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UBC Theses and Dissertations

Demand for freight transportation with a special emphasis on mode choice in Canada Oum, Tae Hoon 1979

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DEMAND FOR FREIGHT TRANSPORTATION WITH A SPECIAL EMPHASIS ON MODE CHOICE IN CANADA B.Comm. (Summa-Cum-Laude), Sung Kyun Kwan Univ., Korea, 1967 M.B.A., University of B r i t i s h Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY -OF GRADUATE STUDIES i n An I n t e r - d i s c i p l i n a r y Program (Between The Faculty of Commerce and Business Administration and the Department of by TAE HOON lOUM Economics) We accept t h i s t h e s i s as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA ^ Tae Hoon Oum, 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permi ssion for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver 8, Canada Date ( i i ) ABSTRACT "Demand for Freight Transportation with a Special Emphasis on Mode Choice i n Canada" This thesis derives a f r e i g h t transportation demand model consistently with neoclassical economic theory: a shipper i s assumed to minimize t o t a l cost of production and d i s t r i b u t i o n with a given output that has to be delivered to various destination markets. With some further assumptions on the shipper's production technology, i t i s possible to express the shipper's transportation sectoral unit cost as a function of f r e i g h t rates and qu a l i t y a t t r i -butes of service and length of haul. Four a l t e r n a t i v e forms of the transportation sectoral unit cost function are hypothesized. These cost functions are sp e c i f i e d i n the translog form, and corresponding modal revenue share functions are derived. Each system of the cost and share functions i s estimated j o i n t l y by a maximum l i k e l i h o o d (ML) method, separately for each of the eight commodity groups selected from the cross-sectional data of Canadian inter-regional f r e i g h t movements during the year 197 0. Results of the hypothesis testing has shown that the qual i t y a t t r i b u t e s of service have s i g n i f i c a n t impact on the mode choice of manufactured products but not of bulk commodities and raw materials. The parameter estimates of the cost and share functions are used to measure the e l a s t i c i t y of s u b s t i t i t i o n and the e l a s t i c i t i e s of demand with respect to f r e i g h t rates and qu a l i t y a t t r i b u t e s of service. Both price and qual i t y e l a s t i c i t i e s of demand vary sub-( i i i ) s t a n t i a l l y from commodity to commodity and from l i n k to l i n k . For each commodity group, the price e l a s t i c i t i e s of the r a i l and truck modes are used to i d e n t i f y the distance range over which an e f f e c t i v e r a i l - t r u c k competition e x i s t s . For the r e l a t i v e l y high-value commodities, the short-haul t r a f f i c i s large l y dominated by the truck mode, and the s i g n i f i c a n t r a i l - t r u c k competition exists only i n the medium and long-haul markets. On the other hand, for the r e l a t i v e l y low-value commodities, the e f f e c t i v e r a i l - t r u c k com-p e t i t i o n exists only i n the short-haul markets leaving the medium and long-haul markets larg e l y rail-dominated. "DEMAND FOR FREIGHT TRANSPORTATION WITH A SPECIAL EMPHASIS ON MODE CHOICE IN CANADA" TABLE OF CONTENTS Chapter Page I. Objectives and Organization of Thesis.. 1 Footnotes for Chapter 1 6 II. Literature Review and Description of Methodology 8 (A) Literature Review 8 (B) Description of Methodology Adopted for t h i s Thesis 30 Footnotes for Chapter II 33 I I I . Model Formulation 35 (A) Derivation of the General Model 38 (B) The Model S t r i c t l y Independent of Distance 54 (.C) The Model with Mode-specific Hedonic Aggregators 57 (D) The Model with Identical Hedonic Aggregators 62 (E) Summary of Alternative Models 65 Footnotes for Chapter III 67 IV. Sources of Data and Construction of the Variables 68 (A) Freight Rate and Commodity Flow Data.... 70 (B) Distance of Link 72 (C) Transit Time and Its V a r i a b i l i t y 73 (D) Construction of the Variables 76 Footnotes for Chapter IV 80 V. Estimation and Hypothesis Testing 81 (A) Econometric Aspects of Estimation 82 (B) The Plan for Hypothesis Testing 85 (C) The Chosen Models 91 Footnotes for Chapter V 101 (iv) (v) Chapter Page VI. General Results..., , 104 (A) Mean Values of Some Important Variables...., , 105 (B) General Observations of the Chosen Models 109 (C) Parameter Estimates and Interpretation of S p e c i f i c Models 112 (D) Summary of the General Findings 119 Footnotes for Chapter VI 120 VII. E l a s t i c i t y Estimates and Inter-mpdal Competition.. 121 (A) E l a s t i c i t y Estimates at the Mean Values of the Variables 123 (B) The E l a s t i c i t y Estimates and Inter-^modal Competition on Some Selected Links... 131 Footnotes for Chapter VII 152 VIII. Summary of Major Findings and Suggestions for Further Research 153 (A) Summary of Major Findings 153 (B) Suggestions for Further Research 159 Footnotes for Chapter VIII 161 BIBLIOGRAPHY 162 APPENDIX 3A Derivation of Linear Homogeneity Condition 170 APPENDIX 4A L i s t of Commodities Included i n the Eight CFTM Commodity Groups 17 3 APPENDIX 5A Tables of Test S t a t i s t i c s and the Results of Hypothesis Testing 176 APPENDIX 7A Derivation of Price & Quality E l a s t i c i t i e s . . 201 (vi) LIST OF TABLES Table Page 4- 1 Selected CFTM Commodity Groups and Number of Observations. 79 5- 1 L i s t of the Chosen Models 100 6- 1 Mean Values of Some Important Variables 106 6-2 Parameter Estimates of the Chosen Models C - l . . . 113 6-3 Parameter Estimates of the Chosen Models C-3... 116 6- 4 Parameter Estimates of the Chosen Models A-3... 117 7- 1 Comparison of E l a s t i c i t i e s (evaluated at means of the variables) 129 7-2 Estimated Parameters of Price and Quality Responsiveness of Demands for CFTM14 (Fruits, vegetables and edible foods), 134 7-3 Price E l a s t i c i t i e s and E l a s t i c i t y of Sub-s t i t u t i o n for CFTM52 (lumber including flooring) « 137 7-4 Price E l a s t i c i t i e s and E l a s t i c i t y of Sub-s t i t u t i o n for CFTM61 (chemicals) 140 7-5 Price E l a s t i c i t i e s and E l a s t i c i t y of Sub-s t i t u t i o n for CFTM66 (fuel o i l other than gasoline) . . 142 7-6 Price E l a s t i c i t i e s and E l a s t i c i t y of Sub-s t i t u t i o n for CFTM69 (Other refined petroleum products) , 143 7^7 Parameters of Price and Quality Responsive-ness of Demands for CFTM75 (metal fabricated basic products) 144 7-8 Parameters of Price and Quality Responsive-ness of Demands for CFTM7 8 (Non-met a l l i c basic products) 147 7-9 The Distance Range for E f f e c t i v e Inter-modal Competition 150 5A-1 Test S t a t i s t i c s for Commodity Group CFTM14: F r u i t s , vegetables and edible foods 177 5A--2 Test S t a t i s t i c s for Commodity Group CFTM52: Lumber including f l o o r i n g . . . . 178 5A-3 Test S t a t i s t i c s for Commodity Group CFTM61: Chemicals 179 5A-4 Test S t a t i s t i c s for Commodity Group CFTM66: Fuel O i l . 180 5A-5 Test S t a t i s t i c s for Commodity Group CFTM69: Other refined petroleum products 181 5A-6 Test S t a t i s t i c s for Commodity Group CFTM71: Steel , irons and a l l o y s . . . . 182 5A-7 Test S t a t i s t i c s for Commodity Group CFTM75: Metal fabricated basic products 183 5A-8 Test S t a t i s t i c s for Commodity Group CFTM78: Non-metallic basic products 184 ( v i i ) T a b l e P a g e 5A-9 H y p o t h e s e s T e s t i n g f o r CFTM14 185 5A-10 H y p o t h e s e s T e s t i n g f o r CFTM52 187 5A-11 H y p o t h e s e s T e s t i n g f o r CFTM61 189 5A-12 H y p o t h e s e s T e s t i n g f o r CFTM66 191 5A-13 H y p o t h e s e s T e s t i n g f o r CFTM69. .. 193 5A-14 H y p o t h e s e s T e s t i n g f o r CFTM71 19 5 5A-15 H y p o t h e s e s T e s t i n g f o r CFTM75 19 7 5A-16 H y p o t h e s e s T e s t i n g f o r CFTM78 199 ( v i i i ) - LIST OF FIGURES Figure Page 5T-1- Plan for Hypothesis Testing 87 5-2 Second-Stage Testing for CFTM14 97 (ix) AC KNOWLED GEMENT S I am grateful to my Thesis Committee: Professors Charles Blackorby, John G. Cragg, Kenneth G. Denike, Karl M. Ruppenthal and i n p a r t i c u l a r to Professors Trevor D. Heaver my chairman and William G. Waters II who offered me constant encouragement throughout my Ph.D. program. In addition, I would l i k e to thank Professors Edwin H. Neave and Hugh V;1 F u l l e r t o n and Mr. Glyn A l l e n for reading and making suggestions on the f i n a l draft of the th e s i s . Professor Ernst R. Berndt has .guided my research p r i o r to > the thesis, which has won therf.irstl-prize from fchehTransportation Research Forum (TRF) i n 1977. For t h i s I am e s p e c i a l l y g r a t e f u l to him. F i n a n c i a l Assistance from the Canadian Federal Government through the Transportation Development Agency (TDA) Fellowship Awards and from the Centre for Transportation Studies at the University of B r i t i s h Columbia during my Ph.D. program are g r a t e f u l l y acknowledged. I am also indebted to Mr. G.W. English and Mrs. M.L. Singer for compiling the data necessary for t h i s research from the CIGGT data base. To Professor Ronald Turner who provided the trucking survey data for t h i s t h e s i s , I o f f e r my sincere thanks. For excellent typing service, I am indebted to Mrs. Miriam-R. Howard, Linda Freeman and Miss Heather Hughes. F i n a l l y , my family, Soo-Ran, Yoosik and Elaine deserve much c r e d i t for enduring hardship and for s p i r i t u a l support during my study period. This thesis i s dedicated to them. A p r i l , 19 79 Tae Hoon Oum CHAPTER I OBJECTIVES AND ORGANIZATION OF THESIS The study of the demand for f r e i g h t transportation i s an important part of the quantitative analysis of many public and managerial decisions concerning f r e i g h t transportation. For example, a demand study i s es s e n t i a l for economic evaluation of a major investment project on transport i n f r a s t r u c t u r e , a regulatory p o l i c y option or a subsidy program. A c a r r i e r who wishes to determine the optimal price-service package for a pa r t i c u l a r class of users, or to examine the f e a s i b i l i t y of introducing a new service or opening up a new market for an exi s t i n g service should also r e l y heavily on the res u l t s obtained from a demand study. Functionally, there are two major purposes of a demand study. The f i r s t i s to forecast future demands. The second i s to analyze the nature of the demand functions facing various transportation modes. Demand forecasts are necessary for planning future transportation systems and t h e i r capacity requirements, as well as for c a r r i e r s ' investment and operational plans. Many public decisions concerning taxes, subsidies and economic regulations, and c a r r i e r s ' decisions on optimal p r i c e - q u a l i t y mix require a precise knowledge of the nature of demand functions as well as demand forecasts. Some demand models are b u i l t so as to serve these two purposes simultaneously whereas others serve primarily one or the other of the two purposes. - 1 -- 2 In an ex post sense, except for a few bulk commodities, t r a f f i c flow between an o r i g i n and a destination depends larg e l y on the geographical d i s t r i b u t i o n of economic a c t i v i t y l e v e l s and distance, and less on price and qual i t y of available transportation services. Therefore, macro-economic s t r u c t u r a l models and gravity--type models turn out to perform reasonably well (in terms of s t a t i s t i c a l f i t ) i n forecasting demands for planning purposes. In spite of an obvious need for improving methodologies for studying the nature of demand functions, r e l a t i v e l y l i t t l e e f f o r t has been devoted to th i s area. Furthermore, the information obtained from the limited e f f o r t i s often r a d i c a l l y d i f f e r e n t among studies,"*" which makes i t d i f f i c u l t to believe the r e s u l t s . The following reasons appear to explain t h i s f a i l u r e : (i) In most previous studies i n t h i s area (such as These ad hoc demand models are r e s t r i c t i v e for studying the nature of demand functions because, i n general, the structure of shippers 1' d i s t r i b u t i o n technologies that they purport to approximate i s not known, nor are the properties of the approximations. of price and qu a l i t y responsiveness of demand. - 3 -( i i ) In most previous demand studies, the demand models were estimated from highly aggregate data primarily because of u n a v a i l a b i l i t y of appropriate disaggregate data. ( i i i ) Cobb-Douglas demand models and l o g i t models which have been used frequently i n the past impose severe r e s t r i c t i o n s on the parameters of price responsiveness of demand and of 3 inter-modal substitution. Consequently, no one r e a l l y knows whether the demand for the r a i l mode by a p a r t i c u l a r class of users i n a p a r t i c u l a r f r e i g h t market i s e l a s t i c or i n e l a s t i c , whether or not i t i s responsive to a qual i t y of service variable, and whether the cross e l a s t i c i t y of demand between railway and trucking modes i s high or low. However, i f regulatory agencies or c a r r i e r s are going to make r a t i o n a l decisions, i t i s imperative for them to have accurate knowledge on the nature of demand functions which various modes are facing i n various f r e i g h t markets. To my knowledge, however, no one has systematically investigated the price and qual i t y responsiveness of demands and cross-mode substitution p o s s i b i l i t i e s using a derived demand model which ensures free v a r i a t i o n of e l a s t i c i t i e s of demand and sub-s t i t u t i o n while at the same time employing s u f f i c i e n t l y detailed 4 data on shippers' mode choice. Therefore, the objective of t h i s thesis i s to demonstrate a method of measuring price and qu a l i t y responsiveness of modal - 4 -demands i n a way that i s consistent with shipper's mode-choice behavior. This objective w i l l be accomplished through the following procedure: (i) The du a l i t y r e l a t i o n between cost and production functions allows a shipper's d i s t r i b u t i o n cost function to describe h i s d i s t r i b u t i o n technology completely. ( i i ) Since disaggregate data of an i n d i v i d u a l shipper's d i s t r i b u t i o n a l and mode-choice data are usually unobtainable, the cost function i s aggregated over shippers of a homogeneous commodity group, ( i i i ) By applying Shephard's lemma [l953] to the cost function, the demand functions for various modes are obtained. (iv) Several alternative forms of the cost functions and corresponding demand functions are hypothesized and tested empirically. For each commodity group, the model which f i t s the empirical data best i s to be chosen for use i n measuring the price and qu a l i t y responsiveness of demands for r a i l and truck modes, (v) E l a s t i c i t i e s of demands for railway and trucking modes with respect to price and q u a l i t y of service variables are computed using the parameter estimates of the cost function. An attempt i s also made, i n t h i s t h e s i s , to compare the resu l t s with those of previous studies, and to evaluate the - 5 -extent of inter-modal competition e x i s t i n g i n various f r e i g h t markets i n Canada. The organization of t h i s thesis i s as follows: A survey of relevant l i t e r a t u r e and an outline of the methodology adopted for t h i s thesis are presented i n chapter I I . In chapter I I I , the shipper's transportation-sectoral-cost function i s derived, and i t s four al t e r n a t i v e forms are hypo-thesized. These cost functions are sp e c i f i e d i n translog form and the corresponding modal revenue share functions are derived. Information about the data i s presented i n chapter IV along with ways of constructing the variables a c t u a l l y used i n estimation. Chapter V presents the econometric aspects of estimation and carr i e s out the hypotheses t e s t i n g . Chapter VI reports the parameter estimates of the cost functions chosen i n chapter V with some explanatory comments. Chapter VII reports and interprets the estimated e l a s t i c i t i e s of substitution and e l a s t i c i t i e s of demand with respect to fre i g h t rates and qual i t y attributes of service. Chapter VIII closes t h i s thesis with a summary of major findings and suggestions for further research. - 6 -x F o o t n o t e s f o r C h a p t e r I : 1. The m o s t c o n s p i c u o u s e x a m p l e o f r a d i c a l d i f f e r e n c e i n r e s u l t s among s t u d i e s c a n be f o u n d i n t h e f o l l o w i n g U.S. s t u d i e s c o n c e r n i n g t h e w e l f a r e l o s s due t o t h e t r a f f i c m i s a l l o c a t i o n w h i c h i s c l a i m e d t o be c a u s e d b y t h e I.C.C. ( I n t e r s t a t e Commerce C o m m i s s i o n ) minimum r a t e r e g u l a t i o n s . C o m p a r i s o n o f W e l f a r e L o s s E s t i m a t e s A u t h o r Y e a r W e l f a r e L o s s ( c u r r e n t $ ) W e l f a r e l o s s % o f f r e i g h t H a r b e s o n 1963 1.1-2.9 b i l l i o n 12-32 F r i e d l a e n d e r 1964 150 m i l l i o n 1.59 B o y e r 1963 125 m i l l i o n 1.37 L e v i n 1972 53-135 m i l l i o n 0.30^0.77 ( S o u r c e : R e p r o d u c e d f r o m T a b l e 11 i n L e v i n [19 7 8 ] ) . N o t e t h a t d i f f e r e n c e s among t h e a b o v e e s t i m a t e s a r e p r i m a r i l y due t o t h e d i f f e r e n c e s i n assumed o r e s t i m a t e d demand e l a s t i -c i t i e s . 2. The t e r m "ad hoc m o d e l " r e f e r s t o a l l demand m o d e l s t h a t a r e s p e c i f i e d a r b i t r a r i l y w i t h o u t r e f e r r i n g e x p l i c i t l y t o t h e s t r u c t u r e o f t h e s h i p p e r s * d i s t r i b u t i o n t e c h n o l o g y . 3. (a) The C o b b - D o u g l a s i n p u t demand m o d e l i s c o n s i s t e n t o n l y t o a Cobb*-Douglas p r o d u c t i o n f u n c t i o n w h i c h i s d u a l t o a C o b b - D o u g l a s c o s t f u n c t i o n . T h e r e f o r e , i t r e s t r i c t s t h e e l a s t i c i t y o f s u b s t i t u t i o n b e t w e e n e v e r y p a i r o f i n p u t s t o u n i t y . The demand m o d e l s p e c i f i e d i n a l i n e a r l o g a r i t h m i c f o r m w h i c h d o e s n o t i m p o s e t h e h o m o g e n e i t y c o n d i t i o n a ssumes c o n s t a n t p r i c e e l a s t i c i t i e s . (b) The l i n e a r l o g i t m o d e l s h a v e t h e f o l l o w i n g i n a d e q u a c i e s i n m e a s u r i n g p r i c e r e s p o n s i v e n e s s o f demand ( s e e Oum [ 1 9 7 8 ] f o r a d e t a i l e d d i s c u s s i o n ) . (1) L i n e a r l o g i t m o d e l s , s p e c i f i e d i n t e r m s o f a " p r i c e d - r a t i o " , a n d o t h e r n o n - p r i c e v a r i a b l e s : ( i ) The 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 a n d t h e p r i c e e l a s t i c i t i e s a r e n o t i n v a r i a n t t o t h e c h o i c e o f b a s e mode M. - 7 -( i i ) A c e r t a i n c h o i c e o f b a s e mode amounts t o i m p o s i n g r i g i d a p r i o r i r e s t r i c t i o n s o n t h e r e l a t i o n s h i p s b e t w e e n t h e 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 a n d t h e c o r r e s p o n d i n g p r i c e r a t i o s , a n d t h e s e r e s t r i c t i o n s a r e c o n t r a d i c t o r y t o t h e o n e s t h a t w o u l d h a v e b e e n i m p o s e d u n d e r a d i f f e r e n t c h o i c e o f b a s e mode. ( i i i ) The p r e f e r e n c e ( o r t e c h n o l o g y ) s t r u c t u r e u n d e r l y i n g t h e m u l t i - n o m i a l l o g i t m o d e l o f t h i s t y p e i s i n -c o n s i s t e n t a n d i r r e g u l a r b e c a u s e t h e r e a r e two d i f f e r e n t m e a s u r e s f o r t h e e l a s t i c i t y o f s u b s t i -t u t i o n b e t w e e n a n y two n o n - b a s e modes i_ a n d j _ : one when i t h p r i c e i s h e l d c o n s t a n t a n d t h e o t h e r when j t h p r i c e i s h e l d c o n s t a n t . T h e r e f o r e , i t i s m e a n i n g l e s s t o m e a s u r e t h e p r i c e r e s p o n s i v e n e s s o f demands u s i n g t h e m u l t i - n o m i a l l o g i t m o d e l o f t h i s t y p e . ( i v ) A l l c r o s s p r i c e e l a s t i c i t i e s w i t h r e s p e c t t o t h e p r i c e o f a n y g i v e n " n o n - b a s e " mode a r e r e s t r i c t e d t o be e q u a l . (2) L i n e a r l o g i t m o d e l s , s p e c i f i e d i n t e r m s o f a p r i c e -d i f f e r e n c e a n d o t h e r n o n - p r i c e v a r i a b l e s : ( i ) The t e c h n o l o g y ( o r p r e f e r e n c e ) s t r u c t u r e u n d e r l y i n g b o t h b i - n o m i a l a n d m u t l i - n o m i a l l o g i t m o d e l s o f t h i s t y p e i s i n c o n s i s t e n t a n d i r r e g u l a r b e c a u s e o f t h e e x i s t e n c e o f two d i f f e r e n t m e a s u r e s f o r t h e same e l a s t i c i t y o f s u b s t i t u t i o n . T h e r e f o r e , i t i s m e a n i n g l e s s t o t r y t o m e a s u r e t h e p r i c e r e s p o n s i v e -n e s s o f demands u s i n g t h e l o g i t m o d e l s o f t h i s t y p e . ( i i ) A l l c r o s s p r i c e e l a s t i c i t i e s w i t h r e s p e c t t o t h e p r i c e o f any g i v e n mode i n c l u d i n g t h e b a s e mode M a r e r e s t r i c t e d t o be e q u a l . T h i s i s a l s o an e x t r e m e l y u n r e a l i s t i c r e s t r i c t i o n . A s w i l l be m e n t i o n e d i n c h a p t e r I I , F r i e d l a e n d e r a n d S p a d y [l9 77_] h a v e u s e d a demand m o d e l t h a t h a s t h e s e d e s i r a b l e p r o p e r t i e s f o r s t u d y i n g i n t e r - m o d a l c o m p e t i t i o n . H o w e v e r , t h e i r d a t a d i d n o t i n c l u d e q u a l i t y a t t r i b u t e s o f s e r v i c e s a n d w e r e h i g h l y a g g r e g a t e d g e o g r a p h i c a l l y . CHAPTER II LITERATURE REVIEW AND DESCRIPTION OF METHODOLOGY Section (A) presents a selec t i v e review of previous demand studies and the approaches taken i n those studies. The methodology adopted for t h i s thesis i s described i n section (B). (A) Review of Previous Studies The study of demand for fr e i g h t transportation i s compli-cated mainly because of the extreme heterogeneity i n qual i t y of service and i n types of cargo and shippers. As a r e s u l t , many studies are not d i r e c t l y comparable because they use di f f e r e n t methodologies and types of data. This makes i t d i f f i c u l t to compare the re s u l t s of the previous works. In order to minimize these d i f f i c u l t i e s , t h i s review of l i t e r a t u r e i s organized according to the approach adopted i n the study. On the basis of the author's subjective opinion, past research on f r e i g h t transport demand can be divided into the following six categories. Normative Approach: (1) System optimizing approach (2) User optimizing approach Empirical Approach: (3) Derived demand modelling approach (4) "Gravity-type" modelling approach (5) Abstract-mode modelling approach (6) Mode-choice modelling approach - 8 -- 9 -In a normative approach'/' the-roocle ^ choice -for- each ''consign-ment i s done a n a l y t i c a l l y by evaluating an objective function. Therefore, the stochastic nature of shippers' mode-choice behavior i s not taken into consideration i n the normative approach. On the other hand, i n empirical approach, the demand model i s imbedded i n a stochastic framework and thus estimated from the observed data. In what follows, each of the above s i x approaches i s described and examples of each approach are c i t e d . Advantages and disadvantages of each approach - i n view of the objectives of t h i s thesis - are to be examined as well. Although the discussion w i l l be r e s t r i c t e d primarily to fre i g h t transport demand studies, some passenger demand studies are to be referred to because t h e i r methodologies are applicable to the study of fr e i g h t transport demand. (1) Normative approach based on system optimization: In t h i s approach, the demand for each mode i n each f r e i g h t market i s determined such that the t o t a l transportation cost over the entire transportation network of a nation or a region i s minimized. Geographic d i s t r i b u t i o n of supply and demand for each commodity and the costs of transportation by alte r n a t i v e modes must be i d e n t i f i e d or forecast ahead of time. Then the demand for each mode i n each f r e i g h t market i s estimated by solving a mathematical programming problem which minimizes the t o t a l transportation cost over the en t i r e network subject to the supply and demand constraints at various locations for each - 10 -commodity. Due to the simultaneity between price (or cost) of a mode's service and the volume of flow i t handles, the problem becomes an i t e r a t i v e one which normally i n v i t e s many d i f f i -c u l t i e s concerning convergence and hence c a l l s for a high computational cost. The approaches taken i n Tavares [ l 9 7 2 J , Dye [ l 9 72J , and Kresge and Roberts JJL9 71 ] belong to t h i s category. The flows of commodities predicted by t h i s approach indicate the c o l l e c t i v e l y i d e a l commodity flow patterns within a region or a nation, and thus provide useful information for strategic planning of future transportation systems. However, thi s approach i s not appropriate for studying the nature of demand functions because of the following shortcomings: 1. The outcome of t h i s approach i s the flow pre-dicted over the transportation network rather than the demand functions. The only way to examine the e f f e c t of a price change i n the i d e a l flow patterns i s to re-solve the entire problem with appropriately changed parameters -a rather expensive option. 2. This approach ignores the e f f e c t of q u a l i t y attributes of service on demand and mode choice primarily due to the d i f f i c u l t i e s i n specifying an appropriate objective function and i n obtaining the necessary data. Yet, both t o t a l demand and mode choice are l i k e l y to depend on the q u a l i t y attributes of available services as well as on the prices of services. 3. An i n d i v i d u a l shipper i s motivated to minimize the t o t a l cost of h i s given transportation requirements rather than the t o t a l transportation costs of an entire region or nation: i . e . , shippers are not system-optimizers. As a r e s u l t , the flows predicted by t h i s approach are not l i k e l y to approximate c l o s e l y the actual flows which are merely an aggregation of the decisions made by i n d i v i d u a l shippers. - 11 -(2) N o r m a t i v e a p p r o a c h b a s e d o n u s e r o p t i m i z a t i o n : T h i s i s an a n a l y t i c a p p r o a c h t o t h e s h i p p e r ' s c h o i c e o f s e r v i c e a t t r i b u t e s ' ' " a nd t h u s h i s c h o i c e o f mode. I n t h i s a p p r o a c h , a s h i p p e r w i t h a g i v e n t r a n s p o r t a t i o n r e q u i r e m e n t i s assumed t o c h o o s e t h e b u n d l e ( o r m i x t u r e ) o f s e r v i c e a t t r i b u t e s w h i c h m i n i m i z e s t h e i m p u t e d t o t a l c o s t o f t r a n s p o r t a t i o n . T h i s a p p r o a c h u t i l i z e s t h e n o t i o n o f a s o - c a l l e d " a b s t r a c t c o m m o d i t y " i n t r o d u c e d f o r m a l l y b y L a n c a s t e r |^1966j . L a n c a s t e r o b s e r v e d t h a t u s e r s p u r c h a s e g o o d s o r s e r v i c e s i n o r d e r t o d e r i v e s a t i s f a c t i o n f r o m t h e i r a t t r i b u t e s , a n d t h e r e -f o r e , a g ood o r a s e r v i c e p e r se h a s no s p e c i a l s i g n i f i c a n c e t o i t s u s e r s o t h e r t h a n a s t h e c o l l e c t i o n o f a t t r i b u t e s i t p o s s e s s e s . C o n s e q u e n t l y , he a d v o c a t e d t h a t t h e u s e r ' s c h o i c e p r o b l e m c a n be h a n d l e d b e t t e r i n t h e a t t r i b u t e s s p a c e t h a n i n t h e g o o d s o r s e r v i c e s s p a c e . T h e r e f o r e , a c o l l e c t i o n o f a t t r i b u t e s i s c a l l e d a n " a b s t r a c t c o m m o d i t y " . T h i s c o n c e p t was i n t r o d u c e d i n t o t h e t r a n s p o r t a t i o n l i t e r a t u r e a s a n " a b s t r a c t mode" t h r o u g h t h e w o r k o f Q u a n d t a n d B a u m o l [ 1 9 6 6 ] o n p a s s e n g e r t r a n s p o r t demand and t h r o u g h a t h e o r e t i c a l c o n t r i -b u t i o n b y B a u m o l a n d V i n o d £l970]j o n f r e i g h t mode s e l e c t i o n . B a u m o l and V i n o d p r o p o s e d t h e n o t i o n o f an a b s t r a c t f r e i g h t t r a n s p o r t mode c h a r a c t e r i z e d b y s e r v i c e a t t r i b u t e s s u c h a s economy, s p e e d , r e l i a b i l i t y a n d p r e v e n t i o n o f l o s s a n d damage. A s s u m i n g t h a t a s h i p p e r m i n i m i z e s t h e t o t a l c o s t o f t r a n s p o r t a t i o n a nd i n v e n t o r y management, t h e a u t h o r s u s e d an i n v e n t o r y m o d e l t o show t h e s h i p p e r ' s t r a d e - o f f s b e t w e e n - 12 -f r e i g h t rate and service a t t r i b u t e s , s p e c i f i c a l l y speed and r e l i a b i l i t y of speed. The shipper i s , therefore, supposed to purchase the bundle of service attributes that y i e l d s the highest value to him per d o l l a r spent on a given transportation requirement. The rationale for this conclusion i s that both speed and r e l i a b i l i t y contribute to save inventory management cost by reducing safety stock requirements and the frequency of stockout occasions. Since inventory costs depend largely on commodity attributes such as the value of the commodity, holding cost, s u s c e p t i b i l i t y of the commodity to loss and damage, cost of a stockout and the nature of the demand for the commodity, so too do the imputed values (or costs) of various q u a l i t y attributes of service. Using the fact that the values of quali t y attributes of service are determined mainly by the commodity a t t r i b u t e s , Roberts |jL9 7 o j attempted to operationalize the concept of an abstract f r e i g h t mode by developing a procedure to compute what he c a l l s a "commodity preference vector". This commodity preference vector e s s e n t i a l l y measures the shipper's imputed costs of service attributes such as t r a n s i t time, waiting time, v a r i a b i l i t y of t r a n s i t time and the p r o b a b i l i t y of loss and damage. Plugging i n appropriate values for the inventory-holding cost, the pri c e of commodity, the i n t e r e s t rate, the p r o b a b i l i t y d i s t r i b u t i o n of the demand, and the loss and damage factors, he arrived at the following commodity preference vectors for two commodity groups: bulk - 13 -commodity and non-bulk commodity groups. Cost Item Bulk General (non-bulk) Travel time $/ton-hour .041 .362 Waiting time $/ton-hour .049 .514 V a r i a b i l i t y of time $/ton-hour .003 ,035 Out-of-pocket cost $/$ 1.0 1.0 (Source: from table 2 i n Roberts [l970}) I n t u i t i v e l y , the figures look reasonable i n the sense that non-bulk commodity shippers value a l l the service attributes far more than do bulk commodity shippers. The combined value of monetary cost and the imputed costs of t r a v e l time, waiting time and the v a r i a b i l i t y of time for using a p a r t i c u l a r mode i s defined as the product of the commodity preference vector and the modal performance vector for a p a r t i c u l a r movement of our concern. Then, the idea i s to choose the lea s t cost mode i n terms of the combined cost. Although Roberts demon-strated t h i s procedure for the two aggregate commodity groups, i n p r i n c i p l e , the same procedure could be applied to as any disaggregate commodity group as i s desired. Cl e a r l y , t h i s approach i s an improvement over the system optimizing approach i n predicting mode-choice i n the senses that the optimizing unit i s an in d i v i d u a l shipper, and the imputed values (or costs) of quality a t t r i b u t e s as well as 14 -prices of various modes' services enter e x p l i c i t l y into the shipper's mode choice decision. However, thi s approach suffers from the following disadvantages: 1. The commodity preference vector i s l i k e l y to depend not only upon the commodity type but also upon such factors as length of haul, shipment si z e , t o t a l size of shipper's operation, etc. However, i t i s not r e a d i l y apparent how t h i s approach could be modified to incorporate these factors, (other than commodity type) and at the same time, maintain the r e l a t i v e s i m p l i c i t y required for p r a c t i c a l applications. 