UBC Theses and Dissertations

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UBC Theses and Dissertations

A study on CBD land value variations Fujiki, Kazuhiko 1989

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A STUDY ON CBD LAND VALUE VARIATIONS by KAZUHIKO FUJIKI A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION in THE FACULTY OF GRADUATE STUDIES Commerce and Business A d m i n i s t r a t i o n We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1989 © Kazuhiko F u j i k i , 1989 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at The U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Commerce and Business A d m i n i s t r a t i o n The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Pl a c e Vancouver, Canada V6T 1W5 Date: August, 1989 ABSTRACT T h i s t h e s i s examines CBD ( C e n t r a l Business D i s t r i c t ) l a n d value v a r i a t i o n s . The o b j e c t i v e s of t h i s study are (1) to compare monocentric and nonmonocentric models and (2) to a n a l y z e the determinants of CBD l a n d value v a r i a t i o n s . T r a n s a c t i o n s of vacant la n d from 1975 to 1987 i n C e n t r a l Ward (Chuo Ku), Tokyo, comprise the data base f o r t h i s study. A monocentric model and nonmonocentric model are compared u s i n g a n e g a t i v e e x p o n e n t i a l f u n c t i o n and t r e n d s u r f a c e a n a l y s i s (based on a double power s e r i e s of l o c a t i o n c o o r d i n a t e s ) . For the comparison, t h r e e - d i m e n s i o n a l p i c t u r e s and contour maps are u t i l i z e d as w e l l as s t a t i s t i c s of g o o d n e s s - o f - f i t and p r e d i c t i v e powers. To analyze determinants of CBD la n d v a l u e v a r i a t i o n s , we employ a h e d o n i c - p r i c e approach. Trend s u r f a c e a n a l y s i s i s s u p e r i o r t o the monocentric model i n terms of g o o d n e s s - o f - f i t and p r e d i c t i v e powers. However, c e n t r a l i t y i s s t i l l an important determinant. P r o x i m i t y to subway or r a i l w a y s t a t i o n s , or to the Ginza shopping a r e a , are a l s o important f a c t o r s . Other i n f l u e n t i a l determinants i n c l u d e time of s a l e , l o t shape, corner l o c a t i o n , road width, and f l o o r area r a t i o . i i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i I. INTRODUCTION 1 A. Background 1 B. Purpose 3 C. O u t l i n e 4 I I . CONCEPTUAL FRAMEWORK 5 A. Land Rent Functions i n Monocentric Models 5 B. Negative E x p o n e n t i a l Land Rent F u n c t i o n and i t s extensions 14 1. Negative E x p o n e n t i a l Land Rent F u n c t i o n . 15 2. C r i t i c i s m s and Extensions of Negative E x p o n e n t i a l F u n c t i o n 18 C. Land Rent Functions i n Nonmonocentric Models . 24 I I I . PAST EMPIRICAL RESEARCH 31 A. E m p i r i c a l T e s t s of Monocentric and Nonmonocentric Models 31 1. T r a d i t i o n a l Monocentric Models 31 2. Trend Surface A n a l y s i s 35 B. Independent V a r i a b l e s i n the Past Hedonic-Price S t u d i e s 38 C. Summary 47 IV. EMPIRICAL ANALYSIS 48 A. Study Area 49 B. The Data ' 52 C. Methodology 54 1. Comparison of Monocentric and Nonmonocentric Models 54 a. Monocentric Model 54 b. Trend Surface A n a l y s i s 55 2. A n a l y s i s of Determinants of CBD Land Value V a r i a t i o n s 57 D. E s t i m a t i o n r e s u l t s 61 1. Comparison of Monocentric and Nonmonocentric Models 61 a. T r a d i t i o n a l Monocentric Model 61 b. Trend Surface A n a l y s i s 66 c. Comparison of the Two Models 68 2. A n a l y s i s of Determinants of CBD Land Value V a r i a t i o n s 80 V. SUMMARY AND CONCLUSIONS . ... 87 BIBLIOGRAPHY 91 i v LIST OF TABLES Table III.1 Summary, of Past E m p i r i c a l Research 32 Table IV.1 I n d u s t r i e s and Employment i n C e n t r a l Ward, 1981 51 Table IV.2 Summary S t a t i s t i c s f o r the Data 53 Table IV.3 Regression r e s u l t s (a) — T r a d i t i o n a l Monocentric Model ...61 Table IV.4 Regression r e s u l t s (b) — Trend Surface A n a l y s i s 67 Table IV.5 Regression r e s u l t s (c) — Hedonic Approach 82 v LIST OF FIGURES F i g u r e 2.1 Land Rent Function by Mohring 9 F i g u r e 2.2 Land Rent Function by Alonso 9 F i g u r e 2.3 Negative E x p o n e n t i a l Land Rent F u n c t i o n 18 F i g u r e 3.1 Hedonic, Buyer B i d , and S e l l e r O f f e r Functions 41 F i g u r e 4.1 Map of Study Area 50 F i g u r e 4.2 Normal P r o b a b i l i t y P l o t s — T r a d i t i o n a l Monocentric Model 64 F i g u r e 4.3 R e s i d u a l P l o t s — T r a d i t i o n a l Monocentric Model 65 F i g u r e 4.4 Normal P r o b a b i l i t y P l o t s — Trend Surface A n a l y s i s 69 F i g u r e 4.5 R e s i d u a l P l o t s — Trend Surface A n a l y s i s 70 F i g u r e 4.6 A c t u a l Land Value i n 1987 - 3-D P i c t u r e 74 F i g u r e 4.7 A c t u a l Land Value i n 1987 - Contour Map 75 F i g u r e 4.8 P r e d i c t e d Land Value i n 1987 (Monocentric Model) - 3-D P i c t u r e 76 F i g u r e 4.9 P r e d i c t e d Land Value i n 1987 (Monocentric Model) - Contour Map 77 F i g u r e 4.10 P r e d i c t e d Land Value i n 1987 (TSA) - 3-D P i c t u r e 78 F i g u r e 4.11 P r e d i c t e d Land Value i n 1987 (TSA) - Contour Map 79 v i ACKNOWLEDGEMENTS I would l i k e to give my thanks to those who commented on e a r l i e r d r a f t s of my t h e s i s and those who have helped me by c r i t i c i z i n g my ideas i n classroom d i s c u s s i o n s . In p a r t i c u l a r , I would l i k e to mention Jim C l y t o n , Reagan P r a t t , Gary Tung, and Yuming Fu, who a l s o helped me with t y p i n g the f i n a l v e r s i o n a f t e r my r e t u r n to Japan. I a l s o wish to thank Dr. Lawrence Jones and Dr. Tae Hoon Oum f o r t h e i r encouragement and thought-provoking comments. I owe a s p e c i a l debt of g r a t i t u d e to Dr. Robert H e l s l e y , who acted as chairman of my t h e s i s committee, f o r a steady supply of v a l u b l e a d v i c e . During my stay at U n i v e r s i t y of B r i t i s h Columbia, I r e c e i v e d f i n a n c i a l support from The Japan Real E s t a t e I n s t i t u t e , which a l s o permitted me to use t h e i r data i n my a n a l y s i s . v i i I. INTRODUCTION A. BACKGROUND Land p r i c e v a r i a t i o n s have f r e q u e n t l y been the t a r g e t of r e s e a r c h . Researchers are c o n t i n u a l l y a t t r a c t e d to the s u b j e c t , because of the p u b l i c i n t e r e s t i n urban land v a l u e s and a widespread i n t e r e s t i n p r e d i c t i n g the f u t u r e p h y s i c a l growth of a c i t y . Land val u e s a f f e c t land uses, which i n tur n a f f e c t the s t r u c t u r e of a c i t y . A small business or f a c t o r y may have to leave a downtown area because the present value of i t s s i t e i n some other use i s g r e a t e r than the present value of p r o j e c t e d cash flows from i t s business a c t i v i t y . A mu n i c i p a l government may have to s e l e c t an econ o m i c a l l y i n e f f i c i e n t highway route to a v o i d a h i g h l y v a l u e d r e s i d e n t i a l a r ea. The detour w i l l a f f e c t subsequent commercial and r e s i d e n t i a l development near the road. An understanding of land p r i c e v a r i a t i o n s i s important to many p u b l i c and p r i v a t e p r o f e s s i o n s . Equipped with such an understanding, m u n i c i p a l o f f i c i a l s are b e t t e r able to eva l u a t e t r a n s a c t i o n s i n v o l v i n g p u b l i c funds. Tax o f f i c i a l s are a b l e to promote f a i r t a x a t i o n on la n d . Real e s t a t e a p p r a i s e r s can b e t t e r estimate l a n d values by knowing the p r i c e l e v e l of the whole c i t y or area i n which the a p p r a i s e d p r o p e r t y e x i s t s . 1 INTRODUCTION / 2 To date, r e s e a r c h e r s have tended toward a monocentric model to e x p l a i n land p r i c e v a r i a t i o n s . The monocentric model assumes that p l a c e s of employment w i l l be found i n the c e n t r a l business d i s t r i c t (the "CBD") and that employees w i l l commute to and from the CBD. I t concludes that land and housing p r i c e s and p o p u l a t i o n d e n s i t y f a l l with the d i s t a n c e from the CBD. T h i s t h e s i s o r i g i n a t e s i n p a r t from the concern that the t r a d i t i o n a l monocentric model does not account f o r e x t e r n a l i t i e s and other p r i c e i n f l u e n c i n g f a c t o r s which d i s t o r t a mon o t o n i c a l l y d e c r e a s i n g land p r i c e g r a d i e n t . We f e l t that the t r a d i t i o n a l monocentric model should be re p l a c e d by another model. Another concern i s the f a c t t h at the CBD has long been n e g l e c t e d as a resea r c h s u b j e c t . U n t i l r e c e n t l y , most s t u d i e s have been i n t e r e s t e d not i n the CBD but i n r e s i d e n t i a l p r o p e r t i e s , u s u a l l y i n o u t l y i n g areas of a c i t y . These s t u d i e s t r e a t the CBD as a p o i n t from which to measure the d i s t a n c e to each p r o p e r t y . In c o n t r a s t , t h i s study t r i e s to i n v e s t i g a t e l a n d p r i c e v a r i a t i o n s w i t h i n the CBD and to t r e a t the CBD as a plane r a t h e r than a p o i n t . INTRODUCTION / 3 B. PURPOSE The l a r g e r purpose of t h i s study i s to examine i n t r a - u r b a n land p r i c e v a r i a t i o n s , with p a r t i c u l a r focus on land p r i c e v a r i a t i o n s w i t h i n the CBD, and to t r e a t the CBD as a plane r a t h e r than a p o i n t . The paper d e a l s with the f o l l o w i n g two s p e c i f i c q u e s t i o n s : (1) Is the t r a d i t i o n a l monocentric model l e s s e f f e c t i v e than nonmonocentric models i n e x p l a i n i n g land p r i c e v a r i a t i o n s ? (2) Can c e n t r a l i t y , e x t e r n a l i t i e s , s i t e - and n e i g h b o r h o o d - s p e c i f i c c h a r a c t e r i s t i c s e f f e c t i v e l y e x p l a i n l a n d p r i c e v a r i a t i o n s w i t h i n the CBD? Both q u e s t i o n s i n v o l v e the search f o r an a l t e r n a t i v e to the t r a d i t i o n a l monocentric model. The d i f f e r e n c e between them i s that the second c o n s i d e r s f a c t o r s other than l o c a t i o n . In other words, the f i r s t q u e s t i o n takes only the l o c a t i o n of each s i t e i n t o account, whereas the second q u e s t i o n i n c l u d e s other p r i c e i n f l u e n c i n g f a c t o r s . The f i r s t q u e s t i o n asks whether or not a complex land p r i c e s t r u c t u r e e x i s t s , one which cannot be e x p l a i n e d by the t r a d i t i o n a l monocentric model. The second q u e s t i o n e x p l o r e s the determinants of the complex p r i c e v a r i a t i o n s w i t h i n the CBD. INTRODUCTION / 4 C. OUTLINE T h i s t h e s i s c o n t a i n s f i v e c h a p t e r s . Chapter II in t r o d u c e s the conceptual framework f o r t h i s study. Chapter III reviews past e m p i r i c a l r e s e a r c h . Chapter IV analyzes land p r i c e v a r i a t i o n s i n the Tokyo CBD. Chapter V summarizes and g i v e s c o n c l u d i n g remarks. 11. CONCEPTUAL FRAMEWORK Th i s chapter examines past t h e o r e t i c a l s t u d i e s on in t r a - u r b a n land p r i c e v a r i a t i o n s . T h i s i s done i n order to provide a t h e o r e t i c a l foundation f o r the e m p i r i c a l a n a l y s i s in Chapter IV. T h i s chapter i s d i v i d e d i n t o three s e c t i o n s . S e c t i o n A examines past s t u d i e s which focused on c e n t r a l i t y and land v a l u e s . S e c t i o n B d i s c u s s e s the negative e x p o n e n t i a l land rent f u n c t i o n and i t s e x t e n s i o n s . S e c t i o n C d i s c u s s e s recent t h e o r e t i c a l s t u d i e s d e a l i n g with land values i n the context of nonmonocentric models. A. LAND RENT FUNCTIONS IN MONOCENTRIC MODELS In monocentric models of urban land use, land rent and land use are completely determined by access to the "cen t e r " of a c i t y . In t h i s s e c t i o n , we review four monocentric models. Mohring (1961) pioneered the land rent g r a d i e n t study. He d e a l t e x c l u s i v e l y with r e s i d e n t i a l land v a l u e s . Alonso (1964) int r o d u c e d and e x p l a i n e d commercial land value v a r i a t i o n s . Two more recent s t u d i e s done by Solow (1973) and Henderson (1985) focused on land v a l u e s w i t h i n a CBD. 1. Mohring (1961) g i v e s us the simplest monocentric model. His model i s based on the f o l l o w i n g assumptions: (1) A l l economic and r e c r e a t i o n a l a c t i v i t i e s , i n c l u d i n g 5 CONCEPTUAL FRAMEWORK / 6 h o u s e h o l d s h o p p i n g , t a k e p l a c e i n a CBD w h i c h i s c o n s i d e r e d a p o i n t . (2) O u t s i d e t h e CBD, l a n d i s u s e d o n l y f o r i d e n t i c a l s i n g l e - f a m i l y d w e l l i n g s w i t h f i x e d l o t s i z e s . (3) L a n d s u r r o u n d i n g t h e CBD i s w o r t h l e s s f o r any p u r p o s e o t h e r t h a n t h e e x i s t i n g home s i t e s . (4) E a c h f a m i l y , w h i c h has u n i f o r m t a s t e and income, p l a c e s t h e same v a l u e on i t s t r a v e l t i m e and makes t h e same number of t r i p s t o t h e CBD. (5) The c o s t of a t r i p t o t h e CBD i s p r o p o r t i o n a t e t o t h e t r a v e l time' r e q u i r e d t o make i t . In e q u i l i b r i u m , e q u a l u t i l i t y must be r e a l i z e d i n e v e r y h o u s e h o l d . S i n c e M o h r i n g assumes f i x e d l o t s i z e s , t h e t o t a l amount o f v a r i a b l e c o s t s i n c u r r e d i n e a c h h o u s e h o l d must a l s o be t h e same a t e v e r y l o c a t i o n i n e q u i l i b r i u m . Land r e n t s and t r a n s p o r t a t i o n c o s t s a r e v a r i a b l e w i t h i n e a c h h o u s e h o l d a c c o r d i n g t o i t s l o c a t i o n . In s h o r t , r e g a r d l e s s of th e l o c a t i o n o f a h o u s e h o l d , t h e sum o f l a n d r e n t s and t r a n s p o r t a t i o n c o s t s i s i d e n t i c a l f o r e v e r y f a m i l y : R. + 2 N T i V T = R. + 2 N T j V T . where Rfc = Rent f o r s i t e " t " ( t = i and j ) ; N = A n n u a l number o f t r i p s t a k e n t o t h e CBD; T. = T r a v e l t i m e f rom s i t e " t " t o t h e CBD ( t = i and j ) ; CONCEPTUAL FRAMEWORK / 7 and V"T = Value p l a c e d on t r a v e l time. T h i s i m p l i e s R. - Rj = 2N(Tj - T . ) V T [2.1] Equation [2.1] i n d i c a t e s that the d i f f e r e n c e i n land r e n t s i n two d i f f e r e n t p l a c e s depends s o l e l y on the d i f f e r e n c e i n t r a v e l time from each p l a c e to the CBD. To determine land rent f o r each s i t e , Mohring uses the assumption that the op p o r t u n i t y c o s t of land i s zero: "those l i v i n g at the c i t y l i m i t s need pay no rent f o r the land they use." 1 By f i x i n g land rent at the c i t y l i m i t at zero, we can d e r i v e Mohring's land rent equation as f o l l o w s : From Equation [2.1], R i " Rmax = 2 N ( T m a x " T i ) V T where Rm ^ „ = Rent f o r c i t y l i m i t s i t e s ; and max •* ' T = T r a v e l time from c i t y l i m i t s i t e s to the CBD. max 2 S i n c e Rmax = °' R i " 2 N ( T m a x " W Because both N (annual number of t r i p s to the CBD) and V T (value of t r a v e l time) are given i n t h i s case, land rent of each s i t e i s determined s o l e l y by t r a v e l time to the CBD. In other words, land rent d e c l i n e s l i n e a r l y as a d i s t a n c e to 'Mohring (1961), p.238. CONCEPTUAL FRAMEWORK / 8 the CBD i n c r e a s e s ( s e e F i g u r e 2.1). 