University of Southern California Dissertations and Theses

A Generalized Economic Derivation Of The ''Gravity Law'' Of Spatial Interaction

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A Generalized Economic Derivation Of The ''Gravity Law'' Of Spatial Interaction

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Content A G E N E R A L IZ E D EC O N O M IC D E R IV A T IO N O F T H E "G R A V IT Y L A W " O F S P A T IA L IN T E R A C T IO N by J o s e f D a v is M o o re h e a d A D i s s e r t a t i o n P r e s e n t e d to the F A C U L T Y O F T H E G R A D U A T E SC H O O L U N IV ER SITY O F S O U T H E R N C A L IF O R N IA In P a r t i a l F u lf illm e n t of th e R e q u ir e m e n ts f o r th e D e g re e D O C T O R O F P H IL O S O P H Y (E c o n o m ic s ) A u g u st 1971 72-573 MOOREHEAD, Josef Davis, 1938- A GENERALIZED ECONOMIC DERIVATION OF THE "GRAVITY LAW" OF SPATIAL INTERACTION. University of Southern California, Ph.D., 1971 Economics, theory University Microfilms, A XERO\ Com pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED , UNIVERSITY O F SO U T H E R N CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 9 0 0 0 7 This dissertation, written by J O S E F . D A V I S ............................ under the direction of hx&... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O CTO R OF P H IL O S O P H Y ............ Dean Date August... 1271. DISSERTATION COMMITTEE PLEASE NOTE: Some Pages have in d istin c t print. Filmed as received. UNIVERSITY MICROFILMS T A B L E O F CO NTEN TS C H A PTER PA G E I. INTRODUCTION TO TH E DISSERTATION . . . 1 1. 1 N a tu re of the G ra v ity M o d e l ........................ . . 1 1. 2 P la n of D i s s e r t a t i o n .................................... 9 1. 3 T h e o ry N eeded to D evelop the D i s s e r t a t i o n ..................................................... 12 II. INDIVIDUAL T R A V E L B E H A V I O R ................... 16 2. 1 A ssu m p tio n s of N ie d e rc o rn -B e c h d o lt M odel . . . . . . . . . . . . . . . . . 16 2. 2 Solution of the N ie d e rc o rn -B e c h d o lt M o d e l ......................... 19 2. 3 E x te n sio n of the N ie d e rc o rn -B e c h d o lt M o d e l .................................................................... 20 2. 4 Solution w ith P o w e r U tility of T r ip - m ak in g R e fo rm u la te d and L in ea r C o st F u n c t i o n ............................................................... 22 2. 5 Solution w ith P o w e r U tility of T r ip - m aking R e fo rm u la te d and P o w e r C o st F u n c t i o n ............................................................... 25 2. 6 Solution w ith P o w e r U tility of T r ip - m ak in g R e fo rm u la te d and E xponential C o st F u n c tio n .................................. 28 2. 7 Solution w ith P o w e r U tility of T r ip - m aking R e fo rm u la te d and L o g a rith m ic C o st F u n c tio n ................................................................. 30 2. 8 Solution w ith L o g a rith m ic U tility of T rip m a k in g R e fo rm u la te d and L in e a r C o st F u n c tio n ................................................................. 32 2. 9 Solution w ith L o g a rith m ic U tility of T rip m a k in g R e fo rm u la te d and P o w e r of C o st F u n c tio n .................................................. 34 2. 10 Solution w ith L o g a rith m ic U tility of T rip m a k in g R e fo rm u la te d and E x p o n e n tia l C o st F u n c tio n ........................................ 35 ii C H A P T E R PA G E 2. 11 Solution w ith L o g a rith m ic U tility of T rip m a k in g R e fo rm u la te d and L o g a rith m ic C o s t F u n c t i o n ............ 36 2. 12 S u m m a ry of In d iv id u al T rav e l B e h a v i o r ..................................................... 37 III. C O M M O D ITY -F LO W GRAVITY M O D E L . . . . 42 3. 1 I n t r o d u c t i o n ............................................ 42 3. 2 D iffe re n tia tio n B etw een the S h o r t- Run V e rsu s L o n g -R u n S itutations . . . . 43 3. 3 A ssu m p tio n s fo r the S ho rt-R u n C o m m o d ity -F lo w G ra v ity M odel . . . . 44 3. 4 S olutions fo r th e S h o rt-R u n C o m m o d ity -F lo w G ra v ity M odel . . . . 49 3. 5 A ssu m p tio n s fo r the Long-R un C o m m o d ity -F lo w G ra v ity M odel . . . . 53 3. 6 Solutions fo r th e L ong-R un C o m m o d ity -F lo w G rav ity M odel . . . . 56 3. 7 S u m m a ry of C o m m o d ity -F lo w G ra v ity M o d e l ..................................... 63 VI. T H E PR O B A B ILIT Y GRAVITY M O D E L . . . . 68 4. 1 I n t r o d u c t i o n .............................................................. 68 4. 2 A ssu m p tio n s of P r o b a b ilis tic G ra v ity M o d e l s ..................................... 68 4. 3 D ev elo p m en t of P r o b a b ilis tic G ra v ity M o d e l s ..................................... 70 4 .4 R e c a stin g the L o w ry and H an sen M odels into a P r o b a b ilis tic F o r m a t . . . 75 4. 5 S u m m a ry of P r o b a b ilis tic G ra v ity M o d e ls ............................................. 82 V. E M P IR IC A L V A LID A TIO N ............................ 84 5. 1 I n t r o d u c t i o n ............................................ 84 5. 2 A n a ly sis of th e Input D a t a . 85 iii C H A P T E R P A G E 5. 3 E m p i r i c a l V a lid a tio n of the In d iv id u a l T r a v e l B e h a v io r M o d e l s ......................... 89 5. 4 S u m m a r y of E m p i r i c a l V a lid atio n . . . . 95 VI. SUM M ARY, C O N C LU SIO N S, AND SU G - . G E ST IO N S F O R F U T U R E R E S E A R C H .................... 96 6. 1 S u m m a r y an d C o n c lu sio n s . 96 6. 2 S u g g e stio n s fo r F u tu r e R e s e a r c h . . . . 99 A P P E N D IX A ............................................................................................................... 101 B IB L IO G R A P H Y .......................................................................................................... 103 LIST O F T A B L E S T A B L E P A G E 2 .1 D e riv a tio n s for Ind iv id u al T r a v e l B e h a v io r . . . 39 3. 1 S h ort-R un C o m m o d ity -F lo w G ra v ity M odel Solutions ............................................................. 64 3. 2 L o n g -R u n C o m m o d ity -F lo w G ra v ity M odel S o lu t io n s ....................................................................................... 66 5. 1 R e g r e s s io n V alues .............................. 92 5 .2 C o efficien ts of D e te r m in a tio n .......................................... 93 5. 3 A n a ly sis of V a r i a n c e .......................................................... 94 v C H A P T E R I IN T R O D U C T IO N T O T H E D IS S E R T A T IO N 1. 1 N A T U R E O F T H E G R A V ITY M O D E L 1. 1. 1 B a c k g ro u n d It is q u ite e v id e n t th a t th e c o n s id e r a tio n of s p a c e as an a s p e c t of e c o n o m ic p h e n o m e n a h a s b e e n e x c lu d e d f r o m th e m a i n s t r e a m of g e n e r a l e c o n o m ic th e o ry . ^ H o w e v e r, th is is n o t to d e n y th a t so m e a tte m p ts h a v e b e e n m a d e to in c lu d e s p a c e . Im m e d ia te ly , th r e e such 2 e f f o r ts c o m e to m in d : (1) L . L e f e b e r , M A llo c a tio n in S p a c e ;" 3 (2) K a r k F o x , " S p a tia l E q u i lib r iu m ;" an d (3) P a u l S a m u e ls o n , 4 " S p a tia l P r i c e E q u ilib r iu m . " E a c h h a s p la c e d th e n o tio n of sp ace in a g e n e r a l e q u ilib r iu m f r a m e w o r k w h ich in c lu d e s a llo c a tio n o v e r s p a c e c o n s id e r in g t r a n s p o r t a t i o n c o s ts . P e r h a p s to c o n c e n tr a te only on s p a c e is to s h o rtc h a n g e th e tim e and u n c e r ta in ty a s p e c t s w h ich w h en ta k e n to g e th e r w ith s p a c e m a k e e c o n o m ic th e o r y t r u l y d e s c r i p tiv e of r e a lity . T h is p a p e r c h ie fly in v o lv e s s p a c e a lth o u g h the e ffe c ts of tim e an d u n c e r ta in t y w ill be m a d e to v a ry in g d e g r e e s w ith in the : m o d e ls d e v elo p e d . B e c a u s e th is d i s s e r t a t i o n is s p e c if ic a lly c o n c e r n e d w ith s p a t ia l in te r a c t io n , it is a p p r o p r ia te th a t p re v io u s w o rk in th e a r e a be B e c h d o lt, B . V. , G e n e r a l D ir e c tio n a l S p a tia l I n te r a c tio n M o d els (U n p u b lish e d M a s t e r s th e s i s ; L o s A n g e les: U n iv e r s ity of S o u th e rn C a lifo rn ia , A u g u st 1967), p. 1. 2 L e f e b e r , L . , A llo c a tio n in S p a c e ( A m s te r d a m : N o rth - H o lla n d P u b lis h in g Co. , 1^58). 3 F o x , K a r l, E c o n o m e tr ic A n a ly s is f o r P u b lic P o li c y (A m es: Iow a S ta te C o lle g e P r e s s , I"9'58). 4 S a m u e ls o n , P a u l, " S p a tia l P r i c e E q u i lib r iu m ," A m e r ic a n E c o n o m ic R ev iew , V ol. X LII, (1952). 2 re v ie w e d . M ost of the p re v io u s w o rk on s p a tia l in te r a c tio n th a t ev o lv ed a ro u n d Von Th'unen, W eb er, L o sc h , and D u n n 's c o n trib u tio n s h a s r e m a in e d is o la te d fr o m g e n e r a l p ric e and d is trib u tio n th e o ry . In fa c t, the d e riv a tio n of th e s e g ra v ity m o d el of s p a tia l in te ra c tio n h ad its ro o ts in N ew to n ian p h y s ic s , sp e c ific a lly , N ew to n 's law of g r a v ita - 5 tio n a s s ta te d a c c o rd in g to S te w a r t's fo rm u la tio n . P .P . F .. = G - ^ - J (1. 1) ij w h e re F y = d e m o g ra p h ic fo r c e b e tw ee n c itite s i and j G = a c o n sta n t P^ = p o p u latio n a t the i^ 1 city P j = p o p u latio n at the j**1 city d.. = d is ta n c e b etw een i and j The a p p lic a tio n of p h y s ic a l law s to so c ia l b e h a v io r c an be tr a c e d to H. C. C a re y , w ho, in 1858 e sp o u s e d the b e lie f th a t m a n b e h av e d in so c ie ty a s a m o le c u le in m a tte r . ^ T h u s, it is a p p a r e n t th a t the g ra v ity m o d e l follow s C a r e y 's id e a of re la tin g s o c ia l p h e n o m en a to N e w to n 's law . 5 S te w a rt, J . Q. , " P o te n tia l of P o p u la tio n an d Its R elatio n sh ip to M a rk etin g , " in R. Cox and W. A ld e rs o n ( e d .), T h e o ry of M a r k e t ing (H om ew ood, Illin o is: R . D. Irw in , 1964). ^ C a re y , H. C . , P r in c ip le s of S o c ia l S c ie n c e (P h ila d e lp h ia : J . B. L ip p in co tt, 1865). 3 F o llo w in g C a r e y , o th e r s a p p lie d th e p h y s ic a l c o n c e p t of g r a v ita tio n to d e s c r ib in g h u m a n in te r a c tio n . T h e f i r s t to s p e c ify th e id e a m a th e m a ti c a lly w a s E. G. R a v e n s te in ; h is m o d e l of m ig r a tio n 7 w a s f(P .) M s - g - L (1.2) J ij w h e re M . = n u m b e r of m ig r a n ts f r o m s o u r c e j to th e c e n t e r of a b s o r p tio n i f(P .) = fu n c tio n of th e p o p u la tio n of c e n te r i D .. = d is ta n c e b e tw e e n s o u r c e j an d c e n t e r i ij N o tic e th a t in the above c a s e th e e x p o n en t on the d is ta n c e v a r ia b le is one. O v e r tim e c o n s id e r a b le d is c u s s io n by lo c a tio n t h e o r i s t s w a s d e v o te d to th e v alu e of th e e x p o n e n t of a lth o u g h th e v a r ia b le s a n d t h e i r fu n c tio n a l f o r m w e r e s till c o n s is te n t w ith th e b a s ic c o n c e p ts o r m o d e l. ® V a rio u s f o r m s of th e g r a v ity m o d e l h a v e b e e n a p p lie d to r e t a i l m a r k e t a n a ly s e s , s tu d ie s of lo c a l r e s i d e n t i a l p a t t e r n s , u r b a n t r a f f i c s tu d ie s , a n d in t e r r e g i o n a l c o m m o d ity flo w s a n d in d u s tr ia l 9 lo c a tio n m o d e ls . 7 R a v e n s te in , E . G . , " T h e L a w s of M ig r a tio n ," J o u r n a l of th e R o y a l S ta tis tic a l S o c ie ty , (V ol. X L V II (1825), a n d V ol. L l l (18^91*) O F o r an h i s t o r i c a l a n a ly s is of the d e v e lo p m e n t of s p a tia l in te r a c t io n m o d e ls s e e B e c h d o lt, op. c it. , pp. 8 -3 3 ; a ls o G e r a ld A. F . C a r r o t h e r s , "A n H i s t o r i c a l R e v ie w of th e G r a v ity a n d P o te n tia l C o n c e p ts of H u m a n I n te r a c tio n , " J o u r n a l of th e A m e r ic a n I n s ti- tu te of P l a n n e r s , XX (1956), (No. 22), pp. 9 4 -1 0 2 . g L o w r y , I r a S. , M ig r a tio n an d M e tr o p o lita n G ro w th ; Two A n a ly tic a l M o d e ls (L o s A n g e le s , C a lifo rn ia : I n s titu te of G o v e rn m e n t a n d P u b lic A ffa irs , U n iv e r s ity of C a lifo rn ia , 1966). An ex a m p le of one of th ese is the m a rk et p o ten tia l m o d el o f C. D. H a r r is : ^ n S. .R = — (i 4 j) (1 .3 ) w h ere j= l 1 J .R = to ta l m a rk et p oten tial at point i i S. = r e ta il s a le s in reg io n j J C .. = tra n sp o rt c o sts b etw een point i and ^ reg io n j In r e a lity the num ber of m o d els and th eir au thors s e e m s to be im m e n se . In ad d ition to th o se w e h ave d isc u s se d , w e fin d m o d els b ein g c o n stru cted by S tou ffer, Z ipf, R e illy , Young, B o ssa r d , C o n v erse, Dodd, A n d erson , C a rro th ers, V o o rh ees, Isa rd , C a rro ll and o th e r s. * ^ In the fo reg o in g d is c u s sio n both g r a v ity and p o ten tial m o d els w e r e p resen ted , w ith no attem p t to d ifferen tia te b etw een th em . T h e re fo r e , b efo re p ro ceed in g it w ou ld be ap p ro p ria te to d istin g u ish the two; f ir s t , the p o ten tia l m od el i s b a sed on the u n d erlyin g funda m en ta l id ea n • * that the d eg ree of a c c e s s ib ility , e a s e , or p o s s ib ility of in ter a c tio n b etw een an in d ivid u al at one g eograp h ic poin t and p e r so n s o r things c o n c e n tra ted in an a r e a and thoughtto be lo c a te d at another point is a fu n ction of (1) the d eg ree of co n cen tra tio n H a r r is, C. C ., "The M arket as a F a c to r in the L o c a liz a tio n of Indu stry in the U nited S ta te s, " A nnals o f the A sso c ia tio n of A m e rica n G eo g ra p h ers, V ol. XLIV, No. 4, (1954), pp. 3 1 5 -3 4 8 . ^ S e e the B ib liograp h y fo r th ese s o u r c e s . 5 of p e r s o n s o r th in g s in the a r e a a ro u n d th e sec o n d p o in t an d (2) a m e a s u r e of the d is ta n c e s e p a ra tin g th e two p o in ts . S econd, the g ra v ity m o d e l is b a s e d on the u n d e rly in g fu n d a m e n ta l id e a " . . . th a t th e d e g re e of in te r a c tio n b e tw e e n two g e o g ra p h ic p o in ts , w h ich a r e c e n te r s of a r e a s in w h ic h th e r e is a c o n c e n tra tio n of p e r s o n s o r th in g s, is a fu n c tio n of (1) th e d e g r e e s of c o n c e n tra tio n of p e r s o n s o r th in g s a t th e two p o in ts and (2) a m e a s u r e of th e d is ta n c e s e p a r a tin g the two p o in ts. So, in the m o s t g e n e r a l s e n s e the above s ta te m e n ts define ithe tw o m o d e ls . H o w e v e r, m a n y tim e s th is e n tir e c la s s of m o d e ls h a s g e n e r a lly b e e n t e r m e d g r a v ity m o d e ls . M o re p r o p e r ly , w hen g e n e r a liz in g , the c la s s sh o u ld be d e s ig n a te d a s s p a tia l in te r a c tio n m o d e ls . D u rin g the c o u r s e of th is p a p e r an e f fo rt is m a d e to u s e th e t e r m s p a tia l in te r a c tio n m o d e ls in the d is c u s s io n s , alth o u g h the n a m e g r a v ity m o d e l a ls o o c c u r s s in c e it is m o r e c o m m o n ly u se d . 1 .1 ,2 D e v e lo p m e n t of th e T h e o ry F r o m its in c e p tio n in s o c ia l p h y s ic s , m a n y a tte m p ts h av e b e e n m a d e to m o d ify an d r e f o r m th e b a s ic s t r u c t u r e of the g ra v ity m o d e l. M o st of th e s e e f f o r ts , w h ile b e in g im a g in a tiv e , h av e done v e r y little to ex p an d o r m o d ify the p r i m a r y fo rm u la tio n . T h e se c h a n g e s e s s e n tia lly re v o lv e a ro u n d r e f o r m in g th e m a th e m a tic a l s tr u c tu r e . T h e y h av e le d to m a n y new f o r m s a s in th e c a s e of S te w a rt 14 w ho u s e d p e r c a p ita in c o m e a s p a r t of th is d e te r m in a n t of fo r c e . 12 B ec h d o lt, op. c it. , p. 23. 13Ibid. , p. 9. 14 S te w a rt, op. cit. 6 (U .P.) (U .P.) F .. = ■ 1 V J (1.4) ij w h ere F . . = fo rc e of d e m o g ra p h ic in te ra c tio n ^ b etw een a r e a s of population c o n c e n tra te d at i and j U ., U. = p e r c a p ita incom e w eights of the p o p ulations c o n c e n tra te d in i and j, re s p e c tiv e ly P^, P j = population in i and j, r e s p e c tiv e ly C. D. H a r r i s , u sin g r e ta il s a le s and tr a n s p o r ta tio n c o st, 15 developed the follow ing m odel: kR = E - ^ r (1- 5) j= i w h e re R. = to ta l m a r k e t p o ten tial at point i Sj = re ta il s a le s in re g io n j C .. = tr a n s p o r t c o st b etw een point i ^ and re g io n j Yet a n o th e r fo rm of the m o d el is one w h ich u s e s p ro b a b ility 16 concepts fo r m ig ra tio n ; an ex am p le is the P r ic e m ethod. ^ H a r r i s , op. cit. 1 / P r ic e , Dan O . , "A M a th e m a tic a l M odel of M ig ra tio n Suitable fo r S im u latio n on an E le c tro n ic C o m p u te r," P ro c e e d in g s of the In te rn a tio n a l P o p u latio n C o n fe re n ce s (1959). 7 P (I A A ) = f(A, B, C, D) + f(X, Y, Z) 1 J k (1.6) + f(A, B, C, D) (X, Y, Z) w h e re P(I.A.A^_) = p ro b a b ility of an in d iv id u a l m o v in g 1 ^ f r o m one a r e a to a n o th e r I. = i ^ in d iv id u a l l A. = j a r e a of o rig in J A ^ = k ^ 1 a r e a of d e s tin a tio n f(A, B, C, D) = fu n c tio n defining the in d iv id u a l's p r o c liv ity to m ig r a te f(X, Y, Z) = fu n c tio n s e le c tin g th e a p p r o p r ia te p o in t of d e s tin a tio n in r e la tio n to the in d iv id u a l p o in t of o rig in f(A, B, C, D)(X, Y, Z) = in te r a c tio n s b e tw e e n th e in d iv id u a l's c h a r a c t e r i s t i c s a n d th o se of h is and o th e r a r e a s A lthough th e r e h av e b e e n m a n y o th e r fo r m u la tio n s in the d e v e lo p m e n t 17 of the m o d e l, th e s e a r e b u t th r e e e x a m p le s of r e c e n t e ffo rt. 1 .1 .3 W e a k n e s s e s of P r i o r S p a tia l I n te r a c tio n M o d els One in h e r e n t w e a k n e s s of s p a tia l in te r a c tio n m o d e ls s te m s f r o m th e ir not b ein g d e v elo p e d f r o m e c o n o m ic th e o r y and e co n o m ic c o n s tr u c ts . A s p r e v io u s ly m e n tio n e d , th e y w e r e b a s e d upon N ew ton ia n p h y s ic s and a p p lie d by a n alo g y to s o c ia l p h e n o m e n a . W hile w e m ig h t a r g u e along the lin e s of C a re y , the u s e of p h y s ic a l a n a lo g ie s 17 F o r s e v e r a l o th e r su c h m o d e ls on m ig r a tio n s e e A n d re i R o g e r s , An A n a ly sis of I n te r r e g io n a l M ig ra tio n in C a lifo rn ia (B e rk e le y ! In stitu te o t U rb a n a n d R eg io n al d e v e lo p m e n t, U n iv e rs ity of C a lifo rn ia , D e c e m b e r 1965). 