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UBC Theses and Dissertations

Factor demands and output supply by the extractive firm : theory and estimation Lasserre, Pierre 1981

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FACTOR DEMANDS AND OUTPUT SUPPLY BY THE EXTRACTIVE FIRM: THEORY AND ESTIMATION by PIERRE LASSERRE M.A., C a r l e t o n U n i v e r s i t y , 1976 Diplome de l ' E c o l e Superieure de Commerce de Par A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept t h i s t h e s i s as conforming to th THE UNIVERSITY OF BRITISH COLUMBIA requ i r e d standard November 1981 P i e r r e L a s s e r r e , 1981 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 advanced degree a t 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 , 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 and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department 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 . Department o f E r ^ H o ^ C ^  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 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 DE-6 (2/79) i ABSTRACT T h i s d i s s e r t a t i o n d e a l s w i t h t h e o r e t i c a l and e m p i r i c a l a s p e c t s of f a c t o r demand and o u t p u t s u p p l y d e c i s i o n s o f f i r m s . I n the t h e o r e t i c a l p a r t of the t h e s i s , some major e x i s t i n g t h e o r i e s o f i n v e s t m e n t a r e d i s c u s s e d and t h e i r f o r m u l a t i o n i s e x t e n d e d t o the case o f f i r m s w h i c h e x t r a c t an e x h a u s t i b l e r e s o u r c e . Those t h e o r i e s a r e t h e n i n c o r p o r a t e d i n t o a model wh i c h e x p l o i t s c o m p l e m e n t a r i t i e s between some of them and can r e f l e c t some well-known h y p o t h e s e s , such as the p u t t y - c l a y h y p o t h e s i s , as s p e c i a l c a s e s . T h i s model r e l i e s on a g e n e r a l n o t i o n of i r r e v e r s i b i l i t y : a d e c i s i o n i s d e f i n e d as i r r e v e r s i b l e i f i t i n t r o d u c e s a new c o n s t r a i n t t o a f i r m . T h i s c o n s t r a i n t may be a non n e g a t i v i t y c o n s t r a i n t , b u t may a l s o mean the ap-pearance of c o s t s o f a d j u s t m e n t s . Such an approach i m p l i e s a d i s t i n c t i o n between ex ante phases and ex p o s t phases i n t h e l i f e of f i r m s , t h o s e phases b e i n g s e p a r a t e d by the i r r e v e r s i b l e d e c i s i o n s . Two e m p i r i c a l a p p l i c a t i o n s a r e p r e s e n t e d . The f i r s t one c o r -responds t o the ex a n t e phase of the t h e o r e t i c a l model, and d e a l s w i t h the c a p a c i t y s e l e c t i o n d e c i s i o n of some N o r t h - A m e r i c a n o p e n - p i t m e t a l mines. A c c o r d i n g t o t h e e v i d e n c e , t h i s d e c i s i o n t a k e s a c c o u n t of economic p a r a m e t e r s , such as e x p e c t e d p r i c e s , as w e l l as g e o l o g i c a l and t e c h n o l -o g i c a l p a r a m e t e r s . The second e m p i r i c a l a p p l i c a t i o n c o r r e s p o n d t o the ex p o s t phase of the t h e o r e t i c a l model, and d e a l s w i t h the s h o r t - r u n p r o d u c t i o n d e c i s i o n s o f some mines. B o t h e m p i r i c a l s t u d i e s p r o v i d e sup-p o r t f o r the p u t t y - c l a y h y p o t h e s i s . TABLE OF CONTENTS i i Page ABSTRACT i TABLE OF CONTENTS i i LIST OF TABLES i v LIST OF FIGURES v i ACKNOWLEDGEMENT v i i GENERAL INTRODUCTION 1 CHAPTER ONE: BRIEF OVERVIEW OF INVESTMENT THEORY 3 1 .1 I n t r o d u c t i o n 3 1.2 The n e o c l a s s i c a l model w i t h m a l l e a b l e and r e s a l a b l e f a c t o r s 4 1.3 Models w i t h c o s t s of adjustment 8 1.4 I r r e v e r s i b l e investment 11 1.5 P u t t y - c l a y technology and v i n t a g e models of investment 16 1.6 P u t t y - s e m i - p u t t y technology 19 1.7 Remarks 22 1.8 Summary 24 Notes to chapter 1 27 CHAPTER TWO : REVERSIBLE INVESTMENT WITH A RESOURCE CONSTRAINT . . 29 2 . 1 I n t r o d u c t i o n 29 2 .2 The n e o c l a s s i c a l model w i t h m a l l e a b l e and r e s a l a b l e f a c t o r s , when the f i r m e x t r a c t s a homogeneous e x h a u s t i b l e resource 30 2 . 3 The c o s t - o f - a d j u s t m e n t model w i t h r e s a l a b l e f a c t o r s , when the f i r m e x t r a c t s a homogeneous e x h a u s t i b l e resource 42 2.4 The case of heterogeneous resources 50 2 .5 A l t e r n a t i v e e s t i m a t i o n procedures 59 2.6 Summary 72 Notes to chapter 2 -JU i i i CHAPTER THREE: IRREVERSIBLE INVESTMENT WITH A RESOURCE CONSTRAINT . 79 3 . 1 I n t r o d u c t i o n 79 3 . 2 Genera l c a s e : F a c t o r demand and e x t r a c t i o n p o l i c y when stock adjustments are c o s t l e s s and non negat i ve 80 3 . 3 S p e c i a l c a s e : no v a r i a b l e f a c t o r s , no s tock i n c r e a s e s d u r i n g e x t r a c t i o n 86 3 .4 C o n c l u s i o n 90 Notes to chapter 3 . 91 CHAPTER FOUR: A GENERALIZED THEORY OF FACTOR DEMANDS 94 4 . 1 I n t r o d u c t i o n 94 4 . 2 Model and b a s i c premises 95 4 . 3 Important s p e c i a l cases 98 4 .4 C h a r a c t e r i z a t i o n of the DNR f u n c t i o n 105 4 . 5 H o t e l l i n g ' s theorem r e v i s i t e d 107 4 .6 Ex ante demands: a p r i m a l approach 109 4 .7 Summary 112 Notes to chapter 4 114 CHAPTER F IVE: EX ANTE DEMAND FOR CAPACITY: THE CASE OF SOME NORTH-AMERICAN OPEN-PIT MINES 116 5 . 1 I n t r o d u c t i o n 116 5 .2 Data and sample 117 5 .3 Model , e s t i m a t i o n , and r e s u l t s 134 5 . 4 Summary and c o n c l u s i o n 157 Notes to chapter 5 159 CHAPTER SIX: THE SHORT-RUN BEHAVIOUR OF SOME NORTH-AMERICAN OPEN-PIT METAL MINES 163 6 . 1 I n t r o d u c t i o n 163 6 .2 Data and sample 164 6 . 3 The model 168 6.4 E s t i m a t i o n and r e s u l t s 173 6 . 5 C o n c l u s i o n 186 Notes to chapter 6 188 GENERAL CONCLUSION 189 BIBLIOGRAPHY 191 ANNEX ONE: P roo fs (chapter 4) 195 ANNEX TWO: Data 214 ANNEX THREE: V a r i a b l e l i s t 236 i v LIST OF TABLES page Table I: Ore grades f o r some o p e n - p i t mines 7g Table I I : Parameter r e s t r i c t i o n s , g e n e r a l i z e d q u a d r a t i c f u n c t i o n 137 Table I I I : Capac i t y demand e s t i m a t i o n . Var ious r e s t r i c t i o n s of equat ion (15) 146 Table IV: Capac i ty demand e s t i m a t i o n s - l i n e a r forms 150 Table V: Capac i t y demand e l a s t i c i t i e s at mean va lues of the observa t ions 155 Table V I : Forms i n v o l v i n g no t e c h n i c a l change 176 Table V I I : Forms i n v o l v i n g " c a p i t a l - a u g m e n t i n g " t e c h n i c a l change 177 Table V I I I : Forms i n v o l v i n g " l a b o u r - a u g m e n t i n g " t e c h n i c a l change 178 Table IX : Forms i n v o l v i n g j o i n t t e c h n i c a l change 179 Table X : I l l u s t r a t i o n of c o l i n e a r i t y problems -^3 Table X I : S e l e c t e d c o r r e l a t i o n c o e f f i c i e n t s ( l o g - l i n e a r forms) 185 Table X I I : Observat ion l i s t and index ; Composit ion of FINAL1, FINAL2, FINAL3, and FINAL4 226 V Table X I I I : Sources of meta l and f a c t o r p r i c e s 228 Table XIV: L o c a t i o n and metal codes 229 Table XV: Data sources and source codes 230 Table XV I : L i s t of mines, o p e r a t i o n p e r i o d s , and data sources ( s h o r t - r u n study) 233 Table X V I I : A l t e r n a t i v e samples and i n f o r m a t i o n used 234 Table X V I I I : Adjustment c o e f f i c i e n t s used to d e r i v e QAMi from p u b l i s h e d output data 235 v i LIST OF FIGURES page F i g u r e 1 : P r i c e and f a c t o r s tock p a t h s . Jo rgenson 's model 7 F i g u r e 2 : Market p r i c e , shadow p r i c e , and investment p a t h s . Model w i t h i r r e v e r s i b l e investment and model w i t h r e v e r s i b l e investment 15 F i g u r e 3 : Campbe l l ' s model (1980) : u n d e r l y i n g technology . . . . 93 F i g u r e 4 : Ex ante and ex post i soquants f o r a l t e r n a t i v e assumptions on the technology 101 F i g u r e 5 : Ex ante and ex post i soquants f o r the g e n e r a l i z e d c o s t - o f - a d j u s t m e n t model : a d i s c r e t e - t i m e i n t e r p r e t a t i o n 102 F i g u r e 6 : C u t - o f f grades f o r some copper mines and some molybdenum mines 160 ACKNOWLEDGEMENT The f i n a n c i a l and i n t e l l e c t u a l s u p p o r t of t h e C a n a d i a n D e p a r t -ment of E n e r g y , Mines and Resou r c e s i s g r a t e f u l l y acknowledged. I have b e n e f i t t e d from the comments of John L i v e r n o i s , Mukesh Eswaran, P a u l B r a d l e y , T r a c y L e w i s , and F r a n k C l a r k e ; o f c o u r s e , the u s u a l w a i v e r a p p l i e s . I e x t e n d s p e c i a l thanks t o John H e l l i w e l l , E r n s t B e r n d t , and W i l l i a m Schworn whose a d v i c e s , a v a i l a b i l i t y , and tremendous competence have immensely h e l p e d me i n w r i t i n g t h i s d i s s e r t a t i o n , and t o Mrs. Lecomte, who d i d much more th a n t y p i n g i t . 1 GENERAL INTRODUCTION T h i s d i s s e r t a t i o n i s d e v o t e d t o t h e o r e t i c a l and e m p i r i c a l a s p e c t s of f a c t o r demand and o u t p u t s u p p l y d e c i s i o n s o f f i r m s . So f a r , most r e s e a r c h i n t h i s a r e a has d e a l t w i t h c o n v e n t i o n a l f i r m s , t h a t i s t o say w i t h f i r m s w h i c h do not f a c e any e x h a u s t i b l e - r e s o u r c e c o n s t r a i n t . Throughout t h i s t h e s i s , on the c o n t r a r y , the emphasis i s put on f i r m s w h i c h a r e r e f e r r e d t o as e x t r a c t i v e f i r m s b ecause t h e y f a c e such c o n s t r a i n t s . However, the t h e o r e t i c a l r e s u l t s o b t a i n e d can be s p e c i a l i z e d t o t h e case of the conven-t i o n a l f i r m . F i r m s a r e assumed t o use two t y p e s of f a c t o r s . S t o c k - f a c t o r s have the d i m e n s i o n o f s t o c k s , b u t may be p e r f e c t l y m a l l e a b l e as w e l l as f i x e d o r q u a s i - f i x e d . V a r i a b l e f a c t o r s have t h e d i m e n s i o n of f l o w s and are assumed t o be p e r f e c t l y and i n s t a n t a n e o u s l y a d j u s t a b l e . The main body o f t h e d i s s e r t a t i o n i s d i v i d e d i n t o a t h e o r e t i c a l p a r t and an e m p i r i c a l p a r t . The t h e o r e t i c a l p a r t c o n t a i n s f o u r c h a p t e r s . C h a p t e r one p r e s e n t s an o v e r v i e w of t h e major e x i s t i n g i n v e s t m e n t t h e o r i e s . C h a p t e r s two and t h r e e i l l u s t r a t e how t h o s e t h e o r i e s c o u l d be a p p l i e d t o the case of the e x t r a c t i v e f i r m and examine some i m p l i c a t i o n s f o r e m p i r -i c a l work. C h a p t e r f o u r combines t h o s e t h e o r i e s i n t o a model w h i c h , a l t h o u g h n o t f u l l y s o l v e d , i s s t u d i e d i n some d e t a i l . The e m p i r i c a l p a r t i s devoted t o the s t u d y of some N o r t h - A m e r i c a n o p e n - p i t m e t a l mines. C h a p t e r f i v e s t u d i e s t h e i r l o n g - r u n , o r ex a n t e , b e h a v i o u r . C h a p t e r s i x d e a l s w i t h t h e i r s h o r t - r u n , o r ex p o s t , b e h a v i o u r . The purpose o f the whole e x e r c i s e i s n o t t o t e s t t he n e o c l a s s i c a l t h e o r y of f a c t o r demand. However, w i t h i n t he n e o c l a s s i c a l framework, t e s t s 2 w h i c h h e l p d i s c r i m i n a t e between a l t e r n a t i v e t h e o r i e s o r h y p o t h e s e s a r e d e s i g n e d . The major t h e o r e t i c a l o b j e c t i v e , however, i s s i m u l t a n e o u s l y to use i n a model the b e s t f e a t u r e s o f a l t e r n a t i v e e x i s t i n g t h e o r i e s , and t o do so i n such a way t h a t the model l e n d s i t s e l f t o e m p i r i c a l a p p l i c a t i o n . 3 CHAPTER 1 BRIEF OVERVIEW OF INVESTMENT THEORY 1 . 1 I n t r o d u c t i o n Th is chapter does not present any of the e x i s t i n g t h e o r i e s of investment i n any d e t a i l . N e i t h e r does i t p rov ide any s u b s t a n t i a l b i b l i o g r a p h y on t h i s a r e a ; on the c o n t r a r y on ly a few key r e f e r e n c e s are g i ven under each head ing . The purpose i s to put a number of important t h e o r i e s i n t o p e r s p e c t i v e , to d i s c u s s and compare t h e i r r e l a t i v e m e r i t s , and to i d e n t i f y the b a s i c components which d r i v e them. A l though the focus i s on theory , e m p i r i c a l re levance i s always c o n s i d -e r e d , both from the p o i n t of v iew of the a b i l i t y of theory to be used i n e m p i r i c a l s t u d i e s of investment and from the p o i n t of v iew of i t s i n t u i t i v e a p p e a l . The approach used i s e c l e c t i c r a t h e r than s e l e c -t i v e : a l l t h e o r i e s are examined from the same p e r s p e c t i v e , and the comparisons made tend to show that they are o f t e n complementary r a t h e r than mut ua l l y e x c l u s i v e , each f a r i n g bes t on a p a r t i c u l a r f a c e t of the complex investment p r o c e s s . S ince the emphasis i s on b a s i c c h a r a c t e r i s t i c s and key assump-t i o n s , a number of important i s s u e s are i g n o r e d : complex e x p e c t a t i o n s , d e l i v e r y l a g s , u n c e r t a i n t y , replacement and d e p r e c i a t i o n , maintenance, f i n a n c i n g and c a p i t a l markets , t a x a t i o n , and most aspects of aggregat ion are among them. S e c t i o n s 1.2 to 1.6 r e s p e c t i v e l y cover the n e o - c l a s s i c a l model w i t h m a l l e a b l e and r e s a l a b l e f a c t o r s , ad jus tment - cos t models , 4 models w i t h i r r e v e r s i b l e investment , the p u t t y - c l a y hypothes i s and i t s a p p l i c a t i o n s i n v i n t a g e models of investment and, f i n a l l y , the p u t t y - s e m i - p u t t y h y p o t h e s i s . S e c t i o n 1.7 g ives a few a d d i t i o n a l remarks and s e c t i o n 1 .8 p rov ides a summary of the chapter . Whi le those f a m i l i a r w i t h investment t h e o r i e s w i l l not f i n d here much more than a few comments and those l e s s f a m i l i a r w i t h t h i s area w i l l p robab ly f i n d such treatment i n s u f f i c i e n t l y d e t a i l e d , i t i s hoped t h a t t h i s overview w i l l he lp c l a r i f y the approach taken i n : the f o l l o w i n g c h a p t e r s . 1.2 The n e o - c l a s s i c a l model w i t h m a l l e a b l e and r e s a l a b l e f a c t o r s Whi le s t a t i c f a c t o r demands have been s t u d i e d e x t e n s i v e l y f o r many decades, problems a s s o c i a t e d w i t h the inescapab ly dynamic e n v i r o n -ment of the f i r m have not been addressed r i g o r o u s l y u n t i l r e c e n t l y . However, i t s u f f i c e s to observe t h a t , from one p e r i o d to the n e x t , f i r m s have to m a i n t a i n a c a p i t a l s tock and a labour f o r c e which they can a f f e c t on ly through investment and h i r i n g , to r e a l i z e that f a c t o r demands are a matter of i n t e r t e m p o r a l o p t i m i z a t i o n . G e n e r a l i z a t i o n s of the s t a t i c case to the m u l t i p e r i o d c a s e , a l though t h e o r e t i c a l l y c o r r e c t , were not very u s e f u l e m p i r i c a l l y because they m u l t i p l i e d the number of v a r i a b l e s by the number of p e r i o d s c o n s i d e r e d , wh ich , s t r i c t l y speak ing , should be i n f i n i t e ( B l i s s (1975) , e s p e c i a l l y chapter 3 ) . A l s o , the s p e c i f i c a t i o n of an i n t e r t e m p o r a l technology set whose components were the f a c t o r l e v e l s i n each p e r i o d conveyed the i m p r e s s i o n that f a c t o r s t o c k s i n two d i f f e r e n t p e r i o d s were e n t i r e l y d i f f e r e n t e n t i t i e s , w h i l e i t was d e s i r a b l e to model them as b e i n g c a r r i e d over from one p e r i o d to the 5 n e x t . Th is impress ion i s c o r r e c t i f p r i c e s are formulated i n terms of a s s e t p r i c e s ; i f p r i c e s are fo rmulated i n terms of i m p l i c i t r e n t a l p r i c e s , i t i s not c o r r e c t . However, i m p l i c i t r e n t a l p r i c e s are not observed and must be c a l c u l a t e d . Jorgenson (1963) used a h i g h l y s i m p l i f i e d technology set to p rov ide the answer. He assumed that the i n t e r t e m p o r a l technology set can be d i v i d e d i n t o independent subsets cor respond ing to each p e r i o d . Thus the l e v e l of any f a c t o r i n any p e r i o d has no e f f e c t on the t e c h n o l o g i c a l c o n s t r a i n t faced by the f i r m d u r i n g other p e r i o d s . Fur thermore , he assumed that undepre-c i a t e d f a c t o r s t o c k s cou ld be r e s o l d , a t any t i m e , at the go ing a c q u i -s i t i o n a s s e t p r i c e . Then, the dynamic o p t i m i z a t i o n problem can be fo rmulated as f o l l o w s : (1) Max J e " r t . [ p t ' f ( x t , L t ] - - O ^ . d t . i V t i o s u b j e c t t o : x f c = I ; X q g i v e n ; x f c > 0 ; L f c > 0 where: p^, w^, <j>t are r e s p e c t i v e l y the p r i c e of the output f l o w , the r e n t a l p r i c e s of the v a r i a b l e f a c t o r s , and the a s s e t p r i c e s of s t o c k f a c t o r s at date t ; x , I , L are r e s p e c t i v e l y a v e c t o r of f a c t o r s t o c k s , t ' t t ^ J ' the v e c t o r of adjustments to those s t o c k s , and a v e c t o r of v a r i a b l e f a c t o r s ; r i s a p o s i t i v e d iscount r a t e . The s o l u t i o n can be w r i t t e n : 6 where: v i s a v e c t o r of i m p l i c i t f a c t o r - s t o c k r e n t a l p r i c e s that s a t i s f i e s (3) Th is i s i n f a c t the f a m i l i a r l i q u i d - a s s e t h o l d i n g c o n d i t i o n which s t a t e s than an e x t r a u n i t of a s s e t s i s worth h o l d i n g i f , i n the absence of d e p r e c i a t i o n , the revenues from the a d d i t i o n a l p r o d u c t i o n which i t permi ts match the i n t e r e s t income foregone by h o l d i n g i t . As i s obvious from (2) and ( 3 ) , f a c t o r l e v e l s are s e l e c t e d a c c o r d i n g to contemporary p r i c e s o n l y ; hence the term "myopic" co ined by Arrow (1968) to q u a l i f y the co r respond ing b e h a v i o u r . Th is r e s u l t depends c r u c i a l l y on the two key assumptions mentioned above, tha t no i n t e r -temporal l i n k i s imposed by the' technology ( f i r s t assumption) nor by the r e s a l e market (second assumpt ion) . Jo rgenson 's model i s i n f a c t on l y h a l f dynamic: i t does account f o r the p e r e n n i a l nature of s t o c k s , s i n c e the i m p l i c i t r e n t a l p r i c e d i s t r i b u t e s a s t o c k ' s asset p r i c e over t i m e ; but i t does not account f o r the " w e i g h t i n e s s " of s t o c k s , s i n c e i t a l l o w s them to a d j u s t i n a p e r f e c t l y v o l a t i l e way to any change i n e c o n -omic parameters . Th is causes s u r p r i s i n g r e s u l t s , such as d i s c r e t e jumps i n s t o c k l e v e l s as a r e s u l t of changes i n the r a t e of v a r i a t i o n of p r i c e s ( e . g . N i c k e l l (1978) pp. 1 4 - 1 6 ) , as i l l u s t r a t e d i n F i g u r e 1 . 8 In the e m p i r i c a l l i t e r a t u r e , the s tock paths i m p l i e d by Jo rgenson 's model were i n t e r p r e t e d as long run e q u i l i b r i a , toward which the f i r m was a iming i n the shor t r u n . Var ious ad-hoc adjustment schemes were d e v i s e d , the most popular of which was the f l e x i b l e a c c e l e r a t o r . At the t h e o r e t i c a l l e v e l , two avenues were a v a i l a b l e to improve on Jo rgenson 's model ; they are presented i n the next two s e c t i o n s . Each can be i n t e r p r e t e d as a removal of one of the key assumptions that under l y the model and make the s t o c k s p e r f e c t l y l i q u i d . . 1 . 3 Models w i t h c o s t s of ad justment . By i n t r o d u c i n g cos ts of a d j u s t i n g s tock l e v e l s , one breaks the assumption t h a t the i n t e r t e m p o r a l technology set i s composed of i n d e - ; pendent subsets that p e r t a i n to each p e r i o d . S ince stock adjustments are c o s t l y , what i s t e c h n o l o g i c a l l y f e a s i b l e at t depends on a l l s t o c k l e v e l s at t - 1 . The dev ice i s a l s o very power fu l i n terms of economizing on the number of v a r i a b l e s r e q u i r e d to d e s c r i b e the technology set or i t s counte rpar t i n revenue terms, at any g i ven d a t e . Indeed, a genera l f o r m u l a t i o n of the i n t e r t e m p o r a l technology of the type mentioned i n s e c t i o n 1 .2 i m p l i e s t h a t , at any t i m e , output i s a f u n c t i o n of s tock  l e v e l s and v a r i a b l e - f a c t o r l e v e l s a t a l l t i m e s . With adjustment c o s t s , output i s a f u n c t i o n of c u r r e n t f a c t o r l e v e l s and c u r r e n t adjustments o r , i n d i s c r e t e terms, of cu r ren t f a c t o r l e v e l s and the stock l e v e l s of the p rev ious p e r i o d . Th is huge s a v i n g i s ach ieved by i m p l i c i t l y assuming that f a c t o r s t o c k s of a c e r t a i n k i n d , cons idered a t d i f f e r e n t t i m e s , have the same n a t u r e . In o ther words, s u c c e s s i v e f a c t o r a c q u i s i t i o n s of a g i ven nature can be aggregated over t ime i n t o what i s c a l l e d the f a c t o r s t o c k . 1 ' 2 9 The various adjustment-cost models which have been proposed i n the l i t e r a t u r e (Eisner and Strotz (1963), Gould (1968), Treadway (1969,1970), Mortensen (1973) p r i n c i p a l l y ) assume various l e v e l s of generality. B a s i c a l l y , adjustment costs may depend on adjustment l e v e l s only, or on adjustment l e v e l s and the l e v e l s of a l l f a c t o r s , and they may be expressed i n d o l l a r terms or i n physical terms (usually i n terms of foregone output). The most advanced versions of the adjustment-cost theory of investment (Treadway (1970), Mortensen (1973)) merge adjust-ment costs and a conventional production function into a s i n g l e technol-o g i c a l constraint. (5) q = f ( x , x, L ) . Treadway (1970) c a l l s "quasi-fixed", any input such as x, whose v a r i a t i o n s , as w e l l as l e v e l , a f f e c t production p o s s i b i l i t i e s . "Such q u a s i f i x i t y may a r i s e as a consequence of planning costs, i n s t a l l a t i o n or break-in costs or other f r i c t i o n s i n the growth process that are i n t e r n a l to the f i r m . " (p. 332). I f f ( - ) i s assumed to have continuous second-order p a r t i a l d e r i v a t i v e s , a necessary optimality condition i n the cha r a c t e r i z a t i o n of the optimal paths i s the concavity of f ( * ) i n (x, L) i n the neighbourhood of any optimum (x , x , L ). I t seems also obvious that f ( * ) should be non-decreasing i n x and L, and i t i s customarily as-sumed that f(«) culminates at some sensible f i n i t e l e v e l of x, for given (x,L) . The intertemporal profit-maximization problem to be solved by a f i r m which operates under the technological constraint (5) i s : 10 (6) Max - r t V q t V L t dt subject to (5) and the usual non-negativity constraints and i n i t i a l con-d i t i o n s . A remarkable result (Treadway (1970)) i s that, under the minimum regularity conditions on f ( - ) just outlined, neither the short-run, nor the long-run, factor demands s a t i s f y the symmetry and own-price monoto-3 n i c i t y properties of thei r s t a t i c counterparts. This result i s obtained i n the pa r t i c u l a r case of (6) where price expectations are assumed to be constant; hence i t holds i n the more general case. Although cost of adjustment models yielded interesting theoreti-cal r e s u l t s , t h e i r popularity stemmed p r i n c i p a l l y from the fact that they provided an i n t u i t i v e l y appealing way to ra t i o n a l i z e the f l e x i b l e accel-erator, which was widely and successively used i n empirical work. As early as 1969, Nadiri and Rosen had used "Interrelated Factor Demand Functions", which could be rationalized by a generalized adjustment-cost model of the type just described, as shown by Treadway (1971). Further re-search, such as that by N i c k e l l (1977), and Treadway (1974) helped "close the gap between theoretical adjustment-cost models and econometric factor demand models." ( i b i d . , p. 18). In pa r t i c u l a r , empirical students of dynamic factor demand, such as Morrison and Berndt (1979), started using endogenous adjustment-coefficient matrices. A survey of the empirical side of this vein i s provided by Berndt e_t a l . (1980) . Despite the high l e v e l of theoretical coherence achieved by recent flexible-accelerator models of factor demand, the f l e x i b l e accelerator remains a specialized t o o l : both i t s i n t u i t i v e appeal - investment i s proportional to the gap between 11 a c t u a l and e q u i l i b r i u m s tock l e v e l s - and i t s r e l a t i v e fo rmal s i m p l i c i t y , r e l y on the e x i s t e n c e of a l o n g - r u n e q u i l i b r i u m . Whi le i t s g e n e r a l i z a t i o n to o ther s i t u a t i o n s i s not i m p o s s i b l e , a s imple look at the m o d i f i e d f o r -mula o b t a i n e d by N i c k e l l (1977, p. 5) w i l l convince the reader of the complex i ty of such an a t tempt . I t i s not yet c l e a r whether t h e o r e t i c a l r e s e a r c h e s , such as E p s t e i n (1981) , or McLaren and Cooper (1980) , which' r e l y on the d u a l i t y between the technology and the net present va lue a c h i e v a b l e by the f i r m , w i l l be more f r u i t f u l i n f u r t h e r pushing back the e m p i r i c a l l y e x p l o i t a b l e l i m i t s of the c o s t - o f - a d j u s t m e n t theory of investment . 1.4 I r r e v e r s i b l e investment As i n d i c a t e d i n 1 . 2 , e a r l y dynamic n e o c l a s s i c a l models of investment such as Jorgenson (1963) f a i l e d to r e f l e c t the " w e i g h t i n e s s " of f a c t o r s t o c k s , which were t r e a t e d as p e r f e c t l y l i q u i d . Ad jus tment -cost t h e o r i e s c o r r e c t e d the problem by i n t r o d u c i n g i n t e r t e m p o r a l l i n k s i n t o the techno logy . An a l t e r n a t i v e way i s to i n c o r p o r a t e the f a c t tha t the market does not permit p e r f e c t f a c t o r - s t o c k l i q u i d i t y . The extreme c a s e , presented by Arrow (1968) , c o n s i s t s i n assuming that f a c t o r s t o c k s cannot be r e s o l d at any p o s i t i v e p r i c e w h i c h , under a s tandard n e o c l a s -s i c a l s p e c i f i c a t i o n of the p r o d u c t i o n f u n c t i o n , i m p l i e s that on ly p o s i -t i v e investment w i l l be observed. Such an approach in t roduces asym-m e t r i e s i n t o the investment behav iou r : when the c o n s t r a i n t i s not b i n d i n g , the s t o c k s a d j u s t i n s t a n t a n e o u s l y upward to any change i n p r i c e s ; when the c o n s t r a i n t i s b i n d i n g , the s t o c k s do not a d j u s t . The mathematics of t h i s behav iour have been g i ven by Arrow, but a more i n t u i t i v e d e r i v a t i o n can be found i n N i c k e l l (1978, pp. 5 1 - 5 8 ) . S t i l l 12 a t h i r d approach i s proposed h e r e , which uses the maximum p r i n c i p l e and i s very i n t u i t i v e , s i n c e i t i n t e r p r e t s the c o s t a t e v a r i a b l e s as shadow p r i c e s of the s t o c k s . Cons ider the f o l l o w i n g problem: (7) Max IVM A ""• [p t-q t - w;-it - *;-i t]-dt s u b j e c t t o : f ( v L t ) = x = I, x > 0 ; x g i ven ; I > 0 ; L > 0 ; t o t t Assume f o r s i m p l i c i t y tha t the n o n - n e g a t i v i t y c o n s t r a i n t s on f a c t o r s are never b i n d i n g . Assume a l s o , but j u s t f o r an i n s t a n t , that the i r r e v e r s i b i l i t y c o n s t r a i n t i s not b i n d i n g . Then the H a m i l t o n i a n i s : H ( 0 = e - r t f ( \ 1 ) p • f x ,L - w -L - d> .I *t [ t ' t j t t t t + V x t wh ere X i s the v e c t o r of p r e s e n t - v a l u e c o s t a t e v a r i a b l e s assoc ia tec w i t h s tocks x f c a t t . F a c t o r i n g 1^ ., one has : ( 8 ) H( - ) = e r t - p t - f [ x t » L t ] " e - r t > ' W t ' L t " e - X t I t appears immediate ly that max imiza t ion w i t h r e s p e c t to g ives the u s u a l e q u a l i t y of m a r g i n a l revenue products w i t h wages. Concerning the max imiza t ion w i t h respec t to I , however, one observes that the H a m i l t o n i a n i s l i n e a r i n 1^ , which i m p l i e s a bang-bang c o n t r o l . Three phases must be d i s t i n g u i s h e d . I f the shadow p r i c e of a s t o c k i s h i g h e r than i t s market p r i c e , expressed i n p r e s e n t v a l u e , then an i n f i n i t e p o s i t i v e investment i s c a l l e d f o r (phase i ) ; i f shadow p r i c e and market p r i c e are e q u a l (phase i i ) the o p t i m a l l e v e l of I seems i n d e t e r m i n a t e but can be determined from the e q u a l i t y of the shadow p r i c e w i t h the market p r i c e , combined w i t h the e q u a t i o n of motion of X ^ , (9); i f the shadow p r i c e i s lower than the market p r i c e (phase i i i ) , an i n f i n i t e n e g a t i v e investment i s d e s i r a b l e , b u t , the i r r e v e r s i b i l i t y c o n s t r a i n t becomes b i n d i n g , so t h a t 1 = 0 . I t i s i n t u i t i v e l y obvious and easy to show t h a t phase i , of i n f i n i t e p o s i t i v e i n v e s t m e n t , i s i n s t a n t a n e o u s . T h i s i s because the shadow p r i c e , X , of the m a r g i n a l u n i t of s t o c k decreases as t h a t s t o c k i s i n c r e a s e d . T h i s r e s u l t depends on the c o n c a v i t y of f ( ' ) i n x , which i s t h e r e f o r e assumed. So, except , p o s s i b l y , a t the i n i t i a l date or when p r i c e l e v e l s or r a t e s of change r e g i s t e r d i s c r e t e jumps, o n l y phases i i and i i i w i l l be observed. D u r i n g both of them, the equat ions of motion of X a r e : H x ' t h a t i s (9) - r t •P t-f x (0 e D u r i n g phase i i , by d e f i n i t i o n , X = tji • e - r t so t h a t (10) - r t - r t e r • e 14 Combining (9) and ( 1 0 ) , one h a s : (11) which i s the i m p l i c i t user cos t r e l a t i o n ( 3 ) . (11) i m p l i c i t l y d e f i n e s the motion that x^ must f o l l o w f o r the system to be i n phase i i . I f t h a t motion n e c e s s i t a t e s a n e g a t i v e investment i n , say , f a c t o r i , i t cannot be f o l l o w e d : I* i s zero i n s t e a d of b e i n g n e g a t i v e , the s t o c k i s not reduced as r e q u i r e d to m a i n t a i n the e q u a l i t y of the shadow p r i c e w i t h the market p r i c e , the shadow p r i c e becomes lower than the market p r i c e . a n d i t s motion i s g iven by ( 9 ) , w i t h x 1 c o n s t a n t , u n t i l i t catches up the market p r i c e and the system switches back to phase i i . : F i g u r e 2 i l l u s t r a t e s such a p a t t e r n f o r the s i m p l e s t p o s s i b l e case where there i s on ly one s tock f a c t o r and no v a r i a b l e f a c t o r s . c o n s t r a i n e d path "x d u r i n g phase i i i . At t^ both paths r e g i s t e r a d i s c r e t e i n c r e a s e which aims at t a k i n g advantage of the c a p i t a l gains tha t are p o s s i b l e d u r i n g t2^|* However, the r i s e i n x^ _ i s much lower because of the a n t i c i p a t i o n tha t the p r i c e w i l l l e v e l o f f at and s t a r t d e c r e a s i n g at t^ . One notes t h a t , d e s p i t e the a n t i c i p a t i o n that the p r i c e w i l l e v e n t u a l l y f a l l below i t s i n i t i a l l e v e l , x^ _ i s i n c r e a s e d at t ^ ; t h i s i s because d i s c o u n t i n g makes the c a p i t a l ga ins of the p e r i o d ft , t„l more, v a l u a b l e than the subsequent l o s s e s a s s o -In F i g u r e 2 , the c o n s t r a i n e d path x d i ve rges from the u n -d a t e d w i t h excess c a p a c i t y and c a p i t a l l o s s e s smoother but remains v o l a t i l e upward. F i g . 2 : Market p r i c e , shadow p r i c e , and investment p a t h s . Model w i t h i r r e v e r s i b l e investment and model w i t h r e v e r s i b l e i n v e s t m e n t . * When d i f f e r e n t from the t r a j e c t o r i e s of the i r r e v e r s i b l e - i n v e s t m e n t models the s tock f a c t o r t r a j e c t o r y (upper graph) and shadow-pr ice t r a j e c t o r y ( lower graph) of the r e v e r s i b l e - i n v e s t m e n t model are g iven by dashed c u r v e s . 16 1.5 P u t t y - c l a y technology and v i n t a g e models of investment A l l three t h e o r i e s of investment j u s t presented assume t h a t f a c t o r s tocks can, to v a r i o u s e x t e n t , be a d j u s t e d over t i m e . T h i s i m p l i e s t h a t those s tocks can be aggregated over t i m e . Al though i t i s not c r u c i a l to the s i m p l e n e o - c l a s s i c a l model of 1.2, t h i s i m p l i c i t assumption i s c r u c i a l i n the models w i t h adjustment c o s t s and the models w i t h i r r e v e r s i b l e investment . I t has been argued t h a t such an assumption i s i n a c c e p t a b l e : Machines purchased at d i f f e r e n t dates are d i f f e r e n t a l t o g e t h e r , even i f they perform i d e n t i c a l t a s k s . Consequent ly , i n a theory of f a c t o r demand, one should keep t r a c k of each v i n t a g e of investment . I n order to o f f s e t the huge i n c r e a s e i n the number of parameters r e q u i r e d by such an approach, v i n t a g e t h e o r i e s have used d r a s t i c s i m p l i f y i n g assumptions on the t e c h n o l o g y . The p u t t y -c l a y assumption i s the major one. I t may have s e v e r a l i n t e r p r e t a t i o n s , but i n a l l of them, f a c t o r s t o c k s , once a c q u i r e d , cannot be adjusted by the f i r m , a l t h o u g h t h e i r l e v e l may be a f f e c t e d by d e p r e c i a t i o n . The s tocks of each v i n t a g e can be combined w i t h a c e r t a i n q u a n t i t y of v a r i -ab le f a c t o r s . The p r o p o r t i o n may or may not be v a r i a b l e , a l though i t i s u s u a l l y assumed to be f i x e d . At t h i s s t a g e , i t should be mentioned t h a t v i n t a g e t h e o r i e s of investment have u s u a l l y been developed w i t h e m p i r i c a l , work i n mind. Thus, the assumptions made r e f l e c t data c o n s t r a i n t s as much as requirements a s s o c i a t e d w i t h the t h e o r y . T h i s e x p l a i n s the p r e -ponderance of the assumption that f a c t o r p r o p o r t i o n s are f i x e d ex p o s t . Given t h i s p r e o c c u p a t i o n w i t h p r a c t i c a l i t y , one o b s t a c l e to be d e a l t w i t h i n any v i n t a g e model i s t h a t of s p e c i f y i n g a p r o d u c t i o n f u n c t i o n which combines a very h i g h number 17 of vintage-factors in.order to produce a single output, while being character-ized by a s u f f i c i e n t l y low number of parameters. It i s usually assumed, as i n N i c k e l l (1978, pp. 247-252), that each vintage f a c t o r , combined with the proper amount of labour, produces a given amount of output, independently of other vintages. T o t a l output at any date i s the sum of each vintage's contribution. In other words, the production function i s a d d i t i v e l y separable i n vintage f a c t o r s . Furthermore, each term i s assumed to be p o s i t i v e l y l i n e a r l y homogeneous i n a l l fa c t o r s , so that a period's output can be formulated i n terms of factor proportions. At each period, p r o f i t maximization r e s u l t s i n the choice of an optimal factor proportion. By combining two successive periods' optimality conditions, one obtains an investment-demand equation which r e l a t e s i n vest-ment to current and previous period's r e l a t i v e p r ices of labour and c a p i t a l . N i c k e l l goes on to i l l u s t r a t e how the approach can be implemented, i n par-t i c u l a r , how i t can be used to test the putty-clay hypothesis or, rather the p a r t i c u l a r r e s t r i c t i o n of the hypothesis under which labour i s assumed not to be v a r i a b l e ex post (pp. 254-256). While there i s ;no formal d i f f i c u l t y i n N i c k e l l ' s excellent account of vintage theories of investment, one should pay some attention to one of h i s preliminary remarks ^p. 249). "... i t i s worth noting im-mediately that there w i l l be a serious d i f f i c u l t y i f the firm wishes to d i s i n v e s t . . . . We s h a l l sidestep t h i s d i f f i c u l t y by simply supposing that the f i r m i s never faced with the circumstances which would lead i t to d i s i n v e s t . " This assumption i s somewhat misleading i n that i t does not imply that investment i s i r r e v e r s i b l e . However, as was noted i n 1.2 (footnote 1 ) , i f investment i s r e v e r s i b l e , the f i r m may r e s e l l i t s 18 whole s tock of equipment at the end of each p e r i o d and be i n a p o s i t i o n to r e p l a c e i t s e n t i r e p l a n t at the b e g i n n i n g of each p e r i o d . When s t o c k s can be aggregated over t i m e , t h i s does not make any d i f f e r e n c e ; when s t o c k s cannot be aggregated over t i m e , t h i s i s a c r u c i a l p r o p e r t y . I n f a c t , u n l e s s there i s no change i n economic parameters between two p e r i o d s , i t i s i n the i n t e r e s t of the f i r m to renew a l l i t s equipment i n each p e r i o d i n order to adopt the most f a v o u r a b l e f a c t o r p r o p o r t i o n s . But then o n l y c u r r e n t v i n t a g e s are observed, and there i s no v i n t a g e theory 5 of i n v e s t m e n t . So i t must be t r u e t h a t v i n t a g e t h e o r i e s of investment i m p l i c i t l y assume t h a t investment i s i r r e v e r s i b l e , whether or not the f i r m i s faced w i t h c i rcumstances t h a t would make i t w i s h i t owned l e s s equipment. As a l a s t remark, the extreme case of the p u t t y - c l a y h y p o t h e s i s , which assumes t h a t labour i s not v a r i a b l e ex p o s t , has been r i g h t l y i n t e r -p r e t e d as the one end of a range of assumptions on the t e c h n o l o g y , whose o ther end i s the n e o c l a s s i c a l model o f Jorgenson, w i t h i t s p e r f e c t l i q u i -d i t y . The i n t e r p r e t a t i o n can be pushed f u r t h e r by c o n t r a s t i n g two r e -s u l t s : i n the pure n e o c l a s s i c a l model , a l l f a c t o r s , i n c l u d i n g s t o c k s , can be i n t e r p r e t e d as r e n t e d f l o w s ; i n the extreme p u t t y - c l a y model , on the c o n t r a r y , a l l f a c t o r s , i n c l u d i n g rented f a c t o r s such as l a b o u r , must be i n t e r p r e t e d as purchased s t o c k s . The f i r s t p a r t of t h i s statement has been s t r e s s e d r e p e a t e d l y and i s i l l u s t r a t e d by the user c o s t f o r -mula ( 3 ) . The second p a r t emerges immediately from the o p t i m a l i t y c o n d i t i o n s which apply to rented f a c t o r s when the technology i s " p u t t y -c l a y " ; f o r a t w o - f a c t o r f i r m the o p t i m a l i t y c o n d i t i o n i s : 19 (12) e r t - p •F ( x , L ) - d t *t L ' - r t . e -w *dt, o r , say , r e " r t . p - F T ( - ) - d t = -t L r 0 Both s i d e s rep resent the c a p i t a l i z e d v a l u e of f lows of revenues or e x p e n d i t u r e s . They have the dimensions of s t o c k s . In g e n e r a l , P t ' F L ( 0 * w t . Because of i t s r i g i d a s s o c i a t i o n w i t h a s tock f a c t o r , the rented f a c t o r 6 takes up the dimension of a s t o c k . 