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A hedonic cost function approach to agricultural land prices Quan, Daniel Wah Choo 1983

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A HEDONIC COST FUNCTION APPROACH TO AGRICULTURAL LAND PRICES By DANIEL WAH CHOO QUAN B . S c , The University of B r i t i sh Columbia, 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE •  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF AGRICULTURAL ECONOMICS We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1983 @ Daniel Wah Choo Quan, 1983 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 a n d s t u d y . I f u r t h e r a g r e e 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 p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t 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 n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 DE-6 f 3 / 8 n ABSTRACT Since Sherwin Rosen's paper on i m p l i c i t markets and hedonic f u n c t i o n s , there has been an increase i n the a p p l i c a t i o n of hedonics theory. These a p p l i c a t i o n s have almost e x c l u s i v e l y d e a l t with consumer choice and the i m p l i c i t consumer demand f o r a t t r i b u t e s . The methodology u s u a l l y e n t a i l s the two stage procedure of i n i t i a l l y o b t a i n i n g i m p l i c i t p r i c e s from the hedonics f u n c t i o n and then using these p r i c e s as dependent v a r i a b l e s i n a system of b i d f u n c t i o n s . In t h i s t h e s i s , a system of b i d f u n c t i o n s f o r land a t t r i b u t e s i s derived from a Generalized L e o t i e f cost f u n c t i o n of the B r i t i s h Columbia d a i r y i n d u s t r y . These bid fu n c t i o n s were estimated with land c h a r a c t e r i s t i c s data from records of d a i r y land t r a n s a c t i o n s . The proposed paper provides several i n t e r e s t i n g extensions to the present a p p l i e d work i n t h i s area. The a p p l i c a t i o n of d u a l i t y r e s u l t s i n the s p e c i f i c a t i o n and d e r i v a t i o n of bid fu n c t i o n s provide an important improvement i n e x i s t i n g methodology since economic s t r u c t u r e can be improsed and consistency of d e r i v a t i o n can be obtained. A l s o , despite i t s p o p u l a r i t y i n consumption a n a l y s i s , the a p p l i c a t i o n of hedonic theory i n production and the derived demand f o r inputs have not been addressed. Two main r e s u l t s are discussed i n the t h e s i s . The f i r s t concerns i d e n t i f i c a t i o n of cost f u n c t i o n parameters (and t h e r e f o r e the dual production technology) through the system of b i d f u n c t i o n s i n the absence of input q u a n t i t y data. The second r e s u l t i s that i n the context of land a t t r i b u t e choices i n a g r i c u l t u r a l production, the question of endogenous and exogenous i m p l i c i t p r i c e s are important i i concerns as some i m p l i c i t a t t r i b u t e p r i c e s are determined by non-production r e l a t e d f a c t o r s . I t i s shown th a t l i n e a r homogeneity of a t t r i b u t e s i n the hedonic f u n c t i o n i s a necessary c o n d i t i o n f o r exogenous i m p l i c i t p r i c e s . i i i TABLE OF CONTENTS ABSTRACT i i ACKNOWLEDGEMENTS v Chapter I INTRODUCTION 1 1.1 Objectives 4 1.2 Plan of Thesis 5 II REVIEW OF HEDONIC LITERATURE 6 I I I THEORETICAL MODEL 12 IV ECONOMETRIC MODEL 26 4.1 F i r s t Stage Model 27 4.2 Cost Function S p e c i f i c a t i o n 32 4.3 Estimation Technique 37 V DATA AND DATA CONSTRUCTION 41 VI DISCUSSION OF RESULTS 45 6.1 F i r s t Stage 45 6.2 Second Stage Results 50 6.3 An A l t e r n a t i v e I n t e r p r e t a t i o n o f the Results . . . 63 VII CONCLUSION 68 REFERENCES 71 i v ACKNOWLEDGEMENT My utmost thanks and a p p r e c i a t i o n goes to my advisor Richard B a r i c h e l l o who introduced the hedonic l i t e r a t u r e to me and als o provided i n v a l u a b l e i n s i g h t s and constant encouragement during the development of my t h e s i s . I w i l l always be indebted to Rick f o r the patience he provided and the continui n g i n t e r e s t he showed f o r my research and academic development. My thanks a l s o extends to the other members of my committee Erwin Diewert and Cameron Short whose i n s i g h t s and observations g r e a t l y c o n t r i b u t e d to the t h e s i s . V CHAPTER I INTRODUCTION An important development i n the economic l i t e r a t u r e has been the hedonics approach to economics or the theory of i m p l i c i t markets. Pioneered by the works of Becker (1965), Lancaster (1966) and Muth (1966), the theory suggests t h a t an economic agent's d e s i r e f o r a good i s not based on the good i t s e l f but on the u t i l i t y derived from the a t t r i b u t e s or c h a r a c t e r i s t i c s which comprise i t . This approach i s appealing i n th a t i t can e x p l i c i t l y account f o r q u a l i t y d i f f e r e n c e s i n goods, an area not addressed i n the conventional l i t e r a t u r e . A l s o , even though one good i s comprised of several c h a r a c t e r i s t i c s , the same a t t r i b u t e s may be included i n d i f f e r e n t goods. The approach can therefore account f o r the demand of many seemingly unrelated goods by suggesting what t h e i r demand i s based on a d e s i r e f o r s e l e c t e d a t t r i b u t e s . An a n a l y s i s employing such an approach possess c e r t a i n problems which are not c h a r a c t e r i s t i c of the t r a d i t i o n a l economic a n a l y s i s . The most obvious i s that since the a t t r i b u t e s themselves are not f r e e l y traded, no e x p l i c i t markets e x i s t thus no e x p l i c i t .1, 2 p r i c e s f o r these a t t r i b u t e s are observed. However, the c h a r a c t e r i s t i c s are c l e a r l y i m p l i c i t l y traded which would r e s u l t i n i m p l i c i t p r i c e s . Though wid e l y used, the t h e o r e t i c a l foundation of these hedonic f u n c t i o n s were not c l e a r l y developed u n t i l the work of Rosen (1974) who suggested th a t the hedonic f u n c t i o n i s an envelope of an underlying system of bid and o f f e r f u n c t i o n s f o r the a t t r i b u t e s . I t was a l s o i n the same a r t i c l e that he suggested an em p i r i c a l procedure which could model the s t r u c t u r e of the demand and supply of these a t t r i b u t e s f o r a p a r t i c u l a r good. This has l e d to a host of e m p i r i c a l studies which employed his procedure with i t s main a p p l i c a t i o n i n consumer demand. The methodology e n t a i l s a two stage procedure of i n i t i a l l y o b t a i n i n g i m p l i c i t p r i c e s from the hedonic f u n c t i o n and then using these p r i c e s as dependent v a r i a b l e s i n a system of b i d f u n c t i o n s . Despite i t s wide range of a p p l i c a t i o n . i n consumption a n a l y s i s , the s p e c i f i c a t i o n and d e r i v a t i o n of b i d fu n c t i o n s i n the l i t e r a t u r e has been l a c k i n g i n consistency and s t r u c t u r e . B i d f u n c t i o n s have t r a d i t i o n a l l y been simple i n i t s s p e c i f i c a t i o n with i t s d e r i v a t i o n being void of important s t r u c t u r a l i m p l i c a t i o n s with respect to the underlying u t i l i t y f u n c t i o n and the o p t i m i z a t i o n procedure. S p e c i f i c a l l y , the recent a p p l i c a t i o n of d u a l i t y theory i n economics which h i g h l i g h t e d the need f o r more s t r u c t u r e i n the s p e c i f i c a t i o n of demand fu n c t i o n s have been ignored i n the t r a d i t i o n a l bid f u n c t i o n a n a l y s i s . Another neglected area i s the a p p l i c a t i o n of hedonic concepts to production or more s p e c i f i c a l l y to f a c t o r demand. Jus t as u t i l i t y f u n c t i o n s can be expressed i n a t t r i b u t e s , so can the production f u n c t i o n 3 of the f i r m . The r e s u l t i n g derived demand would analogously be f o r a t t r i b u t e s r a t h e r than f o r the good i t s e l f . Such an approach can provide a host of r e v e a l i n g comparative s t a t i c r e s u l t s of the f i r m and may provide a b e t t e r understanding of the production technology. The main focus of t h i s t h e s i s i s an attempt at extending the a p p l i c a t i o n of hedonic theory t o analyze the f a c t o r demand de c i s i o n s of the f i r m . Issues such as the consistency and economic i m p l i c a t i o n of bid f u n c t i o n s p e c i f i c a t i o n s w i l l be addressed i n the context of d u a l i t y and f l e x i b l e form a n a l y s i s . S p e c i f i c a l l y , the demand de c i s i o n s regarding the land input i n a g r i c u l t u r a l production w i l l be the input of focus. Due to the important r o l e which land plays i n a g r i c u l t u r e , there has been much.research i n t o the behaviour and determinants of a g r i c u l t u r a l land p r i c e s . Land value have been r e l a t e d to numerous f a c t o r s among which includes the t o t a l farm asset values (Melichar 1979), s o c i o l o g i c a l f a c t o r s (Alonso 1981), landowner c h a r a c t e r i s t i c s (Barnard and Butcher 1971) and expected f u t u r e output p r i c e s (Chavas and Shumway 1981). The d i v e r s i t y of land q u a l i t y and the f l u c t u a t i o n i n land p r i c e s provide a natural avenue to apply our model of the f i r m . To be s p e c i f i c , the production technology and land purchase d e c i s i o n of d a i r y farmers i n B r i t i s h Columbia w i l l be analyzed i n t h i s study. Dairy production i n B r i t i s h Columbia can be a p t l y c h a r a c t e r i z e d by cost minimization behaviour since output, and i n some sense p r i c e s , i s regulated by the market share quota system. Monthly land t r a n s a c t i o n data, input p r i c e s and output l e v e l s w i l l be used to s p e c i f y a cost f u n c t i o n of the in d u s t r y from 1970 to 1977. An i n t e r e s t i n g p o l i c y issue can be addressed i n t h i s study. In 1973, the a g r i c u l t u r a l land 4 reserve program was e s t a b l i s h e d to preserve land f o r the s o l e purpose of a g r i c u l t u r a l production. This measure was deemed necessary because the competing land use problem between the urban and a g r i c u l t u r a l s e c t o r had reached a c r i t i c a l l e v e l . Needless to say, such r e s t r i c t i v e land measures would cause adjustments i n the land market thus knowledge o f the e f f e c t s of the program on a g r i c u l t u r e ' s b i d f o r la n d i s needed in understanding the t o t a l e f f e c t of such a program. I t should perhaps be noted at the outset that the issue of land value i s complicated and i t s determinants are q u i t e d i v e r s e . No claim i s ther e f o r e made as to suggest t h a t the proposed approach addresses a l l the iss u e s r e l e v a n t to land value. I t i s only suggested th a t t h i s e x e r c i s e can perhaps provide another method of an a l y z i n g land demand i n the context of a g r i c u l t u r a l production. 1.1. Objectives There appears to be a lack of a p p l i c a t i o n of hedonic concepts f o r the case of f a c t o r demands i n production. J u s t as i n the conventional hedonic consumer demand a n a l y s i s , i t may be i n s t r u c t i v e to apply the theory of i m p l i c i t markets to the input a t t r i b u t e choices of the f i r m . Since the basis of i m p l i c i t market studies i n consumer demand theory revolves around the system o f b i d and o f f e r f u n c t i o n s , the a p p l i c a t i o n i n the production c o n t e x t ' w i l l centre around the derived demand funct i o n s f o r the a t t r i b u t e s . The o b j e c t i v e of t h i s t h e s i s i s to derive a system of b i d functi o n s based on cost minimizing behaviour of the f i r m . The system must be c o n s i s t e n t w i t h the hedonic theory as w e l l as modern production 5 concepts. The area o f a p p l i c a t i o n w i l l be the land input i n a g r i c u l t u r a l production. Since the demand f o r land s e r v i c e s i n a g r i c u l t u r a l production i s a derived demand, the i m p l i c i t demand f o r i t s a t t r i b u t e s can t h e r e f o r e be seen as being a l s o a derived demand. S p e c i f i c a l l y , the land a t t r i b u t e bidding behavour of the B.C. d a i r y i n d u s t r y w i l l be analyzed. I t i s hoped t h a t t h i s e x e r c i s e would provide us with i n t e r e s t i n g i n s i g h t s of some of the conventional production and hedonic concepts. I t i s a l s o hoped that t h i s a n a l y s i s can reveal i n t e r e s t i n g information with regards to the a t t r i b u t e choices of the d a i r y farmers i n B.C. I . 2 Plan of t h e s i s - The plan of the t h e s i s i s as f o l l o w s . A review of the hedonic l i t e r a t u r e and i t s various a p p l i c a t i o n i s contained i n Chapter I I . Chapter I I I provides the t h e o r e t i c a l development of the model used i n t h i s t h e s i s . The choice of the e m p i r i c a l or econometric model i s o u t l i n e d i n Chapter IV. This Chapter contains a s e c t i o n on the f i r s t stage hedonic f u n c t i o n s p e c i f i c a t i o n s and a second s e c t i o n d e s c r i b i n g the cost f u n c t i o n to be used f o r e s t i m a t i o n . Data i s discussed i n Chapter V while Chapter IV contains a d i s c u s s i o n of the empi r i c a l r e s u l t s obtained and i t s i n t e r p r e t a t i o n i n the context of the d a i r y i n d u s t r y . i n B.C. P a r t i c u l a r l y important i s s e c t i o n 6.3 which captures an important f i n d i n g of t h i s t h e s i s . This s e c t i o n i s followed by conclusions and summary i n Chapter V I I . CHAPTER II REVIEW OF HEDONIC LITERATURE Some: of the e a r l i e r c o n t r i b u t i o n s to the hedonics theory or the c h a r a c t e r i s t i c s approach to consumer theory have been Becker (1965), Lancaster (1966) and Muth (1966). T h e i r basic contention was that the household i s a production u n i t which produces a c t i v i t i e s from p h y s i c a l goods. The u t i l i t y of these households are subsequently functions of these a c t i v i t i e s and not of the goods themselves. Households are then assumed to maximize t h e i r u t i l i t y subject to the budget.constraints f o r goods as well.;as a transformation f u n c t i o n which r e l a t e s the a c t i v i t i e s to the goods. The r e c o g n i t i o n that the conventional consumer.theory- does not encompass the p o s s i b i l i t y o f new goods or changes i n the q u a l i t y of e x i s t i n g ones and ignores the value of consumers' time leads to the development of the household production theory. The household production theory addresses the q u a l i t y concerns by noting that even though a s i n g l e good would l i k e l y embody numerous a c t i v i t i e s , the same a c t i v i t i e s might we l l be produced by a large number of goods. For example, a car as w e l l as a boat could have i n common p r e s t i g e and t r a n s p o r t a t i o n a c t i v i t i e s . The c r u c i a l element of t h i s approach i s the transformation f u n c t i o n which expresses the goods i n terms of the a c t i v i t i e s . An a p p l i c a t i o n of t h i s approach i n v a r i a b l y r e s u l t s i n determining the r e l a t i o n s h i p between the a c t i v i t i e s (which are i n many cases n o n - q u a n t i f i a b l e ) and the p h y s i c a l goods. This approach has been analyzed f u r t h e r by 6 7 Lucas (1975) and i t s a p p l i c a b i l i t y to production has been explored by A r c h i b a l d and Rosenbluth (1978). Though the t h e o r e t i c a l development i n t h i s area i s advanced, the procedure has not l e n t i t s e l f to e m p i r i c a l a p p l i c a t i o n with the exception of King (1976). Other a p p l i c a t i o n s which use the hedonic concept have b a s i c a l l y centered around the hedonic f u n c t i o n , a f u n c t i o n which r e l a t e s the p r i c e of a good to i t s c h a r a c t e r i s t i c s . G r i l i c h e s (1971) employed t h i s approach to derive a p r i c e index f o