8 to d efin e o r p r e d i c t e c o n o m ic p h e n o m e n a is s u s p e c t. T h a t is , w e w ould f e e l m o r e c o m f o r ta b le if s p a tia l i n t e r a c t i o n m o d e ls w e r e d e riv e d f r o m o rth o d o x e c o n o m ic th e o ry . I s a r d s ta t e s th is a ttitu d e : r'T he l a c k of (any) a d e q u a te th e o r y (to ju s tif y th e g r a v ity m o d el) is c le a r ly p e r c e iv e d . T h r e e o th e r c r i t i c i s m s a r e r a i s e d b y B e c h d o lt. F i r s t , th e r e is r e a l l y no g e n e r a l s p a tia l in te r a c t io n m o d e l. M o re s p e c if ic a lly , th e re a r e s p e c ia l fu n c tio n s f o r stu d y in g p a r t i c u l a r p r o b le m s . Second, m a n y w r i t e r s f a il to c o n s id e r p r o b a b ility th e o r y an d s t a tis tic a l m e th o d o lo g y in a n a ly z in g s p a tia l i n t e r a c t i o n s . T h is c r i t i c ism , w h ile v a lid in the p a s t, m ig h t be l e s s a p p lic a b le to d a y s in c e 19 r e c e n tly s e v e r a l a tte m p ts h a v e b e e n m a d e to u s e p r o b a b ility th e o r y . The t h i r d c r i t i c i s m is th a t s p a tia l in te r a c t io n m o d e ls a r e n o n d ir e c - tional; a p r o b le m w h ic h B e c h d o lt a tte m p te d to h a n d le in h is t h e s i s . A n o th e r d iffic u lty w h ic h is e n c o u n te r e d i s th e a g g r e g a tio n p r o b le m . "W ith in s u c h a m a s s it is r e a s o n a b le to a s s u m e th a t th e i r r e g u l a r i t i e s , p e c u l i a r i t i e s a n d i d io s y n c r a s ie s of any in d iv id u a l 20 unit o r s m a l l s u b g ro u p of u n its a r e c a n c e lle d o r a v e r a g e d out. " P e r h a p s so , b u t th is is a n a lo g o u s to the ta c k y p r o b le m e n c o u n te r e d in c a r d in a ll y d e fin in g a to ta l w e lf a r e fu n c tio n . E f f o r ts to r e lie v e th e size p r o b l e m f o r s p a tia l m o d e ls h a v e r e s u l t e d in d is a g g r e g a tio n of data. Y e t,d if f ic u ltie s m ig h t o c c u r w h en p r e c i s i o n an d a d d itio n a l data a r e n o t a tta in a b le , a n d f u r t h e r , w h e n c o h e s io n a n d m e a n in g of the u n ity are d e stro y e d w ith the d iv isio n . 18 I s a r d , W . , M eth o d s of R e g io n a l A n a ly s is ; A n In tro d u c tio n to R e g io n a l S c ie n c e (JNew Y o rk ; W iley , 19&0), p. 515. 19Ibid. 2 0 Ibid. , p. 513. 9 21 Two a d d itio n a l w e a k n e s s e s a r e s e t fo rth b y G o ld n er. F i r s t , w h en u s e d a s a lo c a tio n p r e d ic to r , the m o d e l o v e r p r e d ic ts fre q u e n c ie s g e n e r a te d by s h o r t d is ta n c e s fr o m the o rig in , th e re b y not c o r r e c t ly re fle c tin g th e e x is te n c e of s e p a ra tio n of la n d u s e b e tw een in d u s tr ia l and r e s id e n tia l. T h is p h en o m en o n a lso o c c u rs in the p re d ic tio n of tr a n s p o r ta tio n flow s. T h e se c o n d d e fic ie n c y r e l a t e s to the s p e c ifie d s iz e of the zonal s y s te m up o n w hich the fu n ctio n is being u sed . V a ria tio n s in zonal s iz e m ig h t c r e a te in h e re n t b ia s e s in the r e s u l t s . T h is s te m s fr o m th e zonal s y s te m d e te rm in in g th e siz e of th e f o r c e fa c to r, th e ■ : d is ta n c e value an d the n o rm a liz in g a d ju s tm e n t p r o c e s s to e n s u r e th a t j | o rig in a tin g t r i p s have d e stin a tio n s . E s s e n tia lly , th e o r e tic a l and e m p ir ic a l w e a k n e ss h av e b e en m e r g e d in the above d is c u s s io n . Still, one e m p ir ic a l w e a k n e ss h a s p lag u ed the m o d e l. T h is is the la c k of c o n sis te n c y in p a r a m e tr ic v a lu e s. T h e se h av e v a r ie d depending upon the tr i p p u rp o s e . T his in c o n s is te n c y h a s le d m a n y to be c r itic a l of the p re d ic tiv e c a p a b ilitie s of the m o d e l. 1 .2 PL A N O F D ISSER T A T IO N 1. 2. 1 Scope an d O bjectiv e In the s e c tio n above, the n a tu r e of the g ra v ity m o d el, the b a ck g ro u n d , d e v e lo p m e n t and w e a k n e s s e s of th e g e n e r a l fo r m of the m o d e l w e r e s p e lle d out. A fte r p re s e n tin g the b a s ic m o d e l, s e v e r a l p o in ts of v u ln e ra b ility w e r e in d ic a ted . It is to th e s e a r e a s th a t o u r a tte n tio n h a s b e e n d raw n . 21 G o ld n e r, W. , A n u n p u b lish e d l e t t e r to Jo h n H. N ie d e r c o rn (B e rk e le y : In s titu te of T r a n s p o r ta tio n and T ra ffic E n g in e e rin g , U n iv e rs ity of C a lifo rn ia , F e b r u a r y 24, 1970). 10 I f s p a tia l in t e r a c t i o n m o d e ls a r e to h a v e v ia b ility , e f f o r ts m u s t be m a d e to o v e r c o m e t h e i r w e a k n e s s e s . So, it is only n a tu r a l th a t a s ig n if ic a n t c o n trib u tio n to t h e ir d e v e lo p m e n t lie s in a n e x p a n s io n a n d s tr e n g th e n in g o f the p r e m i s e s and, th e r e b y , th e th e o r y . A s h a s b e e n s ta te d , m u c h e f f o r t h a s b e e n p la c e d o n e v a lu a tio n , e s t i m a tio n a n d te s tin g , b u t l i t t l e h a s b e e n p la c e d on a r ig o r o u s f o r m u la tio n of th e b a s i c th e o r y . T h e r e f o r e , it b e c o m e s q u ite r e le v a n t to fo c u s a tte n tio n on a d e r iv a tio n of s p a tia l in te r a c t io n m o d e ls w h ic h h a s a fo u n d a tio n of v a lid an d a c c e p te d h y p o th e s e s . B e c a u s e th e m o d e ls a r e p r e d i c t o r s o f p h e n o m e n a th a t a r e g e n e r a lly e c o n o m ic in n a tu r e , i t is o b v io u s th a t the b a s e s m u s t b e d e v e lo p e d a r o u n d e c o n o m ic j c o n s t r u c t s . P r o f e s s o r s N ie d e r c o r n a n d B e c h d o lt h a v e m a d e s u c h a I 22 c o n trib u tio n an d s e t th e fo u n d a tio n of a th e o r e t i c a l b a s e . T h is t h e s i s w ill e n c o m p a s s th a t fo u n d a tio n a s i t r e f o r m s an d i e x p a n d s up o n th e e c o n o m ic h y p o th e s e s . H o w e v e r, th e r e a r e s e v e r a l a r e a s w h ic h w ill n o t b e to u c h e d in th is p a p e r . F i r s t , no a tte m p t is m a d e to f o r m a d i r e c tio n a l v e c to r . S eco n d , th e p r o b le m s of a g g r e g a tio n a n d d e te r m in a tio n of z o n a l s iz e w ill n o t be e n c o u n te re d . W hile th e s e a r e te c h n ic a l p r o b le m s w h ic h m e r i t a tte n tio n , it is im p o s s ib le to d is c u s s e v e r y th in g a t th e s a m e tim e . T h e r e f o r e , w e a r e le f t w ith th e c h o ic e of a p r o b le m a r e a to ta c k le . S in ce th e fine tu n in g of a w e a k ly th e o r e t i c a l m o d e l is littl e m o r e th a n a w a s te of tim e , it is u n d e r s ta n d a b le th a t th e t h e o r y m u s t f i r s t b e s tr e n g th e n e d . S p e c ific a lly th e n , th e o b je c tiv e of th is d i s s e r t a t i o n w ill be tw ofold: f i r s t , to e x te n d th e w o rk done by P r o f e s s o r s N ie d e r c o r n a n d B e c h d o lt b y r e fin in g th e a s s u m p tio n s u p o n w h ic h t h e i r w o rk is b a se d ; s e c o n d , to c a r r y th e s e t h e o r e t i c a l e c o n o m ic h y p o th e s e s , a s 22 N ie d e r c o r n , J . H. , an d B . V. B e c h d o lt, "A n E c o n o m ic D e r iv a tio n of th e 'G r a v i ty L a w ' of S p a tia l In te r a c tio n , " J o u r n a l of R e g io n a l S c ie n c e , V ol. IX, No. 2 (1969). 11 w e ll a s new o n e s , in to o th e r a r e a s of t r a n s p o r ta tio n and t r a v e l b e h a v io r. T h e a c c o m p lis h m e n t o f th a t o b je c tiv e w ill a id s u b se q u e n t r e s e a r c h e r s to b e t t e r u n d e rs ta n d th e s e m o d e ls , th u s in tu r n e n h a n c ing th e s ig n ific a n c e of th e ir r o le in lo c a tio n and t r a n s p o r ta tio n th e o r y in o r d e r to im p r o v e eco n o m ic e ffic ie n c y in o p tim u m lan d a llo c a tio n , and le a s t c o s t / m a x i m u m b e n e fit t r a n s p o r ta tio n s y s t e m s , w h ic h in the fin a l a n a ly s is w ill im p ro v e to ta l s o c ia l w e lf a r e . L o w ry a ls o o ffe rs a s i m i l a r c o n c lu sio n : " P r o p e r l y a d a p te d , it (a m o d e l of s p a tia l i n t e r actio n ) sh o u ld b e u s e fu l fo r th e p r o je c tio n of fu tu r e p a tte r n s of land d e v e lo p m e n t and fo r the te s tin g of p u b lic p o lic ie s in th e fie ld s of t r a n s p o r ta tio n p la n n in g , la n d - u s e c o n tro ls , ta x a tio n , and u rb a n r e - 23 n ew al. 1 .2 .2 O r g a n iz a tio n of the D is s e r t a t i o n and A r e a s to b e D ev elo p ed In th e p r e v io u s se c tio n of th is c h a p te r , an in tro d u c tio n to s p a tia l i n te r a c t io n m o d e ls w a s g iv en w h ic h in c lu d e d an o u tlin e of th e ir m a j o r w e a k n e s s e s . T h is is fo llow ed by th e c u r r e n t s e c tio n w h ich s e ts dow n the sco p e and o b je c tiv e of the d is s e r ta tio n . In th e fin a l s e c tio n o f th e c h a p te r, th e a p p ro a c h and m e th o d o lo g y as w e ll as n e c e s s a r y e c o n o m ic a s s u m p tio n s a r e d e v e lo p e d . C h a p te r II c o v e r s the to p ic of in d iv id u a l t r a v e l b e h a v io r . T h e d e v e lo p m e n t f o r th is c h a p te r s t e m s f r o m th e w o r k of P r o f e s s o r s N ie d e r c o r n a n d B ec h d o lt. H o w e v e r, v a rio u s c o n s tr a in in g t r a v e l - c o s t fu n c tio n s w ill b e e v a lu a te d and s e v e r a l t r a v e l b e h a v io r m o d e ls fo r m u la te d . In C h a p t e r III, the c o n c e p ts found in C h a p te r II w ill b e u tiliz e d fo r th e p r o d u c e r 's s e c to r . H e r e c o m m o d ity flow m o d e ls w ill b e d e v el oped fo r b o th s h o r t - r u n and lo n g - r u n e q u ilib riu m . T h e s a m e c o s t con s t r a i n t fu n c tio n u s e d in C h a p te r II w ill b e a p p lie d fo r sh ip p in g c o s ts . 23 L o w ry , I r a S . , A M o d el of M e tr o p o lis , M e m o ra n d u m R M -4 0 3 5 -R C (S an ta M onica, C a lif. : T h e RAND C o rp o ra tio n , A u g u st 1964), p. 2. 12 A d if f e r e n t ta c tic is u se d in C h a p te r IV a s p ro b a b ility th e o ry and a p ro b a b ility d e n sity fu n ctio n fo r d is ta n c e a r e d e v elo p e d and r e la te d to the m o d e l. A lso , in th is c h a p te r s p a tia l in te r a c tio n m o d e ls of L o w ry and H a n se n a r e r e f o r m u la te d in t e r m s of p r o b a b ility c o n c e p ts . N ex t, the in d iv id u a l t r a v e l b e h a v io r a l m o d e ls d e r iv e d in C h a p te r II a r e e m p ir ic a lly te s te d in C h a p te r V. H e re an a tte m p t is m a d e to r e v e a l the c o n s is te n c y of the th e o ry w ith r e a lity . C h a p te r VI s u m m a r iz e s the d i s s e r t a t i o n and s e ts f o r th the r e s u l t s , c o n c lu s io n s and s u g g e stio n s f o r f u r th e r w o rk . T h is d is s e r t a t i o n h a s e x ten d ed th e " G r a v ity Law" of s p a tia l in te r a c tio n in th r e e d ir e c tio n s . T he f i r s t c o n c e r n s i t s e lf w ith the in d iv id u a l t r a v e l e r and h is t r i p s ta k e n . T h e s e c o n d c o n c e r n s i t s e lf w ith the p r o d u c e r who sh ip s to a s p e c ifie d m a r k e t a r e a and th e q u a n tity of goods sh ip p e d . F o r th e th ir d d ir e c tio n , t r i p s ta k e n o r goods sh ip p e d a r e d e te r m in e d b y c o n s tr u c ts of p r o b a b ility th e o ry . B y r e view of th e c h a p te r o rg a n iz a tio n , it is e v id en t th a t e ac h c h a p te r c o n s titu te s one d ir e c tio n o f d e v e lo p m e n t. i . 3 TH E O R Y N E E D E D TO D E V E L O P TH E D ISSER T A T IO N 1. 3. 1 A p p ro a c h and M eth o d o lo g y W hen a p p ro a c h in g a p ro b le m , w e m ig h t c o n s id e r tw o e x t r e m e s : (1) find o u t w h a t has b e e n done and e i t h e r m odify o r ex p an d on it, o r (2) d is r e g a r d a ll p re v io u s w o rk and s t a r t fr o m s c r a t c h . T o ta k e the se c o n d a p p r o a c h r e q u ir e s th e e x p lo r e r w h o se im a g in a tio n th r i v e s on u n c h a r te d c o u r s e s , w h ile th e f i r s t a p p r o a c h u tiliz e s a lr e a d y ex p an d ed e ffo rt to b u ild upon. A lthough th e a d v a n ta g e is c l e a r in th is a p p ro a c h , th e in h e re n t d a n g e r is in b e co m in g tr a d itio n bound o r r u tte d and n o t being a b le to ad v an c e to th e so lu tio n . A l m o s t a ll of th e w o r k done in th e a r e a of the s p a tia l in te r a c tio n m o d e l h a s b e e n of th e f i r s t ty p e . T h e s e a tte m p ts to s tr e n g th e n and e x o r c is e th e w e a k n e s s e s h av e g e n e r a lly b e e n in v a in . H o w e v e r, th is is 13 not to deny the sig n ific a n t im p o rta n c e of the m o d e l and its b a s ic u s e f u ln e s s . The w o rk done in th is d is s e r ta tio n w ill follow , a s a lr e a d y m e n tio n e d , th e le a d s given by P r o f e s s o r s N ie d e r c o r n and B ech d o lt, w h ich fa ll into the sec o n d type. T he o v e r a ll a p p ro a c h c e n te r s a ro u n d a c c e p te d eco n o m ic h y p o th e se s ab o u t m a x im iz a tio n of in d i vidual u tility , an d m a x im a tio n of p r o d u c e r 's re v e n u e and p ro fit. S p ecifically , e a c h c o n cep t w ill be d ev elo p ed into a f o r m a liz e d m o d e l. T h is r e q u ir e d th a t m o d e ls be d e riv e d b a s e d upon d efin ed and f o r m a l a b s tr a c tio n s of e co n o m ic p h e n o m en a . T he ro o t of th e se a b s tr a c tio n s j w ill be found am o n g the g e n e r a lly a c c e p te d p o s tu la te s of eco n o m ic I th e o ry . An a tte m p t w ill be m a d e to r e p ro d u c e a c o lle c tio n of the s e v e r a l p o s s ib le v a ria n ts of the g ra v ity m o d e l of s p a tia l in te ra c tio n . In o r d e r to a c c o m p lis h th is ta sk a m e th o d o lo g y is n eed ed . T h is c o n s is ts of ap p ly in g m a th e m a tic a l o p tim iz a tio n , sin c e we a r e looking f o r the e co n o m ic m a x im u m condition. Since th e a p p ro a c h in v o lv es u tiliz in g a c o n s tr a in t fu n ctio n and the fu n ctio n to be m a x i m iz e d , the m e th o d of L a g ra n g ia n m u ltip lie r s is u s e d to d e riv e the so lu tio n . T h is a p p ro a c h and m eth o d o lo g y h o ld s fo r b o th the in d iv id u a l tr a v e l b e h a v io r c a s e and the p r o d u c e r 's c o m m o d ity sh ip m e n t p ro b le m . In the c a s e of the d e v elo p m en t u tiliz in g p ro b a b ility th e o ry , a so m e w h a t d iffe re n t m e th o d o lo g y w ill be ap p lied . H e re a p ro b a b ility co n d itio n is s ta te d an d f o r m a liz e d w ith a c u m u la tiv e p ro b a b ility d e n sity fu n ctio n fo r d is ta n c e . D iffe re n t f o r m a l r e p r e s e n ta tio n s of the cu m u la tiv e p ro b a b ility fu n ctio n w ill be s u b s titu te d an d so lu tio n s d e riv e d . 1 .3 .2 G e n e ra l E c o n o m ic T h e o ry N eeded T h u s f a r the d is c u s s io n h a s in d ic a te d th a t s p a tia l in te ra c tio n m o d e ls w ould be c o n s tr u c te d f r o m e co n o m ic h y p o th e se s . H o w ev er, th e s e h av e n o t b e e n e s ta b lis h e d y et. In o r d e r to a c c o m p lis h th is 14 m e a n in g fu lly , th e y sh o u ld be c a te g o r iz e d fo r each type of m o d e l, th a t is , in d iv id u a l t r a v e l b e h a v io r, c o m m o d ity flow, a n d p ro b a b ility g r a v ity m o d e l. T h e r e f o r e , e a c h m o d e l w ill be c o n s id e r e d sep arately . 1 .3 . 2 .1 T he In d iv id u al T r a v e l B e h a v io r M odel A s s u m in g u tility th e o r y a s a b a s ic c o n stru c t, it can b e s a id th a t a n in d iv id u a l m a k e s t r i p s in o r d e r to d e riv e p e r s o n a l u tility . T h is e n d e a v o r is b a la n c e d a g a in s t d is u tility of trip m a k in g and n e t u tility r e s u l t s . N ext, w e a s s u m e th a t th e n e t u tility o f in te r a c t io n s at! e a c h t r i p d e s tin a tio n is a g g re g a te d , th a t is , su m m ed in to to ta l n e t | u tility . T h e in d iv id u a l w ill p u r s u e tr ip m a k in g to the p o in t w h e r e n e t j ! m a r g in a l u tility is z e r o . Of c o u r s e th is a s s u m e s an u n c o n s tr a in e d i j e ffo rt. H o w ev e r, it is n o t p o s s ib le to o p e ra te u n d er th is condition; so a l i m i t o r c o n s tr a in t is p la c e d on n e t u tility . T h is l i m i t b e c o m e s e ith e r m o n e y (b u d g eted fo r tra v e l) o r tim e (available f o r tr a v e l) . F o r m a l iz a tio n of th e s e e c o n o m ic a s s u m p tio n s fo r the m o d e l w ill tak e p la c e in C h a p te r II. 1 .3 . 2 .2 T he P r o d u c e r 's C o m m o d ity F lo w M odel In th is m o d e l b o th the s h o r t - r u n and lo n g -ru n e q u ilib r iu m co n d itio n s a r e a s s u m e d . F o r the s h o r t - r u n m a x im iz a tio n of re v e n u e is h y p o th e siz e d , w h ile th e lo n g - r u n a s s u m e s m a x im iz a tio n of p ro fit. In m a x im iz in g re v e n u e , sh ip p in g c o s ts a r e a ss u m e d to b e a c o n s tra in in g f a c to r . On the o th e r h an d , f o r p ro fit m a x im iz a tio n , it is a s s u m e d th a t w e a r e c o n s id e rin g r e v e n u e m inus c o s t o f p ro d u c tio n an d tr a n s p o r ta tio n . 1 .3 . 2. 3 P r o b a b ility M odel F o r th is m o d e l th r e e d if f e r e n t p ro b a b ilitie s a r e d efin ed . F i r s t , th e r e is a p ro b a b ility th a t a n in d iv id u a l (p ro d u c e r) w a n ts to t r a v e l (ship) to a d e s tin a tio n o th e r th a n h is point of o r ig in . S e c o n d ,is th e co n d itio n a l p ro b a b ility th a t th e t r a v e l d istan ce is a c c e p ta b le , 15 g iven th a t he w an ts to m ak e the t r i p s , and th ird , the p ro b a b ility th at the tr a v e l a c tu a lly o c c u r s . T hus, u n c e rta in ty has b e e n ad d ed in th is m o d el. C H A P T E R II IN D IV ID U A L T R A V E L BEHAVIOR 2. 1 A SSU M PTIO N S O F N IE D E R C O R N -B E C H D O L T M O D E L T h e o r ig in a l and c r e a tiv e w o rk done by P r o f e s s o r s N ie d e r c o r n and B e c h d o lt to d e v e lo p th e g ra v ity m o d e l to ta k e into c o n s id e r a tio n e co n o m ic p h e n o m e n a o v e r s p a c e is th e fo u n d atio n upon w h ic h th is c h a p te r h o p efu lly w ill ex p an d . * In t h e ir w o rk , P r o f e s s o r s N ie d e r c o r n and B e c h d o lt h av e m a x im iz e d th e n e t u tility o b ta in e d f r o m the n u m b e r of tr i p s w h ich an in d iv id u a l w ith in a s p e c ifie d re g io n 2 (g e o g ra p h ic a l a r e a ) w ill m a k e to o th e r r e g io n s . T h e ir d e r iv a tio n of th e " G r a v ity Law " d o es p ro v id e a fo u n d atio n w ith in th e f r a m e w o r k of 3 u tility th e o ry by m ak in g th e follow ing a s s u m p tio n s : N ie d e r c o r n , J. H. , and B . V. B e c h d o lt, "A n E c o n o m ic D e riv a tio n of the 'G r a v ity Law ' of S p a tia l I n te r a c tio n ," J o u r n a l of R e g io n a l S c ie n c e , Vol. IX , No. 2 (1969). 2 B y net u tility w e m e a n , n e t of an y d is u tilitie s of tr ip m a k in g . T h e r e f o r e , in th is a n a ly s is a ll in d iv id u a ls ' u tilitie s a r e n e t. 3 In a c o m m e n t on th e N ie d e r c o r n and B e c h d o lt a r t i c l e , M a th u r c r i t i c i z e s the b a s ic h y p o th e sis th a t the n e t u tility is a fu n c tio n of th e n u m b e r of t r i p s . H e s u g g e s ts , " i t w ould b e a p p r o p r ia te to c o n s id e r a m o r e r e a l i s t i c h y p o th e sis w h e re u tility is a fu n c tio n of th e a ttr ib u te s o r c h a r a c t e r i s t i c s of t r i p s , r a t h e r th a n of th e tr i p s t h e m s e l v e s ." N ie d e r c o r n and B e c h d o lt re p ly th a t, " M a th u r 1 s a n a ly s is d e a ls w ith finding th e u tility - m a x im iz in g c o m b in a tio n of t r i p c h a r a c t e r i s t i c s of d iff e r e n t ty p e s of tr i p s f r o m o r ig in to a g iv en d e s tin a tio n , " A lso , th e y p o in t out th a t M a th u r 's c a s e is q u ite lim ite d . F in a lly , th e y d e m o n s tr a te th a t b y a s s u m in g c o n su m a b le a ttr ib u te s a r e c o n s tr a in e d by th e n u m b e r of t r i p s , th e d e r iv a tio n of th e g r a v ity m o d e l c an b e g e n e r a liz e d , b u t w ill s till y ield the s a m e r e s u l t s . F o r th e fu ll c o m m e n t and re p ly r e f e r e n c e s e e th e b ib lio g ra p h y . 16 17 , U.