2. According to t h i s approach, for a given trans-portation requirement, a l l shippers are supposed to choose the same mode, disallowing the stochastic nature of actual mode-choice decisions. Therefore, the demand predicted by t h i s approach may not give a good approximation to the actual demand. 3. The end r e s u l t of t h i s approach i s the mode-choice forecast for a given movement or the modal demand forecasts obtained by aggregating the i n d i v i d u a l mode-choice r e s u l t s . Therefore, as was the case of the previous approach, the demand functions cannot be i d e n t i f i e d through this approach. Although the normative approach based on user optimization i s not appropriate for studying the nature of demand functions, many empirical studies, including t h i s t h esis, have benefited from th i s approach because i t provides t h e o r e t i c a l foundations for the formulation of reasonable hypotheses for empirical research. (3) Empirical approach with derived demand models: This approach uses a demand model derived from, or spe c i f i e d consistently with, neoclassical production theory. Consequently, estimation of the demand functions completely - 15 -i d e n t i f i e s the shape of the underlying technology. In th i s approach, each shipper i s assumed to minimize his t o t a l trans-portation cost subject to his transportation-sectoral technology by responding only to changes'.in f r e i g h t rates of alternative modes. Although the e f f e c t of mean differences i n q u a l i t y attributes of service between modes i s r e f l e c t e d i n his transportation-sectoral technology ( i . e . shapes of isoquants), the e f f e c t of v a r i a t i o n i n r e l a t i v e q u a l i t y attributes of various modes across observational units i s not taken into account i n t h i s approach since the demand models do not include q u a l i t y attributes of service as t h e i r independent variables. The demand models used i n Sloss [19 7 l j , Perle £1964"}, Oum [l977], and Friedlaender and Spady £l977] may be considered to belong to th i s approach. Using time-series data for Canada, Sloss has estimated a highly aggregated demand model i n which the t o t a l tonnage of i n t e r c i t y f r e i g h t t r a f f i c c arried by fo r - h i r e trucks i s 3 expressed as a 'Cobb-Douglas-like 1 function of the average r a i l revenue per ton, the average truck revenue per ton and a variable i n d i c a t i n g the general l e v e l of economic a c t i v i t y . The data for the model were obtained from annual reports published by the Dominion Bureau of S t a t i s t i c s , Canada, Although Sloss termed his study "a macro-economic analysis", the model used i s sim i l a r to the demand model one would obtain by aggregating the i n d i v i d u a l shipper's derived demand model based upon the t r a d i t i o n a l production (or consumption) theory. - 16 -Perle j~1964 ~] conducted one of the f i r s t major studies of fre i g h t demand, and discussed extensively the theory of fre i g h t demand based upon the consumption theory. Among other forms of demand models, he estimated "Cobb-Douglas-like" demand functions for r a i l and truck modes using pooled time-series and cross-sectional U.S. data. The advantages of Perle's study r e l a t i v e to that of Sloss £ l 9 7 l ] are: (i) The models were estimated from r e l a t i v e l y disaggregate data, ( i i ) The t r a f f i c volumes are measured i n ton-miles rather than i n tonnage, and ( i i i ) The differences between geographical regions, between years and between commodities were taken into consideration at least p a r t i a l l y by either including appropriate dummy variables or by estimating separate equations for each commodity group. The "Cobb-Douglas-like" demand model used by Sloss [ l 9 7 l ] and Perle [ 1 9 6 4 J has a major disadvantage i n measuring e l a s t i c i t i e s of demand because i t r e s t r i c t s the price e l a s t i -c i t i e s of demand to constant values (see footnote 3 i n chapter I ) . Consequently, i t i s not adequate for studying the nature of demand functions. The three-mode ( r a i l , truck and ship) demand model estimated i n Oum [ l 9 7 7 ] i s free from t h i s problem because i t was derived from the shipper's "translog" cost 4 function which allows the free v a r i a t i o n of price e l a s t i c i t i e s and e l a s t i c i t i e s of substitution. Oum also imposed l i n e a r homogeneity of the cost function with respect to prices so that the demand functions would be homogeneous of degree zero - 17 -i n p r i c e s . Aggregate time series data of Canadian i n t e r c i t y f r e i g h t transportation, 1945-1974, were used to estimate t h i s derived demand model. Friedlaender and Spady £l9 77j estimated the two-mode ( r a i l and truck) demand model derived from a shipper's "trans-log" cost function using a combination of cross-sectional and time-series data for the seven broad commodity types (durable manufactures, nondurable manufactures, f i e l d crops, other a g r i c u l t u r a l commodities, petroleum and petroleum products, coal, and other bulk commodities) and the three U.S. regions ( O f f i c i a l T e r r i t o r i e s , South, and West) for the years 1961-1972. They d i f f e r from Oum [1977] i n that the additional variables such as truck tons per vehicle, r a i l tons per car, truck average length of haul, r a i l average length of haul, and value of commodity are included i n the model. However, the i n c l u s i o n of the f i r s t four additional variables which b a s i c a l l y represent the c h a r a c t e r i s t i c s of modal outputs can hardly be j u s t i f i e d because the shipper's cost function should r e f l e c t the shipper's choice behavior for a given transportation requirement. What i s relevant for mode choice i s the c h a r a c t e r i s t i c s of shipments required to be transported instead of the c h a r a c t e r i s t i c s of shipments carried by various modes. One advantage of using a derived demand model i s that researchers are given f u l l access to the established economic theory for comparative s t a t i c analysis because the demand - 18 -functions i n combination describe the structure of the shipper's d i s t r i b u t i o n technology. One major disadvantage, however, i s that the s e n s i t i v i t y of demand to v a r i a t i o n i n q u a l i t y -at t r i b u t e s of service cannot be measured from the derived demand models developed so fa r . In order to resolve t h i s problem, q u a l i t y - a t t r i b u t e s of service should be included i n the model. This i s done i n the present t h e s i s . (4) "Gravity-type 5" modelling approach: This approach attempts to specify a demand model purely on empirical grounds rather than deriving i t from the economic theory of the shipper's mode choice. Thousands of studies on the urban t r a f f i c d i s t r i b u t i o n problem have used various forms of gravity models which t e l l e s s e n t i a l l y that the volume of t r a f f i c between a pair of o r i g i n and destination zones i s an increasing function of the t r i p generating factors i n the o r i g i n and the t r i p a t t r a c t i o n factors i n the desti-.--nation and a decreasing function of the impedance factors. Furthermore, Wilson [l967, 1968, 1969] has improved the gravity model so as to solve simultaneously the entire system of t r a f f i c generation, a t t r a c t i o n , d i s t r i b u t i o n , mode-split, and route assignment problems. By adapting the l o g i c under-l y i n g the passenger gravity models, several studies concerning f r e i g h t transport demand have used "gravity-type" models for predicting i n t e r - r e g i onal commodity flows. In these models, the demand for transport of a p a r t i c u l a r commodity between a pair of o r i g i n and destination regions i s usually s p e c i f i e d as - 19 -a function of "push" variables (for example, production of the commodity) at the o r i g i n , " p u l l " variables (for example, the consumption of the commodity) at the destination, and impedance factors such as cost of transportation, distance and t r a v e l time. The models used by Black [ l 9 7 l ] , Chrisholm and 0'Sullivan [1973 ] and the Canadian Transport Commission [l9 76] are of thi s general type. The C.T.C. study i s d i f f e r e n t from the other two studies i n that i t s demand model includes separate demand functions for three f r e i g h t modes ( r a i l , truck and ship). The demand for a mode's service on a l i n k ^ by given commodity shippers was expressed as a "Cobb-Douglas-l i k e " function of excess production of the commodity at the o r i g i n , excess consumption of the commodity at the destination, transport cost by the mode and the movements on a l l l i n k s that are complementary to or i n competition with the l i n k . Note that the C.T.C. model cannot be used to measure c r o s s - e l a s t i c i t i e s of demand, e l a s t i c i t i e s of substitution among modes or e l a s t i c i t i e s of demand to q u a l i t y attributes since i t does not include the prices of al t e r n a t i v e transport modes nor the qual i t y attributes of service. There-fore, the usage of the model i s lim i t e d only to forecasting modal demands. A l l gravity-type models possess t h i s d i s -advantage . - 20 -(5) Abstract-mode modelling approach: The demand model proposed i n Quandt and Baumo 1 [1966 J may be regarded as a natural extension of the gravity-type model so as to measure s e n s i t i v i t y of demands to price and qua l i t y of service variables, at least to a lim i t e d extent. This demand model consists e s s e n t i a l l y of two components: forecasting and mode-split. The forecasting component i s a function of "push and p u l l " variables (populations, mean incomes and i n d u s t r i a l character indices of the o r i g i n and destination) and impedance factors (travel time by the least time mode, cost by the least cost mode and departure frequency of the most frequent mode). The mode-split component i s a function of the number of competing modes on the l i n k , t r a v e l time ( r e l a t i v e to the least time mode), cost of using the mode (r e l a t i v e to the least cost mode) and departure frequency (r e l a t i v e to the most frequent mode). Clearly the model suggested i n Quandt and Baumol [l966] i s constructed by 7 combining the concept of abstract-mode with the gra v i t y -type model. Quandt and Young [l969] applied t h i s model d i r e c t l y to an i n t e r c i t y passenger transport demand study while Mathematica [l96 7^ applied t h i s model with a minor change to the Northeast Corridor f r e i g h t demand study. The model used i n the Mathematica study i s as follows: V. . i jm b 0 - 21 -where V.. = volume of f r e i g h t flow from i to j by mode m, ljm ^ — ~ •* — P^, Pj = population of the o r i g i n and destination, Y., Y. = gross regional product of the o r i g i n and 1 destination, M., M. = i n d u s t r i a l character indices such as the percent ^ of the labor force employed i n mining and manufacturing, T\_. = t r a v e l time from i to j by the l e a s t time mode, r T. = t r a v e l time of mode m from 1 to j divided by that of the least time mode, C..k = cost of shipping from i to j by the least cost mode, C.. = cost of mode m from i to j divided by that of i "i m — J the least cost mode, N.. = number of modes serving' i and j . 13 Notice that t h i s Mathematica model allows one to compute the e l a s t i c i t i e s of demand for a mode with respect to own price and t r a v e l time. It also allows one to compute the cross e l a s t i c i t i e s of demand for a mode with respect to the price of the least-cost mode and the t r a v e l time of the least-time mode. Although the cross e l a s t i c i t i e s of demand with respect to price and quality variables of "non-best" modes are i m p l i c i t l y assumed to be zero i n t h i s model, th i s assumption can be relaxed by including price and q u a l i t y variables of "non-best" modes. However, the major disadvantages of t h i s approach are two-fold. F i r s t , this model r e s t r i c t s the e l a s t i c i t i e s of demand with respect to prices and q u a l i t y variables to be constant values because of i t s "Cobb-Douglas-like - 22 -functional form. Second, the pattern of shipper's transportation-sectoral technology cannot be i d e n t i f i e d d i r e c t l y from the estimated demand model because the model i s not a derived one: i . e . the model was not derived from shippers' production functions. (6) - Mode-choice modelling approach: A demand forecasting study i s often decomposed into two stages: the f i r s t stage i s to forecast the t o t a l f r e i g h t demand between two points, and the second i s to estimate the mode-split p r o b a b i l i t y model and to apply i t to the t o t a l demand i n order to compute the demand for each mode. Although the t o t a l demand and the mode-split may be mutually i n t e r -dependent, for convenience of modelling and estimation these two stages are often treated separately from one another. I f th i s i s the case, a gravity-type model i s normally used for forecasting t o t a l flows. The mode-split stage normally begins with a careful examination of shipper's mode sel e c t i o n behavior, and then uses discriminant, l o g i t or probit analysis to estimate the conditional mode-choice p r o b a b i l i t y functions. Previous studies on fre i g h t mode sel e c t i o n may be grouped into two categories: descriptive studies and mode-choice pr o b a b i l i t y studies. In what follows, an attempt i s made to describe the approaches taken i n these two categories along with a review of some selected examples. - 23 -(i) Descriptive studies: Many studies have attempted to i d e n t i f y the major factors a f f e c t i n g mode^choice using either published (usually aggregate) or survey (usually disaggregate) data. Although the factors considered d i f f e r from study to study, they are usually a sub-set of the following: commodity attr i b u t e s such as commodity type, value and bulkiness; shipment attributes such as shipment l o t size and distance to be shipped; shipper att r i b u t e s such as regular or occasional user, size of shipper's t o t a l operation and location of shipper; service attributes such as rates, t r a v e l time, 9 r e l i a b i l i t y , etc. Church [l967, 19 7 1 J investigated the relationships between truck share and such variables as shipment l o t s i z e , distance shipped and commodity type using the 19 6 3 Commodity Transportation Survey data. Buhl [l967] studied the r e l a t i o n -ships between the truck share and such variables as the number of employees and the SIC (Standard I n d u s t r i a l C l a s s i -f i c a t i o n of Commodities) category. The Economist Intelligence Unit [1967 J conducted a survey of shippers i n the Canadian A t l a n t i c provinces, and from the data, they i d e n t i f i e d s i x important service a t t r i b u t e s : speed, completeness of service, r e g u l a r i t y , frequency, a v a i l a b i l i t y of equipment, size of the unit of service, and safety of shipments. Saleh and LaLonde [ l 9 72j presented a detailed process for se l e c t i n g a motor c a r r i e r using the data obtained from interviewing - 24 -shippers and mail questionnaires. Using waybill samples, Morton [ l 9 7 l ] found that considerable v a r i a b i l i t y e xists i n rates and market shares of truck and r a i l modes which cannot be explained by weight and mileage blocks alone. Evans and Southard I 1974 I conducted a motor c a r r i e r p o l l to determine the r e l a t i v e importance of 28 detailed q u a l i t y attributes of service. They also presented an i n t e r e s t i n g comparison between the perceptions of buyers and s e l l e r s of motor c a r r i e r services. These are only a small f r a c t i o n of the numerous studies which attempted to i d e n t i f y the factors influencing mode sel e c t i o n . Tertiev et a l . 1975J presented in a non-empirical paper a detailed l i s t of a l l service, commodity, market and shipper attributes that may influence mode choice, and subsequently recommended to use multi-nomial l o g i t model which includes a l l these variables. Although the descriptive research such as those mentioned above i s a necessary f i r s t step toward a more rigorous study, i t provides only p a r t i a l information on the nature of the demand. ( i i ) Mode-Choice p r o b a b i l i t y studies: Many studies have estimated the models for predicting conditional p r o b a b i l i t i e s of choosing various modes. D i s c r i -minant analysis and l o g i t analysis are the two major techniques for estimating the mode^choice p r o b a b i l i t i e s . "Probit" models are also used from time to time. Although the variables included i n the model d i f f e r from study to study, v i r t u a l l y a l l previous studies have used a subset of - 25 -service, shipment and shipper a t t r i b u t e s . Following Warner's application of discriminant analysis to a passenger mode-split problem i n Chicago [l9 62j and the subsequent work by the T r a f f i c Research Corporation of Toronto [l965], discriminant analysis has been increasingly applied to both passenger and f r e i g h t mode-choice studies. Examples of passenger mode-choice area are Quamby [ l 9 6 7 ] j , M cGillivary [l970j and the Transportation Development Agency [l976], whereas Miklius £l979], Antle and Haynes [ l 9 7 l ] , Bayliss [1973], Hartwig and Linton [19 74], and Turner [ l 9 7 5 J are those of f r e i g h t mode-choice area. Charles River Associates [ l 9 7 2 J and McFadden [l972] may be considered as among the e a r l i e s t attempts to apply l o g i t model to passenger mode-choice p r o b a b i l i t y study. Since then, the l o g i t model has become an increasingly popular tool for f r e i g h t as well as passenger mode-choice studies. For example, Kullman [l973], Hartwig and Linton [l974^j, Turner [l9 75] , Boyer ^1977^ and Levin [jL978^] applied a l o g i t model to predict f r e i g h t mode-choice. The "probit" model has been less popular than the other two models because a l o g i t model i s far easier and less costly to estimate than a probit model. Note that the l o g i t model approximates quite c l o s e l y the cumulative normal p r o b a b i l i t y function which a probit model attempts to estimate. (See Berkson [l944, 19513 for more details.) Hartwig and Linton [1974] applied a probit model as well as l o g i t and d i s c r i -minant analysis models to a f r e i g h t mode-choice problem. - 26 -In what follows several of the studies c i t e d above are s e l e c t i v e l y described. The emphasis i s placed on the form of data used and the variables included i n the model rather than on t h e i r s p e c i f i c r e s u l t s . Antle and Haynes ^ 1 9 7 1 J used several independent variables i n t h e i r discriminant function to characterize each mode-choice decision; these variables are annual tonnage shipped, distance, average t r a v e l time by the chosen mode, average shipment s i z e , f r e i g h t rate of the chosen mode, fr e i g h t rate of the competing mode, and the handling cost of the chosen mode. As mentioned already, Hartwig and Linton £ l 9 7 4 J used l o g i t , probit and discriminant analysis to model the ind i v i d u a l shipper's mode-choice between f u l l load truck and f u l l load r a i l using the information obtained from 1 2 1 3 waybills for f u l l load truck and r a i l shipments of consumer durables. Their model used the difference i n truck and r a i l t r a n s i t times, the difference i n truck and r a i l f r e i g h t charges, the difference i n standard deviation of the t r a n s i t time d i s t r i b u t i o n between truck and r a i l and the value of the commodity as the independent variables. Kullman £ l 9 7 3 J estimated a binomial l o g i t model to predict the r a i l - t r u c k modal s p l i t using aggregate data for s p e c i f i c c i t y pairs obtained from the 1 9 6 7 Census of Transportation and from cert a i n c a r r i e r s . He used differences i n f r e i g h t rate, transit-time, and r e l i a b i l i t y of t r a n s i t time between r a i l and truck modes, distance, annual volume, - 27 -and value per ton of commodity as the independent variables in his l o g i t model. Turner £ l 9 7 5 J undertook a major study on fre i g h t mode sele c t i o n which seems the most extensive of i t s kind among the Canadian studies. I t seems worthwhile to describe his models b r i e f l y here because b a s i c a l l y the same data set that he used i s to be used for t h i s t h e s i s . Using the 197 0 Canadian inter-regional f r e i g h t transportation data, which i s to be described i n chapter IV, he estimated a regression model, a discriminant function and a l o g i t model for each of the selected 13 commodity groups. His regression models of modal shares ( r a i l , truck and ship) were formulated as a li n e a r function"*"^ of the r e l a t i v e mode attributes ( r e l a t i v e to the three-mode average) such as the price r a t i o , the t r a n s i t time r a t i o , the r a t i o of the standard deviations of t r a n s i t time d i s t r i b u t i o n s , the r a t i o of skewnesses of t r a n s i t time d i s t r i b u t i o n s , and of shipment attributes such as distance and weight. Although the v a l i d i t y of using some variables (especially using both v a r i a b i l i t y and skewness of the t r a n s i t time distribution) i s questionable, more c r i t i c a l problems with his regression model seem to be the following: 1. The shares of modes were not constrained to add up to one. 2. Although cross-equation covariances are not l i k e l y to be zero i n most share models, he did not take these into account by using univariate regression. Therefore, the estimated test s t a t i s t i c s including t and F s t a t i s t i c s are l i k e l y to be biased. - 28 -3. He also reported the share equations estimated from the pooled share data of a l l modes. Although, i n these cases, he added mode dummy-variables, the estimation of a common share equation i s hardly j u s t i f i e d . The regression models were estimated from the data corresponding to the shared ori g i n - d e s t i n a t i o n (0-D) pa i r s , on which at least two modes shared the t r a f f i c . From the same shared 0-D data, Turner also estimated binomial l o g i t models of r a i l - t r u c k competition using the same independent variables that were used i n his regression models. Using the data corresponding to the monopoly O-D's where a l l t r a f f i c was monopolized by a single mode, Turner estimated discriminant analysis models as functions of distance, weight, and service attributes of the mode used such as fr e i g h t rate, t r a n s i t time, and standard deviation and skewness of t r a n s i t time d i s t r i b u t i o n . Except for the usage of l i n k aggregate data (which he claimed was necessary because of otherwise insur-mountable computational cost), the discriminant analysis was well conducted. Overall, although there were several technical or th e o r e t i c a l flaws i n the regression and l o g i t models estimated i n his study, Turner achieved a very important step toward a better f r e i g h t mode sel e c t i o n study by i d e n t i f y i n g the various mode-choice factors considered by shippers of various commodities. A l l mode-choice models such as discriminant analysis, l o g i t and probit models intend to estimate the conditional - 29 -p r o b a b i l i t i e s of choosing various modes with given values of variables included i n the model. Of course, the model can be used for s t a t i s t i c a l tests of whether or not a s p e c i f i c factor a f f e c t s the mode choice p r o b a b i l i t i e s s i g n i f i c a n t l y . However, several recent studies including Boyer £ 1977 J and Levin T l978 3 have used the l o g i t model to estimate price s e n s i t i v i t y of modal demands as i f i t were a demand model. As described i n footnote 3 of chapter I, the usage of l o g i t model as a demand model imposes u n r e a l i s t i c r e s t r i c t i o n s ^ on .e.lasticities . " O finter-modal substitution and thus e l a s t i c i t i e s of demand with respect to price and qu a l i t y of service variables (see Oum £l978^ for a detailed derivation of the r e s t r i c t i o n s imposed by various forms of l o g i t models'!) Since the l o g i s t i c function (the p r o b a b i l i t y function that a l o g i t model t r i e s to estimate) gives a close approximation to the cumulative normal p r o b a b i l i t y function, t h i s r e s t r i c t i o n also c a r r i e s over to "probit" model. Furthermore, since Quamby £l967]| i n his appendix A has derived the binomial l o g i t as a scalar multiple of the score of the discriminant function, this r e s t r i c t i o n i s l i k e l y to carry over to discriminant analysis as well. In conclusion, a l l the mode-choice models which intend to estimate the conditional p r o b a b i l i t i e s of using various modes ( l o g i t , probit and discriminant analysis) are not recommended for measuring price and qu a l i t y responsive-ness of demand. - 30 -(B) Description of Methodology Adopted for t h i s Thesis In the preceding section, the approaches taken i n previous demand studies were grouped into s i x categories. For each category, the advantages and disadvantages were explained. None of the six approaches was considered s a t i s f a c t o r y for studying the nature of demand functions. A l l the disadvantages mentioned i n the previous section would disappear should a demand model s a t i s f y the following three conditions: 1. The model includes both price and qual i t y variables of a l l competing modes so that the price and qual i t y responsiveness of demand can be measured, 2. The functional form of the demand model allows for free v a r i a t i o n of e l a s t i c i t i e s of substitution, and 3. The demand model i s a derived one so that the structure of the shipper's d i s t r i b u t i o n sectoral technology can be in f e r r e d d i r e c t l y from the knowledge of demand functions. The derived demand models used i n Oum [ l 9 7 7 J and i n Friedlaender and Spady [ l 9 7 7 ^ s a t i s f y conditions 2 and 3 but do not meet condition 1 because of the absence of qu a l i t y attributes of service i n the model. However, the abstract mode approach provides us with some i n t u i t i v e rationale for including various quality attributes of service i n the derived models as well. Therefore, the approach taken i n t h i s thesis may be regarded as combining the abstract-mode concept with the derived demand modelling approach. As w i l l be seen l a t e r , the resultant demand model includes the qual i t y attributes of service as well as price variables, and i s derived from the - 31 -shipper's d i s t r i b u t i o n technology consistently with the economic theory of mode^choice. The method of deriving the model i s sketched as follows. On each l i n k , a shipper i s assumed to purchase physical ton-miles associated with certain q u a l i t y attributes of service of his choice. I t i s further assumed that he makes the choices as to the amounts of ton-miles and associated qu a l i t y attributes of service to purchase so that his t o t a l cost of production and d i s t r i b u t i o n i s minimized while s a t i s f y i n g the constraints on the required amounts of his products to be delivered to various market places. As i s shown i n chapter I I I , t h i s allows a shipper's t o t a l cost of production and d i s t r i b u t i o n to be defined as a function of his output l e v e l , the fre i g h t rates and qual i t y attributes of services of alte r n a t i v e modes on a l l l i n k s , and the prices of other factors of production. By imposing some r e s t r i c t i o n s on the shipper's production and d i s t r i b u t i o n technology and by applying the re s u l t s of dual i t y and s e p a r a b i l i t y studies, the shipper's transportation-sectoral-cost function on each l i n k i s derived to be a function of freight rates and quality attributes of alte r n a t i v e modes, and the distance of the l i n k . At t h i s stage, i t i s assumed with some t h e o r e t i c a l j u s t i f i c a t i o n that the transportation-sectoral-cost function i s i d e n t i c a l across a l l shippers of the same commodity on a l i n k . This allows the cost function to be estimated from the data, aggregated over shippers of the same commodity on each l i n k . - 32 -Four a l t e r n a t i v e forms of the shipper's cost function are hypothesized and tested, including one which i s equivalent to the hedonic price hypothesis. The translog function i s chosen to specify these cost functions, and the revenue share of each mode i s derived as a function of fre i g h t rates and the q u a l i t y attributes of a l l modes and the distance of the l i n k . Each set of cost and two modal revenue share functions"*" ( r a i l and truck modes) i s estimated simultaneously from the data on Canadian in t e r - r e g i o n a l f r e i g h t flows during the year 1970, for each of the eight selected commodity groups. For each commodity group, the e l a s t i c i t y of substitution between the two modes and e l a s t i c i t i e s of the modal demands with respect to f r e i g h t rates and qual i t y attributes of service (speed and r e l i a b i l i t y ) are computed for some selected l i n k s . - 33 -Footnotes for Chapter I I : 1. Throughout th i s t h esis, the term "service a t t r i b u t e s " refers to the fr e i g h t rate and the "quality a t t r i b u t e s " of service such as speed (or t r a n s i t time), r e l i a b i l i t y of speed (or v a r i a b i l i t y of transit-time d i s t r i b u t i o n ) , convenience and f l e x i b i l i t y of service, etc. 2. The imputed cost of waiting time i s higher than that of tr a v e l time mainly because the inventory holding cost i s incurred while waiting at the o r i g i n of the cargo whereas i t does not occur while i n t r a n s i t . 3. Throughout t h i s thesis the term "Cobb-Douglas-like" function refers to the Cobb-Douglas demand model without the condition of homogeneous of degree zero i n prices imposed. 4. "Translog" functions belong to a family of " f l e x i b l e " functions that can be used to give a second order Taylor series approximation to any functional form. For more d e t a i l s on " f l e x i b l e " functions, see, for example, Diewert [19 71], H a l l [1973], Christensen et a l . [l973] and Denny [_19 74] . 5. The term "gravity-type model" i s used here i n order to dis t i n g u i s h i t from true gravity models which are being widely used i n urban passenger transportation studies and which require "at t r a c t i o n balancing" i t e r a t i o n and "c a l i b r a t i o n " i t e r a t i o n . 6. Throughout th i s thesis, the term " l i n k " refers to a s p e c i f i c f r e i g h t market (or route) l i n k i n g an o r i g i n region to a destination region. 7. See the discussion i n "normative approach based on user optimization" for an explanation of the concept of abstract-mode. 8. "Non-best" modes are a l l modes other than the least time mode using t r a v e l time c r i t e r i o n but are a l l modes other than the least time mode using t r a v e l cost c r i t e r i o n . 9. See Tertiev et a l . [1975] for a detailed l i s t of factors which are l i k e l y to influence mode-choice. 