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Intertemporal allocation of consumption, savings and leisure : an application using Japanese data Darrough, M. N. 1975

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INTERTEMPORAL ALLOCATION OF CONSUMPTION, SAVINGS AND LEISURE: AN APPLICATION USING JAPANESE DATA by MASAKO NAGANUMA DARROUGH B.A., International C h r i s t i a n University, Tokyo, 1966 A thesis submitted i n p a r t i a l f u l f i l l m e n t the requirements for the degree of DOCTOR OF PHILOSOPHY in the Department of ECONOMICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUlfijfclA August, 1975 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e rm i s s i o n . D e p a r t m e n t o f /.: C c t\ c \ T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e f ) t t 5 ^ i j f -J "t [c\ 7 *T i i INTERTEMPORAL ALLOCATION OF GOODS, SAVINGS AND LEISURE: AN APPLICATION USING JAPANESE DATA Masako Naganuma Darrough Chairman: Professor Erwin Diewert ABSTRACT The purpose of this study i s to investigate t h e o r e t i c a l frameworks used i n formulating models of consumption for i n -dividuals and for the whole economy, which are based on the behavioral postulate of u t i l i t y maximization. Furthermore, the proposed models are subjected to an empirical application using Japanese data. These data are constructed by the author from various sources. Relevant to thi s task are the concepts of a consistent two-stage maximization procedure, functional s e p a r a b i l i t y and aggregation. Four methods of aggregation over goods are extensively discussed. Furthermore, aggregation of i n d i v i d -ual demand functions i s ca r r i e d out over a l l people who pos-sess i d e n t i c a l u t i l i t y functionsbb . u t v Z d r f f-erent^incomes. Then the aggregate share equations become functions of a l l prices, the mean expenditure (wealth) and the d i s t r i b u t i o n of expen-diture (.wealth) i n the economy. Two f l e x i b l e functional forms, i . e . , the translog func-ti o n and the Generalized Leontief function are used to appro-ximate non-homothetic inverse i n d i r e c t u t i l i t y functions. i i i Three models of consumption are s p e c i f i e d and estimated. The f i r s t two models are atemporal models dealing with food, consumer durables, miscellaneous goods and l e i s u r e for the whole economy, based on aggregation by homothetic s e p a r a b i l i t y . The t h i r d model i s the intertemporal model for the represen-t a t i v e consumer, using Leontief aggregation to aggregate goods i n the future. This aggregation method allows one to take into account demographic s h i f t s i n the economy. The computational algorithm i s b a s i c a l l y an i t e r a t i v e version of generalized nonlinear l e a s t squares. An a r b i -t r a r i l y chosen equation i s deleted and the remaining N-l equations are estimated to obtain the maximum l i k e l i h o o d e s t i -mates. Null hypotheses to be tested are: symmetry and homo-t h e t i c i t y conditional on symmetry. The l i k e l i h o o d r a t i o test procedure i s employed to determine the v a l i d i t y of these hypo-theses. In the three good model, homotheticity i s d e c i s i v e l y rejected, while i t i s not rejected i n the l e i s u r e model. Homotheticity i s again d e c i s i v e l y rejected i n the intertempor-a l model. In addition, since both monotonicity and curvature are v i o l a t e d , we impose monotonicity i n order to obtain econ-omically meaningful estimates. One of the more s i g n i f i c a n t findings i n t h i s study re-lates income and l e i s u r e i n the Japanese case. Leisure i s i n -come i n e l a s t i c i n the l e i s u r e model. Moreover, present l e i -sure i n the intertemporal model turns out to be an i n f e r i o r good. i v TABLE OF CONTENTS ABSTRACT LIST OF TABLES LIST OF FIGURES ACKNOWLEDGEMENTS Chapter I INTRODUCTION 1 A. A S t a t i c Case with Labor Supply 4 B. A Dynamic Case 6 Footnotes 13 Chapter II THE THORETICAL FRAMEWORK IN THE NEOCLASSICAL SPIRIT A. Introduction 14 B. Functional Separability 16 C. Four Methods of Aggregation 22 D. Aggregation over People 33 E. Treatment of Durables 35 Footnotes 39 Chapter III THE PROPOSED MODEL A. Introduction 41 B. Functional Forms 41 C. E l a s t i c i t i e s 49 D. Four Aggregation Methods 55 E. The Proposed Models 59 Footnotes 61 V Chapter IV ECONOMETRICS A. Econometric S p e c i f i c a t i o n 63 B. Computational.Algorithm 6 6 C. Hypothesis Testing 68 D. Imposition of Monotonicity and Curvature 69 Footnotes 81 Chapter V DATA A. Introduction 83 B. The Three Good Model 8 6 C. The Leisure Model 87 D. The Intertemporal Model 90 Footnotes j.04 Chapter VI EMPIRICAL RESULTS A. Introduction 106 B. The Three Good.Model 107 C. The Leisure Model 111 D. The Intertemporal Model 117 Footnotes 141 Chapter VII SUMMARY AND CONCLUSIONS 142 BIBLIOGRAPHY 156 Appendix A 167 Appendix B 197 v i LIST OF TABLES page V .1 The Three Good Model (1946-1972): Prices Expenditure Shares Mean Expenditure and Theta 94 95 96 V .2 The Leisure Model (1946-1972): Prices Expenditure Shares Mean Expenditure and Theta 97 98 99 V .3 The Intertemporal Model (1946-1972): Prices Expenditure Shares Future Share and Mean - Wealth. 100 101 102 VI .1 Parameter Estimates: The Three Good Model 125 VI .2 Estimates of E l a s t i c i t i e s : SYM(+6) 126 VI .3 Parameter. Estimates: The Leisure Model 127 VI .4 Estimates of E l a s t i c i t i e s : SYM ( - o ) 129 VI .5 Estimates of E l a s t i c i t i e s : S+H 130 VI .6 Parameter• Model Estimates: The Intertemporal 131 VI .7 Parameter Estimates (BY=19 46): SYM & S+M. 133 VI .8 Parameter Estimates: S+H. 135 VI .9 Estimates of E l a s t i c i t i e s : S+M 136 V3r.; 10 Estimates of E l a s t i c i t e s : S+H 138 VI. 11 Index of i n d i r e c t U t i l i t y 140 v i i L I S T OF TABLES c o n t i n u e d page A . l E x p e n d i t u r e s ( 1 9 1 2 - 1 9 7 1 ) 1 8 6 A. 2 P o p u l a t i o n , H o u s e h o l d s and L a b o r F o r c e ( 1 9 1 2 - 1 9 7 2 ) 1 8 8 A . 3 I n t e r e s t R a t e s , S a v i n g s and T h e t a ( 1 9 7 1 - 1 9 7 2 ) 1 9 0 A. 4 .Consumption Y e a r s and W o r k i n g Y e a r s 1 9 2 A.55 S t o c k E s t i m a t e s ( 1 9 1 2 - 1 9 7 2 ) 1 9 3 v i i i LIST OF FIGURES Figure 1 Likelihood RationTests: The Three;Good Model Figure ,2 Likelihood Ratio Tests: The Leisure Model Figure 3 Likelihood Ratio Tests: The Intertemporal Model ix ACKNOWLEDGEMENTS In the course of my research, I have received i n v a l -uabnet. assistance from many people. F i r s t , I wish to thank the members of my d i s s e r t a t i o n committee: Erwin Diewert, Ernst Berndt and Terence Wales for t h e i r advice. In p a r t i c u l a r , I would l i k e to thank Erwin Diewert for s t a r t i n g me on the r i g h t track -and for providing guidance and encouragement. Ernst Berndt has helped me-a great deal i n the estimation stage of the research. In addition, the comments of Charles Blackorby, Louis Cain and Robert Evans are g r a t e f u l l y acknowledged. I am also indebted to Malcolm Greig and Keith Wales for computaional and s t a t i s t i c a l assitance. Tsuneharu Gonnami of the Asian Studies Library at the University of B r i t i s h Columbia and Emiko Moffat of the. Stanford University Library provided assistance in the c o l l e c t i o n of Japanese data. Special thanks to: Carmen, Donald, Jed and Tammy. Sheila Briggs did an excellent job of typing the entire manuscript. F i n a l l y , I wish to express my deep appreciation to William Darrough for continual encouragement and support and to Shannon Darrough for providing me with a sense of r e s p o n s i b i l i t y . Chapter I INTRODUCTION Economists are interested i n constructing both micro and macro consumption functions that are consistent with an under-lyi n g consumer choice framework and i n empirically estimating these consumption functions. The reasons for t h e i r i n t e r e s t are quite c l e a r . Aggregate consumption functions are impor-tant i n the formulation of national economic p o l i c i e s , since d i f f e r e n t p o l i c i e s may a f f e c t consumption patterns d i f f e r e n t l y . For example, the p o l i c y choice between a tax on l i q u o r and a tax on wages w i l l depend p a r t l y upon the underlying consump-tio n function. Thus, the responses that such p o l i c i e s evoke and the economic e f f e c t s that they produce depend on the underlying decision making process undertaken by the i n d i v i d -ual consumer or household. The usual micro assumption i s that individuals are r a t i o n a l u t i l i t y maximizers. Key v a r i -ables emerge from a consumer choice framework that i s b u i l t upon t h i s premise. The present study follows t h i s approach .and focuses attention on the formulation of a system of con-sumption functions at the microeconomic and macroeconomic l e v e l . The most s i m p l i f i e d choice-theoretic framework i s a single period model i n which the consumer allocates his pur-chases over goods and services subject to a budget constraint which i s given exogeneaxslyi The labor supply decision may be incorporated i n t h i s model. - 2 -A more general approach models intertemporal preferences defined over goods, services and l e i s u r e . This more general approach (due to Fisher [1930], Tintner [1938], and Hicks E194611) allows us to model not only the consumer's current period consumption and labor supply decision, but also his savings or asset holding decision. In t h i s framework, the consumer-worker w i l l invest that portion of his current wealth which i s equal to the discounted value of his planned future period consumptions (plus bequests). Savings or dissavings i n this model equals the change i n assets held for purposes of financing future consumption d e c i s i o n s . 1 Three models of consumption are presented and estimated u t i l i z i n g Japanese data i n t h i s study. The f i r s t , 'the three good model', i s a single period model with three consumption goods, i . e . , food, durables and miscellaneous. Leisure i s added i n the second model, 'the l e i s u r e model'. With both of these models, in d i v i d u a l consumption functions are aggregated over households i n order to derive aggregate consumption func-tions. We assume that a l l individuals (households) possess i d e n t i c a l preferences, and face the same set of p r i c e s , but have d i f f e r e n t l e v e l s of income. The l a s t model named 'the. intertemporal model' i s designed to analyze the intertemporal a l l o c a t i o n of the consumption of goods, services, savings and l e i s u r e f o r the 'representative' consumer. Thus t h i s approach follows c l o s e l y the s p i r i t of a l i f e cycle theory of consump-t i o n . ^ - 3 -The estimation procedure we adopt r e l i e s heavily upon the r e s u l t s of the Shephard duality theorem. A modified version of the Generalized Leontief function proposed by Diewert [1971a] and the translog function due to Christensen, Jorgenson and Lau [1973] are chosen as the functional forms for the inverse or r e c i p r o c a l i n d i r e c t u t i l i t y functions from which we derive a system of derived demand equations to be estimated. These functional forms are more general than most of the more fam-i l i a r functional forms, since they do not a p r i o r i assume either additive or homothetic u t i l i t y functions. In formulating an intertemporal model, substantial e f f o r t i s directed towards the subject of aggregation. Given T per-iods and N goods i n each period, the i n d i v i d u a l must make a l l o -cation decisions over TN goods. If we have s u f f i c i e n t l y many parameters to allow for interdependence among a l l goods, then 2 we require approximately %(TN) parameters, which w i l l be too many to deal with i n an empirical estimation procedure. Thus, the large number of goods, services and l e i s u r e must be aggre-gated into a more manageable number. I t i s now c l e a r that the aggregation problem i s one of the essential parts of an empirically oriented intertemporal consumer model. It w i l l be helpful at t h i s point to further elaborate on s t a t i c and dynamic models of consumption i n c l a r i f y i n g the scope of our study. We s h a l l discuss b r i e f l y the models of Becker [1965] and Fisher-Hicks [1930 and 1946 r e s p e c t i v e l y ] . - 4 -A. A S t a t i c Case with Labor Supply (Becker) Becker's approach follows the c l a s s i c a l path of consumer demand theory assuming consumer u t i l i t y maximization. However, i t s merit l i e s i n the new perspective employed. In his anal-y s i s , the consumer's demand i s for a c t i v i t i e s which are pro-duced by both goods and time instead of by commodities alone. Thus the demand for commodities and time are both derived demands. The demand for a good, therefore, depends on i t s price and the opportunity cost of time. Becker's model of the 'allocation of time' i s b r i e f l y summarized below: (1.1) Maximize U (Z^, i • • • ' Z N ) subject to L: wrt x x >_ 0, . . . ,xN>_ 0 (1) Z n = f n ( x n / T n ) n = 1, • • • ,N T l 1 0, . . . ,T N ^ 0 (2) E P n x n <_ V + wL Z 1,...,Z N; L>_ 0 (3) E T n + L = H where U = the consumer's u t i l i t y function, Z n = the n1"*1 a c t i v i t y which i s produced by the consumer, using inputs of market goods x n and time T , f = the l i n e a r homogeneous "production" function which expresses the "technological" r e l a t i o n s h i p , P n = the (rental) p r i c e for X R , V = non labor income, w = the wage rate, L = the amount of labor service offered during the period, - 5 -H = the number of time units i n the period under consid-eration . Thus, i n addition to the usual budget constraint, more constraints are necessary, i . e . , the time constraint and pro-duction functions for the a c t i v i t i e s . Given s p e c i f i c functional forms for the u t i l i t y function and the technological functions, one may e x p l i c i t l y solve for the demand functions for x and T . n n Diewert Cl971b] c r i t i c i z e s the model for dealing only with the consumer-worker 'labor force p a r t i c i p a t i o n d e c i s i o n 1 and not with his 'occupational choice decision'. Our l e i s u r e model, l i k e Becker's, does not deal with the occupational choice de-c i s i o n problem. The consumer's occupation i s assumed to be given exogenously. However, he i s allowed to vary his work hours at the wage rates which are also determined exogenously. Leisure i s treated symmetrically with other market goods, rather than as a 'complementary' input to produce a c t i v i t i e s . A diminishing marginal rate of substitution between l e i s u r e and income i s assumed. F i n a l l y , the model of i n d i v i d u a l house-hold demand i s aggregated over the entire population of house-holds i n order to obtain a model of aggregate consumption be-havior. - 6 -B. A Dynamic Case (Fisher-Hicks) An intertemporal u t i l i t y maximization model of the Fisher-3 Hicks i s t y p i c a l l y formulated as follows: ( I . 2 ) M a x • ,UU[:C(t).] • s u b j e c t to J Q Y < f c ) " C ( t ) - 0 VW .Yr t t . II1 (1+CK i( t ) ) rr n ^ ^ f U c;tc(t ) > : 0-£=O ' t n Q d + r ( t ) ) where a(t) r(t ) Y(t) C(t) [ 0,T] rate of time preferences market rate of i n t e r e s t income i n period t consumption i n period t l i f e span The one period u t i l i t y function U i s assumed to be time-inva-rian.tt (or the same for each period) with p o s i t i v e f i r s t derivatives and negative second deri v a t i v e s . The rate of time preference a(t) may be considered to be the weight assigned to - u t i l i t y i n period t . This type of analysis enables us to solve the problem of a l l o c a t i o n over time. The t o t a l l i f e t i m e income (or wealth) i s allocated such that the weighted sum of one period u t i l i -t i e s (or l i f e t i m e u t i l i t y ) i s maximized. Saving, S ( t ) , may or may not take place depending upon the shapes of the planned consumption and income streams. I f there i s any discrepancy between the two streams, which i s q u i t e l l i k e l y , the consumer w i l l save i n the periods of r e l a t i v e l y high income and d i s -save i n the periods of low income. - 7 -The i n d i v i d u a l chooses d i f f e r e n t l e v e l s of consumption and saving at each time t with the i d e n t i t y Y(t) = C(t) + S(t) hold-ing for a l l t . Thus the model attempts to solve the problem of intertemporal a l l o c a t i o n of expenditure, rather than the a l -l o c a t i o n a l problem among d i f f e r e n t goods and services. In.other words, the nature of the bundle of goods represented by C(t) i s often not investigated. This kind of a n a l y t i c a l framework i s useful when we are concerned with saving behavior..- Obviously, the Becker type of s t a t i c model cannot explain the existence of saving. Given nonsatiation, there i s no reason for an i n d i v i d u a l to save i n a s t a t i c model unless we introduce some exogenous motivation. However, a dynamic consideration of consumption provides the basis for the ' l i f e cycle theory of saving' such as outlined above. Most work done so far u t i l i z i n g the dynamic framework as-sumes that consumption i s the only source of u t i l i t y , although a few authors (for example, Yaari [1965]) have discussed the issue of bequests. T y p i c a l l y l e i s u r e i s not incorporated into the a n a l y t i c a l framework.^ The budget constraint (wealth) i n -cludes wages, non-wage income and current assets, but i t does not include the opportunity cost of not working. The consumer does not have a choice among occupations, wages rates or work-ing hours. Thus income and wealth are exogenous to the model. - 8 -5 Perfect knowledge and a perfect c a p i t a l market are usu-a l l y assumed i n the Fisher-Hicks approach to modelling the saving decision. Thus, an i n d i v i d u a l may borrow against his human c a p i t a l i n order to supplement his income i n the low i n -come period so as to rais e his consumption to the desired l e v e l . By solving the maximization problem (1.2), we obtain, (1.3) MU[C(t) ] = MU[C(0) ] (l+a.^ ( l + r ) _ t 6 where MU i s marginal u t i l i t y . Therefore, i f r > a, the l e v e l of consumption w i l l i n -crease over time. Given an e x p l i c i t functional form for the u t i l i t y function, we can derive demand equations for C ( t ) . Em-p i r i c a l data could be used to estimate the parameters of the assumed u t i l i t y function and the rate of time preference. The two approaches discussed so far assumed that the sta-t i c and dynamic u t i l i t y maximization problems were independent of each other. The intertemporal decision determines budget constraints for each period. Then solving the s t a t i c problem determines a l l o c a t i o n of the budgets over goods and services. It turns out that t h i s 'two-stage.' maximization i s a special case of a more general u t i l i t y maximization problem. (1.4) Let U* (x) F (x-j^  t^2 ' • * * ' ^N' ^1 '**'"** ' r ' ' ' ' ^ N ^ where there are T periods and N goods i n each period, r e s u l t -ing i n TN goods i n t o t a l , and x^ i s the planned consumption of the nth good i n period t . - 9 -Then the maximization problem becomes: (1.5) Max U*(x) . T st p x = W w.r.t. x > 0 where x i s a vector of dimension TN. S i m i l a r l y , p i s a TN T dimensional vector of discounted p r i c e s , p x i s the inner pro-duct of the two vectors, and W i s the consumer's wealth. But note that only the prices i n the current period are given to the consumer-worker and the rest of the future prices are 'expected future p r i c e s ' which are determined by the current i n t e r e s t rate, expected future i n t e r e s t rates and future ex-pected 'spot' prices (which are presumably affected by current spot p r i c e s ) . Once the a l l o c a t i o n problem i s solved, the demand for goods i n future periods can be aggregated into 'savings' (or 7 asset holdings for purposes of future consumption). Thus, (.1.6) p x -Z p x ^ , ^ n n n=l ^ 1 1 -£,p x n n n n=l W = l i f e t i m e wealth present value of planned future period consumption purchases amount of l i f e t i m e wealth invested during period 1. - 10 -The intertemporal model i n t h i s study i s formulated along these l i n e s , although goods are aggregated into a smaller num-ber of categories. The most serious problem i n a p r a c t i c a l application of models of t h i s nature turns out to be t h e i r s i z e . TN goods, may involve an enormous number of parameters to be estimated i n an econometric study of the intertemporal a l l o c a t i o n model. If one i s allowed to carry out a two-stage maximization procedure, t h i s procedure can reduce the number of goods and parameters to manageable s i z e . A detailed discussion on the necessary and s u f f i c i e n t conditions for a consistent two-stage maximization w i l l be presented i n Chapter I I . I t turns out that t h i s procedure i s c l o s e l y related to the subject of func-t i o n a l s e p a r a b i l i t y and aggregation. In t h i s chapter, the scope of the study has been examined. Our goal i s to formula't'e models of consumption for individuals (households) and for the whole economy, based on the behavior-a l postulate of consumers' u t i l i t y maximization. The labor, supply decision and intertemporal a l l o c a t i o n of^goods are- of s p e c i f i c concern. Two f l e x i b l e functional forms are chosen to approximate the inverse i n d i r e c t u t i l i t y function. These are the translog function and the Generalized Leontief function. - 11 -The organization of the remaining chapters i s as follows. Chapter II i s devoted to the exploration of the t h e o r e t i -c a l material relevant to the task at hand. Background mat-e r i a l on se p a r a b i l i t y and aggregation are discussed i n d e t a i l . Models of consumption are then developed i n a general neo-c l a s s i c a l framework. F i n a l l y a b r i e f discussion on the tr e a t -ment of durable goods i s presented. Chapter III begins with a discussion on the functional forms chosen for the study, namely the translog function and the Generalized Leontief function. We then turn to topics on aggregation over people, r e s t r i c t i o n s imposed on the functional forms and e l a s t i c i t i e s . F i n a l l y , wesspeeify the-modelseto be^ 'implemented©;! „ Econometrics i s the topic of Chapter IV. Methodological problems, estimation algorithm, hypotheses to be tested and methods of imposing monotonicity and concavity are discussed. In Chapter V, necessary data are discussed and presented. Their generation from the existing data for Japan i s described, although the sources for data and the exact procedures employed i n constructing the variables are included i n AppendiX'-A-..'. Chapter VI reports empirical results for the three models. An overview of the study i s provided i n Chapter VII. The reader may treat t h i s chapter as an alternative introduction with a summary of r e s u l t s . - 12 -There are two.appendices: data (Appendix A) and a mathematical note (Appendix B). - 13 -FOOTNOTES to Chapter I. 1. An additional savings motive, i . e . , the precautionary mo-t i v e , may ex i s t i n r e a l i t y due to uncertainty. Under un-certainty, savings may serve as a buffer against unforeseen contingencies i n the future. This may be a reason for in - r eluding savings as one of the arguments i n the u t i l i t y function. However, the present study does not treat the problem of uncertainty. Thus, we omit the p o s s i b i l i t y of savings as a d i r e c t source of u t i l i t y . 2. Here the terminology ' l i f e cycle theory' i s used i n a very broad sense, i . e . , intertemporal a l l o c a t i o n of goods over the l i f e cycle of i n d i v i d u a l s . However, as we s h a l l see l a t e r , i t i s possible to take demographic s h i f t s into con-sideration i n order to obtain a generalization of the M-B l i f e cycle hypotheses. 3. Fisher used a general functional form rather than the ad-d i t i v e form. However, i t i s c l e a r from his discussion on time preference or impatience that, i n f a c t , he had an additive functional form (or strong separability) i n mind. 4. Christensen [1968] i s an exception. 5. Perfect knowledge i s not essential as long as uncertainty can be insured. Yaari [1965; 147] has shown that 'the introduction of insurance i s equivalent to the removal of uncertainty from the a l l o c a t i o n problem,' I in^such a case, the pattern (or p r o f i l e ) or consumption over l i f e t i m e re-mains the same, although the l e v e l of consumption may change. If there exists any uncertainty which i s not i n -surable, then Nagatani [1972] has suggested that one would have to revise his consumption plan continuously. Due to uncertainty, one would discount future income at a higher rate than i n the certainty case. Thus, the expected asset l e v e l and the actual l e v e l w i l l diverge as time passes, and the i n d i v i d u a l w i l l have to revise his consumption plan i n order to optimize consumption according to the new wealth l e v e l . 6. For s i m p l i c i t y , a(t) and r(t) are treated as p o s i t i v e con-stants . 7. See Diewert [1974b; 505-506]. - 14 -Chapter II THE THEORETICAL FRAMEWORK IN THE NEOCLASSICAL- SPIRIT A. Introduction This chapter deals with the general t h e o r e t i c a l framework upon which the models i n th i s study are b u i l t . In p a r t i c u l a r , we.discuss functional s e p a r a b i l i t y , methods'of aggregation over goods and people, and treatment of durable goods. To recapitulate, the consumer a l l o c a t i o n problem with which we are concerned i s : T max U Cx). s.t. p x' < Y x>>0 . ~~ x N ' <_ H, for a l l t = 1, . . . ,T, where x: a -vector of consumption, p: a vector of pri c e s , Y: income or wealth, rt x N : l e i s u r e i n the tthhperiod, H: the maximum number of time units available i n each period, T: l i f e , span. This appears-to be a standard u t i l i t y maximization problem which i s r e l a t i v e l y straightforwar tdc i-to solve. However, problems may aris e when any of the following features are present, i) The number of goods involved i s large, i i ) F l e x i b l e functional forms are used, i i i ) A two-stage maximization procedure i s employed. - 15 -In.any of the above, both t h e o r e t i c a l and p r a c t i c a l (econometric) issues are raised. In,particular, when one attempts to formulate an intertemporal model of consumption, one faces the problem of dealing with a large number of commodities; In the N good, T period model, i f one wishes to use f l e x i b l e functional forms which are capable of hand-l i n g ^irross^commodity interactions, then there w i l l be at 2 least CTN) parameters to be estimated. Fortunately, sym-metry conditions, reduce t h i s number to TN(TN+l ) / 2 . However, th i s may s t i l l be an extremely large number and i n actual estimation, both the computational problems and the cost entailed appear to grow exponentially with the number of parameters to be estimated. Tn t h i s context, 'consistency' of the two-stage maxi-mization procedure 1 i s not only of th e o r e t i c a l i n t e r e s t , but i s also extremely relevant to the p r a c t i c a l aspect of the estimation problem. Thus, our in t e r e s t i n reducing the num-ber of parameters stems from both th e o r e t i c a l and p r a c t i c a l considerations. There are two approaches to handle t h i s m u l t i p l i c i t y of parameters. One way i s to reduce the number of parameters by reducing the numbers of commodities i n question. This can be achieved by aggregation over goods. Three methods of aggregation of t h i s type w i l l be discussed: aggregation by homothetic s e p a r a b i l i t y , Leontief aggregation and Hicks' aggregation. Another approach i s to reduce the number of parameters by imposing a p r i o r i r e s t r i c t i o n s on the values of parameters. This may be achieved by the method of 'aggregation by s t a t i o n -a r i t y ' . The two-stage maximization procedure i s not limited to intertemporal cases, but can be used i n any a l l o c a t i o n problems which involve a hierarchy of decision making, provided that the underlying preferences s a t i s f y c e r t a i n s e p a r a b i l i t y conditions. In other words, th i s procedure i s c l o s e l y related to the concept of functional s e p a r a b i l i t y . The two-stage maximization procedure per se does not o r i -ginate from the problem of too large a number of goods. However, when one faces t h i s problem i t i s l o g i c a l to look into the pos-s i b i l i t y of two-stage maximization procedures. For that matter, one does not have to stop at the second stage; one may consider multi-stage maximization procedures, depending upon the char-a c t e r i s t i c s of preferences, i . e . , on s e p a r a b i l i t y conditions. We shall^now-turnlourcattentionnto a discussion of func-2 " t i o n a l ^ s e p a r a b i l i t y ; io,.Al= to B. Functional Separability 1 1 2 T Let x = [ x ,...,x ] and l e t the u t i l i t y func-. t i o n be U(x) = F*(x). P a r t i t i o n x into T mutually exclusive 1 T and exhaustive categories (or subsets), R = {R ,...,R }. Then corresponding to t h i s decomposition, x may be written as x — r x f • • • f x ~\ j where each x m i s a vector of R m elements and m = 1,...,T. - 17 -Given a continuously twice d i f f e r n t i i a b l e - U ( x ) function wherecF*-(x) i i s s t r i c t l y increasirigrini.its_Toarguments•, we say that the i t h and j t h variables are 'separable from the kth variable, i f and only i f the following condition holds: (11.11 -~r ( =0 n ^  m. ax" 3U / 9 X . k 3 What t h i s means i s that the marginal rate of substitution between the i t h good and j t h good i n the mth category i s inde-pendent of the quantity of the kth good i n the nth category. S i m i l a r l y , the mth category is' separable from the kth variable, i f the above condition holds for a l l i , j eR and for... some Furthermore, the mth category i s separable from the nth category, i f t h i s condition holds for a l l i , j e R m and for a l l keR n, for some m ^ n. The u t i l i t y function i s Weakly separable i n the p a r t i t i o n {R"*~, . . ./RT>, i f every category, Rm, i s separable from a l l the other categories. The function i s v strongly separable i n the 1 T p a r t i t i o n {R ,...,R }, i f every union of categories i s separable from the remaining categories. Moreover, the function U(x) i s weakly separable, i f and 1 T only i f there e x i s t continuous functions F,f , . . . , f such that Cir.2I U (*[•,...,xjjj) = F C f 1 ( x 1 ) , . . . , f T ( x T ) ) , where f^Cx™), m = 1,...,T are referred to as category functions. F o r a s t r o n g l y s e p a r a b l e f u n c t i o n , t h e n e c e s s a r y and s u f r f i c i e n t c o n d i t i o n s a r e t h a t t h e r e e x i s t c o n t i n u o u s f u n c t i o n s F * * , f \ . . . , f T s u c h t h a t ( I I . 3 ) U ( x 1 , . . . ,XjJj) H F * ( f 1 ( x L ) + . . . + f T ( x T ) ) . A n o r m a l i z a t i o n p e r m i t s us t o w r i t e t h e f u n c t i o n as ( I I . 4 ) U(x) = f 1 ( x 1 ) + . . . + f T ( x T ) . T h e r e f o r e , t f e h e u t i l i t y f u n c t i o n i s a d d i t i v e . S t r o n g s e p -3 a r a b i l i t y i s sometimes r e f e r r e d t o as ' a d d i t i v e s e p a r a b i l i t y ' . I t i s e a s y t o s e e t h a t s t r o n g s e p a r a b i l i t y i m p l i e s weak s e p a r a b i l i t y , b u t i s n o t i m p l i e d by weak s e p a r a b i l i t y . E x p a n d i n g e q u a t i o n ( 1 1 . 