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UBC Theses and Dissertations

Resource utilization in oligopolistic markets : the case of exhaustible resources Eswaran, Mukesh 1981

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R E S O U R C E U T I L I Z A T I O N I N O L I G O P O L I S T I C M A R K E T S : T H E C A S E O F E X H A U S T I B L E R E S O U R C E S by MUKESH ESWARAN Ph.D. (Physics), Louisiana State University, 1975 M.A. (Economics) , University of B r i t i s h Columbia, 197i A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies Department of Economics We accept t h i s thesis as conforming to the required standard The University of B r i t i s h Columbia July, 1981 <S> MUKESH EsWARANy 1 9 8 1 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f E c o n o m i c s  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 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e 2 7 t h J u l y , , 1981 nF-fi f?/791 i T h e s i s S u p e r v i s o r : P r o f e s s o r T r a c y L e w i s A B S T R A C T T h i s t h e s i s c o n s i d e r s t h e u t i l i z a t i o n o f an ex-h a u s t i b l e r e s o u r c e i n an o l i g o p o l i s t i c market i n whi c h p r o -d u c e r s a r e assumed t o behave n o n c o o p e r a t i v e l y . W i t h i n a g a m e - t h e o r e t i c framework, t h e amount o f r e s o u r c e r e c o v e r e d by t h e i n d u s t r y i s - endogenized by a l l o w i n g p r o d u c e r s t o u n d e r t a k e , p r i o r t o e x t r a c t i o n , i n v e s t m e n t a c t i v i t i e s w h i c h a l t e r t h e v a r i a b l e c o s t o f r e s o u r c e r e c o v e r y . The open-loop Cournot-Nash e q u i l i b r i u m i s c h a r a c t e r i z e d i n c o n s i d e r a b l e d e t a i l , e s p e c i a l l y i n the symmetric case i n w h i c h p r o p e r t y r i g h t s a r e i d e n t i c a l a c r o s s p r o d u c e r s . I n t h i s c a s e , i t i s shown t h a t an i n c r e a s e i n t h e number o f p r o d u c e r s i n the i n d u s t r y (a) i n c r e a s e s t h e u l t i m a t e amount: of r e s o u r c e r e -c o v e r e d by t h e i n d u s t r y (b) i n c r e a s e s t h e i n i t i a l i n v e s t m e n t u n d e r t a k e n on each d e p o s i t (c) l o w e r s the r e s o u r c e p r i c e , a t l e a s t i n i t i a l l y (d) r a i s e s t h e shadow p r i c e o f t h e r e -s o u r c e , i n i t i a l l y (e) d e c r e a s e s t h e p r e s e n t v a l u e o f i n -d u s t r y p r o f i t s , and ( f ) i n c r e a s e s the p r e s e n t v a l u e o f the t o t a l s u r p l u s g e n e r a t e d i n t h e Cournot-Nash e q u i l i b r i u m . When t h e p r o p e r t y r i g h t s a r e asymmetric, i t i s shown t h a t t h e o u t p u t p r o f i l e o f t h e i n d u s t r y i s i n e f f i c i e n t from s o c i e t y ' s p o i n t o f vi e w : the same stream o f r e s o u r c e o u t p u t can be p r o v i d e d , i n g e n e r a l , a t lo w e r i n v e s t m e n t c o s t and p r e s e n t v a l u e v a r i a b l e c o s t . A C K N O W L E D G E M E N T S I would l i k e t o thank T r a c y L e w i s , my p r i n c i p a l s u p e r v i s o r , f o r h i s enormous h e l p and s u p p o r t which made t h i s t h e s i s p o s s i b l e . H i s keen i n s i g h t and w i l l i n g n e s s t o s h a r e h i s u n d e r s t a n d i n g o f economic t h e o r y w i t h o t h e r s has made t h i s p r o j e c t v e r y p l e a s u r a b l e . I would a l s o l i k e t o thank E r w i n D i e w e r t , D a v i d Donaldson, Hugh Neary, P h i l Neher, Tony S c o t t and Russ U h l e r f o r comments and u s e f u l d i s c u s s i o n s . I g r a t e f u l l y acknowledge t h e f i n a n c i a l s u p p o r t from t h e S o c i a l S c i e n c e s and Hu m a n i t i e s R e s e a r c h C o u n c i l o f Canada i n t h e form o f a D o c t o r a l F e l l o w s h i p . I would l i k e t o thank my w i f e , V i j u , f o r h er s u p p o r t and encouragement d u r i n g t h e p a s t s e v e r a l y e a r s . L a s t l y , I would l i k e t o thank Carmen de S i l v a f o r e f f i c i e n t t y p i n g o f t h i s m a n u s c r i p t . i i i C O N T E N T S Page ABSTRACT • 1 ACKNOWLEDGEMENTS 1 1 CONTENTS i i : L LIST OF FIGURES i v Chapter I INTRODUCTION 1 II REVIEW OF THE LITERATURE 6 Footnotes to Chapter II 23 III THE MODEL AND SOME PRELIMINARY RESULTS . . . 24 Footnotes to Chapter III 58 IV THE CASE OF CONTINUOUS EXPLORATION 60 Footnotes to Chapter IV 104 Appendix to Chapter IV 106 V THE INCORPORATION OF INITIAL INVESTMENT . . . 110 Footnotes to Chapter V 139 Appendix to Chapter V 140 VI SOME SCARCITY IMPLICATIONS OF MARKET STRUCTURE 144 Footnotes to Chapter VI 159 VII CONCLUDING REMARKS 160 REFERENCES 163 i v L I S T O F F I G U R E S P a g e 4 1 1 C h a p t e r I T N T R O D U C T I D N The l a s t two decades have seen a p r o l i f e r a t i o n o f papers on t h e economics o f non-renewable r e s o u r c e e x t r a c -t i o n . Much o f t h i s work has been conducted i n t h e t r a d i -t i o n o f H o t e l l i n g (1931), and has d e a l t w i t h t h e c h a r a c t e r i z a t i o n o f t h e o p t i m a l e x t r a c t i o n p r o f i l e s under p e r f e c t c o m p e t i t i o n and monopoly, and w i t h t h e e f f i -c i e n c y i m p l i c a t i o n s o f t h e fo r m e r . More r e c e n t l y , t h e r e has been a d e f i n i t e t r e n d i n the r e s o u r c e l i t e r a t u r e towards e x p l i c i t c o n s i d e r a t i o n o f market s t r u c t u r e s t h a t a r e i n t e r m e d i a t e between t h o s e o f p e r f e c t c o m p e t i t i o n and monopoly — a t r e n d t h a t has a r i s e n p a r t l y because o f t h e f o r m a t i o n o f a p o w e r f u l c a r t e l i n t h e w o r l d p e t r o l e u m market. I n a p i o n e e r i n g a r t i c l e , S a l a n t (1976) examined, under t h e as s u m p t i o n o f Nash b e h a v i o u r , t h e i m p l i c a t i o n s o f a r e s o u r c e market t h e s u p p l y s i d e o f w h i c h i s c h a r a c t e r i z e d by a dominant f i r m f a c i n g a c o m p e t i -t i v e f r i n g e . G i l b e r t (1978) and U l p h and F o l i e (1980) have c o n s i d e r e d the same problem under t h e von S t a c k e l b e r g h y p o t h e s i s , where t h e dominant f i r m t a k e s a c c o u n t o f the 2 s u p p l y r e s p o n s e o f t h e f r i n g e . More r e c e n t l y , Robson (1980) has c o n s i d e r e d a v a r i a n t o f t h i s model where the c a r t e l (OPEC) f a c e s p r o d u c e r s from t h e consuming n a t i o n s (the West.) who seek t o maximize the n a t i o n ' s w e l f a r e . Lewis and Schmalensee (1979) and (1980), and L o u r y (1981) have dropped the asymmetry o f p r i c e - t a k i n g b e h a v i o u r on the p a r t o f some s u p p l i e r s . Here, each f i r m i s presumed t o have some e f f e c t on t h e market p r i c e o f t h e r e s o u r c e and i n t e r a c t i o n between s u p p l i e r s i s m o d e l l e d as a n o n c o o p e r a t i v e game. W h i l e t h e s t u d i e s mentioned above have p r o v i d e d numerous p r e d i c t i o n s t h a t a r e t e s t a b l e , t h e y a l l r e l a t e t o s i t u a t i o n s where e x t r a c t i o n t a k e s p l a c e from a f i x e d r e s e r v e base. I n r e a l i t y , o f c o u r s e , t h e amount o f the r e s o u r c e t h a t f i r m s can r e c o v e r i s not e x o g e n o u s l y g i v e n . To r e n d e r the t e s t a b l e p r e d i c t i o n s o f such t h e o r e t i c a l models more amenable t o e m p i r i c a l v e r i f i c a t i o n o r r e f u t a t i o n , i t i s n e c e s s a r y t o make the r e s e r v e base endogeneous. The t h e o -r e t i c a l l i t e r a t u r e i s n o t i c e a b l y meagre i n t h i s r e s p e c t , and c o n s i s t s e s s e n t i a l l y o f the p a p e r s by P i n d y c k (1978b), S t e w a r t (1979), and U h l e r ( 1 9 7 8 ) . However, t h e s e papers endogenize r e s o u r c e r e c o v e r y o n l y i n t h e s p e c i a l market s t r u c t u r e s o f p e r f e c t c o m p e t i t i o n and monopoly. I n t h i s t h e s i s , I c o n s i d e r t h e t h e o r e t i c a l i m p l i c a -t i o n s o f i m p e r f e c t l y c o m p e t i t i v e markets f o r nonrenewable r e s o u r c e s when t h e r e s e r v e base i s n o t t a k e n as g i v e n ; t h e t o t a l amount o f r e s e r v e s r e c o v e r e d by each f i r m i s endogenous. I n t e r a c t i o n between s u p p l i e r s i s m o d e l l e d as a n o n - c o o p e r a t i v e game and t h e Cournot-Nash s o l u t i o n i s 3 c h a r a c t e r i z e d . Such a framework i s c o n v e n i e n t i n t h a t one can n o t o n l y c o n s i d e r t h e p o l a r extremes o f monopoly and p e r f e c t c o m p e t i t i o n , b u t a l s o t h e e n t i r e continuum o f i n t e r m e d i a t e market s t r u c t u r e s — a t l e a s t i n s o f a r as t h e s e can be c h a r a c t e r i z e d by t h e number o f f i r m s i n t h e i n d u s t r y and t h e d i s t r i b u t i o n o f i n i t i a l endowments a c r o s s t h e s e f i r m s . W i t h i n t h i s framework, I c o n s i d e r , i n p a r t i -c u l a r , t h e e f f e c t s o f changes i n market s t r u c t u r e on t h e p r i c e and e x t r a c t i o n p r o f i l e s o f the r e s o u r c e , the u l t i m a t e amount o f t h e r e s o u r c e r e c o v e r e d by t h e i n d u s t r y and t h e t o t a l i n d u s t r y p r o f i t s . I a l s o c o n s i d e r t h e w e l f a r e i m p l i -c a t i o n s o f changes i n market s t r u c t u r e . I n a t t e m p t i n g t o endogenize t h e r e s e r v e base o f each f i r m i t i s assumed t h a t t h e r i g h t s t o e x p l o r e and de-v e l o p t h e r e s e r v e s have been h i s t o r i c a l l y d e t e r m i n e d . The i n t e r t e m p o r a l s u p p l y game between s e l l e r s i s t h u s p l a y e d out i n a p a r t l y p r e d e t e r m i n e d s e t t i n g . The model I c o n s i d e r e n v i s a g e s t h e s u p p l y s i d e o f t h e non-renewable r e s o u r c e market as b e i n g c o m p r i s e d o f an a r b i t r a r y number o f f i r m s , each w i t h p r o p e r t y r i g h t s t o e x p l o r e f o r , d e v e l o p and e x t r a c t t h e r e s o u r c e from an e x o g o n o u s l y - g i v e n number o f d e p o s i t s . C o n c e p t u a l l y , t h e s u p p l y p r o c e s s can be t h o u g h t o f as c o n s i s t i n g o f t h r e e s t a g e s . I n the f i r s t s t a g e , f i r m s u n d e r t a k e some i n i t i a l i n v e s t m e n t w h i c h c o u l d be i n t h e form o f i n f o r m a t i o n r e -g a r d i n g t h e d i s t r i b u t i o n o f r e s e r v e s w i t h i n each d e p o s i t o r i n t h e form o f c a p i t a l equipment t o f a c i l i t a t e r e s o u r c e 4 r e c o v e r y . The purpose o f i n c u r r i n g t h i s i n i t i a l o r "up-f r o n t " c o s t i s t o l o w e r t h e subsequent v a r i a b l e c o s t o f r e c o v e r y . N e x t , t h e f i r m e x p l o r e s f o r r e s e r v e s w i t h i n each o f i t s d e p o s i t s — a t a m a r g i n a l c o s t t h a t i s assumed t o be d e c r e a s i n g i n the i n i t i a l i n v e s t m e n t u n d e r t a k e n on the d e p o s i t s and i n c r e a s i n g i n t h e s t o c k o f r e s e r v e s a l r e a d y d i s c o v e r e d i n the s e d e p o s i t s . F i n a l l y , the r e s o u r c e i s e x t r a c t e d a t a m a r g i n a l c o s t t h a t i s assumed t o be co n -s t a n t f o r a l l f i r m s , i n d e p e n d e n t o f t h e i r r e s e r v e base. (Of c o u r s e , t h e second and t h i r d s t a g e s can - and w i l l -o c c u r s i m u l t a n e o u s l y , i n g e n e r a l ) . The demand c u r v e f a c i n g the i n d u s t r y i s assumed t o be s t a t i c , and consumer b a h a v i o u r i s t a k e n t o be n o n - s t r a t e g i c . The o p t i m a l s u p p l y d e c i s i o n s o f t h e f i r m s a r e o b t a i n e d under t h e a s s u m p t i o n o f Cournot-Nash b e h a v i o u r : each f i r m d e t e r m i n e s i t s i n v e s t m e n t and e x t r a c t i o n d e c i s i o n s so as t o maximize t h e p r e s e n t v a l u e o f i t s own p r o f i t s , t a k i n g as g i v e n t h e a c t i v i t i e s o f a l l o t h e r f i r m s . A Cournot-Nash e q u i l i b r i u m i s s a i d t o e x i s t when no f i r m c a n p r o f i t a b l y a l t e r i t s i n v e s t m e n t - e x t r a c t i o n d e c i s i o n s , g i v e n t h o s e o f a l l t h e o t h e r s . W i t h i n t h e framework o f t h e model b r i e f l y d e s c r i b e d above, i t t u r n s o u t t h a t a g r e a t d e a l more can be s a i d about t h e n a t u r e o f t h e Cournot-Nash e q u i l i b r i u m when t h e p r o p e r t y r i g h t s a r e i d e n t i c a l a c r o s s a l l f i r m s than when t h e y a r e n o t . I n t h e former (symmetric) c a s e i t i s found t h a t an i n c r e a s e i n t h e number o f f i r m s i n t h e i n d u s t r y 5 (a) i n c r e a s e s t h e u l t i m a t e amount o f r e s o u r c e r e c o v e r e d by the i n d u s t r y (b) i n c r e a s e s t h e i n i t i a l i n v e s t m e n t under-t a k e n on each d e p o s i t (c) l o w e r s t h e r e s o u r c e p r i c e , a t l e a s t i n i t i a l l y (d) r a i s e s t h e shadow p r i c e o f t h e r e s o u r c e , i n i t i a l l y (e) d e c r e a s e s t h e p r e s e n t v a l u e p r o f i t s o f the i n d u s t r y , and ( f ) i n c r e a s e s t h e p r e s e n t v a l u e o f t h e con-sumer p l u s p r o d u c e r s u r p l u s . I t i s demonstrated t h a t when the p r o p e r t y r i g h t s a r e asymmetric, t h e stream o f r e s o u r c e o u t p u t p r o v i d e d by t h e i n d u s t r y i s i n e f f i c i e n t from s o c i e t y ' s p o i n t o f v i e w — i n t h e sense t h a t t h e same o u t p u t p r o f i l e can be p r o v i d e d a t a l o w e r a g g r e g a t e i n v e s t m e n t c o s t and a lower ( p r e s e n t v a l u e ) v a r i a b l e c o s t . The o u t l i n e o f t h i s t h e s i s i s as f o l l o w s . C h apter I I r e v i e w s t h e r e l e v a n t l i t e r a t u r e . C h a p t e r I I I s p e l l s o u t t h e b a s i c model t o be c o n s i d e r e d i n t h i s t h e s i s . A p r e l i m i n a r y i n v e s t i g a t i o n o f t h i s model i s conducted i n w h i c h i t i s assumed t h a t f i r m s do n o t u n d e r t a k e any i n i t i a l i n v e s t m e n t and t h a t a l l e x p l o r a t i o n w i t h i n d e p o s i t s i s completed p r i o r t o b e g i n n i n g e x t r a c t i o n . I n Chapter IV, t h e case when e x p l o r a t i o n w i t h i n d e p o s i t s i s c o n t i n u o u s l y c o n d u c t e d i s c o n s i d e r e d . C h a p t e r V i n c o r p o r a t e s the p o s s i b i l i t y o f f i r m s u n d e r t a k i n g i n i t i a l i n v e s t m e n t p r i o r t o e x p l o i t i n g t h e i r d e p o s i t s . C h a p t e r V I c o n s i d e r s some s c a r c i t y i m p l i -c a t i o n s o f d i f f e r e n t market s t r u c t u r e s . C o n c l u d i n g r e -marks a r e p r e s e n t e d i n Ch a p t e r V I I . 6 Chapter II R E V I E W O F T H E L I T E R A T U R E The l i t e r a t u r e on the economics of nonrenewable resource extraction i s too vast to be meaningfully reviewed here. Such reviews may be found i n Dasgupta and Heal (1979) and, to a lesser extent, i n Petersen and Fisher (1977). The purpose of t h i s chapter i s to summarize that portion of the l i t e r a t u r e which pertains to the r o l e of market structure i n the e x p l o i t a t i o n of exhaustible resources. The f i r s t systematic treatment of the e f f e c t of market structure on the rate at which a nonrenewable resource i s ex-tracted from the ground i s found i n H o t e l l i n g (1931). He established that i n the absence of extraction costs, the re-source p r i c e must r i s e exponentially with time at the rate of i n t e r e s t i f the resource market i s competitive, a r e s u l t which i s now referred to as the r % rule. The extraction pro-f i l e that arises from competitive markets i s also s o c i a l l y optimal i n that i t maximizes the present value of the consumer plus producer surplus. H o t e l l i n g also noted that i f the p a r t i c u l a r resource industry were monopolised, then i t i s the marginal revenue, not p r i c e , that r i s e s exponentially at the rate of i n t e r e s t . As a r e s u l t of the general tendency for production to be retarded under monopoly, Hot e l l i n g claimed that the e x p l o i t a t i o n of the exhaustible resource i s l i k e l y to be "immensely protracted" as compared to the competitive regime - making the monopolist the conservationist's f r i e n d . Weinstein and Zeckhauser (1975) reconsidered the s o c i a l optimality of the competitive equilibrium. They established that as long as the extraction technology i s convex, the competitive output p r o f i l e of the resource i s optimal from society's point of view. This generalizes, within a p a r t i a l equilibrium framework, the s t a t i c e f f i c i e n c y properties of the competitive outcome to the intertemporal case. I t must be noted, however, that i f the extraction technology i s nonconvex, a competitive equilibrium need not e x i s t , and even i f i t does i t need not be s o c i a l l y optimal. S t i g l i t z (1976) has claimed that there i s only l i m i t e d scope for a monopolist to exercise h i s market power. This claim i s based on a comparison of the rates of extraction from a given stock of resource i n the monopolistic and: the ccr.i-p e t i t i v e regimes when the demand function i s of constant e l a s t i c i t y (greater than unity i n absolute value). When the extraction costs are zero, the extraction p r o f i l e s are i d e n t i -c a l f o r these two polar market structures. This i s so because the marginal revenue i s proportional to price f o r a constant e l a s t i c i t y demand curve. Therefore, i f the p r i c e r i s e s ex-ponentially at the rate of i n t e r e s t i t follows that, the marginal revenue does too, and vice-versa. This and the constraint that the same amount of resource be ultimately 8 extracted by both industries implies that the two time p r o f i l e s of extraction are i d e n t i c a l . S t i g l i t z (1976) argues that the above r e s u l t admits of considerable generalization. The e s s e n t i a l reason i s that the monopolist ultimately extracts the same t o t a l amount of the resource as the competitive industry (since the stock of resource available i s assumed to be exogenously given), i n contrast to the case when the commodity i n question i s an ordinary producible good. Thus the monopolist has f l e x i -b i l i t y only to the extent of manipulating the time p r o f i l e of extraction, the cumulative extraction being exogenously given. However, i t should be noted that S t i g l i t z ' s r e s u l t i s s t r i c t l y v a l i d only when the extraction cost i s zero. I f the marginal extraction cost i s p o s i t i v e (but constant), the mono-p o l i s t i s more conservation-minded than i s s o c i a l l y optimal. This i s also true i f the e l a s t i c i t y of demand increases over time - as might be the case i f substitutes for the resource i n question are discovered with the passage of time. If such i s the case, the r a t i o of marginal revenue to price i s r i s i n g over time; the discounted marginal revenue evaluated at the competitive price i s increasing with time and so i t pays the monopolist to postpone production into the future. Lewis, Mathews and Burness (1979) have considered two extensions of the S t i g l i t z (1976) model which r e s u l t i n the monopolistic outcome being less conservationist than i s s o c i a l l y optimal. The f i r s t extension considers the e x i s t -ence of quasi-fixed costs, i . e . , costs that do not depend on the rate of extraction but can be avoided by stopping 9 p r o d u c t i o n a l t o g e t h e r . ( T h e p r e s e n c e o f q u a s i - f i x e d c o s t s i n t r o d u c e s a n o n - c o n v e x i t y , a s a r e s u l t o f w h i c h t h e c o m -p e t i t i v e e q u i l i b r i u m d o e s n o t e x i s t ' ' " . ) W h e n t h e d e m a n d f u n c t i o n i s o f c o n s t a n t e l a s t i c i t y , t h e i n s t a n t a n e o u s r e -t u r n s t o t h e m o n o p o l i s t a t g i v e n o u t p u t r a t e i s s m a l l e r t h a n t h a t a c c r u i n g t o t h e s o c i a l p l a n n e r ( w h o m a x i m i z e s t h e p r e s e n t v a l u e o f t h e t o t a l s u r p l u s ) , a n d t h e m o n o p o l i s t e x -t r a c t s f a s t e r s o a s t o r e d u c e t h e t o t a l a v o i d a b l e c o s t s . T h e o t h e r e x t e n s i o n c o n s i d e r e d b y L e w i s e t a l ( 1 9 7 9 ) i s t h a t o f a d e m a n d f u n c t i o n , t h e e l a s t i c i t y o f w h i c h i s i n c r e a s i n g i n o u t p u t . S u c h a d e m a n d f u n c t i o n w o u l d b e a p p r o p r i a t e f o r r e s o u r c e s f o r w h i c h a d e c r e a s e i n t h e p r i c e r e s u l t s i n m o r e b e i n g d e m a n d e d b y m a r g i n a l u s e r s a t t h e e x t e n s i v e m a r g i n . I n s u c h a c a s e , t h e p r e s e n t v a l u e o f t h e m a r g i n a l r e v e n u e e v a l u a t e d a t t h e s o c i a l p l a n n e r ' s o u t p u t r a t e i s d e c l i n i n g o v e r t i m e , t h u s p r o v i d i n g i n c e n t i v e t o t h e m o n o p o l i s t t o e x t r a c t f a s t e r r e l a t i v e t o t h e s o c i a l p l a n n e r . T h u s t h e m o n o p o l i s t i s l e s s c o n s e r v a t i o n - m i n d e d a s c o m p a r e d t o t h e s o c i a l p l a n n e r i f e i t h e r t h e d e m a n d f u n c t i o n e x h i b i t s e l a s t i c i t y t h a t i n c r e a s e s w i t h o u t p u t o r i f t h e r e e x i s t f i x e d c o s t s t h a t c a n o n l y b e a v o i d e d b y c e a s i n g p r o d u c t i o n . S w e e n e y ( 1 9 7 7 ) h a s c o n s i d e r e d t h e q u e s t i o n o f i n t e r t e m p o r a l b i a s i n t h e e x t r a c t i o n p r o f i l e o f a n e x h a u s t -i b l e r e s o u r c e i n a v a r i e t y o f s i t u a t i o n s . H e d e m o n s t r a t e s t h a t i n t e r t e m p o r a l p r o d u c t i o n b i a s e s a s s o c i a t e d w i t h v a r i o u s i n s t i t u t i o n s c a n b e e x a m i n e d b y u s i n g w h a t h e c a l l s a " m a r k e t i m p e r f e c t i o n f u n c t i o n " w h i c h c h a r a c t e r i s e s t h e i n -s t i t u t i o n . S i m p l e p r o p e r t i e s o f t h i s f u n c t i o n a r e o f t e n 10 s u f f i c i e n t to determine the dir e c t i o n s ,of the biases. He establishes that monopoly might lead either to over - or under-extraction as compared to the competitive outcome depending on the growth rate of the demand and the shape of the demand function. The pioneering paper dealing with nonrenewable re-source models which are intermediate between the polar extremes of monopoly and perfect competition was that of Salant (1976). In thi s paper, Salant considers a market structure the supply side of which i s comprised of a c a r t e l that faces a competitive f r i n g e . This i s probably the most reasonable description of some resource markets such as that of o i l . Under the assumptions that producers engage i n Cournot-Nash (non-cooperative) behaviour, that the demand e l a s t i c i t y i s increasing i n price and that extraction costs are i d e n t i c a l for a l l producers whether or not they are members of the c a r t e l / Salant derives several i n t e r e s t i n g and important implications from his model. F i r s t l y , pro-duction takes place i n two d i s t i n c t phases. In the f i r s t phase, both the c a r t e l and the fringe produce simultaneously and the pr i c e , net of marginal cost, r i s e s at the rate of in t e r e s t . This phase ends when the fringe exhausts i t s reserves and abandons the market to the c a r t e l . In the second phase, production occurs at a rate that ensures that the net marginal revenue to the c a r t e l r i s e s at the int e r e s t rate; net price r i s e s at less than the rate of i n t e r e s t . The second implication of Salant's model i s that the formation of the c a r t e l increases the present value p r o f i t s 11 of the fringe by a greater percentage than that of the c a r t e l producers. This i s because the fringe producers get to s e l l the entire stock of reserves at the same d i s -counted p r i c e , P , while the c a r t e l has to s e l l part of i t s stock (in the second phase) at a discounted price less than P . This r e s u l t i s the d i r e c t analogue of the f a m i l i a r o r e s u l t i n s t a t i c models: fringe producers benefit to a greater extent than c a r t e l producers since the former reap the benefit of the higher p r i c e without having to r e s t r i c t t h e i r output. F i n a l l y , since both the fringe and the c a r t e l producers gain by the formation of the c a r t e l , t h i s gain i s unambiguously at the expense of the consumers. Lewis and Schmalensee (1979) reconsidered Salant's (1976) model under various s p e c i f i c a t i o n s of the demand e l a s t i c i t y , r etaining the assumption of Cournot-Nash be-haviour. When the e l a s t i c i t y of demand i s decreasing i n output t h e i r r e s u l t s are e s s e n t i a l l y those of Salant. How-ever, when the demand e l a s t i c i t y i s increasing i n output, i t i s possible that there could be an i n i t i a l phase when the c a r t e l alone produces, followed by a phase of simultaneous production by c a r t e l and f r i n g e . Various comparative dynamic r e s u l t s are obtained with respect to the d i s t r i b u t i o n of reserves between the c a r t e l and the fr i n g e . In p a r t i -cular, an increase i n the c a r t e l ' s share of the reserves increases the present value p r o f i t s of the enti r e industry. In the case when the demand e l a s t i c i t y i s constant, the price r i s e s at the rate of in t e r e s t (when costs are zero), i r r e s p e c t i v e of the d i s t r i b u t i o n of reserves between the 12 c a r t e l and the f r i n g e . Lewis and Schmalensee also con-sider the p o s s i b i l i t y of the c a r t e l misrepresenting i t s stock of reserves and i n i t i a l l y supporting i t s b l u f f with a sales path that i s optimal for the announced stock of reserves. The purpose of such a strategy would be to hasten production from, and depletion of, the fringe reserves, en-abling the c a r t e l to acquire monopoly power sooner. I t turns out, quite s u r p r i s i n g l y , that the c a r t e l ' s optimal strategy i s to t e l l the truth; the cost of i n i t i a l l y support-ing a f a l s e announcement by an appropriate sales path over-whelms the possible gains from acquiring monopoly power 2 sooner. Recently, Ulph and F o l i e (1980) have analysed the e f f e c t s of c a r t e l formation when the c a r t e l and fringe producers have d i f f e r e n t (but constant) marginal costs of production. They demonstrate that when the c a r t e l enjoys a s i g n i f i c a n t cost advantage over the fringe, the order of production i n Salant's (1976) o r i g i n a l model i s reversed: the fringe produces i n the second phase. In t h i s case, i t i s possible for the formation of the c a r t e l to lower the present value p r o f i t s of the fringe producers. This r e s u l t a r i s e s from the f a c t that the resource i s exhaustible and the t o t a l stock of reserves i s assumed to be exogenously given. The formation of a c a r t e l r e s u l t s i n higher i n i t i a l p r i ces, but i n order for the same amount of the resource to be ultimately extracted, future prices must be lower. The fringe producers, who extract only i n the second phase, thus receive a lower present value price for t h e i r stock than they 13 would have i n the absence of the c a r t e l . Thus when the c a r t e l has a s i g n i f i c a n t cost advantage over the fringe, the c a r t e l benefits at the expense of the consumers and the fringe producers. Hartwick (1980) has considered the nature of the price p r o f i l e that a r i s e s i n an industry comprised of a dominant firm with a large stock of zero-cost reserves facing a competitive fringe that owns reserves whose costs of ex-t r a c t i o n increase with depletion. He finds that the rate of change of price i n t h i s market structure l i e s between those that would r e s u l t i f a l l the reserves were competitively and monopolistically owned, respectively. Pindyck (19 7 8a) has conducted a simulation study to examine the gains to producers from forming c a r t e l s i n the markets for three exhaustible resources: o i l , bauxite and copper. The incentive to c a r t e l i z e i s , of course, lar g e l y determined by the increase i n present value p r o f i t s of the would-be c a r t e l producers and t h i s , i n turn, i s determined by factors such as the market share of the c a r t e l , the existence' of substitutes and the adjustment lags i n demand and supply to changes i n p r i c e . The large market share of OPEC (Organization of Petroleum Exporting Countries) and the long adjustment lags allow the c a r t e l to reap substantial short-term p r o f i t s ; the r e s u l t i n g incentive to c a r t e l i z e i s , therefore, found to be considerable. The optimal price turns out to be high i n i n i t i a l periods and t h i s i s followed by periods of declining and, ultimately, increasing price s . In the case of bauxite, the optimal p r i c i n g strategy for the c a r t e l IBA (International Bauxite Association) i s that of l i m i t - p r i c i n g , the l i m i t price being determined by the back-stop technology. The gains to c a r t e l i z a t i o n are only moderate. For copper, however, the gains to c a r t e l i z a t i o n are found to be minimal. This i s due to the small market share of the CIPEC (International Council of Copper Exporting Countries) and the short adjustment lags of the fringe supply of copper to price changes. The intertemporal analogue of the s t a t i c von Stackel-berg model was considered i n the exhaustible resource context by G i l b e r t (197 8). Here the dominant firm, acting as the Stackelberg leader, sets prices so as to maximize i t s present value p r o f i t s , taking into account the response of the p r i c e -taking fringe producers. The demand function i s assumed to be everywhere of e l a s t i c i t y less than unity for prices below some P, which i s the marginal cost of production from the backstop technology. I f the marginal extraction costs are constant and same for the c a r t e l and the fringe, the Stackel-berg and Nash e q u i l i b r i a are i d e n t i c a l . In general, however, the two e q u i l i b r i a are not i d e n t i c a l , although the Stackelberg solution, too, exhibits two phases i f the c a r t e l reserves are large r e l a t i v e to those of the fringe. G i l b e r t demonstrates that i f the marginal cost of the fringe i s constant, the optimal price strategy of the c a r t e l i s independent of i t s own production cost and reserves, and i s e n t i r e l y determined by the c h a r a c t e r i s t i c s of the fri n g e . The market price r i s e s monotonically u n t i l the fringe exhausts i t s reserves (when the p r i c e reaches P) and the c a r t e l sets the price at P 1 5 s u b s e q u e n t l y . I f , o n t h e o t h e r h a n d , t h e f r i n g e ' s m a r g i n a l c o s t i s c o n s t a n t up t o s o m e f i n i t e c a p a c i t y t h a t i s s m a l l c o m p a r e d t o t h e r e m a i n i n g r e s e r v e s , t h e c a r t e l m i g h t f i n d i t o p t i m a l t o b e h a v e a s a l i m i t - p r i c i n g f i r m a n d s e t t h e p r i c e a t P . I f t h e f r i n g e c a n i n c r e a s e i t s c a p a c i t y o v e r t i m e , i t i s p o s s i b l e t o o b t a i n a n e q u i l i b r i u m p r i c e p r o f i l e t h a t i s d e c l i n i n g o v e r i n i t i a l p e r i o d s . U l t i m a t e l y , h o w e v e r , t h e c a p a c i t y w o u l d c e a s e t o b e a b i n d i n g c o n s t r a i n t a n d t h e p r i c e w i l l r i s e m o n o t o n i c a l l y u n t i l t h e f r i n g e r e s e r v e s a r e e x -h a u s t e d . R o b s o n ( 1 9 8 0 ) h a s r e c e n t l y c o n s i d e r e d a v a r i a n t o f t h e S a l a n t ( 1 9 7 6 ) m o d e l , i n w h i c h O P E C f a c e s p r o d u c e r s f r o m t h e W e s t w h o m a x i m i z e , n o t t h e i r p r e s e n t v a l u e p r o f i t s , b u t t h e p r e s e n t v a l u e o f t h e t o t a l s u r p l u s o f t h e W e s t . A l l c o n -s u m p t i o n o f t h e r e s o u r c e ( o i l ) i s a s s u m e d t o t a k e p l a c e i n t h e W e s t a n d t h e m a r g i n a l e x t r a c t i o n c o s t s a r e a s s u m e d t o b e c o n s t a n t a n d i d e n t i c a l f o r a l l p r o d u c e r s . I n a t w o - p e r i o d f r a m e w o r k , R o b s o n d e m o n s t r a t e s t h a t i r r e s p e c t i v e o f w h e t h e r t h e W e s t p r o d u c e s i n o n l y o n e o r i n b o t h p e r i o d s , t h e N a s h n o n c o o p e r a t i v e o u t c o m e i s t h e s a m e a s t h a t w h i c h w o u l d b e o b t a i n e d i f t h e p r o d u c e r s o f t h e W e s t w e r e c o m p e t i t i v e . F u r t h e r , t h e S t a c k e l b e r g s o l u t i o n w i t h e i t h e r p l a y e r a s t h e l e a d e r c o i n c i d e s w i t h t h e N a s h o u t c o m e . ( T h e m o d e l i s t h u s " r o b u s t " w i t h r e s p e c t t o t h e s o l u t i o n c o n c e p t u s e d - a s i s t h e S a l a n t ( 1 9 7 6 ) m o d e l u n d e r s i m i l a r c o s t a s s u m p t i o n s . ) T h e r e f o r e , t h e r e i s n o p a r t i c u l a r a d v a n t a g e t o e i t h e r p r o -d u c e r p r e - c o m m i t t i n g h i s s t r a t e g y . I n t h e e v e n t t h a t t h e s o l u t i o n i s i n t e r i o r , i . e . , b o t h p l a y e r s p r o d u c e n o n z e r o 16 amounts i n both periods, the equilibrium i s e f f i c i e n t i n that price r i s e s at the rate of i n t e r e s t , as i t would i n the competitive outcome. However, t h i s l a s t feature of the solution i s an a r t i f a c t of the fixed time-horizon nature of the problem. Were the time horizon free, such an outcome i s u n l i k e l y since part of OPEC's strategy would be to i n i t i a l l y hold back production so as to acquire monopoly power af t e r the producers of the West have exhausted t h e i r reserves. I t i s the fac t that the time horizon i s exogenously fixed i n Robson's model that l a r g e l y accounts for the differences between his r e s u l t s and those of Salant (1976) and Lewis and Schmalensee (1979). In a recent paper, Lewis and Schmalensee (1980) have considered the intertemporal extension of the s t a t i c Cournot-Nash oligopoly equilibrium. Many of the properties of the s t a t i c Cournot model f i n d natural analogues i n t h e i r dynamic model. When marginal extraction costs are zero, the price i s shown to r i s e slower than at the rate of in t e r e s t , while the marginal revenue r i s e s faster than at the rate of i n t e r e s t . So i n t h i s sense, o l i g o p o l i s t i c markets exhib i t properties that are intermediate between those of monopoly and perfect competition. I f a given stock of resource i s evenly d i s t r i b u t e d across a larger number of firms, i t i s shown that the resource i s exhausted sooner, present value p r o f i t s of the industry decline and society's welfare increases. If the d i s t r i b u t i o n of reserves i s un-equal, i t i s shown that equalization of reserve holdings speeds up resource use. When the marginal extraction costs 17 a r e c o n s t a n t b u t unequal a c r o s s f i r m s , t h e c o s t o f p r o -d u c i n g t h e e q u i l i b r i u m t o t a l o u t p u t i s n o t m i n i m i z e d because f i r m s w i t h d i f f e r e n t c o s t s a r e s i m u l t a n e o u s l y p r o d u c i n g o v e r f i n i t e i n t e r v a l s o f t i m e . L o u r y (1981), i n d e p e n d e n t l y o f Le w i s and Schmalensee (1980), has c o n s i d e r e d t h e p o s i t i v e and n o r m a t i v e p r o p e r t i e s o f t h e Cournot-Nash e q u i l i b r i u m o f an N-person o l i g o p o l y i n t h e e x h a u s t i b l e r e s o u r c e c o n t e x t . He demonstrates t h a t i n t h e Cournot-Nash e q u i l i b r i u m , a w e i g h t e d average o f t h e p r e s e n t v a l u e p r o f i t s o f t h e i n d u s t r y and t h e d i s c o u n t e d s u r p l u s i s i m p l i c i t l y maximized. U s i n g t h i s r e s u l t , he d e v e l o p s an i n g e n i o u s a l g o r i t h m f o r l o c a t i n g Cournot-Nash e q u i l i b r i a . L o u r y a l s o g e n e r a l i z e s S t i g l i t z ' s (1976) r e -s u l t by e s t a b l i s h i n g t h a t when e x t r a c t i o n c o s t s a r e z e r o , t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e p r i c e p r o f i l e t o be indep e n d e n t o f the number o f f i r m s i n t h e i n d u s t r y i s t h a t t h e demand f u n c t i o n be o f c o n s t a n t e l a s t i c i t y ; i n such a s i t u a t i o n t h e Cournot-Nash e q u i l i b r i u m c o i n c i d e s w i t h the c o m p e t i t i v e outcome. I n g e n e r a l , o f c o u r s e , t h e r e i s a dead-w e i g h t l o s s a r i s i n g from i m p e r f e c t c o m p e t i t i o n . L o u r y c l a i m s t h a t f o r a v a r i e t y o f r e a l i s t i c v a l u e s o f market p a r a m e t e r s , t h e dead-weight l o s s i s q u i t e s m a l l . T h i s , o f c o u r s e , i s o n l y v a l i d under t h e a s s u m p t i o n t h a t t h e s t o c k o f r e s e r v e s i s e x o g e n o u s l y g i v e n . S i n c e t h e i n c e n t i v e t o a c q u i r e a d d i t i o n a l r e s e r v e s depends on t h e market s t r u c t u r e , i t i s u n l i k e l y t h a t t h i s c l a i m w i l l be v i n d i c a t e d ..when t h e s t o c k o f t h e r e s o u r c e i s made endogenous. 