1 .6 P u t t y - s e m i - p u t t y technology The above p r e s e n t a t i o n of the p u t t y - c l a y hypothes is and i t s use i n v i n t a g e models of investment was concluded by c o n t r a s t i n g i t w i t h i t s p o l a r o p p o s i t e , the assumption that s tock f a c t o r s are p e r f e c t l y m a l l e a b l e . In the former case o p t i m a l i t y c o n d i t i o n s must be cas t i n te rms .o f s t o c k s ; i n the l a t t e r case , they may be cas t i n terms of f l o w s . In o ther words, under the p u t t y - c l a y h y p o t h e s i s , f i r m s make on ly l o n g - r u n d e c i s i o n s ; w i t h p e r f e c t l y m a l l e a b l e f a c t o r s , they make on ly s h o r t - r u n d e c i s i o n s , u n l e s s some market c o n s t r a i n t , the i r r e v e r s i b i l i t y , a l s o comes to b e a r . The a d j u s t m e n t - c o s t theory comes as an i n t e r m e d i a t e ' c a s e , s i n c e 20 intertemporal technological l i n k s require some foresight on the part of the firm. The putty - semi-putty hypothesis i s a d i f f e r e n t kind of intermediate case, where the long-run and the short-run, instead of being "blended", are both introduced, but kept d i s t i n c t . As with putty-clay models, the firm makes a number of irrevocable decisions during a f i r s t step c a l l e d the ex ante phase. Then, ex post, the f i r m has per-f e c t l y malleable factors which i t combines under the technological constraint previously defined by i t s ex ante decisions. D i f f e r e n t ex ante decisions produce d i f f e r e n t firms with p e r f e c t l y malleable f a c t o r s , although the firms were i d e n t i c a l ex ante. The ex ante and ex post phases b a s i c a l l y correspond to the Marshallian notions of "long periods" and "short periods" r e s p e c t i v e l y . (Marshall (1920, p. 374)). However, the ex ante phase must be interpreted as instantaneous, although a f f e c t i n g the long period. Furthermore ex ante decisions need not "set the stock of appliances" (p. 374); they need only irrevocably a f f e c t the ex post technology, as would, for example, a choice among various technologies characterized, say, by d i f f e r e n t rates of s u b s t i t u t i o n between factors of production. As an example, there i s a trade-off between " F l e x i b i l i t y versus E f f i c i e n c y i n Ex Ante Plant Design" (Fuss and McFadden (1978)). An obvious and inescapable complication of t h i s approach i s the existence of a l i n k between ex post technologies and the ex ante choice set. As Fuss (1977) explains, "the l i n k between, long-run and short-run cost curves as functions of output i s given by the well-known Wong-Viner (••) envelope theorem. However, we can also obtain cost curves by holding output and a l l but one f a c t o r p r i c e f i x e d . " In the p a r t i c u l a r case where p r i c e expectations were constant at the time of the ex ante decision, 21 and t u r n out to be v e r i f i e d d u r i n g the ex post phase, the ex post t e c h -nology i s operated at i t s p o i n t of tangency w i t h the ex ante techno logy . When p r i c e s are d i f f e r e n t , the optimum a t t a i n e d under the ex post t e c h n o l -o g i c a l c o n s t r a i n t i s i n f e r i o r to the optimum that was a c h i e v a b l e under the ex ante t e c h n o l o g i c a l c o n s t r a i n t . Th is i s i l l u s t r a t e d i n F i g u r e s (a) and (b) borrowed from Fuss (1977, p. 1801) , a long w i t h , except f o r minor changes, t h e i r e x p l a n a t i o n s (p. 1800) . In F igu re ( a ) , C i s the minimum cos t of p roduc ing one u n i t of output a t v a r i o u s expected p r i c e s of the f i r s t f a c t o r when a l l known techniques are f e a s i b l e . Th is corresponds to the cho ice of technique i n the long 22 r u n . From the d u a l i t y theorems (Diewert (1974, pp. 111-112)) we can represent the cor responding f r o n t i e r of e f f i c i e n t p r o d u c t i o n p o s s i b i l -i t i e s by the ex ante u n i t i soquant AA i n F igu re (b ) . Now suppose the E * producer expects p^ = p^. The ex ante expected minimum cost of p roduc -A * t i o n i s C (p^) a n d the ex ante expected cho ice of i n p u t l e v e l s i s r e p r e -sented by the p o i n t a i n F i g u r e (b ) . Ex p o s t , the producer observes the a c t u a l p r i c e p^, I f he f o r e c a s t s c o r r e c t l y , p^ = p^, and the chosen p ^ A & technique ( represented by "z) i s s t i l l o p t i m a l . C (p^) = C (p^) and expected input l e v e l s a are s t i l l o p t i m a l l e v e l s . I f he f o r e c a s t s i n c o r -r e c t l y , and the technology i s not " p u t t y - p u t t y " , the unexpected o p t i m a l i n p u t l e v e l s (po in t g) are no longer f e a s i b l e . Those i n p u t s a c t u a l l y •p D A A chosen ex post w i l l r e s u l t i n average cost C such that C > C s i n c e C r e p r e s e n t s the min imal a t t a i n a b l e c o s t . Therefore i n F i g u r e (a) we have the r e q u i r e d envelope r e s u l t , w i t h tangency o c c u r r i n g at p*. A second p A a p p l i c a t i o n of d u a l i t y u t i l i z i n g the f a c t that C S C r e s u l t s i n the envelope b e i n g t r a n s f e r r e d to F i g u r e (b ) . The ex post i soquant i s pp and the ex post i n p u t s l e v e l s are represented by the p o i n t y. Tangency occurs a t a r e p r e s e n t i n g the ex ante expected input l e v e l s . 1 .7 Remarks One problem i n i n t e r p r e t i n g and implementing M a r s h a l l ' s d i s t i n c -t i o n between the s h o r t - r u n and the l o n g - r u n has always been to model the . t r a n s i t i o n between the two s i t u a t i o n s . In put ty - s e m i - p u t t y models , t h i s t r a n s i t i o n i s ins tantaneous and the l o n g - r u n i s fundamental ly d i f f e r e n t from the s h o r t - r u n . In c o s t - o f - a d j u s t m e n t models t h i s t r a n s i t i o n i s 23 p e r p e t u a l and there i s no d i f f e r e n c e between the s h o r t - r u n and the l o n g - r u n t e c h n o l o g i c a l c o n s t r a i n t s faced by a f i r m . A d i s t i n c t i o n between the l o n g -run and the s h o r t - r u n can s t i l l be drawn i f , as i s always p o s t u l a t e d i n the l i t e r a t u r e , an e q u i l i b r i u m e x i s t s . In such a case the s h o r t - r u n e f f e c t of a p r i c e change i s that obta ined when a l l s tocks are f i x e d ( i n a d i s c r e t e time f o r m u l a t i o n ) , w h i l e the l o n g - r u n e f f e c t i s the response observed , or computed, when s t o c k s have f u l l y ad jus ted to the new p r i c e s . So : the d i s -t i n c t i o n between s h o r t - r u n and l o n g - r u n c o l l a p s e s i n such important i n -s tances where a f i r m e x t r a c t s an e x h a u s t i b l e resource or does not have any constant p r i c e e x p e c t a t i o n s . With put ty - s e m i - p u t t y models the d i s t i n c -t i o n i s p e r f e c t l y g e n e r a l . However, i f one c r i t i c a l l y looks at the ex post phase i t i s immediate ly apparent that i t s u f f e r s from the drawbacks of any model w i t h p e r f e c t l y m a l l e a b l e f a c t o r s . Even i f adjustments are " r e s t r i c t e d " i n comparison w i t h ex ante p o s s i b i l i t i e s , i t i s d i f f i c u l t to b e l i e v e that the same adjustment can be done on a d a i l y b a s i s as over a f i v e year p e r i o d . The ex post phase of a p u t t y - s e m i - p u t t y model i s not dynamic. Th is r a i s e s a l a s t remark, which has to do w i t h the e m p i r i c a l a p p l i c a t i o n s of p u t t y - s e m i - p u t t y models on one s i d e , and c o s t - o f -adjustment and v i n t a g e models on the other s i d e . The former are m i c r o -economic (Fuss (1977) ) ; the l a t t e r are s e c t o r a l f o r the c o s t - o f - a d j u s t -ment theory and most l y s e c t o r a l f o r p u t t y - c l a y models (Mizon (1974) ) , the except ions b e i n g o f t e n presented as t e s t s of the theory (Fuss (19 7 8 ) ) . With aggregate v i n t a g e models , the drawbacks of the ex post p e r i o d do not matter much because that phase l a s t s on ly one p e r i o d , u n t i l the next v in tage i s a c q u i r e d ; the ex post phase i s t r u l y short run and the model can be viewed as a r a p i d s u c c e s s i o n of ex ante phases. On the c o n t r a r y , 24 w i t h microeconomic a p p l i c a t i o n s , whether the ex post phase i s c h a r a c t e r -i z e d by f u l l r i g i d i t y or p e r f e c t m a l l e a b i l i t y , that phase a p p l i e s to the whole l i f e of a f i r m and the c r i t i c i s m s j u s t made take up t h e i r f u l l i n c i -dence. In such c i r c u m s t a n c e s , i f there i s any ex post f l e x i b i l i t y , i t w i l l be b e t t e r accounted f o r by a c o s t - o f - a d j u s t m e n t model , which a l l o w s a l a r g e r adjustment over a longer p e r i o d then over a s h o r t e r p e r i o d , than by the ex post phase of a p u t t y - s e m i - p u t t y model . In microeconomic a p p l i c a t i o n s , the most v a l u a b l e q u a l i t y of a pu t t y - s e m i - p u t t y model i s not i t s a b i l i t y to account f o r ex post f l e x i b i l i t y , b u t , r a t h e r , the sharp d i s t i n c t i o n drawn between the ex ante ( c o n s t r u c t i o n ) phase and the ex post (operat ion ) phase, a q u a l i t y shared w i t h p u t t y - c l a y models . In s e c t o r a l a p p l i c a t i o n s , t h i s d i s t i n c t i o n i s not as c r u c i a l , as the impact of the c r e a t i o n of a new f i r m w i l l be d i l u t e d , i n the aggregator , w i t h s h o r t - r u n adjustments of e x i s t i n g f i r m s . Another aspect of t h i s aggregat ion e f f e c t i s that the c l e a n d i s t i n c t i o n between the s h o r t - r u n and the l o n g - r u n , which c h a r a c t e r i z e s pu t t y - s e m i - p u t t y models , i s l o s t i n the aggregat ion process where f i g u r e s cor responding to ex ante d e c i s i o n s are added to those that r e f l e c t ex post d e c i s i o n s . Th is makes pu t t y - s e m i - p u t t y models f i t on l y f o r microeconomic a p p l i c a t i o n s , l e a v i n g c o s t - o f - a d j u s t m e n t models and v i n t a g e models f o r s e c t o r a l a p p l i c a t i o n s . 1 .8 Summary Microeconomic t h e o r i e s of inves tment , o r , more g e n e r a l l y , f a c t o r -s t o c k demands, tend to emphasize one p a r t i c u l a r aspect of a r a t h e r complex p r o c e s s . A d j u s t m e n t - c o s t models emphasize smooth adjustments to an ever 25 changing l o n g - r u n e q u i l i b r i u m , on the par t of e x i s t i n g f i r m s . Models w i t h i r r e v e r s i b l e investment s t r e s s the asymmetries a r i s i n g from a p o s i t i v e investment c o n s t r a i n t . P u t t y - c l a y models i n v e s t i g a t e the c r e a t i o n phase of a f i r m and r u l e out any f l e x i b i l i t y on the p a r t of e x i s t i n g f i r m s . P u t t y - s e m i - p u t t y models t r y to remedy t h i s d e f i c i e n c y by endowing e x i s t i n g f i r m s w i t h some f l e x i b i l i t y . However, the ex post phase of p u t t y - s e m i - p u t t y models remains very p r i m i t i v e i n tha t i t i n v o l v e s myopic b e h a v i o u r , a drawback of e a r l y dynamic f a c t o r demand models which p r e c i s e l y s t i m u l a t e d the research that l e d to a d j u s t -ment -cost models and i r r e v e r s i b l e - i n v e s t m e n t models . Such s p e c i a l i z a t i o n can be i n t e r p r e t e d i n terms of the a s -sumptions tha t were found most adequate i n order to economize on , the number of parameters , which i s p r a c t i c a l l y i n f i n i t e i n the b a s i c model obta ined by g e n e r a l i z i n g the s t a t i c theory of f a c t o r demand. Models w i t h m a l l e a b l e f a c t o r s assume that the technology i s p e r f e c t l y separab le from one p e r i o d to the next and that there i s no o b s t a c l e to the r e s a l e of equipment. By r u l i n g out any i n t e r t e m p o r a l l i n k , these a s -sumptions make the models p r a c t i c a l l y s t a t i c . C o s t - o f - a d j u s t m e n t models and i r r e v e r s i b l e - i n v e s t m e n t models i n t r o d u c e an i n t e r t e m p o r a l l i n k r e s -p e c t i v e l y through the technology and through the market , a t the cos t of assuming that s t o c k s can be aggregated over t i m e . In order to a v o i d t h i s l a s t assumpt ion , p u t t y - c l a y models , i n t h e i r v i n t a g e v e r s i o n s , a s -sume that the p r o d u c t i o n f u n c t i o n i s a d d i t i v e i n the c o n t r i b u t i o n to output of each v i n t a g e of s t o c k , combined, i n f i x e d p r o p o r t i o n s , w i t h other f a c t o r s of p r o d u c t i o n . Non -v in tage p u t t y - c l a y models reduce the f i r m ' s l i f e to two b a s i c a l l y s t a t i c p e r i o d s , the ex ante and the ex post 26 p e r i o d s . I n p u t t y - s e m i - p u t t y models, the ex post p e r i o d of p u t t y - c l a y models becomes an ex post phase whose dynamic p r o p e r t i e s have the draw-backs - and r e l y on the assumptions - c h a r a c t e r i s t i c of any model w i t h p e r f e c t l y m a l l e a b l e and r e s a l a b l e f a c t o r s . Al though i n c o m p l e t e , t h i s procedure c o m p l i c a t e s the r e p r e s e n t a t i o n of the ex ante technology , as the l a t t e r must r e f l e c t wider ex post p o s s i b i l i t i e s . Most e m p i r i c a l a p p l i c a t i o n s are s o l i d l y r o o t e d i n the a s -sumption t h a t a s t e a d y - s t a t e e q u i l i b r i u m e x i s t s , a c o n d i t i o n which i s not met when a f i r m uses an e x h a u s t i b l e resource o r has non-constant p r i c e e x p e c t a t i o n s . As the most s o p h i s t i c a t e d i n t e r m e d i a t e cases between p e r f e c t m a l l e a b i l i t y and a b s o l u t e r i g i d i t y , the c o s t - o f - a d j u s t -ment models and the p u t t y - s e m i - p u t t y models s tand up as candidates f o r e m p i r i c a l a p p l i c a t i o n s . The former have the advantage of i m p l y i n g a smooth t r a n s i t i o n between the s h o r t - r u n and the l o n g - r u n , but t r e a t s i m i l a r l y such d i f f e r e n t phases as the c o n s t r u c t i o n phase and the o p e r a t i o n phase of a f i r m ; the l a t t e r have e x a c t l y o p p o s i t e q u a l i t i e s , and t h e i r d i s t i n c t i o n between ex ante and ex post s i t u a t i o n s r e s t r i c t them to p u r e l y microeconomic a p p l i c a t i o n s . 27 Notes to chapter 1 ^Diewert (1977) d i s c u s s e s the c o n d i t i o n s of such an a g g r e g a t i o n . The same assumption i s a l s o made, but i s not so c e n t r a l , i n Jo rgenson 's model . Indeed, w r i t t i n g "x^ = I " i m p l i e s that adjustments of f a c t o r x 1 can be aggregated a d d i t i v e l y . However, Jo rgenson 's model i s e q u i v a -l e n t to s o l v i n g (4) at each d a t e : (4) Max x t , L t s i n c e f a c t o r s t o c k s , b e i n g p e r f e c t l y l i q u i d , can be r e s o l d e n t i r e l y a t each p e r i o d , the f a c t o r l e v e l s i m p l i e d by ( 4 ) , x^, can be viewed as e n t i r e l y a c q u i r e d and r e s o l d a t t . Thus, i n d i s c r e t e t i m e , one needs not i n t e r p r e t x^ + ^ as x , = x + I t+1 t t+1 On the c o n t r a r y , x^ + ^ and x^ _ can be viewed as two d i f f e r e n t f a c t o r s , a c q u i r e d a t t + 1 and t r e s p e c t i v e l y . In the a d j u s t m e n t - c o s t models , a c q u i r i n g and r e s e l l i n g s tocks e n t i r e l y over any p e r i o d would be i n f i n i t e l y c o s t l y ; hence the c e n t r a l r o l e of the aggregat ion assumpt ion . 2 V i n t a g e models of investment e x p l i c i t l y r u l e out the p o s s i b i l i t y of aggregat ing c a p i t a l over t i m e . One such model i s desc r ibed i n 1 . 5 . 3 The symmetry c o n d i t i o n ho lds i f and on ly i f s tock ad justments , I, are s e p a r a -b l e from v a r i a b l e f a c t o r s , L, i n the p r o d u c t i o n f u n c t i o n ; the l o n g - r u n demand [p - f f x , L ) - w ' - L - v '*x *t ( t ' t j f t t t 28 f o r s t o c k - f a c t o r s i s downward-s loping , but the l o n g - r u n supply need not be u p w a r d - s i o p i n g , u n l e s s adjustment c o s t s are n u l l (a s u f f i c i e n t c o n d i t i o n ) ; a s u f f i c i e n t c o n d i t i o n f o r downward-s ioping v a r i a b l e - f a c t o r demands i s aga in the s e p a r a b i l i t y of s tock adjustments from v a r i a b l e f a c t o r s i n the p r o d u c t i o n f u n c t i o n . ' ' A l t e r n a t i v e l y , one cou ld s p e c i f y a deownward-sloping demand c u r v e , to get a s i m i l a r r e s u l t w h i c h , u l t i m a t e l y , guarantees the e x i s t e n c e of a s o l u -t i o n to problem ( 7 ) . 5 0 f course t h i s i s i m p o s s i b l e i n p r a c t i c e : a t the aggregate l e v e l , some f i r m s must h o l d o l d e r v i n t a g e s of equipment; but t h i s i s p r e c i s e l y one of the reasons which i s s t r e s s e d i n t h i s c h a p t e r , to l ook f o r market reasons to l i m i t s t o c k s ' l i q u i d i t y . 6 Except i n the two extreme cases j u s t ment ioned, the d i s t i n c t i o n between f a c t o r s t h a t have the d imension of s t o c k s and those tha t have the dimension of f l o w s i s never c l e a r - c u t . The mode of remunerat ion of the f a c t o r i s c e r t a i n l y a most u n r e l i a b l e a c r i t e r i o n . As a matter of f a c t , Treadway (1970) t r e a t s a l l f a c t o r s i d e n t i c a l l y : they a l l i n c u r a d j u s t -ment c o s t s and they are a l l secured through an a c q u i s i t i o n (asset ) p r i c e and the subsequent payment of r e n t a l f lows (or wages, or m a i n t e -nance c o s t s , e t c . ) . 29 CHAPTER 2 REVERSIBLE INVESTMENT WITH A RESOURCE CONSTRAINT 2 . 1 I n t r o d u c t i o n The overv iew of investment theory presented i n chapter 1 l e d to a d i s t i n c t i o n between two types of approach. The f i r s t type emphasizes and r e f i n e s the c h a r a c t e r i z a t i o n of s tock adjustments by e x i s t i n g f i r m s . I t o r i g i n a t e d w i t h the work of Jorgenson (1963) and evolved i n t o c o s t - o f -adjustment models . The second type emphasizes the i r r e v e r s i b i l i t y of i n -vestment which i m p l i e s a d i s t i n c t i o n between ex ante and ex post phases. I t o r i g i n a t e d w i t h the work of Arrow (1968) and l e d to put ty - s e m i - p u t t y models . In t h i s c h a p t e r , the f i r s t type of approach i s extended to the case of f i r m s which e x t r a c t n a t u r a l r e s o u r c e s . Three s e c t i o n s d e a l each w i t h a d i f f e r e n t s i t u a t i o n . In the model of s e c t i o n 2 . 2 , the f i r m e x t r a c t s a homogeneous resource and does not face any adjustment c o s t ; i n s e c t i o n 2 . 3 , the f i r m e x t r a c t s a homogeneous resource and faces cos ts of ad justment ; f i n a l l y , i n s e c t i o n 2 . 4 , the f i r m e x t r a c t s a heterogeneous resource and faces cos ts of ad justment . Whi le each case can be i n t e r p r e t e d as a p a r -t i c u l a r v e r s i o n of the l a s t model , d i f f e r e n c e s are s u b s t a n t i a l enough that each model deserves b e i n g d e a l t w i t h s e p a r a t e l y . F i n a l l y , 30 s e c t i o n 2 .5 env isages a l t e r n a t i v e procedures and u n d e r l y i n g assumptions f o r the e m p i r i c a l e s t i m a t i o n of the cor responding f a c t o r demands. S e c t i o n 2 .6 summarizes the r e s u l t s o b t a i n e d . 2 .2 The n e o c l a s s i c a l model w i t h m a l l e a b l e and r e s a l a b l e f a c t o r s , when the  f i r m e x t r a c t s a homogeneous e x h a u s t i b l e resource 2 . 2 . 1 Setup Cons ider a f i r m whose t e c h n o l o g i c a l c o n s t r a i n t i s f o r m a l l y s i m i l a r to tha t d e s c r i b e d i n 1 . 2 . I t uses s t o c k - f a c t o r s and v a r i a b l e f a c t o r s to produce i t s ou tput . Both types of f a c t o r s can be ad jus ted i n s t a n t a n o u s l y , which means not on ly tha t there i s no t e c h n o l o g i c a l o b s t a c l e to d i s c r e t e changes i n s tock l e v e l s , but a l s o tha t the r e s a l e market i s such tha t a p i e c e of equipment can be r e s o l d at the going a c q u i s i t i o n p r i c e . As men-t i o n e d i n 1 . 2 , such setup i s on ly h a l f dynamic: i t does take account of the p e r e n n i a l nature of s t o c k s , but i t f a i l s to r e f l e c t t h e i r w e i g h t i n e s s : P i e c e s of equipment are assumed to be as v o l a t i l e as the most l i q u i d paper a s s e t . Now l e t t h i s f i r m face the a d d i t i o n a l c o n s t r a i n t that i t e x t r a c t s an e x h a u s t i b l e homogeneous resource whose i n i t i a l s t o c k at t ime TI i s R l . The dynamic o p t i m i z a t i o n problem can be formulated as f o l l o w s : (1) . Max { V L t } ' T 2 s u b j e c t t o : T2 f - r ( t - T l ) e TI d t + e - r ( T 2 - T l ) . ^ . 31 (2) (a) i t = I t ; (b) x T 1 = x x ; (c) x t * 0, t e [ T I , T2] ; (d) x T 2 = x 2 u n s p e c i f i e d ; (3) Lfc > 0 (4) (a) Rfc =-qt = - f k , L.J ; (b) = R ± ; (c) Rfc > 0, t e [TI, T2] ; (d) R T 2 = 0 ; The i n t e r p r e t a t i o n of (1) when I goes to i n f i n i t y i s given i n C l a r k et a l . (1979). I f one compares (1) w i t h i t s counterpart i n 1.2, one n o t i c e s the presence of an e x t r a term i n the o b j e c t i v e f u n c t i o n a l . That term, -r(T2—TI) T e '^2'X2' r e P r e s e n t s t n e s c r a P value of f a c t o r stocks at c l o s u r e date, T2. I t s i n t r o d u c t i o n i s c o n s i s t e n t w i t h , and required by, the as-sumption that stocks can be r e s o l d at the going a c q u i s i t i o n p r i c e . The same term i s i m p l i c i t l y present i n the corresponding model of chapter 1, but, being discounted over an i n f i n i t e p e r i o d , i t reduces to zero. The other new features i n the problem are the appearance of a new endogenous v a r i a b l e , the c l o s i n g date, T2, which i s of course the counterpart of the second new f e a t u r e , the a d d i t i o n a l resource c o n s t r a i n t (4). 2.2.2 S o l u t i o n For many of the models formulated i n t h i s d i s s e r t a t i o n , the pr o p e r t i e s r e q u i r e d to invoke an existence theorem would be too impover-i s h i n g from the poi n t of view of the economic content of the model. 32 Instead, we g e n e r a l l y simply assume that a s o l u t i o n e x i s t s . In t h i s p a r t i c u l a r case, however, optimal paths can be c h a r a c t e r i z e d i n such a way that e x i s t e n c e i s not i n doubt, f o r i t could be shown by d i r e c t argument that the net present value achieved by the program ( x x , L") corresponding to (9) and (10) below, i s higher than the net present value obtained through any a l t e r n a t i v e program. I t i s assumed that p r i c e paths are smooth and that the production l f u n c t i o n s a t i s f i e s the Inada c o n d i t i o n s : A l f ( - ) i s twice continuously d i f f e r e n t i a b l e ; A2 f ( 0 i s non-decreasing i n x and L; A3 f ( - ) i s s t r i c t l y concave; A4 11m f (•) =0; x A5 l i m f (•) = 0; L->°° A6 1 im f (•) = 0 0; x^O X A7 l i m f (•) = o o . L->0 L Although a l l c o n d i t i o n s f o r the a p p l i c a t i o n the Maximum P r i n c i p l e are not met, the maximization of the Hamiltonian has such an appealing h e u r i s t i c i n t e r p r e t a t i o n here that we choose to apply the Maximum P r i n c i p l e . While i t s a p p l i c a t i o n i s not f u l l y j u s t i f i e d , as mentioned above, the s o l u t i o n i t y i e l d s can be shown to be o p t i m a l . The Hamiltonian of problem ( l ) - ( 4 ) i s : 33 (5) H = e - r - ( t - T l ) P - P F t t f x ,L t t w' -L - fd)' t t |/t where u and A are respectively the costate variable associated with t t the resource stock R and the vector of costate variables asso-t ciated with factor stocks x . t By the Maximum p r i n c i p l e , H must be maximized with respect to 1^ at a l l admissible dates. One notes that, since H i s linear i n 1^, the optimal decisions depend on whether the elements of the vector of switching func-tions , (6) ait) = fX t - <|>tj are positive, negative or n u l l . Consider the case where the switching function i s n u l l . Then, (7) Xt = <f>t and X t = 4>t Also, by the maximum p r i n c i p l e . (8) X t = rX -A - ( p t - u t ) - y - > Substituting (7) into (8), one has (9) fpt-Ut]-f„(-) = r x s ' ~ T t T t ' 34 which i s the conventional user cost r e l a t i o n , except that the output p r i c e has been co r r e c t e d to r e f l e c t the i m p l i c i t value of the resource. Equation (9) defines a s i n g u l a r path f o r x^ _. That such a path i s p r e f e r a b l e to any other one r e s u l t s from the f a c t that the indeterminacy of I i n the maxi-m i z a t i o n of the Hamiltonian, means that no marginal change i n x could a f f e c t the o b j e c t i v e f u n c t i o n a l . Furthermore, by the concavity of f ( " ) , each point so c h a r a c t e r i z e d i s a unique maximum. Hence the s i n g u l a r path •k cannot be improved upon. Now, i f x i s on the s i n g u l a r path and x^ i s d i f -f e r e n t from x^, the optimal p o l i c y i s to b r i n g x instantaneously to x^. That the move i s instantaneous r e s u l t s from the bang-bang nature of the problem, and the absence of bounds on I . That the optimal move i s toward x , not away from i t , again r e s u l t s from the concavity of the production f u n c t i o n . Now the Hamiltonian must a l s o be maximized w i t h respect to L at a l l dates. Hence. (10) [p - W t ] - f L ( - ) = w t The assumption that p r i c e s are continuous over time guarantees the existence of the right-hand s i d e of (9). The assumption that f ( - ) s a t i s f i e s the Inada c o n d i t i o n s guarantees the. e x i s t e n c e of an i n t e r i o r s o l u t i o n T * * L , x , to (9) and (10), provided the right-hand side of (9) t ' t i s p o s i t i v e , i n which case c a p i t a l gains would be such that a f a c t o r stock would be worth h o l d i n g i n i n f i n i t e amount despi t e a n u l l produc-t i v i t y , and provided p - i s p o s i t i v e , i n which case the non n e g a t i v i t y 35 c o n s t r a i n t on f a c t o r l e v e l s would become bi n d i n g and production would be i n t e r r u p t e d . Those d i f f i c u l t i e s could p o s s i b l y be handled but we have prefered to assume them away. The s o l u t i o n L ,x t ' t not only e x i s t s at a l l dates; by the assumption that p r i c e paths are smooth, i t defines smooth t r a i e c t o r i e s f o r L and x . J t t In order to concentrate the argument on stock f a c t o r s , one can now combine (9) and (10) to get (ID n x xt,pt - u t ,w t = r.*t - *t = vt where; (12) n f x t , p t - V w . = pfc - u j - f xT,L x t , p t - V w t - W;.L [ x t , p t - u t , w t j , 7ith L (•) i m p l i c i t l y defined when (10) i s s o l v e d f o r L. I f the Hamiltonian i s i n t e r p r e t e d as d e f i n i n g the i m p l i c i t net revenues of the f i r m , ![(•) can be i n t e r p r e t e d as an i m p l i c i t r e s t r i c t e d p r o f i t f u n c t i o n , expressed i n current terms, and has the p r o p e r t i e s i d e n t i f i e d by Lau (1976), that i s : !!(•) i s decreasing i n w and i n c r e a s i n g i n p-u and x; n( -) i s convex i n w and p-u; n(') i s concave i n x; 36 ^ g i ves the i m p l i c i t a s s e t p r i c e of the s t o c k s x . oX L The l a s t p roper t y of n(-) was e s t a b l i s h e d i n the procedure which y i e l d e d (11). * Now, i f (11) i s so lved f o r x f c , and the s o l u t i o n i s s u b s t i t u t e d i n t o (12), one has the maximized H a m i l t o n i a n , * (6 ) H * ( p t - v v w t ) = ( p t - y t ) ' f x t ( p t - v v t ) L fx ( ' ) , P t - U t , w t - w t - L (•)• A g a i n , s i n c e f(*) i s concave, the maximized H a m i l t o n i a n can be i n t e r p r e t e d as an i m p l i c i t p r o f i t f u n c t i o n w i t h the u s u a l p r o p e r t i e s . H ('•) i s dec reas ing i n w and v , and i n c r e a s i n g i n p - u ; H (•) i s convex; 3H*(-) * , . 8H*(-) * . 3H*(Q * = ~ ( 0 ' = _ X ( 0 ' 9(p-y) = q • where q = f (x*,L*) Now, d i f f e r e n t i a t i n g (11) w i t h respect to t i m e , one o b t a i n s : (13) n x x(.) . i t + n x > p _ p ( . )• ( p t - y t ) + n ^ O - w t = r -* t - ^ 37 a ve ry complex system of non l i n e a r d i f f e r e n t i a l equat ions i n x w h i c h , as was e x p l a i n e d above, has a unique s o l u t i o n under the assumptions made. V a r i o u s s p e c i a l cases can be d e r i v e d from (13) . When ent repreneurs have s t a t i c p r i c e e x p e c t a t i o n s , (13) reduces t o : (13) n (-)-i - n (-)-vi = o xx t x , p - y t Now, s i n c e 11(0 i s a r e s t r i c t e d p r o f i t f u n c t i o n , by H o t e l l i n g ' s lemma, one has r * ] = f x , L (•) J = n X p d i f f e r e n t i a t i n g t h i s e x p r e s s i o n w i t h r e s p e c t to t i m e , (14) q = n ^ ( O - f - y J + n (-)-i. p - y , p - y { t j p - y , x t Now l e f t m u l t i p l y i n g (13) ' by x^ _ y i e l d s : x . n ( « ) - x - i -n ( - ) - y = 0 t xx t t x , p - y t S ince II i s n e g a t i v e s e m i d e f i n i t e , the second term i n the above e x p r e s -xx s i o n must be n e g a t i v e . But , from the theory of dynamic o p t i m i z a t i o n , the motion of obeys the f o l l o w i n g r u l e : 38 ('• \ - r - ( t - T l ) j y — r•y fc|•e - = - H = 0 R Hence, (15) p t = r - y t > 0 As a r e s u l t , (16) n (•)•£,. < 0 p - y , x v t S u b s t i t u t i n g (16) i n t o (14 ) , u s i n g (15) a g a i n , and c o n s i d e r i n g tha t II (•) i s p o s i t i v e , one o b t a i n s : (17) q t < 0 (17) i s a f a m i l i a r r e s u l t i n the theory of e x h a u s t i b l e - r e s o u r c e e x t r a c t i o n : (e .g . Dasgupta and Heal (1979) , chapter 6 ; S c h u l t z e (1974) ; Gray ( 1 9 1 4 ) ) . I f there i s on ly one s tock f a c t o r and no v a r i a b l e f a c t o r s , and ent repreneurs h o l d s t a t i c e x p e c t a t i o n s , by (12) , (18) n (•) = f fx } , so t h a t , x , p - y x{ t j u s i n g (13) and (15) , one o b t a i n s : (19) kt < 0 39 Going back to the problem of e x p r e s s i n g f a c t o r demands, one notes tha t the a p p l i c a t i o n of H o t e l l i n g ' s theorem to the maximized H a m i l t o n i a n (6*) does not p rov ide a complete answer, s i n c e i t g i v e s f a c t o r demands as a f u n c t i o n of u^, the i m p l i c i t m a r g i n a l resource v a l u e . In order to complete the c h a r a c t e r i z a t i o n of the s o l u t i o n , u must be endogenized, and the t e r m i n a l d a t e , T2, must a l s o be determined. Th is i s done by making use of the equat ion of motion of u^, (15 ) , as w e l l as the t r a n s v e r s a l i t y c o n d i t i o n : S ince (11) must h o l d a l s o at the t e r m i n a l t ime T2, one can make the f o l -l ow ing remarks : - Given the assumption tha t the r i g h t - h a n d s i d e of (11) i s p o s i t i v e , by the c o n c a v i t y of 11(0 i n x , x^ > 0=>n^X2,"j > 0 , which c o n t r a d i c t s (20 ) ; hence, (20) (21) C l e a r l y , by the same argument, (22) L 2 = 0 40 Consider the e f f e c t on the net present v a l u e of the f i r m , as determined by s o l v i n g problem (1) - ( 4 ) , of a m a r g i n a l i n c r e a s e i n the reserve s t o c k . S ince the i n i t i a l l y o p t i m a l program remains f e a s i b l e , the i n c r e a s e must be at l e a s t that achieved by ex tend ing tha t program i n such a way as to e x t r a c t the m a r g i n a l u n i t at T2. In tha t case , remembering t h a t , by ( 2 1 ) , (22) , f a c t o r l e v e l s are n u l l , and that the p r o d u c t i o n f u n c t i o n s a t i s f i e s the Inada c o n d i t i o n s , so that the m a r g i n a l cos t i s n u l l , the i n c r e a s e so achieved i s equal to the p r i c e of the e x t r a u n i t produced at T2. S ince u 2 rep resents the v a l u e of the l a s t resource u n i t i n i t s best u s e , and the m o d i f i e d program needs not be o p t i m a l , i t f o l l o w s t h a t : But i f p 2 w a s h i g h e r than p 2 , the i m p l i c i t p r o f i t would be n e g a t i v e , which would c o n t r a d i c t (20) . As a r e s u l t , (23) u 2 = p 2 . Of c o u r s e , the n o t a t i o n used should not d i s s i m u l a t e the f a c t tha t (23) i s an equat ion i n T2. The s o l u t i o n has now been f u l l y c h a r a c t e r i z e d . As a r e c a p i t u l a -t i o n , the equat ions i n v o l v e d are gathered below, i n t o system IA , of equa -t i o n s which apply at a l l dates d u r i n g the o p e r a t i o n p e r i o d , and I B , of i n i t i a l and t e r m i n a l c o n s t r a i n t s . TI and T2, the i n i t i a l and t e r m i n a l d a t e s , have been r e i n t r o d u c e d e x p l i c i t l y to emphasize the f a c t that TI 41 i s g i ven and T2 i s endogenous. The s t a r s used to i n d i c a t e o p t i m a l f a c t o r l e v e l s have been o m i t t e d . IA ft - H (•) v * - H (•) w R. X t L t - H (•) p-u , a t any t . IB x(T2) L(T2) y(T2) R(T1) R(T2) f 0 0 p(T2) R l 0 ,T1 g i v e n ; T2 unknown System IA, IB looks overdetermined s i n c e both IA and IB g i ve f a c t o r l e v e l s at T2. However i t i s e a s i l y seen t h a t , w i t h u(T2) = p (T2) , the equat ions f o r and L f c g ive L(T2) = x(T2) = 0 . The l a s t two equat ions i n IA are f i r s t degree d i f f e r e n t i a l e q u a t i o n s , which are i m p l i c i t l y so l ved i n presence of two c o n d i t i o n s . I f one e l i m i n a t e s the equat ions f o r L(T2) and x ( T 2 ) , IB c o n t a i n s three such c o n d i t i o n s . The e x t r a one a l l o w s the * d e t e r m i n a t i o n of T2. F i n a l l y , one notes t h a t , i f H (•) i s a q u a d r a t i c * f u n c t i o n , - H (•) i s l i n e a r , so that the d i f f e r e n t i a l equat ion f o r R can 42 be so l ved e x p l i c i t l y . 2 . 3 The c o s t - o f - a d j u s t m e n t model w i t h r e s a l a b l e f a c t o r s , when the f i r m  e x t r a c t s a homogeneous e x h a u s t i b l e resource 2 . 3 . 1 Setup Consider a f i r m which uses s t o c k - f a c t o r s and v a r i a b l e f a c t o r s i n order to e x t r a c t and process an e x h a u s t i b l e r e s o u r c e . I t d i f f e r s from the f i r m s t u d i e d i n 2 .2 by i t s t e c h n o l o g i c a l c o n s t r a i n t . F a c t o r s t o c k s are now c o s t l y to a d j u s t . As i s t r a d i t i o n n a l l y done i n modern s t u d i e s of t h i s v e i n (Treadway (1970) ) , the adjustment c o s t s are not assumed to be separab le from the p r o d u c t i o n f u n c t i o n . On the c o n t r a r y , i t i s assumed that adjustments to s t o c k s a f f e c t output n e g a t i v e l y , but i n a way which may vary a c c o r d i n g to s t o c k - f a c t o r and v a r i a b l e - f a c t o r l e v e l s , as w e l l as the l e v e l of other ad justments . In o ther words the p r o d u c t i o n f u n c t i o n has s tock ad jus tments , as w e l l as f a c t o r l e v e l s , as arguments. F o r m a l l y , (24) q = f ( x , L , I ) , where f ( - ) has the f o l l o w i n g p r o p e r t i e s : B l f ( - ) i s non d e c r e a s i n g i n x , and L; B2 f ( - ) i s tw ice c o n t i n u o u s l y d i f f e r e n t i a b l e ; B3 f ( ' ) i s concave i n L, f o r any g iven I, x ; B4 l i m f L ( - ) = 0 ; L-*=° 43 B5 lim f (•) =00; x+0 X B6 lim f (•) = °°; L->G L B7 f(*) i s non increasing i n | l | ; B8 f(») i s concave i n I, for any given x,L; B9 lim f T ( - ) = -°° I I I - 1 BIO f(x,L,0) i s positive for x,L > 0. The problem i s similar to that considered i n 2.2, problem (1) -(4), except that the production function just defined, (24), must be substituted for f(x,L) in (1) and (4). We assume that a solution exists and ignore the problems that might be associated with the non negativity constraints on x and R. Given B9, i t i s reasonable to rule out any jump in x. 2.3.2 Solution The Hamiltonian of the problem i s : ( p t - y t ) - f ( x t , L t , i t ) - w ; . L t - ^ ; - x ; ) . i t , where, again, u and X are respectively the costate variable and costate variable vector associated with R and x . t t As i n 2.2, i t i s assumed that: p - u > 0 , t e [T1,T2] . * t t Then, considering B7, B8, B9 (B5, B7), there exists an i n t e r i o r solution (25) H = e -r-(t-Tl) 44 to the problem of max imiz ing (25) w i t h respec t to I ( t o L ) . Thus by the maximum p r i n c i p l e , (26) (27) (P t -M 'MVV 1 *) " w The Legendre c o n d i t i o n s are a u t o m a t i c a l l y s a t i s f i e d . A g a i n , the maximized Hami l ton ian can be i n t e r p r e t e d as an i m p l i c i t r e s t r i c t e d p r o f i t f u n c t i o n : (28) - r - ( t - T l ) _t , , ^ • n ( X t ' P t " y t * W t ' * t " X t ] = 6 - r - ( t - T l ) ' P t _ V t ) , f l X t » L ^ t j w • L t t t i <}) - A t t !!(•) has the f o l l o w i n g p r o p e r t i e s : - I I ( ' ) i s d e c r e a s i n g i n w and <J>-A, and i n c r e a s i n g i n p - y ; - n ( - ) i s convex i n p - y , w, and <|>-A; - n ( - ) i s i n c r e a s i n g i n x ; - i * = - n . , (•) ; L" = - n ( • ) ; q * = n ( • ) . (j>-A w p ~ y The equat ions of motion of the c o s t a t e v a r i a b l e s must s a t i s f y : - r - ( t - T l ) = - H R ( - ) , o r , (30) ii = r•y , and 45 X - r -X t t - r - ( t - T l ) , s •e = - H (•), or (31) X t = r X t - ( p t - y t ] . f x ( . ) Cons ider now the t r a n s v e r s a l i t y c o n d i t i o n s . F i r s t , s i n c e T2 i s u n s p e c i f i e d , one h a s : H(T2) = 0 , or (32) (P 2 " ^ ( W z ) W 2 ' L 2 ~ 1*2 ~ A 2 j " I 2 = Second, s i n c e i s u n s p e c i f i e d , one h a s : - r - ( T 2 - T l ) X - r - ( T 2 - T l ) <J> e • 2 = e 2' °r (33) X2 = *2 S ince (26) ho lds at T2, remembering the assumption tha t f ( ' , I ) i s maxi -mized at I = 0 , B7, one g e t s : (34) i 2 = o. S u b s t i t u t i n g (34) and (33) i n t o (32 ) , one has: (35) j p 2 - u 2 j - f [ x 2 , L 2 , 0 ] - w 2 - L 2 = 0 . 46 But , from the c o n c a v i t y of f ( # ) i n L, (35) can be s a t i s f i e d s i m u l t a n e o u s l y w i t h (27) o n l y i f : (36) L 2 = 0 Now, i f x 2 > 0, s i n c e f^x2,0,0J = °° when (36) h o l d s , (27) i m p l i e s : (37) (a) p 2 = u 2 i f x 2 > 0. I f x 2 = 0, one can invoke the c o n t i n u i t y of A^, which r e q u i r e s X 2 to be f i n i t e , to deduce t h a t ^p^ -toward i n f i n i t y . As a r e s u l t , must tend toward zero i f f (•) i s to tend x l i m u = p . , so t h a t , t+T2 T (37) (b) p 2 = p 2 i f x 2 = 0. Combining (37)(a) and (37)(b), one has: (37) P 2 = p 2 A l l the c o n d i t i o n s which f u l l y c h a r a c t e r i z e the s o l u t i o n have now been l a i d down. However, some s i m p l i f i c a t i o n s can s t i l l be a c h i e v e d . D i f f e r e n t i a t i n g (26) w i t h respec t to t i m e , one h a s : 47 (38) ;Pt _ i t ) " f I ( ' ) + (Pt " y t ) ' + f I X ( - ) - I t V ^ t + f I L < - > ' L t (26) and (38) are used to e l i m i n a t e X f c and X from (31) to g e t : ( p t - ' , t ) - f i < - ) - * t - ( p t - " t ) - f i ( - > t t f I x ( - > ' X t + f I L ( - > - L t 'hl^'h p t - y t ] - f x ( - ) , or (39) f I x ( ' > - X t + f I L ( - } - L t + r -P t - y t P t - y t fx(-> t t r • <j> One notes t h a t , i f I ' i s not an argument of f ( ' ) , (39) reduces to the e q u a l i t y of the l a s t two terms on the r i g h t - h a n d s i d e w i t h z e r o , a b a s i c user cos t r e l a t i o n w h i c h , not s u r p r i s i n g l y , was found to c h a r a c t e r i z e the o p t i m a l f a c t o r s tock l e v e l s i n the absence of adjustment c o s t s ( r e l a t i o n (11)) (27) i s a l s o d i f f e r e n t i a t e d w i t h r e s p e c t to t ime to g e t : (40) t v t t z - - E — - - f T (•) p t u t p t - y t L 48 (39) and (40), combined with (30), the equation of motion of and (2)(a), the d e f i n i t i o n of 1^, constitute a system of (n + m + 1 + n) fi r s t - d e g r e e non l i n e a r d i f f e r e n t i a l equations i n x, L, I, u, which has a s o l u t i o n under the assumptions made. In order that s o l u t i o n to be f u l l y characterized, (n + m t 1 + 1) dated conditions are needed. Those are (37), (36), (34), derived above, and (2)(b), which gives i n i t i a l stock l e v e l s . Unfortunately, while (2) (b) i s dated at TI, which i s given, (34), (36), and (37) are dated at T2, which i s unknown. In order to close the system, an extra condition i s necessary, which i s given by (4)(a), (4)(b), (4)(c), a d i f f e r e n t i a l equation i n Rfc with two dated conditions. Those equations and conditions are gathered below, into matrix equations IIA and IIB, where time subscripts have been omitted: IIA f l x ( ' > fIL (-> f I I ( - > 0 0 fLx<'> fLL<-> fLI (-> 0 0 i n 0 0 0 0 0 0 • 0 1 0 0 0 0 0 1 ' X ' L i p R r - P~P P-Pj f T ( - ) - — I p-p -M-> + — p-p L p-p r-p - f ( - ) (r-(fr-$) + f (•) x 49 I IB ' x ( T l ) [ X ) L(T2) 0 I (T2) 0 U(T2) P(T2) R(T1) R l R(T2) 4 0 , TI g i v e n , T2 unknown. 2 . 3 . 3 Comments and s p e c i a l cases An important d i f f e r e n c e between the model w i t h adjustment c o s t s and the model w i thout adjustment c o s t s i s that i n the former the p roduc -t i o n f u n c t i o n was not assumed to be j o i n t l y concave i n f a c t o r s , x and L, so that i t i s capable of r e f l e c t i n g the presence of economies of s c a l e , a f e a t u r e which i s hard to r u l e out i n e m p i r i c a l s t u d i e s of m i n i n g . Th is advantage of the cost of adjustment model can be e x p l a i n e d as f o l l o w s : i n presence of i n c r e a s i n g r e t u r n s to s c a l e , the f i r m i s unable to i n s t a n -taneous ly a d j u s t i t s f a c t o r s tock l e v e l s and l e t them tend toward i n f i n i t y i n order to e x t r a c t the resource i n s t a n t a n e o u s l y . I t should be n o t e d , however, that such an advantage r e l i e s on i n i t i a l s tock l e v e l s b e i n g g i v e n . But a model of f a c t o r demand which f a i l s to e x p l a i n i n i t i a l f a c t o r l e v e l s and t r e a t s those a s , somehow, i n h e r i t e d , f a l l s shor t of p r o v i d i n g a com-p l e t e theory of f a c t o r demand. Th is i s s u e i s the focus of chapter 4 below, and concerns c o n v e n t i o n a l n o n - r e s o u r c e models of f a c t o r demand 50 j u s t as w e l l . Bes ides the above genera l remark, a number of p o i n t s should be s t r e s s e d about the c o s t - o f - a d j u s t m e n t model . F i r s t , i t s complex i t y i s r e f l e c t e d i n the f a c t that the system which c h a r a c t e r i z e s i t s s o l u t i o n i n v o l v e s n more d i f f e r e n t i a l equat ions and n more c o n d i t i o n s than i t s ins tantaneous -ad jus tment c o u n t e r p a r t . To make the same p o i n t i n a d i f f e r e n t way, where the s o l u t i o n of the ins tantaneous -ad jus tment model i s c h a r a c t e r i z e d by f i r s t - d e g r e e d i f f e r e n t i a l equat ions i n the s t o c k s , i t i s c h a r a c t e r i z e d by second-degree d i f f e r e n t i a l equat ions i n the adjustment cos t model . Second, the same type of s i m p l i f i c a t i o n s as i n the other model occur when v a r i o u s s p e c i a l cases are c o n s i d e r e d : when p r i c e s are assumed to be c o n s t a n t , s e v e r a l terms d i s a p p e a r , and , moreover, one can so l ve f o r y^ independent ly of t h e . r e s t of the system; when the p r o d u c t i o n f u n c t i o n i s q u a d r a t i c i n a l l i t s arguments, a l l d i f f e r e n t i a l e q u a t i o n s , except the one which g i ves the motion of the r e s e r v e s , become l i n e a r and capable of b e i n g so lved e x p l i c i t l y . Th is f e a t u r e has i n t e r e s t i n g empi r -i c a l i m p l i c a t i o n s , as w i l l be shown i n 2 . 5 . 2 .4 The case of heterogeneous resources 2 . 4 . 