r automobiles. Lucas (1977) a l s o u t i l i z e d the hedonic f u n c t i o n to e x p l a i n the wage rates of workers by i n c l u d i n g v a r i a b l e s r e f l e c t i n g the c h a r a c t e r i s t i c s of the workers as well as the a t t r i b u t e s of the p a r t i c u l a r occupation i n v o l v e d . In the context of land v a l u e s , e s t i m a t i o n of hedonic func t i o n s have been a w e l l used t o o l . I t was not u n t i l the work of Rosen (1974) that an e x p l i c i t theory regarding a t t r i b u t e s o f goods was developed. The hedonic f u n c t i o n was shown to be the envelope of a system of b i d and o f f e r f u n c t i o n s . The b i d f u n c t i o n can be viewed as the amount the consumer would be w i l l i n g to b i d f o r a p a r t i c u l a r a t t r i b u t e . S i m i l a r l y , the o f f e r f u n c t i o n r e f l e c t s the p r i c e . a t which the producer of the good would be w i l l i n g to s e l l the c h a r a c t e r i s t i c . Bid f unctions are derived from u t i l i t y maximizing consumers and o f f e r functions are from p r o f i t maximizing producers where the u t i l i t y and p r o f i t functions of both agents are expressed i n terms of c h a r a c t e r i s t i c s . Assuming e q u i l i b r i u m i n the i m p l i c i t c h a r a c t e r i s t i c s market, the b i d and o f f e r p r i c e s are equal with the hedonic f u n c t i o n being the envelope of these two i n t e r a c t i o n s . The nature of Rosen's model demonstrates the general departure of the c h a r a c t e r i s t i c s approach with t h a t of the conventional.. Since the 8 a t t r i b u t e s are not f r e e l y traded, one has to assume th a t they are i m p l i c i t l y traded which w i l l r e s u l t i n i m p l i c i t p r i c e s f o r these a t t r i b u t e s . The i m p l i c i t p r i c e s are not observable thus some assumptions are required i n order to r e l a t e the free trade o f a good with the i m p l i c i t t r a d i n g of i t s inherent a t t r i b u t e s . Rosen showed that the commonly used hedonic f u n c t i o n i s the l i n k between the p r i c e s u r f a c e , which i s observable i n the market, and the i m p l i c i t p r i c e s of the c h a r a c t e r i s t i c s , which i s not. In the same study, Rosen proposed an e m p i r i c a l procedure to i n v e s t i g a t e s y s t e m a t i c a l l y the b i d , o f f e r and hedonic f u n c t i o n . The suggested method was a two stage procedure whereby the f i r s t stage e n t a i l s the d u p l i c a t i o n of market information by es t i m a t i n g a hedonic f u n c t i o n . The gradient of the estimated f u n c t i o n i s assumed to be the i m p l i c i t p r i c e s . The i m p l i c i t p r i c e s are then c a l c u l a t e d and used as the dependent v a r i a b l e i n the second stage where a system of bid and o f f e r functions are s p e c i f i e d . The b i d fu n c t i o n s have as i t s arguments the c h a r a c t e r i s t i c s as w e l l as b i d function, s h i f t e r s such as income o f the household. S i m i l a r l y , the o f f e r f u n c t i o n s have p r o f i t f u n c t i o n parameters which a f f e c t s the production cost of the c h a r a c t e r i s t i c s producing f i r m . Since Rosen's work, there have been numerous a p p l i c a t i o n s o f his approach. Harrison and Rubinfeld (1978) analyzed the w i l l i n g n e s s to pay f o r c l e a r a i r using housing market data. The choice to purchase a house was assumed to be based on the consumer's d e s i r e f o r the housing c h a r a c t e r i s t i c s which includes the q u a l i t y of a i r . I m p l i c i t p r i c e s f o r cl e a r n a i r were obtained from the hedonic f u n c t i o n . These p r i c e s were i n t e r p r e t e d as a r e f l e c t i o n of the consumer's w i l l i n g n e s s to pay f o r l e s s a i r p o l l u t i o n . The b i d p r i c e f u n c t i o n s f o r c l e a r a i r were then estimated to e x p l a i n the e f f e c t s of bid f u n c t i o n s h i f t e r s on the consumer's b i d f u n c t i o n . 9 Wi t t e , Sumka and Erekson (1979) al s o a p p l i e d t h i s approach to the housing market. A quadratic hedonic f u n c t i o n was estimated f o r four d i f f e r e n t housing regions. A system of l i n e a r bid and o f f e r f u n c t i o n s were estimated simultaneously using the c a l c u l a t e d i m p l i c i t p r i c e s as dependent v a r i a b l e s i n the system. Although reasonable r e s u l t s were obtained, t h e i r study has been c r i t i c i z e d . Brown and Rosen (1981) questioned the procedure on i d e n t i f i c a t i o n grounds. They c r i t i c i z e d the l i n e a r nature of the b i d - o f f e r f u n c t i o n s and showed that s t r u c t u r a l parameters of the second stage are e x a c t l y determined by the f i r s t stage parameters. Epple (1981) addressed the e r r o r s p e c i f i c a t i o n of t h e i r a p p l i c a t i o n . Quigley (1980) looked at two i m p l i c a t i o n s w i t h regard to the u t i l i t y f u n c t i o n and i t s r e l a t i o n s h i p to the hedonic f u n c t i o n . F i r s t , i f the consumer preferences were p r e s p e c i f i e d , what r e s t r i c t i o n s are imposed on the market wide hedonic f u n c t i o n . Second, the converse case of having the hedonic f u n c t i o n determined by some best f i t c r i t e r i a , what r e s t r i c t i o n s would t h i s impose on the u t i l i t y f u n c t i o n . For the former case, i t was shown that p r e - s p e c i f i e d consumer preferences create only weak r e s t r i c t i o n s on the hedonic s t r u c t u r e of market p r i c e s . In the l a t t e r case, the issue becomes one of i d e n t i f i c a t i o n of u t i l i t y contours with n o n l i n e a r c o n s t r a i n t s . Quigley suggested that even the weak c o n d i t i o n of homotheticity of preferences i s not required f o r i d e n t i f i c a t i o n . His f i n d i n g s were subsequently a p p l i e d to housing demand data. Other a p p l i c a t i o n s of the hedonic approach includes Spady (1976) where the c h a r a c t e r i s t i c s of the t r u c k i n g industry'.s output was analyzed. A cost f u n c t i o n was s p e c i f i e d which was a f u n c t i o n of input p r i c e s and output. The output v a r i a b l e was then replaced by a f u n c t i o n which r e f l e c t s 10 the q u a l i t y of the output. Share equations of the t r a n s l o g s p e c i f i c a t i o n were estimated and information regarding the long and short run p r o p e r t i e s of the cost s t r u c t u r e of the t r u c k i n g i n d u s t r y were obtained. Goldman and Grossmen (1978) a p p l i e d the hedonic concept to i n v e s t i g a t e the demand f o r p e d i a t r i c care. A u t i l i t y f u n c t i o n of the parents was assumed which i s a f u n c t i o n of t h e i r c h i l d ' s h e a l t h . A c h i l d ' s health f u n c t i o n i s s p e c i f i e d to be a f u n c t i o n of the number of v i s i t s , q u a l i t y per v i s i t and other v a r i a b l e s . Maximization of the u t i l i t y f u n c t i o n , whdch i n d u c e s the health f u n c t i o n subject to the time and budget c o n s t r a i n t s r e s u l t e d i n demand equations f o r q u a l i t y and the frequency of v i s i t s . These equations were estimated and information p e r t a i n i n g to the e f f e c t s of q u a l i t y of v i s i t s on the consumer were obtained. Despite the many a p p l i c a t i o n s i n the hedonics a r e a , i t i s only r e c e n t l y that researchers have suggested the systematic i n t e g r a t i o n of the hedonics theory with the modern f l e x i b l e form dual approaches. F l e x i b l e form a n a l y s i s based on d u a l i t y r e s u l t s can i n general provide more i n s i g h t i n t o the production technology while imposing economic s t r u c t u r e on the modelling procedure. An i n t e r e s t i n g a n a l y s i s of the q u a l i t y issue i n the context of modern cost and production theory was provided by Berndt (1982). In h i s work, an a n a l y s i s i s made of the t h e o r e t i c a l j u s t i f i c a t i o n s of the various methods f o r i n t r o d u c i n g input a t t r i b u t e s i n t o convention cost and production s t u d i e s . This was done i n the context of a t r a n s l o g cost f u n c t i o n . Factors of production were assumed to be augmented by q u a l i t y a d j u s t i n g i n d i c e s which were defined to be f u n c t i o n s of the input a t t r i b u t e s . Two scenarios based upon the i n d i c e s were considered. The f i r s t was l a b e l l e d the "simple repackaging hypothesis" which suggests that the augmentation f a c t o r s are only f u n c t i o n s of the r e l e v a n t input c h a r a c t e r i s t i c s . The " v a r i a b l e 11 repackaging hypothesis" suggests that the i n d i c e s are f u n c t i o n s of the c h a r a c t e r i s t i c s as well as the input p r i c e s . Functional forms f o r the i n d i c e s under the two hypothesis were s p e c i f i e d and s u b s t i t u t e d i n t o the t r a n s l o g cost f u n c t i o n . Consistent cost share equations f o r the p h y s i c a l inputs were subsequently derived which included the vector of a t t r i b u t e s . As suggested by the author, t h i s approach has the p o t e n t i a l of supplying a host of extremely r e v e a l i n g comparative s t a t i c r e s u l t s w i t h respect to the i n t e r a c t i o n of p h y s i c a l inputs and t h e i r a t t r i b u t e s . Inferences w i t h regards to t e c h n i c a l change and c a p i t a l formation were a l s o proposed. Although the idea of i n t r o d u c i n g input a t t r i b u t e s i n t o the cost f u n c t i o n i s not new ( W i l l s , 1978), Berndt's work provides a valuable t h e o r e t i c a l foundation i n the c o n s i s t e n t i n t e g r a t i o n of q u a l i t y v a r i a b l e s i n the dual f l e x i b l e form a n a l y s i s . However, the problem of d e r i v i n g a system of b i d f u n c t i o n s using d u a l i t y theory was not addressed. In p a r t i c u l a r , the d e c i s i o n choice of the producing agent with regards to inputs and t h e i r a t t r i b u t e s has not r e a l l y been analyzed. His study, however, h i g h l i g h t s the need f o r f u r t h e r research i n t o the p o s s i b i l i t y of merging the two bodies of l i t e r a t u r e i n t o one u n i f i e d model. CHAPTER III THEORETICAL MODEL Consider an aggregate single output per unit of land production technology which is represented by y = f(x, (1) where y is the output level , x is an mxl vector of non-land physical inputs and n-j is an kxl vector of production related land characteristics. It is assumed that this technology possesses the usual regularity conditions with respect to its arguments for a well-behaved technology. That i s , (i) V xf(x,h 1)>0, V^f(x,h 1)>0 (ii) the Hessian of the function f is negative definite, where V f is the gradient or the vector of first partial derivatives of f with respect to the components of the vector x. Although the conventional production analysis does include certain attributes such as the size of the parcel, the suggested specification can be seen to be an extension of the conventional case since the proposed model is not only a function of the usual non-land inputs but is also a function of a host of production related land characteristics including size. Characteristics or attributes of this variety can include the drainage 12 13 quality of land, the fertility level of the soil, or the slope of the land. Concavity in h-j suggests that diminishing returns apply to these production related land attributes. It is not unrealistic to suggest for example that the output capacity of a parcel of land increases with drainage until a situation arises whereby excess water drainage may become a detriment as in the case of porous or sandy soil. The aggregate firm is assumed to have the following cost structure: C = wjx. •+ •w2(h1 ,h2) (2) where C is the total cost, w'-| is an mxl vector of non-land input prices, h 2 is an nxl vector of non-production related land characteristics and w2 is the unit price of land and is a function of h-j and h 2 > The firm employs a unit of land thus w.2 is the unit cost of the parcel. The characteristic space of the land input is separated into two sets: a set for production related attributes,h^ and attributes which do not directly affect production but does however affect the price of the parcel,h2„ Land attributes of the h 2 category would include factors such as the distance from and the population of a nearby centre. In the case of population, an increase in the population of a nearby community would in general result in the increase of land values. Population growth would however unlikely increase the productivity of a nearby farm. In general, the attribute l i s t of w2(.) must include characteristics which reflect the desirable aspects of a parcel from an agricultural 14 production standpoint. These are the production r e l a t e d a t t r i b u t e s o f h-j. On the other hand, since the parcel i s al s o d e s i r e d by the n o n - a g r i c u l t u r a l production s e c t o r , the a t t r i b u t e s of land r e f l e c t e d by t h i s demand must al s o be represented. Since the e q u i l i b r i u m p r i c e of a parcel i s the r e s u l t of the marginal v a l u a t i o n o f each competing agent, the a t t r i b u t e l i s t must r e f l e c t both sectors i n order to avoid a m i s s p e c i f i c a t i o n o f w,,(.). Although t h i s d i s t i n c t i o n i n the a t t r i b u t e l i s t i s necessary i n the cost s t r u c t u r e , the same arguments are not v a l i d f o r the production f u n c t i o n as f i s a t e c h n o l o g i c a l r e l a t i o n s h i p between inputs and outputs. Therefore, f would only be a f u n c t i o n of h^, the land a t t r i b u t e s which would d i r e c t l y a f f e c t the production process. One may argue t h a t a l l c h a r a c t e r i s t i c s has c e r t a i n e f f e c t s on the production technology. For i n s t a n c e , the distance to market could a l t e r the production process through changes i n input p r i c e s due to t r a n s p o r t a t i o n cost and th e r e f o r e a f f e c t i n g h i s p r o f i t . On the other hand, f a c t o r s such as s o i l f e r t i l i t y and c u l t i v a t i o n c a p a b i l i t i e s o f a g r i c u l t u r a l farmland would have a more d i r e c t e f f e c t on the production process as compared to di s t a n c e . Production c h a r a c t e r i s t i c s w i l l t h e r e f o r e be defined to be f a c t o r s which d i r e c t l y a f f e c t the production process and non-production a t t r i b u t e s w i l l be those whose e f f e c t s are i n d i r e c t . The concept of d i r e c t and i n d i r e c t c h a r a c t e r i s t i c s : was employed b y - G r i l i c h e s (1971) in ; his a n a l y s i s of automobile q u a l i t y . Returning to the cost s t r u c t u r e , i t can be seen t h a t the u n i t p r i c e of land f u n c t i o n i s the conventional hedonic p r i c e f u n c t i o n f o r land whereby the p r i c e of a u n i t parcel i s expressed as a f u n c t i o n of i t s a t t r i b u t e s . Given the cost schedule and the production f u n c t i o n , a cost 15 f u n c t i o n can be described as: C(w..1,h21y) = min { w|x + w ^ h ^ h g ) •.: y<f(x,h 1)} (3) (x,h-j) Before d e r i v i n g the comparative s t a t i c s o f t h i s c ost f u n c t i o n , i t i s important to consider the behavioural i m p l i c a t i o n s of the above r e l a t i o n s h i p . The cost f u n c t i o n s t a t e s that given a t a r g e t or a pre-determined l e v e l o f output, the producer would maximize p r o f i t s by minimizing cost to achieve the l e v e l of output. Thus the output l e v e l i s b a s i c a l l y pre-determined. Also i m p l i e d i s a s i n g l e output production process. As we are modelling the d a i r y s e c t o r , both assumptions are a p p l i c a b l e . The d a i r y i n d u s t r y i s one of the more h e a v i l y regulated i n d u s t r i e s i n a g r i c u l t u r e . A program of quotas i s used which d i c t a t e s the amount of milk t h a t can be produced and s o l d at a s