18 Salant (19 80) has developed a computerized model of the i n t e r n a t i o n a l energy market, based on a Cournot-Nash approach. Since real-world complexities can be handled without too much d i f f i c u l t y i n simulation models, he i s able to incorporate such diverse features as capacity constraints, depletion e f f e c t s , the existence of imperfect substitutes, etc. While there i s no t h e o r e t i c a l bench-mark with which the r e s u l t s generated by t h i s computerized model can be compared, i t i s l i k e l y to prove very useful i n the future for p o l i c y purposes. Lewis and Schmalensee (1980), Loury (1981) and Salant (19 80) have a l l assumed that the output strategies of the o l i g o p o l i s t i c producers are "open-loop", that i s , that they depend e x p l i c i t l y only on time and not on the state of the world (as characterized, for example, by the vector of resource stocks owned by the producers) at that time. Such strategies are believable only when producers can precommit t h e i r extraction p r o f i l e s i n advance - as for example when there exists a complete set of futures:.i..a:.;;cts. When such precommitment i s not possible, however, strategies w i l l t y p i c a l l y depend on the state variables also. In other words, strategies i n the absence of precommitment w i l l be "closed loop", and are considerably more d i f f i c u l t to characterize than open-loop strategies. Reinganum (1981) has recently performed the only known closed-loop analysis i n the context of non-renewable resources. She considers a N-person oligopoly, i n which the producers draw the resource 19 a t z e r o c o s t from a common p o o l and f a c e a demand f u n c t i o n o f c o n s t a n t e l a s t i c i t y ( g r e a t e r t h a n u n i t y ) . She f i r s t e s t a b l i s h e s t h a t t h e a g g r e g a t e e x t r a c t i o n p r o f i l e i n a Nash e q u i l i b r i u m i n open-loop s t r a t e g i e s c o i n c i d e s w i t h t h e s o c i a l l y o p t i m a l e x t r a c t i o n p r o f i l e . I n o t h e r words, t h e S t i g l i t z (1976) and L o u r y (1981) r e s u l t i s v a l i d even f o r th e case when t h e r e s o u r c e i s common p r o p e r t y . However, op e n - l o o p s t r a t e g i e s i n t h e c o n t e x t o f a common p r o p e r t y r e s o u r c e a r e n o t v e r y m e a n i n g f u l . Reinganum then demonstrates t h a t t h e Nash c l o s e d - l o o p s o l u t i o n t o t h e problem has f e a t u r e s s h a r p l y d i s t i n c t from t h o s e o f t h e Nash open-loop e q u i l i b r i u m . F i r s t l y , t h e ag g r e g a t e o u t p u t i n t h e c l o s e d -l o o p e q u i l i b r i u m i s n o t s o c i a l l y o p t i m a l and depends ex-p l i c i t l y on the number o f p r o d u c e r s d r a w i n g from t h e common r e s o u r c e p o o l . S e c o n d l y , t h e aggre g a t e o u t p u t r a t e i n t h i s e q u i l i b r i u m i s an i n c r e a s i n g , and not a d e c r e a s i n g , f u n c t i o n o f t i m e . F i n a l l y , t h e a g g r e g a t e e x t r a c t i o n p r o f i l e i s more c o n s e r v a t i o n i s t t h a n t h e s o c i a l l y o p t i m a l one. T h i s l a s t r e s u l t i s p a r t i c u l a r l y s u r p r i s i n g i n view o f t h e f a c t t h e the r e s o u r c e i s assumed t o be common p r o p e r t y . A l t h o u g h some of t h e s e f e a t u r e s a r e a r t i f a c t s o f t h e p a r t i c u l a r demand 3 f u n c t i o n Reinganum has chosen , t h e r e s u l t s s u g g e s t t h a t c l o s e d l o o p e q u i l i b r i a a r e s u f f i c i e n t l y d i f f e r e n t from open-l o o p e q u i l i b r i a as t o w a r r a n t c a r e f u l c o n s i d e r a t i o n i n t h e f u t u r e . A l l t h e papers r e v i e w e d above have assumed t h a t t h e s t o c k o f t h e r e s o u r c e i s exo g e n o u s l y g i v e n and have examined 20 how t h i s given stock i s extracted under various market structures. In such models, the intertemporal i n e f f i c i e n c y r e s u l t i n g from imperfect market structures i s l i k e l y to be underestimated since firms have only l i m i t e d a b i l i t y to exercise t h e i r monopoly power. To my knowledge, there e x i s t only three papers i n the l i t e r a t u r e which have attempted to endogenize the reserve base i n the context of a l t e r n a t i v e market structures. These are now reviewed. Stewart (19 79) develops a dynamic programming model i n which firms simultaneously undertake exploration and extraction. He claims that the burden of monopoly f a l l s mainly on exploration. Firms with market power take ex-p l i c i t account of the f a c t that the a c q u i s i t i o n of additional reserves would lower future prices and therefore undertake less exploration than would a competitive industry. Un-fortunately, Stewart's analysis i s c r u c i a l l y flawed by the fact that he allows the returns to exploration to depend only on the current e f f o r t rate; no depletion e f f e c t exists i n his model. As a r e s u l t , Stewart's claim i s no more than a r e i t e r a t i o n of the s t a t i c r e s u l t that a monopolist under-produces r e l a t i v e to a p e r f e c t l y competitive industry. Uhler (1978) attempts to provide a u n i f i e d theory of petroleum production. He f i r s t develops a theory of o p t i -mal resource exploration, taking into account the fact that the a c q u i s i t i o n of geological information during the early stages of exploration aids reservoir discovery sub-sequently, while discoveries also serve to exhaust the stock of undiscovered reserves. He then develops a model 21 t o examine t h e o p t i m a l e x t r a c t i o n from known r e s e r v e s . However, i n t h e e x p l o r a t i o n s t a g e , t h e w e l l - h e a d p r i c e o f t h e p e t r o l e u m i s assumed t o be e x o g e n o u s l y g i v e n . As P i n d y c k (1978b) p o i n t s o u t , t h i s p r i c e must, f i r s t l y , be endogenous and, s e c o n d l y , be t ime dependent. The o p t i m a l r a t e s o f e x p l o r a t i o n and e x t r a c t i o n must be i n t e r r e l a t e d . P i n d y c k (1978b) goes a s t e p beyond U h l e r (1978) by m o d e l l i n g e x t r a c t i o n and e x p l o r a t i o n as s i m u l t a n e o u s a c t i -v i t i e s . He t r e a t s r e s e r v e s as t h e b a s i s o f p r o d u c t i o n , and e x p l o r a t i o n as an a c t i v i t y t h a t i n c r e a s e s or m a i n t a i n s t h e r e s e r v e base. He i n c o r p o r a t e s a d e p l e t i o n e f f e c t whereby f o r a g i v e n r a t e o f e x p l o r a t o r y e f f o r t , r e s e r v e d i s c o v e r i e s a r e a d e c l i n i n g f u n c t i o n o f t h e c u m u l a t i v e amount a l r e a d y d i s c o v e r e d . The o p t i m a l amount of r e s e r v e s t o h o l d a t any t i m e p a r t l y depends on t h e n a t u r e o f t h e p r o d u c t i o n c o s t s . P i n d y c k assumes t h a t t h e m a r g i n a l e x t r a c t i o n c o s t i s a d e c r e a s i n g f u n c t i o n o f t h e r e s e r v e b a s e . G i v e n t h e s e con-s t r a i n t s , f i r m s must s i m u l t a n e o u s l y d e t e r m i n e t h e i r r a t e s o f e x t r a c t i o n and e x p l o r a t i o n . Such a v i e w o f t h e r e s o u r c e i n d u s t r y f a c i l i t a t e s a d e s c r i p t i o n o f t h e e n t i r e h i s t o r y o f t h e r e s o u r c e use. W i t h some a i d from s i m u l a t i o n , P i n d y c k examines t h e e x t r a c t i o n and e x p l o r a t i o n p r o f i l e s i n the two p o l a r market s t r u c t u r e s o f monopoly and p e r f e c t c o m p e t i t i o n . I n e i t h e r c a s e , i t t u r n s o u t t h a t i f the i n i t i a l endowment i s s m a l l t h e p r i c e p r o f i l e i s U-shaped. A t f i r s t , p r o d u c t i o n i n c r e a s e s as new r e s e r v e s a r e r a p i d l y a c q u i r e d t h rough v i g o r o u s e x p l o r a t o r y a c t i v i t y . L a t e r , b o t h e x p l o r a t i o n 22 e f f o r t and the discovery rate decline and ultimately pro-duction too declines, which r e s u l t s i n pr i c e r i s i n g as i n the H o t e l l i n g model. However, the introduction of exploratory a c t i v i t y has the e f f e c t of reducing the rate of pr i c e i n -crease. On comparing the exploratory a c t i v i t i e s under mono-poly and perfect competition, Pindyck finds that the monopo-l i s t would i n i t i a l l y undertake less exploratory a c t i v i t y than a p e r f e c t l y competitive industry, but l a t e r undertake more 23 Footnotes to Chapter II 1. On t h i s point, see Eswaran, Lewis and Heaps (1981). Proposition 4.6 of Chapter IV below contains a s i m i l a r r e s u l t i n an o l i g o p o l i s t i c framework. 2. This r e s u l t , however, has been established only for a p a r t i c u l a r family of demand functions. 3. In p a r t i c u l a r , the claim that the aggregate extraction rate i n the closed-loop equilibrium i s over-conservationist even when the resource i s common property does not appear to be generally v a l i d . 24 Cha p t e r I I I T H E M O D E L A N D S O M E P R E L I M I N A R Y R E S U L T S I n t h i s c h a p t e r , t h e b a s i c model t o be c o n s i d e r e d i n t h i s t h e s i s and t h e assumptions i n v o k e d a r e s p e l l e d o u t . A p r e l i m i n a r y a n a l y s i s o f t h e i m p l i c a t i o n s o f t h i s model i s con d u c t e d and many t e s t a b l e p r e d i c t i o n s d e r i v e d . I n t h e n e x t two c h a p t e r s t h e model i s e m b e l l i s h e d so as t o make i t more r e a l i s t i c and a more comprehensive a n a l y s i s i s conducted. The model t o be d e s c r i b e d a t t e m p t s t o c a p t u r e the s a l i e n t f e a t u r e s o f some r e a l - w o r l d e x h a u s t i b l e r e s o u r c e m a r k e t s . I n p a r t i c u l a r , t h e f o c u s i s on t h e s u p p l y s i d e o f m arkets i n w h i c h p r o d u c e r - p r o d u c e r i n t e r a c t i o n s a r e i m p o r t a n t . The r e s o u r c e market e n v i s a g e d i s one where t h e r e s o u r c e i s e n t i r e l y c o n c e n t r a t e d i n t h e hands o f a few p r o d u c e r s - few enough t h a t t h e p o s s i b i l i t y o f s t r a t e g i c b e h a v i o u r on t h e s u p p l y s i d e cannot be n e g l e c t e d . Con-sumers, on t h e o t h e r hand, a r e assumed t o be p a s s i v e . The s u p p l y s i d e o f t h e market w i l l be d e s c r i b e d f i r s t . I t w i l l be assumed t h a t t h e r e a re M ( i d e n t i c a l ) r e -s o u r c e d e p o s i t s o r p o o l s i n e x i s t e n c e , the l o c a t i o n s o f whi c h 25 a r e known. These M d e p o s i t s a r e assumed t o be d i s t r i b u t e d a c r o s s N indepe n d e n t p r o d u c e r s . I f i s t h e number o f d e p o s i t s o v e r w h i c h p r o d u c e r i has p r o p e r t y r i g h t s (assumed N t o be e x c l u s i v e ) , t h e n = M. I t w i l l n o t be i n v e s t i -g a t e d as t o how t h e d i s t r i b u t i o n o f p r o p e r t y r i g h t s a r o s e ; i t w i l l s i m p l y be assumed t h a t t h e s e r i g h t s a r e ex o g e n o u s l y g i v e n . T h i s r u l e s o u t s i t u a t i o n s i n w h i c h t h e r e s o u r c e i s common p r o p e r t y , i . e . , s i t u a t i o n s where t h e p r o p e r t y r i g h t s a r e e i t h e r n o t e x c l u s i v e o r a r e ambiguous. I t w i l l a l s o be assumed t h a t t h e number, N, o f p r o d u c e r s i n t h e i n d u s t r y i s exog e n o u s l y g i v e n . T h i s l a t t e r a s s u m p t i o n r u l e s o u t s i t u a -t i o n s where f i r m s c o n t e m p l a t e merging o r t a k i n g o v e r o t h e r f i r m s . W h i l e i t would be i n t e r e s t i n g t o endogenize b o t h t h e number o f p r o d u c e r s i n t h e i n d u s t r y and t h e d i s t r i b u t i o n o f p r o p e r t y r i g h t s a c r o s s them, t h e s e r a m i f i c a t i o n s a r e o u t s i d e t h e scope o f t h i s t h e s i s . ^ " P r i o r t o e x t r a c t i o n from any d e p o s i t , e x p l o r a t i o n f o r r e s e r v e s w i t h i n t h e d e p o s i t must be u n d e r t a k e n . I t w i l l be assumed t h a t t h e c o s t o f u n c o v e r i n g a s t o c k , I , o f the r e s o u r c e from any d e p o s i t i s g i v e n by t h e d e t e r m i n i s t i c f u n c t i o n d ( I ) , assumed t o be smooth and such t h a t (Al) d(0) = 0, d'(•) > 0, d"(•) > 0. The s t r i c t c o n v e x i t y o f d ( 0 r e f l e c t s t h e i n c r e a s i n g s a c r i f i c e s t h a t must be made t o uncover a d d i t i o n a l amounts o f t h e r e s o u r c e w i t h i n each d e p o s i t . I n o t h e r words, i t c a p t u r e s t h e d e p l e t i o n e f f e c t o p e r a t i n g w i t h i n each d e p o s i t , 9 F i n a l l y , i t w i l l be assumed t h a t e x t r a c t i o n o f t h e r e s o u r c e p r o c e e d s a t c o n s t a n t m a r g i n a l c o s t , i n d e p e n d e n t o f t h e r e -s e r v e base. L e t C denote t h e m a r g i n a l e x t r a c t i o n c o s t i n c u r r e d by each f i r m . The l e a s t ( u n d i s c o u n t e d ) c o s t a t w h i c h f i r m i can uncover an amount o f r e s o u r c e e q u a l t o 1^ i s c l e a r l y t h a t i n c u r r e d when an amount I^/M^ i s u n c o v e r e d from each d e p o s i t . T h i s f o l l o w s from t h e s t r i c t c o n v e x i t y o f t h e f u n c t i o n d ( • ) ; t h e c o s t o f an a l l o c a t i o n t h a t u n c o v e r s u n e q u a l amounts from t h e IYL d e p o s i t s can be l o w e r e d by e q u a l i z i n g t h e s e amounts. T h i s m i n i m a l c o s t i s , t h e r e f o r e , g i v e n by t h e e x p r e s s i o n (3.1) D ( I . , M ±) = d(I./M.) M i D i f f e r e n t i a t i o n o f 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 t o M^, y i e l d s 9 0 ( 1 . , M.) = d(I./M.) - d ' ( I . / M . ) ( I . / M . ) . 8M. 1 Upon u s i n g t h e s t r i c t c o n v e x i t y o f d ( - ) , i t f o l l o w s t h a t 9 D ( I , , M.) < 0. 3M. l As one would e x p e c t , a g i v e n amount o f r e s o u r c e can be un-c o v e r e d from a l a r g e r number o f d e p o s i t s a t l o w e r c o s t - a m a n i f e s t a t i o n o f t h e d e p l e t i o n e f f e c t . The demand s i d e o f t h e r e s o u r c e market i s now c o n s i d e r e d . The demand c u r v e f a c i n g t h e i n d u s t r y i s assumed t o be s t a t i c , and consumers a r e assumed t o behave- non-s t r a t e g i c a l l y . The r e s o u r c e demand f u n c t i o n , P ( Q ) , where 27 Q i s t h e q u a n t i t y o f t h e r e s o u r c e demanded i n any p e r i o d , i s assumed t o be smooth and t o s a t i s f y t h e f o l l o w i n g con--. . 3 d x t i o n s : (A2) P(0) = F where F i s a f i n i t e c o n s t a n t s a t i s f y i n g F > C + d' ( 0 ) . (A3) The demand e l a s t i c i t y e = -P/(P'Q) i s s t r i c t l y de-c r e a s i n g i n Q. (A4) The demand, Q, f o r t h e r e s o u r c e a t z e r o p r i c e i s f i n i t e . (A2) i m p l i e s t h a t t h e demand f u n c t i o n has a choke p r i c e t h a t i s f i n i t e b u t y e t l a r g e enough t h a t a l l f i r m s i n t h e i n d u s t r y f i n d p r o d u c t i o n v i a b l e . Assumption (A3) i s s u f f i c i e n t t o ensure t h a t t h e revenue f u n c t i o n o f t h e i n -d u s t r y i s concave i n i t s o u t p u t . T h i s a s s u m p t i o n i s n o t u n r e a l i s t i c i n s i t u a t i o n s where t h e choke p r i c e F i s d e t e r -mined by a b a c k s t o p t e c h n o l o g y . As t h e p r i c e approaches F from below, t h e e l a s t i c i t y o f demand f o r t h i s r e s o u r c e i n -c r e a s e s as s u b s t i t u t e s a r e c a l l e d i n t o p r o d u c t i o n . A t low e r p r i c e s , however, no s u b s t i t u t e s e x i s t and t h e demand f o r t h i s r e s o u r c e becomes more and more i n e l a s t i c . Assump-t i o n (A4) e n s u r e s t h a t t h e s e t o f f e a s i b l e o u t p u t s i s com-p a c t . I t i s a t e c h n i c a l a s s u m p t i o n r e q u i r e d t o ensure t h e e x i s t e n c e o f a Cournot-Nash e q u i l i b r i u m i n t h e problem con-s i d e r e d h e r e . T h i s a s s u m p t i o n e l i m i n a t e s t e c h n i c a l l y 28 troublesome behaviour that might otherwise a r i s e from the demand becoming i n f i n i t e at low enough prices . I t i s convenient i n invoke the additional assumption: (A5) There e x i s t s a f i n i t e stock l e v e l , I, such that d"(!) > F. (A5) posits the existence of a f i n i t e upper bound, I, on the cumulative extraction from each deposit. In other words, i t ensures that the t o t a l amount of the resource that can be economically recovered i s f i n i t e . In t h i s chapter, i t w i l l be assumed that firms complete a l l t h e i r exploration p r i o r to beginning extraction. In general, such a strategy i s suboptimal from each firm's 4 point of view. However, i n a preliminary approach to the problem, the imposition of t h i s simplifying constraint affords considerable i n s i g h t . In subsequent chapters t h i s assumption w i l l be dropped. The optimization problem that each firm faces can now be described. Interaction between producers i s modelled here as a non-cooperative game. In p a r t i c u l a r , i t i s assumed that producers choose Cournot-Nash strategies, i . e . , each producer takes as given the actions of a l l other pro-ducers and chooses his exploration and extraction decisions so as to maximize the present value of h i s own p r o f i t s . Thus the optimization problem facing firm i may be written: (3.2) max J. subject to e~ r t[P(Q)-C] q ± d t " D d ^ K L ) 0 i q i ( t ) dt < 1 ± , •T. 1 0 N where Q i s the industry output , i . e . , Q =^ 1^  3^ a n d the parameter r i s the rate of discount, assumed same for a l l firms. (It i s assumed that r i s also the s o c i a l rate of discount). In the above optimization, firm i takes as given 1^, {q^} for a l l k ^ i . In the Cournot-Nash equilibrium, the amount of exploration and the extraction p r o f i l e that firm i a t t r i b u t e s to firm k, k ^ i , are also what firm k finds optimal. In equilibrium, no firm can p r o f i t a b l y a l t e r i t s own decisions, while taking as given those of the others. Solving for t h i s equilibrium comprises of determining, for each firm, the optimal amount of exploration to undertake, the time p r o f i l e of the extraction that follows and the time horizon over which to produce. The concept of Cournot-Nash equilibrium being used here i s the intertemporal analogue of the corresponding s t a t i c concept. 5 However, i n the intertemporal case the assumption of "taking as given" the decisions of a l l the other agents i s even less believable than i n the s t a t i c case To render t h i s behavioural assumption credible, i t i s necessary to assume the existence of a complete set of futures markets, where consumers and producers get .together and consummate contracts for d e l i v e r i e s of the resource at 30 a l l points i n the future. Thus a l l decisions are made at time t=0 even though production takes place over time. The assumption of a complete set of futures markets allows the (dynamic) game to be resolved at time t=0; as time passes by the producers just extract as much as i s required to honour t h e i r contracts. The assumption that the producers behave non-cooperatively requires some j u s t i f i c a t i o n , since they can always increase t h e i r present value p r o f i t s by colluding and r e s t r i c t i n g t h e i r output. This assumption i s c l e a r l y reasonable when the transaction costs of colluding are high (which would c e r t a i n l y be true i f the number of producers i s s u f f i c i e n t l y large) or when there are laws against c o l l u s i o n . However, even i n the event that c o l l u s i o n i s fea s i b l e and p r o f i t a b l e , i t i s of t h e o r e t i c a l and p r a c t i c a l importance to determine the non-cooperative outcome. For, c o l l u s i o n would determine only the p r o f i t s of the industry; there i s s t i l l the question of how these p r o f i t s are to be di s t r i b u t e d across the various producers. And t h i s l a t t e r question i s one of bargaining. I t i s most l i k e l y that the bargaining powers of the producers are determined by their payoffs i n the non-cooperative outcome. For example, the Nash cooperative solution depends e x p l i c i t l y on the so-c a l l e d threat-point, defined by the pay-offs to the pro-ducers when negotiations break down, and these payoffs would t y p i c a l l y be given by the Nash non-cooperative out-come . 31 F i n a l l y , m e n t i o n must be made o f t h e q u e s t i o n o f s t a b i l i t y o f t h e Cournot-Nash e q u i l i b r i u m . I f t h e r e i s a d e v i a t i o n from e q u i l i b r i u m , i s t h e r e a tendency t o r e t u r n t o t h e e q u i l i b r i u m l e v e l s o f o u t p u t ? I n a model w i t h a degree o f c o m p l e x i t y as t h e one d e s c r i b e d h e r e , t h i s i s a v e r y d i f f i c u l t i s s u e t o r i g o r o u s l y examine. I t i s t r u e t h a t t h e e q u i l i b r i u m i s s t a b l e w i t h r e s p e c t t o u n i l a t e r a l v a r i a t i o n s i n o u t p u t : i f (N-l) p r o d u c e r s f o l l o w t h e i r e q u i -l i b r i u m o u t p u t p r o f i l e s , t h e n any i n a d v e r t e n t d e v i a t i o n from e q u i l i b r i u m by t h e N p r o d u c e r w i l l i m m e d i a t e l y be c o r r e c t e d s i n c e , by d e f i n i t i o n , h i s e q u i l i b r i u m o u t p u t p r o f i l e m a x i -mizes h i s p r e s e n t v a l u e p r o f i t s when t h e o t h e r p r o d u c e r s a r e f o l l o w i n g t h e i r e q u i l i b r i u m s o l u t i o n s . However, i t i s u n l i k e l y t h a t t h e e q u i l i b r i u m i s s t a b l e a g a i n s t d e v i a t i o n s by more t h a n one p r o d u c e r from t h e i r e q u i l i b r i u m l e v e l s o f p r o d u c t i o n . To s i d e - s t e p t h i s t h o r n y i s s u e , i t w i l l be assumed t h a t t h e c o n t r a c t s consummated i n the f u t u r e s m a r kets a t ti m e t=0 a r e b i n d i n g . I n t h a t c a s e , no p r o d u c e r w i l l f i n d i t p r o f i t a b l e t o d e v i a t e from h i s e q u i l i b r i u m l e v e l o f o u t p u t and, i n the d e t e r m i n i s t i c w o r l d b e i n g c o n s i d e r e d h e r e , t h e q u e s t i o n o f s t a b i l i t y becomes i r r e l e -v a n t . ^  R e t u r n i n g t o t h e o p t i m i z a t i o n problem f a c i n g f i r m i , t h e n e c e s s a r y c o n d i t i o n s f o r o p t i m a l i t y may be w r i t t e n down as : ( 3 . 3 a ) e " r t [P (Q)-C - a C P ^ l < X ± (= when q ±>0) 1 (3.3b) 0 q ± ( t ) d t <_ I ± , X i [ I i - q ± ( t ) d t ] = 0 ' 0 (3.3c) X . i d i : (= i f I i>0) (3.3d) l i m H.(t) = 0 , t->T. 1 where (3.4) H ± ( t ) = e - r t [P(Q) - C] q ± - X I Q I , a(P) i s the a b s o l u t e v a l u e o f t h e s l o p e o f t h e demand c u r v e a t p r i c e P, and X^ i s t h e p r e s e n t v a l u e shadow p r i c e o f t h e r e s o u r c e t o f i r m i . X ^ r e p r e s e n t s the i n c r e a s e i n p r e s e n t v a l u e p r o f i t s t o f i r m i due t o the a c q u i s i t i o n o f an a d d i -t i o n a l u n i t o f t h e r e s o u r c e . o p t i m i z a t i o n problem f a c i n g f i r m i . I t r e p r e s e n t s the f l o w o f b e n e f i t s a c c r u i n g t o t h e f i r m a t any i n s t a n t . C o n d i t i o n (3.3a) r e q u i r e s t h a t t h e d i s c o u n t e d m a r g i n a l p r o f i t be i d e n t i c a l i n a l l p e r i o d s o f non-zero p r o d u c t i o n and e q u a l t o the shadow p r i c e o f t h e r e s o u r c e . I f i t i s o p t i m a l f o r the f i r m n o t t o produce i n any p e r i o d , t h e n i t must be t h e case t h a t t h e d i s c o u n t e d m a r g i n a l p r o f i t a t zer o o u t p u t f a l l s s h o r t o f t h e shadow p r i c e o f t h e r e s o u r c e . I n t u i t i v e l y , i t i s c l e a r t h a t an o p t i m a l programme must s a t i s f y t h i s r e q u i r e m e n t o r e l s e t h e r e would be scope f o r i n c r e a s i n g t h e p r e s e n t v a l u e p r o f i t s o f t h e f i r m by r e a l l o c a t i n g t h e e x t r a c t i o n . Con-d i t i o n (3.3b) r e q u i r e s t h e shadow p r i c e t o be n o n - n e g a t i v e (3.4) i s t h e ( p r e s e n t v a l u e ) H a m i l t o n i a n f o r t h e 33 and t h e t o t a l amount o f the r e s o u r c e e x t r a c t e d t o be no g r e a t e r t h a n t h e amount uncovered by e x p l o r a t i o n . The complementary s l a c k n e s s c o n d i t i o n says t h a t i f t h e r e -s o u r c e has p o s i t i v e shadow p r i c e t h e n i t i s o p t i m a l t o e x t r a c t t h e e n t i r e amount o f t h e r e s o u r c e e x p l o r e d f o r , and v i c e - v e r s a . C o n d i t i o n (3.3c) i n d i c a t e s t h a t f o r o p t i m a l i t y , e x p l o r a t i o n must p r o c e e d u n t i l the shadow p r i c e o f the r e s o u r c e e q u a l s t h e m a r g i n a l e x p l o r a t i o n c o s t ; i f the former f a l l s s h o r t o f t h e l a t t e r a t a l l l e v e l s o f the r e s o u r c e s t o c k , t h e n i t i s o p t i m a l n o t t o e x p l o r e a t a l l . F i n a l l y , c o n d i t i o n (3.3d) i s the t e r m i n a l t ime c o n d i t i o n : t h e t i m e h o r i z o n chosen by t h e f i r m must be such t h a t t h e f l o w o f b e n e f i t s a c c r u i n g t o t h e f i r m a t t h a t t ime must v a n i s h ; as l o n g as t h i s f l o w i s p o s i t i v e , i t i s not o p t i m a l t o t e r m i n a t e p r o d u c t i o n . I n g e n e r a l , the shadow p r i c e t o f i r m i o f an a d d i t i o n a l u n i t o f t h e r e s o u r c e would depend on I , the e n t i r e v e c t o r o f r e s o u r c e s t o c k s uncovered by a l l p r o d u c e r s i n t h e i n d u s t r y . I t w i l l be assumed t h a t t h e r e e x i s t s a v e c t o r 1 such t h a t (3.3c) i s s a t i s f i e d f o r a l l f i r m s . Once such a v e c t o r i s d e t e r m i n e d , e x t r a c t i o n p roceeds from f i x e d b u t endogenously d e t e r m i n e d s t o c k s . The q u e s t i o n o f e x i s t e n c e o f a Cournot-Nash e q u i l i b r i u m w i l l be r e l e g a t e d t o subsequent c h a p t e r s . Here i t w i l l s i m p l y be assumed t h a t a unique Cournot-Nash e q u i l i b r i u m e x i s t s , s a t i s f y i n g c o n d i t i o n s (3.3a) - ( 3 . 3 d ) . 34 Some o f t h e c h a r a c t e r i s t i c s o f t h e Cournot-Nash e q u i l i b r i u m a r e summarized i n P r o p o s i t i o n 3.1 below. W h i l e t h e s e p r o p e r t i e s a r e found i n L e w i s and Schmalensee (1980), t h e y a r e p r e s e n t e d h e r e because t h e y w i l l be used i n e s t a b l i s h i n g t h e r e s u l t s t o f o l l o w i n t h i s c h a p t e r . P r o p o s i t i o n 3.1 I n t h e Cournot-Nash e q u i l i b r i u m , a) t h e i n d u s t r y o u t p u t i s c o n t i n u o u s i n time and d e c l i n e s u n t i l i t v a n i s h e s a t t h e t e r m i n a l t i m e b) t h e p r i c e , n e t o f t h e m a r g i n a l e x t r a c t i o n c o s t , r i s e s a t a r a t e s l o w e r t h a n t h e r a t e o f i n t e r e s t c) t h e t i m e h o r i z o n i s f i n i t e . P r o o f a) That t h e i n d u s t r y o u t p u t must be c o n t i n u o u s i n time f o l l o w s p u r e l y from economic c o n s i d e r a t i o n s . A d i s c o n t i n u i t y i n t h e o u t p u t p r o f i l e o f the i n d u s t r y would r e s u l t i n a d i s c o n t i n u i t y i n the p r i c e p r o f i l e . T h i s , i n t u r n , would i m p l y the e x i s t e n c e o f scope f o r the p r o d u c e r s t o i n c r e a s e t h e i r p r e s e n t v a l u e p r o f i t s by r e a l l o c a t i n g t h e i r e x t r a c t i o n o v e r time - w h i c h cannot o c c u r i n an e q u i l i b r i u m . So t h e i n d u s t r y o u t p u t p r o f i l e (and t h e p r i c e ) must be c o n t i n u o u s i n t i m e . From (3.3a) i t f o l l o w s t h a t each f i r m ' s o u t p u t must a l s o be c o n t i n u o u s i n t i m e . 