona l assumption out l ined above considerably s imp l i f i es the problem of determining the appropriate stochast ic spec i f i ca t ion for the model, a number of issues raised by the nature of the nonlinear constraint among the error terms, (5.18), also need to be considered. The f i r s t of these concerns the expected values of the various error terms as well as the covariance between the errors on the hours and weeks equations. Taking the expectation of (5.18) and assuming that K and Z are non-stochast ic , we obtain (5.20) k.E(u h) + Z-E(u h u k ) + E(u k ) + E(u f ) + E(u c ) = 0. From th is expression i t is c lear that i f we assume a l l four errors have - 136 -zero means, the term E(uhU|<), which w i l l then be the covariance between the errors on the equations concerning le isure during working weeks (u^) and le isure during non-working weeks ( u k ) , w i l l also have the value zero. However since these error terms are assumed to be due to errors in opt imizat ion, i t is not desired to presuppose that the two error terms u^ and u k are uncorrelated. S p e c i f i c a l l y , we do not wish to prohib i t the p o s s i b i l i t y that an excess of one type of le isure (as a resu l t of an error in optimization) is at least p a r t i a l l y compensated by under-consumption of the a l ternat ive form of l e i su re . In order to avoid assuming that the covariance between u n and u k i s zero, i t is therefore necessary to assume that not a l l the errors have zero means. In view of the fact that the error term u c is el iminated from our analysis due to the d is t r i bu t iona l assumption made previously, i t is convenient to again treat th is error term asymmetrically. Thus we assume the error on the goods consumption equation to have a non-zero expectat ion; since th is error term does not appear in our ca lcu la t ions i t is not necessary to speci fy a par t icu lar non-zero value for i t s mean. The f i na l problem of stochast ic spec i f i ca t ion which we consider for th i s model concerns the constancy across households of the covariance matrix of the res idua ls . The issue of whether or not the covariance matrix is constant ar ises from the or ig ina l re la t ionsh ip among the error terms described by (5.13). To i l l u s t r a t e the problem here in the non-rationed case we f i r s t add household subscripts to each error term and define the vector of errors for the i th household as u-j, where u . ' = (u. . ,u. . ,u.c. ,u . ) , i = l . . . . . I households. Now the or ig ina l 4x4 covariance matrix of the error terms, fi^, where = E (u^u - ' ) , is by de f i n i t i on constant i f fi-j = n for a l l i = l , . . . , I . The problem of - 137 -non-constancy of th is matrix ar ises when there is a re la t ionsh ip among the error terms for each household of the form a-j'u-j = 0, i = l , . . . , I , where a-j is a 4x1 vector of coe f f i c i en t s , at least one of which varies across households. In such a case, as shown by Pol lak and Wales [1969], i t fol lows that (5.21) a . ' fl. = a . ' E ( u . u . ' ) = E U - ' u - u . ' ) = 0. Hence, since a-j1 ft-j = 0, a-j is an eigenvector of ft-j with a corresponding eigenvalue of zero. Now i f ft-j = ft for a l l i = l , . . . , I , and there are at least four l i nea r l y independent eigenvectors a-j corresponding to four d i f ferent households (which is l i k e l y since I » 4 ) , th is would mean that ft would have zero rank. Consequently we would conclude that ft-j could not be the same for a l l households. The s i tuat ion with the two-ration model formulated here is s imi lar to that described above with a-j1 = (K-j,1,1,1) d i f f e r i ng across households implying that the covariance matrix also must d i f f e r across households. However one important consideration here is that K-j is an endogenous var iab le which is not independent of the error terms. Hence with the vector a-j defined to include such a va r iab le , equation (5.21) w i l l no longer hold. S p e c i f i c a l l y , (5.22) a . ' E ( u . u . ' ) * E ( a . ' u . u . ' ) . Consequently i t is not possible to demonstrate that the covariance matrix i s not constant across households. As with the case of the dependence of the error terms discussed previously , i t appears from (5.18) that non-constancy w i l l again involve covariances together with higher order moments of the d i s t r i bu t i on . Thus in the remainder of our analysis we assume the covariance matrix to be constant. Since the net resul t of our previous d i s t r i bu t iona l assumption is to el iminate the goods equation and - 138 -hence reduce the covariance matrix so that i t does not involve the error term u c , i t is l i k e l y that the consequences of the assumption of constancy, should i t prove to be incor rec t , w i l l be minimal. In summary then, the a l ternat ive stochast ic spec i f i ca t ion consists of appending error terms to the 8 share equations which remain after e l iminat ing the equation for goods consumption in each of the non-rationed and the 3 d i s t i nc t rationed cases. We then assume that these error terms are j o i n t l y normally d is t r ibuted with zero means and constant covariance matrix £ . Hence using Theorem 2.4.3 from Anderson [1958,p23], we know when we consider the marginal d i s t r ibu t ion of any subset of these error terms that they are also normally d is t r ibuted with zero means and with variances and covariances obtained by taking the relevant components of the £ matr ix . Thus, for example, in Case 5 the error vector (u n ,uk ,u f ) is normally d is t r ibu ted with a zero mean vector and covariance matrix £ 5 , where £5 is a 3x3 matrix comprising elements of £ which correspond to the variances of uh, u|< and uf and the i r covariances with each other. S i m i l a r l y , in Cases 1 and 3, the error vector ( u ^ u ^ u ^ ) is normally d is t r ibuted with a covariance matrix which we denote £ 1 , while in Case 2 ( u^jU ^ u ^ ) is normally d is t r ibu ted with covariance matrix £ 2 and in Cases 4 and 6 (u h,uj^,u^) has covariance matrix £^. As a resul t of the jo in t normality of a l l the subsets of error terms, the most complex aspect of the density functions for the various rat ioning cases under th is a l ternat ive stochast ic spec i f i ca t ion w i l l be seen to involve evaluation of double integrals of a b ivar ia te normal density funct ion. Since e f f i c i en t routines for evaluating such integrals are ava i l ab le , estimation of th is model is thus rendered computationally p rac t i cab le . - 139 -5.2.3 Derivation of the Likel ihood Function As in the s ing le- ra t ion model, the l i ke l ihood function here w i l l consist of the product of the density functions for each of the rat ioning cases. To i l l u s t r a t e how th is function is der ived, we consider the density function for the most complex case where both hours and weeks are rat ioned. Density functions for the other cases are obtained analogously and are therefore not derived in d e t a i l . Consider then the density function for Case 1 where the male i s rat ioned to work less than his desired hours in a working week and less than his desired working weeks in a year. Since H° < H* implies s^ > s£, where the ' o ' superscript again indicates observed magnitudes, by expressing (5.11a) in stochast ic form we obtain the inequal i ty (5.23) u h < Zi , where (5.24) Z x = s° - P i L i ( . ) / ( M . K * ) . Also since K° < K* implies s^ > s£, we have from the stochast ic form of (5.11b), (5.25) u k < Z 2 , where (5.26) Z 2 = s°k - p 2 L 2 ( . ) / M • Thus the density function for Case 1, which is obtained by integrat ing out the var iables over the i r unobserved ranges defined by (5.23) and (5.25) is given by (5.27) h = J lf ^ V V ^ ^ ) d V d u h ' where n 3 ( . ) is a t r i v a r i a t e normal density function with a zero vector of means and covariance matrix Z\. - 140 -To s impl i fy th is expression, we begin by wri t ing the jo in t density as the product of a marginal and a condit ional densi ty. Thus we obtain (5.28) hi = n ^ u ^ a 2 ^ ) . ^ ( u ^ u ^ u ^ ^ ) d u k . d u h . — 00 _ 0 0 Next we use Theorem 2.5.1 from Anderson [1958,p29] in order to s impl i fy the expression being integrated. This theorem, which is a mul t ivar ia te analogue of one used previously with the s ing le - ra t ion model, states that i f a vector X of normally d is t r ibuted var iables is divided into 2 groups, Xi and X 2 , with means m and U2» variance matrices E n and E 2 2 a n^ covariance matrix E 1 2, then the condit ional d is t r ibu t ion of Xi given X2 i s also normal, with mean vector u' and covariance matrix E n 1 , where -1 (5.29) u ' = ui + E 1 2 . E 2 2 . (X 2 - u 2) -1 (5.30) Zn' = E n - E 1 2 . E 2 2 . E 2 i . Using th is theorem, (5.28) can now be s imp l i f ied to hk 2hk f l l r l 2 ' (5.31) hi = n i ( u f ; o f ) . J J n 2 ( u h , u k ; Ex) d u k . d u h . where n i ( . ) is an unconditional univar iate normal density function which has zero mean and variance a 2 ^ (which is the (3,3) element of E J , while n 2 ( . ) , which is an unconditional b ivar ia te normal density funct ion, has mean vector and covariance matrix obtained by making the appropriate subst i tu t ions in (5.29) and (5.30). Denoting the elements of the mean vector as y n and m 2 , and the variances and corre la t ion coe f f i c ien t from 0 0 th is resu l t ing covariance matrix by 0 ^ , and p, then (5.31) can be writ ten in estimable form as I* Z 2 (5.32) h x = (l/af ).f(uf/of) . / lJ 2 f 2 ( X i , X 2 , p ) d X 2 . d X i , — OO —00 where (5.33) 1* = (Zi - n u ) / o h l , (5.34) il = (Z 2 - y i 2 ) / *k i . - 141 -and f ( . ) and f 2 ( . ) are standard (0,1) univar iate and b ivar ia te normal density funct ions. Thus (5.32) can be evaluated using well es tab l ished, e f f i c i e n t routines which compute standard univar iate normal densi t ies and double integrals of standard b ivar ia te normal dens i t i es . One problem with the density function for th is case, which also occurs for Case 3 where H° > H* and K° < K*, is that Z l s the l im i t of the integral with respect to u n as defined in (5.23) and (5.24), is a funct ion of the desired quantity K* which is unobserved when weeks are rat ioned. While th is problem is s imi la r in nature to one encountered in the two-equation model of Wales and Woodland [1980], where the unobserved wage rate (for non-working women) is included in the l im i t of an i n teg ra l , the i r solut ion of integrat ing out the unobserved var iable over the ent i re range (-00 , 0 0) cannot read i ly be applied here s ince, by v i r tue of the second of the two in tegra ls , we are already integrat ing th is var iable over a spec i f i c part of th is range, namely (-°°,Z2). One simple method for dealing with th is problem would involve replacing K* by the observed value K°. However such a procedure is not desirable since K° < K* for the observations we are considering in these rationed cases. Hence any estimates obtained in th is way would be biased and inconsis tent . To overcome th is problem the solut ion we adopt consists of replacing K* in (5.23) by an estimate K obtained by unscrambling the determinist ic part of the relevant share equation in the non-rationed case, (5.11b). That i s , K is the estimated value of K in the non-rationed case, obtained as (5.35) K = N - L 2 ( . ) / T , where L 2 ( . ) is the estimated value obtained by evaluating the funct ional form L 2 ( . ) in equation (5.3b) using the estimated parameters. In view of - 142 -the fact that observations for the two cases where th is ca lcu la t ion is required (Cases 1 and 3) account for only 39 of the 492 households in the sample, i t is not expected that th is procedure w i l l s i g n i f i c a n t l y affect the parameter estimates. To complete th is section we present the density functions corresponding to the other 5 cases iden t i f ied in Table 5 .1 . Again using Anderson's theorem 2 .5 .1 , these f i ve densi t ies can be writ ten in terms of unconditional normal density functions as k k hf 7-2 i i (5.36) h 2 = n 2 ( u h , u f ; E 2 ) • / M u ^ l ^ ) du k — 00 (5.37) h 3 = m ( u f ; a f ) . J J M u ^ u ^ E J d u k . d u h , h h k 1 \ r Z l (5.38) h 4 = n 2 ( u k , u f ; £ « • ) . / n 1 (u h ;E l t ) du h —oo (5.39) h 5 = n 3 ( u h , u k , u f ; £ 5 ) h h kf 0 0 ' * (5.40) h 6 = n 2 ( u k , u f ; Ei+ ) . / ni(u h;E4) du h 7-\ Here n2(.,E^' <) is a b ivar ia te normal density function with zero means and covariance matrix comprising the intersect ion of the j th and kth rows and columns of the covariance matrix Z i , while n|(.;E^) and n 2 ( . ;E! ) are univar iate and b ivar ia te normal density functions respect ive ly , with means and variances obtained by making the appropriate subst i tut ions in (5.29) and (5.30). Thus, as indicated previously, the most complex ca lcu la t ions involve double integrat ion of a b ivar ia te normal density function in (5.31) and (5.37). Hence well es tab l ished, e f f i c i en t routines can be used to estimate the l i ke l ihood funct ion: 6 (5.41) L = n . n h j , 3=1 ieCj - 143 -where C j , j = 1 , . . , 6 , are the subsets of households corresponding to each of the s ix d i f ferent cases. 5.2.4 Sample S e l e c t i v i t y Correct ions In order to implement th is model using the avai lable data, we again need a sample s e l e c t i v i t y correct ion of the type defined for the s ing le - ra t ion model in order to take account of the fact that we only include households in which the wife is working. For households in which the male is not rat ioned, the wife works i f H| > 0; that i s , i f Uj; < or , in share form, i f (5.42) s* = p f L* /M < p f T f / M , where Tf is tota l annual hours avai lable to the wi fe. Now defining Sf = P f L ^ ( . ) / M , we see from the stochast ic formulation of (5.11c) in which an error term uf is appended, that the probab i l i t y that the wife works i s given by (5.43) P r ( u f < p f T f / M - s f ) = F ( ( p f T f / M - s f ) / a f ) , where af is the standard error of uf and F(. ) is the standard normal d i s t r i bu t i on funct ion. The corresponding p robab i l i t i es for the various rat ioning cases can be eas i l y obtained by making appropriate subst i tut ions in (5.42). S p e c i f i c a l l y , i f the determinist ic part of the desired female le isure share equation under a par t icu lar form of rat ioning is denoted s^ and the standard error of the stochast ic term appended to that equation is denoted o^, where the superscript R = h,k,hk refers to the various rat ioning cases (hours rat ioned, weeks rationed and both hours and weeks rationed respec t i ve l y ) , then the probab i l i t y that the wife works in each of these cases is given by F ( ( p f T f / M - s^ ) /o^ ) . Now by d iv id ing the - 144 -densi t ies hj for the non-rationed case, (5.31), and for each of the rationed cases, (5.36) - (5.40), by the relevant probab i l i t y that the wife works before incorporating these densi t ies in (5.41), we obtain the corrected l i ke l ihood function which takes account of th is sample s e l e c t i v i t y b ias . In addit ion to th is sample s e l e c t i v i t y bias which resul ts from only including working wives in the sample, an addit ional form of truncation occurs within the hours and weeks formulation used here. S p e c i f i c a l l y , from the share equation for male weeks worked in the non-rationed case, (5.11b), we see that the endogenous var iable in th is equation is truncated at zero when K = N = 52 weeks. While i t is theore t i ca l l y desirable to correct the resu l t ing estimates for th is truncation by d iv id ing the rationed and non-rationed densi t ies by the relevant p robab i l i t y that s£ or s£R, R=h,k,hk, is non-negative, there are two par t icu lar d i f f i c u l t i e s inherent in such an approach. F i r s t , while i t is possible to ca lcu la te these p robab i l i t i es in the non-rationed case or when only hours are rat ioned, i t is not c lear how to determine th is probab i l i t y when weeks are rationed since we do not have a desired weeks equation in th is case. Second, even i f such p robab i l i t i es could be determined, the l i ke l ihood funct ion defined by (5.41) is pa r t i cu la r l y complex, espec ia l l y when the SS correct ions for working wives are included, so that inc lusion of any addi t ional (nonlinear) terms which further complicate th is funct ion i s only l i k e l y to make estimation of the model computationally impract icab le . 7 Due to these reasons and the fact that K = 52 for less than 6% of the sample, no correct ion is made for th is t runcat ion. 7 In f ac t , as detai led in the next chapter, even without these addit ional SS cor rec t ions, we s t i l l require cer ta in parametric res t r i c t i ons to be imposed in order to s u f f i c i e n t l y s impl i fy the model so that parameter estimates can be obtained. - 145 -5.3 Equations for the Stone-Geary Spec i f i ca t ion While i t would be desirable to consider estimating equations derived from several d i f ferent funct ional form representations of household preferences, due to the complexity of the model which is to be estimated and the lack of success in obtaining estimates with other funct ional forms even in the simpler s ing le - ra t ion model, we l im i t our analysis here to equations derived from the Stone-Geary u t i l i t y function formulat ion. Again for notational convenience we refer to the quant i t ies L l 5 L 2 , Lf and X as X i , X 2 , Xf and X c respect ive ly , while for the male wage ra te , Pm, we use interchangeably pi and p 2 ; that is Pi = p 2 = Pm» In the non-rationed case, maximization of (5.1) subject to (5.2) when the u t i l i t y function has the Stone-Geary form 8 (5.44) U = E3i ln (Xi - Y i ) y ie lds the usual equations of the Linear Expenditure Sustem, with £Bi = 1. Using (5.4) and (5.5) to express these equations in the required form, and noting that we drop the equation for goods consumption, the estimating equations for the non-rationed case (Case 5) are obtained where, as indicated prev iously , a l l equations are expressed in share form by d iv id ing both sides by f u l l income, M, pr ior to appending the stochast ic disturbances. To s impl i fy the notation we omit subscripts from summation symbols (E) where i t is not otherwise confusing. Thus in (5.44) summation is with respect to the subscript i over the ent i re range, i = l , 2 , f , c , . as: (5.45) (5.47) (5.46) Pm(T-H) = P iY i /K + M M " 2 P jY j ) /K PmT(N-K) = p m X 2 = p 2 Y 2 + B2(M - Z p.y.) p f X f = p f Y f + 3 f(M - Z PjYj) , - 146 -For Case 2 where male working weeks are rat ioned, maximization of the u t i l i t y function is performed subject to the addit ional rat ioning const ra in t , L 2 = T2 - T(N-K). In th is case the equations obtained, af ter dropping the equation for goods consumption, are given by (5.48) Pm(T-H) = P iY i /K + ( P i / Z P.).(M - p 2T(N-K) - £ p-vJ/K m J K ^ 2 K K (5.49) p fX f = p f Y f + ( 3 f/E ej ) . (M - p 2T(N-K) - Z p ^ ) . J ^  2 2 Here, as in the remaining rationed cases, both sides of the equation are divided by f u l l income before appending the error terms. When both hours and weeks are rat ioned, as occurs in Cases 1 and 3, the u t i l i t y function is maximized subject to the two rat ioning constraints Li = Li and L 2 = ~L 2. Hence in th is case after dropping the equation for goods consumption and subst i tu t ing for T\ and "L 2 in terms of "R" and TT, the only remaining equation is for female l e i su re : (5.50) p f X f = p f Y f + (B f / (B f + 3 C ) ) . (M - P l (NT-KH) - E p j Y j ) . j ^1,2 The most complex equations are obtained for Cases 4 and 6 where hours of the male in a working week are rat ioned. The d i f f i c u l t y in th is case is that while hours are rat ioned, neither Xi nor X 2 , the quant i t ies that ac tua l ly appear as arguments of the household's u t i l i t y funct ion, are i nd i v idua l l y constrained. However due to the rat ion on H, the household cannot choose the levels of both these quant i t ies in order to maximize u t i l i t y . In pa r t i cu la r , as can be seen from (5.8) , once the household chooses a level of X 2 (or K ) , Xi is i t s e l f completely determined. Thus in th is case rather than maximizing the u t i l i t y function (5.44) subject to the budget constraint (5.2) and the rat ioning constraint (5 .8) , to obtain the estimating equations we f i r s t rearrange (5.8) to obtain an expression - 147 -for Xi which we then subst i tute in both the u t i l i t y function and the budget const ra in t . We then choose levels of the quant i t ies X 2 , Xf and X c which maximize th is reformulated u t i l i t y function subject to the amended budget const ra in t . In th is way the estimating equations obtained, after delet ing the goods consumption equation, have the form (5.51) PmT(N-K) = P m X 2 = p 2 Y 2 + T. tj>/ H" A (5.52) p f X f = p F Y F + 3 f / X , where (5.53) I A = (M - Z p . Y , - Pi(T-H)(NT- Y 2 ) / T ) / ( 3 f + 3 + <t>) j * l T C (5.54) <|> = 3 2 - 3i(NT - KT - Y 2 ) / ( K T - T Y I / ( T - H ) ) . From these equations i t is clear that there is an addit ional complexity which ar ises in the estimation when H is rat ioned. This is due to the fact that the endogenous var iable K appears in the rhs of both equations (5.51) and (5.52) s ince, from (5.53), (1/x) depends on 4> which, from (5.54), is i t s e l f a function of K. Thus the Jacobian which is the absolute value of the determinant of J-j, the matrix of der ivat ives of the error terms appended to (5.51) and (5.52) with respect to the endogenous var iab les , is a function of the various parameters which appear in these two equat ions. 9 Hence in addit ion to d iv id ing the l ike l ihood function by the appropriate SS correct ions outl ined in the previous sec t ion , we also need to mul t ip ly by the Jacobian term, n ( a b s j j J ) . i eC i ^Ce In th is case the endogenous var iables are just Xf and K since the endogenous var iable from the excluded equation, X c does not appear in (5.51) or (5.52). - 148 -5.4 Ident i f i ca t ion of Parameters One of the most in terest ing aspects of th is model concerns the fact that two of the arguments in the u t i l i t y function (5 .1) , le isure of the head during working weeks, and his le isure during non-working weeks, L 2 , have the same pr ice which is Pm, the male wage ra te . The usual pract ice in such a s i tuat ion is to aggregate commodities which have the same p r i ce , in th is case y ie ld ing the s ing le - ra t ion model analyzed prev ious ly . However, the adoption of th is procedure is not necessary and in fact needlessly r e s t r i c t s the extent of the economic analysis that can be conducted; despite having non-uniquely priced commodities, there is su f f i c ien t information avai lab le to ident i fy many addit ional ef fects compared with the case in which a l l goods having the same pr ice are aggregated. 1 0 However when prices are not unique i t is not c lear whether estimates of a l l of the parameters can be obtained; that i s , whether a l l of the parameters are i d e n t i f i e d . In general i t is very d i f f i c u l t to ana l y t i ca l l y demonstrate i d e n t i f i a b i l i t y in a nonlinear model, par t icu lary when the model is as complex as the one formulated in th is chapter. As a r esu l t , to examine i den t i f i ca t i on in these models i t is necessary to use an empirical approach which involves estimating the model s tar t ing from many d i f ferent i n i t i a l po in ts . Under th is approach, i f the parameter values obtained at the maximized points are the same, th is is taken as an indicat ion that the model is i d e n t i f i e d . The basis for th is inference is that i f the model were not i d e n t i f i e d , d i f fe r ing parameter vectors could y ie ld the same optimized funct ion value, and we would expect to observe th is resu l t a f ter s tar t ing from many d i f ferent sets of i n i t i a l parameter values. 