UBC Theses and Dissertations

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UBC Theses and Dissertations

Perspectives on oligopoly theory MacLeod, William Bentley 1983

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PERSPECTIVES ON OLIGOPOLY THEORY by WILLIAM BENTLEY MACLEOD B . A . , Queen ' s U n i v e r s i t y , 1975 M S c , Queen ' s U n i v e r s i t y , 1979 A THESIS SUBMITTED IN PARTIAL,FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Depar tment o f Economics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA JANUARY 1984 0 W i l l i a m B e n t l e y MacLeod , 1983 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Ma l l V a n c o u v e r , C a n a d a V6T 1Y3 D a t e T « - 3 / ^ - i i " -PERSPECTIVES ON OLIGOPOLY THEORY A b s t r a c t T h i s t h e s i s i s a c o l l e c t i o n o f f o u r e s s a y s s t u d y i n g t he o l i -g opo l y p rob l em f rom v a r i o u s p e r s p e c t i v e s . In Chap t e r 1 i t i s demon-s t r a t e d t h a t t hough demand c u r v e s a r e g e n e r a l l y c o n t i n u o u s i n d i f f e r e n -t i a t e d p r o d u c t m o d e l s , the n o n - e x i s t e n c e o f a Nash e q u i l i b r i a i s a g e n e r i c p r o p e r t y . C h a p t e r 2 shows t h a t t h e use o f the c l a s s i c a l C o u r n o t -Nash e q u i l i b r i u m i n q u a n t i t i e s i s o f t e n e q u i v a l e n t t o t h e use o f t h e c o r e s o l u t i o n c o n c e p t . C h a p t e r 3 a p p l i e s t h e t h e o r y o f l o c a l games t o t he o l i g o p o l y p rob l em t o show the r e l a t i o n s h i p between t h e l o c a l s t a -b i l i t y o f e q u i l i b r i a and a d j u s tmen t c o s t s . 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 s f o r s t a b i l i t y a r e p r o v i d e d wh i c h w i l l o f t e n y i e l d a un i que l o c a l l y s t a b l e e q u i l i b r i u m . F i n a l l y Chap t e r 4 s t u d i e s t h e r a t i o n a l i t y o f b o t h c o n s c i o u s p a r a l l e l i s m and p r e d a t o r y p r i c i n g i n a dynamic model w i t h f r e e e n t r y . An a x i o m a t i c f o u n d a t i o n f o r c o n s c i o u s p a r a l l e l i s m i s p r o v i d e d wh i c h i s t h en used t o show t h a t p r e d a t o r y p r i c i n g , w i d e l y de -f i n e d , i s a lway s r a t i o n a l i n t he f a c e o f p o t e n t i a l e n t r y . By r a t i o n a l we mean subgame p e r f e c t Nash e q u i l i b r i a i n the sense o f S e l t e n [ 1 9 7 5 ] . - i i i -T a b l e o f C o n t e n t s S e c t i o n Page A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f F i g u r e s v Acknow ledgements v i I n t r o d u c t i o n 1 C h a p t e r 1: On the N o n - e x i s t e n c e o f E q u i l i b r i a i n D i f f e r e n t i a t e d P r o du c t Mode l s 5 1. I n t r o d u c t i o n . 5 2. The Model 7 3 . The F i r m ' s P rob l em 11 4 . C o n c l u d i n g Comments 17 5. No te s . . 20 Chap t e r 2: The Core and Duopo l y Theo r y 21 1. I n t r o d u c t i o n . . . 21 2. The Model 23 3 . The C o m p e t i t i v e E q u i l i b r i u m 25 4 . The Co r e A l l o c a t i o n s 26 5 . D i s c u s s i o n o f t h e Model 29 6 . No tes 31 Chap t e r 3 : On A d j u s t m e n t C o s t s and t h e S t a b i l i t y o f E q u i l i b r i a 33 1. I n t r o d u c t i o n 33 2. The B a s i c Model 36 3 . The D i r e c t i o n a l Core 41 4 . An Example 47 5. The Nash B a r g a i n i n g Core 52 6 . The S t a n d a r d Q u a n t i t y S e t t i n g 0 1 i g o p o l y - P r o b l e m 58 7. C o n c l u d i n g Remarks 61 8 . No te s 64 C h a p t e r 4 : C o n s c i o u s P a r a l l e l i s m and P r e d a t o r y P r i c i n g i n a Con -t e s t a b l e Ma r ke t 73 1. I n t r o d u c t i o n > 73 2. The I n d u s t r y Model 77 3 . The Announcement Game 82 4 . An A x i o m a t i c App roach t o C o n s c i o u s P a r a l l e l i s m 87 5 . The E f f e c t o f E n t r y 92 6 . C o n c l u d i n g Comments 96 7 . N o t e s . 97 - i v -S e c t i o n Page R e f e r e n c e s 99 Append i x t o C h a p t e r 1 1 0 4 Append i x to Chap t e r 3 1 1 4 Append i x t o C h a p t e r 4 1 2 0 - V -L i s t o f F i g u r e s Page Chap t e r 2 - F i g u r e 1 32 C h a p t e r 3 - F i g u r e 1 65 F i g u r e 2 66 F i g u r e 3 67 F i g u r e 4 68 F i g u r e 5 69 F i g u r e 6 70 F i g u r e 7 71 F i g u r e 8 72 - v i -Acknow ledgements T h i s t h e s i s wou ld no t have been p o s s i b l e w i t h o u t t he e n c o u r a g e -ment and s t i m u l a t i n g comments and c r i t i c i s m s t h a t I have r e c e i v e d f rom many p e o p l e . A t the U n i v e r s i t y o f B r i t i s h Co l umb i a I wou ld l i k e t o t hank my t e a c h e r s and f e l l o w g r a d u a t e s t u d e n t s f o r t h e i r i n v a l u a b l e h e l p . In p a r t i c u l a r I am i n d e b t e d t o my commi t tee John Weymark, Hugh Neary and C h a r l e s B l a c k o r b y as w e l l a s ' M u k e s h E s w a r a n , C u r t i s E a t o n , D a v i d D o n a l d s o n , T r a c y L e w i s , Ke' izo N a g a t a n i , M i c h e l P a t r y , A n d r e P l o u r d e , a n d M a r g a r e t S l a d e . I have a l s o b e n e f i t e d f rom d i s c u s s i o n s w i t h my c o l l e a g u e s a t Queen ' s U n i v e r s i t y , p a r t i c u l a r l y R i c h a r d A r n o t t and R i c h a r d H a r r i s . The p r e s e n t v e r s i o n o f t h e t h e s i s wou ld no t have been p o s s i b l e w i t h o u t t he comments r e c e i v e d f rom my c o l l e a g u e s a t CORE and ISE a t the U n i v e r -s i t e C a t h o l i q u e de L o u v a i n , p a r t i c u l a r l y C l aude d ' A s p r e m o n t , M a r i e - P a u l e D o n s i m o n i , J ean G a b s z e w i c z , L o u i s ^ G e . v . e r s , P a u l G e r o s k i , J a c q u e s T h i s s e and Tom T h u r s t o n . F i n a l l y I wou ld l i k e t o t hank S h a r o n A l t o n a n d J e n n i n e B a l l f o r t h e i r e x c e l l e n t t y p i n g . INTRODUCTION The c l a s s i c a l o l i g o p o l y p rob l em i s c on ce r ned w i t h t h e s t u d y o f ma r ke t s c h a r a c t e r i z e d by a r e l a t i v e l y s m a l l number o f s e l l e r s d e a l -i n g w i t h a l a r g e group o f b u y e r s . D e s p i t e t he a p p a r e n t c l a r i t y o f t h e q u e s t i o n , t h e s t a t u s o f t h e p rob l em i s much t h e same t o d a y as t h e s i t u a -t i o n d e s c r i b e d by C h a m b e r l i n [ 1 9 3 3 ] : " I t has been h e l d t h a t t h e c o m p e t i t i o n between two s e l -l e r s w i l l r e s u l t i n a monopo ly p r i c e , a c o m p e t i t i v e p r i c e , a d e t e r m i n a n t p r i c e between them, an i n d e t e r m i n a n t p r i c e between them, a p e r p e t u a l l y o s c i l l a t i n g p r i c e , and no p r i c e a t a l l b e -cause t h e p rob l em i s i m p o s s i b l e " . Today one o n l y needs to see the work by Kurz [ 1982 ] f o r a d i s -c u s s i o n o f t h e p o s s i b i l i t y o f c o l l u s i o n , w h i l e t h e work o f F r i edman [1977 ] and Green [ 1980 ] d emons t r a t e t he b a s i c i n d e t e r m i n a c y o f t h e p r o b -lem t h a t can r e s u l t i n many p o s s i b l e e q u i l i b r i a . S t i l l o t h e r work by B r e snahan [ 1 9 8 1 ] , Grossman [ 1981 ] and Baumo l , P an z a r and W i l l i g [ 1982 ] a r g u e l t h a t t he c o m p e t i t i v e e q u i l i b r i u m i s t he a p p r o p r i a t e s o l u t i o n , w h i l e t h e work o f L e v i t a n and Shub i k [ 1 9 7 8 ] , S h ap l e y and Shub i k [ 1969 ] o r V a r i a n [ 1980 ] s u g g e s t s t h a t m i xed s t r a t e g i e s o r p e r p e t u a l l y v a r y i n g p r i c e s i s t h e e x p e c t e d ou tcome. D e s p i t e the a p p a r e n t v a r i e t y o f p r o -posed s o l u t i o n s , the r e s u l t s f o i l o w e s s e n t i a l l y the same m e t h o d o l o g y , namely a g a m e - t h e o r e t i c a p p r o a c h . By game t h e o r e t i c ou r n o t i o n i s p r o b a b l y c l o s e s t t o t h e i d e a s pu t f o r w a r d by S c h e l l i n g [ 1 9 6 0 ] , t hough i n terms o f i n d u s t r i a l o r g a n i z a -- 1:.-- 2 -t i o n Shub i k [ 1 959 ] i s t h e c l a s s i c r e f e r e n c e . S i m p l y pu t t he me thodo l ogy c o n s i s t s o f f i r s t c a r e f u l l y s p e c i f y i n g a model and the i n f o r m a t i o n t o wh i c h a gen t s have a c c e s s . G i v e n t h i s b a s i c d a t a t h e b e h a v i o u r o f agen t s i s e x p l a i n e d assum ing t h a t not o n l y a r e t h e y m a x i m i z i n g t h e i r payoffs. . . but t h e y a l s o have c o n s i s t e n t o r c o r r e c t c o n j e c t u r e s c o n c e r n i n g t h e i r compe t i t o r s ' behav iour . In t h i s t h e s i s t h i s me thodo l ogy i s a p p l i e d to a s e r i e s o f o l i g o p o l i s t i c mode l s i n an a t t emp t t o a dd r e s s some o u t s t a n d i n g i s s u e s . We b e g i n i n Chap t e r 1 by g e n e r a l i z i n g t h e w e l l known s p a t i a l model o f H o t e l l i n g [ 1 9 2 9 ] . H o t e l l i n g ' s i n s i g h t was t h a t f i r m s chose no t o n l y the p r i c e and o u t p u t o f t h e i r p r oduc t bu t a r e a l s o f r e e t o choose t h e t y pe o f p r o d u c t and i t s l o c a t i o n . In mak ing t h i s o b s e r v a t i o n H o t e l l i n g a l s o f e l t t h a t h i s s p a t i a l model wou ld a d d r e s s a n o n - e x i s t e n c e p rob lem r a i s e d by Edgewor th [ 1 9 2 5 ] . T h i s c h a p t e r e x t ends t h e work o f d ' A sp r emon t e t a l . [ 1979 ] t o . s how t h a t in general t h e r e w i l l no t e x i s t an e q u i l i b r i u m p r o -d u c t i o n p l a n when p r o d u c t s a r e s u f f i c i e n t l y s i m i l a r . Though d i f f e r e n -t i a t e d p r o d u c t mode ls add an e l emen t o f r e a l i s m t h e r e does not seem a t t h i s t i m e a c o m p l e t e l y s a t i s f a c t o r y way t o a d d r e s s t h e e x i s t e n c e p r ob l em . For t h e r e s t o f the t h e s i s we move away f rom t h i s c l a s s o f mode ls t o t h o s e f o r wh i ch e q u i l i b r i a do e x i s t . T y p i c a l l y i n o l i g o p o l y t h e o r y one s i m p l y assumes t h a t buye r s a r e e i t h e r p r i c e o r q u a n t i t y t a k e r s w i t h o u t much j u s t i f i c a t i o n . I n t u i t i v e l y t h e argument i s t h a t f i r m s a r e l a r g e r hence i t seems t h a t buye r s have no marke t power . A way t o a dd r e s s t h i s i s s u e i s t o e x p l i c i t l y model t he t r a d e between s e l l e r s arid b u y e r s , - 3 -w h i c h , f o l l o w i n g Aumann [1964 . ] ' s s u g g e s t i o n , w a s done i n a sequence o f pape r s by Aumann [ 1 9 7 3 ] , Gabs zew i c z and Mer tens [ 1 9 7 1 ] , S h i t o v i t z [ 1973 ] and S h i t o v i t z [1982] t h a t used t h e . c o r e as t h e a p p r o p r i a t e , i n s t i t u t i o n f r e e , s o l u t i o n c o n c e p t . U n f o r t u n a t e l y i n t h e s e a b s t r a c t l y d e f i n e d mode ls t h e y found no c o n s i s t e n t r e l a t i o n s h i p between t he s i z e o f agen t s and t h e ma r ke t power t h ey p o s s e s s . In C h a p t e r 2 we show, u s i n g a r e s u l t o f S h i t o v i t z , t h a t marke t power a r i s e s when f i r m s have t h e a b i l i t y t o p recommi t t h e m s e l v e s t o c e r t a i n c o u r s e s o f a c t i o n before m a r k e t s o pen . In p a r t i c u l a r ou r a n a l y s i s g i v e s f o rma l s u p p o r t t o t h e c l a s s i c a l Cou rno t [ 1838 ] v i ew t h a t f i r m s have power by m a n i p u l a t i n g t he c o m p e t i t i v e e q u i -l i b r i u m t h r ough t h e i r s u p p l y d e c i s i o n s . C h a p t e r 3 e x p l i c i t l y a d d r e s s e s t he i n d e t e r m i n a c y i s s u e . I f f i r m s can m u t u a l l y o b s e r v e each o t h e r s a c t i o n s then i n g e n e r a l t h e r e e x i s t many e q u i l i b r i a . We-suppose howeve r , t h a t q u a n t i t y s e t t i n g f i r m s canno t i n s t a n t a n e o u s l y a d j u s t o u t p u t bu t r a t h e r f a c e a d j u s t m e n t c o s t s . By c omb i n i n g t h i s a s s ump t i o n w i t h some i d e a s f rom l o c a l game t h e o r y , p a r -t i c u l a r l y Mat thews [ 1 9 8 2 ] , i t i s shown t h a t the c r i t e r i o n o f stability w i l l o f t e n s e l e c t a un ique e q u i l i b r i u m . F u r t h e r , a t t h i s e q u i l i b r i u m f i r m ' s ma rke t s h a r e s w i l l be i n v e r s e l y p r o p o r t i o n a l t o t h e i r a d j u s t m e n t c o s t s , a l 1 owing us t o i n t u i t i v e l y i n t e r p r e t a d j u s t m e n t c o s t s as a measure o f b a r g a i n i n g s t r e n g t h . F i n a l l y i n C h a p t e r 4 we move away f rom t h e q u a n t i t y s e t t i n g f ramework t o s t u d y some i s s u e s r e l e v a n t t o t h e i n f o r m a l i n d u s t r i a l o r -g a n i z a t i o n l i t e r a t u r e and " a n t i t r u s t i s s u e s ; In t h e - 4 -work o f Baumo l , P an za r and W i l l i g [1982] i t i s a r gued t h a t i t i s c o s t l e s s e n t r y and e x i t t h a t i s r e s p o n s i b l e f o r e n s u r i n g c o m p e t i t i v e p r i c i n g by f i r m s . In t h i s c h a p t e r we use a r e p e a t e d game f ramework t o show, to t he c o n t r a r y , unde r t he a p p r o p r i a t e i n f o r m a t i o n c o n d i t i o n s c o s t l e s s e n t r y and e x i t i s c o m p l e t e l y c o n s i s t e n t w i t h c o l l u s i v e p r i c i n g b e h a v i o u r and l o n g r un p o s i t i v e p r o f i t . In t h e c h a p t e r we a l s o p r e s e n t a f o rma l t h e o r y o f c o n s c i o u s p a r a l l e l i s m , t h a t y i e l d s a u n i q u e t a c i t l y c o l l u s i v e e q u i l i -b r i u m . What t h e s e r e s u l t s s u g g e s t i s t h a t t h e t h e o r y o f p e r f e c t c o n -t e s t a b i l i t y i s no t a f undamen ta l s t a r t i n g p o i n t f o r t h e s t u d y o f i n d u s -t r i a l o r g a n i z a t i o n as i t s a u t h o r s s u g g e s t . R a t h e r i t r e p r e s e n t s one o f many p o s s i b l e mode ls t h a t one may use i n s t u d y i n g o l i g o p o l i e s . I t i s t h e s t a n d a r d m i c r o e c o n o m i c a s s ump t i o n t h a t f i r m s a r e p r o f i t - m a x i m i z e r s t h a t i s the f o u n d a t i o n s t o n e f o r t he modern s t u d y o f l a r g e f i r m s . T h i s t h e s i s i s s i m p l y an examp le o f an a p p l i c a t i o n o f t h i s t h e o r y t o f o u r d i f f e r e n t , b u t r e l e v a n t m o d e l s . CHAPTER 1 ON THE NON-EXISTENCE OF EQUILIBRIA IN DIFFERENTIATED PRODUCT MODELS 1 . I n t r o d u c t i o n Much o f t h e modern t h e o r y o f m o n o p o l i s t i c c o m p e t i t i o n has i t s r o o t s i n t h e l o c a t i o n model d e v e l o p e d by H o t e l l i n g [ 1 9 2 9 ] . The a im o f t h e s e mode l s i s t o s t u d y ma rke t s t r u c t u r e when f i r m s choose bo th t he cha r -a c t e r i s t i c s ( l o c a t i o n s ) and the p r i c e of the product they are to produce. T y p i c a l l y , the'.-prjbblem.faced by f i rms can be. modelled as a two stage p ro -c e s s : The f i r s t s t a g e p e r t a i n s t o t he s e l e c t i o n o f t he p r o d u c t c h a r a c -t e r i s t i c s and t he second s t a g e to t he p r i c e c h o i c e . To compute the e q u i l i b r i a f o r such a model one f i r s t computes t he Nash e q u i l i b r i u m i n p r i c e s f o r any s e t o f p r o d u c t c h a r a c t e r i s t i c s . T h i s c o m p u t a t i o n d e f i n e s t he p a y o f f o f f i r m s f o r any c h a r a c t e r i s t i c c h o i c e i n t h e f i r s t s t a g e . G i v en t h e s e p a y o f f s one t h en computes t h e Nash e q u i l i b r i u m i n c h a r a c -t e r i s t i c s . The o b j e c t i v e o f t h i s paper i s t o show f o r a v e r y g e n e r a l c l a s s o f s p a t i a l mode ls t h a t t h e r e i s a s i g n i f i c a n t s e t o f l o c a t i o n s f o r wh i c h t he second s t a g e e q u i l i b r i u m does no t e x i s t . Hence i t i s not p o s -s i b l e to s o l v e t he f i r s t s t a ge p rob l em s i n c e t h e r e w i l l e x i s t s t r a t e g y c h o i c e s w i t h no w e l l d e f i n e d ou t come . One o f t h e f i r s t n o n - e x i s t e n c e r e s u l t s i s due t o Edgewor th [ 1 9 2 5 ] . He showed t h a t t h e a d d i t i o n o f c a p a c i t y c o n s t r a i n t s t o the B e r t r a n d [ 1883 ] p r i c e s e t t i n g model wou ld r e s u l t i n i n d e f i n i t e l y f l u c -t u a t i n g p r i c e s . I t was H o t e l l i n g ' s a im t o s o l v e t h i s p rob l em t h r ough the i n t r o d u c t i o n o f p r o d u c t d i f f e r e n t i a t i o n : "The a s s u m p t i o n , i m p l i c i t i n - 5 -i - 6 -t h e i r work ( o f C o u r n o t , Amorose and E d g e w o r t h ) , t h a t a l l b u ye r s dea l w i t h t h e c h e ape s t s e l l e r l e a d s t o a t y p e o f i n s t a b i l i t y wh i c h d i s a p p e a r s when the q u a n t i t y s o l d by each i s c o n s i d e r e d as a c o n t i n u o u s f u n c t i o n o f t h e d i f f e r e n c e s i n p r i c e " . * In o t h e r ' w o r d s , i t was hoped t h a t t he p r o d u c t d i f f e r e n t i a t i o n i n t r o d u c e d t h r o u g h t h e use o f t h e one d i m e n s i o n a l s p a -t i a l model wou ld r e s u l t i n c o n t i n u o u s demands and t h e r e s o l u t i o n o f t h e n o n - e x i s t e n c e p r o b l e m . However , i t was r e c e n t l y shown i n d ' A s p r e m o n t , Gab s z ew i c z and T h i s s e [1979 ] (AGT) t h a t t he demand c u r v e s were i n f a c t n o t c o n t i n u o u s , w i t h t he consequence t h a t when f i r m s l o c a t e s u f f i c i e n t l y c l o s e t o g e t h e r an e q u i l i b r i u m p r i c e does no t e x i s t . I n t h i s e s s a y we p r e s e n t a g e n e r a l s p a t i a l model and show t h a t demand c o n t i n u i t y can be e x p e c t e d i n most c a s e s . I t i s shown t h a t H o t e l -l i n g ' s i n t u i t i o n t h a t demand c o n t i n u i t y depends on consumers a lway s s t r i c t -l y p r e f e r r i n g one f i r m p r o v i d e s a necessary and sufficient condition f o r c o n t i n u i t y . T h i s model i s t h e n a p p l i e d to a problem s i m i l a r - t o Edge-w o r t h ' s . G i v en f i r m l o c a t i o n s we ask whe the r t h e r e e x i s t s a Nash e q u i l i b r i u m when f i r m s choose p r i c e and o u t p u t s i m u l t a n e o u s l y . I f t h e f i r m s a re ' v i ewed as r e t a i l s t o r e s i n g e o g r a p h i c a l s p a c e , t h e n t he o u t p u t v a r i a b l e i s s i m p l y the l e v e l o f i n v e n t o r y h e l d by t h e s t o r e . C e r t a i n l y i f , as A r c h i b a l d , Ea ton and L i p s e y [ 1983 ] s u g g e s t , s u ch mode l s a r e t he n a t u r a l f ramework f o r t h e s t u d y o f p r o d u c t c h o i c e and mar-k e t s t r u c t u r e , t h en one s h o u l d a t l e a s t have e x i s t e n c e of an e q u i l i b r i u m in our s i m p l e ex t ens i on . However, we show that whenever one . f i rm loca tes s u f f i c i e n t l y c l o se t o . ano the r , an equi 1 ibriumi wi 1.1. not e x i s t , - 7 -even with continuous demand. Ea ton and L i p s e y [1978] and Novshek [ 1980 ] s ugge s t t h a t t h i s p rob l em w i l l be r e s o l v e d - i f an a l t e r n a t i v e e q u i l i b r i u m c o n c e p t i s ' emp l o yed "wh " i c h i n c o r p o r a t e s • an a s sump t i on o f no m i l l p r i c e u n d e r -c u t t i n g . .However , -we a l s o show t h a t t h i s app roach does no t a l t e r t h e non -e x i s t e n c e r e s u l t . In t h e f o l l o w i n g , s e c t i o n we p r e s e n t t h e model a l o ng w i t h t h e r e s u l t o f demand c o n t i n u i t y . - S e c t i o n 3 examines t h e f i r m , p r ob l em and p r e s e n t s t h e n o n - e x i s t e n c e r e s u l t . S e c t i o n 4 has ou r c o n c l u s i o n s . 2. The Model L e t t h e marke t be r e p r e s e n t e d by a c onvex , compact s e t U | c=iR r n . The d i s t r i b u t i o n o f consumers o v e r U w i l l be g i v e n by a bounded B o r e l mea su r ab l e f u n c t i o n v ( x ) , x e U. There w i l l be two f i r m s a t f i x e d , bu t d i f f e r e n t , l o c a t i o n s x^, x 2 e U t h a t w i l l s e l l t h e same p r o d u c t a t m i l l p r i c e s ^ and p 2 r e s p e c t i v e l y . The u n i t c o s t o f t r a n s p o r t i n g t h e good f rom p o i n t x t o y w i l l be t ( x , y ) . Hence f o r a consumer l o c a t e d a t x t h e p r i c e o f t h e good bought f rom f i r m i w i l l be P,-(x) = p, + t ( x , x ) . (1) 1 d e f 1 1 I t w i l l be assumed t h a t the t r a n s p o r t a t i o n c o s t f u n c t i o n has t he f o l l o w i n g p r o p e r t i e s . Assumption Tl: t ( x , y ) is continuous for ( x , y ) e U x U. Assumption T2: t ( x , y ) > 0 for all x f- y , x , y e U and t ( x , x ) = 0 for all x e U. Assumption T3: t ( x , y ) + t ( y . , z ) > t ( x , z ) , V x , y e U. - 8 -As sump t i o n T l i s the s t a n d a r d c o n t i n u i t y a s s u m p t i o n w h i l e T2 en su r e s t h a t a l l l o c a t i o n s a r e d i f f e r e n t i a t e d by p o s i t i v e t r a n s p o r t c o s t s . F i n a l l y , T3 c o r r e s p o n d s t o t he t r i a n g l e i n e q u a l i t y . Consumers w i l l a l l have t he same demand c u r v e D(p) w h i c h w i l l be assumed t o have t he f o l l o w i n g p r o p e r t i e s . Assumption DI: D(p) is differentiable. Assumption D2: 3 k and p + > 0 such that 0 < D(p) < k V p e IR + and D(p) = 0 for p > p*. Assumption DZ: 3 a > 0 such that - a < ^ j j ^ < 0 for p e [ 0 , p + ] . These a s s ump t i o n s e n su r e t h a t t h e demand i s o f t h e s t a n d a r d downward s l o p i n g t y p e w i t h bounds on bo th demand and i t s d e r i v a t i v e s . T h i s model c an e a s i l y be ex t ended t o a l l o w demand t o depend on l o c a t i o n 2 w i t h o u t a l t e r i n g any o f the p r o o f s . Consumers w i l l w i s h t o buy f rom t he f i r m o f f e r i n g t h e l o w e s t d e l i v e r e d p r i c e . G i v en t he f i r m l o c a t i o n s , X. - ( x ^ X g ) e"U x U and p r i c e s _P = ( p j , p 2 ) e [ 0 , p + ] x [ 0 , p + ] , t h e l o w e s t d e l i v e r e d p r i c e t o l o c a t i o n x e U i s g i v e n by : p ( x ) = min { p x ( x ) , p 2 ( x ) } , (2 ) where p^(x) i s g i v e n by ( 1 ) . F i rm i ' s marke t a r e a i s fo rmed by those l o c a t i o n s a t wh i c h i t quo t e s t h e l o w e s t p r i c e . I t t h e r e f o r e i s d e f i n e d by : - 9 -U . ( X , P ) = {x e U | p . ( x ) = p ( x ) } , i = 1 ,2 . 1 d e f 1 In g e n e r a l , t h e r e i s no r e a s on no t t o e x p e c t t h e f i r m ma r ke t a r e a s to s i g n i f i c a n t l y o v e r l a p , i n wh i ch case a l l t he consumers l o c a t e d i n \i^{X,P) {] U 2 ( X , P ) wou ld be i n d i f f e r e n t between the two f i r m s . Such o v e r l a p p i n g c r e a t e s d i f f i c u l t i e s . The f i r s t d i f f i c u l t y i s t h a t demand f r om such a r e a s must be a l l o c a t e d i n a n e c e s s a r i l y a d ' h o c f a s h i o n between f i r m s . However , such a r b i t r a r y a l l o c a t i o n s go a g a i n s t t h e s p i r i t o f a d i f f e r e n -t i a t e d p r o d u c t mode l , ;wh ich a r e meant t o p r o v i d e a consumer w i t h a l t e r -na te s o v e r w h i c h he i s not " i n d i f f e r e n t " . C o n s e q u e n t l y we make the f o l -l o w i n g a s s u m p t i o n . Ho telling Assumption: V _P e [ 0 , p + ] x [ 0 , p + ] and X E U x U suah that x1 + x 2 , L u ^ x . p j n u ^ p ) D(P(X ) ) v ( x ) dx = 0. In e s sence H o t e l l i n g a r gued t h a t such an a s s u m p t i o n wou ld e n -3 s u r e ' t h a t t h e demand f a c i n g f i r m s i s c o n t i n u o u s . Not o n l y d i d he f e e l t h a t i n p r a c t i c e demand was n o r m a l l y c o n t i n u o u s b u t a l s o t h a t i t was the d i s c o n t i n u i t y o f demand i n homogeneous p r o d u c t mode ls t h a t was r e s p o n s i b l e f o r t he Edgewor th c y c l e . Somewhat i r o n i c a l l y , as p o i n t e d ou t by AGT, t h e Ho t e l l i n g assump-t i o n i s no t s a t i s f i e d i n H o t e l l i n g ' s o r i g i n a l one d i m e n s i o n a l s p a t i a l mode l . Fo r e xamp l e , l e t U = [ 0 , 1 ] and suppose f i r m l o c a t i o n s a r e x^ = 1/4 and x ? = 3 / 4 . L e t t r a n s p o r t c o s t s be g i v e n by t ( x , y ) = | x - y | , - 10 -v(x) = 1 and D(p) = 1. Fo r p. = 1 and p9•= U a s i m p l e compu ta -t i o n w i l l show U r ( X , P ) = [0 , 1 ] U 2 ( X , P ) = [ f , 1 ] . T h e r e f o r e x e M U v(x) dx = \f 0. G i v e n t he H o t e l l i n g a s s u m p t i o n , t h e demand f a c i n g f i r m l i s u n i q u e l y d e f i n e d by : i s no t s a t i s f i e d . However , i n t h a t c a se no t o n l y w i l l a f i r m ' s demand be o v e r s t a t e d , bu t t h e demand c u r v e s w i l l no t be c o n t i n u o u s . PROPOSITION 1 ling assumption holds. The p r o o f o f t h i s p r o p o s i t i o n i s c o n t a i n e d i n t h e a p p e n d i x . A l t h o u g h t he one d i m e n s i o n a l s p a t i a l : m o d e l w i t h l i n e a r t r a n s p o r t c o s t s has d i s c o n t i n u o u s demands, i t " w i l l , b e - t h e c a s e t h a t the. Ho t e l T i ng a s -sump i t on i s s a t i s f i e d i n a w ide v a r i e t y o f ' c a s e s . : The - f o i l owing;" examp les ( 3 ) F o r m a l l y d . (_X.JP) i s d e f i n e d even i f the H o t e l l i n g a s s ump t i o n If, assumptions Dl and Tl - T3 are satisfied, then the demand function d.