35 Suppose t h a t a t some i n s t a n t m f i r m s a r e o p e r a t i n g . D i f f e r e n t i a t i o n 8 o f (3.3a) w i t h r e s p e c t t o t i m e y i e l d s r t P - a(P) q± - a' (P) Pq± = r\±es A d d i t i o n o f t h e above e x p r e s s i o n a c r o s s a l l t h e m f i r m s p r o d u c i n g a t time t y i e l d s , on s u b s t i t u t i o n f o r t h e \^ from ( 3 . 3 a ) , mP - a(P) Q + Qa'P = r[mP - aQ - mC], so t h a t P - r- I m P " a Q ~ m C ]  e [m+1 - Qa'] T h i s e x p r e s s i o n may be r e w r i t t e n as n «n P - r [ p ( m £ - D ~ cm £] U.u j e [m+1 - Qa' ] P dQ P  w n e r e £ = __ = A s s u m p t i o n (A3) t h a t e i s an i n c r e a s i n g f u n c t i o n o f p r i c e i m p l i e s t h a t de _1 P_ dQ P_ , dP Qa Q2a dP Qa2 i . e . , t h a t ( 3 . 6 ) Qa' < (e+l)/e . Now a d d i t i o n o f (3.3a) o v e r t h e m p r o d u c i n g f i r m s y i e l d s (3.7) P(me-l) = mx"ee r t + mC, 36 where X i s the average shadow price across the m firms. Prio r to exhaustion m can never be zero. For i f m=0 at time t' then from (3.3a) i t follows that e ~ r t ' [F-C^] <. \^ for a l l i , implying that (3.3a) can never hold with equality for any i when t>t'. Since the r i g h t hand side of (3.7) i s p o s i t i v e , i t follows that me>l. This f a c t , together with (3.6), im-p l i e s (3.8) Qa' < (m+1) . t From (3.8) and (3.5) i t follows that P>0 p r i o r to industry shut-down, i . e . , the industry output monotonically declines over time. I t c l e a r l y must be the case that Q(T) = 0, where T i s the terminal time. If not, there would be a d i s -continuous jump i n the pr i c e p r o f i l e at time T from P(Q(T)) to F, which, as argued above, cannot occur i n equilibrium. b) Using (3.5) i t follows that P <. r(P-C) i f and only i f P(me-l) - Gme <_ (P-C) [ (m+1) - Qa'] e, i . e . , i f and only i f (P-C) (e+1) >_ (P-C) Qa'e - C, which i s true i n view of (3.6) . 3 7 c ) F r o m ( 3 . 3 a ) i t f o l l o w s t h a t P - C - a ( P ) q. < A . e r t , i = 1 , . . . , N S i n c e t h e r i g h t h a n d s i d e o f t h e a b o v e e x p r e s s i o n i s i n -c r e a s i n g e x p o n e n t i a l l y w i t h t i m e ^ a n d t h e l e f t h a n d s i d e i s b o u n d e d b y t h e f i n i t e n u m b e r ( F - C ) , i t f o l l o w s t h a t t h e r e e x i s t s a f i n i t e t i m e , T , b e y o n d w h i c h t h i s e x p r e s s i o n m u s t h o l d w i t h s t r i c t i n e q u a l i t y f o r a l l i . T h u s t h e t i m e h o r i z o n m u s t b e f i n i t e . Q . E . D . T h e p i c t u r e t h a t e m e r g e s f r o m t h e a b o v e r e s u l t s i s a s f o l l o w s . A t t i m e t=0, f i r m s d e t e r m i n e t h e r e s o u r c e s t o c k s f r o m w h i c h t h e y w i l l s u b s e q u e n t l y e x t r a c t . I n e q u i l i b r i u m , e x t r a c t i o n p r o c e e d s i n a m a n n e r t h a t e n s u r e s t h a t t h e i n -d u s t r y o u t p u t i s c o n t i n u o u s , a n d m o n o t o n i c a l l y d e c l i n i n g u n t i l i t v a n i s h e s a t t h e t e r m i n a l t i m e . T h e r e s o u r c e p r i c e i s , a s a r e s u l t , c o n t i n u o u s i n t i m e a n d m o n o t o n i c a l l y i n -c r e a s i n g u n t i l i t r e a c h e s t h e c h o k e p r i c e a t t h e t e r m i n a l t i m e . A c c o r d i n g t o p a r t ( b ) o f t h e a b o v e p r o p o s i t i o n , t h e n e t p r i c e a l w a y s r i s e s a t a r a t e l e s s t h a n t h e r a t e o f i n t e r e s t . T h i s i s i n s h a r p c o n t r a s t t o t h e r e s u l t i n c o m p e t i t i v e m a r k e t s w h e r e t h e n e t p r i c e r i s e s a t t h e r a t e o f i n t e r e s t . T h e d i f f e r e n c e , o f c o u r s e , a r i s e s f r o m t h e f a c t t h a t i n t h e c a s e b e i n g c o n s i d e r e d h e r e , p r o d u c e r s e x e r c i s e t h e i r m a r k e t p o w e r a n d d o n o t t a k e t h e p r i c e a s p a r a m e t e r i c a l l y g i v e n . I n p e r i o d s d u r i n g w h i c h a f i r m e x t r a c t s n o n - z e r o a m o u n t s o f t h e r e s o u r c e , i t i s t h e f i r m ' s n e t m a r g i n a l r e v e n u e , n o t t h e 38 n e t r e s o u r c e p r i c e , t h a t r i s e s a t t h e r a t e o f i n t e r e s t . A c c o r d i n g t o p a r t (c) o f P r o p o s i t i o n 3.1, a l l f i r m s f i n d i t o p t i m a l t o cease p r o d u c t i o n i n f i n i t e t i m e . I n t h e r e s t o f t h i s c h a p t e r the c a s e s when the p r o p e r t y r i g h t s a r e e v e n l y and u n e v e n l y d i s t r i b u t e d a c r o s s the p r o d u c e r s a r e c o n s i d e r e d s e p a r a t e l y . As can be e x p e c t e d , a g r e a t d e a l more can be s a i d about t h e n a t u r e o f t h e C o u r n o t -Nash e q u i l i b r i u m i n t h e former c a s e . A. THE SYMMETRIC CASE Here i t i s assumed t h a t the p r o p e r t y r i g h t s a r e s y m m e t r i c a l l y d i s t r i b u t e d a c r o s s a l l p r o d u c e r s , so t h a t M^ = M/N f o r a l l i . 1 0 The a s s u m p t i o n t h a t the Cournot-Nash e q u i l i b r i u m i s u n i q u e i m p l i e s t h a t t h e e q u i l i b r i u m must be symmetric i . e . , the d e c i s i o n s o f a l l t h e p r o d u c e r s must be i d e n t i c a l . As has been mentioned e a r l i e r , t h e shadow p r i c e o f t h e r e s o u r c e t o each f i r m w i l l depend on t h e v e c t o r I o f s t o c k s h e l d by a l l f i r m s i n t h e i n d u s t r y . I n t h e symmetric c a s e , however, t h i s dependence may be summarized as A (I T',N) , where I T i s the t o t a l amount o f r e s o u r c e s t o c k i n t h e i n -d u s t r y c o m p r i s e d o f N f i r m s . An i n c r e a s e i n t h e s t o c k o f d i s c o v e r e d r e s e r v e s , e v e n l y d i s t r i b u t e d a c r o s s a l l f i r m s would c l e a r l y l ower t h e shadow p r i c e . I f , on t h e o t h e r hand, a g i v e n amount o f r e s o u r c e i s a l l o c a t e d e v e n l y a c r o s s a l a r g e r number o f f i r m s , t h e shadow p r i c e would i n c r e a s e . 39 These o b s e r v a t i o n s a r e r e c o r d e d f o r f u t u r e r e f e r e n c e : ( 3 > 9 ) 3_X(I T,N) < 0 , 8 M I T , N ) > 0. 81,p 9N (3.9) c a n be r i g o r o u s l y shown by h o l d i n g P f i x e d i n ( 3 . 5 ) , s e t t i n g m=N and p a r t i a l l y d i f f e r e n t i a t i n g (3.5) w i t h r e s p e c t t o I T and N ( t r e a t e d as a c o n t i n u o u s v a r i a b l e ) , r e s p e c t i v e l y , t o o b t a i n , , v 3P (P) = 0 . 9P (P) > 0 • (3.10) (a) ^ (b) — Now by p a r t (a) o f P r o p o s i t i o n 3.1, t h e i n d u s t r y o u t p u t -and, t h e r e f o r e , t h e o u t p u t o f each o f t h e N i d e n t i c a l f i r m s -v a n i s h e s a t t h e t e r m i n a l t i m e , T. S i n c e (3.3a) must h o l d w i t h e q u a l i t y f o r a l l t such t h a t 0<_t<_T, i t f o l l o w s t h a t (3.11) X ( I T , N ) = e " r T ( F - C ) . I f N i s h e l d f i x e d and 1,^  i s i n c r e a s e d so t h a t e v e r y f i r m ' s r e s o u r c e s t o c k i n c r e a s e s by t h e same amount, t h e new p r i c e p r o f i l e must l i e e n t i r e l y below t h e o l d p r o f i l e . I f i t l i e s e n t i r e l y above, t h e r e s o u r c e c o n s t r a i n t would be v i o l a t e d , whereas i f t h e two p r o f i l e s c r o s s , (3.10) (a) would be v i o l a t e d . I t t h u s f o l l o w s t h a t t h e time h o r i z o n must i n c r e a s e when I T i s i n c r e a s e d ( h o l d i n g N f i x e d ) . The f i r s t i n e q u a l i t y i n (3.9) now f o l l o w s from ( 3 . 1 1 ) . I f , i n s t e a d , a g i v e n amount I T , o f t h e r e s o u r c e i s e q u a l l y d i s t r i b u t e d a c r o s s a l a r g e r number o f f i r m s , t h e r e s o u r c e c o n s t r a i n t i m p l i e s t h a t t h e new p r i c e p r o f i l e c a nnot l i e e n t i r e l y above o r e n t i r e l y below t h e o l d p r o f i l e ; t h e 40 two p r o f i l e s must c r o s s . F u r t h e r , (3.10) (b) i m p l i e s t h a t t h e new p r i c e p r o f i l e must c r o s s t h e o l d p r o f i l e o n l y once and from below. T h i s , i n t u r n , i m p l i e s t h a t when N i s i n c r e a s e d ( h o l d i n g T-T f i x e d ) , t h e time h o r i z o n must d e c r e a s e . The second i n e q u a l i t y i n (3.9) now f o l l o w s from (3.11). The f o l l o w i n g p r o p o s i t i o n c o n s i d e r s t h e e f f e c t o f an i n c r e a s e i n t h e m a r g i n a l e x t r a c t i o n c o s t on the u l t i m a t e amount o f r e c o v e r y o f t h e r e s o u r c e . P r o p o s i t i o n 3.2: An i n c r e a s e i n t h e m a r g i n a l e x t r a c t i o n c o s t l o w e r s t h e amount o f r e s o u r c e r e c o v e r e d by t h e i n d u s t r y . P r o o f : U s i n g ( 3 . 1 ) , (3.3c) (assumed t o h o l d w i t h e q u a l i t y ) may be r e w r i t t e n (3.12) A ( I T , N ) = d* (I T/M) D i f f e r e n t i a t i n g (3.12) t o t a l l y w i t h r e s p e c t t o C, y i e l d s 3 A d I T , 3 A 1 ,» , _ / M v d I T 3 I T SC + 3C = M d U T / M ) dC" ' where J- T i s h e l d f i x e d i n t h e c o m p u t a t i o n o f the p a r t i a l , d e r i v a t i v e 3A/3C. Rearrangement o f t h e above e x p r e s s i o n l e a d s t o d l — (3 13) 1 = i f ( I T / M ) " 3 T T From ( A l ) and (3.9) i t f o l l o w s t h a t t h e denominator o f (3.11) i s p o s i t i v e : 41 (3.14) 3 A I t remains to sign the p a r t i a l d e r i v a t i v e j^. Holding P fixed, p a r t i a l d i f f e r e n t i a t i o n of (3.5) with respect to C y i e l d s rN (3.15) 3P (P) 3 C N+l-Qa' < 0 Now holding I T f i x e d , consider a small increase i n the marginal extraction cost from C to C'. Let {P} and {p'} , respectively, be the price p r o f i l e s corresponding to these two extraction costs. These two price p r o f i l e s must cross. If not, one p r o f i l e would l i e e n t i r e l y above the other -v i o l a t i n g the assumption that the same amount, I T , i s u l t i -mately extracted along both p r o f i l e s . Moreover, i n view of (3.15) the two p r i c e p r o f i l e s can cross only once, with the {p'} p r o f i l e i n i t i a l l y l y i n g above and then below the {P} p r o f i l e (as i n F i g . 3.1). |P(t) Figure 3.1 I t f o l l o w s t h a t t h e t i m e h o r i z o n f o r t h e {P'} p r o f i l e i s h i g h e r , i . e . , t h a t (3.16) § > 0. H o l d i n g I T f i x e d , d i f f e r e n t i a t i o n o f (3.11) w i t h r e s p e c t t o C y i e l d s , on u s i n g ( 3 . 1 6 ) , n \ 2X r dT. -rT,„ -rT _ (3.17) — = - ^ e (F-C) - e < 0. S u b s t i t u t i o n o f (3.14) and (3.17) i n t o (3.13) f i n a l l y e s t a b l i s h e s t h a t (3.18) ^ T " n dC U ' i . e . , an i n c r e a s e i n t h e m a r g i n a l e x t r a c t i o n c o s t l o w e r s t h e t o t a l r e c o v e r y o f t h e i n d u s t r y . Q.E.D. The economic i n t u i t i o n u n d e r l y i n g t h e above r e s u l t i s q u i t e s t r a i g h t f o r w a r d . The i n c e n t i v e t o a c q u i r e a d d i t i o n a l amounts o f t h e r e s o u r c e i s d e t e r m i n e d by i t s shadow p r i c e . From (3.3c) i t f o l l o w s t h a t t h e o p t i m a l l e v e l o f r e s o u r c e r e c o v e r y i s t h a t a t w h i c h t h e m a r g i n a l d i s c o v e r y c o s t i s e q u a l t o t h e shadow p r i c e . An i n c r e a s e i n the m a r g i n a l c o s t o f e x t r a c t i o n , c e t e r i s p a r i b u s , l o w e r s t h e shadow p r i c e and c o n s e q u e n t l y r e s u l t s i n a s m a l l e r amount o f t h e r e s o u r c e b e i n g r e c o v e r e d . The v a r i o u s e f f e c t s o f changes i n market s t r u c t u r e a r e now c o n s i d e r e d . S i n c e a l l f i r m s p o s s e s s t h e same number o f d e p o s i t s (and i n c u r t h e same m a r g i n a l e x t r a c t i o n c o s t s ) i n 43 the symmetric case being considered here, an increase i n the number of firms i n the industry unambiguously implies a decrease i n the supply-side concentration. The following proposition examines the ef f e c t s of d i s t r i b u t i n g the same number of deposits evenly across a larger number of firms. Proposition 3.3: An increase i n the number of firms i n the industry a) increases the ultimate amount of resource recovered by the industry b) decreases the time horizon c) decreases the present value of industry p r o f i t s d) increases the present value of the consumer plus producer surplus. Proof: a) D i f f e r e n t i a t i o n of (3.3c) with respect to N yi e l d s which, on rearrangement gives d I T 3N d N i d " ( I / M ) - — M ^ - V ™ ' 3 I T Using (3.9) and (3.14) i n the above expression, i t follows that d I T (3.20) ^ > 0, i . e . , t o t a l resource recovery increases with N. 44 b) Recognizing i z i n g that the l e f t hand side of (3.19) i s the t o t a l derivative | | , i t follows from (3.19) and (3.20) that d l r (3.21) fA = h d"(I m/M) -g^ > 0. dN M D i f f e r e n t i a t i o n of (3.11) with respect to N y i e l d s dx _ -rT f F _ c x dT = - re (F L) d N dN so that (3.22) dT _ _ e dX dN r(F-C) dN < 0, i . e . , the time horizon decreases with N. nt value p r o f i t s of an industry comprised c) The prese of N firms i s given by rT (3.23) V(N) = r r t [P(Q) " C] Q(t) dt - Md(IT/M) ) i f f e r e n t i a t i o n of (3.23) with respect to N y i e l d s dV(N) dN T - r t dQ(t)dt e r t[P(Q) - C - a(P)Q] ^ T + e - r T [ P ( Q ( T ) ) - C] Q(T) | | - d ' V J - d N 1 Noting that Q(T) = 0 and using (3.12), the above expression reduces to 'T e' r t[P(Q) " C - a(P)Q] § M ^ 0 dV(N) dN - X d I T dN 45 Using the f i r s t order condition (3.3a) (which must hold with equality for t<T) , the above expression can be further reduced to (3.24) where (3.25) dV(N) = _ ( N _ 1 } H + x dN Noting that (3.26) (3.24) becomes (3.27) Now from (3.3a), (3.28) dQ(t).. ,^1 dN - dN 0 r r t a ( P ) Q(t) | | ( t ) d t . d Q ( t ) d t = dN * dN ' dV(N) = _ (N-i) H dN - r t for 0<t<T. ^ e " r t a(P) Q(t) = e " r t [P(Q) - C] - X N Since Q(0)>0, i t follows from (3.28) that (3.29) P(Q(0)) - C - X > 0 Further, since by part (b) of Proposition 3.1 the net price r i s e s at a rate slower than the rate of i n t e r e s t , i t follows that the l e f t hand side of (3.28) i s a non-negative, mono-t o n i c a l l y d e c l i n i n g function of time. 46 Now part (b) above and (3.20) together imply that (3.30) dQ(0) dN > 0, If (3.30) did not hold then by part (b) above, the price p r o f i l e corresponding to an industry with (N+l) firms would l i e e n t i r e l y above that of an industry with N firms - v i o -l a t i n g (3.20), which may be rewritten as f (3.31) gg ( t ) dt > 0. dN 0 Using (3.30), (3.31) and the fa c t that e " r t a(P) Q(t) i s a non-negative, monotonically declining function of time, i t follows from (3.25) that (3.32) H > 0 and t h i s , i n turn, implies, v i a (3.27), that (3.33) dV(N) < dN i . e . , the present value p r o f i t s of the industry decline with N, d) The present value of the t o t a l surplus when the resource industry i s comprised of N firms i s given by (3.34) W(N) = " r t r e i fQ(t) [P(x)-C] dx}dt - Md (IT/M) D i f f e r e n t i a t i o n of (3.34) with respect to N y i e l d s (3.35) dW(N) dN e-r t[P(Q)-C] 3§ ( t )dt - 4^1 , 0 where use has been made of the fact that Q(T) = 0. Use of (3.3a) and (3.26) i n (3.35) y i e l d s ( 3 . 3 6 ) f ( N ) = H > 0 , i . e . , the present value of the t o t a l surplus increases with N. Q.E.D. The ultimate amount of the resource recovered by the industry i s seen from (3.3c) to be determined by the shadow price and the marginal exploration cost - both of which de-pend, i n the model considered here, on the market structure, i . e . , on N. At a given l e v e l of the stock of industry re-serves, T-T, an increase i n N re s u l t s i n an increase i n the shadow price to each firm of an additional unit of the re-source. To maintain equality i n (3.3c), a greater amount of the resource must be recovered - which i s the i n t u i t i o n behind part (a) of Proposition 3.3. According to part (b) of Proposition 3.3, although the t o t a l amount of the resource recovered by the industry increases with N, the extraction i s more rapid; the time horizon decreases. Since the price of the resource increases monotonically with time, t h i s implies that there exists an i n i t i a l phase during which the price i s lower (and the i n -dustry output higher) with N+l firms than with N firms. In a l a t e r phase, the price i n the former case must necessarily be higher. 48 An i n c r e a s e i n N, a c c o r d i n g t o p a r t (c) o f P r o p o s i t i o n 3.3, r e d u c e s t h e p r e s e n t v a l u e o f t h e t o t a l i n d u s t r y p r o f i t s and, t h e r e f o r e , t h e p r o f i t s t o each f i r m . However, t h e c o r r e s p o n d i n g i n c r e a s e i n t h e p r e s e n t v a l u e o f t h e consumer s u r p l u s more t h a n compensates f o r the d e c r e a s e i n t h e p r e s e n t v a l u e o f t h e p r o d u c e r s u r p l u s ; s o c i e t y ' s w e l f a r e 1 1 i n c r e a s e s w i t h N, a c c o r d i n g t o p a r t (d) o f P r o p o s i t i o n 3.3. The t r a d i -t i o n a l r e s u l t t h a t a d e c r e a s e i n market c o n c e n t r a t i o n i n c r e a s e s s o c i e t y ' s w e l f a r e , i s h e r e c o n f i r m e d i n t h e c a s e when the amount o f r e c o v e r e d r e s e r v e s i s endogenous. S t i g l i t z (1976) has d emonstrated t h a t when t h e demand f u n c t i o n i s o f c o n s t a n t e l a s t i c i t y ( g r e a t e r t h a n u n i t y i n a b s o l u t e v a l u e ) and t h e m a r g i n a l e x t r a c t i o n c o s t i s z e r o , t h e n t h e m o n o p o l i s t ' s e x t r a c t i o n p r o f i l e c o i n c i d e s w i t h t h a t o b t a i n e d under p e r f e c t c o m p e t i t i o n i f t h e r e s o u r c e s t o c k i s e x o g e n o u s l y g i v e n . L o u r y (1981) has g e n e r a l i z e d t h i s and has e s t a b l i s h e d t h a t , under t h e same c i r c u m s t a n c e s , the o u t p u t p r o f i l e o f t h e i n d u s t r y i s i n d e p e n d e n t o f the number o f p r o d u c e r s and t h e d i s t r i b u t i o n o f t h e r e s o u r c e s t o c k a c r o s s t h e p r o d u c e r s , g i v e n t h a t t h e p r o d u c e r s engage i n Cournot-Nash b e h a v i o u r . However, when t h e r e s o u r c e s t o c k i s endogenous, as i t i s h e r e , t h i s r e s u l t i s no l o n g e r v a l i d : t h e e x t r a c t i o n p r o f i l e can n ever be i n d e p e n d e n t o f market s t r u c t u r e . To demonstrate t h i s , assume t h a t t h e m a r g i n a l e x t r a c t i o n c o s t i s z e r o (C=0) and t h a t u n i t s a r e chosen so as t o be a b l e t o w r i t e t h e c o n s t a n t - e l a s t i c i t y demand f u n c t i o n as _^ P(Q) = Q~e , e > l . I n t h e symmetric e q u i l i b r i u m , where each f i r m ' s (endo-g e n o u s l y determined) r e s o u r c e s t o c k i s , say, I , the common f i r s t o r d e r c o n d i t i o n (3.3a) y i e l d s e N w h i c h may be r e w r i t t e n as _1 (3.37) P(Q) = Q E = X ( 1 - i j ) " 1 e r t D i f f e r e n t i a t i o n (3.37) w i t h r e s p e c t t o time i m m e d i a t e l y e s t a b l i s h e s t h a t (3.38) P/P = r , Thus t h e p r i c e r i s e s e x p o n e n t i a l l y a t t h e r a t e o f i n t e r e s t . S o l v i n g (3.37) f o r Q, t h e o u t p u t p r o f i l e o f t h e i n d u s t r y i s o b t a i n e d as (3.39) Q(t) = [ 1 _ 1 ^ N e ] " £ e " r e t , i . e . , t he i n d u s t r y o u t p u t d e c l i n e s e x p o n e n t i a l l y w i t h t i m e . Now s i n c e t h e p r i c e a lways r i s e s a t a f i n i t e r a t e and t h e choke p r i c e f o r t h i s demand f u n c t i o n i s i n f i n i t e , i t i s c l e a r t h a t t h e time h o r i z o n must be i n f i n i t e . The r e s o u r c e c o n s t r a i n t Q(t) d t = I T ) y i e l d s , on s u b s t i t u t i o n f o r Q(t) from (3.39) 50 so t h a t _ 1 (3.40) X ( I T , N ) = ( I T e r ) £ (1 - . S i n c e - 1 - 1 (3.41) ^ = " | ( r i T e ) E < 0 , (3.42) l i m A ( I ,N) = », l i m A ( I ,N) = 0, I -*-0 I ->oo T T i t f o l l o w s t h a t t h e r e e x i s t s a s t r i c t l y p o s i t i v e and f i n i t e r e s o u r c e s t o c k I T t h a t s o l v e s ( 3 . 1 2 ) . T o t a l l y d i f f e r e n t i a t -i n g (3.12) and u s i n g (3.41) i t i s e a s i l y seen t h a t ( 3 4 3 ) ^ . ( I T « ^ / N £ | > 0 dN M fi d"(I T/M) + ^ A ( I T , N ) 12 S i n c e the u l t i m a t e r e s o u r c e r e c o v e r y i s n o t independent o f N, the a g g r e g a t e e x t r a c t i o n p r o f i l e can n o t be indepe n d e n t o f market s t r u c t u r e f o r a c o n s t a n t e l a s t i c i t y demand f u n c t i o n B. THE ASYMMETRIC CASE Now the asymmetric case i s c o n s i d e r e d , where t h e p r o p e r t y r i g h t s o f t h e f i r m s i n t h e i n d u s t r y a r e not i d e n t i c a l , i . e . , M^^Mj f o r some f i r m s i and j . I n g e n e r a l , t h e d i s t r i b u t i o n o f r i g h t s a c r o s s p r o d u c e r s w i l l c l e a r l y a f f e c t t h e amount o f r e s o u r c e r e c o v e r y and t h e i n d u s t r y o u t p u t p r o f i l e . An e x c e p t i o n o c c u r s when t h e m a r g i n a l d i s c o v e r y c o s t i s c o n s t a n t w i t h i n each d e p o s i t , i . e . , when d".(«) = 0. T h i s i s e s t a b l i s h e d i n t h e f o l l o w i n g p r o p o s i t i o n : 51 P r o p o s i t i o n 3.4 Suppose t h a t N>1 and t h a t t h e m a r g i n a l d i s c o v e r y c o s t i s c o n s t a n t (=d') f o r a l l d e p o s i t s . Then, market performance i s u n a f f e c t e d by changes i n t h e d i s t r i b u t i o n o f d e p o s i t s a c r o s s t h e f i r m s , and t h e i n d u s t r y e q u i l i b r i u m i s s y mmetric, w i t h a l l f i r m s b e h a v i n g i d e n t i c a l l y . P r o o f : Assuming t h a t a l l f i r m s f i n d i t p r o f i t a b l e t o e x p l o r e and e x t r a c t some amount o f t h e r e s o u r c e , i t f o l l o w s from (3.3c) t h a t t h e shadow p r i c e o f t h e r e s o u r c e i s d e t e r m i n e d by the c o n d i t i o n (3.44) X. = d' i i = l , 2 , . . . N , i . e . , t h e shadow p r i c e i s t h e same f o r a l l t h e f i r m s i n e q u i l i b r i u m . I t f o l l o w s from (3.3a) t h a t e x t r a c t i o n r a t e s a r e the same f o r a l l f i r m s . Thus t h e u l t i m a t e amounts o f t h e r e s o u r c e e x t r a c t e d by t h e f i r m s a r e a l s o i d e n t i c a l . Q.E.D. What d r i v e s t h e r e s u l t o f t h e above p r o p o s i t i o n i s , o f c o u r s e , t h e f a c t t h a t t h e c o s t t o any f i r m o f u n c o v e r i n g a g i v e n amount o f t h e r e s o u r c e i s i ndependent o f t h e number o f d e p o s i t s i t owns. Thus t h e r e i s no advantage t o owning a l a r g e r number o f d e p o s i t s . T h i s b e i n g t h e c a s e , th e e q u i -l i b r i u m i s symmetric and independent o f t h e d i s t r i b u t i o n o f t h e p r o p e r t y r i g h t s . The a ssumption o f c o n s t a n t m a r g i n a l d i s c o v e r y c o s t , however, i s q u i t e u n r e a l i s t i c f o r i t assumes t h a t t h e r e i s 52 no depletion e f f e c t operating: add i t i o n a l amounts of the resource can be acquired at no higher cost. In the more r e a l i s t i c case where the marginal exploration cost i s i n -creasing i n the amount of resource uncovered, the following characterization of the. Cournot-Nash equilibrium i s obtained. Proposition 3.5 Suppose d"(-) >0 and property r i g h t s over deposits are asymmetric i n that M\ > M_. for some firms i and j . Then i n the Cournot-Nash equilibrium, a) a l l firms' begin production at time zero b) x i < Aj c) firm i produces more than firm j at every instant and ceases production l a t e r , so that I. > d) I i/M ± < Ij/Mj Proof a) Suppose firm i chooses not to produce at time t. Then i t s discounted marginal p r o f i t (at zero output) i s given by (3.45) e" r t [ P ( Q ( t ) ) - C] Now from part (b) of Proposition 3.1 i t follows that expression (3.45) i s a monotonically decreasing function of t, since the net price i s r i s i n g at a rate slower than the rate of i n t e r e s t . Thus i f firm i produces at a l l - .(which i t i s assumed to do) then i t i s optimal for i t to begin 53 p r o d u c t i o n a t t i m e z e r o . b) Suppose t h a t a t t i m e t b o t h f i r m s i and j a r e p r o d u c i n g . Then from (3.3a) i t f o l l o w s t h a t A e r t (3.46) q k ( t ) = [ P ( Q ) " a ( p ) " ^ f o r k = i ' j Suppose A ^ > . A j . Then i t f o l l o w s from (3.46) t h a t q ^ i q j as l o n g as b o t h f i r m s a r e p r o d u c i n g and, s i n c e t h e t i m e h o r i z o n i s known t o be f i n i t e from p a r t (c) o f P r o p o s i t i o n 3.1, t h a t T^<T_., i . e . , f i r m i must t e r m i n a t e p r o d u c t i o n sooner t h a n f i r m j . Thus i t must be t h e case t h a t I^<_Ij, so t h a t I i / M i < I j / M j - N o w (3.3c) and (3.1) i m p l y t h a t (3.47) A k = d ' ( I k / M k ) f o r a l l k. S i n c e d 1 i s i n c r e a s i n g i n i t s argument, i t must t h e n be t r u e t h a t A . < A . - c o n t r a d i c t i n g t h e s u p p o s i t i o n t h a t A . j>A . . Thus, i f M.>M. t h e n A . < A . . 1 3 1 3 c) F o l l o w s immediates from t h e r e a s o n i n g used above. d) From p a r t (b) above and (3.47) i t f o l l o w s t h a t (3.48) T i / H i < I j / M j Q. E . D . P a r t (a) o f t h e above p r o p o s i t i o n e s t a b l i s h e s t h a t i t i s n ever o p t i m a l f o r a f i r m t o b e g i n p r o d u c t i o n a t any time o t h e r t h a n time z e r o . T h e r e f o r e , i t n ever pays any f i r m t o h o l d back p r o d u c t i o n w i t h t h e i n t e n t i o n o f a c q u i r i n g monopoly power l a t e r on. A c c o r d i n g t o p a r t (b) o f t h e 54 p r o p o s i t i o n , f i r m s p o s s e s s i n g a l a r g e r number o f d e p o s i t s have a l o w e r shadow p r i c e f o r t h e r e s o u r c e i n e q u i l i b r i u m . T h i s i s i n k e e p i n g w i t h i n t u i t i o n as i s p a r t ( c ) , w h i c h e s t a b l i s h e s t h a t l a r g e r f i r m s r e c o v e r more o f t h e r e s o u r c e . I t i s c l e a r t h a t t h e l a r g e r f i r m s dominate t h e i n d u s t r y i n t h a t t h e y produce more a t each i n s t a n t p r i o r t o ex-h a u s t i o n , and t e r m i n a t e p r o d u c t i o n l a t e r . However, p a r t (d) e s t a b l i s h e s t h a t a l t h o u g h l a r g e r f i r m s r e c o v e r more o f the r e s o u r c e i n a b s o l u t e t erms, s m a l l e r f i r m s e x p l o r e more i n t e n s i v e l y and r e c o v e r more o f t h e r e s o u r c e p e r d e p o s i t . The f a c t t h a t when p r o p e r t y r i g h t s a r e asymmetric the r e c o v e r y p e r d e p o s i t v a r i e s from p r o d u c e r t o p r o d u c e r p o i n t s t o a c e r t a i n i n e f f i c i e n c y i n r e s o u r c e r e c o v e r y from s o c i e t y ' s p o i n t o f v i e w . T h i s i s c o n s i d e r e d i n the f o l l o w i n g p r o p o s i t i o n . P r o p o s i t i o n 3.6 Suppose d">0 and M^M^ f o r some f i r m s i and j . Then th e t o t a l amount o f r e s o u r c e r e c o v e r e d by t h e i n d u s t r y i s n o t a c q u i r e d a t l e a s t c o s t . P r o o f : The l e a s t c o s t o f u n c o v e r i n g a g i v e n amount, I T , o f t h e r e s o u r c e i s g i v e n by t h e s o l u t i o n t o M M (3.49) m y i £ d (y. ) s . t . I y = I Y i = l 1 i = l 1 1 5 5 T h e f i r s t o r d e r n e c e s s a r y c o n d i t i o n f o r t h i s m i n i m i z a t i o n p r o b l e m r e q u i r e s t h a t ( 3 . 5 0 ) d ' ( y i ) = d " ( y 2 ) = . .'. = d*(y M) S i n c e t h e f u n c t i o n d ( . •) i s a s s u m e d t o b e s t r i c t l y c o n v e x , i t f o l l o w s f r o m ( 3 . 5 0 ) t h a t f o r o p t i m a l i t y ( 3 . 5 1 ) y ^ = I T / M f o r a l l i . S i n c e , b y p a r t ( d ) o f P r o p o s i t i o n 3 . 5 , t h e n e c e s s a r y c o n d i t i o n ( 3 . 5 1 ) i s v i o l a t e d w h e n p r o p e r t y r i g h t s a r e a s y m m e t r i c , t h e r e s u l t f o l l o w s . Q.E.D. I f a l l t h e M d e p o s i t s w e r e o w n e d b y a s i n g l e f i r m , i t i s c l e a r t h a t t h e m a r g i n a l d i s c o v e r y c o s t s w o u l d b e e q u a t e d a c r o s s a l l t h e d e p o s i t s . T h i s i s b e c a u s e p r o f i t m a x i m i z a t i o n r e q u i r e s c o s t - m i n i m i z a t i o n , w h i c h , i n t u r n , r e q u i r e s ( 3 . 5 0 ) t o b e s a t i s f i e d . T h u s a m o n o p o l y i s e f f i c i e n t i n t h e s e n s e t h a t i t a c q u i r e s i t s r e s e r v e s a t . . 1 3 l e a s t c o s t . I n t h e o t h e r e x t r e m e o f p e r f e c t c o m p e t i t i o n , w h e r e a l l p r o d u c e r s i g n o r e t h e i r o w n i n f l u e n c e o n t h e p r i c e a n d t a k e t h e p r i c e a s p a r a m e t r i c a l l y g i v e n a t e a c h i n s t a n t , t h e f i r s t o r d e r c o n d i t i o n ( 3 . 3 a ) f o r f i r m i b e c o m e s ( 3 . 5 2 ) e ~ r t[p(Q) - c] < x ± (= i f qi>°) S i n c e a l l f i r m s b e g i n p r o d u c t i o n a t t i m e t = 0 ( b y p a r t ( a ) o f P r o p o s i t i o n 3 . 5 ) , i t f o l l o w s t h a t ( 3 . 5 2 ) m u s t h o l d w i t h 56 e q u a l i t y a t t=0 f o r a l l i . T h e r e f o r e , t h e shadow p r i c e A ^ must be t h e same f o r a l l f i r m s . T h i s , i n t u r n , i m p l i e s v i a ( 3 . 3 c ) t h a t t h e m a r g i n a l d i s c o v e r y c o s t s must be t h e same f o r a l l f i r m s . Thus, i n p e r f e c t c o m p e t i t i o n , t o o , as i n monopoly, t h e i n d u s t r y r e s e r v e s a r e a c q u i r e d a t l e a s t c o s t . P r o p o s i t i o n 3.6 i n d i c a t e s t h a t t o t h i n k o f o l i g o -p o l i s t i c m arkets as i n t e r m e d i a t e between monopoly and p e r f e c t c o m p e t i t i o n i n a l l r e s p e c t s i s t o o v e r l o o k p o t e n -t i a l l y i m p o r t a n t q u a l i t a t i v e d i f f e r e n c e s . Note t h a t t h i s t y p e o f i n e f f i c i e n c y i n o l i g o p o l i s t i c m arkets i m p l i e s t h a t d e c e n t r a l i z e d e x p l o r a t i o n i s s u b o p t i m a l ( i . e . , P a r e t o i n e f f i c i e n t ) from s o c i e t y ' s p o i n t o f v i e w . I n t h e symmetric c a s e , a d e c r e a s e i n t h e number o f f i r m s i n t h e i n d u s t r y unambiguously i m p l i e s an i n c r e a s e i n s u p p l y - s i d e c o n c e n t r a t i o n . When th e p r o p e r t y r i g h t s a r e asymmetric, however, t h e number o f f i r m s i n t h e i n d u s t r y i s n o t an adequate measure o f market c o n c e n t r a t i o n . I n such a s i t u a t i o n , a s i n g l e measure o f c o n c e n t r a t i o n becomes much more d i f f i c u l t t o d e f i n e . I t seems r e a s o n a b l e , how-e v e r , t o r e g a r d an i n d u s t r y i n wh i c h some f i r m s h o l d l a r g e r r e s o u r c e s h a r e s t h a n o t h e r s as b e i n g more co n -c e n t r a t e d t h a n one w i t h t h e same number o f f i r m s b u t i n w h i c h p r o p e r t y r i g h t s a r e symmetric. I t i s d i f f i c u l t t o c h a r a c t e r i z e , i n g e n e r a l , t h e e f f e c t o f c o n c e n t r a t i o n on the u l t i m a t e amount o f r e s e r v e s r e c o v e r e d by t h e i n d u s t r y . I t t u r n s o u t , however, t h a t s m a l l d e v i a t i o n s from symmetry do n o t a f f e c t t h e u l t i m a t e r e c o v e r y o f t h e i n d u s t r y . T h i s 57 i s established i n the following proposition. Proposition 3.7 For small deviations from symmetry i n property r i g h t s , the t o t a l amount of the resource recovered by the industry remains unchanged. Proof: For s i m p l i c i t y assume there are only two firms i n the industry. (The r e s u l t i s e a s i l y extended to an ar b i t r a r y number of firms). Let the firms have exclusive M M property r i g h t s over (-^  + m) and (-^  - m) deposits, res-pectively, where m<<M. Also l e t x^(I^,I 2) and X 2 ( I ^ , I 2 ) be the shadow prices of the resource to the two firms, respectively. The f i r s t order condition (3.3c) y i e l d s (3.53a) X 1 ( I 1 , I 2 ) = d'( (3.53b) X 2 ( I 1 , I 2 ) = d ' / ^ \ D i f f e r e n t i a t i o n of (3.53) with respect to m and evaluation of the. r e s u l t i n g expressions at m=0 y i e l d s (3.54) P ^ l (I°,I°) + !il(I° fI°) " § > _ 8 I 1 3 I 2 2 ...,21° M M ^( I 1 + I 2 ) = 0, dm where 1° i s the (common) recovery of each firm when m=0. I t can e a s i l y be shown that both the p a r t i a l derivatives i n (3.54) are negative, and thus <3-53> a l ( I l + l 2 ) = 0 Q.E.D, 58 FOOTNOTES TO CHAPTER I I I L. N e e d l e s s t o s a y , such c o n s i d e r a t i o n s must be preceeded by w e l l - d e v e l o p e d t h e o r i e s o f c o a l i t i o n - f o r m a t i o n and o f p r e - e m p t i o n . 