2. Alonso (1964) e x p l a i n s land p r i c e v a r i a t i o n s w i t h i n a c i t y u s ing a b i d rent (or p r i c e ) curve concept. In the case of urban firms ( i n c l u d i n g r e t a i l , wholesale, o f f i c e , f i n a n c i a l , and manufacturing f i r m s ) , a b i d p r i c e curve i s an i s o - p r o f i t c u rve. The f i r m i s i n d i f f e r e n t to l o c a t i o n s and rent as long as the po i n t i s on the same b i d p r i c e curve. A b i d p r i c e curve has a downward slope f o r the f o l l o w i n g reasons: (1) Volume of business d e c l i n e s as the d i s t a n c e from the c i t y c e n t e r i n c r e a s e s because " a c c e s s i b i l i t y of the s i t e to p o t e n t i a l customers w i l l decrease with d i s t a n c e from the c e n t e r . " 2 (2) Operating c o s t s i n c r e a s e as the d i s t a n c e from the c i t y c e n t e r i n c r e a s e s because t r a n s p o r t a t i o n c o s t s i n c r e a s e . A l a n d rent g r a d i e n t i n a c i t y i s d e r i v e d by combining the b i d rent (or p r i c e ) curves of v a r i o u s l a n d uses, i n c l u d i n g commercial, i n d u s t r i a l , r e s i d e n t i a l , and a g r i c u l t u r a l uses. Alonso concludes that land use o f f e r i n g the h i g h e s t b i d rent curve can occupy the c i t y c e n t e r , and that land at any po i n t i n a c i t y i s a l l o c a t e d to the h i g h e s t 2 Alonso (1964), p.44. CONCEPTUAL FRAMEWORK / 9 F i g u r e 2.1 Land Rent F u n c t i o n by M o h r i n g Land Rent 2 N T m a x V T D i s t a n c e f r o m CBD F i g u r e 2.2 Land Rent F u n c t i o n by A l o n s o L a n d Rent Maximum B i d Rent F u n c t i o n s ^ E q u i l i b r i u m Land Rent 0 D i s t a n c e f r o m CBD CONCEPTUAL FRAMEWORK / 10 bi d d e r . G e o m e t r i c a l l y , an e q u i l i b r i u m land rent f u n c t i o n i s an envelope of maximum b i d rent f u n c t i o n s f o r v a r i o u s uses (see F i g u r e 2.2). 3. Solow (1973) analyzes land rent v a r i a t i o n s w i t h i n a CBD using a simple monocentric model. As Mohring d i d , Solow a l s o makes s e v e r a l assumptions to keep h i s a n a l y s i s as simple as p o s s i b l e . His assumptions are as f o l l o w s : (1) Everyone works i n the CBD. (2) The CBD i s a f e a t u r e l e s s plane with a s i n g l e c e n t r a l p o i n t . (3) The CBD produces a s i n g l e commodity. (4) T h i s commodity i s produced with labor and l a n d . (5) The pro d u c t i o n f u n c t i o n f o l l o w s constant r e t u r n s to s c a l e . (6) The commodity must be t r a n s p o r t e d to the c e n t r a l p o i n t . (7) A s i n g l e wage r a t e p r e v a i l s i n the CBD. (8) A l l markets are c o m p e t i t i v e . (9) The commodity p r i c e i s the given ' n a t i o n a l ' p r i c e . We can i n f e r from the above assumptions that Solow only examines the manufacturing i n d u s t r y i n the CBD. He in t r o d u c e s the f o l l o w i n g u n i t p r o f i t f u n c t i o n . II = p - C[w, r(x) ] - t ( x ) [2.2] CONCEPTUAL FRAMEWORK / 1 1 where n = P r o f i t per u n i t output, p = Product p r i c e , C = Minimum co s t f u n c t i o n ( e x c l u d i n g t r a n s p o r t a t i o n c o s t s ) , w = Wage r a t e , r(x) = Rent, x = Dis t a n c e from the c e n t r a l p o i n t , t ( x ) = T r a n s p o r t a t i o n c o s t s . In the long-run, p r o f i t i s reduced to zero because of co m p e t i t i o n . In the e q u i l i b r i u m , Equation [2.2] can be w r i t t e n as: p = C[w, r ( x ) ] + t ( x ) [2.3] Since both p (product p r i c e ) and w (wage ra t e ) are constant, the d i f f e r e n t i a t i o n of Equation [2.3] y i e l d s 0 = C r « r 1 ( x ) + t'(x) Rearranging terms, we o b t a i n , r'(x) = - f ( x ) / C r Because t r a n s p o r t a t i o n c o s t s r i s e as the d i s t a n c e from the c e n t r a l p o i n t i n c r e a s e s , t'(x) > 0. Since minimum c o s t s r i s e with input p r i c e s , C r > 0. Th e r e f o r e , r'(x) < 0. Land ren t s f a l l with i n c r e a s i n g d i s t a n c e from the c e n t r a l p o i n t w i t h i n the CBD. CONCEPTUAL FRAMEWORK / 12 4. Henderson (1985) a l s o c o n s i d e r s a land rent f u n c t i o n w i t h i n the CBD. In h i s a n a l y s i s , i d e n t i c a l f i r m s i n the CBD are assumed to produce one type of export good (x) f o l l o w i n g the l i n e a r homogeneous p r o d u c t i o n f u n c t i o n : x(u) = G(N)-x[k(u), n ( u ) , l ( u ) ] where x(u) = Amount of goods produced at a s i t e "u" m i l e s from the c e n t r a l p o i n t of the CBD; G(N) = Hicks n e u t r a l s h i f t f a c t o r (9G/9N > 0) which takes e x t e r n a l s c a l e economies i n t o account; N = P o p u l a t i o n of the c i t y i n which the CBD e x i s t s ; k(u) = C a p i t a l input; n(u) = Labor input; and l ( u ) = Land input. Another important assumption i s that the produced goods must be t r a n s p o r t e d to the c e n t r a l p o i n t of the CBD from which to be exported or to be s o l d l o c a l l y . Henderson assumes t h a t i t c o s t s f i r m s " t " of a u n i t of "x" to s h i p A one u n i t of "x" one u n i t of d i s t a n c e . T h e r e f o r e , the net p r i c e a f t e r t r a n s p o r t a t i o n c o s t s i s p • (1 - t «u) where p A A A i s a u n i t p r i c e of "x." With p e r f e c t c o m p e t i t i o n , the net p r i c e must be equal to u n i t c o s t s i n e q u i l i b r i u m : CONCEPTUAL FRAMEWORK / 13 P x-(1 " t x - u ) = C[N, p n , p k , p 1 ( u ) ] [2.4] where C[ • ] = Minimum u n i t cost f u n c t i o n ; N = P o p u l a t i o n of the c i t y ; p n = U n i t c o s t of l a b o r ; p^ = Un i t c o s t of c a p i t a l ; and p-^(u) = Unit land rent at a s i t e "u" m i l e s from the c e n t r a l p o i n t of the CBD. D i f f e r e n t i a t i n g Equation [2.4], - p x - t x = [9C/3p 1(u)] • [3p 1(u)/3u] [2.5] Using Shephard's lemma,3 ac/9p 1(u) = l ( u ) / x ( u ) [2.6] S u b s t i t u t i n g [2.6] i n t o [2.5], we o b t a i n 9p 1(u)/3u = - x ( u ) - p x . t x / l ( u ) [2.7] M u l t i p l y i n g l ( u ) on both s i d e s of [2.7], l ( u ) • [ 9 p 1 ( u ) / 3 u ] = - x ( u ) - p x - t x [2.8] Equation [2.7] i n d i c a t e s that u n i t land rent decreases with the d i s t a n c e from the c e n t r a l p o i n t of the CBD. From Equation [2.8], we n o t i c e that the in c r e a s e i n a t o t a l rent by moving a u n i t d i s t a n c e towards t h CBD boundary i s equal to the decrease i n t r a n s p o r t a t i o n c o s t s of moving a u n i t d i s t a n c e towards the CBD boundary. 3Shephard's lemma s t a t e s that the d e r i v a t i v e of the u n i t c o s t f u n c t i o n with respect to a pr o d u c t i o n f a c t o r p r i c e equals the per u n i t demand f o r the f a c t o r . CONCEPTUAL FRAMEWORK / 14 The above four s t u d i e s p r o v i d e examples of land p r i c e f u n c t i o n s i n monocentric models. Mohring (1961) d i s c u s s e d a r e s i d e n t i a l land rent f u n c t i o n which has an i n v e r s e l i n e a r r e l a t i o n s h i p to d i s t a n c e from the CBD. Alonso (1964) i n c l u d e d commercial land i n h i s a n a l y s i s . He demonstrated that an e q u i l i b r i u m land rent has a negative s l o p e . Assuming a l l products are t r a n s p o r t e d to a c e n t r a l p o i n t , both Solow (1973) and Henderson (1985) i l l u s t r a t e d a n e g a t i v e l y - s l o p e d land rent f u n c t i o n w i t h i n a CBD. However, these s t u d i e s do not s p e c i f y f u n c t i o n a l forms of land rent g r a d i e n t s . In the next s e c t i o n , we w i l l examine the M i l l s - M u t h type negative e x p o n e n t i a l f u n c t i o n . B. NEGATIVE EXPONENTIAL LAND RENT FUNCTION AND ITS EXTENSIONS As C l a r k (1967) p o i n t e d out i n h i s e a r l y work, "A theory was f i r s t proposed by Winkler ( i n 1957) that l a n d values were at t h e i r h i g h e s t i n the center of the area, and that with i n c r e a s i n g d i s t a n c e from i t they f e l l o f f , not e x p o n e n t i a l l y , as does p o p u l a t i o n d e n s i t y , but i n a double l o g a r i t h m i c r e l a t i o n s h i p . " 4 Since then, however, no study has been done to support Winkler's i d e a . On the c o n t r a r y , s e v e r a l s t u d i e s have s u c c e s s f u l l y i l l u s t r a t e d that l a n d rent f a l l s e x p o n e n t i a l l y with i n c r e a s i n g d i s t a n c e from the "Clark (1967), p.382. CONCEPTUAL FRAMEWORK / 15 c e n t r a l p o i n t . The f i r s t h a l f of t h i s s e c t i o n reviews negative e x p o n e n t i a l land rent f u n c t i o n . The second h a l f examines the c r i t i c i s m s and extensions of the concept of negative e x p o n e n t i a l f u n c t i o n . 1. Negative E x p o n e n t i a l Land Rent Function M i l l s (1972) d e r i v e s an e x p o n e n t i a l l a n d rent f u n c t i o n i n the context of a monocentric model. He assumes that a) urban area i s l o c a t e d on a f e a t u r e l e s s plane and has a predetermined c e n t e r ; b) commuting cost depends only on the s t r a i g h t l i n e d i s t a n c e between the l o c a t i o n of housing and the c i t y c e n t e r ; c) a l l the land that i s l o c a t e d "u" mi l e s from the c i t y center commands the same rent and i s used with the same c a p i t a l / l a n d r a t i o because a l l the land i s homogeneous; d) input and output markets are p e r f e c t l y c o m p e t i t i v e ; e) the c a p i t a l r e n t a l r a t e " r " i s independent of both the d i s t a n c e from the c i t y center and the amount of land used i n the e n t i r e urban area; f) a l l workers r e c e i v e the same income "w" and have the same t a s t e s ; and g) the income e l a s t i c i t y and p r i c e e l a s t i c i t y of demand f o r housing are c o n s t a n t . Based on the above assumptions, M i l l ' s model of urban s t r u c t u r e i s formulated i n the f o l l o w i n g way: (1) The pr o d u c t i o n f u n c t i o n of housing s e r v i c e s takes the CONCEPTUAL FRAMEWORK / 16 form of the Cobb-Douglas f u n c t i o n : X s ( u ) = A - L ( u ) a - K ( u ) 1 a ; 0<a<1 [2.9] where Xg(u) = Production of housing s e r v i c e s at d i s t a n c e u from the CBD, L(u) = Input of land, K(u) = Input of c a p i t a l , and A, a = Constants. (2) In e q u i l i b r i u m , s i n c e households cannot i n c r e a s e t h e i r u t i l i t y by changing t h e i r l o c a t i o n , the decrease or i n c r e a s e in the c o s t of housing s e r v i c e s r e s u l t i n g from such a move i s o f f s e t by the i n c r e a s e or decrease i n commuting c o s t s . where P(u) = P r i c e of housing s e r v i c e s at d i s t a n c e u, X D ( u ) = Demand f u n c t i o n f o r housing s e r v i c e s per worker l i v i n g at u, and t = Cost per two m i l e s of commuting. (3) The demand f u n c t i o n f o r housing s e r v i c e s i s assumed to take the form of a power f u n c t i o n of income and p r i c e . [dP(u)/du]-X D(u) + t = 0 [2.10] x n ( u ) = B.W*1 . p ( u r 2 [2.11] where B = Constant, W = Income, 0, = Income e l a s t i c i t y of demand f o r housing, and CONCEPTUAL FRAMEWORK / 17 62 - P r i c e e l a s t i c i t y of demand f o r housing. (4) S u b s t i t u t i n g [2.11] i n t o [2.10], we o b t a i n , [ d P ( u ) / d u ] - B - W 0 1 - P ( u ) e 2 + t = 0 [2.12] (5) Because of co m p e t i t i v e markets, the value of the marginal product of a f a c t o r should be equal to the f a c t o r ' s p r i c e . Using t h i s r e l a t i o n s h i p and Equation [2.9], we can d e r i v e P(u) and dP(u)/du as f o l l o w s : P(u) = [A«a a-(1 - a ) 1 _ a ] " 1 . R ( u ) a . r 1 _ a [2.13] dP(u)/du = A " 1 . [ a - r / ( 1 - a ) ] 1 ~ a - R ( u ) ~ ( 1 ~ a ) • [ d R ( u ) / d u ] [2.14] (6) S u b s t i t u t i n g [2.13] and [2,14] i n t o [2.12] and s o l v i n g the d i f f e r e n t i a l equation r e s u l t i n g from the s u b s t i t u t i o n , we o b t a i n the f o l l o w i n g two e q u i l i b r i u m land rent f u n c t i o n s : R(u) = [RP + 0-t-E-(u - U ) ] 1 / / 3 i f 0 # 0 [2.15] R(u) = R e t ' E ' ( Q " u ) i f 0 = 0 [2.16] where R(u) = Land rent "u" mil e s from the c e n t e r , R = Land rent of non-urban use, 0 = a-(1 + 62) , t = Cost per two m i l e s of commuting, E = a.B.weMA.aa.O - a ) 1 "cr ( 1 +6>} * r * ( 1 _ a ) * ( 1 } , and u = D i s t a n c e from the c i t y c e n t e r to the edge of the urban a r e a . According to M i l l s , " s t u d i e s of housing demand suggest CONCEPTUAL FRAMEWORK / 18 F i g u r e 2.3 Negative E x p o n e n t i a l Land Rent Fu n c t i o n Land Rent D i s t a n c e from CBD that 6! (income e l a s t i c i t y of demand f o r housing) may be about 1.5 and 62 ( p r i c e e l a s t i c i t y of demand f o r housing). about -1.0." 5 Since 0 = a - d +. 0 2 ) , 62 = -1.0 means 0 = 0. T h i s r e s u l t i m p l i e s that we can estimate land rent v a r i a t i o n s u s i n g Equation [2.16], which takes the negative e x p o n e n t i a l form (see F i g u r e 2.3). t 2. C r i t i c i s m s and Extensions of Negative E x p o n e n t i a l Function (a) F u n c t i o n a l Form Based on M i l l s (1972), the negative e x p o n e n t i a l land rent f u n c t i o n has been d e r i v e d as we saw i n Part 1 of t h i s 5 M i l l s (1972), p.81. CONCEPTUAL FRAMEWORK / 19 s e c t i o n . As we observed, the negative e x p o n e n t i a l f u n c t i o n i s v a l i d i f and only i f /3 i s equal to z e r o . In other words, the negative e x p o n e n t i a l f u n c t i o n i s p l a u s i b l e only when 82 (the p r i c e e l a s t i c i t y of demand f o r housing) i s -1.0. The v a l i d i t y of t h i s assumption i s d o u b t f u l , however. 