1 I n t r o d u c t i o n Students of n a t u r a l - r e s o u r c e e x t r a c t i o n theory have l o n g r e c o g -n i z e d the f a c t that rese rves may not be homogenous (Levhar i and L i v i a t a n (1977) , Puu (1977) ) . Th is may be the case a t the f i r m ' s l e v e l i f a f i r m has access to resources of d i f f e r e n t grades w i t h i n a s i n g l e d e p o s i t , and 51 thus r a t i o n a l l y chooses to e x t r a c t h i g h - g r a d e ore f i r s t ; t h i s i s a l s o the case i f one c o n s i d e r s a m u l t i - f i r m s e c t o r where each f i r m e x t r a c t s a d i f -f e r e n t type of ore (Goldsmith (1974) ) . However, c o n s i d e r i n g a whole s e c t o r r a i s e s the s u b s t a n t i a l aggregat ion problems s t u d i e d by B lackorby and Schworm (1980) . C o n s i s t e n t w i t h the whole f o r e g o i n g r e s e a r c h , on ly the pure microeconomic case w i l l be s t u d i e d h e r e . At that l e v e l , c o m p l i -c a t i o n s may a r i s e i n v a r i o u s ways. In the most genera l c a s e , the cost of e x t r a c t i n g and p r o c e s s i n g one ton of ore i n c r e a s e s as r e s e r v e s get d e p l e t e d , because p i t or s h a f t depth i n c r e a s e s , h a u l i n g d i s t a n c e s i n c r e a s e , the c o n c e n t r a t i o n of one ton of ore becomes more d i f f i c u l t as grade d i -min ishes and, i n a d d i t i o n to t h o s e , and o t h e r , cost e lements , the r e v e -nues from the t reatment of one ton of ore d i m i n i s h as grade d i m i n i s h e s . S ince the same approach i s used to so l ve the problem as i n 2 . 3 , next s u b s e c t i o n 2 . 4 . 2 puts the emphasis on f o r m u l a t i o n and r e s u l t s , w h i l e , i n 2 . 4 . 3 , a number of s p e c i a l cases are examined. 2 . 4 . 2 Set -up and s o l u t i o n The most g e n e r a l way ' to account f o r resource h e t e r o g e n e i t y would be to t r e a t each u n i t of ore as a d i s t i n c t input w i t h i n an i n t e r t e m p o r a l resource a l l o c a t i o n problem. B u t , w i t h an i n f i n i t e number of v a r i a b l e s , the problem would be i n t r a c t a b l e . A " f a i r l y g e n e r a l " f o r m u l a t i o n w i l l be adopted i n s t e a d . I t c o n s i s t s i n assuming t h a t , at any d a t e , the t e c h n o l o g i c a l c o n s t r a i n t i s dependent upon the amount of r e s e r v e s s t i l l i n p l a c e o r , which i s e q u i v a l e n t i n models where reserves are a f f e c t e d by e x t r a c t i o n only, on the amount of cumulated e x t r a c t i o n . As a r e s u l t , 52 d i f f e r e n t reserve l e v e l s w i l l correspond to d i f f e r e n t extraction costs and conditions. So the production function i s written: (41) q t = f(x t,L t,x t,R t) One sensible assumption can be made from the outset on the response of f ( - ) to changes i n R^: since i t i s i n the i n t e r e s t of firms to extract valuable ore f i r s t , f 0 ( * ) i s assumed to be p o s i t i v e . Other properties R of f ( 0 are s p e c i f i e d further below. The problem to solve i s given by (42)-(45) below. (42) Max ,T2 T2 f -r(T2-Tl) J 6 TI p • f (• ) -w'-L - <|>' • I F t t t y t t •dt subject to: -r-(T2-T1) ,' + e '<*)2*X2 (43) (a) 4 = I t ; (b) x T 1 = x n ; (c) xfc > 0,te[Tl,T2]; x^2 unspecified (44) L > 0, t e [T1,T2]; (45) (a) R. = - q t = - f ( - ) ; 0>) RT]_ = \ ; (c) Rfc > 0, t e [ T l , T 2 ] ; 53 Bes ides the form of the p r o d u c t i o n f u n c t i o n , f ( * ) » the s o l e d i f f e r e n c e between (42) - (45) and i t s counte rpar t i n p rev ious s e c t i o n s i s the 2 abandonment of the c o n s t r a i n t tha t rese rves should be exhausted at T2. The problem i s so l ved by s t r a i g h t f o r w a r d a p p l i c a t i o n of c o n t r o l t h e o r y , i n the same manner as i n 2 . 3 . The s o l u t i o n i s c h a r a c t e r i z e d by a system of d i f f e r e n t i a l e q u a t i o n s , I I I A , which i s the counte rpar t of I IA , i n 2 . 3 . I I I A r f i x ( - > f T (•) Lx l n f I L ( - > £ L L ( ' > f I X ( - ) hi" 0 0 f I R ( , ) fLR<'> 0 1 L i r--E=l i p-y j w p-y • £ I ( 0 •fT (•) p-y p-y L r - y - (p - y )• f R (* ) I -f<0 ( r ••-«(.) + f x ( - ) 4 54 A comparison of I IA w i t h I I I A i d e n t i f i e s a d d i t i o n a l terms i n the (n + m) f i r s t equat ions of I I I A , namely p a r t i a l d e r i v a t i v e s of f T ( ' ) and f T ( 0 w i t h r e s p e c t to R. A d i f f e r e n c e which turns out to be more s i g n i f i c a n t i s the appearance of a new term, (p-p)*f (•), i n the equat ion which g ives R the motion of the i m p l i c i t p r i c e of the r e s o u r c e , u . The importance of those d i f f e r e n c e s w i l l be d i s c u s s e d i n 2 . 5 . As i n 2 . 3 , we have a system of (n + m + n + 1 + 1) f i r s t degree d i f f e r e n t i a l equat ions i n x , L, I, u, R, which w i l l be u n i q u e l y so l ved i n the presence of (n + m + n + 1 + 1) dated c o n d i t i o n s . I f i t i s assumed t h a t 2 T2 i s f i n i t e , (n + m + n + 1 + 1 + 1) c o n d i t i o n s can be der i ved from i n i t i a l and t r a n s v e r s a l i t y c o n d i t i o n s , which i s one more than r e q u i r e d ; the e x t r a c o n d i t i o n i s used to determine T2 endogenously. When T2 i s f i n i t e , two p o s s i b i l i t i e s must be env isaged . F i r s t , the r e s o u r c e , a l though h e t e r o -geneous, may be of a q u a l i t y which meets some minimum requ i rements . Then, i f , as assumed i n 2 . 3 , m a r g i n a l p roducts tend toward i n f i n i t y at n u l f a c t o r l e v e l s , i t can be shown that the i m p l i c i t p r i c e of the resource catches up w i t h i t s market p r i c e p r e c i s e l y at T2. The proof f o l l o w s the same argument as was used i n 2 . 3 . The second p o s s i b i l i t y i s that a p o r t i o n of the resource might be of very poor q u a l i t y , a lmost v a l u e l e s s . Then, i n order f o r T2 to be f i n i t e , the r a t e of e x t r a c t i o n must not tend toward zero when the q u a l i t y of the remain ing ore approaches the c u t - o f f l e v e l , as i s i m p l i e d when marg ina l f a c t o r products are maximized at z e r o . Such d i f f i c u l t y i s avoided by imposing assumptions D6 and D7, below, which ensure that marg ina l products are r e l a t i v e l y l o v e r at l o v e r f a c t o r l e v e l s than at cone h igher f a c t o r l e v e l s , so tha t p roduc t ion at such lower f a c t o r l e v e l s w i l l 55 not be observed . A l s o , i f marg ina l products are assumed to be everywhere f i n i t e some of the lower q u a l i t y ore must be l e f t i n the ground. The two p o s s i b i l i t i e s j u s t envisaged imply the f o l l o w i n g a l t e r n a t i v e systems of c o n d i t i o n s : I I I B l x ( T l ) L(T2) I (T2) u(T2) R(T1) R(T2) 0 0 P (T2) R, , TI g i v e n ; T2 unknown but f i n i t e ; f ( - ) s a t i s f i e s C1-CT1 below. I l l B2 ' x ( T l ) ' L(T2) * L 2 I(T2) 0 U(T2) 0 R(T1) R l R(T2) [ K 2 J , TI g i v e n ; T2 unknown but f i n i t e ; R d e f i n e d by (46) be low; f(*) s a t i s f i e s D l - D l l , below. I f some resource i s l e f t i n the ground at c l o s u r e t i m e , the l a s t u n i t e x t r a c t e d must j u s t cover e x t r a c t i o n c o s t s at maximum m a r g i n a l products of the v a r i a b l e f a c t o r s , t h a t i s to say : 56 (46) P 2 . f L | x 2 , L 2 \ o y W 2 As i n p r e v i o u s s e c t i o n s , i t i s assumed that a n t i c i p a t e d p r i c e paths are such tha t i t i s never o p t i m a l to i n t e r r u p t opera t ions t e m p o r a r i l y . Then, i n order I H _ , to a p p l y , i t i s s u f f i c i e n t that f ( x , L , I , R ) B l s a t i s f y the f o l l o w i n g p r o p e r t i e s (C ) : CI f ( - ) i s non d e c r e a s i n g i n x and L; C2 f ( - ) i s tw ice c o n t i n u o u s l y d i f f e r e n t i a b l e ; C3 f ( - ) i s concave i n L, f o r any g iven I, x , R; C4 f ( « ) i s non d e c r e a s i n g i n R; C5 l i m f (•) = 0 ; T Li L-*» C6 l i m f (•) = °°; x+0 X C7 l i m f (•) = °°; L^O L C8 l i m f ( x , L, 0 , R) > 0 , f o r any x > 0 , L > 0 ; R+0 C9 f(*) i s non i n c r e a s i n g i n | l | ; C10 f ( « ) i s concave i n I, f o r any g iven x , L; C l l l i m f T ( - ) = - « Il|-x» 1 S i m i l a r i l y , i n order H l g 2 t o a P P ^ y » i c i s s u f f i c i e n t that f ( x , L, I, R) s a t i s f y the f o l l o w i n g p r o p e r t i e s (D) : 57 Dl f ( 0 i s non decreasing i n x and L; D2 f ( - ) i s twice continuously d i f f e r e n t i a b l e ; D3 f ( \ ) i s concave i n L, for any given I, x, R provided L and x are s u f f i c i e n t l y large; D5 lim f (•) = 0 D6 f (•) i s increasing i n x for small l e v e l s of x and f (•) x x i s f i n i t e f o r any x; D7 f T ( ' ) i s increasing i n L for small l e v e l s of L and f T ( * ) L t. i s f i n i t e f o r any L; D8 lim f ( x , L, 0, R) = 0 for any x > 0, L > 0; R+0 D9 f ( - ) i s non increasing i n | l | ; D10 f ( - ) i s concave i n I, for any given x, L; = +oo D l l lim f T ( 0 Assumptions C and D have been designed i n order to imply a f i n i t e horizon. Cle a r l y , a t h i r d system of conditions, I I I ,j, could be derived from the appropriate i n i t i a l and t r a n s v e r s a l i t y conditions i f T2 were to tend toward i n f i n i t y . F i n a l l y , j u s t as systems IIIA and IIIB are the generalization, when reserves are heterogeneous, of the model with adjustment costs studied i n 2.3, the model with c o s t l e s s adjustments studied i n 2.2 can be extended to accommodate a production function which s h i f t s when re s -serves get depleted. The same complications as were just dealt with, concerning the f i n i t e n e s s of the extraction period, a r i s e also i n that simpler case. 58 2 . 4 . 3 Comments and s p e c i a l cases The i m p l i c i t c h a r a c t e r i z a t i o n of f a c t o r demands was not markedly compl i ca ted by the i n t r o d u c t i o n of heterogeneous r e s e r v e s . As b e f o r e , the s o l u t i o n i n v o l v e s a system of n o n - l i n e a r f i r s t - d e g r e e d i f f e r e n t i a l equa -t i o n s i n x , L, I, y , R. assumption tha t a n t i c i p a t e d p r i c e s are constant and from the assumption tha t the p r o d u c t i o n f u n c t i o n i s quadrat i c* In the presence of the heterogeneous r e s e r v e s , however, the same s i m p l i f i c a t i o n s do not q u i t e a r i s e : the path of y^, as g i ven by (47) , be low, cannot be computed independent ly from the r e s t of the system f o r any a r b i t r a r y t e r m i n a l da te . Th is i s because the va lue of y depends not only on t ime but a l s o on the c u r r e n t rese rve l e v e l which i s not known u n l e s s the whole system has been s o l v e d . P r e c i s e l y In order f o r equat ion (47) to d ichotomize from the r e s t of system I I I A , the f (•), must be c o n s t a n t . Another a d d i t i o n a l c o m p l i c a t i o n a r i s e s when K some reserves are l e f t i n the ground a t c l o s u r e t i m e . In that case c o n d i t i o n s I I Ig2 a P P l y a n c * i t appears that the amount of reserves to be l e f t unex t rac ted i s endogenous and, even i f a n t i c i p a t e d p r i c e s are c o n s t a n t , equat ion (46) cannot be s o l v e d s e p a r a t e l y , un less one assumes that f (•) XJ i s independent of x , at l e a s t when L = I = 0. In p r e v i o u s models, major s i m p l i f i c a t i o n s r e s u l t e d from the (47) s h i f t i n the p r o d u c t i o n f u n c t i o n which r e s u l t s from a change i n reserves 59 2 . 5 A l t e r n a t i v e e s t i m a t i o n procedures 2 . 5 . 1 I n t r o d u c t i o n Is the theory presented i n t h i s chapter amenable to e m p i r i c a l v e r i f i c a t i o n and i s i t a u s e f u l apparatus f o r the purpose of e s t i m a t i n g f a c t o r demands i n the context of n a t u r a l resource e x t r a c t i o n ? The f i r s t q u e s t i o n a r i s e s on ly i f the second one r e c e i v e s a p o s i t i v e answer; f o r the mere p o s s i b i l i t y of u s i n g d a t a i n order to es t imate the parameters of the models does not imply t h a t the theory y i e l d s p r e d i c t i o n s which can be c o n t r a d i c t e d . As mentioned i n the genera l i n t r o d u c t i o n , the main o b j e c t i v e of t h i s r e s e a r c h i s to fo rmulate t h e o r i e s which can be used i n the e s t i m a t i o n of f a c t o r demands. E m p i r i c a l v e r i f i c a t i o n of the n e o c l a s -s i c a l theory of f a c t o r demands i n genera l i s complete ly o u t s i d e of the scope of my t h e s i s ; w i t h i n the n e o c l a s s i c a l framework, however, d i s c r i m i n a t i o n between a l t e r n a t i v e f o r m u l a t i o n s i s seen as an important o b j e c t i v e . In order to i n v e s t i g a t e p o s s i b l e e m p i r i c a l a p p l i c a t i o n s of the models developed i n p rev ious s e c t i o n s , i t i s u s e f u l to envisage a l t e r n a t i v e t reatments f o r y^, the i m p l i c i t v a l u e of the m a r g i n a l rese rve u n i t . One s u b - s e c t i o n w i l l be devoted to each of the f o l l o w i n g a l t e r n a t i v e s : - y f c may be es t imated e x p l i c i t l y ; - u may be d e r i v e d from observed d a t a ; - y^ _ may be es t imated i m p l i c i t l y . 60 2 . 5 . 2 E x p l i c i t e s t i m a t i o n of the i m p l i c i t m a r g i n a l resource va lue A. E s t i m a t i o n procedure Systems I, I I , I I I have a common c h a r a c t e r i s t i c : they i n v o l v e f i r s t degree d i f f e r e n t i a l equat ions which become l i n e a r , thus can be so l ved e x p l i c i t l y when the p r o d u c t i o n f u n c t i o n i s q u a d r a t i c . System I, which corresponds to the case of c o s t l e s s s tock ad justments , i s much s i m p l e r than I I and I I I , as the equat ions which g ive f a c t o r demands are not d i f f e r e n t i a l equat ions i n that system. However, i t i s e a s i l y seen, from e q u a t i o n ( 1 3 ) , t h a t I can be i n t e r p r e t e d as a s p e c i a l case of I I , which i s i t s e l f a s p e c i a l case of I I I . In order not to f u r t h e r expand an e x p o s i t i o n which i s a l ready vo luminous , the e s t i m a t i o n procedure w i l l be examined on ly f o r systems I I and I I I . The same b a s i c p r i n c i p l e s apply to system I, but they can be implemented i n a much s i m p l e r f a s h i o n . Thus, when the p r o d u c t i o n f u n c t i o n i s q u a d r a t i c , the f i r s t n t m f i r s t degree d i f f e r e n t i a l equat ions of I IA and I I I A are l i n e a r . They can be so lved e x p l i c i t l y to get f a c t o r demand t r a j e c t o r i e s , p rov ided t h e i r parameters are known. Besides, p r i c e and q u a n t i t y d a t a , assumed o b s e r v a b l e , one such parameter i s u^, the i m p l i c i t m a r g i n a l resource v a l u e . In I IA , tha t i s to say when the resource i s homogeneous, the d i f f e r e n t i a l equat ion which g i ves i s independent from the r e s t of the system, which means t h a t one can s o l v e f o r up to a constant of i n t e g r a t i o n . As f a r as I I I A i s concerned, tha t i s to say when the resource i s heterogeneous, the same s i t u a t i o n a r i s e s i f f T ) ( ' ) i s c o n s t a n t . The proposed e s t i m a t i o n p r o -cedure s t a r t s w i t h the cho ice of an a r b i t r a r y constant of i n t e g r a t i o n . 61 which i s equivalent to the choice of the closure date, T2, since, at T2, one has eit h e r y 2 = p 2 (IIB, IIIB1) or y 2 = 0 (IIIB2). Then the t r a j e c -tory of y i s immediately derived, given r, the discount rate, and f (•), t K the constant marginal product of reserves, assumed to be known. So u i s now known over the whole period of operations. Other p r i c e s , i n contrast, are observed only up to the current date and are unknown thereafter. I t i s assumed that, beyond the current date, the firm uses a c e r t a i n r u l e to predict future prices from current and past p r i c e s . I d e a l l y , t h i s r u l e should be of the rational-expectations type, for such a rule would imply that the firm has proven to be ri g h t i n the past, which i s why i t has never had to revise i t s estimation of the i m p l i c i t marginal resource value. This l a s t feature i s c r u c i a l to the procedure. At t h i s stage, i f the parameters of the quadratic production function were known, the t r a j e c t o r i e s of xfc and could be computed from the relevant equations i n IA (IIA; IIIA) and the corresponding i n i t i a l or terminal conditions i n IB (IIB; IIIB1 or IIIB2). Those parameters can be estimated by econometric methods, using observed values of x^ _, L^, 1^ as dependent variables and prices, including y^, as independent variables i n 5 an appropriately rearranged form of the d i f f e r e n t i a l equations, for the period over which data are a v a i l a b l e . Once t h i s i s done, the estimated parameters can be used to compute estimated t r a j e c t o r i e s for the variables x and L. Computation of the cor-responding output t r a j e c t o r y i s then a matter of algebra, so that the 6 reserve t r a j e c t o r y can be computed, using Rl as i n i t i a l value. 62 The o v e r a l l c o n s i s t e n c y of the model , g iven the es t imated p r o - d u c t i o n f u n c t i o n , can now be checked from t e r m i n a l c o n s t r a i n t s : at the c l o s u r e date d e f i n e d by the c o n d i t i o n y(T2) = p(T2) (or p(T2) = 0 i f 1IIB 2 a p p l i e s , the der i ved est imated reserve l e v e l may or may not s a t i s f y the t e r m i n a l c o n d i t i o n ( l a s t equat ion i n I I IB) . ; Unless i t does, a c o r r e c t i o n must be done to the constant of i n t e g r a t i o n which was used to compute the t r a j e c t o r y of u, and d e r i v e T2, i n the f i r s t p l a c e : i f rese rves are above the p r e s c r i b e d l e v e l , the constant of i n t e g r a t i o n was too h i g h ; i f r e s e r v e s are below the 7 p r e s c r i b e d l e v e l , i t was too low. Cons is tency can be achieved by a d j u s t i n g the constant of i n -t e g r a t i o n and r e p e a t i n g the whole e s t i m a t i o n procedure u n t i l a v a l u e i s found f o r which the t e r m i n a l resource c o n s t r a i n t i s met a t the t e r m i n a l d a t e . To sum up, the e s t i m a t i o n procedure j u s t d e s c r i b e d i s not very complex: - I t uses econometr ic methods to es t imate the p r o d u c t i o n f u n c t i o n from a v a i l a b l e market p r i c e and q u a n t i t y d a t a , as w e l l as a p o s t u l a t e d i m p l i c i t resource p r i c e s e r i e s ; - I t s o l v e s a system of f i r s t - d e g r e e l i n e a r d i f f e r e n t i a l equat ions i n f a c t o r q u a n t i t i e s , whose parameters a r e , or depend on, the est imated parameters of the p r o d u c t i o n f u n c t i o n , and a se t of a n t i c i p a t e d market p r i c e t r a j e c t o r i e s on one hand, and the p o s t u l a t e d i m p l i c i t resource p r i c e t r a j e c t o r y , on the other hand. - In order to meet a l l t e r m i n a l c o n d i t i o n s , t h i s procedure must be repeated w i t h d i f f e r e n t p o s t u l a t e d i m p l i c i t resource p r i c e s e r i e s , u n t i l 63 the c o r r e c t t r a j e c t o r y i s found ; s i n c e the t r a j e c t o r y depends on ly on one parameter , i t s i n i t i a l v a l u e , and changes i n t h a t parameter have a known e f f e c t on the reserve t r a j e c t o r y , i t i s c o n c e p t u a l l y not d i f f i c u l t to f i n d an a l g o r i t h m which converges. B. C r i t i q u e For the purpose of d i s c u s s i n g the m e r i t s and shortcomings of t h i s p rocedure , a reminder of i t s b a s i c assumptions i s i n o r d e r . Of c o u r s e , on l y a d d i t i o n a l r e s t r i c t i o n s to the models need be ment ioned; those are l i s t e d below. 1 . The p r o d u c t i o n f u n c t i o n , f ( * ) 5 i s q u a d r a t i c ; 2 . f^C") i s c o n s t a n t ; 3 . F i rms are assumed to form t h e i r p r i c e a n t i c i p a t i o n s a c c o r d i n g to a r u l e which i s both s imple enough to generate an a n t i c i p a t e d p r i c e path which can be computed from past o b s e r v a t i o n s , and s o p h i s t i c a t e d enough f o r the f i r m to be approx imate ly r i g h t i n i t s p r e d i c t i o n s . Assumption 1 i s q u i t e a c c e p t a b l e ; f o r e m p i r i c a l a p p l i c a t i o n s , a f u n c t i o n a l form must be s e l e c t e d and, among f l e x i b l e forms, the gener -a l i z e d q u a d r a t i c i s an a c c e p t a b l e c a n d i d a t e . In f a c t a l l f l e x i b l e - a c c e l -e r a t o r type of f a c t o r demand s t u d i e s assume such a f u n c t i o n a l form. Assumption 2 does not make much sense. One would expect the s h i f t i n the p r o d u c t i o n f u n c t i o n r e s u l t i n g from an i n c r e a s e i n r e s e r v e s to tend toward zero as reserves tend toward i n f i n i t y . Even i n the s imple 64 case where such s h i f t r e s u l t s on ly from changes on the revenue s i d e r e s u l t i n g from changes i n grade, that i s to say when f ( x , L , I, R) = Y ( R ) ' F ( x , L , I ) , i t i s e a s i l y checked t h a t , under reasonable assumptions on the grade d i s t r i b u t i o n , Y(R) , f j ^ " ) n o t c o n s t a n t . However, w h i l e assumption 2 i s c l e a r l y not a c c e p t a b l e as a p p l y i n g to the whole range of R , some evidence suggests that i t may have l i t t l e e m p i r i c a l r e l e v a n c e . For example, data gathered on some Nor th -Amer ican o p e n - p i t meta l mines suggest tha t the c u r r e n t c u t - o f f grade does not vary very much over 8 r e l e v a n t p e r i o d s . Assumption 3 i s the most q u e s t i o n a b l e . The i m p l i c i t m a r g i n a l v a l u e of r e s e r v e s , u^, r e f l e c t s the f i r m ' s a n t i c i p a t i o n s . As a r e s u l t , i n p r a c t i c e , i t i s p robably r e v i s e d whenever new p r i c e i n f o r m a t i o n become a v a i l a b l e to the f i r m . . . u n l e s s the new i n f o r m a t i o n on ly conf i rms the f i r m ' s a n t i c i p a t i o n s . In such a case , there i s on ly one t r a j e c t o r y f o r u^, which can be computed from the d i f f e r e n t i a l equat ion i n uand the r e l e -vant t e r m i n a l c o n s t r a i n t . U n f o r t u n a t e l y i t i s hard to assume that f i r m s have p e r f e c t f o r e s i g h t i n a s e c t o r which i s known to be r i s k y and where sudden p r i c e swings o f t e n come as a s u r p r i s e . Even i f the assumption of p e r f e c t f o r e s i g h t i s accepted , i t poses the d i f f i c u l t t e c h n i c a l problem of e s t a b l i s h i n g the s imple r u l e assumed to be s u c c e s s f u l l y used by the f i r m s i n order to d e r i v e f u t u r e p r i c e t r a j e c t o r i e s from past o b s e r v a t i o n s . In v iew of past f l u c t u a t i o n s , the assumption of f i x e d a n t i c i p a t i o n s , which i s s tandard i n e m p i r i c a l f a c t o r demand s t u d i e s of the non - resource f i r m , would be d i f f i c u l t to defend. In p r a c t i c e , one would probably have to 65 use an e x p o n e n t i a l e x t r a p o l a t i o n of the t r e n d . Now that the assumptions which are c r u c i a l to t h i s e s t i m a t i o n procedure have been d i s c u s s e d , i t remains to examine the i s s u e of data a v a i l a b i l i t y . In order to es t imate the parameters of the q u a d r a t i c p r o -d u c t i o n f u n c t i o n , the f o l l o w i n g data are necessary f o r a l l models : p and p, the output p r i c e l e v e l and i t s v a r i a t i o n ; w and w, the p r i c e s of v a r i a b l e f a c t o r s and t h e i r v a r i a t i o n s ; x , x , and x , the s t o c k - f a c t o r l e v e l s , v a r i a t i o n s and changes i n v a r i a t i o n s ; L and L, the v a r i a b l e - f a c t o r l e v e l s and v a r i a t i o n s ; R and R, the reserve l e v e l and v a r i a t i o n . Most of those v a r i a b l e s are e i t h e r d i r e c t l y observab le or can be c o n s t r u c t e d as f i r s t or second d i f f e r e n c e s . Whether or not one can put any conf idence i n a v a r -i a b l e which i s computed as a second d i f f e r e n c e i s a matter of judgment as w e l l as an e m p i r i c a l q u e s t i o n . However, one should note t h a t , under the assumptions made, there i s no e r r o r i n v a r i a b l e s r e s u l t i n g from the f a c t tha t f i r m s might r e v i s e t h e i r programs: two s u c c e s s i v e o b s e r v a t i o n s of I p e r t a i n to the same program; thus * t - * t _ i _ » s a Y j i n d i c a t e s the v a r i a t i o n i n I, w i t h i n the o p t i m a l program, between t - 1 and t , not the change i n I r e s u l t i n g from a r e v i s i o n of the o p t i m a l program. 2 . 5 . 3 Computation of the reserve u n i t v a l u e from observed d a t a . In the above procedure an e s t i m a t e of the i m p l i c i t m a r g i n a l resource v a l u e , y, was d e r i v e d from observed d a t a , by u s i n g v a r i o u s p r o p -e r t i e s of the model and imposing i t s i n t e r n a l c o n s i s t e n c y . An a l t e r n a t i v e route c o n s i s t s i n d i r e c t l y d e r i v i n g y from a v a i l -ab le d a t a . In f a c t , i f the present v a l u e of y i s u n a f f e c t e d by the l e v e l 66 of r e s e r v e s , as i n models i n v o l v i n g homogeneous r e s e r v e s , one has V -R = RR t t t where RR i s the t o t a l resource r e n t . The v a l u e of the share of the f i r m on the s t o c k exchange market r e f l e c t s t r a n s a c t o r s ' e v a l u a t i o n of the f i r m ' s a s s e t s and l i a b i l i t i e s . I t can be ' assumed t h a t l i a b i l i t i e s are c o r r e c t l y recorded i n the books and f i n a n c i a l s ta tements . The same assumption can be made about l i q u i d a s s e t s and " t r a n s i t i o n a l " a s s e t s such as i n v e n t o r i e s or s u p p l i e s . The market e x e r t s i t s judgment when i t comes to e v a l u a t i n g durable a s s e t s , i n c l u d i n g resource r e s e r v e s , i n the l i g h t of f u t u r e p rospects of the f i r m . I f one c a l l s NL the net v a l u e of l i a b i l i t i e s and a s s e t s which are not s u b j e c t to "market i n t e r p r e t a t i o n " but can be assumed to be p r o p e r l y r e f l e c t e d i n f i n a n c i a l s ta tements , one can w r i t e : (48) S -N = T ' -x + y" -R ~ NL , t t t t . t t t where the l e f t - h a n d s i d e represents the t rue s h a r e h o l d e r s ' e q u i t y , and i s equa l to the v a l u e of one share m u l t i p l i e d by the number of s h a r e s ; on the r i g h t - h a n d s i d e , X^ and u^, as b e f o r e , g i ve i m p l i c i t v a l u a t i o n s r e s p e c -t i v e l y of f a c t o r s tocks x^ _ and the rese rves R^. However, u n l i k e and which g ive m a r g i n a l v a l u a t i o n s , A and u a re average v a l u a t i o n s . In f a c t , 67 t x t X (R , X 1-dx ; y = ~— y [R , x 1 t{ t ' t j t t R J t [ t ' t j •dR t Le t us examine under which c o n d i t i o n s (48) can be u s e f u l i n e m p i r i c a l l y e s t i m a t i n g y^. One notes tha t and N are observed v a r i -a b l e s and NL^ can be computed from f i n a n c i a l s ta tements ; x and R are observed as w e l l . Consequent ly , y^ can be computed from (48) i f X i s known. In the v a r i o u s models examined i n t h i s chapter^, A i s equa l to <$>^, the cor respond ing market v a l u e , whenever s tock adjustments are c o s t l e s s . • However1 t h i s does not mean tha t the i m p l i c i t va lue of the i n t r a - m a r g i n a l f a c t o r s t o c k u n i t i s a l s o equa l to i t s market v a l u e . In f a c t , i f m a r g i n a l products decrease w i t h s tock l e v e l s , (49) Afc > A t = <f>t Since the p r o d u c t i o n f u n c t i o n was assumed to be s t r i c t l y concave i n models w i t h c o s t l e s s s t o c k ad justments , (49) h o l d s . Concerning y^, y = y when the resource i s homogeneous. Thus, i n models w i t h c o s t l e s s s tock adjustments and a homogeneous r e s o u r c e , i S -N - 4 y x t + N L t e (50) y f c = u f c < = y t 9 The r i g h t - h a n d s i d e of (50) p r o v i d e s an o v e r e s t i m a t i o n f o r y . In models w i t h heterogeneous r e s e r v e s , s i n c e the most v a l u a b l e rese rves are e x t r a c t e d f i r s t , 68 (51) y < y > t t and S -N - <f> - x + NL (52) e y > y t t The r i g h t - h a n d s i d e of (52) may p r o v i d e a b e t t e r approx imat ion f o r y than when reserves are homogeneous. However i t i s not c l e a r whether t h i s a p -p r o x i m a t i o n overes t imates or underest imates the t r u e v a l u e of y . which c h a r a c t e r i z e s the s o l u t i o n to the problem of choos ing f a c t o r demands when r e s e r v e s are homogeneous and s tock adjustments are c o s t l e s s . Then f a c t o r demands can be est imated from p r i c e and q u a n t i t y o b s e r v a t i o n s . The maximized H a m i l t o n i a n need not be q u a d r a t i c , as i n the p r e v i o u s p rocedure , because y i s no longer d e r i v e d by r e q u i r i n g that the t e r m i n a l c o n s t r a i n t be met under the assumption that f i r m s have p e r f e c t f o r e s i g h t : tha t a p -proach r e q u i r e d s o l v i n g a d i f f e r e n t i a l equat ion i n R (see I A ) , an easy task when the maximized Hami l ton ian i s q u a d r a t i c . Here , on the c o n t r a r y , the approximate v a l u e of y can be computed f o r each o b s e r v a t i o n . At each d a t e , i t may correspond to a d i f f e r e n t t r a j e c t o r y of y^ _, which means that the e s t i m a t i o n i s be ing r e v i s e d by the market as new data become a v a i l a b l e . t In any event , whatever the p r i c e a n t i c i p a t i o n s which produced the market v a l u a t i o n of y , they i m p l y , by d e f i n i t i o n , that t e r m i n a l c o n d i t i o n s are met or tha t economic agents are m i s t a k e n . So t h i s procedure i s very d i f -f e r e n t from the f i r s t one i n that i t works under oppos i te assumptions on a n t i c i p a t i o n s : f i r m s and economic agents are not assumed to h o l d r a t i o n a l Now, suppose tha t y i s used as a proxy f o r y i n system IA, 69 e x p e c t a t i o n s . The drawbacks of the procedure are that i t r e l i e s on an approx imat ion of u 5 and that the s i g n of the e r r o r i s known on ly i n the case of an homogeneous r e s o u r c e , when f a c t o r - s t o c k adjustments are c o s t l e s s . 2 . 5 . 4 I m p l i c i t e s t i m a t i o n of the i m p l i c i t m a r g i n a l resource v a l u e . Whatever the problem cons idered - e x t r a c t i o n of an heterogeneous or homogeneous r e s o u r c e , w i t h or w i thout adjustment cos ts - once the s o l u -t i o n has been found , i t depends on i n i t i a l parameters : x^ , R^, and the output and f a c t o r p r i c e t r a j e c t o r i e s , as a n t i c i p a t e d at T I . S ince the i m p l i c i t m a r g i n a l resource v a l u e , i s p a r t of the s o l u t i o n , i t i s determined as a f u n c t i o n of the same parameters . So one h a s , a t the beg inn ing of o p e r a t i o n s , i n the most genera l c a s e : P l = P l r 1 1 1 ' x , R , p , w , <j> where p \ w \ <j>^" represent the r e l e v a n t p r i c e t r a j e c t o r i e s , as a n t i c i p a t e d at T I . S i m i l a r l y , the op t imized d i scounted net revenue f u n c t i o n , whose max imi -z a t i o n was d e a l t w i t h i n 2 . 2 , 2 . 3 , and 2.4 f o r a l t e r n a t i v e c a s e s , i s g iven by : ~ * T2 *( 1 1 1 R |x . R ^ p ,w ,<f> TI - r - ( t - T l ) e 70 where a "~" i n d i c a t e s an optimum which i s expressed i n s y n t h e t i z e d form. 10 The i n t e r p r e t a t i o n of y^ as the i m p l i c i t m a r g i n a l va lue of the resource comes f rom the r e l a t i o n : 9R (•) 3R, = y (•), ( i n t r i l i g a t o r (1971) , p. 352) Furthermore, s i n c e the max imiza t ion can be c a r r i e d out at any a r b i t r a r y * * i n i t i a l d a t e , p rov ided one uses the i n h e r i t e d s tock l e v e l s x and R as ' v . t t i n i t i a l parameters , i t i s c l e a r tha t (53) VJ = yfc x t » R t > P . w , 4> S i m i l a r i l y , the o p t i m a l i m p l i c i t m a r g i n a l va lues of the f a c t o r s tocks are g iven by the v e c t o r f u n c t i o n : (54) * * x t = X *t> Rt> P > W » <f> In a p p l y i n g the Maximum P r i n c i p l e to the problem of maximiz ing d i scounted net revenues, the maximized Hami l ton ian was i n t e r p r e t e d as an i m p l i c i t r e s t r i c t e d p r o f i t f u n c t i o n , g iven by r e l a t i o n (28) f o r the case of homogeneous r e s e r v e s . For the most genera l c a s e , the maximized H a m i l -t o n i a n i s , i n present v a l u e : 71 .(55) - r - ( t - T l ) n ( x t , R t , p t - y t , w t ^ t - X t ] = - r - ( t - T l ) * * x ,R ,L , 1 t t t t 1 T * - w - L t t t t Of c o u r s e , II(-) i s the maximized H a m i l t o n i a n i n nominal terms, and has the p r o p e r t i e s enumerated i n 2 . 3 . 2 . By H o t e l l i n g ' s theorem, (56) (a) h = V(vVp t-vvV xt) (b) L = n (•) ; t w By s u b s t i t u t i n g (53) and (54) i n t o (56 ) , one o b t a i n s f a c t o r demands i n 10 s y n t h e t i z e d form: (57) ~ft ~ft[ t t t (a) I f c = I x f c , R t , p ,w ,if (b) L = L t x ,R ,p ,w ,<() (57) can be used as the b a s i s f o r e s t i m a t i n g f a c t o r demands. The drawback of the procedure i s tha t i t i s very d i f f i c u l t to d e r i v e , f r o m ~ft ~ft the t h e o r y , any r e s t r i c t i o n on the f u n c t i o n a l form of I (•) and L ( • ) • The advantage i s tha t (57) i s p e r f e c t l y genera l and s i m p l e . In p a r t i c u l a r , the re i s no o b j e c t i o n to assuming tha t the f i r m ho lds s t a t i c p r i c e a n t i c i -72 pations, so that p L = {p}, wL = {w}, and (J)"" = {<)>} and, say, This approach seems to be the least questionable of the three procedures just outlined. I t has been retained i n the empirical application of chapter 6. 2.6 Summary In this chapter, investment models of the Jorgenson (1963) -Treadway (1971) t r a d i t i o n have been adapted to situations where the firm faces a resource constraint. In a l l cases considered, whether the firm faces adjustment costs or not, whether the resource i s heterogeneous or not, the l i k e l y solution, i f any, s a t i s f i e s a system of first-degree d i f f e r e n t i a l equations in factor l e v e l s , and, sometimes, factor adjustment lev e l s , reserve l e v e l s , and the i m p l i c i t marginal value of the resource. The solution of that system i s dependent on i n i t i a l and terminal conditions. In the absence of adjustment costs, an important assumption, which must be introduced i s the concavity of the production function. This assumption guarantees that the solution does not reduce to the t r i v i a l case of instantaneous exhaustion at an i n f i n i t e scale of extraction. In models with adjustment costs, this as-sumption i s not necessary, as adjustment costs w i l l prevent the firm from 73 w i l l prevent the f i r m from i n f i n i t e l y i n c r e a s i n g i t s s c a l e w i t h i n the l i m i t e d t ime p e r i o d a l lowed by the resource c o n s t r a i n t . So economies of s c a l e over the observed p r o d u c t i o n range are compat ib le w i t h the conven-t i o n a l microeconomic theory of the e x t r a c t i v e f i r m . In d e a l i n g w i t h each model , i t was a l s o i n d i c a t e d that the maximized H a m i l t o n i a n of each problem can be i n t e r p r e t e d as an i m p l i c i t p r o f i t f u n c t i o n . When the op t imized shadow p r i c e s are s u b s t i t u t e d i n t o such an i m p l i c i t p r o f i t f u n c t i o n , a f t e r a p p l y i n g H o t e l l i n g ' s theorem, one o b t a i n s a s y n t h e t i z e d form f o r f a c t o r demands. S ince the s y n t h e t i z e d form g ives f a c t o r demands as a f u n c t i o n of observed contemporary v a r i a b l e s and contemporary a n t i c i p a t i o n s o n l y , i t p rov ides a t h e o r e t i c a l base f o r e m p i r i c a l work. The two other p o s s i b l e e s t i m a t i o n procedures d i s c u s s e d have i n common the c h a r a c t e r i s t i c of us ing an e x p l i c i t e v a l u a t i o n of the i m p l i c i t marg ina l va lue of the r e s o u r c e . In 6ne of them i t i s generated from w i t h i n the model , under the assumption that f i r m s have c o r r e c t a n t i c i p a -t i o n s and, t h u s , never r e v i s e t h e i r e v a l u a t i o n of the i m p l i c i t resource v a l u e . In the other one, f i r m s can make m i s t a k e s ; the i m p l i c i t resource v a l u e has to be approx imate ly computed from s tock va lues and o ther f i n a n c i a l i n f o r m a t i o n . A remarkable d i f f e r e n c e between the investment theory of the non - resource f i r m and the same theory a p p l i e d to a f i r m which faces a resource c o n s t r a i n t i s t h a t , i n the l a t t e r case , the s o l u t i o n does not converge toward a l o n g - r u n e q u i l i b r i u m when a n t i c i p a t i o n s are s t a t i c . The f l e x i b l e a c c e l e r a t o r i s no longer a proper apparatus f o r e m p i r i c a l work. 74 Notes to chapter 2 Some of these assumptions cou ld be r e l a x e d , but t h i s would add to the complex i ty of the e x p o s i t i o n , w i thout adding any i n s i g h t i n t o the f a c t o r demand d e t e r m i n a t i o n mechanism. In p a r t i c u l a r , assumption A6 guarantees : - tha t e x t r a c t i o n i s warranted e c o n o m i c a l l y ; - tha t e x t r a c t i o n w i l l be c a r r i e d out over a f i n i t e p e r i o d , and that the i m p l i c i t va lue of the resource at c l o s u r e t i m e , i s equa l to the output p r i c e at that t ime (see (20)} . The assumption that p r i c e paths are smooth guarantees tha t the paths of x^ d e f i n e d by (11) are smooth. L a t e r i n the e x p o s i t i o n , i t i s f u r t h e r assumed that the output p r i c e net of the i m p l i c i t resource v a l u e , i s p o s i t i v e over the whole r e l e v a n t p e r i o d , so tha t p r o d u c t i o n , once i n i t i a t e d , does not get i n t e r r u p t e d . I t i s c l e a r that when the p r o d u c t i o n f u n c t i o n s a t i s f i e s the Inada c o n -d i t i o n i t i s not op t ima l to leave any q u a n t i t y of ore i n the ground a t the end of the o p e r a t i n g l i f e of the mine. With homogeneous r e s e r v e s , as i s shown i n 2 .2 and 2 . 3 , opera t ions cease when the i m p l i c i t resource p r i c e catches up w i t h market p r i c e . In genera l the i m p l i c i t p r i c e t r a j e c t o r y does not tend toward the market p r i c e t r a j e c t o r y a s y m p t o t i c a l l y but cuts i t from below, which i m p l i e s tha t T2 i s f i n i t e . With heterogeneous reserves the i m p l i c i t p r i c e of rese rves may decrease toward zero w h i l e marg ina l e x t r a c t i o n c o s t s i n c r e a s e toward market p r i c e . In that case , i f the p roduct ion f u n c t i o n s a t i s f i e s the Inada c o n d i t i o n s , so t h a t any r e s o u r c e , however poor i t s 75 q u a l i t y can be e x t r a c t e d , p rov ided output i s s u f f i c i e n t l y reduced, the mine may operate f o r e v e r , w i t h r e s e r v e s b e i n g reduced at a r a t e which tends toward z e r o . A s u f f i c i e n t c o n d i t i o n f o r T2 to be f i n i t e i s to have i n c r e a s i n g marg ina l f a c t o r products at low output l e v e l s , say when q < q ,^ and the u s u a l p r o p e r t i e s when q > q. Th is ensures that no output w i l l be produced at a r a t e below q. A d i s c u s s i o n of the e f f e c t s of d i s c o n t i n u i t i e s i n f_(*) on T2 can be found i n Sa lant et a l . (1981) . R ' i n 2 .2 and 2 . 3 , m a r g i n a l p roducts are assumed to be f i n i t e everywhere except at n u l f a c t o r l e v e l s (assumptions A6, A7 ; B5 , B6 ) . , . . . I t i s assumed at t h i s stage that x , L, I are o b s e r v a b l e ; t h i s i s ques -t i o n e d , below, e s p e c i a l l y f o r the case of I. 'Us ing I IA as an example, the d i f f e r e n t i a l equat ions are r e w r i t t e n i n such • • • * f a way tha t the v e c t o r ( x ' , L ' , I 1 , u, R) stands alone on the l e f t - h a n d s i d e . Th is r e q u i r e s on ly l e f t - m u l t i p l y i n g I IA by the i n v e r s e of the square m a t r i x on the l e f t - h a n d s i d e of I IA . Then the equat ions g i v i n g x , L, I may be used as econometr ic e q u a t i o n s . I t i s e a s i l y checked t h a t the system i s e x a c t l y i d e n t i f i e d . ' In, p r a c t i c e , changes i n rese rves may d i f f e r from the negat i ve of output , due to reserve a d d i t i o n s or r e e v a l u a t i o n , a f e a t u r e not d e a l t w i t h i n the f o r e g o i n g r e s e a r c h . As a r e s u l t , when f e a s i b l e , i t i s a d v i s a b l e to use. the l a s t o b s e r v a t i o n of R as s t a r t i n g va lue when computing the reserve , t r a j e c t o r y i m p l i e d by the es t imated p r o d u c t i o n f u n c t i o n . 'with homogeneous r e s e r v e s , the t e r m i n a l l e v e l must be n u l l ; w i t h h e t e r o -geneous r e s e r v e s , the t e r m i n a l l e v e l i s n u l l i f I I IB1 a p p l i e s , but 76 p o s i t i v e i f I I IB2 a p p l i e s . In the l a t t e r case , by (46) , i t i s a f u n c t i o n of x 2 » the t e r m i n a l l e v e l of f a c t o r s t o c k s . 