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DEPARTMENT OF ECONOMICS

E U I W O R K I N G P A P E R No. 8 9 / 3 9 0

Imperfect Information and Financial Markets

A General Equilibrium Model*

Co r r a d o Be n a s s i

* This paper is part o f a doctoral research currently undertaken at the

European University Institute, Florence. I wish to thank Prof. Alan Kirman,

who read and commented on a previous version o f this work. M y greatest debt

is to Prof. Pierre Dehez, whose patience went well beyond that required o f a

supervisor. Any remaining error is obviously mine.

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A ll rights reserved.

No part o f this paper may be reproduced in any form

without permission o f the author.

© Corrado Benassi

Printed in Italy in June 1989

European University Institute

Badia Fiesolana

- 50016 San Domenico (FI) —

Italy

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1. Introduction T h e i n f o r m a t i o n a l a p p r o a c h to e c o n o m i c t h e o r y h a s c h a l l e n g e d s e v e r a l t r a d i t i o n a l v i e w s o n t h e w o r k i n g o f f i n a n c i a l m a r k e t s . It i s b y n o w c o m m o n l y h e l d t h a t t h e r e a r e t y p i c a l i n f o r m a t i o n a l a s y m m e t r i e s w h i c h a r e l i k e l y t o a f f e c t t h e m a r k e t - d e t e r m i n e d a l l o c a t i o n o f c a p i t a l a c r o s s f i r m s : t h e s e a c c o u n t f o r a n u m b e r o f f e a t u r e s - b o t h b e h a v i o u r a l a n d i n s t i t u t i o n a l - w h i c h h a d r e m a i n e d h i t h e r t o u n e x p l a i n e d . A s i s w e l l k n o w n , t w o i n f o r m a t i o n r e l a t e d m e c h a n i s m s s t a n d o u t a s m o s t s i g n i f i c a n t in e x p l a i n i n g w i d e s p r e a d m a r k e t f a i l u r e s : m o r a l h a z a r d a n d a d v e r s e s e l e c t i o n . A s p u t f o r t h b y L a f f o n t a n d M a s k i n ( 1 9 8 0 ) , t h e f o r m e r is g e n e r a t e d b y a n a g e n t ' s b e h a v i o u r b e i n g u n o b s e r v a b l e b y o t h e r s ; t h e l a t t e r is r a t h e r d u e t o t h e i n f o r m a t i o n o f o n e s i d e o f t h e m a r k e t b e i n g i m p e r f e c t l y k n o w n t o t h e o t h e r . T h i s d i s t i n c t i o n i s r e l e v a n t , a m o n g o t h e r t h i n g s , b e c a u s e o f t h e c o m p e t i t i v e a s s u m p t i o n s n e s t e d in e a c h f r a m e w o r k . M o r a l h a z a r d is m a i n l y a n o n c o m p e t i t i v e p h e n o m e n o n : it a r i s e s w h e n s o m e a g e n t C s) c o n t r o l ( s) s o m e v a r i a b l e ( s) w h i c h c a n n o t b e f r e e l y o b s e r v e d b y o t h e r ( s ) , a n d c a n t h e r e f o r e b e b u i l t in i n d e e d , it is f a i r l y t y p i c a l o f - m o d e l s o f t w o - p a r t y c o n t r a c t u a l a r r a n g e m e n t s . H o w e v e r , o n e n e e d h a v e a '’p r o p e r " m a r k e t ( w i t h " m a n y " a g e n t s ) f o r a d v e r s e s e l e c t i o n to w o r k : t h e r e is a c o m p e t i t i v e b e n t i n a g e n t s t a k i n g b o t h t h e i r i n d i v i d u a l q u a l i t y a n d t h e m a r k e t - d e t e r m i n e d a v e r a g e q u a l i t y , 1 as g i v e n . In t h i s p a p e r w e f o c u s o n t h e s i m p l e s t e x a m p l e o f i n f o r m a t i o n a l a s y m m e t r i e s a n d t h e i r e f f e c t o n t h e e q u i l i b r i u m c a p i t a l a l l o c a t i o n o f a c o m p e t i t i v e e c o n o m y . M o r e p r e c i s e l y , w e c o n s i d e r a ( n o n s t o c h a s t i c ) t w o - p e r i o d g e n e r a l e q u i l i b r i u m s e t u p : o n t h e o n e h a n d , c o m p e t i t i v e

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f i r m s f a c i n g a n o n l i n e a r ( c o n v e x ) t e c h n o l o g y c h o o s e t h e i r c a p i t a l r e q u i r e m e n t s , t o b e f i n a n c e d b y s h a r e i s s u i n g ; o n t h e o t h e r h a n d , c o n s u m e r s a l l o c a t e t h e i r e n d o w m e n t s b e t w e e n c u r r e n t a n d f u t u r e c o n s u m p t i o n , u s i n g t h e f i r m s ' s h a r e s as t h e o n l y s t o r e o f v a l u e . T h e e x i s t e n c e o f a p e r f e c t l y c o m p e t i t i v e s h a r e m a r k e t a l l o w s t h e r e s u l t i n g e q u i l i b r i u m c a p i t a l a l l o c a t i o n t o b e j o i n t l y d e t e r m i n e d t o g e t h e r w i t h an e q u i l i b r i u m s h a r e a l l o c a t i o n . It is in t h i s f r a m e w o r k t h a t a s y m m e t r i c i n f o r m a t i o n is i n t r o d u c e d , a n d a d v e r s e s e l e c t i o n e f f e c t s e x a m i n e d . A s y m m e t r i c i n f o r m a t i o n i s b r o u g h t i n u n d e r t h e s i m p l e a s s u m p t i o n t h a t n o s h a r e h o l d e r k n o w s t h e t e c h n o l o g y e a c h f i r m is u s i n g , n e i t h e r c a n he i m p r o v e u p o n h i s k n o w l e d g e . A d v e r s e s e l e c t i o n f o l l o w s f r o m t h i s a s s u m p t i o n , w h i c h a p p e a r s to s u i t t h e c o m p e t i t i v e b e n t o f t h e m o d e l . N o w , t h a t a d v e r s e s e l e c t i o n i t s e l f m a y l e a d t o i n e f f i c i e n t a l l o c a t i o n s is a c t u a l l y a n i n t u i t i v e a n d w e l l k n o w n r e s u l t . H o w e v e r , t h e p u r p o s e o f t h e m o d e l is t o g i v e a g e n e r a l e q u i l i b r i u m o u t l o o k to s u c h r e s u l t . O n t h e o n e h a n d , t h i s a l l o w s to b r i n g o u t e x p l i c i t l y t h e o v e r a l l e f f e c t s o f t h e a g e n t s ' i m p e r f e c t i n f o r m a t i o n o n t h e f i r m s ' c h o i c e s ; o n t h e o t h e r h a n d , it p o i n t s t o t h e r e l a t i o n s h i p b e t w e e n t h e a d v e r s e s e l e c t i o n m e c h a n i s m a n d t h e i n t e r t e m p o r a l p a t t e r n o f r e s o u r c e a l l o c a t i o n e n f o r c e d b y a c o m p e t i t i v e e q u i l i b r i u m o n t h e s h a r e m a r k e t . T h e a b o v e i m p l i c a t i o n s a r e s t u d i e d w i t h i n t h e s t o c k h o l d e r s ' e q u i l i b r i u m n o t i o n , w h i c h D r e z e ( 1 9 7 2 , 1974 ) d e v e l o p e d in a d i f f e r e n t f r a m e w o r k . T h i s n o t i o n is r e q u i r e d to o v e r c o m e a p o t e n t i a l i n c o n s i s t e n c y b e t w e e n t h e f i r m s ' a n d t h e i n v e s t o r s ' b e h a v i o u r s , a r i s i n g w h e n a s h a r e m a r k e t is e x p l i c i t l y m o d e l e d in w h i c h b o t h a c t a s c o m p e t i t i v e p r i c e t a k e r s : w h i l e a n y f i r m ' s o p t i m a l c a p i t a l r e q u i r e m e n t is b a s e d o n m a r g i n a l p r o f i t e v a l u a t i o n , a n y s h a r e h o l d e r ' s i n v e s t m e n t d e c i s i o n is d e t e r m i n e d b y t h e u n i t a v e r a g e r e t u r n o n s u c h i n v e s t m e n t . T h e m a i n r e s u l t o f t h e p a p e r is t h a t t h e a g e n t s '