(X_,_P) is continuous in (_K,P_) if and J only if the Hotel-- 11 -i 1 1 u s t r a t e t h i s c l a i m . Examp l e_ l In t h e paper by A'GT, i t i s shown t h a t f o r t h e o n e - d i m e n s i o n a l ... p model w i t h t ( x , y ) = - c | x - y | - / demands a r e . c o n t i n u o u s T h i s c o n c l u s i o n f o l l o w s f r o m t h e f a c t t h a t . U^ . n w i l l a lways be a s i n g l e p o i n t . Example 2 I t i s ea sy t o v e r i f y t h a t i f t h e d i m e n s i o n o f t h e consumer space i s a t l e a s t two , t hen E u c l i d e a n , t r a n s p o r t c o s t s , " t ( x , y ) ' = . |jx-y|| w i l l a l -ways r e s u l t ' i n n h a v i n g measure z e r o . Fo r m _> 2, n w i l l be a m a n i f o l d o f "d imens ion l e s s t h an rr,. Fo r t h e r ema i nde r o f t h e paper we w i l l assume t h a t t h e H o t e l -l i n g a s s ump t i o n h o l d s . 3 . The F i r m ' s P rob lem In t h i s s e c t i o n , we add r e s s t he second s t a g e p rob l em i n w h i c h f i r m l o c a t i o n s a r e f i x e d and p r i c e s and o u t p u t c h o i c e s a r e m o d e l l e d as a Nash e q u i l i b r i u m . S i n c e one o f t h e i s s u e s to be a d d r e s s e d i s whe the r the r e s t o r a t i o n o f demand c o n t i n u i t y w i l l r e s o l v e t he Edgewor th e x i s t e n c e p r o b l e m , f i r m s w i l l c hoose p r i c e s and o u t p u t s s i m u l t a n e o u s l y . T h i s a s s u m p t i o n - a l l o w s one t o see t h e e f f e c t o f a l l o w i n g f i r m s to^choose c a p a c i t y l e v e l s ex a n t e . More g e n e r a l l y , i f one a c c e p t s the a rguments pu t f o r w a r d by A r c h i b a l d , Ea ton and L i p s e y [1983 ] t h a t a dd r e s s m o d e l s , o f wh i ch s p a t i a l mode ls such as t h i s one form a l a r g e s u b s e t , a r e the n a t u r a l v e h i c l e f o r demand - 12 -models , t h en t h e r e a l i s t i c a s s ump t i o n t h a t f i r m p r o d u c t i o n t a k e s p l a c e ex a n t e t o p roduce an i n v e n t o r y s h o u l d r e s u l t i n w e l l d e f i n e d e q u i 1 i -4 b r i a " . C e r t a i n l y , one i s u n l i k e l y t o have a v e r y g e n e r a l t h e o r y b y r e -s t r i c t i n g o n e s e l f - t o t h e c a s e of i n e l a s t i c demand and z e r o m a r g i n a l c o s t s so u b i q u i t o u s i n s p a t i a l a n a l y s i s . L e t the p a i r o f f i r m l o c a t i o n s be _X = ( x ^ ^ ) ; x^ f x^. Then f i r m i ' s s t r a t e g y i n t h e se cond s t a g e w i l l be a p r i c e - o u t p u t p a i r d e -no ted s . = ( p . , q . ) e [0 , p ] x IR = S . The o u t p u t q . w i l l s i m p l y be an 1 1 1 d e f 1 i n v e n t o r y o f goods p r oduced ex an t e t h a t t h e f i r m w i l l s e l l a t a p r i c e P-. I f a s t o c k - o u t o c c u r s we w i l l suppose t h a t t h e r a t i o n e d consumers w i l l go to t h e c o m p e t i t o r . The two f i r m s w i l l have the same c o s t f u n c t i o n denoted C ( - ) -I t w i l l be assumed t o s a t i s f y t he f o l l o w i n g c o n d i t i o n s . ^ mi eatisfiee <Jm > o, v „ > o. dq" 5 dq " Assumption C2: i n f = 3 > 0. q>0 The f i r s t a s s u m p t i o n , though not t h e most g ene r a l p o s s i b l e f o r ou r r e s u l t s , does i n c l u d e as s p e c i a l c a s e s , c o n s t a n t m a r g i n a l c o s t s and bo t h i n c r e a s i n g and d e c r e a s i n g r e t u r n s t o s c a l e c o s t f u n c t i o n s . The s e -cond a s s u m p t i o n ensu res t h a t p r o d u c t i o n i s a lway s c o s t l y . L e t us now s p e c i f y t he b e h a v i o u r o f r a t i o n e d consumers g i v e n a s t r a t e g y p a i r , S = ( s , , s ? ) e 5 x S . Suppose a t any g i v e n t ime f i r m i d e f 1 f a c e s demand f rom a l l p a r t s o f i t s ma r ke t a r e a , IK U , £ ) , i n p r o p o r t i o n t o the d e n s i t y v ( x ) . Thus i f a f i r m s t o c k s o u t , consumers f r om a l l p a r t s - 13 -o f t h e marke t a r e a w i l l be a f f e c t e d e v e n l y . We w i l l t h e r e f o r e have t h e same, p r o p o r t i o n o f consumers ' r a t i o n e d eve rwhe re i n U-('.X,P_)., w h o - w i l l t hen go t o t h e c o m p e t i t o r t o p u r c h a s e t h e good.-T h i s p r o p o r t i o n o f u n s a t i s f i e d demand w i l l be g i v e n by / U x . s ) = 1 d e f 0 i f q i > d . ( X , P ) > 0 d . (X ,P . ) : . - q [ d i ( X f P ) i f d l ^ . P ) > q 1 > 0 . N o t i c e t h a t due to t h e c o n t i n u i t y o f d . ( X , P j , A.(_X,_S) i s c o n t i n u o u s f o r (_X,Sj e U x S where S = [0,p ] x (0,°°). The d e n s i t y o f rationed consumers w i l l t h e n be X . ( X , S ) v ( x ) f o r x e U.{X,P). These consumers a r e assumed t o demand t he good f r om f i r m . ' j f i a t - t h e . h i g h e r d e l i v e r e d p r i c e . Hence t h e demand c u r v e f o r f i r m j once we a l l o w f o r a s t o c k ou t by f i r m i i s g i v e n by : :,p) + A , (X,S) / D j U . S ) = d ^ X . P ) X . ,  - > x e U . ( X 5 p ) D ( P j . ( x ) ) v ( x ) dx (5 ) where j f i . G i v en t h e d e f i n i t i o n s o f c o s t and t h e r a t i o n e d o r c o n t i n g e n t demand c u r v e s , we can now d e f i n e t h e p r o f i t f u n c t i o n f o r f i r m i : n ( X , S ) = P. Q . ( X , S ) - C(q ) 1 d e f 1 1 1 where Q-(X_,S_) = min { q . , D . ( X , S ) } . 1 d e f 1 1 - 14 -Here {X,S) s i m p l y r e p r e s e n t s t h e q u a n t i t y o f good s o l d wh i c h o f c o u r s e c a n n o t be g r e a t e r t h an t h e l e v e l o f i n v e n t o r y . From ou r a s -2 2 s u m p t i o n s , i t i s c l e a r t h a t n.:(_X,_S) i s d e f i n e d f o r U,S_) e U x S and c o n t i n u o u s f o r (_X,_S) e U x 3" . S i n c e we a r e c o n c e r n e d w i t h t h e e q u i l i -b r i um when both f i r m s a r e p r o d u c i n g , we w i s h t o en su r e t h a t q . f 0, t hu s a l s o a v o i d i n g t h e d i s c o n t i n u i t y p r ob l em when q . =0. r.V I f one f i r m i n f a c t does not p roduce any o u t p u t , t hen i t s com-p e t i t o r wou ld a c t as a m o n o p o l i s t ; We w i l l now r e q u i r e t h a t i t w i l l -a lways be p r o f i t a b l e f o r t h e smal1 f i r m t o - s t a r t up o p e r a t i o n s a t t h e monopo ly e q u i l i b r i u m . Assumption M: There exists an a-> 0 such that p ( x ) > g | (0) + a , V x e U, where p ( x ) solves max p D ( x , p ) - C ( D ( x , p ) ) P and :X,P) = L D ( x , p ) JxcU D(p + t ( x , x ) ) v ( x ) d x . de f T h i s a s s ump t i o n s i m p l y s a y s t h a t t h e monopo ly p r i c e c h o i c e f o r any l o c a t i o n i s s t r i c t l y g r e a t e r than t h e m a r g i n a l c o s t o f p r o d u c t i o n when t h e r e i s no o u t p u t . - 15 -F i rms w i l l now e v a l u a t e t h e i r l o c a t i o n c h o i c e s , _ X , a t t he e q u i -l i b r i u m p r i c e - q u a n t i t y p a i r s^  £ 5 . We w i l l s t u d y t he e x i s t e n c e p r o b -lem f o r two e q u i 1 i b r i u m c o n c e p t s . DEFINITION 1 2 Given a pair of locations _ X , the strategy pair S * £ S will be a Nash Equilibrium (NE) i f and only i f JI , ( X , S * ) = max n ( X , S , , S * ) ; i = 1, 2 and j t i . 1 S . eS 1 J R e c e n t l y , Eaton and L i p s e y [ 1976 ] and Novshek [ 1980 ] a rgue t h a t t h e n o n - e x i s t e n c e p rob l em can be r e s o l v e d by mak ing t h e a s s ump t i o n t h a t f i r m s do no t u n d e r c u t each o t h e r a t t h e mi 1 1 - d o o r . T h i s m o t i v a t e s t h e f o l l o w i n g d e f i n i t i o n . DEFINITION 2 0 2 Given a pair of locations X, the strategy pair S_ £ S will be a no mill-price undercutting equilibrium (ME) i f and only i f n . ( X , S ° ) = max n . ( X , S . , S ° ) for i = 1, 2; j f i and 1 S E S 1 1 0 Pi > Pj° - t ( x r x . ) . U n f o r t u n a t e l y , o u r f i n a l r e s u l t s t a t e s t h a t one c anno t i n g ene r a l e x p e c t t h e e x i s t e n c e o f an e q u i l i b r i u m w i t h e i t h e r s o l u t i o n c o n c e p t . - 16 -PROPOSITION 2 For every e U there will exist a Wg > 0 such that for pairs of locations, ^satisfying t C x ^ ^ ) < WQ there will not exist an NE nor a ME in price and output. The d e t a i l s o f t he p r o o f a r e t o be found i n t he a p p e n d i x . The b a s i c o u t l i n e . o f t h e p r o o f i s as f o l l o w s . G i v en a l o c a t i o n x^ e U, we assume t h a t f o r e v e r y x^ e U, x^ f x^> t h e r e w i l l e x i s t an e q u i l i b r i u m and show t h a t t h i s a s sump t i on r e s u l t s i n a c o n t r a d i c t i o n . I f a NE o r ME e x i s t s , t h e n ' by a s s ump t i o n M both f i r m s w i l l have p o s i t i v e o u t p u t . F u r t h e r t h e r e w i l l be n e i t h e r e x c e s s s u p p l y no r r a t i o n i n g . Then , f o r any sequence • o f e q u i l i b r i a f o r l o c a t i o n s ( v x^ . X j ) such t h a t t(x^,x^) ->• 0 as n ' i t c a n t h en be shown t h a t p r i c e must app roach t h e m a r g i n a l c o s t o f p r o d u c t i o n f o r bo th NE and M E . Hence , i f an e q u i l i b r i u m / e x i s t s , " : i n c r e a s i n g c o m p e t i t i o n w i l l a lway s l e a d to c o m p e t i t i v e p r i c i n g by f i r m s . T h i s w i l l be i m p o s s i b l e f o r one o f t he f o l l o w i n g r e a s o n s . (1) I f we have MC f a l l i n g a s o u t p u t i n c r e a s e s , t hen i f p r i c e app r oa che s MC i t must a t some p o i n t f a l l be l ow the AVC ( ave r age v a r i a b l e c o s t s ) . Hence some f i r m w i l l p r e f e r n o t t o p r o d u c e . Bu t by a s s ump t i o n M, t h i s w i l l be i m p o s s i b l e . Hence , we have a c o n t r a d i c t i o n . ( 2 ) In t he o t h e r c a se , as p r i c e app roaches M C , due t o the p o s s i b i l i t y o f r a t i o n i n g , one f i r m w i l l a lway s w i s h t o r a i s e i t s p r i c e and use t he f a c t t h a t i t s c o m p e t i t o r w i l l r a t i o n consumers t o i t s b e n e f i t . T h i s w i l l a g a i n r e s u l t i n a c o n t r a d i c t i o n . - 17 -In bo th c a s e s m i l l p r i c e u n d e r c u t t i n g i s n e ve r used and t h e p r o o f w i l l a p p l y t o ME as w e l l . Obse rve t h a t r a t i o n i n g i s n eve r used i n c a s e 1; hence t h e argument .^ wi 11 c o n t i n u e t o h o l d even i f f i r m s choose o n l y p r i c e s ex p o s t . Case 2 c o r r e s p o n d more C l o s e l y to t he Edgewor th t ype o f n o n - e x i s t e n c e , wh i ch i s due t o t h e p r e s en c e o f r a t i o n i n g . N o t i c e as w e l l t h a t t he r e s t r i c t i o n o f - t h e a n a l y s i s t o two f i r m s i s no t r e s t r i c t i v e i n t h e sense t h a t f o r - a n y number o f f i r m s ' - i n ; t h e m a r k e t , i f any two l o c a t e c l o s e t o g e t h e r , no e q u i l i b r i u m w i l l e x i s t . 4. C o n c l u d i n g Comments The n o t i o n o f a Nash e q u i l i b r i u m i s c e n t r a l t o the t h e o r y o f i m p e r f e c t c o m p e t i t i o n . I t c a p t u r e s t he n o t i o n t h a t a g e n t s make d e c i -s i o n s ex an te to max im i ze p a y o f f s w i t h e x p e c t a t i o n s t h a t a r e c o n s i s t e n t ex p o s t . 6 The f a c t t h a t e q u i l i b r i a do no t e x i s t i n mode l s w i t h p r o d u c t d i f f e r e n t i a t i o n r a i s e s f undamen ta l q u e s t i o n s c o n c e r n i n g t he p r e d i c t i v e power o f o u r t h e o r y . T h i s i s p a r t i c u l a r l y the c a s e w i t h t h e s e mode l s s i n c e t h e y a r e m o t i v a t e d by an a t t e m p t t o make ou r s t a n d a r d homogeneous mode l s more r e a l i s t i c . As we have shown, a l t h o u g h demands can be e x -p e c t e d t o be w e l l behaved , t h i s f a c t a l o n e canno t r e s o l v e ; n o n - e x i s t e n c e p r ob l ems t h a t a r i s e i n homogeneous p r o d u c t m o d e l s . Of c o u r s e one can a lway s p l a c e r e s t r i c t i o n s on t h e pa rame te r s o f t he model t o r e s t o r e e x i s t e n c e . In t h i s model t h i s may be a c h i e v e d by s u p p o s i n g U i s s u f f i c i e n t l y l a r g e t h a t the f i r m can l o c a t e f a r enough a p a r t t h a t t h e y become l o c a l m o n o p o l i s t s . G i ven t h a t the demand d i s t r i -- 18 -b u t i o n i s u n i f o r m , r e g a r d l e s s o f t h e s o l u t i o n , c o n c e p t chosen t h e r e wou ld e x i s t an e q u i l i b r i u m . However f o r s u f f i c i e n t l y low f i x e d c o s t s , new e n t r a n t s wou ld a lway s be i n a p o s i t i o n o f t r y i n g t o e v a l u a t e the outcome o f l o c a t i n g c l o s e to a n o t h e r f i r m . W i t hou t a c o n s i s t e n t way o f m o d e l l i n g t h i s s i t u a t i o n one w i l l no t have a c o m p l e t e l y s a t i s f a c t o r y m o d e l . The most common way t o r e s t o r e an e q u i l i b r i u m i s t h r o u g h t he use o f ' m i x e d o r random s t r a t e g i e s . ^ Fo r x^ f.x.^; i n our model an e q u i l i b r i u m i n m ixed o r random s t r a t e g i e s w i l l a lways e x i s t s i n c e t h e p r o f i t - f u n c t i o n Q i s c o n t i n u o u s . F u r t h e r t h e r e s u l t s o f Dasgup ta and Mas k i n [ 1982 ] can be used to show e x i s t e n c e when f i r m s l o c a t e t o g e t h e r . On a p u r e l y t e c h n i c a l l e v e l f o r such a p r o c edu r e to make s en se we s h o u l d a t l e a s t have the c l o s e d g raph p r o p e r t y f o r Nash e q u i l i b r i a . That i s f o r t he model t o be w e l l b e -h a v e d , i t s h o u l d be t he case t h a t e q u i l i b r i a f o r t he s l i g h t l y d i f f e r e n t i a t e d model a p p r o x i m a t e e q u i 1 i b r i a f o r t h e homogeneous p r o d u c t c a s e , ' " o t h e r w i s e we g a r e l e f t w i t h a n o t h e r p a t h o l o g y t o e x p l a i n . S e c o n d l y , m i xed s t r a t e g i e s make the most s en se i n r e p e a t e d s i t u a t i o n s where t h e l aw o f l a r g e num-be r s can be a p p l i e d . I t i s no t c l e a r t h a t agen t s f a c e d w i t h a s i n g l e s h o t d e c i s i o n wou l d n e c e s s a r i l y f o l l o w a m ixed s t r a t e g y . However once one a d m i t s some dynamic s t r u c t u r e , g e n e r a l l y t h e s e t o f p o s s i b l e ' e q u i -l i b r i a i n c r e a s e eno rmous l y due to t h e p o s s i b i l i t y o f c o o p e r a t i o n . ^ Such a p o s s i b i l i t y i s q u i t e s e r i o u s f o r d i f f e r e n t i a t e d p r o d u c t mode ls s i n c e f i r m s l o c a t i n g c l o s e to each o t h e r may f i n d i t e a s i e r t o c oope r a t e , and hence p r e f e r to l o c a t e t o g e t h e r r a t h e r t han a p a r t due t o a s h i f t i n t h e solution concept.^ - 19 -A n o t h e r p o s s i b i l i t y i s t o be found i n t h e r e c e n t work o f de Palma e t a l . [1983] who i n t r o d u c e u n c e r t a i n t y i n t o t h e f i r m c h o i c e by i n d i v i d u a l s . Such a model e s s e n t i a l l y assumes t h a t - t h e r e a r e a b s o l u t e d i f f e r e n c e s between f i r m s t h a t a re b e s t m o d e l l e d ' a s b e i n g u n c e r t a i n . One then s i m p l y s t u d i e s the l o c a t i o n p rob l em i n some o b j e c t i v e space w i t h t h e f i r m s b e i n g s u b j e c t i v e l y d i f f e r e n t enough t o g ene r a t e e q u i l i b r i a . F i n a l l y i f one does f e e l t h a t d i f f e r e n t i a t e d p r o d u c t mode ls c a p t u r e some e s s e n t i a l c h a r a c t e r i s t i c s o f ma rke t s , t h e n d e s p i t e the non -e x i s t e n c e o f e q u i l i b r i u m p r o b l e m , one i s s t i l l l e f t w j t h t h e q u e s t i o n o f how t o e x p l a i n agents ' . ' b e h a v i o u r i n such s i t u a t i o n s . ' We. wou ld s ugge s t , tha t t h e s e p r o b l e m s " a r e f undamen ta l and a l s o go t o a c e n t r e o f t h e - n o n - e x i s t e n c e o f equ i 1 i b r i uiti p r ob l em a s s o c i a t e d . w i t h g e n e r a l e q u i l i b r i u m mode l s , as 12 p o i n t e d out by R o b e r t s and Sonnen s che i n [ 1 9 7 7 ] . - 20 -NOTES 1. H o t e l l i n g [ 1 9 2 9 ] , page 471 . The b r a c k e t e d ph r a s e has been added . 2 . See MacLeod [1982] . . 3 . T h i s a s s ump t i o n c o r r e s p o n d s p r e c i s e l y t o H o t e l l i n g [ 1 9 2 9 ] ; he a rgued f o r " t h e s t a b i l i z i n g e f f e c t o f masses o f consumers p l a c e d so as t o have a n a t u r a l p r e f e r e n c e f o r one s e l l e r o r t h e o t h e r " , p. 4 70 . 4 . See L e v i t a n and Shub i k [ 1978 ] f o r an a n a l y s i s o f s u ch mode l s i n a homogeneous p r o d u c t m o d e l . 5 . I t i s q u i t e c l e a r l y no t c o n t i n u o u s a t q^ = 0 . 6. To be p r e c i s e we a r e s t u d y i n g subgame p e r f e c t Nash e q u i l i b r i u m . See S e l t e n [1975 ] on t h i s p o i n t . 7. See Nash [ 1 9 5 1 ] . 8 . T h i s model m i g h t a l s o be seen as a n o t h e r model w h i c h , l i k e V a r i a n [ 1 9 8 0 ] , shows the e x i s t e n c e o f s a l e s . 9 . The r e s u l t s o f R o b e r t s [ 1980 ] i n d i c a t e i n l a r g e economies one does no t o b t a i n su ch r e s u l t s . 1 0 . See F r i edman [ 1 9 7 7 ] . 1 1 . Gab s z ew i c z [1983] a rgues t h a t such i d e a s may fo rm the b a s i s f o r a new way t o model i m p e r f e c t c o m p e t i t i o n . 12 . A l s o see t h e s u r v e y by H a r t [ 1 9 8 3 ] . CHAPTER 2 THE CORE AND DUOPOLY THEORY 1. I n t r o d u c t i on I t was s u g g e s t e d by Aumann [ 1964 ] t h a t m i xed ma rke t m o d e l s , i . e . mode l s w i t h atoms as w e l l as a con t i nuum o f t r a d e r s , wou ld be t h e most a p p r o p r i a t e f o r t he s t u d y o f i m p e r f e c t c o m p e t i t i o n . T h i s s u g g e s -t i o n spawned a l i n e o f r e s e a r c h wh i c h l o o k e d f o r c o n d i t i o n s under w h i c h b e i n g l a r g e o r an atom i n a con t i nuum o f t r a d e r s was " a d v a n t a g e o u s " . By advan tageous we mean t h a t t h e r e a re a l l o c a t i o n s i n t h e c o r e o f an exchange economy t h a t make the l a r g e t r a d e r s b e t t e r o f f t h a n a t t h e com-p e t i t i v e e q u i l i b r i u m . However t h e examples o f Aumann [1973] d emons t r a t e t h a t t h e r e i s no n e c e s s a r y r e l a t i o n s h i p between t h e c o m p e t i t i v e e q u i l i -b r i um and t h e maximum p a y o f f i n t he c o r e t h a t a m o n o p o l i s t m igh t o b t a i n . F u r t h e r , t he r e s u l t s o f Gab s z ew i c z and Me r t en s [1971] and S h i t o v i t z [1973] s i g n i f i c a n t l y e x t e n d t h e c o r e e q u i v a l e n c e theorem t o t h e c a se when t h e r e a r e s e v e r a l l a r g e t r a d e r s w i t h t h e same c h a r a c t e r i s t i c s . We a r e t hen l e f t w i t h t h e p e s s i m i s t i c c o n c l u s i o n s o f H i l d e n b r a n d [ 1974 ] and Aumann [ 1973 ] t h a t no t o n l y i s i t d i f f i c u l t t o p r o v i d e a n a t u r a l economic i n t e r -p r e t a t i o n o f t h e s e mode ls bu t a l s o t he r e s u l t s f rom t h e i r s t u d y s ugge s t t h a t t h e y do no t a d d r e s s i s s u e s r e l e v a n t t o the t h e o r y o f i m p e r f e c t com-p e t i t i o n . In t h i s e s s a y we w i l l s u gge s t t h a t t h e s e mode ls p l a y q u i t e a d i f f e r e n t and p o s s i b l y more i m p o r t a n t r o l e i n t he t h e o r y o f i m p e r f e c t c o m p e t i t i o n . We b e g i n by e x p l i c i t l y a d a p t i n g the model o f S h i t o v i t z [ 1973 ] t o t h e c l a s s i c a l d uopo l y p rob l em o f Cou rno t [ 1838 ] and B e r t r a n d - 21 --.22 -[ 1 8 8 3 ] . The f i r s t p o i n t i s , as a rgued by Aumann [ 1 9 6 4 ] , t h a t t h e use o f t h e c o r e as a s o l u t i o n c on cep t i s d e s i g n e d t o a n a l y z e t he outcome o f mar -k e t c o m p e t i t i o n i n d e p e n d e n t o f any s p e c i f i c t r a d i n g i n s t i t u t i o n . As such i t f o rms a n a t u r a l s t a r t i n g p o i n t f o r t h e a n a l y s i s o f t h e deba te i n i -t i a t e d by B e r t r a n d [ 1883 ] on whe the r p r i c e s o r q u a n t i t i e s a re the a p p r o -p r i a t e s t r a t e g i c v a r i a b l e . One m igh t go f u r t h e r and a rgue t h a t t he co r e i s a c t u a l l y t h e c o r r e c t s o l u t i o n c o n c e p t s i n c e , f o r many c l a s s e s o f p r o -d u c t s , buye r s w i l l i n f a c t b a r g a i n w i t h s e l l e r s . In p a r t i c u l a r , buye r s w i l l o f t e n use a p r i c e quo t ed by one s e l l e r t o o b t a i n a b e t t e r dea l w i t h his.iCompetitor; a p r o c edu r e a k i n t o the b l o c k i n g c on c ep t used to d e f i n e t h e c o r e . In p a r t i a l e q u i l i b r i u m o l i g o p o l y mode ls one t y p i c a l l y assumes t h a t buye r s t a k e t he s e l l e r s s t r a t e g i e s as g i v e n . However t h i s i s m e r e l y an a s s u m p t i o n t h a t , g i v e n t h e Aumann [ 1973 ] e x amp l e s , does n o t f o l l o w f rom m e r e l y h a v i n g a l a r g e t r a d e r f a c i n g a con t i nuum o f s m a l l b u y e r s . The model we p r e s e n t makes a d i s t i n c t s e p a r a t i o n between p r o d u c t i o n and s a l e s a c t i v i t i e s . I t w.iTl be supposed t h a t we have two s e l l e r s t h a t have p r e v i o u s l y bough t o r p r oduced an i n v e n t o r y o f goods . We can then use S h i t b v i t z [ 1973 ] t o model t he s e l l e r s t r a d i n g t h e i r i n v e n t o r i e s f o r s t o c k s o f money h e l d by a con t i nuum o f b u y e r s . In c o n t r a s t w i t h t he i n t e r p r e t a t i o n o f S h i t o v i t z [ 1973 ] t h a t h i s c o r e e q u i v a l e n c e theorem i s i n t he B e r t r a n d [ 1883 ] t r a d i t i o n , we a rgue t h a t t h i s model p r o v i d e s an i m p o r t a n t f o u n d a t i o n f o r t h e r a t h e r c omp l a c en t a s s u m p t i o n o f o l i g o p o l y t h e o r y t h a t t h e b u y e r s a c t c o m p e t i t i v e l y . In t h e f i n a l s e c t i o n o f t he no t e we a rgue t h a t t he "m i xed ma r ke t " mode ls can be q u i t e n a t u r a l l y i n t e r -- 23 -p r e t e d i n t h e C o u r n o t [ 1838 ] q u a n t i t y s e t t i n g app roa ch to i m p e r f e c t com-p e t i t i o n . 