u b s i d i z e d p r i c e with any excess s o l d at a lower p r i c e . Since the ta r g e t output l e v e l s of milk producers are determined by the s i z e of the quota f o r each farm, the l e v e l of output i s b a s i c a l l y pre-determined. An avenue e x i s t s i n some j u r i s d i c t i o n s , notably i n B.C., whereby quotas can be traded. To some ex t e n t , the output l e v e l i s not f i x e d but at l e a s t i n such i n d u s t r i e s , given the high p r i c e o f quotas and some imperfections i n the c a p i t a l markets, the cost o f i n c r e a s i n g output l e v e l s i s more than i n comparable unregulated i n d u s t r i e s without output r e s t r i c t i o n s . I t should be noted t h a t the use of a cost f u n c t i o n does not n e c e s s a r i l y require . the f i r m to have a f i x e d l e v e l of output. Cost minimization subject to a predetermined output l e v e l i s a behavioural assumption on the part of the f i r m . I f the farming d e c i s i o n s are based 16 on a perceived t a r g e t output l e v e l , the a p p l i c a t i o n of the cost f u n c t i o n i s s t i l l v a l i d , even i f the t a r g e t output l e v e l changes. The stronger assumption of a f i x e d output l e v e l can be seen to be an extreme case o f t h i s weaker c o n d i t i o n . Due to the nature of the i n d u s t r y , the second assumption of a s i n g l e output technology i s al s o v a l i d as d a i r y farmers i n general do not indulge h e a v i l y i n other farming a c t i v i t i e s other than milk production. I t i s also assumed that competition p r e v a i l s i n the non-land f a c t o r markets and there f o r e d a i r y farmers are p r i c e takers i n these input p r i c e s . As mentioned.in the l i t e r a t u r e review, the idea of i n t r o d u c i n g input c h a r a c t e r i s t i c s i n t o the cost f u n c t i o n i s not new. However, i t s i n t r o d u c t i o n i n t h i s model i s unique since cost minimization i s performed with respect to x and h-j. When a farmer makes a land purchasing d e c i s i o n w i t h respect t o the land a t t r i b u t e s , i t would seem u n l i k e l y t h a t such choices are made independently of the other non-land inputs. I t i s not u n r e a l i s t i c to suggest that a c e r t a i n degree o f i n t e r a c t i o n i s present between the land c h a r a c t e r i s t i c s and other f a c t o r s of production. For i n s t a n c e , a farmer's d e c i s i o n to purchase a parcel with a lower l e v e l of c u l t i v a t i o n ease that he desires i s i m p l i c i t l y s u b s t i t u t i n g the e x t r a amount of his labour required to work the parcel f o r the land a t t r i b u t e . The actual s u b s t i t u t i o n would be determined by changes i n r e l a t i v e input p r i c e s which i n t h i s context are the changes i n the opportunity cost of the farmer's labour, h i s wages, and the changes i n the i m p l i c i t p r i c e s of the c u l t i v a t i o n a t t r i b u t e . I t i s important to note that the above d e c i s i o n s are s t r i c t l y contained w i t h i n the cost minimization choice framework and are independent of the u t i l i t y f u n c t i o n and personal preferences of the farmer. 17 The attribute choice of the farmers provide an alternative interpretation for the seemingly arbitrary separation of the attribute l i s t into two subgroups. When the farmer embarks on a land purchase decision, he is faced with a menu of land attributes available for each parcel. Ignoring the indivisibility of the attributes for now, he would choose a subset with which he feels is crucial to his farming operation. In the aggregate, this would result in a subset of attributes which are choice variables in the land purchase decisions of the farmers. The implicit prices for the desired attributes would be determined by the market forces exerted by the bidding farmers while the prices for the other attributes will be determined by the marginal valuation of other bidding agents. In this context, the previously defined production related attributes would be those'whose prices are determined by the bids of the farmers. This interpretation has strong implications with regards to the price determination process for the attributes and will be further discussed at the latter part of this chapter. Returning to the cost function, the LaGrangian expression associated with this function is: L = w.Jx + w2(h1,h2) + x[y - f(x,hn)] (4) where A is the LaGrange multiplier. It is assumed that the aggregate firm chooses the optimal level of non-land inputs and relevant land characteristics to minimize cost given the cost structure and the production technology of the firm. That is , the expression is minimized with respect to x, h-, and X resulting i n the f o l l o w i n g set of f i r s t order c o n d i t i o n s : 18 V XL= W l -XV/ (5.1) = V. .w2 - xvh f (5.2) f = y - f-Cx.h,). (5.3) Equation (5.1) i s the f a m i l i a r r e l a t i o n s h i p between the input p r i c e and the marginal product of the input. Equation (5.3) has the i n t e r p r e t a t i o n of X being the shadow p r i c e of an e x t r a u n i t of output or i n our case, the aggregate supply p r i c e o f milk. Equation (5.2) i s s i m i l a r to Equation (5.1) since i t r e l a t e s the p r i c e o f an input to i t s marginal product. From Rosen, we know that the gradient of the hedonic f u n c t i o n i s the vector of i m p l i c i t p r i c e s of the corresponding c h a r a c t e r i s t i c s . I f we l e t the i m p l i c i t p r i c e o f production c h a r a c t e r i s t i c i be where loi<_k, component i of (5.2) becomes This c o n d i t i o n reveals c e r t a i n aspects of the model which are e m p i r i c a l l y important. Consider two elements of the production r e l a t e d land c h a r a c t e r i s t i c vector h.j. C a l l the two elements h.. and h^ where .T<.i,j*k.- The corresponding r a t i o of the two f i r s t order conditions become: 19 P. 3 W 9 / 3 h . 3 f / 3 h . 3w. 2 /3h . 3 f / 3 h i (7) P . J Equation (7) i s the f a m i l i a r tangency c o n d i t i o n f o r minimal cost production whereby the slope o f the budget l i n e (the r a t i o of the i m p l i c i t p r i c e s ) equals to that of the isoquant (the marginal rate of s u b s t i t u t i o n o f h. f o r h.). I f the hedonic f u n c t i o n , w~(.) was assumed to be l i n e a r as i s I J L. tn the case o f several hedonic f u n c t i o n s t u d i e s , the r a t i o P./P. w i l l be "I J a constant. I t i s known that i n the e s t i m a t i o n and i d e n t i f i c a t i o n of demand f u n c t i o n s , r e l a t i v e p r i c e v a r i a b i l i t y i s e s s e n t i a l to trace out the i n d i f f e r e n c e s e t s . In t h i s case, i f a l i n e a r hedonic f u n c t i o n i s adopted, such v a r i a b i l i t y i s not present thus making i t impossible to i d e n t i f y the contours of the production technology. Even i f we were not i n t e r e s t e d i n i d e n t i f y i n g the f a c t o r demand of a production c h a r a c t e r i s t i c , f i x e d r e l a t i v e i m p l i c i t p r i c e s i s a strong assumption to impose on f a c t o r s of production a t t r i b u t e s (Quigley 1980). Since i n g e n e r a l , f i x e d r e l a t i v e p r i c e s i s an unnecessary r e s t r i c t i o n , a non- l i n e a r hedonic f u n c t i o n i s used i n th.is study. For such a case, the problem becomes one of a constrained o p t i m i z a t i o n problem with non-1inear o b j e c t i v e f u n c t i o n and non- l i n e a r c o n s t r a i n t s . The cond i t i o n s f o r o p t i m a l i t y and i d e n t i f i c a t i o n of such a problem i s o u t l i n e d i n I n t r i l l i g a t o r (1971). In the context of hedonic f u n c t i o n s , Quigley (1980) and Edlefsen (1981) have addressed t h i s i s s u e . From the f i r s t order c o n d i t i o n s , optimal values f o r x and h-j can be obtained which are f u n c t i o n s of wn , y and h 9 :: x*(W| , h 2 , y) * h 1(w ], h 2 , y) (8.2) (8.1) 20 where the s t a r s denote optimal values. S u b s t i t u t i n g these values i n t o the o b j e c t i v e f u n c t i o n , we obtain the aggregate cost f u n c t i o n which represents the l e a s t cost method of producing the output l e v e l given exogenous input p r i c e s and non-production c h a r a c t e r i s t i c s . C * ( w r h 2 , y ) =w.j x*(w l sh 2,y) + w 2 ( h 2 , h * ( w i s h 2 , f ) ) (9) I t i s known th a t under general r e g u l a r i t y c o n d i t i o n s , the cost f u n c t i o n i s dual to the production f u n c t i o n and completely c h a r a c t e r i z e s the r e l e v a n t information o f the technology (Diewert 1974)^. Upon a p p l i c a t i o n of the envelope theorem or Shephard's Lemma (Shephard 1953) to equation (9) the f o l l o w i n g comparative s t a t i c r e s u l t s can be obtained: V* = v = **(vh2'y) • ( 1 0 J ) V* V = v V 2 ( h l ' h 2 ) = p2 (v h2- y ) ( 1 0 - 2 ) where P 2 i s ah nxl vector of i m p l i c i t p r i c e s o f the n non-production land a t t r i b u t e s . '! • Equation (10.1) i s the'derived demand f u n c t i o n of f a c t o r inputs which i s a fu n c t i o n o f i t s own p r i c e , output l e v e l and the non-production land c h a r a c t e r i s t i c s . Equation (10.2) s t a t e s that the change i n the optimal cost l e v e l o f the aggregate f i r m i f an e x t r a u n i t of h , / i s employed would be to increase the cost by the i m p l i c i t p r i c e of h 9 . These p r i c e s are 21 functions of w-j , and y. Equation (10.2) provides a system of inverse demand functions whereby i t s p r i c e i s expressed as a f u n c t i o n of the commodity and other f a c t o r s . This r e l a t i o n s h i p when i n t e r p r e t e d as a bid fu n c t i o n reveals an i n t e r e s t i n g i m p l i c a t i o n of the general model with respect to b^-The b i d f u n c t i o n f o r h^, and t h e r e f o r e the cost f u n c t i o n , a r e not functions of the i m p l i c i t p r i c e of the production c h a r a c t e r i s t i c s . This i s not at a l l s u r p r i s i n g since the farmers when making t h e i r land purchasing d e c i s i o n s i m p l i c i t l y choose the c h a r a c t e r i s t i c s that they would l i k e t h e i r parcel of land to have. Thus the production c h a r a c t e r i s t i c s are choice v a r i a b l e s and there f o r e endogenous. Since the p r i c e of the parcel i s a f u n c t i o n of.these c h a r a c t e r i s t i c s , the i m p l i c i t p r i c e s are i n f a c t endogenous and there f o r e would not appear i n Eq. ( 9 ) . This i s s i m i l a r to the case o f a monopsonistic f i r m where the f i r m i s not a p r i c e taker and the input p r i c e s are endogenously determined (Varian 1978, pp.74). Endogenous i m p l i c i t p r i c e s are purely the r e s u l t of the non-linear o b j e c t i v e f u n c t i o n ( E d l e f s e n ) . Consider the conventional case when p r i c e s and q u a n t i t i e s are expressed i n l i n e a r combinations. The gradient of t h i s f u n c t i o n with respect to the inputs i s the vector of input p r i c e s and can be i n t e r p r e t e d as being the shadow p r i c e of the in p u t s . Due to l i n e a r i t y i t has a vanishing Hessian. For t h i s case, the shadow p r i c e s can be i n t e r p r e t e d to be parametric and there f o r e exogenous. This i s co n s i s t e n t with the p r i c e t a k i n g assumption i n the input market f o r cost . fun c t i o n s t u d i e s . However, i f we consider the case of a non-linear o b j e c t i v e f u n c t i o n such t h a t there e x i s t s a non-vanishing Hessian, i t follows that the gradient of such a f u n c t i o n cannot be t r e a t e d p a r a m e t r i c a l l y 22 since i t i s a f u n c t i o n of the arguments. The shadow p r i c e s a r e . " therefore endogenous. In the present model, t h i s t r a n s l a t e s to endogenous i m p l i c i t p r i c e s . There can be a case however, under imperfections i n the c h a r a c t e r i s t i c s market, whereby some o f the production c h a r a c t e r i s t i c s of a parcel are not o p t i m a l l y chosen. For i n s t a n c e , since a given piece of land i s a f i x e d composite of i t s a t t r i b u t e s , cases may a r i s e whereby a producer may purchase a parcel with a non-optimal l e v e l of one c h a r a c t e r i s t i c i n order to have access to the a t t r i b u t e s which he f e e l s i s c r u c i a l to his farm. Because o f the i n d i v i s i b i l i t y , the purchaser may have to be content with such an arrangement. Given t h i s s i t u a t i o n , the i m p l i c i t p r i c e of these non-optimum c h a r a c t e r i s t i c s w i l l be determined by other c o n d i t i o n s and i s ther e f o r e exogenous. This departure from the conventional concept of choosing the optimal f a c t o r input i s a consequence o f the i n d i v i s i b l e nature o f land a t t r i b u t e s with respect to the p a r c e l . Pursuing the idea of an exogenous i m p l i c i t p r i c e , we might ask what i m p ! i c i a t i o n s , i f any, can we make with regards to the general hedonic fu n c t i o n under such a scenario. I f we assume that both the vectors of c h a r a c t e r i s t i c s are exogenous, the new cost s t r u c t u r e Cr , would be very much s i m i l a r to the conventional s p e c i f i c a t i o n : C = wjx + P J h r + P j h 2 (11) where P-| and are the corresponding exogenous p r i c e s . Since the hedonic f u n c t i o n s p e c i f i c a t i o n o f (2) i s expressed i n general terms, an i n t e r e s t i n g e x e r c i s e would be to see i f we can say anything about the general expression. R e c a l l that the o r i g i n a l s t r u c t u r e i s : 23 C = wjx + w 2(h 1,h 2) . (12) Equating equations (11) and (12) and e l i m i n a t i n g the non-land e f f e c t s we have: w 2 ( h 1 5 h 2 ) = PJh ] + P 2 h 2 (13) Since P^  and P 2 are i m p l i c i t prices', they are defined as the gradient of the hedonic f u n c t i o n . R e a l i z i n g t h i s , equation (13) becomes: w. 2(h T,h 2)=(V- h w 2 ) T H ] + ( V ^ ) ^ (14) This i s the f a m i l i a r Euler's theorem and the c o n d i t i o n under which t h i s holds i s i f w'2 i s l i n e a r homogenous i n i t s arguments. I t can there f o r e be seen that a l i n e a r homogenous hedonic f u n c t i o n would imply exogenous i m p l i c i t p r i c e s . This can be a basis f o r a t e s t of exogenous i m p l i c i t p r i c e s since a t e s t f o r l i n e a r homogeneity w i l l give us t h i s r e s u l t . I t should be noted that t h i s t e s t can al s o be used to determine whether a p a r t i c u l a r a t t r i b u t e belongs to the production or non-production subset of a t t r i b u t e s since non-production r e l a t e d land c h a r a c t e r i s t i c s have exogenous i m p l i c i t p r i c e s . In r e a l i t y , however, i t i s seldom the case t h a t a l l of the i m p l i c i t p r i c e s are exogenous as i n the above example. A more r e a l i s t i c p o s s i b i l i t y i s th a t some of the production c h a r a c t e r i s t i c s are o p t i m a l l y chosen and some are not thus r e s u l t i n g i n a small subset of exogenous i m p l i c i t p r i c e s . The analogous t e s t f o r t h i s case of t e s t i n g f o r a 24 small sub-group of exogenous p r i c e s requires stronger assumptions on the hedonic f u n c t i o n . In the context of t h i s study, where an aggregate cost f u n c t i o n i s r e q u i r e d , the p o s s i b i l i t y of a subset of c h a r a c t e r i s t i c s not o p t i m a l l y chosen i s u n l i k e l y even though such m i s a l l o c a t i o n s can occur at the i n d i v i d u a l farm l e v e l i n the context of an imperfect c h a r a c t e r i s t i c market. However, i f we assume t h a t there are many buyers and s e l l e r s i n the market f o r a t t r i b u t e s , t h i s i s no longer a problem since even i f an a t t r i b u t e i s not o p t i m a l l y chosen, the market would assign to i t a lower i m p l i c i t p r i c e to r e f l e c t s i t s l e v e l of u n d e s i r a b i l i t y . Therefore, t h i s lower p r i c e would be endogenous. The assumption of a p e r f e c t market f o r c h a r a c t e r i s t i c s w i l l be kept and the presence of exogenous i m p l i c i t p r i c e s w i l l not be te s t e d . The hedonic f u n c t i o n used i n t h i s study w i l l , however, be non-homogenous. R e f e r r i n g back to the c h a r a c t e r i s t i c demand f u n c t i o n s , i t can be seen t h a t several o f the requirements of t h i s study are met i n t h a t a system of demand equations i s derived which i s c o n s i s t e n t w i t h cost minimization behaviour and are functions of v a r i a b l e s which are not determined at an ad hoc f a s h i o n . 