1 0 See, for example, Caves and Christensen [1981] who examine th is issue in the context of peak pr ic ing for e l e c t r i c i t y consumption. - 149 -While we are unable to ana l y t i ca l l y ascertain whether the overal l model is i d e n t i f i e d , i t is possible to analyze certa in aspects of the i den t i f i ca t i on problem for th is mode l . 1 1 Here we concentrate on two spec i f i c issues. F i r s t we consider whether estimates of the s t ructura l form parameters for the system of equations which forms the basic structure of the rat ioning model can be obtained from the reduced form est imates, pa r t i cu la r l y when two of the prices are i d e n t i c a l . Second, in view of the formulation of the l i ke l ihood function which we use here, we examine which elements of the variance-covariance matrix of the error terms on these equations w i l l be i d e n t i f i e d . Consider the non-rationed case where the equations to be estimated are (5.45), (5.46), and (5.47), expressed in share form by d iv id ing both sides by M. Noting that Pi = P2 = Pm these share equations can be reformulated as fo l lows: (5.55) Pm(T-H)/M = (Yi - MY1+Y2)) . (Pm / M / K) + M l / K ) - 3 iY f (p f /M/K) - BiYc(Pc/M/K) (5.56) pmT(N-K)/M = ( Y 2 - M Y I + Y 2 ) ) . ( P m / M ) + 3 2 - 8 2 Yf(pf/M) - 02Y c (Pc / M ) (5.57) p f X f / M = -3 f (Y i+Y 2 ) . (P m /M) + 3 f + Y f ( 1 - B f ) . ( p f / M ) - 0 fY c (Pc / M ) From the coef f i c ien ts in these reduced form equations we are able to determine which of the st ructura l form parameters can be unscrambled. From (5.55) i t is c lear that we can obtain estimates 3 i , Yf and Y c as well as an estimate of the term ( Y i - 3 i ( Y i + Y 2 ) ) . From the second equation we can obtain B 2 , Y f , Yc and an estimate of ( Y 2 - M Y I + Yz))' F i n a l l y 1 1 Of course when obtaining estimates for th is model (as presented in the next chapter) , optimization of the l i ke l ihood function is performed using several d i f ferent sets of s tar t ing values. - 150 -from (5.57) we w i l l obtain an estimate of (YI + y2) as well as B 0 : , Y°T and Y . Further, by using th is estimate of (YI + Y2) in conjunction with Bi and the estimate of (YI - 3 I (Y I + Y 2 ) )> or with T2 and the estimate of (Y2 - 8 2 ( Y i + Y 2 ) ) , i t is c lear that we w i l l obtain two estimates of Y i and Y2- Thus a l l s t ructura l form parameters can be unscrambled from the reduced form estimates; in the case of the Bi parameters, unique estimates w i l l be obtained whereas for the Y I parameters, mult ip le estimates would be obtained in the absence of appropriate across-equation r e s t r i c t i o n s . 1 2 Since we have no pr ior knowledge that a l l these parameters w i l l have non-zero values, pa r t i cu la r l y in view of our resu l ts with the s ingle ra t ion model, i t is important to determine how th is i den t i f i ca t i on w i l l be affected should some of the parameters be zero. While i t is c lear from the above equations that the res t r i c t i ons B i = 0 or B2 = 0 or B i = B2 = 0 w i l l not af fect i d e n t i f i c a t i o n , in that we w i l l s t i l l be able to unscramble a l l the s t ructura l parameters, such a resul t is not obvious i f Bf = 0. In th is case we w i l l not obtain an estimate of (YI + Y2) f r o m equation (5.57) and as a resu l t i t would appear that we w i l l not be able to unscramble separate estimates of Y i and of Y2- However by adding the estimates of the terms (YI - 3 i ( Y i + Y2)) and (Y2 - M Y I + Y2))» we w i l l obtain an estimate of (YI + Y 2 M I - &i - 82)* Thus, using our estimates of B i and B 2 , we would be able to estimate (YI + Y 2 ) and hence, as before, Y i and 1 2 In fact with the LES, only two d i f ferent prices are required in order to ident i fy a l l the parameters. Thus, using the above methods i t can be seen for the unrationed case considered here that i f , for example, Pf = Pm = Pi = P2> w e would s t i l l be able to unscramble a l l the s t ructura l parameters from the reduced form estimates. It is in terest ing to note that th is resu l t is consistent with the fact that when preferences are add i t i ve , as is the case represented by the Stone-Geary u t i l i t y function from which the LES is der ived, a l l pr ice e l a s t i c i t i e s can also be determined when there is r e l a t i v e l y l i t t l e pr ice va r i a t i on . - 151 -Y 2 . Hence in th is non-rationed case, a l l the parameters appear to be iden t i f i ed regardless of whether or not cer ta in parameters take a value of zero. Simi lar methods to those employed above for the non-rationed case can also be used to examine iden t i f i ca t i on in the various rationed cases. In Case 2 where only weeks are rat ioned, i t is c lear that a l l parameters except B 2 and Y 2 can be obtained from the reduced form estimates. The parameters 3 2 and Y 2 do not appear in equations (5.48) and (5.49) due to the fact that the u t i l i t y function is addit ive and these parameters are only associated with the rationed var iable L 2 , le isure during non-working weeks. S im i la r l y from equation (5.50) we see in Cases 1 and 3 where both hours and weeks (and hence L i and L 2 ) are rationed that a l l parameters other than S i , B 2 , Y i and Y 2 are i d e n t i f i e d . For Cases 4 and 6 where H is ra t ioned, neither of the two u t i l i t y function arguments l_i and L 2 is separately rat ioned, although the household cannot choose the levels of both these var iables i nd i v idua l l y . Thus in th is case although a l l parameters appear in equations (5.51) and (5.52), i t is possible that not a l l the parameters Y i , Y 2 , S i and B 2 w i l l be i den t i f i ed . From an examination of these equations i t seems clear that both Y 2 and Yf are i d e n t i f i e d , but due to the nonlinear nature of the terms in these equations, i d e n t i f i a b i 1 i t y of the other parameters cannot be read i l y determined. Our i n a b i l i t y to determine i d e n t i f i a b i l i t y for a l l these cases proves not to be important for the fol lowing reasons. Since a l l households are assumed to have the same u t i l i t y function parameters regardless of the type of rat ioning constraint they face, and since a l l the d i f ferent cases of rat ioning appear in the l i ke l ihood funct ion, then the fact that the - 152 -parameters are iden t i f ied in at least one case is su f f i c ien t for them to be iden t i f ied in the complete model. For example, suppose that Y I and y2 are both ident i f ied in Case A but that only the i r sum (YI + Y2) "is i den t i f i ed in Case B. From Case B we would conclude that Y I and Y2 are not i den t i f i ed since any changes in these two parameters which leaves the i r sum unchanged w i l l not af fect the value of the l i ke l ihood funct ion. However these parameters also appear in Case A which forms another part of the same l i ke l ihood funct ion. Since both parameters are ident i f ied in th is case, any changes in ei ther Yi or Y2. even i f the i r sum should remain unchanged, w i l l af fect the overal l l i ke l ihood function value. Hence since a l l parameters are iden t i f ied here in the non-rationed case, they are iden t i f ied in the complete model. As indicated at the beginning of th is sec t ion , we refer here to i den t i f i ca t i on in the sense of being able to obtain estimates of the s t ructura l coe f f i c ien ts from reduced form estimates of the share equations. Thus we have ignored the fact that many of these parameters also appear among the terms def ining the l im i ts of various in tegra ls . However, since the st ructura l parameters are iden t i f ied in the share equations, and since they appear in the same form in these integrat ion l im i t s as they do in the share equations, as can be seen from (5.24) and (5 .26) , i t is un l i ke ly that th is w i l l af fect the i r i d e n t i f i a b i l i t y . However, as can be seen from equations (5.33) and (5.34), various elements of the variance-covariance matrix of the error terms from the share equations also appear in the integrat ion l i m i t s . We now turn to a b r ie f examination of the i den t i f i ca t i on of the elements of th is matr ix. From our analysis in Section 5 .2 .2 , i t is c lear that although the variance-covariance matrix of the errors on the 8 remaining share - 153 -equations in the d i f ferent rationed and non-rationed cases has dimension 8x8, many of the elements of th is matrix do not appear in our ca lcu la t ions and hence are not i den t i f i ed . For notational s imp l i c i t y , we rewrite the hk h h k k vector of these error terms ( u ^ u ^ u ^ ,u k ,u^-,u^,u^,u^) as ( u x , u 2 , u 3 , u 4 , U5 ,U6 ,U7 , us ) . Then from Section 5 .2 .2 , the error vectors in the various cases are defined as fo l lows: Cases 1 and 3 (H,K rat ioned): ( u i , u 2 , u 3 ) Case 2 (K rat ioned) : ( u 2 , u 6 , u 7 ) Cases 4 and 6 (H rat ioned): (ui,uit,U5) Case 5 (no ra t i ons ) : ( u i , u 2 , u 8 ) Thus i t is c lear from these vectors that the only corre lat ions which are possib ly iden t i f ied in our model are those between the various errors defined in each case. Hence from Cases 1 and 3 the relevant corre lat ions are P I 2 , P I 3 and p 2 3 , for Case 2 they are p 26, P 27 and p 67, for Cases 4 and 6, P u t , P15 and pi+5 while for the non-rationed Case 5 the corre lat ions that are defined are p i 2 , P i 8 and p 2 8 . In th is way we see that only 11 of the 28 corre lat ions which can be derived from the 8x8 covariance matrix are actua l ly defined in our model. We now examine these remaining 11 corre la t ion coef f i c ien ts in more deta i l in order to determine how and where they enter the l i ke l ihood funct ion and hence whether they are l i k e l y to be i d e n t i f i e d . From the density functions for the d i f ferent cases as speci f ied in (5.31) and (5.36) - (5.40), we see that these coe f f i c ien ts are used in ca lcu la t ing b ivar ia te normal densi t ies ( in (5.36), (5.38) and (5.40)) , t r i v a r i a t e normal densi t ies ( in (5.39)) and in evaluating s ingle and double integrals of univar iate and b ivar ia te normal density functions respect ive ly ( in a l l cases except for the non-rationed density (5.39)) . While in general i t is - 154 -d i f f i c u l t to reach any conclusion concerning iden t i f i ca t ion for most of these parameters, there seems to be no obvious reason to ant ic ipate i den t i f i ca t i on problems with the cor re la t ion coef f i c ien ts which are involved in d i r e c t l y ca lcu la t ing the b ivar ia te and t r i v a r i a t e dens i t i es ; p 6 7 in (5.36), P^5 in (5.38) and (5.40) and p 1 2 , Pis and p 2 8 in (5.39). S im i la r l y since the variances of the eight error terms are also used d i r e c t l y in ca lcu la t ing these and the univar iate densi t ies in (5.31) and (5.37), we have no reason to expect there to be d i f f i c u l t i e s in ident i fy ing these parameters e i the r . It is possible that i den t i f i ca t i on problems may ar ise with the remaining s ix cor re la t ion coe f f i c ien ts which are used only in evaluating the in tegra ls . Recal l from Section 5.2.3 that in order to evaluate these integrals we f i r s t change the var iables of integrat ion and the integrat ion l im i t s so that the functions being integrated are standardized normal univar iate or b ivar ia te dens i t i es . As a resul t the corre la t ion coe f f i c ien ts defined by the or ig ina l Ei matr ices, which v ia (5.29) and (5.30) are used to determine the means and the elements of the variance-covariance matrices E^  of the unstandardized normal density functions being integrated, are transferred to the integrat ion l im i t s for these standardized density funct ions, although in the b ivar ia te case they also are used in forming the cor re la t ion coe f f i c ien t between the standardized var iab les . Thus, for example, we see from (5.33) and (5.34) that the l im i t s of the double integral of the standardized b ivar ia te normal density function for Case 1 in (5.32) are functions of the means and variances of the unstandardized density function which in tu rn , as can be seen by making the appropriate subst i tu t ions in (5.29) and (5.30), - 155 -depend on the variances and in par t icu lar the corre lat ions between the o r ig ina l error terms in Case 1, P13 and P 2 3 . 1 3 While the fact that these cor re la t ion coef f i c ien ts appear in the integrat ion l im i t s does not in i t s e l f imply any lack of i d e n t i f i c a t i o n , pa r t i cu l a r l y in the double integrat ion case where they are also used in forming the cor re la t ion coe f f i c ien t between the standardized var iab les , i t does suggest possible problems in some cases. A par t icu lar instance where i den t i f i ca t i on problems are l i k e l y to ar ise occurs in the s ingle integral in (5.36), the density function for Case 2 where the number of weeks worked (only) is constrained to be less than the desired amount. Since th is integral evaluates the area under a univar iate normal density func t ion , we can re-express i t as F ( (Z 2 - u ) /o ) , where F( . ) is the standard normal d is t r ibu t ion function and p and 0, which both depend on P26> P27 and P67, are the mean and variance calculated from E 2 using (5.29) and (5.30). Now using C 2 to denote the subset of households which have only weeks rat ioned, th is term enters the l i ke l ihood function as n F ( . ) , so that to maximize the l i ke l ihood we want to maximize F(. ) i e C 2 for each household in C 2 . Since 0<_ F( . ) _< 1, th is is achieved by adjusting the avai lable parameters so F( . ) = 1 for a l l households in th i s group. Now in the usual case, maximizing a l i ke l ihood involves t radeof fs , as the parameters appear in other terms in the l i ke l ihood so that choosing them to maximize one term may in fact adversely affect another part of the l i ke l ihood funct ion , the net resu l t being a lower l i ke l ihood funct ion value. While th is is the case for the various u t i l i t y function parameters, 1 3 It can be shown in th is case that p 1 2 does not appear in the integrat ion l i m i t s , although i t is used in ca lcu la t ing the cor re la t ion c o e f f i c i e n t , p, between the standardized var iables in (5.32). - 156 -the va r iance terms and P G 7 , the c o r r e l a t i o n c o e f f i c i e n t s p 2 g and p 2 7 do not appear elsewhere in the l i k e l i h o o d f u n c t i o n and hence are f r e e to ad jus t so that F ( . ) = 1. While t h i s would s t i l l not normal ly be a problem as va lues of these c o e f f i c i e n t s which ensure F ( . ) = 1 f o r one household w i l l cause i t to be l ess than un i t y f o r o t h e r s , so that there would s t i l l be t r a d e o f f s between d i f f e r e n t households in t h i s group, the problem here i s that F ( . ) = 1 f o r any argument which exceeds approx imate ly 3 . 7 . Hence there are a number of va lues which p 2 6 and p 2 7 can take which w i l l ensure F ( . ) = 1 f o r a l l households in C 2 , r ega rd less of the va lues of the other parameters and, in p a r t i c u l a r , r ega rd less of the va lues of the va r ious exp lana to ry v a r i a b l e s f o r the d i f f e r e n t households in t h i s group. C l e a r l y t h e r e f o r e , P 2 6 and p 2 7 are not i d e n t i f i e d . For convenience we set both these parameters equal to zero when maximiz ing the l i k e l i h o o d f u n c t i o n . While the lack of i d e n t i f i c a t i o n of p 2 6 and p 2 7 in Case 2 can be demonstrated, i t i s not p o s s i b l e to reach any conc lus ions concern ing i d e n t i f i c a t i o n of the remaining 4 c o r r e l a t i o n c o e f f i c i e n t s : and p 1 5 in Cases 4 and 6 and P I 3 and P 2 3 in Cases 1 and 3. The d i f f i c u l t y in these cases i s that even though these c o e f f i c i e n t s are on ly used in e v a l u a t i n g i n t e g r a l s , they appear in more than one term. For example p 1 4 and p 1 5 appear in i n t e g r a l s in both (5.38) and ( 5 . 4 0 ) . Rewr i t i ng these two i n t e g r a l s as F ( Z i * * ) and F ( - Z i * * ) , where Z i * * = ( Z i - y * ) / o * t i t i s c l e a r tha t a l though y* and o* depend on these c o r r e l a t i o n c o e f f i c i e n t s , p l k and PIS cannot s imply be chosen to maximize F ( Z x * * ) s ince t h i s may adverse ly a f f e c t the va lue of F ( - Z i * * ) and v i c e - v e r s a ; that i s , t r a d e o f f s do appear to e x i s t f o r these c o e f f i c i e n t s . S i m i l a r arguments can be used with p 1 3 and p 2 3 which appear in both (5.31) and ( 5 . 3 7 ) , so that in general i t i s d i f f i c u l t to conclude whether or not these four parameters are - 157 -i d e n t i f i e d . Thus although the issue of i den t i f i ca t ion is one which, i f poss ib le , should be determined pr ior to est imat ion, in th is case i t seems necessary to use evidence from estimation in order to reach any de f in i te conclusion. A discussion and analysis of the resul ts obtained from estimating the two-ration model is contained in the fol lowing chapter. - 158 -CHAPTER SIX RESULTS AND ANALYSIS FOR THE TWO-RATION MODEL 6.1 Introduction In th is chapter we present and analyze the resul ts obtained from estimation of the hours and weeks model developed in the previous chapter using a sample which includes some households that are rationed in the hours and/or weeks of labour they can supply. We begin by discussing techniques used to deal with various problems encountered in est imat ion. S p e c i f i c a l l y , as out l ined in the fo l lowing sec t ion , in order to keep cer ta in variance-covariance matrices pos i t ive de f i n i t e , i t is not possible to optimize with respect to the elements of these matr ices. Rather, i t is necessary to optimize with respect to the elements of the lower t r iangular matrices into which these variance-covariance matrices can be decomposed. Estimates for the complete hours and weeks rat ioning model are presented in Section 6.3. In order to obtain these estimates i t is necessary to impose cer ta in parametric res t r i c t i ons which are explained in th is sec t ion . As well as parameter estimates, we also present marginal budget shares and implied labour supply e l a s t i c i t i e s for the various rat ion ing cases. The estimates for th is model are then compared to those presented in Chapter 4 for the s ing le - ra t ion model. To evaluate the ef fects and importance of cor rec t ly modelling the behaviour of rationed ind iv idua ls , in Sections 6.4 and 6.5 we re-estimate the model f i r s t using only the non-rationed households and then t reat ing a l l households in the sample as though they are non-rationed. In both these cases the parameters and labour supply e l a s t i c i t i e s are compared to the estimates obtained from - 159 -the cor rec t ly speci f ied model. We conclude th is chapter with a br ie f comparison of our resul ts with those obtained in another study which also estimates an hours and weeks model of labour supply behaviour. 6.2 Lower Triangular Decomposition of Variance-Covariance Matrices Although theore t i ca l l y i t is necessary that the ent i re var iance-covariance matrix of the error terms be posi t ive de f i n i t e , due to the fact that many of the elements of th is matrix are not i d e n t i f i e d , th is condit ion is not checked. However with the formulation of the two-ration model speci f ied in the previous chapter, i t is essent ia l for computational purposes that the relevant variance-covariance matrix be pos i t ive de f in i te in each of the four d i s t i nc t rat ioning cases — hours rat ioned, weeks ra t ioned, both hours and weeks rationed and no r a t i o n s . 1 The reasons for th is requirement concern the nature of the density funct ions for each of the d i f ferent rat ioning cases, as presented in equations (5.32) and (5.36) - (5.40). In order to evaluate the s ingle and double integrals of the density functions in these equations, in each case we standardize the relevant unconditional normal density function and adjust the l im i ts of integrat ion accord ing ly . 2 Essen t ia l l y these adjustments enta i l subtracting the mean of the unstandardized d i s t r i bu t i on from the ex is t ing integrat ion l im i t and then d iv id ing by the standard deviat ion of the same d i s t r i bu t i on . This standard dev ia t ion , which is calculated using Anderson's Theorem 2.5.1 as in (5.30), can be seen to involve a l l the elements of the o r ig ina l variance-covariance matrix of the 1 As explained in Section 5.2.2 of the previous chapter, for each rat ioning case these matrices comprise the rows and columns of the ent i re 8x8 variance-covariance matrix which pertain to that par t i cu la r case. This process is described in deta i l in the previous chapter for Cases 1 and 3 (hours and weeks rat ioned) , with the adjusted integrat ion l im i t s defined in (5.33) and (5.34). - 160 -error terms for the unstandardized d i s t r i b u t i o n . The spec i f i c reason for requir ing th is matrix to be posi t ive de f in i te is that th is condit ion is necessary and su f f i c ien t to ensure that the standard dev ia t ion , which is used as the d iv isor in the adjustment, is s t r i c t l y pos i t i ve . To i l l u s t r a t e th is requirement we consider Cases 4 and 6 where only hours are rat ioned. Here the variance of the unconditional density func t ion , obtained by making the appropriate subst i tut ions in (5.30), is defined as 2 2 (6.1) VAR = O i / U - P45) 1 P u t Pl5 P m 1 P'tS P15 P45 1 where the notation is the same as that used in the previous chapter. Denoting the variance-covariance matrix of the error vector in Cases 4 and 6, (ui.ui+.us), as V, then i t is c lear that (6.1) can be writ ten as (6.2) VAR = IVI / ((1- ol5).ol.ol) 2 2 Hence since we require ah and a 5 to be pos i t ive and -1 < p^5 < 1, i t is necessary that |vj > 0 in order to ensure that VAR > 0 and hence that the adjusted integrat ion l im i t is def ined. To meet a l l these c r i t e r i a , i t is thus necessary that V be a pos i t ive de f in i te matr ix. Simi lar arguments can be used to demonstrate the necessity for the variance-covariance matrix of the error vector in Case 2, ( u 2 , u 6 , u 7 ) , to also be pos i t i ve de f i n i t e . In Cases 1 and 3 which involve integrat ion of a b ivar ia te densi ty , the argument is s l i g h t l y more complex. Here we require 2 2 2 2 two variances defined as o"i( l- p 1 3 ) and a 2 ( l - P 2 3 ) to be pos i t ive and a cor re la t ion coe f f i c ien t defined as 2 2 (6.3) COR = ( p 1 2 - P I 3 P 2 3 ) / A l - P13) • ( ! " P23) to be bounded by pos i t ive and negative uni ty . It is straightforward to show that pos i t ive def ini teness of the variance-covariance matrix of - 161 -( u i , U 2 , U 3 ) i s requ i red in order f o r these c o n d i t i o n s to be met. F i n a l l y in Case 5 , in order to eva lua te the t r i v a r i a t e normal dens i t y f u n c t i o n , i t i s necessary tha t the v a r i a n c e - c o v a r i a n c e mat r ix of ( u l s U 2 , u 8 ) a l so be p o s i t i v e d e f i n i t e . In order to ensure tha t these four v a r i a n c e - c o v a r i a n c e mat r i ces remain p o s i t i v e d e f i n i t e , we make use of a technique o u t l i n e d in Magnus [1982 ] . With t h i s t echn ique , ins tead of op t im i z i ng with respec t to the elements 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 do so with respec t to the elements of a lower t r i a n g u l a r m a t r i x , L-j. As shown by Magnus, provided the d iagonal elements of t h i s lower t r i a n g u l a r mat r ix are a l l kept s t r i c t l y p o s i t i v e , the product L-j L i ' w i l l y i e l d a p o s i t i v e d e f i n i t e m a t r i x . Thus wi th t h i s method we maximize the log l i k e l i h o o d with respec t to the elements of L i , i = l , 2 , 3 , 4 , where (6 .4) L i = 111 0 0 112 122 0 I 1 l 1 I 1 '13 '23 133 and L i L i 1 = Vi which i s the v a r i a n c e - c o v a r i a n c e mat r ix f o r the i t h ( i = l , 2 , 3 , 4 ) r a t i o n i n g c a s e . To ensure l l . > 0 , f o r a l l j = l , 2 , 3 and a l l i = l , 2 , 3 , 4 , we add a pena l ty to the l i k e l i h o o d f unc t i on as f o l l o w s . Observ ing that (6.5a) 1 ^ > 0 -> - l j j - 0 <6-5b> ~> N j l - V 0 -we rep lace L, the l i k e l i h o o d f u n c t i o n which i s to be maximized, by L * , where ( 6 . 