'Most mines g i ve i n f o r m a t i o n about average reserve grade. Some g ive i n f o r -mation about c u t - o f f g rade; however the c u t - o f f grade must be i n t e r p r e t e d i n terms of rese rve e v a l u a t i o n ; i t does not n e c e s s a r i l y i n d i c a t e the c u t - o f f grade which a p p l i e s to cu r ren t o p e r a t i o n s , but to u l t i m a t e e x -t r a c t i o n ; a change i n that f i g u r e on ly i n d i c a t e s that reserves have been r e e v a l u a t e d . One can get an i d e a of cu r ren t c u t - o f f grade by observ ing the average grade of the ore m i l l e d over a g iven y e a r . The l a t t e r i s probably the best proxy-measure of how ore q u a l i t y changes over the l i f e of a mine. Table I p resents some grade data f o r a sample of mines . The ev idence i n d i c a t e s that v a r i a t i o n s are s m a l l over r e l a t i v e l y long p e r i o d s , which suggests that the e r r o r i n v o l v e d i n approx imat ing f ^ ( ' ) as a constant i s of s m a l l magnitude. ' i n chapter 3 , investment i s assumed to be i r r e v e r s i b l e ; then (49) becomes X > X < <j> and (50) no longer h o l d s , t t t 6 5A "*" i s normal l y used , i n t h i s d i s s e r t a t i o n , to express that a v a r i a b l e has been op t imized or that a f u n c t i o n i s eva luated a t o p t i m a l va lues of i t s argument. Here, however, one has to d i s t i n g u i s h between two l e v e l s of o p t i m i z a t i o n : - When a p p l y i n g the Maximum P r i n c i p l e , one obta ins f a c t o r demands as a f u n c t i o n , among o t h e r s , of the c o s t a t e v a r i a b l e s . Th is i s denoted w i t h a ii II Thus: 77 - When the whole max imiza t ion problem has been s o l v e d , the c o s t a t e v a r i a b l e s are expressed as f u n c t i o n s of v a r i o u s parameters , as i n (53) and (54) ; i f those f u n c t i o n s are s u b s t i t u t e d i n t o I (•)> one has the s y n t h e t i z e d form: -*f t t t x t , R f c , p ,w ,<j> x ,R ,p - u t t *t t x f c , R t , p ,w ,<j> ,w ,<j) - A ' t T t t Xt » R t > P » W »<r where i t w i l l be remembered t h a t p represents the p r i c e at t w h i l e p f c r ep resents the whole p r i c e t r a j e c t o r y , from t on, as a n t i c i p a t e d a t t . •'The p r o p e r t i e s of R (•)> the d iscounted net revenue f u n c t i o n , are s t u d i e d i n d e t a i l , i n a more genera l se tup , i n chapter 4. I t turns out t h a t , i f the p r o d u c t i o n f u n c t i o n i s concave i n R^ and f r e e d i s p o s a l i s a l l o w e d , * R (•) i s concave i n R^, so that ft 3 y 1 ( - ) 9R„ < 0 78 Table I : Ore grades f o r some o p e n - p i t Mines ( ( ( 3rade of 3re M i l l e d Dver the Year (%) Average Reserve Grade (%) Reserve C u t - o f f Grade (%) Canada Tungsten 1962 1979 2.74 1.96 2.47 1.55 Graigmont 1961 1979 1.43 .95 2 .08 1.13 Cyprus A n v i l l 1969 1979 4 3 .3 3 .4 3 . 0 Endako 1965 1978 .27 .13 .16 .082 .15 .048 G i b r a l t a r 1972 1979 .46 .42 .37 .36 .25 .25 G r a n i s l e 1967 1977 .78 .44 .53 .42 Source : F i n a n c i a l Reports and Canadian Mines Handbook 79 CHAPTER 3 IRREVERSIBLE INVESTMENT WITH A RESOURCE CONSTRAINT 3.1 I n t r o d u c t i o n In chapter one, a d i s t i n c t i o n was drawn between those t h e o r i e s which present investment as a smooth on-going process, and those which consider i t a once and f o r a l l i r r e v e r s i b l e a c t i o n . Chapter two extended the former approach to the case of firms which e x t r a c t an e x h a u s t i b l e resource. In t h i s chapter, the l a t t e r approach i s adapted to the ex-t r a c t i v e f i r m . In f a c t , the few authors who have s t u d i e d the investment process of the n a t u r a l - r e s o u r c e f i r m have g e n e r a l l y p r e f e r r e d that ap-proach, whether they had e m p i r i c a l preoccupations ( H e l l i w e l l (1978)) , or t h e o r e t i c a l i n t e r e s t s (Masse (1964); Puu (1977); C l a r k et a l . (1979); Campbell (1980)); C l a r k et a l . stand out i n t h i s category as they a l s o envisage the case where stock f a c t o r s can be r e s o l d at a l o s s , a feature which we incor p o r a t e i n the model of chapter 4. Notable exceptions are Burt and Cummings (1970), and Bernard (1979). In s e c t i o n 3.2, i r r e v e r s i b l e investment i s studi e d i n a f a i r l y general case, where the f i r m uses s e v e r a l s t o c k - f a c t o r s and s e v e r a l v a r -i a b l e f a c t o r s . In s e c t i o n 3.3, a simple one s t o c k - f a c t o r case i s used to emphasize the d i f f e r e n c e between the conventional f i r m and the e x t r a c t i v e f i r m . At occasions, some r e s u l t s of the n a t u r a l - r e s o u r c e l i t e r a t u r e are rederived i n a new f a s h i o n . The question of e m p i r i c a l a p p l i c a t i o n s i s postponed to chapters four and f i v e , as a more general type of i r r e v e r s -i b i l i t y i s introduced i n chapter four. 80 3.2 Genera l c a s e : Fac to r demand and e x t r a c t i o n p o l i c y when s tock a d - justments are c o s t l e s s and non n e g a t i v e In t h i s c h a p t e r , i n order to focus on the i r r e v e r s i b i l i t y , no adjustment c o s t s are in t roduced i n t o the p r o d u c t i o n f u n c t i o n , and a n t i c i -pated p r i c e s are assumed^ to be c o n s t a n t . In order to g ive the i r r e v e r s -i b i l i t y c o n s t r a i n t a sound economic c o n t e n t , i t i s assumed that the l a t t e r r e s u l t s from the i m p o s s i b i l i t y of r e s e l l i n g any used s t o c k - f a c t o r at a p o s -i t i v e p r i c e . As a r e s u l t , the term which accounts f o r the scrap va lue of the f i r m at c l o s u r e t ime i n p rev ious models d i s a p p e a r s . The s p i r i t of the model would be best r e f l e c t e d by p o s t u l a t i n g that the asset p r i c e of s t o c k - f a c t o r s i s g iven by a two-way v e c t o r f u n c t i o n cf>(x), which takes up p o s i t i v e v a l u e s , tj>, when the s tock ad justments , x , are non n e g a t i v e , and i s equa l to zero when x i s n e g a t i v e . However, i t i s e q u i v a l e n t , as Arrow (1968) has shown, t o l e t asset p r i c e s be g iven by the v e c t o r , cb, of p o s i t i v e numbers and p o s t u l a t e that x must be non n e g a t i v e , which i s done h e r e . Thus the problem of the f i r m i s : (1) Max {x ,L} ,T2 1 s u b j e c t t o : T2 - r - ( t - T l ) e TI p . f ( x t > L t , R t ] - w ' . L t - * ' . i t •dt (2) (a) Rt = - q t = - f ^ . L ^ j ; (b) R(T1) = R l ; (c) R f c > 81 (3) (a) x t > 0 ; (b) x ( T l ) = x 1 ; (c) x t > 0 ; (d) L . > 0 The H a m i l t o n i a n can be w r i t t e n as (4) H j x t , L t , R _ ; u , A j = - r - ( t - T l ) ( p - y ) . f ( - ) - w ' - L t - ( < r , - ^ j - x t I f L x ^ R ^ . y - A ^ w , the v a l u e of which maximizes H(0> or i t s c u r r e n t v a l u e , a t t , i s s u b s t i t u t e d i n t o (4), the p a r t i a l l y maximized H a m i l t o n i a n can be i n t e r p r e t e d as an i m p l i c i t r e s t r i c t e d p r o f i t f u n c t i o n II ^x^ jR^p -u^ jw j Th is f u n c t i o n w i l l be expressed i n c u r r e n t va lue thereon , and has the p r o p e r t i e s i d e n t i f i e d by Lau (1976) , that i s : 11(0 i s dec reas ing i n w and i n c r e a s i n g i n p - y and x ; n ( 0 i s convex i n w and p - y ; 11(0 i s i n c r e a s i n g i n x ; 3 I U - ) 3w 311(0 L (O; g ives the i m p l i c i t asset p r i c e of the s t o c k s x . OX t 3 n Furthermore, i t i s e a s i l y shown that -r— (O > 0 i f I r ) ( 0 > 0. In order to o R R focus on the demand f o r s tock f a c t o r s , i t i s convenient to use the i m p l i c i t r e s t r i c t e d p r o f i t f u n c t i o n i n what f o l l o w s . Thus, by the maximum p r i n c i p l e , 11(0 must be maximized w i t h respec t to x^ _ a t a l l d a t e s , sub jec t to the non -n e g a t i v i t y c o n s t r a i n t , w i t h : (5) n ( 0 = ( p - y ) - f ( x t , L * ( 0 , R t j - w ' . L * ( 0 - (4> ' - A')-x t 82 Since 7T(») i s linea r i n x^, this i s a bang-bang problem. Three phases must be distinguished for each stock x 1; - Phase i : cp1— X X> 0; x 1 i s i n f i n i t e and x 1 registers a discrete increase. - Phase i i : cp1 — X 1 = 0; &1: i s not determined by the maximum p r i n c i p l e . - Phase i i i : c f ^ - A ^ O ; x 1 should be i n f i n i t e and negative, hence the non negativity constraint i s binding and x 1 = 0. The following remarks are i n order: - Since there are no discrete changes i n x during phases i i and i i i , i t s costate variable vector i s continuous during those phases. Consequently, a t r a n s i t i o n from phase i i i ((j)1 - X 1 < o) to phase i - X 1 > o) must involve a passage i n phase i i (c^ = A"*") . - Since i t involves a discrete change i n x, phase i i s instantaneous. Consider a situ a t i o n where a l l stock-factors are i n either phase i i or i i i , when x i s f i n i t e . C a l l x the vector of elements of x that are non n u l l . The corresponding stocks, x, must be i n phase i i , when A .= cj). Thus 1 = 0 and, from (6) the equation of motion of A, one derives (7). (6) A = Ar - n (•) x (7) A-r = IU(-) x 83 I f the s o l u t i o n of (7) i s s u b s t i t u t e d i n t o ( 5 ) , which d e f i n e s l l ( - ) , one o b t a i n s the maximized H a m i l t o n i a n : (8) H * ( p - y , w , A - r , x , R ] = ( p - u ) - f x * ( 0 , L*( - ) ,R w ' - L (•), where, X : [ X , X j X* s o l v e s ( 7 ) . As a l r e a d y done s e v e r a l t i m e s , H (•) can be i n t e r p r e t e d as a r e s t r i c t e d p r o f i t f u n c t i o n ; t h i s t ime however, the r e s t r i c t i o n concerns on ly those s t o c k - f a c t o r s , x , which are not v a r i a b l e at the date c o n s i d e r e d . By H o t e l l i n g ' s theorem, the output i s : (9) q*-Vy<o. D i f f e r e n t i a t i n g (9) w i t h r e s p e c t to t i m e , remembering t h a t by d e f i n i t i o n x i s f i x e d and t h a t other s t o c k s , x , are i n phase i i , one has : (10) q* = - H * (-)-.V •+ H* P ( - ) ' R p - y , p - y p - y , R I f the l e v e l of reserves does not a f f e c t p r o d u c t i o n c o s t s , the second term on the r i g h t - h a n d s i d e of (10) v a n i s h e s , and y i s p o s i t i v e . S ince is H (•) i s convex i n p - y , i t f o l l o w s tha t output d i m i n i s h e s w i t h t ime i n 84 such a c a s e . Th is r e s u l t i s s l i g h t l y more genera l than i t s counte rpar t i n the n a t u r a l resource l i t e r a t u r e (Levhar i and L i v i a t a n (1977)) i n that i t does not i n v o l v e the assumption tha t c a p a c i t y , as expressed by the l e v e l of 2 s t o c k - f a c t o r s , i s f i x e d . The above a n a l y s i s has been done under the assumption that a l l f a c t o r s t o c k s were e i t h e r i n phase i i i ( f i x e d ) or i n phase i i ( a d j u s t i n g at a f i n i t e speed upward) . Given the p o s s i b i l i t y of a t h i r d c a s e , phase i i i ( d i s c r e t e i n c r e a s e s i n s t o c k s ) , one may wonder, then , about the g e n e r a l i t y of the r e s u l t s . However, i t i s subopt imal to move from phase i i to phase i when, as assumed, p r i c e s do not have d i s c o n t i n u i t i e s . The reason i s o b v i o u s : i n phase i i , the Maximum P r i n c i p l e leaves s tock a d j u s t -ments i n d e t e r m i n a t e , p r e c i s e l y because no marg ina l change i n s t o c k l e v e l s can a f f e c t the H a m i l t o n i a n , hence the o b j e c t i v e f u n c t i o n a l . Th is c h a r -a c t e r i z e s an extremum of the o b j e c t i v e f u n c t i o n a l w i t h r e s p e c t to s tock l e v e l s a t the date c o n s i d e r e d . A s u f f i c i e n t c o n d i t i o n f o r t h i s extremum to be a maximum i s that the p r o d u c t i o n f u n c t i o n be concave. However, such a r e s t r i c t i v e assumption i s by no way necessary i n a model where the scrap v a l u e i s n u l l . I t i s i n t u i t i v e l y c l e a r , and i t i s shown i n chapter 4 f o r a more g e n e r a l c a s e , t h a t , even i n presence of s c a l e economies and i n the absence of adjustment c o s t s , the f a c t that s t o c k - f a c t o r s can be used on ly as l o n g as there i s some resource l e f t to be e x t r a c t e d imposes an upper l i m i t on the e x t r a c t i o n s c a l e . C l e a r l y , the cor responding s t o c k -f a c t o r l e v e l i s the extremum which i s r e l e v a n t i n the fo rego ing argument. S ince any upward s tock adjustment i s f e a s i b l e i n phase i i , i t i s never 85 necessary to s w i t c h from phase i i to phase i i n order to i n c r e a s e a s t o c k . I f i t i s d e s i r a b l e to decrease a s t o c k , which i s not f e a s i b l e , then a s w i t c h from phase i i to phase i i i o c c u r s , but t h i s i s covered i n the above a n a l y s i s . The o p t i m a l f a c t o r demand and e x t r a c t i o n p o l i c y now appears c l e a r l y : the most d e s i r a b l e s i t u a t i o n i s phase i i , when no m a r g i n a l change i n s t o c k - f a c t o r l e v e l s c o u l d i n c r e a s e the o b j e c t i v e f u n c t i o n a l ; i f upward adjustments are r e q u i r e d to m a i n t a i n a s t o c k i n phase i i , as d e f i n e d by ( 7 ) , those are f e a s i b l e and there i s no need to s w i t c h away from the most d e s i r a b l e phase; i f downward adjustments are r e q u i r e d to m a i n t a i n the s i t u a t i o n which c h a r a c t e r i z e s phase i i , those are not f e a s i b l e , hence adjustments are set to zero f o r the r e l e v a n t s t o c k s and a s w i t c h to phase i i i (A' < <j>*; x"*" = Oj o c c u r s . When a l l s t o c k s are e i t h e r i n phase i i or i n phase i i i , output decreases i f the resource i s homogeneous However some s t o c k - f a c t o r s may be i n c r e a s e d . W i t h a scrap v a l u e of z e r o , the t r a n s v e r s a l i t y c o n d i t i o n s r e q u i r e : (ID A 2 = 0 This i m p l i e s t h a t a l l s t o c k s are i n phase i i i (x = 0) not o n l y a t T2 but a l s o , g iven the c o n t i n u i t y of A, toward the end of the e x t r a c t i o n p e r i o d . Thus, w h i l e some s t o c k s may be i n c r e a s e d d u r i n g the e x t r a c t i o n p e r i o d , t h i s may not happen toward the end. I s phase i ( d i s c r e t e i n c r e a s e i n a s t o c k ) ever observed? Only i f a s t o c k has not p r e v i o u s l y been i n phase i i 86 or i i i , tha t i s to say , o n l y , p o s s i b l y , at T I , the beg inn ing of the e x -t r a c t i o n p e r i o d . I f some i n i t i a l s t o c k s are low r e l a t i v e to the d e s i r -ab le l e v e l , they w i l l i n s t a n t a n e o u s l y be brought up to that l e v e l . I f one n e g l e c t s the p o s s i b i l i t y , which d isappears of there i s on ly one f a c t o r , tha t a s t o c k might be i n c r e a s e d d u r i n g the e x t r a c t i o n p e r i o d . The p a t t e r n j u s t d e s c r i b e d i s the t h e o r e t i c a l b a s i s f o r the " p o i n t - i n p u t , f l o w - o u t p u t " investment model of Masse (1962, pp. 3 6 2 - 3 6 9 ) , and f o r the procedure used by H e l l i w e l l (1978) . Of c o u r s e , i t a l s o covers Campbell (1980) . 3 . 3 S p e c i a l c a s e : no v a r i a b l e f a c t o r s , no s tock i n c r e a s e s d u r i n g e x - t r a c t i o n The p rev ious s e c t i o n p rov ided a c h a r a c t e r i z a t i o n of the dynamic p a t t e r n of f a c t o r demand and output f o r an e x t r a c t i v e f i r m , when investment i s i r r e v e r s i b l e . Comparisons were drawn w i t h the e x i s t i n g min ing l i t e r a -t u r e . In order to i l l u s t r a t e f u r t h e r how the e x t r a c t i v e f i r m d i f f e r s from an otherwise s i m i l a r f i r m , i t i s convenient to use the s p e c i a l case of a f i r m whose p r o d u c t i o n f u n c t i o n has on ly one argument, which i s a s t o c k - f a c t o r . Acco rd ing to the p a t t e r n d e s c r i b e d i n 3 . 2 , i f i t - i s assumed that the i n i t i a l f a c t o r s tock l e v e l i s n u l l , the l i f e of the mine beg ins by an ins tantaneous i n c r e a s e i n the s t o c k to the d e s i r e d l e v e l . T h e r e a f t e r , whether or not the p r o d u c t i o n f u n c t i o n i s concave i n i t s s o l e argument, one h a s : q < 0 ; 87 S ince no r e d u c t i o n i n x i s p o s s i b l e , i t f o l l o w s that output i s constant over the l i f e of the mine. The i r r e v e r s i b i l i t y c o n s t r a i n t i s b i n d i n g over the whole p e r i o d , w h i l e the i m p l i c i t va lue of the s t o c k - f a c t o r d i m i n i s h e s from A= <|>, the v a l u e which c h a r a c t e r i z e s the d e s i r e d l e v e l of the s t o c k , to A = 0 , the v a l u e of the s tock when e x t r a c t i o n ceases . Without any l o s s of g e n e r a l i t y , the opening d a t e , T I , can be set to zero and A can be w r i t t e n i n p r e s e n t - v a l u e form: (12) A t * e r < t - A t I t s motion i s (13) A t = - e - r i t - n x ( x t , p - y t ) C o n s i d e r i n g the d e f i n i t i o n of n ( - ) > ( 5 ) , and remembering tha t there i s no v a r i a b l e f a c t o r and that x = 0 , one h a s : n x ( . ) = f ( x ) . ( p - y ) , w i t h x constant on the o p t i m a l t r a j e c t o r y . Th is i s s u b s t i t u t e d i n t o (13) to g e t : (14) A t = -e~r .(p-y).f'(x) . I n t e g r a t i n g (14) between TI and T2, one has : 88 T2 A T 2 - A T 1 = - | e r ' t > ^ p - y f c j « f ' ( x ) « d t , and T1=0 * = f ' ( x ) T2 e >dt -T1=0 T2 f - r - t , e - y ^ ' d t T1=0 where i t was made use of the f a c t s tha t = 0> = <Ji, and that f ' ( x ) and p are c o n s t a n t . Furthermore, s i n c e y f c grows at the constant r a t e r , - r » t e • y i s constant and: - r t - r - T 2 - r - T 2 e -y = e •y2 = e Consequent ly , 4> = p - f ' ( x ) 1 - e - r - T2 - T2-e - r - T 2 or (15) p*f (x) = r-<j>/ 1 - e r " T 2 . ( l + r -T2) (15) w i l l be recogn ized as a m o d i f i e d user cos t f o r m u l a . Indeed i f the c o n v e n t i o n a l f i r m i s viewed as a f i r m which e x t r a c t s a p l e n t i f u l r e s o u r c e , so tha t i t s h o r i z o n , T2, i s i n f i n i t e , (15) reduces t o : ( 1 5 ) ' p . f ' (x) = r-(j) , 89 the user cos t i n absence of p r i c e v a r i a t i o n s . Both (15) and ( 1 5 ) ' g i ve the demand f o r the s t o c k - f a c t o r x i n i m p l i c i t form. By comparing them one can see that the l e v e l choosen by the e x t r a c t i v e f i r m f a l l s shor t of that s e l e c t e d by the c o n v e n t i o n a l f i r m , at which the va lue of the m a r g i n a l product equals the r e n t a l p r i c e of the s t o c k . Th is i s because, g iven the i r r e v e r s i b i l i t y , the f i r m must amort i ze the investment over a f i n i t e p e r i o d , thus adding a f i n a n c i a l component to economic d e p r e c i a t i o n ( p h y s i c a l d e -p r e c i a t i o n i s assumed to be n u l l i n t h i s i l l u s t r a t i o n ) . I t r e s u l t s , and i s e a s i l y checked, that the user cos t of the s tock used by the e x t r a c t i v e f i r m i s c l o s e r to the r e n t a l p r i c e , the longer the e x t r a c t i o n p e r i o d . The optimum d e f i n e d by (15) i l l u s t r a t e d the t r a d e - o f f between a lower cost and a f a s t e r e x t r a c t i o n , a t r a d e - o f f which u n d e r l i e s the paradox of the " d o u b l e - c r o s s " d i s c u s s e d by Neher (1978) . Such a " d o u b l e - c r o s s " i n the e x t r a c t i o n p a t h s , which corresponds to two a l t e r n a t i v e i n t e r e s t r a t e s , r e s u l t s from the ambiguous e f f e c t of the i n t e r e s t r a t e f o r an e x t r a c t i v e f i r m : on one s i d e , a lower i n t e r e s t r a t e reduces the r e n t a l p r i c e of c a p i t a l , thus p e r m i t t i n g a f a s t e r e x t r a c t i o n ; on the other s ide i t lowers the cos t of w a i t i n g , thus making f a s t e x t r a c t i o n l e s s a t t r a c t i v e . Th is ambigui ty can be c l e a r l y i d e n t i f i e d by per forming a comparative s t a t i c e x e r c i s e on the optimum l e v e l of x , as g iven by (15) , a f t e r r e p l a c i n g T2 by i f v a l u e , R/ f ( x ) . Such ambigu i t y , however, i s not the r e s u l t of the i r r e v e r s i b i l i t y i n the investment p r o c e s s . 90 3.A Conc lus ion When s tock adjustments are i r r e v e r s i b l e and c o s t l e s s , a f i r m which faces a resource c o n s t r a i n t w i l l behave i n the f o l l o w i n g f a s h i o n : at i t s c r e a t i o n , i t w i l l b r i n g up s t o c k - f a c t o r s to a c e r t a i n d e s i r a b l e l e v e l . I f p r i c e s are c o n s t a n t , t h i s w i l l be the on ly d i s c r e t e i n c r e a s e i n s t o c k s . T h e r e a f t e r they w i l l be e i t h e r smoothly i n c r e a s e d or main ta ined at a f i x e d l e v e l . I f the r e s e r v e l e v e l i s not an argument of the p r o d -u c t i o n f u n c t i o n , output w i l l decrease over the whole l i f e of the f i r m . T h i s we l l -known r e s u l t i n the n a t u r a l - r e s o u r c e l i t e r a t u r e i s d e r i v e d here as a s imple ex tens ion of d u a l i t y theory to the case of i m p l i c i t p r o f i t f u n c t i o n s . 91 Notes to Chapter 3 1As i n chapter 2, f(x,L,R) i s so chosen that e x t r a c t i o n i s economical (A5, A6) and takes place w i t h i n a f i n i t e p e r i o d (A7, e s p e c i a l l y ) . One notes that f ( - ) i s not assumed to be j o i n t l y concave i n x and L. f ( - ) s a t i s f i e s A: A l f ( - ) i s twice continuously d i f f e r e n t i a b l e ; A2 f ( - ) i s non-decreasing i n x and L; A3 f ( • ) i s s t r i c t l y concave i n L, given x and R; A4 f ( • ) i s non decreasing i n R; A5. l i m f (•) = 0; L-x=° A6 f (•) i s i n c r e a s i n g i n x f o r small l e v e l s of x and f (•) i s x x f i n i t e f o r any x; A7 f T ( • ) i s i n c r e a s i n g i n L f o r small l e v e l s of L and f (•) i s f i n i t e f o r any L; Un l i k e i n chapter 2, i t i s assumed, that output and f a c t o r p r i c e s are constant. 92 Campbell (1980) f i n d s a p a r t i c u l a r case where output i s constant over p a r t * of the l i f e of a mine. By (10) , such a case a r i s e s when H (•) = 0 , J p - y , P - y * that i s to say when u ^") * s l i n e a r i n ( p - y ) , wh ich , not s u r p r i s i n g l y , r e q u i r e s both v a r i a b l e f a c t o r s and s t o c k - f a c t o r s to be i n s e n s i t i v e to the i m p l i c i t output p r i c e (p - y ) ( see ( 8 ) ) . In other words, the m a r g i n a l cos t curve i s v e r t i c a l at the l e v e l of output cons idered and, a l though the l e v e l of output at which the m a r g i n a l - c o s t curve becomes v e r t i c a l i s a f u n c t i o n of f a c t o r s tock l e v e l s ( " c a p a c i t y " i n Campbe l l ' s t e r m i n o l o g y ) , the f i r m does not f i n d i t p r o f i t a b l e to change those l e v e l s . Campbell a l s o f i n d s that output decreases toward the end of the e x t r a c t i o n p e r i o d . A g a i n , by (10 ) , t h i s r e q u i r e s Hp_^ p u (*) to be p o s i t i v e toward the end of the e x t r a c t i o n p e r i o d , when p-y i s low. Remembering t h a t , by the Maximum P r i n c i p l e , m a r g i n a l cost equals p - y , t h i s means that the e q u a l -i t y i s mainta ined on ly i f output d i m i n i s h e s , which means that the m a r g i n a l cost curve i s u p war d - s lo p ing at lower o u t p u t s . So Campbel l 's p e c u l i a r e x -t r a c t i o n p r o f i l e r e s u l t s from the p o s t u l a t e that the marg ina l cost curve i s u p w a r d - s l o p i n g at low output l e v e l s and becomes v e r t i c a l at some l e v e l , q , de f ined as c a p a c i t y . As i s c l e a r i n F i g u r e 1 below, when p-y reaches the minimum m a r g i n a l cost l e v e l compat ib le w i t h an output of q , say c", then output s t a r t s d e c r e a s i n g . 93 F i g u r e 3 : Campbe l l ' s model (1980) : u n d e r l y i n g technology $ unit/?v p-p MC ( q ' q o) MCJq.qJ qo q l y = r . y q = 0 , p -y > c" q < 0 , p-y < "c In F i g u r e 3 , f o r q < "q, the m a r g i n a l cos t curve has been drawn independent of q"; the same curve a p p l i e s whether q" = "q^ or "q = q"^. Th is assumption of Campbell i s not c r u c i a l to h i s q u a l i t a t i v e r e s u l t s . 94 CHAPTER 4 A GENERALIZED THEORY OF FACTOR DEMANDS 4.1 I n t r o d u c t i o n To some extent, v a r i o u s t h e o r i e s of f a c t o r demand, e s p e c i a l l y t h e o r i e s of investment, have been presented as mutually e x c l u s i v e i n the l i t e r a t u r e . However, w h i l e such an a t t i t u d e helps explore the i m p l i c a -t i o n s of i s o l a t e d f e a t u r e s of a model, i t leaves the student with the impression that there i s no agreement i n that area, i n c o n t r a s t , f o r example, with the corresponding area of consumer demand. In t h i s chapter, we attempt to provide a s y n t h e t i c treatment by combining v a r i o u s approaches to investment theory, and arguing that some of them, f a r from e x c l u d i n g each other, o f t e n complement one another i f a p p l i e d at d i f f e r e n t stages of the same process. The approach used can be i n t e r p r e t e d as a new fo r m u l a t i o n of the putty - semi-putty model of Fuss (1977) i n that i t systematizes the d i s t i n c t i o n between ex ante and ex post choices i n -stead of i g n o r i n g i t or a p r i o r i imposing t o t a l r i g i d i t y at e i t h e r level.- 1' I t can a l s o be viewed as an extension of the cost of adjustment theory of investment i n that the model used i s f o r m a l l y s i m i l a r , the ex-tension being that the i n i t i a l f a c t o r stock l e v e l s are made choice v a r i -a b l e s . F i n a l l y , as i n previous chapters, the theory i s a p p l i e d to a f i r m which faces a resource c o n s t r a i n t . The model has been formulated i n such a way that i t reduces to v a r i o u s known models of f a c t o r demand a f t e r a j u d i c i o u s choice of 95 parameters. Except for mutually exclusive hypotheses, i t can also r e f l e c t "mixed flavours" . As appears in section 4.4, where the DNR function, defined below, i s characterized, such an attempt at generality i s not gratuitous; i t reveals that the conditions of the ex ante choice are very similar for a wide class of ex post hypotheses, which has important implications for empirical work. However, this l e v e l of generality i s achieved at a cost: the existence of a solution i s only assumed here, instead of being established as in some part i c u l a r cases of the model already covered in the l i t e r a t u r e ; also, while some of those par t i c u l a r cases have found a complete solution, we provide only a p a r t i a l characterization here. 4.2 Model and basic premises Consider the following problem: (1) Max. (L,I},T2 T2 - r - ( t - T l ) TI f ( x t , I t , L t , R t ; G ) - ^ I t ] .I t-w'.L t . -r-(T2-Tl) . + e -(K-OD) - X 2 , •dt subject to the following constraints: x = I ; x, given ; x 0 unspecified t t 1 2 Rfc =-f(0 ; RL given ; R 2 5 0 I > 0 a- (j) , 0 ^  a 4 1, I ^ 0 While the introduction of I as an argument of the production function has been j u s t i f i e d , the interpretation of the objective function given, and the stock motions explained, in e a r l i e r chapters, a few new features require comments. F i r s t , the stock of reserves R enters as an argument of the production function. This implies that the stock of reserves may affect costs of extraction, an assumption which appears 96 f r e q u e n t l y i n the f i e l d of n a t u r a l resources (Levhar i and L i v i a t a n (1977) ; P indyck (1978) ) . S econd, the asset p r i c e of s tock f a c t o r s i s made a f u n c t i o n of T_t. As should be c l e a r from the d e f i n i t i o n of the f u n c t i o n <j> (1^.) > t h i s i s not to r e f l e c t any market power which might be e x e r c i s e d by the f i r m i n a c q u i r i n g f a c t o r s of p r o d u c t i o n , but r a t h e r to a l l o w f o r the p o s s i b i l i t y tha t a f i r m may be unable to r e s e l l a p i e c e of undepre-c i a t e d equipment at the a c q u i s i t i o n c o s t . 2 By assumption the p r o d u c t i o n f u n c t i o n s a t i s f i e s A l . A l : f ^ x t » l t > L t » R t 5 9j has the f o l l o w i n g p r o p e r t i e s : 1 . non d e c r e a s i n g i n x^ , L^, and ; 2 . concave i n I t and L^.; 3 . Non n e g a t i v e when 1 = 0 ; 4 . Non i n c r e a s i n g i n | l | ; 5 . Twice c o n t i n u o u s l y d i f f e r e n t i a b l e i n x , L R ; cont inuous i n I ; t w i c e , c o n t i n u o u s l y d i f f e r e n t i a b l e i n I e x c e p t , p o s s i b l y , a t I t = 0 ; a k i n k a t 'I = 0 occurs on ly i f j f x C •) I = 0 0 when I :/ 0 . 6. l i m f R ( 0 = 0 ; l i m f £ ( - ) = - «> . s i g n ( I 1 ) R^oo I -j- i | -XJO I 7. M a r g i n a l f a c t o r products are i n c r e a s i n g i n f a c t o r l e v e l s at s m a l l f a c t o r l e v e l s . A g a i n on ly those p r o p e r t i e s which i n v o l v e R^ were not j u s t i f i e d i n e a r l i e r c h a p t e r s , A l . l i m p l i e s that c o s t s of e x t r a c t i o n do not decrease when rese rves d e c r e a s e ; A1.6 i m p l i e s t h a t the case of the c o n v e n t i o n a l non - resource f i r m a r i s e s when R^ tends toward i n f i n i t y . A d m i s s i b l e programs are r e s t r i c t e d to those where I t and L t a re p iecewise cont inuous f u n c t i o n s of t . T h i s i m p l i e s that x f c i s a cont inuous f u n c t i o n of t . Consequent ly , g i ven the c o n t i n u i t y of f ( - ) , Q1. = f(*) .must be 97 piecewise continuous i n t . I t i s a l s o assumed that there e x i s t s a s o l u t i o n to problem ( 1 ) , so that the existence of the Discounted Net Revenue f u n c t i o n (DNR), defined below, i s guaranteed. (2) R* J x ^ R ^ . w . a . e J = TI T2* - r - ( t - T l ) e q t - * M - I , - -L t •dt - r . ( T 2 * - T l ) x i ft + e -<j)(-oo) -x 2 , where a "*" i n d i c a t e s an optimum of problem ( 1 ) . The f a c t o r demands generated i n the process of s o l v i n g problem (1) to o b t a i n the DNR f u n c t i o n , can be c a l l e d "ex post f a c t o r demands" because they are c o n d i t i o n a l on the i n i t i a l l e v e l s of stock f a c t o r s x . Their d e r i v a t i o n has been i l l u s t r a t e d i n previous chapters f o r some p a r t i c u l a r cases. For the general case w i t h i n f i n i t e r e s e r v e s , they f a l l i n the category l a b e l l e d "Third generation dynamics" by Berndt et a l . (1980). As w i l l be i l l u s t r a t e d i n s e c t i o n 4.3, most well-known t h e o r i e s of f a c t o r demand can be cast i n the framework of t h i s model, where they u s u a l l y a r i s e as p o l a r cases. The n o t i o n of "ex ante f a c t o r demands" however, as presented i n t h i s chapter, i s not f a m i l i a r and r e q u i r e s an e x p l a n a t i o n . The DNR f u n c t i o n defined i n (2) i s a f u n c t i o n of the i n i t i a l l e v e l s of stock f a c t o r s , x^. The theory developed here r e l i e s on the assumptions ( i ) that those i n i t i a l l e v e l s r e s u l t from an economic d e c i s i o n of the f i r m and ( i i ) that such a d e c i s i o n i s i r r e v e r s i b l e i n a sense to be explained now. The term i r r e v e r s i b i l i t y has been introduced by Arrow (1968) i n t o the investment l i t e r a t u r e where i t r e f e r s to a non n e g a t i v i t y c o n s t r a i n t imposed on e i t h e r gross or net investment; t h i s meaning, which has been 98 mainta ined up to now, i s u n n e c e s s a r i l y r e s t r i c t i v e . For the purposes of t h i s paper , a d e c i s i o n i s s a i d to be i r r e v e r s i b l e i f i t i n t roduces a new c o n s t r a i n t to the f i r m . A non n e g a t i v i t y c o n s t r a i n t on investment i s an example; the appearance of cos ts of adjustment once a c e r t a i n p l a n t or c a p a c i t y has been s e l e c t e d i s another p o s s i b i l i t y . S t i l l another major branch of the l i t e r a t u r e on f a c t o r demand that f i t s under t h i s u m b r e l l a i s the c l a s s of models which assume d i f f e r e n t ex ante and ex post t e c h -n o l o g i e s : the p u t t y - c l a y models and t h e i r outgrowths, such as Fuss (1977) . The cho ice of x^ i s intended to maximize : (3) R* R r <(>, w; a , ej - ( j , ' . x x The e x i s t e n c e and p r o p e r t i e s of a s o l u t i o n w i l l depend on the p r o p e r t i e s of R*(-) and the p a r t i c u l a r form s e l e c t e d f o r <t> ( I ) . I t i s important to understand c l e a r l y the meaning of the assumptions on f ( ' ) and 'I' (•) tha t l ead to a l t e r n a t i v e forms of R * ( « ) » and, i n p a r t i c u l a r , to r e l a t e them to known t h e o r i e s of investment and f a c t o r demand. S e c t i o n 4 . 3 i l l u s t r a t e s how the assumptions of some important investment models can be embedded i n t o the DNR f u n c t i o n and problem ( 3 ) . 4 . 3 Important s p e c i a l cases A . Simple n e o c l a s s i c a l model ( p e r f e c t l y m a l l e a b l e and r e s a l a b l e f a c t o r s ) The types of models used by Jorgenson (1963) i n h i s e a r l y r e -searches on investment can be cas t i n t o the framework of t h i s model by imposing assumptions A2 . 99 A2: 1 . £ T (0 = 0 2 . a = 1 1 . means t h a t s tocks can be ad jus ted at no t e c h n o l o g i c a l c o s t . 2 . means t h a t the purchase p r i c e and the r e s a l e p r i c e of a u n i t of undeprec ia ted equipment a re i d e n t i c a l . B. Models w i t h c o s t s of adjustment The models w i t h c o s t s of adjustment s t u d i e d by Treadway (1970, 1971) would s a t i s f y A 3 . A 3 : 1 . f x ( . ) t 0, I j i O 2. a = 1 C. I r r e v e r s i b i l i t y a l a Arrow (1968) I t i s e a s i l y shown and i n t u i t i v e l y obv ious , t h a t , i n a model where adjustment c o s t s a re e i t h e r n u l l or min imized a t I = 0 , no n e g a t i v e investment w i l l be o p t i m a l i f the r e s a l e asset p r i c e of s tock f a c t o r s i s n u l l . The l a t t e r c o n d i t i o n i s e q u i v a l e n t to a n o n - n e g a t i v i t y c o n s t r a i n t on I. Thus A4 : 1 . a = 0 100 D. P u t t y - c l a y models P u t t y - c l a y models c o n t r a s t two s u c c e s s i v e s t a t i c - the ex ante and the ex post - s i t u a t i o n s . Ex p o s t , f a c t o r l e v e l s are g i v e n , so t h a t the on ly d e c i s i o n i s to produce at a l l ; 1 * ex a n t e , the f i r m faces a c o n v e n t i o n a l n e o c l a s s i c a l p r o d u c t i o n c o n s t r a i n t . Our model w i l l r e f l e c t such assumptions i f A5 i s imposed. A5 : 1 . A l l f a c t o r s are t r e a t e d as s t o c k - f a c t o r s ; 2 . Adjustment c o s t s are i n f i n i t e . Those assumptions imply tha t f a c t o r p r o p o r t i o n s and the s c a l e of p roduc t ion are chosen ex ante under a t e c h n o l o g i c a l c o n s t r a i n t represented by the p r o d u c t i o n f u n c t i o n , which reduces to f^xt> R t J SJ• The on ly ex post cho ice i n problem (1) i s the cho ice of T2. Concerning the r e s a l e c o n d i t i o n s , a l l p u t t y - c l a y models known to the author be ing models of the c o n v e n t i o n a l f i r m , they are not concerned w i t h the sc rap v a l u e of the p l a n t . However, s i n c e the hypothes i s i m p l i e s that a p l a n t can be operated o n l y i n one s p e c i f i c way, i t might imply tha t the p l a n t i s a l s o e n t i r e l y s i t e and job s p e c i f i c , and cannot be t r a n s f e r r e d a f t e r exhaus-t i o n of the mine , which would mean a = 0. Th is i s on ly cons idered as an extreme case of the p u t t y - c l a y hypothes is h e r e . E. P u t t y - s e m i - p u t t y models P u t t y - s e m i - p u t t y models p rov ide a compromise between the n a i v e n e o c l a s s i c a l assumption o f p e r f e c t f l e x i b i l i t y a t a l l t imes and the p u t t y -c l a y assumption tha t no adjustment can be made ex p o s t . Fuss (1977) i l -l u s t r a t e s the approach d i a g r a m m a t i c a l l y as i n F i g u r e 4 below. 101 F i g u r e A- Ex ante and ex post i soquants f o r a l t e r n a t i v e assumptions on the technology ex ante and ^ ex post i soquants -> L ex ante i soquants X 4 a l t e r n a t i v e I ex post \ I i soquants —> L yy ex ante isoquants a l t e r n a t i v e ex post i soquants a) p e r f e c t l y m a l -l e a b l e and r e s a l a b l e f a c t o r s b) p u t t y - c l a y hypothes is c) put ty - s e m i - p u t t y h y p o t h e s i s E x p l a n a t i o n In case a ) , no th ing d i s t i n g u i s h e s the p e r i o d which precedes a d e c i s i o n from that which f o l l o w s i t . The f i r m can operate at any p o i n t on any isoquant at any t i m e . In case b ) , each p o i n t on any ex ante isoquant can be reached through the cho ice of a s p e c i f i c techno logy ; however no two p o i n t s on an ex ante isoquant can be e f f i c i e n t l y reached through the same techno logy . When such technology i s s e l e c t e d , the cho ice set of the f i r m i s l i m i t e d to motions a long the one p a r t i c u l a r isoquant that corresponds to the t e c h n o l -ogy s e l e c t e d ; g iven i t s r i g h t - a n g l e shape ex post a l t e r n a t i v e s are reduced to produc ing at the p o i n t of tangency w i t h the ex ante isoquant or not producing at a l l . In case c ) , the ex ante s i t u a t i o n can be c h a r a c t e r i z e d as i n case b) However, ex p o s t , the cho ice set of the f i r m i s much w i d e r ; not on ly are movements a long the s e l e c t e d ex post isoquant f e a s i b l e , but a l s o changes from one isoquant to another , p rov ided they belong to the same f a m i l y , as d e f i n e d by the ex post technology s e l e c t e d . The ex post cho ice set of the f i r m i s the isoquant map c h a r a c t e r i s t i c of the technology s e l e c t e d . Such map cannot be q u a l i t a t i v e l y d i s t i n g u i s h e d from that a v a i l a b l e to the f i r m of case a ) , which produces a myopic behav iou r . 102 Figure 5: Ex ante and ex post isoquants f o r the generalized cost-of-adjustment model: A discrete-time i n t e r p r e t a t i o n Explanation The s i t u a t i o n presented i n Figure 5 i s very s i m i l a r to that of Figure 4 c ). However, unlike i n Figure 4c), ex post isoquants are dated i n Figure 5 . The ex post choice set becomes la r g e r , the longer the period allowed to reach a p a r t i c u l a r p o s i t i o n . Under the putty - semi-putty hypothesis, the firm was locked into one p a r t i c u l a r isoquant map ex post. Under the cost-of-adjustment hypothesis, such ex post isoquant map i s parameterized by the i n i t i a l state (beginning of the ex post phase), as represented by A l , and the period of adjustment. The ex ante isoquant i s asymptotically f e a s i b l e over an i n f i n i t e period. The associated ex post behaviour i s not myopic. 103 I t should be s t r e s s e d tha t F u s s ' s model , a l though e x p l i c i t l y dynamic, be longs to the f i r s t (naive) c l a s s of dynamic models i n t h a t i t produces myopic behaviour ex p o s t , a l l f a c t o r s be ing t r e a t e d as i n s t a n -tanous l y a d j u s t a b l e ex p o s t . In c o n t r a s t , cos ts of adjustment models i n t r o d u c e i n t e r t e m p o r a l e f f e c t s . In F i g u r e 5 , a d iagrammat ica l i n t e r -p r e t a t i o n of the model w i t h c o s t s of adjustment i s p r o v i d e d . The ex ante i soquant i s generated by s o l v i n g (4) f ( x , I, L ; R, 6) = q , f o r 1=0, w h i l e t r e a t i n g R and 9 as parameters . I f , as i n models of the f l e x i b l e a c c e l e r a t o r the f i r m does not f a c e any resource c o n s t r a i n t , the ex ante isoquant p rov ides the l o c u s of a l l o u t p u t - c o n s t r a i n e d l o n g - run e q u i l i b r i a a c h i e v a b l e by the f i r m a t v a r i o u s r e l a t i v e f a c t o r r e n t a l p r i c e s . I f , a t t ime T I , the f i r m makes the i r r e v e r s i b l e d e c i s i o n of choosing the f a c t o r p r o p o r t i o n desc r ibed by p o i n t A l , any subsequent change w i l l be c o n s t r a i n e d by c o s t s of ad justment . Consequent ly , over the p e r i o d [ T I , T I+1], the performance a c h i e v a b l e by the f i r m i s g i ven by the ex post i s o q u a n t . I f x + ^ i s to be reached over that p e r i o d , n e c e s s i t a t i n g an investment of x + ^ - x ^ , more l a b o u r , L2', w i l l be r e q u i r e d to m a i n t a i n the same l e v e l of p roduc t ion than i f x + ^ was the s tock l e v e l a t T I , i n which case L2 would s u f f i c e . Th is i n t e r p r e t a t i o n shows the s i m i l a r i t y between the put ty -s e m i - p u t t y h y p o t h e s i s , and an investment model where c o s t s of adjustment a r i s e as a r e s u l t of an i r r e v e r s i b l e d e c i s i o n made a t T I . The on ly d i f -104 ference i s t h a t , i n the l a t t e r case, the ex post isoquant and i t s p o i n t of tangency w i t h the ex ante isoquant are time dependent. Far from representing a departure from the putty - semi-putty hypothesis, the general case of our model should be viewed as a t r u l y dynamic v e r s i o n of i t . F. Returns to s c a l e and optimum s i z e The s p e c i a l cases considered so f a r were drawn from the e x i s t i n g l i t e r a t u r e on the c o n v e n t i o n a l , non resource f i r m . 5 For v a r i o u s reasons which are beyond the scope of t h i s s e c t i o n , e m p i r i c a l and, to a l e s s e r extent, t h e o r e t i c a l s t u d i e s have avoided the determination of output and concentrated on f a c t o r demands while t r e a t i n g output as given. In the model presented here, f a c t o r l e v e l s and output are endogenous. Returns to s c a l e are known to be important f o r the existence and determi-n a t i o n of a f i r m ' s o p t i m a l s i z e . Of course, i n a dynamic model, f a c t o r l e v e l s and output may vary w i t h time so that the n o t i o n of a f i r m ' s s i z e i s time dependent. However, there i s no conceptual d i f f i c u l t y i n d e f i n i n g the e x i s t e n c e of an optimum s i z e , or s c a l e , by the existence of an optimum path f o r output and f a c t o r l e v e l s . Since a s o l u t i o n to the ex post problem (1) has been assumed to e x i s t , t h i s c o n d i t i o n i s s a t i s f i e d i f and o n l y i f the ex ante problem (3) has a s o l u t i o n . I t i s i n t u i t i v e l y c l e a r t h a t , f o r a mine which faces a resource c o n s t r a i n t , the presence of economies of s c a l e does not n e c e s s a r i l y exclude the existence of an optimal s c a l e . S u f f i c i e n t c o n d i t i o n s f o r the existence of an optimal s c a l e under a l t e r n a t i v e r e t u r n s to s c a l e assumptions are s t a t e d i n sub-section 4.4.D. 105 4 .4 C h a r a c t e r i z a t i o n of the DNR f u n c t i o n A . Genera l case The DNR f u n c t i o n , R* ( - ) , s a t i s f i e s p r o p e r t i e s P I . PI : 1 . Continuous i n f a c t o r p r i c e s , w and <j> ; 6 2 . Continuous i n x^ and R^; 3 . Non i n c r e a s i n g i n v a r i a b l e f a c t o r p r i c e s , w, and such tha t R*(*) - 4 > ' « x ^ i s non i n c r e a s i n g i n $ ; 4 . Convex i n f a c t o r p r i c e s ; 5 . Non d e c r e a s i n g i n x^ ; 6 . Non d e c r e a s i n g i n R^. I t i s to be noted that those p r o p e r t i e s are not e x h a u s t i v e , i n the sense tha t the u n d e r l y i n g p r o d u c t i o n f u n c t i o n cannot be un ique ly recovered from them. A complete i n v e s t i g a t i o n of the d u a l i t y between the p roduc t ion f u n c t i o n and the DNR f u n c t i o n can be found i n E p s t e i n (1981) f o r the case of the c o n v e n t i o n a l f i r m . The two s p e c i a l cases envisaged below are p a r t i c u l a r v e r s i o n s of the p u t t y - c l a y h y p o t h e s i s . They d i f f e r by the treatment of. the scrap v a l u e of the p l a n t . B. T e c h n o l o g i c a l r i g i d i t y I f assumptions A6 h o l d s , then the DNR f u n c t i o n R*(*) s a t i s f i e s p r o p e r t i e s P 2 . A6: f T ( 0 - °°, 1 ^ 0 ( i n f i n i t e adjustment c o s t s ) P 2 : 1 . P I ; 2 . Non decreas ing i n s tock f a c t o r p r i c e s <(>. 