25 Footnote A weakness of.the proposed methodology._in j t s present form i s the lack of knowledge of the s t r u c t u r a l i m p l i c a t i o n s i m p l i c i t l y imposed on the production f u n c t i o n as a r e s u l t o f p r e s p e c i f y i n g a n o n l i n e a r o b j e c t i v e f u n c t i o n . S p e c i f i c a l l y , since the cost and the hedonic f u n c t i o n i s p r e s p e c i f i e d , the d u a l i t y r e s u l t s with regards to i t s dual production technology i s somewhat unclear. This i s purely a consequence of the i n t r o d u c t i o n of a p r e s p e c i f i e d nonlinear f u n c t i o n i n t o the cost s t r u c t u r e since the d u a l i t y r e s u l t s f o r the usual l i n e a r case i s w e l l known. The comparative s t a t i c s with regards to the s u b s t i t u t e and output e f f e c t of the expenditure f u n c t i o n has been we l l analyzed by Edlefsen. I t was found that f o r a w e l l behaved production f u n c t i o n with convex l e v e l curves, the degree of s u b s t i t u t a b i 1 i t y of the a t t r i b u t e s vary with the convexity of the l e v e l curves of the cost s t r u c t u r e . I t was a l s o shown that the i n t r o d u c t i o n of n o n l i n e a r c o n s t r a i n t s do not s e v e r l y complicate the comparative s t a t i c a n a l y s i s and the usual r e s u l t s apply. Even though t h i s r e s u l t i s appealing from an a p p l i e d p o i n t of view, i t does not e x p l i c i t l y provide evidence t h a t the underlying production technology would pocess the appropriate r e g u l a r i t y c o n d i t i o n s . I t i s suspected t h a t i n order to derive s p e c i f i c p r o p e r t i e s of the production f u n c t i o n s , a set of l o c a l d u a l i t y r e s u l t s i n the same vein as Blackorby and Diewert (1979) used i n t h e i r paper to show c o n t i n u i t y of the primal production f u n c t i o n under c e r t a i n c o n d i t i o n s of the dual cost f u n c t i o n would have to be derived. A more general approach would be to apply the r e s u l t s of .Epstein (1981) i n h i s d e s c r i p t i o n of g e n e r a l i z e d d u a l i t y a n d . t n t e g r a b i l i t y . Though t h i s i s c l e a r l y a s e r i o u s concern with respect to the d u a l i t y r e s u l t s , i t i s f e l t t h at the d e r i v a t i o n of such d u a l i t y r e s u l t s i s beyond the scope of t h i s paper. CHAPTER IV ECONOMETRIC MODEL With the previous theory and assumptions de s c r i b e d , t h i s and the f o l l o w i n g s e c t i o n s w i l l deal with the s p e c i f i c a t i o n and est i m a t i o n of such a model. In terms o f e s t i m a t i o n , the conventional two stage procedure w i l l be used. The f i r s t stage e n t a i l s the d u p l i c a t i o n o f market information by f i t t i n g a market wide hedonic f u n c t i o n . I m p l i c i t p r i c e s f o r both forms of a t t r i b u t e s w i l l be derived at t h i s stage. The second stage e n t a i l s the estimation of a system of b i d functions which are c o n s i s t e n t with cost minimization behaviour. These w i l l be the b i d functions f o r the non-a g r i c u l t u r a l production r e l a t e d a t t r i b u t e s . s i n c e the cost f u n c t i o n i s a fu n c t i o n o f the input p r i c e s , the output l e v e l and the non-production r e l a t e d land a t t r i b u t e s . Since q u a n t i t y data of the other non-land inputs are not a v a i l a b l e , the f a c t o r demand equations w i l l not be estimated with the system. Model s p e c i f i c a t i o n i s d e a l t w i t h i n the f i r s t two se c t i o n s of t h i s chapter. The f i r s t s e c t i o n i s concerned with the f i r s t stage s p e c i f i c a t i o n and contains a d i s c u s s i o n of some e m p i r i c a l p r o p e r t i e s which are required of the hedonic f u n c t i o n i n the context of t h i s study. The second s e c t i o n w i l l discuss the s e l e c t i o n and the appropriate m o d i f i c a t i o n o f f u n c t i o n a l forms f o r our cost f u n c t i o n . The majo r i t y of these m o d i f i c a t i o n s are required as 26 27 b i d f unctions w i l l be estimated as opposed to the usual f a c t o r demands. The a d d i t i o n of dummy v a r i a b l e s to capture the e f f e c t s o f the a g r i c u l t u r a l land reserve program on production .behaviour w i l l a l s o be discussed. 4.1 F i r s t Stage Model To make the t h e o r e t i c a l model o p e r a t i o n a l , some measure of the e q u i l i b r i u m p r i c e of the c h a r a c t e r i s t i c s i s required. The method used w i l l be to i n i t i a l l y estimate a hedonic f u n c t i o n from which the i m p l i c i t p r i c e s can be obtained as described i n the previous s e c t i o n . The question of f u n c t i o n a l form f o r the hedonic f u n c t i o n a r i s e s and the issue o f s e l e c t i n g the appropriate one has been widely i n v e s t i g a t e d . In t h i s context, the dominant forms used have been l i n e a r (Kain and Quigley 1970, King 1976), l o g , semi-log and double l o g ( G r i l i c h e s 1971, Dhrymes 1971). Best f i t c r i t e r i a s were a l s o used to address t h i s issue (Brereton 1979, Quigley, 1980, Linneman 1980, Halvorsen 1981). The b a s i c requirement f o r the f u n c t i o n r e l a t i o n s h i p i s that i t be no n l i n e a r and nonhomogeneous i n the c h a r a c t e r i s t i c s as suggested i n the l a s t s e c t i o n . For the present a n a l y s i s , the hedonic f u n c t i o n must also be s u f f i c i e n t l y complex such that the r e s u l t i n g i m p l i c i t p r i c e s w i l l vary f o r d i f f e r e n t c h a r a c t e r i s t i c s . The method of Box-Cox was used i n several s t u d i e s (Quigley, Brereton) to obtain i m p l i c i t p r i c e s . Though the r e s u l t i n g f u n c t i o n a l forms w i l l be nonlinear i n the arguments, t h i s technique i s not t o t a l l y a p p l i c a b l e to the present case since v a r i a b i l i t y between r e l a t i v e p r i c e s does not f o l l o w as demonstrated by the f o l l o w i n g r e s u l t . 28 Consider the f o l l o w i n g Box-Cox transformation of the dependent v a r i a b l e (Zarembka 1974) as ap p l i e d to an a r b i t r a r y hedonic f u n c t i o n q wit h J c h a r a c t e r i s t i c s . f o r (15) X log q f o r X=0 (X) J where qKAI = iB^h. + e (16) and q i s the p r i c e o f the commodity, h. i s c h a r a c t e r i s t i c i , B. are the c o e f f i c i e n t s , e i s the e r r o r term which i s assumed to be normally d i s t r i b u t e d with constant variance and X i s a parameter which measures the degree of n o n l i n e a r i t y of the r e l a t i o n s h i p . Sum of square e r r o r s are minimized with respect to the parameters B. and X . Rearranging the terms f o r the case of X?0, eq (15) becomes 1 i>V, q = [X(SB.h. + e) + i f ' ; (17) The i m p l i c i t p r i c e of c h a r a c t e r i s t i c i i s defined to be: i l _ l['A(»,h,••+ e) + 1] A- \B. (18) 3 h i " X 1 29 The r a t i o of the i .th and j i m p l i c i t p r i c e w i l l be!: gq/9h i 3q/3h. B. f o r i 7 j (19) That i s , the v a r i a b i l i t y between the two i m p l i c i t p r i c e s i s e x a c t l y determined by the r a t i o of t h e i r r e s p e c t i v e c o e f f i c i e n t s . Therefore, employing t h i s approach would r e s u l t i n imposing a rather strong i m p l i c i t assumption on the i m p l i c i t p r i c e s . Along with p r i c e v a r i a b i l i t y , the f u n c t i o n a l form s e l e c t e d should have cross terms among the c h a r a c t e r i s t i c s to take account of the i n t e r a c t i o n s between the a t t r i b u t e s . In other words, the hedonic f u n c t i o n should have a non-vanishing Hessian since the existence of i m p l i c i t p r i c e s which are not f u n c t i o n s of other a t t r i b u t e s besides i t s own appears to be an unnecessary r e s t r i c t i o n . I t appears that the n e c e s s i t y f o r p r i c e v a r i a b i l i t y f o r each a t t r i b u t e and the need f o r cross terms would e l i m i n a t e the m a j o r i t y of the hedonic func t i o n s p r e s e n t l y employed i n the l i t e r a t u r e . hedonic s u r f a c e , other than the l i n e a r i t y and homogeneity co n d i t i o n s stated i n the l a s t chapter, i t would seem appropriate to s e l e c t a f u n c t i o n which i s capable of pr o v i d i n g a second order approximation to an a r b i t r a r y f u n c t i o n w hile s t i l l p r o v i d i n g p r i c e v a r i a b i l i t y and cross terms f o r the a t t r i b u t e s . Following W i t t e , Sumka and Erekson, the f o l l o w i n g f u n c t i o n a l form w i l l be used as i t meets a l l our requirements: Since there i s no a p r i o r i theory to determine the shape of the 30 n+k n+k w9 = a. + E a,h. + iZ Ea^-h.h. + e (20) ^ 0 1 1 ^ j 1J 1 J where a'.s, a--'s are parameters and e i s the e r r o r term and i s assumed to "I "I J be e^N(0,a ). I t can be seen that eg. (20) i s quadratic i n i t s arguments and i s also non-homogeneous i n i t s a t t r i b u t e s . Since eg. (20) i s a l s o l i n e a r i n the parameters, conventional OLS estimation techniques can be used to obtain parameter estimates. One assumption of the e r r o r s t r u c t u r e i n the case of OLS a p p l i c a t i o n may however, not be t o t a l l y v a l i d . That i s , the assumption that the data has s u f f i c i e n t range to assume a normal d i s t r i b u t i o n about the mean. Since negative land p r i c e s are not p o s s i b l e , i t i s conceivable that the e r r o r term i s not normally d i s t r i b u t e d but may be a truncated normal with the t r u n c a t i o n l e v e l being 0. For such a case, the r e s u l t i n g d i s t r i b u t i o n w i l l resemble a normal d i s t r i b u t i o n with i t s lower end del e t e d . This property of the data i s c o n v e n t i o n a l l y l a b e l l e d as the case of truncated samples or more g e n e r a l l y , l i m i t e d dependent v a r i a b l e s and has i t s roots i n low wage earning studies where negative wages are not observed. I t has been shown (Hausman and Wise 1977, Goldberger 1973) that f o r the case of truncated samples, a p p l i c a t i o n of OLS w i l l r e s u l t i n a downward bias of the parameter estimates. A maximum l i k e l i h o o d e s t i m a t i o n technique was developed by Hausman and Wise, and modified by Quan and Kerr (1983), was i n i t i a l l y used to address t h i s problem. Maximization of the l i k e l i h o o d f u n c t i o n requires the use of no n l i n e a r o p t i m i z a t i o n techniques which are subject to a high degree of numeric i n s t a b i l i t y f o r the case of c o l l i n e a r data. As eg. (20) i s quadratic and the r e f o r e would i n v a r i a b l y contain cross terms, col l i n e a r i t y of the data i s a r e a l p o s s i b i l i t y . This was in f a c t the case when such a procedure was adopted f o r the hedonic f u n c t i o n . Highly u n r e a l i s t i c values as we l l as an i n d e f i n i t e information matrix were obtained. Various o p t i m i z a t i o n routines were used with the same general r e s u l t . For t h i s reason, OLS was used to estimate the hedonic f u n c t i o n . The problem of m u l t i c o l l i n e a r i t y i s s t i l l present but w i t h i n the context of OLS, i t i s known that m u l t i c o l l i n e a r i t y w i l l bias the standard e r r o r s of the estimates upwards but the parameter estimates w i l l remain unbiased (Johnston 1972). A l s o , the OLS procedures are not subject to the same degree of numeric i n s t a b i l i t y as i n the case of the maximization of a l i k e l i h o o d f u n c t i o n . Since the purpose of t h i s f i r s t stage i s to obtain r e a l i s t i c estimates of the i m p l i c i t p r i c e s f o r the second stage, a p p l i c a t i o n of OLS to col l i n e a r data i s preferred to other estimation techniques such as Ridge regression where the r e s u l t i n g parameters are known to be biased. I t w i l l be assumed that-the bias introduced by the p o s s i b l e t r u n c a t i o n of the sample i s small and w i l l not a f f e c t the r e s u l t s s i g n i f i c a n t l y . From eg. ( 2 0 ) , the i m p l i c i t p r i c e of the i c h a r a c t e r i s t i c w i l l be c a l c u l a t e d as: 9w? n+k • zxr- = a- + Gi'-h. + n a- -h. (21) 3h i a i °h i j 7 1 - " 1 J J I t should be noted at t h i s stage t h a t the estimation of a market wide hedonic f u n c t i o n w i t h rea l market p r i c e s r e s u l t s i n parameter estimates which could r e f l e c t both the o f f e r and the b i d f o r the a t t r i b u t e s As i n the conventional supply land demand a n a l y s i s , i d e n t i f i c a t i o n becomes a relevant i s s u e . I t i s f o r t h i s reason t h a t Rosen's proposed method e n t a i l s the j o i n t e s timation of both the b i d as we l l as the o f f e r f u n c t i o n s However, i t i s known that under c e r t a i n market c o n d i t i o n s , i d e n t i f i c a t i o n of the system of b i d f u n c t i o n s can be unambiguously determined (Berndt). The market f o r land c h a r a c t e r i s t i c s can be viewed i n the short run as being a second hand market. As i n such markets, the supply i s e i t h e r f i x e d or i n e l a s t i c . Market p r i c e f l u c t u a t i o n s can t h e r e f o r e be only a t t r i b u t a b l e to s h i f t s i n demand with the p r i c e s being s t r i c t l y demand determined. Since i t can be argued that the supply of a g r i c u l t u r a l land with p a r t i c u l a r a t t r i b u t e s i s i n e l a s t i c , i t i s reasonable to assume that the i m p l i c i t p r i c e s are b i d determined thus e l i m i n a t i n g the need f o r the j o i n t e stimation of the o f f e r f u n c t i o n s . 4.2 Cost Function S p e c i f i c a t i o n For e m p i r i c a l e s t i m a t i o n , a s p e c i f i c f u n c t i o n a l form i s . r e q u i r e d f o r the cost f u n c t i o n . A host of f l e x i b l e f u n c t i o n a l forms have been developed (Diewert 1971, C h r i s t e n s e n , Jorgenson and Lau 1972, Denny 1974). But due to the p a r t i c u l a r nature of our model, as well as data l i m i t a t i o n s , not a l l of them are appropriate. One requirement of the s p e c i f i c a t i o n i s the need f o r the i n c l u s i o n of the vector of non-production c h a r a c t e r i s t i c s . That i s , u n l i k e the conventional cost f u n c t i o n which i s a f u n c t i o n of input p r i c e s and the output l e v e l , the hedonic cost f u n c t i o n also includes land c h a r a c t e r i s t i c s . P o s s i b l e candidates are the v a r i a b l e cost or p r o f i t f u n c t i o n s which have as one of i t s arguments f i x e d f a c t o r s : However, even w i t h i n t h i s c l a s s , not a l l v a r i a b l e f a c t o r f u n c t i o n a l forms are a p p l i c a b l e . The t r a n s l o g s p e c i f i c a t i o n has been used i n the context of v a r i a b l e cost f u n c t i o n s ( C h r i s t s e n , Cummings ,and Schoech 1981). The • actual estimation procedure i n v a r i a b l y i n v o l v e s the c a l c u l a t i o n of f a c t o r cost shares which req u i r e s data on the p h y s i c a l q u a n t i t i e s of a l l 33 f a c t o r s . Q u a n t i t i e s of non-land f a c t o r inputs are not a v a i l a b l e i n the data s e t . One f u n c t i o n a l form which i s f e a s i b l e i s the v a r i a b l e p r o f i t f u n c t i o n a l form as proposed by Diewert (1973). A v a r i a b l e p r o f i t f u n c t i o n , i n Diewert's n o t a t i o n , i s defined as: •v(p;v) ~. max u(p Tu: (u ;v)eT} (22) where p i s the vector of p r i c e s ( p o s i t i v e f o r input and negative f o r o u t p u t ) , T i s the production p o s s i b i l i t i e s s e t , v i s the vector of f i x e d f a c t o r s and u i s the vector o f v a r i a b l e , i n p u t s and outputs. The v a r i a b l e p r o f i t f u n c t i o n a l form i s : I t has been shown (Diewert) that Eq. (23) i s a second order approximation to an a r b i t r a r y f u n c t i o n and can s a t i s f y a host o f c o n d i t i o n s . Two conditions which are most relevant to the present study are the f o l l o w i n g homogeneity r e s u l t s : ( i ) Tf i s l i n e a r homogeneous i n p., i . e . , f o r every X>0, TT (X.p y) = Atr (.p;v ) and ( i i ) rr i s l i n e a r homogenous i n v, i . e . , f o r every A>0, i r ( p U v ) = ATT(P;V) 34 A l o g i c a l candidate f o r the hedonic cost f u n c t i o n would be to use eq.