Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Consumption, leisure and the demand for money and money substitutes Donovan, Donal John 1977

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

Item Metadata

Download

Media
831-UBC_1977_A1 D65.pdf [ 14.99MB ]
Metadata
JSON: 831-1.0094061.json
JSON-LD: 831-1.0094061-ld.json
RDF/XML (Pretty): 831-1.0094061-rdf.xml
RDF/JSON: 831-1.0094061-rdf.json
Turtle: 831-1.0094061-turtle.txt
N-Triples: 831-1.0094061-rdf-ntriples.txt
Original Record: 831-1.0094061-source.json
Full Text
831-1.0094061-fulltext.txt
Citation
831-1.0094061.ris

Full Text

CONSUMPTION, LEISURE, AND THE DEMAND FOR MONEY AND MONEY SUBSTITUTES by DONAL JOHN DONOVAN B . A . T r i n i t y C o l l e g e , U n i v e r s i t y of Dub l in , 1973 A t h e s i s submitted in p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of DOCTOR OF PHILOSOPHY in the Department of ECONOMICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1977 (?) Donal John Donovan In present ing t h i s thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It i s understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of Economics The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e A p r i l 22, 1977 CONSUMPTION , LEISURE, AND THE DEMAND FOR MONEY AND MONEY SUBSTITUTES Research Superv isor : Professor David E. Rose ABSTRACT The purpose of t h i s research is to develop and tes t a model o f the demand f o r money wi th in a general opt imis ing model of household behaviour. The framework adopted i s the d i r e c t u t i l i t y approach. The serv ices of money and money s u b s t i t u t e s , along with the serv ices o f consumption goods (durable and non durable) and l e i s u r e are assumed to enter as arguments in the representat ive household's u t i l i t y f u n c t i o n . The t h e o r e t i c a l part o f the thes is cons is ts of apply ing the too ls of modern u t i l i t y theory to the p a r t i c u l a r problem of the demand f o r money. The development and so lu t ion o f the model provides a c l e a r basis f o r i n t e r p r e t i n g the demand equations used in es t ima t ion , and a lso makes e x p l i c i t various assumptions i m p l i c i t in previous empir ica l models in th is a rea . In p a r t i c u l a r , der iva t ion o f the rental p r i c e o f money and money subs t i tu tes serves to c l a r i f y the ro le o f expectat ions and the r e l a t i o n s h i p between the rental p r i ces of money and goods within the d i r e c t u t i l i t y model. The major part of the thes is cons is ts of apply ing the model to annual Canadian data fo r the per iod 1947-1974. A substant ia l por t ion o f the empir ica l cont r ibu t ion is the const ruct ion of data s e r i e s cons is ten t with the t h e o r e t i c a l framework o f the model. We d i f f e r from other researchers in t h i s area in using the ARIMA model to take expected cap i ta l gains in to account when const ruc t ing rental p r i c e s e r i e s f o r durable goods. i i • Three d i f f e r e n t groups of models are examined e m p i r i c a l l y . The f i r s t group contains only consumption goods and l e i s u r e . The second group inc ludes aggregate 'money' and aggregate 'near money' along with consumption goods and l e i s u r e , while the t h i r d group contains only ' l i q u i d a s s e t s ' , i . e . , disaggregated components of 'money' and 'near money.' The demand equations f o r each model are der ived from a Gorman po la r form representat ion o f the i n d i r e c t u t i l i t y f u n c t i o n , and are evaluated using a constra ined est imat ion technique. The presence of au tocor re la t ion i s exp lored , and the model tested fo r parametric s t a b i l i t y over t ime. Tests o f the r e s t r i c t i o n s impl ied by the theory o f u t i l i t y maximising behaviour and of homothet ic i ty are performed. The estimated models were found genera l ly to be cons is ten t with the under ly ing theory , and a lso provided some useful in format ion . Money has an expenditure e l a s t i c i t y l ess than one, while near money is a luxury good. There i s no evidence o f s u b s t i t u t a b i l i t y between aggregate money and aggregate near money; however, some s u b s t i t u t a b i l i t y i s reported between chartered bank personal savings d e p o s i t s , and t r u s t and loan company savings depos i ts . ( i i i ) TABLE OF CONTENTS ABSTRACT ( i ) TABLE OF CONTENTS ( i i i ) LIST OF TABLES. (v) ACKNOWLEDGEMENTS ( v i i i ) Chapter 1 INTRODUCTION 1 Footnotes 6 Chapter 2 MODELLING THE DEMAND FOR MONEY A. Theore t i ca l Approaches 7 B. Empir ical Models 22 C. Canadian Studies in the Demand f o r Money 38 D. Conclusions 42 Footnotes 45 Chapter 3 THE DEMAND FOR MONEY WITHIN A GENERALISED UTILITY FRAMEWORK A. The F isher L i f e Cycle Model 49 B. The Treatment of Durable Goods 52 C. Money as a Durable Good 58 D. Funct ional S e p a r a b i l i t y and Aggregation 66 E. Proposed Models 74 Footnotes 76 Chapter 4 FUNCTIONAL FORM AND ESTIMATING EQUATIONS A. Some Resul ts in Dua l i ty Theory 78 B. Aggregation Across Consumers and the Gorman Polar Form 82 C. .A S p e c i f i c Form f o r the Ind i rect U t i l i t y Function 89 (iv) D. E l a s t i c i t i e s and Tests of U t i l i t y Maximising Behaviour 92 Footnotes 97 Chapter 5 DATA: METHODOLOGY AND SOURCES A. The Choice of an Aggregate Index Number Formula 101 B. Non Durable Goods and Serv ices 106 C. Durables and Semi Durables 113 D. Le isure 125 E. Money 138 F. Money Subst i tu tes 154 Footnotes 167 Chapter 6 ECONOMETRICS: ESTIMATION AND HYPOTHESIS TESTING A. Est imat ion 174 B. Hypothesis Tes t ing 180 Footnotes 184 Chapter 7 EMPIRICAL RESULTS A. Consumption-Leisure Models 186 B. Real-Monetary Models 1:97 C. L iqu id Asset Models 218 Footnotes 236 Chapter 8 SUMMARY AND CONCLUSIONS 237 Footnotes 244 BIBLIOGRAPHY 245 STATISTICAL SOURCES 255 Appendix A THE CONSTRUCTION OF FORECAST PRICE SERIES USING AN ARIMA MODEL • 257 Appendix B SUPPLEMENTARY DATA TABLES 266 (v) LIST OF TABLES 5.1 Populat ion of Canada 15 Years and Over 109 5.2 Per Capi ta Quantity Indexes, Non Durable Goods and Serv ices 110 5.3 P r ice Indexes, Non Durable Goods and Services 111 5.4 Aggregate Pr ice and Per Capi ta Quanti ty Indexes, Non Durable Goods and Serv ices 112 5.5 Deprec ia t ion Rates, Discount Rates, and Benchmarks f o r Twelve Durable and Semi Durable Goods 126 5.6 Discount Rates and Property Tax Rates 127 5.7 Per Capi ta Normalised Quantity Indexes, Six Durable and Semi Durable Goods ( S t a t i c Expectat ions) 128 5.8 Normalised Rental P r i c e Indexes, Six Durable and Semi Durable Goods (S ta t i c Expectat ions) 129 5.9 P r i c e and Quantity Indexes, Aggregate Durables and Semi Durables (S ta t i c Expectat ions) 130 5.10 Per Capi ta Normalised Quantity Indexes, Six Durable and Semi Durable Goods (Non S t a t i c Expectat ions) 131 5.11 Normalised Rental P r i c e Indexes, Six Durable and Semi Durable Goods (Non S t a t i c Expectat ions) 132 5.12 P r i c e and Quanti ty Indexes, Aggregate Durables and Semi Durables (Non S t a t i c Expectat ions) 133 5.13 A l t e r n a t i v e Rental P r ice Ser ies fo r Housing 134 5.14 Labour: Net Wages, Hours, Le isure and Tax Rates 137 5.15 Average Year ly Household Holdings of Chartered Bank Savings D e p o s i t s , Personal Chequing Accounts , and Currency 148 5.16 Discount Rate, Aggregate P r i c e L e v e l , and Forecast Aggregate P r i c e Level 149 5.17. Average Year ly Interest Rates, Personal Savings Deposit Categor ies 150 5.18 Rental Pr ices of Chartered Bank Savings Depos i ts , Personal Chequing Accounts , and Currency 151 (v i ) 5.19 Per Capita Quantity Indexes of the Monetary Serv ices of Chartered Bank Savings Depos i ts , Personal Chequing Accounts, and Currency 152 5.20 Normalised Rental Pr ice and Per Capi ta Quantity Indexes f o r the Serv ices of Money 153 5.21 Annual Average Holdings of TML Deposit L i a b i l i t i e s by Category, and Canada Savings Bonds 162 5.22 Annual Average Interest Rates on TML Deposit L i a b i l i -t i e s by Category and Canada Savings Bonds 163 5.23 Rental P r i c e s , TML Deposit L i a b i l i t i e s and Canada Savings Bonds 164 5.24 Normalised Rental P r ice and Per Capita Quantity Indexes f o r Real 'Near Money1 Balances by Category 165 5.25 Normalised Rental P r i c e and Quantity Indexes of the Serv ices of 'Near Money' 166 7.1 Parameter Est imates: The Three Good Consumption Model (Non Homothetic Verson) 198 7.2 E l a s t i c i t y Est imates: The Three Good Consumption Model (Non Homothetic Version) 199 7.3 Test S t a t i s t i c s : The Three Good Consumption Model 201 7.4 Parameter Est imates: The Four Good Consumption-Le isure Model (Non Homothetic Version) 202 7.5 E l a s t i c i t y Est imates: The Four Good Consumption-Le isure Model (Non Homothetic Version) 204 7.6 Test S t a t i s t i c s : The Four Good Consumption-Leisure Model 206 7.7 Comparative E l a s t i c i t y Est imates , Four Good Leisure Model, with (p^0) and without (p=0) F i r s t Order Autoregressive Est imat ion Technique 207 7.8 Parameter Est imates , Four Good Money Model (Non Homothetic V e r s i o n , F i r s t Order Autocor re la t ion Scheme) 219 7.9 E l a s t i c i t y Est imates , Four Good Money Model (Non Homothetic V e r s i o n , F i r s t Order Autocor re la t ion Scheme) 221 7.10 Parameter Est imates , Four Good Money-Money Subst i tu te Model (Non Homothetic V e r s i o n , Non S t a t i c Expectat ions) 223 v i i i . ACKNOWLEDGEMENTS In the course of undertaking t h i s research I have received a great deal o f help from many people. In p a r t i c u l a r , I would l i k e to thank the members o f my d i s s e r t a t i o n committee, c o n s i s t i n g o f Professors D.E. Rose, W.E. Diewert, and C F . Boonekamp. The chairman, Professor Rose, provided i n i t i a l encouragement fo r the p r o j e c t , pa t i en t l y endured innumerable queries and d i s c u s s i o n s , and read in painstaking d e t a i l success ive dra f ts of the t h e s i s . In p a r t i c u l a r , h is suggestions f o r improvements in e x p o s i -t iona l c l a r i t y and s t y l e are much apprec ia ted . The in f luence o f the teachings of Professor Diewert is s e l f - e v i d e n t in my work and I should l i k e to record my apprec ia t ion o f his constant encouragement at var ious stages of the research . Professor Boonekamp made a number o f va luable suggestions f o r improving e x p o s i t i o n . I would a lso l i k e to acknowledge the comments o f Professors Peter C h i n l o y , Robert Evans, John H e l l i w e l l , Ronald Shearer and Kenneth White. Among my fe l low graduate s tudents , Thomas Bennet t , Mohammed Khaled and Theodore Panayotou deserve spec ia l mention f o r put t ing up with a constant stream of d iscuss ion and arguments. Kevin C l in ton and Marg F i t z p a t r i c k of the Bank of Canada, and Dianne Cummings Chr istensen o f the Un ive rs i t y of Wisconsin were kind enough to provide me with unpublished data s e r i e s . Reidun Tvedt d id an e x c e l l e n t job of typing a long and cumbersome manuscript. I wish to acknowledge the f i n a n c i a l support o f a Canada Council doctoral fe l lowship during the per iod 1975-1977. F i n a l l y , I owe a spec ia l debt to Paula Gannon, fo r prov id ing continual a f f e c t i o n and encouragement (as well as inva luable research assistance) throughout the l a s t two years o f my s tudies at U .B .C . Th is thes is is dedicated to her . D . J . D . Washington, D.C. A D H I . 1Q77 Chapter 1 INTRODUCTION A demand f o r money funct ion i s an important part of most aggre-gate models of an economy. Knowledge of the in f luence of var ious explanatory var iab les on the demand fo r money i s essent ia l in order to understand the potent ia l e f f e c t s of monetary p o l i c y act ions designed to a l t e r e i t h e r the supply of money, i n t e r e s t r a t e s , or both. Motivated by these c o n s i d e r a t i o n s , a vast amount of empir ical research on the subject has appeared s ince the 1950's. Despite the volume of research undertaken, two fundamental c r i t i c i s m s can be made of the l i t e r a t u r e as a whole. F i r s t , very r a r e l y has an attempt been made to der ive the demand f o r money funct ion to be estimated from a choice theore t ic model. Second, i t i s u s u a l l y assumed (often without comment) that the demand f o r money d e c i s i o n could be studied without reference to other d e c i s i o n s of the household (or f i r m ) . There i s , t h e r e f o r e , a need f o r an empir ical model of the demand f o r money which is f i r m l y embedded wi th in a general theore t ica l model of household or f i rm behaviour. While the behavioural hypothesis underly ing the t y p i c a l demand f o r money equation i s , at f i r s t s i g h t , u n c l e a r , i t appears that most researchers have, in f a c t , r e l i e d i m p l i c i t l y upon the approach suggested by Mi l ton Friedman [1956]. According to Friedman, money i s assumed to y i e l d ' s e r v i c e s ' which enter as arguments in e i t h e r the household's u t i l i t y f u n c t i o n , or the f i r m ' s production f u n c t i o n . The ob jec t ive of those adopting t h i s approach i s then to estimate how much money the household or f i rm w i l l hold as a r e s u l t of some i m p l i c i t u t i l i t y or p r o f i t maximisation process . However, with very few except ions , t h i s - 2 -maximisation process has not been stated e x p l i c i t l y . We f i n d th is s i t u a -t i o n u n s a t i s f a c t o r y . I f money i s to be regarded as a durable good y i e l d i n g serv ices to e i t h e r the household or f i r m , then i t i s d e s i r a b l e to model e x p l i c i t l y the underly ing u t i l i t y or p r o f i t maximisation problem. Furthermore, the choice problem must be made broad enough to encompass more than j u s t the demand f o r money d e c i s i o n . The main purpose .of : t h i s research i s to develop and tes t such a behavioural model f o r the household sector of the economy. The serv ices of money (and var ious 'near m o n i e s ' ) , as well as the serv ices of real durable goods, consumption goods, and l e i s u r e are assumed to enter as arguments in the household's u t i l i t y f u n c t i o n . Spec i fy ing and so lv ing the associated constra ined opt imisat ion problem r e s u l t s in a system of demand equations which forms the basis f o r the empir ical t es t of the model. In undertaking t h i s task , we make extensive use of a number of important advances in the theory and a p p l i c a t i o n of models of u t i l i t y maximisat ion. F i r s t , the use of the rental p r ice or user cost approach to the demand fo r durable goods enables us to model t h e i r demand s imul -taneously with the demand f o r non durable goods. Since money must be t reated as a durable good y i e l d i n g desi red serv ice flows wi th in t h i s framework, an important issue to be dea l t with i s the de r iva t ion of the renta l p r i c e of money. Second, funct iona l forms f o r household u t i l i t y funct ions have been developed which place few a p r i o r i r e s t r i c t i o n s on preferences . Furthermore, the recogn i t ion of var ious d u a l i t y r e l a t i o n -ships g rea t ly f a c i l i t a t e s the de r iva t ion of demand equations cons is tent with such general representat ions of pre ferences . T h i r d , r e s u l t s in the theory of aggregation across consumers enable us to obtain market demand equations d i r e c t l y from a model of ind iv idua l consumer behaviour. - 3 -The major part of the thes is c o n s i s t s of applying the model to Canadian data f o r the period 1947-1974. A substant ia l por t ion of the empir ica l con t r ibu t ion i s the const ruct ion of data se r ies cons is ten t with the t h e o r e t i c a l framework of the model. P r i c e and quant i ty indexes f o r non durables and s e r v i c e s , l e i s u r e , and the serv ices of durables ( inc lud ing semi durab les ) , money and money subst i tu tes are c a l c u l a t e d . ^ Ke d i f f e r from other researchers in th is area in attempting to take expected cap i ta l gains into account when const ruc t ing rental p r ice se r ies f o r durable goods. The ARIMA model of a time s e r i e s (Box and Jenkins [1970]) i s used to obtain estimates of the expected rate of i n f l a t i o n fo r d i f f e r e n t 2 durable goods. Three d i f f e r e n t groups of models are examined e m p i r i c a l l y . The f i r s t group contains only consumption goods and l e i s u r e . The second group includes aggregate 'money' and aggregate 'near money1 along with consumption goods and l e i s u r e , whi le the t h i r d group contains only ' l i q u i d a s s e t s ' , i . e . , disaggregated components of 'money' and -'near money 1. Thus, while i n t e r e s t in the demand f o r money funct ion i s our i n i t i a l mot iva t ion , the con t r ibu t ion of t h i s research i s of a much more general nature. An important feature of the model is that i t allows an e x p l i c i t t es t of the fundamental hypothesis of u t i l i t y maximising behaviour underly ing the model. As w e l l , d i f f e r e n t estimated models may be compared in order to e s t a b l i s h the extent to which, f o r example, the subst i tu t ion /complementar i ty r e l a t i o n s h i p s among real consumption • goods a re . a l te red by :the . - inc lus ion of l e i s u r e and money (or near money) as add i t iona l va r iab les in the u t i l i t y f u n c t i o n . Vie a lso explore the s e n s i t i v i t y of our r e s u l t s to whether the est imat ing technique - 4 -takes in to account the poss ib le presence of a u t o c o r r e l a t i o n . The remainder of the thes is is organised as f o l l o w s . In Chapter 2, we survey d i f f e r e n t t h e o r e t i c a l and empir ica l approaches to model l ing the demand fo r money, paying p a r t i c u l a r a t tent ion to the ' d i r e c t u t i l i t y ' approach. The d i s c u s s i o n deals p r i n c i p a l l y with recent models of the demand f o r money as e a r l i e r work i s adequately surveyed elsewhere. A b r i e f summary of Canadian demand fo r money studies is a lso prov ided. Chapter 3 o u t l i n e s a general intertemporal model of household u t i l i t y maximisat ion, and develops the rental p r ice concept fo r the serv ices of both durable goods and money. We then consider the model l ing of expectat ions and issues of funct iona l s e p a r a b i l i t y and aggregat ion. The f i n a l sec t ion of Chapter 3 contains a summary of the models to be est imated. In Chapter 4, we der ive the est imat ing equations from an e x p l i c i t form f o r the household's i n d i r e c t u t i l i t y funct ion using Roy's Ident i ty . Careful a t tent ion i s paid to the issue of aggregating over consumers to obta in market demand equat ions. P r i c e and expenditure 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 s u b s t i t u t i o n are d e r i v e d . The chapter concludes by summarising the var ious tes ts of the funct iona l form and of the theory of demand. Chapter 5 is concerned with data . The ob jec t ive i s to obtain p r i c e and quant i ty indexes f o r d i f f e r e n t c l a s s e s of expenditure on consumption s e r v i c e s , inc lud ing expenditure on monetary s e r v i c e s . The procedure adopted i s to commence with disaggregated da ta , and using a s p e c i f i c aggregation method descr ibed in sec t ion A of Chapter 5 , b u i l d up .the aggregated s e r i e s . Chapter 6 is devoted to econometric c o n s i d e r a t i o n s , s p e c i f i c a l l y , - 5 -est imat ion and hypothesis t e s t i n g . In Chapter 7, we report r e s u l t s from the three groups of models descr ibed above. F i n a l l y , Chapter 8 assesses the overa l l r e s u l t s of the model, and d iscusses the scope fo r fu r ther research in t h i s area . There are two appendices. Appendix A descr ibes the use of ARIMA models to const ruct fo recas t p r i c e s e r i e s , while Appendix B contains some supplementary data t a b l e s . - 6 -Footnotes - Chapter 1 1. Ser ies f o r consumption goods and l e i s u r e already have been constructed f o r Canada by Gussman [1972] and Cummings and Meduna [1973]. However, f o r a number of reasons, to be explained in Chapter 5, t h e i r work was considered u n s a t i s f a c t o r y , and i t was found necessary to reconstruct e n t i r e l y these s e r i e s . 2. The f o r e c a s t i n g procedure adopted uses r e s u l t s developed by Rose [1976]. Chapter 2 MODELLING THE DEMAND FOR MONEY The purpose of t h i s chapter is to review some a l t e r n a t i v e approaches to model l ing the demand f o r money. A voluminous l i t e r a t u r e , both t h e o r e t i c a l and e m p i r i c a l , e x i s t s on the s u b j e c t , and t h i s review does not c la im to be, in any sense, exhaust ive. Rather, our approach i s s e l e c t i v e , and seeks to h i g h l i g h t a number of key issues in the a rea , issues which a r i s e , , below, in the development of our theore t i ca l and empir ical model. In sec t ion A of t h i s chapter , a l t e r n a t i v e t h e o r e t i c a l frameworks are d i s c u s s e d . Sect ion B deals with e m p i r i c a l l y or iented models of the demand f o r money, w h i l e , in sect ion C , a b r i e f review of Canadian studies on the top ic i s presented. In sec t ion D, we summarise the main conclusions of the survey, and expla in our mot ivat ion f o r the model to be developed subsequently. Throughout t h i s chapter , our main concern i s models of household behaviour, although in p r a c t i c e , much of the work to be surveyed does not make any sectora l d i s t i n c t i o n . A. Theore t ica l Approaches^ A large and r a p i d l y growing l i t e r a t u r e e x i s t s on the t h e o r e t i c a l issues of why and in what amount money i s demanded. Broadly speaking, three well known approaches may be d i s t i n g u i s h e d : 1) the Tobin-Markowitz mean.-variance model o f . p o r t f o l i o c h o i c e , 2) the Baumol-Tobin t ransact ions cost model, and 3) the d i r e c t u t i l i t y approach. Although these three approaches are i n t e r r e l a t e d ( e s p e c i a l l y 2 and 3 ) , we begin by consider ing each separa te ly . - 8 -1) The Tobin-Markowitz Mean-Variance Model This model i s a spec ia l case of a more general model of expected u t i l i t y maximisation (Markowitz £ 1 9 5 2 ] , Tobin [1958]). The expected u t i l i t y model considers the problem faced by a f i rm or household (no sec tora l d i s t i n c t i o n i s made) of how to a l l o c a t e i t s to ta l 'wealth' among d i f f e r e n t forms of assets at the beginning of a per iod . 'Wealth' in the model usua l ly cons is ts of the sum to be held in f i n a n c i a l assets such as money, bonds, or e q u i t i e s . However, at l eas t in p r i n c i p l e , the model could be extended to include other forms of wealth such as durable goods or phys ica l c a p i t a l held by households or f i r m s , and even human c a p i t a l . It i s assumed that the dec is ion maker assoc ia tes with the set of assets a s u b j e c t i v e l y perceived m u l t i v a r i a t e p r o b a b i l i t y d i s t r i b u t i o n of returns over the p e r i o d , and possesses a u t i l i t y funct ion def ined over the end of per iod wealth. The dec is ion maker i s assumed to maximize 3 the expected u t i l i t y of end period wealth. Solv ing t h i s maximization problem r e s u l t s in a system of der ived demand equations fo r f i n a n c i a l assets on the part of the ind iv idua l ' i n v e s t o r ' . These demand funct ions a r e , in g e n e r a l , complex funct ions of both the moments of the perceived p r o b a b i l i t y d i s t r i b u t i o n of asset returns and the parameters of the i n d i v i d u a l ' s u t i l i t y f u n c t i o n . The expected u t i l i t y model as descr ibed above has been analysed ex tens ive ly and appl ied to a wide range of i n t e r r e l a t e d problems. It has been used to inves t iga te problems such as stock p o r t f o l i o s e l e c t i o n (Markowitz [1952]), the ro le of r i s k and uncerta inty in the consumption-saving and p o r t f o l i o a l l o c a t i o n dec is ions within a l i f e time consumption flow model (Samuelson [1969]), and the theore t ica l foundations of a - 9 -general equ i l ib r ium model of f i n a n c i a l markets (Sharpe [1964], L in tner [1965]). Under c e r t a i n assumptions, the expected u t i l i t y maximisation hypothesis reduces to the mean-variance model proposed by Markowitz [1952] and Tobin [1958]. For example, i f the u t i l i t y funct ion i s quadrat ic in weal th, or i f the perceived p r o b a b i l i t y d i s t r i b u t i o n of returns is normal, then the p o r t f o l i o a l l o c a t i o n problem can be solved by consider ing only the mean and var iance of the p r o b a b i l i t y d i s t r i b u t i o n . In t h i s case , the maximisation problem may be envisaged as a two stage procedure. F i r s t , the investor der ives the ' e f f i c i e n t f r o n t i e r ' (defined as the set of a l l o c a t i o n s g iv ing maximum mean returns f o r any standard d e v i a t i o n ) . Second, he superimposes h is i n d i f f e r e n c e curves drawn in mean-standard dev ia t ion space in order to der ive the optimal p o r t f o l i o . Tobin [1958] f i r s t proposed t h i s model in order to expla in the simultaneous holding (by an i n d i v i d u a l ) of p o s i t i v e amounts of a r i s k l e s s a s s e t , 'money' (the nominal c a p i t a l value of which is known with c e r t a i n t y ) , and another r i s k y a s s e t , ' bonds ' . Given the existence of the r i s k l e s s a s s e t , money, the e f f i c i e n t f r o n t i e r in mean-standard dev ia t ion space 4 5 w i l l be a s t r a i g h t l i n e , and d i v e r s i f i c a t i o n . w i l l normally occur . Tobin was not the f i r s t to suggest the existence of uncer ta inty as a motive f o r holding money. The theory of l i q u i d i t y preference ou t l ined by Keynes [1936] a lso r e l i e d in part upon uncer ta inty in order to expla in the demand fo r money. However, Keynes' theory d id not permit i n d i v i d u a l p o r t f o l i o d i v e r s i f i c a t i o n to occur . His model requires the assumption of e i t h e r d i f f e r i n g (cer ta in ) views among investors regarding some 'normal ' rate of i n t e r e s t on bonds, or a l t e r n a t i v e l y , d i f f e r i n g degrees of uncer ta inty concerning one (commonly held) 'normal 1 r a te . - 10 -However, recent ly the use of the mean-variance model to expla in the demand f o r money has come under seri.ou ;s c r i t i c i s m , The quadrat ic u t i l i t y funct ion is l i m i t e d in i t s range of a p p l i c a b i l i t y as a u t i l i t y f u n c t i o n , as f o r c e r t a i n values of wealth marginal u t i l i t y becomes negat ive . Furthermore, within the app l i cab le range, i t d isp lays the implausib le property of increas ing absolute r i s k avers ion . On the other hand, there does not e x i s t strong evidence to support the assumption that f i n a n c i a l returns are charac te r ised by a normal d i s t r i b u t i o n . The mean-variance model has a lso been j u s t i f i e d as an approximation to the more general expected u t i l i t y model. S . C . ' T s i a n g [1972] using a T a y l o r ' s s e r i e s expansion to der ive expected u t i l i t y as an ( i n f i n i t e ) weighted sum of the moments of the re levant p r o b a b i l i t y d i s t r i b u t i o n , examined the condi t ions under which the i n f i n i t e sum may be s a f e l y t run -cated beyond the f i r s t two moments, namely, the mean and the var iance . T s i a n g ' s p r i n c i p a l r e s u l t i s the f o l l o w i n g . He considered two u t i l i t y funct ions of weal th , namely, the negative exponential funct ion and the fami ly of constant e l a s t i c i t y of marginal u t i l i t y f u n c t i o n s , both of which s a t i s f y the four ' acceptab le 1 r i s k proper t ies of Arrow [1965].^ Tsiang demonstrates that fo r both, of these f u n c t i o n s , the standard dev ia t ion of expected wealth (a) must be a ' s m a l l ' proport ion of the mean expected wealth (W) in order that moments higher than the second order may be neglected s a f e l y . Of course , what const i tu tes 'small enough to be neglected ' i s a r b i t r a r y . However, T s i a n g ' s r e s u l t s served to c l a r i f y the i m p l i c i t assumptions under ly ing the use , as an approximation, of the mean-variance model, and a lso helped resolve paradoxes ra ised by previous c r i t i c s (Borch [1969], F e l d s t e i n [1969]) o f the model. - 11 -Furthermore,. Ts-i/ang shows that provided the u t i l i t y funct ion i s everywhere increas ing in weal th , the i n d i f f e r e n c e curves which may be drawn l e g i t i m a t e l y in mean-standard dev ia t ion space (that i s , i f the 'o/W s m a l l 1 assumption i s warranted) w i l l nowhere have a s lope greater than un i ty . Th is r e s u l t has strong impl ica t ions fo r model l ing the demand f o r money wi th in the mean-variance framework. Suppose there e x i s t s any one r i s k y a s s e t , the expected y i e l d of which is at l eas t as great as the standard dev ia t ion of that y i e l d . Tsiang demonstrates t h a t , so long as the y i e l d of the asset i s not p e r f e c t l y p o s i t i v e l y cor re la ted with the y i e l d of the e x i s t i n g p o r t f o l i o , the ind iv idua l always can move to a higher i n d i f f e r e n c e curve by switching from money into that r i s k y asse t . This conclus ion fo l lows from the assumed slope of the i n d i f f e r e n c e curve drawn in mean-standard dev ia t ion space. Thus, according to the Tsiang argument, an ind iv idua l ac t ing in accordance with the mean-variance approximation to the expected u t i l i t y c r i t e r i o n , w i l l not possess any demand f o r money, def ined as the r i s k l e s s asse t . It would appear as though the Tobin explanat ion of the demand f o r money has been weakened to a ser ious extent. E s s e n t i a l l y , T s i a n g ' s argument s tates that we can only draw mean-standard dev ia t ion ind i f f e rence curves l e g i t i m a t e l y under c e r t a i n assumptions. However, the f u l f i l l m e n t of these assumptions impl ies that there w i l l never e x i s t a p o s i t i v e demand f o r money from t h i s source. This r e s u l t i s extremely important and o does not seem to have been s e r i o u s l y chal lenged as y e t . Of course , T s i a n g ' s conclus ion does not preclude the existence of a -demand fo r money based on the more general expected u t i l i t y model. Apart from the t h e o r e t i c a l impl ica t ions of the mean-variance approximation assumptions, there are a number of other aspects of the - 12 -Tobin p o r t f o l i o model of the demand f o r money which should be s t ressed at t h i s s tage . F i r s t , the model ignores the real consumption and product ion g dec is ions of the household or f i r m , Second, the t ransact ions aspect of the demand f o r money i s not taken into account - money i s t reated simply as one way of holding wealth. T h i r d , at best the model can expla in only the d i v i s i o n between to ta l non r i s k y assets (that i s , 'money') and other r i s k y a s s e t s . It provides no explanat ion as to why households or f i rms hold d i f f e r e n t forms of 'money 1 , such as cash and d i f f e r e n t kinds of bank and near bank d e p o s i t s . However, t h i s l a t t e r problem could be a l l e v i a t e d i f t ransact ions costs were introduced into the model. This is the next t h e o r e t i c a l approach to be considered. 2) The Baumol-Tobin Transact ions Cost Model The t ransact ions cost approach (Baumol [1952], Tobin [1956]) is based on: (a) the recogni t ion that payments are made in the form of money, and (b) the ex is tence o f t ransact ions costs in switching between money and i n t e r e s t bearing bonds. At l e a s t f o r simple assumptions regarding the time path of rece ip ts and expendi tures , the behavioural postu late of cost minimisat ion (or p r o f i t maximisation) on the part of the dec is ion u n i t ^ leads to a unique s o l u t i o n fo r '.the optimal demand f o r money. According to the theory developed by Baumol and Tobin and extended by Feige and Parkin [1971], the demand f o r real cash balances i s shown to be inverse ly propor-t i o n a l to the square root, of the i n t e r e s t rate earned on bonds, and d i r e c t l y proport ional to the square root of both the income (that i s , rece ip ts or expenditures) v a r i a b l e , and the cost of f i n a n c i a l t r a n s a c t i o n s . The model i s usua l l y presented in a two asset (money-bond) framework. In p r i n c i p l e , the model could be extended to inc lude vectors - 13 -of d i f f e r e n t kinds of monies, bonds, and real a s s e t s . However, in p r a c t i c e t h i s i s extremely d i f f i c u l t . The work of Brunner and Mel tzer [1967] ind ica tes that even the two asset (money-bond) model becomes very complex when the simple assumptions regarding the time path of rece ip ts and expenditures are re laxed . A model p a r t l y along these l i n e s has been developed by Gray and Parkin [1973]. The i r model i s a combination of the t ransact ions costs and uncer ta inty approaches - the authors term i t a 'precaut ionary demand f o r money1 model. They assume a simple t ransac t ion cost technology where the conversion of any f i n a n c i a l asset into money i s assumed to cost a f i x e d d o l l a r amount, and where net cash requirements are u n c e r t a i n . ^ Maximi-sat ion of expected p r o f i t r e s u l t s in non zero demand equations f o r a l l a s s e t s , provided that c e r t a i n inequal i ty ; 1 r e l a t i o n s h i p s hold among the i n t e r e s t rates earned on the a s s e t s . An important conc lus ion i s that at most three i n t e r e s t rates w i l l appear in each asset equation - the own rate and the two 'ad jacent 1 r a t e s , where the assets have been ranked according to t h e i r r e l a t i v e i n t e r e s t r a t e s . In applying the model to the U.K. (Bar re t t , Gray and Parkin [1972]), a problem with i t became apparent. The ranking of the assets changed f requent ly over the est imat ion p e r i o d , and hence the inequa l i t y r e l a t i o n s h i p required f o r p o s i t i v e demands fo r a l l assets in every time per iod did not ho ld . In t h i s case the model would p red ic t cont inual disappearance and appearance of assets over time, which d id not occur . However, t h i s t h e o r e t i c a l conclus ion of the model fo l lows only i f a l l d e c i s i o n makers face i d e n t i c a l l i q u i d a t i o n c o s t s . Thus a way out of the d i f f i c u l t y , which was adopted by the authors , i s to assume that l i q u i d a t i o n costs d i f f e r across i n d i v i d u a l s in some unspec i f ied manner. - 14 -An a l t e r n a t i v e way of extending the t ransact ions cost approach is to re ta in the two asset framework, but expand the model to include as a choice v a r i a b l e the time path of real va r iab les when car ry ing costs of holding inventor ies of goods are present . Feige and Parkin [1971] inves-t i g a t e a four good model conta in ing money, bonds, commodities and c a p i t a l . A l s o included are parameters to take account of the car ry ing costs of each good, and the costs of bond market and commodity market t r a n s a c t i o n s . The model y i e l d s (complex) demand funct ions f o r average cash balances, commodities, and bonds, each contain ing as arguments, d isposable human and nonT-fiuman income, i n t e r e s t r a t e s , t ransact ions costs and inventory car ry ing c o s t s . Feige and Parkin show that t h e i r model, with appropriate s i m p l i f y i n g assumptions imposed, reduced to the o r i g i n a l Baumol-Tobin inventory model. The model i s used by Feige and Parkingto inves t iga te t h e o r e t i c a l issues in the optimal quant i ty of money debate. However, the empir ica l a p p l i c a t i o n of the model i s severe ly r e s t r i c t e d by the (general ly unobservable) carrying-and t ransact ions cost parameters in the f i n a l demand equat ions. Somewhat s i m i l a r problems are addressed by Saving ([1971], [1972]). However, while Saving a lso includes parameters to take account of car ry ing c o s t s , the t ransact ions cost funct ion is made more s p e c i f i c . For the household (Saving [1971]) , t ransact ions costs are assumed to c o n s i s t of time c o s t s , • and are hypothesised to be a funct ion of the l e v e l s of income and consumption, "the average leve l of work serv ices 12 de l ivered but not c o l l e c t e d f o r " , the average leve l of consumption goods h e l d , and average real holdings of the var ious media of exchange. Saving assumes that the optimal paths of consumption and l e i s u r e are - 15 -predetermined, and der ives so lu t ions f o r the optimal stock of goods, work se rv ices and money forms. Imposing add i t iona l r e s t r i c t i o n s on the form of 13 the t ransact ions cost f u n c t i o n , Saving es tab l ishes that an increase in des i red income or l e i s u r e is accompanied by an increase in the average leve l of the stocks of money and goods demanded. However, the quant i ty e f f e c t s of i n t e r e s t rate changes are indeterminate, and depend upon the r e l a t i v e storage costs of the var ious s tocks . While Sav ing 's model appears too complex to solve f o r e x p l i c i t t ransact ions costs or u t i l i t y f u n c t i o n s , i t nonetheless provides some i n t e r e s t i n g q u a l i t a t i v e r e s u l t s . Furthermore, the model s t resses the interdependence of the consumption, l e i s u r e , and money holding dec is ions of the household. Th is interdependence a r i s e s because of the assumed ro le of money in reducing t ransact ions t ime, and hence i n d i r e c t l y a f f e c t i n g the demand f o r l e i sure -and a lso the demand f o r consumption. Sav ing 's [1972] model of the demand f o r money by f irms is s i m i l a r in s t r u c t u r e , and corresponding q u a l i t a t i v e r e s u l t s are obta ined. Recent ly , Saving [1976] has shown that the existence and proper t ies of the t ransac t ion • cost f u n c t i o n , which he had merely assumed in h is e a r l i e r work, can be der ived from a genera l ised Baumol-Tobin inventory theore t ic model. Saving es tab l ishes that the t ransac t ion cost func t ion w i l l be a monotonical ly decreas ing , concave funct ion of the average stocks of money and goods h e l d , and a monotonical ly increas ing funct ion of the l e v e l s of income and consumption. The Saving approach l i n k s the real and monetary dec is ions of the households by in t roducing time t ransact ions c o s t s . Diewert [1970] has proposed a model f o r households along s i m i l a r l i n e s . He assumes the ex is tence of a 'purchasing technology' f o r each good whereby a - 16 -purchased good i s re la ted to three ' f a c t o r s of p r o d u c t i o n ' , namely, the unpurchased good, time spent purchasing the good, and real balances used in order to purchase the good. That i s , the consumer combines his t ime, money, and the 'good in the s t o r e ' in order to obtain a good ready f o r consumption. With an add i t iona l assumption of constant returns to s c a l e , the shadow p r i c e of a uni t of the 'purchased good' equals the uni t cost funct ion corresponding to the purchasing technology. Assuming that the consumer has preferences def ined over purchased goods and l e i s u r e , the model may be solved f o r the u t i l i t y maximising quant i t i es of unpurchased goods, l e i s u r e , t ransact ions time and real balances used to purchase each good. While the l a s t set of var iab les i s not observable , we may aggregate over a l l goods to obtain aggregate t ransact ions time and aggregate real balances demanded. Each demand equation w i l l conta in the p r i c e o f each purchased good, the wage r a t e , and to ta l expenditure. The Diewert approach is extremely i n t e r e s t i n g , but there are some problems assoc ia ted with the model. F i r s t , from the point of view of empir ica l implementation, the actual est imat ing equations w i l l be of a very complex form, as each o f the independent purchased good p r i c e va r iab les are themselves f u n c t i o n s , that i s , un i t cost funct ions repre-sent ing the under ly ing purchasing technology. Second, as Diewert himself points out , i t i s not very reasonable to impose constant returns to sca le on the purchasing technology. Diewert shows that the technology cons is ten t with the Baumol-Tobin inventory model in f a c t w i l l e x h i b i t increas ing returns to s c a l e . However, in th is c a s e , the shadow p r i c e of the purchased good no longer w i l l be unique, but w i l l depend upon the quant i t i es demanded. Pol lak and.Wachter [1975] recent ly have demonstrated - 17 -that t h i s problem i s present in a l l models attempting to incorporate the concept of households 'producing ' a c t i v i t i e s in to a u t i l i t y maximi-sing model, and they argue that i t e f f e c t i v e l y precludes the useful empir ica l a p p l i c a t i o n of t h i s type of m o d e l . 1 4 As can be seen from the above d i s c u s s i o n , the t ransact ions costs approach has progressed some considerab le way beyond the simple Baumol-Tobin inventory theore t i c model. Our understanding of how money in te rac ts with the real d e c i s i o n making by households and f i rms has been enhanced great ly by the development of these models. However, the models share two ser ious problems: f i r s t , considerable complexi ty , and second, the large part played by va r iab les d i f f i c u l t to observe and measure, such as storage and t ransact ions c o s t s . As a consequence, these models have ra re ly been developed s u f f i c i e n t l y to provide e x p l i c i t funct iona l so lu t ions in terms of observable va r iab les which could form the basis f o r empir ical 15 implementation. Because of t h i s , most e m p i r i c a l l y or iented researchers in the area have r e l i e d on another approach, the d i r e c t u t i l i t y approach, to which we now t u r n . 3) The D i rec t U t i l i t y Approach Many economists have considered the average stock of real money balances as an argument in the household's u t i l i t y funct ion (or , c o r r e s -1 c pondingly , viewed moneycas an input in the f i r m ' s production f u n c t i o n ) . Walras [1926; Lesson 29] , Hicks [1935], Samuelson [1947; 117-124], Friedman [1956], and Pat ink in [1965; Ch.5] are among the many wr i te rs who have d iscussed and defended the approach. E s s e n t i a l l y , the model can be regarded as a very general statement of the t ransact ions cost models discussed in the previous s e c t i o n . Households are assumed to hold money f o r i t s 'convenience y i e l d ' - 18 -which cons is ts of a reduct ion in t ransact ions costs in terpreted in a very broad sense. That i s , there is assumed to be some r e l a t i o n between the average stock of purchasing power held in the form of real balances (which may be of d i f f e r i n g types) and c h a r a c t e r i s t i c s such as 't ime and t r o u b l e 1 , the des i re f o r safekeeping, and accounting s e r v i c e s . Real balances then enter the u t i l i t y funct ion because of these serv ice f l o w s . 1 7 To quote Samuel son: " . . . Possession of an average amount of i t [money] y i e l d s convenience in permit t ing the consumer to take advantage of o f f e r s of s a l e , in f a c i l i t a t i n g exchanges, in br idging the gap between r e c e i p t of income and expenditure e t c . . . . Possession of t h i s balance then y i e l d s a real se rv ice which can be compared with the d i r e c t u t i l i t i e s from the consumption of sugar , tobacco, e t c . in the sense that there is some margin at which the i n d i v i d u a l would be i n d i f f e r e n t between having more tobacco and less of a cash balance with a l l the inconvenience which the l a t t e r cond i t ion i m p l i e s " . (Samuelson [1947; 118]) Friedman, in h is famous "Restatement of the Quantity Theory of Money" a r t i c l e (Friedman [1956]) o f f e r s an e s s e n t i a l l y s i m i l a r argument. It i s observed that households hold money balances and, s ince they are not forced to do s o , possession of real balances must y i e l d some kind of u t i l i t y or s a t i s f a c t i o n . According to t h i s view, the d i r e c t u t i l i t y approach does not seek to analyse the nature of t h i s u t i l i t y ( for the same reason that economists do not ask what i t i s that cons t i tu tes t h e . u t i l i t y y i e l d e d by a r e f r i g e r a t o r ) but rather seeks to determine how much money an ind iv idua l w i l l want to ho ld . A var iant of t h i s argument r e l i e s on the axiomatic basis of u t i l i t y theory. Provided that a consumer's preferences s a t i s f y t h e axioms of t r a n s i t i v i t y , r e f l e x i v i t y , and symmetry over a choice set inc lud ing consumption bundles and real ba lances, then i t i s p e r f e c t l y leg i t imate to represent h is preferences by a u t i l i t y funct ion conta in ing - 19 -real balances as an argument. What tends to cause some unease at t h i s approach i s the unspec i f i ed and vague nature of the ' s e r v i c e s ' y i e l d e d by money. As we noted, one view, namely that of Friedman, would regard t h i s as an i r r e l e v a n t and unhelpful ques t ion , at l eas t f o r c e r t a i n purposes. However, some w r i t e r s , notably Pat ink in [1965] have t r i e d to e laborate a l i t t l e more p r e c i s e l y on the i s s u e . P a t i n k i n ' s model deals with an ind iv idua l during a Hicksian 'week 1 , who can hold his wealth in the form of e i t h e r money or i n t e r e s t bearing bonds. It i s assumed that encashment of bonds can be undertaken only at f i x e d i n t e r v a l s during the week, but payments and rece ip ts occur randomly. Hence, in order to avoid the 'embarrassment and bother' of being without cash fo r unforeseen payments, the ind iv idua l w i l l plan to hold an average stock of real balances during the week. Avoidance of the 'embarrassment;and bother ' of running short of cash const i tu tes the u t i l i t y of holding real money balances. It might be suggested that P a t i n k i n ' s explanat ion only begs the quest ion of the nature of the 'embarrassmentand bother ' invo lved . However, his d i s c u s s i o n does help to focus on the s t o c h a s t i c nature of payments and r e c e i p t s , as well as the existence of t ransact ions costs ( 'bother ' in P a t i n k i n ' s terminology) as being the source of the u t i l i t y of money. This supports the argument that the u t i l i t y model is r e a l l y a very general and purposely vague statement of the t ransact ions cost approach. However, the d i f f e r e n c e between the two models is that the u t i l i t y approach in essence assumes that c e r t a i n v a r i a b l e s , which may a f f e c t the u t i l i t y der ived from the average stock of real balances h e l d , remain 18 constant . If these v a r i a b l e s , such as the time path of r e c e i p t s and - 20 -expendi tures , t ransact ions c o s t s , (both pecuniary and non-pecuniary in nature) and storage c o s t s , a l t e r , then the d i r e c t u t i l i t y model can say only that a 'change in t a s t e s ' , in terpre ted in the broadest sense, has o c c u r r e d , s h i f t i n g the u t i l i t y funct ion in an unspec i f i ed manner. Such arguments are unnecessary in the case o f a completely s p e c i f i e d t ransact ions cost model. Since the above mentioned var iab les enter e x p l i c i t l y into t h i s model, the e f f e c t s of t h e i r a l t e r i n g over time can be.analysed r e a d i l y . However, as we have seen, the t ransact ions models have not in general been developed s u f f i c i e n t l y to provide e x p l i c i t funct iona l s o l u -t ions conta in ing observable va r iab les as arguments. The 'unobservable v a r i a b l e ' problem i s p a r t l y , but not e n t i r e l y , a data d e f i c i e n c y , as there are conceptual d i f f i c u l t i e s associa ted with measuring ce r ta in types of ' c o s t s ' . Thus, i t could be argued, with some f o r c e , that s ince the f u l l t ransact ions cost model i s not o p e r a t i o n a l , the best that can be done at t h i s stage is to adopt the d i r e c t u t i l i t y model, and hope that the va r iab les a f f e c t i n g the u t i l i t y of money remain constant . This is a methodology s i m i l a r to that adopted in most ordinary consumer demand s t u d i e s , where va r iab les such as a d v e r t i s i n g and q u a l i t y v a r i a t i o n may a l t e r the u t i l i t y der ived from c e r t a i n commodities. However, data and measurement problems r e s t r i c t the ana lys is to e s t a b l i s h i n g whether or not a s t r u c t u r a l change, of an unspec i f ied nature , has occurred. On the other hand, the d i r e c t u t i l i t y model does possess a number o f major advantages over the other approaches, mainly with respect to empir ica l implementation. F i r s t , proper ly s p e c i f i e d in a s u i t a b l y general way, the v a l i d i t y of the model i t s e l f may be t e s t e d . That i s , the parameters o f the estimated u t i l i t y funct ion may be used to tes t whether actual observed behaviour over time i s cons is ten t with the e x i s -- 21 -tence of a 'well behaved' preference order ing contain ing money as an argument. By 'well behaved V we mean, fo r example, that marginal u t i l i t i e s are p o s i t i v e , and that i n d i f f e r e n c e curves are convex. Th is point is o f considerable i n t e r e s t in view of the use made of the d i r e c t u t i l i t y approach to money in growth models such as those by Levhari and Pat ink in [1968] and S idrausk i [1967]. Second, s ince va r iab les such as consumption and l e i s u r e a lso enter as arguments in the household's u t i l i t y f u n c t i o n , in terdependences between the real and f i n a n c i a l dec is ions of the household can be captured e a s i l y . The common approach of modell ing the choice among l i q u i d assets without cons ider ing the ro le real va r iab les may p l a y , i s seen to be a spec ia l case of a more general model. Furthermore, the p r a c t i c e of model l ing the demand fo r one a s s e t , money, in i s o l a t i o n , i s an even more spec ia l case of the general model. With the d i r e c t u t i l i t y model, the s e p a r a b i l i t y r e s t r i c t i o n s on the general preference order ing required fo r 19 these sub models to be v a l i d may be e x p l i c i t l y derived and recognised. T h i r d , as w i l l be seen s h o r t l y in the d iscuss ion of empir ical models, the funct iona l s p e c i f i c a t i o n of the estimated demand fo r money equation i s of ten a r b i t r a r y . However, i f instead we s t a r t with an e x p l i c i t (general) form f o r the u t i l i t y f u n c t i o n , the demand funct ions fo r assets corresponding to t h i s u t i l i t y funct ion may be derived as the s o l u t i o n to a constra ined maximization problem. Furthermore, once the parameters of 20 the preference funct ion have been est imated, corresponding expenditure e l a s t i c i t i e s , own and cross p r i c e 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 s u b s t i t u t i o n may be computed. Four th , re la ted to the f i r s t point above, researchers often are faced with the dilemma of choosing both the appropr iate dependent - 22 -v a r i a b l e (the ' d e f i n i t i o n of money1 problem) and the most ' s u i t a b l e ' i n t e r e s t ra te (s ) and measure(s) of wealth or income as independent v a r i a b l e s in t h e i r demand f o r money equat ion. However, once the u t i l i t y maximisation problem has been s p e c i f i e d , the dependent and independent 21 va r iab les emerge n a t u r a l l y . F i n a l l y provided a s u f f i c i e n t l y general funct iona l form f o r 22 the u t i l i t y funct ion is adopted, r e s t r i c t i o n s on preferences such as fo r example, homothet ic i ty (which impl ies uni tary expenditure e l a s t i c i t i e s ) may be tested s t a t i s t i c a l l y . '>> A disadvantage o f the d i r e c t u t i l i t y approach should a lso be noted. One should not expect the model to work well f o r r i s k y assets such as government and corporate bonds. Even i f i t i s assumed that these assets y i e l d some 'monetary s e r v i c e s ' , the d i r e c t u t i l i t y model is not well equipped to deal with a s i t u a t i o n where the consumer faces an uncerta in set of p r i c e s . In t h i s case , some element of the mean-variance or expected u t i l i t y model needs to be inc luded . We have discussed above the advantages and disadvantages of three a l t e r n a t i v e t h e o r e t i c a l approaches to the demand f o r money. The remainder of t h i s chapter attempts to summarise the main threads running through the empir ical l i t e r a t u r e and t h e i r r e l a t i o n s h i p to the t h e o r e t i c a l models with which we have j u s t d e a l t . B. Empir ica l Models The empir ical l i t e r a t u r e on the demand fo r money is vast and the f l o o d of research continues unabated. Several good surveys e x i s t . For e a r l i e r work, the reader is re fe r red to L a i d l e r [1969], while Gold fe ld [1973] and Feige and Pearce [1976] summarise the more recent -.23 -1 i t e r a t u r e . Our p r i n c i p a l concern i s to r e l a t e the methodology used in these studies to the t h e o r e t i c a l 1 i te ra ture discussed in the previous s e c t i o n . However, as Feige and Pearce point out ; "The theore t i ca l l i t e r a t u r e in the demand f o r money exh ib i ts a greater d i v e r s i t y of approach than is to be found in the empir ical l i t e r a t u r e " . (Feige and Pearce [1976; 3]) Indeed, Feige and Pearce, dea l ing with U.S. s t u d i e s , go fur ther and s t a t e : "Every one of the [approximately f o r t y ] empir ical s tudies surveyed in t h i s paper r e l i e s upon the demand theory [ that i s , d i r e c t u t i l i t y ] approach fo r s e l e c t i o n of the re levant va r iab les to be included in the demand funct ion fo r money". (Feige and Pearce [1976; 4]) T h i s . c o n c l u s i o n i s perhaps an overstatement, as some of the models surveyed in t h e i r paper (as well as some others which were not included) are based upon the t ransact ions cost approach, a l b e i t in a h ighly s i m p l i f i e d form. However, t h e i r conclus ion gives an accurate impres-s ion of the approach taken in the vast major i ty of empir ical s t u d i e s . The reasons why the mean-variance model has not been used more often are not e n t i r e l y c l e a r . It should be noted that the c r i t i c i s m by Tsiang of the t h e o r e t i c a l underpinnings of the model appeared only r e l a -t i v e l y r e c e n t l y , in 1972. For empir ical a p p l i c a t i o n s , the d i s t i n g u i s h i n g feature of the mean-variance approach i s the in t roduct ion of uncer ta inty regarding the rate of return on bonds as an argument in the demand f o r money f u n c t i o n . One of the major reasons why the mean-variance approach has not been used more widely in empir ica l work i s p r e c i s e l y the d i f f i c u l -t i e s involved in measuring t h i s r i s k . In a d d i t i o n , as was noted in sec t ion A , for many households (and f i rms) there e x i s t a l t e r n a t i v e assets to money - 24 -(narrowly def ined as non i n t e r e s t bearing chartered bank l i a b i l i t i e s ) , the nominal c a p i t a l value of which i s known with c e r t a i n t y . The mean-variance model cannot e a s i l y expla in the simultaneous holding of such a s s e t s . However, many researchers have considered i n t e r e s t rates on these a l t e r n a -t i v e s to be the re levant p r i c e var iab les in est imat ing a demand f o r (narrowly def ined) money equat ion. Nonetheless, given the great t h e o r e t i c a l i n t e r e s t in . the mean-variance model, i t i s s l i g h t l y puzz l ing to note , with Feige and Pearce, that "None of the empir ica l s tudies have . . . e x p l i c i t l y introduced r i s k f a c t o r s " . (Feige and Pearce [1976; 4]) It i s eas ie r to see why the t ransact ions cost model has been used only r a r e l y as the basis for empir ical work. Although the simple Baumol-Tobin model y i e l d s an exact funct iona l so lu t ion f o r the quant i ty o f money demanded, the cons t ra in ts impl ied by the so lu t ion have not been imposed on the est imat ing equat ion. Furthermore, the presence of the 23 t r a n s a c t i o n s - c o s t . v a r i a b l e has been c o n s i s t e n t l y ignored. There are a number o f explanat ions f o r the apparent neglect of the model. F i r s t , the exact funct iona l so lu t ion re fe rs only to an i n d i v i -dual d e c i s i o n u n i t . The d i r e c t p r o p o r t i o n a l i t y r e l a t i o n s h i p between money and the square root of income would be a p p l i c a b l e in the aggregate only under s t r ingent r e s t r i c t i o n s on the expenditure d i s t r i b u t i o n . Another reason i s that the o r i g i n a l Baumol-Tobin model may be regarded as too s i m p l i s t i c to approximate r e a l i t y . However, re lax ing some of t h e i r assumptions makes the model complex and qui te d i f f i c u l t to s o l v e . Perhaps the most important p o i n t , however, i s that in general data are not a v a i l -able on t ransact ions c o s t s . Thus models such as those by Barro and Santo-mero [1972] and Kami [1974] (both based e x p l i c i t l y on the t ransact ions r 25 -cost approach) must be estimated in simpler form ignoring the t ransact ions f a c t o r . Thus, in p r a c t i c e , the vas t major i ty of empir ica l research has 24 adopted the d i r e c t u t i l i t y , or demand theory , framework. Within th is category , two groups may be d i s t i n g u i s h e d : studies using an ad hoc s p e c i -f i c a t i o n of the demand f u n c t i o n ; and studies which der ive the demand f o r money funct ion from an e x p l i c i t model of u t i l i t y maximisat ion. Each of these groups may be subdivided fu r ther into (a) models which consider only a r e s t r i c t e d subset of f i n a n c i a l assets (often c o n s i s t i n g of one element, money), and (b) those deal ing with a l l f i n a n c i a l (and sometimes rea l ) asset dec is ions s imul taneously . The pragmatic approach of the f i r s t broad group of studies was, u n t i l qu i te r e c e n t l y , the only v e r s i o n . Fol lowing the impetus given by Friedman ([1956], [1959]) in the t y p i c a l equation estimated over the l a s t two decades, the des i red quant i ty of real balances (employing d i f f e r e n t 'broad' and/or 'narrow' d e f i n i t i o n s ) i s hypothesised to be some l i n e a r or log l i n e a r funct ion of in te res t r a t e s , wealth or (permanent) income, and, p o s s i b l y , other v a r i a b l e s as w e l l . This methodology provided the basis f o r an extensive ana lys is of the importance of a l t e r n a t i v e explana-tory v a r i a b l e s , the length and nature of adjustment lags which may be present , and the parametric s t a b i l i t y over time of var ious demand f o r money s p e c i f i c a t i o n s . The problem with a l l of these studies is that although they c la im to be based on the d i r e c t u t i l i t y approach to the demand f o r money, none of the advantages inherent in using t h i s framework have been exp lo i ted to any extent . F i r s t , by est imat ing a s i n g l e demand f o r money equation in i s o l a t i o n , the model in f a c t makes two extremely r e s t r i c t i v e assumptions: . 