2. The M o d e l 1 C o n s i d e r a two good exchange economy w i t h a s e t o f t r a d e r s g i v e n by T = [ 0 , 1 ] u { a , b } where [ 0 , 1 ] r e p r e s e n t s a c on t i nuum w h i l e { a , b } r e p r e s e n t s two a t oms . An a l l o c a t i o n o f t he two goods w i l l be the map, x : T -*IRJ de f where x ( t ) = ( y ( t ) , m ( t ) ) . The p r e f e r e n c e s o f t r a d e r s o v e r the goods a r e r e p r e s e n t e d by a u t i l i t y f u n c t i o n U t s a t i s f y i n g : ( i ) U t ( y , m ) f o r t e [ 0 , 1 ] i s assumed t o be c o n t i n u o u s and s t r i c t l y i n c r e a s i n g i n (y ,m) e M+; d e f ( i i ) U t ( y , m ) = 6y + m f o r t e {a ,b> and some 6 > 0 . The i n i t i a l a l l o c a t i o n i o f goods h e l d by t h e t r a d e r s i s g i v e n by ( i ) i ( t ) = ( 0 , m ) , f o r t e [ 0 , 1 ] and ( i i ) i ( t ) = ( 1 , 0 ) , f o r t e { a , b } . F i n a l l y t o comp l e t e the s p e c i f i c a t i o n o f t h e model we need to d e f i n e t h e measure o r s i z e o f t h e t r a d e r s . L e t A deno te the s m a l l e s t - 24 -a - a l g e b r a c o n t a i n i n g t h e Lebesgue mea su r ab l e s e t s on [ 0 , 1 ] and t h e atoms {a} and { b } . D e f i n e a measure p ( . ) on t h e s e t s i n A by ( i ) y ( . ) i s Lebesgue measure when r e s t r i c t e d t o [ 0 , 1 ] ; d e f ( i . i ) y ( a ) = q a d e f " W h e r e V % > ° -y ( b ) = q b T h i s model i s s p e c i a l c a s e o f S h i t o v i t z [ 1 9 7 3 ] . I t i s i n t e r -p r e t e d as a f o r m a l r e p r e s e n t a t i o n o f a d u o p o l i s t i c ma r ke t i n wh i c h a c on t i nuum o f b u y e r s , [ 0 , 1 ] , w i s h to max im i z e t h e i r u t i l i t y by t r a d i n g some o f t h e i r h o l d i n g s ' o f m, a c o m p o s i t e commodi ty denomina ted i n money u n i t s , f o r t h e good y . The s e l l e r s , r e p r e s e n t e d by t he two atoms { a , b } , wou ld on t h e i r p a r t T i k e to max im i ze t h e i r p r o f i t s o r money h o l d i n g s . The 6y te rm i n t h e u t i l i t y f u n c t i o n o f t h e d u o p o l i s t s r e p r e s e n t s t h e v a l u e o f u n s o l d s t o c k . I t w i l l n o r m a l l y be s m a l l bu t can be assumed t o 2 be p o s i t i v e by s u p p o s i n g t h a t y a lway s has a s c r a p v a l u e . F i n a l l y t he measure o r s i z e o f the s e l l e r s {a} and {b} r e p r e s e n t s t h e i r i n v e n t o r y h o l d i n g s o f good y . By s u p p o s i n g i ( t ) = ( 1 , 0 ) f o r t e { a , b } t h e t o t a l amount o f good y a v a i l a b l e i s t £ { a , b } ^ ( t ) = % + V The re a r e two r ea sons f o r p r o c e e d i n g t h i s way. The f i r s t i s s i n c e t h e s e l l e r s have t h e same p r e f e r e n c e s and endowments t h e y w i l l b e , to use S h i t o v i t z 1 s t e r m i n o l o g y , o f t he same type. T h i s w i l l be i m p o r t a n t when one a p p l i e s h i s r e s u l t s . S e c ond l y i t p r o v i d e s a n o t h e r i n t e r p r e t a -- 25 -t i o n o f t h e s i z e o f an atom i n t e rms o f a t r a d e r ' s i n v e n t o r y h o l d i n g s . N o t i c e t h a t , i n c o n t r a s t w i t h S h i t o v i t z [ 1 9 8 2 ] , p r o d u c t i o n o c c u r s ex an t e to p roduce t h e i n v e n t o r y o f goods . 3 . The C o m p e t i t i v e E q u i l i b r i u m L e t us suppose t h a t t h e p r i c e o f good y i s p w h i l e t he p r i c e o f m i s n o r m a l i z e d to 1. Now, 1 ' fu r ther suppose t h a t a l l agen t s i n T a c t as p r i c e t a k e r s . S i n c e the a g e n t s i n [ 0 , 1 ] have no i n i t i a l h o l d i n g s o f y , t hen t h e y w i l l be bu ye r s o f y f o r any p r i c e . The demand D t ( p ) f o r good y by t e [ 0 , 1 ] i s f ound by s o l v i n g max U t ( y , m ) such t h a t py - m < m. y ,m The t o t a l demand f o r y w i l l t hus be : d e f f D ( P } = 4 [ O , I ] ¥ P > D ^ ) -L e t us assume t h a t t h e p r e f e r e n c e s a r e s u f f i c i e n t l y r e g u l a r t o g ene r a t e a c o n t i n u o u s , downward s l o p i n g demand c u r v e . The s u p p l y c o r r e s p o n d e n c e , S t ( p ) , t e { a , b } , f o r good y i s o f c o u r s e g i v e n by s o l v -i n g : max 6y + m such t h a t py + m < p • 1, y ,m the s o l u t i o n o f wh i ch i s 1 1 i f p > 6 [ 0 , 1 ] i f p = 6 0 i f p < 6 . - 26 -Hence t he a g g r e g a t e s u p p l y o f good y i s : d e f f S ( P > = ^ e { a , b } S t ( P > q a + q b i f p > 6 [ 0 , q a + q b ] i f p = 6 0 i f p < 6 The competitive equilibrium i s g i v e n by t h e p r i c e , p*, a t wh i ch t h e marke t c l e a r s o r S ( p* ) = D ( p * ) . T h i s i s shown d i a g r a m m a t i c a l l y i n F i g u r e 1. N o t i c e t h a t t h e r e a r e b a s i c a l l y two c a s e s t o c o n s i d e r . W i t h demand c u r v e D^(p) we have 0-^(6) > q f l + q^., hence a t the c o m p e t i t i v e * e q u i l i b r i u m p^  t h e two atoms s e l l a l l o f t h e i r i n v e n t o r y a t a p r i c e * g i v e n by D^(p^) = q a + q^. In t h e second c a s e , t h e demand c u r v e i s such t h a t D 2(6) < q a + q^. T h i s i s e s s e n t i a l l y a case o f e x c e s s c a p a c i t y f rom t h e s e l l e r s p o i n t o f v i ew ; ' w i t h t h e e q u i l i b r i u m c h a r a c t e r i z e d by P 2 = <5, t h e s c r a p v a l u e o f y , w h i l e the t o t a l s a l e s i s D 2(6). N o t i c e t h a t t h i s e q u i l i b r i u m i s not un i que i n t h e sense t h a t t h e s a l e s o f each s e l l e r i s i n d e t e r m i n a t e . 4 . The Core A l l o c a t i o n s Aumann [ 1964 ] a rgued t h a t " I n t u i t i v e l y , one f e e l s t h a t money and p r i c e s a r e no more t han a d e v i c e t o s i m p l i f y t r a d i n g . " In c o n t r a s t the c o r e c o n c e p t i s seen as a f undamen ta l s o l u t i o n c o n c e p t t h a t i s no t o n l y i n s t i t u t i o n f r e e bu t i s a l s o w e l l d e f i n e d ' r e g a r d l e s s o f t he s i z e o f - 27 -t he t r a d e r s . F u r t h e r i n t h e o l i g o p o l y p rob l em i t i s assumed t h a t t he s e l l e r s s e t p r i c e s o r q u a n t i t i e s w h i l e -buyers p a s s i v e l y a c c e p t t h e s e c h o i c e s . Yet as Aumann [ 1973 ] a r g u e s , t h i s i s f u n d a m e n t a l l y a symmet r i c and does no t e x p l a i n why a s ymmet r i e s i n s i z e a l o n e can endow f i r m s w i t h 4 ma rke t power . A c c o r d i n g l y t h e c o r e c o n c e p t can a c t as a benchmark a g a i n s t wh i ch i n s t i t u t i o n s p e c i f i c s o l u t i o n c o n c e p t s can be compa red . L e t us now f o r m a l l y d e f i n e t h e c o r e . A s e t S e A w i l l be c a l l e d a c o a l i t i o n . An a l l o c a t i o n , x ( . ) , i s f e a s i b l e f o r S i f and o n l y i f x ( t ) d y ( t ) = iteS i ( t ) d u ( t ) . A c o a l i t i o n S can improve upon an a l l o c a t i o n x ( . ) i f t h e r e e x i s t s a f e a s i b l e a l l o c a t i o n f o r S , x ( . ; ) , s u ch t h a t U t ( x ( t ) ) > U t ( x ( t ) ) , f o r a lmo s t a l l t e S. The oove i s t h e s e t o f f e a s i b l e a l l o c a t i o n s wh i ch canno t be improved upon by any c o a l i t i o n S £ A . For ou r model t h e c o r e a l l o c a t i o n i s p a r t i c u l a r l y e a s y t o compute s i n c e t h e model s a t i s f i e s a l l t he assump-t i o n s o f Theorem B o f S h i t o v i t z [ 1 9 7 3 ] . Hence~we have : PROPOSITION'(Shitovitz). The oove allocations are exactly the set of competitive allocations. Us i n g t h e c o m p u t a t i o n s o f the p r e v i o u s s e c t i o n we can e x p l i -c i t l y compute t h e c o r e a l l o c a t i o n f o r t h e two c a s e s . - 28 -Case 1: The e q u i l i b r i u m p r i c e i s P* = D l 1 ( q a + V ' where D ^ ( . ) i s t he i n v e r s e demand c u r v e . I f x ( . ) i s a c o r e a l l o c a t i o n , t hen x ( t ) = ( D t ( p * ) , m - p * D t ( p * ) ) , t e [ 0 , 1 ] x ( t ) = ( 0 , , . p * ) , t e { a , b } . N o t i c e t h a t the t o t a l amount o f money s t o c k h e l d by each a t om , t e { a , b } , d e f j • V q a ' V = K=t P * d ^ T ) = P * q t = D j 1 ( q a + q b ) q t . Hence t h e r e venue o f t h e s e l l e r s has e x a c t l y t he same fo rm as the Cou r no t [ 1 8 3 8 ] p r o f i t f u n c t i o n s when t h e i n v e n t o r i e s a r e i n t e r p r e t e d as o u t p u t c h o i c e s . Case 2: In t h i s c a s e t he c o m p e t i t i v e p r i c e i s 6 ; hence t h e c o r e a l l o c a t i o n s x ( . ) , a r e g i v e n by : x ( t ) = ( D t ( 6 ) „ m - 6 D t ( 6 ) ) , t e [ 0 , 1 ] x ( t ) = (1 - y ( t ) , 6 y ( t ) ) , t e { a , b } where y ( a ) , y ( b ) a r e any s a l e s l e v e l s by s e l l e r s { a , b } s a t i s f y i n g q a y ( a ) + q b y ( b ) = D (6 ) - 29 -and 0 < y ( t ) < 1, t e { a , b } . In t h i s c a s e , g i v e n t h a t t h e f i r m s s e l l t h e i r unused i n v e n -t o r y a t t h e s c r a p p r i c e 6 , t h e t o t a l r e venue f o r each s e l l e r w i l l now be: M t ( q a , q b ) - 6 q t , t e {a,b:}.. 5 . D i s c u s s i o n o f t h e Model Though we have p r e s e n t e d a r a t h e r s p e c i f i c example o f the S h i t o v i t z [ 1973 ] m o d e l , i t i s however a q u i t e g e n e r a l r e p r e s e n t a t i o n o f 5 t h e d u o p o l y p r o b l e m . The model s i m p l y r e p r e s e n t s two p r o f i t m a x i m i z i n g agen t s s e l l i n g t h e i r i n v e n t o r y t o many buye r s r e p r e s e n t e d by an a r b i -t r a r y demand c u r v e . I f we now suppose t h a t t h e agen t s buy o r p roduce t he good y a t a c o n s t a n t m a r g i n a l c o s t c > 6 then t he t o t a l p r o f i t s o f each s e l l e r i s g i v e n b y : % } = M t ( V % r ~ C V t e { a ' b K I f t he i n v e n t o r i e s must be chosen i n d e p e n d e n t l y ex a n t e t hen t he outcome can be d e s c r i b e d as a Nash [ 1951 ] e q u i l i b r i u m i n : o u t p u t s . 6 * * Hence t he e q u i l i b r i u m ( q , q , ) i s a s o l u t i o n t o a D * max T T , ( q , q , ) q 3 b * max TT. (q , q ) . q b 3 T h i s i s o f c o u r s e s i m p l y t h e s t a n d a r d Cou r no t -Na sh e q u i l i b r i u m f o r t h e d u o p o l y p r o b l e m . - 30 -T h i s model t hu s shows t h a t S h i t o v i t z [ 1973 ] q u i t e n a t u r a l l y f i t s i n t o t h e C o u r n o t q u a n t i t y - s e t t i n g f r amework . One o f t h e ma jo r c o n t r i -b u t i o n s o f t h i s work t h en i s t o p r o v i d e an e x p l i c i t i n s t i t u t i o n f r e e f o u n d a t i o n f o r the a s s ump t i o n i n much o f o l i g o p o l y t h e o r y t h a t buye r s behave c o m p e t i t i v e l y . ^ The c o r e app roa ch a l s o has a d i s t i n c t advan tage ove r t h e mode ls w i t h ; p r i c e , s e t t i n g f i r m s , i n t h a t t h e c o r e . w i 1 1 e x i s t , whenever a c o m p e t i t i v e e q u i l i b r i u m e x i s t s . On t h e o t h e r hand , ma r ke t s w i t h p r i c e s e t t i n g f i r m s , though i n t u i t i v e l y c o m p e t i t i v e , i n f a c t o f t e n g s u f f e r f rom t he n o n - e x i s t e n c e o f an e q u i l i b r i u m . We c o n c l u d e t h a t "m ixed ma r ke t " mode ls no t o n l y have a n a t u r a l economic i n t e r p r e t a t i o n bu t they; . .a lso h i g h l i g h t t h e c l a s s i c a l " C o u r n o t " v i ew o f i m p e r f e c t c o m p e t i t i o n i n wh i ch the fundamenta l i s s u e i s t h e way i n wh i c h economic a gen t s can a f f e c t t he c o m p e t i t i v e e q u i l i b r i u m to t h e i r a d v a n t a g e . F i n a l l y the B e r t r a n d , " c o m p e t i t i v e " , r e s u l t w i l l o n l y a r i s e when f i r m s i n the marke t have no way t o p recommi t t h emse l v e s t o r e s -t r i c t e d o u t p u t l e v e l s , a r e s u l t c o r r e s p o n d i n g t o o u r c a s e 2 above . - 31 -NOTES 1. See S h i t o v i t z f o r the c omp l e t e t e c h n i c a l d e t a i l s o f t h e m o d e l . 2 . F o r examp le an u n s o l d c a r a lways has v a l u e due to i t s meta l c o n -t e n t . The main r e a s on f o r t h e i n c l u s i o n o f 6 i s t o s a t i s f y t h e a s sump t i on s made by S h i t o v i t z [ 1 9 7 3 ] . 3 . Aumann [ 1 9 6 4 ] , page 4 0 . 4 . Aumann [ 1 9 7 3 ] , page 10 . 5. Of c o u r s e t h e e x t e n s i o n o f t h e m o d e l , g i v e n theo rem B o f S h i t o v i t z [ 1 9 7 3 ] , t o t he o l i g o p o l y p r ob l em i s q u i t e s i m p l e . 6. B y ' i n d e p e n d e n t c h o i c e we mean t h a t each f i r m must choose i t s i n v e n -t o r y l e v e l w i t h o u t be i ng a b l e t o o b s e r v e the c o m p e t i t o r ' s c h o i c e . A Nash e q u i l i b r i u m i s t h e c a s e i n wh i ch f i r m s max im i ze t h e i r p a y o f f s g i v e n correct e x p e c t a t i o n s . 7 . One can a r g u e once p r o d u c t i o n c o s t s a r e i n c l u d e d t h a t t h i s a p p l i e s even to the h o t e l example o f S h i t o v i t z [ 1 9 7 3 ] . G i v e n t h a t one has p e r f e c t i n f o r m a t i o n t hen t he m o n o p o l i s t wou ld b u i l d c a p a c i t y equa l t o t h e monopo ly l e v e l . . Hence ex po s t we wou ld have a c o m p e t i t i v e ' e q u i -l i b r i u m a t t h e monopo ly p r i c e . 8 . See Edgewor th [ 1925 ] f o r t he f i r s t p r o o f o f t h i s . - 32 -Figure 1 CHAPTER 3 ON ADJUSTMENT COSTS AND THE STABIL ITY OF EQUILIBRIA 1. I n t r o d u c t i o n I t i s w e l l known t h a t when f i r m s i n an o l i g o p o l i s t i c i n d u s t r y can ob se r v e d i r e c t l y each o t h e r s a c t i o n s t h e r e e x i s t s a l a r g e c l a s s o f Nash e q u i l i b r i a t h a t y i e l d h i g h e r p r o f i t s f o r a l l f i r m s t h an t ho se a s -s o c i a t e d w i t h 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 . As F r i edman [ 1977 ] has shown, t h e s e e q u i l i b r i a can a r i s e when f i r m s use t h r e a t s a g a i n s t compe-t i t o r s t h a t t a k e t h e fo rm o f m a i n t a i n i n g an ag reed upon l e v e l o f o u t p u t u n l e s s some f i r m d e v i a t e s f rom t h i s ag reement . I f d e v i a t i o n o c c u r s then a l l f i r m s s u b s e q u e n t l y p roduce t h e i r Cou r no t -Na sh o u t p u t s . Green [ 1980 ] has f u r t h e r shown t h a t t h e s e e q u i l i b r i a a r e a l s o p e r f e c t i n t h e s en se o f S e l t e n [ 1 9 7 5 ] . T h i s m u l t i p l i c i t y o f e q u i l i b r i a l i m i t s ones a b i l i t y t o make q u a n t i t a t i v e p r e d i c t i o n s c o n c e r n i n g b e h a v i o u r i n such ma r ke t s and p r o v i d e s l i t t l e b a s i s f o r a d e s c r i p t i o n o f d i s e q u i l i b r i u m b e h a v i o u r . In t h i s paper we s ugge s t a s o l u t i o n t o t h i s p rob l em by r e q u i r i n g e q u i l i -b r i a t o have t h e p r o p e r t y o f local stability when f i r m s f a c e a c o s t o f a d j u s t m e n t . In p r a c t i c e one wou ld e x p e c t f i r m s to f a c e a d j u s t m e n t c o s t s t h a t wou ld l i m i t t h e i r a b i l i t y t o a d j u s t o u t p u t i n s t a n t a n e o u s l y . The q u e s t i o n t hen i s the r e l a t i o n s h i p between t h e s e c o s t s and t h e s t r a t e g i c b e h a v i o u r o f f i r m s . We w i l l p o s t u l a t e a c o s t o f a d j u s t m e n t t h a t w i l l r e -s t r i c t f i r m s t o s m a l l , i n f i n i t e s i m a l o u t p u t changes a t e a ch t ime t . T h i s w i l l i n a sense r e s t r i c t the p e r s p e c t i v e o f f i r m s t o e v a l u a t i n g t he e f -- 33 -- 34 -f e e t o f s m a l l changes i n o u t p u t on p r o f i t s and a l l o w us t o assume t h a t f i r m s choose o u t p u t changes t o max im i z e t h e r a t e o f change o f p r o f i t s . T h i s app roach i s m o d e l l e d f o r m a l l y as a l o c a l game, a t h e o r y t h a t t i l l now has been a lmo s t e x c l u s i v e l y used i n t he p u b l i c goods a l -l o c a t i o n l i t e r a t u r e i n , f o r e x amp l e , papers by Dreze and de l a V a l i n e P o u s s i n [ 1 9 7 1 ] , R o b e r t s [ 1 9 7 9 ] , Schoumaker [1979] and , most r e c e n t l y , Mat thews [ 1 9 8 2 ] . Such an app roach i s a n a l y t i c a l l y much more t r a c t a b l e t h an t a k -i n g a d i f f e r e n t i a t e game a p p r o a c h . * I t a l s o a l l o w s ; o n e t o s t u d y i n a s i m p l e manner d i f f e r e n t s o l u t i o n c o n c e p t s t h a t embody v a r i o u s n o t i o n s o f l o c a l r a t i o n a l i t y . A s o l u t i o n c o n c e p t w i l l i d e n t i f y a l o c a l l y s t a b l e e q u i l i b r i u m as b e i n g a v e c t o r o f o u t p u t s f o r wh i ch any s m a l l o u t p u t change by a f i r m w i l l not make i t b e t t e r o f f g i v e n c o r r e c t l y a n t i c i p a t e d r e s -ponses by c o m p e t i t o r s . The l o c a l game app roach can a l s o be v i ewed as a way to e x t e n d t h e c o n j e c t u r a l v a r i a t i o n s app r oa ch t o t he o l i g o p o l y p r o b -lem t o any a r b i t r a r y v e c t o r o f o u t p u t s . I n s t e a d o f j u s t i d e n t i f y i n g e q u i l i b r i a t h r o u g h t he c r i t e r i o n o f l o c a l s t a b i l i t y o u r a n a l y s i s a l s o p r o v i d e s t h e b a s i s f o r t h e d i s e q u i l i b r i u m a n a l y s i s o f f i r m b e h a v i o u r based on t he strategic interaction o f f i r m s . In t he e s s a y f o u r s o l u t i o n c o n c e p t s a r e s t u d i e d , two o f w h i c h a r e new t o t h e l i t e r a t u r e . The f i r s t o f t h e s e i s t h e d i r e c t i o n a l c o r e due to Mat thews [ 1 9 8 2 ] . The d i r e c t i o n a l c o r e i s i n many r e s p e c t s t he n a t u r a l d e f i n i t i o n - 35 -o f l o c a l r a t i o n a l i t y f o r i t c o n s i s t s o f t h o s e o u t p u t changes wh i c h no g roup o r c o a l i t i o n o f f i r m s wou l d w i s h t o a l t e r . However , as..;shown i n Ma t t hews , t h i s s o l u t i o n w i l l n o t i n g ene r a l e x i s t . I t w i l l be shown t h a t when the d i r e c t i o n a l c o r e does e x i s t f o r some v e c t o r o f o u t p u t s t h en t h i s p o i n t i s n o t l o c a l l y s t a b l e . The r e s u l t i s t h a t t h e u l t i m a t e e q u i l i b r i u m f o r t he o l i g o p o l y w i l l l i e i n a r e g i o n where the d i r e c t i o n a l c o r e does no t e x i s t . T h i s w i l l r e s t r i c t e q u i l i b r i a t o r e g i o n s t h a t y i e l d p a y o f f s t o f i r m s somewhere between the Cou r no t -Na sh s o l u t i o n and t he j o i n t - p r o f i t m a x i m i z i n g s o l u t i o n . We can a lway s ensu re e x i s t e n c e f o r t he s e cond s o l u t i o n c o n c e p t , t he Nash e q u i l i b r i u m f o r t he l o c a l game, wh i ch we w i l l c a l l t he Nash d i r e c t i o n a l c o r e . We show t h a t t h e s e t o f C ou r no t -Na sh e q u i l i b r i a f o r the i n d u s t r y and t h e l o c a l l y s t a b l e o u t p u t s w i t h r e s p e c t t o t h e Nash d i r e c t i o n a l c o r e a r e t h e same. The Cou r no t -Na sh e q u i l i b r i u m i s w e l l known not t o be r a t i o n a l f rom t he i n d u s t r y ' s p e r s p e c t i v e s i n c e t h e r e w i l l g en -e r a l l y e x i s t o t h e r v e c t o r o f o u t p u t s t h a t w i l l ' m a k e e v e r y f i r m b e t t e r o f f . To model t h e i n c e n t i v e s t h a t f i r m s have t o a c t as a g roup and a t t h e same t ime en su r e t h a t f i r m s have no i n c e n t i v e t o p l a y C o u r n o t -Nash , t he n o t i o n o f a Nash bargaining core i s i n t r o d u c e d t o l o c a l game t h e o r y . I t s i m p l y c o n s i s t s o f t h o s e o u t p u t changes t h a t a r e r a t i o n a l f rom an i n d u s t r y p e r s p e c t i v e and a r e b e t t e r than t h e Nash e q u i l i b r i u m . F o r m a l l y i t i s shown t h a t t h i s s o l u t i o n i s t h e l o c a l e q u i v a l e n t o f t h r e a t s t r a t e g i e s f o r t h e g l o b a l game. .< Our ma in r e s u l t i s a c omp l e t e c h a r a c t e r i -z a t i o n o f t h e l o c a l l y s t a b l e e q u i l i b r i a under t h i s s o l u t i o n c o n c e p t . - 36 -These e q u i l i b r i a w i l l o f t e n be un ique w i t h t h e ma r ke t s h a r e s o f t h e f i r m s b e i n g i n v e r s e l y p r o p o r t i o n a l t o t h e i r c o s t s o f a d j u s t m e n t . Thus the i n -t r o d u c t i o n o f a d j u s t m e n t c o s t s no t o n l y g r e a t l y r e s t r i c t s t he s e t o f e q u i l i b r i a bu t a l s o has a n a t u r a l i n t e r p r e t a t i o n i n te rms o f t he b a r -g a i n i n g power o f f i r m s . The f i n a l s o l u t i o n c o n c e p t i s t h e b a r g a i n i n g c o r e wh i ch c a p t u r e s t h e p o s s i b i l i t y o f s o p h i s t i c a t e d c o a l i t i o n f o r m a t i o n by f i r m s . Though i t w i l l no t e x i s t i n g e n e r a l , i t w i l l e x i s t a t t h e l o c a l l y s t a b l e e q u i l i -b r i um g i v e n by t h e Nash b a r g a i n i n g co re ' . These o u t p u t s w i l l a l s o be s t a b l e w i t h r e s p e c t t o t h e b a r g a i n i n g c o r e p r o v i n g t h a t r e g a r d l e s s o f t he c o a l i t i o n s t h a t ' m i g h t form t h e y c anno t improve on the l o c a l l y s t a b l e o u t -pu t s i d e n t i f i e d w i t h t he Nash b a r g a i n i n g c o r e . The agenda o f t h e e s s a y i s as f o l l o w s . In S e c t i o n 2 the model i s d e s c r i b e d a l o n g w i t h an e x p l a n a t i o n o f t he s t r a t e g y space to be u s e d . In S e c t i o n 3 t h e n o t i o n o f a d i r e c t i o n a l c o r e i s i n t r o d u c e d and i t s i m -p o r t a n t p r o p e r t i e s a r e c h a r a c t e r i z e d . S e c t i o n 4 goes t h r o u g h a d uopo l y examp le t o i l l u s t r a t e t he s t r u c t u r e o f t he m o d e l . The n o t i o n o f a Nash b a r g a i n i n g c o r e i s i n t r o d u c e d i n S e c t i o n 5 a l o n g w i t h o u r ma in r e s u l t on t h e c h a r a c t e r i z a t i o n o f l o c a l l y s t a b l e o u t p u t s / S e c t i o n 6 s p e c i a l i z e s t h e s e r e s u l t s f o r t he s t a n d a r d q u a n t i t y s e t t i n g o l i g o p o l y w h i l e the f i n a l s e c t i o n c o n t a i n s o u r c o n c l u d i n g r ema r k s . 2 . The B a s i c Model C o n s i d e r an o l i g o p o l i s t i c i n d u s t r y c o n s i s t i n g o f n f i r m s i n -- 37 -dexed by t he s e t N = { l , . . . , n } . Each f i r m i e N s e l l s an amount o f a s i n g l e u n d i f f e r e n t i a l p r o d u c t t o a l a r g e number o f consumer s . I f q ( t ) e 1R+ i s a v e c t o r o f o u t p u t s a t t ime t , then l e t t he p r o f i t s pe r p e r i o d be g i v e n by V . ( t ) = TT . (q ( t ) ) - C . ( q . ( t ) , q . ( t ) ) . Here C . . (q . ( t ) , q . ( t ) ) r e p r e s e n t s a c o s t o f a d j u s t m e n t where t h e r a t e o f dq ( t ) change o f o u t p u t i s q^( t ) = — ^ — . I t i s supposed t h a t C . ( q ^ , 0 ) = 0 hence i f q* i s an e q u i l i b r i u m , t h en iT | (q*) r e p r e s e n t s t he p r o f i t s pe r p e r i o d . I f i t were no t f o r t h e a d j u s t m e n t c o s t s , t hen unde r the assump-t i o n t h a t f i r m s can o b s e r v e each o t h e r ' s a c t i o n s i m m e d i a t e l y t h e r e wou ld e x i s t many e q u i l i b r i a t h a t can be s u p p o r t e d by t he a p p r o p r i a t e t h r e a t 3 s t r a t e g i e s . In t h i s pape r a s sump t i on s w i l l be p l a c e d on the a d j u s t m e n t c o s t s t h a t w i l l r e s t r i c t f i r m s t o c o n t i n u o u s o u t p u t c h ange s . Hence I t w i l l be supposed t h a t a t t i m e t t h e f i r m ' s s t r a t e g y w i l l be to choose i t s o u t p u t change deno ted by x . ( t ) ' = g . ( t ) , i e N. However r a t h e r t h an a n a l y z e t he i n t e r a c t i o n o f t h e f i r m as a f u l l - f l e d g e d d i f f e r e n t i a l game, the 1oca l s t r a t e g i c b e h a v i o u r o f t h e l o c a l game g e n e r a t e d by t h i s p r ob l em w i l l be a n a l y z e d . By t h i s we mean t h a t f i r m s w i l l a t each t ime t choose a s t r a t e g y x . ( t ) " , i N t o max im i ze t h e i r change i n p r o f i t s V . ( t ) , i e N, g i v e n t h a t t h e y c o r r e c t l y a n t i c i -pa te t h e i r c o m p e t i t o r s r e a c t i o n s . The re a r e s e v e r a l r e a sons f o r d o i ng t h i s . F i r s t l y i f q* i s a l o n g run e q u i l i b r i u m , then i t i s r e a s o n a b l e f o r i t t o have t h e p r o p e r t y t h a t t h e r e e x i s t s no a c t i o n by a f i r m t h a t - .38 -w i l l r e s u l t i n an immed i a t e i n c r e a s e i n p r o f i t s , o r f o r m a l l y r e s u l t i n V . ( t ) > 0. S e c o n d l y , t h i s a n a l y s i s w i l l p r o v i d e s o l u t i o n s , d e f x ( t ) = Xj(t) x'(t) f o r e v e r y q e K + and hence p r o v i d e a b a s i s f o r t h e d i s -e q u i l i b r i u m a n a l y s i s o f t h e o l i g o p o l y p r o b l e m . F u r t h e r S h a p i r o [1980] has r e c e n t l y s u g g e s t e d a dynamic p r o c e s s t h a t i f f o l l o w e d by f i r m s w i l l a c h i e v e a P a r e t o - e f f i c i e n t ou t come . T h i s l o c a l game f ramework can p r o -v i d e an a n a l y s i s o f t h e l o c a l i n c e n t i v e p r o p e r t i e s ' o f such schemes . L e t us now comp l e t e t he d e s c r i p t i o n o f t he f r amewo r k . S i n c e a l l t h e a n a l y s i s w i l l o c c u r a t a g i v e n t ime t f u r t h e r e x p l i c i t t ime de-pendence w i l l ' be s u p p r e s s e d . I t w i l l be assumed t h a t t he a d j u s t m e n t c o s t s w i l l be o f t he f o l l o w i n g r a t h e r e x t r eme f o rm : 0 , i f x. c [-a.Cq.J.b^q.)] °°, i f no t V v x , ) where a ^ . ) , b . . ( . ) . a r e mea su r ab l e n o n - n e g a t i v e v a l u e d maps. I t w i l l be assumed t h a t f i r m s can a lways a d j u s t o u t p u t ; hence suppose b^(q^) > 0 f o r a l l q . > 0 . S i n c e n e g a t i v e o u t p u t s w i l l n o t be p e r m i t t e d then we need o n l y assume a ^ O ) = 0 whil:e>.a.. (q.