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i n t e r t e m p o r a l p r e f e r e n c e s d o i n t e r a c t w i t h a d v e r s e s e l e c t i o n , so t h a t t h e r e m a y b e p o s i t i v e t r a d e e q u i l i b r i a e v e n w h e n a g e n t s a r e v e r y p e s s i m i s t i c a b o u t t h e f i r m s ' ( u n k n o w n ) q u a l i t y . M o r e o v e r , t h e a g e n t s ' a t t i t u d e t o w a r d s r i s k a f f e c t s t h e e q u i l i b r i u m o u t c o m e i n s u c h a w a y t h a t d i f f e r e n t ( s u b j e c t i v e l y h e l d ) q u a l i t y e v a l u a t i o n s a r e c o n s i s t e n t w i t h e q u i l i b r i u m o n l y w h e n t r a d e r s h a v e d i f f e r e n t , r i s k a v e r s e p r e f e r e n c e s . T h e p a p e r i s o r g a n i z e d a s f o l l o w s . In t h e n e x t s e c t i o n t h e b a s i c m o d e l i s p r e s e n t e d . I n a n o n s t o c h a s t i c c o m p e t i t i v e e c o n o m y , a s e t o f a g e n t s o w n i n g " t e c h n o l o g i e s " t r y t o s e l l t h e m t o a s e t o f c o n s u m e r s . T h e l a t t e r e x p l o i t t h e s e t e c h n o l o g i e s b y s e t t i n g up e q u i t y f i n a n c e d f i r m s . S e c t i o n 3 e x a m i n e s t h e n a t u r a l r e s u l t t h a t , u n d e r c o m p l e t e i n f o r m a t i o n , b a d q u a l i t y t e c h n o l o g i e s m a y n o t b e u s e d . S e c t i o n 4 t a k e s u p t h e a s y m m e t r i c i n f o r m a t i o n c a s e : if c o n s u m e r s c a n n o t d i s t i n g u i s h g o o d f r o m b a d q u a l i t y , t h e e q u i l i b r i u m c a p i t a l a l l o c a t i o n i s i n d e p e n d e n t o f f i r m s ' ( a n d t e c h n o l o g i e s ' ) t y p e s b u t d e p e n d s o n t h e a g e n t s ' q u a l i t y e v a l u a t i o n s ; s o m e e f f i c i e n c y a s p e c t s a r e a l s o d i s c u s s e d . F i n a l l y , s e c t i o n 5 p r o v i d e s s o m e c o n c l u d i n g r e m a r k s . A l l p r o o f s a r e g a t h e r e d i n t h e A p p e n d i x .

2. The basic model

W e c o n s i d e r a o n e c o m m o d i t y , t w o - p e r i o d e c o n o m y , w h e r e p r o d u c t i o n h a s t o b e i m p l e m e n t e d in t h e f i r s t p e r i o d t o 2 y i e l d o u t p u t i n t h e s e c o n d . S i n c e c o n s u m p t i o n t a k e s p l a c e in b o t h p e r i o d s , a n i n t e r t e m p o r a l a l l o c a t i o n p r o b l e m a r i s e s : c o n s u m e r s e x c h a n g e p a r t o f t h e i r w e a l t h i n p e r i o d t= 1 f o r o u t p u t to b e d e l i v e r e d at t = 2 , so t h a t t h e t r a n s f e r o f r e s o u r c e s o v e r t i m e i n v o l v e s t r a d i n g i n a s s e t s . H e r e f o l l o w s a m o r e d e t a i l e d d e s c r i p t i o n o f t h e e c o n o m y . T h e r e a r e t w o c l a s s e s o f a g e n t s : t e c h n o l o g y s e l l e r s ( T S ' s ) a n d c o n s u m e r s . B o t h o w n c o m m o d i t y e n d o w m e n t s , b u t t h e

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f o r m e r a r e a l s o e n d o w e d w i t h " t e c h n o l o g i e s " , i . e . b l u e p r i n t s w h i c h s p e c i f y h o w o u t p u t c a n b e p r o d u c e d . A s w a s p o i n t e d o u t , o n l y o n e p e r i s h a b l e g o o d ( " c o r n " ) i s t r a d e d in t h i s e c o n o m y : a s i n p u t i n t h e f i r s t p e r i o d a n d o u t p u t in t h e s e c o n d , a s c o n s u m p t i o n g o o d in b o t h p e r i o d s .

2.1. Preferences, technologies, resources

In t h e e c o n o m y t h e r e a r e m T S ' s , b e l o n g i n g t o t h e se t M : =*{ 1 , . . . , m } . E a c h o f t h e m , J E M , l i v e s t w o p e r i o d s , t * 1 , 2 , a n d i s i d e n t i f i e d b y h i s p r e f e r e n c e s a n d b y t h e t e c h n o l o g y h e o w n s p r i o r t o t r a d e . F o r e a c h J E M , a c o n s u m p t i o n p l a n is 2 a n o n n e g a t i v e v e c t o r c j ; c j 1 > c j2^ eF* + > h i s c o n s u m p t i o n se t. H i s i n t e r t e m p o r a l p r e f e r e n c e s a r e t h e n d e s c r i b e d b y a

2

u t i l i t y f u n c t i o n U j : R + - > R + , w h i c h is m o n o t o n i c a l l y i n c r e a s i n g , t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e a n d s t r i c t l y q u a s i - c o n c a v e . E a c h a g e n t J E M a l s o o w n s a t e c h n o l o g y , w h i c h is d e c r i b e d b y a p r o d u c t i o n s e t , Yj. P r o d u c t i o n t a k e s o n e p e r i o d t o b e c a r r i e d o u t . E a c h p o i n t i n Yj i s a p a i r o f i n p u t ( xj) e m p l o y e d at t*1 a n d o u t p u t C y j ) d e l i v e r e d at t = 2. F o r e a c h T S J E M , t h e r e f o r e , w e d e f i n e Y j : “ { C x j , y j J | ( x j , y j 3 e R 5; y j 5 f j ( X j ) } C D w h e r e i n p u t s a r e m e a s u r e d b y n o n n e g a t i v e r e a l n u m b e r s . T h e p r o d u c t i o n f u n c t i o n f j ( x j ) : R + - > R + is u s e d a s a s h o r t h a n d t o d e s c r i b e t e c h n o l o g y j. W e a s s u m e t h a t a n y p r o d u c t i o n f u n c t i o n e x h i b i t s d e c r e a s i n g r e t u r n s t o s c a l e . W e a l s o a s s u m e t h a t t h e r e is n o p o s i t i v e i n p u t v a l u e at w h i c h t w o d i f f e r e n t t e c h n o l o g i e s y i e l d t h e s a m e ( a b s o l u t e as w e l l as m a r g i n a l ) l e v e l o f o u t p u t : a s w i l l b e c l e a r , t h i s s i m p l i f i e s t h e c r i t e r i o n b y w h i c h t e c h n o l o g i e s c a n b e r a n k e d . O u r a s s u m p t i o n s o n p r o d u c t i o n f u n c t i o n s a r e s u m m a r i z e d as f o 1 l o w s .

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A . 1 . F o r a l l J E M , f j ( x j ) l e a s t t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e a n d s u c h t h a t : (a) f j ( 0 ) « 0 ; (b) 0 < -FjCxj) < f o r a l l x j > 0 ; (c) f ’j( x j) < □, f o r a l l x j > 0 ; ( d) f o r a n y j , k E M ( jt=K) , t h e r e is n o x > 0 s u c h t h a t f j( x) = fK( x) o r f j( x) = f £ ( x ) . L i k e T S ' s , c o n s u m e r s - b e l o n g i n g to t h e s e t N := { m + 1 , . . . , n } - l i v e t w o p e r i o d s . E a c h o f t h e m is a c c o r d i n g l y e n d o w e d w i t h a m o n o t o n i c a l l y i n c r e a s i n g , t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e a n d s t r i c t l y q u a s i - c o n c a v e 2 u t i l i t y f u n c t i o n , Uj[:R+ - > R + , d e f i n e d o v e r c o n s u m p t i o n p l a n s

2

c i E R + , i E N . It i s a s s u m e d t h a t , f o r a t l e a s t o n e c o n s u m e r , t h e m a r g i n a l r a t e o f s u b s t i t u t i o n b e t w e e n c u r r e n t a n d f u t u r e c o n s u m p t i o n v a n i s h e s a s t h e l a t t e r g o e s t o z e r o . O u r 3 a s s u m p t i o n s o n p r e f e r e n c e s a r e s u m m a r i z e d a s f o l l o w s . A . 2. F o r a n y a E A : = M u N , u a ^ c a 1 > c a2^ is t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e a n d s u c h t h a t : (a) ( 6 u a / 6 c a t ) >0, f o r a l l c a ^ > 0 , t = 1,2; ( b) f o r a n y c a a n d c a s u c h t h a t u a ( c a ) > u a ( c a ) , M c a + t 1 “ M ) c ' ] > u a ( c ^ ) , f o r a l l p, 0 < p < 1 ; ( c) f o r at l e a s t o n e i E N lim( 6 u ^ / 6 c ^ y)/( S u i / S c ^ g ) =0. W e f i n a l l y c o n s i d e r t h e r e s o u r c e e n d o w m e n t o f o u r e c o n o m y . It is a s s u m e d t h a t s u c h e n d o w m e n t is m a d e up b y a n a m o u n t W^. o f r e s o u r c e s a v a i l a b l e in e a c h p e r i o d ( t = 1 , 2 ) s u c h t h a t W-j = W > 0 a n d W 2 = 0. S o m e c o m m e n t s a r e n o w in o r d e r a b o u t t h e s e a s s u m p t i o n s . A s f a r a s A . 1 is c o n c e r n e d , p o i n t s Cb ) a n d ( d) a r e n o t s t a n d a r d . T h e f o r m e r v i o l a t e s t h e s o - c a l l e d I n a d a c o n d i t i o n s . It is m a d e h e r e b e c a u s e it a l l o w s t o u s e t h e m a r g i n a l p r o d u c t at z e r o as a q u a l i t y i n d e x , a s w i l l be c l e a r in t h e s e q u e l . P o i n t ( d) is t h e n r e q u i r e d to p r e v e n t