2. T h i s a b s t r a c t s from t h e p o s s i b i l i t y t h a t t h e geo-l o g i c a l i n f o r m a t i o n a c q u i r e d d u r i n g t h e e a r l y s t a g e s o f e x p l o r a t i o n w i t h i n a d e p o s i t c o u l d l ower th e c o s t o f u n c o v e r i n g a d d i t i o n a l amounts o f t h e r e s o u r c e . (See U h l e r (1978) on t h i s p o i n t ) . T h i s phenomenon i s m o d e l l e d i n a s i m p l e f a s h i o n i n C h a p t e r V. 3. A s s u m p t i o n s (A2)-(A4) a r e m o d i f i c a t i o n s o f t h o s e found i n L e w i s and Schmalensee (1980). 4. T h i s a s s u m p t i o n may n o t be e n t i r e l y u n r e a l i s t i c when e x p l o r a t i o n w i t h i n one d e p o s i t y i e l d s g e o l o g i c a l i n -f o r m a t i o n r e l e v a n t t o o t h e r d e p o s i t s . I n such a s i t u a t i o n , i n f o r m a t i o n has t h e n a t u r e o f a p u b l i c good. I t i s n o t i n c o n c e i v a b l e , t h e n , t h a t f i r m s m u t u a l l y a g r e e on u n d e r t a k i n g e x p l o r a t i o n s i m u l t a n e o u s l y w i t h i n a l l t h e i r d e p o s i t s w i t h t h e e x p l i c i t p urpose of s h a r i n g i n f o r m a t i o n t o a v o i d the f r e e - r i d e r p roblem. 5. Note t h a t t h i s i s an e q u i l i b r i u m i n e x p l o r a t i o n and o u t p u t d e c i s i o n s and n o t an e q u i l i b r i u m i n d e c i s i o n r u l e s ( i . e . , r e a c t i o n f u n c t i o n s ) . I n o t h e r words, the s t r a t e g i e s b e i n g c o n s i d e r e d h e r e a r e "open-loop" Nash s t r a t e g i e s r a t h e r t h a n " c l o s e d - l o o p " o r "feedback" Nash s t r a t e g i e s . The l a t t e r a r e i n t r a c t i b l e i n the model b e i n g c o n s i d e r e d . See Reinganum (1981) f o r a t r e a t m e n t o f c l o s e d - l o o p s t r a t e g i e s i n a s i m p l e r e x h a u s t i b l e r e s o u r c e model. 6. The a l t e r n a t i v e a s s u m p t i o n o f r a t i o n a l e x p e c t a t i o n s , i . e . , p e r f e c t f o r e s i g h t , w o u l d , o f c o u r s e , a l s o s e r v e e q u a l l y w e l l . 7. I m p l i c i t l y , i t i s b e i n g assumed t h a t t h e dynamic game b e i n g c o n s i d e r e d i s one o f " f u l l i n f o r m a t i o n " , i . e . , each p r o d u c e r has complete i n f o r m a t i o n r e g a r d i n g t h e c o s t s , number o f d e p o s i t s , e t c . o f a l l p r o d u c e r s . T h i s i s an u n r e a l i s t i c a s s u m p t i o n t o make i n g e n e r a l , b u t i t can be dropped i f t h e f u t u r e s c o n t r a c t s a r e b i n d i n g . F o r i n t h a t c a s e , i t i s n o t p r o f i t a b l e f o r any p l a y e r t o m i s r e p r e s e n t t h e t r u e s t a t e o f a f f a i r s . 8. Time d e r i v a t i v e s a r e denoted by d o t s . Of c o u r s e , t h e f o l l o w i n g e x p r e s s i o n i s v a l i d o n l y a t t h e p o i n t s o f d i f f e r e n t i a b i l i t y o f Q ( t ) and q i ( t ) . 59 C l e a r l y , i t must be t h e case t h a t Xj_>0 i f f i r m i f i n d s i t o p t i m a l t o u n d e r t a k e any p r o d u c t i o n a t a l l . I f Xj=0 f o r some j t h e n (3.3c) i m p l i e s t h a t f i r m j would n o t a c q u i r e any r e s e r v e s a t a l l and, t h e r e f o r e , would u n d e r t a k e no p r o d u c t i o n w h a t s o e v e r . The s e t o f f i r m s c o m p r i s i n g t h e i n d u s t r y can be r e d e f i n e d so as t o e x c l u d e such f i r m s . T h i s , o f c o u r s e , assumes t h a t N must be a d i v i s o r o f M. I t must be n o t e d t h a t i n o r d e r f o r t h e a r e a under the demand c u r v e t o be an a c c u r a t e measure o f consumers' s u r p l u s , t h e income e f f e c t a s s o c i a t e d w i t h a change i n t h e r e s o u r c e p r i c e must be n e g l i g i b l e . W i l l i g (1976) has shown t h a t t h e a p p r o x i m a t i o n i s an e x c e l l e n t one under a wide range o f c i r c u m s t a n c e s . A l t h o u g h r e s u l t (3.43) i s f o r m a l l y i d e n t i c a l t o p a r t (a) o f P r o p o s i t i o n 3.3, t h e former r e s u l t i s e x p l i c i t l y d e m onstrated because the c o n s t a n t - e l a s t i c i t y demand f u n c t i o n v i o l a t e s Assumptions (A2), (A3) and (A4) used i n e s t a b l i s h i n g P r o p o s i t i o n 3.3. The l i m i t o f p e r f e c t c o m p e t i t i o n can be s i m u l a t e d i n t h i s model by assuming t h a t M>>1 and M^<<M f o r a l l i . 6 0 C h a p t e r IV THE CASE OF CONTINUOUS EXPLORATION I n t h e l a s t c h a p t e r i t was assumed t h a t f i r m s complete a l l t h e i r e x p l o r a t i o n p r i o r t o b e g i n n i n g e x t r a c t i o n . As was p o i n t e d o u t , such a s t r a t e g y would be s u b o p t i m a l i n g e n e r a l b u t was i n v o k e d i n o r d e r t o s i m p l i f y t h e problem. Equipped w i t h t h e i n s i g h t t h e r e b y o b t a i n e d , t h i s a s sumption i s dropped i n t h e p r e s e n t c h a p t e r . Here f i r m s a r e a l l o w e d t o e x p l o r e and e x t r a c t as and when t h e y f i n d o p t i m a l . S i n c e t h e model b e i n g c o n s i d e r e d i s a d e t e r m i n i s t i c one, t h e r e t u r n s t o e x p l o r a t o r y e f f o r t a r e c e r t a i n . F u r t h e r , t h e e x t r a c t i o n c o s t has been assumed t o be independent o f t h e r e s e r v e base. I n such a s i t u a t i o n i t i s n o t p r o f i t a b l e f o r any f i r m t o h o l d a non-zero base o f e x p l o r e d r e s e r v e s . F o r t o do so would r e q u i r e t h e f i r m t o i n c u r e x p l o r a t i o n ex-p e n d i t u r e s t o a c q u i r e r e s e r v e s w h i c h , a t t h e moment, p r o v i d e no b e n e f i t s . As l o n g as t h e d i s c o u n t r a t e i s p o s i t i v e (which i s assumed t o be t h e c a s e ) , t h e p r e s e n t v a l u e c o s t o f ex-p l o r a t i o n i s m i n i m i z e d when e x p l o r a t i o n and e x t r a c t i o n a re un d e r t a k e n s i m u l t a n e o u s l y and i n i d e n t i c a l amounts. I n any p e r i o d , i t i s s u b o p t i m a l t o i n c u r t h e expense o f u n c o v e r i n g 61 an amount o f r e s e r v e s t h a t exceeds t h e amount t o be ex-t r a c t e d i n t h e p e r i o d . Thus i f I ^ ( t ) be t h e t o t a l amount o f t h e r e s o u r c e f i r m i has uncovered up u n t i l t i m e t and q ^ ( t ) i t s r a t e o f e x t r a c t i o n , p r e s e n t - v a l u e c o s t m i n i m i z a t i o n r e q u i r e s (4.1) i ± ( t ) = q i ( t ) Now t h e ( u n d i s c o u n t e d ) c o s t t o f i r m i , p o s s e s s i n g iVL d e p o s i t s , o f u n c o v e r i n g an amount o f r e s o u r c e e q u a l t o 1\ i s g i v e n by ( 3 . 1 ) . The r a t e a t w h i c h e x p l o r a t i o n c o s t s a r e i n c u r r e d i n p e r i o d t , o b t a i n e d by d i f f e r e n t i a t i n g (3.1) w i t h r e s p e c t t o t , i s g i v e n by f — D(I.,M.) = d'(I./M. ) i . , w h i c h , upon i n v o k i n g ( 4 . 1 ) , r educes t o (4.2) d , ( I i / M i ) q ± . T h i s i s one component o f t h e v a r i a b l e c o s t i n c u r r e d by f i r m i i n p e r i o d t . The o t h e r component i s t h e m a r g i n a l e x t r a c t i o n c o s t C. I n t h i s and subsequent c h a p t e r s i t w i l l be assumed t h a t C=0 s i n c e no g e n e r a l i t y i s l o s t by t h i s . Note t h a t i n t h i s model t h e r e a r e no f i x e d c o s t s - e i t h e r o f t h e one-shot s e t - u p c o s t t y p e o r t h e " q u a s i " f i x e d c o s t t y p e w h i c h a r e i n c u r r e d i n e v e r y p e r i o d i n w h i c h a p o s i t i v e amount o f r e s o u r c e i s e x t r a c t e d and z e r o o t h e r w i s e . The o p t i m i z a t i o n problem f a c i n g f i r m i may be w r i t t e n (4.3) r T i maximize T. ,{q,> e " r t [P(Q) " d' ( 1 ^ ) ] q ± ( t ) d t i ' i ' 0 s u b j e c t t o = q^ . I n what f o l l o w s , i t w i l l be seen t h a t t h e t ime h o r i z o n , T\ , i s n o t f i n i t e . I t i s w e l l known t h a t i n f i n i t e h o r i z o n o p t i m a l c o n t r o l problems a r e p l a g u e d by problems p e r t a i n i n g t o the e x i s t e n c e o f s o l u t i o n s . 1 I t does n o t appear p o s s i b l e t o e s t a b l i s h t h e e x i s t e n c e o f a s o l u t i o n t o an i n d i v i d u a l f i r m ' s o p t i m i z a t i o n problem as s t a t e d i n (4.3) by a p p e a l i n g t o any theorems d e v e l o p e d i n the a p p l i e d mathematics l i t e r a t u r e o f c o n t r o l t h e o r y . However, on i n t u i t i v e grounds one would ex-p e c t t h a t , g i v e n t h e o u t p u t p r o f i l e s o f a l l o t h e r f i r m s , a s o l u t i o n t o (4.3) ought t o e x i s t i n v i e w o f t h e f a c t s t h a t t h e d i s c o u n t r a t e i s p o s i t i v e , t h e c o n t r o l s e t i s compact by Assumption (A4) and t h e demand and c o s t f u n c t i o n s a r e assumed smooth. W h i l e t h e s e c o n d i t i o n s a r e n o t , i n t h e m s e l v e s , s u f f i c i e n t t o ensure e x i s t e n c e , t h e y do l e n d c r e d i b i l i t y t o t h e a s s u m p t i o n t h a t (A6) A s o l u t i o n e x i s t s t o t h e o p t i m i z a t i o n problem f a c i n g an i n d i v i d u a l p r o d u c e r , g i v e n t h e o u t -p u t p r o f i l e s o f t h e o t h e r p r o d u c e r s . As i n t h e p r e v i o u s c h a p t e r , i t w i l l be assumed t h a t each p r o d u c e r f o l l o w s an open-loop Cournot-Nash s t r a t e g y , i . e . , t h a t he chooses h i s e x t r a c t i o n p r o f i l e so a s " t o maxi-mize t h e p r e s e n t v a l u e o f h i s own p r o f i t s , t a k i n g as g i v e n the extraction p r o f i l e s of a l l other producers. Under t h i s assumption, the f i r s t order necessary conditions for problem A2 (4.3) a r e g i v e n by t h e r e s o u r c e c o n s t r a i n t and (4.4a) (4.4b) (4.4c) P(Q) - d' ( I j / M ^ - a(P) q ± ± y ± (= i f ^  > 0 ) y. - r y . = - d - d i / M . ) q ±/M. f v ± > 0 l i m e _ r t [ P ( Q ) - d ' d ^ M . ) - v ± ] q ± = 0 t->Ti - r t , . n 3 (_ i f T. =» ) (4.4d) l i m e i i i ( t ) > 0 ! t->Ti (4.4e) l i m e ~ r t y . ( t ) [M.I - I . ( t ) ] = 0, t+T. l where T i s d e f i n e d i n Assumption (A5) and i s t h e c o s t a t e v a r i a b l e c o r r e s p o n d i n g t o t h e r e s o u r c e c o n s t r a i n t i n ( 4 . 3 ) . I n what f o l l o w s , v ^  w i l l be r e f e r r e d t o as t h e s c a r c i t y r e n t o f t h e r e s o u r c e t o f i r m i . (4.4a) r e q u i r e s t h e m a r g i n a l revenue t o be e q u a l t o ( l e s s t h a n ) t h e m a r g i n a l c o s t p l u s s c a r c i t y r e n t i n any p e r i o d when p r o d u c t i o n i s p o s i t i v e ( z e r o ) . C o n d i t i o n (4.4b) can be i n t e r p r e t e d as an a s s e t market e q u i l i b r i u m c o n d i t i o n by r e -w r i t i n g i t as (4.4b) 7T + a u i / r V K7v The f i r s t term on t h e l e f t hand s i d e o f (4.4b) r e p r e s e n t s the p r o p o r t i o n a l i n c r e a s e i n t h e s c a r c i t y r e n t o f t h e m a r g i n a l u n i t w h i c h c o u l d be earned by h o l d i n g t h i s u n i t i n t h e ground f o r one more p e r i o d . The second term i s non-zero when c u r r e n t e x t r a c t i o n i n c r e a s e s f u t u r e c o s t s due t o a de-p l e t i o n e f f e c t , t h e p r e s e n c e o f w h i c h i s assumed h e r e . I n p a r t i c u l a r , t h e second term r e p r e s e n t s t h e r e t u r n s , i n terms o f d i m i n i s h e d r e c o v e r y . c o s t s i n t h e f u t u r e , o f w i t h o l d i n g t h e m a r g i n a l u n i t from c u r r e n t p r o d u c t i o n . The t o t a l r e t u r n s from h o l d i n g t h e m a r g i n a l u n i t i n t h e ground f o r one more p e r i o d i s g i v e n by t h e sum o f t h e s e two terms. (4.4b) says t h a t i n e q u i l i b r i u m t h i s must e q u a l th e r a t e o f i n t e r e s t . C o n d i t i o n (4.3c) i s a t r a n s v e r s a l i t y c o n d i t i o n t h a t r e q u i r e s t h e t e r m i n a l t i m e t o be such t h a t t h e f l o w o f bene-f i t s t o the f i r m (the H a m i l t o n i a n ) v a n i s h a t t h a t i n s t a n t . (4.3d) i s t h e t e r m i n a l s t o c k c o n d i t i o n , w h i l e (4.3e) i s t h e complementary s l a c k n e s s c o n d i t i o n . Note t h a t i n v i e w o f A s s u m p t i o n (A5), M^I r e p r e s e n t s an upper bound on t h e amount of s t o c k t h a t f i r m i can r e c o v e r . The v a r i o u s r e s u l t s e s t a b l i s h e d i n t h i s c h a p t e r a r e p r o p e r t i e s o f t h e Cournot-Nash e q u i l i b r i u m . To ensure t h a t t h e s e r e s u l t s a r e n o t vacuous, i t i s n e c e s s a r y t o e s t a b l i s h t h a t an e q u i l i b r i u m e x i s t s . B e f o r e p r o c e e d i n g t o do t h i s , however, i t i s c o n v e n i e n t t o c a s t c o n d i t i o n s ( 4 . 4 c ) -(4.4e) i n t o a l t e r n a t i v e , b u t e q u i v a l e n t , forms. I f an e q u i l i b r i u m e x i s t s , Q(t) must be c o n t i n u o u s i n t . Any d i s c o n t i n u i t y i n Q(t) would r e s u l t i n a d i s -c o n t i n u i t y i n t h e p r i c e p r o f i l e . T h i s , i n t u r n , would i n -duce p r o d u c e r s t o r e a l l o c a t e t h e i r p r o d u c t i o n i n o r d e r t o b e n e f i t from t h e jump i n p r i c e - t h e r e b y e l i m i n a t i n g t h e 65 d i s c o n t i n u i t y . From Assumption (A4) i t i s c l e a r t h a t t h e i n d u s t r y o u t p u t must always s a t i s f y Q(t) < Q < 0 0. Thus even i f t h e r e i s a jump d i s c o n t i n u i t y i n q ^ ( t ) , f o r some i , t h e jump must be f i n i t e . C o n d i t i o n (4.4b) t h e n e n s u r e s t h a t y^(t) i s always c o n t i n u o u s i n t . The c o n t i n u i t y o f Q(t) and y . ( t ) i m p l y , from ( 4 . 4 a ) , t h a t q . ( t ) must a l s o c on-1 1 t i n u o u s . Now i t w i l l be e s t a b l i s h e d i n P r o p o s i t i o n 4.8 below t h a t t h e t e r m i n a l t i m e T^ i s i n f i n i t e , f o r a l l i . I f l i m q . ( t ) > 0, t h i s would i m p l y t h a t t h e t o t a l amount o f t->-°o 1 r e s o u r c e r e c o v e r e d by f i r m i i s i n f i n i t e , v i o l a t i n g A s s u m p t i o n (A5). Thus i t must be t r u e t h a t (4.4f) l i m q . ( t ) = 0 , V i . t-*°° I t must a l s o be t h e c a s e t h a t (4.4g) l i m y . ( t ) = 0 , V i . t-*-°° I f (4.4g) were not t r u e , t h e n (4.4f) and (4.4b) would r e -q u i r e t h a t y^(t) be u l t i m a t e l y r i s i n g e x p o n e n t i a l l y . T h i s would v i o l a t e c o n d i t i o n ( 4 . 4 d ) , w h i c h must h o l d w i t h equa-l i t y s i n c e T^ i s i n f i n i t e . E v a l u a t i n g (4.4a) i n t h e l i m i t t->T^ and u s i n g (4.4f) and (4 . 4 g ) , i t f o l l o w s t h a t (4.5) l i m d ' ( I . ( t ) / M . ) = F. t->Ti 1 1 L e t I * = l i m I . ( t ) . Then s i n c e d"> 0 i t f o l l o w s t h a t I ? i s u n i q u e l y d e t e r m i n e d i n terms o f F and M^. Thus t h e f i r s t o r d e r c o n d i t i o n s (4.4c) - (4. 4e) i m p l y (4 . 4f) , (4 . 4g) and (4.4h) r T i q i ( t ) d t = IL 66 I t i s easy t o v e r i f y t h a t , c o n v e r s e l y , ( 4 . 4 f ) - ( 4 . 4 h ) i m p l y ( 4 . 4 c ) - ( 4 . 4 e ) . H a v i n g c a s t some o f t h e f i r s t o r d e r c o n d i t i o n s i n t o a more c o n v e n i e n t form, t h e b a s i c e x i s t e n c e r e s u l t may now be s t a t e d . P r o p o s i t i o n 4.1 Under Assumptions ( A 1 ) - ( A 6 ) , Cournot-Nash e q u i l i b r i a e x i s t t h a t a r e d e r i v e d as t h e s o l u t i o n s t o (4.3) and a r e c h a r a c t e r i z e d by t h e e x i s t e n c e o f c o n t i n u o u s f u n c t i o n s q ^ ( t ) , y ^ ( t ) s a t i s f y i n g t h e c o n s t r a i n t s and t h e c o n d i t i o n s ( 4 . 4 a ) , (4.4b) and ( 4 . 4 f ) - ( 4 . 4 h ) f o r i = l , 2 , . . . , N . P r o o f : See Appendix t o t h i s c h a p t e r . The p r o o f o f P r o p o s i t i o n 4.1 pro c e e d s by u s i n g a f i x e d - p o i n t argument. I t i s demo n s t r a t e d t h a t t h e f i x e d -p o i n t o f a c e r t a i n c o n t i n u o u s mapping o f t h e v e c t o r o f i n i t i a l shadow p r i c e s g e n e r a t e s p r o d u c t i o n p r o f i l e s t h a t s i m u l t a n e o u s l y s a t i s f y t h e f i r s t o r d e r c o n d i t i o n s ( 4 . 4 a ) , (4.4b) and ( 4 . 4 f ) - ( 4 . 4 h ) f o r a l l i . S i n c e , by Assumption (A6 ) , a s o l u t i o n e x i s t s t o each f i r m ' s o p t i m i z a t i o n problem f o r ( a r b i t r a r i l y ) g i v e n p r o d u c t i o n p r o f i l e s o f a l l o t h e r f i r m s , t h e e x i s t e n c e o f a Cournot-Nash e q u i l i b r i u m f o l l o w s . H a v i n g d i s p o s e d o f t h e q u e s t i o n o f t h e e x i s t e n c e o f a Cournot-Nash e q u i l i b r i u m , some o f t h e b a s i c f e a t u r e s 67 o f t h e e q u i l i b r i u m can now be d e s c r i b e d . W i t h o u t p u t t i n g more s t r u c t u r e on t h e problem, o n l y a few r e -s u l t s o f a g e n e r a l n a t u r e a r e a v a i l a b l e . However, i t i s easy t o show t h a t i n e q u i l i b r i u m t h e i n d u s t r y o u t p u t d e c l i n e s and the market p r i c e i n c r e a s e s m o n o t o n i c a l l y o v e r t i m e . D i f f e r e n t i a t i o n o f ( 4 . 4 a ) . w i t h r e s p e c t t o time (assuming f i r m i i s p r o d u c i n g a p o s i t i v e amount a t t i m e t ) y i e l d s P - a(P)q± - a ' ( P ) P q . - d" (I±/M±) q±/H± = y\, w h i c h upon u s i n g (4.4b) beomes (4.6) P - a ( P ) q i - a ' t P ) ^ = r y ± Suppose m f i r m s a r e o p e r a t i n g a t time t . A d d i t i o n o f (4.6) a c r o s s t h e m f i r m s y i e l d s (4.7) (m+1 - Qa')P = rmy(t) where y ( t ) i s t h e average s c a r c i t y r e n t o f t h e m o p e r a t i n g f i r m s . S i n c e by (3.8) m+1 - Qa'>0, i t f o l l o w s from (4.7) t h a t as l o n g as m>0. Thus t h e p r i c e must be m o n o t o n i c a l l y r i s i n g as l o n g as a t l e a s t one f i r m i s o p e r a t i n g . E q u i -v a l e n t l y , the i n d u s t r y o u t p u t must be m o n o t o n i c a l l y 68 d e c l i n i n g and p r o d u c t i o n c e a s e s once p r i c e r i s e s t o F, the choke p r i c e . I n what f o l l o w s t h e symmetric case (where p r o p e r t y r i g h t s a r e i d e n t i c a l a c r o s s a l l p r o d u c e r s ) w i l l be con-s i d e r e d i n some d e t a i l . Even i n t h i s c a s e i t i s n o t gu a r a n t e e d t h a t t h e e q u i l i b r i u m i s unique o r t h a t a l l e q u i -l i b r i a a r e n e c e s s a r i l y s y m m e t r i c , i . e . , a r e e q u i l i b r i a i n w h i c h a l l f i r m s produce i n an i d e n t i c a l f a s h i o n . However, when t h e p r o p e r t y r i g h t s a r e symmetric i t i s r e a s o n a b l e t o r e s t r i c t a t t e n t i o n t o t h e s e t o f symmetric e q u i l i b r i a . I t i s p o s s i b l e t o e s t a b l i s h t h a t t h i s r e s t r i c t e d s e t o f e q u i l i b r i a i s a s i n g l e t o n ; t h e r e i s a unique symmetric e q u i l i b r i u m . I n a d d i t i o n , i n t h i s c a s e a c e r t a i n f u n c t i o n a l i s i m p l i c i t l y maximized a t t h e ag g r e g a t e l e v e l i n the Cournot-Nash e q u i l i b r i u m . T h i s i s e s t a b l i s h e d i n t h e f o l l o w i n g p r o p o s i t i o n , t a k e n from Eswaran and Lewis (1980), and i n s p i r e d by Spence (1976) and L o u r y (1981). P r o p o s i t i o n 4.2 C o n s i d e r t h e f u n c t i o n a l Y(T,{q}) d e f i n e d by 'T rNq (4.9) Y(T,{q}) = — - r t { e 0 T [P (z) - d' (I/M)]dz) d t 0 e ~ r t { P ( N q ) - d'(I/M)}Nq d t 0 Then t h e s o l u t i o n [T*,{q*>] t o t h e a u x i l i a r y problem: (4.10) maximize Y ( T , { q ) ) . s u b j e c t t o I=NqiO 69 i s unique, and there i s a one-to-one correspondence between the solution to the a u x i l i a r y problem (4.10) and the symmetric Cournot-Nash equilibrium programme characterized by (4.4). Proof: The Hamiltonian for problem (4.10) i s given by r N q (4.11) H(t) = N-l N [P(z) - d (I/M)] dz 0 + i [P(Nq) - d'(I/M)] Nq - yNq The necessary conditions for problem (4.10) are given by (4.12a) (4.12b) (4.12c) (4.12d) (4.12e) P(Nq) - a(P)q - d' (I/M) <_\i (= i f q>0) = ry - d"(I/M) Nq/M , y^O lim e ~ r t H(t) = 0 t+T lim e r t y (t) >. 0 t+T lim e " r t u(t) [MI - I(t ) ] = 0. t+T (= i f T = °°) Suppose T i s f i n i t e . Then substitution of the left-hand-side of (4.12a) (which must hold with equality at t=T) for y into (4.12c) y i e l d s (4.13) N-l N Nq (T) P(z)dz - Nq (T) P(Nq(T)} + a (P)Nq (T) =0 J o Now i f q(T)>0 then (4.13) could never hold since a(P)>0 and Nq (T) P(z) dz > P(Nq(T) ) Nq (T) 70 Thus, i t must be true that (4.12f) lim q(t) = 0 t->T Even i f T i s i n f i n i t e (4.12f) must hold, otherwise i t would imply that an i n f i n i t e amount of the resource i s re-covered - v i o l a t i n g Assumption (A5). [It i s shown below that T i s , i n fact, i n f i n i t e ] . Further, i t must also be the case that (4.12g) lim y(t) =0 t-*T I f not, (4.12b) and (4.12f) would contradict (4.12e) which -r t requires that lim e y(t) = 0. Now evaluation of (4.12a) t+T i n the l i m i t t+T and use of (4.12f) and (4.12g) y i e l d s (4.14) F = lim d'(I(t)/M) t+T Let I* = lim I ( t ) . I t then follows that (4.12c) - (4.12e) t-»-°° hold i f and only i f (4.12f), (4.12g) and rT (4.12h) hold. q(t) dt = I*/N 0 Setting q i ( t ) = q(t) and I ^ t ) = I(t)/N the necessary conditions (4.4a), (4.4b) and (4.4f)-(4.4h), characterizing the symmetric Cournot-Nash equilibrium, are seen to be i d e n t i c a l to the f i r s t - o r d e r conditions (4.12a), (4.12b) and (4.12f)-(4.12h) characterizing the solution to problem (4.10). In view of t h i s , each Cournot-Nash 71 e q u i l i b r i u m i s a l s o a s o l u t i o n t o the a u x i l i a r y problem (4.10) i f t h e n e c e s s a r y c o n d i t i o n s a r e a l s o s u f f i c i e n t . I t i s now shown t h a t t h i s i s i n d e e d t h e c a s e by e s t a b l i s h i n g t h a t t h e s o l u t i o n t o (4.10) i s u n i q u e . Se t Nq = Q i n (4.12a), w h i c h must h o l d w i t h e q u a l i t y f o r t<T. L e t (4.15) U = P(Q) - a(P) Q/N D i f f e r e n t i a t i o n o f t h e above e x p r e s s i o n w i t h r e s p c t t o Q y i e l d s dU a(P) r_ T J , _ i , dQ = " ~~N IN+I-Qa ] , w h i c h upon u s i n g (3.8) ( w i t h m=N) e s t a b l i s h e s t h a t (4.16) § < 0. S i n c e U i s m o n o t o n i c a l l y d e c l i n i n g i n Q, (4.12a) may be i m p l i c i t l y s o l v e d f o r Q t o o b t a i n (4.17) Q = f ( I , y ) w i t h (4.18) | | < 0 and f £ < 0. Thus t h e c o n s t r a i n t f o r problem (4.10) and t h e f i r s t o r d e r c o n d i t i o n (4.12b) may be w r i t t e n as (4.19a) I = f ( I , y ) and (4.19b) y = r y - d"(I/M) f ( I , y ) / M , a system o f autonomous d i f f e r e n t i a l e q u a t i o n s i n I and u, t h e s o l u t i o n s t o w h i c h i n c l u d e t h e s o l u t i o n s t o (.4.10). Now t h e r e s t - p o i n t o f t h e d i f f e r e n t i a l e q u a t i o n s (4.19a) and (4.19b) s a t i s f y i n g (4.12f) and (4.12g) i s e a s i l y shown t o be a s a d d l e p o i n t . S i n c e t h e r e i s o n l y one p a t h i n t h e I-y phase p l a n e t h a t can l e a d t o a s a d d l e p o i n t , i t f o l l o w s t h a t (4.12a) - (4.12e) admit o f a u n i q u e s o l u t i o n . F u r t h e r , s i n c e by A s s umption (A6) a s o l u t i o n e x i s t s t o ( 4 . 1 0 ) , i t f o l l o w s t h a t t h e n e c e s s a r y c o n d i t i o n s must a l s o be s u f f i c i e n t . T h i s c o m p l e t e s t h e p r o o f . Q.E.D. Note t h a t what P r o p o s i t i o n 4.2 i s i m p l y i n g i s t h a t i n t h e Cournot-Nash e q u i l i b r i u m , a l t h o u g h i n d i v i d u a l p r o d u c e r s a r e s e e k i n g t o maximize t h e p r e s e n t v a l u e o f t h e i r own p r o f i t s , t h e y i n a d v e r t e n t l y end up a l s o m a x i m i z i n g a n o t h e r f u n c t i o n a l a t t h e a g g r e g a t e l e v e l - t h a t g i v e n by ( 4 . 9 ) . Now (4.9) i s a w e i g h t e d average o f two terms: th e f i r s t i s t h e p r e s e n t v a l u e o f t h e consumer p l u s p r o d u c e r s u r p l u s , and t h e second i s t h e p r e s e n t v a l u e o f t h e i n d u s t r y p r o f i t s . Thus a c c o r d i n g t o P r o p o s i t i o n 4.2, i n a symmetric C o u r n o t -Nash e q u i l i b r i u m t h e p r o d u c e r s behave as i f t h e y were m a x i m i z i n g a w e i g h t e d sum o f d i s c o u n t e d t o t a l s u r p l u s and d i s c o u n t e d i n d u s t r y p r o f i t s . The w e i g h t s g i v e n t o the two terms depend on t h e market s t r u c t u r e , w h i c h i n the symmetric case can be c h a r a c t e r i z e d by t h e number o f f i r m s i n t h e i n d u s t r y . I n t h e c a s e o f monopoly (N=l) a l l the i i g h t i s p l a c e d on d i s c o u n t e d p r o f i t s . T h i s i s as one we: 73 would e x p e c t s i n c e t h e m o n o p o l i s t maximizes t h e p r e s e n t v a l u e o f t h e i n d u s t r y (and hence h i s own) p r o f i t s . I n th e o t h e r extreme o f p e r f e c t c o m p e t i t i o n (N->-°°, M->-°°, M/N+l) a l l t h e w e i g h t i s p l a c e d on t h e d i s c o u n t e d t o t a l s u r p l u s and s o c i e t y ' s w e l f a r e i s i m p l i c i t l y maximized, i . e . , t h e I n v i s i b l e Hand i s o p e r a t i n g . I n what f o l l o w s t h e p r o p e r t i e s o f t h e Cournot-Nash e q u i l i b r i u m and t h e e f f e c t o f market s t r u c t u r e on i n d u s t r y performance i s i n v e s t i g a t e d . F o r t h i s purpose i t i s c o n v e n i e n t t o s e p a r a t e l y c o n s i d e r t h e ca s e s when t h e p r o -p e r t y r i g h t s a r e symmetric and asymm e t r i c , r e s p e c t i v e l y . A. THE SYMMETRIC CASE Here i t i s assumed t h a t t h e p r o p e r t y r i g h t s o f a l l the f i r m s a r e i d e n t i c a l , so t h a t M^ = M/N. I n t h i s c a s e t h e f o l l o w i n g p r o p o s i t i o n c h a r a c t e r i z e s the Cournot-Nash e q u i l i b r i u m i n some d e t a i l . P r o p o s i t i o n 4.3 When the p r o p e r t y r i g h t s a r e symmetric, t h e r e e x i s t s a unique symmetric Cournot-Nash e q u i l i b r i u m [T,{q>] s a t i s f y i n g (4.4a)-(4.4e) and i s such t h a t (a) Q ( t ) > 0 , Q(t)<0 f o r 0<t<T and i i m Q(t) = 0, t+T (b) T i s i n f i n i t e • _ l (c) l i m I (t) = Md (F) t-*T 74 Cd) l i m y ( t ) = 0. t+T P r o o f : The e x i s t e n c e o f a unique symmetric Cournot-Nash e q u i l i b r i u m was p r o v e d i n P r o p o s i t i o n 4.2. (a) That Q(t) ( = N q ( t ) ) i s c o n t i n u o u s and s t r i c t l y p o s i -t i v e f o r 0<t<T was e s t a b l i s h e d i n P r o p o s i t i o n 4.2. S e t t i n g q ^ t ) = Q ( t ) / N and I ± ( t ) = I ( t ) / N i n ( 4 . 4 a ) , w h i c h must h o l d w i t h e q u a l i t y f o r 0<_t<T, and d i f f e r e n t i a t i n g w i t h r e s p e c t t o t i m e , one o b t a i n s -a(P)Q - a(P)Q + a' (P)a(P) QQ N N - d"(I/M) Q/M = y , w h i c h upon u s i n g (4.4b) r e d u c e s t o - | (N+1 - Qa') Q = ry so t h a t ( 4' 2 0> Q = - a(N+l-Qa') S i n c e N+1 - Qa'>0 by ( 3 . 8 ) , i t f o l l o w s from (4.20) t h a t (4.21) Q(t) < 0 f o r 0^t<T. F u r t h e r , from (4.4f) i t f o l l o w s t h a t (4.22) l i m Q(t) = 0. t+T 7 5 ( b ) I n P r o p o s i t i o n 4 . 2 i t w a s e s t a b l i s h e d t h a t t h e C o u r n o t - N a s h e q u i l i b r i u m m a y b e o b t a i n e d a s t h e s o l u t i o n t o t h e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s ( 4 . 1 9 ) . N o w t h e r e s t p o i n t o f t h i s s y s t e m , s a t i s f y i n g ( 4 . 4 f ) a n d ( 4 . 4 g ) i s a s a d d l e p o i n t . S i n c e t h e t i m e t a k e n t o r e a c h a s a d d l e p o i n t i s i n f i n i t e i t f o l l o w s t h a t t h e t i m e h o r i z o n i n t h e C o u r n o t - N a s h e q u i l i b r i u m i s a l s o i n f i n i t e . ( c ) S i n c e d " > 0 , e q u a t i o n ( 4 . 5 ) m a y b e i n v e r t e d , w h i c h a f t e r s e t t i n g I ^ = I / N , y i e l d s ( 4 . 2 3 ) l i m I ( t ) = M d ' _ 1 ( F ) t + T ( d ) T h i s r e s u l t f o l l o w s i m m e d i a t e l y f r o m ( 4 . 4 g ) . Q . E . D . A c c o r d i n g t o p a r t ( a ) o f t h e a b o v e p r o p o s i t i o n , t h e i n d u s t r y o u t p u t d e c l i n e s m o n o t o n i c a l l y t o w a r d z e r o o v e r t i m e . P a r t ( b ) s t a t e s t h a t f i r m s n e v e r f i n d i t o p t i m a l t o c e a s e p r o d u c t i o n i n f i n i t e t i m e ; e x t r a c t i o n n e v e r c e a s e s . T h i s i s a s u r p r i s i n g r e s u l t . I n r e s o u r c e m o d e l s w h e r e o n l y a f i n i t e a m o u n t o f t h e r e s o u r c e i s r e c o v e r e d t h e t i m e h o r i z o n h a s b e e n k n o w n t o b e i n f i n i t e o n l y w h e n t h e c h o k e p r i c e i s i n f i n i t e . H e r e , h o w e v e r , t h e c h o k e p r i c e , b y A s s u m p t i o n ( A 2 ) , i s f i n i t e . I f t h e a m o u n t t h a t i s u l t i -m a t e l y r e c o v e r e d w e r e e x t r a c t e d a t c o n s t a n t m a r g i n a l c o s t , t h e t i m e h o r i z o n w o u l d h a v e n e c e s s a r i l y b e e n f i n i t e ( s e e f o r e x a m p l e , L e w i s a n d S c h m a l e n s e e ( 1 9 8 0 ) ) . T h e f a c t t h a t c u r r e n t e x p l o r a t i o n a n d e x t r a c t i o n i n c r e a s e s t h e c o s t o f 76 u n c o v e r i n g a d d i t i o n a l r e s e r v e s i n t h e f u t u r e causes f i r m s t o s l o w down t h e r a t e o f d e p l e t i o n . That th e l e n g t h o f t h e t i m e h o r i z o n i s i n c r e a s e d t o i n f i n i t y i s n o t m e r e l y an a r t i f a c t o f i n c r e a s i n g c o s t s ; i t h i n g e s on t h e (assumed) c o n t i n u i t y o f t h e m a r g i n a l c o s t f u n c t i o n d'(.•)• More p r e -c i s e l y , t h e t i m e h o r i z o n i s i n f i n i t e i f t h e m a r g i n a l c o s t a t t h e choke p r i c e i s c o n t i n u o u s (see F i g u r e 4 . 1 ( a ) ) . I n t h i s c a s e , t h e r e s t p o i n t o f t h e d i f f e r e n t i a l e q u a t i o n s (4.19) i s n e c e s s a r i l y a s a d d l e p o i n t and the time t a k e n t o r e a c h i t i s i n f i n i t e . I f , however, t h e m a r g i n a l c o s t f u n c t i o n were d i s c o n t i n u o u s a t t h e choke p r i c e as shown i n F i g u r e 4 . 1 ( b ) , t h e t i m e h o r i z o n would be f i n i t e . T h i s i s because t h e m a r g i n a l revenue o f t h e l a s t u n i t e x t r a c t e d (which must be e q u a l t o F) i s s t r i c t l y g r e a t e r t h a n i t s c o s t o f r e c o v e r y . F i g u r e 4.1(a) d' (I) F i g u r e 4.1(b) 77 C o n s e q u e n t l y , i t would never pay a f i r m t o postpone i t s r e c o v e r y i n t o the i n f i n i t e f u t u r e . Note t h a t t h i s a s p e c t o f p r o d u c t i o n from a r e s e r v e base t h a t e x h i b i t s i n c r e a s i n g 4 c o s t s i s i n d e p e n d e n t o f market s t r u c t u r e . P a r t (c) o f P r o p o s i t i o n 4.3 e x p r e s s e s t h e c u m u l a t i v e e x t r a c t i o n o f t h e i n d u s t r y o v e r t h e e n t i r e t i m e h o r i z o n e x p l i c i t l y i n terms o f e x o g e n o u s l y g i v e n q u a n t i t i e s . F i n a l l y p a r t (d) i m p l i e s t h a t t h e s c a r c i t y r e n t o f t h e r e s o u r c e must u l t i m a t e l y v a n i s h . The l a s t u n i t r e c o v e r e d has a m a r g i n a l c o s t t h a t e q u a l s t h e choke p r i c e and hence has no s c a r c i t y v a l u e . Note t h a t such would n o t be th e c a s e i f r e c o v e r y proceeded under c o s t c o n d i t i o n s de-p i c t e d i n F i g u r e 4 . 1 ( b ) . There, i n f a c t , t h e s c a r c i t y r e n t a t t h e t e r m i n a l t ime would be g i v e n ( i n t h e n o t a t i o n used i n t h e f i g u r e ) by (.4.24) u (T) = F - d' > 0 . The v a r i o u s e f f e c t s o f a change i n t h e market s t r u c t u r e a r e now c o n s i d e r e d . Suppose t h e M d e p o s i t s a r e e q u a l l y d i s t r i b u t e d a c r o s s a l a r g e r number o f f i r m s . T h i s would unambiguously i m p l y a d e c l i n e i n the s u p p l y - s i d e con-c e n t r a t i o n . The f o l l o w i n g p r o p o s i t i o n examines t h e e f f e c t o f t h i s on t h e r a t e o f e x t r a c t i o n , the u l t i m a t e r e s o u r c e r e c o v e r y o f t h e i n d u s t r y , t h e p r e s e n t v a l u e p r o f i t s o f t h e i n d u s t r y and s o c i e t y ' s w e l f a r e . 78 Proposition 4.4 An increase i n the number of firms i n the industry (a) does not a l t e r the stock of resource ultimately recovered by the industry (b) increases the cumulative output of the industry as of any time a f t e r production begins (c) decreases the present value of industry p r o f i t s , and (d) increases the present value of the t o t a l surplus. Proof: (a) The ultimate recovery of the industry i s given by (4.23), which i s independent of N. (b) Let Q(t,N) and I ( t , N ) , respectively, denote the industry output rate and the cumulative extraction up u n t i l time t when the industry i s comprised of N (identical) firms. Now substituting for y(t) from (4.4a) (which holds with equality for a l l t>_0) into (4.20) y i e l d s the following expression for Q as a function of Q , I and N: (A nro T MI - M T.