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The demand for site-specific recreational activities : a characterics approach Morey, Edward Rockendorf 1978

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THE DEMAND FOR SITE-SPECIFIC RECREATIONAL ACTIVITIES: A CHARACTERISTICS APPROACH by EDWARD ROCKENDORF MOREY B.A., Un iver s i ty of Denver, 1971 M.A., Un iver s i ty o f Ar izona, 1973 A thes i s submitted in pa r t i a l f u l f i l l m e n t of the requirements f o r the degree of The Faculty of Graduate Studies in the Department of ECONOMICS We accept th i s thes i s as conforming to the required standard DOCTOR OF PHILOSOPHY - i n THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1978 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requ i rement s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I ag ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Edward R o c k e n d o r f Morey Department o f ' ECONOMICS The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e August 1, 1978 i i THE DEMAND FOR SITE-SPECIFIC RECREATIONAL ACTIVITIES: A CHARACTERISTICS APPROACH Research Supervisor: Professor John G. Cragg ABSTRACT A model of constra ined u t i l i t y maximizing behaviour is deve-loped to expla in how a representat ive ind iv idua l a l l oca tes his ski days amongst a l t e r n a t i v e s i t e s . The physical c h a r a c t e r i s t i c s of the ski areas and the i n d i v i d u a l ' s sk i ing a b i l i t y are e x p l i c i t arguments in the u t i l i t y func t i on ; the budget a l l o c a t i o n i s given along with the parametric costs to ski ( inc lud ing t rave l cos t s , entrance fees , equipment costs and the opportunity cost o f his t ime). Shares (a s i t e ' s share being the propor-t i on of ski days that the ind iv idua l spends at that s i t e ) are der ived and assumed mult inomia l ly d i s t r i b u t e d , a s tochast ic s p e c i f i c a t i o n which maintains the inherent propert ies o f the shares. Maximum l i k e l i h o o d est imat ion confirms the basic hypothesis that cos t s , a b i l i t y and charac-t e r i s t i c s a l l are important determinants of the s i t e s ' shares. i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS v i i Chapter 1 INTRODUCTION 1 Chapter 2 MODELLING THE DEMAND FOR RECREATIONAL ACTIVITIES: THE LITERATURE 7 Footnotes 21 Chapter 3 A DETERMINISTIC MODEL OF SKIER BEHAVIOUR A. The Determinis t ic Model 24 -B. The S p e c i f i c Form of the U t i l i t y Function 2 3 C. The Cost Functions f o r the Sk i ing A c t i v i t i e s and the Result ing Shadow Pr ices 37 D. Pr ice and Cha rac te r i s t i c E l a s t i c i t i e s 38 E. The Hypothesis of U t i l i t y Maximizing Behaviour 43 Footnotes 44 Chapter 4 A STOCHASTIC MODEL OF SKIER BEHAVIOUR A. The Stochast ic Model 46 B. Maximum L ike l ihood Estimation of the Parameters in the Stochast ic Model 49 C. S i gn i f i cance Tests and Hypothesis Test ing 53 Footnotes 56 Chapter 5 DATA A. A Cross-Sect ional Survey of Skiers 57 iv B. L i f t T icket Pr ices and Charac te r i s t i c s o f the F i f teen Colorado Ski Areas 61 C. Construct ion of Cost and E f f e c t i v e Physical Cha rac te r i s t i c Data for the F i f teen Ski Areas 64 Footnotes 71 Chapter 6 EMPIRICAL RESULTS AND THEIR INTERPRETATIONS A. The Maximization Procedure f o r the L ike l ihood Function 72 B. Maximum L ike l ihood Estimates and Hypothesis Test ing 75 C. Pr ice and Cha rac te r i s t i c E l a s t i c i t y Estimates 87 Footnotes 108 Chapter 7 SUMMARY AND CONCLUSIONS 109 BIBLIOGRAPHY 113 Appendix A SUPPLEMENTARY DATA AND RELATED INFORMATION 121 V LIST OF TABLES I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI A b i l i t y Levels of Student Skiers by Residence A l l o c a t i o n of Student Ski Days by A b i l i t y Level L i f t T i cke t Pr ices and Ski Area Ter ra in 1967/68 Season Distances from C i ty of Residence to Ski Area Shadow Pr ices of the Sk i ing A c t i v i t i e s The E f f e c t i v e Physical Charac te r i s t i c s o f the Ski Areas s . . , The Predicted Shares -s, DIST "S Y • m ' j "s a, . m l j m 2j a 2 j = 0 1 j=0 Colorado Student Sk ier Data Supplementary Ski Area Charac te r i s t i c s 62 63 65 57 59 70 80 83 89 92 95 93 102 105 123 138 vi LIST OF FIGURES 1. Some predicted shares as a funct ion of sk i i ng a b i l i t y 85 vi i ACKNOWLEDGEMENTS I would l i k e to thank my superv i sor , J . G . Cragg, and other committee members, H.F. Campbell, A.D. Woodland, and T . J . Wales. My thanks also go to G.C. Arch iba ld and W.E. Diewert fo r helpfu l suggestions and encouragement. The accumulation of the data required the help of many people: C.R. Goeldner at the Business Research D iv i s i on at the Un ivers i ty of Colorado; G. Lodders at Colorado Ski Country USA; A. Everson and R. Ferdinandsen at the State of Colorado's D iv i s ion of Parks and Outdoor Recreat ion; D. Coddington of B i cker t , Brown, Coddington & Assoc iates , Inc., Denver, Colorado; and f i n a l l y the help of most of Colorado's ski areas. My fe l low students devoted many hours of t h e i r valuable time to help me with my research. Of these Brenda Lundman, Mohammed Khaled, and Theodore Panayotou warrant my specia l thanks. F i n a l l y , but most important ly, I want to thank Reidun Tvedt f o r expos i t iona l and grammatical help, typing a. d i f f i c u l t manuscript countless t imes, research a s s i s tance, and encouragement. E.R.M. Bergen, Norway J u l y , 1978 - 1 -Chapter 1 INTRODUCTION The purpose of th i s research is to model and estimate a repre-sentat ive i n d i v i d u a l ' s demand funct ions f o r s i t e - s p e c i f i c sk i ing a c i t i v -t i e s . The model adopted assumes u t i l i t y maximizing behaviour: faced with a l im i ted sk i ing budget, the ind iv idua l attempts to a l l oca te his time amongst competing s i t e s so as to maximize the u t i l i t y he derives from sk i i n g . This u t i l i t y i s hypothesized to be a funct ion of : (1) the amount the ind iv idua l sk i s at each of the ava i l ab le s i t e s ; (2) ce r ta in physical c h a r a c t e r i s t i c s of those s i t e s ; and (3) his sk i ing a b i l i t y , i . e . whether he is a novice, an intermediate or advanced sk ie r . The cost of s k i i ng at a s p e c i f i c s i t e i s hypothesized to be the sum of: (1) the pr i ce of the. 1 i f t - t i c k e t ; (2) the cost o f equipment r e n t a l ; (3) veh ic le t ranspor ta t ion cos t ; and (4) the opportunity cost of the i n d i v i d u a l ' s t ime, both while t r a v e l l i n g and sk i i ng . A system of share equations i s derived from th i s model, a s i t e ' s share being the proport ion of ski days that the ind iv idua l decides to spend at that p a r t i c u l a r s i t e . A stochas-t i c component is then added to each of these determin i s t i c shares. Maxi-mum l i k e l i h o o d estimates are obtained by apply ing the model to a c ross -sect iona l sample o f Colorado sk ie r s . This model of sk ie r behaviour was developed because there are a number of d e f i c i e n c i e s in the models cur rent ly used to estimate a repre-sentat ive i n d i v i d u a l ' s demand for s i t e - s p e c i f i c recreat iona l a c t i v i t i e s . The demand equations in many recreat iona l demand studies are not derived e x p l i c i t l y from a well s p e c i f i e d model of consumer behaviour. This has led to two problems: (1) the funct iona l forms of the estimated demand equations are often t h e o r e t i c a l l y implaus ib le , or i f p l a u s i b l e , they - 2 -impose questionable r e s t r i c t i o n s on the i n d i v i d u a l ' s preferences; and (2) the costs o f producing recreat iona l a c t i v i t i e s are often i n c o r r e c t l y def ined, s ince the production funct ions f o r those a c t i v i t i e s are not adequately s p e c i f i e d . A method pioneered by Clawson (1959), often re fe r red to as the t r ave l - co s t approach, i s the one most extens ive ly used to estimate recrea -t iona l demand. App l i ca t ions of t h i s technique genera l ly assume that the p r o b a b i l i t y that an ind iv idua l w i l l v i s i t a s i t e on a given day i s a l i n e a r funct ion of : (1) the cost of v i s i t i n g that s i t e ; (2) the costs of i t s sub s t i tu te s ; (3) his s k i i ng budget; and (4) other socio-economic c h a r a c t e r i s t i c s of the i n d i v i d u a l . This l i n e a r i t y r e s t r i c t i o n implies that the i n d i v i d u a l ' s a l l o c a t i o n a l behaviour is sens i t i ve to equ i -pro -port iona l changes in the sk i ing costs and the sk i ing budget, that i s , the quant i t ie s demanded are not homogeneous of degree zero in pr ices and income. This s e n s i t i v i t y v i o l a te s the standard assumption of most consumer theory that pr ices are not arguments in the u t i l i t y funct ion . These p r o b a b i l i t y or share-equations are therefore t h e o r e t i c a l l y implau-s i b l e within the framework of conventional demand theory. Whenever share equations are . spec i f ied ad hoc, there is a high p robab i l i t y that they w i l l e i t h e r be t h e o r e t i c a l l y implaus ib le, or i f p l a u s i b l e , that they w i l l i m p l i c i t l y impose quest ionable r e s t r i c t i o n s on preferences. To avoid these r i s k s , the determin i s t i c components of my ski areas ' shares are derived e x p l i c i t l y from the model of sk ier behaviour. A s p e c i f i c funct iona l form fo r the representat ive i n d i v i d u a l ' s u t i l i t y funct ion i s presented. The l im i t a t i on s imposed on the preferences by th i s form are then d iscussed. Clawson (1959) suggested that va r i a t i ons in cos t s , both across - 3 -s i t e s and across i n d i v i d u a l s , were l a rge ly a t t r i bu tab le to var ia t ions in t rave l costs . For a given ind iv idua l t rave l costs w i l l vary from s i t e to s i t e because the s i t e s are geographical ly d ispersed. For a given s i t e t rave l costs w i l l vary across i nd iv idua l s because they must t rave l d i f f e r e n t distances to reach the same s i t e . Repeated empir ical a p p l i -cat ions of the technique have re in fo rced the hypothesis that t rave l costs are an important determinant of demand. Unfortunately, though, the technique has tended to ignore many of the other costs of producing recreat iona l a c t i v i t i e s , due to the f a c t that the production functions f o r those a c t i v i t i e s are often unspec i f ied. To r e c t i f y th i s d e f i c i e n c y , I have completely s p e c i f i e d the production technology and cost f o r a s k i i n g a c t i v i t y . Few of the demand studies f o r recreat iona l s i t e s e x p l i c i t l y consider the s i t e s ' physical c h a r a c t e r i s t i c s . I consider th i s to be a de f i c i ency . Even those studies which consider s i t e c h a r a c t e r i s t i c s do not e x p l i c i t l y incorporate the c h a r a c t e r i s t i c s into the model of consumer behaviour, i.e.. the c h a r a c t e r i s t i c s do not appear in the u t i l i t y func t ion . I am e x p l i c i t l y inc lud ing ce r t a in physical c h a r a c t e r i s t i c s o f the s i t e s d i r e c t l y in to the u t i l i t y funct ion because, a p r i o r i , i t seems very reason-able that an i n d i v i d u a l ' s choice amongst competing ski areas i s inf luenced by the c h a r a c t e r i s t i c s of those areas. Length and var ie ty of ski runs, the uph i l l capac i ty of the l i f t s at an area, the snow cond i t i ons , and the weather, a l l seem to in f luence the s k i e r ' s cho ice. I have a l so hypothe-s ized that an i n d i v i d u a l ' s preferences over s i t e s depend on his s k i i n g a b i l i t y . The enjoyment an ind iv idua l derives from sk i i ng at a p a r t i c u l a r area i s dependent on his a b i l i t i e s in the sense that his s k i i n g a b i l i t y can remove ce r t a in ski runs from his f ea s i b l e choice set . For example, a - 4 -beginner cannot take advantage of the expert runs at a s i t e , consequently one would not expect the acres of expert runs to play an important explanatory ro le in how that ind iv idua l chooses amongst competing s i t e s . It i s a l so surmised that the s k i e r ' s enjoyment of an area i s re l a ted to the amount of t e r r a i n s p e c i f i c a l l y designed f o r his a b i l i t y l e v e l . For these reasons a substant ia l part of th i s d i s se r t a t i on i s devoted to inc lud ing the i n d i v i d u a l ' s s k i i n g a b i l i t y and the c h a r a c t e r i s t i c s of the d i f f e r e n t s i t e s as arguments in the u t i l i t y func t ion . The model of sk ie r behaviour was completed by spec i f y ing the density funct ion for the s tochas t i c component of the shares. The density funct ion had to be chosen quite c a r e f u l l y . Shares, by d e f i n i t i o n , possess ce r t a i n p roper t i e s , and the density funct ion chosen f o r the s tochas t i c component of these shares must in no way be incons i s tent with these proper t ie s . The dens i ty funct ion for the s tochas t i c share must l i m i t th i s random var iab le to only (T^+l) d i s c re te values in the 0 -1 range, where T. i s the number of days ind iv idua l i sk ied during the season. The shares must a l so sum to one. For these reasons, i t i s i nco r rec t to assume that the s tochas t i c components of the shares are normally d i s t r i -buted. Thus, the maximum l i k e l i h o o d funct ion fo r the parameters in th i s system of s tochas t i c share equations was based on the multinomial d i s t r i -but ion, a d i s t r i b u t i o n which is cons i s tent with a l l the required proper-t i e s o f my system of equations. The a v a i l a b i l i t y of the fo l lowing data made est imation of the parameters in the s tochas t i c share equations pos s ib le : (1) a c ross -sect iona l sample of Colorado student s k i e r s ; and (2) data on physical c h a r a c t e r i s t i c s of a l l the Colorado ski areas. The estimated share equations were found to be genera l ly cons i s tent with the underlying - 5 -theory. E l a s t i c i t i e s est imat ing the e f f e c t o f changes in pr ices as well as i n c h a r a c t e r i s t i c s on a s i t e ' s share were der ived from the system o f estimated share equations. These e l a s t i c i t y estimates can be used to pred ic t how a representat ive i n d i v i d u a l ' s a l l o c a t i o n amongst s i t e s w i l l change when r e l a t i v e cos t s , physical c h a r a c t e r i s t i c s , or s k i i n g a b i l i t y , change. The remainder of the thes i s i s organized as follows.^ In Chapter 2 the recreat iona l demand l i t e r a t u r e is reviewed and discussed. The d iscuss ion concentrates on the methods used to estimate the represen-t a t i ve i n d i v i d u a l ' s demand fo r s i t e - s p e c i f i c recreat iona l a c t i v i t i e s . The d e f i c i e n c i e s of these models are d i scussed, and the chapter concludes with a d i scuss ion of the propert ies which should be included in a model o f s k i e r behaviour. A determin i s t i c model o f sk ier behaviour i s presented in Chapter 3. The chapter f i r s t out l ines the complete model with i t s corresponding system of determin i s t i c share equations f o r s k i i n g a c t i v i t i e s . The d iscuss ion then backtracks to consider in turn the u t i l i t y f unc t i on , the cost funct ions fo r sk i i ng a c t i v i t i e s , and the budget cons t ra in t . Pr ice and c h a r a c t e r i s t i c s e l a s t i c i t i e s are then def ined with in the frame-work of the model. In Chapter 4 the model o f s k i e r behaviour i s made s tochas t i c by adding random e r ro r terms onto the determin i s t i c components of the shares. The chapter mostly concerns i t s e l f with choosing a density funct ion fo r those s tochast ic components which i s cons i s tent with the propert ies o f a system of share equations. To complete the model, the parameters in th i s system must be estimated. The maximum l i k e l i h o o d technique of e s t i -mating these parameters i s then proposed, because i t can be shown that - 6 -under ce r t a in r egu l a r i t y cond i t i ons , maximum l i k e l i h o o d estimates possess des i rab le asymptotic p roper t ie s . The chapter concludes with a d i scuss ion of a method that can be used to te s t the s t a t i s t i c a l s i g n i f i -cance of these maximum l i k e l i h o o d est imates. Chapter 5 discusses the data: a c ros s - sec t iona l sample of s k i e r s ; data on the c h a r a c t e r i s t i c s of ski areas; and pr i ce data ( l i f t t i c k e t p r i c e s , t ransportat ion cos t s , e t c . ) . The sources are o u t l i n e d , and the appropriate data are constructed. The actual s k i e r data and a copy of the quest ionnaire completed by the sk iers are contained in Appendix A. Chapter 6 consider the actual est imation and the r e s u l t i n g empir ica l maximum l i k e l i h o o d estimates of the parameters. F i r s t , the maximization procedure fo r the l i k e l i h o o d funct ion is ou t l i ned . The empir ica l re su l t s then are presented with the d iscuss ion concentrat ing on the parameters' consistency with the hypothesis of u t i l i t y maximizing behaviour, and on the s t a t i s t i c a l s i gn i f i c ance of those parameters. The f i na l sect ion of the chapter considers the e l a s t i c i t y est imates: p r i ce e l a s t i c i t i e s , c h a r a c t e r i s t i c e l a s t i c i t i e s , and the e l a s t i c i t y o f s u b s t i -tu t i on . Chapter 7 assesses the overa l l re su l t s of the model, and considers i t s cont r ibu t ions . Emphasis i s placed on the importance of inc lud ing the c h a r a c t e r i s t i c s o f the s i t e s and the i n d i v i d u a l 1 s ' s k i i n g a b i l i t y in a model which attempts to pred ic t how the ind iv idua l w i l l a l l o ca te ski days amongst s i t e s . In add i t i on , the l im i t a t i on s of the model and the scope fo r fur ther research are noted. - 7 -Chapter 2 MODELLING THE DEMAND FOR RECREATIONAL ACTIVITIES: THE LITERATURE The purpose of t h i s chapter is to consider the method most com-monly used to estimate a representat ive i n d i v i d u a l ' s demand f o r s i t e -s p e c i f i c recreat iona l a c t i v i t i e s . The method was pioneered by Clawson (1959) and i s often re fer red to as the t r ave l - co s t technique. An attempt is made to present the t r a v e l - c o s t technique in two parts . The behavioural postulates o f the method are f i r s t d iscussed. These postulates are d i f f i -c u l t to i d e n t i f y because the technique was designed to pred ic t the demand fo r v i s i t s to a recreat iona l s i t e , not as a theory of a l l o c a t i o n a l beha-v iour . The d i scuss ion then proceeds to consider the empir ica l implemen-ta t i on of the model. The d e f i c i e n c i e s inherent in the use of the t r a v e l -cost technique are then examined. The chapter concludes with a d i scuss ion of the propert ies which should be included in a model o f recreator behavi-our, or in our p a r t i c u l a r case, s k i e r behaviour. Many studies have u t i l i z e d the t r a v e l - c o s t technique in an attempt to estimate the demand fo r s i t e - s p e c i f i c recreat iona l a c t i v i t i e s by a representat ive i nd i v i dua l .^ The intent of th i s review is not to ou t l i ne these a r t i c l e s in d e t a i l , but rather to give an accurate repre-sentat ion of the approach taken in the majority of studies that u t i l i z e the t r a v e l - c o s t technique. In b r i e f , the technique s p e c i f i e s the i n d i v i -dua l ' s demand and supply funct ions (the ind iv idua l both produces and consumes the a c t i v i t i e s ) f o r a group of s i t e - s p e c i f i c recreat iona l a c t i v i t i e s . The reduced form of th i s system is then obtained. The reduced form is then made s tochas t i c , a f t e r which i t s parameters are estimated by applying i t to a c ros s - sec t iona l sample of recreators . The - 8 -demand equations f o r the recreat iona l a c t i v i t i e s are s t a t i s t i c a l l y i d e n t i f i e d by va r i a t i ons in the data generated by c ro s s - sec t iona l v a r i a -t ions in supply,; The t r a v e l - c o s t technique derives i t s name from the fac t that i t e x p l i c i t l y assumes that a l l these var ia t ions in supply are generated by va r i a t i ons in t rave l costs to the s i t e s . Most recreat iona l demand studies which u t i l i z e the t r a v e l - c o s t technique commence by assuming that the i n d i v i d u a l ' s demand for v i s i t s to a recreat iona l s i t e is a l i n e a r funct ion of : (1) the cost of v i s i t i n g that s i t e ; (2) the costs of i t s subs t i tu te s ; (3) his recreat ion budget; and (4) other socio-economic c h a r a c t e r i s t i c s of the i n d i v i d u a l . The supply funct ions f o r v i s i t s to a recreat iona l s i t e are a lso s p e c i f i e d . The producer suppl ies v i s i t s to himself at t h e i r marginal cos t , the i n d i v i d u a l ' s market f o r v i s i t s i s therefore always in equ i l i b r ium. The t r ave l r co s t l i t e r a t u r e makes two assumptions about the costs o f these v i s i t s : (1) the only va r i a t i on in the cost of producing a v i s i t to a s p e c i f i c s i t e is due to var ia t ions in the pecuniary cost of t rave l to that s i t e ; and (2) i t i s assumed that the cos t .o f t r a v e l l i n g to a given s i t e from a given geographic region i s constant. The cost of a t r i p is not assumed to be a funct ion of the number of t r i p s made. An i n d i v i d u a l ' s marginal cost f o r producing a v i s i t i s therefore constant and equal to the pecuniary cost o f t r a v e l l i n g to that s i t e . These marginal cost/supply curves vary across i nd i v idua l s fo r a given s i t e because d i f f e r e n t i nd iv idua l s must t rave l d i f f e r e n t distances to reach the s i t e . For a given i n d i v i d u a l , marginal costs of producing a v i s i t vary across s i t e s because those s i t e s are not a l l located the same.distance from the i n d i v i d u a l ' s res idence. The demand funct ions can be s t a t i s t i c a l l y i d e n t i -- 9 -f i ed from a c ros s - sec t iona l sample of recreator behaviour due to the var ia t ion, in the cost of producing v i s i t s . I f an i n d i v i d u a l l s demand fucntions for v i s i t s to rec rea t iona l , s i t e s are to be estimated using the t r a v e l - c o s t technique as we have so far descr ibed i t , the fo l lowing data are requ i red. A group of recreat iona l s i t e s must be i d e n t i f i e d and a c ros s - sec t iona l sample of v i s i t o r s to those s i te s must be ava i l ab l e . For each ind iv idua l in the sample the fo l lowing information i s requ i red: (1) the number of t r i p s he made to each of the recreat iona l s i t e s in the period under cons iderat ion ; (2) his recreat ion budget f o r that group of s i t e s ; (3) the locat ion of the i n d i v i d u a l ' s res idence; and (4) any other socio-economic information that the resear -chers deem important. UnfoYtunately data of t h i s de ta i l are not genera l ly a va i l ab l e . The usual data ava i l ab le contains only a l im i ted amount of information about the i n d i v i d u a l . " For a given day or weekend the resear -cher knows the number of v i s i t o r s to each of the recreat iona l s i te s under cons idera t ion . The only information about each of these i nd i v idua l s i s the geographical zone from which t h e i r t r i p to the s i t e o r i g ina ted. Data are normally c o l l e c t e d in the fo l lowing manner. On a chosen weekend sur-veyors count the number of cars that enter each of the recreat iona l s i t e s . The number o f occupants in each car and the c a r ' s l i cence p late number i s recorded. In many states in the U.S. the county in which the automobile was reg i s tered can be ascerta ined from i t s l i cence plate number. The t r ave l - co s t technique has reacted to th i s l i m i t e d amount of data by modifying some of i t s behavioural postulates so est imation of the demand for v i s i t s to recreat iona l s i t e s i s s t i l l poss ib le . At the r i s k of some o v e r s i m p l i f i c a t i o n , one might argue that a f t e r these mod i f i ca t ions , the t r a v e l - c o s t technique has degenerated into a mere recogni t ion of a s t a -- 10 -t i s t i c a l r e l a t i on sh ip between the p robab i l i t y that an ind iv idua l w i l l v i s i t a s i t e on a p a r t i c u l a r day and the locat ion of that i n d i v i d u a l ' s res idence. Given l i m i t e d data, the equations that are normally estimated 3 hypothesize that the i n d i v i d u a l ' s "expected leve l of demand' for a v i s i t to a recreat iona l s i t e i s a l i nea r funct ion of : (1) the cost of v i s i t i n g that s i t e ; (2) the costs o f i t s sub s t i tu te s ; (3) the average income in the geographical zone in which the ind iv idua l re s ides ; and (4) a number of other average per cap i ta socio-economic c h a r a c t e r i s t i c s of the i n d i v i -dua l ' s zone of res idence. An observation of the quant ity var iab le ex i s t s for each of the geographical zones. The "expected leve l o f demand" f o r an ind iv idua l i s the proport ion of the zone's population which v i s i t s the s i t e on the given day. Since the ind iv idua l can v i s i t at most one s i t e during the period under cons idera t ion , I p re fer to i n te rp re t a quant ity var iab le def ined in th i s way as the p r o b a b i l i t y that an ind iv idua l w i l l v i s i t a p a r t i c u l a r s i t e on a given day, or as that s i t e ' s share. Est imation of the c o e f f i c i e n t s in the system of l i n e a r probar.", b i l i t y / sha re -equa t i ons requires that the v i s i t data be grouped into geographic c e l l s . For example, the C i c c h e t t i , F isher and Smith (.1976) study designates each county in C a l i f o r n i a as a geographic c e l l . Counties are chosen as population centres in most of the U.S. studies because: (1) l i cence p late number only designate the county in which the veh ic le was reg i s te red ; (2) data are ava i l ab le on the population s ize o f each county; and (3) data are ava i l ab le on the average income, age and other socio-economic c h a r a c t e r i s t i c s o f i nd iv idua l s :by county. A quantity observation for each s i t e i s constructed f o r each geographical zone. The ana lys i s proceeds by assuming that every ind iv idua l l i v i n g in the same - 11 -geographical zone must incur the same trave l costs to v i s i t a s i t e . By assuming a constant cost per mile of t r a v e l , one can estimate the cost to v i s i t each of the recreat iona l s i t e s from the population centre of each of the geographical zones. The t r a v e l - c o s t technique assumes that the i n d i v i d u a l ' s socio-economic c h a r a c t e r i s t i c s in f luence his a l l o c a t i o n , but they do not have data on the ind iv idua l so they use instead the i n d i v i d u a l ' s zone's average per cap i ta .va lues o f these c h a r a c t e r i s t i c s . The zone data are a poor proxy f o r data on the c h a r a c t e r i s t i c s o f one of i t s res idents because these var iab les vary widely with in the zone. Th i s ,quant i t y , cost and socio-economic data are then used to estimate the c o e f f i c i e n t s in the system of l i n e a r equations. As might be expected, the c o e f f i c i e n t s o f the socio-economic c h a r a c t e r i s t i c s of the geographical zones are general ly found to be 4 s t a t i s t i c a l l y i n s i g n i f i c a n t . However, repeated empir ica l app l i ca t ions of the technique have re in forced the hypotheses that the p robab i l i t y that an ind iv idua l w i l l v i s i t a s i t e s trongly depends on the cost to t rave l to that s i t e and the cost to t rave l to i t s subs t i tu tes . The actual econo-metric est imation of the share equations has become progress ive ly more soph i s t i ca ted . Clawson in his o r i g i na l work (1959), estimated the share equation f o r only one s i t e using ordinary l ea s t squares, whereas C i c c h e t t i , F isher and Smith ,(1976 ) s imultaneously estimate the share equations fo r a l l the ski areas in the state of C a l i f o r n i a , using Z e l l n e r ' s technique 5 fo r seemingly unrelated regress ions. Share equations for s i t e - s p e c i f i c recreat iona l a c t i v i t i e s should be estimated as a system. There are numerous cross equation r e s t r i c t i o n s which can only be considered when the system is estimated in th i s way. Shares must sum to one; th i s means that only N-l o f the share must be est imated, and that the covariances - 12 -amongst the shares are not a l l equal to zero. Cross equation r e s t r i c -t ions are also implied i f the equations are to be cons is tent with an underlying theory of u t i l i t y maximizing behaviour. One o f the factors that has not been considered by the t r a v e l -cost l i t e r a t u r e i s the fac t that a share can only be randomly d i s t r i bu ted between