\ = f(. T ..) k I J k i j ' (2 . 1) w h e r e . U.1 . k i j T k ij n e t u tility of in d iv id u a l k a t o r ig in i of in te r a c t in g w ith p e r s o n s o r th in g s a t d e s tin a tio n j, p e r u n it tim e n u m b e r of t r i p s ta k e n b y in d iv id u a l k f r o m o r ig i n i to d e s tin a tio n j n w h e r e u . 1 k 1 ,uf = £ f { , T..) k l 'k ij' = to ta l n e t u tility of in d iv id u a l k a t o r ig i n i of in te r a c t in g w ith p e r s o n s an d th in g s at a ll d e stin a tio n s , p e r u n it tim e ( 2. 2) O ne m ig h t e x p e c t th a t the n u m b e r of p o s s ib le i n t e r a c t i o n s a v a ila b le in s u b a r e a j is p r o p o r tio n a l to its p o p u la tio n ; th e n , i n t e r p r e tin g f (, T ..) a s n e t u tility of tr ip m a k in g to s u b a r e a j p e r a v a ila b le in te r a c t io n ( a v e r a g e u tility ), in d iv id u a l k 's to ta l u tility c a n b e w r itt e n as n k U i i V P . f(, T . .) L j j k i y j= i (2 .3 ) w h e r e P . = J a = T = k ij to ta l n e t u tility of in d iv id u a l k a t o r ig i n i o f in te r a c t in g w ith p e r s o n s o r th in g s a t a ll d e s tin a tio n s , p e r u n it tim e p o p u la tio n of d e s tin a tio n j c o n s ta n t of p r o p o r tio n a lity s a m e a s p r e v io u s ly d e fin e d 18 H o w ev er, the n u m b e r of t r i p s is lim ite d by the a v a ila b ility of m o n ey o r tim e the in d iv id u al w is h e s to a llo c a te to th is e n d e a v o r. So the a u th o rs of th is m o d e l p r e s e n te d the follow ing c o n s tr a in t eq u atio n s: r = c o st p e r m ile of d is ta n c e tr a v e lle d d . . = d is ta n c e b e tw ee n o rig in i and ^ d e stin a tio n j b. F o r the tr a v e l tim e c o n s tr a in t lo c a te d at o rig in i is w illin g to sp en d on tr a v e l, p e r unit tim e s = a v e r a g e sp ee d at w h ich p eo p le in the re g io n tr a v e l The p r o c e s s is , then, one of d e riv in g so lu tio n s w h ich w ill m a x im iz e c o n s u m e r s ’ u tility lim ite d by th e se c o n s tr a in ts . It should be p o in ted out h e r e th a t th e c h o ice of w h ich of the above c o n s tr a in ts , tim e o r m oney, is a p p lic a b le in d e riv in g the o p tim u m so lu tio n dep en d s a. F o r the m o n e ta ry c o n s tr a in t n (2.4) w h e re = to ta l am o u n t of m o n ey th a t in d iv id u a l k lo c a te d a t o rig in i is w illin g to sp en d on tr a v e l, p e r u n it tim e n (2 .5 ) j= l w h e re . H. = to ta l am o u n t of m o n ey th a t in d iv id u al k K 1 i , l . • ■ • • „ . i n 4 . . . i 19 upon w h e th e r M . / r § s , H.. If M . / r < s H ., th e n m o n ey is the a p p ro - i £ l K 1 K 1 K 1 p ria te c o n stra in t; h o w e v er, if the r e v e r s e w e re tru e , th en w ould be a p p licab le. 2 .2 SOLUTION O F TH E N IE D E R C O R N -B E C H D O L T M ODEL M ax im ized solutions w h ich give us an equation fo r d e te rm in in g the n u m b e r of tr i p s ta k e n by in dividual k re s id in g in re g io n i depend upon the fo rm of and th e f o r m of the c o n s tra in t function, be it , M. o r , H ,. T h e so lu tio n s p r e s e n te d in the N ie d e rc o rn -B e c h d o lt k i k l a r tic le u s e d m o n e y in the f o r m given as E q u a tio n (2 .4 ). E s s e n tia lly T he sec o n d is the " P o w e r U tility of T rip m a k in g F u n ctio n " and is as follow s: Itwo f o r m s of f(^T ..) a r e p re s e n te d . The f i r s t is the " L o g a rith m ic i ^ jU tility of T rip m a k in g F u n c tio n " and is as follow s: If (2 . 6 ) th en the so lu tio n w ould be (0<b < 1) 4 (2 . 8) ^Note th a t th is r e s tr ic tio n im p lie s d im in ish in g m a rg in a l u tility . 20 th e n the so lu tio n w ould be , T k ij . . . ( s H ) ij \ r / n p; E -JL 1/(1 -b) ij (i * j) (2. 9) j= l db / ( l - b ) ij A s s ta te d by N ie d e r c o r n and B echdolt, "It is co m m o n , e s p e c ia lly w hen s ta tis tic a l m e th o d s su ch a s the m e th o d of le a s t s q u a r e s a r e u sed , to find th a t the p o p u latio n w eig h t P . is r a i s e d to 3 a n e s tim a te d p o w er th a t is u s u a lly d iffe re n t fr o m th e p o w er of d „ . T he v a lu e s of P. in the u tility fu nction c an be r a i s e d to a p o w er c, i S ^ Iw here c > 0. " T h is m o d ific a tio n le a d s to the follow ing r e s u lts : . T .. k ij p c/(l-b ) J _______ 1 d l/( l- b ) . ij (i 4 j) (2 . 10) 2. 3 n P ^ 1^ 2 --------- j= l db /( l- b ) ij EX T EN SIO N O F TH E N IE D E R C O R N -B E C H D O L T M ODEL B e fo re f o r m a l so lu tio n s a r e d e riv e d fo r th e b a s ic m o d el, s e v e r a l m o d ific a tio n a s s u m p tio n s m u s t be added. F i r s t , th e e n tire su b n atio n al re g io n is c o m p o se d of n + 1 a r e a s , w h e re i = 0 and j = 1, 2, . . . , n and, th e r e f o r e , i 4 j. N ie d e r c o rn and B ech d o lt so lv ed th e p ro b le m of d e te rm in in g the to ta l n u m b e r of t r i p s ta k e n in the re g io n if the o ^ o r i* re g io n is ex clu d ed . If w e w a n t to include th tr a v e l w ith in the i a r e a , w e m u s t som ehow define d . . fo r the c a s e 9 w h e re i = j. The m o s t p la u sib le so lu tio n is to a s s u m e th a t d is e q u al }N ie d e r c o rn , J . H. an d B. V. B ech d o lt, op. cit. : i 21 j i \ to the a v e r a g e d is ta n c e f r o m the c e n te r of the a r e a to its o u te r | p e r ip h e r y . T h is , th e n , is c o n s is te n t w ith th e g e n e r a lly u s e d a s s u m p - j tio n th a t d is ta n c e b e tw e e n tw o a r e a s is m e a s u r e d f r o m th e ir c e n te r s , th u s g iv in g an a v e r a g e d is ta n c e . T h e d e r iv a tio n s w h ic h w ill follow f r o m th is p o in t f o r w a r d w ill a ll d efin e th e i = j c a s e in th is m a n n e r . T h e r e f o r e , w e can c o n s id e r a re g io n c o m p o s e d of n s u b a r e a s in w h ich b o th i a n d j c a n ta k e on any v a lu e f r o m 1 to n. T he s e c o n d m o d ific a tio n w ill w e ig h t the p o p u la tio n v a lu e s by r a is in g P to th e p o w e r of c. T h is is n o t f o r s t a t is tic a l p u rp o s e s , b u t I is a n alo g o u s to in c r e a s in g o r d e c r e a s i n g r e t u r n s to s c a le . If c is I g r e a t e r th a n o ne, th e n in c r e a s in g p o p u la tio n s iz e m e a n s th a t the ! n u m b e r of p o s s ib le in te r a c t io n s i n c r e a s e s f a s t e r th a n p opulation, b ut if c is l e s s th a n one, th e n th e o p p o site is tr u e . A th i r d c h an g e e x p lic itly m a k e s a v e r a g e u tility a fu n ctio n of the r a tio of n u m b e r of t r i p s to the a v a ila b le n u m b e r of in te r a c tio n s ; th e re b y , c o r r e c t i n g the w o rk of N ie d e r c o r n a n d B ec h d o lt w ho h ad a s s u m e d the d e n o m in a to r of th is f r a c tio n to b e unity. T h e s e m o d ifi c a tio n s p ro v id e the fo llo w in g r e v is e d u tility fu n ctio n : w h e re i can e q u a l j* T he d e r iv a tio n s w h ic h follow w ill u s e a c o s t fu n c tio n (budget c o n s tr a in t) a s a c o n s tr a in t on the to ta l u tility fu n c tio n s. Two u tility fu n c tio n s w ill b e e v a lu a te d w ith f i r s t th e " L o g a r ith m ic U tility of T r ip m a k in g F u n c tio n R e f o r m u la te d " an d s e c o n d th e " P o w e r U tility of T r ip m a k in g F u n c tio n R e f o rm u la te d . " T h e w o rd " r e f o r m u la te d " h a s b e en a d d e d to in d ic a te th a t E q u a tio n (2, 11) is b ein g u s e d r a t h e r th a n E q u a tio n (2. 3). To e a c h of th e s e , th e fo llo w in g c o s t c o n s tr a in t fu n c tio n s w ill be a p p lie d : 1. L in e a r C o st F u n c tio n n , M , = r V d . . ( . T . . ) (2 .1 2 ) k 1 i j j i j ' ' ' j= l 2. P o w er C o st F u n c tio n k M r r E d i j < k T i j ) ( 0 < a < 1 > <2 - 1 3 > j =i 3. E xponential C ost F u n ctio n k ^ i ±[« - K exp(-Q'd.j)J k T _ (u>0) (2.14) j= l 4. L o g a rith m ic C o st F u n ctio n k M. = r £ ( l n d..) k T .. (2.15) j= l 2. 4 SOLUTION WITH PO W ER U TILITY O F TRIPMAKING R E FO R M U L A T E D AND LIN EA R COST FUNCTION B a se d upon the fo reg o in g a ss u m p tio n s , the to ta l n et u tility w hich an individual k w ill d e riv e fro m in te ra c tio n s (with p e r s o n s or things) at a ll d e stin atio n s w hen c o n s tra in e d by an ex p licit tr a v e l co st function is m a x im iz ed fo r the follow ing a u g m en ted objective function [using E q u atio n (2, 11) and (2. 12)]: 23 w h e re \ = a L a g ra n g ia n (u n d e te rm in e d ) m u ltip lie r N ext, the f i r s t o r d e r c o n d itio n fo r m a x im iz a tio n of E q u a tio n (2. 16) w ith r e s p e c t to ^.T.. (n u m b e r of t r i p s to e a c h d e s tin a tio n p e r .K I J u n it tim e ) and X . c o n s is ts of the follow ing s e t of sim u lta n e o u s e q u a - ,. 6 tio n s: \ Ui c af<kT i l / a P ? srrr = aPi — sttt:-------xrdi i = 0 k m k ll a. U. „ 9f( T . / a P i k i = a p c k m n _x r d = 0 (2j 1?) k in n k in in If ❖ U. n = r V d . l T ..) - , M. = 0. xj k 1] k x j= l f(. T . . / a P f ) ={, T ../ a P ? ) (2. 18) k ij j k ij j ' the f i r s t p a r tia l d e riv a tiv e is b-1 a f ( , T . . / a P . ) . , i b( T . .) ■ V r ^ - ■ V / - V K ) " - - M b - <2 - 19> k ij (a P .) J A lthough the sec o n d o r d e r co n d itio n s a r e not s ta te d , it is c le a r a s in the N ie d e r c o r n - B e c h d o lt m o d e l th a t th e se f i r s t o r d e r e q u atio n s a r e a t th e m a x im u m u tility p o s itio n u n d e r the sp e c ifie d conditions,, 24 S ubstituting E q u atio n (2. 19) into the f i r s t n p a rtia l d e riv a tiv e s of E quation (2. 17) and a lso e lim in a tin g \ r as defined by th e f i r s t equation of (2. 17) g iv es the follow ing n - 1 equations V i z ' 1 " 1' “ P C / M i----------------: i ’ rt H i . <a P 2 ) b J 1 \ 1 1 / ( a P * ) b = 0 ( 2 . 20 ) P c V i n * ” ' 1’ - P C V i / ' 1 ' n (aPc)b - n 1 v W 1 | U < - 1 ! & i d ^ - I = 0 Solving th ese eq u atio n s fo r , T .., w h e re j = 2, . . . , n g iv es I -K IJ N ext su b stitu tio n of E q u a tio n (2. 21) into the l a s t eq u atio n in E qu atio n (2. 17) g iv es k ij 1' j j=2 Solving fo r , T g iv e s iC 1L ( 2.21) = , M. (2,22) k x 1 1- b / b - h 1/ b - l (2.23) F in a lly , su b stitu tin g E q u a tio n (2. 23) into E q u a tio n (2 .2 1 ). T .. = ( P < r / P ? )1" b / b " 1( d . . / d .1)i / b “1 k ij 1 y ' ij i l ' (2. 24) X< d . . + £ d . . ( P ^ / P ? )1_ b /b “ 1( d . . / d . 1) 1 /b " i| 11 j=2 1 J 1 J 1 J 11 J S im p lify in g p ro d u c e s T k ij ■ m p c £ p c /(d..)b/1“b j=i ij l /l - b (2 .2 5 ) ( j -1, 2 , . s o , n , ) T h u s, the n u m b e r of t r i p s - - b a s e d upon the a s s u m p tio n s of E q u a tio n s (2. 11) and (2. 12)— ta k e n by in d iv id u al k to d e stin a tio n j f r o m o rig in i is given by E q u a tio n (2. 25); and w h en i=j th e d ista n c e w ould be g iven a s the m e a n d is ta n c e s f r o m the c e n te r to the p e r ip h e r y of the o r ig in a r e a , a s p re v io u s ly d is c u s s e d . SO LUTIO N W ITH PO W E R U TILITY O F TRIPM AKING 2. 5 R E F O R M U L A T E D AND PO W E R COST FU N C TIO N In th is c a s e , the u tility fu n ctio n s till u s e s the p o w er f o r m u la tio n , b u t th e re a r e ch an g e s in the c o s t function. T he c o st fu nction is E q u a tio n (2. 13) an d a s s u m e s th a t c o s ts a r e no lo n g e r c o n sta n t w ith r e s p e c t to d is ta n c e tr a v e le d . T he change is defined a s d is ta n c e to so m e p o w e r, th a t is , a p a r a m e t e r a. W hen th e p a r a m e te r is 26 b o u n d ed b y 0 < o < l , th e n up to th e p o in t th a t a = 1 (c o n s ta n t c o sts) the c h an g e in c o s t a s s o c i a t e d w ith a c h a n g e in d is ta n c e w o u ld b e l e s s th a n p r o p o r tio n a l. In o th e r w o r d s , th e p e r u n it d is ta n c e c o s ts p e r u n it tim e w o u ld be d e c r e a s i n g a s th e to ta l d is ta n c e w a s in c r e a s in g . A l th o u g h a w ill n o t b e n u m e r i c a l l y d e fin e d in th is c h a p te r , th e p re c e d in g d is c u s s io n is only to in d ic a te its im p a c t on th e c o s t c o n s tr a in t fu n ctio n . W ith th is in m in d , th e n , th e fo llo w in g a u g m e n te d fu n c tio n is p r e s e n t e d u s in g E q u a tio n s (2. 11) a n d (2. 13): D iff e re n tia tin g th is w ith r e s p e c t to , T .. a n d X . g iv e s the follow ing s e t K I J of e q u a tio n s: n n (2 . 26) (2 .2 7 ) 27 N ext, su b stitu tin g E q u a tio n (2. 19) into th e n eq u atio n s in E q u a tio n (2. 27) and e lim in a tin g k r gives n - 1 e q u atio n s of the f o r m "b <kT i2 > b ' r - P c ' A® i2 b <kT i l ' b ' r (a P ^ jh 1 Aa A l J . ( a P i )b = 0 C ' b ‘u T in>b ' r _ p c [d ? 1 in b 'k T n >b_1 P n ( a P c ) b L ' n *1 . i l . ( a P ? b ] = 0 T h en solving fo r , T .., w h e re j = 2, . . . , n gives K 2 J (2 .28) , T . . = ( P = / P C) 1- b / b " 1- ' “ / b - ' k ij ' 1 j ' ' n i l 'k xl (2. 29) Next, s u b stitu tin g E q u a tio n (2.29) fo r T .. in the l a s t e q u atio n of K I J E q u a tio n (2. 27) a n d so lv in g fo r g iv e s T = k i l k Mi n a0 ; + V d“(p^/p?) i i ij i j a ,„ c ,_.c. 1- b / b - l , ,ot /jQ? , 1/ b - l (d“ /d“)- j=2 (2. 30) F in a lly , su b stitu tin g E q u atio n (2. 30) in to E q u atio n (2. 29) g iv e s . o / l - b 'k Mi k T ij = \V n L U * i c / Jcvb/l-b) j / ij i ij (j = l , 2 , . . . n ) (2.31) 2. 6 SO L U T IO N W ITH P O W E R U T IL IT Y O F T R IPM A K IN G R E F O R M U L A T E D AND E X P O N E N T IA L C O ST FU N C T IO N A g a in th e u tility fu n c tio n r e m a i n s a s b e fo re , an d only the c o s t fu n c tio n is r e v is e d . T h e c o s t- f u n c tio n a s s u m p tio n m a d e i n t h i s c a s e is th a t it ta k e s on a n e x p o n e n tia l f o r m , the f o r m b ein g K - K exp(-o,d^l T h e a u g m e n te d o b je c tiv e fu n c tio n to be m a x im iz e d is (2.32) T ak in g th e f i r s t p a r t i a l d e r iv a tiv e w ith r e s p e c t to . T . . a n d \ g iv e s the fo llo w in g n + 1 e q u a tio n s : . (2.33) - \ | k - K exp(o?din )J = 0 29 E lim in a tin g \ a n d su b stitu tin g E q u a tio n (2. 19) into the n e q u atio n s of E q u a tio n (2. 33) g iv e s n - 1 eq u atio n s. n K V ' 1' - P c K - K expf-ofd^) V u > b " r . <*P 2>b • • K - K exp(-ocLj) ( a P l )b V i / " 1 ' - P c K - K exp(-od. ) c m W 1' . V b J K - K exp(-o'<l.1) ( a p l )b = 0 = 0 Next, so lv in g f o r T .., w h e re j = 2, n g iv es k ij (2. 34) v T -- = K - K ex p (-o d ..) _________ ^ ij K - K expt-cvd^) 1/ b - l / c \ l / b - l A PJ C *kT il* (2. 35) S u b stitu tin g E q u a tio n (2. 35) into th e la s t e q u atio n of E q u a tio n (2. 33) and so lv in g fo r g iv e s T = k il + 4 * 6 ) I T [l - expt-o-d.^j £ [ * • exp< V ’ V j=2 \ i C \ 1/ b - l 1 - exp(-o;d,.) ij 1 / b - l 1 - e x p (-a d u ) ■ 1 (2. 36) S u b stitu tin g into E q u atio n (2. 35) gives 30 iT - - = k ij 1/1-b S p c j= 1 j [1 " exp(-cvd..)j ( 2 .3 7 ) (j — 1, 2 , . . . j n) 2. 7 SO L U T IO N W ITH P O W E R U T IL IT Y O F TRIPMAKING R E F O R M U L A T E D AND L O G A R IT H M IC COST FUNCTION T he fin a l c o s t fu n c tio n to be e v a lu a te d w ith th e power u t i l i t y fu n c tio n is a lo g a r ith m ic f o r m . A s s u m e th a t t r a v e l c o sts a re p r o p o rtio n a l to the n a tu r a l lo g a r ith m of d is ta n c e . T his assu m p tio n c o r r e s p o n d s to the c o s t c o n s tr a in t fu n c tio n E q u a tio n (2. 15). T h e a u g m e n te d o b je c tiv e fu n c tio n is f o r m e d by u s in g E q uation (2. 11): j= i ' j n r V ( l n d..) . T .. - . M. IJ k lj k 1 j= l D iffe re n tia tin g th is e q u a tio n g iv es df<kT i i / a P J> ak u i - N r = a P i k in T ~ T — k il - \ r In d ,, = 0 il (2. 38) > ! < a, u. k x _ O T " n k in of{. T. / a P C) k m n' s r r k in - \ r In d. = 0 m (2 .3 9 ) a, u. k l n f t j j= i 31 E lim in a tin g \ r in the f i r s t n e q u a tio n s of E q u a tio n (2. 39) g iv e s n - 1 e q u a tio n s . A g a in u s in g E q u a tio n {2 . 19) a n d s u b s titu tin g it into th e n - 1 e q u a tio n s g iv e s bt T «> b - n <*P2>b P? X In d._ i2 i n a:\ xl b <kT n> b- 1 = 0 • i (2 .4 0 ) n P C In d. m 1 ( a P ^ _ *1 In d . . i l ,b- 1 k il ( a P f ) b Now, so lv in g f o r , T .., w h e r e j=2, . . . , n g iv e s K Ij k ij p c \ l - b / b - l / l n d ,1 / b - l 4 ) ( " S ( k T i i ) (2 .4 1 ) A nd s u b s titu tin g in to the l a s t e q u a tio n o f E q u a tio n (2. 39) a n d so lv in g £or k T n g iv e s T k i l c xi-b /b -iAnd ;v n b -i In d ^ 4 ~y ~ In d^. j = 2 aJV p c j ij (2 .4 2 ) S u b s titu tin g E q u a tio n (2 .4 2 ) into E q u a tio n (2 .4 1 ) f o r g iv e s T .. = -kM i k ij \ r P ? n P . r -----------1 [j= 1 (In d..) F 7TTB a / l - b In d . . V (j= 1, 2, . . . , n . ) (2 .4 3 ) 32 2. 8 SOLUTION W ITH LO G A RITH M IC U T IL IT Y O F TRIPM A K IN G R E F O R M U L A T E D AND L IN E A R COST FU N C TIO N In the p re v io u s so lu tio n s u tility w a s a s s u m e d a p o w er fu nction. H o w ev er, in th is so lu tio n and the o n es to follow , u tility w ill be r e p r e s e n te d by a lo g a rith m ic function. To m a k e th is m o d ific a tio n it is n e c e s s a r y to a s s u m e th a t P ro c e e d in g on this h y p o th e sis, the f i r s t so lu tio n w ill m a x im iz e the lo g a rith m ic u tility of trip m a k in g r e f o r m u la te d a s c o n s tr a in e d by the l in e a r c o st function. T he a u g m e n te d L a g r a n g ia n fu n c tio n is E q u atio n (2. 16). W orking th ro u g h the f i r s t o r d e r co n d itio n g iv es the E q u a tio n (2. 17). Now, if w e a s s u m e E q u atio n (2 .4 4 ), th en th e f i r s t p a r tia l d e riv a tiv e is af< k V * p j> _ sk T ij “ ' 1 TT (2.45) S u b stitu tin g E q u a tio n (2.45) into th e f i r s t n e q u atio n s of E q u a tio n (2 . 17) and e lim in a tin g \ r as defin ed by th e f i r s t eq u atio n in E q u atio n (2. 17) g iv es the follow ing s e t of n - 1 e q u atio n s: 33 2\kTi2, P > = 0 nl k in , c / ^ in - o (2 .4 6 ) S o lv in g th e n e q u a tio n s in E q u a tio n (2. 46) f o r j ^ i j g iv e s k ij i r / w | k 'r “ ) i ' 1 - , n (2 .4 7 ) N e x t s u b s titu tin g E q u a tio n (2. 47) in to the l a s t e q u a tio n of E q u a tio n (2. 17) an d so lv in g f o r g iv e s T = k i l 'k M i n d.. + £ d..(d../d..)(p?/p'r) 11 i = 2 1J 11 J 1 (2 .4 8 ) F in a lly , s u b s titu tin g , T in to E q u a tio n (2 .4 7 ), an d s im p lify in g t e r m s K X X g iv e s k M i \ / p j i T . . = k ij \ r /I n E ? j = i 1 , d.. c / V ij (j= l, 2, . n) (2 .4 9 ) T h is s o lu tio n is a l m o s t th e s a m e a s th e one d e r iv e d b y P r o f e s s o r s N i e d e r c o r n an d B e c h d o lt in t h e i r lo g a r ith m i c u tility of tr ip m a k in g 34 function. The d iffe re n c e lie s in the u n sp e c ifie d p a r a m e t e r c w hich is ta k en on by the p o p u latio n v a ria b le . T h is r e s u l t r e v e a ls th e e ffe c t of the a s s u m p tio n m ad e in the re f o r m u la te d u tility fu nction, th a t is , the p o p u latio n e ffe ct is w e ig h ted by so m e p a r a m e t e r c. 2. 9 SOLUTION W IT H L O G A R IT H M IC U T ILIT Y O F TRIPM A K IN G ! R E F O R M U L A T E D AND PO W E R O F COST F U N C T IO N T h is so lu tio n w ill m odify the c o s t fu n ctio n w ith the f o r m of u tility re m a in in g the s a m e a s the p re v io u s so lu tio n . T h e r e f o r e , E q u a tio n (2. 13) w ill be u s e d as the c o s t c o n s tr a in t fu n ctio n . The au g m en ted L a g ra n g ia n fu n ctio n w ill be E q u a tio n (2. 26) an d th e set of E q u atio n (2. 27) is d e riv e d . E lim in a tin g \ r as d e fin e d by th e f i r s t eq u atio n of E q u atio n (2. 27) and su b stitu tin g E q u a tio n (2 .4 5 ) g iv e s a r e v is e d s e t of s im u lta n e o u s e q u atio n s. <2 . 50) Solving fo r T .., w h e re j=2, . . . , n g iv e s k ij S u b stitu tin g E q u a tio n (2. 