10. Turner reported that he also t r i e d to use a Cobb-Douglas-l i k e regression model. - 34 -This r e s t r i c t i o n could have introduced some bias i n the ca l c u l a t i o n of welfare loss by Boyer [1977J and Levin [1978] which they claimed i s due to the t r a f f i c mis-a l l o c a t i o n caused by the I.C.C. minimum rate regulation. Due to the s i n g u l a r i t y of the two share equations, only one revenue share function i s ac t u a l l y estimated together with the cost function. CHAPTER III MODEL FORMULATION The plan for t h i s chapter i s as follows: a general form of shipper's transportation-sectoral-cost function that appropriately characterizes the structure of the shipper's transportation-sectoral-technology i s derived i n r e l a t i o n to a firm's o v e r a l l optimization of production and d i s t r i b u t i o n a c t i v i t i e s . Some plausible r e s t r i c t i o n s are then imposed on the general structure to generate three alternative forms of the .transportation-sectoral-cost function. Each form of the cost function i s s p e c i f i e d i n a translog form for the case of two-mode ( r a i l and truck) competition, and the corresponding demand functions of the two modes are derived. In modelling the demand for fr e i g h t transportation, i t i s important to r e a l i z e that transportation i s used as a factor of production. Therefore, the demand model should be derived from the shipper's underlying production or cost function. There are two d i s t i n c t methods of deriving such an input demand model. One method would be to postulate a functional form for the production function s a t i s f y i n g c e r t a i n r e g u l a r i t y conditions"'" and then solve an output-constrained cost minimization problem for the derived input demands. The other method would be to postulate a d i f f e r e n t i a b l e functional form for the shippers' cost function again s a t i s f y i n g the reg u l a r i t y conditions, and obtain the derived input demands by - 35 -- 36 -applying Shephard's lemma [shephard, 1953, p. l l j . The d i f f i c u l t y with the f i r s t method i s that i f the production function i s s p e c i f i e d i n a ' f l e x i b l e functional form', i t i s in general impossible to obtain the derived demand functions as e x p l i c i t functions of the "unknown" parameters of the production function. As w i l l be explained l a t e r i n t h i s chapter, the usage of a 'fle x i b l e ' functional form i s e s s e n t i a l for studying inter-modal competition. Thus, the l a t t e r method of deriving demand models w i l l be used i n thi s t h e s i s . The method i s based on the dua l i t y r e l a t i o n that exists between production and cost functions, a r e s u l t established o r i g i n a l l y by Shephard [l953 1 and Samuelson [1953-4 J and re f i n e d by Uzawa [ 1 9 6 4 ] , McFadden [l966, 1 9 7 0 J , Shephard [l970 j , Diewert [l971,1974] and Blackorby-Primont-Russell [l978]. Duality theory implies that i f producers minimize input costs of producing given outputs, and i f competition p r e v a i l s i n factor markets, then the cost 3 function s a t i s f y i n g the usual r e g u l a r i t y conditions contains s u f f i c i e n t information to describe completely the corresponding production technology, and vice versa. Thus, rather than specifying a functional form for the production function and deriving the input demand functions therefrom (as i n the f i r s t method for deriving input demands), one can specify a cost function d i r e c t l y and then apply Shephard's lemma to obtain the input demands. Since our primary objective i s to derive the demand model for f r e i g h t transportation rather than for a l l inputs of - 37 -production and d i s t r i b u t i o n , our i n t e r e s t l i e s primarily i n examining the structure of the shipper's technology for the transportation sector of his t o t a l a c t i v i t i e s (called here-afte r as transportation-sectoral-technology). - 38 -(A) Derivation of the General Model (Model A) To derive a demand model for fr e i g h t transportation, i t i s e s s e n t i a l to look into how shippers value various 4 c h a r a c t e r i s t i c s of service and how these c h a r a c t e r i s t i c s serve shippers i n achieving t h e i r objectives. Since i n d u s t r i a l or commercial shippers are major users of fr e i g h t services, f r e i g h t transportation services can be considered as an intermediate input to shippers' production and d i s t r i b u t i o n a c t i v i t i e s . Then i t i s reasonable to say that a shipper maximizes his p r o f i t by using the optimal combination of inputs and at the same time by producing the optimal amount of output. By the same token, a shipper i s also motivated to use optimal combinations of various modes of fr e i g h t services, the c h a r a c t e r i s t i c s of which d i f f e r from one mode to another and from one l i n k to another. Therefore, a shipper's demand for a mode of f r e i g h t service depends upon the level s of c h a r a c t e r i s t i c s b u i l t into the service and the r e l a t i v e contributions of these c h a r a c t e r i s t i c s to the shipper's production and d i s t r i b u t i o n a c t i v i t i e s . A shipper's demand for each mode of fre i g h t service can then be derived from the shipper's production and d i s t r i b u t i o n technology by maximizing p r o f i t . A l t e r n a t i v e l y , for a given requirement for shipper's output i n various geographical locations, the demands can be derived equally well by minimizing the t o t a l cost of production and d i s t r i b u t i o n . Of course, the shipper's t o t a l demand for fre i g h t service can be obtained by summing the demands of a l l modes. - 39 -To maintain the generality of the discussion, i t i s assumed throughout t h i s chapter,'unless mentioned otherwise, that: (i) The number of modes competing on each l i n k i s M, ( i i ) The qual i t y of a service can be completely described by N dimensions, and ( i i i ) A shipper's d i s t r i b u t i o n network i s composed of L l i n k s . In order for a model to be estimated, i t should be expressed i n terms of only the variables for which data are available. Therefore, i t i s es s e n t i a l to consider the kinds and form of the available data before formulating the model. As w i l l be explained i n chapter IV, only the following data aggregated by each l i n k (directed route) were available for t h i s t h e s i s : 1. Yearly t r a f f i c volume (in tons) of each mode for each commodity group, 2. Average f r e i g h t rate (per ton) of each commodity group charged by each mode, 3. Average t r a n s i t time and i t s v a r i a b i l i t y of each mode, and 4. Distance of the l i n k . Therefore, the objective of t h i s section i s to derive a l i n k -s p e c i f i c unit transport cost function for shippers of a pa r t i c u l a r commodity group, as a function only of fr e i g h t rates and qual i t y a t t r i b u t e s 1 of services of various modes and the distance of the l i n k . Since the demand for fr e i g h t transportation service i s generated as a r e s u l t of i n d i v i d u a l shipper's optimization of the production and d i s t r i b u t i o n - 40 -a c t i v i t i e s , t h e p r i m a r y t a s k h e r e i s t o i d e n t i f y t h e s e t o f r e s t r i c t i o n s ( r e q u i r e d t o be i m p o s e d o n t h e s h i p p e r ' s m a c r o p r o d u c t i o n f u n c t i o n a n d / o r c o s t f u n c t i o n ) w h i c h w i l l a l l o w u s t o e x p r e s s t h e l i n k - s p e c i f i c u n i t t r a n s p o r t c o s t f u n c t i o n i n t e r m s o f o n l y t h e a v a i l a b l e d a t a m e n t i o n e d i n t h e a b o v e . The c o s t f u n c t i o n f o r a s h i p p e r ' s e n t i r e p r o d u c t i o n a n d d i s t r i b u t i o n a c t i v i t i e s c a n be d e f i n e d a s : r T m m (3,1) C ( Y , P , P X ,Z) E w . r . t . C T w h e r e P 1 C C L f M S P7X7 + E £ P -X .. ^=1 1 1 .£=1 \m=l m £ m £ j s u b j e c t t o t h e f o l l o w i n g p r o d u c t i o n t e c h n o l o g y , (3,2) f ( X C , X T , Z ) > Y Y = s h i p p e r ' s t o t a l o u t p u t t h a t n e e d s t o be d e l i v e r e d t o v a r i o u s d e s t i n a t i o n m a r k e t s , = [ p ^ , P 2 , . . .'','Pj J w h e r e P^ = t h e p r i c e o f i t h i n p u t o t h e r t h a n t r a n s p o r t a t i o n s e r v i c e , i = l , 2 , . . . , I , T T P = [P .1 a m a t r i x o f o r d e r MxL r e p r e s e n t i n g t h e u mx,J f r e i g h t r a t e s o f M modes o n L l i n k s ; i . e . , P „ = mth mode's f r e i g h t r a t e p e r t o n - m i l e o n mil — l i n k i l , m=l,2,...,M, £=1,2,...,L. c X = p r o d u c t i o n f a c t o r s o t h e r t h a n t r a n s p o r t a t i o n s e r v i c e s u c h a s l a b o u r (L) and c a p i t a l ( K ) , X T = [ x ^ ] a m a t r i x o f o r d e r MxL r e p r e s e n t i n g q u a n t i t i e s o f M modes u s e d o n L l i n k s ; i . e . , X „ = t o n - m i l e s mx, s h i p p e d b y mode m o n l i n k m=l,2,...,M, £=1,2, . . . ,L. - 41 -Z = , Z 2 , . . . , , . . . , Z J where each element of Z i s a matrix of order MxN representing the amounts of N qua l i t y attributes of M modes on a p a r t i c u l a r l i n k ; i . e . , for l i n k I, Z F L = | Z . J and Z „ = nth ' — Jl u mn£ J mn£ — quali t y a t t r i b u t e of mode m on l i n k Z_, m=l,2,...,M n=l,2,...,N. In order to be able to write the l i n k - s p e c i f i c unit transport cost function as a function only of the prices and qual i t y attributes of various modes serving the l i n k , the following r e s t r i c t i o n s are required to be imposed on the (macro) cost function (3,1). C T 1. The cost function C(Y,P ,P , Z ) i s completely s t r i c t l y s e p a r a b l e i n link-wise p a r t i t i o n of the transport-related variables {(p , Z ) , (P_ , Z _ ) , . . . , (P £, Z £ ) , . . . , ( P L , Z L ) } . 2. The cost function C i s p o s i t i v e l y l i n e a r l y homogeneous (PLH) i n output Y. 3. The cost function C i s d i f f e r e n t i a b l e and s t r i c t l y p o s i t i v e l y monotonic i n ( Y , P G , P T , Z ) , and PLH and concave i n (P C , P T ) . Then, Theorem 4.8 (p. 136), Corollary 4.8.4 (p. 142), Theorem 4.9 (p. 143) and Corollary 4.9.4 (p. 156) of Blackorby-Primont-Russell [ l 9 78] allow cost function C to be re-written as: (3,3) C = Y-C(P C,C T ( E C £ ( P „ , Z J ) ) 1=1 * L -— = Y-C(P C, ( E { C £ ( P „ , Z 0 ) } P ) P ) £ = 1 * * 0 ± V < 1 where — A£ C i s an increasing function of i t s arguments, and each C (•) i s d i f f e r e n t i a b l e , s t r i c t l y p o s i t i v e l y monotonic, PLH and concave i n P.. - 42 -Therefore, the l i n k - s p e c i f i c unit transport cost function, say for l i n k can be written independently of the output l e v e l (Y) and the prices of other inputs (P ) as the following: (3.4) UC £ = C £(P £,Z £) £=1,2,...,L where UC^ ; unit transport cost per ton on l i n k £_. In interpreting the meaning of the r e s t r i c t i o n s , i t i s some-times easier to work with a production function rather than cost function. Applying theorem 4.12 of Blackorby-Primont-Russell JjL978, p. 157], i t i s obvious that i n order for cost function C to s a t i s f y r e s t r i c t i o n s 1 and 2 l i s t e d above, i t s dual production function f(*) i n equation (3,2) must s a t i s f y C T the following r e s t r i c t i o n s : PLH i n (X ,X ,Z), complete s t r i c t s e p a r a b i l i t y i n link-wise p a r t i t i o n of transport-related variables { (X 1,Z 1) , (X 2 ,Z2) ,. . . , (X^  ,\ ) ,. . . , (X ,Z ) } , p o s i t i v e C T monotonicity and quasi-concavity i n (X ,X ), and d i f f e r e n t i a b i l i t y i n a l l variables. By theorem 4.8 of Blackorby-Primont-Russell [l978, p. 1 5 2 J , the production function f(•) can be re-written as follows: L 1. (3.5) f(X C,X T,Z) = f (X C, ( E f £(X„;Z„) P) P) 0 ft p < 1 1=1 * * where f i s an increasing function i n i t s second argument, and each f (•) i s d i f f e r e n t i a b l e , p o s i t i v e monotonic, quasi-concave and PLH i n X^. - 43 -Among the r e s t r i c t i o n s imposed on the production function f ( - ) f the following items have empirically important i m p l i -cations that deserve special attention. C T 1. Positive l i n e a r homogeneity of f i n (X ,X ,Z): This implies that a shipper's production technology i s characterized by constant returns to scale i n T both amounts (X ) and q u a l i t i e s (Z) of transportation C services and i n other inputs (X ) such as labour and c a p i t a l : i . e . , a proportionally i d e n t i c a l . C T increase i n a l l inputs (X ,X ,Z) w i l l r e s u l t i n an increase i n output (Y) by the same proportion. Note that homotheticity of the production function i s implied by PLH of f ( - ) . 2. Complete s t r i c t s e p a r a b i l i t y of the transport-related T C variables (X ,Z) from other inputs (X ): This means that every union of link-transport sectors i s s t r i c t l y separable from a l l variables i n the remaining link-transport sectors and non-transport inputs. This implies that each l i n k - s p e c i f i c transportation technology i s unaffected by the transportation a c t i v i t y l e v e l s of a l l other l i n k s and the amounts of non-transport inputs used i n the production. Note that t h i s complete s t r i c t s e p a r a b i l i t y implies s t r i c t T C se p a r a b i l i t y of (X ,Z) from X i n the production function f ( • ) . The two assumptions together imply that a shipper changes the amount of his product transported by each mode on each l i n k i n exactly the same proportion as the change i n his t o t a l output i f prices and quality attributes of a l l modes on a l l l i n k s remain unchanged. This may well be an u n r e a l i s t i c assumption. However, i n a l l the transport demand models estimated to date using data aggregated by each route, exactly the same set of assumptions as made i n t h i s thesis have been imposed without mentioning them e x p l i c i t l y . Whether or not these assumptions are v a l i d i s an empirical question which cannot be tested here due to lack of necessary data. Turning our attention to equation (3,4), i t should be noticed that the l i n k unit transportation cost function C^(') could be estimated from any time series data of an i n d i v i d u a l firm's unit cost of transportation (UC^), prices (P^) and q u a l i t y attributes of service (Z^) of various modes on l i n k Since cross-sectional data are to be used i n t h i s thesis, i t i s e s s e n t i a l to make the functional form of the unit cost function independent of the l i n k index I f a shipper's choice c r i t e r i o n i n the p r i c e - q u a l i t y a t t r i b u t e s space i s consistent from l i n k to l i n k , i t would be possible to write the unit cost function (3,4) without the subscript However, the choice behaviour i s l i k e l y to depend on the distance of l i n k (D^) because the imputed costs of q u a l i t y attributes of service, such as t r a n s i t time and i t s v a r i a b i l i t y , are l i k e l y to depend on distance due to t h e i r 5 implications on inventory management cost. The only way to handle t h i s problem i s to parameterize the difference i n the cost function by including the distance variable (D^) as an argument of the cost function as i n (3,6): (3,6) UC £ = 6(P A,Z £,D A) £=1,2,...,L where UC^ = unit cost per ton moving on l i n k For convenience of interpretation of empirical r e s u l t s , both sides of equation (3,6) are divided through by distance of the l i n k (D^) so that the image of the new function becomes "unit cost per ton-mile" as follows: (3,7) UC, = C(P.,Z.,D„) where UC = unit cost per ton-mile, = Mxl vector of prices of M modes on l i n k £, = MxN matrix of quality attributes of service of M modes on l i n k I, = distance of l i n k I i n miles. This l i n k unit cost model i n ( 3 , 7 ) i s referred to as the "general model (Model A)" throughout t h i s t h e s i s . Several al t e r n a t i v e models w i l l be hypothesized l a t e r i n th i s chapter by imposing various r e s t r i c t i o n s on t h i s general model. Since the functional form of the l i n k unit cost function ( 3 , 7 ) i s independent of the l i n k index £ = 1 , 2 , . . . , L , i t can be estimated from cross-sectional data of shipments on L d i f f e r e n t l i n k s . Furthermore, the type of analysis presented by Baumol and Vinod £ l 9 7 o ] j u s t i f i e s using the same unit cost function for a l l shippers of a commodity and across a l l l i n k s . Using inventory analysis, they treated q u a l i t y attributes of fre i g h t service, such as t r a n s i t time and i t s r e l i a b i l i t y as the major determinants of safety stock requirements. They concluded that shippers would use a mixture of transport modes whose pr i c e - a t t r i b u t e combination renders the minimum t o t a l cost of inventory and transportation. Since the key parameters of an inventory model are commodity attributes such as value of the commodity, cost of storage and inventory holding cost. - 46 -f o r a g i v e n c o m m o d i t y t h e r e l a t i v e v a l u a t i o n o f v a r i o u s s e r v i c e a t t r i b u t e s i s l i k e l y t o be s i m i l a r a c r o s s s h i p p e r s o n a g i v e n l i n k a n d a c r o s s c r o s s - s e c t i o n a l l i n k s . T h i s i m p l i e s t h a t t h e l i n k u n i t c o s t f u n c t i o n , ( 3 , 7 ) , c a n be e s t i m a t e d f r o m c r o s s -s e c t i o n a l l i n k d a t a , e a c h o f w h i c h i s a g g r e g a t e d o v e r a l l s h i p p e r s on t h a t l i n k . F o r t h e p u r p o s e o f e s t i m a t i o n , t h e c o s t f u n c t i o n (3,7) i s p o s t u l a t e d i n a s p e c i f i c f u n c t i o n a l f o r m . F o r t h e s t u d y o f i n t e r - m o d a l s u b s t i t u t i b i l i t y , t h e f u n c t i o n a l f o r m s h o u l d a l l o w f o r f r e e v a r i a t i o n o f A l l e n p a r t i a l e l a s t i c i t i e s o f s u b s t i t u t i o n (APES) a n d be s u f f i c i e n t l y ' f l e x i b l e * t o p r o v i d e a v a l i d s e c o n d o r d e r a p p r o x i m a t i o n t o an a r b i t r a r y d i f f e r e n -t i a b l e f u n c t i o n . I n r e c e n t y e a r s , t h e r e h a s b e e n c o n s i d e r a b l e w o r k d e v e l o p i n g s o - c a l l e d ' f l e x i b l e ' f u n c t i o n s w h i c h s a t i s f y t h e s e p r o p e r t i e s . The g e n e r a l i z e d L e o n t i e f f u n c t i o n 6 [ D i e w e r t , J19 7 l J , the quadratic mean of order-r function [Denny, 1 9 7 2 & a n d t h e g e n e r a l i z e d C o b b - D o u g l a s f u n c t i o n [ D i e w e r t , 1 9 7 3 ] b e l o n g t o t h e f a m i l y o f f l e x i b l e f u n c t i o n s . The t r a n s l o g f u n c t i o n was c h o s e n t o be u s e d t h r o u g h o u t t h i s t h e s i s s i n c e i t g e n e r a t e s t h e s y s t e m o f c o s t a n d demand f u n c t i o n s t h a t a r e t h e most c o n v e n i e n t t o e s t i m a t e u s i n g t h e a l g o r i t h m d e v e l o p e d by B e r n d t - H a l l - H a l l - H a u s m a n [ l 9 7 4 ] . To s p e c i f y t h e u n i t c o s t f u n c t i o n s i n t r a n s l o g f o r m , i t i s n e c e s s a r y t o d e c i d e w h i c h modes a n d w h a t k i n d o f q u a l i t y v a r i a b l e s a r e t o be i n c l u d e d i n t h e e m p i r i c a l e s t i m a t i o n . As - 47 -w i l l be seen i n chapter IV, data on q u a l i t y attributes were severely limited and the accuracy of the available data could be disputed i n many aspects. Nevertheless, i t was possible to obtain the data on l i n k - s p e c i f i c average f r e i g h t rates per ton-mile, l i n k - s p e c i f i c average t r a n s i t times i n days and l i n k - s p e c i f i c v a r i a b i l i t y of t r a n s i t times for both railway and highway (truck) modes. I t was also possible to obtain l i n k s p e c i f i c t o t a l ton-miles c a r r i e d by each mode and the distance of each l i n k i n miles. In order to follow the convention of defining q u a l i t y attributes such that, c e t e r i s p a r i b u s , more of an a t t r i b u t e i s preferred to l e s s , average t r a n s i t time and v a r i a b i l i t y of t r a n s i t time were used to generate 'average speed i n miles per day' and ' r e l i a b i l i t y of t r a n s i t time' (reciprocal of the c o e f f i c i e n t of v a r i a t i o n in the transit-time d i s t r i b u t i o n ) . Due to data l i m i t a t i o n s , therefore, the empirical implementation i s l i m i t e d to the case of two modes (railway and highway modes) and two q u a l i t y a t t r i b u t e s . With the Hicks-Samuelson symmetry condition imposed, the translog function corresponding to the l i n k unit cost function (3,7) can be written as: - 48 -(3,8) In C(P £,Z £ ,D£) where (i) t 1 t .= Zna + a «v + ^ V *S*V o 2 InP, = l n a Q + [a t,b t,c t,d] ZnZ InZ InD, 1£ 21 + | [(ZnP £) t, (lnZ1SL) t , UnZ 2 £) tflnDJ A E F g E B G h F f c Gfc C i _ t ,_t , t _ ln?l l n Z l l lnZ2l J lnDn V = 7x1 vector of variables which i s , for convenience, partitioned into four components as follows: V lnPn E InZ 1£ L^ n Pr£ ' ^ n Ph£^ t ' 2x1 vector of logarithms of r a i l (Pj.^) and truck (P^) fr e i g h t rates on l i n k l_, i ^InZ 1 p,ZnZ,, j ] t ; 2x1 vector of logarithms £nZ 2£ of r a i l (Zr-L£) and truck ( Z ^ i ^ speed on l i n k [ZnZ r 2^,ZnZ^ 2^] t; 2x1 vector of logarithms of r a i l ( z r2£) a n d truck ( z h2£^ r e l i a b i l i t y of speed on l i n k £, = distance of l i n k _£ i n miles. - 49 -(ii) a 7x1 vector of first order parameters of translog cost function which is partitioned into four components: a = Ca^a^l ; 2x1 vector of first order para-meters corresponding to price vector (P), b = D3 r't )h^ t' 2 x^ v e c t o r °f first order para-meters corresponding to speed vector (Z^), c = C c r ' c i j t ' 2x1 vector of first order para-meters corresponding to reliability vector ( Z 2 ) , d = the first order parameter corresponding to distance (D). (iii) 7x7 symmetric matrix of second-order parameters of translog cost function which is partitioned into 16 components as follows: S = A E F E B G h F f c Gt C i t x t . t ,, g h l d d A E a a , rr rh L arh ar\h ; corresponding to the price variables - 50 -B b r r b r h 1 L b r h bhh J ; corresponding to the speed variables C = c r r c r h c r h chh ; corresponding to the r e l i a b i l i t y v ariables E = a b r r a b r h a b h r a b h h ; corresponding to the products of p r i c e and speed variables F = a c r r a c r h a c h r a c h h corresponding to the products of p r i c e and r e l i a b i l i t y v a riables G = g = be be . r r rn i corresponding to the products b c h r b c h h of speed and r e l i a b i l i t y var-iables [ a d r , a d h ] t ; corresponding to the products of p r i c e and distance variables h = [bd r,bd h] ; corresponding to the products of speed and distance variables i = [cd r,cdj i] ; corresponding to the products of r e l i a b i l i t y and distance variables ? dd = the second order parameter corresponding to distance variable . - 51 T Although the Hick-Samuelson symmetry conditions were already imposed i n the translog cost function (3,8), the li n e a r homogeneity conditions of the cost function with respect to freight rates are yet to be imposed. The derivation i n Appendix 3A shows that the l i n e a r homogeneity conditions impose the following r e s t r i c t i o n s on the parameters of the translog cost function (3,8): (3,9) (a) a r + a h = 1 ( b ) a r r + a r h = °' a r h + ahh = 0 implying = - a r h = a h h (c) a b r r + a b h r = 0, a b r h +. a b ^ = 0 (d) ac + ac, = 0, ac , + ac, , = 0 r r hr rh hh (e) ad r + ad h = 0 The imposition of these l i n e a r homogeneity conditions to equations (3,8) gives: - 52 -P r £ (3,10a) lnc^9 lfZl,ul) = Ina + a^ln (- ) + £ " P h £ + b r Z n Z r l £ h£ + b, ZnZ, , „ + c ZnZ o n + c.ZnZ, o n + dZnD„ h hlx- r r2£ h h2£ £ p + % a U n ( ^ ) ] 2 + h b (ZnZ 1 0 ) 2 + % b.. (ZnZ, . 0) 2 r r L Ph& r r rl£ hh hlx, + b r h Z n Z r l £ Z n Z h l £ + h c r r U n Z r 2 £ ) 2 + % c h h ( Z n Z h 2 £ ) 2 + C r h Z n Z r 2 £ Z h 2 £ + h *d(ZnD £> 2 + a b r r * n Z r l £ Z n fe* h£ + -abrhlnZhUZn(gg) + ' a c r r Z n Z r 2 £ Z n ( ^ ) • p „ -+ a c r h Z n Z h 2 A Z n ( ^ i ) . . + . b c r r Z n Z r i a l n Z r 2 £ h£ + b c h r Z n Z h l A Z n Z r 2 A + b c r h Z « Z r l £ Z n Z h 2 £ + b c ^ Z n Z ^ Z n Z ^ P r £ + a d r Z n D £ Z n ( — — ) + b d r Z n Z r ^ £ Z n D £ + b&^lnZ^^lnD ^ h£ + cdrlnZr2llnTJz + c d h Z n Z h 2 £ Z n D £ The demand f u n c t i o n s f o r the two modes can be d e r i v e d by-a p p l y i n g Shephard's lemma t o the c o s t f u n c t i o n (3,10a) as f o l l o w s : = 3C(«) = C(-) UnCj') r i l ~ 9 P r £ " P r £ . 9 Z * P r £ = F T 1 K + a r r ^ ^  + a b r r ^ Z r l £ + a b r h Z * Z h l £ r£ h£ + a c r r Z n Z r 2 £ + a c r h Z n Z h 2 J l + ad rZnD £] * P X h £ = KT- [ ( 1 " a r ) " a r r l w ( P T 7 > " a b r r ^ Z r l £ " a b r h M h l £ h£ n£ ' a c r r Z n Z r 2 £ - a c r h Z n Z h 2 £ - a d rZnD £ ] . - 53 -Therefore, the share of expenditure on r a i l and truck modes are: (3,10b) S r £ = a r + a r r Z » < ^ > + ^ l n \ ^ + ab^ZnZ hx. + ac ZnZ „ „ + ac , ZnZ, o n + ad ZnDn r r r2£ rh h2£ r Z (3,10c) S h £ = ( l - a r ) - a r r Z n ( ^ ) - a b ^ Z n Z ^ - ^ I n Z ^ hZ - a c r r Z n Z r 2 £ - a c r h Z n Z h 2 J i - a d r Z n D £ where S r and are shares of expenditures on r a i l and truck modes, respectively. Since the system of these two share equations i s singular, only the r a i l share equation S ^ i n (3,10b) i s to be estimated together with the translog cost function (3,10a) using a 7 nonlinear multivariate system estimator. There are 2 8 unknown parameters to be estimated. - 54 -(B) The Model S t r i c t l y Independent of Distance (Model B) I t may be of i n t e r e s t to test whether the shipper's l i n k transportation sectoral technology i s s t r i c t l y independent of the distance of the l i n k ; that i s to say, whether shipper's choice behaviour i n the p r i c e - q u a l i t y space remains unchanged as the length of haul varies. In the context of production technology, this implies that the rate of technical substitution (or the r a t i o of marginal productivities) between any pair of elements i n the vector ( X^£r X2£'"* * , X M £ ' Z l l £ ' Z 1 2 J r * •* , Z1N£' Z21£' Z22JI'- ' " ' Z2N£' ' ' * ' ZMN£ ) ± S i n v a r i a n t to the distance of l i n k (D^). In the context of the cost function, t h i s implies that the monetary values of various service attributes do not depend on the length of the l i n k . Thus, the shipper's valuation of service attributes i s exactly the same between short and long-haul l i n k s . This s t r i c t independence holds i f and only i f the l i n k unit cost function can be written i n a multiplicatively-decomposable form as i n (3,11) [Theorem 3.15, Blackorby-Primont-Russel, 1978]: (3,11) C(P £,Z £,D £) = h (D£) • C (P £,Z £) The cost function (3,11) means that cost per ton-mile can be expressed as a product between a scale factor of distance and a function of prices and service attributes of a l l modes. I f the scale factor h(D^) i s s p e c i f i e d as an exponential function and the function C(») i s a translog form, then i n the two-mode context the l i n k unit cost function can be written as: - 55 (3,12) ZnC(P £,Z £,D £) = dZnD % + ZnC (P £, , Z 2 £) where ZnC(P^,Z 1^,Z 2^) i s obtained from equation (3,8) by sett i n g the following parameters to zero: g = [ a d r , a d h ] t , h = [ b d ^ b d j S i = T ucd r,cd h] t, dd Imposition of the l i n e a r homogeneity condition l i s t e d i n (3,9) gives the following cost function and corresponding modal revenue share functions. (3,13a) ZnC(P £,Z £,D £) P r £ = dZnD£ + Zna Q + a rZn ( _ ) + ZnP h £ + V n Z r l £ + V n Z h l £ h£ + c ZnZ o n + c, ZnZ, _„ + % a rZn(=^-)~|2 r r2£ h h2£ r r L P, „ J 2 2 + h b (ZnZ , 0) + J5 b,, (ZnZ, l 0) + b ,ZnZ l 0 Z n Z , l 0 r r rl£' hh hl£ rh rl£ hl£ 2 2 + H c (ZnZ „„) + J5 c',, (ZnZ, _„) + c , ZnZ „ nZnZ, o 0 r r r2£ hh h2£ rh r2£ h2£ ^r£ ^r£ + ab ZnZ , „ Zn (-—) + ab , ZnZ, .. „ Zn (-—) r r rl£ ^ rh h i t Pr£ Pr£ + * c r r Z n Z r 2 £ Z n ( — ) + a c r h Z n Z h 2 £ Z n ( P ^ } + b c r U n Z r l J l Z n Z r 2 £ + b c h r Z n Z ^ Z n Z ^ + b c r h Z n Z r l J i Z n Z h 2 J l + b c ^ n Z ^ Z n Z ^ - 56 -(3,13b) .S r £ = a r - + . a r r Z » & ) + ab^lnZ^ + a b r h Z n Z h l £ + a c r r Z n Z r 2 £ + a c r h Z n Z h 2 £ Pr£ (3,13c) S h £ = (1-a ) - a r r Z n ( p — ) - ab I«Z u - ab^ZnZ hi - ac ZnZ - ac , ZnZ, o n r r r2£ rh h2£ The system of equations (3,13) i s the same as that obtained from the system (3,10) by setting the parameters ad^., bd r, bd^, cd^, cd^ and dd to zero, and has 22 parameters to be estimated. - 57 -(C) The Model with Mode-Specific Hedonic Aggregators (Model C) In the two models discussed so f a r , shippers are assumed to choose the le v e l s of qual i t y attributes of service for each mode as well as the amount of each mode to use: i . e . , every qu a l i t y variable as well as every quantity variable was treated as a choice variable. However, i t may be plausible to hypothesize that shippers make decisions only about the quantities of various modes used by valuing each mode as a combined en t i t y of price and qual i t y variables. In t h i s case, q u a l i t y variables are no longer choice variables, but they s t i l l a f f e c t mode-choice i n d i r e c t l y through t h e i r imputed prices. Under t h i s hypothesis, the l i n k unit cost function can be written as: (3,14) C(P £,Z £,D j i) = c V t P ^ Z ^ W V ' M - i C ( P M £ ' Z M 1 £ " *'* 'ZMNA' IV-' The structure of t h i s cost function i s p a r t i c u l a r l y i n t e r e s t i n g because i t has M symmetrically separable modal aggregators, one for each mode, as i t s arguments. Each modal aggregator i s defined as a function of price and the qual i t y attributes of the mode and the distance of the l i n k , which i s why i t i s g c a l l e d here a "hedonic" aggregator. I f hedonic aggregators ex i s t , shippers would base t h e i r mode-choice decision on prices adjusted for q u a l i t y v a r i a t i o n s . Then, the hedonic aggregator of a mode may be regarded as the quality-adjusted price function for the p a r t i c u l a r mode. - 58 -Blackorby-Primont-Russel [l9 77] have shown that i n order for a (macro) translog function to have non-additively separable micro-aggregators, the micro-aggregators nested i n the macro translog function are l i n e a r l y logarithmic. Since, among others, our objective i s to study inter-modal competition, an a d d i t i v e l y separable cost function i s of no i n t e r e s t . Therefore, i n the two-mode context, the translog s p e c i f i c a t i o n of cost function (3,14) r e s t r i c t s the two hedonic aggregators r h C (•) and C (•) to be of l i n e a r logarithmic form as i n (3,15). Moreover, the l i n e a r homogeneity of each hedonic aggregator with respect to i t s fr e i g h t rate r e s t r i c t s the exponent of the price variable to unity. Therefore, the translog s p e c i f i c a t i o n of the cost function (3,14) becomes: (3,15) ZnC[P £,Z £,D £] = Z n C [ C r ( P r £ , Z r l £ , Z r 2 £ , D £ ) , C h ( P h £ , , Z h 2 £ ,D£) ^r Y r 6 r 8 h y h 6 h = Zna Q + a r Z n ( P r £ Z r l £ Z r 2 £ D £ ) + a hZn ^ h Z \ l z \ 2 ^ z ) 3 r Y r 6 r Bh ^h 6 h + h a r r [ Z n ( P r £ Z r l £ Z r 2 £ D £ ).] + H a h h [in ^ h l \ l l \ 2 l ^ l >] B r ^ r 5 r 8 h ^h 6 h + a r h Z n ( P r £ Z r l £ Z r 2 £ D £ ) ^ ( P h £ Z h l £ Z h 2 £ D £ ) where a Q = a constant of pro p o r t i o n a l i t y , (8 r,Y rf<5 r ) and (P^'^h'^h^ a r e P a r a m e' l : : e r s o r" hedonic aggregator functions for railway and highway modes, respectively, and - 59 -^ a r , a h ' a r r ' a h h , a r h ^ a r e P a r a m e t e r s °f macro translog cost function C. The l i n e a r homogeneity conditions of translog function C i n r h i t s arguments C (•) and C (•) are: (3,16) (a) a r + a h = 1 (b) a _ + a , = 0 and a . + a, , = 0 r r rh rh hh implying ^ = ^ = a h h Imposition of these r e s t r i c t i o n s and some straightforward manipulation give the following system of equations: - 60 -( 3 ' 1 7 a ) 6 r Y r 5 r B h Y h 6 h l n C [<Pr£Zrl£ Zr2£ D£ > ' <PhAlJl Zh2£ D£>3 = Zna o + a r ( Z n P r £ + 6 r Z n Z r l £ + Y r * n Z r 2 £ + 6 rZnD £) + (l-ar) (ZnP h £ + B h Z n Z h l £ + Y h Z n Z h 2 £ + ^InD£) + * a r r[Zn(pfV + e 2 ( Z n Z r l £ ) 2 + B 2 ( Z n Z h l £ ) 2 + Y * U » Z r 2 £ > 2 h I + \ ( l n \ l i ) 2 + ( 6 r - 6 h ) 2 ( ^ D £ ) 2 l + a rrfc l M P r£ Z B Z r l£ + V n P r ZlnZr21 + ( V 6 h U n P r £ ^ D £ + M r * n ! W n Z r 2 A + 6r ( 5 r " V l n \ l l l n X > l + \ ( 6 r " V l n Z r 2 l l ^ l + 3 h Z n P h £ Z n Z h l £ + Y h Z n P h £ Z n Z h 2 £ - ( V 6 h ) Z n E W n D £ ~ W n Z h l £ Z n Z h 2 £ " V V V ^ Z h l £ ^ D £ " V f i r - V * » Z h 2 A * » D £ - B r Z n P h £ Z n Z r l £ - Y r ^ n P h £ Z n Z r 2 £ - ^ln?r%lnZ^% ' Mh Z n Zrl£ Z n Zhl£ ~ Y r B h Z n Zr2£ Z n Z hl£ " V n P r £ ^ Z h 2 £ - e r Y n £ n Z r l £ Z n Z h 2 £ " Y r V n Z r 2 £ Z w Z h 2 £ 1 (3,17b) S r £ = a r + a r r Z n ( ^ ) + * T ^ r l n < L ^ % - Bh*nZ > h£ + a r r ( V n Z r 2 £ " V n i W ' + a r r ( W *«D* (3,17c) S h £ = ( l - a r ) - *rr*«<^> " a r r ( ^ Z r l £ " V * W - a r r ( Y r ^ n Z r 2 £ - Y h ^ Z h 2 £ ) - a r r ( 6 ^ ) Z n D £ Note that, a l t e r n a t i v e l y , the system of equations i n (3,17) can be obtained by imposing a set of r e s t r i c t i o n s on the para-meters of the general model i n (3,10). Therefore, Model C i s nested i n the general model. I t has 9 parameters to be estimated. In t h i s model, hedonic aggregators are allowed to take d i f f e r e n t parameter values across the modes although the functional form i s constrained to be i d e n t i c a l . Since, as mentioned previously, the hedonic aggregator of a mode can be regarded as the quality-adjusted price function, a difference in value of parameters between modes implies that shippers evaluate the imputed values of qu a l i t y attributes of service d i f f e r e n t l y from mode to mode. This inconsistent evaluation would, of course, contradict the assumption of optimal behaviour postulated i n any economic study. However, t h i s could happen for one or more of the following reasons: (i) Shippers may wrongly perceive the le v e l s of q u a l i t y attributes of various modes, yet t h e i r mode choice decision i s made on the basis of perceived q u a l i t y attributes rather than of the actual l e v e l s , ( i i ) Since the dimension of qual i t y a t t r i b u t e s of service i s numerous, and most qual i t y a t t r i b u t e s are unmeasurable, the omitted variables could cause the difference i n parameter values. - 62 -(D) The Model with Identical Hedonic Aggregators (Model D) Under some r e s t r i c t i v e conditions, Rosen [19 74 ] has shown a n a l y t i c a l l y that each consumer who faces a choice among a set of d i f f e r e n t i a t e d products defines a well-behaved value (hedonic) function over the q u a l i t y of product space as a r e s u l t of u t i l i t y maximizing behaviour. Then, the product with the highest value per d o l l a r spent i s chosen for actual purchase. However, his analysis assumes a pe r f e c t l y r a t i o n a l consumer whose choice depends not on i n s t i t u t i o n a l or psychological b a r r i e r s , such as brand preference and brand insistence, but only on the true (actual) contents of q u a l i t i e s . In the context of transport mode choice, t h i s implies that shippers choose a p a r t i c u l a r mode, not as a physical e n t i t y of the mode but as a c o l l e c t i o n of quality attributes b u i l t i n the mode, through an inter-modal comparison of the true contents of quality attributes of service. Therefore, i n the framework of Rosen's model, the physical e n t i t y of a mode has no p r a c t i c a l s ignificance to shippers other than as a c o l l e c t i o n of q u a l i t y a t t r i b u t e s . Consequently, i f the model i s comprehensive and correct and i f there i s no deviation between the perceived and the actual le v e l s of q u a l i t y attributes of various modes, then the hedonic aggregators are expected to have an i d e n t i c a l set of parameters. In the context of the l i n k unit cost function (3,14), t h i s means that both the functional form and the parameters of M hedonic - 63 -aggregators [c (•),C («),..., C C*)] are i d e n t i c a l . Therefore, under Rosen's framework, the cost function can be written as: (3,18) C £(P £,Z £,D £) E C [ f i ( P l t , Z 1 A , D A ) , . . . f e ( P M A f Z M A , D A ) ] Notice that the form of the hedonic aggregator functions, A C(-)r are independent of the modal index. For the two-mode case, cost function (3,18) can be sp e c i f i e d i n translog form, and corresponding modal revenue shares can be obtained as follows: (3,19a) *„C(P £,Z A,D A> = C [ ( P r £ Z r ^ Z r 2 £ M / ) , ( P h £ Z J A 2 £ D ^ h ) ] = Ina + a „ p n ( ^ i ) + B Z n ( ^ i ) + Y Z n ( ^ ^ ) + ( S - 6 ) InD J ° r L Ph£ Zhl£ Zh2£ r h 1 + ZnP h £ + BZnZ h l £ .