6 ) L* - L -PEW..CI - i j j ) ] , where PENN i s a la rge p o s i t i v e number. In t h i s way l1._> 0 as r e q u i r e d , J 3 - 162 -and by scal ing the parameters i t is quite straightforward to ensure l i . > 0 for a l l Here the requirement that l l . b e s t r i c t l y pos i t ive is necessary to ensure pos i t ive def in i teness of the matr ix, since i f 11.= 0 for any i , j , the determinant of Vj w i l l also equal zero as J <J ( 6 - 7 ) | V i | = | L i L i ' | = Oll) 2-(l22) 2 . ( l 33) 2 • A par t i cu la r d i f f i c u l t y which ar ises when we apply th is lower t r iangu lar decomposition technique to the hours and weeks rat ioning model concerns the fact that cer ta in variances appear in more than one of the covariance matrices that are to be decomposed. S p e c i f i c a l l y , a2 appears in the covariance matrices for hours ra t ioned, hours and weeks rationed and no ra t ions , while a2 appears in covariance matrices for the weeks ra t ioned, hours and weeks rationed and non-rationed cases. As a r e s u l t , four covariance matrices cannot be decomposed separately. Rather i t is necessary to impose res t r i c t i ons so that these variances w i l l take the same value in each matr ix. To accomplish th is we f i r s t decompose the variance-covariance matrices for hours rationed and for weeks rat ioned. Then the lower t r iangular elements which form a2 and a2 in these matrices 3 1 2 are also used to form these two variances in the matrices for the cases where both hours and weeks are rationed and where there are no ra t ions . In th is way, rather than 24 d i s t i nc t lower t r iangular elements we only use 19 elements which, as explained in the previous chapter, is the same as the to ta l of the number of variances (8) and the number of cor re la t ion coef f i c ien ts (11) defined in the model . 3 In fact opt imizat ion is only performed using 17 lower t r iangular elements. As outl ined in the previous chapter, p26 and p 2 7 are not i d e n t i f i e d . In order to set these two corre la t ion coef f i c ien ts equal to zero, i t is necessary to set two lower t r iangular elements equal to zero. - 163 -6.3 Empirical Results for the Complete Rationing Model We now turn to a consideration of the resu l ts obtained for the complete hours and weeks model using the Michigan data. We begin by discussing various parametric res t r i c t i ons which are imposed in order to s impl i fy the model. Next we examine the parameter estimates obtained for the model. As with the s ing le - ra t ion model estimated in Chapter 4, households are separated into four groups according to various demographic p ro f i l es and estimates are obtained for each group separately. In addit ion we also examine the ef fect and importance of including the SS correct ion in the model by performing the estimation for each household group both with and without th is correct ion included. Estimates of marginal budget shares and the implied labour supply e l a s t i c i t i e s are analyzed in Section 6 .3 .3 , while in the f i na l section we compare these estimates to those obtained for the s ing le - ra t ion model. 6.3.1 Parametric Rest r ic t ions A major d i f f i c u l t y which is encountered when we attempt to maximize the l i ke l ihood function derived in the previous chapter concerns the fact that we are unable to concentrate th is function with respect to the parameters of the variance-covariance matrix of the error terms. Essen t i a l l y th is is due to two fac to rs . F i r s t , various elements of th is matrix appear in the integrat ion l im i t s and in the sample s e l e c t i v i t y correct ions in the l i ke l ihood funct ion . Second, many other elements of th is matrix are not determined in the model and therefore cannot be ca lcu la ted . As a resul t i t is necessary to estimate 17 parameters of the variance-covariance matrix of the error terms in addit ion to the 7 s t ructura l parameters of the model. Due to th is large number of parameters - 164 -and the complexity of the likelihood function, i t is therefore not surprising that convergence of the procedure used to maximize the likelihood function could not readily be achieved. In several instances, problems of convergence occurred as various elements of the lower triangular matrices, which decompose the variance-covariance matrices of the error terms, approached zero. As outlined in Section 6.2, when the diagonal elements of these lower triangular matrices approach zero, so also does the determinant of the variance-covariance matrix which they decompose, defined in equation (6.7). Consequently since this determinant is used as a divisor in calculating limits of integration, i t is clear that the likelihood function will be unstable should any of these diagonal elements approach zero. Further i t is clear that convergence cannot be achieved by fixing these parameters at zero. Rather i t is necessary either to f i x these parameters at some small positive value, or to set other off-diagonal elements — and hence correlation coefficients in the various variance-covariance matrices ~ at zero. However not a l l convergence d i f f i c u l t i e s are due to problems with diagonal elements of the lower triangular matrices. As explained in the previous chapter, correlation coefficients which appear only in the integration limits can possibly be identified only i f there is a tradeoff due to these coefficients forming part of more than one integration lim i t . Thus while P 26 and p27 are not identified, i t is possible that the remaining correlation coefficients are identified, since in each of the hours rationed and the hours and weeks rationed cases, there are two integrals corresponding to the states where observed hours exceed, and where they are less than, desired hours. However with the division of the - 165 -observations into four groups according to demographic cha rac te r i s t i c s , i t is not c lear that there are su f f i c ien t observations in each of the separate rat ioning cases for these cor re la t ion coef f i c ien ts to be ident i f ied. 1* For example, the group of households which have chi ldren and education contains only 7 observations which have both hours and weeks ra t ioned . 5 Of these 7 observations, 5 refer to the case where hours worked are below desired levels while only 2 households have hours worked which exceed the desired l e v e l . Thus i t is not c lear that there are su f f i c i en t observations in each of the two groups for P13 and p 23 to be i d e n t i f i e d . Simi lar d i f f i c u l t i e s occur with the other groups of households both for th is hours and weeks rationed case and for the hours rationed case, so that problems in ident i fy ing P ^ and p 1 5 are also l i k e l y to contr ibute to convergence d i f f i c u l t i e s . In an attempt to overcome these various convergence and i den t i f i ca t i on problems, four addit ional off-diagonal elements of the lower t r iangular matrices are each set equal to z e r o . 6 Consequently a fur ther four cor re la t ion coef f i c ien ts — P14. and p 1 5 in the cases where only hours are rationed and P13 and p 2 3 in the cases where hours and weeks are both rationed — are f ixed at zero. Hence in addit ion to the 8 var iances, we now only estimate the 5 cor re la t ion coe f f i c ien ts p 1 2 , pk5, P67> Pis a n d P 28 • Despite the s imp l i f i ca t i on of the optimization procedure due to th is reduction in the number of parameters which remain to be estimated, convergence of the maximizing algorithm could s t i l l not be k The four groups of households correspond to those adopted in the s ing le - ra t ion model. Detai ls are provided in the fol lowing sec t ion . Detai ls of the data used for the two-ration model are contained in Appendix 1. These are in addit ion to the two elements which are set equal to zero in order to ensure that p 2 6 and p 2 7 , which are not i d e n t i f i e d , take zero values. - 166 -achieved. Addit ional experimentation which involved sett ing various other of f-diagonal elements equal to zero proved to be of l i t t l e assistance in th is regard. Further, by set t ing these parameters equal to zero, much of the underlying interest of the model is l o s t , as these res t r i c t i ons imply that the error terms in the d i f ferent rat ioning cases are uncorrelated, thereby reducing the model to a simple switching regression model. In order to avoid further res t r i c t i ons on the variance-covariance matrix and yet achieve convergence, i t is necessary to also impose res t r i c t i ons on the st ructura l parameters. Since one of the major d i f f i c u l t i e s in convergence was observed to involve an extremely large gradient on Y I , i t was decided to set th is parameter at a par t icu lar value. In view of the fact that the Stone-Geary u t i l i t y function (5.44) used with th is model is only defined i f Xi > Y i , the value at which Y I has been set is the minimum value of Xi ( le isure of the household head during working weeks) for the relevant subsample. For households with chi ldren the minimum values of Xi are 2560 hours and 2880 hours for households with and without education respect ive ly . For households without chi ldren the corresponding minimum Xi values are s i g n i f i c a n t l y lower at 858 hours for households with education and 1024 hours for those without education. One d i f f i c u l t y which resu l ts from imposing d i f ferent YI values for the four groups of households involves the test ing of hypotheses concerning the s ign i f icance of di f ferences between parameter estimates for the d i f ferent household groups. In addit ion to preventing the use of Likel ihood Ratio tests to determine whether the parameters as a whole d i f f e r from one group to another, such an approach also makes i t d i f f i c u l t to e f fec t i ve l y compare the values of par t icu lar parameters for the - 167 -d i f fe rent groups. S p e c i f i c a l l y , i t is not c lear to what extent di f ferences in parameter values across groups are due to the d i f ferent demographic charac te r i s t i cs of the groups and to what extent they are a r t i f i c i a l l y caused by the d i f ferent Y i values imposed. While i t is possible to impose the same Y i value for each group, th is procedure is also undesirable due to the fact that the minimum Xi value of 858 hours for the group of households with education but no chi ldren is only approximately one-third of the values for households with ch i l d ren . However since the two values for households with chi ldren are of a s imi lar magnitude, as are the two values for households without ch i l d ren , i t is possible to impose Y I values of 2560 and 858 for households with and without chi ldren respect ive ly . In th is way comparisons among the two groups with chi ldren and among the two groups without chi ldren are made less confusing. To determine the ef fect of assuming the same Y i value for both groups of households with chi ldren and the same Y i value for both groups of households without ch i ld ren , estimates are obtained for the non-educated households in each of these groups using two d i f ferent assumed values of Y i in each case. Thus for households with ch i l d ren , estimates for the subset of non-educated households are obtained f i r s t with YI = 2560 hours, th is being the minimum Xi value for households with chi ldren and education, and then with Y i = 2880 hours, which is the minimum Xi value for households with chi ldren and no education. For households with no chi ldren and no education, estimates are obtained using Y I = 858 and Y i = 1024. By comparing the parameter estimates which resu l t from the d i f fe rent Y i values, we are able to determine the s e n s i t i v i t y of these estimates to the par t i cu la r Y i value which is imposed. - 168 -The parameter estimates obtained for households with no education using the d i f ferent Y i values are presented in Table 6 . 1 . 7 For each of the two groups of households, estimation is performed f i r s t excluding and then including the sample s e l e c t i v i t y correct ion for working wives. As can be seen from the tab le , the di f ference in Y i values appears to cause no s ign i f i can t di f ferences in the estimates of the other parameters. For each of the four cases, pairwise comparisons between the estimates obtained using the two Y i values reveal that the same parameters are s i g n i f i c a n t , while the largest di f ferences in magnitudes, which occur for Bi and B c , are s t i l l much smaller than the standard errors on these coe f f i c i en t s . Thus i t appears that parameter estimates obtained using a Y i value of 2560 hours for the two groups of households with ch i ld ren , and a Y i value of 858 hours for both groups of ch i l d less households, w i l l not d i f f e r s i g n i f i c a n t l y from those obtained for each group by set t ing Y i at the minimum Xi value for that group. We now turn to an examination of the parameter estimates obtained using these Y i values. 6.3.2 Parameter Estimates In th is section we discuss the estimates obtained with the f u l l two-ration model after incorporating the various res t r i c t i ons described in the previous sec t ion . Thus Y i is f ixed at 2560 for households with chi ldren and at 858 for households without ch i ld ren , while a l l except f i ve cor re la t ion coef f i c ien ts are set equal to zero by f i x i ng various off-diagonal elements of the lower t r iangular matrices at zero. 7 Note that even with the res t r i c t i ons imposed, to obtain these estimates i t is necessary to scale cer ta in parameters so that in the optimizat ion procedure a l l parameters are of the same order of magnitude. - 169 -TABLE 6.1: COMPARISON OF PARAMETER ESTIMATES WITH DIFFERENT Y i VALUES PARAMETER c NE NC NE NO SS SS NO SS SS Y l .256 ( ) .288 ( ) .256 ( ) .288 ( ) .0858 ( ) .1024 ( ) .0858 ( ) .1024 ( ) Y 2 -.020 (.081) -.020 (.084) -.205 (.163) -.192 (.131) .249* (.030) .246* (.031) .250* (.029) .246* (.026) Y f .817* (.014) .816* (.014) .926* (.017) .926* (.017) .731* (.009) .731* (.010) .723* (.012) .723* (.012) Y c .230* (.098) .238* (.096) .123* (.053) .133* (.052) .711* (.029) .714* (.030) .720* (.030) .723* (.028) h .516* (.060) .492* (.062) .377* (.084) .363* (.076) .863* (.049) .852* (.052) .862* (.047) .850* (.044) .072 (.113) .075 (.123) .298* (.133) .296* (.114) -.307* (.074) -.308* (.079) -.309* (.071) -.310* (.066) 3 f -.072* (.018) -.075* (.019) .353 (.309) .329 (.274) -.005 (.012) -.005 (.012) .016 (.016) .017 (.017) P c .484* (.070) .508* (.077) -.027 (.291) .013 (.269) .449* (.031) .461* (.033) .431* (.031) .443* (.029) Code: C = Youngest c h i l d less than 6 years o l d NC = No c h i l d r e n younger than 6 years o l d E = Head or spouse completed at l e a s t one year of col lege NE = Neither head nor spouse attended c o l l e g e SS = Includes sample s e l e c t i v i t y c o r r e c t i o n f o r working wives NO SS = Does not include sample s e l e c t i v i t y c o r r e c t i o n Notes: Numbers in parentheses are asymptotic standard e r r o r s . S i n c e Y i i s held f i x e d , i t has no standard e r r o r . 3 C i s obtained as 1 - $y - 3 2 " 3 f Y j are a i l s c a l e d by 1/10000 * i n d i c a t e s t h a t the estimate i s s i g n i f i c a n t l y d i f f e r e n t from zero at the 5% I eve I. - 170 -As in the s i n g l e - r a t i o n model , i t i s des i red to inco rpora te va r i ous demographic v a r i a b l e s in our a n a l y s i s s i nce we expect the labour supp ly behaviour of household members to d i f f e r accord ing to va r ious demographic c h a r a c t e r i s t i c s such as whether or not young c h i l d r e n are present in the househo ld . Due to the complex i ty of the model , ra the r than adding e x t r a parameters we again choose to d i v i d e the sample accord ing to p a r t i c u l a r demographic p r o f i l e s . In view of the r e s u l t s of our a n a l y s i s in the s i n g l e - r a t i o n model , and the f a c t tha t any f u r t h e r breakdown of obse rva t i ons would r e s u l t in some r a t i o n i n g ca tego r i es with no o b s e r v a t i o n s , we again s p l i t households in to four groups accord ing to whether or not there are young c h i l d r e n present in the household and whether or not the head or spouse completed at l e a s t one year of c o l l e g e . While i t i s d e s i r a b l e to conduct L i k e l i h o o d Ra t io (LR) t e s t s to determine whether the parameters are s i g n i f i c a n t l y d i f f e r e n t f o r a l l f ou r household groups, due to the complex i ty of the l i k e l i h o o d f u n c t i o n es t imates cou ld not be obta ined using the e n t i r e sample. However s i n c e Y I i s f i x e d at a p a r t i c u l a r va lue f o r both groups of households wi th c h i l d r e n and at another va lue f o r both groups of households wi thout c h i l d r e n , and s i nce the l i k e l i h o o d f u n c t i o n can be maximized f o r each of these two groups s e p a r a t e l y , some l i m i t e d t e s t i n g i s p o s s i b l e . In p a r t i c u l a r , we can use the LR technique to t e s t the hypothes is tha t f o r each of these two groups of households ~ with and wi thout young c h i l d r e n ~ the parameters d i f f e r depending on whether or not the head or spouse completed at l e a s t one year of c o l l e g e . To c a l c u l a t e the LR s t a t i s t i c we compute the negat ive of tw ice the d i f f e r e n c e between the f u n c t i o n va lue f o r the pooled data set — a l l households wi th c h i l d r e n — and the sum of the f u n c t i o n va lues obta ined - 171 -f o r households wi th c h i l d r e n and educat ion and those wi th c h i l d r e n and no e d u c a t i o n . Th is process i s then repeated f o r households wi th no c h i l d r e n . Here these LR s t a t i s t i c s w i l l have a c h i - s q u a r e d i s t r i b u t i o n wi th 19 degrees of freedom s ince there are 6 f r ee s t r u c t u r a l parameters and 13 remain ing elements of the v a r i a n c e - c o v a r i a n c e matr ix of the e r ro r terms. Hence the c r i t i c a l va lues at the 1% and 5% s i g n i f i c a n c e l e v e l s are 36.19 and 30 .14 , r e s p e c t i v e l y . For households wi th c h i l d r e n the c a l c u l a t e d LR s t a t i s t i c i s 104.92 when the SS c o r r e c t i o n i s excluded and 126.83 when i t i s i n c l u d e d . For households wi thout c h i l d r e n the cor responding s t a t i s t i c s are 59.81 and 65 .74 . S ince these va lues exceed the 1% c r i t i c a l va lue we conclude that f o r households with c h i l d r e n and households wi thout c h i l d r e n , the parameter es t imates d i f f e r s i g n i f i c a n t l y depending on whether or not the head or spouse attended c o l l e g e . Hence in Table 6.2 we present separate es t imates f o r the d i f f e r e n t household t y p e s . As wi th the s i n g l e - r a t i o n model , in order to examine the e f f e c t s and importance of the SS c o r r e c t i o n f o r working w ives , es t imates are presented both wi th and wi thout t h i s c o r r e c t i o n inc luded in the l i k e l i h o o d f u n c t i o n . Comparing the SS-co r rec ted and SS-uncor rec ted es t imates in Table 6 . 2 , i t i s c l e a r that the main e f f e c t s of the c o r r e c t i o n occur f o r households wi th both c h i l d r e n and educa t i on . Here wi th the add i t i on of the SS c o r r e c t i o n , both Y c and Bf become 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 . In a d d i t i o n f o r t h i s group the magnitudes of Y f and B c decrease when the c o r r e c t i o n i s i n c l u d e d . Changes in s i g n i f i c a n c e of parameters a l so occur f o r households with c h i l d r e n and no e d u c a t i o n , wi th 3 2 becoming s i g n i f i c a n t and 3f and 3 C becoming i n s i g n i f i c a n t when the SS c o r r e c t i o n i s added to the model . For t h i s group of households, the magnitudes of and &2 both inc rease wh i le &i decreases in the - 172 -TABLE 6.2: PARAMETER ESTIMATES FOR THE TWO-RATION MODEL PARAMETER c E c NE NC E NC, NE NO SS SS NO SS SS NO SS SS NO SS SS y\ .256 ( ) .256 (____) .256 ( ) .256 ( ) .0858 ( ) .0858 ( ) .0858 ( ) .0858 ( ) Y 2 .271* (.036) .234* (.040) -.020 . (.081) -.205 (.163) .270* (.032) .268* (.029) .249* (.030) .250* (.029) Yf .738* (.018) .640* (.038) .817* (.014) .926* (.017) .705* (.009) .686* (.012) .731* (.009) .723* (.012) Yc -.071 (.325) .364* (.099) .230* (.098) .123* (.053) .758* (.141) .791* (.134) .711* (.029) .720* (.030) h .553* (.096) .486* (.080) .516* (.060) .377* (.084) .845* (.061) .827* (.052) .863* (.049) .862* (.047) -.381* (.1 13) -.278* (.103) .072 (.113) .298* (.133) -.352* (.077) -.341* (.067) -.307* (.074) -.309* (.071) 8f .015 (.025) .214* (.048) -.072* (.018) .353 (.309) .024* (.012) .055* (.016) -.005 (.012) .016 (.016) 3c .813* (.067) .578* (.058) .484* (.070) -.027 (.291) .484* (.040) .458* (.039) .449* (.031) .431* (.031) For Notes and Codes, see Table 6.1. - 173 -SS-corrected case. For households with no ch i ld ren , no changes in s ign i f icance are recorded although with the inclusion of the SS cor rec t ion , Y f decreases and Sf increases for those households which also have education. Examining the SS-corrected estimates for the four household groups we observe that while Y f and Yc are s ign i f i can t for a l l four groups, Y2 is not s ign i f i can t for households with chi ldren and no education. However the Y2 values for the other three groups are a l l s imi la r in magnitude. In fac t a l l s ign i f i can t Y i parameters are pos i t i ve , lending credence to the i r in terpretat ion as subsistence or necessary quant i t ies . The subsistence quantity of female l e i su re , Y f , is largest for the two groups of households with no education. Since " l e i su re " here is defined as non-market time-, i t is not surpr is ing that the largest of these two values occurs for households which also have ch i ld ren . On the other hand, Yc is largest for the two groups with no chi ldren and is smallest for households with chi ldren and no education. In general the largest values for the Y i parameters occur for Y f and Yc> with the imposed Y I value being the smallest magnitude for households with no chi ldren while Y2 is the smallest for households with ch i ld ren . As is the case with the s ing le - ra t ion model, the Si parameters can be interpreted as the marginal budget shares (mbs) for non-rationed households. While S i and S2 are s ign i f i can t for a l l four household groups, Sf is not s ign i f i can t for e i ther group of households with no education and S c is not s ign i f i can t for households with chi ldren but no education. Values of S i are s imi la r at approximately 0.85 for both groups of households with no ch i l d ren , but are considerably lower at 0.49 and 0.38 for households with chi ldren and education and those with chi ldren - 174 -and no educa t i on , r e s p e c t i v e l y . Here B 2 i s negat ive at around - 0 . 