51) into the la s t e q u atio n of E q u a tio n (2. 27) and so lv in g fo r ^ T .^ g iv es N ext, s u b s titu tin g E q u a tio n (2. 52) in to E q u a tio n {2, 51) a n d s im p lify in g , (j= l, 2 • * * J n) (2. 53) j 2. 10 SO LU TIO N W ITH L O G A R IT H M IC U T IL IT Y O F T R IP M A K IN G R E F O R M U L A T E D AND E X P O N E N T IA L COST F U N C T IO N A g ain th e f o r m of th e u tility fu n c tio n w ill r e m a in the s a m e a n d the c o s t fu n c tio n w ill b e r e f o r m u la te d . T h e e x p o n e n tia l c o s t fu n c tio n u s e d is E q u a tio n (2. 14). T h u s, the a u g m e n te d L a g r a n g ia n fu n c tio n w ill be g iv e n a s E q u a tio n (2 .3 2 ). D iff e re n tia tin g E q u a tio n (2. 32) w ith r e s p e c t to a n d ^ g iv e s th e s e t of s im u lta n e o u s e q u a tio n s w h ic h a r e d e fin e d by E q u a tio n (2. 33). E lim in a tin g X . a n d su b stitu tin g E q u a tio n (2. 45) in the n e q u a tio n s of E q u a tio n (2. 33) g iv e s n - 1 e q u a tio n s. 0 (2. 54) 0 Next, so lv in g f o r T . . t w h e r e j=2, . . . , n g iv e s K I J 36 K - K exp(-acLj) K - K ex p (-ad ..) ^ ij *kT il* (2. 55) If w e tak e E q u atio n (2. 55) and su b stitu te it into the l a s t equation of E q u atio n (2. 33), the follow ing eq u atio n fo r r e s u lts : . T ., = . M. k i l k l K - K exp(-od.j) + 5 Z (Pj / P 1} [K " K e x P ^ d i i j | j=2 (2. 56) Now su b stitu tin g E q u a tio n (2. 56) into E q u atio n (2. 55) and sim plifying gives v T -- = k U kMA / l [ I T n X j= l 1 1 - exp(-acLj)j (2.57) (j- If 2, ■■■) n) 2. 11 SOLUTION WITH LOGARITHM IC U TILITY O F TRIPM AKING R E F O R M U L A T E D AND LOGARITHM IC COST FU N CTIO N T h is, the fin al so lu tio n of th is c h a p te r, d e a ls w ith the sa m e lo g a rith m ic u tility fu n ctio n and th e lo g a rith m ic c o s t function, E q u a tio n {2. 15). F o r th is se c tio n the au g m en ted L a g ra n g ia n function is E q u atio n (2. 38). D iffere n tia tin g it w ith r e s p e c t to VT .. and \ g iv es K IJ E q u atio n (2. 39). E lim in a tin g \ r and su b stitu tin g E q u atio n (2.45) into the n eq u atio n of E q u atio n (2. 39) g iv es E q u atio n (2. 58). 37 ,c / 1 - P ik i2j c / l n d i2 1 y in cL - = 0 (2. 58) ,c / 1 n - P / I n d. ' c l m k in; 1 \ l n d il = 0 S olving fo r , T . . , w h e r e j=2, . . . , n g iv es /In d ., \ / P ' k T ij = [ml. ) ( p c ) <kT i i ) (2 . 59) S u b stitu tin g E q u a tio n (2. 59) into th e l a s t e q u a tio n of E q u a tio n (2. 39) and so lv in g f o r g iv e s 'k M i k il n In d .. + Z J In d. .(In d . , / l n d ..) ( P ? /P ^ ) 11 p z 1J 11 1J J 1 ( 2. 60) S u b stitu tin g E q u a tio n (2. 60) into E q u a tio n (2. 59) a n d sim p lify in g t e r m s giv es k u i n E p - > i J In d.. (j= l, 2 ................n) (2 . 61) 2. 12 SUM M ARY O F INDIVIDUAL T R A V E L BEH A V IO R In th is c h a p te r, th e e s s e n s e of the m o d e l a n d a s s u m p tio n s m a d e b y the N ie d e r c o r n a n d B e c h d o lt a r t i c l e w a s s e t f o r th and I 38 j ! i g e n e ra liz e d . F i r s t , the a v e ra g e u tility of trip m a k in g function w as re fo rm u la te d to sa y th at a v e ra g e u tility w as a fu nction of th e ra tio j of the n u m b e r of tr ip s th a t individual k m a d e to h is n u m b e r of p o ssib le in te ra c tio n s . A fte r re fo rm u la tin g the u tility function, tw o a ssu m p tio n s about its fo rm w e re m a d e . F i r s t , it w as a pow er function th a t w as a ss u m e d to have d e c re a s in g m a rg in a l u tility . Second, it took on a lo g a rith m ic fo rm . T h ese two a ssu m p tio n s a re the s a m e as th o se u sed in the N ie d e r c o rn and B echdolt d e riv a tio n . H o w ev er, one p ro b le m a r o s e in th e ir fo rm u la tio n , and it is th th at the o rig in o r i a r e a w as not c o n sid e re d . T h u s, trip s tak en w ithin the o rig in a r e a w e re not c o n sid e re d . In o r d e r to get a ro u n d \ th is p ro b le m , we have a ss u m e d th a t the o rig in a r e a be c o n sid e re d and th at d is ta n c e be c alc u la te d as the a v e ra g e len g th of d is ta n c e fro m the c e n te r of the a r e a to its o u te r b o u n d ary . T h e re fo re , a ll of the so lu tio n s in this c h a p te r hold fo r i = j as w ell as i 4 j. T h is c h a p te r then p ro c e e d s to give so lu tio n s c o n sis te n t w ith the b a s ic fo r m of the ! " G ra v ity Law" of s p a tia l in te ra c tio n fo r c alc u la tin g the n u m b e r of in te rre g io n a l and in tr a re g io n a l tr ip s tak en by in d iv id u als. The ; so lu tio n s im p ly m a x im iz e d utility of trip m ak in g f o r the in dividual ] as c o n stra in e d by sp ecific co st fu n ctio n s. The so lu tio n eq u atio n fo r ; e ac h c o st function and u tility function is lis te d in T ab le 2. 1. T h e se solutions have given a n u m b e r of tr ip s p e r individual; h o w ev er, the ! 7 to tal n u m b e r of tr ip s tak en by a ll in d ividuals is m T ij = £ k Tij (k=l, 2, . . . , m ) (2 . 62 ) k=l 7 N ie d e r c o rn and B ech d o lt, op. cit. T ab le 2 .1 g D e riv a tio n s fo r Individual T r a v e l B eh a v io r U tility ^ s'" v F u n c t io n C o st F u n c tio n P o w e r of U tility - E q u atio n (2. 8) L o g a rith m ic U tility - E q u atio n (2. 6) L in e a r C o st E q u atio n (2.12) E q u atio n (2.1 ^ - ( £ ) 15) P C J E p c /(d ..)b / 1 "b J = i J 1J 1/ 1-b ( v ) E q u atio n (2. 49) j= l J P o w e r C o st E q u atio n (2. 13) E q u atio n (2.1 T k ij \ r / 51) / ■ > P C I.K'-r-i I . J h r E q u atio n (2. 53) j= l J 8 uj N ote: F o r a ll c a s e s (j= l, 2, . n). 'O T ab le 2. 1 D e riv a tio n s fo r Individual T r a v e l B e h a v io r (Continued) U tility F u n c tio n P o w e r of U tility - E q u a tio n (2. 8) L o g a rith m ic U tility - E q u atio n (2. 6) E q u atio n (2. 57) E q u a tio n (2. 37) M. P. M \ / p : E x p o n e n tia l C o st E q u a tio n (2 . 14) E q u a tio n (2. 43) E q u a tio n (2. 61) 1 / 1-b M p : L o g a rith m ic C o st E q u a tio n (2 . 15) In d . . In d . . P. . £ p c j= i J b / l - b j= l (In d .) 41 T hen, w ith E q u a tio n {2. 62), it is p o s s ib le to d e riv e a s q u a r e m a t r i x of t r i p s ta k e n w ith in s o m e d e fin e d s u b n a tio n a l re g io n by p e r s o n s re s id in g th e re in . C H A P T E R III C O M M O D IT Y - F L O W G R A V ITY M O D E L 3. I IN T R O D U C T IO N In C h a p te r II a n a t t e m p t w a s m a d e to e x te n d th e w o r k done b y P r o f e s s o r s N i e d e r c o r n a n d B e c h d o lt th ro u g h the f o r m u la t io n of s e v e r a l m a t h e m a t i c a l m o d e ls b a s e d u p o n v a r y in g a s s u m p tio n s . T h e e s s e n t i a l ta s k , th e n , w a s to m a t h e m a t i c a l l y a n s w e r th e q u e s tio n of how m a n y t r i p s a n in d iv id u a l w ill m a k e b e tw e e n g e o g r a p h ic a l a r e a s , g iv e n th a t he w a n ts to m a x im iz e h is n e t u tility of t r i p m a k in g w h ic h is c o n s t r a i n e d by th e c o s t of tr i p m a k in g . To a c c o m p lis h th is o p tim iz in g j | o r m a x im iz in g p r o c e s s , th e m a t h e m a t i c a l te c h n iq u e of L a g r a n g ia n i m u l t i p l i e r s an d d if f e r e n tia tio n w a s u s e d . In th is c h a p te r th e t a s k is j s i m i l a r e x c e p t th e q u e s ti o n w ill be d i r e c t e d to p r o d u c e r s o r s h ip p e r s i r a t h e r th a n in d iv id u a ls o r c o n s u m e r s . S p e c ific a lly , th is m e a n s th e e m p h a s is is p la c e d on q u a n tity s h ip p e d r a t h e r th a n t r i p s ta k e n . T h e r e f o r e , m a x im iz a t io n of p r o f it a n d to ta l re v e n u e a s c o n s t r a i n e d b y sh ip p in g c o s ts is th e e c o n o m ic k e y r a t h e r th a n m a x im iz a t io n of u tilit y a s c o n s t r a i n e d b y t r a v e l c o s ts . A n o th e r u n d e r ly in g c o n s id e r a tio n m u s t b e m a d e , an d , th a t is , th e s h o r t - r u n v e r s u s th e l o n g - r u n e q u ilib r iu m p o s itio n . So th e d e v e l o p m e n t of the c o m m o d ity - f lo w g r a v ity m o d e l w ill be s p e c if ic a lly tw o fo ld . A s h o r t - r u n m o d e l w ill b e d e fin e d f i r s t , w ith th e lo n g - r u n v e r s i o n to fo llo w . H o w e v e r, b e f o r e e a c h o f t h e s e m o d e ls is d e v e lo p ed, t h e i r g e n e r a l a s s u m p tio n s w ill b e s p e c if ie d . T o th e t a s k of d i s tin g u is h in g b e tw e e n s h o r t - r u n v e r s u s l o n g - r u n a s s u m p tio n s , w e now t u r n o u r a tte n tio n . 42 43 3. 2 D IF F E R E N T IA T IO N B E T W E E N T H E SH O R T-R U N VERSUS L O N G -R U N SITUATIONS As the m o d e ls to b e dev elo p ed a re s h o r t ru n and long ru n , it b e c o m e s n e c e s s a r y to d efin e th e s e te rm s . A lfr e d M a r s h a ll in h is P r in c ip le s d is tin g u is h e s th e s h o r t ru n as th a t p e rio d w h e re " . . . the a p p ro p ria te in d u s tr ia l o r g a n iz a tio n has not tim e to b e fully a d a p te d to d em an d . On the o th e r h a n d , the long ru n is a p e r io d w h e re " . . . a ll in v e s tm e n ts of c a p ita l and e f f o r t is prov id in g the m a t e r i a l p la n t and the o rg a n iz a tio n of a b u s in e s s , and in a c q u irin g tr a d e know ledge and ; s p e c ia liz e d a b ility , have tim e to be a d ju sted to the in c o m e s w h ich 2 ; a r e e x p e c te d to be e a r n e d b y th e m . 1 1 In th e tr a d itio n a l s e n s e , m a x i- ! ' i j m iz a tio n of p ro fit is the o b je c tiv e of the f i r m . T h is o b je ctiv e is ! a s s u m e d fo r both the s h o r t r u n and long ru n . In re c a p itu la tin g the tra d itio n a l th e o r y of th e f ir m , it c an be s ta te d th a t although p ro f it is m a x im iz ed , th e d ic h o to m y b e tw ee n s h o r t- r u n a n d the lo n g - r u n s itu a tio n is as fo llo w s: (1) S h o rt ru n is w h e re c a p ita l is fix e d w h ile o th e r fa c to rs a r e v a ria b le ; and (2) L ong ru n is w h e re a ll f a c to r s a r e v a ria b le . H o w ev er, f o r the c a s e of s p a tia l e q u ilib riu m of th e fir m , the tr a d itio n a l th e o r y of s h o r t ru n is not quite a p p ro p ria te . In s te a d , the sh o rt r u n fo r the s p a tia l s itu a tio n m u s t be re d e fin e d in t e r m s of the tra d itio n a l d efin itio n of the v e ry s h o r t ru n . T h is is d e fin e d a s the situ atio n w h e r e ou tp u t of the f ir m is fixed, an d the o b je ctiv e i s m a x im iz a tio n of re v e n u e . S till, fo r th e lo n g - r u n s p a tia l s itu a tio n th e tra d itio n d e fin itio n w ill be u se d . M a r s h a ll, A lfre d , P r in c ip le s of E c o n o m ic s , ed. by C. W. G u illeb au d , Vol. I (N ew Y ork: M a c m illia n a n d Co. , 1961), p. 376. 2 Ibid., p. 377. 44 : 3- 3 ASSUM PTIONS F O R T H E SH O RT-RU N COMMODITY - FLOW GRAVITY M ODEL T u rn in g now to the s h o r t- r u n m o d el, w e m u s t s e t fo rth the a ssu m p tio n s w h ich u n d e rlie the follow ing fo rm a liz a tio n . F i r s t , in th is m o d e l the f ir m w ill a tte m p t to m a x im iz e re v e n u e as its o b jectiv e function (see S ection 3 .2 above). Second, the to ta l output fo r the su b n atio n al re g io n is p r e d e te r m in e d and c o n sta n t fo r the tim e in te rv a l t^, w hich w ill not be sp e c ific a lly s ta te d o th e r th a n as s h o r t run. H ow ever, p ric e of goods w ill be allow ed to vary. F u r th e r , a s s u m e th a t we a r e c o n sid e rin g an e n tire sub- ! n atio n al re g io n c o m p o se d of n a r e a s (shipping d e stin a tio n s),w h e re ! I j= 1, 2, . . . , n. N ext the p h y sica l lo c atio n of the firm , th a t is , sh ip - ! th j ping-origination point, w ill be c a lle d the i point. The d ista n c e co m - i p u tatio n fo r the i ^ d e stin a tio n a r e a w ithin w hich the i* p oint is j th p h y sica lly lo c a te d w ill be m e a s u r e d a s the a v e ra g e d ista n c e fr o m i th point to its o u te r p e rip h e ry . D istan c e m e a s u r e d fr o m i point to o th e r a r e a s (that is , excluding i ^ a re a ) w ill be to the c e n te r of th at a re a . th A gain, p opulation v a lu es fo r each j a r e a w ill be w eig h ted by th e p o w er of the p a r a m e t e r c, w h e re c> 0 . A lso , the functio n al re la tio n s h ip of re v e n u e p e r a r e a w ill be view ed as re v e n u e e a r n e d p e r c u s to m e r in the j^1 a r e a . T h e se a s s u m p tio n s , then, p ro v id e the follow ing re v e n u e function: » * ■ • £ r > • & ) (3.1) 45 w h e re R. k = to ta l re v e n u e a c c ru in g to the f i r m f o r c o m m o d ity k a = f r a c tio n of the p e r s o n s in the a r e a who a re in te r e s t e d in buying the p ro d u c t, a s s u m e d the s a m e fo r all a r e a s , th at is , a c o n sta n t of p r o p o r tio n a lity P . = p o p u latio n in a r e a j c= a p o s itiv e c o n sta n t, a w eig h tin g p a r a m e t e r k = c o m m o d ity k re v e n u e e a r n e d p e r p o te n tia l c u s to m e r in s u b a r e a j a P 7 f = to ta l re v e n u e e a r n e d in a r e a n = to ta l n u m b e r of sh ipping d e s tin a tio n a r e a s L ik e th e t r a v e l b e h a v io r m o d e l in the p re v io u s c h a p te r, th e c o m - m o d ity -flo w m o d e l w ill be e v a lu a te d fo r two re v e n u e fu n ctio n s: the " P o w e r R ev en u e F u n c tio n " and th e " L o g a rith m ic R evenue F u n c tio n . " T he f o r m of th e s e fu n c tio n s is , re s p e c tiv e ly , a s follow s: f(k q j / a P p = (k q . / a P p D (3 .2 ) £( q ./a P ? ) = In K J J a P . J (3 .3 ) 46 It is n e c e s s a r y to a n a ly z e th e a c c e p ta b il ity of the lo g a r ith m i c a n d p o w e r d e m a n d fu n c tio n s a s fittin g th e e c o n o m ic c r i t e r i a fo r d e m a n d fu n c tio n s, th a t is , c h a n g e s in q u a n tity le a d to in v e r s e c h a n g e s in p r i c e an d v is a v e r s a . So, b e f o r e p r o c e e d in g E q u a tio n s (3. 2) a n d (3. 3) w ill be e v a lu a te d to d e te r m in e if th e y s a t i s f a c t o r i l y m e e t th is c r i t e r i o n . 1. E v a lu a tio n of E q u a tio n (3 .2 ) If f(k q ./ a P ? ) = (k q j/a P 7 ) (3 .4 ) fo r re v e n u e p e r p o te n tia l c u s to m e r , a n d to ta l re v e n u e is T R . = a P ? (, q . / a P ? ) b (3 .5 ) k j j k j j th en a v e r a g e re v e n u e is , T R . a 1- b P ? (1 "b) , A R . = — -i - = j-JL (3 .6 ) k i , q. 1- b k M j . q. J k ^ j w h ic h is c l e a r l y a d e c r e a s i n g fu n c tio n of , q. K J g iv e n th a t 0 < b < l . T h u s , th e a v e r a g e re v e n u e c u rv e a p p e a r s to h a v e th e p r o p e r n e g a tiv e slope a s n e c e s s i t a t e d b y th e ab o v e c r i t e r i a . 47 2. E v a lu a tio n of E q u atio n (3. 3) If f(k q ./a P ? ) = In (k q j/a P j) (3.7) fo r re v e n u e p e r p o ten tial c u s to m e r, and to ta l rev en u e is ■ “ ( S t ) fcTR. = aPT- In p M r ) (3. 8) then a v e ra g e re v e n u e is , TR. a P c In . q. - a P ? In a P ? AR. » S 1 = — i U ----------1----------J- (3.9) k J k^j k*j w hich is a lso a d e c re a s in g fu n ctio n of , q., J and th e re b y ex h ib its the p r o p e r tie s of a d em an d function. C ontinuing, then, to each of the above re v e n u e functions, the follow ing sh ip m e n t c o st c o n s tr a in t functions will be applied: 1. L in e a r T r a n s p o rta tio n C o st F u n ctio n sk Q = r E (dij)(k'5j> j-1 (3. 10) 48 2. P o w e r T r a n s p o r ta tio n C o st F u n c tio n skQ= j= i 3. E x p o n e n tia l T r a n s p o r ta tio n C o st F u n c tio n SkQ = E [K ‘ K exP(- D 'dij>] <kqj> 4. L o g a r ith m ic T r a n s p o r ta tio n C o st F u n c tio n n sk Q = r E (ln V W w h e re , Q = to ta l c o m m o d ity k o u tp u t of the f ir m , a c o n s ta n t fo r the s u b n a tio n a l re g io n an d g iv e n n , Q = q., w h e re q. is q u a n tity so ld k “ / j k J k j j= l in j, i. e. , q u a n tity s h ip p e d f r o m i to j t h d . . = d is ta n c e f r o m the f i r m to j a r e a , ^ see ab o v e f o r sp e c ific m e a s u r e m e n t a s s u m p tio n s r = c o s t of tr a n s p o r tin g a u n it of p r o duct a u n it of d is ta n c e s = a v e r a g e t r a n s p o r t c o s t p e r u n it of p ro d u c t, a s s u m e d e te r m in e d by the s iz e of the m a r k e t a r e a f o r a ty p ic a l f i r m in the in d u s try , a n d is c o n sta n t _______a t l o n g - ru n e q u ilib r iu m . ___________ (3 .1 1 ) (3 .1 2 ) I I t ! i (3. 13) 49 Next, we w ill p ro c e e d to solve fo r q u antity sh ip p ed u sin g in d ep en d en t ly the two h y p o th e size d d em an d functions in t e r m s of each of the above c o st c o n s tra in t fu n ctio n s. The m a th e m a tic s w ill follow th a t of C h ap ter II. 3 .4 SOLUTIONS F O R THE SH O R T-R U N C O M M O D ITY -FLO W GRAVITY M OD EL 3 .4 .1 S h o rt-R u n Solution w ith P o w e r R evenue F u n c tio n and L in e a r T r a n s p o rta tio n C ost F u n ctio n B a s e d upon the fo reg o in g a ss u m p tio n s , the to ta l re v e n u e ! e a r n e d fr o m co m m o d ity k, w hich w ill be d e riv e d fr o m s a le s in a r e a j ! w hen c o n stra in e d by an e x p lic it s h ip m e n t-c o s t function, is m a x im iz e d : fo r the follow ing au g m en ted o b jectiv e fu n ctio n u sin g E q u atio n s (3. 1) and (3. 4): n r n w h e re k R ' = a £ P ] * f<k y * P - > - X - sk Q j= l L 3= 1 k = a L a g ra n g ia n (u n d eterm in ed ) m u ltip lie r. (3.14) T aking the f i r s t o r d e r condition fo r m a x im iz a tio n of E q u atio n (3. 14) w ith r e s p e c t to , q. (quantity shipped to d e stin a tio n j p e r u n it tim e), ^ J k w ill y ie ld a se t of sim u lta n eo u s e q u atio n s. T h e se eq u atio n s w hen so lv ed w ill p ro d u ce a r e s u lt s im ila r to E q u atio n (2.25) in C h ap ter II. T h e re fo re , fo r the m a th e m a tic a l m an ip u latio n , re v ie w E q u atio n s (2. 16) th ro u g h (2. 25) in C h ap ter II. T h e re s u lta n t so lu tio n g iv es the g follow ing equation: 8 Note th at the p o w er re v e n u e fu n ctio n p ro d u c e s the sam e f i r s t d e riv a tio n as the p o w e r u tility function, E q u atio n (2. 19), by su b stitu tin g , q. fo r 1 T ... A lso, a f i r s t o r d e r m a x im u m p o sitio n for all equations^.£p assufn^A to e x ist fro m th e h y p o th esized condition, although the seco n d o r d e r condition is not stated . 50 (3. 15) (j™ 1> 2, .»», n) 3. 4. 2 S h o rt-R u n S olution w ith P o w e r R ev en u e and P o w e r T r a n s p o r ta tio n C o st F u n c tio n s A gain, th is so lu tio n w ill give s i m i l a r r e s u l t s to E q u atio n (2. 31) of C h a p te r II, w ith m a th e m a tic a l m a n ip u la tio n p r e s e n te d in E q u a tio n s (2.26) th ro u g h {2. 31). F i r s t , h o w e v er, le t u s sp e c ify the j a u g m en ted function, and, then, the so lu tio n a s follow s: 1. A u g m e n ted F u n c tio n (3. 16) 2. S olution c (3 . 17) 51 3. 4. 3 S h o r t-R u n S o lu tio n w ith P o w e r R e v e n u e an d E x p o n e n tia l T r a n s p o r t a t i o n C o st F u n c tio n s I n th is s o lu tio n th e a n a ly s is of th e c o s t fu n c tio n is th e s a m e a s s p e c ifie d in C h a p te r II, w ith th e m a th e m a ti c a l s o lu tio n a s s e t f o r th in E q u a tio n s (2. 32) th ro u g h ( 2. 37) . T h u s, the a u g m e n te d fu n c tio n fo r th is c o m m o d ity flow s o lu tio n is: 1. A u g m e n te d F u n c tio n ❖ R = a' £ p j c f ( i ^ ) ■ | £ i K ■ k exp<- ffdij 3 j = l V V j = l x ( k V - sk Q (3. 18) 2. S o lu tio n c \ 1 /1 - b j^l - e x p ( - o ? d )J (3. 19) (j — 1, 2, . . . , n) 3„ 4. 4 S h o r t-R u n S o lu tio n w ith P o w e r R ev e n u e a n d L o g a r it h m ic C o st F u n c tio n s R e f e r r in g b a c k to S e c tio n 2. 