26 ^ 25 f i r s t , that l i qu i d assets are weakly separable from consumption goods and 26 l e i su re in the household's u t i l i t y func t ion ; and fur ther , that the group of l i qu i d assets , excluding money (however defined) is weakly separable from money. Thesec.assumptions rare ly are stated e x p l i c i t l y , nor can they be tested given a s ingle equation model. Second, the l inear addi t ive or log l inear form is used without any e x p l i c i t j u s t i f i c a t i o n . The l i near addi t ive form implies that absolute changes associated with uni t changes in the independent var iables are constant, while the log l inear form implies constant e l a s t i c i t i e s . Note that even when a l te rnat ive funct ional forms are used in order to capture the ef fect of lagged adjustments, the above behavioural res t r i c t i ons often are imposed with respect to desired quant i t ies . 27 Th i rd , the de f i n i t i on of money and the choice and form of the explanatory var iables to be used remain a rb i t ra ry . For example, should leve ls of in terest rates : or in terest rate d i f f e ren t i a l s be employed? How should in f la t ionary expectations be handled? It fol lows from the reasoning behind the d i rec t u t i l i t y approach that since money essen t ia l l y i s being treated as a durable good, the correct pr ice var iab le to employ i s the rental pr ice of monetary serv ices . When defined properly, the re la t i ve pr ices of monetary services and services from real assets are affected by the expected rate of i n f l a t i o n . This rental pr ice concept w i l l be explained in Chapter-3 in .some ; d e t a i l . 1 • However-, i t su f f ices t o ~ ~ 28 note here that , with one except ion, none of the s ingle equation studies have in fact contained the rental pr ice of monetary services as an independent var iab le . F i n a l l y , the model general ly has been estimated for aggregate money holdings without regard for the sectoral d i s t i nc t i on between house-r 27 -holds.and f i r m s . However, i f money i s viewed as a var iab le -in the house-ho ld 's u t i l i t y f u n c t i o n , and at the same time as a product ive input in a f i r m ' s production f u n c t i o n , then one would suspect that the demand equations from each model would be qui te d i f f e r e n t . Neglect ing th is sec tora l d i s t i n c t i o n may cause obvious est imation problems. A number of models have deal t with the demand f o r money within a more general framework, a l b e i t using the same pragmatic approach. Most of these studies ( for example, De Leeuw [1965], Teigen [1964] and Goldfe ld [1966]) have been concerned with model l ing the supply of money simultane-ously with a demand f u n c t i o n . While t h i s i s , of course , an important i s s u e , i t does not a l t e r our c r i t i c i s m s of the demand s ide of t h e i r models. One study, however, by Gramlich and Kalchenbrenner [1970] has estimated a simultaneous system of l i q u i d asset demand equat ions, and imposed symmetry and adding up r e s t r i c t i o n s s i m i l a r to those of the d i r e c t 29 u t i l i t y model. While t h i s represents a s i g n i f i c a n t departure from the standard approach, s ince the estimated model was not der ived e x p l i c i t l y from a u t i l i t y maximisation framework, most of the problems mentioned above are s t i l l present . Quite recent ly a number of papers have appeared in which the authors s p e c i f i e d an e x p l i c i t u t i l i t y funct ion and attempted to obtain estimates of preferences d i r e c t l y . Feige [1964] was the f i r s t author to consider t h i s p o s s i b i l i t y . However, although he discussed the use of a quadrat ic u t i l i t y funct ion in the t h e o r e t i c a l part o f h is paper, he assumed l i n e a r demand funct ions f o r convenience in his empir ical work, and so did not estimate preferences d i r e c t l y . However, he was the f i r s t to estimate simultaneously a system of l i q u i d asset demand equat ions. He was a lso the f i r s t to impose cross - 28 -equation r e s t r i c t i o n s impl ied by the theory of consumer c h o i c e . The condi t ions he imposed were not the usual symmetry r e s t r i c t i o n s , but those imposed a lso by Gramlich and Kalchenbrenner [1970], namely, the simpler ' H o t e l l i n g c o n d i t i o n s ' , der ived by assuming that income e f f e c t s were e i t h e r zero or symmetric. In a well known a r t i c l e , Chetty [1969] attempted to extend the Feige model and apply i t to U.S. time s e r i e s data. Chetty considered two models. The f i r s t i s descr ibed by the fo l lowing equations f o r the repre-senta t ive household: (2.2) subject t o : MQ= M + T 1+i where U in (2.1) i s a C . E . S . u t i l i t y f u n c t i o n , conta in ing M (next p e r i o d ' s cash holdings = money) and T (next p e r i o d ' s time deposi ts ) as arguments, and 3 i , g 2 , and p are parameters. Mo is t h i s p e r i o d ' s f i n a n c i a l wealth and i i s the i n t e r e s t rate earned on time d e p o s i t s . ^ Solv ing the maximisation problem given by (2.1) and (2.2) y i e l d s : (2.3) = x (2-4) - f f = ^ (2.5) M Q = M+I l+i where X i s a Lagrange m u l t i p l i e r . Using (2,3) and (2 .4 ) , s u b s t i t u t i n g f o r - 29 -! i j and -^p from (2.1) and rear rang ing , we obta in : 3 r l a I ( 2 . 6 ) , o g H . . _I_ l o g fe + _ 1 _ Chetty estimates (2 .6 ) , using quar ter ly data on M, T and i , in order to obtain an estimate of 1/1+p which equals the e l a s t i c i t y of s u b s t i t u t i o n between M and T. There are a number of problems assoc ia ted with th is f i r s t vers ion of the Chetty model. F i r s t , one might expect that the u t i l i t y y i e l d e d by money i s o l a t e d to the average stock of money balances h e l d , not the end of per iod s tock . Chetty does not s ta te whether the data he used r e f e r to average or end per iod asset holdings and i n t e r e s t r a t e s . Second, Chetty does not use the 'budget c o n s t r a i n t ' (2.5) in order to der ive demand equations f o r M and T each as funct ions of i and M 0 . The l a t t e r i s , of course , the conventional approach of appl ied demand theory. Chet ty 's procedure i s v a l i d only i f the u t i l i t y funct ion is homothetic. Only in the l a t t e r case is i t poss ib le to express the r a t i o of the quant i t i es demanded as a func t ion o f p r ices alone as in (2 .6 ) . Thus Chet ty , by using a homothetic u t i l i t y funct ion such as the C . E . S . , has i m p l i c i t l y c o n s t r a i n -31 ed the wealth e l a s t i c i t i e s of both money and time deposi ts to equal u n i t y . F i n a l l y , while the d i r e c t u t i l i t y model can (at best) , r e f e r only to households, i t i s not c l e a r whether Chetty used data f o r households on ly . The second model estimated by Chetty i s of a more general form, as f o l l o w s : , i_ ( 2 7 ) Maximise U = feM"p+ M i P l + + e n X n ~ P n f p v * ; w . r . t . M, X i , . . . ,X - 30 r (2.8) subject to : M0 = M t -A- + ,,, + ^ J -where U i s a general ised C .E .S . u t i l i t y funct ion, and i k , X k , ( B ^ p ^ ) , the in terest ra te , do l la r value, and parameters, respect ive ly , associated with the k non money a s s e t , ( k = l , n ) . M 0 and M are defined as in Chetty 1 s f i r s t model. As in his f i r s t model, Chetty solves the n+1 f i r s t order 'marginal u t i l i t y 1 condit ions for the problem given by (2.7) and (2.8) to obta in: (2.9) log Xj - log £ - - A r log & log M j = l , . . . , n In th i s case, since the general ised C .E .S . u t i l i t y function i s not homothetic, the f i r s t order condit ions cannot be solved for log ^ as a funct ion of in terest rates only. In order to estimate (2 .9) , an estimate of M i s used, as fo l lows: Subst i tute (2.9) into (2.8) to obtain a re la t ion (2.10) M0 = g( i i n , M) Now combine (2.10) with what Chetty terms (without any explanation) an ' e x p l i c i t r e l a t i o n ' between M 0 , in terest rates and income Y, given by (2.11) (2.11) MQ = h( i i n , Y) - 31 r to der ive an i m p l i c i t r e l a t i o n between i i , . , . , i n , M and Y. Solv ing e x p l i c i t l y fo r M and approximating the so lu t ion by a f i r s t order T a y l o r ' s s e r i e s expansion, we get the f o l l o w i n g : n (2.12) log M = a 0 + z log ( l -+ 1.) + a n + 1 log Y j=l where a . , j = 0 , . . . , n+1 are parameters. From (2.12) a regress ion estimate J of M i s der ived which is then used in place of M in (2 .9 ) . Th is second vers ion of the Chetty model removes one of the problems present in the f i r s t v e r s i o n , namely, the homothetici ty of the u t i l i t y f u n c t i o n . The genera l ised C . E . S . u t i l i t y funct ion i s not neces- . s a r i l y homothetic, and furthermore, the e l a s t i c i t y of s u b s t i t u t i o n between a l l pa i rs of assets w i l l not be i d e n t i c a l . However, t h i s u t i l i t y funct ion s t i l l imposes a strong r e s t r i c t i o n , namely, strong s e p a r a b i l i t y . That i s , Chetty i m p l i c i t l y assumes that the marginal rate of s u b s t i t u t i o n between any two assets i and j i s independent df the leve l of any other a s s e t , k. The genera l ised model a lso r a i s e s two new i s s u e s . F i r s t , as in the f i r s t model, Chetty does not estimate demand equat ions , but rather n 'marginal u t i l i t y ' equations descr ibed by (2 .9) . The labor ious procedure Chetty uses in order to circumvent the obvious s imul tanei ty problem created by the presence of M (an endogeneous choice var iab le ) on the r igh t hand s ide of (2.9) i s not very s a t i s f a c t o r y . It would seem much more pre ferab le to so lve the e n t i r e set of (n+2) f i r s t order condi t ions (2.7) and (2.8) simultaneously in order to obtain a set of (n+1) e s t i -mating equations f o r M, Xi,..,, X n, each contain ing as arguments the exogeneous v a r i a b l e s , p r ices ( i n t e r e s t rates) and wealth. Using d u a l i t y concepts to be explained in the next chapter , t h i s can be achieved very - 32 -e a s i l y f o r any genera l ised preference o rder ing . A second problem with t h i s model i s that the system of equations (2.9) i s not estimated s imul taneously . That i s , from an econometric point of view, no account i s taken of poss ib le contemporaneous c o r r e l a t i o n ? among the res idua ls in the n equat ions. Furthermore, and t h i s i s an economic i s s u e , Chetty does not impose the cross equation r e s t r i c t i o n s on 32 g and p c a l l e d fo r by the system (2 .9) . However, th is defeats one of the main purposes of using an e x p l i c i t u t i l i t y f u n c t i o n , namely, to obtain est imates , v i a the demand equat ions , of one under ly ing preference o rder ing . We have descr ibed the Chetty paper in some de ta i l fo r two reasons. F i r s t , apart from the e a r l i e r work of Fe ige , i t was the f i r s t attempt to use t h i s approach, and, despi te the problems discussed above, has been widely regarded as a major innovat ive piece of research . Second, a number of wr i ters have s ince wr i t ten papers adopting the Chetty model wi thout , however, removing any of the d e f i c i e n c i e s of the model. model which, they c l a i m , leads to the same est imat ing equations as Chet ty , but which is based on an a l t e r n a t i v e t h e o r e t i c a l assumption. They argue 33 that the fo l lowing two problems are equiva lent : Moroney and Wi lbra t te [1976] estimate a l i q u i d asset demand Problem I: Maximise U = f (X i w . r . t . X i , . . . ,X 5 • • (2.13) n subject ..to: W n Problem II: Maximise W = E r . X . , rl w . r . t . X i , . . , X n i=l 1 = 1 (2.14) subject t o : T = f ( X i , . . . , X p ) - 33 -Problem I i s a vers ion of the Chetty model, where f ( ) i s the u t i l i t y f u n c t i o n , W i s weal th, X . are d o l l a r amounts of each asset i and r. i s (one plus) t h e ! i n t e r e s t rate on the i t n asset ( i = l n ) ; Xj i s money, and thus r{ = 1. Problem It states that the household maximises 35 wealth (W) subject to a ' t ransac t ions c o n s t r a i n t ' . The f i r s t order condi t ions f o r problems I and II are given by (2.15) - (2 .18) . (2.15) 3 f 9X. 1 n Ar . = 0, i = l n (2.16) W - z r . X . = 0 i=l 1 1 0, i = l , . . . , n (2.18) T - f ( X l s . . . , X n ) = 0 : l l Moroney and Wi lbrat te s ta te that provided the f ( ) funct ions in (2.13) and (2.14) are i d e n t i c a l , s ince (2.15) and (2 i17) are equ iva len t , i t makes no d i f f e r e n c e whether we use the theore t i ca l formulat ion I or II. Th is i s t r u e , in. so fa r as only the 'marginal c o n d i t i o n s ' (2.15) and (2.17) are considered . However, the asset demand equations from each model w i l l be qu i te d i f f e r e n t . That i s , (2.15) and (2 .17) , when s o l v e d , y i e l d s a system o f demand funct ions f o r optimal X ' s (denoted by a s t e r i s k s ) given by (2.19) . (2.19) X* n. = g ( r 1 } . . . , r n , W) 1=1 n - 34 -whi le (2.17) and (2 .18) , combined, y i e l d (2.20) X * . = h C r j , . . . , r n , T) 1=1 n Furthermore.(and t h i s point appears to have escaped Moroney and Wi lbrat te comple te ly ) , there is no ' d u a l i t y ' between the problems I and II, as the authors c la im [1976; 183]. Such a d u a l i t y would e x i s t only i f in problem II , the o b j e c t i v e was to minimise W = .E r-\^y Another way; of expressing the same point i s to note that the second order (curvature) condi t ions fo r the f( ) funct ion in problems I and II (as they now stand) w i l l be e n t i r e l y d i f f e r e n t . Moroney and Wi lbrat te do not estimate (2.19) or (2.20) but rather the system: (2.17) where f ( ) i s a genera l ised C . E . S . funct ion and where each X.j, i=2, . . . , n , i s expressed as a funct ion of r l 5 . . . , r n and Xx. This is i d e n t i c a l to the Chetty procedure where the budget const ra in t was i g -nored a l s o . Note that Moroney and Wi lbrat te r e f e r repeatedly to (2.17) as a system of 'asset demand e q u a t i o n s ' . This i s i n c o r r e c t , at l e a s t i f by demand equations we mean r e l a t i o n s expressing endogeneous quant i t i es demanded as a funct ion of exogeneous pr ices and 'income' (or weal th , in t h i s c a s e ) . Unl ike Chet ty , the authors make no attempt to deal with the s imul tane i ty problem caused by the presence of Xi on the r igh t hand s ide of the est imat ing equations (2.17) . However, they fo l low his procedure in neg lect ing to impose any cross equation r e s t r i c t i o n s on the system (2.17) . Moroney and Wi lbrat te inc lude short term government bonds and long term corporate bonds as assets in t h e i r model. As noted already in our d i s c u s s i o n of the t h e o r e t i c a l merits and demerits of the u t i l i t y - 35 -model, th is may be a dangerous procedure due to considerations of r i sk possibly associated with the demand for these assets which is ignored by the d i rec t u t i l i t y model. The Chetty type of model considers only f i nanc ia l assets as var iables in the u t i l i t y funct ion. A number of attempts have been made to extend the model to include real var iables a lso . Bisignano [1974] assumes that the household maximises the sum of discounted u t i l i t y over an i n f i n i t e time horizon subject to an accumulation constraint and an intertemporal wealth const ra in t . Included in his u t i l i t y function are ' r e a l ' f i nanc ia l assets (namely, the do l l a r value of demand deposits and currency (money), time deposi ts , U.S. government s e c u r i t i e s , and household debt, each divided by the pr ice leve l ) and consumer holdings of durable goods. Bisignano solves the intertemporal wealth maximisation problem to obtain f i r s t order condit ions which must hold at each point in time. He estimates these 'marginal u t i l i t y ' condit ions using both C .E .S . and 36 generalised C.E .S . funct ional forms for the u t i l i t y funct ion. With the generalised C .E .S . function the same simultaneity problem faced by Chetty a r i s e s , and Bisignano deals with i t by using an instrumental var iable estimate for money. The Bisignano model i s an in terest ing extension of the Chetty approach but some d i f f i c u l t i e s should be noted. F i r s t , the usual problem of not estimating demand equations is , present. Second, the separab i l i t y r es t r i c t i ons implied by his use of C .E .S . type funct ions, mean that , for example, the marginal rate of subst i tu t ion between money and time deposits is independent of the consumer's holdings of durable goods. This may be unduly r e s t r i c t i v e . Th i rd , Bisignano assumes that the intertemporal - 36 -u t i l i t y funct ion i s s t rong ly separable with respect to consumption in each per iod . Four th , whi le i t appears that expected c a p i t a l gains have been included in the t h e o r e t i c a l . f o r m u l a t i o n of the problem, there is no i n d i c a t i o n given as to whether an actual or expected c a p i t a l gains se r ies was used in his empir ica l work. F i n a l l y , f o r the same reason as in the Moroney-Wilbratte model, one may question the i n c l u s i o n of government s e c u r i t i e s and long term corporate bonds in the u t i l i t y f u n c t i o n . The recent paper by P a r k i n , Cooper, Henderson and Danes [1975] app l ies a model to A u s t r a l i a n data which is s i m i l a r in s p i r i t to that of 37 Bis ignano. Parkin et a l . assume that households maximise an i n t e r -temporal u t i l i t y funct ion subject to a binding f i n a n c i a l cons t ra in t at each point in t ime. Included as var iab les in the u t i l i t y funct ion are 38 var ious f i n a n c i a l and real assets (fourteen in a l l ) as well as t h e i r rates o f change. The l a t t e r innovat ion is designed to capture the ad just -ment process e x p l i c i t l y wi thin the theore t i ca l model (the rate of change var iab les are assumed to y i e l d negative marginal u t i l i t y ) . Parkin et a l . so lve the model using c a l c u l u s of va r ia t ions techniques and a f t e r employing a number o f s i m p l i f y i n g assumptions, der ive est imating equations f o r non durable consumption goods and stocks of real and f i n a n c i a l a s s e t s . Each demand equation is a l i n e a r funct ion of the lagged values of a l l the endogeneous a s s e t s , current r e l a t i v e p r ices and i n t e r e s t r a t e s , real income net o f tax (assumed exogeneous), and a vector of assets a lso assumed to be exogeneous. This represents the most ambitious attempt, among the models surveyed so f a r , to deal with a l l aspects of the household sector s imul -taneously . However, we should note some d i f f i c u l t i e s which are present (a number of these points are recognised e x p l i c i t l y by the authors ) . - 37 -F i r s t , u t i l i t y is assumed to be in ter tempora l ly s t rongly separable (as with the Bisignano model). Second, Parkin et a l , assume a quadrat ic approximation (containing second order terms only) fo r the u t i l i t y f u n c t i o n . While th is form imposes no r e s t r i c t i o n s with respect to e i t h e r s e p a r a b i l i t y or the s i zes of the e l a s t i c i t i e s of s u b s t i t u t i o n , unfortunate ly i t does 39 imply homothet ic i ty , which in turn means that a l l the wealth e l a s t i c i -t i e s of the optimal l e v e l s and rates of change of assets i m p l i c i t l y are constra ined to equal un i ty . T h i r d , although the est imat ing equations are a l l der ived e x p l i -c i t l y from the under ly ing u t i l i t y f u n c t i o n , i t i s not poss ib le to i d e n t i f y separate ly the parameters of the preference f u n c t i o n . Thus, f o r example, wealth 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 s u b s t i t u t i o n cannot be computed. A l s o , the model throughout assumes s t a t i c p r ice expectat ions. F i n a l l y , the model su f fe rs perhaps from being too ambitious in scope. With fourteen endogeneous var iab les present , there are over four hundred c o e f f i c i e n t s to be est imated. It i s very d i f f i c u l t to in te rpre t meaningful ly such a large body of empir ica l r e s u l t s . It would seem preferab le to f i r s t estimate a much more aggregative model, and then proceed s e l e c t i v e l y in stages to examine var ious poss ib le sub models der ived through d isaggregat ion . The f i n a l model to be considered in t h i s sect ion adopts t h i s approach, in e f f e c t going to the opposite extreme to that of Parkin et a l . Diewert [1974a] estimated an intertemporal u t i l i t y funct ion f o r the representa t ive U.S. household conta in ing l e i s u r e , consumption serv ices (of both durable and non durable goods) and aggregate monetary serv ices as arguments ( i n t e r e s t bearing near monies were not cons idered) . Many of the problems re fe r red to in e a r l i e r s tudies were not present in t h i s - 38 -model. U t i l i t y was not assumed to be temporal ly separable , a h ighly genera l ised preference funct ion was employed which imposes no a p r i o r i 40 c o n s t r a i n t s , and a set of wealth and pr ice e l a s t i c i t i e s were der ived . In a d d i t i o n , the rental p r ices of durable consumption and monetary serv ices were c a r e f u l l y formulated and const ructed . However, there is one major problem present in t h i s model. Unl ike a l l the other studies we have surveyed, Diewert assumed that mone-tary serv ices were proport ional to the nominal stock of money he ld . This formulat ion might work qui te well in a time per iod of very low i n f l a t i o n r a t e s , but i t s relevance f o r a good part of the post war per iod must be in some doubt. Furthermore, at the est imat ion s tage , Diewert assumes s t a t i c p r ice expecta t ions . I f a model conta in ing durable goods and money is to have relevance in times of i n f l a t i o n , t h i s assumption needs to be re laxed . In t h i s sec t ion we have surveyed at some length the var ious approaches to e m p i r i c a l l y model l ing the demand f o r money, concentrat ing mainly on the d i f f e r e n t i m p l i c i t or e x p l i c i t vers ions of the d i r e c t u t i l i t y model. Before drawing some conclusions as regards the appropr iate model to be used in t h i s t h e s i s , we w i l l summarise b r i e f l y the s p e c i f i c a l l y Canadian studies on the t o p i c . C. Canadian Studies in the Demand fo r Money Canadian research in the demand f o r money usua l ly has adopted the methodology and techniques of models developed f o r use with U.S. data . Thus, whi le the empir ica l r e s u l t s f o r Canadian models are often qui te 41 d i f f e r e n t , the general approach tends to be very s i m i l a r . E a r l i e r s tudies by Breton [1968], Goodhart [1969] and Shearer - 39 -[1970] explored the poss ib le ex istence of a s tab le v e l o c i t y funct ion f o r Canada, using chart techniques, Actual estimates of demand f o r money 42 funct ions are contained in the work of Salyzn [1966], Laumas and Formuzis, [1968], Short [1972], Clark [1973] and C l in ton [1973], These authors genera l l y fo l low what was re fe r red to in the previous sec t ion of t h i s chapter as the 'pragmatic ' vers ion of the d i r e c t u t i l i t y approach. Within the s i n g l e equation framework, var ious sets of explanatory v a r i a b l e s , together with a l t e r n a t i v e d e f i n i t i o n s of money, are used in an attempt to i s o l a t e a s tab le demand f o r money funct ion which al lows f o r d i f f e r e n t forms of lagged response. To i l l u s t r a t e , the model by C l in ton [1973] c o n s i s t s of the fo l lowing behavioural equat ions: (2.21) M* = e a Y B RC P ( 2 22) ^ — = ( ^ - ) g where M* i s the des i red nominal money s tock , M, the actual money s tock , M_i i s the lagged (one period) money s tock , Y i s aggregate real expendi-43 t u r e , R i s a (net) i n t e r e s t r a t e , and P re fe rs to the aggregate p r i c e l e v e l . S u b s t i t u t i n g (2.21) in to (2 .22) , rear ranging , and taking logarithms (denoted by lower case l e t t e r s ) , C l in ton obta ins : (2.23) m - p = ga +•• (gb)y + (gc)r +• (1 - g)(m_! - p) C l in ton estimates (2.23) using d i f f e r e n t d e f i n i t i o n s of m, and a l t e r n a t i v e p o s s i b i l i t i e s f o r r. A l s o , wealth is t r i e d instead of expendi ture , y . In a d d i t i o n , he conducts s t a b i l i t y tes ts between d i f f e r e n t r 40 T. sub per iods . The model of Clark [1973J i s s i m i l a r in s t ruc ture to that of C l i n t o n , except that she invest iga tes ex tens ive ly the ro le of permanent income and the long run or permanent demand f o r money. The model of the demand f o r l i q u i d assets contained in the large sca le econometric model of the Canadian economy, RDX2, (He l l iwe l l et a l . [1971]) fo l lows the approach suggested by Brainard and Tobin [1968] which had been used in the previous U.S. study by Gramlich and Kalchenbrenner [1970]. The basic behavioural hypothesis underlying the model i s summarised in (2.24): n (2.24) A ( i ) / A = a ( i ) + b ( i ) Y/A + z c ( i , j ) R , i = l , . . . , n j=l J where A ( i ) / A is the proport ion of t o t a l l i q u i d assets (A) held in each category ( A ( i ) ) , R( i ) i s the rate of return on-each asset ( i ) , and Y i s income ( included to r e f l e c t the r o l e of t r a n s a c t i o n s ) . A f t e r a number of add i t iona l m o d i f i c a t i o n s , the system (2.24) is est imated, v i a a constra ined est imat ion procedure, f o r two groups of a s s e t s , The f i r s t group, held predominantly by households, c o n s i s t s of personal savings and chequing accounts in chartered banks, to ta l currency, savings deposi ts in t r u s t and loan companies, and Canada Savings Bonds. The rate o f i n t e r e s t on government bonds was allowed to enter as an add i t iona l explanatory v a r i a b l e . The second group of assets contained demand d e p o s i t s , non personal term and not ice d e p o s i t s , and swapped deposi ts ( a l l l i a b i l i t i e s of chartered banks), and t r u s t and loan company term d e p o s i t s . Two other s tudies deal ing with l i q u i d assets other than the - 41 -l i a b i l i t i e s of chartered banks are those by Laumas [1969] and C l in ton [1974]. The former i s a s t ra ight forward a p p l i c a t i o n to chartered banks, the Quebec Savings Bank, and t r u s t and loan companies of the Timberlake 44 and Fortson [1967] t e s t of 'moneyness' descr ibed e a r l i e r . The C l in ton a r t i c l e examines the demand f o r the l i a b i l i t i e s of t rus t and loan companies d iv ided by c a t e g o r i e s ; savings d e p o s i t s , and term deposi ts broken down by term to matur i ty . His est imat ing equations express the demand f o r each l i a b i l i t y as a l i n e a r funct ion of current and lagged weal th , i n t e r e s t r a t e s , and the lagged value of the endogeneous v a r i a b l e . A l l the above studies have dea l t e x c l u s i v e l y with the demand s ide and no attempt was made to take into account poss ib le simultaneous equation bias by u s i n g , f o r example, instrumental v a r i a b l e est imating techniques. However, the paper by Courchene and Ke l ly [1971] c o n s i s t s o f an extensive twenty s ix equation supply and demand model f o r the Canadian f i n a n c i a l s e c t o r . Demand equations f o r four types of assets a r e : s p e c i f i e d (currency, demand d e p o s i t s , personal savings d e p o s i t s , and corporate not ice d e p o s i t s ) , and the model contains a lso a react ion .. funct ion f o r the Bank of Canada, a money supply f u n c t i o n , and var ious term s t ruc ture equat ions. No cons idera t ion is given to the fore ign s e c t o r . The demand equations f o r assets are of the conventional C l in ton type discussed above. The f i n a l Canadian study to be mentioned is a recent a p p l i c a t i o n by Short and V i l l enueva [1976] of the Chetty model to Canadian data . A l l the problems assoc ia ted with the use of th is approach, with which we have already d e a l t , are present in t h e i r work. A l s o , Short and V i l l enueva inc lude assets which c l e a r l y are not held by households, such as non personal term and not ice deposi ts in chartered banks, and - 42 -demand deposits other than personal chequing accounts, This concludes our br ie f survey of Canadian demand for money s tud ies. As can be seen, the typ ica l approach adopted i s very s im i la r to that used elsewhere. However, the numerical r esu l t s . a re , of course, d i f fe ren t . We have not attempted to summarise any of the resul ts concerning issues such as s u b s t i t u t a b i l i t y , the s ize of wealth e l a s t i c i t i e s , and the parametric s t a b i l i t y over time of the estimates. This i s l e f t un t i l Chapter 8 ;where we w i l l make some comparisons between the resul ts of our models and those contained in the Canadian l i t e ra tu re c i ted above. D. Conclusions The purpose of th is chapter has been to review the various approaches to the demand for money with par t i cu la r emphasis on the methodology underlying empir ical models. In the f i r s t sect ion,- we discussed a l te rnat ive theoret ica l frameworks. Each of the three considered: the mean-variance, transactions cost , and d i rec t u t i l i t y models, captures d i f fe rent aspects of the ro le of money. The mean-variance model deals with the d i v i s ion between aggregate r i s k l ess assets , 'money 1, and a l ternat ive ' r i s k y ' assets. The 45 t ransact ions cost model, on the other hand, ignores uncerta inty, and focuses instead on the ro le of money in reducing transactions costs (both pecuniary and non pecuniary in nature, depending upon the par t i cu la r model). F i n a l l y , the d i rec t u t i l i t y approach can be viewed as a very general statement of the transactions cost model. Each of these approaches possesses theoret ica l advantages and disadvantages. On the other hand, when one examines the empirical l i t e r a t u r e , i t appears that for a number of p rac t ica l reasons, the great majority of - 43 -empir ica l s tudies have been based e i ther e x p l i c i t l y or i m p l i c i t l y , on the d i r e c t u t i l i t y (or.-demand theory) model. However* most of these studies have not been b u i l t on any e x p l i c i t opt imis ing model. Thus, at the r i s k of some o v e r s i m p l i f i c a t i o n , i t seems best to regard these models as expressing a s t a t i s t i c a l r e l a t i o n s h i p between money and var ious explanatory v a r i a b l e s , rather than as behavioural hypotheses derived from a choice theoret ic: framework. A l im i ted number of s tudies have attempted recent ly to e x p l o i t the potent ia l advantages to be gained from using the d i r e c t u t i l i t y approach e x p l i c i t l y . However, these studies have f a i l e d , in varying degrees, to u t i l i s e f u l l y many recent advances in demand theory. Fur ther -more, fo r Canada, there e x i s t s only one such attempt, by Short and V i l l enueva [1976], which is l im i ted in scope and which contains a number of ser ious problems. Our p r i n c i p a l o b j e c t i v e , in the theore t i ca l and empir ica l models to be developed in subsequent chapters of t h i s t h e s i s , i s to formulate and est imate , f o r Canada, a model of the demand fo r money (and money subst i tu tes ) within- a general opt imis ing model of household behaviour, based upon the f u l l e x p l o i t a t i o n of the d i r e c t u t i l i t y approach. In so do ing , we s h a l l u t i l i s e a number of extremely useful concepts and t e c h -niques in demand theory such as: rental p r ices f o r the serv ices of durable goods ( inc lud ing money and near money); the d u a l i t y between d i r e c t and i n d i r e c t u t i l i t y and expenditure f u n c t i o n s ; r e s u l t s in the theory of aggregation across households; and constra ined est imat ion and hypothesis t es t ing techniques. The v a l i d i t y of the theory i t s e l f may a lso be t e s t e d . An important aspect considered a lso in t h i s thes is is the e x p l i c i t model l ing of p r i c e expectat ions held by the consumer, using - 44 -recent ly developed time s e r i e s f o r e c a s t i n g models. These concepts and techniques are by now qui te common in empir ical research deal ing with consumer preferences f o r ' r e a l ' goods such as consumption and l e i s u r e , but have not yet been appl ied to the problem of the demand f o r l i q u i d a s s e t s . Our aim i s to e x p l i c i t l y l i n k these two (h i ther to separated) strands in the l i t e r a t u r e more c l o s e l y . As w e l l , qu i te apart from the i n c l u s i o n of the demand f o r money, a p p l i c a t i o n of the model is of i n t e r e s t in i t s own r igh t as an exerc ise in appl ied demand theory. It can provide some useful i ns igh ts into the i n t e r r e l a t i o n s h i p s among real v a r i a b l e s . In p a r t i c u l a r , the r o l e of p r ice e x p e c t a t i o n s - ™ a f f e c t i n g the demand f o r durable goods r e l a t i v e to non durable goods and serv ices can be explored. This chapter has explained the motivat ion for the t h e o r e t i c a l and empir ica l model to be developed i n ' t h e remainder of the t h e s i s . Our purpose, in the next chapter , i s to explore in more d e t a i l the d i r e c t u t i l i t y approach to the demand f o r money wi th in a general model of house-hold u t i l i t y maximisation. - 45 -Footnotes - Chapter 2 1. An exce l len t survey of the t o p i c i s contained in Nagatani [1977], 2. We ignore , fo r the moment, issues surrounding the appropr iate d e f i -n i t i o n of 'money 1 . The question i s d iscussed l a t e r in t h i s chapter . 3. See Arrow [1965] f o r a statement of the condi t ions under which t h i s basic assumption of expected u t i l i t y maximisation w i l l be v a l i d . 4. Th is conc lus ion requires the add i t iona l assumption that only non-negative proport ions of each asset can be h e l d , i . e . , the ind iv idua l cannot issue e i t h e r cash or bonds. Note a l s o , that the s t r a i g h t l i n e r e s u l t w i l l not n e c e s s a r i l y hold i f there e x i s t more than two a s s e t s . 5. Though not n e c e s s a r i l y . See Tobin [1958; 77-78] fo r a d iscuss ion of the var ious p o s s i b l e types of s o l u t i o n s . 6. For evidence that stock pr ices are more appropr ia te ly descr ibed by a log normal d i s t r i b u t i o n , see Cootner [1964]. 7. The negative exponential funct ion d isp lays constant absolute r i s k a v e r s i o n , while the constant e l a s t i c i t y of marginal u t i l i t y funct ions exh ib i t constant r e l a t i v e r i s k avers ion . Hence, Tsiang argues, these two funct ions form the 'bounds' f o r acceptable u t i l i t y f u n c t i o n s . 8. However, f o r some d iscuss ion of T s i a n g ' s method, see the comments by Bierwag [1974], Borch [1974] and Levy [1974] together with a rep ly by Tsiang [1974]. 9. With one important except ion , the model by Samuel son [1969], which examines the condi t ion under which the consumption-saving d e c i s i o n can be divorced from the p o r t f o l i o a l l o c a t i o n problem. 10. The inventory - theore t ic vers ions of the model usua l ly do not d i s t i n g u i s h between household and f i rm behaviour. An exception is the model by M i l l e r and Orr [1966], dea l ing s p e c i f i c a l l y with f i rms . 11. The i r model is s i m i l a r in concept to that developed by Edgeworth [1888] f o r the case of a bank fac ing uncerta in cash needs. 12. The terminology i s that of Saving [1971; 410]. 13. Namely, that the cross p a r t i a l d e r i v a t i v e s of the funct ion with respect to a l l i t s arguments are ze ro . 14. However, a recent paper by Barnett [1975] claims that the model in f a c t could be estimated using a simultaneous equation approach. - 46 -15. One poss ib le approach i s to use proxies f o r the unobservable t ransact ions cost v a r i a b l e s . For example, Kami [1974] assumes the 'brokerage fee ' in the Baumol-Tobin model i s proport ional to the real wage ra te . 16. Our treatment here is l im i ted to the household u t i l i t y model. For d iscuss ion of the 'money in the production f u n c t i o n ' model, see Moroney [1972] and F ischer [1974]. 17. A d i f f e r e n t r a t i o n a l e has been proposed recent ly in a paper by Morishima [1973]. Morishima assumes that the 'expected future l i v i n g standard ' , U) , enters as an argument in the u t i l i t y funct ion along with present per iod consumption. In t u r n , s i s assumed to be re la ted to the present p e r i o d ' s holdings of money, bonds, and present p e r i o d ' s p r i c e s . Thus i n d i r e c t l y , money and bonds appear in the u t i l i t y f u n c t i o n . 18. For fu r ther d i s c u s s i o n of t h i s i n t e r p r e t a t i o n in the context of both household and f i rm models, see F ischer [1974]. 19. It i s a lso poss ib le to tes t r i g o r o u s l y the s e p a r a b i l i t y assumption by using a very general u t i l i t y f u n c t i o n . However, as w i l l be explained in Chapter 3 , there are problems assoc ia ted with these t e s t s . In any case , we can examine the v a l i d i t y of the s e p a r a b i l i t y assumption i n f o r m a l l y , by comparing f o r example, the estimated r e l a t i o n s h i p s among real goods in two d i f f e r e n t models, one i n c l u d i n g , and one exc lud ing , money. 20. By 'expendi ture ' i s meant the t o t a l expenditure on a l l goods and s e r v i c e s , inc lud ing monetary s e r v i c e s , which enter as arguments in the household u t i l i t y f u n c t i o n . 21. Of course , i t i s s t i l l a r b i t r a r y as to which var iab les enter the u t i l i t y f u n c t i o n , and a lso whether they appear in aggregate or disaggregated form. However, proper ly speaking, these are issues r e l a t i n g to the theory of funct iona l s e p a r a b i l i t y and aggregat ion. This l a t t e r t o p i c i s d iscussed in sect ion D of Chapter 3. 22. Or f o r the i n d i r e c t u t i l i t y or expenditure funct ion (see sect ion A of Chapter 4 below). 23. An exception is the paper by Edwards [1972]. 24. The term 'demand theory framework 1 encompasses both the d i r e c t u t i l i t y model i forvhouseholds, and the production funct ion model f o r f i r m s . Our concern i s mainly with the former. For empir ical a p p l i c a t i o n s of the production funct ion model see Nadir i [1969], S ina i and Stokes [1972], and f o r fu r ther d i s c u s s i o n of the l a t t e r a r t i c l e , Pra is [1 975], Kahn and KourJi[1975], and SiriaTand Stokes [1975]. 25. That i s , the marginal ra te o f s u b s t i t u t i o n between any two l i q u i d assets i s independent of the leve l of any real v a r i a b l e . - 47 -26. Analogous s e p a r a b i l i t y r e s t r i c t i o n s hold f o r the production funct ion i n t e r p r e t a t i o n of the model. 27. One way of reso lv ing th is problem i s to use a procedure.suggested by Friedman and Meiselman [1963]. They proposed that a money subst i tu te should be added to the narrow d e f i n i t i o n of money i f income i s co r re la ted more h ighly with the sum of narrow money and the money subs t i tu te than with each component separa te ly , The i r method has been modif ied subsequently by Timberlake and Fortson [1967] and Kaufman [1969].using changes, and changes in logar i thms, r e s p e c t i v e l y , as the v a r i a b l e s . It i s r e a d i l y admitted by the proponents of t h i s approach that i t is a purely s t a t i s t i c a l e x e r c i s e , and i s not designed to provide any t h e o r e t i c a l i n s i g h t s . 28. K le in [1974], using an e x p l i c i t model of u t i l i t y maximizat ion, invest iga ted the appropr iate rental p r i c e concept. However, he does not der ive the demand equations from any s p e c i f i c u t i l i t y f u n c t i o n , but instead uses a log l i n e a r approximation f o r the est imat ing equat ions. Furthermore, K le in assumed s t a t i c p r i c e expecta t ions . 29. The imposi t ion of r e s t r i c t i o n s of t h i s nature was suggested f i r s t by Brainard-and Tobin [1968]. However, i t should be noted that the d i r e c t u t i l i t y model does not imply that the cross p a r t i a l p r ice e l a s t i c i t i e s are symmetric. Rather, symmetry re fe rs to the compen- sated (that i s , holding u t i l i t y constant) cross p r ice p a r t i a l d e r i -vat ives of the demand f u n c t i o n s . G e n e r a l l y , these r e s t r i c t i o n s cannot be imposed d i r e c t l y on demand equations without f i r s t s p e c i -fy ing the u t i l i t y f u n c t i o n . 30. It i s not necessary to d iv ide M and T by the p r ice leve l in the Chetty model s ince t h i s w i l l have the e f f e c t merely of sea l ing both by the same number. 31. It i s not c l e a r how the cons t ra in t (2.2) should be in te rp re ted . Chetty e x p l i c i t l y t rea ts i t as a budget c o n s t r a i n t . However, a budget cons t ra in t i s normally def ined in terms of the expenditures on the se rv ice flows of goods, rather than as an accounting i d e n t i t y as in (2 .2) . See sec t ion 3, Chapter 3 below f o r development of the rental expenditure concept. 32. That i s , the estimates of e and p w i l l be d i f f e r e n t depending on the p a r t i c u l a r equation est imated. 33. We have t rans la ted the verbal argument advanced by Moroney and Wi lbrat te into more e x p l i c i t mathematical form. 34. Note that apart from the change in n o t a t i o n , (2.13) i s not i d e n t i c a l to (2 .8 ) , s ince r-j, rather than 1/(1 + r.).. appears in - the ^'budget ~ c o n s t r a i n t ' . -Moroney and Wi lbrat te do noV d iscuss any issues c f t iming t h e i r paper, 35. T i s def ined as "the an t ic ipa ted volume of t ransact ions that can be accomplished during a given per iod" (Moroney and Wi lbrat te [1976; 185]). - 48 -36. The dynamic c o n s t r a i n t , analogous to the budget cons t ra in t in the s t a t i c Chetty model, i s ignored in es t imat ion . 37. A s i m i l a r a n a l y s i s i s c a r r i e d out fo r f i rms . 38. S i m i l a r l y to B is ignano, these are measured in ' r e a l ' terms, that i s , t h e i r d o l l a r value is d iv ided by the p r i c e l e v e l , 39. In f a c t , the quadrat ic funct ion with second order terms only is p o s i t i v e l y homogeneous of degree two-':: 40. This is not qui te t r u e . I n i t i a l l y , a homothetic funct ion was postu la ted . To avoid t h i s r e s t r i c t i o n , Diewert employed the device of 'committed expenditure' on each good. However, rather than est imat ing the committed expendi tureparameters , he chose t h e i r values to be .8 of the minimum per cap i ta amounts of the goods consumed during the time period (1930-1965). 41. The.reader i s re fe r red to F isher and Sparks [1976] fo r a d e t a i l e d survey of Canadian demand f o r money research . 42. See a lso comments on S a l y z n ' s paper by Smithy [1967],, Laudadio [1967] and a rep ly by Salyzn [1968]! 43. That i s , R equals some a l t e r n a t i v e i n t e r e s t rate ( for example, the government bond rate) minus the i n t e r e s t rate on M. 44. See footnote 27 of t h i s chapter . 45. Except f o r the Gray and Parkin [1973] model re fer red to e a r l i e r . Chapter 3 THE DEMAND FOR MONEY WITHIN A GENERALISED UTILITY FRAMEWORK We begin t h i s chapter by descr ib ing ( in sect ion A) the pure F i s h e r -ine l i f e c y c l e consumption-savings model o f household behaviour , and the treatment of durable goods wi th in t h i s framework (sect ion B) . In sec t ion C, the serv ices o f money and money subst i tu tes are introduced into the u t i l i t y f u n c t i o n , and the associa ted rental p r ice concepts impl ied by the use of various s i m p l i f i e d vers ions o f the general F isher ine model. F i n a l l y , in sect ion E , a b r i e f summary o f the models to be estimated in the thes is i s presented. A. The F isher L i f e Cycle Model This well known model, along the l i n e s o f F isher [1930] and extended fur ther by Modig l ian i and Brumberg [1954] and Ando and Modigl iani [1963] has the fo l lowing s t r u c t u r e . Suppose that the representa t ive consumer at age on o f his economic l i f e and expecting t o l i v e T y e a r s J has a set of preferences def ined over the serv ices o f durable and non-durable goods and l e i s u r e during each per iod in his l i f e t i m e . It i s assumed tha t : ( i ) The flow o f serv ices y i e l d e d by each good bears a funct ional r e l a t i o n s h i p to the stock of each good he ld . Normally, p r o p o r t i o n a l i t y i s assumed, but th is may be relaxed to the assumption of a convex 'product ion p o s s i b i l i t i e s ' set 1 i hMngss tocks and the flow o f serv ices (see Pol lak and Wachter [1975;268]). ( i i ) The representat ive consumer has formed point expectat ions about the path o f a l l future spot pr ices o f durable and non-durable consumption goods, futureenominal i n t e r e s t rates and future wage r a t e s . Consumers are assumed to act as i f these point expectat ions were held with c e r t a i n t y . Present good's p r i c e s , wage Kates and i n t e r e s t r a t e s are known w i t h c e r t a i n t y . - 50 -( i i i ) He may borrow or lend at the same i n t e r e s t rate by buying or s e l l i n g uni ts of homogeneous 'bonds' in each per iod , ( iv ) He i s q u a l i f i e d to o f f e r labour serv ices in each per iod at the p r e v a i l i n g wage r a t e , (v) The ind iv idua l has some i n i t i a l holdings of durable consumption goods and bonds (which may be negat ive , that i s , outstanding debt ob l iga t ions ) at the beginning of the f i r s t p e r i o d . Under these assumptions, the consumer's problem is to choose goods and labour suppl ies which maximise h is index of intertemporal preferences subject to t h i s intertemporal wealth const ra in t and his time c o n s t r a i n t s . That i s , we assume that the ind iv idua l solves the fo l lowing problem: U(Xj J . . . j X ^ . , . . . , X - | - i % > . . . 5 ^ ^ . J . . . 5 ^ ' j ) subject to the fo l lowing c o n s t r a i n t s : T T (3.2) ( i ) Wealth c o n s t r a i n t ; z (p t« x t + w t £ t ) <_ B + p f - x i + z w t H t E W: t 1 "t 1 (3.3) (ii.") Time c o n s t r a i n t ; <_ H t = l , . . . , t , . . . ,T (3.4) ( i i i ) Non-negat iv i ty c o n s t r a i n t ; x^ ^ 0 ; & t •>. 0 t ^ l , , , , , t , , . . ,T (* denotes the inner product .o f" twOvvectors) where: (3.1) Maximise w . r . t . X[ ) • • • )X^, . . . ; X-p , - 51 -U( ) = a quasi-concave u t i l i t y func t ion possessing the usual 2 r e g u l a r i t y p r o p e r t i e s , = an nxl vector of the serv ices of consumption goods (durable and non durable) used in per iod t , P t = a lxn vector of discounted pr ices fo r the serv ices of the n consumption goods in per iod t , H t = the to ta l number of hours a v a i l a b l e in per iod t , & t = l e i s u r e (= H minus hours worked) in period t , 3 w^ = the discounted wage rate in per iod t , B~ = the consumer's i n i t i a l holdings of bonds, xf = a vector of the consumer's i n i t i a l holdings of consumption goods ( in general t h i s w i l l be of dimension less than n, i f not a l l goods are durab le ) , P i * =. a : v e c t o r . ' a f . spot (purchase) pr ices o f the durable consumer goods (xi ) held by the consumer in period 1. W1; = the discounted value of fu l 1 l i f e t i m e wealth. The term f u l l l i f e t i m e wealth i s used s ince the value of the consumer's to ta l t ime, whether i t i s spent at work or not , i s inc luded . The wealth cons t ra in t (3.2) s tates that the present value of the i n d i v i d u a l ' s consumption inc lud ing consumption of l e i s u r e , must equal the present value of his to ta l human and non human wealth. Assuming that the time const ra in t (3.3) i s s a t i s f i e d as an i n e q u a l i t y , we may solve the problem def ined by (3 .1) - (3 .4) to obtain demand equations f o r present consumption df the serv ices of x i , present l e i s u r e , fu ture consumption of the serv ices of x^ and future l e i s u r e Jt ( x = 2 , . , . , T ) e a c h ' a s a funct ion of a l l present and discounted expected future p r i c e s , wage r a t e s , and wealth. Note that because of assumption ( i ) - 52 -above, the demand fo r the flow of serv ices from may be transformed to and represented as a demand for x^, i . e . , fo r the stock of x^ he ld . For each subsequent p e r i o d , s i m i l a r demand equations may be der ived as funct ions of present ( for that period) and expected future pr ices and wage r a t e s , and wealth in that p e r i o d . Note that in the absence of complete futures markets of the Arrow-Debreu type , the consumer can only plan his future consumption on the basis of expected; future p r i c e s . He cannot t r a n s l a t e his future planned demands into r e a l i s a b l e observable demands. However, the sum of future demands i s equal to the supply of savings in the per iod . That i s , assuming the f u l l l i f e t i m e wealth const ra in t i s s a t i s f i e d as an e q u a l i t y , we may wr i te : T (3.5) E ( p t • x t + w t £ t ) = Wi - (p i '« X! + Vfili) The r i g h t hand s ide of (3 .5 ) , the amount of f u l l l i f e t i m e wealth invested in per iod one i s , in p r i n c i p l e , observable , and hence could be used, along with the demand equations fo r X! and ix, to estimate the parameters of the u t i l i t y f u n c t i o n . However, t h i s requires that we form some numerical estimate of the future pr ices and wage rates expected by the consumer. Such an approach was taken in two recent s t u d i e s , by Diewert [197~4a] and-Darrough [1975] fo r the United States and Japan, r e s p e c t i v e l y . 4 B. The Treatment of Durable Goods In formulat ing the above model of household behaviour, we have treated t h e demand f o r the serv ices of durable goods as an ordinary demand. - 53 -However, the assoc ia ted p r ice for the serv ices of the stock of the durable good held must take into account the fac t that in general durable goods are not j u s t rented f o r the p e r i o d , but t y p i c a l l y are purchased and then held from period to per iod . The purchase of the durable by the consumer then c o n s i s t s of a rental part and an investment in an asset part . Consider the n consumer good x with which i s associa ted a one 3 n 5 per iod deprec ia t ion rate • 5 n , 0 <_ fi ± 1. If 6 = 1, then x n i s an ordinary consumption good.. We assume that a depreciated durable good i s considered equivalent to the new good except that i t i s smal ler in quant i ty . Denote the current ( in per iod one) p r i c e fo r the serv ices of x n t ^ x n t o k e de l i ve red during per iod t) by p Let p *^ t be the expected purchase (or resa le ) p r i c e fo r x n in per iod t . Let R* t be the expected nominal rate of return on bonds during per iod t . Then the discounted rental p r i c e , p ^ , i s given by the discounted^expected cost of the purchase of x p during per iod t minus the discounted expected resa le (poss ib ly s e l l i n g to himself ) value of the depreciated good during per iod t+1, P * ^ t + 1 » i - e . [3 61 o - ^ ( 1 - 6 r X , t + l l J - D ; p n t ( 1 + R * 1 ) . . . ( l + R * t _ 1 ) " (1+R*x) . . . (T+R* t ) For t=l (the rental p r i c e of the serv ices of a durable good to be d e l i v e r e d in the current p e r i o d ) 7 * ( U 6 n ) P * n , t + l  ( 3 ' 7 ) Pnt B P * n t ' 1 + R t " whe re both p * n t and R t , the current per iod purchase pr ice and i n t e r e s t - 54 -r a t e , r e s p e c t i v e l y , are known with c e r t a i n t y , Define e ^ , the expected rate of increase of P * n t ? by; ( 3 . 