6 Kau and Sirmans (1979) and Okawara (1981) examined whether (3 = 0 i s an a p p r o p r i a t e assumption or not. Using the Box-Cox t r a n s f o r m a t i o n , the Kau and Sirman's study found that 0 d i f f e r s s i g n i f i c a n t l y from zero at the 5% l e v e l f o r four out of s i x sets of land value data i n Chicago. Ap p l y i n g the same technique to Tokyo data s e t s , Okawara concluded that /3 d i f f e r s s i g n i f i c a n t l y from zero at the 1% l e v e l . The assumption that j3 = 0 i s mostly r e j e c t e d at a very s i g n i f i c a n t l e v e l i n both s t u d i e s . Aside from i t s f u n c t i o n a l form, the M i l l s type land rent f u n c t i o n has many drawbacks. One of them a r i s e s from the assumption that the c i t y i s l o c a t e d on a f e a t u r e l e s s plane. " F e a t u r e l e s s " means the nonexistence of both n a t u r a l and man-made f e a t u r e s . The l a t t e r f e a t u r e s i n c l u d e neighborhood environments as w e l l as the laws and r e g u l a t i o n s at work w i t h i n the c i t y . The M i l l s type land 6 F o r i n s t a n c e , Kau and Lee (1976) found that h a l f of 50 United S t a t e s c i t i e s which they examined had the p r i c e e l a s t i c i t y of demand f o r housing g r e a t e r than -1.0. CONCEPTUAL FRAMEWORK / 20 rent f u n c t i o n i s regarded as l o c a t i o n rent f u n c t i o n and does not account f o r the e f f e c t s of environment, laws and other r e g u l a t i o n s on land v a l u e s . (b) E x t e r n a l i t i e s 1. P o l i n s k i and S h a v e l l (1976) and Richardson (1977) added the concept of "amenities" to the l o c a t i o n rent f u n c t i o n . The two s t u d i e s suggest the p o s s i b i l i t y of p o s i t i v e l y - s l o p e d land rent f u n c t i o n s as the d i s t a n c e from the c i t y center i n c r e a s e s . In both s t u d i e s , a household determines i t s l o c a t i o n to maximize i t s u t i l i t y f u n c t i o n s u b j e c t to a budget c o n s t r a i n t . Richardson i n c l u d e s p o p u l a t i o n d e n s i t y as a proxy f o r neighborhood c o n d i t i o n s . In Richardson's model, the l e s s populated a neighborhood i s , the b e t t e r the environment becomes. Since P o l i n s k i and S h a v e l l i n c l u d e a l l amenities i n one general e x p r e s s i o n , t h e i r study i s more u s e f u l f o r i n v e s t i g a t i n g i n d u s t r i a l or commercial land value v a r i a t i o n s . T h e r e f o r e the f o l l o w i n g e x p l a n a t i o n s w i l l use the P o l i n s k i and S h a v e l l model. In t h e i r paper, the i n d i r e c t u t i l i t y f u n c t i o n V i s expressed as: V(k) = V [ p ( k ) , y - T ( k ) , a ( k ) ] [2.17] where k = D i s t a n c e from the CBD, CONCEPTUAL FRAMEWORK / 21 p(k) = P r i c e per u n i t of housing (housing rent) at l o c a t i o n "k" y - T(k) = Net income before t r a n s p o r t a t i o n c o s t s at l o c a t i o n "k", and a(k) = Index of amenities at l o c a t i o n "k". In e q u i l i b r i u m , a household cannot i n c r e a s e i t s u t i l i t y by changing l o c a t i o n . In other words, every household achieves the same l e v e l of u t i l i t y , implying dV(k)/dk = 0. D i f f e r e n t i a t i n g [2.17] with respect to k and s o l v i n g f o r p ' ( k ) , we o b t a i n the slope of housing r e n t s as f o l l o w s : p'(k) = T'»[9V(k)/9(y - T ( k ) ) ] / [ 9 V ( k ) / 9 p ( k ) ] -a ' ( k ) • [ 9 V ( k ) / 9 a ( k ) ] / [ 9 V ( k ) / 9 p ( k ) ] [2.18] 9V(k)/9(y - T(k)) > 0 because an i n c r e a s e i n net income before t r a n s p o r t a t i o n c o s t s i n c r e a s e s u t i l i t y given housing rent and am e n i t i e s . 3V(k)/9p(k) < 0 s i n c e an i n c r e a s e i n housing rent decreases u t i l i t y given income and a m e n i t i e s . 9V(k)/9a(k) > 0 because an i n c r e a s e i n amenities i n c r e a s e s u t i l i t y given income and housing r e n t . T h e r e f o r e the f i r s t term i n the r i g h t hand s i d e of [2.18] i s n e g a t i v e . The s i g n of the second term, - a ' ( k ) • [ 9 V ( k ) / 9 a ( k ) ] / 9 V ( k ) / 9 p ( k ) ] i s p o s i t i v e . The i n c o r p o r a t i o n of amenities r e s u l t s i n an ambiguous slope of the rent f u n c t i o n . The c o n d i t i o n f o r a n e g a t i v e l y - s l o p e d rent f u n c t i o n i s : i CONCEPTUAL FRAMEWORK / 22 T'•[3V(k)/9(y - T ( k ) ) ] / [ 9 V ( k ) / 9 P ( k ) ] > a' . [ 9 V ( k ) / 9 a ( k ) ] / [ 9 V ( k ) / 9 p ( k ) ] i . e . T'•[9V(k)/9(y - T ( k ) ) ] > a'•[9V(k)/9a(k)] [2.19] Equation [2.19] shows that the slope of the housing rent f u n c t i o n i s negative "as long as a small movement away from the CBD r e s u l t s i n a gr e a t e r u t i l i t y l o s s from higher t r a n s p o r t a t i o n expenses than i t does i n a gain from improved a m e n i t i e s . " 7 2. S t u l l (1974) analyzed the e f f e c t s of zoning r e g u l a t i o n s on land v a l u e s . H i s model c o n t a i n s manufacturing and r e s i d e n t i a l s e c t o r s . The manufacturing s e c t o r i s l o c a t e d i n the CBD surrounded by the r e s i d e n t i a l a r e a . Assuming a l i n e a r c i t y , he examines the i n f l u e n c e of land zoned f o r manufacturing on r e s i d e n t i a l land v a l u e s . The r e s u l t of h i s a n a l y s i s shows that the land rent g r a d i e n t has a p o s i t i v e l y - s l o p e d p o r t i o n c l o s e to the CBD. T h i s r e s u l t , c o n t r a d i c t o r y to the negative e x p o n e n t i a l f u n c t i o n , appeared because i n d u s t r i a l use e x e r t s negative e x t e r n a l i t i e s on r e s i d e n t i a l use. 3. Henderson (1977) analyzed how a i r p o l l u t i o n a f f e c t s the r e s i d e n t i a l land rent g r a d i e n t . Firms l o c a t e d i n the CBD ' P o l i n s k i and S h a v e l l (1976), p.122. CONCEPTUAL FRAMEWORK / 23 a r e assumed t o p r o d u c e two g o o d s : consumer goods and p o l l u t a n t s . H e n d e r s o n a l s o assumes t h a t h o u s e h o l d s w i t h i d e n t i c a l incomes and p r e f e r e n c e s a r e l o c a t e d a r o u n d t h e CBD. E a c h h o u s e h o l d f a c e s t h e f o l l o w i n g i n d i r e c t u t i l i t y f u n c t i o n : V = V [ I , p ] _ ( u ) f p x , p , e ( u ) , a ( u ) ] [2.20] where I = Income; p-^(u) = L a n d r e n t o f a s i t e "u" m i l e s from t h e c i t y c e n t e r ; p = H o u s i n g p r i c e ; A. Py = P r i c e s of o t h e r consumer goods; e ( u ) = L e i s u r e ; and a ( u ) = D i s a m e n i t y c a u s e d by a i r p o l l u t i o n . In e q u i l i b r i u m , dV/du = 0. D i f f e r e n t i a t i n g [ 2 . 2 0 ] , we o b t a i n 0 = [dV/dp1(u)]>[dp1(u)/du] + [ 3 V / 3 e ( u ) ] • [ 3 e ( u ) / 3 u ] + [ 9 V / 9 a ( u ) ] • [ 9 a ( u ) / 9 u ] M o v i n g t h e f i r s t t e r m on t h e r i g h t - h a n d s i d e t o t h e l e f t - h a n d s i d e , - [ 3 V / 3 p 1 ( u ) ] • [ 9 p 1 ( u ) / 9 u ] = [ 3 V / 3 e ( u ) ] - [ 9 e ( u ) / 9 u ] + [ 3 V / 3 a ( u ) ] • [ 3 a ( u ) / 3 u ] [2.21] U s i n g Roy's I d e n t i t y , 8 we c a n w r i t e t h e amount o f l a n d u s e d , 1 ( u ) , a s f o l l o w s : 8 R o y ' s I d e n t i t y i n d i c a t e s t h e f o l l o w i n g e q u a l i t y : The amount of c o n s u m p t i o n of a good i s e q u a l t o t h e n e g a t i v e o f t h e r a t i o o f " t h e m a r g i n a l u t i l i t y o f t h e good" a g a i n s t " t h e m a r g i n a l u t i l i t y o f income." CONCEPTUAL FRAMEWORK / 24 l ( u ) = -[3V/9p 1(u)] / [9V/9I] There f o r e we can rewrite. [2 .21 ] as 9p 1(u)/9u = [(3V/3e(u))•(9e(u)/9u) + 9V/9a(u))•(9a(u)/9u)] [ l ( u ) • ( 9 V / 9 I ) ] [2.22] When there i s no p o l l u t i o n , the s i g n of Equation [2.22] i s always n e g a t i v e . T h i s i s because we can d e l e t e the second term of the numerator on the ri g h t - h a n d s i d e i n t h i s case. A f t e r the d e l e t i o n , the numerator i s negative and the denominator i s p o s i t i v e . However, with p o l l u t i o n , the sign i s i n c o n c l u s i v e because the second term of the numerator i s p o s i t i v e . Equation [2.22] i n d i c a t e s that the i n c l u s i o n of the negative e x t e r n a l i t y of a i r p o l l u t i o n made the slope of land - r e n t g r a d i e n t ambiguous. In summary, the s t u d i e s done by P o l i n s k i and S h a v e l l , Richardson, S t u l l , and Henderson demonstrate that e x t e r n a l i t i e s such as neighborhood amenities and p o l l u t i o n make the slope of land rent f u n c t i o n ambiguous. C. LAND RENT FUNCTIONS IN NONMONOCENTRIC MODELS In S e c t i o n s A and B, we reviewed land rent f u n c t i o n s of monocentric models. Those models have the f o l l o w i n g two p o i n t s i n common: a) a c i t y has a predetermined c e n t r a l p o i n t ; and CONCEPTUAL FRAMEWORK / 25 b) la n d values (or housing p r i c e s ) vary i n order to o f f s e t the d i f f e r e n c e s of t r a n s p o r t a t i o n c o s t s from v a r i o u s l o c a t i o n s to the c e n t r a l p o i n t . In nonmonocentric models, there i s no c e n t r a l p o i n t to which a l l people t r a v e l and a l l products are shipped. Rather these models t r y to e x p l a i n why the CBD e x i s t s . Some asp e c t s of agglomeration economies provide a b a s i s f o r e x p l a i n i n g the e x i s t e n c e of the CBD. Agglomeration economies can be d e f i n e d as pr o d u c t i o n b e n e f i t s a r i s i n g from the s c a l e and pr o x i m i t y of the economic a c t i v i t i e s i n an urban a r e a . T h i s s e c t i o n examines how nonmonocentric models analyze land value v a r i a t i o n s w i t h i n a CBD. I t i s d i v i d e d i n t o two p a r t s a c c o r d i n g to the source of agglomeration economies. Two s t u d i e s to be reviewed i n the f i r s t p a r t regard i n t e r a c t i o n among fi r m s as the source of agglomeration economies. The source f o r the study i n the second part i s the l e v e l of knowledge i n each f i r m . 1. O'Hara (1979) and Tauchen and Witte (1984) examine o f f i c e l o c a t i o n s w i t h i n a square CBD. The two s t u d i e s assume that a f i r m b e n e f i t s s o l e l y from i t s employees c o n t a c t s with other f i r m s ' employees. Although the two papers have many common p o i n t s , the l a t t e r paper d e a l s only with o f f i c e rent CONCEPTUAL FRAMEWORK / 26 f u n c t i o n s . T h e r e f o r e , most of the d e s c r i p t i o n s i n t h i s p a r t are based on O'Hara's work, which e x p l a i n s land rent f u n c t i o n s as w e l l . The CBD i s assumed to be a square, centered at (x=0, y=0), with a g r i d road system. I d e n t i c a l firms are u n i f o r m l y d i s t r i b u t e d w i t h i n the CBD. (This assumption of u n i f o r m i t y i s r e l a x e d l a t e r . ) Employees in each f i r m t r a v e l ( i n a r e c t i l i n e a r manner) to c o n t a c t employees of other f i r m s . The frequency of t r a n s a c t i o n s between f i r m s i s c o n s t a n t . Both models n e g l e c t congestion c o s t s i n c u r r e d from t r a v e l l i n g . The b u i l d e r s who provide the o f f i c e space pay l a n d r e n t s to absentee l a n d l o r d s . F i r s t of a l l , O'Hara analyzes the case of u n i f o r m l y d i s t r i b u t e d f i r m s . In t h i s case, he f i n d s that employees' t r a v e l c o s t s r i s e as the square of a firms r a d i a l d i s t a n c e from the c e n t e r of the CBD i n c r e a s e s . T h i s can be expressed as: T(x,y) = [ OS/2] + [ ( C / S ) - ( x 2 + y 2 ) ] [2.23] where T(x,y) = Average t r a v e l c o s t per t r a n s a c t i o n f o r a f i r m l o c a t e d at ( x , y ) , C = Round-trip c o s t per u n i t d i s t a n c e , S = Length of one s i d e of the square CBD, and CONCEPTUAL FRAMEWORK / 27 x, y = x (east-west) and y (north-south) c o o r d i n a t e s (the o r i g i n i s the c e n t r a l p o i n t of the CBD). In e q u i l i b r i u m , a f i r m ' s p r o f i t , 7 r(x,y), equals zer o . T h e r e f o r e , 7 r(x,y) = q(F) - N-T(x,y) - R(x,y) = 0 which i n tu r n i m p l i e s , R(x,y) = q(F) - N-T(x,y) [2.24] S u b s t i t u t i n g [2.23] i n t o [2.24], we o b t a i n R(x,y) = q(F) - (N-C-S/2) - (N-C/S)-(x 2 + y 2 ) [2.25] where R(x,y) = O f f i c e rent at ( x , y ) , q(F) = Seminet revenue ( net revenue before t r a v e l c o s t s and o f f i c e r e n t ) , and N = Number of t r a n s a c t i o n s . F u r t h e r , i n e q u i l i b r i u m , the p r o f i t of o f f i c e b u i l d i n g c o n s t r u c t o r s , n(x,y), i s a l s o reduced to zero. Thus n(x,y) = G-R(x,y) - K(G) - W(x,y) = 0. which i m p l i e s W(x,y) = G-R(x,y) - K(G) [2.26] S u b s t i t u t i n g [2.25] i n t o [2.26], we o b t a i n W(x,y) = G-{q(F) - (N-C-S/2)- (N-C/S)-(x 2 + y 2 ) } - K(G) [2.27] where W(x,y) = Land rent at ( x , y ) , G = D e n s i t y of f i r m s , and CONCEPTUAL FRAMEWORK / 28 K(G) = Cost of p r o v i d i n g o f f i c e space of d e n s i t y G. Equations [2.25] and [2.27] represent o f f i c e rent f u n c t i o n and land rent f u n c t i o n , r e s p e c t i v e l y . Since q ( F ) , N, C, S, G, K(G) are a l l co n s t a n t , R' < 0, R" < 0, W' < 0, and W" < 0 are e a s i l y d e r i v e d . Hence, both o f f i c e rent and land rent f u n c t i o n s c l e a r l y have downward slop e s which are concave from below. Now the assumption of uniform f i r m d i s t r i b u t i o n i s r e l a x e d . In t h i s case, the d e n s i t y of o f f i c e space, G, becomes another endogenous v a r i a b l e . The o f f i c e space market i s in e q u i l i b r i u m when the marginal cost of p r o v i d i n g o f f i c e space equals o f f i c e r e n t : K'[G(x,y)] = R(x,y) [2.28] For s i m p l i c i t y , o f f i c e c o n s t r u c t i o n cost i s w r i t t e n as: K(G) = a-G b, b > 1 [2.29] The zero p r o f i t c o n d i t i o n becomes: W(x,y) = G(x,y)-R(x,y) - K[G(x,y)] [2.30] Using Equations [2.28], [2.29], and [2.30], we can d e r i v e the f o l l o w i n g e x p r e s s i o n s f o r the f i r s t and second order d e r i v a t i v e s of the land rent f u n c t i o n with r e s p e c t to x: 9W/9x = (b - l)-R-(9G/9x) [2.31] 92W/9x2 = (b - 1)•[(9R/9x)•(9G/9x) + R-(9 2G/9x 2)] [2.32] Since b > 1, R > 0, and 9G/9x < 0, 9W/9x i s always n e g a t i v e . CONCEPTUAL FRAMEWORK / 29 Th e r e f o r e , the land rent g r a d i e n t has a negative s l o p e . However, s i n c e b > 1, 9R/9x < 0, 9G/9x < 0, R > 0, and 9 2G/9x 2 < 0, we cannot determine i f [2.32] i s p o s i t i v e or ne g a t i v e . In other words, the cu r v a t u r e of land rent f u n c t i o n cannot be e x a c t l y determined i n the case of non-uniform f i r m d i s t r i b u t i o n . Tauchen and Witte (1984) a r r i v e d at a s i m i l a r c o n c l u s i o n . A n a l y z i n g o f f i c e rent f u n c t i o n , they conclude t h a t : a) at the center of the CBD, the slope i s zero; b) at the edges of the CBD, the slope i s upward towards the cente r of the CBD; and c) at other p o i n t s the slope depends on the d i s t r i b u t i o n of f i r m s , the d i s t r i b u t i o n of c o n t a c t s , and the seminet revenue 9 f u n c t i o n . From the d i s c u s s i o n i n t h i s p a r t , we found that the assumption of uniform f i r m d i s t r i b u t i o n guarantees the concave lan d rent f u n c t i o n , but that r e l a x i n g the assumption makes the cu r v a t u r e of the slope ambiguous. The