+ YZnZ h 2 £ + S^nD^ + h a \jnU&)2 + 32^n(5£M)2 + Y 2 ^ ( ! £ 2 £ ) 2 + ( 6 6 ) 2 ( I n D ) 2 ] r r Ph£ Zhl£ Zh2£ r h * + a r r . [ 3 Z n ( - ^ ) Z n ( - ^ ) + yln(Aln(^-) + Zyln (^-) In (-^) r r Fh£ *hl«. Fh£ *h2£ ^hl£ h2£ + (8 -6 ) l n D 9 l n ( ^ ) + 3lnZ,gln(A + 6 (6 -6.) In (^ M) InD. r h ' Ph£ r i Ph£ r h Zhl£ 2 ' + Y (6 -<*h) Z n D „ Z n ( ? ^ ) ] . r h * Zh2£ (3,19b) S = a + a [j n(^i) + ez„(^ £M) + y Z n (^ H) r £ r r r Ph£ Zhl£ Zh2£ + (S r-6 h)ZnD £] (3,19c) S. . = (l-a)-ar.r[ln{~) + B Z n ( ^ ^ ) + YZn(^ 2-^) h l r r r Ph£ Zhl£ Zh2£ + (6 -6. ) ZnD„] r h £ - 64 -The l i n e a r homogeneity conditions are already imposed i n equations (3,19). This system of equations can also be obtained from (3,17) by imposing two r e s t r i c t i o n s ; 3^ = 3^ = 3 and y " Y h = Y- I t has 7 parameters to be estimated. - 65 -(E) Summary of Alternative Models In the preceding sections, four alternative forms of the l i n k transport unit cost function were sp e c i f i e d i n translog form for the case of two-mode ( r a i l and truck) and two-quality attributes (speed and r e l i a b i l i t y of speed). The corresponding revenue share functions of the two modes were also derived. The following i s a summary of the alternative models: Model A: general model i n (3,10), discussed i n section (A), Model B: model s t r i c t l y independent of distance i n (3,13), discussed i n section (B), Model C: model with mode-specific hedonic aggregators i n (3,17), discussed i n section (C), Model D: model with i d e n t i c a l hedonic aggregators i n (3,19), discussed i n section CD). Note that models B, C and D are nested i n model A, and model D i s nested also i n model C, but there i s no nested r e l a t i o n between model B and either model C or model D. The qu a l i t y a t t r i b u t e variables (speed and r e l i a b i l i t y of speed) may or may not play an empirically s i g n i f i c a n t role i n mode-choice decisions, depending on the commodity. Consequently, for each commodity group, i t i s necessary to decide whether speed and/or r e l i a b i l i t y variables should be included i n the model as well as to decide the best model to use. Therefore, each of the above four models i s to have the following three sub-models : - 66 -Sub-model 1: includes both speed and r e l i a b i l i t y , Sub-model 2: includes speed only, Sub-model 3: does not include any q u a l i t y variable. Therefore, for each commodity group, there are eleven d i f f e r e n t 9 sub-models that need to be estimated. Footnotes for Chapter I I I : 1. The r e g u l a r i t y conditions required here are that the production function be d i f f e r e n t i a b l e , increasing and concave i n i t s arguments. 2. The term ' f l e x i b l e functional form' w i l l be explained l a t e r i n t h i s chapter. 3. The r e g u l a r i t y conditions that are required to determine uniquely the corresponding production function are that the cost function be increasing, l i n e a r l y homogeneous and quasi-concave i n the input p r i c e s . 4. The term ' c h a r a c t e r i s t i c s ' represents both qu a l i t y attributes of service such as t r a n s i t time (speed) and r e l i a b i l i t y of service, and fre i g h t rates. 5. In fa c t , i t i s to be tested l a t e r whether the distance a f f e c t s shippers' choice behavior i n p r i c e - q u a l i t y attributes space. 6. Notice that the generalized Leontief function i s a special case of a quadratic mean of order-r function where r = 1. 7. See chapter IV for more d e t a i l s on the s i n g u l a r i t y of the system and the properties of the estimator. 8. Hedonic Price Theory i s o r i g i n a l l y due to Court [_19 39], and added and formalized by Stone [1956], Lancaster [196 6] and Fisher and S h e l l (l968 J . Hedonic Price Theory has been widely used to construct true price indices of i n d u s t r i a l c a p i t a l goods and household durable goods which are usually subject to quality change. Some t y p i c a l applications can be found i n Cragg^ and Uhler [19 70] , H a l l [1971], Ohta [1975], G r i l i c h e s [l97l] and Terleckyj [1976] . 9. Although three sub-models for each of the four macro models make twelve, model C-3 (sub-model 3 of model C) and model D-3 (sub-model 3 of model D) are i n fact i d e n t i c a l . CHAPTER IV SOURCES OF DATA AND CONSTRUCTION OF THE VARIABLES In order to estimate the unit cost functions and modal revenue share functions s p e c i f i e d i n chapter I I I , i t i s necessary to construct the following variables: C £ = weighted average unit cost to shippers i n cents per ton-mile on l i n k Z_, % = 1,2,...,L. S r £ = revenue share of railway mode on l i n k Z_, S^ £ = revenue share of highway mode on l i n k l_, P r £ = average railway f r e i g h t rate i n cents per ton-mile on l i n k l_, P^£ = average trucking f r e i g h t rate i n cents per ton-mile on l i n k Z r^ £ = average speed of railway services i n miles per day on l i n k Zhl£ = average speed of trucking services i n miles per day on l i n k - r e c i p r o c a l of c o e f f i c i e n t of v a r i a t i o n i n t r a n s i t time d i s t r i b u t i o n of railway services on l i n k l_ (a measure of r e l i a b i l i t y of t r a n s i t time), Zh2£ = r e c i P r o c a x °f c o e f f i c i e n t of v a r i a t i o n i n t r a n s i t time d i s t r i b u t i o n of trucking services on l i n k l_, D £ = distance of l i n k l_ i n miles. Since the empirical implementation has to be done separately for each commodity group, i t i s necessary to con-- 68 -t r u c t these variables for a l l commodity groups to be studied. Furthermore, both for railway and for trucking modes the same de f i n i t i o n s of commodity groups and of the system of l i n k s should be employed i n constructing the variables i n order to achieve consistency of data between the two modes. To construct the above variables, i t i s es s e n t i a l to obtain the following data for each commodity group and for each l i n k : (1) Distance of the l i n k (D£) , (2) Total tons moved by railway mode (V ^ ) , (3) Total tons moved by trucking mode (V^) , (4) Average railway f r e i g h t rate (P g ) , r x. (.5) Average trucking f r e i g h t rate (Pj^) r (6) R a i l mode's average t r a n s i t time (t „), r A/ (.7) Truck mode's average t r a n s i t time ( t ^ ) , (8) Standard deviation of the r a i l mode's t r a n s i t -time d i s t r i b u t i o n ( K r £ ) , (9) Standard deviation of the truck mode's t r a n s i t -time d i s t r i b u t i o n (K^ £). In the remainder of t h i s chapter, the sources from which these data were obtained and the ways these data were used to construct the variables included i n the models are discussed i n d e t a i l . - 70 -(A) Freight Rate and Commodity Flow Data Waybill records of actual f r e i g h t shipments are kept i n Canada by various government agencies and by c e r t a i n c a r r i e r companies as c o n f i d e n t i a l information. Most of the available data are inappropriate for a multi-modal demand study such as t h i s , primarily due to the inconsistency of the data between modes with respect to the c l a s s i f i c a t i o n of commodities, d e f i n i t i o n s of l i n k s , units of measurements and methods of sampling. For example, both major railways (CN and CP) c l a s s i f y t h e i r i n t e r n a l records by the Standard Transportation Commodity Code (STCC) while S t a t i s t i c s Canada uses Standard Commodity C l a s s i f i c a t i o n (SCC) system for t h e i r annual survey of domestic fo r - h i r e trucking. Again both major railways use th e i r own systems of station numbers to record o r i g i n and destination of cargo whereas S t a t i s t i c s Canada uses Census Divisions defined primarily on the basis of the Standard Geographic Code (SGC). Fortunately, Peterson [1972] has developed the 'Canadian Freight Transportation Model (CFTM) data base' which uses common systems of c l a s s i f y i n g commodities and of designating geographical regions of o r i g i n and destination. The CFTM data base employs-; 78 commodity groupings (CFTM commodity codes) with cross-references to the STCC and SCC systems and 69 geographic regions (CFTM Canadian regions) with cross-references to the S t a t i s t i c s Canada Census Divisions, SGC and CN/CP station numbers. The CFTM data base has been maintained and updated by the Canadian I n s t i t u t e of - 71 -Guided Ground Transport. (See Graham [1975] for more technical d e t a i l s on the data base.) For each of the eight selected CFTM commodity groups that w i l l be l i s t e d l a t e r i n thi s chapter, the following information was developed from the CFTM data base using 1970 data. (.1) Total tons carried by railway mode on each l i n k , (2) Total tons ca r r i e d by trucking mode on each l i n k , (.3) Average railway f r e i g h t rate i n cents per ton-mile on each l i n k , (.4) Average trucking f r e i g h t rate i n cents per ton-mile on each l i n k . - 72 -(B) Distance of Link A major c i t y was chosen i n each CFTM region and was regarded as the centroid of the region. Each of a l l possible pairs of CFTM regions was treated as a l i n k composing the Canadian f r e i g h t transport network. 1 Both railway and highway distances were measured between the major c i t i e s of each 2 pair of regions from handbook sources. Shipments within each CFTM region were eliminated from the data set because (i) the measure of distance i s meaningless i n t h i s case and ( i i ) the intra-regional flow patterns cannot be represented well by a l i n k . Furthermore, those l i n k s having distances s i g n i f i c a n t l y d i f f e r e n t between the two modes were also eliminated from the data set since on these l i n k s distance would be the dominant factor deter-mining modal f r e i g h t rates per ton and t r a n s i t times and thus mode-choice decision. For each of the remaining l i n k s , the average of the 3 railway and highway distances was used as the distance measure of the l i n k (D^). - 73 -(C) Transit Time and Its V a r i a b i l i t y Since t r a n s i t time information i s not available i n the CFTM data base, other sources had to be employed. While the proper measure of o v e r a l l t r a n s i t time would be the time span between the receipt by the c a r r i e r of a request for service and the delivery of the shipment to consignee, no data was available that could provide a l l t h i s information. The railway's d a i l y car and t r a i n movement records show only the time from entry of a loaded car into the o r i g i n yard to the e x i t of the car from the destination yard on i t s way to delivery. The source for the trucking performance data also lacked information on elapsed time during pickup and delivery and the time between a shipper's request for service and actual pickup. Since t h i s non-transit service time required for railway service i s normally longer than that for trucking service, there would have been a downward bias of railway service time i f i t were omitted. However, as t h i s bias seems more or less constant across the cross^-sectional l i n k s , on the basis of the information obtained from selected interviews of shippers and railway o f f i c i a l s , the r a i l t r a n s i t times on a l l l i n k s were increased by two days to account for the p o t e n t i a l difference i n non-transit service time between the two modes. The t o t a l population of actual car movements recorded during October, 1970 for Canadian National Railways and during March, 19 71 for Canadian P a c i f i c Railways were used to calculate the average t r a n s i t time and standard deviation of - 74 -4 t r a n s i t time for each l i n k . Since i t was d i f f i c u l t to c l a s s i f y railway cars by the CFTM commodity groups, these were not computed separately for each commodity group. Rather a l l commodities were grouped into two categories; 'bulk' and 'nonrbulk^ and only two sets of average t r a n s i t time and standard deviation of t r a n s i t time were computed, one for the 'bulk' group and another for the 'non-bulk' group. Since, for some l i n k s , there was no car movement record during the observation periods, the following regression equations estimated from the data were used to generate average t r a n s i t times and the standard deviation of t r a n s i t 5 times for those l i n k s : (4.1) l n ( t 0) = -1.835 + 0.448 In(D „) ; R 2 =0.5019 (4.2) K r £ = 0.332 + 0.1727 t ^ ; R 2 = 0.3745 sample size = 1524 lin k s where t r £ = average r a i l t r a n s i t time i n days on l i n k l_, D r £ = railway distance of l i n k £_ i n miles, K r £ = standard deviation of r a i l t r a n s i t time d i s t r i b u t i o n on l i n k The trucking survey information obtained from Turner [l9 7 was used to generate the trucking performance data. Turner has obtained the actual t r a n s i t times of 12 74 truck shipments over 12 d i f f e r e n t l i n k s with varying distances from the record of two large trucking companies. Using t h i s information i t was possible to estimate the following regression equations: (4.3) ^ ( t h j ^ = -4.056 + 0 .7858 £n(D h £) ; R 2 = 0.9418 (4.4) K h £ = 0.3672 + 0.3617 t h £ ; R 2 = 0.7164 where t ^ £ = average truck t r a n s i t time i n days on l i n k l_, = highway distance of l i n k l_ i n miles, K, „ = standard deviation of truck t r a n s i t time h£ d i s t r i b u t i o n on l i n k These regression equations were used to generate the average t r a n s i t time and the standard deviation of the t r a n s i t time d i s t r i b u t i o n for the truck mode, for a l l l i n k s . - 76 -(D) Construction of the Variables Having explained the sources and the ways to obtain necessary data i n the preceding sections, t h i s section presents the formulae to compute the variables a c t u a l l y used as arguments of the cost and share functions s p e c i f i e d i n chapter I I I . CD Distance of l i n k : D = Dr£ + Dh£ a 2 where D ^ = railway distance of l i n k £_ i n miles, Dh£ = n i 9 n w a Y distance of l i n k _£ i n miles. C2) Railway average f r e i g h t rate per ton-mile on l i n k £_: „ r£ r £ Vr£* D£ where R . = t o t a l revenue of r a i l mode on l i n k £, r£ —' V „ = t o t a l tons moved by r a i l mode on l i n k £ (.3) Average trucking f r e i g h t rate per ton-mile on l i n k £_: Rh£ h £ Vh£' D£ where R^£ = t o t a l revenue of truck mode on l i n k £_, V^ £ = t o t a l tons moved by truck mode on l i n k _£_ (4) Railway share of revenue on l i n k £_: R. q = r£  br£ R„„+K r£'"h£ - 77 -(5) Trucking share of revenue on l i n k £ Sun = 1 - S „ h£ rl (6) Average r a i l speed i n miles per day on l i n k _£: D £ Z - * Til t „ Tl where . t . = average r a i l t r a n s i t time i n days on l i n k £_ (7) Average truck speed i n miles per day on l i n k l_i Dl Z *" hll t, „ h£ where t ^ £ = average truck t r a n s i t time i n days on l i n k (8) Reciprocal of c o e f f i c i e n t of v a r i a t i o n of r a i l t r a n s i t 7 time d i s t r i b u t i o n on l i n k I: 7 - ^l LTll ~ K Tl where K „ = standard deviation of r a i l t r a n s i t time Tl d i s t r i b u t i o n on l i n k £. (9) Reciprocal of c o e f f i c i e n t of v a r i a t i o n of truck t r a n s i t 7 time d i s t r i b u t i o n on l i n k l_: h 2 £ " Kh£ where K, „ = standard deviation of truck t r a n s i t time h J l d i s t r i b u t i o n on l i n k I, - 78 -So far i n t h i s chapter, the sources of data and the methods of computing the variables included i n the model have been discussed. Before c l o s i n g t h i s chapter, i t seems worthwhile to summarize the q u a l i f y i n g conditions for an observation. As mentioned previously i n various places, a l i n k should s a t i s f y the following conditions to q u a l i f y as an observation: (i) Both railway and trucking modes should a c t u a l l y share the t r a f f i c of the p a r t i c u l a r commodity group on that l i n k . ( i i ) Since a common measure of distance (Dn) i s to be used, distances by the two modes should be s i m i l a r to each other. ( i i i ) Both o r i g i n and destination regions of the l i n k should have a single major c i t y so that the major portion of the t r a f f i c a c t u a l l y flows between the two c i t i e s . 8 Eight d i f f e r e n t CFTM commodity groups were chosen for analysis from the 78 CFTM commodity groups such that they represent a wide variety of commodity a t t r i b u t e s . The number of l i n k s s a t i s f y i n g the above conditions among the t o t a l 4692 ( i . e . , 69 x 68) Canadian inter-regional l i n k s was d i f f e r e n t from commodity to commodity. These are l i s t e d i n Table (4-1). These observations are used to estimate the cost functions and modal share functions that are to be reported i n chapters V and VI. - 79 -Table 4-1, Selected CFTM Commodity Groups and Number of Observations CFTM Commodity Name of Commodity Number of Code No. : Group - ' Observations CFTM14 F r u i t s , vegetables 133 l i n k s and edible foods CFTM52 Lumber including 52 li n k s f l o o r i n g CFTM61 Chemicals 86 l i n k s CFTM6 6 Fuel O i l 65 li n k s CFTM69' - Refined petroleum 77 l i n k s products CFTM71 Stee l , irons and 151 li n k s a l l o y s CFTM75 Metal fabricated 137 li n k s basic products CFTM78 Non-metallic basic 156 l i n k s products - 80 -Footnotes for Chapter IV: 1. Due to the d e f i n i t i o n of l i n k adopted i n t h i s study, there - i s a possible danger that r a i l and truck t r a f f i c may originate at d i f f e r e n t c i t i e s i n an o r i g i n region and/or go to d i f f e r e n t c i t i e s i n a destination region. To avoid t h i s problem as much as possible, l i n k s having no major c i t y or else two or more c i t i e s of s i m i l a r size at either end of the l i n k were eliminated from the data. 2. Mileage information l i s t e d i n CN/CP Regional Time Tables was used to measure railway distances of l i n k s . An O f f i c i a l Canadian Highway Map was used to measure highway distances of l i n k s . 3. As long as a shipper gets his cargo moved from one location to another, the fact that a c e r t a i n mode has a longer or shorter distance than the other mode has no d i r e c t s ignificance to him other than i t s impacts on the ton-mile f r e i g h t rate and q u a l i t y attributes which are also computed using the common measure of distance (D^). However, the l i n k s whose railway and highway distances are s i g n i f i c a n t l y d i f f e r e n t from one another were eliminated from the data because the mode with a shorter distance would generally dominate t r a f f i c due to i t s favorable impact on f r e i g h t rates per ton and quality a t t r i b u t e s . 4. When the author t r i e d to c o l l e c t the data for t h i s research, these were the only t r a n s i t time records available i n the CFTM data base. Fortunately, there was no labor dispute during the periods. 5. Several a l t e r n a t i v e functional forms were estimated from the sample of 1524 observed l i n k s and the equations (4,1) and (4,2) were chosen primarily on the basis of goodness of f i t . 6. As i n the case of the railway mode, several a l t e r n a t i v e functional forms were t r i e d , and the equations (4,3) and (4,4) were chosen on the basis of goodness of f i t . 7. As another indicator of transit-time r e l i a b i l i t y , the variable c a l l e d "percentage of shipment that took longer than 3/2 times of mean t r a n s i t time" was obtained. However, some experiments indicated the superiority of the variable "reciprocal of c o e f f i c i e n t of v a r i a t i o n of t r a n s i t time d i s t r i b u t i o n ( Z ^ ^ ) " over t h i s variable. 8. A detailed l i s t of commodities included i n the eight CFTM commodity groups i s available i n Appendix 4A with cross-references to the Standard Commodity C l a s s i f i c a t i o n (SCC) code. CHAPTER V ESTIMATION AND HYPOTHESIS TESTING This chapter i s organized as follows. The econometric aspects of estimating the models and the algorithms chosen for the estimation are discussed i n section (A). Section (B) lays out the plan for hypothesis testing and discusses a the o r e t i c a l problem i n tes t i n g amongst the non-nesting models. A summary table of the models chosen for the eight CFTM commodity groups as a r e s u l t of the hypothesis t e s t i n g i s presented i n section (C). F i n a l l y , the tables of test s t a t i s t i c s and the detailed reports on hypothesis t e s t i n g , including the l i s t s of intermediate re s u l t s are presented i n Appendix 5A. 81 -- 82 -(A) Econometric Aspects of Estimation The four al t e r n a t i v e models derived and s p e c i f i e d i n chapter III are similar i n that each cost function has the natural logarithm of average transportation cost per ton-mile as i t s dependent variable and each demand function has modal expenditure share (revenue share from the c a r r i e r ' s viewpoint) as the dependent variable. Empirical implementation requires that the cost and share functions be imbedded i n a stochastic framework because, i n practice, there are errors i n adjustment to the cost-minimizing expenditure shares and thus to the cost-minimizing unit transportation cost. For each alternative model, the additive disturbance i n the ith equation at l i n k l_ i s defined as E^(£) . Further, the column vector of disturbances at l i n k £_ i s defined as: (5,1) E*(£) = [ E c (£), E r U ) , E h (£)], £ = 1,2,...,L. and the associated disturbance covariance matrix as 9,*. Since the two modal shares always sum to one at each l i n k , the sum of the disturbances of the two modal share equations i s zero at each observation £. This implies that the disturbance covariance matrix of the f u l l three-equation system, fi*, i s singular and non-diagonal. I f the estimators of parameters are to be e f f i c i e n t , t h i s disturbance covariance matrix must be taken into account. Due to the s i n g u l a r i t y , the determinant of the disturbance covariance matrix i s zero, and consequently the l i k e l i h o o d function i s undefined."*" There-- 83 -fore, either one of the two modal share equations should be dropped so that the remaining two equations can be estimated. Since the maximum l i k e l i h o o d estimates are inv a r i a n t to the equation deleted, the disturbance of the trucking revenue share equation, E^(£), i s to be dropped from a l l the alt e r n a t i v e models. Define the new disturbance vector as E(£) = [E c(£), E r ( £ ) ] , and assume that the disturbance vector E(£) i s independently joint-normally d i s t r i b u t e d with a mean vector of zeroes and non-singular covariance matrix 2, for a l l £ = 1,2,...L: i . e . NIID. The logarithm of the l i k e l i h o o d 2 function can be written as: (5,2) lnX.= - L [ln(2t) + l ] - | In |0|. where L i s the number of oberservations (links) used. The !parameters of the four a l t e r n a t i v e models are estimated using a nonlinear m u l t i v a r i a t e maximum l i k e l i h o o d procedure. Statistica-1 inference i s based on the asymptotic l i k e l i h o o d r a t i o c r i t e r i o n . Of course, the t e s t r e s u l t s are in v a r i a n t to the equation deleted. The computation i s done by the algorithm developed by Berndt-Hall-Hall-Hausman [1974]. The algorithm used f o r the nonlinear models i s e s s e n t i a l l y a combination of the i t e r a t i v e Zellner e f f i c i e n t (IZEF) procedure with the Gauss-Newton method of nonlinear l e a s t squares, and i s equivalent to maximum l i k e l i h o o d (ML) estimation. Since estimation of the nonlinear systems i s c o s t l y , i t was i n practice essential, to set somewhat loose convergence c r i t e r i a . ' - 84 -The convergence c r i t e r i a used are that: (i) The largest change i n parameter estimates from one i t e r a t i o n to another should be no greater than 0.5%, and ( i i ) The largest absolute deviation of the the elements 3 of the transformed residual covariance matrix from the i d e n t i t y matrix should be no greater than 0.005, Convergence was achieved for a l l alternative models. This convergence c r i t e r i o n i s far looser than those normally used i n nonlinear programming (.NLP) algorithms. Therefore, 4 as a means to check the accuracy of computation, FLETCH and 5 SIMPLX at the University of B r i t i s h Columbia were used to estimate the parameters of the cost functions by the c l a s s i c a l equation-by-equation least squares method. The convergence c r i t e r i a used were 10.ET-10 for the FLETCH algorithm and 10.E-6 for the SIMPLX algorithm. Convergence was achieved and the Hessian matrix of second order p a r t i a l derivatives s a t i s f i e d p o s i t i v e definiteness for a l l cases. By t h i s experiment, i t was confirmed that at least the signs of parameter estimates were exactly the same between Berndt-Hall-Hall^Hausman algorithm and the NLP algorithms. - 85 -(B) The Plan for Hypothesis Testing Eleven d i f f e r e n t sub-models are hypothesized i n section (E) of chapter III for each commodity group. Recall that (i) models B, C and D are nested i n model A, and ( i i ) model D i s nested also i n model C, but C i i i ) there i s no nested r e l a t i o n between model B and either model C or model D. Due to the non-nestedness amongst some models, the following complications arise i n te s t i n g hypotheses: 1, C l a s s i c a l testing procedures cannot be used without a q u a l i f i c a t i o n . 2. There i s no guarantee that i n t r a n s i t i v i t y amongst the r e s u l t s of i n t e r - r e l a t e d tests does not occur. Therefore, hypothesis t e s t i n g i s designed such that the occurence of non-nestedness i s minimized. Three a l t e r n a t i v e plans are considered: 1. A complete test requires t e s t i n g a l l possible combinations of two sub-models, i . e . , 55 separate tests i n t o t a l . 2. A two-stage test f i x i n g the type of sub-model at the f i r s t stage: This would require 12 tests i n the f i r s t stage, and another 6 tests i n the second stage. 