1 ) , we o b t a i n ( I I . 5) f . f ... = . f . f . . where f .-= - S u/ax^x f ™ = S f ^ / S x ? . l k i / k The e q u a t i o n (IT.5) r e v e a l s t h a t t h e a s s u m p t i o n o f s e p a r -a b i l i t y i n v o l v e s r e s t r i c t i o n s on t h e p a r t i a l d e r i v a t i v e s o f t h e u t i l i t y f u n c t i o n . I t , f u r t h e r i m p l i e s r e s t r i c t i o n s on t h e ( A l l e n p a r t i a l ) e l a s t i c i t i e s o f s u b s t i t u t i o n among c o m m o d i t i e s . We w i l l come back t o t h i s p o i n t s h o r t l y . The p r e c e d i n g d i s c u s s i o n on s e p a r a b i l i t y a p p l i e s e q u a l l y t o i n d i r e c t u t i l i t y f u n c t i o n s . I n s t e a d o f v e c t o r s o f commod-i t i e s , p r i c e v e c t o r s a r e u s e d a s a r g u m e n t s . However, i t s h o u l d be n o t e d t h a t i n g e n e r a l d i r e c t and i n d i r e c t s e p a r a b i l i t y do . 4 h o t i m p l y one a n o t h e r . - 19 -A u t i l i t y f u n c t i o n i s d e f i n e d t o be homo t h e t i c a l l y s e p a r -a b l e , i f t h e f u n c t i o n i s s e p a r a b l e and e a c h c a t e g o r y f u n c t i o n 5 i s h o m o t h e t i c . The u t i l i t y f u n c t i o n i t s e l f , however, d o e s n o t have t o be h o m o t h e t i c . W i t h t h i s d e f i n i t i o n , we have c o v e r e d enough t h e o r e t i c a l b a c k g r o u n d t o examine c o n s i s t e n c y o f t h e t w o - s t a g e m a x i m i z a t i o n p r o c e d u r e . ^ I f a u t i l i t y f u n c t i o n i s w e a k l y s e p a r a b l e , t h e n t h e n e c -e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a c o n s i s t e n t f i r s t s t a g e m a x i m i z a t i o n p r o c e d u r e i s t h a t t h e u t i l i t y f u n c t i o n be homo-7 t h e t i c a l f r y w e a k l y s e p a r a b l e . I n o t h e r w o r d s , i f U i s homo-t h e t i c a l l y s e p a r a b l e , t h e r e w i l l e x i s t a g g r e g a t e p r i c e and q u a n t i t y i n d i c e s f o r t h e c a t e g o r y f u n c t i o n s s u c h t h a t t h e p r o d u c t s o f p r i c e and q u a n t i t y i n d i c e s f o r a l l c a t e g o r i e s add up t o t h e b u d g e t c o n s t r a i n t . T h i s c o n d i t i o n c a n be ex-p r e s s e d as f o l l o w s . ( I I . 6 ) E g ™ ( p m ) . f m ( x m ) = y m = l , . . . , T , . m where gm i s t h e p o s i t i v e l y l i n e a r l y homogeneous (PLH) p r i c e TQ 8 i n d e x and f i s t h e q u a n t i t y i n d e x f o r t h e Tilth c a t e g o r y . T h e r e f o r e , t h e consumer m a x i m i z e s - h i s u t i l i t y w i t h r e s p e c t t o 1 T t h e c o m p o s i t e c o m m o d i t i e s , f ,...,£ , s u b j e c t t o t h e b u d g e t c o n s t r a i n t (11.6) e x p r e s s e d i n t h e a g g r e g a t e p r i c e and q u a n t i t y i n d i c e s . S i n c e g 1 L l i s PLH, we c a n n o r m a l i z e p r i c e s by d i v i d i n g them by t h e t o t a l b u d g e t . T h e n t h e b u d g e t c o n s t r a i n t becomes /• -p -j- „ _m, m« £m, m, , m „m / C I I . 7 ) I g (v ) - f Cx ) = 1 v = p / y , m m 9 where v i s a v e c t o r o f n o r m a l i z e d p r i c e s i n t h e mth c a t e g o r y . I t i s c l e a r t h a t a s u f f i c i e n t c o n d i t i o n f o r ( I I . 7 ) i s a l s o h o m o t h e t i c weak s e p a r a b i l i t y , s i n c e (IT.7) i s i m p l i e d b y ( I T . 6 ) . However, n e c e s s a r y c o n d i t i o n s h a v e n o t b e e n e s t a b l i s h e d . A s s u m i n g t h a t t h e f i r s t s t a g e m a x i m i z a t i o n i s p e r f o r m e d c o r r e c t l y , what t h e n a r e t h e n e c e s s a r y and s u f f i c i e n t c o n d i -t i o n s f o r t h e s e c o n d s t a g e a l l o c a t i o n t o be s o l v e d o p t i m a l l y ? " ^ I t t u r n s o u t t h a t t h e s e c o n d s t a g e m a x i m i z a t i o n p r o c e d u r e r e ^ q u i r e s an e v e n weaker c o n d i t i o n t h a n t h e f i r s t s t a g e . Weak s e p a r a b i l i t y o f t h e o v e r a l l d i r e c t ( o r i n d i r e c t ! u t i l i t y f u n c -t i o n i s b o t h n e c e s s a r y and s u f f i c i e n t f o r a c o n s i s t e n t s e c o n d -s t a g e m a x i m i z a t i o n . As was p o i n t e d o u t e a r l i e r , t h e a s s u m p t i o n o f s e p a r a b i l i t y i m p l i e s r e s t r i c t i o n s on t h e v a l u e s o f p a r t i a l d e r i v a t i v e s . S p e c i f i c a l l y , i t i n v o l v e s e q u a l i t y r e s t r i c t i o n s on t h e A l l e n p a r t i a l e l a s t i c i t i e s o f s u b s t i t u t i o n . B e r n d t and C h r i s t e n s e n ID-973a, 1 have shown t h a t g i v e n a w e a k l y s e p a r a b l e h o m o t h e t i c f u n c t i o n , t h e e l a s t i c i t y o f s u b s t i t u t i o n b e t w e e n two e l e m e n t s i n two d i s t i n c t g r o u p s i s t h e same f o r any p a i r f r o m t h e s e two g r o u p s . C o n s i d e r t h e i t h good i n t h e mth g r o u p and t h e k t h good i n the nth group. Then cr™£ = for any good jeR111 and any good heR n, where m ^ n. Furthermore, i f the category functions were l i n e a r homogeneous, i t can be shown that o-™£ = a m n , where a m n i s the p a r t i a l e l a s t i c i t y of substitution between the two groups, m and n. Russell E1975] extended the proposition to the non-homothe-t i c case. He showed that homothetic weak se p a r a b i l i t y i s suf-f i c i e n t , but not necessary, for the equality r e s t r i c t i o n s to hold. That i s category functions may be non-homothetic. I f the equality holds and, i n addition, i f the o v e r a l l function i s weakly separable, then the o v e r a l l function i s necessarily homo-t h e t i c a l l y separable. However, i t i s pointed out by Diewert Hl974c; 152-1533 that i n t h i s case we f i n d crm£ = a™ 1, . only i f the category functions are l i n e a r homogeneous. If the o v e r a l l function i s homothetic and strongly separ-able i n the p a r t i t i o n , t h e n the equality holds for any good j i n the mth category, and good h i n the nth category and the 1th good i n any of the remaining categories. That i s (II. 8) a ? J=-o™ j eR m h £R n l £ R ^ U R n I t i s noteworthy, however, that the strong s e p a r a b i l i t y analogue of the above equality does not 'go through' when the o v e r a l l function i s not homothetic. "Thus, unless the o v e r a l l - 22 -(production) function i s homothetic, i n which case the aggrega-tors can be chosen p o s i t i v e l y l i n e a r l y homogeneous, strong sep-a r a b i l i t y does not generate implications that are not obtained 12 with weak s e p a r a b i l i t y . " C. Four Methods" of Aggregation We now move on to a discussion of d i f f e r e n t methods of ag-gregation. Four methods of aggregation are examined here. These are: aggregation by homothetic s e p a r a b i l i t y , Leontief aggrega-t i o n , Hicks' aggregation and aggregation by s t a t i o n a r i t y . The f i r s t three methods are designed to reduce the number of goods involved, whereas the l a s t method attempts to reduce the number of unknown parameters involved. 1) Aggregation by Homothetic Separability I t has been stated that i f U i s weakly separable, then homothetic weak se p a r a b i l i t y i s necessary and s u f f i c i e n t for the existence of aggregate p r i c e and quantity indices for the cate-gories. In-addition, the a l l o c a t i o n of the t o t a l budget among the categories can be performed c o r r e c t l y . Therefore, the as-sumption of homothetic weak se p a r a b i l i t y of the d i r e c t or i n -d i r e c t u t i l i t y functions alone w i l l enable us to aggregate goods according to the p a r t i t i o n . This i s referred to as aggregation  by' homothetic s e p a r a b i l i t y . Note t h i s does not require homo-t h e t i c i t y of the o v e r a l l functions. Thus, i t i s possible to maintain a non-homothetic o v e r a l l function. Furthermore, homothetic weak se p a r a b i l i t y i s s u f f i c i e n t for the second-stage maximization fc© be carr i e d out consistently. If the o v e r a l l function i s homothetic, then weak separa-b i l i t y i s s u f f i c i e n t for both stages to be carr i e d out consis-tently, since a homothetic o v e r a l l function implies homothetic category functions (Lau[1969; 385]). An implication of homothetic s e p a r a b i l i t y i s that the (.category) income e l a s t i c i t i e s of a l l elementary commodities i n a l l categories are unitary. I f each of the categories i s taken to be one period, as i s i n the 'conventional' two-stage maximization procedure, then the re s u l t i n g unitary income 13 e l a s t i c i t i e s contradict h i s t o r i c a l evidence. Therefore, the conventional two-stage maximization procedure presents a ser-ious disadvantage, as long as the aggregation of commodities i s based upon the assumption of homothetic s e p a r a b i l i t y . The assumption of homotheticity for the category functions per se may not always be a problem, as long as the grouping i s '•• done properly. The problem with the 'conventional' p a r t i t i o n i s that i t includes luxuries and necessities i n the same group, i . e . , i n the same period. In t h i s case, i t i s unreasonable to assume that the composition of consumption i s independent of the l e v e l of income (or wealth). In.this respect, we believe that the conventional way of p a r t i t i o n i n g i s inappropriate. We suggest that the p a r t i t i o n be altered. What we propose to do i s to aggregate future goods, instead of aggregating goods in each period. By thi s i t i s meant that the same good i n a l l future periods (starting next period u n t i l the end of the l i f e span) i s aggregated into a single commodity with an aggregate price index of Pt and a guantity index 'Xt for the i t h good. Thus, there would be only two periods, present and future, each containing N goods. The weak s e p a r a b i l i t y assumption implies that the marginal rate of substitution of the same commodity between two d i s t i n c t future periods i s not influenced by the consumption of other goods i n either present or future periods. Homotheticity im-p l i e s unitary (category) income e l a s t i c i t y for each good. The aggregate price indices for categories may be calculated 14 by constructing u n i t cost functions for each category. How-ever, i n order to derive the unit cost functions (aggregate pr i c e i n d i c e s ) , we need to know the optimum a l l o c a t i o n of com-modities i n each group. Otherwise, i t i s impossible to obtain aggregate price indices. Nor i s the two-stage maximization possible. Consequently, our method of p a r t i t i o n (present vs* future) also turns out to be inappropriate, i f aggregation i s based on homothetic s e p a r a b i l i t y . Furthermore, i f we make a p r i o r i assumptions as to the optimum a l l o c a t i o n , then t h i s method w i l l collapse into the Leontief aggregation method, which w i l l be - 25 -discussed shortly. Hicks' aggregation based on price propor-t i o n a l i t y can also provide the future aggregate p r i c e indices without the knowledge of optimum a l l o c a t i o n within each category. Thus, i n order to formulate an empirically testable model based upon aggregation by homothetic s e p a r a b i l i t y , we have to p a r t i t i o n i n the conventional two-stage manner. However, the problem of unitary income e l a s t i c i t i e s may be eliminated by introducing the notion of 'precommitted expenditures' c^ for 15 a l l i and t . Thus, CII. 9) U** = F** (X* , . . . ,.X*) , where X* = f t ( x t - c 1) , x t and ay~ are N by 1 vectors; and X* i s a vector of aggregate quantity indices for a l l t = 1,...,T. Thus, homotheticity applies only to 'supernumerary' i n -16 come. This i s referred to as 'marginal homotheticity'. This way, the Engel curves no longer go through the o r i g i n , although they are l i n e a r . The income e l a s t i c i t i e s now can take on any values. These values, of course, are dependent upon the mag-nitude of the precommitted expenditures. Therefore, t h i s raises another problem: how to determine these l e v e l s . - 26 -2) L e o n t i e f A g g r e g a t i o n I f a g r o u p o f c o m m o d i t i e s a r e a l w a y s consumed i n f i x e d p r o -p o r t i o n s ( i . e . , q u a n t i t y p r o p o r t i o n a l i t y ) , t h e n we c a n t r e a t t h e b u n d l e as a s i n g l e commodity. T h i s i s r e f e r r e d t o as L e o n t i e f  a g g r e g a t i o n ( L e o n t i e f [ 1 9 3 6 ] ) . O b v i o u s l y , t h e s e c o n d s t a g e a l l o -c a t i o n p r o b l e m i s assumed away h e r e . I f q u a n t i t y p r o p o r t i o n a l i t y i s due t o t e c h n o l o g i c a l c o n d i t i o n s i n t h e U f u n c t i o n , t h e n t h e L e o n t i e f a g g r e g a t i o n m e thod may be c o n s i d e r e d a s a s p e c i a l c a s e 17 o f a g g r e g a t i o n by h o m o t h e t i c s e p a r a b i l i t y . G i v e n f i x e d p r o p o r t i o n s , i t i s now p o s s i b l e t o i m p l e m e n t - b u r / p a r t i t i o n i n g ( p r e s e n t v s . f u t u r e ) by a g g r e g a t i n g goods c o n -sumed i n f u t u r e by t h e L e o n t i e f a g g r e g a t i o n method. T h i s method i s d e s c r i b e d b y a demand e q u a t i o n f o r t h e j t h commodity. CII.10) Xj = p^X* f o r t = 2,...,T, j = i , . . . , N , * t where i s a p o s i t i v e c o n s t a n t , X t t h e a g g r e g a t e q u a n t i t y i n -dex o f t h e g r o u p . T t The a g g r e g a t e p r i c e i n d e x i s P* = E.p.p.. 3 t=2 3 3 The q u a n t i t y demanded o f t h e j t h good i n t h e p e r i o d t i s a f i x e d p r o p o r t i o n o f X*. T h i s i m p l i e s t h a t w i t h i n t h e g r o u p , t h e ' e x p e n d i t u r e ' e l a s t i c i t y i s u n i t a r y f o r a l l e l e m e n t a r y g o o d s . An a d v a n t a g e o f t h i s a g g r e g a t i o n method i s t h a t i t a l l o w s us t o t a k e t h e d e m o g r a p h i c a s p e c t i n t o c o n s i d e r a t i o n i n a s i m p l e , manner. F o r example, u s i n g t i m e s e r i e s d a t a , t h e p o p u l a t i o n - 27 -under consideration faces d i f f e r e n t l i f e expectancies i n d i f f e r -ent periods. The l i f e expectancy of a Japanese boy of 15 years of age i n 1921 to 1925 was 42.3 years, while i t was 53.1 years 18 i n 1955.- Accordingly, one would most l i k e l y incorporate t h i s d;emog^ a$hu\c:i consideration i n planning one's future consumption. Since the prices of the aggregated future goods are the weighted sum of the expected future prices over the expected l i f e span, a d i f f e r e n t l i f e expectancy can be used each year as the plan-ning horizons That i s , T may be d i f f e r e n t . Thus, the aggregate p r i c e index w i l l take'the changes i n demographic d i s t r i b u t i o n into account through the changes i n l i f e expectancy. 3) Hicks' Aggregation Another approach to a consistent two-stage maximization pro-cedure i s based upon the behavior of independent variables rather than on the form of the u t i l i t y function. I f a l l p r i c e changes are proportional within a group of commodities, then such a group can be treated as a single commodity. This i s referred to as' Hicks' aggregation theorem (Hicks[1946jl) . Green[1964; 313 points out that i n t h i s case "there i s no reason to suppose, that i n the absence of pr o p o r t i o n a l i t y , the group would be weakly separable." Thus, t h i s i s a form of aggregation which does not involve any s e p a r a b i l i t y condition i n the u t i l i t y function. How-ever, the second-stage maximization presumably cannot be performed i f the u t i l i t y function i s not weakly separable. - 28 -If we were to aggregate future goods, Hicks' aggregation through a 'price p r o p o r t i o n a l i t y 1 assumption requires that the price of the i t h good i n the period t i s equal to P^P^f where p^T i s a p o s i t i v e constant, and Pt i s the aggregate p r i c e index 3 3 T T for the i t h group. Consequently, p^ = p^Pt,.../p^ = P^ p t . For any change i n P t , each price changes i n the same propor-2 t i o n . I f we are to take p^ as Pt (static.expectations) then the quantity index Xt i s obtained i n the following way: -3 T 2 Pj 3 P-! T CII.11J. X* = xf + - | x^+. . .+-| x! • p. p. 1 Let r*" be the expected one. period i n t e r e s t rate for period t and e^ T the rate of expected p r i c e change i n period t for the i t h good. In addition, l e t the expected discounted price i n the future periods be s o l e l y a function of the e^ and r f c and the current p r i c e . Then, the present value of the price of the fu-ture goods i n the tth period: . .(l.+.ej).(1+e2).......Cl+.e^"1) (11.12) p\ = ^ - i : r r r - p j , t = l , . . . , T , 1 (1+r) (1+r) • • • (1+r X) 1 i = 1, . . . , N. It i s clear from the above that any change i n e^, r"^, p^, w i l l change the prices i n periods 2 to T proportionally, since they are contained i n each equation. However, any o t h e r c h a n g e , i . e . , i n e ^ , . . . , r , . . . , r w i l l n o t r e s u l t i n p r i c e p r o p o r t i o n a l i t y . T h e r e f o r e , i n o r d e r t o m a i n t a i n t h e p r i c e p r o p o r t i o n a l i t y a s s u m p t i o n , we must f i r s t o f a l l assume t h a t : i). t h e e x p e c t e d f u t u r e i n t e r e s t r a t e s ( i . e . , l o n g - t e r m i n t e r e s t r a t e s ) r e m a i n c o n s t a n t and a r e n o t i n f l u e n c e d by any change i n t h e s h o r t - t e r m i n t e r e s t r a t e , i i ) t h e r a t e o f e x p e c t e d f u t u r e p r i c e change i s n o t i n -f l u e n c e d by t h e r a t e o f change i n p r i c e s i n t h e c u r -r e n t p e r i o d . Of t h e two, t h e s e c o n d a s s u m p t i o n may- r a i s e a p r o b l e m , un-l e s s t h e c u r r e n t change c a n be c o n s i d e r e d a s a t r u e s h o r t - r u n f l u c t u a t i o n , r a t h e r t h a n an i n d i c a t i o n o f t h e f u t u r e t r e n d . T h u s , l e t P* = ( l + e 1 ) ? 1 / ( 1 + r 1 ) . Then, t h e t a b l e b e l o w summarizes t h e p r i c e s o f goods i n e a c h p e r i o d . 1 2 3 T 1 1 I (1+e 2) ' ( 1 + e 2 ) , . . . , ( 1 + e ? - 1 ) 2 1 (1+r ) L ,, . 2, .. , T - l ) *1 (1+r ) , . . . , (1+r... N P N (1+e 2) .^l+4)' • ' ( 1 + e N _ 1 ) r + N ( 1 + r 2 ) N (1+r ) , . . . , (1+r ) 1 e^ may be t h e same f o r a l l i o r may be d i f f e r e n t f o r d i f -f e r e n t goods i n t h e t t h p e r i o d . A l i m i t i n g c a s e i s when we assume s t a t i c p r i c e e x p e c t a t i o n s , i . e . , e^ = 0 f o r a l l i and t . - 30 -Note, t h a t b y t h i s f o r m u l a t i o n , a l t h o u g h t h e r e l a t i v e p r i c e s t r u c t u r e may r e m a i n t h e same, t h e q u a n t i t y demanded o f e a c h e l e m e n t a r y good do e s n o t n e c e s s a r i l y c h ange p r o p o r t i o n a l l y . The ( c a t e g o r y ) income e l a s t i c i t i e s a r e n o t r e s t r i c t e d t o u n i t y when we assume a n o n - h o m o t h e t i c - o . v e r a l l f f u n c t i o n . I f one c a n make a h e r o i c a s s u m p t i o n t h a t t h e r e l a t i v e p r i c e s move p r o p o r -t i o n a l l y i n e a c h p e r i o d , t h e n t h e ' c o n v e n t i o n a l ' ( i . e . , where we a g g r e g a t e o v e r goods w i t h i n a t i m e p e r i o d ) p a r t i t i o n i n g c a n be a d o p t e d w i t h o u t t h e p r o b l e m o f u n i t a r y income e l a s t i c i t i e s , a I t s h o u l d be n o t e d t h a t t h e H i c k s ' a g g r e g a t i o n m ethod c a n n o t a c c o u n t f o r dynamic d e m o g r a p h i c c h a n g e s . P* d o e s n o t i n v o l v e T, s i n c e P* i s t h e i n d e p e n d e n t v a r i a b l e and X* i s t h e d e p e n d e n t v a r i a b l e . F u r t h e r m o r e , P* i s a r b i t r a r i l y c h o s e n s u c h t h a t P|X| = Zp^x^. C o n s e q u e n t l y , we c a n n o t accommodate t h e d e m o g r a p h i c f a c t o r i n t h i s f o r m o f a g g r e g a t i o n . 4} A g g r e g a t i o n b y S t a t i o n a r i t y T h i s m e thod a t t e m p t s t o r e d u c e t h e number o f p a r a m e t e r s , " h o t by a g g r e g a t i n g g o o d s , b u t b y p l a c i n g a p r i o r i r e s t r i c t i o n s o n t h e v a l u e s o f p a r a m e t e r s . An a d v a n t a g e o f t h i s method i s t h a t i t e n a b l e s us t o t e s t h y p o t h e s e s o f f u n c t i o n a l s e p a r a b i l i t y . F u n c t i o n a l s e p a r a b i l i t y i mposes r e s t r i c t i o n s on t h e v a l u e o f e l a s t i c i t i e s o f . s u b s t i t u t i o n (ES) w h i c h , i n t u r n , impose r e -s t r i c t i o n s on t h e v a l u e s o f c r o s s - c o m m o d i t y p a r a m e t e r s . - 31 -We s t a r t with TN goods and (TN+l)TN/2 A l l e n p a r t i a l e l as-t i c i t i e s of substitution a... LetiZ be a TN by TN dimensional ID symmetric matrix of ^. Then, p a r t i t i o n t h i s matrix into T by T submatrices, each of which has N by N elements. A sub-mn 2 matrix Z represents the mnth submatrix which has N elements JT nin ^ - i n of cr^ k, where i,k = 1,...,N. It has been shown .earlier that homothetic s e p a r a b i l i t y -, • .. mn mn _ ,, _m , , ^n , , _, implies = a j h ' for a l l i,jeR ; h,keR and m f- n. These r e s t r i c t i o n s can be translated into equality r e s t r i c t i o n s on the values of the corresponding cross-commodity parameters for the base period. Linear homogeneous category functions imply that = a m n . Strong s e p a r a b i l i t y i n the case of homothetic o v e r a l l function -i • ., . mu nu implies that a . , = - a . , . ^ i l -jl Thus, both homothetic.weak .separability and l i n e a r homoge-neous category functions reduce TN(TN+1)/2•cross-commodity para-meters to (T-l)T/2 + TN(N+l)/2. If the o v e r a l l function i s homo-thet i c , then strong s e p a r a b i l i t y results i n TN(N+3)/2 parameters. Note that the main diagonal submatrices within the. same periods) are not affected by these s e p a r a b i l i t y hypotheses. In order to perform, a ' p a r t i a l ' test of.functional s e p a r a b i l i t y , a simplifying assumption of 'st a t i o n a r i t y ' i s made to reduce the number of parameters. More s p e c i f i c a l l y , i t i s designed to reduce the number of submatrices to be estimated,.thus reducing parameters. S t a t i o n a r i t y assumes that taste remains the same, - 32 -i n t h e s e n s e t h a t t h e ES between a g i v e n p a i r o f goods r e m a i n s t h e same when t h e p a i r o f goods a r e s e p a r a t e d by a common num-b e r o f p e r i o d s . T h i s a p p l i e s t o a l l g o o d s , and t h e number o f p e r i o d s o f s e p a r a t i o n r a n g e s f r o m 1 t o T - l . Of c o u r s e , t h e ES may s t i l l d i f f e r b etween d i f f e r e n t p a i r o f g o o d s . A more d e t a i l e d d e s c r i p t i o n o f t h i s method r e q u i r e s t h e d e f i n i t i o n o f a p s e u d o e l a s t i c i t y o f s u b s t i t u t i o n . F u r t h e r m o r e , f u n c t i o n a l f o r m s h ave t o be i d e n t i f i e d . T h e r e f o r e , we s h a l l p o s t p o n e t h i s d i s c u s s i o n u n t i l C h a p t e r I I I . 5) A C o m p a r i s o n o f D i f f e r e n t A p p r o a c h e s I n t e r m s o f t h e t h e o r e t i c a l and o p e r a t i o n a l i m p l i c a t i o n s , t h e f o u r a p p r o a c h e s t o a g g r e g a t i o n a r e by no means e q u i v a l e n t . P r i c e p r o p o r t i o n a l i t y ( H i c k s ' a g g r e g a t i o n ) d o e s n o t r e s t r i c t t h e f u n c t i o n a l f o r m f o r t h e c a t e g o r y f u n c t i o n s , w h i l e q u a n t i t y p r o p o r t i o n a l i t y ( L e o n t i e f a g g r e g a t i o n ) assumes f i x e d p r o p o r t i o n s ' f o r t h e f u n c t i o n . Of c o u r s e , a g g r e g a t i o n by h o m o t h e t i c s e p a r -a b i l i t y i m p l i e s h o m o t h e t i c c a t e g o r y f u n c t i o n s . A c c o r d i n g l y , t h e l a t t e r two p r o d u c e u n i t a r y income e l a s t i c i t i e s o f demand w i t h r e s p e c t t o t h e ("supernumerary") g r o u p e x p e n d i t u r e , w h e r e a s H i c k s ' a g g r e g a t i o n a l l o w s f l e x i b l e e l a s t i c i t i e s . The e x p e c t e d f u t u r e p r i c e s o f t h e e l e m e n t a r y goods a r e computed i d e n t i c a l l y f o r e a c h method, however, t h e a g g r e g a t e p r i c e i n d i c e s f o r e a c h method a r e n o t t h e same. P r i c e i n d i c e s a r e s e l e c t e d a r b i t r a r i l y i n t h e c a s e o f p r i c e p r o p o r t i o n a l i t y , and t h e q u a n t i t y i n d i c e s w i l l a d j u s t i n o r d e r -to s a t i s f y t h e - 33 -budget constraint. Aggregate prices are derived by a weighted sum of the expected future prices i n the case of quantity pro-p o r t i o n a l i t y , weights being the fixed proportions. On the con-trary, the homothetic s e p a r a b i l i t y method requires that the unit cost functions be estimated i n the preliminary stage. Leontief's aggregation method gives r i s e to a r b i t r a r i n e s s , since i t i s assumed that the second stage was solved before-hand, i . e . / the quantities consumed were fixed proportions re-gardless of r e l a t i v e p r i c e s . But then, how does one decide on these 'fixed proportions'? An advantage of t h i s approach i s that i t can incorporate dynamic changes i n demographic d i s t r i -bution into the analysis of i n d i v i d u a l demand functions. In,contrast to these three methods,. the ' s t a t i o n a r i t y ' approach does not attempt to aggregate commodities at a l l . Instead i t enables us to impose a p r i o r i , r e s t r i c t i o n s on the unknown, parameters. F i n a l l y , we should note that the expected future prices depend upon the kind of p r i c e expectations we adopt, which may be s t a t i c or dynamic. D. '^Aggregation oyer" People Given expenditure functions (p^x..) of a 'representative' consumer (household). , we attempt to derive 'aggregate' (market), expenditure equations f o r the whole economy. The following assumptions' are made i n order to aggregate over people: - 34 -i) Everybody possesses the i d e n t i c a l u t i l i t y function, ii). Everybody faces the same prices, i i i ) The expenditure d i s t r i b u t i o n i s known. Let tyCy) be the frequency function of the expenditure d i s -t r i b u t i o n . Then, Ity (y)dy =-1. The aggregate expenditure (Y^) function for the i t h good i s calculated as follows: &T.13) Y ± = Q?p 1x i(y,p ) i(;(y)dy where p^x^ i s the i n d i v i d u a l expenditure on good i , and Q i s the number of independent decision making units i n the economy. Al t e r n a t i v e l y , the average aggregate expenditure equations are defined as follows: oo (11.14) y ± = Y±/Q = ^(p 1x j , ) i J J(y)dy The aggregate expenditure share (S^) equation for the i t h good i s : S". ^ Y./Y ^:(Pixi)-4)(y)dyy (11.15) = y * ^ P i x i ) - ^ y ) d y = k* £ C v i x i } Y ^ ( y } d y y * = Y / @ v i = p i / y - 1 0 0 . 5 5 p^s iCp,y)y*(y)dy - 35 -where Y i s the t o t a l expenditure for the whole economy, y* i s the mean expenditure and i s the normalized price for good i and s^ i s the expenditure share of the each i n d i v i d u a l . Given s p e c i f i c functional forms for x^ and ty Ly) , i t w i l l be possible to evaluate the aggregate functions. The next chapter w i l l derive e x p l i c i t aggregate functions, using the translog and the Generalized Leontief functional forms. E. ' Treatment of Durable Goods In developing a model of consumer theory, consumer dur-ables deserve special attention. Suppose we define consumer durables as goods which are not completely 'used up' i n the act of consumption or depreciated by being held i n one period. Then demand for new consumer durable goods w i l l be influenced by the past purchases and thus by the exis t i n g stock l e v e l . Furthermore, the present stock and new purchases w i l l , i n a depreciated form, be carried over to the next period and a f f e c t future consumption. The c l a s s i f i c a t i o n of goods into nondurables and durables, of course, depends upon the d e f i n i t i o n of the unit period and €herd repreeiatiohera€efiofheachmcommodity. An intertem-poral consumer model i s , therefore, e s p e c i a l l y suitable for trea t i n g the inherent intertemporal nature of consumer durable goods. One of the simplifying assumptions i n modelling the demand for consumer durables i s that the goods are p e r f e c t l y d i v i s i b l e . Another simplifying assumption i s that a depreciated consumer durable good i s considered equivalent to the new good except that i t i s smaller i n quantity. I f the rate of depreciation i s constant, then the goods decay exponentially (reducing-balance depreciation). What i s most relevant to each consumer may be the quantity 'available for use'. This consists of two parts: the stock car r i e d over from the previous periods and the new a c q u i s i t i o n in the current periods.. In l i g h t of t h i s observation, the con-sumer choice problem may be viewed i n two ways: the 'acquisi-tion-goods p r i c e ' approach and the 'availability-~rent' approach These two approaches are equivalent, i f perfect second markets e x i s t . The acquisition-goods p r i c e approach focuses on the relationship between stock and a c q u i s i t i o n of consumer durables since new a c q u i s i t i o n depends upon the past a c q u i s i t i o n , i . e . , stock. The t y p i c a l examples of t h i s approach are the stock adjustment model by Stone and Rowe[1957j;|?1960 J and the state adjustment model by Houthakker and Taylor[1970]. The'cavailablility-^rent approach emphasizes the flow of services derived from the quantity availabflef for use and at-tempts: tootreat durables i n an i d e n t i c a l way as nondurables. Obviously we would have to construct appropriate 'user cost' prices for the durables for t h i s approach to be operational. The a v a i l a b i l i t y - r e n t approach turns out to be p a r t i c u -l a r l y suitable for our model of intertemporal a l l o c a t i o n of the Fisher-Hicks-type, since we can treat durables i n an iden-t i c a l manner as non-durables, without having to assume addition-a l behavioral assumptions as to the consumer's purchase pattern of durable goods. Rental prices, therefore, instead of pur-chase prices w i l l be used as price variables for durables. The a v a i l a b i l i t y - r e n t approach focuses on the consumption side rather than the a c q u i s i t i o n side of demand for durable goods. In each period, a consumer receives a flow of services from his stock of durables. He w i l l be i n d i f f e r e n t as to whe-ther he owns the durables or rents them, assuminggperfect second-hand markets. If the consumer purchased a new durable good, we must decompose the purchase into two parts; the rental (consumption) and the investment. Whatever i s not consumed through depreciation i s available for l a t e r use or can be sold i n the next period. Thus i t i s an investment i n an asset. The relevant p r i c e for r e n t a l of consumer durable goods, i . e . , user cost or rental p r i c e may be calculated as follows. F i r s t we assume that the consumer purchases a good at the be-ginning of the period t at p r i c e p ( t ) , and then s e l l s the re-mainder at the beginning of the next period for p(t+l) (pos-s i b l y to himself) . Since the non-consumption part is-' consid-ered to be an investment, we assume that he receives an average rate of return r ( t ) on the asset. Letting W as' the depreciation rate per period and p(t) as the r e n t a l p r i c e , we obtain the following r e l a t i o n s h i p for the investment part. - 38 -(11.16) investment = p C t ) - p (t) = ( (11.17) pCt) = p(.t) - ( i+ r "( t j ) P ( j : + 1 ) , : Thus the rental p r i c e i s the difference between the pur-chase price i n the current period and the discounted s e l l i n g price of the depreciated good. Note i f we expect the next period price to be equal to the current one, we obtain CIT.18) p e t ) = P(t) g f f i f f i On the other hand, i f we expect the price i n the next per-iod to be, e% higher (or lower) than the current p r i c e , then Cri.19); p(t) = P(t) ' where r*Ct) i s defined as the r e a l r a t e of i n t e r e s t and r * (t) l+r(,t) 1+eCtJ. ~ 1 ' - 39 -FOOTNOTES to Chapter II 1. See Green[1964] . 