3 f o r a l l households except those with c h i l d r e n but no educa t i on , where i t takes a va lue of +0.3. Al though the mbs f o r female l e i s u r e , B f , i s p o s i t i v e f o r a l l househo lds , i t i s on ly s i g n i f i c a n t f o r households with educat ion and, l i k e B c , i s l a r g e r f o r those households that have both c h i l d r e n and e d u c a t i o n . Note tha t s ince at l e a s t one Bi va lue is negat ive f o r each group of households, r e g u l a r i t y c o n d i t i o n s are not s a t i s f i e d f o r t h i s model . Comparing the r e l a t i v e s i z e s of the va r ious non- ra t ioned mbs fo r the d i f f e r e n t groups of househo lds , we see tha t both groups wi th no c h i l d r e n e x h i b i t s i m i l a r behav iour : 8 i i s approx imate ly tw ice the s i z e of B c , Bf i s approx imate ly zero and B 2 i s negat ive at - 0 . 3 . For households wi th c h i l d r e n and educa t i on , B i and B c are of s i m i l a r s i z e s , with Bf being p o s i t i v e at approx imate ly 0.2 and B 2 again negat ive at - 0 . 3 . Thus wh i le B c f o r t h i s group i s l a rge r than f o r households wi th no c h i l d r e n , B i i s on l y about h a l f the s i z e of the cor respond ing mbs in c h i l d l e s s househo lds . F i n a l l y , the mbs es t imates f o r households wi th c h i l d r e n but no educat ion are qu i t e d i f f e r e n t from those fo r the other households in that on l y B i and B 2 are s i g n i f i c a n t and both these mbs have s i m i l a r p o s i t i v e va lues of 0.38 and1 0 .30 . Thus t h i s i s the on ly group of households f o r which the mbs of male l e i s u r e dur ing non-working weeks i s p o s i t i v e . Before examining the mbs f o r the va r ious ra t i oned c a s e s , i t i s use fu l to b r i e f l y cons ide r the es t imates of the va r ious elements of the v a r i a n c e - c o v a r i a n c e matr ix of the e r r o r terms obta ined f o r the d i f f e r e n t household groups. In terms of the va r i ances of the e r ro r s on the va r i ous l e i s u r e demand equa t i ons , the e f f e c t of the SS c o r r e c t i o n i s to inc rease the va r iances on the equat ions f o r female l e i s u r e in the d i f f e r e n t - 175 -r a t i o n e d c a s e s . However s i g n i f i c a n t inc reases in these va r iances on ly occur f o r households with c h i l d r e n , p a r t i c u l a r l y those which a l so have no e d u c a t i o n . While a l l the va r iances are 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 , the va r i ances on the hours equat ions are much sma l le r than the other v a r i a n c e s . Th is r e s u l t i s expected s ince the dependent v a r i a b l e in these equat ions i s a sha re , l y i n g between zero and u n i t y , which i s then d i v i ded by K, the number of weeks worked. S ince the average value of K i s between 30 and 40 weeks, the dependent v a r i a b l e s and d e t e r m i n i s t i c rhs of these equat ions are c o n s i d e r a b l y sma l le r than those in the other equa t i ons . One other i n t e r e s t i n g aspect of these va r i ances concerns t h e i r r e l a t i v e s i z e s in the d i f f e r e n t household groups. For households which have c h i l d r e n but no educat ion - - the group wh ich , as d iscussed above, y i e l d s c o n s i d e r a b l y d i f f e r e n t es t imates of the s t r u c t u r a l parameters to the other groups — the es t imates of the va r iances on the four equat ions f o r female l e i s u r e are l a r g e r than the cor responding va r i ances f o r the other groups. F u r t h e r , f o r t h i s group, es t imates of the va r iances of the e r ro r s on the equat ions f o r the two types of male l e i s u r e are sma l l e r than those f o r the other household groups. In Table 6.3 we present es t imates of the f i v e non-zero c o r r e l a t i o n c o e f f i c i e n t s f o r each of the four groups of househo lds , both wi th and wi thout i n c l u d i n g the SS c o r r e c t i o n . While the i n c l u s i o n of the SS c o r r e c t i o n has no e f f e c t on the s i g n i f i c a n c e of these c o e f f i c i e n t s f o r the two groups of households with no c h i l d r e n , d i f f e r e n c e s do occur f o r households wi th c h i l d r e n . In p a r t i c u l a r , when the SS c o r r e c t i o n i s i n c l u d e d , both pt^ and P^7 become s i g n i f i c a n t f o r the two groups of households wi th c h i l d r e n , wh i le f o r households with c h i l d r e n and e d u c a t i o n , P i s and P 2 8 a l so become s i g n i f i c a n t . Examining the SS-co r rec ted - 176 -TABLE 6.3: CORRELATION ESTIMATES FOR THE TWO-RATION MODEL COEFFICIENT c E C,NE NC E NC, NE NO SS SS NO SS SS NO SS SS NO SS SS Pl2 -.843* (.051) -.816* (.065) -.555* (.151) -.460* (.149) -.892* (.022) -.894* (.021) -.919* (.015) -.919* (.015) Pi+5 -.340 (.262) -.720* (.154) -.300 (.302) .759* (.279) -.019 (.191) -.108 (.209) -.016 (.162) -.117 (.187) P67 -.138 (.526) -.966* (.039) .128 (.383) -.731* (.140) -.312 (.376) -.568 (.292) -.206 (.260) -.264 (.276) P18 -.126 (.158) -.370* (.154) .341 (.175) -.239 (.204) -.188* (.088) -.260* (.094) -.016 (.095) -.008 (.103) P28 .131 (.159) .352* (.152) -.055 (.216) -.134 (.183) .174* (.087) .234* (.092) -.009 (.095) -.033 (.105) For Notes and Codes, see Table 6.1. - 177 -coe f f i c i en t s , only P12 is s ign i f i can t for a l l four household groups. This co r re la t i on , which is negative for a l l groups, is largest in absolute value for households with neither chi ldren nor education (-0.919) and smallest in absolute value for those with chi ldren but no education (-0.460). While pG7 is also negative for a l l household groups, i t is only s ign i f i can t for households with ch i l d ren . S im i la r l y p 4 5 is only s ign i f i can t for these two groups, but th is cor re la t ion is negative for households with chi ldren and education and pos i t ive for those with chi ldren but no education. F i n a l l y , the two corre la t ion coef f i c ien ts defined only in the non-rationed case ~ p1Q and p 28 ~ are only s ign i f i can t for household groups which are educated. However while p 1 8 , the cor re la t ion between male le isure in working weeks and female l e i s u r e , is negative, P28> the cor re la t ion between male le isure in non-working weeks and female l e i su re , is pos i t i ve . 6.3.3 Estimates of E l a s t i c i t i e s and Marginal Budget Shares We now turn to an analysis of the e l a s t i c i t i e s and marginal budget shares calculated from the parameter estimates in the preceding section for the various rat ioning cases. Except for Cases 4 and 6 where only hours are rat ioned, the mbs in th is model depend only on the parameter est imates. 8 Thus these mbs are constant for a l l ind iv iduals in each group and are the same for ind iv iduals in Case 1 and in Case 3; that i s , the two cases where both hours and weeks are ra t ioned. However when only hours are rat ioned, the mbs d i f f e r across ind iv iduals as they depend on both data and parameters. Consequently, in order to evaluate these indiv idual mbs at the estimated point , i t is necessary to obtain estimated values of the 8 Expressions for the mbs and e l a s t i c i t i e s for th is model are contained in Appendix 4. - 178 -endogenous var iab les . However, as out l ined in the previous chapter, in the hours rationed case the endogenous var iable K (weeks worked by the male) cannot be determined e x p l i c i t l y . Thus i t is necessary to use a gr id search procedure to f ind the value of K which, when used with the estimated parameters, w i l l sa t i s f y (5.51). A further problem with having mbs (and e l a s t i c i t i e s ) which d i f f e r across ind iv iduals within any rat ioning category is that i t makes comparisons of these measures for the d i f ferent rat ioning categories quite awkward. In order to f a c i l i t a t e analysis i t is therefore necessary to compute a s ingle set of mbs and e l a s t i c i t y estimates which can be considered representative of the ind iv iduals in each rat ioning category. In th is regard two a l ternat ive measures which can be used are the median of the indiv idual values and a weighted average of the indiv idual values. A par t i cu la r advantage of using the median is that unl ike a weighted average, i t is not affected by extreme indiv idual values. Thus even i f some observations have large predict ion errors and consequent extreme estimated values of mbs and e l a s t i c i t i e s , the median w i l l be representative of the majority of the indiv idual values in the par t i cu la r ra t ion ing category. Hence whenever mbs or e l a s t i c i t i e s vary across ind iv idua ls , median values are presented. However when the number of observations in any category is less than 5, no median is calculated since with so few observations even the median cannot necessar i ly be considered representat ive of the indiv idual values. Estimated marginal budget shares for th is model are presented in Table 6.4. Although the mbs in the NR case are simply the 3i estimates, we reproduce them in th is table for comparison purposes. Comparing the values in the SS-corrected and SS-uncorrected cases, we observe that the - 179 -TABLE 6.4: MARGINAL BUDGET SHARES FOR THE TWO-RATION MODEL MARGINAL c, E c, NE NC, E NC, NE BUDGET SHARE NO SS SS NO SS SS NO SS SS NO SS SS HK f .018 .270* -.174* 1.084 .047* .107* -.012 .036 (.030) (.054) (.043) (.891) (.023) (.031) (.026) (.036) c .982* .730* 1.1 74* -.084 .953* .893* 1.012* .964* (.030) (.054) (.043) (.891) (.023) (.031) (.026) (.036) H(4) 1 .589* .474* -.137 -.735* .696* .659* .879* .879* (.188) (.161) (.224) (.330) (.161) (.131) (.196) (.182) 2 -.773* -.606* .184 .983* -.962* -.911* -1.206* -1.209* (.247) (.211) (.301) (.441) (.222) (.181) (.271) (.255) f .021 .305* -.165* .813 .059* .134* -.015 .046 (.035) (.058) (.040) (.660) (.029) (.039) (.034) (.046) c 1.1 46* .826* 1.115* -.063 1.207* 1.118* 1.303* 1.242* (.066) (.079) (.100) (.666) (.065) (.062) (.074) (.076) TT(6) 1 T t # # .776* .744* .638* .636* (.181) (.151) (.138) (.127) 2 # # t t -1.073* -1.029* -.872* -.868* (.250) (.209) (.189) (.174) f t t § t .059* .135* -.015 .044 (.029) (.039) (.032) (.044) c t t t # 1.213* 1.125* 1.255* 1.193* (.064) (.062) (.063) (.067) 1 .400* .381* .555* .537* .624* .617* .661* .658* (.045) (.042) (.025) (.045) (.023) (.022) (.010) (.010) f .011 .167* -.077* .502 .017* .041* -.004 .012 (.018) (.041) (.018) (.446) (.009) (.012) (.009) (.012) c .589* .452* .522* -.039 .358* .342* .343* .330* (.046) (.030) (.031) (.415) (.024) (.023) (.012) (.014) NR 1 .553* .486* .516* .377* .845* .827* .863* .862* (.096) (.080) (.060) (.084) (.061) (.052) (.049) (.047) 2 -.381* -.278* .072 .298* -.352* -.341* -.307* -.309* (.113) (.103) (.113) (.133) (.077) (.067) (.074) (.071) f .015 .214* -.072* .353 .024* .055* -.005 .016 (.025) (.048) (.018) (.309) (.012) (.016) (.012) (.016) c .813* .578* .484* -.027 .484* .458* .449* .431* (.067) (.058) (.070) (.291) (.040) (.039) (.031) (.031) Notes: Marginal budget shares for H cases are median values t i n d i c a t e s fewer than 5 observations; mbs > 0 # i n d i c a t e s fewer than 5 observations; mbs < 0 For Codes and a d d i t i o n a l notes, see Table 6.1 and t e x t - 180 -major ef fects of th is correct ion occur for groups of households with ch i l d ren . For those households with both chi ldren and education, the mbs for female le isure increases in s ize and becomes s ign i f i can t for a l l rat ioning cases when the SS correct ion is included. This increase in the female le isure mbs is of fset by a decrease in the mbs for goods consumption, although th is mbs remains s ign i f i can t both with and without the SS cor rec t ion . For households with chi ldren but no education the mbs for female le isure and for goods consumption, which are both s ign i f i can t when there is no SS cor rec t ion , change sign and become ins ign i f i can t in a l l rat ioning cases when the SS correct ion is included. However for th is household group, the mbs for Good 2 ~ le isure during non-working weeks ~ is only s ign i f i can t in the SS-corrected case as is the mbs for Good 1 ~ le isure during working weeks ~ for those households in th is group whose desired hours exceed the i r observed hours (Case 4) . Despite these changes however, except for households with chi ldren and no education, the re la t i ve orderings of the d i f ferent mbs within each rat ioning category are unchanged when the SS correct ion is included. Concentrating now on the SS-corrected mbs for the four groups of households we see in the two cases where both hours and weeks are rationed (which both have the same mbs) that except for households with chi ldren and no education, an extra do l la r of income is al located predominantly to consumption. Here the largest mbs for female le isure occur for households with education, pa r t i cu la r l y those which also have ch i ld ren . This resul t is not surpr is ing in view of the fact that le isure is defined simply as non-market t ime, so that with young chi ldren present a larger demand for such time could be expected. For Case 4 where male hours are rationed to be less than the i r desired l e v e l , we see that except for households with - 181 -chi ldren and no education, the re la t i ve s izes of the mbs are the same for each household group. 9 S p e c i f i c a l l y , in each of these three groups, a l l mbs are pos i t ive except for male le isure in non-working weeks. Of the three pos i t ive values, the largest mbs is for goods consumption, while female le isure has the smallest mbs in each case. In addi t ion, apart from th is mbs for female l e i su re , which is only s ign i f i can t for households with education and which is largest for those households which also have ch i l d ren , the remaining mbs are largest in absolute value for households with no ch i ld ren . For households with chi ldren but no education, the only s ign i f i can t mbs in Case 4 are those for the two types of male l e i su re , although here the mbs for male le isure during working weeks is negative, while the mbs for male le isure during non-working weeks is pos i t i ve . For Case 6 where observed hours exceed desired hours, su f f i c ien t observations are only avai lab le for households without ch i ld ren . For these households, the re la t i ve ordering of mbs is the same as in Case 4, with the only s ign i f i can t mbs for female le isure again occurring for households which also have education. Relat ionships among the SS-corrected mbs are s imi lar in both the weeks rationed and the non-rationed cases. For households with no ch i l d ren , the mbs for male le isure in working weeks exceed those for goods consumption. For households with chi ldren and education, the re la t i ve s ize of these two mbs is reversed, and in th is case the mbs for female le isure takes i t s largest values of 0.17 and 0.21 in the weeks rationed and non-rationed cases respect ive ly . While a l l these mbs are pos i t i ve , the mbs for male le isure in non-working weeks, defined in the non-rationed case, 9 Note that since the mbs presented in the hours rationed cases are medians of the indiv idual observations, they do not necessar i ly sum to un i ty . However for each indiv idual in these groups, mbs do sum to uni ty . - 182 -i s again negative for these three groups of households. For households with chi ldren but no education, the only s ign i f i can t mbs, which occur for the two types of male l e i su re , are both pos i t i ve . To complete our analysis of the mbs for th is model we compare the values of par t i cu la r mbs under the d i f ferent rat ioning s i tua t ions . Examining f i r s t the mbs for male le isure during working weeks, we see that for the two groups of households which have education, th is mbs is largest for non-rationed households, next largest for the two groups of hours rat ioned households, and is smallest for households which have weeks rat ioned. These same educated households are also the only groups which have s ign i f i can t mbs for female l e i su re . Among these households th is mbs is largest for households with hours rationed and with hours and weeks ra t ioned, is next largest for non-rationed households and is again smallest for those households which have weeks rat ioned. The mbs for goods consumption fol lows the same pattern as the female le isure mbs described above, except for households with chi ldren and no education where i t is not s i g n i f i c a n t . F i n a l l y , the mbs for male le isure during non-working weeks, which is negative except for households with chi ldren and no education, is larger in absolute value in the hours rationed cases than in the non-rationed case. We now turn to an analysis of the implied labour supply e l a s t i c i t i e s for th is model. Since these e l a s t i c i t i e s vary across ind iv idua ls , the f igures in Table 6.5 refer to median values for each of the s ix rat ioning categor ies. Further, since par t icu lar e l a s t i c i t i e s may be negative for some ind iv iduals and posi t ive for others, medians are calculated separately for the pos i t ive and negative values of each e l a s t i c i t y , provided there are at least f i ve observations of each type. Separate -183 -T A B L E 6 . 5 : M E D I A N E L A S T I C I T I E S FOR THE T W O - R A T I O N MODEL C , E c N E N C , E N C . N E F l A S T I T I T Y NO S S S S NO S S S S NO S S S S NO S S S S H K U ) HFPM . 5 3 4 1 . 0 0 4 . 0 3 0 - - . 0 3 0 - . 4 9 2 t t - . 0 8 9 H F P F + . 0 4 0 . 2 7 2 t t t . 0 0 9 ( 6 ) . 0 3 8 ( 1 0 ) - - . 4 6 8 - 1 . 0 1 0 t t - . 0 1 3 ( 1 0 ) - . 0 2 6 ( 6 ) H K ( 3 > H F P M + t t t t - t t t t t H F P F + t t t t t - t t T t H U ) WM=M . 0 0 3 ( 5 ) t t t . 0 3 2 ( 1 8 ) . 0 3 8 (1 7) - - . 0 5 9 - . 0 4 3 - . 0 0 1 ( 5 ) - . 0 1 0 - . 0 2 3 - . 0 2 5 - . 0 1 3 ( 1 2 ) - . 0 1 4 ( 1 3 ) W f P F + . 0 3 0 . 0 4 4 . 0 1 5 . 0 4 3 . 0 4 5 . 0 3 3 . 0 3 4 - - . 0 0 3 t t H F P M t . 5 1 9 . 9 4 9 . 0 3 2 t - - . 0 4 9 - . 8 9 9 - . 0 9 2 - . 2 1 8 t - . 1 0 3 H F P F + . 0 7 0 . 8 9 8 . 0 5 6 . 1 2 7 . 0 0 6 ( 5 ) . 0 6 2 ( 2 5 ) - t - . 4 0 4 - . 9 9 9 - . 0 1 9 ( 2 5 ) - . 0 1 7 ( 5 ) H ( 6 > W W M + t t t t t - t t t - . 0 2 0 - . 0 1 9 - . 0 1 4 - . 0 1 4 W W F + t t t . 0 4 2 . 0 4 5 . 0 2 5 . 0 2 6 H F P M + t t t . 0 2 9 - t t - . 1 1 5 - . 2 7 5 - . 0 9 1 H F P F + t t . 0 7 0 . 1 5 9 . 0 6 1 - t t t - . 0 1 9 K < 2 ) HMPM t t t t t . 1 1 1 ( 7 ) . 1 2 8 ( 7 ) - t - . 0 7 1 - . 1 5 0 t t - . 1 8 1 ( 7 ) - . 1 7 3 ( 7 ) HMPF *• . 0 9 8 t t - t t - . 1 3 4 - . 3 4 2 - . 3 7 4 - . 2 7 3 - . 2 8 8 H F P M t . 4 2 3 . 9 2 6 . 0 2 3 - t t - . 0 8 2 - . 1 9 5 - . 0 7 2 H F P F + t t . 0 7 3 .1 7 0 . 0 5 2 - t - . 3 6 9 - . 9 0 7 t t - . 0 1 7 NR HMPM . 4 0 3 . 2 8 9 . 1 0 5 ( 2 1 ) . 0 1 9 ( 9 ) . 2 9 4 ( 9 7 ) . 2 9 5 ( 1 0 0 ) ' . 2 0 4 ( 5 5 ) . 2 1 3 ( 5 6 ) - - . 0 4 4 ( 7 ) - . 0 3 1 (1 9 ) - . 1 0 6 ( 4 1 ) - . 1 0 6 ( 3 8 ) - . 1 0 4 ( 5 8 ) - . 0 9 9 ( 5 7 ) HMPF + t . 0 5 1 t t t t - - . 1 9 4 - . 2 7 8 - . 0 8 4 - . 3 7 4 - . 4 0 1 - . 2 8 2 - . 2 9 7 WM=M + . 0 0 0 1 ( 2 2 ) . 0 0 0 1 ( 9 ) . 0 0 0 3 ( 3 8 ) . 0 0 0 3 ( 3 5 ) . 0 0 0 2 ( 5 7 ) . 0 0 0 2 ( 5 6 ) - - . 0 0 2 - . 0 0 1 - . 0 0 0 0 3 ( 6 ) - . 0 0 0 2 ( 1 9 ) - . 0 0 1 ( 1 0 0 ) - . 0 0 1 ( 1 0 3 ) - . 0 0 0 4 ( 5 6 ) - . 0 0 0 4 ( 5 7 ) WKPF -f . 0 0 1 . 0 0 1 . 0 0 0 2 . 0 0 1 . 0 0 1 . 0 0 1 . 0 0 1 - - . 0 0 0 1 H F P M + 2 . 7 0 0 ( 1 8 ) . 5 2 6 . 9 0 2 t t . 0 3 3 - - . 0 6 6 - . 8 3 1 ( 2 6 ) - . 1 2 6 - . 3 1 1 - . 1 0 8 H F P F + . 0 9 3 . 8 6 8 ( 2 6 ) . 1 1 7 . 2 8 1 . 0 8 9 - 2 . 9 5 8 ( 1 8 ) - . 5 2 5 - . 9 2 9 t t - . 0 2 7 N o t e s : t I n d i c a t e s f e w e r t h a n 5 o b s e r v a t i o n s N u m b e r s o f h o u s e h o l d s a r e c o n t a i n e d I n p a r e n t h e s e s F o r C o d e s a n d a d d i t i o n a l n o t e s , s e e T a b l e 6 . 1 a n d t e x t . - 184 -e l a s t i c i t i e s are presented for each of the two cases which have both hours and weeks rationed and for each of the two cases which have hours rationed in order to allow the p o s s i b i l i t y that responses of indiv iduals d i f f e r according to whether they are rationed to work fewer or more hours than they desire in a working week. When both hours and weeks of the male are rat ioned, the only e l a s t i c i t i e s that can be calculated are for female hours with respect to the male and female wage ra tes , HFPM and HFPF respect ive ly . When only male hours are ra t ioned, we are also able to compute e l a s t i c i t i e s of weeks worked by the male with respect to these two wage ra tes , WMPM and WMPF. On the other hand, when only male weeks are rationed we calcu late e l a s t i c i t i e s of male hours worked in a working week with respect to the two wage ra tes , HMPM and HMPF, as well as the two female hours e l a s t i c i t i e s . F i n a l l y , when there are no ra t ions , a l l s ix of these e l a s t i c i t i e s can be ca lcu la ted . In th is case we can also ca lcu late e l a s t i c i t i e s of male annual hours worked with respect to the two wage ra tes , where annual hours are the product of weeks worked and hours worked per week. However since th is e l a s t i c i t y is simply the sum of the e l a s t i c i t i e s of hours worked in a working week and of weeks worked in a year, and since the weeks worked e l a s t i c i t i e s are very small in magnitude, the annual hours e l a s t i c i t i e s do not d i f f e r s i g n i f i c a n t l y from the hours worked e l a s t i c i t i e s and are therefore not p resented . 