zero and one. This poses two potent ia l problems. (1) The l i n e a r s p e c i f i c a t i o n s of the share equations means that there i s a po s i t i ve p robab i l i t y that shares greater than one or less than zero w i l l be p red i c -ted , pred ic t ions obviously in cont rad i c t i on with r e a l i t y . Idea l ly the funct ional form of the determin i s t i c component of the shares should res -t r i c t those components so they are a l l between zero and one and together sum to one. (2) The density funct ion f o r the s tochas t i c components of the shares must r e s t r i c t th i s var iab le to the zero to one range. Other-wise, i t i s not the cor rec t density f unc t i on , and therefore nothing can be s a i d , in genera l , about the s t a t i s t i c a l propert ies o f any estimates based on i t . This ru les out the normal and other continuous d i s t r i bu t i on s that allow fo r shares outs ide of the zero to one range with a pos i t i ve p r o b a b i l i t y , unless one has a large sample where the average shares are not near zero or one. In that case the normal w i l l be a good approxima-t i o n to the true density func t ion . One way to guarantee that one's estimates have the des i red asymptotic propert ies i s to choose a density funct ion cons i s tent with a l l the propert ies implied by the d e f i n i t i o n of shares and then obtain the corresponding maximum l i k e l i h o o d estimates of the parameters. One should be uneasy about any est imation technique, and the r e s u l t i n g est imates, when that technique does not take e x p l i c i t account of the r e s t r i c t i o n s imposed on the s tochas t i c component of the var iab les that are being est imated. - 13 -The t r a v e l - c o s t technique suf fers from a number of d e f i c i e n c i e s , but should not be judged too harshly on a p r ac t i c a l l e v e l . It was deve-loped to do appl ied work ;and thus cannot be fau l ted f o r omitt ing important explanatory var iab les f o r which data does not e x i s t . Many of i t s less reasonable assumptions were required i f the demand f o r recreat iona l v i s i t s was to be estimated with such data. Data were simply not ava i l ab le on the socio-economic c h a r a c t e r i s t i c s o f the user. The t r ave l - co s t l i t e r a t u r e has also been forced to assume that the ind iv idua l has a one day s t a t i c planning hor izon. Researchers in th i s area d id not make th i s i m p l i c i t assumption because of i t s p l a u s i b i l i t y , rather i t was made because data were ava i l ab le only on the i n d i v i d u a l ' s choice of s i t e on a given day. Improvements brought about by th i s d i s s e r t a t i on in the techniques used to estimate an i n d i v i d u a l ' s demand fo r v i s i t s to recreat ional .?s i tes are made poss ib le by my access to data that are more deta i l ed than those prev ious ly a v a i l a b l e . My data give a complete record of where a large number of i nd iv idua l s ski during an en t i re season, along with a cons ider -able amount o f socio-economic data on those i n d i v i d u a l s . The major f a u l t o f the t r a v e l - c o s t technique is that i s su f fer s from the lack o f a sound theore t i ca l foundation. The hypothesized demand or share equations f o r the recreat iona l a c t i v i t i e s are not e x p l i c i t l y derived from a model o f consumer behaviour, therefore the choice of t h e i r funct iona l forms and the d e f i n i t i o n s of t h e i r var iab les i s a r b i -t r a r y . ^ Most of the studies never suggest that the share equations e s t i -mated are the re su l t o f any opt imiz ing behaviour on the part o f the 8 i n d i v i d u a l . Without any e x p l i c i t underlying theory, i t i s quite d i f f i -c u l t to judge the share funct ions f o r v i s i t s to recreat iona l s i t e s . Obviously they can be judged on t h e i r a b i l i t y to pred ic t behaviour, but - 14 -th i s i s not enough. One must ask whether the t r a v e l - c o s t share equa-t ions are cons i s tent with an underlying theory of consumer behaviour and i f so, are the impl ied r e s t r i c t i o n s on preferences reasonable. We pro-ceed by examining the l im i t a t i on s the t r a v e l - c o s t technique places on the i n d i v i d u a l ' s preferences and on his constra ints when the technique is analyzed wi th in the u t i l i t y maximizing framework. F i r s t , by assuming that the demand for v i s i t s to a recreat iona l s i t e is a funct ion of the cost of v i s i t i n g that s i t e , the costs of v i s i -t ing the a l t e r n a t i v e s i t e s , and the to ta l budget a l l o c a t i o n to those v i s i t s , but not a funct ion of the costs of non-recreat ional a c t i v i t i e s , the model i s making a s e p a r a b i l i t y assumption. The group of s i t e - s p e c i -f i c recreat iona l a c t i v i t i e s considered i s i m p l i c i t l y assumed weakly 9 separable from a l l other a c t i v i t i e s in the u t i l i t y funct ion . The •  a l l o c a t i o n of a recreat iona l budget amongst a l im i ted number of s i t e s can only be analyzed in i s o l a t i o n from the other a c t i v i t i e s in the con-sumer's choice set when th i s weak s e p a r a b i l i t y assumption i s invoked. The t r a v e l - c o s t technique does not make th i s assumption e x p l i c i t . Second, a l i n e a r form fo r the demand or share equations is t h e o r e t i c a l l y implaus ib le. Such demand equations are not homogeneous of degree zero in costs and recreat iona l budget. The l i n e a r s p e c i f i -cat ion of the demand equations impl ies that the i n d i v i d u a l ' s choice amongst s i t e s i s sens i t i ve to equi -proport iona l changes in the t rave l costs and in the recreat iona l budget. The consumer suf fers from a form o f money i l l u s i o n . This v i o l a te s one of the standard assumptions o f most consumer theory, namely that pr ices and/or costs are not argu-ments in the u t i l i t y f u n c t i o n . ^ Conventional demand theory makes ce r t a i n pred ic t ions about a l l o c a t i o n a l behaviour: (1) the demand - 15 -funct ions are homogeneous of degree zero in pr ices and income; (2) the subs t i tu t i on matrix i s negative s em i -de f i n i t e ; (3) the subs t i tu t i on e f f ec t s are symmetrical; (4) the sum of the income e l a s t i c i t i e s of demand, each weighted by that p a r t i c u l a r a c t i v i t y ' s share of the to ta l budget, must equal one; and (5) the sum of a l l the subs t i tu t i on e f f e c t s , each weighted by i t s p r i c e , must equal z e r o . ^ If a system of demand equations i s incons i s tent with any of these f i v e p roper t ie s , i t obv i -ously could not have been derived from a conventional theory of consumer behaviour. It i s important to ask whether the preferences impl ied by an a r b i t r a r i l y def ined funct ion are t h e o r e t i c a l l y p laus ib le ,and i f so, to i d e n t i f y the re su l t i n g preference order ing and assess i t s reasonable-ness. T h i r d , the t r a v e l - c o s t l i t e r a t u r e i s obviously cor rec t in hypothesiz ing that socio-economic var iab les such as age and family s i ze play a ro le in the i n d i v i d u a l ' s choice of recreat iona l s i t e s . But the assumption that those var iab les enter as l i n e a r arguments in the demand funct ion is d i f f i c u l t to j u s t i f y i f those demand funct ions are derived from u t i l i t y maximizing behaviour. If socio-economic var iab les a f f e c t one's a b i l i t y to enjoy recreat iona l s i t e s , these var iab les should appear as e x p l i c i t arguments in the i n d i v i d u a l ' s u t i l i t y func t ion . They should not be inserted in the demand funct ions in an a r b i t r a r y manner. I have incorporated sk i i n g a b i l i t y d i r e c t l y into the u t i l i t y funct ion in a systematic way because I fee l i t inf luences the preference order ing f o r d i f f e r e n t s k i i n g a c t i v i t i e s . Fourth, the t r a v e l - c o s t technique does not recognize the fac t that the.amount of u t i l i t y derived from v i s i t i n g a recreat iona l s i t e is a funct ion of the physical c h a r a c t e r i s t i c s o f that s i t e . I consider - 16 -th i s a de f i c iency o f the technique. The technique has e i t he r completely overlooked the importance of the s i t e ' s c h a r a c t e r i s t i c s , or i s i m p l i c i t l y assuming that a l l the d i f ferences in c h a r a c t e r i s t i c s across s i t e s have been accounted f o r by va r i a t ions in entrance p r i ce s . It i s the hypothe-s i s of th i s d i s se r t a t i on that the c h a r a c t e r i s t i c s of a s i t e and i t s sub-s t i t u t e s have a strong e f f e c t on how a l im i ted budget i s a l l oca ted amongst the s i t e s . There has been some appl ied work on the c h a r a c t e r i s t i c s of s i t e s , but i t has been of a rudimentary nature. Burt and Brewer (1971) estimated the demand funct ions fo r s i x categor ies of lakes. The lakes were grouped in categories by t h e i r s i ze and physical c h a r a c t e r i s t i c s . The serv ices from each category were then treated as separate but re la ted goods. Even though the lakes were categor ized by s i z e , the importance of s i ze on preferences f o r lakes was not made e x p l i c i t in the u t i l i t y func t ion . Grimes (1974) bui lds and tests an empir ical model to estimate the va lues, s i z e , and bu i l d ing density of recreat iona l l o t s as a funct ion o f the l o t ' s distance from population centres and the c h a r a c t e r i s t i c s of the l o t s . Grimes' model i s not e x p l i c i t l y a.demand study, but charac-t e r i s t i c s of the s i t e s are cons idered, and the explanatory var iab les are i d e n t i f i e d using a model of consumer behaviour. There r e a l l y has been l i t t l e appl ied work which incorporates 13 the c h a r a c t e r i s t i c s o f goods or a c t i v i t i e s into an ana lys i s of demand. Baumol and Quandt (1966) out l ined and estimated a demand equation to exp la in the demand fo r t r i p s between c i t i e s by d i f f e r e n t t ransportat ion modes, where the c h a r a c t e r i s t i c s of the d i f f e r e n t modes appear in the demand equation. Unfortunately th i s demand equation is not derived from the so lu t ion to a constrained u t i l i t y maximization problem. The work of - 17 -McFadden and others has a stronger theore t i ca l foundat ion. McFadden (1974a, 1974b), McFadden and Reid (1974), Donencich and McFadden (1975), and Kraft and Kraft (1974) have formulated a behavioural aoproach that has been used to estimate the demand fo r t ransportat ion modes as a func-t i on of the c h a r a c t e r i s t i c s of the d i f f e r e n t modes. Log i t , prob i t and other p r o b a b l i s t i c models are used to estimate the p robab i l i t y that an ind iv idua l w i l l choose a ce r ta in opt ion. The p robab l i s t i c equations are derived from u t i l i t y maximizing behaviour, but the form of the r e -quired u t i l i t y funct ion general ly i s assumed l i n e a r (McFadden 1974a: 1 "13, and McFadden and Reid 1974:5-6). Once the dec i s ion to recreate has Been made, a mu l t i - cho ice l o g i t or prob i t model could poss ib ly be used to estimate the i n d i v i d u a l ' s choice of recreat iona l s i t e s i f one is w i l l i n g to accept the preference order ing required by a u t i l i t y funct ion l i n e a r in i t s arguments, or i f 'one i s capable and w i l l i n g to incorporate a less r e s t r i c t i v e f o r m i n t o t h e l o g i t ana ly s i s . The Hedonics mul t ip le regress ion approach to the est imation of p r i ce ind ices considers the c h a r a c t e r i s t i c s o f goods. The technique hypothesizes a funct iona l r e l a t i on sh ip between a good's pr ice and i t s c h a r a c t e r i s t i c s . Attempts are made to separate changes in pr ices due to qua l i t y changes from pure pr i ce changes. An extensive review of the Hedonics pr ice indices l i t e r a t u r e can be found in G r i l i che s (1971). One method used by the Hedonics approach is to regress the pr ice of a good on i t s c h a r a c t e r i s t i c s . The estimated c o e f f i c i e n t s are considered the shadow pr ices o f the c h a r a c t e r i s t i c s . These estimated "shadow pr i ces " are then used to construct pr ice and q u a l i t y i nd i ce s . Unfortunate ly, these estimated c o e f f i c i e n t s are not in general shadow p r i ce s . Mue l l -bauer (1974), Lucas (1975), and Pol lak and Wachter (1975) have a l l shown - 18 -that when c h a r a c t e r i s t i c s are j o i n t l y produced by a good, the shadow pr ices and the u t i l i t y maximizing vector of c h a r a c t e r i s t i c s are s imul -taneously determined by the i n d i v i d u a l . The shadow pr ices of the char-a c t e r i s t i c s are not exogenous and i den t i ca l across i n d i v i d u a l s , but d i f f e r e n t fo r each as a funct ion of that i n d i v i d u a l ' s preferences. Boyle, Gorman and Pudney (1975) model the demand fo r food using a char-a c t e r i s t i c s approach. They u t i l i z e the Hedonics technique to estimate the shadow pr ices f o r the c h a r a c t e r i s t i c s of the good. Given the recent works mentioned above, which quest ion the existence of a unique vector of shadow pr ices f o r c h a r a c t e r i s t i c s , I doubt whether the Hedonics tech -nique w i l l prove f r u i t f u l as a method to estimate the demand for an ac-t i v i t y as a funct ion of the c h a r a c t e r i s t i c s of that a c t i v i t y . Lancaster (1966a,1971) helped to incorporate c h a r a c t e r i s t i c s into demand theory, but the empir ical a p p l i c a b i l i t y o f his model is quest ionable. I know of no d i r e c t empir ical app l i ca t ions of the model. Lancaster assumes that the ind iv idua l derives u t i l i t y from the j o i n t c h a r a c t e r i s t i c s of goods (or a c t i v i t i e s ) . A system of demand equations fo r these c h a r a c t e r i s t i c s is derived as a funct ion of the budget and the shadow pr ices of the c h a r a c t e r i s t i c s by .max im iz ing ' a -u t i l i t y funct ion defined over the c h a r a c t e r i s t i c s subject to ahbudget cons t ra in t , and to the parametric costs o f producing a c t i v i t i e s . Unfortunately, est imation o f these demand equations would be qu i te d i f f i c u l t . Pol lak and Wachter (1975), among.others, have shown that the vector of these shadow pr ices is not parametric to the ind iv idua l but j o i n t l y determined with the u t i l i t y maximizing vector o f c h a r a c t e r i s t i c s . Therefore, est imation of the demand equations requires that the u t i l i t y maximizing vector of shadow pr ices be f i r s t determined fo r each ind iv idua l in the sample. - 19 -This involves so lv ing a l i n e a r programming problem for each i n d i v i d u a l . For some ind iv idua l s th i s l i n e a r programming problem w i l l l i k e l y have mul t ip le s o l u t i on s , i . e . i t w i l l probably not be poss ib le to ca l cu l a te a unique vector o f shadow pr ices fo r a l l i nd i v i dua l s . F i n a l l y , many of the costs in producing s i t e - s p e c i f i c rec rea -t iona l a c t i v i t i e s are ignored m the t r a v e l - c o s t l i t e r a t u r e . It assumes that only va r i a t i ons in t rave l costs produce var ia t ions in an a c t i v i t y ' s production cos t s . Entrance fees , the cost o f recreat iona l equipment, and the opportunity cost o f one's t ime, a l l in f luence the cost of pro-ducing a recreat iona l a c t i v i t y . Travel cost i s not the only varying component of production cos t s . Entrance fees vary across s i t e s , the costs of required equipment and the opportunity cost of time vary across i nd i v i dua l s . The t r a v e l - c o s t technique ignores the v a r i a b i l i t y in these other components o f cos t s . These omissions would not have occurred i f the production funct ions fo r the s i t e - s p e c i f i c recreat iona l a c t i v i t i e s had been well s p e c i f i e d . Because of the i l l def ined nature of cos t s , constructed r e l a t i v e costs o f v i s i t i n g the d i f f e r e n t s i t e s are b iased. Knetsch (1963), Cesario and Knetsch (1970), and Brown and Nawas (1972) have a l l recognized the bias generated by ignoring time costs . They have each proposed methods of e l iminat ing or reducing th i s b ias . If one hopes to estimate a system of share equations f o r s i t e -s p e c i f i c s k i i n g a c t i v i t i e s that do not su f fe r from the aforementioned d e f i c i e n c i e s , one must der ive that system d i r e c t l y from a model of u t i l i t y maximizing behaviour. A model descr ib ing the a l l o c a t i o n a l beha-v iour of the s k i e r must be constructed. The share equations f o r the s i t e - s p e c i f i c sk i ing a c t i v i t i e s are those shares that maximize a s p e c i f i c u t i l i t y funct ion f o r sk i i ng a c t i v i t i e s subject to a l im i ted sk i i ng - 20 -budget and the costs to ski at the d i f f e r e n t s i t e s . The model must, at a minimum, assume that the u t i l i t y produced by sk i i ng a c t i v i t i e s i s weakly separable from the u t i l i t y produced by other a c t i v i t i e s . This assumption is required i f the a l l o c a t i o n of the sk i i ng budget amongst a number of competing s i t e s i s to be analyzed in i s o l a t i o n from the a l l o -cat ion to other a c t i v i t i e s . The funct ional form of the u t i l i t y funct ion must be made e x p l i c i t and i t s arguments must inc lude: (1) the amount the ind iv idua l sk is at each of the ava i l ab le s i t e s ; (2) the important physical c h a r a c t e r i s t i c s of those s i t e s ; and (3) any personal character -i s t i c s of the ind iv idua l that w i l l s t rong ly a f f e c t that i n d i v i d u a l ' s r e l a t i v e a b i l i t y to enjoy s k i i n g at the d i f f e r e n t s i t e s . In add i t i on , the funct iona l form of t h i s u t i l i t y funct ion must be simple enough to al low fo r the der i va t i on of e x p l i c i t share funct ions from the model. The model of s k i e r behaviour must a lso cons ider the product ion/cost funct ions fo r the d i f f e r e n t s i t e - s p e c i f i c s k i i ng a c t i v i t i e s . The pro-duction functions must cons ider a l l the inputs required to produce a sk i i ng a b i l i t y . These are: (1) the serv ices of a ski area; (2) s k i i n g equipment; (3) veh ic le t ransportat ion cos t s ; and (4) the opportunity cost of the i n d i v i d u a l ' s t ime, both while t r a v e l l i n g and sk i i ng . Before the parameters in the u t i l i t y funct ion in th i s determin i s t i c model o f s k i e r behaviour can be est imated, the model must be made s tochas t i c . Shares by d e f i n i t i o n possess ce r ta in p roper t ie s , and the density funct ion chosen fo r the s tochast ic component of these shares must in no way be incons i s tent with these proper t ie s . F i n a l l y , the required data must be a v a i l a b l e . - 21 -Footnotes - Chapter 2 1. To c i t e j u s t a few of these s tud ies : Clawson (1959); Knetsch (1963); Pearse (1968); Cesario and Knetsch (1970); Burt and Brewer (1971); Smith, R.J. (1971); Ranken and Sinden (1971); Brown and Nawas (1973); Grimes (1974); Smith, V.K. (1975); Rausser and O lve i ra (1976); and C i c c h e t t i , F i sher and Smith (1976). The bib l iography contains a more complete l i s t . 2. Data have j u s t not been c o l l e c t e d on how a group of i nd iv idua l s a l l o ca ted t h e i r recreat iona l v i s i t s amongst competing s i t e s during a season. Considerable data have been c o l l e c t e d by recreat iona l s i t e s on the users of t h e i r s i t e s , but these studies are not i n t e r -ested in the other s i t e s the ind iv idua l has or might v i s i t . These s i t e - s p e c i f i c surveys c o l l e c t considerable data on the t r i p in progress and the personal c h a r a c t e r i s t i c s of the i n d i v i d u a l , which usua l ly does not include where the ind iv idua l res ides . Since these studies only consider the v i s i t s to one s i t e , there i s no va r i a t i on in the choice of s i t e s across the sample, re su l t i ng in a sample that i s useless f o r demand ana l y s i s . Examples of sk i ing studies o f t h i s type are: L i f t Interview Study of the Aspen Sk ier 1972 (Tourism Research Associates 1972); L i f t Interview Study of the  Aspen Sk ier (Tourism Research Associates 1975); The Aspen Skier  1974, vo l s . 1 and 2 (Goeldner 1975); and The Breckenridge Skier XGoeldner and S l e t t a 1975). Some i n s t i t u t i o n s and organizat ions such as the Center f o r Business and Economic Research, Brigham Young Un ivers i ty (1971) have simultaneously conducted on - s i t e interviews at a large number of Utah ski areas. Considerable data on the c h a r a c t e r i s t i c s of the sk iers were c o l l e c t e d and th i s data could be u t i l i z e d fo r a t r a v e l -cost study, but to my knowledge i t has not been. Recreation or ientated magazines such as Ski sometimes survey t h e i r subscr ibers concerning recreat ion s i t e s they v i s i t e d during a season (Goeldner, Dicke and S le t ta 1973:2,13), but these studies are not genera l ly publ i shed. Studies of th i s type could t h e o r e t i c a l l y be used fo r a t r a v e l - c o s t study, but there would be large problems because the ind i v idua l s are so dispersed that many s i t e s would have to be cons idered. 3. Clawson-(1959), C i c c h e t t i , F i sher and Smith (1976) and most o f the other studies using the t r a v e l - c o s t technique descr ibe the dependent var iab les that are ac tua l l y estimated as the expected leve l of demand/part ic ipation by a representat ive i n d i v i d u a l . 4. See fo r example Burt and Brewer (1971), or C i c c h e t t i , F i sher and Smith (1976). 5. C i c c h e t t i , F i sher and Smith (1976) suggest two reasons why t h e i r equations should be estimated as a system: (1) they impose the cross equation r e s t r i c t i o n that the gross cross p r i ce e f f ec t s on the shares are a l l symmetrical; and (2) they assume that the e r ro r terms are contemporaneously co r re l a ted across equations. The - 22 -o r i g i n of the contemporaneous c o r r e l a t i o n is not discussed. The symmetry r e s t r i c t i o n was imposed to f a c i l i t a t e benef i t est imation (1976:1265). One should not confuse the symmetry r e s t r i c t i o n imposed .in th i s study with the p red ic t i on of conventional demand theory that : ^1 3 P . 8=0 3x. 3 3 P i 8=0 6. See page 14 fo r a d i scuss ion of the cross equation r e s t r i c t i o n s impl ied by conventional theory. 7. One is unsure whether to c a l l the equations out l ined by the studies demand equat ions, share equat ions, or p r o b a b l i s t i c equations. When the technique i s descr ibed, the dependent var iab les f o r each s i t e : are def ined as the quant i ty of v i s i t s demanded by a representat ive, i n d i v i d u a l . However, when the equations are ac tua l l y est imated, an observat ion of the dependent var iab le f o r each s i t e is the propor-t i o n o f a geographical zone's population which v i s i t s the s i t e on the given day. The estimated dependent var iab le could thus be bet ter descr ibed as a share or p r o b a b i l i t y . When d iscuss ing the technique, the cir itc. isms made are appropriate f o r both the demand equations s p e c i f i e d and f o r the share equations a c tua l l y est imated. 8. The study by C i c c h e t t i , F i sher and Smith (1976) is an except ion. It hypothesizes that Becker 's (1965) theory of consumer behaviour forms the foundation of the t r a v e l - c o s t approach. However, i t does not take advantage of t h i s t heo re t i c a l model to der ive i t s share equations. 9. That i s , the marginal rate of subs t i tu t i on between any two s i t e -s p e c i f i c recreat iona l a c t i v i t i e s i s independent of the leve l of consumption of any non-recreat ional a c t i v i t y . 10. The demand theor ies o f H icks -A l len (see Samuelson (1947)), Becker (1965), and Lancaster (1966b, 1971), a l l e x p l i c i t l y assume that ne i ther pr ices nor costs are arguments in the u t i l i t y funct ion . 11. The demand theor ies o f H icks -A l len (see Samuelson (1947)), Becker (1965), and Lancaster (1966b, 1971) a l l maintain these f i v e proper-t i e s ; H i cks -A l l en in goods space, Becker in a c t i v i t i e s space, and Lancaster in c h a r a c t e r i s t i c s space. 12. The demand studies by Lau, L in and Yolopoulos (1975) and Pol lak and Wales (1976) are examples of two demand studies which introduce soc io-economic c h a r a c t e r i s t i c s of the household d i r e c t l y into the u t i 1 i t y funct ion . 13. The reader should note that the in tent of t h i s present sect ion of Chapter 2 is not to survey a l l the appl ied work on c h a r a c t e r i s t i c s , but rather to mention and discuss the two most popular techniques fo r incorporat ing c h a r a c t e r i s t i c s into demand ana ly s i s . The reader in teres ted in a more extensive l i s t i n g of studies in th i s area can - 23 -re fe r to my b ib l iography which.includes. references f o r a l l the appl ied works.on the c h a r a c t e r i s t i c s of goods and a c t i v i t i e s that I encountered while doing th i s research. - 24 -Chapter 3 A DETERMINISTIC MODEL OF SKIER BEHAVIOUR The purpose of th i s chapter i s to develop a determin i s t i c model which describes how a representat ive ind iv idua l a l l oca tes a f i xed sk i i n g budget amongst competing ski areas. F i r s t ( sect ion A) the com-plete model o f s k i e r behaviour i s presented. Then I w i l l backtrack and cons ider in turn the u t i l i t y funct ion (sect ion B) and the cost functions and r e s u l t i n g shadow pr ices fo r the s k i i n g a c t i v i t i e s ( sect ion C). The corresponding pr i ce and c h a r a c t e r i s t i c e l a s t i c i t i e s are determined in sect ion D. Sect ion E considers the hypothesis o f u t i l i t y maximizing behaviour. A. The Determin is t ic Model A model i s required which describes how the representat ive ind iv idua l a l l o ca te s a predetermined sk i i ng budget amongst a l t e rna t i ve s i t e s . The a l l o c a t i o n i s hypothesized to depend in part on the para-metric costs o f s k i i n g at d i f f e r e n t s i t e s . The s k i e r a l l oca te s his budget amongst s i t e s so as to maximize the u t i l i t y he receives from his s k i i n g given these sk i i ng co s t s . The model must exp la in what determines the u t i l i t y an ind iv idua l derives from sk i ing a c t i v i t i e s , i . e . what determines why he prefers some s i t e s over others . This i s hypo-thes ized to depend in part on the amount and types of t e r r a i n at the d i f f e r e n t s i t e s . Ski t e r r a i n is designed fo r s p e c i f i c a b i l i t y l e v e l s , thus one's a b i l i t y to u t i l i z e and enjoy an area i s determined by sk i ing a b i l i t y in conjunction with the amounts of novice, intermediate and advanced t e r r a i n at the s i t e . It is also assumed that the ind iv idua l des i res va r ie ty in his sk i ing a c t i v i t i e s , i . e . that the marginal u t i l i t y - 25 -he receives from sk i i n g at a s i t e diminishes as the amount of time he spends s k i i n g there increases . Therefore, the ind iv idua l preference order ing f o r sk i i ng a c t i v i t i e s must depend on the quant i t ie s of the d i f f e r e n t s k i i n g a c t i v i t i e s consumed, the i n d i v i d u a l ' s sk i ing a b i l i t y , and the amounts and types of t e r r a i n at the d i f f e r e n t s i t e s . The model w i l l not cons ider how the to ta l sk i ing budget was determined or how the rest o f the i n d i v i d u a l ' s income i s a l l o ca ted amongst non-ski ing a c t i v i t i e s . The a l l o c a t i o n of the sk i i ng budget can be sepa-rated, or i s o l a t e d , from the i n d i v i d u a l ' s complete choice problem, i f the to ta l s k i i ng budget is known, and i f the u t i l i t y produced by sk i i ng a c t i v i t i e s i s weakly separable from the u t i l i t y produced by other a c t i -v i t i e s (Phl ips 1974:72-77). Then, the a l l o c a t i o n amongst a l t e rna t i ve s i t e s can be completely described as a funct ion of: (1) the pr ices of the s k i i n g a c t i v i t i e s ; (2) the e f f e c t i v e physical c h a r a c t e r i s t i c s o f the s i t e s ; (3.) the sk i i ng budget; and (4) tha t i po r t i on of the preference map which re la tes to s k i i n g . Information on the pr ices of non-sk i ing a c t i v i t i e s , the c h a r a c t e r i s t i c s o f those a c t i v i t i e s , non-sk i ing pre-ferences, and to ta l income, is not required to determine the a l l o c a t i o n of the sk i i ng budget amongst s i t e s . The model o f s k i e r behaviour i s based on three postu lates . The i r support ive arguments w i l l be considered in sect ion B. Postulate I: Sk i ing i s an a c t i v i t y . The s k i e r combines sk i i ng equip-ment, the serv ices o f a sk i a rea , t ranspor ta t ion s e r v i c e s , and some o f his own time,,to produce a s i t e - s p e c i f i c sk i ing a c t i v i t y . Postulate 2: The arguments o f the u t i l i t y funct ion are: (1) the amount the ind iv idua l sk is at each o f the ava i l ab le s i t e s ; (2) the important phys ica l c h a r a c t e r i s t i c s of each of those s i t e s , i . e . the acres of - 26 -beginner, intermediate and expert t e r r a i n ; and (3) the i n d i v i d u a l ' s s k i i ng a b i l i t y . If the u t i l i t y funct ion has as i t s arguments these va r i ab le s , the share funct ions (or demand funct ions) fo r the d i f f e r e n t s i t e - s p e c i f i c s k i i ng a c t i v i t i e s w i l l be i d e n t i c a l . Postulate 3: The quant i ty o f sk i ing a c t i v i t y j and the quant i t ie s of s i t e j ' s c h a r a c t e r i s t i c s are assumed weakly separable from the amount consumed of any other sk i i ng a c t i v i t y , and i t s cha rac te r i s t i c s J The ra t iona l s k i e r solves the fo l lowing problem: (3.1) Maximize U = U(Y,A) w . r . t . Y (3.2) s . t . x = T'Y where Y = [y.;]> where y . = the amount of sk i ing a c t i v i t y j produced J J and demanded by the unindexed i n d i v i d u a l , y . i s the number of units o f ski a c t i v i t y j produced per season, where one un i t o f y- i s one day of s k i i ng at s i t e j . r = [Y-J]» where y. = the shadow pr ice (measured in units of time) of s k i i n g a c t i v i t y j . y. i s the hours required to produce one day of s k i i ng at s i t e j . It includes sk i i ng t ime, t ransportat ion time, and the time required to earn the money that i s needed to purchase (or rent) the sk i i ng equipment and the l i f t t i c k e t . x = the representat ive i n d i v i d u a l ' s to ta l time al lotment to J s k i i ng a c t i v i t i e s , i . ; : ^ y . y . . j=l J J A = [a • ] ^ kj j = l , . . . , J where - 27 -= the amount of e f f e c t i v e physical c h a r a c t e r i s t i c k that the ind iv idua l can u t i l i z e at s i t e j . S p e c i f i c a l l y : a-jj = the acres of ski runs at s i t e j which the ind iv idua l i s capable o f s k i i n g . For example, the intermediate s k i e r i s l im i ted to the beginner and intermediate runs at s i t e j . a^j = the acres of ski runs at s i t e j s p e c i f i c a l l y designed f o r the i n d i v i d u a l ' s s k i i ng a b i l i t y . For example, fo r the intermediate s k i e r , would be the acres of intermediate t e r r a i n at s i t e j . a -j j and are re fe r red to as e f f e c t i v e physical character -i s t i c s o f the s i t e because they do not depend on the physical c h a r a c t e r i s t i c s on ly , but are determined also by the i n d i v i -dua l ' s sk i ing a b i l i t y . The ind iv idua l character izes an area on the basis of the t e r r a i n that he can u t i l i z e . Sk i ing a b i l i t y removes some of the t e r r a i n from the fea s ib le choice se t , and determines the extent to which the physical charac-t e r i s t i c s can be e f f e c t i v e l y u t i l i z e d . One can define a^j = a- j j -a2j , which i s the acres of t e r r a i n at s i t e j on which the ind iv idua l can ski but that are not s p e c i f i c a l l y designed f o r the i n d i v i d u a l ' s s k i i ng a b i l i t y , a^j adds no new informat ion, but l a t e r i t w i l l be useful in expressing some e l a s t i c i t i e s . A s p e c i f i c form of the u t i l i t y funct ion was chosen so as to be cons i s tent with postulates 1, 2 and 3. 