2. For a more complete discussion, see Blackorby, Primont and Russell[1975] . \ 3. Strotz[1959; 485]. 4. A special case i s that of homothetically separable d i r e c t and i n d i r e c t functions. In t h i s case, s e p a r a b i l i t y of the d i r e c t function implies and i s implied by the corresponding i n d i r e c t function. See Lau[l969]. 5. A function i s homothetic, i f i t can be written i n the form U = F[f(x^,...,x ) ] , where F i s a p o s i t i v e , f i n i t e contin-uous and s t r i c t l y monotonically increasing function of one variable with F(0) = 0, and f i s homogeneous function of m variables. Lau[1969; 377]. 6. "Aggregation w i l l be said to be consistent when the use.of information more detailed than that contained i n the aggre-gate would make no difference to the results of the analysis of the problem at hand." Green[1964; 3]. 7. If we do not assume a weakly separable u t i l i t y function to begin with, the necessary condition i s not known (Blackorby, Primont and Russell[1975; 34]). 8. (.11.6) i s referred to as 'strong additive price aggregation.' See Blackorby, Primont and Russell[1975; 32]. 9. (II.7) i s referred to as 'weak additive price aggregation.' Ibid. 10. If the consumer can carry out the second-stage maximization with information only about i t s category expenditure and prices of the elementary goods (without the prices.of goods i n other categories, then the .consumer's preferences are said to be 'strongly decentralized' (Blackorby, Primont and Russell[1975; 35]). Any change which takes place outside the category under consideration affects a l l o c a t i o n within, the category only through a scalar, i . e . , the category ex-penditure. S i m i l a r l y , weak d e c e n t r a l i z a b i l i t y i s defined in terms of normalized prices and expenditure shares. 11. Then the function i s homothetically strongly separable (LauI1969; 385]). 12. Blackorby and Russell[1974; 7], FOOTNOTES to Chapter II (con't) 13. Mizoguchi[1970] has calculated both time-series and cross-section income e l a s t i c i t i e s for Japan and has shown that the income e l a s t i c i t i e s varied widely from goods to goods and time to time. Houthakker[1957] also 'confirmed' Engle's law i n a study of international comparison of household expenditure patterns. 14. For methods of constructing unit cost functions, see Blackorby, Nissen and Russell[1970] and Diewert[1974; 152]. 15. In order to estimate econometrically the unit cost function corresponding to f , one may assume a l i n e a r homogeneous cost function using a s p e c i f i c functional form. Then the expenditure share functions derived using Shephard's Lemma (Diewert[1974c; 112]) w i l l be l i n e a r i n the unknown parame-ters, thus simplifying the estimation procedure. The ag-gregate price index can be constructed using the parameter estimates and p r i c e data. The aggregate quantity index i s determined such that the product of two indices exhausts the t o t a l expenditure. 16. See Afriat[1972] on the concept of marginal homotheticity. 17. E.R. Berndt has pointed out that Leontief aggregation may be made possible due to market rather than technological conditions (in consumption behavior). I f the market forces fixed proportions upon the consumer, i t can be seen that relevant goods need not be homothetically separable. In t h i s case, Leontief aggregation i s not a special case of homothetic weak s e p a r a b i l i t y . However, t h i s s i t u a t i o n i s only conceivable for a Robinson Crusoe economy where there i s only one consumer. Since t h i s i s not what we deal with, we w i l l treat Leontief aggregation as i f i t i s a special case of homothetic s e p a r a b i l i t y . 18. Ohsat©;B9.66;;.; 17].;.r w.«- c .. : "j 19. The terminology i s due to Martin[1973]. - 41 -Chapter III THE PROPOSED MODELS A. Introduction In the l a s t chapter, the necessary t h e o r e t i c a l material relevant to th i s study has been outlined. S p e c i f i c a l l y , d i s -cussions on functional s e p a r a b i l i t y , d i f f e r e n t aggregation meth-ods over goods, aggregation over people and the treatment of durables were presented i n a general neoclassical framework. In t h i s chapter, we f i r s t introduce two s p e c i f i c functional forms: the translog function and the GeneralizedL'Leontief . func-t i o n . The properties of these functional forms are examined and then the estimating equations are sp e c i f i e d for these func-t i o n a l forms. Secondly, price e l a s t i c i t i e s , income e l a s t i c i t i e s and e l a s t i c i t i e s of substitution are derived. In addition, the con-cept of the pseudo e l a s t i c i t y of substitution i s introduced. Thirdly, a further discussion on the four methods of aggre-gation over goods i s presented i n r e l a t i o n to the s p e c i f i c func-t i o n a l forms. F i n a l l y , we define the models to be used for our empirical estimation. B. Functional Forms Instead of following the conventional method of deriving demand functions from the d i r e c t u t i l i t y function, our approach u t i l i z e s the i n d i r e c t u t i l i t y function which, by Roy's Identity [1947; 222] s i m p l i f i e s the derivation a great deal. Duality between the d i r e c t u t i l i t y function and the i n d i r e c t u t i l i t y function has been discussed extensively by Hanoch[1970], Lau [1969] and Shephard[19<70] . Furthermore, Diewert[1974c] has shown that the inverse i n d i r e c t u t i l i t y function h(v) defined as the reci p r o c a l of the i n d i r e c t u t i l i t y function presents a ' s u f f i c i e n t ' s t a t i s t i c for the d i r e c t u t i l i t y function F(x), provided that the following r e g u l a r i t y conditions are satisfied."*" (III.l) Regularity conditions: i) F(x) i s continuous and f i n i t e for x>>0N. i i ) F(x) i s non-decreasing for x>>0N. i i i ) F(x) i s quasiconcave for x>>0N. The condition i i ) w i l l be referred to as the 'monotonicity' condition and the condition i i i ) as the 'curvature' condition. Moreover, h(v) exhibits the i d e n t i c a l c h a r a c t e r i s t i c s as F(x), whereas the i n d i r e c t u t i l i t y function does. not. Thus, i t i s convenient to use the inverse i n d i r e c t u t i l i t y function, since i t s properties are more f a m i l i a r . Given that the inverse i n d i r e c t u t i l i t y function h(v) i s at least once d i f f e r e n t i a b l e , Roy's Identity enables us to de-rive a system of demand equations by d i f f e r e n t i a t i n g h(v) with respect to each commodity p r i c e . Thus, the solution to the T u t i l i t y maximization problem max{F(x) : v x < 1, x > OJJ} i s ex-pressed as follows: - 43 -x(v) = Vh(v)/v TVh(v) where x i s a vector of consumptions, v i s a vector of normalized prices and Vh(v) i s the gradient vector of h(v). For the inverse i n d i r e c t u t i l i t y function, we assume two functional forms, the translog function due tbeChristensen, Jorgenson and Lau[1973] and the Generalized Leontief function proposed by Diewert[1971a]. These functional forms do not a p r i o r i assume either a d d i t i v i t y or homotheticity. These l a t t e r r e s t r i c t i o n s l i m i t the properties of model severely. For example, d i r e c t a d d i t i v i t y rules out i n f e r i o r goods as well as complemen-tary goods. Indirect a d d i t i v i t y r e s t r i c t s the relationships 2 among cross price e l a s t i c i t i e s . In t h i s respect, the two func-t i o n a l forms we chose'are more a t t r a c t i v e than some of the more popularly used forms such as the l i n e a r expenditure system, the d i r e c t addilog system (Houthakker[1960j), the Rotterdam model. (Theil[1965]) and the i n d i r e c t addilog system (Houthakker[1960j). Furthermore, they are ' f l e x i b l e ' enough to provide a second order approximation to any a r b i t r a r y twice d i f f e r e n t i a b l e i n -verse i n d i r e c t u t i l i t y function at a given set of p r i c e s . 1. The Translog Inverse Indirect U t i l i t y Function This function i s of the form ; (III. 2) l n h (v) = ct 0 + Ha, l n v . + -^ EEY/ • • lnv . lnv . i , j = l , . . . ,N. i 1 1 i j 1 3 1 3 - 44 -The function i s l i n e a r homogeneous when Za. = 1, y. • = Y - -and Ey-- =0 for every i . S i m i l a r l y , i t i s non-homogeneous i f j ± J either one or both of Ea. 4 1 and Zy. . ¥• 0 hold. l j If every y^j equals zero, we have the usual Cobb-Douglas function. I f , i n addition, a l l > 0, then the function sat-i s f i e s the r e g u l a r i t y conditions, g l o b a l l y . Of course, i f some parameter y^ ^ 0, then h (v) w i l l not s a t i s f y the r e g u l a r i t y conditions g l o b a l l y . But we can look for a neighbourhood of prices where the conditions are s a t i s f i e d . Therefore, the trans-log functional form may provide a good 'local approximation' for both => homothetic and non-homothetic inverse i n d i r e c t u t i l i t y functions. By Roy's Identity, the system of expenditure shares i s : a. + Ev..lnv. l ' ij j (III.3) s. = v.x. = — — 1 for i , j = 1,...,N. Za. + EEY- lnv D D™ Three normalizations are made: N N i) E E = 0. i=l j=i 1 - l N i i ) £ a. = 1. i= l 1 i i i ) a 0 = °« The reason for these particular forms of normalization is given below. Using the normalizations, the aggregate expendi-ture share equation for the ith good is derived as: S . = Y./Y 1 I ' / s ± (p,y)y^(y)dy (III.4) a . + Zv..lnP.. - l n y Z v . . 0 0 i ID ~ J . . i j = / — J = J y ^ ( y ) d y ° laK + 5^kmlnPm ~ l n y ^ k m k km km a i + E Y i i l n P i ~ 9 l n y * E Y ; L I  j J J j 1 + E Z Y , InP , ' km m km where y* = /y^(y)dy is mean incomer k,m = 1,...,N> (III. 5 ) and 0 = fylnyty(y)dy/y*lny*. 6 i s referred to as the expenditure distribution variable. If every individual had the mean income, then 6 = y*lny* / i j j (y) dy/ y*lny* = 1 . If the u t i l i t y function is homothetic, the level of income is irrelevant in determining individual.consumption and thus the expenditure distribution pattern of the economy also becomes irrelevant in determining the aggregate level of consumption. This is what we have here, for i f lnh(v) is homothetic, then Ey^j = 0 « Thus the term involving the income distribution variable vanishes. As long as the integral for 0 can be evaluated, a nonlin-ear regression technique can be used to estimate the unknown parameters and y^j f° r i / j = 1,...,N. The f i r s t of the three normalizations was made in order to eliminate the term lnyXZy, in the denominator. Jxn'km km Note that (III.2) is homogeneous of degree zero in.the parameters and y^ _. . Thus in order to determine these para-meters by an econometric estimation, i t is necessary to impose at least one normalization on the values of parameters. The normalization i i ) proves to be extremely convenient for this purpose.. Since CXQ does not appear in the equation to be estimated and thus cannot be estimated from data, the third normalization CXQ = 0 is added. The three normalizations together make h(v) homogeneous of degree one along the rays of equal normalized prices; For ex-ample, lnh (v) = 1, when v^ = 1 for i = 1,...J>,N. lnh(v) = Ink, when v. = k for i = 1,...>N and k is a constant. - 47 -Therefore, the three normalizations imply a convenient c a r d i n a l i z a t i o n of u t i l i t y (or r e a l income). 2. The Generalized Leontief Inverse Indirect U t i l i t y Function The inverse i n d i r e c t u t i l i t y function i s of the form: (III.6) h(v) = E E ^ . ( v ± v . ) ^ + £ e o i l n v . + B Q 0 i , j = 1,...,N, i j J J i where £.. = £... ID ] i I f = 0 for a l l i , then h(v) reduces to a homothetic function. And i f , in. .addition, = 0 for a l l i ^  j , then F(x) reduces to the fa m i l i a r fixed proportions Leontief function, and Mvfriw.il.le.be A i n d i r e c t l y ' additive. 1 I f a l l 8.. > 0 with at least one B . . > 0, then h(v) defined above s a t i s f i e s the r e g u l a r i t y conditions g l o b a l l y . However, i f not a l l >_ 0 then h (v) may generate a l o c a l approximation in a neighbourhood of prices where the conditions are s a t i s f i e d . Applying Roy's Identity, we obtain derived demand functions x^ ^ (v) : Eg.. (v^vT 3 5) + £ n .vT 1 . i j j r Oi i (III.7) x.(v) = J p i , j = 1,...,N. ^ B k j ( v k v j ) 2 + = B 0 j The i n d i v i d u a l expenditure share functions are written as: f i j ( v i v j ) J s + 3 0 i (III. 8) s. = E V . X . = 1 5- . A normalization such that i) N Z 3 i = l Oi = 0 i s imposed to simplify the denominator. The aggregate expenditure share equation for the i t h good i s derived as follows: We need an additional normalization i n order to determine the values of unknown parameters, since the share equations are homogeneous of degree zero i n the parameters 3^j and B Q ^ for i , j = 1,..,N. Consequently, the second normalization i s of the form: The parameter 3QQ cannot be i d e n t i f i e d , since, i t does not appear i n the estimating share equation. Thus, we set ( I I I . 9 ) S ± = Y ± / Y X3..(p.p.) 2 + 3 n.e*y l j r a r j Oi J oo 2 2 where 0* = fy ^(y)dy/(y*) . i i i ) 3 00 = 0. - 49 -The e f f e c t of these three normalizations i s to make h(v) homogeneous of degree one along the rays of equal normalized prices as i n the case of the translog function. Thus, h(v) = 1 when v^ = 1 for i = 1,...,N. h(v) = k when v. = k for a l l i = 1,...,N and k i s a constant. The inverse i n d i r e c t u t i l i t y function i s homothetic i f and only i f = 0 for a l l i , i = 1,...,N. In th i s case, the consumption behavior of the consumers does not depend upon the l e v e l of income, and furthermore, aggregate expenditure i s inde-pendent of the expenditure d i s t r i b u t i o n i n the economy. Indeed, the income d i s t r i b u t i o n variable 0* drops out when 3Q^ = 0 . Another simplifying assumption often made i n empirical studies i s that the consumers have i d e n t i c a l mean income. Then the variance becomes zero and 0* becomes unity. C. E l a s t i c i t i e s In t h i s section, we define relevant e l a s t i c i t i e s . These are the own and cross (uncompensated) pr i c e e l a s t i c i t i e s r ^ j ) , the income e l a s t i c i t i e s • ("n^  ) , the A l l e n p a r t i a l e l a s t i -c i t i e s of substitution (c^j) and the pseudo e l a s t i c i t i e s of substitution (a..). R e c a l l . h... x . ( v ) = — : i , k = 1,...,N, k , , , , 3h (v) where v. = p./y and h. = i * i ' J 1 3v. 1 Then, t h e c r o s s - p r i c e e l a s t i c i t y f o r t h e . i t h good w i t h r e s p e c t t o a change i n t h e p r i c e o f t h e j t h good i s d e f i n e d a s : Ev n_h ( I I I . 10) n . • = 1 = 3 ± 3 - 3 3 -3 l n x . v. .h. . v . h . j . k k j -1 - 1 - 1 X 1 3 3 l n p . h. Z v k h k E v k h k S i m i l a r l y > t h e o w n - p r i c e e l a s t i c i t y f o r t h e i t h good f o l l o w s f r o m t h e aboveeequation^-when i = j . -u v- v. Ev, h. . 3 l n x . v . h . . v . h . i , k k i , T T T -i -i \ _ i _ l n l l k  ( I I I . 11) n • • = = - ~ • 1 1 3 l n p . h. Ev, h, Ev, h. F u r t h e r m o r e , t h e income e l a s t i c i t y f o r t h e i t h good i s ., ZEv, v Eh., v, •3lnx. , k km m , l k k (III.1 2 ) T I . = — = 1 + — - . i y 3 l n y 2 v.h. h. k k k i . When h(v) i s a h o m o t h e t i c f u n c t i o n , Eh. v, = 0 by E u l e r ' k i k k t h e o r e m . Then t h e l a s t two t e r m s d r o p o u t and we o b t a i n The A l l e n p a r t i a l e l a s t i c i t y of substitution (c^j) between the i t h and j t h commodities i s derived from the following r e l a -tionship (Allen[1938; 512]). (HI.13) n . • = s. ( 0 . . - n . ) , i ] .] ij l y ' where s^ i s the (individual) expenditure share for the j t h good Solving for , we get: CT.. = r|../s. + n. ID ID D i y (III.14) h..Zv,h. E v . h . , Zv , h . . E Z v . h . v ^ k k k k k J k - k k l k . km k m h . h . h . h . Ev. h, I D D 1 k k 2 where h. . = — — a n d k = 1,..;,N. 1 3 8 v . 3 v . D 1 I f the o v e r a l l function i s homothetic, c r^j becomes: h . . Ev, h, 1 3 , k k (III.15) CT., = - . 1 J h.h. 1 D The two goods are substitutes, i f > 0 and are comple-ments^ i f c r^j < 0. The N by N matrix of :CT^J i s a symmetric and negative semi-d e f i n i t e matrix with rank equal to at most N-l.' In p a r t i c u l a r the curvature conditions require that own e l a s t i c i t i e s of sub-s t i t u t i o n (cr^^, i = l , . . . ,N) must a l l be nonpositive. - 5 2 -The following relationships are .also well known and are used during the estimation process of the entire model as checks for correct c a l c u l a t i o n s . i) En.. = -n. (homogeneity condition) -3 1 3 X Y i i ) Es.n. =1 (Engel aggregation) i 1 i y i i i ) Es^n^j = -Sj (Cournot aggregation) iv) Es . a. . =0 j 3 1 3 Recall the cross-price e l a s t i c i t y between goods i and j , i . e . , ^ i 1- 1- v. Ev. h.. . .3.1nx. h..v h.v. j , k k: (III. 10) n-, = 3 1 - + — ) 3lnp. h. E v k h k E v k h k = ( s p e c i f i c effeet) + (general effect) • The f i r s t term can be interpreted as a s p e c i f i c e f f e c t on good i and the l a s t two terms as a general e f f e c t common to a l l goods due to a change in p.. . Furthermore, define the pseudo e l a s t i c i t y of substitution (PES) as follows: h. . Ev, h. . l-j, . k k (III. 16) a.. = . 1 3 h.h. 1 3 - 53 -Then the cross-price e l a s t i c i t y defined i n (III.10) can be rewritten as n.. = s . a . . - (general e f f e c t ) . ID D ID Therefore, the P E S i s a normalization of the 's p e c i f i c e f f e c t ' above. By substituting the d e f i n i t i o n of the P E S into (III.14), we obtain: (III. 17) a-±j = o L . - Zs k2 j k - Zsk3.k + ^ s k 2 k m s m . In other words, the A l l e n e l a s t i c i t i e s of substitution are weighted sums of the pseudo e l a s t i c i t i e s of substitution. In the base period, we.can impose a normalization such that £.Vkhk = 1. Thus, h... (III.18) a. 1 3 h.h. i D This way, i t i s easier to r e l a t e the y^ and B^j parame-ters to a. ., since v.. and B - . d i r e c t l y r e l a t e to a. .. Further---.} i j ijI . i j i j more, >ns l i n e a r l y related to the a^ . In theecase of the translog function, i f we set y^j = 0/ Oj O J we obtain a . . = Zv.h./h. Therefore, a . . = 1 for the base year, since E v k h k = 1 and h(v) = 1 by the normalization = 0. For the Generalized Leontief function, a. . = 0 i f and only ID J i f B ^ j =0 and v>>0N. In Chapter I I , the equality r e s t r i c t i o n s on the ES were discussed. We r e c a l l that given a homothetically weakly sep-arable o v e r a l l function, we have a™£ = a™^ for a l l m,n,h,i,j,k, where i .and j belong to the mth group and k and h to the nth group. • Furthermore, i f the o v e r a l l function i s l i n e a r l y homogen-eously separable, then we have a1?^ = a m n , where a m n i s the ES I K between the mth group and the nth group. It i s easy to show that s i m i l a r equality r e s t r i c t i o n s hold for the PES. (III.19) Let h(v\..,v T) = -Hdi 1 (v 1) , . . . ,hT (vT) }, where v*" i s a N dimensional vector of normalized p r i c e s . Then, (III.20) a ™ TTmn,m, n T ^ t, t . . H h.h, E E v h ^mn,^ t ^ t N 1 k, .......... .s. s ir. (EEVch ) t=l s=l s s ik H ^ H ^ f 1 H V * I k m^n c . _m , _.n = O j j ^ for any j eR , h e R . Since EEv^h^ i s common, for any PES, we fi n d that a1!1^ s s X J C depends only upon the two groups involved. Thus, a3?^ i s the same for any two goods from two d i s t i n c t groups. Note that the equality r e s t r i c t i o n s depend only upon weak sep a r a b i l i t y of the inverse i n d i r e c t u t i l i t y function. If the o v e r a l l function were nonhomothetic, then the^pseudo e l a s t i c i t i e s of substitution are much easier to re l a t e to the - 55 -g . . and Y • • terms. This i s why the PES instead of the ES are used for aggregation by ' s t a t i o n a r i t y 1 . D. Four Aggregation Methods Before moving to a discussion on the models to be imple-mented empirically a few more comments are i n order concerning the four methods of aggregation i n r e l a t i o n to the functional forms employed i n t h i s study. i) Aggregation by homothetic s e p a r a b i l i t y p a r t i t i o n s TN goods into T periods. This w i l l r e s u l t i n T(T+1) unknown parameters, i f we were to te s t for symmetry, or T(T+l)/2 + T parameters, i f we were to maintain the symmetry hypothesis. Precommitted expenditures may be introduced so as to avoid unitary expenditure e l a s t i c i t i e s . Then the relevant budget con-s t r a i n t now i s the 'supernumerary' expenditure y*" i n time t, i . e . , -*t t N t t ¥ = y s p T c ^ t = 1,...,T, i=l 1 1 where y f c i s the t o t a l expenditure, c^ i s the precommitted quantity on good i i n time t. For the f i r s t stage of the two-stage maximization pro-cedure, we need to have aggregate p r i c e indices for each of the T periods involved. As pointed out e a r l i e r , we cannot derive the aggregate price indices for the future periods, unless we make.an assumption as to- the optimum a l l o c a t i o n of - 56 -goods i n each period. Therefore, the f i r s t stage cannot be properly solved. However, the second stage maximization for "1 the present period can be c a r r i e d out consistently, given y for t h i s period. Then we now deal with N present goods. The number of unknown parameters i s N(N+1) + N (without assuming symmetry) or N,;(N+l)/2 + N (assuming symmetry) . One problem s t i l l to be solved i s the determination of the supernumerary expenditure. We may incorporate the pre-committed quantities d i r e c t l y into our models by treating them as a new set of parameters. However, t h i s w i l l lead to a rather complicated structure, when we aggregate the i n d i v i d u a l 4 share equations over people. Furthermore, t h i s implies addi-t i o n a l N parameters to be estimated. I f we were to te s t for symmetry, we w i l l have N(N+2) parameters. i i ) Since we assume fixed proportions for future quantities with Leontief aggregation, i t i s possible to derive the .expected aggre-gatete- p r i c e indices for the future goods. Thus, we p a r t i t i o n TN goods into 2N goods, i . e . , N present goods and N future goods. The number of unknown parameters i s 2N(2N+1) + 2N (without symmetry assumed) or 2N(2N+l)/2 + 2N (with symmetry assumed). For t h i s method to be operational, we need to a r b i t r a r i l y decide on the proportions. From equation (11.12), we have: x 3 " p i x r We assume that p.. = 1 for a l l j and t . Then the aggregate price index for the j t h future good i s calculated as: T>* T t P* = E P r  3 t=2 3 I t should be pointed out that the assumption that = 1 cannot be tested i n our model. A unique advantage attached to thi s technique i s i t s a b i l i t y to take a non-constant l i f e span into account, since T may vary. However, th i s feature raises a problem when we attempt to aggregate the micro demand functions over people (of d i f f e r e n t ages), since they do not have the same expected l i f e span. Our method of aggregation over people i m p l i c i t l y assumes that a l l consumers have an i d e n t i c a l average l i f e span. In theory, we should have double i n t e -gration of i n d i v i d u a l demand functions over l i f e span (or age) and over l i f e t i m e wealth. i i i ) Hicks' aggregation also reduces TN goods to 2N. Substi-t u t i o n i s possible between two commodities i n the same per-iods, between the same commodity but i n two d i f f e r e n t periods (present and future) and between two d i s t i n c t commodities in two d i f f e r e n t periods. This also applies to Leontief aggregation. Furthermore, the use of f l e x i b l e functional forms such as the translog function and the Generalized Leontief function allows for i n f e r i o r goods and complimentary relationships among goods. - 58 -iv) A l l the equality relationships about the ES hold for the PES. For example, homothetic weak se p a r a b i l i t y implies a ™ = 0 ™ , for a l l i , j e R m and for a l l k,heRn. This equality r e s t r i c -t i o n can be translated into equality r e s t r i c t i o n s on the values of and f ° r the base period. In the base period/, we have, by normalization, v, = v 0 = v 0 = ... = 1 and Ev h = 1. ' 1 2 3 „ m m m The l a t t e r normalization r e s u l t s i n h^ = x^ for a l l i , where i = T n™ • , ,„ , m^n m^n . n . . , 1,...,TN, since x. = h./Zvh. a . , = a . , then implies that m b i k = b j h C x i x k / x j x h } a n d Y i k = Y j h ( x i x k / x j x h } w h e r e L'i£Rm a n d h,k £R n. Given observed values for x^,x^ ,Xj^,x k, we can impose .above r e s t r i c t i o n s for a l l i,k,m,n, where i,k = 1,...,N; m,-n = 1,...,T, This equality r e s t r i c t i o n i s very simple to incorporate, as long as we are given observations on the quantities demanded (pur-chased) , i . e . , x^,x^ ,x^,;.x^. In. other words, i t i s no problem i f we are dealing with only current consumption. However, i n a dynamic model l i k e ours, t h i s raises a serious problem, since we cannot observe future demand for each commodity i n each period. We.only have the sum of future demands, i . e . , the current investment i n the future. One way to get around th i s problem i s to assume the ex-pected future demand for each commodity and use the value for the purpose of imposing equality r e s t r i c t i o n s . A l t e r n a t i v e l y , a p r i o r i assumptions on the values of a ^ _ . may be imposed i n order to reduce TN(TN+l ) / 2 + TN (symmetry assumed) parameters to a fewer number. Consider a square 2 matrix £ of TN x TN dimension which i s pa r t i t i o n e d into T submatrices i;™11. Each of the submatrics contain N x N elements. The assumption of 'stationary tastes'may be made as to the relationships among submatrices. For example, submatrices which are separated by the same number of periods may be assumed i d e n t i c a l . ~* E. The Proposed Models For the actual empirical estimation, three al t e r n a t i v e mod-els are formulated. These are: the three good model, the l e i -sure model and the intertemporal model. The f i r s t two models deal only with the current period consumption for the whole economy, while the l a s t model attempts to deal with the intertemporal a l l o c a t i o n of goods for the 'representative' consumer. The t h e o r e t i c a l framework for the two atemporal models i s based upon the assumption of a homothetically separable i n t e r -temporal (inverse indirect) u t i l i t y function for the in d i v i d u a l household. Thus, the category function for the present period i s assumed to be homothetic. Instead of attempting to incor-porate precommitted expenditures into the. models, we formulate - 60 -the models without them so that we can test the hypothesis of homotheticity. In addition, we test for symmetry. The three good models include . food, durables and miscel-laneous. The l e i s u r e model incorporates l e i s u r e as the fourth commodity. Therefore, t h i s model i s designed to deal with l a -bor supply decisions.. In both models, the expenditure share functions for i n d i -vidual households are aggregated over the entire population of households to derive the aggregate ((or market) expenditure share functions. The l a s t model i s the intertemporal model and includes two future commodities i n addition to four current goods from the l e i s u r e model. The future goods' are the aggregates of the goods i n a l l future periods using the Leontief aggregation method. Ideally, four future goods, i . e . , future food, future durables, future miscellaneous and future l e i s u r e shoudld be con-sidered. However, in.order to contain the size of the model, the f i r s t three future goods are aggregated into one single commodity e n t i t l e d 'the future compositee•good.^ Only the expenditure share function for the representative consumer w i l l be estimated for t h i s model. It i s f e l t that proper aggregation over households requires double integration over 'age' as well as l i f e t i m e wealth. Unfortunately, our model i s not capable of handling double integration of t h i s nature at t h i s stage. Furthermore, i t i s extremely d i f f i c u l t to obtain time series data on the j o i n t d i s t r i b u t i o n of age and wealth of households. - 61 -FOOTNOTES to Chapter III 1. See Diewert[1974c]. 2. A d d i t i v i t y i s used to mean ' a d d i t i v i t y i n coordinate-wise p a r t i t i o n . A d d i t i v i t y or homotheticity assumption does lead to substantial economies i n the number of parameters involved. But these economies are c o s t l y . For further discussion, see Goldberger[1967] and Phlips[1974]. 3. Hicks[1939] and Samuelson[1947] have shown that the N by N matrix of Slutsky substitution terms(k..) i s symmetric and negative semidefinitecy., i f the consumer i s maximizing u t i l i t y subject to his budget constraint. The converse.is true under c e r t a i n r e g u l a r i t y condition. Now the r e l a t i o n s h i p between k.. and a. . i s as follows: x.,x.>>0 y>>0 1 3 -Although the two matrices £ (of ) and K(of are not the same, i t i s a t r i v i a l exercise to show that the signs of the determinants of a l l p r i n c i p a l minors are i d e n t i c a l . The determinant of the mth p r i n c i p a l minor of the matrix £ i s k. . = 13 x i x j 13 m l a i j l = (x n!..x )2 l k i j l ' J 1 m J Therefore, we w i l l use £ rather than K i n order to examine the curvature conditions of the inverse i n d i r e c t u t i l i t y , function i n the empirical testing procedure of our models. 4. Using the translog inverse i n d i r e c t u t i l i t y function, the i n d i v i d u a l expenditure shares for the i t h good i s now defined i n terms of v, where v = p/y. Thus, a. + Ev..lnv. l - 1 3 3 p. c. _ j J J + * i l S i T _i_ v v i * i , j ,m = 1,. . . ,N. 1 + E E Y . lnv y , § m Y 3 m m The aggregate expenditure share equation becomes: a i + ^ Y i i l n p i ~ 6 1 n ( Y * _ S. = Ep.c . )Ey. . j 3 3 3 1 3 1 + EE Y k i nlnp. • i io y km • m FOOTNOTES to Chapter III (con't) ', where we assume that c. , i = 1,...,N i s the same for every i n d i v i d u a l . Therefore, the in t e g r a l / dy i s an additional variable and must be constructed from the data. A l t e r n a t i v e l y , i f we assume a log-normal d i s t r i b u t i o n of y, then 0 0it Cv) a -8/2 f———dy = e , where a i s the mean and 6 i s the variance o y • J ' of z and -z = lny. See the mathematical note • \( Appendix A) for the derivation of a similar i n t e g r a l . A l l C j'sCj = 1,...,N) are treated as unknown parameters to be estimated. '5. For an empirical application of a similar method, see Diewert[1974b]. %. See Chapter V for the method of aggregating these three goods. - 63 Chapter IV ECONOMETRICS This chapter i s devoted to econometric considerations. We.discuss: the s p e c i f i c a t i o n of the model and econometric problems associated with our models; the computational algor-ithm and i t s properties; hypotheses te s t i n g and s t a t i s t i c a l inference. In addition, a simple i l l u s t r a t i o n of methods for t e s t i n g and imposing monotonicity and curvature i s presented. A. Econometric Spec i f i c a t i o n The equations to be estimated can be represented i n ma-t r i x notation as follows. where S Ct) i s an N by 1 vector of dependent variables, .x-.