8 ) ; e ="P ' " P " n t Then, (3,9) p * e = (1 + e . )p* . ' y n,t+l n t / K nt Subst i tu t ing fo r p*^ ^ in (3,7) and rear rang ing , we obta in: p* . (Rv, + 6 + S e - e .) n m\ ^ - nt t n n nt n t ' ( 3 - 1 0 ) p n t TT-R-For the spec ia l case of s t a t i c p r ice expecta t ions , where & = 0, (3.10) reduces t o : P* n t(R* + V) I3-11* P n t - ^ A ; / In the case of two durable goods, housing and l a n d , the rental p r i c e formula (3.7) must be modif ied s l i g h t l y in order to take account of property taxes. Assuming an e f f e c t i v e property tax rate on the current p e r i o d ' s purchase p r i c e , T^ , the rental p r i c e becomes: M - O P * ! - t + i '"'> -Pnt ' P*nt{1 * V - ' l " R ' which a f t e r rear rang ing , may be wr i t ten"as : - 55 -m n n - nt t t t t n n nt n t 7 u . i u ) p n t - 1 + R t — Note that t h i s formulat ion assumes that property taxes are paid at the s t a r t o f the p e r i o d . A l t e r n a t i v e l y , we could have assumed, fo r example, that they are paid at the end of the period (but are s t i l l measured as a proport ion of P* t)> in which case they .should be d iscounted. Using t h i s assumption we have: u - / ' P n t p nt V + R t 1 •+ R t which in turn reduces to : (R . + T . + 6 + 6 e , - e .) (3 in") D = D * 1—t t _n n_nt n t ! [ 6 ' l u ; p n t p nt 1 + R' The d i f f e r e n c e between (3.10' ) and (3.10") i s the presence of the cross term, R V T T ^ . in brackets in the former. However, th is term i s l i k e l y to be extremely small r e l a t i v e to the other terms, and thus i t makes l i t t l e d i f f e r e n c e which assumption i s used. In const ruct ing the rental p r ices for housing and l a n d , the formulat ion (3.10' ) was adopted. It should be noted that t h i s approach d i f f e r s from the 'stock adjustment 1 and 'hab i t format ion ' models o f the demand f o r durables proposed by Stone and Rowe [1960] and Houthakker and Tay lor [1970], r e s p e c t i v e l y . The l a t t e r models (they are o p e r a t i o n a l l y very s i m i l a r ) assume a simple l i n e a r demand funct ion f o r the des i red stock of the durab le , and concentrate on model l ing the consumer's ' investment demand' as a response to reducing the gap between the des i red and actual s tock, - 56 -The rental p r i c e approach, on the other hand, does not take in to account adjustment costs (although the model c o u l d , in p r i n c i p l e , do so) but attempts to model the under ly ing preference s t ructure in a more d e t a i l e d way. The issue of which approach is most appropr iate i s l a r g e l y an empir ical one depending on the importance of adjustment costs in the market fo r the p a r t i c u l a r durable in quest ion . However, a major advantage of the rental p r ice approach i s that i t allows us to t rea t consumer durable and non durable goods ( inc lud ing l e i s u r e ) i d e n t i c a l l y , provided that the appropriate rental p r ice se r ies given by (3.10) or (3.11) i s used. In order to achieve our ob jec t ive of model l ing several aspects of the consumer's general choice problem s imul taneously , we therefore adopt t h i s approach. I f s i g n i f i c a n t ad just -ment: costs a r e , in f a c t , present , then one would expect some evidence of t h i s to appear i f we attempt;to estimate the r e s u l t i n g demand f u n c t i o n s . In the empir ical models reported in Chapter 7, we explore t h i s issue to some extent . In an empir ica l model, the use of the rental p r ice approach involves the const ruc t ion of the renta l p r ice se r ies fo r each good (s ince o f f i c i a l s t a t i s t i c a l sources t y p i c a l l y publ ish purchase p r i c e se r ies only) as well as the const ruct ion of stock est imates. This work forms an important part of the t h e s i s , and i s dea l t with in Chapter 5,.below. F i n a l l y , we have not yet deal t with the issue of whether the formula (3 .11) , which assumes s t a t i c expecta t ions , or (3 .10) , which allows fo r the p o s s i b i l i t y of cap i ta l g a i n s , should be used. The rental p r i c e formula (3,11) with s t a t i c expectat ions imposed, was employed by Gussman [1972] and Cummings and Meduna [1973] fo r Canada, and by Diewert [1974a] -for the U . S . , in const ruc t ing renta l p r i c e s e r i e s . However, in times of - 57 -c o n t i n u a l l y r i s i n g p r i c e s , the consequent omission of c a p i t a l gains p o s s i -b i l i t i e s could be^misleading. , If changes in the nominal i n t e r e s t ra te are merely a r e f l e c t i o n of i n f l a t i o n a r y expecta t ions , then from (3 .7 ) , a model with s t a t i c expectat ions imposed would tend to overstate the rental p r i c e of durablejgoods. One a l t e r n a t i v e to imposing s t a t i c expectat ions is to assume that c a p i t a l gains are p e r f e c t l y a n t i c i p a t e d , i . e . , P * ^ + 1 equals the actual purchase p r ice in per iod t+1. However, t h i s approach causes problems e m p i r i c a l l y when i t comes to a l lowing fo r p o s s i b l e fo recas t e r r o r s . The method we have chosen to model expectat ions fo rmat ion . i s rather more general and i s based upon the Box and Jenkins [1970] ARIMA fo recas t ing model. In essence, t h i s approach assumes that the time ser ies in quest ion ( in our case a purchase p r i c e se r ies ) or some d i f f e r e n c e of some transform of i t , represents the outcome of an under ly ing s ta t ionary s t o c h a s t i c process. Given knowledge of t h i s s t o c h a s t i c process on the part of the consumer, i t may be shown that the optimal f o r e c a s t ( in the sense of minimising the mean square er ror involved) cond i t iona l upon the sample information to date , i s simply the cond i t iona l expectat ion of the process which is given by the point p r e d i c t i o n of the ARIMA model. Such well known suggestions fo r expectat ions modell ing as s t a t i c expectat ions and adapt ive expectat ions may be shown (see Rose [1976; Chapter 9]) to be spec ia l cases of a general ARIMA model, where the underlying s t o c h a s t i c process i s of a p a r t i c u l a r kind in each case. For each of the durable goods we cons idered , an ARIMA model of the pur-c h a s e r p r i c e s e r i e s was i d e n t i f i e d , est imated, and used to obtain estimates of e ^ . . These estimates were then used fo r rental p r i c e c a l c u l a t i o n s based on (3 .11) . The model est imates , as well as a d e t a i l e d - 58 -d i s c u s s i o n of the ARIMA model technique are a l l to be found in Appendix A, As w e l l , we constructed an a l t e r n a t i v e rental p r i c e based on s t a t i c expecta t ions . In Chapter 7, we report empir ical r e s u l t s based on the two a l t e r n a t i v e s p e c i f i c a t i o n s , C. Money as a Durable Good In t h i s s e c t i o n , we der ive the renta l p r ice formula for the s e r -v ices of non i n t e r e s t bearing and i n t e r e s t bearing money. The problem is examined under two a l t e r n a t i v e assumptions about household behaviour: ( i ) households are assumed to hold s t a t i c expectat ions as regards the general p r ice l e v e l ; ( i i ) households expect some p o s i t i v e (or negative) rate of i n f l a t i o n to occur . An i n t e r e s t i n g r e s u l t i s that the rental p r i c e of monetary:services (as def ined in our model) i s i d e n t i c a l under both assumptions. That i s , the renta l p r i c e is unaffected by the existence of i n f l a t i o n a r y expectat ions. 1) The Case of S t a t i c Expectat ions Consider f i r s t non i n t e r e s t bearing money. Let us assume that o there i s only one good, ' f o o d 1 , consumed by households. This 'one good 1 assumption i s not ininoGuous, and the problems ra ised by i t s re laxa t ion are d iscussed l a t e r in t h i s s e c t i o n . Denote the pr ice of food at time t by p ^ t > p^ t i s expressed in d o l l a r s per uni t of food. By assumption, p^ ^ , the expected p r i c e of food in per iod t+1, equals p ^ t . Suppose the consumer holds M nominal d o l l a r s of money form i in per iod t , denoted M^ t < The serv ices of money are assumed to be proport ional to the real value of M, i . e . , to M ^ / p ^ . The quest ion we ask i s the f o l l o w i n g . What i s the rental p r i c e to the consumer of holding one un i t of monetary se rv ice during the period? - 59 -Recal l the expression fo r the renta l p r i c e , p n t (.expressed in d o l l a r s ) , of the serv ices of an ordinary durable good, n, given in the previous sect. ion: n - n* 0 " 6 n ) p * n , t + 1 , , 7 ) Pnt ~ P nt " ( 1 H - R t ) ( 3 ' 7 ) Consider t h i s formula in the case of money. The flow of monetary serv ices is assumed to be proport ional to the real value of M ^ , W-^/p^ = m.^-M . ^ / p ^ is expressed in the same uni ts as food . For exposi t iona l convenience, we may assume a value of uni ty f o r the f a c t o r of p ropor t ion - ' a l i t y l i n k i n g monetary serv ices to M . ^ / p ^ . The d o l l a r purchase p r i c e , p * m i - t » of a uni t of monetary serv ices i s then equal to p .^ , the p r i c e of food . That i s , in order to be able to buy one uni t of food (or e q u i v a l e n t l y , to hold one food uni t of monetary s e r v i c e s ) , the consumer must pay p^ t d o l l a r s . Now l e t us examine the second term in (3.7). In the next p e r i o d , s ince the p r i c e of food is expected to remain the same, the consumer expects to s t i l l hold M . ^ / p ^ food un i ts of monetary serv ice (that i s , he expects to be able to buy M ^ / p . ^ un i ts of food ) . The value of each un i t i s expected to be unchanged, equa l l ing p.^. Thus, un l ike ordinary durable goods where we t y p i c a l l y assume that the serv ice flow is reduced in the next per iod due to physica l d e p r e c i a t i o n , in the case of nominal money ba lances , there is no d e p r e c i a t i o n . Since the p r i c e of food p^ t i s expected to remain unchanged, the consumer foresees no a l t e r a t i o n in the se rv ice f low from the M.. d o l l a r s which he holds and furthermore the value of each un i t i s expected to remain constant , equa l l ing p^. For non i n t e r e s t bearing money, the rental p r i c e of monetary s e r v i c e s , p . T l l i t» 1 b - 60 -then p f t ( 3 J 2 ) p m i t = Pf t ' '(1 + R t T ( 3 J 3 ) = ( ? + R t ) Suppose that the money in ques t ion , M. , earns nominal i n t e r e s t at a rate r - t per pe r iod . In t h i s case , the consumer at the end of the period does not hold M j t / p ^ t (= m.^ .) un i ts of monetary s e r v i c e s , but rather M^ (^1 + r .^. ) /p^ t u n i t s , s ince M j t has increased to M ^ O + M.. therefore appreciates in terms of i t s a b i l i t y to y i e l d monetary J i> s e r v i c e s . However, each uni t of monetary serv ices i s s t i l l valued at p ^ . For i n t e r e s t bearing money M- the rental p r ice expression then becomes ( 3 - 1 4 > p mjt = P f t , ' ( 1 \Tititft (3.15) P f t ( R t " r j t l 1: + R t 2) Non S t a t i c Expectat ions Consider now the s i t u a t i o n where the pr ice of food p^ . i s not expected to remain constant . Let us a lso assume i n i t i a l l y that the money M. earns no i n t e r e s t . In the current p e r i o d , the purchase p r i c e of a food uni t of monetary serv ices remains p ^ . However, in the next p e r i o d , while the consumer w i l l s t i l l hold M.^ d o l l a r s , he does not expect to hold the same number of un i ts of monetary s e r v i c e s , or (which is the same thing) to be - 61 -able to buy the same number of un i ts of food. That i s , .in the next p e r i o d , he expects to hold M ^ / P f un i ts which w i l l be less than M . j t /P f t (his current holdings) i f the expected i n f l a t i o n rate i s p o s i t i v e . To der ive the exact expression f o r the expected deprec ia t ion rate on M.^., we may solve (3,16) f o r 6 t M i t M i t (3.16) - — = (1 - 6 n L P f t P f , t+1 T t to obtain (3.17) 6 p f t p f , t + l or (3.18) 6 = — -p f , t + l Thus 6^ i s the expected change in the pr ice of f o o d , expressed as a proport ion of the expected pr ice of food. The one per iod deprec ia t ion rate on money, 6 ^ , i s given by (3.17) or (3 .18) . Consider now the expected value of each uni t of monetary serv ices in the next pe r iod . This is no longer p^ t a s ; i n (3.12) fo r the case of s t a t i c expecta t ions . Although the consumer holds l ess un i ts of monetary serv ices due to the deprec ia t ion f a c t o r , < 5 ^ , each uni t of monetary se rv ice is now worth more to the consumer, due to the expected r i s e in p.^. Thus the rental p r i c e becomes < 3- i 9> p . i t • • Vffi i t + 1 - 62 -Subst i tu t ing f o r (J - <$t) from. £ 3 , 1 7 ) , we obtain e n - n P f t P f , t H ( 3 - 2 0 ) p m i t " p f t P f , t + l ( 1 + ' R t < (3.21) = p f t - ( 1  f+\j (3.21) i s i d e n t i c a l to (3 .12) , the rental p r ice expression f o r non i n t e r e s t bearing monetary serv ices when s t a t i c p r i c e expectat ions are imposed. By a s i m i l a r argument to that ou t l ined above, the rental p r ice fo r i n t e r e s t bearing monetary s e r v i c e s , M-, becomes .e ( 3 - 2 2 > p mjt = p f t " 0 (IVR^*1 where 8^ i s defined by (3.23) M j i - ( 1 + r.it»H.it p f t Pf t+i (3.23) s * - 1 ^ — P ? , t + 1 (3.22) reduces to ( 1 - r i t ) p f t ( 3 - 2 4 > p mjt B Pf t " ' ' 1 + R t which i s a l s o i d e n t i c a l to the s t a t i c expectat ions formula (3,14) . This r e s u l t , namely, that the rental p r ices of monetary serv ices are unaffected by i n f l a t i o n a r y expecta t ions , may seem at f i r s t glance rather c o u n t e r i n t u i t i v e . However, a l l that we are r e a l l y saying is that - 63 -the d o l l a r value of the goods the consumer can buy with a f i xed d o l l a r amount w i l l always remain constant . If p r ices r i s e , consumers can buy less goods, but each good w i l l be worth correspondingly more. Note that t h i s i s a ceteris -paribus r e s u l t . I f r i s e s due to i n f l a t i o n a r y expec-t a t i o n s , then the rental p r ice of monetary serv ices w i l l rise..,(from (3.13)) . However, while i t i s t rue that expected i n f l a t i o n does not a l t e r the own rental p r i c e of monetary s e r v i c e s , assuming R^  i s constant , we must remember that the rental p r i c e of real durable goods w i l l f a l l r e l a t i v e to that of money. Suppose our one good was not food , but a durable such as housing. Then from (3 .10) , we know that a p o s i t i v e expected rate of i n f l a t i o n w i l l , c e t e r i s par ibus, lower the rental p r i c e of housing, and thus w i l l tend to lead to s u b s t i t u t i o n on the part of the consumers away from money into the durable . This r e l a t i v e p r i c e change s t i l l occurs i f we assume that R^  moves exact ly in accordance with i n f l a t i o n a r y expecta t ions . In t h i s case , the renta l p r ice of housing remains the same, while that of monetary serv ices r i s e s . Presumably, t h i s is the reason why demand f o r money researchers have considered the expected rate of i n f l a t i o n as a v a r i a b l e in t h e i r est imat ing equat ions. However, the prec ise r o l e of the expectat ions va r iab le wi thin the frame-work of the d i r e c t u t i l i t y approach has not been explained e x p l i c i t l y . 3) The Index Number Problem, In the above d i s c u s s i o n , we assumed, f o r expos i t iona l convenience, that only one good, ' f o o d ' was consumed by the household. The existence of a vector of goods, other than money, as arguments in the u t i l i t y f u n c t i o n , leads to considerable d i f f i c u l t i e s . The p r ice index used to de f l a te nominal money balances must now contain these goods as 'we igh ts ' . However, the weights are themselves choice v a r i a b l e s . - 64 -The u t i l i t y maximising problem i s the fo l lowing in the case of two goods, x i and x 2 , and one money, Mi Mi (3.25) Maximise U = U ( x i , X 2 , = T _ _ _ _ ^ ) x i , x 2 , m i subject to Mi (3.26) P l x i + p 2 x 2 + p ^ ^ p i f p 2 t X l f X 2 ) = y where U i s the u t i l i t y f u n c t i o n , p i , p 2 , are the ( renta l ) p r ices of x i and x 2 , r e s p e c t i v e l y , p~(pi , p 2 , x i , x 2 ) i s some aggregate pr ice index, M i / p ( p i , P 2 > * i , x 2 ) (= mi) i s the real value of money balances M i , p m l i s the nominal rental p r i c e fo r the serv ices of M i , and y i s to ta l expenditure. S i n c e , from (3 .13) , assuming Mi does not earn i n t e r e s t , and rep lac ing p.^ by p( ), we have (dropping the time subscr ip t f o r convenience) ( 3 - 2 7 > p . , = p ' ' r n ? S u b s t i t u t i n g (3.27) into (3 .26) , the budget cons t ra in t may be rewri t ten equ iva len t ly as (3.28) P iX i + p 2 x 2 + Mi y4" The f i r s t order condi t ions f o r the problem def ined by (3.25) and (3.26) or (3.28) are then 65 -(3.29) 8U 3U 9 x i " aMi /sp l ) 1 -p- ( ) Mi(3p( ) / 9 x . 1 j _ X p . = 0 1 = 1.2 (3.30) 3U 8Mi/p( ) " p ml *Pml = o Ml (3 .3 i ) P l X l + P 2 x 2 + P m l p ( p i 5 P 2 5 X l j X 2 ) - y = 0 The d i f f i c u l t y a r i s e s in the f i r s t order u t i l i t y maximising condi t ions (3.29) fo r x x and x 2 . A s o l u t i o n may ex is t fo r the system of equations (3 .29) - (3 .31) , but i t would appear to be a major task to d iscover i t s p r o p e r t i e s . ^ C e r t a i n l y , we could no longer appeal to the standard r e s u l t s of d u a l i t y theory (to be 'explained in the next chapter) in order to der ive e x p l i c i t demand funct ions from a genera l ised u t i l i t y f u n c t i o n . The problem is caused by the presence of present per iod 'weights' in the p r i c e index p ( p i , p 2,X! , x 2 ) . A poss ib le s o l u t i o n i s to choose a r b i t r a r i l y the p r i c e of one good, x : or x 2 , as a d e f l a t o r . This is the procedure adopted by Parkin et a l . [1975] and Bisignano [1974]. However, ne i ther of these papers d iscuss the problem e x p l i c i t l y . The model by Diewert [1974] avoids the problem e n t i r e l y by assuming that monetary se rv ices are proport ional to the nominal value of money balances. The s o l u t i o n adopted in t h i s study is to use a ' fo recast ' . " pr ice s e r i e s fo r p. That i s , p t (the actual general p r i c e l e v e l ) i s replaced by p t = E ( p t ) t _ ^ , the optimal fo recas t fo r p"t made in the previous per iod . Thus we are assuming that consumers, when computing the real value o f t h e i r nominal money balances, use as a d e f l a t o r the - 66 -p r i c e leve l they expected to occur in the present p e r i o d , rather than that which a c t u a l l y occurs . While t h i s i s a r b i t r a r y , i t i s no more a r b i t r a r y than the procedure o f using one good as a numeraire. Indeed, i t i s very d i f f i c u l t to assign any cons is tent i n t e r p r e t a t i o n to the l a t t e r procedure.^ The renta l p r i c e formulae adopted in t h i s study fo r non i n t e r e s t bearing and i n t e r e s t bearing monetary serv ices are stated in (3.32) and (3.33) r e s p e c t i v e l y . P t ( R t ) ( 3 ' 3 2 ' Pmit = l M ; P t ( R t - r . ) ( 3 ' 3 3 ) Pmjt = 1 I R t J The method of const ruc t ing the aggregate p r i c e index p~t i s explained in Chapter 6 along with the estimated s e r i e s fo r p .^. D. Funct ional S e p a r a b i l i t y and Aggregation The F isher model of sect ion A , modif ied to include the serv ices of money and money s u b s t i t u t e s , may be summarised as f o l l o w s : (3.34) Maximise U(x1 , . . . , x t , . . . , x T . m x , . . . , m t , . . . ,m-p J ^ ' . . . ,1^,... w . r . t . ' Xl,...,X^.,...,Xy, " • • ' ^"p' m i , . . . , m ^ . , . . . ,m-|-subject t o : T (3.35) ( i ) Wealth c o n s t r a i n t : I ( P t - * t + + P m t - ™ t ) < B + pfx i + Mi - 67 -( 3 .36 ) ( i i ) Time c o n s t r a i n t : s,^ <_ H, t = l , . . . , T ( 3 . 3 7 ) ( i i i ) Non nega t iv i t y c o n s t r a i n t : >_ 0; >_ 0; m t > 0 t = l , . . . ,T where: m. = a kxl vector of real money and money subs t i tu te balances M t ( E where M, is a vector of d o l l a r holdings of d i f f e r e n t money p t ^ t forms and p t i s the general ( forecast ) p r ice l e v e l ) , P m t = a kxl vector of rental p r ices f o r m^, each element of which i s def ined by (3.32) and(3.33) , Mi = the d o l l a r value of the consumer's i n i t i a l holdings of real „ Mi money and money subs t i tu te balances (= p^  -g—), T U( ), x t , p^, a^, w .^, B, x i , p * i , H are def ined as in the model of sec t ion A. The model (3.34)-(3.37) in i t s present form i s too general to be appl ied e m p i r i c a l l y . Previous appl ied demand theory studies (both inc lud ing and excluding money) usua l ly have estimated more h igh ly aggregated sub models. Our purpose in t h i s sect ion is to state e x p l i c i t l y the assumptions necessary fo r the var ious appl ied models to be v a l i d , e s p e c i a l l y those dea l ing with money. F i r s t , a b r i e f summary of some background material on funct iona l 12 s e p a r a b i l i t y i s requ i red . Consider a d i r e c t u t i l i t y funct ion def ined over an n dimensional vector of goods, x: (3.38) U(x) = f ( x i , . . . , x n ) - 68 -P a r t i t i o n the set of ind ices [ l , . . . , n ] into m mutually exc lus ive and exhaustive sectors [ x 1 , . . . , x m ] . The r t n s e c t o r , r = l , . . . , m , i s separable from the k va r iab le i f and only i f a 9U/9X. fo r a l l i , j e x and f o r some k £ x . That i s , the marginal rate of s u b s t i t u t i o n between any two goods in the r sector does not depend upon the value of x^. Consumer pre ferences , represented by f ( x ) , are sa id to be d i r e c t l y weakly separable i f every sector x r i s separable from the va r iab les in a l l the other s e c t o r s . Preferences are d i r e c t l y weakly separable i f and only i f U(x) can be wr i t ten (3.40) U(x) = k^U1),..., Ax" 1 ) ] 13 where F[ ] i s s t r i c t l y increas ing in each of i t s m arguments. I f a funct ion i s weakly separab le , and, in a d d i t i o n , each f m funct ion (ca l l ed an aggregator funct ion) i s homothetic, then the funct ion is sa id to be homothet ical ly weakly separable . The funct ion U(x) i s sa id to be d i r e c t l y s t rong ly separable i f the union of any number of sectors i s separable from i t s complement in U. Preferences are d i r e c t l y s t rong ly separable i f and only i f U(x) can be wr i t ten in the form (3.41) U(x) = F f f U x 1 ) + . . . + f m ( x m ) ] - 69 -o r , by a s u i t a b l e norma l isa t ion , (3.42) U(x) = f ^ x 1 ) + . . . + f m ( x m ) For s i m p l i c i t y of e x p o s i t i o n , l e t us suppose that there are only three goods, x l 5 x 2 , and x 3 . The u t i l i t y maximisation problem is then: (3.43) Maximise U = f ( x l 5 x 2 , x 3 ) w . r . t . x l 5 x 2 , x 3 subject t o : 3 (3.44) E p .x . = y i=l 1 1 where y is def ined as aggregate expenditure on the three goods. Two questions now a r i s e : 14 (1) Under what condi t ions i s i t poss ib le to solve the problem (3 .43 ) - (3 .44 ) , fo r example, fo r the aggregate of x r and x 2 , and x 3 , i . e . to rewri te the problem as : (3.45) Maximise U = f [ g ( x x , x 2 ) , x 3 ] w . r . t . g ( x ! , X 2 ) . x 3 subject t o : (3.46) p g ( x l s x 2 ) + P 3 x 3 = y where g ( x l 5 x 2 ) i s some aggregate quant i ty index, and p the corresponding 70 -aggregate p r i c e index, r e f e r r i n g to combined x i and x 2 . Th is is re fe r red to as 'Strong Pr ice Aggregation ' (Blackorby, Primont and Russel l [1975; 31] . (2) Under what condi t ions i s is p o s s i b l e 1 4 to solve a submodel r e f e r r i n g t o , fo r example, X i and x 2 o n l y , that i s , to rewri te (part of ) problem (3.43)-(3.44) as : (3.47) Maximise U = f ( x i , x 2 ) w . r . t . Xi,X 2 subject t o : (3.48) p i x i + p 2 x 2 = y where y is aggregate expenditure on the two goods. Blackorby, Primont and Russel l [1975; 32] r e f e r to t h i s as 'Strong D e c e n t r a l i z a b i l i t y ' . Taken together , problems (1) and (2) c o n s t i t u t e the condi t ions under which a 'two stage maximisation' procedure is poss ib le (see Strotz [1957; 271]). However, the set of necessary and s u f f i c i e n t cond i t ions f o r each problem taken separate ly are not n e c e s s a r i l y i d e n t i c a l . The d i s t i n c t i o n is important. We s ta te the condi t ions for each problem below. S e p a r a b i l i t y re fe rs to the p a r t i t i o n [ (x x , x 2 ) , x 3 ] . (1) S u f f i c i e n t condi t ions fo r 'Strong Pr ice Aggrega t ion ' : U(x) i s e i t h e r (a) homothet ical ly weakly separable o_r (b) s t rong ly separable , where the aggregator 15 funct ions are of the c l a s s known as the Gorman polar form. (2) Necessary and s u f f i c i e n t condi t ions fo r 'Strong D e c e n t r a l i z a b i l i t y ' : - 71 -U(x) i s weakly separable . Note that the homothetic s e p a r a b i l i t y . c o n d i t i o n fo r problem 1 g (1) ;i:S- s u f f i c i e n t , not necessary. A l t e r n a t i v e s u f f i c i e n t condi t ions f o r aggregation over goods (that i s , the formation of an aggregator g ( x 1 , x 2 ) ) are the Leont ie f condi t ions (goods are assumed to be consumed in f i x e d proport ions) or Hicks Pr ice Aggregation ( r e l a t i v e pr ices remain constant ) . Note a lso that the homothet ic i ty requirement a r i s e s in e f f e c t because of the d i f f i c u l t i e s otherwise of de f in ing an aggregate p r i c e index. However, i t i s poss ib le to def ine an approximate p r ice index even when the underly ing aggregator funct ion i s not homothetic. This pr ice index (the Tornquist index) i s d iscussed in sec t ion A of Chapter 5, and in fac t i s used in const ruc t ing our aggregate indexes. We return now to d iscuss ing the general intertemporal model def ined by (3 .34) - (3 .37) . The great major i ty of appl ied demand studies have invest iga ted the a l l o c a t i o n among present period consumption goods on ly . That i s , they have estimated a model of the form (3.47)-(3.48) where x x and x 2 r e f e r to present per iod consumption. Thus present consumption goods are assumed to be weakly separable from future goods ( i . e . , savings) and a lso to be weakly separable from both present and future monetary s e r v i c e s . Two exceptions are the recent s tudies by Diewert [1974a]' f o r the U . S . , and Darrough [1975] fo r Japan. Diewert assumed no s e p a r a b i l i t y r e s t r i c t i o n s with respect to future consumption and a lso considered one money form. However, money subst i tu tes were not included in his model, thereby implying weak s e p a r a b i l i t y with respect to l i q u i d assets other than money. Darrough a lso estimated the intertemporal model (3.34)-(3.37) with monetary serv ices excluded. In order to construct an - 72 -aggregate p r ice and quant i ty index of future consumption, she adopted the method of Leont ie f aggregat ion. Those demand fo r money studies which are based i m p l i c i t l y on the d i r e c t u t i l i t y approach, genera l ly have imposed qui te s t r ingent separa-b i l i t y r e s t r i c t i o n s . Studies deal ing only with'1 iquidassets*assume'"'t 'Rat;-they are weakly separable from present consumption (that i s , they are of the form (3 .47) - (3 .48) , where x1 and x 2 r e f e r to l i q u i d a s s e t s , and x 3 i s a real var iab le ) as well as from future per iods ' consumption. The models of Feige [1964], Chetty [1969] and the RDX2 model (He l l iwe l l et a l . [1971]) are examples. Furthermore, the genera l ised C . E . S . u t i l i t y func t ion used by Chetty and others imposes the r e s t r i c t i o n of strong s e p a r a b i l i t y among l i q u i d a s s e t s . In s i n g l e equation s t u d i e s , the procedure of 'adding ' moneys to form a broad money aggregate as the dependent v a r i a b l e impl ies r e s t r i c t i o n s of the form out l ined in problem (1) above. F i n a l l y , (with the exception of Diewert [1974]) the demand f o r money models which deal with real assets and a lso future period consumption, namely, those by Parkin et a l . [1975] and Bisignano [1974], by wr i t ing u t i l i t y as the sum of discounted u t i l i t y in each per iod ( i . e . , in a form equivalent to (3.42) above) have imposed strong s e p a r a b i l i t y with respect to consumption in each p e r i o d . Our approach in t h i s t h e s i s i s to adopt a middle ground. While i d e a l l y , we would have l i k e d to estimate the f u l l intertemporal model (3 .34) - (3 .37) , data d i f f i c u l t i e s as regards the measurement of expected pr ices and wage rates fo r an extended period into the f u t u r e , as well as time l i m i t a t i o n s , prevented us from doing so . Instead, we have assumed weak s e p a r a b i l i t y with respect to future consumption and estimated atemporal models on ly . These models are of the fo l lowing form: - 73 -(3.49) Maximise U(x, £ , m) w . r . t . x , £ , m subject t o : (3 .50) ( i ) Budget c o n s t r a i n t : p • x + w£ + p • m <_ y , ( 3 .51 ) ( i i ) Time c o n s t r a i n t : £ <_ H, ( 3 . 5 2 ) ( i i i ) Non nega t iv i t y c o n s t r a i n t : x >_ 0; £ >_ 0; m >_ 0. where: x = an nxl vector of present per iod consumption goods (durable and non durable) only (we have dropped the time subscr ip t fo r convenience) , p = the nxl rental p r ice vector associa ted with x , £ = fl .eisure o'n the present p e r i o d , w = present period wage r a t e , m = a kxl vector of the serv ices of money and money s u b s t i t u t e s , P m = the kxl vector of rental p r ices assoc ia ted with m, y E : t o t a l expenditure on current per iod x , £ , and m, H = to ta l number of hours a v a i l a b l e in the current pe r iod . Within the present consumption model def ined by (3 .49) - (3 .52) , a number of d i f f e r e n t models, der ived by imposing a l t e r n a t i v e s e p a r a b i l i t y assumptions, are i n v e s t i g a t e d . A b r i e f summary of the models to be appl ied e m p i r i c a l l y appears in the fo l lowing s e c t i o n . r 74 r-E. Proposed Models Three groups of models are est imated, based on annual (1947-1974) Canadian da ta , in t h i s t h e s i s . The f i r s t group fo l lows the t r a d i t i o n of Gussman [1972] and Berndt, Darrough and Diewert {1977] and i s der ived by assuming that consumption and l e i s u r e are weakly separable from monetary s e r v i c e s . Within t h i s group, we invest iga te a number of i s s u e s , such as the extent to which the i n c l u s i o n of cap i ta l gains in the rental p r ices of durable goods (see sect ion B above) leads to r e s u l t s very d i f f e r e n t from the model with s t a t i c expectat ions imposed, and whether the demand fo r l e i s u r e can be modelled s u c c e s s f u l l y wi thin t h i s framework. As w e l l , estimates of p r ice and 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 s u b s t i t u t i o n f o r d i f f e r e n t kinds of consumption goods and l e i s u r e are obta ined. The second group, termed ' rea l -monetary 1 models, includes aggregate 'money' (defined as chartered bank l i a b i l i t i e s and currency , both held by households) and aggregate 'near money1 (defined as chequable and non chequable deposi ts and over one year term deposi ts of t rus t and loan companies, and Canada Savings B o n d s ) 1 7 as well as consumption and l e i s u r e . The purpose of these models is to examine issues such as the poss ib le degree o f s u b s t i t u t a b i l i t y between money and near money, in a model where real va r iab les are present , and the r e l a t i o n s h i p between consumption (of both durable and non durable goods) , l e i s u r e , and the serv ices of money and near money. Some evidence on the question of whether money and near money are ' luxury g o o d s ' , in the sense of having expenditure e l a s t i c i t i e s greater than u n i t y , i s presented a l s o . This model assumes that the condi t ions necessary f o r 'Strong P r i c e Aggregat ion' (see Problem (1) in the previous sect ion) are s a t i s f i e d with respect to disaggregated money and - 75 -near money forms, r e s p e c t i v e l y . The f i n a l group, which we c a l l the ' l i q u i d a s s e t ' models, re laxes the l a t t e r r e s t r i c t i o n , but i s based on the assumption that l i q u i d assets are weakly separable from consumption and l e i s u r e . Aggregate money and near money are disaggregated into f i v e sub groups: currency and personal chequing accounts of chartered banks; personal savings deposi ts of chartered banks; t r u s t and loan company chequable and non chequable d e p o s i t s ; over one year term deposi ts of t r u s t and loan companies; and Canada Savings Bonds. Th is permits us a more d e t a i l e d ana lys is of the subst i tut ion-complementar i ty r e l a t i o n s h i p s among these a s s e t s , and as w e l l , of t h e i r ind iv idua l expenditure e l a s t i c i t i e s . While our data base i s from 1947 to 1974V f o r each o f the above three groups, d i f f e r e n t models, corresponding to d i f f e r e n t time p e r i o d s , are est imated, and the models tested f o r parametric s t a b i l i t y between these var ious sub per iods . A l s o , fo r each group, we conduct two other types of t e s t s . The f i r s t are t e s t s of the theory of demand to examine whether the data are cons is tent with the model's assumptions of u t i l i t y maximising behaviour on the part of the household. The second set , of tes ts examines s t a t i s t i c a l l y the v a l i d i t y of the r e s t r i c t i o n of homothet ici ty ( implying uni tary expenditure e l a s t i c i t i e s ) on the preference o r d e r i n g . In the next chapter , the est imat ing equations corresponding to a s p e c i f i c form f o r the u t i l i t y funct ion are d e r i v e d , as well as the corresponding set of e l a s t i c i t i e s . - 76 -Footnotes - Chapter 3 1. The l a s t per iod of his l i f e may be t reated as his en t i re retirement p e r i o d , when he consumes (or passes on as an inher i tance) the remainder of his wealth. 2. These r e g u l a r i t y proper t ies are enumerated as condi t ions A in Chapter 4 (sect ion A ) . 3. For one p e r i o d , the 'wealth c o n s t r a i n t ' ( i ) i s : (3 .2 ' ) Pt • xt + w t y < w tH t Denoting E. as the number of hours worked (E^ = - i ^ ) , t h i s may be rewri t ten as : (3.2") P' tx t < w tE t We use the formulat ion ( 3 . 2 ' ) , where Jl. i s a choice v a r i a b l e , rather than (3.2") in order to avoid the problem caused by the presence in the u t i l i t y funct ion of a good, work ( E f ) , which y i e l d s negative u t i l i t y . 4. The approach taken here fo l lows that of Diewert [1974a], which, as that author points out , resembles c l o s e l y that of Walras [1926]. 5. Note that without loss of genera l i t y the subscr ip t n now re fe rs to the rr" good, rather than to the dimension of the consumption goods vector . 6. We assume that the discount rate households use equals the i n t e r e s t rate on homogeneous 'bonds ' . 7. Rather than use p * n ] l and p * n 2 . , henceforth in t h i s sect ion the subscr ip t t r e f e r s ' t o the current pe r iod . 8. Whether or not the one good in quest ion is a durable is• .. immaterial fo r the moment. 9. Note that 6. in the case of monetary serv ices i s an expected rate which may vary over t ime. Th is d i f f e r s from an ordinary durable where i t was i m p l i c i t l y assumed that 6 (the physica l deprec ia t ion rate) i s a known constant independent of t ime. 10. There i s a lso the quest ion of e s t a b l i s h i n g the second order condi t ions fo r the problem. 11. It would be s e l f con t rad ic to ry to argue, on the one hand, fo r example, that the pr ices of a l l consumption goods move together , and, on the other hand, to then use the model in order to expla in how r e l a t i v e p r i c e changes a f f e c t consumer a l l o c a t i o n s among d i f f e r e n t consumption goods. - 77 -12. For an extensive review of the sub jec t , see Blackorby, Primont and Russel l [1975]. 13. In a recent paper, Blackorby, Primoqt and Russel l [1976] point out that the increas ing r e s t r i c t i o n on F i s qui te important. If such an P cannot be found, they show that U w i l l not be separable in the sense of (3 .39) . The authors r e f e r to s e p a r a b i l i t y in the sense defined by (3.39) and (3.40) as s t r i c t s e p a r a b i l i t y . They d i s t i n g u i s h a weaker n o t i o n , simply.termed s e p a r a b i l i t y , where the assumption of F being an increas ing funct ion is weakened to that of a non decreasing F. The authors show that s e p a r a b i l i t y is not s u f f i c i e n t to e s t a b l i s h a number of important r e s u l t s , such as the r e l a t i o n s h i p between s e p a r a b i l i t y of the d i r e c t and i n d i r e c t (see sect ion A , Chapter 4) u t i l i t y f u n c t i o n s . Under a loca l non s a t i a t i o n assumption, the two d e f i n i t i o n s c o i n c i d e . Henceforth, we use the t e r m ' s e p a r a b i l i t y 1 as r e f e r r i n g to ' s t r i c t s e p a r a b i l i t y 1 . 14. By ' p o s s i b l e ' or ' c o n s i s t e n t 1 , i s meant that exact ly the same s o l u -t ion w i l l be generated as i f we had solved the en t i re problem (3 .43. ) - (3 .44) ins tead . 15. This a n t i c i p a t e s our d i s c u s s i o n of funct ional forms in sect ion B of Chapter 4. It s u f f i c e s to note here that the Gorman polar form is homothetic only beyond a c e r t a i n minimum leve l of 'committed expend i tu res ' . 16. Necessary condi t ions are known only i f U(x) i s weakly separable . In t h i s case , the s u f f i c i e n t condi t ions are a lso necessary. 17. It i s assumed that these assets are held only by households. The j u s t i f i c a t i o n f o r t h i s assumption i s d iscussed in Chapter 5. Chapter 4 FUNCTIONAL FORM AND ESTIMATING EQUATIONS In t h i s chapter we der ive the s p e c i f i c demand equations to be est imated. F i r s t (sect ion A) some well known r e s u l t s in d u a l i t y theory are presented. Sect ion B introduces; a c l a s s of preferences known as the Gorman polar form and derives the associa ted aggregate market demand f u n c t i o n s . In sec t ion C, a s p e c i f i c funct iona l form is chosen f o r the representat ive consumer's i n d i r e c t u t i l i t y f u n c t i o n , w h i l e , i n sect ion D, the corresponding p r i c e and expenditure 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 o f s u b s t i t u t i o n are c a l c u l a t e d . Tests of the u t i l i t y maximising hypothesis are summarised a lso in t h i s s e c t i o n . A. Some Results in Dua l i ty Theory 1 Given an e x p l i c i t funct iona l form f o r the u t i l i t y funct ion i n , f o r example, the model descr ibed by (3.49)-(3.52) of the previous chapter , the usual ' p r i m a l ' approach to obta in ing demand equations f o r est imat ion purposes i s to d i f f e r e n t i a t e the assoc ia ted Lagrangean funct ion and solve the r e s u l t i n g f i r s t order condi t ions fo r optimal quant i t i es demanded as funct ions of the exogenous v a r i a b l e s , namely, p r ices and to ta l expen-d i t u r e . However, i f we wish to use a genera l ised funct iona l form fo r the u t i l i t y funct ion which imposes a p r i o r i as few r e s t r i c t i o n s on preferences as p o s s i b l e , then t h i s primal approach becomes very unwieldy and cumbersome. It i s often very d i f f i c u l t , i f not imposs ib le , to solve e x p l i c i t l y the system of f i r s t order condi t ions corresponding to the genera l ised f u n c t i o n . Th is was the source of the d i f f i c u l t i e s present in the model used by Chetty and others (see the d iscuss ion in sect ion J3 of - 79 -Chapter 2). Instead, i t i s convenient to use the d u a l i t y r e l a t i o n s between ( i ) d i r e c t and i n d i r e c t u t i l i t y funct ions and ( i i ) d i r e c t u t i l i t y funct ions and expenditure or cost funct ions in order to der ive e x p l i c i t demand equat ions. These r e l a t i o n s h i p s are summarised below. 1) Dua l i ty between D i r e c t and Ind i rec t U t i l i t y Functions Let the consumer's u t i l i t y funct ion be u = f (x) where x > 0 p i s 2 and nxl dimensional vector of consumption s e r v i c e s . Suppose the consumer faces s t r i c t l y p o s i t i v e ( renta l ) p r ices given by the nxl dimensional vector p and has 'expendi ture ' y to spend on the n goods. Then def ine the normalised nxl vector v = p / y , and the consumer's i n d i r e c t u t i l i t y  funct ion by: (4.1) g(v) = max {f (x ) ; v • x ± 1; x >_ o } w . r . t . x n Suppose that the d i r e c t u t i l i t y funct ion f (x) s a t i s f i e s the fo l lowing condi t ions A: f (x ) i s : ( i ) cont inuous, ( i i ) non decreas ing , ( i i i ) subject to loca l non s a t i a t i o n , ( iv ) quasi-concave f o r x > 0 p . If f s a t i s f i e s condi t ions A , then i t can be shown that the i n d i r e c t u t i l i t y funct ion g(v) def ined by (4.1) s a t i s f i e s the fo l lowing condi t ions B: g(v) i s ; ( i ) cont inuous, ( i i ) non i n c r e a s i n g , - 80 -( i i i ) subject to loca l non s a t i a t i o n , ( iv ) quasi-convex fo r v >> 0 n , Define the d i r e c t u t i l i t y funct ion corresponding to g(v) by (4.2) f * (x) = min {g(v); v • x < 1; v > 0 } v .3 It can be shown that f * (x ) i s continuous fo r x >_ 0 n , and f u r t h e r , that i f f (x ) s a t i s f i e s condi t ions A , then f * (x ) = f ( x ) . Thus there i s a d u a l i t y between f (x ) and g (v ) . Given one f u n c t i o n , the other i s uniquely deter -mined. The usefulness of t h i s r e l a t i o n s h i p is apparent when we consider the theorem known as Roy's Ident i ty (Roy [1947; 222]): Roy's Ident i ty : Given that g(v) s a t i s f i e s condi t ions B, that f * (x) i s def ined by (4 .2 ) , and g(v) i s once d i f f e r e n t i a b l e at v*>> Q n with vg(v*) < 0 n , then the s o l u t i o n to the problem: (4.3) max f f * (x ) ; v* • x <. 1; x >_ 0 } X is unique and is equal to (4 4) x(v*) = v 9 ( v * ) y ^ '*> M V 1 v* • vg(v*) Thus simple d i f f e r e n t i a t i o n of an i n d i r e c t u t i l i t y funct ion s a t i s f y i n g condi t ions B y i e l d s the same set of demand funct ions which would have been obtained had we generated f and then solved the r e s u l t i n g maximisation problem. This l a t t e r approach can be f a i r l y i n t r a c t a b l e , while the a p p l i c a t i o n of Roy's Iden t i t y , on the other hand, i s qui te - 81 -st ra ight forward . Our approach in so lv ing the u t i l i t y maximisation problem w i l l be to s t a r t from an i n d i r e c t u t i l i t y funct ion and then use Roy's Ident i ty to generate demand funct ions f o r est imat ing purposes, 2) Dua l i ty between D i rec t U t i l i t y and Expenditure Functions The expenditure f u n c t i o n , corresponding to a u t i l i t y funct ion f i s def ined by (4.5) m(u; p) = min {p ••:x; f (x ) >_ u} w . r . t . x where u is some.given leve l of u t i l i t y . If f (x ) s a t i s f i e s condi t ions A, then m(u; p) def ined by (4.5) w i l l s a t i s f y the f o i l owing condi t ions C: m(u; p) i s : ( i ) def ined f o r a l l p 0 n and u such that u" <_ u <_ u , where LT = f (o) and u = l im f ( n l ) where I = D ^ ] N ' S 0 T N A T U C A N eciua^ +0O» YY+OO n' n n and is continuous on t h i s domain, ( i i ) increas ing in u , f o r u~ _< u _< u, .and where m(u"; p) = 0, ( i i i ) f o r every u~ <_ u <^u, m(u; p) i s nondecreasing, l i n e a r l y homogeneous and concave in p » 0 n -We may construct the d i r e c t u t i l i t y funct ion f * (x) corresponding to m(u; p) as the s o l u t i o n to the fo l lowing problem: (4.6) f * (x ) =. max {u; m(u; p) <_ p x; x >_ 0 } w . r . t . u As be fore , i t may be shown that f * (x ) def ined by (4.6) w i l l s a t i s f y condi t ions A and w i l l equal f ( x ) . Thus an expenditure funct ion s a t i s f y i n g condi t ions C can charac te r ise preferences j u s t as well as a d i r e c t or an - 8 2 -i n d i r e c t u t i l i t y funct ion s a t i s f y i n g condi t ions A or B r e s p e c t i v e l y . Our purpose in in t roducing the expenditure funct ion is p a r t l y one of pedagogical convenience. As w i l l be seen below, the c l a s s to which the funct iona l form used in t h i s study belongs can best be i l l u s t r a t e d i n i t , i a l l y - u s i n g the expenditure f u n c t i o n . B. Aggregation across Consumers and the Gorman Polar Form Using Roy's Ident i ty (4.4) a system of demand equations f o r the ind iv idua l consumer may be d e r i v e d , given an e x p l i c i t funct iona l form fo r g (v ) . However, even i f a l l consumers are assumed to possess i d e n t i c a l p re fe rences , then the corresponding market demands, der ived by aggregation over consumers, w i l l in general depend upon the parameters (other than the mean) of the d i s t r i b u t i o n of expenditure across consumers. Unfor tunate ly , data on the l a t t e r are usua l ly unava i l ab le . A well known s u f f i c i e n t cond i t ion to avoid the problem is to impose homothetici ty on the ind iv idua l preference o r d e r i n g s , s i n c e , in t h i s case , the d i s t r i b u t i o n of expenditure w i l l not a f f e c t the aggregate market demands. However, t h i s assumption is extremely r e s t r i c t i v e . A less r e s t r i c t i v e a l t e r n a t i v e i s to use the c lass of preferences known as the Gorman polar form which does.not impose homothet ic i ty , but which a lso e l i m i n a t s the necess i ty f o r consider ing the d i s t r i b u t i o n of expenditure. This c l a s s we proceed t o ' i l l u s t r a t e , f i r s t in terms of the expenditure f u n c t i o n , and then v i a the associa ted i n d i r e c t u t i l i t y f u n c t i o n . Consider the fo l lowing condi t ions C on the expenditure funct ion m(u; p ) : Condi t ions C ' : Let P be a c losed convex subset of the p o s i t i v e orthant in R n -.83 -with a non empty i n t e r i o r and l e t (Ku~ <, u, Then ( i ) m(u; p) > 0 and i s twice cont inuously d i f f e r e n t i a t e fo r a l l p e P and u~ «_ u <_u, ( i i ) 3m(u; pj /3u > 0 f o r a l l p e P and u s u < J, ( i i i ) m(u; p) i s non decreas ing , l i n e a r l y homogeneous and concave in p t P f o r every u such that u~ £ u <_ u", ( iv ) m(Au*; p*) = Am(u*; p*) f o r every A > 0 such that u" <_ Au* <_ u where u < u* < u and the f i x e d vector of p r ices p* e i n t e r i o r of P. Condit ions C on the expenditure funct ion are stronger than the previous condi t ions C , in two respec ts . F i r s t , with condi t ions C , the expenditure funct ion m(u; p) was def ined fo r a l l u ^ 0. However, condi t ions C imply that i t i s def ined only f o r u t i l i t y l e v e l s equal to or greater than some p o s i t i v e leve l uT. Second, ( iv) above i s a s c a l i n g convent ion. It impl ies that the u-m locus indexed by a f i xed p r ice vector p* is a s t r a i g h t l i n e segment which passes through the o r i g i n when extended. The i n d i r e c t u t i l i t y func t ion associa ted with m(u; p) can be der ived as f o l l o w s . We may wr i te (4.7) y = m(u; p) where y i s , d e f i n i t i o n a l l y , expenditure. Since m(u; p) i s l i n e a r homogeneous in p, t h i s can be rewr i t ten as (4.8) 1 = m(u; y) - 84 -where v = p /y , or (4.9) u = g(v) (4.9) i s the i n d i r e c t u t i l i t y funct ion der ived from m(u; p ) . I f m(u; p) s a t i s f i e s condi t ions C , then i t may be shown that g(v) der ived as in (4.9) w i l l s a t i s f y the fo l lowing condi t ions B 1 . Condit ions B 1 : Over the c l o s e d , connected subset V which is a subset of the p o s i t i v e orthant in R n with a non empty i n t e r i o r , g(v) i s : ( i ) twice cont inuously d i f f e r e n t i a t e , ( i i ) non i n c r e a s i n g , ( i i i ) subject to loca l non s a t i a t i o n , ( iv ) quasi-convex fo r v » 0 n , (v) . homogeneous of degree -1 along a ray ; i . e . , g(xv*) = A - 1 g(v * ) > 0 fo r a l l x > 0 such that Xv* e V. As with the expenditure f u n c t i o n , condi t ions B 1 are stronger than condi t ions B, due to the s c a l i n g convent ion, condi t ion (v) . Consider the fo l lowing form for the expenditure f u n c t i o n : (4.10) m(u; p) = b(p) + u c(p) u >_ u", p e P where c s a t i s f i e s the fo l lowing c o n d i t i o n s : c(p) i s twice cont inuously d i f f e r e n t i a t e , s t r i c t l y i n c r e a s i n g , l i n e a r homogeneous, concave and has Hessian matrix of rank n-1 fo r a l l p: and where b(p) is twice d i f f e r e n t i -a t e , and' . l inear homogeneous f o r p e P with b(p*) = 0 f o r some p* e i n t e r i o r of P. It can be shown that i f IT i s chosen large enough, m(u; p) defined by (4.10) w i l l s a t i s f y condi t ions C The i n t e r p r e t a t i o n of b(p) is - 8 5 -i s that i t represents the cost of achieving the minimum leve l of u t i l i t y u", when the consumer faces pr ices p. The i n d i r e c t u t i l i t y funct ion g(v) corresponding to m(u; p) in (4.10) is der ived from: (4:11) y = b(p) + u c(p) D iv id ing through by y and so lv ing f o r u y i e l d s (4.12) u = g(v) = c ( v ) - 1 - d(v) where c(v) s a t i s f i e s the same condi t ions as for c(p) above, while d(v) {= b (p) /c (p) ) i s a twice d i f f e r e n t i a t e homogeneous of degree zero funct ion with d(p*) = 0 f o r some p*e P. Since m(u; p ) , def ined by (4.10) s a t i s f i e s condi t ions C , the corresponding i n d i r e c t u t i l i t y funct ion def ined by (4.12) w i l l s a t i s f y condi t ions B ' . The reasons fo r our i n t e r e s t in the expenditure and i n d i r e c t u t i l i t y f u n c t i o n s , def ined by (4.10) and (4.12) r e s p e c t i v e l y , are twofold . 4 F i r s t , they can provide second order d i f f e r e n t i a l approximations to a r b i t r a r y expenditure and i n d i r e c t u t i l i t y funct ions s a t i s f y i n g condi t ions B 1 and C , r e s p e c t i v e l y . Second, the aggregate market demand equationscorresponding to these funct ions contain only the mean of the expenditure d i s t r i b u t i o n as a v a r i a b l e . Consider the f i r s t po in t , For the case of the i n d i r e c t u t i l i t y func t ion (4 .17) , we have the fo l lowing theorem due to Diewert [1976a;'27}: - 86 -Theorem: Let the i n d i r e c t u t i l i t y funct ion g*(v) s a t i s f y condi t ions B ' ; l e t h(v) be a f l e x i b l e homogeneous of degree minus one funct iona l form, * n ^ and l e t a be a vector of constants such that z a- = 0. Then i=l 1 (4.13) g(v) =. h(v) + z a . ln ( v . / v . * ) i=l 1 1 1 can provide a second order d i f f e r e n t i a l approximation to g* at the ' point v * . The . indirect u t i l i t y funct ion (4.13) corresponds to (4 .12) , where n ^ h(v) = c (v ) . " 1 , and d(v) = - z an- ^ ( v ^ / v ^ * ) = O.when v = v * . A s i m i l a r approximation theorem may be proved f o r the expenditure funct ion (4.10) (see Diewert [1976a; 26]) . We consider now the second i s s u e , the de r iva t ion of aggregate market demand equations based on (4.12) or (4.13) . For expos i t iona l convenience, we s h a l l use the l a t t e r representa t ion , (4 .13) . Applying Roy's Ident i ty (4.4) to the i n d i r e c t u t i l i t y f u n c t i o n , (4 .13) , we obtain (4.14) x(v*) = ^ L V 1 * v ' v • vh(v) since Vg(v) = vh(v) + v _ 1 c( n v-vg(v) = v -vh(v) + 0, as z a- = 0 i=l 1 and where " means d iagona l i se a vector into a diagonal matr ix , Since h(v) i s homogeneous of degree - 1 , v -vVhCv) = - h ( v ) , and (4.14) becomes: - 87 -(4.15) x(v) . ' W & f 9 ' 1 3 " Now mul t ip ly the numerator and denominator of (4,15) by u , and wr i te v as p /y , on the r i g h t hand s i d e , (4.16) x(v) * y , y — - ( | : vh(^) + p-1^.) ( 4 J 7 ) - - ^ - i ^ again using the homogeneity of degree minus one of h (v ) . Fur ther , s ince vh(v) i s homogeneous of degree - 2 , vh(^ . -y ) = p vh(^) or (4.18) 1 Vh(^) = y vh(p) Subs t i tu t ing from (4.18) into (4 .17) , we obtain f i n a l l y (4.19) x W . 4L2ffiL±£**l The reason for transforming the demand equations (4.14) into the form (4 .19) , where they are wr i t ten as a funct ion o f unnormalised pr ices p, and expendi ture, y , i s to f a c i l i t a t e aggregation across consumers. - 8 8 -Suppose that ( i ) each consumer has expenditure y between y~and y , ( i i ) each consumer has the same set of preferences which are represent -able by the demand f u n c t i o n s r i n (4 .19) , ( i i i ) each consumer faces the same commodity rental p r ices p, and ( iv) expenditure i s d i s t r i b u t e d according to the dens i ty funct ion <j>(y); i . e . , the number of consumers y j having expenditure between y 0 and y 2 i s M / - c|>(y)dy, where M is the number of consumers in the group. Then the market demand fo r the i t n commodity can be wr i t ten as : (4.20) x.(p;<j>) = - M /^ { (y^M + p . - ^ i ) / h ( p ) } * (y)dy y H i (4.21) = - M ( y * ^ 1 + P i " 1 ^ ) / h ( p ) i = 1 - . . . , n y where y * = J-_ y<f>(y)dy is the average expenditure of the M consumers. y Rewrit ing (4.21) in per cap i ta terms, t h i s becomes (4.22) = - ( y * 1 ^ * P i " 1 a i ) / h ( p ) i=l n The s i g n i f i c a n c e of (4.22) is that the per cap i ta demand f u n c t i o n s , x^(p; <j))/M, are i d e n t i c a l to the demand funct ions generated by a ' r e p r e s e n t a t i v e ' consumer fac ing pr ices p and average to ta l expen-d i t u r e y * , i r r e s p e c t i v e of the d i s t r i b u t i o n of income. Note that t h i s form has been obtained without having to impose the strong r e s t r i c t i o n of homothet ic i ty (namely a . = 0, a l l i ) on the i n d i r e c t u t i l i t y funct ion (4.13) .^ However, the ind iv idua l expenditure funct ion (4 .10) , from which the i n d i r e c t u t i l i t y funct ion (4.12) was d e r i v e d , assumed that the - 89 -i n d i v i d u a l ' s expendi ture , b(_p), was large enough to achieve the minimum leve l of u t i l i t y , u". When aggregating across consumers, i t i s necessary to assume that the lowest leve l of expendi ture , y , i s at l eas t as great as b (p ) , and thus that each consumer a t ta ins t h i s minimum u t i l i t y l e v e l . C. A S p e c i f i c Form fo r the Ind i rect U t i l i t y Function The; per c a p i t a market demand equations are stated in (4.22) . A l l that remains is to choose a s p e c i f i c f l e x i b l e funct iona l form fo r h(v) . We have chosen the fo l lowing form, given by (4.23): (4.23) h(v) = E E b . .v " % v , " % ; b = b , . , a l l . ; i , j i=l j=l 1 J 1 J 1 J J 1 This funct ion i s homogeneous of degree - 1 , and has been employed recent ly in a study of uncerta in p r i c e expectat ions by Epstein [1977]. Combining (4.23) and (4.13) the i n d i r e c t u t i l i t y funct ion i s then: (4.24) g(v) = E E b . . v " % v " % + E a . = ln(v / v .*) b. . 5 b.= ,: al 1 i , j i=l j=l 1 J J i=l 1 1 1 J J E %'.=0 i = l The per cap i ta market demand equations corresponding to t h i s funct iona l form, are then, from (4 .22) , (4.25) - ^ T — - - f L -1 E bkm P k ^ pm 2 k=l m=l K m K m The system (4.25) may be converted in to a system of expenditure share equat ions , by m u l t i p l y i n g each equation by p ^ y * to ob ta in : - 90 -(4,26) My* " n n S * bkm Pk k=l m=l m K Km 1 S 1 . , >n The equat ion, system (4.26) is homogeneous o f degree zero in the para-meters b... and a \ . Thus, in order to i d e n t i f y econometr ica l ly these parameters i t i s necessary to impose a normal isa t ion . We chose to assume that E 7 E b . . = 1. The r e s t r i c t i o n s on the system (4.26) may be i j ^ summarised as f o l l o w s . (4.27) ( i ) b . . = b . . (symmetry) a l l 1, j (4.28) ( i i ) >z a . = 0 i = l 1 n n (4.29) ( i i i ) E E b . . = 1 i=l j = l 1 J With these r e s t r i c t i o n s imposed, there are (n(n+l ) /2) - l b . . and n-1 a^ independent c o e f f i c i e n t s in the system (4.26) . The r e s t r i c t i o n s ( i i ) and ( i i i ) have a convenient i n t e r p r e t a t i o n . Together they make g(v) homogeneous of degree minus.one.along a . ray of equal normalised p r i c e s . For example, g(v) = 1 when a l l v^ = 1, as in the base year . g(v) = \ when a l l v . = k, some p o s i t i v e constant . We note that the set of preferences def ined by (4,24) w i l l be homothetic i f a^ - 0, a l l i , This impl ies that a l l expenditure e l a s t i c i t i e s are u n i t y . The advantage of using a f l e x i b l e form f o r the i n d i r e c t u t i l i t y funct ion is tha t , a p r i o r i , no r e s t r i c t i o n s are placed on the e l a s t i c i t i e s - 91 -of s u b s t i t u t i o n between goods. In t u r n , t h i s impl ies that no s e p a r a b i l i t y assumptions are imposed on preferences . Thus, our s p e c i f i c a t i o n is: more general than that used, fo r example, by Chetty [1969] and other wr i ters who have adopted h is model. I d e a l l y , we could use our genera l ised i n d i r e c t u t i l i t y funct ion to t e s t s t a t i s t i c a l l y the assumption of s e p a r a b i l i t y between, fo r example, consumption goods and l i q u i d a s s e t s . Cer ta in r e l a t i o n s h i p s among the e l a s t i c i t i e s of s u b s t i t u t i o n between goods are implied by s e p a r a b i l i t y which in turn can be t rans la ted in to equa l i t y r e s t r i c t i o n s on the parameters of the u t i l i t y f u n c t i o n . Berndt and Christenson [1974] used the i d e n t i c a l approach, in the case of production to tes t fo r s e p a r a b i l i t y in a three good model of the U.S. manufacturing s e c t o r . However, there i s a major problem associa ted with t e s t i n g f o r s e p a r a b i l i t y in t h i s manner. Blackorby, Primont and Russel l [1977] have shown that with ' f l e x i b l e ' funct iona l forms, t e s t i n g fo r s e p a r a b i l i t y impl ies t e s t i n g , in a d d i t i o n , a much stronger hypothes is , namely, that the under ly ing aggregator , funct ions (see sec t ion D, Chapter 3) are of Cobb Q Douglas form. Since the l a t t e r funct ion would be an extremely r e s t r i c -t i v e representat ion of consumer preferences def ined over , fo r example, consumption goods, we have not undertaken any s t a t i s t i c a l t e s t s f o r s e p a r a b i l i t y . However, some useful information may be obtained from examining the empir ica l estimates to see i f , f o r example, the s u b s t i t u t i o n -complementarity r e l a t i o n s h i p s among consumption goods a l t e r when a new good, money, i s in t roduced. This is the approach we adopt below in Chapter 7, where the empir ica l r e s u l t s are presented. - 92 -D. E l a s t i c i t i e s and Tests of U t i l i t y Maximising Behaviour 1) P r ice e l a s t i c i t i e s Define the p r i c e e l a s t i c i t y of the i t n demand x . ( p / y ) with respect to a change in p . * as J (4.30) £ * . . = p j * x i*- 18x . * ( p * / y*)/9p*- i , j = l , , . . , n where x - * denotes the f i t t e d demand corresponding to pr ices p* and expen-d i tu re y * . From Roy's Iden t i ty , (4.4) we may der ive n 3x. g . . m=1 V m 9 m J (4.31) —- = L L r - r — r - v - x . x . i , j= l , ' . . . , n v ' 3V, v • vg(v) 1 v . vg(v) l j where (4.32) g.(v) = ag(v)/3v. i , j= l n (4.33) g^-tv) = 3 2 g ( v)/3v i3v j. i , j = l , . . . , n and the a s t e r i s k s have been omitted for notat ional convenience. Def in ing n (4.34) d = v • vg(v) = E v.g. 1=1 1 1 n ( 4 ' 3 5 ) " j ' ^ V m j ^ (4.31) may be rewri t ten as : - 93 -3x. g . - x . d . Since (4.37) 9 P j y 9 V j' 9x. p. 1 9x. p. g . . v . v . d . (4.38) E . - ^ T - 1 - - " 1 = 1 r - L ' - J - = - - 4 J - - x . v . i , j=l n v ' U 9Pj x . y 9Vj x . x .d d j j (4.38) i s used, together with (4.3,4.) and (4 .35) , to c a l c u l a t e the nxn matrix of own and cross p r i c e e l a s t i c i t i e s . In (4.34) and (4.35) , g^ and g . . corresponding to the i n d i r e c t u t i l i t y funct ion def ined by (4.24) •J J» - •" " . are given by n 1 i_ (4.39) g . = - E b . . v "2 v , " 2 + a . / v . i = l , . ' . . , n i j = 1 U I J i i A 1 (%b. . v . " 2 V , " 2 i^j i,3=1 n ^ 'J i J (4-40) g.. =[ " n • 5 1 ( 2 b i j v i " ? > i k ^ = t v i " 2 - v k - l a t V 2 ' - ^ , . i ? j = n . . . , n 2) Expenditure E l a s t i c i t i e s Define the e l a s t i c i t y of the i f i t t e d demand with respect to t o t a l expendi ture, y , as ax-Cp/y) (4.41) E. = —l ' *— i = l , . . . , n v ' i y 9y x . - 94 -A well known r e s u l t i s that (4.42) E. = - S E . . i = l , . . , , n iy j s 0 l j where E . . i s def ined by (4.38) , If the funct ion g(v) i s homothetic, ^ 3 then E.. def ined by (4.42) w i l l be u n i t y , a l l i . This serves as a useful check on the accuracy of our computations. Another useful check is Enge l 's Aggregation which states that the share weighted sum of the expenditure e l a s t i c i t i e s must equal u n i t y , i . e . , (4.43) E S i E = 1 1=1 J 3) E l a s t i c i t i e s of Subs t i tu t ion The A l l e n p a r t i a l e l a s t i c i t y of s u b s t i t u t i o n between two goods, i and j , i s der ived from the fo l lowing r e l a t i o n s h i p (Al len [1938; 512]). ( 4 ' 4 4 ) E i j - S j [ o i j " V (4.45) = s,[a.. + \ E .] where s . (E p -x . / y ) is the j ( f i t t e d ) share of to ta l expenditure y . 