3. A two-stage test f i x i n g the type of macro model at the f i r s t stage: This would require 12 tests i n the f i r s t stage, and another 6 tests i n the second stage, The f i r s t plan i s the worst because i t has the largest number of non^ nested hypotheses and a high p r o b a b i l i t y of i n t r a n s i -t i v i t y of test r e s u l t s . Under the second plan, non-nested cases are present both i n the f i r s t and i n the second stages, whereas under the t h i r d plan, non-nested cases are avoided - 86 -completely i n the f i r s t stage but some arise i n the second stage. There i s almost no danger of i n t r a n s i t i v i t y of test r e s u l t s i n either the second or t h i r d plan. The t h i r d plan was chosen for te s t i n g hypotheses about the models. The plan i s summarized i n Figure (5-1). In the f i r s t stage, decisions have to be made about which of the three sub-models i s most appropriate to represent each (macro) model for the second stage t e s t s . This amounts to determining the independent variables to be included i n each (macro) model: For a given (macro) model, te s t (i) determines whether or not "speed" and " r e l i a b i l i t y " variables together are s t a t i s t i c a l l y s i g n i f i c a n t . Tf the test r e s u l t i s negative, then test ( i i ) i s conducted to see i f "speed" alone i s a s t a t i s t i c a l l y s i g n i f i c a n t factor. I f the r e s u l t of tes t ( i i ) i s also negative (or positive) then sub-model 3 (sub-model 2) i s chosen to represent the (macro) model i n the second-stage te s t s . Were the r e s u l t of test (i) p o s i t i v e , then test ( i i i ) i s conducted to see i f " r e l i a b i l i t y " i s s t a t i s t i c a l l y s i g n i f i c a n t even aft e r "speed" i s included i n the model. I f the r e s u l t of test ( i i i ) i s negative (or p o s i t i v e ) , then sub-model 2 (sub-model 1) i s chosen to represent the (macro) model i n the second-stage t e s t s . This procedure i s repeated for each of the four alternative (macro) models hypothesized i n chapter I I I . In the second-stage te s t s , a decision i s made about which of the four sub-models chosen i n the f i r s t - s t a g e tests i s the - 87 -Figure (5-1) ^ p ' x a n f Hypothesis Testing 1 1 F i r s t Stage ** For each (macro) model, the following sub-models are to be tested. Sub-model 1 (including speed and| r e l i a b i l i t y ) ( i i i ) H, Sub-model 2 (excluding r e l i a b i l i t y ) H Sub-model (excluding speed and| r e l i a b i l i t y ) Second Stage *** Model A (general model) (4) Model C (wi,th mode-specific hedonic aggregators) (6) (3) •Model B (independent of distance) (2) Model D (with i d e n t i c a l hedonic aggregators)| ** Note that some of the tests w i l l become redundant as the r e s u l t s of the preceding tests are obtained. An arrow l i n k s a p a i r of n u l l hypothesis (at origin) and al t e r n a t i v e hypothesis (at destination). *** In each te s t , the model with fewer parameters than the other becomes the n u l l hypothesis. Therefore, the n u l l hypothesis depends on which sub-models were chosen i n the f i r s t stage tests. - 88 -most appropriate model to use. Since the number of parameter estimates of each (macro) model depends on the sub-model chosen i n the f i r s t - s t a g e tests, i n each t e s t , the model with fewer parameters becomes the n u l l hypothesis. Therefore, the procedure for conducting the second—stage tests depends largely on the sub-models chosen i n the f i r s t - s t a g e t e s t s . This i s i l l u s t r a t e d l a t e r i n section (C) using the empirical results of CFTM14 ( f r u i t s , vegetables and edible foods) as an example. Note that some of the tests i n both stages w i l l become redundant depending on the re s u l t s of the preceding te s t s . The asymptotic l i k e l i h o o d r a t i o c r i t e r i o n i s used to discriminate amongst the three sub-models 1, 2, and 3, i n the f i r s t stage tests. Note that the sub-models 2 and 3 are nested i n sub-model 1, and sub-model 3 i s again nested i n sub-model 2. I t requires a q u a l i f i c a t i o n to use the same test method i n the second stage tests because of the possible presence of non-nested cases depending on the sub-models chosen i n the f i r s t stage t e s t s . In the non-nested cases, a p r i o r i , i t i s not i n general possible to choose one from a l t e r n a t i v e models on the basis of c l a s s i c a l t e s t s t a t i s t i c s . However, one can discriminate between the non-nested models a p o s t e r i o r i using Bayesian c r i t e r i o n . Let 0 1 and 0 2 denote the parameter vectors corresponding to models (1) and (.2) , respectively, and l e t fi^ and denote the associated disturbance covariance matrices. - 89 -Let P(l) and P(2) denote the p r i o r p r o b a b i l i t i e s that each model holds. Then P [6^, ^ || model (1)J and P [0 2, ^ || model (2)] are the p r i o r d i s t r i b u t i o n s for the parameters of each model. The j o i n t posterior d i s t r i b u t i o n s of the models and t h e i r parameters, given data X, can be written as: (5,3) p ( e i , n i , i |x) ccX(x|e i, n ±) p ( e ± , Q±\\) P ( i ) i » 1,2. Furthermore, Press [l972, p. 167 ] has shown that a d i f f u s e 7 prxor density'makes the mode of the posterior d i s t r i b u t i o n correspond to the maximum l i k e l i h o o d estimator. Therefore, i f one employs the same d i f f u s e p r i o r d i s t r i b u t i o n s for the two models involved i n a hypothesis t e s t , then the l i k e l i h o o d r a t i o t e s t i s equivalent to comparing the modes of the two posterior d i s t r i b u t i o n s . In addition, when the number of parameters of the two alternative models i s the same, the maximum mode of the two posterior d i s t r i b u t i o n s can be obtained simply by comparing the values of the l i k e l i h o o d functions evaluated at the maximum l i k e l i h o o d estimates of 6. and Q.. l l In conclusion, using Bayesian c r i t e r i o n , one can choose a p o s t e r i o r i which of the models i s most l i k e l y to have generated the observed data, through a l i k e l i h o o d r a t i o test for the case involving d i f f e r e n t numbers of parameters, and by comparing the values of the l i k e l i h o o d functions for cases involving the same number of parameters. - 90 -The values of the logarithm of the l i k e l i h o o d functions obtained by estimating eleven alternative sub-models are reported i n Appendix 5A i n tables (5A-1) to (5A-8) , one for each CFTM commodity group. T h e i l [l9 71, p. 3961 has shown that, asymptotically, -2Z-nA (A being the l i k e l i h o o d ratio) has a Chi-square d i s t r i b u t i o n with appropriate degrees of 2 freedom. These x ~ s t a t i s t i c s and the r e s u l t s of the hypothesis t e s t i n g are reported i n Appendix 5A i n tables (5A^9) to (5A-16). A significance l e v e l (probability of type I error) of 0.05 i s used for a l l hypothesis t e s t i n g . - 91 -(C) The Chosen Models In this section, (i) the t e s t i n g procedure for choosing the model i s i l l u s t r a t e d using the f r u i t s , vegetables and edible foods (CFTM14) as an example, and ( i i ) the l i s t of the finally-chosen models i s presented. For convenience of the discussion, Tables (5A-1) and (5A-9) are reproduced here. Table (5A-1) reports for each of the eleven sub-models estimated from the CFTM14 data, the number of parameters estimated, the logarithm of the l i k e l i h o o d function evaluated at the ML parameter estimates (hereafter c a l l e d "log of l i k e l i h o o d " or InX) and R 2 values of the cost and revenue share functions. The f i r s t - s t a g e tests ( r e c a l l the schematics for the f i r s t - s t a g e hypothesis te s t i n g outlined i n Figure 5-1) are conducted i n part (A) of Table (5A-9). This f i r s t - s t a g e procedure involves, for each (macro) model, t e s t i n g amongst the three sub-models, i n which d i f f e r e n t sets of independent variables are included. For a given (macro) model, the f i r s t -stage tests are composed of three t e s t s : sub-model 3 vs. sub-model 1, sub-model 3 vs. sub-model 2, and sub-model 3 vs. sub-model 1. For (macro) model A, test (i) compares sub-models (A-3) and ( A - l ) . The test s t a t i s t i c 30.14 {-2ln\ where X i s the l i k e l i h o o d ratio) reported i n table (5A-9) i s computed as: - 92 -Table 5A-1, Test S t a t i s t i c s for Commodity Group CFTM14: F r u i t s , Vegetables and Edible  Foods (using 133 l i n k observations) Model and No. of free 7 Y, R 2 R 2 Sub-model Parameters n** c s General Model (A) (A-l) 28 -635.822 0 .8714 0 .3754 (A-2) 15 -644.605 0 .8596 0 .3564 (A-3) 6 -650.892 0 .8460 0 .3536 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -641.1 0 .8595 0 .3590 (B-2) 11 -647.103 0 .8552 0 .3502 (B-3) 4 -682.177 0 .7994 0 .0215 Model with Mode -S p e c i f i c Hedonic Aggregators (C) (C-l) 9 -646.74 0 .8589 0 .3636 (C-2) 7 -658.267 0 . 8525 0 .3007 CC-3) * 5 -670.689 0 .8270 0-.2722 Model with Identical Hedonic Aggregators (D) CD-I) 7 -657.646 0 .8511 0 .3016 (D-2) 6 -658.369 0 .8523 0 .2999 (D-3)* 5 -670.689 0 .8270 0 .2722 ln3C = The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 2 R - R value for the translog cost function, c 2 2 R = R value for the modal share functions. s *Models (C-3) and (D-3) are i d e n t i c a l . Table 5A-9, Hypotheses Testing for CFTM14: F r u i t s , Vegetables and Edible Foods (A) Test amongst the three sub-models ( F i r s t stage t e s t s ) : t e s t s t a t i s t i c (-2ZnX) and degrees of freedom Degrees X~ c r i t i c a l Model A Model B Model C Model D of freedom value at a=.05 (i) H Q : sub-model H N : sub-model 3 (22) * (18) (4) (2) 1 3.841 1 30.140** 82.154 47.89'8 26.086 2 5.991 ( i i ) 1 H Q : sub-model H ^ : sub-model 3 2 (9) 12.574 (7) 70.148 (2) 24.844 (1) 24.64 4 7 9 9.488 14 .067 16.919 i ( i i i ) H Q : sub-model 2 (13) (11) (2) (1) 11 19.675 H ^ : sub-model 1 17.566 12 .006 23.054 1.446 13 22.362 i Chosen sub-model (A-3) (B-2) (C-l) (D-2) 18 28.869 No. of free parameters 6 11 9 6 22 33.924 In'i, -650.892 -647,103 <--646.74 -658.369 R 2 c ,8460 ? 8552 .8589 .8523 * Figures reported i n parentheses are the degrees of freedom for the respective t e s t s . ** Figures reported on the same l i n e as H^^ are the test s t a t i s t i c s for the respective te s t s . (B) Tests amongst the chosen sub-models (Second stage tests) 2 Test (1) choice between models (D-2) and (A-3) (2) V H l : model model (D-2) (B-2) (3) H o : model model (D-2) (C-l) (4) V H l : model model (A-3) (C-l) (5) choice between models (C-l) and (B-2) (6) H Q : model (A-3) 1^: model (B-2) Test s t a t i s t i c (-2ln\) * * * 22.532 23.258 8. 304 * * * 6,126 Model (C-l) i s f i n a l l y chosen for use. Degrees of Freedom X- c r i t i c a l value at a=.05 Test Result favours (A-3) due to higher InX value. 9.236 6.251 6.251 9.236 favours (B-2) favours (C-l) favours (C-l) favours (C^l) because i t has higher InaC-value and smaller number of parameters favours (A-3) *** The two models that are compared have the same number of parameters. In th i s case, the model with a larger value of the l i k e l i h o o d function was chosen. ^"95 -- 2 l n \ = 2 (ZnJ£for alternative hypothesis (H^) - Zn«C for n u l l hypothesis (H^)) = 2 (Zn<£for model (A-l) - Zn£ for model (A-3)) = 2 (-635.822 - (-650.892)) = 30.14 The test s t a t i s t i c has 22 degrees of freedom because the alte r n a t i v e hypothsis has 22 more parameters to estimate than the n u l l hypothesis. Since the test s t a t i s t i c 30.14 i s 2 smaller than the c r i t i c a l value of the x d i s t r i b u t i o n with 22 degrees of freedom at the 5% l e v e l of si g n i f i c a n c e , one cannot r e j e c t sub-model (A-3). Exactly the same procedure i s used to conduct test ( i i ) , which favours sub-model (A-3) over sub-model (A-2). Since the two test r e s u l t s consistently favoured sub-model (A-3) over sub-models (A-l) and (A-2), the sub-model (A-3) was chosen to represent (macro) model A i n the second-stage t e s t s . Note that t e s t ( i i i ) i s i n f a c t redundant i n t h i s p a r t i c u l a r case. The re s u l t s of the f i r s t - s t a g e tests amongst the three sub-models of (macro) model B are as follows: te s t (i) : model (B-l) i s favoured over model (B-3), test ( i i ) : model (B-2) i s favoured over model (B-3), t e s t ( i i i ) : model (B-2) cannot be rejected i n favour of model (B-l ) . Therefore, sub-model (B-2) was chosen to represent (macro) model B i n the second-stage t e s t s . Using a similar procedure, sub-models (C-l) and (D-2) were chosen to represent (macro) - 96 -models C and D, respectively, i n the second-stage t e s t s . The purpose of the second-stage tests i s to choose the most appropriate model to use among the four a l t e r n a t i v e models chosen i n the f i r s t - s t a g e t e s t s . According to the schematics for the second-stage hypothesis t e s t i n g outlined i n Figure 5-1, the tests for t h i s p a r t i c u l a r commodity group (CFTM14), are carr i e d out as i n Figure (5-2). 2 The test s t a t i s t i c s (-2£nA), degrees of freedom, x -c r i t i c a l values at the 5% l e v e l of si g n i f i c a n c e , and the test r e s u l t s for the second-stage tests are presented i n part (B) of Table (5A-9). The results of the hypothesis te s t i n g are summarized at the bottom of Figure 5-2. Test (1) favours model (A-3) over model (D-2) because the former has the higher Zn^,than the l a t t e r , and these two models have the same number of parameters. Tests (2), (3), (4) and (6) are conducted according to the standard procedure for 2 X - t e s t . Test (5) favours model (C-l) over model (B-2) because the former has fewer parameters but attained a higher ln<L than the l a t t e r . Since model (C-l) i s favoured over a l l three other models, (D-2), (A-3) and (B-2), the f i n a l choice i s model ( C - l ) . Model (C-l) i s used to obtain various empirical r e s u l t s about CFTM14 (fruits., vegetables and edible foods) i n the following chapters. E s s e n t i a l l y the same testing procedure i s used to make the choice of the model for each of the remaining commodity groups. The models f i n a l l y chosen are shown at the bottom of - 97 -(Figure (5-2) Second-Stage Testing for CFTM14 Test* Test Result (1) (A-3) (D-2) favours (A-3) (2) CD-2) (B-2) favours (B-2) (3) (D-2) (C-l) favours (C-l) (4) (A-4) (C-l) favours (C-l) (.5) (C-l) (B-2) favours (C-l) (6) (A-3) (B-2) favours (A-3) * Note.that some tests are redundant; for t h i s p a r t i c u l a r commodity, tests (.1) , (2) and (6) are redundant. - 98 -Tables A5-9 to A5-16 i n Appendix 5A. Some models are too complicated to see i n t u i t i v e l y whether or not the signs of some parameter estimates are reasonable. However, the reasonable-ness of those models can be judged a p o s t e r i o r i after computing the e l a s t i c i t i e s of demand with respect to price and q u a l i t y variables which are reported i n chapter VII. As w i l l be noticed i n chapter VII, a l l the chosen models other than that for CFTM71 (ste e l , iron and a l l o y s , etc.) seem to give reasonable estimates for various e l a s t i c i t i e s . For commodity group CFTM71, model A-2 (the general model with p r i c e , speed and distance variables only) was chosen as the r e s u l t of hypothesis t e s t i n g . However, the chosen model i s found to be unusable because a l l the speed e l a s t i c i t y estimates computed from the model have the wrong signs. A careful examination of Tables 5A-6 and 5A-14 indicates that model A-3 (the general model with price and distance variables only) i s the next best model. Therefore t h i s model i s used instead of model A-2. Fortunately, the values of parameter estimates of model A-3 were very close to the corresponding parameters of model A-2, and consequently, the results about the intermodal s u b s t i t u t i -b i l i t y and price responsiveness of demands which are to be discussed i n chapter VII do not vary s i g n i f i c a n t l y between the two models. (As w i l l be mentioned i n chapter VII, eventually the empirical re s u l t s for t h i s commodity group (CFTM71) w i l l not be used i n this thesis because of the c o u n t e r - i n t u i t i v e l y high e l a s t i c i t i e s of r a i l - t r u c k substitution predicted by the model.) - 99 -The models f i n a l l y chosen for use i n the remaining chapters are summarized i n Table 5-1. - 100 -Table (5-1), L i s t of the Chosen Models CFTM14 Model (C-l) ( f r u i t s , vegetables & edible foods) CFTM52 (lumber including flooring) Model (C-3) CFTM61 Model (C-3) (chemicals) CFTM66 Model (C-3) (fuel o i l excluding gasoline) CFTM69 Model (A-3) (other refined petroleum products) CFTM71 (steel, iron & alloys) CFTM75 (metal fabricated basic products) CFTM78 (non-metallic basic products) Model (A-3) Model (C-l) Model (C-l) cost function with mode-s p e c i f i c aggregators as functions of price, speed, r e l i a b i l i t y and distance. cost function with modal aggregators as functions of price and distance only. same as the case of CFTM52. same as the case of CFTM52. general model, defined. i n prices and distance" only. same as the.case of CFTM69 ; « _ .;s -• and ^ * P » » - :• ? same as the case of CFTM14. same as the case of CFTM14. - 101 -Footnotes f o r Chapter V: 1. The l i k e l i h o o d function for the f u l l model i s undefined because the inverse of the cross-equation covariance matrix i s undefined as i s shown below: ft = r-t t .e e e e t _ _ £ £, c c c r c h t t E E £ £ t c'r " r _ r e r e h t t EE, £ E t c"h "r"h e h e h 1 L t t t E E £ £ -£ £ :c~c c r c r „t t t E E E E -EE c r r r r r t-£ £ -£ £ -£ £ ^ c r r r r r where e' = [e (1) , E (2 ) , . . . , E (L)] C *o e l = t£ r(l), e r ( 2 ) , . . . f e r ( L ) ] e£ [£ h(.D, £ h(2) , . . .,£ r(L) ] and E, U) = -£ U) h r f o r a l l I =1,2,...,L . ft l r t , . , t .2 . t ,2, t f t , t , 7 3 [ e c e c { C e r e r ) " C " e r e r ) } ~ e ~ e - { e ~ £ - ( e ~ £ - ) c r c r r r ete (£ t£ -) } c r r r -E^ E {-£t£ (£^"ej) + £T£ (ET£ )}] c r c r r r c r r r / J 1 * - —=• [0] = 0 ft i s singular IT * - l Therefore, ft i s undefined, and consequently, oC = (2TT) 2 1 (3)L + - | , L +-1 |ft | zexp{- i Z q ft e £} £=1 i s undefined - 102 2. The logarithm of the l i k e l i h o o d function i n (5,2) i s derived below: For a two-equation model, the l i k e l i h o o d function f o r L_ observations can be written as: - 1 - L (1) X = (2TT) 2 | n | 2 exp{- y E eU)^'^,}. 1=1 where elH)t = [e1U),e2U)] By taking the natural log of both sides, i t can be written as: (2) In o£= - Lin (2TT) - y An| ft| - y Z e (A) • ' e (£) z 1=1 = - L[ln (2TT) + 1] - j'2n-|-.fi| The above equality holds because of the following: r Let .-I 11 12 w w , 12 22 , then Z eU)t£f1e(L£) = Z (e U) 2* 1 1 + 2e U)e-U)w12 + e ?(£) 2w 2 2} a=i i=i 1 i. ^ t 11 x . t 12 ^ t 22 = e^ e^ w + 2e^e2w + e2e2 Meanwhile, ft = 1 L e^ 1 1 e l £ 2 e l e 2 t e2 e2 1 r T l L rete 2 2 t L ~ e l e 2 El f c2 e l 1 ^2 He^) < e2 e2 } (Ele2) L Z £=1 t L eU) J T l e C U = • t t 2 (e 1e 1) (e 2 e2^ " ^ e ^ t t t t [ e 1 e 1 ( e 2 e 2 ) + 2 e 1 e 2 ( - e 1 12' + 4 s 2C ci £l^ = 2L - 103 r The transformed covariance matrix i s the product between the inverse of the covariance matrix from the previous i t e r a t i o n and the covariance matrix from the current i t e r a t i o n . FLETCH i s a subroutine for minimizing a function by_ a quasi-Newton method based on Fletcher's algorithm | 1 9 6 7 ] . SIMPLX i s a subroutine for function minimization using a Simplex algorithm developed by Nedler and Meads [1965] . For example, the phrase "model B i s nested to model A" means that model B i s a special or l i m i t i n g case of Model The term "diffuse p r i o r density" means a uniform p r i o r density over the r e a l l i n e . When the analyst does not have any p r i o r information about the parameters of model he may assume a uniform p r i o r d i s t r i b u t i o n for P ( 9 i , f t i | i ) P ( i ) . CHAPTER VI GENERAL RESULTS This chapter examines the mean values of the important variables, and describes the general results of t h i s study using the chosen models reported i n Table 5-1. Section A reports the mean values of some important variables with explanatory notes. Section B comments on the general implications of the chosen models without s p e c i f i c reference to the parameter estimates. In Section C, the para-meter estimates for the models are reported, and the signs and s t a t i s t i c a l significance of some important parameter estimates are examined as well. And f i n a l l y , Section D summarizes the discussion i n SectionsB and C i n order to present general findings from the chosen models. - 104 - 105 -(A) Mean Values of Some Important Variables I t i s b e n e f i c i a l to examine the data before any s p e c i f i c r e s u l t s of a model are studied. Although, eventually, the usage of a model i s es s e n t i a l because of the j o i n t r e l a t i o n s amongst the independent variables, the data themselves are often useful for i n t u i t i v e interpretations of the conclusions that can be drawn from the model. Since i t i s neither necessary nor possible"'' to l i s t the entire data, only the mean values of some important variables such as shares of revenue, shares of tonnage, shares of ton-miles, average f r e i g h t rates per ton-mile, average lengths of haul and the simple averages of speed and r e l i a b i l i t y variables are l i s t e d i n Table 6-1 by each commodity group. In the remainder of thi s section, the information contained i n the table i s discussed. The following observations may be made from the tonnage shares and average lengths of haul of the two modes: (i) The r a i l mode carr i e d a r e l a t i v e l y small portion of the t o t a l tonnages of • CFTM14 ( f r u i t s , vege-tables and edible foods), CFTM69 (refined petro-leum products), CFTM75 (metallic basic products) and CFTM78 (non-metallic basic products ) . More-over, the r a i l and truck modes tended to concen-trate on long-haul and short-haul t r a f f i c , r espectively. ( i i ) The t o t a l tonnages of CFTM61 (chemicals) and CFTM71 (stee l , iron and a l l o y s / etc>) were shared almost equally by the two modes, with a heavy concen-t r a t i o n of the railway mode on the longer haul movements. As w i l l be mentioned formally i n chapter VII, the r a i l and truck shares of t r a f f i c for CFTM^yj reported i n Table 6-1 misrepresent ' - . what has r e a l l y happened i n the fr e i g h t market-Table 6-1, Mean Values of Some Important Variables Commodity Group CFTM14 CFTM52 CFTM61 CFTM66 CFTM69 CFTM71 CFTM75 CFTM78 Variables Fruits, Vegetables & Edible Foods Lumber, Including Flooring Chemicals Fuel Oil Except Gasoline Refined Petroleum Products Steel, Iron & Alloys Basic Metallic Products Non-Metallic Products Revenue Railway 0.31 0.49 0.60 0.73 0.32 0.44 0.22 0.32 share Trucking 0.69 0.51 0.40 0.27 0.68 0.56 0.78 0.68 Tonnage Railway 0.24 0.66 0.56 0.68 0.20 0.47 0.35 0.34 share Trucking 0.76 0.34 0.44 0.32 0.80 0.53 0.65 0.66 Ton-mile Railway 0.50 0.62 0,70 0,75 0.32 0.63 0.49 0.51 share Trucking 0.50 0.38 0.30 0.25 0.68 0.37 0.51 0.49 Average freight rate in cents per ton-mile Railway Trucking 2.43 5.40 2.16 3.70 2,39 3.86 1,98 2.11 2.17 2.15 2.18 4.70 2.27 7.71 2.30 5.25 * speed in Railway 126.0 77.0 77.0 51.0 92.0 109.0 126.0 107.0 miles per day Trucking 230.0 204.0 202.0 188.0 209.0 219.0 228.0 219.0 *relia- Railway 4.39 4.14 4.14 4.24 4.10 4.25 4.33 4.34 bility Trucking 1.90 1,66 1.65 1.45 1.70 1,80 1.89 1.80 average Railway 941,0 351.0 389.0 228.0 450.0 546.0 640.0 552.0 length of haul in Trucking 296,0 401.0 207.0 165.0 236.0 292.0 351.0 270.0 miles *These figures may be misleading in some respects since they are the unweighted mean over a l l links. "Reliability" is measured in terms of "mean transit-time/standard deviation of transit-time distribution." - 107 -i n t e r m s o f i n t e r - m o d a l c o m p e t i t i o n . T h i s was c a u s e d b y t h e a g g r e g a t i o n o f a w i d e v a r i e t y o f h e t e r o g e n e o u s c o m m o d i t i e s r a n g i n g f r o m p r i m a r y m e t a l s t o c o n s t r u c t i o n h a r d w a r e s i n t o t h i s c o m m o d i t y g r o u p ( s e e A p p e n d i x 4A f o r a d e t a i l e d l i s t o f c o m m o d i t i e s i n c l u d e d i n CFTM71). T h i s i s a d e f e c t o f t h e CFTM d a t a b a s e . ( i i i ) The r a i l mode c a r r i e d a l a r g e r p r o p o r t i o n o f t o t a l t o n n a g e s o f CFTM52 ( l u m b e r i n c l u d i n g f l o o r i n g ) a n d CFTM66 ( f u e l o i l ) . S u r p r i s i n g l y , t h e a v e r a g e l e n g t h o f h a u l o f t h e r a i l w a y t r a f f i c f o r CFTM52 was s h o r t e r (351 m i l e s ) t h a n t h a t o f t r u c k t r a f f i c (401 m i l e s ) . A c a r e f u l e x a m i n a t i o n o f t h e raw d a t a r e v e a l e d t h a t t h i s may h a v e h a p p e n e d b e c a u s e t r u c k s c a r r i e d a m a j o r p o r t i o n o f medium-/ l o n g - h a u l " f l o o r i n g " t r a f f i c . C l e a r l y t h i s i s a l s o a d e f e c t o f t h e d a t a a g g r e g a t i o n i n t h e CFTM d a t a b a s e . E x c e p t f o r t h e c a s e o f CFTM69 ( o t h e r r e f i n e d p e t r o l e u m p r o d u c t s ) , t h e a v e r a g e f r e i g h t r a t e o f t r u c k i n g s e r v i c e was h i g h e r t h a n t h a t o f r a i l w a y s e r v i c e . The a v e r a g e t r u c k i n g r a t e s f o r CFTM14, CFTM71, CFTM75 and CFTM78 w e r e more t h a n t w i c e a s h i g h a s t h e a v e r a g e r a i l w a y r a t e s . T h i s i s p r o b a b l y due t o t h e t r u c k mode's c o n c e n t r a t i o n on r e l a t i v e l y s h o r t - h a u l t r a f f i c . A s u r p r i s i n g a s p e c t o f t h e d a t a i s t h a t t h e a v e r a g e r a i l f r e i g h t r a t e p e r t o n - m i l e f o r CFTM69 ( o t h e r r e f i n e d p e t r o l e u m p r o d u c t s ) was m a r g i n a l l y h i g h e r (£2.17) t h a n t h e a v e r a g e t r u c k i n g r a t e (£2.15) a l t h o u g h t h e a v e r a g e l e n g t h o f h a u l f o r t h e r a i l t r a f f i c was l o n g e r (450 m i l e s ) t h a n t h e t r u c k i n g t r a f f i c (236 m i l e s ) . An e x a m i n a t i o n o f t h e raw d a t a showed t h a t t h e m o s t p r o b a b l e e x p l a n a t i o n f o r t h e h i g h e r r a i l w a y a v e r a g e r a t e i s t h a t r a i l w a y s moved t h e m a j o r p o r t i o n o f l u b r i c a t i n g o i l a n d g r e a s e s , w h i c h a r e n o r m a l l y more e x p e n s i v e t h a n t h e o t h e r c o m m o d i t i e s b e l o n g i n g t o CFTM69. T h i s i s a n o t h e r p r o b l e m o f d a t a a g g r e g a t i o n i n t h e CFTM d a t a b a s e . - 108 -Although the average speed of the r a i l mode was sub-s t a n t i a l l y slower than that of the truck mode, the r e l i a b i l i t y measure (reciprocal of the c o e f f i c i e n t of v a r i a t i o n of t r a n s i t -time d i s t r i b u t i o n , i . e . , mean/standard deviation) was higher for the railway mode. Note that, because of the way i n which the r e l i a b i l i t y variables was constructed, the higher r e l i a b i l i t y for r a i l mode does not necessarily mean that for a given l i n k , r a i l t r a n s i t time d i s t r i b u t i o n i s less dispersed from i t s mean than that of truck mode. - 109 -(B) General Observations About the Chosen Models The following observations may be made about the chosen models reported i n Table 5-1: (1) The models with the qu a l i t y variables (speed and r e l i a b i l i t y ) are chosen for the r e l a t i v e l y high-value (per ton) commodities such as CFTM14 ( f r u i t s , vegetables and edible foods), CFTM75 (metal f a b r i -cated basic products such as b o l t s , nuts, n a i l s , screws, etc.) and CFTM78 (non-metallic basic products such as glass products, t i l e s , gypsum products, etc.) whereas the models without the quali t y variables are chosen for the r e l a t i v e l y low-value (per ton) i n d u s t r i a l raw materials such as CFTM52 (lumber), CFTM61 (chemicals) 2, CFTM66 (fuel o i l ) , CFTM69 (other refined petroleum products) and CFTM71 (s t e e l , iron and a l l o y s , e t c . ) . This shows that shippers of the former commodity groups place higher values on the qu a l i t y attributes of f r e i g h t service than do shippers of the l a t t e r commodity groups. (2) The model with mode-specific hedonic aggregators (C-l) i s chosen for a l l the above three commodity groups whose model included the speed and r e l i a -b i l i t y variables. As was mentioned i n chapter I I I , t h i s model implies that shippers base t h e i r mode-choice decision on prices adjusted for qu a l i t y - no -v a r i a t i o n s . Since t h i s model has mode-specific hedonic aggregators as i t s arguments, each hedonic aggregator could take d i f f e r e n t values of the parameters from those of the other hedonic 3 aggregator. The difference between the two aggregators i s l i k e l y to have been caused by the following factors: (i) shippers' p o t e n t i a l mis-perception of the qual i t y attributes of the two modes, and ( i i ) errors i n the model s p e c i f i c a t i o n including the omitted q u a l i t y variables such as convenience, f l e x i b i l i t y and completeness of service, etc. (.3) The revenue share functions i n a l l the models other than model B include the distance (D£) as an independent variable. In these models, the l i n k unit cost function i s not independent of the distance. Since model B was not chosen for any commodity group, i t may be generalized that the choice p o s s i b i l i t y sets i n shippers' transport-sectoral-technology space depend on the distance to transport. As can be noticed from chapter VII, this i s the reason why the e l a s t i c i t y of substitution between the two modes tends to vary with distance. (4) The model with an i d e n t i c a l hedonic aggregator for the two modes was not chosen for any commodity - I l l -group. Therefore, i t appears that the conditions for Rosen-like (see Rosen £l974]) j o i n t application of hedonic price theory to a family of d i f f e r e n t i a t e d products are not s a t i s f i e d i n the f r e i g h t mode-choice decision. - 112 -(C) Parameter Estimates and Interpretation of S p e c i f i c Models The maximum l i k e l i h o o d (ML) estimates of parameters of the chosen models are reported i n Table 6-2 f o r CFTM14, CFTM75 and CFTM78, i n Table 6-3 f o r CFTM52, CFTM61 and CFTM66, and i n Table 6-4 for CFTM69 and CFTM71. In what follows, an attempt w i l l be made to i n t e r p r e t the estimated models. (1) Model C - l Reported i n Table 6-2: The parameter estimates of model C - l for commodity groups CFTM14 ( f r u i t s , vegetables and edible foods), CFTM75 (metal fabricated basic products) and CFTM78 (non-metallic basic products) are reported i n Table 6-2. The model implies that both speed and r e l i a b i l i t y v a r i a b l e s , as well as f r e i g h t rates, influence the mode s e l e c t i o n d e c i s i o n . In a l l the three commodity groups, the parameter estimate a r h ( t n e second-order translog parameter) i s s t a t i s t i c a l l y s i g n i f i c a n t even at the 1% l e v e l of s i g n i f i c a n c e . This implies that the Cobb-Douglas model i s inappropriate to use. As w i l l be noted i n chapter VII, the p o s i t i v e value of a r h indicates that 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 the two 5 modes i s larger than one. In order to examine the parameter estimates of the hedonic aggregators of the two modes, i t seems worthwhile to write the quality-adjusted p r i c e functions. By looking back to the derivation of model C i n chapter I I I , these functions can be written as (6,1) . - 113 -Table 6-2 Parameter Estimates for the Chosen Models ( C - l ) * * (asymptotic t - s t a t i s t i c s i n parentheses) Parameter Commodity Group CFTM14 CF.TM75 CFTM78 ( f r u i t s , vege-tables & edible foods) (metal f a b r i -cated basic products) (non-metallic basic products) Zna o -2.9471 (0.810) 1.5044 (0.823) -0.3052 (0.091) a r -0 .5246 (2.201) -0.0650 (0.399) -0.1718 (0.497) a h 1.5246 (6.288) -0.0650 (6.424) 1.1173 (3.339) a r r -0.0981 (4.926) -0.0872 (3.738) -0 .1173 (6.191) ahh -0.0981 (4.926) -0.0872 (3.738) -0.1173 (6.191) a r h * 0.0981 (4.926) 0.0872 (3.738) 0.1173 (6.191) g r -0.1340 (0.447) 0.0973 (0.318) -0.2572 (0.883) eh -0.8963 (1.455) -0.9771 (2.159) -1.2283 (1.638) Y r -0.0340 (0.302) -0.1450 (1.281) -0.0829 (0.772) -2.4209 (3.959) -0.9740 (2.387) -2.4212 (3.365) 6 r -0.9504 (3.142) -0.8664 (2.718) -0.5283 (2.071) 6 h 1.3408 (.4.040). 0.7430 (2.976) 1.4371 (3.858) * Denotes that the parameter estimates were computed using the l i n e a r homogeneity conditions. ** Model with mode-specific hedonic aggregators s p e c i f i e d i n terms of price, speed, r e l i a b i l i t y and distance. - 114 -(6,1) P r = P r Z r i Z r 2 D r f o r r a i l m o d e * 3 h Y h 6 h P . = P , Z , , Z " D for truck mode n n n l rz * * where P r and P h are the quality-adjusted prices of r a i l and truck modes, respectively, and a l l other variables are defined as i n (3,17). The signs of parameters & r,Y r,6 h and Y h are expected to be negative because, for a given observed p r i c e , the q u a l i t y -adjusted p r i c e should be lower for a service with a higher l e v e l of qual i t y variable. The t - s t a t i s t i c s reported i n Table 6-2 show that a l l the parameter estimates for the r a i l hedonic aggregators 6 r (corresponding to "speed" variable) and Y r (corresponding to " r e l i a b i l i t y " variable) are not s t a t i s t i c a l l y d i f f e r e n t from zero whereas those for the truck hedonic aggregators ( 8 ^ and y^) are s t a t i s t i c a l l y less than zero at the 10% l e v e l of sig n i f i c a n c e (one-tail t e s t ) . This implies that speed and i t s r e l i a b i l i t y of the truck mode influence the demands for both modes but those of the r a i l mode do not a f f e c t the demands s i g n i f i c a n t l y . Aside from the s t a t i s t i c a l i n s i g n i f i c a n c e of the parameter estimates of the r a i l hedonic aggregators, the parameter estimates of the truck hedonic aggregators are consistently larger i n absolute value than the corresponding parameters of the r a i l hedonic aggregators. The assumption of shippers' r a t i o n a l behaviour rules out the p o s s i b i l i t y that shippers i n t e n t i o n a l l y place a higher - 115 -value on the trucking service with the same qu a l i t y attributes as the railway service. The probable explanations for the difference between the two hedonic functions are: (i) The shippers of the three ( r e l a t i v e l y high-value) commodity groups may have over-perceived the qu a l i t y a t t r i b u t e s of truck mode and/or under-perceived those of r a i l mode. ( i i ) The omitted quality variables such as convenience, f l e x i b i l i t y , and completeness of service are l i k e l y to favour the truck mode. (2) Model C-3 Reported i n Table 6-3: Table 6-3 reports the parameter estimates of model C-3 for commodity groups CFTM52 (lumber), CFTM61 (chemicals) and CFTM66 (fuel o i l ) . Model C-3 i t s e l f implies that neither speed nor r e l i a b i l i t y of service s i g n i f i c a n t l y influences mode sel e c t i o n . The po s i t i v e values of (the second-order translog parameter) mean that the e l a s t i c i t y of substitution between the two modes i s greater than one. The value of a ^ for CFTM52 i s not s t a t i s t i c a l l y d i f f e r e n t from zero implying that i t i s possible to use an appropriate constant e l a s t i c i t i e s of substitution (CES) model i n place of the translog cost function. This point w i l l become clearer i n chapter VII. (.3) Model A-3 Reported i n Table 6-4: Table 6-4 reports the parameter estimates of model A-3 for commodity groups, CFTM69 (other refined petroleum products) and CFTM71 (steel, iron and a l l o y s ) . Since model A-3 does not - 116 -Table 6-3 Parameter Estimates of the sChosen Models (C-3)** (asymptotic t - s t a t i s t i c s i n parentheses) Parameter Commodity Group CFTM52 CFTM61 CFTM66 (lumber, i n c . (chemicals) (fuel o i l other flooring) than gasoline) Ina o -1.0222 (1.024) -0.8298 (1.613) -1.533 (1.224) a r 0.5711 (4.532) 0.0294 (0.202) -0.0574 (0.288) a h * 0.4289 (3.320) 0.9706 (6.566) 1.0574 (5.207) a r r -0.0111 (.0. 