|) > 0 f o r q. > 0. S i n c e f i r m s w i l l n o t choose a c t i o n s w i t h i n f i n i t e c o s t s t h en one m i gh t as w e l l assume t h a t t he v e c t o r o f s t r a t e g i e s , x , i s chosen f rom t h e s e t d e f X(q) = X [ - a . ( q . ) , b (q ) ) ] . i eN 1 1 1 1 - 39 -Under t h e s e c o s t a s s u m p t i o n s t h e r a t e o f change o f p r o f i t s w i l l be where i t i s assumed t h a t ir. Cq) i s d i f f e r e n t i a b l e w i t h i t s g r a d i e n t g i v e n by and x e X ( q ) . Thus t h e l o c a l game a t a v e c t o r o f o u t p u t s q i s g i v e n by ( i ) the s t r a t e g y space X ( q ) _ = I R n , ( i i ) t he p a y o f f s t o f i r m i f o r i e N, deno ted by U . ( q , x ) . w i l l have s u f f i c i e n t s t r u c t u r e f o r t h e d i s c u s s i o n o f t h e s o l u t i o n con-; c e p t s t o be i n t r o d u c e d . More g e n e r a l s t r a t e g y spaces c an be i n t r o d u c e d , however t h a t wou ld g r e a t l y c o m p l i c a t e t h e a n a l y s i s w i t h o u t s u b s t a n t i a l l y a l t e r i n g t h e ma in r e s u l t s . t h a t w i l l a s s o c i a t e f o r a v e c t o r o f o u t p u t s , q , a s e t S ( q ) <= X (q ) o f f e a s i b l e s t r a t e g i e s . In t h i s paper t h e e x i s t e n c e and s t r u c t u r e o f S (q ) w i l l be s t u d i e d f o r s e v e r a l solution concepts. G i v en a s o l u t i o n t h en d i s e q u i -l i b r i u m b e h a v i o u r i s d e s c r i b e d by consistent trajectories. A c o n s i s t e n t g i v e n by d e f U , ( q , x ) = V TT.(q)x = V . ( t ) Though t h e s t r a t e g y space X ( q ) i s o f a r a t h e r s p e c i a l form i t A solution f o r t h e l o c a l game w i l l s i m p l y be a c o r r e s p o n d e n c e - 40 -t r a j e c t o r y i s a t ime dependent pa th i n o u t p u t space deno ted by q ( t ) t h a t s a t i s f i e s : M ^ - e S ( q ( t ) ) , q ( 0 ) = q Q , Vt > 0. T h i s pape r w i l l no t be c on c e r n ed w i t h t h e e x i s t e n c e and c h a r a c t e r i z a t i o n o f such t r a j e c t o r i e s but w i t h t h e n a t u r e o f t h e s o l u -t i o n c o n c e p t s and how t h e y d e f i n e S ( q ) . E x i s t e n c e r e s u l t s f o r t h i s t y p e o f s y s t em can be found i n C a s t a i n g , C. and M. V a l a d i e r [ 1969 ] and Champsaur , D reze and Henry [ 1 9 7 7 ] . In p a r t i c u l a r i t w i l l be o f i n t e r e s t t o d e s c r i b e t h e n o t i o n o f local stability. DEFINITION 1. An o u t p u t v e c t o r q* i s locally stable i f and o n l y i f 0 e S ( q * ) . L o c a l s t a b i l i t y can be i n t e r p r e t e d as i m p l y i n g t h a t t h e r e e x i s t s no i n c e n t i v e s .to d e v i a t e f rom t h e v e c t o r o f o u t p u t s q* unde r t h e s o l u t i o n c o n c e p t d e f i n i n g S ( q * ) . By r e q u i r i n g t h a t e q u i l i b r i a a r e l o c a l l y s t a b l e w i l l r e s u l t i n c o n s i d e r a b l e r e s t r i c t i o n s on t he s e t o f p o s s i b l e e q u i l i b r i a . The f i r s t s o l u t i o n c o n c e p t t o be s t u d i e d i s t h e d i r e c t i o n a l 4 c o r e . When i t e x i s t s i t has t h e a p p e a l i n g p r o p e r t y t h a t no s t r a t e g y v e c t o r i n t h e d i r e c t i o n a l c o r e can be improved upon by any s i n g l e f i r m o r group o f f i r m s . Such s o l u t i o n s can c e r t a i n l y q u a l i f y as " r a t i o n a l " a c t i o n s . - 41 -3 . The D i r e c t i o n a l Core To d e f i n e t h i s s o l u t i o n a number o f d e f i n i t i o n s a r e r e q u i r e d . A c o a l i t i o n o f f i r m s w i l l be deno ted by a s u b s e t M c ± N . I f t h e i n d u s t r y has chosen a v e c t o r o f o u t p u t changes x e X (q ) t hen the c o a l i t i o n M can change t h i s d i r e c t i o n t o any p o i n t i n d e f F ( q , x , M ) = {y e X(q) = x f i f i t M}. F ( q , x , M ) i s t h e s e t o f f e a s i b l e d i r e c t i o n s f o r c o a l i t i o n M. T h i s s e t i s shown f o r t h e two f i r m c a s e i n F i g u r e 1 when X (q ) 2 — i s a s qua r e i n IR . In the f i g u r e , x i s a " p r o p o s e d " o r c o n s i d e r e d s t r a -t e g y c h o i c e w h i l e the s e t F ( q , x , { 2 } ) c o n t a i n s a l l t h o s e s t r a t e g i e s s t a r t -i n g a t x t h a t f i r m 2 can b r i n g a b o u t . One s h o u l d p r o b a b l y no t v i ew t h e s e s t r a t e g y c h o i c e s as r e a c t i o n s t o some c h o i c e x bu t r a t h e r t h e s e a r e i n a s en se c o n c e p t u a l e x e r c i s e s c a r r i e d o u t by f i r m s as t h e y t r y t o e v a l u a t e t h e i r a c t i o n s i n the l i g h t o f a n t i c i p a t e d r e s pon se s by com-p e t i t o r s . Hence i f f i r m 2 b e l i e v e s t h a t f i r m l ' s s t r a t e g y i s such t h a t x i s a p o t e n t i a l s o l u t i o n , t h en F ( 2 , x , { 2 } ) r e p r e s e n t s t he a l t e r n a t i v e s open to f i r m 2 t h r o u g h the c h o i c e o f i t s s t r a t e g y x 2 > We W i l l now i n t r o -duce a s e t t h a t w i l l r e p r e s e n t a c o a l i t i o n ' s e x p l i c i t e v a l u a t i o n o f t h e s t r a t e g i e s open to i t and p r o v i d e s t h e b a s i s f o r the d e f i n i t i o n o f t h e d i r e c t i o n a l c o r e . DEFINITION 2. The s e t o f M-undomina ted d i r e c t i o n s a t a v e c t o r o f o u t p u t s q i s g i v e n by - 42 -K(q ,M) = {x e X ( q ) |^y e F ( q , x , M ) such t h a t U . ( q , y ) > U . ( q , x ) V i e M}. , A v e c t o r o f changes x i s undomina ted w i t h r e s p e c t t o a c o a l i -t i o n M i f t h e r e does n o t e x i s t any f e a s i b l e a l t e r n a t i v e t h a t w i l l make a l l f i r m s i n t he c o a l i t i o n b e t t e r o f f . DEFINITION 3. The d i r e c t i o n a l c o r e f o r the l o c a l game a t q w i l l be g i v e n by K(q) = H K ( q , M ) . I f an o u t p u t change i s i n K ( q ) , then no c o a l i t i o n can by i t -s e l f improve on t h i s s t r a t e g y . When i t e x i s t s t h e d i r e c t i o n a l c o r e p r o -v i d e s an a p p e a l i n g d e f i n i t i o n o f l o c a l l y r a t i o n a l b e h a v i o u r . F o r i f x e K ( q ) , t hen no t o n l y w i l l no f i r m by i t s e l f be a b l e t o improve i t s l o t bu t a l s o f rom t he i n d u s t r y ' s p o i n t o f v i ew t h e r e w i l l e x i s t no o t h e r s t r a t e g y t h a t w i l l make e v e r y one b e t t e r o f f . Hence such a s o l u t i o n c o n -c e p t s a t i s f i e s t he c r i t e r i a o f bo th g roup and i n d i v i d u a l r a t i o n a l i t y . Our f i r s t r e s u l t i s t h a t t he M-undo i i i inated d i r e c t i o n s a lways e x i s t . PROPOSITION 1. K (q ,M) f <j> for all M <=• N. The p r o o f o f t h i s as w e l l a s t h e r e m a i n i n g p r o p o s i t i o n s a r e to be found i n the A p p e n d i x . In many ca se s t h e M-undominated d i r e c t i o n s can be q u i t e e a s y to compute . B e f o r e d o i n g so d e f i n e t he cone o f l o c a l l y P a r e t o - i m p r o v i n g d i r e c t i o n s by - 43 -+ d e f n C (q ,M) = {x e ]R |v IT. ( q ) x > 0 f o r e v e r y i e M}. T h i s s e t r e p r e s e n t s a l l t ho se changes i n o u t p u t t h a t wou l d make a l l mem-bers o f t h e c o a l i t i o n M s t r i c t l y b e t t e r o f f compared w i t h r e m a i n i n g a t t h e i n i t i a l o u t p u t v e c t o r q . The r e a d e r i s r e f e r r e d back t o F i g u r e 1 on wh i c h t h i s s e t i s shown f o r t h e two d i m e n s i o n a l c a s e . N o t i c e t h a t u s i n g t h i s s e t one can d e f i n e t h e M-undominated d i r e c t i o n s by K(q ,M) = {x e X(q) |# y £ F ( q , x , M ) ( 1 ) such t h a t ( y - x ) £ C + ( q , M ) } . T h i s c h a r a c t e r i z a t i o n w i l l make i t e a s i e r t o compute K (q ,M) f o r t he two d i m e n s i o n a l example o f t he n e x t s e c t i o n ; . A l s o t he f o l l o w i n g p r o p o s i -t i o n s f o l l o w a l m o s t i m m e d i a t e l y f rom t h i s c h a r a c t e r i z a t i o n . PROPOSITION :2. If C + ( q , M ) = and q £ ft" then K(q ,M) = X ( q ) . I f C + ( q , M ) = cb t hen t h e v e c t o r o f o u t p u t s q i s a l o c a l l y s t r i c t -l y P a r e t o e f f i c i e n t p o i n t f o r t he c o a l i t i o n M. Tha t i s , i t i s no t p o s -s i b l e t o v a r y o u t p u t s i n a way to make e v e r y member o f M b e t t e r o f f , hence a l l f e a s i b l e d i r e c t i o n s a r e undom i na t ed . PROPOSITION 3. (i) If j e.M and for all i £ M we have 3TT (q ) —4— > o. 3q. then if x £ K(q ,M) we must have x . = b . ( q . ) . - 44 -( i i ) I f j £ M and for a l l i £ M we have 3Tf,(q) t h e n i / x £ K (q ,M) we must have x . = - a . ( q . ) . J J J T h i s s i m p l e p r o p o s i t i o n s t a t e s t h a t i f a l l f i r m s i n a c o a l i t i o n wou ld w i s h t o change an o u t p u t q . i n t h e same d i r e c t i o n , t h en t hey w i l l J w i s h t o do so as q u i c k l y as p o s s i b l e . H a v i n g c h a r a c t e r i z e d the M-undominated d i r e c t i o n s i t i s a s i m p l e m a t t e r t o compute t he d i r e c t i o n a l c o r e u s i n g D e f i n i t i o n 3. How-e v e r , the d i r e c t i o n a l c o r e w i l l no t a lway s e x i s t due to an e s s e n t i a l c o n -f l i c t between i n d i v i d u a l and group r a t i o n a l i t y as d e f i n e d u s i n g t h e no -t i o n o f an M-undomi na t ed d i r e c t i o n . T h i s f a c t w i l l be d emon s t r a t e d i n t h e example o f the nex t s e c -t i o n . We may s t i l l wonder i f t h e r e a r e o u t p u t s f o r wh i c h K(q) e x i s t and a r e l o c a l l y s t a b l e . In such a s i t u a t i o n , d e s p i t e the l a c k o f a g e n e r a l e x i s t e n c e p r o o f , we wou ld s t i l l have o u t p u t s f rom wh i c h no group o f f i r m s wou ld w i s h t o d e v i a t e . U n f o r t u n a t e l y , t h e answer to t h i s q u e s t i o n i s g e n e r a l l y n e g a t i v e . I f q* i s l o c a l l y s t a b l e , t hen 0 £ K ( q ) . I t w i l l Sir-f o l l o w t h a t - J C — ; (q*) = 0 f o r a l l i £ N. Bu t t h e s e a re s i m p l y t he f i r s t o r d e r c o n d i t i o n s f o r t h e Cou r no t -Na sh e q u i l i b r i u m w h i c h g e n e r a l l y do no t y i e l d l o c a l l y P a r e t o e f f i c i e n t o u t p u t s and hence 0 E K (q) i s i m p o s s i b l e . T h i s i s summar i zed 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 . - 45 -9TT. PROPOSITION 4: Suppose that for every q* s a t i s f y i n g - (q*) = 0, V i e N q i 3TT. ( i ) - r - 1 (q*) < 0 for i £ j , i , j e N; 3q. 9TT. C i i J there exist V j e N an i ' ^ j suc/z t/zat ^ — (q*) < 0. Then there exist no vector of outputs q that are l o c a l l y stable with respect to K(q). N o t i c e t h a t one can i n g e n e r a l e x p e c t the a s s ump t i o n s o f t h i s p r o p o s i t i o n t o be s a t i s f i e d . A s s u m p t i o n ( i ) s i m p l y means^that a f i r m i s no t made b e t t e r o f f i f a c o m p e t i t o r i n c r e a s e s o u t p u t . A s s ump t i o n ( i i ) s i m p l y r e q u i r e s t h a t t h i s i n e q u a l i t y be s t r i c t f o r some c o m p e t i t o r , t h a t i s t h e f i r m s a r e a c t u a l l y i n e f f e c t i v e c o m p e t i t i o n w i t h ea ch o t h e r . Hence the d i r e c t i o n a l c o r e appea r s to be too s t r o n g a s o l u t i o n c on cep t f o r t h e c h a r a c t e r i z a t i o n o f l o c a l l y s t a b l e o u t p u t s . One way t o o b t a i n e x i s t e n c e i s t o weaken ou r r a t i o n a l i t y r e q u i r e m e n t by h a v i ng f i r m s e v a l u a t e o n l y t h e i r own c o n t r i b u t i o n t o p r o f i t s . T h i s i s the Nash -e q u i l i b r i u m f o r t h e l o c a l game wh i ch we d e f i n e as f o l l o w s : DEFINITION 4. The Nash d i r e c t i o n a l c o r e f o r t h e l o c a l game a t q w i l l be g i v e n by K(q) = n K ( q , { i } ) . i eN The f o l l o w i n g p r o p o s i t i o n c o m p l e t e l y c h a r a c t e r i z e s K ( q ) . PROPOSITION 5. x e K (q ) for q e IR" i f and only i f x . i s given by - 46 -3TT. (i) x. e C - a ^ q . ) , b ^ . ) ] when ~ (q) = 0; 3-rr. (ii) x. = b. ( q . ) when (q ) > 0; l l i dqi aw. (Hi) x . = - a . ( q . ) w f r e n T — - ( q ) < 0. The Nash d i r e c t i o n a l c o r e c a p t u r e s the e s s e n t i a l f e a t u r e s o f the Cou r no t -Na sh e q u i l i b r i u m c o n c e p t . S t r a t e g i e s i n the Nash d i r e c t i o n a l c o r e a re e v a l u a t e d by f i r m s t a k i n g i n t o a c c oun t t h e i r i n d i v i d u a l a c t i o n s o n l y . Under t h i s b e h a v i o u r a l a s s ump t i o n l o c a l s t a b i l i t y i s e q u i v a l e n t t o the f i r s t - o r d e r c o n d i t i o n s c h a r a c t e r i z i n g t he Cou r no t -Na sh e q u i l i -b r i u m . PROPOSITION 6. A vector of outputs q* is locally stable with respect to the Nash directional core if and only if 3 i T . ( q * ) — \ = 0 V i e N. 3q. T h i s r e s u l t shows t h e d i r e c t r e l a t i o n s h i p between t h e Nash d i r e c t i o n a l c o r e and t he Cou rno t -Na sh e q u i l i b r i u m . F u r t h e r i t a l l o w s one t o a rgue t h a t t h e Cou rno t -Na sh e q u i l i b r i u m i s no t " r a t i o n a l " i n t h e sense t h a t i t r e s u l t s f rom f i r m s e v a l u a t i n g o n l y t h e i r own c o n t r i b u t i o n to t h e i r p r o f i t s and i g n o r e s t he b e n e f i t s f rom a c t i n g as a g r o u p . T h i s w i l l be seen w i t h i n t h e c o n t e x t o f the f o l l o w i n g e x a m p l e , w h i c h w i l l a l s o h e l p i l l u s t r a t e t h e c o n c e p t s t h a t have been i n t r o d u c e d . - 47 -4. An Example C o n s i d e r an i n d u s t r y w i t h two f i r m s and N = { 1 , 2 } . L e t us s up -pose t h a t c o s t s a r e z e r o and t he demand cu r ve i s l i n e a r and g i v e n by d e f P(Q) = 1 - Q where Q = i n d u s t r y o u t p u t and P i s p r i c e . The p r o f i t s pe r p e r i o d o f f i r m i ( w i t h o u t a d j u s t m e n t c o s t s ) a r e : d e f ^ ( q 1 , q 2 ) = ( 1 - • ( q 1 + q 2 ) > q i -We : jw i l l assume t h a t t h e a d j u s t m e n t c o s t pa rame te r s a r e g i v e n by f o r a . ^ . ) = b . t q . ) = 1 , i e { 1 , 2 } q . > 0. Thus t h e s e t o f f e a s i b l e changes i s X = [ - 1 , 1 ] x [ - 1 , 1 ] _=]R . The g r a d i e n t s o f t h e p r o f i t f u n c t i o n s a r e V T r 1 ( q 1 , q 2 ) = ( l - q 2 - 2 q 1 , -q^ V T T 2 ( q 1 , q 2 ) = ( - q 2 , l - q 1 - 2 q 2 ) . R e c a l l t h a t f i r m i ' s Cou rno t r e a c t i o n f u n c t i o n i s i m p l i c i t l y d e f i n e d by 3TT. 8 q T ( q l ' q 2 > - 0 - 48 -f rom wh i c h i t i s an ea sy m a t t e r t o compute 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 um i s (-j , . T h i s i s d e p i c t e d i n F i g u r e 2 a l o n g w i t h the l i n e P (q^+q 2 ) = 0 and the c o n t r a c t c u r v e f o r t h e two f i r m s g i v e n by VTT^ + VTT^ = 0 . These l i n e s d i v i d e t he o u t p u t space i n t o f i v e r e g i o n s t h a t w i l l be used to c o m p l e t e l y c h a r a c t e r i z e the l o c a l game s t r u c t u r e . 9TT, 3TT?  m G i m 1 : 3 ^ <q> < 0. 3 ^ <<> < °' For a t y p i c a l v e c t o r o f o u t p u t s q i n t h i s reg ime we have : K ( q , { l } ) = { -1} x [ - 1 , 1 ] K ( q , { 2 } ) = [ - 1 , 1 ] x { - 1 } . C o n s e q u e n t l y the Nash d i r e c t i o n a l c o r e i s K(q) = ( - 1 , - 1 ) . From P r o p o s i t i o n 3 i t w i l l a l s o f o l l o w t h a t K (q ,N ) = ( = 1 , - 1 ) . Hence t h e d i r e c t i o n a l c o r e i s un i que and i s g i v e n by K(q) = ( - 1 , - 1 ) f o r any q i n Regime 1. In t h i s reg ime K (q ) = K ( q ) . These s e t s a r e de-p i c t e d i n F i g u r e 3 . REGIME 2: (q) > 0 , (q) < 0 . dq-^  d q 2 - 49 -For a t y p i c a l v e c t o r o f o u t p u t s we have: K ( q , { l } ) = { 1 } x [ - 1 , 1 ] . K(q,{2}) = [ - 1 , 1 ] x { - 1 } . K ( q ,N ) = K(q,{2}). Hence t h e Nash d i r e c t i o n a l c o r e i s g i v e n by K (q ) = ( 1 , - D and the d i r e c t i o n a l c o r e i s g i v e n by K(q) = ( 1 , - 1 ) . These s e t s a r e i l l u s t r a t e d i n F i g u r e 4. BIT. 3 7 T p REGIME 3: j± (q) < 0, -~ (q) > 0. Here we ge t r e s u l t s a na l ogou s t o t h o s e o b t a i n e d i n Regime 2; t he r o l e s o f f i r m s 1 and 2 a r e s i m p l y r e v e r s e d . •REGIME 4: ^ (q) > 0, -r-^ (q) > 0. dq^ d q 2 Here we have two s ub ca s e s i n wh i c h t h e P a r e t o i m p r o v i n g d i r e c -t i o n s a r e q u i t e d i f f e r e n t . _ 5 ( i ) ^GIME_4a: Vir^q) + VTr 2 (q) > 0. Fo r a t y p i c a l v e c t o r o f o u t p u t s we have - 50 -K ( q , { l } ) = {1} x [ - 1 , 1 ] K ( q , { 2 } ) = [ - 1 , 1 ] x {1} K ( q ,N ) = K ( q , { l } ) u K ( q , { 2 } ) . Hence t he Nash and d i r e c t i o n a l c o r e s a r e g i v e n by K (q ) = K (q ) = ( 1 , 1 ) as d e p i c t e d i n F i g u r e 5 . ( i i ) REGIMEJb: V f T j ( q ) + V 7 T 2 ( q ) < 0. Fo r a t y p i c a l v e c t o r o f o u t p u t s we have K ( q , { l } ) = {1} x [ - 1 , 1 ] K ( q , { 2 } ) = [ , 1 , 1 ] x {1} K (q ,N ) = [ - 1 , 1 ] x { - 1 } U { - 1 } x [ - 1 , 1 ] . However i n t h i s c a s e the Nash d i r e c t i o n a l c o r e i s g i v e n by t ( q ) = ( 1 , 1 ) . Hence , as d e p i c t e d i n F i g u r e 6 t he d i r e c t i o n a l c o r e i s empty: K (q ) = K(q) n K ( q ,N ) = <J>, Note t h a t i n Regimes 4a and 4b t h e Nash d i r e c t i o n a l c o r e s a r e i d e n t i c a l . We l e a v e as an e x e r c i s e t o the r e a d e r t he d e t e r m i n a t i o n o f t h e d i r e c t i o n a l c o r e s f o r o u t p u t s on t he boun 'dar ies o f t h e s e r e g i o n s . - 51 -An i m p l i c a t i o n o f t h i s a n a l y s i s o f t he d i r e c t i o n a l c o r e i s t h a t no v e c t o r o f o u t p u t s i n r eg imes 1 , 2 , 3 o r 4a a r e l o c a l l y s t a b l e . F u r t h e r any t r a j e c t o r i e s c o n s i s t e n t w i t h K(q) must end up i n r e g i o n 4b . Thus t h i s s o l u t i o n c on cep t though not y i e l d i n g any l o c a l l y s t a b l e o u t -p u t s , i t does s u g g e s t t h a t i f f i r m s a r e l o c a l l y n a t i o n a l t h e y w i l l end up i n r e g i o n 4 b . T h i s i s a p p e a l i n g s i n c e t h i s r e g i o n can be i d e n t i f i e d as the a r e a between the c o n t r a c t c u r v e and the C o u r n o t - N a s h e q u i l i b r i u m o f t e n s ugge s t e d as t h e a r e a where one most e x p e c t s an o l i g o p o l i s t i c e q u i -1 i b r i u r n t o e x i s t . The p rob l em t hen i s t o i d e n t i f y l o c a l l y r a t i o n a l b e h a v i o u r i n r e g i o n 4 b . C o n s i d e r some o u t p u t i n t he i n t e r i o r o f r e g i o n 4b . Suppose t hen t h a t f i r m 1 i g n o r e s t he b e h a v i o u r o f f i r m 2 and chooses x-^  = 1 so t h a t x e K ( q , { l } ) . C e r t a i n l y one wou l d e x p e c t f i r m 2 to do l i k e w i s e r e -s u l t i n g i n a s t r a t e g y x £ K ( q , { l } ) Pi K ( q , { 2 } ) = ( 1 , 1 ) . T h i s outcome i s c l e a r l y worse t han f o r example r e m a i n i n g a t q and c h o o s i n g x = ( 0 , 0 ) . One must a s k i f i t i s i n f a c t rational f o r the f i r m s to choose x . = 1 . Under t h e i n f o r m a t i o n a l a s s ump t i o n s the ' n o t i o n o f " c h e a t i n g " p l a y s no r o l e i n t h i s a n a l y s i s f o r f i r m s can a lway s i m m e d i a t e l y r e spond to s t r a -t e g y c h a n g e s . I f we were to s t a r t a t q and f i r m 1 choo se s x-^  = 0 t h en f i r m 2 c o u l d no t g a i n by c h o o s i n g = 1 s i n c e f i r m 1 wou ld i m m e d i a t e l y r e a c t w i t h x^= 1 . The p r ob l em t hen i s how t o f o r m a l i z e t h e c o n j e c t u r e s t h a t f i r m s have c o n c e r n i n g each o t h e r ' s a c t i o n s . T h i s a rgument s u g g e s t s t h a t t h e Nash d i r e c t i o n a l c o r e , K ( q ) , c anno t be e x p e c t e d to p r o v i d e a d e s c r i p t i o n o f r a t i o n a l b e h a v i o u r i n r e -gime 4b . In t h e f o l l o w i n g s e c t i o n we s ugge s t a s o l u t i o n c o n c e p t t h a t - 52 -no t o n l y i s r a t i o n a l f r om the i n d u s t r y ' s v i ew b u t a l s o c a p t u r e s t he i d e a t h a t f i r m s can c r e d i b l y choose r e s p o n s e s i n t h e i r own undomina ted s e t s , K ( q , { i } ) , i e N. 5 . The N a s h " B a r g a i n i n g Core In t h i s s e c t i o n we i n t r o d u c e a s o l u t i o n c on cep t t o l o c a l game t h e o r y t h a t does no t s u f f e r f rom t he n o n - e x i s t e n c e p rob lem a s s o c i a t e d w i t h t he d i r e c t i o n a l c o r e y e t has s t r o n g e r r a t i o n a l i t y p r o p e r t i e s t h an t h o s e a s s o c i a t e d w i t h t h e Nash d i r e c t i o n a l c o r e . I f the i n d u s t r y i s l o -c a l l y r a t i o n a l t h en a l l f i r m s w i l l w i s h t o choose some d i r e c t i o n i n K ( q , N ) , t he s e t o f undomina ted d i r e c t i o n s f o r the who le i n d u s t r y . How-e v e r t h e d i r e c t i o n a l c o r e K ( q ) , c ea se s t o e x i s t p r e c i s e l y when t h e r e a r e i n c e n t i v e s f o r f i r m s to d e v i a t e f rom e l emen t s i n K ( q , N ) . Suppose we s t a r t w i t h an x-'e K (q ,N ) such t h a t x'. £ K ( q , { i } ) f o r some f i r m i . T h i s f i r m , c a n n o t e x p e c t t o a d j u s t i t s o u t p u t t o a c h i e v e a d i r e c t i o n y £ K ( q , { i } ) w i t h o u t i n v o k i n g a response f rom the o t h e r f i r m s . I f a f i r m t r i e s t o b l o c k x t h en t h e o t h e r f i r m s must c o n c l u d e t h a t t h i s f i r m i s a c t i n g non -c o o p e r a t i v e l y and hence t h e y w i l l do l i k e w i s e r e s u l t i n g i n a v e c t o r z £ i ( q ) , t h e Nash d i r e c t i o n a l c o r e . S i n c e f i r m s w i l l d e v i a t e o n l y i f after t he r e spon se o f c o m p e t i t o r s t h e y a r e made b e t t e r o f f , t h i s mo-t i v a t e s t h e f o l l o w i n g s o l u t i o n c o n c e p t . ^ DEFINITION 5. The Nash Bargaining Core i s g i v e n by ~ d e f B ( q ) = {x £ X ( q ) | x £ K (q ,N ) and 3y £ K(q) f o r wh i c h I K t q . x ) > U ^ q . y ) V i £ N} . - 53 -A d i r e c t i o n x e B ( q ) i s one i n wh i c h f i r m s c anno t e x p e c t t o be made b e t t e r o f f by d e f e c t i n g . In f a c t when K (q ) n K ( q ,N ) = cj>, as i s g e n e r a l l y t h e c a s e i f K(q) = <j), t hen t h e y can e x p e c t t o be worse o f f by p l a y i n g Nash'. N o t i c e t h a t each e l emen t i n B (q ) i s an e q u i l i b r i u m i n l o c a l t h r e a t s t r a t e g i e s . I f x e B (q ) t h e t h r e a t s t r a t e g y f o r . f i r m i i s s i m p l y : x . , i f x . = x . f o r a l 1 j f1 i i J J y . , i f x . f x \ f o r some j $ i J J where y E K (q ) such t h a t IK ( q , x ) > U. . ( 'q ,y ) , V i e N. PROPOSITION 7. The set B (q ) always exists. S i n c e B (q ) e x i s t s i t can p r o v i d e a b a s i s f o r a d i s e q u i l i b r i u m a n a l y s i s o f the o l i g o p o l y p r o b l e m . The q u e s t i o n t hen i s ' t h e c h a r a c t e r i -z a t i o n o f t h e l o c a l l y s t a b l e o u t p u t s . •PROPOSITION 8. Suppose at q eK" ^-M ' q ) < 0 V i f j ; i , j E N then q + J)q. is locally stable with respect to B ( q ) , that is 0 e B ( q ) , if and only if (i) C + ( q , N ) = <f> (ii) x = ' ( ! b j ( q j ) , . . . , b n ( q ) ) satisfies VTT^ (q )«x = 0 V i £ N. T h i s p r o p o s i t i o n s t a t e s t h a t i f q i s l o c a l l y s t a b l e w i t h r e s -p e c t t o B (q ) t h en i t must be ( i ) P a r e t o e f f i c i e n t f rom t h e f i r m s p o i n t o f v i ew and ( i i ) t h e g r a d i e n t s o f t he p r o f i t f u n c t i o n s must be o r t h o g o n a l - 54 -t o x . C o n d i t i o n ( i ) i s i n t e r p r e t e d as P a r e t o e f f i c i e n c y s i n c e C + ( q , N ) = <j> i m p l i e s t h a t no x wou ld make a l l f i r m s s i m u l t a n e o u s l y b e t t e r o f f and must h o l d s i n c e B(q)<= K ( q , N ) . The second c o n d i t i o n r e s u l t s f rom t he f a c t t h a t i f i t d i d no t h o l d , t h en by P a r e t o e f f i c i e n c y i t must be t h e 3Tr. .