106 D e s p i t e the absence of stock adjustments ex p o s t , the p r i c e s of s t o c k - f a c t o r s matter because they a f f e c t the scrap va lue of the p l a n t . C. T e c h n o l o g i c a l r i g i d i t y and i r r e v e r s i b i l i t y I f and on ly i f assumptions A7 h o l d , the DNR f u n c t i o n R * ( 0 s a t i s f i e s 7 p r o p e r t i e s P 3 . A7 : 1 . A6 ( t e c h n o l o g i c a l r i g i d i t y ) ; 2 . E i t h e r A4 ( i r r e v e r s i b i l i t y a l a A r row) , or R^ = » P 3 : 1 . P I ; 2 . <p i s not an argument of R (•)• Here i t i s not on ly assumed that s tock adjustments are i m p o s s i b l e ex post but a l s o that the scrap va lue of the p l a n t i s n u l l . In t h i s c a s e , and on ly i n t h i s c a s e , s t o c k - f a c t o r p r i c e s do not a f f e c t the DNR f u n c t i o n , a p roper ty which can be t e s t e d . D. E x i s t e n c e I f assumptions A8 h o l d , then problem (3) has a f i n i t e non n u l l s o l u t i o n x*. A8 : !• There e x i s t s x-^  such tha t R* (x^ , ' ) 2 < r ' " x » x~ > 0 .. 2 . The f o l l o w i n g p r o p e r t i e s are not s a t i s f i e d s i m u l t a n e o u s l y : f (*) e x h i b i t s non n e g a t i v e r e t u r n s to s c a l e when x^ tends to i n f i n i t y . 8 a = 1 E. Decreas ing r e t u r n s to reserves volume. I f assumptions A9 h o l d , then the DNR f u n c t i o n R*(« ) s a t i s f i e s p r o p e r t i e s P 4 . 107 A9: Besides A l , the underlying production function f ( - ) s a t i s Free d i s p o s a l . P 4 ; 1. .PI I 2. Concave in, R. 4 . 5 H o t e l l i n g ' s theorem r e v i s i t e d Problem ( 3 ) i s reminiscent of the conventional problem of determining f a c t o r demands i n s t a t i c theory. As a matter of f a c t , i f * the production function f ( ' ) i s chosen i n such a way that R (x^;*) i s twice d i f f e r e n t i a b l e w.r.t. x^, then assumption A8, which imply that ( 3 ) has an i n t e r i o r s o l u t i o n , implies the l o c a l quasi-concavity of R (x^;*)» * In that case, R (x^;*) s a t i s f i e s the standard r e g u l a r i t y conditions imposed on the production function i n s t a t i c microeconomic theory. I t follows that theorems developed i n the framework of s t a t i c theory should apply to problem ( 3 ) . Hotelling's theorem i s one of them but, as w i l l be shown now, i t cannot be generally extended to problem ( 3 ) . The same applies to Shephard's lemma. - R*(*) i s defined by ( 2 ) ; - Assumptions A8 are imposed to guarantee the existence of n (•) I t i s further assumed that Il(<f>) i s d i f f e r e n t i a b l e with respect The ex ante p r o f i t function i s defined as : (5) where to <}>. D i f f e r e n t i a t i n g !!(•) with respect to (f*1, one has : 108 311(0 3 R * ( 0 _ x i * . ! ! i + 3 R * ( 0 ' . ! ! l ( 6 ) 3 C 0 1 " 3 * 1 1 3 * 1 • 9 X 1 3 * 1 A Now the f i r s t o rder c o n d i t i o n s f o r problem (5) are A (7) -^4^ - ^ = 0 , j = 1 . . . m i * 1 M u l t i p l y i n g by — r , and w r i t t i n g the system i n m a t r i x fo rm, one has : 3R\O' 3 X 1 ' K (8) 1 " •'" ' " I s ° 9 x i s * 1 a r S u b s t i t u t i n g (8) i n t o (6) and r e a r r a n g i n g , one has 3 n ( 0 3R*(Q _ x i * Si})1 34) 1 1 w h i c h , rearranged and w r i t t e n i n m a t r i x fo rm, y i e l d s the f i n a l r e s u l t A One observes t h a t , when <j> i s not an argument of R ( 0 » (9) reduces to the f a m i l i a r form of H o t e l l i n g ' s theorem : Assumptions A 7 , g i ven i n s u b - s e c t i o n A . 4 , are s u f f i c i e n t f o r t h i s p a r t i c u l a r case to a r i s e . However, i n g e n e r a l , (9) h o l d s , and , f o r the 109 purpose of d e r i v i n g e x p l i c i t demand f u n c t i o n s , (9) i s even more i n c o n -v e n i e n t to use than the d i r e c t approach which d e r i v e s them from (7), the f i r s t o r d e r c o n d i t i o n s f o r problem (5). 4 . 6 Ex ante demands: a p r i m a l approach To c h a r a c t e r i z e ex ante f a c t o r demands, one d i f f e r e n t i a t e s (7), the f i r s t - o r d e r c o n d i t i o n s f o r problem (5), w i t h r e s p e c t to the d e s i r e d s t o c k f a c t o r p r i c e , say (f)1, to o b t a i n : (10) R* v (•) X1 X1 3x; x 1 ^ 1 - R i . i ( 0 <j>x x x 4>x where x , = 1 i - l ] t f i+1 m [*1 • . • ^ 1 J x l = By the l o c a l q u a s i - c o n c a v i t y of R * (0 i n the neighbourhood of i t f o l -lows from (10) t h a t tfx j * (11) i s unsigned and 3<j> 3x (12) s i g n - s i g n 3<t> 1 " R * i ^ i ( # ) x 1 <?± eva luated at x . 110 (12) does not exc lude the d i s t u r b i n g p o s s i b i l i t y that the ex ante demand f o r a s t o c k f a c t o r might be upwards lop ing . Th is problem i s r e m i n i s c e n t of a we l l -known p u z z l e i n comparat ive dynamics, that of de te rmin ing whether the impact e f f e c t on a f low has the same d i r e c t i o n as the e q u i -l i b r i u m e f f e c t on the cor responding s tock ( O n i k i (1973) , Nagatan i (1979)) In c o n v e n t i o n a l comparat ive dynamic problems 3cf> I 1 - d t t TI 3cj> _ i — i x 2 - X ] _ ax: 3<j> where x_ i s the e q u i l i b r i u m of the s t o c k s , whose g i ven i n i t i a l l e v e l s 1 3 l i a re x_ . But i t seldom f o l l o w s tha t the s i g n of — r should be the same 1 . 3(f)1 3x^ as the s i g n of r at a l l d a t e s . In the theory presented h e r e , the 3d)1 problem i s compounded by the f a c t s tha t r i s no longer g i ven and t h a t , 34)1 3X 1 i n the absence of any l o n g run e q u i l i b r i u m , the s i g n of — v cannot be 3d) e s t a b l i s h e d w i thout d i f f i c u l t y . S i n c e , by d e f i n i t i o n , T2 X l + I t dt = x 2 > one has TI T2* 3x 1 + 9 3x i * 3d)1 3d)1 TI 3d) b u t , i n g e n e r a l no term i n t h i s e x p r e s s i o n can be s i g n e d . From ( 1 0 ) , or ( 9 ) , i t i s c l e a r , however, that when <|> i s not an argument of R (•), * i 3x - < 0. 3<(> I l l From (10) a l s o , I t f o l l o w s that 3x i * m 3 ^ k = l ± k where E., i s the k1"*1 element i n the i1"*1 row of R* (•) \ l k xx S ince R x k < j ) j ( ' ) ^ s n o t e1ua^- t o R -K-kcbi^*^ ^ n g e n e r a l , 3 x 3 x 3<fr 3<f>-112 4.7 Summary By generalizing the notion of i r r e v e r s i b i l i t y to apply to any s i t u a t i o n where a decision narrows the choice set of a firm, one can construct a theory of investment which covers most well-known theories as s p e c i a l cases. That theory distinguishes between an ex ante phase and an ex post phase, separated by the i r r e v e r s i b l e decision. Two basic types of i r r e v e r s i b i l i t y are b u i l t into the model. The f i r s t one i s the assump-t i o n that adjustment costs become present a f t e r the i r r e v e r s i b l e decision. The second one i s the assumption of a discrepancy between the resale p r i c e and the purchase p r i c e of c a p i t a l equipment. Those two basic i r r e v e r s i -b i l i t i e s are combined at various i n t e n s i t y l e v e l s , during the ex post phase: adjustment costs can be anything between zero and i n f i n i t y ; the resale p r i c e can be anything between zero and the purchase p r i c e . Various well-known theories of investment can be interpreted as either the i n s t a n -taneous ex ante phase of t h i s model or i t s ex post phase, provided the required combination of the two basic constraints, i s selected. Since the ex post behaviour of the model has been discussed i n previous chapters, the emphasis i s put on the ex ante phase. The Discounted Net Revenue function, DNR, i s characterized and turns out to have many of the properties of a production function, at l e a s t i n the neighbourhood of an ex ante equilibrium, defined as the choice of i n i t i a l f a c t o r stocks which maximizes the difference between the DNR and the a c q u i s i t i o n cost of the factor stocks. However, i t d i f f e r s from a production function by i t s s e n s i t i v i t y to economic parameters. As a r e s u l t , while the ex ante decision can be formally reduced to a s t a t i c problem, the r e s u l t i n g factor demands do not have the same properties as s t a t i c f a c t o r demands. In 113 p a r t i c u l a r , they need not be downward s l o p i n g nor do they need to s a t i s f y the symmetry c o n d i t i o n . Exceptions a r i s e i n p a r t i c u l a r cases, which e n t a i l s the p o s s i b i l i t y of i n f e r r i n g p r o p e r t i e s of the ex post phase from the ex ante behaviour. For t h i s purpose, a d d i t i o n a l p r o p e r t i e s of the DNR f u n c t i o n were e s t a b l i s h e d f o r v a r i o u s s p e c i a l assumptions on the ex post behaviour. F i n a l l y under a dynamic d e f i n i t i o n of optimal s c a l e , s u f f i c i e n t c o n d i t i o n s f o r the existence of an optimal s c a l e were e s t a b l i s h e d . They tu r n out to be very m i l d and to be compatible, i n most r e l e v a n t circum-stances, w i t h the existence of p o s i t i v e p h y s i c a l r e t u r n s to s c a l e over the whole range of the production f u n c t i o n , and, i n a l l circumstances, w i t h the existence of p o s i t i v e p h y s i c a l r e t u r n s to s c a l e at the observed production l e v e l . A l l those r e s u l t s are c o n d i t i o n a l on the existence of a s o l u t i o n to the ex post o p t i m i z a t i o n problem. 114 Notes to Chapter 4 1Dynamic models of investment have t r a d i t i o n a l l y emphasized the ex post period by a n a l y s i n g the adjustment of a stock whose i n h e r i t e d l e v e l , or i n i t i a l l e v e l , was not e x p l a i n e d (e.g. Jorgenson (1963); Treadway (1970, 1971); Berndt and Wood (1975)). Models w i t h i r r e v e r s i b l e i n v e s t -ment (Arrow (1968)) and other bang-bang models (Clark, C l a r k e , and Munro (1979)) may sometimes be exceptions, i n that an i n i t i a l phase of instantaneous adjustment, f o l l o w e d by a phase during which the system folloxtfs a s i n g u l a r path, may be i n t e r p r e t e d as the ex ante choice of a f a c t o r l e v e l , f o l l o w e d by ex post short-term adjustments. H e l l i w e l l (1978), Campbell (1980), Masse (1962, pp. 362-369) emphasize the ex-ante ch o i c e . So do p u t t y - c l a y models. 2A r a t i o n a l e f o r such an assumption i s given i n chapter 3. 3See, e.g. C l a r k , C l a r k e , and Munro (1979). ^Problem (1) reduces to a maximization with respect to T2. Note that t h i s does not imply a steady production unless i s i n f i n i t e (case of the c onventional f i r m ) , or f ^ ( ' ) = 0. 5 F o r an assessment of v a r i o u s dynamic models, see Berndt et a l . (1980). 6 I n problem ( 1 ) , p r i c e s are expressed i n r e l a t i v e terms. C l e a r l y , R*(*) i s l i n e a r l y homogeneous i n absolute p r i c e s . C o n s i s t e n t w i t h the o p t i o n of not r e s t a t i n g A l each time a new property of R*(-) i s e s t a b l i s h e d , n e c e s s i t y i n t h i s property does not i n v o l v e A l : 115 P3 does not necessa r i l y imply A l ; but, given that A l i s s a t i s f i e d , necessity means that P3 implies A7. Tor the purposes of A8, - x^ tends to i n f i n i t y i f and only i f at le a s t one stock f a c t o r , say x 1, tends to i n f i n i t y ; - Non negative returns to scale are defined as a s i t u a t i o n where f (•)/4>'-x^  does not tend toward zero when at l e a s t one stock f a c t o r , x^, say, tends to i n f i n i t y . I t w i l l be noted that negative returns are then defined as a s i t u a t i o n where f (•)/<i>' .x^ tends toward zero when a l l stock factors approach i n f i n i t y . The d i s t i n c t i o n between cases where only some factors tend to i n f i n i t y , instead of a l l f a c t o r s , would be useless i f , i n A l , decreasing returns to stock factors had been imposed i n presence of at l e a s t one f i x e d stock f a c t o r . However, the only assumption made i s that f ( * ) i s non decreasing i n the stock factors ( A l . l ) . 116 CHAPTER 5 EX ANTE DEMAND FOR CAPACITY: THE CASE OF SOME NORTH-AMERICAN OPEN-PIT MINES 5 . 1 I n t r o d u c t i o n Chapter 5 c o n s i s t s i n an e m p i r i c a l implementat ion of the theory developed i n chapter 4 . That theory invoked a g e n e r a l i z e d n o t i o n of i r -r e v e r s i b i l i t y to d e r i v e ex ante f a c t o r - s t o c k demands. I t has been shown that the ex ante o p t i m i z a t i o n problem i s f o r m a l l y very s i m i l a r to a s t a t i c p r o f i t max imiza t ion problem f o r a wide range of assumptions on the ex post techno logy . However a c r u c i a l d i f f e r e n c e i s that ex ante f a c t o r -s tock demand d e c i s i o n s are not myopic i n g e n e r a l , and , as a r e s u l t , demands do not have the f a m i l i a r p r o p e r t i e s found i n s t a t i c theory . The theory i s implemented on a sample of Nor th -Amer ican open-p i t n o n - f e r r o u s - m e t a l mines . S e c t i o n 5 .2 d e s c r i b e s the d a t a , the sample, and the i n t e r a c t i o n s between the cho ice of the p a r t i c u l a r v e r s i o n of the theory to be used and the sample data a v a i l a b l e . S e c t i o n 5 .3 d e s c r i b e s the model u s e d , the e s t i m a t i o n s and hypothes is t e s t s c a r r i e d o u t , and goes over the r e s u l t s o b t a i n e d . A b r i e f summary f o l l o w s i n s e c t i o n 5 . 4 . 117 5.2 Data and Sample 5.2.1 A Microeconomic approach The theories presented i n Part 1 may be very diverse, but they a l l have i n common the c h a r a c t e r i s t i c of s t r e s s i n g a microeconomic ap-proach. Empirical applications dealing with the non-resource f i r m are plenty; f o r t h e i r vast majority, they use a s e c t o r a l approach. 1 The aggregation problems ra i s e d by such an approach are serious and researches i n the theory of aggregation have contributed more to showing problems than to circumventing them (Nataf (1948), Gorman (1968), Blackorby and Schworm (19SCa)). In a very h e u r i s t i c way, one can say that they become worse, the more firms are i n d i v i d u a l i z e d . Since extractive firms are characterized not only by t h e i r l e v e l s of stock-factors, but also by the orebodies from which they produce, i t i s to be ex-pected that t h e i r aggregation should cause serious problems. Blackorby and Schworm (1980b) investigate those issues and are led to introduce the resource stocks and c a p i t a l stocks of each fi r m as arguments of the aggregate technology set i f i n d i v i d u a l technologies are not to be reduced to t r i v i a l forms. But, from the point of view of empirical a p p l i c a t i o n s t h i s leaves the researcher with the need to c o l l e c t some microeconomic data even i f he, or she, plans to use a s e c t o r a l approach; or e l s e , he, or she, can keep with a long t r a d i t i o n of ignoring aggregation questions altogether. A f t e r a l l , aggregation problems encountered i n the study of extr a c t i v e firms may be compounded by the existence of resource stocks, but they are of the same t h e o r e t i c a l nature as those associated with non-extractive firms, and empirical studies of the l a t t e r have been 118 s u c c e s s f u l i n i n f e r r i n g meaningfu l s t a t i s t i c a l r e l a t i o n s h i p s (Berndt et a l . (1980) ) . However, non - resource s e c t o r s can be thought of as always i n a per turbed l o n g - r u n e q u i l i b r i u m ; such l o n g - r u n e q u i l i b r i u m does not e x i s t w i t h e x t r a c t i v e s e c t o r s where i t i s n o r m a l , even i n the absence of p r i c e p e r t u r b a t i o n s , to observe es tab l i shments e n t e r i n g and l e a v i n g the s e c t o r as a r e s u l t of d i s c o v e r i e s and d e p l e t i o n . Th is v o l a t i l i t y suggests t h a t e x t r a c t i v e s e c t o r s shou ld be t r e a t e d i n a more d e t a i l e d way than non c' e x t r a c t i v e ones . So f a r , the argumentat ion has not been s p e c i f i c to the theory developed i n P a r t I. I f one now l o o k s a t the s p e c i f i c requirements a s -s o c i a t e d w i t h the model of chapter 4 , i t appears immediate ly t h a t ex ante f a c t o r demands can be s t u d i e d on ly w i t h microeconomic data'. As f a r as ex post demands are concerned, they r a i s e the problems j u s t d i s c u s s e d . So reasons to adopt a s t r i c t microeconomic approach appear to be overwhelming and, indeed t h i s approach i s r e t a i n e d h e r e . However, a r g u -ments i n favour of a s e c t o r a l study are very s t r o n g , t o o . S e c t o r a l data are r e a d i l y a v a i l a b l e f o r s u b s t a n t i a l t ime p e r i o d s ; a s e c t o r a l study would be l e s s complex; f i r m l e v e l data are sca rce and l a c k u n i f o r m i t y . A l l t h i s must be kept i n mind when going through the l o n g l i s t of compromises which must be accepted i n order to r e a l i z e t h i s e m p i r i c a l s tudy . 5 . 2 . 2 Sample c h o i c e , u n i f o r m i t y , and s i z e The model of chapter 4 a p p l i e s to any f i r m . The t e c h n o l o g i c a l s e t which s u s t a i n s the p r o d u c t i o n f u n c t i o n f ( « ) and , through i t , the DNR f u n c t i o n R (•), i s common to a l l f i r m s . The c o n t r a r y would imply tha t d i f f e r e n t f i r m s do not have access to the same knowledge p o o l , a problem which i s r u l e d out as beyond the scope of t h i s r e s e a r c h . However, d i f -119 ferent firms operate at d i f f e r e n t points of the technological set, which necessitate the in t r o d u c t i o n of parameters to allow f o r those d i f -ferences. The more s i m i l a r the firms, the les s damaging the assumption that a given parameter takes i d e n t i c a l values for a l l , that i s to say the lower the number of necessary parameters. Obviously, there do not e x i s t two i d e n t i c a l mines. Even i f they use i d e n t i c a l techniques to produce the same metal concentrate, they w i l l be distinguished by d i f -ferent l o c a t i o n s , d i f f e r e n t ore reserves, d i f f e r e n t ore grades, etc. In order to be able to s e l e c t a sample of reasonable s i z e r e l a t i v e to the number of va r i a b l e s and parameters, i t has been assumed that the following major factors could be neglected: ( i ) Location, provided i t i s r e s t r i c t e d to North America; ( i i ) Output composition, provided i t i s r e s t r i c t e d to non ferrous metals; ( i i i ) Ore conformation, and s i t e geology, provided open-pit extrac-ti o n i s warranted, although not necessa r i l y e x c l u s i v e l y . Assumption ( i ) eliminates such important f a c t o r s as t a x a t i o n 2 and proximity to markets and transportation networks. Given that i t i s impossible to f i n d a large enough sample of mines i n any province or state, i t s sole j u s t i f i c a t i o n i s the desire to avoid too much complexity. Assumption ( i i ) i s j u s t i f i e d by the f a c t that the techniques and equipment used i n the extraction and concentration of non ferrous m e t a l l i c ores are very s i m i l a r (Canadian I n s t i t u t e of Mining and Metallurgy (1978)) . As f a r as assumption ( i i i ) i s concerned, i t i s clear that open-pit mines face widely d i f f e r i n g conditions. The slope of the p i t and the excavating and grinding equipment vary according to the nature of the rock which con-tains the ore; under maximum slope constraint, the shape of the p i t w i l l be 120 planned a c c o r d i n g to the shape and grade d i s t r i b u t i o n of the o r e b o d y . 3 This w i l l determine the t ime p r o f i l e of the s t r i p p i n g r a t i o , the r a t i o of waste over ore e x t r a c t e d . The orebody i s a l s o covered w i t h an overburden whose s i z e and compos i t ion va ry w i d e l y a c r o s s mines , and which has to be removed and d isposed of p r i o r to the b e g i n n i n g of normal o p e r a t i o n s . D e s p i t e those d i f f e r e n c e s , i t i s s t r i k i n g t h a t o p e n - p i t mines operate i n v e r y s i m i l a r f a s h i o n s and use the same k i n d of equipment. Data on s t r i p -p i n g r a t i o s are a v a i l a b l e , but on a s c a t t e r e d basis:J i t i s d o u b t f u l that they would be adequate f o r i n t e r - m i n e compar isons . ' 1 S i m i l a r • r e s e r v a t i o n s had to be a p p l i e d to i n f o r m a t i o n on the overburden removed or the p i t ' s s l o p e . The cho ice of non f e r r o u s m e t a l l i c o p e n - p i t mines has a l s o the advantage of s e l e c t i n g what appears to be a new g e n e r a t i o n o f mines . Th is i s e s p e c i a l l y s t r i k i n g i n the case of Canada where t h i s type of mine was absent p r i o r to the sample p e r i o d (1960-1979) and has become f a i r l y widespread s i n c e then . Th is phenomenon has been a t t r i b u t e d to the appearance of new l a r g e s c a l e equipment which gave a c o s t advantage to o p e n - p i t min ing over underground methods i n many s i t u a t i o n s , and promoted a number of p r e v i o u s l y known low-grade d e p o s i t s to the s t a t u s of economic d e p o s i t s . Th is research does not aim a t c o n f i r m i n g o r r e j e c t i n g such an e x p l a n a t i o n . However, i f one accepts i t , the re i s no reason to b e l i e v e t h a t t h i s was a once and f o r a l l s c a l e e f f e c t which then l e f t the technology unchanged over the s i x -t i e s and s e v e n t i e s . Indeed some people suggest (Mular (1978)) and some data seem to c o n f i r m , that new s c a l e economies were exper ienced over the whole p e r i o d . For example, a t P i n e P o i n t M i n e s , the average c a p a c i t y of the h a u l i n g equipment was 58 tons i n 1973 and 77.9 tons i n 1979; a t Lo rnex , i t was 120 tons i n 1972 and 151 tons i n 1979; a t Brenda, 90.7 tons i n 1969 and 97 tons i n 1979; a t Cyprus A n v i l , 66.3 tons i n 1971 and 120 tons i n 1979. 121 Of c o u r s e , economic f o r c e s were a t work d u r i n g the same p e r i o d , which may p rov ide an a l t e r n a t i v e e x p l a n a t i o n , but i t seems d i f f i c u l t to assume a p r i o r i t h a t . f a c t o r demands were independent of t h e i r dates over the s e l e c t e d p e r i o d . A c o n s t r a i n t tha t g r e a t l y a f f e c t e d the sample cho ice was the n e c e s s i t y to f i n d data on s i n g l e es tab l i shments whatever the data s o u r c e ; a r e c u r r i n g problem was to d i s t i n g u i s h the e s t a b l i s h m e n t , or p l a n t , from the co rpora te e n t i t y . A g a i n , i n s e l e c t i n g e s t a b l i s h m e n t s , every e f f o r t was made to s e l e c t comparable e n t i t i e s , which l e d to the e l i m i n a t i o n of any es tab l i shment which was not a m i n e - m i l l o p e r a t i o n . Data were gathered on a t o t a l of 42 mines b u t , f o r the p u r -pose of e m p i r i c a l l y e s t i m a t i n g the model of chapter 4 , one s i n g l e mine may p rov ide s e v e r a l o b s e r v a t i o n s . The i n i t i a l development of a m i n e - m i l l o p e r a t i o n , as w e l l as any major expansion of an e x i s t i n g m i n e - m i l l , were cons idered as o b s e r v a t i o n s f o r the purposes of the ex ante model of f a c t o r demands, and t h i s whether o r not the p r o j e c t was completed, or even under taken . As a r e s u l t , the body of data c o l l e c t e d i n c l u d e s 57 o b s e r v a - t i o n s . However, the same v e c t o r of data was not a v a i l a b l e f o r a l l 57 : o b s e r v a t i o n s ; as a r e s u l t , the cho ice of the v a r i a b l e s i n c l u d e d i n the a p p l i e d . v e r s i o n of the model a f f e c t s the sample s i z e . The four major data f i l e s generated f o r e s t i m a t i o n purposes i n c l u d e d 4 0 , 4 2 , 5 7 , and 57 o b s e r -v a t i o n s ' r e s p e c t i v e l y . The next s e c t i o n p resents the v a r i a b l e s inc luded i n the v a r i o u s v e r s i o n s of the model . 122 5.2.3 Selection of the v a r i a b l e s : c r i t e r i a and constraints In the next sections, I go over the variables which were used i n the empirical model, j u s t i f y the choices made and, when necessary, describe and explain the construction of some p a r t i c u l a r v a r i a b l e s . The objective i s to give the reader enough information to understand c l e a r l y the empir-i c a l research and the constraints that i t faced, while sparing him, or her, minute d e t a i l s about sources, u n i t s , or intermediate steps. Those are given i n Annex 2. Variables were selected according to the following c r i t e r i a and constraints. - Use the c l o s e s t possible substitutes to the va ri ab le s of the t h e o r e t i c a l model; ; - Avoid excessive reduction in the number of observations, as a r e s u l t of the i n c l u s i o n of a p a r t i c u l a r v a r i a b l e as some variables are not a v a i l -able for a l l observations; - keep the number of variables low r e l a t i v e to the number of observations, while t r y i n g to incorporate as much of the data a v a i l a b l e as possible. 5.2.A Stock factors and ^capacity i The variables which are the focus of t h i s research, stock i f a c t o r s , were not a v a i l a b l e i n a s u f f i c i e n t number of instances. When ———"—•* \ they could be found, i t was often impossible to d i s t i n g u i s h between mining and m i l l i n g equipment, or between machinery and b u i l d i n g s , or between buildings that housed the m i l l or mining equipment as opposed to r e s i d e n t i a l housing. Furthermore^ data on c a p i t a l were always expressed i n d o l l a r terms which could have ra i s e d d i f f i c u l t index problems i f they 123 had had t o be expressed i n p h y s i c a l terras. On the contrary, capacity was most frequently reported, usually without ambiguity. Under the assumption that capacity can be treated as an aggregator f o r a l l stock f a c t o r s , t h i s v a r i a b l e was used here as a substitute for the l a t t e r . Of course, since a mine-mill operation transforms ore into metal concentrate of a given q u a l i t y , i t s output varies according to the metal content of the ore treated. Two s i m i l a r m i l l s could produce widely d i f f e r e n t quantities of concentrate i f they treated ores of d i f f e r e n t grades. So output i s a poor i n d i c a t o r of the work performed by a mine-mill operation. Indeed, capacity i s always reported i n terms of the quantity of ore that can be processed over a given period, and was used i n t h i s form. However, i f capacity i s to be used to construct an index of stock f a c t o r s , another consideration must be taken into account. The quantity of stock factors required to extract and concentrate one ton of ore i s l i k e l y to increase as the metal content of the ore diminishes. This can best be seen by considering extreme cases: i f the ore i s of very high grade, i t can be 5 sent to the smelter without being previously concentrated. At the other end, i f the ore does not contain any metal, i n f i n i t e amounts of factors w i l l not be s u f f i c i e n t to process the ore to the desired l e v e l of con-centration. So, i f cD(?(x) i s an aggregator f o r the vector of stock factors x, capacity i s proportional t o S l S ( ' ) but the c o e f f i c i e n t of p r o p o r t i o n a l i t y increases as the metal content of the ore increases. Thus, (1) CAP = A(AGG)-OS(x) , where: 6 124 CAP stands for capacity, A ( 0 i s a c o e f f i c i e n t of p r o p o r t i o n a l i t y , AGG i s an index of metal content, described below. Furthermore, the same h e u r i s t i c argument as was used to j u s t i f y (1) requires that A(>) be close to zero f o r ores of low grade. Since the change i n A(-) has to do wi th changes i n the concentration process, f o r ores of high metal content, when no concentration i s required, there i s no reason f o r conditions of ext r a c t i o n to be affected as AGG increases, so that A(-) should become almost constant when AGG tends to 100 percent. A f u n c t i o n a l form that approximates t h i s behaviour i s A l (2) A(AGG) = 3. AGG , where B i s a p o s i t i v e parameter, A l i s a parameter that belongs to the i n t e r v a l (0,1) . 5.2.5 Reserves Reserves, another c r u c i a l v a r i a b l e i n the model of chapter 4, were found f o r a l l mines i n the sample. This notion, however, i s not without ambiguities. Reserves can be known with various l e v e l s of cer t a i n t y . The industry and governments have adopted a terminology that re f e r s to three basic l e v e l s of ce r t a i n t y . However, standardization i s not perfect, and there are s i x major designations f o r those three categories (Zwartendyck (1972)). Like the e n t i t i e s that they cover, those designations are not p e r f e c t l y accurate and categories are subject 125 to overlap. Proven, or measured, reserves designate those reserves that have been delineated, i n terms of quantities and l o c a t i o n , with the highest accuracy. They can be considered as c e r t a i n . Probable, or  indicated, reserves are known with c e r t a i n t y to be a v a i l a b l e at a spe-c i f i c l o c a t i o n , but t h e i r amount, as w e l l as the exact shape of the ore-body, have not been determined exactly. When one follows a time ser i e s giving the probable reserves of a mine, one notices that the l a t t e r are not a f f e c t e d by e x t r a c t i o n and that they most frequently diminish at the completion of an exploration program. The opposite i s true of proven reserves. What happens i s that current e x t r a c t i o n i s associated only with proven reserves, although the reverse does not n e c e s s a r i l y hold. This explains the f i r s t phenomenon. Concerning the second one, as exploration on the property proceeds, reserves that were recorded as probable are trans-ferred to the f i r s t category. So sudden changes i n a s e r i e s need not s i g n i f y a major r e v i s i o n i n estimates but rather a transfer from one ore category to another. S i m i l a r i l y the category of possible, or i n f e r r e d , reserves, which are known with l e a s t accuracy, may feed into the f i r s t two categories but, unlike probable reserves, they are i r r e l e v a n t to a mine's operations and u s u a l l y ' r e f e r to a property which i s not under e x p l o i t a t i o n . A mine, i n operation always has proven reserves. Sometimes those are the only reported reserves. This i s also the best defined cate-gory, and the one which was adopted here, i n f u l l awareness that other r e -serves play a r o l e i n f a c t o r demand decisions. Another problem associated with the d e f i n i t i o n of reserves i s the d i s t i n c t i o n between economic and p h y s i c a l reserves. Economic reserves are that portion of p h y s i c a l reserves that are worthwhile extracting. Physical reserves are affected only by discoveries, estimation e r r o r s , 126 and extraction. Economic reserves also reflect present market conditions and the state of the technology, as well as anticipations about their evolution. Physical reserves are perennial; economic reserves are time dependent. What companies and statistical sources report must be inter-preted as economic reserves. Indeed, in normal circumstances, when a mine shuts its operations, reserves, as reported, have been exhausted. In several instances reserves of a mine were revised as a result of changes in economic parameters. For some mines in the sample, the cut-off grade was reported, and in some instances, i t was modified during the mine's l i f e . If such changes in the definition of reserves can be observed over the l i f e of any particular mine, differences of definition must exist between cross-sections taken at different dates as are the data used for the empirical estimation of the ex ante factor demand model. This causes a problem of errors in variables in the estimation. However, the brief discussion of technological change given in 5.2.7 suggests that technological change augments reserves. If this phe-nomenon has been important over the period, the use of economic reserves instead of physical reserves might be an excellent way to standardize the model for technological change. ' Reserves are also characterized by their metal content, both, qualitatively (which metals), and quantitatively (grades). Several mines in the sample produced more than one metal; in those instances major metals were easy to rank by order of importance. Grades reported were averages for the deposit, but, of course, there is in reality a distribu-tion of reserves according to grade. This distribution has been studied, among others, by Bradley (1979, p .39) who estimates the parameters of a log normal distribution of reserves for two British Columbia copper deposits, 1 2 7 and Musgrove (.1976), who studies some of its mathematical aspects. It would have been conceivable to modify the model of chapter 4 to make * R (•) a function of Rl and the two parameters of a log normal distribu-tion, instead of a function of Rl only. However, in the absence of suf-ficient data to estimate the parameters for each deposit, such an attempt would not have been productive. So, for purposes of empirical estimation deposits have been treated as homogeneous and characterized by their size and average metal content. Since recoveries were not treated as decision variables, pro-portions of various metals in a particular mine's output were not consid-ered variable, so that both output value and metal content could be expressed as one variable each, in terms of the principal metal. AGG, the indicator of metal content, was constructed according to the fol-lowing heuristic argument. First define COMG, the combined grade of the ore as the grade that the ore would rate i f i t contained only its principal metal, but in quantities such that the value of the metal content was unchanged. The combined grade is frequently used as an indicator of metal content. Now, i f a mine produces several metals, the metal content, expressed in terms of the principal metal, must be higher than Gl, the grade of the deposit in that particular metal. It must also be lower than COMG, expressed in terms of the same metal because, other things equal, i t is cheaper to produce an output of given value in the form of one single concentrate rather than several concentrates. As a result, the indicator of metal content was constructed as a weighted average, (3) AGG = 1/2 • Gl + 1/2 • COMG 128 Another difficulty associated with reserves is of a more theoretical nature. It is often argued that mines "produce" reserves via exploration, and that they tend to "maintain" a level of reserves adequate for their production plans. As a result, published reserves data would be irrelevant as a determinant of factor demand. Indeed, it can be shown that, when reserves are "produced" at a cost, i t is in the interest of the firm to postpone exploration expenditures until such a time when discoveries will be immediately put under ex-ploitation. As a corollary, the firm keeps low or null reserves, but produces according to what i t knows will be ultimately recovered. Of course i f uncertainty is introduced, one expects that a mine will indeed maintain some reserves in order to protect its extraction operations from the risks inherent to its exploration activities. This may mean that capacity determines reserves ex post but the model of chapter 4 predicts an ex ante statistical relationship between reserves and capacity. Indeed, in the spirit of the model and precisely because investment has an irre-versibility attached to i t , a firm will be willing to incur high pre-production exploration costs to guarantee that the capacity to be acquired is backed by sufficient reserves. Furthermore, since there is an opportu-nity cost in acquiring too l i t t l e capacity, i t is in the interest of the firm to find out what the reserves are before selecting its scale of operation. So, ex ante, the causality goes from reserves to capacity, not the contrary, and measured reserves are probably a good explanatory variable for the capacity investment decision. 129 5.2.6 Prices For the reasons invoked in the construction of AGG, output of a multi-metal mine could be expressed in terms of the principal metal. The same holds for output value, ORP, which was computed in the following straightforward way : m (4) ORP = I PSi • Gi • REi , i=l where : PSi is the price of metal i , REi is the recovery of metal i . Consistent with the model, ORP gives the revenues from the treatment of one unit of ore, not revenues per unit of metal or concentrate produced. Only two factors, labour, the variable factor, and capital, the stock-factor, were used, although other variable factors such as energy could have been considered as well. Price indices were used for their respective prices, W and KP. As far as the index of the price of capital is concerned, Energy, Mines, and Resources Canada (1980) recently reviewed some of the major North American material and equip-ment price indices, and found i t necessary to construct a special index for Metal Mining Capital goods. The new index adds an open-pit-equipment component to the Chemical and Mineral Process Plant Index of Statistics Canada. This index was used for Canadian observations; unfortunately, for United States' observations, a less specialized index of capital costs had to be used. In the theoretical model of chapter 4, prices are anticipated prices, expected to be stable over the planning period. For empirical 130 purposes, anticipated prices were assumed to be a weighted average of the current price, the four most recent past prices, and the following-year's price. The introduction of information which has not been observed by the decision maker at the date considered is justified by the fact that managers have an idea of future prices and the naive belief that their guess is more often right than wrong. The weight structure emphasized present and future prices as determinants of expected prices. Thus, (5) ORPE = (6-0RPP1 + 5 ORP + 4 0RPL1 + 3 0RPL2 + 2 0RPL3 + 0RPL4)/21 , where : ORPE is the current expected price of output; ORP is the current price of output; 0RPP1 is the price of output during the following year;7 ORPLi is the price of output lagged i years. The expected prices of labour, WE, and capital, KPE were constructed in the same way. One output price and two factor prices leave room for two relative prices. Most of the empirical work was done using expected factor prices, normalized by the expected output price, KPE3 and WE3. However, as will be seen in the next section, a few experiments, some without much theoretical backing, involved other relative prices, or nominal prices, or an output price expressed per unit of metal rather than per unit of ore. Also, since data on metal recoveries were often missing, the following alternative ore price formulae were also used. ? 131 - (4) , with REi = 1 f o r any i ; - (4) , with REi as observed i f d i f f e r e n t from zero, and REi set equal to the mean of the non n u l l observations i f equal to zero 1 (zeros replace missing data). Those a l t e r n a t i v e formulas avoided discarding observations which had no figures f o r recoveries altogether and, for those frequent cases where only the recovery of the p r i n c i p a l metal was a v a i l a b l e , they avoided giving excessive weight to the p r i n c i p a l metal and discarding precious information on the grades and prices of secondary metals. F i n a l l y , f o r some metals, two important a l t e r n a t i v e p r i c e s e r i e s were ava i l a b l e , e.g. a serie s of the London Metal Exchange (LME) and a North-American series . While a l l mines i n the sample were North-Amer-ican mines, many traded with Europe or Japan and, i n some instances, had sales contracts which s p e c i f i e d prices i n terms of the London Metal Exchange quotations. However, i t was not possible to determine which seri e s applied to which p a r t i c u l a r mine. So one data f i l e was constructed using formula (4) only, and the LME series or, for zinc, Cana-dian p r i c e s , when applicab l e . That f i l e had 40 observations. In three other data f i l e s , . US pri c e series were used with formula (4) and i t s two a l t e r n a t i v e s . Those f i l e s had 42, 57, and 57 observations respec-t i v e l y . 