assumption of uniform f i r m d i s t r i b u t i o n i s i n a p p r o p r i a t e f o r a c t u a l CBD's. T h e r e f o r e , the s t u d i e s we have examined l e a d us to the c o n c l u s i o n that i t i s d i f f i c u l t to e s t a b l i s h a land rent f u n c t i o n with an unambiguous c u r v a t u r e w i t h i n a CBD. 2. H e l s l e y (1986) c o n s i d e r s a nonmonocentric model with an endogenous f i r m d i s t r i b u t i o n , and where the p r o d u c t i o n 3The seminet revenue means "revenue net of s p a t i a l l y i n v a r i a n t c o s t s (Tauchen and Witte (1984), p.72)." CONCEPTUAL FRAMEWORK / 30 and exchange of knowledge i s the source of agglomeration economies. In h i s model, a f i r m ' s p r o d u c t i v i t y depends on the l e v e l of knowledge at the l o c a t i o n of the f i r m w i t h i n the CBD, where knowledge i s accumulated from other f i r m s . The q u a n t i t y of knowledge exchanged among firms decreases with i n c r e a s e d d i s t a n c e between f i r m s because of i n c r e a s i n g contact c o s t s . In e q u i l i b r i u m , the l e v e l of knowledge and the most p r o d u c t i v e l o c a t i o n s c l a i m the h i g h e s t land r e n t , l a n d rent a l s o a t t a i n s i t s maximum at the c e n t e r and d e c l i n e s as the d i s t a n c e from the center i n c r e a s e s . In summary, the three nonmonocentric s t u d i e s reviewed i n t h i s s e c t i o n a l l show that the c e n t r a l p o i n t c l a i m s the highest land value w i t h i n the CBD even without the monocentric assumption. The f i r s t two s t u d i e s showed that the c u r v a t u r e of land rent f u n c t i o n i s not d e f i n i t e l y determined i n the case of non-uniform f i r m d i s t r i b u t i o n . The t h i r d study demonstrated that a l a n d rent curve d e c l i n e s a c c o r d i n g to the l e v e l of knowledge. These three nonmonocentric models c o n f i r m the importance of c e n t r a l i t y and r a i s e q u e s t i o n s concerning the shape of l a n d rent f u n c t i o n w i t h i n a CBD. I I I . PAST EMPIRICAL RESEARCH Th i s chapter reviews past e m p i r i c a l s t u d i e s on i n t r a - u r b a n l a n d p r i c e v a r i a t i o n s . I t s purpose i s a) to i n t r o d u c e how monocentric and nonmonocentric models were t e s t e d i n these s t u d i e s and b) to examine independent v a r i a b l e s i n the hedonic r e g r e s s i o n equations. In s e l e c t i n g e m p i r i c a l s t u d i e s , f i r s t p r i o r i t y was given to s t u d i e s of commercial or i n d u s t r i a l land p r i c e v a r i a t i o n s and t h e i r determinants. Because the number of e m p i r i c a l s t u d i e s i n t h i s category i s very l i m i t e d , t h i s chapter a l s o i n c l u d e s s e v e r a l s t u d i e s which examine o f f i c e rent v a r i a t i o n s and r e s i d e n t i a l l a n d value v a r i a t i o n s . However, the i n c l u s i o n of these s t u d i e s r e i n f o r c e s t h i s c hapter's arguments because t h e i r methods of s e l e c t i n g independent v a r i a b l e s are s t i l l u s e f u l when a n a l y s i n g CBD land value v a r i a t i o n s . The past e m p i r i c a l s t u d i e s to be reviewed i n t h i s chapter are summarized i n c h r o n o l o g i c a l order i n Table 111.1 . A. EMPIRICAL TESTS OF MONOCENTRIC AND NONMONOCENTRIC MODELS 1. T r a d i t i o n a l Monocentric Models The t r a d i t i o n a l monocentric model attempts to e x p l a i n l a n d value v a r i a t i o n s i n terms of d i s t a n c e from a c e n t r a l p o i n t to the s i t e . The t h e o r e t i c a l papers by Mohring (1961), Alonso (1964), Solow (1973), Henderson (1985), and M i l l s 31 Tab 1e 111. 1 Summary of Past Empirical Research STUDY M i l l s (1969) M i l l s (19S9) Downing (1973) Dewees (1978) Jackson (1979) McDonald (1979) Clapp (1980) Asabere (1981) Hembd and Infanger (1981) Schmenner (1981) Hough and Cratz (1983) STUDY AREA Chicago Chicago The City of Milwaukee, Wi scons i n Toronto The c i t y of Milwaukee, Wisconsin Chicago Los Angeles Accra, Ghana Jessamine County, Kentucky Cincinnati metropolitan area, Ohio Chicago DATA Estimated land values in Hoyt* ( 1836, 1857, 1873. 1910, and 1928) 132 vacant land values in Olcott** (1966) 406 vacant land transactions data (1958 -1962) 1864 home sales from M.L.S. 1istings Average housing rents in 147 census tracts 253 observations (1960) and 258 observations (1970) in Olcott** 105 o f f i c e rents 211 vacant land transactions (1974 - 1978) 2016 vacant land transactions (1973 - 1978) 47 rents and 71 transactions (1975 - 1976) 139 o f f i c e rents (1978) FEATURES His t o r i c a l data study, distance from the center onl y. Zoning and distance from the center as independent var i ables. Variations in land values for reta i1 use. Changes in land rent gradients before and after the construction of a subway 1i ne. Trend surface analysis as an alternative access measure. Polynomial functions of distance from the CBD. Office rent variations in a decentralized c i t y . Ordinary hedonic regression anlysis applied to an African c i t y . Trend surface analysis as a check on the orthodox hedonic equation. Insignificant r e l a t i o n s h i p between rent (or value) of manufacturing s i t e and cen t r a l i ty. Hedonic rental equation within a CBD. Table III.1 Summary of Past E m p i r i c a l Research (Continued) Brennan et al . (1984) Chicago Cannaday and Kang (1984) Urbana-Champaign, I l l i n o i s Asabere and Harvey (1985) Halifax, Nova Scotia Kowalskl and Cplwell (198S) Wayne County, Michigan Johnson and Ragas (1987) New Orleans, Louisiana Peiser (1987) North Dal las NOTES: 29 o f f i c e rents (3 years) 19 o f f i c e rents (1979 -1980) 123 lot sales from M.L.S. l i s t i n g s (over 7.5 years) 24 undeveloped land transactions (1975 - 1984) 110 actual vacant land transactions (1972 - 1983) 466 vacant land transact ions Distance from the main street 1s included as an independent variable. Office rent variations in a small c i t y . Ordinary hedonic regression anlysis applied to a Canadian c i t y . Scale and frontage e f f e c t s on industrial land value. CBD land data, trend surface analysis, and three-dimensional pictures. Separate analysis for ind u s t r i a l , commercial, and o f f i c e land values. Homer Hoyt, One Hundred Yeras of Land Values in Chicago, (University of Chicago Press, 1933) Olcott's Land Value Blue Book of Chicago . (Chicago: G.C. Olcott & Co.) 13 > C/l •-3 S i—i O > w h in M > o x 00 PAST EMPIRICAL RESEARCH / 34 (1972) reviewed i n Chapter II form the conceptual framework of the t r a d i t i o n a l monocentric model. M i l l s (1969) conducted two e m p i r i c a l s t u d i e s of Chicago lan d v a l u e s . In the f i r s t study, he used h i s t o r i c a l Chicago land value data i n order to t e s t a negative e x p o n e n t i a l land value f u n c t i o n . The e x p l a n a t o r y power in terms of R 2 was q u i t e high f o r the 19th century data. However, i t d e c l i n e d d r a s t i c a l l y f o r the 20th century data. The second a n a l y s i s i n h i s paper u t i l i z e d O l c o t t ' s d a t a 1 0 and i n c l u d e d zoning dummy v a r i a b l e s as w e l l as d i s t a n c e to the CBD from each s i t e i n the r e g r e s s i o n equation. The g o o d n e s s - o f - f i t was g r e a t l y improved in t h i s case. The d e c r e a s i n g explanatory power of the negative e x p o n e n t i a l f u n c t i o n a l form was confirmed by McDonald (1979). He a l s o used O l c o t t ' s Chicago land v a l u e s data i n h i s study. In h i s a n a l y s i s of r e s i d e n t i a l l a n d value data, a fourth-degree polynomial f u n c t i o n of d i s t a n c e from the CBD turned out to be the best model both f o r 1960 and 1970 data s e t s . N o n r e s i d e n t i a l land values were a l s o examined i n the study. McDonald found that a fourth-degree polynomial f u n c t i o n of d i s t a n c e dominated the monocentric model i n 1 0 See the notes of Table III.1 f o r the complete names of the data sources u t i l i z e d by M i l l s (1969) and McDonald (1979). PAST EMPIRICAL RESEARCH / 35 e x p l a i n i n g n o n r e s i d e n t i a l land value v a r i a t i o n s i n 1960. The study p o i n t e d out that there are p o s i t i v e l y - s l o p e d p o r t i o n s i n both r e s i d e n t i a l and n o n r e s i d e n t i a l land value g r a d i e n t s . The above two s t u d i e s demonstrated the i n a p p l i c a b i l i t y of the negative e x p o n e n t i a l f u n c t i o n to recent data i n Chicago. The i n c l u s i o n of zoning dummy v a r i a b l e s improved the performance of the r e g r e s s i o n a n a l y s e s , as d i d the employment of higher degree polynomial f u n c t i o n s . T h i s i m p l i e s the e x i s t e n c e of e x t e r n a l i t i e s and/or other p r i c e i n f l u e n c i n g f a c t o r s such as zoning r e g u l a t i o n s . 2. Trend Surface A n a l y s i s Three e m p i r i c a l s t u d i e s i n Table III.1 [Jackson (1979), Hembd and Infanger (1981), and Johnson and Ragas (1987)] used a technique c a l l e d t r e n d s u r f a c e a n a l y s i s . As we have a l r e a d y seen i n Chapter I I , a model i s c a l l e d monocentric when i t has a p r e s p e c i f i e d c e n t r a l p o i n t to which a l l people t r a v e l and a l l products are sent. I f a model does not have such a predetermined c e n t r a l p o i n t , i t i s c o n s i d e r e d a nonmonocentric model. Because the s t u d i e s which u t i l i z e t r e n d s u r f a c e a n a l y s i s do not s p e c i f y a c e n t r a l p o i n t , they can be regarded as nonmonocentric. The nonmonocentric models of O'Hara (1979), Tauchen and Witte (1984), and H e l s l e y (1986) form the conceptual framework of t r e n d s u r f a c e PAST EMPIRICAL RESEARCH / 36 a n a l y s i s . Trend s u r f a c e a n a l y s i s i s "a s p e c i a l case of o r d i n a r y l e a s t square r e g r e s s i o n a n a l y s i s with r e s t r i c t i o n of the independent v a r i a b l e to l o c a t i o n a l i n d i c e s . " 1 1 In other words, t r e n d s u r f a c e a n a l y s i s , l i k e the t r a d i t i o n a l monocentric model, t r i e s to e x p l a i n land value v a r i a t i o n s based s o l e l y on v a r i a t i o n s i n l o c a t i o n or ac c e s s . However, in t h i s type of a n a l y s i s , a c c e s s i b i l i t y from each s i t e to the c e n t r a l p o i n t or other p o i n t s i s not measured. The l o c a t i o n of each s i t e i s i d e n t i f i e d by C a r t e s i a n c o o r d i n a t e s . More s p e c i f i c a l l y , a land value r e g r e s s i o n using t r e n d s u r f a c e a n a l y s i s takes the f o l l o w i n g form: Z=bo+b1x+b2y+b3x2+b(|xy+b5y2+b6x3+b7x2y+b8xy2 + - • • where Z i s land value and x and y are the C a r t e s i a n c o o r d i n a t e s of each o b s e r v a t i o n . Jackson's (1979) study of housing p r i c e v a r i a t i o n s i n the c i t y of Milwaukee compared two a c c e s s i b i l i t y measures. One employs commonly used monocentric measurements, such as d i s t a n c e from each s i t e to the CBD. The other method i s tren d s u r f a c e a n a l y s i s . Using the average housing rent i n each census t r a c t as the dependent v a r i a b l e , the o r d i n a r y r e g r e s s i o n a n a l y s i s and t r e n d s u r f a c e a n a l y s i s were ^ S c h r o e d e r and S j o q u i s t (1976), p.383. PAST EMPIRICAL RESEARCH / 37 performed. The monocentric a n a l y s i s f a i l e d to e x p l a i n housing rent v a r i a t i o n s i n the study area. On the other hand, a fourth-degree polynomial equation of t r e n d s u r f a c e a n a l y s i s produced a h i g h R 2 and s i g n i f i c a n t c o e f f i c i e n t s with expected s i g n s . The f a i l u r e of monocentric a c c e s s i b i l i t y measure i s a r e s u l t of the l o c a t i o n of employment c e n t e r s and expressways. Hembd and Infanger (1981) used t r e n d s u r f a c e a n a l y s i s to complement as w e l l as examine the t r a d i t i o n a l monocentric land value r e g r e s s i o n model. They demonstrated that land values peak i n areas other than the two urban c e n t e r s that they examined i n the t r a d i t i o n a l r e g r e s s i o n model. T h i s f i n d i n g g i v e s an i n d i c a t i o n of the complexity of the a c t u a l l a n d value s u r f a c e . T h e i r study area l i e s i n a u r b a n - f r i n g e a r e a . In such an area, b e h a v i o r a l f a c t o r s such as s u b d i v i s i o n approval seem to have a great impact on land value v a r i a t i o n s . Johnson and Ragas (1987) i n v e s t i g a t e d land value v a r i a t i o n s w i t h i n the New Orleans CBD using t r e n d s u r f a c e a n a l y s i s . Based on the a d j u s t e d R 2 and maximum-likelihood c r i t e r i a , a s i x t h - d e g r e e polynomial equation of t r e n d s u r f a c e a n a l y s i s was a b e t t e r p r e d i c t o r than the b e h a v i o r a l PAST EMPIRICAL RESEARCH / 38 models they s t u d i e d . 