1 0 We begin our examination of the e l a s t i c i t i e s in Table 6.5 by comparing the SS-corrected and SS-uncorrected e l a s t i c i t i e s for each household type in order to determine the ef fect of omitt ing th is cor rec t ion . While the inc lusion of the SS correct ion causes no sign 1 0 See Appendix 4 for de ta i l s of a l l these e l a s t i c i t i e s . - 185 -changes in any of the estimated e l a s t i c i t i e s for the two groups of households with education, some changes do occur for the other groups of households. In par t icu lar for households with neither chi ldren nor education, in a l l rat ioning categories HFPF changes from being pos i t ive or predominantly pos i t ive to being negative or predominantly negative, while HFPM changes in the opposite d i r ec t i on . In the group of households with chi ldren but no education, sign changes occur for the e l a s t i c i t i e s of male weeks worked and male hours worked in a working week, although HMPM only changes from predominantly pos i t ive to predominantly negative for the non-rationed households in th is group. In addit ion to these sign changes, the inc lusion of the SS correct ion also causes a l l female hours of work e l a s t i c i t i e s to increase in absolute value for a l l household types. For households with chi ldren and no education, changes in s ize also occur for the male weeks worked e l a s t i c i t i e s in Case 4 and the male hours worked in a working week e l a s t i c i t i e s , pa r t i cu la r l y in the weeks rationed case. Confining our attention to the SS-corrected e l a s t i c i t i e s in Table 6.5 we see that in a l l the rat ioning cases and for a l l households except those with chi ldren and no education, the response of female hours worked is pos i t i ve with respect to her own wage rate but negative with respect to the wage of her spouse. S im i la r l y for these same three household groups the e l a s t i c i t y of male hours worked in a working week with respect to the female wage ra te , defined in the weeks rationed and non-rationed cases, is negative or predominantly negative and with respect to the male wage rate is pos i t ive or predominantly pos i t ive except for households with neither chi ldren nor education where HMPM consists of an even number of posi t ive and negative values. For households with chi ldren but no education the signs of these e l a s t i c i t i e s are reversed, with own-wage e l a s t i c i t i e s - 186 -negative and cross-wage e l a s t i c i t i e s pos i t i ve . F i n a l l y , male weeks worked e l a s t i c i t i e s tend to be pos i t ive with respect to the female wage rate and negative with respect to male wage ra te , although for households with neither chi ldren nor education, there is again a mixture of pos i t ive and negative WMPM values. Examining the re la t i ve sizes of these e l a s t i c i t i e s we see that for the three groups of households in which these e l a s t i c i t i e s have the same s ign , the absolute values of the female hours e l a s t i c i t i e s are largest for households with both chi ldren and education and smallest for households with neither chi ldren nor education. On the other hand the e l a s t i c i t i e s of male hours worked in a working week with respect to the female wage rate tend to be largest in absolute value for households with education but no chi ldren and smaller in absolute value for the other two groups of households, while HMPM appears to be of s imi la r magnitudes for a l l three household groups. For the remaining households which have chi ldren but no education, the female hours e l a s t i c i t i e s are general ly larger in absolute va lue, and the male hours in a working week e l a s t i c i t i e s smaller in absolute value, than the corresponding e l a s t i c i t i e s for the other three groups. F i n a l l y , the male weeks worked e l a s t i c i t i e s appear to be of s imi la r magnitudes for a l l four groups of households. Next we compare the values of the various e l a s t i c i t i e s in the d i f ferent rat ioning cases. Here we see that the male weeks worked e l a s t i c i t i e s WMPM and WMPF are larger in absolute value in the two hours rationed cases than in the non-rationed case. While HMPF is s imi lar in the weeks rationed and non-rationed cases, no conclusion can be reached in th is regard with HMPM as i t involves a mixture of pos i t ive and negative values. The re la t ionsh ip between the female hours e l a s t i c i t i e s under - 187 -d i f fe rent rat ion ing condit ions appears to vary considerably for the d i f ferent types of households. For households with chi ldren and education, HFPM and HFPF are largest in absolute value for the hours rationed households in Case 4 and for non-rationed households, and are smallest fo r households in Case 1 where hours and weeks are both rat ioned. However for households with chi ldren but no education these e l a s t i c i t i e s are larger in the hours rationed and hours and weeks rationed cases than when ei ther weeks are rationed or there are no ra t ions . For households with no ch i l d ren , the female hours e l a s t i c i t i e s are larger in absolute value for non-rationed households, although for those households which also have no education, these e l a s t i c i t i e s have s imi lar magnitudes under a l l types of ra t ion ing . It is in terest ing to note that despite any di f ferences in magnitudes, the signs of a l l these e l a s t i c i t i e s remain the same regardless of the par t icu lar type of ra t ion ing . We conclude our analysis of the e l a s t i c i t i e s in Table 6.5 by examining the re la t i ve sizes of the d i f ferent e l a s t i c i t i e s in each rat ion ing category. In the hours and weeks rationed case, HFPM exceeds HFPF in absolute value except for households with chi ldren and no education where the two values are s im i la r in magnitude. When only hours are rationed the absolute value of the female cross-wage e l a s t i c i t y again exceeds the absolute value of the female own-wage e l a s t i c i t y except in Case 4 for households with chi ldren where the two e l a s t i c i t i e s have s im i la r absolute values. The two male weeks worked e l a s t i c i t i e s have s imi lar absolute values in Case 4, but in Case 6, |WMPF| > |WMPM|. However in both these hours rationed cases the absolute values of the female hours e l a s t i c i t i e s exceed those for the male weeks worked e l a s t i c i t i e s . When - 188 -only weeks are rationed, j HFPM j > | HFPF | for all household groups, although the relationship between absolute values of HMPM and HMPF differs between households with children and no education and those with no children and no education. However, apart from households with children and no education, these male hours in a working week elasticities tend to exceed the corresponding female hours ela s t i c i t i e s in absolute value. Finally for non-rationed households, the male weeks worked e l a s t i c i t i e s with respect to both the male and female wage rates are similar in absolute value, while JHFPMJ exceeds |HFPF| for households with no children but is smaller for the two groups with children. In addition, |HMPM j < |HMPF j except for households with children and education. Thus, summarizing the relationship between the absolute values of the el a s t i c i t i e s , we see that for households with children, the female hours el a s t i c i t i e s tend to exceed the male hours worked in a working week el a s t i c i t i e s , which in turn exceed the male weeks worked e l a s t i c i t i e s . In general, for households without children, the relative sizes of the female hours worked and male hours worked in a working week elasticities are reversed. To complete this section, in Table 6.6 we present values of the various cross-rationing effects that can be calculated for this model. Thus when male hours worked in a working week are rationed we have elast i c i t i e s of female hours with respect to the male hours ration, HFHM, and of male weeks worked with respect to the male hours ration, WMHM. When only weeks are rationed, elasticities with respect to the level of the weeks worked ration can be calculated for both female hours (HFWM) and male hours (HMWM). While elasticities of female hours with respect to the male hours ration and the male weeks ration can be calculated in the case - 189 -TABLE 6.6: MEDIAN CROSS RATIONING ELASTICITIES FOR THE TWO-RATION MODEL C.E C,NE NC,E NC,NE F l A Q T 1 C 1 T Y L L A o 1 IT NO SS SS NO SS SS NO SS SS NO SS SS "HU) HFHM + t .556 1.361 .022 t - -.027 -.558 -.058 -.140 t -.073 WMHM + .131 .095 t .131 .127 .129 .129 - t -.021 -.183 t t IH(6) HFHM + t t t .020 - t t -.076 -.186 -.063 WMHM + t t .136 .133 .109 .109 - t t T t K*(2) HFWM + t .656 1.434 .026 - t t -.097 -.229 -.083 HMWM + t t .526 .372 t t .215 .228 - t t t t t Notes: t Indicates fewer than 5 observations For Codes and a d d i t i o n a l notes, see Table 6.1 and t e x t . - 190 -w h e r e b o t h h o u r s a n d w e e k s a r e r a t i o n e d , f o r t h i s m o d e l t h e s e e l a s t i c i t i e s a r e n u m e r i c a l l y e q u a l t o H F P M , t h e e l a s t i c i t y o f f e m a l e h o u r s w i t h r e s p e c t t o t h e m a l e w a g e r a t e . 1 1 F r o m an e x a m i n a t i o n o f t h e v a l u e s o f t h e c r o s s - r a t i o n i n g e l a s t i c i t i e s p r e s e n t e d i n T a b l e 6.6, i t i s e v i d e n t t h a t t h e e f f e c t s o f c h a n g e s i n t h e m a l e h o u r s o r w e e k s r a t i o n s o n f e m a l e h o u r s w o r k e d a r e l a r g e r i n a b s o l u t e v a l u e i n t h e S S - c o r r e c t e d c a s e s . I n a d d i t i o n , t h e i n c l u s i o n o f t h e SS c o r r e c t i o n t e n d s t o c a u s e c h a n g e s i n t h e m a g n i t u d e s o f WMHM a n d HMWM f o r t h o s e h o u s e h o l d s w i t h c h i l d r e n . C o n c e n t r a t i n g o n t h e S S - c o r r e c t e d v a l u e s , we s e e t h a t e x c e p t f o r t h e g r o u p o f h o u s e h o l d s w i t h c h i l d r e n b u t n o e d u c a t i o n , i n c r e a s e s i n t h e l e v e l s o f t h e m a l e r a t i o n s c a u s e f e m a l e h o u r s w o r k e d t o d e c r e a s e , w h i l e a n i n c r e a s e i n t h e m a l e h o u r s r a t i o n c a u s e s m a l e w e e k s w o r k e d t o i n c r e a s e . F o r a l l h o u s e h o l d s , an i n c r e a s e i n t h e l e v e l o f t h e m a l e w e e k s w o r k e d r a t i o n c a u s e s m a l e h o u r s w o r k e d i n a w o r k i n g w e e k t o i n c r e a s e . I n g e n e r a l t h e r e s p o n s e o f f e m a l e h o u r s t o a c h a n g e i n e i t h e r t y p e o f m a l e r a t i o n i s l a r g e r i n a b s o l u t e v a l u e f o r t h e t w o g r o u p s o f h o u s e h o l d s w h i c h h a v e c h i l d r e n . O f t h e t w o g r o u p s w i t h o u t c h i l d r e n , t h e r e s p o n s e t e n d s t o b e l a r g e r i n a b s o l u t e v a l u e f o r t h o s e h o u s e h o l d s w h i c h a l s o h a v e e d u c a t i o n . H o w e v e r f o r C a s e 4 w h e r e WMHM i s c a l c u l a t e d f o r a l l f o u r h o u s e h o l d g r o u p s , we s e e t h a t t h e r e s p o n s e o f m a l e w e e k s w o r k e d t o a c h a n g e i n t h e m a l e h o u r s r a t i o n i s s i m i l a r i n m a g n i t u d e f o r t h e t w o g r o u p s w i t h n o c h i l d r e n a n d e x c e e d s , i n a b s o l u t e v a l u e , t h e m a g n i t u d e o f t h i s e l a s t i c i t y f o r h o u s e h o l d s w i t h c h i l d r e n a n d e d u c a t i o n . A s u r p r i s i n g a s p e c t o f t h e r e s u l t s i n T a b l e 6.6 c o n c e r n s t h e r e l a t i o n s h i p b e t w e e n t h e e l a s t i c i t i e s i n C a s e 4, w h e r e r a t i o n e d h o u r s a r e l e s s t h a n d e s i r e d h o u r s , a n d C a s e 6, w h e r e r a t i o n e d h o u r s e x c e e d t h e 1 1 F o r d e t a i l s , s e e A p p e n d i x 4. - 191 -d e s i r e d amount. From the t a b l e we see tha t the responses of female hours and male weeks each have the same s ign and s i m i l a r magnitudes in the two cases even though an inc rease in the r a t i o n l e v e l imp l ies an eas ing of the r a t i o n in Case 4 , wh i le in Case 6 i t imp l i es a t i g h t e n i n g of the r a t i o n c o n s t r a i n t . F i n a l l y , wh i le comparisons are l i m i t e d to the two groups with no c h i l d r e n due to an i n s u f f i c i e n t number of observa t ions in the other g roups , i t appears that f o r households with e d u c a t i o n , the abso lu te value of the response of female hours worked i s l a r g e r to a change in the male weeks r a t i o n than to a change in the male hours r a t i o n . For households wi th no educa t i on , the magnitude of the response is s i m i l a r f o r changes in e i t h e r type of male r a t i o n . 6 . 3 . 4 Comparison wi th R e s u l t s of the S i n g l e - R a t i o n Model At t h i s po in t i t i s use fu l to compare the est imated va lues of the parameters , marginal budget shares and labour supply e l a s t i c i t i e s obta ined wi th the hours and weeks model to those presented in Chapter 4 f o r the s i n g l e - r a t i o n m o d e l . 1 2 The e s s e n t i a l d i f f e r e n c e s between the models are t w o f o l d . F i r s t , in the hours and weeks model , male l e i s u r e , Lm, i s d isaggregated in to l e i s u r e dur ing working weeks, and l e i s u r e dur ing non-working weeks, l 2 . Consequent ly the Model I parameters Ym (which, i f p o s i t i v e , may be i n t e rp re ted as a subs i s tence quan t i t y of male l e i s u r e ) and 3 m „ the mbs fo r male l e i s u r e , are rep laced in Model II by YI and y2 and by B i and & 2 , r e s p e c t i v e l y . The second d i f f e r e n c e i s that the number of r a t i o n i n g ca tego r i es i s expanded from three in Model I to s i x in Model I I . Thus the underemployed case in Model I, where des i red hours of work exceed observed hours of work, i s s p l i t i n to Cases 1 and 4 in Model II 1 2 For n o t a t i o n a l convenience we r e f e r to the s i n g l e - r a t i o n model as 'Model I' and the hours and weeks model as 'Model I I ' . - 192 -according to whether weeks worked are rationed or non-rationed, respec t ive ly . S im i la r l y the overemployed case in Model I is s p l i t into Cases 3 and 6 while non-rationed households in Model I are separated into Cases 2 and 5 in Model I I , according to whether or not weeks worked are constrained. In Table 6.7 we present the SS-corrected parameter estimates for the two models for each of the four demographic g roups . 1 3 From th is tab le , i t is c lear that one of the main di f ferences between the two models concerns the s ign i f icance of the parameters. S p e c i f i c a l l y , while Y c is s i gn i f i can t only for households with education but no chi ldren in Model I, i t is s ign i f i can t for a l l four groups of households in Model I I . S im i la r l y Bi and B2 , the two components of Bm which are estimated in Model I I , are both s ign i f i can t for a l l four groups while in Model I, Bm is only s ign i f i can t for households with chi ldren and education. In terms of magnitudes of the parameters, another di f ference between the two models is that the sum of Y i and Y2 in Model II is in a l l cases considerably smaller than Ym» the corresponding parameter in Model I. However, apart from Y f which increases in Model II for households with chi ldren and no education, and Y c which decreases in Model II for households with education but no ch i ld ren , other changes in the Y i parameters between the two models tend not to be s i g n i f i c a n t . Possib ly the most important di f ferences between the parameter estimates for the two models concern the marginal budget shares for male le isure in the non-rationed case. F i r s t , as noted above, in Model II g! and B2 are both s ign i f i can t for a l l households, while in Model I, Bm is only s ign i f i can t for one household group. Second, in Model II the sum of 1 3 Model I parameters are taken from Table 4.1 while Model II parameters are from Table 6 .2 . - 193 -TABLE 6.7: COMPARISON OF PARAMETERS IN THE SINGLE-RATION AND TWO-RATION MODELS PARAMETER c, E c, NE NC,E NC, NE MODEL 1 MODEL 11 MODEL I MODEL 11 MODEL 1 MODEL 11 MODEL I MODEL .1 1 Y l .256 ( ) .256 ( ) .0858 ( ) .0858 ( ) Y 2 .234* (.040) -.205 (.163) .268* (.029) .250* (.029) Ym .707* (.020) .490 .722* (.027) .051 .650* (.006) .354 .720* (.013) .336 Yf .715* (.035) .640* (.038) .822* (.021) .926* (.017) .704* (.016) .686* (.012) .725* (.017) .723* (.012) Yc .494 (.477) .364* (.099) -.321 (.254) .123* (.053) 1.530* (.305) .791* (.134) -2.213 (1.548) .720* (.030) 8 l .486* (.080) .377* (.084) .827* (.052) .862* (.047) 8 2 -.278* (.103) .298* (.133) -.341* (.067) -.309* (.071) 8m -.296* (.146) .208 -.249 (.137) .675 .049 (.040) .486 -.081 (.048) .553 8 f .130 (.082) .214* (.048) -.103 (.069) .353 (.309) .135* (.031) .055* (.016) .010 (.013) .016 (.016) 8 C 1.166* (.188) .578* (.058) 1.352* (.185) -.027 (.291) .816* (.048) .458* (.039) 1.072* (.049) .431* (.031) Notes: A l l values r e f e r +0 the case where the SS c o r r e c t i o n i s included. For Model I I , Y m i s obtained as Y l + Y2» w h i l e g m i s obtained as 3i+ 8 2 « For codes and a d d i t i o n a l notes, see Table 6.1 and t e x t . - 194 -Si and B 2 i s pos i t ive for a l l four groups of households, while in Model I, 3 m is negative in the only case where i t is s i g n i f i c a n t l y d i f ferent from zero. Thus in contrast to Model I and many other studies in the l i t e r a t u r e , in Model II we no longer have the unappealing resul t that male le isure is an in fe r io r good. 1 1* Although B 2 1 S negative for three household groups, suggesting that le isure during non-working weeks may be i n f e r i o r , the large pos i t ive values for Bi outweigh these ef fects so that overal l the mbs for male le isure is pos i t i ve . Further, apart from households with chi ldren and education, the magnitude of the sum of Si and B 2 in Model II is considerably larger than the Bm values in Model I, suggesting that male le isure is much more important at the margin than we would conclude from our Model I r e s u l t s . Of course since the mbs sum to un i ty , t h i s increase in (Si + S 2 ) r e l a t i ve to S m must be of fset by decreases in e i ther or both of Bf and B c . Here we see that the major i ty of the adjustment involves a decrease in the mbs for goods consumption, S c , pa r t i cu la r l y for households with chi ldren where the mbs for female le isure is larger in Model II than in Model I. While i t would also be of in terest to compare the marginal budget shares in the various rationed cases for the two models, such a comparison here is of l imi ted use since in Model I the rationed mbs are the same for both underemployed and overemployed cases, and are only avai lable for female le isure and goods consumption. However, as can be seen from Table 6.4, in Model II we are able to ca lcu la te separate mbs for 4 of the 5 rationed s i tua t ions , and in addit ion to Bf and S c , we can ca lcu la te both Bi and B 2 when hours are rationed and Si when only weeks are See Section 4.5.1 for a discussion of studies in which le isure has been found to be an in fe r io r good. - 195 -ra t ioned. We now turn to a consideration of the implied labour supply e l a s t i c i t i e s in the two models. Due to the fact that households in each of the Model I rat ioning cases are separated in Model II according to whether or not male weeks are ra t ioned, each Model I e l a s t i c i t y has two counterparts in Model I I . For example, each of the female own-wage and cross-wage e l a s t i c i t i e s , HFPF and HFPM, for the underemployed case in Model I, is calculated separately in Model II for Case 1 where weeks are rationed and Case 4 where weeks are not rat ioned. S im i la r l y HFPF and HFPM for the overemployed case in Model I are each separated into two values in Model II depending on whether the household also has weeks rationed (Case 3) or non-rationed (Case 6) . Of course in Model II we are also able to ca lcu late male weeks worked e l a s t i c i t i e s for households which have hours rationed but weeks non-rat ioned; these e l a s t i c i t i e s have no counterpart in Model I. While the female hours e l a s t i c i t i e s for the non-rationed case in Model I are comparable to the values in the weeks rationed (Case 2) and non-rationed (Case 5) categories in Model I I , the fact that we can also compare the male labour supply e l a s t i c i t i e s in the two models is not obvious. S p e c i f i c a l l y , while HMPM and HMPF in Model I refer to e l a s t i c i t i e s of annual hours of male labour with respect to the male and female wage ra tes , in Model II HMPM and HMPF are e l a s t i c i t i e s of male hours worked in a working week while WMPM and WMPF are e l a s t i c i t i e s of male weeks worked in a y e a r . 1 5 However as outl ined in the previous sec t ion , for each i nd i v i dua l , annual hours e l a s t i c i t i e s in Model II are calculated as the sum of the weeks worked and the hours worked per week e l a s t i c i t i e s . Further, since the indiv idual weeks worked e l a s t i c i t i e s are 1 5 To avoid confusion with notat ion, we henceforth denote the annual hours e l a s t i c i t i e s from Model I as AHMPM and AHMPF. - 196 -very small in magnitude, di f ferences between median values of hours worked per week e l a s t i c i t i e s and annual hours e l a s t i c i t i e s in Model II are neg l i g i b l e . Hence AHMPM and AHMPF values in the non-rationed case in Model I can be compared to values of HMPM and HMPF in the weeks rationed and non-rationed cases in Model I I . An addit ional d i f f i c u l t y in comparing the e l a s t i c i t y values for Model II in Table 6.5 and those for Model I in Table 4.2 is that for many household groups and rat ioning types in Model I I , median values are not presented due to an insu f f i c ien t number of observations. Despite these l im i t a t i ons , some general observations can s t i l l be made. In pa r t i cu l a r , we observe that in general the comparable SS-corrected e l a s t i c i t i e s in the two models have the same s igns. Further, the change in signs of the female hours e l a s t i c i t i e s which occur in Model I for households with no chi ldren and no education when the SS correct ion is included, is repeated in Model I I . However the corresponding change, for households with chi ldren and education, only occurs for Model I. In terms of magnitudes, e l a s t i c i t i e s for the two household groups with chi ldren are general ly two to four times larger in absolute value in Model I I , although for households with no chi ldren and education, the magnitudes of the female hours e l a s t i c i t i e s are general ly s imi la r in the two models. For households with neither chi ldren nor education, the own-wage e l a s t i c i t i e s in Model II are smaller in absolute value than those in Model I, while the cross-wage e l a s t i c i t i e s are approximately four times larger in absolute value in Model I I . In order to summarize the essent ia l di f ferences between the SS-corrected e l a s t i c i t y estimates for the two models, in Tables 6.8 and 6.9 we present the median labour supply e l a s t i c i t i e s obtained in each model for the four household groups combined. These estimates are obtained - 197 -by f i r s t ca lcu la t ing the indiv idual e l a s t i c i t i e s for each household group separately, and then determining the median value of each e l a s t i c i t y across the four groups for each of the rat ioning cases. One of the f i r s t things we notice from Table 6.8 is the amount of extra information avai lab le in Model I I ; we obtain two values of each of the Model I e l a s t i c i t i e s , depending on whether or not weeks are rat ioned, ca lcu la te weeks worked e l a s t i c i t i e s when weeks are not constrained, and separate annual male hours worked e l a s t i c i t i e s into hours worked per working week and weeks worked per year e l a s t i c i t i e s . Comparing the e l a s t i c i t y estimates for the d i f ferent models we see that in general for Model I and both cases of Model I I , HFPF is predominantly pos i t ive while HFPM is predominantly negative. However di f ferences in sign occur for the two annual male hours e l a s t i c i t i e s , AHMPF and AHMPM. S p e c i f i c a l l y , AHMPF is predominantly pos i t ive in Model I but predominantly negative in the two Model II cases, while AHMPM, which is predominantly negative in Model I, is predominantly negative in Model II only in the weeks rationed case. Comparing the re la t i ve s izes of the e l a s t i c i t i e s for the two models, we see from Table 6.8 that apart from the negative values for AHMPM, the e l a s t i c i t i e s in the weeks free case of Model II are larger in absolute value than the corresponding e l a s t i c i t i e s in Model I. In 8 of the 12 comparable cases, the weeks rationed e l a s t i c i t i e s are also greater in absolute value than those in Model I. Thus in general the re la t ionsh ip between the three sets of e l a s t i c i t i e s appears to be that they are larger in absolute value in the weeks free case of Model II than in the weeks rationed case of the same model, while the Model I values are usual ly the smallest in absolute value. - 198 -TABLE 6.8: COMPARISON OF MEDIAN ELASTICITIES IN THE SINGLE-RATION AND TWO-RATION MODELS. MODEL 1 1 RATIONING CASE Fl A S T i n i T Y MTinFl 1 WEEKS RATION WEEKS FREE H° < H* HFPF + .106 (69) .040 (16) .117 (48) Model 1 - underemployed - -.369 (29) -.542 (18) -.994 (16) Model I I : HFPM + .272 (19) 1.004 (9) .949 (12) - weeks r a t i o n e d : Case 1 - -.053 (79) -.113 (25) -.141 (52) - weeks f r e e : Case 4 WM°F + .038 - t WM=M + .025 (21) - -.020 (43) H° > H* HFPF + .088 (27) t .116 Model 1 - overemployed - -.056 (10) t t Model I I : HFPM + t t t - weeks r a t i o n e d : Case 3 - -.162 t -.175 - weeks f r e e : Case 6 WMPF + .041 WM=M + .007 (6) - -.019 (26) H° = H* HFPF + .097 (251) .084 (21) .188 (274) Model 1 - non-rationed - -.168 (106) -.907 (13) -.961 (49) Model I I : HFPM + .224 (41) .960 (1 1) .957 (49) - weeks r