[P(Q)-a(P)Q/N - d' ( I / M ) ] (4.25) Q(Q,I,N) - - Nr a ( P ) ( N + 1 _ Q a . ) From (4.25) i t i s e a s i l y v e r i f i e d that for I<l' and N > N " , (4.26) Q(Q,I,N) <Q(Q,I,N') <Q(Q,I',N') <05. Now from part (a) above i t follows that (4.27) lim I(t,N) = lim I(t,N ). t->-°o t"*"°° 79 Though b o t h i n d u s t r i e s u l t i m a t e l y r e c o v e r t h e same amount, t h e i r o u t p u t p r o f i l e s c a nnot be i d e n t i c a l i n v i e w o f (4.26). F u r t h e r , one o u t p u t p r o f i l e c a n not l i e e n t i r e l y above t h e o t h e r - o r e l s e , (4.27) would be v i o l a t e d . S i n c e the ex-t r a c t i o n p r o f i l e s a r e c o n t i n u o u s i n t i m e , i t f o l l o w s t h a t {Q(t,N)) and {Q(t,N')} must i n t e r s e c t a t l e a s t once. L e t one such i n t e r s e c t i o n o c c u r a t time t * . Suppose t h a t Q ( f , N ) = Q(t',N') and I ( t ' , N ) <_ I ( t ' , N ' ) . Then from (4.26) i t f o l l o w s t h a t Q(t,N) i s d e c l i n i n g f a s t e r t h a n i s Q( t , N 1 ) a t t i m e t ' so t h a t I ( t , N ' ) > Kt,N) f o r t > t ' . However, t h i s i s i m p o s s i b l e s i n c e i t c o n t r a d i c t s ( 4 . 2 7 ) . Thus, i t must be t r u e t h a t (4.28) Q ( t , N 1 ) = Q(t,N) i m p l i e s I ( t , N ' ) < I ( t , N ) . I f Q ( t ,N') < Q(t,N) and I ( t , N ' ) >_ I ( t , N ) , t h e n by c o n t i n u i t y o f the o u t p u t p r o f i l e s i t f o l l o w s t h a t t h e r e must have been a t i m e t ' p r i o r t o t when Q(t',N') = Q(t',N) and I ( t ' , N * ) > I ( t ' , N ) . But t h i s has been r u l e d o u t by (4.28). Thus, i t must be t r u e t h a t (4.29) Q(t,N') < Q(t,N) i m p l i e s I ( t , N ' ) < I ( t , N ) . N e x t , suppose t h a t a t some time t , Q(t,N') > Q(t,N) and I ( t , N ' ) >_ I ( t , N ) . Then i n v i e w o f (4.27) and t h e c o n t i n u i t y o f t h e o u t p u t p r o f i l e s , i t f o l l o w s t h a t t h e r e must be some i n s t a n t t " a f t e r t when Q(t",N") = Q(t",N) and I ( t " , N ' ) > I ( t " , N ) . B u t t h i s i s i n v i o l a t i o n o f (4.28). 80 T h e r e f o r e , i t must a l s o be t r u e t h a t (4.30) Q ( t ,N ') > Q(t,N) i m p l i e s I ( t , N " ) < I ( t , N ) From ( 4 . 2 8 ) , (4.29) and (4.30) i t f o l l o w s t h a t when N>N 1, (4.31) I ( t , N ' ) < I ( t , N ) f o r te(0,»). (c) and ( d ) . T h i s p r o o f e x p l o i t s P r o p o s i t i o n 4.2 and f o l l o w s a f t e r Eswaran and Le w i s (1980). L e t {Q(t,N_.)> c h a r a c t e r i z e t h e u n i q u e symmetric Cournot-Nash p r o f i l e when t h e r e a r e N.. f i r m s i n t h e i n d u s t r y . I n P r o p o s i t i o n 4.2 i t was e s t a b l i s h e d t h a t ( Q ( t , N j ) } s o l v e s t h e a u x i l i a r y p r o b lem < 4 " 3 2 ) " t i i . !^1VtQ))+^V,Q,) 5 * < V W ( where W_. and V\ ( a b b r e v i a t e d f o r convenience) a r e t h e p r e s e n t v a l u e s o f t o t a l s u r p l u s and i n d u s t r y p r o f i t s , r e s p e c t i v e l y , f o r an i n d u s t r y w i t h N j f i r m s . C o n s i d e r two i n d u s t r i e s , one w i t h N.^  ( i d e n t i c a l ) f i r m s and a n o t h e r w i t h N 2 ( i d e n t i c a l ) f i r m s , where N^>N2- Then" i t f o l l o w s from (4.32) t h a t (4.33a) Y(N 2,W 2,V 2) > Y f N ^ W ^ V ^ (4.33b) Y(N 1,W 1,V 1) >_ Y(N 1,W 2,V 2) I f , f o r example,(4.3 3a) were v i o l a t e d , i t would i m p l y t h a t {Q(t,N^)} p r o v i d e s a h i g h e r v a l u e f o r t h e f u n c t i o n a l i n (4.32) w i t h N j = N 2 t h a n does {Q(t,N 2)} - w h i c h c o n t r a d i c t s t h e r e s u l t o f P r o p o s i t i o n 4.2. S i m i l a r l y (4.33b) must h o l d . 8 1 N o w s u p p o s e t h a t ( 4 . 3 3 a ) h e l d w i t h e q u a l i t y , i m p l y i n g t h a t { Q C t , ^ ) } a n d ( { Q ( t , N 2 ) } b o t h s o l v e ( 4 . 3 2 ) w i t h N . . = N 2 . S i n c e P r o p o s i t i o n 4 . 2 e s t a b l i s h e d t h a t t h e s o l u t i o n i s u n i q u e a n d p a r t ( b ) a b o v e i m p l i e s t h a t { Q ^ ( t , N ^ ) } a n d { Q ( t , N 2 ) } c a n n o t b e i d e n t i c a l , t h i s i s i m p o s s i b l e . T h u s b o t h ( 4 . 3 3 a ) a n d ( 4 . 3 3 b ) m u s t h o l d w i t h s t r i c t i n e q u a l i t y . A d d i t i o n o f ( 4 . 3 3 a ) a n d ( 4 . 3 3 b ) a n d r e a r r a n g e m e n t y i e l d s ( 4 . 3 4 ) Y ( N 1 , W 1 , V 1 ) - Y ( N 2 , W 1 , V 1 ) > Y ( N 1 , W 2 , V 2 ) - Y ( N 2 , W 2 , V 2 ) , w h i c h , u p o n s u b s t i t u t i o n f o r Y f r o m ( 4 . 3 2 ) , r e d u c e s t o ( 4 . 3 5 ) (-N i " 1 N 2 ~ 1 . 1 1 _ _!_) ( W i-w 2) + ( i f ^ ( V ^ ) > 0 . N o w s u p p o s e > W 2 a n d V-^ > V 2 . T h e n i t m u s t b e t r u e t h a t N - 1 N - 1 N 2 1 N 2 1 N 2 2 N 2 2 i . e . , t h a t Y ( N 2 , W 1 , V 1 ) > Y ( N 2 , W 2 , V 2 ) , w h i c h v i o l a t e s ( 4 . 3 3 a ) . T h u s i t c a n n o t b e t h e c a s e t h a t W 1 > W 2 a n d V 1 > V 2 . S i m i l a r l y , i t i s e a s i l y s e e n t h a t o n e c a n n o t h a v e W x < W 2 a n d V 1 < V 2 - T h u s (^^^^^2) a n d ( V 1 - V " 2 ) m u s t b e o f o p p o s i t e s i g n . I f ^ - V ^ < 0 a n d V 1 ~ V 2 > 0 t h e i n e q u a l i t y i n ( 4 . 3 5 ) w o u l d b e v i o l a t e d s i n c e N 1 > N 2 " T n u s x t f o l l o v / s t h a t W 1 > W 2 a n d V 1 < V 2 -Q . E . D . 82 P a r t (a) o f t h e above p r o p o s i t i o n i s s u r p r i s i n g s i n c e i t i m p l i e s t h a t market s t r u c t u r e has no impact on u l t i m a t e r e s o u r c e r e c o v e r y . However, p a r t (b) i n d i c a t e s t h a t t h e time p r o f i l e o f e x t r a c t i o n I s a f f e c t e d by market s t r u c t u r e ; c u m u l a t i v e e x t r a c t i o n t o d a t e i s always h i g h e r i n l e s s c o n c e n t r a t e d i n d u s t r i e s . As a c o r o l l a r y , i t f o l l o w s t h a t a m o n o p o l i s t would u l t i m a t e l y r e c o v e r the same amount o f t h e r e s o u r c e as a p e r f e c t l y c o m p e t i t i v e i n d u s t r y , b u t t h e c u m u l a t i v e o u t p u t p r o f i l e o f t h e com-p e t i t i v e i n d u s t r y would dominate t h a t o f t h e m o n o p o l i s t . P a r t (a)above i s i n s h a r p c o n t r a s t w i t h t h e r e s u l t o f S t e w a r t (1979) , w h i c h c l a i m s t h a t t h e m o n o p o l i s t r e c o v e r s a s m a l l e r amount o f t h e r e s o u r c e t h a n does a c o m p e t i t i v e i n d u s t r y . As was p o i n t e d o u t i n C h a p t e r I I , S t e w a r t n e g l e c t s t o i n c o r p o r a t e any d e p l e t i o n e f f e c t w h atsoever; i n h i s model t h e r e t u r n s t o e x p l o r a t i o n a r e independent o f the c u m u l a t i v e d i s c o v e r i e s o f t h e p a s t . He, t h e r e f o r e , draws no d i s t i n c t i o n between an o r d i n a r y p r o d u c i b l e commodity and an e x h a u s t i b l e r e s o u r c e . C o n s e q u e n t l y , h i s r e s u l t m e r e l y r e i t e r a t e s t h e well-known s t a t i c r e s u l t t h a t a m o n o p o l i s t underproduces r e l a t i v e t o a c o m p e t i t i v e i n -d u s t r y . P a r t s (c) and (d) o f P r o p o s i t i o n 4.4 i m p l y t h a t w h i l e an i n c r e a s e i n t h e number o f f i r m s i n t h e i n d u s t r y d e c r e a s e s t h e p r e s e n t v a l u e o f i n d u s t r y p r o f i t s , the i n -c r e a s e i n t h e p r e s e n t v a l u e o f t h e consumers' s u r p l u s more than compensates f o r i t ; s o c i e t y ' s w e l f a r e i n c r e a s e s . An 83 i n t u i t i v e u n d e r s t a n d i n g o f t h i s r e s u l t may be o b t a i n e d by i n t e r p r e t i n g P r o p o s i t i o n 4.2 i n t h e l i g h t o f t h e Le C h e t e l i e r p r i n c i p l e : an i n c r e a s e i n N i n c r e a s e s the w e i g h t g i v e n t o t h e p r e s e n t v a l u e o f t h e t o t a l s u r p l u s and d e c r e a s e s t h a t g i v e n t o t h e p r e s e n t v a l u e o f i n d u s t r y p r o f i t s i n ( 4 . 9 ) ; t h u s i n t h e m a x i m i z a t i o n o f ( 4 . 9 ) , the p r e s e n t v a l u e o f t h e t o t a l s u r p l u s i s i n c r e a s e d a t the expense o f the p r e s e n t v a l u e o f i n d u s t r y p r o f i t s when N i n c r e a s e s . A t t e n t i o n i s now b r i e f l y t u r n e d t o t h e v a r i o u s e f f e c t s o f t h e i m p o s i t i o n o f t a x e s on t h e i n d u s t r y s i n c e no a n a l y s i s o f t h e e x p l o i t a t i o n o f e x h a u s t i b l e r e s o u r c e s can be complete w i t h o u t a t r e a t m e n t o f such a p e r v a s i v e phenomenon as t a x e s . However, i t i s n o t t h e i n t e n t i o n h e r e t o a n a l y s e t h e e f f e c t s on e x t r a c t i o n and r e s o u r c e r e c o v e r y o f a whole spectrum o f r e a l - w o r l d t a x e s ; an e x t e n s i v e t r e a t m e n t o f t h i s s u b j e c t can be f o u n d , f o r ex-ample, i n Burness (1976) , a l t h o u g h t h e a n a l y t i c framework w i t h i n w h i c h he embeds h i s t r e a t m e n t o f t a x e s i s l e s s comprehensive t h a n t h e one b e i n g c o n s i d e r e d h e r e . F o r b r e v i t y , o n l y two t a x e s w i l l be c o n s i d e r e d i n what f o l l o w s : a c o n s t a n t r o y a l t y (per u n i t ) t a x and a c o n s t a n t lump-sum ( o r f r a n c h i s e ) t a x p e r p e r i o d i n each p e r i o d d u r i n g w h i c h the o u t p u t i s n o n - z e r o . C o n s i d e r a t i o n o f t h e r o y a l t y t a x d e m o n s t r a t e s t h e e f f e c t s , i n a dynamic c o n t e x t , o f i n f l u e n c i n g p r o d u c e r s ' m a r g i n a l d e c i s i o n s . Con-s i d e r a t i o n o f a f r a n c h i s e t a x , on t h e o t h e r hand, 84 i l l u s t r a t e s a serious problem - not previously recognized i n the th e o r e t i c a l l i t e r a t u r e that can arise from the imposition of ce r t a i n taxes. The following proposition examines the comparative dynamics of an increase i n the royalty tax • Proposition 4.5 An increase i n the royalty tax (a) decreases the ultimate amount of resource recovered by the industry (b) decreases the cumulative extraction of the industry as of any time a f t e r production begins. Proof; When a royalty tax of k per unit of output i s imposed on the industry, the dynamic optimization problem facing firm i may be written: (4.36) max T . 1e" r t[P(Q) " d'(I i/M i)-k] q ± dt 0 subject to 1\ = In a symmetric Cournot-Nash equilibrium, where Q=Nq^, I=NI. and T.=T for a l l i , the f i r s t order necessary i i conditions for problem (4.36) are given by the resource constraint and (4.37a) P(Q) - a (P) Q/N - d'(I/M) - k £ \i (=ifQ>0) 85 (4.37b) y = ry - d"(I/M) Q/M ; y^O (4.37c) l i m e " r t [ P ( Q ) - d'(I/M) - k-y] Q/N = 0 (4.37d) l i m e " r t y (t) >. 0 (= 0 i f T = ») t+T (4.37e) l i m e " r t y ( t ) [ M I - I ( t ) ] = 0. t-*-T As i n P r o p o s i t i o n 4.3 i t i s e a s i l y shown t h a t T must be i n f i n i t e , t h a t (4.37a) must h o l d w i t h e q u a l i t y f o r a l l t>_0, t h a t (4.37f) l i m Q ( t ) = 0 t+T and t h a t (4.37g) l i m v (t) = 0. t-*T (a) E v a l u a t i n g (4.37a) i n the l i m i t t->-°° and u s i n g ( 4 . 3 7 f ) , (4.37g) i t f o l l o w s t h a t l i m d ' ( I ( t ) / M ) = F-k , t.-9-oo w h i c h , upon i n v e r t i n g , y i e l d s ' -1 (4.38) l i m I ( t ) = Md (F-k) S i n c e d' - 1(«) i s m o n o t o n i c a l l y i n c r e a s i n g i n i t s argument, i t f o l l o w s t h a t (4.39) & £ m > < ° 86 (b) D i f f e r e n t i a t i o n o f (4.37a) w i t h r e s p e c t t o t i m e and use o f (4.37b) y i e l d s Q as a f u n c t i o n o f Q,I and k: (4 40) 6(0 I k) = - n r [ P ( Q ) ~ a ( P ) Q / N " d ' ( I / M ) ~ k 1 ^ • 4 U J U I U # J - # K ; wr A ( P ) ( N + l - Q a 1 ) L e t I ( t , k ) and I ( t , k ' ) , r e s p e c t i v e l y , denote t h e c u m u l a t i v e e x t r a c t i o n o f t h e i n d u s t r y when t h e r o y a l t y t a x e s a r e k and k'. From (4.40) i t i s easy t o v e r i f y t h a t f o r k<k' and I I I ' , (4.41) Q(Q,I,k) < Q(Q,I,k') < Q(Q,l',k') < 0. M i m i c i n g t h e argument t h a t l e d t o (4.31) o f P r o p o s i t i o n 4.4, i t i s easy t o see t h a t f o r k<k', (4.42) I ( t , k ' ) < I ( t , k ) f o r t>0. Q.E.D. The r e s u l t o f t h e above p r o p o s i t i o n i s q u i t e i n t u i t i v e : an i n c r e a s e i n t h e r o y a l t y t a x sl o w s down t h e r a t e o f e x t r a c t i o n and, by dampening t h e i n c e n t i v e t o a c q u i r e a d d i t i o n a l r e s e r v e s , r e d u c e s t h e amount o f r e -s o u r c e u l t i m a t e l y r e c o v e r e d by t h e i n d u s t r y . T h i s l a t t e r p o i n t i s n o t b r o u g h t o u t i n models l i k e t h o s e o f Burness (1976) s i n c e t h e y i n v a r i a b l y assume t h a t t h e r e s o u r c e s t o c k i s e x o genously g i v e n . I n t h o s e s i t u a t i o n s , a r o y a l t y t a x may slow down e x t r a c t i o n d u r i n g i n i t i a l p e r i o d s b u t i n l a t e r p e r i o d s t h e r a t e o f e x t r a c t i o n must n e c e s s a r i l y be h i g h e r . When t h e r e s o u r c e s t o c k i s made endogenous, how-e v e r , t h e t a x n o t o n l y a f f e c t s t h e r a t e o f e x t r a c t i o n b u t 87 a l s o , t h r o u g h a l o w e r i n g o f t h e shadow p r i c e o f t h e r e -s o u r c e , r e d u c e s th e u l t i m a t e r e s o u r c e r e c o v e r y . The p r o p o s i t i o n t o f o l l o w examines the e f f e c t o f t h e i m p o s i t i o n o f a c o n s t a n t lump-sum (or f r a n c h i s e ) t a x i n e v e r y p e r i o d o f p o s i t i v e p r o d u c t i o n . I n s t a t i c models a t a x o f t h i s s o r t does n o t a l t e r a f i r m ' s o u t p u t r a t e ( p r o v i d e d p r o d u c t i o n i s s t i l l e c o n o m i c a l l y v i a b l e ) s i n c e i t does n o t a f f e c t t h e f i r m ' s m a r g i n a l d e c i s i o n s . As B u r n e s s (1976) p o i n t s o u t , t h e same t a x i n a dynamic con-t e x t does a f f e c t a f i r m ' s o u t p u t d e c i s i o n s s i n c e the t a x can be a v o i d e d by t e r m i n a t i n g p r o d u c t i o n sooner; what i n a s t a t i c c o n t e x t i s an u n a v o i d a b l e f i x e d c o s t becomes, i n a dynamic c o n t e x t , an a v o i d a b l e c o s t . However, B u r n e s s ' s (1976) model i s a v e r y p a r t i a l e q u i l i b r i u m one where the p r i c e o f t h e r e s o u r c e i s assumed c o n s t a n t o v e r t i m e . As a r e s u l t , B u r n e s s f a i l s t o r e a l i z e t h a t i n t h e p r e s e n c e o f a f r a n c h i s e t a x a c o m p e t i t i v e e q u i l i b r i u m does not e x i s t ! T h i s s t a r t l i n g r e s u l t i s a s p e c i a l c a s e o f t h e r e s u l t e s t a b l i s h e d i n t h e f o l l o w i n g p r o p o s i t i o n . P r o p o s i t i o n 4.6 When a c o n s t a n t f r a n c h i s e t a x i s imposed on a l l f i r m s i n t h e i n d u s t r y , a Cournot-Nash e q u i l i b r i u m f a i l s t o e x i s t . P r o o f : The p r o o f p r o c e e d s by d e m o n s t r a t i n g t h a t i t i s 88 i m p o s s i b l e f o r p r o d u c t i o n p r o f i l e s s a t i s f y i n g t h e n e c e s s a r y c o n d i t i o n s t o be i n d i v i d u a l l y p r o f i t m a x i -m i z i n g . When a lump-sum t a x o f K per p e r i o d i s imposed on each f i r m o f t h e i n d u s t r y , t h e o p t i m i z a t i o n problem o f f i r m i may be w r i t t e n : 1 e " r t { [P(Q)-d' ( I i / M i ) ] q± - K) d t rT (4.43) max T.,{q.> J 0 s u b j e c t t o 1^ = q^. I n a symmetric Cournot-Nash e q u i l i b r i u m t h e f i r s t o r d e r n e c e s s a r y c o n d i t i o n s a r e g i v e n ( i n u s u a l n o t a t i o n ) by t h e r e s o u r c e c o n s t r a i n t i n (4.43) and (4.44a) P(Q) - a(P)Q/N - d'(I/M) ^ y (= i f Q>0) (4.44b) y = ry - d"(I/M) Q/M ; y^O (4.44c) l i m e " r t { [ P ( Q ) - d'(I/M) - y ] Q ( t ) / N - K> = 0 t+T l i m e ~ r t y (t) >. 0 (= 0 i f T = ») t+T (4.44d) (4.44e) l i m e ~ r t y ( t ) [MI - I (t) ] = 0. t+T As i n P r o p o s i t i o n 4.3 i t i s easy t o show t h a t i n e q u i l i -b r i u m - i f one e x i s t s - Q(t) must be m o n o t o n i c a l l y de-c l i n i n g o v e r t h e i n t e r v a l 0<_t<T. Now when K>0, T cannot be i n f i n i t e . I f T i s i n f i n i t e , A s s u mption (A5) would r e -q u i r e t h a t Q ( t ) / N < K/F f o r a l l t > T, where T i s f i n i t e . S i n c e [P(Q) - d 1 ( I / M ) ] i s bounded above by F i t f o l l o w s t h a t [P(Q) - d'(I/M)]Q/N - K < 0 f o r t > T ., 89 i . e . , each f i r m ' s p r o f i t s a r e n e g a t i v e f o r a l l t>T. T h i s c l e a r l y cannot o c c u r i n e q u i l i b r i u m s i n c e each f i r m has t h e o p t i o n o f t e r m i n a t i n g p r o d u c t i o n i n f i n i t e t i m e . Thus T must be f i n i t e . Now s u b s t i t u t i o n f o r u(T) from (4.44a) i n t o (4.44c) y i e l d s a(P) Q 2 ( T ) / N 2 = K, w h i c h r e q u i r e s t h a t Q(T)>0, o r t h a t (4.45) P(Q(T))<F . Now (4.45) i m p l i e s t h a t as soon as p r o d u c t i o n c eases a t t i m e T, t h e p r i c e must jump t o the choke p r i c e , F. However, a d i s c o n t i n u i t y i n t h e p r i c e p r o f i l e i s i n c o n s i s t e n t w i t h t h e c o n c e p t o f an i n t e r t e m p o r a l e q u i l i b r i u m where each p r o d u c e r i s m a x i m i z i n g h i s ( p r e s e n t v a l u e ) p r o f i t s . F o r , such a d i s c o n t i n u i t y i m p l i e s t h e e x i s t e n c e o f an o p p o r t u n i t y f o r each p r o d u c e r t o i n c r e a s e h i s p r o f i t s by h o l d i n g back p r o d u c t i o n u n t i l a f t e r T. S i n c e each p r o d u c e r t a k e s as g i v e n t h e d e c i s i o n s o f t h e o t h e r s , i t f o l l o w s t h a t i t i s i n h i s i n t e r e s t t o postpone some p r o d u c t i o n u n t i l a f t e r T. However, s i n c e a l l o t h e r p r o d u c e r s would behave l i k e w i s e , t h i s i m p l i e s t h a t no Cournot-Nash e q u i l i b r i u m e x i s t s f o r N>1. Q.E.D. 90 The above r e s u l t i s q u i t e s t a r t l i n g . As a c o r o l l a r y t o t h i s p r o p o s i t i o n , one has t h e r e s u l t t h a t a c o m p e t i t i v e e q u i l i b r i u m does n o t e x i s t i n t h e p r e s e n c e o f a f r a n c h i s e t a x . The e s s e n t i a l r e a s o n f o r t h i s n o n e x i s t e n c e r e s u l t i s t h e n o n - c o n v e x i t y i n t r o d u c e d by t h e lump-sum t a x . I t can be v i ewed as a " q u a s i - f i x e d " c o s t - one t h a t i s i n c u r r e d o n l y i n p e r i o d s o f non-zero p r o d u c t i o n . T h i s n o n - c o n v e x i t y , w h i c h i s h a r m l e s s i n s t a t i c models, i s v e r y s e r i o u s i n t h e dynamic model b e i n g c o n s i d e r e d , s i n c e i t s p r e s e n c e r e q u i r e s the e x t r a c t i o n r a t e i n t h e l a s t p e r i o d t o be n o n - z e r o . T h i s , i n t u r n , r e s u l t s i n a d i s c o n t i n u o u s p r i c e p r o f i l e - w h i c h i s i n c o n s i s t e n t w i t h t h e c o n c e p t o f an i n t e r t e m p o r a l e q u i l i -b r i u m . Burness (1976) f a i l s t o r e a l i z e t h i s s e r i o u s i m p l i c a t i o n o f t h e p r e s e n c e o f a f r a n c h i s e t a x s i n c e he assumes t h e r e s o u r c e p r i c e t o be c o n s t a n t o v e r t i m e , an a s s u m p t i o n w h i c h , even a t t h e a g g r e g a t e l e v e l ? i s n o t c o n -s i s t e n t w i t h t h e e x h a u s t i b l e n a t u r e o f t h e r e s o u r c e s t o c k . B e f o r e p r o c e e d i n g t o c o n s i d e r the case when the p r o p e r t y r i g h t s a r e asymmetric t h e f o l l o w i n g p r o p o s i t i o n , w h i c h examines t h e c o m p a r a t i v e dynamics o f t h e a g g r e g a t e ex-t r a c t i o n p r o f i l e w i t h r e s p e c t t o t h e d i s c o u n t r a t e , i s c o n s i d e r e d . P r o p o s i t i o n 4.7 An u n i f o r m i n c r e a s e i n t h e r a t e o f d i s c o u n t of a l l f i r m s (a) does not a l t e r t h e u l t i m a t e r e s o u r c e r e c o v e r y o f t h e i n d u s t r y 91 (b) i n c r e a s e s t h e c u m u l a t i v e e x t r a c t i o n o f t h e i n d u s t r y as o f any d a t e a f t e r p r o d u c t i o n b e g i n s . P r o o f : L e t r and r ' be two d i s c o u n t r a t e s , w i t h r > r ' , and l e t I ( t , r ) , I ( t , r ' ) be t h e c o r r e s p o n d i n g c u m u l a t i v e ex-t r a c t i o n s o f t h e i n d u s t r y as o f time t . (a) The u l t i m a t e r e s o u r c e r e c o v e r y o f t h e i n d u s t r y i s g i v e n by ( 4 . 2 3 ) , w h i c h i s independent o f t h e d i s c o u n t r a t e . Thus (4.46) l i m I ( t , r ) = l i m I ( t , r ' ) . (b) E x p r e s s i o n (4.25) f o r Q may be r e - w r i t t e n as a f u n c t i o n o f Q,I and r as (4 47) Q(Q I r ) = - N r [ P ( Q ) - a ( P ) Q / N - d ' (I/M)] ( q ' q / ) giy,±,r; wr a ( P ) ( N + l - Q a 1 ) From ( 4 . 4 7 ) , i t i s r e a d i l y v e r i f i e d t h a t f o r r > r ' and I < I ' , (4.48) Q(Q,I,r) < Q(Q,I,r') < Q(Q,I',r') < 0. Comparing ( 4 . 2 6 ) , (4.27) w i t h (4.40) and (4 . 4 6 ) , r e s p e c t i v e l y , t h e analogue o f (4.31) may a t once be w r i t t e n down as (4.49) I ( t , r ' ) < I ( t , r ) f o r t e ( 0 , ~ ) . Q.E.D. The above p r o p o s i t i o n v e r i f i e s f o r o l i g o p o l i s t i c m arkets t h e f a m i l i a r r e s u l t t h a t an i n c r e a s e i n the 92 d i s c o u n t r a t e h a s t e n s p r o d u c t i o n i n t h e sense t h a t t h e c u m u l a t i v e o u t p u t t o d a t e i s i n c r e a s e d . The r e a s o n f o r t h i s , o f c o u r s e , i s t h a t p r o d u c e r s now d i s c o u n t f u t u r e 9 p r o f i t s more h e a v i l y and so e x t r a c t f a s t e r . B. THE ASYMMETRIC CASE I t i s now supposed t h a t t h e M d e p o s i t s a r e d i s t r i -b u t e d a c r o s s t h e N p r o d u c e r s i n an asymmetric f a s h i o n . The f o l l o w i n g p r o p o s i t i o n e s t a b l i s h e s some f e a t u r e s o f t h e r e s u l t i n g Cournot-Nash e q u i l i b r i u m . P r o p o s i t i o n 4.8 Suppose N>1 and r e s o u r c e o w n e rship s h a r e s a r e d i s t r i b u t e d u n e v e n l y so t h a t M^>M^ f o r f i r m s i and j . Then an e q u i l i b r i u m s a t i s f y i n g (4.4a) - (4.4e) e x i s t s and i s such t h a t (a) Q (t) > 0, Q(t) < 0 f o r 0 <_ t < T and v a n i s h e s a t t i m e T, (b) A l l f i r m s s t a r t p r o d u c i n g a t t i m e t = 0. (c) = T = 0 0 f o r a l l i . (d) I . ( t ) > I . ( t ) f o r a l l t > 0. 1 j (e) l i m I . (t) = M. d ' ^ f F ) . t-*T. 1 1 l P r o o f : The e x i s t e n c e o f a Cournot-Nash e q u i l i b r i u m i s e s t a b l i s h e d i n P r o p o s i t i o n 4.1. (a) The p r o o f o f t h i s i s s i m i l i a r t o t h a t o f p a r t (a) o f P r o p o s i t i o n 4.3 and hence i s d e l e t e d . (b) I f m f i r m s a r e p r o d u c i n g a t t i m e t , t h e r a t e o f p r i c e i n c r e a s e i s g i v e n by (4.8) and (4.4a) as P ( t ) = rm [ p ( t ) " a ( P ) Q -°d'(f), , F ( t ; r m 1 (m+1 - Qa') J where d ^ t ) i s t h e average o f t h e m a r g i n a l c o s t s o f t h e m p r o d u c i n g f i r m s . R e c a l l i n g t h a t e = P/(Qa), t h e above e x p r e s s i o n may be r e w r i t t e n it*.\ - r [ P ( m e - l ) - d!roe] ~ e(m+l-Qa') From t h i s e x p r e s s i o n i t f o l l o w s t h a t (4.50) P ( t ) < r [ P ( t ) - d' ( t ) ] i f and o n l y i f P ( t ) (me-1) - d'(t) me> [P ( t ) - d ' ( t ) ] [m+l-Qa'] e i . e . , i f and o n l y i f [P(t) - d' (t) ] (e+1) > [P(t)-d'(,t) ] Qa'e - d' ( t ) , w h i c h i s t r u e i n v i e w o f ( 3 . 6 ) . I t f o l l o w s from (4.50) t h a t P ( t ) < r [ P ( t ) - d' C 0 )1 , s i n c e ST'Ct) >. d' (0) . S i n c e t h e m a r g i n a l p r o f i t a t z e r o o u t p u t ( i . e . , n e t p r i c e ) i s always r i s i n g a t a r a t e s l o w e r t h a n t h e r a t e o f i n t e r e s t , a l l f i r m s w i l l f i n d i t o p t i m a l t o b e g i n p r o d u c t i o n a t t i m e t=0. 94 (c) S i n c e f i r m i t a k e s as g i v e n {q.. (t) , j f ^ i ) , (4.4a) can be i m p l i c i t l y s o l v e d t o o b t a i n f i r m i ' s o u t p u t , q ^ ( l \ , u ^ , t ) . Then t h e e q u a t i o n s (4.4b) and 1\ = q^ can be c o l l a p s e d i n t o t h e s i n g l e non-autonomous d i f f e r e n t i a l e q u a t i o n (4.51) y ± = <f> ( v ^ t ) , t±< 0 Now from p a r t ( a ) , i t f o l l o w s t h a t t h e p r i c e must r i s e c o n t i n u o u s l y u n t i l a t t h e i n d u s t r y ' s shut-down t i m e , T, P(T) = F. Suppose T^<T f o r some f i r m i . Then from (4.4a) and (4.5) i t f o l l o w s t h a t y i ( T i ) = P ( Q ( T ± ) ) - a ( P ) q i ( T i ) - d ' (I ± (T ± ) / M ± ) < F - d ' ( I i ( T ) / M i ) = 0, i m p l y i n g t h a t y^<0, w h i c h i s n o t p o s s i b l e . C o n s e q u e n t l y , T.=T f o r a l l i . l Now f o r y^ s u f f i c i e n t l y c l o s e t o z e r o , (4.51) may be T a y l o r expanded and a p p r o x i m a t e d by (4.52) y\ = * (0 , t ) + y i<() 1(0,t) . S i n c e q i ( t ) i s c o n t i n u o u s i n t f o r a l l i (by P r o p o s i t i o n 4.1), i t i s easy t o see, u s i n g (4.4a) and (4.4b), t h a t <t>1(0,t) i s bounded f o r a l l t , so t h a t 0£<x^ - $ ( 0 , t ) <. 3 Thus T must exceed t h e t i m e t a k e n by t h e system d e s c r i b e d by (4.53) y ± = <j> ( 0 , t ) - B p ± 95 t o r e a c h y^ = 0. S i n c e a l l f i r m s c ease p r o d u c t i o n ( c o n t i n u o u s l y ) a t ti m e T, i t f o l l o w s t h a t <)>(0,t) -* 0 as t-*T. Thus, (4.53) may be r e p l a c e d by (4.54) y i = " 6 y i as t-*T. S i n c e (4.54) t a k e s an i n f i n i t e amount o f time t o r e a c h y^=0, so does t h e system ( 4 . 5 1 ) . (d) Suppose t h a t a t some t i m e t ' , I ^ t ' ) = I j ( t ' ) and q i ( t ' ) = q j ( t ' ) . Then from (4.4a) i t f o l l o w s t h a t , s i n c e M^>Mj , (4.55) v ^ t ' ) = P ( Q ) ~ (*LML) - a ( P ) q i ( f ) > P(Q) - d' ( I j / M j ) - a(P) q j ( t ' ) = U j ( t ' ) D i f f e r e n t i a t i o n o f (4.4a) and use o f (4.4b) f o r f i r m s i and j t h e n y i e l d s , upon i n v o k i n g ( 4 . 5 5 ) , P - r y . - aq. P - ry . - aq. * i ( t ' > = ir h < 1 1 = V^' w h i c h i m p l i e s t h a t q . ( t ) < q . ( t ) and I . ( t ) < I . ( t ) f o r t > t ' , M ] l 3 v i o l a t i n g p a r t (e) o f t h i s p r o p o s i t i o n (see b e l o w ) . Thus i t must be t r u e t h a t (4.56) I i ( t ) = I j ( t ) i m p l i e s qAt) ? q^ ( t ) . Next suppose t h a t a t some t i m e t ' , I ^ ( t ' ) = I j ( t ' ) and 9 6 q^(t') < q Ct ' ) . Then i n view of part (e) below and the continuity of the output p r o f i l e s of a l l firms, i t follows that there must e x i s t a time t">t* at which I i ( t " ) < I^(t") and q.(t") = q . ( t " ) . As above, t h i s implies that y.(t") > y j ( t " ) and q ^ t " ) < q^(t") so that I ^ t ) < I., (t) for a l l t>t" - v i o l a t i n g part (e) below. Thus i t follows that (4.57) I i ( t ) = I j ( t ) implies q ± ( t ) > q^ (t) . Since at t = 0, 3^(0) = 1^(0) = 0, i t follows that (4.58) I i ( t ) > I j ( t ) for a l l t > 0. (e) This r e s u l t follows d i r e c t l y from (4.5), inversion of which y i e l d s (4.59) lim I.(t) = M. d' - 1(F) Q.E.D. Parts (a) through (c) imply that a l l firms begin production at time t=0 and industry output declines mono-t o n i c a l l y to zero over the i n f i n i t e horizon. According to parts (d) and (e) larger firms dominate the industry i n that t h e i r cumulative extraction at any time t>0 always exceeds those of smaller firms. Addition of (4.59) across a l l firms y i e l d s N (4.60) lim I I.(t) =Md'"-L(F). t+~ i = l 1 Thus the ultimate resource recovery by the industry i s independent of the d i s t r i b u t i o n of property rights across f i r m s . Thus market s t r u c t u r e has no i m p a c t on t h e t o t a l r e s o u r c e r e c o v e r y o f t h e i n d u s t r y . I t i s i n t e r e s t i n g t o e n q u i r e i f t h e r e a r e f o r c e s o p e r a t i n g i n t h e i n d u s t r y w h i c h ensure t h a t t h e a g g r e g a t e o u t p u t p r o f i l e i n e q u i l i b r i u m i s p r o v i d e d a t the l e a s t p r e s e n t - v a l u e c o s t . I t i s c l e a r t h a t t h i s would be a n e c e s s a r y , b u t n o t s u f f i c i e n t , c o n d i t i o n f o r P a r e t o o p t i -m a l i t y . To examine t h i s q u e s t i o n i t i s c o n v e n i e n t t o make the f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n : The o u t p u t p r o f i l e o f an i n d u s t r y i s s a i d t o be e f f i c i e n t i f i t i s p r o v i d e d a t l e a s t p r e s e n t v a l u e c o s t . I n a c o m p e t i t i v e e q u i l i b r i u m - w h i c h i s known t o be s o c i a l l y o p t i m a l i n the absence o f e x t e r n a l i t i e s o r non-c o n v e x i t i e s i n the t e c h n o l o g y - p r o d u c t i o n o c c u r s so as t o equate the u s e r c o s t s ( i . e . , m a r g i n a l e x t r a c t i o n c o s t p l u s s c a r c i t y r e n t ) a c r o s s a l l d e p o s i t s o p e r a t i n g a t t h e same i n s t a n t . ( T h i s can be seen by d r o p p i n g t h e term - a ( P ) q ^ i n (4.4a) i n o r d e r t o s i m u l a t e t h e c o m p e t i t i v e l i m i t ) . However, Solow (1974) and Hanson (1978) have shown t h a t t h e c o m p e t i t i v e e q u i l i b r i u m i s e f f i c i e n t i n the sense de-f i n e d above. Now i t i s i n t u i t i v e l y c l e a r t h a t e f f i c i e n c y would r e q u i r e t h a t m a r g i n a l r e c o v e r y c o s t s be equated a c r o s s a l l d e p o s i t s o p e r a t i n g a t t h e same i n s t a n t . T h i s s u g g e s t s t h a t i f e f f i c i e n c y be t h e o n l y r e q u i r e m e n t p l a c e d on t h e o u t p u t p r o f i l e o f t h e i n d u s t r y , t h e n the r u l e o f 98 e q u a t i n g t h e u s e r c o s t a c r o s s a l l o p e r a t i n g d e p o s i t s s h o u l d i m p l y t h e r u l e o f e q u a t i n g t h e m a r g i n a l c o s t o f a l l o p e r a t i n g p o o l s . T h i s i s v e r i f i e d i n the f o l l o w i n g p r o p o s i t i o n . P r o p o s i t i o n 4.9 (a) E f f i c i e n t p r o d u c t i o n r e q u i r e s t h a t m a r g i n a l e x t r a c t i o n c o s t s be equated a c r o s s a l l d e p o s i t s a t a l l p o i n t s o f t i m e . (b) The p r o d u c t i o n p r o f i l e t h a t maximizes the p r e s e n t v a l u e o f t h e t o t a l s u r p l u s i s e f f i c i e n t . P r o o f ; F o r c o n v e n i e n c e i t w i l l be assumed t h a t t h e r e a r e o n l y two d e p o s i t s (M=2). (a) L e t ( Q ( t ) } be an e x o g e n o u s l y g i v e n o u t p u t p r o f i l e t o be p r o v i d e d by t h e i n d u s t r y over the i n f i n i t e h o r i z o n . F o r c o n v e n i e n c e i t w i l l be assumed t h a t t h e c u m u l a t i v e e x t r a c t i o n r e q u i r e d t o p r o v i d e the g i v e n stream o f r e s o u r c e o u t p u t i s f i n i t e . The minimum p r e s e n t v a l u e c o s t o f p r o v i d i n g t h e g i v e n s t r e a m o f o u t p u t i s o b t a i n e d by s o l v i n g (4.61) mm { q U { q 2 > j r e " r t [ d ' ( I 1 ( t ) ) q 1 ( t ) + d ' ( I 2 ( t ) ) q 2 ( t ) ] d t 0 s u b j e c t t o I 1 = q^O f o r i = l , 2 ; and q X ( t ) + q 2 (t) = Q(t) , where q 1 i s t h e r a t e o f e x t r a c t i o n and I 1 t h e c u m u l a t i v e e x t r a c t i o n from d e p o s i t i , i = l , 2 . N o t i n g t h a t 99 d ' ( I i ( t ) ) q 1 ( t ) = d d ( I x ( t ) ) , i = 1,2, d t and i n t e g r a t i n g t h e e x p r e s s i o n i n (4.61) by p a r t s , i t i s easy t o see t h a t t h e e f f i c i e n t p r o f i l e s o l v e s e " r t [ d ( I 1 ( t ) ) + d ( I 2 ( t ) ) ] d t (4.62a) min { q 1 } , ^ 2 } 0 (4.62b) s u b j e c t t o I 1 ( t ) + I 2 ( t ) = Q ( t ) , I 1 , I 2 i 0. 