(t) i s an N by K matrix of the exogenous variables, r i s a K by 1 vector of unknown parameters and e(t) i s an N by 1 vector of random disturbances. T* i s the number of observations for each equation. More s p e c i f i c a l l y , the expenditure share equa-tions, for the i t h good derived from the two functional forms are presented i n Chapter III as equations (III.4) and ( I I I . 9 ) . Share equation for the three good model and the l e i s u r e model are derived from the translog u t i l i t y function. The share for the i t h good at time t:_ i s (IV. 1) S(t) = F ( x ( t ) , D + e(t) t = 1 , . . . , (IV.2 ) S i ( t ) = + e ±(t) , - 64 -for i = 1,2,3, i n the three good model and for i = 1, ...,4 i n the l e i s u r e model. The i t h share i n the intertemporal model for the 'rep-resentative' consumer i s derived from the Generalized Leontief function and i s of the form:"'" .. (p.p.)^ +, e n . e * y * (IV. 3) S . (t) = ^-3 ^ + e . (t) for i = 1, . . . , 6 , 1 ZZB. (p. p ) 2 1 km km k*m We assume that the inverse i n d i r e c t translog and Gener-a l i z e d Leontief u t i l i t y functions approximate the 'true' mean u t i l i t y function. The share equations are then interpreted as conditional expectations of the shares S^, given y*,0,9*, p^, . . . f P j j - Additive disturbance terms are assumed to be due to random errors i n approximation, i n aggregation over goods, households, time and or i n u t i l i t y maximizing behavior. Define an N by 1 vector of additive disturbances at time t as (.IV ;4) e(t) = [ E l (t) ,e 2 (t) , . . . , e N(t) 1 t = l , . . . , T * . We assume that t h i s vector i s random and i s independent of t. Furthermore, e(t) i s assumed to be normally d i s t r i b u t e d with mean zero, i . e . , (IV.5) E { E i ( t ) } = 0 for a l l i and t . - 6 5 -The N by N c o v a r i a n c e m a t r i x o f t h e d i s t u r b a n c e t e r m s i s s p e -c i f i e d as Q* f o r s = t (IV.6) E [ e ( s ) e ( t ) ] = '{ } f o r a l l s and t . 0 f o r s jt t The p r o b l e m i n t h e e s t i m a t i o n o f t h e s e e x p e n d i t u r e s h a r e e q u a t i o n s a r i s e s f r o m t h e f a c t t h a t t h e s h a r e s , by d e f i n i t i o n , add up t o u n i t y i n e a c h p e r i o d and t h a t e ( t ) a r e i n d e p e n d e n t w i t h . E ( e ( t ) ) = 0. T h i s i m p l i e s ^ e . ( t ) = 0 f o r a l l t . T hus t h e d i s -1 1 u r b a n c e c o v a r i a n c e m a t r i x f o r e a c h model i s s i n g u l a r and non-d i a g o n a l . G o l d b e r g e r and G a m a l e t s o s [1970] have shown t h a t t h e N by N e s t i m a t e d c o v a r i a n c e m a t r i x w i l l a l s o be s i n g u l a r . Be-cause-;, o f t h e s i n g u l a r i t y , i t i s n o t p o s s i b l e t o e s t i m a t e t h e f u l l s y s t e m o f N e q u a t i o n s d i r e c t l y by t h e t r a d i t i o n a l g e n e r -a l i z e d l e a s t s q u a r e s method.^ T h i s p r o b l e m i s s o l v e d as f o l l o w s . An a r b i t r a r i l y c h o s e n e q u a t i o n i s d r o p p e d i n e a c h m o d e l . Then a new t r u n c a t e d ( N - l ) by ( N - l ) d i s t u r b a n c e c o v a r i a n c e m a t r i x 0, i s d e f i n e d as m f o r s = t (IV.7) E [ e * ( s ) e * ( t ) ± ] ='{ } . f o r a l l s and t , 0 f o r s ^ t where e * ( s ) and e * ( t ) a r e ( N - l ) by 1 v e c t o r o f a d d i t i v e d i s -t u r b a n c e s a t t i m e s and t . I t i s assumed t h a t t h e r e a r e ( N - l ) i n d e p e n d e n t e q u a t i o n s , w h i c h i m p l i e s t h a t t h e ( N - l ) by ( N - l ) d i s t u r b a n c e c o v a r i a n c e - 66 -matrix fi i s nonsingular. We denote the i j t h element of fi by A A A o)^j , the estimate of by fi, and the i j t h element of fi by • Maximum l i k e l i h o o d estimates of OK., are computed from the res-idual cross products, - T _ i. e . e . A- - ]_ -| where e^ and e_. are1 T* by 1 vectors of residuals. B. Computational Algorithm Assuming further that the vector e*(t) i s i d e n t i c a l l y multivariate normally d i s t r i b u t e d , the logarithmic l i k e l i h o o d function i s : CIV. 8 ) InL = - ln (2r r + 1 ) - ^- l n | Q, \ . I t i s clear from the above equation that maximizing InL i s equivalent to minimizing l n [ f i | . I t i s also evident that f i r s t - o r d e r conditions for a maximum of the logarithmic l i k e -lihood function are nonlinear i n the estimated parameters, therefore, a nonlinear generalized l e a s t squares procedure must be used. The computational algorithm i s , b r i e f l y , the following. A The .iife'rative process begins by assuming £2 = 1. Given th i s A fi, the parameters are estimated by nonlinear generalized l e a s t A squares i n order to obtain a new estimate of fi. This i t e r a t i v e -67 -procedure i s continued u n t i l 'convergence 1 i s achieved. Con-vergence i s attained when the parameter estimates and elements of the estimated covariance matrix change by less than a spe-c i f i e d tolerance l e v e l , which isnormally set at 1%. This computational algorithm has been programmed into TSPr1 (Time Series Processor) . The nonlinear optimization algor-ithm i s based on the methods of Gauss-Newton with varying step size s . The actual computation was car r i e d out at the computer terminal at the University of B r i t i s h Columbia on the IBM 370-Model 168. Properties of t h i s computational algorithm have been d i s -cussed by Malinvaud [1972; 343-348], Berndt, H a l l , H a l l and Hausman [.1974] and Oberhofer and Kmenta [1974]. The following property i s of p a r t i c u l a r importance. I f convergence i s at-tained, the r e s u l t i n g estimator converges numerically to i) Zellner's minimum distance estimator and i i ) the maximum l i k e l i h o o d estimator, provided that the disturbances are multi-2 variate normally d i s t r i b u t e d . Further, Barten [1969] has shown that maximum l i k e l i h o o d (ML) estimates of the f u l l N by N matrix can be derived from (N-l) equations. These ML e s t i -mates are invariant to the equation dropped. This i s so be-cause the l i k e l i h o o d function i s invariant to the equation de-leted and the deleted equation i s a l i n e a r combination of the equations included. Thus the parameter estimates are the same regardless of the equation deleted. - 68 -C. Hypothesis Testing Each of the three models i s estimated with d i f f e r e n t con-s t r a i n t s imposed. These constraints are symmetry (SYM), homotheticity (HOM), and homotheticity conditional on symmetry (S+M). In addition, an unconstrained version (NS+NH) i s e s t i -mated. Each version, of course, includes the normalizations sp e c i f i e d i n Chapter III.. In.summary, the hypotheses to be tested are: Unconstrained, Unconstrained, Unconstrained, Symmetry., The v a l i d i t y of these n u l l hypotheses i s inferred from l i k e l i h o o d • ratio's... .test. s t a t i s t i c s , computed as twice the d i f f e r -ence i n the log of the l i k e l i h o o d under the n u l l and alternative hypotheses. This s t a t i s t i c i s d i s t r i b u t e d asymptotically as a chi-square variable, the degrees of freedom being equal to the difference in the numbers of free parameters under HQ and H^. As well as the parameter estimates, asymptotic.t s t a t i s -t i c s adjusted by (T*-K)>degrees of freedom are computed, T* being the number of observations and K being the.number of free parameters i n each equation.^ i) V Symmetry H l : i i ) V Homotheticity H l : i i i ) V Symmetry and Homotheticity H l ! iv) H Q: Symmetry and Homotheticity Based on the maximum l i k e l i h o o d parameter estimates, cross and own (uncompensated) p r i c e e l a s t i c i t i e s and A l l e n par-t i a l e l a s t i c i t i e s of substitution are calculated for each an-nual observation. In actual computations, a set of converged parameter e s t i -mates from a r e s t r i c t i v e version i s used as s t a r t i n g values i n the estimation of the next less r e s t r i c t i v e version. We do th i s because n o n l i n e a r i t i e s and the large number of parameters make the attainement of convergence more d i f f i c u l t for less r e s t r i c t i v e versions. The most r e s t r i c t i v e version, S+H, i s l i n e a r i n the para-meters. Thus convergence fo r . t h i s - y e r s i g n i s not d i f f i c u l t . Once convergence i s achieved for the S+H version, the r e s u l t -ing parameter estimates are used as the i n i t i a l values for SYM and HOM. The converged estimates from SYM or HOM are then used as st a r t i n g values for NS+NH. D. Imposition of Monotonicity and Curvature The monotonicity condition may be checked by ca l c u l a t i n g the vector of f i r s t p a r t i a l derivatives (Vh(v)) of the inverse i n d i r e c t u t i l i t y function h(v) for each observation period, given the parameter estimates and the observed exogenous var-ia b l e s . I f a l l the elements of Vh(v) are p o s i t i v e at every sample point, the monotonicity condition i s said to be s a t i s -5 f i e d g l o b a l l y . The curvature property of a model may be investigated by various equivalent methods. F i r s t l y , one may check the deter-minantal conditions of the bordered Hessian matrix H* or the matrix E of e l a s t i c i t i e s of substitution. As . i s well known, i f the u t i l i t y function i s quasiconcave, H* and E are negative semidefinite. Thus, the determinants of the p r i n c i p l e minors obtained by deleting the l a s t (n-k) rows and (n-k) columns must be zero or alternate i n sign, f i r s t being negative. Another method of determining i f the matrix E i s negative semidefinite i s to calculate the eigenvalues of the matrix. One eigenvaluee : must be zero (since the matrix i s singular) and the rest must be nonpositive. Calculating the determinants of a l l p r i n c i p a l minors i s not p r a c t i c a l when the si z e of the matrix i n question becomes large. The determinantal condition of the matrix E i s used for 1 the three good model' and 'the l e i s u r e model 1, since the d i -mension of the matrices are within manageable s i z e . The _ eigenvalue© method, however, i s used for 'the intertemporal model' i n which the dimension of the matrix E i s six by s i x . Again, i f these tests are s a t i s f i e d at every sample point, we say that the curvature condition i s s a t i s f i e d " g l o b a l l y . I f monotonicity and or curvature conditions are not sat-i s f i e d g l o b a l l y , but are "'hot s t a t i s t i c a l l y rejected at the point of expansion', i t i s possible to impose either or both at the point of expansion. This way one can obtain econometric r e s u l t s which are l o c a l l y consistent with economic theory. - 71 -The methods of testing and imposing monotonicity or curv-ature i s respectively based on the concept of squared transfor-mation and the Cholesky f a c t o r i z a t i o n of a r e a l symmetric matrix. The test and imposition are performed at a point of expansion. One may choose for the point of expansion an observation at which the conditions appear to be v i o l a t e d most severely. An observation with a largest negative h^ or the largest p o s i t i v e eigenvalue i s an example. I t must be noted, however, that im-posing monotonicity and or curvature at a point of expansion does not i n general guarantee global s a t i s f a c t i o n of the r e l e -vant condition. 1. Imposition of monotonicity conditions. Now we move to a discussion of a method for te s t i n g and imposing monotonicity using the Generalized Leontief function. The t e s t i s performed by reparameterizing the model and carry-ing out tests on the transformed model. The imposition of monotonicity employs the method of a squared transformation. It was pointed out e a r l i e r that monotonicity requires a l l the f i r s t p a r t i a l derivatives of the inverse i n d i r e c t u t i l i t y function hCv) with respect to the normalized prices to be non-negative, i . e . , h iCv)\> 0 for a l l i = 1, ,N. Then, 6 h i C v ) = . s 1 e ± j + ^ o i = 1 ' - - " 6 at the point of expansion where v = [1]. Let h.(v) = x . . Then, (IV.9) h ±(v) H T ± = Z g ^ + h&0± i , j = 1,...,6. The notion here i s that by l e t t i n g h^ = x^, the n u l l hy-pothesis to be tested i s H Q : X ^ >_ 0 for a l l i , against the a l -ternative hypothesis : x^ .< 0 for some i . This method i s another way of t e s t i n g the n u l l hypothesis H N : Z g . . + hftn- > 0, U j X^ J u x ~— for a l l i against the a l t e r n a t i v e hypothesis : + 3 s ^ 0 i < ^ for some i . I t i s clear that the t e s t for the former i s com-putationally less cumbersome, than the test for the l a t t e r , which would require summing of variances and covariances among seven parameters. Therefore, the parameters in/.-a-model, w i n be.; rede-fined i n terms of T ^ , i = 1, ...,6 so that we can perform a t e s t on a l l x. terms. One-tailed Bonferroni t e s t s t a t i s t i c s are 7 used i n place of usual t s t a t i s t i c s . Reparameterization of the model i s performed as follows. Solve the equation (III. 9) for 3 ^ , for i = 2, ... ,6 i n terms of 6 i j t i f j ) , 6 Q i and T±. Then, 6 E (IV. 10} 6 . . = x. - l^±. - -%B 0 1 i _= 2,...,6, where g . . = 8... i } ] i From the normalization Z Z g . . = 1, we obtain: i j 1 3 (IV.11) g n l = 1 - z g . . = 1 - ( x 2 + . . . + x 6 ) - ( g 1 2 + . . . + g 1 6 ) - IjB and (IV.12) T1 = 1 - ( T 2 + . . . + T 6 ) . Thus, six terms are replaced by equations (IV. 10) and (IV.11). Note that the number of unkown parameters has not been changed. Upon estimation of the transformed model, the sign and s t a t i s t i c a l s i gnificance of T ^ , i =1,...,6vare inves-tigated. If the n u l l hypothesis HQ : >_ 0 i s ' not rejected, monotonicity can v a l i d l y be imposed at the point of expansion. Imposition of monotonicity i s achieved by replacing each 2 T ^ , i = 2,.../6 by x* for i = 2,..;,6 i n equations (IV.10) and CIV.11). Thus, CIV.13) h iCv) = T | 2 2 ° i = 2, . . . ,6. Therefore, the monotonicity condition i s imposed at the sp e c i f i e d point of expansion by reparameterizing six terms and replacing f i v e x^ terms by squared terms. 2 The remaining term,' x* i s not e x p l i c i t l y estimated and i s calculated as a r e s i d u a l , due to the normalization such that: T * 2 = 1 - ( T * 2 + . . . + T * 2 ) . 2 ;'"" This implies, however, that T£ may indeed take on a negative 6 2 value, i f H T * > 1. Therefore, there i s no way of quaran-i=2 x 2 teeing that a l l six T * w i l l be non-negative simultaneously. - 74 -Consequently, one i s advised to use d i s c r e t i o n i n selecting the residual parameter. One should t r y to avoid selecting any x | as the residual parameter, i f i s negative at the point of expansion. 2. Imposition of curvature conditions. The curvature condition may be tested and imposed by a reparameterization of ( i ^  j) and B Q ^ terms for i , j = 1,.. These are terms which remain untransformed i n the process of imposing monotonicity. Before "going into the actual method, b r i e f discussion on Cholesky f a c t o r i z a t i o n of r e a l symmetrix matrices i s required. Lau [1974; 17, 21] has shown that every p o s i t i v e semi-d e f i n i t e real-valued symmetric matrix A has a unique Cholesky f a c t o r i z a t i o n of the form: (IV.14) A = LDL T, where L i s a unit lower triangular matrix and D i s a diagonal matrix. It follows that every negative semidefinite matrix C has a unique Cholesky f a c t o r i z a t i o n of the form: C = LD*LT, where C = (-l)A and D* = (-l)D. The diagonal elements i n the matrix D and D* are c a l l e d Cholesky values, These elements are denoted as d^ and -d^, i = 1,...,N. It i s shown by Lau [1974; 19] that a r e a l symmetric matrix A i s p o s i t i v e (semi)definite^, i f and only i f a l l i t s Cholesky values are po s i t i v e (nonnegative). A c o r o l l a r y to the Lau theorem i s that a r e a l symmetric matrix C i s negative (semi) d e f i n i t e , i f and only i f a l l i t s Cholesky values are negative (nonpositive). In other words, a concave function implies and i s implied by nonpositive Cholesky values, i . e . , (IV.15) -d. < 0 or d ± >_ 0 i = l , . . . , N . A quasiconcave function, however, implies at le a s t (N-l) nonpositive Cholesky values. I f i n addition, they are a l l s t r i c t l y negative and the gradient i s nonzero, then a quasi-9 concave function i s implied by (N-l) negative Cholesky values. Therefore, i n order to tes t (quasi)concavity» of a twice d i f f e r e n t i a b l e function, one need only check i t s Cholesky v a l -ues. This-provides a r e l a t i v e l y simple way of imposing the appropriate curvature. One need only constrain the Cholesky values.,-'-:;!>-£--/one wishes to impose nonpositive Cholesky values, the diagonal elements -d^ should be replaced by a set of 2 10 squared terms, -dt for i = 1,...,N. - 7 6 -I t turns out that the r e a l symmetric matrix to be i n v e s t i -gated i n th i s study i s the Hessian matrix H derived from h(v). The 'global' curvature conditions are tested using the bordered Hessian matrix H* or the matrix of e l a s t i c i t i e s of substitution S. But for l o c a l imposition of curvature at the point of expan-sion, i t i s s u f f i c i e n t to check i f h(v) i s concave, rather than if\/h(v)> is,l .quasiconcave. The reason for t h i s i s explained below. Recall that the p a r t i c u l a r normalizations we employ have made hCv) l i n e a r homogeneous along the rays of equal normalized p r i c e s . Thus, h(v) = 1 when v = [ 1 ] and h(v) = k when v = [k]. It i s obvious that any l i n e a r homogeneous function which i s quasiconcave w i l l also be concave. In other words, h(v) i s concave at the point of expansion. I t follows that we need only check i f the Hessian matrix H i s negative semidefinite rather than H*. In summary, one Cholesky value should be zero and others should be nonpositive. Thus, we heed hot look into the bordered Hessian matrix H* or the matrix of e l a s t i c i t i e s of substitution Z . Testing concavity at the point of- expansion i s now re-duced to testing the n u l l hypothesis HQ : >_ 0 for a l l i against the alternative hypothesis H^: d^ < 0 for some i . If the n u l l hypothesis i s not rejected, then we can v a l i d l y impose curvature at the point of expansion where v = [ 1 ] . The Cholesky f a c t o r i z a t i o n of a r e a l symmetric matrix H when H i s generated by the Generalized Leontief function i s now i l l u s t r a t e d for the case of s i x commodities. At the point of expansion where v = [1], the diagonal elements of the Hessian matrix H are written as: (IV.16) h. . = 11 6 I • E. t j=l ID - hi Oi for i , j = 1,...,6, The off-diagonal elements are (IV.17) h.. = $.. ID ID where h.. = h... ID . ] i Since H i s a r e a l symmetric matrix, H i s Cholesky fa c t o r i z a b l e . Thus, H can be represented as follows: (IV. 18) H = LD*L , where L i s a unit lower traingular matrix of the form: (IV. 19) L = 1 0 A . , , 1 Jl '21 32 '61 '62 and D* i s a diagonal matrix of the form: -d. (IV.20) - 78 -T The matrix product C = LD*L i s symmetric and i s of the form: (IV.21) C = -d-* 2 1 d l - £ 2 1 d l - d 2 * 3 1 d l -? 9 d -*31 21 1 £32 d2 * 4 1 d l " £ 4 1 £ 2 1 d l ~ $ d *42 2 £ 5 1 d l ~ £ 5 1 £ 2 1 d l ~ a d *52Q2 * 6 1 d l ~ £ 6 1 £ 2 1 d l " *62d2 2 2 - j d -I d -d 31 1 32 2 3 2 2 **" 5/ r ^ d -, • • • ~~ f r d r — d. ^  61 1 65 5 6 In order to reparameterize S ^ j , i ^  j and BQ^ terms, the matrix ii i s Broken down into two' matrices. Let H = Y + Z CjOV'P where Y is a 6 by 6 matrix with zero diagonal elements, B-jCi^j) as the off-diagonal elements and I D D i ' (IV.22) Y = 0 B 12 * ' ' p16 e f 2 0 J13 p23 iB-j^ g • . . . 0 and Z i s a diagonal matrix of h ^ = - £ £ . • - ^oi' i i i j = 1 I D i ,6, i . e . , (IV.23) Z- = !- £ B, - %& n , 0 . . . 0 j = 2 ^ 0 1 0 0 " . E / i 6 " * B 0 6 i=l - 79 -Since H, = C = Y + Z , Y = C - Z . Then, each off-diagonal element i n the matrix Y i s solved as: B. . = y . . , I D 13 where y^j i s the i j t h element i n the matrix Y. Thus, .2 (IV,24) • &±. = - Z £ . k £ i k d k i ? j , j > i for i = 1,...,5, J k=l J j =2,...,6, where £ ^ = 1. From the main diagonal of Y, 0 - c.. = z.. i = l , . . . , 6 , I D I D where c.^ .. and.z... are the ij.th elements of 'Cand Z respectively, Sovling for BQ^, we obtain: 1 2 6 (IV.25) B Q I = 2 Z £ i k d k ( l + Z i t i k ) for i = 1,...,5, k=l j=k+l J where £. . = 1, and: 11 ' 5 S06 = 0 - . z . b 0 i 1=1 5 , 5 6 (IV.26) = 0 - 2 Z Z A ..) (1 + Z £. ) k=l K j=k+l D * j=k+l J K - 8 0 -In t h i s s i x commodity model, t h i s i n v o l v e s r e p a r a m e t e r i z a t i o n of twenty— one parameters. I f one wishes to impose both monotonicity and concavity simultaneously, then a l l ( i n c l u d i n g i = j) and terms f o r i , j =•!,...,6 must be reparameterized as above. - 81 -FOOTNOTES to Chapter IV The choice of the Generalized Leontief function i s esse n t i a l i n our intertemporal model. Since we can only estimate four out of six equations of expenditure shares, i t turns out that the translog form cannot estimates-all the parame-ters i n the model. In contrast, the Generalized Leontief form, when modified, can estimate a l l of the parameters d i r e c t l y (except the i m p l i c i t parameters due to normali-zations) . The modification i s done as follows: CIII.6)' h(v) = EEg..(v.v.)^ + Eg-.v.*5 . . in 1 j O i l i l J J Eg. ,. ( v . ^ v / 5 ) + ^ 3 0 i v h (1X1.7)' x. (v) = j 1 3 1 3 0 1 1 1 k . £ £ 8 i • (v, v . ) 2 + %EB~•v. 2  k j k] k j' ^ Oj j £ 8 , • ( v . v . ) ^ + h 8 . . V , 3 5 ij i j Oi l (III.8)' s. = v.x. = ^ l i l EEg. . (v, v.) 2 + %£g„ k . j k ] k ] ' j o : : 2 I t should be noted that i n thi s formulation the normali-zation Eg . = 0 does not simplify the denominator. Thus oj * J i t w i l l not y i e l d an aggregate share funtion s i m i l a r to (III.9). I t should be noted further that because the denominator of ,.:(III. 8) ' contains a l l 8 Q j , j = 1,...,6, we.can estimate^ a l l of the parameters even when we drop two equations. 2. See Berndt, H a l l , H a l l and Hausman[1974: 664]. 3. Ibid., 659. FOOTNOTES to Chapter IV (con't) 4. Gallant[l974; 43] recommends that t s t a t i s t i c s should be computed using T*-K rather than T* degrees of freedom when the sample size i s not large (for example, 50). His Monte Carlo experiments shows that t s t a t i s t i c s are overestimated when they are not adjusted by T*-K. 5. 'Globally' i s used here to mean ' l o c a l l y at each sample point'. There i s no simple computational method of cheking the s t a t i s t i c a l s i g n i f i c a n c e of the values of Vh(v), even i f a l l (or some) of them turn out to be neg-ative. 6. See. Lau[1974; 25] and Katzner[1974; 210-211]. 7. For a further discussion, see Lau[1974; 40-42]. 8. See Theorems 4, 6 and 9 i n Lau[1974; 22, 24 and 28]. 9. This was pointed out verbally by Diewert. J_0. For quasiconcavity, there arises a problem of determining which d. should be l e f t unconstrained. - 83 -Chapter V REQUIRED DATA A. Introduction The commodities considered i n t h i s study are food, consu-mer durables, miscellaneous, l e i s u r e , the future composite good and future l e i s u r e . In the'three good model', food, consumer dur-ables and miscellaneous are dealt with, while l e i s u r e i s added i n the 'leisure model'. A l l six commodities are included i n the intertemporal model. Consumer durables consist'of housing; furniture and ap-pliances; clothing; automobiles and other vehicles. Included i n the miscellaneous category are: f u e l and l i g h t ; medical and health care; transportation (other than vehicles) and communication; s o c i a l expenses and recreation; services as well as education. The future composite good i s an aggregate index of future, food, future consumer durables and future miscellaneous. For the actual estimation of the three models, the annual Japanese post-war time series data ranging from 1946 to 1972 are used. However, for most of the required data, extensive e f f o r t s were made to c o l l e c t and construct reasonably consis-tent series for both pre-war (1912-1940) and post-war periods. The data covering World War II years are extremely poor and unreliable.^ The aggregate (or individual) expenditure share equation to be estimated for the i t h good i s i n the form below: - 84 -Y. 1 f(p;y*,e) Y or s . 1 f(p;w*) for i = 1 f • • • / where Y. i s the aggregate expenditure on good i and Y = ZY. . i> y^ i s the ind i v i d u a l expenditure on good i ; p i s a vector of (rental) prices of a l l goods; y* i s the mean expenditure of households and w* i s the mean ' f u l l ' l i f e t i m e wealth of the 1 representative'consumer. The f i r s t equation of applies to both the three good model and the l e i s u r e model. The i n t e r -temporal model uses the second equation. Furthermore, 0 i s a measure of expenditure d i s t r i b u t i o n and i s of the form: where y i s the t o t a l expenditure of each household. Therefore', the data required i n order to implement the models are: the aggregate expenditures on a l l commodities (Y^), the (rental) prices of a l l commodities (p), the mean expenditure (y*), the mean l i f e t i m e wealth (w*) and the expen-diture d i s t r i b u t i o n variable ( 8 ) . 'Households' are chosen as independent decision making units for the f i r s t two models of aggregate consumption. The i m p l i c i t assumption i s that household members, i n general, pool t h e i r resources and j o i n t l y experience the benefits of consump-t i o n . On the contrary, the intertemporal model i s formulated 00 0 = /ylnyij; (y) dy/y*lny*, - 85 -for the 'mean' or the 'representative 1 consumer. Thus, i n t h i s case, the f u l l l i f e t i m e wealth estimates are divided by popu-l a t i o n to obtain w*. This model i s equivalent to a model of aggregate consumption i n which everybody i s assumed to have mean wealth. Either way, the d i s t r i b u t i o n variable i s no lon-ger relevant. One reason for formulating the intertemporal model for the 'representative' consumer i s that the d i s t r i b u -tion variable of f u l l l i f e t i m e wealth of the households i n the economy over time cannot, at thi s stage, be adequately estimated due to the lack of relevant data. Another reason i s that a proper aggregation of households i n an intertemporal context requires double integration of micro functions over age as well as wealth. This point w i l l be discussed again l a t e r i n t h i s chapter. Since, savings are not included i n our atemporal models, i d e a l l y we ought to use a measure of the expenditure d i s t r i b u -t i o n . However, time series data on the expenditure d i s t r i b u t i o n are not available for a l l years. Therefore, i t i s assumed that the expenditure d i s t r i b u t i o n can be approximated by the income d i s t r i b u t i o n . The mean expenditure, however, i s no problem. I t i s estimated by d i v i d i n g the aggregate t o t a l expenditure, by the number of households. We-will now deal with the three models separately. How-ever, a detailed discussion on the methods adopted to construct estimates and the sources for data i s presented i n Appendix A. - 86 -B. The Three Good Model The three goods considered i n t h i s model are food, consu-mer durables and miscellaneous. Aggregate expenditures and price indices on food and miscellaneous are re a d i l y available from the national income s t a t i s t i c s . For the consumer durables, 'rental p r i c e s ' instead of purchase prices are estimated, given market in t e r e s t rates, depreciation rates and purchase p r i c e s . S i m i l a r l y 'rental expenditures' instead of investment expenditure represent the consumption on the 'flow of service' that consumers enjoy from given stocks of consumer durable goods. The flow of service that one receives i s assumed to depend upon the size of c a p i t a l stock; therefore, the value of the ex i s t i n g stock for each consumer durable i s estimated from gross investment series data, given depreciation rates and bench mark year stock estimates. The y i e l d s on long-term government bonds are chosen as the opportunity cost for purchases of consumer durables. Other i n t e r e s t rates such as i n t e r e s t rates paid on time de-posits shorter than a year, however, are inappropriate since durables, by d e f i n i t i o n , l a s t longer than a year i n our model. Rental prices are estimated by using the expected pur-chase prices of the consumer durablesgood i n question. There-fore, the expected prices must be e x p l i c i t l y formulated. In th i s study, i t was decided to use the assumption of s t a t i c e x p e c t a t i o n . T h i s i s t h e s i m p l e s t and p o s s i b l y most n a i v e a p p r o a c h , however, i t t u r n e d o u t t o be t h e o n l y method o f de-r i v i n g p o s i t i v e and n o t n e g a t i v e r e n t a l p r i c e s . A more d e t a i l e d d i s c u s s i o n on t h i s p o i n t w i l l be p r e s e n t e d i n A p p e n d i x A. The a n n u a l s t o c k . e s t i m a t e s f o r e a c h t y p e o f consumer d u r -a b l e a r e b u i l t upon t h e b e n c h mark y e a r v a l u e s , u s i n g g r o s s i n v e s t m e n t s e r i e s and d e p r e c i a t i o n r a t e s . The d e p r e c i a t i o n r a t e s u s e d a r e .2748, .25 and .025 f o r c l o t h i n g ; f u r n i t u r e and a p p l i a n c e s and h o u s i n g r e s p e c t i v e l y . A p p e n d i x A d e a l s w i t h t h e p r e c i s e method o f s t o c k e s t i m a t i o n and t h e s e l e c t i o n o f t h e s e p a r t i c u l a r r a t e s o f d e p r e c i a t i o n . Due t o l a c k o f a p p r o p r i a t e t i m e s e r i e s d a t a on a u t o m o b i l e s (and o t h e r v e h i c l e s ) , t h e y c a n n o t be e s t i m a t e d as an i n d e p e n -d e n t c a t e g o r y . T h ey a r e i n c l u d e d i n ' f u r n i t u r e and a p p l i a n c e s ' c a t e g o r y , a s s u m i n g t h e same d e p r e c i a t i o n r a t e . The t h r e e t y p e s o f consumer d u r a b l e goods ( c l o t h i n g ; f u r -n i t u r e and a p p l i a n c e s ; and h o u s i n g ) a r e a g g r e g a t e d i n t o one c a t e g o r y e n t i t l e d 'consumer d u r a b l e s ' , by c o n s t r u c t i n g F i s h e r 2 i d e a l q u a n t i t y and p r i c e i n d i c e s . T h e s e a r e g e o m e t r i c means o f t h e more f a m i l i a r L a s p e y r e s and P a a c h e i n d i c e s . The y e a r 1965 i s c h o s e n as t h e ( f i x e d ) b a s e y e a r . C. The L e i s u r e M o d e l F o r t h e l e i s u r e m o d e l , an a t t e m p t i s made t o c o n s t r u c t a t i m e s e r i e s e s t i m a t e o f l e i s u r e c o n s u m p t i o n e n j o y e d by h o u s e -h o l d s . T h i s e s t i m a t e o f l e i s u r e e x p e n d i t u r e i s added t o t h e - 88 -consumer expenditures on the three goods to give an estimate of Becker's ' f u l l income' (or ' f u l l expenditure'). Two assumptions are made as to the number of hours i n d i v i -duals have per day to all o c a t e between labor supply and l e i s u r e . The f i r s t assumption i s that those who are i n the labor force have twelve hours per day as free time. The rest of the day i s spent for maintenance a c t i v i t i e s , such as sleeping, eating and personal care. Thus, the maximum number of 'working' hours for any ind i v i d u a l i s twelve hours per day. Furthermore, i t i s assumed that there are twenty six days per month and twelve months per annum to be allocated between labor supply and l e i s u r e . The second assumption i s that each of those who are not . i n the labor force (and are over 15 years of age) has a fixed number of hours per day for l e i s u r e . The model i s estimated using 0, 2, 4, 6 and 8 hours (out of 12 hours). As w i l l be made clear i n Chapter VI, the econometric re s u l t s are not very sen-s i t i v e to the p a r t i c u l a r number used, except for the case of 'zero hours. The assumption of zero hour implies that those who are not i n the labor force, on the average, perform 'unpaid' work for twelve hours a day with no time for l e i s u r e . A l t e r -natively, t h i s assumption may be interpreted as evidence that the opportunity costs of not being i n the labor force are non-existent. Either interpretation seems to be t o t a l l y unreason-able except for a small segment of population. This i l l u s t r a t e s an example i n which an 'unreasonable' a p r i o r i assumption y i e l d s - 89 -d i f f e r e n t r e s u l t s . We w i l l dismiss the r e s u l t s on the ground of 'unreasonableness' of the assumption. Therefore, i t i s concluded that the econometric re s u l t s are more or less independent on the choice of p a r t i c u l a r number for l e i s u r e time for those who are not i n the labor force. For the price of l e i s u r e , the average wage rates are used as the opportunity cost of hot working i n the labor market. The average wage rates are defined as the sum of 'annual compen-sation of employees and income earned by proprietors, divided by the t o t a l annual manhours worked by the labor force. The assumption of average wage rates being the opportunity cost of l e i s u r e for every i n d i v i d u a l i n the economy may not seem to be consistent with the observed skewed income d i s t r i b u t i o n pattern. I f the wage rates are the same for everyone, i t may not be possible to explain the degree of skewness by the d i f -f e r e n t i a l amount of labor supply. Therefore, i n essence, we are assuming that the variance i n the income d i s t r i b u t i o n i s p a r t l y due to the unequal d i s t r i b u t i o n of the c a p i t a l stock rather than wage rate d i f f e r e n t i a l s . The same expehditureadistribution variable 6 i s used i n t h i s model as a proxy for ' f u l l expenditure' d i s t r i b u t i o n . - 30 -D. The Intertemporal Model The t h i r d model to be estimated i s the intertemporal con-sumption model for a representative consumer (or an average individual) i n the economy. The representative consumer i s studied here instead of the aggregate economy as i n the pre-vious models, since i t was f e l t that aggregation over people cannot be performed adequately. F i r s t of a l l , the estimates of l i f e t i m e wealth d i s t r i b u t i o n are almost impossible to make. A f a i r l y detailed estimate was attempted by the author for 1970, using the reports of the  Wealth Survey of the Household Sector :as well as the Savings  Survey for the year. However, there i s no other wealth survey of the household sector for any other year. I t i s d i f f i c u l t , therefore, to conjecture on the trend of the d i s t r i b u t i o n var-i a b l e . 