a t i o n e d : Case 2 - -.167 (316) -.096 (23) -.218 (274) - weeks f r e e : Case 5 WNPF + .001 WMPM + .0002 (100) - -.001 (223) HMPF + .116 (11) .054 (32) - -.292 (23) -.340 (291) HNPM + .120 (14) .261 (209) - -.161 (20) -.091 (1 14) AHMPF + .040 (212) .116 (1 1) .054 (32) - -.026 (145) -.292 (23) -.340 (291) AHM=M + .011 (36) .120 (14) .261 (209) - -.159 (321) -.161 (20) -.091 (1 14) Notes: t i n d i c a t e s fewer than 5 observations A l I values are with the SS- c o r r e c t i o n included Numbers of households are contained in parentheses For Codes and a d d i t i o n a l notes, see t e x t . - 199 -From an examination of the values of HFPF and HFPM in the three rat ioning cases, further di f ferences between the e l a s t i c i t i e s in Model I and the two cases in Model II are evident. In Model I, the absolute value of HFPF tends to be largest for underemployed households (H° < H*) and smallest for overemployed households (H° > H*). However in the weeks rationed case of Model II we see that HFPF is larger in absolute value for non-rationed households than for underemployed households. A s imi lar re la t ionsh ip is found for the weeks free case of Model II for the pos i t i ve values of th is e l a s t i c i t y , although for the negative values, HFPF is larger in absolute value for underemployed households. Examining the female cross-wage e l a s t i c i t y , HFPM, we see that the posi t ive values of th is e l a s t i c i t y in Model I and the Model II weeks rationed case are larger for underemployed households than for non-rationed households, although in the Model II weeks free case th is re la t ionsh ip is reversed. However for both Model I and Model II with weeks f ree , the negative values of HFPM are largest in absolute value in the non-rationed case and smallest in the underemployed case, while in the weeks rationed case of Model I I , they are larger in absolute value for underemployed households. To conclude th is sec t ion , in Table 6.9 we present the medians for a l l households of the cross-rat ion ing e l a s t i c i t i e s that are defined in the various models. As can be seen from th is tab le , Model II provides much more information concerning the ef fects on unrationed quant i t ies of changes in the rat ion l eve l s . In pa r t i cu la r , while in Model I we can determine the ef fect on female hours of work of a \% change in the rat ion on annual hours worked by the male, HFAHM, in Model II we can ca lcu la te the separate ef fects af an increase in the weeks worked ra t i on , HFWM, and in the hours worked in a working week ra t i on , HFHM. In Model II when weeks - 200 -TABLE 6.9: COMPARISON OF MEDIAN CROSS-RAT I ON ING ELASTICITIES IN THE SINGLE-RATION AND TWO-RATION MODELS. RATIONING CASE ELASTICITY MODEL 1 MODEL 1 1 WEEKS RATION WEEKS FREE H° < H* Model 1 - underemployed Model I I : - weeks r a t i o n e d : Case 1 - weeks f r e e : Case 4 HFHM + HFWM + WMHM + HFAHM + .272 (19) -.053 (79) 1.004 (9) -.113 (25) 1.004 (9) -.113 (25) 1.361 (12) -.096 (52) .126 (53) -.185 (11) H° > H* Model 1 - overemployed Model I I : - weeks r a t i o n e d : Case 3 - weeks f r e e : Case 6 HFHM + HFWM + WMHM + HFAHM + t -.162 t t t t t -.119 .1 14 t H° =H* Model 1 - non-rationed Model I I : - weeks r a t i o n e d : Case 2 - weeks f r e e : Case 5 HFWM + HMWM + 1.468 (11) -.109 (23) .350 (26) -.272 (8) Notes: t i n d i c a t e s fewer than 5 observations A l I values are with the SS - c o r r e c t i o n included Numbers of households are contained in parentheses In Model I, HFAHM = HFPM In Model I I, hours and weeks r a t i o n e d cases, HFHM = HFWM = HFPM For Codes and a d d i t i o n a l notes, see t e x t . - 201 -are not ra t ioned, we can also ca lcu la te the e l a s t i c i t y of weeks worked by the male with respect to the level of the rat ion on his hours worked in a working week, WMHM. When only weeks are rat ioned, Model II also allows us to determine HFWM and HMWM, the e l a s t i c i t i e s of female hours and male hours in a working week with respect to the level of th is ra t i on . 6.4 Results using only Non-Rationed Households and Comparison with  Estimates from the Complete Model In Tables 6.10 and 6.11 we present the parameter estimates and implied labour supply e l a s t i c i t i e s which are obtained when we estimate the hours and weeks model using only those households in which the hours worked by the head in a working week, and the weeks worked by the head in a year, are both unrationed. By comparing these parameters and e l a s t i c i t i e s to those presented in the previous section for the complete hours and weeks model, we are able to determine the ef fects of excluding a l l rationed indiv iduals from the sample, and hence the importance of co r rec t l y modelling labour market ra t ion ing . When estimating th is simpler non-rationed model, i t is not necessary to impose a f ixed value of Y i in order to obtain convergence of the optimizat ion procedure. However i f we estimate th is model with Y i f r ee , i t w i l l not be clear to what extent di f ferences between these estimates and those for the f u l l model are due to the d i f ferent treatment of Y i and to what extent they are due to including a l l observations and cor rec t ly modelling the behaviour of those ind iv iduals who are rat ioned. Since the estimates obtained in the non-rationed case are not of interest in themselves — as they are inconsistent since no correct ion is made for the fact that only non-rationed households are used ~ in order to avoid - 202 -confusing these ef fects we obtain estimates in th is non-rationed case with Y i held f ixed at the same values as for the f u l l mode l . 1 6 For households both with and without ch i l d ren , Likel ihood Ratio tests are again used to determine whether the parameter estimates d i f f e r i f the head or spouse completed at least one year of co l lege. For households with chi ldren the LR s t a t i s t i c is calculated as 44.55 in the SS-corrected case and 41.56 in the SS-uncorrected case, while for households without chi ldren the corresponding s t a t i s t i c s are 92.17 and 87.19. Since the c r i t i c a l chi-square value with 12 degrees of freedom at the 1% s ign i f icance level is 26.22, we conclude that the parameters are s i g n i f i c a n t l y d i f ferent according to whether or not the household is "educated". Thus, as for the f u l l model, we present separate estimates for a l l four household groups, again both with and without including the SS correct ion for the presence of working wives. Comparing the parameter estimates obtained using only non-rationed households (NR-only) in Table 6.10 with those for the f u l l model contained in Table 6.2, we see that in the SS-corrected case for households with both chi ldren and education, Yc a n d 3f are s ign i f i can t only in the f u l l model, while for households with chi ldren but no education, 3 2 i s only s ign i f i can t in the f u l l model. However, for th is same group of households, both 8f and 8 C are s ign i f i can t in the SS-corrected case only in the model estimated using non-rationed households. For households with no ch i l d ren , the same parameters are s i g n i f i c a n t l y d i f ferent from zero in both models. While some of the parameters have d i f ferent signs in 1 6 The main di f ferences in the parameter estimates when Y I is not f ixed are that Y i is s ign i f i can t and larger in magnitude (0.33 for households with neither chi ldren nor education and between 0.53 and 0.59 for other households), Y 2 ^ s smaller in magnitude, and fewer 8i and 8 2 values are s i g n i f i c a n t . - 203 -TABLE 6.10: PARAMETER ESTIMATES WHEN ONLY NON-RATIONED OBSERVATIONS ARE USED PARAMETER C.E C,NE NC,E NC,NE NO SS SS NO SS SS NO SS SS NO SS SS Yl .256 ( ) .256 ( ) .256 ( ) .256 ( ) .0858 ( ) .0858 ( ) .0858 ( ) .0858 ( ) Y2 .287* (.038) .268* (.042) .138 (.073) .136 (.075) .280* (.027) .281* (.026) .492* (.021) .490* (.022) Yf .772* (.021) .738* (.058) .820* (.018) .826* (.022) .697* (.011) .674* (.015) .746* (.016) .733* (.020) Yc .273 (.439) .478 (.411) .298* (.124) .288* (.123) .849* (.141) .868* (.137) -1.125* (.262) -1.157* (.272) Pi .720* (.164) .695* (.179) .675* (.096) .678* (.098) .898* (.057) .871* (.056) .701* (.060) .687* (.059) P2 -.535* (.171) -.470* (.192) -.175 (.171) -.173 (.175) -.395* (.066) -.384* (.063) -.599* (.066) -.585* (.065) Pf -.030 (.035) .142 (.106) -.100* (.031) -.098* (.034) .031* (.013) .067* (.020) -.018 (.015) -.001 (.018) Pc .844* (.102) .632* (.158) .600* (.107) .592* (.109) .467* (.042) .446* (.040) .916* (.034) .899* (.036) For Notes and Codes, see Table 6.1. - 204 -the two tab les , pa r t i cu la r l y for the two groups of households with ch i l d ren , the only case where a parameter which is s ign i f i can t in both sets of estimates changes sign occurs for households with neither chi ldren nor education. Here Yc» which is pos i t ive in the f u l l model, becomes negative in the model estimated using only non-rationed households. While the magnitudes of several parameters d i f f e r quite not iceably in the two tab les , in general the standard errors of the estimates are larger in the model estimated using only non-rationed households. Consequently di f ferences between corresponding estimates, while large, do not appear to be s ign i f i can t as they rare ly exceed more than one standard er ror . However for households with neither chi ldren nor education, 0i appears to be s i g n i f i c a n t l y larger in the f u l l model, while Yc (which changes s ign ) , Y 2 > 02 a n d 0c a r e a ^ s i g n i f i c a n t l y smaller in absolute value in the f u l l model. Also for households with chi ldren but no education, Yf and Bi appear to change s i g n i f i c a n t l y between the f u l l model and the model estimated using only non-rationed households. A comparison of the cor re la t ion coe f f i c ien ts defined in both the NR-only and f u l l models, P I 2 , P i e and P28» reveals few d i f fe rences. However for households with chi ldren and education, P i e and p 2 8 a<"e s ign i f i can t in the SS-corrected case only in the f u l l model, while for households with chi ldren but no education, P i 8 i s only s ign i f i can t in the NR-only model. F i n a l l y , as was the case with the f u l l model, regu lar i ty condit ions are never sa t i s f i ed in the NR-only model. Turning to a comparison of the e l a s t i c i t i e s obtained in the NR-only model, in Table 6.11, with the non-rationed e l a s t i c i t i e s for the f u l l model, contained in Table 6 .5 , we see that the female hours e l a s t i c i t i e s HFPM and HFPF d i f f e r in the two models except for households with TABLE 6.11: MEDIAN ELASTICITIES WHEN ONLY NON-RATIONED OBSERVATIONS ARE USED ELASTICITY c ,E C,NE NC E NC,NE NO SS SS NO SS SS NO SS SS NO SS SS HMPM + .232 .318 -.043 (34) (10) .097 (20) -.117 (8) .091 -.116 (20) (8) .308 (89) -.152 (49) .332 (93) -.129 (45) 1.076 t 1.091 t HM°F + -.194 -.247 t -.105 t -.094 t -.427 t -.461 t -.222 t -.240 WMPM + -.001 .0002 -.001 (10) (34) .0002 (7) -.0001 (21) .0002 -.0001 (7) (21) .0004 (46) -.001 (92) .0004 (42) -.001 (96) -.005 -.005 WM=F + .001 .001 .0001 .0001 .001 .001 .001 .001 HFPM + .128 2.827 -.789 (16) (28) .558 .584 t -.158 t -.355 .062 .002 HFPF + -.148 .717 -3.206 (28) (16) -.538 -.566 .144 t .313 t -.129 -.005 Notes: t Indicates fewer than 5 observations Numbers of households are contained in parentheses For Codes and a d d i t i o n a l notes, see Table 6.1 and t e x t . - 206 -education but no ch i ld ren . For households with neither chi ldren nor education, the SS-corrected female hours e l a s t i c i t i e s are smaller in absolute value in the NR-only model where they also have d i f ferent signs to the i r counterparts in the f u l l model. However the SS-uncorrected female hours e l a s t i c i t i e s for these households are larger in absolute value in the NR-only model. For households with chi ldren but no education, the SS-corrected values of HFPM and HFPF are also smaller in absolute value in the NR-only model, while for households with chi ldren and education, the SS-uncorrected female hours e l a s t i c i t i e s in the NR-only model are larger in magnitude and have a d i f ferent sign to the corresponding e l a s t i c i t i e s in the f u l l model. In comparison to the female hours e l a s t i c i t i e s , the male weeks worked e l a s t i c i t i e s appear to be reasonably s imi lar in the two models. For households with neither chi ldren nor education, WMPM is negative in the NR-only model although in the f u l l model i t consists of an even number of pos i t ive and negative values. For households with chi ldren and education, the SS-corrected value of WMPM changes from negative in the f u l l model to a mixture of pos i t ive and negative values in the NR-only model. However in the SS-corrected case, for households with chi ldren and no education, WMPM changes from predominantly pos i t ive in the f u l l model to predominantly negative in the NR-only model, while WMPF changes in the opposite d i r ec t i on . F i n a l l y , the e l a s t i c i t i e s of male hours worked in a working week, HMPF and HMPM, are general ly larger in absolute value in the NR-only model, pa r t i cu la r l y for the two household groups with no education. In the group of households with chi ldren and no education, the SS-corrected male hours e l a s t i c i t i e s change s ign , while in households with neither chi ldren - 207 -nor education, HMPM changes from an even mixture of pos i t ive and negative values in the f u l l model to predominantly pos i t ive in the NR-only model. Thus, although the estimates are inconsistent i f estimation is performed using only the subsample of non-rationed households, in general i t appears that the magnitudes do not d i f f e r great ly from those obtained when the f u l l model is estimated using a l l observations. However since some values d i f f e r , and since much more information is avai lable in the f u l l model — in terms of mbs and e l a s t i c i t i e s in the various rationed cases - - there seems l i t t l e point in estimating the hours and weeks model using only the non-rationed households. 6.5 Results Treating A l l Households as Non-Rationed and Comparison with  Estimates from the Complete Model In th is section we compare the f u l l model estimates with those obtained i f we estimate the hours and weeks model using a l l households in the sample, but ignoring the fact that some of the households have ei ther or both of these var iables rat ioned. As with the NR-only model, in order to f a c i l i t a t e comparisons, the model is estimated with Y I f ixed at the same values as in the f u l l model. LR tests are again used to determine whether the parameter estimates d i f f e r i f the head or spouse attended co l lege . For households with ch i l d ren , LR s t a t i s t i c s of 30.16 and 29.66 are obtained in the SS-corrected and SS-uncorrected cases, while for households without ch i l d ren , the corresponding s t a t i s t i c s are calculated as 273.53 and 261.29. Since for both groups these s t a t i s t i c s exceed 26.22, the c r i t i c a l chi-square value with 12 degrees of freedom at the 1% s ign i f icance l e v e l , - 208 -we again conclude that the parameters d i f f e r according to whether or not the head or spouse completed at least one year of co l lege. From a comparison of the estimated parameters in Table 6.12 with those for the f u l l model contained in Table 6.2, i t is c lear that many of the estimates are quite d i f ferent in the two models. Most of the di f ferences in s ign i f icance of the parameters occur for households with ch i l d ren . S p e c i f i c a l l y , with the SS-correct ion included, for households with chi ldren and education both Yc and Bf are s ign i f i can t only in the f u l l model. For households with chi ldren but no education, Yc is only s ign i f i can t in the f u l l model while Y2» both Bf and B c in the SS-corrected case and 8 2 in the SS-uncorrected case are only s ign i f i can t in the model in which a l l households are treated as non-rationed (A l l - as -NR) . In addi t ion, for households with education but no ch i ld ren , Bf is s i gn i f i can t in the SS-corrected case only in the f u l l model, while there are several changes in the sign of the parameters in the two models, these also occur only for the two household groups with chi ldren and in general are associated with the changes in s ign i f icance reported above. However for households with chi ldren and no education, B 2 which is s ign i f i can t in both models in the SS-corrected case, is only pos i t ive in the f u l l model. Confining our attention to parameters that are s ign i f i can t in both models, and using a di f ference exceeding twice the larger of the two standard errors as an indicat ion of s ign i f i can t d i f ferences, we see that for households with both chi ldren and education and those with neither chi ldren nor education, Y2 i s s i g n i f i c a n t l y smaller in the f u l l model than in the Al l -as-NR model. In the SS-corrected case, Yf is also s i g n i f i c a n t l y smaller in the f u l l model for households with chi ldren and - 209 -TABLE 6.12: PARAMETER ESTIMATES WHEN ALL OBSERVATIONS ARE TREATED AS NON-RATIONED c, E c, NE NC, E NC,NE PARAMFTFR r f l n r t r l L 1 Cr\ NO SS SS NO SS SS NO SS SS NO SS SS Yi .256 .256 .256 .256 .0858 .0858 .0858 .0858 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Y2 .393* .377* .412* .410* .315* .311* .401* .398* (.029) (.030) (.030) (.030) (.024) (.024) (.025) (.024) Yf .766* .737* .818* .817* .716* .703* .733* .721* (.017) (.030) (.014) (.017) (.010) (.012) (.011) (.014) Yc -.597 -.889 -.135 -.141 .941* .994* .799* .816* (.711) (1.004) (.118) (.125) (.113) (.109) (.055) (.056) P i .551* .475* .882* .874* .982* .970* 1.135* 1.113* (.182) (.187) (.124) (.128) (.061) (.060) (.066) (.066) P2 -.583* -.475* -.946* -.931* -.486* -.472* -.735* -.716* (.226) (.216) (.200) (.203) (.072) (.071) (.101) (.099) Pf -.021 .051 -.134* -.111* .013 .042* -.009 .030 (.025) (.039) (.042) (.049) (.014) (.018) (.018) (.024) Pc 1.052* .949* 1.198* 1.167* .491* .460* .610* .573* (.081) (.071) (.125) (.129) (.041) (.040) (.054) (.053) For Notes and Codes, see Table 6.1. - 210 -education, although for households with chi ldren and no education, Yf is s i g n i f i c a n t l y smaller in the Al l -as-NR model. Turning to the marginal budget shares, B]_ is s i g n i f i c a n t l y smaller in the f u l l model for a l l households except those with chi ldren and education, while 3 C is smaller in the f u l l model except for households with education but no ch i ld ren . For households with neither chi ldren nor education, B 2 is s i g n i f i c a n t l y smaller in absolute value in the f u l l model, while in the SS-corrected case for households with chi ldren but no education, 3 2 changes from s i g n i f i c a n t l y negative in the Al l -as-NR model to s i g n i f i c a n t l y posi t ive in the f u l l model. One other in terest ing di f ference between the parameters in the two models concerns the cor re la t ion coe f f i c ien ts between the error terms. When a l l households are treated as non-rat ioned, P I 2 i s s ign i f i can t for a l l household groups while P i 8 is only s ign i f i can t in the SS-corrected case for households with education but no chi ldren and p 2 8 i s never s i g n i f i c a n t . However in the f u l l model, the cor re la t ion coe f f i c ien ts P I 8 and P 2 8 are both s ign i f i can t for these households and for households with chi ldren and education in the SS-corrected case. Further, for the two groups of households with ch i l d ren , the p 1 2 values in the Al l -as-NR model are considerably smaller in absolute value than the corresponding coe f f i c ien ts in the f u l l model. F i n a l l y , one s i m i l a r i t y between the parameters in the two models is that in neither case are the regu la r i t y condit ions sa t i s f i ed at any point . ' In view of the large di f ferences between the parameters in the two tab les , we would also expect the e l a s t i c i t i e s in Table 6.13 to d i f f e r considerably from the non-rationed e l a s t i c i t i e s for the f u l l model, contained in Table 6 .5 . However, apart from changes in absolute magnitude, TABLE 6.13: MEDIAN ELASTICITIES WHEN ALL OBSERVATIONS ARE TREATED AS NON-RATIONED ELASTIC TY C,E C,NE NC.E NC NE NO SS SS NO SS SS NO SS SS NO SS SS HMPM + .575 .655 .470 .474 .261 (93) -.191 (85) .260 (93) -.197 (85) .217 (59) -.186(128) .207 (65) -.180(122) HMPF + -.146 -.160 -.137 -.137 t -.412 t -.436 t -.336 t -.357 WM=M + -.004 -.004 -.003 -.003 .001 (83) -.001 (95) .001 (83) -.001 (95) .001 (127) -.001 (60) .001(121) -.001 (66) WM=F + .001 .001 .001 .001 .001 .001 .002 .002 HFPM + .069 3.549 (7) -.209 (62) .336 .307 t -.066 t -.219 .042 -.145 HFPF + -.126 .433 (62) -6.225 (7) -.481 -.441 .054 t .178 t t -.031 .108 t Notes: t Indicates fewer than 5 observations Numbers of households are contained in parentheses For Codes and a d d i t i o n a l notes, see Table 6.1 and t e x t . - 212 -the e l a s t i c i t i e s in the two tables are quite s imi la r in sign and re la t i ve s i z e . In pa r t i cu la r , the only e l a s t i c i t i e s which have d i f ferent signs in the two models are the SS-uncorrected female hours e l a s t i c i t i e s for households with chi ldren and education, and the SS-corrected male hours e l a s t i c i t i e s and SS-uncorrected male weeks e l a s t i c i t i e s for households with chi ldren but no education. In add i t ion , for households with neither chi ldren nor education, changes also occur in the male hours and male weeks own-wage e l a s t i c i t i e s . While both these e l a s t i c i t i e s consist of an even number of pos i t ive and negative values in the f u l l model, HMPM is predominantly negative and WMPM is predominantly pos i t ive in the Al l -as-NR model. Comparing the absolute magnitudes of the e l a s t i c i t i e s in the two tab les , we see that the male hours e l a s t i c i t i e s are general ly larger in absolute value in the Al l -as-NR model. However for households with chi ldren and education, HMPF is smaller in the Al l -as-NR model, as are the the pos i t ive values of HMPM for households with education but no ch i l d ren . S im i la r l y WMPM is larger in absolute value for a l l households, and WMPF is larger in absolute value for the two groups of households without education, in the Al l -as-NR model. However for households without education, values of WMPF are s imi lar in the two tab les . The behaviour of the female hours e l a s t i c i t i e s is s l i g h t l y d i f ferent to the male hours and weeks e l a s t i c i t i e s . Both HFPM and HFPF are larger in absolute value in the f u l l model for households with chi ldren and no education and those with education but no ch i ld ren . In add i t ion , in the SS-corrected case for households with both chi ldren and education, the predominantly negative value of HFPM and the predominantly pos i t ive value of HFPF are also larger in absolute value in the f u l l model. However for these same households in - 213 -the case where the SS-correct ion is omitted, HFPF is smaller in absolute value in the f u l l model. Also in the SS-corrected case, both female e l a s t i c i t i e s are smaller in absolute value in the f u l l model for households with neither chi ldren nor education. Thus from these resul ts i t would appear that the main consequences of incor rec t l y t reat ing a l l households as non-rationed are s ign i f i can t changes in many of the parameter estimates, a general overpredict ion of male labour supply e l a s t i c i t i e s and a general underprediction of the female labour supply e l a s t i c i t i e s . However as is evident from the above comparisons, rather than fol lowing a general pat tern, the ef fects of misspec i f ica t ion vary among the d i f ferent e l a s t i c i t i e s and among the d i f fe rent household groups. Thus, as was the case with the model estimated using only non-rationed households, i t would appear to be quite important to estimate the complete model, in which each type of rat ioning is modelled appropr iately, using the ent i re sample of rationed and non-rationed households. 