790) -0.1368 (5.813) -0.0846 (3.176) ahh * -0.0111 (0.790) -0.1368 (5.813) -0.0846 (3.176) a r h * 0 .0111 (0.790) 0.1368 (5. 813) 0.0846 (3.176) 5 r 1.18636 (2.633) -0.2991 (2.765) -0.6499 (2.275) 5 h -0.8913 (2.299) 0.2661 (2.479) 0.5190 (1.988) Denotes that the parameter estimates were computed using the l i n e a r homogeneity conditions. Note that model (C-3) i s equivalent to model (D-3): model with mode-specific hedonic aggregators s p e c i f i e d i n terms of price and distance only. - 117 -Table 6-4 Parameter Estimates of the Chosen Models(A-3)** (asymptotic t - s t a t i s t i c s i n parentheses) Parameter Commodity Group CFTM69 CFTM71 (other refined petro- (s t e e l , iron and leum products) a l l o y s , etc.) Ina. o -0 .0325 (0.027) -0.0575 (0.115) a r -0.6841 (4.497) -0.6238 (5.356) d 0.0559 (0.139) 0.0091 (0.053) a r r -0.0883 (3.755) -0.2740 (16.0) dd -0.0209 (0.312) -0.0054 (0.187) ad r 0.1733 (6.869) 0.1626 (8.799) a h * 1.6841 (10.852) 1.6238 (7.658) ahh * -0.0883 (3.755) -0.2740 (16.0) a r h * 0.0883 (3.755) 0.2740 (16.0) ad h * -0.1733 (6.869) -0.1626 (8.799) Denotes that the parameter estimates were computed using the l i n e a r homogeneity conditions. General model s p e c i f i e d i n terms of prices and distance only. - 118 -include speed and r e l i a b i l i t y variables, the shippers of these commodities do not seem to place s i g n i f i c a n t values on the quality variables. The p o s i t i v e signs of (the second-order translog parameter) imply that the e l a s t i c i t y of substitution i s greater than unity for both of the commodities. Since the value i s larger for CFTM71 ( a r h = .274) than for CFTM69 (= .0883), c e t e r i s -paribus, the inter-modal s u b s t i t u t i b i l i t y i s higher f o r the shipments of CFTM71 than those of CFTM69. The lower inter-modal s u b s t i t u t i b i l i t y for CFTM69 may be j u s t i f i e d i n t u i t i v e l y because, normally, s p e c i a l types of r o l l i n g stocks and equipment are required to handle i t . - 119 -(D) Summary of the General Findings Based on the discussions i n sections (B) and (C), the general findings from the chosen models may be summarized as follows: (i) Speed and r e l i a b i l i t y variables s i g n i f i c a n t l y influence mode selection for the r e l a t i v e l y high-value commodities but not for the low-value i n d u s t r i a l raw materials. For those commodities whose mode-choice i s s i g n i f i c a n t l y influenced by the q u a l i t y variables, the parameter estimates of the truck hedonic aggregators are consistently larger i n absolute value than the corresponding parameters of r a i l hedonic aggregators. Moreover, the parameter estimates of the r a i l hedonic aggregators are not s t a t i s t i c a l l y d i f f e r e n t from zero. ( i i ) For a l l commodity groups, the c h o i c e - p o s s i b i l i t y sets faced by the shippers depend upon the distance to transport. ( i i i ) For a l l the commodity groups, the e l a s t i c i t y of r a i l -truck substitution i s greater than unity because of the p o s i t i v i t y of the second-order parameter ( a ^ ) . (iv) Since the second-order translog parameter ( a ^ ) i s s i g n i f i c a n t l y d i f f e r e n t from zero,r the Cobb-Douglas model - i s inappropriate - for fr e i g h t demand studies for a l l the commodity groups other than CFTM52 (lumber including f l o o r i n g ) . - 120 -Footnotes for Chapter VI: 1. Users of the CFTM data base are not allowed to quote the price and qu a l i t y variables for s p e c i f i c l i n k s . 2. As can be seen from Appendix 4A, commodity group CFTM61 consists of chemicals mainly for i n d u s t r i a l use. 3. I f (i) the functional form c o r r e c t l y represents the quality-adjusted price function over the entire range of a l l possible l e v e l s of q u a l i t y a t t r i b u t e s , ( i i ) i f shippers are consistent i n evaluating the values of quali t y attributes of the two modes, and ( i i i ) i f there i s no misperception of the level s of qu a l i t y attributes of the two modes, then the parameter estimates of the two hedonic aggregators should not be d i f f e r e n t , at least s t a t i s t i c a l l y , as i n model D-l. 4. The existence of th i s misperception does not v i o l a t e the basic postulate of the optimization models, because shippers s t i l l behave optimally but only on the basis of the perceived levels of qu a l i t y attributes of the two modes, whereas the models are estimated using the actual l e v e l s . 5. See the formula for the e l a s t i c i t y of substitution i n equation (7,3). 6. Heaver and Oum [l977] reported a s l i g h t l y d i f f e r e n t form of hedonic price function estimated from more aggregated Canadian data. CHAPTER VII ELASTICITY ESTIMATES AND INTER-MODAL COMPETITION Having discussed the general findings of t h i s study i n the preceding chapter, i t i s now appropriate to examine some findings concerning s p e c i f i c segments of the Canadian i n t e r -c i t y f r e i g h t market. For some time, Transport Canada has been preparing for a major r e v i s i o n of the 1967 National Transportation Act (NTA) which would empower the government with the f l e x i b i l i t y to apply d i f f e r e n t sets of regulatory p o l i c i e s to d i f f e r e n t segments of the transport market depending on the "maturity" of transportation service and the extent of competition i n each s p e c i f i c market.^" Undoubtedly, the correct i d e n t i f i c a t i o n of the extent of competition e x i s t i n g i n various segments of the f r e i g h t market i s an es s e n t i a l pre-requisite for implementing such a f l e x i b l e regulatory p o l i c y . Realizing t h i s importance, Heaver and Nelson [19 77] have studied the workings of competitive forces under the commercial freedom underlying the 196 7 NTA, by examining primarily the process of shipper-carrier negotiations and mutual adaptations to the changing market conditions. Their major conclusion was that, although the extent of v i s i b l e competition varies from1 market to market, there i s a s i g n i f i c a n t l e v e l of "dynamic competition" throughout the Canadian i n t e r c i t y transport market. While t h e i r study i d e n t i f i e s some descriptive facts about the nature and process of competition, so far no one has attempted to - 121 -- 122 -measure systematically the extent of inter-modal competition e x i s t i n g i n various segments of the Canadian i n t e r c i t y f r e i g h t market. In view of the r i s i n g current i n t e r e s t on t h i s issue, therefore, the discussion i n th i s chapter w i l l be focused upon i d e n t i f y i n g the extent of inter-modal competition e x i s t i n g i n various segments of the fre i g h t market. To achieve t h i s objective, t h i s chapter i s organized as follows: In section A, the formulae for the e l a s t i c i t i e s of demand with respect to the price and qual i t y variables and for the e l a s t i c i t y of substitution are presented. (The detailed derivations are presented i n Appendix 7A.) In addition, the e l a s t i c i t i e s are evaluated at the mean values of the variables, and compared across the eight CFTM commodity groups. Section B reports the e l a s t i c i t y estimates for each of the major l i n k s . These e l a s t i c i t y estimates are used to determine, for each commodity group, the range of distance over which e f f e c t i v e inter-modal competition e x i s t s . - 123 -(A) E l a s t i c i t y Estimates at the Mean Values of the Variables Chapter VI showed that shippers of the low-value i n d u s t r i a l raw materials (note that these are mainly bulk commodities) make th e i r mode-choice decision primarily on the basis of freight rates, whereas shippers of the high-value commodities (note that these are mainly manufactured commodities) base t h e i r mode-choice on both f r e i g h t rates and q u a l i t y attributes of service. C a r r i e r s ' management as well as government regulators are normally interested i n seeing what would happen to demands for the two modes i f a certain change in f r e i g h t rate or quality of service i s to be introduced. The extent of the e f f e c t of such a change can best be measured by the A l l e n 3 p a r t i a l e l a s t i c i t i e s of substitution and e l a s t i c i t i e s of demand with respect to price and q u a l i t y variables. In t h i s section, therefore, the formulae for these e l a s t i c i t i e s are derived, along with an investigation of t h e i r properties, and applied to estimate the e l a s t i c i t i e s evaluated at the mean values of the variables. A l l e n [ l 9 3 8 j defined the p a r t i a l e l a s t i c i t y of s u b s t i -tution between two inputs of production, X.. and X.. , as: M E k=l 1 3 (7,1) a . 3 for a l l i , J = 1,2 M. where denotes the f i r s t p a r t i a l derivative of the production function f(X) with respect to kth input (X k), - 124 -F denotes the determinant of the bordered Hessian matrix of second-order derivatives of the production function f ( X ) , F _ denotes the ( i , j ).th co-factor of F, and X^ and Xj denote ^th and j_th inputs, respectively. Furthermore, A l l e n £l9 38] has shown the following additive property of the p a r t i a l e l a s t i c i t i e s of substitution: M (7,2) £ a..S. = 0 for a l l i=l,2,...,M. j=l 1 3 3 where Sj = the share of expenditures (revenue share from c a r r i e r ' s viewpoint) for the j_th mode. Later, Uzawa [1962] derived the following expression for the Al l e n p a r t i a l e l a s t i c i t i e s of substitution i n terms of cost function: C C. . (7,3a) a = -^±2- for a l l i ^ j where C = the cost function, C\ = f i r s t p a r t i a l derivative of C with respect to the price of input X^, C\j = second p a r t i a l derivative of C with respect to prices of inputs X^ and X.. . For a two-mode translog cost function, the A l l e n e l a s t i c i t y of substitution (AES) can be written as: (7,3b) 125 -C C , S' S- + a , = rh _ r h rh rh C C. S • S u r h r h where = the revenue share of the d^ th mode, a r ^ = a second-order parameter of the translog cost function. The additive property (7,2) and equation (7,3b) together allow one to write the following equation: -a. .S . ( 7 ' 3 c ) a i i = - i p - 1 i a.. + S 2 - S. i i l l s 2 l i , j = r,h where a ^ i s a second-order parameter of the translog cost function and i s equal to -a ^  due to li n e a r homogeneity of the cost function discussed i n chapter I I I . As i s shown i n Appendix 7A, the e l a s t i c i t y of Hicksian (compensated) demand for the i t h mode with respect to the fre i g h t rate of the j t h mode and of the _ith mode can be written as (7,4a) and (7,4b), respectively. 3 X. P . (7,4a) E = ±- • =J-1 3 O P.) . . l 3 isoquant a. . + S . • S . = 3 0 c 1 = S. • a. . using (7,3b) S i j 13 for a l l i ^  j a.. + S2 - S. (7,4b) E = ^ i - = S . c . using (7,3c) n i i n fo r a l l i = r,h Since the Hicksia n (compensated) demand function only takes s u b s t i t u t i o n e f f e c t s i n t o account, a new measure of p r i c e responsiveness of demand i s required to include the e f f e c t of a change i n f r e i g h t rate on the shippers' output l e v e l . Adapting the Allen's formula {Allen,1938] to our se c t o r a l l y separable structure, the e l a s t i c i t y of Marshallian (ordinary) demand f o r the i t h mode with respect to the price of 4 the ;jth mode, . , can be written a s (7,5) F . . = S . ( a . . + A . n ) "' • • • r r ( r a i l ) 1 3 3 1 3 3 x'3 = < h(truck) where dP pi A. = -=— —**- i s the proportion of change i n the commodity's 3 Pj P pri c e (p) with respect to a change i n the p r i c e of the j t h mode (p_^ ) . n = p^- £ i s the price e l a s t i c i t y of demand for the p commodity (shiper's product). Note that the p r i c e e l a s t i c i t y of ordinary demand f o r a mode depends, among other things, on the competition i n the destination commodity market, which i s the so-called "market competition" i n transportation l i t e r a t u r e . If n and.A_.'s f o r a commodity group are known to us, the formula (7,5) can be used to compute the pri c e e l a s t i c i t i e s of ordinary demand for each mode. Since these were not r e a d i l y a v a i l a b l e , the p r i c e e l a s t i c i t i e s F^_.'s reported i n t h i s chapter were computed under the rather a r b i t r a r y assumptions of n = -1 and A. = 0.1 fo r a l l j ' s . D -was not r e a d i l y available, the price e l a s t i c i t i e s of ordinary demands (F^'s) for r a i l and truck modes reported i n t h i s chapter were computed under a rather a r b i t r a r y assumption that the commodity price e l a s t i c i t y n i s unity; i . e . , n = -1. Turning attention to the quality responsiveness of demand, the e l a s t i c i t y of demand for the i t h mode with respect to the nth q u a l i t y a t t r i b u t e of the jth mode may be defined as: ( 7 ,6 ) E. . IP n dZnX. 1 dZnZ . Dn 3X. 1 Dn 3Z . X. Dn l i , j s r,h n = 1,2 , . . . , N. As i s shown i n Appendix 7A, i n the context of our translog cost function (3,17a) for model C - l , E.. n can be written as (7,7) E, n ID B. (a. . + S. • S .) Pn 3-D i D S . l = B . E. . for a l l i 4 j B . (a. . + S . " i n i i I S i ) S . l = B. E.. for a l l i = j i n n J where B. = pn 8 , 3^ when n = 1 (speed) Y , Yy, when n = 2 ( r e l i a b i l i t y ) This completes the derivation of the formulae for computing the e l a s t i c i t y of r a i l - t r u c k substitution and e l a s t i c i t i e s of demand with respect to f r e i g h t rates and q u a l i t y attributes of service. The formulae i n equations (7,3), (7,4) and (7,5) are used here to evaluate various e l a s t i c i t i e s at the mean values of - 128 -the variables reported i n Table 6-1, and the res u l t s are reported i n Table 7-1. The estimated e l a s t i c i t y of substitu-t i o n i s the lowest for CFTM52 (lumber including f l o o r i n g : = 1.044) and the highest for CFTM71 (s t e e l , i r o n and a l l o y s , etc.: °ryl = 2.132). This implies that as the price r a t i o of the two modes, p j j / p r / increases by one percent, the r a t i o of the demands for the two modes, X^/X^, increases by 1.044% for the case of CFTM52 and by 2.132% for the case of CFTM71. Clea r l y , the highest inter-modal s u b s t i t u t i b i l i t y obtained for CFTM71 (ste e l , iron and a l l o y s , etc.) i s counter-i n t u i t i v e . This e l a s t i c i t y of sub s t i t u t i o n (°" ^  = 2.132) i s l i k e l y to have been over-estimated because a wide variety of heterogeneous commodities ranging from primary metals to construction hardwares are aggregated into the commodity group CFTM71 (see Appendix 4A for a detailed l i s t of the commodities included i n thi s commodity group). As a r e s u l t ,:• one1 cannot r e l y on the e l a s t i c i t y r e s u l t s estimated from the CFTM71 data. Therefore, the model for t h i s commodity group w i l l not be discussed further i n the remainder of t h i s thesis. Equation (7,3b) implies that the e l a s t i c i t y of substitution, a ^ , i s greater than one i f the translog parameter a ^ i s po s i t i v e . Since, for a l l the chosen models reported i n Tables (6-2) , (6-3) and (6-4) , the estimate a ^ i s p o s i t i v e , °rh "*"s 9 r e a t e r than one for a l l the commodity groups. This implies that the two modes are highly s u b s t i t u t i b l e . Note an a n a l y t i c a l - fact that, i n the two-mode model, the Table 7-1, Comparison of Elasticities (Evaluated at Means of Variables) \- CcOTnodity \, Group Elasticx ities *CFTM14 Fruits, Vegetables & Edible Foods CFTM52 Lumber, Including Flooring CFTM61 Chemicals CFTM66 Fuel Oil Except Gasoline CFTM69 Refined Petroleum Products CFTM71 Steel, Iron & Alloys CFTM75 Basic Metallic Products CFTM78 Non-Metallic Products 0 r h 1.458: 1.044 1.57 1.429 1.4'. 2.132 1.508 1.539 a r r -3.2466 -1.087 -1.047 - .5286 -2.987 -2.714 -5.347 -3.271 ahh - .6553 -1.003 -2.355 -3.864 - .6615 -1.675 - .4254 - .7243 E r r -1.006 - .5324 - .6282 - .3858 - .9560 -1.194 -1.176 -1.047 E r h 1.006 .5324 .6282 .3858 .9560 1.194 1.176 1.047 \ r .4522 .5116 .942 1.043 .4499 .9381 .3318 .4925 *hh - .4522 - .5116 - .942 -1.043 - .4499 - .9381 - .3318 - .4925 F' rr -1.037 - .5814 - .6882 - .4588 - . 988 -1.238 -1.198 -1.079 Fhh - .5212 - .56 26 - .-98 2 -1.07 - .5179 - .9 941 - .4098 - :. 56 05 o\ j = elasticity of substitution between modes i and j . E. . = compensated elasticity of demand for ith mode with respect to freight rate of jth mode. F.. = ordinary elasticity of demand for ith mode with respect to its own freight rate computed 1 1 assuming unitary elasticity of demand for the comaiodity and value of 0.1 for a l l L ' s -*"CFTM" stands for Canadian Freight Transportation Model commodity group. Subscripts "r" and "h" stand for r a i l and highway (truck) modes, respectively. - 130 -cross p r i c e e l a s t i c i t y , E ^ (or E h r ) , i s the negative of the own-price e l a s t i c i t y , E^^ (or E h h ) because the compensated pr i c e e l a s t i c i t i e s of a mode sum to zero, i . e . . E + E , = 0 c ' ' r r rh and E ^ + E ^ = 0. The (Hicksian) compensated demand f o r the r a i l mode i s p r i c e - e l a s t i c f o r CFTM14, CFTM75 and CFTM78, and p r i c e - i n e l a s t i c f o r the other four commodity groups. The compensated demand f o r the truck mode i s p r i c e - e l a s t i c only for CFTM66 (fu e l o i l other than g a s o l i n e ) . Generally, the ordinary demand for the r a i l mode i s own-price-elastic for the r e l a t i v e l y high-value commodities such as CFTM14, CFTM75 and CFTM78, and i s own-price-inelastic for the r e l a t i v e l y low-value commodities such as CFTM52, CFTM61 and CFTM66. The absolute values of the own-price e l a s t i c i t i e s for the truck mode are close to unity for the commodity groups CFTM61, CFTM69 and CFTM71, but are between 0.41 and 0.56 for a l l the other commodities. Of course, one should keep i n mind that these estimates are subject to the highly a r b i t r a r y assumptions of n=-l and A^ = 0.1 for a l l j_'s. - 131 -(B) The E l a s t i c i t y Estimates on Some Selected Links and  Inter-modal Competition In section A, a single set of various e l a s t i c i t i e s were computed for each commodity group as the aggregate indicators of the competition e x i s t i n g i n the p a r t i c u l a r commodity fr e i g h t market i n general. However, the extent of competition i s l i k e l y to be d i f f e r e n t not only from commodity to commodity but also from l i n k to l i n k . In t h i s section, therefore, the e l a s t i c i t i e s of demand with respect to the price and q u a l i t y variables and the e l a s t i c i t y of substitution are computed separately for each of the major l i n k s . The r e s u l t s are reported i n Tables 7-2 to 7-8. For an e f f e c t i v e i n t e r p r e t a t i o n of the information i n the tables, i t i s b e n e f i c i a l to know the following relations between each of the e l a s t i c i t y measures and the d i v i s i o n of the revenue shares between the two modes: 1. From the formula i n equation (7,3b), i t i s easy to see that the e l a s t i c i t y of substitution, o- ^, increases as the absolute deviation between the two shares | s r - S^| i s increased because the translog parameter a ^ i s p o s i t i v e i n a l l the chosen models. As a r e s u l t , i s minimized when S r = S^ = 0.5. 2. I f l s r *" s n l l S large, and S^ > S^ i n a p a r t i c u l a r f r e i g h t market, then there are two forces acting to increase the compensated e l a s t i c i t y of demand for the trucking service as shown below: E h r ^ = ~ Ehh^ = a r h S r 9 e t s x a r c J e because of both the large o*^ caused by the large value of |S r - S^ | and the large S . ^ r This condition occurs normally on the long-haul l i n k s where the r a i l mode dominates the t r a f f i c . - 132 -3. If | s - S^l i s large and S r < i n a p a r t i c u l a r f r e i g h t market, the compensated e l a s t i c i t y of the demand for r a i l mode, E u (= —E ) = a , S, , becomes ' rh r r rh h large because of both the large cr and the large S^. This condition occurs normally on the short-haul l i n k s where the truck mode dominates the t r a f f i c . The above a n a l y t i c a l r e s u l t s show that as the distance increases the e l a s t i c i t y of demand for the r a i l mode decreases and that for the truck mode increases, and vice versa. This can be i n t u i t i v e l y j u s t i f i e d i n terms of the r e l a t i v e cost structures of the two modes because as the length of haul increases, the truck mode becomes increasingly disadvantageous r e l a t i v e to the r a i l mode, and as the length of haul decreases, the r a i l mode becomes increasingly disadvantageous r e l a t i v e to the truck mode. On the li n k s where one mode dominates a major portion of the t r a f f i c and no s i g n i f i c a n t inter-modal competition e x i s t s , the demand for the other mode i s l i k e l y to be p r i c e - e l a s t i c . The c a r r i e r s of the l a t t e r mode have to operate on the e l a s t i c portion of t h e i r demand curve for the following reasons: 1. An increase i n price would reduce the t r a f f i c proportionally more than the price increase, and thus reduce the revenue. 2. Although a reduction of the price would increase the t r a f f i c proportionally more than the price reduction, the pressure of cost eliminates such a p o s s i b i l i t y because the c a r r i e r s are presumably o f f e r i n g a fre i g h t rate quite close to th e i r marginal cost i n the r e l a t i v e l y disadvantageous market. Even i f i t were possible, the reduction of price would invoke a" price war with the competitive mode which has a cost advantage. - 133 -Although service competition among trucking firms has been f a i r l y high i n some corridor routes, the intra^modal p r i c e competition has been n e g l i g i b l e i n the railway industry and marginal i n the trucking industry i n Canada, Therefore, for a given commodity, the extent of inter-modal competition i s the single most important factor which determines the compensated price e l a s t i c i t i e s of demand i n various f r e i g h t markets. Due to the reasons stated previously, on those l i n k s where inter-modal competition i s not s i g n i f i c a n t , at lea s t one of the modes should have a p r i c e - e l a s t i c demand. Since the compensated p r i c e - e l a s t i c i t y of r a i l demand E R R decreases with distance whereas that of truck demand increases with distance, r a i l and truck modes w i l l dominate the long-haul and short-haul l i n k s , respectively, leaving the medium-haul link s as the pot e n t i a l markets for the inter-modal competition. In what follows, for each commodity group, an attempt i s made to i d e n t i f y the upper-bound of distance up to which truck mode p r a c t i c a l l y dominates the t r a f f i c and the lower-bound of distance beyond which r a i l mode dominates the t r a f f i c . The c r i t e r i o n used for the i d e n t i f i c a t i o n i s as follows: a market i s regarded truck-dominated i f |E I > 1 and |E I > 2 • IE , , I, •o 1 r r 1 1 r r 1 1 hh 1 and rail^dominated i f IE, , 1 > 1 and IE, , I > 2 • | E I. 1 hh 1 1 hh 1 1 r r 1 (.1) The Results f o r CFTM14 (Fru i t s , Vegetables and edible foods) : Table 7-3 reports the e l a s t i c i t y of substitution and the e l a s t i c i t i e s of demand with respect to pr i c e s , speed and r e l i a b i l i t y computed for the selected l i n k s . The distances of Table 7-2, Estimated Parameters of Price and Quality Responsiveness of Demands for CFTM14 (Fruits, vegetables and edible foods) Link Compensated price elasticities Speed elasticities Reliability elasticities X ** Origin* Dest'n.* Miles E rr ! *hh °rh E 1 rr E , 1 rh E 2 rr E , 2 rh *hh2 11 . P.E.I. 53. Toronto 1067 -0 .75 -0.64 1.39 0.10 -0.67 0.57 -0.09 0.02 -1 .83 1.55 -0 .02 33. St. John 46. Montreal 590 -0 .91 -0 .51 1.42 0.12 -0 .82 0.46 -0.07 0.03 -2 .21 1.24 -0.02 33. St. John 53. Toronto 925 -0 .84 -0 .57 1.41 0.11 -0 .75 0.51 -0.08 0.03 -2 .02 1.38 -0.02 46. Montreal 43. Quebec 173 -1 .13 -0.39 1.52 0.15 -1 .01 0.35 -0.05 0.04 -2 .73 0.94 -0 .01 46. Montreal 53. Toronto 335 -1 .10 -0 .40 1.50 0.15 -0.98 0.36 -0 .05 0.04 -2.66 0.97 -0 .01 46. Montreal 71. Regina 1783 -0 .44 -1 .03 1.47 0.06 -0.39 0.92 -0.14 0.01 -1 .06 2.49 -0.04 46. Montreal 95. Vancouver 2908 -o:4o -1 .11 1.51 0.05 -0.35 1.0 -0 .15 0.01 -0.96 2.69 -0.04 53. Toronto 25. Halifax 1113 -0 .70 -0 .69 1,39 0.09 -0 .62 0.62 -0.09 0.02 -1.69 1.67 -0 .02 53. Toronto 46. Montreal 335 -0 .95 -0 .49 1.44 0.13 -0 .85 0.43 -0.07 0.03 -2 .31 1.17 -0.02 53. Toronto 56. Windsor 229 -1 .13 -0 .39 1.52 0.15 -1 .01 0.35 -0 .05 0.04 -2 .73 0.93 -0 .01 53. Toronto 67. Winnipeg 1256 -0 .70 -0 .69 1.39 0.09 -0 .63 0.62 -0.09 0.02 -1.70 1.66 -0 .02 53. Toronto 95. Vancouver 2736 -0 .51 -0 .92 1.43 0.07 -0.46 0.82 -0 .12 0.02 -1 .23 2.22 -0 .03 54. Hamilton 57. Kitchener 78 -1 .70 -0 .23 1.93 0.22 -1 .53 0.21 -0 .03 0.05 -4 .12 0.55 -0 .01 67 Winnipeg 46. Montreal 1428 -0 .65 -0 .75 1.40 0.09 -0.58 0.67 -0.10 0.02 -1.56 1.80 -0 .03 67. Winnipeg 53. Toronto 1256 -0 .84 -0.56 1.40 0.11 -0 .75 0.50 -0.07 0.03 -2.04 1.36 -0.02 67. Winnipeg 71. Regina 356 -1 .03 -0.44 1.47 0.13 -0.92 0.39 -0.06 0.03 -2.49 1.06 -0 .01 67. Winnipeg 95. Vancouver 1480 -0 . 51 -0 .91 1.42 0.06 -0.46 0.81 -0 .12 0.01 -1.24 2.20 -0 .03 71. Regina 73. Saskatoon 161 -1 .67 -0 .23 1.90 0.22 -1 .50 0.21 -0 .03 0.06 -4 .05 0.56 -0 .01 86. Edmonton 83. Calgary 190 -1 .13 -0 .38 1.51 0.15 -1 .01 0.35 -0.05 0.04 -2.74 0.93 -0 .01 95. Vancouver 53. Toronto 2736 -0 .40 -1 ,10 1.50. 0.05 -0.36 0.98 -0 .15 0.01 -0 .97 2.65 -0 .03 95. Vancouver 67. Winnipeg 1480 -0 .66 -0.74 1.40 0.09 -0.59 0.65 -0.09 0.02 -1.59 1.77 -0 .03 * Numbers preceding names of origin and destination are the CFTM region codes. ** These elasticities may be regarded as zero because the parameter estimates B and different from zero (see Table 6 - 2 ) . r Yr are not statistically - 135 -the l i n k s are also l i s t e d in the table. As i s expected from the previous discussion on the rel a t i o n s h i p between the e l a s t i c i t i e s and the distance, the compensated own-price e l a s t i c i t y of demand for the railway service (E ) i s roughly inversely related to distance of the l i n k while the reverse i s true for the trucking service. The li n k s on which |E I > 1 and |E I > 2 • IE. . I are: 1 r r 1 1 r r 1 1 hh 1 Link name Miles E r r E hh Hamilton-Kitchener 78 -1.70 -0.23 Regina-Saskatoon 161 -1.67 -0.23 Montreal-Quebec 173 -1.13 -0.39 Edmonton-Calgary 190 -1.13 -0.38 Toronto-Windsor 229 -1.13 -0.39 Montreal-Toronto 335 -1.10 -0.40 Winnipeg-Regina 350 -1.03 -0.44 The l i n k s on which I E , , I > 1 and IE.,1 > 2 • | E I are: 1 hh 1 1 hh 1 1 r r 1 E E Link name Miles r r hh Montreal-Regina 1783 -0.4 4 -1.0 3 Toronto-Vancouver 2736 -0.51 -1.10 Montreal-Vancouver 2908 -0.40 -1.11 Although there are a few exceptions, a ca r e f u l examination of the above l i s t s and Table 7-2 allows the following general remarks on the inter-modal competition for t h i s t r a f f i c : (i) The trucking mode tends to dominate the t r a f f i c moving up to about 400 miles, whereas the r a i l mode dominates the t r a f f i c moving longer than 1800 miles. - 136 -( i i ) Therefore, the e f f e c t i v e inter^modal competition for t h i s t r a f f i c i s l i k e l y to e x i s t only on those l i n k s whose distance i s between 400 and 1800 miles. Table 7-2 shows also that the signs of the e l a s t i c i t i e s of demand with respect to quality variables conform to our expectation. Generally, the e l a s t i c i t i e s of demand for r a i l -way service with respect to speed and r e l i a b i l i t y of trucking 1 2 service, E ^ and E r ^ , are i n absolute value very high on short-haul t r a f f i c but decrease gradually with distance. E l a s t i c i t i e s of demand for trucking service with respect to i t s own speed and r e l i a b i l i t y , E ^ 1 and E ^ > follow a pattern that i s exactly opposite to those of railways. These also show that e f f e c t i v e inter-modal competition exists only for medium-haul t r a f f i c . The q u a l i t y e l a s t i c i t y measures, E r r ^ f 1 2 2 E h r ' E r r a n ( ^ E h r a r e v e r ^ small i n absolute value implying that a (small) change i n the quality attributes of railway service i s not much appreciated by the shippers, and consequently i s not an e f f e c t i v e means to compete against the trucks. Note that these e l a s t i c i t y estimates may be regarded as zero because the parameter estimates 8^, and y r are not s t a t i s t i c a l l y d i f f e r e n t from zero as mentioned i n chapter VI. (2) The Results for CFTM52 (Lumber including f l o o r i n g ) : Table 7-3 reports the compensated price e l a s t i c i t i e s and e l a s t i c i t y of substitution between the two modes for t h i s commodity group. Notice from the table that the e l a s t i c i t y of substitution between the two modes i s almost same = 1.04 or 1.05) on Table 7-3, Price E l a s t i c i t i e s and E l a s t i c i t y of Substitution  for CFTM52 (Lumber including flooring) Compensated : E l a s t i c i t y of • price e l a s t i c i t i e s Substitution Origin* Dest*n.* Miles (1) E r r (2) hh (3) rh 32. Moncton 53. Toronto 949 -0.62 -0.43 1.05 33. St. John 53. Toronto 925 -0.62 -0.4 3 1.05 43. Quebec City 46. Montreal 173 -0.56 -0.48 1.04 46. Montreal 25. Halifax 778 -0.59 -0.46 1.05 46. Montreal 43. Quebec 173 -0.56 -0.48 1.04 46. Montreal 45. Sherbrook 102 -0.55 -0.49 1.04 53. Toronto 25. Halifax 1113 -0.61 -0.44 1.05 53. Toronto 46. Montreal 335 -0.59 -0.46 1.05 53. Toronto 56. Windsor 229 -0.57 -0.47 1.04 53. Toronto 59. Sudbury 248 -0.57 -0.47 1.04 54. Hamilton 59, Sudbury 289 -0,59 -0.46 1.05 59. Sudbury 46. Montreal 436 -0,56 -0.48 1.04 59. Sudbury 53. Toronto 248 -0,57 -0.47 1.04 59. Sudbury 57. Kitchener 308 -0,59 -0.46 1.05 67. Winnipeg 50. T.^Bay 434 -0,56 -0.48 1.04 86, Edmonton 83. Calgary 190 -0.59 -0.46 1.05 46. Montreal 53, Toronto 335 -0.57 -0.47 1.04 * Numbers preceding names of o r i g i n and destination are the CFTM region codes. - 138 -a l l l i n k s . The s i m i l a r i t y of a ^ across the l i n k s i s caused by the s t a t i s t i c a l i n s i g n i f i c a n c e of the second-order parameter, a ^, of the translog function mentioned previously i n chapter VI. As indicated previously, therefore, an appropriate constant e l a s t i c i t y of substitution (CES) model can be used for t h i s commodity group i n place of the translog function. Notice also from Table 7 -1 that the e l a s t i c i t y of substitution for this commodity i s the lowest among a l l the eight commodity groups. Furthermore, even t h i s low e l a s t i c i t y of substitution Co" ^  = 1 . 