1 2 Using t h r e e - d i m e n s i o n a l p i c t u r e s , Johnson and Ragas demonstrated that the e f f e c t of major s t r e e t c o r r i d o r s overwhelms that of the c e n t r a l p o i n t of the CBD. The above three s t u d i e s a p p l i e d t r e n d s u r f a c e a n a l y s i s to three d i f f e r e n t areas. Jackson (1979) used r e s i d e n t i a l rent data i n a c i t y . Hembd and Infanger (1981) a p p l i e d the a n a l y s i s to ur b a n - f r i n g e area data. Johnson and Ragas (1987) surveyed CBD land v a l u e s . In s p i t e of the d i f f e r e n t study areas, a l l three s t u d i e s r e v e a l e d : 1) the e x i s t e n c e of complex land value s u r f a c e s which cannot be dete c t e d by the t r a d i t i o n a l monocentric models and 2) the us e f u l n e s s of tre n d s u r f a c e a n a l y s i s i n land value r e s e a r c h . B. INDEPENDENT VARIABLES IN THE PAST HEDONIC-PRICE STUDIES The p r e v i o u s two models, i . e . , the t r a d i t i o n a l monocentric models and t r e n d s u r f a c e a n a l y s i s , e x p l a i n land value v a r i a t i o n s i n a p u r e l y s p a t i a l c o n t e x t . Both of them 1 2 I n the b e h a v i o r a l models, the independent v a r i a b l e s i n c l u d e d the d i s t a n c e from the c e n t r a l p o i n t of the CBD to the s i t e , dummy v a r i a b l e s f o r major s t r e e t c o r r i d o r s , d i s t a n c e s from e x t e r n a l i t i e s , and zoning and time dummy v a r i a b l e s . The v a r i a b l e s i n the b e h a v i o r a l models w i l l be f u r t h e r e x p l o r e d i n Part B of t h i s s e c t i o n . PAST EMPIRICAL RESEARCH / 39 c o n s i d e r o n l y t h e r e l a t i o n s h i p between l a n d v a l u e and l a n d l o c a t i o n . In o t h e r words, t h o s e m o d e ls assume h o m o g e n e i t y o f n e i g h b o r h o o d s and i d e n t i c a l p h y s i c a l c h a r a c t e r i s t i c s o f l o t s w i t h i n t h e CBD. In r e a l i t y , however, e a c h n e i g h b o r h o o d has i t s own f e a t u r e s . As w e l l , we r a r e l y f i n d two l o t s e x a c t l y t h e same i n t e r m s o f p h y s i c a l c h a r a c t e r i s t i c s . The h e t e r o g e n e i t y o f b o t h n e i g h b o r h o o d s and l o t s w i t h i n t h e CBD l i m i t s t h e e x p l a n a t o r y power of b o t h t r a d i t i o n a l m o n o c e n t r i c models and t r e n d s u r f a c e a n a l y s i s . In a d d i t i o n t o t h e d i v e r s e c h a r a c t e r i s t i c s of n e i g h b o r h o o d s and l o t s , a s t a t i s t i c a l p r o b l e m stems from t r e n d s u r f a c e a n a l y s i s . B e c a u s e o f i t s s p e c i f i c a t i o n ( i . e . , t h e d o u b l e power s e r i e s ) , t r e n d s u r f a c e a n a l y s i s s u f f e r s f r o m a c o l l i n e a r i t y p r o b l e m . The p r o b l e m d o e s n o t c o n c e r n us a s l o n g a s t h e a n a l y s i s i s r e s t r i c t e d t o d e r i v i n g p r e d i c t e d v a l u e s . However, b e c a u s e o f t h e c o l l i n e a r i t y p r o b l e m , t h e c o e f f i c i e n t s o f i n d e p e n d e n t v a r i a b l e s of t r e n d s u r f a c e a n a l y s i s p r o v i d e us w i t h no s i g n i f i c a n t i n f o r m a t i o n . T h e r e f o r e , we must r e s o r t t o a n o t h e r model t o o b t a i n m e a n i n g f u l c o e f f i c i e n t s t h a t can e x p l a i n d e t e r m i n a n t s o f CBD l a n d v a l u e . The h e d o n i c - p r i c e a p p r o a c h i s t h e most w i d e l y employed t e c h n i q u e t o a n a l y z e l a n d v a l u e d e t e r m i n a n t s . "The word PAST EMPIRICAL RESEARCH / 40 hedonic has to do with p l e a s u r e ; i . e . , a hedonic p r i c e i s r e l a t e d to the ple a s u r e d e r i v e d from the v a r i o u s a t t r i b u t e s of a given commodity." 1 3 In other words, the hedonic approach examines the r e l a t i o n s h i p between the p r i c e s and a t t r i b u t e s of a commodity to e x p l a i n d i f f e r e n c e s i n the p r i c e of the commodity. The h e d o n i c - p r i c e approach assumes that market p a r t i c i p a n t s accept a hedonic p r i c e f u n c t i o n as g i v e n . 1 8 A h e d o n i c - p r i c e f u n c t i o n , f o r i n s t a n c e , p ( z , , z 2 , z 3 , z a,...) in F i g u r e 3.1, rep r e s e n t s market c l e a r i n g c o n d i t i o n s f o r an a t t r i b u t e of a commodity. The f u n c t i o n can be d e f i n e d as a set of p o i n t s of tangency between buyers' b i d f u n c t i o n s [0, and 0 2 i n F i g u r e 3.1] and s e l l e r s ' o f f e r f u n c t i o n s and <t>2 in F i g u r e 3.1]. The e m p i r i c a l s t u d i e s reviewed i n t h i s p a r t used the simple hedonic approach. In t h i s approach, a c o e f f i c i e n t i s assumed to be the marginal w i l l i n g n e s s to pay f o r a p a r t i c u l a r a t t r i b u t e . Using the h e d o n i c - p r i c e approach, we can untangle the "bundle of a t t r i b u t e s . " The e m p i r i c a l s t u d i e s i n Table III.1 g i v e s us an i n s i g h t f o r choosing independent v a r i a b l e s i n the hedonic r e g r e s s i o n equation. The a t t r i b u t e s c o n s i d e r e d i n these 1 3Cannaday and Kang (1984), p.68. 1"The d e s c r i p t i o n i n t h i s paragraph i s based on the argument i n F o l l a i n and Jimenez (1985), pp.78-81. PAST EMPIRICAL RESEARCH / .41 Fi g u r e 3.1 Hedonic, Buyer B i d , and S e l l e r O f f e r F u n c t i o n s p ( z ) 4>2 0i 0 I : ; . • Notes .z,, z 2 , z 3 , z 0 , ... = A t t r i b u t e s of a commodity, P( • ) = Hedonic p r i c e f u n c t i o n , 0, and 6'2 = Buyers' b i d f u n c t i o n s , and • 0., and <t>2 = S e l l e r s ' o f f e r f u n c t i o n s . PAST EMPIRICAL RESEARCH / 42 s t u d i e s can be c l a s s i f i e d as f o l l o w s : 1 5 (1) A c c e s s i b i l i t y ( i ) C e n t r a l i t y , ( i i ) P r o x i m i t y to t r a n s p o r t a t i o n f a c i l i t i e s , and ( i i i ) A c c e s s i b i l i t y to e x t e r n a l i t i e s . (2) S i t e - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s ( i ) S i z e , ( i i ) Lot shape, and ( i i i ) L o c a t i o n on a b l o c k . (3) N e i g h b o r h o o d - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s ( i ) Road width, ( i i ) P r e s t i g i o u s address and major s t r e e t c o r r i d o r s , and ( i i i ) Zoning. The h e d o n i c - p r i c e approach c o n s i d e r s b e h a v i o r a l f a c t o r s [items (2) and (3)] as w e l l as l o c a t i o n f a c t o r s [item ( 1 ) ] , whereas the t r a d i t i o n a l monocentric models and trend s u r f a c e a n a l y s i s c o n s i d e r a c c e s s i b i l i t y o n l y . Each category i s examined in turn i n the r e s t of t h i s c h a pter. (1) A c c e s s i b i l i t y ( i ) C e n t r a l i t y Although s e v e r a l s t u d i e s r e f u t e d the importance of c e n t r a l i t y [e.g., Schmenner (1981), Cannaday and Kang 1 5 T h e a t t r i b u t e s are l i m i t e d to those which can be a p p l i e d to r e s e a r c h on CBD l a n d v a l u e s . T h e r e f o r e , the l i s t d i d not i n c l u d e f a c t o r s such as income and p o l l u t i o n . PAST EMPIRICAL RESEARCH / 43 (1984), Kowalski and C o l w e l l (1986), and P e i s e r (1987)], a l l CBD land value s t u d i e s i n Table III.1 obtained a s i g n i f i c a n t c o e f f i c i e n t of the v a r i a b l e of d i s t a n c e to the c e n t r a l p o i n t [Hough and Kratz (1983), Brennan et a l . (1984), and Johnson and Ragas (1987)]. A study of o f f i c e r e n t s i n Los Angeles demonstrated t h a t c e n t r a l i t y i s the most important v a r i a b l e i n the r e g r e s s i o n equation [Clapp (1980)]. ( i i ) P r o x i m i t y to t r a n s p o r t a t i o n f a c i l i t i e s Hough and C r a t z (1983) i n c l u d e d " d i s t a n c e to the nearest commuter r a i l w a y s t a t i o n " i n t h e i r independent v a r i a b l e s , but the c o e f f i c i e n t of the v a r i a b l e was i n s i g n i f i c a n t . To the c o n t r a r y , the f o l l o w i n g two s t u d i e s confirmed the importance of p r o x i m i t y to t r a n s p o r t a t i o n f a c i l i t i e s . Schmenner (1981) found that " d i s t a n c e to expressways" i s s i g n i f i c a n t . Dewees (1978) i n v e s t i g a t e d the impact of subways on r e s i d e n t i a l land v a l u e s i n Toronto. The study a s c e r t a i n e d the importance of d i s t a n c e from the s i t e to the subway l i n e i n determining pr o p e r t y v a l u e s . ( i i i ) A c c e s s i b i l i t y to E x t e r n a l i t i e s Among the e m p i r i c a l s t u d i e s summarized i n Table 111.1 , only Johnson and Ragas e x p l i c i t l y i n c l u d e d PAST EMPIRICAL RESEARCH / 44 e x t e r n a l i t y v a r i a b l e s i n the r e g r e s s i o n e q u a t i o n . 1 6 According to t h e i r study, three sources of e x t e r n a l i t i e s i n the New Orleans CBD are a) the L o u i s i a n a Superdome, b) the M i s s i s s i p p i R i v e r , and c) a l a r g e housing p r o j e c t . E x t e r n a l i t y b e n e f i t s are d e r i v e d from the Superdome and the M i s s i s s i p p i R i v e r . The Superdome o f f e r s inexpensive p u b l i c p a r k i n g space and a l a r g e shopping area and the M i s s i s s i p p i R i v e r enhances the a e s t h e t i c s of nearby s i t e s . On the other hand, the housing p r o j e c t i s g e n e r a l l y c o n s i d e r e d to be a disadvantage as i t widely c o n c e i v e d as a source of crime. Although the c o e f f i c i e n t s of the r i v e r v a r i a b l e had an expected negative s i g n , i t was not s i g n i f i c a n t . The other two v a r i a b l e s were h i g h l y s i g n i f i c a n t . (2) S i t e - s p e c i f i c P r i c e I n f l u e n c i n g F a c t o r s ( i ) S i z e Some s t u d i e s examined the r e l a t i o n s h i p between land area and u n i t p r i c e . Downing(1973), Asabere (1981), Asabere and Harvey (1985), and Kowalski and C o l w e l l (1986) a l l observed a h i g h l y s i g n i f i c a n t i n v e r s e r e l a t i o n s h i p between s i z e and u n i t p r i c e . However, the i n v e r s e r e l a t i o n s h i p was not s i g n i f i c a n t i n the o f f i c e 1 6Downing (1973) t r i e d to i n c o r p o r a t e the disadvantages of brewing or tanning o p e r a t i o n s i n t o h i s study but he was not s u c c e s s f u l . PAST EMPIRICAL RESEARCH / 45 land study done by P e i s e r (1987). ( i i ) Lot shape Only Asabere and Harvey (1985) i n v e s t i g a t e d the i n f l u e n c e of l o t shape on land v a l u e . They obtained a s i g n i f i c a n t c o e f f i c i e n t with a negative s i g n f o r an " i r r e g u l a r shape" dummy v a r i a b l e . ( i i i ) L o c a t i o n on a block P e i s e r (1987) i n c l u d e d "corner l o t " dummy v a r i a b l e s i n h i s r e g r e s s i o n equations. The c o e f f i c i e n t s of the v a r i a b l e s were s i g n i f i c a n t only i n the i n d u s t r i a l land r e g r e s s i o n s . Downing (1973) used the d i s t a n c e from the corner l o t to a s i t e on the same block to a s c e r t a i n the advantages of pr o x i m i t y to the corner l o t . As a r e s u l t , he found a s i g n i f i c a n t c o e f f i c i e n t of the v a r i a b l e . (3) N e i g h b o r h o o d - s p e c i f i c P r i c e I n f l u e n c i n g F a c t o r s ( i ) Road width P e i s e r (1987) employed three kinds of dummy v a r i a b l e s f o r road width, depending on the number of t r a f f i c l a n e s . He obtained s i g n i f i c a n t c o e f f i c i e n t s on the road width dummies f o r a l l h i s i n d u s t r i a l land r e g r e s s i o n equations and for two out of four o f f i c e land r e g r e s s i o n equations. Instead of road width, Downing PAST EMPIRICAL RESEARCH / 46 (1973) measured t r a f f i c volume. Using 24-hour weekdays volume , he observed that the amount of t r a f f i c has an important p o s i t i v e i n f l u e n c e on r e t a i l l a n d v a l u e s . ( i i ) P r e s t i g i o u s address and major s t r e e t c o r r i d o r s Clapp (1980) found that l a n d i n Bev e r l y H i l l s commanded a higher land p r i c e than land elsewhere i n Los Angeles. Johnson and Ragas (987) i n c l u d e d dummy v a r i a b l e s f o r two major s t r e e t c o r r i d o r s w i t h i n the CBD to determine d i f f e r e n c e s i n land v a l u e s . The r e s u l t s v a r i e d a c c o r d i n g to f u n c t i o n a l forms. G e n e r a l l y speaking, the d i f f e r e n c e i n land values was more s i g n i f i c a n t f o r one major c o r r i d o r than f o r another. ( i i i ) Zoning Zoning r e g u l a t i o n s are a l s o expected to a f f e c t land v a l u e . Almost a l l r e g r e s s i o n equations i n the s t u d i e s i n Table III.1 c o n t a i n e d zone c l a s s i f i c a t i o n dummy v a r i a b l e s . D e n s i t y v a r i a b l e s were i n c l u d e d i n two s t u d i e s [Johnson and Ragas (1987) and P e i s e r (1987)] i n order to i n c o r p o r a t e development e x p e c t a t i o n s . In Johnson and Ragas' study, the d e n s i t y v a r i a b l e s were h i g h l y s i g n i f i c a n t , whereas i n P e i s e r ' s study, the v a r i a b l e s were s i g n i f i c a n t only i n the commercial land r e g r e s s i o n s . PAST EMPIRICAL RESEARCH / 47 C. SUMMARY In t h i s chapter, we reviewed 16 e m p i r i c a l s t u d i e s on land value v a r i a t i o n s . Using Chicago data, two s t u d i e s demonstrated the d e c r e a s i n g e x p l a n a t o r y power of the negative e x p o n e n t i a l f u n c t i o n a l form. Trend s u r f a c e a n a l y s i s i l l u s t r a t e d complex lan d value v a r i a t i o n s i n v a r i o u s areas ( i n c l u d i n g a CBD). The t e s t of monocentric and nonmonocentric models proved that the t r a d i t i o n a l monocentric model cannot adequately e x p l a i n i n t r a - u r b a n l a n d value v a r i a t i o n s . The examination of the past e m p i r i c a l s t u d i e s l e a d s us to b e l i e v e that i t i s worthwhile to c o n s i d e r a c c e s s i b i l i t y , and s i t e - and n e i g h b o r h o o d - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s when we analyze the determinants of CBD l a n d v a l u e s . IV. EMPIRICAL ANALYSIS The purpose of t h i s chapter i s to i n v e s t i g a t e l a n d p r i c e v a r i a t i o n s w i t h i n the Tokyo CBD by a n a l y z i n g vacant land s a l e s data d u r i n g the p e r i o d from 1975 to 1987. Two t o p i c s are c o n s i d e r e d i n t h i s c h a pter: a) Can nonmonocentric models perform b e t t e r than, monocentric models? and b) Can a c c e s s i b i l i t y and s i t e - and n e i g h b o r h o o d - s p e c i f i c f a c t o r s e x p l a i n land p r i c e v a r i a t i o n s w i t h i n the CBD? P r e d i c t e d land p r i c e v a r i a t i o n s are d e r i v e d using the t r a d i t i o n a l monocentric model and t r e n d s u r f a c e a n a l y s i s , which were both examined in Chapter I I I . The p r e d i c t i v e powers of the two models w i l l be compared s t a t i s t i c a l l y i n t h i s c h a pter. The hedonic r e g r e s s i o n a n a l y s i s w i l l uncover other important determinants of CBD land p r i c e v a r i a t i o n s . The chapter c o n t a i n s four s e c t i o n s . S e c t i o n A d e s c r i b e s the study a r e a . S e c t i o n B e x p l a i n s the data used. S e c t i o n C d e a l s with the methodologies of e m p i r i c a l the analyses of t h i s c h a pter. S e c t i o n D presents and i n t e r p r e t s the e s t i m a t i o n r e s u l t s . 48 EMPIRICAL ANALYSIS / 49 A. STUDY AREA When we look at the c e n t r a l p a r t of Tokyo, we f i n d one major i n d u s t r i a l and business c o n c e n t r a t i o n c e n t e r e d around Tokyo S t a t i o n . F i g u r e 4.1 shows the area c o n s i d e r e d i n t h i s study. In the f i g u r e , however, the area l o c a t e d to the west of the r a i l w a y l i n e s i s not i n c l u d e d i n the study a r e a . The reasons the western p a r t of the CBD i s excluded a r e : a) data u n a v a i l a b i l i t y ( b e c a u s e most of the land to the west of Tokyo S t a t i o n i s owned by major companies f o r t h e i r headquarters, very few t r a n s a c t i o n s are recorded) and b) the western area belongs to another m u n i c i p a l i t y . The study area i n c l u d e s a l l of C e n t r a l Ward ( Chuo Ku) e x c l u d i n g a small i s l a n d i n Sumida R i v e r . C e n t r a l Ward i s one of 23 wards i n Tokyo S p e c i a l Ward A r e a . 