(q) case t h a t for.v. some i e N, V i T . ( q ) x > 0 s i n c e - - — — - < 0 i m p l i e s — > 0 l dq.j dq^ and hence x e K (q ) w i l l be p r e f e r r e d t o 0 . I n F i g u r e 7 t h e s t r u c t u r e o f B (q ) i s shown f o r an a r b i t r a r y o u t p u t i n r eg ime 4b o f the p r e v i o u s e xamp l e . N o t i c e t h a t t h i s r e q u i r e m e n t t h a t B (q ) be undomina ted w i t h r e s p e c t t o ' K (q) does p l a c e c o n s i d e r a b l e r e s t r i c t i o n s on the s e t K ( q , N ) . F i g u r e 8 shows t h e s t r u c t u r e o f B (q ) a t a l o c a l l y s t a b l e v e c t o r o f o u t p u t s . S i n c e B ( q ) i s c h a r a c t e r i z e d by c o n -d i t i o n s on t he d e r i v a t i v e s o f ir.-C'q) t hen i t w i l l n o t a l w a y s e x i s t o r be u n i q u e . However , l i k e t he f i r s t - o r d e r 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 , t h e r e w i l l e x i s t many examp les f o r wh i c h t h i s i s the c a s e , the p r e v i o u s example b e i n g one p o s s i b i l i t y . T h i s i s s u e w i l l be d i s c u s s e d a t g r e a t e r l e n g t h f o r t he s t a n d a r d o l i g o p o l y p rob lem i n the nex t s e c t i o n . I t s h o u l d a l s o be no ted t h a t i n r eg imes 1, 2 , 3 , 4a we have B (q ) = K(:q). Tha t i s when i t e x i s t s t h e d i r e c t i o n c o r e and t he Nash b a r -g a i n i n g c o r e w i l l o f t e n be t h e same, s u g g e s t i n g t h a t t h i s s o l u t i o n i s no t o v e r l y l e s s r e s t r i c t i v e than the d i r e c t i o n a l c o r e . A c r i t i c i s m t h a t can be made o f t h i s s o l u t i o n i s t h a t we have supposed t h a t i f , say f i r m i , a c t e d i n d e p e n d e n t l y , t h en so wou l d a l l t h e o t h e r f i r m s . However i t may be the case t h a t t he f i r m s i n N / { i } r e -main b e t t e r o f f a c t i n g as a g r o u p , o r i n o t h e r words i t i s no t c r e d i b l e - 55 -f o r them t o a c t i n a Nash f a s h i o n . We w i l l now show t h a t , r e g a r d l e s s o f t h e c o n j e c t u r e d b e h a v i o u r o f f i r m s , t h e y w i l l no t f i n d i t p r o f i t a b l e t o b l o c k s o l u t i o n s i n B (q ) i n a. ' ;ne ighborhood o f a l o c a l l y s t a b l e v e c t o r o f o u t p u t s . T h i s w i l l be done u s i n g t h e n o t i o n o f a c o a l i t i o n s t r u c -t u r e . L e t N = { l , . . . , n } deno t e the s e t o f f i r m s and d e f i n e t h e s e t o f p a r t i t i o n s o f N by 3 ( N ) = ( 3 e P(N) I ( i ) u M = N and ( i i ) i f M,L e 3 , M f L, Me3 t hen M O L = <j>} where P(N) = power s e t o r s e t o f s u b s e t s o f N. A p a r t i t i o n , 3 e 3 ( N ) , w i l l be c a l l e d a c o a l i t i o n s t r u c t u r e . To see what t h i s means l e t us c o n s i d e r f i r s t t he n o t i o n o f a c o a l i t i o n . W i t h i n t he c o n t e x t o f a pure t r a d e model a c o a l i t i o n has t h e n a t u r a l i n t e r p r e t a t i o n o f - a g e n t s t r a d i n g w i t h i n the c o a l i t i o n o n l y . However , i n a n o n - c o o p e r a t i v e game where each a g e n t ' s a c t i o n a f f e c t s all o t h e r agen t s p a y o f f s , t h i s i n t e r p r e t a t i o n i s no t p o s s i b l e . We p r opo se a n o t i o n o f a c o a l i t i o n to d ea l w i t h t h i s i n t e r d e p e n d e n c e p rob l em and a l s o c l a r i f y t h e n o t i o n o f a c o a l i t i o n s t r u c t u r e . L e t us v i ew a c o a l i t i o n as a g roup o f agen t s who communica te f r e e l y among t h emse l v e s and hence may use s t r a t e -g i e s t h a t a r e c o n t i n g e n t upon a c t i o n s by o t h e r members o f t h e c o a l i t i o n . G i ven t h a t t h i s - i s what i s meant by a c o a l i t i o n t hen agen t s o u t s i d e t h e c o a l i -t i o n c anno t use s t r a t e g i e s t h a t . a r e c o n t i n g e n t on a c t i o n s o f a g e n t s i n - 56 -t h e c o a l i t i o n . Fo r t h i s r e a s o n , members o f a c o a l i t i o n must t a ke the a c -t i o n s o f a gen t s o u t s i d e t h e c o a l i t i o n as given. Hence i f M i s a c o a l i -t i o n , t h en d i r e c t i o n s o f change w i l l be chosen f rom t he s e t o f M-undomi -na t ed d i r e c t i o n s K ( q , M ) . I t i s s t i l l n e c e s s a r y t h a t f i r m s i n M f o rm e x p e c t a t i o n s on how f i r m s no t i n M w i l l choose t h e i r a c t i o n s and they must a l s o form a c o n -j e c t u r e on wh i ch c o a l i t i o n s o t h e r t h an M w i l l f o r m . S i n c e e v e r y f i r m by d e f i n i t i o n must be i n some c o a l i t i o n , i f o n l y a c o a l i t i o n c o n t a i n i n g a s i n g l e f i r m , then c o n j e c t u r e s a r e formed on t he b e h a v i o u r o r c o a l i t i o n s t r u c t u r e o f t h e i n d u s t r y and not the p a r t i c u l a r a c t i o n s t a k e n by f i r m s . G i ven a c o n j e c t u r e 3 e 3 ( N ) , t hen we wou ld e x p e c t t he outcome t o l i e i n t he s e t B ( q,3) d e f i n e d by B ( q,3) = H K ( q , M ) . Me3 Tha t i s x e B ( q,3) i f and o n l y i f i t i s undomina ted w i t h r e s p e c t t o each c o a l i t i o n i n 3. We no te t h a t t h i s s e t w i l l a lways b e non -empty . PROPOSITION 9. B ( q,3) t <$>. The s e t B ( q,3) c o n s i s t s o f t h o s e a c t i o n s t h a t wou ld o c c u r i f t he c o a l i t i o n s t r u c t u r e i s 3. A way o f v i e w i n g t h i s s e t i s t o suppose t h a t i f a c o a l i t i o n M forms t h e n i t w i l l c hoose x e K ( q , M ) , however i f i t c o n -j e c t u r e s t h a t t h e c o a l i t i o n s t r u c t u r e 3 w i l l fo rm then t h e f i r m s w i l l r e spond so t h a t t h e f i n a l d i r e c t i o n chosen w i l l l i e i n B ( q , 3 ) . These a r e m e r e l y t h e c r e d i b l e r e sponse s t o an a c t i o n x e K(q ,M) t a k e n by c o a l i t i o n M. Hence a d i r e c t i o n x e B (q ) can be b l o c k e d by a c o a l i t i o n M o n l y - 57 -i f t h i s d i r e c t i o n can be b l o c k e d by some c o a l i t i o n s t r u c t u r e . We now d e f i n e o u r f o u r t h s o l u t i o n ' c o n c e p t , the b a r g a i n i n g c o r e . DEFINITION 6: The Bargaining Core a t a v e c t o r o f o u t p u t s q i s g i v e n b y : d e f B (q ) = {x £ X ( q ) | V 3 £ B ( N ) , By £ B(q,g) such t h a t U ^ q . x ) > U \ ( q , y ) , V i £ N} . N o t i c e t h a t s i n c e {N} and {(1), ( 2 ) , . . . , ( n ) } a r e p o s s i b l e c o a l i -t i o n s t r u c t u r e s , t hen c l e a r l y B (q ) •<= B ( q ) . We d i d no t s t a r t w i t h B (q ) as t h e s o l u t i o n c o n c e p t s i n c e t h e n o t i o n i s f a r too s t r o n g and w i l l no t e x i s t i n g e n e r a l J F u r t h e r t he Nash B a r g a i n i n g s e t has t h e a p p e a l i n g p r o p e r t y o f b e i n g a f a i r l y weak n o t i o n i n t he s en se o n l y Nash b l o c k i n g b e h a v i o u r i s used w h i l e s t i l l p r o v i d i n g a s t r o n g c h a r a c t e r i z a t i o n o f l o c a l s t a b i l i t y . We w i l l now show t hen even the s t r o n g n o t i o n s o f b l o c k i n g i n c o r p o r a t e d i n t h e s e t B (q ) w i l l n o t u p s e t t h e l o c a l s t a b i l i t y a t a v e c t o r o f o u t p u t s . PROPOSITION 10; Suppose that q is locally stable with respect to B (q ) and i f V i £ N we have 3TT. d) ( q ) < 0 for j f i ( i i ) VTT.. (q) and b ^ q ) are continuous at q then B ( q ) = B (q ) in a neighborhood of q . C o n d i t i o n {i) s i m p l y r e q u i r e s t h a t f i r m s a r e i n d i r e c t compe-t i t i o n w i t h each o t h e r . The r e s u l t f o l l o w s f rom t he f a c t t h a t i n the - 58 -ne i ghbo rhood o f a l o c a l l y s t a b l e v e c t o r o f o u t p u t s t h e r e w i l l e x i s t no i n c e n t i v e s f o r c o a l i t i o n s s m a l l e r t h an N to f o r m . Hence f i r m s must com-pa re t h e outcome f rom a c t i n g as a g roup w i t h t h e outcomes i n K ( q ) . N o t i c e t h a t the a n a l y s i s u n t i l now has been q u i t e g e n e r a l and a p p l i c a b l e to any n o n - c o o p e r a t i v e game f o r wh i ch t h e a d j u s t m e n t c o s t a s -sumpt i on seems r e a s o n a b l e . In the f o l l o w i n g s e c t i o n we s p e c i a l i z e t h e r e s u l t s f o r the o l i g o p o l y p r o b l e m . 6. The S t a n d a r d Q u a n t i t y S e t t i n g O l i g o p o l y P rob lem In t h i s s e c t i o n we w i l l assume t h a t t he p r o f i t f u n c t i o n s t a k e on the u sua l fo rm f o r t h e q u a n t i t y s e t t i n g o l i g o p o l i s t and p r o v i d e a s i m p l e c h a r a c t e r i z a t i o n o f l o c a l s t a b i l i t y . We a l s o show the r e l a t i o n -s h i p between f i r m s ' c o s t o f a d j u s t m e n t and t h e i r ma r ke t s h a r e s . Suppose t hen t h a t p r o f i t s a r e g i v e n by T T . ( q ) = q . P(Q) - C . ( q . ) where Q = X q . - i s a g g r e g a t e o u t p u t , i eN P(Q) i s a d i f f e r e n t i a b l e downward s l o p i n g demand c u r v e , C^'(q^) i s a t w i c e d i f f e r e n t i a t e c o s t f u n c t i o n . Under t h i s f u n c t i o n a l form t h e c o n d i t i o n s f o r l o c a l s t a b i l i t y can be f u r t h e r s i m p l i f i e d . PROPOSITION 11. Suppose the profit functions are of tne standard form given above, then - 59 -i f VTT.. ( q ) x ( q ) = 0 V i e N (call this condition ( B ) J where x ( q ) • = ( b j ( q ) . . , b ( q ) ) , then C + ( q , N ) = <j>. Thus i f t he p r o f i t f u n c t i o n s a r e o f t he s t a n d a r d f o r m , then c o n d i t i o n (B) w i l l i m p l y t h a t q i s a l o c a l l y P a r e t o e f f i c i e n t p o i n t . Hence w i t h P r o p o s i t i o n 8 c o n d i t i o n (B) w i l l be a 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 c h a r a c t e r i z i n g l o c a l s t a b i l i t y . The r e s t r i c t i o n s i m p l i e d by l o c a l s t a b i l i t y a r e q u i t e s t r o n g and hence one can e x p e c t s i g n i f i c a n t r e s t r i c t i o n s on the s e t o f e q u i l i b r i a . S i n c e we have a l l o w e d t h e a d j u s t m e n t c o s t s t o depend on q , a g e n e r a l e x i s t e n c e and un i quene s s theorem o f s i g n i f i c a n t i n t e r e s t wou ld be ha r d t o f i n d . I n s t e a d we w i l l c o n s i d e r e x i s t e n c e a n d ' u n i q u e n e s s f o r a s p e c i a l c a s e . PROPOSITION 12. Suppose that (i) Firms have identical marginal costs: c ; ( i i ) The adjustment cost parameters are independent of q and given by b = { b p . . . , b n > and a = {a^ , . . . , a n } ; ( i i i ) P(Q) is twice differentidble with P ' (Q ) < 0, P"(Q) < 0, c < P(0) < °° and 1 im P(Q) = 0. Q-*». Then there exists a unique vector of outputs that -is locally stable with respect to the Nash bargaining core. In F i g u r e 8 we have shown the s t r u c t u r e o f t he l o c a l game f o r - 60 -ou r d u o p o l y example when t he o u t pu t v e c t o r q i s l o c a l l y s t a b l e . In t h i s case t he P a r e t o e f f i c i e n t p o i n t s w i l l be g i v e n by = Q = q^ + q 2 . Fo r su ch p o i n t s g r a d i e n t s o f the p r o f i t f u n c t i o n s w i l l b e : V T ' j ( q ) = {\ - q p - q j ) V T T 2 (q ) = ( - q 2 , j - q 2 ) . S i n c e V iTj(q-) = - V I T ^ C q ) f o r Q = j, t hen to i d e n t i f y t h e l o c a l l y s t a b l e o u t p u t v e c t o r we need o n l y s o l v e VTTjCq) (b1}b2) J = 0. S imp l e m a n i p u l a t i o n s y i e l d : b l 1 q l = b~fb^ * 7 • In o u r e x amp l e , b^  = b 2 = 1,Whence the l o c a l l y s t a b l e p o i n t i s 11 ne f rom I 3 > 3 J t o ( 7j » 4 J I f we had s t a r t e d a t t h e Nash e q u i l i b r i u m ^ , t hen the s t r a i g h t i s e a s i l y seen t o be a t r a j e c t o r y c o n s i s t e n t w i t h t h e Nash B a r g a i n i n g C o r e . N o t i c e t h a t i n t h i s examp le the marke t s h a r e o f each f i r m w i l l be d e t e r m i n e d by t he r e l a t i v e s i z e s o f the a d j u s tmen t c o s t p a r a m e t e r . T h i s r e s u l t i s q u i t e a p p e a l i n g i n t h a t t h e l o w e r i s o n e ' s a d j u s t m e n t c o s t i n r e l a t i o n to a c o m p e t i t o r , as g i v e n by a h i g h e r b^, t hen the l a r g e r i s o n e ' s marke t s h a r e . We can a l s o show t h a t t he l owe r a f i r m ' s m a r g i n a l c o s t , then t h e l a r g e r i t s ma rke t s h a r e . These r e s u l t s a r e summar i zed i n - 61 -t h e f o l l o w i n g p r o p o s i t i o n . PROPOSITION IS. Suppose we have an oligopolistic industry with differen-tiable profit functions in the standard form given above. Further suppose q is locally stable with respect to the Nash Bargaining Core B ( q ) J then i t will be the case that: (1) If for two firms i and j C j t q ) = C j ( q ) then q . > q . i f and only i f b . ( q . ) > b . ( q . ) . (2) If for two firms i and j b , ( q , ) - b j U j ) then q . > q . i f and only i f C . ( q . ) > C . (q . ) . T h i s p r o p o s i t i o n d emons t r a t e s t h a t t h e a d j u s t m e n t : c o s t , p a r ame t e r s can be i n t e r p r e t e d as r e p r e s e n t i n g t h e b a r g a i n i n g power o f t he f i r m s . F i rms W i t h l owe r a d j u s t m e n t c o s t s w i l l have l a r g e r m a r k e t s . One a l s o h a s , as e x p e c t e d , t h a t a l o w e r m a r g i n a l c o s t w i l l ' r e s u l t i n a l a r g e r marke t s h a r e . 7. C o n c l u d i n g Remarks Through t he i n t r o d u c t i o n o f e x p l i c i t a d j u s tmen t c o s t s i n the o l i g o p o l y p r ob l em we have been a b l e t o a n a l y z e t he l o c a l s t r a t e g i c b e -h a v i o u r o f f i r m s . Not o n l y does t he f ramework p r o v i d e an a n a l y t i c a l l y t r a c t a b l e way o f s t u d y i n g i n d u s t r y d y n a m i c s , i t a l s o a l l o w s one t o e x a -- 62 -mine t h e consequences o f u s i n g d i f f e r e n t s o l u t i o n c on c ep t s i n a . d ynam i c s e t -t i n g . Whether one uses t h e Nash c o r e o r t h e Nash b a r g a i n i n g c o r e , . the s e t o f o u t p u t s t h a t a r e l o c a l l y s t a b l e w i t h r e s p e c t t o t h e s e s o l u t i o n c o n -c e p t s a r e h i g h l y r e s t r i c t e d . In p a r t i c u l a r , t he Nash b a r g a i n i n g c o r e i s s een as one s o l u t i o n t o t h e p rob l em o f s e l e c t i n g a un i que e q u i l i b r i u m when f i r m s r e c o g n i z e t he b e n e f i t s f rom c o o p e r a t i v e b e h a v i o u r . Our ma j o r r e s u l t i s t h a t such e q u i l i b r i a a r e l o c a l l y P a r e t o e f f i c i e n t and f i r m s have marke t s h a r e s t h a t a r e i n v e r s e l y p r o p o r t i o n a l t o t h e i r m a r g i n a l c o s t s and a d j u s t m e n t c o s t s . In t h i s way, a d j u s tmen t c o s t s p l a y an i m p o r t a n t r o l e i n c h a r a c t e r i z i n g t he r e l a t i v e b a r g a i n i n g s t r e n g t h s o f f i r m s . A p o s s i b l e i n t e r p r e t a t i o n o f t h i s model i s as g e n e r a l i z a t i o n o f t h e c h a r a c t e r i z a t i o n o f Nash e q u i l i b r i a by f i r s t - o r d e r c o n d i t i o n s . I t was shown t h a t when the r e spon se s o f c o m p e t i t o r s t o a f i r m ' s a c t i o n s were no t t a k e n i n t o a c c o u n t , t h i s model y i e l d e d t h e same s o l u t i o n as p r e -d i c t e d by C o u r n o t . Once t h e s e r e s pon s e s were c h a r a c t e r i z e d u s i n g more " r a t i o n a l " c r i t e r i o n t h en we o b t a i n e d l o c a l l y s t a b l e s o l u t i o n s t h a t were l o c a l l y P a r e t o e f f i c i e n t . Our app r oa ch d i f f e r s f rom t h e r e c e n t work on c o n s i s t e n t c o n j e c t u r a l v a r i a t i o n s (CCV) as p r e s e n t e d f o r example by B r e s h -nahan (1981) o r P e r r y ( 1 9 8 2 ) . F i r s t l y f o r a c o n j e c t u r a l v a r i a t i o n s a p -p r oa ch t o make s en se i t i s assumed t h a t f i r m s can ob s e r v e i m m e d i a t e l y com-p e t i t o r s ' a c t i o n s . Hence i t i s r e a s o n a b l e t o s i m p l y assume, as we have , t h a t e x p e c t a t i o n s a r e a lways c o n s i s t e n t . The r e a l i s s u e t h en i s no t whe the r e x p e c t a t i o n s a r e c o n s i s t e n t o r no t bu t r a t h e r , g i v e n the l o c a l v a r i a t i o n s a p p r o a c h , how does one c h a r a c t e r i z e r a t i o n a l b e h a v i o u r . In the CCV app r oa ch an a t t emp t i s made to c h a r a c t e r i z e e q u i l i b r i a under the a s -- 63 -sump t i on t h a t s h o u l d e i t h e r f i r m e x p e r i m e n t t h e o t h e r i s , assumed to be p r o f i t m a x i m i z i n g . Q u i t e c l e a r l y such an e x e r c i s e i s open to q u e s t i o n . As an a l t e r n a t i v e t h e ' a p p r o a c h p r e s e n t e d he re d e f i n e s a s o l u t i o n c o n c e p t t h a t i n c o r p o r a t e s an e x p l i c i t n o t i o n o f r a t i o n a l i t y and uses i t t o d e t e r -mine r a t i o n a l o u t p u t changes f o r any v e c t o r o f o u t p u t s . The r e s u l t s c o n -t r a s t s h a r p l y w i t h some o f t h e CCV r e s u l t s . Fo r e xamp l e , B reshnahan (1982) a n a l y z e s t he examp le o f S e c t i o n 4 . and ge t s t he c o m p e t i t i v e r e s u l t i n c o n t r a s t t o t he P a r e t o - e f f i c i e n t r e s u l t p r e d i c t e d by t h e Nash b a r g a i n -i n g c o r e . To c a r r y t h i s p o i n t f u r t h e r one s h o u l d r e c o g n i z e t h a t i n p r a c -t i c e f i r m s w i l l no t a lways be a t t h e P a r e t o - o p t i m a l p o i n t ; however t h i s does no t imp l y t h a t t h i s model i s u n r e a s o n a b l e . The a n a l y s i s he re s u g -g e s t s t h a t when the number o f f i r m s i s f i x e d and i n f o r m a t i o n t r a n s m i s s i o n i s i n s t a n t a n e o u s , then a P a r e t o - e f f i c i e n t outcome s h o u l d be e x p e c t e d . The i n t r o d u c t i o n o f a d j u s t m e n t c o s t s s i m p l y y i e l d s a un i que s o l u t i o n . I f one w i s h e s t o u n d e r s t a n d why f i r m s a r e no t on t h e P a r e t o f r o n t i e r , t h en i t seems t h a t t h e assumptions o f t he model must change t o a l l o w , f o r e x a m p l e , g f o r e n t r y and e x i t and i m p e r f e c t i n f o r m a t i o n . In f u t u r e work i t wou ld be i n t e r e s t i n g t o see how such a s sump t i o n s w i l l a f f e c t t h e s t a b i l i t y a n a -l y s i s p r e s e n t e d h e r e . - 64 -NOTES 1. In Case (1979) t h e r e i s some a n a l y s i s o f o l i g o p o l y u s i n g d i f f e r e n -t i a l games. Howeve r ,Case does e x p r e s s r e s e r v a t i o n s on the e f f i c a c y o f su ch an a p p r o a c h . 2. L e t IR" = {x e F n | x . > 0 , 1 < i < n} 1R" + = {x e I R n | x i > 0 , 1 < i < n } . 3 . See F r i edman (1977) f o r a g ene r a l d i s c u s s i o n o f t h i s p o i n t and Spence ( 1 9 7 8 ) . 4 . T h i s n o t i o n was f i r s t i n t r o d u c e d t o t h e l i t e r a t u r e by Matthews [ 1 9 8 2 ] . 5 . F o r x , y e IR n , x > y means t h a t x^ > y^, V i e N. 6 . The i m p o r t a n c e o f e v a l u a t i n g outcomes a f t e r a l l p l a y e r s have r e sponded has been p o i n t e d ou t by Ma r s chak and S e l t e n [ 1 9 7 8 ] . Our a n a l y s i s has t h e advan tage t h a t i t i s c o n s i d e r a b l y e a s i e r to a p p l y . 7 . As an examp le o f n o n - e x i s t e n c e , l e t N - { 1 , 2 , 3 } and l e t VTT x = ( 1 , - 2 , 0) V T T 2 = ( - 2 , 1, 0) V T T 3 = ( 1 , 1, 1) and X = [ - 1 , 1] x [ - 1 , 1] x [ - 1 , 1] Note t h a t e v e r y f e a s i b l e d i r e c t i o n i s b l o c k e d by e i t h e r 3^ = { ( 1 , 2 ) , ( 3 ) } o r 3 1 = { ( 1 ) , ( 2 ) , ( 3 ) } . 8 . See S t i g l e r [ 1964 ] on t h i s p o i n t . - 65 -X = [ - 1 , 1 ] x [ - 1 , 1 ] / 4-F ( q , X , { l » / C + ( q , N ) V ^ 2 ( q ) / \ t 1 \ F ( q , x , { 2 } ) ^ Nftr^q) t N = { 1 , 2 } F ( q , x , N ) = X F i g u r e 1 F e a s i b l e D i r e c t i o n s Fo r a T y p i c a l L o c a l Game - 66 -Figure 2 - 67 -Figure 3 The Local Game in Regime 1 - 68 --1 r l . O x 2 K ( q , { l » -'7 1.0 X l \ V T T 2 ( q ) - C + ( q , N ) ^ K ( q , { 2 } ) = K ( q , N ) ( l , - l ) = K(q) = K(q) . 1. 0 F i g u r e 4 The Lo ca l Game i n Regime 2 - 69 -F i g u r e 5 The L o c a l Game i n Regime 4a - 70 -! / 1.0 * x2 / J K(q,{2}) (l ,D=K(q) [ V 7 T 2 ( q ) K ( q { l } ) - ^ ^ r' \ ' ' C +(q,N) / : N/ , K(q,N) / L___V _-/... \ " V ^ ( q ) -1.0 F i g u r e 6 The L o c a l Game i n Regime 4b - 71 -F i g u r e 7 The B a r g a i n i n g Core i n Regime 4b F i g u r e 8 The B a r g a i n i n g Core a t an E q u i l i b r i u m CHAPTER 4 CONSCIOUS P A R A L L E L I S M AND PREDATORY P R I C I N G IN A CONTESTABLE MARKET 1. I n t r o d u c t i o n T h e c o n c e p t s o f c o n s c i o u s p a r a l l e l i s m a n d p r e d a t o r y p r i c i n g h a v e l o n g b e e n p a r t o f t h e f o l k l o r e o f t h e s t r u c t u r e - c o n d u c t - p e r f o r m a n c e 1 p a r a d i g m o f i n d u s t r i a l o r g a n i z a t i o n . H o w e v e r t h e y h a v e r e c e i v e d l i t t l e a t t e n t i o n a s c o n s t r u c t s o f f o r m a l t h e o r y a n d a s a r e s u l t t h e r a t i o n a l i t y o f s u c h p r a c t i c e s i s o f t e n c a l l e d i n t o q u e s t i o n . F o r e x a m p l e , S h u b i k ( 1 9 5 9 ) s t a t e s t h a t " a c o m p l e t e g e n e r a t i o n o f e c o n o m i s t s h a s p o i n t e d o u t t h e i d i o c y o f t h e c h a r g e o f c o n s c i o u s p a r a l l e l i s m " . T h e p u r p o s e o f t h i s p a p e r i s t o f i t t h e s e n o t i o n s i n t o a game t h e o r e t i c s t r u c t u r e t h a t w i l l e l u c i d a t e t h e r e l a t i o n s h i p b e t w e e n i n f o r m a l d e s c r i p t i o n s o f f i r m b e h a v i o u r a n d t h e i d e a o f r a t i o n a l i t y e m b o d i e d i n t h e d e f i n i t i o n o f an e q u i l i b r i u m . To b e g i n w i t h c o n s c i o u s p a r a l l e l i s m , n o t e t h a t t h i s i s n o t a p r e c i s e l y d e f i n e d c o n c e p t , b u t r e f e r s t o t h e g e n e r a l s i t u a t i o n i n w h i c h f i r m s s e e k t o a t t a i n c o l l u s i v e p r o f i t l e v e l s b y a d o p t i n g a s t r a t e g y o f m a t c h i n g e a c h o t h e r ' s b e h a v i o u r . O u r f i r s t o b j e c t i v e i s t o s e e k a r a t i o n a l e f o r s u c h b e h a v i o u r . I t i s w e l l k nown t h a t , i n o l i g o p o l y s e t t i n g s , c o l l u s i v e b e h a v i o u r c a n l e a d LO h i g h e r f i r m p r o f i t s t h a n d o e s n o n - c o o p e r a t i v e b e h a v i o u r . F u r -t h e r , t h e r e l a t i o n b e t w e e n t h e i n f o r m a t i o n a v a i l a b l e t o f i r m s a n d t h e s e t • • 3 o f ' c o l l u s i v e ' e q u i l i b r i a i s now w e l l u n d e r s t o o d . . H o w e v e r , a r e a l d r a w -b a c k o f s u c h r e s u l t s i s t h e l a r g e n u m b e r o f e q u i l i b r i a t h a t t h e y t y p i c a l l y a l l o w . T h e i s s u e we a d d r e s s i s n o t w h e t h e r c o l l u s i o n i s p o s s i b l e , b u t - 73 -- 74 -r a t h e r , how i t i s t h a t f i r m s a c t u a l l y m i g h t a r r i v e a t a p a r t i c u l a r e q u i l -i b r i u m . One f e a t u r e o f i m p o r t a n c e i n t h i s r e s p e c t i s t h e p r o c e s s w h e r e b y f i r m s ' e x p e c t a t i o n s o n e a c h o t h e r ' s b e h a v i o u r a r e f o r m e d . E x p e r i m e n t a l s t u d i e s o f t h e t y p e d o n e b y R o t h a n d S c h o u m a k e r [ 1 9 8 3 ] d e m o n s t r a t e , f i r s t , t h a t c o n s i s t e n c y o f e x p e c t a t i o n s i s i n s t r u m e n t a l i n t h e a t t a i n m e n t o f c o l -l u s i v e o u t c o m e s , a n d s e c o n d l y , t h a t p l a y e r s t e n d t o a c h e i v e s u c h c o n s i s -t e n c y b y a d o p t i n g r u l e - o f - t h u m b b e h a v i o u r w i t h r e s p e c t t o o b s e r v a b l e v a r -i a b l e s i n t h e a b s e n c e o f c o m p l e t e i n f