0 4 4) may be considered as an over-estimated figure for the true s u b s t i t u t i b i l i t y due to the aggregation of two d i f f e r e n t products: lumber moved primarily by r a i l mode, and f l o o r i n g , a major portion of which i s moved by truck mode. The reason for the low s u b s t i t u t i b i l i t y r e l a t i v e to the other commodity groups may be that the lumber shippers who have the access to r a i l system may not consider the trucking service as an e f f e c t i v e a l t e r n a t i v e . None of the l i n k s l i s t e d i n Table 7 - 3 has an e l a s t i c demand for either one of the two modes, and the compensated price e l a s t i c i t i e s are quite stable from l i n k to l i n k ; 0 , 5 6 < IE I < 0 . 6 2 , 0 . 4 3 < IE, , I < 0 . 4 9 . These e l a s t i c i t i e s r r • — 1 hh 1 — are not related to the distance of the l i n k unlike the relationships found for other commodity groups. This strange behaviour of the e l a s t i c i t i e s may be pa r t l y due to the low s u b s t i t u t i b i l i t y , and pa r t l y due to the aggregation problem. Therefore, i t may be that there i s no s i g n i f i c a n t - 139 -inter-modal competition i n t h i s f r e i g h t market, and shipper's mode-choice i s determined largely by the a c c e s s i b i l i t y to r a i l s ervice. (3) The Results for CFTM6T (chemicals): The e l a s t i c i t y estimates for t h i s commodity group are reported i n Table 7-4 along with the e l a s t i c i t y of substitution between the two modes. As i n the case of CFTM14, the compensated e l a s t i c i t y of demand for the r a i l mode tends to decrease with the distance of a l i n k whereas that for the truck mode increases with i t . Generally on those l i n k s whose distance exceeds 500 miles, the compensated demand for truck mode was p r i c e - e l a s t i c ( i . e . , I E , , I > 1) and IE, . I > 2 • | E I. * ' 1 hh 1 1 hh 1 1 r r ' This implies that the r a i l mode generally dominated the t r a f f i c moving beyond 500 miles. However, no l i n k had a p r i c e - e l a s t i c compensated demand for r a i l mode. Therefore, i t can be concluded that the e f f e c t i v e inter-modal competition s t a r t s at a very short distance (probably 100 miles), and ends at the distance of around 500 miles. (4) --The Results for CFTM66 (fuel o i l other than gasoline) : This i s the commodity group for which the inter-modal price competition seems to be quite strong. The average revenue per ton-mile was £1.98 for the railway mode carrying the average of 228 miles, and £2.11 for the truck mode carrying the average distance of 165 miles. With a few exceptions, however, most medium-/long-haul t r a f f i c was moved - 140 -Table 7-4, Price E l a s t i c i t i e s and E l a s t i c i t y of Substitution ,, for CFTM61 (chemicals) . . Compensated ;. E l a s t i c i t y of 1 , 1 price e l a s t i c i t i e s Substitution Origin* ^ Dest'n.* Miles E r r hh 0 r h 33. St. John 53. Toronto 925 -.88 - .68 1.55 46. Montreal 25. Halifax 778 -.51 -1.12 1.63 46. Montreal 33. St. John 590 -.55 -1.06 1.61 46. Montreal 43. Quebec 137 -.81 - .74 1.55 46. Montreal 53. Toronto 335 -.68 - .87. 1.56 46. Montreal 71. Regina 1783 -.49 -1.17 1.66 53. Toronto 46. Montreal 335 -.69 - .86 1.55 53. Toronto 59. Sudbury 248 -.71 - .84 1. 55 53. Toronto 67. Winnipeg 1256 -.45 -1.26 1.71 56. Windsor 46. Montreal 564 -.59 - .99 1.58 56. Windsor 53. Toronto 229 -.71 - .84 1.55 56. Windsor 86. Edmonton 2132 -.56 -1.04 1.60 67. Winnipeg 73. Saskatoon 493 -.20 -3.58 3.77 67. Winnipeg 86. Edmonton 822 -.49 -1.18 1.66 83. Calgary 86. Edmonton 190 -.62 - .95 1.57 * Numbers preceding CFTM region codes names of o r i g i n and destination are the - 141 -by railways. The railways moved 68% of the t o t a l tonnage of t h i s t r a f f i c . Table 7-5 shows that the demand for railway service i s not p r i c e - e l a s t i c on any of the li n k s l i s t e d i n the table whereas the demand for truck mode i s p r i c e - e l a s t i c generally on those l i n k s , the distance of which exceeds about 400 miles. This implies that the e f f e c t i v e inter-modal competition st a r t s from very short distance but ends when the distance reaches 400 miles. Beyond th i s distance, the r a i l mode dominates the t r a f f i c . (5) The Results for CFTM69 (Refined petroleum products other  than gasoline, f u e l 'oily coke and gas) : The e l a s t i c i t i e s of demand and e l a s t i c i t y of substitution for t h i s commodity group are reported i n Table 7-6. The demand for railway service i s generally p r i c e - e l a s t i c on short-haul lin k s whose distance i s less than 300 miles while the demand for trucking service i s p r i c e - e l a s t i c on long-haul l i n k s over 1500 miles. Therefore, the e f f e c t i v e inter-modal competition i s l i k e l y to e x i s t for medium-haul t r a f f i c over the distance between 300 and 1500 miles. (6) The Results for CFTM75 (Metal fabricated basic products): Shippers of t h i s commodity group demonstrated a strong preference for the truck mode over railways even on many long-haul l i n k s . On those l i n k s whose distance i s less than 400 miles, the two conditions for truck-domination were s a t i s f i e d (see Table 7-7): i . e . , |E I > 1 and |E I > 2 • IE.. I . This ' 1 r r 1 1 r r 1 ' hh 1 - 142 -Table 7-5, Price E l a s t i c i t i e s and E l a s t i c i t y of Substitution  for CFTM66 (Fuel o i l other than gasoline) L i n k Compensated E l a s t i c i t y of price e l a s t i c i t i e s Substitution Origin* Dest'n.* Miles E - r r Ehh rh 25 . Halifax 32. Moncton 180 -.62 - .72 1.34 46 . Montreal 43. Quebec 173 -.76 - .58 1. 34 46 . Montreal 45. Sherbrooke 102 -.71 - .63 1. 34 46 . Montreal 53. Toronto 335 -.39 -1.04 1.43 46 . Montreal 59. Sudbury 436 -.34 -1.13 1.47 53 . Toronto 46. Montreal 335 -,5.3 - .83 1.36 53 . Toronto 56. Windsor 229 -.83 - .53 1.36 53 . Toronto 59. Sudbury 248 -.54 - .81 1. 35 53 . Toronto 67. Winnipeg 1256 -.38 -1.06 1.44 56 . Windsor 46. Montreal 564 -.37 -1.07 1.44 56 . Windsor 53. Toronto 229 -.58 - .76 1.34 67 . Winnipeg 50. T.-Bay 434 -.21 -1.65 1.86 67 . Winnipeg 86. Edmonton 822 -.43 - .96 1.40 71 . Regina 67. Winnipeg 356 -.91 - .47 1.38 95 . Vancouver 83. Calgary 649 -.53 - .83 1.36 * Numbers preceding CFTM region codes names of o r i g i n and destination are the - 143 -Table 7-6, Price E l a s t i c i t i e s and E l a s t i c i t y of Substitution for CFTM69 (Other refined petroleum products) L i n k Compensated E l a s t i c i t y of Price e l a s t i c i t i e s Substitution Origin* Dest'n.* Miles E r r hh rh 25. Halifax 21. Sydney 271 -1.00 -0.42 1.42 46. Montreal 25. Halifax 778 -0.69 -0.66 1. 35 46. Montreal 33. St. John 590 -0.41 -1.02 1.43 46. Montreal 43. Quebec 173 -1.14 -0. 35 1.49 46. Montreal 45. Sherbrooke 102 -1.64 -0.22 1.86 46. Montreal 53. Toronto 335 -0.79 -0.57 1. 36 46. Montreal 67. Winnipeg 1428 -0.52 -0.86 1.38 46. Montreal 95. Vancouver 2908 -0.20 -1.78 1.98 53. Toronto 33. St. John 925 -0.59 -0.77 1.36 53. Toronto 46. Montreal 335 -0.85 -0.52 1.37 53. Toronto 56. Windsor 229 -1.28 -0.30 1.58 53. Toronto 59. Sudbury 248 -0.90 -0.49 1.39 53. Toronto 67. Winnipeg 1256 -0.57 -0.79 1.36 53. Toronto 86. Edmonton 2077 -0.40 -1.04 1.44 67. Winnipeg 50. T.-Bay 434 -0.81 -0.56 1.37 83. Calgary 86. Edmonton 190 -1.35 -0.27 1.62 86. Edmonton 67. Winnipeg 882 -0.71 -0.64 1.35 86. Edmonton 73. Saskatoon 329 -0.83 -0.54 1.37 86. Edmonton 95. Vancouver 772 -1.02 -0.41 1.43 95. Vancouver 86. Edmonton 772 -0. 85 -0.52 1. 37 95. Vancouver 96, Vancouver Island 65 -2.42 -0.15 2.58 *Numbers preceding names of o r i g i n and destinations are the CFTM region codes. Table 7-7, E l a s t i c i t i e s o f Demand w i t h Respect t o P r i c e and Q u a l i t y V a r i a b l e s f o r CFTM75 (Metal f a b r i c a t e d b a s i c products) L i n k Compensated p r i c e e l a s t i c i t i e s Speed e l a s t i c i t i e s ** N O r i g i n * Dest'n.* M i l e s E r r 3* r h E 1 r r 25. H a l i f a x 53. Toronto 1113 -0.80 -0.56 1.36 -0.07 33. S t . John 46. Montreal 590 -0.94 -0.45 1.39 -0.09 46. Montreal 33. S t . John 590 -0.90 -0.48 1.38 -0.08 46. Montreal 43. Quebec 173 -1.15 -0.34 1.49 -0.11 46. Montreal 53. Toronto 335 -1.02 -0.41 1.43 -0.09 46. Montreal 67. Winnipeg 1428 -0.80 -0.56 1.36 -0.07 46. Montreal 95. Vancouver 2908 -0.60 -0.76 1.36 -0.05 53. Toronto 56. Windsor 229 -1.00 -0.42 1.42 -0.09 53. Toronto 67. Winnipeg 1256 -0.81 -0.55 1.36 -0.07 53. Toronto 83. Calgary 2087 -0.74 -0.61 1.35 -0.07 53. Toronto 95. Vancouver 2736 -0.65 -0.70 1.35 -0.06 67. Winnipeg 53. Toronto 1256 -0.84 -0.53 1.37 -0.08 67. Winnipeg 83. Calgary 831 -0.92 -0.47 1.39 -0.08 67. Winnipeg 95. Vancouver 1480 -0.65 -0.70 1.35 -0.06 95. Vancouver 53. Toronto 2736 -0.62 -0.74 1.36 -0.06 95. Vancouver 67. Winnipeg 1480 -0.73 -0.62. 1.35 -0.07 95. Vancouver 86. Edmonton 772 -0.87 -0.51 1.38 -0.08 E ,1 r h -0.78 -0.92 -0.88 -1.12 -1.00 -0.78 -0.58 -0.98 -0.80 -0.73 -0.64 -0.82 -0.90 -0.64 -0.60 -0.71 -0.85 ** 0.54 0.44 0.47 0.34 0.40 0.55 0.74 0.41 0.54 0.60 0.68 0.52 0.46 0.68 0.72 0.61 0.50 * Numbers preceding names o f o r i g i n and d e s t i n a t i o n are the CFTM r e g i o n codes. 0.04 0.04 0.04 0.03 0.03 0.05 0.07 0.04 0.05 0.05 0.06 0.05 0.04 0.06 0.07 0.06 0.04 R e l i a b i l i t y e l a s t i c i t i e s E 2 r r 0.12 0.14 0.13 0.17 0.15 0.12 0.09 0.15 0.12 0.11 0.09 0.12 0.13 0.09 0.09 0.11 0.13 E ,2 r h -0.78 -0.92 -0.88 -1.12 -1.00 -0.97 -0.79 -0.72 -0.64 -0.81 -0.89 -0.64 -0.60 -0.71 -0.85 0.55 0.44 0.47 0.34 0.40 -0.78 0.54 -0.58 0.74 0.41 0.53 0.59 0.68 0.52 0.46 0.68 0.72 0.61 0.49 -0.08 -0.07 -0.07 -0.05 -0.06 -0.08 -0.11 -0.06 -0.08 -0.09 -0.10 -0.08 -0.07 -0.10 -0.11 -0.09 -0.07 ** These e l a s t i c i t i e s may zero (see Table 6-2). be regarded as zero s i n c e the parameter estimate B r i s not s t a t i s t i c a l l y d i f f e r e n t from - 145 -means that the truck mode was dominant on the short-haul l i n k s . On the other hand, there i s not any l i n k which s a t i s f i e s the two conditions for rail-domination. Note that even on extremely long-haul l i n k s such as Montreal-Winnipeg and Toronto-Calgary l i n k s , the compensated demand for truck mode i s p r i c e - i n e l a s t i c . This implies that even the long-haul markets are not dominated by r a i l mode. For example, more than two-thirds of the t o t a l t r a f f i c moved from Montreal to Winnipeg was carr i e d by trucking mode even at the high average rate of 3.9 3 cents per ton-mile as compared to the railways' 2.60 cents per ton-mile. A sim i l a r s i t u a t i o n occurred on Toronto-Calgary l i n k : more than 60% of the t o t a l t r a f f i c was moved by trucks at an average rate of 5.04 cents per ton-mile as opposed to railways' average rate of 3.36 cents per ton-mile. From the above discussion, i t i s possible to conclude that the trucks dominate the t r a f f i c moving up to about 400 miles, and an e f f e c t i v e inter-modal competition exists for the t r a f f i c moving beyond i t . For thi s commodity group, there seems to be no railr-dominant distance range. This i s because of the e f f e c t of the difference i n qu a l i t y attributes of service between the two modes. Among the four d i f f e r e n t e l a s t i c i t i e s of demand with 1 2 respect to speed l i s t e d i n Table 7-7, E r r and E ^ r have wrong signs. This was caused due to the wrong sign of parameter - 146 estimate 8 r = 0.0973 reported i n Table 6-2, However, since the parameter estimate 8 r had asymptotic t^value of only 1 2 0.318, the E^^ and for thi s commodity group can be regarded as zero. The e l a s t i c i t y of demand for railway service with respect to trucking speed ( E ^ ) decreases generally as distance increases. On the other hand, the e l a s t i c i t y of demand for trucking service with respect to i t s own speed (E h^) tends to increase with distance of l i n k . A l l the estimated e l a s t i c i t i e s of demand with respect to r e l i a b i l i t y of service have correct signs. A comparison of 2 2 2 2 E r r and E ^ to E ^ and E ^ , respectively, shows that a change i n the r e l i a b i l i t y of trucking t r a n s i t time has far more influence on demands of both modes than same proportionate change i n r e l i a b i l i t y of railway t r a n s i t time. This can be explained by comparing the parameter estimates y^ = -0.1450 and y h = -0.8664 reported i n Table 6-2. Note also that the 2 2 parameter y^ used for computing E and E^ has asymptotic 2 2 t-value of only 1.281. The absolute values of E and E , J r r rh 2 2 tend to decrease with distance while those of E, , and E, tend hh hr to increase with distance. (7) The Results for CFTM78 (Non-metallic basic and fabricated  products): Similar to other commodities, the absolute value of E r r decreases with distance and of E,, increases with distance. hh An examination of Table 7-8 shows that, with a few exceptions, Table 7-8, Parameters of Price and Quality Responsiveness of Demand for CFTM78 (Non-metallic basic & fabricated products) Link Compensated price elasticities Speed elasticities Reliability elasticities ** ** ** Origin* ; Dest'n.* Miles E rr 5* arh E 1 rr E ,1 rh E 2 rr E ,2 rh ."W Ehr 2 25. Halifax . 24. Yarmouth 217 -0.98 -0.53 1.51 0.25 -1.21 0.65 -0.14 -0.08 -2.38 1.29 0.04 43. Quebec city 46. Montreal 173 -1.45 -0.33 1.78 0.37 -1.79 -0.40 -0.08 -0.12 -3.52 0.79 0.03 46. Montreal 25. Halifax 778 -0.54 -0.97 1.51. 0.14 -0.67 1.19 -0.25 -0.04 -1.31 2.34 0.08 46. Montreal 43. Quebec 173 -1.29 -0.38 1.67 0.33 -1.58 0.47 -0.10 -0.10 -3.11 0.92 0.03 46. Montreal 53. Toronto 335 -0.83 -0.65 1.48 0.21 -1.02 0.79 -0.17 -0.07 -2.01 1.56 0.05 46. Montreal 67. Winnipeg 1428 -0.52 . -1.00 1.52 0.13 -0.64 1.23 -0.26 -0.04 -1.26 2.43 0.08 46. Montreal 83. Calgary 2559 -0.35 -1.38 1.73 0.09 -0.43 1.69 -0.35 -0.02 -0.85 3.33 0.11 46. Montreal 95. Vancouver 2908 -0.29 -1.60 1.89 0.08 0.36 1.97 0.41 -0.02 -0.71 3.88 0.13 53. Toronto 46. Montreal 335 -0.81 -0.67 1.48 0.21 -0.99 0.82 -0.17 -0.06 -1.95 1.61 0.05 53. Toronto 56. Windsor 229 -1.01 -0.52 1.53 0.26 -1.24 0.63 -0.13 -0.08 -2.44 1.25 0.04 53. Toronto 67. Winnipeg 1256 -0.61 -0.87 1.48 0.16 -0.75 1.07 -0.22 -0.05 -1.48 2.11 0.07 53. Toronto 83. Calgary 2087 -0.67 -0.81 1.48 0.17 -0.82 0.99 -0.21 -0.05 -1.62 1.95 0.06 53. Toronto 95. Vancouver 2736 -0.24 -1.93 2.17 0.06 -0.30 2.37 -0.50 -0.02 0.59 4.67 0.16 95. Vancouver 71. Regina 1125 -0.25 -1.88 2.13 0.06 -0.31 2.31 -0.48 -0.02 -0.61 4.55 0.15 95. Vancouver 86. Edmonton 772 -0.66 -0.82 1.48 0.17 -0.80 1.01 -0.21 -0.05 -1.59 1.99 0.06 ** —] * Numbers preceding origin and destination are the CFTM region codes. ** These elasticity estimates may be regarded as zero because the parameter estimates 8^  and Yr are not statistically different from zero (see Table 6-2). - 148 -the railway demand i s p r i c e - e l a s t i c on the l i n k s whose distance i s l e s s than 200 miles, and the truck mode has a p r i c e - e l a s t i c demand on the l i n k s longer than 1200 miles. This implies that e f f e c t i v e inter-modal competition e x i s t s on those l i n k s , the distance of which i s between 200 and 1200 miles, leaving the t r a f f i c moving l e s s than 200 miles and farther than 1200 miles p r i m a r i l y to the truck and r a i l modes, r e s p e c t i v e l y . One exception i s the Toronto-Winnipeg l i n k (1256 miles) on which trucks moved more than 75% of the t o t a l tons transported during the year 1970. Unlike the rate on other l i n k s , the average rate charged by trucking mode on Toronto-Winnipeg l i n k (2.84 cents per ton-mile) was s l i g h t l y lower than the average rate charged by railways (2.87 cents per ton-mile). Another exception i s that truckers c a r r i e d more than two-thirds of t o t a l t r a f f i c on the Montreal-Toronto l i n k (335 miles) whereas railways c a r r i e d about 80% of the t o t a l t r a f f i c moving i n the opposite d i r e c t i o n . A l l the estimated e l a s t i c i t i e s of demand with respect to qu a l i t y v a r i a b l e s had the correct signs. As i n CFTM14 and 1 1 2 2 CFTM75,the absolute values of E ^ , E h r , E r r-and E h r are 1 1 2 2 fa r less than the absolute values of E r h , ^hh' E r h a n d Ehh' resp e c t i v e l y , meaning that the e f f e c t s on the demands f o r the two modes caused by a change i n the q u a l i t y a t t r i b u t e s of the r a i l mode i s r e l a t i v e l y smaller than those caused, by a si m i l a r change i n the q u a l i t y a t t r i b u t e s of the. truck mode. This can be explained by comparing the parameter estimates of quality—adjusted p r i c e functions of the two modes reported i n Table 6-2. - 149 -B r = -Q.2575 6 h = -1,2283 Y r = -0.0829 Y h = -2.4212 1 1 2 2 Furthermore, the e l a s t i c i t y estimates E , E, , E and E, r r hr' r r hr may be regarded as zero because the parameter estimates 8 r and y r are not s i g n i f i c a n t l y d i f f e r e n t from zero (see Table 6-2). 1 1 2 2 Generally, the absolute values of E , E , , E and E , r r ' rh' r r rh 1 1 2 2 decrease with distance whereas those of E.r, E, , E, , and E, hh' hr' hh hr increase with distance. (8) Summary about Inter-modal Competition: So far i n t h i s section, the l i n k - s p e c i f i c e l a s t i c i t y estimates were reported separately for each commodity group, and attempts were made to i d e n t i f y the range of distance over which an e f f e c t i v e inter-modal competition appears to e x i s t . Table 7-9 summarizes the previous discussions about the inter-modal competition. The re s u l t s roughly conform with expectations i n the following sense: (i) Normally for high-value (per ton) commodities such as CFTM14 ( f r u i t s , vegetables and edible foods), CFTM69 (other refined petroleum products), CFTM75 (metal fabricated basic products) and CFTM78 (non-metallic basic products), the truck mode dominates the short-haul t r a f f i c , and the r a i l - t r u c k competition exists for medium-haul and f a i r l y long-haul t r a f f i c . - 150 -Table 7-9, The Distance Range for E f f e c t i v e Inter-Modal Coirtpetition Commodity Group CFTM14 (Frui t s , vegetables and edible foods) CFTM52 (Lumber, including flooring) CFTM61 (Chemicals) CFTM66 (Fuel o i l except gasoline) CFTM69 (Other refined petroleum products) CFTM7 5 (Metal fabricated basic products) CFTM78 (Non-metallic basic products) Distance Range 400 - 1800 miles There i s no s i g n i f i c a n t inter-modal competition Up to 500 miles Up to 400 miles 300 - 1500 miles From 400 miles with no upper bound 200 - 1200 miles - 151 -( i i ) F o r CFTM61 (chemicals) and.CFTM66 ( f u e l o i l ) , r a i l - t r u c k competition i s active on short-distance l i n k s and the r a i l mode dominates medium- and long-haul t r a f f i c . This i s because shippers of t h i s commodity group are very sensitive to f r e i g h t rates, ( i i i ) Because of the r a i l mode's e f f i c i e n c y of handling lumber and f l o o r i n g (CFTM52), a c c e s s i b i l i t y to r a i l service i s the major determinant of shippers' mode-choice. Therefore, e f f e c t i v e r a i l - t r u c k competition does not seem to ex i s t even on short-haul routes. Since there i s no previous study of a similar type, i t i s not possible to compare these r e s u l t s with those of others. - 152 -Footnotes f o r C h a p t e r V I I : 1. For more d e t a i l on the proposed r e v i s i o n , see t h e p o l i c y -documents, T r a n s p o r t Canada [1975a, 1975b and 1975c]. 2. G i v i n g a t t e n t i o n t o a l l types o f c o m p e t i t i v e a p p e a l s i n s t e a d o f p r i c e a l o n e , C l a r k [1961 ] found t h a t c o m p e t i t i o n can be dynamic and e f f e c t i v e i n s p i t e o f market i m p e r f e c t i o n i n the modern economy. 3. Diewert [1974] i n t e r p r e t e d the e l a s t i c i t y o f s u b s t i t u t i o n as a n o r m a l i z a t i o n o f the c o r r e s p o n d i n g p r i c e - e l a s t i c i t y o f demand so t h a t a symmetric r e l a t i o n s h i p h o l d s , i . e . , o~ij = Ojq/ where a ^ j i s the e l a s t i c i t y o f s u b s t i t u t i o n between i n p u t s ii and j_. 4. The term " M a r s h a l l i a n ( o r d i n a r y ) demand", borrowed from consumption t h e o r y , r e f e r s t o the i n p u t demand when the l e v e l o f s h i p p e r s ' o u t p u t i s a l l o w e d t o v a r y i n response to changes i n f r e i g h t r a t e s ; i t t h e r e f o r e i s d i s t i n g u i s h e d from the i n p u t demand a l o n g an i s o g u a n t . P r o f e s s o r J.H.E. T a p l i n p r o v i d e d some i n s i g h t f u l s u g g e s t i o n s f o r m o d i f y i n g the A l l e n ' s formula t o f i t t o our s i t u a t i o n . 5. Suppose, f o r example, t h a t the s t a n d a r d commodity code (SCC) No. 476 ( w i r e s , i r o n o r s t e e l ) i s moved m a i n l y by the t r u c k mode, whereas SCC 456 ( f e r r o - a l l o y s ) i s moved by the r a i l mode. The a g g r e g a t i o n o f the two commodities i n t o a commodity group (CFTM72: s t e e l , i r o n and a l l o y s ) , would g i v e a f a l s e i m p r e s s i o n as i f the t r a f f i c i s shared between the r a i l and t r u c k modes, and thus l e a d t o o v e r - e s t i m a t i o n o f the i n t e r - m o d a l s u b s t i t u t i b i l i t y . CHAPTER VIII SUMMARY OF MAJOR FINDINGS AND SUGGESTIONS FOR FURTHER RESEARCH This chapter i s organized as follows: In Section A, the major findings of t h i s study are summarized on the basis of the discussions i n the preceding two chapters. In the process, an attempt i s made to compare these findings with those of others wherever i t i s appropriate to do so. x Section B presents several suggestions for further research and future research needs. (A) Summary of Major Findings The major findings of th i s study may be grouped into the following f i v e items. (1) The appropriate functional form for a f r e i g h t  demand model: The second-order parameter a ^ of the translog cost function was s t a t i s t i c a l l y s i g n i f i c a n t i n a l l the chosen models except that for the commodity group CFTM52 (lumber). Consequently, the e l a s t i c i t y of substitution between the two modes varies with the shares of expenditure as indicated by the formula i n equation (7,3b). Therefore, CES (constant e l a s t i c i t y of substitution) models including the Cobb-Douglas model are not appropriate to use as a fr e i g h t demand model. Logit models, which have been used most frequently i n fre i g h t demand studies, impose u n r e a l i s t i c a p r i o r i r e s t r i c t i o n s - 154 -both on the e l a s t i c i t y of substitution and on the price e l a s t i c i t i e s (see footnote 3 i n chapter I for the d e t a i l s of the r e s t r i c t i o n s ) . Throughout t h i s thesis we have seen that " f l e x i b l e " functions are appropriate to use to approximate the shippers' cost function and thus the demand functions, because they allow for a free v a r i a t i o n of the e l a s t i c i t i e s of substitution and the price e l a s t i c i t i e s of demand. (2) The variables to include i n a demand model: The r e s u l t s of hypothesis t e s t i n g i n chapter V have shown that the mode selection by the shippers of the r e l a t i v e l y high-value (per ton) commodities i s influenced not only by the fr e i g h t rates but also by the qu a l i t y attributes such as speed and r e l i a b i l i t y of speed, whereas prices are the single major mode-choice factor for the r e l a t i v e l y low-value (per ton) commodities. Turner [ l 9 75] has obtained more or less s i m i l a r r e s u l t s i n thi s regard from his l o g i t analysis: i . e . , the parameters associated with the t r a n s i t time and the v a r i a b i l i t y of t r a n s i t time were s t a t i s t i c a l l y s i g n i f i c a n t for most of the manufactured goods but they were s t a t i s t i c a l l y i n s i g n i f i c a n t for most i n d u s t r i a l raw materials. Recently, Levin [ l 9 78^ J estimated a l o g i t model as a function only of the differences i n fre i g h t rates, t r a n s i t time and v a r i a b i l i t y of t r a n s i t time between a pair of modes. In order to j u s t i f y the l o g i t model, which does not include the distance variable, he asserted that shipment mileage aff e c t s the mode selection only i n d i r e c t l y through changing - 155 -the f r e i g h t rates, t r a n s i t time and v a r i a b i l i t y of t r a n s i t 2 time. However, this thesis has shown, by choosing models which depend upon; the distance, that the shipper's choice p o s s i b i l i t y sets in the transportation-sectoral-technology space depend on the distance to transport a s p e c i f i c cargo. The distance a f f e c t s shipper's transportation-sectoral-technology d i r e c t l y as well as i n d i r e c t l y through i t s influence on the freight rates, speed and r e l i a b i l i t y of speed, implying that Levin's assumption postulated i n his l o g i t model does not seem to hold empirically. This i n turn implies that the distance variable should enter d i r e c t l y i n the demand model. A l l the eight chosen models reported i n chapter VI are d i f f e r e n t from one another. This implies that the shippers' transportation sectoral technology depends on the commodity type. Therefore, even without a formal s t a t i s t i c a l t e s t , i t may be concluded that the commodity att r i b u t e variables such as value and density of the commodity should be included i n a demand model i f the model i s to be estimated from the data which include heterogeneous commodities. By integrating the discussions so f a r , i t can be said that a demand model should include prices and distance i n any case, with an addition of the q u a l i t y a t t r i b u t e variables for the manufactured or high-valued goods and the commodity at t r i b u t e variables when i t i s estimated from the aggregate data over heterogeneous commodities. - 156 -(3) Mode-specific hedonic aggregators: For the r e l a t i v e l y high-value (per ton) commodities, the mode-choice of which i s s i g n i f i c a n t l y influenced by the qua l i t y attributes of service, the model with mode-specific hedonic aggregators (model D-3) was chosen as the r e s u l t of hypothesis t e s t i n g i n chapter V. This implies that shippers perceive a mode as i t s i n s t i t u t i o n a l e n t i t y rather than as a mere combination of the c h a r a c t e r i s t i c s of service i t has. The comparisons of the two hedonic aggregators have shown that the parameter estimates for the'truck mode are consistently larger i n absolute value than those for the r a i l mode. The author attributed t h i s to the following factors: (i) Shippers may have overperceived the qual i t y attributes of trucking service and/or underperceived those of railway service. ( i i ) The omitted q u a l i t y variables such as convenience, f l e x i b i l i t y , and completeness of service are l i k e l y to favouY* the truck mode. (4) Estimates of e l a s t i c i t y of substitution: The e l a s t i c i t y of substitution between the two modes reported i n chapter VII are a l l greater than one as a r e s u l t of the p o s i t i v e parameter estimate This also t e l l s that Cobb-Douglas model should not be used to estimate the fre i g h t demand functions. Table 7-1 shows that the e l a s t i c i t y of substitution evaluated at the mean values of the variables - 157 -v a r i e s f r o m 1.044 f o r CFTM52 ( l u m b e r ) t o 2.132 f o r CFTM71 ( s t e e l , i r o n a nd a l l o y s , e t c . ) . A s m e n t i o n e d i n c h a p t e r V I I , an a g g r e g a t i o n o f two o r more c o m m o d i t i e s o f a h e t e r o g e n e o u s n a t u r e c a u s e s an o v e r - e s t i m a t i o n o f s u b s t i t u t i b i l i t y . The same h o l d s f o r t h e c a s e o f an a g g r e g a t i o n o v e r h e t e r o g e n e o u s g e o g r a p h i c a l r e g i o n s . S i n c e t h e d a t a u s e d i n t h i s s t u d y s u f f e r s f r o m a g g r e g a t i o n p r o b l e m s , a l t h o u g h t o a l e s s e x t e n t t h a n m o s t o t h e r s t u d i e s , t h e 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 r e p o r t e d i n c h a p t e r V I I may h a v e b e e n O o v e r - e s t i m a t e d . The 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 c o m p u t e d f r o m t h e t r a n s l o g c o s t f u n c t i o n r e p o r t e d i n F r i e d l a e n d e r a n d S p a d y [ l 9 77] a r e , i n g e n e r a l , s u b s t a n t i a l l y h i g h e r t h a n t h o s e o f 3 t h i s s t u d y . T h i s may be b e c a u s e t h e y e s t i m a t e d t h e i r m o d e l f r o m t h e d a t a t h a t i s more h i g h l y a g g r e g a t e d c o m m o d i t y - w i s e and r e g i o n - w i s e : The e n t i r e U.S.A. was d i v i d e d o n l y i n t o t h r e e r e g i o n s a n d t h e n o n - a g r i c u l t u r a l . p r o d u c t s , . i n t o -t h e f o u r c o m m o d i t y g r o u p s . (5) I n t e r - m o d a l c o m p e t i t i o n : I n c h a p t e r V I I i t was shown t h a t a s t h e d i s t a n c e i n c r e a s e s , t h e c o m p e n s a t e d p r i c e e l a s t i c i t y f o r t h e r a i l mode d e c r e a s e s w h e r e a s t h a t f o r t h e t r u c k mode i n c r e a s e s . The r e l a t i v e v a l u e s o f p r i c e e l a s t i c i t i e s o f t h e two modes w e r e u s e d t o i d e n t i f y t h e r a n g e o f d i s t a n c e o v e r w h i c h e f f e c t i v e i n t e r - m o d a l c o m p e t i t i o n i s l i k e l y t o e x i s t . The r e s u l t s a r e r o u g h l y a s f o l l o w s : ( i ) F o r t h e r e l a t i v e l y l o w - v a l u e c o m m o d i t i e s s u c h a s - 158 -chemicals (CFTM61) and fu e l o i l (CFTM66), the inter-modal competition seems to e x i s t only for the short-haul t r a f f i c leaving the medium- and long-haul t r a f f i c primarily r a i l -dominated. One exception i s lumber and f l o o r i n g (CFTM52) for which no s i g n i f i c a n t inter-modal competition seems to ex i s t even i n short-haul markets. ( i i ) For the r e l a t i v e l y high-value products such as foods (CFTM14) , refined petroleum products (CFTM69), metal fabricated products (CFTM75) and non-metallic basic products (CFTM78), the inter-modal competition i s l i k e l y to e x i s t over a f a i r l y wide range of medium-distance markets. - 159 -(B) Suggestions for Further Research For any empirical research such as t h i s study, the qua l i t y of data and the choice of model are c r u c i a l l y impor-tant. In thi s study, for each (CFTM) commodity group, the shipper's transport unit cost function and the corresponding expenditure share functions were estimated from the data aggregated over the shippers of the commodity group on each l i n k . As explained i n chapter I I I , since the decision making unit for mode selection i s an i n d i v i d u a l shipper, i d e a l l y the model should be estimated from the disaggregated data on the i n d i v i d u a l shipper's production and d i s t r i b u t i o n a c t i v i t i e s over i t s entire d i s t r i b u t i o n network. Although t h i s i s the ide a l way to eliminate the p o t e n t i a l aggregation bias, the data are almost impossible to obtain because of the c o n f i d e n t i a l i t y of shipper's business information. The only p r a c t i c a l way to reduce the aggregation bias i s , therefore, to use data that i s as disaggregated as possible. The data used i n t h i s study, perhaps the lea s t aggregated one among the 4 studies which did not use survey or interview information, s t i l l suffers from the following aggregation problems: 1. Some of the CFTM commodity groups include a f a i r l y heterogeneous range of products as can be seen from Appendix 4A. For example, CFTM71 includes a d i v e r s i t y of items ranging from primary s t e e l and iron to the i n d u s t r i a l hardwares such as pipes, tubes, wires, etc. - 160 -2. The commodity flow data are compiled from region to region rather than from c i t y to c i t y . The true v a r i a b i l i t y i n mode-choice may have been concealed by the data aggregation, and thus the inter-modal s u b s t i t u t i b i l i t y may have been over-estimated. To reduce aggregation bias, therefore, the true i n t e r - c i t y flow data should be compiled separately for each homogeneous commodity, and each commodity-specific model should be estimated. In modelling f r e i g h t demand, a derived demand model should be used i n order to treat the f r e i g h t demand as an i n t e r -mediate input for production and d i s t r i b u t i o n a c t i v i t i e s of the firms. More empirical models should be estimated using " f l e x i b l e " functions which do not impose a p r i o r i r e s t r i c t i o n on the e l a s t i c i t y of substitution and can serve as the second order approximation to the a r b i t r a r y true function. So far i n the f r e i g h t transport area, only the translog function has been used i n Oum [l977] and Friedlaender and Spady [l977^ as well as i n ' t h i s t h e s i s . Other forms of f l e x i b l e function such as generalized Leontief function, generalized Cobb-Douglas function and quadratic mean of order-r functions should also be used i n future f r e i g h t demand studies to compare with the res u l t s of t h i s study. - 161 -Footnotes for Chapter VTTT: Since the r e s u l t s are not comparable between the studies which use both d i f f e r e n t models and d i f f e r e n t data, the r e s u l t s of t h i s study are compared only with the following studies: (i) Friedlaender and Spady [19 77], i n which the demand model derived from the translog cost function was estimated, ( i i ) Turner [l975], i n which the two-mode ( r a i l , truck) l o g i t model was estimated from data b a s i c a l l y the same as those i n t h i s thesis, and ( i i i ) Levin [1978], i n which the three-mode (truck, box car and piggyback) l o g i t model was estimated as a function of d i f f e r e n t i a l f r e i g h t rate, t r a n s i t time and v a r i a b i l i t y of t r a n s i t time. Levin f_19 78 ] c i t e d the r e s u l t s of the two shipper surveys conducted by Wood and Domencich [ l 9 7 l j and Kullman [1973] for the j u s t i f i c a t i o n of t h i s assumption. However, the authors of the two surveys did not test whether or not distance influences the mode selection only i n d i r e c t l y . The e l a s t i c i t i e s of substitution computed by summing the compensated price e l a s t i c i t i e s reported i n t h e i r paper across the two modes are: Durable manufactured: 1.715 Non-durable manufactured: 1.757 Petroleum and related: 1.709 Mineral, chemical and others: 1.935 Notice that the above figures are sub s t a n t i a l l y higher than those reported i n table 7-1 except that of CFTM71 ( 6 r n = 2.132) which also suffers from aggregation. 