3 3 3 Solv ing fo r a--, we obta in : Subs t i tu t ing for E. .from (4.38.), a f te r some manipulation, 0 . . may be expressed - 95 -d i r e c t l y in terms of the d e r i v a t i v e s of the i n d i r e c t u t i l i t y func t ion as 9 i i d - d . d j r V j ( 4 .47 ) a . . = J J - ^ - J - —L + J " ' The n x n matrix of e l a s t i c i t i e s of s u b s t i t u t i o n [a..] is Q symmetric and has rank n-1. These r e s t r i c t i o n s a lso are useful as computational checks. 4) Tests of U t i l i t y Maximising Behaviour The fo l lowing r e s t r i c t i o n s on the form of the i n d i r e c t u t i l i t y funct ion are implied by the theory of u t i l i t y maximising behaviour and may be tested using our estimates of p r e f e r e n c e s . 1 ^ ( i ) P o s i t i v i t y : x^ £ 0 i = l , . . . , n ( i i ) Monotonic i ty: g^(v) < 0 i = l , . . . , n ( i i i ) Curvat ive : g(v) i s a quasi convex f u n c t i o n . We check the p o s i t i v i t y and monotonicity r e s t r i c t i o n s by d i r e c t computation of the values of the f i t t e d demands, and the gradient vector of the estimated i n d i r e c t u t i l i t y f u n c t i o n , r e s p e c t i v e l y . The curvature condi t ions may be tested in a number of equivalent ways. It can be shown that quas i -convex i ty of the i n d i r e c t u t i l i t y funct ion (and hence quas i -concav i ty of the d i r e c t u t i l i t y funct ion) impl ies that the Slutsky matrix of s u b s t i t u t i o n e f f e c t s w i l l be negative s e m i - d e f i n i t e . In t u r n , i t can be shown (see Diewert [1977]) tha t , provided the monotonicity condi t ions h o l d , t h i s impl ies that the matrix '.'3 -•" of e l a s t i c i t i e s of s u b s t i t u t i o n [a..] w i l l be negative semi d e f i n i t e . We check quas i -convex i ty by examining the computed [a..] matrix f o r negative semi d e f i n i t e n e s s , Th is may be done using e i t h e r - 96 -of two equivalent methods. The determinantal condi t ions f o r negative semi -def in i tehess require that the leading p r i n c i p l e minors of [a..] "I3 a l te rna te in sign with the f i r s t order minor negat ive. Thus a necessary cond i t ion is that the.own e l a s t i c i t i e s of s u b s t i t u t i o n a., be < 0 . A l t e r n a t i v e l y , the eigenvalues of the l a ^ l matrix may be computed. Negative semi -de f in i teness requires that the n-1 eigenvalues be $ 0 . One eigenvalue must be zero s ince <?.• i s s i n g u l a r . 3 For smal ler models, the determinantal condi t ions are r e l a t i v e l y easy to eva luate , and thus we used th is method f o r the three and four good models est imated. _ - 97 -Footnotes - Chapter 4 1. .Sect ions A and B of t h i s chapter are based, on Diewert [1974b], [1976a]. 2. This d i s c u s s i o n df course c a r r i e s over to the case of (3,49) where there i s an (n+k)xl vector of arguments in the u t i l i t y f u n c t i o n . 3. S t r i c t l y speaking, f * (x) i s continuous fo r x >> 0 , and i t s exten-s ion to the nonnegative orthant x 2 0 is a lso continuous. See Diewert [1976a; 4 ] , n 4. "A funct ion of n var iab les f d i f f e r e n t i a l l y approximates f* to the second order at a point x* i f f (x* ) = f * (x* ) and the f i r s t and second order p a r t i a l de r iva t i ves of the two funct ions a lso co inc ide at x * ; i . e . , Vf (x* ) = v f * (x * ) and v 2 f ( x * ) = v 2 f * ( x * ) " . (Diewert [1976a;26, footnote 29]) . 5. The reason why the mean expenditure only appears in (4.21) i s because the ind iv idua l demand funct ions (4.19) are l i n e a r in expendi ture, y . 6. See Berndt and Christenson [1973]. 7. Berndt and Chr istenson [1973]. 8. Khal id [1977] has recent ly developed a very general (and complex) funct iona l form based on the Box-Cox transformation method which permits the aggregator funct ions to be of C . E . S . form. 9. From (4 .44) , n n n E E . - = E S - 0 . . - E . E S • j = i I J J = 1 J U i y j = 1 J n -E-j = l S -O . . J U n E ' U = 0, from (4.42) Thus, each column in [ajjj] can be expressed as a weighted sum of elements in the other columns. There fore , from the theory of determinants, the matrix [ a . - ] i s s i n g u l a r . 1 j 10. In t h i s study, we have chosen to impose the symmetry requirement as a maintained hypothes is , and have concentrated our a t tent ions on examining the proper t ies l i s t e d . This is mainly fo r computational reasons, as the unsymmetric vers ion of the model can be extremely d i f f i c u l t to estimate economet r ica l l y , due to the large number of parameters invo lved . Chapter 5 DATA: METHODOLOGY AND SOURCES In t h i s chapter , we descr ibe the const ruct ion of ( renta l ) p r ice and quant i ty indexes f o r f i v e categor ies of consumer expendi ture, based on annual Canadian data f o r the per iod'1947-74. The f i v e categor ies are: ( i ) non durable goods and s e r v i c e s ; ( i i ) durable ( inc lud ing semi durable) goods; ( i i i ) l e i s u r e ; ( iv) money (defined to include currency plus c e r t a i n categor ies of chartered bank l i a b i l i t i e s ) ; and (v) money subst i tu tes (defined to inc lude c e r t a i n l i a b i l i t i e s of t r u s t and loan companies as well as Canada Savings Bonds). For each of these f i v e c a t e g o r i e s , rental p r ice and quant i ty indexes f o r the disaggregated goods wi th in the group are const ructed . We then proceed in stages to form an aggregate p r ice and quant i ty index f o r that group. Referr ing to the d i s c u s s i o n in sect ion E of Chapter 3, of the models to be est imated, some involve using data ser ies at a more aggregated leve l than o thers . The f i r s t group, the consumpt ion- le isure models, permits some disaggregat ion within these c a t e g o r i e s . The second group, the ' rea l -monetary ' models, use h ighly aggregated data , while the t h i r d set of models, dea l ing with l i q u i d assets o n l y , examines the disaggregated components of aggregate money and money s u b s t i t u t e s . The f i r s t sect ion of th is chapter deals b r i e f l y with the r a t i o n a l e f o r the p a r t i c u l a r index number formula adopted in aggregating sub cate -gor ies of goods. In the fo l lowing f i v e s e c t i o n s , the rental p r ice and quant i ty indexes fo r the f i v e categor ies l i s t e d in the f i r s t paragraph above are der ived and presented. The const ruct ion of rental p r i c e and quant i ty indexes f o r money i and money subst i tu tes has not , to our knowledge, been attempted before . - 99 -However, the papers by Gussman [1972] and Cummings and Meduna [1973] deal with the const ruc t ion of Canadian time s e r i e s data f o r consumption goods and le isure (categor ies ( i ) - ( i i i ) above). The Gussman paper provides annual data f o r the per iod 1946-1969. His se r ies were updated to 1971 by Cummings and Meduna. These l a t t e r estimates a lso incorporate a substan-t i a l number of conceptual changes, as well as data r e v i s i o n s fo r the l a t t e r part of the 1947-1969 period based on more recent National Income Accounts est imates . Some comments are in order as to the r e l a t i o n s h i p between the s e r i e s contained in the Cummings and Meduna paper and those der ived and presented in t h i s chapter . I n i t i a l l y , our ob jec t ive was simply to update the Cummings and Meduna s e r i e s f o r the per iod 1972-74. However, when we attempted to do s o , a number of ser ious problems arose. F i r s t , as w i l l be explained below (sect ion C) we required 1971 benchmark stock data f o r disaggregated durable goods c l a s s e s . Unfor tu-n a t e l y , these were not publ ished in the Cummings and Meduna paper, and were not a v a i l a b l e from the authors. Simple updating of t h e i r s e r i e s thus proved i m p o s s i b l e . : Second, Cummings and Meduna [1973; 3] employ what they term a D i v i s i a aggregation method in order to der ive aggregate p r i c e and quan-t i t y s e r i e s . However, t h e i r index number formula i s not a conventional o D i v i s i a index, and lacks a s p e c i f i c t h e o r e t i c a l j u s t i f i c a t i o n in terms of the under ly ing preference s t r u c t u r e . Super ior indexing procedures are a v a i l a b l e , and we make use of one of these in order to a r r i v e at our aggregate s e r i e s . Of i n t e r e s t i s whether the two methods provide very d i f f e r e n t answers in p r a c t i c e , and some evidence on t h i s issue i s presented in sec t ion A of t h i s chapter , and in Appendix B. - 100 -T h i r d , a number of ambigu i t ies , as well as apparent computational e r r o r s , are contained in the Cummings and Meduna paper, which could not be resolved s a t i s f a c t o r i l y . For example, we show below that t h e i r per ' !capita normalized housing stock se r ies (which forms a very large component of aggregate durables) i s overstated by a f a c t o r of between two and three . Given a l l these d i f f i c u l t i e s , i t was decided to reconstruct completely the consumption and l e i s u r e s e r i e s . In doing s o , we have t r i e d to be most care fu l to provide s u f f i c i e n t data and accompanying explanat ion so that the se r ies may be a l t e red and/or updated by o thers . It should be noted a lso that in the case of durable and semi durable goods, we have der ived two a l t e r n a t i v e renta l p r ice s e r i e s . The. f i r s t ser ies is based upon the methodology of Gussman and Cummings and Meduna and assumes that the purchase pr ices of durable goods are expected to remain constant . The second, a l t e r n a t i v e se r ies which we have c a l c u -l a t e d , includes the in f luence of expected cap i ta l ga ins . As one might expect , the two sets of rental p r ice s e r i e s , f o r both aggregate and d i s -aggregated durable and semi durable goods, behave qui te d i f f e r e n t l y . For comparison purposes, both sets of s e r i e s are tabulated .below (sect ion -C) . One other important extension to the work of Cummings and Meduna and (Gussman) should be noted at t h i s stage. We have reconstructed the net wage rate (opportunity cost of l e i s u r e time) va r iab le using a procedure qui te d i f f e r e n t from that employed prev ious ly by these authors. As well as making use of some recent employment and wage rate data publ ished by Diewert [1975], we attempt to capture the heterogeneity of labour inputs and the p r o g r e s s i v i t y of the income tax s t ruc ture in const ruct ing net wage ser ies by occupat ion. These net wage se r ies are then aggregated across occupations to provide estimates of the aggregate net wage rate and - 101 -quant i ty of manhours worked. We der ive an estimate of the aggregate marginal tax rate impl ied by our procedure, and compare i t with the tax rate se r ies constructed by both Gussman and Cummings and Meduna, who used a rather simpler approach. A. The Choice of an Aggregate Index Number Formula In t h i s s e c t i o n , we o u t l i n e the r a t i o n a l e fo r the p a r t i c u l a r index number formula used in the const ruc t ion of these data s e r i e s . F i r s t , some notat ion is requ i red . Define an aggregate p r ice index r e l a t i n g periods 0 (the base period) and 1 as some funct ion P ( p ° , p 1 , x ° , x 1 ) where p ° , x ° , and p 1 , x 1 , are n-dimensional p r ice and quant i ty vectors f o r periods 0 and 1, r e s p e c t i v e l y . S i m i l a r l y , an aggregate quant i ty index may be def ined as some funct ion Q ( p ° , x ° , p 1 , x 1 ) . Many suggestions have been made as to the appropr iate funct iona l forms f o r P and Q. However, i t seems that u n t i l r e l a t i v e l y r e c e n t l y , there ex is ted no basis on which to d iscr imina te among a f a i r l y large subset of ' reason-ab le ' formulae. The aggregation method used here i s based on some recent advances in the theory of exact and super la t i ve index numbers (Diewert [1976b]).. A quant i ty index, Q* , i s termed exact f o r an aggregator f u n c t i o n , f , i f : (5.1) fl*A = Q * ( p 0 } p i , X 0 } x i ) f ( x ° ) where x 1 i s the s o l u t i o n to the problem (5.2) Max { f (x ) ; p x - x < p ^ x 1 ; x >_ 0 , p 1 » 0 } w . r . t . . x - 102 r and x° i s the s o l u t i o n to the problem (5.3) Max • ( f ( x ) ; p ° - x < p ° - x 0 ; x > 0 . p° » 0 1 w . r . t . n n x where p«x denotes the inner product of two column v e c t o r s , p and x. In our consumer context , f may be taken to be a d i r e c t u t i l i t y f u n c t i o n . As we have seen in the previous chapter , preferences may a lso be character -ized by the expenditure f u n c t i o n , m(u; p) def ined as: (5.4) m(u; p) = min {p*x; f (x) >_ u; x >_ 0n> w . r . t . where u i s some given index leve l of u t i l i t y . A p r ice index, P* , i s def ined as exact f o r the expenditure (or cost ) funct ion m(u; p) i f 1 (5-5) I f e £iI = P * ( p ° , p i , x ° , x i ) m(u; p u ) Consider the fo l lowing forms fo r quant i ty and p r i c e indexes: (5.6) Q D = n [x i / x o ] * £ s l i + s ° i ] i=l (5.7) P n = I [p i / p 0 j ^ [ s l i + S ° i ] where D - T\'V n-r r r p i x i s i = T~T ' r = 0 , 1 , p • x i s the i t h good's share of to ta l expenditure in per iod r. Diewert [1976b] has shown that the quant i ty index is exact - 103 -f o r a l i n e a r l y homogeneous Translog d i r e c t u t i l i t y f u n c t i o n , f ( x ) . Furthermore, the p r i c e index, Pg, i s exact f o r a Translog expenditure 2 f u n c t i o n , m(u; p ) , where the reference u t i l i t y leve l i s taken to be u* = ( u ° u 1 ) ^ . Since the Translog funct ion can provide a second order d i f f e r e n t i a l approximation to any a r b i t r a r y u t i l i t y or expenditure f u n c t i o n , we term (5.6) and (5.7) super la t i ve index numbers. Thus these index numbers have the considerable advantage of being exact (to the second order) fo r a wide range of consumer preferences. To der ive the p r i c e and quant i ty indexes corresponding to (5.6) and (5 .7 ) , we make use of F i s h e r ' s [1922] weak f a c t o r reversa l t es t which states that the product of the p r i c e and quant i ty indexes should equal the r a t i o of to ta l expenditures in the two p e r i o d s , and def ine (5.8) Pp = p i . x V ( p ° - x ° ) Q D (5.9) ftD s p i . x i / ( p o . x O ) p D E i t h e r ( Q D , Pp) or (Pp, ?JD), def ined by (5.6) - ( 5 .9 ) , may then be used to construct p r i c e and quant i ty indexes. These are of ten re fe r red to as d i s c r e t e D i v i s i a indexes. Note that Pp and Qp do not in general s a t i s f y the weak f a c t o r reversal t e s t : i . e . , PpQp f p 1 « x 1 / p ° ' X 0 . This i s qui te reasonable , as Q Q i s cons is ten t with a Translog u t i l i t y f u n c t i o n , whi le Pp i s exact fo r a Translog expenditure f u n c t i o n , and these two funct ions w i l l in general correspond to d i f f e r e n t consumer preferences . Note a lso that i t i s not true in general that a D i v i s i a index of D i v i s i a indexes i s a D i v i s i a index of the components, where the d i s c r e t e D i v i s i a index i s def ined by (5.6) or (5 .7 ) . - 104 -(5.6) and (5.7) may be wr i t ten in s l i g h t l y d i f f e r e n t form by taking logs as n (5.10) log Q n = % z ( s 1 . + s ° . ) ( l o g x 1 . - log x ° . ) i = l 1 (5.11) log P n = % " ( s 1 , + s ° . ) ( l o g p 1 . - log p 0 , ) o r , more g e n e r a l l y , noting that Qp and Pp represent r a t i o s of current to base per iod quant i t i es and pr ices as (5.12) log - log Q p t _ 1 = ( s * . + s ^ - K l o g x 1-. - log x t _ 1 . ) (5.13) log - log P p t _ 1 = % " ( s * . + s t _ 1 . ) ( l o g p*. - log p * " 1 . ) The index (5.12) or (5.13) is a lso known as the Tornquist [1936] index. It i s considered by Christenson and Jorgenson [1975] as a d i s c r e t e approximation to a continuous D i v i s i a index. However, there e x i s t many other p o s s i b l e approximations, and the major advantage of using the exact index numbers approach is that i t al lows us to d iscr imina te among these var ious approximations. In const ruc t ing aggregate p r i c e and quant i ty s e r i e s from d i s -aggregated da ta , we have used the p r i c e index P^, def ined by (5.13) together with the i m p l i c i t quant i ty index, ftp, def ined by (5 .9 ) . Our p r i n c i p a l reason f o r choosing Pp rather than Qp i s to f a c i l i t a t e compari-sons with the work of Cummings and Meduna who a lso der ive the quant i ty index r e s i d u a l l y , v ia the f a c t o r reversa l t e s t . Cummings and Meduna [1973; 3] use an aggregate p r i c e index, P r , - 105 -def ined as : t n t t (5.14) PCZ = Z sZ.pZ. i=1 1 1 where t t t / t t s i ; = p ..x . / P -x which may be rewri t ten more g e n e r a l l y , assuming p ° . = 1, a l l i , as : (5.15) P r ( p ° , p l , q ° , ql)= Z — l -L i=l p i - x n P 1 ^ ^ . p 1 . ^ x 1 p°.j Cummings and Meduna r e f e r to (5.14) as a D i v i s i a index: however, the r e l a t i o n s h i p between t h e i r index and the conventional D i v i s i a indexes d i s c u s s e d , f o r example, by Diewert [1975] and Christenson and Jorgenson [1975] i s not c l e a r . Note that t h e i r index (5.15) does not correspond to e i t h e r a Paasche (Pp) or Laspeyres (P^) p r ice index which a r e , respec-t i v e l y , n P ° - x i . p i , (5.16) Pp = z 1 i=l pO-x 1 p ° . n p 1 . (5.17) P. = z i=l p ° . x ° p°. Of some i n t e r e s t i s the numerical magnitude of the d i f f e rence between a l t e r n a t i v e p r i c e indexes. In Appendix B, we present some compari-sons of the two p r i c e indexes, Pp and P^> (the Cummings and Meduna index) based on disaggregated data . As one would expect , the s i z e of the d i f f e r -ence depends on the degree of p r i c e v a r i a t i o n over the per iod . In genera l , - 106 -however, the d i f f e r e n c e is not l a rge . The average annual percentage d i f f e r e n c e turns out to be 1.31 per cent per y e a r , where f o r each y e a r , the percentage d i f f e r e n c e i s taken as the average of the percentage d i f f e rences fo r each of four aggregate p r i c e s e r i e s : non durab les ; s e r v i c e s ; semi durab les; and durables . B. Non Durable Goods and Serv ices Fol lowing Cummings and Meduna, we consider seven d i f f e r e n t ca te -g o r i e s : food , a l c o h o l , tobacco, energy, other non durab les , medical s e r v i c e s , and other s e r v i c e s . For each of these seven groups, p r i c e and 3 quant i ty data are required f o r 1947-74. Data f o r these categor ies are taken from the National Accounts. The most recent National Accounts summary p u b l i c a t i o n , National Income and Expenditure Accounts , V o l . 1, The Annual Est imates , 1926-1974 4 (13-531) presents se r ies in 1971 d o l l a r u n i t s . However, s ince both Gussman and Cummings and Meduna used s e r i e s based on 1961 = 100 p r i c e indexes, f o r comparison purposes we wished to use the same indexing pro-cedure. The Cummings and Meduna s e r i e s f o r the per iod 1947-71 come from National Income and Expenditure Accounts, H i s t o r i c a l R e v i s i o n , 1926-1971, and t h i s i s our source a lso f o r t h i s p e r i o d . The 1972-1974 estimates are taken from (13-531), where the 1971 constant d o l l a r quant i t i es f o r each category have been converted to 1961 d o l l a r u n i t s . Hereaf ter , we s h a l l r e f e r to these two sources j o i n t l y as NIEA. The two pub l ica t ions are i d e n t i c a l in terms of tab le numbering and l i n e c a t e g o r i e s . Real quant i ty data f o r the f i r s t three categor ies ( food, a lcohol and tobacco) are taken from l i n e s 2, 3, and 4 of Table 54, NIEA, f o r 1947-74. The corresponding i m p l i c i t p r i c e s e r i e s are der ived by d i v i d i n g current d o l l a r expenditure f o r each of the three items ( l i n e s 2, 3 , and - 107 -4 of Table 53, NIEA) by the real quant i ty f i g u r e s . We d iv ide the quant i ty s e r i e s by the populat ion of Canada aged 15 years and over (see Table 5.1) 5 to obtain per cap i ta f i g u r e s . The pr ice and quant i ty s e r i e s are l i s t e d in the f i r s t three columns of Tables 5.2 and 5.3 r e s p e c t i v e l y . The next category considered i s energy. Disaggregated quant i ty and ( impl ied)^ p r i c e se r ies are a v a i l a b l e fo r four energy components: ( i ) e l e c t r i c i t y , ( i i ) gas , ( i i i ) other f u e l s , and ( iv) g a s o l i n e , o i l and grease. These are taken from l i n e s 13, 14, 15 and 32 of Tables 53 and 54, NIEA. To obtain an aggregate energy p r i c e s e r i e s we used the Tornquist approximation (5.13) to a continuous D i v i s i a index discussed in sect ion A. The energy quant i ty s e r i e s was then der ived r e s i d u a l l y using (5 .9 ) . These two s e r i e s , in per cap i ta terms, are tabulated in the fourth column of Tables 5.2 and 5.3. The f i n a l non durable category considered is 'other non d u r a b l e s ' , c o n s i s t i n g o f : ( i ) non durable household s u p p l i e s , ( i i ) drugs and sundr ies , and ( i i i ) t o i l e t a r t i c l e s , cosmet ics . Data come from l i n e s 19, 28 and 43 of Tables 53 and 54, NIEA. Again we employ the D i v i s i a aggregation method, the (per capi ta ) quant i ty s e r i e s being der ived r e s i d u a l l y . The r e s u l t i n g 'o ther non durables ' p r i c e and quant i ty se r ies are reported in Tables 5.2 and 5.3 (column f i v e ) . We now turn to s e r v i c e s . Two groups are d i s t i n g u i s h e d : 'medical s e r v i c e s ' , and 'other s e r v i c e s ' . The medical serv ices se r ies is a D i v i s i a aggregation of ( i ) medical c a r e , ( i i ) hospi ta l care and the l i k e ( l i n e 26, NIEA), ( i i i ) other medical care expenses ( l i n e 27, NIEA), and ( iv ) govern-ment hosp i ta l ca re . The current d o l l a r value of the f i r s t component, medical c a r e , i s given in l i n e 25 of NIEA. However, from 1969 onwards, t h i s se r ies shows - 108 r a dramatic f a l l due to the commencement of p r o v i n c i a l medicare schemes. To preserve approximate consistency with e a r l i e r y e a r s , the current d o l l a r f igures from 1969 onward are taken from sources other than the National Accounts. The 1969 f i g u r e comes from Table 1, Evans [1976], and cons is ts of to ta l expenditure on physic ians and d e n t i s t s ' s e r v i c e s . From 1970-1973 i n c l u s i v e , the f igures are taken from Table 1 (health expenditure on pro-f e s s i o n a l se rv ices ) of National Health Expenditures in Canada 1960-1973, publ ished by Health and Welfare Canada. We estimated the 1974 f i g u r e by assuming the same percentage growth (8.3 per cent) between 1973 and 1974 as occurred from 1972-19737 The p r i c e index f o r th is (adjusted) category i s the i m p l i c i t p r i c e index contained in l i n e 25, NIEA. The reason f o r the i n c l u s i o n of ( i v ) , government hosp i ta l c a r e , i s that in 1961 government t r a n s f e r payments to hosp i ta ls were switched from the consumer expenditure accounts to the government expenditure s e c t i o n . Fol lowing the procedure of Gussman [1972; 45] and Cummings and Meduna [1973; 3 ] , we have added back government hospi ta l care expenditure from 1961 onwards. The current d o l l a r value of ( iv ) comes from l i n e 38, Table 43, NIEA. For the p r i c e index we used the p r i c e index i m p l i c i t in o category ( i i ) , hospi ta l care and the l i k e . The 'other s e r v i c e s ' category is qui te s t ra ight forward . We aggregate: ( i ) laundry and dry c l e a n i n g , ( i i ) domestic s e r v i c e s , ( i i i ) other household s e r v i c e s , ( iv ) other auto re la ted s e r v i c e s , (v) purchased t r a n s p o r t a t i o n , (v i ) communication, ( v i i ) recrea t iona l s e r v i c e s , ( v i i i ) educat ional^ and c u l t u r a l s e r v i c e s , ( ix) personal c a r e , (x) expenditure on restaurants and h o t e l s , (x i ) f i n a n c i a l , legal and other s e r v i c e s , and ( x i i ) operat ing expenses of non p r o f i t o rgan iza t ions . The data fo r these groups comes from l i n e s 21, 22, 23, 33, 34, 35, 39, 40, 44, 45, 46 and 47,NIEA. - 109 -TABLE 5.1 POPULATION OF CANADA 15 YEARS AND OVER YEAR POPULATION (mi l l ions of persons) 1947 9.0126 1948 9.1447 1949 9.5089 1950 9.6415 1951 9.7587 1952 10.0063 1953 10.2169 1954 10.4523 1955 10.6591 1956 10.8555 1957 11.1534 1958 11.3948 1959 11.6253 1960 11.8400 1961 12.0464 1962 12.2733 1963 12.5131 1964 12.7917 1965 13.0877 1966 13.4233 1967 13.8115 1968 14.1785 1969 14.5424 1970 14.9112 1971 15.1874 1972 15.4383 1973 15.9090 1974 16.3490 Source: 1947-1971; Cummings and Meduna [1973], Table 6 1972-1974; Populat ion of Canada and the Provinces by Sex and Age Group (91-202) - no -TABLE 5.2 PER CAPITA QUANTITY INDEXES, NON DURABLE GOODS AND SERVICES OTHER MEDICAL OTHER YEAR FOOD ALCOHOL TOBACCO ENERGY N-DUR SERVICES SERVICES 1947 363. 158 61. 581 47. 045 81. 405 59, .150 66. 124 341. 373 1948 339. 869 62. 441 48. 006 80. 872 57, .689 64. 993 341. 610 1949 325. 590 64. 150 48. 271 82. 367 54, .725 65. 573 348. 299 1950 343. 100 65. 031 48. 644 83. 534 57, .472 68. 793 347. 737 1951 343. 488 64. 763 43. 858 87. 957 55, .782 69. 480 350. 485 1952 345. 482 71. 755 45. 272 92. 187 56, .860 72. 286 360. 285 1953 359. 111 72. 135 49. 232 96. 815 58, .246 75. 452 371. 045 1954 370. 541 69. 841 50. 898 103. 422 58, .358 78. 830 364. 708 1955 378. 925 72. 333 53. 663 112. 602 60 .967 81. 343 374. 868 1956 394. 823 75. 630 56. 193 121. 163 65 .406 89. 946 388. 028 1957 397. 009 76. 120 59. 533 127. 300 68 .265 90. 474 391. 154 1958 392. 109 76. 438 62. 836 132. 265 69 .094 97. 333 390. 460 1959 399. 646 77. 503 63. 740 140. 408 69 .191 103. 558 399. 790 1960 407. 095 78. 125 62. 922 146. 381 69 .471 I l l . 999 408. 080 1961 399. 373 79. 692 65. 580 148. 675 72 .055 125. 681 415. 892 1962 399. 404 81. 152 67. 871 155. 549 75 .444 129. 918 423. 105 1963 398. 942 84. 152 68. 089 164. 070 79 .573 133. 288 432. 308 1964 408. 312 84. 195 68. 091 167. 912 84 .310 139. 368 443. 050 1965 411. 302 90. 161 70. 142 172. 368 88 .425 146. ,979 459. 281 1966 402. 882 94. 016 71. 070 174. ,241 93 .886 152. 147 473. 677 1967 413. 351 98. 252 70. 955 175. 886 97 .357 158. 188 490. 596 1968 405. ,262 95. 638 67. 850 177. 482 101 .202 162. 192 500. 414 1969 413. ,756 98. 815 66. 220 180. ,334 104 .198 169. ,344 507. 002 1970 423. ,306 105. 223 69. ,279 185. ,402 104 .830 190. ,543 506. 526 1971 440. ,233 111. 869 69. 400 186. ,495 110 .379 197. ,949 506. on 1972 452. ,446 117. 111 70. ,798 194. ,184 119 .024 207. ,443 534. ,668 1973 453. ,193 127. 032 71. ,216 194. ,913 132 .487 202. ,673 537. ,582 1974 455. ,247 127. ,164 71. ,992 201. ,720 143 .006 205. ,802 533. 392 - Ill -TABLE 5.3 PRICE INDEXES, NON DURABLE GOODS AND SERVICES YEAR FOOD ALCOHOL TOBACCO 1947 0 .6517 0 .8090 0. 8467 1948 0 .7976 0 .8301 0. 8907 1949 0 .8120 0 .8262 0. 9085 1950 0 .8289 0 .8469 0. 9318 1951 0 .9475 0 .9161 1. 0234 1952 0 .9488 0 .9178 1. 0508 1953 0 .9084 0 .9172 0. 9284 1954 0 .9078 .0 .9205 0. 9135 1955 0 .9077 0 .9222 0. 9161 1956 0 .9183 0 .9245 0. 9180 1957 0 .9580 0 .9517 0. 9172 1958 0 .9966 0 .9644 0. 9176 1959 0 .9877 0 .9822 0. 9798 1960 0 .9909 0 .9924 0. 9973 1961 1 .0000 1 .0000 1. 0000 1962 1 .0218 1 .0201 1. 0036 1963 1 .0531 1 .0237 1. 0035 1964 1 .0653 1 .0520 1. 0149 1965 1 .0933 1 .0610 1. 0458 1966 1 .1601 1 .0753 1. 0901 1967 1 .1640 1 .0921 1. 1357 1968 1 .2052 1 .1733 1. 2713 1969 1 .2373 1 .2122 1. 3333 1970 1 .2552 1 .2097 1. 3514 1971 1 .2626 1 .2531 1. 3805 1972 1 .3665 1 .3175 1. 4154 1973 1 .5854 1 .3211 1. 4687 1974 1 .8360 1 . 4084 1. 5480 OTHER MEDICAL OTHER ENERGY N-DUR SERVICES SERVICES 0.7851 0.6640 0. 4732 0.5506 0.8654 0.7412 0. 5334 0.6012 0.8963 0.7841 0. 5677 0.6323 0.9337 0.7832 0. 5805 0.6559 0.9716 0.8634 0. 6371 0.7113 0.9919 0.8893 0. 7092 0.7442 1.0039 0.8990 0. 7407 0.7676 1.0046 0.9050 0. 7865 0.7943 0.9973 0.9125 0. 8108 0.8169 1.0081 0.9197 0. 8457 0.8497 1.0255 0.9417 0. 8978 0.8878 1.0185 0.9602 0. 9377 0.9269 1.0102 0.9971 0. 9627 0.9542 0.9988 1.0139 0. 9819 0.9775 1.0000 1.0000 1. 0000 1.0000 0.9958 1.0033 1. 0317 1.0241 0.9859 1.0033 1. 0589 1.0496 0.9912 1.0088 1. 0916 1.0875 0.9921 1.0291 1. ,1416 1.1371 1.0103 1.0553 1. 1937 1.1964 1.0402 1.0798 1. 2660 1.2891 1.0801 1.0886 1. 3398 1.3423 1.1066 1.1219 1. ,4164 1.4256 1.1441 1.1554 1. 4469 1.5132 1.1919 1.1662 1. 5354 1.6143 1.2332 1.1880 1. 6390 1.6537 1.3474 1.2231 1. ,7448 1.7703 1.5504 1.3263 1. 9642 1.9619 - 112 -TABLE 5.4 AGGREGATE PRICE AND PER CAPITA QUANTITY INDEXES, NON DURABLE GOODS AND SERVICES NON DURABLES SERVICES YEAR PRICE QUANTITY PRICE QUANTITY 1947 0.7051 609.129 0.5378 407.693 1948 0.8149 587.080 0.5899 406.926 1949 0.8327 573.993 0.6215 414.265 1950 0.8518 596.157 0.6433 416.616 1951 0.9480 594.217 0.6990 420.038 1952 0.9564 609.829 0.7384 432.590 1953 0.9269 634.016 0.7631 446.481 1954 0.9264 651.868 0.7931 443.435 1955 0.9262 677.773 0.8160 456.101 1956 0.9350 712.730 0.8492 477.832 1957 0.9646 727.972 0.8899 481.484 1958 0.9874 732.489 0.9292 487.713 1959 0.9917 750.404 0.9561 503.297 1960 0.9952 763.947 0.9785 520.060 1961 1.0000 765.374 1.0000 541.573 1962 1.0132 779.336 1.0259 553.029 1963 1.0274 794.421 1.0518 565.605 1964 1.0391 812.360 1.0884 582.441 1965 1.0589 831.669 1.1381 606.292 1966 1.1036 834.621 1.1957 625.857 1967 1.1202 854.425 1.2834 648.814 1968 1.1697 845.329 1.3417 662.630 1969 1.2040 860.892 1.4233 676.354 1970 1.2255 885.744 1.4959 696.667 1971 1.2477 915.971 1.5936 703.296 1972 1.3190 950.450 1.6510 741.447 1973 1.4541 973.525 1.7648 739.642 1974 1.6424 991.718 1.9643 738.559 - 113 -The per cap i ta quant i ty and the p r i c e se r ies f o r medical and other se rv ices are reported in columns 6 and 7 of Tables 5.2 and 5.3. F i n a l l y , we aggregate columns 1-5 and columns 6-7 of Tables 5.2 and 5.3 to form pr ice and quant i ty s e r i e s fo r aggregate 'non durables ' and 'aggregate s e r v i c e s ' , r e s p e c t i v e l y . ^ These se r ies appear in Table 5.4. C. Durables and Semi Durables In sec t ion B of Chapter 3, we descr ibed the treatment of durable goods in our model. In order to apply the model e m p i r i c a l l y , we r e q u i r e , f i r s t , an estimate of the stock of the durable in each p e r i o d , and second, an estimate of the rental p r i c e of the serv ice flow y i e l d e d by the stock of the durable good. There are s ix categor ies of durables to be considered: c l o t h i n g , 'other semi d u r a b l e s ' , 'o ther d u r a b l e s ' , housing, and r e s i d e n t i a l land. We f i r s t der ive the stock of each of these durables (or t h e i r d isaggrega-ted components, where app l icab le ) over the p e r i o d , and then estimate the corresponding rental p r i c e s e r i e s . Since the const ruct ion of p r i c e and quant i ty se r ies f o r housing and land involved some spec ia l data and methodological problems, they are dea l t with separate ly below. 1) The Construct ion of Stock Estimates To estimate a c a p i t a l stock s e r i e s , a perpetual inventory method with geometric deprec ia t ion is used. The current y e a r ' s s tock , « t , equals the undepreciated por t ion of l a s t p e r i o d ' s s tock , . ( T-6 ) K t _ 1 , where $ is the one per iod (constant) deprec ia t ion r a t e , plus th is y e a r ' s investment, 1^. Thus - 114 -(5.18) K t = (1-6) K t - 1 • + I t o r , e q u i v a l e n t l y , oo (5.19) K = I (1-6)T I 1 x=0 Z ~ T To estimate c a p i t a l stocks using (5.18) or (5 .19) , we r e q u i r e : ( i ) a time s e r i e s of real investment, I^ , ( i i ) deprec ia t ion rates f o r each durab le , and ( i i i ) a benchmark observat ion of the c a p i t a l stock (s ince we do not have an i n f i n i t e se r ies of past investments) . The real investment se r ies fo r 1947-1974 are taken from Table 53, NIEA. For ' c l o t h i n g 1 , there are three sub components, 'men's and boys' c l o t h i n g ' , women's and c h i l d r e n ' s c l o t h i n g ' , and 'footwear and r e p a i r ' ( l i nes 6, 7, and 8) . 'Other semi durables ' c o n s i s t of 'semi durable house-hold f u r n i s h i n g s ' , books, newpapers and magazines ' , and ' jewel lery , watches and r e p a i r s ' ( l i n e s 19, 38, and 42). "Automobiles" equals the sum of l i n e s 30 and 31, 'new and used (net) automobiles plus repa i rs and p a r t s 1 . 1 1 The three components of 'other durables ' are ' f u r n i t u r e , carpets and other f l o o r c o v e r i n g s ' , 'household app l iances ' and ' r e c r e a t i o n , spor t ing and camping equipment 1 ( l i nes 17, 18 and 37). Stock ser ies were const ruc-ted separate ly f o r each of these sub components. The deprec ia t ion rates assumed fo r a l l of these categor ies are i d e n t i c a l to those used by Cummings and Meduna and Gussman (with the exception of housing) and are l i s t e d in Table 5.5. The sources f o r these var ious rates are d iscussed in Gussman [1972; 25-30]. F i n a l l y , we require a benchmark f igure f o r the stock of each durable in 1947. Cummings and Meduna (p. 7) s ta te that they used - 115 -investment data from the National Accounts fo r the per iod 1921 to 1946 in order to estimate the benchmark, K ^ g , using formula (5.18) and assu-ming K-|g2Q equals zero . However, these c a l c u l a t e d benchmarks were not publ ished in t h e i r paper, and a r e , in f a c t , no longer obta inable from the authors. Since they do not. provide disaggregated stock data fo r 1971 (the l a s t year in t h e i r se r i es ) t h i s means that t h e i r se r ies cannot be updated. We therefore constructed our own benchmarks fo r 1947, using i n -12 vestment data fo r the per iod 1926-1946. However, a problem arose immediately, namely, that the pre 1947 s e r i e s employ a d i f f e r e n t c l a s s i f i -ca t ion than the post 1947 s e r i e s . We have no way of knowing how Cummings and Meduna dea l t with t h i s problem ( i t i s not d iscussed in t h e i r paper) 13 and we used our own methods of approximation to l i n k the two per iods . With the pre 1947 investment da ta , 1946 benchmarks were constructed from (5 .18) , assuming = 0. The next stage was to estimate the stocks fo r the per iod 1947-1974, again using the same perpetual inventory method, (5 .18) . The disaggregated benchmarks fo r 1946, along with the 1974 stock f igures (which may be used to update these accounts in the future) are l i s t e d in Table 5.5. 2) Construct ion of Rental Pr ices In Chapter 3 , we der ived the rental p r ice formula (3 .10) , repeated here f o r convenience: n _ P * n t ( R t + 6 n + e n t ' 5 n " e n t ) n i m p n t ~ [TT-Rp ( 3 J 0 ) o r , in the case of s t a t i c expecta t ions , - 116 r. n _ P * n t ( R t + ^ n i n Pnt ~ (1 + R t ) { 3 J 1 ) In order to apply (3.10) or (3 .11) , we requi re data on p * n t (the purchase p r i c e of the durab le ) , .R t (the discount r a t e ) , 8^ (the deprec ia t ion rate) and, in the case of (3 .10) , an estimate of e ^ (the expected rate of increase in the purchase p r i c e during the p e r i o d ) . The purchase pr ice se r ies fo r each of the ten disaggregated durable and semi durable components were der ived i m p l i c i t l y from NIEA in the same manner as the p r i c e indexes fo r non durable goods and s e r -v i c e s . These are tabulated in Appendix B. The one per iod deprec ia t ion rates appear in Table 5.5. For R^, the re levant discount rate f o r each durab le , we used the same set of s e r i e s as Cummings and Meduna, updated to 1974. Unfor tunate ly , the rates on automobiles and 'other d u r a b l e s ' , RMV and RCFP, r e s p e c t i v e l y , were not a v a i l a b l e a f t e r 1972. We therefore employed a Box-Cox maximum l i k e l i h o o d f o r e c a s t i n g technique, developed by Professor Kenneth J . White of the U n i v e r s i t y of B r i t i s h Columbia, with RSBY (the government f i v e year bond rate) as the independent v a r i a b l e , in order to extrapolate these two s e r i e s f o r 1973 and 1974. Table 5.6 contains a l l the discount rates used. Note that our procedure v i o l a t e s the assumption made in sec t ion A of Chapter 3 , namely, that there e x i s t s one d iscount r a t e , R^, the same borrowing or lending rate fac ing a l l consumers. The re laxa t ion of the assumption does not a l t e r the s t a t i c vers ion of the model, provided that only one discount rate is used in the case of each durable . However, in r e a l i t y , consumers may face d i f f e r e n t discount rates f o r purchasing the same durab le , a compl icat ion which we have indeed - 117 -assumed away. I f , f o r example, a consumer borrows to f inance the purchase of the durable , and uses the durable as c o l l a t e r a l , then the appropr iate R t i s the i n t e r e s t rate on that type of loan ( e . g . , the automobile r a t e , the mortgage rate on house purchases e t c . ) . On the other hand, the consumer could be a net lender , in which case the re levant should be equal ized across a l l h is durable goods purchases, and would equal his average p o r t f o l i o y i e l d . The imp l ica t ion of t h i s i s that in r e a l i t y , d i f f e r e n t consumers face d i f f e r e n t rental p r ices f o r the i d e n t i c a l durable . T h e o r e t i c a l l y , in order to obtain aggregate market demands, we would have to integrate not j u s t over the d i s t r i b u t i o n of expenditures (as was discussed in sec t ion B of Chapter 4) but a lso ( j o i n t l y ) over the d i s t r i b u t i o n of p r i c e s . Th is is l i k e l y to be extremely complex. In any case , data do not e x i s t on consumers c l a s s i f i e d j o i n t l y by t h e i r expenditures and t h e i r net c r e d i t p o s i t i o n s . Thus, at t h i s s tage , not very much can be done about the problem. In order to construct a time s e r i e s f o r e ^ , we used the Box and Jenkins [1970] ARIMA fo recas t ing m o d e l . 1 4 In essence, t h i s approach assumes that the time s e r i e s in quest ion ( in our case , a purchase p r i c e s e r i e s ) or some d i f f e r e n c e o f some transform&of i t , - represents -a sample r e a l i z a t i o n from an i n f i n i t e populat ion of such samples, which could have been generated by a s ta t ionary s t o c h a s t i c process . The term ' s t o c h a s t i c process 1 r e fe rs to a model descr ib ing the p r o b a b i l i t y s t ruc ture of a sequence of observat ions . A s ta t ionary process i s assumed to be in a s p e c i f i c form of s t a t i s t i c a l e q u i l i b r i u m , and in p a r t i c u l a r , to vary about a f i x e d mean. Given knowledge of t h i s s t o c h a s t i c process on the part of the - 118 -consumer, one may show that the optimal ( in the sense of minimising the mean square e r ror involved) f o r e c a s t , condi t iona l upon the sample i n f o r -15 mation to date , i s the predicted value from the ARIMA model. Such well known suggestions fo r expectat ions model l ing as s t a t i c expectat ions and adaptive expectat ions may be shown (see Rose [1976; Ch.9]) to be spec ia l cases of a general ARIMA model, where the under ly ing s t o c h a s t i c process i s of a spec ia l kind in each case . The use of an ARIMA model to construct an expected pr ice s e r i e s proceeds in a number of s tages. F i r s t , given the time s e r i e s in ques t ion , i d e n t i f i c a t i o n techniques are appl ied in order to obtain some idea of the appropriate ARIMA process underly ing the data . Second, the chosen model i s est imated, usua l l y v ia non l i n e a r techniques, and var ious c r i t e r i a such as the goodness of f i t , s i g n i f i c a n c e tes ts on parameters and sets of parameters, and e s p e c i a l l y whether there is evidence of au tocor re la t ion among the r e s i d u a l s , are used to determine the adequacy of the model. U s u a l l y , a number of d i f f e r e n t models must be estimated u n t i l a s a t i s f a c -tory candidate is found. T h i r d , the predic ted values from the estimated model are used as the expected p r i c e s e r i e s . In our case , s ince we are in te res ted in the expected rate of change of p * t , we transformed the s e r i e s by taking natural logarithms and used the fo l lowing approximation P*t+1 " p * t e (5.20) e t E — ^ - — * d o g p * t + 1 ) e - log p* t ( log P * t + i ) i s the predicted value from the estimated ARIMA model. For each of the ten durable and semi durable goods l i s t e d e a r l i e r (excluding housing and l and , f o r the p resen t ) , an expected rate of change of the purchase p r i c e s e r i e s , e^ , was constructed using an - 119 -ARIMA model and (5.20) . The ARIMA model est imates , as well as a more d e t a i l e d o u t l i n e o f the technique i n v o l v e d , are contained in Appendix A. Two a l t e r n a t i v e rental p r i c e se r ies were then c a l c u l a t e d f o r each of the ten disaggregated durable and semi durable components using (3.10) ( s t a t i c expectat ions imposed) and (3.11) (al lowing f o r expected c a p i t a l g a i n s , where e ^ is def ined by (5 .20) ) . 3) Housing and Land Since these two categor ies pose some spec ia l data problems, they are deal t with separa te ly . F i r s t , consider housing. The investment se r ies in the sum of l i n e s 5 and 10 of Table 6, NIEA, c o n s i s t i n g of 16 government and business r e s i d e n t i a l c o n s t r u c t i o n . The benchmark f o r 1947, 11,805 comes from Cummings and Chr istensen [1976], which is almost i d e n t i c a l to that used by Gussman. The deprec ia t ion rate was taken to be .025, a lso from the Cummings and Christensen p a p e r . 1 7 The 1947 and 1974 benchmark f igures are in Table 5.5. The renta l p r ice formula f o r housing and l a n d , i t was noted in Chapter 3, must be a l t e red s l i g h t l y in order to take in to account the property tax rate ( T ^ ) . The cor rec t expression f o r the rental p r ice in the case of these two durables was determined as ( 3 . 1 0 ' ) , repeated here f o r convenience. (R . + T + + R . T . + 6 + 6„e - e ) _ t t t t n n nt n t 7 l n , p . = p* . -i , n l o . o ^nt * n t 1 + R t The purchase p r i c e se r ies f o r housing i s from l i n e 8, Table 7, NIEA. We take RNHA, the housing discount r a t e , as the NHA rate on 'approved lenders ' home ownership loans ' from Table 75, Canadian Housing  S t a t i s t i c s , publ ished by the Central Mortgage and Housing Corpora t ion . - 120 -Estimates of the r e s i d e n t i a l property tax rate are given in Gussman [1972; Table IV -4 ] , and are updated by Cummings and Meduna. These estimates were der ived using a rather complex method explained in o u t l i n e by Gussman (pp. 53-54). Th is method, we f e l t , was too d i f f i c u l t to extend fu r the r . However, more important, we be l ieve that Gussman's se r ies provides extremely u n r e a l i s t i c est imates. From 1959 onwards his r a t e , PROPTAXG, is in the order of .08 (see Table 5 .6) . This i m p l i e s , f o r example, that an 'average' householder owning property valued at $50,000 1 o would pay the sum of $4,000 per year in property taxes. This order of magnitude was f e l t to be qui te unreasonable. Instead, we obtained our property tax r a t e , PR0PTAX, f o r the per iod 1947-1973 from unpublished to ta l property tax s e r i e s provided by E. Dianne Cummings, which had been used in the work of Cummings and 19 Christenson [1976]. Th is s e r i e s has the advantage of provid ing r e a l i s t i c est imates , in the order of .02 throughout the time per iod . The 1974 property tax rate estimate is based upon the work of Chinloy [1977], which contains actual e f f e c t i v e property tax rates r e f e r r i n g to the c i t y of London, Ontar io . We c a l c u l a t e d the percentage change in C h i n l o y ' s rate from 1973-1974, and then appl ied i t to our 1973 f i g u r e to obtain an estimate f o r 1974. PR0PTAX is l i s t e d in Table 5 .6 , along with PROPTAXG, the Gussman ra te . We consider next the data f o r r e s i d e n t i a l l and . Unfor tunate ly , no d i r e c t estimates of the stock of r e s i d e n t i a l land e x i s t . The f a i r l y complicated method employed by Gussman [1972; 32-34] involved using minimum legal l o t s i z e requirements fo r the c i t y of Vancouver, and assumptions about the s i z e of the ' t y p i c a l ' apartment b lock , as well as other da ta , to approximate a ' r e s i d e n t i a l land increment' s e r i e s . We - 12i: -d id not use th is approach s ince (a) i t was based on l o c a l i s e d information and (b) i t would have proved extremely d i f f i c u l t , i f not imposs ib le , to update. The a l t e r n a t i v e approach which was adopted fol lowed the method of Cummings and Christensen [1976]. Using the U.S. study by A l l e n Manvel [1968], we assumed that the value of r e s i d e n t i a l land was 32.5% of the value of r e s i d e n t i a l housing. We appl ied t h i s r a t i o to the current d o l l a r housing va lue , and then d iv ided by a r e s i d e n t i a l land p r i c e index (obtained from Table 112, Canadian Housing S t a t i s t i c s , Central Mortgage and Housing Corporat ion) to der ive a land stock se r ies in each per iod . The r e s u l t i n g se r ies d i f f e r s s l i g h t l y from that of Cummings and Christenson sriince (a) t h e i r s e r i e s f o r the d o l l a r value of the r e s i d e n t i a l housing stock i s d i f f e r e n t and (b) , they do not use the CHMC land p r i c e index as a d e f l a t o r . Note that our estimates d i f f e r s u b s t a n t i a l l y from those of Gussman. In the absence of any more d i r e c t in format ion , i t i s impossible to judge which method is the most appropr iate (except, perhaps, on grounds of computa-t iona l f e a s i b i l i t y ) . The land purchase pr ice s e r i e s , as noted a l ready , comes from Canadian Housing S t a t i s t i c s . The deprec ia t ion rate on land i s , of course , z e r o , while and PROPTAX are the same as fo r the housing category. Our next step was to construct a l t e r n a t i v e rental p r i c e s e r i e s fo r housing and l a n d , based on s t a t i c and non s t a t i c expecta t ions . The s t a t i c model (3.10' ) where P*^ t +-j = P* nt> 1 S qui te s t ra ight forward . The 20 normalized rental p r ices fo r housing and l and , with s t a t i c expectations:: imposed, appear in Table 5.8. However, f o r both of these c a t e g o r i e s , the const ruct ion of rental p r ices a l lowing f o r the in f luence of expected c a p i t a l gains ra ised some - 122 -i n t e r e s t i n g problems. In the case of housing, the optimal fo recas t ing model turned out to be of adaptive expectat ions form (see Appendix A fo r the actual es t imates) . The corresponding normalised rental p r i c e s e r i e s appears as RPH2 in Table 5.13. For comparison purposes, the ' s t a t i c ' renta l p r i c e s e r i e s is l i s t e d a l s o , as RPH1. As can be seen, the se r ies RPH2 d isp lays f a i r l y extreme f l u c t u a -t ions between time p e r i o d s , due to large r e l a t i v e changes in the expected c a p i t a l gains term from year to year . We considered that the behaviour o f th is s e r i e s was incons is ten t with a model such as ours , which assumes per fect c a p i t a l markets, absence of r i s k , and zero t ransact ions c o s t s . Consequently, i t was decided to explore other poss ib le expectat ions models. As a poss ib le a l t e r n a t i v e , a constant rate of i n f l a t i o n model was estimated over the en t i re time p e r i o d . The expected rate turned out to be 4.57 per cent per annum. The corresponding normalised rental p r i c e se r ies based on t h i s model, RPH3, i s given a lso in Table 5.13. The problem with t h i s index is that the growth of the s e r i e s over the time 21 per iod is l a rger than that of the se r ies with s t a t i c expectat ions imposed. However, i n t u i t i v e l y , one might expect that i f c a p i t a l gains p o s s i b i l i t i e s were higher in the l a t t e r part of the p e r i o d , t h i s should reduce, rather than i n c r e a s e , the 'spread ' of the 'non s t a t i c ' r e l a t i v e to the ' s t a t i c ' s e r i e s . Th is led us to inves t iga te the p o s s i b i l i t y of a s t r u c t u r a l change occur r ing in the s e r i e s over t ime. We estimated a constant rate model fo r the per iod 1947-1961, g iv ing an annual rate of 3.28 per cent . This r e s u l t confirms the hypothesis that c a p i t a l gains p o s s i b i l i t i e s were greater in the s i x t i e s and ear ly s e v e n t i e s , as the average i n f l a t i o n rate over the e n t i r e 1947-1974 per iod was 4.57 per cent . Furthermore, c lose inspect ion - 123 -of the data fo r the 1963-1974 period revealed a marked autoregressive s t ruc tu re in the s e r i e s , which was absent in the e a r l i e r pe r iod . In the end, we s e t t l e d on a ' s p l i c e d 1 fo recas t ing model. For the 1947-1961 p e r i o d , the constant rate of i n f l a t i o n (3.27 per cent) model was used. For 1962-1974, we estimated a f i r s t order autoregressive 22 model using the en t i re data s e r i e s (see Appendix A f o r the parameter estimates of both models) . The corresponding rental p r i c e s e r i e s , RPH4, which was used in actual es t ima t ion , appears a lso in Table 5.13. Of some i n t e r e s t i s the comparison between our imputed rental p r i c e s e r i e s and the o f f i c i a l housing cost s e r i e s . Apart from the housing purchase p r i c e s e r i e s , S t a t i s t i c s Canada publ ishes two other re levant p r i c e s e r i e s , a 'home ownership c o s t 1 index, and a 'tenant rent 23 c o s t 1 index. These two ser ies are l i s t e d f o r the per iod 1952-1974 in Table 5.13, as RPH5 and RPH6, r e s p e c t i v e l y . The former index includes the e f f e c t of the housing purchase p r i c e , the property tax r a t e , and the 24 mortgage r a t e , and i s thus roughly equivalent to our rental p r i c e s e r i e s , RPH1, constructed using the assumption of no expected c a p i t a l ga ins . The tenant cost index, on the other hand, i s S t a t i s t i c s Canada's index of the actual rent paid by tenants over the p e r i o d . The i n t e r e s t i n g point to note is that .over the per iod 1950-1974 the home ownership index exh ib i ts higher o v e r a l l growth than the tenant cost index. Between 1950 and 1974, RPH5 increases by a f a c t o r of 3 .23, compared to an increase of 1.73 f o r RPH6. Thus some element of expected c a p i t a l gains may have been 'passed on 1 to tenants in the form of lower actual ren ts . I f we compare our two imputed indexes, RPH1 and RPH4, they e x h i b i t the same pat te rn . From 1950-1974, RPH1 (the s t a t i c s e r i e s ) - 124 -increases by a f a c t o r of 4.34, while RPH4 (the non s t a t i c s e r i e s ) increased by 2.17. Furthermore, these r a t i o s , based on imputed s e r i e s , are not too fa r o f f the corresponding r a t i o s fo r the actual home owner-25 ship and tenant cost indexes, 3.23 and 1.73, r e s p e c t i v e l y . Thus i t seems reasonable to conclude t e n t a t i v e l y (a) that expected c a p i t a l gains do a f f e c t the actual rents p r e v a i l i n g in the market, and a lso (b) that our method of const ruct ing the imputed se r ies does provide r e s u l t s cons is tent with actual observat ion . Turning to l a n d , a d i f f e r e n t problem arose when we included the ARIMA model estimates of e . in the rental p r i c e formula (3 .10 ' ) . e , nt v x ' nt turned out to be a constant r a t e , equa l l ing 7.34 per cent per year . For a l l of the ear ly y e a r s , up to and inc lud ing 1955, the renta l p r ices of land were negat ive. Th is s i t u a t i o n arose because the expected c a p i t a l gains term was large enough to more than o f f s e t the mortgage rate (which was at a legal c e i l i n g during most of the period) and the property tax payment. Remember, the deprec ia t ion rate on land is zero . However, zero or negative rental p r ices are incons is ten t with an empir ica l consumer model of our type in so fa r as they imply that the consumer w i l l buy an i n f i n i t e amount of the good in quest ion . C l e a r l y , what happens in p r a c t i c e is that there e x i s t i n s t i t u t i o n a l b a r r i e r s such as c r e d i t r a t i o n i n g , r i s k f a c t o r s , imperfect resa le markets, brokerage f e e s , as well as c a p i t a l gains tax which is a p p l i c a b l e in a ( l imi ted) 26 number of cases . A l l of these fac to rs prevent the consumer from taking f u l l advantage of the s i t u a t i o n . Besides data 1 imi ta t ions, the model is not equipped to deal with the problems of the type d iscussed above. Thus we decided to employ a compromise, and assume an expected i n f l a t i o n rate of 4%. While t h i s procedure i s , of course , a r b i t r a r y , the aggregate - 125 -durable goods ser ies may not be a f fec ted very much, due to the small con t r ibu t ion of land (about 9 per cent of the t o t a l ) . The r e s u l t i n g normalised land rental p r i c e ser ies appears in Table 5.11. 4) Aggregation Using the constructed renta l p r ice and stock se r ies fo r the sub components of ' c l o t h i n g 1 , 'other semi durables ' and 'other d u r a b l e s ' , we construct aggregate D i v i s i a indexes in the usual way f o r these ca te -g o r i e s . Together with normalised per cap i ta indexes f o r automobiles, 27 housing and l a n d , these three indexes are l i s t e d in Tables 5.7 and 5.10 (per cap i ta stocks) and Tables 5 .8 and 5.11 (rental p r ices ) where we present the s t a t i c and non s t a t i c cases separa te ly . F i n a l l y , ' c l o t h i n g ' and 'other semi durables ' are combined to give a 'semi durables ' s e r i e s , while housing, l a n d , automobi les, and 'other durables ' are aggregated into the category ' d u r a b l e s ' . Aga in , the s t a t i c and non s t a t i c s e r i e s are presented separate ly in Tables 5.9 and 5.12 r e s p e c t i v e l y . D. Le isure In t h i s s e c t i o n , p r i c e and (per capi ta ) quant i ty indexes are constructed f o r ' l e i s u r e 1 , def ined as the d i f f e r e n c e between the to ta l hours ' a v a i l a b l e ' and the number of hours worked. For our purposes we therefore requ i re : ( i ) an index of hours worked per year ( in order to subtract t h i s from our ' t o t a l hours' es t imate ) ; ( i i ) an index of the opportuni ty cost of l e i s u r e t ime, namely, the marginal wage rate net of income tax. Our method d i f f e r s from that used by Gussman [1972] (Cummings and Meduna [1973] merely update his work using i d e n t i c a l methodology). - 126 -TABLE 5.5 DEPRECIATION RATES, DISCOUNT RATES, AND BENCHMARKS FOR TWELVE DURABLE AND SEMI DURABLE GOODS DISCOUNT 1946 1974 CATEGORY 6 RATE BENCHMARK BENCHMARK Men's and boys' c l o t h i n g .500 RSBY 1,071.807 2,467.203 Women's and c h i l d r e n ' s c l o t h i n g .500 RSBY 1,329.280 4,818.121 Footwear and repa i r .500 RSBY 712.178 1,199.172 Semi durable house-hold fu rn ish ings .400 RCFP 1 ,607.229 3,901.907 Books, newspapers, and magazines .500 RCFP 264.777 1 ,199.049 Jewe l le ry , watches and repa i rs .400 RCFP 0.000 696.257 F u r n i t u r e , carpets and other f l o o r coverings .193 RCFP 1,210.933 4,915.836 Household appl iances .229 RCFP 561.110 4,552.938 Recreat ion , spor t ing and camping equipment .210 RCFP 699.829 7,773.215 Automobiles .280 RMV 391.453 14,151.227 Housing .025 RNHA 11,805.000 50,244.000 Land* .000 RNHA - 18,277.000 * S i n c e the land stock i s c a l c u l a t e d i n d i r e c t l y , ra ther than v ia the perpetual inventory method (see t e x t ) , the 1946 benchmark was not computed. Discount rate s e r i e s : See Table 5.6 - 1 2 7 -TABLE 5.6 DISCOUNT RATES AND PROPERTY TAX RATES YEAR RNHA RMV RSBY RCFP PROPTAXG PROPTAX 1947 .0450 .1300 .0175 .1560 .0599 .0199 1948 .0450 .1300 .0227 .1560 .0564 .0178 1949 .0450 .1300 .0224 .1560 .0566 .0178 1950 .0450 .1300 .0222 .1560 .0573 .0174 1951 .0450 .1300 .0258 .1560 .0618 .0181 1952 .0525 .1300 .0323 .1560 .0643 .0192 1953 .0525 .1500 .0344 .1690 .0653 .0192 1954 .0550 .1485 .0271 .1690 .0674 .0198 1955 .0525 .1530 .0275 .1860 .0678 .0190 1956 .0550 .1580 .0372 .2010 .0714 .0200 1957 .0600 .1580 .0460 .2010 .0746 .0205 1958 , .0600 .1580 .0347 .1860 .0752 .0208 1959 .0606 .1585 .0490 .1860 .0818 .0220 1960 .0675 .1585 .0455 .1860 .0848 .0227 1961 .0673 .1585 .0438 .1965 .0874 .0237 1962 .0650 .1585 .0456 .1985 .0898 .0249 1963 .0635 . 