8 5.2.7 Observation dates; technological change The determination of an observation's date was important i n two respects. F i r s t , f o r the purposes of t h i s empirical research, an observation i s the combination of mine-specific data such as reserves 132 and capacity, with market parameters which are drawn from time series. The date attributed to the mine-specific data determines with which element in the price series they are combined. Second, as was mentioned earlier, there are reasons to believe that technology has not been invariant during the sample period. Factor demands may have shifted with time and, as a result, i t is reasonable to test for any explanatory power of time in the model. The question of technological change will be discussed in the second part of this sub-section. The choice of an observation's date involves some personal judgment; in fact, the ex ante phase is not instantaneous as the theory of chapter 4 implies. Several years separate the development decision from the completion of the project. Obviously the theory to be tested focuses on decisions rather than their manifestations. However, at the moment a development decision is made, i t is s t i l l entirely reversible; between the development decision and the project's completion, the deci-sion is restlessly questioned, qualified, confirmed, modified in the light of new information. Each step, each action (phone calls, contracts signatures with contractors or suppliers, press releases, various construc-tion phases) "build up" the irreversibility until, at completion date, i t can be said that the ex post period sets up. The reduction of the ex ante period to an instantaneous phenonenon is a simplification that can be justified on the grounds that i t is brief, relative to the mine's l i f e , and that data are not available to describe i t , but i t remains that the choice of a date to summarize this period involves some arbitrariness. The criterion used was consistency across observations. From that point of view, completion date is the best choice since i t coincides.with the beginning of production. This choice caused a few problems in the 133 cases of non-completed projects. For those, the date was set at 1980 and, when conflicting information was available on a project, the most recent data were used and the conflict attributed to revisions of the project. One should note that the choice of the completion date as observation date accentuates the foresight component in the expected price data. In fact, i f the ex ante phase actually lasts four years, only one of the six prices used to determine expected price was observed at the beginning of the phase. It was also given the lowest weight. Empirical studies often use time as a variable to pick up trend effects not explained by other variables, technological change being one of them. On those heuristic grounds, the observation date TI was introduced in some of the regressions performed. One remark should be made at this stage. First assume that technological change affects only ex ante technological possibilities, as is consistent with the spirit of the model. Then, for two identical deposits developed at different dates but under identical price anticipations, the plants actually selected will not be identical. The more recent mine will use a more advanced technology. The cut -off grade will be lower for the high technology mine, more ore will be extracted before closure, the production paths will look as i f the high technology mine had had more i n i t i a l reserves. Indeed, since economic reserves are defined as that part of physical reserves which will eventually be extracted, economic reserves are higher in the case of the high technology mine than in the case of the low technology mine. So the use of economic reserves data instead of physical reserves data might be one way to take account of technological change. If technological change was the focus of the study, this approach would be inadequate because economic reserves react 134 to prices as well. As a way to standardize a model which uses prices as variables anyway, i t looks quite attractive. 1 0 5.3 Model, estimation, and results 5.3.1 The model Given the variables selected and the simplifications made, the Discounted Net Revenue function, DNR, used for empirical purposes was R (R1,0G(X1), KPE3, WE3, Tl) where11 - a l l arguments are scalars; - Rl represents economic reserves; -9G(xl) is an aggregator for stock factors xl which satisfies (6) c£(xl)- 3 • AGGA1 = CAP, 0 < Al < 1, B > 0 ; - KPE3 and WE3, the expected relative prices of the factors in terms of output, could be replaced by any other set of two normalized * prices, subject to the required modifications in R (•); - Tl, the completion date, is optional; - R (•) satisfies PI at least; 1 2 A - R (•) is furthermore assumed to be at least twice differentiable in its first four arguments; - Conditions A8 for the existence of an optimum vector of initial stock-135 1 3 factors, are met by assumption. Given (6), the problem of choosing 0Q(xl), or CAP, takes the form (7) Max CAP R 1 CAP Rl, —~—T=- , KPE3, WE3, TI a-AGG K P E 3 CAP' a-AGGM The first-order condition for optimality is 9R*(Q 1 KPE3 a•AGG a•AGG (8) ^ - KPE3 . The second order condition is 2 * 902 5.3.2 Functional forms and parameter restrictions The model to be estimated can be represented by conventional flexible forms such as the Generalized Quadratic, the Translog, or the Generalized Cobb Douglas which a l l have the advantage of generating demand functions that are either linear, or linear in some transform of the variables. Since econometrics is not the main focus of this empirical work, using any more sophisticated form might appear super-fluous. However, two of the properties established in chapter 4 pre-cisely call for a test of the linearity of the demand function in KPE3. 136 In fact, property P3 provides for a test of technological rigidity combined with Arrowian irreversibility, while property P2 provides for a partial test of technological rigidity. The implementation of these tests is described in 5.3.3. It suffices here to mention that they collapse in the special case where R (•) is a priori restricted to be quadratic in KPE3. With this particularity of the model in mind, demand * functions were derived for two particular functional forms of R (•): the Generalized Quadratic form, and a form which could be called the Generalized Power form by analogy with the Generalized Quadratic function. The latter is : (9) R (•) = a + fa., ' a s ) Rl 06 KPE3 WE3 TI •j (R1,9S, KPE3, WE3, TI)-A-Rl 93 KPE3 WE 3 TI where - A is a 5 x 5 symmetric matrix whose elements are written a „; - a, a,,...,a_, and the a..'s are scalars. 1 5 xi 12 In order for R* (• ) to satisfy PI, P4 , and sufficient conditions for the existence of an optimum i n i t i a l capacity to exist, the parame-ters must meet the restrictions given in Table II below. 137 Table II : Parameter restrictions, Generalized Quadratic function Property Restrictions PI.3: Non-increasing in WE3 9R (•) < Cva 8KPE3 PI.4: Convex in KPE3 and WE3 PI.5: Non-decreasing in 9 S PI.6: Non-decreasing in Rl Second-order condition: concave in 08 P4.2: Concave in Rl Locally imposed: a^ +a^ j Rl + a^-DOM-a, -KPE3+a. . -WE3+a.-T1<0 43 44 45 Globally imposed: a^<0,a i^<0,i=l-5 Locally imposed: a^+a^'Rl-t-^a^-lj + a.o-KPE3+a0.-WE3+a„,-T1<0 33 34 35 Globally imposed: a3<0; a 2 3 < l ; a33 a34 a34 344 positive semi-definite, that is : a33-° ; a44"° ; a33 a44- a34-° Locally imposed: a2+a2.j-Rl+a2203+ a23-KPE3+a24-WE3+a25•T1>0 Globally imposed: a2^0; a2i>0 Locally imposed: a^+a^^•Rl+a 2^*3g+ a., 0-KPE3+a.,.-WE3+3, c-TlsO 13 14 15 Globally imposed: a^O; a^-® a 2 2.0 . all^° • 1 3 8 Relation (8) becomes: 31 + a22'^ + a 2 l ' R l + a 2 3 , K P E 3 + a2A < W E 3 + a25* T 1 = KPE3 Using (6) to substitute forSff, and rearranging, one has : Al (10) CAP = AGG •(A2 + A3-R1 + A4-KPE3 + A5-WE3 + A6-T1) with: -a «a -a_n«a (l-a^-J-a A2 — ; A3 = / X ; A4 = — a22 a22 a22 « " a24' a A f t " a25' a A5 «• ; A6 *> a22 a22 The only restrictions that can be derived from those in Table 5.1 are: A2 > 0 and, i f PI.3 is globally imposed, A4 < 0 Furthermore, from (6), we have: 0 < Al < 1 Besides the fact that TI is optional, special cases of (10) can be obtained by varying the treatment of AGG. Two linear special cases are obtained by imposing Al = 0 or Al = 1. Al = 0 means that capacity is the aggregator for stock factors no matter the grade. If this is assumed, one may, or may not, want to allow for effects of AGG on physical productivity by introducing this variable into the production * 3R*(«) function, thus making i t an argument of R (•), with %&QQ > 0. This gives the following equation, where TI and AGG are optional: 139 (ID CAP = A2 + A3-R1 + A4-KPE3 + A5-WE3 + A6T1 + A7-AGG with A2 > 0 and, i f PI.3 is globally imposed, A4 < 0. Al = 1 means that the coefficient of proportionality between capacity and the aggregator£B is linearly proportional to AGG. This gives another linear form: (12) CAP = A2-AGG + A3-R1-AGG + A4-KPE3-AGG + A5-WE3-AGG + A6-T1-AGG with A2 > 0 and, i f PI.3 is globally imposed, A4 < 0 The other basic functional form used, the Generalized Power function i s : (13) R (•) = a + (a^ . .a5) Rl KPE3 WE 3 TI a2 ,„„a3 „ V 1 . R ,£)8,KPE3 , WE3 , TI -A-9 " t. 1 ) V 1 a, KPE3 ' a3 WE 3 a4 TI H where - a, a.'s, A and its elements a..'s are defined as in the case of the x xj Generalized Quadratic form; 140 - 1 s are real exponents; - i t should be noted that R (•) is quadratic in26, which allows the derivation of a simple explicit functional form for CAP, as will be noted below. * Parametric restrictions for R (•) to satisfy PI, P4, and the existence condition turn out to be more complex than in the previous case, and have no implications which can be exploited at the level of the demand function. So restrictions are imposed on the parameters of the demand function, according to the properties discussed in 4.6. (8) becomes: a i a2 a3 a4 A l + A 2 2 3 e + a 2 l ' R 23* K P E 3 + a 2 4 * W E 3 a25* T 1 = K P E 3 Using (6) to substitute for3S and rearranging, one has M A? A-} (14) CAP = AGG • (B1 + B2-R1 +B3-KPE3 +B6-KPE3 + B4-WE3M +B5-T1A5) with: -a..-a - a 9 i ' a ~ a ? V a Bl = — - — ; B2 = — — — ; B3 = — — — a22 a22 a22 -a„,-a -a„r«a T,/ 24 _c 25 a B4 = ; B5 = ; B6 = a22 a22 a22 A2 = ; A3 = a 2 ; A4 = c*3 ; A5 = The following restrictions must be imposed. 141 From (6), 0 < Al< 1 . 9 CAP From the discussion in 4.6, since 3KPJ?3 ^ 0 i f KPE3 is not an argument * of R (•), we must have: A3 B6 < 0 i f the term B3-KPE3 is not significant. From an econometric point of view, (14) has the drawback of containing two exogenous terms in KPE3 which become more collinear the closer A3 is to one. Since A3 tended to take on values close to one, the term B6'KPE3 was removed from (14), which gave the second basic form used for empirical estimation: (15) CAP = AGGA1-(B1 + B2-R1A2 + B3-KPE3 A 3 + B4-WE3 M + B5-T1A5) . . . , , « - . c 9CAP The parameter restriction associated with the sign of a^p^ then becomes: B3 < 0 i f A3 = 1 As before, the term in TI aims at picking up effects that are not re-flected in other variables and is optional. Before outlining, in 5.3.3, the hypotheses that the model can test, and indicating what particular statistical tests were used, i t is important to make the following remark about the choice of functional forms for R (•)• As already mentioned, the Generalized Power form was selected because, unlike the Generalized Quadratic form, i t allows one to detect, in the demand equation, the component which results from the 142 presence of KPE3 as an argument of R (•)• An advantage of using (15) as the basic form for the demand equation, is that a l l hypotheses subject to tests are particular cases of (15), including the linear form (11), that is to say, a l l hypotheses are nested, which is the case where the statistical tests used are least subject to controversy. 1 4' 1 5 5.3.3 Testable hypotheses and estimations. It is important to stress that, in (15), unless R (•) happens to be quadratic in KPE3, the exponent A3 takes a value which is different * from one i f KPE3 is an argument of R (•)• On the contrary, i f KPE3 is not an argument of R (•), A3 = 1 and B3 £ 0. In view of property P3, in * 4.4, which states that the absence of KPE3 in R (•) is a necessary and sufficient condition for ex post technological rigidity and Arrowian i r -reversibility, a test on whether A3 is significantly different from one indicates whether such ex post properties may hold. If A3 = 1, a test of whether B3 is non positive provides a test of the validity of the model (since i t is its sole prediction). Another obvious test is whether the technology is of a "clay-clay" nature, that is whether economic variables affect the choice of the capacity level or bear only on the decision to create the operation at a l l . Such a test is performed by comparing an unrestricted form of (15) with a form where a l l coefficients of economic variables have been set equal to zero. The significance of the parameters attached to Tl indicates any trend effect. However, as was argued in 5.2.7, interpreting Tl as an indicator of technological change would be rather dubious, in view of the fact that technological change effects are likely to be embedded in 143 the reserves variable, Rl. If Al takes on a value which is not significantly outside the interval [0,1] , the argument made in 5.2.4 is validated. This argument states that, at equal capacity, a higher investment is required to treat low-grade ore than high-grade ore. Although they are accompanied by significance tests, parameter values must be interpreted with caution for two major reasons. The first one is that, as appears in the result statements, parameter values and significance levels are sensitive to model specification. It is normal, in general, to expect changes in estimated values as a result of changes in specification; a better basis for inter-model comparisons is the elasticity, which was computed at mean values of the variables. However changes in t-ratios across models remain difficult to interpret. The second reason has to do with the basic deficiencies of the model and data. The chief ones are, I believe, the use of capacity as an aggregator for factor stocks, and the fact that price data were not adjusted for tax changes. While they have their shortcomings, the tests just mentioned were performed in a statistically meaningful way. Various special cases of (15) were estimated by a maximum likelihood procedure, using White's (1980) Shazam statistical package. While t-ratios were auto-matically provided as tests on individual parameters, likelihood-ratio tests were used whenever an hypothesis involved comparisons between restricted and unrestricted forms. The likelihood-ratio test is based on the following property: 144 (16) 2(LI - L2) ~ -X(r) where - LI is the value of log-likelihood function achieved using the less restricted form of the model; - L2 is the value of the log-likelihood function achieved using the more restricted form of the model; - r is the number of restrictions added between the two forms of the model. When a non-linear form is estimated, both the t-test and the likelihood-ratio test are only asymptotically valid. With linear forms, which were estimated by OLS, they also apply to small samples, provided the specification of the model is exact. 1 5 Of course, the specification which is considered "exact" in this research is equation (15), since i t was derived from the least restricted functional form of R (•)> the Generalized Power function. Besides statistical tests, a number of "casual tests" consisted in visually comparing the results of various OLS estimations. For those equation (11) was chosen, as i t is the simplest linear form.16 The "casual tests" involved the use of alternative relative and nominal prices; the use of contemporary instead of expected prices; the use of Gl, the grade of the principal metal, instead of AGG; comparisons between the four data sets; regressions on subsets of the samples; the use of metal capacity and metal prices instead of milling capacity and value per ton processed. 145 5.3.4 Results Those results which are considered most meaningful and less subject to conflicting interpretations are presented first. Then come those results which correspond to the "casual tests" mentioned in 5.3.3, and, to finish with, some comments on the significance and robustness of the estimators. Initial parameter values for non-linear estimations were generated by the Box Cox regression reported as # 1 in Table III Equation # 1 is a special case of (15) where the variable TI has been excluded (B5 = 0), as well as the variable AGG (Al - 0). The latter restriction implies that capacity is assumed to be directly proportional to the factor-stocks aggregator 2G. The exponent of the reserves variable Rl, A2, was set equal to 0.75, its last value in a Box-Tidwell estimation which did not converge (not reported). A3 and A4, the exponents of KPE3 and WE3, were estimated subject to the restriction: A3 = A4. A l l variables except the intercept, Bl, were found to be highly significant. The R^  was .97; the value of the log-likelihood function was -364.05. In the least constrained of the non-linear estimations, that value was only increased to -361.94. Run # 2 is the least constrained case. Both parameters attached to TI are insignificant; so is Bl, which can be interpreted as the constant term of run # 1, in view of the very low value of Al, Al which dampens variations of AGG . A somewhat puzzling result is the fact that B2, the coefficient of Rl, is not significant. This obser-vation must be compounded by the remark that A2, the exponent of Rl, is highly significant. All other parameters are significant and, while 146 Table 111 : Capacity demand est imation. Various r e s t r i c t i o n s of equation (15). ,„A1 Basic equation: (15) CAP = AGG t B2-R1 A 2 i- B3-KPE3 A 3 t B4-WE3 M t B5-T1A5J Number of observations: 40; Data set H 1. Run It Estim. procedure Box-Cox Max. l i k e l i -hood R e s t r i c t . A1-0;A2..75 B5=0;A3«A4 B5=0 B3=B4=B5-0 B5=0;A3-1 B5=0;A3=1 A2.1 Max. l i k e l i -hood PLS B5-0;A3=1 A2=1;A4.1 A1=A2=A3-A4=l =A5 A1=0;A2-A3. A4-1;B5=0 Kst imated A l .0615 (2.07) .0624 (2.15) 232.0 (225.16) -.0208 (.57) .0622 (2.13)1 .0799 (2.31) 0.0797 (2.50) B l 232.0 (.42) -253.6 (.24) -480.0 (.41) 232.0 (231.16) 907.08] (1.78) 820.62 (1.03) -5362. ( .39) 1057.8 (1.66) B2 1.724 (20.63) 1.727 (1.17) 1.648 (1.76) 11.76 (1.08) 1.366 (1.86) .1373 (5.01) .137 (5.50) 7.449 (2.75) .0887 (18.53 A2 e l f ic i e l l t s ( t - r n t L o s ) B3 . 7976 (11.61) .8018 (16.56) -939.4 (554.67) .63 (8.48) .8158 (17.33) -940.1 (3.22) -708.7 (1.70) 939.0 (754.55) -1390.1 ( 2.81) -1421.6 (2.64) 32042. ( .11) -967.4 (2.65) A3 B4 1.050 1.188 (5.13) 1.077 (12.42) 1 070.1 (713.33) 1 068.8 (4.13) 998. i (3.56) A4 1.050 1.142 (9.79) B5 1.101 (17.13) 1 070.0 (1000.3) 1 795.0 (3.77) 1.052 (42.08) 1.006 (25.92) -25.89 (.57) A5 1.97 -85.63 (.59) 1862.0 (3.43) 90578. (.34) 1204.3 (3.68) -271.85 (.01) L o g - l i k e l i -hood f u n c t i o n ^ Lj= -364.05 To generate a set of i n i t i a l parameter values for subsequent non-l inear runs. -361.94 -362.05 |L4= -380.64 |L5= -362.33 |r = -364.90 6 .20 L 0 = -430.25 L ? = -364.90 96 Remarks Least r e s t r i c t e d case Follow up on Run 0 2; confirms that TI i s not an important de-terminant of CAP Compare with Run tf 3 for a test of the s igni f icance of the economic v a r i a b l e s . The Clay-clay hypoth-es is i s rejected. Compare with Run 0 3 for a test of "A3-1"; The " r i g i d i t y - i r r e -v e r s i b i l i t y " hypoth-e s i s i s not rejected. B3 i s negative as required. _ Compare with Run t 5 for a test of "A2=l" the hypothesis i s re jected.  Compare with Run II 6 for a test of "A4 = l " the hypothesis i s not re jected.  Compare with Run it 7 for a test of "Al»l"; the hypothesis i s rejected.  Linear form. a) For some c r i t i c a l values of the t - r a t i o , see Table 5 .3 . 2 b) C r i t i c a l values of the x d i s t r i b u t i o n CAP AGG Rl KPF. 3 WE 3 C a p a c i t y M e t a l content Ore r e s e r v e Expected r e l a t i v e p r i c e of c a p i t a l Expected r e l a t i v e p r i c e of la b o u r 147 the theory does not give any predictions on their signs, they have sen-sible signs. The poor explanatory power of Tl calls for its removal from the model, which is done in run # 3. That estimation looks close to perfect, with a l l estimators significant at levels above 95 %, except for B2 (90 % ) . However, the increase in some t-ratios is puzzling, especially in the case of Bl ?that takes up the value i t took in the Box-Cox run (# 1), instead of its very different (insignificant) level of run #2, at a t-ratio level of 225! This suggests that the l i k e l i -hood function might have several maxima in Bl and that t-ratios may not be very reliable tests in those non-linear estimations. This impression is reinforced by the most puzzling results of run # 4, which seem to contradict the general impression acquired from linear regressions that physical data were of primary importance in determining capacity. Fortu-nately, the likelihood function behaves in the expected fashion, registering only a very mild decrease between run 2 and run 3, and a substantial decrease between run 3 and run 4. For those heuristic but compelling reasons, conclusions based on likelihood-ratio tests are believed to be more reliable, in this study, than those that rely on t-ratios. Comparison between run if 3 and run # 4 leads, through a likelihood ratio test, to the conclusion that economic variables contrib-ute very significantly to the determination of capacity. The "clay-clay" hypothesis is rejected at the 99 % level. In view of this result, the surprising estimator values and t-ratios obtained in run # 4 may be attributed to a specification error of the " ommitted variables" type. The test of "technological rigidity" with "Arrowian irrevers-i b i l i t y " , of which the putty-clay hypothesis is a special case, is 148 performed by comparing run # 3 and run // 5, through a likelihood-ratio test. In run # 5, A3 has been set equal to one, which corresponds to the situation where KPE3 is not an argument of R (•)• It is found that the hypothesis cannot be rejected. In fact, the estimated value of A3 is very close to one and significant in all relevant runs. Run// 5 allows a test of the overall validity of the theory, using the condition that B3 should be negative i f A3 = 1. Since A3 is found not to be significantly different from one, this condition applies and one can see that i t is unambiguously met. It is also remarkable that B4 is positive, which indicates that capacity is substituted for labour when wages increase. In a static model with two factors, this result is a necessity. Here i t is only a possibility, although the opposite result would be intuitively disturbing. Other runs in Table III. can be used to test for the significance of some parameters. However, except in the case of run # 8, they cannot be used to test any other economic hypothesis. Run # 6, compared with run.# 5 establishes that A2, the exponent of Rl, is significant at the 95 % level, and run # 7, compared with run # 6, that A4, the exponent of WE3, is not significantly different from one. The result that both ex-ponents of the price variables are not significantly different from one is a l i t t l e disturbing in that i t suggests a symmetry which is not to be expected from the model. In particular, whatever the* value of A4, WE3 * is and should be an argument of R (•). Run # 8, compared with run # 7, shows that Al is very significantly different from 1. The constraint 0 2 Al ^ 1 is met, as is evident from a l l runs where Al is not;constrained However, while this confirms the hypothesis that the coefficient of pro-portionality between the factor-stock aggregator,and capacity CAP 149 increases with the grade index, AGG, run # 1 establishes that the simple assumption that = a«CAP can be considered a good approximation. Finally, to complete the array of restrictions on the curvature of the basic equation in its significant variables, run # 9 corresponds to the special case where CAP is linear in a l l its variables as is. the case if R'c( •) 2 is quadratic and CAP is perfectly proportional to OG . With an R of .96, this model turns out to give a good f i t . The estimated parameters are quite different from those in run # 3, but are not comparable. To get a feeling for the biases that may be involved in specifying a linear equation in-stead of the non linear form used in run # 3, one should compare the elas-ticies given in Table V,., rather than coefficient values. Table IV- presents the results of some linear regressions which were carried out in order to check what happened under various circumstances. Runs # 9 (also in Table III) to # 12 estimate the same model with each of the data sets which were constructed. As indicated in 5.2.6, the four data sets differed by the number of complete observa-tions which could be included under alternative ways to compute ORP, the variable which represents output value. In set 1, London Metal Ex-change prices or Canadian (zinc) prices were preferred to their US counterpart, whenever two alternatives were available. In set 2, the US price series were used and two more observations could be included because unlike London Metal Exchange series, North-American series were complete up to 1979. In sets 3 and 4, the same price series were used, however, whenever the recovery of the principal or the second metals was not available, the observation was not dropped but RE1, or RE2, was set equal to one (set 3) or to the sample mean (set 4). It seems that the explanatory power of the model diminishes as the data sets used change, Table IV : Capacity demand estimators - l i n e a r forms Run II Equation Data set (0 obs. used) R e s t r i c t i o n s Estimated c o e f f i c i e n t s ( t - r a t i o s ) a R 2 Remarks A l A2 A3 A4 A5 A6 A7 9 CAP . A l + A2-R11 A3-KPE3 + A4-WE3 # 1 (40) None 1057.8 (1.66) .0887 (18.53) -967.4 (2.65) 1204.3 (3.68) .96 Comparisons of data sets 10 II ( 4 0 ) 1 ' II 1180.8 (.94) .1083 (13.21) -1479.3 (2.38) 1591.0 (2.81) .92 11 II II 3 (57) it 850.9 (.58) .0811 (10.79) -1410.8 (1.42) 1808.1 (1.99) .86 12 II t 4 (57) II 886.7 (.59) .0818 (10.74) -1084.7 (1.29) 1412.2 (1.84) .86 13 II II 4 (23) M i n e - M i l l creations 2161.3 (.77) .1222 (10.25) -1722.0 (1.15) 1808.9 (1.38) .90 Comparisons between creat ions and major expansions. 14 It 9 4 (34) M i n e - M i l l expansions -42.33 (.05) .0562 (10.69) -1095.0 (2.07) 1546.1 (3.04) .96 15 CAP-A1 +A2-R1 +A3-KPE2 +A4-ORPE2 # 1 (40) None 15 882 (2.38) .1079 (17.87) -12 910 (1.92) -381.0 (1.46) .91 Use of a l t e r n a t i v e p r i c e d e f i n i t i o n s . 16 CAP = A l + A2-R1 t A3-KPE + A4-WE + A5-0RPE II 1 (40) t i 4426.6 (.11) .1030 (16.01) -462.3 (1.85) 121.13 (.36) -1.063 (.53) -227.2 (.91) 459.6 (.58) .92 17 CAP o A l + A2-R1 + A3-KPE1 + A4-WE1 + A5-AGG + A6-T1 II 1 (40) it -4360.8 (.40) .1037 (16.40) -36 775. (.38) 65 481 (.77) -1014 (2.44) 96.66 (.66) .92 18 CAP = A l + A2 • R l + A3 • KPE3 + A4 • WE3 + A5-AGG // 1 (40) I  -122.0 (.15) .0879 (19.13) -790.5 (2.19) 1087.0 (3.42) 354.56 (2.06) .96 To be compared with Runs It 9, #5, and #3; the model uses form (11), not a s p e c i a l case of (14). a) C r i t i c a l values of the t - r a t i o , for 2 - t a i l tes ts : s igni f icance l e v e l degrees of freedom AGG CAP Rl KPE3(WE3) KPE2(ORPE2) KPE.WE.ORPE KPE1 (WEI.) 20 30 Metal content Capacity Ore reserves Expected rel a t i v e price of cap i t a l (Labour) i n terms of output Expected r e l a t i v e price of cap i t a l (output) in terms of labour Expected prices of c a p i t a l , labour, output Expected rel a t i v e price of capital (labour) in terms of metal 90% 95% 99% 1.725 1.697 2.086 2.042 2.845 2.750 151 from set # 1 to set #4. As far as the difference between runs # 9 and # 10 is concerned, there has been a significant gap between the London Metal Exchange prices and corresponding prices on major North-American markets, over part of the seventies. Since the few mines for which information on long-term sales was available used London quotations as reference in their contracts, the difference in goodness of f i t between the two runs may simply mean that London Metal Exchange prices are the adequate series for most of the mines in the sample. As to the differ-ence between sets # 3 and # 4 on one hand, and sets # 1 and if 2 on the other hand, the explanation is probably different. Sets if 3 and # 4 were constructed in an attempt to avoid discarding incomplete observa-tions when recoveries were the sole missing variables. Since recoveries of principal metals did not vary much across mines, i t was thought that there was l i t t l e damage in assuming reasonable arbitrary values when the information was not available. However, this probably also caused the inclusion of data of poorer quality for the following reasons. First, observations that did not include recoveries were most frequently taken, from directories, or newspaper clippings, or the annual report of a multiple-establishment firm, while those that did include recoveries were mostly derived from annual reports, or Financial Post cards of single-establishment firms. Second, beside the difference in sources, the newly included observations often corresponded to incomplete  projects. Since projects are often substantially modified during the construction phase, this may have caused the inclusion of observations which would prove incorrect in a few years. Also, expected prices being a weighted average of actual prices prevailing before, on, and after the completion date, the formula had to be modified- to use the last 152 observed price for subsequent years. This applied to uncompleted proj-ects. In contrast, with the possible exception of uranium (3 observa-tions in sets if 1 and 2, 6 observations in sets if 3 and 4), the observa-tions which were included in sets if 3 and 4, and not in sets // 1 and 2, did not seem to be drawn in abnormal proportions from any particular metal producing sector. Another issue was whether i t was legitimate or not to treat identically the first capacity creation and major expansions of existing firms. Since there was not enough observations in sets # 1 or # 2 to divide the sample into two parts, linear equations were estimated from two sub-samples of set if 4. The results are reported under Runs #13 and # 14 of Table IV. The coefficient of Rl, when estimated from the "creations" sample, is more than twice as high as when i t is estimated from the "expansions" sample, with values of the t-ratio which suggest that the difference is probably significant. If the results of this study were to be used for purposes of policy formulation or prediction, this result would certainly warrant the use of a rigorous test of the hypothesis that a l l observations belong to the same population. Since none of the qualitative results already mentioned would be affected, such a test was not carried out. KPE3 and WE3 are respectively the relative expected prices of the stock factor and variable factor, in terms of output, when the price of output is defined as the value realized in processing one ton of ore. Other prices can be used. In run if 15, ORPE2 and KPE2 respec-tively represent the expected relative prices of output and the stock factor in terms of the variable-factor price. Again, the sole predic-tion of the model is that A3 should be negative, and this condition is 153 met. The coefficient of Rl is not affected, and there is no particular interpretation of the fact that the intercept, Al, is much higher than in run #9. No run was done with the third possible set of relative prices, those expressed in terms of the stock factor price because the 9CAP derivation of the comparative statics result corresponding to g^g^ < 0 was a l i t t l e more involving. Other prices were used, but simply out of curiosity, since no theoretical justification was available. This resulted in regressions which used relative prices in terms of metal output (run it17), or even nominal prices (run it 16) . It is surprising that the f i t is higher in run//9 than in run #16, while the introduction of two relative prices instead of three nominal prices could be interpreted as the imposition of positive linear homogeneity in nominal prices. A model that has two more 2 variables and satisfies fewer restrictions should yield a higher R . The explanation probably involves a specification error: the function that gener-ates a demand function which is linear in relative prices is a better approximation * of the true R («)than the function which generates a demand function which is linear in nominal prices. Another intriguing result is that the coefficient of AGG is significant in run it 17, and not in run it 16. Here, the explanation rests with the definition of PE, PE = ORPE/AGG. Since the model of run it 17, which includes "wrong" prices, is misspeci-fied, a bias is to be expected in the coefficients of the relevant var-iables. This may explain the fact that A5 has a different sign than in other runs. However, without appealing to econometric theory, the change in sign can be explained by the fact that, in other models, AGG enters the demand equation because of a technological relationship. The price effect of changes in AGG is taken into account in the construction 154 of the price variable, ORPE. On the contrary, in the model of run # 17, this price effect is not provided for in price variables, so that, quite understandably, i t is picked up by AGG, to the detriment of the technol-ogical relationship just mentioned. This explanation is confirmed by the results of run # 18, which uses form (11), where the technological relationship between AGG and production is introduced at the level of the production function instead of being introduced at the level of the factor-stocks aggregator, as done in a l l special cases of (15), including the models of runs # 1 to 14. To terminate this result presentation, Table V gives elasticities of the demand for capacity with respect to changes in its major arguments, as estimated for various model specifications. In Table V , run numbers refer to the runs of Tables III and IV. The figures produced suggest a certain robustness of the results. 155 Table V : Capacity Demand Elasticities at mean values of the observations ^"\Variable Run // Rl AGG KPE3 WE 3 1 .9967 0 -1.166 1.406 2 .7352 .0615 -1.089 1.406 3* .7365 .0624 -.995 1.307 4 .7126 -.0208a 0 b 0 b 5 .7270 .0622 -.766 1.106 9 .5836 0b -1.012 1.332 10 .6651 0b -1.050 1.715 11 .5519 0b -1.030 1.420 12 .5568 0 b -.954 1.337 13 .6926 0b -1.345 1.544 14 .4596 0b -1.092 1.637 15 .7096 0b 18 .5780 .0572 -.8267 1.203 Preferred specification and results, ^ o t significant. bImposed. 156 In view of the result that in non-linear forms, the exponent of Rl, A2, and the exponent of AGG, Al, are significantly different from one and zero respectively, i t can be argued that the proper speci-fication is non linear. In view of the result that the exponents of KPE3 and WE3, A3 and AA, are not significantly different from one, the preferred non linear form might be that used in runs # 5 or 6. It will be noted that the price elasticities of run # 5 are substantially lower than those of run # 3, the unconstrained form, which indicates a high sensitivity of the elasticities to minor changes in parameters. Since the uncon-strained form yields estimates that are consistent i t is safer to choose run # 3 as best estimation of the true model. As to the significance of the elasticities given in Table V, for linear forms, elasticities are derived from one estimated parameter only, so that the t-ratios of Table IV can be used as significance tests, subject to the restrictions associated with specification error. Most significance levels are quite high; for example a l l estimators for runs # 9 and?y.8 pass the two-tail test at a significance level which exceeds 95 %. For non-linear forms, some elasticities depend on two parameter estimates, so that t-ratios on individual estimators do not give definite information on the significance of the corresponding elasticities. Besides that, t-tests or likelihood-ratio tests are only asymptotically valid. However, in both run # 3 and run # 5, the parameters of the price variables have t-ratios whose levels exceed the 99 % level of significance and the l i k e l i -hood-ratio test of the joint significance of price variables is passed at a.level which also exceeds the 99 % level of significance. For AGG, the t-test is repeatedly at levels which exceed 95 % and for Rl, although the 157 coefficient B2 has a significance level of only 90 % in non linear runs, i t significance exceeds 99 % when the exponent of Rl, A2, is constrained to be equal to one. To sum up, examination of the t-ratios and l i k e l i -hood ratios suggest that, unless otherwise mentioned, the figures of Table V are highly significant, that differences in estimated elastici-ties between runs are attributable to specification errors, and that the best values are those of run # 3. 5.4 Summary and conclusion The theory developed in chapter 4 implies that the planned li f e of a firm can be divided into two phases. In the first one, or ex ante phase, i t selects the factor-stock levels at which i t will begin operations. During the second one, or ex post phase, i t adjusts stock levels and selects optimal flows of variable-factor services. A major unforseen change in the firm's environment will put i t in a situation where a major revision of its plant must be considered, that is to say in a new ex ante phase. Ex ante decisions of a sample of North-American open-pit metal mines were studied according to that theory. As implied by the model, the capacity selected reacts to physical variables such as reserves and metal content, as well as economic parameters such as factor prices. The clay-clay hypothesis is rejected. The elasticity of capacity with respect to the price of capital relative to output price is around -1. There is a strong substitution effect between capital and labour; the elasticity 158 with respect to the relative wage being +1.3. The null hypothesis that the ex post phase might be characterized by fixed factor stocks and a null resale price of those stocks could not be rejected, a result compatible with the observed negative reaction of capacity to its own price. The significantly positive, and small, elasticity with respect to metal content confirms the hypothesis that a given capacity to treat low-grade ore involves more equipment than the same capacity to treat high-grade ore. The elasticity of capacity with respect to reserves is about .7, which suggests that, even if economies of scale are present at observed operation levels, their impact becomes lower, the higher the reserves. Finally, the date of the observation has no explanatory power. Since there is l i t t l e doubt that some technological progress occurred during the period considered, this result gives some weight to the hypothesis that technological progress might be reflected in re-serves, as reported by firms. 159 Notes to chapter 5 *A notable exception is Fuss (1978) who uses electricity generation data. 2Mining taxation has been a very hot issue during the seventies. The same decade also witnessed substantial price fluctuations whose effect is likely to have been preponderant. However, the only excuse to exclude taxation from the study is the importance and complexity of the issue, which justifies a research of its own, on the line suggested by Bernard (1979) for example. 3For a careful modelling of this decision, see Bradley (1979). ^Data sources are given in Annex 2. Stripping ratios reflected either the situation for a given year, or an average over a mine's entire l i f e or activity phase. Sometimes, the overburden removed were also included in the computation of the ratio. 5For one of the mines in the sample, Pine Point Mines, operations began before completion of the mill, with shipments of ore from selected zones of the deposit for direct smelting! 6 A l i s t of variables and symbols, including functions, is provided as Annex 3. 7A few observations were dated 1979 or 1980; uncompleted projects were dated in 1980. In such cases 0RPP1 and, sometimes, ORP was not avail-able and had to be arbitrarily set equal to the last observed figure, that of 1979. 160 8The 1979 figure for Copper was missing in the London Metal Exchange series. This explains the difference of two observations between two files which differ only by the choice of some price series (40 observa-tions vs 42 observations). 9Steady technological progress ex post might also cause some technical problems to the model. For one thing, the mine's l i f e would become infinite. 10Besides the scale trends illustrated in section 5.2.2 some data on cut-off grades can be interpreted as evidence of technological change. cut-off grade (%) A .7 .6 ,35 .25 (a) Copper 63 Bethlehem 67 Craigmont Beth. 72 75 Project dates Beth Beth Gibraltar cut-off grade (%) .08 .048 * .014 (b) Molybdenum 65 67 Endako End. 70 Lornex 75 Lornex 78 79 Proj.dates End Lornex Figure 6: Cut-off grades for some copper mines and some molybdenum mines 161 Figure 6 gives some feeling for the phenomenon. Cut-off grades are plotted against development or expansion dates for some copper mines (a) and some molybdenum mines (b). Interpretation should be cautious, as those figures pertain to firms of widely different scales. For example, Endako's capacity after its 1978 expansion was 32 500 t.p.d. while Lornex increased its capacity to 80 000 t.p.d. from 47 000 t.p.d. in 1979. Expected prices also affect cut-off grades. However the trend is observed over a period where prices do not seem to have been a major factor. In 1963 the price of copper was $ 834.33 per ton in 1970 Cana-dian dollars, while in 1975, i t was $ 889.95. Corresponding figures were 81.7 and 118.1 for labour and 90.5 and 107.0 for capital. Molybdenum prices were higher in 1965 than in 1976 but registered a strong increase thereafter. 1 1For a discussion of each variable, see Section 5.2; for a l i s t of var-riables and symbols, see Annex 3; for a derivation of the properties of * R (•)» in general and special cases, see chapter 4. 12P1: 1. Continuous in factor prices, w and cb; 2. Continuous in x^ and R^ ; 3. Non increasing in variable factor prices, w, and such that R (•) - 4>'"X^  is non increasing in cb; 4. Convex in factor prices; 5. Non decreasing in x^; 6. Non decreasing in R^ . If one assumes decreasing returns to reserves volume, R (•) is also concave in Rl (see 4.5.E). 1 6 2 13See 4.5.D. ^Specification error tests are discussed in Ramsey (1977). Since a l l functional forms estimated are special cases of (14), the problem of distinguishing alternative models in this work can be viewed as one of distinguishing between two classes of models, Ml and M2, where Ml C M2. This implies that (16) is a valid test for comparisons between various forms of (15), although only asymptotically. 1 Properties of likelihood-ratio tests are described in Kendall and Stuart (1967), Vol. II, pp. 224-247. 16When AGG is not an argument of the demand function, that is to say when A7 = 0, (11) is a special case of the function derived from the Generalized Power form, (15). 17A1though set # 2 contains 42 observations, as a result of a programming mistake, only 40 observations were used; of those 40 observations, two were not included in set #1. . 163 CHAPTER 6 THE SHORT-RUN BEHAVIOUR OF SOME NORTH AMERICAN OPEN-PIT METAL MINES 6 . 