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t h i s q u a l i t y o r d e r i n g f r o m d e p e n d i n g o n t h e a c t i v i t y l e v e l . A n e x a m p l e o f a p r o d u c t i o n f u n c t i o n s a t i s f y i n g A . 2 is t h e l o g a r i t h m i c f o r m f j ( x j ) “ a j l o g ( x j + 1), a j >0 A s s u m p t i o n A . 2 i s s t a n d a r d : d i f f e r e n t i a b i l i t y a n d s t r i c t q u a s i - c o n c a v i t y e n s u r e t h e e x i s t e n c e o f a u n i q u e 4 s o l u t i o n f o r t h e a g e n t ' s i n t e r t e m p o r a l a l l o c a t i o n p r o b l e m . P o i n t ( c) r u l e s o u t c o r n e r s o l u t i o n s , a t w h i c h n o c o n s u m e r is w i l l i n g t o c o n s u m e in t h e s e c o n d p e r i o d . A n o b v i o u s e x a m p l e o f a u t i l i t y f u n c t i o n s a t i s f y i n g A . 2 i s t h e s t a n d a r d C o b b - D o u g 1 a s , o f t h e t y p e u a ( c a 1 > c a 2 ) = C a 1 c a 2 > P e ( 0 , 1 ) F i n a l l y , t h e w a y r e s o u r c e s a r e m o d e l e d a l s o d e s e r v e s s o m e c o m m e n t s . F i r s t , w e n e e d n o t s p e c i f y h o w t h e e c o n o m y ' s e n d o w m e n t is d i s t r i b u t e d a c r o s s a g e n t s , s i n c e o p t i m a l i t y is o u r o n l y c o n c e r n . S e c o n d , s h a r e s in ( f u t u r e ) p r o d u c t i o n a r e a s s u m e d t o b e t h e o n l y a s s e t - a s w i l l b e e x p l a i n e d l a t e r in m o r e d e t a i l . T h i s i m p l i e s t h a t t h e r e a r e n o c r e d i t m a r k e t s o p e n , i . e . n o e x c h a n g e o f f u t u r e r e s o u r c e s is a l l o w e d . H e n c e , w e d o n o t l o o s e i n g e n e r a l i t y b y a s s u m i n g t h a t W 2 = 0 •

2.2. Further t echnological and institutional c o ns traints

A s s u m i n g W 2 = 0 e n t a i l s t h a t s e c o n d p e r i o d c o n s u m p t i o n m u s t b e w h o l l y f i n a n c e d b y i m p l e m e n t i n g t h e a v a i l a b l e t e c h n o l o g i e s . U n d e r t h i s r e s p e c t , w e c h a r a c t e r i z e m o r e s h a r p l y o u r e c o n o m y b y a s s u m i n g t h a t T S ' s a r e n o t a l l o w e d to i n v e s t t h e i r f i r s t p e r i o d w e a l t h : in t h e s e c o n d p e r i o d , e a c h T S o n l y c o n s u m e s o u t o f t h e p r o c e e d s f r o m t h e s a l e o f h i s t e c h n o l o g y ^ . T h e l a t t e r is m o d e l l e d a s f o l l o w s : e a c h T S

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i m p o s e s a f e e i n e x c h a n g e f o r t h e t e c h n o l o g y he is s e l l i n g . T h i s f e e is p r o p o r t i o n a l t o t h e q u a n t i t y o f o u t p u t a c t u a l l y p r o d u c e d , a n d is e x o g e n o u s l y g i v e n i n t h e m o d e l . L e t t h e q u a n t i t y o f c a p i t a l f o r t e c h n o l o g y J E M b e xj; t h e n a g e n t j w i l l g e t a f e e 6 j f j ( x j ) , t o b e p a i d at t = 2 o u t o f t h e a c t u a l p r o d u c t i o n d e l i v e r e d in t h e s e c o n d p e r i o d . S i n c e 6j is t a k e n a s g i v e n o u t s i d e t h e m o d e l , t h e r e is n o l o s s o f g e n e r a l i t y in a s s u m i n g t h a t it is t h e s a m e f o r a l l T S ' s : f o r a l l J E M , g 6 j = 6 > 0 . E a c h T S c a n s e l l h i s t e c h n o l o g y o n l y o n c e , s o t h a t n o m o r e t h a n o n e f i r m c a n u s e J ' s t e c h n o l o g y . T h u s , t h e T S ' s b e h a v i o u r c a n b e d e s c r i b e d a s f o l l o w s : i n t h e f i r s t p e r i o d , t h e y c o n s u m e t h e i r w e a l t h a n d t r y to s e l l t h e i r t e c h n o l o g y . T h o s e w h o s u c c e e d g e t a f e e in t h e s e c o n d p e r i o d , w h i c h is t h e n ( i m m e d i a t e l y ) c o n s u m e d . T h i s c o n s t r a i n t o n t h e T S ' s b e h a v i o u r c a n b e f o r m a l i z e d a s C J2* 6 J f x j> > 6 j “ 6 > 0 -h o l d i n g f o r a l l J E M . O n e b a s i c p i c t u r e e m e r g e s f r o m t h e a b o v e d e s c r i p t i o n . In o r d e r t o a c h i e v e a s a t i s f a c t o r y i n t e r t e m p o r a l p a t t e r n o f c o n s u m p t i o n , t h e T S ' s h a v e to s e l l t h e i r t e c h n o l o g i e s . C o n s u m e r s , o n t h e o t h e r h a n d , h a v e to i n v e s t t h e i r w e a l t h in t h e a v a i l a b l e t e c h n o l o g i e s . H o w e v e r , t h e s e t o f p r o d u c t i o n p o s s i b i l i t i e s m a d e a v a i l a b l e b y t h e T S ' s d o e s n o t e x h a u s t t h e r a n g e o f i n v e s t m e n t o p p o r t u n i t i e s f o r t h e c o n s u m e r s . It is a l s o a s s u m e d t h a t t h e r e is a t e c h n o l o g y j = 0 w h i c h p r o v i d e s c o n s t a n t r e t u r n s to s c a l e ( C R S ) . T h e s e t o f a v a i l a b l e t e c h n o l o g i e s is t h e r e f o r e T := { 0 , 1 , . . . , m } . T h e C R S t e c h n o l o g y , u n l i k e a n y o t h e r , is f r e e l y a v a i l a b l e . N o f e e b e i n g i m p o s e d ( i . e . , 6 0 = 0 ) , it g r a n t s a g i v e n r e t u r n at t = 2 f o r e a c h u n i t i n v e s t e d at t = 1 . T h u s t h e f o l l o w i n g d e f i n i t i o n c a n b e g i v e n . 2 Y 0 : * { ( x 0 , y 0 ) | C x 0 , y 0 ) E R + ; y 0 < r x 0 , r > 1 } (2)