o r m a t i o n o n t h e game. T h e s e o b s e r -v a t i o n s s u g g e s t t h a t , on t h e o n e h a n d , t h e e x p e c t a t i o n s f o r m a t i o n p r o c e s s m u s t be an i n t e g r a l p a r t o f a n y c o m p l e t e o l i g o p o l y m o d e l , a n d , on t h e o t h e r h a n d , t h a t ' s o c i a l ' o r ' h i s t o r i c a l ' f a c t o r s may be a n e c e s s a r y c o n s t r u c t i n a c h i e v i n g d e t e r m i n a t e o u t c o m e s i n s u c h m o d e l s . We s u g g e s t t h a t t h e m a t c h i n g -b e h a v i o u r r e f e r r e d t o a s c o n s c i o u s p a r a l l e l i s m c a n be i n t e r p r e t e d a s an e a s i l y v e r i f i a b l e r u l e - o f - t h u m b t h a t f i r m s c a n t a c i t l y a g r e e o n , a n d t h e f u n c t i o n o f w h i c h i s t o r e n d e r m u t u a l l y c o n s i s t e n t t h e e x p e c t a t i o n s o f f i r m s on how t h e y w i l l r e a c t t o e a c h o t h e r ' s p r o f i t - i n c r e a s i n g m o v e s . To d e v e l o p t h i s l o g i c f o r m a l l y we m o d e l f i r s t an a b s t r a c t p r o c e s s o f e x p e c t a t i o n s - f o r m a t i o n , b y p r e s e n t i n g a n ' a n n o u n c e m e n t ' game i n w h i c h f i r m s c o m m u n i c a t e w i t h e a c h o t h e r t h r o u g h t h e m e d i u m o f a n n o u n c e d p r i c e c h a n g e s . T h e n , s i n c e t h e r e i s i n g e n e r a l no way t o d e f i n e u n i q u e l y t h e a n n o u n c e m e n t s t r a t e g i e s o f f i r m s i n s u c h a game we m o d e l s u c h b e h a v i o u r a s f o l l o w i n g a r u l e - o f - t h u m b t h a t i s d e r i v e d f r o m an a x i o m a t i c a l l y s p e c i f i e d s o c i a l c o n v e n t i o n . T h e d e r i v e d r u l e - o f - t h u m b i s t h e m a t c h i n g o f a n n o u n c e d p r i c e c h a n g e s , o r c o n s c i o u s p a r a l l e l i s m . We show t h a t o u r m o d e l o f e x p e c t -- 75 -a t i o n s f o r m a t i o n , t o g e t h e r w i t h t h e s o c i a l c o n v e n t i o n , r e s u l t s i n a u n i q u e c o l l u s i v e o u t c o m e f o r t h e i n d u s t r y . C o n s c i o u s p a r a l l e l i s m i s t h u s s e e n a s a p l a u s i b l e ( n o n - i d i o t i c ) means f o r t h e i n d u s t r y t o a c h i e v e a u n i q u e c o l l u s i v e o u t c o m e . We t u r n now t o t h e q u e s t i o n o f p r e d a t o r y p r i c i n g , w h i c h r e f e r s t o p r i c e c u t t i n g b y f i r m s i n t h e f a c e o f new e n t r y . A w e l l known w e a k n e s s w i t h m o d e l s t h a t p r o v i d e c o l l u s i v e o u t -c o m e s c o n c e r n s t h e r o b u s t n e s s o f t h e s e o u t c o m e s w i t h r e s p e c t t o new e n t r y . T h e g e n e r a l p e r c e p t i o n i s t h a t b a r r i e r s t o e n t r y a r e a n e c e s s a r y c o n d i t i o n f o r p o s i t i v e l o n g r u n p r o f i t s . T h i s p e r c e p t i o n h a s b e e n c a r r i e d e v e n f u r t h e r i n t h e r e c e n t w o r k o f B a u m o l , P a n z a r a n d W i l l i g [ 1 9 8 2 ] . T h e y a r g u e t h a t w i t h c o s t l e s s e n t r y a n d e x i t t h e n 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 s t r u c -t u r e s a r e e s s e n t i a l l y t h e o n l y s u s t a i n a b l e p o s s i b i l i t y . We a d d r e s s t h i s i s s u e b y f o r m a l l y m o d e l l i n g e n t r y i n t h e c o n t e x t o f o u r m o d e l o f c o l l u s i v e b e h a v i o u r . We e x p l i c i t l y a s s u m e t h a t n o t o n l y i s e n t r y a n d e x i t c o s t l e s s b u t t h a t a l l p r o d u c i n g i n d u s t r y members a r e known b e f o r e p r i c e s a r e s e t . I n c o n t r a s t w i t h B a u m o l , P a n z a r a n d W i l l i g [ 1 9 8 2 ] we s h ow t h a t c u r r e n t i n d u s t r y m embers w i l l h a v e e q u i l i b r i u m p r i c i n g s t r a t e g i e s t h a t a r e e n t r y d e t e r r i n g ; t h e s e e q u i l i b r i u m s t r a t e g i e s a r e a f o r m o f p r e d a t o r y p r i c i n g . T h e c o n s e q u e n c e i s t h a t t h e e x i s t e n c e o f l o n g - r u n p o s i t i v e p r o f i t s i n a c o n t e s t a b l e m a r k e t i s a l o g i c a l p o s s i b i l i t y . T h e r e a s o n f o r t h i s s t r i k i n g -l y d i f f e r e n t r e s u l t i s d u e t o t h e a s s u m p t i o n made on t h e i n f o r m a t i o n a v a i l -a b l e t o f i r m s . I f p o t e n t i a l e n t r a n t s c a n b e i d e n t i f i e d b e f o r e t h e y a c t u -a l l y s t a r t p r o d u c t i o n t h e n t h e e q u i l i b r i u m r e s p o n s e f o r e x i s t i n g f i r m s i s : t o a d o p t p r i c e c u t s t h a t w i l l make e n t r y u n p r o f i t a b l e . H e n c e i t c a n be - 76 -c o n c l u d e d t h a t t h e n a t u r e o f m a r k e t i n s t i t u t i o n s i s a s i m p o r t a n t a s 4 t h e c o s t o f e n t r y a n d e x i t i n d e t e r m i n i n g m a r k e t s t r u c t u r e . A s a f i n a l p o i n t we c o n s i d e r t h e e f f e c t o f s u n k c o s t s on i n -d u s t r y s t r u c t u r e . I n t h e w o r k o f B a u m o l , P a n z a r a n d W i l l i g [ 1 9 8 2 ] i t i s a r g u e d t h a t t h e p r e s e n c e o f s u n k c o s t s c a n l e a d t o e n t r y b a r r i e r s a n d s u p e r n o r m a l p r o f i t s . H o w e v e r i n o u r m o d e l we a r e l e d t o t h e o p -p o s i t e c o n c l u s i o n . To a p o t e n t i a l e n t r a n t s u n k c o s t s h a s t h e e f f e c t o f c o m m i t t i n g a c a p i t a l t o t h e m a r k e t , t h u s m i t i g a t i n g t h e e f f e c t o f e n t r y d e t e r r i n g s t r a t e g i e s . T h e c o n s e q u e n c e i s t h a t i n d u s t r y p r o f i t s d e c r e a s e w h i l e t h e n u m b e r o f e n t r a n t s i n c r e a s e s a s s u n k c o s t s r i s e . T h e p l a n o f t h e c h a p t e r i s a s f o l l o w s . I n s e c t i o n 2 t h e b a s i c m o d e l i s p r e s e n t e d a l o n g w i t h a d e f i n i t i o n o f a s u b g a m e p e r f e c t N a s h e q u i l i b r i u m . S e c t i o n s 3 a n d 4 s t u d y t h e p r o b l e m o f m o d e l l i n g t h e d o c -t r i n e o f c o n s c i o u s p a r a l l e l i s m s . I t i s a r g u e d t h a t s i n c e i t i s g e n e r -a l l y v i e w e d a s a s i g n a l l i n g p h e n o m e n a t h e n i t s h o u l d b e s e e n a s a c o -o r d i n a t i o n p r o b l e m . T h i s i s f o r m a l l y m o d e l l e d a s an announcement..game I n s e c t i o n 4 , c o n s c i o u s p a r a l l e l i s m i s v i e w e d a s a social convention w h i c h i s b e s t m o d e l l e d a s r e s u l t i n g f r o m s i m p l e p r i n c i p l e s t h a t a r e r e -p r e s e n t e d a x i o m a t i c a l l y . T o g e t h e r t h e a n n o u n c e m e n t game a l o n g w i t h t h e e q u i l i b r i u m c o n c e p t o f s e c t i o n 2 d e f i n e a u n i q u e c o l l u s i v e e q u i l i b r i u m . F i n a l l y , i n s e c t i o n 5 t h e e f f e c t s o f e n t r y a r e s t u d i e d . H e r e p r e d a t o r y p r i c i n g i s d e f i n e d i n t h e b r o a d s e n s e a s r a d i c a l p r i c e d r o p s i n t h e f a c e - 77 -o f e n t r y . I t i s a rgued t h a t t h e Cou r no t -Na sh e q u i l i b r i u m i n p r i c e s fo rms t h e a p p r o p r i a t e s o l u t i o n c on c ep t f o r d e f i n i n g " t h e e n t r y e q u i l i b r i u m . Com-b i n i n g t h i s w i t h t h e c o o p e r a t i v e b e h a v i o u r o f t h e p r e v i o u s s e c t i o n s , i t i s shown t h a t f i r m s can be i n l o n g - r u n e q u i l i b r i u m a t " c o l l u s i v e " ' p r i c e s u s i n g an e q u i l i b r i u m e n t r y d e t e r r i n g s t r a t e g y ! 2. The I n d u s t r y Model In t h i s s e c t i o n a d y n a m i c o l i g o p o l y model s i m i l a r t o t h a t f ound i n F r i edman [ 1 9 7 7 ] o r Green [ 1980 ] i s p r e s e n t e d . L e t the s e t N = { l , . . . , n } i ndex f i r m s i n an o l i g o p o l i s t i c i n d u s t r y , each o f w h i c h p roduce a s i n g l e d i f f e r e n t i a t e d p r o d u c t t h a t i s s o l d f o r a p r i c e p. > 0 , i e N. Le t IT.(•£) be t h e p r o f i t s pe r p e r i o d o f f i r m i w h i l e i s t he v e c t o r o f p r i c e s c h a r g e d . I f {P_ }^ r e p r e s e n t s t he t ime pa th o f p r i c e s f rom t ime T t o the i n d e f i n i t e f u t u r e t hen t he f i r m ' s d i s c o u n t e d p r o f i t a t t h i s t ime i s g i v e n by where & i s a s p e c i f i e d c o n s t a n t d i s c o u n t r a t e . F o r t h e p r e s e n t a n a l y s i s i t w i l l be assumed t h a t f i r m s c a n , s t a r t i n g a t t ime t=0 , o n l y a d j u s t t h e i r p r i c e s e v e r y Y p e r i o d s . T h i s d e f P = - 78 -p e r i o d can e i t h e r be i n t e p r e t e d as an i n d i v i s i b i l i t y i n a f i r m ' s a b i l i t y t o a d j u s t p r i c e s o r , l i k e Spence [ 1 9 7 8 a ] , i t can be seen as t h e amount o f t ime r e q u i r e d to a c q u i r e enough i n f o r m a t i o n i n o r d e r t o j u s t i f y a p r i c e 5 change . As a n o t a t i o n a l c o n v e n i e n c e r e s c a l e t so t h a t t = l r e p r e s e n t s Y p e r i o d s . Thus i n t h e p e r i o d t e [m ,m+ l ) , m an i n t e g e r , p r i c e s w i l l r ema in f i x e d . The d i s c o u n t e d p r o f i t s can now be w r i t t e n i n summat ion f o rm , where T i s an i n t e g e r . r t + 1 1 z 1 t=T t 1 t=T 1 0 d e f -5Y t where 3 = e and J__ i s t h e p r i c e i n p e r i o d [ t , t + l ) . N o t i c e t h a t as t h e t ime r e q u i r e d to change p r i c e s , Y, d e c r e a s e s , 3 app roa che s 1 * When f i r m s choose i t w i l l be assumed t h a t t h e y have been a b l e t o o b s e r v e a l l p a s t p r i c e s bu t c a n n o t o b s e r v e t h e i r c o m p e t i t o r s ' c u r r e n t p r i c e c h o i c e s . Hence t h e i r i n f o r m a t i o n s e t c o n s i s t s o f t h e h i s -t o r y o f p a s t p r i c e s deno ted by : I t = { P T | 0 < x < t } . L e t H t r e p r e s e n t t h e s e t o f p o s s i b l e h i s t o r i e s o r i n f o r m a t i o n s e t s a t t ime t . A decision a t t ime t by f i r m i w i l l s i m p l y be t h e map: - 79 -Sl: H. + IR i t + Tha t i s f o r each h i s t o r y I t e H t , S * ( l t ) = P | , w i l l g i v e the p r i c e c h o i c e o f f i r m i . L e t r e p r e s e n t the s e t o f p o s s i b l e d e c i s i o n s by f i r m i , t hen t he strategy o f f i r m i a t t i m e T w i l l be a sequence o f d e c i s i o n s g i v e n by T °° t d 6 f T 1 t=T *• L e t = (aj,...,a^) be t he v e c t o r o f s t r a t e g i e s f o r t h e i n -d u s t r y . The e q u i l i b r i u m c o n c e p t t h a t w i l l be used i s the Subgame Perfect Nash Equilibrium ( S P E ) . T h i s c o n c e p t w i l l en su re t h a t f o r e v e r y t ime t f i r m s a r e a lways a c t i n g " r a t i o n a l l y " . DEFINITION 1. A s t r a t e g y a T i s a subgame p e r f e c t e q u i l i b r i u m i f and o n l y > max v j ( a * 5*11*) a i e n i V I * e H t , V t > T, and V i e N. where V.{a | l ) = V. ({P_ }^) and P_T i s t he sequence o f p r i c e s g e n e r a t e d by the s t r a t e g y a* g i v e n an i n i t i a l i n f o r m a t i o n r s e t 1^ a t t ime t . The n o t a t i o n ( a ^ a ^ ) r e p r e s e n t s w i t h f i r m i ' s s t r a t e g y r e p l a c e d by a*. 1 In g e n e r a l , as shown i n Green [ 1 9 8 0 ] , t h e r e w i l l e x i s t many S P E . - 80 -We now p o i n t ou t two c l a s s e s o f such e q u i l i b r i a t h a t w i l l be e x t e n s i v e l y used i n the f o l l o w i n g s e c t i o n s . B e f o r e d o i n g so suppose t h a t f l o w p r o -f i t s s a t i s f y t h e f o l l o w i n g a s s u m p t i o n s . Assumption Al. TT. ( • ) i s c o n t i n u o u s l y d i f f e r e n t i a t e w i t h t he bound J T T . ( P > | < A,V £ e IR*, V i e N. T h i s a s s ump t i o n i s c o n s i s t e n t w i t h o u r o r i g i n a l a s s e r t i o n t h a t f i r m s p roduce a d i f f e r e n t i a t e d p r o d u c t . ' 7 F u r t h e r s i n c e the p r o d u c t s a r e p roduced by t h e same i n d u s t r y , by d e f i n i t i o n t hey must be s u b s t i t u t e s . Hence we have the f o l l o w i n g s i g n s on t h e p a r t i a l d e r i v a t i v e s : 3TT. Assumption A2. ( i ) — - ( _ P ) > 0, P j 3TT. ( i i ) I f TT. ( P ) > 0, V k £ N, t h en ^ ( P ) > 0 k - 3 P j -V i , j £ N, i t j. The second cond i t i on , . ; r e q u i r e s t h a t the p r o d u c t s be s t r i c t s u b -s t i t u t e s i f a l l f i r m s a r e mak ing p o s i t i v e p r o f i t s . F i n a l l y i t w i l l be g assumed t h a t a Cou r no t -Na sh e q u i l i b r i u m i n p r i c e s e x i s t s . Assumption A3. There e x i s t s a un i que Cou rno t -Na sh e q u i l i b r i u m , P ^ , d e -f i n e d by •n.{P°) = max T T . ( P . , P ° . ) , V i £ N. 1 P.>0 1 - 1 l -For t he r e m a i n d e r o f the e s s a y i t w i l l a lway s be supposed t h a t p r o f i t s s a t i s f y t h e s e a s s u m p t i o n s . - 81 -The f o l l o w i n g w e l l known p r o p o s i t i o n shows t he i m p o r t a n t r e -l a t i o n s h i p between Cou r no t -Na sh e q u i l i b r i a and SPE f o r t h e dynamic game. PROPOSITION 1. Suppose that firms ' decisions are independent of past t d e f t t history. That is S . ( I . ) = P., for some P. e IR and V I . e H. . Then the I w 1 1 T L t unique SPE in this set at time T is given by the following decision se-quence: t d e f 0 S | ( l t ) = P", V t > T , i e N. where P^ is the unique Cournot-Nash equilibrium given in Assumption A3. The p r o o f o f t h i s r e s u l t and s ub sequen t r e s u l t s a re t o be f ound i n the a p p e n d i x . The r e a d e r may r e f e r t o Green [1980] f o r a d e t a i l e d t r e a t -men t , i n c l u d i n g t h e e f f e c t s o f u n c e r t a i n t y . I t s h o u l d a l s o be no t ed t h a t i f Y i s s u f f i c i e n t l y l a r g e , t h en t h a t t h i s s o l u t i o n i s t h e o n l y S P E . Our ma j o r i n t e r e s t now l i e s i n s t u d y i n g t h e c a s e i n w h i c h " c o o p e r a t i v e " e q u i l i b r i a a re p o s s i b l e . Fo r Y s m a l l , t h a t i s i f f i r m s can a d j u s t p r i c e s q u i c k l y , we have t h e f o l l o w i n g s p e c i a l c a s e o f a w e l l known r e s u l t on r e p e a t e d games. PROPOSITION 2. Let {P J^._Q be a sequence of prices 'and suppose that (i) P^ = at most f i n i t e l y many times ( i i ) If f _P° then ir. (P1) > TI\ ( P ^ ) + 6, V i £ N and for some constant 6 > 0 . - 82 -Let the strategy o be defined by the decisions: r i» i f P t - 1 = ~ t - 1 i f P t - 1 f1 P t - 1 Then o is a SPE for sufficiently small Y. H e n c e i f t h e r e e x i s t p r i c e s t h a t a l l f i r m s c a n t a c i t l y a g r e e u p o n a n d t h a t l e a v e t h e m b e t t e r o f f t h a n 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 , c f T i s a S P E . C o n d i t i o n ( i ) w i l l be u s e d i n a l a t e r s e c t i o n when t h e i s s u e o f p r e d a t o r y p r i c i n g i s a d d r e s s e d . 3. T h e A n n o u n c e m e n t Game A l t h o u g h i t i s w e l l known t h a t a c o o p e r a t i v e o u t c o m e c a n be v i e w e d a s an e q u i l i b r i u m f o r an a p p r o p r i a t e l y d e f i n e d r e p e a t e d game, an i s s u e t h a t h a s r e c e i v e d r e l a t i v e l y - l i t t l e a t t e n t i o n i n o l i g o p o l y t h e o r y i s t h e q u e s t i o n of which.of t h e many p o s s i b l e o u t c o m e s s h o u l d b e s e l e c t e d a s t h e " s o l u t i o n " . T h e d i f f i c u l t y i s t h a t i n g e n e r a l t h e a s s u m p t i o n t h a t f i r m s a r e i n d i v i d u a l p r o f i t - m a x i m i z e r s i s n o t s u f f i c i e n t t o s e l e c t an u n i q u e e q u i l i b r i u m . F r o m a p u r e l y game t h e o r e t i c v i e w p o i n t t h e r e h a v e b e e n many p r o p o s e d s o l u t i o n s t o t h i s p r o b l e m w i t h i n t h e f r a m e w o r k o f n - p e r s o n c o o p e r a t i v e g a m e s . I f o n e a l l o w s p r e - p l a y c o m m u n i c a t i o n t h e n t h e p r o b l e m c a n a l s o b e v i e w e d a s a b a r g a i n i n g p r o b l e m , i n t h i s c a s e m o s t s o l u t i o n s a r e f o u n d b y u s i n g a p p r o p r i a t e s e l e c t i o n c r i t e r i a , - 83 -9 w i t h i n t u i t i v e l y a p p e a l l i n g p r o p e r t i e s . The d i f f i c u l t y w i t h t h e s e a p -p r o a che s i s t h a t t h e y r e q u i r e e x p l i c i t c o o r d i n a t i o n o f a s o r t t h a t i s i l l e g a l i n many c o n c e n t r a t e d i n d u s t r i e s . F u r t h e r , t h e s e app r oa che s p r e -suppose t h a t p r o f i t s o r p a y o f f s a r e common know l edge ; i n p r a c t i c e t h e s e v a r i a b l e s a r e o f t e n d i f f i c u l t t o o b s e r v e o r known o n l y w i t h a l a r g e l a g . In t h i s and t h e f o l l o w i n g s e c t i o n a s o l u t i o n i s p r opo s ed w h i c h i s ba sed on o b s e r v e d p r i c i n g b e h a v i o u r . Even i f f i r m s do n o t d i r e c t l y commun i ca te , t h e y do t r a n s m i t i n f o r m a t i o n i n t h e way t h e y a d j u s t p r i c e s . Here i t i s assumed t h a t p r i c e announcements a r e e a s i l y o b s e r v a b l e and can t h u s p r o v i d e a b a s i s f o r t a c i t c o o p e r a t i v e b e h a v i o u r . F o r e x amp l e , i n a d u o p o l i s t i c i n d u s t r y when one f i r m announces a p r i c e i n c r e a s e , u s u a l l y t h e c o m p e t i t o r r e sponds w i t h e i t h e r a s i m i l a r i n c r e a s e o r no i n c r e a s e a t a l l . I f t h e second f i r m does no t i n c r e a s e i t s p r i c e , t h en t h e f i r s t f i r m u s u a l l y r e s c i n d s i t s p l a nned i n c r e a s e . I t s h o u l d be no t ed t h a t s u ch announcements and r e s pon s e s can o c c u r before p r i c e s have a c t u a l l y been a d -j u s t e d . T h i s b e h a v i o u r w i l l be f o r m a l l y mode led as an announcement game. S i n c e t h e announcement game.can be- -p layed each p e r i o d w e - w i l l s u p p r e s s t ime i n d i c e s f o r - t h e p r e s e n t d i s c u s s i o n . L e t _P_ s i m p l y r e p r e s e n t t h e c u r r e n t p r i c e v e c t o r wh i c h w i l l be t he s t a r t i n g p o i n t o r s t a t u s quo p o i n t f o r t he announcement game. F u r t h e r t o a v o i d e x c e s s c o m p l i c a t i o n s , we w i l l suppose o n l y one f i r m each p e r i o d i n i t i a t e s a p r i c e c hange . I t s h o u l d be s t r e s s e d t h a t t h e p l a y e r s a r e m e r e l y announc i ng p l a n n e d p r i c e s and a r e n o t a c t u a l l y c omm i t t ed t o c a r r y i n g o u t t h e s e c h a n g e s . I t w i l l be shown l a t e r t h a t i n g e n e r a l " c h e a t i n g " w i l l no t o c c u r . - 84 -S u p p o s e t h a t f i r m i a n n o u n c e s a p r i c e c h a n g e o f A P ^ . T h e o t h e r f i r m s j e N / i w i l l either n o t r e s p o n d a n d f o r m a l l y s e t A P . = 0 o r t h e y 3 w i l l s e t A P j = r j ( I P , A P . j ) . T h e r e a c t i o n f u n c t i o n r l(_P,AP^) r e p r e s e n t s t h e u n i q u e r e s p o n s e t o a p r i c e c h a n g e b y f i r m i g i v e n t h e s t a t u s q u o P_. As h a s a l r e a d y b e e n p o i n t e d o u t , o n c e f i r m s b e g i n c o o p e r a t i v e b e h a v i o u r t h e i r " o p t i m a l " r e s p o n s e s a r e no l o n g e r u n i q u e l y d e f i n e d . R a t h e r r 1 . (P_,AP.) w i l l r e p r e s e n t a c o n v e n t i o n o f b e h a v i o u r t h a t i s t a c i t l y r e -c o g n i z e d b y t h e i n d u s t r y . B e f o r e f u r t h e r s p e c i f y i n g t h e s e r e s p o n s e s t h e n o t i o n o f a s t r a t e g y w i l l b e d i s c u s s e d . B y d e f i n i t i o n i t w i l l b e s u p p o s e d t h a t rJ(_P_,AP.j) = A P i a n d r l(P,0) = 0 V j e N. T h e s e c o n d a s -s u m p t i o n i s p u r e l y f o r m a l s i n c e i t s i m p l y r e q u i r e s t h a t no p r i c e c h a n g e s w i l l o c c u r w i t h o u t an a n n o u n c e m e n t . An announcement strategy f o r f i r m j , j e N i s g i v e n b y : / r j ( P , A P . ) ; i f j e K i d e f i i R J ( _ P , A P i ,K) = { a n d R^ = f o r k e K / j l b ; i f n o t . A s was p o i n t e d o u t , i n p r a c t i c e f i r m s may r e s c i n d p l a n n e d p r i c e c h a n g e s s h o u l d t h e y n o t be f o l l o w e d b y o t h e r f i r m s . T h e s e t K r e p r e s e n t s t h o s e f i r m s t h a t m u s t a l l c h a n g e t h e i r p r i c e s i n o r d e r t o s u s t a i n an a n -n o u n c e d p r i c e c h a n g e . T h e s e t K may n o t i n c l u d e a l l o f N s o t h a t l a t e r we may e x p l i c i t l y s t u d y t h e i n c e n t i v e s f o r f i r m s t o t a k e p a r t i n t h e a n -n o u n c e m e n t g a m e . 1 ^ L e t R"" (P_,AP-,K) r e p r e s e n t t h e v e c t o r o f a n n o u n c e m e n t s t r a t e g i e s . - 85 -Definition 2. The s e t { A P . . R 1 (_P , A P . , K )} w i l l be c a l l e d an equilibrium announcement i f and o n l y i f ( i ) T r J ( P + R 1 { P , A P i , K ) ) > max T T . ( P + ( A P . , R ^ . ( P , A P . , K ) ) ) , V j e N A P . e { o V . ( P , A P ) } ( i i ) ^ . ( P + R ^ P . A P ^ K ) ) > T T . ( P ) i f A P I ? Q . An equilibrium announcement s i m p l y d e s c r i b e s a s e t o f announce-ment s t r a t e g i e s wh i c h ( i ) a r e o p t i m a l f o r each f i r m w i t h r e s p e c t to i t s f l o w p r o f i t s g i v e n t he s t r a t e g i e s o f the o t h e r f i r m s and ( i i ) wh i ch make t h e f i r m i n i t i a t i n g t h e p r i c e change s t r i c t l y b e t t e r o f f . T h i s s e cond c o n d i t i o n s i m p l y r e q u i r e s t h a t no f i r m w i l l i n i t i a t e a p r i c e change un -l e s s i t i s i n i t s i n t e r e s t . L e t A (R ) r e p r e s e n t a l l t he p o s s i b l e equili-brium announcements. In g e n e r a l A(F_) w i l l no t be u n i q u e . Fo r any K <= N, we1 h a v e " ( O . R 1 ( _ P , 0 , K ) ) e A ( P ) ; by d e f i n i t i o n r e m a i n i n g a t t h e s t a t u s quo o r c u r r e n t p r i c e F_ w i l l a lway s be an e q u i l i b r i u m announce -ment. L e t us now d e s c r i b e how the announcement game f i t s i n t o t h e dynamic game o f t h e p r e v i o u s s e c t i o n . G i ven t h a t f i r m s a t t=0 have no h i s t o r i e s t o draw upon i t i s o n l y n a t u r a l t h a t _ P ^ , t he Cou r no t -Na sh e q u i -l i b r i u m , be t h e - f i r s t p e r i o d p r i c e . Suppose t h a t t h e announcement game i s p l a y e d once each p e r i o d . L e t us now d e f i n e r e c u r -s i v e l y the r e s u l t i n g sequence o f p r i c e s . D e f i n e = and a t p e r i o d t l e t { A P . , R 1 ' ( _ P T ' , A P * , K - t ) } e A(J__t) be the e q u i l i b r i u m announcement s e l e c t e d . ~ t + i d e f ~. * ~+ t t Now d e f i n e _ P L = P L + R (F_ , A P . T , K ) . I t w i l l be assumed - 86 -t h a t a c t u a l p r i c e s w i l l b e c h o s e n a c c o r d i n g t o t h e s t r a t e g i e s p r e s e n t e d i n p r o p o s i t i o n 2. T h a t i s e a c h f i r m w i l l s e t p r i c e e q u a l t o P* a s l o n g a s o t h e r f i r m s h a v e d o n e s o i n p r e v i o u s p e r i o d s . T h e c o n s e q u e n c e i s t h a t i f p l a y i n g t h e a n n o u n c e m e n t game m a k e s all f i r m s b e t t e r o f f c o m p a r e d t o p l a y i n g C o u r n o t - N a s h ; b y p r o p o s i t i o n 2 no f i r m w i l l d e v i a t e o r c h e a t f r o m i t s a n n o u n c e d p r i c e c h a n g e . A l t h o u g h f i r m s a r e n o t f o r c e d t o f o l l o w s t a t e d p r i c e c h a n g e s , t h e f a c t t h a t t h e a n n o u n c e m e n t s a s s i s t i n m a k i n g e x p e c t a t i o n s c o n -s i s t e n t i n t h e f o l l o w i n g p e r i o d c r e a t e s an i n c e n t i v e f o r " g o o d b e h a v i o u r " . B e f o r e g o i n g on t o r p e c i s e l y s p e c i f y t h e r ^ . , . ) u s e d i n t h e a n n o u n c e m e n t game, l e t u s d e f i n e a n e q u i l i b r i u m p r i c e f o r t h e i n d u s t r y . T h e i n d u s t r y w i l l h a v e r e a c h e d an e q u i l i b r i u m p r i c e P* i f ~ t P_ = P*, t >_ T f o r some T > 0. By p l a y i n g t h e a n n o u n c e m e n t game f i r m s w i l l g e n e r a l l y b e b e t t e r o f f t h a n p l a y i n g C o u r n o t - N a s h a n d t h u s c a u s e a c h a n g e i n p r i c e s o v e r t i m e . T h e r e w i l l c e a s e t o b e an i n c e n t i v e t o c h a n g e p r i c e s ( r e m e m b e r t h e a n n o u n c e r i s a l w a y s made b e t t e r o f f ) when t h e r e no l o n g e r e x i s t n o n t r i v i a l e q u i l i b r i u m a n n o u n c e m e n t s . T h i s d i s c u s s i o n m o t i v a t e s t h e f o l l o w i n g d e f i n i t i o n . DEFINITION 3. A - - p r i c e . v e c t o r P* e w i l l b e an equilibrium price i n t h e a n n o u n c e m e n t game i f a n d , o n l y i f \ / {AP . , R 1 ( P _* , AP . , K ) } e A ( P * ) , A P i = 0. T h a t i s P* i s an e q u i l i b r i u m i f o n l y t r i v i a l e q u i l i b r i u m a n -n o u n c e m e n t s a r e p o s s i b l e . I n t e r m s o f t h e i m p l e m e n t a t i o n o f t h e a n n o u n c e -m e n t game t h i s w i l l r e s u l t i n c o n s t a n t p r i c e s s e t a t P* o v e r t i m e . Ob-- 87 -s e r v e t h a t we have two e q u i l i b r i u m c o n c e p t s , namely e q u i l i b r i u m announce -ments and e q u i l i b r i u m p r i c e s . The e q u i l i b r i u m announcements i s m e r e l y 11 t he f o rma l model o f t h e way one may move to an e q u i l i b r i u m p r i c e . 