28 (3.3) U = I y A ( a j=l J where l j ' a 2 j > (3.4) n ( a i j ' a 2 j ) = a 0 + a l a l j + a 2 ^ a l j a 2 j ^ + a 3 a 2 i + a 4 a l j 2 + a 5 a 2 j S The parameters in (3.3) and (3.4) are a Q, , a 9, a^, a^, and 3 . The fo l lowing system of share equations can be obtained by maximizing the u t i l i t y funct ion (3.3) subject to the budget constra int (3.2). These shares which express the proport ion o f to ta l ski days to a s p e c i f i c s i t e 2 are: (3.5) v .* /T* = 1 / Y . ' J J . . Y3 / T ykl, LYJ * ( a 1 k , a 2 - R ) j 1 -1 a . . . » J where T* = I y k * k=l K It should be noted that a l l the share functions (3.5) are i d e n t i c a l . The only thing that varies from one s i t e ' s share funct ion to another s i t e ' s share funct ion is the value of the exogenous var iab les (Y j , a-^ ^ and a 2.j) in the funct ion (3.5). If two s i t e s (j and k) are i d e n t i c a l , that i s i f Y J = V a l j = a l k a n d a 2 j a 2 k , then s .* w i l l equal s^*, B. The S p e c i f i c Form of the U t i l i t y Function The u t i l i t y funct ion U = i . y 1 - B h ( a l j . , a 2 . ) j=l (3.3) i s a var iant o f the d i r e c t addi log u t i l i t y funct ion introduced by Houthakker (1960): - 2 9 -0 R • (3.6) U=-Xy. BJh. j=l J J I have constra ined a l l the 8 • in the addi log to be equa l , but have d i s -<J aggregated the h. and made each an i den t i c a l funct ion of the e f f e c t i v e physical c h a r a c t e r i s t i c s o f s i t e j (hj = " ( a ^ a , ^ . ) ) . This technique fo r incorporat ing the e f f e c t i v e physical c h a r a c t e r i s t i c s into the d i r e c t u t i l i t y funct ion is s i m i l a r to the method used by Pol lak and Wales (1976) to incorporate demographic c h a r a c t e r i s t i c s of the household d i r e c t l y in to the household's u t i l i t y funct ion . Pol lak and Wales assumed that a sub-set of the parameters in the i n d i r e c t funct ion was a funct ion of demographic va r i ab le s . The i r demographic var iab les depend", only on the . indiv idual , whereas mine depend on both the s i t e s and the i n d i v i d u a l . By r e s t r i c t i n g the Houthakker funct ion such that 3- = 3^ f o r a l l j and k, the u t i l i t y funct ion (3.3) jo in s the Bergson fami ly o f u t i l i t y funct ions , that i s , u t i l i t y functions which are both d i r e c t l y add i t i ve and homothetic (see Samuel son (1961 : 787-788 ) and Pol lak (1971 : 403)). (3.3) i s homogeneous of degree 8, or in other words homothetic. Chipman (1965a.:485 ) argues that "maximizing a quasi-concave funct ion U which i s homogeneous of degree 8 where 1 ^ 8 ^ 0 i s equivalent to the problem of maximizing an appropriate concave funct ion which i s po s i t i ve homogeneous of degree 1 (the new funct ion being the o ld one ra i sed to the power 1/8) " . Therefore, the i nd i f f e rence maps corresponding to the preference order ing (3.3) are i den t i c a l to the isoquant maps of the c o n s t a n t - e l a s t i c i t y - o f - s u b s t i t u t i o n (CES) production funct ion which in our notat ion i s : - 30 (3.7) u - [.Iiy/h(a1J,a2.) A deta i l ed explanat ion o f the CES production funct ion along with i t s isoquant ( i nd i f fe rence ) maps can be found in Chipman (1966:57-70). The u t i l i t y funct ion (3.3) i s d i r e c t l y add i t i ve in y . and h(a-|j 5 a2j). This implies that the marginal u t i l i t y derived from any of the sk i i ng a c t i v i t i e s i s independent of the quant i t ie s consumed of the other sk i i ng a c t i v i t i e s . Since the u t i l i t y produced by a c t i v i t y j (3.8) U j - y j B h ( . 1 J . . y ) i s independent of the quant i t i e s of the other a c t i v i t i e s produced, i t i s not improper to r e f e r to (3.8) as the s ing le a c t i v i t y u t i l i t y func t ion . D i rect a d d i t i v i t y requires tha t : 9 y * 3y * (3.9) jJ- = L^L- Phi ips (1974:62) k where 8 y i * / 3 Y k This r e s t r i c t i o n impl ies that the change in the demand fo r a c t i v i t y j induced by the change in the p r i ce of s k i i ng a c t i v i t y k i s proport ional to the change in demand fo r a c t i v i t y j induced by a change in the sk i i ng budget (T,.). The f a c to r o f p r o p o r t i o n a l i t y , y, does not depend on the a c t i v i t y whose quant i ty response we are cons ider ing ( a c t i v i t y j ) , but i t does depend on the pr i ce of the a c t i v i t y that has changed. (act iv i ty k). I hope to show that the d i r e c t a d d i t i v i t y of the u t i l i t y funct ion (3.3) i s more p l aus ib le than i t might i n i t i a l l y appear. 4 - 31 -Nomothetic preferences f o r sk i i ng a c t i v i t i e s require that a s i t e ' s share i s i n sens i t i ve to changes in the s k i i n g budget..(t), i . e . the share e l a s t i c i t i e s with respect to T equal zero. (3.10) E. * = 0 fo r a l l j This i s a r e s t r i c t i v e assumption that was made fo r p r ac t i c a l reasons. The combination of the d i r e c t a d d i t i v i t y and homotheticity r e s t r i c t i o n s means that the A l l en (1938) e l a s t i c i t y of subs t i tu t i on between any two sk i i ng a c t i v i t i e s i s constant and equal to : ( 3 J " " J K - - 1 ^ 1 ) 3'k.i o This fol lows from the fac t that : (1) the ind i f fe rence maps f o r Bergson u t i l i t y funct ions are i d e n t i c a l to the CES isoquant maps; and (2) the e l a s t i c i t y o f sub s t i tu t i on f o r the CES funct ion is (3.11) (Uzawa 1962). In terms o f preferences, C j k measures the responsiveness of the r e l a t i v e demand f o r a c t i v i t i e s j and k to a change in the marginal rate of sub-s t i t u t i o n (or r e l a t i v e p r i ce s ) between these two a c t i v i t i e s . The rest of th i s sect ion i s organized as fo l lows. Postulates 2 and 3 o f the model are discussed. The d i scuss ion then proceeds to i den t i f y a s p e c i f i c form for the u t i l i t y func t ion : J (3.12) u = F[ ^ u ( y j J a l j , a 2 j ) ] which i s cons i s tent with the three postu la tes . The sect ion concludes with a d i scuss ion of the propert ies of the s ing le a c t i v i t y u t i l i t y func t i on : - 32 -j = l , . . . , J Postulate 2 l i s t e d the arguments in the u t i l i t y func t ion . Conventional demand theory hypothesizes that the amount the ind iv idua l sk i s at each of the ava i l ab le s i t e s are arguments in the u t i l i t y func t ion . But why should the physical c h a r a c t e r i s t i c s of the s i t e s and the i n d i -v i dua l ' s s k i i ng a b i l i t y a lso appear as arguments? One would genera l ly expect that each sk i i n g a c t i v i t y would inf luence the production of u t i l i t y d i f f e r e n t l y . For example, one would not expect the u t i l i t y produced by the f i r s t t r i p to s i t e j to equal the u t i l i t y produced by the f i r s t t r i p to s i t e k. But why do the marginal u t i l i t i e s produced by the two a c t i -v i t i e s d i f f e r ? It i s because they are produced at d i f f e r e n t s i t e s . The s i t e s have d i f f e r e n t amounts o f the physical c h a r a c t e r i s t i c s , and one can reasonably assume that t h i s w i l l in f luence the amount of u t i l i t y that can be produced by s k i i n g at the d i f f e r e n t s i t e s . If d i f ferences amongst a c t i v i t i e s are q u a n t i f i a b l e , and i f they exp la in va r i a t i ons in the a b i l i t y of a c t i v i t i e s to produce u t i l i t y , then these quan t i f i ab le d i f ferences ( i . e . the physical c h a r a c t e r i s t i c s o f the s i t e s ) should appear as argu-ments in the u t i l i t y func t ion . It i s my contention that a ski area can be descr ibed in terms of three quan t i f i ab le physical c h a r a c t e r i s t i c s : (1) the acres o f novice t e r r a i n at the s i t e ; (2) the acres of intermediate t e r r a i n at the s i t e ; and (3) the acres of expert t e r r a i n at the s i t e . However, i t i s . a l s o my contention that an i n d i v i d u a l ' s s k i i ng a b i l i t y can remove somerof the ski t e r r a i n from the f ea s i b l e choice set . Ski area j can therefore be completely described in terms of the two e f f e c -t i ve phys ica l c h a r a c t e r i s t i c s , a 1 . and a?.. D i f ferences in ski areas, - 33 -for a given i n d i v i d u a l , are a t t r i b u t a b l e to var ia t ions in the quant i ty o f these two e f f e c t i v e physical c h a r a c t e r i s t i c s . The u t i l i t y the ind iv idua l derives from sk i ing i s therefore a funct ion of: (1) the amount the ind iv idua l skis at each of the ava i l ab le s i t e s ; and ( 2 ) the e f f e c t i v e physical c h a r a c t e r i s t i c s of the s i t e s . If the u t i l i t y funct ion has as i t s arguments the amount the ind iv idua l sk is at each of the ava i l ab le s i t e s and the quant i t i e s o f the e f f e c t i v e physical c h a r a c t e r i s t i c s o f the s i t e s , then the share funct ions (or demand funct ions) for the d i f f e r e n t s i t e - s p e c i f i c sk i ing a c t i v i t i e s w i l l a l l be i d e n t i c a l . This can be demonstrated by cons ider ing why the share funct ions are not normally expected to be i d e n t i c a l . Share func-t ions are genera l ly not i den t i ca l because not a l l the fac tors ( i . e . c h a r a c t e r i s t i c s ) that expla in var ia t ions in the demand for a c t i v i t i e s are included as exogenous va r i ab le s . Therefore, var i a t ions in demand across a c t i v i t i e s have to be, at l eas t p a r t i a l l y , accounted fo r by v a r i a -t ions in the form of the share equations. If a l l the factors that expla in var ia t ions in the demand for a c t i v i t i e s appear as e x p l i c i t argu-ments in the share funct ions , there i s no reason f o r those share funct ions not to be i d e n t i c a l . In the case of sk i ing there are only two reasons why the demand for one sk i ing a c t i v i t y might d i f f e r from the demand for another: t h e i r pr ices d i f f e r ; or they possess d i f f e r e n t quant i t ie s o f a-jj and a 2 j . Since these both are e x p l i c i t l y included as arguments in the share funct ions , a l l the functions are i d e n t i c a l . An a c t i v i t y ' s name ( s i t e name) w i l l not help expla in the demand for that a c t i v i t y . The names of a c t i v i t i e s have l o s t t h e i r importance because the e f f e c t i v e physical c h a r a c t e r i s t i c s of the a c t i v i t i e s have been e x p l i c i t l y included in the u t i l i t y func t i on . The ind iv idua l d i s t ingu i shes between a c t i v i t i e s - 34 -on the basis of d i f ferences in these c h a r a c t e r i s t i c s . The names supply no add i t iona l information so are extraneous to the a l l o c a t i o n problem. This i s the advantage gained when one e x p l i c i t l y includes the e f f e c t i v e physical c h a r a c t e r i s t i c s o f a c t i v i t i e s as arguments in the u t i l i t y func t ion . One can account fo r d i f ferences in the demand for a c t i v i t i e s by va r i a t i ons in the values of the indpendent var iab les (the pr ices and the e f f e c t i v e physical c h a r a c t e r i s t i c s ) in the share func t ion , rather than having the var i a t ions appear in the form of d i f f e r e n t share func-t i on s , each s p e c i f i c to only one name s p e c i f i c a c t i v i t y . Postulate 3 states that the quant ity of sk i ing a c t i v i t y j and the quant i t i e s of s i t e j ' s c h a r a c t e r i s t i c s are assumed weakly separable from the amount consumed of any other sk i ing a c t i v i t y , and i t s charac-t e r i s t i c s . The three postulates imply that the u t i l i t y funct ion i s of the form: (3.13) U = F [ u 1 ( y 1 , a - | 1 , a 2 1 ) , u 2 ( y 2 , a - | 2 , a 2 2 ) , . . . , u J ( y J , a - 1 J , a 2 j ) ] Each a c t i v i t y i s assumed to produce an intermediate product in the pro-duction of u t i l i t y , a l l of which are weakly separable from one another. In terms of s k i e r preferences the weak s e p a r a b i l i t y r e s t r i c t i o n implies that : both (1) the rate at which the ind iv idua l would be w i l l i n g to subs t i tu te an e f f e c t i v e physical c h a r a c t e r i s t i c for another at the same s i t e , and (2) the rate at which the ind iv idua l would be w i l l i n g to sub-s t i t u t e units of one of a s i t e ' s e f f e c t i v e physical c h a r a c t e r i s e s f o r s k i i n g days at the s i t e , do not depend on the quant i t ie s o f the other a c t i v i t i e s consumed or on t h e i r e f f e c t i v e physical c h a r a c t e r i s t i c s . This weak s e p a r a b i l i t y r e s t r i c t i o n was imposed to s imp l i f y the model. One must now determine what fur ther r e s t r i c t i o n s on th i s form - 35 -w i l l be s u f f i c i e n t to generate i den t i ca l demand (or share) funct ions . A l l the demand funct ions wil 1.be i den t i ca l i f , J U = F[J u ( y j , a l j , a 2 j ) ] (3.12) because the y . are treated symmetrical ly, i . e . the funct ions (3.14) u. = u(y j . ,a l j . ,a 2 j.) j = l , . . , , J are i d e n t i c a l , and they each enter the u t i l i t y funct ion a d d i t i v e l y . The var iab les in each group ( - ^ j ' a ] j » a 2 j ) °^ (3.12) are strongly separable from the var iab les in the other g roups . 0 . This strong s epa rab i l i t y across groups implies that the funct ion is d i r e c t l y add i t i ve in the u t i l i t y produced by each s i t e - s p e c i f i c sk i ing a c t i v i t y . I do not know whether t h i s c lass of preference orderings is the only c lass that w i l l generate i den t i ca l share func t ions , but i t i s a reasonable preference order ing given that i t e x p l i c i t l y considers the e f f e c t i v e physical c h a r a c t e r i s t i c s of the s i t e s . The r e s t r i c t i o n s that a l l the funct ions (3.14) are i den t i ca l i s not unreasonable because each includes as exogenous var iab les (y^a-^., a 2 j ) a l l the fac tors which might contr ibute to va r i a t ions in the amounts of intermediate output produced by the d i f f e r e n t s i t e s . When the e f f e c -t i ve physical c h a r a c t e r i s t i c s o f s i te s are e x p l i c i t l y included in the u t i l i t y func t i on , these c h a r a c t e r i s t i c s , not the a c t i v i t i e s ' names, should be the basis on which the ind iv idua l d i s t ingu i shes between a c t i v i t i e s . That i s , the u t i l i t y funct ion should have the property that the amount of t o ta l u t i l i t y produced by consuming a given bundle of J sk i ing a c t i v i t i e s i s i nvar i an t to how those a c t i v i t i e s are named. The u t i l i t y funct ion w i l l - 36 -possess th i s property when the s ing le a c t i v i t y funct ions (3.14) are i den t i ca l and when the funct ion is add i t i ve across them. The s ing le a c t i v i t y u t i l i t y funct ion i s now discussed. It has been assumed that: U j = U ( y j ' a l j ' a 2 j ) = g ( y j ) h ( a l j 5 a 2 j ) = y / h ^ . , a 2 j ) (3.8) where i h ( a l j » a 2 j ) = a 0 + a l a l j + a 2 ^ a l j a 2 j ^ + a 3 a 2 j (3.4) + a 4 a ^ 2 + a 5 a 2 j 2 The e f f e c t i v e physical c h a r a c t e r i s t i c s of s i t e j (a-|j,a 2 j) are weakly separable from the quant ity of y . consumed. This implies that the rate at which the ind iv idua l would be w i l l i n g to subs t i tu te a^j fo r a 2 ^ while holding the u t i l i t y produced by s i t e j constant, is independent of the amount the ind iv idua l has skied at the s i t e , y .^ was chosen as an appro-J p r i a te form for g(y.) because i t is a simple non- l inear funct ion of y-J J with only one parameter. The amount of d i v e r s i t y there w i l l be in the choice of sk i ing a c t i v i t i e s depends on the value of 8. The s p e c i f i c form for h(a-|j,a 2 j) i s the General ized Leont ief funct iona l form which was introduced by Diewert (1971a). This form was chosen because: (1) i t i s a simple non- l inear funct ion of the two c h a r a c t e r i s t i c s - i t has only s ix parameters; (2) i t is a second order approximation to any twice d i f f e r e n t i a t e funct ion in the a^-'s in the sense that i t does not place any a p r i o r i r e s t r i c t i o n s on e i t he r the f i r s t or second order pa r t i a l der i va t i ves of n ( a ] j » a 2 j ) a ^ ^ n e P ° i ; n t o f approximation; and (3) the function.remains f l e x i b l e but is homogeneous - 37 -of degree one in the a^j ' s i f ag, and are a l l equal to zero. In general , the s p e c i f i c form for the s ing le a c t i v i t y u t i l i t y funct ion (3.3) was chosen so as to make the model simple enough.that i t would be emp i r i c a l l y t r a c t a b l e . C. The Cost Functions fo r the Ski ing A c t i v i t i e s and The i r Result ing  Shadow Pr ices The budget cons t ra in t i s described by: J T = r ' Y I y .y, (3.2) j=l J J The pr ices of the J sk i ing a c t i v i t i e s are assumed parametric to the i n d i v i d u a l . This assumption w i l l be es tab l i shed with a d i scuss ion of the production and cost funct ions for sk i ing a c t i v i t i e s . The inputs required to produce a sk i ing a c t i v i t y at s i t e j are well s p e c i f i e d : (1) one day's use of ski area j ; (2) sk i ing equipment ( sk i s , boots, poles,, e t c . ) ; (3) t ranspor ta t ion serv ices to and from s i t e j ; and (4) the time required to s k i , and to t rave l to and from the s i t e . These four non-subst i tutable inputs are required of every ind iv idua l who wants to produce a sk i ing a c t i v i t y . The fac t that these inputs do not subs t i tu te fo r one another suggests that a Leont ie f process w i l l c l o se l y approximate the " t r u e " production funct ion fo r a s k i i n g / a c t i v i t y . Thus i t i s assumed that: (3.15) y j - n^TsneT- W V b 2 j - VCJ> where x^. = units of sk i ing equipment a l l oca ted by the ind iv idua l to the production of sk i ing a c t i v i t y j . - 38 -b-jj = the minimum number of units o f sk i ing equipment that i s required to produce one uni t of sk i ing a c t i v i t y j . One uni t of s k i i ng equipment can be ca l i b r a ted so that b - j j - l -= units (miles) of t ranspor ta t ion serv ices a l l oca ted by the ind iv idua l to the production of s k i i ng a c t i v i t y j . b 2 j = the minimum number of miles the ind iv idua l must t rave l to produce one uni t of sk i ing at s i t e j , i . e . twice the distance from the i n d i v i d u a l ' s residence to s i t e j . t . = the time devoted by the ind iv idua l to the production of sk i ing a c t i v i t y j . c . = the minimum amount of time required by the ind iv idua l to ski and to t rave l to and from s i t e j . Given that sk i i ng a c t i v i t i e s are produced by th i s Leont ief process, the p r i ce (marginal cost measured in units of time) of sk i ing a c t i v i t y j is parametric to the ind iv idua l and equals L i f t t i c k e t Ski Per mile (3.16) y . = [ (p r i ce at ) + (equipment ) + b~-(transportat ion)]/w + c-s i t e j rental fee J costs J where w = the opportunity cost (measured in $) of the i n d i v i d u a l ' s time. D. Pr i ce and Cha rac te r i s t i c s E l a s t i c i t i e s One can gain fu r ther ins ights into the nature of our p a r t i c u l a r u t i l i t y funct ion (3.3) by examining the share and demand e l a s t i c i t i e s with respect : to pr ices and c h a r a c t e r i s t i c s . The j o i n t assumptions of d i r e c t a d d i t i v i t y and homothetic ity imply l im i t a t i on s on the e l a s t i c i t i e s . The i l (1967) has shown that a d d i t i v i t y requires that the e l a s t i c i t y of - 39 -demand for a c t i v i t y j with respect to a change in the sk i ing budget (T): EV *-T J ' - l • • » J i s s t r i c t l y p o s i t i v e . This rules out i n f e r i o r sk i ing a c t i v i t i e s . Goldberger (1967) has shown that a d d i t i v i t y requires that the p r i ce e l a s t i c i t y of demand fo r y^* with respect to a change in y^, where T i s allowed to adjust so that the u t i l i t y leve l does not change: E i J7k g = g J » k = l , . . . , J i s s t r i c t l y p o s i t i v e . D i rect a d d i t i v i t y the re fo re , rules out complemen-tary sk i ing a c t i v i t i e s . Houthakker (1960) has demonstrated that a d d i t i -v i t y impl ies that the r e l a t i v e percentage response of any two sk i ing a c t i v i t i e s to a pr ice change must be the same as t h e i r r e l a t i v e response to a change in the tota l sk i ing budget, i\.e.: (3.17) V Y k _ V T _ f j k J7k, ir-k E y . * Y k E y _„ x a i k i , j , k = l , . . . , J As was noted e a r l i e r , homotheticity impl ies that : E ^ = 0 j=l J (3.10) j or a l t e r n a t i v e l y in terms of quant i ty demanded: (3.18) E i = 1 j = l , . . . , J The combination of d i r e c t a d d i t i v i t y and homothet ic i ty, as e x p l a i n e d . e a r l i e r , impl ies that the A l l en (1938) e l a s t i c i t y of subs t i tu t ion between any two sk i ing a c t i v i t i e s i s constant and equal to : - 40 -This combination therefore a l so impl ies that (3.17), the r e l a t i v e percentage response of any two sk i ing a c t i v i t i e s to a pr i ce change at another s i t e , equals one. I am p a r t i c u l a r l y in teres ted in how the s i t e ' s share responds to a change in the p r i ce or one of the c h a r a c t e r i s t i c s of that s i t e or one of i t s subs t i tu tes . For th i s reason I report the s p e c i f i c share, rather than demand, e l a s t i c i t i e s required by the preference order ing (3.3). S i te j ' s share e l a s t i c i t y with respect to a change in the cost to ski at s i t e j i s : (3.19) E = ( l / ( 3 - l ) ) [ l - s , * ] j Y J J S i te m's share e l a s t i c i t y with respect to a change in the cost to ski at s i t e j i s : (3.20) E = - s , * ( l / ( 3 - l ) ) m Y j J S i te j ' s share e l a s t i c i t y with respect to a change in the amount of sk iab le t e r r a i n at the s i t e holding a 9 ^ constant i s : (3.21) E c „ s r- au '-3 3 h ( a 1 , , a 2 , ) = - ( l / ( B - l ) ) [ l-Sj*] ^ / h ( a i r a 2 j ) a 2 j-u u 9 h ( a l i 5 a 2 i ) = "S* Yj ^j /h(alj' a2j> Where 9h(a a ) 7 h ( a 1 j a a 2 j . ) 41 -can be in terpreted as a monotonic transformation of the proport ionate amount by which the u t i l i t y of a ski day increases when the sk iable t e r r a i n at the s i t e increases byvone acre. S i te j ' s share e l a s t i c i t y with respect to a change in the amount of a 2 j at an area, holding sk iab le t e r r a i n constant, i s : a h ( a i j , a 2 j ) (3.22) "s .* a 0 . J 2j - - (1 / (3 -1 ))[l-s-*] V 0 3 -3a 2j / . • h ( a l j f a 2 j ) = -E 3 h ( a l j f a 2 j ) - 3 a 2 j 1 h ( a l j ' a 2 j ) ' lj the acres of t e r r a i n at s i t e j on which the ind iv idua l can s k i . a 2 j = the acres o f t e r r a i n at s i t e j designed s p e c i f i c a l l y f o r the i n d i v i d u a l ' s s k i i ng a b i l i t y . E a r l i e r a ^ was defined as the acres of t e r r a i n at s i t e j on which the ind iv idua l can s k i , but that ;are not s p e c i f i c a l l y designed fo r his sk i ing a b i l i t y . (3.23) l l j a 2 j + a 3 j Therefore Y a i j (3.24) This resu l t s because E "s .* a 0 . J 3j o measures the response a t t r i b u t a b l e a?i=r "j J l j 1 a 2j=0 to increas ing sk iab le t e r r a i n (a-jj) while holding a 2 j . constant, therefore the Aa-j.- = A a 3 j . Another c h a r a c t e r i s t i c e l a s t i c i t y can therefore be i d e n t i f i e d . S i te j ' s share e l a s t i c i t y with respect to a change in the amount o f a 2 ^ at an area while holding a^- constant ( i . e . Aa 2 ^ = Aa-jj) i s : - 42 -(3.25) V 3 2 j § 3 j =(T E s j* a l j ! i2f-0 + S* a 2 j § i r o S i te m's share e l a s t i c i t y with respect to a change in the amount of sk iab le t e r r a i n at s i t e j (a-^) holding a,,., constant i s : 3h (a , . , a 2 . ) 5-1)) —11 J (3.26) m I j = s . * ( l / ( 3 - l ) J § 9 . = 0 J 2j 9a l j / h (a , . ,a l j ' u 2 j = -E sm * Y-i 3 h ( a 1 j 9 a 2 j ) 9a l j / . h ( a i j , a 2 j . ) S i te m's share e l a s t i c i t y with respect to a change in the amount of a 2 j hold ing a ^ constant i s : 3 h ( a l j , a 2 j ) (3.27) m 2 j = s . * ( l / ( 0 - l ) ) 3 a S l j = 0 J d 3 2 j / h ( a l j 9 a 2 J ) "S * Y . m ' j 3 h ( a l i 5 a 2 i ) 9 ^ - / h ( a l j ' a 2 j ) S i te m's share e l a s t i c i t y with respect to a change in the amount of a 2 j while holding a^j constant i s : (3.28) V a 2 j § 3 j = 0 = \ * a l j | § 2 j = 0 + V a 2 j § i r ° Examination of these share e l a s t i c i t y funct ions leads to a number of general conc lus ions: (1) t h e i r magnitudes are a l l inverse ly re l a ted to . the absolute value of B; (2) the absolute value of the own e l a s t i c i t i e s are inverse ly re la ted to the magnitude of t h e i r predicted shares; (3) the cross e l a s t i c i t i e s f o r s i t e m are d i r e c t l y re l a ted to the magnitude of the pred icted share whose p r i ce or c h a r a c t e r i s t i c i s - 43 -changing ( s i t e j ) , but not d i r e c t l y re l a ted to the magnitude o f s i t e m's predicted share; and (4) the c h a r a c t e r i s t i c s e l a s t i c i t i e s are a l l d i r e c t l y re la ted to a monotonic transformation of the proport ionate amount by which the . .u t i l i t y of a ski day increases when the quant ity of the \ appropriate e f f e c t i v e physical c h a r a c t e r i s t i c increases by one acre. E. The Hypothesis of U t i 1 i t y Maximizihg Behaviour The fo l lowing r e s t r i c t i o n s on the form of the d i r e c t u t i l i t y funct ion are impl ied by the theory of u t i l i t y maximizing behaviour: (1) U(Y*,A) is a continuous f i n i t e funct ion f o r Y * » 0 j ; (2) Monotonicity: 9 U ( Y * ' A ) >0 for a l l j , j = l , . . . , J ; (3) U(Y*,A) is a quasi concave funct ion f o r Y*>>0j, i . e . the, i nd i f f e rence . curves must be quasi convex. The chosen u t i l i t y funct ion (3.3) imposes r e s t r i c t i o n (1), but f u l f i l l m e n t of r e s t r i c t i o n s (2) and (3) depends on the values of the parameters ( 3 and oig) in the funct ion . The monotonicity and curvat ive propert ies w i l l be f u l f i l l e d i f and only i f 1>B^U and h ( a 1 j , a 2 j ) i s of the same sign f o r a l l j . Pol lak (1971:402) notes that the Bergson f u n c t i o n , U = £ y •^h(a 1 . , a 9 •), w i l l f u l f i l l the monotonicity j=l J J J and curvat ive propert ies i f e i t h e r 3 i s negative and h ( a ^ j , a 2 j ) is negative f o r a l l j , or i f 1>3>0 and h (a^ - , a 2 j ) i s po s i t i ve f o r a l l j . But these two condit ions can always be f u l f i l l e d by the system of share equations (3.5) i f 1>3^0 and i f h ( a^ - , a 2 j ) i s o f the same sign fo r a l l j because the shares are invar iant to mul t ip ly ing a l l of the h funct ions by e i the r plus or minus one. - 44 -Footnotes -•Chapter 3 A funct ion i s weakly separable across groups i f the marginal rate of subs t i tu t i on between any two var iab les belonging to the same group is independent of the value of any var iab le in any other group (Phi ips 1974:68). The system of demand equations (y.