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T h e C R S t e c h n o l o g y p r o v i d e s a s i m p l e b e n c h m a r k w i t h i n t h e q u a l i t y r a n k i n g o f t h e a v a i l a b l e t e c h n o l o g i e s t h a t w e a r e g o i n g to i m p o s e . T h e r e a l n u m b e r r i s a s s u m e d t o be g r e a t e r t h a n u n i t y . A l t h o u g h n o t s t r i c t l y n e c e s s a r y , t h i s is h o w e v e r c o n v e n i e n t : it c a p t u r e s t h e i d e a t h a t a g e n t s c a n a l w a y s i n v e s t t h e i r w e a l t h at a g i v e n p o s i t i v e i n t e r e s t r a t e , ( r - 1 ) . F o r n o t a t i o n a l c o n v e n i e n c e o n e c a n d e f i n e f Q ( x Q ) := r x Q , a n d s u m m a r i z e a l l t h i s in t h e f o l l o w i n g a s s u m p t i o n : A . 3. T h e r e e x i s t s a C R S t e c h n o l o g y Y 0 s u c h t h a t o( x o) : “ r x o> r > 1 a n d 6 o “ 0-T h e C R S t e c h n o l o g y b e i n g a l w a y s f r e e l y a v a i l a b l e h a s c l e a r l y a b e a r i n g u p o n t h e i n v e s t m e n t c o s t . A s w i l l t u r n o u t , t h e e x i s t e n c e o f t h i s f e e s y s t e m a m o u n t s t o s a y i n g t h a t t h e u n i t c a p i t a l c o s t t o i n v e s t o r s f o r j e M i s R j = r/( 1 — 6 j ) . It is m a d e u p b y t h e o p p o r t u n i t y c o s t r o f f o r e g o i n g i n v e s t m e n t in t h e C R S t e c h n o l o g y , a n d t h e a c t u a l c o s t 6j of h a v i n g a c c e s s t o a n y t e c h n o l o g y j e M . T h e e x o g e n o u s f e e b e i n g t h e s a m e f o r a l l T S ' s ( i . e . , 6j = 6 f o r a l l je M) a n d n i l f o r t h e C R S t e c h n o l o g y ( i . e . , 6 o = 0 ) , t h e u n i t c a p i t a l c o s t w i l 1 be R j = r/( 1- 6) := R, f o r a l l J E M ; R 0 = r-E a c h t e c h n o l o g y i s i d e n t i f i e d b y a p r o d u c t i o n f u n c t i o n : w e c a n f o r m a l i z e t h e i d e a o f q u a l i t y d i f f e r e n c e s a c r o s s t e c h n o l o g i e s b y g i v i n g a s i m p l e r a n k i n g c r i t e r i o n f o r s u c h f u n c t i o n s . In o r d e r t o d o so, l e t us c o n s i d e r t h e f o l l o w i n g d e f i n i t i o n : q j C x j ) f j ^ x j ) / R j , J E T (3) q j ( x j ) is j u s t t h e r a t i o o f t h e m a r g i n a l p r o d u c t t e c h n o l o g y

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j y i e l d s at t h e i n p u t l e v e l xj, t o t h e u n i t c o s t o f 7 c a p i t a l . U s i n g t h e a b o v e d e f i n i t i o n o f c a p i t a l c o s t , w e h a v e t h a t q j ( x J 3 = f j C x J 3 / R » J eM C 3') w h i l e q 0 ( x 0 )= 1 f o r a n y a c t i v i t y l e v e l . W e c a n u s e q j ( 0) to r a n k d i f f e r e n t t e c h n o l o g i e s ^ . W i t h n o l o s s o f g e n e r a l i t y , o n e c a n o r d e r t h e m a c c o r d i n g a s t h e q j( 0) 's a r e g r e a t e r ( o r s m a l l e r ) t h a n q Q *1 , a n d t h e r e f o r e a s s u m e t h a t q 1 ( 0) > . . • > q k(°) > 1 > q k + l ( ° 3 > • • • > q m (0) W e n o w d e f i n e t h e s e t s G : = { j E M | q j ( 0 ) > 1 } , B : = { j E M | q j ( 0 ) < 1 } , B : * { J E M | q j ( 0 ) = 1 } . W e a s s u m e t h a t B is e m p t y , so t h a t G a n d B m a k e up a p a r t i t i o n o f M s u c h t h a t G : = { 1 , . . . , k } a n d B := { k + 1, . . . ,m > . By ( d) o f a s s u m p t i o n A . 2 it f o l l o w s t r i v i a l l y t h a t , if q j( 0) >q j + -j C 0) , t h e n f j( x) / x > f j + l ( x ) / x , f o r a n y x > 0 a n d a n y J E M . T h u s w e c a n u n a m b i g u o u s l y s a y t h a t t e c h n o l o g y j is " b e t t e r " t h a n t e c h n o l o g y j + 1 . q □) is t h e r e f o r e a n u n a m b i g u o u s " q u a l i t y i n d e x " i n t h i s f r a m e w o r k . C l e a r l y , a l l t h i s a l s o a p p l i e s to a n y p a i r ( j , j + 1 ) s u c h t h a t j E G a n d ( j + 1 ) £ B . R e m a r k 1: T h e l a s t a s s u m p t i o n s b r i n g i n t h e m o d e l t h e m a i n s t r u c t u r a l d e p a r t u r e f r o m D r e z e ' s o r i g i n a l w o r k . T h i s is p a r t i c u l a r l y t r u e a s f a r a s t h e e x i s t e n c e o f an e x o g e n o u s f e e s y s t e m is c o n c e r n e d . I t s u s e to e v a l u a t e t e c h n o l o g i e s is a r o u g h s h o r t c u t t o m o d e l i n g a m a r k e t f o r t e c h n o l o g i e s . In t h e p r e s e n t f r a m e w o r k t h i s is u s e d o n l y t o i n t r o d u c e a s y m m e t r i c i n f o r m a t i o n ( s e c t i o n 4) . U n d e r t h i s r e s p e c t , T S ' s n o t i n v e s t i n g in t h e f i r s t p e r i o d is a n i n n o c u o u s a s s u m p t i o n , w h i c h s i m p l i f i e s t h e a l g e b r a . H o w e v e r , m o d e l i n g e x p l i c i t l y a m a r k e t f o r t e c h n o l o g i e s ( i . e . e n d o g e n i z i n g 6)

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is, in p r i n c i p l e , a n i n d e p e n d e n t i s s u e . A s f a r a s t h e q u a l i t y r a n k i n g i s c o n c e r n e d , t h e c r i t e r i o n w e a d o p t e d is c o n s i s t e n t w i t h a n y m o d e l w i t h s t r i c t l y c o n v e x p r o d u c t i o n s e t s : it is i n t r o d u c e d h e r e s o a s t o h a v e a r e a d y b e n c h m a r k i n t h e a n a l y s i s o f t h e a s y m m e t r i c i n f o r m a t i o n c a s e . In t h e p r e v i o u s e x a m p l e w h e r e f j ( x j ) = a j l o g ( x j + 1), q j ( G ) = a j / R , s o t h a t a j > R = > j £ G . U n d e r o u r a s s u m p t i o n s t h e s t a n d a r d m a r g i n a l p r o d u c t i v i t y ( o p t i m a l i t y ) c o n d i t i o n f o r a n y f i r m J E M is o b v i o u s l y g i v e n b y q j ( x j ) = 1 , w h i c h i n t h e s a m e e x a m p l e g i v e s X j = ( a j - R ) / R = q j ( 0 ) - 1 .

3. Full information equilibria

In t h i s s e c t i o n t h e p o l a r c a s e o f c o m p l e t e i n f o r m a t i o n w i l l b e e x a m i n e d . At t h e b e g i n n i n g o f t h e f i r s t p e r i o d t h e m a r k e t s o p e n . E a c h T S e x h i b i t s h i s t e c h n o l o g y ; c o n s u m e r s o b s e r v e t h e a v a i l a b l e t e c h n o l o g i e s a n d d e c i d e w h i c h f i r m s s h o u l d b e s e t u p a n d w h a t is t h e a m o u n t o f c a p i t a l r e q u i r e d . In d o i n g so, t h e y t a k e i n t o a c c o u n t t h a t a n y f i r m s h o u l d p a y g a f e e o n o u t p u t w h e n p r o d u c t i o n is o v e r a n d t h a t i n v e s t m e n t in t h e C R S t e c h n o l o g y is a l w a y s p o s s i b l e . E a c h f i r m a c q u i r e s i t s c a p i t a l b y i s s u i n g c l a i m s o n o u t p u t t o b e d e l i v e r e d in p e r i o d t = 2 , d e f i n e d a s s h a r e s o f t h e a c t u a l p r o d u c t i o n ; c o n s u m e r s g r a n t t h a t c a p i t a l b y p u r c h a s i n g t h o s e a s s e t s ( c h o o s i n g a p o r t f o l i o w h i c h m a y i n c l u d e i n v e s t m e n t in t h e C R S t e c h n o l o g y ) t r a d i n g o f f c u r r e n t a n d f u t u r e c o n s u m p t i o n , w h i l e T S ' s c o n s u m e t h e i r e n d o w m e n t . W h e n t h e s h a r e m a r k e t h a s c l e a r e d , f i r m s b e g i n t h e i r p r o d u c t i o n . T h i s is c o m p l e t e d at t = 2, w h e n o u t p u t is d i s t r i b u t e d t o b o t h a s s e t o w n e r s a n d s u c c e s s f u l T S ' s , a n d s e c o n d p e r i o d c o n s u m p t i o n t a k e s p l a c e ; u n s u c c e s s f u l T S ' s - w h o c o u l d n o t s e l l t h e i r t e c h n o l o g i e s i n t h e f i r s t p e r i o d - a r e a l l o c a t e d a z e r o ( " s u b s i s t e n c e " ) l e v e l o f c o n s u m p t i o n . W e m o d e l t h i s s t o r y b y l o o k i n g at c a p i t a l a l l o c a t i o n s a c r o s s f i r m s : a n y o f t h e m c o r r e s p o n d s t o a s h a r e a l l o c a t i o n