1 2 Suppose ( I ( t ) } and { I ( t ) } a r e c u m u l a t i v e e x t r a c t i o n p r o f i l e s w h i c h a r e n o t i d e n t i c a l b u t w h i c h s a t i s f y the c o n s t r a i n t (4.62b). L e t { T ^ t ) } and { I 2 ( t ) } be two i d e n t i c a l c u m u l a t i v e e x t r a c t i o n p r o f i l e s d e f i n e d by (4.63) I ^ t ) = I 2 ( t ) = [ I X ( t ) + I 2 ( t ) ] / 2 V t ^ 0 —1 —2 C l e a r l y { I ( t ) } and {I ( t ) } a r e f e a s i b l e s i n c e , by con-s t r u c t i o n t h e y s a t i s f y t h e c o n s t r a i n t (4.62b). S i n c e d(-) i s a s t r i c t l y convex f u n c t i o n , i t f o l l o w s t h a t (4.64) d ( I 1 ( t ) ) + d ( I 2 ( t ) ) ^ d ( I 1 ( t ) ) + d ( I 2 ( t ) ) , t>0 1 2 Now by a s s u m p t i o n / t h e p r o f i l e s { I ( t ) } and {I ( t ) } a r e not i d e n t i c a l . Thus (4.64) must h o l d w i t h s t r i c t i n e q u a l i t y f o r a non-zero i n t e r v a l o f t i m e , i m p l y i n g t h a t { I 1 ( t ) } and { I ( t ) } a r e e x t r a c t i o n p r o f i l e s t h a t a r e f e a s i b l e b u t p r o v i d e the g i v e n o u t p u t stream a t a l o w e r p r e s e n t v a l u e c o s t t h a n do { I 1 ( t ) } and { I 2 ( . t ) } . I t f o l l o w s t h a t t h e s o l u t i o n t o (4.62) o r , e q u i v a l e n t l y , t o (4.61) must be such as t o equate the m a r g i n a l e x t r a c t i o n c o s t s a c r o s s the two d e p o s i t s 100 at a l l points i n time, (b) Present value maximization of the t o t a l surplus re-quires the deposits to be exploited so as to solve (4.65) max 0 ql(t)+q 2 (t) P(x)dx -[d' ( I 1 ( t ) )q1(t) + d' ( I 2 (t))q 2 (t) ] 1 dt subject to I 1 = q^O , i=l,2. The Hamiltonian for t h i s optimization problem i s (4.66) H(t) = ,q 1 + q 2 P(x)dx - [d' (I 1)+u 1]q 1-[d' ( I 2 ) + U 2 ] q 2 , 0 1 2 where y and y are the costate variables (or s c a r c i t y rents) corresponding to the two deposits. The Maximum P r i n c i p l e immediately implies that for optimality i t must be the case that, for i=l,2, (4.67) P(q 1+q 2) ± d'(I 1) + y 1 (= i f q > 0) I t follows from (4.67) that i f both deposits are operating at any instant then th e i r marginal user costs must be equal to one another and to the price of the resource. Now the dynamic optimization problem i n (4.65) may be rewritten as ,Q(.t) P(x)dxJ dt - C(T,{Q}) , (4.68) max (Q(t)),T - r t e [ 0 where 101 (4.69) C(T,{Q}) = min s u b j e c t t o q V r t fcl' (I i)qidt 0 1 = 1 1 ( t ) + q 2 ( t ) = QCt) liX^qX>.0f i = l , 2 . From (4.68) i t i s c l e a r t h a t f o r any { Q ( t ) } , p r e s e n t v a l u e m a x i m i z a t i o n o f t h e t o t a l s u r p l u s r e q u i r e s t h a t t h e p r e s e n t v a l u e c o s t o f p r o v i d i n g t h a t stream o f o u t p u t be m i n i m i z e d . I n p a r t i c u l a r , t h i s i s t r u e a l s o o f t h e o p t i m a l o u t p u t p r o f i l e . Q.E.D. P a r t (a) o f t h e above p r o p o s i t i o n e s t a b l i s h e s t h a t a n e c e s s a r y c o n d i t i o n f o r e f f i c i e n c y i s t h a t a l l d e p o s i t s i n o p e r a t i o n a t any i n s t a n t be o f t h e same m a r g i n a l c o s t . S i n c e a l l d e p o s i t s a r e assumed t o have i d e n t i c a l c o s t s t r u c t u r e s , i t f o l l o w s t h a t e f f i c i e n t p r o d u c t i o n r e q u i r e s them t o be e x p l o i t e d i d e n t i c a l l y . P a r t (b) shows t h a t t h e w e l f a r e m a x i m i z i n g o u t p u t p r o f i l e ( w h i c h i s a l s o t h e com-p e t i t i v e p r o f i l e ) r e q u i r e s e f f i c i e n t p r o d u c t i o n . I n s o f a r as o n l y e f f i c i e n c y i s c o n c e r n e d , t h e r u l e t h a t d e t e r m i n e s w h i c h d e p o s i t s s h o u l d o p e r a t e a t any g i v e n i n s t a n t by e q u a t i n g t h e m a r g i n a l u s e r c o s t s a c r o s s a l l such d e p o s i t s i m p l i e s t h e r u l e t h a t a l l o w s t h o s e p o o l s t o o p e r a t e w h i c h have i d e n t i c a l (and l o w e s t ) m a r g i n a l c o s t a t t h a t i n s t a n t . From (4.5) i t f o l l o w s t h a t even i f t h e p r o p e r t y r i g h t s o f t h e N p r o d u c e r s over t h e M d e p o s i t s a r e asymmetric, the m a r g i n a l c o s t o f r e c o v e r i n g t h e l a s t u n i t i s t h e same f o r a l l t h e p r o d u c e r s s i n c e T\ = 0 0 f o r a l l i . I t i s i n t e r -e s t i n g t o i n q u i r e i f t h i s i s t r u e o f t h e Cournot-Nash 102 (asymmetric) e q u i l i b r i u m a t a l l p o i n t s i n t i m e ; i f i t i s , p r o d u c t i o n would be e f f i c i e n t . The f o l l o w i n g p r o p o s i t i o n e s t a b l i s h e s t h a t t h i s i s n o t t h e c a s e . P r o p o s i t i o n 4.10 I n an asymmetric Cournot-Nash e q u i l i b r i u m , t h e o u t -p u t p r o f i l e o f t h e i n d u s t r y i s i n e f f i c i e n t . P r o o f : The p r o o f p r o c e e d s by c o n t r a d i c t i o n . Suppose t h a t (4.70) d' ( I i / M i ) = d ' d j / M j ) V i , j , Vt>.0 S i n c e d'(-) i s a monotone ( i n c r e a s i n g ) f u n c t i o n (4.70) i m p l i e s t h a t f o r a l l i , j and t>.0 (4.71) I i / M ± = I j / M j ' q - j / M ^ q j / M j , ^/M^q./M.. D i f f e r e n t i a t i o n o f (4.4a) w i t h r e s p e c t t o time and use o f (4.71) y i e l d s (4.72) v± ~ V j = " ( a q i + a q ± ) (1 - M../*^ ) Now from (4.4b) and (4.7 0) i t f o l l o w s t h a t p . - y j = r ( U i - W j ) , which/upon u s i n g (4.4a) and (4.71), can be r e w r i t t e n as (4.73) y . - p . = _ r a q . (l-M./M.) 1 i i 3 l 103 Comparison o f (4.72) and (4.73) y i e l d s a q i + a q ± = xaq± Summing t h e above e x p r e s s i o n o v e r a l l i y i e l d s (4.74) ^ (aQ) = raQ , whi c h i m p l i e s t h a t aQ r i s e s e x p o n e n t i a l l y w i t h t i m e . But t h i s i s i m p o s s i b l e , s i n c e by p a r t (a) o f P r o p o s i t i o n 4.8 l i m Q (t) = 0. 1 0 Thus (4.70) c a n n o t be t r u e and t h e p r o -t-»-°o p o s i t i o n i s p r o v e d . Q.E.D. Symmetric Cournot-Nash e q u i l i b r i a a r e c l e a r l y e f f i c i e n t , i r r e s p e c t i v e o f t h e number o f p r o d u c e r s com-p r i s i n g t h e i n d u s t r y ; i n such e q u i l i b r i a t he m a r g i n a l e x t r a c t i o n c o s t s a r e t h e same f o r a l l the d e p o s i t s . P r o -p o s i t i o n 4.10 e s t a b l i s h e s t h a t t h i s p r o p e r t y i s n o t s h a r e d by asymmetric e q u i l i b r i a : s o c i e t y can be made b e t t e r o f f by p r o v i d i n g t h e e q u i l i b r i u m o u t p u t p r o f i l e a t a lower p r e s e n t v a l u e c o s t . I t may be n o t e d i n p a s s i n g t h a t a l t h o u g h symmetric e q u i l i b r i a a r e e f f i c i e n t i n the sense d e f i n e d above, by no means a r e t h e y n e c e s s a r i l y P a r e t o e f f i c i e n t . The r e a s o n , o f c o u r s e , i s t h a t w h i l e t h e m a r g i n a l u s e r c o s t s a r e equated a c r o s s a l l f i r m s , t hey a r e e q u a l t o t h e m a r g i n a l revenue t o each f i r m and n o t t o t h e p r i c e o f the r e s o u r c e . O n l y i n p e r f e c t c o m p e t i t i o n i s P a r e t o e f f i c i e n c y g u a r a n t e e d . 104 F o o t n o t e s t o C h a p t e r j y 1. See, f o r example, H a l k i n (1974) on t h i s p o i n t . 2. See, f o r example, Takayama (1974), Theorem 8.C.4 on page 658. 3. I t i s w e l l - known t h a t t h e t r a n s v e r s a l i t y c o n d i t i o n (4.4d) need n o t h o l d w i t h e q u a l i t y when t h e t e r m i n a l t i m e i s i n f i n i t e (see H a l k i n ( 1 9 7 4 ) ) . However, S e i e r s t a d (1977) has shown t h a t t h i s c o n d i t i o n i s n e c e s s a r y i f more s t r u c t u r e i s imposed on t h e c o n t r o l problem. The problem under c o n s i d e r a t i o n h e r e s a t i s f i e s t h e a d d i t i o n a l c o n d i t i o n s t h a t S e i e r s t a d has shown t o be s u f f i c i e n t f o r (4.4d) t o h o l d . 4. A d e t a i l e d t r e a t m e n t o f t h e c o n d i t i o n s d e t e r m i n i n g whether th e t i m e h o r i z o n i s f i n i t e o r i n f i n i t e i s c o n t a i n e d i n S a l a n t , Eswaran and L e w i s (1980). 5. T h a t Q ( Q , I , N ' ) < Q ( Q , I ' , N ' ) f o r I < l ' i s immediate from (4.25). H o l d i n g Q and I c o n s t a n t , d i f f e r e n t i a t i o n o f (4.25) w i t h r e s p e c t t o N y i e l d s 3Q(Q,I,N) _ r.[P(Q) - d' ( I / M ) ] rN [P (Q) -a (P) Q/N-d' (I/M) ] 3N a(N+l-Qa') a ( N + l - Q a ' ) 2 so t h a t . j - i Q(Q,I,N) < Q(Q,I,N') f o r N>N . 6. F o r an e x p o s i t i o n o f t h i s p r i n c i p l e , see E i c h h o r n and O e t t l i (1972). 7 8 An e x h a u s t i v e t r e a t m e n t of t h e s u b j e c t - i n v a r i a b l y w i t h i n t h e framework o f p e r f e c t c o m p e t i t i o n - may be found i n t h e volume e d i t e d by G a f f n e y (1967) . More r e c e n t l y , U h l e r (1978) has c o n s i d e r e d the e f f e c t o f r o y a l t y and p r o f i t s t a x e s on t h e r a t e s o f e x p l o r a t i o n and e x t r a c t i o n . A s i m i l i a r r e s u l t , e s t a b l i s h i n g t h e n o n e x i s t e n c e o f a c o m p e t i t i v e e q u i l i b r i u m i n t h e p r e s e n c e o f i n c r e a s i n g r e t u r n s t o s c a l e i n i t i a l l y , may be found i n Eswaran, L e w i s and Heaps (1981). Note t h a t t h i s r e s u l t assumes t h a t t h e e x t r a c t i o n c o s t s a r e i ndependent o f t h e r a t e o f i n t e r e s t . I f . s u c h i s not t h e c a s e , t h e r e s u l t can be r e v e r s e d (see Neher ( 1 9 7 8 ) ) . 105 10. T h i s can be seen as f o l l o w s . F i r s t n o t e t h a t l i m aQ = P/e e x i s t s . As Q i s d e c r e a s e d t o z e r o , P Q+0 approaches th e f i n i t e c o n s t a n t F w h i l e e mono-t o n i c a l l y i n c r e a s e s and i s t h e r e f o r e s t r i c t l y p o s i t i v e i n t h e l i m i t . L e t l i m aQ=A. Now t h e Q+0 m a r g i n a l and t o t a l revenue are g i v e n , r e s p e c t i v e l y , by MR = P - aQ and TR (Q) = P(Q) Q . S i n c e T R ( Q ) = 0 MR(q) dq i t f o l l o w s t h a t aq dq = 0 0 Q P(q) dq - P(Q)Q T a k i n g t h e l i m i t Q-*-0 i t f o l l o w s t h a t Q A = l i m aQ = l i m Q->0 Q->0 P(q) dq - P(Q)> = l i m FQ - F) Q+0 = 0, 106 Appendix t o C h a p t e r IV T h i s Appendix p r o v i d e s the p r o o f o f P r o p o s i t i o n 4.1. The f i r s t o r d e r n e c e s s a r y c o n d i t i o n s f o r f i r m i ' s o p t i m i z a t i o n problem (4.3) a r e g i v e n by ( 4 . 4 ) . By A s s umption (A 6 ) , a s o l u t i o n e x i s t s t o ( 4 . 3 ) , g i v e n the o u t p u t p r o f i l e s {q^ (t) , j j ^ i } . To e s t a b l i s h t h e e x i s t e n c e o f a Cournot-Nash e q u i l i b r i u m , i t w i l l be d e m o n s t r a t e d t h a t t h e r e e x i s t [ { y ^ } , {q^)3 t h a t s i m u l t a n e o u s l y s a t i s f y ( 4 . 4 a ) , ( 4 . 4 b ) , (4.4f) - (4.4h) f o r a l l i . T L e t y = ( y ^ , . . . , y ) and M be t h e s e t o f a l l v e c t o r s y f o r w h i c h 0<_y^ <_ F-d'(O), i = l , . . . N . By A s s u m p t i o n (A2), M i s a non-empty, convex and compact s e t , T A l s o , l e t I = ( I ^ , . . . , I N ) and L be t h e s e t o f v e c t o r s I f o r w h ich I.>0, i = l , . . . , N . l — F i r s t , f o r a r b i t r a r y yeM and I e L a p r i c e P ( t ) i s o b t a i n e d t h a t s a t i s f i e s (4.4a) f o r a l l i . D e f i n e t h e f o l l o w i n g f u n c t i o n t h a t i s c o n t i n u o u s i n P,I and y: (4A..1) Z i ( P , I , y ) = max {0 ,P-d' {I^/VL^) -\i ±} , w h i c h i s n o n - n e g a t i v e i n g e n e r a l and s t r i c t l y p o s i t i v e i n e q u i l i b r i u m i f and o n l y i f f i r m i i s p r o d u c i n g . Now d e f i n e the f u n c t i o n N (4A.2) Y ( P , I , y ) = I Z ( P , l , y ) - a(P) Q ( P ) , i = l 1 w h i c h i s c o n t i n u o u s i n i t s arguments and v a n i s h e s i n e q u i l i b r i u m . 107 L e t m be the number o f f i r m s f o r w h i c h (4A.1) i s p o s i t i v e f o r g i v e n y ( t ) , I ( t ) and P ( t ) . I f P=F, t h e n s i n c e Q(F) = 0 i t f o l l o w s t h a t Y(F,I,y)>.0 from (4A.2). Assumption (A4) i m p l i e s t h a t 3 P such t h a t P [1-1/Ne (P) )] = d'(0). I f m=0 a t t h i s p r i c e , Y=-a(P)Q(P)<0. I f m>0, th e n w r i t i n g -a(P)Q(P) as - P / e ( P ) , i t f o l l o w s from (4A.2) t h a t (4A.3) Y ( P , I , y ) = m [ P ( l - ) - d ' - y ] , where d' and y a r e t h e averages o f d^ and y^ f o r f i r m s w i t h p o s i t i v e Z^. Now p ( 1 - n T w r ] i p ^ - NTTV-) = D,<°>^'-Thus from (4A. 3) i t now f o l l o w s t h a t Y (P, I , y) <_ 0.. S i n c e Y i s c o n t i n u o u s i n i t s arguments, i t f o l l o w s t h a t t h e r e e x i s t s a p r i c e , P e ( I , u ) e [ P , F ] , such t h a t (4A.4) Y ( P e , I , y ) = 0. F u r t h e r , w i t h a p r o c e d u r e i d e n t i c a l t o t h a t used by Lewis and Schmalensee (1980) i t i s p o s s i b l e t o show t h a t P ( I , y ) i s unique and c o n t i n u o u s i n i t s arguments. Now choose an a r b i t r a r y i n i t i a l v e c t o r y°eM and i t e r a t i v e l y compute (4A.5) P(t,y°) = P e [ I ( t , y ° ) , y (t,y°) ] , (4A.6) q i(t,y°) = Z i[P(t,y°) , I(t,y°) ,y (t,y°) ]/a'(P) , C4A.7) I i(t,y°) = t - o q i ( s , y ) ds , 108 and (4A.8) y\ (t,y°)=e r tmax{0,y?-' ' 0 e " r s q i ( s ) d " ( I i ( s ) / M i ) ds}. Since P (I,y) i s continuous i n i t s arguments so are the variables on the l e f t hand sides of (4A.5)-(4A.8). By con-struction, (4A.5)-(4A.8) s a t i s f y (4.4a) and (4.4b) for ar b i t r a r y u° E M . I t remains to est a b l i s h that there exists a p°eM which generates production p r o f i l e s that also s a t i s f y (4.4f)-(4.4h). Now define D i(y°) = max {li(co,y°) _ j* 0} E i(y°) = max {I*- i i(» fp°), 0} , where i t i s such that d' (I*±/HL) = F . . • Consider a mapping, y, whose components are given by (4A.9) YL(V°) = y j + [F-d 1 (0)-y?] [1 -exp,(- D i(y 0) ) ] - y ? [ l - exp(-E i(y°))] , i=l,...N This i s a continuous mapping that maps the compact, convex set M into i t s e l f . By Brouwer's fixed point theorem, i t follows that the mapping has at le a s t one fixed point y*°, such that (4A.10) Y i(y*°) = y*° for a l l i . I t w i l l now be demonstrated that these fixed points generate production p r o f i l e s , v i a (4A.5) - (4A.8), that are Cournot-Nash e q u i l i b r i a . Suppose D^(y )>0, which implies 109 that y*° = F-d'(O). But by (4A.1), (4A.6) and (4A.8) t h i s would require that q^(t,y °) = 0 for a l l t>_0, since P ee[P,F]. This, i n turn, would imply thatl^(°°,y °) = 0<I£, *o contradicting the supposition that (y )>0. Thus, at a fixed point of (4A.9) i t must be true that Di=0, V i . Next, suppose that E ^ ( y )>0, which implies that y ^ =0. From (4A.8) i t follows that y \ ( t , y * ° ) = 0 for a l l t>0. Then (4A.1) and (4A.6) imply that production w i l l proceed u n t i l d 1 (I^/M^) = F, so that i^(°°,y*°) = i t , a contradiction. Thus, at a fixed point of (4A.9) i t must be true that V i , D. = E. = 0. This implies that (4.4h) holds and, f] (4A.1) and (4A.6), that (4.4f) also holds when y=y Further, i t must also be true that lim y . ( t , y * ° ) = 0, t+T£ 1 ^ Q i . e . , (4.4g) must hold. If not, (4A.1) implies that q^(t,y ) would f a l l to zero (and remain zero) when d* (1^(T^,y°)/M^)<F, * o * so that I^(T^,y )<I^ ~ contradicting the r e s u l t that E^(y*°) = 0. Thus the existence of a Cournot-Nash equi-librium i s proved. In the special case where property ri g h t s are symmetric, i t i s easy to see that there exists a fixed point, y * ° , of (4A.9) with i d e n t i c a l components, implying that there e x i s t s a symmetric Cournot-Nash equilibrium. :rom *o 110 C h a p t e r V T H E I N C O R P O R A T I O N O F I N I T I A L I N V E S T M E N T I n t h e p r e v i o u s c h a p t e r i t was assumed t h a t t h e o n l y c o s t s i n c u r r e d by each f i r m were o f t h e v a r i a b l e c o s t v a r i e t y . T h i s , however, i s u n r e a l i s t i c . I t i s u s u a l l y t h e case t h a t p r i o r t o b e g i n n i n g e x t r a c t i o n , f i r m s need t o i n c u r some lump-sum expense o r u p - f r o n t c o s t . T h i s would be s o , f o r example, i f each f i r m u n d e r t a k e s some i n i t i a l i n v e s t m e n t i n i n f o r m a t i o n r e g a r d i n g t h e d i s t r i b u t i o n o f r e s e r v e s w i t h i n each d e p o s i t . A more p r e c i s e knowledge o f t h i s d i s t r i b u t i o n would f a c i l i t a t e a more j u d i c i o u s a l l o c a t i o n o f e x t r a c t i o n o v e r t h e d e p o s i t 5 and would c o n s e q u e n t l y lower t h e v a r i a b l e c o s t s o f r e s o u r c e r e c o v e r y . A l t e r n a t i v e l y , i n t h e c a s e o f o i l r e s e r v o i r s , f o r example, t h i s i n i t i a l i n v e s t m e n t might t a k e t h e form o f i n s t a l l a t i o n o f equipment w h i c h augments th e n a t u r a l p r e s s u r e t h a t b r i n g s t h e o i l t o t h e s u r f a c e . 1 Whatever t h e s p e c i f i c n a t u r e o f t h e i n i t i a l i n v e s t m e n t a c t i v i t y , t h e purpose o f such i n v e s t m e n t i s t o reduce t h e subsequent v a r i a b l e c o s t o f r e c o v e r i n g t h e r e s o u r c e . There i s t h u s a t r a d e o f f betwen t h e i n c r e a s e i n p r e s e n t v a l u e c o s t due t o any i n i t i a l i n v e s t m e n t u n d e r t a k e n and t h e I l l d e c r e a s e i t b r i n g s about i n t h e p r e s e n t v a l u e o f t h e v a r i a b l e c o s t s . The o p t i m a l amount o f i n i t i a l i n v e s t m e n t t o u n d e r t a k e i s endogenous, i n g e n e r a l , and i s t h e s u b j e c t o f i n v e s t i g a t i o n i n t h i s c h a p t e r . L e t X denote an a g g r e g a t e i n p u t v a r i a b l e t h a t measures t h e e x t e n t o f i n v e s t m e n t u n d e r t a k e n on any d e p o s i t . X m i g h t be, f o r example, t h e number o f e x p l o r a t o r y d r i l l i n g s u n d e r t a k e n on t h e d e p o s i t . I t i s c o n c e i v a b l e t h a t , i n g e n e r a l , X i s a f u n c t i o n o f t i m e , t h a t i s , t h a t r e c o v e r y e n h a ncing a c t i v i t y o c c u r s n ot j u s t a t t=0 b u t a l s o o ver t i m e . However, such a g e n e r a l i z a t i o n r e n d e r s t h e problem i n t r a c t a b l e and so i t i s assumed he r e t h a t X i s a c o n t r o l 2 parameter, t h e c h o i c e o f w h i c h i s made a t t i m e t=0. The c o s t o f p u r c h a s i n g i n p u t X i s assumed t o be g i v e n by a f u n c t i o n g(X) w i t h t h e f o l l o w i n g p r o p e r t i e s : (A7) g(0) = 0, g' (•) '> 0, g"( •) > 0. The s t r i c t c o n v e x i t y o f g(-) i m p l i e s t h a t the m a r g i n a l i n v e s t m e n t c o s t i s i n c r e a s i n g i n X. S i n c e a l l i n v e s t m e n t i s c o m p leted i n an i n s t a n t ( a t t = 0 ) , t h i s can be r a t i o n -a l i z e d i n terms o f a d j u s t m e n t c o s t s . I f g(-) were non-convex, i t would s e r i o u s l y c o m p l i c a t e t h e a n a l y s i s t o f o l l o w . F o r , i n t h a t c a s e , a p r o d u c e r w i t h s e v e r a l d e p o s i t s i s n o t l i k e l y t o t r e a t them s y m m e t r i c a l l y . I t w i l l a l s o be assumed t h a t (A8) The i n i t i a l i n v e s t m e n t t h a t any f i r m f i n d s p r o f i t a b l e t o u n d e r t a k e on a d e p o s i t l i e s i n 112 t h e non-empty, convex and compact s e t [ 0 , X ] , where X i s s t r i c t l y p o s i t i v e . T h i s a s s u m p t i o n i s e n t i r e l y r e a s o n a b l e , e s p e c i a l l y i n v i e w o f ( A 7 ) ; i t r u l e s o u t t h e tro u b l e s o m e (and u n r e a l i s t i c ) p o s s i b i l i t y o f f i r m s u n d e r t a k i n g i n f i n i t e amounts o f i n i t i a l i n v e s t m e n t . S i n c e t h e e x t e n t o f i n i t i a l i n v e s t m e n t u n d e r t a k e n on a d e p o s i t a l t e r s t h e v a r i a b l e c o s t o f r e c o v e r i n g t h e r e s o u r c e , A s s u m p t i o n ( A l ) must now be m o d i f i e d t o t a k e t h i s i n t o a c c o u n t . L e t d ( I , X ) denote t h e t o t a l ( u n d i s -counted) c o s t o f u n c o v e r i n g a s t o c k , I , o f the r e s o u r c e from a d e p o s i t on w h i c h X i s t h e amount o f i n i t i a l i n -vestment u n d e r t a k e n . A ssumption ( A l ) i s now r e p l a c e d by d( I , X ) i s s t r i c t l y convex i n I and X ( j o i n t l y ) , (Al ) * w i t h d 1>0 , d 1 ; L>0 , d i ; L 1 ^ 0 , d 2<o d 1 2<o d 2 2 >0 , d(0,X) = 0 f o r a l l X>0. I n ( A l ) ' , d 1>0 i m p l i e s t h a t t h e m a r g i n a l c o s t o f r e s o u r c e r e c o v e r y i s p o s i t i v e and d^>0 i m p l i e s t h a t a d e p l e t i o n e f f e c t i s i n o p e r a t i o n . The ass u m p t i o n t h a t d^-^C 1 s t a t e s t h a t t h e m a r g i n a l d i s c o v e r y c o s t i n c r e a s e s a t a non-i n c r e a s i n g r a t e . I n the n e x t c h a p t e r i t w i l l be shown t h a t t h i s i s e q u i v a l e n t t o the more i n t u i t i v e a s s u m p t i o n t h a t w i t h i n each d e p o s i t , t h e amount o f t h e r e s o u r c e a v a i l a b l e 113 at a given l e v e l of marginal cost i s an increasing function of the marginal cost. The assumptions d2<0 and d 1 2 < ^ ) r e f l e c t the trade-off between higher i n i t i a l investment and lower cost of recovery: the higher the i n i t i a l investment the lower are the t o t a l cost of recovering a stock I and the marginal cost of recovery at any stock l e v e l . F i n a l l y , the inequality d 2 2>0 implies diminishing returns to additional i n i t i a l investment i n the sense that an additional unit of i n i t i a l investment lowers the t o t a l recovery cost at a slower and slower rate as X increases. From (Al)' i t follows that an increase i n X lowers the en t i r e marginal cost schedule and therefore increases the t o t a l amount of the resource that can be economically recovered. Assumptions (A2) and (A5) also need to be modified. (A2) i s replaced by: (A2)' The demand function has a f i n i t e choke pr i c e F, with F>d 1(0,0). Assumption (A5) i s modified to: (A5)' Within each deposit, there exists a f i n i t e stock l e v e l I such that d., (I,X) > F. 114 T h i s a s s u m p t i o n ensures t h a t the amount o f r e s o u r c e t h a t i s e c o n o m i c a l l y r e c o v e r a b l e from each d e p o s i t i s bounded. B e f o r e p r o c e e d i n g t o f o r m u l a t e an i n d i v i d u a l p r o d u c e r ' s o p t i m i z a t i o n p roblem, one f u r t h e r s i m p l i f i -c a t i o n needs t o be i n t r o d u c e d . When f i r m s u n d e r t a k e i n i t i a l i n v e s t m e n t , i t i s c o n c e i v a b l e t h a t a f i r m w i t h s e v e r a l d e p o s i t s may s t a g g e r t h e development o f t h e s e d e p o s i t s , t h a t i s , t h a t i t might c a l l f o r t h o n l y a few o f i t s d e p o s i t s i n t o p r o d u c t i o n a t t=0 and i n c u r the i n i t i a l i n v e s t m e n t on the r e s t o f t h e d e p o s i t s a t a l a t e r s t a g e . T h i s s t r a t e g y w i l l be r u l e d o u t i n the a n a l y s i s t o f o l l o w . I t w i l l be assumed t h a t each f i r m f i n d s i t o p t i m a l t o c a l l f o r t h a l l t h e d e p o s i t s i n t o p r o d u c t i o n a t th e same time and t h a t t h e i n i t i a l i n v e s t m e n t on a l l de-p o s i t s i s u n d e r t a k e n a t t=0. Though t h i s a s s u m p t i o n i s no t i n n o c u o u s , i t i s n e v e r t h e l e s s n o t u n r e a s o n a b l e . W h i l e t h e s t a g g e r i n g o f t h e development o f d e p o s i t s might lower t h e p r e s e n t v a l u e o f t h e i n v e s t m e n t c o s t s , i t would i n -c r e a s e t h e p r e s e n t v a l u e o f t h e v a r i a b l e c o s t s o f p r o -v i d i n g a g i v e n o u t p u t p r o f i l e because t h e p r o d u c e r must a l l o c a t e h i s o u t p u t a c r o s s a fewer number o f d e p o s i t s t o b e g i n w i t h . The o p t i m i z a t i o n problem f a c i n g an i n d i v i d u a l f i r m can now be w r i t t e n down. L e t denote t h e s e t o f d e p o s i t s owned by f i r m i , and q?, I ? t h e o u t p u t r a t e and c u m u l a t i v e o u t p u t , r e s p e c t i v e l y , from d e p o s i t j e lSL . S i m i l a r l y , l e t 115 (5.1) max I { X i , { q i ) r T i J e N i J 0 X? denote t h e i n i t i a l i n v e s t m e n t u n d e r t a k e n on d e p o s i t jelSk. F i r m i seeks t o s o l v e T i e ~ r t [ P ( Q ) - d j ^ C l ^ x j j l q ^ t ) d t - gcxj)} s u b j e c t t o I ? = q? > 0. The p r o p o s i t i o n below e s t a b l i s h e s t h a t under t h e assumptions made, i t i s o p t i m a l f o r a p r o d u c e r t o t r e a t i k a l l h i s d e p o s i t s s y m m e t r i c a l l y , so t h a t XV = X^ and q ? ( t ) = q k ( t ) f o r a l l j , k e N i and f o r a l l t>0. The p r o o f h i n g e s on t h e c o n v e x i t y o f t h e c o s t f u n c t i o n d ( I , X ) . S i n c e t h e r e s o u r c e i s homogeneous a c r o s s a l l d e p o s i t s , i t i s i r r e l e v a n t from t h e p o i n t o f v i e w o f revenue w h i c h o f t h e d e p o s i t s a u n i t o f o u t p u t comes from. Whatever t h e o u t p u t p r o f i l e p r o d u c e r i f i n d s o p t i m a l t o pro d u c e , t h i s stream o f o u t p u t he w i l l s u p p l y a t t h e l e a s t p r e s e n t v a l u e c o s t . The c o n v e x i t y o f the c o s t f u n c t i o n d(-,') t h e n r e -q u i r e s t h a t t h e p r o d u c e r r u n a l l h i s d e p o s i t s i n an i d e n t i c a l f a s h i o n . T h i s i s f o r m a l i z e d below. P r o p o s i t i o n 5.1 Each p r o d u c e r w i l l f i n d i t o p t i m a l t o e x p l o i t a l l h i s d e p o s i t s s y m m e t r i c a l l y i n the s o l u t i o n t o ( 5 . 1 ) . P r o o f : F o r s i m p l i c i t y , suppose p r o d u c e r i owns o n l y two d e p o s i t s . L e t { q \ ( t ) } be t h e o u t p u t s t r e a m t h a t p r o d u c e r 116 i f i n d s o p t i m a l t o produce o v e r , say, t h e i n f i n i t e time h o r i z o n . A n e c e s s a r y c o n d i t i o n f o r s o l v i n g (5.1) i s c l e a r l y t h a t [ g \ ( t ) } s h o u l d be produced a t l e a s t p r e s e n t v a l u e c o s t . C o n s i d e r t h e o p t i m i z a t i o n p r oblem [ d ^ I ^ X 1 ) q\(t\ +. d ^ I ^ X ^ q ^ t ) ] d t + g{x}) + g ( x 2 ) } s u b j e c t t o ql + q? = q ^ U ) , q ?>_0, j = l , 2 , Vt>_0 N o t i n g t h a t i ? = q?, j = 1,2, and i n t e g r a t i n g by p a r t s , t h e 3 above problem can be t r a n s f o r m e d t o t h e e q u i v a l e n t one: (5.3) ^ min { r f e ~ r t [ d ( I 1 , x } ) + d ( I 2 , X 2 ) ] d t (5.2) r-.mirw { { q ^ , xi-. s u b j e c t t o i f + I 2 = q ± ( t ) , i h j i , 3=1,2. Now suppose t h a t t h e d o u b l e t s [ x f , { q f } ] and 2 2 [X7,{q7}] a r e not i d e n t i c a l . C o n s i d e r an a l t e r n a t i v e v e c t o r l ^ i o f i n i t i a l i n v e s t m e n t s , X , and o u t p u t p r o f i l e s ( q ^ l d e f i n e d by (5.4a) X^ 1 = X^ 2 = ( X 1 + X 2 ) / 2 (5.4b) q [ 1 ( t ) = q ^ 2 ( t ) = [q\(t) + q 2 ( t ) ] / 2 Vt>0-C l e a r l y , i f {q^} s a t i s f i e s t h e c o n s t r a i n t s i n ( 5 . 3 ) , o r e q u i v a l e n t l y i n ( 5 . 2 ) , so does iq\}. Thus {q^} i s a f e a s i b l e v e c t o r o f o u t p u t s from t h e two d e p o s i t s . 117 By t h e c o n v e x i t y o f g(•) i t f o l l o w s t h a t (5.5) g C x ! 1 ) + g ( x ! 2 ) <. g(.xj) + g ( x 2 ) Now i n v i e w o f ( 5 . 4 b ) , i t must be t r u e t h a t (5.4c) l [ 1 ( t ) = l'±2{t) = [ l j ( t ) + I 2 ( t ) ] / 2 Vt>0 From ( 5 . 4 a ) , (5.4c) and the s t r i c t ( j o i n t ) c o n v e x i t y o f d(-,') i t f o l l o w s t h a t f o r a l l t>0, (5.6) dd!1 !^1) + d ( i ^ 2 , x ^ 2 ) < d( i] ; ,x];) + d ( i 2 , x 2 ) . From (5.5) and (5.6) i t i s c l e a r t h a t X i , { q p p r o v i d e s t h e o u t p u t p r o f i l e {q~p a t l o w e r p r e s e n t v a l u e c o s t t h a n does x \ ' { q \ } . Thus f o r o p t i m a l i t y , p r o d u c e r i must e x p l o i t h i s d e p o s i t s i d e n t i c a l l y . Q.E.D. W h i l e i t i s t r u e t h a t each p r o d u c e r w i l l f i n d i t o p t i m a l t o t r e a t a l l h i s d e p o s i t s i d e n t i c a l l y i t i s by no means t h e case t h a t d i f f e r e n t p r o d u c e r s w i l l i n c u r t h e same i n i t i a l i n v e s t m e n t s and f o l l o w i d e n t i c a l e x t r a c t i o n p r o f i l e s . That t h i s i s so w i l l be e x p l i c i t l y d emonstrated l a t e r i n t h i s c h a p t e r when t h e case o f asymmetric p r o p e r t y r i g h t s i s c o n s i d e r e d . I n v i e w o f t h e r e s u l t o f P r o p o s i t i o n 5.1 t h e o p t i m i z a t i o n problem f a c i n g p r o d u c e r i may be r e w r i t t e n 1 e " r t [ P ( Q ) - d 1 ( I i / M i , X i ) ] q i ( t ) d t 0 - M ± g(X.) s u b j e c t t o 1^ = q^ >i 0, r T . (5.7) max X i ^ i ' T i 118 where q. » £ q j , I , = .J i j and X? - X. f o r a l l j e N . . 1 . _T X 1 » »T X X X X As i n the p r e v i o u s c h a p t e r , t h e t i m e h o r i z o n w i l l be seen t o be i n f i n i t e f o r a l l f i r m s and t h e e x i s t e n c e o f a s o l u t i o n t o p r o blem (5.7) o f f i r m i , g i v e n t h e output p r o f i l e s o f a l l o t h e r f i r m s appears i m p o s s i b l e t o e s t a b l i s h . Now, i n v i e w o f a s s u m p t i o n ( A 5 ) ' , t h e t o t a l amount o f r e s o u r c e t h a t f i r m i w i l l f i n d p r o f i t a b l e t o e x t r a c t i s f i n i t e . T h i s and t h e f a c t s t h a t t h e d i s c o u n t r a t e i s s t r i c t l y p o s i t i v e and the c o n t r o l s e t i s compact p r o v i d e some i n -t u i t i v e r a t i o n a l e f o r t h e a s s u m p t i o n (A6) 1 A s o l u t i o n e x i s t s t o t h e o p t i m i z a t i o n problem (5.7) f a c i n g an i n d i v i d u a l p r o d u c e r , g i v e n t h e o u t p u t p r o f i l e s o f a l l o t h e r p r o d u c e r s . The n e c e s s a r y c o n d i t i o n s f o r a Nash e q u i l i b r i u m 4 a r e g i v e n by t h e r e s o u r c e c o n s t r a i n t i n (5.7) and (5.8a) (5.8b) (5.8c) (5.8d) (5.8e) (5.8f) P(Q) - d 1 ( I i / M i , X i ) - a ( P ) q i < y ± (= i f q ±>0) v± - r y ± = - d 1 1 ( I i / M i , X i ) qL/VLL ; v ^ O l i m e ~ r t [ P ( Q ) - d 1 ( I i / M i , X i ) - y ± ] qj_ = 0 t+T . l l i m e ~ r t y i ( t ) > 0 t+T^ l i m e " r t y i ( t ) [m± - 1 ^ = 0 t-^ -T . l (= i f T i= ») i - r t d 1 2 ( l . / M i f X . ) q.dt - M-gMX.) ± 0 (= i f X i> 0) , f o r i = 1,2, ....N 119 C o n d i t i o n s (5.8a)-(5.8e) a r e , b y now, f a m i l i a r and r e q u i r e no comment. C o n d i t i o n (_5.8f), however, i s new. Now d^2^±/^i/X^) r e p r e s e n t s t h e d e c r e a s e i n t h e m a r g i n a l c o s t o f r e c o v e r y b r o u g h t about by an a d d i t i o n a l u n i t o f i n i t i a l i n v e s t m e n t . C o n d i t i o n ( 5 . 8 f ) , t h e r e f o r e , s t a t e s t h a t i n v e s t m e n t must be u n d e r t a k e n on each d e p o s i t owned by t h e p r o d u c e r u n t i l t h e m a r g i n a l i n v e s t m e n t c o s t e q u a l s the d e c r e a s e i n p r e s e n t v a l u e v a r i a b l e c o s t ( a l o n g the o p t i m a l p r o f i l e ) t h a t t h i s m a r g i n a l u n i t o f i n v e s t m e n t b r i n g s about. I f t h e former always exceeds t h e l a t t e r , t h e n i t i s o p t i m a l n o t t o u n d e r t a k e any i n i t i a l i n v e s t m e n t a t a l l . F o l l o w i n g t h e r e a s o n i n g used i n t h e p r e v i o u s c h a p t e r , i t i s easy t o see t h a t i n a Cournot-Nash e q u i l i -b r i u m ( i f one e x i s t s ) , the o u t p u t p r o f i l e ( Q ( t ) } o f t h e i n d u s t r y must be c o n t i n u o u s and m o n o t o n i c a l l y d e c l i n i n g . F u r t h e r , e x a c t l y as i n t h e p r e v i o u s c h a p t e r , the n e c e s s a r y c o n d i t i o n s (5.8c)-(5.8e) can be shown t o be e q u i v a l e n t , f o r a l l i , t o (5.8g) l i m q . ( t ) = 0 t-*-» (5.8h) l i m ^ ( t ) = 0 t-*-<*> and ( 5 . 8 i ) q i ( t ) d t = I * ^ ) , 120 where I*(X.) i s t h a t v a l u e o f I± w h i c h s o l v e s 1 x (5.9) d 1 ( I i / M i , X ±) = F. From ( 5 . 8 i ) and (5.9) i t i s c l e a r t h a t t h e u l t i m a t e amount o f t h e r e s o u r c e r e c o v e r e d by p r o d u c e r i i s d e t e r -mined e n t i r e l y by t h e amount o f i n i t i a l i n v e s t m e n t he u n d e r t a k e s . S i n c e a change i n X. a l t e r s t h e m a r g i n a l c o s t s c h e d u l e , d 1 (I\/M^,X^) , i n i t i a l i n v e s t m e n t i s t h e e s s e n t i a l i n s t r u m e n t a v a i l a b l e t o t h e p r o d u c e r t o d e t e r m i n e how much o f t h e r e s o u r c e i s t o be u l t i m a t e l y r e c o v e r e d . T o t a l d i f f e r e n t i a t i o n o f (5.9) w i t h r e s p e c t t o X i y i e l d s d l * d ,, ( 5 - 1 0 ) dxT = - M i d7 f > 0 ' u s i n g ( A l ) ' . Thus an i n c r e a s e i n i n i t i a l i n v e s t m e n t i n c r e a s e s u l t i m a t e r e c o v e r y . The f o l l o w i n g p r o p o s i t i o n e s t a b l i s h e s t h a t under t h e assumptions made, a Cournot-Nash e q u i l i b r i u m e x i s t s , t h a t i s , t h e r e e x i s t i n v e s t m e n t and e x t r a c t i o n s t r a t e g i e s t h a t maximize the p r e s e n t v a l u e p r o f i t s o f each p r o d u c e r i f he t a k e s as g i v e n t h e i n v e s t m e n t - e x t r a c t i o n d e c i s i o n s o f a l l o t h e r p r o d u c e r s . P r o p o s i t i o n 5.2 Under Assumptions (A l ) ', (A2) ', ( A 3 ) , (A4)'-(A6)' , (A7) and (AB), Cournot-Nash e q u i l i b r i a e x i s t w h i c h a r e d e r i v e d as t h e s o l u t i o n s t o (5.7) and a r e c h a r a c t e r i z e d by t h e v e c t o r s o f c o n t i n u o u s f u n c t i o n s q ( t ) , y (t) and a v e c t o r X, s a t i s f y i n g t h e c o n s t r a i n t s and the n e c e s s a r y c o n d i t i o n s ( 5 . 8 a ) , ( 5 . 8 b ) , (5.8f) - ( 5 . 