6.6 Comparison with Results from Other Studies While i t is of general interest to compare the resul ts we obtain with the two-ration model to those reported elsewhere, there are no other studies in which an hours and weeks model of labour supply, estimated for households, takes account of any rat ioning constraints which the household heads may face in the labour market. In fact the only study in which estimates are obtained for an hours and weeks model of labour supply is by Hanoch [1980b]. While the model estimated by Hanoch does not take account of ra t ion ing , and thus in th is respect is s imi la r to the model estimated in the previous section where a l l ind iv iduals are treated as non-rationed, - 214 -by comparing his resul ts to those obtained for our model, we can evaluate the ef fects of cor rec t ly modelling the behaviour of those households which are rat ioned. Unfortunately a number of other features of Hanoch's study severely l im i t the effect iveness of such a comparison. Apart from Hanoch's f a i l u r e to model any rat ioning which may ex is t in the labour market, other aspects of Hanoch's study which d i f f e r from our own concern the structure of the underlying model and the sample which is used. Rather than estimating labour supply and commodity demand equations which resu l t from a jo in t optimization procedure by the household members, Hanoch simply spec i f ies l inear labour supply equations for annual hours worked and annual weeks worked, and estimates them using a sample of married women. Thus he ignores the issue of jo in t a l locat ion of time by the household members and consequently cannot ca lcu late marginal budget shares, although he does compare the ef fect of an increase in income, defined as the sum of non-wage income and husband's earnings, on annual weeks and on annual hours. In add i t ion , Hanoch estimates his model using the non-random Survey of Economic Opportunity sample of married women. Instead of the ML approach adopted in th is study, Hanoch applies 3SLS and includes a Heckman-type correct ion to deal with the SS b ias . F i n a l l y , rather than obtaining separate estimates for groups of households with d i f fe rent demographic cha rac te r i s t i c s , Hanoch includes various demographic var iables in the hours and weeks equations which are then estimated for the ent i re sample. Despite these s ign i f i can t di f ferences in Hanoch's study, i t is s t i l l possible to make some comparisons between Hanoch's resul ts and those presented in th is chapter. F i r s t , Hanoch f inds that the s ign i f i can t var iables have the same signs in both the annual hours worked and annual - 215 -weeks worked equa t i ons . While Hanoch's r e s u l t s pe r t a i n to equat ions f o r female hours and weeks worked, in our model we can perform s i m i l a r comparisons f o r the male weeks and hours equa t i ons . From the mbs in Table 6 . 2 , we see tha t B i > 0 and B 2 < 0 except f o r households wi th c h i l d r e n and no e d u c a t i o n , wh i le (S i + B 2 ) > 0 f o r a l l househo lds . Now s ince ( S i + S 2 ) represen ts the mbs f o r annual male l e i s u r e , an inc rease in f u l l income w i l l decrease annual hours worked. However wi th S 2 < 0 , an inc rease in f u l l income w i l l cause l e i s u r e in non-working weeks to f a l l , and hence weeks worked to i n c r e a s e . Thus the e f f e c t s of income on annual hours worked and weeks worked d i f f e r in s ign in our model . S i m i l a r l y , i f we examine the non- ra t ioned male hours and weeks worked e l a s t i c i t i e s in Table 6 . 5 , we see that except f o r households wi th c h i l d r e n and no e d u c a t i o n , HMPM and WMPM have d i f f e r e n t s i g n s , as do HMPF and WMPF. Thus, r e c a l l i n g tha t annual hours e l a s t i c i t i e s d i f f e r n e g l i g i b l y from HMPM and HMPF, we have the r e s u l t tha t the e f f e c t s of changes in the wage ra tes on annual hours worked and on annual weeks worked a l so d i f f e r in s i g n . A second conc lus ion from Hanoch's study i s that an inc rease in income, de f ined as non-wage income p lus husband's e a r n i n g s , w i l l r e s u l t i n a decrease in hours and weeks worked by the w i f e . While the income v a r i a b l e used in our model i s f u l l income which a l so inc ludes the w i f e ' s e a r n i n g s , s ince an inc rease in non-wage income w i l l i nc rease f u l l income, we can use the negat ive of the s ign of the mbs f o r female l e i s u r e in Table 6.2 to i n d i c a t e the d i r e c t i o n of the e f f e c t on female hours worked. Con f in ing our a t t e n t i o n to the SS-co r rec ted v a l u e s , we see tha t Sf i s o n l y s i g n i f i c a n t f o r households wi th educa t i on , and in both these cases i t i s p o s i t i v e . Thus, as in Hanoch's model , an inc rease in non-wage income w i l l decrease hours worked by the fema le . - 216 -In terms of the demographic var iables included in his model, Hanoch f inds that an increase in the level of education w i l l increase hours and weeks worked by the female. In addi t ion, an increase in the number of chi ldren w i l l decrease hours worked for females with high levels of education but increase hours worked for those with less education. However he notes that such women w i l l s t i l l work less than comparable women who have no ch i l d ren . From the resu l ts presented in Table 6.2, we see that for our model 3f is only s ign i f i can t for households with education, so that an increase in income w i l l decrease female hours worked for these households, but w i l l have no s ign i f i can t ef fect on female hours for households without education. Further, a comparison of the non-rationed e l a s t i c i t i e s in Table 6.5 for educated and non-educated households reveals that for households without ch i l d ren , educated households have female hours e l a s t i c i t i e s larger in absolute value but of the same sign as non-educated households. However for households with ch i l d ren , educated households have female hours e l a s t i c i t i e s which have a s imi la r magnitude but a d i f fe rent sign to the corresponding e l a s t i c i t i e s for non-educated households. Thus for households with ch i l d ren , HFPF changes from negative in educated households to predominantly pos i t ive in non-educated households, while the sign of HFPM changes in the opposite d i r e c t i o n . Hence we would conclude from our model that although females in educated households may be more responsive to changes in income or wage ra tes , they do not in general tend to work more hours than the i r counterparts in households without education. An examination of the mbs and e l a s t i c i t i e s in Table 6.2 and Table 6.5 lends only par t ia l support to Hanoch's conclusion concerning the ef fects of chi ldren on female hours worked. The female mbs is larger for - 217 -households with chi ldren and education than those with education but no ch i l d ren , suggesting that for educated households, the presence of young chi ldren decreases hours worked by the female. However, for households with no education the female mbs are not s i g n i f i c a n t , so that no conclusion can be reached for them on th is issue. From the e l a s t i c i t i e s in Table 6.5 we see that for households with education, the presence of ch i ldren increases the absolute value of the female hours worked e l a s t i c i t i e s but HFPF remains pos i t ive and HFPM negative. For households without education, the presence of ch i ldren also increases the absolute value of the e l a s t i c i t i e s , but here they change s ign . Thus from our analysis of these e l a s t i c i t i e s we would conclude simply that females in households with chi ldren tend to be more responsive to changes in economic var iab les than females in households with no ch i ld ren . F i n a l l y , Hanoch concludes from his study that an increase in the female wage rate w i l l increase both annual hours and annual weeks worked by the female. Thus he f inds evidence of a pos i t i ve l y sloped labour supply curve for women, and calcu lates annual hours and weeks e l a s t i c i t i e s of 0.64 and 0.55 respect ive ly . From the resu l ts in Table 6.5 we f ind that in the SS-corrected case, apart from households with chi ldren but no education where HFPF = -0 .93 , for our model th is e l a s t i c i t y is also pos i t i ve or predominantly pos i t ive for non-rationed households. However here HFPF ranges between 0.09 for households with neither chi ldren nor education to 0.87 for households with both chi ldren and education. - 218 -CHAPTER SEVEN CONCLUSIONS In th is thesis we formulate and estimate two models of household labour supply behaviour which take account of the fact that due to constraints in the labour market, some household heads are prevented from supplying the i r desired amount of labour. In order to be consistent with the economic theory of consumer behaviour, we derive these models by assuming that households, based on the i r preferences, perform an opt imizat ion procedure to determine the i r desired quant i t ies of male le isure (provided i t is not ra t ioned) , female le isure and goods consumption. Since we model household behaviour, by j o i n t l y estimating demand equations for rationed and non-rationed households, we are able to examine the ef fects that rat ioning of the labour supply of the household head has on the labour supply of his spouse and on household goods consumption. In the f i r s t model, which is developed in Chapter 3, households are c l a s s i f i e d into three groups according to whether the male household head is underemployed, overemployed or non-rationed in his hours of work. In order to evaluate the ef fects of th is rat ioning on the non-rationed female, marginal budget shares and labour supply e l a s t i c i t i e s are compared for these three household groups. Detai led analysis of these measures in Chapter 4 reveals that there is no general pattern to the way in which they d i f f e r among the rationed and non-rationed groups. Further, the re la t ionsh ip between the rationed and non-rationed marginal budget shares - 219 -and e l a s t i c i t i e s d i f f e r s accord ing to the demographic c h a r a c t e r i s t i c s of the househo ld . To determine the e f f e c t s and importance of c o r r e c t l y mode l l ing the behaviour of r a t i oned househo lds , we re -es t ima te the model f i r s t us ing on l y the subsample of non- ra t ioned households and then t r e a t i n g a l l obse rva t i ons as though they are n o n - r a t i o n e d . Al though the es t imates obta ined us ing these approaches d i f f e r from those obta ined using the c o r r e c t l y s p e c i f i e d model , there again appears to be no general pa t te rn to these d i f f e r e n c e s . Thus, we are unable to make any general statements concern ing the d i r e c t i o n or magnitude of the b ias which i s present in the es t imates obta ined using the m i s s p e c i f i e d models. In view of t h i s r e s u l t , and the f a c t tha t marginal budget shares and labour supp ly e l a s t i c i t i e s f o r ra t i oned households cannot be c a l c u l a t e d us ing the m i s s p e c i f i e d models , i t would appear to be qu i t e important to use the e n t i r e sample o f r a t i oned and non- ra t ioned households to es t imate the model in which each type of r a t i o n i n g i s model led a p p r o p r i a t e l y . S ince the demand equat ions we est imate r equ i r e both male and female wage r a t e s , and s i n c e wage ra te in fo rmat ion i s not a v a i l a b l e f o r females who are not work ing , es t ima t ion i s performed using on ly those households in which the female i s employed. In order to c o r r e c t f o r the sample s e l e c t i v i t y b ias that such a procedure would o therwise i n t r oduce , we d i v i d e the l i k e l i h o o d f u n c t i o n by the p r o b a b i l i t y that the female i s work ing . In Chapter 4 we show tha t f o r our r a t i o n i n g model , the d i r e c t i o n of the e f f e c t s of t h i s c o r r e c t i o n on the parameter es t imates cannot be a n a l y t i c a l l y determined. Thus in order to f u r t h e r examine t h i s i ssue we compare es t imates obta ined when t h i s c o r r e c t i o n i s f i r s t omi t ted and then inc luded in the l i k e l i h o o d f u n c t i o n . While we f i n d that the e f f e c t s of - 220 -t h i s correct ion are quite s ign i f i can t for some parameters, involving changes in sign and/or s i z e , other parameters are not af fected. Further, the par t icu lar parameters which are af fected, and the nature of these e f fec t s , d i f f e rs for households with d i f ferent demographic cha rac te r i s t i c s . In the second model, which is developed in Chapter 5, we take account of the fact that many of the household heads are unemployed for some time during the year even though they claim to be non-rationed in the i r hours of work. Here we view th is unemployment as resu l t ing from a constraint on the number of weeks that the household head can work during the year, and assume that the rat ioning information used in the f i r s t model re lates s p e c i f i c a l l y to constraints on hours worked by the head in a working week. Thus, in th is second model we consider separate rat ions on the weeks worked by the household head and on the hours that he works in a working week. Thus since weeks worked in a year are ei ther less than or equal to the i r desired l e v e l , we have s ix rat ioning cases to consider in th is second model. While th is complicates estimation to the extent that i t is necessary to introduce certa in parametric res t r i c t i ons in order to obtain any estimates, we are able to obtain much more information than in the f i r s t model. S p e c i f i c a l l y , we estimate marginal budget share's and labour supply e l a s t i c i t i e s in the d i f ferent rat ioning cases and in addit ion ca lcu la te many more cross-rat ion ing e l a s t i c i t i e s which show the ef fect on the unrationed quant i t ies of changes in the level of the weeks worked rat ion or in the level of the rat ion on hours worked in a working week. From the estimates for th is model which are presented and analyzed in Chapter 6, we again observe that the re la t ionsh ip between the marginal budget shares (and between the e l a s t i c i t i e s ) in the d i f ferent rat ioning - 221 -cases d i f f e r s accord ing to the demographic c h a r a c t e r i s t i c s of the househo ld . The e f f e c t s of the sample s e l e c t i v i t y c o r r e c t i o n , which are a l so examined in t h i s model , again tend to d i f f e r in d i r e c t i o n and magnitude f o r d i f f e r e n t parameters and f o r households wi th d i f f e r e n t demographic c h a r a c t e r i s t i c s . A comparison of the es t imates obta ined f o r t h i s model wi th those obta ined us ing on l y non- ra t ioned househo lds , or those obta ined when a l l households are t r ea ted as n o n - r a t i o n e d , r evea l s tha t the e f f e c t s of m i s s p e c i f i c a t i o n are qu i t e la rge in some cases but again f o l l o w no general p a t t e r n , and t he re fo re cannot be asce r ta ined p r i o r to e s t i m a t i o n . Thus from t h i s a n a l y s i s we again conclude tha t i t i s important to inc lude the appropr ia te sample s e l e c t i v i t y c o r r e c t i o n and to c o r r e c t l y model the behaviour of ra t i oned househo lds . One i n t e r e s t i n g comparison between the r e s u l t s obta ined with our two models concerns the est imated marginal budget shares f o r male l e i s u r e . In the f i r s t model t h i s marginal budget share i s n e g a t i v e , imply ing tha t male l e i s u r e i s an i n f e r i o r good. In order to determine p o s s i b l e reasons f o r t h i s r e s u l t we examine a number of other labour supply s t u d i e s , many of which have a lso found male l e i s u r e to be i n f e r i o r . While there appears to be no s i n g l e f ea tu re common to a l l these s tud ies which can account f o r t h i s r e s u l t , we show that l i n e a r i t y of the est imated labour supply or labour earn ings f u n c t i o n i s p o s s i b l y one of the p r i n c i p a l c o n t r i b u t i n g f a c t o r s . However in our second model where male l e i s u r e i s d i v i d e d i n to l e i s u r e dur ing working weeks and l e i s u r e dur ing non-working weeks, we no longer ob ta in t h i s r e s u l t . S p e c i f i c a l l y , wh i le male l e i s u r e in non-working weeks i s i n f e r i o r , the e f f e c t of inc reased income on male l e i s u r e in working weeks i s p o s i t i v e and outweighs the negat ive e f f e c t on male l e i s u r e in non-working weeks. Thus i t would appear that in order to avoid - 222 -the theo re t i ca l l y unappealing resul t that male le isure is i n f e r i o r , i t is necessary to consider the d i f ferent dimensions of male l e i su re . A number of avenues for further research are suggested by the resul ts and analysis presented in th is t hes i s . In re la t ion to the s ing le - ra t ion model, i t would seem worthwhile to pers is t with the use of other funct ional form representations of household preferences. As is well documented, the Stone-Geary u t i l i t y funct ion is very r e s t r i c t i v e , so that use of some other more f l e x i b l e funct ional forms may y ie ld much better r e s u l t s . However, in order to estimate equations such as those derived from the Indirect Translog spec i f i ca t i on , i t is necessary to decide how to continue with the optimizat ion procedure when ei ther no v i r tua l wage solut ion or more than one v i r tua l wage solut ion is obtained. While i t would also be of interest to use other funct ional forms with the hours and weeks model, the complexity of the l i ke l ihood function for th is model, even when the Stone-Geary u t i l i t y function is used, would appear to preclude any general izat ions in th is d i r ec t i on . It would also be of interest to extend the analysis by u t i l i z i n g addit ional information which is contained in the Michigan data set . S p e c i f i c a l l y , use could be made of the fact that th is data set contains observations on the same ind iv iduals over a number of years. Thus, as in Ham [1980], a rat ioning model which contains indiv idual ef fects could be estimated. In th is way account could be taken of the fact that ind iv idua ls in certa in households may have a higher l i ke l ihood of being rat ioned. A l t e rna t i ve l y , in the case where a s ingle cross-sect ion is used, information on the number of weeks that an indiv idual is unemployed could also be u t i l i z e d , pa r t i cu la r l y in conjunction with a model s imi lar to the hours and weeks model developed here. - 223 -One aspect of the Michigan data set which may hinder further rat ioning analysis concerns the nature of the rat ioning information i t contains. S p e c i f i c a l l y , many of the questions which are used to determine whether the household head is rationed in the labour market are ambiguous in nature. For example, i t is l i k e l y that many of the ind iv iduals who claim that they want to work more hours only do so because they expect to receive a higher wage rate for these extra hours. Of course, i f th is information is known, and de ta i l s of the overtime wage rates are ava i l ab le , i t would be possible to construct an appropriate model for th is s i t ua t i on . Another ambiguity concerns the fact that many household heads who are unemployed for some time during the year also claim to be non-rat ioned. We have attempted to deal with th is problem in our hours and weeks model by assuming that the re la t ionsh ip between desired hours and actual hours worked concerns a constraint on hours worked in a working week, while unemployment during the year resu l ts from a constraint on weeks worked. However, i t is possible that information on these two rat ion ing var iables is condit ional in nature. For example, i t is l i k e l y that an i nd i v idua l ' s response to questions asking whether he wants to work more hours in a working week d i f f e r according to whether or not he is unemployed for any time during the year. F i n a l l y , one of the most in terest ing areas for further research l i e s in the development of a model of labour supply behaviour which simultaneously takes account of d isequi l ibr ium in the aggregate labour market and the way in which households adjust the i r labour supplies in response to th is d isequ i l ib r ium. In th is way, an in te rna l l y consistent model of indiv idual and aggregate labour market behaviour in the presence of rat ioning would be obtained. In order to develop such a model, we would - 224 -need to general ize the models constructed here to deal with the choice an indiv idual makes between working at the rationed level and not working at a l l . While a recent study by Mof f i t t [1982] represents an i n i t i a l considerat ion of th is issue, much research remains to be done before a meaningful model of th is type is developed. - 225 -BIBLIOGRAPHY Abbott, M. and 0. 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J . , "Labour Supply and Commuting Time: An Empirical Study", Journal of Econometrics, 8-2, October 1978, 215-226. Wales, T . J . and A.D. Woodland, "Estimation of Household U t i l i t y Functions and Labour Supply Response", Internat ional Economic Review, 17-2, June 1976, 397-410. Wales, T . J . and A.D. Woodland, "Estimation of the A l locat ion of Time for Work, Leisure and Housework", Econometrica, 45-1, January 1977, 115-132. Wales, T . J . and A.D. Woodland, "Sample Se lec t i v i t y and the Estimation of Labour Supply Funct ions", Internat ional Economic Review, 21-2, June 1980, 437-468. - 231 -APPENDIX ONE DATA The data used in th is study are drawn from the Univers i ty of Michigan Survey Research Centre's A Panel Study of Income Dynamics, Wave XI , 1978. In order to obtain a sample which is appropriate for the rat ioning models which we estimate, households are only selected i f they sa t i s f y cer ta in c r i t e r i a . S p e c i f i c a l l y , 44.5% of the or ig ina l sample of 6154 households is excluded as these households form part of the non-random 1967 Survey of Economic Opportunity subsample. In add i t ion , since we require households in which both the head and spouse are present, a further 19.2% of the o r ig ina l sample is e l iminated. The requirement that the wife works, so that her wage rate information is ava i l ab le , resul ts in a further loss of 15.4% of the o r ig ina l sample. Since the rat ioning models estimated here are not appropriate for households in which the head earned addit ional income from overtime, bonuses or commissions, or had other ways of making money apart from his main job , the sample is reduced by a fur ther 10.9%. 