1 7 As the name of the ward i n d i c a t e s , C e n t r a l Ward i s l o c a t e d almost at the center of Tokyo S p e c i a l Ward Area. I t i s almost r e c t a n g u l a r i n shape with a l e n g t h of 3,500 meters and a width of 1,600 meters. According to the 1981 Establishment Census of Japan, the number of business establishments i n C e n t r a l Ward t o t a l s 42,247, with a t o t a l employment of 690,000. As Table IV.1 i n d i c a t e s , C e n t r a l Ward has a d i v e r s i f i e d i n d u s t r i a l 1 7 A s p e c i a l ward i s one type of m u n i c i p a l i t y i n Tokyo P r e f e c t u r e . I t has almost the same a d m i n i s t r a t i v e and l e g i s l a t i v e powers as a c i t y . EMPIRICAL ANALYSIS / 50 F i g u r e 4.1 Map of Study Area 250 meters 500 1000 Table IV.1 I n d u s t r i e s and Employment i n C e n t r a l Ward, 1981 In d u s t r y A g r i c u l t u r e , F o r e s t r y , and F i s h i n g Mining C o n s t r u c t i o n Manufacturing Wholesale and R e t a i l Finance and Insurance Real E s t a t e T r a n s p o r t a t i o n and Communication U t i l i t i e s S e r v i c e s * * P u b l i c Sector TOTAL Number of Establishments 32(0.08)* 26(0.06) 1,244(2.94) 4,138(9.80) 22,612(53.52) 1,323(3.13) 2,016(4.77) 1,494(3.54) 23(0.05) 9,298(22.01) 41(0.10) 42,247(100.00) Number of Workers 1 ,126(0.16) 1,4310.21) 37,018(5.36) 103,070(14.39) 297,053(43.02) 73,763(10.68) 16,718(2.42) 40,669(5.89) 1,922(0.28) 113,263(16.40) 4,500(0.65) 690,533(100.00) Notes: * Percentages are i n paranthses. ** " S e r v i c e s " i n c l u d e s both p r i v a t e and business s e r v i c e s . Source: 1981 Establishment Census of Japan. EMPIRICAL ANALYSIS / 52 s t r u c t u r e . "Wholesale and R e t a i l " has the l a r g e s t share of workers and e s t a b l i s h m e n t s while " S e r v i c e s " and "Manufacturing" are second and t h i r d i n s i z e r e s p e c t i v e l y . Each d i s t r i c t i n the ward can be c l a s s i f i e d a c c o r d i n g to i t s major i n d u s t r y . For example, the Ginza, which i s l o c a t e d i n the south-western corner of the map, i s the most p r e s t i g i o u s shopping area. O f f i c e s are c o n c e n t r a t e d i n the Yaesu, Nihon-bashi, and Kabutocho areas. Ningyocho and the area north of i t (not shown on the map) s p e c i a l i z e i n the garment i n d u s t r y . A c o n c e n t r a t i o n of p r i n t i n g e s t a b l i s h m e n t s can be found i n Kayabacho and H a t c h o b o r i . The area to the south of the Ginza d i s t r i c t s p e c i a l i z e s i n food p r o c e s s i n g and food w h o l e s a l i n g . In the study area, the p u b l i c t r a n s p o r t a t i o n system i s densely developed. Besides the r a i l w a y l i n e s shown on the map, there are 7 subway l i n e s and numerous bus r o u t e s . B. THE DATA The data u t i l i z e d i n t h i s study c o n s i s t s of vacant land t r a n s a c t i o n s that took p l a c e i n the study area between 1975 and 1987. I t was compiled by the Japanese A s s o c i a t i o n of Real E s t a t e A p p r a i s e r s . The data c o n t a i n s i n f o r m a t i o n such as vacant l a n d p r i c e per square meter, address, date of EMPIRICAL ANALYSIS / 53 Table IV.2 Summary S t a t i s t i c s f o r the Data VARIABLE N MEAN STANDARO MINIMUM MAXIMUM DEVIATION VALUE VALUE P 836 5296568.69617 6145409.94638 211750.00000 57579000.000 SIZE 836 140.09809 261 .99944 15.00000 6708.000 ROAD 836 12.08732 9.08500 2.00000 45.000 CORNER 836 0. 1 1842 0. 32330 0.00000 1.000 SHAPE 836 0.03349 0. 1B003 0.00000 1 .000 FAR 836 6. 3421 1 1.01141 4.00000 9.000 CENDIS 836 1365.03589 446.80490 27 3.00000 2303.000 GINZA 836 0. 17344 0.37886 0.00000 1 .000 STAOIS 836 255.08852 134.50328 14.00000 763.000 FINDIS 836 1000.63278 514. 17610 154.00000 2310.000 GINZADIS 836 1652.36124 865.36274 91.00000 3912.000 P = land p r i c e i n Japanese Yen per square meter. SIZE = Lot s i z e i n square meters. ROAD = Road width i n meters. CORNER = Dummy v a r i a b l e (1 = corner l o t ) . SHAPE = Dummy v a r i a b l e (1 = i r r e g u l a r l o t ) . FAR = F l o o r area r a t i o . CENDIS = D i s t a n c e from the c e n t r a l p o i n t of the CBD ( i n meters). GINZA = Dummy v a r i a b l e (1 = l o t i n the G i n z a ) . STADIS = D i s t a n c e from the nearest s t a t i o n ( i n meters). FINDIS = D i s t a n c e from the f i n a n c i a l c e n t e r ( i n meters). GINZADIS = D i s t a n c e from the Ginza shopping area ( i n meters). t r a n s a c t i o n , names of vendor and purchaser, width of road on which the s i t e abuts, shape, s i z e , d i s t a n c e to the nearest s t a t i o n , and zoning r e g u l a t i o n s . There i s not, however, any inf o r m a t i o n on the f i n a n c i n g of each t r a n s a c t i o n . A l l the t r a n s a c t i o n s are fo r vacant (or to-be - c l e a r e d ) l a n d . Only "arm's l e n g t h " t r a n s a c t i o n s are i n c l u d e d . The number of ob s e r v a t i o n s t o t a l s 836. Table IV.2 summarizes the data. EMPIRICAL ANALYSIS / 54 C. METHODOLOGY T h i s s e c t i o n d i s c u s s e s t h e methods o f s o l v i n g two p r o b l e m s : 1) C o m p a r i s o n of m o n o c e n t r i c and n o n m o n o c e n t r i c m o d e l s ; and 2) A n a l y s i s o f d e t e r m i n a n t s of CBD l a n d v a l u e v a r i a t i o n s . 1. Comparison of Monocentric and Nonmonocentric Models M o n o c e n t r i c and n o n m o n o c e n t r i c models a r e compared i n terms of e x p l a n a t o r y power f o r l a n d v a l u e v a r i a t i o n s w i t h i n t h e Tokyo CBD. In t h i s s t u d y , " t r e n d s u r f a c e a n a l y s i s " r e p r e s e n t s n o n m o n o c e n t r i c m o d e l s . a. M o n o c e n t r i c Model The e x p l a n a t o r y v a r i a b l e s i n t h e t r a d i t i o n a l m o n o c e n t r i c model a r e l i m i t e d t o " d i s t a n c e t o t h e c e n t r a l p o i n t of t h e CBD from t h e s i t e " and " t i m e o f s a l e . " M i l l s ' n e g a t i v e e x p o n e n t i a l f u n c t i o n i s a p p l i e d , a s w e l l a s l i n e a r and p o l y n o m i a l f u n c t i o n s . • N e g a t i v e e x p o n e n t i a l f u n c t i o n LN(P) = 0 O + /J,T + |S-2d + e • L i n e a r f u n c t i o n p = |30 + 0,T + p2a + e • P o l y n o m i a l f u n c t i o n EMPIRICAL ANALYSIS / 55 P = 0 O + /3,T + (32d + 0 3 d 2 + /3«d3 + • • • + e where P = Land p r i c e per square meter; T = Time v a r i a b l e (e.g., 1=1975 1st q u a r t e r , 2=1975 2nd q u a r t e r , . . . ) ; d = D i s t a n c e to the c e n t r a l p o i n t of the CBD; and e = E r r o r term. The e r r o r terms are assumed to be normally d i s t r i b u t e d , u n c o r r e l a t e d random v a r i a b l e s with common v a r i a n c e . We u t i l i z e the o r d i n a r y l e a s t squares method. b. Trend Surface A n a l y s i s As d i s c u s s e d i n Chapter I I I , t r e n d s u r f a c e a n a l y s i s a l s o uses the o r d i n a r y l e a s t square method. (X,Y) l o c a t i o n c o o r d i n a t e s and time v a r i a b l e s are both t r e a t e d as independent v a r i a b l e s , whereas land value i s t r e a t e d as the dependent v a r i a b l e . X and Y c o o r d i n a t e s are as s i g n e d based on a C a r t e s i a n plane which i s a r b i t r a r i l y p l a c e d i n the CBD. Six r e g r e s s i o n equations a c c o r d i n g to the degree of power can be expressed as f o l l o w s : • L i n e a r P = 0 O+ 0T +02Y+e • Second degree polynomial P = /3Q+ 0T +0 2Y+0 3X 2+ 0 4XY+ 0 5Y 2+e • T h i r d degree polynomial EMPIRICAL ANALYSIS / 56 P = 0 O+ 0T + 0 ^ + ... +0 6X 3 +/3?X2Y +/3gXY2 +0 gY 3+e • Fourth degree polynomial P = 0 O+ 0T + 0 ^ +... +/3 1 0X 4 +/3 nX 3Y +/3 1 2X 2Y 2 +0 1 3XY 3 • F i f t h degree polynomial P = |30+ |3T +0}K + ... +/315X5 + 0 1 6 X 4 Y + 0 1 ? X 3 Y 2 + 0 1 8 X 2 Y 3  + /3 1 9XY 4 +/3 2 QY 5 +e • S i x t h degree polynomial P = 0 O+ 0T + 0 ^ +... +/3 2 1X 6 +/3 2 2X 5Y + 0 2 3 X 4 Y 2 +/3 2 4X 3Y 3  + /3 2 5X 2Y 4 +/3 2 6XY 5 + /3 2 ?Y 6 +e where P = Land p r i c e per square meter, T = Time v a r i a b l e ( e.g., 1=1975 1st q u a r t e r , 2=1975 2nd qu a r t e r , ... ) or time dummy v a r i a b l e s , X = X c o o r d i n a t e , Y = Y c o o r d i n a t e , and e = E r r o r term(normally d i s t r i b u t e d , u n c o r r e l a t e d random v a r i a b l e s with common v a r i a n c e ) . The l i n e a r equation r e p r e s e n t s a plane, whereas the maximum numbers of extrema from the f i v e subsequent equations are 1, 4, 9, 16, and 25, r e s p e c t i v e l y . The l a s t number i s l a r g e enough to e x p l a i n land p r i c e v a r i a t i o n s i n a narrow area such as a CBD. The above s i x r e g r e s s i o n equations w i l l be used to a s c e r t a i n which degree of EMPIRICAL ANALYSIS / 57 p o l y n o m i a l e q u a t i o n i s most s a t i s f a c t o r y s t a t i s t i c a l l y . 2. A n a l y s i s of Determinants of CBD Land Value V a r i a t i o n s The model we employ t o c l a r i f y d e t e r m i n a n t s o f CBD l a n d v a l u e v a r i a t i o n s i s h e d o n i c r e g r e s s i o n a n a l y s i s . The i n d e p e n d e n t v a r i a b l e s i n t h e a n a l y s i s c a n be d i v i d e d i n t o t h e f o l l o w i n g t h r e e g e n e r a l c a t e g o r i e s as i n t r o d u c e d i n C h a p t e r I I I : P = f (A; S; N) where P = L a n d p r i c e p e r s q u a r e m e t e r ; A = A c c e s s i b i l i t y ; S = S i t e - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s ; and N .= N e i g h b o r h o o d - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s . (a) A c c e s s i b i l i t y A c c e s s i b i l i t y v a r i a b l e s w i l l i n c l u d e t h e d i s t a n c e s f r o m t h e s i t e t o t h e c e n t e r o f t h e C B D , 1 8 t h e n e a r e s t s t a t i o n , t h e c e n t e r of G i n z a S h o p p i n g A r e a , and t h e c e n t e r o f t h e f i n a n c i a l d i s t r i c t . 1 9 The s i g n s o f t h e c o e f f i c i e n t s of t h e s e f o u r v a r i a b l e s a r e e x p e c t e d t o be n e g a t i v e . B e c a u s e t h e r e a r e no slums or r e s i d e n t i a l d e v e l o p m e n t p r o j e c t s i n t h e CBD, 1 8 T h e s i t e o f Tokyo S t a t i o n i s r e g a r d e d as t h e c e n t e r o f t h e CBD i n t h i s s t u d y b e c a u s e o f i t s i m p o r t a n c e as a t r a n s p o r t a t i o n node. 1 9 T h e s i t e o f t h e Tokyo S t o c k E x c h a n g e i s r e g a r d e d as t h e c e n t e r o f t h e f i n a n c i a l d i s t r i c t . EMPIRICAL ANALYSIS / 58 no negative e x t e r n a l i t i e s are s p e c i f i e d . (b) S i t e - S p e c i f i c P r i c e I n f l u e n c i n g F a c t o r s S i t e - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s determine the development s u i t a b i l i t y of a l o t and, hence, a f f e c t i t s p r i c e . V a r i a b l e s i n t h i s category are s i z e , shape, and corner l o c a t i o n . Dummy v a r i a b l e s are designed to determine the p o t e n t i a l e f f e c t s of shape and corner l o c a t i o n . "1" i s assigned to i r r e g u l a r s h a p e d - l o t s , while "0" i s assig n e d to r e c t a n g u l a r or square l o t s . In the same way, "1" i s assig n e d to corner l o c a t i o n s and "0" to middle l o c a t i o n s w i t h i n a block. Signs of the c o e f f i c i e n t s of " s i z e " and "shape" are expected to be ne g a t i v e , while a p o s i t i v e s i g n i s expected for the "corner l o c a t i o n . " (c) Neighborhood-Price I n f l u e n c i n g F a c t o r s Road width i s used as a proxy f o r both the p r o f i t a b i l i t y and development s u i t a b i l i t y of s i t e s i n a neighborhood. O f f i c e s or shops on a wider road enjoy much more a d v e r t i s i n g b e n e f i t s than those on a narrower road because of v i s i b i l i t y to passengers and t r a f f i c . The zoning bylaw l i m i t s development depending on road width. S t r i c t e r b u i l d i n g r e g u l a t i o n s are imposed on a s i t e which abuts on a narrower road than a s i t e on a wider road. Thus, a p o s i t i v e sign i s expected f o r t h i s v a r i a b l e . EMPIRICAL ANALYSIS / 59 A dummy v a r i a b l e i s designed to con f i r m the d i f f e r e n c e s i n land value between the Ginza, the most p r e s t i g i o u s shopping area i n Tokyo, and a l l areas elsewhere i n the study. "1" i s ass i g n e d to l o t s i n the Ginza, while "0" i s ass i g n e d to l o t s elsewhere. A p o s i t i v e s i g n i s expected f o r t h i s dummy v a r i a b l e . A l l the land i n the CBD i s designated as "Commercial Area" a c c o r d i n g to the zoning bylaw i n the study area. The f a c t o r that d i f f e r e n t i a t e s each neighborhood i n terms of zoning r e g u l a t i o n s i s F l o o r Area R a t i o , which can be d e f i n e d as the r a t i o of the maximum t o t a l f l o o r area i n a b u i l d i n g to the area of the b u i l d i n g s i t e . F l o o r area r a t i o ranges from 4 to 9 i n t h i s study area. A p o s i t i v e s i g n i s expected f o r t h i s v a r i a b l e . (d) Summary V a r i a b l e s i n the hedonic r e g r e s s i o n a n a l y s i s can be summarized as f o l l o w s : P = Land p r i c e i n Japanese Yen per square meter (Land p r i c e s range from 211,750 Yen per sq. meter to 57,579,000 Yen per sq. meter with a mean value of 5,296,569 Yen per sq. meter.); T = Time v a r i a b l e (e.g., 1=1975 1st q u a r t e r , 2=1975 2nd qu a r t e r , . . . ) ; EMPIRICAL ANALYSIS / 60 DISCEN = Dis t a n c e from the c e n t r a l p o i n t of the CBD (Tokyo S t a t i o n ) to each s i t e (a s t r a i g h t - l i n e d i s t a n c e i n meters); DISGIN = Dis t a n c e from the ce n t e r of the Ginza Shopping Area to each s i t e (a s t r a i g h t - l i n e d i s t a n c e i n meters); DISFIN = Dis t a n c e from the F i n a n c i a l Center (Tokyo Stock Exchange) to each s i t e (a s t r a i g h t - l i n e d i s t a n c e i n meters); DISSTA = Di s t a n c e from the nearest s t a t i o n to each s i t e (a s t r a i g h t - l i n e d i s t a n c e i n meters); SIZE = S i z e of the l o t i n square meters; SHAPE = Dummy v a r i a b l e ( 1 = i r r e g u l a r l o t . 28 out of 836 obs e r v a t i o n s are i r r e g u l a r l y - s h a p e d l o t s ) ; CORNER = Dummy v a r i a b l e ( 1 = corner l o t . 