Asset Ownership and Market Structure in Oligopoly - Further Results

8/9/2019 Asset Ownership and Market Structure in Oligopoly - Further Results

1/6

ELSEVIER Ec ono m ics Le t ters 50 (1996) 437 -442e c o n o m i c sl e t t e r s

A s s e t o w n e r s h i p a n d m a r k e t s t r u c t u r e i n o l i g o p o l y :F u r t h e r r e s u l t s

S a n g - S e u n g Y iDepartment of Economics, Dartmouth College, Hanover, N H 03755, U SA

Rec eived 14 June 1995; accep ted 2 October 1995

A b s t r a c tHow does the sa le o f asse t s f rom a smal l f i rm to a l a rge f i rm af fec t the equ i l ib r ium pr ice in o l igopoly? Us ing the' c ro s s - s ec ti o n a l d i f f e r en t i a ti o n ' t e ch n i q u e i n t ro d u ced b y Fa r r e l l an d Sh ap i ro (Rand Journal o f Economics, 1990, 21,2 7 5 -2 9 2 ) , I s h o w t h a t t h e eq u i l ib r i u m p r i ce r is e s i f t h e co m m o n p ro d u c t i o n t ech n o l o g y is h o mo g en eo u s o f d eg ree t ,0 < t ~ < l .

Keywords: O l i g o p o l y ; A s s e t s a l e s ; O u t p u t p r i ceJEL classif ication: L 13

1 . I n t r o d u c t i o nI n a r e c e n t p a p e r , F a r r e l l a n d S h a p i r o ( 1 9 9 0 ) p r e s e n t a n e l e g a n t g e n e r a l a n a l y s i s o f t h ee f f e c ts o f c h a n g e s i n a s s e t o w n e r s h i p i n o l i g o p o l is t ic m a r k e t s . S e c t i o n 5 o f t h e i r p a p e re x a m i n e s t h e e f fe c t s o f c a p it a l s al es b e t w e e n f ir m s o n p r i c e a n d w e l f a r e . A k e y r e s u l t in t h a ts e c t i o n i d e n t if i e s c o n d i t io n s o n t h e c o s t fu n c t i o n u n d e r w h i c h a s m a l l c a p it a l s a le t h a t i n c r e a s e st h e c o n c e n t r a t i o n o f c a p i ta l r a is e s p r i ce :

Propos i t i on 3 ( F a r r e l l a n d S h a p i r o ) . Sup pose that a l l f i rm s hav e access to the sam etechnolo gy. Increasing the concentra t ion o f capital own ership necessarily reduces outpu t i fmarg ina l cos t i s a convex f unc t ion o f ou tpu t and cap i ta l .

U n f o r t u n a t e l y , t h is c o n v e x i t y c o n d i t i o n is q u i te r e s tr ic t iv e . F o r e x a m p l e , a s I s h o w b e l o w ,t h i s c o n d i t i o n i s v i o l a t e d f o r c o s t f u n c t i o n s t h a t a r e d u a l t o t h e p r o d u c t i o n f u n c t i o n s w i t hc o n s t a n t r e t u r n s t o s c al e ( C R S ) . C R S p r o d u c t i o n f u n c t io n s in c l u d e t h e w i d e l y s t u d i e d ,c o n s t a n t e l a s t i c i t y o f s u b s t i t u t i o n ( C E S ) p r o d u c t i o n f u n c t i o n s , w h i c h i n t u r n i n c l u d e t h eC o b b - D o u g l a s , l i n e a r a n d L e o n t i e f p r o d u c t i o n f u n c t io n s as sp e c i a l c a se s .0165-1765/96 /$12 .00 1996 Elsev ier Science S .A . Al l r igh t s reserv edS S D I 0 1 6 5 - 1 7 6 5 ( 9 5 ) 0 0 7 7 6 - 8


2/6

438 S.-S. Yi / Economics Letters 50 (1996) 437-442

N o n e t h e l e s s , F a r r e l l a n d S h a p i r o ' s ' c r o s s - s e c t i o n a l d i f f e r e n t i a t i o n ' t e c h n i q u e p r o v e s u s e f u li n e x t e n d i n g t h e i r r e su l t . B y a p p l y i n g t h e i r t e c h n i q u e , I f in d a n o t h e r s u f f i ci e n t c o n d i t i o n t h a tg u a r a n t e e s t h a t t h e e q u i l i b r i u m p r i c e i n c r e a s e s as a re s u l t o f a n a s s e t t r a n s a c t i o n f r o m a f ir mw i t h a s m a l l s to c k o f c a p i ta l t o a f i rm w i t h a l a rg e s t o c k o f c a p i ta l . M y c o n d i t i o n s t a t e s t h a t t h ec o m m o n p r o d u c t i o n f u n c t i o n i s h o m o g e n e o u s o f d e g r e e t , 0 < t ~< 1. T h e C R S p r o d u c t i o nf u n c t i o n s a r e o b t a i n e d w h e n t = 1 . W h e n t < 1, t h e 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 gr e t u r n s t o s c al e . H e n c e , t h is p a p e r s i g n i fi c an t ly i n c r e a s e s t h e a p p l i c a b i li t y o f F a r r e l l a n dS h a p i r o ' s P r o p o s i t i o n 3 .I a ls o g e n e r a l iz e F a r r e l l a n d S h a p i r o ' s P r o p o s i t i o n 4 , w h i c h s h o w s t h a t , i f t h e c o m m o nt e c h n o l o g y ex h i b i ts C R S , t h e n t h e d i r e c t e f f ec t o f i n c r e a s i n g t h e c o n c e n t r a t i o n o f c a p it a l is t oi n c r e a s e t h e f i r m s ' c o s ts . I e x t e n d t h e i r r e su l t to t h e p r o d u c t i o n f u n c t i o n s t h a t a r e h o m o g e -n e o u s o f d e g r e e t , 0 < t ~< 1 .