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a c r o s s c o n s u m e r s , w h i c h c a n b e s u p p o r t e d b y a s h a r e p r i c e v e c t o r . C o n s i d e r a g i v e n s e t o f e x i s t i n g f i r m s ( a c t i v e t e c h n o l o g i e s ) , t h e s h a r e a l l o c a t i o n w h i c h is a s s o c i a t e d to it a n d t h e s h a r e p r i c e s y s t e m w h i c h s u p p o r t s t h a t a l l o c a t i o n . T h i s is a n e q u i l i b r i u m s i t u a t i o n i f t h e f o l l o w i n g h o l d s : (a ) n o s h a r e h o l d e r is w i l l i n g to i n v e s t m o r e w e a l t h i n a n y e x i s t i n g f i r m , ( b) t h e r e a r e n o i n a c t i v e t e c h n o l o g i e s w h i c h c o n s u m e r s m i g h t w a n t t o f u n d w i t h p o s i t i v e c a p i t a l a m o u n t s a n d ( c) n o c o n s u m e r is w i l l i n g t o t r a d e s h a r e s at t h e g o i n g p r i c e . T h i s e q u i l i b r i u m is d e c r i b e d b y a n a r r a y o f v e c t o r s in 2 R + , c o r r e s p o n d i n g t o c o n s u m p t i o n a n d p r o d u c t i o n o f t h e n a g e n t s i n t h e t w o p e r i o d s , t o w h i c h a s e t o f e x i s t i n g f i r m s i s a s s o c i a t e d . S u c h a n e q u i l i b r i u m h a s t o s a t i s f y c e r t a i n c o n d i t i o n s . In o r d e r t h a t t h e s e b e d e r i v e d , w e t a k e a d v a n t a g e o f a w o r k b y D r e z e ( 1 9 7 4 ) , w h e r e t h e f o l l o w i n g s t r a t e g y is s u g g e s t e d . F i r s t , t a k i n g a s g i v e n t h e a l l o c a t i o n o f s h a r e s a m o n g a g e n t s a n d t h e p r o d u c t i o n p l a n s o f a l l e x i s t i n g f i r m s b u t o n e , t h e o p t i m a l c h o i c e o f t h e r e m a i n i n g f i r m is c o n s i d e r e d a n d a p s e u d o e q u i l i b r i u m f o r t h a t f i r m is d e f i n e d . S e c o n d , t a k i n g as g i v e n t h e e x i s t i n g f i r m s ' d e c i s i o n s a b o u t p r o d u c t i o n p l a n s , t h e o p t i m a l a l l o c a t i o n o f s h a r e s a m o n g a g e n t s is e x a m i n e d , a n d a p r i c e e q u i 1 i b r i u m f o r t h i s p r o b l e m i s d e f i n e d . T h i r d , t h e s e t w o e q u i l i b r i u m c o n c e p t s a r e p u t t o g e t h e r t o y i e l d a s t o c k h o l d e r s ' e q u i 1 i b r i u m . T h e e c o n o m y c o n s i d e r e d i n t h i s p a p e r is a c t u a l l y d i f f e r e n t f r o m D r e z e ' s . In p a r t i c u l a r , t h e w o r l d d e s c r i b e d h e r e is n o n s t o c h a s t i c , w h i c h s i m p l i f i e s s o m e w h a t t h e a n a l y s i s . O n c e a s t o c k h o l d e r s ' e q u i l i b r i u m h a s b e e n d e f i n e d , i t s e x i s t e n c e c a n q u i t e s i m p l y b e e s t a b l i s h e d f o r t h i s e c o n o m y . A s y m m e t r i c i n f o r m a t i o n w i l l b e i n t r o d u c e d in t h e n e x t s e c t i o n .

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3.1. Feasibility A n y e q u i l i b r i u m m u s t o b v i o u s l y b e f e a s i b l e . T h e f e a s i b l e s e t f o r t h i s e c o n o m y i s d e c r i b e d b y t h e s e t o f p r o d u c t i o n ( xj) a n d c o n s u m p t i o n d e c i s i o n s ( c a ^-) w h i c h s a t i s f y : EjXj + £ a c a1 < W (4 a ) Ea°a2 5 z x j) C 4b) x j > 0 , c a t > 0 , J E T , a E A , t - 1 , 2 ( 4 c ) A s i s c l e a r , t h e f e a s i b l e s e t e x h i b i t s s o m e s t a n d a r d p r o p e r t i e s . I n p a r t i c u l a r , it is c o m p a c t a n d c o n v e x . H o w e v e r , t h i s se t d o e s n o t e x h a u s t i v e l y d e s c r i b e t h e r e l e v a n t c o n s t r a i n t s u n d e r w h i c h t h e e c o n o m y is o p e r a t i n g . A c t u a l l y , o n e h a s t o t a k e i n t o a c c o u n t t h a t a n y c o n s u m p t i o n a n d p r o d u c t i o n p l a n m u s t b e i m p l e m e n t e d w i t h i n t h e e x i s t i n g f e e s y s t e m a n d t h r o u g h j o i n t s t o c k o w n e r s h i p o f t h e f i r m s . In o t h e r w o r d s , w e h a v e t o l i m i t o u r s e l v e s t o c o n s i d e r i n g t h e s t o c k o w n e r s h i p f e a s i b l e s e t o f t h i s e c o n o m y , w h i c h is a ( p r o p e r ) s u b s e t o f t h e f e a s i b l e s e t d e s c r i b e d b y (4 ) . T h i s s t o c k o w n e r s h i p f e a s i b l e s e t is t h e s e t o f p r o d u c t i o n a n d c o n s u m p t i o n d e c i s i o n s w h i c h s a t i s f y : eJ E T x J + r a E A c a1 < w CSa) Cj 2 < 6fj(xj), all JEM (5b) °12 i E jET0 i j C x j ) , all i E N (5c) E iEN0 ij 5 1 > a l l JET ( 5d)

c ati°. x j?°. 0 ij>°. all iEN, all JET, t=1,2 (5e)

w h e r e 5 j( x j) : =( 1-6) f j( x j) , J E M , a n d 5 0 ( x D ) : = f D ( x 0 ) : = r x Q .

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C l e a r l y , 3j( .) i s t h e a g g r e g a t e a m o u n t t o b e p a i d o u t a s C n e t ) d i v i d e n d s ; 0 ^ j is a n e l e m e n t o f t h e ( m + 1 ) x ( n - m ) m a t r i x

i eN

10

0 : * [ 0 ^ j ] j e M , w h i c h d e s c r i b e s a n a l l o c a t i o n o f s h a r e s a m o n g c o n s u m e r s ; f i n a l l y , t h e p a r a m e t e r 6 is c o n s t r a i n e d t o l i e in t h e i n t e r v a l [ 0 , 1 ] i f ( Be) is t o b e s a t i s f i e d . U s i n g t h e v e c t o r n o t a t i o n c a := C c a ^ ,c a 2 ) > w e c a n d e f i n e t h e s e t o f s t o c k o w n e r s h i p f e a s i b l e p r o g r a m m e s a s Z : = { z : = ( x j , c a ,0) | J E T , a E A ; ( S) i s s a t i s f i e d ) (6) T h i s s e t i s c l e a r l y c o m p a c t . It is h o w e v e r n o t g e n e r a l l y c o n v e x , d u e to t h e i n d i v i d u a l c o n s t r a i n t ( 5 c ) , i m p o s i n g t h a t t h e e c o n o m y ' s r e a l a l l o c a t i o n s m u s t b e a t t a i n e d t h r o u g h s h a r e a l l o c a t i o n s . T h i s n o n c o n v e x i t y m a y b e i l l u s t r a t e d in t h e f o l l o w i n g w a y ^ V o o C o n s i d e r t w o p o i n t s z , z EZ, s u c h t h a t x o > x o > 0 a n d o o 0 i j > © i j > O. A s s u m e f u r t h e r t h a t , f o r a l l J E M , X j = X j = 0 , a n d [ c i 2 " 0 i o f o ( x o ) 3 * £ c i 2 “ e i o f o ( x o)3 = °- T h e n , f o r a n y p E ( 0 , 1 ) o o o P<=i2 + ( 1 - p ) c i 2 - [ p 0 i o + C 1 - p ) 0 i o ] [ p x o + ( 1 - p ) x 0 ] r o o o — p C 1 — p ] [ 0 i o ^ x o ~ x io^ — 0 i o ^ * i o — x io) ^ ^ = p ( 1 - p ) ( x 0 - x o ) ( 0 i o - 0 i o ) r > 0 o H e n c e , t h e ( s t r i c t l y ) c o n v e x c o m b i n a t i o n o f z a n d z d o e s n o t b e l o n g t o Z . R e m a r k 2 : T h e n o n c o n v e x i t y j u s t c o n s i d e r e d g i v e s a r a t i o n a l e f o r t h e t w o - s t a g e p r o c e d u r e s u g g e s t e d b y D r è z e : w h e n t h e s h a r e a l l o c a t i o n p r o b l e m is n o t e x p l i c i t l y a n d s e p a r a t e l y t a k e n c a r e of, t h e f i r m s ' c h o i c e s m a y n o t b e c o n s i s t e n t w i t h