* = j = l , . . . , J ) corresponding to the preference order ing (3.3) i s reported by Pol 1ak (1971:403). The share equations are derived by d i v i d ing y . * by j l y k * . j=i k • Many of the terms, inc lud ing the sk i ing budget T , cancel out in the equat ion. It should be noted that (3.5) i s homogeneous of degree zero in the s ix a parameters. A funct ion is homothetic in the y^'s i f i t can be wr i t ten : U - G [ g ( y r . . . , y j ) ] (Phi ips 1974:86-87) where G i s a f i n i t e , continuous a n d . s t r i c t l y monotonical ly increas ing funct ion of one va r i ab le with G(0)=0, and where g i s a homogeneous funct ion of the J va r i ab le s , y - | , . . . , y j . R e s t r i c t i o n (3.9) can a l t e r n a t i v e l y be expressed in terms of the r e s t r i c t i o n s i t places on the Slutsky equation. The Slutsky equation, in genera l , i s : 9y k 8 = 0 3y k J k 3T 3y j *^y k * K ^ ' ^ 2T where A = marginal u t i l i t y o f the sk i ing budget.. The f i r s t term: 8 y J * 8 y k is r e fe r red to as the " s p e c i f i c subs t i tu t i on e f f e c t " ; the second term j u - . ^ i * 8 y k * -ST") i s c a l l e d the "general subs t i tu t i on e f f e c t " . The terminology was introduced by Houthakker (1960:248), but the concepts are more c l e a r l y explained by Phl ips (1974:47-53). When the u t i l i t y funct ion i s d i r e c t l y a d d i t i v e , the " s p e c i f i c subs t i tu t i on e f f e c t " i s zero and ?a 3 y ^ oy* . 0 = ~ X ^ ^ ~ ~ W ~ Houthakker (1960:248) - 45 -The cross subs t i tu t i on e f f e c t does not vanish, i t i s non-zero because a l l the sk i ing a c t i v i t i e s compete fo r the i n d i v i d u a l ' s sk i ing budget. The u t i l i t y funct ion (3.3) belongs to the family of u t i l i t y funct ions defined by (3.12). The share equations cons is tent with (3.12) are invar iant to the form of the F func t ion . A funct ion is s t rong ly separable across groups i f the marginal rate of subs t i tu t i on between two var iab les belonging to d i f f e r e n t groups is independent of the quant i ty o f any var iab les in another group (Phl ips 1974:69). - 46 -Chapter 4 A STOCHASTIC MODEL OF SKIER BEHAVIOUR The purpose of th i s chapter ( sect ion A) i s to develop a stochas-t i c model o f s k i e r behaviour which describes how an ind iv idua l a l l oca te s a f i xed sk i i n g budget amongst competing areas. The model w i l l be used to pred ic t the proport ion of ski days that ind iv idua l i w i l l spend at s i t e j , i . e . the shares which express the proport ion of v i s i t s to s i t e j . Sect ion B considers the maximum l i k e l i h o o d estimates of the parameters in th i s s tochas t i c model. The chapter ends ( sect ion C) with a d i scuss ion o f a method that can be used to t e s t the s t a t i s t i c a l s ign i f icance, o f these maximum l i k e l i h o o d est imates. A. The Stochast ic Model The proport ion of ski days that the ind iv idua l spends at s i t e j i s s. = y - / T , where T i s the number of days the ind iv idua l skied at the J s i t e s during the season, and y . i s the number of days the ind iv idua l skied at s i t e j . s. can be d iv ided into two components: a determin i s t i c component, s-*, and a s tochas t i c component. One might a t t r i b u t e the s tochas t i c nature of s^ . to the inherent i r r e p r o d u c i b i l i t y of soc ia l and b i o l o g i c a l phenomena. Given the complexity of behaviour, i t i s impossible to cons ider e x p l i c i t l y in a determin i s t i c model a l l of the thousands of neg l i g i b l e factors that in f luence the i n d i v i d u a l ' s dec i s i on . These omitted va r i ab l e s , each with an i n d i v i d u a l l y small e f f e c t , together introduce a s tochas t i c component into the model. Chapter 3 was devoted to a d i scuss ion of S j * , the determin i s t i c component of the model. There i t was argued that s^ .* was a funct ion of the pr ices and e f f ec t i ve , physical c h a r a c t e r i s t i c s o f the J ski areas, such - 47 -that : J - • Y k h ( a l i , a 2 . ) g r r (4.1) s.* = s * ( Y . 9 a l i 5 a 2 i ; r ; A : 9 ) = 1/ I [\, I J J * j = l , . . . . J J J U k = 1 Y j h ( a 1 k , a 2 k ) where the vector o f parameters, 0, = [aQ,a-j.,a 2,a2,a^,ag,6]. Let 0^ re fe r to the element of 0, r = l , . . . , 7 , where (4.2) n ( a T j » a 2 j ) = a 0 + a l a l j + a 2 ^ a l j a 2 j ^ 2 + a 3 a 2 j » a 4 a l j 2 + a 5 a 2 j 2 s.* i s the expected proport ion of his to ta l ski days that the ind iv idua l w i l l spend at s i t e j . The actual proport ion often d i f f e r s from i t s expected value because o f random errors by the i n d i v i d u a l . We complete th i s model of s k i e r behaviour by spec i fy ing the i n d i v i d u a l ' s dens i ty funct ion fo r the s tochas t i c component o f the share equations. The density funct ion has to be chosen quite c a r e f u l l y . Shares, be d e f i n i t i o n , possess ce r t a i n p roper t ie s , and the density funct ion chosen f o r t h e i r s tochas t i c components must in no way be incons i s tent with these proper t ie s . Our d e f i n i t i o n of the share, s- = y - / T , requires that i t can J 3 take onionly one of (T+1) d i s c re te values in the 0-1 range, where J £ s- = 1. This fol lows because ski days can only be consumed in integer j=l 3 J (day) increments and because T = £ y - . For example, i f T=4, s- can take j=l J J on only one of f i v e d i s c re te values (.00, .25, .50, .75, 1.00). Each s. i s a lso p e r f e r c t l y co r re l a ted with the other J - l s i t e - s p e c i f i c shares. A standard assumption in econometric work is that the random var iab le is normally d i s t r i b u t e d . Unfortunately, such an assumption i s incons i s tent with the propert ies of the s . ' s . The normality assumption 3 requires that the random var iab le be continuously d i s t r i b u t e d from - o o to +°° where there is a po s i t i ve p robab i l i t y that shares w i l l be outside the 0-1 range. This i s incons i s tent with the requirement that each s. - 48 -is d i s c r e t e l y d i s t r i b u t e d between zero and one. The normal d i s t r i b u t i o n also assumes that the shares are symmetrical ly d i s t r i b u t e d . This seems un l i ke l y f o r shares with expected values near zero or one. Even when the populat ion i s not normally d i s t r i b u t e d , the normal d i s t r i b u t i o n can often be j u s t i f i e d as the appropriate density funct ion by an appeal to the Central L imit Theorem. The Central L imit Theorem s ta tes , in essence, that the d i s t r i b u t i o n of sample means from a sample of T independent ob-servat ions w i l l approximate a normal d i s t r i b u t i o n fo r large values of T (Hays and Winkler 1970:292-296). If T is large enough, the normal d i s -t r i b u t i o n is a good approximation o f the sampling d i s t r i b u t i o n of the mean, but unfortunate ly , the average s k i e r does not ski enough times in a season to invoke th i s j u s t i f i c a t i o n fo r the use of the normal density funct ion.^ The standard normality assumption must thus be re jec ted . It i s assumed that the i n d i v i d u a l ' s density funct ion fo r s., j = l , . . . , J , i s a multinomial where: (4.3) f ( s r s 2 S J ; T : 0 ) = - j l i — IT ( S J ) Y J n y . l j = 1 j=l J where s j * = s * ( Y j , a l j . , a 2 j ; r ; A ; e ) j=l J (4.1) Wilks (1962:139), amongst o thers , has shown that i f the s. are d i s t r i -3 buted as a multinomial then: (4.4) E ( S j ) = s » * J=1 , . . . , J (4.5) v a r ( S j * ) = ( S j * ) ( l - s j * ) / T j= l .••• .J - 49 -(4.6) cov ( s . s k ) - - ( s ^ ) ( s k * ) / T . The d i s t r i b u t i o n of the s. w i l l be skewed, except in the case where s.* = X f o r a l l j . The variance of s. f a l l s toward zero as s. approaches e i the r i t s upper l i m i t of one, or i t s lower l i m i t of zero. As the number of t r i p s increases , the variances and covariances of the s. 3 decrease.. The covariance matrix s a t i s f i e s the condi t ion that J 1 E(s.s. )=0, but the signs on a l l the covariances (j^k) are required k=l J K to be negat ive. In th i s respect, the multinomial i s more r e s t r i c t i v e than the normal d i s t r i b u t i o n . This i s one of the few unappealing aspects o f the multinomial d i s t r i b u t i o n . The multinomial was chosen as an appropriate density funct ion for the i n d i v i d u a l ' s shares because i t is s imple, and because i t i s cons i s tent with a l l the aforementioned propert ies of the density funct ion of s.. The multinomial places the fo l lowing required r e s t r i c t i o n s on the 3 random var iab le s.: (1) the expected value of s. is s.*; (2) s. i s J J J J l im i ted to (T+1) d i s c re te values, a l l of which are in the 0-1 range; J and (3) the s. are cor re l a ted across s i t e s in such a way that £ s. = 1. J j=l J . B. Maximum L ike l ihood Estimation of the Parameters in the Stochast ic Model Presented with a c ros s - sec t iona l sample of s k i e r s , I want to use that sample to obtain estimates of the Q parameters in my s tochas t i c model o f s k i e r behaviour. Maximum l i k e l i h o o d estimates are appropriate because under qu i te general condit ions they possess des i rab le asymptotic s t a t i s -t i c a l p roper t ie s . These propert ies w i l l be discussed in a moment, but f i r s t l e t me b r i e f l y descr ibe the maximum l i k e l i h o o d technique. Define a density f unc t i on , f (X :G) , fo r the vector of random var iab les X charac-- 50 -t e r i z e d by a vector o f parameters 0. Given a sample of N independent observations of the vectors (X-|,...,X^) from th i s density func t i on , the maximum l i k e l i h o o d estimate of 9 i s the value of 0 that would most often generate the observed sample. In other words, the maximum l i k e l i h o o d est imate, 0, i s that value of 0 for which the p r o b a b i l i t y ( i . e . the l i k e l i h o o d ) of a given set o f sample values i s at a maximum. The probabi -l i t y o f observing a sample of N independent observations (X^, . . . ,X N ) from f(X:0) is def ined by the j o i n t p robab i l i t y d i s t r i b u t i o n : N (4.7) L = L(X, ,...,X.,:0) = n f (X- :0) 1 I N i=l 1 For a given vector of parameters 0, th i s j o i n t density funct ion defines the p r o b a b i l i t y of observing a s p e c i f i c sample of the X ' s , but (4.7) can a l so be in terpreted as a l i k e l i h o o d funct ion . 0 can be interpreted as the vector o f random var iab les in the funct ion (4.7) ..conditional on a given sample of IM observations of X. Given th i s i n t e r p r e t a t i o n , the funct ion def ines the l i k e l i h o o d ( i . e . p robab i l i t y ) that a s p e c i f i c 0 generated the sample. The maximum l i k e l i h o o d estimate is that value of 0 which maximizes the l i k e l i h o o d funct ion (4.7) , a property which I f i n d i n t u i -t i v e l y appeal ing. In add i t i on , to th i s i n t u i t i v e appeal , under su i tab le regu l a r i t y cond i t i ons , the maximum l i k e l i h o o d est imator possesses the 2 fo l lowing des i rab le p roper t ie s : (1) The maximum l i k e l i h o o d est imate, 0, i s a cons i s tent estimate of the 3 true value of the parameter o . (2) The maximum l i k e l i h o o d est imate, 0, i s asymptot ica l ly normally d i s t r i b u t e d with mean 0 Q and covariance matrix (T ) given by R~^(0) where (4 .8) R(0) = -U^°£S :Qh - 51 -If des i red , these facts can be used to obtain asymptotic t s t a t i s t i c s for each of the estimated parameters. (3) The maximum l i k e l i h o o d est imate, § , i s asymptot ica l ly e f f i c i e n t in the sense that i t asymptot ica l ly at ta ins the minimum variance bound R ~ \ Q ) , Prev ious ly i t was assumed that the i n d i v i d u a l ' s density funct ion for the s i t e ' s share of the to ta l number of ski days was a multinomial where f ( s l i , a 2 i , . . . , s J i ; T i : G ) i= l , . . . ,N (4.3) T, I J y . . n ' ( s . * ) J" 1 n ! 3 j=i J where s j i * = s * ^ i - a u i ' a 2 j i ; r i - A r 6 ) i:i;:::;JN c- 1) The i subscr ipt i s added to make the density funct ion s p e c i f i c to the i n d i v i d u a l , where i = l , . . . , N . It i s now assumed that the choice of shares by one ind iv idua l i s completely.independent of .any other i n d i v i d u a l ' s 4 cho ice. < 4- 9) E ( s j i s k * . ) = 0 i^;i;i ; ^ , N Given a l l the information required f o r the s p e c i f i c a t i o n of the j o i n t density funct ion fo r the s . . , the l i k e l i h o o d funct ion fo r a sample of N sk iers i s : M (4.10) L - n f ( s - | i , s 2 i , . . . , s J i ; T i : 0 ) i=l - 52 -The maximum l i k e l i h o o d estimates of the parameters in the s tochas t i c model o f s k i e r behaviour f o r a p a r t i c u l a r sample is that 0 which g l o -ba l l y maximizes the l i k e l i h o o d funct ion (4.10). Rao (1965:295-296) has shown that these estimates w i l l have a l l the aforementioned des i rab le 5 asymptotic p roper t ie s . The parameters that maximize the l i k e l i h o o d funct ion (4.12) a lso maximize the log o f that l i k e l i h o o d func t i on , so for reasons of s i m p l i c i t y , the maximum l i k e l i h o o d estimates w i l l be obtained by maximizing the log of the l i k e l i h o o d funct ion (4.11) rather than the l i k e l i h o o d funct ion i t s e l f . N (4.11) l = £og .L = J log f (s-, . , s 2 i , . . . , S j . ;T. :0) N T. ! J = I U o g f - j - i — ) + I y . - aog(s..*)] i=l i=l J J n y...| J j=l J 1 Since the f i r s t expression in (4.11): T i ! ^og( j ),. n y ^ i j=l J i s not a funct ion of 0, the maximum l i k e l i h o o d est imates, 0, i s therefore that value o f 0 which maximizes the funct ion 1*. N J (4.12) I* = .L .1 Y j i wg(s..,*). i - l j = 1 J where S j i * = S * ^ j i ' a l j i ' a 2 j i ' r i ^ A i : 0 ) (4.1) - 53 -and G = [ a 0 > a ^ , a 2 , a 3 , a ^ , a 5 > 3 ] (4.1) i s homogeneous of degree zero with respect to the a parameters, therefore the maximum l i k e l i h o o d estimates are not uniquely i d e n t i f i e d . To r e c t i f y t h i s s i t u a t i o n , one of the a parameters w i l l be set equal to one. It does not matter which i s set equal to one. C S i gn i f i cance Tests and Hypothesis Test ing I would l i k e to use the maximum l i k e l i h o o d estimates of the parameters in my s tochas t i c model of s k i e r behaviour to tes t s t a t i s t i c a l l y whether pr ices and c h a r a c t e r i s t i c s play an important ro le in how the ind iv idua l a l l o ca te s his ski days amongst s i t e s . The basic hypothesis o f t h i s d i s s e r t a t i o n i s that pr ices and c h a r a c t e r i s t i c s do play an important ro le in the s k i e r ' s a l l o c a t i o n . The corresponding nu l l -hypo-thes i s i s that pr ices and c h a r a c t e r i s t i c s play no r o l e , or in other words, the nu l l -hypothes i s assumes that the ind iv idua l randomly a l l oca tes his ski days amongst s i t e s independent of the s i t e ' s pr ices or charac-t e r i s t i c s , such that : i=.l. N j=l • s.. • »J where J is the to ta l number of ski areas. I. would l i k e to compare s t a -t i s t i c a l l y these two hypotheses in*a r igorous manner in the hopes of concluding that the pr i ce and c h a r a c t e r i s t i c determining model of sk ie r behaviour pred ic t s a l l o c a t i o n a l behaviour s i g n i f i c a n t l y bet ter thanothe model impl ied by the nul1-hypothes is . Since (4.13) is a nested hypothesis o f the determin i s t i c model of s k i e r behaviour (4.1) and (4.2) , a l i k e l i -(4.13) s * ^ - 54 -hood r a t i o tes t can be used to compare s t a t i s t i c a l l y the r e l a t i v e exp la -natory power o f these two models. I f i t is assumed that a^=l, and a^=a2=a2=a^=ag^(l/(8-l ))^0,.then (4.1) reduces to (4 .13) . The l i k e l i h o o d r a t i o tes t works as fo l lows. Advantage is taken of the fac t that : (4.14) -2 in U X 2 4- f o r large samples (Mood and Grayb i l l 1963:301 ) where (4.15) A - j—Q-LH where L u = the value of the l i k e l i h o o d funct ion fo r the constrained case (the nu l l - hypo thes i s ) . = the value of the . l ikel ihood funct ion f o r the unconstrained case. t = the number of r e s t r i c t e d parameters. If -2 £n A i s s i g n i f i c a n t l y d i f f e r e n t from zero, the nul1-hypothesis i s r e j e c t e d . There are a number o f i n te re s t i ng hypotheses nested in (4.1) and ( 4 .2 ) : the nested hypothesis that a^^ l , and =a,2=a3=0^=0^=0 implies that pr ices play a ro le in a l l o ca t i ona l behaviour, but the e f f e c t i v e phys ica l c h a r a c t e r i s t i c s do not; the nested hypothesis that CXQ=1, and c ^ a - ^ a ^ O implies that only e f f e c t i v e physical c h a r a c t e r i s t i c a2j has no in f luence ; and the nested hypothesis that 6=0 impl ies that the A l l en e l a s t i c i t y o f subs t i tu t i on is one f o r a l l pa irs of a c t i v i t i e s ( i . e . that the u t i l i t y funct ion i s of the Cobb-Douglas form). L ike l ihood r a t i o tests - 55 -can be used to tes t i f any or a l l of these nul l -hypotheses should be re jec ted . Unfortunate ly, the nu l l -hypothes i s that the e f f e c t i v e physical c h a r a c t e r i s t i c s matter but that pr ices do not, i s not nested in (4.1) and (4.2). The assumption that pr ices do not matter requires that (1/(3-1))=0 in (4.1), th i s guarantees that the e f f e c t i v e physical c h a r a c t e r i s t i c s also have no e f f e c t ((4.1) reduces to (4.13) when (1/(3-1))=0 independent of the values of the a parameters). Therefore no way ex i s t s to tes t sepa-ra te l y th i s poss ib le nu l l -hypothes i s . - 56 -Footnotes - C h a p t e r 4 1. The average ind iv idua l in my sample took only 9 ski t r i p s during the season. This i s not a large enough value to j u s t i f y an appeal to the Central L imi t Theorem. 2. See e i t he r Kendall and Stuart (1962:39-46, 55) or Wilks (1962:359-363, 379-381) fo r a d i scuss ion of the proofs and the regu l a r i t y condi t ions necessary f o r the proofs. 3. e i s a cons i s tent estimate of 0 O i f § r i s a cons i s tent estimator of 8 r f o r a l l r. 9 r i s cons i s tent i f f o r any a r b i t r a r y pos i t i ve number P ( | § r - 6 |<e)-*-l, as N-x» Hays and Winkler (1970:311 ). 4. The problem of poss ib le congestion i s ignored. 5. The regu l a r i t y condi t ions on (4.1) ( s - - * = s * (YjM >ai j -j >a2j i » r i : Q ) s u f f i c i e n t fo r theimaximum l i k e l i h o o d estimates of 0 to possess these propert ies when S j i s mult inomia l ly d i s t r i b u t e d (Rao 1965:295-296) are: (1) The funct ion (4.1) , . i = l , . . . ,N i s continuous in 0. (2) The funct ion (4.1), i=l N admits f i r s t order p a r t i a l de r i va -t i ve s with respect to 9 which are continuous at 0 O . (3) The funct ion (4.1), i=l N is t o t a l l y d i f f e r e n t i a t e with respect to 0 at 0 Q . (4) The information matrix I = i^rs^ 1 S non-s ingular at 0 Q where J •, 3(s,.*) 3(s..*) i = y _J J 1 J 1 i=i i\i V s A s..* 30 30 -j=l j i r s (5) There i s a unique ( i d e n t i f i a b l e ) 0 O which generates the true vector o f shares, S 0 , i . e . the vector S 0 cannot be generated with a number of d i f f e r e n t values o f 0. This i d e n t i f i a b i 1 i t y condi t ion b a s i c a l l y requires that (4.1) be responsive to changes in 0. In terms of the l i k e l i h o o d func t ion , th i s condi t ion requires that a unique global maximum ex i s t s . See Rao (1965:295-299) for more d e t a i l s and a d i scuss ion o f the proofs . Condit ion (1) i s g l oba l l y v i o l a ted because (4.1) i s discontinuous at the point where 3=1. If 3=1, then (4.1) i s undefined, but condi t ion (1) w i l l hold l o c a l l y i f goJn. Condit ions (2) and (3) w i l l be f u l -f i l l e d i f 3Q^1. The f u l f i l l m e n t of the f i v e condit ions i s genera l ly sample s p e c i f i c . Condit ions (4) and (5) depend on how responsive (4.1) i s to changes in e, which w i l l in turn depend on the amount of c o r r e l a t i o n amongst the independent var iab les in the funct ion . - 57 -Chapter 5 DATA Estimation of the share equations f o r J s i t e - s p e c i f i c s k i i ng a c t i v i t i e s requires three types o f data: (1) a c ro s s - sec t iona l survey of sk iers which de ta i l s t h e i r s k i i ng a c t i v i t i e s f o r an en t i re season at the J s i t e s ; (2) data on the acres of nov ice, intermediate and advanced t e r r a i n a t each of the s i t e s ; and (3) p r i ce data ( l i f t t i c k e t p r i c e s , t ransportat ion cos t s , e t c . ) . A. A Cross-Sect ional Survey of Skiers The required c ros s - sec t iona l sample of sk iers must include a complete record of each i n d i v i d u a l ' s s k i i n g a c t i v i t i e s during a pre-s p e c i f i e d time per iod. A complete record embodies the number o f ski t r i p s made, the duration o f each, and the ski s i t e or s i t e s v i s i t e d during each t r i p . The s k i e r ' s planning horizon i s assumed to be one season. A season i s short enough so there i s not s u f f i c i e n t time fo r most i n d i v i d u a l s ' s k i i n g a b i l i t y to vary apprec iab ly , but long enough to allow fo r the fac t that numerous sk iers do not r e s t r i c t themselves to one s i t e , but plan on v i s i t i n g a number of s i t e s throughout the season. The survey must a lso c o l l e c t data on the personal c h a r a c t e r i s t i c s of each ind iv idua l sampled. S p e c i f i c a l l y we need to know: (1) the i n d i v i d u a l ' s s k i i ng a b i l i t y ; (2) the i n d i v i d u a l ' s occupation and hourly wage ra te ; and (3) the l oca t ion of the i n d i v i d u a l ' s res idence. The best ava i l ab le s k ie r survey i s the one done in 1968 by the Denver Research Ins t i tu te (DRI). This survey was part o f an extensive ana lys i s o f the Colorado t o u r i s t market t i t l e d A Prof i1e of the Tour i s t  Market in Colorado 1968 (Denver Research Ins t i tu te 1968a). Five thousand - 58 -skiers were surveyed by mail at the end of the 1967/68 season. This resulted in approximately three thousand returned and completed interviews. A copy of the questionnaire is included in Appendix A. The names and addresses of the f i ve thousand skiers were i n i t i a l l y obtained through the use of short contact interviews conducted by DRI personnel on a random basis at Colorado ski areas during the 1967/68 season. Information ob-tained from each mail interview was coded and placed on computer tape along with the results of the other surveys undertaken as part of the overal l tour i s t study. Colorado Parks and Outdoor Recreation, an agency of the State of Colorado, has furnished me with a copy of these tapes. The DRI survey co l lected the fol lowing data on each indiv idual sampled: (1) a, complete record of each i nd i v i dua l ' s sk i ing a c t i v i t i e s during the 1967/68 season (question No. 13 on the questionnaire); (2) the i nd i v i dua l ' s sk i ing a b i l i t y (question No. 10); (3) information on the i nd i v i dua l ' s family and the i r sk i ing habits (question No. 11); (4) the i nd i v i dua l ' s occupation and approximate earning a b i l i t y (questions No. 3 and No. 5); and (5) the locat ion of the i nd i v idua l ' s residence. Skiing a c t i v i t i e s are recorded for an entire season. It should be noted that the question ascertaining sk i ing a b i l i t y did not c l a s s i f y a s k i e r ' s a b i l i t y as necessari ly equivalent to the type of te r ra in he most enjoyed, but rather equivalent to the type of te r ra in he is capable of s k i i n g J The residences of the indiv iduals sampled are widely dispersed across North America, and th i s dispersion is ref lected in the i r choice of ski s i t e s . My estimation of the share equations w i l l be based on a sub-sample of the skiers questioned by DRI. This sub-sample w i l l be re s t r i c ted to include only s ingle post-secondary Colorado students, who do not belong - 59 -to a ski club and whose family does not own a dwel l ing at a ski area. There are 163 i nd i v idua l s in t h i s group. My sample does not inc lude Colorado adu l t s , Colorado c h i l d r e n , and i nd i v idua l s who e i the r res ide or attend school outs ide of the State of Colorado. The reasons f o r examining the behaviour of only Colorado post-secondary students are as fo l lows: (1) The technology I s p e c i f i e d fo r producing sk i ing a c t i v i t i e s most accurate ly describes the production of one day t r i p s . Therefore, I would l i k e to l i m i t the sample to a group that predominantly takes day t r i p s . S ingle post-secondary Colorado students come c lo ses t to meeting th i s c r i t e r i o n . Colorado students averaged 1.55 days of sk i ing per t r i p , Colorado adults averaged 1.70 days, ou t -o f - s t a te students averaged 4.03 days, and o u t - o f - s t a t e adults averaged 4.25 days of sk i ing per t r i p (Denver Research In s t i tu te 1968a:74). (2) I have modelled the behaviour of the i n d i v i d u a l , not the behaviour of the household. The ind iv idua l s are i m p l i c i t l y assumed s e l f i s h , so the consumption of other family members does not enter t h e i r u t i l i t y func t ion . The s k i e r sample should therefore be l im i ted to include only i nd i v idua l s whose dec i s ion to ski i s made by the ind iv idua l alone and not by the fami ly . In add i t i on , est imation i s made simpler i f one can assume that the i n d i v i d u a l ' s choice of s i t e s i s independent of any other i n d i -v i d u a l s choice. The group that come c lo ses t to meeting t h i s c r i t e r i o n i s again s ing le post-secondary Colorado students, who do not belong to a ski c lub , and whose fami ly does not own a ski dwel l ing. The average number of family members that accompanied a Colorado student on a ski t r i p was 0.34; i t was 2.1 f o r Colorado adults (Denver Research In s t i tu te 1968c:6). One might a l t e r n a t i v e l y argue that the student ' s dec i s ion as to where to ski i s h ighly inf luenced by his peers, so the i m p l i c i t - 60 -assumption of s e l f i s h preferences i s not appropr iate, but I would disagree and make the unsubstantiated claim that the peer group.is made up of i nd iv idua l s who a l l want to ski at the same s i t e . Students who want to ski at other s i t e s ski with other groups: -(3) The value of each i n d i v i d u a l ' s time must be estimated so that the costs o f v i s i t i n g each of the s i t e s by each of the ind iv idua l s can be determined. The DRI data include annual income data, but not hourly wage ra tes . It i s quest ionable whether income data alone are s u f f i c i e n t to construct r e l i a b l e estimates of the value of each i n d i v i -dua l ' s time. The a l t e r n a t i v e to construct ing such an opportunity cost var iab le which var ies across i nd i v i dua l s , is to choose a sub-group of the populat ion within which i t i s not unreasonable to assume that every i n -d i v i d u a l ' s time has the same d o l l a r value. The opportunity cost of time should be r e l a t i v e l y s table across s ing le post-secondary Colorado students. It can hopefu l ly be approximated using the 1968 hourly U.S. Federal minimum wage ra te , which was $1.15 an hour (U.S. Departement of Commerce,1976:382). The s e n s i t i v i t y o f the est imation to th i s assumption was tested by re -est imating the share equations assuming d i f f e r e n t values fo r the i n d i v i -dua l ' s time ( s p e c i f i c a l l y $1.00 and $1.25). (4) Other questions in the survey suggests that post-secondary students, more so than other s k i e r s , v i s i t an area predominantly to s k i . Students in my sub-sample ski approximately s ix hours per day (Denver Research In s t i tu te 1968a:75). Other groups ski on the average fewer hours per day and have a greater i n te re s t in non-ski ing a c t i v i t i e s at the s i t e (Denver Research In s t i tu te 1968a:75, 1968c:13). (5) Colorado post-secondary students t rave l to the ski s i t e s almost exc lu s i ve l y by car. 39.3% of the students ' t r i p s were taken by - 61 -car , 9.3% by bus, and the res t were mostly by t r a i n (Denver Research In s t i tu te 1968b). There i s much more va r i a t i on in t rave l modes amongst the other groups (Denver Research In s t i tu te 1968a:74). This makes i t s implest to estimate t rave l costs for the student group. (6) Many var iab les that poss ib ly in f luence the choice of s i t e s but that are not e x p l i c i t l y included as independent var iab les in the share equations, vary l i t t l e within the student group. Var iat ions within the student group in age, t a s tes , e t c . , are genera l ly much less than the comparable va r i a t i ons in the population as a whole. Each of the sk iers in my sample attends school ( res ides) in one of the fo l lowing eleveni,Colorado c i t i e s : Denver, Boulder, Ft . C o l l i n s , Greeley, Golden, The A i r Force Academy, Colorado Springs, Pueblo, Alamosa, Gunnison, and Durango. The i r ski t r i p s were l im i ted almost exc lu s i ve l y to the fo l lowing f i f t e e n ski areas: the Aspen areas, cons i s t ing of Aspen Highlands, Aspen Mountain, Buttermilk and Snowmass; V a i l ; Arapahoe-Basin; Breckenridge; Loveland; Winter Park; Broadmoor; Crested Butte; Lake E ldora; Monarch; Mount Werner (Steamboat Spr ings) ; Wolf Creek; Purgatory; Cooper; and Hidden Va l ley (Estes Park). Tables I and II summarize the s k i e r s ' c h a r a c t e r i s t i c s and t h e i r ski t r i p s , while Table XV contains a l l the student s k i e r data that i s required fo r est imat ion. B. L i f t T icket Pr ices and Charac te r i s t i c s of. the F i f t een Colorado Ski Areas Est imation of the share equations f o r the f i f t e e n ski areas mentioned in the previous sect ion requires that we know t h e i r l i f t t i c k e t pr ices and the amount of novice, intermediate, and advanced t e r r a i n at each during the 1967/68 season. In 1967/68 there were more than 25 ski areas in the State of Colorado, but we have chosen to reduce th i s number - 62 -Table I ABILITY LEVELS OF STUDENT SKIERS BY RESIDENCE Total Novice Intermediate Advanced Denver 29 2 11 16 Boulder 61 6 27 28 Ft . C o l l i n s 28 1 15 12 Greeley 10 1 3 6 Golden 6 2 2 2 A i r Force Academy 4 0 3 1 Colorado Springs 10 0 4 6 Pueblo 3 1 1 1 Alamosa 1 0 1 0 Gunnison 6 1 3 2 Durango 5 0 2 3 Total 163 14 72 77 - 63 -Table II ALLOCATION OF STUDENT SKI DAYS BY ABILITY LEVEL Total Novice Intermediate Advanced Aspen Vai l A-Basin Breckenridge Loveland Winter Park Broadmoor Crested Butte Lake Eldora Monarch Mt. Werner Wolf Creek Purgatory Cooper Hidden Va l ley 297 241 167 96 137 221 7. 