7 of C h a p te r II g iv e s th e m a th e m a t i c a l d e r iv a tio n f o r th e fo llo w in g a u g m e n te d fu n c tio n : 1. A u g m e n te d F u n c tio n * n R j= l ; < £ ) ■ n M r £ < l n d . . ) ^ . ) - sk Q j= l (3. 20) 52 2. S o lu tio n n - m i / i - b (3 .2 1 ) (j — 1, 2, . . . i n) 3. 4. 5 S h o r t-R u n S o lu tio n w ith L o g a r ith m ic R ev en u e and C o n sta n t T r a n s p o r ta tio n C o st F u n c tio n s T h e fo u r p re v io u s s o lu tio n s d e a lt w ith a p o w e r re v e n u e fu n ctio n . T h e n ex t fo u r so lu tio n s w ill u se a lo g a r ith m ic re v e n u e fu n ctio n . A s in the p re v io u s so lu tio n s, the o n es w h ic h follow w ill be c o n s is te n t w ith the m a th e m a tic a l so lu tio n s found in C h a p te r II. L o o k in g b a c k to S e c tio n 2. 8 of C h a p te r II, w e fin d th e d e r i v a tio n s s i m i l a r in E q u atio n s (2. 44) th ro u g h (2 .4 9 ) fo r th e lo g a rith m ic u tility fu n c tio n . T h e r e f o r e , E q u a tio n (3. 14) is a p ro p o s and w e can g a s s u m e a lo g a r ith m ic re v e n u e fu n ctio n . (3. 22) ( j— l, 2, n) 3 . 4 . 6 S h o r t-R u n S o lu tio n w ith L o g a r ith m ic R ev en u e and P o w e r T r a n s p o r ta tio n C o st F u n c tio n s F o r th is so lu tio n , the m a th e m a tic s of E q u a tio n s (2. 50) th ro u g h (2. 53) a r e a p p r o p r ia te . T h e r e s u l t is 9 N ote th a t th e lo g a r ith m ic re v e n u e fu n ctio n p ro d u c e s th e s a m e f i r s t d e r iv a tiv e a s th e lo g a r ith m ic u tility fu n c tio n in E q u a tio n (2 .4 5 ) b y s u b s titu tin g ^q . f o r ^ T ... A lso , E q u a tio n (3. 14) is a s s u m e d a s th e a u g m e n te d o b je c tiv e fu n c tio n fo r th e r e m a in d e r of the s h o r t- r u n so lu tio n . 53 (j= l, 2, . n) (3 .2 3 ) 3. 4 . 7 S h o r t- R u n S o lu tio n w ith L o g a r it h m ic R e v e n u e a n d E x p o n e n tia l C o st F u n c tio n s A g a in the r e f e r e n c e is to C h a p te r II, a n d th is tim e to E q u a tio n s (2. 54) th ro u g h (2. 57). T he so lu tio n is -£ p j. 5=1 J ' |l - expf-ud^H (j — 1» •••» n) (3 .2 4 ) 3. 4 . 8 S h o r t- R u n S o lu tio n w ith L o g a r ith m ic R e v e n u e a n d L o g a r i t h m i c T r a n s p o r t a t i o n C o st F u n c tio n s 3. 5 A SSU M PT IO N S F O R TH E L O N G -R U N C O M M O D IT Y -F L O W G R A V IT Y M O D E L Now w e a r e r e a d y to tu r n o u r a tte n tio n to the lo n g - r u n c o m m o d ity m o d e l. A s d e fin e d p r e v io u s ly , th is m o d e l w ill u tiliz e th e p r o f it m a x im iz a t io n o b je c tiv e f o r th e f i r m . W e m a k e th e a d d i ti o n a l a s s u m p tio n of c o n s ta n t l o n g - r u n a v e r a g e c o s ts of p ro d u c tio n . T h e e s ta b lis h m e n t of p r i c e a n d o u tp u t in e a c h of th e v a r io u s s u b - a r e a s is d e te r m in e d b y th e p r o f it m a x im iz a tio n c r i t e r i a of m a r g in a l c o s t e q u al to m a r g in a l re v e n u e (M C = MR) o v e r a tim e i n te r v a l t ^ w h e r e t^ (lo n g run) > t j ( s h o r t ru n ). 54 In ad dition, the s a m e a s s u m p tio n s a s sp e c ifie d in S ectio n t i l 3 .3 s till apply to defining the su b n atio n al re g io n and a r e a s (j a r e a s ) a s w e ll a s th e i ^ shipping o rig in a tio n point. T h is in c lu d e s the m e a s u r in g of d is ta n c e b etw een the i ^ point a n d a r e a . H e re , as b e fo re , the w e ig h ted value of p o p u latio n w ill be a s s u m e d . The lo n g - r u n m o d el w ill m a x im iz e p ro f it a s s ta te d above. It follow s th a t if w hen an d th e n , tt= TR - TC k k k (3.26) n (3.27) n ^ T C = K E , q. + (tr a n s p o r ta tio n cost) j= l J (3 .28) n n k " = E P ' flk q : / a P “ ) - K £ q k j = 1 J k j j j — 1 J (3.29) w h e re - (tr a n s p o r ta tio n cost) k T R rr = p ro fit on co m m o d ity k to ta l re v e n u e e a r n e d f r o m c o m m o d ity k in su b n atio n al re g io n T C = to ta l c o s t of p ro d u c in g and tr a n s p o r tin g c o m m o d ity k w ith in su b n a tio n a l re g io n 55 I I F u r t h e r fo r a ll j a = f r a c tio n of the p e r s o n s in the a r e a w ho a r e i n te r e s t e d in buying the p ro d u c t, a s s u m e d the s a m e f o r a ll a r e a s , th a t is , a c o n s ta n t of p r o p o r tio n a lity P . = p o p u la tio n in a r e a j J c = a p o s itiv e c o n sta n t, a w e ig h tin g p a r a m e t e r k = c o m m o d ity k til jO . = q u a n tity of c o m m o d ity k so ld to the j 3 a r e a , q u a n tity so ld a s s u m e d e q u a l to q u a n tity sh ip p ed K - a v e r a g e p ro d u c tio n c o s t p e r u n it re v e n u e e a r n e d p e r p o te n tia l c u s to m e r in s u b a r e a j > # ) a p r f = to ta l re v e n u e e a r n e d in a r e a n = to ta l n u m b e r of sh ip p in g d e s tin a tio n a r e a s . , if m a x im u m p ro f it is w h e re MC = MR, th e n a. tt* a (. t r - . t c ) k _ = k k = 0 { 3 3 Q ) d. q. o, q. k J k Mj = 1, 2, n a n d w h e re ❖ T 7 7 - ! 7 7 k m a x . = k In d e riv in g the so lu tio n s f o r , q., w h ic h follow f r o m the above a s s u m p - tio n s, p r o f it w ill be m a x im iz e d by in clu d in g v a r io u s h y p o th e s iz e d 56 tr a n s p o r ta tio n cost v a ria b le s . S p ecifically , tr a n s p o rta tio n c o sts w ill be d e fin e d in four w ay s as sp ec ifie d in E q u atio n s (3. 10) th ro u g h * (3. 13). A lthough, in the p re v io u s d e riv a tio n s it w a s n e c e s s a r y to u tiliz e th e u n d eterm in ed L a g ra n g ia n m u ltip lie r , in th is c a se the tr a n s p o r ta tio n cost v a ria b le s a re in clu d ed in the o b jectiv e function. A lso, th e re v e n u e function w ill be sp e c ifie d in the sam e fa sh io n as E q u atio n (3. 2) and (3. 3). T ra n s p o rta tio n C o st F u n ctio n s A pplying E quations (3.29) and (3. 10), the follow ing objective fu n ctio n is fo rm ed Next, ta k in g the f ir s t d e riv a tiv e of rrand settin g it to z e ro y ie ld s a s e t of sim u lta n e o u s equations 3. 6 SOLUTIONS F O R TH E LO N G -R U N C O M M O D ITY -FLO W GRAVITY MODEL 3. 6. 1 L o n g -R u n Solution w ith P o w e r R evenue and L in e a r (3.31) S f < k V a P l> (3. 32) - K - rd . = 0 in 57 Now le t f(, q ./a P ? ) be d e fin e d a s E q u a tio n (3. 2) fo r a ll th e so lu tio n s J J w ith the p o w e r re v e n u e fu n ctio n , an d tak in g its f i r s t d e r iv a tiv e and su b stitu tin g into E q u a tio n (3. 32) p r o d u c e s ^ a P ? (b, qb _ 1 / a bP b c ) = K + r d . . (3 .3 3 ) J k J J ij C ontinuing to s im p lify an d so lv e fo r , q. g iv e s k J b1/1"b(aP?) k q, = ---------------- . j . , (j= 1. 2, n| (3 .34) K J (K + r d . .) ij 3.6 .2 L o n g -R u n S o lu tio n w ith P o w e r R ev e n u e and P o w e r T r a n s p o r ta tio n C o st F u n c tio n s In th is s e c tio n E q u a tio n s (3. 29) and (3. 11) w ill be f o r m e d a s follow s k n = a E P c f( q ./ a p " ) - K E kq. - r E (<£)(kq.) (3. 35) j= l 3 k J J j= l j= l 1J k 3 T a k in g the f i r s t d e r iv a tiv e of th is e q u a tio n p ro d u c e s the fo llow ing s e t of s im u lta n e o u s e q u a tio n s ^ S e e fo o tn o te 8 fo r th is f i r s t d e riv a tiv e . Now l e t f(, q ./a P 7 ) a n d its f i r s t d e riv a tiv e be d e fin e d a s in S e c tio n k J J 3. 6. 1; s u b s titu tin g th is in E q u a tio n (3. 36) g iv e s a P C(b ,q b _ 1 / a b P b c ) = K + r d “ (3. 37) J ^ J J ij S im p lify in g a n d so lv in g f o r , q. p r o d u c e s K J b 1 / 1 " b {aP °) , q. = — Y T \ K (j= l» 2, . „ „, n) (3. 38) k 3 (K+ r d . .) ° ij 3. 6. 3 L o n g - R u n S o lu tio n w ith P o w e r R e v e n u e an d E x p o n e n tia l T r a n s p o r t a t i o n C o st F u n c tio n s A s s u m e th e o b je c tiv e fu n c tio n a s fo llo w s f r o m E q u a tio n s (3. 29) an d (3. 12): k" = *£*3 ' Kj| k^j - 1 £ t1 ' exP^dij)] W (3‘ 391 3 59 The f i r s t d e riv a tiv e of E q u a tio n (3. 39) gives a s e t of sim u lta n e o u s eq u atio n s: V . 3f(k q i / a P ^> „ r. . , . 1 ------ ^ ----------- K - r j^l - e x p (-a d .^ )Js ■(3. 40) a. n „ 3f( q / a P ° ) -x~a— ~ a F n ----- ^ y ~ —------ K - r fl - e x p (-o d . )"]= 0 \ qn n V n L Now l e t f( q ./ a P f ) be s p e c ifie d th e s a m e as in S e c tio n 3. 6. 1 and K J J tak e th e f i r s t d e riv a tiv e w hich s u b s titu te d into E q u a tio n (3. 40) g iv e s a P j ( ^ E ^ = K + r [ l - expf-ort..)] (3 .41) S im plifying an d so lv in g fo r . q. p ro d u c e s ■ K J b ' / ' - V p 0) ^ [ k + r ( l - e x p ( - a l J..)|1 / 1 - b (3_ 42) (j — 1 1 3, . * o , n) 3. 6. 4 L o n g -R u n S o lution w ith P o w e r R evenue and L o g a rith m ic T r a n s p o r ta tio n C o st F u n c tio n s The m a x im iz in g e q u a tio n h e r e u tiliz e s E q u a tio n s (3. 29) and (3. 13) T h e f i r s t d e r iv a tiv e a ls o p r o d u c e s a s e t of s im u lta n e o u s e q u a tio n s V c = a P ? B k q l J a f ^ q j / a P j ) ■ w K - r In d ., = 0 ll a, T T 3 f ( , q / a P ) k t -,c k n n ' T q “ = a n STq K in k n K - r In d. = 0 m Now le t f f ^ q ./a P ? ) be s p e c ifie d as b e fo re in S e c tio n 3 .6 . s u b s titu te its f i r s t d e r iv a tiv e in to E q u a tio n (3 .4 4 ) g iv e s a P C( b q ^ " 1 / a b P ^ C) = K + r In d .. J k J J 13 A g a in sim p lify in g a n d so lv in g f o r ^ q . g iv e s b 1^ 1_b(aP.C) , q. — "" ^ 1 /k— r (j = ^9 • • • » 1 k J (K + r In d..) / ij (3 .44) 1 an d (3 .4 5 ) ) (3 .4 6 ) 61 3. 6. 5 L o n g -R u n S olution w ith L o g a rith m ic R evenue and L in e a r T r a n s p o r ta tio n C o st F u n c tio n s F o r th is so lu tio n , the d em an d fu n ctio n is lo g a rith m ic w ith an o b je ctiv e fu n c tio n the s a m e a s E q u a tio n (3. 31) and a f i r s t d e r iv a tiv e s e t of sim u lta n e o u s eq u atio n s lik e E q u atio n (3. 32)„ Now le ttin g £ f{ q . / a P . ) be d e fin ed as E q u a tio n (3. 3) fo r all the so lu tio n s u sin g a ^ J J lo g a rith m ic d e m an d function; su b stitu tin g its f i r s t d e riv a tiv e into e q u atio n (3. 32) g iv e s '^ a P ? ( 1 /. q.) = K + rd .. (3. 47) | J k V xj S im p lify in g and solving fo r , q. gives ■K J a P ? ~ j£“h - (j= 1» . . . , n) (3. 48) J ij 3. 6. 6 L o n g -R u n S olution w ith L o g a rith m ic R ev en u e and P o w e r T r a n s p o r ta tio n C o st F u n c tio n s T h is so lu tio n w ill u tiliz e E q u a tio n (3. 35) a n d E q u a tio n (3. 36). L e t fh q ./a P ? ) be d efin ed a s b e fo re in E q u a tio n (3. 3); s u b stitu tin g its J J f i r s t d e riv a tiv e into E q u a tio n (3. 36) g iv es a P ° ( l / q.) = K + r d" (3.49) J K J i j **See footnote 9 fo r th is f i r s t d e riv a tiv e . 62 Solving f o r ^q . y ie ld s J a P ? k q . -----------------------------(j= 1, 2, . . . , n) (3.50) k ] K + r d a. ij 3. 6. 7 L o n g -R u n S o lu tio n w ith L o g a r ith m ic R ev en u e a n d E x p o n en tial T r a n s p o r ta tio n C o st F u n c tio n s T h e e q u a tio n s to be u s e d fo r th is so lu tio n a r e E q u a tio n s (3.39) a n d (3. 40). A g ain le t f(, q ./a P ? ) be d e fin e d a s E q u a tio n (3. 3 3 J s u b s titu tin g its f i r s t d e riv a tiv e into E q u a tio n (3. 40) p ro d u c e s a P j ( l / k q.) = K + r [l - e x p f-a d ..)] (3. 51) F in a lly , so lv in g fo r , q. g iv e s K J a P ? ~ i- i — — — (j = l, 2, . . . , n) K + r 1 - exp(-Q'd^j) (3.52) 3, 6. 8 L o n g -R u n S o lution w ith L o g a r ith m ic R ev en u e a n d L o g a r it h m ic T r a n s p o r ta tio n C o st F u n c tio n s T h is is the l a s t d e r iv a tio n to be d e v e lo p e d in th e lo n g -ru n c o m m o d ity m o d e l. So fo r th is s o lu tio n E q u a tio n s (3. 43) and (3. 44) a r e u tiliz e d . A s b e fo re , le t f(. q ./a P ? ) be s p e c ifie d a s E q u a tio n (3.3) K J J a n d th en s u b s titu te its f i r s t d e r iv a tio n into E q u a tio n (3 .4 4 ): a P ? ( l / , q.) = K + r In d .. (3.53) j K J N ext, solve fo r . q.: k j a P C q = -----------J (3.54) J K + r Ind.. l J 3 - 7 SUMMARY O F C O M M O D ITY -FLO W GRAVITY M ODEL In th is c h a p te r an a tte m p t w as m ad e to develop a co m m o d ity - flow m o d el b a se d upon eco n o m ic a ssu m p tio n s about the fir m . T h e se j a ssu m p tio n s w e re : (1) m a x im iz a tio n of re v e n u e in the s h o rt ru n , and (2) m a x im iz a tio n of p ro fit in the long ru n . T he en su in g d e riv a tio n s w e re b a se d upon two ty p es of a v e ra g e rev en u e functions: (1) a p o w er ! re v e n u e function, and (2) a lo g a rith m ic re v e n u e function. T he p r o b le m th at w as ta ck led w as to o p tim ize the sh ip m e n t of c o m m o d itie s to ; a r e a s to a c c o m p lish e ith e r the s h o r t- r u n o r lo ng-run ob jectiv e fo r the f ir m , w hen faced w ith s e v e r a l p o ssib le a lte rn a tiv e tr a n s p o r ta tio n - c o s t functions. H e re , we w e re looking fo r , q., the quantity of c o m - th ^ ' • m o d ity k sh ip p ed (also sold) to the j a r e a . T h is w as an e ffo rt to ; develop m a th e m a tic a lly a lo g ical definition of sh ip m e n t b e h a v io r fo r | the fir m , w hile in C h ap ter II, it w as in d iv id u al tr a v e l b e h a v io r w h ich i iwe w e re try in g to explain. ; i T he r e s u lts (quantity shipped) for th e s h o rt ru n a re id e n tic a l j ; m a th e m a tic a lly to those dev elo p ed (trip s taken) in C h ap ter II for th e ! in dividual, w ith , q. being s u b stitu te d for , T ... H o w ev er, the re s u lts k J k ij ; fo r the long ru n w e re d iffe re n t, and involve an additive co n stan t t e r m ! in the d en o m in a to r as w e ll as the d ista n c e v a ria b le . A lthough the m a th e m a tic a l a p p ro a ch w as e s s e n tia lly the s a m e , it w as not n e c e s s a r y to u se L a g ra n g ia n m u ltip lie r s in the lo n g -ru n c a s e . In o r d e r to f a c ili ta te the finding of the q u antity sh ip m e n t so lu tio n s, ta b le s w ith the a p p ro p ria te eq u ations a r e given in T a b le s 3. 1 and 3. 2. T ab le 3. 1 S h o rt-R u n C o m m o d ity -F lo w G ra v ity M odel S olutions 12 A v e ra g e R evenue „ F u n c tio n T r a n s p o rta tio n C o st F u n c tio n P o w e r R evenue E q u a tio n (3. 2) L o g a rith m ic R evenue E q u atio n (3. 3) L in e a r C o st E q u a tio n (3. 10) E q u a tio n (3. 15) k qj = , £ ( p f / d b./ 1 - b ') .J" 1\ 3 ij / 1 /1 -b E q u a tio n (3. 22) k qj P o w e r C o st E q u a tio n (3. 11) E q u a tio n {3. 17) E q u a tio n (3. 23) - m p c ( t ) a / 1-b 1 2 N ote: F o r a ll c a s e s (j= l, 2, . „ „, n). O' T ab le 3. 1 S h o rt-R u n C o m m o d ity -F lo w G ra v ity M odel S olutions ( Continued) ^ s ^ v e r a g e R evenue m \ F u n c tio n T r a n s - ^ v p o rta tio n C o st F u n c tio n L o g a rith m ic R evenue E q u a tio n (3.3) P o w e r R evenue E q u a tio n (3. 2) E q u a tio n (3. 24) E q u a tio n (3. 19) E x p o n e n tia l C o st E q u a tio n (3. 12) k^i = E q u a tio n (3, 25) E q u a tio n (3.21) 1/1— b L o g a rith m ic C o st E q u a tio n (3. 13) O' U 1 T a b le 3. 2 L o n g -R u n C o m m o d ity -F lo w G ra v ity M odel Solutions ''''■'^A verage R evenue F u n ctio n T r a n s - p o rta tio n C o st F u n c tio n P o w e r R evenue E q u a tio n (3, 2) L o g a rith m ic R evenue E q u a tio n (3. 3) E q u atio n (3. 34) E q u a tio n (3. 48) L in e a r C o st b ^ - h u p ? ) a P ? a - J E q u a tio n (3. 10) kq- > (K + r d . . ) 1/1_b ij K J K + r d . . ij E q u a tio n (3. 38) E q u a tio n (3. 50) P o w e r C ost E q u a tio n (3. 11) b l / U b ( a P |) a P ? a - J k ’ j (K+ rd ? .)1 /1 - b k j K + r d?. Ij N ote: F o r a ll c a s e s {j=l, 2, n). O' T able 3. 2 L o n g -R u n C o m m o d ity -F lo w G ra v ity M odel Solution (C ontinued) ' ^ v >A v e ra g e R evenue _ F u n c tio n T r a n s - ^ ^ p o rta tio n C o st F u n c tio n N w P o w e r R evenue E q u atio n (3. 2) L o g a rith m ic R evenue E q u atio n (3. 3) E x p o n en tial C ost E q u atio n (3. 12) E q u a tio n (3. 42) ^1 / i-b (ap C ) E q u atio n (3. 52) a P ? q " 3 ^ j K + r ( l - exp(-o'd..) ^ ^ ij ^ ^ K + r 1 - exp(-o'd^) L o g a rith m ic C o st E q u atio n (3. 13) E q u a tio n (3. 46) . 1 /1 -b , c b ( a P j ) its < i j xl / b - l J (K + r In d . .) ij E q u atio n (3. 54) a P C k qj = J K J K + r In d . . ij O' -j C H A PTER IV TH E PR O B A BILITY GRAVITY M O D EL 4 - 1 INTRODUCTION In the p re v io u s c h a p te rs , the d e riv a tio n of the g ra v ity e q u a tions has b e en b a se d upon o p tim iz a tio n of d e te r m in is tic u tility , p ro fit and re v e n u e m o d e ls. A d ifferen t a p p ro a c h w ill b e tak en in th is c h a p te r . E s s e n tia lly , the g ra v ity m o d e l w ill be c o n stru c te d u tilizin g sto c h a s tic a ssu m p tio n s and p ro b a b ility th e o ry . M o re sp e c ific a lly , a ssu m p tio n s about tr a v e l d ista n c e s w ill be fo rm u la te d in t e r m s of sp ecific p ro b a b ility -d e n sity fu n ctio n s. Upon co m p letio n of th is effo rt, an a tte m p t w ill b e m ad e to re d e riv e s e v e r a l g ra v ity m o d els b a s e d on p ro b a b ility c o n s tr u c ts . T h is p ro b a b ility d e riv a tio n w ill be ap p lied to two e x istin g m o d els: (1) L ow ry M ig ra tio n M odel; (2) H an sen A c c e s s i b ility M odel. 4. 2 ASSUMPTIONS O F PROBABILISTIC GRAVITY M O D ELS H e re the fa c to rs u n d erly in g the d ev elo p m en t of the p r o b a b ilis tic a s s u m p tio n s . T he f i r s t is the " d e s ir e p r o b a b ility ," w hich is defined as the p ro b a b ility th at the in dividual w ants to go to a r e a j. T he fo rc e w hich pulls o r d riv e s the in dividual to m ak e a t r i p fro m a r e a i to a r e a j is b a se d upon so m e a ttrib u te p r o c e s s e d by j b ut not av a ila b le in i. T h e se a ttrib u te s m ig h t take m an y fo rm s , fo r e x a m ple, finding so m e in d iv id u al to in te r a c t w ith, e m p lo y m en t p o ten tial, T h e d ev elo p m en t of a p ro b a b ilis tic g ra v ity m o d el a s s u m e s a given t im e in te rv a l, and the r e s u lts w ill be d e stin a tio n s p e r unit tim e , although not s p e c ific a lly sta te d . 68 69 shopping p o te n tia l, o r r e c r e a t i o n p o te n tia l. If w e w e r e d is c u s s in g c o m m o d ity s h ip m e n ts , the " d e s i r e p ro b a b ility " m ig h t c o n s is t of p o ten tial m a r k e t s , th a t is , p ro b a b ility of d e m a n d . So le t the s im p le event be P r (j) = " d e s i r e p ro b a b ility " th a t in d iv id u a l (o r p ro d u c e r) w a n ts to t r a v e l (o r ship) to a r e a j T h e sec o n d p ro b a b ility a s s u m p tio n is th a t of " a c c e p ta b le d is ta n c e " f o r the in d iv id u a ls (o r p r o d u c e r s ) to t r a v e l (o r ship) the d ista n ce to a r e a j. F o r th is p ro b a b ility , w e w a n t to s p e c ify a c o n d i tional c a s e and a s s u m e th e e v e n ts to be in d e p en d e n t, b u t m u tu a lly in c lu s iv e . T h e r e f o r e , l e t the c o n d itio n a l p ro b a b ility be P r ( j s |j ) = " a c c e p ta b le d is ta n c e p ro b a b ility " th a t in d iv id u a l (o r p ro d u c e r) is w illin g to t r a v e l (o r ship) th e d is ta n c e to a r e a j g iv e n th a t he w a n ts to go to So f r o m th e above two a s s u m p tio n s , the b a s is f o r the fin a l c o n d itio n is f o r m e d ; th a t is , w h a t is the p ro b a b ility th a t th e in d iv id u a l d o es go to a r e a j ? In o r d e r to f o r m th is p ro b a b ility , it fo llo w s th a t a c o m pound e v e n t of the " d e s ir a b le and a c c e p ta b le d is ta n c e p r o b a b ilitie s " is n e c e s s a r y . T h u s, P r ( j f ) j s | j) = P r ( j) - P ( js | j) = p ro b a b ility th a t in d iv id u a l (o r p ro d u c e r) a c tu a lly t r a v e l s (o r sh ip s) to a r e a s j. F in a lly , g iv e n the a s s u m p tio n th a t the in d iv id u a l w ill m a k e one trip w ith in the re g io n , the p r o b a b ility th a t th e in d iv id u a l w ill go to a sp e c ifie d j*'*1 a r e a w ith in th e re g io n c a n be g iv e n by th e follow ing: 2 I n h e r e n t in su c h a p ro b a b ility is s o m e g iv e n m o d e of t r a n s p o rta tio n , fo r e x a m p le , foot, b ic y c le , a u to m o b ile , tr a in , a ir p la n e o r w h a te v e r e ls e . It m ig h t b e r e a s o n a b le to h y p o th e siz e th a t th e f o r m of the p r o b a b ility d e n sity fu n c tio n c o u ld w e ll d ep en d upon th e p a r t i c u l a r m ode o f tr a n s p o r t a t i o n a s s u m e d . 