1 Addi t ional observations are excluded i f the household received supplementary welfare payments ( loss of 0.1%), i f the head's income was zero or undefined (0.9%), i f information was not avai lable on his hours and weeks worked (0.1%) and i f values were assigned to the hours or income var iables of the head or spouse by the Survey Research Centre (0.3%). As a resul t of the d iv i s ion of households into various rat ioning categor ies, as described below, a fur ther 0.3% of the sample is lost due 1 Note that as a resul t of these c r i t e r i a , the heads of a l l households remaining in the sample are male. - 232 -to households providing inconsistent responses to the questions on which the i r rat ioning status is based. In order to el iminate other households with inconsistent responses, observations are excluded i f reported annual hours worked d i f fe rs from the product of weeks worked and hours worked per week by more than ei ther the number of weeks worked or the number of hours worked per week, whichever is la rger . Thus th is c r i t e r i a excludes households in which the di f ference between these two measures of annual hours worked exceeds the equivalent of working one more (or less) hour than reported in each week worked, or the equivalent of working the same number of hours per week for one more (or one less) week than reported, whichever is greater. This c r i t e r i on resu l ts in a loss of 0.1% of the o r ig ina l sample. The f i n a l reason for exc lus ion, incomplete data on the required demographic var iab les , resul ts in a further loss of 0.2%. Hence, as a resu l t of a l l these c r i t e r i a , the f i na l sample used in estimating the ra t ion ing models comprises 492 households. The basic var iables needed to estimate the system of share equations are the af ter- tax marginal wage rates of the head and his spouse, hours of le isure for both labour force par t i c ipan ts , and f u l l income. In add i t ion , for the weeks and hours rat ion model, we also require information on the number of hours which the male works in a working week, H, and his number of weeks worked, K. For both the male and female, the gross wage rate is calculated as annual income from wages divided by annual hours of work for money.2 Net wages, Pm and pf, are then computed as the gross wage rate mul t ip l ied by (1 - t ) , where t is the household's marginal tax ra te . Note that the 2 While reported annual hours are used in the s ing le- ra t ion model, in order to be consistent with the formulation of the hours and weeks model, annual hours of the male in th is model are defined as the product of K and H. - 233 -pr ice of the consumption good, p c , is set equal to unity throughout the ana lys i s . In the s ing le - ra t ion model, the annual hours of le isure of the head and his spouse, Xm and Xf, are calculated as the di f ference between the to ta l number of annual hours avai lab le (8760 = 365 x 24) and the i r hours of employment. In the hours and weeks model, male le isure during working weeks, X i , is calculated as K(T-H), where T = 168 is the number of avai lab le hours in a week. Male le isure during non-working weeks, X 2 , is computed as T(N-K), where N = 52 is the number of weeks in a year. For th is model, female le isure is again computed as the di f ference between annual hours avai lable and hours of employment; to be consistent with the male hours va r iab les , annual hours avai lab le in th is model are defined as NT = 8736. F i n a l l y , the amount of household goods consumption, X c , is computed by subtracting income taxes and adding any income tax credi ts to tota l taxable income. Thus, f u l l income is obtained as M = p„X + p.X,. + p X , ' rm m r f f r c c where in the hours and weeks model, Xm = Xi + X 2 = NT - KH. Descr ipt ive s t a t i s t i c s for the required var iables in each of the two models are presented in Table Al for the whole sample. For the s ing le - ra t ion model, households are c l a s s i f i e d into the three categories non-rat ioned, underemployed and overemployed according to the response of the head to various questions. A household is c l a s s i f i e d as underemployed i f the head answered "no" or "don't no" to the question "Was there more work avai lable (on your job / any of your jobs) so that you could have worked more i f you had wanted t o ? " , and also answered "yes" to the question "Would you have l iked to work more i f you could have found more work?". A l te rna t i ve l y , households are c l a s s i f i e d as overemployed i f - 234 -TABLE A l : DESCRIPTIVE STATISTICS FOR ALL DATA VARIABLE MEAN STANDARD DEVIATION MINIMUM MAXIMUM 5.1293 1.9897 0.54622 13.567 3.2890 2.0523 0.15625 22.917 6596.4 553.04 3864.0 8681.0 7360.4 665.96 5016.0 8748.0 19068. 7231.1 2200.0 47043. 77514. 27457. 19161. 249050. 5.1276 1.9906 0.54167 13.567 3.2890 2.0523 0.15625 22.917 46.380 6.9126 6.0 52.0 46.480 9.4015 5.0 96.0 5627.3 917.72 858.0 8058.0 944.15 1161.3 0.0 7728.0 7336.4 665.96 4992.0 8724.0 19068. 7231.1 2200.0 47043. 77297. 27395. 19054. 248370. Model 1: Model 2: Pm Pf Xm Xf Xc Pi ,P2 Pf K H Xi X 2 Xf X C M the head responds "no" to the question "Could you have worked less i f you wanted t o ? " , but rep l ies "yes" to the question "Would you have preferred to work less even i f you had earned less money?". F i n a l l y , households are c l a s s i f i e d as non-rationed i f they were neither overemployed nor underemployed. The d is t r ibu t ion of these households among the unrationed and two rationed groups for the s ing le - ra t ion model is shown in Table A2 for each of the four data subsets, as well as for a l l observations. The f i n a l column of the table shows the d is t r i bu t ion of observations when households in which the head was unemployed for a non-zero number of weeks during the year are grouped with the underemployed households, as in the studies of Ham [1977], [1980]. - 235 -TABLE A2: DISTRIBUTION OF HOUSEHOLDS IN THE SINGLE-RATION MODEL CLASSIFICATION C,E C,NE NC,E NC,NE ALL Ham's Grouping Underemployed 15 19 18 46 98 137 Overemployed 6 2 15 14 37 32 Non-Rationed 48 37 145 127 357 323 Total 69 58 178 187 492 492 Codes: C = youngest ch i ld less than 6 years old NC = no chi ldren younger than 6 years E = head or spouse completed at least one year of col lege NE = neither head nor spouse attended col lege In the hours and weeks model of labour supply, each of the three ra t ion ing categories in the s ing le - ra t ion model is divided into two groups according to whether or not the weeks worked by the household head are constrained. Since household heads were not e x p l i c i t l y asked i f they wanted to work more weeks in a year, they are c l a s s i f i e d as desir ing to work more weeks i f they spent at least one week during the year unemployed. Unfortunately there is no information which allows us to determine i f a household head wished to work fewer weeks during the year. The d is t r ibu t ion of households among the 6 rat ioning categories in the hours and weeks model is presented in Table A3 for each of the four data subsets. Note that Cases 5 and 6 in th is table correspond to the non-rationed and overemployed categories in the s ing le- ra t ion model when households are grouped as in Ham's s tud ies . - 236 -TABLE A3: DISTRIBUTION OF HOUSEHOLDS IN THE HOURS AND WEEKS MODEL CLASSIFICATION C,E C,NE NC,E NC,NE WEEKS RATIONED: -Underemployed (Case 1) 5 9 4 16 -Overemployed (Case 3) 2 1 1 1 -Non-Rationed (Case 2) 4 9 7 14 WEEKS UNCONSTRAINED: -Underemployed (Case 4) 10 10 14 30 -Overemployed (Case 6) 4 1 14 13 -Non-Rationed (Case 5) 44 28 138 113 TOTAL 69 58 178 187 For Codes, see Table A2. Case numbers in parentheses are those used in Chapters 5 and 6. - 237 -APPENDIX TWO FORMULATION OF LABOUR SUPPLY ELASTICITIES IN THE SINGLE-RATION MODEL In th is appendix we speci fy the formulae for the labour supply e l a s t i c i t i e s for the s ing le- ra t ion model estimated using le isure and commodity demand equations derived from the Stone-Geary u t i l i t y func t ion . While there are two rat ioning cases ~ underemployed and overemployed ~ as can be seen from Section 3.3.1 the demand equations, and hence the e l a s t i c i t i e s , have the same formulation in both these cases. In the non-rationed case we ca lcu la te e l a s t i c i t i e s of male hours, HM = Tm - X m , and of female hours, HF = Tf - Xf, with respect to the male and female wage ra tes , Pm and pf, respect ive ly . In the rat ioned case we calcu late only the female e l a s t i c i t i e s , although we are also able to compute the cross-rat ion ing e l a s t i c i t y . To avoid confusion with the notation used in the hours and weeks model, we replace Tm and Tf , the number of hours in a year, by NT. Note that the wage e l a s t i c i t i e s for th is model d i f f e r from those in the usual Stone-Geary model which does not include le isure among the arguments of the u t i l i t y funct ion. These di f ferences are caused by the fact that changes in the wage rates in th is model have an addit ional e f fec t on le isure demands in that they cause a revaluat ion of f u l l income, M. - 238 -1. Rationed Households Using equation (3.31), the female hours elasticities for this case are calculated as: HFPM = - ( 3 f / ( l - 0m)).(NT - X m).p m / (pf(NT - X f)) HFPF = ( 8 f / ( l - P m ) ) . ( M - . ^ P J Y J - p / J / (Pf(NT - X f ) ) - (P f/(1 - Sm)).(NT - Y f) / (NT - X f ) For these households we can also calculate the cross-rationing elasticity, HFHM. However, since 3Xf/9Xm = -(S f/(1 - B m)). P | n / P f , i t follows that HFHM = HFPM. 2. Non-Rationed Households Using equation (3.29), we f i r s t form the derivatives 9Hi/3pi = 8.(M - ^PjYjO/p2 - ^ (NT - y.J/p. , i=m,f 3H./3p. = - 8^ (NT - Y j)/p i , i,j=m,f; i * j Hence we obtain the e l a s t i c i t i e s : HMPM = 8m(M - Zp.Yj)/(pm(NT - Xm>) - 8m(NT - Ym)/(NT - X j HMPF = -Bm(NT - Y f).p f / (pm(NT - X j ) HFPM = -0f(NT - Y m).p m / (pf(NT - X f)) HFPF = 3f(M - £pjYj)/(pf(NT - X f)) - 8f(NT - Yf)/(NT - Xf) - 239 -APPENDIX THREE REGULARITY CONDITIONS FOR THE SINGLE-RATION MODEL In order for the estimated demand functions to be integrable into a consistent preference order ing, i t is necessary that they sa t i s f y cer ta in regu la r i t y condi t ions. These condi t ions, as shown by Deaton and Muellbauer [1980,p50] and others, are simply the requirements that the Slutsky matrix of compensated pr ice responses be symmetric and negative sem i -de f i n i t e . 1 In th is appendix we speci fy these res t r i c t i ons for the s ing le - ra t ion model of household labour supply when the u t i l i t y function is of the Stone-Geary form, (3.28). While these condit ions have been determined elsewhere in the l i t e ra tu re for the non-rationed case when the u t i l i t y function depends only on consumption of commodities, we speci fy them here for the case where two of the arguments of the u t i l i t y function are le isure var iab les . 1. Non-Rationed Households We begin by speci fy ing the demand equations in the non-rationed case. Thus, using the notation from Chapter 3, we have X i = Y i + P^M - S p K Y k ) / p i i=m,f,c. Taking der ivat ives with respect to pr ices and income, we obtain { - e (M - E p . Y j / p 2 + MNT - Y i ) / p i i=m,f aXj /Spi = { 1 . { - p.(M - E P k \ ) / P - - V i / p i i = c 1 Of course i t is also necessary that the demand functions sa t i s f y the adding-up condit ion and are homogeneous of degree zero in pr ices and incomes. However, when we derive demand functions by maximizing a u t i l i t y function subject to a budget const ra in t , these two requirements are automatical ly s a t i s f i e d . - 240 -t B.(NT - y j / p , j=m,f; i=m,f,c { - 3 i Y j / p i j=c; i=m,f,c aX^aM = p. /p . i=m,f,c. The Slutsky equation which re lates the compensated pr ice responses to the uncompensated pr ice and income der ivat ives d i f f e rs depending on whether the response is to a change in a le isure pr ice or to a change in a commodity p r i ce . This is due to the fact that when we change a le isure pr ice (that i s , a wage ra te ) , f u l l income is revalued. Hence we o b t a i n : 2 { 3X. /3p . - (NT - X.) . (9X./3M) j=m,f; i=m,f,c s . = 3X./3p |u = { 1 J J 1 J 0 1 { n./zp. + x . ^ a x ^ a M ) j=c; i=m,f,c We are now in a posi t ion to obtain the Slutsky matrix for the Stone-Geary u t i l i t y funct ion. Using the der ivat ives defined above, and noting that at the estimated po in t , 3 (A3.1) X i - Y i = P . ( M - ^ P k Y k ) / p i i=m,f,c, we obtain s.. = - (X i - y . ) ( l - 3 i ) / p i i=m,f,c s ^ = 3 i ( X j - Y J - ) / p i i , j=m,f,c; i*j Thus, regardless of whether the arguments in the u t i l i t y function are le isure or goods consumption, the Slutsky compensated price responses are the same. It is well known that the Linear Expenditure System (LES) automatical ly s a t i s f i e s the Slutsky symmetry condi t ions. Here th is can be demonstrated quite simply by subst i tu t ing for (Xj - Yj) in s-jj using (A3.1) . Thus we obtain 2 See, for example, Deaton and Muellbauer [1980,p91-92]. 3 Since i t is not confusing, for notat ional convenience we make no d i s t i nc t i on between actual and estimated values of the endogenous va r iab les . - 241 -S i j = S j i = ^ ^ ( M - 2 V k ) / ( P i P j ) -To sa t i s f y the negative semi-definiteness property of the Slutsky matrix i t is necessary that a l l compensated own pr ice e f fec ts , s-j-j, be non-posi t ive and that successive pr inc ipa l minors of the Slutsky matrix al ternate between being non-posit ive and non-negative. Here where there are 3 arguments of the u t i l i t y funct ion, we thus require that s-j-j _< 0 for a l l i=m,f,c, and that a l l the 2x2 determinants, ( s . . s . . - s . . s . . ) , I I J J I J J I be non-negative for i , j=m,f ,c, i * j . Here there is no need to check the determinant of the f u l l 3x3 Slutsky matr ix, since by v i r tue of Eu le r ' s Theorem, Ep . s . . = 0 for i=m,f,c, so that the 3x3 determinant equals zero, j J i j Subst i tut ing in the expression for the 2x2 pr inc ipa l minors, we see that the condit ions for the Slutsky matrix to be negative semi-def in i te are therefore that s-ji _< 0, i=m,f,c and (A3.2) (X. - Y i ) ( X j - y . ) ( l - 8. - e j)/(p iP j) > 0 i , j=m,f ,c; i*j In order for s ^ j< 0, we require that (X i - Y.J)(1 - B^) _> 0, i=m,f,c. Hence we require ei ther (A3.3) (Xi - Y i ) > 0 and Bi <. 1 , or (A3.4) (Xi - Y i ) < 0 and Bi > 1 . However in order to derive the demand functions from the Stone-Geary u t i l i t y funct ion, we assume that the budget constraint holds with equa l i t y . This equal i ty is ensured provided that preferences sa t i s f y the axiom of non-sat ia t ion; that i s , provided that the u t i l i t y function is non-decreasing in each of i t s arguments and is increasing in at least one of i t s arguments. Thus in order to derive the demand functions i t i s necessary that the marginal u t i l i t i e s be non-negative. With the Stone-Geary u t i l i t y function th is implies that &i/(X-j - Yi) ^ 0, and hence that Bi and (Xi - Y i ) cannot have d i f ferent s igns. Thus th is - 242 -ru les out condit ion (A3.4) and strengthens (A3.3) to (A3.5) (Xi - Y i ) > 0 and 0 < Bi < 1 , for a l l i=m,f,c. In add i t ion , as can be seen by observat ion, th is condit ion w i l l also ensure that (A3.2) is s a t i s f i e d . Thus in order to determine whether the Slutsky matrix is negative semi-def in i te , we only need check whether condit ion (A3.5) holds. 2. Rationed Households In the rationed case, the demand equations are given by X i = Y i M 3 / ( l - 3 j ) . ( M - Z p k Y k - P m X m ) / p i i = f , C k*m Taking der ivat ives with respect to pr ices and income, and subst i tu t ing in the appropriate Slutsky equations, we obtain the compensated pr ice responses as S i i = - (X i - Y i ) [ l - ( 3 i / ( l - B m ) ) ] /P i i=f,c s i j = [S i / (1 - 3 m ) ] . (X j - Y j ) / P i i , j= f , c ; i*j Since the Slutsky matrix in the rationed case only has dimension 2x2, and since the determinant of the f u l l matrix is again zero, to check that the matrix is negative semi-def in i te we only need ensure that S i i _< 0 for i= f , c . Thus we require ei ther (A3.6) (Xi - Y i ) > 0 and ( 3 i / ( l - 3m)) < 1 , or (A3.7) (Xi - Y i ) < 0 and ( 3 i / ( l - 3 m ) ) > 1 . However (ei / (1 - 3 m)) >_ 1 implies Bi _> 0, since the parameters are the same for a l l households ~ rationed and non-rationed — and from the non-rationed condi t ions, 0 < (1 - 3m) _< 1. Since Si and (Xi - Y i ) must again have the same sign in order that the non-sat iat ion axiom is s a t i s f i e d , we thus require (A3.6) to hold for i= f ,c . Further, since - 243 -(1 - B M ) = 0f + 8 C , the r e s t r i c t i o n (3^/(1 - B M ) ) < 1 f o r i = f , c i m p l i e s 0 <_&i <_ 1, i = f , c . Thus the c o n d i t i o n s to be checked in the r a t i oned case are the same as those fo r the non- ra t ioned c a s e . I t i s i n t e r e s t i n g to note that wi th the LES, the cond i t i ons (A3.5) are a l so s u f f i c i e n t to ensure that the u t i l i t y f u n c t i o n i s concave. Taking second d e r i v a t i v e s of the Stone-Geary u t i l i t y f u n c t i o n we ob ta in 3 2 U / 3 X 2 = - B I / ( X I - Y.J ) 2 3 2 U / 3 X , 3 X i = 0. Thus the mat r ix of second d e r i v a t i v e s of the u t i l i t y f u n c t i o n i s d i a g o n a l , and w i l l be negat ive s e m i - d e f i n i t e prov ided - 3 i / ( X i - Y ^ 2 < 0 i=m,f ,c B I B J / ( X I - Y I ) 2 / ( X J - Y j ) 2 > 0 i , j = m , f , c ; i* j " Wc^ rn" V 2 / < X f " V 2 / ( X c - Y c ) 2 < 0 . C l e a r l y these c o n d i t i o n s are a l l s a t i s f i e d i f 0 <_ 3-j <_ 1, i =m, f , c . - 244 -APPENDIX FOUR FORMULATION OF THE MARGINAL BUDGET SHARES AND LABOUR SUPPLY ELASTICITIES FOR THE HOURS AND WEEKS MODEL In th is appendix, we speci fy the formulae for the marginal budget shares (mbs) and labour supply e l a s t i c i t i e s for the four d i s t i nc t cases in the hours and weeks model of labour supply. These cases correspond to the rat ion ing s i tuat ions where both hours and weeks are rat ioned, only hours are rat ioned, only weeks are rat ioned, and where there are no ra t ions . Since the systems of equations are the same in the hours and weeks rat ion case whether observed hours exceed or are less than desired hours, the mbs and e l a s t i c i t i e s have the same formulation for these two cases. S im i la r l y the mbs and e l a s t i c i t i e s have the same formulation in the two hours rationed cases. For each of the four cases, mbs are obtained by ca lcu la t ing the der ivat ive 3p-jX-j/9M for the appropriate equation systems in Section 5 .3 . Where poss ib le , we calcu late e l a s t i c i t i e s of male hours in a working week (HM), male weeks worked (WM) and female hours worked (HF) with respect to both the male wage rate (PM) and the female wage rate ( P F ) . 1 In add i t ion , we also ca lcu late cross-rat ion ing e l a s t i c i t i e s , and conclude with some comments concerning annual hours e l a s t i c i t i e s . The notation used here corresponds to that adopted in Chapters 5 and 6; in pa r t i cu la r , the subscripts 1, 2, f and c , refer to the le isure of the male during working weeks, his le isure during non-working weeks, 1 In determining these e l a s t i c i t i e s , we take account of the fact that in addit ion to i t s d i rect e f fec ts , changes in the wage rates also affect le isure demands i nd i rec t l y v ia the i r revaluat ion of f u l l income, M. - 245 -female le isure and goods consumption, respect ive ly . In order to s impl i fy various expressions, we make use of the re la t ionsh ip z3j = 1, and al ternate between using the male le isure var iab les , Xi and X 2 , and the male hours and weeks worked var iab les , H and K, where Xi = K(T -H) and X 2 = T(N-K), with N defined as the number of weeks in a year, and T as the number of hours in a week. 1. Hours and Weeks Rationed mbsi = mbs2 = 0 mbsf = Bf / (Bf + B c) mbsc = S c / ( B f + B c) HFPM = - ( B f / ( B f + e c ) ) . ( N T - X ! - X 2 ) . P l / ( p f ( N T - X f ) ) HFPF = - ( B f / ( B f + 3 C ) ) . ( N T - Y f ) / ( N T - X f ) + ( B f / ( S f + 3 C ) ) . (M - P l X i - p 2 X 2 - E P j Y ^ . P i / t P f x N T - X ^ ) j _ f ,c HFWM = HFPM HFHM = HFPM Here, the equal i ty of HFWM and HFHM ar ises due to the fact that Pi = P2 = Pm.; t n a t 1S> t n e P r i c e of male le isure (the wage rate) is the same in working weeks and in non-working weeks. 2. Hours Rationed For notat ional convenience, we repeat here equations (5.53) and (5.54): (5.53) I A = (M - I p,y. - P i(T - H)(NT - Y 2 ) / T ) / (B f+ B + <t») j ^ l 3 3 T C (5.54) 4> = B 2 - Bi (NT - KT - Y 2 ) / (KT - T Y I / ( T - H)). - 246 -Then, mbs! = -((T - TT)/H"). <t>/(0f + 3 C + <t>) mbs2 = (T/TT). <t>/(3f + B c + <t>) mbsf = 0 f / (3 f + B c + •) mbsc = 3 c / ( 3 f + 8 C + <)>) In order to formulate the various e l a s t i c i t i e s for th is model, i t is convenient to make use of the fol lowing expressions: a<t>/3K = (8! + 3 2 - <r)/(K - Y i / ( T - TT)) DEN = - (34> /aK).(lA ) . (B f + 0 c ) / ( 3 f + &c + •) " P 2 H Then we obtain the fol lowing der iva t i ves : 3K/3p f = 4>(NT - Y f ) / ( 8 f + 8 C + 4>)A>EN SK/apn, = ({.((NT - Y 2 ) - ( H / T ) / ( 0 f + B c + •) - (1/Xj/pJ /DEN 3 ( l A ) / 3 p f = (NT - Yf - (lA)(3<j)/3K)(3K/3pf))/(0f + 8 C + •) 3 ( l / ^ ) / 3 p m = ((NT - Y 2 ) (H/T) - (1/X) (3<t»/3K)( 3K/3p m ) ) / (0f + 0C+ <t>) Using these der iva t i ves , we obtain the wage e l a s t i c i t i e s : WMPM = (3K/3p m ) .p m /K WMPF = (3K/3pf).pf/K HFPM = - 0 f ( 3(l / X ) / 3 p m ) . p 1 / ( p f ( N T - Xf)) HFPF = - 0 f ( 3(l / X ) / 3 p f ) / ( N T - Xf) + 0 f ( l A ) / ( p f ( N T - Xf)) In order to s impl i fy the expressions for the cross-rat ion ing e l a s t i c i t i e s , we def ine: TERM1 = pi(NT - Y 2 ) / ( 8 f + 0 C + <t>)/T TERM2 = (0 2 - * ) . Y i / (K(T - H) - Y i ) / ( T - TT) TERM3 = - (3f + 0 c ) / ( 8 f + 0 C + <t>) NUM = (MTERM1 + (1A)/TT) + (1/X) .TERM2.TERM3)/(p2TT) DENOM = (34>/3K) .TERM3.(lA)/(p 2TT) - 1. - 247 -Using these expressions, we obtain the der iva t ives : 3K/3"H = NUM/DENOM 3cp/aTT = (3K/3TT) .(3+/3K) - TERM2 3 (1A ) /3H = TERM1 - (1/X) .(3+/3TT) / (Bf + B c + •) Hence we obtain the cross-rat ion ing e l a s t i c i t i e s : WMHM = (3K/3TT) -TT/K HFHM = - Bf(3( lA) /3H). lT/ (pf (NT - Xf)) 3. Weeks Rationed mbsi = B i / (Bi + Bf + B c) mbS2 = 0 mbsf = Bf/ (Bi + Bf + B c) mbsc = B c / (Bi + Bf + B c) Define: SUPINCOME = M - p2T(N - K) - £ p.y. Then: HMPM = (Bi /(1 - B 2)).SUPINCOME/(K.p 1.H) - (Bi / (1 - B 2 ) ) . (KT - Y i ) / (K .H) HMPF = - (Bi /(1 - B 2 )) .(NT - Y f ) - P f / ( K - P i .H) HFPM = - (Bf/(1 - B 2 ) ) . (KT - Y i ) .P i / (P f (NT - Xf)) HFPF = (Bf/(1 - B 2)).SUPINCOME/(p f(NT - Xf)) - (Bf/(1 - B 2 )) .(NT - Yf)/(NT - X f ) HFWM = - (B f / (1 - B 2 ) ) .P iKT / (pf(NT - Xf)) HMWM = (Bi / (1 - B 2)).SUPINCOME/(K.pi.H) + Y i / (K .H) - (Bi /(1 - B 2 ) ) .T /H - 248 -Here HFWM and HMWM d i f f e r from HFPM and HMPM respect ive ly . This occurs because pi = p 2 , so that the e l a s t i c i t i e s with respect to pr ice take account of the fact that a change in pr ice af fects desired values of both K and H. 4 . No Rations mbs-j = 3-j, i = l , 2 , f , c WMPM = 8 2 ((M - 2 P j Y j ) / p i " (NT - Yi - Y 2 ) ) / (KT) WMPF = -3 2(NT - Y f ) .P f / (piKT) HMPM = 3i(M - Z p j Y j ) . ( l + WMPM) / (piKH) + (Yi.WMPM - 3i(NT - Yi - Y 2 ) ) / (KH) HMPF = 8i((M - EpjYj).WMPF - Pf(NT - Yf)) / (PiKH) + Y i .WMPF / (KH) HFPM = - 3f(NT - Yi - Y 2 ) . P i / (pf(NT - Xf)) HFPF = 3 f (M - E P j Y j ) / (P f (NT - X f ) ) - 3f(NT - Yf)/(NT - X f ) Annual Hours E l a s t i c i t i e s Annual male hours worked are defined as NT - Xi - X 2 = KH. Thus we can calcu late annual hours e l a s t i c i t i e s , AHMPi, by d i f fe ren t ia t ing KH with respect to the wage ra tes . However, 3KH/3pi = K.3H/3pi + H.3K/3pi . Hence, annual hours e l a s t i c i t i e s , defined as the product of 3KH/3p-j and p-j/KH, are equal to the sum of WMPi and HMPi, i=M,F, that i s , the sum of the e l a s t i c i t i e s for male hours worked in a working week and for male weeks worked in a year, with respect to the wage ra te , p-j. Thus, when weeks are rat ioned, AHMPi = HMPi, and when hours are rat ioned, AHMPi = WMPi. 

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