3 Secondly, the aggregate expenditure share equations i d e a l -l y , should be derived by aggregating micro functions over age groups as well as over weaLtfr groups, since consumers are of d i f f e r e n t ages and have d i f f e r e n t expected time horizon. Fu-ture prices are dependent upon the consumer's age and, there-fore, are no longer the same for everybody. This double in t e -gration requires knowledge of j o i n t d i s t r i b u t i o n of age and l i f e t i m e wealth over time. Unfortunately, no estimate ex i s t s . Four current goods are the same as those i n the l e i s u r e model, namely food, consumer durables, l e i s u r e and miscellan-eous. In addition, two future goods are included. These are the - 91 -future composite good which i s an aggregate of future consump-tio n goods and future l e i s u r e . The expenditure and p r i c e data on current goods are taken from the l e i s u r e model. Since the expenditures on the two future goods are not observable, only the f i r s t four share equations are to be e s t i -mated. Therefore, we need only have data on prices of future goods and the average l i f e t i m e wealth. The Leontief aggregation method i s used to aggregate con-sumption of each of the three consumption goods i n a l l future periods into three future goods.- By setting the quantity weights as one., the aggregate price indices become the sum of discounted future p r i c e s . These three future price indices are aggregated, again, into a new aggregate price index of the 'future composite good'../ by the Fisher ideal index method, using current expenditures as weights. Present market interest rates are used as the discount rates. Furthermore, a s t a t i c expectation of prices i s assumed for the future p r i c e s . An.important feature of the Leontief aggregation method i s , as discussed e a r l i e r , that the choice of time horizon can r e f l e c t longitudinal demographic changes of the population. For each observed period, estimates are made of the average remaining 'consumption years'. I t i s calculated as the aver-age l i f e expectancy, which i s the weighted sum of l i f e expect-ancy of the population i n each age and sex bracket (for census years), weights being the number of people i n each bracket. Thus, the aggregate price index of the future composite good i s calculated as the discounted sum of future prices over the entire l i f e horizon. In.contrast, the aggregate p r i c e index of future l e i s u r e i s constructed over the average remaining 'working years'. Since individuals expect to r e t i r e completely at the age of s i x t y - f i v e , the opportunity cost of l e i s u r e extends only that f a r . Instead of a s t a t i c expectation, however, individuals are assumed to expect a modest increase, i . e . , 2 per cent per annum, i n wage rates over the entire working years. The average ' f u l l l i f e t i m e wealth' for the representative consumer i s estimated as follows. F i r s t , i t i s defined as the sum of a l l present assets, the present and future ' f u l l labor income'. Present assets consist of a l l c a p i t a l stock (pro-ducer durables, inventories, private r e s i d e n t i a l housing and a l l consumer durables) and personal savings. Net foreign assets are ignored i n t h i s study, due to lack of consistent data as well as the small magnitude involved. The present f u l l labor income may be estimated from the income side as the product of the average wage rate and the t o t a l available hours (adjusted by the labor force composition). The future f u l l labor income for the representative consumer of the mean age, i n turn, i s the discounted sum of a l l expected future f u l l labor income, over the remaining working years. - 93 -The mean age i s estimated as the weighted average of the t o t a l population, weights being the percentage of the population i n each age bracket. A l t e r n a t i v e l y the present f u l l labor income can be estima-ted from the expenditure side as the sum of a l l expenditures (on food, consumer durables, miscellaneous and leisure) and current savings minus income from c a p i t a l . This i s the portion of ' f u l l ' current income hot attributed to capital'. Theoreti-c a l l y , t h i s estimate i s equivalent to that from the f i r s t method. The future f u l l labor income i s also the discounted sum of a l l expected future f u l l labor income. In either ap-proach, we assume that the expected labor force p a r t i c i p a t i o n rates of an i n d i v i d u a l at d i f f e r e n t age i n the future are i d e n t i c a l to those of the present population of the corresponding age. The second method i s chosen i n t h i s study. The reason i s as follows. Our estimate of expenditures on consumer durables i s on rental expenditures and not on investment expenditures which the national income s t a t i s t i c s report. Our wage rate i s based upon the national income s t a t i s t i c s . Thus, there w i l l be a discrepancynihhthesestimate of f u l l income between the two methods. Since we consider the rental expenditure as a better estimate of consumption than the investment expenditure, i t i s more consistent to follow the method from the expenditure side. - 94 -TABLE V.I THE THREE GOOD MODEL PRICES YEAR FOOD DURABLES MISCELLANEOUS 1946 0. 1330 0. 1170 0.0439 1947 0.2670 0.2854 0.1074 1948 0.44 70 0.5035 0.2654 1949 0.56 10 0.656 9 0. 3 9 3 3 1950 0.5020 0.5584 0.4170 1951 0.5770 0.7558 0.4861 1952 0.6460 0.7483 0.5917 19 53 0.6830 0.7979 0.6466 1954 0.7340 0.7992 0 . 6 8 1 0 1955 0.7160 0.7758 0.6949 1956 0.70 20 0.7881 0.7088 19 57 0.7250 0.8081 0.7229 1958 0.7150 0.7956 0.7245 1959 0.7170 0.8014 0.7411 1960 0.74 00 0.8226 0.7669 1961 0.7800 0.8742 0.8012 1962 0.8300 0.9107 0.8829 1963 0.8970 0.9448 0.9024 1964 0.9260 0.9751 0.9357 1965 1.0000 1.0000 0.9982 1966 - 1.0340 1.0447 1.0630 1967 1.07 80 1.0912 1.0941 1968 1.1590 1.1391 1. 1444 1969 1.2360 1.1898 1. 1844 1970 1.3530 1.2780 1.2622 1971 1.4360 1.3532 1.3343 1972 1.4870 1.4241 1.4173 - 9 5 -TABLE V.1 CONTINUED EXPENDITURE SHARES XEAR FOOD DURABLES MISCELLAN: 1946 0 . 6 5 9 8 0.1718 0. 1684 1947 0.6266 0. 1747 0.1987 1948 0. 61 06 0. 1609 0.2285 1949 0.6231 0.1501 0.2269 1950 0.6334 0.1307 U.2359 1951 0.6120 0. 1645 0.22 35 1952 0.60 27 0.1793 0.2180 19 53 0 . 5 7 1 6 0.2021 0.2263 1954 0- 5 6 3 1 0.2024 0.2345 1955 0.5489 0.2057 0 . 2 4 5 3 1956 0 . 5 3 5 8 0.2162 0.2480 1957 0.5228 0 . 2 2 3 5 0.2537 1958 0.5150 0.2254 0.2596 1959 0.50 14 0.2304 0.2682 1960 0.4796 0.2376 0.2828 1961 0.46 12 0.2482 0.2906 1962 0.4441 0.2488 0.3071 1963 0.4347 0.2477 0.3177 1964 0.4223 0.2499 0.3278 1965 0.4194 0.2494 0.3311 1966 0.4101 0.2523 0.3376 1967 0.4063 0 . 2 5 5 5 0.3383 1968 0. 3971 0.2582 0.3447 1969 0.3899 0.2618 0.3484 1970 0.37 98 0.2681 0.3521 1971 0.37 37 0.2751 0 . 3 5 1 2 1972 0.3652 0.2758 0.3590 - 9 6 -TABLE V.1 CONTINUED YEAR HEAN EXP THETA 1946 23.6584 1.0516 1947 52.9319 1.0280 1948 100.7135 1.0238 1949 134.1812 1.0190 1950 141.0 739 1.0195 1951 169.5722 1.0180 1952 201.1361 1.0170 1953 245.0298 1.0167 1954 271.16 33 1.0164 1955 286.5237 1.0128 1956 301.3601 1.0127 1957 322.7207 1.0146 1958 335.2026 1.0150 1959 351.8538 1.0168 1960 384.2817 1.0186 1961 426.7983 1.0209 1962 484.3665 1.0270 1963 550.4399 1.0257 1964 613-0086 1.0248 1965 675.7212 1.0214 1966 754.6091 1.0220 1967 835.1321 1.0208 1968 939.3066 1.0205 1969 1066.9609 1.0241 1970 1180.8572 1.0246 1971 1308.1428 1.0279 1972 1466.5586 1.0280 - 97 -TABLE V.2 THE LEISURE HGDEL . PRICES YEAR FOOD DURABLES LEISURE aiSCELL&MJ 1946 0.1330 0.1170 0.0245 0.0439 1947 0.2670 0.2854 0.0585 0.1074 1948 0.4470 0.5035 0.1207 0.2654 1949 < 0.5610 0.6569 0.1664 0.3933 1950 0.502 0 0.5584 0.1897 0.4170 1951 . 0.5770 0.7558 0-2381 0.4861 1952 0.6460 0.,7483 0.2757 0.5917 1953 0.6830 0.7979 0.3010 0.6466 1954 0.7340 0.7992 0.3302 0.6810 1955 0.7160 0.7758 0.3466 0.6949 1956 0.7020 0-7881 0.3639 0.7088 1957 0.7250 0.8081 0.3961 0.7229 1958 0.7150 0.7956 0.4126 0.7.245 1959 i 0.7170 0.8014 0.4406 0.7411 1960 0.7400 0.8226 0.4904 0.7669 1961 0.7800 0.8742 0.5728 0-8012 1962 0.8300 0.9107 0.6667 0.8829 1963 0.8970 0.9448 0.7741 0.9024 1964 0.9260 0.9751 0.8771 0.9357 1965 1.0000 1.0000 1.0000 0.9982 1966 1.0340 1.0447 1.1115 1.0630 1967 1.0780 1.0912 1.2820 1.0941 1968 1.1590 1.1391 1.4774 1.1444 1969 1.2360 1. 1898 1.7049 1.1844 1970 1.3530 1.2780 2.0113 1.26 22 1971 1.4360 1.3532 2.2947 1.3343 1972 1.4870 1.4 241 2.6702 1-4173 TABLE V.2 CONTIHUED EXPENDITURE SHARES YEAR FOOD DURABLES LEISURE MISCELLANl 1946 0.3440 0.0895 0.4787 0.0878 1947 0.3037 0.0847 0.5154 0*0963 1948 0.2809 0.0740 0.5399 0.1051 1949 0.2807 0-0676 0.5495 0.1022 1950 0.2762 0-0570 0.5640 0.1028 1951 0*2623 0.0705 0.5714 0.0958 1952 0-2609 0*0776 0.5671 0.0944 1953 0*2648 0-0936 0.5367 0.1049 1954 0.2606 0-0937 < 0.5372 0.1085 1955 0-2557 0.0959 0.5341 0-1143 1956 0.2536 0-1023 0.5267 0-1174 1957 0-2455 0-1049 0.5304 0.1192 1958 0-2417 0.1058 0.5307 0-1218 1959 0.2353 0.1081 0.5308 0.1258 1960 0-2272 0-1126 0.5262 0.1340 1961 0.2135 0.1149 0.5370 0.1346 1962 0.2012 0.1127 0.5470 0.1391 1963 0.1942 0.1106 0.5533 0.1419 1964 0.1865 0.1104 0-5583 0.1448 1965 0.1811 0.1077 0.5682 0.1430 1966 0-1808 0.1112 0.5593 0-1488 1967 0.1758 0.1105 0.5674 0.1464 1968 0.1700 0.1105 0-5720 0.1475 1969 0.1637 0.1099 0.5801 0.1463 1970 0.1548 0.1093 0.5924 0.1435 1971 0-1498 0.1103 0.5992 ©.1408 1972 0.1436 0.1084 0.6068 0.1412 - 9 9 -TABLE V . 2 COHTIHOED YEAR HEAS EXP THETA 1946 4 5 . 3 8 3 3 1.0516 1947 109 .2261 1 .0280 1948 2 1 8 . 8 9 2 7 1 .0238 1949 2 9 7 . 8 3 7 2 1.0190 1950 3 2 3 . 5 7 9 6 1.0195 1951 3 9 5 . 6 1 6 2 1.0180 1952 4 6 4 . 6 1 4 0 1.0170 1953 5 2 8 - 8 6 6 7 1.0167 1954 5 8 5 - 8 9 7 7 U 0 1 6 4 1955 6 1 4 . 9 7 5 6 1.0128 1956 6 3 6 - 7 5 4 1 1.0127 1957 6 8 7 - 2 6 8 3 1.0146 1958 7 1 4 . 3 2 1 1 1-0150 1959 7 4 9 . 9 2 1 1 1.0168 1960 8 1 1 . 0 9 3 0 1.0186 1961 9 2 1 . 7 1 5 6 1. f0209 1962 1 0 6 9 . 2 9 2 7 1.0270 19B3 1 2 3 2 . 2 8 9 8 1.0257 1964 1 3 8 7 . 9 6 3 1 1.0248 1965 1 5 6 4 . 9 5 0 7 1-0214 1966 1712w,1538 1.0220 1967 1 9 3 0 . 3 2 7 2 1.0208 1968 2 1 9 4 . i 4 6 2 4 . 1.0205 1969 2 5 4 1 - 2 6 6 1 1.0241 . 1970 2896*8701 v. 1.0246 1971 3263^9353 1.0279 1972 3 7 2 9 . 5 5 9 8 1.0280 0 - 100 -T A B L E V . 3 T H E I N T E R T E H P O R A L H O B E L ' P R I C E S YEAR F O O D D U R A B L E S L E I S U S . 1 9 4 6 0 . 1 3 3 0 0 . 1 1 7 0 0 . 0 2 4 5 1 9 4 7 0 . 2 6 7 0 0 . 2 8 5 4 0 . 0 5 8 5 194 8 0 * 4 4 7 0 0 . 5 0 3 5 0 . 1 2 0 7 1 9 4 9 0 . 5 6 1 0 0 . 6 5 6 9 0 . 1 6 6 4 1 9 5 0 0 . 5 0 2 0 0 . 5 5 8 4 0 . 1 8 9 7 1951 0 . 5 7 7 0 0 . 7 5 5 8 0 . 2 3 8 1 1 9 5 2 0 . 6 4 6 0 0 . 7 4 8 3 0 . 2 7 5 7 1 9 5 3 0 . 6 8 3 0 0 . 7 9 7 9 0 . 3 0 1 0 1 9 5 4 0 . 7 3 4 0 0 . 7 9 9 2 0 . 3 3 0 2 1 9 5 5 0 . 7 1 6 0 0 . 7 7 5 8 0 . 3 4 6 6 1 9 5 6 0 . 7 0 2 0 0 . 7881 0 . 3 6 3 9 1 9 5 / 0 . 7 2 5 0 0 . 8 0 8 1 0 . 3 9 6 1 1 9 5 8 0 . 7 1 5 0 0 . 7 9 5 6 0 . 4 1 2 6 1 9 5 9 0 . 7 1 7 0 0 . 8 0 1 4 0 . 4 4 0 6 1 9 6 0 0 . 7 4 0 0 0 . 8 2 2 6 0 . 4 9 0 4 1961 0 . 7 8 0 0 0 . 8 7 4 2 0 . 5 7 2 8 1 9 6 2 0 . 8 3 0 0 0 . 9 1 0 7 0 . 6 6 6 7 1 9 6 3 0 . 8 9 7 0 0 . 9 4 4 8 0 . 7 7 4 1 1964 - 0 . 9 2 6 0 0 . 9 7 5 1 0 . 8 7 7 1 1 9 6 5 1 . 0 0 0 0 1 . 0 0 0 0 1 . 0 0 0 0 1 9 6 6 1 . 0 3 4 0 1 . 0 4 4 7 1 . 1 1 1 5 1 9 6 7 - - - 1 . 0 7 8 0 1 . 0 9 1 2 1 . 2 8 2 0 1 9 6 8 1 . 1 5 9 0 1 . 1 3 9 1 1 . 4 7 7 4 1 9 6 9 1 . 2 3 6 0 1 . 1 8 9 8 1 . 7 0 4 9 1 9 7 0 1 . 3 5 3 0 1 . 2 7 8 0 2 . 0 1 1 3 1 9 7 1 1 . 4 3 60 1 . 3 5 3 2 2 . 2 9 4 7 1 9 7 2 1 . 4 8 7 0 1 . 4 2 4 1 2 . 6 7 0 2 - 101 -T A B L E V . 3 C O N T I N 0 E D P R I C E S YEAR H I S C E F O T COMP F O T L E I S O RE 1 9 4 6 0 . 0 4 3 9 0 . 1 4 3 7 0 . 0 2 7 5 1 9 4 7 0 . 1 0 7 4 0 . 2 7 8 1 0 . 0 6 3 2 1 9 4 8 0 . 2 6 5 4 0 . 4 4 3 8 0 . 1 2 2 7 1 9 4 9 0 . 3 9 3 3 0 . 5 9 0 6 0 . 1 7 0 2 1 9 5 0 0 . 4 1 7 0 , 0 . 5 4 8 5 0 . 1 9 5 2 1951 0 . 4 8 6 1 0 . 6 6 0 9 0 . 2 4 6 2 1 9 5 2 0 . 5 9 1 7 0 . 7 3 6 6 0 . 2 8 6 5 1 9 5 3 0 . 6 4 6 6 0 . 6 3 2 9 0 . 2 8 39 1 9 5 4 0 . 6 8 1 0 0 . 6 6 5 9 0 . 3 2 9 8 1 9 5 5 0 . 6 9 4 9 0 . 7 3 7 2 0 . 3 4 7 3 1 9 5 6 0 . 7 0 8 8 0 . 7 3 7 6 0 . 3 6 5 3 1 9 5 7 0 . 7 2 2 9 0 . 7 5 7 6 0 . 3 9 8 2 1 9 5 8 0 . 7 2 4 5 0 . 7 5 0 5 0 . 4 1 5 6 1 9 5 9 r 0 . 7 4 1 1 0 . 7 5 8 9 0 . 4 4 4 6 1 9 6 0 0 . 7 6 6 9 0 . 7 7 2 0 0 . 4 9 2 3 1 9 6 1 0 . 8 0 1 2 0 . 8 1 0 0 0 . 5 7 4 6 1 9 6 2 0 . 8 8 2 9 0 . 8 6 7 1 0 . 6 6 8 3 1 9 6 3 0 . 9 0 2 4 0 . 9 1 1 2 0 . 7 7 5 3 1 9 6 4 ' • 0 . 9 3 5 7 0 . 9 4 1 9 0 . 8 7 7 H 1 9 6 5 0 . 9 9 8 2 1 . 0 0 0 0 1 . 0 0 0 0 1 9 6 6 1 . 0 6 3 0 1 . 0 2 4 2 1 . 0 9 1 3 1 9 6 7 1 . 0 9 4 1 1 . 0 6 4 3 1 . 2 6 2 7 1 9 6 8 1 . 1 4 4 4 1 . 0 8 4 / 1 . 4 5 0 4 1 9 6 9 1 . 1 8 4 4 1 . 1 3 9 0 1 . 6 7 8 8 1 9 7 0 1 . 2 6 2 2 1 . 2 1 3 2 1 . 9 7 3 8 1 9 7 1 - 1 . 3 3 4 3 1 . 3 1 8 9 2 . 2 6 1 1 1 9 7 2 1 . 4 1 7 3 1 . 3 9 6 8 2 . 6 8 3 5 - 102 -TABLE V .3 CONTINUED E X P E N D I T U R E YEAR FOOD DURABLES 1 9 4 6 0 . 0 2 7 0 0 . 0 0 7 0 1 9 4 7 0 . 0 2 5 2 0 . 0 0 7 0 1 9 4 8 0 . 0 2 4 3 0 . 0 0 6 4 1 9 4 9 0 . 0 2 4 3 0 . 0 0 5 8 1 9 5 0 0 * 0 2 2 4 0 . 0 0 4 6 1 9 5 1 0 . 0 2 0 8 0 . 0 0 5 6 1 9 5 2 0 . 0 2 0 7 0 . 0 0 6 2 1 9 5 3 - 0 . 0 2 3 1 0 . 0 0 8 2 1 9 5 4 0 . 0 2 1 5 0 . 0 0 7 7 1 9 5 5 0 . 0 2 0 6 0 . 0 0 7 7 1 9 5 6 0 . 0 2 0 2 0 . 0 0 8 1 1 9 5 7 0 . 0 1 9 2 0 . 0 0 8 2 1 9 5 8 0 . 0 1 9 0 0 . 0 0 8 3 1 9 5 9 - 0 . 0 1 8 2 0 . 0 0 8 4 1 9 6 0 0 . 0 1 7 4 0 . 0 0 8 6 1 9 6 1 0 . 0 1 6 1 0 . 0 0 8 7 1 9 6 2 0 . 0 1 5 3 0 . 0 0 8 6 1 9 6 3 ' 0 . 0 1 4 9 0 . 0 0 8 5 1 9 6 4 0 . 0 1 4 5 0 . 0 0 8 6 1 9 6 5 0 . 0 1 4 1 0 . 0 0 8 4 1 9 6 6 0 . 0 1 4 2 0 . 0 0 8 8 1 9 6 7 ' 0 . 0 1 3 7 0 . 0 0 8 6 1 9 6 8 ' 0 . 0 1 3 2 0 . 0 0 8 6 1 9 6 9 ;: 0 . 0 1 2 7 0 . 0 0 8 5 1 9 7 0 0 . 0 1 2 0 0 . 0 0 8 4 1 9 7 1 0 . 0 1 1 6 0 . 0 0 8 5 1 9 7 2 ' - 0 . 0 1 0 9 0 . 0 0 8 2 SHARES LEISURE MISCELLANEOUS 0 . 0 3 7 5 0 . 0 0 6 9 0 . 0 4 2 8 0 . 0 0 8 0 0 . 0 4 6 6 0 . 0 0 9 1 0 . 0 4 7 5 0 . 0 0 8 8 0 . 0 4 5 7 0 . 0 0 8 3 -0 . 0 4 5 4 0 . 0 0 7 6 0 . 0 4 51 0 . 0 0 7 5 0 . 0 4 6 7 0 . 0 0 9 1 0 . 0 4 4 4 0 . 0 0 9 0 0 . 0 4 3 1 0 . 0 0 9 2 0 . 0 4 2 0 0 . 0 0 9 3 0 . 0 4 1 5 0 . 0 0 9 3 0 . 0 4 1 7 0 . 0 0 9 6 0 . 0 4 1 0 0 . 0 0 9 7 0 . 0 4 0 3 0 . 0 1 0 3 0 . 0 4 0 4 0 . 0 1 0 1 0 . 0 4 1 7 0 . 0 1 0 6 0 . 0 4 2 5 0 . 0 1 0 9 0 . 0 4 3 4 0 . 0 1 1 2 0 . 0 4 4 2 0 . 0 1 1 1 0 . 0 4 4 0 0 . 0 1 1 7 0 . 0 4 4 1 0 . 0 1 1 4 0 . 0 4 4 3 0 . 0 1 1 4 0 . 0 4 4 9 0 . 0 1 1 3 0 . 0 4 5 7 0 . 0 1 1 1 0 . 0 4 6 4 0 . 0 1 0 9 0 . 0 4 6 1 0 . 0 1 0 7 ; - 103 -T A B L E V . 3 C O N T I N U E D Y E A R F U T U R E S H A R E MEAN W E A L T H 1 9 4 6 0 . 9 2 1 6 1 1 7 . 7 1 6 4 1 9 4 7 0 . 9 1 7 0 2 6 7 . 3 0 5 4 1 9 4 8 0 . 9 1 3 6 5 0 9 . 7 3 8 0 1 9 4 9 0 . 9 1 3 5 6 9 4 . 3 3 5 7 1 9 5 0 0 . 9 1 8 9 . 7 9 5 . 2 3 9 5 1 9 5 1 0 . 9 2 0 5 9 9 7 . 5 5 8 4 1 9 5 2 0 . 9 2 0 6 1 1 7 1 . 9 4 6 3 1 9 5 3 0 . 9 1 2 9 1 2 1 9 . 5 1 7 8 1 9 5 4 0 . 9 1 7 4 1 4 2 4 . 5 2 8 8 1 9 5 5 0 . 9 1 9 3 1 5 3 3 . 8 9 1 4 1 9 5 6 0 . 9 2 0 4 1 6 3 8 . 3 2 7 9 1 9 5 7 0 . 9 2 1 9 1 8 3 6 . 0 5 7 1 1 9 5 8 0 . 9 2 1 5 1 9 3 6 . 5 2 7 3 1 9 5 9 0 . 9 2 2 7 2 1 0 4 . 1 6 7 2 1 9 6 0 0 . 9 2 3 3 2 3 3 9 . 1 3 0 4 1961 0 . 9 2 4 7 2 7 6 2 . 1 3 6 5 1 9 6 2 0 . 9 2 3 8 3 2 3 3 . 4 5 2 2 1 9 6 3 0 . 9 2 3 1 3 7 7 0 . 9 0 2 6 1 9 6 4 0 . 9 2 2 3 4 2 9 6 . 7 2 2 7 1 9 6 5 0 . 9 2 2 3 4 9 3 2 . 4 1 0 2 1 9 6 6 0 . 9 2 1 2 5 4 2 2 . 1 2 5 0 1 9 6 7 0 . 9 2 2 3 6 3 0 8 . 4 6 0 9 1 9 6 8 0 . 9 2 2 6 7 2 9 1 . 2 6 9 5 1 9 6 9 0 . 9 2 2 6 8 4 5 2 . 8 2 4 2 1 9 7 0 0 . 9 2 2 8 1 0 0 3 9 . 6 4 5 0 1971 0 . 9 2 2 6 1 1 4 0 2 . 6 7 6 0 1 9 7 2 0 . 9 2 4 0 1 3 2 6 5 . 4 3 8 0 - 104 -FOOTNOTES to Chapter V 1. The pre-war data and the post-war data are not necessarily consistent, for the sources, d e f i n i t i o n s , and methods of derivations d i f f e r over time. Where possible, we have t r i e d to estimate series which are consistent. Dummy variables may be used to account for possible inconsis-tencies i n data and changes i n taste, but .will seriously complicate the nonlinear estimation procedure. 2. Fisher ideal index numbers for price index (Pj^) and quantity index (XT-.) are defined as follows: n i n i • al'Txl I T 0 , P I d t p , p ,x ,x ) _ L Q T ± • Q T Q J p x, p X C P E . P L ] J S n i \r, i , ITT. 1 . OT 1 , X I d C p ,P -,x ,x ) - l - . 0 T Q J p x p X where p^ ,p"^" ,x^ ,x"*" are vectors of prices and quantities for N goods i n the base period 0 and the period i n question 1. Pp and Xp are Paache price and quantity indices and P L and X^ are Laspeyres indices. In t h i s study, a fixed base period i s used; thus, t h i s i s hot a chain index. It can be shown that Fisher i d e a l indices are exact for a homogeneous quadratic u t i l i t y function. See Diewert[1974a], I t i s of i n t e r e s t to note that the ranking of households according to l i f e t i m e wealth (which)includes expected fu-ture income from human c a p i t a l , but does-not include l e i -sure) i s i d e n t i c a l to the ranking of seventeen groups c l a s s i f i e d by present income l e v e l . Therefore, i f one wishes to maintain that the l i f e t i m e wealth d i s t r i b u t i o n Banki-ng.:isrthemsame as thatiofoincome for a l l years, the l i f e t i m e wealth d i s t r i b u t i o n may be approximated by the income d i s t r i b u t i o n . If one i s w i l l i n g to further assume - 105 -FOOTNOTES for Chapter V (con't) that the l i f e t i m e wealth d i s t r i b u t i o n can approximate the f u l l l i f e t i m e wealth (including current and future leisure) d i s t r i b u t i o n , then the income d i s t r i b u t i o n variable may be used as a proxy for the f u l l l i f e t i m e wealth d i s t r i b u t i o n . - 106 -Chapter VI EMPIRICAL RESULTS A. Introduc t ion This chapter reports empirical results obtained from e s t i -mating the expenditure share functions for the aggregate econ-omy, derived from the inverse i n d i r e c t translog u t i l i t y func-t i o n ; and the share functions for the 'representative' consumer, derived from the inverse i n d i r e c t Generalized Leontief u t i l i t y function, based on time series data for 1946-1972. Three models were i l l u s t r a t e d i n previous chapters. They are: 'the three good model 1 which deals with food, consumer durables and miscellaneous; 'the l e i s u r e model' i n which l e i -sure i s added to the above three goods; and 'the intertemporal model' which includes the above mentioned four and, i n addition, a future composite good and future l e i s u r e . The present chapter reports the estimates of parameters; 2 asymptotic t s t a t i s t i c s ; log of the l i k e l i h o o d functions; R ; l i k e l i h o o d r a t i o tests; p r i c e , income and substitution e l a s t i -c i t i e s and an index of i n d i r e c t u t i l i t y . The f i r s t two atemporal models are estimated i n four ver-sions, imposing d i f f e r e n t constraints. Furthermore, each version i s estimated with or without expenditure d i s t r i b u t i o n variable. We denote these (+6) and (-0) respectively. The version without any constraint i s denoted by NS+NH ( i . e . , non-symmetric and non-homothetic). The version with symmetry im-posed i s denoted by SYM, that with homotheticity imposed by HOM, and the version with both symmetry and homotheticity im-posed by S+H. - 107 -B. The Three Good Model Three goods are denoted by numbers one to three; food by 1, consumer durables by 2 and miscellaneous by 3. F i r s t , we estimate the most r e s t r i c t i v e version, i . e . , S+H, of the three good model. This involves nonlinear estimation of six parameters, since s i x out of the twelve o r i g i n a l parameters are not free to vary due to the normalizations and the constraints of symmetry and homotheticity. Both the symmetry (SYM) and homo-t h e t i c i t y (HOM) versions contain eight free parameters. The unconstrained (NS+NH) version has ten free parameters. The parameter estimates are presented i n Table VI.1.- The values i n the parentheses are asymptotic t s t a t i s t i c s . The parameter estimates seem to be r e l a t i v e l y stable, re-gardless of the constraints imposed. For example, the sign of the parameters appear to be p a r t i c u l a r l y stable for parameter estimates which are s t a t i s t i c a l l y s i g n i f i c a n t (at .05). The values of the log of the l i k e l i h o o d functions (InL) show that estimates with the expenditure d i s t r i b u t i o n variable have attained s l i g h t l y higher InL values than ones with the same constraints but without the d i s t r i b u t i o n variable. The InL values are f a i r l y close except for S+H, suggesting a pos-s i b i l i t y of accepting n u l l hypotheses of symmetry or homothe-t i c i t y . The l i k e l i h o o d r a t i o test r e s u l t s are reported schemati-c a l l y i n Figure 1. The test for symmetry with the expenditure - 10 8 -FIGURE 1 THE LIKELIHOOD RATIO TESTS THE THREE GOOD MODEL k = 10 kk -•= 5 C r i t i c a l Values .001 .01 13.815 9.210 16.268 11.341 20.517 15.086 ks nu.ios:- of f r „• Chi-Square k: numBer o f f r e e q p a i P a m e t e r s f roa:'1'.'. 2 q: n u m b e r of i n d e p e n d e n t 3 p a r a m e t e r r e s t r i c t i o n s 5 - 1 0 9 -d i s t r i b u t i o n variable ( S Y M(+6)) i s accepted at the . 1 % l e v e l of sig n i f i c a n c e . The l i k e l i h o o d r a t i o t e s t s t a t i s t i c i s 1 3 . 7 7 5 , while the chi-square c r i t i c a l value with -three degrees of -free-dom i s 1 . 6 . 2 6 . 8 at . 1 % l e v e l . S Y M ( - e ) i s also accepted at . 1 % . The te s t for homotheticity conditional on symmetry ( S + H ) i s de c i s i v e l y rejected. The l i k e l i h o o d r a t i o test s t a t i s t i c s are 9 7 . 3 3 6 ( H Q : S + H and E ± : S Y M ( £ 0 ) ) and 9 9 . 3 0 7 ( H Q : S + H and H^ : S Y M ( + G ) ) . i ^ e j T h e .chi-square c r i t i c a l value withifiv'e degreesobf freedom i s . 2 0 . 5 1 7 . a t . 1 % . The loop on the r i g h t hand side of Figure 1 , i . e . , the homotheticity hypothesis, does not present any t h e o r e t i c a l l y i n t e r e s t i n g r e s u l t . Without symmetry, i t i s meaningless to discuss homotheticity alone. Returning to. Table V I l l , r the measurement of goodness of 2 f i t , R , are l i s t e d at the bottom of the table. Our f i t seems 2 to be excellent, except for S + H . S Y M(+6) obtained the R figures of . 9 8 4 4 , . 9 0 1 3 and . 9 4 9 1 for equations 1 , 2 and 3 re-spectively . The estimates of e l a s t i c i t i e s are presented i n Table V I . 2 . for selected years ( 1 9 5 0 , 1 9 5 5 , 1 9 6 0 , 1 9 6 5 and 1 9 7 0 ) . These estimates are computed using the parameter estimates of S Y M ( + 8 ) G ^ J stands for the Allen e l a s t i c i t y of substitution between the i t h good and the j t h good, for the expenditure e l a s t i -c i t y of good i , and n ^ j for the cross-price e l a s t i c i t y of good i i n response to a change i n the price of good j , where i , j = l , 2 - 1 1 0 -All...own e l a s t i c i t i e s of substitution are negative at a l l sample points. This is necessary but not sufficient for a correct curvature. A l l three goods turn out to be. substitutes, since a^j (i ^ j) are a l l positive. Food and durables ( o ^ ) are, however, relatively poor substitutes, while durables and miscellaneous (^23) seem to be more substitutable. Monotonicity is satisfied 'globally 1, since the gradient of the inverse indirect u t i l i t y function is positive at a l l sample points. The curvature condition is checked by investi-gating the matrix of e l a s t i c i t i e s of substitution. Determin-ants of principal minors of a l l order are calculated and are found to alternate in sign properly at every sample point. Thus, the curvature condition i s satisfied globally. The expenditure elasticity for food i s , as expected, sig-nificantly less than one. The values decrease slightly over the years. This is reasonably consistent with the rapid rate of economic growth and the resulting rise in real income in Japan over the 1 9 4 6 - 7 2 period. Both durables and miscellaneous are found to be luxury goods, with the expenditure elas t i c i t y estimates ranging from 1 . 3 2 to 1 . 5 3 . It is interesting to note that in each year, the elasticity estimates are very close for these two goods. The own-price e l a s t i c i t i e s are a l l negative. Food and durables are own-price inelastic, while miscellaneous is only slightly elastic. A l l cross-price e l a s t i c i t i e s are negative except n-|^, which is very close to zero. - I l l -C. The Leisure Model In t h i s model, food and durables are denoted by subscripts 1 and 2 respectively as i n the three good model. However, i t should be noted that subscript 3 i s used for l e i s u r e instead of miscellaneous which now i s denoted by 4. The parameter estimates presented i n Table VI.3 are from the model when 6 hours are assumed as l e i s u r e time for those who are not i n the labor force. The estimating procedure i s the same as the previous model. The estimates of parameters do not seem to be as stable as those i n the three good model. In the unconstrained version (NS+NH), half (10) of the parameters change sign, when the expen-diture d i s t r i b u t i o n variable 6 i s introduced, although only f i v e of them are s i g n i f i c a n t i n both (-6) and (+6). When symmetry i s imposed, only one parameter (y^) changes sign. The parameter y-^  i s not s t a t i s t i c a l l y s i g n i f i c a n t i n SYM(-e), but s i g n i f i c a n t i n SYM(+6). There are two sign re-versals i n the homotheticity version. Although the number of reversals i s small and the parameter estimates are r e l a t i v e l y stable for SYM and HOM, the r e s u l t from NS+NH i s rather disturb-ing. This could r e s u l t from the estimated expenditure d i s t r i -bution variable being a poor proxy for expenditure d i s t r i b u t i o n when l e i s u r e i s introduced i n consumption. - 112 -2 The log of the l i k e l i h o o d functions (InL) and the R f i g -ures are presented i n the second page of Table VL.3. Unlike the case i n the three good model, InL for S+H i s r e l a t i v e l y close, to those for the other versions. 2 The R values for both consumer durables and l e i s u r e are lower than those for food and miscellaneous. However, consid-ering elements of a r b i t r a r i n e s s i n the way l e i s u r e consumption 2 i s estimated, the r e s u l t i n g R figures are s a t i s f a c t o r y . For example, these values are .9720, .7232, .6682, .8969 for equa-tions 1, 2, 3 and 4 respectively i n SYM(-e). Tests of n u l l hypotheses are schematically presented i n Figure 2. Symmetry i s rejected. Homotheticity may be accepted conditional on symmetry,; (HQ : S+H and H^: S(-0)) when the ex-penditure d i s t r i b u t i o n variable i s ignored. However, i f we do not hold symmetry as a maintained hypothesis, simultaneous sym-metry and homotheticity are d e c i s i v e l y rejected (HQ: S+H and H 1: NS+NH). The monotonicity condition i s found to be s a t i s f i e d glo-b a l l y i n SYM ( - e ) , SYM(+0) and S+H. This r e s u l t holds when a l -ternative l e i s u r e hours are assigned to those not i n the labor force. The curvatureccondition i s s a t i s f i e d at most but not at a l l observations. The sample periods for which the condition i s s a t i s f i e d does depend upon the choice of l e i s u r e time (x^) and are summarized below:-- 113 -.FIGURE. 