70 Pr(j) • Pr(je|j) £ P r ( j ) • P r ( js |j) j-1 Thus f a r a ssu m p tio n s of a g e n e ra liz e d p ro b a b ilis tic a p p ro a c h to the g ra v ity m o d el have b een la id down; h o w ev er, if ap p licab le m o d e ls w e re to be developed, th en w ell defined sp e c ific a tio n s a re n e c e s s a r y . It is to this ta s k th at the d is c u s s io n now tu r n s . j 4. 3 D E V E L O PM E N T O F PR O B A BILISTIC GRAVITY MODELS j To each of the above two p ro b a b ility a s s u m p tio n s , sp ecific fo rm u la tio n s w ill be given. The seco n d a ss u m p tio n (a ccep tab le d istan ce) w ill be dev elo p ed a s su b se c tio n s, fo r th is p ro b a b ility can be d efin ed in s e v e r a l d ifferen t w ay s. By th is is m e an t, it is p o s sib le to find s e v e r a l p ro b a b ility d e n sity fu n ctio n s w hich m e e t a g e n e ra lly a c c e p te d c r ite r io n fo r tr a v e l d ista n ce . T h is c r ite r io n is sim p ly s ta te d th a t d ista n c e o v e r sp ace fo r m s an in v e rs e re la tio n s h ip to d e s ir e to tra v e l a d ista n ce w hen s e t w ith in bounds (th at is , bounded 3 in an a c tu a l and m a th e m a tic a l se n se ). A lthough s e v e r a l d istrib u tio n s m ig h t be in v e stig ate d , th is w o rk w ill only ev alu a te two: (1) the P a r e to p ro b a b ility -d e n s ity function and (2) th e n a tu ra l ex p onential p ro b a b ility -d e n s ity function. B efo re continuing, le t us sp ec ify the " d e s ir e p ro b a b ility . " T h is d efin itio n w ill hold fo r both p ro b a b ility d e n sity fu n ctio n s of tr a v e l d ista n c e . S p ecifically , then, p o p u latio n siz e h a s b een u s e d by C a r r o th e r s , S tew art, Zipf, and s e v e r a l o th e r s in d e te rm in in g the d e s ir e of an individual to tr a v e l f r o m h is own (o rig in a re a ) to a n o th e r a r e a . F ollow ing th is lead , it is p la u sib le to a ssu m e th a t the " d e s ir e 3 Is a rd , W. , M ethods of R egional A n a ly sis; An In tro d u ctio n to R eg io n al S cience (New Y ork: W iley, I960), pp. 496-499. 71 p ro b a b ility " could be s ta te d a s P . / S P . . T h a t is , the d e s ir e p r o b a th ^ ^ ^ b ility to go to the j a r e a is eq u al to th e r a tio of its p o p u latio n r a i s e d to th e c p o w e r to the s u m of p o p u latio n s r a i s e d to the c p o w er s u m m e d o v e r a ll a r e a s . On the o th e r hand, the d e s ir e to s ta y w ith in o n e 's own a r e a h a s a r e ta r d in g o r n e g ativ e e ffe c t w hich c o u n te r b a l- th a n c e s the d e s ir e to go to j a r e a . The c o u n te rv a ilin g fo r c e h e r e w ill be p o p u latio n in the in d iv id u a l's own a r e a , th at is P ., and th e d e s ir e n 1 ,c s ta b - p ro b a b ility to r e m a in in o n e 's h o m e a r e a is /,Z^ P^ . H aving e lis h e d the " d e s ir e p ro b a b ility , " it is p o s s ib le to tu r n to developing the " a c c e p ta b le d is ta n c e p ro b a b ility " an d fo r m u la te p ro b a b ilis tic g ra v ity m o d e ls . i j 4. 3. 1 A P r o b a b ilis tic G ra v ity M odel w ith a P a r e t o P r o b a b ility D e n sity F u n c tio n of T r a v e l D is ta n c e s 4 G iven a P a r e to d is trib u tio n , a p ro b a b ility d e n sity fu n c tio n oi 5 tr a v e l d is ta n c e s can be w r itte n a s r A r f(x ) a f o r x> A x - 0 f o r x < A (4 .1 ) w h e re x = d is ta n c e the d e c is io n m a k e r is w illin g to tr a v e l, a s s u m e d to be a continuous ra n d o m v a ria b le A = a p o s itiv e co n stan t^ r = a p a r a m e t e r on d is ta n c e and is <1 by a ssu m p tio n ^ 4 T in tn e r , G . , M a th e m a tic s and S ta tis tic s fo r E c o n o m is ts (New Y ork: W iley, 196o)^ p. 37. ’ 5 P a r z e n , E . , M o d ern P r o b a b ility T h e o ry and Its A p p licatio n (New Y ork: W iley, I960), p. 211. £ One d ifficu lty w ith u sin g a P a r e to d is trib u tio n is th a t the r a n d o m v a ria b le d is ta n c e can n o t be z e r o , sin ce A m u s t be a p o sitiv e v alu e g r e a t e r th an z e ro ; th e r e f o r e , it is im p o s s ib le to h av e a non-zero p r o b a b ility w h e re the d is ta n c e i s z e ro , th a t is , a t the o rig in point. •7 T he p a r a m e t e r r is a s s u m e d to be im p lic itly d e riv e d f r o m c o st, tim e , and p h y s ic a l-p s y c h o lo g ic a l a s p e c ts a s s o c ia te d w ith tr a v e l o v e r d is ta n c e . 72 X r F(x) - J f(x) d x = J dx (4 .2 ) and F(®) = - A r x r = 1 (4. 3) Now l e t d^j be the d is ta n c e b e tw e e n o r ig in i and d e s tin a tio n j. A lso , l e t x b e th e m a x im u m d i s t a n c e p e o p le a r e w illin g to tr a v e l. In a d d itio n , a s s u m e the ra n d o m v a r ia b le x c o n f o r m s to th e above P a r e t o d is tr i b u t i o n . Then d.. P r ( x < d .,) - J ' ^ f(x )d x = - A r - r A x d .. ij A - r = -A r d .. + 1 ij (4 0 4) P ( x > d ) = 1 - P ( x < d ) = — 1J J d7. ij (4. 5) H a v in g d e r iv e d the " a c c e p t a b le d is ta n c e p ro b a b ility , " an d c o u p lin g it w ith th e " d e s ire p r o b a b ility , " the P a r e t o p r o b a b ility - g r a v ity m o d e l c a n be d ev elo p ed . It is d y j (4. 7) (j — 1 , 2, < g «j n) w h e re is the to ta l n u m b e r of t r i p s p e r u n it tim e m a d e by in d iv id u al k r e s id in g in a r e a i. T h is n u m b e r is d e te rm in e d o u tsid e the m o d e l and can a s a f i r s t a p p ro x im a tio n be a s s u m e d p ro p o rtio n a l to in c o m e . It is in te r e s tin g to note th a t th is fo r m u la tio n is e s s e n tia lly the s a m e a s E q u a tio n s (2. 25), (2. 31), and (2. 53) and is , th e re b y , c o n s is te n t w ith the h y p o th e se s u n d e rly in g th o se r e s u lts . 4. 3. 2 A P r o b a b ilis tic G ra v ity M odel w ith a N a tu ra l E x p o n en tial P r o b a b ility D e n sity F u n c tio n of T r a v e l D ista n c e In th is s e c tio n w e w ill a s s u m e the sec o n d p ro b a b ility d e n sity fu n ctio n fo r a c c e p ta b le t r a v e l d is ta n c e . F o llow ing th e outline given fo r the above P a r e to d is trib u tio n , w e s h a ll p ro c e e d to d evelop a m o d e l c o n fo rm in g to a n a tu r a l ex p o n en tial function. T h a t is, f(x) = r e x p (-rx ) fo r x>0 (4. 8) = 0 fo r x=0 w h e re the t e r m s a r e as p re v io u s ly sp e c ifie d , T .. = k ij . T . k l i ( p c / d r .) j= i j ij 74 L e t f(x) b e a d e n s ity f o r one v a r i a t e , a n d the c u m u la tiv e d is tr ib u tio n is x F(x) = / f ( x ) d x = / r e x p ( - r x ) d x (4. 9) F(°°) ~ - e x p (-rx ) = 1 (4 .1 0 ) N ext, l e t d ^ be th e d is ta n c e b e tw e e n o r ig in i a n d d e s tin a tio n j, an d x be th e m a x im u m d is ta n c e p e o p le a r e w illin g to t r a v e l . A lso , a s s u m e v a r ia b le x c o n f o r m s to th e ab o v e d is tr ib u tio n . d . . •ij / X J f(x)dx = - e x p (-rx ) ij (4 .1 1 ) - - e x p ( - r d . .) + 1 P ( x > d . .) = 1 - P ( x < d ..) = e x p ( - r d . .) (4. 12) v ij 1 ij ^ ij H aving d e r iv e d th e " a c c e p ta b le d is ta n c e p ro b a b ility , " it is now p o s s ib le to jo in it w ith th e " d e s i r e p r o b a b ility " an d p ro d u c e the n a t u r a l e x p o n e n tia l- p r o b a b ility g r a v ity m o d e l. T h en , exp ( r d.j) (4. 14) P . _ 2 (j=l, 2, . . ., n) w h e re is as p re v io u s ly defined. 4 .4 RECA STIN G THE LOW RY AND HANSEN MODELS. INTO A PR O B A B ILISTIC FO RM A T In re d e fin in g s p a tia l m o d e ls a ro u n d a p ro b a b ility concept, the e s s e n tia l change, fr o m o th e rs who h a v e ap p ro ach ed th e se sp a tia l in te ra c tio n m o d e ls fr o m p ro b a b ilis tic th e o rie s, is the conceptual d ev elo p m en t of the "a c c e p ta b le d is ta n c e p ro b ab ility , " T his b ro u g h t fo rth tra v e l d ista n ce not ju s t as a v a ria b le , but as a s to c h a s tic v a r i a b le. On the o th e r hand, fo rc e v a r ia b le s of the p u s h -p u ll th e o ry , as R o g e rs c a lls it, a re g e n e r a lly of a s im ila r fo rm as th o se fo rc e p r o - g b a b ility v a ria b le s w hich have b een a lr e a d y presented. H ow ever, w ith of U rb an and R eg io n al D ev elo p m en t, U n iv e rsity of C a lifo rn ia , 1965). 8 R o g e rs, A n d rei, A n aly sis of In te rre g io n a l M ig ra tio n in C alifo rn ia (B e rk e le y : R e p o rt on M ig ra tio n p re p a re d fo r the In stitu te 76 the d e v e lo p m e n t of the fo reg o in g s e c tio n on p ro b a b ility c o n c e p ts, it s e e m s r e a s o n a b l e to p ro p o se the r e c a s tin g of a lr e a d y s p e c ifie d s p a tia l i n t e r a c t i o n m o d e ls in to su ch a p r o b a b ility f o r m a t. T he r e c a s tin g w ill u tiliz e tw o a lr e a d y a c c e p te d m o d e ls . T he f i r s t of th e s e m o d e ls to b e r e c a s t w ill be th e L o w ry m ig r a tio n m o d e l. U pon r e - j f o r m u la tin g th e L o w r y m o d el, a se c o n d m o d e l w ill be r e c a s t; fo r th is : i c a s e , th e H a n s e n a c c e s s ib ility m o d e l w ill b e u se d . | 4. 4. 1 T h e L o w r y M ig ra tio n M odel I ! i j B a s ic a l ly , the L o w ry m ig r a tio n m o d e l a s it e x is ts fo llo w s the j j tr a d itio n a l g r a v ity m odel. B u t a s L o w r y s ta t e s , "T h e m o d e l (L ow ry's) e m p lo y e d is c l o s e s t in s p ir it to (the D utch d e m o g ra p h e r) 9 S o m e r m e i j e r 's . " In his (L o w ry 's) m o d e l th e d ire c tio n a l flow s b e tw e e n e a c h p a i r of a r e a s , i and j, a r e a m e a s u r e m e n t of e c o n o m ic o p p o rtu n ity a t i v e r s u s j, s iz e of a re a s , an d in te rv e n in g d is ta n c e . T he m o d e l u s e s u n e m p lo y m e n t r a t e s in th e a r e a s a s w e ll a s m a n u fa c tu r in g w ag e r a t e s a s th e v a ria b le s fo r e c o n o m ic o p p o rtu n ity . F o r the s iz e v a r ia b le of o r ig i n s and d e stin a tio n s , n o n a g r ic u ltu r a l la b o r f o r c e s a r e c a te g o r iz e d in to c iv ilia n an d m ilit a r y g ro u p in g s . T he re m a in in g v a r ia b le d is ta n c e is m e a s u r e d by a i r l i n e m i l e s . ^ F u rth e r, th e tim e in te r v a l, w h ile g iv en , c o v e rs a p e r io d g r e a t e r th an one day and g e n e r a lly a p p lie s to a m onth, a y e a r o r s e v e r a l y e a r s . T h e m o d e l h a s m a d e, e s s e n tia lly , r e a s o n a b le a s s u m p tio n s 11 about th e d y n a m ic conditions c£ m ig r a tio n . T h e s e a r e as fo llo w s: g L o w r y , I r a , M ig ra tio n and M e tr o p o lita n G row th: Tw o A n a ly tic a l M o d e ls (L os A n g eles: In s titu te of G o v e rn m e n t a n d P u b lic A tr a ir s , U n iv e r s ity of C a lifo rn ia , 1966), p. 11. ^ U s e o f a ir lin e d is ta n c e is a c c e p ta b le in th is m o d e l a s L o w ry is w o rk in g w ith SM SA 's ( s ta n d a r d m e tr o p o lita n s ta t is tic a l a r e a s ) ; h o w e v e r, if th e m o d e l w e re to be u s e d f o r t r a v e l b e h a v io r on s m a l l e r c o n tig u o u s a r e a s , th e n a n o th e r m e a s u r e of d is ta n c e (su c h a s fr e e w a y m ile s ) w o u ld h a v e to be found. ^ L o w r y , op. cit. , p. 13. 77 1. P e o p le m ig r a te in s e a r c h of jo b s fr o m lo w -w ag e to h ig h -w ag e a r e a s . 2. P e o p le m ig r a te fro m a r e a s of su rp lu s la b o r to those w ith la b o r sh o rta g e s. 3. The m ig r a n ts , o v er tim e , w ill a ffe ct the la b o r m a r k e t of the re c e iv in g a re a , and as its la b o r supply is au gm ented, its re la tiv e a ttr a c tiv e n e s s is d im in ish e d , o r vice v e rs a . 4. G iven id e n tic al co efficien ts fo r h o u rly w age and un em p lo y m en t, the e q u ilib riu m condition is one in w h ich the h o u rly w age in i and j a r e eq u al and u n e m p lo y m en t in i and j a re a ls o equal. 5. T h e re still re m a in s a ran d o m exchange b etw een i and j even u n d e r conditions of e q u ilib riu m a s s ta te d above, and the volum e of the ra n d o m v a ria b le depends upon size of i and j as w e ll as d is ta n c e betw een i and j, but w h o se n et e ffe ct on p o pulation d is trib u tio n is nil. 6. A lso, the in te rc h a n g e betw een each p a ir of a r e a s is independent of th at betw een e ac h o th e r p a ir of a r e a s . T h is l a s t a ss u m p tio n allow s the m o d el to be a p p lied in le s s th an a c lo se d s y s te m of a r e a s , th at is, "we can d eal w ith the c r o s s - f lo w s of m ig ra tio n am ong n a r e a s even though so m e o r all of th e se a r e a s a r e 12 a lso en g ag ed in in te rc h a n g e of population w ith m o th e r a r e a s . " A lso, L o w ry su g g e sts th at the to ta l am ount of m ig r a tio n fr o m and to e ac h a r e a in the s y s te m w ould in c r e a s e as additio n al o rig in s and d e stin a tio n s a re included. 12 L o w ry , op. cit. , pp. 13-14. 78 L o w ry g o es on to p o in t out th a t, b e c a u s e . . . th e m o d e l is d e sig n e d fo r u s e w ith c r o s s - s e c tio n a l d a ta a g g re g a te d o v e r a f i v e - y e a r p e rio d , the r e la tio n s h ip s b e tw e e n m ig r a tio n an d th e in d e p e n d e n t v a r ia b le s c an n o t be i n t e r p r e t e d s t r i c t l y in t e r m s of u n id ir e c tio n a l c a u s a tio n . If the c o n d itio n of la b o r m a r k e t s a t o r ig in an d d e s tin a tio n 'c a u s e s ' the flow of m ig r a tio n , m ig r a tio n in t u r n a l t e r s l a b o r - m a r k e t c o n d itio n s. T he m o d e l is in th is s e n s e i n c o m p le te ly sp ec ifie d ; b u t th e u n s p e c ifie d fe e d b a c k r e l a t i o n sh ip s sh o u ld n o t b e of g r e a t im p o r ta n c e , sin c e w e a r e d e a lin g w ith r e la tiv e ly s m a ll flow s of m ig r a tio n b e tw e e n p a i r s of r e la tiv e ly la r g e la b o r m a r k e ts . The L o w r y m o d e l, th en , ta k e s on the follow ing f o r m w h en the i n o n a g r ic u ltu r a l la b o r fo r c e is d iv id e d b e tw e e n c iv ilia n an d m ilita r y : w h e re M .. = K j? \ / w . A C? c Ta . A* (4. 15) (i, j=l, 2, n) M.. = ij U., U = i J W o w . = 1 J c., c. = 1 J A .., A. = ij J D .. = ij n u m b e r of m ig r a n ts from a r e a i to j u n e m p lo y m e n t a s a p e r c e n ta g e of the c iv ilia n n o n a g r ic u ltu r a l la b o r f o r c e at i a n d j, r e s p e c tiv e ly h o u r ly m a n u fa c tu r in g w ag e, in d o lla r s , a t i a n d j, r e s p e c tiv e ly n u m b e r of p e r s o n s in the c iv ilia n n o n a r g r ic u ltu r a l la b o r f o r c e at i and j, r e s p e c tiv e ly n u m b e r of p e r s o n s in the m i l i t a r y n o n a g r ic u ltu r a l la b o r f o r c e a t i and j, r e s p e c tiv e ly a ir lin e d is ta n c e f r o m i to j, in m ile s ; f o r i=j, a s s u m e d is ta n c e d e te r m in e d a s in C h a p te r II K = a c o n s ta n t of p r o p o r tio n a lity ^ L o w r y , op. c it. , p. 16. 79 4. 4. 2 R e f o rm in g th e L o w r y M ig r a tio n M o d el H e re a n a tte m p t w ill b e m a d e to r e f o r m the L o w r y m o d e l, E q u a tio n (4. 15), in to th e p r o b a b ility d e r iv a tio n a s c o n c e iv e d in S e c tio n 4.3. T h is w ill give a p r o b a b ilis tic L o w r y m o d e l, a s s u m in g th a t C an d A a r e c o n s o lid a te d a n d c a lle d L . w h e r e B is d e fin e d a s A in E q u a tio n (4. 1). N ex t th e p r o b a b ility th a t a w o r k e r w ill go to j if h e m o v e s f r o m i is If the d e s i r e p r o b a b ility f o r le a v in g i is T he d e s i r e p r o b a b ility is (4. 16) T h e a c c e p ta b le d is ta n c e p r o b a b ility is <BX/D * ) ij (4 .1 7 ) Ui Wi 6 \ "n “ n 7 1 , £ , u ? x w r s / 1 = 1 i 1=1 l / (4. 19) 80 T hen, the p ro b a b ility th a t a w o r k e r w ill le a v e i is u? w :6 1 l n n c E u “ E w : 6 'i = 1 1 i= 1 1 - 2 ^ / u“ \ / w ;6 s ' ■ (4.20) n J I n -6 a lso , le t g L . = to ta l la b o r fo rc e r a i s e d to s p o w e r i i Then, th e p ro d u c t of E q u a tio n s (4. 18), (4. 20) an d L . sh o u ld give the ! n u m b e r m ig r a tin g f r o m i to j. M . . = ij u?wr6u7p wyL?i> i i J J i J J D x ij (4.21) (4) —r r \ (i, j=l» 2, . . . , n) W h ere th e d e m o n in a to r w ith in the f i r s t s e t o f b r a c k e ts is a c o n sta n t. So, w e have d e riv e d the L o w ry m o d el. We s e e th a t it a s s u m e s a P a r e to a c c e p ta b le d is ta n c e p ro b a b ility . 4. 4. 3 T he H a n se n A c c e s s ib ility M odel H a n se n lik e o th e r s f e lt the n e c e s s ity fo r q u a n tita tiv e in fo r m a tio n upon w h ich a n u r b a n tr a n s p o r ta tio n p lanning p r o c e s s co u ld be 81 d e v e lo p e d . T he e s s e n c e o f s u c h a to o l w o u ld e n a b le f o r e c a s t s of c h a n g e s in t r a v e l d e m a n d s w h ic h r e s u l t e d f r o m p o s s ib le ( a n tic ip a te d o r p ro p o s e d ) c h a n g e s in th e la n d u s e p a t t e r n s an d tr a n s p o r t a t i o n s y s te m s of th e s e a r e a s . E s s e n t i a l l y , " th e h e a r t of s u c h a p r o c e s s is a p r o c e d u r e c a p a b le of s y n th e s iz in g z o n e - to - z o n e m o v e m e n ts f o r 14 a lte r n a t iv e c o n fig u ra tio n s of la n d u s e a n d t r a n s p o r t a t i o n f a c i l i t i e s . ,r F o r h is m o d e l, H a n s e n h a s c h o s e n th e g r a v ity - la w f o r m a t w ith a g iv e n tim e in te r v a l. T h is g e n e r a l f o r m u la tio n H a n se n s p e c if ie s a s A . / d 1 ?. T . . = P . rr (4 .2 2 ) 1J 1 A ./d ? . + A ./d ? . + . . . + A /d ? w h e re i li j ij n m T ^ = n u m b e r o f^ trip s p ro d u c e d a t zone i a t t r a c t e d by zone - to ta l t r i p s p ro d u c e d by zo n e i A. = to ta l a t t r a c t i o n of zo n e j 3 d . . = m e a s u r e m e n t of the s e p a r a tio n b e tw e e n ^ z o n es i a n d j n o r m a lly e x p r e s s e d in t e r m s of d riv in g tim e b = e m p i r i c a l l y d e te r m in e d e x p o n e n t T h is m o d e l is a lm o s t e x a c tly th e s a m e a s E q u a tio n (4. 7). If w e a s s u m e c = l, th e n th e y a r e e x a c tly id e n tic a l. ^ H a n s e n , W a lte r G . , " E v a lu a tio n of G ra v ity M o d el T r i p D is tr ib u tio n P r o c e d u r e s , " H ig h w ay R e s e a r c h B o a r d B u lle tin , No. 347 (1962), pp. 6 7 -7 5 . 15 In th is c a s e th e t e r m zone is in te r c h a n g e a b le w ith th e t e r m a r e a w h ich h a s b e e n u s e d th ro u g h o u t th is p a p e r . 82 4. 4. 4 R e fo rm in g the H a n se n A c c e s s ib ility M odel We h av e ju s t c o m p le te d re f o r m in g the L o w ry m odel; t h e r e fo re , the H a n se n m o d el w ill be a ls o r e f o r m e d a ro u n d a p ro b a b ilis tic fo rm a t. A lthough the d e ta il e q u a tio n s a r e o m itte d , th e y follow the sa m e c o n c e p tu a l fo r m a s E q u a tio n s (4. 16) th ro u g h (4. 18). T h e p ro b a b ility th a t a n in d iv id u al w ill m a k e a t r i p f r o m i to j is ! T his p ro b a b ility e q u atio n h a s an a lm o s t id e n tic a l f o r m to the o rig in a l H a n se n eq u atio n , E q u a tio n (4. 