36 89 43 69 10 20 10 10 9 13 4 15 16 20 1 0 6 4 0 0 0 6 1 101 70 65 38 73 8T 5 26 40 15 43 4 4 3 4 187 158 98 43 48 120 1 10 43 24 26 6 16 1 6 Total 1453 95 572 787 Average no. of ski days per season 8.9 6.79 7.94 10.22 by aggregating the four Aspen areas into one and r e f e r r i n g to i t as Aspen, and not cons ider ing at a l l a number of the smal ler areas. The small areas, such as Ski Id lewi ld , Sunl ight , Berthoud Pass, Geneva Basin, and Stoner, were not considered in the ana lys i s because they were v i s i t e d by very few people in my sample. For example, from my group o f student sk ier s there was only one v i s i t to Ski Id lewi ld and two v i s i t s to Sunl ight. The Aspen areas were aggregated f i r s t because aggregation was pos s ib le , second because Aspen is considered by many to be a s ing le sk i i ng complex, and t h i r d to reduce the number of s i t e s to be considered. Aggregation of the Aspen areas i s poss ib le because, f o r a l l p r ac t i v a l purposes, they are adjacent to one another; the aggregation is s i m p l i f i e d because t h e i r l i f t - t i c k e t pr ices are a l l i d e n t i c a l . 1967/68 l i f t - t i c k e t pr ices and data on the t e r r a i n of the f i f t e e n 3 ski areas during the same period are l i s t e d in Table III. C. Construct ion of Cost and E f f e c t i v e Physical Cha rac te r i s t i c Data  for the F i f t een Ski Area"? In sect ion C o f Chapter 3 i t was argued that the cost to ski at s i t e j was a funct ion of: (1) s i t e j ' s l i f t t i c k e t p r i c e ; (2) the cost of sk i ing equipment; (3) t ransportat ion cos t s ; and (4) the value of the i n d i v i d u a l ' s time both while t r a v e l l i n g and sk i i ng . l i f t - t i c k e t ski per mile (5.1) Y.;.: = [ (p r i ce at ) + (equipment ) + b 2 i i ( t ransportat ion)]/w + c-J s i t e j renta l fee J costs J where ^2j i ~ ^ e m i ' n i m u r n n u m D e r of miles that ind iv idua l i must t rave l to produce one uni t of sk i ing at s i t e j ; i . e . twice the distance from the i n d i v i d u a l ' s residence to s i t e j . The - 65 -Table III LIFT TICKET PRICES AND SKI AREA TERRAIN 1967/68 SEASON Acres Acres Acres Novice Intermediate Advanced Ski Areas Pr ices Terra in Ter ra in Terra Aspen 6.50 624 1559 722 Vai l 7.00 1024 3072 1024 A-Basin 4.75 100 160 140 Breckenridge 5.00 70 140 140 Loveland 5.00 122 220 73 Winter Park 5.00 127 158 59 Broadmoor 3.00 12 4 4 Crested Butte 5.00 98 24 35 Lake Eldora 4.00 22 70 16 Monarch 3.50 20 65 15 Mt. Werner 5.00 70 160 29 Wolf Creek 3.00 10 20 17 Purgatory 4.50 25 25 50 Cooper 2.75 86 108 22 Hidden Va l ley 3.50 10 16 50 Sources: L i f t T i cket P r i ce s : Ter ra in Data: Colorado V i s i t o r s Bureau (1967). Data on the t e r r a i n at a l l of the areas except Aspen Corporation areas, Hidden Va l l ey , Loveland, and Wolf Creek was provided d i r e c t l y by the ski area manage-ments at my request. Estimates fo r the other areas were constructed on the basis of data provided by Colorado Ski Country U.S.A., Denver, Colorado. - 66 -distances are l i s t e d in Table IV. w = the opportunity cost (measured in do l l a r s ) of the i n d i -v i dua l ' s time, w = $1.15 fo r our sample of s k i e r s , as per assumption in the l a s t sec t ion . c . . = the minimum amount of time required by ind iv idua l i to ski and to t rave l to and from s i t e j . c..j = ( ^ 2 j i ^ ( a v e r a 9 e speed of an automobile)) + sk i ing time. In the l a s t sect ion i t was noted that the average student sk ied approx i -mately s ix hours per day. I w i l l now assume that the average d r i v i ng speed during the 1967/68 ski t r i p s was 45 miles per hour. Therefore c - - = (b ? . . /45 ) + 6. The 1967/68 l i f t t i c k e t pr ices were included in Table III. The rental fee fo r s k i i n g equipment averaged approximately $3.50 per day during the 1967/68 season (Colorado V i s i t o r s Bureau 1967). In the previous sect ion i t was noted that the ski t r i p s were made almost exc lu s i ve l y by automobile. The per mile var i ab le cost of operat ing an automobile in 1968 was $0,064 (U.S. Department of Commerce 1971:537). 4 In my sample there were 3.8 members in each sk i i ng par ty , so average per mile t ranspor ta t ion costs f o r each ind iv idua l was $0.0017.per mi le. This combination of fur ther assumptionsand add i t iona l information allows us to make ind iv idua l i ' s cost funct ion ( in units o f time) for ski a c t i v i t y j more e x p l i c i t . l i f t t i c k e t (5.3) v . . = [ (p r i ce at ) + ($3.50) +b ? . . ($0 .0017) ] /$ l .15+ (b 9 . ./45) +6 J 1 s i t e j A vector o f shadow pr ices fo r the f i f t e e n ski areas was ca l cu la ted for each of the eleven poss ib le c i t i e s o f res idence. For my p a r t i c u l a r sample the only f ac to r generating var ia t ions in these vectors across i nd iv idua l s 00 CT 3=-o C — J c -5 3 Cu -s Co 3 3 o O fD l O (/> (/) * • O on 0) -o fD cr cu 3 O -Co CO CO w n > > "O O O -J. -S — 1 CO ~i _•. O CL 3 -s -n to cu 3 o to Q - ^< -S o o cn T | CO o OO o -s o fD 7T — 1 fD • 3 ,r+ — i . Q . fD < << fD — J o Q. fD • • CU 3 o fD -s << —1 3 (A -s ea: ro ro ro ro ro ro — i ro ro ro ro cn o oo ro oo to cn CTl —t — i o to •^j o o CTl OJ o CTl o ro co CTl oo -p* cn to cn CTl .—I i CO 00 cn co oo CTl oo O cn o co , , , , , , , o CT) CTl CO ro cn o —i CTl -P=> -P=» to o o ro to CT) ro CTl ro , , , , , , , 00 on o — i CTl ro CO CO CO o O CTl ro cn cn ~J -p=> 1 1 OJ , ' oo cn ro — • -p* to o CTl cn -p=» -p=» to o O ro to CTl ro CTl OJ ro ro , , , , , cn — ' oo CTl OJ ro cn — i — - i CTl cn cn o oo oo o oo OJ ,- , . , , OJ CTl cn OJ — i oo 1 OJ to o CO 00 -P=> 4^ co to -P* CTl 4=> , , , , ro ro ro ro ro ro to ro CTl oo to O oo CO to cn 00 cn oo -P» CO ro ro oo co 1 00 OJ ro ro , , OJ -P* -P=> o to oo Cn ro 4^ » O o cn 4=» ro CO CO oo ro , , , , ro ro , , — 1 00 —i ro CO on o ro CTl cn -P» CTl ~-J cn ro cn ro -P» ro ro ro ro ro , ro ro , ( ro cn CTl — ' —-J o — i CTl CTl 00 to ro CTl CO cn ro 00 ro , , ro ro ro ro ro ro ro 00 cn CTl 00 ro oo ro 00 oo 00 CTl OO 00 cn cn cn to cn ro _, _ , ro CO oo oo oo oo CO ro cn to 00 oo 00 o cn -p* cn ro CTl cn oo 00 oo cn o oo o ro oo CTl -P=- cn o cn CTl ro —1 O o CTl o O 1 CO cn 1 cn oo ro ro , , , CD 00 to CO ro CTl CTl cn -P» O cn CTl CTl to to CTl o O CTl cn Aspen Vai l A-Basin Brecken-ridge Loveland Winter Park Broad-moor Crested Butte Lake Eldora Monarch Mt. Werner Wolf Creek Purgatory Cooper Hidden Va l ley I—I CO —I o m oo 73 O - < - 1 „ CO o o-m oo i—i CT 0O 7^. 3=> fD Z9 -- 68 -i s the d i spers ion of the i n d i v i d u a l s ' res idences, which resu l t s in v a r i a -t ions i n auto t r ave l costs and t r a v e l l i n g time. Table V contains these shadow pr i ce vectors . The s k i e r i s not d i r e c t l y concerned with the amount of novice, intermediate and advanced terrain.>at each of the f i f t e e n s i t e s . Rather the ind iv idua l judges the s i t e on the basis of a ^ - the acres of ski runs at the s i t e which the ind iv idua l i s capable of s k i i n g , and a^j - the acres of ski runs at the s i t e s p e c i f i c a l l y designed f o r the i n d i v i d u a l ' s sk i ing a b i l i t y . Therefore, the i n d i v i d u a l ' s perception of the s i t e and his a b i l i t y to enjoy the s i t e depends both on the actual c h a r a c t e r i s t i c s of the s i t e and on the i n d i v i d u a l ' s sk i ing a b i l i t y (tastes or production technology). For a given s i t e , i t s e f f e c t i v e phys ica l c h a r a c t e r i s t i c s vary depending on whether the ind iv idua l i s a novice, intermediate or advancedsk i e r . These var i a t ions are noted in Table VI which l i s t s the amounts o f a^. and a 2 j - possessed by the f i f t e e n ski areas fo r sk iers of the three a b i l i t y l e v e l s . O 3 2 |— o CO r — CO > < 3> 00 o —j* O O O D> -s -s o -s to 1. Q- o -s — 1 • TT fD o 3 < fD 00 —i. T3 CI-CL t d -h o> fD CO 0) r+ fD O 0J —i fD SC fD 05 0) e : -s r+ CL fD —i to 3 3 -s r+ O fD o m fD 3 -s Q> fD —1. -s O -s -s 3" —j CL o 3 3 3 fD < ,-s n> CL o CL -s << fD fl) o CO -s 0) — 1 -s -s c -s CL — 1 0) c+ tO fD r+ fD << fD —1 —• oo ro ro ro — 1 GO — • — 1 —> — • — 1 ro co s i ^ c o c o c r i - p a c r i O s i o o s i i j D c o c o O C T I W D O - ^ J O O O O I I O — • oo on oo o ro —• ro —1 co o> *• w «3 * —1 co ro cr> -Pa -p=. co —1 ro co ro ro ro — • co —1 —1 —1 —1 —1 ro co C n O C » V £ i C n ^ . 4 S " — ' O O O O - ^ J U O C O C O O * to W s i — l l O N ^ M M l O C a ^ O l O l a m a i N C O i o i o o u v j o i - i c o a i r o —' w 4^ u w N - ' c o r o r o r o r o r o r o c o c n o o r o r o i o o o s j c n —> ro —1 oo —' cr> oo • ~ j c r i - f 5 » i o o - P » c n c o c n o r v > o - v j i o c o s i o c n c 7 > r o a ~ > - p a c T > c o r o —1 on oo oo s i —1 ro -pa co ro ro — ' c o r o r s o r o r o r o r o c o c r i c o — ' r o c o s i c o - p i o — ' o r o — ' c r i c o u n o o o o c n o j r o r o - p i c r i c r i c n r o - f i c o —1 B ^ Co —1 cn cti co o co —1 —> on —| —1 co ro ro ro —1 co — ' — • — • —< —> ro ro C ^ C O s J C O - P a c O - p a O C O s l C T l O O s i r o ^ O < £ > o o c n r o c o c n O D 4 = > — 1 oo -P* to1 o ro —• e n c o — ' j ^ o r o o o r o - p i O v o c o —• —1 cn — ' r o o o c o i o r o t o c o r o r o — • —> M m - ' cn cn ro 10 oo — ' O ^ r o c n o o c n c o o s i c r i O O o n a > r \ j c o s i u D C D — ' O s i r o o o c n r o r o c o r o r o r o r o r o —1 rv> ro ro ro ro co M - J s i c o i a - ' O M - i u r o —> ro cr> o c o s i c n r o o - P a > - P » c n ^ o O r o —> s i o 10 ^ O l - J ^ M W - ' t n u l U l ^ C O O l ^ N r o r o c o r o c o r o r o r o —1 ro ro ro ro ro co c n o o - C * c n — ' O o o s | . t » c n - P » o o c n c o r o o s i s i cn s i cr> — ' r o o s j c o c o - p a s i i j D C T i C T l O O — ' 0 1 * . ^ 0 1 i D 0 3 s l 0 1 l D s J —i c o r o r o — ' c o — ' c o r o r o c o r o r o r o r o c o c o r o c n o ^ o o c o o c n r o o s i - p a c r i s i r o v o r o ^ o c n o o c n - p a k o c o c o c o c o r o c o — ' O O O s i O o c n ^ o c o o o c n c o i O s i r o r o oo ro ro ro oo — ' r o — • ro ro ro ro ro ro ro co —1 co — i c n u D c n c o ^ c r t c o c n c T i v o —i o —• — ' u D c o - p a - p a - p a r o r o s i r o o i - o o r o c T i c n o c o C O C T i c n - p i r o —i s * cn -pa oo —1 — i - p a r o o o r o c o c o c o c o c o c o o o o - ' ^ M ^ s i i D ' s i a i i o a i ^ m o i w C » o o O D s i < x ) i o c o s i c n c n c n o c n c o — 1 ^ O SI Ul * O S I O s l - p a - p a < X > S l C 0 Denver Boulder Ft. oo Co l l i n s § o Greeley o m oo -—. o Golden 3 DJ — I D-o m fD -s oo to 7^ A i r Force 5 Academy ° o Colorado Springs Pueblo Alamosa Gunnison Durango - 69 -- 70 -Table VI THE EFFECTIVE PHYSICAL CHARACTERISTICS OF THE SKI AREAS Novice Intermediate Advanced Aspen Va i l A-Basin Breckenridge Loveland Winter Park Broadmoor Crested Butte Lake Eldora Monarch Mt. Werner Wolf Creek Purgatory Cooper Hidden Val ley a 2 j a n . a 2 j a 1 ^ 3 2 j a 2 j a, . a 2 j a 2 j a, . a 2 j a, . a 2 j 3 2 j a , . a 2 j 2j J l j 2j 2j 2j ! ] J l2J 624 2183 2905 624 1559 722 1024 4096 5120 1024 3072 1024 100 260 400 100 160 140 70 210 350 70 140 140 122 343 415 122 220 73 127 285 344 127 158 59 12 16 20 12 4 4 98 122 157 98 24 35 22 92 108 22 70 16 20 85 100 20 65 15 70 230 259 70 160 29 10 30 47 10 20 17 25 50 100 25 25 50 86 194 216 86 108 22 10 26 76 10 16 50 - 71 -Footnotes - Chapter 5 1. The exact wording of the,quest ion a scer ta in ing sk i ing a b i l i t y (question No. 10) defines sk i i ng in terms of the type of ski turn one uses. This i s a measure of the type of t e r r a i n one is capable of s k i i n g . For example, i t i s un l i ke l y that an ind iv idua l can succes s fu l l y navigate through intermediate and advanced t e r r a i n using a snow-plough turn . 2. It i s assumed that the ind iv idua l thinks of Aspen as one s ing le ski area. 3. Other data on the c h a r a c t e r i s t i c s of the ski areas during the 1967/68 season were c o l l e c t e d but not u t i l i z e d in the est imat ion. The fo l lowing data for each ski area are l i s t e d in Appendix A, Table XVI: (1) the average annual snowfa l l ; (2) the v e r t i c a l transport feet at the area , i . e . the number of people that can be l i f t e d 1000 v e r t i c a l feet per hour by the l i f t system at the area; (3) the number o f l i f t s ; (4) the hourly uph i l l capac i ty of the area; (5) the v e r t i c a l r i s e of the ski area; (6) the number of runs at the area; (7) the longest run in f e e t ; (8) the shortest run in f e e t ; (9) the e leva t i on at the base; and (10) the e levat ion at the summit. Out of t h i s l i s t o f ten c h a r a c t e r i s t i c s , the two that I considered most se r i ous l y fo r i nc lu s i on as important explanatory c h a r a c t e r i s t i c s in the model were the v e r t i c a l transport feet of the areas and the average annual snowfall at the areas. I had once conjectured that snow condit ions are d i r e c t l y re la ted to the average annual snowfall at the s i t e , but fu r ther thought on the matter has led me to conclude that the landscaping of the:;ski t e r r a i n , the snow making equipment at the area, and the grooming of the snow are a l l at l eas t as impor-tant as the actual snowfa l l . An area ' s v e r t i c a l t ransport feet could place a binding cons t ra in t on the amount of sk i ing ava i l ab l e at an area. If t h i s was the case, i t would d e f i n i t e l y in f luence the i n d i -v i d u a l ' s choice of areas. But i f an area i s well managed, the v e r t i c a l t ransport feet should be equivalent to that amount necessary to u t i l i z e the skn..ruhs -:.to'.their":fuT-1 est. capac i ty , in::whieh case i t i s the ski t e r r a i n that plays the important r o l e , not the v e r t i c a l t ransport f e e t . Therefore, one would expect the v e r t i c a l t ransport feet and the amount of sk iab le acreage at a s i t e to be highly c o r r e -l a ted . The v e r t i c a l transport feet was not included as an explana-tory v a r i a b l e , because (1) i f the area is well managed, i t should not play an important r o l e ; and (2) i t s i nc lu s i on would make est imation much more d i f f i c u l t , that i s (a) the number of parameters would be increased by four , and (b) given the.high degree of c o l i n e a r i t y between v e r t i c a l t ransport feet and the var iab les already inc luded, est imation by an i t e r a t i v e search technique becomes much more d i f f i -c u l t . 4. Per mile var i ab le costs '•= ( to ta l cost of the car..- deprec iat ion - parking fees) / !00,000. For de t a i l s see the source. - 72 -Chapter 6 EMPIRICAL RESULTS AND THEIR INTERPRETATIONS In t h i s chapter the maximum l i k e l i h o o d estimates o f the para-meters in the s tochas t i c model of s k i e r behaviour are reported and d i s -cussed. The chapter is d iv ided into three sect ions . F i r s t (sect ion A) the method used to obtain the maximum l i k e l i h o o d estimates of the para-meters i s d iscussed. It i s an i t e r a t i v e numerical a lgorithm which r e l i e s on the information provided by the gradients o f the funct ion to f i n d the maximum. Sect ion B reports the resu l t s of the est imat ion. Hypothesis tests are performed to ascer ta in whether pr ices and charac-t e r i s t i c s play a s t a t i s t i c a l l y s i g n i f i c a n t ro le in the s k i e r ' s a l l o c a t i o n of ski days amongst s i t e s . The estimated preference orderings are a lso examined.to determine i f they are cons i s tent with the underlying hypo-thes i s o f u t i l i t y maximizing behaviour. The t h i r d and f i n a l sect ion (sect ion C) reports the e l a s t i c i t y estimates and attempts to i n te rp re t them in a way that gives the reader more ins ights into the a l l o ca t i ona l behaviour impl ied by the estimated model. A. The Maximization Procedure fo r the L ike l ihood Function The DRI sample of student sk iers i s used to obtain estimates o f the parameter vector 0 = [e^J = [ O Q - a ] in the s tochas t i c model of s k i e r behaviour. Maximum l i k e l i h o o d estimates are obtained by f i nd ing those values of G which maximize: 163 15 (6.1) 1* = I I y i n - Aog(s._.*) i=l j=l J 1 J l . where - 73 (6.2) s.^ = sHy..,aw,a2..;v.,A.:e) J Yi -h(a.n .. ,a„ ..) = ] / I C k l h a 3 1 a 2 ' 1 1 3"° k=l a j i n U l k i ' a 2 k i j and (6.3) h ( a l j i 5 a 2 j i ) = aQ + ^ a ^ . + c ^ a , ..^? + a 3 a 2 j i 1 i 2 . , 2 4 l j i 5 2 j i where (6.4) -a - 0 / ( 3 - 1 ) . - a , rather than 3, was the parameter a c tua l l y estimated. As a reminder, subscr ipt i re fers to the i t n i nd iv idua l where: (1) a^.^ i s the amount of t e r r a i n ind iv idua l i can ski on at s i t e j ; (2) a,,^. i s the amount o f t e r r a i n at s i t e j designed s p e c i f i c a l l y f o r the i n d i v i d u a l ' s reported sk i i n g a b i l i t y ; (3) y . . i s the amount of s k i i ng a c t i v i t y j produced and consumed by ind iv idua l i ; (4) y... i s ind iv idua l i ' s parametric cost to v i s i t s i t e j ; and (5) s . .* i s his predicted share f o r s i t e j . (6.2) i s homogeneous o f degree zero with respect to the s i x a parameters, therefore the maximum l i k e l i h o o d estimates are not uniquely i d e n t i f i e d . To r e c t i f y th i s s i t u a t i o n , w i l l be set equal to one. A s u f f i c i e n t condi t ion for a l oca l maximum of (6.1) i s : (6.5) ^ = 0 r= l , . . . ,6 8 9 r where the Hessian matrix - 74 -( 6- 6) [ 9 r § r ] r s i s negative semi d e f i n i t e . The largest l oca l maximum of (6.1) i s the global maximum, the parameters of which are the maximum l i k e l i h o o d est imates. One could attempt to obtain these maximum l i k e l i h o o d estimates by f i r s t proceeding to solve the system of s ix equations (6.5) a n a l y t i c a l l y f o r the 9 parameters, but such an approach would be very d i f f i c u l t , because the system of equations i s qu i te non- l inear in the s ix 0 r parameters. A l t e r n a t i v e l y one can locate the maximum of (6.1) with respect to the 9 parameters by.using an i t e r a t i v e technique which searches for the maximum by examining d i f f e r e n t values of the 0 vector . Numerous search methods ex i s t f o r maximizing an unconstrained funct ion . They vary in the l i m i t a t i o n s they place on the set o f 0 vectors a c t u a l l y considered as poss ib le so lu t i ons . One set o f these search methods, the gradient methods, u t i l i z e s the property that the gradient vector o f the funct ion with respect to the 0 i nd ica tes the d i r e c t i o n of the maximum increase of that funct ion at a po int . This c lass of numerical methods cons is ts o f choosing an i n i t i a l 0 vector and then i t e r a t i n g according to the scheme: (6.7) 0 X + 1 = 0 X + s X D x Go ldfe ld and Quandt (1972:2) where Qx i s a n approximation to the maximum at the x^*1 i t e r a t i o n , s x i s the step s i ze which determines the amount the parameter estimates w i l l be th x changed at the x i t e r a t i o n , and D determines the d i r ec t i on in which the 0 parameters w i l l be changed. D x depends on the gradient vector o f the funct ion with respect to the 8 X parameters. The U.B.C. Computer Centre (B ird and Moore 1975:2) supports two gradient algorithms which - 75 -vary as to the method of c a l c u l a t i n g D x: (1) a Newton method, formulated by F letcher (1972), and (2) the Quadratic Hi 11-CIimbing method (see Go ld fe ld , Quandt and T r o t t e r (1966:514-551 )). The Newton method was used because i t requires less computer time per i t e r a t i o n . A general descr ip t ion of the Newton method can be found in Goldfe ld and Quandt (1972:1-38) or M i l l e r ,(1972 : 397-400). For the pa r t i cu l a r s of the F letcher ver s ion , see his a r t i c l e c i t e d above. As the Newton method was succes s fu l , the Quadratic H i l l - C l i m b i n g technique was not employed.^ B. Maximum L ike l ihood Estimates and Hypothesis Test ing The bas ic hypothesis of th i s d i s se r t a t i on i s that pr ices and e f f e c t i v e physical c h a r a c t e r i s t i c s play an important ro le in the s k i e r ' s a l l o c a t i o n . The corresponding nu l l -hypothes i s i s that pr ices and char-a c t e r i s t i c s play no r o l e , or in other words, the nu l l -hypothes i s assumes that the ind iv idua l randomly a l l oca tes his ski days amongst s i t e s , independent of the s i t e ' s pr ices or c h a r a c t e r i s t i c s , such that: (6.8) s . .* = J = TV F O R A 1 1 1 A N D J '* (6.8) i s a nested hypothesis of the determin i s t i c model o f s k i e r behaviour (6.2) and (6.3). I f i t i s assumed that cu = a 0 = a 0 = a. = ac = -a = 0, I c 5 4 o then (6.2) reduces to (6.8). -a approaches 0 as g approaches +<*>. The log of the l i k e l i h o o d func t i on , £* , fo r th i s r e s t r i c t e d case of the model (the nu l l -hypothes i s ) i s -4021.454. The model i s made less r e s t r i c t i v e by a l lowing pr ices to play an explanatory ro le in the s k i e r ' s a l l o c a t i o n amongst s i t e s . I f the bas ic hypothesis that pr ices and c h a r a c t e r i s t i c s are important is c o r r e c t , - 75 -th i s i n c lu s i on of pr ices should s i g n i f i c a n t l y increase the explanatory power o f the model. For the moment we w i l l assume that c h a r a c t e r i s t i c s play no ro le and examine only the ro le played by pr i ce s . This can be accomplished by determining the maximum l i k e l i h o o d estimate of - a , given the r e s t r i c t i o n that =0:2=012=014 =ag=0. Estimation of th i s model (model 1) led to the fo l lowing r e s u l t s : I* = -4017.232 (-0) -0.287942 -2.91907 (asymptotic t s t a t i s t i c ) On the basis o f a l i k e l i h o o d r a t i o t e s t , as descr ibed in sect ion C of 2 Chapter 4, and the asymptotic t s t a t i s t i c s , model 1 pred ic t s the a l l o c a t i o n of the s k i e r ' s budget s i g n i f i c a n t l y bet ter (at the .005 leve l ) than the model impl ied by the nu l l -hypothes i s (6.8). The l i k e l i h o o d r a t i o s t a t i s t i c i s 8.44, whereas the c r i t i c a l Y 2 i s 7.88. A . 0 0 5 The model i s now genera l ized by al lowing e f f e c t i v e physical c h a r a c t e r i s t i c a - ^ (the acres on which the ind i v idua l i s capable o f sk i ing at s i t e j ) to play an explanatory ro le in the s k i e r ' s choice of s i t e s . The hypothesis underlying such a model, that e f f e c t i v e physical c h a r a c t e r i s t i c 82^ - does not matter, can be described in terms of the r e s t r i c t i o n that 0^  =013=065=0'.:. This model, hereafter re fer red to as model 2, was est imated, and the fo l lowing resu l t s were obtained: I* = -3563.518 a 1 a 4 ( - a ) 0.026943 -0.329859 -0.344674 39.8030 -86.1106 26.3112 (asymptotic t s t a t i s t i c s ) - 77 -Model 2, according to the l i k e l i h o o d r a t i o t e s t , pred ic t s the s k i e r ' s a l l o c a t i o n of ski days s i g n i f i c a n t l y better than model 1. Therefore, i t can be concluded that both pr ices and c h a r a c t e r i s t i c a ^ play important explanatory ro les in the s k i e r ' s a l l o c a t i o n . The model i s made even less r e s t r i c t i v e by hypothesiz ing that pr ices and both of the e f f e c t i v e physical c h a r a c t e r i s t i c s (a-|jn- and a 2 j.j) in f luence the s k i e r ' s a l l o c a t i o n amongst s i t e s . In terms of the para-meters, th i s hypothesis places no a p r i o r i r e s t r i c t i o n on the values of the s i x parameters. The fo l lowing maximum l i k e l i h o o d estimates of these s i x parameters a re :^ £* = -3472.695 a-| a-j (-5) -0.131272 0.371265 -0.281522 3.21047 -1.39646 -2.12766 -2.90477 2.83732 -2.81757 3.11085 -2.78069 -20.4070 (asymptotic t s t a t i s t i c s ) The i nc lu s ion of e f f e c t i v e phys ica l c h a r a c t e r i s t i c a,,^. s i g n i f i c a n t l y 4 increases the explanatory power of the model. Of a l l the models te s ted , th i s model, the most general model, pred icts the s k i e r ' s a l l o c a t i o n of v i s i t s amongst s i t e s s i g n i f i c a n t l y bet ter than any of i t s r e s t r i c t i v e cases. Therefore, the maximum l i k e l i h o o d estimate of h (6.3) i s : (6.9) n ( a i j i > a 2 j i ) = " L 0 " •131272a 1 J i + .371265(3^ . a ^ . ) * - .281522a 2 j- i + 3 .210478^^ - 1.39646a2j^ and the maximum l i k e l i h o o d estimate of § i s .53. o is the A l l en e l a s t i -c i t y o f subs t i tu t i on between any two sk i i n g a c t i v i t i e s , but i n te rp re ta t i on of the separate a parameters i s more d i f f i c u l t . h(a^ .. > a 2j.j) i s a mono-ton ic transformation of the amount of u t i l i t y produced by one day of - 78 -sk i ing at s i t e j . The a 's are the parameters in th i s func t ion , but s ince the funct ion i s non- l inear in a^.. and j ^ > the a 's cannot i n d i v i d u a l l y be given i n t u i t i v e and meaningful i n te rp re ta t i on s . This estimated model of s k i e r behaviour i s cons i s tent with the hypothesis o f u t i l i t y maximizing behaviour fo r each ind iv idua l in the sample. For each ind iv idua l i : (1) U(Y.,A.) i s a continuous f i n i t e funct ion in Y . » 0 n i r . i i l i b (2) M | i A i l > o f o r a l l j t j = 1 and 9 y j i ,15 (3) U(Y\ ,Aj) i s a quasi-concave funct ion in Y i >>0 1 5 -This fol lows because 1 > s f 0 and h(a-| j-j » a 2j-j ) ^ o r a 9 i v e n i "is ° f the same sign fo r a l l j . For more de ta i l s see Chapter 3, sect ion C. 2 A modif ied R can be ca l cu l a ted to give us an i nd i ca t i on o f the model's goodness o f f i t . Goodness o f f i t i s a r e l a t i v e concept. In our p a r t i c u l a r case we would l i k e an i nd i ca t i on of how well our complete model f i t s the data r e l a t i v e to the model s p e c i f i e d by the nu l l -hypothes i s . 2 The fo l lowing modif ied R s t a t i s t i c w i l l give us a measure of th i s r e l a -t i ve goodness of f i t . (6.10) R2 = 1 - e ( 2 ^ * H o " ^ * H ^ / T ^ Baxter and Cragg (1970.) where £ * H 0 = the log of the l i k e l i h o o d funct ion i f the nu l l -hypothes i s i s co r rec t . In our case a*^ = -4021.451. = the log of the l i k e l i h o o d funct ion i f the hypothesis i s co r rec t . In our case = -3472.695. T = number of observations =. 