(23) and s u b s t i t u t e the subset of c h a r a c t e r i s t i c s f o r the f i x e d f a c t o r s . Though appealing, the r e s u l t i n g f u n c t i o n a l form i s not a l t o g e t h e r appropriate as c e r t a i n m o d i f i c a t i o n s are required. By s u b s t i t u t i n g the input p r i c e w. f o r p., r e p l a c i n g the f i x e d f a c t o r v- by c h a r a c t e r i s t i c h., and i n c l u d i n g the appropriate m o d i f i c a t i o n s , J J the f o l l o w i n g f u n c t i o n a l form i s proposed: I I J o 9 1 I J p C(w,h,y) = z S r a , ( i v / + i v / j 5 h . + yE Ui-uY. + . _ i m j i j J J T3'j?l.' , , I J 2 (24) « ^ i h ? k + ^ f i j w i h ? An important departure of eq. (24) from eq. (23). i s the a d d i t i o n of the square term of the c h a r a c t e r i s t i c s i n the second expression. This would v i o l a t e the homogeneity c o n d i t i o n of the f u n c t i o n with respect to the c h a r a c t e r i s t i c s . However, l i n e a r homogeneity i n the input p r i c e s i s preserved. L i n e a r homogenous i n the input p r i c e s suggest t h a t as input p r i c e s are increased by a proportionate amount, the cost would also increase by t h i s amount. This c o n d i t i o n must be s a t i s f i e d f o r any cost f u n c t i o n . The same c o n d i t i o n i s not appropriate f o r the c h a r a c t e r i s t i c s since i f a l l c h a r a c t e r i s t i c s i n the cost f u n c t i o n are increased by the same p r o p o r t i o n , the t o t a l cost w i l l not i n general increase by the same f a c t o r . Total cost w i l l be a f f e c t e d i n some fashion as d i c t a t e d by the hedonic f u n c t i o n . In gene r a l , i t i s only under r e s t r i c t i v e c o n d i t i o n s of the hedonic f u n c t i o n , namely l i n e a r homogeneous 1 i n the cost f u n c t i o n .attributes that the cost f u n c t i o n w i l l be homogeneous of degree one i n these a t t r i b u t e s . 3 5 Another m o d i f i c a t i o n i s the need f o r the i n t r o d u c t i o n of the output term i n eq. ( 2 3 ) since the cost f u n c t i o n i s a l s o a f u n c t i o n of the output l e v e l . The method f o r i t s i n c l u s i o n i s ' i n the same vein as Park's ( 1 9 7 1 ) adaptation o f the Generalized L e o n t i e f f u n c t i o n a l form. This would c o n t r o l f o r the e f f e c t of the a g r i c u l t u r a l land reserve program which was enacted w i t h i n the period under i n v e s t i g a t i o n . The proposed method of i n c l u d i n g D would allow f o r the e f f e c t s of the land reserve program on the input p r i c e s , the c h a r a c t e r i s t i c s as wel l as on output. I t w i l l be shown l a t e r that i t s i n c l u s i o n i n eq ( 2 4 ) can con t r o l f o r the land reserve e f f e c t on each b i d f u n c t i o n as w e l l as on the p r i c e f l e x i b i l i t i e s . A t h i r d m o d i f i c a t i o n i s the i n c l u s i o n of a dummy v a r i a b l e D. From eq. ( 2 4 < ) i , the f o l l o w i n g r e s u l t s can be obtained: 2 n ( 2 5 . 1 ) I 2y£d, .w.h. ( 2 5 . 2 ) 8w 9w. ( 2 5 . 3 ) ( 2 5 . 4 ) 36 i i 2j:dijwihj + 2Edi.w1h.D (25.6) ayah. where I and J are the number of input p r i c e s and c h a r a c t e r i s t i c s r e s p e c t i v e l y . In the present study, where F represents feed p r i c e s , W i s the wage r a t e , H i s the p r i c e of hay and C i s the r e n t a l p r i c e of c a p i t a l , j , & = M, D, P, V where M i s the month a t t r i b u t e D i s distance of nearest c e n t r e , P i s the population o f the nearest centre and V i s the distance from Vancouver. Equation (25.2) w i l l provide the system o f es t i m a t i n g equations. Equation (25.1) i s the r e l a t i o n s h i p of the e f f e c t s on cost due to a change i n input p r i c e s . These are the c o n v e n t i o n a l l y estimated derived demand c o n d i t i o n s . The symmetry c o n d i t i o n s o f the cost f u n c t i o n with respect to input p r i c e s w i l l r e s u l t i n a. = a .. These symmetry co n d i t i o n s have i m p l i c a t i o n s with lm mi regards to the s t r u c t u r a l d e r i v a t i o n of the cost f u n c t i o n . I t has been shown by Hurwicz and Uzawa (1971) that i f the Hessian matrix of the cost f u n c t i o n i s symmetric, then the system of derived demand con d i t i o n s i s i n t e g r a b l e , that i s , there e x i s t an aggregate cost f u n c t i o n and a production i , m - F, W, H, C 37 technology c o n s i s t e n t with the o p t i m i z i n g procedure. I n t e g r a b i l i t y of a cost f u n c t i o n can be a t e s t a b l e hypothesis as i n Binswanger (1979). In the present case, symmetry w i l l be imposed as a maintained hypothesis s i n c e t h i s w i l l have the nice feature of imposing s t r u c t u r e i n t o the problem as well as reducing the number of free parameters. Another set of normalizations which are useful and are imposed are: a i i = 0 ' b j j = 0 a n d b j k = b k j These normalizations are required as without them, the r e s u l t i n g f u n c t i o n a l form would have parameters which are not i d e n t i f i a b l e . Equation (25.5) describes the change i n the i m p l i c i t p r i c e s of the c h a r a c t e r i s t i c s as a r e s u l t of the change i n the input p r i c e s . This w i l l provide an avenue to analyze the e f f e c t s of input p r i c e changes on the bid p r i c e s of the non-production c h a r a c t e r i s t i c s . Equation (25.6) w i l l record the e f f e c t on the b i d p r i c e s due to a change i n the output l e v e l . Since ea.h of the above comparative s t a t i c r e s u l t s d i f f e r . f r o m each : o b s e r v a t i o n , t h e i r values w i l l be evaluated at the mean of the observations. Given the econometric model, we can not proceed to estimate the system of bi d f u n c t i o n s . 4.3 Estimation Techniques _ - - ' , , ' ' A p p l i c a t i o n of the.envelope theorem to the cost f u n c t i o n (24) w i l l r e s u l t i n the f o l l o w i n g system of i m p l i c i t p r i c e dependent bid f u n c t i o n s P. = I Ljiw? + i w 2)* + 2 y L i jw ih j + iz j k w ^ h T V / i m i IK. I (26) 2yZd i jw ih jD 38 f o r i , m = F, W, H, C and j , k, = M, D, P, V. where P. i s the i m p l i c i t p r i c e o f c h a r a c t e r i s t i c j from the f i r s t stage J e s t i m a t i o n . A l l other terms are as p r e v i o u s l y defined. I t i s assumed that the explanatory v a r i a b l e s are non-stochastic and that they also s a t i s f y the usual rank c o n d i t i o n . Each equation a l s o contains an a d d i t i v e Txl vector of disturbances which has a zero mean 2 vector with covariance matrix a I where I i s a TxT i d e n t i f y matrix with T being the number of observations. Let e. be the disturbance vector f o r equation i . I t i s known t h a t under these i d e a l c o n d i t i o n s , a p p l i c a t i o n of OLS w i l l r e s u l t i n BLUE estimates f o r each equation. OLS however i m p l i c i t l y assumes that a l l information relevant to each equation are present w i t h i n the model. In the present case where a system of equations e x i s t , the assumption of complete information may not be v a l i d as the across equation e r r o r s may be c o r r e l a t e d . I f the e r r o r s are c o r r e l a t e d , e f f i c i e n c y gains can be had by recognizing these r e l a t i o n s h i p s e x p l i c i t l y i n the es t i m a t i o n procedure. Let E be the covariance matrix of the system of equations where the diagonal contains the variance o f each equation and the o f f - d i a g o n a l s would contain the covariance of the d i f f e r e n t equation e r r o r s ( i . e . , element j , k of E f o r j^k would contain E(e.e,*)) . A simple t e s t f o r across equation c o r r e l a t i o n would be to t e s t the hypothesis t h a t E i s a diagonal matrix. I f E i s d i a g o n a l , t h i s would suggest that no across equation c o r r e l a t i o n s are present thus a p p l i c a t i o n of OLS would be appropriate. This can therefore be viewed as a t e s t f o r the appropriateness of OLS. 39 A l i k e l i h o o d r a t i o t e s t was used to determine whether Z i s diagonal. The r e s t r i c t e d l o g l i k e l i h o o d f u n c t i o n value was -2841.1 and i s obtained by f o r c i n g £ to be diagonal. -2 times the d i f f e r e n c e of t h i s value and the corresponding value f o r the free case r e s u l t e d i n a value o f 31.4. I t i s known that minus twice the logarithm of the l i k e l i h o o d r a t i o i s asymptotically a Chi-square d i s t r i b u t i o n with the number o f degrees of freedom being the number of r e s t r i c t i o n s imposed i n the r e s t r i c t e d case ( T h e i l ,1971). Since & (5) = 18.55<31.4, the hypothesis that Z i s diagonal i s r e j e c t e d 95 at the 95 percent confidence l e v e l . Since we cannot r e j e c t that across equation c o r r e l a t i o n of the e r r o r s are present, the chosen estimator must address t h i s i n t e r r e l a t i o n s h i p . An estimator which takes t h i s information i n t o c o n s i d e r a t i o n and also allows f o r across equation r e s t r i c t i o n s of the parameters i s Z e l l n e r ' s Seemingly Unrelated Regression.., (SUR) estimator ( Z e l l n e r 1962 , 1963 and Z e l l n e r and Huang 1962). Although the n o n - i t e r a t i v e SUR w i l l f u l f i l l the e r r o r s t r u c t u r e as w e l l as the parameter r e s t r i c t i o n requirements, i t has been shown by Kmenta and G i l b e r t (1968) t h a t ' i f one i t e r a t e s t h i s estimator with the r e s t r i c t i o n s imposed at each i t e r a t i o n , the converged parameter estimates w i l l i n f a c t be a s y m p t o t i c a l l y maximum l i k e l i h o o d . The i t e r a t i v e SUR w i l l t h e r e f o r e be used i n t h i s study as i t meets a l l the requirements as well as p r o v i d i n g a r e l a t i v e l y easy way to obtain maximum l i k e l i h o o d estimates. Another source o f e f f i c i e n c y gains would be to estimate the hedonic f u n c t i o n j o i n t l y w i t h the system of bid f u n c t i o n s . This would be analogous to i n c l u d i n g the cost f u n c t i o n i n t o the estimation of the system 40 of derived demand functions i n cost f u n c t i o n e s t i m a t i o n s . E f f i c i e n c y gains i n the cost context has been suggested by Green and Christensen (1976). However, f o r the present study where the b i d f u n c t i o n s have as t h e i r dependent v a r i a b l e s r e l a t i v e l y complex func t i o n s of the f i r s t stage parameter and v a r i a b l e s , the j o i n t e s timation of the hedonic f u n c t i o n with the b i d functions would considerably complicate the estimation procedure. Therefore, the two stage procedure w i l l be adopted. CHAPTER V DATA AND DATA CONSTRUCTION In the f i r s t stage e s t i m a t i o n , observations on land values and t h e i r corresponding land c h a r a c t e r i s t i c s are r e q u i r e d . Records of monthly s a l e s of d a i r y farmland throughout B r i t i s h Columbia were obtained from the Farm C r e d i t Corporation from January 1970 to December 1977. A t o t a l of 215 t r a n s a c t i o n s were used. The c h a r a c t e r i s t i c s of these d a i r y l a n d s which changed hands were: ( i ) P r i c e of the parcel of land i n thousands of d o l l a r s , ( i i ) Size of the parcel i n acres, ( i i i ) Number of acres c u l t i v a t e d , ( i v ) Distance of the parcel i n miles to the nearest centre with a population greater than 500. (v) Population of the nearest centre with a population greater than 500. ( v i ) Distance from Vancouver i n miles and ( v i i ) An index of the month which the t r a n s a c t i o n took place. The p r i c e of the parcel was d e f l a t e d by the Consumer P r i c e Index i n order to express the value i n re a l terms. From the l i s t , the s i z e and the number of acres c u l t i v a t e d w i l l represent the production c h a r a c t e r i s t i c s with the remaining c l a s s i f i e d as land a t t r i b u t e s not d i r e c t l y r e l a t e d to production. 41 42 Since the parameters estimated at the f i r s t stage play a very c r u c i a l r o l e i n the second stage, i t i s important that a l l r e l e v a n t c h a r a c t e r i s t i c s be present to minimize s p e c i f i c a t i o n e r r o r . A casual survey of the v a r i a b l e s suggest that non-production c h a r a c t e r i s t i c s are reasonably wel1-represented. A p o t e n t i a l source of l e f t out v a r i a b l e bias may however be present f o r the production a t t r i b u t e s as only two measures are used, s i z e and the acreage c u l t i v a t e d . Depending on the l e v e l of d e t a i l , the l i s t of a t t r i b u t e s can be a r a t h e r lengthy one thus some e r r o r s from omitted v a r i a b l e s i s i n e v i t a b l e . However, G r i l i c h e s pointed out that i f the l e f t out v a r i a b l e s were separable from the included c h a r a c t e r i s t i c s l i s t and are constant over time, the l e f t out v a r i a b l e s w i l l not pose a serious problem. The approach of using a small subset to represent the spectrum of c h a r a c t e r i s t i c s i s p a r t i a l l y supported by Kain and Quigley. In t h e i r hedonic study of housing q u a l i t y , a l i s t of 39 a t t r i b u t e s were compiled i n d i c a t i n g the p h y s i c a l or v i s u a l q u a l i t y of the bundle of r e s i d e n t i a l s e r v i c e s . Through f a c t o r a n a l y s i s , i t was shown th a t i n d i v i d u a l households evaluated r e s i d e n t i a l q u a l i t y i n terms of several broad aggregates. Analogously, i t i s assumed th a t the two production c h a r a c t e r i s t i c measures are h i g h l y c o r r e l a t e d with the omitted ones. I t i s important to note t h a t since t h i s study i s designed to i n v e s t i g a t e bid f u n c t i o n f o r nonproduction c h a r a c t e r i s t i c s , the purpose of the production a t t r i b u t e s are only to c o n t r o l f o r the e f f e c t s of production c h a r a c t e r i s t i c s i n the p r i c e of land. The p a r c e l ' s t o t a l value was d i v i d e d by the s i z e of the parcel and m u l t i p l i e d by 1000 to obtain the p r i c e per acre of land i n r e a l d o l l a r s . A l s o , acres of c u l t i v a t e d land were expressed as a percentage of t o t a l acreage c u l t i v a t e d . 