22). H o w ev er, if w e u s e H a n s e n 's n o tatio n and c a ll P . th e to ta l n u m b e r of tr ip s g e n e r a te d p e r u n it tim e , then In th is c h a p te r a new a p p ro a c h w as ta k e n to dev elo p a g ra v ity m o d el f o r s p a tia l in te ra c tio n . T h e a p p ro a c h a s s u m e s th a t th e d is tan ce an in d iv id u al is w illin g to tr a v e l c o n fo rm s to a co n tin u o u s - p ro b a b ility d e n sity fu n ctio n . T he r e s u lts w e r e o b tain ed fo r tw o d is trib u tio n s , the P a r e t o an d the ex p o n en tial. m ig r a tio n m o d e l and the H a n se n a c c e s s ib ility m o d el w e r e r e c a s t into a p ro b a b ility fo r m a t. T he s im ila r itie s b e tw ee n eq u atio n s d e riv e d in C h a p te r II fo r in d iv id u a l tr a v e l b e h a v io r, a n d the P a r e t o (4.23) (i, j= l, 2, . . . , n) (4. 24) w hich is id e n tic a l to the H a n se n a c c e s s ib ility m o d el. 4. 5 SUMMARY O F PR O B A B IL IS T IC GRAVITY M ODELS A fte r d ev elo p in g the b a s ic p ro b a b ility m o d e ls , the L o w ry 83 p ro b a b ility m o d e ls of th is c h a p te r w e re ev id en t. A lso , th e r e s u l t s r e v e a le d th a t it is p o s s ib le to r e c o n s t r u c t the H a n se n a n d L o w ry m o d e ls a ro u n d p r o b a b ility c o n c e p ts . In th e c a s e of th e L o w ry m o d e l, th e p ro b a b ility d e r iv a tio n a s s u m e d a P a r e t o a c c e p ta b le d is ta n c e p ro b a b ility . In a d d itio n , th e L o w ry m o d e l, s tr i c t l y sp ea k in g , c an n o t b e d e riv e d f r o m u tility a n a ly s is , b e c a u s e m ig r a tio n is u s u a lly a n o n r e c u r r in g d e c is io n . T h e r e f o r e , a p e r s o n m a k e s o n ly one m o v e to a unique a r e a . H o w e v e r, auto t r i p s p e r y e a r a r e u s u a lly r e c u r r i n g and both u tility and p ro b a b ility f o r m a ts m a k e s e n s e . T h e H a n se n m o d e l fa lls in to th is l a t t e r c a te g o ry . A lso the H a n se n p r o b a b ility : m o d e l w a s the s a m e a s its o r ig in a l m o d e l. T h is in tu r n w a s th e s a m e a s the t r i p p ro b a b ility m o d e l w ith a P a r e t o d is ta n c e fu n ctio n . T he p ro b a b ility g ra v ity m o d e ls d e v elo p e d in th is c h a p te r a r e a s fo llo w s: 1. P a r e t o D is trib u tio n - E q u a tio n (4. 7) P7 J 2. N a tu ra l E x p o n e n tia l D is trib u tio n - E q u a tio n {4. 14) * • • 9 C H A P T E R V E M P IR IC A L VALIDATION i 5 .1 IN TRO D U CTIO N T h u s fa r the w o rk of th is d is s e r ta tio n has e n c o m p a s s e d a th e o r e tic a l d e v e lo p m e n t of tr a v e l d em an d m o d e ls . T h e s e d em an d fu n ctio n s have b e e n c o n s tru c te d by e co n o m ic re a s o n in g . H o w ev e r, the a p p lic a b ility of th e se m o d e ls to th e r e a l w o rld b e c o m e s a p e r t i n ent is s u e . In o r d e r to find th e o r e tic a lly d ev elo p ed h y p o th e se s u sefu l in the p re d ic tiv e o r d e s c rip tiv e s e n s e , th e n , th ey m u s t be v a lid a te d b y e m p ir ic a l e v id en c e . It is to th is ta s k th a t th is c h a p te r is d ire c te d . A lthough in d iv id u al t r a v e l b e h a v io r , c o m m o d ity flow and proba-; b ility g ra v ity m o d e ls h av e b e e n d e riv e d , th is c h a p te r a tte m p ts to p e r - sue th e v a lid a tio n of the in d iv id u a l tr a v e l b e h a v io r m o d e ls . H o w ev e r, the ex ce p tio n is th e ex p o n en tial t r a v e l c a s e w h e re th e e x p o n en tial p ro -, b a b ility m o d e l is su b stitu te d . T he s u b s titu tio n is m a d e b e c a u s e the la t t e r m o d e l len d s its e lf m o r e re a d ily to lo g a rith m ic tr a n s f o r m a tio n . S till, it m ig h t b e a p p ro p ria te to v a lid a te a ll th e m o d e ls ; although, the | a rg u m e n t re m a in s th a t not e v e ry th in g c a n b e u n d e rta k e n in one d is s e r- i ta tio n . Y et, e m p ir ic a l s u b s ta n tia tio n of the in d iv id u al tr a v e l b e h a v io r | m o d e l w ill be an in d ic a to r th at m e th o d o lo g ic a lly th e u se of e co n o m ic c o n s tr u c ts to d e riv e tr a n s p o r ta tio n m o d e ls is a p p ro p ria te . A s o u rc e of c r o s s - s e c t i o n a l d a ta , th e I960 T u c s o n A re a T r a n s - p o rta tio n Study w ill be u se d . ^ T he s u c c e e d in g se c tio n w ill o u tlin e the j so u rc e and m e th o d s u s e d in c o lle c tin g th e s e d ata, in clu d in g sa m p lin g j te c h n iq u e s. T h is se c tio n w ill b e follow ed b y the e m p ir ic a l v a lid a tio n w hich u tiliz e d lo g a r ith m ic a lly - lin e a r m u ltip le r e g r e s s io n , p e r f o r m e d in a s te p w is e fash io n . ^ 1960 T u c so n A r e a T r a n s p o r ta tio n Study (T u cso n , A rizo n a: C ity of T u c so n , I960). 84 85 5. 2 ANALYSIS O F T H E IN P U T D A TA 5 .2 .1 M e th o d s of D a ta C o lle c tin g As w a s s ta te d in the In tro d u c tio n , th e d a ta w e r e ta k e n f r o m the I960 T u c s o n A r e a T r a n s p o r ta tio n S tu d y , w h ic h w a s s p o n s o re d by the C ity of T u c s o n , th e C ounty of P im a , an d the S ta te of A riz o n a in c o o p e ra tio n w ith the U n ited S ta te s D e p a r tm e n t of C o m m e r c e , B u r e a u of P u b lic R o a d s . T h e a r e a c o v e r e d by th e stu d y f o r m s , e s s e n tia lly , a re c ta n g le . T h e le n g th of r a d iu s v e c to r s ru n n in g f r o m the c e n tr o id to the b o u n d a ry s id e s an d at 9 0 ° f r o m e a c h o th e r a r e a s fo llo w s: N o rth V e c to r » 15 m ile s E a s t V e c to r = 17 m ile s South V e c to r ~ 10 m ile s ! W est V e c to r 5 = 12 m ile s P h y s ic a l A r e a C o v e re d by the Study: »725 sq. m ile s T he c e n tr o id in th is stu d y is th e CBD (C e n tr a l B u s in e s s D is tric t). W ith the C B D c o n stitu tin g zo n e one, th e e n tir e stu d y a r e a is divided in to one h u n d re d an d fifty one z o n e s . E a c h zone is n u m b e r ed in a c lo c k w is e s p i r a l s ta r tin g a t the CBD an d ex p an d in g u n til the b o u n d ary of th e stu d y a r e a is r e a c h e d . E a c h zone is d e fin e d to c o in cide w ith the c e n s u s t r a c t s . T h u s, th e p o p u la tio n a n d o th e r v ita l in fo rm a tio n c o lle c te d b y th e c e n s u s w e r e a v a ila b le on th e s a m e g e o g ra p h ic a l b a s i s a s th e t r a v e l z o n e s, th e r e b y , allo w in g fo r c o n s is te n c y in a n a ly sis. O nce th e z o n a l c o m p o s ite w a s d e v elo p e d , th e c o lle c tio n of t r a v e l data w a s r e a d y to be u n d e rta k e n . T h is p o r tio n of the stu d y w a s defined a s th e o r ig i n - d e s tin a tio n s u rv e y , o r s im p ly th e O -D s u r - v e ry . E s s e n tia lly , th r e e m e th o d s of d a ta c o lle c tio n w e r e e m p lo y e d fo r in d iv id u al t r a v e l b e h a v io r . F i r s t , the in te r n a l d w e llin g -u n it s u rv e y w as u tiliz e d . T h is te c h n iq u e w a s to s e le c t a r e p r e s e n ta tiv e c r o s s s e c tio n ( o r s a m p le ) of dw ellin g u n its w ith in th e stu d y a r e a , and o b tain d e s ir e d in f o r m a tio n b y in te rv ie w in g th e o c c u p a n ts of th e s e dw elling u n its . F r o m th e s e s a m p le d a ta , a g e n e r a liz a tio n w a s th e n m a d e about th e e n tir e p o p u la tio n in th e stu d y a r e a . T h u s, b a s e d upon 86 n a tio n a l sam p lin g r a te s ta n d a rd s , a dw elling unit sa m p le size of ten p e r c e n t w as u s e d fo r the T u cso n u rb a n a r e a . A dw elling u n it a s defined by the B u re a u of the C en su s is: A g ro u p of ro o m s o r a single ro o m , o ccu p ied or inten d ed fo r o ccu p an cy as s e p a ra te liv in g q u a r te r s , by a fa m ily o r o th e r g ro u p of p e rs o n s living to g e th e r o r by a p e r s o n liv in g alone. ^ In te rv ie w s w e re conducted o v e r a six m onth p e rio d fo r each day of the w eek, M onday th ro u g h F r id a y . T r a v e l in fo rm a tio n w as obtain ed only fo r th o se w e ek d a y s. A lso , each tr ip w a s defined a s "th e o n e-w ay 3 1 (tra v e l fro m one point to a n o th e r fo r a p a r tic u la r p u rp o se . " T h e r e f o r e , a d iv e rs io n fr o m o rig in to final d e stin atio n , if it w e re in the d ire c tio n of the ro u te of tra v e l, w ould be c o n s id e re d a trip ; fo r ex am p le, v isitin g a fr ie n d w h ile e n ro u te fro m w ork. T he seco n d s o u rc e of data w as a ta x i su rv e y . The technique em p lo y ed a sam p lin g of ta x is and the su rv e y w a s conducted in the s a m e m a n n e r a s the in te rn a l dw elling unit s u rv e y (IDUS). T he sam p le siz e n e c e s s a r y to a ch ie v e a c c u r a c y equal to th a t o b tain ed in the IDUS w as a 50 p e rc e n t sam p le of ta x is . A gain, the in te rv ie w s c o v e re d the m o n th s of J a n u a r y th ro u g h M ay w ith d aily d ata co llectio n ru n n in g M onday th ro u g h F r id a y . E a c h tr i p h ad its o rig in point at the point w h e re the f a re began, and its d e stin a tio n w h e re each fa re te rm in a te d . The th ird , and fin al, data c o lle ctin g tech n iq u e w as the e x te rn a l c o rd o n -lin e su rv e y . T h is w as co n d u cted by e sta b lish in g in te rv ie w statio n s w h e re m a jo r s tr e e ts and highw ays c r o s s e d the study a r e a b o u n d a rie s (which have b e en p re v io u s ly defin ed a s synon- m o u s w ith c e n su s t r a c t b o u n d a rie s) th u s c re a tin g the e x te rn a l co rd o n ^Ibid. , p. 6. ^Ibid. , p. 9. 87 lin e . A t th e s e in te r v ie w p o in ts (sta tio n s) v e h ic le s w e r e sto p p e d and th e d r i v e r s w e r e q u e s tio n e d a b o u t the t r i p b ein g m a d e . T h e s e p o in ts w e r e s e le c te d in s u c h a fa s h io n to a s s u r e th a t the v o lu m e of tr a f f ic c r o s s in g th e c o rd o n lin e on th e p r e s e l e c t e d r o u te s is m o r e th a n 99 p e r c e n t of the tr a f f ic c r o s s in g the c o rd o n lin e o v e r a ll p o s s ib le r o u te s . A s s ta te d in th e Study, in te rv ie w s w e r e p e r f o r m e d a s fo llo w s: In te rv ie w in g a t th e s e lo c a tio n s w as c o n d u c te d by m e n in e ig h t-h o u r s h ifts . A t e a c h of the four m a jo r lo c a tio n s w h e re 87 p e r c e n t of th e to ta l tr a f f ic v o lu m e a t a ll lo c a tio n s w a s r e c o r d e d , s ix sh ifts w e re w o r k e d - - r e p r e s e n t - ing two fu ll d a y s. A t e a c h of th e o th e r s e v e n lo c a tio n s , only tw o sh ifts p e r day w e r e w o rk e d . j T h e in te rv ie w in g p e r io d e x te n d e d f r o m the m id d le of J a n u a r y , I960, to la te in A p ril, I960. A t the m a jo r lo c a tio n s the w o r k sc h e d u le w a s a r r a n g e d so th a t no p a r t i c u l a r lo c a tio n w a s c o m p le te d in a n y g iv e n m o n th , b u t r a t h e r w a s e q u a lly r e p r e s e n t e d in e a c h of th e m o n th s of the in te rv ie w in g p e rio d . ^ T h e fin a l in te rv ie w r e s u l t s sh o w ed th a t 94 p e r c e n t of the to ta l tr a f f ic p a s s in g a ll in te rv ie w lo c a tio n s w a s in te rv ie w e d . 5. 2. 2 A d e q u ac y of th e S a m p lin g D ata In re v ie w in g th e a d e q u a c y of the sa m p lin g d a ta , tw o c a t e g o r ie s w ill be c h e c k e d , p o p u la tio n d a ta an d t r i p d a ta . A c h e c k of th e a d e q u a c y of the d w ellin g u n it s a m p le s e le c tio n f o r p o p u latio n , r e v e a l e d th a t stu d y d a ta w h en p r o je c te d a c c o u n te d f o r 99. 2 p e r c e n t of the c e n s u s p o p u la tio n . T he s u r v e y to o k p la c e c o n c u r r e n tly w ith the U. S. d e c e n n ia l p o p u la tio n c e n s u s . T h e v e r y c lo se a g r e e m e n t b e tw e e n the s u r v e y a n d the c e n s u s , a ls o a tte s ts to th e c o m p le te n e s s w ith w h ic h th e s e d a ta w e r e r e p o r te d . F o r t r i p d ata, b o th a p a s s e n g e r c a r c h e c k and a b u s p a s s e n g e r c h e c k w e r e m a d e . Tw o c h e c k s f o r c a r t r i p s w e r e m a d e . F i r s t , a s c r e e n lin e w a s e s ta b lis h e d and a c o u n t m a d e of a ll tr a f f ic c r o s s in g ^Ibid. , p. 1 1. 88 it. B ec a u se the s c r e e n lin e w as s itu a te d in su ch a w ay th at a t r i p m ig h t c r o s s it tw ice o r not at all, it b e c a m e a p p a re n t th a t the check w ould not be s a tis f a c to r y to su b sta n tia te the su rv e y d ata. The s c r e e n lin e u s e d in the c h e c k w a s the m ain r a ilr o a d lin e w hich ru n s diagon a lly a c r o s s the stu d y a r e a . B ecau se of the in c o n c lu siv e n e ss of this check, a second c h e c k w as m ade. T h is c o n sis te d of a c o n tro l point check. T h re e c o n tro l p o in ts w ere c h o se n to au g m en t the s c r e e n line check. A q u e ry of the tr ip m a k er w a s m ade in o r d e r to d e te rm in e w h e th e r the tr ip r e p o r te d had p a ss e d any of the o th e r c o n tro l p o in ts, i W hile the volum e p r o je c te d by the O -D survey, sam p le d ata w a s only | 75 p e rc e n t of the ground counts ta b u la te d at the s c r e e n lin e (no doubt | re s u ltin g fro m a double counting e r r o r at the s c r e e n line), it, O -D d ata, p ro je c te d 86. 5 p e rc e n t of the count at th e c o n tro l points. A cco rd in g to the Study, th is p a s s e n g e r c a r ch eck w as s a tis fa c to ry fo r s u rv e y s of th is ty p e. T he o th e r p a s s e n g e r trip c h ec k w as m a d e on bus p a s s e n g e r tra ffic . The O -D s u rv e y had c o lle c te d bus p a s s e n g e r sa m p le data as a p a r t of the IDUS. T he check c o n s is te d of, " . . . obtaining fro m the lo c a l bus c o m p a n ie s and the sch o o l d is tr ic ts the to ta l d aily a v e ra g e n u m b e r of bus p a s s e n g e rs tr a n s p o r te d during the s u rv e y 5 p e r i o d . 1 1 The r e s u l t s of the check re v e a le d th at the su rv e y d a ta p ro je c te d 99 p e r c e n t of the a v erag e d a ily bus p a s s e n g e r volum e. W hile a check of b u s p a s s e n g e r tra ffic w as fe lt n e c e s s a r y , the high le v e l of tax i sa m p lin g (50 percen t) m a d e a ta x i check u n n e c e s s a ry . T he a d eq u a c y of the sam p lin g d ata is s u m m a riz e d in the r e p o r t a s follow s: It m u s t be r e a liz e d that the v e ry n a tu re of the O -D s u rv e y is su ch th at som e d e g re e of u n d e r - r e p o r tin g of tr a v e l e x is t s . W ith full know ledge of th is fact, the study s ta ff h a s concluded, on the b a s is of the ch eck s 5 Ibid. ,p. 19. 89 re v ie w e d ab o v e, that the s a m p le s e le c te d in the O -D s u r v e y w a s ad eq u ate, and th a t th e d ata o b ta in e d tr u ly r e f le c t v e h ic u la r m o v e m e n ts of p e r s o n s and ^ goods in the stu d y a r e a d u rin g th e su rv e y p e rio d . 5. 3 E M P IR IC A L VALIDATION O F T H E INDIVIDUAL T R A V E L B EH A V IO R M ODELS 5. 3. 1 S e le c tio n of the D ata fo r Use in E m p ir ic a l V alid atio n T he d a ta in th e T ucson study w a s v o lu m in o u s, so it w as n e c e s s a r y to s e le c t an a p p ro p ria te a m o u n t of d ata fo r u se in v alidation. S ince th e g e o g ra p h ic a l a r e a c o v e re d by the study e n c o m p a s s e d o u t lying and s p a r s e l y p o p u lated re g io n s, it w a s d e te r m in e d to u s e the | |a r e a d efin ed a s the u r b a n region. T he u r b a n re g io n c o n s is te d of 40 c e n su s t r a c t s (zones) out of an a p p ro x im a te to ta l of 150 t r a c t s . T he to ta l p e rm a n e n t, n o n p e rm a n e n t and t r a n s i e n t r e s id e n t p o p u latio n in the u rb a n re g io n w a s 221, 064 w hich is 88. 9 p e r c e n t of the to ta l stu d y - a r e a p o p u latio n . T he n o n p e rm a n en t an d tr a n s ie n t r e s id e n ts a c c o u n te d f o r a p p ro x im a te ly 10. 4 p e rc e n t of the to ta l p o p u latio n f o r th e u rb a n re g io n . F r o m th e u r b a n region, th r e e o r ig in zo n es w e r e s e le c te d w ith the 40 d e s tin a tio n zo n es. T h e se w e r e s e le c te d f r o m a tra ffic v olum e g r id in o r d e r to give m e d iu m to m e d iu m -h ig h vo lu m e le v e ls . D is ta n c e f r o m the o rig in to the d e stin a tio n zone w a s m e a s u r ed a t the g e o g ra p h ic a l c e n te r of the z o n e s . F o r th e o rig in e q u als d e s tin a tio n zone c a s e , d ista n c e w as the m e d ia n d is ta n c e d e riv e d fr o m th e s u m m a tio n of m e a s u r e d d is ta n c e s f r o m the c e n te r of the zone to s e v e r a l (8-12) p o in ts along the b o u n d a ry of the zone. A lso , d is ta n c e s w e re c a lc u la te d f r o m s tra ig h t-lin e type m e a s u r e m e n ts , th a t is , p o in t to point. ^ I b i d ., p. 2 0 . 5. 3. 2 E m p i r i c a l R e s u lts of In d iv id u al T r a v e l B e h a v io r M o d e ls S te p w ise r e g r e s s i o n s w e re p e r f o r m e d on th e lo g a r ith m ic 7 tr a n s f o r m a t i o n of th e e ig h t d e riv e d e q u a tio n s in C h a p te r II. T h e lo g a r ith m i c t r a n s f o r m a t i o n s , the c o n s ta n ts an d th e c o e f f ic ie n ts r e s u ltin g f r o m the r e g r e s s i o n s w ill b e p r e s e n t in th is s e c tio n . T h e 2 e q u a tio n s w ill b e o r d e r e d b y the v a lu e of R , a c c o m p a n ie d b y th e 2 v a lu e of th e s te p w is e c h a n g e s in R . In a d d itio n , th e s ig n ific a n c e of th e F r a t i o w ill b e te s te d a t the 0. 001 and 0. 005 le v e ls of c o n fid e n c e . T h e s a m p l e s iz e u s e d in th is r e g r e s s i o n is 40. We w ill a s s u m e e r r o r s :of o b s e r v a tio n to b e a b s e n t. A lth o u g h w e a r e u s in g c r o s s - s e c t i o n a l d a ta and h av e in d e p e n d e n c e of o b s e r v a tio n s , a u to c o r r e la ti o n c a n e x is t in s p a c e j u s t as it does o v e r tim e; h o w e v e r, w e w ill a s s u m e th a t it c a u s e s no s e r io u s p r o b le m s in th is a n a l y s i s . F u r t h e r , th e h ig h e s t c o r r e l a t i o n c o e ffic ie n t b e tw e e n the d e te r m in in g (in d e p e n d en t) v a r i a b l e s is 0. 259 w ith m o s t of the v a lu e s l e s s th a n 0. 100; t h e r e f o r e , w e w ill n o t w o r r y about th e p r o b le m of m u ltic o lli n e a r ity . F in a lly , w ith only o ne s t r u c t u r a l e q u a tio n , w e a re n o t fa c e d w ith th e p r o b le m of i d e n t i f i c a tio n . b y o r d e r of g o o d n e ss of fit; h o w e v e r, E q u a tio n s (2 .2 5 ) , (2 .3 1 ), and (2 .5 3 ) e c o n o m e tr ic a lly f o r m a s e t, th a t i s , e a c h o f w h ic h h a s th e s a m e r e g r e s s i o n r e s u lts : T h e fo llo w in g a r e th e lo g a r ith m ic t r a n s f o r m a t i o n e q u a tio n s E q u a tio n (2. 25) In T .. = In ij + c In P . - -5 — T- In d . . J 1-D ij 7 D ata u s e d is l is te d in A p p e n d ix A. 91 E q u a tio n (2. 31) In T „ - l ( Mi \ 1 ,---------- •i Z (pc/d fb/1- b) J = iV J 1 J J + c In P. J a , t In d . . 1-b ij E q u a tio n (2. 53) In T . . = In ij + c In P . - cvln d .. 3 iJ E q u a tio n (4. 14) In T .. = In ij j ? ! P j C H -r d i j > l + c In P . - r d .. J iJ E q u a tio n (2, 43) In T .. = In IJ ( 4 - ) ^ ( P f /ln J ? / l - b) j=V J 1 J ' + c In P . ITS ln(ln dij> E q u a tio n (2. 49) In T .. + In d . . = In ij / M . n \j=l J + c In P . J 92 E q u a tio n (2. 61) In T .. + ln (ln d..) = In ij iJ T he r e s u l t s of th e r e g r e s s i o n a r e in T a b le 5. 1 below T a b le 5. 1 R e g r e s s io n V alu es E q u a tio n s (R an k ed by ^ D e sc e n d in g O r d e r of R ) g V a lu es by O rig in Z one 7 13 18 1. E q u a tio n s (2 .2 5 ), (2 .3 1 ) an d (2. 