2 THE LIKELIHOOD RATI.O TEST" THE LEISURE MODEL k = 18 S+H k = 9 number of free parameters number of independent parameter r e s t r i c t i o n s Chi-Square C r i t i c a l Values q .001 .01 3 16.268 11.345 6 22.457 16.812 9 27.877 21.666 - 114 -x 3 = 0 = 2 x 3 = 4 x 3 = 6 = 8 1957-72 1955-72 1954-72 1953-72 1953-72 The estimates of e l a s t i c i t i e s from SYM(-e) when x^ = 6 are presented i n Table VT.4;, A comparison of e l a s t i c i t y estimates with d i f f e r e n t values of x^ reveals that the estimates are not very sensitive to the choice of x^ except when x^ = 0. However, when x^ = 0, the underlying assumption i s that those who are not i n the labor force do not have any l e i s u r e which commands posi t i v e opporutnity costs. This i s an unreasonable assumption and we may say that 'unreasonableness' i s confirmed by the re-sul t s . The own e l a s t i c i t i e s of substitution are of the proper sign g l o b a l l y , except a 2 2 for the years 1946 and 1948-1950. Since the e a r l i e r years do not s a t i s f y the curvature con-d i t i o n , we may ignore the e l a s t i c i t y estimates for 1950. Cross e l a s t i c i t i e s of substitution indicate that food and l e i s u r e (cx-j^ ) , food and miscellaneous (tf-^) , durables and l e i s u r e (^23) are substitutes. In pa r t i c u l a r / food and miscellaneous are found to be good substitutes. On the other hand, food and durables (tf^) a P P e a r to be complements. The relationship be-tween durables and miscellaneous ( a24^ seems to be changing rapidly from that of complements to substitutes. Both food and l e i s u r e are. found to be nec e s s i t i e s . Small expenditure e l a s t i c i t i e s for l e i s u r e are rather surprising. One may interpret? t h i s r e s u l t , however, as the c h a r a c t e r i s t i c - 115 -work ethic of the Japanese people and as the dr i v i n g force of a. remarkable economic growth during the period. Durables and mis-cellaneous are found to be luxury goods. It appears that the r i s i n g r e a l income i s lar g e l y channeled into consumption of 'services' from durable goods and other types. A large propor-tion of miscellaneous consumption i s on so c a l l e d 'services' such as medical and health care; transporation and communication; s o c i a l expenses and recreation; and education. If one assumes that l e i s u r e i s enjoyed by j o i n t l y consuming other consumption goods, then the production of l e i s u r e a c t i v i t i e s may have be-come more ' c a p i t a l ' and 'service' intensive, without a propor-tionate increase i n the input of time by the consumer. Consistent with economic theory, a l l own-price e l a s t i c i t i e s are negative. Food, durables and l e i s u r e are own-price i n e l a s -t i c , while miscellaneous i s s l i g h t l y own-price e l a s t i c . It i s not s u r p r i s i n g that the f i r s t three goods are own-price i n e l a s -t i c , i n view of lack of good substitutes due to a high degree of aggregation (of goods). Recall that the same results were obtained i n the three good model. A l l cross-price e l a s t i c i t i e s are negative except for n - ^ and n24 - Since a l l goods are normal, p o s i t i v e ri-^ 4 and are consistent with p o s i t i v e a-^ and cr-^ - Notice that and 0^4 are p o s i t i v e and r e l a t i v e l y large, while n - ^ and are r e l a t i v e l y smalls- I t i s useful to r e c a l l the rel a t i o n s h i p be-tween the three e l a s t i c i t i e s , i . e . , - 116 -^ i j = s j ( C T i j " n i y ) i ' j = i " - " 4 -Thus, i f the i t h good i s normal, then CT.. > n. > 0 for ' ^ I J l y n^ j to be p o s i t i v e . Our e a r l i e r suspicion that the estimate of the expenditure d i s t r i b u t i o n variable 0 may not be a good proxy i s further sup-ported by the r e s u l t on the curvature condition for SYM(+0). Only a few sample years s a t i s f y the curvature condition, the longest span being 1968-1972 when x^ = 6 and the shortest being zero when x^ = 8. When both symmetry and homotheticity (S+H) are imposed, both monotonicity and curvature are s a t i s f i e d g l o b a l l y . The estimates of e l a s t i c i t i e s are presented i n Table VT.5. The own,elasticities of substitution are, of course, a l l negative. The relationships of substitutes and complements among goods are exactly the same as i n SYM(-0). However, s u b s t i t u t a b i l i t y between durables and l e i s u r e (^23) / a n d between l e i s u r e and miscellaneous. (0^4) i s found to be much greater i n S+H than i n SYM ( - e ) Since homotheticity i s imposed, a l l expenditure e l a s t i c i -t i e s must be unity. The own-price e l a s t i c i t i e s are a l l negative, Unlike SYM(-0) i n which a l l but two cross-price e l a s t i c i t i e s (n-j^ and T I 3 4 ) are negative, six out of twelve i ^ j ' s ( i ^ j ) are p o s i t i v e i n S+H. In addition to the above two, • ^^2' ^23 - 117 -and n ^ 3 are p o s i t i v e . The net e f f e c t of the larger values of a n <3 CT43 a n <3 the smaller values of r^y a n c ^ r i ^ i s to change the signs of these four parameters from negative i n SYM(-O) to p o s i t i v e i n S+H. D. The Intertemporal Model In t h i s intertemporal model, numerical': subscripts one. to six are used to represent food, durables, l e i s u r e , miscellaneous, future composite good and future l e i s u r e respectively. Due to p r a c t i c a l d i f f i c u l t i e s , the model i s estimated only i n two versions. The f i r s t one imposes symmerty (SYM). Homo-t h e t i c i t y i s added i n the second version (S+H). The unconstrained version could not be estimated because of i t s enormousrsize (42 parameters). Since the model i s for the 'representative 1 consumer with the mean ' f u l l l i f e t i m e wealth', there i s no wealth d i s t r i b u -t i o n variable involved i n t h i s model. The parameter estimates for SYM and S+H are presented i n Table VI.6. Only the f i r s t four equations are estimated i n this model, since we do not observe the expenditure shares for 2 the future goods.. The R figures indicate that our f i t for the four estimates i n the model i s quite good. The l i k e l i h o o d r a t i o t e s t of homotheticity conditional on symmetry res u l t s i n the r e j e c t i o n of homotheticity. The test s t a t i s t i c i s 66.138, whereas the chi-square c r i t i c a l value at .001 i s 20.517 (FIGURE 3). - 118 -FIGURE 3 THE LIKELIFOOD RATIO TESTS THE INTERTEMPORAL MODEL fc = 25 -y-S 66.138 S&HH k = 20 number of free parameters er -'off independent parameter r e s t r i c t i o n s Chi-Square C r i t i c a l Value q .001 .01 5 20.517 15.086 - 119 -A problem with SYM turns out to be that both monotonicity and curvature conditions are violated everywhere. S+H s a t i s f i e d monotonicity global l y and exhibits proper curvature for 1950-1972. By checking the monotonicity and curvature conditions for SYM, i t i s found that the f i r s t p a r t i a l derivatives of h with respect to future l e i s u r e (hg) are a l l negative. A Bon-f e r r o n i t test shows that hg i s indeed negative and s t a t i s t i c -a l l y s i g n i f i c a n t (t =-9.46763). The own e l a s t i c i t i e s of sub-s t i t u t i o n for future composite good ( a ^ t j ) a n d future l e i s u r e (agg) are p o s i t i v e instead of the desired negative sign. Ideally, both monotonicity and curvature should be im-posed i n order to ensure a proper set of estimates as outlined i n Chapter IV. Reparameterization of t h i s model requires re-placing a set of parameters by lengthy equations consisting of an equal number of new parameters. Although, t h i s i s an alge-braic problem, i n practice, i t i s computationally i n f e a s i b l e to apply to a model as large as ours. Therefore, instead of attempting to impose both monotonicity and curvature, we s e t t l e d for just imposing monotonicity. We choose 194 6 to be the new base year, since v i o l a t i o n seems to be more serious i n the e a r l i e s t years. Another Bon-f e r r o n i t test shows that when 194 6 i s used as the point of expansion, h,- i s negative but i s not s t a t i s t i c a l l y s i g n i f i c a n t (t = -.9619) . Therefore, monotonicity is_ not rejected."'" - 120 -For purposes of comparison, symmetry (SYM) and symmetry with homotheticity (S+H) are reestimated with the rescaled data. The parameter estimates and asymptotic t s t a t i s t i c s for SYM and S+M (symmetry with monotonicity) are presented i n Table VI. 7 „ while those of S+H are presented i n Table VI. 8.., When monotonicity i s imposed, four out of s i x x*(i.e., T ^ , T | , TJ?,, T^) ; are found to be s i g n i f i c a n t and p o s i t i v e . i s p o s i t i v e but not s t a t i s t i c a l l y s i g n i f i c a n t . The l i k e l i -hood function i s maximized when T | equals to zero. This i s not surprising since T,- was negative p r i o r to the imposition of monotonicity. The R 2 estimates are .9956, .9075, .7145 and .8820 for the f i r s t four equations i n SYM, while they are .9909, .9019, .7994 and .9010 i n S+M. The imposition of monotonicity helped the curvature prop-erty of the model su b s t a n t i a l l y . The curvature condition i s now s a t i s f i e d for the periods between 1962 and 1972 i n c l u s i v e . The second p a r t i a l derivatives of h with respect to durables ( 1 ^ 2 2 ) a n d i t s own e l a s t i c i t y of substitution ( ^ 2 2 ^ a r e o r the wrong s i g m f o r 1946 and 1947, but they are negative for the rest of the periods (1948-1972). The estimates of e l a s t i c i t i e s based on the parameter e s t i -mates >ffomlS+Miare 4presented i n Table VI. 9V for selected years. Five out of f i f t e e n cross commodity relationships are found to be complements. These are food and durables ( a n o ) , food and - 121 -future composite good (o-^ g) / durables and future l e i s u r e (°2S^ ' present l e i s u r e and future composite good (0-35) r and miscellan-eous and future composite good (^45) • ^2 a n d a35 a r e P a r ^ i c -u l a r l y large i n absolute values. Food and durables (tf-j^) w e r e also found to be complements i n the l e i s u r e model. The remaining e l a s t i c i t i e s are p o s i t i v e . Of these, food and l e i s u r e ( 0 ^ 3 ), food and miscellaneous ( ^ 4 ) / durables and l e i s u r e ( 0 ^ 3 ), durables and miscellaneous ( 0 2 4 )/ durables and future composite good (^25) a n c ^ future composite good and fu-ture l e i s u r e (^55) appear to be. es p e c i a l l y good substitutes. It i s of p a r t i c u l a r i n t e r e s t to examine res u l t s from ( f u l l lifetime) wealth e l a s t i c i t y estimates. As we expected, present food has e l a s t i c i t i e s s i g n i f i c a n t l y less than unity. Durables, miscellaneous, future composite good and future l e i s u r e are found to be luxury goods, although the e l a s t i c i t y estimates of the two future commodities are only s l i g h t l y l a r -ger than unity. The most inter e s t i n g finding i s that present l e i s u r e appears to be an i n f e r i o r good for 1957-1972. The estimates for selected years are .149, .031, -.122, -.258 and -.367 for 1950, 1955, 1960, 1965 and 1970 respectively. This r e s u l t may be interpreted as follows. The 'representative' consumer in Japan feels that the higher f u l l l i f e t i m e wealth becomes, thernmore consumption of food, durables, miscellaneous, future composite good and future l e i s u r e he can afford. But he w i l l - 12/ -actually choose to transfer part of his expenditure from present l e i s u r e to the others goods. To further investigate t h i s r e s u l t , we examine time series labor-supply data. This data reveals a slow but steady decline i n labor supply per worker. In other words, l e i s u r e consumption per worker has increased over the post-war period. Therefore, one may conclude that the (nega-tive) wealth'effect isomeretthan compensated by complementary e f f e c t from the future composite good (^35/ ^35 < 0 ) • The r e l a -t i v e price of present l e i s u r e v i s - a - v i s other goods has gone up considerably over time. An a l t e r n a t i v e interpretation i s that labor force p a r t i c i -pation rates of the population increase with the l e v e l of f u l l l i f e t i m e wealth. Even though labor supply per worker i n the labor force decreases, t o t a l labor supplied i n the economy can be higher, i f the labor force participationsrates^become s u f f i -c i e n t l y high. Our 'representative consumer' can be considered as a mixture of a p a r t i c u l a r population composition. Therefore, his labor supply i s the average supply of the t o t a l 'popula-t i o n ' rather than that of the t o t a l labor force. Thus, i t i s possible to have r i s i n g per.Qapita•labor supply,while per. 2 worker labor supply may decrease. A l l own—price e l a s t i c i t i e s are negative. Two future goods, are own-price e l a s t i c ( n ^ are -1.149 and -2.054 and rigg are -1.098 and -1.072 for 1965 and 1970 r e s p e c t i v e l y ) , while a l l present goods are found to i n e l a s t i c . - 123 -Twelve out of t h i r t y cross-price e l a s t i c i t i e s are p o s i t i v e ; most, however, are r e l a t i v e l y small i n absolute value. Relative-l y large e l a s t i c i t y estimates are obtained for 1 1 2 3 (-666 a n d «550) .ri3gC.766 and .891) and n^g(.239 and . 1 2 5 ) , where two estimates i n parentheses are for 1965 and 1970 respectively. This paren-t h e t i c notation i s followed below. Among negative cross-price e l a s t i c i t i e s , r i 4 r ) ( - . 5 2 1 and -.468) n 2g (-3.683 and -3.230) and r i 4 g ( - . 6 7 5 and -.656) are r e l a t i v e l y large i n absolute value. Present durables and future l e i s u r e ( r ^ g ) are the only e l a s t i c r e l a t i o n s h i p s . Thus when the price of future l e i s u r e i s increased, the consumer w i l l decrease the consumption of present durables to a considerable degree, since these two goods are found to be r e l a t i v e l y good complements ( a 2 6 are -1.631 and - 1 . 4 3 9 ) . I t i s useful to r e c a l l that the cross-price e l a s t i c i t i e s are p o s i t i v e l y related to the expenditure share of the commodity whose price i s changing. This i s more relevant to an i n t e r -temporal model i n which the shares of two future goods are very large. I t follows that the estimates of and'ri^g, i = 1,...,6 are r e l a t i v e l y large. F i n a l l y , indices of i n d i r e c t u t i l i t y estimated by three d i f f e r e n t models are presented for selected years i n Table VI.11. The index i s set at unity for 1965. The i n d i r e c t u t i l i t y index i s defined by Diewert 1974; 1207] as 'the maximum u t i l i t y ' , the household (or the consumer) can a t t a i n , facing a given set of - 1 2 4 -prices and t o t a l expenditure (or wealth). I t i s reassuring to f i n d that the index estimates consistently increased over the years i n a l l three models. However, since the index i s ord i n a l , the absolute values do not provide any i n t e r e s t i n g inference. TAtSXiil, VX . X +NH SYM P a r a m e t e r ( - 6 ) ( + 6 ) ( - 6 ) . ( + 6 ) 2 .0404 1 .8668 1 .8513 1 .7490 (20 .2178) (19 .8032) (22 .2877) (22 .3379) — .3797 — .2966 _ .4403 _ .3420 (-3 .3753) (-2 .5569) (-6 .8532) (-3 23413) — .6607 — .5702 — .4403 — .4070 (-4 .8417) (-4 .7111) (-6 .8532) (-7 .2113) 1 .1718 1 .0367 .3028 .2968 (5 .3387) (4 .8092) (5 .9105) (5 .6572) — .4066 — .1250 _ .0771 — .0704 (-1-.1997) (- .3198) (-1 .34 68) (-1 .2106) — .5183 — .6961 _ .0080 _ .0288 (-1 .3986) (-1 .7194 (- .3623) (-1 .3843) .1807 .1451 (1 .9428) (1 .4156) — .0830 _ .0220 .0358 .0238 (- .6659) (- .1531) ( .5628) ( .3690) — .1944 — .2054 _ .0600 _ .0421 (-2 .0334) (-2 .0070) (-2 .3424) (-1 .7353) .1794 11252 (1 .5489) (1 .0503) — .0440 .0432 (- .2743) ( .2428) - .2855. — .3018 — .0487 — .0382 (-2 .3139) (-2 .3120) (-2 .4505) (-2 .1470) 82 .5085 82 .9267 75 .0639 76 .0491 ,9894 .9900 .9837 .9844 .9295 .9200 .9056 .9013 .9426 .9515 .9430 .9491 "12 13 21 Y-22 T23 Y31 Y'32 Y 3 3 I n L R 2 e q . l 2 3 HOM S+H (-0) (+8) 1 (17 .9516 .9085) 1 (18 .8311 .5566) (31 .4240 .0378) (-3 .3450 .0402) ( -2 .2874 .6995) (38 .2396 .8141) (-5 .6066 .6449) ( -5 .5436 .7694) (27 .2865 .1345) (3 .2351 .8751) (3 .2330 .9146) (1 .1353 .6877) (-1 .0931 .3751) ( -1 .0844 .2787) (1 .0455 .0081) (3 .0908 .4864) (2 .0611 .5809) (-3 .1808 .5098) (-1 .0649 .0281) (-.0587 .9145) ( .0032 .1459) (-.0063 .0886) (-3 .1263 .1149) (-1 .0295 .0894) (-.0156 .6114) (4 .0809 .5473) (-2 .1702 .8447) ( -3 .1742 .0658) (1 .0899 .3466) (1 .0907 .4391) (-2 .0612 .3851) ( -2 .0455 .0117) (2 .0999 .9090) 75 .6651 76 .5624 26 .3958 ,9827 .9025 .9517 .9832 .8985 .9561 .4306 .6089 .2050 - 1 2 6 -TABLE VI*2 T A U L J B « ESTIMATES OF E L A S T I C I T I E S BASED ON SYM(H-O) : THE THREE.GOOD MODEL 1950 1955 1960 1965 1970 a l l -.474 — .542 -.589 -.630 -.648 a 2 2 -3.102 -2.614 -2.276 -2.061 -1.856 §33 -3.642 -2.828 -2 .381 -1.993 -1.705 a12- .546 .457 .379 .300 .208 CT13 .987 .884 .804 .725 .641 a 2 3 .844 1.015 1.097 1.147 1.178 n i y .687 .647 .608 .571 .525 n 2 y 1.527 1.461 1.415 1.376 1.341 n 3 y 1.518 1.447 1.406 1.359 1.324 n n -:726 -.668 -.612 -.554 -.482 n 2 i -.614 -.564 -.530 -.496 -.466 n 3 l -.332 -.316 -.307 -.292 -.281 n 1 2 -.024 -.037 -.050 -.064 -.081 n 2 2 -.771 d.787 -.799 -.811 -.821 n 3 2 -.112 -.083 -.067 -.050 -.038 n l . 3 ' .062 .058 -.053 .047 .039 n 2 3 -.142 -.109 -.087 -.069 -.054 n 3 3 -1.074 -1.047 -1.032 -1.017 -1.006 (1) F o o d , (2) D u r a b l e s , (3) Leisure;, C x 3 - 6) NS+NH SYM ( - 6 ) ( + 8 ) ( - 6 ) ( + 9 ) a l ( -.1537 .3906) (2 .8522 .5324) (2 .8417 .3672) 1 (4 .1435 .2201) a 2 -1 (-4 .6477 .4056) -2 (-8 .2522 .5702) -1 (-5 .2752 .4186) (-3 .9834 .9810) a 3 3 (6 .4432 .7384) 3 ( .6420 » ) 2 (5 .2431 .2993) 1 (4 .7845 .5370) °4 (-1 .6419 .8043) -1 (-3 .2420 .0480) (-3 .8096 .2482) (-3 .9446 .5907) Y l l (-1 .1442 .3747) (-3 .2818 .0845) (3 .1001 .8278) (6 .1461 .2742) Y 1 2 (-1 .1452 .0965) (4 .4080 .8703) (-4 .0572 .4678) (-4 .0672 .4607) Y 1 3 X-2 .0990 .6554) XI .0796 .9329) (1 .0420 .3545) (2 .0667 .3974) Y 1 4 (6 .3418 .2324) (^5 .1171 .6351) ( .0042 .6104) (-2 .0183 .3882) Y 2 1 (-1 .0423 .4300) (-3 .1827 .9347) Y 2 2 (-2 .0910 .5435) (2 .1376 .6735) (2 .0276 .0090) (2 .0400 .7438) Y 2 3 (-3 .1787 .9090) (-6 .2401 .9239) (-4 .1071 .5544) (-3 .0887 .3878) Y 2 4 (4 .0731 .4703) (-1 .0293 .5630) (-6 .0515 .6369) (-3 .0293 .9407) (4) M i s c e l l a n e o u s HOM . S+H -H-:e) .(+8) (1 .6701 .2401) (2 .9723 .2139) (11 .1901 .0549) -2 (-3 .7853 .9902) ( -2 -3 .0933 .2589) (32 .1064 .4875) 4 (4 .4423 .0558) 3 (3 .3447 .3083) (116 .5654 .3550) -1 (-2 .3271 .1733) ( -1 -2 .2237 .4057) (70 .1381 .6422) (5 .1682 .2028) (5 .1859 .9503) (8 .1039 .9167) (-2 .0979 .9237) ( -3 .1083 .4989) (-2 .0242 .0282) (-.0463 .7734) (-.0059 .1117) (-26 .0871 .9174) ( (2 .0410 .7132) (3 .0321 .1502) (1 .0074 .0518) (-1 .0435 .0418) (-.0354 77751) (-.0074 .1711) (-.0268 .5916) (2 .0278 .0157) (-4 .3156 .0845) ( -3 .2593 .3673) (6 .3183 .4566) (-1 .0273 .3963) (1 .0283 .9008) (-6 .0355 .7359) NS+NH .(-e) ,(+e) SYM .(-6) .( + 6) BOM He) r+e) S.+H Y 3 1 -.1627 (-.8523) -.9767 (46 .8281) -'.0760 (-1.1598) -.0946 (-1.3154) Y 3 2 -.4098 (-1.6296) 1.1427 (8.5707) .1011 (1.4903) .1327 (1.8621) Y 3 3 .4213 (4.9776) .6640 (17.5377) .3489 (4.3573) .2705 (3.5387) .5002 (4.1255) .4045 (3.3370) .0292 (4 .4381) Y 3 4 .5429 (5.7402) -.4202 ( 0 0 ) -.0555 (-1.9844) -.0863 (-2.8546) .0027 (.0885) -.0720 (-3.0667) .0261 (7.9155) Y 4 1 -.0312 .'(-.8 795) -.2365 (-5.3333) -.0487 (-1.3328) -.0059 (-1.5439) Y 4 2 -.1130 (-2.4920) .2092 (4.5424) .0042 (.1097) .0024 (.0669) i Y 4 3 -.0537 (-1.1832) -.0922 (-1.8248) -.0165 (-2.0463) -.1393 (-2.2838) H to 00 Y 4 4 .0918 C7.6568) -.0644 (-1.4945) -.0263 (-3.0099) -.0103 (-1.5891) -.0165 (-.9635) .0115 U9751) .0019 (.2990) 1 I n L 175.575 178.589 160.375 169.937 169.697 169.724 152.693 R 2: 1 2 3 4 .9847 .8037 .9017 .8657 .9933 .7693 .7758 .9119 .9720 .7232 .6682 .8969 .9766 .6537 .6285 .9069 .9871 .7285 .8120 .8889 .9883 .7474 .7804 .8942 .9836 .3841 .5159 .8512 TABLE VI.4 - 129 -ESTIMATES OF E L A S T I C I T I E S BASED ON SYM (-9) : THE LEISURE MODEL ( x 3 = 6) 11 22 ' 33 r44 12 13 14 23 24 34 '1Y '2Y '3Y '4Y 11 ^21 ^31 n 4 1 n l 2 n 2 2 n32' n 4 2 *13 n 2 3 ^33 n 4 3 n!4 n 2 4 n34 n 4 4 1950 1955 1960 1965 197 0 -1.7611 -1.928 -2 .117 -2 .442 -2 .732 .339 -.982 -1.590 -2.394 -2 .866 -.404 -.425 -.450 -.485 -.501 -8.859 -6.818 -6.501 -5.611 -5.292 .293 -.020 -.271 -.572 -.827 .409 .4 30 .464 .517 .552 2.408 2.026 1.878 1.601 1.448 .322 .322 .475 .580 .665 -2.782 -.743 -.368 .286 .528 .808 .738 .794 .815 .846 .590 .564 .542 .521 .498 4 .191 3.421 3.157 2 .723 2.474 .464 .483 .533 .596 .644 2.627 2.292 2 .164 1.919 1.789 -.653 -.618 -.587 -.551 -.517 -1.084 -.844 -.756 -.612 -.528 -.015 -.013 -.015 -.015 -.015 -.061 -.066 -.063 -.059 -.054 -.022 -.051 -.080 -.119 -.152 -.291 -.416 -.470 -.559 -.614 -.011 -.014 -.006 -.002 -.002 -.408 -.286 -.251 -.178 -.145 -.099 -.072 -.043 -.002 -.031 *2.109 -1.656 -1.487 -1.209 -1.045 -.473 -.486 -.545 -.610 -.661 -.991 -.833 -.759 -.623 -.545 .185 .178 .168 .152 .140 -.708 -.506 —.-444 -.343 -.287 .035 .031 .033 .031 .030 -1.167 -1.107 -1.091 -1.058 -1.045 TABLE V I . 5 - 130 -ESTIMATES OF ELASTICITIES BASED ON S+H THE LEISURE MODEL 1950 119.55 1960 1965 1970 a 11 -1.276 -1.335 -1.395 -1.386 -1.148 a 22 -6.918 -6 .840 -6*446 -5.940 -5.406 a 33 -.766 -.734 -.699 -.678 -.655 a 44 -8.050 -7.461 -6.961 -6.141 -5.567 a 12 -.050 -.111 -.138 -.194 -.289 a 13 .400 .365 .302 .189 .029 a 14 1.253 1.254 1.269 1.283 1.316 a 23 1.700 1.673 1.601 1.529 1.463 o 24 -2.847 -2.524 -2.020 -1.413 -.968 a 34 1.449 1.412 1.379 1.335 1.303 n 1Y 1.0 1.0 1.0 1.0 1.0 n 2Y 1.0 1.0 1.0 1.0 1.0 n 3Y 1.0 1.0 1.0 1.0 1.0 n 4Y 1.0 1.0 1.0 1.0 1.0 n 11 -.617 -.586 -.536 -.454 -.336 n 21 -.284 -.279 -.254 -.227 -.202 n 31 -.163 -.160 -.156 -.154 -.152 n 41 .684 .064 .060 .054 .049 12 -.089 -.096 -.108 -.127 -.155 n 22 -.673 -.679 -.707 -.739 -.768 n 32 .059 .058 .057 .056 .055 n 42 -.327 -.305 -.287 -.257 -.236 n 13 -.322 -.347 -.390 -.458 -.557 n 23 .375 .367 .335 .299 .266 Tl 33 -.946 -.947 -.948 -.948 -.949 n 43 .241 .225 .211 .189 ..174 n 14 .027 .030 .0 33 .039 .047 n 24 -.418 -.410 -.373 -.333 -.296 n 34 .049 .048 .047 .046 .046 n 44 -.982 -.983 -.984 -.986 -.987 TABLE VI.6 - 131 -PARAMETER ESTIMATES, InL AND : THE .INTERTEMPORAL MODEL (Base Year = 1965) (1) Food, (2) Durables, (3) Leisure, (4) Miscellaneous, (5) Future Composite Good, (6) Future Leisure 3 \SYM parameter t - s t a t i s t i c .03474 2.1708 .09408 6.9300 -.14877 -3.4403 .04544 2.9324 4.04702 3.0993 -4.07252 -2.6431 -.00319 -1.4997 -.00185 -1.3624 .01299 2.2660 .00662 7.4067. .00810 2.6040 -.02575 -2.1892 .00104 .7409 .00554 1.1409 -.00004 -.0466 .00080 .3599 -.04403 -4.5604 .01284 .4249 -.00057 -.1335 -.00233 -.2609 .08829 2.6769 -.00522 -5.8072 -.00507 -3.3971 -.00763 -.8070 -.06455 -1.0163 -.61166 -1.7175 2.21209 2.7721 parameter t - s t a t i s t i c - .00258 -1.4628 - .00156 -1.3396 .01842 2.2995 .00649 8.0304 .00464 1.4532 .01117 -1.2315 - .00483 -1.9799 .02778 2.5934 - .00260 -3.0055 .00400 1.2509 - .01440 -1.2763 - .062744 -1.1602 .00585 1.2419 -.03396 -2 .5450 .08744 1.5984 - .00587 -8.0643 - .00456 -3.2150 .01161 2.5108 - .19814 -4.0781 .45998 8.0030 .15824 1.2101 - 132 -TABLE VI.6 ( c o n t i n u e d ) SYM S+H I n L 519.109 486.040 2 R 5. 1 .9921 .9949 2 .8750 .6357 3 .8533 .4958 4 .9122 .8747 TABLE VI.7 - 133 -PARAMETER ESTIMATES,vInL AND : THE INTERTEMPORAL MODEL (Base Year = 1946) (1) Food, (2) Durables, (3) Leisure, (4) Miscellaneous, (5) Euture Composite Good, (6) Future Leisure SYM S+M: Monotonicity parameter t - s t a t i s t i c parameter t - s t a t i s t B o i .01804 .9386 -.04147 -1.9116 h i .11024 7.6265 .11179 7.1636 B03 -.10494 -2 .8354 -.14186 -4.1219 e04 .03419 2.8266 .02697 2.2066 305 -2.03810 -1.3801 .02905 .0427 606 1.98057 1.2090 .01553 .0207 h i -.01045 -1.4790 .00287 .2521 h i -.00465 -1.3154 -.00686 -1.4041 h3 .01911 2.2253 .00598 .6068 P14 .01340 4.8769 .01101 3.8631 h s .02147 2.8248 .00335 .2977 g16 -.02122 -1.5229 .03105 1.6904 B22 .00101 .2656 .00109 .1695 B23 .01226 1.7579 .01470 2 .0038 B24 -.00025 -.1661 .00111 ,.7409 g25 .00229 .3924 -.00404 -.5872 he -.05921 -5.6324 -.06189 -5.4931 333 .00802 .3804 .01445 .5842 B34 .00164 .4702 .00099 .2579 S35 -.01475 -.9706 -.01000 -.5823 he .06605 2.1535 .08430 2.5176 644 -.00537 -3.7458 -.00640 -3.4707 h s --.01156 -2.8668 -.01211 -2.7116 he ri.00723 -.8243 -.00047 -.0515 h s -.93475 -3.4536 -.80343 -3.0768 he 1.50227 3.0633 .81171 6.2694 he -1.09769 -1.2192 .05376 .1151 - 134 -TABLE VI.7 ( c o n t i n u e d ) SYM S+M t - s t a t i s t i c p a r a m e t e r t - s t a t i s t i c T l - .02669. - 54.4671 x* .02666 54 .8060 T 2 .00657 20.0431 x 2 .00102 .00043 T 3 .03985 36.1922 T * .19872 84.2315 T 4 .00772 19.5764 T | .08731 42.3285 T 5 -.45334 -1.0955 T * 0.0 OO T 6 1.37252 3.3193 x* 6 .96241 387.547 I n L 517.348 507.553 2 R : 1 2 3 4 .9956 . 9075 .7145 .8820 .9909 .9019 .7994 .9010 TABLE VI.8 PARAMETER ESTIMATES , InL AND R' THE INTERTEMPORAL MODEL (Base Year = 1946) SYM .parameter t - s t a t i s t i c 611 §12 6 3 3 13 514 ! 1 5 '16 !22 !23 >24 !25 !26 •33 S34 535 36 '44 545 !55 !56 66 InL R2: 1 2 3 4 -..00912 -.00542 .02883 .01353 .01761 -.01872 -.01552 .04075 -.00503 .01446 -.02260 -.04231 .00508 -.05499 .06218 -.00700 -.00979 .01108 -.76659 .11731 486.041 . 9949 .6349 .4949 .8749 -1.3305 -1.4665 2.5036 4.2478 1.8043 -1.4096 -2.0192 2.6584 -2.3166 1.4032 -1.3676 -1.1909 1.0741 -2.1743 1.5717 -4.5114 -2.2607 2.6024 -2.5017 5.7160 1.0178 TABLE' VI . 9 - 136 -ESTIMATES OF E L A S T I C I T I E S BASED ON S+M: THE INTERTEMPORAL MODEL 1950 1955 1960 1 . 9 6 5 1970 -21.574 -23.885 -28.352 -34 .127 -39.418 130.551 -86.959 -66.684 -65.389 -62 .248 -8.708 -8.674 -8.691 -8 .270 -7.615 136.000 -104.898 -82.767 -67.205 -58 .041 -11.471 -6.676 -4.583 -3.256 -2.634 -.594 — .618 - .632 -.635 -.624 -38.928 -15.617 -10.404 -8.262 -6.699 1.227 1.282 1.427 1.484 1.404 29.027 24.700 21.200 17.443 14 .507 -.053 -.158 -.242 -.320 -.385 .370 .416 .479 .567 .646 88 .988 37.233 24 .962 19.718 15.709 50.490 18.591 10 .852 7.624 5.816 8 .283 4 .111 3.027 2.655 2.396 -6.088 -2.703 -1.863 -1.631 -1.439 1.985 1.584 1.145 .655 .260 -1.371 -1.365 -1.451 -1.516 -1.573 .528 .633 .752 .905 1.019 -3.170 -1.579 -.751 -.220 .046 .993 .898 .802 .704 .643 2 .575 1.997 1.668 1.403 1.243 .653 .629 .587 .546 .505 13.322 6.237 4 .565 3.961 3.588 .149 .031 -.122 -.258 -.367 2.036 1.936 1.834 1.730 1.664 1.038 1.035 1.035 1.041 1.049 1.010 1.015 • 1.021 1.032 1.044 TABLE V I . 9 ( c o n t i n u e d ) - 137 -1950 1955 " l l "21 n 3 1 n 4 1 -.494 -.494 -1.160 -.44 0 .240 . 025 .599 .458 " 5 1 -.024 -.024 "61 -.014 -.012 " l 2 -.076 -.071 " 2 2 -.277 -.409 n32 .171 .163 n42 "52 .093 .073 .014 .013 \2 -.014 -.016 n i 3 .026 .028 "23 "33 3.386 1.347 -.396 -.378 "43 "53 -.002 -.015 -.108 -.104 "63 "14 -.022 -.017 .222 .222 "24 "34 "44 "54 .291 .114 .014 .014 -1.080 -.986 -.033 -.024 "64 "15 "25 "35 -.0001 -.001 -.116 -.162 -.827 -.437 -.250 -.287 "45 "55 "65 -.855 -.723 -2.054 -1.586 .257 .202 "16 -.215 -.152 "26 -14.734 -6.411 "36 "46 .288 .432 -.792 -.744 "56 1.167 .690 "66 -1.218 -1.171 1960 1965 1970 -.493 -.493 -.494 -.255 -.174 -.127 .026 .025 .022 .330 .223 .159 -.022 -.019 -.018 -.009 -.007 -.005 -.068 -.059 -.052 -.443 -.461 -.471 .156 .133 .115 .056 .039 .030 .012 .011 .010 -.018 -.018 -.018 .035 .040 .041 .849 .666 .550 -.357 -.339 -.329 -.029 -.045 -.064 -.103 -.011 -.119 -.011 -.005 -.001 .211 .184 .159 .064 .040 .025 .013 .010 .007 -.867 -.750 -.678 -.018 -.014 -.011 -.002 -.004 -.005 -.197 -.231 -.250 -.367 -%34 9 -.335 -.317 -.337 -.339 -.616 -.521 -.468 -1.339 -1.149 -1.054 .154 .099 .064 -.074 -.014 .091 -4.414 -3.683 -3.230 .600 .766 .891 -.709 -.675 -.656 .435 .239 .125 -1.135 -1.098 -1.072 TABLE- VI .10 - 138 -ESTIMATES OF E L A S T I C I T I E S BASED ON S+H: THE INTERTEMPORAL MODEL 1950 1955 1960 1965 1970 -27.860 -29.693 -34 .386 -41.528 -48.154 183.179 -145.789 -118 .7801 -93.835 -80.727 -25.595 -25.826 -26 .868 -28.813 -28.368 132.639 -109.468 -90.468 -70.429 -60.304 -15.727 -8.637 -5.754 -3.997 -3 .208 -.535 -.552 -.560 -.557 -.542 -10.797 -9.057 -7.914 -6.535 -5.602 12.373 12.522 13.465 15.126 15.828 32 .743 28.659 25.341 20.876 17.703 1.645 1.133 .891 .703 .579 -.407 -.437 -.489 -.567 -.641 62.451 53.533 46.679 38.677 32.231 -43.916 -32.557 -23.344 -14.184 -9.579 4.740 2.765 1.764 1.027 .674 -1.757 -1.5991 -1.448 -1.240 -1.116 9.594 8.587 7.571 6.259 5.160 -3.910 -2.754 -2.160 -1.710 -1.370 1.036 1.138 1.267 1.477 1.622 -3.949 -2.405 -1.552 -.901 -.585 1.034 .982 .900 .769 .684 2.875 2.149 1.751 1.433 1.239 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 T A B L E v i . i o ' ( c o n t i n u e d ) - 139 -'11 n21 n31 n41 n51 n61 \ 2 ^22 n32 n42 n52 162 n13 ^ 3 ^ 3 n43 ^ 3 ^63 \ 4 ^24 ]34 ^4 ^ 4 ^ 5 ^ 5 ^ 5 ^ 5 ^ 5 ^ 5 ]16 ]26 ]36 ]46 ]56 ]ee 1950 -.641 -.262 .253 .705 .014 -.031 -.083 -1.289 .430 -.314 .026 -.019 .478 2 .582 -1.117 .361 -.2.06 .002 .263 -.372 .071 -1.108 -.041 .0003 .087 .506 -.665 -.670 -2.265 .254 -1.105 -2.165 .028 .027 1.472 •1.205 1955 -.629 -.206 .236 .567 .003 -.029 -.075 -1.105 .395 -.252 .013 -.020 .494 2.254 -1.151 .325 -.161 .006 .259 -.314 .071 -1.033 -.032 -.0002 .023 .308 -.654 -.594 = 1.680 .200 -1.072 -1.937 .103 -.013 .857 -1.157 I960 -.617 -.155 .217 .424 -.002 -.026 -.071 -.9 58 .365 -.195 .006 -.020 .534 1.957 -1.194 .281 -.135 .011 .244 -.244 .066 -.917 0.026 -.001 -.022 .156 -.647 -.522 — 1 . 3 8 2 .154 -1.068 -1.756 .192 -.072 .539 -1.119 1965 -.605 -.107 .201 .283 -.004 -.022 -.063 -.796 .316 -.127 .0002 -.019 .605 1.612 -1.276 .225 - .116 .020 .217 -.166 .057 -.779 -.021 -.003 .069 .006 -.629 -.441 -1.159 .101 -1.084 -1.550 .330 -.160 .300 -1.077 1970 -.595 -.080 .180 .202 -.005 -.020 -.058 -.713 .272 -.092 -.003 -.018 .678 1.427 1.34 2 .190 -.108 .028 .190 -.121 .047 -.699 -.018 -.004 .103 -.080 -.579 -.387 -1.028 . 058 -1.112 -1.434 .421 -.214 .162 -1.045 - 140 -TABLE VI.11 INDEX OF INDIRECT UTILITY TherTEree Good Model The Leisure Model The Intertemporal Model SYM(+0) SYM(-e) S+M(BY = 1946) 1946 .327835 .660881 .615555 1950 .424883 .742981 .627257 1955 .587519 .830081 .721314 I960 .744124 .874991 .841458 1965 1.000000 1.000000 1.000000 1970 1.341605 1.114991 1.206593 1972 1.501431 1.171591 1.240183 - 141 -FOOTNOTES to Chapter VI 1. Of course, r e s u l t s depend on the base year selected. 