99 out of 836 obs e r v a t i o n s are corner l o t s . ); RD = Road width i n meters; GINZA = Dummy v a r i a b l e (1 = l o t i n the Ginza, 0 = l o t elsewhere); FAR = F l o o r area r a t i o ; The o r d i n a r y l e a s t squares method i s a l s o u t i l i z e d i n the hedonic r e g r e s s i o n a n a l y s i s . The e r r o r terms i n the r e g r e s s i o n equations are assumed to be normally d i s t r i b u t e d , u n c o r r e l a t e d random v a r i a b l e s with common v a r i a n c e . EMPIRICAL ANALYSIS / 61 Tabl e IV.3 Regr e s s i o n R e s u l t s (a) T r a d i t i o n a l Monocentric Model L i n e a r P Negative Exp. LN(P) 13.901 (166.83)*** 0.058098 (37.539)*** -0.00068429 (-13.885)*** 0.6417 -13302.5 Box-Cox Dep. V a r i a b l e CONSTANT TIME CENDIS Adjusted R 2 Log of L i k e l i h o o d Funct ion X 1849400 (2.8728)*** 250790 (21.014)*** -3760.4 (-9.8948)*** 0.3719 -14058.0 p* 10.015 (254.70)*** 0.027616 (37.812)*** -0.00032388 (-13.926)*** 0.6449 -13299.8 -0.05 NOTES: • t - s t a t i s t i c s are i n parenthses. • *** i n d i c a t e s s i g n i f i c a n t at 1% l e v e l . • N = 836 f o r a l l e s t i m a t e s . • P = Land p r i c e (Yen per square meter) • TIME = Time v a r i a b l e s (1 = 1975 1st q u a r t e r , 2 = 1975 2nd q u a r t e r , 52 = 1987 4th quater) • CENDIS = D i s t a n c e t o the c e n t r a l p o i n t of the CBD D. ESTIMATION RESULTS 1. Comparison of Monocentric and Nonmonocentric Models a. T r a d i t i o n a l Monocentric Model Table IV.3 shows the r e g r e s s i o n r e s u l t s f o r the *°Other r e g r e s s i o n equations, such as the q u a d r a t i c and higher degree polynomial equations, were a l s o attempted. There were no s i g n i f i c a n t d i f f e r e n c e s between these t r a d i t i o n a l monocentric m o d e l s . 2 0 A l l c o e f f i c i e n t s are EMPIRICAL ANALYSIS / 62 s i g n i f i c a n t at the 1% l e v e l i n every r e g r e s s i o n e q u a t i o n . In the l i n e a r equation, the average q u a r t e r l y i n c r e a s e i n land p r i c e s d u r i n g the p e r i o d from 1975 to 1987 i s 250,790 Yen per square meter. By moving one meter from the c e n t r a l p o i n t , land p r i c e s are decreased by 3,760 Yen per square meter. In the negative e x p o n e n t i a l r e g r e s s i o n , on the other hand, land p r i c e s r i s e about 5.8% every q u a r t e r year and decrease 0.07% every meter from the c e n t r a l p o i n t . To choose the b e t t e r f u n c t i o n a l form between the l i n e a r equation and the negative e x p o n e n t i a l f u n c t i o n , we u t i l i z e the Box-Cox t r a n s f o r m a t i o n and the l o g - l i k e l i h o o d r a t i o t e s t . Since the dependent v a r i a b l e s are d i f f e r e n t , R 2 cannot be used f o r t h i s purpose. The l i k e l i h o o d r a t i o t e s t f o r s i g n i f i c a n c e i s based on the theory t h a t , under n u l l h y p o t h e s i s , twice the d i f f e r e n c e i n the l o g a r i t h m i c l i k e l i h o o d between a n u l l h y p o thesis and . 2 an a l t e r n a t i v e hypothesis i s d i s t r i b u t e d as x with one degree of freedom. In order to get 99% confi d e n c e i n t e r v a l f o r X , we should c o n s i d e r a l l the va l u e s of X Q which s a t i s f y the f o l l o w i n g c o n d i t i o n : Lmax ( * > " Lmax ( X0 > < < \ ™ ' *2< « > ' where, 2 0 ( c o n t ' d ) equations and the negative e x p o n e n t i a l equation in terms of the g o o d n e s s - o f - f i t and p r e d i c t i v e power. EMPIRICAL ANALYSIS / 6 3 L ,„ : L o g - l i k e l i h o o d F u n c t i o n ; max 3 X Q : Value s p e c i f i e d i n H Q ; and X : Value which maximizes the l i k e l i h o o d . For a = 0 . 0 1 , ( 1 / 2 ) X 2 = 3 . 3 1 5 . Using the above i n e q u a l i t y , we can t e s t the corresponding value ( X Q ) to determine whether or not i t f a l l s w i t h i n the a p p r o p r i a t e confidence i n t e r v a l . Based on Box-Cox t r a n s f o r m a t i o n , i t was found that the maximum l o g a r i t h m i c l i k e l i h o o d o c c u r r e d at X = - 0 . 0 5 and L™o„{ ) i s equal to - 1 3 2 9 9 . 8 (see Table I V . 3 ) . When X Q = 0 ( i . e . , Negative Exp. ), Lmax ( - ° ' 0 5 > " Lmax ( 0 > = - 1 3 2 9 9 . 8 - ( - 1 3 3 0 2 . 5 ) = 2 . 7 < ( 1 / 2 ) x 2 ( 0 . 0 1 ) = 3 . 3 1 5 When X Q = 1 ( i . e . , L i n e a r ) , L ( - 0 . 0 5 ) - L ( 1 ) max max = - 1 3 2 9 9 . 8 - ( - 1 3 3 0 2 . 5 ) = 7 5 8 . 2 > ( 1 / 2 ) x 2 ( 0 . 0 1 ) = 3 . 3 1 5 T h e r e f o r e , X Q = 0 i s w i t h i n the 9 9 % c o n f i d e n c e i n t e r v a l , whereas X Q = 1 i s o u t s i d e the i n t e r v a l . In other words, the l i n e a r equation i s r e j e c t e d at 0 . 0 1 l e v e l . The l o g a r i t h m i c l i k e l i h o o d r a t i o t e s t demonstrated that the negative e x p o n e n t i a l f u n c t i o n i s the b e t t e r f u n c t i o n a l form f o r the monocentric models. F i g u r e 4 . 2 N o r m a l P r o b a b i l i t y P l o t s - T r a d i t i o n a l M o n o c e n t r i c M o d e l PLOT OF RANKRES'RANKRES SYMBOL USED IS N PLOT OF S'RANKRES SYMBOL USED IS R 3 + 2 + R A I N 1 • K F 0 R 0 • V A R I -1 + A B L E -2 • •3 • R R N RN RRRR RRRR RRRN RRRR RRRR NRRR RRRRR NRRRR NRRRR RRRRR RRRRR RRRRN RRRRN RRRR RRRR RRRR RRRRR RRRRN RRRN RRRR RRRR RRRR NNR NRRRRR NNNRRR NNRRR N R RR R £ i—i w (—1 o > > z > K in •-) -5 + -1 0 RANK FOR VARIABLE S CTn EMPIRICAL ANALYSIS / 65 O a u •H J-l 4-> c QJ U 0 C O s • rrj C O •H u 4-) 3 •H T ) •rH (0 CM EH •P O rH P-l •H W QJ t / l r - D f J U i Z I— H N U J Q ( C U J [/) H O 3 < I EMPIRICAL ANALYSIS / 66 The normal p r o b a b i l i t y p l o t s ( F i g u r e 4.2) and r e s i d u a l p l o t s ( F i g u r e 4.3) of the negative e x p o n e n t i a l equation confirmed the n o r m a l i t y and randomness of the e r r o r term. No o u t l i e r s were found based on the c r i t i c a l value f o r the o u t l i e r u s i n g "R-student." b. Trend Surface A n a l y s i s The r e g r e s s i o n r e s u l t s of t r e n d s u r f a c e a n a l y s i s (1st to 6th degree polynomial) appear on Table IV.4. A l l r e g r e s s i o n equations are s i g n i f i c a n t i n terms of t h e i r F - v a l u e s . The c h o i c e among the s i x polynomial equations estimated i n t h i s s e c t i o n i s based on c o n t r i b u t e d F-values. A c o n t r i b u t e d F-value i s d e f i n e d as: F = [ R * / d f 1 ] / [ ( 1 - R j ) / d f 2 ] . where R| = e x t r a R 2 given by s u r f a c e of order n+1 over one of order n. R 2 = T o t a l R 2 accounted f o r by s u r f a c e . d f ! = degrees of freedom a s s o c i a t e d with the added components, 3 f o r a q u a d r a t i c over a l i n e a r , 4 f o r a cubic over a q u a d r a t i c , and so on. d f 2 = degrees of freedom a s s o c i a t e d with the r e s i d u a l s c a l c u l a t e d as t o t a l degrees of freedom l e s s the number of independent v a r i a b l e s . When we look at " C o n t r i b u t e d F-values" i n Table IV.4, we Table IV.4 Regression Results (b) Trend Surface Analysis 1st Degree 2nd Degree 3rd Degree 4th Deqree 5th Deqree INTERCEPT 14 .2466* * * 15 .0739*** 14.4132*** 11.3179*** 15.0469*** TIME 0. 5859E-01*** 0. 5929E-01*** 0.5891E-01*** 0.5944E-01*** 0.5957E-01*** X -0 .8445E -03*** -0 .2029E -02*** -0.1387E-02*** 0.5714E-02** -0.6269E-02 Y 0. 2691E-03*** 0. 6663E-03*** 0.1626E-02*** 0.2786E-02** 0.1916E-02 X' 0. 1771E-06*** -0.3707E-06 -0.5293E-05** 0.9494E-05 XY 0. 2528E-06*** 0.4633E-06 -0.3776E-05 -0.6481E-06 Y» -0 .2484E -06*** -0.8802E-06*** 0.8157E-06 -0.1107E-05 X' 0.2068E-09* 0.1609E-08 -0.8555E-08 X'y -0.3598E-09** 0.1733E-08 0.3952E-08 XY' 0.3494E-09*** 0.8771E-10 -0.3525E-08 Y 3 -0.1489E-10 -0.5044E-09** 0.1848E-08 X' -0.1668E-12 0.3798E-11 x'y -0.1597E-12 -0.4485E-11 X' Y« -0.2746E-12 0.4849E-11* XY' 0.2519E-12 0. 1919E-11** Y * -0.7824E-14 -0.1052E-12 X» -0.6603E-15* X'Y 0.1341E-14* X'Y« -0.1678E-14*** X'Y' 0.8348E-15** XY* -0.1220E-16 Y* -0.2702E-16 X • X'Y X'Y' X'Y> X'Y' XY* Y« F-value Adjusted R * Contributed F Notes: *** 742.225 0.7267 Si g n i f i c a n t at X = X coordinate. Y = Y 438.324 0.7586 36.516*** 0.01 l e v e l . coordinate. 271.484 0.7641 4.809*** 187.463 0.7701 4.280*** Significant at 0.05 l e v e l . 135.624 0.7722 1.131 Signif leant at 0.1 6th Deqree 6.3460 0.5987E-01*** 0.1877E-01 0.1541E-01 -0.3019E-04 0.4726E-05 -0.3314E-04*** 0.3060E-07 -0.4352E-07 0.6122E-07** 0.2630E-08 -0.1658E-10 0.3405E-10 -0.3567E-10 -0.6089E-11 0. 1521E-12 0.3787E-14 -0.6908E-14 0.2262E-14 0.8892E-14 -0.2591E-14 0.4331E-15 -0.8359E-18 -0.4685E-18 0.2697E-17 -0.4285E-17*** 0.2264E-17*** -0.6705E-18* 0.7444E-19 104.501 0.7763 2.216** l e v e l . S 13 t-H w r-H o > > z > w i—i EMPIRICAL ANALYSIS / 68 f i n d that the value of the 5th degree polynomial i s not s i g n i f i c a n t ( 1 . 1 3 1 ). T h e r e f o r e , the 4th degree polynomial i s chosen as the most r e l i a b l e s p e c i f i c a t i o n f o r t r e n d s u r f a c e a n a l y s i s . The normal p r o b a b i l i t y p l o t s and r e s i d u a l p l o t s appear i n F i g u r e 4.4 and 4.5, r e s p e c t i v e l y . There i s no i n d i c a t i o n t h a t they v i o l a t e the n o r m a l i t y and randomness assumptions of the e r r o r term. No o u t l i e r s were d e t e c t e d i n the data s e t . c. Comparison of the Two Models 2 1. Adjusted R , Akaike Information C r i t e r i o n , and PRESS s t a t i s t i c s are u t i l i z e d to s e l e c t between the t r a d i t i o n a l monocentric model and t r e n d s u r f a c e a n a l y s i s . • Akaike Information C r i t e r i o n can be d e f i n e d as f o l l o w s 2 1 : AIC = [(-2/T) • l o g L ^ y , 0^)] + [2K 1/T] [4.1] where T = Number of o b s e r v a t i o n s ; K 1 = Number of independent v a r i a b l e s ; and L ^ Y r ^ 1 ) = L i k e l i h o o d f u n c t i o n . For the l i n e a r s t a t i s t i c a l model, Equation [4.1] can be reduced to 2 1Amemiya, T. (1980), p343. F i g u r e 4.4 Normal P r o b a b i l i t y P l o t s - T r e n d S u r f a c e A n a l y s i s PLOT OF RANKREG'RANKRES SYMBOL USED IS N PLOT OF S-RANKRES SYMBOL USED IS R 3 + 2 + R R R A . N 1 K F 0 R 0 V A R I -1 A B L E -2 + •3 + NN NNNN R NNN RRR NNN RRRR NNNN RRRR NNNRRRRR NNRRRRR RRRRRR RRRRR RRRRR RRRRRN RRRRNN RRRNN RRRRRN RRRN RRRN RRRR RRRR RRRRR RRRR RRRR NRRR NRRR NNRRR NNNRRR NNNNRRRR NN R N N R RRRR R R R H 3 TJ n » i—i O > > Z > f K in t-H •A - 1 0 RANK FOR VARIABLE S EMPIRICAL ANALYSIS / 70 in U •H 03 •H W >i rH rd C < o m CO c 0) -p O CM r0 QJ PS o o [ f l ^ D O U J Z h H N U J D ( £ I U I / 1 M Q D < J EMPIRICAL ANALYSIS / 71 AIC = log[(y'M,y)/T] + 2K,/T [4.2] where y = Vector of o b s e r v a t i o n s ; M, = I - X,(X,'X,)" 1X,•; and y'M,y = R e s i d u a l sum of squares. If K, (number of independent v a r i a b l e s ) i s c o n s t a n t , a smaller AIC i n d i c a t e s a smaller, r e s i d u a l sum of squares (y'M,y). In other words, a smaller AIC means b e t t e r g o o d n e s s - o f - f i t . By i n c l u d i n g the second term on the r i g h t - h a n d s i d e of Equations [4.1] and [4.2], AIC e x p l i c i t l y i n c o r p o r a t e s the p e n a l t y f o r i n c r e a s i n g the number of independent v a r i a b l e s . Using AIC, we can compare models which have d i f f e r e n t numbers of independent v a r i a b l e s . Akaike Information C r i t e r i o n r e q u i r e s us to s e l e c t the model fo r which the value of the above AIC i s minimum. • PRESS ( P r e d i c t i o n Sum of Squares) s t a t i s t i c i s d e f i n e d a s 2 2 : PRESS = .L ( y. - y. . ) i = 1 1 11 1 where n = Number of o b s e r v a t i o n s ; y^ = A c t u a l value of " i t h " o b s e r v a t i o n ; and y- _• = P r e d i c t e d value of " i t h " o b s e r v a t i o n using the M 1 data " i t h " o b s e r v a t i o n d e l e t e d . 2 2 M y e r s , R.H. (1986), p.107. EMPIRICAL ANALYSIS / 72 The PRESS s t a t i s t i c r e f l e c t s the p r e d i c t i o n c a p a b i l i t i e s of a model. I t i s the sum of p r e d i c t i o n e r r o r s (y. - y. _ : ) • 1 l , i The p r e d i c t e d v a l u e , y. _•, i s d e r i v e d by not u s i n g the 1 , 1 o b s e r v a t i o n y^. "Thus, in t h i s way, the o b s e r v a t i o n y^ was not s i m u l t a n e o s l y used for f i t and model assessment, t h i s being the true t e s t of v a l i d a t i o n . " 2 3 The best model i s the one with the s m a l l e s t PRESS. Adju s t e d R 2, AIC, and PRESS are shown below i n order to compare a) the t r a d i t i o n a l monocentric model (negative e x p o n e n t i a l f u n c t i o n ) and b) t r e n d s u r f a c e a n a l y s i s (4th degree p o l y n o m i a l ) . MODEL Rf_ AIC PRESS Monocentric 0.6417 -0.9154 334.57 TSA 0.7701 -1.3293 229.39 Comparing these s t a t i s t i c a l f i g u r e s , we f i n d t h a t t r e n d s u r f a c e a n a l y s i s i s c o n s i s t e n t l y s u p e r i o r to the t r a d i t i o n a l monocentric model. We can conclude that t r e n d s u r f a c e a n a l y s i s i s the b e t t e r model f o r e x p l a i n i n g land value v a r i a t i o n s i n the Tokyo CBD based on the three s t a t i s t i c s . 2. In a d d i t i o n to the above s t a t i s t i c s , p r e d i c t e d land values based on the t r a d i t i o n a l monocentric model and t r e n d s u r f a c e a n a l y s i s are c a l c u l a t e d and compared with a c t u a l 2 3 M y e r s , R.H. (1986), p.106. EMPIRICAL ANALYSIS / 73 land values i n the f o l l o w i n g way: i ) The study area i s d i v i d e d i n t o 45 subareas. i i ) A l l p r e v i o u s land t r a n s a c t i o n p r i c e s are converted to 1987 p r i c e s using l a n d p r i c e i n d i c e s which were compiled by the Japan Real E s t a t e I n s t i t u t e . The i n d i c e s r e c o r d semi-annual land p r i c e i n c r e a s e s f o r t y p i c a l commercial land i n the study area. i i i ) The average of the 1987 land v a l u e s obtained i n Step i i ) i s c a l c u l a t e d f o r each subarea. iv ) Two kinds of p r e d i c t e d 1987 land v a l u e s are c a l c u l a t e d f o r each subarea. One i s based on the best f u n c t i o n a l form of the t r a d i t i o n a l monocentric model, using the d i s t a n c e from the c e n t r a l p o i n t of each subarea to the center of the CBD. The other i s based on the p r e f e r r e d degree of polynomial equation of t r e n d s u r f a c e a n a l y s i s , u t i l i z i n g l o c a t i o n c o o r d i n a t e s of the c e n t r a l p o i n t of each subarea. v) Three-dimensional p i c t u r e s and contour maps are produced. In each case a comparison w i l l be drawn between the a c t u a l land values and the p r e d i c t e d land v a l u e s . A software package c a l l e d "DISSPLA" i s u t i l i z e d to o b t a i n these p i c t u r e s . F i g u r e s 4.6 and 4.7 represent a c t u a l average land values i n 1987, whereas F i g u r e s 4.8 through 4.11 represent EMPIRICAL ANALYSIS / 74 F i g u r e 4 . 7 ACTUAL LAND VALUE IN 1987 16 \ — CO CD ft a -0 2 ( N ^ J F i g u r e 4 . 