163. (6.11) 5 R2 = .9988. - 79 -My s tochas t i c model of s k i e r behaviour f i t s the data s i g n i f i -cant ly bet ter than the model suggested by the nu l l -hypothes i s . Pr ices and the e f f e c t i v e physical c h a r a c t e r i s t i c s of the s i t e play a s i g n i f i c a n t ro le in how the student sk ie r a l l oca tes his ski days amongst the a l t e r -nat ive s i t e s . I f i n d th i s r e su l t very encouraging. Conventional theory has suggested, and empir ical t e s t i ng has confirmed the f a c t , that pr ices play an important explanatory ro le in the consumer's a l l o ca t i ona l beha-v iour. My model suggests that pr ices are not the only f a c to r . Charac-t e r i s t i c s of the a c t i v i t i e s and the consumer's a b i l i t y to u t i l i z e those c h a r a c t e r i s t i c s a lso help to expla in a l l o ca t i ona l behaviour. The impor-tance of inc lud ing the e f f e c t i v e physical c h a r a c t e r i s t i c s of the s i t e in a s tochas t i c model of s k i e r behaviour w i l l become more apparent when we examine the pr i ce and c h a r a c t e r i s t i c e l a s t i c i t i e s corresponding to our model. Table VII l i s t s the shares pred icted by the complete s tochast ic model of s k i e r behaviour. The vectors of pred icted shares for the f i f t e e n sk i ing a c t i v i t i e s vary - across ind iv idua l s in my sample only because of var ia t ions in sk i i ng a b i l i t y and d i spers ion in the l oca t ion of res idence. Given that there were only three leve l s of sk i ing a b i l i t y and eleven poss ib le locat ions of res idence, there are only 33 separate vectors o f predicted shares f o r the s k i i n g a c t i v i t i e s . A number of a l l o ca t i ona l patterns can be discerned by examining the table of predicted shares. F i r s t , an, i n d i v i d u a l ' s predicted share fo r a s p e c i f i c s i t e tends to decrease as the distance increases between the s i t e and the i n d i v i d u a l ' s res idence. To check to see how universal th i s tendency i s and to get some ind i ca t i on of i t s magnitude, I ca l cu la ted the s i t e ' s share e l a s t i c i t y with respect to a change in the distance to the s i t e . These estimated e l a s t i c i t i e s were a l l negat ive, f a l l i n g - 80 -Table VII s . , , THE PREDICTED SHARES Brecken- Love- Winter Broad- Crested J = .Aspen Va i l A-Basin r idge land Park moor Butte Denver N I A .08207 .12900 .17120 .09156 .13540 .21010 .12710 .11730 .11660 .08437 .07586 .08537 .15480 .15150 .12080 .14430 .13650 .09385 .02635 .01330 .00817 .03976 .02641 .01593 Boulder N I A .08194 .12880 .17130 .09058 .13400 .20720 .12430 .11480 .11370 .08280 .07450 .08354 .15120 .14810 .11760 .14130 .13370 .09159 .02237 .01131 .00692 .03837 .02550 .01532 Ft . Col 1 ins N I A .08963 .14060 .18520 .09342 .13780 .21120 .11940 .11000 .10800 .08126 .07293 .08103 .14390 .14060 .11070 .13640 .12870 .08739 .02254 .01136 .00689 .04176 .02768 .01648 Greeley N I A .08920 .13990 .18470 .09376 .13830 .21250 .12100 .11150 .10970 .08213 .07370 .08209 .14610 .14270 .11260 .13820 .13040 .08874 .02429 .01224 .00744 .04303 .02852 .01702 Golden N I A .07986 .12530 .16740 .09113 .13450 .20910 .13020 .11990 .11940 .08565 .07688 .08667 .15920 .15550 .12420 .14750 .13930 .09593 .02102 .01059 .00652 .03724 .02469 .01491 A i r Force Academy N I A .15300 .20370 .13150 .20360 .10080 .09995 .07687 .08632 .12870 .10240 .11800 .08093 .03302 .02024 !04196 .02525 Colorado Spri ngs N I A .15910 .21050 .13800 .21230 .09294 .09158 .08180 .09125 .11850 .09364 .10890 .07422 .03710 .02259 .04391 .02625 Pueblo N I A .10510 .16800 .22190 .08951 .13460 .20680 .09379 .08807 .08667 .08329 .07617 .08487 .11220 .11170 .08815 .10760 .10350 .07044 .06135 .03151 .01916 .08019 .05417 .03234 Alamosa N I A .17870 .15500 .08383 .07124 .09432 .07484 .01143 .06395 Gunnison N .1 A .10480 .17110 .22900 .09033 .13880 .21600 .07800 .07483 .07462 .06890 .06438 .07268 .08251 .08393 .06713 .06716 .06600 .04552 .01693 .00889 .00548 .21880 .15100 .09137 Durango N I A .24820 .31540 .12250 .18110 .06486 .06141 .05361 .05747 .07441 .05651 .06232 .04081 .00623 .00364 .07800 .04480 cont!d - 8T. -Table VII (cont 'd) J Lake = Eldora Monarch Mt. Werner Wolf Creek Purga-tory Cooper Hidden Va l ley Denver N I A .05236 .04817 .04781 .01954 .01802 .01803 .04762 .04391 .03409 .00750 .00539 .00540 .00885 .00581 .00544 .09251 .07933 .04870 .02129 .01415 .01766 Boulder N I A .06579 .06056 .05990 .01851 .01708 .01704 .04726 .04361 .03374 .00721 .00518 .00517 .00864 .00567 .00530 .09091 .07801 .04772 .02887 .01920 .02388 Ft, Co l l i n s N I A .05770 .05297 .05192 .01903 .01752 .01731 .04977 .04581 .03511 .00784 .00562 .b0556 .00977 .00670 .00592 .08995 .07699 .04666 .03759 .02493 .03070 Greeley N I A .05091 .04674 .04592 .01995 .01835 .01819 .04981 .04584 .03522 .00800 .00573 .00569 .00996 .00652 .00605 .09081 .07772 .04722 .03285 .02179 .02692 Golden N I A .04864 .04864 .04837 .01702 .01702 .01707 .04327 .04327 .03365 .00505 ;00505 .00507 .00539 .00539 .00506 .08014 .08014 .04928 .01385 .01385 .01732 A i r Force Academy N I A .03909 .03872 .02803 .02800 .04226 .03274 .00696 .00696 .00753 .00704 .08037 .04923 .01204 .01500 Colorado Springs N I A .03576 .03518 .02980 .02957 .04272 .03287 .00728 .00723 .00776 .00722 .08535 .05194 .01109 .01372 Pueblo N I A .03528 .03301 .03244 .04158 .03900 .03865 .04535 .04254 .03270 .01490 .01089 .01081 .01647 .01099 .01020 .09792 ' .08542 .05191 .01548 .01046 .01293 Alamosa N I A .01866 .05303 .03991 .02758 .02100 .10090 .00563 Gunnison N I A .01720 .01645 .01638 .06390 .06123 .06150 .03666 .03514 .02737 .01496 .01117 .01123 .02919 .01989 .01872 .10360 .09236 .05688 .00699 .00483 .00605 Durango N I A .01560 .01475 .03068 .02926 .03043 .02250 .03522 .03363 .10140 .09061 .07093 .04147 .00555 .00660 - 82 -in the range of 0 to - 1 . They are l i s t e d in Table VIII. This i s the s t a t i s t i c a l property which was f i r s t recognized by Clawson and now forms the basis of what is known as the t r a v e l - c o s t technique. This inverse re l a t i on sh ip between distances and pred icted shares i s not unexpected when cons ider ing that automobile operating costs and the value of the i n d i v i -dua l ' s time while t r a v e l l i n g are the major var iab le components of s k i i ng costs at the d i f f e r e n t s i te s (see Tables IV and V). Another f ac to r that comes to l i g h t by examination of the pre-d ic ted shares i s that the share is d i r e c t l y re l a ted to the s i ze of the area in terms of t e r r a i n . For example, Broadmoor, Hidden Va l l ey , Monarch and. Wolf Creek are a l l qu i te smal l , and th i s i s r e f l e c t e d in t h e i r pred icted shares. These f i r s t two observations are not at a l l unexpected. The f ac to r which I found most i n te re s t i n g upon examining the pred icted shares, was the tendency fo r them to depend on the leve l of the i n d i v i d u a l ' s s k i i n g a b i l i t y . Some of these re l a t i onsh ip s are i s o -lated in Figure 1. The sharels dependence on sk i ing a b i l i t y var ies from s i t e to s i t e in a not a l together unpredictable way. For a l l r e s i den t i a l locat ions Aspen's and V a i l ' s shares are highest f o r the advanced s k i e r , decreasing as one moves to the intermediate and then to the novice l e v e l . Apparent ly, the greater your a b i l i t y l e v e l , the greater the appeal o f areas such as Aspen and V a i l , areas which are character ized by immense amounts o f t e r r a i n . The advanced s k i e r does not become bored with the lack o f var ie ty of t e r r a i n . On the other hand, novice and intermediate sk ier s seem to l i k e Winter Park and Loveland r e l a t i v e l y more than advanced sk iers do. This should not be unexpected. Most of the t e r r a i n at the two areas i s designed fo r the novice and the intermediate sk ie r . Neither area has much advanced t e r r a i n (see Table II I). Novices are - 83 -Table VIII E s , , DIST J Brecken- Love- Winter Broad- Crested J .= Aspen Vai l A-•Basin r idge land Park moor Butte Denver N I A -0 . -0 . -0 . 6442 6113 5810 -0 -0 -0 .4336 .4127 .3771 -0 -0 -0 .3221 .3257 .3259 -0 . -0 . -0 . 3863 3899 3859 -0 - 0 -0 .2724 .2735 .2834 -0 -0 -0 .3154 .3183 .3340 -0. -0. -0. 4247 4304 4327 -0. -0 . -0 . 7449 7553 7634 Boul der N I A -0 . - 0 . - 0 . 6531 6197 5895 -0 -0 -0 .4492 .4278 .3916 -0 -0 -0 .3440 .3478 .3482 -0 . -0 . - 0 . 4063 4100 4060 -0 -0 -0 .2955 .2966 .3072 -0 -0 -0 .3367 .3397 .3562 -0. -0 . -0 . 5053 5111 5133 -0 . - 0 . - 0 . 7631 7733 7814 Ft. C o l l i n s N I A -0 . - 0 . - 0 . 7049 6654 6309 -0 -0 -0 .5433 .5166 .4727 -0 -0 -0 .4742 .4792 .4803 -0 . -0 . - 0 . 5266 5314 5267 -0 -0 -0 .4315 .4332 .4483 -0 -0 -0 .4628 .4669 .4891 -0 . -0 . -0 . 6135 6205 6233 -0 . - 0 . -0 . 8144 8263 8358 Greeley N I. A -0 . -0 . -0 . 6968 6581 6237 -0 -0 -0 .5294 .5034 .4601 -0.4555 -0.4605 -0.4614 -0 . -0. - 0 . 5094 5141 5094 -0 -0 -0 .4121 .4137 .4282 -0 -0 -0 .4446 .4487 .4702 -0 . -0 . -0 . 5733 5803 5832 -0 . - 0 . -0 . 7973 8094 8189 Golden N I A -0 . - 0 . - 0 . 6240 5932 5746 -0 -0 -0 .3962 .3773 .3448 -0 -0 -0 .2682 .2714 .2715 -0 . - 0 . - 0 . 3370 3403 3367 -0 -0 -0 .2160 .2169 .2250 -0 -0 -0 .2634 .2660 .2793 -0 . -0 . -0 . 4789 4840 4860 -0 . -0 . -0 . 7372 7460 7543 A i r Force Academy N I A -0 . -0 . 6207 5835 -0 -0 .5168 .4739 -0 -0 .4942 .4946 -0 . -0. 4895 4845 -0 -0 .4495 .4631 -0 -0 .4824 .5026 -0 . -0 . 1076 1091 -0. - 0 . 6900 7020 Colorado Springs N I A -0 . - 0 . 6035 5666 -0 -0 .4940 .4515 -0 -0 .5225 .5233 -0 . -0 . 4601 4554 -0 -0 .4797 .4932 -0 -0 .5107 .5306 -0 . -0 . 0325 0330 -0 . - 0 . 6720 6844 Pueblo N I A -0 . -0 . - 0 . 6774 6298 5890 -0 -0 -0 .5904 .5612 .5144 -0 -0 -0 .5988 .6026 .6035 -0 . - 0 . -0 . 5502 5545 5492 -0 -0 -0 .5630 .5634 .5783 -0 -0 -0 .5865 .5892 .6110 -0. -0 . - 0 . 2218 2288 2318 -0 . -0 . -0 . 6399 6580 6732 Alamosa N I A -0 . 6126 -0 .5134 -0 .6271 --Q. 5870 -0 .6317 -0 .7071 -0 . 6651 -0 . 6075 Gunnison N I A -0 . - 0 . -0 . 6238 5776 5372 -0 -0 -0 .5214 .4936 .4493 -0, -0, -0, .6036 .6056 .6058 T0. - 0 . - 0 . 5544 5571 5521 -0 -0 -0 .6140 .6131 .6243 -0 -0 -0 .6917 .6925 .7077 -0 . - 0 . -0 . 6765 6820 6844 -0. -0 . -0 . 1427 1551 1660 Durango N I A -0 . - 0 . 5722 5210 -0, -0, .7008 .6540 -0 -0. .8056 .8086 -0 . -0 . 7852 782--0 -0 .8021 .8176 -0 -0 .8485 .8679 -0 . -0 . 9253 9277 -0 . -0 . 6530 6765 con t ' d - 34 -Table VIII (cont 'd) Lake Mt. Wolf Purga- Hidden J . = . Eldora Monarch Werner Creek tory Cooper Va l ley Denver N I A -0 -0 -0 .2619 .2630 .2631 -0 -0 -0 .6666 .6676 .6676 -0 -0 -0 .6141 .6165 .6229 -0 -0 -0 .3075 .8092 .8092 -0 -0 -0 .3938 .8965 .8968 -0 -0 -0 .5285 .5362 .5540 -0 -0 -0 .4202 .4233 .4218 Boulder N I A -0 -0 -0 .1582 .1591 .1592 -0 -0 -0 .6932 .6942 .6942 -0 -0 -0 .6262 .6286 .6351 -0 -0 -0 .8255 .8272 .8272 -0 -0 -0 .9053 .9080 .9084 -0 -0 -0 .5450 .5527 .5709 -0 -0 -0 .2911 .2940 .2926 Ft . C o l l i n s • N I A -0 -0 -0, .3685 .3704 .3708 -0 -0 -0 .7713 .7730 .7732 -0 -0 -0, .7000 .7029 .7108 -0 09 -0 .3773 .8793 .8793 -0 -0, -0, .9410 .9442 .9447 -0 -0, -0, .6417 .6508 .6722 -0 -0 -0 .3077 .3117 .3099 Greeley N I A -0 -0, -0, .4086 .4104 .4107 -0 -0 -0, . 7481 .7493 .7494 -0, -0, -0, .6891 .6920 .6997 -0 -0 -0 .8640 .8659 .8660 -0, -0, -0, .9298 .9330 .9335 -0, -0, -0, .6276 .6366 .6577 -0 -0 -0 .3545 .3585 .3566 Golden N I A -0, -0. -0. .2054 .2063 .2064 -0, -0, -0, .6525 .6534 .6534 -0. -0. -0. .5853 .5876 .5935 -0, -0, -0, .7978 .7994 .7994 -0, -0, -0, .8880 .8905 .8903 -0, -0. -0, .4890 .4962 .5129 -0, -0, -0, .3844 .3872 .3858 A i r Force Academy N I A -0. -0. .4778 .4780 -0, -0, .6052 .6053 -0. -0, .7116 .7187 -0, -0, .8121 .8121 -0. -0. .8926 .8921 -0. -0. .6184 .6394 -0, -0, .5951 .5933 Colorado Springs N I A -0. -0. .5098 ,5101 -0. -0. .5793 .5794 -0. -0. .7052 .7125 -0. -0, .7976 .7977 -0. -0. .8384 .8889 -0. -0. .5936 .6152 -0. -0, .6207 .6190 Pueblo N I A -0. -0. -0. ,6055 6069 ,6073 -0. -0. -0. ,5432 .5447 ,5449 -0. -0. -0. ,7550 ,7573 ,7650 -0. -0. -0. .7320 ,7350 ,7351 -0. -0. -0. ,8441 ,3488 ,8495 -0. -0. -0. ,6398 ,6486 ,6724 -0. -0. -0. .6972 .7008 .6990 Alamosa N I A -0 . 7928 -0. ,4284 -0. 7816 -0. ,4001 -0. 6701 -0 . 5963 -0. ,3753 Gunnison N I A -0 . - 0 . - 0 . 7133 7739 7739 -0. -0. -0 . 2709 2717 2716 -0 . -0 . -0 . 7645 7657 7719 -0. -0. -0 . ,6693 6719 6718 -0 . -0 . -0 . 6159 6218 6225 -0 . -0 . -0 . 5591 5661 5882 -0. -0 . -0 . 8623 8641 8631 Durango N I A -0 . - 0 . 9229 9237 -0 . - 0 . 7523 7534 -0 . - 0 . 9311 9387 -0 . -0 . 4544 4552 -0 . - 0 . 1531 1549 -0 . -0 . 8072 8327 -0 . -0 . 9577 9567 - 85 -Figure 1 SOME PREDICTED SHARES AS A FUNCTION OF SKIING ABILITY Aspen 1s predicted shares .17 .12 .08 Denver Residents . * V a i l ' s pred ic ted shares .201 .1 .081 Boulder Residents Cooper 1s predicted shares .09 .07 .04 Denver Residents N I A Figure 1 (cont 'd) Winter Park 1 s predicted shares Ft. Co l l i n s Residents .13 .12 .08 A Crested Butte ' s predi cted shares .21 .151 Gunnison Residents .09 A Breckenridge 's predicted shares .08 .07 Boulder Residents N I A - 87 -more a t t rac ted to the smal ler areas than are the intermediate and ad-vanced sk i e r s . This i s probably due to the f ac t that there i s s u f f i c i e n t t e r r a i n at the small areas to chal lenge a novice s k i e r , whereas the i n t e r -mediate and advanced s k i e r , who general ly skies much more and much f a s te r during a day of s k i i n g , would qu ick ly become bored with the small amount of t e r r a i n at these areas. Breckenridge seems to be an area that i s equal ly preferred by a l l a b i l i t y l e v e l s . A b i l i t y l eve l s in conjunction with the amount and types of t e r r a i n at the d i f f e r e n t areas, seem to play an important ro le in the consumer's a l l o c a t i o n of ski days amongst s i t e s . C. Pr ice and Cha rac te r i s t i c E l a s t i c i t y Estimates Further ins ights can be gained into the nature of our estimated share equations and t h e i r underlying preference order ing by examining the share e l a s t i c i t i e s with respect to pr ices and c h a r a c t e r i s t i c s . The e l a s -t i c i t y formulas p a r t i c u l a r to our share equations were discussed and t h e i r funct iona l forms determined in sect ion C of Chapter 3. There i t was noted that: (6.12) E~ = 0 Homothetic preferences lead to share e l a s t i c i t i e s , with respect to the budget, equal to zero. The f ac t that the predicted shares are i n s e n s i -t i ve to changes in the s k i i n g budget i s one of the more r e s t r i c t i v e aspects of the model, a r e s t r i c t i o n that I hope someday to e l iminate. The A l len (1938) e l a s t i c i t y of subs t i tu t i on measures the r e l a -t i ve change in the demand for two sk i i n g a c t i v i t i e s a t t r i b u t a b l e to a change in t h e i r r e l a t i v e p r i ce s . For my p a r t i c u l a r preference order ing , - 38 -the e l a s t i c i t y o f subs t i tu t i on is the same fo r a l l pa irs of sk i ing a c t i v i t i e s , and i s equal to (6.13) a = - ( -a ) = 2.12766 If the preference order ing were of the Cobb-Douglas form, the A l l en e l a s -t i c i t y of subs t i tu t i on would be equal to one ( i . e . -a - -1 ) . On the basis of the asymptotic t s t a t i s t i c : (6.14) asymptotic ( -a ) - (a 0 ) = _ 1 0 .81573 v ' t asym. std-. e r r o r we can re jec t th i s nu l l -hypothes i s and conclude that preferences are not of the Cobb-Douglas form. S i te j ' s share e l a s t i c i t y with respect to a change in the pr i ce at s i t e j , i s : (6.15) - = - a ( l - s . ) The estimates of these e l a s t i c i t i e s are l i s t e d in Table IX. They are a l l negat ive, and there i s considerable v a r i a t i o n , th i s va r i a t i on being completely a t t r i b u t a b l e to var ia t ions in the s i t e s ' predicted shares. It should be noted that these estimated pr i ce e l a s t i c i t i e s are a l l greater in absolute value than the corresponding distance e l a s t i c i t i e s , i . e . the sk ie r i s more responsive to a change in the to ta l cost of a ski t r i p than he is to a change in the distance to the s i t e . This i s not unexpec-ted, but the magnitude of the d i f ferences re in forces the importance.of i nc lud ing a l l the components o f costs (not ju s t t ravel costs) in the ana ly s i s . S i te m's share e l a s t i c i t y with respect to a change in the pr i ce at s i t e j i s : - 39 -Table IX E~ Brecken- Love- Winter Broad- Crested J = Aspen Vai 1 A-Basin r i dge 1 and Park moor Butte N -1 .9530 -1 .9330 -1 v8570 -1 .9480 -1 17980- -1 .8210 -2 .0720 -2 .0430 I -1 .8530 -1 .8400 -1 .8780 -1 .9660 -1 .3050 -1 .8370 -2 .0990 -2 .0710 A -1 .7620 -1 .6810 -1 .8800 -1 .9460 -1 .8710 -1 .9280 -2 .1100 -2 .0940 fl -1 .9530 -1 .9350 -1 .8630 -1 .9510 -1 .8060 -1 .8270 T2 .0800 -2 .0460 I -1 .8540 -1 .8430 -1 .8830 -1 .9690 -1 .8130 -1 .8430 -2 .1040 -2 .0730 A -1 .7630 -1 .6870 -1 .3860 -1 .9500 -1 .8770 -1 .9330 -2 .1130 -2 .0950 N -1 .9370 -1 .9290 -1 .3740 -1 .9550 -1 .8210 -1 .8370 -2 .0800 -2 .0390 I -1 .8290 -1 .8340 -1 .8940 -1 .9720 -1 .8290 -1 .8540 -2 .1030 -2 .0690 A -1 .7340 -1 .6730 -1 .8980 -1 .9550 -1 .8920 -1 .9420 -2 .1130 -2 .9030 ri -1 .9380 -1 .9280 -1 .8700 -1 .9530 -1 .8170 -1 .8340 -2 .0760 -2 .0360 i -1 .8300 -1 .8330 -1 .8900 -1 .9710 -1 .8240 -1 .8500 -2 .1020 -2 .0670 A -1 .7350 -1 .6760 -1 .8940 -1 .9530 -1 .8880 -1 .9390 -2 .1120 -2 .0910 N -1 .9580 -1 .9340 -1 .8510 -1 .9450 -1 .7890 -1 .3140 -2 .0830 -2 .0480 I -1 .8610 -1 .8420 -1 .8720 -1 .9640 -1 .7970 -1 .8310 -2 .1050 -2 .0750 A -1 .7710 -1 .6830 -1 .8740 -1 .9430 -1 .8630 -1 .9240 -2 .1140 -2 .0960 fl I -1 .8020 -1 .8480 -1 .9130 -1 .9640 -1 .8540 -1 .8770 -2 .0570 -2 .0380 A -1 .6940 -1 .6950 -1 .9150 -1 .9440 -1 .9100 -1 .9550 -2 .0850 -2 .0740 Ft . C o l l i n s A i r Colorado N SnHnn<; 1 "1-7890 -1.8340 -1.9300 -1.9540 -1.8760 -1.8960 -2.0490 -2.0340 oLHiriyb A _i.6800 -1.6760 -1.9330 r.l .9340 -1.9280 -1.9700 -2.0800 -2.0720 N -1.9040 -1.9370 -1.9280 -1.9500 -1.8890 -1.8990 -1.9970 -1.9570 Pueblo I -1.7700 -1.8410 -1.9400 -1.9660 -1.3900 -1.9070 -2.0610 -2.0120 A -1.6560 -1.6880 -1.9430 -1.9470 -1.9400 -1.9780 -2.0870 -2.0590 N Alamosa I -1.7470 -1.7980 -1.9490 -1.9760 -1.9270 -1.9680 -2.1030 -1.9920 A N -1.9050 r l .9350 -1.9620 -1.9810 -1.9520 -1.9850 -2.0920 -1.6620 Gunnison I -1.7640 -1.8320 -1.9680 -1.9910 -1.9490 -1.9870 -2.1090 -1.8060 A -1.6400 -1.6680 -1.9690-1.9730 -1.9850 -2.0310 -2.1160 -1.9330 N Durango I -1.6000 -1.8670 -1.9900 -2.0140 -1.9690 -1.9950 -2.1140 -1.9620 A -1.4570 -1.7420 -1.9970 -2.0050 -2.0070 -2.0410 -2.1200 -2.0320 cont 'd - 90 -Table IX (cont 'd) Lake Mt. Wolf Purga- Hidden J = Eldora Monarch Werner Creek tory Cooper Va i ley Denver N I A -2 -2 -2 .0160 .0250 .0260 -2 -2 -2 .0860 .0890 .0890 -2 -2 -2 .0260 .0340 .0550 -2 -2 -2 .1120 .1160 .1160 -2 -2 -2 .1090 .1150 .1160 -1 -1 -2 .9310 .9590 .0240 -2 -2 -2 .0820 .0980 .0900 Boulder N I A -1 -1 -2 .9880 .9990 .0000 -2 -2 -2 .0880 .0910 .0910 -2 -2 -2 .0270 .0350 .0560 -2 -2 -2 .1120 .1170 .1170 -2 -2 -2 .1090 .1160 .1160 -1 -1 -2 .9340 .9620 .0260 -2 -2 -2 .0660 .0870 .0770 Ft. Co l l i n s N I A -2 -2 -2 .0050 .0150 .0170 -2 -2 -2 .0870 .0900 .0910 -2 -2 -2 .0220 .0300 .0530 -2 -2 -1 .1110 .1160 .1160 -2 -2 -2 .1070 .1140 .1150 -1 -1 -2 .9360 .9640 .0280 -2 -2 -2 .0480 .0750 .0620 Greeley N I A -2 -2, -2, .0190 .0280 .0300 -2 -2 -2 .0850 .0890 .0890 -2 -2 r'2. .0220 .0300 .0530 -2 -2 -2 .1110 .1150 .1160 -2 -2 -2 .1060 .1140 .1150 -1 -1 -2 .9340 .9620 .0270 -2 -2 -2 .0580 .0810 .0700 Golden N I A -2.0150 -2.0240 -2.0250 -2, -2 -2, .0880 .0910 .0910 -2 -.2. -2, .0280 .0360 .0560 -.2 -2 -2 .1130 .1170 .1170 -2 -2, -2 .1100 .1160 .1170 -1 -1 -2 .9280 .9570 .0230 -2 -2 -2 .0830 .0980 .0910 A i r Force Academy N I A -2. -2. .0440 .0450 -2. -2. .0680 .0680 -2, -2. .0380 .0580 -2, -2, .1130 .1130 -2, -2, .1120 .1130 -1 -2 .9570 .0230 -2 -2 .1020 .0960 Colorado Springs N I A -2. -2. .0520 .0530 -2. -2. .0640 .0650 -2. -2. .0370 .0580 -2, -2, .1120 .1120 -2, -2, .1110 .1120 -1 -2 .9460 .0170 -2, -2, .1040 .0980 Pueblo N I A -2. -2. -2. .0530 .0570 ,0590 -2. -2. -2. .0390 ,0450 .0450 -2. -2. -2. .0310 .0370 .0580 -2. -2. -2. .0960 .1040 .1050 -2. -2. -2. .0930 .1040 .1060 -1 -1 -2 .9190 .9460 .0170 -2, -2, -2. .0950 .1050 .1000 Alamosa N I A -2. ,0880 -2. ,0150 -2. ,0430 -2. .0690 -2. ,0830 -1, .9130 -2, .1160 Gunnison N I A -2. -2 . -2 . ,0910 0930 0930 -1 . - 1 . - 1 . 9920 9970 9970 -2. -2. -2. ,0500 0530 0690 -2. -2. -2. .0960 ,1040 .1040 -2. -2. -2. ,0660 ,0850 ,0880 -1, -1, -2, .9070 .9310 .0070 -2. -2. -2. .1130 .1170 ,1150 Durango N I A -2 . -2 . 0940 0960 -2 . -2 . 0620 0650 -2. -2 . 0630 0800 -2. -2. 0530 0560 -1 . - 1 . ,9120 9350 -1 . -2. .9770 .0390 -2. -2. ,1160 1140 - 91 -(6.16) Y • a One should note that these e l a s t i c i t i e s are independent of the magnitude of s i t e m's pred icted share. The estimates of these e l a s t i c i t i e s are l i s t e d in Table X. The estimates possess some d i s t i n c t i v e p roper t ie s . (6.16) r e s t r i c t s a l l these c r o s s - e l a s t i c i t i e s to be p o s i t i v e , but they vary extens ive ly depending on the loca t ion of the i n d i v i d u a l ' s residence and the i n d i v i d u a l ' s s k i i n g a b i l i t y . One can see the general r e l a t i on sh ip that these c r o s s - e l a s t i c i t i e s are l a rger when the s i t e at which the pr i ce i s changing i s important to the i n d i v i d u a l . That i s , as the i n d i v i d u a l ' s predicted share f o r a given s i t e increases , the more responsive his a l l o -cat ion amongst s i t e s becomes to changes in that s i t e ' s p r i c e . For example, expert sk iers enjoy Aspen and V a i l , therefore t h e i r demand f o r s k i i n g at one of the other s i t e s i s r e l a t i v e l y responsive to changes in the p r i ce of s k i i n g at e i the r Aspen or V a i l . A l t e r n a t i v e l y , novices seem to enjoy s k i i n g at Winter Park, and as a r e su l t t h e i r demand for s k i i n g at s i t e m is quite responsive to a change in the pr i ce of sk i ing at Winter Park. For areas such as Broadmoor, where very few people s k i , the demand f o r s k i i n g at the other s i t e s i s quite unresponsive to changes in i t s shadow p r i ce . respect to a change in the amount of a , - at that s i t e , while holding a 0 . constant. We now examine s i t e j ' s pred icted share e l a s t i c i t i e s with (6.17) s j a i j 82J=o " 3 [ 1 - S j ] / h ( a i r a 2 . ) - 92 -Table X S Y • m r j J . =. Aspen Va i l A' Brecken- Love- Winter Broad- Crested •Basin r idge land Park moor Butte N 0.1746 0.1948 0. 2704 0. 1795 0.3294 0.3071 0 .0560 0.0846 Denver I 0.2744 0.2880 0. 2496 0. 1614 0.3224 0.2904 0 .0283 0.0561 A 0.3661 0.4469 0. 2481 0. 1816 0.2570 0.1997 0 .0173 0.0338 N 0.1743 0.1927 0. 2644 0. 1762 0.3217 0.3006 0 .0476 0.0816 Boulder I 0.2741 0.2851 0. 2443 0. 1585 0.3150 0.2844 0 .0240 0.0542 A 0.3645 0.4409 0. 2420 0. 1777 0.2503 0.1949 0 .0147 0.0326 N 0.1907 0.1988 0. 2540 0. 1729 0.3062 0.2902 0 .0479 0.0888 r t . I 0.2991 0.2933 0. 2341 0. 1552 0.2991 0.2739 0 .0241 0.0588 oo111 ns A 0.3940 .04493 0. 2297 0. 1724 0.2355 0.1859 0 .0146 0.0350 N 0.1898 0.1995 0. 2575 0. 1748 0.3108 0.2940 0 .0516 0.0915 Greeley I 0.2976 0.2943 0. 2373 0. 1568 0.3036 0.2775 0 .0260 0.0606 A 0.3930 0.4521 0. 2334 0. 1747 0.2396 0.1888 0 .0158 0.0362 N 0.1699 0.1939 0. 2769 0.1822 0.3387 0.3139 0.0447 0.0792 Golden I 0.2665 0.2861 0. 2552 0. 1636 0.3309 0.2963 0 .0225 0.0525 A 0.3562 0.4448 0. 2541 0. 1844 0.2643 0.2041 0 .0138 0.0317 A i r N Force I Academy A Colorado Springs A N Pueblo I A N Alamosa I A 0.3255 0.2797 0.2144 0.1636 0.2738 0.2501 0.0702 0.0892 0.4334 0.4331 0.2127 0.1837 0.2178 0.1722 0.0430 0.0537 0.3386 0.2936 0.1977 0.1740 0.2521 0.2317 0.0789 0.0934 0.4478 0.4517 0.1949 0.1942 0.1992 0.1579 0.0480 0.0558 0.2236 0.1905 0.1995 0.1772 0.2387 0.2289 0.3574 0.2864 0.1874 0.1621 0.2376 0.2202 0.4722 0.4400 0.1844 0.1806 0.1876 0.1499 0.1305 0.1706 0.0670 0.1153 0.0407 0.0688 0.3802 0.3297 0.1784 0.1516 0.2007 0.1592 0.0243 0.1361 N 0.2229 0.1922 0.1660 0.1466 0.1756 0.1429 Gunnison I 0.3641 0.2953 0.1592 0.1370 0.1786 0.1404 A 0.4873 0.4597 0.1588 0.1546 0.1428 0.0968 0.0360 0.4656 0.0189 0.3213 0.0116 0.1944 Durango I A 0.5281 0.6711 0.2607 0.3854 0.1380 0.1307 0.1141 0.1223 0.1583 0.1202 0.1326 0.0868 0.0132 0.0077 0.1660 0.0953 cont 'd - 93 -Table X (cont 'd) Lake Mt. Wolf Purga- Hidden Eldora Monarch. Werner Creek tory Cooper Va l ley N 0.1114 0.0415 0.1013 0.0159 0.0188. 0.1968 0.0452 Denver I 0.1025 0.0383 0.0934 0.0114 0.0123 0.1688 0.0301 A 0.1017 0.0383 0.0725 0.0115 0.0115 0.1036 0.0375 N 0.1400 0.0393 0.1006 0.0153 0.0183 0.1934 0.0614 Boulder I 0.1288 0.0363 0.0927 0.0110 0.0120 0.1660 0.0408 A 0.1275 0.0362 0.0717 0.0110 0.0112 0.T015 0.0508 F. N 0.1228 0.0405 0.1059 0.0166 0.0207 0.1914 0.0799 rniiinc 1 0.1127 0.0372 0.0974 0.0119 0.0136 0.1638 0.0530 0 l l i n s A 0.1105 0.0368 0.0747 0.0118 0.0126 0.0992 0.0653 N 0.1083 0.0424 0.1060 0.0170 0.0211 0.1932 0.0698 Greeley I 0.0994 0.0390 0.0975 0.0122 0.0138 0.1654 0.0463 A 0.0977 0.0386 0.0749 0.0121 0.0128 0.1005 0.0572 N 0.1127 0.0393 0.1000 0.0149 0.0175 0.1992 0.0444 Golden I 0.0135 0.0362 0.0920 0.0107 0.0114 0.1705 0.0294 A 0.1029 0.0373 0.0716 0.0107 0.0107 0.1049 0.0368 A i r N Force I 0.0831 0.0596 0.0899 0.0148 0.0160 0.1701 0.0256 Academy A 0.0823 0.0595 0.0696 0.0148 0.0149 0.1047 0.0319 N w ° n n c ° 1 0.0760 0.0634 0.0908 0.0154 0.0165 0.1816 0.0236 borings A 0 > 0 7 4 8 Q.0629 0.0699 0.0153 0.0153 0.1105 0.0292 N 0.0750 0.0884 0.0965 0.0317 0.0350 0.2083 0.0329 Pueblo I 0.0702 0.0829 0.0905 0.0231 0.0233 0.1817 0.0222 A 0.0690 0.0822 0.0695 0.0229 0.0217 0.1105 0.0275 N Alamosa I 0.0397 0.1128 0.0849 0.0586 0.0446 0.2147 0.0119 A N 0.0366 0.1360 0.0780 0.0318 0.0621 0.2205 0.0148 Gunnison I 0.0349 0.1303 0.0746 0.0237 0.0423 0.1965 0.0102 A 0.0348 0.1308 0.0582 0.0238 0.0398 0.1210 0.0128 N Durango I 0.0331 0.0652 0.0647 0.0749 0.2158 0.1509 0.0118 A 0.0313 0.0622 0.0478 0.0715 0.1928 0.0882 0.0140 - 94 -can be in terpreted as a monotonic transformation of the proport ionate amount by which the u t i l i t y o f a ski day increases when the amount of sk iab le t e r r a i n at the s i t e increases by one acre. As was noted (Chapter 3, sect ion D), these e l a s t i c i t i e s are equal to s i t e j ' s pre-d ic ted share e l a s t i c i t y with respect to a change in the amount of a^. at the s i t e while holding a 2 j constant. (6.13) E. § 2 - 0 E ' j a 3 j | § 2 j = 0 where a^. = the acres of t e r r a i n at s i t e j on which the ind iv idua l can s k i , but that are not s p e c i f i c a l l y designed fo r the i n d i v i d u a l ' s s k i i ng a b i l i t y . It fol lows that the Aa-jj = A a ^ because a2j i s being held constant. These e l a s t i c i t i e s measure how responsive a s i t e ' s predicted share is to an increase in sk iab le t e r r a i n at a s i t e when the increase i s completely in terms of t e r r a i n that i s designed f o r i nd iv idua l s of l e s se r s k i i n g a b i l i t y . These e l a s t i c i t y estimates are l i s t e d in Table XI. With two except ions, these c h a r a c t e r i s t i c e l a s t i c i t i e s are a l l p o s i t i v e . An i n d i v i d u a l ' s pred icted share f o r a given s i t e increases as the amount of t e r r a i n at that s i t e on which he i s capable of sk i ing increases. This seems qu i te reasonable. The more t e r r a i n there is on which he can s k i , the more options he has as to where to s k i , and the less bored he should become. This should increase his enjoyment of the area. The f igures in the table ind ica te that these c h a r a c t e r i s t i c e l a s t i c i t i e s fo r a given s i t e are la rges t f o r the novice sk iers and then decreases as one moves from the intermediate to the advanced l e v e l s . The other thing to note about the f i gures in th i s t a b l e , with the exception of the e l a s t i c i t i e s - 95 -Table XI E g j . a U V 0 Brecken- Love- Winter Broad- Crested J = Aspen Va i l A-Basin r idge land Park moor Butte Denver N I A 7. 4. -0 . 0780 3510 4913 12. 9. -2. 4900 4040 9260 2 1 0 .6020 .7200 .9686 2 1 1 .4950 .9180 .1760 2 1 0 .7020 .8220 .4245 2 1 0 .7020 .5420 .5179 1 0 1 .8320 .9726 .0430 2. 0. 0. 8810 8548 9065 Boul der N I A 7. 4. -0 . 0790 3520 4918 12. 9. -2 . 5100 4190 9370 2 1 0 .6100 .7250 .9718. 2 1 1 .5000 .9200 . 1780 2 1 0 . 71 30 .8300 .4260 2 1 0 .7110 .5470 .5192 1 0 1 .8390 .9745 .0450 2. 0. 0. 8850 8466 9071 Ft. C o l l i n s N I A 7. 4. -0 . 0190 2930 4835 12. 9. -0 . 4700 3770 2922 2 1 0 .6250 .7350 .9781 2 1 1 .5040 .9240 .1820 2 1 0 .7270 .8460 1-4294 2 1 0 .7270 .5510 .5216 1 0 1 .8390 .9745 .0405 2. 0. 0. 8750 8445 9060 Greeley N I A 7. 4. -0 . 0230 2960 4838 12. 9. -2 . 4600 3720 9170 2 1 0 .6200 .7320 .9762 2 1 1 .5010 .9220 : isoo 2 1 0 .7300 .8410 .4285 2 1 0 .7210 .5530 .5208 1 0 1 .8360 .9736 .0440 2. 0. 0. 8710 8440 9055 Golden N I A 7. 4. -0 . 0950 3690 4940 12. 9. -2. 5000 4140 9300 2 1 0 .5930 .7150 .9655 2 1 1 .4920 .9150 .1740 2 1 0 .6880 .8140 .4228 2 1 0 .6920 .5370 .5167 1 0 1 .8420 .9752 .0450 2. 0. 0. 