2. The average labor force p a r t i c i p a t i o n rates (labor force divided by population)' have r i s e n s l i g h t l y over time. - 142 -Chapter VII SUMMARY AND CONCLUSIONS In t h i s f i n a l chapter, a summary of the study and con-clusions are presented and areas for future research are sug-gested. In Chapter I, the scope of t h i s study i s examined. Our goal i s to formulate models of consumption for individuals (or households) and for the whole economy, based on the beha-v i o r a l postulate of consumer's u t i l i t y maximization. The l a -bor supply decision (a l a Becker) and intertemporal a l l o c a t i o n of goods (a l a Fisher-Hicks) are of s p e c i f i c concern. Moreover, two f l e x i b l e functional forms are to be applied. These are the translog function and the Generalized"Leontief function. The Japanese economy i s selected for the empirical application ofro.urustudy. In Chapter I I , relevant t h e o r e t i c a l materials for the formulation of our model are investigated. In p a r t i c u l a r , materials on functional s e p a r a b i l i t y , methods of aggregation over goods and over people and the treatment of durable goods are covered. The discussion on functional s e p a r a b i l i t y highlights the concept of a consistent two-stage maximization procedure. When a consumer faces an eno-crmous number of goods, he may wish to make a l l o c a t i o n decisions i n two stages. I f a u t i l i t y function i s weakly separable, i t i s stated that the necessary and s u f f i c i e n t condition for a consistent f i r s t stage maxi-mization procedure i s that the u t i l i t y function be homothetically - 143 -weakly separable. Furthermore, t h i s i s s u f f i c i e n t (but not necessary) for a consistent second stage maximization. I f th i s condition holds, then a l l category functions can be ag-gregated to form aggregate p r i c e and quantity indices. This method of aggregation i s referred to as' aggregation by homo-the t i c s e p a r a b i l i t y . The Leontief aggregation method based on quantity proportionality may be regarded as a special case of t h i s method. In contrast, tlxe Hicks aggregation method does not r e l y on functional s e p a r a b i l i t y at a l l . r t i s based on the beha^ v i o r of independent varia b l e s . I f prices within a group of commodities move proportionally, then these commodities can be aggregated into a single commodity. These three -methods of aggregation, i n e f f e c t , reduce the number of commodities one' must deal with i n each stage. The l a s t method proposed does not attempt to reduce the number of goods, but t r i e s to re-^ duce th.e number of parameters, involved by placing a p r i o r i r e s t r i c t i o n s on the values o f parameters. In prac t i c e , t h i s can be performed by constraining the values of e l a s t i c i t i e s of substitution according to functional s e p a r a b i l i t y assump-tions. This method i s referred to as'aggregation by 'station^ a r i t y ' . The aggregation o f individuals (households) demand func-ti o n oyer a l l people (households) i s carr i e d out by making the following assumptions,: - 144 -i) Everybody possesses the i d e n t i c a l u t i l i t y function, i i ) Everybody faces the same prices, i i i ) The expenditure (or income) d i s t r i b u t i o n i s known. Then, the aggregate share function becomes a function of a l l p r i c e s , the mean income (wealth) and expenditure (wealth) d i s t r i b u t i o n i n the economy. Chapter II i s concluded by a b r i e f discussion on the treatment of durable goods. These durables are treated sym-met r i c a l l y with nondurables. Thus, rental prices instead of purchase prices are formulated. Assuming a perfect second hand market, a consumer i s i n d i f f e r e n t to a purhcase or rental ofaa p a r t i c u l a r consumer durable good. Chapter III i s devoted to formulating models of consump-ti o n i n terms of s p e c i f i c functional forms. The translog func-t i o n and the Generalized Leontief function are introduced and t h e i r properties are examined. The i n d i v i d u a l share equations are derived from nnpnhomothetic inverse i n d i r e c t translog and Generalized Leontief u t i l i t y functions using Roy's Identity. Integrating the share equation over expenditure, we derive the aggregate share equations to be estimated. Relevant e l a s t i c i t i e s are now defined i n terms of normal-ized p r i c e s , the u t i l i t y function, i t s gradient and the second p a r t i a l d erivatives. Furthermore, pseudo e l a s t i c i t i e s of sub-s t i t u t i o n are defined i n order to f a c i l i t a t e the actual imple-mentation of the 'stationary' method. - 145 -F i n a l l y , three models are proposed and spe c i f i e d for an empirical investigation. The f i r s t two models are atemporal models. The three good model 1 deals with three present goods: food, consumer durables and miscellaneous. The l e i s u r e model, in addition, incorporates present l e i s u r e consumption. Both models are formulated for the whole economy. Aggregation by homothetic s e p a r a b i l i t y i s employed i n these models so that we can deal with the second stage maximiza-t i o n problem. Thus, the hypothesis of a homothetic category function for the present period i s to be tested. The l a s t model i s the intertemporal model for the repre-sentative consumer. Two future goods are added. These are the future composite good and future l e i s u r e which are aggregates of a l l future goods by the Leontief aggregation method. Chapter IV discusses relevant econometrics issues. A stochastic s p e c i f i c a t i o n of the models i s f i r s t presented. A problem we encounter originates from the nature of our depen-dent variables and the assumption made on the disturbance terms. By d e f i n i t i o n , the shares add up to one, thus rendering the disturbances no longer independent. An a r b i t r a r i l y chosen equation i s deleted and the remaining N-l equations are estimated to obtain the maximum l i k e l i h o o d estimates. The computational algorithm i s b a s i c a l l y an i t e r a t i v e version of generalized non-li n e a r l e a s t squares. When convergence i s attained, the r e s u l t -ing estimator converges numerically to the maximum l i k e l i h o o d estimator provided the stochastic s p e c i f i c a t i o n used i s correct. - 146 -Null hypotheses to be tested are: symmetry and homotheticity conditional on symmetry. The l i k e l i h o o d r a t i o tests are employed to determine the v a l i d i t y of these hypotheses. This chapter i s concluded with a b r i e f discussion on the method of te s t i n g and imposing monotonicity and or curvature conditions. If these conditions are not s a t i s f i e d g l o b a l l y , but not rejected at a point of expansion, one may wish to impose one or both i n order to obtain economically meaningful^ estimates. Imposition of monotonicity employs the method of a squared transformation, whereas imposition of curvature uses the technique of Cholesky f a c t o r i z a t i o n . I In Chapter V we discuss data: required data for estimation and the procedure employed for t h e i r construction. In p a r t i c u -l a r , a l l expenditure shares, a l l p r i c e indices (purchase and r e n t a l ) , the mean expenditure, the expenditure d i s t r i b u t i o n variable and the mean wealth of the Japanese economy are pre-sented for the three models. Time.series data are used for 1946-1972. Our estimation of the p r i c e of l e i s u r e deserves a comment. The price of l e i s u r e i s interpreted as the opportunity cost of not working i n the 'labor market', rather than working at home. The average hourly wage rate i s chosen as the opportunity cost. It i s calculated as the sum of a l l wages and s a l a r i e s divided by the t o t a l manhours worked. Those who are i n the labor force are assumed to consume l e i s u r e for the number of hours which - 147 -are not spent at work out of the maximum available hours (12 hours per day). Those who are not i n the labor force are also assumed to face opportunity costs and they consume l e i s u r e for a fixed number of hours per day. For the price estimates of future goods, the Leontief aggregation method i s adopted i n order to aggregate' goods i n the future. The same good i n a l l future periods, i s aggregated into a single good.. Two future goods are constructed: future composite good and future l e i s u r e . The future composite good i s aggregated over the entire future consumption years', where-as future l e i s u r e i s aggregated over the entire future working years. By incorporating demographic variables of l i f e expec-tancy, the mean age and the age-specific labor force p a r t i c i -pation rates i n the construction of aggregate p r i c e s , we are able to allow for demographic s h i f t s i n the economy. In order to calculate shares, an estimate of the average l i f e t i m e f u l l wealth i s made as the sum of a l l c a p i t a l stock and both, current and future f u l l labor income. Chapter VI reports the r e s u l t s obtained from econometric estimation of the three models. The estimates of parameters, 2 asymptotic t s t a t i s t i c s and R are presented. In general, the models obtained excellent f i t . The l i k e l i h o o d r a t i o test s t a t i s t i c s are used to t e s t the v a l i d i t y of n u l l hypotheses of symmetry and homotheticity conditional on symmetry. - 148 -Using a set of parameter estimates, price and income e l a s t i c i t i e s , e l a s t i c i t i e s of substitution and index of i n -d i r e c t u t i l i t y are computed. Xn the three good model, symmetry i s accepted at the .1% l e v e l of significance with orewithout theaexpenditure-distri-butionivariable. However, homotheticity conditional on symme-try i s d e c i s i v e l y rejected. This implies a nonhomothetic cate-gory function. Therefore, aggregation by homothetic separable l i t y cannot be used to j u s t i f y an atemporal model of t h i s sort. If one started with an o v e r a l l u t i l i t y function over the entire l i f e t i m e of an i n d i v i d u a l as we have, the 'conventional' two-stage maximization procedure Ceach period i s treated as a cat-egory! must be rejected as 'inconsistent' according to t h i s r e s u l t . An alternative approach may be to assume that i n d i -viduals possess a tnonhomothetic) u t i l i t y function for the present period only, i.e.,'they are myopic. An obvious extension of t h i s study i s to introduce pre-committed expenditures so that the category function may be homothetic only i n the supernumerary goods. This can be based on a model of habit formation. In addition, one may wish to t r e a t the precommitted expenditures as unknowns to be estimated i n the system. The expenditure e l a s t i c i t y for food i s s i g n i f i c a n t l y less than one. This supports Engel's law. Moreover, the values have declined over time. Both durables and miscellaneous are found to be luxuries. Estimates of e l a s t i c i t y of substitution reveal that a l l three goods are substitutes. - 1 4 9 -For the l e i s u r e model, the results are reported for the model: when six hours are assumed to be l e i s u r e consumption for those not i n the labor force. I t i s concluded that the p a r t i -cular hour chosen i s not very c r u c i a l for r e s u l t s . Symmetry i s rejected. However, homotheticity may be ac-cepted (without the expenditure d i s t r i b u t i o n v a r i a b l e ) , i f we wish to maintain symmetry as a maintained hypothesis. ,e monotonicity condition i s - a c i s : : - 9 ^ a . - - . , • i : * * * cThesmonoto'nici^ while the cur-vature condition i s s a t i s f i e d at most but not a l l observa-tions i n the symmetry version (without the expenditure d i s t r i -bution v a r i a b l e ) . In the symmetry plus homotheticity -version, both conditions are s a t i s f i e d g l o b a l l y . However, the symmetry version' with the expenditure d i s t r i b u t i o n did not perform very well i n terms of the curvature condition. I t . i s suggested that an estimate of the income d i s t r i b u t i o n may be a poor proxy for the ' f u l l " expenditure d i s t r i b u t i o n . Two sets of e l a s t i c i t y estimates are made for symmetry (without a d i s t r i b u t i o n variable) and for both symmetry and homotheticity imposed. Under the symmetry version, the expenditure e l a s t i c i t y of l e i s u r e turned out to be s i g n i f i c a n t l y less than one. This r e s u l t may be interpreted as the 'char a c t e r i s t i c work ethic' of the Japanese people. An alternative interpretation i s that the ef f e c t of a r i s i n g expenditure i s to increase the - 150 -consumption of durables and miscellaneous (consisting of mostly service) proportionally more, than l e i s u r e . In other words, l e i s u r e becomes more ' c a p i t a l ' and 1 s e r i v c e intensive'. The estimates of e l a s t i c i t i e s of substitution reveal that both, substitute and complementary relationships e x i s t . In p a r t i c u l a r , food and durables are now complements. Recall that these are substitutes i n the three good model. Relationships i d e n t i c a l to the symmetry case i s obtained when both symmetry and homotheticity are imposed. In the intertemporal model, the symmetry hypothesis i s maintained, since i t i s not p r a c t i c a l l y possible to estimate the unconstrained model. Only share equations for the present goods are estimated. Homotheticity conditional on symmetry i s d e c i s i v e l y rejec-ted. When symmetry alone i s maintained, both, monotonicity and curvature are vi o l a t e d at every sample point. Thus, we select a new base year (1946) as a point of expansion i n order to impose monotonicity. I t i s found that monotonicity i s not re-jected at the point of expansion. The imposition of curvature i s not carr i e d out, since the model becomes too large for the computer. When monotonicity i s imposed, the curvature condi-ti o n i s s a t i s f i e d for 1962 to 1972 i n c l u s i v e . Substitutes and complements are also found among goods. Themmost intere s t i n g r e s u l t obtained i s that the wealth e l a s t i c i t i e s of present l e i s u r e are negative for 1957 to 1972. - 1 5 1 -Two alternative but not necessarily mutually exclusive interpre-tations are presented. One i s that present l e i s u r e i s indeed an i n f e r i o r good. Another attempts to explain the r e s u l t i n terms of the average labor force p a r t i c i p a t i o n rate. A r i s i n g l i f e t i m e f u l l wealth can be consistent with the actual decline i n the average monthly hours worked per worker. It i s not surprising thattthe wealth e l a s t i c i t i e s of the future goods are close to zero, since aggregation of the goods are based on theoLeontief aggregation method with weights of one.. I t has been pointed out that double integration of i n d i -viduals over age as well as income w i l l be required, i f one wishes to formulate a s a t i s f a c t o r y intertemporal consumption model for the aggregate economy. This is';a.J-topic~-whichodeserves future research. Another area of i n t e r e s t i s i n r e l a t i o n to intertemporal consistency of taste or a l t e r n a t i v e l y changing taste. In our study, the consumer's taste (or u t i l i t y function) i s assumed to remain the same during the period of investigation. At every sample point, the consumer examines the economic environ-ment i n which he l i v e s and makes an a l l o c a t i o n decision for goods'in the future as well as i n the present period. The past consumption i s i r r e l e v a n t . However, i n the next period, i f the expected future condition i s not r e a l i z e d , the con-sumer's new decision may not be consistent with the one made. - 1 5 2 -in the l a s t period. Obviously, t h i s i s not a problem,_if one assumes perfect foresight into a l l future periods. A popular way of incorporating past consumption i s to f o r -mulate a habit formation model. Habit formation has been iden-t i f i e d as an important factor i n explaining consumption (Brown and Heien [ 1 9 7 3 i l , Pollak and Wales [ 1 9 6 9 3 and Houthakker and Taylor [ 1 9 7 0 3 ) . Manser [ 1 9 7 5 ] has incorporated habit forma-t i o n into the translog functional form. In an intertemporal context, t h i s type of consideration results i n an asymmetrical structure of the o v e r a l l u t i l i t y function. Furthermore, t h i s asymmetry may work i n the opposite d i r e c t i o n . It i s conceivable that the consumer's present be-havior depends upon the expected future and not the past. Then any period i n the future depends upon a l l the periods which l i e ahead. In t h i s case, one l i v e s for the future, where-as one l i v e s i n the past glory i n the case of habit formation. 5 This .sort of asymmetry i n the structure of the u t i l i t y function may be investigated along the l i n e of 'recurvise' structures by Blackorby, Primont and Russell [ 1 9 7 5 ] . How t h i s sort of recursive structure can be formulated into an empirically testable framework remains to be investigated i n the future. F i n a l l y , we wish to address ourselves to the question of c c how we may assess the results of t h i s study v i s - a - v i s Japanese data used for the empirical application of the models. The neoclassical t h e o r e t i c a l basis employed i n formulating the - 153 -models i s often considered to be oriented towards the econ-mies i n the 'Western World', Then, i s the neoclassical frame-work capable of s a t i s f a c t o r i l y modeling other economies? Japan i s of p a r t i c u l a r i n t e r e s t to many development econ-omists , partly due to her remarkable performance in recovery and growth i n the post-war period. On.the consumption side, one outstanding c h a r a c t e r i s t i c of the.Japanese people i s persistent high rates of current.savings.. These high rates of savings have been oberved not only over the years but also cr o s s - s e c t i o n a l l y at d i f f e r e n t levels of-income. The d e f i n i t i o n of savings used i n t h i s study i s not i d e n t i c a l to the ' t r a d i t i o n a l ' one ( i . e . , current savings), but used to mean the portion of expected f u l l l i f e t i m e wealth reserved for the future periods. However, the implications obtained from the results on work-leisure choice are', quite consistent with the high,rates of current savings. The estima-; tes of e l a s t i c i t i e s also seem to be reasonable. 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Ohkawa, K. and e t . a l . / Estimates of Long-Term Economic S t a t i s t i c s of Japan since 18 68, Tokyo: Toyo k e i z a i Shinpo-Sha. Volume 1 - National Income, Volume 3 - Capital Stock, Volume 4 - Capital Formation, Volume 6 - Personal Consump-ti o n Expenditure, Volume 7 - Government Expenditures, Volume 8 - Prices, Volume 9 - Agriculture and Forestry, Volume 12 - Railroads and E l e c t r i c U t i l i t i e s . - 164 -Appendix A DATA: METHODOLOGY AND SOURCES Econometric estimation of the three models i s c a r r i e d out with the Japanese data constructed on annual basis for the post-war period (1946-1972) . However, a considerable amount of data are also c o l l e c t e d and constructed for the pre-war period (1912-1940), although i t was not possible to construct a com-plete set of estimates which are consistent with the post-war data. An attempt was made to extend data as far back as pos-s i b l e , and thus, this appendix presents data for a l l the years for which estimates haveobeen constructed. The following i s a l i s t of data sources frequently c i t e d i n the appendix. The t i t l e s are abbreviated for notational conve-nience as follows': 100YR OneeHundred Year S t a t i s t i c s of the Japanese Economy. N I 7 R Revised National Income S t a t i s t i c s . NI-( ) Annual Report on the National Income S t a t i s t i c s , T(year) . ALS-( ). Annual National Living S t a t i s t i c s , (year). ELTS-( ) Estimates of the Long-term S t a t i s t i c s of Japan, (vol. 1 - 13). SYJ-( ) S t a t i s t i c a l Yearbook of Japan, (year). HSJE H i s t o r i c a l S t a t i s t i c s of the Japanese Economy. EI-( ) Economic Indicators/ (date). NWS National Wealth Survey. - 165 -1. The Three Good Model a) Food The pre-war (1912-1940) data on the expenditure on food are drawn from the estimates by Shinohara (ELTS-6; 134). The e s t i -mates are. i n b i l l i o n s of yen and for calendar years. The 1946-50 data are taken from "personal consumption expenditure" by the Economic Planning Agency (EPA) compiled i n 100YR(48). The 1946 figure i s for the f i s c a l year, but f i s c a l year estimates are converted into calendar year estimates for the years 1947-50."'" For the remaining years (1951-1972) national income accounts data on household consumption expenditure on food are used (NI-R; 78-9, NI-73; 54, NI-74; 50-1). Consumer price indices (CPI) on food for urban areas are used as prices for 1912 to 1938 (ELTS-8; 13566) ?.rwhereas impl i -c i t deflators from (ELTS-6: 106) are used for 1939 and 1940. For the years 1946-51, CPI for c i t i e s are supplied by (ALS-70; 168). EPA's national income, data provided deflators for 1952-72. A l l p rice indices are set at unity i n the base year, i . e . , the 1965 calendar year.. The l i n k between pre- and post-war prices i s made by using post-war CPI whose base year i s 1934-1936 (ELTS-8; 136). b) Consumer Durables A b r i e f discussion on the treatment of durables was made i n Chapter I I . I t was explained that i n our model, we are primarily concerned with the 'flow of service' consumers enjoy from given - 166 -stocks of consumer durables at a given time. Therefore, pur-chase prices of a unit of the goods are no longer the relevant prices . 'Rental pr i c e s ' must be calculated to measure the true opportunity cost, given purchase pr i c e s , the r e a l or market in t e r e s t rate and depreciation rates. What one spends or con-sumes™ then, i s the product of re n t a l p r i c e and quantity of stock available for consumption. This i s defined as 'rental expenditure'. The flow of service that one receives depends upon the size of ±herc 'apUfecJl stock; therefore, i t i s necessary to estimate the value of e x i s t i n g stock for each year. This i s estimated from gross investment (or current expenditure) data, given a depre-c i a t i o n rate and a bench mark year (BMY) stock estimate. Thus, i n vector notation, (A.l) S*(t) = ( l - 6 ) S * ( t - l ) + I(t)/D(t) where S*(t) 5 K t ) D(t) value of stock (in 1965).yen) i n time t, depreciation rate, gross investment i n time t, deflator or CPI (1965 = 1.0), (a.2) and S* (t) = i | | f + + ...+ ( l - a ) " " 1 + (l-6) nS*(BMY) 1 6 7 -Furthermore, l et Q(t): quantity of stock in time t p(t): purchase price per unit of the durable in time t x(t): quantity purchased in time t p(t): rental price (per period) in time t r * ( t ) : real rate of interest in time t (- 1 + r(t) K ~ 1 + e(t) " ± ; r ( t ) : market rate of interest in time t e(t): rate of expected rise in the purchase price in time t. A l l the above variables (except r(t) and r*(t)) are vectors of N by 1 dimension. The current expenditure on the durable in t is (A. 3) Y*(t) = p(t) -x(t) , (A.4) and the rental expenditure is Y Ct) = p(t)-Q(t). From equation ( I I . 1 , 9 ) = -r,\ U.'-+• r*(t)) p(t) = p(t) ( 1 + r*(t)) ' The value of stock in constant yen is -N (A. 5) S*Ct) = I Y*tt) „ v - t t=0 D t t > ( 1-6) - 168 -The quantity of stock i s : -N (A.6) Q(t) = I t=0 P(t) -X(t) P(t) (1-6) - t = D(t)-S*(t) P(t) F i n a l l y , the r e n t a l expenditure ( i n c u r r e n t yen) i s i-5 • + r* (t) (A.7) Y(t) = p(t) -Q(t) = [ 1 + r*(t) •] • D(t) -S* (t) . Before proceeding to s p e c i f i c consumer durables, a few comments are i n order concerning the choice of i n t e r e s t rate and the role of expectation. In order to calculate rental prices, an i n t e r e s t rate must be chosen from a range of market i n t e r e s t rates. There i s usually a great discrepancy between lending and borrowing rates for an average consumer due to imperfect c a p i t a l markets. Which i s the true opportunity cost? We have chosen the y i e l d s on long-term government bonds as the opportunity cost for the purchase of consumer durables. These are lending rates. Interest rates paid for time deposits shorter than a year are inappropriate; since durables, by def-i n i t i o n , l a s t longer than a year i n t h i s study. Average bond yi e l d s are reported i n the Bank of Japan's HSJE-62(83-4) and SYJ. For the years p r i o r to 1930, the highest y i e l d s are used instead of the average rates, since there i s no data available on the average y i e l d s . - 169 -Furthermore, we have decided to ignore.the tax factor i n estimating.the opportunity cost. To be precise, most tax sys-tems penalize people for saving instead of consuming by 'double t a x a t i o n 1 , and i t should be taken into consideration. However, in the case of Japan, (small) i n t e r e s t incomes were either com-pl e t e l y exempted from tax or were subject to small tax rates as a part of economic p o l i c i e s designed to encourage savings and investments. Therefore, as long as small to medium scale dur-able purchases are involved, i t would be j u s t i f i a b l e to d i s -regard the tax aspect. A problem may a r i s e i n the case of housing, i f a consumer were making decisions as to an outright purchase of a house. The second point to be discussed here i s the r o l e of ex-pectations . The estimation of rental prices of a consumer durable requires a formulation of expected future prices of the durable for the next period. The simplest and possibly the most naive approach i s to assume that people expect no change i n p r i c e s : the assumption of s t a t i c expectations. This would be p e r f e c t l y adequate when we observe rather stable prices or price movements which seem random. Although t h i s i s not exactly what we observe, i t was decided that the s t a t i c expectation model should be used i n t h i s study. The reasons.are presented below following a b r i e f d i s -cussion on dynamic expectations. - 170 -It may be more, r e a l i s t i c to conjecture that people learn to adjust t h e i r expectations, according to past price r i s e s (or declines) i n a period of persistent i n f l a t i o n (or d e f l a t i o n ) . In t h i s case, a theory of dynamic expectations must be devel-oped. In f a c t , we have t r i e d to use a d i s t r i b u t e d lag model to estimate the expected p r i c e s . Here i t i s assumed that people p a r t i a l l y adjust t h e i r ex-pectations to past experiences. Thus, there w i l l be lags i n response. A d i s t r i b u t e d lag model was t r i e d using a Pascal d i s t r i b u t i o n of the form: CA.8) w(t) = '[y '* ( l - X ) v A t t = 0 , l , . . . where w(t) are weights, y i s a p o s i t i v e integer and A i s a oo parameter of speed of adjustment. Furthermore, Z w(t) = 1 t=0 and w(t) >_ 0. I f we assume that py = 1, then the d i s t r i b u t i o n i s reduced to a simple geometric d i s t r i b u t i o n . Ideally, one should t r y to estimate the parameter A by choosing a vailiuewwhichmmax'imizesRR2 (maximum l i k e l i h o o d estimate) . In a model l i k e ours, however, th i s leads to an enormously com-pl i c a t e d procedure. Thus, we assumed a 'reasonable' value for A 2 and t r i e d A = .4 to calculate the d i s t r i b u t e d lags. The r e s u l t s , however, are inconsistent with our model. The rental prices and rental expenditures became negative for some, years, while our model requires p o s i t i v e prices for a l l - 171 -years. Therefore, i t i s not possible to accept these r e s u l t s . What happened i s that during a period of rapid i n f l a t i o n (es-p e c i a l l y immediately aft e r the World War I I , when prices were r i s i n g at 30-60% per annum), any depreciation one faced on the durable good was more, than compensated by the expected appreci-3 ation. This phenomenon was p a r t i c u l a r l y pronounced for housing, since the depreciation rate used i s very low r e l a t i v e to other consumer durables. Even i f people face negative rental prices i n r e a l i t y , im-perfect c a p i t a l markets do not allow everybody to take advantage of t h i s opportunity. The desired l e v e l of consumption for a good whose price i s negative should be i n f i n i t e . Therefore, our' model cannot incorporate both negative prices and imperfect cap-i t a l markets at the same time. Another reason for the r e j e c t i o n of a d i s t r i b u t e d lag model i s that a simple geometric d i s t r i b u t e d lag model i s not capable of incorporating any c y c l i c a l trend i n expectations or price s . Pascal d i s t r i b u t i o n s with d i f f e r e n t parameter values y i e l d more sophisticated models. I t i s not p r a c t i c a l , however, to estimate the parameters i n our model. Furthermore, there i s no a p r i o r i j u s t i f i c a t i o n for selecting an appropriate lag form from a possible set. Consequently, the assumption of s t a t i c expecta-tions i s adopted i n t h i s study. Now we w i l l discuss b r i e f l y each consumer durable good. - 172 -i) Clothing In order to estimate the stock for the whole period, two periods are linked by adding an estimate, of the war damage for 4 the year of 194 5. There are two estimates of clothing stock from the NWS data for 1955 and 1970. The depreciation rate which best l i n k s these two estimates turned out to be 0.2748. A higher value w i l l not give us s u f f i c i e n t spread between the two e s t i -mates and too much divergence would r e s u l t i f the rate i s lower. However, t h i s depreciation rate re s u l t s i n unreasonably high estimates of stock for the e a r l i e r years, i f we started from recent period back to the early years. An in c r e d i b l e rate of decumulation i s implied. Therefore, we decided to s t a r t from 1912 using .2748 as the depreciation rate ( 6 ) . To do t h i s , a bench mark year value <£or 1911 (or any other year i n the early period) i s required. In the end, 546 b i l l i o n yen i s used as 5 the stock value for 1911. This estimate i s quite a r b i t r a r y but can be considered innocuous, since small changes i n t h i s estimate w i l l minimally a f f e c t the stock estimates a f t e r sev-era l years. However, we must bear i n mind that the estimates for the f i r s t several years are less than i d e a l . The sources for clothing data are i d e n t i c a l to those' for food, except for the years 1941-1950. The expenditure data for the l a t t e r years are the EPA estimates reported i n 100YR(48). The price data are drawn from Ohuchi[1958; 260] on the ' r e t a i l p r i c e s ' i n Tokyo. - 173 -i i ) Furniture and appliances The same estimate for the war damage i s also applied to l i n k the two periods. Consumer pr i c e indices for 1912-1920 (except for 1913, 1917 and 1919), and 1939-45 are not available, so indices are interpolated for these years. Furthermore, the expenditure data for 1941-1944 are not available, thus 1.0 b i l l i o n yen i s substituted for each year. The best estimate seems to r e s u l t when a depreciation rate of .25 i s used. However, a l l stock estimates by the NWS ( i . e . , for 1924, 1930 and 1935) seem to be.unreliably high. For the pre-war years, gross expenditures are less than 100 b i l l i o n yen for each year and the NWS estimates are i n the range of 2,000 to 3,500 b i l l i o n yen. This result, i s achieved only when an ex-tremely low depreciation rate, such as 3 per cent i s assumed. Even for the post-war year, the values appear to be over-estimated, e s p e c i a l l y for 1970. A depreciation rate of 10 per cent or so i s required, i f the d i f f e r e n t estimates are to be. linked. Ten per cent s t i l l seems to be low. In the end, i t was decided that an 'arbitrary' but more reasonable rate of 25% should be used as the depreciation rate and a bench mark year estimate for 1911 should be 80 b i l l i o n yen. The data on 'furniture and appliance' are more d i f f i c u l t to obtain than other data. Up to 1940, ELTS-6(230) i s the source for the expenditure data. For the recent years (1958-72), - 174 -r e l i a b l e data are available i n the national accounts (NI?>R; 236-7, NI-7 3; 220 and NT-74; 142). However, for the interim period, c e r t a i n proportions, of expenditure on housing are used as proxies. The proportion of expenditure on furniture and appliances in the 'housing' expenditure was around 80 per cent p r i o r to the war and i t dropped to 70 per cent i n the 60's. Therefore, we used proportions s t a r t i n g at 75 per cent and gradually de-creasing to 70 per cent over the interim period. S i m i l a r l y , the data on prices are fragmented and disj o i n t e d . For the early years, 1913, 1917 and 1919 data are reported i n 100YR(4 33). The values up to 1922 are interpolated. Weighted averages of price indices for several durable items are c a l c u l -ated for 1923-38 from ELTS-8(146, 151) as the price for ' f u r n i -ture and appliance'. Further interpolation i s made for 1939-1945. CPI for urban area are used for 194 6-72 (ALS-70; 69, ALS-72; 145, EX-6/74; 76). i i i ) Housing stock and housing service A depreciation rate of .025 i s used for r e s i d e n t i a l b u i l d -ings. The r e s u l t i n g estimates of r e s i d e n t i a l stock for the pre-war years seem f a i r l y reasonable, when the lowest bench mark year value ( i . e . , for 1917) estimated by the NWS (lOOYR; 433) i s used. A l l the others seem to be overestimates, e s p e c i a l l y for 1924, 1930 and 1935. According to these estimates, the - 175 -stock i s supposed to have more than doubled during these years. This, however, does not seem reasonable i f we look at the cap-i t a l formation data. The magnitude of investment does not warrant t h i s implication. Therefore, the most conservative estimate i s chosen as the BMY value. In order to l i n k the stock estimates of the two periods, we have estimated the damage caused by the war. The 1935 r a t i o of r e s i d e n t i a l to a l l buildings (100YR; 433) i s applied to take a proportion of the damage to the 'buildings and structures' for 1945 (TO'OYR; 27). 