1 I n t r o d u c t i o n The theory developed i n chapters two to four of t h i s d i s s e r t a t i o n i m p l i e s t h a t a d i s t i n c t i o n should be made between ex ante and ex post f a c t o r demands of f i r m s . The e m p i r i c a l study of ex ante f a c t o r demand was the ob jec t of chapter f i v e . Chapter s i x i s an e m p i r i c a l study of the ex post phase. Whi le there i s a l i n k between the two e m p i r i c a l a p p l i c a t i o n s , t h i s l i n k i s not p e r f e c t . For one t h i n g , the mines whose s h o r t - r u n behaviour i s s t u d i e d here were a l l p a r t of the sample used f o r the ex ante s tudy ; however, the converse i s not t r u e . A l s o , s t r i c t l y s p e a k i n g , the f u n c t i o n a l forms used to model the ex post phase determine the f u n c t i o n a l form of the Discounted Net Revenue (DNR) f u n c t i o n used to model the ex ante phase . However, no attempt was made to d e r i v e one from the o t h e r . The f u n c t i o n s used i n t h i s chapter should be viewed as approx imat ions of the t r u e f u n c t i o n which generated the t r u e DNR f u n c t i o n . The l a t t e r was i t s e l f approximated by the f u n c t i o n used f o r the e m p i r i c a l work of chapter . f i v e . T h i s chapter i s d i v i d e d i n t o three s e c t i o n s , f o l l o w e d by a c o n c l u -s i o n . S e c t i o n 6 .2 d e s c r i b e s the sample and the v a r i a b l e s u s e d . S e c t i o n 6 .3 i s a reminder of the theory which i s used and g i v e s a s p e c i a l i z a t i o n of that theory to the p a r t i c u l a r case to which i t i s a p p l i e d . The r e s u l t s are presented i n s e c t i o n 6 . 4 . 164 6.2 Data and sample Yearly data on the operations of 12 open-pit non-ferrous-metal mines were collected, mainly from financial reports.1 Depending on the mine, the period covered could extend to a maximum of nineteen years, from 1961 to 1979. All those mines were also part of the sample used for the ex ante factor demand study of chapter 5; however, short-run data could not be found for some of the mines of the ex ante study, which had to be excluded. The number of years over which short-run data could be found raised another problem, as all the mines considered were created, and sometimes, expanded, between 1961 and 1979, some of them quite late in that period. Under the assumption that those mines faced the same tech-nological constraint and differed only by a number of parameters to be included as independent variables in the estimation, joint estimation 2 was in order; but as the period covered differed from mine to mine, this raised a problem of missing observations in the sample. While such a difficulty can be dealt with (Schmidt (1977)), the gains in the quality of estimation do not seem to be very high in view of the programming costs 3 involved m implementing the required procedures. Instead, a sample without missing observations was constructed, by selecting a subset of mines, and a subperiod, in such a way as to include the highest possible number of mine-year observations. This number was maximized when five mines were included over a period of fifteen years, from 1965 to 1979, for a total of 75 observations. The dependent variable was output, QArL, expressed in thousands of tons of ore processed over a year. Actual output data were corrected 165 for such uncontrollable events as strikes or fire. For the first year of production, the quantity of ore processed during the tune-up period was not taken into account; thus if the tune-up period ended in August, production was recorded for the remaining period and then put on a yearly basis. Similar corrections were carried out when a change in financial-year-end reduced or lengthened the normal financial year. However, there may be some differences in the periods covered by any specific year from one mine to the other and no attempt was made to adjust the corresponding price data accordingly. The hypothesis that stock-factors are fixed ex post passed the test carried out in chapter five, and indeed, the data showed that capacity, CAPMi, registered only infrequent jumps but of a substantial magnitude, which corresponded to major expansions. Such expansions fa l l into the category of ex ante decisions studied in chapter five. In the short-run, or ex post, phase, stock-factors, as represented by capacity, were consequently treated as fixed. As this study deals with only one variable factor, labour, this left room for only one relative price to determine output. NPMi, the relative price of output in terms of wage, was computed as the ratio of ORPMi, the output value, over w, the wage, both expressed in Canadian dollars of 1970. As in chapter five, the output value is given by the formula: (1) ORPMi = 1 GMij-REMij-Pj , j where : GMij is the grade of the ore extracted at mine i in metal j ; REMij is the recovery of metal j at mine i ; Pi is the price of metal j . 166 When the grade of the ore currently extracted was not available, the aver-age reserve grade was used instead. As already mentioned, the metal price, Pj, is a weighted average over a calendar year, while grade and recovery, GMij and REMij, correspond to the financial year of the mine, as does output. However, discrepancies between financial and calendar years are exceptional in the sample, as most mines adopted the calendar year as financial year during the 60's if they had not previously done so. Other variables in the restricted output supply function must reflect individual technological parameters of the mines. The major one is capacity, CAPMi, which is meant to reflect stock-factors. While i t is usually expressed in tons of ore processed per day, it was translated into the same units as output in order to facilitate comparisons and parameter restrictions. The level of reserves, RMi, is another characteristic which is mine specific. As published by firms, reserves are to be understood as "economic reserves" and refledt price anticipations, as well as the state of the technology and its anticipated evolution, at the time of the  evaluation. Reserve data used for the ex ante study of chapter five could be considered as most up to date, since an evaluation had precisely been carried out in order to help in the capacity development decision. The situation is different for existing mines. Exploration is carried out on the property on a more or less permanent basis, but fluctuates widely in 167 magnitude and does not usually involve the ore which is being depleted. Known reserves are not corrected in the short run for changes in the economic or technological environment but, if such changes occur, there are reasons to believe that they do not go unnoticed and are reflected in the operations long before an update is finally made. So, as extrac-tion proceeds, reductions in reserves may not mean that the ore is getting depleted and increases in reserves, whether due to exploration or updates, may acknowledge conditions which had been known for a long time to man-agers. The available data were used despite those drawbacks. As in the ex ante study of chapter five, it was postulated that processing is cheaper when the metal value is concentrated into one single metal than when the same value is split between several metals. Thus, the indicator of metal content used, AGGMi, was an average between the grade in the principal metal and the combined grade, as expressed in terms of the principal metal. This indicator varied widely accross mines but, for any given mine, registered only a minor but steady decrease as the firm optimally extracted the best ore first. Finally, many firms report in their financial statements of an internal learning process which extends much beyond the tune-up period. In order to reflect this process, as well as the possibility of ex post technological change which could improve the way the capacity was ex-ploited, the age of the firm, AGEMi, was used as an independent variable. As shown further below, it could be used in such a way as to reflect factor specific technological change or learning process. In order to avoid any problems of vanishing logarithmic functions, AGEMi was set at one plus the number of years of production of the most recent capacity.when used in logarithmic form. 168 6 .3 The model The theory which i s b e i n g implemented here i s a p a r t i c u l a r case of the g e n e r a l i z e d i r r e v e r s i b i l i t y theory of f a c t o r demands developed i n chapter f o u r . Chapter f i v e was an e s t i m a t i o n of the ex ante demand f o r c a p a c i t y . T h i s chapter d e a l s w i t h the ex post phase i m p l i e d by tha t t h e o r y . S ince i t was e s t a b l i s h e d i n chapter f i v e t h a t s t o c k - f a c t o r s cou ld be t r e a t e d as f i x e d ex p o s t , and s i n c e data on c a p a c i t y conf i rmed t h i s r e s u l t , the ex post problem reduces to tha t of choos ing the l e v e l of v a r i a b l e f a c t o r s ; s i n c e labour i s assumed to be the only important v a r -i a b l e f a c t o r , the ex post problem reduces to de te rmin ing labour demand o r , g iven c a p a c i t y , output supp l y . Th is ex post prob lem, aga in i s a p a r t i c -u l a r case of any of those s t u d i e d i n chapters two and t h r e e . Among the three e s t i m a t i o n procedures envisaged i n chapter two, the one which uses the s y n t h e t i c form of the H a m i l t o n i a n was s e l e c t e d . Th is approach i s b r i e f l y d e s c r i b e d aga in now, f o r the s p e c i a l case cons idered i n t h i s c h a p t e r . The s h o r t - r u n problem of the f i r m b e i n g to s e l e c t an optimum o u t -put t r a j e c t o r y , output i s be ing chosen so as to maximize the Hami l ton ian at any t i m e . The maximized c u r r e n t - v a l u e H a m i l t o n i a n can be i n t e r p r e t e d as an i m p l i c i t r e s t r i c t e d p r o f i t f u n c t i o n (Lau (1976) ) : (2) n(CAPMi, RMi , NPMi - y i ) = (NPMi - y i ) - f ( C A P M i , RMi , LMi*) - LMi* where y i ( C A P M i , RMi , NPMi) i s the o p t i m a l i m p l i c i t r e l a t i v e p r i c e of the o r e , f o r mine i ; 169 (3) LMi*(>) is the optimal quantity of labour resulting from the maxi-mixation of the Hamiltonian, given that the implicit relative price of the ore, yi»is at an intertemporal optimum, pi. By Hotelling's theorem, the output supply function is: qi(CAPMi, RMi, NPMi) = <*HilL_ E N , = f (cAPMi, RMi, LMi*) 3(NPMi-yi) From this, one has: 3 q i ( - ) = 9 3NPMi 3NPMi n33 N 3NPMi It follows that output supply has a positive price elasticity. To prove i t , suppose that the opposite holds. Since Jl^^ > 0 (Lau (1976)), this implies: 3NPMi But this means that the value of the marginal unit of ore increases more, as a result of a price increase, than the price itself, which is absurd. In general, a change in CAPMi, which represents the fixed stock-factors, has an ambiguous effect on output. However, given the close link 170 between the concept of capacity and that of output in practice, a positive  almost one to one relationship is to be expected. The effect of a change in reserves, RMi, raises difficult inter-pretation issues. Dropping, for the purposes of this paragraph, the mine specific symbols "Mi" or " i " at the end of the variable names, one can write the effect of a change in reserves on output as: = f (•) + f (•) • — 3R Rv ' L v ' 3R From the Maximum principle, (NP - y)-f L(-) = 1 Differentiating this equation with respect to R, one has: - ^ • V 0 + (NP - y)-f L R ( 0 + ( N P - q ) . f L L . | | ! l - 0 3L* It appears that is positive i f the first two terms in this expression o R are positive. This is clearly the case when the resource is homogeneous so that fTr>(') =0 and 3y/3R < 0. Consequently, if the resource is homo-geneous, 3q/3R > 0. If the resource is not homogeneous, the "Levhari-Liviatan effect" (1977) may invert this result. The first term is a rent effect; the second term is a current-marginal-cost effect. A change in the type of ore currently being extracted, as, for example, is observed in the data when extraction proceeds, would affect both terms. The dis-covery, of an orebody of some economic value but of poorer quality than 171 the ore currently being extracted would affect the rent but not current extraction costs. Unfortunately, the model is not able to distinguish between those two types of changes in reserves. As formulated i t implies a non nul second term whatever the type of reserve change which actually occurs, unless the reserve is homogeneous. Consequently, if i t is found that output reacts negatively to the level of reserves, a l l that can be said is that a "Levhari-Liviatan effect" is at play. Since, of the two effects mentioned, the first one affects output through the implicit rent, y, while the second one affects output through current extraction and processing costs, one can heuristically argue that the second one could be picked up by the index of metal content, AGGMi. In particular a strong current-cost effect of depletion would imply a negative relationship between output and the metal content of the ore currently being extracted. Simultaneously, the effect of discoveries, which may not affect current costs in practice, would s t i l l be picked up through the level of reserves, RMi. To sum up, the model predicts a positive reaction of output to relative output price. A negative reaction of output to either the level of reserves or the index of metal content, or both variables, indicates that reserve heterogeneity is so high that the implicit rent is negatively related to the level of reserves (a strong "Levhari-Liviatan effect"). For obvious reasons which are not directly related to the theory, one also expects a positive association between output and capacity. However, most of those predictions must be compounded by a discussion of technological progress, which may affect both the ex ante and the ex post technological constraints. 172 In presence of ex ante technological change the short-run prod-uction functions of the mines will differ according to their creation dates. This could be captured by the inclusion of the creation date as an argument of the production function. However, ex ante technolog-ical change may be more specific and affect certain factors individually; thus, we may have "capital augmenting", "labour augmenting" and "reserve augmenting" ex ante technological progress. Here, a few remarks are in order. First, capacity means capacity, whatever the amount of capital which is behind i t ; so capital-augmenting ex ante technological progress is reflected in the corresponding variable. Second, as was argued earlier, in 6.2, data on reserves are to be understood as economic reserves, at least at the creation date, which means that they reflect reserve aug-menting ex ante and, for that matter, anticipated ex post technological progress. Third, the function to be estimated is an output supply function, not a labour demand function. If ex ante "labour augmenting" technological progress occurs, the reported capacity may correspond to different levels of labour per unit of capacity, according to creation dates, but this should not affect significantly, the way output diverges from capacity in response to short-run changes in prices. The best assumption to make about ex ante "labour augmenting" technological change is that it is also reflected in the capacity variable. Capacity does refer to machinery and equipment, but that machinery and that equipment have been selected, ex ante according to anticipated prices and the state of technology (Fuss (1977)), which reflects the number of efficiency units of labour produced by one worker at the creation date and over the relevant sub- . sequent period. Fourth, the empirical evidence of chapter five indicates 173 that the creation date, Tl, does not affect significantly the ex ante capacity choice; this can be interpreted to mean that technological change, in its effects on the ex ante capacity choice, is basically "reserve augmenting". Those four remarks justify the assumption adopted here, that ex ante technological progress is fully reflected in the var-iables used. Ex post technological change does occur, and may take the form of a learning process, often mentioned in financial reports. Again, this progress may be factor specific. But i t is not reflected in the data: reported capacity is not adjusted when the workers become more acquainted with i t or when minor reorganisations of the production process permit a better use of the equipment; the training experienced by the labour force is not reflected in the aggregate wage data used; adjustments to reserves, other than resulting from extraction or discoveries, are not frequently done. Consequently, the age of the most recent major plant was used as an index of ex post technological progress. As can be seen in 6.3, attempts were made to determine whether such progress was factor specific or not. 6.4 Estimation and results Since the same wage index was involved in the determination of the relative output price of each mine,, the five output supply models constitute a system of seemingly unrelated equations (Zellner (1962)). This system was estimated by the Full-Information Maximum likelihood method using the TSP statistical package (Time Series Processor (1973)). Several different functional forms were tested. A major dif-174 ference with the empirical work of chapter five is that, here, l i t t l e infor mation could be derived from the theory as to which functional form was most appropriate. So the choice was made on intuitive grounds. In particular most forms were selected in such a way that a basic proportional between capacity and output could be established, leaving i t for other var-iables to explain discrepancies between those two variables. Also, a major preoccupation was to determine whether economic parameters affect output; so the price variable entered most models in a way which left the curvature of its relationship with output be determined endogenously. It appears immediately from the results reported in Tables VI-IX below that capacity is the major determinant of output and that the effects of other variables are not as clear. Consequently, the presentation of the results was organized in such a way as to help better identify what other variables might affect the output decision and how. It was thought that technical change, during the lif e of a mine, might play an important role and explain such oddities as the significant negative relation between output and price, which is reported in Table VI (run // 3). Each Table cor-responds to a particular form of technological change; in TableVI, there is no provision for technological change; in Tables VII and VIII, technolo-gical change is assumed to be "capital augmenting", and "labour augmenting" respectively; finally, in Table IX, technical change is not as-sumed to affect any factor specifically. No attempt was made to envisage "reserve-augmenting" technological change; this is not to deny its pos-sibility but because ambiguities in the meaning of reserves and metal content arise if one assumes "reserve-augmenting" technical progress. In fact, reserves are reevaluated periodically, at dates which cannot be easily identified, so that they reflect the current state of technology 175 Variable key AGEMi : Age of the most recent plant at mine i ; AGElMi : AGEMi +1; AGGMi : Metal content of the ore, at mine i ; CAPMi ' : Capacity of mine i , in thousands of tons of ore processed over a year; NPMi : Relative price of output in terms of wage; QAMi : Output of mine i ; RMi : Ore reserves of mine i ; ui : Implicit value of the ore in terms of wage. YEAR : Creation or expansion year minus 1960. Table VI: Forms involving no technical change; 5 mines; 1965-79; FIML Basic equation: LnQAMi = C + Bl-LnRMi + B2-LnCAPMi + B3-NPMiA3 + B4-LnAGGMi Run # Restric-tions Estimated Coefficients (t-ratios between brackets) Log likelihood function Remarks C Bl B2 B3 A3 B4 1 B2=l 0 -.0387 1 .1781 -.0547 -.0211 33.55 C=0 (2.39) (1.88) (.66) (.91) B2=l 2 C=0 0 -.0019 . t . . . .0001 10.43 0 32.36 B4=0 (1.18) (.00) (.01) 3 C=0 0 -.0022 -.9367 -.6435 .1316 0 33.87 B4=0 (.14) (22.69) ( 3.79) (1.08) 4 B4=0 -.1063 .0001 .9350 -.5418 .1550 0 33.74 ( .103) (.00) (21.26) ( .57) (.42) Table VII: Forms involving "capital-augmenting" technical change; 5 mines; 1965-79 A2 A3 Basic equation: LnQAMi = C + Bl-LnRMi + AGEMi •LnCAPMi + B3•NPMi + B4'LnAGGMi Restric-tions Estimated Coefficients (t-ratios between brackets) Log Run # C Bl A2 B3 A3 B4 likelihood function Remarks 5 -.3132 (.00 .0049 (.44) .0141 (4.55) .1950 (.00) .0189 (.00) -.0206 (.86) 27.66 Runs 5 and 6 differ only by initial values of A2 and A3 6 .0542 (.47) -.0264 (3.75) .0162 (10.27) .0339 (.41) .6991 (.24) -.0340 ( 1.54) 50.03 Convergence was not reached 7 C=0 0 -.0195 ( 2.22) .0159 (10.78) -.0000 (.00) -.4369 (.00) -.0073 ( .36) 48.89 Table VIII: Forms involving "labour-augmenting" technical change; 5 mines; 1965-79 a) Basic equation: LnQAMi = C + Bl-LnRMi + B2-LnCAPMi + B3•(AGEMi-NPMi] + B4-LnAGGMi Run // Restric-tions Estimated Coefficients (t-ratios between brackets) Log likelihood function Remarks C Bl B2 B3 A3 B4 8 B2=l C=0 0 -.01627 (-2.04) 1 .1562 (1.91) .2132 (1.56) -.06596 (2.95) 36.33 b) Basic equation: LnQAMi = C + Bl-Ln-RMi + B2•LnCAPMi + B3-LnAGElMi-NPMi + B4-AGGMi Restric-tions Estimated Coefficients (t-ratios between brackets) Log Run it C Bl • B2 B3 A3 B4 likelihood function Remarks 9 B2=l C=0 0 -.0316 (10.09) 1 .1757 (12.15) .0133 (.51) -.0755 (4.68) 54.27 10 B2=l C=0 B4=0 0 -.0279 (15.19) 1 .1258 (8.71) -.0982 (3.48) 0 49.60 11 C=0 0 -.0282 (3.39) .9950 (84.90) .1735 (11.69) .0006 (.02) -.0735 (4.45) 54.33 12 -.2100 (2.65) -.0335 (3.62) 1.027 (54.49) .2039 (10.54) .0475 (1.18) -.0420 (2.31) 55.73 Table IX: Forms involving joint technical change; 5 mines; 1965-79 a) Basic.equation: LnQAMi = Bl•LnRMi + B2-LnCAPMi + B3-NPMi + B4•LnAGGMi + B5-LnAGElMi Run # Restric-tions Estimated Coefficients (t-ratios between brackets) Log likelihood function Remarks Bl B2 B3 B4 B5 13 -.0181 .9885 -.0532 -.0238 .1529 52.69 (2.38) (90.96) (1.62) (-1.40) (9.88) 14 B1=0 0 .9662 -.0903 -.0081 .1532 51.08 (293.90) (3.35) (.51) (9.26) 15 B1=B3= 0 .9811 0 0 .0803 45.57 B4=0 (386.75) (5.36) b) Basic equation: LnQAMi = AGElMi •(B2»LnCAPMi + B3-NPMi) Run # Restric-tions Estimated Coefficients (t-ratios between brackets) Log likelihood function Remarks Bl B2 B3 B4 A2 16 .9602 (280.21) -.0020 (.09) .0234 (10.42) 47.52 c) Basic equation: LnQAMi = AGElMi -(B2-CAPMi + B3-NPMi) Run // R e s t r i c -t i o n s Est imated C o e f f i c i e n t s ( t - r a t i o s between b r a c k e t s ) Log l i k e l i h o o d f u n c t i o n Remarks B l B2 B3 B4 A2 17 .0071 (46.26) 5.2515 (19.72) .1253 (12.93) - 9 8 . 9 6 Convergence was not reached 180 at some dates, especially at and near creation and expansion dates, but lag behind at other dates. The index of metal content has the same drawback. As was just mentioned, among the forms which do not provide for technological change, run # 3 indicates a possible negative effect of output price, NPMi, on QAMi, the quantity produced (B3 < 0); however, the reality of this effect is dubious, since A3, the exponent of NPMi is not significant. Given the low value assumed by A3, which dampens varia-tions in NPMi, the proper interpretation is probably that the term in NPMi takes the role of a constant term in that run, where the constant happens to be constrained to be nul. Indeed the existence of a significant negative relation between QAMi and NPMi appears to be ruled out in view of the results of the less constrained run #4. A more meaningful negative relation is identified in run // 14 (Table IX) , where joint technological progress is postulated. This relation does not seem to resist minor specification changes, however; i t does not pass the 90 % significance test in run # 13, which is less constrained. Even if this negative relationship is significant, one notes that the implied elasticity is low. In the same Table, run // 17 seems to imply a very significant positive association between QAMi and NPMi; however a comparison with run # 16, which involves a very similar functional form, should make one cautious. Indeed, run # 17 associates the logarithm of QAMi with the untransformed capacity variable, which creates a gap between the two variables at some values; chance probably explains that this gap is apparently " f i l l e d " by the price variable. Indeed, since runs 16 and 17 differ only by their curvature in one variable, a comparison of the corresponding likelihood functions is meaningful; such a comparison implies that run // 16, which does not identify any 181 relation between output and price, should be preferred. Runs // 5-7, in Table VII imply that no price-output relation can be identified when "capital augmenting" technical progress is provided for. To finish this search of a possible relation between output and price, one should look at the results of Table-.VII where "labour augmenting" technological change is modelled. Coefficient B3 is positive and significant in all runs. However, any interpretation of this result should take account of the value taken by A3, the exponent of NPMi, in those runs. It is always low in absolute value, never significantly positive, once significantly negative (run it 10). Again the result of run it 10 is too accidental to be meaningful. So what-ever the model, it seems that no significant statistical relation between price  and output can be identified from the results presented in Tables VI-IX In contrast, if one looks for manifestations of technological change, the evidence seems overwhelming. Coefficients A2, in Table VII, B3, in Table VIII and B5 in Table IX-, are in a l l instances positive, small, and significantly different from zero. The implied elasticity lies between .08 and .15 for model (a) of Table IX, where the log-log functional form implies a constant elasticity. This means that output may increase by about 10 % each time the age of the firm doubles (starting from a minimum of one year). When computed for the case of run it 5 in Table VII, the implied age elasticity of output, for a 5000 ton per day capacity mine, is about .11 at 5 years, and almost identical at 10 years. In run it 11, the implied elasticity is about .17 at one year and .85 at 5 years. Obviously those figures must be accepted with caution. In fact, the functional forms of both Tables VIII and IX impose a curvature to the output-age relation. The imposed curvature implies a constant elasticity in the Table IX results and an'in-ising elasticity in the Table VII results. In that respect, the least con-creas 182 strained runs provide the most reliable evidence; they are to be found in Table 2. Indeed, the .11 elasticity figure given as an example (run it 5) seems quite  reasonable. While the existence of technological change appears to be fairly well-established, according to the evidence presented, no infor-mation was obtained on the specific nature of that progress. The evidence appears to be just as convincing whatever the specialization introduced in the formulation of the models. The last type of evidence which this empirical study deals with is the effect of geological variables such as reserves, RMi, and metal content, AGGMi, on output. If one looks at the values and significance levels of Bl and B4 in Tables VI-IX,.there is some evidence of a negative relation. As implied by the theory, this relation would indicate that a strong "Levhari-Liviatan" effect is being observed. Such an effect obtains when the resource is heterogeneous and extraction costs increase as extraction proceeds. It could be invoked to deny the apparent irrelevance of Hotelling's (1931) theory of extraction to real world situation. More recently i t was used as a basic assumption by Cairns (1981) in his evaluation of rents in the nickel industry. So its identification here is another step toward a reconciliation of the pure theory of extraction with the realities of mining. The results outlined above should be contrasted with those of Table X, which correspond to a log-linear formulation of the model. Runs it 18 and 20 seem to indicate a significant negative effect of price on out-put, as well the absence of any impact of the geological variables, reserves and metal content, on output. However, a likelihood ratio test using run it 19 and the (less constrained) run it 18 indicates that B3 is not as sig-Table X: Illustration of colinearity problems (log-linear forms); 5 mines; 1965-79 Basic equation: LnQAMi = C + Bl-LnRMi + B2-LnCAPMi + B3-LnNPMi + B4-LnAGGNi + B5-LnAGElMi + B6-LnYEAR Restric- Estimated coefficients (t-ratios between brackets) Log Run # C Bl B2 B3 B4 B5 B6 Likelihood Remarks tions function 18 -.2210 .89-10-3 .9530 -.0453 -.0150 .1739 .0664 57.68 Geological variables (2.64) (.07) (37.61) (1.94) (.71) (12.67) (2.77) are not significant 19 B3-0 -.4294 -.0086 .9945 -.0299 .1824 .0976 56.93 According to likelihood (5.62) (1.09) (87.30) (1.78) (13.55) (4.03) ratio test. B3 does not pass the 90% significance test in run # 18 20 B6=0 .0134 -.0087 .9500 -.0607 -.0173 .1753 56.68 Despite its high t-ratio (.18) (.90) (38.56) (2.47) (.78) (15.71) in run # 9, B6 is not sig-nificantly different from zero 21 B6=0 -.1461 -.0261 1.0090 -.0404 .1862 55.30 A likelihood ratio test B3=0 (2.33) (4.01) (94.31) (2.20) (15.26) shows that B3 is different from zero at the 90% signi-ficance level in run // 20. A comparison of t-ratios for Bl and B4 in runs t 20 and 21 illustrates colin-earity problems. 184 nificantly different from zero as implied by the t-test of run # 18. Obvi-ously, there is a problem of colinearity between some of the independent variables, which allows a strong shift of explanatory power from the price variable, LnNPMi, to the constant, capacity, the time trend, and the geo-logical variables, when LnNPMi is removed from the model. Indeed, the correlation between some of the independent variables is quite high, as Table XI indicates. The fact that capacity is highly correlated with time in 3 mines does not raise any problem, as the effect of that variable is well identified in a l l runs and robust to specification. But the high correlation of time with price LnNPMi, and metal content, LnAGGMi, in two instances implies that any of those variables may reflect a price effect, a time trend which could result from ex ante technological change, or a strong "Levhari-Liviatan effect" . What is clear3 from the results is that at least one of those effects is being identified by the data. As the sign of B3 is "wrong" in Table X, one can appeal to faith in standard economic theory to eliminate the price effect as an interpretation. The evidence does not allow one to conclude in favour, or against, a strong "Levhari-Liviatan effect" as opposed to a time trend of another origin. However, if ex ante technological change is already reflected in the data, as argued in 6.3, the relationship identified is more likely to reflect a "Levhari-Liviatan effect". The high correlation between time and metal content is an indication that conditions are present for this effect to arise and, indeed, that firms order their selection of heterogeneous extractive mate-ri a l . As noted by Bradley et al. (1981) this is important in determining the effects of taxes and royalties. Finally, one notes that the identi-fication of ex post technological change is not in question, as LnYEAR Table XI: Selected correlation coefficients LnYEAR LnCAPM2 . LnCAPM7 LnCAPMll LnNPM2 LnNPMll LnAGGM2 LnAGGM7 LnRM5 LnAGElM5 LnYEAR LnCAPM2 1 .899 1 .798 .924 -.893 -.835 -.858 -.937 -.925 .902 -.883 .807 LnCAPM7 LnCAPMll 1 1 -.879 -.927 LnNPM2 LnNPMll 1 1 .938 LnAGGM2 LnAGGM7 1 1 LnRM5 LnAGElM5 1 -.852 1 Comments: Table XI gives a selection of high correlation coefficients between independent variables for each mine individually; all variables whose correlation coefficient with any independent variable pertaining to the same mine exceeded .85 were selected. 186 shows low correlations with LnAGElMi (except in the case of mine 5) and the value and significance of B5 is remarkably robust to specification. 6.5 Conclusion The results of this empirical study of the short-run behaviour of supply confirm of reinforce three important hypotheses. The first one is the "putty-clay" hypothesis. According to the findings of chapter 5, the hypothesis that stock-factors were fixed ex post could not be rejected. From that hypothesis to the "putty-clay" hypothesis, there is only one step: the proportions of variable factors to stock-factors must also be fixed ex post. The failure of this study to establish any significant or meaningful positive relationship between output and price must be interpreted as more evidence that the "putty-clay" hypothesis holds for the type of mines under investigation. As a result , the use of micro foundations based on finite costs of adjustments should be seriously questioned in empirical studies of mining. The second hypothesis is the existence, and importance, of the "Levhari-Liviatan" effect, which results from the heterogeneity of the reserves. While no one has doubted the fact that resources differ from one mine to the other, the evidence that such heterogeneity might also be relevant at the microeconomic level comes as a surprise. However, more work should be done in that respect, as the effect is not clearly distinguishable from a mere time trend. 187 The third hypothesis is the importance of ex post technological progress, or learning by doing. This phenomenon, often reported by firms, is clearly identified in this study and can be unambiguously distinguished from technological progress dependent on time alone. 188 Notes to chapter 6 XDetails on data sources, units, or construction are given in Annex 3. 2The output of each mine had to be treated as seemingly unrelated to the output of any other mine, as the same wage variable was used to determine relative output price for a l l mines. 3Schmidt (1977) studies alternative estimators of the coefficients of two " seemingly unrelated regressions", one of which has T observations and the second T + E observations. He finds that neither analytic criteria nor the results of Monte Carlo experiments justify using an estimator which exploits the T + T + E observations rather than the 'usual' estimator based on T + T observations. T^he correlation coefficients of LnAGElMi with LnYEAR are .437, .401, .807, .550, and .579 for mines 2, 4, 5, 7, and 11 respectively. 189 GENERAL CONCLUSION The t h e o r e t i c a l part of t h i s d i s s e r t a t i o n provides a s y n t h e t i c , although mathematically p a r t i a l , treatment of the n e o c l a s s i c a l theory of f a c t o r demand. Investment t h e o r i e s which had been presented as mutually e x c l u s i v e are incorporated i n t o a s i n g l e model w i t h i n which they e i t h e r complement one another or a r i s e as p o l a r cases. Conventional f a c t o r demand th e o r i e s are a l s o g e n e r a l i z e d to the case of a f i r m which e x t r a c t s an e x h a u s t i b l e resource. Throughout the e x p o s i t i o n , the o c c a s i o n a l i n t e r -p r e t a t i o n of maximized Hamiltonians as i m p l i c i t p r o f i t f u n c t i o n s provides a p p l i c a t i o n s f o r standard d u a l i t y theory to the theory of the e x t r a c t i v e f irm. The t h e o r e t i c a l model used makes a d i s t i n c t i o n between two phases i n the l i f e of a f i r m : the ex ante phase and the ex post phase, separated by an i r r e v e r s i b l e d e c i s i o n . The " p u t t y - c l a y " hypothesis a r i s e s as a s p e c i a l case which i s t e s t a b l e . Indeed, from the e m p i r i c a l r e s u l t s obtained, t h i s hypothesis cannot be r e j e c t e d . The mines included i n the samples appear to plan t h e i r operations w i t h a long-run h o r i z o n . Economic v a r i a b l e s i n f l u e n c e only t h e i r ex ante d e c i s i o n s i n a s t a t i s t i c a l l y mean-i n g f u l way. Thus, ex ante, the l e v e l of c a p a c i t y s e l e c t e d i s s e n s i t i v e to the r e l a t i v e p r i c e s of c a p i t a l and labour, as w e l l as the qua n t i t y and q u a l i t y of the ore. Ex post, p r i c e s do not seem to i n f l u e n c e output d e c i s i o n s i n a meaningful way. However, ex post t e h n o l o g i c a l progress turns out to be an important f a c t o r , and there i s a l i n k between the l e v e l 190 of reserves, their quality, and output. Those results have important policy implications. In particular, they imply that tax incentives should be designed with the long run in mind, and they show that capital labour substitution possibilities could be exploited for the purpose of promoting employment. 191 BIBLIOGRAPHY Aitchison, J., and Brown, J.A.C. (1963), The Lognormal Distribution, Cambridge University Press, Cambridge. Arrow, K.J. (1968) "Optimal Capital Policy with Irreversible Investment", in J.N. Wolfe (ed.) Value, Capital, and Growth, Edinburgh, University Press, Chicago. Bernard, J.-T. (1979) An Economic Analysis of the Taxation of the Canadian  Mining Industry, Center for Resource Studies, Queen's University, Kingston. Berndt, E.R., Morrison, C.J., and Watkins, G.C. (1980), "Dynamic Models of Energy Demand: an Assessment and Comparison", Resources Paper No. 49, Department of Economics, University of British Columbia, Vancouver. Blackorby, C. and Schworm, W. (1980a), "Intertemporal Technologies and the existence of an Aggregate Investment Function", Discussion Paper 80-18, Department of Economics, University of British Columbia, Vancouver. Blackorby, C. and Schworm, W. (1980b), "Rationalizyng the Use of Aggregates in Natural Resource Economics", Discussion Paper No. 80-19, Department of Economics, University of British Columbia, Vancouver. Bliss, C. (1975), "Capital Theory and the Distribution of Income", North-Holland Publishing Company, Amsterdam. Bradley, P. (1979), "Modelling Mining: Open Pit Copper Production in British Columbia", Resources Paper No. 31, Department of Economics, University of British Columbia, Vancouver. Bradley, P., Helliwell, J., and Livernois, J. (1981), "Efficient Taxation of Resource Income: The Case of Copper Mining in British Columbia", forthcoming in Resources Policy. Cairns, R.D. (1981), "An Application of Depletion Theory to a Base Metal: Canadian Nickel", Canadian Journal of Economics, forthcoming. Campbell, H.F. (1980), "The Effect of Capital Intensity on the Optimal'Rate of Extraction of a Mineral Deposit", Canadian Journal of Economy, 13,(2), 349-56. Canadian Mines Handbook, Northern Miner Press, Toronto, annual. Canadian Institute of Mining and Metallurgy (1978), "Milling Practice in  Canada" , Montreal. Clark, C.W., Clarke, F.H., and Munro, G.R. (1979), "The Optimal Exploitation of Renewable Resource Stocks: Problems of Irreversible Investment", Econometrica, 47, 25-47. 192 Dasgupta, P.S. and Heal, G.M. (1979), Economic Theory and Exhaustible Resources, Cambridge University Press, Oxford Diewert, W.E. (1974) "Applications of Duality Theory" in Frontiers of  Quantitative Economics, Vol. II, ed. by M.D. Intrilligator and D.A. Kendrick, North-Holland Publishing Company, Amsterdam. Diewert, W.E. (1977), "Aggregation Problems in the Measurement of Capital", Discussion Paper No. 77-09, Department of Economics, University of British Columbia, Vancouver. Eisner, R. and Strotz, R.H. (1963), "Determinants of Business Investment" in Commission on Money and Credit: Impacts of Monetary Policy, Englewood Cliffs, N.J.: Prentice-Hall, 60-138. Energy, Mines, and Resources Canada (1980), "Capital Cost Escalator in the Non-energy Mineral Industry in Canada" , Mineral Division, EPAS, Ottawa. Epstein, L.G. (1981), "Duality Theory and Functional Forms for Dynamic Factor Demands", Review of Economic Studies, 48(1), 81-96. Fuss, M.A. (1977), "The Structure of Technology Over Time: A Model for Testing the 'Putty-clay* Hypothesis", Econometrica, 45(8), 1797-1821. Fuss, M.A. (1978), "Factor Substitution in Electricity Generation: A Test of the Putty-clay Hypothesis", in Fuss, M.A. and McFadden, D., editors, Production Economics: A Dual Approach to Theory and Ap- plications " , North-Holland Publishing Company, Amsterdam. Fuss, M.A. amd McFadden, D. (1978), "Flexibility versus Efficiency in Ex Ante Plant Design" in Fuss, M.A. and McFadden, D., editors, Production Economics: A Dual Approach to Theory and Applications, North-Holland Publishing Company, Amsterdam. Goldsmith, O.S. (1974), "Market.Allocation of Exhaustive Resources", Journal of Political Economy,82(5), 1035-1040. Gorman, W.E. (1968), "Measuring the Quantities of Fixed Factors", in J.N. Wolfe, ed., Value, Capital, and Growth, Papers in honour of  Sir John Hicks, Edimburgh University Press, Chicago. Gould, J.P. (1968), "Adjustment Costs in the Theory of Investment of the Firm", Review of Economic Studies,35, 47-55. Gray, L.C. (1914), "Rent Under the Assumption of Exhaustibility" , Quarterly Journal of Economics, 28, 466-89. Rpt. in Extractive  Resources and Taxation, Mason, M. Gaffney, ed., Madison, Wise, the University of Wisconsin Press, 1967, 423-46. Helliwell, J.F. (1978), "Effects of Taxes and Royalties on Copper Mining Investment in British Columbia", Resources Policy, 4, 35-44. Hotelling, H. (1931), "The Economics of Exhaustible Resources", Journal  of Political Economy, 39, 137-75. 193 Tntriligator, M.D. (1971), Mathematical Optimization and Economic Theory, Prentice-Hall, Inc., Englewood Cliffs, N.J. Jorgenson, D.W. (1963), "Capital Theory and Investment Behaviour", American  Economic Review, Papers and Proceedings, 53, 247-259. Kendall, M.G. and Stuart, A. (1967), Advanced Theory of Statistics, Griffin, London. Lau, L.J. (1976), "A Characterization of the Normalized Restricted Profit Function", Journal of Econometric Theory, 12(1), 131-163. Levhari, D. and Liviatan, N. (1977), "Notes on Hotelling's Economics of Exhaustible Resources", Canadian Journal of Economics, 10, 177-192. Lucas, R.E. (1967), "Optimal Investment Policy and the Flexible Accelerator", International Economic Review, 8, 78-85. McLaren, K.R. and Cooper, R.J. (1980), "Intertemporal Duality: Application to the Theory of the Firm", Econometrica, 48(7), 1755-1762. Marshall, A. (1920), Principles of Economics, 8th edition, MacMillan and co. Ltd., London. Masse, P. (1962), Le choix des investissements, lere edition, Dunod, Paris. Mizon, G.E. (1974), "The Estimation of Non-Linear Econometric Equations: An Application to the Specification and Estimation of an Aggregate Putty-Clay Relation for the U.K.", Review of Economic Studies, 41(128), 353-370. Mortensen, D.T. (1973), "Generalized Costs of Adjustment and Dynamic Factor Demand Theory", Econometrica, 41, 657-667. Morrison, C.J. and Berndt, E.R..(1979), "Short Run Labour Productivity in a Dynamic Model", Department of Economics, University of British Columbia, Vancouver (typewritten). Mular, A.L. (1978), Mineral Processing Equipment Costs and Preliminary  Capital Costs Estimations, Special Volume 18, Canadian Institute of Mining and Metallurgy, Montreal. Musgrove, P.A. (1976), "Mathematical Aspects of the Grade-Tonnage Distri-bution of Metals", in Vogely, W.A., ed., Economics of the Mineral  Industries, 3rd edition, American Institute of Mining, Metallurgical and Petroleum Engineers, Inc., New York, 192-207. Nagatani, K. (1979), "Notes on Comparative Dynamics", Discussion Paper No. 79-14, Department of Economics, University of British Columbia, Vancouver. 