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Uzawa, H., "Duality P r i n c i p l e s i n the Theory of Cost and Production," International Economic Review, Vol. 5 (1964), pp. 216-299. Warner, S.L. "Stochastic Choice of Mode i n Urban Travel: A Study i n Binary Choice," Northwestern University Press, 1962. - 169 -Wilson, A.G., "A S t a t i s t i c a l Theory of S p a t i a l D i s t r i b u t i o n Models," Transportation Research, Vol. 1, No. 3 (1967) . Wilson, A.G., "Inter-regional Commodity Flows: Entropy Maximizing Approaches," Center for Environmental Studies, Working paper No. 19, 1968. Wilson, A.G., "The Use of Entrophy Maximizing Models i n the Theory of Tr i p D i s t r i b u t i o n and Modal S p l i t , " J . of  Transport Economics and Policy, V o l . 3, No. 1 (1969) . Woods, D.W. and Domencich, T.A., "Competition Between R a i l and Truck i n I n t e r c i t y Freight Transportation," Transportation  Research Forum, Proceedings of the 12th Annual Meeting, 19 71, pp. 25 7-288. - 170 -APPENDIX 3A Derivation of Linear Homogeneity Condition The l i n e a r homogeneity condition for the translog cost function (3,8) i s derived i n t h i s appendix. (1) InC [inPflnZ1,lnZ2,lnD] Zna + a^'lnP + b^'lnZ, + c t«ZnZ 0 + dlnD O 1 2. + 's(ZnP) t»A»lnP + H(lnZ1)t'B-lnZ1 + h (lnZ2)t-C' lnZ2 + hdd(lnD)2 + HXln-p)t'E'lnZ1 + h (InZ^) fc 'E^ • InV + h(lnV)t'F*lnZ2 + % {lnZ2)t *Ft • InT? + h (InZ^) t «G« lnZ2 + H(lnZ2)t*Gt*lnZ1 + InDigt'lnP) + Inu (n^ > InZ ^  + InDi^-lnZ^ where a = ( a r / a h ) t b = ( b r , b h ) t c = ( c r , c h ) t A = a a , r r rh a a rh hh B = b b . r r r h b , b, . rh hh C = c c , r r rh c r h chh E = ab ab . r r rh a b h r a b h h F = ac ac , r r rh a c h r a c h h G = be be , r r rh be, be, , hr hh ••= ( a d r , ad h) f c h = (bd,., b d j f c i = ( c d r , c d h ) t p = c p r l ; hi (Z r a }  Z h U z 2 - ( ^ ) - 171 -Linear homogeneity of the translog function with respect to prices (P) holds only i f equality (2) holds for any p o s i t i v e scalar X. (2) ZnC [ZnXP, ZnZ^ lnZ2, Inu"] = ln\C^\lnV, InZ^, ZnZ 2, InD] where ZnC \ l n P, InZ^, ZnZ,,, ZnD] = ZnAQ + a t«ZnP + A*(ZnXe) + b t«ZnZ 1 + c t*ZnZ 2 + dZnD + h (ZnP) t*A«ZnP + h (ln\e)t»A«(ZnXe) + h (ZnXe)t•A*ZnP + h (.ZnP) t«A» (ZnXe) + h ( Z n Z ^ t »B • ZnZ x + h (ZnZ 2) t«C•ZnZ 2 + h dd(ZnD) 2 + h (ZnP) t«E«ZnZ 1 + h (ZnXe) t»E•ZnZ^ + J5 ( ZnZx) 1'E 1 1 • ZnP + Js (ZnZ 1) t « E t • (ZnXe) + % ( ZnP) fc «F« ZnZ 2 + h (ZnXe) T « F « ZnZ 2 + 3j (ZnZ 2) fc•Ft* InP + h (ZnZ,,) «F^ ~ • (ZnXe) + h (ZnZ x) t«G*ZnZ 2 + h {lnZ2)t'Gt'ZnZ1 + ZnD (gt*ZnP) + ZnD (g^ZnXe) + ZnD (h t'ZnZ 1) ZnD (i t*ZnZ 2) where e = [ l , ! ] 1 " - 172 -In order for the q u a l i t y (2) to hold, the following conditions must be met: (a) a*"*ZnXe = InX (b) (ZnAe) t«A = 0_ and A'ZnXe = 0 (c) (ln\e)t'E = 0 (d) •(ZnXe) t-P = 0 (e) gt'{ln\e) = 0 where 0 = (0,0) or = My Therefore, the l i n e a r homogeneity condition imposes the following r e s t r i c t i o n s on the parameters of translog function (3,8) : (3) : (a) e*r + a h = 1 ( b ) a r r + a r h = °' a r h + *hh = ° ^ P 1 * 1 ^ * r r = " a r h = ahh' CO a b r r + ab h r= 0, a r h + ab f a h= 0 Cd) a c r r + a c h r = 0, a c r h + a c h h = 0 Ce) . a d r + a d h = 0 - 173 -APPENDIX 4A L i s t of Commodities Included i n  the Eight CFTM Commodity Groups CFTM 14 (Fruits, Vegetables and Edible Foods): SCC code Description 076 Dried and dehydrated f r u i t s 078 F r u i t juices, and f r u i t juice concentrates, not frozen 080 F r u i t juice concentrates, frozen 082 F r u i t s and f r u i t preparation, n.e.s. 084 Nuts, except o i l nuts 104 Vegetables, dried 106 Vegetables and preparations, n.e.s. 112 Sugar preparations (inc. confectionery), n.e.s. 114 Coffee 116 Cocoa and chocolate, tea, spices and vinegar 118 Margarine and sim i l a r products 120 Shortening and l a r d 122 Soups and infant and junior foods 124 Pre-cooked frozen food preparations 126 Food preparations and materials for food preparations, n.e.s. CFTM 52 (Lumber including f l o o r i n g ) : SCC code Description 30 8 Lumber 310 Flooring CFTM 61 (Chemicals): SCC code Description 378 Carbon blacks 380 Chemical elements 384 Inorganic acids and oxygen compounds of non-metals or metalloids, n.e.s. 386 Sodium hydroxide 388 Inorganic bases and me t a l l i c oxides, hydroxides and peroxides, n.e.s. 390 Sodium sulphate 392 Sodium carbonate 394 Me t a l l i c s a l t s and peroxy-salts of inorganic acids, n.e.s. 396 Calcium carbide 398 Inorganic chemicals, other, n.e.s. CFTM 61 (Chemicals) continued: SCC code Description 400 Hydrocarbons and the i r derivatives 402 Alcohols and t h e i r derivatives 404 Phenols, ethers, aldehydes, ketones and t h e i r derivatives 406 Organic acids, t h e i r anhydrides, halides, peroxides, peracids, and derivatives 408 Nitrogen function compounds 410 Organic chemicals, n.e.s. CFTM 66 (Fuel o i l other than gasoline): SCC code Description 436 Aviation turbine f u e l 4 38 Diesel fuel 440 Kerosene 442 Fuel o i l , n.e.s. CFTM 69 (Refined petroleum products other than coke and gases) SCC code Description 444 Lubricating o i l s and greases 452 Asphalts and road o i l s 4 54 Other petroleum and coal products CFTM 71 (Steel, iron and a l l o y s , etc.) : SCC code Description 456 Ferro-alloys 458 Pig iron 460 Ingots, blooms, b i l l e t s and slabs, iron and st e e l 4 61 Primary iron and s t e e l , n.e.s. 462 Castings and forgings, iron or s t e e l 464 Bars and rods, s t e e l 466 Plates, s t e e l , fabricated 468 Sheet and strip,, s t e e l 4 70 Structural shapes and sheet p i l i n g , i r o n or ste e l 472 Rails and railway track materials 474 Pipes and tubes, iron and s t e e l 4 76 Wire, iron or s t e e l - 175 -CFTM 75 (Metal fabricated basic products): SCC code Description 496 Tanks 498 Bolts, nuts, n a i l s , screws and basic hardware 500 Metal fabricated basic products, n.e.s. CFTM 78 (Non-metallic basic products): SCC code Description 502 Natural stone basic products, c h i e f l y s t r u c t u r a l 504 Bricks and t i l e s , clay 506 F i r e brick and similar shapes 508 Dolomite and magnesite, calcined 510 Refractories, n.e.s. 512 Glass basic products 514 Asbestos and asbestos-cement basic products 518 Concrete pipe 520 Cement and concrete basic products, n.e.s. 522 Plaster 52 4 Gypsum wallboard and sheathing 52 6 Gypsum basic products, n.e.s. 52 8 Lime, hydrated and quick 5 30 Non-metallic mineral basic products, n.e.s. 5 32 Bituminous pressed or molded fabricated materials 534 Miscellaneous fabricated materials - 176 -APPENDIX 5A Tables of Test S t a t i s t i c s and the Results of Hypotheses Testing The values of the logarithm of l i k e l i h o o d functions evaluated 2 at the maximum l i k e l i h o o d estimates and R values for the cost and revenue share functions are reported i n tables (5A-1) to (5A-8). Tables (5A-9) to (5A-16) report the detailed r e s u l t s of the f i r s t and second stage t e s t s . The following i s a l i s t of the tables by commodity group: Commodity group Tables CFTM 14 5A-1, 5A-9 5A-2, 5A-10 5A-3, 5A-11 5A-4, 5A-12 5A-5, 5A-13 5A-6, 5A-14 5A-7, 5A-15 5 A-8, 5A-16 CFTM 52 CFTM 61 CFTM 66 CFTM 69 CFTM 71 CFTM 75 CFTM 78 - 177 -Table 5A-1, Test S t a t i s t i c s for Commodity Group CFTM14: F r u i t s , Vegetables and Edible  Foods (using 133 l i n k observations) _ 2 Model and Sub-model General Model (A) (A-l) (A-2) (A-3) No. of free Parameters 28 15 6 -635.822 -644.605 -650 .892 0. 8714 0.8596 0.8460 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -641.1 0.8595 (B-2) 11 -647.103 0.8552 (B-3) 4 -682.177 0.7994 Model with Mode-Specific Hedonic Aggregators (C) (C-l) 9 -646.74 0.8589 (.02) 7 -658.267 0.8525 (C-3)* 5 -670.689 0.8270 Model with Id e n t i c a l Hedonic Aggregators (D) (D-l) 7 -657.646 0.8511 (D-2) 6 -658.369 0.8523 (D-3)* 5 -670.689 0.8270 R 0.3754 0.3564 0.3536 0.3590 0.3502 0.0215 0.3636 0.3007 0 .2722 0.3016 0.2999 0.2722 2 R = The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 = R value for the translog cost function. R = R value for the modal share functions, *Models (C-3) and (D-3) are i d e n t i c a l . - 178 -Table 5A-2, -Test S t a t i s t i c s for Commodity Group CFTM5 2: Lumber Including Flooring (using 52 l i n k observations) Model and No. of free Sub-model Parameters ln<L R 2 c R 2 s General Model (A) (A-l) 28 -236.250 .7007 .0932 (A-2) 15 -242.445 .7396 .0202 (A-3) 6 -249.925 .6609 .0135 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -237.373 .6980 .0821 (B-2) 11 -244.155 .6527 .1042 (B-3) 4 -252.165 .6546 .0001 Model with Mode-specific Hedonic Aggregators (C) (C-l) 9 -247.023 .6870 .0305 (C-2) 7 -249.783 .6784 .0312 (C-3) * 5 -249.935 .6621 .0153 Model with Identical Hedonic Aggregators (D) CD-I) 7 -247.791 .6725 .0072 (D-2) 6 -248.557 .6732 .0190 (D-3)* 5 -249.935 .6621 ,0153 The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 R value for the translog cost function. 2 R value for the modal share functions. *Models (C-3) and (D-3) are i d e n t i c a l . Table 5A-3, \Test S t a t i s t i c s for Commodity Group CFTM61 Chemicals (using 86 l i n k observations) Model and Sub-model General Model (A) (A-l) (A-2) (A-3) No. of free Parameters 28 15 6 ln£> -397.607 -401.526 -409.718 R 0 .7558 0 .7504 0 .7008 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -400.357 0.7470 (B-2) 11 -405.466 0.7211 (B-3) 4 -416.264 0.6718 Model with Mode-specific Hedonic Aggregators (C) (C-l) 9 -408.965 0.7068 (C-2) 7 -409.982 0.7060 (C-3)* 5 -411.166 0.7002 Model with Id e n t i c a l Hedonic Aggregators (D) (D-l) 7 -410.077 0.7054 (D-2) 6 -410.140 0.7049 (D-3)* 5 -411.166 0.7002 R s: .2247 .1947 ,2041 .2210 .2115 .0897 .2038 .1911 .1890 .1920 .1915 .1890 ln& =• The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 2 R = R value for the translog cost function, c 3 2 2 R = R value for the modal share functions. *Models (C-3) and (D-3) are i d e n t i c a l . - 180 -Table 5A-4 Test S t a t i s t i c s for Commodity Group CFTM66: Fuel O i l Model and Sub-model General Model (A) (A-l) (A-2) (A-3) (using 65 l i n k observations) No. of free Parameters R 28 15 6 -323.210 -326.629 -334.696 Q .7248 0 .7106 0 .6721 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -324.928 0.7205 (B-2) 11 -333.609 0.7199 (B-3) 4 -338.515 0 .6312 Model with Mode-specific Hedonic Aggregators (C) (C-l) 9 -332.386 0.6926 (C-2) 7 -332.720 0.6827 (C-3)* 5 -335.264 0.6623 Model with I d e n t i c a l Hedonic Aggregators (D) (D-l). 7 -334.200 0 .6831 (D-2) 6 -334.747 0.6755 (D-3)* 5 -335.264 0.6623 0 0 0 0 0 0 0 0 0 0 . 0 . 0 . .1015 .1012 ,0431 0873 0834 0320 0546 0550 0447 0537 0528 0447 ln<L = The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 2 R = R value for the translog cost function, c ^ 2 2 R = R value for the modal share functions, s *Models (C-3) and (D-3) are i d e n t i c a l . - 181 -Table 5A-5, Test S t a t i s t i c s for Commodity Group  CFTM69: Refined Petroleum Products (using 77 l i n k observations) Model and No. of free 7- f „ 2 2 Sub-model Parameters _ c s General Model (A) (A-l) 28 -356.286 0.9102 0.2836 (A-2) 15 -358.271 0.9007 0.2838 (A-3) 6 -362.732 0.8859 0.2835 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -357.542 0 .9089 0 .2719 (B-2) 11 -361.320 0 .8968 0 .2572 (B*3) 4 -383.776 0 .8389 0 .0442 Model with Mode-specific Hedonic Aggregators (C) (C-l) 9 -360.967 0 .8943 0 .2691 (C-2) 7 -365.806 0 .8701 0 .2465 (C-3)* 5 -366.266 0 .8802 0 .2479 Model with Identical Hedonic Aggregators (D) CD-I) 7 -364.704 0 .8881 0 .2422 (D-2) 6 -366.150 0 .8804 0 .2487 (D-3) * 5 -366.266 0 .8802 0 .2479 = The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 2 R c = ' R value for the translog cost function. 2 2 R g = R value for the modal share functions. *Models (C-3) and (D-3) are i d e n t i c a l . - 182 -Table 5A-6, Model and Sub-model General Model (A) (A-l) (A-2) (A-3) Test S t a t i s t i c s for Commodity Group  CFTM71: S t e e l / Irons and Alloys (using 151 l i n k observations) No. of free Parameters 28 15 6 R -754.384 -754.919 -770.533 0 .9230 0.9167 0.9031 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -759.230 0.9189 (B-2) 11 -760.100 0.9159 (B-3) 4 -803.767 0.8537 Model with Mode-specific Hedonic Aggregators (C) (C-l) 9 -771.330 0.9055 (C-2) 7 -774.156 0.9045 (C-3)* 5 -774.303 0.9039 Model with I d e n t i c a l Hedonic Aggregators (D) (D-l) 7 -774.104 0.9052 (D-2) 6 -774.297 0.9040 (D-3)* 5 -774.303 0.9039 0 0 0 0 0 0, 0, 0, 0, 0 , 0 , 0. 3810 3923 3560 3782 3605 0981 3550 3322 3334 3331 3326 3334 ln& = The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates, 2 2 R = R value for the translog cost function, c 3 2 2 R = R value for the modal share functions. s *Models (C-3) and (D-3) are i d e n t i c a l . - 183 -Table 5A-7, Test S t a t i s t i c s for Commodity Group CFTM75: Metal Fabricated Basic Products (using 137 l i n k observations) Model and No. of free In it " 2 ~ 2 Sub-model Parameters R R c s General Model (A) (A-l) 28 -645.499 0.8377 0.2698 (A-2) 15 -656.541 0.8309 0.2742 (A-3) 6 -751.292 0.8060 0.3038 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -652.478 0.8350 0.2577 (B-2) 11 -662.574 0.8265 0.2491 (B-3) 4 -770,771 0.8178 0.3206 Model with Mode-specific Hedonic Aggregators (C) (C-l) 9 -658.974 0.8226 0.3111 (C-2) 7 -664.516 0.8215 0.3111 (C-3)* 5 -759.663 0.7980 0.3291 Model with Identical Hedonic Aggregators (D) CD-I) 7 -662.851 0.8130 0.2920 (D-2) 6 -665.333 0.8178 0.3206 (D-3)* 5 -759.663 0.7980 0.3291 Int. = The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 2 R «= R value for the translog cost function. c 2 2 R = R value for the modal share functions. *Models (C-3) and (D-3) are i d e n t i c a l . - 184 -Table 5A-8, Test S t a t i s t i c s for Commodity Group CFTM7 8: Non-metallic Basic Products (using 156 l i n k observations) No. of free Sub-model Parameters Model and e , n 2 2 General Model (A) (A-l) 28 -844.632 0.7548 0.3702 (A-2) 15 -850.886 0.7258 0.3543 (A-3) 6 -859.013 0.7191 0.3453 Model S t r i c t l y Independent of Distance (B) (B-l) 22 -845.685 0.7416 0.3584 (B-2) 11. -854.979 0.7124 0.3515 (B-3) 4 -889.667 0.6198 0.1230 Model with Mode-specific Hedonic Aggregators (C) (C-l) 9 -853.032 0.7216 0.3498 (C-2) 7 -860.165 0.7052 0.3388 (C-3)* 5 -861.214 0.7018 0.3344 Model with Identical Hedonic Aggregators (D) (D-l) 7 -860.135 0.7074 0.3394 (D-2) 6 -860,219 0.7053 0.3384 (D-3)* 5 -861.214 0.7018 0.3344 Zn<£ = The value of natural logarithm of l i k e l i h o o d function evaluated at the ML parameter estimates. 2 2 R c = R value for the translog cost function. 2 2 R = R value for the modal share functions, s *Models („C-3) and (D-3) are i d e n t i c a l . Table 5A-9, Hypotheses Testing for CFTM14: F r u i t s / Vegetables and Edible Foods (A) Test amongst the three sub-models ( F i r s t stage t e s t s ) : test s t a t i s t i c (-2ln\) and degrees of freedom 2 Degrees X ~ c r i t i c a l Test Model A Model B Model C Model D of freedom value at a : (i) H Q : sub-model 3 (22) * (18) (4) (2) 1 3 .841 H ^ : sub-model 1 30.140 ** 82.154 47.898 26.086 2 5 .991 ( i i ) H Q : sub-model H , : sub-model 3 (9) (7) (2) (1) 4 9 .488 2 12.674 70.148 24.844 24 .64 7 14 .067 l 9 16 .919 ( i i i ) H Q : sub-model 2 (13) ( I D (2) (1) 11 19 .675 H ^ : sub-model 1 17.566 12.006 23.054 1.446 13 22 . 362 Chosen sub-model (A-3) (B-2) (C-l) (D-2) 18 28 . 869 No. of free parameters 6 11 9 6 22 33 .924 -650.892 -647.103 -646.74 -658.369 R 2 c .8460 .8552 .8589 • ,852 3 * Figures reported i n parentheses are the degrees of freedom for the respective tests, ** Figures reported on the same l i n e as H, are the test s t a t i s t i c s for the respective te s t s . (B) Tests amongst the chosen sub-models (Second stage t e s t s ) : 2 Test Test s t a t i s t i c (-2ZnX) Degrees of Freedom X~ c r i t i c a l value at a=.05 Test Result. (1) choice between models (D-2) and (A-3) * * * favours (A-3) due to higher Int, value. (2) V model (D- 2) model (B- 2) (3) model (D- 2) H l ! model (C-1) (4) H o : model ( A - 3) H-: model ( c - 1) (5) choice between models (C-l) and (B-2) (6) H H, 0 model (A-3) model (B-2) 22,532 23.258 8. 304 *** 6.126 9.236 6.251 6.251 favours (B-2) favours (C-l) favours (C-l) oo 9 .236 favours (C-l) because i t has higher InSj value and smaller number of parameters favours (A-3) Model (C-l) i s f i n a l l y chosen for use, *** The two models that are compared have the same number of parameters. In t h i s case, the model with a larger value of the li k e l i h o o d function was chosen. Table 5A-10, Hypotheses Testing for CFTM52; Lumber Including Flooring (A) Tests amongst the three sub-models ( F i r s t stage tests) Test Model A Model B Model C Model D (i) H Q : sub-model 3 (22) * (18) (4) (2) H ^ : sub-model 1 27.35** 29.584 5.824 4. 288 ( i i ) HQ : sub-model 3 (9) (7) (2) (1) H ^ : sub-model 2 14.96 16.02 .304 2.756 ( i i i ) H Q : sub-model 2 (13) ( I D (2) (1) H ^ : sub-model 1 12. 39 13.564 5.52 1.532 Chosen sub-model (A-3) (B-3) (C-3) (D-3) No. of free parameters 6 4 5 5 InlL -249 .925 -252.165 -249.935 same as i R 2 c .6609 .6546 .6621 Degrees X - c r i t i c a l of freedom value at a=.05 1 3.841 2 5.991 4 9 .488 7 14.067 9 16 .919 11 19 .675 13 22.362 18 28.869 22 33.924 00 * Figures reported i n parentheses are the degrees of freedom for the respective t e s t s . ** Figures reported on the same l i n e as H.. are the test s t a t i s t i c s for the respective tests. (B) Tests amongst the chosen sub-models (Second stage tests) Test Test s t a t i s t i c Degrees of X - c r i t i c a l value i-2ln\) Freedom at a=.05 Test Result (1) H o : model (D- 3) H l : model (A- 3) (2) H o : model (D- 3) H l : model (B- 3) (3) H o : model (D- 3) H l : model (C- 3) (4) H o : model (C- 3) H l : model (A- 3) (5) H o : model ( c - 3) H l : model (B- 3) (6) H o : model (B-3) H l : model (A- 3) Model (D-3) = (C-3) 02 4.46 these are an i d e n t i c a l model, same r e s u l t as i n (1) same r e s u l t as i n (2) 4.48 3.841 3.841 5.991 favours (D-3) favours (D-3) favours (B-3) Table 5A-11, Hypotheses Testing for CFTM61: Chemicals (A) Tests amongst the three sub-models ( F i r s t stage tests) test s t a t i s t i c (~2lnX) and degrees of freedom -Test Model A Model B Model C Model D Degrees X~ c r i t i c a l of freedom value at a=.05 (i) H Q : sub-model 3 (22) * (18) (4) (2) 1 3 .841 H ^ : sub-model 1 24.222 ** 31.814 4.402 2.178 2 5 .991 ( i i ) H _ : sub-model 3 (9) (7) (2) (1) 4 9 .488 H , : sub-model 2 16.384 21.596 2.368 2.052 7 14 .067 X 9 16 .919 ( i i i ) H Q : sub-model 2 (13) (11) (2) (1) 11 19 .675 H , : sub-model 1 7.838 10.218 2.034 0.126 13 22 .362 1 Chosen sub-model (A-3) (B-2) (C-3) (D-3) 18 28 .869 No. of free parameters 6 11 5 5 22 33 .924 -409.718 -405.466 -411.166 -411.166 R 2 .7008 .7211 .7002 .7002 00 ** Figures reported i n parentheses are the degrees of freedom for the respective tests, Figures reported on the same l i n e as H. are the test s t a t i s t i c s for the respective t e s t s . IB) Tests amongst the chosen sub-models (Second stage tests) Test (1) H H. (2) H H. 0 0 model (D-3) model (A-3) model (D-3) model (B-2) Test s t a t i s t i c (-2ZnA) 2.896 11.4 (3) models (D-3) and (C-3) are i d e n t i c a l (4) H H. (5) H H. (6) H 0 0 0 model (C-3) model (A-3) model (C-3) model (B-2) model (A-3) model (B-2) Degrees of Freedom X~ c r i t i c a l value at a=.05 r e s u l t i s exactly same as i n (1) r e s u l t i s exactly same as i n (2) 8.504 5 3.841 12.592 Test Result favours (D-3) favours (D-3) 11,071 favours (A-3) Model (D-3) = model (C-3) i s f i n a l l y chosen for use. Table 5A-12, Hypotheses Testing for CFTM66: Fuel O i l (A) Tests amongst the three sub-models ( F i r s t stage tests) test s t a t i s t i c (-2ZnA) and degrees of freedom Test (i) ( i i ) H ( H, H ( H, ( i i i ) H H, No. Z n o C R 2 0 Degrees X~ c r i t i c a l Model A Model B -Model C Model D of freedom value at < sub-model 3 (22) * (18) (4) (2) 1 3 .841 sub-model 1 22.972 ** 27.174 5.756 2.128 2 5 .991 sub-model 3 (9) (7) (2) (1) 4 9 .488 sub-model 2 16.134 9 .812 5.088 1.034 7 14 .067 9 16 .919 sub-model 2 (13) (11) (2) (1) 11 19 .675 sub-model 1 6.838 17.362 .668 1.094 13 22 .362 model (A-3) (B-3) (C-3) (D-3) 18 28 .869 : parameters 6 4 5 same as (C-3) 22 33 .924 -334.696 -338.515 -335.264 .6721 .6312 16623 *. Figures reported i n parentheses are the degrees of freedom for the respective tests, ** Figures reported on the same l i n e as H1 are the test s t a t i s t i c s for the respective te s t s . (B) Tests amongst the chosen sub-models (Second stage tests) 2 Test (1) V model (D- 3) H l : model (A- 3) (2) V model (D- 3) H l : model (B- 3) (4) H Q : model (C-3) H l : model (A-3) (5) H Q : model (B-3) H l : model (C-3) (6) H Q : model (B-3) H l : model (A-3) Model (D-3) = (C-3) i s Test s t a t i s t i c (-2ln\) 1.136 Degrees of Freedom-6.502 1 (3) Models (4,11C) and (4,18C) are i d e n t i c a l same r e s u l t as i n (1) same r e s u l t as i n (2) 7.638 2 X- c r i t i c a l value at a=.05 3.841 3.841 5.991 Test Result favours (D-3) favours (D-3) favours (A-3) Table 5A-l3, Hypotheses Testing for CFTM69: Refined Petroleum Products (A) Tests amongst the three sub-models ( F i r s t stage tests) test s t a t i s t i c (-2ln\) and degrees of freedom 2 . . Degrees X - c r i t i c a l Test Model A Model B Model C Model D of freedom value at i (i) HQ : sub-model 3 (22) * (18) (4) (2) 1 3 .841 H ^ : sub-model 1 12.892 ** 52.468 10.598 3.124 2 5 .991 ( i i ) H Q : sub-model H , : sub-model 3 (9) (7) (2) (1) 4 9 .488 2 8.922 44.912 0.92 .232 7 14 .067 ± 9 16 .919 ( i i i ) H Q : sub-model 2 (13) (11) (2) (1) 11 19 .675 H ^ : sub-model 1 3.97 7.556 9.678 2 .892 13 22 . 362 Chosen sub-model (A-3) (B-2) (C-l) (D-3) 18 28 .869 No. of free parameters 6 11 9 5 22 33 .924 ln£. -362.732 -361.320 -360.967 -366.266 R 2 c . 8859 .8968 .8943 . 8802 * Figures reported i n parentheses are the degrees of freedom for the respective t e s t s . ** Figures reported on the same l i n e as H^ are the test s t a t i s t i c s for the respective t e s t s . IX> (B) Tests amongst the chosen sub-models Test s t a t i s t i c Test (-2ln\) (1) V model (D-3) model (A-3) 7.068 (2) H o : model (D-3) H l : model (B-2) 9.892 (3) model (D-3) model (C-l) 10.598 (4) H o : model (A-3) model (C-l) 7.068 (5) H o : model (C-l) H 1; model (B-2) .706 (6) H o : model (A-3) model (B-2) 2.824 Model (A-3) i s f i n a l l y chosen for use. (Second stage t e s t s ) : Degrees of X - c r i t i c a l value Freedom at a=.05 Test Result 1 3.841 favours (A-3) 6 12.592 favours (D-3) 4 9.488 favours (C-l) 3 7.815 favours (A-3) 2 5.991 favours (C-l) 5 11.071 favours (A-3) Table 5A-14, Hypotheses Testing for CFTM71; Steel, Irons and Alloy (A) Tests amongst the three sub-models ( F i r s t stage t e s t s ) : t e s t s t a t i s t i c {-2ln\) and degrees of freedom Test Model A Model B Model C Model D Degrees of freedom 2 . . X - c r i t i c a l value at a=.05 (i) H Q : sub-model 3 (22) * (18) (4) (2) 1 3.841 H ^ : sub-model 1 32.398 ** 89.074 5.946 .398 2 5.991 ( i i ) H Q : sub-model H , : sub-model 3 (.9) (7) (2) CD 4 9.488 2 31.228 87.334 0.294 .012 7 14.067 , ( i i i ) i H Q : sub-model H ^ : sub-model 2 1 (13) 1.07 (11) 1.74 (2) 5.652 (D .386 9 11 13 16.919 M CD 19 .675 0 1 22.362 ' Chosen sub-model (A-2) (B-2) (C-3) (D-3) 18 28 .869 No. of free parameters 15 11 5 5 22 33.924 -754.919 -760.1 -774.303 same as (C-3) R 2 c .9167 .9159 .9039 *-Figures reported i n parentheses are the degrees of freedom for the respective t e s t s . ** Figures reported on the same l i n e as H are the test s t a t i s t i c s for the respective tests. (B) Tests amongst the chosen sub-models (Second stage tests) 2 Test Test s t a t i s t i c (-2ln\) Degrees of - Freedom X- c r i t i c a l value at a=.05 \ Test Result (1) V model model (D-3) (A-2) 38.768 10 18 .307 favours (A-2) (2) H o : H l : model model (D-3) (B-2) 28.406 6 12 .592 favours (B-2) (3) Models (4,11C) and (4,18C) are i d e n t i c a l (4) H o : H l : model model (C-3) (A-2) same r e s u l t as i n (1) (5) H o : H l : model model (C-3) (B-2) same r e s u l t as i n (2) (5) H o : H l : model model (B-2) (A-2) 10.362 4 9 .488 favours (A-2) Model (A-2) i s f i n a l l y chosen for use. Table 5A-15, Hypotheses Testing for CFTM75: Metal Fabricated Basic Products (A) Tests amongst the three sub-models ( F i r s t stage t e s t s ) : test s t a t i s t i c (-27-nX) and degrees of freedom Degrees ;x~ c r i t i c a l Test Model A Model B Model C Model D of freedom value at a= .05 (i) HQ: sub-model 3 (22) * (18) (4) (2) 1 3 .841 H^: sub-model 1 211.586' **236.586 201.378 193.694 2 5 .991 ( i i ) H N: sub-model 3 (9) (7) (2) (1) 4 9 .488 u H,: sub-model 2 189.502 216.394 190.294 188.73 7 14 .067 i i 9 16 .919 i- 1 ( i i i ) HQ: sub-model 2 (13) (11) (2) (1) 11 19 .675 -J H N: sub-model 1 22.084 20.192 11.084 4.964 13 22 .362 1 1 Chosen sub-model (A-2) (B-l) (C-l) (D-l) 18 28 .869 No. of free parameters 15 22 9 7 22 33 .924 -656.541 -652.478 -658.974 -662.851 R 2 c .8309 .8350 < .8226 ,8130 * Figures reported i n parentheses are the degrees of freedom for the respective t e s t s . ** Figures reported on the same l i n e as H1 are the test s t a t i s t i c s for the respective te s t s . (B) Tests amongst the chosen sub-models Test s t a t i s t i c Test (~2ln\) -(1) H 0 ! model (D-l) model (A-2) 12 .6.2 (2) Ho : model (D-l) model (B-l) 20.746 (3) V model (D-l) H l : model (C-l) 7. 754 (4) Ho : model (C-l) H l : model (A-2) 4.866 (5) V model (C-l) model (B-l) 12.992 (.6) Ho : model (A-2) H l : model (B-l) 8.126 Model (C-l) i s f i n a l l y chosen for use. (Second stage t e s t s ) : Degrees of X - c r i t i c a l value Freedom at g=.05 Test Result 8 15.507 favours (D-l) 15 24.996 favours (D-l) 2 5.991 favours (C-l) 6 12.592 favours (C-l) 13 22.362 favours (C-l) 7 14.067 favours (A-2) Table 5A-16, Hypotheses Testing for CFTM78: Non-metallic Basic Products (A) Tests amongst the three sub-models ( F i r s t stage tests) test s t a t i s t i c (,-2ln\) and degrees of freedom Test Model A Model B Model C Model D Degrees X - c r i t i c a l of freedom value at a-.05 (i) HQ: sub-model 3 (22) * (18) (4) (2) 1 3 .841 H^: sub-model 1 28.762 87.964 16.364 2.158 2 5 .991 (i i ) H_: sub-model 3 (9) (7) (2) (1) 4 9 .488 u H,: sub-model 2 16.254 69.376 2.098 1.99 7 14 .067 ± 9 16 .919 ( i i i ) HQ : sub-model 2 (13) (11) (2) (1) 11 19 .675 H^: sub-model 1 12.508 18.588 14.266 0.168 13 22 .362 Chosen sub-model (A-3) (B-2) (C-l) CD-I) 18 28 .869 No. of free parameters 6 11 9 5 22 33 .924 Inij -859.013 -854.979 -853.032 -861.214 R 2 .7191 .7124 .7216 .7018 * Figures reported i n parentheses are the degrees of freedom for the respective tests ** Figures reported on the same l i n e as H.. are the test s t a t i s t i c s for the respective t e s t s . (B) Tests amongst the chosen sub-models Test s t a t i s t i c Test {-2lnX) (1) H o : model (D-3) H l : model (A-3) 4.402 (2) H o : model (D-3) H l : model (B-2) 12.47 (3) H o : model (D-3) H l : model (C-l) 16.364 (4) H o : model (A-3) H l : model (C-l) 11.962 (5) H o : model (A-3) H l ! model (B-2) 3.894 (6) H o : model (A-3) H l : model (B-2) 8.068 Model (C-l) i s f i n a l l y chosen for use. (Second stage t e s t s ) : Degrees of X" c r i t i c a l value Freedom at a-.05 Test Result 1 3.841 favours (A-3) 6 12.592 favours (D-3) 4 9.488 favours (C-l) 3 7.815 favours (C-l) 2 5.991 favours (C-l) 5 11.071 favours (A-3) - 201 -APPENDIX 7A Derivation of P r i c e and Quality E l a s t i c i t i e s E l a s t i c i t y of demand f o r the iLth mode with respect to the fr e i g h t rate of the jth mode can be written as: 3X. P. (1) E = — J i j 9P. X. f o r a l l i=r,h -1 1 and a l l j=r,h Shephard's lemma allows us to write the demand for i t h mode as (o\ v = - 3Zn.C C .._ S i - , C K " A i 9P. ~ 8ZnP. T. ¥~ I i i I where C i s the shipper's transportation s e c t o r a l t o t a l cost function. Therefore, 9 X i 1 3 S i 3C (3) — ± . = _ ( ± . r + 6 • q ) K ' 9Pj P ± (dP. C + 9Pj S i ) i a..-C " ^ ( - ^ T — ) + * j • V P. P. ; P B ± > 1 3 3 C (a + S S ) f o r a l l i ^ j P.P.. - i j - i - j 3 X i a S.-C 3(S.-C) P , 1 9 S -• [ p i <5fr • c + Hr si ) - s± • « l l P z a . .-C S. 2«C P l [ P i '~PT + -T— ) - s. • C] ^ 1 X • X P . 1 C • 4- S . 2 - S. ) X - 202 -Substitution of equations (3) and (4), r e s p e c t i v e l y , i n t o equation (1) gives: 3X P P ( 5 ) E i j " SPTX* " I T P T X 1 <•« + s i V j l X j X J J = ~ - (a. . + S. • S .) S ± ID i D = S.. • using equation (7,3) for i ^ j a x . P . 9 p . E,, = ^  ~ = -K (a,, + S / - S;) ' 1 ' i i 3P. X, _ 2 v " i i 1 " i ~ i ' X, IT ( a i i + s i 2 " ' Si> l = S.^  • using equation (7,3) for i=j E l a s t i c i t i e s of demand f o r i t h mode with respect to nth qu a l i t y a t t r i b u t e of mode j. c a n ^ e written as: „ j 7 „ „ , 3X. Z . _ S.'C Z. m F n d^ -nx- i _ jn = _3 , i x pn 1 1 i j Z. 3Z. X. 3Z. P . X. jn jn x un . x x For the translog cost function (3,19a) f o r model C - l , (7) 3X, i 3Z Dn 3Z S. 'C Dn 3Z Dn C + 3C 3Z nD S.) x j n J s ) k f o r 3S. x 3Z Dn a. . 3 • Z . Dn - 203 -, 3C _ dln.C C a n C l 3Z . " JT*Z. Z . Dn Dn jn = B [a. + a . (ZnP. + E B. k£nZ + 6 z„:;'p) D n ] !D i k = 1 x* ik i 2 + A . (Z.n.p + Z , a R Z . + 6. £„D)] DD D j ^ - ^ D*- J K J = B • ' S. Dn D Substitution of equation (7) i n t o equation (6) gives: «> E i : n - rr £=• % + s i • V " % <•« + 8 J x ;jn x J J x J = 3- E.. using equation (5) ]n l ] for a l l i ^ j . S i m i l a r l y , 3X. « S.-C (9) 1_ = _° ( _ i — ) K y ' 3Z . 9Z. v P. ' i n xn x , 3(S.'C) 3P. 1 r ' 1 . p - i _ c • Cl _ 2 1 3Z. F i 3Z. x U J P. xn xn x ITF "*!^" c + WT" • si» p i " JTT- si P. xn xn i n x (a.. + S. 2 - S.) P. •Z. xx x x x xn - 204 -for 3S. a . . 3 . 1 _ n i n 3Z . Z . i n i n 3C _ dln.C C _ U c . . . . , _ 3 Z - " 3ZnZ. ~ " $ i n S i s i m i l a r l y as before, i n i n i n and 3P. 3ZnP. P. 3 . p. 3Z = dln-.Z Z ^ — = 1 invoking the hedonic price i n i n i n i n r e l a t i o n . Substitution of (9) i n t o (6) gives (10) _ . n _ P i n C Z i n (a.. + s. 2 - S.) - E i i ~ P T Z : XT 1 1 1 1 I i n I 3 i n E i i again using (5). 

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