7 Contrary to the pre-war period, the Wealth Survey estimates of housing stock for the post-war period seem to be grossly un-derestimated. There are two reasons to suspect t h i s . F i r s t l y , the depreciation rate which i s implied by the d i f -ferent survey estimates i s i n the neighborhood of 5 per cent, or twice the pre-war rate. One may argue, for example, that 5 per cent i s reasonable for the postriwar data and that 2.5 per cent for tbe pre-war data i s a gross underestimation. However, a 5 per cent depreciation rate i s equivalent to the durable years of 45 years (which implies that i n 45 years, there i s only 10.5 per cent of the i n i t i a l stock l e f t ) . In f a c t , t h i s i s the rate the NWS used for 'block, brick and stone buildings'. We f e e l , however, that 4 5 years i s too short as the durable years and therefore, 5 perceehtiis not acceptable. At any rate, the - 176 -depreciation rate for the post-war period must be equal to or less than that for the pre-war period, since more concrete and reinforced steel was used i n the post-war period. Secondly, i t i s f e l t that the Wealth Survey estimates of housing stock are too low to be compared reasonably with the estimates of other consumer durables. According to the 1970  Wealth Survey: Household, the assessed values of owned houses are i n the neighborhood of o n e - f i f t h of the t o t a l value of con-sumer durables for each of the seventeen income classes. Fur-thermore, the values for owned houses are estimated to be ap-proximately one-half of the values for clothing. I t i s f e l t , therefore, that the housing estimates are grossly underestimated, probably due to depreciation rates which are too high. Thus, we have decided to use 2.5 per cent for the whole, period and to use r e s u l t i n g stock, estimates which are much higher, but hopefully more r e a l i s t i c than the national wealth estimates. For the post-war period, there exist two kinds of housing: public and private. Since our interests l i e i n the flow of service one. receives from r e s i d e n t i a l buildings, p u b l i c l y owned (that i s government owned) housing i s also included for rental expenditure estimation. There i s no data available on govern-ment c a p i t a l formation i n r e s i d e n t i a l buildings before 1950; therefore, i t i s assumed that no public housing existed p r i o r to t h i s date. - 177 -The estimate of gross investment i n private r e s i d e n t i a l buildings for 1912-1929 i s by Emi and Rosovsky reprinted i n 100YRC34-5). The rest of the estimates are by EBA i n 100YR(48) and national accounts reports (NI-R; 7 8-9, NI-73; 54 and NI - 7 4; 50). The same national accounts reports also provided the government housing estimates for 1951-1972. For the deflator series of private housing for 1912-194 5, 'investment good price indices' out :of ELTS-8(158-9) are. used. The deflators for the period immediately a f t e r the war (1946-1951) are approximated by the weighted average of the invest-ment good pr i c e indices. National income s t a t i s t i c s are the sources for the rest of the postwar period for both private and government housing (NI-R; 8 6-7/. NI-73; 62 and' NI-74; 58-9). Using the investment s e r i e s , the de f l a t o r s e r i e s , a de-pre c i a t i o n rate of 2.5 per cent and the bench mark year (1917) estimate, the stock' series for both types of housing are e s t i -mated. In the case of public housing, the stock value i s as-sumed to be zero for the years p r i o r to the f i r s t year of i n -vestment. In order to aggregate the two types of housing, the Fisher Ideal p r i c e and quantity indices are calculated. Using these indices, r e n t a l prices and rental expenditures on housing are estimated according to the equations discussed e a r l i e r i n t h i s appendix. - 1 7 8 -F i n a l l y , housing and other types of consumer durables are also aggregated by the Fisher method to form a category of 'consumer durables'. c) Miscellaneous The t h i r d commodity to be considered i s an aggregate of a l l residual commodities which do not f i t i n the f i r s t two categories or l e i s u r e . To,be precise, i t contains two groups from usual national income account s t a t i s t i c s , i . e . / 'fuel and l i g h t ' and 'miscellaneous'. These are aggregated into 'miscellaneous' by the Fisher Ideal indices. The'data sources are roughly the same as those for food. Price data are missing for a few years around the war period; thus, an inte r p o l a t i o n i s made for these years. d) Expenditure D i s t r i b u t i o n Variable To recapitulate, the aggregate translog expenditure.equa-ti o n for the i t h good i s written as: (III.5 ) S . = . + EY..Inp. -l , .... . i j . j 6 l n y * E where (A. 9 ) e = /ylny i H y ) d y / y * l n y * , y i s the t o t a l expenditure of each household and y* i s the mean t o t a l expenditure. - 179 -Since i t i s impossible to obtain data on actual expenditure d i s t r i b u t i o n for more than a li m i t e d sample years, 0 i s approx-imated by the income d i s t r i b u t i o n i n the economy. For the pre-war period; there are no data available on actual d i s t r i b u t i o n of income. Consequently, i t i s impossible to estimate 0 accurately. However, data on the mean and the standard deviation of income are estimated by Takahashi[1959; 23]. Therefore, i n order to use t h i s information, we assume a log-normal d i s t r i b u t i o n of income for the period. Given ty (y) , the r i g h t hand side of (A.9) can be integrated to obtain: 0 = l n ( y * 2 +cra2)/21ny*, where y* i s the mean and--a i s the standard deviation of y. See Appendix B for the derivation of 0 when y i s assumed to be log-normally d i s t r i b u t e d . For the post-war period, there are better data available on income d i s t r i b u t i o n . The National Taxation Bureau provides frequency d i s t r i b u t i o n and t o t a l income of taxpayers according to income brackets. Thus, the expenditure (income) d i s t r i b u t i o n variable 0 i s d i s c r e t e l y approximated i n the following way. K _ _ _ _ (A.10) 0 = E y.lny.-w./y*lny*, i=l 1 1 1 where K i s the t o t a l number of income brackets, yn. i s the mean - 1 8 0 -income and w^  i s the frequency (or weight) of the taxpayers i n the i t h bracket. These weights add up to one. Moreover, y* i s the mean income for a l l taxpayers, calculated by d i v i d i n g the aggregate income by the t o t a l number of taxpayers. The assumption i s that the expenditure d i s t r i b u t i o n var-iable can be c o r r e c t l y approximated by an income d i s t r i b u t i o n v a r i a b l e . The data are reported i n the Annual S t a t i s t i c a l Report of  the National Taxation Bureau for the years 1 9 6 1 - 1 9 7 1 on s e l f -assessed income and earned income. The e a r l i e r data are ob-tained from S t a t i s t i c a l Abstracts of Japanese Finance and Economy, 1 9 5 0 - 1 9 5 8 _ . Interpolated estimates are used for 1 9 5 9 and 1 9 6 0 , whereas the estimate for 1 9 7 2 i s an extrapolation of the e a r l i e r years. 2 . The Leisure Model Every person over 1 5 years of age i n the economy i s assumed to have 1 2 hours per day, 2 6 days a month and 1 2 months per annum as time available to a l l o c a t e to labor supply and l e i s u r e . Those^whoaagennotidinltherlabor force are assumed to be enjoying l e i s u r e for a fixed number of hours (less than 1 2 hours) per day. The entire model i s estimated for the case with six hours of l e i s u r e and the e l a s t i c i t i e s estimates are made for 0 , 2 , 4 , 6 and 8 hours separately (only 6 hour case i s reported). - 181 -The estimates of g a i n f u l l y occupied population was made by Ohkawa for the pre-war period (100YR; 57), whereas the post-war estimates of labor force employedriare obtained from JSY-52 C48) , JSY-61 (44) , JSY-69 (47) and' JSY-73/74 (4 9) . Geometric means are used for the missing years (1946, 48 and 49) based upon 1947 and 1950 estimates. The estimates of population over 15 years of age for 1920-1969 are compiled i n Population Estimates of Japan. JSY-73/74 (401 contains the.estimates for 1970-1973. About 5 to 7% of the population over 15 years of age are persons, of 65 years of age-. They are treated symmetrically with the r e s t of population not i n the labor force. In other words, they are assumed to be engoying l e i s u r e out of th e i r choice and actually do face a p o s i t i v e opportunity cost of l e i s u r e . The average wage rates are adopted as the opportunity cost of not working. The sum of annual compensation of employees and income earned by proprietors comprises the t o t a l wage b i l l . Compensation of employees include wages and s a l a r i e s , other pay and allowances and s o c i a l insurance contribution by employ-8 ees. For 1919-1929, the estimates by Yamada of compensation of employees and proprietors income are co l l e c t e d i n 10 0YR(31). 1930-1941 data are estimated by EPA and are also compiled i n lQQYR(38). S i m i l a r l y , the estimates for the f i r s t f i v e years - 182 -af t e r the war are by EPA i n 100YR(38). The f i s c a l year figures are converted into calendar year estimates. The estimates for the rest of the period are from the national income s t a t i s t i c s by EPA i n NI-R(74-5), NI-72(38-9), NI-7 3(50) and NI-74(34-5). The t o t a l wage b i l l i s divided by the t o t a l annual manhours worked by the labor force. There i s no d i r e c t estimate of the t o t a l manhour worked, therefore, i t i s estimated by obtaining the product of the t o t a l labor force and average hours worked per month m u l t i p l i e d by 12 months. The estimates of monthly hours worked for the pre-war period (1923-1946) are calculated from the data i n Ohuchi [1958; 285-290] by multiplying d a i l y average hours worked by average days worked per month. The post-war data are from JSY-71(74) and JSY-73/74(70). 3. The Intertemporal Model. Two future goods are added i n t h i s intertemporal model of consumption for the representative consumer. These are the future composite good and future l e i s u r e . When iihenLebntief aggregation method i s adopted, the price of future composite good becomes the sum of discounted future prices over the entire 'remaining consumption years'. Furthermore, the assumption of s t a t i c expectation implies that a l l future prices are i d e n t i c a l to the present p r i c e s . Current market i n t e r e s t rates ( i . e . , y i e l d s on long-term government bonds) are used as the discount rate. - 183 -The average remaining consumption years are defined to be the weighted sum of l i f e expectancy for a l l age and sex cate-gories. L i f e expectancy tables are compiled i n TOOYR(17) and JSY-73/74 (36-8) for census years. The estimates are interpola-ted for the intervening., period. On,the contrary, the price of future l e i s u r e i s calculated over the average 'remaining working years', which are estimated as the weighted sum of remaining working years of individuals of p a r t i c u l a r age and sex, weights being the frequency d i s t r i -bution of population by age and sex. The remaining working years for an i n d i v i d u a l i s the number of years l e f t p r i o r to his re-tirement at the age of 65, each year being adjusted by the l a -bor force p a r t i c i p a t i o n rates (LFPR); bylage^and sex. The average age of the population i s the weighted average of the entire population with the same wedlghts. The data on population and i t s composition by age and sex are also from 100YR(16) and JSY-73/74(38-39). The labor force p a r t i c i p a t i o n data arettaken from 1971 Employment Status: Survey Summary (37) and Employment Structure of Japan (38). The J f u l l l i f e t i m e wealth' consists of a l l private c a p i t a l stock (producer durables, inventories, r e s i d e n t i a l buildings, consumer durables), 'present f u l l labor income' and expected 'future f u l l labor income'. - 1 8 4 -Stock estimates for producer durables (including business buildings) are made using the expenditure data i n ELTS-1(187) and national income s t a t i s t i c s (NI-R; 78-9, NIV73; 55, NI-74; 50-1) and deflators. in"ELTS-8 (159, 162-6), NI(78-9) and NI-74 (58-9). A depreciation rate of .25 i s applied. The expenditure data for inventories are also found i n the same sources, whereas.s deflators are from ELTS-1 (233) , NI-R(78-9) and NIV74(58-9). A higher rate of depreciation, i . e . , .40 i s used for inventories. The present f u l l labor income i s estimated from the ex-penditure side rather than the income side. I t i s the t o t a l expenditure (on food, consumer durables, l e i s u r e and m i s c e l l a -neous) plus current saving minus income from c a p i t a l . The income from c a p i t a l i s approximated by the product of the value of a l l c a p i t a l stock and current i n t e r e s t rate as the rate of return on c a p i t a l . The data on current saving are found i n 100YR(38), NI-R(64-5),,NI-73(40-1) and NI-74(36-7)., This estimate of the present f u l l labor income r e f l e c t s the LFPR of the population, since the maximum free hours are set d i f f e r e n t for those i n the labor, force and for those who are not. The' future f u l l labor income i s the sum of a l l discounted future f u l l labor income over the remaining working years. The wage rate i s expected to increase at 2% per annum. Thus, the future f u l l labor income i n each period i n the future i s the - 185 -present f u l l labor income adjusted by a 2 % annual increase and, in addition, by the expected LFPR of the representative i n d i -vidual i n each period. The expected LFPR of the i n d i v i d u a l when he i s t * years old i s assumed to be, i d e n t i c a l to that of current population of age t * . The" average f u l l l i f e t i m e wealth i s obtained by di v i d i n g the aggregate f u l l l i f e t i m e wealth estimated above, by popula-t i o n . - 186 -T A B L E A . l EXPEND!TORES BILLION YEN YEAR FOOD DURABLES LEISURE HISC EL LA N E OUS 1912 2.6420 0.5613 0.0 0.7510 1913 2.8750 0.6022 0*0 0.7890 1914 2.2820 0.6115 0.0 0.7880 1915 2.2680 0.6149 0. 0 0.7910 1916 2.4680 0.7787 0*0 0.9040 1917 3.3240 1.0616 0.0 1.1140 1918 4.8520 1.5383 0.0 1.4520 1919 6.9930 2.3181 0*0 1.54 30 1920 7.2990 2.4572 0 . 0 2.2910 1921 6*9080 2.2866 0 .0 2-3210 1922 7.0290 2.2172 0.0 2.5580 1923 7 .1210 2.5007 3.2455 2.6710 1924 7.3750 2.4143 3.9701 2.5490 1925 7.8430 2.2909 4.0934 2.6580 1926 7.4050 2. 1931 3.8685 2.7090 1927 7.0900 2.0707 4.2972 2.7810 1928 6.9230 2 . 0 « 8 8 4.4346 2.8010 1929 6.7040 2.0486 4 .4 889 2.7580 1930 6*0570 1.7334 4.5096 2.6180 1931 5.1200 1.6098 4.6161 2.4290 1932 5*1710 1.6854 4.8257 2.3960 1933 5* 7320 1.8387 4.8532 2.7750 1934 6.2220 2.0248 4.6285 3.1460 1935 6.5750 2.2492 4.9044 3.3010 1936 6.8940 2.3916 5.5832 3.3010 1937 7^5220 2.9159 6.1583 3.8030 1938 8.1140 3.8232 6.8425 4.2950 1939 9*1290 4.2502 8^6028 4.7460 1940 9.9550 • 4*945 8--. • 10.2884 5.4990 1941 7*9000 5.1201 i 0.0 Oi 0 1942 7.7000 5.4641 ,, 0 .0 0.0 ~ 1943 8.3000 5.8865 0.0 0.0 1944':^| 9*4000 6.5014 0*0 0-0 1945 0 .0 34.1223 0 .0 0.0 - 187 -T A B L E a*: 1 C O N T I N U E D EXPENDITURES YEAR POOD DURABLES L E I S U R E HISC ELLASEOU 1 9 4 6 2 4 0 . 5 0 0 0 6 2 . 6 0 6 0 3 3 4 . 7 1 4 8 6 1 . 4 0 0 0 1 9 4 7 5 2 6 . 4 0 0 0 1 4 6 . 7 8 2 7 8 93i- 4 4 5 3 1 6 6 * 9 0 0 0 1 9 4 8 9 8 9 * 4 0 0 0 2 6 0 . 7 8 0 3 1 9 0 1 . 3 8 6 2 3 7 0 . 2 0 0 0 1 9 4 9 1 3 7 8 . 5 0 0 0 3 3 1 . 9 8 1 2 2 6 9 8 . 3 6 5 2 5 0 1 . 8 9 9 9 1 9 5 0 1 4 8 1 . 6 0 0 0 3 0 5 . 7 0 5 8 3 0 2 5 . 9 5 0 7 5 5 1 . 7 0 0 0 1951 1 7 5 7 . 3 0 0 0 4 7 2 . 4 3 6 5 3 8 2 7 . 8 3 4 7 6 4 1 . 8 0 0 0 1 9 5 2 2 0 8 4 . 3 0 0 0 6 2 0 . 2 3 5 1 . 4 5 3 0 . 2 4 2 2 7 5 3 . 7 9 9 8 1 9 5 3 2 4 4 6 * 8 0 0 0 8 6 4 . 9 7 4 9 4 9 5 8 . 6 3 2 8 9 6 8 . 8 9 9 9 1 9 5 4 2 7 0 4 . 5 0 0 0 9 7 1 . 9 7 7 8 5 5 7 4 . 2 6 1 7 1 1 2 6 . 0 9 9 9 1 9 5 5 2 8 2 4 . 7 0 0 0 •10.S8U7693 5 8 9 9 . 0 0 0 0 1 2 6 2 . 5 0 0 0 1 9 5 6 2 9 8 3 . 8 0 0 0 1 2 0 3 v 9 4 0 4 6 1 9 7 . 4 0 6 3 1 3 8 0 . 8 0 0 0 1 9 5 7 3 2 0 2 . 6 0 0 0 1 3 6 9 . 1 1 2 6 6 9 2 0 . 2 0 7 0 1 5 5 4 . 5 0 0 0 1 9 5 8 3 3 7 0 . 7 0 0 0 1 4 7 5 - 3 3 8 4 7 4 0 2 . 2 8 5 2 1 6 9 8 . 8 0 0 0 1 9 5 9 3 5 4 5 * 6 0 0 0 1 6 2 8 . 9 6 2 2 7 9 9 9 . 5 5 8 6 1 8 9 6 . 3 0 0 0 1 9 6 0 3 8 0 7 . 0 0 0 0 1 8 8 6 . 3 3 3 0 8 8 1 6 . 2 1 4 8 2 2 4 4 . 3 9 9 9 1 9 6 1 4 1 8 9 . 1 0 0 0 2 2 S 4 i 7 5 : 4 9 T 0 5 3 3 . 3 2 4 0 2 6 3 9 . 7 0 0 0 1 9 6 2 4 7 1 7 . 8 0 . 0 0 2 6 4 2 * ; 7 5 8 8 1 2 8 2 7 * 4 3 0 0 3 2 6 1 . 5 9 9 9 1 9 6 3 5 4 1 3 . 1 0 0 0 3 0 8 4 . 3 1 1 8 1 5 4 2 6 . 8 5 2 0 3 9 5 6 . 3 0 0 0 1964 6 0 4 7 . 6 0 0 0 3 5 7 9 . 3 0 9 6 1 8 1 0 4 . 4 8 8 0 4 6 9 4 . 1 9 9 2 1 9 6 5 6 8 2 5 . 3 0 0 0 4 0 5 8 . 9 2 1 6 2 1 4 1 4 . 4 2 6 0 5 3 8 8 . 5 0 0 0 1 9 6 6 7 6 4 4 f . 7 0 0 0 4 7 0 2 . 8 5 9 4 2 3 6 5 3 . 2 6 6 0 6 2 9 2 . 7 9 6 9 1 9 6 7 8 6 3 2 . 0 0 0 0 5 4 2 7 . 7 3 8 3 2 7 8 6 3 * 9 5 3 0 7 1 8 7 . 6 9 5 3 1 9 6 8 9 7 2 4 . 0 0 0 0 6 3 2 1 . 8 9 0 6 3 2 7 2 0 . 6 5 2 0 8 4 4 0 . 8 9 8 4 1 9 6 9 1 0 9 7 6 . 7 0 0 0 7 3 6 9 . 5 7 4 2 3 8 9 0 4 . 0 0 4 0 9 8 0 8 - 6 9 5 3 1 9 7 0 1 2 4 4 9 . 6 0 0 0 8 7 8 7 * 6 5 6 3 4 7 6 3 1 . 4 4 1 0 1 1 5 3 9 . 8 0 0 8 1 9 7 1 1 3 8 7 3 . 2 0 0 0 1 0 2 1 3 * 7 0 7 0 5 5 5 0 5 . 4 2 6 0 1 3 0 3 8 * 1 9 9 2 1 9 7 2 1 5 5 3 7 . 2 0 0 0 1 1 7 3 1 . 6 0 6 0 6 5m 7 * 5 0 0 0 ^ 1 5 2 7 4 . 5 9 7 7 - 188 -T A B L E A.2 M I L L I O N YEAR P O P U L A T I O N HQOSEHOLDS LABOR F O R C E 1912 50.5770 10.1360 26.3470 1913 51.3050 10.2820 26.4220 1914 52.0390 10.4290 26.4710; 1915 52.7520 10.5720 26.5270 1916 53.4960 10.7210 26.5570 1917 54.1340 10.8480 26.5890 -1918 54.7390 10.9700 26.6180 1919 55.0330 11.0290 26.6230 1920 ~ 55.3910 11.1010 27.2630 1921 56*1200 11.2460 27.4980 1922 56.8400 11.3910 27.7330 1923 57.5800 11.5390 27.9690 1924 58.3500 11.6930 28.2060 1925 59.1790 11*8790 28.4420 1926 60.2100 12.0420 28.6760 1927 . 61.1400 12.1790 28.9130 1928 62.0700 12.3150 29.1480 1929 62.9300 12.4370 29.3840 1930 63.8720 12-5820 29*6190 1931 64*8700 12.7450 28.9900 1932 65.8900 12.9200 29.1760 1933 66*a8:oo- 13.0880 29.7770 1934 67.6900 13.2210 30*7940 1935 68*6620 13.3780 31.4000 1936 69.5900 13.5650 30.8550 1937 ;70.0400 13.6800 31.1620 1938 70.5300 13.8020 31.47 30 19 39 v 70.8500 13.8920 31.7800 1940. 71.4000 14.2190 32.4780 1941 0. 0 0.0 32.5770 1942 0*0 T 0. 0 32.5970 1943 0.0- 0*0 • 0.0 1944 , 0*0 0.0 0.0 1945 ; 72.2000 14.6750 0.0 - 1 8 9 -T A B L E A . 2 G O H f l H O B D Y E A B P O P O L A T I O U H O U S E H O L D S LABOR F O R C E 1 9 4 6 7 5 . 8 0 0 0 1 5 . 4 0 7 0 3 2 . 5 9 7 0 1 9 4 7 7 8 . J 0 1 0 1 5 - 8 7 1 0 3 3 . 3 2 9 0 1 9 4 8 8 0 . 0 1 0 0 1 6 . 0 8 9 0 3 4 . 0 7 8 0 1 9 4 9 8 1 . 7 8 0 0 1 6 . ; 4 8 8 0 3 4 . 8 4 3 0 1 9 5 0 8 3 . 2 0 0 0 1 6 . 5 8 0 0 3 5 . 6 2 6 0 1 9 5 1 8 4 . 5 0 0 0 1 6 . 9 3 4 0 3 6 . 2 2 0 0 1 9 5 2 8 5 - 8 0 0 0 1 7 * 1 3 4 0 3 7 . 2 9 0 0 1 9 5 3 8 7 - 0 0 0 0 1 7 . 4 7 0 0 3 9 . 1 2 0 0 1954 8 8 . 2 0 0 0 1 7 . 7 1 1 0 3 9 . 6 2 0 0 • 1 9 5 5 8 9 . 2 7 6 0 1 7 . 9 6 0 0 4 1 . 1 9 0 0 1 9 5 6 9 0 . 1 7 2 0 1 8 * 4 7 8 0 4 1 . 7 2 0 0 1 9 5 7 9 0 . 9 2 8 0 1 8 . 9 8 3 0 4 2 . 8 4 0 0 ~ 1 9 5 8 9 1 . 7 6 7 0 1 9 . 5 2 5 0 4 3 . 1 2 0 0 1 9 5 9 9 2 . 6 4 1 0 2 0 . 0 9 6 0 4 3 ^ 7 0 0 0 1 9 6 0 9 3 . 4 1 9 0 2 0 . 6 5 6 0 4 4 * 6 1 0 0 1 9 6 1 9 4 . 2 8 5 0 2 1 . 2 8 3 0 4 5 . 1 8 0 0 1 9 6 2 9 5 . 1 7 8 0 2 1 . 9 3 0 0 4 5 . 7 4 0 0 1 9 6 3 9 6 . 1 5 6 0 2 2 . 6 2 5 0 4 6 . 1 3 0 0 1 9 6 4 9 7 . 1 8 6 0 2 3 . 3 6 2 0 4 6 . 7 3 0 0 1 9 6 5 9 8 . 2 7 5 0 2 4 . 0 8 2 0 4 7 . 4 8 0 0 1 9 6 6 9 9 * 0 3 6 0 2 4 . 7 0 2 0 4 8 . 4 7 0 0 1 9 6 7 1 0 0 . 1 9 6 0 2 5 . 4 4 2 0 4 9 . 2 0 0 0 1 9 6 8 1 0 1 . 3 3 1 0 2 6 . 0 6 9 0 5 0 . 0 2 0 0 1 9 6 9 1 0 2 . 5 3 6 0 2 6 . 3 8 8 0 5 0 * 4 0 0 0 1 9 7 0 1 0 3 . 7 2 0 0 2 7 . 7 5 7 0 5 0 * 9 4 0 0 r 1 9 7 1 1 0 5 . 0 1 4 0 2 8 . 3 8 0 0 5 1 . 1 4 0 0 1 9 7 2 1 0 7 . . 3 3 2 0 2 9 . 0 0 9 0 5 1 . 0 9 0 0 - 1 9 0 -T A B L E A . 3 YEAR I N T E R E S T B A T E T H E T A 1912. 5 . 0 0 0 0 1. 1 1 5 2 1 9 1 3 5 . 0 0 0 0 1. 1 0 5 5 1 9 1 4 5 . 0 0 0 0 1 . 1 1 7 5 1 9 1 5 5 . 0 0 0 0 - 1 . 1 1 9 5 1 9 1 6 5 . 0 0 0 0 1. 1 4 1 7 1 9 1 7 5 . 0 0 0 0 1 . 1 9 2 6 1 9 1 8 5 . 0 0 0 0 1. 1 9 2 1 1 9 1 9 5 . 0 0 0 0 1 . 1 6 9 6 1 9 2 0 5 . 0 0 0 0 1 . 1 3 3 1 1 9 2 1 5 . 0 0 0 0 1 . 1 3 6 8 1922 , 5 . 0 0 0 0 1 . 1 4 0 6 1 9 2 3 5 * 0 0 0 0 1 . 1 4 4 3 1 9 2 4 5 . 0 0 0 0 1 . 1 3 8 8 1 9 2 5 5 . 0 0 0 0 1. 1 5 7 5 1 9 2 6 6 * 2 2 4 0 1 . 1 3 8 6 1 9 2 7 5 . 8 0 0 0 1 . 1 5 1 6 1 9 2 8 5 * 3 3 1 0 1 * 1 3 7 7 1 9 2 9 5 * 3 8 2 0 - 1 . 1 4 5 1 1 9 3 0 5 . 5 5 0 0 1 . 1 4 7 5 1 9 3 1 5 . 1 8 1 0 1 . 1 4 0 4 1 9 3 2 5 . 4 6 5 0 1 . 1 2 6 6 1 9 3 3 4* 1 2 2 0 1 . 1 2 9 6 1 9 3 4 4 . 1 1 8 0 1. 1 5 5 3 1 9 3 5 4 . 1 1 6 0 1 . 1 4 1 3 1 9 3 6 3 . 7 0 5 0 1 . 1 5 2 2 1 9 3 7 3 . 7 0 5 0 1 . 1 7 7 2 1 9 3 8 3 . 7 0 5 0 1 . 2 0 3 9 1 9 3 9 3 . 7 0 5 0 1 . 1 8 4 5 1 9 4 0 3 . 6 8 9 0 i ) 1 . 1 0 4 5 1 9 4 1 3 * 6 8 9 0 ; 1 . 1 0 0 3 1 9 4 2 3 . 6 8 9 0 , 1 * 1 0 2 1 1 9 4 3 3 - 6 8 9 0 1 . 1 0 0 6 1 9 4 4 , 3 * 6 8 9 0 1 . 0 9 8 7 1 9 4 5 : 3 * 6 8 9 0 : 0 . 0 - 191 -TABLE A . 3 CONTINUED YEAR INTEREST RATE SAVINGS TBETA; 1946 3 .68 90 8.H3000 1 .0516 1947 4 . 4 3 9 0 • : - 2 8 . 3 0 0 0 1.0280 1948 5 . 5 3 6 0 : - 1 3 . 4 0 0 0 1.0238 1949 5 . 5 1 6 0 - 3 3 . 7 0 0 0 1.0190 1950 • 5 . 5 0 0 0 2 9 6 . 6 0 0 0 1.0195 1951 5 . 5 0 0 0 3 8 8 . 6 0 0 0 1.0180 1952 5 .5000 4 4 2 . 4 0 0 0 1.0170 1953 7 . 2 3 3 0 3 9 4 . 8 0 0 0 1.0167 1954 6 . 3 2 4 0 S48*i6060 1.0164 1955 6 *3420 8 5 2 . 5 0 0 0 1.0128 1956 6 - 3 3 6 0 9 5 1 . 5 0 0 0 1.0127 1957 6 . 3 3 6 0 1 2 1 8 . 5 0 0 0 1*0146 1958 6 . 3 3 6 0 1247.8000< 1.0150 1959 6 . 3 2 4 0 1546 .7000 1.0168 1960 6 . 4 3 2 0 1863 .5000 1*0186 1961 6 . 4 3 2 0 2 4 0 1 . 9 0 0 0 1.0209 1962 6 . 4 3 2 0 2 6 9 1 . 0 0 0 0 1.0270 1963 6 . 4 3 2 0 3 0 2 2 . 4 0 0 0 1.0257 1964 6 . 4 3 2 0 3 1 5 6 . 1 0 0 0 1.0248 1965 6 . 4 3 2 0 3 8 3 5 . 9 0 0 0 1.0214 1966 6 . 7 9 5 0 4 3 4 1 . 4 0 0 0 1.0220 1967 6 . 7 9 5 0 5 5 4 7 . 9 0 0 0 1.0208 1968 6 . 9 0 2 0 6 7 6 4 . 1 0 0 0 1.0205 1969 6 . 9 0 2 0 7 6 1 8 . 5 0 0 0 1.0241 1970 7 . 0 1 1 0 9 4 7 4 . 5 0 0 0 - 1.0246 1971 6 . 9 9 4 0 10460 .8000 1.0279 1972 6 . 7 1 7 0 1 2 5 5 7 . 6 0 0 0 1.0280 - 192 -TABLE A. 4 YEAR COHSUSPTIOH YEARS 10BKIHG YEARS 1 9 4 6 2 1 . 1 8 4 0 1 2 . 0 6 6 0 1 9 4 7 1 8 . 9 2 7 3 1 2 . 1 4 4 0 1 9 4 8 1 6 . 2 3 4 4 1 2 . 2 2 3 0 1 9 4 9 1 6 v 3 0 6 4 1 2 . 3 0 1 0 1 9 5 0 1 6 . 3 6 4 6 1 2 i 3 7 9 0 1 9 5 1 1 6 . 4 4 2 6 1 2 . 4 5 7 0 1 9 5 2 1 6 . 4 1 8 0 1 2 . 5 3 5 0 1 9 5 3 1 3 . 1 7 9 7 1 2 . 6 1 4 0 1 9 5 4 1 3 . 1 9 1 1 1 2 . 6 9 2 0 1 9 5 5 1 4 . 8 0 0 5 1 2 . 7 7 0 0 1 9 5 6 1 4 . 8 0 3 6 1 2 . 7 9 5 0 1 9 5 7 1 4 . 7 9 5 1 .. 1 2 . 8 2 0 0 1 9 5 8 1 4 . 7 9 4 8 1 2 . 8 5 0 0 1 9 5 9 1 4 . 8 0 8 8 1 2 . 8 7 0 0 1 9 6 0 1 4 . 5 9 5 6 1 2 . 8 9 5 0 1 9 6 1 1 4 . 5 2 9 6 1 2 . 8 8 1 0 1 9 6 2 1 4 . 5 2 7 9 1 2 . 8 6 7 0 1 9 6 3 1 4 . 5 2 6 3 1 2 . 8 5 4 0 1 9 6 4 1 4 . 5 2 4 6 1 2 . 8 4 0 0 1 9 6 5 1 4 . 5 2 3 0 1 2 . 8 2 6 0 1 9 6 6 1 4 . 2 0 8 5 1 2 . 8 8 3 0 1 9 6 7 1 4 . 2 1 5 6 1 2 . 9 3 9 0 1 9 6 8 1 3 . 7 0 3 2 1 2 . 9 9 6 0 1 9 6 9 1 3 . 7 0 3 8 1 3 . 0 5 2 0 1 9 7 0 1 3 . 5 2 5 1 1 3 . 1 0 9 0 1 9 7 1 1 3 . 8 8 3 9 1 3 * 1 6 6 0 1 9 7 2 1 4 . 0 2 0 1 1 3 . 2 2 2 0 - 193 -TABLE A.5 STOCK ESTIMATE BILLION YEN YEAB PRIVATE HOUSE PUBLIC HOUSE CLOTHING FURNITURE 1912 3593.5493 0.0 609.7090 101.9731 1913 ,. 3722.4612 0.0 665.6174 .105.3453 1914 3698.2314 0.0 H 682.7056 101.2762 1915 3650.647 9 0.0 725.5615 101.6876 1916 , 3595.3372 0.0 . . . '. 837.6526 106.3501 1917 - 3530.9087 0.0 919.3508 115.9669 1918 , 3504.3381 0.0 1038.5315 131.5235 1919 3584.3767 0.0 1265.152 V 148.6398 1920 3632.6274 o.o 1275.8823 162.1154 1921 ; 3633. 8472 0.0 13 83.194 8 170.6389 1922 3619.3403 0.0 1476.5557 179.7187 1923 3616.1931 0.0 1449.8865 184.4 813 1924 3697.7883 0.0 1454.4228 207.4939 1925 •< 3687.7300 0.0 1423.4121 231.7914 1926 -,. 3717.3547 0.0 14 48.2459 260.93,73 1927. 3728.7419 0.0 1518.0999 298.0579 1928 3687.4717 0.0 1639.7749 322.4084 1929 3693.2161 0.0 ;  1646.3603 335.2852 1930 3731.2134 0.0 1698.9863 352.8145 1931 .3763.9495 0.0 1836.9177 . 377.4055 1932 3808.0620 0.0 1949.6191 . 397.9419 1933 3870.2249 0.0 . 2005.8066 413.4951 1934 3945.2434 0.0 . 2198.1892 446.3335 1935 4020.6582 0.0 2353.6719 44 8.8848 1936 4108.1094 0.0 2545.7222 463.9561 1937 4 249.0273, 0.0 2838.5269 496.2334 1938 4297.0859 0.0 3039.1345 511.4072 1939 4348.3086 0.0 2701.3833 565.3479 1940 4402.0078 0.0 2433.0942 646.0781 1941 4449.8359 0.0 2195.6919 698.6804 1:942 4463.8164 0.0 1942.5105 725.6108 1943 '4454.3203 0.0 1750.2649 734.3345 1944 , • •r , 4400.6836 0.0 1474.5164 730.3245 1945 3447.4861 0.0 817.3196 502.8293 - 194 -T A B L E A . 5 C O N T I N U E D S T O C K E S T I M A T E B I L L I O N YEN YEAR P R I V A T E H O U S E P U B L I C H O U S E C L O T H I N G F U R N I T U R E 1 9 4 6 3 6 0 5 . 2 9 1 5 0 . 0 7 0 8 . 8 0 4 0 4 5 0 . 0 1 1 0 1 9 4 7 3 6 9 9 . 6 0 5 7 0 . 0 6 3 4 . 7 3 1 4 4 0 6 . 4 8 7 5 1 9 4 8 3 8 2 0 . 8 1 6 2 0 . 0 5 8 5 . 3 7 5 0 3 7 9 . 1 9 5 8 1 9 4 9 3 8 4 8 . 5 7 9 6 0 . 0 5 5 8 . 5 1 3 9 3 6 9 . 9 4 1 2 1 9 5 0 3 9 5 2 . 3 6 5 2 0 . 0 6 4 3 . 2 6 1 7 3 6 8 . 7 6 5 6 1 9 5 1 4 0 7 4 . 5 7 9 6 2 5 . 3 5 9 2 8 4 0 . 3 5 0 1 3 6 1 . 9 5 9 7 1 9 5 2 4 2 8 7 . 5 9 7 7 5 1 . 8 2 6 4 r 1 3 2 0 . 8 1 9 8 4 0 7 . 8 0 5 2 1 9 5 3 4 5 1 4 . 6 0 9 4 8 8 . 4 9 3 7 : 1 7 7 7 . 4 5 9 5 4 8 9 . 8 1 3 7 1 9 5 4 4 7 7 7 . 7 0 3 1 1 2 7 . 9 9 8 2 2 1 3 0 . 1 6 2 6 5 6 6 . 5 8 0 6 1 9 5 5 , 5 0 7 2 . 0 0 7 8 1 6 9 . 6 5 2 3 2 4 4 6 . 1 2 5 5 6 4 4 . 9 6 7 3 1 9 5 6 5 4 2 5 . 7 6 5 6 2 1 7 . 5 1 1 8 2 7 7 9 . 3 5 8 4 7 3 4 - 3 3 2 0 19 57 5 8 1 9 . 6 9 9 2 2 6 5 . 3 4 5 0 - 3 1 2 4 . 6 1 6 2 • 8 1 4 . 0 6 8 8 1 9 5 8 6 2 6 3 . 8 2 4 2 3 2 5 . 3 7 7 7 3 4 1 6 . 8 2 9 6 9 1 6 . 4 6 7 8 1 9 5 9 6 7 3 9 . 8 2 0 3 3 8 2 . 8 4 3 0 3 7 0 9 . 9 3 9 2 1 0 7 4 . 4 0 0 9 1 9 6 0 7 3 8 2 . 0 9 7 7 4 3 5 . 8 3 5 9 4 1 6 0 . 1 5 2 3 1 2 7 8 . 0 2 3 2 1 9 6 1 8 0 9 5 . 6 0 1 6 4 9 5 . 4 1 5 0 4 6 3 5 . 1 1 3 3 1 5 4 5 . 8 2 9 1 1 9 6 2 8 9 1 2 . 6 6 0 2 5 7 4 . 8 8 3 5 5 1 6 7 . ,1914 1 8 4 8 . 6 6 7 0 1 9 6 3 9 9 0 5 . 5 1 9 5 6 5 7 . 3 6 0 1 ..: . 5 6 8 8 . 2 4 6 1 2 2 6 5 . 5 1 6 8 1964 1 1 1 9 7 . 2 4 2 2 7 4 6 . 3 2 3 0 6 2 0 6 . 7 1 8 8 2 7 6 3 . 1 7 6 5 1 9 6 5 1 2 7 2 6 . 7 9 6 9 , 8 5 8 . 5 6 4 7 6 6 7 9 . 2 5 3 9 3 1 6 2 . 1 4 1 4 1 9 6 6 1 4 3 6 7 . 5 8 5 9 9 8 4 . 3 5 4 2 - 7 1 1 4 . 9 8 4 4 3 5 9 2 . 1 5 1 6 1 9 6 7 1 6 2 7 6 . 5 0 3 9 11 1 8 . 4 4 4 8 7 6 0 2 . 5 7 4 2 , 4 1 4 0 . 5 1 5 6 1 9 6 8 1 8 5 3 2 . 5 9 7 7 1 2 8 0 . 0 3 7 1 8 1 5 8 . 3 4 7 7 4 8 1 6 . 3 9 0 6 1 9 6 9 2 1 1 2 2 . 3 5 9 4 1 4 6 5 . 7959, 8 7 2 1 . 5 0 0 0 5 7 0 5 . 8 7 5 0 1 9 7 0 2 3 9 7 6 . 7 2 6 6 1 7 0 6 . 4 5 4 3 9 2 4 7 . 9 7 2 7 6 6 1 2 . 2 9 3 0 1 9 7 1 . 2 6 8 4 9 . 5 1 5 6 1 9 9 2 . 4 0 5 3 9 7 8 6 . 2 5 0 0 7 6 1 6 . 6 9 5 3 1 9 7 2 3 0 2 1 6 . 3 8 2 8 2 2 6 4 . 3 1 6 6 1 0 4 1 2 . 7 0 3 1 8 6 5 2 - 7 0 7 0 - 195 -FOOTNOTES t o A p p e n d i x A 1. T h i s i s d o n e b y t a k i n g t h e sum o f o n e - q u a r t e r o f t h e p r e -v i o u s y e a r ' s e s t i m a t e a n d t h r e e - q u a r t e r s o f t h e e s t i m a t e f o r t h e c u r r e n t y e a r . 2. T h e e x p e c t e d r a t e s o f p r i c e r i s e a r e f o r m u l a t e d a s e ( . t + l ) = t X 1(1-A)p ( t - i ) w h e r e p i s t h e a c t u a l r a t e o f i=0 p r i c e c h a n g e a n d A i s a p a r a m e t e r o f s p e e d o f a d j u s t m e n t ( t o b e e s t i m a t e d o r a s s u m e d ) . U p o n t r a n s f o r m a t i o n , we g e t a n e q u a t i o n b e l o w . e ( t + l ) = ( l - A ) p ( t ) +.aeCt). 3 . T h e r e n t a l p r i c e i s d e f i n e d a s f o l l o w s . p C t ) - *"•+•**<<pet), 1 + r * ( t ) w h e r e r * ( t ) i s t h e r e a l i n t e r e s t r a t e a n d r * ( t ) = ^ ' +" e ( t ) ~ 1 ' 6 i s t h e d e p r e c i a t i o n r a t e a n d p ( t ) <is t h e p u r c h a s e p r i c e o f t h e c o m m o d i t y i n q u e s t i o n a t t i m e t . S o i f r * ( t ) < 0 a n d S £ f r * ( t ) | , t h e n p ( t ) < 0. T h e h i g h e r t h e e x p e c t e d p r i c e r i s e a n d t h e l o w e r t h e d e p r e c i a t i o n r a t e , t h e m o r e p r o b a b l e ' i t i s t o h a v e n e g a t i v e r e n t a l p r i c e s . 4. T h e d a t a we h a v e a r e o n t h e d a m a g e t o f u r n i t u r e a n d u t e n -s i l s . So i t i s a s s u m e d t h a t t h e damage t o c l o t h i n g o c c u r r e d t o t h e same e x t e n t a s t o f u r n i t u r e a n d u t e n s i l s . I n a d d i -t i o n , we u s e d 1955 r a t i o o f c l o t h i n g s t o c k t o t h a t o f f u r -n i t u r e a n d a p p l i a n c e s f o r 194 5, s i n c e t h e r e i s n o e s t i m a t e o f c l o t h i n g s t o c k f o r 1945. 5. T h i s f i g u r e i s d e r i y e d d a s a p r o x y e s t i m a t e . A s s u m i n g t h a t t h e c u r r e n t e x p e n d i t u r e s ( i n c o n s t a n t y e n ) h a s b e e n i n t h e n e i g h b o r h o o d o f 150 b i l l i o n y e n f o r a, l o n g t i m e , t h e n t h e s t o c k c a n b e a p p r o x i m a t e d b y 150/.2748 = 54 6. 6. ' H o u s i n g e x p e n d i t u r e ' i n t h e n a t i o n a l i n c o m e s t a t i s t i c s i n c l u d e s e x p e n d i t u r e o n r e n t , f u r n i t u r e a n d a p p l i a n c e s , w a t e r a n d l i g h t . I t d o e s n o t i n c l u d e m o r g a g e p a y m e n t s o r a n y o t h e r p a y m e n t d o r p u r c h a s e o f h o u s e s . T h i s i s i n c l u d e d i n c a p i t a l f o r m a t i o n . FOOTNOTES to Appendix A (con't) 7. 22.220 b i l l i o n yen x 77.6% = 17.242 b i l l i o n yen. In 1965 p r i c e , 17.242/.0205 = 843.1648 b i l l i o n yen. 8. 'Wages and s a l a r i e s ' include bonus, family allowance and goods i n kind as well as regular wages and s a l a r i e s . 'Other pay and allowances' include t i p , honorarium, retirement fund and housing subsidy. - 197 -Appendix B A MATHEMATICAL NOTE The evaluation of 9, when y i s lognormally d i s t r i b u t e d , i s performed as follows: where 6 = /ylny (y)dy/y*lny*. Let z = lny, then y = e . Furthermore, g ( z l = ( 2 ' i f B ) - V ( z - a , 2 / 2 e , where a i s the mean and £ i s the variance of z. In order to evaluate 8, we f i r s t integrate the i n t e g r a l A, i. e . , A . = |ylny^ C y)dy =_/e Zzg (z) dz . Then, A = ( 2 , g ) - ^ z e z - C z - a ) 2 / 2 p d z + Let k = z - (z - a ) 2 / 2 B = - Iz- - ( a + B ) ] 2 / 2 B + (a+|)', then dk = ( a + B-z ) /6 . -3*- ^ « v -V [z- (aft[3)] 2 A =.-(.2.*) "e^e^dK + (2.&S) % ( a + B)^e d - z z -e ( z - a r ~ 2 l dz (2 TJB) 2 ( a + B ) £ e z d j Thus, - 198 -Let x = [z-(a+6)]/(23) i s, then dz = (20) ^ dx. £ 2 A = 0 + (2Tr-.6)"35(a+6) (2g) V +^e~ x dx 2 2 = .1f"35(a+e)ea+2tje"x dx + / e _ x dx] . From a table of i n t e g r a l s , we f i n d /e x dx = * 2/2. 0 2 Notice that e i s an even function, i . e . , f(x) = f(-x) 2 , °° - X , _//e dx = TT' • Therefore, £ A = ( a + e ) e a + 2 £ a+2 j . However, e = y* and (a+6) = ^ l n ( a 2 + y* 2) . In ( a 2 .+. y* 2) Thus, 9 = — : -21ny* where a y i s the standard deviation and y* i s the mean of y, 

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