8890 8473 9074 A i r Force Academy N I A 4.2310 ^0.4725 9. -2. 4470 9500 1 0 .7530 .9865 1 1 .9160 .1750 1 0 .8710 .4334 1 0 .5760 .5253 0 1 .9531 .0310 0. 0. 8323 8979 Colorado Springs N I A 4. -0 . 2000 4685 9. -2. 3760 9180 1 0 .7680 .9961 1 1 .9050 .1680 1 0 .8930 .4376 1 0 .5920 .5291 0 1 .9491 .0280 0. 0. 8306 8970 Pueblo. N I A 6. 4. -0 . 9000 1560 4617 12. 9. -0 . 5200 4130 2938 2 1 1 .7010 .7770 .0010 2 1 1 .4980 .9170 .1770 2 1 0 .8380 .9080 .4403 2 1 0 .8180 .6010 .5313 1 0 1 .7660 .9546 .0320 2. 0. 0. 7600 8217 8914 Alamosa N I A 4. 1030 9. 1910 1 .7860 1 .9270 1 .9450 1 .6530 0 .9744 0. 81 32 Gunnison N I A 6. 4. -0 . 9030 1400 4575 12. 9. -2. 5100 3670 9040 2 1 1 .7480 .8030 .0151 2 1 •1 .5370 .9410 .1920 2 1 0 .9330 .9670 .4504 2 1 0 .9450 .6680 .5455 1 0 1 .8490 .9769 .0460 2. 0. 0. 3440 7376 8370 Durango N I A 3.7550 -0.4062 9. - 3 . 5440 0330 1 1 .8230 .0290 1 1 .9640 .2120 1 0 .9880 .4555 "1 0 .6750 .5482 0 1 .9795 .0480 0. 0. 8010 8799 cont 'd - 96 -Table XI (cont 'd) Lake Mt. Wolf Purga- Hidden J = Eldora Monarch Werner Creek tory Cooper Va l ley N Denver I A N Boulder I A Ft. N C o l l i n s ^ N Greeley I A M Golden I A A i r N Force I Academy A Colorado j Springs A N Pueblo I A N Alamosa I A N Gunnison I A N Durango I A 1 .8770 2 .0150 2 .5950 1 .6700 2 .1010 2 .5740 1 .6470 2 .0110 2 .0320 2 .1060 1 .6050 1 .4320 1 .5490 1 .4670 0 .7935 0 .8450 0 .4091 .1 .2600 1 .5200 0 .4955 1 .7470 1 .8500 2 .0170 2 .5960 1 .6710 2 .1020 2 .5780 1 .6340 1 .9850 2 .0340 2 .1060 1 .6050 1 .4330 1 .5520 1 .4590 0 .7834 0 .8458 0 .4093 1 .2610 1 .5200 0 ,4960 1 .7360 1 .8660 2 .0160 2 .5900 1 .6700 2 .0990 2 .5810 1 .6200 2 .0010 2 .0330 2 .1010 1 .6050 1 .4320 1 .5530 1 .4510 0 .7901 0 .8456 0 .4087 1 .2600 1 .5190 0 .4965 1 .7230 1 .8790 2 .0140 2 .5900 1 .6690 2 .0990 2 .5790 1 .6280 2 .0140 2 .0310 2 .1010 1 .6050 1 .4310 1 .5220 1 .4550 0 .7951 0 .8448 0. .4086 1 .2600 1 .5190 0 .4962 1 .7300 1 .8750 2, .0170 2 .5970 1 .6710 2 .1030 2 .5710 1 .6480 2, .0100 2, .0340 2, .1070 1 .6060 1 .4330 1, .5480 1 .4670 0, .7931 0. .8458 0, .4093 1 .2610 1 .5200 0 .4952 1 .7470 2. .0310 2. .0110 2. .1090 1 .6030 1 .4300 1. .5480 1 .4700 0. .8011 0. ,8364 0. .4097 1 .2580 1, .5170 0. .4952 1 .7510 2. ,0380 2. 0070 2. ,1080 .1, .6020 1. .4300 1. .5390 1 .4710 0. ,8040 0. 8350 0. 4096 1, .2580 1. .5170 0. ,4938 1 .7540 1. 9100 1. 9700 2. 6020 1. .6580 2. .0850 2. ,5580 1, .6570 2. 0430 1. 9880 2. 1090 1. .5960 1. ,4250 1. 5390 1. .4720 0. 8063 0. 8272 0. 4097 1. .2530 1. ,5120 0. 4938 1. .7550 2. 0740 1. 9590 2. 1140 1. ,5690 1. 4100 1. 5130 1. ,4800 1. 9460 1. 9240 2. 6250 1. 6580 2. 0580 2. 5420 1. ,6710 2. 0780 1. 9420 2. 1250 1. 5960 1. 4120 1. 5280 1. 4810 0. 8197 0. 8076 0. 4120 1. 2530 1. 4990 0. 4912 1. 7670 0. 2080 0. 2206 2. 1350 1. 5570 1. 2950 1. 5640 1. 4800 0. 8211 0. 8353 0. 4140 1. 2240 1. 3890 0. 4992 1. 7660 - 97 -fo r the advanced sk iers at Aspen and V a i l , is that the e l a s t i c i t i e s seem to increase with the amount of a-j^ at the area. Vai l has the largest amount of novice t e r r a i n , and i t a lso has the corresponding l a rges t c h a r a c t e r i s t i c e l a s t i c i t y with respect to a-jj for novice sk ie r s . The area with the fewest novice acres has the lowest share e l a s t i c i t y f o r novices with respect to a-^. One could almost rank the e l a s t i c i t i e s fo r the novices on the basis o f the number of acres of novice t e r r a i n at each of the f i f t e e n s i t e s . The same re l a t i on sh ip seems to hold fo r the intermediate s k i e r s , but does not hold f o r the advanced sk i e r s . The i n d i v i d u a l ' s share e l a s t i c i t y fo r s i t e j with respect to a change in the amount of a 2 j . while holding tota l sk iab le t e r r a i n (a-j^) constant, i s : < 6 - 1 9 ) E s , a j °2j 3 h ( a 1 . , a 2 . ) „ 3 f i ( a l i » a 2 . ) „ = - E ? li £ J / h ( a , . , a 9 . ) S j Y J 9 a 2 j l j These are the e l a s t i c i t i e s that I found to be the most i n t e r e s t i n g . They are l i s t e d in Table XII. Examination w i l l show that they are predominantly negative. A negative c h a r a c t e r i s t i c e l a s t i c i t y of th i s type says that as the proport ion of sk iab le t e r r a i n at an area designed s p e c i f i c a l l y f o r your sk i i ng a b i l i t y increases , while holding tota l sk iab le t e r r a i n constant, your demand for sk i i ng at that area decreases. These e l a s t i c i t y estimates are quite reasonable when we remember that sk i ing a b i l i t y was measured in terms of the i n d i v i d u a l ' s c a p a b i l i t i e s rather than preferences. When an intermediate sk ie r i s def ined as a s k i e r who has the a b i l i t y to ski on both novice and intermediate t e r r a i n , we should not be too sur-pr i sed i f he happens to enjoy the novice t e r r a i n r e l a t i v e l y more than - 98 -Table XII 8, -=o i j Brecken- Love-Aspen Vai l A-Basin r idge land Winter Broad- Crested Park moor Butte Denver N I A - 7 . -4 . 0. 3650 1170 9040 -14. -10. .2. 0100 2900 5270 -1 . -0 . -0 . 9820 9491 1263 -1 -1 -0 .7610 .0650 .2822 -2. - 1 . 0. 1390 0930 2822 -2. -0 . 0. 2510 6914 1883 -0. -0. 0. 9158 1945 0 -2. 0. 0. 1950 0 0 Boulder N I A - 7 . -4. 0. 3660 1170 9048 -14. -10. 2. 0300 3100 5360 - 1 . -0 . - 1 . 9880 9518 2680 -1 -1 -2 .7640 .0670 .8280 -2. - 1 . 0. 1480 0930 2130 -2. -0. -T . 2590 6937 8880 -0. -0. 0. 9196 1949 0 -2. 0. 0. 1980 0 0 Ft. Col 1 ins N I A -7 . -4. 0. 3040 0620 8897 -13. -10. 2. 9800 2600 5340 -2. -0 . -0 . 0000 9570 1276 -1 -1 -0 .7670 .0690 .2836 -2. - 1 . 0. 1670 1070 2147 -2. -0 . 0. 2720 6976 1897 -0 . -0. 0. 9194 1949 0 -2. 0. 0. 1910 0 0 Greeley N I A r7. -4 . 0. 3080 0650 8902 -13. -10. 2. 9800 2500 5300 - 1 . - 0 . -0 . 9960 9554 1273 -1 -1 -0 .7660 .0680 .2832 -2 . - 1 . 0. 1610 1050 2142 -2. -0 . 0. 2680 6963 1894 -0 . -0 . 0. 9178 1947 0 -2. -0. 0. 1880 0 0 Golden N .1 A -7 . -4. 0. 3830 1340 9090 -14. -10. 2. 0200 3000 5030 - 1 . - 9 . - 0 . 9750 4630 1259 -1 -1 -0 .7950 .0640 .2818 -2. - 1 . -0. 1280 0880 2114 -2. -0 . • 0. 2430 6892 1879 -0 . -0 . 0. 9208 1950 0 -2. 0. 0. 2010 0 0 A i r Force Academy N I A -4 . 0. 0003 8695 -10. 2. 3400 5480 -0 . -0. 9669 1287 -1 -0 .0640 .2819 -1 . •••o. 1230 2167 -0 . 0. 7063 1910 -0 . 0. 1906 0 0. 0. 0 0 Colorado Springs N I A -3 . 0. 9740 8621 -10. 2. 2600 5200 -0 . -0 . 9754 1299 -1 -0 .0580 .2804 - 1 . 0. 1360 2188 -0 . 0. 7135 1924 -0 . 0. 1898 0 0. 0. 0 0 Pueblo N I A - 7 . -3 . 0. 1800 9320 8496 -14. -10. 2. 0400 3000 5380 -2. -0 . -0. 0580 9806 1306 -1 -1 -0 .7630 .0650 .2824 -2. - 1 . 0. 2470 1450 2201 -2. -0 . 0. 3480 71 79 1932 -0 . -0 . 0. 8829 1909 0 -2. 0. 0. 1030 0 0 Alamosa N I A -3 . 8820 -10. 0600 -0 . 9852 -1 .0710 - 1 . 1167 -0. 7408 -0 . 1949 0. 0 Gunnison N I A - 7 . - 3 . 0. 1830 9180 8418 -14. -10. :.2. 0300 2500 5080 -2 . -0 . - 0 . 0940 9948 1323 -1 -1 -0 .7910 .0790 .2861 -2. - 1 . 0. 3220 1180 2252 -2 . -0 . 0. 4540 7479 1984 -0 . -0 . 0. 9247 1954 0 - 1 . 0. 0. 1786 0 0 Durango N I A - 3 . 0. 5530 7475 -10. 2. 4400 6200 - 1 . -0 . 0060 1342 -1 -0 .0910 .2908 - 1 . 0. 1930 2278 -0 . 0. 7508 1994 -0 . 0. 1959 0 0. 0. 0 0 cont ' d - 99 -Table XII (cont 'd) Lake Mt. Wolf Purga- Hidden J = Eldora Monarch Werner Creek tory Cooper Va l ley Denver N I A -1 -1 0 .1730 . 1-170 .0 -1 -1 0 .2590 .1950 .0 -1 -1 0 .8320 .3070 .2338 -1 . -0 . -0 . 0020 7134 2520 - 1 . -0 . -0 . 1670 3906 4748 - 1 . - 0 . •0. 8960 6737 1858 -0 -0 -0 .9882 .5501 .8151 Boulder M I A -1 -1 0 .1560 .1030 -.0 -1 -1 0 .2610 .1960 .0 -1 -1 0 .8330 .3070 .2339 - 1 . -0 . -0 . 1020 7135 2521 - 1 . -0. -0 . 1680 3907 4749 - 1 . -0 . •0. 9000 6746 1860 -0 -0 -0 . 9806 .5472 .8099 Ft. Co l l ins N I A -1 -1 0 .1660 .1120 .0 -1 -1 0 .2600 .1960 .0 -1 -1 0 .8280 .3040 .2335 - 1 . -0 . -0 . 0020 7132 2520 - 1 . - 0 . -0 . 1660 3904 4746 -1 . -0. 0. 9200 6754 1863 -0 -0 -0 .9718 .5440 .8043 Greeley N I A -1 -1 0 .1750 .1190 .0 -1 -1 0 .2590 .1950 .0 -1 -1 0 .8280 .3040 .2335 -1 . -0 . -0 . 0020 7131 2520 - 1 . - 0 . -0 . 166-3904 4746 - 1 . - 0 . 0. 9000 6748 1861 -0 -0 -0 .9766 .5458 .8074 Golden N I A -1 -1 0 .1720 .1170 .0 -1 -1 0 .2610 .1960 .0 -1 -1 0 .8330 .3080 .2339 - 1 . -0 . - 1 . 0030 7136 2521 -1 . - 0 . -0. 1680 3908 4750 - 1 . - 0 . 0. 8940 6731 1857 -0 -0 -0 .9887 .5502 .8154 A i r Force Academy N I A -1 0 .1280 .0 -1 0 .1830 .0 -1 0 .3090 .2341 -0 . - 0 . 7123 2517 -0 . -0. 3900 4741 -0 . -0. 6729 1857 -0 -0 .5512 .8173 Colorado Springs N I A -1 0 .1320 .0 -1 0 .1810 .0 -1 0 .3090 .2341 -0 . -0 . 7120 2516 -0. -0 . 3899 4740 -0 . 0. 6693 1852 -0 -0 .5518 .8184 Pueblo N I A -1 -1 0 .1940 .1350 .0 -1 PI 0 .2310 .1700 .0 -1 -1 0 .8360 .3090 .2341 -0 . -0. -0 . 9947 7094 2507 - 1 . - 0 . -0 . 1580 3866 4726 - 1 . -0 . 0. 8850 6992 1852 -0 -0 -0 .9941 .5521 .8190 Alamosa N I A -1 .1.520 -1 .1530 -1 .3120 -0 . 6975 -0 . 3847 -0 . 6579 -0 .5548 Gunnison N I A -1 -1 0 .2160 .1550 .0 -1 -1 0 .2020 .1430 .0 -1 -1 0 .8530 .3190 .2354 -0 . -0 . - 0 . 9946 7092 2506 - 1 . -0 . - 0 . 1430 3851 4685 - 1 . - 0 . 0. 8730 6641 1842 -1 -0 -0 .0030 .5553 .8247 Durango N I A -1 0 .1560 .0 -1 0 .1800 .0 -1 0 .3250 .2366 -0 . -0 . 6920 2449 -0 . - 0 . 3531 4342 -0 . -0 . 6798 1872 -0 -0 .5549 .8243 - 100 -the intermediate t e r r a i n . The intermediate t e r r a i n might be fo rc ing him to the l i m i t s o f his a b i l i t i e s - a s i t ua t i on which every ind iv idua l does not necessar i l y enjoy. If th i s i s the case, given our d e f i n i t i o n of a b i l i t y l e v e l s , l e t us consider what happens when the amount of at an area increases , while holding the amount of a-^ constant. For example, f o r an intermediate sk ie r i t would mean decreasing the amount of beginner t e r r a i n . If the ind iv idua l a c tua l l y enjoyed ski t e r r a i n that required less than his utmost a b i l i t y , then one would expect his enjoyment of the s i t e to in fac t decrease when the amount of &2j increases holding a-|j constant. This argument might a lso expla in the fac t that the e l a s t i c i t i e s for advanced sk iers at Aspen and Vai l have the opposite signs from a l l the other s k i e r s ' e l a s t i c i t i e s . Maybe the advanced sk iers who ski at Aspen and Va i l are in f ac t advanced s k i e r s , who gain t h e i r most enjoyment by.sk i ing on advanced t e r r a i n . Given the fac t that the ind iv idua l genera l ly reacts adversely when the proport ion of the t e r r a i n at a s i t e designed s p e c i f i c a l l y for h is leve l o f s k i i ng a b i l i t y increases (a-^ constant ) , a question a r i ses concerning what happens to s i t e j ' s predicted share when the amount of and a-jj increase in equal amounts. In Chapter 3, sect ion C, the e l a s t i c i t y measuring th i s response was i d e n t i f i e d and i t was noted that i t equals the sum o f the c h a r a c t e r i s t i c e l a s t i c i t i e s (6.17) and (6.19). (6.20) Eg S j 9 2 j + E-I s i a 2 i l2j=0 J ^ J These e l a s t i c i t y estimates are not reported separately because they are ju s t the sum of the other two. If one examines Tables XI and XII one w i l l see that the E § . , . i g are predominantly po s i t i ve except f o r J [ a3j=0 - 101 -Aspen and V a i l . These c h a r a c t e r i s t i c e l a s t i c i t i e s genera l ly lead one to the conclus ion that a s i t e ' s predicted share increases when a 3 j . or a 2 j . increases , but that novice and intermediate sk iers respond more p o s i -t i v e l y to an increase in a^. than an increase in a 2 j . For advanced sk iers t h e i r r e l a t i v e preferences fo r a 2 j . and a.^. are mixed. They seem to prefer a^. over a 2 j , except at t h e i r f avour i te s i t e s (Aspen and Va i l ) and at s i te s designed more fo r the novice and intermediate sk ie r (Loveland, Winter Park, Cooper). S i te m's share e l a s t i c i t y with respect to a change in the quant i ty o f sk iab le t e r r a i n at s i t e j while holding a 2 j . constant, i s : 3h(a 1 . - ,a 2 , ) A (6-21) Es„«1J|8^ --sJ°-V-/h "^y» 8h(a , , , a 2 , ) These e l a s t i c i t i e s are l i s t e d in Table XIII. The things to note about these e l a s t i c i t i e s are as fo l lows. S i te m's share e l a s t i c i t y i s indepen-dent of the magnitude of the predicted share fo r that s i t e . As one would expect these e l a s t i c i t i e s , with the exception of those f o r advanced sk iers at Aspen and V a i l , are a l l negative. As with the cross p r i ce e l a s t i c i t i e s (6.16), these e l a s t i c i t i e s are larger when the s i t e at which the charac-t e r i s t i c s are changing i s important to the i n d i v i d u a l . If an i n d i v i d u a l ' s predicted share i s l a rge , the i n d i v i d u a l ' s a l l o c a t i o n of ski days amongst s i t e s w i l l be responsive to changes in the c h a r a c t e r i s t i c s of that s i t e . Again, as with the cross p r i ce e l a s t i c i t i e s f o r the shares, these e l a s -t i c i t i e s seem to increase in absolute value as one moves from the leve l of expert to intermediate and then novice s k i e r s . - 102 -Table XIII I sm a 1 j | § 2 j . = 0 Brecken- Love- Winter Broad- Crested Aspen Va i l A-Basin. r idge land Park moor Butte Denver N I A -0.6328 -0.6442 0.1021 -1.2590 -1.4720 0.7781 -0 . - 0 . - 0 . 3788 2286 1279 -0 . - 0 . - 0 . 2299 1574 1098 -0.4949 -0.3254 -0.0583 -0 . - 0 . - 0 . 4558 2438 0536 -1.4956 -0.0131 -0.0085 -0.1193 -0.0229 -0.0146 Boul der N I A .r0.6318. -0.6436 0.1016 ,-1.2460, -1.4570 0.7676 ,-0. - 0 . - 0 . 3704, 2238 1247 ,-0. - 0 . - 0 . 2256. 1546 1074 -0.4833 -0.3180 -0.0567 -0 . - 0 . - 0 . 4461 2388 0523 -0.0420 -0.0111 -0.0072 -0.1151 -0.0221 -0.0141 F t . Col 1 ins N I A -0.6911 -0.7022 0.1099 -1.2850 -1.4990 0.7823 -0.3559 -0.2144 -0.1184 -0 . - 0 . - 0 . 2215 1513 1042 -0.4601 -0.3019 -0.0534 -0 . - 0 . - 0 . 4307 2300 0499 -0.0424 -0.0112 -0.0072 -0.1253 -0.0240 -0.0151 Greeley N I A -0.6878 -0.6988 0.1096 -0.1289 -1.5050 0.7871 -0 . -0 . - 0 . 3608 2174 1203 -0.2238 -0.1529 -0.1055 -0.4670 -0.3064 -0.0543 -0 . - 0 . - 0 . 4363 2329 0507 -0.0456 -0.0120 -0.0078 -0.1291 -0.0247 -0.0156 Golden N I A -0.6157 -0.6257 0.0993 -1.2530 -1.4630 0.7745 -0 . - 0 . 3880 2338 1310 - 0 . - 0 . - 0 . 2334 1595 1114 -0.5088 -0.3340 -0.0599 -0 . - 0 . - 0 . 4659 2488 0548 -0.0395 -0.0104 -0.0068 -0.1117 -0.0214 -0.0137 A i r Force Academy N I A -0.6742 0.1209 -1.4300 0.7541 -0 . -0 . 1964 1096 - 0 . - 0 . 1595 1110 -0.2764 -0.0494 - 0 . - 0 . 2107 0462 -0.0325 -0.0212 -0.0364 -0.0232 Colorado Springs N I A -0.7948 0.1249 -0.1501 0.7864 -0 . - 0 . 1811 1004 -0 . - 0 . 1697 1173 -0.2545 -0.0452 -0 . - 0 . 1945 0424 -0.0365 -0.0237 -0.0381 -0.0241 Pueblo N I A -0.8104 -0.8392 0.1317 -1.2310 -0.1464 0.7661 -0 . - 0 . - 0 . 2795 1716 9504 - 0 . - 0 . - 0 . 2270 1581 1091 -0.3586 -0.2398 -0.0425 -0 . - 0 . -0 . 3397 1849 0402 -0.1154 -0.0310 -0.0201 -0.2406 -0.0470 -0.0297 Alamosa N I A -0.8926 -1.6860 -0 . 1634 - 0 . 1478 -0.2026 -0 . 1337 -0.0112 -0.0555 Gunnison N I A -0.8079 -0.8547 0.1359 -1.2420 -1.5090 0.8003 -0 . - 0 . - 0 . 2325 1458 0818 -0 . - 0 . - 0 . 1878 1336 0934 -0.2638 -0.1802 -0.0324 - 0 . - 0 . - 0 . 2121 1179 0260 -0.0318 -0.0087 -0.0057 -0.6566 -0.1312 -0.0841 Durango N I A -1.2400 0.1872 -1.3330 0.6710 -0 . - 0 . 1264 0673 -0.1112 -0.0738 -0.1598 -0.0272 - 0 . - 0 . 1113 0233 -0.0061 -0.0038 -0.0677 -0.0412 cont ' d - 103 -Table XIII (cont 'd) Lake Mt. Wolf Purga-Eldora Monarch Werner Creek tory Hidden Cooper Va l ley N Denver I A N Boulder I A Ft . Col 1 ins N I A N Greeley I A N Golden I A 0. 1037 -0.0401 -0.1298 -0 .0126 -0.0187 -0 . 2624 -0 . 0358 0. 1018 -0.0372 -0.0967 -0 .0086 -0.0083 -0 . 1335 -0 . 0210 0. 0398 -0.0155 -0.0144 -0 .0068 -0.0083 -0.0253 -0 . 0314 0. 1)303 -0.0380 -0.1288 -0 .0121 -0.0183 - 0 . 2578 -0 . 0485 0. 1280 -0.0353 -0.0960 -0 .0083 -0.0081 - 0 . 1313 -0 . 0285 0. 0499 -0.0146 -0.0142 -0 .0065 -0.0080 - 0 . 0248 -0 . 0424 0. 1143 -0.0391 -0.1356 -0 .0131 -0.0207 - 0 . 2551 -0 . 0632 0.1119 -0.0362 -0.1009 -0 .0096 -0.0092 -0 . 1296 -0 . 0371 0. 0432 -0.0149 -0.0148 -0 .0070 -0.0090. -0.0243 -0 . 0546 0. 1008 -0.0409 -0.1357 -0.0134 -0.0211 -0 . 2576 -0.0552 0. 0987 -0.0379 -0.1010 -0 .0092 -0.0093 -0 . 1308 -0 . 0324 0. 0382 -0.0156 -0.0149 -0 .0072 -0.0092 -0 . 0245 - 0 . 0478 0. 1049 -0.0380 -0.1281 -0 .0118 -0.0174 -0 . 2655 -0 . 0351 0. 1028 -0.0352 -0.0953 -0 .0081 -0.0077 -0 . 1349 -0.0206 0. 0403 -0.0146 -0.0142 -0 .0064 -0.0077 - 0 . 0256 -0 . 0307 A i r Force Academy Colorado Springs N I A N I A Pueblo I A N Alamosa I A -0 .0826-0.0580 -0.0930 -0.0112 -0.0108 -0.1353 -0.0179 -0.0322 -0.0240 -0.0138 -0.0088 -0.0107 -0.0256 -0.0266 -0.0755 -0.0616 -0.0940 -0.0117 -0.0111 -0.1436 -0.0165 -0.0293 -0.0254 -0.0139 -0.0091 -0.0110 -0.0270 -0.0244 -0.0698 -0.0854 -0.1236 -0.0250 -0.0349 -0.2777 -0.0260 -0.0697 -0.0806 -0.0937 -0.0175 -0.0158 -0.1438 -0.0155 -0.0270 -0.0332 -0.0138 -0.0136 -0.0155 -0.0270 -0.0230 -0.0394 -0.1097 -0.0878 -0.0445 -0.0302 -0.1698 -0.0083 N Gunnison I A -0.0340 -0.1313 -0.0347 -0.1267 -0.0136 -0.0529 -0.0999 -0.0251 -0.0618 -0.0773 -0.0180 -0.0286 -0.0115 -0.0142 -0.0286 -0.2939 -0.0117 -0.1554 -0.0071 -0.0296 -0.0107 N Durango I A -0.0329 -0.0634 -0.0122 -0.0251 -0.0670 -0.0568 -0.1461 -0.1194 -0.0082 -0.0095 -0.0426 -0.1384 -0.0216 -0.0117 - 104 -S i te m's share e l a s t i c i t y with respect to a change in the amount of t e r r a i n at s i t e j designed s p e c i f i c a l l y fo r the i n d i v i d u a l ' s sk i ing a b i l i t y while holding to ta l sk iab le t e r r a i n at s i t e j (.a-jj) cons tan t : i s : (6.22) E~ =-s.S 1 ^ — / h ( a 1 - , a 2 i ) sm a 2 j 8^=0 J 9 a 2 j I J ^ 3 h ( a 1 , , a 2 . ) „ •These appear in Table XIV. For the novice and intermediate sk ie r these cross e l a s t i c i t i e s with respect to c h a r a c t e r i s t i c a 2 j . are a l l p o s i t i v e . This i s not unreasonable i f one accepts our i n te rp re ta t i on of sk i ing a b i l i t y , i . e . that the ind iv idua l a c tua l l y enjoys acreage below his reported sk i ing a b i l i t y more than acreage at his s k i i ng a b i l i t y . The signs on these e l a s t i c i t i e s fo r the expert sk iers are mixed, and i t appears that they are negative for the s i t e s experts enjoy most and fo r those s i t e s designed predominantly f o r novices and intermediates. These e l a s t i c i t i e s are negative f o r Aspen and V a i l , s i t e s the advanced sk ie r l i k e s , but they are a lso negative f o r areas such as Loveland, Winter Park and Cooper. I f i n d th i s p a r t i c u l a r re su l t quite i n t e r e s t i n g , but a reasonable explanat ion eludes me. Examination of the e l a s t i c i t y formula and the estimated share e l a s t i c i t i e s leads to a number of general conclus ions. The magnitude and the signs of a l l the e l a s t i c i t i e s c r u c i a l l y depend on the estimate of (-a). The absolute value of .the own e l a s t i c i t i e s are inverse ly re l a ted to the magnitude of t h e i r pred icted shares. A l l of the cross e l a s t i c i t i e s depend l a r ge l y on the magnitude of the predicted share fo r - 105 -Table XIV m 2j J = Aspen Va i l Brecken- Love-A-Basin r idge land Winter Broad- Crested Park moor Butte N 0.6585 1.4120 0.2886 0.1623 0.3198 0.3798 0.0247 0.0909 Denver I 0.6095 1.6110 0.1261 0.0874 0.1952 0.1093 0.0026 0.0 A -0.1879 -0.6720 0.0166 0.0263 -0.0291-0.0195 0.0 0.0 N 0.6574 1.3970 0.2822 0.1593 0.3826 0.2718 0.0210 0.0877 Boulder I 0.6089 1.5950 0.1235 0.0858 0.1908 0.1070 0.0022 0.0 A -0.1870 -0.0162 0.0162 0.0257 -0.0284 -0.0190 0.0 0.0 F . N 0.7191 1.4410 0.2712 0.1563 0.3642 0.3589 0.0212 0.0954 r n i i i n c 1 0.6644 1.6400 0.1183 0.0840 0.1812 0.1031 0.0022 0.0 o o i n n s A _ 0.2022 -0.6756 0.0154 0.0250 -0.0267 -0.0181 0.0 0.0 N 0.7157 1.4460 0.2749 0.1580 0.3697 0.3636 0.0228 0.0983 Greeley I 0.6611 1.6460 0.1199 0.0849 0.1839 0.1044 0.0024 0.0 A -0.2017 -0.6797 0.0156 0.0253 -0.0271 -0.0184 0.0 0.0 N 0.6407 0.1405 0.2956 0.1648 0.4028 0.3882 0.0197 0.0851 Golden I 0.5920 1.6000 0.1290 0.0886 0.2004 0.1115 0.0020 0.0 A -0.1828.-0.6689 0.0170 0.0267 -0.0299 -0.0199 0.0 0.0 A i r N Force I 0.7230 0.1564 0.1084 0.0886 0.1658 0.0944 0.0065 0.0 Academy A -0.2224 -0.6512 -0.0143 0.0266 -0.0247 -0.0168 0.0 0.0 N Colorado l Q J 5 2 0 1.6420 0.0999 0.0943 0.1527 0.0872 0.0073 0.0 springs A _ 0 > 2 2 9 8 -0.6791 0.0131 0.0281 -0.0226 -0.0154 0.0 0.0 N 0.8433 1.3810 0.2130 0.1602 0.2839 0.2831 0.0577 0.1833 Pueblo I 0.7940 0.6020 0.0947 0.0878 0.1439 0.0828 0.0062 0.0 A -0.2423 -0.6616 0.0124 0.0261 -0.0212 -0.0146 0.0 0.0 N Alamosa I 0.8445 1.8440 0.0901 0.0821 0.0121 0.0599 0.0022 0.0 A N 0.8407 1.3930 0.1771 0.1325 0.2088 0.1767 0.0159 0.5003 Gunnison I 0.8087 1.6520 0.0804 0.0742 0.1081 0.0528 0.0017 0.0 A -0.2501 -0.6912 0.0106 0.0224 -0.0162 -0.0094 0.0 0.0 N Durango I 1.1730 1.4580 0.0697 0.0618 0.0958 0.0499 0.0012 0.0 A -0.3444 -0.5795 0.0087 0.0177 -0.0136 -0.0084 -0.00 0.0 con t ' d - 106 -Table XIV (contld) Lake Mt. Wolf Purga- Hidden Eldora Monarch Werner Creek tory Cooper Va l ley N 0.0648 0.0251 0.0916 0.0075 0.0104 0.1933 0.0215 Denver I 0.0565 0.0219 0.0600 0.0038 0.0022 0.0580 0.0078 A 0.0 0.0 -0.0082 0.0013 0.0025 -0.0095 0.0146 N 0.0814 0.0237 0.0909 0.0072 0.0101 0.H900 0.0291 Boulder I 0.0710 0.0207 0.0596 0.0037 0.0022 0.0570 0.0107 A 0.0 0.0 -0.0081 0.0012 0.0025 -0.0093 0.0198 F t N 0.0714 0.0244 0.0957 0.0079 0.0115 0.1880 0.0379 C n i l i n c : 1 ° - 0 6 2 1 0.0213 0.0626 0.0040 0.0025 0.0563 0.0139 A 0.0 0.0 -0.0084 0.0014 0.0028-0.0091 0.0255 N 0.0630 0.0256 0.0958 0.0080 0.0117 0.1898 0.0331 Greeley I 0.0548 0.0223 0.0626 0.0041 0.0025 0.0568 0.0121 A 0.0 0.0 -0.0085 0.0014 -0.0028 -0.0092 0.0223 N 0.0655 0.0237 0.0904 0.0071 0.0096 0.1956 0.0210 Golden I 0.0571 0.0207 0.0591 0.0036 . 0.0021 0.0536 0.0077 A 0.0 0.0 -0.0081 0.0012 0.0024 -0.0096 0.0143 Ai r N Force I 0.0458 0.0341 0.0577 0.-0049 .0,0029 .0,0588 0,0067 Academy A 0.0 0.0 -0.0079 0.0017 0.0033 -0.0096 0.0124 N E°!j n r a nJ° I 0.0419 0.0362 0.0583 0.0052 0.0030 0.0624 0.0061 w r i n g s A 0 > 0 Q^Q -0.0079 0.0018 0.0034 -0.0101 0.0113 N -0,0436 0.0534 0.-0872 0.0150 0.0194 0.2046 0.0156 Pueblo I 0.0387 0.0474 0.0581 0.0078 0.0043 0.0625 0.0058 A 0.0 0.0 -0.0079 0.0027 0.0048-0.0101 0.0107 N Alamosa I 0.0219 0.0645 0.0545 0.0197 0.0082 0.0738 0.0031 A N 0.0212 0.0820 0.0705 0.0151 0.0343 0.2166 0.0070 Gunnison I 0.0193 0.0745 0.0480 0.0080 0.0078 0.0675 0.0026 A 0.0 0.0 -0.0066 0.0028 0 . 0 0 8 9 - 0 . G i l l 0.0050 Durango I 0.0183 0.0373 0.0416 0.0252 0.0398 0.0519 0.0030 A 0.0 0.0 -0.0054 0.0085 0.0432 -0.0081 0.0054 - 107 -t h e . s i t e whose p r i ce or c h a r a c t e r i s t i c i s changing, and the c h a r a c t e r i s t i c e l a s t i c i t i e s depend on how responsive the u t i l i t y produced by a day of sk i ing is to changes in the e f f e c t i v e phys ica l c h a r a c t e r i s t i c s of the s i t e . - 108 -Footnotes - Chapter 6 1. The Newton method required approximately 150 i t e ra t i on s with eight funct ion evaluat ions per i t e r a t i o n , to converge with the f u l l model ( s ix parameters) when ai-015 were i n i t i a l l y set equal to one and -a was set equal to -2.0.(which corresponds to a 8 of .5). The model was res tar ted from another point and converged to the same so lu t i on . The convergences were a l l qu i te smooth and no d i f f i c u l t i e s were encountered. There was no tendency f o r f a l s e convergence to e i the r a minimum or a sadd le-po int . The algorithm used did not ca l cu la te der i va t i ves a n a l y t i c a l l y but ca l cu la ted the numerical approximations. The estimated covariance matrix of the parameter estimates i s the inverse of the matrix of expected values of the second der iva t i ves of the log of the l i k e l i h o o d funct ion with respect to the parameters (equation (4.8) ) . The expected values of these second der i va t i ves are assumed equal to t h e i r numerical, approximations. 2. Since the der i va t i ves of I* with respect to the 0 parameters were ca l cu la ted numerica l ly , the asymptotic t s t a t i s t i c s are subject to approximation e r ro r . But s ince the t s t a t i s t i c and the l i k e l i h o o d r a t i o s t a t i s t i c are both report ing that - a i s s i g n i f i c a n t l y d i f -ferent from zero at approximately the same leve l s of s i gn i f i c ance (.002 and .004 r e s p e c t i v e l y ) , one might therefore conclude that the approximation e r ro r for at l eas t th i s p a r t i c u l a r asymptotic t s t a t i s t i c i s qu i te smal l . 3. The model was a l so estimated assuming that the constant term in h(a-[j,a2j) w a s zero. The I* f o r that model was -3472.963. On the basis of a l i k e l i h o o d r a t i o t e s t , one cannot re jec t th i s n u l l -hypothesis. Both of the f u l l models p red ic t behaviour equal ly w e l l . I have chosen to report the resu l t s f o r the model invo lv ing the constant term because the more r e s t r i c t i v e models (models 1 and 2) are nested in th i s model. 4. It was thought that the parameter estimates might be sens i t i ve to the assumption that the opportunity cost of the students ' time was $1.15 per hour. To tes t t h i s , the f u l l model was re-est imated f i r s t assuming that the opportunity cost of the students ' time was $1.25 per hour, and then assuming i t was $1.00 per hour. Neither a l t e r n a -t i v e wage assumption had any appreciable e f f e c t on e i t he r the para-meter estimates or the value of the l i k e l i h o o d funct ion. 5. This modif ied R s t a t i s t i c i s i nd i ca t i ng how well the model explains the average behaviour in each o f the 33 data groups. A l l of the i nd i v i dua l s in my sample can be placed in one of the 33 groups on the basis o f t h e i r sk i ing a b i l i t y (three c lasses) and t h e i r c i t y of residence (eleven p o s s i b i l i t i e s ) . The model i s exp la in ing the behaviour of each of these 33 representat ive i nd i v idua l s extremely w e l l . To get an i nd i ca t i on of how well the model pred ic t s the behaviour of each of the 163 i n d i v i d u a l s , one should make T the t o t a l number of ski t r i p s which is 1453. The modif ied R^  based on t h i s T value i s .5301. - 109 -Chapter 7 SUMMARY AND CONCLUSIONS The purpose of th i s research was t h r e e - f o l d . F i r s t , I wanted to obtain a system of share equations f o r s i t e - s p e c i f i c recreat iona l a c t i v i t i e s which are cons i s tent with an underlying theory of constrained u t i l i t y maximizing behaviour. The second object i ve was to incorporate the important phys ica l c h a r a c t e r i s t i c s of the recreat iona l s i t e s d i r e c t l y into the u t i l i t y funct ion in such a way that the i n d i v i d u a l ' s production technology ( in our case sk i ing a b i l i t y ) can l i m i t the i n d i v i d u a l ' s a b i l i t y to u t i l i z e the c h a r a c t e r i s t i c s . F i n a l l y , I wanted to estimate emp i r i c a l l y such a model for a group of recreat iona l s i t e s in the hopes of confirming my basic hypothesis that both pr ices and the e f f e c t i v e physical c h a r a c t e r i s t i c s of the s i t e s play an important explanatory ro le in the i n d i v i d u a l ' s a l l o c a t i o n of recreat ion days amongst those s i t e s . I fee l that the d i s se r t a t i on has been reasonably successful in accomplishing these three goals. Most of the previous work on recreat iona l demand has u t i l i z e d the t r a v e l - c o s t technique, a technique which recognizes the strong s t a t i s t i c a l r e l a t i on sh ip between a s i t e ' s predicted share and the d i s -tance from that s i t e to the i n d i v i d u a l ' s res idence, but lacks a strong foundation in basic consumer theory. The model of sk ie r behaviour out-l i ned in the d i s s e r t a t i o n recognizes and confirms the hypothesis that t rave l costs are important and also gives th i s hypothesis a strong theore t i ca l foundation. Even though my underlying preference ordering is qu i te r e s t r i c t i v e , I fee l that I have made a small step by making the share equations fo r my recreat iona l a c t i v i t i e s cons i s tent with an underlying theory of u t i l i t y maximizing behaviour. I a lso d i r e c t l y - no -include the c h a r a c t e r i s t i c s of the s i t e s in the .ana l y s i s , whereas charac-t e r i s t i c s have not prev ious ly played a major ro le in recreat iona l demand s tud ies . Inclus ion of the c h a r a c t e r i s t i c s resu l ted i n i i d e n t i c a l share equations, a useful r e s u l t . The f ac t that we cou ld : inc lude the charac-t e r i s t i c s , along with the other improvements brought about by th i s research in the techniques used to estimate an i n d i v i d u a l ' s demand for v i s i t s to recreat iona l s i t e s , must be a t t r i b u t a b l e in part to my access to data that are more de ta i l ed than those p rev i ou s l y . ava i l ab l e . A s i g n i f i c a n t part of my research was the c o l l e c t i o n and construct ion of th i s data. The Denver Research Ins t i tute survey contains the most deta i led data of i t s type on recreat iona l use, but no one working in the area of rec rea -t iona l demand seems to have been aware of i t s ex i s tence. Data on the c h a r a c t e r i s t i c s of the ski areas f o r the period covered by the DRI cross sect iona l sample of sk iers were obtained from numerous sources with the help of many people and organizat ions associated with the sk i ing industry. Hopeful ly other researchers in the area of recreat iona l demand w i l l want to u t i l i z e the data. The estimated system of share equations confirms my basic hypotheses ;that the c h a r a c t e r i s t i c s of the ski areas, the i n d i v i d u a l ' s sk i ing a b i l i t y , and the p r i ce s , a l l play aniimportant ro le in how the sk ie r a l l oca te s his ski days amongst . s i tes . The e l a s t i c i t y - e s t i m a t e s are not unreasonable, and o f f e r ce r t a in ins ights into the behaviour of the s k i e r . My model of sk ie r behaviour confirms and quant i f i e s the non-quant i f ied operating postulates of the managements of most ski..areas. I f i n d i t very encouraging that a formal model cons i s tent with u t i l i t y maximizing behaviour i s cons i s tent with t h e i r hypotheses. My model of sk ie r behaviour i s more l im i ted and r e s t r i c t i v e - I l l -than I had intended when I o r i g i n a l l y s ta r ted the research. The process has been one of adding more and more r e s t r i c t i v e assumptions to the model. The preference order ing impl ied by my u t i l i t y funct ion is qu i te r e s t r i c -t i v e , preferences are both d i r e c t l y add i t i ve and homothetic. It i s my hope that the model can eventua l ly be general ized to allow for non-homo-t h e t i c preferences. This might be accomplished by genera l i z ing my Bergson funct ion to admit non-homothetic preferences. One p o s s i b i l i t y i s : (7.1) U.