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t h e s h a r e h o l d e r s ' . T h a t is, a n e q u i l i b r i u m w h e r e t h e s h a r e a n d c a p i t a l a l l o c a t i o n s a r e s i m u l t a n e o u s l y P a r e t o o p t i m a l c a n n o t b e a c h i e v e d ^ . L e t e a c h s h a r e h o l d e r b e h a v e c o m p e t i t i v e l y a n d t a k e as g i v e n t h e u n i t r e t u r n o n c a p i t a l p a i d o u t b y a n y s h a r e . T h e n , i n v e s t m e n t i n a g i v e n f i r m is o p t i m a l t o h i m w h e n t h e a v e r a g e r e t u r n a n d t h e u n i t c o s t o f c a p i t a l a r e e q u a l ( " a r b i t r a g e " s o l u t i o n ) , w h i c h i s i n c o n s i s t e n t w i t h p r o f i t m a x i m i z i n g ( P a r e t o o p t i m u m ) . T h i s i n c o n s i s t e n c y is f o r m a l i z e d t h r o u g h a s i m p l e g a m e in a p p e n d i x A. E a c h s h a r e h o l d e r t a k e s i n t o a c c o u n t t h e d e p r e s s i n g e f f e c t t h a t h i s o w n i n v e s t m e n t h a s o n t h e a v e r a g e r a t e o f r e t u r n , b u t t a k e s as g i v e n t h e o t h e r i n v e s t o r s ' b e h a v i o u r . I n d e e d , t h e ( N a s h ) s o l u t i o n t o t h i s g a m e t e n d s t o t h e a r b i t r a g e s o l u t i o n a s t h e n u m b e r o f s h a r e h o l d e r s g o e s t o i n f i n i t y , w h i l e t h e P a r e t o o p t i m u m f o l l o w s i f s u c h n u m b e r is s e t e q u a l t o o n e ( c o o p e r a t i v e s o l u t i o n ) . T h e n o n c o n v e x i t y o f t h e ( s t o c k h o l d e r s ' ) f e a s i b l e s e t e n t a i l s t h a t a n y e q u i l i b r i u m a l l o c a t i o n w i l l be , at b e s t , c o n s t r a i n e d P a r e t o e f f i c i e n t . A s t h e g a m e t h e o r e t i c e x a m p l e s h o w s , t h i s is a c o n s e q u e n c e o f t h e m o d e l ' s c o m p e t i t i v e s e t u p . F u r t h e r c o m m e n t s o n t h i s p o i n t w i l l b e o f f e r e d in r e m a r k 3. 3.2. Pseud o eq u i li b r i um L e t us c o n s i d e r a g i v e n p o i n t z = ( [ x j ] j e T >t c a ] a £ A > E Z • T o t h i s p o i n t t h e r e c o r r e s p o n d s a g i v e n s e t o f f i r m s , L(z)c_M, d e f i n e d as t h e s e t o f f i r m s " e x i s t i n g " at z: a f i r m u s i n g t e c h n o l o g y J E M ( f i r m J E M f o r s h o r t ) " e x i s t s " a t z i f at l e a s t o n e c o n s u m e r h o l d s p o s i t i v e s h a r e s in it , t h a t is L ( z ) : = { J E M | I i e N s . t . 0 i J > O } 1 3 . F o r a n y J E L ( z ) , c o n s i d e r t h e c o n s u m p t i o n a n d p r o d u c t i o n p l a n s w h i c h m a y b e a t t a i n e d b y r e d i s t r i b u t i n g t h e i n v e s t e d c a p i t a l b e t w e e n t e c h n o l o g y j a n d t h e s h a r e h o l d e r s ' c u r r e n t c o n s u m p t i o n . T h e s e t F j ( z ) i n c l u d e s a l l s u c h p l a n s , w h i c h

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c a n b e c a r r i e d o u t w h i l e l e a v i n g u n c h a n g e d b o t h t h e s h a r e a l l o c a t i o n a n d t h e a m o u n t o f c a p i t a l i n v e s t e d in t e c h n o l o g i e s o t h e r t h a n j. G i v e n z, a n y o t h e r p o i n t in Z g e n e r a t e d b y r e a l l o c a t i n g r e s o u r c e s t o C o r f r o m ) f i r s t p e r i o d c o n s u m p t i o n m u s t t h e n l i e i n F j ( z ) , t o w h i c h z i t s e l f o b v i o u s l y b e l o n g s . W e t h e r e f o r e d e f i n e F j ( z ) : - { z E Z | x h « x h , a l l he T , h*> j ; 0 = 0 } (7) - 14 T h e s e t F j ( z ) i s c o m p a c t a n d c o n v e x W i t h i n s u c h a s e t , w e a r e n o w l o o k i n g f o r f i r m j ' s p r o d u c t i o n p l a n s t h a t w o u l d be a c c e p t e d a s o p t i m a l b y i t s s h a r e h o l d e r s . T o t h i s e n d , g i v e n m o n o t o n i c i t y o f p r e f e r e n c e s , w e c a n d e f i n e a n i n d u c e d u t i l i t y f u n c t i o n , r e p r e s e n t i n g t h e c o n s u m e r s ' p r e f e r e n c e s o v e r a l t e r n a t i v e p o i n t s in F j( z) , t h a t is v ± C c i 1 » x j J := S i j O j C x j ) + r h e T 0 i h 5 h ( x h ) J ( 8 ) w h i c h , u n d e r o u r a s s u m p t i o n s , is a c o n t i n u o u s , s t r i c t l y 15 q u a s i - c o n c a v e m o n o t o n i c ( i n c r e a s i n g ) f u n c t i o n T h e P a r e t o o p t i m a l a l l o c a t i o n o f c a p i t a l w i t h i n F j ( z ) c a n n o w be c h a r a c t e r i z e d b y n o t i n g t h a t , g i v e n t h e s h a r e a l l o c a t i o n 0, a n y v a r i a t i o n in Xj a f f e c t s s e c o n d p e r i o d c o n s u m p t i o n o f a l l s h a r e h o l d e r s . T h u s , Xj c a n b e t r e a t e d a s a p u b l i c g o o d f o r t h e s o l u t i o n o f o u r p r o b l e m . F i n d i n g a P a r e t o o p t i m a l a l l o c a t i o n w i t h i n F j ( .) t h e n a m o u n t s t o c o n s i d e r i n g a p u b l i c g o o d e c o n o m y , w h e r e (a) V ( . , . ) d e s c r i b e s t h e c o n s u m e r s ' p r e f e r e n c e s o v e r o n e p r i v a t e g o o d ( c-j) a n d o n e p u b l i c g o o d ( x j ) ; ( b) t h e t r a n s f o r m a t i o n s e t b e t w e e n t h e p r i v a t e a n d t h e p u b l i c g o o d is d e s c r i b e d b y t h e l i n e a r c o n s t r a i n t E a e A c a1 + »j < W " E h e T x h , h»-j (9) A P a r e t o o p t i m a l a l l o c a t i o n c a n t h e r e f o r e be c h a r a c t e r i z e d b y s o l v i n g t h e f o l l o w i n g p r o g r a m m e w . r . t . ( c ^ ^ , X j ) , w i t h ( i £ N ) a s a r b i t r a r y p o s i t i v e w e i g h t s .