9 PREDICTED LAND VALUE IN 1987 (MONOCENTRIC MODEL) I S EMPIRICAL ANALYSIS / 79 EMPIRICAL ANALYSIS / 80' values p r e d i c t e d by the t r a d i t i o n a l monocentric model ( F i g u r e s 4.8 and 4.9) and t r e n d s u r f a c e a n a l y s i s ( F i g u r e s 4.10 and 4.11) f o r the same year. Comparing p r e d i c t e d land v a l u e s , we n o t i c e that t r e n d s u r f a c e a n a l y s i s produced a t h r e e - d i m e n s i o n a l shape and contour map more s i m i l a r t o a c t u a l land values than the t r a d i t i o n a l monocentric model. Both a c t u a l land values and land v a l u e s p r e d i c t e d by t r e n d s u r f a c e a n a l y s i s show a peak i n the Ginza area, whereas the t r a d i t i o n a l monocentric model does not capture the h i g h l a n d p r i c e s i n the Ginza a r e a . From t h i s a n a l y s i s of s t a t i s t i c a l f i g u r e s and p r e d i c t e d land v a l u e s , we recognize the s u p e r i o r i t y of trend s u r f a c e a n a l y s i s . T h i s r e s u l t v e r i f i e s that there are e x t e r n a l i t i e s and other f a c t o r s which a f f e c t land values w i t h i n the CBD. In other words, the t e s t of monocentric and nonmonocentric models proved the e x i s t e n c e of complex land value v a r i a t i o n s w i t h i n the CBD. 2. A n a l y s i s of Determinants of CBD Land Value V a r i a t i o n s S e v e r a l combinations of independent v a r i a b l e s d i s c u s s e d i n Part 2 of S e c t i o n C i n t h i s chapter were regr e s s e d on the n a t u r a l l o g of land p r i c e per square meter. Because we i n c l u d e d some d i s t a n c e v a r i a b l e s , we encountered c o l l i n e a r i t y problems. EMPIRICAL ANALYSIS / 81 In order to determine the e x i s t e n c e of c o l l i n e a r i t y , we used two s t a t i s t i c a l d e v i c e s : V a r i a n c e I n f l a t i o n F a c t o r ( V I F ) and C o n d i t i o n Number. VIF i s d e f i n e d a s 2 " : VIF = 1/(1 - R ^ ) where 2 i s R 2 produced by r e g r e s s i n g the independent v a r i a b l e x^ a g a i n s t other independent v a r i a b l e s . VIF has a l i m i t a t i o n t h a t i t cannot d i s t i n g u i s h between s e v e r a l simultaneous c o l l i n e a r i t i e s . VIF which i s g r e a t e r than 10 i n d i c a t e s harmful c o l l i n e a r i t y . 2 5 C o n d i t i o n Number i s based on eigenvalues and d e f i n e d as: C o n d i t i o n Number 1 = [ ( L a r g e s t E i g e n v a l u e ) / ( S m a l l e s t E i g e n v a l u e ) ] ' C o n d i t i o n Number of 30 or more i n d i c a t e s moderate to strong c o l l i n e a r i t y . 2 6 We employed "10" f o r VIF and "30" f o r C o n d i t i o n Number as c u t - o f f v a l u e s f o r d e c i d i n g whether or not c o l l i n e a r i t y e x i s t s i n a r e g r e s s i o n a n a l y s i s . Three equations are presented i n Table IV.5. They c l e a r e d the above c u t - o f f v a l u e s . Other s p e c i f i c a t i o n s , such as r e g r e s s i o n equations c o n t a i n i n g both DISCEN and FAR v a r i a b l e s , produced very 2"Myers (1986), p.78. 2 5 N e t e r et a l . (1985), p.392 and Kennedy (1985), p153. 2 6 W e i s b e r g (1985), p.200. EMPIRICAL ANALYSIS / 82 Table IV.5 Regression R e s u l t s (c) Hedonic Approach Equation (1) Equation (2) Equation (3) INTERCEPT 13.3344 13.6203 11.4231 (170.926)*** (166.359)*** (74.202)*** TIME 0.06003 0.05983 0.05897 (48.505)*** (44.895)*** (47.099)*** CENDIS -0.00049 -0.00031 — (-11.779)*** (-5.343)*** GINZADIS — -0.00023 — (-7.966)*** STADIS -0.00067 -0.00085 -0.00065 (-5.002)*** (-5.799)*** (-4.634)*** SI ZE 0.00010 0.00008 0.00009 (1.577) (1.077) (1.384) CORNER 0.24457 0.25708 0.25014 (.4.274)*** (4.174)*** (4.293)*** SHAPE -0.54022 -0.57593 -0.48385 (-5.640)*** (-5.584)*** (-4.952)*** ROAD 0.02202 0.02276 0.01326 (10.763)*** (10.343)*** (6.017)*** GINZA 0.66347 — 0.58148 (14.256)*** (11.625)*** FAR — — 0.21973 (10.215)*** Adjusted R 2 0.7773 0.7423 0.7690 C o n d i t i o n 11.694 12.718 26.085 Number NOTES: • t - s t a t i s t i c s are i n parenthses. • *** i n d i c a t e s s i g n i f i c a n t at 1% l e v e l . • N = 835 f o r a l l e s t i m a t e s . • P = Land p r i c e (Yen per square meter) • TIME = Time v a r i a b l e s (1 = 1975 1st q u a r t e r , 2 = 1975 2nd q u a r t e r , ..., 52 = 1987 4th quater) • CENDIS = Di s t a n c e to the c e n t r a l p o i n t of the CBD • GINZADIS = D i s t a n c e to the c e n t r a l p o i n t of the Gi n z a . • STADIS = D i s t a n c e to the nea r e s t s t a t i o n . • SIZE = Lot s i z e . • CORNER = (1 = Corner l o t , 0 = Otherwise) • SHAPE = (1 = I r r e g u l a r shape, 0 = Regular shape) • ROAD = Road width. • GINZA = (1 = l o t i n the G i n z a , 0 = Lot elsewhere) • FAR = F l o o r Area R a t i o EMPIRICAL ANALYSIS / 83 l a r g e VIF's and C o n d i t i o n Numbers. FAR and DISCEN i n d i c a t e h i g h c o r r e l a t i o n because the c e n t r a l part tend to be assign e d a higher FAR than the p e r i p h e r a l area of the CBD. The c o e f f i c i e n t s i n a l l three equations are h i g h l y s i g n i f i c a n t with the ex c e p t i o n of the SIZE v a r i a b l e . The si g n s of a l l c o e f f i c i e n t s except those of the SIZE v a r i a b l e had the r e s u l t s as we expected. The hedonic r e g r e s s i o n analyses e x p l a i n 74% - 78% of land p r i c e v a r i a t i o n s of the study area. Based on the c r i t i c a l value f o r the o u t l i e r t e s t , one o u t l i e r was d e t e c t e d . Table IV.5 g i v e s the r e s u l t s of the r e g r e s s i o n a n a l y s i s which employed the data with the o u t l i e r d e l e t e d . The normal p r o b a b i l i t y p l o t s and r e s i d u a l p l o t s d i d not show any v i o l a t i o n of the assumption of the e r r o r term. (a) B a s i c Equation F i r s t we w i l l c o n s i d e r Equation (1) i n Table IV.5 as a b a s i c model. According to the a n a l y s i s , land p r i c e s i n c r e a s e about 6% every q u a r t e r , i . e . , about 26% a year, on the average d u r i n g the study p e r i o d . T h i s annual f i g u r e i s l a r g e r than the a c t u a l l a n d p r i c e i n c r e a s e , which was c a l c u l a t e d as 20% based on the land p r i c e i n d i c e s compiled by the Japan Real E s t a t e I n s t i t u t e . EMPIRICAL ANALYSIS / 84 Land p r i c e s decrease about 0.05% every meter from the c e n t r a l p o i n t , other t h i n g s being e q u a l . They f a l l 0.07% every meter from a s t a t i o n w i t h i n the CBD, other v a r i a b l e s being c o n s t a n t . The l o c a t i o n of the r a i l w a y or subway s t a t i o n s were confirmed to be important. T h i s i s an expected r e s u l t because most of commuters use r a i l w a y s or subways and because employees t r a v e l w i t h i n the CBD^using them. In every equation which i n c l u d e d the DISFIN v a r i a b l e , the s i g n of the v a r i a b l e was i n c o r r e c t . I t i s c o n j e c t u r e d that t h i s i n c o r r e c t s i g n was a r e s u l t of the overwhelming impacts of DISCEN and DISGIN. SIZE i s the only v a r i a b l e which had an i n s i g n i f i c a n t c o e f f i c i e n t . The i n s i g n i f i c a n c e of l o t s i z e can be e x p l a i n e d by the f a c t that even piecemeal l o t s tend to be developed i n the study a r e a . Even small b u i l d i n g s b u i l t on those s i t e s can charge market r e n t s . T h e r e f o r e , s c a l e economies i n l o t s i z e are not r e a l i z e d . Since both CORNER and SHAPE are dummy v a r i a b l e s , we cannot employ t h e i r c o e f f i c i e n t s d i r e c t l y as percentage e f f e c t s on the dependent v a r i a b l e . We used the f o l l o w i n g formula i n Halvorsen and Palmquist (1980) to convert the c o e f f i c i e n t s i n t o percentage e f f e c t s : g = 100 • [(exp C) - 1] EMPIRICAL ANALYSIS / 85 where g = Percentage e f f e c t on the dependent v a r i a b l e and C = C o e f f i c i e n t of a dummy v a r i a b l e . Employing the above formula, we found that a corner l o t i s about 2 8 % 2 7 more v a l u a b l e than a s i t e i n the middle of a block and that an i r r e g u l a r - s h a p e d l o t i s about 4 2 % 2 B l e s s v a l u a b l e than a regular-shaped l o t . Each meter of road width adds 2.2% to land p r i c e s . For in s t a n c e , a l o t a b u t t i n g on a 15 meter wide road c o s t s 4 2 % 2 9 more than a l o t a b u t t i n g on a 4 meter wide road. Another v a r i a b l e i n c l u d e d i n n e i g h b o r h o o d - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s i s GINZA. Using the above Halvorsen and Palmquist's formula, we can estimate that l a n d i n the Ginza c o s t s about 9 4 % 3 0 more than land elsewhere. (b) Other Equations Equation (2) d e l e t e d the GINZA dummy v a r i a b l e from Equation (1) and i n c l u d e d the DISGIN v a r i a b l e . Because now we have three a c c e s s i b i l i t y v a r i a b l e s i n s t e a d of two, the value of the c o e f f i c i e n t s changed. Other v a r i a b l e being equal, land p r i c e s decrease about 0.02% every meter away from the center of the Ginza. 2 7 1 0 0 • [exp(0.24457) - 1] = 27.7%. 2 8 1 00 • [exp(-0.54022) - 1 ] = -41.7% 2 9100 • (1 + 0.022) 1 6 = 41 .6% 3 0 1 00 • [exp(0.66347) - 1] = 94.2% EMPIRICAL ANALYSIS / 8 6 Equation ( 3 ) d e l e t e d the DISCEN v a r i a b l e from Equation ( 1 ) and i n c l u d e d FAR. Since both ROAD and FAR v a r i a b l e s are p r o x i e s f o r development s u i t a b i l i t y of a neighborhood, the i n c l u s i o n of FAR made the c o e f f i c i e n t smaller than that of Equation ( 1 ) . The high land value in the Ginza area was p a r t l y e x p l a i n e d by FAR because most of the area i s as s i g n e d to high FAR v a l u e s . T h e r e f o r e , the c o e f f i c i e n t of FAR v a r i a b l e i s smaller than that of Equation ( 1 ) . The c o e f f i c i e n t of the FAR v a r i a b l e i n d i c a t e s that a one u n i t i n c r e a s e i n FAR r e s u l t s i n about a 2 2 % i n c r e a s e i n land p r i c e s . V. SUMMARY AND CONCLUSIONS Th i s t h e s i s has examined lan d p r i c e v a r i a t i o n s w i t h i n the CBD. I t s general m o t i v a t i o n stemmed from a) the ex i s t e n c e of e x t e r n a l i t i e s and other p r i c e i n f l u e n c i n g f a c t o r s which cannot be e x p l a i n e d by the t r a d i t i o n a l monocentric model and b) the f a c t that very few s t u d i e s have been done on CBD land p r i c e s . T r a d i t i o n a l monocentric model l i t e r a t u r e s t r e s s e d the importance of c e n t r a l i t y . Rents d e c l i n e m o n o t o n i c a l l y with d i s t a n c e from the c e n t r a l p o i n t . In the context of CBD land value v a r i a t i o n s , the model predetermines the c e n t r a l p o i n t from which produced goods are exported. T r a n s p o r t a t i o n c o s t s to the c e n t r a l p o i n t are c r u c i a l i n order to d e r i v e a land r e n t - g r a d i e n t of the model. A negative e x p o n e n t i a l f u n c t i o n i s o f t e n u t i l i z e d to estimate l a n d r e n t s because of i t s simple form. However, the f u n c t i o n a l form was c r i t i c i z e d by both U.S. and Japanese r e s e a r c h e r s . In a d d i t i o n , e x t e r n a l i t i e s such as a i r p o l l u t i o n d i s t o r t the simple monotonic f u n c t i o n . Nonmonocentric models i l l u s t r a t e land v a l u e s without s p e c i f y i n g the c e n t r a l p o i n t . The concept of agglomeration economies p l a y s an important r o l e i n demonstrating the ambiguity of the cu r v a t u r e of land rent 87 SUMMARY AND CONCLUSIONS / 88 g r a d i e n t s . Using these models as the conceptual framework, past e m p i r i c a l s t u d i e s were reviewed f o c u s i n g on a p p r o p r i a t e s p e c i f i c a t i o n s f o r the purpose of e x p l o r i n g : a) comparison of monocentric and nonmonocentric models and b) a n a l y s i s of determinants of CBD land value v a r i a t i o n s . A negative e x p o n e n t i a l f u n c t i o n was one s p e c i f i c a t i o n f o r the monocentric models. The double power s e r i e s of l o c a t i o n c o o r d i n a t e s , c a l l e d t r e n d s u r f a c e a n a l y s i s , was the s p e c i f i c a t i o n f o r nonmonocentric models. Trend s u r f a c e a n a l y s i s employs d i r e c t e d d i s t a n c e r a t h e r than one-dimensional s t r a i g h t d i s t a n c e which i s inherent to the t r a d i t i o n a l monocentric model. We do not need to s p e c i f y the c e n t r a l p o i n t i n trend s u r f a c e a n a l y s i s . Each l o c a t i o n i s i d e n t i f i e d based on a C a r t e s i a n plane p l a c e d on the CBD. However, n e i t h e r a negative e x p o n e n t i a l f u n c t i o n or tre n d s u r f a c e a n a l y s i s can e f f e c t i v e l y analyze the second q u e s t i o n , i . e . , determinants of CBD land value v a r i a t i o n s . T h i s i s because both f u n c t i o n s i n c l u d e only the l o c a t i o n f a c t o r s of these determinants i n the a n a l y s i s . Employing the h e d o n i c - p r i c e approach, we examined three c a t e g o r i e s of determinants: 1) a c c e s s i b i l i t y ; 2) s i t e - s p e c i f i c p r i c e i n f l u e n c i n g f a c t o r s ; and 3) n e i g h b o r h o o d - s p e c i f i c p r i c e SUMMARY AND CONCLUSIONS / 89 i n f l u e n c i n g f a c t o r s . The data of the Tokyo CBD was u t i l i z e d to answer the above two q u e s t i o n s . The data c o n t a i n s 836 vacant la n d t r a n s a c t i o n s t a k i n g p l a c e from 1975 to 1987. Trend s u r f a c e a n a l y s i s turned out to be s u p e r i o r to a negative e x p o n e n t i a l f u n c t i o n i n terms of a d j u s t e d R 2, AIC, and PRESS s t a t i s t i c s . P r e d i c t e d v a l u e s of t r e n d s u r f a c e a n a l y s i s were much more r e a l i s t i c than those of a negative e x p o n e n t i a l f u n c t i o n . The nonmonocentric s p e c i f i c a t i o n was b e t t e r able to e x p l a i n l a n d value v a r i a t i o n s w i t h i n the CBD. Among the three c a t e g o r i e s a l l v a r i a b l e s were h i g h l y s i g n i f i c a n t except the c o e f f i c i e n t of l o t s i z e . The • g o o d n e s s - o f - f i t of the r e g r e s s i o n equation was s a t i s f a c t o r y . C e n t r a l i t y was important f o r the Tokyo CBD l a n d v a l u e . A c c e s s i b i l i t y to a s t a t i o n and p r o x i m i t y to the Ginza area were a l s o c r u c i a l determinants. As we expected, corner l o c a t i o n , l o t shape, road width, and f l o o r area r a t i o appeared as i n f l u e n t i a l determinants on the CBD land v a l u e s . In c o n c l u s i o n , we b e l i e v e that t h i s t h e s i s c o n t r i b u t e s to r e s e a r c h on land value v a r i a t i o n s . Trend s u r f a c e a n a l y s i s was proved to be a u s e f u l approach f o r p r e d i c t i n g land value v a r i a t i o n s . Another major f i n d i n g of t h i s t h e s i s i s that SUMMARY AND CONCLUSIONS-^/ 90 three c a t e g o r i e s of land value determinants are a p p l i c a b l e to the CBD data. We b e l i e v e that p r o f e s s i o n s such as a s s e s s o r s and a p p r a i s e r s can u t i l i z e the techniques e x p l o r e d i n t h i s t h e s i s as a support when they determine s p e c i f i c land v a l u e s . o BIBLIOGRAPHY Alonso, W. (1964), L o c a t i o n and Land Use, Cambridge, Mass.: Harvard U n i v e r s i t y P r e s s . Amemiya, T. (1980), " S e l e c t i o n of Regressors," I n t e r n a t i o n a l  Economic Review, Vol.21, pp.331-354. Asabere, P.K. 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