194 Nataf, A. (1948), "Sur la possibilite de contribution de certains macro-modeles, Econbmetfica, 16, 232-244. Neher, P.A. (1978), "Rent-a-Rig", Resources Paper No. 29, University of British Columbia, Vancouver. Nickell, S.J. (1978), The Investment Decisions of Firms, Cambridge Economic Handbooks, Cambridge University Press, Oxford. Oniki, H. (1973), "Comparative Dynamics (Sensitivity Analysis) in Optimal Control Theory", Journal of Economic Theory, 6(3), 265-283. Puu, T. (1977), "On the Profitability of Exhausting Natural Resources", Journal of Economic and Environmental Management, 4, 185-199. Ramsey, (1977), "Classical Model Selection Through Specification Error Tests", in Zarembka (ed.) Frontiers of Econometrics, Salant, S., Eswaran, M., and Lewis, T. (1981), "The Length of Optimal Extraction Programs when Depletion Affects Extraction Costs", Resources Paper No. 61, Department of Economics, University of British Columbia, Vancouver. Schmidt, Peter (1977), "Estimation of Seemingly Unrelated Equations with Unequal Number of Observations", Journal of Econometrics, 5, 365-377. Schulze, W.D. (1974), "The Optimal Use of Non-Renewable Resources: the Theory of Extraction", Journal of Economic and Environmental Management. 1(1), 53-73. Time Series Processor, Version 3.0 (1973) by Bronwyn H. Hall (adapted to CDC by John Breslaw). Treadway, A.B. (1969), "On Rational Entrepreneurial Behaviour and the Demand for Investment", Review of Economic Studies, 36, 227-239. (1970), "Adjustment Costs and Variable Inputs in the Theory of the Competitive Firm", Journal of Economic Theory, 2, 329-347. (1971), "The Rational Multi-variate Flexible Accelerator, Econometrica, 35(5), 845-855. (1974), "The Globally Optimal Flexible Accelerator", Journal of Economic Theory, 7, 17-39. White, K.J. (1980), UBC Shazam - An Econometrics Computer Program, Computing Center, University of British Columbia, Vancouver. Zwartendyck, J. (1972), "What is 'Mineral Endowment' and 'How Should we Measure It?", Mineral Bulletin MR 126, Energy, Mines, and Resources Canada, Ottawa. 195 ANNEX 1 PROOFS (Chapter 4) 196 Proof of PI ft 1. R (•) is continuous in factor prices w and 4>. Suppose that there exists a vector of arguments of R*(0 ft i at which R (•) has a discontinuity in at least one price, say w • After reordering the factor prices vector so that w1 comes first, i t can be written (w^M). To contradict the definition of continuity, proposition (*1) must hold. (*1) There exist two positive scalars, e and X such that for any positive scalar, X , smaller than X, either (a) R*(w1, M) - R*(w1 + X, M) < - e , or (b) R^ Cw1, M) - R*(w1 - X, M) > e / Suppose that (*1)(b) holds. Call {2} the optimal path ft i which solves problem (1), thus defining R (•)> at prices (w , M). {z} is feasible at (w1 -X, M). Call R the net present value achieved by using program {z} at (w1 -X,'M). Since {z} may not be optimal at (w1 -X, M), R < R*(w1 -X, M) Hence, using (*l)(b), (*2) R*(w1, M) - R > e VX, 0 < X < X" * i Now, since R (w , M) and R differ only by the price paid for factor i , 197 (*3) R (w , M) - R = + Tl T2 —r t 1 e L • A -dt t = X • B T2 where B = Tl t 1 -rt , . Lfc • e -dt > 0 Combining (*2) and (*3), one has (*4) X'B>e , ¥ X , 0 < X < X But, since B and e are positive, there exists X > 0 such that (*4) is contradicted. Hence (*l)(b) is contradicted. (*1)(a) can be contradicted in a similar fashion. Hence there is no discontinuity in w1 at (w1, M). Since i and (w1, M) are * 1 arbitrary, R (•) must be continuous in factor prices. QED. 2. R (•) is continuous in x^ and R^  The same notation is used as in the proof of P l . l . Suppose that there is a discontinuity at (x 1, M), then (*5) must hold. (*5) There exist two positive scalars, e and X, such that, V X, 0<A<X, either, (a) R (x x + x » M) - R (x^, M) > e , or, (b) R*(x^ - X , M) - R*(x^, M) < - e . 198 ~ ft ~ Suppose that (*5)(b) holds and that {z) defines R (x , M). {z} is feasible at (x^ - X , M) since f(-) is non decreasing in x; but the corresponding net present value R, is not a maximum in general. Hence, R < R (x 1 - X, M) consequently, using (*5)(b), (*6) R - R*(xa, M ) < - £ VX,0<X<X Now T2 (*7) R - R (xj, M) -rt f(x - x » z ) - f ( x , z ) v t t y v t' V •dt -r(T2-Tl) . - e • A From the continuity of f(-) in x 1 (A1.5), for any positive scalar e , there exists A > 0 such that (*8) f ( x t ~ V V : f ( v V > ~ v Consider the sequence {EJ.} = {e} . There exists a sequence{ X } such that (*8) is satisfied at a l l time, with e = e. Let X be the smallest element in{A t>. Then, For 0<{X }<{X), (*8) is satisfied at a l l time. Hence, combining (*7) and (*8), T2 (* 9 ) R - R M) > TI -rt e -e-dt = - Z Since I can be choosen to be equal to e, i t follows that (*6) can 199 always be contradicted. A similar contradiction can be established * i if (*5)(a) holds. Hence R (•) is continuous in x^. The same approach can be used for any i as well as to establish continuity in R^ * Hence R (•) is continuous in x and R . QED. 3. A. R (•) is non increasing in variable factor prices, w. The same notation is used, when applicable, as in the proofs of P l . l and PI.2. Suppose that (*10) R*(w1-A, M) < R*(wX, M) , X > 0 -Consider {z} which defines R (w , M). {z} is feasible at (w -X, M) and R > R*(w1, M) which contradicts (*10). QED. * 3.B. R (•) - cb'-x^ is non increasing in d>. Suppose that there exists cb° and cb"^  such that : cb0 < c])l * 1 1' * o o' R (<j>X) - cb *x^> R ( c b ) - c b -xx i 1 * 1 Let {zx} be the optimal program corresponding to cb . R (cb ) can be written as : 200 * 1 i ' 1' 1' 1 R ((f.-1) = CI, w1, * , cr<|> )• MU^) where MCz 1) = T2 TI T2 TI T2 TI T2 TI -rt 1 , e »q «dt e 'L -dt -rt Tl+ -e •! -dt -rt 1- -rT2 1 e •! 'dt t e * 1 M2 M3 M, . I t refers to any positive (negative) element of d 1 } 1^" and L^ " are the optima which define {z^}; is the corresponding output. Now suppose that program {z^ } is applied at prices <j>°. The corresponding net present value achieved by the firm is R (^.{z1}) = (l,w\ f \ a - * ° > M(zl), 1' .o' and, 201 (*11) f * 1 1' ^ 'R ( O - cb1 . X l fM 3-x 1]" M, (*12) £ (a(<f>°' - f 1 ' ) , * (<b°' - / ' ) ) 1\ M3~ X1 M, , since is negative and 0 > a • (<j>° - cb"'') > cb°- cb^ . The sign of the difference on the left hand side of (*11) is positive i f the right hand side, RHS, of inequality (*12) is positive, which is shown now. Considering the definition of and i t appears that RHS is an inner product whose terms can be grouped by stock factors. If we consider, say, factor i , the corresponding term is (*13) ,,ci . l i . a • (<J> - d> ) T2 -rT2 i e * x2 ~ -rt T i I e -I dt - x. t 1 Tl Now the amount of stock i left at closure is equal to the i n i t i a l amount plus the cumulated adjustments made during.the firm's l i f e , that is : i i X2 " X l + T2 f r dt , Tl (*14) or, T2 x 2 -Tl I*-dt - x : = o 202 Given that x* is non-negative, and that T2 > t, (*14) implies: T2 (*15) e -x2 - e ".I^-dt - ^ 0 , -rt ,.1 e T I with equality only if T2 = T I . (*15) implies that (*13), as the product of two negative terms, is positive. Since this result holds for any arbitrary i , the RHS of (*12), as a sum of positive terms, is positive which, considering (*11) and (*12) implies: [R(<), 0,{ Z1}) - <0°'-xJ- [ R V ) - ^ ' - x j > 0 But, as R(<j>°,{zi}) is not optimal at prices <J>°, we also have [ R V ) " " [ RV) - ^ ' x j ^ 0 , which contradicts the i n i t i a l assumption. So we have * r IN 1 T *i-ON O 1 1 o R (<{> J - cf> • X ; L < R [<S> J - <)> • x 1, i f <j> > <(> QED 4. R (») is convex in factor prices Let g' = [l , w', <t»(It)'] Let g x = X'gQ + (1-X)'g 1, 0<A<1 . Let | Z Q } » \zi\> \z-^\ b e t n e optimum ft paths that solves problem ( 1 ) at g , g , g respectively. Let R (g ), O X A O ft ft R (g..), R (g,) be the corresponding values of the DNR function. 1 A Let R^jz^|,gjj be the net present value achieved by using program j z ^ with prices g., i = 0, 1, A; j = 0, 1, A; i * j . Clearly j z ^ J is feasible 203 whatever the price vector, including g_.. Now R g^_.j can be written follows: as where: M h) -T2 Tl -rt T2 Tl -rt T2 -rt Tl * qt-dt * L -dt T* . -r-(T2 - Tl) I «dt + e * * * q^, L^, 1^  are precisely the optima that define Z j t -Similarily, For j = X, R * K ) = X g o ' - M ( { z x i ) + ( l - x ) - 4 - M ( { z x i ) = X-R < X-R J g Q j + (l-x).R J g ^ , since | z^J is not optimal at g Q or g^ QED. 204 5. R (•) is non-decreasing in x^. o 1 Consider < x^ If |z°|, which is optimal for x^ = x°, is used when x^ = x^, cumulated output will be higher than for x^ = x°, because the same quantity of variable factors and investment efforts will be combined with more stock factors. As a result |x°} might break the terminal constraint of non negative reserves. Let \%°\ be the program that replicates \z°\ until reserves are exhausted, i f at a l l . Clearly jz°| * o f 1 o 1 is feasible. If one compares R (jz }) with R x-,{z } , variable factor xl,{2°|); the i s and investment costs are either identical or lower for Rj scrap values are identical in nominal terms in both cases but realization may occur earlier in the second case; revenues from sales are realized earlier in the second case. Consequently, R:({z°}) , R(XJ, |Z O|) But since {Z q } is n o t optimal, R*(|z°}) < R ^ z H ) QED. A 6. R (•) is non-decreasing in R^ . The proof uses the same argument as proof 5. 205 Proof of P2 1. PI PI holds in any particular case. 2. R (•) is non decreasing in stock factor prices <j). If A6 holds, no stock adjustment occurs during the mine's l i f e ; the only term involving <|> in R (•)» as defined by (2), is the last one, which accounts for the scrap value of the stocks. Since that term is non negative, i t follows immediately that an increase in (j> cannot reduce R*(-)-3 QED. Proof of P3 A. Sufficiency: 1. PI PI holds in any particular case. * 2. $ is not an argument of R (•). In the definition of R (•), (2), a l l terms involving <J> vanish if A7 holds. QED. 206 B. Necessity: Consider (2), the definition of R (•); (2) can be written as T2 (*16) R (•) = -rt J.' e • q dt + cb Tl • T 2 - a Tl _ r t T— . e • I -dt -t T2 -rt T+ „ -rT2 e •It« dt + e • a -x2 Tl T2 - w e~ r t-L t dt Tl Let IP represent the investment and stock factor management program described by the term between square brakets. i) Suppose that IP * 0; then i f IP < 0, a reduction in cb will permit a strict increase in the net discounted value of the firm with the same * program, a fortiori i f anew, optimum, program is used. . Hence R (•) is a strictly decreasing function of cb. Similarly, i f IP > 0, R (•) is a strictly increasing function of cb. So, i f R (•) is not a function of cb, IP e 0 for any value of cb. ii) Suppose that IP = 0 for any <J> and that there exists a value of cb, — i * and a date, t, at which I * 0. Since f(*) is smooth in I f c when 1^  * 0 i * (A1.5), I_ occurs as an interior solution to the maximization of a t i * . Hamiltonian. In economic terms, in the case where I is positive, i t is set at a level such that the combined cost of acquisition and of the resulting immediate effect on production (cost of adjustment), will exactly offset the gain from a faster extraction path made possible by a higher level of the corresponding stock, x^, at a l l dates between t i * and T2. Since a change in cb affects the acquisition cost, I will be 207 i * affected. In fact, i f I_ > 0 and <f> increases from <f> to (J) , the same t o ± program as was optimal at <j>Q, except for a slightly lower investment at t, will increase the discounted net present value of the firm. Hence * * R (•) is an increasing function of <)>. So, i f R (•) is not a function * of <j>, I is a corner solution for any date and for any value of <f>. But, by A1.5, this requires I = 0 at any date and This establishes A7.1. fI(*> _ oo for 1*0. * i i i ) Suppose that I = 0 for a l l t. Then IP = 0 requires -rT2 (*17) e -a-x2 = 0. If we exclude the trivial case where x^ = 0 and the firm does not exist, x^ > 0 and 1^ = 0 at a l l dates imply that = x^ > 0. Then by (*17), either T2 = + 00, which means Rl = °°, or a = 0. This establishes A7.2. QED. Proof of existence LetSSbe the set of a l l physically and economically admissible vectors x^, that is to say of a l l vectors x^ that satisfy both * r (*18) x 1 > 0 and R (x^-) - (f> *x^  ^  0 no Since R (x^,) is continuous, i t has a maximum over JS provided 36 is compact and non empty. is riot empty. By A8.1 i t contains x^; i t 208 also contains the null vector. Clearly, by (*18), i f the null vector belongs to a?. i t is also its lower bound. So&is closed and boundedJbelow. We now investigate what happens when x_^ , tends toward infinity, that is to say when at least one element of x^, say x^, tends to infinity. lim ( V f r J - <K»)'-xJ = lim T2 TI -rt * 1 * e qt-w L t J T2 r — lim -rt 1 TI c(> (I )-It-dt + f -r-(T2-T1) ,, . ' s • lim e . <j>(-°°) -x^  - 4>(+°°) -x^  (*19) K< R + 0 + lim e r -a - 1 1 ^ The above inequality follows from the arguments that: (i) the revenues from extraction, net of variable costs, cannot exceed the value of the ore, 1 • R^ ; (ii) The extraction period, T2-T1, defined as A, tends to zero, so that the investment scheme can be approximated as one which consists in acquiring x.^  at TI and selling x^ at TI + A. Assumption A8.2 has the following implications. - If ct < 1, r -rA lim [e -a-lj'tji -x = - » - r 00 1 Hence lim j^ R*(x^ ) - 4>(m) '-x^ J = - ~, using (*19) . This means that, i f X l x^->-°° 209 x^ tends to infinity, x^ does not belong to , as defined by (*18). - Alternatively. If a = 1. the limit of I R (x^) - cb(«>) is deter-mined as follows. Since x^ tends to infinity, the production path can be approximated as constant over the extraction period, A : Also Hence q- = q = 't => A -—• , where R is the total amount of ore extracted R q -rA - 1 ~ - A-r , since A ~ 0 lim |e r A - 1  - 1 • cb I J - lim f—A • r • «J>T -x^l X^°° = lim -r R ^ q/9 •x1 (*20) - lim x,->°° R -r • — where q is the average output per dollar of stock factor. 210 5 _ i With non negative returns to scale, q does not vanish as <j> -x^  ap-proaches infinity, the limit in (*20) is finite, possibly null. Conse-quently, i t is possible that, considering (*20), and (*19), lim |R*(x ) - <t>'(«>).x 1 > 0 . 1 In such a case,36 is not bounded above. With negative returns to scale, on the contrary, "q tends toward zero as cb -x^  approaches °°; the limit in (*20) is -00, so that, using (*19), lim [R*(X ) - 9'(°°)-x = — Consequently, whenever A8.2 holds, lim |~R*(x , •) - $'(x)«x 1= z ->• oo L x l But this implies that06 is bounded above. So i t has been shown that R (x^,*) is a continuous function defined over a non-empty, compact set,«Ju . This implies that R (x^,*) reaches a maximum over that set. Finally, by A8.1, the maximum of R (x^-) cannot be the null vector; so A8 implies that problem (3) has a non null solution x^ QED. 211 Proof of P4 Let R* = AR° t (l-A)-R* , R° < R* and Using the same notation as in previous proofs, let z^ = ^1^, L^j {z°}, {z1}, be optimal at R^  = R° and R^  = R^  respectively. Let \ zX} = {\z° + (l-X)z^"} ; this definition is made coherent by the extension of the shortest program, say jz° \ to the end of the longest one, by a stream of zeros. At Tl, by the concavity of f(-) in R and the definition of RX, the efficient output corresponding to z X, q^ = f x^ R1> z is higher than the convex combination of the outputs corresponding to z° and z^, ° A 1 q 1 and q±. By free disposal the output associated with z X can be reduced A , o , \ 1 to q 1 = Aq1 + (1 - A)q1 . If this is done, at Tl t e, e small, R , - AR° + (1 - A)R1, , Tl+e Tl+e Tl+e ' and free disposal can be invoked to set output at q T l + E = Kl+c + ( 1 " X ) 4 l + £ Since this construction can be repeated sequentially, a feasible program can be constructed , which is such that 212 z* = Xz° + (1- X)zJ V t q* = Xq° + ( l - X ) q J V t R A = A R ° + (1- A ) R 1 t t t V t I f we c a l l R({Z^|, jqMJ the net present value associated with t h i s *( \\ X program, and R R the optimal program corresponding to R^ = R ^ , i t xs clear that R R, R , + (1- A ) - R 1R1J QED. 213 Notes to annex 4 1The direction of the inequalities results from PI.3. For the proof of continuity in <}>, some extra care must be used in * i stating ( l ) , as R (•) may either be increasing or decreasing in <f> . 2Note that, while Pl . l does not rely on any properties of f(*) (provided a solution to (1) exists), PI.2 relies on the continuity of f(*) in R^  and x^. 3For a formal proof, one can use the approach of proof PI.3. am indebted to Mukesh Eswaran for suggesting this proof. 5"Non negative returns to scale" and "negative returns to scale" are defined in footnote 7, chapter 4. 214 ANNEX 2 DATA 215 This data annex is divided into two parts. The first part de-scribes data files FINAL1, FINAL2, FINAL3, FINAL4 which were used for the empirical study of ex ante factor demands (chapter 5). The second part describes the data used for the empirical study of ex post output supply (chapter 6). As several mines provided data for both studies, and some variables were constructed in similar fashions in both studies as well, the two parts are not entirely independent and some of the accompanying tables pertain to both of them sources are given in code form. Table XV gives the code key. 1. Description of FINAL!, FINAL2, FINAL3, FINAL4 (ex ante data) 1.1 File format and content The data in files FINAL1, FINAL2, FINAL3, FINAL4 were used for the estimations of ex ante factor demands presented in chapter 5. Those files use the same format: (5 (E16.8,l), 4 E16.8). Thus one observation occupies 5 lines of 5 fields each and 1 line of 4 fields; for example, observation # n uses lines n + 1 to n + 6; i t contains the following variables : 216 Line ft Variables n+ 1 TI DE Rl Gl G2 n+ 2 G3 G4 G5 G6 REI n + 3 RE2 RE 3 RE4 RE5 RE 6 n + 4 CAP S ORP ORPE P n+ 5 PE KP KPE W WE n+ 6 AGG MC RC J See variable l i s t for symbol key and see further below for units, sources and modes of computation. Each observation corresponds to the creation or expansion of a mine. Depending on the definition used for some variables such as REi, the recov-ery of metal i , or ORP, the value of output, some observations could or could not be completed; in the latter case they were dropped. Thus files FINAL1, FINAL2, FINAL3, and FINAL4 basically contain the same information except for differences in the definition of some variables and the number of observations which could be included. FINAL1 contains 40 observations; FINAL2 contains 42 observations; FINAL3 and FINAL4 contain 57 observations. The last variable in each observation; J , is an index of mine creations or expansions; this index is common to the four files. For example J = 5 refers to the creation of Island Copper Mine in al l files and the observa-tion corresponding to J = 4, which refers to the creation of the Highmont mine, is to be found only in files FINAL3 and FINAL4. Thus the index J is discontinuous in files FINALl and FINAL2 which contain less than 57 observa-tions and is continuous in the other two files where i t ranges from 1 to 57, 217 for 57 observations. Table XII gives a li s t of mine creations or expansions, as indexed by J, and indicates whether or not any particular observation is included in files FINAL1 - FINAL4. 1.2 Variable description, units, and sources Tl gives the last two digits of the creation, or expansion, year of the mine. That year is taken to be the first year of production for the newly created capacity. Source^ (1), (2), (4), (5), (6), (7), (9). DE is a dummy variable which takes on a value of 1 when the observa-tion pertains to the creation of a mine and a value of 2 in the case of a major expansion. Source^ (1), (2), (4), (5), (6), (7), (9). Rl gives the amount of proven reserves of ore, in thousands of short ton. Source^ (1), (2), (4), (5), (6), (7), (9). Gl - G6 give the grade of the ore in each of the metals contained in significant quantities, by order of decreasing importance; grades are expressed in percentage (e.g. 4.1 means 4.1%), with the exception of gold and silver grades which are expressed in fine troy ounces per ton of ore. No practical difficulty was encountered in defining the order of importance of the metals for each mine; this information is given in Ta-ble XIIX. When less than six metals are present, the last grade figure(s) is(are) set to zero. 218 RBI - RE6 give the recovery, after processing, of the metals contained in significant quantities in the ore, by order of importance; when less than 6 metals are present, the last recovery figure(s) is(are) set to zero; recoveries are expressed in percentage (e.g. 90 means 90% of the metal is recovered). Sources or determination^ The data recorded in those fields differ ac-cording to the f i l e : - In FINAL1 and FINAL2, only those mines for which the recovery of the principal metal at least was available were included as observations; when no figure was available for metals other than the principal one, the recovery was set to zero. Sources were (1), (2), (3), (5), (6). - In FINAL3, REi, i = l - 6 , was arbitrarily set to 100% for al l mines whether or not data on the true recovery of some metals were available. - In FINAL4, al l mines which were included as observations in FINAL3 were also included; however, when data on recovery were available (the same as were used in FINAL1 and FINAL2), they were used, while when no figure was available, REi was set to the mean observed for metal of impor tance i in those mines for which data were available. CAP is the capacity of the mine-mill, expressed in short tons of ore processed per day. Sources: (1), (2), (4), (5), (6), (7), (9). S is the current stripping ratio, the ratio of waste to ore in current extraction; it is not believed to be very reliable, as it was oft hard to distinguish between the current stripping ratio and the average stripping ratio; when missing altogether this variable is set to zero. Sources: (1), (2), (3), (5), (6). 219 ORP is the value recovered from one ton of ore, after processing, at the metal prices prevailing during year Tl. Precisely, PSi is the price of metal i in 1970 Can. $ per short ton (or per ounce in the case of gold and silver) at Tl. REi is the recovery of metal i . Thus ORP is expressed in Canadian dollars of 1970 per short ton of ore processed. Sources: Sources for Gi and REi are given above; Sources for PSi depend on the metal and are given in Table XI.LI below; for copper, zinc, lead, and nickel, two alternative price series were used, according to the f i l e : FINAL1 uses sources (12), (12 and 13), (12 and 13), (12) for copper, zinc, lead, and nickel respectively, while other files use (10), (10), (10), and (15) respectively for the same metals; - The l i s t of the metals produced by each mine is given in Table XII below. ORPE is the expected value of one ton of ore, after processing, in 1970 Can. $, as anticipated at Tl, the creation or expansion date. ORPE is a weighted arithmetic average of ore values corresponding to price prevailing from 4 years before Tl to one year after Tl. Precisely: ORP = l PSi-Gi-REi, i=l where: ORPE = (6-ORPPl + 5-ORP + 4 0RPLl + 3 0RPL2 + 2 0RPL3 + l-ORPL4)/21 220 where: ORPP1 is the value recovered from one ton of ore, after proces-sing, at the metal prices prevailing during year Tl + 1 (see ORP for computation and sources). ORPLi is the value recovered from one ton of ore, after proces-sing, at the metal prices prevailing during year Tl - i , i = l - 4 (see ORP for computation and sources). ORPE is expressed in Canadian dollars of 1970 per short ton of ore processed. P is the value of the metal produced, in 1970 Can. $ per short ton of metal. It is derived from ORP and the index of metal content, AGG (see below), as follows: P = 100- ORP j AGG PE is the expected value of one ton of metal, after processing, at Tl, expressed in 1970 Can. $/short ton. It is obtained as a weighted average of metal values at various dates in the same way as ORPE is derived from ORP. PE is expressed in 1970 Can. $/short ton of metal. KP is the value of an index of capital equipment prices at Tl; a different index is used for Canadian mines than for US mines (see Table XIII for sources). The base year for both indices is 1971; the US index is cor-rected for fluctuations in the exchange rate; both indices are furthermore adjusted for inflation using the Canadian Consumer Price index. So they must be interpreted as indices of capital equipment prices in Canadian dollars of 221 1970. A Canadian mine is defined as a mine which is located in Canada; a US mine is located in the United States. Information on the location of mines is given in Table XII below. KPE is the expected price of capital equipment at TI, expressed in Canadian dollars of 1970. It is a weigthed average of capital equip-ment prices over six years: 4 years before TI, TI, and the following year. As in the formula which gives ORPE, the weights increase each year by 1/21, starting at 1/21 in Tl-4 to finish at 6/21 in Tl + 1. W is the value of a wage index at TI; a different index is used for Canadian mines than for US mines (see Table XIII.for sources; see Table XII for mine location). The base year for both indices is 1970; the US index is corrected for fluctuations in the exchange rate; both indices are adjusted for inflation using the Canadian Consumer Price index. So they must be interpreted as indices of hourly earnings in manufacturing in Canadian dollars of 1970. WE is the expected level of hourly earnings at TI in Canadian dollars of 1970. It is obtained as a weighted average in the same fashion as KPE, above. AGG is the index of metal content of the ore; i t is defined as the arithmetic average between Gl, the grade of the ore in the principal metal, and COMG, the combined grade of the ore. COMG is the percentage grade that the ore would rate in its principal metal if it contained only that metal but in such quantities that its value was equal to the total value, at TI, of al l metals actually contained in the ore in significant, quantities. Thus: 222 COMG = I PSi-Gi i = l / PS1-D1 where: PSi is the price of metal i in 1970 Can. $ per short ton (or per ounce in the case of gold and silver) at TI; Gi is the grade of the ore in metal i in percentage (or in ounces per short ton in the case of gold and silver); PSI is the price of metal 1, the principal metal extracted by the mine, in 1970 Can. $ per short ton (or per ounce in the case of gold and silver) at TI; Dl is a dummy variable whose value is 1 when PSI is expressed in $/short ton and whose value is 29 167 when PSI is expressed in $/oz (as 1 short ton — 29 167 troy ounces). Then COMG is expressed in percentage and AGG is defined as: AGG = Gl • D2 + -| (C0MG-G1-D2) where D2 is a dummy variable whose value is 1 when metal 1 is neither gold nor silver and whose value is 3.4285.10 ~* otherwise. Thus AGG is expressed as a percentage. jJource^: Sources for Gi are given above. Sources for PSi depend in the metal and are given in Table XIII below for copper, zinc, lead, and nickel, two alternative price series were used according to the f i l e : FINAL1 uses sources (12), (12 and 223 13), and (12) for copper, zinc, lead, and nickel respectively, while other files use (10), (10), (10), and (15) respectively for the same metals; - The l i s t of the metals produced by each mine is given in Table XII below. MC is the code of the principal metal produced by the mine; RC is the regional code of the mine; J is the observation index (a given mine may generate several observations if expanded). j>ources_: M^ a n c* a r e g i - v e n i - n Table XIV, J is given in Table XII. 2. Description of the data used in the short-run study 21. The sample Time-series data were collected on the operations and character-istics of 12 C a n a d i a n open-pit non-ferrous-metal mines, whose lis t is given in Table XVI. Those mines operated over different periods, between 1961 and 1979; those periods are also given in Table XVI. In selecting a unique time period and a subset of a l l mines in such a way as to maximize the number of observations used (where an observation consists in data about one specific mine at a specific date), there was a trade-off between the number of mines which could be included and the length of the period studied. As Table XVII indicates the amount of information used was maximized when five mines were included and the sample period was 1965-79 (see "Total number of obervations") . 224 2.2 Variable description, units, and sources The variables used were constructed from two basic kinds of data: market data such as metal prices, factor prices, and indices; mine specific data such as ore reserves, grades, recoveries, capacity. The sources of market data are given in Table XIII; the sources of mine-specific data are given in Table XVI. Source codes are given in Table XV As far as notation is concerned, al l variables use the suffix Mi which refers to the mine index given in Table XVI, or simply the suffix i . QAMi is the yearly output of mine Mi, in thousands of short tons per year. When, for reasons other than conjonctural, production does not proceed over the whole year, the published figure is adjusted in such a way as to put i t on a yearly basis. This is done in the case of fire, when a plant becomes operational during a specific year or, which is more questionable, in case of a strike. A similar adjustment is done in case of a change in the financial year. The corresponding adjust-ment coefficients are given in Table XVIIL.whare empty spaces correspond to instances where the adjustment coefficient is one (no adjustment). CAPMi is the capacity of mine Mi in thousands of short tons of ore processed over a 360 day year. RMi gives the proven ore reserves of mine Mi, in thousands of short tons of ore. AGEMi is the age, in years, of the most recent plant at mine Mi; it is set at one when the plant whose capacity is CAPMi starts operating. As a result, for mines which have undergone major capacity expansions, 225 AGEMi is lower than the age of the mine itself. AGGMi is the current index of metal content of the ore at mine Mi. It is constructed exactly as AGG was constructed for the ex-ante study (see 1.2), using contemporary prices, with that sole differ-ence that, whenever the grade of the ore currently extracted is avail-able, i t is used instead of the average grade. AGGMi is expressed as a percentage. NPMi is the relative price of output in terms of wage, in Can., $ per ton of ore processed. NPMi = ORPMi / W where: W an index of hourly earnings in Canadian manufacturing, with base year 1970; ORPMi is the nominal price of output in Can. $ per ton of ore processed. Precisely, ORPMi = 10 4- m I Pij-Gij-REij where: Pij is the price of metal j in Can. $ per ton; Gij is the grade of the ore in metal j , in percentage; REij is the recovery of metal j , in percentage; m is the total number of metals produced at mine Mi. Table XII: Observation l i s t and index; Composition of FINAL1, FINAL2, FINAL3, and FINAL4. Index1 (J) Mine Name Loca-tion1»2 (RC) Date1 (Tl) Metals produced2 Observation included FINAL1 FINAL2 FINAL3 in FINAL4 01 Bell Copper Mine 01 72 1, 7 X X X X 02 Brunswick Mining 09 66 5, 3, 1, 8 X X 03 Equity Silver Mine 01 80 8, 1, 7 X X 04 Highmont Mines 01 80 1, 9 X X 05 Island Copper Mine 01 71 1, 9 X X X X 06 Mattabi Mines 07 72 3, 1, 5, 8, 7 X X X X 07 Phoenix (a) 01 61 1, 7, 8 X X X . X 08 (b) 01 63 1, 7, 8 X X X X 09 (c) 01 69 1, 7, 8 X X X X 10 Ruttan Mine 06 73 1, 3 X X 11 Similkameen (a) 01 72 1,7,8 . X X 12 (b) 01 75 1, 7, 8 X X 13 Sturgeon Lake 07 74 1, 3, 5, 7, 8 X X 14 Valley Copper Mines 01 80 1, 9 X X 15 Granisle Copper (a) 91 66 1 X X X X 16 I I I I (b) 01 72 1 X X X X 17 Gibraltar Mines 01 72 1, 9 X X X X 18 Lornex (a) 01 72 1, 9 X X X X 19 " (b) 01 74 1, 9 X X X X 20 " (c) 01 79 1, 9 X X X 21 Bethlehem Copper (a) 01 63 1 X X X X 22 ti I I (b) 01 67 1 X X X X 23 ti I I (c) 01 72 1 X X X X 24 I I n (d) 01 77 1 X X X X 25 Brenda Mines (a) 01 70 1, 9 X X X X 26 01 75 1, 9 X X X X 27 Pine Point Mines (a) 03 64 5, 3 X X X X 28 I I it I I (b) 03 68 5, 3 X X X X The symbols between brackets refer to variables which are included in each observation. See Table XIV for location and metal codes; the principal metal appears first. Table XII contin.: Observation li s t and index; Composition of FINAL!, FINAL2, FINAL3, and FINAL4. Index1 (J) Mine Name Loca-tion 1, 2 (RC) Date1 (TI) Metals produced2 Observation FINAL1 FINAL2 included FINAL3 in FINAL4 29 Pine Point Mines (c) 03 74 5, 3 X X X X 30 Afton Mines 01 78 1, 7 X X X 31 Craigmont Mines (a) 01 61 1 X X X X 32 " (b) 01 67 1 X X X X 33 Cyprus Anvil (a) 02 70 5, 3, 8 X X X X 34 » (b) 02 73 5, 3, 8 X X X X 35 Endako Mine (a) 02 65 9 X X X X 36 " (b) 02 67 9 X X X X 37 " (c) 02 78 9 X X X X 38 Canada Tungsten (a) 03 62 13, 1 X X X X 39 (b) 03 67 13, 1 X X X X 40 (c) 03 69 13, 1 X X X X 41 (d) 03 79 13, 1 X X X X 42 St-Lawrence Columbium and Metals (a) 08 64 14 X X X X 43 » TI ft (b) 08 71 14 X X X X 44 " » ! » l (c) 08 75 15 X X X X 45 Blizzard 01 83 10 X X X X 46 Brinex 10 81 10 X X X X 47 Cluff Lake 05 80 10 X X X X 48 Denison Mines 01 77 10 X X X • X 49 Key Lake 05 83 10 X X 50 Midwest Lake 05 84 10 X X X X 51 Vekol Copper Mining 20 78 1, 9 X X 52 Cities Service Co. 20 74 1 X X 53 Intermountain Exploration Co. 20 78 7, 8 X X 54 Pinson 20 81 7 X X X X 55 Anaconda (Berkeley pit) 20 69 1 X X 56 Cyprus Pima Mining 20 72 1 X X 57 Chevron Resources Co. 20 79 10 X X 2The symbols between brackets refer to variables which are included in each observation. See Table XIV for location and metal codes; the principal metal appears f i r s t . 228 Table XIII: Sources of metal and factor prices 1 Variable /ariable Symbol Unit Source Code3 Copper price i Monthly average settle- 0P1 US $/m.t. 12 ment price, London Metal Exchange 1000 Copper price; US Products and Producers; up to February 1st, 1970: Net, Domestic Refinery; l/2/70-l/2/73:f.o.b. Domestic Refinery; from December 73: US Producer Cathodes 0P2 US :c/lb 10 Zinc price; Monthly average settlement price, London Metal Exchange 0P3 US $/m.t. 12 13 Zinc price; up to January 71: US prime Estern, f.o.b.; from Jan.71: US prime Western, delivered 0P4 US c/lb 10 Lead price; Monthly average settlement price, London Metal Exchange 0P5 US $/m.t 12 13 Lead price; US price, New York 0P6 US c/lb 10 Gold price; based on value of US pro-duction divided by quantity for years up to 78; based on quotations of Handy and Harman; New York, for 79 OP 7 US $ per fine troy ounce 12 Silver price; Handy and Harman, New York 0P8 US c/fine troy ounce 10 Molybdenum in concentrate; as quoted by major US producers 0P9 US $/lb of contained MO , 15 Uranium price; Values for immediate delivery to domestic [us] market; Uo0Q in concentrate (Yellowcake) 3 o OP10 US $/lb of U3O3 : 14 Nickel price; C a n a d i a n electrolytic 0P11 US $/m.t. 12 cathodes; f.o.b. shipping point, US duty included (contract prices) 1000 Nickel prices; average producer listed price for cathode nickel as reported in the Engineering and Mining Journal 0P12 US c/lb 15 229 Table XIII: contin.: Sources of metal and factor prices Variable Variable Symbol Unit Source Code3 Tungsten price; average quoted price, as derived from London Metal Bulletin 0P13 US $/lb 16,18 Columbium price; average annual price of contained columbium, in concentrate 0P14 US $/lb 17,18 Canadian capital goods price index; open-pit mining and milling OCANKP 1971=100 19, Table 5.4, # 1. US wage index, hourly earnings in manufacturing OUSW 1970=100 20, (code 11165M). Canadian US exchange rate EX 20 (code 156RF) Canadian consumer price index CP I 1970=100 20 (Code 15664) . Canadian wages, hourly earnings OCANW 1970=100 20 (code 15665) US capital equipment price (Marshall and Swift) OUSKP 1971=100 19 (Table 3.1) 1The original price series and indices had to be expressed in the units indicated in the text. 2For operations or projects whose completion date was 1979 or later, the ' prices or index levels for years subsequent to 1979 were arbitrarily set to 1979 levels, if available. The absence of observations 20 and 30 in FINAL1, but not in FINAL2 results from the absence of the London Metal Exchange figure for copper in 1979, at the time of collection (early 1980). 3See Table XV. Table XIV : Location and metal codes Metal Copper Zinc Lead Gold Silver Molybdenum Uranium Nickel Tungsten Columbium Code 01 03 05 07 08 09 10 11 13 14 Region British Columbia Yukon N-W-T. Alberta Saskat. Manitoba Ontario Quebec N-Bruns NFL P.E.I. N.-S. USA Code 01 02 03 04 05 06 07 08 09 10 11 12 20 231 Table XV: Data sources and source codes Source Code Financial Post Corporation Service, Maclean Hunter Ltd, yellow cards, occasional Financial Reports, annual I 2 Canadian Mining Journal, Reference Manual and Buyers I 3 Guide, Toronto, annual Engineering and Mining Journal International Directory  of Mining and Mineral Processing Operations, New York, annual Special Company Publications, occasional | 5 Special issues of the Northern Miner, Toronto, I 6 occasional Canadian Mines Handbook, Northern Miner Press, Toronto, annual Mining International Yearbook, the Financial Times, London, annual Engineering and Mining Journal, March 1980, McGraw H i l l , | 10 New York Non-Ferrous Metal Data, American Bureau of Metal Statistics, | 11 Washington Commodity Trade and Price Trends, Report No. EC-166/79, I 12 World Bank, Washington Lead and Zinc Statistics, Monthly Bulletin of the Inter- I 13 national Lead and Zinc Study Group, April 1980 Metal Statistics, American Bureau of Metal Statistics, I 14 Washington Minerals and Materials, a Monthly Survey, Bureau of Mines, | 15 US Department of the Interior, April 1980, Washington 232 Table XV contin.: Data sources and source codes Code MCP-21, Mineral Commodity Profiles, Tungsten, Bureau of Mines, US Department of the Interior 16 MCP-10, Mineral Commodity Profiles, Columbium, Bureau of Mines, US Department of the Interior 17 Phone calls to US Department of the Interior 18 Capital Cost Escalation in the Non-energy Mineral Industry in Canada, Energy, Mines and Resources Canada, Mineral Division, Ottawa, 1980 19 International Financial Statistics, International Monetary Fund, Washington, annual 20 233 Table XVI: List of mines, operation periods, and data sources (short-run stu Mine Index Operation period Sources of mine spe-cific data Afton Mines Ml 1978-79 (1), (2) Bethlehem Copper* M2 1963-79 (1), (2) Brenda Mines M3 1970-79 (1), (2) . Canada Tungsten* M4 1962-79 (1), (2) Craigmont Mines* M5 1961-79 (1), (2) Cyprus Anvil M6 1969-79 (1), (2) Endako Mine* M7 1965-79 (1), (2) Gibraltar-Mines M8 1972-79 (1), (2) Granisle Copper M9 1966-77 (1), (2) Lornex M10 1972-79 (1), (2) Pine Point Mines* Mil 1965-79 (1), (2) St-Lawrence Columbium & Metals Ml 2 1964-74 (1), (2) Included in the sample for which the amount of information used is maximized (see Table XVII) Table XVII: Alternative Samples and information used Sample period Number of oberva-tions per mine Number of mines1 61-79 19 1 62-79 18 2 63-79 17 3 64-79 16 3 65-79* 15* 5* 66-79 14 5 67-79 13 5 68-79 12 5 69-79 11 6 70-79 10 7 71-79 9 7 72-79 8 9 Total number of observations 19 36 51 48 75* 70 1 63 i 60 66 70 63 72 Sample period Number of observa-tions per mine Number of mines1 73-79 7 9 74-79 6 9 75-79 5 9 66-77 12 6 67-77 11 6 68-77 10 6 69-77 9 1 — 70-77 8 8 71-77 7 8 72-77 6 10 73-77 5 10 Total number of observations 63 54 45 72 66 60 63 64 56 60 50 Sample period Number of observa-tions per mine Number of mines1 64-74 11 4 65-74 10 6 66-74 9 7 67-74 8 7 68-74 7 7 69-74 6 8 70-74 5 9 71-74 4 9 72-74 3 11 73-74 2 11 Total number of observations 44 60 63 56 49 48 45 36 33 22 Sample which uses the maximum quantity of information. *A mine can be included in the sample when its operation period, given in Table XVI contains the sample period. Table XVIII: Adjustment coefficients used to derive QAMi from published output data ~~"~--~^year Mine index -. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 Ml 1.23 M2 4.05 .55 1.05 1.14 M3 1.33 1.13 1.10 M4 11.0 1.50 2.99 1.01 11.77 1.01 1.05 M5 1.09 2.18 1.03 M6 1.22 1.10 1.86 1.07 M7 1.05 1.49 1.43 1.21 M8 1.33 1.04 1.10 1.55 2.52 1.11 M9 8.0 1.33 M10 1.33 Mil 8.0 Ml 2 1.55 1.11 1.09 1Equal to one unless otherwise mentioned. ANNEX 3 VARIABLE LIST 237 Variable l i s t ; Roman symbols1 A(-): AGG: AGE: AGE1: CAP: COMG: DE: DNR: Gi: J : KP, KPE, KPE3: L: NP: Coefficient of proportionality between CAP andt£(0; Index of metal content; Age of the most recent major plant, in a mine (in years); AGE plus one year; Capacity, in tons of ore processed per day (except in chapter 6 where CAPMi is in thousands of tons of ore processed per year); Combined grade, i.e. total metal content, expressed in terms of the principal metal; Dummy variable which takes a value of 1 when it refers to the creation of a mine and a value of 2 when i t refers to a major expansion of an existing mine; Discounted net revenue function, R*(-); Production function, output in terms of ore processed; Grade in the ith metal contained in the ore reserves of a mine; Vector of adjustments to stocks x; Mine index (in ex ante data files); Price, expected price, relative (in terms of output) expected price of capital equipment; Vector of variable factors; Relative price of output in terms of wage; LIn chapter 6, the addition of the suffix "Mi" to any variable symbol refers to mine "Mi " . 238 ORP, ORPE: PSi: V QA: Rl, R2, Rt: R*(0: R, R(-): REi: t, Tl, T2: W, WE, WE3: YEAR: Price, expected price, of a mine's output in terms of ore processed; Price of the i t n metal produced by a mine; Output of a mine in terms of ore processed; Output of a mine, in thousands of tons of ore processed over a year, adjusted for the duration of the actual operative period (in chapter 6) ; Ore reserves at Tl, T2, t; Discounted net revenue function (DNR), a maximized function; Discounted net revenues associated with any feasible program; Recovery of metal i ; Date, date of start-up, date of close-up; Price, expected price, relative (in terms of output) expected price of labour; Observation date minus 1960 (in chapter 6). i 239 Variable l i s t , Greek symbols a : Ratio of the resale prices over the acquisition prices of factor stocks x; 8 : A positive real number; 9,9^ ,92 : Vector of acquisition prices of stocks x; at Tl, at T2; 9(1) : Asset price of stocks x (acquisition i f x ^ 0; resale if x < 0); Y(R) : Grade distribution of reserves; X : Implicit value of the marginal factor-stock unit; X : Implicit value of the average factor-stock unit; u : Implicit value of the marginal resource unit; "y : Implicit value of the average resource unit; J2 : The set of admissible control trajectories; 11(0 : Ex ante profit function; 6 : Parameters. 

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