1585 .0448 .1985 .0874 .0250 1964 .0625 .1490 .0471 .1985 .0871 .0247 1965 .0625 .1490 .0488 .1955 .0848 .0237 1966 .0692 .1395 .0555 .1995 .0822 .0239 1967 .0744 .1395 .0561 .2020 .0842 .0249 1968 .0884 .1390 .0668 .2010 .0894 .0258 1969 .0938 .1565 .0762 .2095 .0871 .0263 1970 .1007 .1545 .0721 .2070 .0862 .0263 1971 .0905 .1395 .0558 .2010 .0857 .0251 1972 .0895 .1465 .0626 .2160 . - .0239 1973 .0940 .1504 .0699 .2184 - .0210 1974 .1090 .1550 .0812 .2238 - .0160 Sources f o r RNHA, RMV, RSBY, RCFP, PROPTAXG: 1947-1971: Table l b , Cummings and Meduna [1972]. 1972- : RNHA; Table 75, Canadian Housing S t a t i s t i c s . RMV; 1972 f i g u r e from Canadian Consumer Cred i t Factbook, Table 46 (mean of high and low r a t e s ) . 1973-1974 f igures were der ived by extrapo-l a t i o n (see t e x t ) . RCFP; as f o r RMV. RSBY; 1972-1974; from ser ies B14010, Bank of Canada  Review; ar i thmet ic mean of monthly r a t e s . Source f o r PROPTAX: see tex t . - 128 -TABLE 5.7 PER CAPITA NORMALISED QUANTITY INDEXES, SIX DURABLE AND SEMI DURABLE GOODS (STATIC EXPECTATIONS) OTHER OTHER YEAR CLOTHING SEMI DURABLES DURABLES AUTOS HOUSING LAND 1947 179. 143 124. 160 101. 050 35. 691 149. 858 83. 942 1948 172. 416 128. 026 105. 493 48. 174 155. 112 89. 031 1949 163. 536 128. 182 109. 305 63. 370 157. 808 90. 900 1950 161. 120 133. 873 116. 262 85. 435 165. 679 89. 639 1951 158. 102 132. 871 117. 868 99. 057 170. 043 97. 474 1952 158. 916 132. 406 121. 424 109. 009 173. 033 89. 501 1953 161. 346 134. 472 126. 585 122. 770 179. 813 92. 636 1954 161. 716 134. 935 132. 027 127. 723 187. 584 67. 978 1955 166. 118 138. 305 140. 712 143. 441 199. 073 68. 221 1956 173. 297 142. 564 151. 504 158. 836 210. 086 65. 635 1957 177. 206 142. 397 158. 465 162. 959 216. 248 62. 257 1958 180. 936 142. 389 164. 696 166. 664 227. 120 59. ,618 1959 185. 264 143. 808 171. 884 172. 383 237. 840 60. 843 1960 188. 245 143. 148 177. 735 176. 371 244. 526 65. ,318 1961 190. 251 143. 233 184. 607 180. 583 250. 694 64. ,158 1962 193. 588 143. 126 192. 480 190. 283 256. 564 61. 180 1963 195. 243 143. 346 201. 132 203. 527 262. 274 59. ,898 1964 198. 289 145. 711 211. 499 219. 483 269. 480 61. 801 1965 202. 072 148. 506 223. 036 238. 729 276. 563 66. 790 1966 204. 728 151. 842 235. 803 253. 562 280. 788 64. 281 1967 207. 577 154. 486 247. 040 263. 129 283. 786 66. 768 1968 211. 358 155. 874 258. 776 274. 007 289. 311 65. 711 1969 216. 092 157. 648 271. 640 282. 482 296. 813 62. 885 1970 217. 502 156. 633 280. 932 276. 108 301. 562 66. 151 1971 224. 922 . 159. 077 296. 420 282. 536 310. 531 65.993 1972 238. 326 165. 166 319. 831 296. 056 322. 431 67. 430 1973 253. 784 174. 999 345. 238 314. 815 331. 017 86. 975 1974 269. 538 182. 295 370. 485 327. 624 338. 672 97. 004 - 129 -TABLE 5.8 NORMALISED RENTAL PRICE INDEXES, SIX DURABLE AND SEMI DURABLE GOODS (STATIC EXPECTATIONS) OTHER OTHER YEAR CLOTHING SEMI DURABLES DURABLES AUTOS HOUSING LAND 1947 • 0.6969 0.5752 0.7361 0.6295 0.4810 0.2023 1948 0.8405 0.6547 0.8007 0.6929 0.5503 0.2237 1949 0.8770 0.7001 0.8293 0.7374 0.5730 0.2321 1950 0.9027 0.7112 0.8495 0.7687 0.5990 0.2578 1951 0.9718 0.8129 0.9548 0.8712 0.7323 0.3047 1952 0.9874 0.8302 0.9776 0.8933 0.7733 0.3616 1953 0.9796 0.8468 0.9968 0.9113 0.7798 0.3661 1954 0.9624 0.8499 0.9733 0.9054 0.7963 0.5371 1955 0.9519 0.8719 0.9842 0.8519 0.7866 0.5546 1956 0.9648 0.9137 1.0011 0.8761 0.8529 0.6866 1957 0.9707 0.9447 1.0109 0.9520 0.8926 0.7716 1958 0.9705 0.9686 1.0105 0.9628 0.8923 0.8467 1959 0.9821 0.9749 1.0081 0.9994 0.9068 0.8877 1960 0.9893 0.9831 0.9974 1.0123 , ^0.9842 0.9410 1961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1962 1.0113 1.0311 1.0049 0.9902 0.9900 1.0599 1963 1.0383 1.0586 1.0086 0.9956 1.0020 1.1162 1964 1.0658 1.0703 1.0128 0.9623 1.0305 1.1393 1965 1.0854 1.0849 1.0121 0.9617 1.0796 1.1305 1966 1.1322 1.1273 1.0387 0.9484 1.2152 1.3650 1967 1.1866 1.1833 1.0837 0.9760 1.3480 1.4930 1968 1.2294 1.2390 1.1044 1.0052 1.5075 1.7759 1969 1.2727 1.2918 1.1376 1.0501 1.6372 2.0861 1970 1.2876 1.3419 1.1476 1.0956 1.7593 2.1851 1971 1.2857 1.3783 1.1422 1.0886 1.7407 2.1963 1972 1.3074 1.4673 1.1783 1.1298 1.7937 2.2914 1973 1.3762 1.5332 1.2131 1.1516 2.0581 2.2113 1974 1.5089 1.7564 1.3294 1.2653 2.6042 2.4665 - 130 -TABLE 5.9 PRICE AND QUANTITY INDEXES, AGGREGATE DURABLES AND SEMI DURABLES (STATIC EXPECTATIONS) PRICE INDEXES QUANTITY INDEXES YEAR SEMI DURABLES DURABLES SEMI DURABLES DURABLES 1947 0.6409 0.5460 306.235 340.495 1948 0.7568 0.6074 302.250 367.305 1949 0.7970 0.6334 292.567 392.956 1950 0.8161 0.6603 294.881 434.325 1951 0.9001 0.7690 290.694 459.139 1952 0.9164 0.8062 291.151 474.149 1953 0.9198 0.8185 295.641 503.591 1954 0.9118 0.8437 296.642 509.718 1955 0.9161 0.8315 304.247, 547.360 1956 0.9421 0.8674 315.745 584.261 1957 0.9593 0.9281 319.559 599.502 1958 0.9696 0.9383 323.306 618.164 1959 0.9790 0.9568 329.058 643.039 1960 0.9866 0.9911 331.390 663.875 1961 1.0000 1.0000 333.484 680.042 1962 1.0198 1.0005 336.700 700.356 1963 1.0469 1.0121 338.563 726.345 1964 1.0677 1.0159 343.976 761.416 1965 1.0853 1.0319 350.553 803.730 1966 1.1302 1.1004 356.541 830.928 1967 1.1853 1.1761 362.034 855.995 1968 1.2336 1.2654 367.200 880.396 1969 1.2809 1.3543 373.696 902.911 1970 1.3105 1.4184 374.058 916.484 1971 1.3246 1.4093 383.826 944.817 1972 1.3741 1.4570 403.126 991.185 1973 1.4417 1.5528 428.335 . 1 ,065'. 784. 1974 1.6115 1.8132 451.084 1 ,118.635 - 131 -TABLE 5.10 PER CAPITA NORMALISED QUANTITY INDEXES, SIX DURABLE AND SEMI DURABLE GOODS (NON STATIC EXPECTATIONS) OTHER OTHER YEAR CLOTHING SEMI DURABLES DURABLES AUTOS HOUSING LAND 1947 175.773 119.407 98.621 34. 350 109. 061 47 .689 1948 169.473 123.556 102.977 46. 364 112. 884 50 .580 1949 160.819 123.844 106.911 60. 989 114. 846 51 .642 1950 158.471 129.395 113.917 82. 224 120. 575 51 .328 1951 155.566 128.447 115.524 95. 334 123. 750 55 .377 1952 156.400 127.982 119.016 104. 913 125. 926 50 .848 1953 158.811 129.973 124.104 118. 156 130. 860 52 .629 1954 159.198 130.421 129.476 122. 923 136. 516 38 .620 1955 163.554 133.679 138.002 138. 051 144. 877 38 .758 1956 170.629 137.798 148.600 152. 867 152. 891 37 .289 1957 174.483 137.635 155.446 156. 834 157. 376 35 .370 1958 178.154 137.629 161.543 160. 400 165. 289 33 .871 1959 182.425 139.000 168.568 165. 904 173. 090 34 .566 1960 185.369 138.364 174.284 169. 743 177. 956 37 .109 1961 187.354 138.450 181.019 173. 796 182. 445 36 .450 1962 190.648 138.348 188.764 183. 131 186. 716 34 .758 1963 192.283 138.561 197.284 195. 878 190. 872 34 .030 1964 195.290 140.847 207.507 211. 234 196. 116 35 .111 1965 199.018 143.549 218.878 229. 757 201. 271 37 .945 1966 201.636 146.773 231.467 244. 033 204. 346 36 .520 1967 204.455 149.327 242.539 253. 240 206. 528 37 .933 1968 208.201 150.669 254.106 263. 709 210. 548 37, .332 1969 212.894 152.386 266.798 271. 866 216. 008 35 .726 1970 214.306 151.408 275.968 265. 731 219. 464 37 .584 1971 221.627 153.772 298.224 271. 917 225. 991 37 .492 1972 234.838 159.660 314.288 284. 930 234. 652 38 .308 1973 250.077 169.163 339.354 302. 983 240. 901 46, .869 1974 265.611 176.216 364.303 315. 311 246. 472 55, .110 - 132 -TABLE 5.11 NORMALISED RENTAL PRICE INDEXES, SIX DURABLE AND SEMI DURABLE GOODS (NON STATIC EXPECTATIONS) OTHER OTHER YEAR CLOTHING SEMI DURABLES DURABLES AUTOS HOUSING LAND 1947 0.6264 0.4942 0.6860 0.6233 0.4277 0.1397 1948 0.7897 0.6234 0.7358 0.6454 0.4838 0.1462 1949 0.8558 0.6836 0.8267 0.7281 0.5037 0.1515 1950 0.8499 0.6998 0.7864 0.7174 0.5249 0.1654 1951 0.9643 0.8165 0.8754 0.8111 0.6633 0.2252 1952 0.9930 0.8323 0.9859 0.9025 0.7145 0.2863 1953 0.9777 0.8474 0.9632 0.8851 0.7207 0.2897 1954 0.9638 0.8486 1.0034 0.9497 0.7469 0.4468 1955 0.9555 0.8712 0.9840 0.9017 0.7257 0.4374 1956 0.9704 0.9170 1.0090 0.8713 0.7756 0.5397 1957 0.9749 0.9493 0.9915 0.9387 0.8591 0.6934 1958 0.9656 0.9710 1.0012 0.9524 0.8592 0.7635 1959 0.9849 0.9755 0.9991 0.9517 0.8803 0.8178 1960 0.9801 0.9829 1.0167 1.0484 0.9813 0.9340 1961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1962 1.0014 1.0331 1.0090 1.0325 0.9875 1.0504 1963 1.0312 1.0615 1.0008 0.9916 0.9900 1.0928 1964 1.0579 1.0720 1.0046 1.0104 0.9202 1.0102 1965 1.0779 1.0862 1.0050 0.9452 0.8628 1.0818 1966 1.1158 1.1305 1.0227 0.9932 0.9613 1.3880 1967 1.1676 1.1891 1.0502 0.9369 1.1596 1.5889 1968 1.2112 1.2455 1.0845 1.0290 1.6808 2.0525 1969 1.2530 1.2981 1.1134 1.0134 1.6035 2.4739 1970 1.2829 1.3479 1.1378 1.0957 1.8579 2.6593 1971 1.2792 1.3829 1.1341 1.0788 1.5673 2.5542 1972 1.3076 1.4743 1.1673 1.1196 1.7132 2.6363 1973 1.3525 1.5433 1.1877 1.1609 1.2132 2.5610 1974 1.4675 1.7832 1.2409 1.1860 1.1409 2.9685 - 133 -TABLE 5.12 PRICE AND QUANTITY INDEXES, AGGREGATE DURABLES AND SEMI DURABLES (NON STATIC EXPECTATIONS) PRICE INDEXES. QUANTITY INDEXES YEAR SEMI DURABLES DURABLES SEMI DURABLES DURABLES 1947 0.5675 0.5350 298.002 266.120 1948 0.7156 0.5799 294.664 289.210 1949 0.7789 0.6348 285.386 312.665 1950 0.7828 0.6287 287.738 350.492 1951 0.8983 0.7357 283.747 371.064 1952 0.9212 0.8196 284.222 386.244 1953 0.9195 0.8102 288.622 411.828 1954 0.9124 0.8664 289.445 - 422.307 1955 0.9181 0.8386 297.074 455.961 1956 0.9468 0.8607 308.330 490.084 1957 0.9637 0.9136 312.094 504.658 1958 0.9680 0.9256 315.776 521.136 1959 0.9808 0.9350 321.421 542.205 1960 0.9813 1.0010 323.735 559.028 1961 1.0000 1.0000 325.805 573.709 1962 1.0148 1.0019 328.973 593.371 1963 1.0440 0.9997 330.804 618.014 1964 1.0640 0.9861 336.098 650.070 1965 1.0815 0.9467 342.527 688.591 1966 1.2221 1.0131 348.377 716.818 1967 1.1768 1.0690 353.754 740.622 1968 1.2258 1.2797 358.823 763.816 1969 1.2721 1.2826 365.201 785.370 1970 1.3103 1.3976 365.585 796.265 1971 1.3225 1.3058 375.163 822.175 1972 1.3766 1.3747 .394.056 864.830 1973 1.4321 1.2585 418.704 927.341 1974 1.5967 1.2895 440.921 985.494 - 134 -TABLE 5.13 ALTERNATIVE RENTAL PRICE SERIES FOR HOUSING YEAR RPHl RPH2 RPH3 RPH4 RPH5 RPH6 1947 0.4810 0.3046 0.3862 0.4277 _ _ 1948 0.5503 0.0129 0.4313 0.4838 - -1949 0.5730 0.5917 0.4489 0.5037 - -1950 0.5990 0.2585 0.4673 0.5249 0.7028 0.7551 1951 0.7323 0.0605 0.6098 0.6633 0.7762 0.7990 1952 0.7733 0.8842 0.6678 0.7145 0.8094 0.8437 1953 0.7798 0.5917 0.6733 0.7207 0.8223 0.8751 1954 0.7963 0.8418 0.7073 0.7469 0.8313 0.9058 1955 0.7866 0.5748 0.6779 0.7257 0.8440 0.9302 1956 0.8259 0.7603 0.7349 0.7756 0.8711 0.9462 1957 0.8926 0.7116 0.8313 0.8591 0.8874 0.9630 1958 0.8923 0.9390 0.8327 0.8592 0.9199 0.9805 1959 0.9068 0.8486 0.8573 0.8803 0.9539 0.9909 1960 0.9842 0.8930 0.9797 0.9813 0.9837 0.9965 1961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1962 0.9900 1.0067 0.9823 0.9875 1.0285 1.0028 1963 1.0020 0.8598 0.9852 0.9900 1.0590 1.0063 1964 1.0305 0.8519 1.0004 0.9202 1.1045 1.0119 1965 1.0796 0.7955 1.0459 0.8628 1.1499 1.0188 1966 1.2152 0.9348 1.2305 0.9613 1.2008 1.0363 1967 1.3480 1.1403 1.4130 1.1596 1.2687 1.0712 1968 1.5075 1.6581 1.6955 1.6808 1.3610 1.1180 1969 1.6372 1.4366 1.8857 1.6035 1.4830 1.1630 1970 1.7593 1.8644 2.0762 1.8579 1.6130 1.2030 1971 1.7407 1.4429 1.9704 1.5673 1.7430 1.2250 1972 1.7937 1.7420 2.0127 1.7132 1.8830 1.2430 1973 2.0581 1.0497 2.3302 1.2132 2.0700 1.2640 1974 2.6042 1.4249 2.8675 1.1409 2.2700 1.3010 RPHl; s t a t i c expecta t ions . RPH2; adaptive expectat ions model I. RPH3; constant rate of i n f l a t i o n model. RPH4; ' s p l i c e d ' expectat ions model (ser ies used in est imation - see text RPH5; home ownership cost index. RPH6; tenant cost index. Source f o r RPH5, RPH6; Pr ices and Pr ice Indexes (62-002). - 135 -Gussman [1972; 46-47] m u l t i p l i e s average hours worked by employment (both unpublished s e r i e s ) to obtain to ta l annual hours worked. This l a t t e r f i g u r e he d iv ides into an estimate of the to ta l wage b i l l in order to der ive an average gross wage ra te . This ser ies is then converted into a net wage rate se r ies using an average tax rate (equal to the r a t i o of to ta l d i r e c t taxes minus succession dut ies and estate taxes to to ta l personal income). Gussman's procedure, by using to ta l annual hours and to ta l employment does not al low f o r d i f f e r e n t kinds of work. More important, however, as Gussman r e a d i l y admits, h is use of an average tax rate s e r i e s ignores the p r o g r e s s i v i t y in the tax schedule due to exemptions and changes in the marginal rates a p p l i c a b l e at d i f f e r e n t income l e v e l s . We make some attempt to come to gr ips with these d i f f i c u l t i e s . Diewert [1975] l i s t s (Table 6) estimates of annual average gross wage Of rates f o r twelve d i f f e r e n t occupations in Canada fo r the per iod 1947-1974. This paper a lso contains (Table 2 ) , fo r the same twelve occupat ions, data on the average annual labour earnings (these d i f f e r considerably across occupat ions , n a t u r a l l y ) as well as t o t a l annual hours worked (Table 23). Our procedure i s as f o l l o w s . F i r s t , fo r each occupation in each y e a r , we obtain an estimate of the a p p l i c a b l e marginal income tax ra te . Taxat ion S t a t i s t i c s , publ ished annual ly by the Department of National Revenue, l i s t s fo r each tax bracket the to ta l tax payable on taxable 29 income. These tax brackets r e f e r to t o t a l income from a l l sources , however. In order to determine the bracket into which the representat ive worker 's labour earnings would f a l l , we adjusted the upper and lower end of each tax bracket downwards by the r a t i o of labour income to to ta l 30 income recorded. If a worker's average labour earnings f a l l into a p a r t i c u l a r downward adjusted tax bracket , the re levant tax rate fo r him - 136 -i s computed as the r a t i o of to ta l tax paid to taxable income f o r the corresponding unadjusted tax bracket . One fur ther adjustment was made. The computed tax rate was m u l t i p l i e d by the r a t i o of the to ta l income reported from taxable returns to to ta l income ( inc lud ing income of non taxable returns) f o r that bracket . In t h i s manner, a representa t ive tax rate was der ived f o r each 31 32 occupation in each year . Note that our rate i s s t i l l some way away from a representat ive marginal r a t e . However, two of the main biases of Gussman's approach have been removed. F i r s t , we have not ignored exemp-t ions (probably, fo r many people, the major source of divergence between t h e i r average and marginal tax r a t e s ) . Second, the existence of non taxable returns has been taken into account. Furthermore, we are in e f f e c t assuming that there are twelve d i f f e r e n t workers (types of work), along with twelve corresponding tax r a t e s , which is some improvement over the homogeneous worker and uniform tax rate assumption implied by Gussman's method. A net wage s e r i e s was der ived fo r each occupation by mu l t ip ly ing the gross wage rate f o r that occupation by (one minus) the corresponding tax rate as computed above. Then, using the manhours by occupation s e r i e s (Diewert [1975; Table 23] ) , together with the net wage rate s e r i e s , we constructed a D i v i s i a index of the aggregate net wage r a t e , NW, and aggregate manhours, MNN (as before , t h i s was der ived r e s i d u a l l y ) . To convert the manhours se r ies to a l e i s u r e s e r i e s , we assume, along with Gussman, that the average worker would probably not work more 33 than 60 hours per week fo r 52 weeks. Le isure per c a p i t a , LPC, is then def ined as: (5.21) LPC = 3120 - (MNN/population) - 137 -TABLE 5.14 LABOUR: NET WAGES, HOURS, LEISURE, AND TAX RATES YEAR NW MNN LPC TAX PTAX 1947 0.4175 17117.08 1220.76 .1725 .0818 1948 0.4756 17657.92 1189.05 .1469 .0763 1949 0.5367 17615.38 1267.48 .0948 .0698 1950 0.5590 17574.68 1297.18 .1150 .0620 1951 0.6023 18167.34 1258.34 .1430 .0748 1952 0.6312 18348.89 1286.27 .1737 .0846 1953 0.6719 18557.58 1303.64 .1703 .0884 1954 0.7067 18197.89 1378.96 .1616 .0894 1955 0.7409 18396.24 1394.13 .1533 .0839 1956 0.7977 19290.21 1343.00 .1475 .0868 1957 0.8475 19524.68 1369.44 .1481 .0907 1958 0.8927 19228.73 1432.50 .1347 .0806 1959 0.9210 19710.94 1424.48 .1426 .0843 1960 0.9561 19756.50 1451.38 .1481 .0911 1961 1.0000 19666.86 1487.41 .1507 .0951 1962 1.0213 20453.55 1453.49 .1515 .0943 1963 1.0562 20947.80 1445.93 .1558 .0942 1964 1.0933 21775.14 1417.71 .1649 .1020 1965 1.1776 22631.18 1390.81 .1508 .1039 1966 1.2792 23364.56 1379.40 .1527 .1224 1967 1.3672 23868.22 1391.86 .1628 .1353 1968 1.4559 23950.34 1430.80 .1763 .1451 1969 1.5298 24676.91 1423.11 .2032 .1597 1970 1.6420 24740.31 1460.82 .2065 .1686 1971 1.7830 25036.97 1471.47 .2056 .1723 1972 1.8592 25617.50 1460.65 .2329 -1973 2.0849 26306.64 1466.59 .2129 -1974 2.3786 26788.50 1481.46 .2128 -NW; aggregate net wage NMM; aggregate man hours. LPC; . l e i s u r e per c a p i t a . TAX; impl ied tax ra te . PTAX; Gussman's tax r a t e ; taken from Cummings and Meduna [1973, Table l b ] . - 138 -NW, MNN, and LPC are l i s t e d in the f i r s t three columns of Table 5.14. Of some i n t e r e s t is whether the aggregate tax rate impl ied by our procedure, TAX, d i f f e r s from PTAX, that derived by Gussman (described above). To obtain TAX, we f i r s t of a l l need the aggregate gross wage r a t e , GW, corresponding to the index MNN. We der ive GW by d i v i d i n g MNN • 1 2 into the current gross d o l l a r wages b i l l , ^E^ GW^  MN.., where GW^  and MN.j are the gross wage rate and manhours worked indexes, r e s p e c t i v e l y , fo r occupation i . The i m p l i c i t aggregate tax r a t e , TAX, is then def ined as: (5.22) TAX = 1 - (NW/GW) TAX and PTAX (Cummings and Meduna, Table lb) appear in columns 4 and 5 of Table 5.14. As one would expect , the estimate of the marginal tax rate from our procedure, TAX, is c o n s i s t e n t l y higher than PTAX, the average rate constructed by Gussman. E. Money Our ob jec t ive in t h i s and the fo l lowing sec t ion i s to construct rental p r ice and quant i ty s e r i e s fo r the serv ices of money and money subst i tu tes held by households. In t h i s part we consider only l i a b i l i t i e s of chartered banks ( 'money' ) , and, in the next s e c t i o n , deal with the l i a b i l i t i e s of t r u s t and loan companies and Canada Savings Bonds. In Chapter 3 (sect ion C ) , we d iscussed the treatment of money as a durable good wi th in our model. The serv ices of money are assumed to be proport ional to the real value of money, i . e . to M.^/p^., where M^ t i s the d o l l a r value of the stock of money form i h e l d , and p t i s ' the ' aggregate p r i c e l e v e l , in per iod t . As was explained in Chapter 3, - 139 -complex index number problems a r i s e on account of the fac t that p^ i s , in genera l , a funct ion of ( x ^ , . . . , x t ) , the vector of real goods in the model. In order to circumvent t h i s problem, we decided to use as a d e f l a t o r , an optimal fo recas t f o r p* t, p^, made at per iod t - 1 . Thus, fo r the purposes of the model, we assume the serv ices of money form i to be proport ional to rtK = M . ^ / p t , where m^  i s the ' rea l va lue 1 of money M.. In Chapter 3, the rental p r ices for monetary serv ices were derived a lso as equations (3.32) and (3 .33) , repeated here fo r con-venience, Pmit = T T \ (3.32) P t ( R t " r i t } Pmjt = l \ ( 3 - 3 3 ) (3.32) re fe rs to non i n t e r e s t bearing money, = ^ . / p ^ , while (3.33) i s a p p l i c a b l e to the case of money m. = M^'/p., earning i n t e r e s t at a rate r , on the d o l l a r va lue , M.. J J Our procedure i s f i r s t , to c a l c u l a t e the average stock of real money balances held in d i f f e r e n t forms, and second, to construct the corresponding rental p r i c e s e r i e s based on (3.32) and (3.33) . However, before commencing t h i s task , some d i s c u s s i o n on the ownership of chartered bank deposi ts is requ i red . 1) Ownership of Chartered Bank Deposits Our ob jec t ive i s to obtain estimates of the average annual amount of each form of money held by households, together with the corresponding i n t e r e s t r a t e s . - 140 -Unfor tunate ly , no r e l i a b l e d i r e c t information on household money holdings is a v a i l a b l e as yet f o r Canada. Since 1967, F inanc ia l Flows (13-002) has publ ished quar te r ly estimates of the changes in f i n a n c i a l assets and l i a b i l i t i e s by sec to r . P r i o r estimates reaching back to 1962 are a v a i l a b l e in F inanc ia l Flow Accounts, 1Q1962-4Q1973, recent ly publ ished by S t a t i s t i c s Canada. Furthermore, stock f igures are now provided in recent issues of F inanc ia l Flows and these could presumably be l inked with the flow data to obtain stock estimates from 1962 onwards. However, f o r a number of reasons, t h i s data source i s unsui table f o r our purposes. F i r s t , data fo r the household sector are not l i s t e d separa te ly , but are included with unincorporated bus iness , and there • 34 appears to be no method whereby these sectors may be d isentangled . . Also included are corporat ions engaged in a g r i c u l t u r e , f i s h i n g , and t rapp ing . Second, even f o r t h i s j o i n t s e c t o r , the estimates are der ived r e s i d u a l l y , 35 using estimates of corporate holdings of f i n a n c i a l a s s e t s . Not s u r p r i -s i n g l y , perhaps, the estimates are f requent ly rev ised and r e - r e v i s e d f o r some years afterwards often by very considerable amounts. T h i r d , only one f i g u r e i s reported f o r the aggregate of currency and demand and savings deposi ts at chartered banks. Construct ing disaggregated s e r i e s , however, i s an important part of our ob jec t ive in t h i s study. F i n a l l y , as already mentioned, the data e x i s t only as f a r back as 1962, and there appears t o p -be no meaningful way of extending the s e r i e s back any f u r t h e r . For these reason's i t was decided to adopt an a l t e r n a t i v e approach. We a l l o c a t e d the d i f f e r e n t types of deposi ts to d i f f e r e n t ownership categor ies by examining a l l the a v a i l a b l e i n s t i t u t i o n a l evidence. S p e c i f i c a l l y , personal savings deposi ts and personal chequing accounts are assigned to the household s e c t o r , while demand deposi ts (defined to exclude personal chequing accounts ) , non personal term and not ice deposi ts and - 141 -swapped deposi ts are assumed not to be held by the household s e c t o r . Obviously excluded from the household sector are government deposi ts ( federal and p r o v i n c i a l ) and deposi ts of other banks. The treatment of currency ( i . e . , notes and coin outstanding) w i l l be explained below. These assumptions are reasonable,based on our present knowledge of the i n s t i t u t i o n a l c h a r a c t e r i s t i c s of chartered bank l i a b i l i t i e s . In the document, Ins t ruct ions with respect to the Return of Assets and  L i a b i 1 i t i e s (Pursuant to sect ion 103 of the Bank Act) chartered banks are ins t ruc ted (p. 8) to s p e c i f i c a l l y exclude from the category of personal savings deposi ts " . . . deposi t accounts of (e) i n d i v i d u a l s i f i t i s known that the funds belong to other than those l i s t e d above [ i . e . , i n d i v i d u a l s , estates of i n d i v i d u a l s , or t rustees act ing f o r them or t h e i r e s t a t e ] ; (f) f i r m s , business par tne rsh ips , and personal corpo-r a t i o n s ; (g) pension funds; and (h) r e l i g i o u s , c h a r i t a b l e , f r a t e r n a l , labour , r e c r e a t i o n a l , educational and welfare o r g a n i s a t i o n s , i n s t i t u t i o n s and c o r p o r a t i o n s . " Thus the personal savings deposi t concept seems to approximate qui te well the holdings of the household s e c t o r . Since May 1, 1967, personal savings deposi ts data are d iv ided in to three c a t e g o r i e s , chequable, non chequable, and f i x e d term. Personal chequing accounts were introduced by the chartered banks in 1957 to cater s p e c i f i c a l l y to i n d i v i d u a l s who d id not wish to use the regular current account but who wished to have more economical chequing f a c i l i t i e s than were a v a i l a b l e with t h e i r regular chequable personal savings deposi t (see Royal Commission on Banking and Finance Report [1964; 118], Ga lbra i th [1970; 83] ) . Personal chequing accounts a r e , by d e f i n i t i o n , held by the personal s e c t o r . - 142 -Turning to the categor ies not a l loca ted to the personal s e c t o r , demand deposi ts other than personal chequing accounts are ' e s s e n t i a l l y current accounts held by businesses ' (Binhammer [1968; 116]). See a lso Ga lbra i th [1970; 83] , Bond and Shearer [1972; 173] f o r s i m i l a r d e s c r i p t i o n s . Demand deposi ts are held in general by "spending un i ts which r e g u l a r l y make a large volume of payments" (Bond and Shearer [1972; 173]) and some ind iv idua l households may f a l l into t h i s category. However, the evidence we have c i t e d above ind ica tes that t h e i r numbers w i l l be s m a l l . It is not iceab le that there was no sudden d e c l i n e in the rate of growth of demand deposi ts a f t e r the in t roduct ion of personal chequing accounts in 1957, which tends to ind ica te that the growth of th is deposi t type was due e i t h e r to a s h i f t away from chequable savings deposi ts or to increases in the s i z e of the to ta l l i q u i d asset, .por t fo l io h e l d . Non personal term and not ice deposi ts are excluded from the household sector by d e f i n i t i o n . Swapped deposi ts ( i . e . , chartered bank deposi ts held in U.S. d o l l a r form) are a lso assumed not to be held by households as the t r a d i t i o n a l users of t h i s f a c i l i t y are large corpora-t ions and investment houses (see Bond and Shearer [1972; 180] and Ga lbra i th [1970; 303]). Although the above assumptions regarding ownership are only working approximations, they represent , we f e e l , some improvement over the more t r a d i t i o n a l approach of aggregating a l l forms of money without regard to ownership. P a r t i c u l a r l y i f one wishes to model the demand for monetary se rv ices wi th in a s p e c i f i c t h e o r e t i c a l model, then the owner-ship quest ion becomes extremely important. The assumptions which we have made in order to der ive household money holdings fo r use wi thin such a t h e o r e t i c a l model, a r e , we b e l i e v e , supported s t rongly by the - 143 -a v a i l a b l e i n s t i t u t i o n a l evidence. 2) Average Stocks of Money Balances In t h i s s e c t i o n , we construct data se r ies fo r the average y e a r l y household holdings of d i f f e r e n t deposi t types. ( i ) Personal Savings Deposi ts . We consider f i r s t the per iod 1947-1967. Month end quant i ty f igures are tabulated in Canadian Currency and Chartered Bank Depos i ts , 1926 to date , publ ished by the Bank of Canada (hereaf ter re fe r red to as Currency and Depos i ts ) . Two minor adjustments were made to t h i s s e r i e s . F i r s t , in September 1957, the d e f i n i t i o n of 'personal savings d e p o s i t s ' was changed to exclude ce r ta in business and i n s t i t u t i o n a l d e p o s i t s , which were subsequently t reated as non personal 37 term and not ice d e p o s i t s . The amount in question as of September 1, 1957 was $140 m i l l i o n amounting to 2.232 per cent of to ta l personal savings deposi ts outstanding at that date. In an attempt to preserve c o n s i s t e n c y , we therefore adjusted the s e r i e s p r i o r to September 1957 downwards by the f a c t o r .9768. The second adjustment concerns the treatment of ' f l o a t 1 . The problem a r i s e s because cheques wr i t ten on an i n d i v i d u a l ' s account may be c red i ted to another deposi t before being debited from the drawer's account. An element of double counting w i l l therefore be present . Total ' f l o a t ' ( i . e . , the value of net items in t r a n s i t ) f igures at month end are a lso l i s t e d in Currency and Depos i ts . However, t h i s f igure does not d i s t i n g u i s h between cheques drawn on demand and savings accounts. To make the necessary adjustment we u t i l i s e d monthly f igures fo r the to ta l value of cheques wr i t ten by type of account which are reported in Cheques  Cashed in C lear ing Centres (11-002). For each month, the value of cheques wr i t ten on personal savings deposi ts was ca lcu la ted as a proport ion - 144 -of the to ta l value of cheques wr i t t en . The to ta l f l o a t f i g u r e was then m u l t i p l i e d by t h i s proport ion and the r e s u l t i n g se r ies subtracted from the savings deposi t se r ies to obtain estimates of personal savings deposi ts net of f l o a t . To der ive annual average f igures we proceeded in two stages. F i r s t , an average f o r each month was computed as an ar i thmet ic average of month end f igures f o r the month in question and the immediately preceding month. Second, these monthly averages were summed and d iv ided by twelve to obtain an annual average. For the per iod 1968-1974, personal savings deposi ts are d iv ided into three ca tegor ies : ( i ) chequable, ( i i ) non chequable, and ( i i i ) f i xed term. While t h i s breakdown i s not given in Currency and Depos i ts , the three se r ies are a v a i l a b l e in the Bank of Canada Review as se r ies B 452, B 453 and B 454, r e s p e c t i v e l y . Note, however, that these data r e f e r to, the l a s t Wednesday of the month, and thus are .not^jbrdet ly. comparable to the calendar month end f i g u r e s . To adjust fo r th is we m u l t i p l i e d each of the three s e r i e s in the Bank of Canada Review by the r a t i o of the month end f igures f o r to ta l personal savings deposi ts ( i n c l u s i v e of f l o a t ) given in Currency and Deposits to the sum of the three se r ies B 452, B 453 and B 454. Then we subtracted the f l o a t adjustment f igure from the chequable savings deposi t s e r i e s . Monthly and y e a r l y averages f o r the disaggregated savings deposi t types were constructed in a manner s i m i l a r to that 39 descr ibed above f o r 1947-1967. ( i i ) Personal Chequing Accounts. Month end f igures (ser ies BS 676) were made a v a i l a b l e to us by the Bank of Canada. A s i m i l a r f l o a t adjustment was employed here a l s o , using the monthly data on the value of cheques wr i t ten against personal chequing accounts , a lso l i s t e d in - 145 -Cheques Cashed in C lear ing Centres, While the proport ion of to ta l f l o a t a t t r i b u t a b l e to t h i s type of account i s extremely small (varying between .000 and .022 over the p e r i o d ) , the absolute s i z e r e l a t i v e to the value of personal chequing accounts i s qui te s i g n i f i c a n t . We then computed a monthly and year ly average s e r i e s as before. ( i i i ) Currency. Unfor tunate ly , no d i r e c t information is a v a i l a b l e as regards the proport ion of currency outstanding held by households. We therefore proceed as f o l l o w s . F i r s t , annual average f igures fo r to ta l cur rency , C , were constructed in the usual way, using the month end data in Currency and Deposi ts . From the same p u b l i c a t i o n , annual average f igures f o r demand deposi ts ( f i r s t adjusted fo r f l o a t ) and non personal term and not ice deposi ts were der ived . A currency adjustment f a c t o r , CADJ, was obtained by assuming that the proport ion of currency held by households equal led the proport ion of household to to ta l (pr ivate sector ) chartered bank d e p o s i t s , i . e . , (personal savings deposi ts ( to ta l ) + -.QJ _ personal chequing accounts)  (personal savings deposi ts + personal chequing accounts + demand deposi ts + non personal term and not ice deposi ts ) Household holdings of currency are estimated as CADJ times C ( tota l currency outs tanding) . CADJ remained f a i r l y constant over the post war per iod ranging between .58 and .66. Average y e a r l y f igures f o r to ta l personal savings d e p o s i t s , chequable, non chequable, and f i x e d term savings d e p o s i t s , personal chequing accounts , and currency are l i s t e d in Table 5.15. 3) Rental Pr ices For each type of bank deposi t and currency , we apply the formula - 146 -(3.32) or (3 .33) . p t , the general p r ice l e v e l , i s a D i v i s i a index of the p r i c e indexes fo r non durable consumption and s e r v i c e s , durable and semi 40 durable consumption (using non s t a t i c renta l p r i c e s ) , which have been constructed already (see Tables 5.4 and 5.12). We chose R t , the general 41 discount r a t e , as the average y i e l d on ten i n d u s t r i a l bonds, obtained from Selected Canadian and Internat ional In terest Rates Including Bond  Y ie lds and Interest A r b i t r a g e , publ ished by the Bank of Canada (hereafter re fe r red to as Interest Rates) . The annual rate i s an ar i thmet ic average of the (average) monthly r a t e s . To construct a se r ies fo r p^, the expected leve l of p t , we used an ARIMA model, and transformed the se r ies by taking logar i thms. In te res t ing ly enough, the optimal forecast turned out to be a constant ra te of i n f l a t i o n over the per iod equa l l ing 3.67 per cent per annum. Since the forecasts from the model were in the form ( log^p^) , we untrans-formed the se r ies to obtain p^., using a method der ived by Nelson [1973] and descr ibed in Appendix A. The parameter estimates of the constant rate model are l i s t e d in Appendix A a l s o . Table 5.16 contains the se r ies f o r R, p , and p . The next stage is to construct average nominal i n t e r e s t rates f o r each of the money forms. For currency and personal chequing accounts , 42 of course , the rate i s always zero . In the per iod 1947-1967, the monthly i n t e r e s t rate on personal savings deposi ts was provided by the Bank of Canada. This rate changed r e l a t i v e l y in f requent ly . The same ser ies is a p p l i c a b l e a lso to the chequable savings deposi t category from 1968 onwards. The ra te on non chequable deposi ts (1968-1974) i s a v a i l a b l e in Interest Rates. For f i xed term savings deposi ts no d i r e c t l y app l i cab le rate is pub l ished . However, these deposi ts a r e , apart from ownership, - 147 -very s i m i l a r in c h a r a c t e r i s t i c s to non personal term and not ice d e p o s i t s , and thus we used the rate appropr iate fo r the l a t t e r , i . e . the chartered bank 90 day deposi t r a t e , a lso l i s t e d in Interest Rates. Annual average i n t e r e s t rates fo r each asset were constructed as a weighted sum of monthly r a t e s , the weights being equal to the average month's holding of the asset (using the average monthly data constructed already) d iv ided by (the average y e a r l y holdings times twelve) . The r e s u l t i n g year ly i n t e r e s t rate s e r i e s may be found in Table 5.17. F i n a l l y , we constructed rental pr ices f o r a l l chartered bank l i a b i l i t i e s and currency using the formulae ( 3 . 3 2 ) and ( 3 . 3 3 ) . Note that the rental p r ices of currency and personal chequing accounts from 1959 onwards are i d e n t i c a l . Table 5.18 contains these s e r i e s . 4) Aggregation and Conversion to Real Balances F i r s t , each of the d o l l a r s e r i e s in Table 5.15 was d iv ided by ( p t times populat ion) to obtain per cap i ta quant i ty se r ies of monetary serv ices f o r each l i a b i l i t y . These s e r i e s are l i s t e d in Table 5.19. Then, using the corresponding rental p r i c e s e r i e s in Table 5.18 we aggregate, using a D i v i s i a procedure, in two s tages . F i r s t , personal savings d e p o s i t s , chequable, non chequable, and f i xed term deposi ts are aggregated to provide a normalised p r i c e index fo r chartered bank i n t e r e s t bearing l i a b i l i t i e s and a corresponding quant i ty index (derived r e s i d u a l l y ) . Personal chequing accounts and currency are a lso aggregated in an i d e n t i c a l manner. Second, these two subaggregates are fur ther aggregated to obtain a p r i c e and quant i ty index f o r money s e r v i c e s . This index, QM, corresponds to the usual d e f i n i t i o n of 'M2' (held by households) . However, our method of const ruct ion i s considerably d i f f e r e n t and seeks to capture the rental p r i c e aspect of money as a durable good. Furthermore, - 148 -TABLE 5.15 AVERAGE YEARLY HOUSEHOLD HOLDINGS OF CHARTERED BANK SAVINGS DEPOSITS, PERSONAL CHEQUING ACCOUNTS, AND CURRENCY (mi l l ions of $) YEAR SD FTS NCS CHSC PCA CU 1947 3256.300 _ 0 705.472 1948 3544.710 - - - 0 689.635 1949 3877.970 - - - 0 720.371 1950 4057.360 - - - 0 710.727 1951 4125.150 •- - - 0 739.831 1952 4335.660 - - - 0 791.692 1953 4654.550 - - - 0 852.908 1954 4920.700 - - - 0 884.203 1955 5383.800 - - - 0 894.515 1956 5701.740 - - 0 978.818 1957 6009.160 - - - 0 1055.700 1958 6549.270 - - - •0 1102.880 1959 7061.080 - - - 58.009 1182.690 1960 7116.090 - - - 72.687 1204.270 1961 7523.580 - - - 91.144 1213.560 1962 7925.740 - - - 113.383 1264.280 1963 8324.930 - - - 137.507 1292.030 1964 8795.930 - - - 162.968 1341.220 1965 9926.820 - - - 190.876 1386.730 1966 10115.200 - - - 223.492 1473.920 1967 11087.900 - - - 291.300 1581.440 1968 - 2004. 960 3702. 340 7071. 310 460.933 1714.320 1969 - 3181. 580 5138. 330 6214. 610 653.471 1944.920 1970 - 4120. 340 6463. 380 5376. 960 797.785 2156.380 1971 - 4417. 710 7666. 690 5478. 600 1010.680 2266.380 1972 - 4752. 860 8410. 820 6046. 640 1271.290 2406.640 1973 - 6527. 950 8987. 430 6551 . 530 1607.410 2804.580 1974 - 11183. 500 10464. 700 6487. 700 1960.650 3359.520 SD: personal savings deposi ts ( to ta l ) FTS: f i xed term savings deposi ts NCS: non chequable savings deposi ts CHS: chequable savings deposi ts PCA: personal chequing accounts CU: estimated currency held by households - 149 -TABLE 5.16 DISCOUNT RATE, AGGREGATE PRICE LEVEL, AND FORECAST AGGREGATE PRICE LEVEL YEAR RN P P 1947 .0338 0.5992 0.6644 1948 .0353 0.6888 0.6219 1949 .0357 0.7253 0.7150 1950 .0351 0.7372 0.7528 1951 .0396 0.8287 0.7652 1952 .0429 0.8642 0.8601 1953 ,0450 0.8575 0.8970 1954 .0410 0.8767 0.8900 1955 .0399 0.8765 0.9099 1956 .0461 0.8975 0.9097 1957 .0537 0.9336 0.9316 1958 .0500 0.9547 0.9690 1959 .0562 0.9970 0.9909 1960 .0570 0.9928 1.0037 1961 .0548 1.0000 1.0306 1962 .0545 1.0162 1.0380 1963 .0537 1.0284 1.0548 1964 .0550 1.0406 1.0675 1965 .0568 1.0511 1.0800 1966 .0650 1.1045 1.0910 1967 .0709 1.1553 1.1465 1968 .0792 1.2534 1.1991 1969 .0875 1.2921 1.3010 1970 .0918 1.3560 1.3411 1971 .0835 1.3628 1.4075 1972 .0830 1.4277 1.4145 1973 .0847 1.4719 1.4819 1974 .1017 1.6118 1.5278 RN: McLeod, Young, Weir bond y i e l d averages - 10 i n d u s t r i a l s Source; ar i thmet ic mean of month end r a t e s , obtained from Interest Rates P: aggregate p r i c e leve l P: fo recas t aggregate p r i c e leve l - 150 -TABLE 5.17 AVERAGE YEARLY INTEREST RATES, PERSONAL SAVINGS DEPOSIT CATEGORIES YEAR RSD RFTS RNCS RCHS 1947 .0150 _ 1948 .0150 - _ 1949 .0150 - - -1950 .0150 - - -1951 .0150 - - _ 1952 .0150 - - -1953 .0154 - - -1954 .0200 - - -1955 .0200 - - -1956 .0219 - - -1957 .0273 - - -1958. .0275 - - -1959 .0275 - - -1960 .0275 - - -1961 .0275 - . _ _ 1962 .0288 • - . - -1963 .0300 - - -1964 .0300 - - • -1965 .0300 - - -1966 .0300 - - -1967 .0334 - - -1968 - .0651 .0494 .0350 1969 - .0718 .0600 .0350 1970 - .0690 .0615 .0338 1971 - .0474 .0453 .0300 1972 - .0534 .0400 .0296 1973 - .0706 .0547 .0292 1974 - .0945 .0859 .0300 RSD: rate on personal savings deposi ts ( tota l RFTS: rate on f i x e d term savings deposi t RNCS: rate on non chequable savings deposi ts RCHS: rate on chequable savings deposi ts - 151 -TABLE 5.18 RENTAL PRICES OF CHARTERED BANK SAVINGS DEPOSITS, PERSONAL CHEQUING ACCOUNTS, AND CURRENCY YEAR PSD PFTS PNCS PCHS PPCA POT, 1947 .0121 _ _ _ .0127 1948 .0122 - - - - .0212 1949 .0143 - - - - .0246 1950 .0146 - - - - .0255 1951 .0181 - - - - .0291 1952 .0230 - - - - .0353 1953 .0254 - - - - .0386 1954 .0180 - - - - .0351 1955 .0174 - - - - .0349 1956 .0210 - - - - .0401 1957 .0233 - - - - .0475 1958 .0208 - - - - .0461 1959 .0270 - - - .0527 .0527 1960 .0280 - - - .0542 .0542 1961 .0267 - - - .0535 .0535 1962 .0253 - - - .0536 .0536 1963 .0237 - - - .0538 .0538 1964 .0253 - - .0557 .0557 1965 .0273 - - - .0581 .0581 1966 .0359 - - - .0666 .0666 1967 .0401 - - - .0759 .0759 1968 - .0157 .0332 .0491 .0880 .0880 1969 - .0188 .0329 .0628 .1047 .1047 1970 - .0280 .0372 .0713 .1128 .1128 1971 - .0469 .0497 .0695 .1086 .1086 1972 _ .0387 .0562 .0698 .1084 .1084 1973 - .0193 .0410 .0758 .1157 .1157 1.974 - .0100 .0219 .0994 .1410 .1410 PSD: savings deposi ts ( to ta l ) PFTS: f i xed term savings deposi ts PNCS: non chequable savings deposi ts PCHS: chequable savings deposi ts PPCA: personal chequing accounts PCU: currency - 152 -TABLE 5.19 PER CAPITA QUANTITY INDEXES OF THE MONETARY SERVICES OF CHARTERED BANK SAVINGS DEPOSITS, PERSONAL CHEQUING ACCOUNTS AND CURRENCY YEAR QSD QFTS QNCS QCHS . QPCA QCU 1947 543.377 _ 117.739 1948 623.119 - 121.143 1949 570.258 - 105.903 1950 558.929 - 97.816 1951 552.395 - 98.962 1952 503.691 91.908 1953 507.827 - 92.970 1954 528.870 - 95.025 1955 555.011 - 92.175 1956 577.279 - 99.032 1957 578.345 - 101.540 1958 593.129 - 99.806 1959 612.935 5.035 102.604 1960 598.795 6.059 101.314 1961 605.985 7.330 97.708 1962 622.070 8.887 99.217 1963 630.695 10.381 97.893 1964 644.092 11.864 98.207 1965 666.843 13.442 98.053 1966 690.685 15.227 100.581 1967 700.173 18.377 99.844 1968 - 117. 869 217. 740 415. 895 27.056 100.812 1969 - 168. 138 271. 579 328. 453 34.516 102.754 1970 - 206. 025 323. 189 268. 832 39.855 107.813 1971 - 206. 634 358. 628 256. 270 47.249 106.007 1972 - 217. 603 385. 109 276. 857 58.201 110.175 1973 - 276. 862 381. 210 277. 880 68.166 118.940 1974 - 447. 727 418. 941 259. 716 78.471 134.482 QSD: savings deposi ts ( to ta l ) QFTS: f i xed term savings deposi ts QNCS: non chequable savings deposi ts QCHS: chequable savings deposi ts QPCA: personal chequing accounts QCU: currency - 153 -TABLE 5.20 NORMALISED RENTAL PRICE AND PER CAPITA QUANTITY INDEXES FOR THE SERVICES OF MONEY YEAR PMl QMl PM2 QM2 PM QM 1947 0.4360 5.886 0.4530 14.504 0.4489 20.331 1948 0.4256 6.036 0.4572 16.620 0.4491 22.639 1949 0.4947 5.277 0.5358 15.210 0.5252 20.485 1950 0.5124 4.874 0.5481 14.908 0.5389 19.796 1951 0.5850 4.931 0.6788 14.734 0.6552 19.670 1952 0.7101 4.579 0.8627 13.435 0.8241 18.009 1953 0.7753 4.632 0.9522 13.545 0.9075 18.170 1954 0.7036 4.735 0.6731 14.106 0.6810 18.836 1955 0.7007 4.593 0.6528 14.804 0.6649 19.375 1956 0.8046 4.934 0.7886 15.398 0.7931 20.315 1957 0.9529 5.059 0.8751 15.426 0.8948 20.473 1958 0.9261 4.973 0.7785 15.820 0.8151 20.759 1959 0.9848 5.763 1.0095 16.349 1.0031 22.112 1960 1.0109 5.749 1.0502 15.971 1.0400 21.717 1961 1.0000 5.624 1.0000 16.163 1.0000 21.787 1962 1.0020 5.787 0.9498 16.592 0.9633 22.379 1963 1.0040 5.797 0.8894 16.822 0.9190 22.615 1964 1.0394 5.893 0.9484 17.180 0.9719 23.066 1965 1.0842 5.969 1.0269 17.786 1.0419 23.742 1966 1.2437 6.200 1.3443 18.422 1.3920 24.608 1967 1.4177 6.330 1.5045 18.675 1.4835 24.988 1968 1.6436 6.846 1.5305 19.273 1.5612 26.102 1969 1.9551 7.349 1.8225 17.948 1.8584 25.333 1970 2.1061 7.906 2.1390 17.274 2.1243 25.232 1971 2.0259 8.205 2.5812 17.557 2.4007 25.802 1972 2.0248 9.015 2.6223 18.830 2.4282 27.853 1973 2.1612 10.018 2.1596 19.467 2.1643 29.429 1974 2.6341 11.402 1.9309 20.441 2.1809 31.869 PMl, p r i c e and quant i ty indexes f o r aggregate currency and QMl personal chequing accounts PM2j p r i c e and quant i ty indexes f o r aggregate QM2 personal savings accounts PM i p r i c e and quant i ty indexes fo r the aggregate QM ' of (PMl, QMl, PM2, QM2) - 154 -in using a D i v i s i a index aggregation method, we have not fol lowed the usual procedure of simply adding d o l l a r values of d i f f e r e n t forms of money. The three aggregate p r i c e and quant i ty se r ies are given in Table 5.20. E. Money Subst i tu tes In t h i s s e c t i o n , we d e r i v e , in a manner s i m i l a r to that used f o r money, estimates of the se rv ice flow from cer ta in near bank l i a b i l i t i e s , as well as the corresponding rental p r i c e s . We include two main types , the l i a b i l i t i e s of t r u s t and mortgage loan (TML) companies, and Canada Savings Bonds. The nominal c a p i t a l values o f both these assets are known with c e r t a i n t y , and hence cons idera t ions of r i s k a r i s i n g from t h i s source need not be incorporated into the a n a l y s i s . TML l i a b i l i t i e s and Canada Savings Bonds together form the great major i ty of l i q u i d nominally c a p i t a l c e r t a i n a s s e t s . However, excluded from our estimates are ( i ) the Quebec Savings Bank, ( i i ) Government Savings I n s t i t u t i o n s and ( i i i ) C red i t Unions and Caisses Popu la i res . These have not been considered owing to the u n a v a i l a b i l i t y of information on the i n t e r e s t rates o f fered on t h e i r l i a b i l i t i e s . Although one f inds stated (see, fo r example, Bond and Shearer [1972; 206] and the Report of the Royal Commission on Banking and Finance [1964; 150]) that the Quebec Savings Bank o f f e r s the same i n t e r e s t as chartered banks, we have no information as regards which chartered bank rate(s) i s / a r e r e l e v a n t , or on the composition of Quebec Savings Bank personal savings depos i ts . We could f i n d no publ ished information on the rates o f fered by Government Savings I n s t i t u t i o n s . However, the omission of these two categor ies is not of very great concern, as they are r e l a t i v e l y s m a l l , amounting in 1973, f o r example, to less than 2% of chartered bank l i a b i l i t i e s . - 1 5 5 -The omission of Cred i t Unions and Caisses Populaires is more important, as these i n s t i t u t i o n s , although s m a l l , have grown rap id ly in recent y e a r s . Unfor tunate ly , what i n t e r e s t rate information we have extends back only as fa r as 1966. Annual i n t e r e s t paid is reported in Cred i t Unions (61.209) along with scat tered information on the range of i n t e r e s t rates on c e r t a i n c lasses of deposi ts for some prov inces . There i s no information a v a i l a b l e at a l l p r i o r to 1966. For the time be ing , the re fo re , a reasonably long time s e r i e s on Cred i t Union rates cannot be const ructed . no 1) T rus t and Mortgage Loan Company L i a b i l i t i e s TML l i a b i l i t i e s f a l l into two c a t e g o r i e s , savings deposi ts 'de f a c t o ' withdrawable upon demand, and term d e p o s i t s . The f i r s t group i s d iv ided fu r ther in to chequable and non chequable components, while term deposits are d is t ingu ished by term to matur i ty , namely, less than 44 one y e a r , one to f i v e y e a r s , and over f i v e y e a r s . As with chartered bank l i a b i l i t i e s , no d i r e c t information is a v a i l a b l e as regards the ownership of these d i f f e r e n t c a t e g o r i e s . We proceeded therefore on the basis of some reasonable i n s t i t u t i o n a l assumptions. In l i n e with our previous assumption that households own chartered bank personal savings d e p o s i t s , i t i s assumed that TML savings depos i t l i a b i l i t i e s are held e n t i r e l y by households. While most probably t h i s w i l l be true in the case of non chequable deposits (which formed about 74% of to ta l savings deposi ts in 1974), there may be an e r ror i n v o l -45 ved as regards the chequable category. The ownership quest ion i s eas ie r to reso lve in the case of term d e p o s i t s . The empir ica l and i n s t i t u t i o n a l evidence contained in C l in ton [1974] suggests that under one year term deposi ts are very c lose - 156 -subst i tu tes fo r the chartered bank category , non personal term and not ice d e p o s i t s . We therefore assume that t h i s deposi t type i s not held by households. Over one year d e p o s i t s , i t seems, are held by a va r ie ty of i n v e s t o r s , with households and small savers forming a high propor t ion . Thus we decided to regard a l l the over one year term deposits as held by 46 the household s e c t o r . ( i ) Chequable and Non Chequable Deposi ts . As with monetary s e r v i c e s , our f i r s t step is to obtain annual average y e a r l y holdings of each of these components. From 1963 to 1974, end of quarter f igures f o r TML chequable and non chequable deposi ts are l i s t e d in F i n a n c i a l I n s t i t u -47 t ions (61.006). F i r s t , average quar ter ly f igures were der ived as the ar i thmet ic mean of the d o l l a r values outstanding at the end of the 48 quarter and the end of the preceding quar ter . The average year ly stock then equals the sum of the quar te r ly averages d iv ided by f o u r . No data on the chequable/non chequable breakdown are a v a i l a b l e , p r i o r to 1963 in F inanc ia l I n s t i t u t i o n s . However, data f o r the per iod 1951-1960 are contained in the U n i v e r s i t y of Western Ontario [1965] p u b l i -c a t i o n , The Role of the Trus t and Loan Companies in the Canadian Economy, prepared f o r the 1964 Royal Commission on Banking and Finance. Table 11-35 of t h i s study contains quar ter ly deposi t data by chequable and non chequable types f o r 15 TML companies. Since these companies do not cover a l l f i rms in the TML i n d u s t r y , f o r each y e a r , 1951-1960, we computed an adjustment f a c t o r , as f o l l o w s . F i r s t , sum over four quarters the combined chequable and non chequable data in Table 11-35, and d i v i d e by four to obtain average y e a r l y savings deposi ts (chequable and non chequable) , WESTASD^.. Year end f igures f o r savings deposi ts f o r all TML companies are a v a i l a b l e in Bank of Canada S t a t i s t i c a l Summary fo r 1950-1960. - 157 -Taking the mean of adjacent year end f igures provides us with average y e a r l y savings deposi ts ( to ta l ) f o r a l l TML companies, BSASD^.. The adjustment f a c t o r , AF^ , was then c a l c u l a t e d as A F t = BSASD t/WESTASD t A F t was then used to 'blow up' the o r i g i n a l quar ter ly data in the Western 49 Ontario study. Using t h i s cons is tent quar ter ly s e r i e s , we then computed f i r s t , q u a r t e r l y , and then annual , average f i g u r e s , as be fore , f o r 50 chequable and non chequable components f o r 1951-1960. The chequable/non chequable breakdown does not appear to be a v a i l a b l e f o r 1947-1950, or f o r 1961-1962. We therefore used i n t e r p o l a t i o n techniques as f o l l o w s . F i r s t , using the data constructed already fo r 1951-1960, and 1963-1974, the average annual proport ion of chequable to t o t a l savings deposi ts was c a l c u l a t e d . This s e r i e s exh ib i t s a steady down-ward dec l ine over the p e r i o d . We obtained estimated proport ions f o r 1961-1962 by f i t t i n g a l i n e a r time trend using the 1951-1960 and 1963-1974 observa t ions . In a s i m i l a r manner, proport ions f o r 1947-1950 were e s t i -mated using the data f o r 1951-1960. 5 1 These proport ions were then appl ied t o : ( i ) the 1947-1950 annual average estimates which were der ived from adjacent year end f igures f o r to ta l savings deposi ts f o r 1946-1950, as l i s t e d in Bank of  Canada S t a t i s t i c a l Summary, ( i i ) the 1961-1962 annual average estimates der ived ( in the usual way) from the quar te r ly data in F inanc ia l I n s t i t u - t ions fo r to ta l savings depos i ts . The r e s u l t i n g annual se r ies f o r the en t i re per iod 1947-1974 are l i s t e d in Table 5.21. Next we requi re i n t e r e s t rates fo r each of the deposi t types . F i r s t , chequable d e p o s i t s . From 1951 - June 1963, quar ter ly rates are - 158 -taken from Table V , Appendix E , Royal Commission on Banking and F inance,  Appendix Volume [1964]. From J u l y 1963 onwards, the rates were provided by the Bank of Canada. P r i o r to 1951, we used a se r ies of a well known Trust and Loan company suppl ied by Brock K. Short (see Short and V i l leneuva [1976; 3 9 ] ) . 5 2 Annual average i n t e r e s t rates were obtained in the same manner as fo r chartered bank l i a b i l i t i e s , namely, as a weighted sum of quar ter ly r a t e s , the weights equa l l ing the proport ion of the annual t o t a l held in each quar ter . The i n t e r e s t rate on non chequable TML deposi ts proved to be rather more d i f f i c u l t to ob ta in . From 1968-1974, the quar ter ly rate was provided by the Bank of Canada. For the t h i r d quarter of 1956 to 1967, we r e l i e d on the unpublished ser ies provided by Brock Short . P r i o r to 53 1956, there seemed to be no publ ished data a v a i l a b l e . We e x t r a p o l a t e d - t h i s s e r i e s back to 1947 by f i r s t c a l c u l a t i n g the average annual rates fo r 1956-1974, using the same technique as used above for chequable deposi t r a t e s . The annual chequable rate was then used to pred ic t the non chequable rate from 1947-1956, using an ordinary l eas t squares regress ion . The r e s u l t i n g annual i n t e r e s t rate se r ies for both savings deposi t types may be found in Table 5.22. ( i i ) Term Deposi ts . We requi re the breakdown between over one year and under one year ca tegor ies . Our method of const ruct ion of the over one year s e r i e s proceeds as f o l l o w s . F i r s t , for 1947-1960, we com-puted the average year ly holdings of to ta l term d e p o s i t s , as the mean of adjacent year end f igures obtained from the Bank of Canada S t a t i s t i c a l  Summary. For 1961-1974 we a l s o der ived s i m i l a r annual averages, using end of quarter data l i s t e d in F inanc ia l I n s t i t u t i o n s . From 1967-1974 - 159 -quarter end f igures f o r over one year deposi ts are a v a i l a b l e from the l a t t e r pub l i ca t ion and annual averages f o r th is category were computed. . For 1967-1974, we then c a l c u l a t e d the proport ion of over one year to to ta l term d e p o s i t s , based on the above annual average data . For 1951-1960, a s i m i l a r s e r i e s on the term to maturity composition is contained in the U n i v e r s i t y of Western Ontario study (chart 11-1) and these proport ions were appl ied to the to ta l deposi t s e r i e s in order to obtain a s e r i e s for over one year term deposi ts f o r 1951-1960. There appears to be no data a v a i l a b l e on the term to maturi ty from 1947-1950 and 1961-1966. From 1951-1960, the Western Ontario estimates ind ica te that the over one year category as a percentage of the to ta l increased only s l i g h t l y over the p e r i o d , from 55.83 in 1951 to 62.5 in 1960. However, by 1967, the F inanc ia l I ns t i tu t ions data show that t h i s percentage had r isen considerably to 83.5 We assumed that over the in tervening y e a r s , the percentage rose by equal amounts each year . For 1947-1950, we assumed that the percentage fo r 1951, 55.83, was the same fo r the preceding y e a r s . The r e s u l t i n g estimated proport ions were then appl ied to the annual s e r i e s for to ta l term deposi ts constructed a l ready . The se r ies fo r the en t i re per iod is l i s t e d in Table 5.21. The i n t e r e s t rate on over one year term deposi ts was taken to be the f i v e year G . I . C . (Guaranteed Investment C e r t i f i c a t e ) r a te . This rate is a v a i l a b l e (quar ter ly from 1951-1963, and on a month end basis from 1964-1974) in Interest Rates. From 1951-1960, the annual rate i s 54 the a r i thmet ic mean of the quar te r ly r a t e s . For 1961-1963, the annual rate is a weighted average of quar ter ly r a t e s , using quar ter ly average data f o r to ta l term depos i ts . For 1964 to 1966, the same procedure - 160 -was used, except that the quar ter ly rates are now the ar i thmet ic mean of the month end rates as l i s t e d in Interest Rates. F i n a l l y , f o r 1967-1974, we constructed the annual rate again as a weighted average of (average) i n t e r e s t r a t e s , where the weights are based on the quar te r ly average f igures fo r greater than one year term d e p o s i t s . Unfortunately the G . I . C . rate is not a v a i l a b l e p r i o r to 1951. We therefore extrapolated the ser ies using RSBY (the f i v e year government bond r a t e , given in Table 5.6) as the independent 'var iable . The r e s u l t i n g annual i n t e r e s t rate se r ies is l i s t e d in Table 5.22. F i n a l l y , using i n t e r e s t rates and the data on P and RN (Table 5.16) rental p r ice se r ies f o r each of these se r ies were const ruc ted , which appear in Table 5.23. . 55 2) Canada Savings Bonds End of quarter f igures f o r the stock of Canada Savings Bonds (CSB's) outstanding are a v a i l a b l e as se r ies B2406 from the CANSIM data system. At any point in time the average y i e l d earned on CSB's depends on the composi t ion, by date of i s s u e , and length of per iod h e l d , of the to ta l stock outstanding. The Bank of Canada publ ishes a quar ter ly n k s e r i e s , BS700, which equals £ I r . . s . . , where r . . i s the annual i=l j=l 1 J ^ U coupon rate on the bond issue of date of issue i and holding per iod j . ( i = l , . . . , n ; j = l , . . . , k ) , and s . . i s the corresponding d o l l a r value at 56 the end of the quar ter . To obtain an average coupon rate fo r the end of the quar te r , we d iv ided B2406 in to BS700. For the per iod 1954-1972, t h i s equals EACRSB, used in the Bank of Canada econometric model RDX2 (He l l iwe l l et a l . [1971]). Our s e r i e s extends back to 1947 and continues up to 1974. Using these end o f q u a r t e r f igures f o r rates and quan t i t i es to - 161 r 57 approximate quar te r ly averages, we then constructed annual average f igures fo r quant i t i es and rates (employing the usual weighting proce-dure) . These s e r i e s appear in Tables 5.21 and 5.22. The corresponding renta l p r ice i s l i s t e d in Table 5.23. 