8 i ) f o r i = 1, ... , N. P r o o f : See Appendix t o t h i s C h a p t e r . As i n p r e v i o u s c h a p t e r s , t h e p r o p e r t i e s o f t h e Cournot-Nash e q u i l i b r i u m w i l l be s e p a r a t e l y examined f o r the c a s e s o f symmetric and asymmetric p r o p e r t y r i g h t s . A. THE SYMMETRIC CASE Here i t w i l l be assumed t h a t M^ = M/N f o r i = 1,...,N, i . e . , t h a t t h e p r o p e r t y r i g h t s a r e i d e n t i c a l a c r o s s a l l p r o d u c e r s . Even when t h e p r o p e r t y r i g h t s a r e symmetric, however, asymmetric Cournot-Nash e q u i l i b r i a , i n w h i c h n o t a l l f i r m s produce i n an i d e n t i c a l f a s h i o n , cannot be r u l e d o u t . However, i n t h i s c a s e i t i s n a t u r a l and c o n v e n i e n t t o r e s t r i c t a t t e n t i o n t o t h e symmetric Nash e q u i l i b r i a . Some o f t h e p r o p o s i t i o n s t o f o l l o w examine t h e c o m p a r a t i v e dynamics o f t h e symmetric Cournot-Nash e q u i l i b r i a w i t h r e s p e c t t o v a r i o u s p a r a m e t e r s . The performance o f com-p a r a t i v e dynamics (and o f c o m p a r a t i v e s t a t i c s , f o r t h a t m a t t e r ) i s m e a n i n g f u l o n l y when t h e e q u i l i b r i u m i s u n i q u e . T h i s g u a r a n t e e s t h a t t h e same e q u i l i b r i u m i s b e i n g " t r a c k e d " when t h e r e a r e changes i n t h e parameter w i t h r e s p e c t t o wh i c h c o m p a r a t i v e dynamics i s b e i n g p e r f o r m e d . Tn t h e p r e v i o u s c h a p t e r , where i t was assumed t h a t f i r m s d i d not engage i n any i n i t i a l i n v e s t m e n t , i t was e x p l i c i t l y d e -m o n s t r a t e d t h a t t h e s y m m e t r i c C o u r n o t - N a s h e q u i l i b r i u m i s u n i q u e . I n t h e p r e s e n t c a s e , however, t h i s I s n o t p o s s i b l e and i t i s n e c e s s a r y t o assume t h a t ( A 9 ) T h e r e i s a u n i q u e s y m m e t r i c s o l u t i o n t o c o n d i t i o n s (5.8a) - ( 5 . 8 f ) when t h e p r o p e r t y r i g h t s a r e s y m m e t r i c . A s s u m p t i o n ( A 9 ) w o u l d , o f c o u r s e , b e r e d u n d a n t i f e a c h p r o -5 d u c e r ' s o p t i m i z a t i o n p r o b l e m were c o n c a v e . The d i f f i c u l t y i s t h a t an i n d i v i d u a l p r o d u c e r ' s H a m i l t o n i a n i s n o t j o i n t l y c o n c a v e i n t h e c o n t r o l p a r a m e t e r , X, t h e c o n t r o l v a r i a b l e , q, and t h e s t a t e v a r i a b l e , I . However, w h i l e c o n c a v i t y o f t h e p r o b l e m i s a s u f f i c i e n t c o n d i t i o n f o r u n i q u e n e s s o f t h e s o l u t i o n , i t i s n o t n e c e s s a r y . The i m p o r t o f A s s u m p t i o n (A9) i s t h a t i t e n s u r e s t h a t t h e s y m m e t r i c C o u r n o t - N a s h e q u i l i b r i u m i s u n i q u e . The f o l l o w i n g p r o p o s i t i o n summarizes some o f t h e b a s i c f e a t u r e s o f t h i s e q u i l i b r i u m . P r o p o s i t i o n 5.3 The u n i q u e , s y m m e t r i c C o u r n o t - N a s h e q u i l i b r i u m [T,x,{q>] o b t a i n e d when t h e p r o p e r t y r i g h t s a r e s y m m e t r i c i s s u c h t h a t (a) Q (t)>0 , Q(t)<0 f o r 0<t<T and l i m Q ( t ) = 0 t+T (b) T i s i n f i n i t e 123 (c) l i m d CNI(t)/M,X) = F fc+T (d) l i m p (t) = 0, t->-T P r o o f : The p r o o f o f t h i s p r o p o s i t i o n proceeds e x a c t l y as t h a t o f P r o p o s i t i o n 4.3 o f t h e p r e v i o u s c h a p t e r and i s t h e r e f o r e d e l e t e d t o a v o i d d u p l i c a t i o n . Q.E.D. I n t h e p r e v i o u s c h a p t e r i t was d e m o n s t r a t e d t h a t t h e symmetric Cournot-Nash e q u i l i b r i u m i m p l i c i t l y maximizes a w e i g h t e d average o f t h e p r e s e n t v a l u e s o f t h e t o t a l s u r p l u s and t h e i n d u s t r y p r o f i t s . The f o l l o w i n g p r o p o s i t i o n g e n e r a l i z e s t h a t r e s u l t t o t h e c a s e where f i r m s u n d e r t a k e i n i t i a l i n v e s t m e n t . P r o p o s i t i o n 5.4 C o n s i d e r t h e f u n c t i o n a l Y ( T , X , { q } ) d e f i n e d by (5.11) Y(T,X,{q}) = ^ L p •It r T - r t r e [ Nq(t) ( P ( z ) - d (I/M,X) )dz] d t 0 - Mg(X) } T e " r t [ p ( N q ) - d 1(I/M,X) ]Nq(t) d t 0 - Mg(X) } The s o l u t i o n [T*,X*,{q*}] t o t h e a u x i l i a r l y problem (5.12) maximize Y(T,X,{q}) T,X,{q} s u b j e c t t o I = Nq >_ 0 i s a l s o t h e u n i q u e , symmetric Cournot-Nash e q u i l i b r i u m . 124 P r o o f : The p r o o f r u n s on l i n e s p a r a l l e l t o t h o s e o f P r o p o s i t i o n 4.2. By A s s u m p t i o n ( A 6 ) ' , a s o l u t i o n e x i s t s t o problem ( 5 . 1 2 ) . I t i s e a s i l y v e r i f i e d , as i n P r o p o s i t i o n 4.2, t h a t t h e f i r s t o r d e r n e c e s s a r y c o n d i t i o n s f o r problem (5.12) a r e i d e n t i c a l , when p r o p e r t y r i g h t s a r e symmetric, t o co n -d i t i o n s ( 5 . 8 ) , w h i c h c h a r a c t e r i z e t h e u n i q u e , symmetric Cournot-Nash e q u i l i b r i u m . S i n c e , by Assumption ( A 9 ) , con-d i t i o n s (5.8) admit o f a u n i q u e s o l u t i o n i n t h e symmetric c a s e , i t f o l l o w s t h a t t h e s o l u t i o n t o (5.12) i s , i n f a c t , t h e Cournot-Nash e q u i l i b r i u m . Q.E.D. The above p r o p o s i t i o n w i l l be used below i n e v a l u a t i n g t h e e f f e c t o f an i n c r e a s e i n t h e number o f f i r m s i n t h e i n d u s t r y on t h e p r e s e n t v a l u e s o f t h e t o t a l s u r p l u s and i n d u s t r y p r o f i t s . T h i s and t h e e f f e c t o f changes i n t h e market s t r u c t u r e on t h e e x t r a c t i o n p r o f i l e and t h e u l t i m a t e r e c o v e r y o f t h e r e s o u r c e a r e c o n s i d e r e d i n the f o l l o w i n g p r o p o s i t i o n . P r o p o s i t i o n 5.5 An i n c r e a s e i n t h e number o f f i r m s i n t h e i n d u s t r y (a) i n c r e a s e s t h e c u m u l a t i v e e x t r a c t i o n o f t h e i n d u s t r y as o f any d a t e a f t e r p r o d u c t i o n b e g i n s (b) i n c r e a s e s t h e i n i t i a l i n v e s t m e n t on each d e p o s i t (c) i n c r e a s e s t h e u l t i m a t e amount o f t h e r e s o u r c e r e c o v e r e d from each d e p o s i t (d) d e c r e a s e s the p r e s e n t v a l u e o f i n d u s t r y p r o f i t s , and (e) i n c r e a s e s t h e p r e s e n t v a l u e o f t h e t o t a l s u r p l u s . P r o o f : ( a ) , (b) and (c) . L e t Q(t,N,X) and I ( t , N ,X), r e s p e c t i v e l y , denote t h e e x t r a c t i o n r a t e and t h e c u m u l a t i v e e x t r a c t i o n a s ; o f t i m e t when t h e i n d u s t r y i s c o m p r i s e d o f N ( i d e n t i c a l ) f i r m s and an i n i t i a l i n v e s t m e n t o f X i s u n d e r t a k e n on each d e p o s i t . C o n s i d e r two i n d u s t r i e s , one w i t h N f i r m s and a n o t h e r w i t h N 1(>N) f i r m s . L e t X and X' be t h e amounts o f i n i t i a l i n v e s t m e n t s u n d e r t a k e n on each d e p o s i t i n t h e r e s p e c t i v e Cournot-Nash e q u i l i b r i a . (X and X' a r e , o f c o u r s e , endogenously d e t e r m i n e d ) . I t w i l l f i r s t be e s t a b l i s h e d t h a t (5.13) X' > X . Assuming an i n t e r i o r s o l u t i o n f o r X, t h e f i r s t o r d e r c o n d i t i o n (5.8f) may be w r i t t e n (5.14) G(N,X) - g'(X) = 0 where (5.15) G(N,X) = - ^ 1 e "r t d,.(Kt,N,X)/M,X)Q(t,N,X) d t W r i t i n g Q y i e l d s 12 0 (t,N,X) = i(t,N,X) and i n t e g r a t i n g (5.15) by p a r t s /•OO (5.16) G(N,X) * - r e "r t d 0 ( I ( t , N , X ) / M , X ) d t . ) o 2 126 T o t a l d i f f e r e n t i a t i o n o f t h e f i r s t o rder c o n d i t i o n (5.14) w i t h r e s p e c t t o N y i e l d s , on rearrangement, 9G(N,X) dX 3N (5.17) -^ - r3G(N,X)_^, ( x. ) 1 [9X J Now the second o r d e r s u f f i c i e n t c o n d i t i o n f o r t h e m a x i m i z a t i o n o f each p r o d u c e r ' s p r e s e n t v a l u e p r o f i t s w i t h r e s p e c t t o X i s (5.18) f f ( N , x ) - g"(x) < 0 , wh i c h s i g n s t h e denominator o f the r i g h t hand s i d e o f (5.17) . F u r t h e r , d i f f e r e n t i a t i o n o f (5.16) p a r t i a l l y w i t h r e s p e c t t o N y i e l d s , n o x 3G(N,X) _ r (5.19) g-jj M e -r t d 1 2 ( K t , N , X ) / M , X ) f | ( t ' N ' X ) ^ From p a r t (b) o f P r o p o s i t i o n 4.4 o f the p r e v i o u s c h a p t e r , i t f o l l o w s t h a t ( 5 - 2 0 ) i l ( t , N , X ) > Q f o r a l l t > 0_6 A l s o , s i n c e d 1 2 < 0, (5.19) i m p l i e s t h a t (5.2D f f ( N ' X ) > 0. From ( 5 . 1 7 ) , (5.18) and (5.21) i t f o l l o w s t h a t (5.22) H > 0 , w h i c h , i n t u r n , i m p l i e s (5.13) s i n c e N<N*. T h i s e s t a b l i s h e s p a r t ( b ) . From t h i s and (5 . 1 0 ) , p a r t (c) f o l l o w s ^ -127 Now d i f f e r e n t i a t i o n o f (5,8a) (which h o l d s w i t h e q u a l i t y f o r a l l t ) w i t h r e s p e c t t o time y i e l d s Q as a f u n c t i o n o f Q,X,I and N: Nr [P(Q)-a(P)Q/N-d (I/M,X) ] (5.23) Q(Q,X,I,N) = a ( P ) ( N + l - Q a ' ) where, as u s u a l , N+1 - Qa'>0. S i n c e d-j^O a n d d i 2 < 0 ' i t f o l l o w s t h a t (5.24) Q(Q,X' ,I*,N')<Q(Q,X',1 ,N')<Q(Q,X,I,N') <0 when X'>X and I'<I. Note t h a t i n (5.24) a l l c o m p a r i s o n s ar e b e i n g made w i t h t h e number o f f i r m s i n t h e i n d u s t r y b e i n g f i x e d a t N'. S i n c e X'>X i t f o l l o w s from (5.10) t h a t (5.25) l i m I ( t , N ' , X ) < l i m I ( t , N ' , X ' ) . Suppose t h a t a t some time t ' , (5.26) I ( t ' ,N',X) >JE ( t ' ,N ' ,X ' ) and Q ( t ' ,N *, X) = Q ( t ' ,N' , X' ) . Then from (5.24) i t f o l l o w s t h a t Q(t',N',X) > Q(t',N ,,X') , w h i c h i m p l i e s t h a t I ( t , N ' , X ) > I ( t , N ' , X ' ) V t > t ' . But t h i s i s i m p o s s i b l e i n view o f (5.25). Thus (5.26) c a n n o t h o l d . F u r t h e r , i f f o r some t (5.27) I ( t , N ' , X ) ^ I ( t , N ' , X ' ) and Q(t,N',X) > Q ( t , N ' , X ) , t h e n t h e r e must e x i s t an i n s t a n t t'>t a t w h i c h (5.26) must h o l d because o f (5.25) and t h e f a c t t h a t t h e e x t r a c t i o n 128 mu p r o f i l e s a r e c o n t i n u o u s . B u t (5.26) has been r u l e d o u t as i m p o s s i b l e , so (5.27) cannot h o l d . S i m i l a r l y , i f a t some time t (5.28) I ( t , N J X) >_I ( t , N ' , X 1 ) and Q(t,N',X) <Q(t,N',X') , th e n t h e r e must have been an i n s t a n t t '<t a t wh i c h (5.26) s t have h e l d - w h i c h i s i m p o s s i b l e . The i m p o s s i b i l i t y o f (5 . 2 6 ) , (5.27) and (5.28) i m m e d i a t e l y i m p l i e s t h a t f o r X'>X (5.29) K t . N ' ^ 1 ) > I ( t , N ' , X ) f o r a l l t > 0 • A l s o , (5.20) i m p l i e s t h a t when N'>N (5.30) I ( t , N ' , X ) > I ( t , N , X ) f o r a l l t > 0. F i n a l l y , from ( 5 . 1 3 ) , (5.29) and (5.30) i t f o l l o w s t h a t when N' >N (5.31) K t f N ' j X ' ) > I ( t , N , X ) V t > 0, wh i c h e s t a b l i s h e s p a r t ( a ) . (d) and ( e ) . The p r o o f s o f t h e s e two p a r t s p r o c e e d e x a c t l y as do t h e p r o o f s o f p a r t s (c) and (d) o f P r o p o s i t i o n 4.4 o f th e p r e v i o u s c h a p t e r and a r e t h u s d e l e t e d . Q.E.D, P a r t s ( a ) , (d) and (e) o f t h e above p r o p o s i t i o n a r e , by now, f a m i l i a r and r e q u i r e no comment. P a r t s (b) and (c) are new, however, and (c) s t a n d s i n c o n t r a s t t o the c o r r e s p o n d i n g r e s u l t i n t h e p r e v i o u s c h a p t e r . The economic 129 i n t u i t i o n b e h i n d t h e s e r e s u l t s i s r o u g h l y as f o l l o w s . The p r e s e n t v a l u e shadow p r i c e o f t h e r e s o u r c e a t t=0 i s g i v e n by (5.8a) as t h e m a r g i n a l revenue n e t o f t h e m a r g i n a l c o s t a t t h a t i n s t a n t . An i n c r e a s e i n N l o w e r s t h e i n i t i a l 7 p r i c e o f t h e r e s o u r c e , as e x p e c t e d . However, t h e i n c r e a s e i n c o m p e t i t i o n between p r o d u c e r s tends t o c l o s e t h e gap between m a r g i n a l revenue and p r i c e . The l a t t e r e f f e c t dominates t h e former and as a r e s u l t , the m a r g i n a l revenue t o each f i r m a t time t=0 i n c r e a s e s when N i n c r e a s e s . F o r a g i v e n X (and t h e r e f o r e f o r a g i v e n m a r g i n a l r e c o v e r y c o s t s c h e d u l e ) , t h i s i m p l i e s an i n c r e a s e i n t h e ( p r e s e n t v a l u e ) shadow p r i c e o f t h e r e s o u r c e a t t=0. S i n c e a l l d e c i s i o n s a r e e s s e n t i a l l y made a t t=0 i n t h i s model, each p r o d u c e r would seek t o i n c r e a s e t h e u l t i m a t e r e c o v e r y o f t h e r e s o u r c e i n v i e w o f i t s h i g h e r shadow p r i c e . T h i s , however, can o n l y be a c c o m p l i s h e d by an i n c r e a s e i n t h e i n i t i a l i n v e s t m e n t w h i c h , by l o w e r i n g t h e m a r g i n a l r e c o v e r y c o s t s c h e d u l e f a c i n g t h e p r o d u c e r , enhances t h e u l t i m a t e amount o f t h e r e s o u r c e t h a t can be. p r o f i t a b l y r e c o v e r e d . The n e x t p r o p o s i t i o n c o n s i d e r s t h e e f f e c t on i n i t i a l i n v e s t m e n t and on t h e t i m e p r o f i l e o f the c u m u l a t i v e ex-t r a c t i o n o f t h e i n d u s t r y o f a r o y a l t y t a x . P r o p o s i t i o n 5.6 An i n c r e a s e i n t h e r o y a l t y t a x (a) d e c r e a s e s t h e i n i t i a l i n v e s t m e n t on each d e p o s i t (b) d e c r e a s e s t h e c u m u l a t i v e e x t r a c t i o n o f t h e i n d u s t r y 130 as o f any d a t e a f t e r p r o d u c t i o n b e g i n s and l e s s o f th e r e s o u r c e i s u l t i m a t e l y r e c o v e r e d . P r o o f : L e t {Q(t,k,X)} and { I ( t , k , X ) } denote the t i m e p r o f i l e s o f t h e i n d u s t r y e x t r a c t i o n r a t e and c u m u l a t i v e o u t p u t t o d a t e , r e s p e c t i v e l y , when t h e r o y a l t y t a x i s k per u n i t o f r e s o u r c e e x t r a c t e d and X i s t h e amount o f i n i t i a l i n v e s t m e n t u n d e r t a k e n on each d e p o s i t . I n t h e pr e s e n c e o f th e r o y a l t y t a x , the n e c e s s a r y c o n d i t i o n (5.8a) i s m o d i f i e d , i n t h e symmetric c a s e , t o (5.32) P ( Q ) - d 1 ( I / M , X ) - a ( P ) Q / N - k<v ( = i f Q > 0 ) The o t h e r f i r s t o r d e r c o n d i t i o n s a r e unchanged, (a) T o t a l d i f f e r e n t i a t i o n o f (5.14) w i t h r e s p e c t t o k and rearrangement y i e l d s dG ( 5 - 3 3 ) i f = - - T G - 1 J S — t | | - g"] where X i s h e l d f i x e d i n e v a l u a t i n g t h e p a r t i a l d e r i v a t i v e The denominator o f (5.33) i s n e g a t i v e by (5.18). P a r t i a l d i f f e r e n t i a t i o n o f (5.16) w i t h r e s p e c t t o k y i e l d s 0 0 (5.34) | f ( N ' X ) = - r f e - r t d 1 2 ( I ( t , k , X ) / M , X ) f p ' k ' X ) d t Now by p a r t (b) o f P r o p o s i t i o n 4.5, i t f o l l o w s t h a t | I ( t , k , X ) < Q f o r t > Q > 8 T h u s ( 5 - 3 4 ) y i e i d s 131 C5.35) | | < 0 , w h i c h , t o g e t h e r w i t h (5.18) and ( 5 . 3 3 ) , i m p l i e s t h a t (5.36) | | < 0. (b) L e t k and k'(<k) be two v a l u e s o f t h e r o y a l t y t a x and l e t X, X' be t h e c o r r e s p o n d i n g v a l u e s o f the ( o p t i m a l ) i n i t i a l i n v e s t m e n t p e r d e p o s i t u n d e r t a k e n by the f i r m s . Then from (5.36) i t f o l l o w s t h a t (5.37) X < X' E v a l u a t i n g (5.32) i n t h e l i m i t t-*-°°, y i e l d s upon u s i n g (5.8g) and (5.8h) , l i m d 1 ( I ( t , k , X ) / M , X ) = F-k . t->°° D i f f e r e n t i a t i o n o f 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 t o k y i e l d s , i n v i e w o f (5 . 3 6 ) , l i m | f ( t ' k - X ) < 0 . t-*-°° From t h i s , (5.37) and (5.10) i t f o l l o w s t h a t (5.38) l i m I ( t , k , X ) < l i m I ( t , k ' , X ' ) . H o l d i n g t h e r o y a l t y t a x f i x e d a t k*, d i f f e r e n t i a t i o n o f (5.32) w i t h r e s p e c t t o time and use o f (5.8b) y i e l d s Q as a f u n c t i o n o f Q,k',I and X: 132 Nr fP (0) -a (P)Q/N-di (T/M,X)-k'] (5.39) Q(Q,k',I,X) = - , a ( F ) ( N + l - Qa') From (5.39) i t f o l l o w s t h a t (5.40) Q(Q,kM',X') < Q(Q,k',I ,X') < Q(Q,k',I ,X) <0 f o r X*>X and I'<I. Comparing (5.40) t o (5.24) and m i m i c i n g the argument t h a t l e d t o (5 . 2 9 ) , i t f o l l o w s t h a t (5.41) I ( t , k \ X ) < I(t,k',X') Vt>0 • A l s o , by p a r t (b) o f P r o p o s i t i o n 4.5, (5.42) I ( t , k , X ) < I ( t , k ' , X ) Vt>0. From (5.41) and (5.42) t h e r e s u l t f o l l o w s . Q.E .D. W h i l e t h e r e s u l t t h a t an i n c r e a s e i n t h e r o y a l t y t a x d e c r e a s e s u l t i m a t e r e c o v e r y i s i n t u i t i v e , i t must be no t e d t h a t t h i s i s t h e outcome o f two e f f e c t s . F i r s t l y , an i n c r e a s e i n t h e t a x l o w e r s t h e choke p r i c e . S i n c e the u l t i m a t e r e c o v e r y from each d e p o s i t i s d e t e r m i n e d as t h a t s t o c k l e v e l a t w h i c h t h e m a r g i n a l r e c o v e r y c o s t i s e q u a l t o the choke p r i c e , t h e u l t i m a t e r e c o v e r y would now be l o w e r . S e c o n d l y , t h e i n c r e a s e d t a x l o w e r s t h e m a r g i n a l v a l u e o f i n v e s t m e n t and so l e s s i n i t i a l i n v e s t m e n t i s u n d e r t a k e n . T h i s r a i s e s t h e m a r g i n a l r e c o v e r y c o s t s c h e d u l e - w h i c h f u r t h e r r e d u c e s t h e u l t i m a t e r e c o v e r y o f t h e r e s o u r c e from each d e p o s i t . The d e c r e a s e d i n c e n t i v e t o e x t r a c t as a r e s u l t 133 o f t h e h i g h e r t a x a l s o m a n i f e s t s i t s e l f as a s l o w i n g down i n t h e r a t e o f e x t r a c t i o n ; c u m u l a t i v e e x t r a c t i o n i s l o w e r as o f any t i m e w i t h h i g h e r t a x r a t e s . I n t h e p r e v i o u s c h a p t e r , i t was shown t h a t an i n -c r e a s e i n t h e d i s c o u n t r a t e h a s t e n s p r o d u c t i o n , a r e s u l t w h i c h i s i n d e p e n d e n t o f t h e number o f f i r m s i n t h e i n d u s t r y and, t h e r e f o r e , o f market s t r u c t u r e . However, i n t h e more r e a l i s t i c c a s e when f i r m s engage i n i n i t i a l i n v e s t m e n t , such an unambiguous r e s u l t i s n o t a v a i l a b l e . The r e a s o n i s t h a t when t h e d i s c o u n t r a t e r i s e s , l e s s i n i t i a l i n v e s t m e n t may be u n d e r t a k e n and t h i s , t h r o u g h ' i t s e f f e c t on t h e m a r g i n a l c o s t s c h e d u l e , o f f s e t s t h e tendency t o d e p l e t e t h e r e s o u r c e f a s t e r . To see how t h i s m ight a r i s e , c o m p a r a t i v e dynamics w i t h r e s p e c t t o t h e d i s c o u n t r a t e i s now b r i e f l y c o n s i d e r e d . D i f f e r e n t i a t i o n o f (5.14) w i t h r e s p e c t t o r y i e l d s 9G dX = 3 r d r r3G _ l3X y where X i s held f i x e d i n the evaluation of the p a r t i a l aerivate |f . " s i n , (5.18), the above expression S p i r e s t h a t ,dX - 9 G (5.43) s i g n (||) = s i g n (j^) I n e v a l u a t i n g t h e r i g h t hand s i d e o f ( 5 . 4 3 ) , i t w i l l be assumed, f o r s i m p l i c i t y , t h a t d l 2 i s a c o n s t a n t (of c o u r s e , n e g a t i v e ) . T h i s amounts t o t h e a s s u m p t i o n t h a t an a d d i t i o n a l 134 u n i t o f i n i t i a l i n v e s t m e n t l o w e r s t h e m a r g i n a l c o s t everywhere by a c o n s t a n t amount c t - ^ f say. Then (5.15) may be w r i t t e n d f 0 0 G(N,X) = - e " r t Q(t,N,X) d t , J 0 w h i c h upon p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o r y i e l d s ( 5 44) 3G(N,X) = _ ^12 f - r t ,9Q _ fcQ) d t J 0 9 0 where by — i s meant t h e change i n Q a t a g i v e n i n s t a n t , a r h o l d i n g X f i x e d . Now from p a r t (b) o f P r o p o s i t i o n 4.7 i t f o l l o w s t h a t (9Q/9r) must be p o s i t i v e a t l e a s t i n i t i a l l y . However, s i n c e t h e u l t i m a t e r e s o u r c e r e c o v e r y must be t h e same (because X i s b e i n g h e l d f i x e d ) , (9Q/9r) must be n e g a t i v e f o r non-zero i n t e r v a l s o f t i m e . Thus the s i g n o f t h e r i g h t hand s i d e o f (5.44) i s ambiguous, and so i s t h a t o f ( d X / d r ) . The r e a s o n f o r t h i s i s t h a t t h e r e a r e two o p p o s i n g f o r c e s t h a t , t o g e t h e r , d e t e r m i n e th e n e t change i n X. F i r s t l y , an i n c r e a s e i n t h e d i s c o u n t r a t e t i l t s t h e e x t r a c t i o n p r o f i l e towards t h e p r e s e n t , so t h a t t h e r e t u r n s t o i n i t i a l i n v e s t m e n t a c c r u e e a r l i e r t han b e f o r e . T h i s t e n d s t o i n c r e a s e t h e o p t i m a l amount o f i n v e s t m e n t p e r d e p o s i t u n d e r t a k e n by t h e f i r m s . S e c o n d l y , s i n c e t h e d i s -c o u n t r a t e i s now h i g h e r , f u t u r e r e t u r n s o f t h e m a r g i n a l i n v e s t m e n t a r e d i s c o u n t e d more h e a v i l y , and t h i s tends t o d e c r e a s e th e o p t i m a l amount o f i n i t i a l i n v e s t m e n t . The n e t 135 e f f e c t i s t h u s ambiguous. I f the second e f f e c t d o m i n a t e s , th e i n i t i a l i n v e s t m e n t u n d e r t a k e n would d e c r e a s e when t h e d i s c o u n t r a t e i n c r e a s e s and t h i s , i n t u r n , would d e c r e a s e t h e u l t i m a t e amount o f r e s o u r c e r e c o v e r e d . Thus t h e cumu-l a t i v e e x t r a c t i o n o f t h e i n d u s t r y cannot be h i g h e r a t a l l i n s t a n t s o f t i m e , c o n t r a r y t o t h e r e s u l t o f t h e p r e v i o u s c h a p t e r . Nor i s i t g u a r a n t e e d t h a t t h e r e e x i s t s some i n -s t a n t o f time a t w h i c h c u m u l a t i v e e x t r a c t i o n o f t h e i n d u s t r y 9 i s l a r g e r when t h e d i s c o u n t r a t e i s i n c r e a s e d . Thus when the u l t i m a t e r e c o v e r y o f f i r m s i s made endogenous, t h e i n t u i t i o n d e r i v e d from f i x e d - s t o c k r e s o u r c e models i s n o t n e c e s s a r i l y v a l i d . B. THE ASYMMETRIC CASE Now t h e c a s e when t h e p r o p e r t y r i g h t s o f the f i r m s i s asymmetric i s c o n s i d e r e d . I n v i e w o f the c o m p l e x i t y o f the model i n t h e absence o f symmetry, i t i s d i f f i c u l t t o c h a r a c t e r i z e t h e Cournot-Nash e q u i l i b r i u m i n v e r y g r e a t d e t a i l . I n f a c t , t h e o n l y r e s u l t a v a i l a b l e i n t h i s c a s e i s the l a c k o f e f f i c i e n c y (as d e f i n e d i n t h e p r e v i o u s c h a p t e r ) o f t h e i n v e s t m e n t and p r o d u c t i o n u n d e r t a k e n by t h e i n d u s t r y a t t h e ag g r e g a t e l e v e l . B e f o r e e s t a b l i s h i n g t h i s , two n e c e s s a r y c o n d i t i o n s f o r e f f i c i e n c y a r e d e r i v e d i n the p r o -p o s i t i o n below. 136 P r o p o s i t i o n 5.7 E f f i c i e n t p r o d u c t i o n a t t h e i n d u s t r y l e v e l r e q u i r e s (a) t h e m a r g i n a l r e c o v e r y c o s t s t o be e q u a l a c r o s s a l l d e p o s i t s a t a l l p o i n t s i n t i m e , and (b) the m a r g i n a l i n v e s t m e n t c o s t s t o be e q u a l a c r o s s a l l d e p o s i t s . P r o o f : E f f i c i e n c y a t t h e a g g r e g a t e l e v e l r e q u i r e s t h a t t h e v e c t o r s £ and {q} o f i n i t i a l i n v e s t m e n t s and o u t p u t p r o f i l e s , r e s p e c t i v e l y , s o l v e t h e dynamic o p t i m i z a t i o n problem M . . . M I d, ( I ^ X 1 ) q 1 ( t ) d t + I gCX 1) i = l 1 i = l (5.45) min X,(q) - r t e 0 M . _ s u b j e c t t o I q 1 ( t ) = Q(t) Vt>0 i = l and I 1 = q 1 >_ 0 V i , where X 1 , q 1 and I 1 denote t h e i n i t i a l i n v e s t m e n t , o u t p u t r a t e and c u m u l a t i v e o u t p u t t o d a t e , r e s p e c t i v e l y , o f the i t h d e p o s i t , and ( Q ( t ) } i s some e x o g e n o u s l y g i v e n o u t p u t p r o f i l e o f t h e i n d u s t r y . Now i t was p r o v e d i n t h e c o u r s e o f P r o p o s i t i o n 5.1 t h a t t h e s o l u t i o n t o ( 5 . 2 ) , w h i c h i n p r i n c i p l e , i s i d e n t i c a l t o t h e o p t i m i z a t i o n b e i n g performed h e r e , s a t i s f i e s X 1 = X3 f o r a l l i , j and q 1 ( t ) = q ^ ( t ) f o r a l l i , j and t>0. 137 From t h i s , i t follows that (5.46a) d ^ d S x 1 ) = d ^ I ^ X 3 ) for a l l i , j and t>0 and V. (5.46b) g'(X 1) = g'(X j) for a l l i , j , as was required to be shown. Q.E.D. Thus e f f i c i e n c y at the aggregate l e v e l requires a l l deposits i n the industry to be run i d e n t i c a l l y . Any de-v i a t i o n from t h i s rule would leave scope for a reduction i n the present value cost of providing a given stream of output at the aggregate l e v e l by r e a l l o c a t i n g investment and/or extraction across deposits. The following proposition establishes the i n e f f i c i e n c y of asymmetric Cournot-Nash e q u i l i b r i a . Proposition 5.8 When the property rights are asymmetric, the Cournot-Nash e q u i l i b r i a are i n e f f i c i e n t . Proof: The proof proceeds by establishing that i t i s im-possible to simultaneously s a t i s f y conditions (a) and (b) of Proposition 5.7, both of which are necessary for e f f i -ciency. Since the proof runs on l i n e s p a r a l l e l to those of Proposition 4.10, the d e t a i l s are deleted. Q.E.D. 138 The above p r o p o s i t i o n s u g g e s t s t h a t a l l o w i n g f o r t h e p o s s i b i l i t y o f i n i t i a l i n v e s t m e n t i n t r o d u c e s a n o t h e r d i m e n s i o n t o the i n e f f i c i e n c y o f asymmetric Cournot-Nash e q u i l i b r i a : n o t o n l y i s t h e r e i n e f f i c i e n c y i n e x t r a c t i o n (as b e f o r e ) , b u t now t h e r e can a l s o be i n e f f i c i e n c y , a t t h e a g g r e g a t e l e v e l , i n i n v e s t m e n t . 139 F o o t n o t e s t o C h a p t e r V I 1. See U h l e r (1978) on t h i s p o i n t . 2. Campbell(1980) has c o n s i d e r e d t h e c a s e o f f i r m s u n d e r t a k i n g a o n c e - f o r - a l l i n v e s t m e n t i n c a p a c i t y a t t=0. J a c o b s o n and Sweeny (19 80) have d e v e l o p e d a model o f a c o m p e t i t i v e f i r m u n d e r t a k i n g i n v e s t m e n t i n c a p i t a l c o n t i n u o u s l y d u r i n g t h e c o u r s e o f e x t r a c t i o n . 3. Use has been made o f t h e f a c t t h a t d(0,X) = 0 and t h a t l i m d ( I ( t ) , X ) i s f i n i t e . 4. See, f o r example, Takayama(1974), Theorem 8.C.4, pp. 658. 5. F o r t h i s would ensure t h a t t h e n e c e s s a r y c o n d i t i o n s (5.8) admit o f o n l y one s o l u t i o n . 6. R e c a l l t h a t i n P r o p o s i t i o n 4.4, X was i d e n t i c a l l y z e r o . However, even i f X were p o s i t i v e b u t e x o g e n o u s l y f i x e d , i t i s c l e a r t h a t p a r t (b) o f P r o p o s i t i o n 4.4 would remain v a l i d . 7. T h i s f o l l o w s by e v a l u a t i n g (5.20) i n t h e l i m i t t+0; the r e s u l t i n d i c a t e s t h a t t h e i n d u s t r y o u t p u t must be i n i t i a l l y h i g h e r when N i n c r e a s e s . 8. A remark i d e n t i c a l t o t h a t i n f o o t n o t e 6 above a p p l i e s h e r e . 9. A s i m i l a r p o s s i b i l i t y a r i s e s i n the model o f J a c o b s o n and Sweeny (1980). 140 A ppendix t o C h a p t e r V T h i s Appendix p r o v i d e s t h e p r o o f o f P r o p o s i t i o n 5.2. The p r o o f runs on l i n e s p a r a l l e l t o t h o s e o f t h e p r o o f o f P r o p o s i t i o n 4.1 o f t h e l a s t c h a p t e r . By Assumption (A6)' t h e r e e x i s t s a s o l u t i o n t o f i r m i ' s o p t i m i z a t i o n p r o blem ( 5 . 7 ) , g i v e n t h e o u t p u t p r o f i l e s , { ( t ) , j ^ i } , o f a l l o t h e r f i r m s . The f i r s t o r d e r n e c e s s a r y c o n d i t i o n s f o r t h e m a x i -m i z a t i o n problem (5.7) a r e g i v e n by ( 5 . 8 ) . The e x i s t e n c e o f a Cournot-Nash e q u i l i b r i u m w i l l be e s t a b l i s h e d by de-m o n s t r a t i n g t h a t t h e r e e x i s t [ X i , { y ^ } f { q ^ l t h a t s i m u l t a n -e o u s l y s a t i s f y ( 5 . 8 a ) , ( 5 . 8 b ) , (5.8f) - ( 5 . 8 i ) f o r a l l i . T L e t y= ( y ^ , . . . / P N ) and M be t h e convex, compact and non-empty s e t o f a l l v e c t o r s , y, w i t h 0<_yi<_ F - d 1 ( 0 , X ) , where — T X has been d e f i n e d i n t h e t e x t . L e t I = ( I ^ , . . . , I N ) and L be t h e s e t o f a l l v e c t o r s , I , w i t h I ^ O f o r a l l i . T F i n a l l y , l e t X=(x 1,...,X N) and S be the s e t o f a l l v e c t o r s , X, w i t h 0<X^<X. S i n c e X i s s t r i c t l y p o s i t i v e b u t f i n i t e , S i s a non-empty, convex and compact s e t . F i r s t , a p r i c e P ( t ) i s o b t a i n e d f o r any yeM, I e L and XeS such t h a t (5.8a) i s s a t i s f i e d f o r a l l i . D e f i n e t h e f o l l o w i n g f u n c t i o n t h a t i s c o n t i n u o u s i n P , I , y and X: (5A.1) Z i ( P , I , y , X ) = max { 0 ,P-d 1 (1\ ,X i) -y ±} , w h i c h i s n o n - n e g a t i v e i n g e n e r a l and s t r i c t l y p o s i t i v e i n e q u i l i b r i u m i f and o n l y i f f i r m i i s p r o d u c i n g . Now d e f i n e t h e f u n c t i o n N C5A.2) Y(P,I,y,X) = £ Z (P,I,y,X) - a ( P ) Q ( P ) , i = l w h i c h i s c o n t i n u o u s i n i t s arguments, and v a n i s h e s i n e q u i l i b r i u m . As i n t h e Appendix o f the p r e v i o u s c h a p t e r , i t i s p o s s i b l e t o show t h a t t h e r e e x i s t s a unique and con-t i n u o u s f u n c t i o n P e ( I , y , X ) e[P,F] i m p l i c i t y d e f i n e d by (5A.3) Y(P,I,y,X) = 0 . Now choose a r b i t r a r y v e c t o r s o f i n i t i a l i n v e s t m e n t s , XeS, and i n i t i a l s c a r c i t y r e n t s , y°eM, and i t e r a t i v e l y compute (5A.4) P(t,y°,X) = P e[I(t,y°,X),y(t,y°,X),X], (5A.5) q i(t,y°,X)= Zi[P(t,y°,X) ,i(t,y' G,)0,v (t,y°,X) ,X]/a(P) rt (5A.6) I i(t,y°,X)= q i(s,y°,X) ds J 0 and (5A.7) y.(t,y°,X)=e r t max { 0,y°- Vr s d 1 1 ( i i ( s ) / M . ,X.) '0 q.(s) ds) S i n c e P e ( I , y , X ) i s c o n t i n u o u s i n i t s arguments, so a r e the v a r i a b l e s on t h e l e f t hand s i d e s o f (5A.4) - (5A.7). By c o n s t r u c t i o n , c o n d i t i o n s (5.8a) and (5.8b) a r e s a t i s f i e d by ( 5 A . 4 ) - ( 5 A . 7 ) . I t remains t o e s t a b l i s h t h a t t h e r e e x i s t s a XeS and a y°eM w h i c h g e n e r a t e p r o d u c t i o n p r o f i l e s t h a t a l s o s a t i s f y ( 5 . 8 f ) - ( 5 . 8 i ) . Now d e f i n e 142 D i C y u , X ) = max { I i ( « » f y u , X ) - I i (X ±) , 0} , E i(y°,X) = max ' { I ^ X ^ - I ± (.»,yu,X) , 0} , G. (v°,X) = max { K ±, 0) , H i(v°rX) = max {-Ki,0}, where I i ( x ^ ) solves the equation d l ( I i ' X i ) = F and K. l e r t d 1 2 ( i i ( t ) / M i , X i ) q i ( t ) dt - M ig'(X i). 0 Consider the mapping ( ) whose components are defined by (5A.8a) B i(y°,X)=y° + [F-d^ (0,X)-y?] [1-exp (-D^ (y°,X) ) ] - y°[l-exp(-E i(y°,X))] (5A.8b) y-(v?X) = X. + X [l-exp(-G.(y°,X))] l l l - X ± [ l - exp(-H i (y°,X) ) ] This i s a continuous mapping which maps the compact, convex set MxS into i t s e l f . Thus by Brouwer's fixed point theorem, *° the mapping has at least one fixed point(^* ) such that (5A.9a) B i(y*°,X*) = y*° (5A.9b) Y i(v*°,X*) = X* . I t i s now demonstrated that the investment-production p r o f i l e s generated by such a fixed point i s a Cournot-Nash equilibrium. That D i(y°*,X*) = E i(y°*,X*) = 0 follows i4a from an argument i d e n t i c a l t o t h a t used i n the Appendix t o C h a p t e r IV and i s n o t r e p e a t e d h e r e . From t h i s , (5.8g) -( 5 . 8 i ) a r e seen t o h o l d . I t remains t o show t h a t (5.8f) a l s o h o l d s . To see t h i s , suppose t h a t G i ( y*°,X*)>0. Then (5A.8b) i m p l i e s t h a t X = 0, v i o l a t i n g t h e a s s u m p t i o n t h a t X i s s t r i c t l y p o s i t i v e . Thus i t must be t r u e t h a t (y ,X )=0. S i m i l a r l y , i t f o l l o w s from (5A.8b) t h a t H i ( y*°,X*) >0 i m p l i e s X? = 0. Thus (5.8f) i s s a t i s f i e d and t h e p r o o f i s c o m p l e t e . I n t h e s p e c i a l c a s e where p r o p e r t y r i g h t s a r e symmetric, i t i s easy t o see t h a t t h e r e e x i s t s a f i x e d p o i n t * o o f the mapping (5A.8) i n w h i c h a l l t h e components o f y a r e * i d e n t i c a l as a r e t h e components o f X . T h i s e s t a b l i s h e s t h a t t h e r e e x i s t s a symmetric Cournot-Nash e q u i l i b r i u m . 