4 . An A x i o m a t i c App roach to C o n s c i o u s P a r a l l e l i s m In t h e p r e v i o u s s e c t i o n an a b s t r a c t "announcement game" was d e s c r i b e d whose r o l e was t o communicate intent among i ndus t r y . .membe r s . In d o i n g so i t p r o v i d e d a way to en su r e t h a t a l l f i r m s h e l d the same e x -p e c t a t i o n s w i t h r e s p e c t to nex t p e r i o d s ' p r i c e s . The impo r t a n c e o f c o n -s i s t e n t e x p e c t a t i o n s has r e c e n t l y been h i g h l i g h t e d i n t h e e x p e r i m e n t a l work o f Ro th and Schoumaker [ 1 9 8 3 ] . They f i n d t h a t a key i n g r e d i e n t f o r t h e a t t a i n m e n t o f agreements between agen t s wh i ch dom ina te t he d e f a u l t p a y o f f s was t h e e x i s t e n c e o f c o n s i s t e n t e x p e c t a t i o n s . These were o f t e n ba sed on w i d e l y a c c e p t e d p r i n c i p l e s . For examp le i f one examines t h e r e -s u l t s o f e a r l i e r work by Roth and Murn i ghan [ 1 9 8 2 ] , one i s s t r u c k by t h e f a c t t h a t ag reements were o f t e n equa l d i v i s i o n outcomes w i t h r e s p e c t t o e a s i l y o b s e r v a b l e p r o x y v a r i a b l e s when p a y o f f s were not d i r e c t l y o b s e r v -a b l e . I t has been e x p l i c i t l y assumed t h a t p r i c e : forms t h e most e a s i l y o b s e r v a b l e i n d u s t r y w i d e pa r ame t e r and as such t h e above e x -p e r i m e n t a l work s u g g e s t s t h a t f i r m s wou l d match each o t h e r ' s p r i c e a n -nouncements u s i n g e a s i l y i d e n t i f i a b l e p r i n c i p l e s . In the announcement game t he f u n c t i o n r 1 (P_,AP.) r e p r e s e n t s • t h e way the f i r m s wou ld c a r r y ou t t h e s e c h ange s . In t h i s s e c t i o n i t w i l l be supposed t h a t t h e s e f u n c -- 88 -t i o n s s a t i s f y w i d e l y a c c e p t e d s o c i a l c o n v e n t i o n s o r norms t h a t a r e r e -p r e s e n t e d a x i o m a t i c a l l y . A l t h o u g h f i r m s a r e i n d i v i d u a l p r o f i t m a x i m i z e r s , s i n c e t he outcome o f t a c i t c o l l u s i o n i s i n d e t e r m i n a n t , the f i n a l e q u i -l i b r i u m must be mode led u s i n g " s o c i a l l y " g e n e r a t e d common e x p e c t a t i o n s . Once we have c h a r a c t e r i z e d r ^ P j A P . ) a x i o m a t i c a l l y , t h e r e s u l t i n g " c o o p e r a -t i v e " e q u i l i b r i a w i l l : t hen be e x am i ned . The f i r s t ax iom i s a s t r o n g fo rm o f c o n t i n u i t y . When s t a t i n g t h e a x i o m s , t h e s e t ( P j w i l l r e p r e s e n t t h e f e a s i b l e p r i c e , changes.. AP . j , i . e . . P + r ^ P . A P . ) C I R " , V A P . e B . ( P ) . Axiom 1 (Continuity): For a l l P elR-", A P i e B.. (P) and i e N; r j ( P , A P . . ) i s c o n t i n u o u s l y d i f f e r e n t i a b l e i n (P^.AP^.), Vj e N. T h i s a s s ump t i o n i s n a t u r a l i n the sense t h a t one does not w i s h the r e s p o n s e c o n v e n t i o n t o change r a d i c a l l y w i t h s m a l l changes i n t h e p a r a m e t e r s . Note t h a t p r i c e s i n t h i s model a r e p r i m a r i l y signals t h a t i n g ene r a l w i l l be n o t i o n a l r a t h e r than r e a l p r i c e s . Hence one s h o u l d no t e x p e c t phenomena such as i n f l a t i o n t o b i a s t h e r e s p o n s e c o n v e n t i o n . Axiom 2 (Independence of Seale Change): Fo r a l l P_ e I R " , A P . e B^(£ ) , i e N and a > 0 t h en r ! ( a P , o i A P . ) = a r ! ( P , A P . ) , V j e N. j — "i j — i F i n a l l y . , we impose the a s s ump t i o n o f s ymmet ry , i . e . t h e l a b e l s o f f i r m s s h o u l d have no e f f e c t on t h e i r r e s p o n s e s . Note - 89 -t h a t t h i s a s s u m p t i o n d o e s not i m p l y t h a t t h e p r i c e s m u s t be t h e same. R a t h e r , s i n c e a t P^, 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 , f i r m s may be c h a r g i n g d i f f e r e n t p r i c e s , t h i s s y m m e t r y a s s u m p t i o n w i l l h a v e t h e e f f e c t o f m a i n -t a i n i n g s u c h d i f f e r e n c e s . Axiom 3 (Symmetry): F o r a l l £ e K " , AP^ e (_P), d e N, a n d f o r a n y p e r m u t a t i o n , o f N, g i v e n b y 0 ( * ) » w h e r e T Q ( P ) = ( P e ( 1 } , P Q ( 2 ) , . . . , P 0 ( n ) ) a n d A P . e B . ( T 0 ( P ) ) , t h e n r e ( j ) ( P » A P i ) = r j ( T 0 ( P ) ' A P i ) , e N. PROPOSITION 3. If the response convention s a t i s f i e s Axiom 1 to Axiom 3, then i t i s uniquely defined by r ] ( P , A P . ) = A P r T h i s p r e p o s i t i o n s t a t e s t h a t f i r m s w i l l r e s p o n d w i t h equal p r i c e c h a n g e s , a b e h a v i o u r c o m p l e t e l y a n a l o g o u s t o c o n s c i o u s p a r a l l e l i s m . T h i s a n a l y s i s t h u s v i e w s c o n s c i o u s p a r a l l e l i s m n o t a s c o l l u s i v e b e h a v i o u r per se b u t a s a " r u l e o f t h u m b " o r s o c i a l c o n v e n t i o n t h a t s e r v e s t o e n s u r e 12 t h e c o n s i s t e n c y o f e x p e c t e d p r i c e c h a n g e s . I t r e m a i n s t o sh o w how t h i s p r i n c i p l e c a n r e s u l t i n t a c i t c o l l u s i o n . F i r s t l e t u s a d a p t o u r p r e v i o u s n o t a t i o n f o r t h e a b o v e r e s p o n s e c o n v e n t i o n . S u p p o s e now t h a t R(P_,AP-,K) i s d e f i n e d b y - 9 0 -A P i i f j e K a n d R k = A P i f o r k e K / j 0 i f n o t . I f a l l f i r m s e i t h e r f o l l o w a p r i c e i n i t i a t i v e a s a g r o u p o r do n o t , t h e n w i t h 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 a s t h e t h r e a t p o i n t i n d u s t r y p r i c e s w i l l b e g i v e n b y : d e f 0 P.(a) = P + a 1 f o r some a e JR;., I n g e n e r a l o n e c a n a l s o e x p e c t f i r m s , i f r e s t r i c t e d t o P ( a ) , t o p r e f e r an a > 0. T h i s i s g i v e n f o r m a l l y i n t h e f o l l o w i n g a s s u m p t i o n . Assumption A4. ( i ) T h e r e e x i s t s a u n i q u e > 0 s o l v i n g Tr.(P_(a^)) = TT-; ( P . ( a - j ) ) = m a x Tr,(_P(a)) 1 1 a e F 1 d u . ( i i ) - ^ T ( P ( a ) ) > 0 f o r a e [ ° > a 1 ) C o n d i t i o n ( i ) e n s u r e s t h a t e a c h f i r m h a s a u n i q u e m o s t p r e f e r r e d p o i n t o n P ( a ) , a e ]R. C o n d i t i o n ( i i ) s i m p l y r e q u i r e s t h a t TT. ( P _ ( a ) ) b e s t r i c t l y i n c r e a s i n g i n a f o r a e [ 0 , a ^ ) . N e i t h e r c o n d i t i o n p l a c e s v e r y s e v e r e r e s t r i c i t o n s o n t h e p r o f i t f u n c t i o n s s i n c e a s a i n c r e a s e s , d emand f o r a l l p r o d u c t s w i l l i n g e n e r a l b e d e c r e a s i n g . N o t i c e a s w e l l t h a t b y a s -s u m p t i o n A2 d T T i ( P ( a ) ) d a a=0 > 0. F i n a l l y d e f i n e - 91 -a * = mm a . i eN 1 and l e t M = { j e N | a . = a * } . PROPOSITION 4. For all i e N, a e [0 , a * ) if firm i announces a price change AP . e ( 0 , a * - a ] then R(P_(a) ,AP . ,N) is an equilibrium announcement. T h i s e a s y p r o p o s i t i o n s i m p l y p o i n t s o u t t h a t i n c r e a s i n g p r i c e s a l o n g P (a ) i s a lway s an e q u i l i b r i u m p r i c e change i n t h e announcement game. We now show t h a t P (a* ) can be an e q u i l i b r i u m . D e f i n e R ( -P ,AP . ,K ) by R . ( P , A P . , K ) = A P . , j e K 0, j e K PROPOSITION 5. Suppose Va > a * , VK = N , K f N and Vj e H/K;the following condition holds Tt J ( P ( a ) + R ( P _ , A P i , K ) ) < T T J ( P ( a ) + R ( P , A P i , K - u { j } ) ) whenever i e K and AP . e [ P^ -P^ (a ),0). (*) Then _P(a*) is an equilibrium price in the announcement game. C o n d i t i o n (*) r e q u i r e s t h a t s h o u l d some f i r m o r s e t o f f i r m s c u t p r i c e s f rom a p r i c e v e c t o r P ( a ) , t hen i t w i l l a lway s pay f o r the r e m a i n i n g f i r m s t o match such c u t s . The p o i n t i s o f c o u r s e t h a t s h o u l d any f i r m a t -tempt t o o b t a i n h i g h e r p r o f i t s by p r i c e c u t t i n g i t w i l l a lways pay t h e o t h e r f i r m s t o f o i l s u ch i n i t i a t i v e s . W i th t h i s c o n d i t i o n P (a* ) w i l l t hen be an e q u i l i b r i u m p r i c e . N o t i c e t h a t t h e e q u i l i b r i u m i s s i m i l a r t o 13 t he k i n k e d demand c u r v e t h e o r y o f Sweezy . At P (a* ) f i r m s w i l l match - 92 -p r i c e d e c r e a s e s bu t no t i n c r e a s e s ; t hu s the i n d u s t r y demand c u r v e f rom a f i r m ' s p e r s p e c t i v e i s k i n k e d . In c o n t r a s t w i t h Sweezy , t h e announcement game has p r o v i d e d a way t o e x p l i c i t l y model t h e movement t o an e q u i l i -b r i um s t a r t i n g a t t he t h r e a t p o i n t d e f i n e d by t h e C o u r n o t - N a s h e q u i l i -b r i um P^. 5 . The E f f e c t o f E n t r y To c o m p l e t e the d e s c r i p t i o n o f an i n d u s t r y e q u i l i b r i u m one must c o n s i d e r t h e p o s s i b i l i t y o f e n t r y by new f i r m s . T h i s p o i n t has been a ma j o r weakness o f p r e v i o u s mode ls o f c o o p e r a t i v e b e h a v i o u r where i t i s t y p i c a l l y assumed e n t r y i s b l o c k a d e d and hence t he number o f f i r m s i s 14 f i x e d . In p r a c t i c e few ma rke t s a r e c o m p l e t e l y f r e e f rom e n t r y . Fo r e x amp l e , P o r t e r ' s [ 1982 ] a n a l y s i s o f an Ame r i c a n r a i l r o a d c a r t e l i n t h e 19 th c e n t u r y shows t h a t f i r m s o f t e n c o n t i n u e t o c o l -l u d e even i n . t h e f a c e o f e n t r y . On the o t h e r hand , as S c h e r e r [1980] p o i n t s o u t , f i r m s may o f t e n d r o p p r i c e s i n t h e f a c e o f new e n t r y , i n s t i g a t -i n g a p e r i o d o f c u t t h r o a t c o m p e t i t i o n . I t i s c l e a r l y i m p o r t a n t t o u n d e r -s t a n d t h e s e outcomes and t h e i r r e l a t i o n t o c o l l u s i v e b e h a v i o u r . Suppose now a t t h e b e g i n n i n g o f each p e r i o d t new f i r m s a r e a b l e t o e n t e r t he i n d u s t r y . L e t N be the s e t o f a l l p o t e n t i a l f i r m s and l e t c= N deno te t h e s e t o f f i r m s p r o d u c i n g i n p e r i o d "t. F u r t h e r l e t t t TT.J(£>N ) , i e N be t h e f l o w p r o f i t s , w i t h £ r e p r e s e n t i n g -t he v e c t o r o f p r i c e s f o r the i n d u s t r y i n d e x e d by N t . I t w i l l be sup -posed t h a t a l l t he a s s ump t i o n s we have made c o n t i n u e t o h o l d . Now l e t - 93 -fP deno te the Cou r no t -Na sh e q u i l i b r i u m f o r t h e p a y o f f s .ir. ( £ , N t ) , i e ^. By a s s ump t i o n a l l f i r m s can e a r n z e r o p r o f i t s o u t s i d e t he i n d u s t r y , hence ir. = 0 f o r i e N/.N*. The f o l l o w i n g c o n d i t i o n w i l l e x p l i c i t l y mo-de l t h e e n t r y / e x i t c o n d i t i o n . Assumption A5. L e t E > 0 deno te the l e v e l o f sunk c o s t s . I f i n any p e r -i o d , [ t , t + l ) , ^ i ( P t , N t ) < -E t h e n i ft N t + 1 . N o t i c e t h a t we a r e impo s i n g a c o s t o f e x i t i n terms o f t he sunk c o s t E. I f any f i r m ' s p r o f i t i s l e s s t h an sunk c o s t s i t w i l l be supposed t h a t i t goes b an k r up t and must e x i t t h e f o l l o w i n g p e r i o d . A l -t hough t h i s a s s u m p t i o n i s i n some sense n a i v e ( s i n c e i t does not t a k e an i n t e r t e m p o r a l p e r s p e c t i v e ) , as we s h a l l see i t i s no t p a r t i c u l a r l y r e s t r i c t i v e . . F i rms now, i n a d d i t i o n to c h o o s i n g p r i c e , w i l l be f r e e s u b j e c t t o t h e c o n s t r a i n t imposed by A s sump t i o n A5 t o e n t e r o r e x i t the i n d u s t r y . The n o t i o n o f a s t a b l e i n d u s t r y s t r u c t u r e wi11 be d e f i n e d . I t w i l l t h en be shown how t h i s s t r u c t u r e can be s u s t a i n e d as a c o o p e r a t i v e e q u i l i b r i u m u s i n g a s t r a t e g y ana l o gou s t o p r e d a t o r y p r i c i n g . DEFINITION 4. A s e t N* e= N d e f i n e s a stable industry structure i f and o n l y i f © ( i ) T T J C P ,N*) > - E , V i e N* © where P_ i s t h e Cou r no t -Na sh e q u i l i b r i u m . © ( i i ) V j £ N/N* T T . ( P , N* U { j } ) < 0 J where P ^ i s t h e Cou rno t -Na sh e q u i l i b r i u m f o r N* U { j } . - 94 -Obse rve t h a t t h e d e f i n i t i o n i s asymmetric i n the s en se t h a t e n -t r y r e q u i r e s n o n n e g a t i v e p r o f i t s ( c o n d i t i o n ( i i ) ) w h i l e t h e p r e s en ce o f sunk c o s t s r e q u i r e s n e g a t i v e p r o f i t s t o e x i t . Thus ceteris paribus the number o f p o s s i b l e s t a b l e s t r u c t u r e s (and f i r m s ) w i l l i n c r e a s e as E i n c r e a s e s . I f E = 0, we have c o s t l e s s e x i t , w h i c h c o r r e s p o n d s t o t h e c o n t e s t a b i l i t y = n o t i o n o f Baumo l , P an za r and W i l l i g [ 1 9 8 2 ] . I t w i l l now be shown t h a t t h i s s t r u c t u r e i s s u s t a i n a b l e i n the r e p e a t e d game o f s e c t i o n 2. L e t us suppose t h a t the f i r m s i n t he i n -d u s t r y i n p e r i o d t a r e known before p r i c e s a r e s e t f o r t h i s p e r i o d . T h i s i s q u i t e r e a s o n a b l e s i n c e a l l t h a t i t r e q u i r e s i s f o r i n cumben t s t o o b -s e r v e the new f i r m s e t t i n g up i t s p l a n t and equ ipment b e f o r e a c t u a l p r o -d u c t i o n b e g i n s . A t t ime t t he i n f o r m a t i o n s e t w i l l t hu s be lt = {-^T|° - T - t - 1 } u { ^ T | Q - T " - t } Now suppose a t t=0 t h a t N^=N* i s a s t a b l e i n d u s t r y s t r u c t u r e . L e t P^, as b e f o r e , r e p r e s e n t a sequence o f p r i c e s r e s u l t i n g from the a n -* nouncement game. C o n s i d e r the s t r a t e g y f o r f i r m i , a., ' d e f i n e d u s i n g t h e f o l l o w i n g d e c i s i o n s a t t ime t . / i f { P t _ 1 = P t _ 1 ( i ) I f i e N*, S * ( I t ) = < and N t = N*} o r { N t _ 1 f N* and N^ = N*} PT, i f no t i ( i i ) I f i e N/N*, do no t e n t e r . - 95 -The s t r a t e g y v e c t o r a* r e q u i r e s t h a t f i r m s o u t s i d e N* do no t e n t e r w h i l e f i r m s i n N* d e t e r bo th p r i c e c u t s and e n t r y by a t h r e a t t o go t o t h e Cou rno t -Na sh e q u i l i b r i u m . PROPOSITION 6. Suppose the sequence {P_ )^._Q s a t i s f i e s the conditions of proposition 2, then a* defines a SPE. T h i s p r o p o s i t i o n has s e v e r a l i m p l i c a t i o n s . F i r s t l y i f E=0 t hen one can say t h a t t he marke t i s c o n t e s t a b l e i n t h a t f i r m s a r e f r e e to e n t e r and e x i t . However , t h e f a c t t h a t p r i c e s can a d j u s t i n the f a c e o f e n t r y means t h a t t he marke t i s not p e r f e c t l y c o n t e s t a b l e and t h e Cou r no t -Na sh e q u i l i b r i u m d e f i n e s t h e e n t r y e q u i l i b r i u m . ^ F u r t h e r despite the p o s s i b i l i t y o f f r e e e n t r y and e x i t , t he t h r e a t o f a move t o p r o v i d e s an incentive . fo r c o l l u s i v e b e h a v i o u r . T h i s t h r e a t " does no t c o r r e s p o n d t o . " c l a s s i -c a l " p r e d a t o r y p r i c i n g i n t h e sense o f i r r a t i o n a l p r i c i n g be low c o s t i n t h e f a c e o f e n t r y . As S c h e r e r . . [ l 9 8 0 ] p o i n t s o u t , t he l a t t e r f o rm o f p r i c i n g has l i t t l e e m p i r i c a l s u p p o r t , w h i c h i s no t t he case f o r t h e t y p e o f p r i c e c u t s d e f i n e d by a*. Mov ing to t he Cou rno t -Na sh e q u i l i b r i u m when t h e r e i s e n t r y i s c o m p l e t e l y " r a t i o n a l " a n d s e r v e s t o d e t e r e n t r y t h a t may a t -tempt t o r eap , some o f t he b e n e f i t s e n j o y e d by an i n d u s t r y p r a c t i c i n g t a c i t c o l l u s i o n . P r o b a b l y t h e most i n t e r e s t i n g r e s u l t i s the e f f e c t o f sunk c o s t s . Baumo l , P an z a r and W i l l i g [ 1982 ] a rgue t h a t sunk c o s t s a c t as a b a r r i e r to entry and hence g e n e r a t e r e n t s f o r t h e i n d u s t r y members. We i n f a c t have t he opposite r e s u l t . Namely as E i n c r e a s e s t h e r e can be a larger number o f f i r m s i n a s t a b l e i n d u s t r y s t r u c t u r e , w h i c h c a n o n l y - 96 -imply lower p r o f i t s per firm. The e x p l i c i t welfare e f f e c t s are ambi-guous since the excess capacity generated by a larger number of firms may be o f f s e t by the i m p l i c i t increase in product variety. 6. Concluding Comments In this paper a simple dynamic oligopoly model has been pre-sented to demonstrate that the mainly i n t u i t i v e notions of conscious p a r a l l e l i s m and predatory p r i c i n g are consistent with r a t i o n a l behaviour on the part of firms. However, our main point i s not to argue that con-scious p a r a l l e l i s m is necessary f o r c o l l u s i v e behaviour but rather the announcement game is meant to demonstrate the importance of consistent expectations in any t a c i t agreement. In many respects t h i s work can be seen as a formalization of F e l l n e r [I960] 1s seminal contributions to the problem. Certai n l y our model also demonstrates, in contrast with Bain [1956], that barriers to entry are i n no way a necessary condition for the attainment of non-competitive p r o f i t s . At the same time the model does not imply a lack of competition in concentrated i n d u s t r i e s . The cooperative equilibrium, P(a*) ultimately depends on the Cournot-Nash e q u i l i b r i u m ^ . Hence such a model suggests that there e x i s t incentives for firms to use some of t h e i r p r o f i t s for non-price competition such as advertising and R&D. Such expenditures can lead to larger market shares at the Cournot-Nash equilibrium and thus ultimately to higher p r o f i t s at the cooperative equilibrium. - 97 -NOTES 1. S e e S c h e r e r [ 1 9 8 0 ] f o r a d e t a i l e d s u r v e y o f t h e s e i s s u e s . 2. S h u b i k [ 1 9 5 9 ] , p a g e 2 8 5 - 8 6 . 3. T h e r e i s now a l a r g e l i t e r a t u r e o n t h i s p r o b l e m . S e e f o r e x a m p l e , F r i e d m a n [ 1 9 7 7 ] , G r e e n [ 1 9 8 0 ] , G r e e n a n d P o r t e r [ 1 9 8 1 ] , a n d S t i g l e r [ 1 9 6 4 ] . 4. S e v e r a l o f t h e s e p o i n t s h a v e b e e n made i n p u b l i s h e d c r i t i c i s m s o f t h e c o n t e s t a b i 1 i t y n o t i o n , t h o u g h s o m e w h a t l e s s f o r m a l l y t h a n we h a v e h e r e . S e e f o r e x a m p l e , K n i e p s a n d V o g e l s a n g [ 1 9 8 2 ] , a n d S c h w a r t z a n d R e y n o l d s [ 1 9 8 3 ] . 5. T h e e f f e c t o f l a r g e y o r e q u i v a l e n t l y l a r g e u n c e r t a i n t y i n p r i c e o b s e r v a t i o n s i s a w e l l s t u d i e d p h e n o m e n a , f o r e x a m p l e S t i g l e r [ 1 9 6 4 ] , F r i e d m a n [ 1 9 7 7 ] , G r e e n a n d P o r t e r [ 1 9 8 2 ] , a n d G r e e n [ 1 9 8 0 ] . 6. T h o u g h S e l t e n [ 1 9 7 5 ] o r i g i n a t e d t h e n o t i o n o f s u b g a m e p e r f e c t n e s s , t h e d e f i n i t i o n we u s e h e r e f o l l o w s m o r e c l o s e l y G r e e n [ 1 9 8 0 ] . 7. S e e C h a p t e r 1 f o r a d i s c u s s i o n o f c o n t i n u i t y a n d b o u n d e d n e s s o f t h e p r o f i t f u n c t i o n . 8. A s s h o w n i n C h a p t e r 1 t h i s i s n o t a v e r y g e n e r a l a s s u m p t i o n . L e t u s s u p p o s e t h a t t h e u n d e r l y i n g m o d e l i s l i k e d e P a l m a e t . a l . [ 1 9 8 3 ] w h e r e a l l t h e s e a s s u m p t i o n s a r e s a t i s f i e d . 9. S e e N a s h [ 1 9 5 0 ] a n d N a s h [ 1 9 5 3 ] f o r t h e s e m i n a l c o n t r i b u t i o n s on t h i s p r o b l e m . R o t h [ 1 9 7 9 ] p r o v i d e s an e x c e l l e n t s u r v e y o f t h e l i t e r a t u r e . R e c e n t l y O s b o r n e a n d P i t c h i k [ 1 9 8 3 ] c a r r y o u t an e x p l i c i t a p p l i c a t i o n o f b a r g a i n i n g t h e o r y t o t h e d u o p o l y p r o b l e m . - 98 -1 0 . N o t e t h a t we a r e a l l o w i n g f o r m o r e g e n e r a l s t r a t e g i e s t h a n s i m p l y all f i r m s d o o r do n o t f o l l o w . T h i s i s i m p o r t a n t i f we w i s h t o a d d r e s s t h e i s s u e o f w h e t h e r c o n s c i o u s p a r a l l e l i s m i s an i n d u s t r y -w i d e p h e n o m e n a . 11. A l s o s e e C y e r t a n d De G r o o t [ 1 9 7 0 ] a n d [ 1 9 7 1 ] f o r a s i m i l a r a p p r o a c h . T h e y d i f f e r b y a s s u m i n g t h a t f i r m s c h a r g e t h e same p r i c e . T h e p r o b l e m t h e n i s how o n e l e a r n s w h e r e P* i s l o c a t e d . I n o u r a n a l y s i s , a s l o n g a s t h e a n n o u n c e m e n t game r e s u l t s i n P a r e t o i m p r o v e m e n t s f o r a l l f i r m s , t h e p r o c e s s w i l l q u i c k l y c o n v e r g e t o P*. 1 2 . S e e S t a n b u r y a n d R e s c h e n t h a l e r [ 1 9 7 7 ] f o r a n i c e d i s c u s s i o n o f c o n -s c i o u s p a r a l l e l i s m a n d s o c i o l o g i c a l a p p r o a c h e s t o c o l l u s i o n . 1 3 . S e e S w e e z y [ 1 9 3 9 ] . I t s h o u l d be n o t e d t h a t t h e t h e o r y we p r e s e n t h e r e a d d r e s s e s m o s t o f t h e c r i t i c i s m s a g a i n s t t h e t h e o r y r a i s e d f o r e x a m p l e i n S c h e r e r [ 1 9 8 0 ] . 1 4 . F o r e x a m p l e K u r z [ 1 9 8 2 ] e x p l i c i t l y j u s t i f i e s h i s m o d e l b a s e d o n t h e a s s u m p t i o n t h a t t h e m a r k e t i s not c o n t e s t a b l e . 1 5 . T h u s i n c o n t r a s t t o B a u m o l , P a n z a r a n d W i l l i g [ 1 9 8 1 ] t h e m a r k e t i s n o t perfectly c o n t e s t a b l e . H o w e v e r t h i s a s s u m p t i o n i s m o r e c o n s i s -t e n t w i t h t h e g e n e r a l p e r c e p t i o n t h a t p r i c e s a d j u s t m o r e q u i c k l y t h a t f i r m s ' s e t u p s p e e d s . 1 6 . I t s h o u l d be n o t e d t h a t t h i s e n t r y e q u i l i b r i u m i s a n a l o g o u s t o C h a m b e r l i n ' s l a r g e g r o u p e q u i l i b r i u m c o n c e p t . - 99 -REFERENCES A r c h i b a l d , G. C , B. C. Ea ton and R. G. L i p s e y [ 1 9 8 3 ] , " A d d r e s s Mode l s o f V a l u e T h e o r y " , i n New Developments in the Analysis of Market Structure, F. Mathewson and J . S t i g l i t z ( e d s . ) , f o r t h c o m i n g . Aumann, R. J . [ 1 9 6 4 ] , " M a r k e t s w i t h a Con t i nuum o f T r a d e r s " , Econome-t r i c a , 32, 3 5 - 5 0 . Aumann, R. J . 