3) Aggregation and Conversion to Real Balances F i r s t , we aggregated TML chequable and non chequable d e p o s i t s , and d iv ided the r e s u l t i n g quant i ty se r ies by (p (the fo recas t p r i c e l eve l ) times popula t ion) . The rental p r i c e and quant i ty se r ies fo r TML term deposi ts were normalised so that P-jgg-] = 1.0000 and were a lso d iv ided by (p times popula t ion) . These normalised p r i c e and quant i ty se r ies are l i s t e d in Table 5.24. F i n a l l y , the three sub components of near money, TML savings deposi ts ( t o t a l ) , TML term deposi ts (over one y e a r , and CSB's) were aggregated into p r i c e ,and quantity, indexes-of . the serv ices from . money s u b s t i t u t e s , which appear, in Table 5.25 as PNM and QNM, r e s p e c t i v e l y . - 162 -TABLE 5.21 ANNUAL AVERAGE HOLDINGS OF TML DEPOSIT LIABILITIES BY CATEGORY, AND CANADA SAVINGS BONDS (mi l l ions of $) YEAR TNC TC TD SB 1947 15.000 166.000 133.000 1293.750 1948 18.000 175.000 138.000 1311.750 1949 22.000 188.000 154.000 1124.750 1950 27.000 204.000 172.000 1128.250 1951 34.375 213.625 196.000 1067.750 1952 39.625 218.375 233.000 1115.750 1953 39.250 235.875 246.000 1268.000 1954 48.375 261.750 303.000 1646.500 1955 67.000 310.000 374.000 2081.000 1956 71.500 334.375 375.000 2357.750 1957 65.375 345.250 416.000 2403.250 1958 91.000 356.875 490.000 2577.250 1959 105.250 386.500 643.000 2874.000 1960 134.750 380.500 758.000 3199.500 1961 229.000 404.000 942.000 3628.250 1962 315.000 485.000 1140.000 4129.500 1963 415.500 545.500 1439.500 4642.500 1964 600.625 616.875 1857.000 5151.250 1965 751.375 697.750 2374.000 5543.250 1966 764.925 715.837 2932.000 5698.500 1967 789.537 740.762 3494.000 6007.750 1968 872.800 715.100 3940.000 5958.750 1969 1081.450 641.375 4380.000 6111.250 1970 1264.940 557.862 5264.000 6750.000 1971 1579.250 591.600 6157.000 8256.250 1972 1736.910 655.037 7087.000 9948.000 1973 1968.450 726.975 8188.000 10754.000 1974 2088.940 703.387 10496.000 10633.000 TNC: TML non chequable savings deposi ts TC: TML chequable savings deposi ts TD: TML over one year term deposi ts SB: Canada Savings Bonds - 163 -TABLE 5.22 ANNUAL AVERAGE INTEREST RATES ON TML DEPOSIT LIABILITIES BY CATEGORY AND CANADA SAVINGS BONDS YEAR RTNC RTC RTD RSB 1947 .0314 .0200 .0225 .0109 1948 .0314 .0200 .0282 .0142 1949 .0314 .0200 .0278 .0192 1950 .0314 .0200 .0276 .0225 1951 .0311 .0195 .0340 .0260 1952 .0314 .0200 .0364 .0296 1953 .0318 .0209 .0399 .0309 1954 .0333 .0240 .0365 .0327 1955 .0340 .0254 .0365 .0332 1956 .0350 .0275 .0406 .0323 1957 .0350 .0313 .0464 .0322 1958 .0350 .0317 .0466 .0330 1959 .0374 .0323 .0528 .0355 1960 .0400 .0331 .0527 .0434 1961 .0400 .0333 .0495 .0426 1962 .0400 .0359 .0520 .0432 1963 .0400 .0367 .0514 .0461 1964 .0400 .0372 .0526 .0462 1965 .0400 .0388 .0547 .0468 1966 .0400 .0400 .0596 .0474 1967 .0438 .0400 .0635 .0491 1968 .0526 .0400 .0701 .0516 1969 .0609 .0400 .0803 .0555 1970 .0634 .0400 .0851 .0690 1971 .0503 .0366 .0771 .0721 1972 .0481 .0350 .0764 .0724 1973 .0584 .0379 .0823 .0718 1974 .0840 .0400 .0974 .0738 RTNC: rate on TML non chequable savings deposi ts RTC: rate on TML chequable savings deposi ts RTD: rate on TML over one year term deposi ts RSB: rate on Canada Savings Bonds - 164 -TABLE 5.23 RENTAL PRICES, TML DEPOSIT LIABILITIES AND CANADA SAVINGS BONDS* YEAR PTNC PTC PTD PSB 1947 .0016 .0089 .0073 .0147 1948 .0024 .0092 .0043 .0127 1949 .0030 .0184 .0054 .0114 1950 .0027 .0198 .0055 .0092 1951 .0063 .0148 .0041 .0100 1952 .0095 .0189 .0054 .0109 1953 .0113 .0207 .0044 .0121 1954 .0066 .0147 .0038 .0071 1955 .0052 .0128 .0030 .0059 1956 .0096 .0162 .0048 .0120 1957 .0165 .0198 .0065 .0190 1958 .0138 .0169 .0031 .0156 1959 .0176 .0225 .0032 .0195 1960 .0161 .0227 .0041 .0129 1961 .0145 .0210 .0052 .0120 1962 .0143 .0184 .0025 .0111 1963 .0137 .0170 .0023 .0076 1964 .0152 .0180 .0024 .0089 1965 .0172 .0184 .0021 .0102 1966 .0256 .0256 .0055 .0180 1967 .0291 .0331 .0079 .0238 1968 .0296 .0436 .0101 .0307 1969 .0318 .0568 .0086 .0382 1970 .0349 .0636 .0083 .0280 1971 .0431 .0609 .0083 .0148 1972 .0456 .0627 .0086 .0138 1973 .0359 .0640 .0033 .0176 1974 .0246 .0856 .0060 .0386 * The s e r i e s are l i s t e d to four s i g n i f i c a n t places of decimals . However, in the computation of normalised rental p r ices (see Table 5.24) and in es t ima t ion , more than four f igures were used. PTNC: rental p r i c e of TML chequable deposi ts PTC: rental p r i c e of TML non chequable deposi ts PTD: rental p r i c e of TML over one year term deposi ts PSB: rental p r i c e of Canada Savings Bonds - 165 -TABLE 5.24 NORMALISED RENTAL PRICE AND PER CAPITA QUANTITY INDEXES OF REAL 'NEAR MONEY1 BALANCES BY CATEGORY TML SAVINGS DEPOSITS OVER 1 YEAR CANADA SAVINGS ( to ta l ) TML TERM DEPOSITS BONDS YEAR P Q P Q P Q 1947 0.3936 0.6347 1.4024 0.1150 1.2306 2.5851 1948 0.4111 0.7062 0.8237 0.1256 1.0593 2.7594 1949 0.4857 0.6370 1.0532 0.1173 0.9526 1.9793 1950 0.4905 0.6499 1.0534 0.1227 0.7681 1.8597 1951 0.6778 0.6669 0.7960 0.1359 0.8396 1.7107 1952 0.8766 0.5967 1.0353 0.1402 0.9148 1.5511 1953 0.9668 0.6009 0.8454 0.1390 1.0081 1.6554 1954 0.6728 0.6625 0.7430 0.1687 0.5917 2.1176 1955 0.5831 0.7619 0.5745 0.1997 0.4898 2.5671 1956 0.7659 0.8060 0.9237 0.1966 1.0023 2.8564 1957 0.9782 0.7791 1.2463 0.2073 1.5873 2.7674 1958 0.8322 0.7934 0.6059 0.2298 1.3074 2.7927 1959 1.0977 0.8330 0.6160 0.2890 1.6263 2.9849 1960 1.0890 0.8354 0.7885 0.3303 1.0773 3.2212 1961 1.0000 0.9503 1.0000 0.3929 1.0000 3.4967 1962 0.9074 1.1159 0.4752 0.4634 0.9256 3.8782 1963 0.8519 1.1332 0.4446 0.5646 0.6383 4.2086 1964 0.9180 1.1613 0.4690 0.7042 0.7432 4.5136 1965 0.9859 1.1849 1.4415 0.8697 0.8520 4.6920 1966 1.4202 1.1823 1.0683 1.1037 1.5027 4.6555 1967 1.7211 1.1743 1.5299 1.1143 1.9540 4.5394 1968 2.0076 1.1670 1.9527 1.2000 2.5647 4.1933 1969 2.3912 1.1566 1.6634 1.1198 3.1960 3.8648 1970 2.6482 1.1504 1.5894 1.1363 2.3387 4.0385 1971 2.9591 1.1647 1.6055 1.1492 1.2394 4.6212 1972 3.0993 1.1777 1.6648 1.1680 1.1530 5.4503 1973 2.6861 1.1851 0.6332 1.1798 1.4741 5.4577 1974 2.5160 1.1775 1.1515 2.1760 3.2291 5.0934 P and Q: normalised rental p r ice and quant i ty s e r i e s , r e s p e c t i v e l y - 166 -TABLE 5.25 NORMALISED RENTAL PRICE AND QUANTITY INDEXES OF THE SERVICES OF 'NEAR MONEY' YEAR PNM QNM 1947 1.0261 3.5011 1948 0.8838 3.7529 1949 0.8314 2.7887 1950 0.7028 2.6699 1951 0.7867 2.5379 1952 0.9027 2.3119 1953 0.9775 2.4219 1954 0.6126 2.9775 1955 0.5105 3.5578 1956 0.9403 3.8941 1957 1.4923 3.7872 1958 1.1584 3.8420 1959 1.4449 4.1158 1960 1.0567 4.3914 1961 1.0000 4.8400 1962 0.8837 5.5012 1963 0.6686 6.0906 1964 0.7570 6.8236 1965 0.8380 7.3751 1966 1.4435 7.4076 1967 1.8675 7.2919 1968 2.3815 6.9071 1969 2.8024 6.4557 1970 2.3331 6.6833 1971 1.7360 7.4855 1972 1.7234 8.4649 1973 1.6215 8.7300 1974 2.7728 8.4460 PNM: normalised renta l p r i c e s e r i e s fo r aggregate of TML savings deposi ts ( t o t a l ) , TML over one year term d e p o s i t s , and Canada Savings Bonds QNM: corresponding normalised quant i ty index - 167 -Footnotes - Chapter 5 1. The p r i c e index P* is more c o r r e c t l y re fer red to as a t rue cost of l i v i n g index evaluated at a u t i l i t y leve l u. If the expenditure funct ion in (5.4) i s not homothetic, then the reference u t i l i t y leve l in (5.5) may be taken as an average of the u t i l i t y l e v e l s u° (reached when pr ices p° p r e v a i l ) and u 1 (reached when p* p r e v a i l ) . 2. That i s , P D ( p ° , pi , x ° , x 1 ) = m(u*; p1)/m(u*; p ° ) where u* = (u° u 1 ) 2 and m is a Translog expenditure f u n c t i o n . 3. Due to changes in c l a s s i f i c a t i o n , a cons is tent consumer expenditure s e r i e s is not a v a i l a b l e p r i o r to 1947. 4. Henceforth, in t h i s chapter , numbers in round parentheses r e f e r to S t a t i s t i c s Canada catalogue numbers. A l i s t of a l l the s t a t i s t i c a l sources used may be found at the end of the b ib l iography . 5. Note that t h i s impl ies that we are t rea t ing the basic dec is ion making un i t as the i n d i v i d u a l , ra ther than the household. Berndt, Darrough, and Diewert [1977] adopted the l a t t e r approach in an e a r l i e r vers ion of t h e i r paper, using estimates of the number of households and unattached i n d i v i d u a l s as a d e f l a t o r . However, in the f i n a l v e r s i o n , they def la ted by populat ion aged f i f t e e n and over . 6. Unless otherwise s t a t e d , from now on we mean by the impl ied purchase p r i c e s e r i e s of a good, the current d o l l a r value (Table 53) d iv ided by the corresponding quant i ty ser ies (Table 54) , NIEA. 7. Upon examining the aggregate medical se rv ices se r ies contained in Cummings and Meduna [1973], i t appears that they must have under-taken some s i m i l a r adjustment. However, the issue i s not d iscussed in t h e i r paper. Note a lso that no reference is made in NIEA to the r e c l a s s i f i c a t i o n of medical ca re . However, the reader is a le r ted to the next adjustment to be d iscussed below, namely, the r e c l a s s i -f i c a t i o n of hospi ta l care expenditure from 1961. The l a t t e r ad jus t -ment i s taken in to account by Cummings and Meduna. 8. Our p r ice and quant i ty se r ies fo r medical serv ices must be regarded with extreme cau t ion . E s p e c i a l l y fo r the l a t t e r part of the p e r i o d , there are major conceptual and data problems involved in measuring the pr ices consumers (may) face f o r medical ca re . Furthermore, the value of government cap i ta l , subs id ies to hosp i ta ls are not included in our data. Indeed, given a l l these d i f f i c u l t i e s , a strong case could be made f o r leav ing out medical serv ices a l toge ther , at l eas t f o r the l a t t e r part of the time per iod . However, some component of medical se rv ices probably should be reta ined fo r e a r l i e r y e a r s . Sor t ing t h i s en t i re matter out proper ly would be a major task in i t s e l f , and so we decided to adopt the r e l a t i v e l y s i m p l i s t i c approach descr ibed above. - 168 -9. Note that the same problem as was present in the medical se rv ices sector occurs here a l s o , namely, the existence of large government subs id ies to the educational s e c t o r . We have not attempted to take th is in to account. 10. Note that the aggregate p r ice se r ies ca lcu la ted in t h i s way is not i d e n t i c a l to that which would have been obtained had we used the D i v i s i a method d i r e c t l y on the sub c a t e g o r i e s , i . e . , i f , f o r example, in the case of s e r v i c e s , we had aggregated d i r e c t l y the s ix teen p r i c e and quant i ty components. However, Diewert [1975] has shown that the numerical d i f f e r e n c e between the l a t t e r method and the 'two stage' approach which we adopted, w i l l genera l ly be extremely s m a l l . 11. Unl ike other c a t e g o r i e s , c l o t h i n g , 'other semi durab les ' and 'other d u r a b l e s ' , where a D i v i s i a aggregation of the components was per-formed (see 4) below), these two components were added a r i t h m e t i c a l l y . 12. The National Accounts pub l ica t ions (13-351) and the H i s t o r i c a l  R e v i s i o n , 1926-1971, do not contain any disaggregated expenditure data p r i o r to 1926. 13. The l inkage scheme i s the f o l l o w i n g . The l i n e numbers on the r igh t hand s ide r e f e r to the old pre 1947 c l a s s i f i c a t i o n given in National Accounts (13-531), Table 54a, whi le the l i n e numbers on the l e f t r e f e r to the post 1947 data in NIEA. New C l a s s i f i c a t i o n Old C l a s s i f i c a t i o n Men's and boys' c lo th ing (6) = 6+9(6/(6+7+8)) Women's and c h i l d r e n ' s c l o t h i n g (7) = 7+9(7/(6+7+8)) Footwear and r e p a i r (8) = 8+9(8/(6+7+8)) Books, newspapers and magazines (38) = 35 New and used (net) autos and repa i rs and parts (30+31) = 28 F u r n i t u r e , carpets and other f l o o r coverings (17) = 0.214(18+19+21+34) Household appl iances (18) = 0.115(18+19+21+34) Semi durable household fu rn ish ings (19) = 0.538(18+19+21+34) Recreat ion , spor t ing and camping equipment (37) = 0.133(18+19+21+34) Jewel lery = 0 The f i r s t f i v e categor ies are r e l a t i v e l y s t ra ight forward . For the next four i tems, the r a t i o s .214, .115, .538, and .133 are der ived from NIEA data , Table 54, by d i v i d i n g , f o r 1947, l i n e s (17), (18), (19), and (37), r e s p e c t i v e l y , into the sum of these four c a t e g o r i e s . E s s e n t i a l l y , we are assuming that the sum of (17), (18), (19), and (37) in the post 1947 NIEA c l a s s i f i c a t i o n equals the sum of (18), (19), (21), and (24) in the pre 1947 Accounts. This assumption i s not unreasonable, given the d e s c r i p t i o n of the items and the orders of magnitude of the d o l l a r f i g u r e s invo lved . Furthermore, we assume that these four categor ies were ..consumed in the same proport ion during 1926-1946 as in 1947, F i n a l l y , s ince jewel le ry ( l i n e 42, NIEA) does not appear e x p l i c i t l y anywhere in the pre 1947 Accounts , we were forced to assume that the stock of jewel le ry - 169 -equals zero in 1947, These assumptions a r e , of course , only working approximations, However, in the absence of any other da ta , t h i s is the best that can be done. In any case , we have good reason to be l ieve that the r e s u l t s , e s p e c i a l l y in aggregate, w i l l not be grea t ly a f fec ted due to (a) the high deprec ia t ion rates on the ' troublesome' c a t e g o r i e s , and (b) the e l im ina t ion of d i f f e rences through aggregat ion, 14. The fo l lowing is a very b r i e f summary of a large and r a p i d l y growing l i t e r a t u r e in t h i s area. For more d e t a i l the reader is re fe r red to Appendix A , and the work by Box and Jenkins [1970], Nelson [1973], and Rose [1976], 15. So c a l l e d ' r a t i o n a l ' expecta t ions , where the actual ex post value i s used as the fo recas t is a lso opt ima l , but only when a l l i n f o r -mation needed to determine the value of the v a r i a b l e , other than that contained in the sample se r ies i t s e l f , i s a v a i l a b l e f o r use. An a l t e r n a t i v e e r ror learning i n t e r p r e t a t i o n ' of ARIMA f o r e c a s t i n g are provided by Rose [1976; C h . 8 ] . 16. Th is is a lso the source fo r the Cummings and Meduna s e r i e s , although not e x p l i c i t l y stated by them. Note that t h i s se r ies includes repa i rs to the e x i s t i n g s tock . 17. There appears to be some confusion as regards the appropriate housing deprec ia t ion ra te . Cummings and Meduna [1972; 6] using a census benchmark f o r 1940, and a time ser ies of past investments s t re tch ing back to 1970, solve [5.19] to obtain a deprec ia t ion rate of .022. However, in const ruc t ing t h e i r post 1947 est imates , they appear to have used (Table la) a rate e q u a l t o .02. Furthermore, Cummings and Christensen [1976; 23] s ta te they "have taken t h e i r rate from Cummings and Meduna". However, the rate they in f a c t used was .025. 18. Note that the rate we want, from (3.10') i s the effective rate on the market va lue , p . * , and not the announced rate used in conjunc-t i o n with some assessed va lue . The l a t t e r , announced r a t e , could indeed be qui te h igh , but is not re levant f o r our purposes. 19. The Cummings s e r i e s c o n s i s t i n g of to ta l property taxes on r e s i d e n t i a l housing and land fo r 1947-1973, is based on ext rapola t ion of a se r ies provided in an e a r l i e r study by Glenn Jenkins [1972]. We d iv ided t h i s s e r i e s in to our estimates of the d o l l a r value of the stock of housing and land (purchase pr ice times stock) to obtain an e f f e c t i v e r a t e , PR0PTAX, f o r 1947-1973. 20. That i s , a l l rental p r ices are d iv ided by the 1961 rental p r i c e , and a l l stocks are m u l t i p l i e d by t h i s same value to leave to ta l expen-d i tu re unchanged. 21. It may be shown e a s i l y that the 'spread ' between two periods w i l l always i n c r e a s e , the higher the constant rate of i n f l a t i o n assumed. - 170 -22. I n i t i a l l y , the autoregressive model was estimated f o r the 1963-1974 per iod on ly . However, t h i s led to extreme f l u c t u a t i o n s in rental p r ices due, p a r t l y , to the r e l a t i v e l y few observat ions a v a i l a b l e f o r use in es t imat ion . The procedure which we adopted assumes that the pre 1961 constant rate experience s t i l l exerc ises a 'dampening' e f f e c t on consumer expectat ions during the s i x t i e s . 23. A cons is tent s e r i e s is not a v a i l a b l e p r i o r to 1952. 24. We do not know how exact ly t h i s s e r i e s was const ructed . 25. Part of the d i f f e r e n c e in overa l l growth between the actual and imputed rent indexes may be explained by the presence of rent cont ro ls during 1974 in some Canadian prov inces . 26. Canadian tax regula t ions permit each taxpayer to r e a l i s e c a p i t a l gains on one ' p r i n c i p a l res idence ' without being taxed. This exemp-t i o n extends to up to one acre of surrounding l a n d , or more, i f the taxpayer can e s t a b l i s h that more than one acre i s necessary fo r the use and enjoyment of the p r i n c i p a l res idence . Note t h a t , fo r example, a spouse may a lso c la im an exemption on a ' p r i n c i p a l r e s i d e n c e ' , even though he/she has not l i v e d in i t during the p e r i o d . See the Master Tax Guide [1976] f o r d e t a i l s . The above d i s c u s s i o n re fe rs to the present c a p i t a l gains tax law. For most of the per iod up to the ear ly 1970's , the law was even more favourable as regards exemptions. 27. Note that our normalised per cap i ta housing index s t a t i c expectat ions i s s u b s t a n t i a l l y less than that o f Cummings and Meduna [1973; Table 5 ] . A small part of the d i f f e rence can be explained by the f a c t that we used a d i f f e r e n t property tax , which lowered the 1961 (unnormalised) rental p r i c e . However, there s t i l l appears to be a major computational e r ro r present in t h e i r work. M u l t i p l y i n g the 1960 per cap i ta un-normal ised housing stock computed by Gussman [1972; Table IV-7] by the Gussman unnormal i s e d r e n t a l p r i c e value f o r 1961 (Gussman [1972; Table IV-7]) y ie lds^aper cap i ta normalised quant i ty value f o r 1961 of 361, compared to the f i g u r e of 780 reported by Cummings and Meduna. The l a t t e r authors c la im to have used the i d e n t i c a l method to that of Gussman, a l b e i t with a s l i g h t l y lower deprec ia t ion rate (see footnote 17 above). However, the use of t h e i r lower deprec ia t ion rate cannot expla in such a large d iscrepancy. 28. The Diewert study deals with the per iod 1926-1974, and is a cont inua-t i o n of an e a r l i e r study by Diewert and Woodland [1975]. The const ruc t ion of the s e r i e s is based on a wide va r i e ty of labour data s e r i e s , p r i n c i p a l l y the Labour Force Survey, the Employment Survey, and National Accounts p u b l i c a t i o n s . Estimates of employment, manhours, earnings and wages, cross c l a s s i f i e d by industry and occupation are c a l c u l a t e d and presented. 29. For both taxable and non taxable r e t u r n s , reported income by source i s given a l s o . - 171 -30. The prec ise items included in labour income var ied over time as the c l a s s i f i c a t i o n system a l t e r e d , Throughout, we attempted to presume consistency with the d e f i n i t i o n of labour earnings contained in Diewert [1975]. 31. An a l t e r n a t i v e is to fo l low the same procedure f o r wage rates and labour earnings c l a s s i f i e d on an industrial basis (using Diewert [1975; Tables 7 and 3 ] ) , However, we decided to use the occupat ional rather than the i n d u s t r i a l c l a s s i f i c a t i o n as there was considerably more v a r i a t i o n in earnings across occupat ions , and the e s s e n t i a l purpose of our method was to capture such v a r i a t i o n . 32. These tax rates are l i s t e d in Appendix B. 33. Th is assumes that p h y s i o l o g i c a l l y required time (e .g . f o r s leeping) i s not l e i s u r e . 34. In e a r l i e r y e a r s , F inanc ia l Flows contained estimates f o r some assets f o r household and unincorporated business sectors separa te ly . However, (see Shearer [1972]) t h i s procedure impl ied that S t a t i s t i c s Canada possessed more information than was a c t u a l l y a v a i l a b l e . In recent years the d i s t i n c t i o n between the two sectors has been dropped. In any case , in the e a r l i e r est imates , the sectora l breakdown was not provided fo r e i t h e r chartered bank (or t r u s t and loan company) d e p o s i t s . 35. De ta i l s of the scope and method of c o l l e c t i o n of the corporate sector data are provided in F inanc ia l Flows (13-002, IQ1971). 36. However, there i s some ambiguity (which we could not resolve) as to whether t h i s inc ludes unincorporated businesses or not . Most l i k e l y , the same types of accounts which were excluded from personal savings deposi ts (see previous paragraph) are excluded from th is category a l s o . 37. Namely, 'depos i t balances of r e l i g i o u s , educational and welfare i n s t i t u t i o n s and personal accounts used mainly f o r business p u r p o s e s ' , Bank of Canada S t a t i s t i c a l Summary, Supplement [1960; 28, foo tnote ] . 38. Note that our procedure impl ies that households i m p l i c i t l y subtract the value of cheques wr i t ten against t h e i r account before the char-tered bank o f f i c i a l l y does so . That i s , we are assuming no ' f l o a t i l l u s i o n ' on the part of the households. 39. The December 1967 f igures were used to construct the January 1968 average. 40. Our choice of p. should r e f l e c t the actual t ransact ions undertaken by the household. Using the non s t a t i c se r ies f o r durable and semi durable good impl ies that (a) consumer purchases of durable goods are -paid for on a rental basis and (b) the rental p r ices consumers face take expected cap i ta l gains in to account. 41. We chose th is i n t e r e s t rate because i t represents the highest a l t e r -nat ive a v a i l a b l e among o f f i c i a l l y publ ished i n t e r e s t r a t e s . It - 172 -should be noted that i f we had used a very much lower r a t e , some rental p r ices f o r i n t e r e s t bearing moneys ( for example, the guar-anteed investment c e r t i f i c a t e s of t r u s t and loan companies, to be discussed in the next sect ion) would become negative in c e r t a i n years (see (3 .33) ) . However, the use of the i n d u s t r i a l bond y i e l d captures the in f luence of the major non l i q u i d asset investment a l t e r n a t i v e (apart from government bonds exc lus ive of Canada Savings Bonds) fac ing households. 42. One problem not deal t with in these estimates concerns the existence of bank serv ice charges. Unfor tunate ly , the only s t a t i s t i c a v a i l a b l e in t h i s regard is the to ta l se rv ice charges pa id . Apart from charges f o r chequable savings deposi ts th is f i g u r e includes a l l current account charges, as well as l e v i e s f o r such items as t r a v e l l e r s ' cheques, safety deposi t boxes, e t c . Rather than attempt any a r b i t r a r y apport ioning among these c a t e g o r i e s , we have ignored the problem a l together . Note, however, that se rv ice charges fo r banks (and other f i n a n c i a l i n s t i t u t i o n s ) are not excluded e n t i r e l y from the complete set of consumer accounts. They appear in ' f i n a n c i a l , l e g a l , and other s e r v i c e s ' in the category 'other s e r v i c e s ' (see sect ion B of t h i s chap te r ) . 43. While the assets of t r u s t c o . s and loan c o . s d i f f e r to some extent , the c h a r a c t e r i s t i c s of t h e i r l i a b i l i t i e s are s u f f i c i e n t l y s i m i l a r to be t reated as one homogeneous s tock . In what f o l l o w s , f o r each category , the l i a b i l i t i e s of both i n s t i t u t i o n s have been added a r i t h m e t i c a l l y , and one i n t e r e s t rate i s used for both. 44. For some periods (and data sources) the data d i s t i n g u i s h 1-6 y e a r s , and over 6 years c a t e g o r i e s . 45. On the other hand, i t i s not c l e a r why a f i rm would hold a current account with a TML company rather than with a chartered bank. One poss ib le reason might be to obtain favourable lending f a c i l i t i e s . However, the great major i ty of TML loans are to the household sec tor . 46. An e r ro r is l i k e l y to be present here, as TML term deposi ts probably form a s i g n i f i c a n t part of ' i n s t i t u t i o n a l ' p o r t f o l i o s . 47. Note that with TML l i a b i l i t i e s , owing to the survey nature of the data sources , major r e v i s i o n s occur often in the publ ished f i g u r e s . Our data se r ies r e f l e c t s these r e v i s i o n s . 48. The f i r s t quarter 1963 average was taken to be the f i g u r e fo r the end of the f i r s t quarter . 49. AFt var ied between 1.1 in 1951 and 1.2 in 1960. 50. The f i r s t quarter 1951 average was taken to be the f igure fo r the end of the f i r s t quar ter . 51. Using the en t i re sample provided unacceptable r e s u l t s , as the estimated proport ions exceeded u n i t y . - 173 -52. This s e r i e s was a lso a v a i l a b l e fo r the en t i re per iod . For the per iod 1951-1974, during which comparisons can be made, the Royal Commission - Bank of Canada se r ies and t h i s s e r i e s were almost i d e n t i c a l . 53. The U n i v e r s i t y of Western Ontario [1965] study in several places re fe rs to the behaviour of the non chequable rate over the per iod 1951-1960. However, the rate i s not l i s t e d anywhere in the Report. 54. Although quar te r ly data on to ta l term deposi ts i s a v a i l a b l e in the Western Ontario study, i t has been already deseasona l ised , and so could not be used to construct a weighted average annual ra te . 55. There are few ownership problems here as Canada Savings Bonds cannot be held by f i r m s , and can be held only to a l im i ted extent ( recent ly ) by i n s t i t u t i o n s . 56. BS700 equals (approximately) four times the actual i n t e r e s t paid/100. 57. Although month end CSB f igures are a v a i l a b l e , s ince BS700 was constructed using quarter end da ta , to preserve cons is tency , we used only the corresponding quar te r ly quant i ty data . Chapter 6 ECONOMETRICS: ESTIMATION AND HYPOTHESIS TESTING In sect ion A of th is chapter, we discuss the stochast ic s p e c i f i -cat ion of the model. We deal f i r s t with the so ca l led ' c l a s s i c a l ' case, where the disturbances are assumed free of any autocorre la t ion, and second, with a model where f i r s t order autocorrelated disturbances are taken into account. In sect ion B we consider the test ing of hypotheses regarding ( i ) the funct ional form of the ind i rec t u t i l i t y funct ion, ( i i ) the s t a b i l i t y of the function over t ime, and ( i i i ) the presence of autocorre lat ion. A. Estimation 1) . ' C l a s s i c a l ' Stochastic Spec i f i ca t ion • The market expenditure share equation for the i t n good at time t derived in Chapter 4 as equation (4.26) s b i j P i %. * - a y * 1:1::::;? ( 4- 2 6> k^i m i^ b ^ " ^ where the time subscripts on p. ( i = l , . . . , n ) and y have been omitted for notational convenience. We assume that the actual shares deviate from the " t rue" shares given by (4.26) by an addi t ive disturbance term, e ^ t ) . That i s , n _ i i b i j P j « P j ' - afly* z 1 bkm pk \ ' - 175 -where T is the to ta l number of observat ions (years) in the sample, (t) is assumed to be due to errors in the u t i l i t y maximising process or in aggregation over goods and consumers. Define an nxl column vector of a d d i t i v e disturbances at time t as (6.2) eft) = [ £ l ( t ) , e 2 ( t ) , . . . , e ^ t ) ] ' t=l , . . . J where ' denotes vector t r a n s p o s i t i o n . It i s assumed that e(t) i s d i s t r i b u t e d normal ly , independent of the var iab les on the r igh t hand s ide of (4 .26) . The fo l lowing propert ies are a lso assumed as regards e ( t ) : (6.3) ( i j E [ E i ( t ) ] = o j : ] ; ; ; ; ; " ( i i ) E [ e ( S ) e ( t ) ' ] = f0 ff00rrl?l 311 Sj z where n i s an nxn covariance matr ix . Assumption ( 6 . 3 ) ( i i ) permits contemporaneous c o r r e l a t i o n among the disturbances at time t , but ru les out any intertemporal , c o r r e l a t i o n . Th is is the so c a l l e d ' c l a s s i c a l ' s t o c h a s t i c s p e c i f i c a t i o n . Since the shares s . ( t ) on the l e f t hand s ide of (4 .26) , by d e f i n i t i o n , add up to un i ty at each observa t ion , i t fo l lows that n E e - ( t ) = 0 at each observa t ion . Furthermore, the disturbance covariance i=l 1 matrix w i l l be s i n g u l a r at each observa t ion , as w i l l the estimated covar iance matr ix . To solve t h i s problem, we a r b i t r a r i l y drop one share equat ion, and def ine the t runctated (n-1) x ( n - l ) disturbance covariance matrix n* as - 176 -(6.4) E [ e * ( s ) £ * ( „ t ) ' ] = { n* s j t a „ S j t where e*(s) and e*(t) are (n-1) x 1 vectors of a d d i t i v e disturbances at times s and t . Def ine the i j * * 1 element of ft* by to... A maximum l i k e l i h o o d "1 3 estimate of t o . . , t o . . , i s given by e, • e . (6.5) = -V—L where e^ and e^ are Txl vectors of computed residuals from the i t n and j t n equat ions , r e s p e c t i v e l y . Q* i s the r e s u l t i n g estimate of ft*. 2) Computational Algori thm Since e*(t) i s normally d i s t r i b u t e d , the log l i k e l i h o o d funct ion may be wr i t ten as (6.6) In L = ^ ln (2TT + 1) - I ln |ft*| Maximising the l i k e l i h o o d funct ion is equivalent to minimising |ft*| . Furthermore, i t may be shown'(Barten [1969]) that the l i k e l i h o o d funct ion i s invar ian t to the equation d e l e t e d , in a s ingu la r system. Thus the parameter est imates , estimates of the various e l a s t i c i t i e s , and a lso the t e s t s t a t i s t i c s are invar ian t to the equation de le ted . The computational a lgori thm we used to minimise |ft*| proceeds as f o l l o w s . The i t e r a t i v e process commences by assuming that ft* = I. Using the Gauss-Newton method with varying step s i z e s , the parameters are estimated by non l i n e a r leas t squares to obtain a new estimate of ft*. - 177 -Using t h i s estimate of ti*, the parameters are reestimated using a genera l ised inverse procedure and a fur ther estimate of ti* i s obta ined. This i t e r a t i v e process continues u n t i l 'convergence' i s a t t a i n e d . 'Convergence' i s declared when both the parameter estimates and the elements of the estimated covariance matrix change by less than some s p e c i f i e d to lerance leve l from one i t e r a t i o n to the next. The to lerance level i s set at 1% in the work reported below. This computational a lgori thm has been programmed into TSP (Time Ser ies Processor ) . The actual computations were c a r r i e d out on an IBM 370-168 model at the U n i v e r s i t y of B r i t i s h Columbia Computing Centre . The proper t ies of the algori thm have been d iscussed by Malinvaud [1972] and Berndt, H a l l , Hal l and Hausman [1974]. It can be shown that i f convergence i s ach ieved , ' the r e s u l t i n g estimator converges numerical ly to ( i ) Z e l l n e r ' s minimum distance est imator , and ( i i ) the maximum l i k e l i h o o d est imator , provided the disturbances are normally d i s t r i b u t e d . One problem with the computational a lgori thm should be noted. O c c a s i o n a l l y , i t may happen that the ' t ransformed' matrix of independent v a r i a b l e s ( i . e . , transformed by a 'weight ing' matr ix , der ived from ti*) i s almost s i n g u l a r . The programme cannot then inver t t h i s matrix and proceeds by c a l c u l a t i n g a 'pseudo i n v e r s e 1 i ns tead , obtained a f t e r de le t ing the 'o f fend ing ' row and column from the matr ix . The change in the parameter corresponding to the omitted element is set to zero at that i t e r a t i o n . Furthermore, there i s no estimate obtained of the standard e r r o r of that parameter. If t h i s 'genera l ised inverse ' problem occurs repeatedly throughout the a lgor i thm, and e s p e c i a l l y i f i t occurs at the l a s t i t e r a t i o n , the r e s u l t i n g estimates must be regarded as qui te - 178 -u n r e l i a b l e . We mention t h i s problem here s ince i t arose in two of the models we estimated (see Chapter 7) . However, apart from these two i s o l a t e d ins tances , the problem was not present . 3) Au tocor re la t ion Assumption ( 6 . 3 ) ( i i ) ru led out the p o s s i b i l i t y of autocorre la ted r e s i d u a l s . Th is s t o c h a s t i c s p e c i f i c a t i o n fo l lows the t r a d i t i o n of most appl ied demand studies using the ' f l e x i b l e ' funct iona l form approach. However, in many of the models we estimated using the ' c l a s s i c a l ' s t o -c h a s t i c s p e c i f i c a t i o n , the computed equation by equation Durbin-Watson s t a t i s t i c s were low. While a r igorous s t a t i s t i c a l i n t e r p r e t a t i o n of these Durbin-Watson s t a t i s t i c s i s not poss ib le in a mult i equation system such as o u r s j an attempt was made to al low f o r au tocor re la t ion in a l imi ted form, as f o l l o w s . F i r s t , rewri te equation (6.1) as (6.7) s , ( t ) = F . [ p ( t ) , y ( t ) ] + e . ( t ) i=l n . 1 1 1 t = l , . . . , T Let us assume the fo l lowing f i r s t order autocorre la ted s tochas t ic s t r u c t u r e , (6.8) e . ( t ) = p . e . ( t - l ) + u . ( t ) i = l , . . . , n 1 1 1 1 t = l , . . . , T where u(t) i s an nxl column vector of a d d i t i v e disturbances u . ( t ) , with the fo l lowing p r o p e r t i e s : (6.9) ( i ) E C u ^ t ) ] = 0 i = l , . . . ,n t = l , . . . , T - 179 -(ID E[u( S)u(t)'] - i*0f&i;i That i s , u.(t) i s assumed to be a ' c l a s s i c a l ' disturbance term with an nxn covariance matrix $ . Writ ing (6.7) for period t - 1 , and mul t ip ly ing by p . , we obtain: (6.10) p . s . f t - l ) = p . F . [ p ( t - l ) , y ( t - l ) ] + e . ( t - l ) Subtracting (6.10) from (6.7) , and rearranging: (6.11) S i ( t ) = F . [p ( t ) , y ( t ) ] - p . F . [ p ( t - l ) , y ( t - l ) ] + P i s i ( t - 1 ) + e . ( t ) - .p.e . f t- l ) o r , using (6.8) : (6.12) s . ( t ) = F . [p ( t ) , y ( t ) ] - P i F i [ p ( t - l ) , y ( t - l ) ] + P i s . ( t - 1 ) + u.( t) i = l , . . . ,n t=l T The transformed equation system (6.12) may now be estimated using the same procedure as described above for the ' c l a s s i c a l ' model, since u^(t) possesses the required c l ass i ca l propert ies. The parameters p.. may be estimated d i r ec t l y along with the parameters of the F^[ ] funct ion. However, there is an important constraint on the system (6.12). Berndt and Savin [1975] have shown that since (6.12) is a s ingular system - 180 -(the shares s t i l l add up to u n i t y ) , p. must have the same value (=p) in a l l equat ions. This r e s t r i c t i o n may be inappropr iate in our model, where the d i f f e r e n t equations represent demands for goods as heterogeneous a s , fo r example, non durab les , durab les , l e i s u r e or money. It i s poss ib le to re lax the r e s t r i c t i o n of the 'd iagona l ' model by al lowing f o r more complex au tocor re la t ion schemes. However, at present t h i s i s computat ional ly i n f e a s i b l e with our model, due to the extreme non l i n e a r i t i e s present . Even in est imat ing the equation system (6 .12) , convergence was at ta ined only with d i f f i c u l t y f o r some models. For most of the models reported in Chapter 7, we reestimated the model in the form (6 .12) , a l lowing f o r a u t o c o r r e l a t i o n . Using the l i k e l i -hood r a t i o tes t (discussed below) the hypothesis that p=0 may be tes ted . In some but not a l l cases the parameter estimates and e l a s t i c i t y estimates changed considerably compared to those obtained using the ' c l a s s i c a l ' est imat ion method. However, fo r the reason j u s t mentioned the estimates in the autocorre la ted vers ion should be regarded with cau t ion . As noted above, most appl ied demand researchers do not appear to have considered poss ib le au tocor re la t ion problems. Our e f f o r t s to deal with i t should be regarded merely as exploratory f i r s t s teps . B. Hypothesis Test ing Three hypotheses were tested f o r our model, as fo l lows: ( i ) H 0 : homothet ici ty (a^ = 0, a l l i ) . H ^ non homothet ic i ty . ( i i ) H 0 : parametric s t a b i l i t y over t ime. H x : unconstra ined. : ( i i i ) H 0 : no f i r s t order au tocor re la t ion (p=0). H1: f i r s t order a u t o c o r r e l a t i o n . For each hypothes is , we c a l c u l a t e d the l i k e l i h o o d r a t i o (LR) - 181 -tes t s t a t i s t i c , computed as minus twice the d i f f e r e n c e between the log of the l i k e l i h o o d funct ion under the nul l hypothesis (H 0) and under the a l t e r n a t i v e hypothesis (H i ) . Th is method thus requires that both constra ined and unconstrained versions be est imated. 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 d i s t r i b u t e d asymptot ica l ly as X 2 with degrees of freedom equal to the d i f f e rence in the number of f ree parameters under H 0 and H x . Hypotheses ( i ) and ( i i ) are qui te s t ra ight forward . However, hypothesis ( i i ) requi res some e l a b o r a t i o n . The problem of maximising the log of the l i k e l i h o o d funct ion in constrained and unconstrained vers ions may be wr i t ten as f o l l o w s : Hi (unconstrained) : (6.13) Max t in L ( e t l , e t 2 , D t l , D t 2 ) 91 »©2 Ho (constrained) : (6.14) Max In L ( e . , , 9. , D. , D . J w . r . t . 1 1 L 1 l z © 1 > ©2 subject to : G t l = 6 t 2 e t i and % denote the set of parameters in the two sub periods indexed by t1 ( r e fe r r ing to the per iod 1 to t j and t 2 ( r e fe r r ing to the per iod t 2 to T ) . The time periods 1 to t1 and t 2 to T are mutually e x c l u s i v e . and D^2 are the corresponding data sets fo r the two sub per iods . Since the two time periods are mutual ly e x c l u s i v e , i f the - 182 -disturbances are assumed to be u n c o r r e c t e d over t ime, the unconstrained problem (Hx) may be rewri t ten as : (6.15) Max ln L(e. , D. ) + Max ln L (e , D. ) w . r . t . Z 1 Z l w . r . t . Z 2 Z 1 9t l 6 t 2 Thus in order to der ive the maximum of the log l i k e l i h o o d funct ion without any const ra in ts imposed, we estimate the model in the two sub p e r i o d s , and add the r e s u l t i n g two log l i k e l i h o o d va lues . The constra ined value of the log l i k e l i h o o d funct ion i s der ived simply by est imat ing one model f o r the e n t i r e time per iod . The d i f f e r e n c e in the number of f ree parameters under H 0 and H x i s , of course , equal to the dimension of e i t h e r e or Q^2 that i s , the number of f ree parameters in the model. In Chapter 7, we report t e s t r e s u l t s of the parametric s t a b i l i t y hypothesis f o r two sub p e r i o d s . It should be noted that the procedure of adding the two log l i k e l i h o o d values as in (6.15) is v a l i d only i f the time periods are mutually e x c l u s i v e . I f th is i s not the ease, then the log l i k e l i h o o d funct ion would be considerably more complex to der ive . Th is s i t u a t i o n a r i s e s in the models with a f i r s t order au tocor re la t ion s t o c h a s t i c s t r u c t u r e , as lagged dependent and independent var iab les appear in the est imat ing equations (see (6 .12) ) . Thus, our tes ts of parametric s t a b i l i t y are confined to models estimated using ' c l a s s i c a l ' assumptions. A fu r ther l i m i t a t i o n of the parametric s t a b i l i t y tes t i s that i m p l i c i t l y t h e , t e s t assumes constancy of the var iance of the disturbance term over the time p e r i o d , i . e , the absence of h e t e r o s c e d a s t i c i t y . Thus, the nul l hypothesis i s r e a l l y composed of two par ts : ( i ) parametric - 183 -s t a b i l i t y , and ( i i ) homoscedast ic i ty . This point should be borne in mind when evaluat ing the r e s u l t s of the t e s t . 3 - 184 -Footnotes - Chapter 6 1. I d e a l l y , one would l i k e to der ive one Durbin-Watson s t a t i s t i c f o r the e n t i r e equation system. Chapter 7 EMPIRICAL RESULTS In th is chapter , empir ical r e s u l t s are reported fo r three groups of models, as summarised in Chapter 3 (sect ion E ) ; A) Consumption- le isure models, B) 'Real-monetary 1 models, and C) L iqu id asset models. Within each of these groups, a number of d i f f e r e n t s p e c i f i c a t i o n s were est imated, using a l t e r n a t i v e groupings of v a r i a b l e s . Furthermore, while our data were constructed on an annual basis f o r the per iod 1947-1974, bet ter r e s u l t s 'were obtained in a l l cases when the models were estimated fo r a shorter time p e r i o d , 1952-1974. Not a l l of the est imat ion experiments w i l l be formal ly presented. However, the main features of the 'o ther ' models ( inc lud ing those estimated f o r the en t i re sample period) are summarised where re levan t . For each model, the expenditure share equations (6.1) were used to estimate the parameters of the under ly ing i n d i r e c t u t i l i t y f u n c t i o n . We report these parameter estimates (and t h e i r associated t - s t a t i s t i c s ) , the log of the l i k e l i h o o d funct ion ( ln L) and the R 2 and Durbin-Watson (D-W) s t a t i s t i c of the (n-1) share equat ions. We a lso l i s t estimates of the p r ice (E . - ) and expenditure ( E . v ) e l a s t i c i t i e s of demand, and the H i c k s - 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 s u b s t i t u t i o n (a..)- These estimates are based on the f i t t e d share values and are evaluated at se lected years in the sample pe r iod . The parameter estimates are examined to d iscover whether the three r e g u l a r i t y condi t ions (namely, p o s i t i v i t y , monotonicity and quas i -convex i ty ) on the i n d i r e c t u t i l i t y funct ion are s a t i s f i e d at each observa t ion . We a lso perform s t a t i s t i c a l tes ts to determine (a) whether the nu l l - 186 -hypothesis of homothet ic i ty of the i n d i r e c t u t i l i t y funct ion can be r e j e c t e d , and (b) whether there is evidence of parametric i n s t a b i l i t y between d i f f e r e n t sub per iods . A l s o , f o r most models, the hypothesis of a f i r s t order au tocor re la t ion scheme i s inves t iga ted . In evaluat ing the reported r e s u l t s from the various models, a number of d i f f e r e n t c r i t e r i a should be borne in mind. From an econometric point of view, the goodness of f i t as measured by the R 2 of ind iv idua l equations i s of concern.^ It i s a lso reassur ing i f the reported Durbin-Watson s t a t i s t i c s do not ind ica te the poss ib le presence of f i r s t order a u t o c o r r e l a t i o n . However, f o r the reasons explained in Chapter 6, in a m u l t i v a r i a t e equation system, ind iv idua l Durbin-Watson s t a t i s t i c s should be in terpreted with c a u t i o n . From an economic point of view, i t i s des i rab le that the estimated i n d i r e c t u t i l i t y funct ion s a t i s f y the regu-l a r i t y condi t ions required f o r consistency with the theory of u t i l i t y maximizing behaviour under ly ing the model. It i s a lso d e s i r a b l e that the estimated e l a s t i c i t i e s not be t o t a l l y 'unreasonab le ' , in some loose sense. However, i t should be noted that no overa l l s i n g l e c r i t e r i o n e x i s t s f o r evaluat ing the 'performance' (absolute o r , r e l a t i v e ) of a p a r t i c u l a r model. Furthermore, as w i l l be s e e n , o c c a s i o n a l l y : d i f f e r e n t c r i t e r i a provide c o n f l i c t i n g assessments. A. Consumption-Leisure Models 1) Consumption Models We began our empir ical i nves t iga t ions by est imating a four good consumption model conta in ing the fo l lowing v a r i a b l e s ; (1) non durab les , (2) s e r v i c e s , (3) semi durab les , and (4) durab les . For t h i s model - 187 -(and those f o l l o w i n g ) , two versions were examined. The f i r s t (the SE model) uses renta l p r i c e data fo r semi durable and durable goods constructed on the assumption of s t a t i c purchase pr ice expecta t ions . The second model (termed the NSE model) al lows for the p o s s i b i l i t y of expected cap i ta l gains in c a l c u l a t i n g semi durable and durable rental p r ice s e r i e s . The four good model (both SE and NSE vers ions) d id not provide 2 good r e s u l t s fo r the 1947-1974 p e r i o d , nor f o r the 1952-1974 per iod . The curvature condi t ions were re jected at a l l observat ions ( a 2 2 was p o s i t i v e in a l l y e a r s ) , and the Durbin-Watson s t a t i s t i c s were extremely low. S i m i l a r r e s u l t s were obtained by Berndt, Darrough and Diewert [1977] f o r t h e i r four good model based a lso on Canadian data fo r the per iod 1947-1971. Thus we decided to explore a three good model derived by aggregating two of the four goods l i s t e d above. The use of a three good model at t h i s stage a lso f a c i l i t a t e d comparisons; with the r e s u l t s of models with add i t iona l goods such as l e i s u r e and money, s ince compu-ta t iona l l i m i t a t i o n s on the to ta l number of var iab les which could be handled forces some aggregation at l a t e r s tages. Inspection of the c o r r e l a t i o n s between the pr ices of the four goods did not provide any strong a p r i o r i grounds (based on H icks ' Aggregation Theorem) f o r d i s c r i m i n a t i n g among var ious poss ib le three good aggregates, as the c o r r e l a t i o n s were a l l roughly the same. Aggregating categor ies considered to be most 'homogeneous' ( in some sense) , i t was f e l t , d i d not lead to any c l e a r l y preferred grouping e i t h e r . Thus i t was decided to inves t iga te in est imat ion the fo l lowing three a l t e r -nat ive models, each conta in ing the three goods l i s t e d in parentheses: model I ((1) non durables + s e r v i c e s , (2) semi durab les , (3) d u r a b l e s ) ; - 188 -model II ((1) non durables + semi durab les , (2) s e r v i c e s , (3) d u r a b l e s ) ; model III ((1) non durab les , (2) s e r v i c e s , (3) durables + semi durab les ) . In each case , the aggregate s e r i e s was constructed using the D i v i s i a aggregation method descr ibed in Chapter 5. Of the three p o s s i b i l i t i e s examined, model I c l e a r l y provided the most favourable r e s u l t s , f o r both SE and NSE v e r s i o n s . While the f i t s t a t i s t i c s in models I - III were genera l ly about the same, the Durbin-Watson s t a t i s t i c s were much lower in II and III ( less than .5 in a l l c a s e s ) . Furthermore, in terms of s a t i s f y i n g the curvature c o n d i t i o n s , model I was f a r super ior to e i t h e r of the two other a l t e r n a t i v e s . Reinspect ion of the r e s u l t s from the four good model of fered add i t iona l support f o r choosing model I, as the p r i n c i p a l ' c u l p r i t ' in the former model was s e r v i c e s . In the four good model, a22,was always greater than z e r o , thus v i o l a t i n g the curvature c o n d i t i o n s . There are good reasons why one might expect serv ices not to perform very w e l l . As was es tab l ished in Chapter 5, there are ser ious conceptual and measurement problems associated with two large components of the serv ices s e r i e s , namely, educational s e r v i c e s , and medical s e r v i c e s . By inc lud ing serv ices with non durables as in model I, the poss ib le adverse e f f e c t s of the 3 serv ices s e r i e s on the r e s u l t s have been d imin ished. Model I was used as the basis fo r extended models contain ing l e i s u r e and money and near money balances. The above aggregation experiments revealed a feature common to a l l v e r s i o n s : the r e s u l t s were much improved when the ear ly post war observat ions f o r 1947-1951 were removed. I n . f a c t , t h i s phenomenon was observed in the more elaborate models as w e l l . . Th is is perhaps not too s u r p r i s i n g . Post war p r ice cont ro ls and t h e i r removal, r e s t r a i n t s on - 189 -consumer durable good purchases and the advent of the Korean war could a f f e c t adversely the a p p l i c a b i l i t y of the model. Furthermore, as was noted in Chapter 5, there are data problems associated with measuring the stock of durable and semi durable goods fo r the ear ly part of the sample pe r iod . Reported r e s u l t s f o r model I (and a l l subsequent models) a r e , t h e r e f o r e , those obtained a f t e r dropping the 1947-1951 observa t ions . The parameter estimates fo r model I appear in Table 7.1 f o r both SE and NSE v e r s i o n s . The model was estimated in unconstrained ( i . e . , non homothetic) form. The f i t as measured by the R 2 s t a t i s t i c s i s qui te high fo r both models, and most c o e f f i c i e n t s are s i g n i f i c a n t (based on ind iv idua l t e s t s ) . However, the Durbin-Watson (D-W) s t a t i s t i c s are r e l a t i v e l y low. We invest iga ted the poss ib le presence of a u t o c o r r e l a -t i o n by reest imat ing the model assuming a f i r s t order autocorre la ted s t o c h a s t i c s t r u c t u r e . For both models, using a l i k e l i h o o d r a t i o t e s t , we could not r e j e c t , at the 99% l e v e l , the hypothesis that p (the f i r s t order au tocor re la t ion c o e f f i c i e n t ) was ze ro . In both c a s e s , the parameter estimates and e l a s t i c i t i e s were a l te red only s l i g h t l y , while the D-W s t a t i s t i c s increased only m a r g i n a l l y . 4 Consider now a comparison of the r e s u l t s from SE and NSE models. In terms of the absence of f i r s t order a u t o c o r r e l a t i o n , the NSE model appears to do bet te r . As regards f i t , as measured by R 2 , the r e s u l t s are ambiguous, depending on which equation is considered. For both models, 5 p o s i t i v i t y and monotonicity were s a t i s f i e d at a l l observa t ions . The curvature requirement (namely, that the matrix of e l a s t i c i t i e s of s u b s t i t u t i o n [a.-] be negative semi d e f i n i t e ) passed f o r a l l years in the SE model. For the NSE model, the i n d i r e c t u t i l i t y funct ion is quasi-convex at a l l observat ions except two, 1973 and 1974, where a 3 3 - 190 -was > 0. This i s qui te i n t e r e s t i n g , as the behaviour of the durable rental p r ice se r ies fo r SE and NSE versions is e n t i r e l y d i f f e r e n t in those years (see Tables 5.9 and 5.12). The SE rental p r i c e s e r i e s continues to r i s e , due mainly to the higher purchase p r i c e of durables ( in p a r t i c u l a r , housing and l a n d ) , while the NSE rental p r i c e f a l l s qui te cons iderab ly , due to the large expected cap i ta l ga ins . However, the quant i ty s e r i e s f o r durables does not exh ib i t any marked d i s c o n - . t i n u i t y . Thus, speak ing) !oose ly , i t would appear that consumers • •: d id not act according to the pred ic t ions of the NSE model at "'least on the evidence presented here. - -r; A p o s s i b l e weakness of the model i s i n d i c a t e d : the model does not consider t ransact ions costs and other fac to rs which may prevent consumers from taking advantage of the f a l l in the rental p r i c e of durab les . It i s i n t e r e s t i n g to note, comparing the f i r s t two columns of Table 7 .1 , that the parameter estimates of the two models appear to be qui te s i m i l a r . With one except ion , e 1 2 (which is s t a t i s t i c a l l y i n s i g n i f i -can t , in any c a s e ) , the signs are the same, and the numerical d i f f e rences are genera l ly s m a l l . The same i s true:of the var ious e l a s t i c i t i e s evaluated at 1955, 1965, and 1974, reported in Table 7.2. A l l the signs are i d e n t i c a l (with the exception of a 2 3 in 1955 and Ogg in 1974). Thus we can conclude that in t h i s model, while the absence of autocorre-l a t i o n appears more pronounced in the NSE v e r s i o n , the cons idera t ion of expected c a p i t a l gains genera l ly does not lead to parameter estimates and e l a s t i c i t i e s numerical ly very d i f f e r e n t from those obtained in the s t a t i c expectat ions case. However, when the expected c a p i t a l gains are very l a r g e , as in 1973 and 1974, the models provide c o n f l i c t i n g r e s u l t s . - 191 -Turning to the issue of temporal s t a b i l i t y , we d iv ided the time per iod in to two approximately equal sub p e r i o d s , 1952-1962 and 1963-1974. The parameter estimates and various e l a s t i c i t i e s from these sub per iod models are reported in Tables 7.1 and 7.2. The l i k e l i h o o d r a t i o s t a t i s t i c s f o r the t e s t of the nu l l hypothesis of i d e n t i c a l parameter values in each per iod are reported in Table 7.3 . The hypothesis that tastes have not a l t e red over time i s r e j e c t e d . Furthermore, while the own e l a s t i c i t i e s of s u b s t i t u t i o n were a l l negative (except a 3 3 i n 1973 and 1974) in the sub period models, the s u f f i c i e n t condi t ions that the matrix [a.-] be negative semi d e f i n i t e were re jected f o r some years in both vers ions of the 1952-1962 m o d e l ? However, i t should be noted that the e l a s t i c i t i e s fo r the same yearscomputed from a l t e r n a t i v e models based on d i f f e r e n t time p e r i o d s , do not appear to d i f f e r much at a l l . For example, the t h i r d and four th columns of Table 7.2 l i s t the e l a s t i c i t i e s f o r 1965 from the 1952-1974 model and the 1963-1974 model. Scanning the two columns ind ica tes a s t r i k i n g numerical s i m i l a r i t y . Thus the r e s u l t s of the l i k e l i h o o d r a t i o tes t may exaggerate the extent of parametric s t r u c t u r a l change. For both models, the e l a s t i c i t i e s in Table 7.2 reveal a s i m i l a r pa t te rn . Durables and non durables + serv ices are own p r i c e i n e l a s t i c , durables more s o . Semi durables are genera l ly un i tary p r ice e l a s t i c . The cross p r ice e l a s t i c i t i e s ( E . . ) i nd ica te that apart from non durables + s e r v i c e s and semi durab les , a l l goods are (gross) complements. These r e l a t i o n s h i p s appear not to have a l te red great ly over t ime. The expenditure e l a s t i c i t i e s ( E ^ ) reveal a c l e a r cut pat tern . Durables are luxury goods, as one might have suspected; however, the expenditure e l a s t i c i t y has decl ined s l i g h t l y over t ime. The expenditure - 192 -e l a s t i c i t y of non durables + serv ices has remained very s tab le throughout the p e r i o d , at j u s t under one, while that of semi durables is qui te low. Turning tof the e l a s t i c i t i e s of s u b s t i t u t i o n , a.., a l l own a., are less • "ij n than zero (except fo r a 3 3 in 1974). Def in ing subst i tu tes and complements in the Hicksian sense ( i . e . , holding u t i l i t y cons tan t ) , non durables + serv ices are subst i tu tes f o r both semi durables and durab les . The sub-s t i t u t i o n r e l a t i o n s h i p between semi durables and consumption + serv ices i s qui te s t r o n g , cons is tent with the p o s i t i v e cross p r ice e l a s t i c i t y , E 1 2 , noted above. The complementarity r e l a t i o n s h i p between.durables and semi durables appears to be qu i te weak. F i n a l l y , we tested f o r homothet ic i ty . For both models in the three time p e r i o d s , 1952-1974, 1952-1962, and 1963-1974, the r e s t r i c t i o n o was d e c i s i v e l y r e j e c t e d . The values of the l i k e l i h o o d r a t i o tes t s t a t i s t i c s appear in Table 7.3. 