43 In the second stage model, observations on input p r i c e s and output l e v e l s are re q u i r e d . Four input p r i c e s were used; the p r i c e of feed, p r i c e of hay, wages and a c a p i t a l term which includes c a p i t a l , b u i l d i n g s and s t r u c t u r e . The feed, wage and hay p r i c e s t a t i s t i c s are from the B r i t i s h Columbia M i l k Board's Annual Reports. A p r i c e index of 16 percent d a i r y feed was used to r e f l e c t the p r i c e of feed. The p r i c e of a l f a l f a hay was obtained and i s expressed i n d o l l a r s per ton. Farm wages with board i n B r i t i s h Columbia was used f o r the wage r a t e . These wages were expressed i n d o l l a r s per month. Rental rates of c a p i t a l and s t r u c t u r e were obtained through y e a r l y f i g u r e s on farm stock values of c a p i t a l i n B r i t i s h Columbia a g r i c u l t u r e ( S t a t i s t i c s Canada Cat. No. 21-003 Quarterly B u l l e t i n of A g r i c u l t u r e S t a t i s t i c s ) , d e p r e c i a t i o n on machinery, r e p a i r s to b u i l d i n g s , and d e p r e c i a t i o n of b u i l d i n g ( S t a t i s t i c s Canada Cat..No. 21-003-P Farm Net Income) f o r the period 1971 to 1977. The corresponding values f o r 1970 were obtained by l i n e a r e x t r a p o l a t i o n from the 1971 values. This was necessary since the accounting system f o r these f i g u r e s were changed i n 1971. The d e p r e c i a t i o n f i g u r e s take i n t o account the d e c l i n i n g value of c a p i t a l as a r e s u l t of i t s u t i l i z a t i o n . B u i l d i n g d e p r e c i a t i o n s are based on farm b u i l d i n g s which includes the farm business share of the house or owner-occupied farms. Repairs to b u i l d i n g s are the estimated expense of cost a s s o c i a t e d with r e p a i r s and maintenance of farm b u i l d i n g s which include the farm business share of the farm house. The t o t a l value of farm stock i n B r i t i s h Columbia r e f l e c t s the replacement value of the stock of machinery and implements. 44 The r e n t a l rates of farm machinery and implements are obtained by the f o l l o w i n g formula: R = r + 6 V (27) m j - ^ m where R m i s the r e n t a l value of machinery and implements r i s the i n t e r e s t r a t e , 6 i s the net d e p r e c i a t i o n rate (net of a p p r e c i a t i o n ) V m i s the t o t a l value of machinery and implements. r i s assumed to be a constant 6% f o r the period under i n v e s t i g a t i o n . The net d e p r e c i a t i o n r a t e , 6 i s estimated by d i v i d i n g the t o t a l machinery d e p r e c i a t i o n f i g u r e s by the t o t a l farm c a p i t a l stock. For the analogous measure f o r b u i l d i n g and s t r u c t u r e , the values of r e p a i r s to and d e p r e c i a t i o n of b u i l d i n g s were summed. The output v a r i a b l e i s measured i n t o t a l value terms. Due to the present milk marketing system, each d a i r y producer can s e l l milk at a set p r i c e , as determined by a cost of production formula, up to a q u a n t i t y as determined by each producer's quota. Sales of milk i n excess of the allowable quota amount are p o s s i b l e but at a lower p r i c e . Total value of milk production of both quota as well as excess production were c a l c u l a t e d f o r B r i t i s h Columbia (excluding Vancouver Island) and are expressed i n p r i c e per l i t r e . A l l value f i g u r e s were subsequently d e f l a t e d by the consumer p r i c e index. The r e s u l t s of using the above data i n the econometric model can now be discussed. CHAPTER VI DISCUSSION OF RESULTS With the estimation performed, the r e s u l t s of the estimates and t h e i r i m p l i c a t i o n s can now be discussed. The f i r s t s e c t i o n of t h i s chapter w i l l deal with the estimates of the hedonic f u n c t i o n and the r e s u l t i n g i m p l i c i t p r i c e s . Since a subset of these p r i c e s w i l l be used as dependent v a r i a b l e s i n the second stage, i t i s q u i t e important that they take on r e a l i s t i c values. The v a l i d i t y of a l l i m p l i c i t p r i c e s w i l l be discussed i n t h i s s e c t i o n . The second s e c t i o n of t h i s chapter w i l l d i s c u s s the estimates from the system of i m p l i c i t p r i c e dependent bid bunctions. Results p e r t a i n i n g to the e f f e c t s of input p r i c e changes on- the b i d p r i c e s f o r a t t r i b u t e s w i l l be evaluated. This w i l l be followed by a d i s c u s s i o n of the e f f e c t s of the land reserve program on the i m p l i c i t p r i c e s . An a l t e r n a t e i n t e r p r e t a t i o n of the b i d f u n c t i o n parameter estimates w i l l be proposed at the end of t h i s chapter. 6.1 F i r s t Stage The estimated c o e f f i c i e n t s f o r the f i r s t stage model are presented i n Table 1. An R-square value of .6841 i m p l i e s that the model has captured a reasonable amount of t o t a l v a r i a t i o n i n land p r i c e s . An adjusted R-square value of .6385 i n d i c a t e s that a large number of i r r e l e v a n t v a r i a b l e s were not i n c l u d e d . S i g n i f i c a n c e at the 95% confidence l e v e l were obtained by 11 of the parameters while at the 90% l e v e l , 15 were s i g n i f i c a n t . 45 TABLE 1 46 Estimates of the Hedonic Function ( t - r a t i o s are i n parentheses) V a r i a b l e Coeff i c i e n t s i . V a r i a b l e C o e f f i c i e n t s Month -0.1112 (Month)*(Size) -.096* (-.485) (2.160) Si z e -28.917* (Month)*(Cult.) .298** (-4.357) (1.665) C u l t . -60.018* (Month)*(Dist.) -.195 (-2.103) (-.941) D i s t . -91.953* (Month)*(Pop.) -.018** (-1.965) (-1.458) Pop. 3.599 (Month)*(Dist.Van.) .002 (.932) (.108) ( D i s t . Van) -10.335* ( S i z e ) * ( C u l t . ) .157 p (-2.418) (2.970) Month .196** ( S i z e ) * ( D i s t . ) .104 (1.372) (.895) S i z e £ .105* (Size)*(Pop.) .007 p (3.206) (.812) C u l t / .316 (S i z e ) * ( D i s t . V a n . ) .006 p (.955) (.508) D i s t . .793 ( C u l t . ) * ( D i s t . ) .697 p (.642) (2.013) P o p / -.008** (Cult.)*(Pop.) .020 (-1.397) (.526) ( D i s t . V a n / .027 (C u l t . ) * ( D i s t . V a n . ) .033 (2.695) (1.089) (Dist.)*(Pop.) -.037 p (-.442) R = .6841 ( D i s t . ) * ( D i s t . V a n ) .087 p (1.008) R = .6385 (Pop.)*(Dist.Van) .049* (-3.054) Intercept 6115.5* (3.945) * s i g n i f i c a n t at the 95% confidence l e v e l . * * s i g n i f i c a n t at the 90% confidence l e v e l . 47 Due to the nature of the f u n c t i o n a l form used, i t i s suspected that a c e r t a i n degree of m u l t i c o l 1 i n e a r i t y e x i s t s . This would create an upward bias i n the variance of the estimates and subseqently r e s u l t i n g i n lower t - s t a t i s t i e s . Even with t h i s b i a s , the model i n general s t i l l a t t a i n e d a reasonable number of s i g n i f i c a n t v a r i a b l e s . The f u n c t i o n a l form, being quadratic i n the arguments, provides c o e f f i c i e n t e s which are not, i n themselves, e a s i l y i n t e r p r e t a b l e . The purpose of t h i s f i r s t stage estimation i s , once again, to obtain meaningful i m p l i c i t p r i c e s f o r the non-production r e l a t e d land a t t r i b u t e s . These i m p l i c i t p r i c e s were c a l c u l a t e d f o r each of the s i x c h a r a c t e r i s t i c s according to equation (21) of Chapter IV and are presented i n Table 2. The expression f o r the i m p l i c i t p r i c e of c h a r a c t e r i s t i c i i s once again: .Wo n+k f f i f - »i + « i i h i + * J , V J - - ( 2 8 ) Since these p r i c e s vary f o r each o b s e r v a t i o n , summary s t a t i s t i c s are a l s o provided. A l s o , since the a t t r i b u t e s are measured i n such a way as to allow f o r both p o s i t i v e and negative e f f e c t s on the land value, p o s i t i v e as well as negative i m p l i c i t p r i c e s are p o s s i b l e . The i m p l i c i t p r i c e f o r month i s $12.32 per acre. This r e s u l t suggests that i f a parcel of land was held f o r an e x t r a month, f a c t o r s which increase the land p r i c e s over time would r a i s e the land value by t h i s amount. On average, a parcel l a r g e r by one acre would decrease the p r i c e per acre by $8.91. Several circumstances can provide t h i s r e s u l t . One i s the p o s s i b i l i t y of c a p i t a l r a t i o n i n g . A s c a r c i t y of c a p i t a l w i l l r e s u l t i n a decrease i n the demand f o r Targe p l o t s of land and w i l l t herefore lower the per acre p r i c e of large p a r c e l s . 48 TABLE 2 Table of I m p l i c i t P r i c e s 225 observations I m p l i c i t p r i c e of Mean Std.Dev. Min. Max. Month 12.318 7.938 -19.703 27.179 Siz e (acres) -8.915 5.913 -23.432 26.224 C u l t . (% of t o t a l ) 3.663 11.384 -27.550 47.588 D i s t . (miles) -19.251 19.198 -68.629 56.216 Pop. - 1.249 4.828 -20.498 3.756 D i s t . Van. (miles) -10.603 9.031 -36.053 6.777 49 A simpler explanation would be that the buyers simply want smaller s i z e p a r c e l s . On average, an e x t r a percentage of acreage c u l t i v a t e d would increase land p r i c e s by $3.66 per acre. This value appears to be q u i t e small i f one considers the purpose which t h i s a t t r i b u t e serves i n the hedonic f u n c t i o n . Once again, the reader i s reminded that the a t t r i b u t e s s i z e and c u l t i v a t i o n are the two aggregate production c h a r a c t e r i s t i c s which h o p e f u l l y provide an adequate r e p r e s e n t a t i o n f o r - t h e spectrum of land a t t r i b u t e s r e l e v a n t to production. For t h i s reason, i t may be s l i g h t l y misleading to i n t e r p r e t these r e s u l t s only i n terms of s i z e and c u l t i v a t i o n . Since the i m p l i c i t p r i c e s of the non-production c h a r a c t e r i s t i c s are the v a r i a b l e s which w i l l be h o p e f u l l y e x p l a i n e d , i t i s only necessary that these two a t t r i b u t e s c o n t r o l f o r the production c h a r a c t e r i s t i c ' s e f f e c t on land value. The two distance v a r i a b l e s , the distance from a nearby centre and the distance from Vancouver, b a s i c a l l y captures the l o c a t i o n a l i n f l u e n c e on land p r i c e s . Average i m p l i c i t p r i c e s f o r the distance from the nearest centre and from Vancouver are -$19.25 and -$10.60 r e s p e c t i v e l y . That i s , a parcel of land a mile f u r t h e r away from a nearby centre would be cheaper by $19.25 per acre, given a l l other a t t r i b u t e s remain constant. S i m i l a r l y , a parcel of land being f u r t h e r away from Vancouver would be cheaper by $10.60 per acre. Both p r i c e s suggest that l o c a t i o n and t r a n s p o r t a t i o n cost to the nearest town i s on average approximately twice as important i n a f f e c t i n g the land values that the more i n d i r e c t e f f e c t of proximity to Vancouver. I t i s only reasonable t h a t since the m a j o r i t y of the s e r v i c e s and inputs are provided by nearby centres r a t h e r than by Vancouver, the a t t r a c t i v e n e s s of being c l o s e r to the nearby centre would i n f l u e n c e land values more. 50 The population of a nearby centre on average has a small negative e f f e c t on the land p r i c e s . A mean value of -$1.25 per acre was obtained. This i s of some concern as population increases would tend to bid up the p r i c e of land i n general. The mean value would the r e f o r e appear to have obtained the wrong s i g n . There i s some evidence however t h a t the d i s t r i b u t i o n of t h i s p r i c e i s somewhat skewed as the minimum p r i c e obtained was -$20.49 while the maximum value was $3.75. With the f i r s t stage r e s u l t s d i s c u s s e d , we can now proceed to evaluate the second stage r e s u l t s . 6.2 Second Stage Results The system of bid f u n c t i o n s f o r the four c h a r a c t e r i s t i c s ( i . e . , month, distance from nearby c e n t r e , distance from Vancouver and the population of the nearest centre) were estimated with the symmetry r e s t r i c t i o n s and the norm a l i z a t i o n s imposed. These bid fu n c t i o n s are provided by equation (25.2) of Chapter IV. For ease of referen c e , equation (25.2), or the b i d p r i c e f u n c t i o n f o r a t t r i b u t e j i s : f£'= I I a,mUw2 + iw2)* + 2yL,,w,h, + . . (29) 3h.~ \ J a i m V 5 W i 5 V V ^ i j w i " j *Z Zb. kw ih|h-- +-2y|d i.w ih.D A l l estimated c o e f f i c i e n t s , other than the a.,-mAs) are arranged.in a matrix format i n Table 3. For in s t a n c e , the '£„ c o e f f i c i e n t corresponding to the cross term between the distance to nearby centre and the p r i c e of feed i n the distance from centre equation has the value of -.0210. Since the a.^.is TABLE 3 BID FUNCTION ESTIMATES ( t - s t a t i s t i c s i n p a r e n t h e s i s ) Equ. 'H M D V 1 -.0022 M (-.27) .0001 -.00005 -.0005 0 -.0109 .0008 .0011 .0038 -.0001 .00004 .0005 (.43) (-.11) (-.27) (12.18) (3.22) (2.65) (.46) (-.79) (.08) (.23) 2 -.0210 -.0003 -.0002 .0063 D (-1.21) (-.97) (-.28) (1.91) .0027 -.0037 .0144 .0005 .00032 -.0071 (8.72) (-3.85) (.67) (1.40) (.35) (-2.07) 3 .00006 -.000006 -.000003 .00003 P (.37) (-3.30) (-.51) (1.41) 0 -.0019 -.0001 (-9.04) (-.57) .000002 .000009 -.00003 (.87) (1.21) (-1.26) 4 -.0002 -.00002 -.000004 .0001 V (-.17) (2.13) (-.22) (2.01) 0 -.0002 (-.15) .00002 .000003 -.0002 (2.24) (.12) (-2.29) a - =• -6.5985 F W (-3.61) a = .8529 (4.01). a p _ = -.2408 ^ (-1.49) a T =• .5285 W H ( 3 . 2 0 ) a = 6.1244 W C (3.46) a = -1.0690 rtC (-4.12) C h a r a c t e r i s t i c s Input P r i c e s "where M = month = d i s t a n c e t o nearby c e n t r e = p o p u l a t i o n D P V = d i s t a n c e t o Vancouver F = f e e d W = wage H = hay C = c a p i t a l 52 are i d e n t i c a l f o r each equation, they are l i s t e d under the other c o e f f i c i e n t s . Once again, due to the p a r t i c u l a r f u n c t i o n a l form used, each of the c o e f f i c i e n t s i n Table 3 does not have a s t r a i g h t f o r w a r d i n t e r p r e t a t i o n . I t i s however, worth noting t h a t a l l the own cross terms f o r the input p r i c e s , the a / m c o e f f i c i e n t s were s i g n i f i c a n t at the 95 percent confidence l e v e l . This s t r o n g l y i m p l i e s t h a t f a c t o r p r i c e s play an important r o l e i n the bid p r i c e determination process and supports our model as these p r i c e s are included as a consequence of the t h e o r e t i c a l model. Since one of the o b j e c t i v e s of t h i s t h e s i s i s to look at the underlying determinants of the b i d f u n c t i o n s f o r our four a t t r i b u t e s , t h i s r e l a t i o n s h i p w i l l be discussed f i r s t . The e f f e c t s of a change i n input p r i c e i on the bid p r i c e of c h a r a c t e r i s t i c j i s provided by equation (25.5) of Chapter IV. These r e l a t i o n s h i p s were c a l c u l a t e d at the sample mean f o r each of the 16 combinations and are presented i n Table 4. As the b i d f u n c t i o n s include dummy v a r i a b l e s to capture the p o s s i b l e s h i f t s i n the b i d f u n c t i o n s due to the land reserve program, two sets of c a l c u l a t i o n s are provided. Column 1 of Table 3 would be the e f f e c t s of input p r i c e changes on the various i m p l i c i t p r i c e s before the program. Since each of the four i m p l i c i t p r i c e s d i f f e r i n magnitude, the second and f o u r t h column of values are the absolute percentage changes i n the corresponding i m p l i c i t p r i c e s . R e s t r i c t i n g our a t t e n t i o n to the pre-land f i g u r e s ( c o l . 