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Max

s t - Ea £ A c a1 + X j < W - £heT*h> C 10) It is w e l l K n o w n ( e . g . , M a l i n v a u d , 1 9 7 2 , p . 2 1 2 ) t h a t a n y P a r e t o a l l o c a t i o n f o r a p u b l i c g o o d e c o n o m y m u s t s a t i s f y t h e e q u a l i t y b e t w e e n t h e s u m o f t h e c o n s u m e r s ' m a r g i n a l r a t e s of s u b s t i t u t i o n a m o n g p r i v a t e a n d p u b l i c g o o d s a n d t h e m a r g i n a l r a t e o f t r a n s f o r m a t i o n o f p r i v a t e i n t o p u b l i c g o o d s . In t h i s p a r t i c u l a r i n s t a n c e , t h i s c o n d i t i o n b e c o m e s : 5 '( x j) J-ieNn i j 0 i j “ 1 (1 1 ) w h e r e n i j : = n i( c ± -| , x j) =( 6 u 6 c ^ 2 ) / ( <5u ± / i 1) r e p r e s e n t s c o n s u m e r ( s h a r e h o l d e r ) i ' s m a r g i n a l s u b s t i t u t i o n r a t e o v e r t i m e , e v a l u a t e d at (cj_-j,Xj), g i v e n x ^ ( ht= j) a n d 0. E q u a t i o n ( 1 1 ) is a n o p t i m a l i t y c o n d i t i o n o n t h e a m o u n t o f c a p i t a l Xj r e q u i r e d b y a n y f i r m j e L ( z ) to m a x i m i z e i t s s h a r e h o l d e r s ' w e l f a r e , g i v e n t h e s h a r e a l l o c a t i o n 0: it is t a k e n a s t h e p s e u d o e q u i l i b r i u m c o n d i t i o n f o r f i r m j. T h i s e q u a t i o n c a n be s e t i n t e r m s o f q j ( .): b y s t r a i g h t f o r w a r d m a n i p u l a t i o n s o n e o b t a i n s : q j C x j ) - C r E i E N n i j B i j ) 1 ( 1 2 ) w h i c h a l l o w s a n e a s i e r c o m p a r i s o n w i t h t h e s t a n d a r d o p t i m a l i t y c o n d i t i o n , q j ( . ) * 1 . O n l y f i r m s j G L ( z ) h a v e b e e n c o n s i d e r e d t i l l n o w . W e n o w b r i e f l y t u r n to t h e r e m a i n i n g c a s e s . A f i r m n o t e x i s t i n g at z, K E M \ L ( z ) , is f o r m a l l y i d e n t i f i e d w i t h 0 ^ * 0 f o r a l l i E N . W h e n t h i s is so, xj^ d i s a p p e a r s f r o m a l l a g e n t s ' u t i l i t y f u n c t i o n s : it is n o t t r a d e d in t h e f i c t i t i o u s p u b l i c g o o d e c o n o m y . H e n c e , 0 i k = ^» ^ o r a 11 i E N , i m p l i e s x ^ = 0 . F i n a l l y , a p s e u d o e q u i l i b r i u m c a n b e d e f i n e d a l s o f o r t h e C R S t e c h n o l o g y : t h i s c a n b e s e e n a s a f i r m f o r w h i c h n o T S

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c h a r g e s a f e e . T h e p s e u d o e q u i l i b r i u m c o n d i t i o n f o r j = 0 is:

r E i 8 N n i o a i o = 1 C 1 3 J

s o t h a t q 0 ( x o ) = 1 - W e c a n at t h i s p o i n t g i v e a n e x p l i c i t d e f i n i t i o n o f a p s e u d o e q u i l i b r i u m ( D r è z e , 1 9 8 7 , p . 2 7 2 ) .

G i v e n a n y p o i n t z E Z , a p s e u d o eq u i li b r i um f o r J E T is

a point zEFj(z)cZ, such that

Ca) V U i , , x j) > V C c l1 , x j) _> c i1+ n ija lj5jCxj) C i1 + "i ja ij5 x j) . for all 16N;

Cb) q J( x J)-CrI:i ENn i.|0ij) ~ 1 for JELC z) ;

Cc) xj = 0 for j E M \ L (z) , and cr '"o X 0 II A p s e u d o e q u i l i b r i u m is t h e r e f o r e s u c h t h a t t h e s h a r e h o l d e r s ' w e l f a r e i s m a x i m i z e d (a) b y c h o o s i n g a n a p p r o p r i a t e a c t i v i t y l e v e l ( b , c ) . F o r m a l l y , t h i s is r e a l l y t h e s t a n d a r d d e f i n i t i o n o f a " L i n d a h l e q u i l i b r i u m " f o r a p u b l i c g o o d e c o n o m y , w h e r e n ij a r e " L i n d a h l p r i c e s " ( e . g . , M i l l e r o n , 1 9 7 2 ) . O n e h a s n o w t o s h o w t h a t , g i v e n z e Z ( a n d t h e r e l a t e d s h a r e a l l o c a t i o n 8) , t h e r e i n d e e d e x i s t s a s e t := { n ^ j | i E N > o f w e i g h t s w h i c h w a r r a n t e x i s t e n c e o f a p s e u d o e q u i l i b r i u m f o r a n y J E T . T h i s is e s t a b l i s h e d in t h e f o l l o w i n g p r o p o s i t i o n . Prop o s it i o n 1. U n d e r a s s u m p t i o n s A . 1 t o A . 3, a n d g i v e n a n y p o i n t z E Z s u c h t h a t £ i e N c i 1 > 0 > ^ o r a n y J E T : (a ) t h e r e e x i s t s a p o i n t z E F j ( z ) w h i c h is a p s e u d o e q u i l i b r i u m ; ( b) z is a P a r e t o o p t i m u m w i t h i n F j ( z ) ; ( c) t o a n y s u c h o p t i m u m o n e c a n a s s o c i a t e a s e t o f w e i g h t s n J , s u c h t h a t ( z , n ^ ) i s a p s e u d o e q u i l i b r i u m . c o n v e x i t y a n d c o m p a c t n e s s o f F j ( .), t h i s G i v e n r e s u l t

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f o l l o w s b y s t a n d a r d a r g u m e n t s 1 6 . It is s i m p l y a v e r s i o n o f D r è z e ' s T h e o r e m 3. 1 ( D r è z e , 1 9 0 7 , p . 2 7 3 ) . 3.3. Price e q u i l i br i u m W e n o w t u r n t o t h e i n v e s t o r s ' o p t i m a l c h o i c e s g i v e n t h e f i r m s ' p r o d u c t i o n p l a n s . S t a r t i n g f r o m a n y p o i n t z E Z , a l t e r n a t i v e p o i n t s c a n b e g e n e r a t e d v i a a l t e r n a t i v e a l l o c a t i o n s o f s h a r e s , i . e . t h r o u g h c h a n g e s in 0 m a t c h e d b y v a r i a t i o n s in t h e c o n s u m e r s ' f i r s t p e r i o d c o n s u m p t i o n . F o l l o w i n g a g a i n D r è z e ( 1 9 8 7 , p .2 75 ) w e d e n o t e b y E( z) t h e s e t o f s t o c k o w n e r s h i p f e a s i b l e p r o g r a m m e s w h i c h c a n be o b t a i n e d b y r e a l l o c a t i n g f i r s t p e r i o d c o n s u m p t i o n a n d s h a r e s : E ( i ) : = { z £ Z | x j » X j , a l l J E T } ( 1 6 ) W e c a n t r e a t E ( z ) a 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 f o r a n e x c h a n g e e c o n o m y , w h e r e f i r s t p e r i o d c o m m o d i t y a n d s h a r e s a r e t h e c o m m o d i t i e s b e i n g t r a d e d . T h i s s e t is c o m p a c t a n d - 17 c o n v e x , Xj ( J E T ) b e i n g g i v e n . W e n o w l o o k f o r t h e c o n s u m p t i o n - s h a r e a l l o c a t i o n s w h i c h a r e P a r e t o o p t i m a l w i t h i n E ( z ) . L i k e in t h e f o r m e r c a s e , w e c a n i d e n t i f y e a c h c o n s u m e r ' s p r e f e r e n c e s o v e r a l l o c a t i o n s in E( z) b y t h e i n d u c e d u t i l i t y f u n c t i o n W l ( c i 1 > 0 i ) := U i C c i , , E j E T B i j B j C J j ) ] ( 1 7 ) w h e r e ^ i : ^ i j ]j gT t h e C m + 1) v e c t o r o f s h a r e s t r a d e d b y c o n s u m e r i. T h i s f u n c t i o n is c o n t i n u o u s , s t r i c t l y q u a s i ­ c o n c a v e a n d ( m o n o t o n i c a 1 ly) i n c r e a s i n g . T h e s e t o f P a r e t o o p t i m a l s h a r e a l l o c a t i o n s , g i v e n a p o i n t z E Z at w h i c h x j = x j f o r a l l J E T , c o i n c i d e s w i t h t h e s e t o f P a r e t o o p t i m a l a l l o c a t i o n s o f a f i c t i t i o u s e c o n o m y s u c h t h a t : (a) E ( z ) d e s c r i b e s i t s f e a s i b l e s e t ; ( b) W 1 , i £ N , d e s c r i b e s i t s