2) The Le isure Model Our next step was to add l e i s u r e to the three good model descr ibed above and estimate a four good model conta in ing the fo l lowing v a r i a b l e s ; (1) non durables + s e r v i c e s , (2) semi durab les , (3) durab les , (4) l e i s u r e . The parameter estimates of the model are l i s t e d in Table 7.4 and the corresponding e l a s t i c i t i e s in Table 7.5. Consider f i r s t the estimates based on the 1952-1974 data . Aga in , the f i t was good, but in t h i s model, the r e l a t i v e performance of the SE and NSE versions in terms o f both R 2 and Durbin-Watson s t a t i s t i c s was ambiguous. For both models, the curvature condi t ions were passed at a l l observa t ions , i n c l u d i n g , f o r the NSE model, the years 1973 and 1974. This - 193 -is an i n t e r e s t i n g r e s u l t , as the curvature condi t ions were re jected prev ious ly fo r these two years in the NSE model without l e i s u r e . Upon reexamination of the l e i s u r e index we f i n d that in 1973 there was.a small r i s e in per cap i ta l e i s u r e consumed due to a s l i g h t dec l ine in per cap i ta manhours worked (see Table 5.14). The increased demand for l e i s u r e appears to have had an e f f e c t upon the choice among consumption goods, in p a r t i c u l a r upon the demand f o r durable goods. Including l e i s u r e removes the 1973-1974 'problem' with the NSE model. It seems that the omission of l e i s u r e , rather than omission of t ransact ions cost f ac to rs (as we had surmised e a r l i e r ) may well have been the cause of t h i s problem. In genera l , i t appears that the i n c l u s i o n of l e i s u r e has not a l te red q u a l i t a t i v e l y the estimated (gross) subst i tut ion/complementar i ty r e l a t i o n s among the three consumption goods. For example, the 1965 estimates f o r E . . ( i , j = l , 2 , 3 ) a l l have the same sign in the three good as in the four good model (compare Tables 7.2 and 7 .5 ) , except fo r E 23 in the SE model. However, the s i z e of the e l a s t i c i t i e s d i f f e r s somewhat in both models. In p a r t i c u l a r , the own p r i c e e l a s t i c i t y of semi durables is considerably more negative in t h e . l e i s u r e model. More v a r i a t i o n i s apparent when we examine the e l a s t i c i t i e s of s u b s t i t u t i o n . Again looking at the 1965 f i g u r e s , d 2 3 changes sign in both SE and NSE models, while the s ign of a a l t e r s in the SE model. The e l a s t i c i t i e s of s u b s t i t u t i o n are a l l l a rger in absolute value in the four good model. However, t h i s i s to be expected. Recal l the expression fo r a., der ived in Chapter 4, - 194 -In a la rger model, w i l l be smal ler and thus tend to increase the s i ze of a.. fo r given E. . . As a general point i t should be s t ressed that a l l the e l a s t i c i t y estimates discussed are point est imates. It would be necessary to der ive a corresponding p r o b a b i l i t y d i s t r i b u t i o n before making any r i g o r o u s : s t a t i s t i c a l statements about the d i f fe rences in the e l a s t i c i t y estimates between the two models. The expenditure e l a s t i c i t i e s (which are not s t r i c t l y comparable s ince y , to ta l expendi ture, i s d i f f e r e n t ) exh ib i t the same pattern as in the three good model, a l b e i t to a more marked extent . The expenditure e l a s t i c i t y f o r durables is now h igher , exceeding 2 (but s t i l l d e c l i n i n g over time) while that of semi durables i s lower. The expenditure e l a s t i c i t y of non durables + serv ices remains about the same, j u s t less than one. Le isure i t s e l f i s always a gross complement f o r non durables + s e r v i c e s . The r e l a t i o n s h i p between l e i s u r e and durables is asymmetric. An increase in the p r ice of l e i s u r e (the wage rate) i s accompanied by a decrease in the demand f o r durab les , while an increase in the rental p r i c e df durab les ' l eads to an increase in the demand fo r l e i s u r e . The own pr ice e l a s t i c i t y f o r l e i s u r e i s around -.8. Turning to the e l a s t i c i t i e s of s u b s t i t u t i o n , l e i s u r e is a subs t i tu te fo r a l l other goods. It i s useful to r e c a l l the r e l a t i o n s h i p between the var ious e l a s t i c i t i e s stated in Chapter 4: Only i f cKj (corresponding to the ' s u b s t i t u t i o n ' e f f ec t ) > E.. (the 'income e f f e c t ' ) > 0 w i l l . have the same s ign as a^.. The d i f f e rence in many - 195 -cases between the signs of the e l a s t i c i t i e s of s u b s t i t u t i o n and the gross pr ice e l a s t i c i t i e s i l l u s t r a t e s the strength of the 'income e f f e c t ' in t h i s model. We next invest igated the model f o r s t a b i l i t y . The nul l hypothesis that the parameters were the same in both sub periods was re jec ted d e c i s i v e l y . The parameter estimates f o r the two sub periods and the re levant l i k e l i h o o d r a t i o tes t s t a t i s t i c s are in Tables 7.4 and 7.6. We note that curvature was re jected f o r the 1952-1962 models in both SE and NSE v e r s i o n s , while i t passed in both models fo r the 1962-1974 per iod . The homothet ici ty t es t s t a t i s t i c s are summarised in Table 7.6 . The r e s t r i c t i o n of un i tary expenditure e l a s t i c i t i e s was re jected fo r a l l periods except in the SE model for the per iod 1963-1974. This corresponds to the c a l c u l a t e d expenditure e l a s t i c i t i e s based on that sub period model (which we have not l i s t e d ) . ^ However even looking at the estimates based on the 1952-1974 model (Table 7.5) we can see that they are tending towards a value of un i ty over t ime. Since the Durbin-Watson s t a t i s t i c s f o r both SE and NSE vers ions of the 1952-1974 model were low, we invest igated the poss ib le presence of f i r s t order au tocor re la t ion in the model. The r e s u l t s proved i n t e r -e s t i n g . Unl ike the three good consumption model, where we could n o t . r e j e c t the hypothesis that p=0, in the l e i s u r e model, p turned out to be s i g n i -f i c a n t l y d i f f e r e n t from z e r o , in both SE and NSE v e r s i o n s . ^ In a d d i t i o n , the R 2 and Durbin-Watson s t a t i s t i c s improved n o t i c e a b l y , although the l a t t e r should be in terpreted with some caut ion owing to the presence of the lagged dependent v a r i a b l e in the est imat ing equat ions. However, while the curvature condi t ions continued to be s a t i s f i e d in a l l years - 196 -for the SE au tocor re la t ion model, they were re jected in the NSE model fo r the years 1970-1974. For i l l u s t r a t i v e purposes, in Table 7 .7 , we l i s t the 1965 e l a s t i c i t i e s f o r both the au tocor re la t ion (p^O) and ' c l a s s i c a l ' (p=0) v e r s i o n s . E s p e c i a l l y for the SE model, the estimates are very s i m i l a r (only E 2 t + changes s i g n ) . On the other hand, in the NSE model, there are some changes: E 1 2 , E 2 1 , E 2 1 f and a 2 3 a l t e r s i g n . As w e l l , E 2 2 i s much lower, and E 2 y much h igher , in the NSE au tocor re la t ion model. It would appear that v a r i a b l e 2, semi durab les , i s a f f e c t e d , but the other e l a s t i c i t i e s remain about the same. It i s i n t e r e s t i n g to note that equation 2 had the highest Durbin-Watson s t a t i s t i c in the ' c l a s s i c a l ' model (see Table 7 .5 ) . This tends to confirm the view that ind iv idua l Durbin-Watson s t a t i s t i c s are not a r e l i a b l e guide as to what may happen when a model is reestimated to al low fo r a u t o c o r r e l a t i o n . Apart from the actual numerical est imates , we may summarise the main q u a l i t a t i v e conclusions from est imat ing the consumpt ion- le isure models as f o l l o w s : ( i ) Both the 'consumption on ly ' and the consumpt ion- le isure model perform remarkably well over the en t i re 1952-1974 time p e r i o d , in that the r e s t r i c t i o n s required f o r the theory to be v a l i d are passed at v i r t u a l l y a l l observat ions . ;The only two years in which the assumptions of the model are re jec ted are 1973 and 1974 in the NSE vers ion of the consumption model. However, the i n c l u s i o n of l e i s u r e removes t h i s problem. ( i i ) The SE and NSE models do not provide v a s t l y d i f f e r e n t numerical r e s u l t s , p a r t i c u l a r l y with respect to the consumption good model. However, the a u t o c o r r e l a t i o n - s t a t i s t i c s f o r the consumption model are - 197 -somewhat higher with the NSE v e r s i o n , ( i i i ) E s p e c i a l l y in the case of the NSE model, the estimated r e l a t i o n s h i p s between consumption goods are very s tab le with respect to the in t roduct ion of the fourth good, l e i s u r e . ( iv) There i s some evidence of parametric i n s t a b i l i t y . As w e l l , i t appears that ne i ther the consumption nor l e i s u r e models do very well f o r the 1952-1962 period a lone , in that the curvature condi t ions are re jected f o r most.years in models estimated f o r that sub per iod . (v) There i s no evidence in the consumption good model to support the hypothesis Of a f i r s t order au tocor re la t ion s t ruc ture which could ind ica te the need, f o r example, to incorporate d i s e q u i l i b r i u m adjustments in to the model. On the other hand, f i r s t order au tocor re la t ion does appear to be present in the l e i s u r e model. Incorporat ing t h i s into our est imat ing technique does not lead to q u a l i t a t i v e l y v e r y . d i f f e r e n t estimates of pre ferences , as ind icated by the e l a s t i c i t y measures. However, the actual numbers do a l t e r somewhat. B. Real-Monetary Models In t h i s s e c t i o n , we inves t iga te the r e s u l t s obtained when monetary serv ices are added as a va r iab le to the real models j u s t d iscussed . We begin by cons ider ing aggregate 'money', def ined as currency plus personal chequing accounts and personal savings deposi ts held in chartered banks. 1) Aggregate Money Model We f i r s t contemplated the add i t ion of money to the four good consumpt ion- le isure model. However, i t was f e l t that t h i s would be - 198 -Table 7.1 PARAMETER ESTIMATES: THE THREE GOOD CONSUMPTION MODEL (NON HOMOTHETIC VERSION) (1) Non durables + s e r v i c e s , (2) Semi durab les , (3) Durables. SE; s t a t i c expectat ions imposed NSE; non s t a t i c expectat ions i i 1 2 1 3 ' 2 2 J 2 3 3 3 1952- •1974 1952- -1962 1963- -1974 SE NSE SE NSE SE NSE - .102 - .106 - .114 -.098 -.132 -.132 (-5.919) ( T 7 . 0 3 8 ) (-5.671) (-10.322) (-2.234) (-2.287) - .203 T . 1 9 2 -.141 -.151 -.204 T . 1 8 0 (-17.209) (-16.636) (-7.287) (-11.270) (-3.1993) (-4.902) .305 .298 .254 .249 .336 .312 (23.097) (28.165) (13.963) (25.856) (6.811) (6.462) - .148 - .150 - .252 -.153 - .216 - .159 (-1.413) (-2.962) (-2.473) (-3.332) (-1.511) ( T 1 . 5 8 1 ) - .063 - .011 .160 .101 - .070 -.017 (-1.234) (.241) (2.199) (2.641) (-.861) (-.198) .728 .681 .606 .599 .791 .706 (10.714) (31.036) (11.507) (31.794) (8.041) (17.717) - .032 - .113 - .299 -.253 -.001 ^.069 (-.810) (-2.411) (-4.619) (-6.220) (-.002) (-.766) .153 .163 .222 .231 .128 .151 (5.205) (14.469) (6.136) (11.451) (2.618) (7.988) - .454 -.447 - .426 - .456 -.481 - .455 (-8.300) (-26.317) (-8.846) (-27.962) (-5.616) (-15.973) ln L 147.089 148.998 101 .512 110.736 84 .613 86 .793 R 2 i .912 .903 .975 .994 .837 .891 R 2 2 .986 .990 .993 .995 .935 .957 D-Wj .873 1.048 1 .368 2.316 1 .030 1 .143 D-W2 1.294 1.478 1 .128 2.023 1 .164 1 .697 Note: ( i ) t s t a t i s t i c s are in parentheses. n n n ( i i ) due to rounding, the r e s t r i c t i o n s E a". = 0, and E E 3 = i=l 1 k=l m=l km may not be s a t i s f i e d exact ly in t h i s and fo l lowing t a b l e s . - 199 -Table 7.2 ELASTICITY ESTIMATES: THE THREE GOOD CONSUMPTION MODEL (NON HOMOTHETIC VERSION) (1) Non durables + s e r v i c e s , (2) Semi durab les , (3) Durables SE; s t a t i c expectat ions imposed NSE; non s t a t i c expectat ions 1955* 1965* 1974* rr\ ce E l a s t i c i t i e s 1952-1974 1952-1962.; 1952-1974 1963-1974 1952-1974 1963-1974 E i i SE NSE - .804 - .783 - .717 -.781 -.813 -.787 - .755 -.781 - .826 -.783 - .765 -.778 E12 SE NSE .113 .056 -.041 .006 .116 .060 .120 .086 .119 .069 .128 .095 E i 3 SE NSE -.221 - .182 -.143 - .140 -.233 - .204 - .276 - .218 - .236 - .233 . - .287 - .250 E 2 i SE NSE .716 .518 . .053 .258 . .748 .525 .764 .608 .787 .547 .800 .633 E 2 2 SE NSE -1.034 - .809 - .323 - .456 -1.046 -.797 -1.152 -.951 -1.036 - .800 -1 .178 - .971 , E 2 3 SE NSE - .055 - .102 - .290 -.327 - .139 -.202 - .004 - .160 -.231 -.312 - .108 -.257 E31 SE NSE - .876 - .945 - .649 - .763 - .699 - .812 - .818 - .873 - .549 -.843 - .649 - .906 E 3 2 SE NSE -.228 - .286 - .328 - .410 - .197 -.262 - .158 - .234 - .175 - .284 -.137 -.253 E 3 3 SE NSE - .477 - .447 -.507 - .395 - .496 -.397 - .456 - .385 -.548 -.247 -.514 -.232 continued - 200 -Table 7.2 (continued) 1955* 1965* 1974* 1952-1974 1952-1962 1952-1974 1963-1974 1952-1974 1963-1974 Expenditure E l a s t i c i t i e s E ly SE NSE .912 .909 .902 .915 .931 .930 .910 .913 .943 .947 .925 .934 E 2 y / SE NSE .373 .392 .567 .525 .437 .474 .432 .503 .480 .565 .486 .594 E 3 y SE NSE 1.580 1.679 1 1 .484 .568 1.391 1.471 1.432 1.492 1 1 .273 .374 1.299 1.391 E l a s t i c i t i e s of S u b s t i t u t i o n a SE NSE - .479 - .386 _ .345 .378 -.514 - .376 - .425 - .379 - .592 .325 - .504 - .330 o SE NSE 1.612 1.249 .646 .952 1.766 1.346 1.785 1.510 1 1 .941 .453 1.980 1.623 °13 S E NSE .065 .116 .357 .304 .149 .122 - .016 .046 .252 .005 .088 - .080 a SE NSE 6.040 4.494 -1 -2 .424 .217 -7.117 -5.005 -7.953 -6.107 -8 -5 .184 .272 -9.247 -6.472 a„, SE NSE .164 -.051 _ .538 .897 - .030 - .329 -.283 -.131 _ .194 .696 .171 - .448 oso SE 3 NSE - .244 - .270 - .451 .151 - .268 - .103 - .100 - .036 - .330 .375 - .198 .450 * The e l a s t i c i t i e s are evaluated at each of these y e a r s . The two columns underneath the year are the e l a s t i c i t i e s ca lcu la ted from models estimated f o r d i f f e r e n t time per iods . - 201 -Table 7.3 TEST STATISTICS, THE THREE GOOD CONSUMPTION MODEL (1) Non durables + s e r v i c e s , (2) Semi durab les , (3) durables . Null Hypothesis Ho L ike l ihood Degrees c r i t i c a l r a t i o tes t of value s t a t i s t i c freedom (99%) 1. Parametric S t a b i l i t y a. U1952-1962) 1 (1963-1974)' j (1952-1962) = B i j (1963-1974) a l l i ) ) , a l l i , j ) SE NSE 78.072 95.062 7 7 18.48 18.48 2. Homotheticity a. = 0, a l l i SE NSE 1952-1974 96.876 1952-1962 40.922 1963-1974 36.038 1952-1974 117.462 1952-1962 61.926 1963-1974 37.590 2 2 2 2 2 2 9.21 9.21 9.21 9.21 9.21 9.21 - 202 -Table 7,4 PARAMETER ESTIMATES, THE. FOUR GOOD CONSUMPTION-LEISURE MODEL (NON HOMOTHETIC VERSION) (1) Non durables and s e r v i c e s , (2) Semi durab les , (3) Durables, (4) Le isure a3 '11 ' 1 2 13 'lk 322 523 5 2 4 J33 534 1952- •1974 1952- 1962 1963- 1974 SE NSE SE NSE SE NSE - .014 .119 .378 .723 - .099 -.677 (-.071) (.459) (1.935) (6.267) (-.367) (11.863) - .235 - .244 -.178 .027 -.044 .095 (-2.443) (-2.338) (-1.474) (.292) (-.274) (1.865) .418 .440 .688 .884 .281 .517 (2.689) (3.785) (3.475) (8.790) (.915) (3.160) - .169 - .314 - .888 -1.635 - .138 .065 (-.474) (-.777) (-1.953) (-7.984) (-.255) (.120) - .248 - .219 - .389 r.337 - .206 -.091 (-3.886) (-3.309) (-7.649) (-14.720) (-2.423) (-1.033) - .118 - .102 .010 .016 - .119 -.044 (-4.770) (-3.032) (.219) (.613) (-2.744) (-rl .507) .413 .320 .339 .335 .415 .261 (11.528) (12.380) (9.261) (20.516) (6.818) (6.632) .300 .396 .486 .540 .234 .056 (3.782) (4.266) (5.505) (10.055) (1.870) (.411) .048 .013 - .105 - .155 -.003 - .135 (1.556) (.289) (-1.519) (-2.661) (-.035) (-4.439) .033 .064 .091 .144 .048 .088 (1.958) (5.225) (5.586) (9.217) (1.445) (10.135) .066 .049 .045 .122 .153 .205 (1 .616) (1.144) (.700) (2.131) (2.509) (11.223) -.351 - .310 - .449 -.491 -.331 -.331 (-7.592) (-13.111) (-7.291) (-18.109) (-3.756) (-9.513) .202 .208 .381 .440 .129 .285 (2.754) (3.769) (4.132) (8.267) (.957) (5.187) -.238 -.352 -.762 -1.152 - .179 - . 147 (-1.359) (-1.737) (-3.190) (-9.083) (-.626) (-.513) continued - 203 -Table 7.4 (continued) 1952-SE -1974 NSE 1952-SE •1962 NSE 1963 SE -1974 NSE In L 218.527 211.427 148.648 156.803 133.601 141.197 R 2 i .920 .839 .916 .850 .916 .760 *\ .974 .980 .984 .969 .941 .985 R 2 3 .939 .919 .875 .860 .873 .901 D-Wi 1.227 1.054 1.952 1.937 2.574 2.036 D-W2 1.035 1.562 2.050 1.864 1.013 1.255 D-W3 .742 .806 2.111 1.880 1.057 .873 - 204 -Table 7,5 ELASTICITY ESTIMATES, THE FOUR GOOD CONSUMPTION-LEISURE MODEL (NON HOMOTHETIC VERSION) (1) Non durables and s e r v i c e s , (2) Semi durab les , (3) Durables , (4) Le isure 1955 1965 1974 Pr ice E l a s t i c i t i e s SE NSE SE NSE SE NSE E l l . .818 - .863 - .792 - .850 -.754 -.841 El 2 .179 .152 .197 .164 .233 .178 E13 .126 .006 - .212 - .085 - .296 -.182 Ei *+ .010 - .258 - .078 - .200 - .087 - .130 E 2 l .895 .872 1 .046 .967 1.248 1.057 E22 -1 .337 -1.163 -1 .416 -1.196 -1.537 -1.247 E23 .262 .123 .194 - .005 .134 - .155 E 2 4 .006 .038 - .004 .047 - .095 .060 E31 .824 -.652 - .807 -.707 - .739 -.891 E32 .062 - .180 - .065 - .194 - .064 - .239 E33 -1 .162 -1.431 - .939 -1.068 - .835 -.641 E34 .316 - .476 - .211 - .379 -.104 - .386 Em .054 - .124 - .038 -.082 - .047 - .034 Ei+2 .058 - .038 - .058 - .033 -.061 - .026 E « .127 .125 .125 .107 .138 .067 Ei+u .755 T . 5 7 0 - .813 -.677 - .849 - .753 Expenditure E l a s t i c i t i e s Eiy .864 .962 .886 .971 .903 .975 E 2 y .187 .131 .214 .187 .250 .285 E 3 y 2 .364 2.740 2 .023 2.347 1.742 2.156 Ei+v .740 .608 .784 .684 .819 .745 continued - 205 -Table 7,5 (continued) 1955 1965 1974 SE E l a s t i c i t i e s of Subs t i tu t ion NSE SE NSE SE NSE a n -1.374 -1.330 -1.363 -1.353 -1.376 -1.310 CT12 2.636 2.447 3.174 2.831 4.023 3.156 °13 .109 1.007 - .262 .415 -.491 -.263 °lk .592 .277 .677 .462 .676 .653 CT22 13.021 -11.236 -16.244 -13.349 -20.314 -14.986 <?23 1.756 .979 1.263 .154 .881 - .766 a24 .169 .232 .11:3 .306 .002 .433 a3 3 -4.609 -7.172 -3.056 -4.622 -2.194 -2.203 ° 3 k 1.502 1.476 1.461 1.382 1.469 1.200 -1.320 - .905 -1.381 -1.309 -1.400 -1.122 - 206 -Table 7.6 TEST STATISTICS, THE FOUR GOOD CONSUMPTION-LEISURE MODEL (1) Non durables + s e r v i c e s , (2) Semi durab les , (3) Durables, (4) Le isure „2 Null Hypothesis Ho 1. Parametric S t a b i l i t y ?i(1952-1962) ? a i ( 1 9 6 3 - 1 9 7 4 ) ' A 1 1 1 J B i j (1952-1962) " B i j (1963-1974)' a 1 1 1 , J ' * SE NSE L ike l ihood r a t i o tes t s t a t i s t i c 127.444 173.146 Degrees of freedom 12 12 c r i t i c a l value (99%) 26.22 26.22 2. Homotheticity a.j = 0, a l l i SE NSE 1952-1974 1952-1962 1963-1974 1952-1974 1952-1962 1963-1974 22.260 20.381 1 .696 29.660 16.178 15.488 11.34 11.34 11.34 11.34 11.34 11.34 - 207 -Table 7.7 COMPARATIVE ELASTICITY ESTIMATES, FOUR GOOD LEISURE MODEL, WITH (pfO) AND WITHOUT (P=0) FIRST ORDER AUTOREGRESSIVE ESTIMATION TECHNIQUE 1 9 6 5 Pr ice E l a s t i c i t i e s SE (P=0) (p#3) NSE (P=0) (p/0) E n - .792 -.762 - .850 - .569 E12 .197 .165 .164 - .025 El3 - .212 -.222 - .085 - .036 E ^ - .078 - .072 - .200 -.127 E 2 l 1.046 .924 .967 - .176 E 2 2 -1.416 -1.294 -1.196 -.288 E 2 3 .194 .107 - .005 -.202 E 2 t t - .004 .083 .047 -.288 E31 - .807 -.752 -.707 - .615 E 3 2 - .065 - .093 - .194 - .225 E 3 3 - .939 - .905 -1.068 - .939 E 3 4 -.211 - .076 - .379 - .415 E m - .038 - .066 -.082 - .124 E1+2 - .058 -.042 -.033 - .049 E 4 3 .125 .137 .107 .056 Ei+i+ - .813 -.914 - .677 -.653 Expenditure E l a s t i c i t i e s E ^y E * y E - y hy .886 .889 .971 .758 .214 .180 .187 .955 2.023 1.826 2.347 2.194 .784 .884 .684 .771 continued - 208 -E l a s t i c i t i e s of Subs t i tu t ion a 1 2 ° 1 3 alk ° 2 2 a 2 3 a2k a 3 3 a 3 4 Table 7.7 (continued) 1 9 6 5 SE NSE (p=0) (p^O) (p=0) (pfO) -1.363 -1.266 -1.353 -.792 3.174 2.796 2.831 .475 -.262 -.304 .415 .521 .677 .698 .462 .433 -16.244 -14.752 -13.349 -2.314 1.263 .757 .154 -.361 .113 .402 .306 .218 -3.056 -3.044 -4.622 -3.939 1.461 1.622 1.382 1.345 -1.381 -1.558 -1.309 - .899 - 209 -unwise, due to the number of parameters (.18) in a f i v e good model r e l a t i v e to the maximum number of observa t ions , 23, Limited experimentation confirmed the d i f f i c u l t i e s of obta in ing convergence. Furthermore, with a f i v e good model, i t would have been d i f f i c u l t to al low f o r autocorre-l a t i o n , and v i r t u a l l y impossible to obtain estimates fo r shorter time per iods . >As~ w e l l , \money subst i tu tes remained to be added to the model and some aggregation would then have been abso lu te ly necessary, as a s ix good model was computat ional ly i n f e a s i b l e . The f i r s t four good monetary model we attempted to estimate was obtained by combining semi-durables with non durables + s e r v i c e s , and adding money as the new fourth good. However, th is model could not be estimated s a t i s f a c t o r i l y , as the 'genera l ised i n v e r s e 1 problem c o n t i n u a l l y occurred. The estimates obtained were thus extremely u n r e l i a b l e . We then s e t t l e d on adding money to the pure consumption model, and omit t ing l e i s u r e . The model then consis ted of four goods; (1) non durables + s e r v i c e s , (2) semi durab les , (3) durab les , (4) money. While t h i s model does not al low us to examine poss ib le r e l a t i o n s h i p s between the demand f o r l e i s u r e and the demand f o r monetary s e r v i c e s , i t has the advantage of f a c i l i t a t i n g d i r e c t comparison between the r e s u l t s of the three good consumption model, and those of the i d e n t i c a l model, but with money i n -cluded as an add i t iona l v a r i a b l e . We f i r s t estimated t h i s model f o r 1952-1974 assuming no auto-c o r r e l a t i o n . In genera l , the r e s u l t s were d i s a p p o i n t i n g . The curvature condi t ions were s a t i s f i e d in only s ix of the twenty three y e a r s , and in the remaining y e a r s , the own p r i c e e l a s t i c i t y o f money was p o s i t i v e . Furthermore, the e l a s t i c i t y estimates appeared to be qui te unstable over t ime. - 210 -Since the Durbin-Watson s t a t i s t i c s were very low in both models, i t was decided to reestimate them with a f i r s t order autoregressive e r ror s t r u c t u r e . The parameter estimates of the au tocor re la t ion models are reported in Table 7 .8 . p (the f i r s t order au tocor re la t ion c o e f f i c i e n t ) was s i g n i f i c a n t ^ on the basis of both an ind iv idua l t t e s t and the l i k e l i -hood r a t i o t e s t . The corresponding e l a s t i c i t y estimates are l i s t e d in Table 7 .9 . The allowance f o r a u t o c o r r e l a t i o n , in t h i s model, made a substant ia l d i f f e r e n c e to the r e s u l t s . The curvature condi t ions were s a t i s f i e d f o r each year except 1974 in the SE model, and f o r a l l years in the NSE model. Th is overa l l r e s u l t lends considerable support to the approach of analys ing the demand f o r money within the d i r e c t u t i l i t y framework. Furthermore, the use of the NSE rental p r i c e se r ies appears to have captured something of the r e l a t i v e p r ice change between monetary serv ices and durable goods which occurred as a r e s u l t of i n f l a t i o n a r y expectat ions in 1974. The f i r s t issue to consider i s whether the i n c l u s i o n of money a f f e c t s the estimated r e l a t i o n s h i p s among consumption goods. Comparing the e l a s t i c i t y estimates f o r E . . ( i , j = l , 2 , 3 ) in Tables 7.2 and 7 .9 , the answer is y e s , to a l im i ted extent . For example, in 1965, f o r both SE and NSE models (1952-1974), E i 2 , E 2 i and a 23 change s i g n , depending on whether or not money i s included in the ' r e a l ' consumption model., A lso the s ign of E 2 3 a l t e r s in the SE model. However, i t should be remembered that the l a t t e r model was estimated without any allowance f o r auto-c o r r e l a t i o n (p proved to be i n s i g n i f i c a n t ) and th is may d i s t o r t the comparison to some extent . Th is note of caution is re in forced by the f a c t that in the l e i s u r e model, as was seen e a r l i e r , many of the - 211 -e l a s t i c i t y estimates invo lv ing va r iab le 2, semi durab les , a l te red s ign when au tocor re la t ion was taken into account, In add i t ion ...to the 1974 curvature issue mentioned above, the SE and NSE vers ions provide qui te d i f f e r e n t numerical est imates. The signs of E 1 3 , E 23, a 1 2 , o"23, °2k d i f f e r between the two models, f o r one or more of the years 1955, 1965, and 1974. We turn now to consider the c h a r a c t e r i s t i c s of the demand f o r money funct ion which has been est imated. Of i n t e r e s t in t h i s regard is how our r e s u l t s compare with those obtained by other researchers ( in p a r t i c u l a r those deal ing with Canadian data) . The own p r i c e e l a s t i c i t y of monetary serv ices i s negative throughout the p e r i o d , which i s r e a s s u r i n g , and tends to d e c l i n e through.t ime. The average e l a s t i c i t y over the 1952-1974 per iod (ca lcu la ted as the ar i thmet ic mean of the e l a s t i c i t i e s in each year) was - .867 in the SE model and - .850 in the NSE model. The c l o s e s t Canadian r e s u l t s with which to compare our f igures are those obtained by C l in ton [1973; Table 1] , Using quar ter ly data fo r the 12 13 period 1955-1970, he estimated long run i n t e r e s t e l a s t i c i t i e s of money of - . 3 0 , - . 5 0 , and - . 7 9 , depending on whether the rate on 90 day sales f inance paper, the rate on one to three year government bonds or the rate on 10 year and over government bonds was used. Money was def ined as- to ta l (Currency,- outstanding plus to ta l demand deposi ts ( inc lud ing personal chequing accounts) plus personal savings d e p o s i t s . Our d e f i n i t i o n of money is d i f f e r e n t s ince we include only that por t ion of currency and demand deposi ts estimated to be held by households. Our rental p r i c e s e r i e s was constructed using a ' l o n g ' r a t e , the i n d u s t r i a l bond y i e l d as a discount ra te . It i s i n t e r e s t i n g to note that the average i n t e r e s t e l a s t i c i t y we obtained is qui te c lose to C l i n t o n ' s ten - 212 -year bond rate e l a s t i c i t y . The expenditure e l a s t i c i t y f o r monetary serv ices behaves in a very i n t e r e s t i n g manner in our model, E 4 i s always much less than one and dec l ines sharply over t ime. In f a c t , in the SE model, during 1972-1974, money has become an i n f e r i o r good. However, t h i s r e s u l t i s not that s u r p r i s i n g . Our model is fundamentally a t ransact ions cost model and a l l models of the Baumol-Tobin type pred ic t (at l e a s t fo r the i n d i v i d u a l ) economies of sca le in money ho ld ings . It would be s l i g h t l y worrying i f , in f a c t , our model produced the r e s u l t that money was a luxury good. The f a c t that the expenditure e l a s t i c i t y has been d e c l i n i n g over time i s qui te p l a u s i b l e a l s o , as the growth o f , f o r example, c r e d i t cards would tend to reduce g rea t ly the need f o r money f o r t ransact ions purposes. On t h i s i s s u e , i t i s worth noting the comments by L a i d l e r [1969], surveying e a r l i e r empir ical work on the demand f o r money, "There i s only the s lenderest evidence coming v i r t u a l l y e x c l u s i v e l y from post war da ta , that there might be economies of sca le in money holding . . . on the whole then , the propos i t ion that there are economies of sca le in money holding that in f luence the aggregate demand f o r money funct ion i s one that waits to be es tab l ished in the face o f what, at the moment, appears to be a f a i r amount of contrary ev idence." ( L a i d l e r [1969; 90, 107]) I f any economies of sca le are present , one might expect that they would become more pronounced in recent y e a r s . Our r e s u l t c e r t a i n l y supports t h i s view. Although our expenditure e l a s t i c i t i e s are much lower than the long run e l a s t i c i t y obtained f o r Canada by C l in ton [1973], i t should be remembers that h is model imposed the r e s t r i c t i o n of a constant income e l a s t i c i t y , and thus f a i l s to capture the poss ib le e f f e c t of growing economies of sca le in money t r a n s a c t i o n s . Clark [1973], using annual data f o r the per iod 1927-1965 (excluding 1941-1945) reports low e l a s t i c i t i e s - 213 -of a s im i la r order of magnitude to ours with respect to current income. However, i t i s d i f f i c u l t to compare her resul ts d i r e c t l y , since her model makes extensive use of a 'permanent income' concept as an explanatory var iab le . In both models, E 1 ( + , E 2 i+ , and are a l l extremely smal l . Thus a change in the re la t i ve pr ice of monetary services has v i r t u a l l y no ef fect on the demand for consumption goods. However, E 4 2 and E 4 3 are strongly negative, implying that money are semi durables, and money and durables are gross complements. That i s , a f a l l in the pr ice of semi durables or durables leads to an increased demand fo r monetary serv ices. This i s the resu l t one would i n t u i t i v e l y expect from a transact ions demand model. A f a l l in the pr ice of e i ther semi durables or durables leads to an increase in the i r quantity demanded. In order to f a c i l i t a t e the transact ions required to purchase these goods, a larger stock of real balances are held by households. On the other handy both Em and akl are pos i t i ve , ind icat ing gross subs t i t u t ab i l i t y between non durables + services and money. That i s , an increased demand for th i s category of consumption is accompanied by a f a l l in the demand for monetary serv ices . This resu l t i s s l i g h t l y puzzl ing for prec ise ly the same reason as stated in the previous paragraph. One would expect monetary services and non durables + services to be complements (perhaps even more so than money and semi durables or durables) . An answer to th is apparent paradox is not readi ly ava i l ab le . . We attempted to tes t for homotheticity in the four good money model. However, the homothetic model could not be estimated successfu l ly due to convergence problems. This i s caused probably by the fact that the estimated p began to exceed unity during the algori thm. In any case, - 214 -the r e j e c t i o n of homothet ici ty is ind ica ted by the general s i g n i f i c a n c e of the a*, c o e f f i c i e n t s in the unconstrained v e r s i o n . We did not t es t the model fo r s t a b i l i t y between the two sub p e r i o d s , as the use of an est imat ion technique taking account of au tocor re la t ion renders the usual simple l i k e l i h o o d r a t i o tes t i n a p p l i c a b l e . We conclude t h i s examination of the money model by sounding a note of cau t ion . The r e s u l t s obtained in t h i s model were very s e n s i t i v e to whether or not the est imat ion technique took account of a u t o c o r r e l a t i o n . We have noted already the d i f f e r e n t performance of the two models (with and without allowance f o r au tocor re la t ion ) with respect to whether the curvature condi t ions were s a t i s f i e d . However, the e l a s t i c i t i e s (which we have not reported) were a lso qui te d i f f e r e n t in the model which ignored a u t o c o r r e l a t i o n . In p a r t i c u l a r , the expenditure e l a s t i c i t y of money behaved in a d i a m e t r i c a l l y opposite f a s h i o n , r i s i n g over time from about .7 to .9 in the 1952-1974 time per iod . However, th is s t i l l supports the hypothesis of the ex is tence o f economies of s c a l e in money h o l d i n g s , a l b e i t to a l e s s e r extent . Although E 1 4 , E2i+, E 3 ( t and E 4 3 were roughly the same in the two models, E 4 1 was negative in the ' c l a s s i c a l ' model. As noted e a r l i e r , th is complementarity r e l a t i o n s h i p is i n t u i t i v e l y what one would have expected. E 4 2 was a lso negative in the NSE model, but p o s i t i v e in the SE model. These d i f f e rences between the ' c l a s s i c a l ' and the 'au tocorre -l a t i o n ' model are somewhat d i s t u r b i n g , but perhaps not too s u r p r i s i n g in view of the general complexity of the model, and our r e l i a n c e on point e l a s t i c i t y est imates. Most researchers est imat ing complex demand systems do not appear to have undertaken the experiments which we have performed with respect to the issue of a u t o c o r r e l a t i o n . However, i t i s - 215 -important to rea l i se the potent ial s e n s i t i v i t y of the resul ts to such experiments, 2) Aggregate Near Money Model In order to reta in a four good model, i t was decided to combine semi durables with non durables + serv ices . The model which we estimated contained the fol lowing va r iab les ; (1) non durables + services + semi durables, (2) durables, (3) money, and (4) near money. Since the NSE model had provided s l i g h t l y better resu l ts for the money model, and, in add i t i on , appeared to be more appropriate t heo re t i ca l l y , we estimated only the NSE version here. The parameter estimates of the model appear in Table 7.10 and the corresponding e l a s t i c i t i e s in Table 7.11. As was the case in previous models, the Durbin-Watson s t a t i s t i c s were low, and a f i r s t order auto-cor re la t ion scheme was invest igated. While p turned out to be s i g n i f i c a n t , the resu l t ing e l a s t i c i t y estimates seemed unreasonable (both money and near money had highly negative expenditure e l a s t i c i t i e s throughout the per iod, and as well a 3 3 and 0 ^ were always pos i t i ve ) . Thus we decided to ignore these resu l t s . In general , the resul ts from the aggregate near money model were d isappoint ing, at least from the point of view of sa t i s fy ing the res t r i c t i ons of the d i rec t u t i l i t y model. In the 1952-1974 model, for the f i r s t time in any of the models reported to date, p o s i t i v i t y and mono-ton i c i t y were both v io la ted at one observat ion, 1955. Since the resu l t ing e l a s t i c i t y estimates for that year are meaningless, we report instead ( in Table 7.11) those obtained for 1956. Furthermore, although the own pr ice 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 subst i tu t ion of near money were - 216 -negative f o r 17 out of the 23 y e a r s , the overa l l curvature condi t ions were s a t i s f i e d in only 6 y e a r s , from 1952-1972, In the 1952-1962 sub model, curvature passed in 1953, 1959 and 1960, while f o r the 1963-1974 p e r i o d , the model s a t i s f i e d the condi t ions in 1967 and 1969-1971. However, the estimates do provide some i n t e r e s t i n g information (apart from the fac t that the d i r e c t u t i l i t y model appears to be genera l ly i n a p p l i c a b l e ) . F i r s t , in terms of both the p r i c e e l a s t i c i t i e s and the e l a s t i c i t i e s of s b u s t i t u t i o n , there is no evudence of s u b s t i t u t a b i l i t y between money and near money. E3t+, E 4 3 , and a 4 3 are negative in a l l y e a r s . Second, the expenditure e l a s t i c i t i e s of near money are very h igh , in both the 1952-1974 model and the sub models. C l e a r l y , near money i s a luxury good. This i s i n t u i t i v e l y p l a u s i b l e , as near monies may be regarded as more ' s o p h i s t i c a t e d ' f i n a n c i a l instruments than the l i a b i l i t i e s of chartered banks. If one examines the quant i ty indexes f o r money and near money (Tables 5.20 and 5.25) , between 1952 and 1974, near money has grown more than twice as rap id ly as money, by a f a c t o r of 3.65 compared to 1.73. Our evidence c l e a r l y ind ica tes that t h i s i s due not to r e l a t i v e p r i c e changes in favour of f i n a n c i a l i n s t i t u t i o n s other than chartered banks, but ra ther to a higher expenditure e l a s t i c i t y . The behaviour of the expenditure e l a s t i c i t y of money in t h i s model i s i n t e r e s t i n g . Unl ike the model of the previous s e c t i o n , i t i s r i s i n g over time (while always remaining less than one) . We be l ieve s t rongly that t h i s i s not due to the add i t iona l presence of near money in the model. The behaviour of the expenditure e l a s t i c i t y of money here i s very c lose to that of the four good consumption money-model, estimated without taking account of a u t o c o r r e l a t i o n . The l a t t e r model was re jected in the previous sec t ion in favour of the model with a u t o c o r r e l a t i o n . Thus, un for tunate ly , i t appears that the estimate we obtain of t h i s important - 217 -e l a s t i c i t y var ies g rea t ly depending on the s tochas t ic s t ruc ture assumed. As was the case with money ( in both t h i s and the previous model) , the e f f e c t on the demand f o r real goods of a change in the r e l a t i v e p r i c e of near money serv ices (as measured by E 1 ) + and E2i+) appears n e g l i g i b l e . On the other hand, a f a l l in the rental p r i c e of the category non durables + serv ices + semi durables increases the demand f o r near money, support ing the t ransact ions demand hypothesis . However, Ei+2 i s p o s i t i v e , a somewhat implausib le r e s u l t . I n t e r e s t i n g l y , both E 3 1 and E 3 2 are negat ive , the r e l a t i o n s h i p one would expect. Thus the i n c l u s i o n of near .money has, i t appears, improved the ' reasonableness' of the estimated r e l a t i o n s h i p s between money and goods, as compared to the previous model. However, we suspect that t h i s i s in large measure due to the d i f f e r e n t est imat ion technique used. We tested the model formal ly f o r parametric s t a b i l i t y and homo-t h e t i c i t y . The tes t s t a t i s t i c s are l i s t e d in Table 7.12. Both hypotheses were re jected d e c i s i v e l y . We may summarise the main conclusions from the real-monetary models as fo l lows: ( i ) The r e s t r i c t i o n s of the d i r e c t u t i l i t y model are s a t i s f i e d at a l l observat ions in the NSE vers ion of the money-goods model, and were re jected at only one observat ion (1974) in the SE model. ( i i ) The expenditure e l a s t i c i t y of money i s low and dec l ines over time. ( i i i ) Results ( i ) and ( i i ) are extremely s e n s i t i v e to whether the model i s estimated with or without an au tocor re la t ion s t r u c t u r e . ( iv) The i n c l u s i o n of money a l t e r s the estimated r e l a t i o n s h i p s between - 218 -consumption goods only to a f a i r l y l im i ted extent . (v) The aggregate near money model genera l ly does not s a t i s f y the curvature condi t ions in e i t h e r autocorre la ted or ' c l a s s i c a l ' models. Formal ly , our ' theory ' i s r e j e c t e d . This may be due, in p a r t , t o the 14 heterogeneous nature of the d i f f e r e n t near money components. The p o s s i b i l i t y that the r e s u l t s may improve upon disaggregat ion is invest igated in the fo l lowing s e c t i o n . (v i ) Near money i s a luxury good. ( v i i ) Money and near money are complementary in both compensated and uncompensated senses. ( v i i i ) The r e l a t i o n s h i p between both money and near money and consumption goods in most models seems p l a u s i b l e . However, some puzz l ing paradoxes are present . C. L i q u i d Asset Models In t h i s s e c t i o n , we estimate the disaggregated l i q u i d asset model which assumes that l i q u i d assets are weakly separable from consump-t i o n s e r v i c e s . O r i g i n a l l y , f i v e assets were cons idered , namely; (1) currency + personal chequing accounts of chartered banks, (2) personal savings deposi ts of chartered banks (PSD's ) , (3) t r u s t and loan company (TML) savings d e p o s i t s , (4) TML over one year term d e p o s i t s , and (5) Canada Savings Bonds (CSB 's ) . For reasons discussed above i t was decided not to proceed with the f i v e good model, but to drop one v a r i a b l e . The obvious choice was (1) , currency + personal chequing accounts. The reason is because t h i s category was dominated f o r most of the per iod by the currency component. However, our s e r i e s fo r currency held by households i s only a very crude - 219 -Table 7.8 PARAMETER ESTIMATES, FOUR GOOD MONEY MODEL (NON HOMOTHETIC VERSION, FIRST ORDER AUTOCORRELATION SCHEME) (1) Non durables + s e r v i c e s , (2) Semi durab les , (3) Durables , (4) Money 1952-1974 SE NSE ai - .577 - .313 (-3.870) (-2.228) a2 .110 .042 (2.139) (.864) a 3 , .518 .317 (3.913) (2.659) -.052 - .047 (-2.330) (-1.738) 011 - .612 -.397 (-5.180) (-4.702) 012 .215 .207 (4.865) (5.795) 013 .801 .664 (9.513) (15.481) 014 - .028 - .005 (-2.000) (-.376 022 - .238 -.261 (-5.513) (-6.811) 023 .154 .178 (5.816) (14.220) 024 .012 .004 (2.878) (.963) 033 - .498 -.451 (-7.299) (-20.028) 034 - .026 .014 (2.661) (3.054) 044 - .015 - .016 (-2.998) (-2.629) continued - 220 -Table 7,8 (continued) 1952-1974 SE NSE P .937 .938 (72.383) (71.847) In L 267.780 269.106 R 2 i .950 .929 R 2 2 .996 .996 R 2 3 .988 .985 D-Wi 2.308 1.540 D-w2 1.776 1.324 D-W3 2.532 1.587 - 221 -Table 7,9 ELASTICITY ESTIMATES, FOUR GOOD MONEY MODEL (NON NOMOTHETIC VERSION, FIRST ORDER AUTOCORRELATION SCHEME) (1) Non durables + s e r v i c e s , (2) Semi durab les , (3) Durables , (4) Money 1955 1965 1974 P r i c e • SE NSE SE NSE SE NSE E l a s t i c i t i e s -.501 -.621 -.464 -.617 -.428 -.627 El 2 - .098 - .085 - .078 -.058 -.097 -.038 E l 3 - .022 .002 -.071 - .056 -.123 - .115 E i - .032 .010 .028 .008 .024 .006 E21 - .164 - .096 - .328 - .176 - .525 - .176 E 2 2 - .273 - .259 - .319 -.307 . -.211 -.331 E23 .209 .026 .082 - .124 - .036 - .263 E2i+ - .037 -.007 -.041 -.008 -.048 - .009 E31 -1.061 -.947 -.921 - .846 - .739 -.877 E 3 2 - .208 -.302 -.152 - .258 - .115 - .280 E33 -1.042 -1.007 -.877 - .784 - .786 - .559 E 3 - - .053 -.031 - .042 - .025 - .034 - .025 E m 2.645 .880 2.007 .752 1.525 .671 Ei+2 - .839 -.184 -.572 - .076 - .390 .009 E 4 3 -1.401 -.531 -.861 - .260 -.403 - .037 Ei+4 - .812 - .687 - .708 - .744 - .546 -.727 Expenditure E l a s t i c i t i e s E i y .588 .694 .585 .722 .625 .774 E * y E 3 y E » y .264 .336 .606 .617 .820 .784 2.364 2.287 1.993 1.913 1.674 1.741 .398 .521 .133 .329 - .186 .084 continued - 222 -Table 7.9 (continued) 1955 1965 1974 SE E l a s t i c i t i e s of Subs t i tu t ion NSE SE NSE SE NSE o n - .290 - .344 - .246 - .310 -.191 - .256 a 1 2 - .023 .177 .018 .320 -.181 .495 <*13 .505 .705 .344 .498 .266 .300 a l k 5.049 1.992 3.728 1.586 2.721 1.186 a22 -1.437 -1.243 -1.713 -1.521 -.181 -1.662 0"23 1.064 .448 .884 .117 .716 - .324 a2i+ -4.838 - .599 -4.030 -.202 -3.427 .150 a 3 3 .505 -2.097 - .979 -1.236 - .612 - .569 a 3 4 -4.953 -1.791 -2.782 - .717 -1.359 - .069 a 4 4 -111.448 -90.903 -79.678 -77.584 -48.701 -53.422 - 223 -Table 7.10 PARAMETER ESTIMATES, FOUR GOOD MONEY-MONEY SUBSTITUTE MODEL (NON HOMOTHETIC VERSION, NON STATIC EXPECTATIONS) (1) Non durables + serv ices + semi durab les , (2) Durables, (3) Money, (4) Near money 1952-1974 1952-1962 1963-1974 <»! -.301 -.272 -.293 (-29.371 ) (-29.432) (-7.594). a 2 .294 .271 .288 (31.593) (32.942) (7.258) a 3 -.004 -.005 .002 (-1.165) (-2.321) (.258) a 4 .012 .006 .002 (4.443) (1.917) (.483) e n -.258 -.237 -.265 (-15.762) (-11.896) (-6.352) g 1 2 .807 .819 .810 (60.107) (39.861) (25.975) B 1 3 .028 .021 .039 (5.578) (5.935) (5.117) B l t t .017 .006 .012 (5.070) (2.472) (2.868) 3 2 2 -.433 -.456 -.437 (-36.873) (-21.028) (-19.583) B 2 3 .016 .010 .014 (3.858) (2.497) (3.953) 3 2 1 + .004 .005 .0003 ' (.138) (2.590) (,148), g 3 3 -.038 -.024 -.050 (-17.864) (-17.892) (-20.143) g 3 1 t -.003 -.001 .006 (2.237) (1.642) (5.964) -.012 -.007 -.013 (-8.540) (-4.908) (-7.083) continued - 224 -Table 7.10 (continued) 1952-1974 1952-1962 1963-1974 ln L 277.569 181.982 173.258 R 2 j .986 / .998. .966; R 2 2 .988. .998 .964" R 2 3 .914: .960. .977;'. D-Wi 1.179 1.720 1.293 D-W2 1.303 2.076 1.342 D-W3 .903, 1.442 2.174 - 225 -Table 7;11 ELASTICITY ESTIMATES, FOUR GOOD MONEY-MONEY SUBSTITUTE MODEL (NON HOMOTHETIC VERSION, NON STATIC EXPECTATIONS) Pr ice 1956 1965 1974 E l a s t i c i t i e s E l i - .629 -.634 -.632 E l 2 . - .158 -.191 - .239 El3* - .015 -.011 - .005 E i - - .003 - .006 -.001 E21 -1.115 -1.028 -1.094 E 2 2 - .447 - .419 -.264 E23 -.031 - .256 - .020 E 2 - .007 .006 .007 E31 -1.078 - .655 - .215 E32 - .592 - .405 -.232 E33 1.331 .525- - .285 E 3 - - .149 -.131 -.058 E- i -2.722 -3.158 . - .928 E1+2 .307 .277 .293 E-3 - .537 -.581 - .204 Ei+it - .324 .626 -1.060 Expenditure E l a s t i c i t i e s .805 .842 .877 X J E2 y 1.616 1.466 1 .371 .489 .666 .789 E, „ 3.277 2.836 1.899 ^y continued - 226 -Table 7,11 (continued) 1956 1965 1974 E l a s t i c i t i e s of S u b s t i t u t i o n a n - .030 - .017 ,023 a 1 2 .135 .075 - .106 a 1 3 - .944 -.221 .499 a m - .339 -1.441 .646 a 2 2 - .405 - .220 .283 a 2 3 -2.021 - .964 - .165 a 2 4 4.578 3.950 3.106 a 3 3 154.875 50.602 -21.225 a 3 h -59.073 -52.393 -13.872 aw -126.211 256.349 -268.173 - 227 -Table 7,12 TEST STATISTICS, FOUR GOOD NEAR MONEY MODEL (NSE VERSION) Null Hypothesis H 0 Like l ihood Degrees r a t i o t e s t of s t a t i s t i c freedom c r i t i c a l value (99%) 1. Parametric S t a b i l i t y a i (1952-1962) = a i ( 1 9 6 3 - 1 9 7 4 ) ' a 1 1 1 j 1 5 5 3 4 2 B i j (1952-1962) = 3 i j (1963-1974)' a 1 1 1 , J ' ^ 12 26.22 2. Homotheticity a. = 0, a l l i 1952-1974 95.764 3 11.34 1952-1962 54.578 3 11.34 1963-1974 25.006 3 11.34 - 228 -' guess t imate ' , obtained by assuming that the proport ion of to ta l currency outstanding held by households i s the same as that of the sum of personal savings deposi ts and personal chequing accounts to to ta l chartered bank l i a b i l i t i e s (see Chapter 5) , We f e l t that a se r ies constructed in t h i s way when used v i r t u a l l y alone ( for most of the period) could not be expected to convey any useful behavioural in format ion. The parameter estimates of the four asset model, obtained by dropping currency and personal chequing accounts from the f i v e assets l i s t e d above, are given in Table 7.13. The corresponding e l a s t i c i t i e s appear in Table 7.14. The model was estimated f o r ; t h e three time periods 1952-1974, 1952-1962, and 1963-1974. Since the D-W s t a t i s t i c s were low, we attempted to estimate the model a l lowing f o r a u t o c o r r e l a t i o n . However, despi te repeated attempts, a f t e r a large number of i t e r a t i o n s , 15 a maximum value of the l i k e l i h o o d funct ion could not be a t t a i n e d . In any case , the 1952-1974 model, estimated without consider ing a u t o c o r r e l a t i o n , provided qui te favourable r e s u l t s . Although p o s i t i v i t y and monotonicity were re jected f o r two y e a r s , 1952 and 1973, the own p r i c e 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 s u b s t i t u t i o n were genera l ly negat ive. The f u l l set of curvature condi t ions were s a t i s f i e d in the years 1953, 1956-1962, 1964-1970, and 1974, that i s , in s ixteen of the twenty three y e a r s . The.hypothesis of parametric s t a b i l i t y was re jected (see Table 7 .15) . We next tes ted f o r homothet ic i ty . This was re jected in a l l three per iods . The expenditure e l a s t i c i t i e s conform to the pattern revealed in the aggregate money-near money model of the previous s e c t i o n . PSD's have an e l a s t i c i t y less than one, but which r i s e s g r a d u a l l y . The 'near money' expenditure e l a s t i c i t i e s are a l l h igh , with TML term deposi ts (perhaps the most ' s o p h i s t i c a t e d ' of the l i q u i d asset l i a b i l i t i e s which - 229 -we consider ) being the h ighest . A l l the near money expenditure e l a s t i c i t i e s appear to dec l ine over t ime, e s p e c i a l l y that of TML term depos i ts . The own p r i c e e l a s t i c i t i e s of the assets are a l l negat ive. The f i g u r e f o r E 2 2 seems to be roughly constant at - , 5 , and i s a l s o f a i r l y c lose to the estimates from the previous two aggregate money models. TML l i a b i l i -t i e s are more p r i c e s e n s i t i v e , e s p e c i a l l y term d e p o s i t s . It i s i n t e r e s t i n g to note that EI+I+J the i n t e r e s t e l a s t i c i t y of CSB's ,has been growing over t ime. Th is could well be a r e s u l t (or cause?) of the increas ing adver t i s ing e f f o r t undertaken each year to boost CSB s a l e s . Turning to the poss ib le subs t i tu tab i l i t y / complementa r i t y r e l a t i o n s h i p s , E i 2 and E 2 1 have d i f f e r e n t s i g n s . A f a l l in the rental p r i c e of TML savings deposi ts leads to s u b s t i t u t i o n into t h i s category from PSD's . This i s the f i r s t evidence so f a r support ing the Gurley and Shaw [1960] hypothesis of money-near money s u b s t i t u t a b i l i t y . However, the e l a s t i c i t y estimate i s extremely sma l l . On the other hand, an increased demand f o r PSD's i s associa ted with a r i s e in the demand f o r TML savings deposi ts ( E 2 1 < 0 ) . A l l other gross p r i c e e l a s t i c i t i e s are genera l ly negat ive , i n d i c a t i n g complementarity,except f o r E 2 3 . There is a lso evidence of s u b s t i t u t a b i l i t y between TML term deposi ts and CSB's in the l a t t e r ha l f of the per iod . The e l a s t i c i t i e s of s u b s t i t u t i o n between assets are genera l ly p o s i t i v e , i n d i c a t i n g s u b s t i t u t a b i l i t y . The apparent c o n f l i c t between these estimates and the complementarity evidence c i t e d above i s resolved e a s i l y when we consider the very high expenditure e l a s t i c i t i e s f o r a l l of the near money a s s e t s . R e c a l l i n g the d i s c u s s i o n in sect ion A of t h i s chapter , E . . and a . , w i l l have the same s ign only i f a.. > E . > 0. Given our evidence that E . i s qui te high fo r 'near mon ies ' , ana lys is - 2 3 0 -of money-near money s u b s t i t u t a b i l i t y re la t ionships in terms of the e l a s t i c i t i e s of s u b s t i t u t i o n alone (the procedure of Chetty [1969], Moroney and Wi lbra t te [1976] and Bisignano [1974;]) could be qui te mis lead ing . Overa l l the r e s u l t s of the l i q u i d asset model are encouraging. The curvature r e s t r i c t i o n s implied by the d i r e c t u t i l i t y theory are passed in most y e a r s . F a i r l y ' reasonable ' estimates can be obtained of the p r ice e l a s t i c i t i e s and expenditure e l a s t i c i t i e s . Furthermore, these estimates are r o u g h l y ' i n accordance with those obtained in the aggregate money and near money models. However, the disaggregat ion of near money has improved the r e s u l t s cons iderab ly . - 231 -Table 7,13 PARAMETER ESTIMATES, LIQUID ASSET MODEL (1) Chartered bank personal savings deposi ts ( t o t a l ) , (2) T rus t and loan co . savings deposi ts ( t o t a l ) , (3) Trust and loan co . over one year term d e p o s i t s , (4) Canada Savings Bonds 1952-1974 1952-1962 1963-1974 a x - .006 - .005 - .293 (-15.906) (-13.297) (-7.594) az .002 .0003 .288 (6.996) (1.663) (7.258) a 3 .001 .0004 .002 (4.492) (2.732) (.258) ak .003 .004 .002 (7.676) (7.855) (.483) 3 n - .416 - .353 - .265 (-8.420) (-6.536) (-6.352) 3 1 2 .159 .040 .810 (6.231) (1.214) (25.975) 3 1 3 .129 .054 .040 (7.888) (3.502) (5.117) 3m .569 .786 .012 (17.800) (23.268) (2.868) 3 2 2 - .120 - .023 -.437 (-4.505 (-.699) (-19.583) 3 2 3 .007 .006 .014 (.755) (.996) (3.953) 3 2 i + .089 .035 .0003 (7.806) (2.886) (.148) 3 3 3 - .806 -.411 - .049 (-9.273) (-4.658) (-20.143) 33it .035 .017 .006 (4.127) (2.141) (5.964) 3 4 4 - .356 - .458 - .013 (-15.946) (-17.705) (-7.083) continued - 232 -Table 7,13 (continued) 1952-1974 1952-1962 1963-1974 ln L 128.525 111.772 88,295 R 2 i .939 .986 .966 R22 .834 .412 .964 R23 .733 .460 .977 D-Wi .888 2.288 1.293 D-W2 .792 .624 1.342 D-W3 .893 .710 2.174 - 233 -Table 7,14 ELASTICITY ESTIMATES, LIQUID ASSET MODEL (1) Chartered bank personal savings deposi ts ( t o t a l ) , (2) Trust and loan co . savings deposi ts ( t o t a l ) , (3) Trust and loan c o . over one year term d e p o s i t s , (4) Canada Savings Bonds Pr ice E l a s t i c i t i e s 1955 1965 1974 E n - .586 - .595 -.527 E 1 2 . . .042 .033 .006 E l 3 .083 - .002 -.017 E i . -.061 - .059 -.031 E21 -1.034 - .546 -.698 E 2 2 -1.016 -1.022 -1.047 E 2 3 .090 .049 .082 E21+ - .686 - .333 - .089 E 3 1 -3.594 -5.770 -2.240 E 3 2 - .087 - .199 .019 E 33 -5.768 - .717 -1.281 E 3 4 - .995 -1.576 T-.176 Em -1.970 - .880 -.633 E 4 2 - .273 - .663 - .004 E « - .005 .054 .058 E1+4 -.171 - .700 -.871 Expenditure E l a s t i c i t i e s E . y hy hy hy .521 .623 .568 2.650 1.852 1.752 10.444 8.264 3.677 2.419 1.657 1.450 - 234 -Table 7.14 (continued) 1955 1965 1974 E l a s t i c i t i e s of Subs t i tu t ion s n - .195 - .183 - .266 s i 2 1.380 1.112 .647 s i 3 6.046 .447 .132 Sm .009 .297 .448 s 2 2 -17.987 -13.266 -12.020 s 2 3 8.668 5.317 3.925 s 2 1 + -3.138 .003 1.400 s 3 3 -373.579 -42.599 -30.181 s 3 i + 2.056 - .489 2.987 s k k .981 -1.178 -1.975 - 235 -Table 7.15 TEST STATISTICS, LIQUID ASSET MODEL Null Hypothesis HQ L ike l ihood r a t i o t e s t s t a t i s t i c Degrees c r i t i c a l of value freedom (99%) 1. Parametric S t a b i l i t y a i (1952-1962) = a i (1963-1974) ) ) 143.084 'i j(1952-1962) " ' i j (1963-1974) , a l l i , j ) 12 26.22 2. Homotheticity a . = 0, a l l i 1 1952-1974 51.570 3 11.34 1952-1962 33.986 3 11.34 1963-1974 20.732 3 11.34 - 236- -Footnotes - Chapter 7 1. However, i t should be noted that there is no constant present in the model. Thus the R 2 s t a t i s t i c may be mis lead ing , 2. See the d i s c u s s i o n below regarding the appropr iate time per iod . 3. The same would a lso be true of models in which serv ices were aggregated with e i ther semi durables or durab les . These poss ib le aggregation schemes were not explored due mainly to the heterogeneous nature of the r e s u l t i n g aggregate. 4. 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 s and D-W s t a t i s t i c s f o r the autocorre la ted models were as f o l l o w s : Test s t a t i s t i c ( H 0 ; p=0) D-W s t a t i s t i c s D-Wx D-W2 SE 5.12 1.024 1.351 NSE .87 1.063 1.495 The x 2 c r i t i c a l value at the 99% l e v e l , with 1 degree of freedom, i s 6.63. 5. Except f o r a couple of c a s e s , the p o s i t i v i t y and monotonicity condi t ions were s a t i s f i e d in each year f o r a l l the models in t h i s chapter . To avoid r e p e t i t i o n , in our subsequent d i s c u s s i o n , the reader may take f o r granted that these condi t ions always were s a t i s f i e d unless otherwise i n d i c a t e d . 6. However, as d iscussed in Chapter 6, the nul l hypothesis a lso i m p l i c i t l y assumes a constant e r ro r var iance through t ime, which may be i n -appropr ia te . Th is caveat should be borne in mind when cons ider ing the r e s u l t s of a l l the s t a b i l i t y t es ts reported in t h i s chapter . 7. In the NSE model curvature f a i l e d in a l l y e a r s , while in the SE v e r s i o n , i t was re jec ted f o r 1959-1962. 8. Th is i s not unexpected, in view of the high t - s t a t i s t i c s associa ted with the ind iv idua l a . c o e f f i c i e n t s (Table 7 .1 ) . 9. However, the c o e f f i c i e n t s f o r the model l i s t e d in Table 7 .4 , a l l appear i n s i g n i f i c a n t . 10. 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 s and value of p were the f o l l o w i n g : SE: L.R. = 7.640 p = .407 (2.992) NSE: L.R. = 27.484 p = .909 (55.905) 11. 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 s fo r the hypothesis that p=0 were 38.48 (SE) and 43.488 (NSE), 12. The long run e l a s t i c i t i e s are obtained by d i v i d i n g the short run quar te r ly e l a s t i c i t i e s by the estimated speed of adjustment. - 237 * 13. The i n t e r e s t rate i s measured as the 90 day rate or bond rate minus the own rate on money. 14. The reader i s re fe r red to the d iscuss ion of the condi t ions necessary f o r a v a l i d aggregation procedure (sect ion D, Chapter 3) . 