144 C h a p t e r V I SOME SCARCITY IMPLICATIONS OF MARKET STRUCTURE T h i s b r i e f c h a p t e r i s d e v o t e d t o t h e e x a m i n a t i o n o f the s c a r c i t y i m p l i c a t i o n s o f market s t r u c t u r e . There has r e c e n t l y been a surge o f t h e o r e t i c a l i n t e r e s t i n t h e s e a r c h f o r v a r i o u s i n d i c e s t h a t would a d e q u a t e l y r e f l e c t t h e s c a r c i t y o f nonrenewable r e s o u r c e s . The OPEC o i l embargo and t h e Forrester-Meadows p r e d i c t i o n o f a c u t e r e s o u r c e s h o r t -ages i n t h e f u t u r e have been p a r t l y r e s p o n s i b l e f o r t h i s r e -newed i n t e r e s t i n s c a r c i t y i n d i c e s . The c u r r e n t l i t e r a t u r e on t h e s u b j e c t i s summarized i n v a r i o u s p apers c o n t a i n e d i n the volume S c a r c i t y and Growth R e c o n s i d e r e d , e d i t e d by V. K e r r y S m i t h . There a r e e s s e n t i a l l y two s c h o o l s o f tho u g h t on the i s s u e o f t h e s c a r c i t y o f nonrenewable r e s o u r c e s . The neo-c l a s s i c a l s c h o o l , as e x e m p l i f i e d by Kay and M i r r l e e s (1975) and S t i g l i t z ( 1979), f o r example, t a k e s t h e vi e w t h a t non-renewable r e s o u r c e s a r e j u s t l i k e o t h e r i n p u t s used i n t h e p r o d u c t i o n o f manu f a c t u r e d goods. T h e r e f o r e , t h e r e . i s no more r e a s o n t o be co n c e r n e d w i t h t h e s c a r c i t y o f ex-h a u s t i b l e r e s o u r c e s t h a n w i t h t h e s c a r c i t y o f o t h e r i n p u t s . G i v e n a complete s e t o f f u t u r e s m a r k e t s , under c o m p e t i t i v e c o n d i t i o n s n a t u r a l r e s o u r c e s w i l l be c o n v e r t e d i n t o o t h e r forms o f c a p i t a l a t t h e s o c i a l l y o p t i m a l r a t e . F u t u r e g e n e r a t i o n s w i l l be l e f t w i t h s m a l l e r s t o c k s o f n a t u r a l r e s o u r c e s b u t w i t h l a r g e r s t o c k s o f produced c a p i t a l . I t i s a l s o argued t h a t t h e i n c r e a s e i n p r i c e o f an e x h a u s t i b l e r e s o u r c e d u r i n g t h e c o u r s e o f i t s use w i l l n o t o n l y i n c r e a s e e x p l o r a t i o n a c t i v i t y b u t w i l l a l s o b r i n g about t h e use o f s u b s t i t u t e s t h a t were n o t e c o n o m i c a l l y v i a b l e a t lower p r i c e s . F u r t h e r m o r e , i t has been proposed - e s p e c i a l l y by B a r n e t t and Morse (19 63) - t h a t f o r most m i n e r a l s , l o w e r grade o r e i s found i n g r e a t e r abundance th a n h i g h grade o r e . Thus t h e r e i s no cause f o r a n x i e t y o v e r the e x h a u s t i b i l i t y o f t h e s t o c k o f non-renewable n a t u r a l r e s o u r c e s . A d i f f e r e n t v i e w i s t a k e n by some e c o n o m i s t s , e s p e c i a l l y Georgescu-Roegen (1979) . T h i s v i e w c l a i m s t h a t t h e i m p l i c a t i o n s drawn from n e o c l a s s i c a l models a r e based upon t h e a s s u m p t i o n o f some a g g r e g a t e p r o d u c t i o n f u n c t i o n s w h i c h a r e n o t n e c e s s a r i l y c o m p a t i b l e w i t h p h y s i c a l l a w s , and t h a t t h e s e i m p l i c a t i o n s , t h e r e f o r e , a r e s u s p e c t . F u r t h e r m o r e , t h e r e i s t h e q u e s t i o n o f i n t e r g e n e r a t i o n a l e q u i t y . F u t u r e g e n e r a t i o n s w i l l be l e f t w i t h t h e o p t i m a l mix o f n a t u r a l r e s o u r c e s and produced c a p i t a l o n l y i f t h e i r p r e f e r e n c e s a re a d e q u a t e l y r e p r e s e n t e d a t t h e p r e s e n t t i m e . When t h i s i s n o t t h e c a s e , t h e damage done t o f u t u r e 146 g e n e r a t i o n s i s i r r e v o c a b l e s i n c e t h e t r a n s f o r m a t i o n from n a t u r a l r e s o u r c e s t o produced c a p i t a l i s i r r e v e r s i b l e . A n o t h e r s t r a n d o f t h e l i t e r a t u r e has s i d e - s t e p p e d t h e i s s u e o f why t h e s c a r c i t y o f e x h a u s t i b l e r e s o u r c e s i s o f more c o n c e r n t h a n t h a t o f o t h e r r e s o u r c e s , and has a d d r e s s e d i t s e l f t o t h e u s e f u l n e s s o f t h e u s u a l s c a r c i t y i n d i c e s from an e m p i r i c a l p o i n t o f v i e w . Thus Brown and F i e l d (1979) and F i s h e r (1979) , f o r example, have i n v e s t i -g a t e d t h e advantages and t h e d i s a d v a n t a g e s o f u s i n g t h e t h r e e t r a d i t i o n a l measures o f s c a r c i t y : u n i t c o s t o f e x t r a c t i v e o u t p u t , t h e r e n t a l v a l u e o f t h e r e s o u r c e , and t h e p r i c e o f t h e e x t r a c t e d r e s o u r c e . The p r e s e n t c h a p t e r i s c o n c e r n e d w i t h t h i s s t r a n d o f t h e l i t e r a t u r e . The u n i t c o s t measure o f s c a r c i t y was used by B a r n e t t and Morse (19 63) i n t h e i r monumental s t u d y o f t r e n d s i n n a t u r a l r e s o u r c e s c a r c i t y i n the U.S. s i n c e the C i v i l War. T h i s i n d e x , w h i c h i s t h o r o u g h l y c l a s s i c a l i n i t s n a t u r e , has t h e d e f e c t o f v i e w i n g s c a r c i t y e n t i r e l y from s u p p l y -s i d e c o n s i d e r a t i o n s and i g n o r i n g t h e demand s i d e c o m p l e t e l y . F i s h e r (1979) has r e q u i r e d t h a t any s c a r c i t y i n d e x s h o u l d have t h e p r o p e r t y t h a t i t must "summarize th e s a c r i f i c e s , d i r e c t and i n d i r e c t , made t o o b t a i n a u n i t o f t h e r e s o u r c e " . The u n i t c o s t i n d e x o f s c a r c i t y measures o n l y the d i r e c t c o s t o f e x t r a c t i n g a n o t h e r u n i t o f t h e r e s o u r c e . The e x t r a c t i o n o f t h e m a r g i n a l u n i t i n v o l v e s , i n a d d i t i o n t o t h i s , an " i n d i r e c t " c o s t t h a t r e f l e c t s the v a l u e o f f u t u r e consumption f o r e g o n e by c u r r e n t e x t r a c t i o n . The t r u e c o s t 147 o f r e s o u r c e u t i l i z a t i o n must i n c l u d e t h i s i n d i r e c t c o s t , o r s c a r c i t y r e n t as i t has been c a l l e d i n t h e p r e v i o u s c h a p t e r s . The s c a r c i t y r e n t i s i t s e l f a measure o f s c a r c i t y - not o f t h e e x t r a c t e d r e s o u r c e , b u t o f t h e r e -s o u r c e i n s i t u , i . e . , i n t h e ground. I t r e p r e s e n t s t h e ( c u r r e n t ) v a l u e o f an a d d i t i o n a l u n i t o f t h e r e s o u r c e i n t h e ground. The most comprehensive i n d e x o f s c a r c i t y , o f c o u r s e , i s t h e p r i c e o f t h e e x t r a c t e d r e s o u r c e s i n c e i t i n c o r p o r a t e s b o t h the d i r e c t and t h e i n d i r e c t c o s t s o f r e -s o u r c e u t i l i z a t i o n . An i n c r e a s e i n t h e p r i c e o f t h e ex-t r a c t e d r e s o u r c e ( r e l a t i v e t o o t h e r commodities) unambig-u o u s l y i m p l i e s i n c r e a s e d s c a r c i t y o f t h e r e s o u r c e , whatever t h e u n d e r l y i n g cause o f t h e p r i c e i n c r e a s e . I n t h e c o n t e x t o f r e s o u r c e a v a i l a b i l i t y , p h y s i c a l measures such as "proven r e s e r v e s " a r e o f t e n q u o t e d . By "proven r e s e r v e s " i s meant the amount o f r e s e r v e s e c o n o m i c a l l y r e c o v e r a b l e a t t h e c u r r e n t p r i c e l e v e l . B u t , as d e p l e t i o n o f t h e r e s o u r c e p r o c e e d s i t s p r i c e r i s e s , t h e r e b y i n c r e a s i n g t h e volume o f r e c o v e r a b l e r e s e r v e s . Thus the volume o f proven r e s e r v e s i s a t b e s t a l o w e r bound on t h e amount o f r e c o v e r a b l e r e s e r v e s . I n models o f t h e t y p e c o n s i d e r e d i n t h e p r e v i o u s c h a p t e r s , t h e amount o f the r e s o u r c e u l t i m a t e l y r e c o v e r a b l e i s d e t e r m i n e d endogenously by t h e i n t e r a c t i o n o f t h e demand f o r t h e r e s o u r c e , t h e c o s t c o n d i t i o n s p e r t a i n i n g t o r e s o u r c e r e c o v e r y and market s t r u c t u r e . W h i l e t h e volume o f u l t i m a t e l y r e c o v e r a b l e 148 r e s e r v e s i s a p h y s i c a l measure, w h i c h by i t s e l f i s n o t m e a n i n g f u l , t h e comparison o f u l t i m a t e r e c o v e r i e s o f t h e r e s o u r c e i n d i f f e r e n t c i r c u m s t a n c e s I s m e a n i n g f u l . S i n c e s c a r c i t y i s a r e l a t i v e c o n c e p t , t h e volume o f u l t i m a t e r e c o v e r y recommends i t s e l f as a v a l i d and p o t e n t i a l l y use-f u l s c a r c i t y i n d e x . Much o f t h e r e c e n t c o n c e r n o v e r the a v a i l a b i l i t y o f e x h a u s t i b l e r e s o u r c e s was t r i g g e r r e d by the o i l embargo o f 1973, f o l l o w i n g OPEC's newly d i s c o v e r e d monopoly power. I n s p i t e o f t h i s , no s y s t e m a t i c s t u d y o f the e f f e c t o f market s t r u c t u r e on s c a r c i t y i n d i c e s e x i s t s . The model de v e l o p e d i n t he p r e v i o u s c h a p t e r s i s w e l l - s u i t e d f o r t h i s purpose and i s now b r i e f l y examined f o r i t s s c a r c i t y i m p l i c a t i o n s . F i r s t , t he assu m p t i o n made i n the p r e v i o u s c h a p t e r t h a t d^-^;iO, t h a t i s , t h a t t h e m a r g i n a l r e c o v e r y c o s t i n -c r e a s e s a t a n o n - i n c r e a s i n g r a t e , i s g i v e n an a l t e r n a t i v e i n t e r p r e t a t i o n . T h i s assumption i s used i n P r o p o s i t i o n 6.1 below. Lemma 6.1 The a s s u m p t i o n t h a t d^^I ,X) <_0 i s e q u i v a l e n t t o the a s s u m p t i o n t h a t w i t h i n each d e p o s i t , the r e s o u r c e becomes no l e s s abundant as the m a r g i n a l r e c o v e r y c o s t i n c r e a s e s , P r o o f : F o r a g i v e n i n i t i a l i n v estment, X, l e t u ( c ) dc be t h e amount o f r e s o u r c e a v a i l a b l e i n a d e p o s i t i n the m a r g i n a l recovery cost i n t e r v a l [c,c+dc]. Letting c be the lowest marginal recovery cost, i t follows that the t o t a l amount of the resource available i n the marginal cost range [c,c] i s given by (6.1) I = u(.c) dc = F (c) , c where F(c) i s the d i s t r i b u t i o n function corresponding to the density function u(c), and s a t i s f i e s (6.2) F' (c) = u(c) > 0 , F"(c) 1 0 as u* (c) ^ 0. D i f f e r e n t i a t i o n of (6.1) with respect to I y i e l d s . / N dc 1 = F ' ( C ) so that dc 1 (6.3) d I F . ( c ) i . e . , the marginal recovery cost increases at a rate that i s inversely proportional to the density of the ore at that l e v e l of marginal cost. D i f f e r e n t i a t i o n of (6.3) with respect to I re s u l t s i n i r d 2c _ F" (c) dc _ F" (c) ( 6 . 4 ) j - - 2 rff ~ ~ 3 d l ^ [F« (c)r [F' ( c ) ] J On noting that c= d 1(.I,X), i t follows from (6.2) and (6.4) that (6.5) d i ; L 1(I,X) 1 0 according as u' (c) 1. 0. Q.E.D. 150 I n v i e w o f t h e above lemma, t h e assu m p t i o n t h a t d^ 1^(.I,X) i 0 seems q u i t e r e a s o n a b l e . I n f a c t , B a r n e t t and Morse (1963) suggest t h a t one o f t h e r e a s o n s why, a c c o r d i n g t o t h e i r s t u d y , e x h a u s t i b l e n a t u r a l r e s o u r c e s a r e not growing s c a r c e r o v e r t i m e i s t h a t l o w e r grade o r e s o f most m i n e r a l s a r e a v a i l a b l e much more a b u n d a n t l y t h a n h i g h e r grade o r e s . 1 The f o l l o w i n g p r o p o s i t i o n examines t h e t e m p o r a l b e h a v i o u r o f t h e p r e s e n t v a l u e shadow p r i c e and t h e s c a r c i t y r e n t o f t h e r e s o u r c e t o each f i r m i n t h e symmetric C o u r n o t -Nash e q u i l i b r i u m . As p o i n t e d o u t above, t h e s e s c a r c i t y i n d i c e s a r e the r e l e v a n t ones f o r t h e r e s o u r c e i n t h e i n s i t u c o n t e x t . L e t \ ( t ) and y ( t ) be t h e p r e s e n t v a l u e shadow p r i c e and the s c a r c i t y r e n t , r e s p e c t i v e l y , o f t h e r e s o u r c e a t t i m e t . These two q u a n t i t i e s a r e , o f c o u r s e , r e l a t e d by (6.6) X ( t ) = e " r t y ( t ) i . e . , t h e p r e s e n t v a l u e shadow p r i c e i s t h e s c a r c i t y r e n t d i s c o u n t e d back t o time z e r o . P r o p o s i t i o n 6.1 The p r e s e n t v a l u e shadow p r i c e and t h e s c a r c i t y r e n t o f t h e r e s o u r c e t o each f i r m i n t h e symmetric Cournot-Nash e q u i l i b r i u m a r e m o n o t o n i c a l l y d e c l i n i n g over t i m e . P r o o f : D r o p p i n g t h e s u b s c r i p t s i n t h e n e c e s s a r y c o n d i t i o n (5.8b) o f t h e p r e v i o u s c h a p t e r and u s i n g (6.6) i t f o l l o w s t h a t 151 (6.7) H t ) = - e " r t d i : L C l ( t ) / M , X ) Q ( t ) / M , where Q(t) and I ( t ) a r e t h e o u t p u t r a t e and c u m u l a t i v e out-p u t t o d a t e o f t h e i n d u s t r y , r e s p e c t i v e l y . S i n c e d^>0 and Q(t)>0 f o r a l l t^O, i t f o l l o w s t h a t (6.8) X (t) < 0 f o r a l l t>:0. F u r t h e r , t h e s o l u t i o n t o the d i f f e r e n t i a l e q u a t i o n ( 5 . 8 b ) , may be w r i t t e n i n t h e i n t e g r a l form (6.9) V (t) = e r t e " r S d 1 1 ( I ( s ) / M , X ) 9151 ds t D i f f e r e n t i a t i o n o f (6.9) w i t h r e s p e c t t o time y i e l d s M t ) = - d,, ( I ( t ) / M , X ) Q ( t ) / M l l l + r e e - r s d i ; L ( I ( s ) / M , X ) ds t I n t e g r a t i n g t h e second term on the r i g h t hand s i d e by p a r t s , t h e above e x p r e s s i o n may be c a s t i n t h e form (6.10) y ( t ) = t 111 M2 11 M where use has been made o f t h e f a c t t h a t l i m Q(t) = 0. S i n c e by p a r t (a) o f P r o p o s i t i o n 5.3 Q(s) <0 f o r a l l s^O and by Assumption (Al) ' d i ; L>0, d ^ ^ O , i t f o l l o w s from (6.10) t h a t (6.11) v ( t ) < 0 f o r a l l t^O. Q.E.D. 152 The r e s u l t t h a t t h e p r e s e n t v a l u e shadow p r i c e o f t h e r e s o u r c e d e c l i n e s o ver t i m e i s n o t s u r p r i s i n g . As t h e low c o s t r e s e r v e s w i t h i n each d e p o s i t a r e d e p l e t e d , t h e f i r m s a r e f o r c e d t o r e c o v e r a d d i t i o n a l amounts o f t h e r e -c o u r c e a t h i g h e r and h i g h e r c o s t and, as a r e s u l t , t h e shadow p r i c e o f t h e r e s o u r c e d e c l i n e s o v e r t i m e . That th e s c a r c i t y r e n t s h o u l d a l s o e x h i b i t t h i s m o n o t o n i c i t y p r o p e r t y , however, i s n o t t r a n s p a r e n t . W h i l e p a r t (d) o f P r o p o s i t i o n 5.3 e s t a b l i s h e s t h a t t h e s c a r c i t y r e n t must u l t i m a t e l y d e c l i n e t o z e r o , P r o p o s i t i o n 6.1 f u r t h e r de-m o n s t r a t e s t h a t t h e s c a r c i t y r e n t must, i n f a c t , monotoni-c a l l y d e c l i n e t o z e r o . T h i s m o n o t o n i c i t y i s g u a r a n t e e d by t h e a s s u m p t i o n t h a t t h e m a r g i n a l r e c o v e r y c o s t w i t h i n each d e p o s i t i n c r e a s e s a t a n o n - i n c r e a s i n g r a t e w i t h r e s p e c t t o c u m u l a t i v e r e c o v e r y . S i n c e , by Lemma 6.1, more o f t h e r e -s o u r c e becomes a v a i l a b l e as t h e grade of t h e o r e d e c r e a s e s , i t i s i n k e e p i n g w i t h i n t u i t i o n t h a t t h e s c a r c i t y r e n t de-c l i n e s o v e r t i m e . I t must be emphasized t h a t t h e r e s u l t s o f t h e above p r o p o s i t i o n a r e v a l i d f o r a l l market s t r u c t u r e s - as l o n g as p r o p e r t y r i g h t s a r e symmetric. The r e a s o n f o r t h i s i s t h a t t h e s e r e s u l t s have been d e r i v e d by l o o k i n g o n l y a t ( 5 . 8 b ) , v i z . , t h e e q u a t i o n o f m o t i o n f o r t h e c o s t a t e v a r i a b l e , y (t) , w h i c h depends o n l y i m p l i c i t l y on t h e market s t r u c t u r e ( t h r o u g h i t s dependence on t h e r a t e o f e x t r a c t i o n ) . M arket s t r u c t u r e e n t e r s e x p l i c i t l y o n l y i n t o c o n d i t i o n ( 5 . 8 a ) . d 1 ( I , X ) = -j 153 The r e s u l t s o f P r o p o s i t i o n 6.1 a r e i n s h a r p c o n t r a s t t o t h o s e o b t a i n e d i f t h e m a r g i n a l r e c o v e r y c o s t i s c o n s t a n t . Suppose, f o r example, t h a t t h e m a r g i n a l r e c o v e r y c o s t has t h e form r f ( X ) f o r I < J °° f o r I>I where f ( - ) > 0 , f'(-)<0 and f " (-)>0. I n t h i s c a s e i t f o l l o w s from (6.7) t h a t t h e p r e s e n t v a l u e shadow p r i c e o f t h e r e s o u r c e i s i n d e p e n d e n t o f t ime and from (6.6) t h a t t h e s c a r c i t y r e n t r i s e s e x p o n e n t i a l l y w i t h t i m e . T h i s d e m o n s t r a t e s what F i s h e r (1979) has c a l l e d a " d u a l i t y " i n t h e b e h a v i o u r s o f t h e u n i t e x t r a c t i o n c o s t and t h e s c a r c i t y r e n t . When the u n i t e x t r a c -t i o n c o s t remains c o n s t a n t - t h e r e b y f a i l i n g t o s i g n a l im-p e n d i n g e x h a u s t i o n - t h e s c a r c i t y r e n t r i s e s and s i g n a l s r e s o u r c e s c a r c i t y . On t h e o t h e r hand, i f t h e u n i t e x t r a c t i o n c o s t i n c r e a s e s w i t h t ime - s i g n a l l i n g r e s o u r c e d e p l e t i o n -t h e s c a r c i t y r e n t does n o t n e c e s s a r i l y r i s e . (However, when t h e r e i s a d e p l e t i o n e f f e c t o p e r a t i n g , p h y s i c a l e x h a u s t i o n i s not o f p r i m a r y c o n c e r n ; w e l l b e f o r e t h e r e s o u r c e becomes p h y s i c a l l y e x h a u s t e d i t i s l i k e l y t o become une c o n o m i c a l t o e x p l o i t , and a d e c l i n i n g s c a r c i t y r e n t does i n d i c a t e t h i s ) . The f o l l o w i n g p r o p o s i t i o n i n v e s t i g a t e s how changes i n market s t r u c t u r e r e g i s t e r on the v a r i o u s s c a r c i t y i n d i c e s . I n p a r t i c u l a r , t h e e f f e c t on t h r e e s c a r c i t y i n d i c e s i s i n v e s t i g a t e d , namely, t h e u l t i m a t e r e s o u r c e r e c o v e r y o f t h e i n d u s t r y , t h e p r i c e o f the e x t r a c t e d r e s o u r c e and t h e 154 s c a r c i t y r e n t o f t h e r e s o u r c e . I n o r d e r t o s i m p l i f y t h e a n a l y t i c s i n t h e d e t e r m i n a t i o n o f t h e e f f e c t on t h e s c a r c i t y r e n t , i t i s c o n v e n i e n t t o assume (AlO) The c o s t f u n c t i o n d ( I , X ) i s such t h a t d 1 2 ( I , X ) = - d 1 ] L ( I , X ) f o r a l l I,X>0 The above ass u m p t i o n i m p l i e s t h a t t h e u n d e r t a k i n g o f i n i t i a l i n v e s t m e n t s h i f t s t h e m a r g i n a l c o s t s c h e d u l e l a t e r a l l y t o the r i g h t : an i n c r e a s e i n i n i t i a l i n v e s t m e n t by one u n i t r e d u c e s the m a r g i n a l c o s t by e x a c t l y as much as would a r e d u c t i o n i n c u m u l a t i v e e x t r a c t i o n by one u n i t . 3 The e f f e c t o f an i n c r e a s e i n the number o f f i r m s i n th e i n d u s t r y on t h e s c a r c i t y i n d i c e s i n a symmetric Cournot-Nash e q u i l i b r i u m may now be s t a t e d : P r o p o s i t i o n 6.2 An i n c r e a s e i n t h e number o f f i r m s i n the i n d u s t r y (a) i n c r e a s e s t h e u l t i m a t e amount o f r e s o u r c e r e c o v e r e d by t h e i n d u s t r y (b) d e c r e a s e s the r e s o u r c e p r i c e - a t l e a s t i n i t i a l l y (c) i n i t i a l l y i n c r e a s e s t h e s c a r c i t y r e n t o f t h e r e s o u r c e . P r o o f : (a) A l r e a d y e s t a b l i s h e d i n p a r t (c) o f P r o p o s i t i o n 5.5. (b) L e t {Q(t,N)} and { I ( t , N ) } denote the o u t p u t and c u m u l a t i v e e x t r a c t i o n p r o f i l e s , r e s p e c t i v e l y , . , o f an i n d u s t r y c o m p r i s e d o f N f i r m s . I n p a r t (a) o f 155 Proposition 5.5 i t was established that (6.12) d ^ C t , N ) > 0 f ° r a l l t>0. dN Since t I(t,N) = Q(s,N) ds , ^0 i t follows, upon taking the l i m i t t+0 i n (6.12), that (6.13) >0. O.N Since the output i s higher at t=0 for an industry with (N+1) firms than for one with N firms, the price must be lower i n the former case. Further, since the price p r o f i l e s are continuous i n time i t follows that there must e x i s t a non-zero time i n t e r v a l , beginning at t=0, during which the resource price i s lower when N i s increased. (c) Under Assumption (A10), expression (6.9) for the s c a r c i t y rent may be rewritten /, \ r t y (t) = -e e " r S d 1 2 ( I ( s ) / M , X ) 3 1 ^ - d s . t The s c a r c i t y rent of the resource at time t=0 i s thus given by (6.14) y (0) = - e" r Sd 1 2(I(s)/M,X) ^ } d s . 0 The f i r s t order condition (5.8f) becomes (assuming an i n t e r i o r solution for X i n the symmetric Cournot-Nash equilibrium) 156 (6.15) y (0) = g< CX) C o n d i t i o n (6.15) r e q u i r e s i n i t i a l i n v e s t m e n t on each d e p o s i t t o be u n d e r t a k e n up to a l e v e l a t w h i c h t h e m a r g i n a l i n v e s t m e n t c o s t e q u a l s t h e i n i t i a l shadow p r i c e o f t h e r e -s o u r c e . (Note t h a t a t t=0, the s c a r c i t y r e n t and the shadow p r i c e o f t h e r e s o u r c e a r e seen t o be i d e n t i c a l from ( 6 . 6 ) ) . T o t a l d i f f e r e n t i a t i o n o f (6.15) w i t h r e s p e c t t o N y i e l d s dy(0) _ „ . . dX dN " g ( X ) dN ' w h i c h , upon u s i n g Assumption ( A 7 ) and p a r t (b) o f P r o p o s i -t i o n 5.5, e s t a b l i s h e s t h a t dy(0) (6.16) dN U ' The c o n t i n u i t y o f t h e i n d u s t r y o u t p u t i n time i m p l i e s t h e c o n t i n u i t y o f the s c a r c i t y r e n t ( v i a ( 6 . 9 ) ) . From t h i s f a c t and (6.16) i t f o l l o w s t h a t t h e r e e x i s t s a non-zero i n t e r v a l o f t i m e , s t a r t i n g a t t=0, d u r i n g w h i c h the s c a r c i t y r e n t i s h i g h e r f o r an i n d u s t r y w i t h N+l f i r m s t h a n f o r one w i t h N f i r m s . Q.E.D. I f t h e u l t i m a t e r e s o u r c e r e c o v e r y o f t h e i n d u s t r y i s used as a s c a r c i t y i n d e x , t h e n a c c o r d i n g t o p a r t (a) o f P r o p o s i t i o n 6.2, an i n c r e a s e i n c o m p e t i t i o n i n t h e i n d u s t r y would r e g i s t e r a d e c r e a s e i n r e s o u r c e s c a r c i t y . T h i s c o n c l u s i o n would be r e i n f o r c e d i f the p r i c e o f the e x t r a c t e d r e s o u r c e i s chosen as t h e r e l e v a n t s c a r c i t y i n d e x : 157 a move towards g r e a t e r c o m p e t i t i o n r e s u l t s i n l o w e r r e s o u r c e p r i c e s - a t l e a s t d u r i n g t h e i n i t i a l p e r i o d s 4 o f e x t r a c t i o n . S i n c e t i m e d i s c o u n t i n g p u t s a g r e a t e r w e i g h t on t h e e a r l i e r p e r i o d s , t h i s i m p l i e s t h a t i n c r e a s e d c o m p e t i t i o n l o w e r s r e s o u r c e s c a r c i t y . However, i f the s c a r c i t y r e n t i s chosen as t h e r e l e v a n t s c a r c i t y i n d e x , p a r t (c) o f t h e above p r o p o s i t i o n i m p l i e s t h a t e x a c t l y t h e o p p o s i t e c o n c l u s i o n w i l l be r e a c h e d : i n c r e a s e d com-p e t i t i o n r a i s e s t h e s c a r c i t y r e n t o f t h e r e s o u r c e i n the e a r l y phase o f e x t r a c t i o n . T h i s a p p a r e n t l y p a r a d o x i c a l r e s u l t a r i s e s from t h e f a c t t h a t w h i l e i n c r e a s e d com-p e t i t i o n l o w e r s t h e i n i t i a l p r i c e o f t h e r e s o u r c e , i t l e s s e n s t h e gap between p r i c e and t h e m a r g i n a l revenue t o each f i r m . The l a t t e r e f f e c t dominates and t h e n e t m a r g i n a l revenue ( i . e . , t h e s c a r c i t y r e n t ) i n c r e a s e s i n t h e i n i t i a l p e r i o d s . I t i s , . i n f a c t , the i n c r e a s e i n the i n i t i a l shadow p r i c e o f t h e r e s o u r c e t h a t r e s u l t s i n more o f t h e r e s o u r c e b e i n g r e c o v e r e d as t h e i n d u s t r y becomes more c o m p e t i t i v e . I n c o n c l u s i o n , t h e main f i n d i n g s o f t h i s c h a p t e r w i l l be b r i e f l y summarized. F i r s t l y , i t has been shown t h a t t h e t e m p o r a l b e h a v i o u r o f t h e s c a r c i t y r e n t i n t h e p r e s e n c e o f a d e p l e t i o n e f f e c t d i f f e r s d r a s t i c a l l y from t h a t when r e c o v e r y can p r o c e e d a t c o n s t a n t m a r g i n a l c o s t . S e c o n d l y , t h e u l t i m a t e amount o f t h e r e s o u r c e r e c o v e r e d by t h e i n d u s t r y has been proposed as a m e a n i n g f u l and p o t e n t i a l l y u s e f u l s c a r c i t y i n d e x . F i n a l l y , i t was 158 d e monstrated t h a t changes i n market s t r u c t u r e may r e g i s t e r d i f f e r e n t l y on t h e v a r i o u s s c a r c i t y i n d i c e s . I n p a r t i c u l a r , i n c r e a s e d c o m p e t i t i o n was shown t o l o w e r t h e r e s o u r c e p r i c e b u t r a i s e i t s s c a r c i t y r e n t . Thus i t i s v e r y i m p o r t a n t t o d i s t i n g u i s h between t h e e x t r a c t e d r e s o u r c e and t h e r e s o u r c e i n s i t u . I t must be mentioned t h a t an e n t i r e l y s a t i s f a c t o r y t r e a t m e n t o f r e s o u r c e s c a r c i t y must n o t o n l y be f o r m u l a t e d w i t h i n t h e framework o f a r e a s o n a b l y g e n e r a l and f l e x i b l e market s t r u c t u r e , b u t must a l s o m e a n i n g f u l l y i n c o r p o r a t e t h e e x i s t e n c e o f i m p e r f e c t s u b s t i t u t e s f o r t h e r e s o u r c e i n q u e s t i o n . As Solow (1974) has p o i n t e d o u t , i f i t i s easy t o s u b s t i t u t e f o r t h e r e s o u r c e b e i n g d e p l e t e d , i t s e x h a u s t i o n would m e r e l y be an e v e n t , n o t a c a t a s t r o p h e ; r e s o u r c e s u b s t i t u t a b i l i t y l i e s a t t h e h e a r t o f t h e problem o f s c a r c i t y . The model p r o v i d e d i n t h i s t h e s i s i s n o t e n t i r e l y adequate f o r a n a l y s i n g q u e s t i o n s o f r e s o u r c e s c a r c i t y s i n c e t h e p r e s e n c e o f i m p e r f e c t sub-s t i t u t e s f o r t h e r e s o u r c e has been assumed away. T h i s drawback, however, p o i n t s t o a d e f i n i t e a r e a i n w h i c h 5 f u t u r e r e s e a r c h s h o u l d prove f r u i t f u l . 159 ypwfwntps t o C h a p t e r VI 1. 3. 5. R e c e n t l y , B r o b s t (1979) has s u g g e s t e d an a l t e r n a t i v e (but as y e t unproven) h y p o t h e s i s i n w h i c h , a t t h e a g g r e g a t e l e v e l , t h e o r e o f c e r t a i n elements i s n o t a v a i l a b l e i n g r e a t e r q u a n t i t i e s a t l o w e r g r a d e s . R e s u l t s s i m i l a r t o t h o s e o f P r o p o s i t i o n 6.1 may be found i n H e a l (1976) and Hanson (1979) and (1980) i n t h e s p e c i a l c a s e o f p e r f e c t c o m p e t i t i o n . The method o f p r o o f , e s p e c i a l l y i n Hanson (1980), e x p l i c i t l y r e -l i e s : on t h e a s s u m p t i o n t h a t t h e agents a r e p r i c e - t a k e r s . No such a s s u m p t i o n i s made i n t h e p r o o f o f P r o p o s i t i o n 6.1 (A10) would be s a t i s f i e d , f o r example, f o r a c o s t f u n c t i o n o f t h e form d ( I , X ) = d ( I - X ) T h i s form i s p a r t i c u l a r l y r e l e v a n t t o t h e c a s e o f o i l , where w a t e r i s o f t e n pumped i n t o t h e r e s e r v o i r s i n o r d e r t o m a i n t a i n t h e p r e s s u r e t h a t b r i n g s t h e o i l t o the s u r f a c e . I f X be t h e volume o f w a t e r pumped i n , t h e n t h e p r e s s u r e w i t h i n t h e r e s e r v o i r and/ t h e r e -f o r e , t h e m a r g i n a l r e c o v e r y c o s t w i l l be a f u n c t i o n o f I-X, where I i s t h e c u m u l a t i v e amount of o i l w i t h -drawn from t h e r e s e r v o i r . See U h l e r (1978) f o r d e t a i l s . I t i s p o s s i b l e t h a t t h e e n t i r e p r i c e p r o f i l e f o r an i n d u s t r y c o m p r i s e d o f (N+1) f i r m s l i e s below t h a t f o r an i n d u s t r y w i t h N f i r m s . However, i t does n o t appear p o s s i b l e t o d emonstrate t h i s i n g e n e r a l . Eswaran and L e w i s (1981) a t t e m p t s t o r e c t i f y p r e c i s e l y t h i s drawback by e x p l i c i t l y c o n s i d e r i n g t h e e x i s t e n c e o f r e s o u r c e s t h a t are i m p e r f e c t s u b s t i t u t e s , o r even complements, i n p r o d u c t i o n . 160 C h a p t e r V I I CONCLUDING REMARKS T h i s t h e s i s has examined i n some d e t a i l t h e u t i l i z a t i o n o f an e x h a u s t i b l e r e s o u r c e i n o l i g o p o l i s t i c m a r k e t s . The number o f p r o d u c e r s i n t h e i n d u s t r y has been a l l o w e d t o be a r b i t r a r y , and t h e i n t e r a c t i o n s between t h e s e p r o d u c e r s has been m o d e l l e d as a n o n - c o o p e r a t i v e dynamic game. The o u t s t a n d i n g f e a t u r e o f t h e model de-v e l o p e d i n t h i s t h e s i s i s t h e f a c t t h a t t h e amount o f r e -s o u r c e t h a t each f i r m u l t i m a t e l y r e c o v e r s i s a l l o w e d t o be endogenously d e t e r m i n e d w i t h i n a g a m e - t h e o r e t i c framework. I n s p i t e o f t h e c o m p l e x i t y o f t h e r e s u l t i n g model, many t e s t a b l e p r e d i c t i o n s have been d e r i v e d w h i c h enhance our u n d e r s t a n d i n g - a t l e a s t a t t h e t h e o r e t i c a l l e v e l - o f t h e m a j o r i t y o f t h e e x h a u s t i b l e r e s o u r c e markets t h a t can n e i t h e r be c l a s s e d as p e r f e c t l y c o m p e t i t i v e nor as pure m o n o p o l i e s . As might be e x p e c t e d , some r e a l - w o r l d p o s s i b i l i t i e s have had t o be r u l e d o u t i n o r d e r t o keep t h e problem 161 manageable. I m e n t i o n o n l y two o f t h e s e h e r e . F i r s t l y , a l t h o u g h t h e u l t i m a t e r e c o v e r y o f each f i r m was made endogenous i n t h i s t h e s i s , t h e number of r e s o u r c e d e p o s i t s t h e f i r m e x p l o i t s was assumed t o be e x o g e n o u s l y g i v e n . I n t h e r e a l w o r l d , o f c o u r s e , f i r m s c o n t i n u o u s l y engage i n e x p l o r a t i o n f o r new d e p o s i t s w h i l e e x t r a c t i n g from known d e p o s i t s . However, t h e i n c o r p o r a t i o n o f such a phenomenon i n t o t h e model p r e s e n t e d r e n d e r s t h e dynamic game i n -t r a c t a b l e . S i n c e t h e p r i m a r y f o c u s o f t h i s t h e s i s was t h e r o l e o f market s t r u c t u r e on r e s o u r c e u t i l i z a t i o n , r e a l i s m i n t h e m o d e l l i n g o f t h e e x p l o r a t i o n a c t i v i t i e s o f f i r m s has been s a c r i f i c e d t o a c e r t a i n e x t e n t . S e c o n d l y , i t has been assumed t h a t p r o d u c e r s ' s t r a t e g i e s can be a d e q u a t e l y r e p r e s e n t e d by "open-loop" Cournot-Nash s t r a t e g i e s . The a s s u m p t i o n o f n o n - c o o p e r a t i v e b e h a v i o u r on the p a r t o f p r o d u c e r s has been j u s t i f i f e d i n C h a p t e r I I I . However, t h e a s s u m p t i o n t h a t p r o d u c e r s * s t r a t e g i e s a r e open-loop i s l e s s b e l i e v a b l e . Such an a s s u m p t i o n might be a p p r o p r i a t e when f i r m s can precommit t h e i r o u t p u t p r o f i l e s i n advance a s , f o r example, when t h e r e e x i s t s a complete s e t o f f u t u r e s m a r k e t s . I n the r e a l w o r l d - w h i c h i s t y p i c a l l y c h a r a c t e r i z e d by a l a c k o f f u t u r e s markets - t h e r e l e v a n t s t r a t e g i e s a r e l i k e l y t o be " c l o s e d - l o o p " , i . e . , depend n o n - t r i v i a l l y on t h e c u r r e n t s t a t e o f t h e w o r l d . However, w h i l e c l o s e d - l o o p s t r a t e g i e s a r e more m e a n i n g f u l t h a n open-loop s t r a t e g i e s , t h e y a r e a l s o much l e s s t r a c t a b l e . C o n s i d e r a t i o n o f c l o s e d - l o o p 162 s t r a t e g i e s i n a model as comprehensive as t h e one de-v e l o p e d i n t h i s t h e s i s i s o u t o f t h e q u e s t i o n , g i v e n the c u r r e n t s t a t e o f our knowledge o f dynamic games. None-t h e l e s s , I f o r e s e e t h a t i n t h e near f u t u r e such s t r a t e g i e s w i l l be g i v e n c a r e f u l c o n s i d e r a t i o n i n t h e t h e o r e t i c a l l i t e r a t u r e - even i f i n much s i m p l e r c o n t e x t s . i6a REFERENCES B a r n e t t , H.J. and C. Morse (1963). Sca.fic.lty and Growth: The Economics o& Natural Re-60uA.ce A v a i l a b i l i t y , Johns Hopkins U n i v e r s i t y P r e s s f o r Resources f o r t h e F u t u r e , B a l t i m o r e . B r o b s t , D.A. 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