1 of Table 4 ) , the f i r s t i n t e r e s t i n g r e s u l t s are the signs of these e f f e c t s . The f i r s t e f f e c t s considered are the ones associated w i t h the month equation. I t should be noted that the i n t e r p r e t a t i o n of changes i n the bid p r i c e f o r month i s not at a l l c l e a r . This i s due to the r o l e TABLE 4 Table of Changes i n Bid Pri c e s due to Changes i n Input P r i c e s Pre ALR Program \ % A | A f t e r ALR Program 3M/3F -4.77 38.80 3.23 26.24 9M/3W .29 2.32 -.05 .38 3M/3H - .11 .86 -.03 .23 9M/3C -1.21 9.79 -.23 1.84 9D/9F 9.86 51.21 -3.21 16.69 9D/9W - .15 .80 .07 . .36 3D/9H - .11 .55 .05 .23 9D/9C 2.92 15.18 -.44* 2.26 3P/3F .36 28.84 -.31 25.02 3P/9W - .03 2.59 -.02 1 .62 9P/9H - .01 1.18 .04 3.41 9P/3C .14 10.92 -.03 2.71 9V/3F - .84 7.89 -1.52 14.33 9V/3W - .07 .69 .01* .12 9V/9H - .01 .12 .01 .11 9V/9C .64 6.08 - .16* 1.53* * S i g n i f i c a n t at the 95% confidence l e v e l . which time plays i n the e s t i m a t i o n . I t s purpose i s to represent the combined e f f e c t s of f a c t o r s which increase over time. This i s analogous to the time trend v a r i a b l e commonly used i n regression a n a l y s i s . Since the four b i d functions were j o i n t l y estimated, i t i s hoped that the i n c l u s i o n o f t h i s equation would c o n t r o l f o r these time r e l a t e d elements. The remaining three bid funct i o n s w i l l b a s i c a l l y capture the e f f e c t s o f urban pressures on a g r i c u l t u r a l production. Since the two distance b i d functions capture the same f a c t o r s , the e f f e c t s of an increase i n the same f a c t o r p r i c e should a f f e c t the two b i d p r i c e s i n the same d i r e c t i o n . This was i n f a c t the case f o r changes i n the wage r a t e , hay p r i c e and the p r i c e o f c a p i t a l . The feed p r i c e however d i f f e r s . The next r e s u l t addressed w i l l be the e f f e c t s of feed and hay p r i c e increases on the two distance b i d p r i c e s . From the r e s u l t s , the e f f e c t of an increase i n feed and hay p r i c e s on the i m p l i c i t p r i c e of the distance from nearby centre a t t r i b u t e are $9.86 and -$.11 r e s p e c t i v e l y . S i m i l a r l y , the same p r i c e increases would a f f e c t the b i d p r i c e f o r the distance to Vancouver a t t r i b u t e by -$.84 and -$.01 r e s p e c t i v e l y . The r e l a t i o n s h i p between feed and hay p r i c e s on the b i d p r i c e o f the two distances have i n t e r e s t i n g i n t e r p r e t a t i o n s with regards to the feed and hay a l l o c a t i o n choices o f the d a i r y farmer. A d a i r y herd i s t y p i c a l l y fed a l f a l f a hay and a feed mix which i s comprised of several s e l e c t e d g r a i n s . P r o t e i n supplements may be added but they are n e g l i g i b l e compared to the other two components. Hay can be purchased but many d a i r y operations i n B.C. grow much of that which they require on t h e i r farms. In c o n t r a s t , feed mixes are g e n e r a l l y purchased from feed o u t l e t s l ocated i n nearby centres. Since the feed supply companies i n each region tend to be q u i t e concentrated, 55 they appear to compete on the c h a r a c t e r i s t i c s o f the feed than on the p r i c e . Feed p r i c e s would not ther e f o r e vary w i t h i n d i f f e r e n t regions. Some d a i r y operations do grow and mix t h e i r own grains but such p r a c t i c e s are not t y p i c a l . I d e a l l y , an optimal r a t i o n f o r a d a i r y herd would comprise both hay and feed. At the usual mean l e v e l s , a certain(degree of s u b s t i t u t i o n i s p o s s i b l e with the optimal mix being determined by r e l a t i v e p r i c e s . An increase i n the p r i c e of purchased feed would lead the farmer to purchase l e s s feed and use more hay. Since hay i s grown on the farm, each farm could reduce i t s cost by l o c a t i n g i n areas f u r t h e r away from populated centres to take advantage of cheaper land to grow more hay. Since the b i d p r i c e f o r distance i s negat i v e , the above d i s c u s s i o n would suggest that the b i d p r i c e would become l e s s negative given an increase i n the feed p r i c e . This was i n f a c t obtained f o r the case o f distance from nearby centre. This r e s u l t , however was not obtained f o r the distance from Vancouver a t t r i b u t e p r i c e which would suggest that a wrong value was obtained f o r t h i s value. For the same feed p r i c e increases i n the bid p r i c e f o r population equation, the r e s u l t s suggest that the bid p r i c e f o r population would increase by $.36 per acre. As suggested i n the preceeding d i s c u s s i o n , a feed p r i c e increase would lead d a i r y producers to grow more hay. An increase i n hay production w i l l g e n e r a l l y r e q u i r e more labour thus i t would be to the farm's advantage to be lo c a t e d c l o s e r to a populated centre to take advantage of a l a r g e r labour market. I t i s i n t e r e s t i n g to note that of a l l the absolute percentage changes of the b i d p r i c e s , the e f f e c t s due to feed increases i s g e n e r a l l y the l a r g e s t f o r a l l four b i d f u n c t i o n s . 56 This r e s u l t appears c o n s i s t e n t as purchased feed c o n s t i t u t e s a large component of the v a r i a b l e cost of a d a i r y operation. For the case of an increase i n hay p r i c e s , the r e s u l t s i n d i c a t e that cost minimization behaviour would tend to favour purchases of p l o t s c l o s e r to nearby centres as w e l l as to Vancouver. The r e s u l t s a l s o suggest that the same forces would favour a decrease i n the population of a nearby centre. For the i n t e r p r e t a t i o n of these r e s u l t s , i t may be i n s t r u c t i v e to d i s t i n g u i s h between the consumption mix of the d a i r y herd. As p r e v i o u s l y s t a t e d , d a i r y herds i n general consumes a feed mix and hay. Hay can be purchased or i t can be grown on the farm. The feeding mix of the herd can therefore be s a i d to be comprised of feed mixes, purchased hay and hay grown on the farm. Given an increase i n the p r i c e of purchased hay, the farmer can s u b s t i t u t e the r a t i o n by the other two components. Since feed p r i c e s do not i n general vary f o r d i f f e r e n t r e g i o n s , cost minimization behaviour would not r e s u l t i n any s p a t i a l preferences as f a r as feed consumption i s concerned. Therefore, the l a s t a l t e r n a t i v e would be to grow more hay on the own operation. As the population of nearby centres i n c r e a s e , the l i k e l i h o o d of urban sprawl pressures from nearby communities would increase. The e f f e c t s of such a c t i v i t i e s would, among o t h e r s , increase the surrounding land values. This would t h e r e f o r e a f f e c t the a v a i l a b i l i t y of farms to purchase lower p r i c e land to grow more hay. Using the same argument, i t would appear that cost minimization behavour would favour lands .whi:ch are l o c a t e d f u r t h e r away from populated centres f o r the above reasons. This would suggest that the two distance e f f e c t s due to hay p r i c e increases would have the wrong s i g n . 57 An increase i n the wage rate was found to have a decreasing e f f e c t on the b i d p r i c e f o r population and an i n c r e a s i n g one f o r the two di s t a n c e s . The distance r e s u l t can be r a t i o n a l i z e d i f one considers the opportunity cost of the farmer. As the wage rate of the farmer i n c r e a s e s , t h i s would imply that the opportunity cost of h i s time f o r off- f a r m a c t i v i t i e s would be higher. As the operator must make c e r t a i n t r i p s to e i t h e r nearby centres or to Vancouver, cost minimization behaviour'would favour the l o c a t i o n of farms c l o s e r to these centres to minimize the t r a v e l l i n g time. Given such a wage i n c r e a s e , i t would t h e r e f o r e not be s u r p r i s i n g to observe an i n c r e a s i n g e f f e c t on the b i d p r i c e f o r di s t a n c e . Extending the above argument, an increase i n the wage rate would al s o r a i s e his value f o r r e c r e a t i o n a l time s i n c e the income of the operator w i l l a l s o be increased. Assuming that a community which has a l a r g e r population w i l l g e n e r a l l y have b e t t e r and more abundant r e c r e a t i o n a l a c t i v i t i e s , there would be a c e r t a i n d e s i r e f o r such a d d i t i o n a l a c t i v i t i e s i f the worker's wage increases. However, since the r e s u l t s suggest that the r e l a t i o n s h i p has an inverse e f f e c t on the d e s i r e f o r more people i n a nearby c e n t r e , t h i s would imply that a wrong sign was obtained f o r t h i s value. As the r e n t a l p r i c e o f c a p i t a l i n c r e a s e s , the cost minimization model would suggest t h a t land purchase d e c i s i o n s would favour purchases of land f u r t h e r away from both the populated nearby centre as we l l as from Vancouver. This p r i c e increase would a l s o favour a l a r g e r population o f the nearby centre. The c a p i t a l term used i n the model includes a measure f o r machinery and equipment. As the p r i c e of c a p i t a i n c r e a s e s , forage a c t i v i t i e s would become more expensive since a c e r t a i n amount of machinery i s required f o r hay production. Given that the farmer would have a land 58 purchasing d e c i s i o n to make at the time of an increase i n the r e n t a l p r i c e of c a p i t a l , cost minimization behaviour would lead the choice to lower p r i c e lands to compensate f o r the increase i n r e n t a l p r i c e . As lands which are f u r t h e r away from populated centres are g e n e r a l l y cheaper, the above r e s u l t appears to be c o n s i s t e n t . One explanation f o r the population r e s u l t obtained would be based on the a v a i l a b i l i t y o f c a p i t a l . As a populated centre increases i n p o p u l a t i o n , the a v a i l a b i l i t y of c a p i t a l i n general would perhaps be relaxed since more lending agents would be i n the market. So the de s i r e to have access to a more competitive c r e d i t market would r e s u l t i n the farmers to favour a l o c a t i o n which would be c l o s e r to a populated centre with more people. Although the above r e s u l t s are not unanimously c o n s i s t e n t with the expected i n d u s t r y behaviour, there does e x i s t a la r g e number of r e s u l t s which are c o n s i s t e n t . As the previous r e s u l t d i s c u s s i o n can als o be in t e r e p r e t e d as being a rought form o f model v a l i d a t i o n , there would appear to be some support f o r the proposed model. With the r e s u l t s evaluated, we can now proceed to evaluate the e f f e c t s which the land reserve program had on the model i n general and the bid funct i o n s i n p a r t i c u l a r . A l i k e l i h o o d r a t i o t e s t was used to t e s t f o r any e f f e c t s the land reserve program had on the i n d u s t r y . A dummy v a r i a b l e was included i n the model to c o n t r o l f o r such an e f f e c t . The r e s t r i c t e d l i k e l i h o o d f u n c t i o n value was obtained by esti m a t i n g the system o f equations without these v a r i a b l e s and i s equivalent to r e s t r i c t i n g a l l dummy v a r i a b l e c o e f f i c i e n t s to 0. As there are 4 binary v a r i a b l e s i n each equation, there w i l l be 16 degrees of freedom. The u n r e s t r i c t e d log l i k e l i h o o d 59 f u n c t i o n value was -2569.15 while f o r the r e s t r i c t e d case, i t i s -2589.0. Minus twice the d i f f e r e n c e of the two values i s 33.0. The Chi-square value at the 99.5% confidence l e v e l f o r 16 degrees of freedom i s 34.27 and at the 99% l e v e l , i t i s 32.0. I t appears that the program had a marginal e f f e c t on the ind u s t r y since the n u l l hypothesis i s re j e c t e d at the 99.5% l e v e l and cannot be re j e c t e d at the 99% l e v e l . This should not r e a l l y be t h a t s u r p r i s i n g i f the parameter estimates are i n t e r p r e t e d i n the f o l l o w i n g context. In the course of est i m a t i n g the bi d f u n c t i o n s , we have a l s o i d e n t i f i e d the cost f u n c t i o n parameters since both can be expressed as functio n s of the same parameters. An a l t e r n a t i v e i n t e r p r e t a t i o n of the estimates i s that we have al s o i d e n t i f i e d a cost f u n c t i o n . In t h i s i n t e r p r e t a t i o n , a t e s t f o r the j o i n t s i g n i f i c a n c e of the land reserve program i s a l s o a t e s t f o r the s i g n i f i c a n c e o f t h i s program on the cost s t r u c t u r e of the f i r m . (This i n t e r p r e t a t i o n has i n t e r e s t i n g i m p l i c a t i o n s and w i l l be discussed i n s e c t i o n 6.3). In terms of the t o t a l cost of a da i r y o p e r a t i o n , the r e n t a l p r i c e of land i s q u i t e small i n r e l a t i o n to other f a c t o r s such as feed cost and wage payments. Although the j o i n t s i g n i f i c a n c e i s marginal, t h i s does not r u l e out i n d i v i d u a l s i g n i f i c a n c e . By c o n s t r u c t i o n , the dummy v a r i a b l e s were introduced such that each input p r i c e has a d i f f e r e n t dummy v a r i a b l e f o r each b i d f u n c t i o n . That i s , there are 16 d i s t i n c t binary v a r i a b l e s . From equation (25.5), i t can be seen that the t e s t f o r the s i g n i f i c a n c e of d.. would t e s t f o r the program's e f f e c t of the i input p r i c e on the j bid f u n c t i o n . Only three of the si x t e e n dummy v a r i a b l e s were s i g n i f i c a n t . The r e s u l t s that were a f f e c t e d were the changes i n the two distance functions 60 due to changes i n the p r i c e of c a p i t a l and the e f f e c t of a wage rate increase on the b i d p r i c e f o r the distance from Vancouver. S t a r t i n g with the changes i n the two b i d p r i c e s f o r distance and how they are a f f e c t e d by increases i n the p r i c e of c a p i t a l , we can see that a f t e r the program, cost minimization behaviour would favour lands which are c l o s e r to the nearby centre as w e l l as to Vancouver. Re c a l l t h a t before the program, the r e s u l t s suggest that increases i n c a p i t a l cost would favour l o c a t i o n s f u r t h e r away to take advantage of lower p r i c e land f o r forage a c t i v i t i e s . I m p l i c i t i n t h i s r e s u l t i s the assumption t h a t as a parcel of land i s f a r t h e r away from a c e n t r e , the p r i c e decreases. This would be p r i m a r i l y due to the urban pressures of the populated centres and i s the basis f o r the c o n c e n t r i c c i r c l e theory of land p r i c e s and development. However, i f c e r t a i n p a r c e l s are r e s t r i c t e d f o r a g r i c u l t u r a l use by law, then the p r i c e of land c l o s e r to the centre may not be higher. So given an increase i n the p r i c e of c a p i t a l and the subsequent increase i n the cost of forage a c t i v i t i e s , the farm may reduce i t s t o t a l cost by being c l o s e r to a populated centre. The next bid p r i c e which was s i g n i f i c a n t l y i n f l u e n c e d was the e f f e c t of wage increases on the b i d p r i c e from Vancouver. From a d e s i r e to be c l o s e r to Vancouver before the program, the r e s u l t suggests that a f t e r the program, the land purchase d e c i s i o n would favour parcels f u r t h e r away from Vancouver. This r e s u l t appears to have the wrong sign as whether the program was i n e f f e c t or not, farmers would cost minimize by l o c a t i n g c l o s e r to populated centres to reduce t h e i r opportunity cost f o r t r a v e l l i n g f o r instance. As we have estimated a system of b i d or demand f u n c t i o n s , i t may be i n s t r u c t i v e to c a l c u l a t e the p r i c e f l e x i b i l i t i e s f o r each b i d f u n c t i o n s . The own p r i c e f l e x i b i l i t i e s were c a l c u l a t e d f o r the: four b i d functions and are defined i n the present study as: - 9P. h. J.r h. , , f . ; i = 1 • _J_ - L J L • _^ 30 3h. P. 3h^ 9C/9h, J J J <J 2 where 9 C i s expression (25.4) and 9C_ i s (25.2). P r i c e f l e x i b i l i t y 9h,2 9h, J J i s a measure of the percentage e f f e c t on the p r i c e due to a one percent increase i n the q u a n t i t y . In the present study of p r i c e dependent b i d f u n c t i o n s , t h i s measure i s more appropriate than the conventional e l a s t i c i t y measure as f . . can a l s o be i n t e r p r e t e d as being the responsiveness o f J J changes i n an endogenous v a r i a b l e due to a change i n an exogenous v a r i a b l e . Although i t i s known that only under very r e s t r i c t i v e c o n d i t i o n s that the inverse o f the p r i c e f l e x i b i l i t y i s equal to the demand e l a s t i c i t y (Houck 1965), a f l e x i b l e p r i c e i s g e n e r a l l y c o n s i s t e n t with an i n e l a s t i c demand and an i n f l e x i b l e p r i c e i s c o n s i s t e n t with an e l a s t i c demand (Tomek and Robinson 1972). Own f l e x i b i l i t i e s are expected to be negative and i s unaffected by negative i m p l i c i t p r i c e s . The p r i c e f l e x i b i l i t i e s f o r the four a t t r i b u t e s before and a f t e r the land reserve program i s presented i n Table 5. Only two of the dummy v a r i a b l e s were s i g n i f i c a n t a f t e r the land reserve program so only two a f t e r program f l e x i b i l i t i e s were c a l c u l a t e d . 