15. Although convergence was not achieved the e l a s t i c i t i e s corresponding to the estimates at the f i n a l i t e r a t i o n were c a l c u l a t e d . Curvature was re jected at a l l observat ions . Furthermore, p was i n s i g n i f i c a n t , on the basis of an ind iv idua l t t e s t . Chapter 8 SUMMARY AND CONCLUSIONS The purpose of t h i s research was to develop and tes t a model of the demand f o r money wi th in a general opt imis ing model of household beha-v iour . In t h i s chapter , we evaluate the main r e s u l t s of the research , note some l i m i t a t i o n s of the model, and suggest some ways in which the model might be extended. The theore t i ca l part of the thes is cons is ted of applying the too ls of modern u t i l i t y theory to the p a r t i c u l a r problem of the demand fo r money. The development and so lu t ion of the model provided a c l e a r basis f o r i n t e r p r e t i n g the demand equations used in es t imat ion . Furthermore, our work has made e x p l i c i t var ious assumptions which were i m p l i c i t in previous empir ical models of the demand fo r money. In p a r t i c u l a r , the d e r i v a t i o n of the rental p r i c e of money concept c l a r i f i e d the r o l e of expectat ions and the r e l a t i o n s h i p between the rental pr ices of money and goods wi th in 'the d i r e c t u t i l i t y ' model. A l s o , r e s u l t s in the theory of funct iona l s e p a r a b i l i t y and aggregation enabled us to recognise the condi t ions under which var ious commonly used 'sub models' of the demand f o r money are v a l i d . A s i g n i f i c a n t part of the empir ical cont r ibu t ion of the research was the c o l l e c t i o n and const ruc t ion of the data . Rental p r i c e and quant i ty ser ies fo r the serv ices of consumption goods (durable and non d u r a b l e ) , l e i s u r e , money, and money subst i tu tes cons is tent with the t h e o r e t i c a l framework of the model were prepared. These data se r ies may be used in the future by other researchers working within th is a n a l y t i c framework. As was noted in Chapter 7, there e x i s t s no s i n g l e c l e a r cut - 239 -c r i t e r i o n by which to evaluate the 'performance' (absolute or r e l a t i v e ) of the p a r t i c u l a r models which were est imated. However, the estimated models were found genera l ly to be cons is tent with the under ly ing theory , and a lso provided some useful in format ion . In p a r t i c u l a r , the r e s t r i c t i o n s implied by the theory, in most c a s e s , have not been r e j e c t e d . When we tested the consumption model, the consumpt ion- le isure model, the aggregate money model, and the l i q u i d asset model, in almost a l l y e a r s , the observed data were found to be cons is tent with the maximisation of a preference order ing possessing c e r t a i n r e g u l a r i t y p r o p e r t i e s . Th is i s qui te a strong r e s u l t . In the case of the demand f o r money, i t i n d i c a t e s , at the very least , that the d i r e c t u t i l i t y approach should not be summarily r e j e c t e d . However, i t should be noted that these r e s t r i c t i o n s were re jected f o r the aggregate near money model. In a d d i t i o n , the models genera l ly have provided i n t e r e s t i n g and ' reasonable ' r e s u l t s . Concerning the ' r e a l ' models, our r e s u l t s ind ica te that the demand f o r l e i s u r e can be model led, with some degree of s u c c e s s , s imultaneously with the demand f o r consumption goods. In the monetary and l i q u i d asset models, the own p r i c e e l a s t i c i t y estimates are in the range reported by other researchers . There appear to be economies of sca le in money ho ld ings ; on the ofiher hand, near money i s a luxury good. There i s no evidence of s u b s t i t u t a b i l i t y between aggregate money and aggregate near money; however, some s u b s t i t u t a b i l i t y is reported between chartered bank personal savings d e p o s i t s , and t rus t and loan company savings d e p o s i t s . While previous researchers have undertaken i s o l a t e d attempts to quant i fy these r e l a t i o n s h i p s , our model has the advantage of provid ing a comprehensive set of r e s u l t s within a h ighly st ructured - 240 -t h e o r e t i c a l framework. For example, our r e s u l t s ind ica te that attempts to measure l i q u i d asset s u b s t i t u t a b i l i t y by est imating e l a s t i c i t i e s of s u b s t i t u t i o n could be qui te m is lead ing , in view of the presence of strong 1 income e f f e c t s 1 . The r e s u l t s do not provide a c l e a r cut answer as to the importance of model l ing a l l ' r e a l ' and ' f i n a n c i a l ' dec is ions by households s i m u l -taneously . As was noted in Chapter 3, formal i n v e s t i g a t i o n of the problem (at l eas t wi th in the context of the d i r e c t u t i l i t y model) awaits the development of t e s t procedures f o r s e p a r a b i l i t y which do not involve as a maintained hypothesis an extremely r e s t r i c t i v e form f o r the 'aggregator f u n c t i o n 1 . Thus, our d i s c u s s i o n has been l im i ted to comparisons of an informal nature. Some r e s u l t s are worth not ing . The i n c l u s i o n of l e i s u r e and money, r e s p e c t i v e l y , in the consumption goods model, a l te red to some extent the estimated r e l a t i o n s h i p s among consumption goods. On the other hand, the r e s t r i c t i o n s implied by the theory were genera l ly s a t i s f i e d in the l i q u i d asset model (where l i q u i d assets were assumed separable from consumption goods and l e i s u r e ) , while the theory was c l e a r l y re jected in the case of the aggregate near money model where consumption goods were inc luded . Turning to the r o l e of expectat ions within the model, a number of points emerge. In almost a l l c a s e s , the estimated ARIMA models f o r durables and semi durables were of a f a i r l y simple nature - some of them were constant rate of i n f l a t i o n models. Thus a poss ib le c r i t i c i s m of the use of the ARIMA model f o r fo recas t ing purposes, namely, that i t assumes economic agents possess unreasonably complex powers of c a l c u l a t i o n , does not appear to be v a l i d in t h i s ins tance . No strong case emerges f o r assuming a p r i o r i s t a t i c expecta t ions , which is a spec ia l case of the - 241 -ARIMA model, and which was dec is ive ly rejected for the time period in question. Nevertheless, i t should be noted that some problems arose in using ARIMA models for two large components of aggregate durables, namely, housing and land. These problems were discussed in Chapter 5. In pa r t i cu la r , the use of the ARIMA model led to negative rental pr ices for land, a s i tua t ion inconsistent with the model. In order to be able to proceed fur ther , i t was necessary to employ an ad hoc a l te rna t i ve . For the remaining ten durable goods, the rental pr ice data were constructed using a ser ies for the expected i n f l a t i on rate based on the optimal forecast ing propert ies of the ARIMA model. The resul ts from the estimated models indicate that the ARIMA approach can be used successfu l ly to capture the ro le of expectat ions, in that the NSE model performs at least as w e l l , i f not bet ter , than the SE model. Thus on both theoret ica l and prac t ica l grounds, i t would seem general ly preferable to use the forecast ing approach, as opposed to a r b i t r a r i l y imposing the s ta t i c expectations assumption. Several l im i ta t ions in the work presented should be noted. F i r s t , the index number problem of choosing a correct index for def la t ing nominal balances was avoided by means of a s l i g h t l y a r t i f i c i a l device. As was noted in the discussion of the problem, no easy solut ion i s apparent. Second, there are measurement problems associated with the estimated household f inanc ia l data which we have constructed, pa r t i cu la r l y with some of the near money se r ies . It i s l i k e l y to be some time before the Canadian Flow of Funds Accounts are s u f f i c i e n t l y well developed to provide r e l i a b l e household f inanc ia l ser ies to cover a long enough time per iod. However, our work represents the f i r s t attempt to come to grips - 242 -with the ownership issue which must be dea l t with i f : e m p i r i c a l demand f o r money models are to be der ived from an e x p l i c i t t h e o r e t i c a l s t r u c t u r e . There are a number of ways in which the model could be extended. F i r s t , as i t s tands, the model ignores uncer ta in ty . Economic agents are assumed to act as i f a l l expectat ions are held with c e r t a i n t y . The incorpora t ion of uncer ta inty regarding the p r i c e leve l could a l t e r s i g n i f i c a n t l y the estimated r e l a t i o n s h i p s between goods and l i q u i d a s s e t s . A re la ted point i s that the model does not consider the choice among r i s k y f i n a n c i a l a s s e t s , or al low f o r d i r e c t s u b s t i t u t a b i l i t y between these assets and durable goods. A p o t e n t i a l l y important l i m i t a t i o n of the model is the neglect of t ransact ions cost f a c t o r s . The model used here i s in sharp contrast to some other approaches to the demand f o r durable goods ( for example, Stone and Rowe [I960]) and money ( for example, Brainard and Tobin [1968]). E s s e n t i a l l y , these l a t t e r approaches assume a very simple 'pragmatic ' model of optimal holdings of the stocks of the var ious goods and money, but attempt to model the adjustment process in considerable d e t a i l . Our model, on the other hand, concentrates on model l ing the opt imis ing process in a very d e t a i l e d way, but ignores (at l e a s t in i t s present form) adjustment f a c t o r s . The importance of d i s e q u i l i b r i u m adjustments due to t ransact ions cost f a c t o r s i s d i f f i c u l t to assess . These costs w i l l probably be l ess important, f o r example, in the demand f o r food than in the demand f o r housing. As regards the demand f o r l i q u i d assets by households, few a p r i o r i judgements can be made. Judging from our r e s u l t s , the consumpt ion- le isure models do not appear to ind ica te an immediate need to incorporate ad jus t -ment f a c t o r s . On the other hand, the presence of s i g n i f i c a n t au tocor re la t ion - 243 -in the aggregate money model, and the re ject ion of the theory in the near moriey model indicate a cer ta in misspec i f ica t ion which may be associated with d isequi l ibr ium behaviour. The ideal model should contain a theory of adjustment costs wi th in a general optimising model ofhousehold behaviour. Such a model has yet to be developed. A possible way of approaching the problem might be to develop a ' f i xed fac to r ' u t i l i t y funct ion analogous to the var iab le p ro f i t function which has been used in the case of product ion. 1 Another p o s s i b i l i t y is to explore some kind of a habit formation hypothesis 2 in conjunction with a f l e x i b l e u t i l i t y funct ion. F i n a l l y , one could attempt to take account of misspec i f i ca t ion econometrically by al lowing for a more complex,error s t ructure. In conclus ion, the intent of th is thes is was to formulate an e x p l i c i t theoret ica l structure in order to test a model of the household demand for money. Notwithstanding the problems which s t i l l remain with the model, th is research const i tu tes a reasonably successful attempt to achieve that end. Footnotes - Chapter 8 1. See Diewert [1974b; 133-141]. 2. See Manser [1975]. BIBLIOGRAPHY A l l e n , R.G.D. [1938], Mathematical Ana lys is fo r Economists, Toronto:; MacMillan Co. of Canada, Ando, A . and M o d i g l i a n i , F. [1963], "The ' L i f e C y c l e ' Hypothesis of S a v i n g " , American Economic Review, 53, 55-84. Arrow, K . J . [1965], Aspects of the Theory of Risk Bear ing , Yrjo Johnsson Lec tu res , He ls ink i (The Yrjo Johnsson Foundation) . Barnet t , W.A. [1975], "Pol lak and Wachter on the Household Production Function Approach", D i v i s i o n of Research and S t a t i s t i c s , Federal Research Board, for thcoming, Journal of P o l i t i c a l Economy. B a r r e t t , R . J . , Gray, M.R. and P a r k i n , J . M . [1972], "The Demand f o r F i n a n c i a l Assets by the Personal Sector of the U.K. Economy", mimeo, Department of Economics, U n i v e r s i t y of Manchester. Bar ro , R . J . and Santomero, A . J . [1972], "Household Money Holdings and the Demand and Deposit Rate", Journal of Money, Cred i t and  Banking, 4, 397-413. Barten, A . P . [1969], "Maximum L ike l ihood Est imat ion of a Complete System of Demand Equat ions" , European Economic Review, 1, 7-73. Baumol, W.J . [1952], "The Transact ions Demand fo r Cash: An Inventory Theore t ic Approach", Quarter ly Journal of Economics, 66, 545-556. Berndt, E.R. and Chr is tenson , L.R. [1973], "The Internal St ructure of Funct ional R e l a t i o n s h i p s : S e p a r a b i l i t y , Subs t i tu t ion and Aggregat ion" , Review of Economic S t u d i e s , 40, 403-410. Berndt, E.R. and Chr is tenson , L.R. [1974], "Test ing fo r the Existence of a Consistent Aggregate Index of Labour Inputs", American  Economic Review, 64, 391-404. Berndt , E . R . , Darrough, M.N. and Diewert , W.E. [1977], " F l e x i b l e Funct ional Forms and Expenditure D i s t r i b u t i o n s : An A p p l i c a t i o n to Canadian Consumer Demand Funct ions" , Internat ional Economic Review ( forthcoming) . Berndt, E . R . , H a l l , B . H . , H a l l , R .E . and Hausman, J . A . [1974], "Est imat ion and Inference in Nonl inear S t ructura l Models", Annals of Economic  and Socia l Measurement, 3 /4 , 653-665. Berndt, E.R. and S a v i n , N .E. [1975], "Est imation and Hypothesis Test ing in S ingular Equation Systems with Autoregressive D is turbances" , Econometrica, 43, 937-957. Bierwag, G.O. [1974], "The Rat ionale of the Mean-Standard Deviat ion A n a l y s i s : Comment", American Economic Review, 64, 431-433. - 246 -Binhammer, H,H, £ 1 9 6 8 ] , Money, Banking and the Canadian F inanc ia l System, Toronto; Methuen, B is ignano, J , £ 1 9 7 4 ] , "Real Money S u b s t i t u t e s " , Working Paper No, 17, Federal Reserve Bank of San F r a n c i s c o , for thcoming, Internat ional Economic Review, Blackorby, C , Primont, D, and Russe l l , R.R. [1975], "Budgeting, D e c e n t r a l i s a t i o n and Aggregat ion" , Annals of Economic and  Soc ia l Measurement, 4, 23-44. Blackorby, C , Primont, D. and R u s s e l l , R.R. [1976], " S e p a r a b i l i t y Versus Functional S t ruc tu re : A Charac te r isa t ion of The i r D i f f e r e n c e s " , D iscuss ion Paper 76-20, Department of Economics, U n i v e r s i t y of B r i t i s h Columbia. Blackorby, C , Primont, D. and R u s s e l l , R.R. [1977], "On Test ing S e p a r a b i l i t y R e s t r i c t i o n s with F l e x i b l e Funct ional Forms", Journal of Econometr ics, forthcoming. Bond, D.E. and Shearer , R.A. [1972], The Economics of the Canadian  F inanc ia l System: Theory, P o l i c y and I n s t i t u t i o n s , Scarborough, Ontar io : Prent ice Hal l of Canada, L t d . Borch, K. [1969], "A Note on Uncerta inty and Indi f ference Curves" , Review of Economic S t u d i e s , 36, 1-4. Borch, K. [1974], "The Rat ionale of the Mean-Standard Deviat ion A n a l y s i s : Comment", American Economic Review, 64, 428-430. Box, G.P. and J e n k i n s , G.M. [1970], Time Ser ies A n a l y s i s , Forecast ing  and C o n t r o l , San F r a n c i s c o : Holden Day. Bra ina rd , W.C. and T o b i n , J . [1968], " P i t f a l l s in F inanc ia l Model B u i l d i n g " , American Economic Review, 58, 99-122. Breton, A. [1968], "A Stable V e l o c i t y Function f o r Canada?", Economica, 35, 451-453. Brunner, K. and Me l t ze r , A . [1967], "Economies of Scale in Cash Balances Reconsidered", Quarter ly Journal of Economics, 81, 422-436. Chetty , V .K . [1969], "On Measuring the Nearness of Near-Moneys", American Economic Review, 59, 270-281. C h i n l o y , P.T. [1977], "An Hedonic Rental Pr ice Index f o r Owner Occupied Housing", unpublished memorandum. Chr is tensen , L.R. and Cummings, D. [1976], "Real Product , Real Factor Input, and P r o d u c t i v i t y in Canada, 1947-1973", D iscuss ion Paper 7604, Soc ia l Systems Research I n s t i t u t e , U n i v e r s i t y of Wisconsin - Madison. - 247 -Chr is tensen , L.R. and Jorgenson, D.M. [1975], "Measuring Economic Performance in the .Private S e c t o r " , in The Measurement of Economic and Socia l Performance, (M, Moss, e d , ) , Studies in Income and Wealth, 38, National Bureay of Economic Research, C l a r k , C, [1973], "The Demand f o r Money and the Choice of a Permanent Income Est imate: Some Canadian Evidence, 1926-65", Journal  of Money, Cred i t and Banking, 5, 773-793. C l i n t o n , K. [1973], "The Demand f o r Money in Canada, 1955-70: Some S ing le Equation Estimates and S t a b i l i t y T e s t s " , Canadian  Journal of Economics, 6, 53-61. C l i n t o n , K. [1974], "The Demand f o r L i a b i l i t i e s of Trust and Loan Companies", Canadian Journal of Economics, 7, 191-204. Cootner, P.H. [1964], The Random Character of Stock Market P r i c e s , Cambridge, Mass. Courchene, T . J . and K e l l y , A . K . [1971, "Money Supply and Money Demand: An Econometric Ana lys is f o r Canada", Journal of Money, Cred i t  and Banking", 3 , 219-244. Cummings, E.D. and Meduna, L. [1973], The Canadian Consumer Accounts , Research Projects Group, S t ra teg ic Planning and Research, Department of Manpower and Immigration, Ottawa. Darrough, M.N. [1975], Intertemporal A l l o c a t i o n of Consumption, Savings  and L e i s u r e : An -App l ica t ion Using Japanese Data, unpublished Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia. De Leeuw, F. [1965], "A Model of F inanc ia l Behaviour" , in Duesenberry, J . S . , Fromm, G . , K l e i n , L.R. and Kuh, E. ( e d s . ) , The Brookings v Quarter ly Model of the United S t a t e s , Amsterdam: North Hol land . Diewert , W.E. [1970], "A Note on Becker 's Theory of the A l l o c a t i o n of Time and the Consumer's Derived Demand f o r Real Balances" , unpublished memorandum. Diewert, W.E. [1974a], "Intertemporal Consumer Theory and the Demand f o r Durables" , Econometrica, 42, 497-516. Diewert, W.E. [1974b], "App l ica t ions of Dua l i ty Theory", in F ron t ie rs of Quant i ta t ive Economics (M.D. I n t r i l i g a t o r and D.A. Kendr ick, e d s . ) , Amsterdam: North Hoi 1 and. Diewert , W.E. [1975], Annual Employment, Industry By Occupation in  Canada, Research Projects Group, S t ra teg ic Planning and Research, Department of Manpower and Immigration, Ottawa. Diewert , W.E. [1976a], "General ised Slutsky Condit ions f o r Aggregate Consumer Demand Func t ions" , D iscuss ion Paper 76-05, Department of Economics, U n i v e r s i t y of B r i t i s h Columbia. - 248 -Diewert, W.E. [1976b], "Exact and Super la t ive Index Numbers", Journal of Econometr ics, 4, 115-145. Diewert, W.E. [1977], "Nine Kinds of Quasi -Concavi ty and Concav i ty" , unpublished memorandum. Diewert, W.E. and Woodland, A .D. [1975], Annual Employment, Industry  by Occupat ion, in Canada 1926-1965 on a Labour Force Survey  B a s i s , Research Projects Group, S t ra teg ic Planning and Research, Department of Manpower and Immigration, Ottawa. Edgeworth, F.Y. [1888], "The Mathematical Theory of Banking", Journal  of the Royal S t a t i s t i c a l S o c i e t y , 51, 113-127. Edwards, J . R . [1972], "More on S u b s t i t u t a b i l i t y Between Money and Near Monies", Journal of Money, C r e d i t and Banking, 4, 551-571. E p s t e i n , L. [1977], Essays in the Economics of Uncer ta in ty , unpublished Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia. Evans, R.G. [1976], "Beyond the Medical Market P lace: Expenditure, U t i l i s a t i o n , and P r i c i n g of Insured Health Care in Canada", in The Role of Health Insurance in the Health Serv ices  Sector (R.N. Roset t , e d . ) , New York: National Bureau of Economic Research. Fe ige , E . L . [1964], The Demand fo r L i q u i d A s s e t s : A Temporal Cross - Sect ion A n a l y s i s , Englewood C l i f f s , N . J . : Prent ice H a l l . Fe ige , E . L . and Park in , M. [1971], "The Optimal Quantity of Money, Bonds, Commodity Inventor ies , and C a p i t a l " , American  Economic Review, 61, 335-349. Fe ige , E . L . and Pearce, D.K. [1976], " S u b s t i t u t a b i l i t y Between Money and Near-Monies: A Survey of the Time Ser ies Ev idence" , Soc ia l Systems Research I n s t i t u t e , U n i v e r s i t y of Wiconsin - Madison. F e l d s t e i n , M.S. [1969], "Mean-Variance Ana lys is in the Theory of L i q u i d i t y Preference and P o r t f o l i o S e l e c t i o n " , Review of  Economic S t u d i e s , 36, 5-12. F i s c h e r , S. [1974], "Money in the Production Funct ion" , Economic  Inquiry , 12, 517-533. F i s h e r , G.R. and Sparks, G.R. [1976], "A Survey of Empir ical Evidence on the St ructure of the Canadian Monetary S e c t o r " , paper de l i ve red at the Queens U n i v e r s i t y Conference on Monetary Issues, K ingston, Ontar io . F i s h e r , I. [1922], The Making of Index Numbers, Boston: Houghton M i f f l i n . - 249 -F i s h e r , I. [1930], The Theory of In te res t , New York: Macmillan Co. Friedman, M. [1956], "The Quantity Theory of Money: A Restatement", in Studies in the Quantity Theory of Money (M. Friedman, e d . ) , Chicago: Chicago Un ive rs i t y Press . Friedman, M. [1959], "The Demand f o r Money: Some Theoret ica l and Empir ical R e s u l t s " , Journal of P o l i t i c a l Economy, 67, 327-351. Friedman, M. and Meiselman, D.I. [1963], "The Rela t ive S t a b i l i t y of Monetary V e l o c i t y and the Investment M u l t i p l i e r in the United S t a t e s , 1897-1958", in S t a b i l i s a t i o n P o l i c i e s , E. Carey Brown et a l , Englewood C l i f f s , N . J . : Prent ice H a l l , 182-185, 242-246. G a l b r a i t h , J , A . [1970], Canadian Banking, Toronto: The Ryerson Press . G o l d f e l d , S.M. [1966], Commercial Bank Behaviour and Economic  A c t i v i t y , Amsterdam: North Hol land. G o l d f e l d , S.M. [1973], "The Demand fo r Money R e v i s i t e d " , Brookings  Papers on Economic A c t i v i t y , 577-638. Goddhart, C . A . E . [1969], "A Stable V e l o c i t y Function f o r Canada? A Note", Economica, 36, 314-315. Graml ich, E.M. and Kalchenbrenner, J . H . [1970], "A Constrained Est imat ion Approach to the Demand f o r L i q u i d Assets" , , Specia l Studies Paper No. 3, D i v i s i o n of Research and S t a t i s t i c s , Federal Reserve Board, Washington, D.C. Gray, M.R. and P a r k i n , J . M . [1973], " P o r t f o l i o D i v e r s i f i c a t i o n as Optimal Precaut ionary Behaviour" , in Theory of Demand:  Real and Monetary, M. Morishima and o t h e r s , Oxford: Clarendon Press . Gur ley , J . G . and Shaw, E . S . [1960], Money in a Theory of F inance, Washington: The Brookings I n s t i t u t i o n . Gussman, T .K . [1972], The Demand f o r Durables, Nondurables, Serv ices  and the Supply of Labour in Canada: 1946-1969, Research Branch, Department of Manpower and Immigration, Ottawa. H e l l i w e l l , J . F . et a l . [1971], The Structure of RDX2, Bank of Canada S t a f f Research Study No. 7, Ottawa. H i c k s , J . R . [1935], "A Suggestion f o r S i m p l i f y i n g the Theory of Money", Economica, 2, 1-19. Houthakker, H. and T a y l o r , L. [1970], Consumer Demand in the United  States (2nd e d . ) , Cambridge: Harvard U n i v e r s i t y Press . \ i - 250 -Jenk ins , G .P . [1972], Ana lys is of Rates of Return from Capi ta l in  Canada, unpublished Ph.D. D i s s e r t a t i o n , U n i v e r s i t y o f Chicago. K a m i , E. [1974], "The Value of Time and the Demand f o r Money", Journal of Money, Cred i t and Banking, 6, 45-64. Kaufman, G .C . [1969], "More on an Empir ica l D e f i n i t i o n o f Money", American Economic Review, 59, 78-87. Keynes, J . M . [1936], The General Theory of Employment, Interest and  Money, New York: Harcourt Brace. Khan, M.S. and K o u r i , P . J . J . K . [1975], "Real Money Balances as a Factor of Product ion: A Comment", Review of Economics and  S t a t i s t i c s , 57, 244-246. K h a l i d , M.S. [1977], "Choice Among Functional Forms: A Parametric Approach Based on the Genera l ised Box-Cox Functional Form", Ph.D. D i s s e r t a t i o n Prospectus, Department of Economics, U n i v e r s i t y of B r i t i s h Columbia. K l e i n , B. [1974], "Cdmpetit ive Interest Payments and the Long Run Demand fo r Money", American Economic Review, 64, 931-949. L a i d l e r , D. [1969], The Demand fo r Money: Theories and Evidence, Scranton: Internat ional Textbook Co. Laudadio, L. [1967], "On the Competition f o r Sav ings" , Canadian  Journal of Economics and P o l i t i c a l S c i e n c e , 33, 295-296. Laumas, G.S. and Formuzis, P.A. [1968], "The Demand fo r Money in Canada", Canadian Journal o f Economics, 1, 640-644. Laumas, P .S. [1969], "The Role of Savings Deposits as Money", Journal of Money, C r e d i t and Banking, 1, 789-795. L e v h a r i , D. and P a t i n k i n , D. [1968], "The Role of Money in a Simple Growth Model", American Economic Review, 58, 713-753. Levy, H. [1974], "The Rat ionale of the Mean-Standard Deviat ion A n a l y s i s : Comment", American Economic Review, 64, 434-441. L i n t n e r , J . H . [1965], "The Valuat ion of Risk Assets and the Se lec t ion of Risky Assets in Stock P o r t f o l i o and Capi ta l Budgets", Review of Economics and S t a t i s t i c s , 47, 13-37. Mal invaud, E. [1972], S t a t i s t i c a l Methods of Econometrics (2nd e d . ) , New York: American E l s v i e r Pub l ish ing Co. Manser, M:E. [1976], " E l a s t i c i t i e s of Demand fo r Food: An Ana lys is Using Non-Addit ive U t i l i t y Functions Al lowing f o r Habit Formation", Southern Economic J o u r n a l , 43, 879-891. - 251 -Manvel, A. [1968], "Three Land Research S t u d i e s " , Research Report No. 12, prepared for the National Commission on Urban Problems, Washington D.C. Markowitz, H.M. [1952], " P o r t f o l i o S e l e c t i o n " , Journal of F inance, 7, 77-91. Master Tax Guide: A Guide to Canadian Income Tax [1976], C . C . H . Canadian L imi ted . M i l l e r , M.M. and Or r , D. [1966], "A Model of the Demand f o r Money by F i rms" , Quarter ly Journal of Economics, 80, 413-435. M o d i g l i a n i , F. and Brumberg, R .E . [1954], " U t i l i t y Ana lys is and A the Consumption Funct ion: An In terpreta t ion of Cross Sect ion Data", in Post-Keynesian Economics ( K i r i h a r a , K.K. e d . ) , New Brunswick, N . J . : Rutgers U n i v e r s i t y Press . Morishima, M. [1973], "Consumer Behaviour and L i q u i d i t y Pre ference" , in Theory of Demand: Real and Monetary, M. Morishima and o thers , Oxford: Clarendon Press . Moroney, J . R . [1972], "The Current State of Money and Production Theory", American Economic Review, Papers and Proceedings, 62, 335-343. Moroney, J . R . and W i l b r a t t e , B . J . [1976], "Money and Money S u b s t i t u t e s : A Time Ser ies Ana lys is of Household P o r t f o l i o s ' ! , Journal  of Money, Cred i t and Banking, 8, 191-208. N a d i r i , M.I. [1969], "The Determinants of Real Cash Balances in the U.S. Total Manufacturing S e c t o r " , Quarter ly Journal  of Economics, 83, 173-196. Nagatani , K. [1977], Theories of a Monetary Economy (Japanese e d i t i o n ) , Tokyo: Sabunsha Publ ish ing Company. Nelson, C. [1973], Appl ied Time Ser ies Ana lys is f o r Managerial  F o r e c a s t i n g , San F ranc isco : Hoi den-Day. P a r k i n , J . M . , Cooper, R i J . , Henderson, J . F . and Danes, M.K. [1975], "An Integrated Model of Consumption, Investment and P o r t f o l i o D e c i s i o n s " , in Papers in Monetary Economics, V o l . II , Reserve Bank of A u s t r a l i a . P a t i n k i n , D. [1965], Money, In terest and Pr ices (2nd e d . ) , New York: Macmillan Co. P o l l a k , R.A. and Wachter, M.K. [1975], "The Relevance of the Household Production Function and i t s Impl icat ions f o r the A l l o c a t i o n of Time", Journal of P o l i t i c a l Economy, 83, 255-277. - 262 -P r a i s , Z. [1975], "Real Money Balances as a Var iab le in the Production Funct ion: A Comment", Journal o f Money, Cred i t and Banking, 7, 535-543. Rose, D.R. [1976], Expectat ions Formation as Optimal Forecast ing and  E r r o r Learn ing , unpublished Ph.D. D i s s e r t a t i o n , Un ive rs i t y of Manchester. Roy, R. [1947], "La D i s t r i b u t i o n du Revenue entre les D i f f e r s B i e n s " , Econometr ica, 15, 205-225. Royal Commission on Banking and Finance [1964], Report , Ottawa: Queenis P r i n t e r . Royal Commission on Banking and Finance, Appendix Volume [1964], Ottawa: Queen's P r i n t e r . S a l z y n , V. [1966], "The Competition f o r Personal Savings Deposits in Canada", Canadian Journal of Economics and P o l i t i c a l  S c i e n c e , 32, 327-337. Sa l zyn , V. [1968], "The Behaviour of Personal Savings Deposi tors: A Rejo inder" , Canadian Journal of Economics, 1, 110-113. Samuelson, P.A. [1947], Foundations o f Economic A n a l y s i s , Cambridge, Mass. : Harvard U n i v e r s i t y Press . Samuelson, P.A. [1969], "L i fe t ime P o r t f o l i o S e l e c t i o n " , Review of  Economics and S t a t i s t i c s , 51, 239-247. Saving, T .R. [1971], "Transact ions Costs and the Demand f o r Money", American Economic ..Review, 61, 407-419. Sav ing , T.R. [1972), "Transact ions Costs and the Firm Demand f o r Money", Journal of Money, Cred i t and Banking, 4, 245-259. Sav ing , T.R. [1976], "Transact ions Cost Functions and the Inventory-Theoret ic Approach to Money Demand", Journal of MOney,  C r e d i t and Banking, 8, 339-346. Sharpe, W.F. [1964], "Capi ta l Assets P r i c e s : A Theory of Market Equ i l ib r ium Under Condit ions of R i s k " , Journal of F inance, 19, 425-442. Shearer , R.A. [1970], "The Income V e l o c i t y of Money in Canada, 1960-68: A Further Comment", Economica, 37, 409-419. Shearer , R.A. [1972], "A C r i t i c a l Note on Two Sectors of the F inanc ia l Flow Accounts" , Canadian Journal of Economics, 5, 541-553. Shor t , B.K. [1972], "The Demand f o r Money in Canada: A Comment", Economica, 39, 442-446. - 253 -Shor t , B.K. and V i l l e n e u v a , D.P. [1976], "The E l a s t i c i t i e s and S t a b i l i t y of Subs t i tu t ion Between Money and Near Monies in Canada", paper de l i ve red to the Annual Meeting of the Canadian Economics A s s o c i a t i o n , Quebec C i t y . S i d r a u s k i , M. [1967], "Rational Choice and Patterns o f Growth in a Monetary Economy", American Economic Review, Papers and  Proceedings, 57, 534-544. S i n a i , A. and Stokes, H.H. [1972], "Real Money Balances: An Omitted Var iab le from the Production Funct ion?" , The Review of  Economics and S t a t i s t i c s , 65, 290-296. S i n a i , A. and Stokes, H.H. [1975], "Real Money Balances: An Omitted Var iab le from the Production Function? A Reply" , Review of  Economics and S t a t i s t i c s , 57, 247-252. Smith, L . B . [1967], "The Competition fo r Personal Savings Deposits in Canada: A Comment", Canadian Journal of Economics and  P o l i t i c a l Sc ience , 33, 291-294. Stone, R. and Rowe, D.A. [1960], "The D u r a b i l i t y of Consumers' Durable Goods", Econometrica, 28, 407-416. S t r o t z , R.H. [1957], "The Empir ica l Impl icat ions of a U t i l i t y T r e e " , Econometrica, 25, 269-280. Te igen , R.L . [1964], "Demand and Supply Functions f o r Money in the United S t a t e s , Some St ruc tura l Est imates" , Econometr ica, 32, 477-509. Timberlake, R.H. and For tson , J . [1967], "Time Deposits in the D e f i n i t i o n of Money", American Economic Review, 57, 190-194. Tob in , J . [1956], "The Interest E l a s t i c i t y of Transact ions Demand f o r Cash" , Review of Economics and S t a t i s t i c s , 38, 241-247. Tob in , J . [1958], " L i q u i d i t y Preference as Behaviour Towards R i s k " , Review of Economics and S t a t i s t i c s , 38, 241-247. Tornqu is t , L. [1936], "The Bank of F i n l a n d ' s Consumption Pr ice Index", Bank of F in land Monthly B u l l e t i n , 10, 1-8. T s i a n g , S . C . [1972], "The Rat ionale o f the Mean-Standard Deviat ion A n a l y s i s , Skewness Preference, and the Demand f o r Money", American Economic Review, 62, 354-371. T s i a n g , S . C . [1974], "The Rat ionale of the Mean-Standard Deviat ion . , A n a l y s i s : Reply and Erra ta for Or ig ina l A r t i c l e " , American  Economic Review, 64, 442-450. U n i v e r s i t y of Western Ontario [1965], The Role of the Trust and Loan Companies in the Canadian Economy (study prepared f o r the Royal Commission on Banking and F inance) , School of Business A d m i n i s t r a t i o n , London, Ontar io . - 254 -Walras, L. [1926], Elements of Pure Economics, P a r i s . Homewood, 111'.; 1954. i - 25:5 -STATISTICAL SOURCES STATISTICS CANADA (Catalogue number in brackets) CANSIM - Canadian Socio-Economic Information Management System. Cheques^Cashed in C lear ing Centres (11-002). C r e d i t Unions (61-209). Estimated Populat ion by Sex and Age Group f o r Canada and  the Provinces (91-202). F inanc ia l Flow Accounts, System of National Accounts (13-002). F inanc ia l I n s t i t u t i o n s , F inanc ia l S t a t i s t i c s (formerly Business  F inanc ia l S t a t i s t i c s , Balance Sheets, Selected F inanc ia l  I n s t i t u t i o n s ) (61-006). National Income ?and Expenditure Accounts, V o l . 1, The Annual  Est imates , 1926-1974 (13-531). National Income and Expenditure Accounts, H i s t o r i c a l R e v i s i o n ,  1926-1971 (mimeo, Gross National Product D i v i s i o n ) . Pr ices and Pr ice Indexes (62-202). BANK OF CANADA Bank of Canada Review. Bank of Canada S t a t i s t i c a l Summary. Canadian Currency and Chartered Bank Deposits,^1926 to Date (Dept. of Banking and F inanc ia l Ana lys is (DBFA)). I n s t i t u t i o n s with respect to the Return of Assets and L i a b i l i t i e s (Pursuant to sect ion 103 of the Bank Act) (copy a v a i l a b l e from DBFA). Se lected Canadian and Internat ional Interest Rates, Including  Bond Y ie lds and Interest Arb i t rage (DBFA). OTHER National Health Expenditures in Canada, 1960-1973, Health and Welfare Canada, Ottawa. Taxation S t a t i s t i c s , Department of National Revenue, Government of Canada, Ottawa. - 256 -Canadian _Housing S t a t i s t i c s , Central Mortgage and Housing Corporat ion (formerly HousTng in Canada). Canadian Consumer Cred i t Factbook, Canadian Consumer Loan Assoc ia t ion and Federated Council of Sales Finance Companies, Montreal . Appendix A THE CONSTRUCTION OF FORECAST PRICE SERIES  USING AN ARIMA MODEL The general l i n e a r s t o c h a s t i c process model of a time s e r i e s , z^, may be wr i t ten as: ( A . l ) z t = y + a t + a t _ 1 + e 2 a t _ 2 + . . . oo = y + I 0, a . , 0 = 1 j = 0 where: y i s any reference l e v e l , usua l l y taken to be the mean of the uncondi t ional d i s t r i b u t i o n of z , i f the process i s s ta t ionary (the s t a t i o n a r i t y concept i s explained below); the " a ' s " are genera l ly assumed to be random drawings from a d i s t r i b u t i o n with zero mean and constant va r iance ; and 0-j, • • • » are f i x e d c o e f f i c i e n t s . A s t o c h a s t i c process def ined by (A . l ) i s s ta t ionary i f the mean, va r iances , and autovariances of are f i n i t e and independent of t ime, i . e . , i f the fo l lowing condi t ions hold: ( i ) E ( z t ^ ~ y ( s o m e cons tan t ) , ( i i ) E ( z ' t z ' ^ ) = a z 2 (some constant , independent of t ) , ( i i i ) E ( z ' t z ' t _ s ) = Y s (a constant , independent of t , i . e . E ( z ' t + r z ' t + j _ s ) = y s , f o r a l l j ) , where z ' t i s the dev ia t ion of z^ from i t s mean. Sta t ionary processes e x h i b i t a c e r t a i n kind of s t a t i s t i c a l e q u i l i b r i u m , possessing proper t ies which are independent of t ime. The general l i n e a r process def ined by (A . l ) w i l l be s ta t ionary i f and only i f - 258 r the sequence , e 2 , . . . . i s e i t h e r f i n i t e or i n f i n i t e and convergent. We may rewri te (A.T) using the lag operator , Lz^ E ^ , as fo l lows: (A.2) z t = I - t + e(L) a t , where (A.3) e(L) = I + 9 -j L + e 2 L 2 + . . . The s ta t ionary condi t ions on (A.2) can then be restated as the requirement that e(L), viewed as the generating funct ion of the e c o e f f i c i e n t s , converges on or wi th in the un i t c i r c l e . I f e(L) i s i n v e r t i b l e (the " i n v e r t i b i l i t y cond i t ions" are met) then we may express (A.2) in an autoregressive form as (A.4) e^1 (L) z t = e-1 (L) y + a t o r , de f in ing (A.5) e - 1 (L) = <j,(L) = I - - c ^ L 2 . . . we may rewri te (A.5) as co (A.6) z t = y ( l - I + j ) + a t + + <f>2 z t _ 2 + . . . 3 * where l im <(). = 0 (A.6) i s ( A . l ) reexpressed in autoregressive form. A l t e r n a t i v e l y , we could have s ta r ted out with a representat ion of the form (A.6) and - 259 -assuming the corresponding s ta t ionar i t y condit ions hold, reexpressed i t in the form ( A . l ) , ca l led the moving average representat ion. . I t may be shown that i f both the s ta t ionar i t y and the i n v e r t i b i l i t y condit ions are met, which we shal l henceforth assume, then the two forms (A. l ) and (A.6) are equivalent, being d i rec t unique transforms of each other. The special case of (A.6) where thep coe f f i c ien t i s non zero and a l l succeeding coef f i c ien ts are zero i s ca l led the autoregressive  process of order p, or AR(p): (A.7) <|>(L) z t = y + a t , where (A.8) <|,(L) = I - - <)»2L2- . . . - <f, pL P and y i s now redefined to be any constant. t h The specia l case of (A . l ) where the q coe f f i c ien t i s non zero and a l l succeeding coe f f i c ien ts are zero i s ca l led the moving average  process of order q , or MA(q): (A.9) z t = y + 6(L) a t , where (A.10) e(L) = I + e 1 + e 2 L 2 + . . . + e q L q An obvious general izat ion i s the mixed autoregressive moving  average model of order (p,q) referred to as the ARMA(p,q) process: (A.11) <f»(L) z t = y + e(L) a t , where <j>(L) and e(L) are defined by (A .8) and (A. 10) respect ive ly . - 260 -Most economic time ser ies do not exh ib i t s t a t i o n a r i t y and therefore the ARMA(p,q) model (A.11) i s inappropr ia te . However, we may genera l ise (A.11) by al lowing z to r e f e r , not to the leve l of the s e r i e s , but t o some d i f fe rence of some transform of the s e r i e s : (A.12) <j>(L) w t = y + 8 (L) a t (A.13) w t = (I - L ) d l ' t (A.14) z ' t = f(3: t) The model def ined by (A.12) - (A.13) i s c a l l e d the autoregressive  integrated moving average process of order (p ,d ,q) where d is the appropr iate order of d i f f e r e n c i n g required to reduce the (transformed, i f necessary) s e r i e s to s t a t i o n a r i t y . In empir ica l a p p l i c a t i o n s in economics, logar i thmic transformations are often useful where rates of change e x h i b i t s t a t i o n a r i t y . The f i r s t stage in using the ARIMA model fo r fo recas t ing purposes i s to undertake some pre l iminary i d e n t i f i c a t i o n of the appropr iate order of p, d , and q , and a lso obtain some approximate parameter va lues . The basic too ls used in t h i s i d e n t i f i c a t i o n process are the a u t o c o r r e l a -t i o n and p a r t i a l au tocor re la t ion f u n c t i o n s . The au tocor re la t ion funct ion cons is ts of the c o r r e l a t i o n s , p Q , between and z t _ ^ as q v a r i e s . The p a r t i a l au tocor re la t ion funct ion e x p l o i t s the fac t that whereas an AR(p) process has an i n f i n i t e au tocor re la t ion f u n c t i o n , that funct ion can be represented with p non zero funct ions of the a u t o c o r r e l a t i o n s . We have: (A.15) P j = * p l P . _ } + . . . + * p l p _ l P j - _ p + 1 + * p p P j _ p j = l , . • • ,p - 261 -We may so lve the system (A.15) fo r a>^, <f >22'"''' ^pp' c a ^ e < ^ the P p a r t i a l a u t o c o r r e l a t i o n s . Knowledge of the t h e o r e t i c a l au tocor re la t ion and p a r t i a l auto-c o r r e l a t i o n funct ions f o r the var ious forms of ARIMA models provides the key to i d e n t i f i c a t i o n . For example, an MA(q) process has at most q non zero a u t o c o r r e l a t i o n s , while the corresponding p a r t i a l a u t o c o r r e l a t i o n funct ion of the s e r i e s t a i l s o f f g radua l l y . Converse ly , an AR(p) process has at most p non zero p a r t i a l a u t o c o r r e l a t i o n s , and the corresponding a u t o c o r r e l a t i o n funct ion t a i l s o f f . I f the computed au tocor re la t ions as p a r t i a l au tocor re la t ions reveal one o f these pa t te rns , one can i d e n t i f y t e n t a t i v e l y the s t ruc ture of the model, and i n f e r some approximate s t a r t i n g values (see Box and Jenkins [1970; 517-520]). Mixed models are rather more d i f f i c u l t to i d e n t i f y - Box and Jenkins [1970, Ch.6] d iscuss some simple cases . F i n a l l y , i f the a u t o c o r r e l a t i o n funct ion f o r a s e r i e s f a i l s to die out qu ick ly , e s p e c i a l l y when the autocor re la t ions are a l l p o s i t i v e and dec l ine approximately l i n e a r l y , the need to d i f f e r e n c e the s e r i e s in order to achieve s t a t i o n a r i t y i s i n d i c a t e d . The second stage o f the model l ing process c o n s i s t s of est imat ing the parameters of the se lec ted model. Except f o r pure AR models, non l i n e a r techniques w i l l be requ i red . We use a maximum l i k e l i h o o d e s t i -mation procedure employed by Rose [1976]. Once the model has been e s t i -mated, a number of t e s t s of the model 's adequacy may be performed. F i r s t , the s i g n i f i c a n c e o f ind iv idua l c o e f f i c i e n t s may be t e s t e d . Second, we can examine the res idua ls to see i f they have been reduced to 'white n o i s e ' . There are several forms in which such t e s t i n g can proceed ( e . g . Box-Pierce t e s t s , Kolmogorov-Smirnov t e s t s , and tes ts on ind iv idua l res idua l auto-c o r r e l a t i o n s ) . Based on these t e s t s , a number of a l t e r n a t i v e poss ib le models - 262 -may be considered and est imated. The process continues u n t i l a s a t i s -fac tory model i s found. The f i t t e d values from the estimated model, z ^ + ^ , provide the f o r e c a s t s . A f i t t e d value z^ + ^ provides the optimal fo recas t of z t + - | , cond i t iona l on the information z ^ , zt_-|,... . In our case , we require fo recas ts f o r 1948-1975. However, in est imat ing an ARIMA model of order ( p , d , q ) , the f i r s t p+d observat ions must be dropped from the sample i f extra data is not a v a i l a b l e . In our case , f i r s t d i f f e r e n c i n g was required f o r a l l twelve durable good purchase pr ice s e r i e s . With three s e r i e s , autoregressive terms were present in the model, and hence needed observat ions were l o s t . In order to recover these observa t ions , we used a backcast ing technique, explained in Box and Jenkins [1970; C h . 6 . 2 ] , The ARIMA model estimates f o r the purchase p r i c e s e r i e s fo r twelve durable goods, and the aggregate p r i c e s e r i e s , p, are given in Table A . l . In a l l c a s e s , we transformed the data by taking logar i thms. The au tocor re la t ion funct ion fo r the estimated res idua ls i s not shown fo r space reasons. However, in the models we used, there was no evidence to i n d i c a t e the presence of s i g n i f i c a n t au tocor re la t ion among the r e s i d u a l s . In the case of the twelve durable good p r i c e s e r i e s , we required a fo recas t f o r the log of the s e r i e s , log z ^ + ^ , in order to c a l c u l a t e the expected rate of change of the s e r i e s . However, in the case of "p, the aggregate p r ice s e r i e s , we required a fo recas t f o r the Z - eyeZ . , not the rate of change, of p. Thus i t was necessary to "untransform" the f o r e -cast f o r log p, namely log p ~ + 1 , in the manner der ived by Nelson [1973; ] , as f o l l o w s : (A.16) p t + 1 = exp [(lrw^+1) + ( s 2 + l ) / 2 ] - 263 -where s 2 i s the var iance of the forecasts of Ini P t + ] » f ° r a l l t , in the sample pe r iod , s 2 i s approximated by s 2 , which is simply the sum of the squared res idua ls between the fo recas t se r ies and the actual In p^ . s e r i e s . The term ( s 2 + l ) / 2 a r i s e s as a r e s u l t of the fo l lowing r e l a t i o n -s h i p . Let y be a va r iab le ( in our case ( ln *^ ' + ^) ) normally d i s t r i b u t e d with mean y and var iance a 2 . (A.17) y a, n ( y , a 2 ) Then (A.18) x B e y * l n ( e ( x ) , v(x)) where (A.19) e(x) = e v + a 2 / 2 (A.20) (x) = e ( 2 p + a 2 ) [ e ° 2 - l ] - 264 -TABLE A . l ARIMA MODEL ESTIMATES FOR TWELVE DURABLE GOODS PURCHASE PRICE SERIES AND THE AGGREGATE PRICE LEVEL ( a l l se r ies were f i r s t transformed by taking logarithms) SERIES PFOOT PRECE PHAPP PSDHF PAUTO PJEWL PWCLH PHOUS**I II III PLAND PMCLH PBOOK MODEL PARAMETER ESTIMATES (s . e r rors in brackets) 4>2 *1 9 0 e l 6 3 67 (p d q)e£ (013 )1=e2 (no (017 e2+e6 (111 (013 e2=0 (010 (013 02=O (on (110 (oio (010 (211 <h=o (010 'o =0 e0 .360 (.042) 1 0o .044 (.012) .416 . .012 (.184) (.006) .500 .014 (.167) (.007) .027 (.013) .035 (.008) .028 (.012) .050 (.013) .461 .022 (.194) (.011) .033 (.013) .071 (.014) .015 (.006) .041 (.007) ,666 -.811 (.153) .634 (.321) - .465 .260 (.161) -.361 (.191) .330 - .419 .853 .9720 .1.551 .9407 .1442 .7935 .2665 .9632 .0094 .9080 .1220 .7100 -.8708 .1563 .9646 .1126 .9697 .1563 .7802 -.9836 -.9837 .5742 .9846 -continued - 265 -TABLE A . l (continued) SERIES MODEL PARAMETER ESTIMATES (s . er rors in brackets) R 2 R 2 f (p d q ) e * * 2 <h 0o e i '3 °7 PFURN (013)6 0 .013 -.327 - .377 .9520 .1029 62=0 (.010) (.196) (.207) P (010)6 0 .037 .9847 -(.011) * (p d . q ) e 0 : re fe rs to an ARIMA model of order (p d q) with the presence of a constant term, e 0 . * * Models I and II were based on the e n t i r e sample p e r i o d . Model III was estimated f o r the period 1947-1961 (see sec t ion C, Chapter 5, f o r d i s c u s s i o n ) . t R 2 re fe rs to the W of the d i f fe rences s e r i e s . PF00T: Footwear and repa i r PRECE: Recreat ion , spor t ing and camping equipment PHAPP: Household appl icances PSDHF: Semi durable household fu rn ish ings PAUT0: Automobiles PJEWL: Jewel lery PWCLH: Women and c h i l d r e n ' s c l o t h i n g PH0US: Housing PMCLH: Men's c l o t h i n g PLAND: Land PB00K: Books, newspapers and magazines PFURN: F u r n i t u r e , carpets and other f l o o r coverings P: Aggregate p r i c e leve l Appendix B SUPPLEMENTARY DATA TABLES Th is Appendix contains a number of data tab les too lengthy to inc lude in Chapter 5. Tables B. l - B.3 are s t ra igh t fo rward , and the reader is re fer red to the re levant sect ions of Chapter 5 f o r f u r t h e r e l a b o r a t i o n . Table B.4 provides a comparison of the Cummings and Meduna [1973] and the D i v i s i a methods of const ruc t ing aggregate p r i c e indexes, based on i d e n t i c a l disaggregated data . The f i r s t four columns of Table B.4 l i s t , f o r four consumption good s e r i e s , the d i f f e r e n c e between the Commings and Meduna formula (5.15) and the D i v i s i a formula (5 .7 ) , expressed as a percentage of the l a t t e r . The f i n a l column l i s t s the average absolute per cent d i f f e r e n c e over the four s e r i e s . There are d i f f e r e n c e s , but they do not appear to be very great . Inspecting the l a s t column, in t h i r t e e n of the 28 years the average d i f f e r e n c e was less than one percent . For s ix years i t ranged between one and two percent , and was between two and three percent f o r another s ix y e a r s . F i n a l l y , in only two years out of twenty eight d id i t exceed three percent . The average d i f f e r e n c e over the en t i re period ( i . e . , the sum of the elements in the l a s t column d iv ided by twenty eight) was 1.31 percent . Over the sub per iod 1968-1974, however, when i n f l a t i o n rates were h igh , the d i f f e r e n c e was cons iderab ly l a r g e r , averaging 2.3%. The conc lus ion to be drawn from th is comparison seems to be that these two a l t e r n a t i v e methods do not provide very d i f f e r e n t answers in p r a c t i c e . However, when pr ices are changing r a p i d l y from period to p e r i o d , s i g n i f i c a n t d i f f e rences begin to appear (see f o r example, the 1974 d i f f e r e n c e ) . In these c i rcumstances, due caut ion should be exerc ised as regards the t h e o r e t i c a l r a t i o n a l e f o r the p a r t i c u l a r index number used. - 267 -TABLE B. l STOCK SERIES, TWELVE DURABLE GOODS Semi Men's pH Women's & Footwear durable Books, "Jewel lery , boys' c h i l d r e n ' s and household newspapers ; watches YEAR c lo th ing • c l o t h i n g repa i r f u r n i t u r e & magazines & ' repa i r 1947 1063. ,903 1350. 640 745. ,089 1777. .337 330. 388 87. ,000 1948 1003. .952 1355. 320 716. ,544 1800. .402 364. 194 122. ,000 1949 959. .976 1363. 660 704. ,272 1815. .241 386. ,096 174. ,320 1950 954. ,988 1376. 830 687, .136 1879, .145 419. ,048 214. ,592 1951 944. .494 1394. 415 649. .568 1930, .487 450. ,524 239. ,755 1952 964, .247 1475. 208 630, .784 1832, .292 470. ,262 272. ,853 1953 988, .124 1559. 604 629, .392 1881, .376 489. ,131 300. ,712 1954 992, .062 1630. 807 630, .696 1921 ,826 506. ,565 313. ,427 1955 1023, .031 1736. 401 645, .348 2004, .095 533. ,283 328. ,056 1956 1074, .516 1870. 201 669, .674 2091, .458 572. ,641 342, .834 1957 1107, .258 1995. 101 692, .837 2126, .875 606. ,321 351 , .700 1958 1136, .629 2098. ,551 723, .418 2183 .125 611. .160 358, .831 1959 1171, .314 2217. ,276 745, .709 2252.875 628. .580 366, .812 1960 1205, .657 2308. ,638 763, .855 2272 .725 648, .290 370. .087 1961 1233 .829 2388. ,319 776 .927 2299 .635 675, .145 373, .052 1962 1287 .914 2473. ,160 799 .464 2320 .781 700, .573 384, .831 1963 1 349 .947 2520. ,580 818 .732 2344 .469 731, .286 399, .898 1964 1430 .979 2593. ,291 844 .366 2408 .682 775, .643 424, .939 1965 1511 .490 2696. ,646 868 .183 2482 .209 824, .822 453 .963 1966 1579 .745 2813, .092 882 .092 2575 .326 882 .411 483 .378 1967 1631 .873 2961 , .662 910 .046 2666 .196 942 .206 514 .027 1968 1696 .937 3121, .831 935 .023 2769 .718 969 .103 532 .416 1969 1776 .469 3303, .916 955 .511 2895 .831 987 .552 551 .449 1970 1843 .234 3427, .958 960 .756 2986 .499 982 .776 555 .870 1971 1935 .617 3648, .980 983 .378 3127 .900 992 .388 569 .522 1972 2072 .809 3968, .490 1036 .689 3354 .740 1010 .194 600 .713 1973 2274 .405 4384, .242 1112 .344 3644 .844 1116 .097 655 .428 1974 2467 .203 4818, .121 1199 .172 2901 .907 1199 .049 696 .257 continued - 268 -TABLE B. l (continued) YEAR F u r n i t u r e , carpets & other f l o o r coverings Recre-a t i o n , ' ' spor t ing Household Scamping appl iances equipment New & used (net) automobiles 8i repa i rs 8< oarts Housing Land 1947 1301. 263 606, .615 753, .864 849, .846 12255. 900 8718, .048 1948 1356. 119 652, .700 802, .552 1163, .889 12871. 500 9382, .058 1949 1396. 388 756, .232 842, .017 1592, .000 12616. 700 9960, .544 1950 1443. 885 873, .055 877, .193 2176, .240 14495. 300 9959, .367 1951 1458.216 900, .125 914, .982 2553, .893 15057. 900 10961, .534 1952 1524. 780 953, .997 975, .836 2881 .803 15711. 400 10320, .343 1953 1603. 498 1018, .531 1051, .910 3313 .899 16670. 600 10906, .586 1954 1684. 023 1090 .288 1142 .009 3527 .007 17791. 900 8187, .876 1955 1804. 007 1183 .612 1267 .187 "4039 .446 19255. 100 8379 .671 1956 1945. 833 1290 .565 1429 .078 4555 .398 20694. 700 8210 .586 1957 2055. 288 1372 .026 1588 .971 4801 .887 21886. 300 8001 .670 1958 2156. 617 1430 .832 1745 .287 5017 .355 23484. 200 7828 .465 1959 2265. 390 1495 .171 1923 .777 5294 .496 25090. 100 8150, .914 1960 2351. 170 1546 .777 2093 .784 5517 .035 26271. 800 8912 .069 1961 2436. 395 1613 .565 2284 .089 5747 .266 27404. 000 8906 .300 1962 2533. 171 1703 .059 2492 .431 6170 .031 28573. 900 8652 .953 1963 2636. 269 1816 .059 2714 .020 6728 .422 29780. 600 8637 .139 1964 2761. 469 1958 .181 298? .076 7417 .461 31280. ,100 9109 .882 1965 2911. 506 2120 .758 3274 .050 8254 .570 32845. ,100 10073 .086 1966 3084. 586 2309 .104 3611 .500 8992 .289 34201. 900 9943 .381 1967 3244. 261 2493 .320 3969 .085 9601 .449 35566. ,900 10626 .744 1968 3410. 119 2693 .350 4333 .574 10264 .043 37222. ,700 10736 .388 1969 3596. ,966 2911 .573 4727 .523 10853 .109 29168. ,100 10538 .390 1970 3740. ,752 3094 .823 5078.742 10877 .238 40803. ,900 11366 .770 1971 3933. ,786 3339 .109 5530 .203 11336 .609 42795. ,800 10854 .051 1972 4216. ,566 3676 .454 6149 .859 12075 .359 45169. .900 11996 .108 1973 4574. ,770 4120 .543 6929 .387 13232 .258 47787. ,700 15124 .289 1974 4915. ,836 4552 .938 7773 .215 14151 .227 50244. .000 18277 .263 - '269 -TABLE B.2 PURCHASE PRICE SERIES, TWELVE DURABLE GOODS YEAR Men's Women's & Footwear & boys' c h i l d r e n ' s and c lo th ing c lo th ing repa i r Semi durable household f u r n i t u r e Books, newspa-pers & magazines Jewe l le ry , watches & repa i rs 1947 0.6666 0.8178 0.5347 0.5818 0, .6212 0. .7126 1948 0.8114 0.9666 0.6541 0.6757 0, .6583 0, .7714 1949 0.8472 1.0117 0.6763 0.7320 0, .6863 0, .7624 1950 0.8463 1.0072 0.6836 0.7380 0. .7124 0, .7909 1951 0.9529 1.0790 0.8007 0.8634 0. .7510 0. .8829 1952 0.9695 1.0656 0.8529 0.8787 0, .8041 0, .8450 1953 0.9585 1.0499 0.8599 0.8824 0, .8268 0, .8467 1954 0.9578 1.0294 0.8576 0.8802 0, .8435 0. .8571 1955 0.9506 1.0109 0.8606 0.8884 0, .8536 0, .8786 1956 0.9538 1.0120 0.8761 0.9325 0 .8431 0 .9110 1957 0.9719 0.9934 0.8911 0.9679 0 .8656 0 .9315 1958 0,9794 1.0004 0.9019 0.9813 0 .9708 0 .9592 1959 0.9834 0.9880 0.9375 0.9841 0 .9876 0 .9671 1960 0.9887 0.9892 0.9821 0.9913 0 .9970 0 .9800 1961 1.0000 1.0000 1.0000 1.0000 1 .0000 1 .0000 1962 1.0164 1.0055 1 .0122 1.0319 1 .0275 1 .0186 1963 1.0382 1.0421 1.0215 1.0609 1 .0446 1 .0592 1964 1.0556 1.0743 1.0391 1.0709 1 .0561 1 .0811 1965 1.0766 1.0843 1.0762 1.0858 1 .0755 1 .1055 1966 1.1104 1.1181 1.1473 1.1215 1 .1170 1 .1564 1967 1.1675 1.1646 1.2154 1.1731 1 .1717 1 .2188 1968 1.2009 1.1846 1.2813 1.2162 1 .2550 1 .2857 1969 1.2317 1.2140 1.3300 1.2439 1 .4000 1 .3319 1970 1.2513 1.2247 1.3747 1 .2762 1 .4458 1 .3778 1971 1.2682 1.2315 1 .4175 1.2987 1 .5389 1 .4153 1972 1.2805 1.2481 1.4257 1.3430 1 .6174 1 .4672 1973 1.3352 1.3000 1.5236 1.4314 1 .6743 1 .5593 1974 1.4654 1.4094 1.6376 1.6268 1 .8892 1 .8580 continued - 270 -TABLE B.2 (continued) Furn i tu re , New & carpets Recreation, used (net) & other spor t ing automobiles f l o o r Household & camping & repa i rs YEAR coverings appl iances equipment & parts ' Housing Land 1947 0.6821 0.9655 0.7413 0.6567 0.6100 0.2787 1948 0.7647 1.0270 0.7923 0.7228 0.7150 0.3188 1949 0.7947 1.0395 0.8413 0.7692 0.7450 0.3310 1950 0.8107 1.0724 0.8585 0.8019 0.7820 0.3699 1951 0.9181 1.2519 0.9414 0.9088 0.9020 0.4027 1952 0.9339 1.2423 0.9763 0.9319 0.9180 0.4542 1953 0.9463 1.2014 0.9964 0.9225 0.9260 0.4600 1954 0.9487 1.1279 0.9904 0.9185 0.9180 0.6483 1955 0.9393 1.0962 0.9808 0.8587 0.9360 0.6990 1956 0.9469 1.0741 0.9743 0.8767 0.9500 0.7782 1957 0.9771 1.0716 0.9783 0.9527 0.9770 0.8685 1958 0.9839 1.0965 1.0122 0.9635 0.9740 0.9496 1959 0.9923 1.0816 1.0092 0.9994 0.9730 0.9434 1960 0.9885 1.0508 1.0087 1.0123 0.9920 0.9504 1961 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1962 1.0123 0.9913 1.0000 0.9902 0.9970 1.0700 1963 1.0270 0.9682 1.0148 0.9956 1.0200 1.1430 1964 1.0410 0.9534 1.0251 0.9755 1.0610 1.1840 1965 1.0527 0.9525 1.0261 0.9749 1.1210 1.1890 1966 1.0857 0.9585 1.0468 0.9751 1.1960 1.3370 1967 1.1404 0.9789 1.0914 1.0035 1.2650 1.3760 1968 1.1629 0.9883 1.1227 1.0343 1.2780 1.4400 1969 1.1858 0.9976 1.1449 1.0531 1.3370 1.6150 1970 1.2029 0.9988 1 .1659 1 .1015 1.3800 1 .6100 1971 1.2153 0.9968 1.1706 1.1238 1.4640 1.7630 1972 1.2399 1.0091 1 .1690 1.1495 1.5330 1 .8760 1973 1.3055 1.0350 1.1782 1 .1651 1.7480 1.7950 1974 1.4616 1.1250 1.2553 1.2716 2.0940 1.8710 - 271 v TABLE B.3 ESTIMATED TAX RATES ON EARNINGS BY OCCUPATION, 1947-1974 0 C C U P A T I 0 N YEAR 1 2 3 4 5 6 1947 .2127 .1698 .1698 .1576 .1173 .1963 1948 .1806 .1477 .1477 .1365 .0833 .1614 1949 .1495 .0923 ,0831 .0632 .0932 .0834 1950 .1560 .1019 .0927 .0777 .0956 .1237 1951 .1678 .1382 .1284 .1095 .1087 .1552 1952 .1970 .1742 .1662 .1459 .1208 .1795 1953 .1914 .1752 .1646 .1524 .1193 .1796 1954 .1796 .1666 .1567 .1519 .1083 .1691 1955 .1720 .1577 .1496 .1473 .1010 .1597 1956 .1632 .1550 .1456 .1437 .0925 .1550 1957 .0640 .1568 .1469 .1419 .0926 .1561 1958 .1540 .1434 .1263 .1263 .0885 .1371 1959 .1641 .1518 .1357 .1324 .0978 .1489 1960 .1693 .1560 .1415 .1415 .1046 .1528 1961 .1706 .1589 .1471 .1444 .1059 .1550 1962 .1698 .1612 .1429 .1429 .1131 .1577 1963 .1706 .1618 .1462 .1462 .1216 .1590 1964 .1793 .1736 .1589 .1589 .1319 .1688 1965 .1649 .1582 .1460 .1460 .1232 .1557 1966 .1654 .1592 .1436 .1436 .1240 .1527 1967 .1892 .1571 .1515 .1534 .1312 .1695 1968 .1990 .1917 .1618 . 161;8 .1386 .1785 1969 .2204 .2155 .1886 .1886 .1658 .2058 1970 .2239 .2186 .1901 .1901 .1707 .2075 1971 .2184 .2184 .1900 .1900 .1726 .2088 1972 .2459 .2459 .2169 .2169 .2104 .2328 1973 .2485 .2369 .1940 .1940 .1799 .2022 1974 .2258 .2258 .1928 .1928 .1596 .2258 continued - 272 -TABLE B.3 (continued) 0 C C U P A T I 0 N YEAR 7 8 9 10 11 12 1947 .1078 .1946 .1227 .2081 .1946 .1576 1948 .0902 .1614 .1279 .1806 .1614 .1477 1949 .0949 .0834 .0838 .1476 .0834 .0831 1950 .0927 .1237 .0670 .1560 .1325 .0927 1951 .1056 .1575 .0974 .1678 .1575 .1382 1952 .1180 .1864 .1302 .1970 .1848 .1742 1953 .1179 .1892 .1405 .1888 .1815 .1721 1954 .1083 .1747 .1409 .1409 .1747 .1615 1955 .1030 .1642 .1326 .1699 .1642 .1544 1956 .0954 .1570 .1259 .1632 .1570 .1516 1957 .0975 . 1568 .1165 .1640 .1568 .1533 1958 .0824 .1449 .0978 .1540 .1469 .1342 1959 .0823 .1498 .1166 .1611 .1518 .1424 1960 .0869 .1535 .1219 .1693 .1560 .1496 1961 .0972 .1589 .1281 . 1684 .1593 .1517 1962 .0989 .1005 .1377 .1679 .1590 .1488 1963 .1086 .1618 .1394 .1706 .1618 .1516 1964 .1210 .1708 .1597 .1782 .1708 .1641 1965 .1146 .1582 .1465 .1629 .1582 .1524 1966 .1208 .1575 .1509 .1633 .1573 .1495 1967 .1264 .1779 .1571 .1929 .1695 .1571 1968 .1348 .1878 .1674 .1982 .1785 .1674 1969 .1607 .2155 .2000 .2213 .2097 .1945 1970 .1652 .2186 .2008 .2278 .2131 .2008 1971 .1639 .2184 .1900 .2265 .2118 .2027 1972 .2031 .2495 .2241 .2564 .2365 .2241 1973 .1799 .2485 .2151 .2545 .2022 .2151 1974 . .1487 .2565 .1928 .2565 .2258 .2026 OCCUPATIONS (Diewert [1975; 106]) 1. Managerial 7. 2. Profess iona l and techn ica l 8. 3. C l e r i c a ! 9. 4. Sal es 10. 5. Serv ice and recrea t ion 11. 6. Transport and communication 12. Farmers and farm workers Loggers and re la ted workers Fishermen, trappers and hunters Miners and quarrymen Craftsmen, production process and re la ted workers Labourers - 273 -TABLE B.4 PER CENT AND AVERAGE ABSOLUTE PER CENT DIFFERENCES BETWEEN TWO ALTERNATIVE PRICE INDEXES NON SEMI YEAR DURABLES SERVICES DURABLES DURABLES AVERAGE 1947 -.0063 -.0222 -.0719 -.0214 .0305 1948 - .0000 -.0197 -.0623 -.0227 ,0262 1949 -.0010 -.0161 -.0704 -.0189 .0266 1950 -.0010 -.0144 -.0761 -.0168 .0271 1951 .0007 -.0133 -.0624 -.0069 .0208 1952 .0007 -.0099 -.0522 -.0058 .0172 1953 .0005 -.0068 -.0507 -.0041 .0155 1954 .0001 -.0038 -.0249 -.0030 .0079 1955, - .0005 -.0024 -.0247 -.0009 .0071 1956 -.0010 -.0022 -.0176 -.0007 .0053 1957 -.0011 -.0015 -.0097 -.0014 .0034 1958 -.0001 -.0007 -.0082 .0002 .0024 1959 -.0001 -.0006 -.0061 .0003 .0018 1960 -.0001 -.0003 -.0020 .0001 .0016 1961 . 0000 .0000 .0000 .0000 .0000 1962 .0000 -.0001 .0003 -.0003 .0002 1963 .0003 -.0002 .0007 -.0003 .0004 1964 .0006 -.0005 .0004 -.0006 .0005 1965 . 0012 -.0011 .0001 -.0009 .0008 1966 .0024 -.0020 -.0092 -.0014 .0037 1967 .0018 -.0043 -.0163 -.0018 .0060 1968 .0025 -.0049 -.0322 -.0028 .0106 1969 .0027 -.0070 -.0446 -.0037 .0145 1970 .0029 -.0091 -.0565 -.0055 .0185 1971 .0029 -.0141 -.0540 -.0077 .0197 1972 .0033 -.0227 -.0504 -.0117 .0220 1973 .0079 -.0303 -.0675 -.0066 .0280 1974 .0123 -.0372 -.1313 -.0081 .0472 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0094061/manifest

Comment

Related Items