53) C o n sta n t 9. 240 2. 910 56. 700 C o e ffic ie n t fo r P 0. 584 0. 683 0.4 5 1 C o e ffic ie n t fo r d -1 . 861 -1 . 113 -2 .0 5 0 2. E q u a tio n (4. 14) C o n sta n t 7. 500 1. 740 45. 500 C o e ffic ie n t fo r P 0. 642 0. 746 0. 494 C o e ffic ie n t f o r d - 0 .7 1 4 - 0 .3 4 5 -0 .6 9 1 3. E q u a tio n (2 .4 3 ) C o n sta n t 2. 830 6. 790 8 .0 6 0 C o e ffic ie n t f o r P 0. 489 0 .4 2 2 0. 399 C o e ffic ie n t fo r d - 0 .4 8 5 -0 . 344 - 0 .6 9 5 E q u a tio n (2 .4 9 ) C o n sta n t 4. 860 3. 300 13.700 C o e ffic ie n t f o r P 0. 566 0. 653 0 .4 8 2 5. E q u a tio n (2. 61) C o n sta n t 1. 020 3 .4 9 0 11.300 C o e ffic ie n t fo r P 0. 585 0. 477 0. 352 2 2 The v a lu e s of R a n d th e a m o u n ts of s te p w is e change in R by in cluding th e in d e p e n d e n t v a r ia b le d is ta n c e w ith p o p u la tio n a r e lis te d below a s w e ll a s a n a n a ly s is of the v a r ia n c e in T a b le 5. 3. " " " O T he u n its u s e d f o r p o p u la tio n a r e n u m b e r of p e o p le and the u n its f o r d is ta n c e a r e in m ile s . + c In P. J I T ab le 5. 2 C o efficien ts of D e te rm in a tio n E q u atio n s R^(AR^) B y O rig in Zone S tepw ise R e g r e s s io n (R anked by D escen d in g O rd e r of R ) 7 13 18 N u m b er of Steps 1. E q u a tio n s (2. 25), (2.31) and (2. 53) 0. 5563(0. 0570) 0. 5192(0. 1138) 0. 7571(0. 0275) 2 2. E q u atio n (4. 14) 0. 5172(0. 0682) 0 .4 8 9 0 (0 . 1324) 0. 7978(0. 0330) 2 3. E q u atio n (2.43) 0. 3582(0. 0402) 0. 3429(0. 0454) 0 .4 2 7 8 (0 .0 2 1 4 ) 2 4. E q u atio n (2. 49) (d ista n ce w as not in clu d ed a s an in d ep en d en t v a ria b le) 0. 0928(0. 0000) 0. 1830(0. 0000) 0. 0677(0. 0000) 1 5. E q u a tio n (2. 61) (d ista n ce w as not inclu d ed a s an in d ep en d en t v a ria b le ) 0. 0516(0. 0000) 0. 0342(0. 0000) 0. 0254(0. 0000) 1 v O 00 T ab le 5. 3 A n a ly sis of V a ria n c e E q u atio n s Zone T e s t of S ignificance F R atio L e v e l of C onfidence 0. 01 0. 05 A ccept R e je c t A ccept R e je c t 1. E q u atio n s (2. 25), (2.31) 7 23.191 X X and (2. 53) 13 19.975 X X 18 57. 678 X X 2. E q u atio n (4. 14) 7 19. 820 X X 13 17. 707 X X 18 73. 000 X X 3. E q u atio n (2. 43) 7 10.327 X X 13 9.6 5 6 X X 18 13.833 X X 4. E q u atio n (2. 49) 7 3. 887 X X 13 8. 513 X X 18 2. 759 X X 5. E q u a tio n (2. 6 l) 7 2. 067 X X 13 1. 347 X X 18 0. 992 X X 95 15.4 SUMMARY O F E M P IR IC A L VALIDATION I I ! In th is c h a p te r the eq u ations of C h a p te r II w e re e m p ir ic a lly j v alid ated . The r e s u lts in d icated th at the r e g r e s s io n p e r f o r m e d fo r j E q u atio n s (2. 25), (2. 31), (2. 53), and (4. 14) had a c c e p ta b le fits. i | The a n a ly sis of the v a ria n c e fo r th e s e equations w as te s te d j w ith the F R atio. The r e s u lt in d ic a ted re je c tio n of E q u atio n (2. 61) at I the 0. 01 and 0. 05 confidence le v el fo r a ll th r e e trip zo n es te s te d . | E quation (2.49) w as re je c te d in tr i p zones 7 and 18, w h ile fo r zone 13 j it w as a c c e p te d at b o th confidence lev els u sed . T h o se eq u ations w hich re lie d on only one ind ep en d en t v a ria b le , th at is, E q u atio n s (2 .49) and (2.61) w e re u n accep tab le fr o m a ll s t a t i s tic a l stan d p o in ts. E quation (2. 43) would b e c o n sid e re d on the b o r d e r line of a c c e p ta b le . E quations (2. 25), (2 .3 1 ), (2. 53), and (4. 14) a r e ! a c c e p ta b le fro m a ll s ta tis tic a l stan d p o in ts u sed in th is a n a ly s is . T hat 2 is , fo r five of th e s e s ix eq u atio n s, the R 's w e re at le a s t 0 .5 o r b e tte r and the co m p u ted value of F ex ceed ed the value given in the F - d is tr ib u tio n ta b le at the sig n ific a n ce le v els used. T h u s , the null h y p o th esis th a t (3=0 w as re je c te d . T h is e m p ir ic a l a n aly sis not only show s the v a lid ity of the m ethodology u se d in th is d is s e r ta tio n to d e riv e g ra v ity m o d e ls, but a s s is ts us in se le c tin g those m o d e ls w hich give the b e s t r e s u lts and a r e r e p r e s e n ta tiv e of the b a sic "G ra v ity L a w ." E q u atio n s (2 .2 5 ), (2. 31) and (2. 53) p ro v id e in two out of th r e e o rig in zone c a s e s th e 2 h ig h e st value of R . The c a s e in w hich th e y do not is so m e w h a t d i s c o n c e r tin g b e c a u s e it im p lie s th a t w hen the d istan ce v a ria b le a p p e a rs in the eq u atio n ex p o n en tially (a re s u lt as y e t not d e riv e d fro m u tility m a x im iz atio n ) the r e s u lts a r e roughly c o m p a ra b le w ith th o se o btained fro m o u r b e s t th e o r e tic a l m o d e l. F u r th e r r e s e a r c h is obv io u sly | needed. i C H A P T E R VI SUM MARY, CONCLUSIONS, AND SUGGESTIONS F O R F U T U R E R E S E A R C H 6. 1 SUM MARY AND CONCLUSIONS - — ' — " 1 j In su m , w e h av e in v e s tig a te d th e d e v e lo p m e n t of s p a tia l in te r a c tio n m o d e ls w h ich w e r e p r e d ic a te d upon the "G ra v ity L a w . " T h e b a s ic o b je c tiv e w a s to d e riv e m o d e ls of s p a tia l in te r a c tio n w h o se r e q u is ite w as e c o n o m ic h y p o th e s e s . T r a d itio n a lly , s p a tia l in te r a c t io n m o d e ls h a d n o t h a d a b a s is in o rth o d o x e c o n o m ic th e o ry , b u t r a t h e r in s o c ia l p h y s ic s . T h e r e f o r e , if a v iab le m o d e l w e r e to be c o n s t r u c t ed w h ic h r e p r e s e n t e d p h e n o m e n a u n d e r e c o n o m ic s u p p o sitio n s, th en th a t m o d e l m u s t in c lu d e e c o n o m ic c o n s tr u c ts . T h is d is s e r t a t i o n s e t out to a c c o m p lis h the above o b je c tiv e by d e lv in g into tw o a r e a s : (1) in d iv id u a l t r a v e l b e h a v io r and (2) c o m m o d ity flow b e h a v io r . E s s e n tia lly , w e w e r e e v a lu a tin g the c o n s u m e r an d th e p ro d u c t s id e s of th e e co n o m ic s c h e m a tic . F o r e ac h , e c o n o m ic a s s u m p tio n s w e r e n e c e s s a r y in o r d e r to c o n s tr u c t th e m o d e ls . In the c a s e of in d iv id u a l c o n s u m e r of tr a v e l, he m a x im iz e d n e t u tility of tr a v e l . T h is w a s e x p r e s s e d a s an o b je c tiv e fu n c tio n c o n s tr a in e d by a v a ila b le m o n e y fo r tr a n s p o r ta tio n . To id e n tify p o s s ib le m o d e ls , v a r io u s s p e c ific a tio n s w e r e g iv en to d efin e th e f o r m s of the u tility fu n c tio n an d t r a v e l c o s t fu n c tio n . F o r th e u tility fu n c tio n , a l o g a r i t h m ic an d p o w e r f o r m w e r e a s s u m e d . T he c o n s tr a in in g t r a n s p o r t a t i o n - c o s t fu n c tio n w a s p o s tu la te d in fo u r m a th e m a tic a l f o r m s : (1) lin e a r , (2) p o w e r, (3) lo g a r ith m ic an d (4) e x p o n en tial. 96 A s i m i l a r a p p r o a c h w a s ta k e n fo r the p r o d u c e r . H e re th e j c o n s tr a in e d m a x im iz a tio n so lu tio n f o r th e c o m m o d ity s h ip m e n t p r o b le m w a s e v a lu a te d u n d e r th e a s s u m p tio n of m a x im iz a tio n of re v e n u e f o r th e s h o r t - r u n , a n d m a x im iz a tio n of p ro f it f o r the lo n g - r u n s i t u a tio n . T he t r a n s p o r t a t i o n c o s t functions used a r e id e n tic a l to th o s e s e t f o r th in th e in d iv id u a l tr a v e l- b e h a v io r c a s e . A g ain , a s s u m p tio n s a b o u t th e f o r m of th e re v e n u e fu n c tio n a r e the s a m e a s th o s e f o r th e in d iv id u a l t r a v e l - b e h a v i o r fu n c tio n , th a t is , lo g a r ith m i c an d p o w e r. T h e s e c o n d c a te g o r y of d e r iv a tio n of th e " G r a v ity L a w " w a s th e p r o b a b ilis tic m o d e ls . H e r e th e c o n c e p t of s to c h a s tic v a r ia b le s iw as c o n s id e r e d . T h a t is , p r o b a b iliti e s a r e e x p r e s s e d f o r the fo llo w in g e v e n ts : (1) the d e s i r e of a n in d iv id u a l to t r a v e l to a n a r e a ; an d (2) th e d is ta n c e to th a t a r e a is an a c c e p ta b le t r i p le n g th . G iv en th a t th e in d iv id u a l is p la n n in g a t r i p , the p r o b a b ility th a t it w ill be m a d e to a n y p a r t i c u l a r s u b a r e a is th e n d e r iv e d . In the p r o b a b ility d e r iv a tio n of th e in d iv id u a l's t r i p m a k in g b e h a v io r , two s p e c if ic fu n c tio n s f o r t r a v e l p r o b a b ility w e r e h y p o th e s iz e d . T he f i r s t w a s th e P a r e t o d i s tr ib u tio n , a n d the s e c o n d w a s the n a t u r a l e x p o n e n tia l d is tr ib u tio n . T h e fin a l s e c tio n of th e d i s s e r t a t i o n u n d e r ta k e s to e m p ir ic a lly v a lid a te a p o r tio n of the d e r iv a tio n s . T he in d iv id u a l t r a v e l - b e h a v i o r m o d e ls w e r e th e s u b je c t of th is e v a lu a tio n . T h e s e m o d e ls w e r e t r a n s f o r m e d l o g a r ith m i c a lly an d a m u ltip le r e g r e s s i o n p e r f o r m e d to e s ti m a te th e p a r a m e t e r s a n d t e s t t h e i r fit. T h e stu d y d a ta u s e d w e r e ta k e n f r o m th e I960 T u c s o n A riz o n a T r a n s p o r t a t i o n S tu d y . F o r the in d iv id u a l t r a v e l b e h a v io r c a s e , e ig h t m o d e ls w e r e d e r iv e d f r o m the c o m b in a tio n of th e h y p o th e s iz e d u tility a n d t r a n s p o r ta tio n c o s t fu n c tio n s . B e c a u s e of th e s h o r t - r u n an d l o n g - r u n s itu a tio n , s ix te e n m o d e ls w e r e d e r iv e d fo r th e c o m m o d ity s h ip m e n t c a s e . T h e n , f o r the p r o b a b ility d e r iv a tio n , two m o d e ls w e r e d e r iv e d , one f o r th e P a r e t o p r o b a b ility d e n s ity fu n c tio n a n d th e o th e r f o r th e e x p o n e n tia l 98 p ro b a b ility d e n sity fu n c tio n of tr a v e l d is ta n c e s . B a s e d upon the ! b e fo re d is c u s s e d a s s u m p tio n s , a ll of the above so lu tio n s y ie ld e d r e s u lts c o n s is te n t w ith th e " G ra v ity L aw " fo r m a t. As p r e s e n te d p re v io u s ly , an e m p ir ic a l a n a ly s is of the C h a p te r II m o d e ls w as p e r f o r m e d . The r e s u lts r e v e a le d th a t five out 2 of the e ig h t h a d R of 0 .5 0 o r b e tte r . An a n a ly s is of the v a ria n c e w hen using th e F s ta tis tic r e s u lte d in the a c c e p ta n c e of six of the so lu tio n s j at the 0 .0 1 sig n ific a n c e le v e l. T h e se e m p ir ic a l r e s u l t s a ls o p ro d u c e d j a co effic ien t f o r d is ta n c e g r e a t e r th an one fo r the th r e e b e s t fit, 2 ■ h ig h e st R , e q u atio n s. In c o n clu sio n , w e have se e n th a t it is p o s sib le to d e riv e ig e n e ra liz e d m o d e ls of s p a tia l in te r a c tio n w h ich r e s e m b le the " G ra v ity j |L aw " and a r e p r e d ic a te d upon e c o n o m ic h y p o th e se s . T h e s e m o d e ls re fu te the c r i t i c i s m th a t, "w hile the ’G ra v ity L a w 1 h a s b e en a u sefu l tool fo r p re d ic tin g s p a tia l in te ra c tio n , it h a s no fo u n d atio n in e co n o m ic th e o ry . " It is the o b je ctiv e of th is d is s e r ta tio n to p ro v id e s e v e r a l th e o re tic a lly d e riv e d m o d e ls w hich give good d e s c rip tio n s of tr a v e l p a tte r n s . Since the " G ra v ity L aw " has b e e n the m o s t p ro m is in g con cept, th e e x te n s io n of i t to in clu d e e c o n o m ic p ro p o s ito n s m u s t be c o n s id e re d a s te p f o r w a r d in its d e v elo p m en t. As I s a r d p o in ts out, "O nly w ith su ch r e s e a r c h can w e e x ten d b a s ic know ledge of the f o r c e s u n d erly in g g ra v ity m o d e ls , c o n s tr u c t m o r e m e an in g fu l and u sefu l d e s c rip tiv e an d p ro je c tiv e m o d e ls, and d e s ir a b ly a ch ie v e a f ir m sy n th e sis w ith o th e r te c h n iq u e s of a n a ly s is an d b o d ie s of th e o ry . " F in a lly , the m o d e ls d e riv e d in th is d is s e r ta tio n have added to the body th e o r y of s p a tia l in te r a c tio n m o d e ls . T he r e s u l t s of th a t *Tsard, W ., M eth o d s of R eg io n al A n a ly sis: An In tro d u ctio n to R eg io n al A n a ly sis (New Y ork: W iley, I960), p. 566. 99 a c c o m p lis h m e n t r e v e a le d th e b a s ic f o r m of th e g r a v ity m o d e l in b o th th e m a x im iz a tio n p r o c e s s a n d the p r o b a b ility f o r m a t. T h e s ig n ifi c a n c e is q u ite e v id e n t, f o r i t r e ta in s a n a s s e m b la n c e of c o n s is te n c y in th e fo r m , n o t o n ly b e tw e e n the tw o c o n c e p ts , b u t w ith th e f o r m a li ty of th e d e fin e d " G r a v ity L a w . " T h e r e f o r e , w e h a v e a tte m p te d to a c h ie v e in th is ta s k a n e x te n s io n of th e b a s ic c o n c e p t u p o n w h ich th is d e s c r ip ti v e m o d e l h a s fo u n d its g e n e s is . A ls o , th e f a c t th a t th ro u g h a ‘ d iv e r s e ro u te w e w e r e a b le to r e t a i n a s i m i l a r i t y in th e th e o r e tic a l s o lu tio n s w ith th is b a s ic c o n c e p t, h o p e fu lly , s tr e n g th e n s th e f u r t h e r u s e o f the s p a tia l i n t e r a c t i o n m o d e ls . j i | 6. 2 S u g g e s tio n s f o r F u t u r e R e s e a r c h No m a t t e r how f a r r e s e a r c h g o e s, it c o u ld go f u r th e r . S uch is th e c a s e w ith th is d i s s e r t a t i o n . W e h av e o n ly s c r a t c h e d th e s u r f a c e in r e s e a r c h . E s s e n ti a lly , tw o o p tio n s m ig h t b e o ff e re d . T he f i r s t is d e a lin g w ith th e c o n s tr u c ti o n of the b a s ic m o d e l. F o r e x a m p le , the b a s ic e c o n o m ic s p e c if ic a tio n s m ig h t b e r e v is e d . T h is w ould e n c o m p a s s a re v ie w of o th e r f o r m s of the u tility fu n c tio n , th e t r a v e l c o s t fu n c tio n , th e d e m a n d fu n c tio n , and, of c o u r s e , the p r o b a b ility d e n s ity fu n ctio n ; w e h av e n o t e x h a u s te d the p o s s i b i l i t i e s . M a th e m a tic a lly , a n o th e r to o l m ig h t b e s u g g e s te d , th a t is , l i n e a r p r o g r a m m in g . P e r h a p s ev en d y n a m ic p r o g r a m m i n g sh o u ld be in v e s tig a te d a s a p o s s i b ility fo r so lv in g a d y n a m ic c o n c e p t. T h u s, m o r e in v e s tig a tio n of the d y n a m ic a s p e c ts of the c o n c e p t sh o u ld be q u e s tio n e d . F o r e x a m p le : sh o u ld th e tim e in te r v a l b e g iv e n o r sh o u ld it be a d d e d a s a v a r ia b le p a r a m e t e r ? D oes t r a v e l b e h a v io r a s s u m e a p u r e ly r a n d o m ( s to c h a s tic ) p r o c e s s o r c a n it b e d e fin e d by th a t ra n d o m p r o c e s s , s u c h a s B ro w n ia n m o v e m e n t a b o u t t ? N ex t, w h e n lo o k in g a t th e p r o b a b ility m o d e l, w h a t a r e o th e r p o s s ib le d e n s ity fu n c tio n s ? O ne, f o r e x a m p le , m ig h t be th e l o g a r i t h m i c - n o r m a l (lo g n o rm a l) d is tr ib u tio n . T h e s e a r e b u t a few s u g g e s tio n s w h ic h a r e p u t f o r th in stu d y in g a d d itio n a l w o r k th a t m ig h t b e done o n the b a s i c m o d e l. i 5 ! 100 i T he seco n d av enue, w h e re added r e s e a r c h m ig h t be undertaken,; is in e m p iric a l v a lid a tio n of the m o d els developed h e re . T he e m p ir i- j c a l r e s e a r c h u n d e rta k e n in th is d is s e r ta tio n involved only th o se m o d els developed fo r the in dividual tr a v e l b e h a v io r c a s e as w ell as the ex p o n en tial p ro b a b ility m o d e l. T he co m m o d ity -flo w m o d e l as w ell as : the o th e r p ro b a b ility m o d els should b e te s te d in a s im ila r fashion. A lso , o th e r study d a ta should be u sed to c r o s s ch eck the r e s u lts d e riv e d fro m using the I960 T u c so n T r a n s p o rta tio n Study d ata. P e rh a p s the m o s t in te re s tin g m o d els to e c o n o m e tric a lly ev alu a te a r e the p r o b a b ility m o d e ls, sin c e they a r e a lre a d y b a se d on sto c h a s tic e le m e n ts . T his a u th o r, like I s a rd , fe e ls th at the g ra v ity type m o d el d o es hold p r o m is e as a u sefu l tool in m eth o d s of re g io n a l a n a ly s is . T h e re fo re , the su g g estio n s for fu r th e r r e s e a r c h a r e im p o rta n t to a continued evolution of the co n cep t, an evolution so n e c e s s a r y to p r o - ! vide in sig h t fo r so lu tio n of u rb a n and re g io n a l econom ic p ro b le m s , as th ey re la te to tra n s p o rta tio n , ta x atio n , land u s e , in v e stm e n t in so c ia l j o v e rh e a d c a p ita l and the o v e ra ll eco n o m ic w e lfa re o r quality of life. 101 A PP E N D IX A T u cso n I960 T r a n s p o rta tio n Study D ata^ D e s ti n atio n Z ones O rig in Zone 7 O rig in Zone 13 O rig in Zone 18 T r ip s P o p u latio n D istan ce T r ip s P o p u latio n D istan ce T r ip s P o p u latio n D istan ce 1 1103 1221 2 .0 1440 1221 2 .0 859 1221 3 .4 2 168 1772 3 .0 136 1772 1 .9 94 1772 4 .2 3 153 3430 2 .6 230 3430 1 .2 94 3430 3 .6 4 785 4064 2 .0 1341 4064 1. 5 592 4064 2 .9 5 1045 8189 1. 5 990 8189 1 .9 1146 8189 2. 3 6 864 6838 1. 3 411 6838 2 .6 1737 6838 1. 3 7 3083 6313 0. 5 491 6313 3 .2 613 6313 2 .2 8 572 2600 1 .2 388 2600 2 .9 243 2600 3 .2 9 229 4565 2. 0 295 4565 2 .8 58 4565 3. 9 10 105 3229 2 .2 230 3229 2. 5 102 3229 4. 1 11 11 4717 3 .5 75 4717 1 .7 42 4717 4. 5 12 40 2141 4. 0 216 2141 1 .4 52 2141 4 .6 13 449 9711 3 .2 4788 9711 0. 5 459 9711 3 .4 14 156 3476 2. 5 631 3476 0 .9 229 3476 2. 5 15 335 5555 2 .2 226 5555 1 .5 392 5555 2 .0 16 336 2850 2. 2 437 2850 2 .0 1102 2850 1 .2 17 47 2436 3 .2 94 2436 2. 7 440 2436 1 .5 18 653 7004 2 .2 350 7004 3 .4 5555 7004 0 .5 ^1960 T u c so n A re a T r a n s p o rta tio n Study (T u cso n , A rizo n a: C ity of T ucson, i 960). T u cso n I960 T r a n s p o r ta tio n Study D ata (Continued) D e s ti n ation Z ones O rig in Zone 7 O rig in Z one 13 O rig in Zone 18 T r ip s P o p u latio n D istan ce T r ip s P o p u latio n D istan ce T rip s P o p u latio n D istan ce 19 1463 3297 1.0 321 3297 4 .0 851 3297 1.9 20 1010 5477 1. 5 236 5477 4 .7 203 5477 3. 2 21 144 5876 2 .2 74 5876 5. 3 124 5876 4. 1 22 107 4394 2. 5 142 4394 4 .9 71 4394 4. 9 23 177 6906 2 .2 153 6906 4 .0 184 6906 4 .6 24 166 7211 3 .0 313 7211 5 .0 103 7211 5. 3 25 132 4763 3 .6 105 4763 4. 5 21 4763 5 .9 27 142 7828 4 .0 2775 7828 1.2 415 7828 3 .4 28 60 3616 3. 7 332 3616 2. 3 449 3616 2. 3 29 84 4557 3 .2 192 4557 3. 5 1042 4557 1. 1 30 96 7220 3 .8 332 7220 5 .0 2256 7220 1. 8 31 354 4937 4 .5 184 4937 6. 5 912 4937 3 .2 32 503 5330 2 .8 373 5330 4 .9 2074 5330 1 .5 33 269 6594 2 .4 178 6594 4 .7 615 6594 1. 7 34 303 9446 4. 1 326 9446 6 .6 339 9446 3. 5 35 451 7691 2 .4 72 7691 5. 3 550 7691 2. 5 36 728 15493 3. 5 343 15493 6. 5 704 15493 3. 8 37 354 3520 4 .2 43 3520 7 .9 194 3520 5 .5 38 10 3481 5 .2 62 3481 7. 7 31 3481 7. 3 39 41 7344 4. 5 168 7344 6. 5 32 7344 6 .9 40 51 4498 6. 1 72 4498 7 .7 11 4498 8. 3 45 137 11474 5. 5 1088 11474 2 .4 153 11474 5. 5 T o tal 16889 221064 20653 221064 25143 221064 o ro B IB LIO G R A PH Y ; i i j BOOKS ! 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Creator Moorehead, Josef Davis (author)

Core Title A Generalized Economic Derivation Of The ''Gravity Law'' Of Spatial Interaction

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Economics, theory,OAI-PMH Harvest

Language English

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Advisor Niedercorn, John H. (committee chair), Schultz, George P. (committee member), Tintner, Gerhard (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-533573

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Rights Moorehead, Josef Davis

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