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[ 1 9 8 2 ] , "Some Notes on t h e Core o f a P r o d u c t i o n Economy W i t h Some L a r g e T r a d e r s and a Con t i nuum o f Sma l l T r a d e r s " , Journal of Mathematical Economics, 9, 9 9 - 1 0 5 . - 103 -S h u b i k , L. [ 1 9 5 9 ] , Strategy and Market Structure, W i l e y , New Yo r k . S pen ce , M. [ 1 9 7 8 a ] , " T a c i t C o o r d i n a t i o n and I m p e r f e c t I n f o r m a t i o n " , The Canadian Journal of Economics, 11, 5 27 - 533 . Spen ce , M. [ 1 9 7 8 b ] , " E f f i c i e n t C o l l u s i o n and R e a c t i o n F u n c t i o n s " , The Canadian Journal of Economics, 12, 4 9 - 7 0 . S t a n b u r y , W. T . and G. B. R e s c h e n t h a l e r [ 1 9 7 7 ] , " O l i g o p o l y and C o n s c i o u s P a r a l l e l i s m : T h e o r y , P o l i c y and t h e Canad i an C a s e s " , Osgood Ball Law Journal, IS, 6 17 - 700 . S t i g l e r , G. [ 1 9 6 4 ] , "A Theo r y o f O l i g o p o l y " , Journal of P o l i t i c a l Eco-nomy, 72, 4 4 - 6 1 . Sweezy , P. M. [ 1 9 3 9 ] , "Demand Under C o n d i t i o n s o f O l i g o p o l y " , Journal of P o l i t i c a l Economy, 47, 5 6 8 - 7 3 . T u l k e n s , H. and S. Z am i r [ 1 9 7 9 ] , " S u r p l u s - s h a r i n g L o c a l Games i n Dynamic Exchange P r o c e s s e s " , Review of Economic Studies, 46, 3 05 - 314 . V a r i a n , H. [ 1 9 8 0 ] , "A Model o f S a l e s " , American Economic Review, 70, 6 5 1 -- 104 -APPENDIX FOR CHAPTER 1 Proof of Proposition 1. S u f f i c i e n c y : Suppose t h e H o t e l l i n g a s s u m p t i o n h o l d s . G i v en ( X , P ) e U x [ 0 , p ] , choose any h e l R 2 m + 2 such t h a t ( X , P ) + h e U 2 x [ 0 , p + ] 2 . L e t Z = ( X , P ) and Z(e) = Z + eh. D e f i n e : u (e) = ( u ^ z ) rti u 2 ( z ( e ) ) } u i u ^ i U ) ) n u 2 ( z ) } T h i s i s i n some s en se t he d i s p u t e d marke t boundary between f i r m s 1 and 2 under t he p e r t u r b a t i o n h. The i n d i c a t o r f u n c t i o n f o r U(e) i s g i v e n by : I £ ( x ) 1 , i f x e U(e) 0 , i f x t U(e). For f i r m 1 we can w r i t e down t he f o l l o w i n g i n e q u a l i t i e s , where p^(x) i s the component _Z(e) c o r r e s p o n d i n g t o f i r m 1 ' s p r i c e . I d j CZ ) - d ^ Z j e ) ) ! = U x e U ( z ) D( P l (x)) v ( x ) dx " ' x e U ^ Z U ) ) D ( P l ( x ) ) v ( x ) dx | < ^ x e U ^ n u ^ e ) ) D ( P l ( x ) ) ~ D ( P l ( x ^ v ( x > d x l + l J xeU(c) S u p { D ( p 1 ( x ) ) ' D ( P l ( x ) ) } v (x) dx| K- ' x e U ^ Z ^ t f U ) ) i D ( P l ( x> > " ° ( P l ( x ) ) l v < x ) d x + / x e U I £ ( x ) sup {D(p 1(x)) > D(pJ(x))} v ( x ) dx . - 105 -S i n c e t ( - , - ) and D ( - ) a r e c o n t i n u o u s , l i m | D ( p , ( x ) ) - D ( p f ( x ) ) | = 0 e-K) A l s o l i m I e ( x ) = I ° ( x ) where I ( x ) = 1 , i f x e u x ( z ) n u 2 ( z ) 0 . , i f x ^ U j t Z ) n u 2 ( z ) Hence, t a k i n g l i m i t s as e->0 we ge t (by t h e Lebesgue dom ina ted con-ve rgence t heo r em) : l i m I d^Z) - d1(l(e))\ « 0 + J " ^ I ° ( x ) D ( P l ( x ) ) v(x) d x . But by the H o t e l l i n g a s s ump t i o n the t e rm on t h e r i g h t i s z e r o , hence l i m | d , ( Z * - c M Z ( e ) ) | = 0, £+0 and t h e demand c u r v e s a r e c o n t i n u o u s . N e c e s s i t y : Suppose t he demand c u r v e s a r e c o n t i n u o u s , bu t t h e H o t e l l i n g a s -sump t i on does no t h o l d a t (_X,J__). L e t 3 = / X £ U 1 ( X , P ) ^ U 2 ( X , P ) D ( P l ( x ) ) v ( x ) d x > °" Then by a s s u m p t i o n T2 f o r any £ > 0 we w i l l have ( X , ( p ^ + e , p 2 )) n UgCX.P.) = 0. Thus a l o n g w i t h a s s u m p t i o n D3 we have - 106 -d ^ X . ^ + e . p g ) ) < d 1 (X ,P) + 3 and thus d^{XtP) i s not continuous at (_X,P). Proof of Proposition 2. The proof w i l l proceed by c o n t r a d i c t i o n . Take X j e U. I f no 0 k k w e x i s t s , then fo r a sequence w > 0 such that w -> 0, as k -* °° there ~k ~k k w i l l always e x i s t an x 2 such that '•t ' (x 1 ,x 2 ) < w and a NE(ME) e x i s t s for X = ( X j . x * ) . Let S k - ( s ^ , s 2 ) denote the NE(ME). Now def ine a subse-~k ~k quence of {X ,S_ ) as fo l lows, ( i ) X 1 = X 1 ( i i ) Define _X £ r e c u r s i v e l y from X ^ - 1 . For some k, X_ J l r -•= _Xk k1 then choose X_. , k1 > k such that k1 k 1 t ( x 1 , x 2 ) < min { t ( x 1 , x 2 ) , j} . Let X = X . Thus t ( x ^ , x 2 ) < -j for a l l I > 1. Denote the corresponding NE(ME) by S £ = {s[,s\). Si We w i l l now derive some o f the propert ies of S for large I. In p a r t i c u l a r , our arguments w i l l be constructed so as to apply to the case o f ME as well as NE. It w i l l be shown that fo r some large % one gets a cont rad ic t ion and our o r i g i n a l assumption that no w^  ex is ted must be f a l s e . That w i l l complete the proof . We now proceed with a ser ies of lemmas. When we use the word - . 1 0 7 -" e q u i l i b r i u m " , t h e r e s u l t a p p l i e s t o bo th t h e c a s e o f a NE and ME. LEMMA 1 In equilibrium neither rationing nor excess production will occur. That is 4 = d . ( X A , P A ) , V I > 0 , i• = 1 , 2 . Proof: (a ) Suppose q^ < d . ( X A , P A ) . P r o f i t i s t h e n : n , ( X £ , S £ ) = p^q^ - C ( q * ) . S i n c e demand i s c o n t i n u o u s , t h e r e e x i s t s , a p^ > p^ such t h a t d.i^Ap^pp) > q* Thus a t t h i s p r i c e f i r m i ' s p r o f i t i s , p^q| - C(q^) > I I . ( X , S A ) . Hence S^  was no t an e q u i l i b r i u m , a c o n t r a d i c t i o n . Thus no r a t i o n i n g oc -c u r s a t e q u i l i b r i u m and hence (b) Now suppose q^-> d 1 . ( X A , P £ ) . From (a ) q j > d . ( X A , P A ) , so d ^ X ^ . P 3 1 ) = D . ( X A , S A ) . Then , by a s s u m p t i o n C2 , f i r m i can o f f e r l owe r p r o d u c t i o n t o l o w e r c o s t s w i t h o u t d e c r e a s i n g r e v enue . Hence S a g a i n was no t an e q u i l i b r i u m and a c o n t r a d i c t i o n r e s u l t s . LEMMA 2 The sequence S has a convergent subsequence. Let this sequence be S n with limit S*. . - 108 -£ Proof: We need o n l y show t h a t S_ i s bounded . P r i c e s a r e t a k en f rom [ 0 , p + ] and hence bounded . By lemma 1 q{ = d . ( x \ p £ ) and s i n c e demand i s bounded, t h e o u t p u t s must a l s o be bounded . Hence £ S_ i s bounded above and be low, f rom w h i c h one c o n c l u d e s t h e r e e x i s t s a c o n v e r g en t s ub s equen ce . LEMMA 3 The outputs of both firms are always positive, i.:e. q " > 0, i = 1, 2, V n > 0. Proof: Suppose q " = 0. Then f i r m 2 i s a c t i n g as a m o n o p o l i s t and w i l l c ha rge a p r i c e pn{x^) > C ' (0) + a ( b y - a s s u m p t i o n M).. But t hen f i r m 1 c o u l d c h a r g e a p r i c e p" e ( C ( 0 ) , p n ( x 2 ) ) f a c e p o s i t i v e demand and pro-duce , o u t p u t q " such t h a t n n p ^ ' - C(q^) > -C(0). Hence q? = 0 i s i m p o s s i b l e . We w i l l now s t u d y some o f t h e p r o p e r t i e s o f t h e l i m i t S*. S i n c e t h e r e i s no r a t i o n i n g , JI. (X ,S ) = p . q . - C ( q . ) = IT, . 1 1 1 1 d e f 1 N o t i c e t h a t t h e p r o f i t s have a l i m i t as w e l l , n * * , *\ * l i m n" = p q - C(q ) = n n ^ 1 1 1 1 d e f 1 - 109 -LEMMA 4 In the limit,'prices are the same, i.e. * * P i = Po = P * . 1 ^def Proof: S i n c e q1? > 0 , i = 1, 2 and t h e r e i s no r a t i o n i n g , we must have P i - p 2 | < t ( x 1 , x 2 ) < w p , Hence -i • I n n i „ l i m |p, - p J = 0 . N o t i c e t h a t t h e s e r e s u l t s show t h a t i f a NE e x i s t s , i t a l s o s a t i s f i e s t h e p r o p e r t i e s r e q u i r e d f o r a ME. Fo r t he r e s t o f t h e p r o o f m i l l p r i c e u n d e r c u t t i n g w i l l neve r be used and hence t h e / r e s u l t s w i l l a p p l y t o bo th c a s e s . LEMMA 5 * In the limit both outputs are positive, q . > 0 , i = 1, 2 . Proof: Due t o a s s ump t i o n M and t h e f a c t t h a t t h e r e i s no r a t i o n i n g , we must have q x + q 2 t o . Suppose q 2 = O.and q 1 V 0 . By a s sump t i on M, i t e a s i l y f o l -_ . _ * l ows t h a t we must have p* > C ' ( 0 ) . Choose q 2 > 0 such t h a t q 2 e (0,%q^) and 110 n 2 = P *q 2 - c(q2) >-C(0) = n*. de f _ * T h i s i s p o s s i b l e s i n c e p* > C'(0) and C ' ( q ) i s c o n t i n u o u s . L e t Y = q ^ / q ^ . Fo r s u f f i c i e n t l y l a r g e n we can choose a sequence c -+ 0 such . that p —n p. - e s a t i s f i e s r l n d 2 ( X n s ( P ; , p n 2 ) ) = Yd^xV") = Y q J . T h i s i s due t o t h e c o n t i n u i t y o f demand and t h e f a c t t h a t f o r l a r g e n, p 2 can be chosen so t h a t d 2 ( X n , P _ n ) t a k e s on v a l u e s i n ( 0 , q " - e ) f o r sma l l e . Hence t h e c o n s t r u c t i o n o f pj0, f o l l o w s f rom t he i n t e r m e d i a t e v a l u e t h e o r e m . Note l i m Y p " q j - C ( Y q J ) = f j > JT* n-*» Hence _5n c anno t be a NE(ME) f o r some l a r g e n, a c o n t r a d i c t i o n . T h e r e f o r e * * q 2 > 0, and s i m i l a r l y q^ > 0. We now show t h a t t h e sequence o f e q u i l i b r i u m p r i c e s niust app roach MC. LEMMA 6 * dC * P i = Hq ( q i ) j 1 = U 2 -* * * Proof: Suppose p^  > G ' ( q ^ ) . Then t h e r e e x i s t s a 6 e ( 0 , q 2 ) such t h a t n i = p i q i " C ( q i ) < P i ^ V 6 ) - c ( q i + < $ ) = n i -d e f L e t 3 = 6 / q 2 - • By t h e c o n t i n u i t y o f -demand and t h e i n t e r m e d i a t e - 111 -v a l u e theorem, t h e r e e x i s t s a sequence en and > 0 such t h a t • /vn / n n>, n , „ n ~n . ^ d ^ X .(pj-e ,p2)) = q x + 6q = q f o r n > H y d e f S i nee e -»- 0 as n -* °°, n ' l i m ( ( p j - e n ) q n - C(q" n ) ) = H > H* Hence 3 n such t h a t / n x~n r / ~ n x ^ n n ( p 1 - £ n ) q - C(q ) > n . T h e r e f o r e S^  was not an e q u i l i b r i u m , , a c o n t r a d i c t i o n . S i m i l a r l y f o r p^, q 2 N o t i c e t h a t due t o t h e c o n t i n u i t y o f demand we ge t p r i c e = MC i n . the l i m i t even f o r ME : / Thus m e r e l y t h e a s sump t i o n o f e x i s t e n c e .o f an e q u i l i -b r i um wi11 y i e l d t h e c o m p e t i t i v e r e s u l t as d i f f e r e n t i a t i o n - b e c o m e s s m a l l . We now show t h a t f o r ou r model t h i s i s i m p o s s i b l e . N o t i c e . - i f one supposes i n Lemma 3 t h a t p.. = p + i f q . = 0 as a m a t t e r o f c o n v e n t i o n , t hen we have not used u n t i l now any p r o p e r t i e s o f the r a t i o n i n g scheme o r t he f a c t p r o d u c t i o n t a k e s p l a c e ex ante." LEMMA 7 * dC *\ It i s not possible in our model to have p^  = ( q ^ ) , i = 1, 2. Proof: We have two c a se s * (1) I f p* < A V C ( q i ) f o r e i t h e r i = 1, 2 where = C (q ) - C (0 ) AVC(q) d e f q - .112 -t hen t h e r e must be an n such t h a t p? < AVC(q"?), i n wh i c h c a s e q? = 0 , a c o n t r a d i c t i o n o f Lemma 3. ( 2 ) p. > A V C ( q . ) , i = 1 , 2 . From d e f i n i t i o n o f TJ on p. 14 and t h e assump-t i o n on demand we have 9p D e f i n e 9 D * / * n^p) = P D(x 1 5p) * q i * - clrKxpp) * q : + q 2 \ q 2 + q . l l L i a [ p * _ c i ( q * ) ] 9 P ( X - P * ) ^ + q * > 0 dp L P • ^ q l ; j 3p n * + „ * q l S i n c e U(x l sp*) =' qx + q 2 , Sjtp*) = n*. F u r t h e r , p* = C (q*) t hu s Si.(p*) - C p * - C ! ( q i ) ] ^ ^ q l + q 2 Hence t h e r e e x i s t s a p-^  > p* such t h a t n ^ p j ) > n^. Now choose q n = D x ( X . n , ( s J , s 2 ) ) where s" = ( p ^ q ^ ) . As p 2 -> p* t h e n f o r some N 2 > 0 we must have P j - P 2 > w > t ( X j , x 2 ) f o r n > N,,. For t h e s e s t r a t e g i e s f i r m 2 i s u n d e r c u t t i n g f i r m 1. Note t h a t s i n c e i t i s f i r m 1 who i s d e v i a t i n g , t h e c o n d i t i o n f o r ME a r e s t i l l s a t i s -f i e d . Thus -113 -d 2 ( x n , ( s j , s 5 ) ) * TJ (x^p^) T h e r e f o r e f i r m 1 f a c e s o n l y r a t i o n e d demand o f t h e f o rm Tr/ n n \ n ~n n / Y n ,=n c n A P ( x 2 * p 2 ) " q2 ^ ~ * q = D l f X , ( S l , S )j - • n n , • D ( x l i P l ) By demand c o n t i n u i t y , we.must have * ~n - q l Tim q1 = D ( x , p * ) • * Thus l i m n ^ x ^ s " s")) = n ^p j ) >n n-x» Hence f o r some l a r g e n, S_n i s no t an e q u i l i b r i u m , NE o r ME Thus p.j = C 1 (q^) i s i m p o s s i b l e . These lemmas i m p l y , - b y c o n t r a d i c t i o n , t h a t t h e r e - m u s t e x i s t a WQ > 0 such t h a t no-NE o r ME " e x i s t s when t ( , ) <_ WQ. - 114 -APPENDIX FOR CHAPTER 3 Proof of Proposition 1. L e t x s o l v e max I V i T . ( q ) x . Such a x e x i s t s x eX (q ) i eM 1 s i n c e X (q ) i s compac t and the o b j e c t i v e f u n c t i o n i s c o n t i n u o u s . C l e a r l y x i s undom ina t ed ; h e n c e x -e K ( q , M ) . Proof of Proposition 2. F o l l o w s i m m e d i a t e l y f rom t h e d e f i n i t i o n o f K(q>M) g i v e n by t h e e x p r e s s i o n i n e q u a t i o n 1. Proof of Proposition 3. S t r a i g h t f o r w a r d a p p l i c a t i o n o f t he d e f i n i t i o n . Proof of Proposition 4. I f 0 £ K ( q ) , t hen "0 e K ( q , { i } ) , V i e N. H e n c e 3-rr, by P r o p o s i t i o n 3 we must have ~ - (q) = 0 V i e N. L e t x = "(.-.a^ (q.y),'... , - a n - ( ' q n ) ) , t hen by a s s u m p t i o n s ( i ) and ( i i ) V i T ^ q J x > 0 V i £ N- H e n c e : TJ £ K ( q , N ) . T h e r e f o r e 0 e K (q ) i s i m p o s s i b l e . Proof of Proposition 5. S i m p l y no te t h a t x e K (q , { i J.) i f and o n l y i f 3 7 F i (q> 3 7 T - ( q ) — ^ x . = max — s — - y . 3 q i 1 y c [ - a i ( q 1 ) , b i ( . q n ) ] 8 q i Proof of Proposition 6. D i r e c t a p p l i c a t i o n o f P r o p o s i t i o n 5 . Proof of Proposition 7. From P r o p o s i t i o n 5 K (q ) a lway s e x i s t s . I f t h e r e e x i s t s an x e K (q ) n K ( q , N ) , t h en x e B (q ) and we a r e done . I f n o t , t hen i f x e K(q) i t i s , by t h e d e f i n i t i o n o f K ( q , N ) , dom ina t ed by some y e K ( q , N ) . Tha t i s VTT. ( q ) ( y - x ) > 0 V i £ N. Hence y £ B (q ) and we a re done . Proof of Proposition 8. Suppose ( i ) and ( i i ) a r e s a t i s f i e d . S i n c e C + ( q , N ) = <f>, t h e n 0 £ K ( q , N ) . We have by t h e a s s ump t i o n s on TT. ( q ) , - 115 -3TT1 3TT1 T ^ J - (q) < 0 f o r i f j . Thus j^- (q) > 0 be cause V T K ( q ) - x = 0. Hence by P r o p o s i t i o n 5 we have x e K ( q ) . S i n c e V " n \ ( q ) - x = 0 V i e N, t hen w i t h 0 e K ( q , N ) , we have 0 e B(q). C o n v e r s e l y , l e t 0 e B (q ) wh i ch i m p l i e s 0 e K ( q , N ) . As a c o n s e -quence C + ( q , N ) = <b f o r o t h e r w i s e any x e X ( q ) .f\ C + ( q , N ) wou ld dom ina te 0 and we wou ld have TJ f. K ( q , N ) . n We now w i s h t o show t h a t t h e r e e x i s t A . > 0, £ A . > 0 s u ch t h a t 1 " i = l 1 Z A . V T T . ( q ) = 0. Suppose . n o t , t h en we must have i eN 1 1 •0 t C^(q) = { x | x = Z A . V T r . ( q ) , A . > 0, Z A . > £ } , f o r £ > 0. i eN 1 i eN 1 ' Note t h a t C^(q) i s a c l o s e d convex s e t t h a t i s s t r i c t l y s e p a r a t e d f r om 0. Hence by an e x t e n s i o n o f t h e s e p a r a t i n g h y p e r p l a n e theorem t h e r e e x i s t s a v e c t o r p such t h a t p-x > 0 f o r a l l x e C^('q). Bu t .. e C^(q) V i e N o r V T T . - p > 0 V i e N. Thus p e C + ( q , N ) . Hence C + ( q , N ) f cj> wh i c h i s a c o n t r a d i c t i o n . T h e r e f o r e t h e r e e x i s t A . > 0 such t h a t Z A . V n - . ( q ) = 0". ~ ~ i e N S i n c e ^ — < 0 i t must be t h e c a s e t h a t A . > 0 and T C — - > 0. By P r o p o s i t i o n a q . r dq . ~ def^ 1 5 i t w i l l f o l l o w t h a t K (q) = ( b ^ q ) , . . . ,b (q)0 = xv. S i n c e 0 e B ( q ) , V i T . ( q ) x < 0 f o r a l l i e N. Suppose VTT. ( q )x < 0. S i n c e Z ' A . V - r r . ( q ) x = 0 1 1 j eN J J and A . > 0, t h e r e must be a k such t h a t VTT. ( q ) x > 0, wh i c h i s no t p o s s i b l e . J K Hence V i r ^CqJx = 0 V i e N. Proof of Proposition 9. N o t i c e t h a t i f x e K (q ,M) t h e n so i s any y e X (q ) such t h a t y. = x. f o r i e M. Thus s i n c e K(q ,M) f <j> f o r a l l M = N and 116 s i n c e 3 i s a partition o f N, i t t h en f o l l o w s t h a t n K(q ,M) $ c|>. MeS Proof of Proposition 10. F i r s t we d e f i n e a ne i ghbou rhood r i j o f q such t h a t x ( q j = ( b j ( q ' p , . . . i b r ) ( q ' n ) ) e K (q) f o r q~ e n . T h i s i s p o s s i b l e s i n c e 3TT by P r o p o s i t i o n 6 we know - r — (q) > 0 , V i £ N and hence we can choose m dq. i dir. 1 such t h a t — (q) > 0 , Vq c r i j» i £ N. By P r o p o s i t i o n 2 we w i l l have K(q) = { ( b 1 ( q 1 ) , . . . , b - ( q n ) } f o r q e ^ . We now c l a i m we can f i n d a n o t h e r n e i ghbou rhood such t h a t x ( q j e K(q~,M)' f o r any H 9 N, M ^ N. By P r o p o s i t i o n 8 we have V i T . ( q ) . x ( q ) = 0 V i £ N. S i n c e we a l s o have TT—- (q) < 0 V i f j ; i , j e N t h en we can f i n d a n e i ghbou rhood o f q such VM 5= N, M f N and i £ M we have 3 T T , ( q ) £ a - • b , ( q . ) > 0 , Vq £ n ? . J£M 3 q j 3 J " 2 We c l a i m t h a t , f o r q £ n 1 n n 2 > x ( q j = ( b 1 ( q 1 ) J . . . ,b (q^)) £ K ( q , M ) , H e N, M / I . I f t h i s were no t t h e c a s e , t h en t h e r e wou ld e x i s t an x £ F ( q , x , M ) , x f x ( q j , s u ch t h a t ( x - x ( q j ) e C + ( q , M ) . Choose i £ N such t h a t x- - b . ( q . ) x . - b . ( q \ ) 1 J 1 < - J V j £ N. W " b.(q j) x - b ( q . ) S i n c e x £ F ( q , x , M ) t h en we must have i £ M and — * — - = 0 f o r a l l X i " b i ( q i > j $ M. L e t a = — ' — - < 0 . - 117 -Now 3TT . VTT (q ) • ( x - x ( q ) ) = E -^- l (q) ( x . - b ( q . ) ) 1 j eN 8 q j J J J 3T T X - b . ( q ) _ z _ L _ J L i b . ( q . ) . jeM d q j b ( q . ) J J J J 3TT. But by a s s u m p t i o n T — - (q) < 0 f o r i f j ; hence 3 q . _ „ 3TT. 3TT. VTT (q ) • (x - x ( q ) ) < E A _ 1 b. = a E —1 b . < 0. 1 " jeM 9 q j J j eM 3 q j J " + ~ d e f Hence ( x - x ( q ) ) f. C (q ,M) and thus x E K (q ,M) f o r q e n_ = L T h e r e f o r e x ( q ) £ B ( q,3) f o r e v e r y .3 t {N} 'and q~£ n. Hence B (q ) <=• B (q ) f o r q £ n. Bu t B ( q ) = P, B ( q , 3 ) , so we a l s o have Beg(N) B ( q ) ^ B(q") f r om wh i ch we c o n c l u d e B ( q ) = B(q^) f o r q^  £ n • Proof of Proposition 11. Note ' t h a t V i r i ( q ) x = b. (P'(-.Q) - C . ( q . ) ) + ^ b.) q . P ' ( Q ) = 0. b. D e f i n e A . = — > 0, so 1 q i n 3 T T , , I X ^ (q) = A.(P(Q) - C (q )) + ( E U P ' (Q ) j = l J 3 q i J J J \ j = l J J = ^ (P (Q) - C ] ( q i ) ) + f ^ b j ) P ' ( Q ) = 0 f rom above .118 -n _ Hence Z X.Vir.(q) = 0. L e t t e l R n , t f- 0. We must have j=l J J n X A . V T T . ( q ) . t = 0. j=l J J Hence e i t h e r V i r . ( q ) - t = 0, V j e N o r t h e r e e x i s t s a k e N such t h a t V7T^(q)• t < 0. In e i t h e r c a se t t C + ( q , N ) . S i n c e t was a r b i t r a r y , i t t h en f o l l o w s t h a t C + ( q , N ) = <$>. Proof of Proposition 12. The v e c t o r o f l o c a l l y s t a b l e o u t p u t s q a r e c h a r a c t e r i z e d by n b-(P(Q) - c ) + q. I b . P ' ( Q ) = 0 f o r i e N. 1 j = l J T h i s e x p r e s s i o n c an be w r i t t e n as n L b . q i / V ( Q ) "\ Q \ P ( Q ) - cj b i ' v ' v v y 1 f o r i e N. Summing bo th s i d e s y i e l d s QP ' (Q ) _ i From t h e a s sump t i o n s on P(Q) i t e a s i l y f o l l o w s t h a t t h e r e i s a un i que Q* s a t i s f y i n g t h i s c o n d i t i o n . The l o c a l l y s t a b l e o u t p u t s a r e t hen u n i q u e l y g i v e n by - 119 -Proof of Proposition 13. (1) Since 0 e B(q), we have n 3TT, Vi e N, E b. = 0 or b.(q.P'(Q) + P(Q) - C:(q.)) + E b.q.P'(Q) = 0. 1 1 1 jYi J 1 Let b = E b., so ieN 1 b^PCQ) - C.(q.)) + bq.P'(Q) = 0. For f i rms i , j we have b i qj qj h. = P(Q) - c;.(q.)  1 P(Q) - C'.( q j) Define f ^ Q , ^ ) = p(..Q) J 1 ^ . ) . By assumption f.(Q,q) = f .(Q,q). Since C.(-q) > 0, ' J 1 3q Hence 3f. r (Q,q) > 0. q. > q. i f and only i f b. > b.. (2) In t h i s case b. = b.. From above we get q. P(Q) - C'(q.) = P(Q) " C j ( q j ) Thus C.(q) > C,(q) i f and only i f q. > q.. - 120 -APPENDIX FOR" CHAPTER 4 Proof of Proposition 1. S i n c e i s i n dependen t o f pa s t h i s t o r y t h en i t must be a c o n s t a n t f u n c t i o n say P^. F i r m i ' s p rob l em i s t o s o l v e a t t ime t max ^ ( P . ^ . ) + g V*+ 1({P*}"= t + 1) P.>0 S i n c e f u t u r e p r i c e s a r e i n dependen t o f h i s t o r y t hen the above i s e q u i v a l e n t t o max T r . ( P . , P t . ) . P.>0 1 1 By a s s u m p t i o n . A 3 , t h e un i que s o l u t i o n t o t h i s p r ob l em f o r a l l i i s P^, t h e Cou r no t -Na sh e q u i l i b r i u m . Proof of Proposition 2. L e t m be the number o f t i m e s P^  = P^. To show t h a t o^- ' is a SPE we must show t h a t f o r any h i s t o r y f i r m s a re f o l l o w i n g o p t i m a l d e c i s i o n s . Suppose lj f {p* |.t < T} then p j = p^ \/- i e N wh i ch i s by p r o p o s i t i o n 1 .a SPE f o r t > T.-> C o n s i d e r t h e c a s e i n wh i ch I T = { P f c | 0 <_ I < T } . I f f i r m 1. p l a y s " s { ( I T ) . and oth.er f i r m s f o l l o w a"'", t hen t h e p a y o f f i s : \ co -t=T 0 I f f i r m i does no t f o l l o w a j , t h e n the b e s t i t can do i n p e r i o d T i s A , as g i v e n i n a s sump t i o n A l . Subsequen t p e r i o d s w i l l have p a y o f f s - 121 -g i v e n by t h e Cou r no t -Na sh e q u i l i b r i u m . H e n c e , i f Vi i s t he maximum pro-f i t when f i r m i does no t f o l l o w a j , w e have t h e f o l l o w i n g i n e q u a l i t i e s : V i - Vl" > (TT 1 ( P T ) - A) 00 + S [ 7 r , ( P T ) - T T . ( P ° ) ] 3 t - T t=T+l 1 ~ 1 ~ 6 ~ 1 6 t=T+m 6 ( s i n c e p} = P_° a t most-m t i m e s ) ( . . . ( p t ) . . A ) l l i 3 ) + | 6m_ S i n c e 3 = e 6 Y , 3 -* 1 as y -> 0 wh i ch i m p l i e s v| - ?T > 0 f o r s m a l l y. Hence sj i s o p t i m a l f o r f i r m i f o r . s m a l l y. Thus c T i s a SPE f o r s u f f i c i e n t l y s m a l l y. . . ' * -a r ! Proof of Proposition 3. L e t Fl(P_,w) = (P ,w) wh i ch by-Axiom--1 e x i s t s and i s c o n t i n u o u s . By ax iom 2 3 r ! a r 1 . a (aP ,aw) = a (P ,w) and hence F l f a P . a w ) = F1. (JP,w). S i n c e F 1 ^ - , - ) i s c o n t i n u o u s t hen J J 3 - 1 2 2 -F(P_,w) = l i m Fl(aP_,aw) = F ^ O . O ) . J By t h e symmetry ax iom r 0 ( j ) ( £ » w ) = r j ( T 0 ( O ) > w ) f o r a l l p e r m u t a t i o n s 8. Bu t r l ( 0 , w ) = w. Hence r l ( 0 , w ) = w . f o r j e ,M. Thus F^O .w ) = 1 V w > 0, Hehte f]0\AP,)"'= 1 tf P e Rn, A ? , e B ( P _ ) ; . ; By de f in i t i on , . r . l . (P_,0)" = 0* so i n t e g r a t i n g F N . ( P _ , w ) we ge t r l ( P , A P . ) - A P , , P c K J . A P . e B ^ P ) . _ Proof of Proposition 4'. By a s s ump t i o n s A l and A2 a* > 0 . S i n c e TT..(P_(CO) i s s t r i c t l y q u a s i - c o n c a v e f o r a < a * , Tr . j (P(a)) < T T i ( P ( a ) + R 1 ( _ P ( a ) , A P . - , N ) ) V i e N. By t h e d e f i n i t i o n o f R 1 ( ' • . , • , • ) > i t " any f i r m does no t f o l l o w t h e change A P . a l l f i r m s p l a y A P , = 0 . ' C l e a r l y R 1 ( P ( a ) , A P . N ) i s an I J 1 e q u i l i b r i u m announcement . Proof of Proposition 5. To show t h a t _P(a*) i s an e q u i l i b r i u m i t must be demon s t r a t e d t h a t t h e i r e x i s t no n o n t r i v i a l e q u i l i b r i u m announcements . There a r e two c a s e s : ( i ) I f some f i r m i d e c i d e s to announce a. ' !pr ice change A P ^ < 0 t hen c o n d i t i o n ('*) en su r e s t h a t a l l f i r m s ' w i l l match t h e p r i c e c u t . S i n c e TT^(_P(a)) i s i n c r e a s i n g f o r a e [0 , a * ] , f i r m i w i l l r e -s c i n d i t s p r i c e c u t . ( i i ) Suppose t h a t i t i s f i r m k t h a t s a t i s f i e s ^ 123 -i Tr^(_P(a*)) = max T r ^ ( £ ( a ) ) . Then i f f i n n i announces a ; p r i c e i n c r e a s e aelR AP.. > 0 , t hen i t i s neve r i n f i r m k ' s i n t e r e s t t o match such an i n c r e a s e . Then a g a i n by c o n d i t i o n (*) f i r m i w i l l r e s c i n d t h e p l anned i n c r e a s e . Proof of Proposition 6. By p r o p o s i t i o n 2 one need o n l y c o n s i d e r t he c a s e o f a p o t e n t i a l e n t r y by some f i r m j e N/N*. The re a r e two c a se s to c o n s i d e r . ( i ) Suppose f i r m j e n t e r s a t t ime T > 0 , and f u r t h e r Tr.(_Fp,N* \3• { j } ) > " S i n c e a l l f i r m s p l a y Cou r no t -Na sh i n t h e f ace o f e n t r y t hen f i r m j ' s o p t i m a l p r i c e r e spon se i s a l s o to p l a y C o u r n o t -Nash . S i n c e "TT. > - E , a t t h e s e p r i c e s t h e f i r m w i l l no t e x i t . By p r o -3 p o s i t i o n 1 _P i s a ' S P E . Hence g i v e n t h a t f i r m j en t e r s . , t h e C o u r n o t -Nash" s o l u t i o n i s a. SPE. But t h e i n d u s t r y N* i s a ' s t a b l e s t r u c t u r e , hence TT. < 0 and t h e r e f o r e V T <"0, so f i r m j i s b e t t e r - o f f o u t s i d e t h e i n d u s t r y . - J ( i i ) Suppose TT .(-F^,N* { j } ) < - E . As b e f o r e f i r m j must J c hoose t h e C o u r n o t - N a s h p r i c e upon e n t r y and c o n s e q u e n t l y i s o u t t h e nex t p e r i o d . C l e a r l y t he p r i c i n g s t r a t e g i e s g i v e n j ' s a c t i o n s a r e SPE . Hence o* i s a SPE . 

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