- j [ y J - f ( « 1 j . a z j ) ] B h ( « 1 J . . y ) 3 ' Another candidate, the-K le in-Rubin li inear expenditure system,-is a specia l case of (7.1) where 6=0. These preference orderings assume d i r e c t add i -t i v i t y , but the funct ions are not homothetic to the o r i g i n (see Pol lak (1971:403-404) fo r more d e t a i l s ) . I think that the e l im inat ion of the homotheticity r e s t r i c t i o n would improve the p red i c t i ve power of the model. The model can a lso be genera l ized by inc lud ing more character -i s t i c s o f ski areas d i r e c t l y into the. ana lys i s to determine i f th i s increases i t s explanatory power, but I doubt i t would. The added charac-t e r i s t i c s w i l l be qu i te cor re la ted with the ones already inc luded, making est imation d i f f i c u l t and making i t un l i ke l y that the model's explanatory power would improve. One might l i k e to model the behaviour of the house-hold rather than the i n d i v i d u a l , but th i s i s not f ea s i b l e with my data source r because i t does not record the complete f am i l y ' s sk i ing a c t i v i t i e s . F i n a l l y , my production technology does not consider mult ip le-day t r i p s . This i s not a major de f i c iency in a study that considers only student s k i e r s , but i t would become important i f o u t - o f - s t a t e adult sk ier s had been included in the sample. 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The Breckenridge Sk ier . Boulder: Business Research D i v i s i o n , Un ivers i ty of Colorado. - 117 -Goldberger, A.S. 1967. "Funct ional Form and U t i l i t y : A Review of Consumer Demand Theory". Workshop Paper 6703. Madison: Socia l Systems Research D i v i s i o n , Un ivers i ty of Wisconsin. Go ld fe ld , S.M. and Quandt, R.E. 1972. Nonlinear Methods in Econometrics. Amsterdam: North-Holland Publ i sh ing Company. Go ld fe ld , S.M., Quandt, R.E. and T r o t t e r , R. 1966. "Maximizing by Quadratic H i l l C l imbing" . Econometrica 34:514-551. G r i l i c h e s , Z. 1971. " In t roduct ion: Hedonic Pr ice Indexes R e v i s i t e d " . In Pr ice Indexes and Qual i ty Change, Studies in New Methods  o f Measurement. Boston' Harvard Un ivers i ty Press. Grimes, O.F. 1974. "P r i va te Access Recreation Near Urban Centers: A Land-Use Approach". Land Economics 5:2-7. H a l l , R.E. 1973. "The S p e c i f i c a t i o n o f Technology with Several Kinds of Output". 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Kendal l , M.G. and S tuart , A. 1967. The Advanced Theory of S t a t i s t i c s , v o l . 2. 2nd e d i t i o n . London: Charles G r i f f i n and Company L imi ted. Knetsch, J . L . 1963. "Outdoor Recreation Demands and Bene f i t s " . Land Economics 39:385-396. - 118 -Knetsch, J . L . 1972. " I n terpret ing Demand f o r Outdoor Recreat ion" . Economic Record 48:429-432. Kra f t , A. and Kra f t , J . 1974. "Empir ica l Estimation of the Value of Travel Time Using Multi Mode Choice Models". Journal of  Econometrics 2:317-326. K r u t i l l a , J .V . 1972. Introduction to National Environments Studies in  Theoret ica l and Appl ied Ana ly s i s . Balt imore: Johns Hopkins Press. Lancaster, K. 1966a. "Change and Innovation in the Technology of Consumption". American Economic Review 56:567-585. Lancaster, K. 1966b. "A Mew Approach to Consumer Theory". Journal of  P o l i t i c a l Economy 74:132-157. Lancaster, K. 1971. Consumer Demand: A New Approach. 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London: Academic Press. McFadden, D. 1974b. "The Measurement of Urban Travel Demand". Working Paper No. 227. Berkeley: I n s t i tu te of Urban and Regional Development, Un ivers i ty o f C a l i f o r n i a . McFadden, D. and Reid, F. 1974. "Aggregate Travel Demand Forecast ing from Disaggregated Behavioural Modes". Working Paper No. 228. Berkeley: I n s t i tu te of Urban and Regional Development, Un iver s i t y o f C a l i f o r n i a . Michael , R.T. and Becker, G.S. 1973. "On the New Theory of Consumer Behaviour". Swedish Journal of Economics.75:378-396. - 119 -M i l l e r , R.E. 1972. Modern Mathematical Methods fo r Economics and  Business. New York: Holt Rinehart and Winston, Inc. Mood, A.M. and G r a y b i l l , F.A. 1963. Introduction to the Theory of S t a t i s t i c s . 2nd e d i t i o n . New York: McGraw-Hill Book Company. Muellbauer, J . 1974. "Household Production Theory, Qual i ty and the Hedonic Technique". American Economic Review 64:977-994. Pearse, P.H. 1968. "A New Approach to the Evaluat ion of Non-priced Recreational Resources". Land Economics 44:88-99. Ph l i p s , L. 1974. Appl ied Consumption Ana ly s i s . Amsterdam: North-Holland Publ i sh ing Company. Po l lak , R.A. 1971. "Add i t ive Functions and L inear Engel Curves". Review of Economic Studies 38:401-414. Po l l ak , R.A. and Wachter, M.L. 1975. "The Relevance of the Household Production Function and Its Implications f o r the A l l o ca t i on of Time". Journal of P o l i t i c a l Economy 83:255-278. Po l lak , R.A. and Wachter, M.L. 1977. "Reply: IPol lak and Wachter on the Household Production Approach ' " . Journal o f P o l i t i c a l  Economy 85:1083-1086. •Pollak, R.A. and Wales, T . J . 1975. "Est imat ion of Complete Demand Systems from Household Budget Data". Discussion Paper No. 345. Ph i l ade lph ia : Department of Economics, Un ivers i ty o f Pennsylvania. Rausser, G. and O l v e i r a , R. 1976. "An Econometric Analys i s of Wi lder-ness Area Use". Journal of the American S t a t i s t i c a l Asso- c i a t i o n 71:276-28lT Rand McNally & Co. 1973. Road A t l a s . New York: Rand McNally & Co. Ranken, R.L. and Sinden, J .A . 1971. "Causal Factors in the Demand fo r Outdoor Recreat ion" . Economic Record 47:418-426. Rao, C.R. 1965. L inear S t a t i s t i c a l Inference and Its App l i ca t i on s . New York": John Wiley and Sons, Inc. Samuelson, P.A. 1947. Foundations of Economic Ana ly s i s . Cambridge: Harvard Un iver s i t y Press. Samuelson, P.A. 1961. "Using Fu l l Dual i ty to Show that Simultaneous Add i t i ve D i rect and Ind i rect U t i l i t y Implies Unitary Pr ice E l a s t i c i t y o f Demand". Econometrica 33:781-796. S i l v e r b e r g , E. 1972. "Dua l i ty and the Many Consumer Surpluses" . American Economic Review :62:942-952. - 120 -Smith, R.J. 1971. "The Evaluat ion of Recreation Benef i t s : The Clawson Method in P r a c t i c e " . Urban Studies 8:89-102. Smith, V.K. 1975. "Travel Cost Demand Models fo r Wilderness Recreat ion: A Problem on Non-Nested Hypotheses". Land Economics 51:103-111. Smith, V.E. and Koo, A.Y.C. 1973. "A General Consumption Technology in New Demand Theory". Western Economic Journal 11:243-259. T e r l e c k y j , N.E.,ed. 1975. Household Production and Consumption. New York: Columbia Un ivers i ty Press. T h e i l , H., ed. 1967. Economics arid Information Theory: Studies in  Mathematic and Managerial Regressions, v o l . 7. Amsterdam: North-Holland Publ i sh ing Company. Tourism Research Assoc iates . 1972. L i f t Interview. Study of the Aspen  Sk ier . Amherst, Mass.: Tourism Research Assoc iates . Tourism Research Assoc ia tes . 1975. L i f t Interview Study of the Sugarloaf  Sk ier 1975. Amherst, Mass.: Tourism Research Assoc ia tes . U.S. Department of Commerce. 1971. S t a t i s t i c a l Abstract of the U.S. Bureau of the Census. U.S. Department of Commerce. 1976. S t a t i s t i c a l Abstract of the U.S. Bureau of the Census. Uzawa, H. 1962. "Production Functions with Constant E l a s t i c i t i e s of S b u s t i t u t i o n " . Review of Economic Studies 29:291-299. Wales, T . J . 1976. "On the F l e x i b i l i t y of F l e x i b l e Functional Forms: An Empir ical Approach". Discuss ion Paper 76-3. San..Di;ego: Department.of Economics; Un ivers i ty of C a l i f o r n i a . Wales; T . J . and Woodland, A.D. 1974. "Est imation of the A l l o c a t i o n of Time fo r Work, Leisure and Housework". Discuss ion Paper 74-19. Vancouver: Department of Economics, Un ivers i ty of B r i t i s h Columbia. Wilks, S.S. 1962. Mathematical S t a t i s t i c s . New York: John Wiley and Sons, Inc. Woodland, A.D. 1976. "Model l ing the Production Sector of the Economy: A Se lec t i ve Survey and Ana l y s i s " . Discussion Paper 76-21. Vancouver: Department of Economics, Un iver s i t y o f B r i t i s h Columbia. Woodland, A.D. 1977. "S tochast ic S p e c i f i c a t i o n and the Estimation of Share Equat ions". Working Paper. Vancouver: Department of Economics, Un ivers i ty of B r i t i s h Columbia. - 121 -Appendix A SUPPLEMENTARY DATA AND RELATED INFORMATION STATE OF COLORADO SURVEY OF SKIERS Dear Skier: During the past season you were contacted by one of our interviewers at a Colorado ski area. A t that t i m e you w e r e asked to give your name and address, and also to agree to answer some questions concerning your skiing experiences. This survey i s an important part of a large-scale effort being conducted by the State of Colorado to i m p r o v e recreational facilities in the State. Therefore, as a skier your opinions and experiences a r e important to future planning in Colorado. A t the end of the study in 1969 a drawing wil l be held and ten $100 U . S . Savings Bonds will be awarded to individuals who have participated in the surveys. If you wish your name to be included i n the drawing, do not forget to check the box above your name and address on the back page of this torn-.. We w o u l d g r e a t l y a p p r e c i a t e your thoughtful replies to the following questions. Please try to answer a l l of the q.-.estions as completely as possible. Your answers wil l remain in the strictest c o n f i d e n c e . C L A S S I F I C A T I.O .\ IN F O R M A T IP N : Q u e s t i o n s 1 t h rough 5 a r e needed solely for the purpose of analysis. Your answers, un-less otherwise i n d i c a t e d , shou ld pertain only to you and not to the other members of your family. 1. Sex: f~jMale 1 | Female 2. Mari ta l status: [^JMarried r~JSingle (includes divorced, widowed, etc.) is vour occupation. f ~ z > ~ — ' I i f S T U D E N . T : Name of school: 4. Education: r~j8th grade or less [ | Some.college (""JSome high school CH College graduate [TjHigh school graduate | | Post-graduate work 5. Please indicate which group best approximates the total annual income for your family. f~JUnder $5, 000 fJJ$7, 500 - $9, 999 [J$15,000 - $24, 999 f~J $5, 000 - $7, 499 Q S l O , 000 - $14, 999 Q$25, 000 and over 6. How far is the nearest ski area from your place of residence? .miles 7. Do you own a second home or condominium near a ski area? r~jYes | | No IF Y O U C H E C K E D " Y E S " IN Q . 7: 7a. Where is this dwelling located? Ski area: State: 8. A r e you a member of a ski club? r~J Yes [ | No 9. In what year did you first begin skiing? 10. How would you Classify yourself as to your skiing ability? r~J Novice (snow-plow) | |Between novice and intermediate r~J Intermediate (stem-Christie) r~jBetween intermediate and advanced ^ A d v a n c e d (linked parallel turns) (Please Continue to the Next Page) 11. Now we w o u l d l i k e to have some i n f o r m a t i o n about the o ther m e m b e r s of your f a m i l v l i v i n g at y o u r p l a c e of r e s i d e n c e , as w e l l as y o u r s e l f . In the f i r s t co lumn p l ease ind ica te the r e l a t i o n s h i p of each f a m i l y m e m b e r to y o u r s e l f , and then for each i n d i v i d u a l i n your f a m i l y , i n c l u d i n g your se l f , answer the ques t ions i n each succeed ing c o l u m n . (a) (b) RELATIONSHIP OF FAMILY MEMBER TO SELF AGE OF FAMILY MEMBER (0 DOES INDIVIDUAL SKI? AT WHAT AGE DID INDIVIDUAL FIRST SKI? (e) DOES INDIVIDUAL OWN OR RENT THE FOLLOWING? SKIS BOOTS (f) HOW MUCH DID EACH FAMILY MEMBER SPEND ON THE PUR-CHASE OF THE FOLLOWING ITEMS DURING 1967-68 SEASON? SKIS AND OTHER 1. Se l f YES NO n n OWNS RENTS • • OWNS RENTS • • BINDINGS $ BOOTS $ POLES • $ ' CLOTHING $ ' ' EQUIPMENT s . 2. n n • • s S $ • . $ ' s 3. n n • i n • • $ s $ S • $ 4. n n • • $ $ $ s 5. n n • • • • $ s ' $ $ s • 6. n n • • $ $ $ $ s 7. n n • • • • $ $ $ $ $ HOW MANY TOTAL DAYS DID EACH MEMBER SKI DURING 1967-68? 12. ro o j E v e r y s k i e r has c e r t a i n expecta t ions r e g a r d i n g future equipment p u r c h a s e . Tha t i s , he o r she knows that boots w i l l be r e p l a c e d e v e r y X n u m b e r of y e a r s , and so fo r th . N o r is it unusual for the s k i e r to ant ic ipate , buy ing a p a r t i c u l a r b r a n d of equ ipment . T h e ' s ame ho lds t rue for c lo th ing . In the spaces below p l e a s e m a k e your best e s t i m a t e s c o n c e r n i n g the p u r c h a s e of the i t e m s l i s t e d in C o l u m n A . T r y to be as spec i f i c as p o s s i b l e . DURING WHICH FUTURE SKI SEASON WOULD YOU EXPECT YOUR NEXT PURCHASE OF THIS EQUIPMENT TO OCCUR? HOW OFTEN DO YOU USUALLY PURCHASE THIS ITEM? (E.G., EVERY YEAR, EVERY TWO. YEARS, FIVE) S k i s and b i n d i n g s . Boo t s P a r k a • _ S k i P a n t s D w e l l i n g p u r c h a s e s . (e. g. , c o n d o m i n i u m ) In what a r e a wou ld this an t i c ipa ted d w e l l i n g p u r c h a s e be located? A r e a : State: HOW MUCH DO YOU EXPECT TO PAY FOR THIS ITEM WHEN YOU DO PURCHASE IT? $-. s ,s s $ (Please Continue to the Next Page) 13. 1. 2. 3. 4. 5. 6. 7 . 8 . 9 . 10. 1 1. 12. The next s e r i e s of quest ions p e r t a i n on l y to your s k i i n g a c t i v i t i e s du r ing the 1967-68 s e a s o n . We a re i n t e r e s t e d in l e a r n i n g s o m e -thing about each s k i i n g t r i p that you took away f r o m h o m e . In the f i r s t c o l u m n under " D A T E " i nd i ca t e on ly the mon th and i n w h i c h ha l f of the mon th the m a jo r p o r t i o n of the t r ip took p l a c e . MONTH DATE OF TRIP. FIRST LAST •A6) AREA(S) SKIED (Name and State) TYPE OF TRANSPORTATION USED TO GET FROM HOME TO SKI AREA ' 41) NUMBER OF DAYS AWAY FROM RESIDENCE -flf NUMBER OF DAYS SPENT SKIING NUMBER OF FAMILY MEMBERS IN THE PARTY TO SKI AREA WAS TRIP PART OF A CLUB ACTIVITY AMOUNT SPENT ON YOURSELF AND YOUR FAMILY ON TRIP FOR: (Total column below should include all trip expenditures.) HALF HALF " T l YES NO n n TOTAL s LODGING s DRINK s LIFTS s . LESSONS s • n n n $ s s s s • n n n $ $ s S • $ • n . • n n $ s s s s • n n n $ $ s s s • n n n $ s s s s • • n n n $ $ s s - s • n • n n $ s $ s s • n n n $ s $ s s • n n n $ $ s s s • n n n s $ s s s • n n n $ s s s s / F O R A D D I T I O N A L T R I P S D U R I N G T H E 1967-68 S E A S O N , P L E A S E U S E T H E I N S E R T S H E E T P R O V I D E D . 13i . Of a l l the a r ea s w h i c h you s k i e d th is pas t season, w h i c h one d id you enjoy the mos t? 13j. What was it about that a r e a w h i c h you found p a r t i c u l a r l y enjoyable? •  (Please Continue to the Next Page) IF YOU TOOK MORE THAN 12 SKI TRIPS DURING T H E 1967-68 SEASON, P LEASE CONTINUE Q. 13 ON THIS PAGE, (a) " . ( b ) (c) (d) (e) (f) (g) (h) SKI TRIP NO.: 13. 14. 15. 16. 17. 18. 19. 20. .21. 22. 23. 24. 25. 26. 27. 28. 29. 30. MONTH DATE OF TRIP FIRST LAST • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • AREA(S) SKIED {Name and State) TYPE OF TRANSPORTATION USED TO GET FROM HOME TO SKI AREA NUMBER OF DAYS AWAY FROM RESIDENCE NUMBER OF DAYS SPENT SKIING NUMBER OF FAMILY MEMBERS IN THE PARTY TO SKI AREA WAS TRIP PART OF. A CLUB ACTIVITY • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • AMOUNT SPENT ON YOURSELF AND YOUR FAMILY ON TRIP FOR: (Total column below should include alt trip expenditures.) YES NO FOOD AND DRINK LESSONS $ . s . s . $. $. $ . $ ; s . $ . S . $ $ $ . $ . $ . $ . $ . $ . s $ . $ _ s $ . $ . s _ s . $ s . s s s . s s . $ . 5 . s . $ . s . s . s . s . s . s . $ s . s $ $ $ $, $. $•. $ . s . s . s . s . $ . s r o on (Please Continue to the Next Page) NOV. 1967 S M T W T F S 1 2 3 4 5 0 7 8 910 11 12 13 14 IS IS 17 18 19 2 0 21 22 23 24 25 26 2728 29 30 DEC. 1967 S M T W T F S 1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18 19 20 2122 23 242528 27 2829 30 31 JAN. 1968 S M T W T F S 1 2 3 4 S 6 7 8 9 10 1112 13 14 15 16 17 18 19 20 21 22 23 24 252627 28 29 30 31 FEB. 1968 S M T W T F S 1 2 3 4 5 6 7 8 9 10 1112 13141518 17 181920 212223 24 12526272829 MAR. 1968 S M T W T F S 1 2 3 4 5 8 7 8 9 10 1112 13 14 15 16 17 18 19 20 2122 23 24 25 26 2728 29 30 31 APR. 1968 5 M T W T F S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 17 18 1920 21 22 2324 25 26 27 28 29 30 MAY 1963 S M T W F S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2122 23 24 25 262728 29 30 31 T h e a c c o m p a n y i n g c a l e n d a r and m a p o f C o l o r a d o s k i a r e a s has been p r o -v i d e d to he lp y o u to r e c a l l the s k i t r i p i n f o r m a t i o n r e q u e s t e d in Q . 13. H o w e v e r , p l e a s e note that the q u e s -t i o n r e f e r s to a l l s k i t r i p s d u r i n g the 1967-68 s e a s o n , not j u s t t hose i n the state of C o l o r a d o . STEAMBOAT SPRINGS. HOWLSf H KILL Q ( P l e a s e C o n t i n u e to the N e x t Page ) • - 127 -1 4: A s far as y o u ' a r e eonce r n c c l , ' w h a t is the p r i n c i p a l d r a w b a c k of C o l o r a d o sk i a r e a s ' rjPoor snow cond i t ions r~J P o o r wea the r [""J-C rowded l if t l i n e s (~J Shor tage of lodging space [~Jl-Iigh p r i c e of l i f t t i cke t s [ ~ J P o o r a c c e s s to s k i a r e a s ( roads , e t c . ) ("J C r o w d e d s k i s lopes J O the r (please spec i fy) : Inadequate base f a c i l i t i e s 14a. P l e a s e name the C o l o r a d o s k i a reas hav ing the d r a w b a c k you jus t noted . 15. How many annual p a i d v a c a t i o n days do you have at the p r e s e n t t ime? (Note: If you have no p a i d v a c a t i o n days , a n s w e r by i n d i c a t i n g the number of vaca t ion days w h i c h you u s u a l l y take in a y e a r . ) days pe r year 16. D u r i n g the 1967-68 s k i season how many p a i d v a c a t i o n days d id you spend sk i ing? days 16a. How m a n y of these d a y s w e r e spent s k i i n g in C o l o r a d o ? days 17. How many a n n u a l . p a i d v a c a t i o n days do you e s t i m a t e that you w i l l have f ive y e a r s f r o m  now? days per yea r 18. A r e you p lann ing to take any m a j o r (one week o r longer) s k i t r i p s next season (1968-69)? [ | None p lanned P l a n n e d D e s t i n a t i o n . L en g th of T r i p (in days) 19- On w h i c h w e e k - day(s) a r e you m o s t l i k e l y to be able to take one -day s k i - t r i p s d u r i n g the next season (1968-69)? (Note: M o r e than one day m a y be c h e c k e d if a p p r o p r i a t e . ) M o n d a y ["J Wednesday | ~| F r i day r~J'Tucsday [ ^ T h u r s d a y [~_]Not able to s k i on weekdays unless in connec t ion wi th a vaca t ion o r h o l i d a y . 20. What p u b l i c a t i o n s do you read on a r e g u l a r b a s i s ? M A G A Z I N E S N E W S P A P E R S j - - jP le .ase check he re if you w i s h to be i n c l u d e d in the d r a w i n g for one of the ten $100 U . S . Sav ings B o n d s . The State of C o l o r a d o w i s h e s to thank you for comple t ing , the above q u e s t i o n n a i r e . A n y i n q u i r i e s r e g a r d i n g the study should be made to.the f o l l o w i n g a d d r e s s : I n d u s t r i a l E c o n o m i c s D i v i s i o n D e n v e r R e s e a r c h Institute, U n i v e r s i t y of Denver • D e n v e r , C o l o r a d o 80210 - 128 -Table XV COLORADO STUDENT SKIER DATA Indiv. No.: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 C i ty of residence Ski ing a b i l i t y Total -no. o f ski days Aspen Vai l A-Basin Brecken-ridge Love-land s- £_ S- s_ s_ s. S- S- s- s_ s_ s_ s-CD OJ CU CD CD d) cu cu CD CD CD CD > > > > > > > > > > > > > > > > a E C c c c c CD OJ CD CD CU CU cu cu CD CD cu CD CD CD CD Q Q o Q Q Q Q Q o Q Q O Q Q o N N I I I I I I I I I I I A A A 1 3 12 4 19 13 8 9 9 11 5 8 12 17 12 51 5 6 17 7 6 5 3 2 4 29 1 13 2 1 2 5 1 5 22 1 1 Winter Park 1 1 2 4 7 6 Broad-moor Crested Butte Lake El dora Monarch Mt. Werner 19 Wolf Creek Purgatory Cooper Hidden Va l ley - 129 -cont 'd Indiv. No.: 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 City of residence Ski ing a b i l i t y Total no. of; ski days Aspen Vai l A-Basin Brecken-ridge Love-1 and Winter Park Broad-moor Crested Butte Lake El dora Monarch Mt. Werner Wolf Creek Purgatory Cooper Hidden Valley s- S- s-s- S- s_ s_ s_ s- s_ o> QJ cu cu CL) cu CU cu cu cu cu cu cu cu cu -o T3 XI. > > > > > > > > > > > > > , — i — rz fZ c c rz fz rz sz fZ Z5 Z5 Z5 OJ CD cu cu O) cu cu cu cu cu cu cu o o o Q Q Q o Q Q Q Q Q Q Q o ca cn A. A A A A A A A A A A A A N N N 7 12 13 4 5 25 11 4 12 36 .4 7 2 7 11 9 4 4 8 2 6 9 3 1 5 2 2 2 7 7 11 1 2 4 8 7 1 2 4 4 6 7 4 4 1 1 2 - 130 -cont!d Indiv. No.: 33 34 35 36 37 38 39 40 41 42 43 44 45 46 .47 .48 s -CD - o s -QJ T3 S-a> - o %. CD T3 s -CD X3 s -CD T3 s -CD "O S-CD T3 S-CD • a S-CD - o s -CD - a s -CD T3 s -<D "O s -CD • a S-<D T3 s_ CD T3 C i ty of ri Z5 Z3 13 Z3 ZS Z3 Z5 Z3 rs Z5 Q res i dence o CO O CQ o CQ O CO o CO O CO O CQ O CQ o CO o CQ o CQ O CQ O CO O CO o CO CO Ski ing ab i1 i t y N N N I I I I I I I I I I I I I Total no. of ski days 8 2 12 10 11 11 4 5 10 21 11 4 7 11 10 13 Aspen 6 9 1 8 11 Vai l 10 2 2 4 3 A-Basin 1 2 2 1 1 1 2 Brecken-ridge 4 2 2 4 2 1 Love-land 1 6 3 1 3 6 3 Winter Park 8 1 6 6 3 Broad-moor Crested Butte Lake Eldora Monarch Mt. Werner Wolf Creek Purgatory Cooper 3 2 1 4 Hidden Va l ley - 131 -con t ' d Indiv. No.: 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 S-CD "O S-OJ "O s--a aj S- s-a> -a S-<D T3 S-C1J T3 s-d) T3 5-a> TD i -a> -a S-QJ T3 a» -a s-T3 s_ a> -a s_ a) T3 C i t y of residence 3 O CO 3 o CO 3 o CO 3 o ca 3 o CO 3 o CO 3 O CO 3 o CO 3 o CO 3 O CO 3 o CO 3 o CO 3 O CO 3 o CO 3 o CO 3 o CO Ski ing a b i l i t y I I I I I I I I I I I I I I A A Total no. of ski days 6 14 9 12 4 11 12 13 2 2 1 1 4 12 IT Aspen 3 6 3 12 7 Va i l 2 3 11 A-Basin 1 1 1 T 2 Brecken-ridge 2 3 Love-land 1 1 3 1 1 Winter Park 5 2 2 1 3 1 Broad-moor Crested Butte 9 3 3 Lake Eldora 5 11 4 2 1 1 1 Monarch 1 Mt. Werner Wolf Creek Purgatory Cooper Hidden Va l ley - 132 -con t ' d Indiv. No.: 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 cu -a S-CU T5 cu -a s_ cu TJ s-"O s_ cu -a s-cu X3 s--a cu . -o a> s-cu ~o s-cu T3 s-cu -a s-a> XJ s-cu T3 cu -a C i t y o f residence 3 o CQ 3 o CQ 3 o CQ 3 o CQ 3 o CQ 3 o CQ 3 o CQ 3 o CO 3 o CQ 3 o CQ 3 o CQ 3 O CQ 3 o CQ 3 O CQ 3 o CQ 3 o CQ Ski ing abi 1 i t y A A A A A A A A A A A A A A A A Total no. o f " ski days 10 8 4 3 7 16 5 5 6 13 8 7 4 6 12 11 Aspen 6 1 6 1 3 5 Va i l 1 1 4 1 3 6 6 A-Basin 3 1 2 3 Brecken-ridge 1 1 3 3 Love-1 1 land 5 Winter Park 1 2 1 1 1 2 5 5 3 3 5 Broad-moor Crested Butte Lake Eldora Monarch Mt. Werner Wolf Creek Purgatory Cooper Hidden Va l ley 1 1 - 133 -cont 'd Indiv. No.: 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 9 6 <£> CO 00 (/) CO CO .= E SZ SZ C ^ S ^ S I S S S £ t f £ « j C i t y of residence Z3 O CO 3 o CO Sk i ing a b i l i t y A A Total ; . no. o f ski days 15 10 Aspen Va i l 5 A-Basin 1 1 Brecken-ridge 1 2 Love-land 2 Winter Park 7 Broad-moor Crested Butte Lake Eldora Monarch Mt. Werner Wolf Creek 6 Purgatory Cooper Hidden Va l ley 1 8 17 14 3 2 1 3 2 2 9 3 1 2 6 3 5 2 1 1 1 1 1 6 2 3 3 1 2 3 9 1 - 134 -cont ' d Indiv. No.: 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 ( / i i / i C i c / i c / i i / i c / i m m t / i c A t f t i / i i A t / ) ! / ) o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O C J C J residence L J _ L J _ L J _ U . U . L J _ L J _ U . L I _ L L : U : L I L L C i ty of r Sk i ing ab i1 i t y I I I I I I I I I I A A A A A A Total no. of 6 3 3 13 13 2 2 2 1 1 7 7 20 7 6 5 ski days Aspen 3 13 3 5 Vai l 2 A-Basin 1 1 Brecken-ridge ' ° Love-land 1 1 1 2 1 Winter Park 6 1 2 1 4 15 4 3 Broad-moor Crested Butte 2 Lake ETdora 1 1 5 1 Monarch 2 Mt. Werner ^ 2 2 Wolf Creek Purgatory Cooper Hidden Va l ley 1 •< O -a O s : 3 IS m r~ CO o 3 ca -o SZ CU — i . O "5 o ro r+ o —J CU SZ -s o -s 01 - J . —I CL o -s ro 1 • CL 7T c+ ro o o -S 3 —J Q- "O CO ro -h 3 CU O ro r+ CO -s Cu 7T r+ ro fD fD Co fD -s -s ro c+ CL ro << 3 -s -s o 11 Cu pd 1 -s en cri ro ro cri en —1 i— -5 co cu o —' • -s 3 < CL ro CL ro to o i ro TT ro 3 CO Cu en 3> co T3 ro 3 1/13 — I CU CO 7=r O O D-_ i . . <-h _.. _ i . CU — i _ i . Q- O —' -'•3 Cu .-(, c+CQ << << en -s o ro -•• CO <-+ -•••< CL ro o 3 -h o ro 3 CL o o 3 en ro cr> ro —1 co ro —j ro oo o co oo ro en 3» F t . C o l l i n s > Ft . C o l l i n s 3=. F t . C o l l i n s D> Ft . C o l l i n s 3> F t . C o l l i n s 3=- F t . C o l l i n s Greeley >— Greeley Greeley >-< Greeley s» Greeley 3> Greeley 3= Greel ey 3= Greeley 3=> Greeley 3S Greeley ro o ro ro ro ro 4* ro en ro en ro ro co < nz O o 2 m i — CO O 3 CO TJ s : Cu ~J. O c: -s o fD <-+• O —• CU c -5 O -s CU - i . — • CL o -s fD —1 ~s • 3 CL 7C r+ fD O o -5 3 — • CL T3 to fD -h 3 CU O fD c+ </> -s Cu r+ fD (D fD CU 7"T fD -s -s fD r+ CL fD << 3 -s to ry -s ch Cu ed 1 -s C O cn -j* ro —> r— -5 ca 3= < to 3 —1 Cu OO -s o i—) o CU O -•• -s 1 CU to 7T O o CT* 7T fD —1. 3- o 3 < CL fD CO _J. • a « — • r+ —i. —J. to e+ CL 3 Q. fD IQ O Cu 1 fD CO — i — i . i. << 1 fD 7T (/> 3 CL O — i -•• 3 CL < — fD —1. CU .-:+, r+ia fD O • • CL 3 3 <<-to *< nee -h No ro ro co ro -P* -»J co oo 00 i—i cn i—i 00 ro ro Golden Golden Golden Golden Golden Golden A i r Force Academy A.F.A. A.F.A. A.F.A. Col orado Springs C. -Springs C. C. C. C. Springs Springs Springs Springs ro C O O C O C O ro C O C O C O CO cn co co co co C O o ro co - F i C O C T l - 137 -cont 'd Indiv. No.: 145 146 147 148 149 150-151 152 153 154 155 156 157 158 159 160 to to CO CO CD CD CD CD sz SZ SZ SZ SZ SZ SZ SZ SZ SZ •1— • r - • r - •T— o o o o o o o o S- S- S- S- o o o CO CO CO to CO CO to CD CT. Q- CL D_ CL r — i — 1— o • 1— • r - "•I— • 1— •1— • r— SZ SZ OO OO 00 CO JD _Q -0 E SZ sz SZ SZ SZ SZ rd fd OJ OJ OJ fd SZ SZ SZ SZ SZ SZ S- S-• . • . Z3 Z3 Z5 i — Z5 Z5 Z5 Z3 3 3 3 3 o c_> o c_> D_ Cu o_ CD CD CD CD CD CD o Q A A A A N I A I M I I I A A I I C i t y o f res i dence Ski ing a b i l i t y Total no. of 6 6 8 3 4 3 25 10 7 8 6 2 5 1 5 1 ski days Aspen 6 6 2 3 5 Vai l 3 3 A-Basin 1 1 2 1 1 3 2 1 Brecken-r i dge Love-1 and Winter Park Broad-moor Crested Butte 2 Lake El dora Monarch 2 4 3 20 6 Mt. Werner Wolf Creek ^ Purgatory Cooper 1 1 1 2 1 1 Hi dd nVa l ley cont ' d Indiv. No.: 161 162 163 - 138 -o o o CD e n CD SZ sz c ro (O rd S- i- s-3 3 3 Q o A A A 15 12 5 C i ty of res i dence Sk i ing a b i l i t y Total no. of ski days Aspen 5 Vai l A-Basin Brecken-r i dge Love-land Winter Park 4 8 Broad-moor Crested Butte Lake Eldora Monarch Mt. Werner 1 1 4 Wolf Creek Purgatory Cooper Hidden Va l ley Source: Denver Research In s t i tu te (1968). - 139 -Table XVI SUPPLEMENTARY SKI AREA CHARACTERISTICS Average annual snowfal1* VTF No. 1 i f ts Hourly uphill capacity Vertical No. runs Longest run (ft.) Shortest run (ft.) Base elevation Summit elevation Aspen 250 - Highlands 200 3682 9 6000 1500 30 26400 -- Mountain 277 5571 6 5500 3500 45 10500 - 7930 11212 - Buttermilk 211 3685 5 4000 2000 22 - - 7868 9840 - Snowmass 244 6338 5 5200 3500 - 15840 - 8230 11700 Va i l 301 8849 9 8390 3050 45 31680 5280 8200 11250 A-Basin 280 2913 7 6000 1700 20 16000 2000 - -Breckenridge 285 4100 8 6356 1900 22 11250 2000 9950 11900 Loveland 280 4512 8 7000 1430 22 7920 3960 10800 12230 Winter Park 250 5130 6 7700 1700 30 7920 1000 9000 10700 Broadmoor 40 480 1 850 600 2 3000 2700 6500 7140 Crested Butte 210 1607 3 1600 2000 15 30000 2500 9300 11900 Lake Eldora 150 1484 3 3000 .._ 19 7000 1500 - -Monarch 354 192 2 - 890 12 .6500 400 - -Mt. Werner 325 2914 3. 3000 2100 19 15840 900 - 10600 Wolf Creek 435 384 2 - - 9 - - - -Purgatory 300 1507 3 1750 1500 16 10000 600 8950 10450 Cooper 250 856 3 1500 1150 6 6200 850 - -Hidden Va l ley 150 1106 2 2300 - 8 - - - -Source: Colorado Ski Country U.S.A. (1974) *Source: Colorado Ski Country U.S.A. (1975) 

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