62 •TABLE 5 Own P r i c e F l e x i b i l i t i e s Before Program A f t e r Program Month 1.70 Distance to Nearby Centre - .31 -12.02 Population 5.50 Distance to Vancouver .68 - .08 The r e s u l t s are somewhat discouraging as only one of the four f l e x i b i l i t i e s obtained the expected negative s i g n . For the case of the distance t o nearby, c e n t r e , the r e s u l t suggests that the demand f o r t h i s d i s t a nce i s r e l a t i v e l y e l a s t i c as i t s value i s l e s s than one i n absolute terms. That i s , the market wide bid p r i c e of t h i s a t t r i b u t e i s i n s e n s i t i v e to changes i n q u a n t i t y . I t i s i n t e r e s t i n g to note however that a f t e r the program, there i s a s i g n i f i c a n t change i n the demand as i t became i n e l a s t i c . One explanation can perhaps be based on the p o s s i b i l i t y that the r e s t r i c t i v e land measure may be r e l a x e d . I f urban land p r i c e s f o r a p a r t i c u l a r community reaches a high l e v e l , t h i s would i n v a r i a b l y r e s u l t i n pressure on the p o l i c y makers to take land out of the reserve. Since such an a c t i v i t y would undoubtedly r e s u l t i n huge p r o f i t s f o r the land owner., t h i s would provide a strong i n c e n t i v e f o r farmers to purchase land c l o s e r to nearby populated centres. This can perhaps a l s o e x p l a i n the s i g n i f i c a n t change i n the f l e x i b i l i t y f o r the distance to Vancouver bid f u n c t i o n . 63 6.3 An A l t e r n a t i v e I n t e r p r e t a t i o n of the Results As p r e v i o u s l y mentioned, a l l cost f u n c t i o n parameters are a l s o contained w i t h i n the system of b i d f u n c t i o n s . The cost f u n c t i o n can therefore be recoverable from these b i d f u n c t i o n estimates and can be i n t e r p r e t e d i n some sense as being analogous to the conventional case o f es t i m a t i n g a system of derived demand functions to obtain cost f u n c t i o n parameters. This r e s u l t suggests that j u s t as the input choices of a f i r m can i d e n t i f y i t s cost c o n d i t i o n s , input choices of the input a t t r i b u t e s can al s o c h a r a c t e r i z e such c o n d i t i o n s . This i n t e r p r e t a t i o n o f the parameters, i f v a l i d , has f a r reaching e m p i r i c a l i m p l i c a t i o n s . The most important o f which i s the p o s s i b i l i t y of i d e n t i f y i n g a cost f u n c t i o n without q u a n t i t y data. C h a r a c t e r i s t i c ; composition of i n p u t s , input p r i c e s and the output l e v e l are a l l t hat i s required. This i s an important concern as i t i s we l l - r e c o g n i z e d among a p p l i e d economists t h a t although the modern f l e x i b l e form d u a l i t y approaches to production are appealing from a t h e o r e t i c a l and e m p i r i c a l s t a n d p o i n t , t h i s approach i s extremely demanding i n i t s data requirements. In the context o f the cost f u n c t i o n s , both p r i c e and q u a n t i t y data are required to obtain cost f u n c t i o n parameters e i t h e r through derived demand functions (Lopez 1980), cost shares (Burgess 1979, Ray 1982) or by c o n s t r u c t i n g the t o t a l cost and es t i m a t i n g the s i n g l e equation cost f u n c t i o n . Though p r i c e data are u s u a l l y a v a i l a b l e , q u a n t i t y data i s g e n e r a l l y much more d i f f i c u l t to o b t a i n . I f input a t t r i b u t e s can be used instead of qu a n t i t y data i n cost f u n c t i o n e s t i m a t i o n , the proposed method would c e r t a i n l y be of i n t e r e s t to anyone under the undesirable s i t u a t i o n of r e q u i r i n g cost f u n c t i o n estimates without q u a n t i t y data. The cost of such an approach would of course r e s t on the approximating a b i l i t y of the hedonic f u n c t i o n which l i n k s the input p r i c e and the a t t r i b u t e s . The idea of recovering the cost f u n c t i o n from the system of b i d functions i s u n t u i t i v e l y f e a s i b l e . J u s t as input choices of the f i r m w i l l reveal i t s cost minimizing behaviour, a t t r i b u t e choices can i n t u i t i v e l y be used to i d e n t i f y the cost c o n d i t i o n s since each input i s a composite of these a t t r i b u t e s . Whether we deal i n the parent u n i t or i n the a t t r i b u t e s which comprise i t , i t would seem reasonable to suggest that l o o k i n g at production from an a t t r i b u t e standpoint would provide us with s i m i l a r i n f o r m a t i o n . The idea of l o o k i n g at production from an a t t r i b u t e p o i n t of view " i s , of course, not new. I t can however be seen as being an extension of the present e m p i r i c a l work. For i n s t a n c e , i n a g r i c u l t u r a l production s t u d i e s , i t i s w e l l recognized that operator labour and h i r e d labour are d i s t i n c t inputs with d i f f e r e n t behaviours i n the production process. There have been attempts to f u r t h e r disaggregate the labour component i n t o educated and uneducated labour s i n c e educated labour i s hypothesized to be more productive. This disaggregation of the inputs can be i n t e r p r e t e d as being an attempt at d e a l i n g with the various a t t r i b u t e s which comprise the i n p u t s . I t would appear that the need f o r disaggregating the inputs i s a r e s u l t of the r e c o g n i t i o n that each a t t r i b u t e has d i f f e r e n t p r o p e r t i e s and t h e r e f o r e would a f f e c t production i n d i f f e r e n t ways. Disaggregating the inputs i n t o subsets would address t h i s i s s u e , e m p i r i c a l l y speaking, but i s only a p p l i c a b l e when these subsets are f r e e l y traded and therefore r e s u l t i n g i n observable e x p l i c i t market p r i c e s . The d i f f e r e n c e of the proposed approach i s that the land a t t r i b u t e s are not f r e e l y traded e x p l i c i t l y . 65 The second i m p l i c a t i o n which a r i s e s as a r e s u l t of the f i r s t i s that i f the cost f u n c t i o n can be recovered, then we have a l s o i d e n t i f i e d the f a c t o r demand parameters. A simple way to measure the r e l i a b i l i t y of t h i s r e s u l t would be to derive the e l a s t i c i t i e s of these demand fu n c t i o n s and see i f they are c o n s i s t e n t with the indu s t r y which i s being modelled. Since q u a n t i t y data i s not a v a i l a b l e , exact e l a s t i c i t y values cannot be obtained. However, the signs of these e l a s t i c i t i e s can be i n f e r r e d . Applying Shephard's Lemma and r e c a l l i n g (25.3), we know that ^ 1 = = _|*a. (iw? + iw 2) 1 s+w.w: -s h. ( 2 5 ' 3 ) 9w m 3wm 9w i 5 amK2 i m1 l m . j where i s the q u a n t i t y demanded of input i . Since x i , wi, h,> 0 Vi , j 8 x i ' a 2 r s 9 n e im = s9" -ST = S 9 n 9w3wT ( 3 1 ) m m i where e. is the cross e last ic i ty between input i and m. I t follows that: sgnOe im] =-sgn[ ia i n i (^ + ^ l ^ m f f l J (32) or sgn[e i m] = -sgn[a i m] . That i s , the sign o f .the cross e l a s t i c i t i e s can be determined. These signs can therefore be used to determine whether any two inputs are s u b s t i t u t e s or complements. From Table 3, the f o l l o w i n g r e l a t i o n s h i p s are suggested by the model: 66 Inputs TABLE 6 Table of Inferred Input R e l a t i o n s h i p s -sgn Ca- ) Input R e l a t i o n s h i p s Feed and Labour Feed and Hay Feed and C a p i t a l Labour and Hay Labour and C a p i t a l Hay and C a p i t a l >0 <0 >0 <0 <0 >0 S u b s t i t u t e Complement S u b s t i t u t e Complement Complement S u b s t i t u t e From the s t r u c t u r e of the d a i r y i n d u s t r y , i t i s known that given a r i s e i n the feed p r i c e , the d a i r y producer would use l e s s feed and s u b s t i t u t e the r a t i o n w i t h more hay. Hay can e i t h e r be purchased or grown on the farm. I t i s l i k e l y t hat the increase i n feed p r i c e would r e s u l t i n the operation producing more hay and t h e r e f o r e would r e q u i r e more labour. Feed and labour i n t h i s case would therefore be s u b s t i t u t e s . This r e s u l t was obtained. The feed and hay a l l o c a t i o n choices have been discussed p r e v i o u s l y and have been l a b e l l e d as being s u b s t i t u t e s . However, i n a c t u a l i t y , these two components are only s u b s t i t u t e s w i t h i n a s p e c i f i c r e g i o n . The r a t i o n f o r d a i r y c a t t l e must contain a minimum l e v e l of f i b r e f o r a proper d i e t . That i s , the herd cannot e x i s t i n a healthy s t a t e i f i t was s o l e l y fed by hay. Since f i b r e can only be a t t a i n e d through the feed or g r a i n component, a f t e r a c r i t i c a l l e v e l of feed content i s reached, the hay and feed components can be i n t e r p r e t e d as being complements. Since the c a p i t a l term includes machinery and equipment, as 67. the p r i c e of c a p i t a l i n c r e a s e s , forage a c t i v i t i e s such as hay production would become r e l a t i v e l y more expensive. This would lead the farmer to e i t h e r purchase h i s hay or s u b s t i t u t e h i s r a t i o n w i t h feed. For the case of purchasing more fee d , c a p i t a l and feed can be seen as s u b s t i t u t e s . Labour and hay were estimated to be complements. This r e s u l t appears to have the wrong sign as labour and hay should be s u b s t i t u t e s . As hay p r i c e s i n c r e a s e , the farmer would attempt to obtain more hay from h i s farm so as to minimize the purchase hay. Since more forage a c t i v i t i e s are r e q u i r e d , t h i s would lead the operator to re q u i r e more labour. Labour and hay would t h e r e f o r e be s u b s t i t u t e s . The labour wage rate used i n the present study i s the farm wage r a t e with board. The r e p r e s e n t a t i v e farmer can ther e f o r e be c l a s s i f i e d as s k i l l e d labour. I t i s g e n e r a l l y accepted that s k i l l e d labour are complements with c a p i t a l as s k i l l e d labour are required to operate the machinery. Complementarity was obtained i n our r e s u l t . Hay and c a p i t a l can be seen as s u b s t i t u t e s since as hay p r i c e s i n c r e a s e , the operation would produce more of i t s own hay which would r e q u i r e more extensive use of the c a p i t a l i n the farm. This r e l a t i o n s h i p was r e f l e c t e d i n our r e s u l t s . Of the s i x r e l a t i o n s h i p s a r r i v e d a t , only one has a su s p i c i o u s s i g n . I t would appear that t h i s approach of loo k i n g at the r e l a t i o n s h i p s among inputs have some support i n that r e a l i s t i c r e s u l t s were obtained i n f i v e of the s i x cases. 68 CHAPTER VII CONCLUSION This t h e s i s , through the i n t e g r a t i o n of hedonic and production theory, provided a method of d e r i v i n g a system of bid fu n c t i o n s f o r land a t t r i b u t e s f o r the B.C. d a i r y i n d u s t r y . The system of b i d fu n c t i o n s were derived from cost minimizing behaviour of the i n d u s t r y . Estimates of the .system were obtained and evaluated with respect to the r e l a t i o n s h i p between bid p r i c e s of the various a t t r i b u t e s and how they are a f f e c t e d by non-land f a c t o r p r i c e changes. The r e s u l t s suggest that feed p r i c e changes on average, played a dominant r o l e i n the bid p r i c e determination f o r land a t t r i b u t e s . A l s o , the feed and hay a l l o c a t i o n choices appear to be very much r e l a t e d to the land a t t r i b u t e choices of the farm. I t was a l s o shown th a t the a g r i c u l t u r a l land reserve program had only a m a r g i n a l l y s i g n i f i c a n t e f f e c t on the b i d p r i c e s . Perhaps the most i n t e r e s t i n g r e s u l t s a r i s i n g from t h i s study are the various a p p l i e d i m p l i c a t i o n s of the proposed model. As i t turns out, i n the context of production, endogenous as well as exogenous i m p l i c i t p r i c e s are important c o n s i d e r a t i o n s i n the systematic d e r i v a t i o n of b i d fu n c t i o n s from the cost f u n c t i o n . I t was suggested that l i n e a r homogeneity of the hedonic f u n c t i o n i s a necessary c o n d i t i o n f o r exogenous i m p l i c i t p r i c e s . This r e s u l t can ther e f o r e e i t h e r be the 69 basis of a s t a t i s t i c a l t e s t to determine whether c e r t a i n i m p l i c i t p r i c e s are exogenous or endogenous or i t can be used to impose economic s t r u c t u r e i n t o the s p e c i f i c a t i o n of the hedonic f u n c t i o n . This i s an important r e s u l t since i n the e x i s t i n g l i t e r a t u r e , no economic s t r u c t u r e i s a v a i l a b l e to determine the f u n c t i o n a l form of the hedonic f u n c t i o n . Researchers have t r a d i t i o n a l l y used s t a t i s t i c a l c r i t e r i a s to determine the appropriate s p e c i f i c a t i o n (Halvorsen). Imposition o f a p r i o r i s t r u c t u r e derived from theory would undoubtly strengthen the model. Another r e s u l t which has f a r reaching i m p l i c a t i o n s i s the p o s s i b i l i t y of i d e n t i f y i n g cost f u n c t i o n parameters from the system of b i d f u n c t i o n s . I f such a procedure i s v a l i d , then the hedonics approach i n production can be seen as another method of o b t a i n i n g production parameters. A l s o , i d e n t i f i c a t i o n of cost f u n c t i o n parameters would necessary a l s o i d e n t i f y the f a c t o r demand f u n c t i o n parameters. Since t h i s would suggest that cost f u n c t i o n parameters can be estimated without q u a n t i t y data, the e m p i r i c a l i m p l i c a t i o n s of t h i s r e s u l t would appear to be of some importance. Although t h i s r e s u l t i s suggested, i t i s suspected t h a t more t h e o r e t i c a l developments are r e q u i r e d before such a procedure can be o p e r a t i o n a l i z e d . Although t h i s t h e s i s d i d provide several i n t e r e s t i n g r e s u l t s and t h a t i t met the o b j e c t i v e s o r i g i n a l l y set out, i t i s not without i t s shortcomings. The most important and p o t e n t i a l l y damaging of which are the i m p l i c i t and unknown assumptions placed on the production f u n c t i o n when both the cost and hedonic f u n c t i o n s are p r e s p e c i f i e d . I t i s not c l e a r t h a t i n such a circumstance, the usual d u a l i t y r e s u l t s with respect to the underlying production f u n c t i o n s t i l l apply. The answer 70 can be obtained by d e r i v i n g a set of l o c a l d u a l i t y r e s u l t s as i n Blackorby and Diewert (1979) f o r the case of nonlinear budget c o n s t r a i n t s . I t i s suspected that such an e x e r c i s e would be a t h e s i s i n i t s e l f . Acknowledging that t h i s i s indeed a serious concern, t h i s problem cannot be f e a s i b l y addressed i n the present study. Another downfall i s the r e l a t i v e complexity of the f u n c t i o n a l form used. This r e s u l t s i n several problems with the dominant one being t h a t the comparative s t a t i c expressions are extremely complicated. Their f u n c t i o n a l forms does not allow f o r an easy c a l c u l a t i o n f o r the s t a t i s t i c a l p r o p e r t i e s of these r e s u l t s . I t was therefore not p o s s i b l e to s t a t i s t i c a l l y t e s t f o r the s i g n i f i c a n c e of some of the e f f e c t s . This i s an important concern as the e f f e c t s can, i n some cases, be neutral or i s not s i g n i f i c a n t l y d i f f e r e n t from zero. Despite the previous mentioned shortcomings, the approach provided does address an important area of economics, the c h a r a c t e r i s t i c choices o f f a c t o r s of production. Through the proposed model, i t can be seen that such an a n a l y s i s has the p o t e n t i a l of supplying more d e t a i l with respect to the behaviour of the f i r m . 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