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t r a d e r s ' p r e f e r e n c e s . G i v e n t h e o u t l i n e d f e a t u r e s o f a n d E ( z ) , s u c h a n a l l o c a t i o n e x i s t s , a n d c a n b e d e c e n t r a l i z e d b y a n a p p r o p r i a t e p r i c e s y s t e m . It is t h e r e f o r e t e r m e d p r i c e e q u i l i b r i u m . W e n o w , f i r s t , d e f i n e a p r i c e e q u i l i b r i u m , t h e n e s t a b l i s h i t s e x i s t e n c e , a n d f i n a l l y l o o k m o r e p r e c i s e l y i n t o i t s f e a t u r e s . T o d e f i n e a p r i c e e q u i l i b r i u m , c o n s i d e r t h e f i c t i t i o u s e c o n o m y r e f e r r e d t o a b o v e , w h e r e ( m + 2) c o m m o d i t i e s a r e t r a d e d . W e d e f i n e a p r i c e v e c t o r p £ R + + + 1” f o r t h i s e c o n o m y , a n d n o r m a l i z e t h e p r i c e o f t h e c o n s u m p t i o n g o o d p c = 1 1 0 . T h e m a r k e t c l e a r i n g c o n d i t i o n s a r e g i v e n b y *-ieN0 i j - 1 (18a) E a 8 A c a1 * w ~ E J 8 T X J M B b ) T h e f o l l o w i n g d e f i n i t i o n c a n n o w b e g i v e n : ■B-A price e q ui l i b r i u m is a p a i r (z , -> -p c =1 , z e Z , s u c h t h a t : (a) f o r a l l i 8 N , W 1 [ c i 1 ( B i ) > ■H- # * c i 1 + E ^ 8 T s i j P j > c i l + E J 8 T 0 lj Pj; (b ) E i E N c i 1 * w -eJ E T Xj; ( c) f o r a l l J E T s u c h t h a t x j > 0 , ^ i £ N 0 i j = 1 * T h i s is t h e d e f i n i t i o n o f a c o m p e t i t i v e e q u i l i b r i u m f o r t h e e x c h a n g e e c o n o m y d e s c r i b e d b y (1 6) a n d ( 1 ? ) . W e n o w e s t a b l i s h i t s e x i s t e n c e , i . e . t h a t of a P a r e t o o p t i m a l s h a r e a l l o c a t i o n ( a n d o f a p r i c e v e c t o r w h i c h s u p p o r t s i t ) , g i v e n t h e f i r m s ' p r o d u c t i o n p l a n s . T h i s is d o n e in t h e f o l l o w i n g p r o p o s i t i o n : Prop o s it i o n 2. U n d e r a s s u m p t i o n s A . 1 to A . 3, g i v e n a n y z E Z s u c h t h a t £■}.g^c;i. i > 0 : (a) t h e r e e x i s t s a p r i c e e q u i l i b r i u m ; ( b) a n y p r i c e e q u i l i b r i u m is a P a r e t o o p t i m u m ; p * ) 8 E ( ^ ) x R + + + 2 , i •}£ w ( c n . B j ) =>

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( c) to a n y P a r e t o o p t i m u m z E E (z) o n e c a n a s s o c i a t e * m + 2 * * a p r i c e v e c t o r p E R + + s u c h t h a t ( z ,p ) is a p r i c e e q u i l i b r i u m . G i v e n c o n v e x i t y a n d c o m p a c t n e s s o f E ( z ) , t h i s r e s u l t f o l l o w s b y s t a n d a r d c o m p e t i t i v e a r g u m e n t s . W e n o w c h a r a c t e r i z e a p r i c e e q u i l i b r i u m . T o d o so, n o t i c e t h a t , at s u c h a n e q u i l i b r i u m , t h e o p t i m a l s h a r e p o r t f o l i o f o r e a c h s h a r e h o l d e r m u s t s a t i s f y ( D r è z e , 1 9 8 7 , p . 2 7 5 ) :

<

.

6 W 1 / 6 c i 1 ” PJ if-w i t h e q u a l i t y if 0 ^ j > O , a s t h e s t a n d a r d m a r g i n a l u t i l i t y p r i c e r a t i o o p t i m a l i t y r u l e . In t h e p r e s e n t f r a m e w o r k t h i s b e c o m e s n i S j Û j ) < P j ( 19) w h e r e n i : “ n i( c-^-j , 0-^) is t h e u s u a l m a r g i n a l r a t e o f

«-s u b «-s t i t u t i o n o v e r t i m e , e v a l u a t e d at ( c i 1 > 0 i ) > g i v e n x :■[ xj] j 6 j . T h e e q u i l i b r i u m c o n d i t i o n (1 9 ) e n t a i l s t h a t p o s i t i v e t r a d e s o f s h a r e s , 0 i j > Q , t a k e p l a c e at a p r i c e p j = ri iS j t x j ) , a l l J E T , a l l i £ N . if- * H e n c e , pj a n d Xj b e i n g g i v e n t o a n y a g e n t i E N , n-j^ i_s^ e q u a l ^ if-f o r a l l c o n s u m e r s at a^ p r i c e e q u i l i b r i u m , n ^ = n , a l l i s N . T h i s is n o t s u r p r i s i n g : in e q u i l i b r i u m , t h e m a r g i n a l r a t e o f s u b s t i t u t i o n o v e r t i m e m u s t b e t h e s a m e t o a l l c o n s u m e r s . S i n c e 5 j ( 0 ) = 0 , t h e s h a r e s J E T s u c h t h a t X j = 0 d i s a p p e a r f r o m t h e a g e n t s ' u t i l i t y f u n c t i o n a n d a r e n o t t r a d e d : x j= 0 i* i m p l i e s 0 ^ j = O f o r a l l i E N .

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3.4. Full information equi l i br i u m W e n o w s p e c i f y a f u l l i n f o r m a t i o n s t o c k h o l d e r s ' e q u i l i b r i u m ( F I E ) f o r o u r e c o n o m y . A F I E is b o t h a p s e u d o e q u i l i b r i u m f o r e a c h f i r m a n d a p r i c e e q u i l i b r i u m f o r t h e c o n s u m e r s . It c a n t h e r e f o r e b e d e f i n e d a s f o l l o w s : A ( f u l l i n f o r m a t i o n ) stockholders' e q ui l i b r i u m is a p o i n t z * E Z s u c h t h a t : 1 * (a) f o r e v e r y J E T , t h e r e e x i s t s a s e t o f w e i g h t s n J s u c h t h a t ( z ,n ) i s a p s e u d o e q u i l i b r i u m f o r j; -a- m + 2 ( b) t h e r e e x i s t s a p r i c e s y s t e m p E R + + , s u c h t h a t •a- * (z ,p ) is a p r i c e e q u i l i b r i u m . W e h a v e t o e s t a b l i s h t h e e x i s t e n c e o f a F I E , a n d t h e n c h a r a c t e r i z e i t . B e f o r e d o i n g so, h o w e v e r , l e t us r e c a p t h e g e n e r a l a r g u m e n t f o l l o w e d so f a r . C o n s i d e r a n y p o i n t z £ Z s u c h t h a t E i £ N C i i > 0 a n d a n y J E T . U n d e r p r o p o s i t i o n 1, o n e c a n a s s o c i a t e at z a s e t o f w e i g h t s := { n ^ j | i e N } s u c h t h a t ( z ,n J ) , z e F j ( z ) , i s a p s e u d o e q u i l i b r i u m . O n t h e o t h e r h a n d , - a- -b y p r o p o s i t i o n 2, t o z t h e r e c o r r e s p o n d a l s o a p o i n t z E E ( z ) m + 2 -K- * a n d a p r i c e v e c t o r p E R + + s u c h t h a t ( z ,p ) is a p r i c e e q u i l i b r i u m . T h e l a t t e r a l s o i d e n t i f i e s a n e q u i l i b r i u m s e t ■H- # * * o f w e i g h t s n : = { n i | i 6 N > , h a v i n g t h e p r o p e r t y t h a t n ^ = n f o r a l l i E N . A s a n a t u r a l c o n s i s t e n c y r e q u i r e m e n t , w e t h e r e f o r e ~ -j -a-e x p -a-e c t t h a t fl - (1 f o r a l l J G T at a F I E . T h e f o l l o w i n g p r o p o s i t i o n e s t a b l i s h e s e x i s t e n c e o f a F I E : Prop o s it i o n 3. U n d e r a s s u m p t i o n s A . 1 t o A . 3, t h e r e e x i s t s a f u l l i n f o r m a t i o n ( s t o c k h o l d e r s ' ) e q u i l i b r i u m . S e e D r è z e ( 1 9 0 7 ) , T h e o r e m 3 . 3 , f o r a p r o o f o f t h i s p r o p o s it i o n .

*

C o n s i d e r n o w a F I E , z EZ. It c a n b e c h a r a c t e r i z e d b y u s i n g t h e F I E d e f i n i t i o n a n d p r o p o s i t i o n s 1 a n d 2. T h e

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