2 . T h e r e s u ltT o f a ci li ta t e c o m p a r i s o n w i t h F a r re l l a n d S h a p i r o ' s m o d e l , I e m p l o y t h e i r n o t a t i o n . L e t

p ( X ) b e t h e i n v e rs e d e m a n d , w h e r e p i s p r ic e , X is i n d u s t r y o u t p u t a n d p ' ( X ) < 0 . T h ec o m m o n c o s t f u n c t i o n is gi v e n b y c ( x e, k i ) , w h e r e x i is f i rm i ' s o u t p u t a n d k i is f i r m i ' s c a p i t a ls t o c k . I d e n o t e p a r t i a l d e r i v a t i v e s b y s u b s c r i p t s a n d a s s u m e t h a t Cxk < 0 a n d C x k k ~ O ' -a d d i t i o n a l c a p i t a l l o w e r s t h e m a r g i n a l c o s t c u r v e , b u t a t a d e c r e a s i n g r a t e .

2c x is c o n v e x i n ( x , k ) i f a n d o n l y i f Cxxx >t O , Cxk k >! O, a n d CxxxCxk k - Cxx k >I 0 . I f i rs t s h o w t h a tc o s t f u n c t i o n s d u a l t o t h e C R S p r o d u c t i o n f u n c t i o n s v i o l a t e t h i s c o n v e x i t y c o n d i t i o n .W h e n t h e p r o d u c t i o n f u n c t i o n is h o m o g e n e o u s o f d e g r e e t , t h e d u a l c o s t fu n c t i o n s a ti sf ie s

c ( ' x , A k ) = k ) . (1 )W h e n t = 1, E q . ( 1 ) r e d u c e s t o t h e c a se o f C R S . D i f f e r e n t i a t i n g ( 1 ) w i t h r e s p e c t to A a n ds e t t i n g A = 1 y i e ld s

t x c x + k c k = c . ( 2 )D i f f e r e n t i a t i o n o f ( 2 ) w i t h r e s p e c t t o x a n d k y i e l d s , r e s p e c t i v e l y ,

a n d( t - 1)c x + tXCxx + kC~k = 0 ( 3 )

t x c x , + k c ~ = 0 .S i m i l a r l y , d i f f e r e n t i a t i o n o f ( 3 ) w i t h r e s p e c t t o x a n d k y i e l d s , r e s p e c t i v e l y ,

( 2 t - 1 ) C x x + t X C x x~ + k c x x k = 0a n d

tCxk + IXCxxk d- kCxk k = 0 .

( 4 )

(5 )

( 6 )


3/6

s.-s . Yi / Economics Letters 50 (1996) 437-442 439U s i n g t h e s e r e s u l ts f o r t = 1 , I e st a b li s h t h a t t h e C R S t e c h n o l o g y v i o l a te s t h e c o n v e x i t y

c o n d i t i o n u s e d i n F a r r e l l a n d S h a p i r o ' s P r o p o s i t i o n 3 .

L e m m a . F o r a C R S t ec h no l og y ( t = 1 ) , m a r g i n a l c o s t i s n o t c o n v e x i n o u t p u t a n d c a p i t a l .P r o o f . I s h o w t h a t CxxxCxk -- C2xk < 0 fo r t = 1 .

- 1CxxxCxkk--Cxxk-- k CxxxCxk . .~ XCxx xCxxk] 2- - - Cxxk ( b y ( 6 ) )- 1- k [CxxxCxk+ Cxxk(XCxxx + kCxxk)]--1-- k [CxxxCxk--CxxkCxx] ( b y ( 5 ) )Cxxk2 [XC~x + kCx~k] ( b y ( 3 ) )

2- - C x xk2 < 0 ( b y ( 5 ) ) . O . E . D .E x a m p l e s o f C R S t e c h n o l o g y i n c lu d e t h e C E S p r o d u c t i o n f u n c t i o n , x = ( a l k P + a 2 vP ) l/ p,w h e r e k i s c a p i t a l, v i s t h e v a r i a b l e i n p u t a n d 1 / ( 1 - p ) i s t h e e l a s t ic i ty o f s u b s t i t u ti o n . T h e

C E S p r o d u c t i o n f u n c t i o n c o n t a in s s e v e r a l w e l l - k n o w n p r o d u c t i o n f u n c t i o n s , s u c h as t h e l i n e a rp r o d u c t i o n f u n c t i o n ( x = a l k + a 2 v) , t h e C o b b - D o u g l a s p r o d u c t i o n f u n c t i o n ( x = k ~v 1 ~) , a ndt h e L e o n t i e f p r o d u c t i o n f u n c t i o n ( x = m i n [ k , v ] ) ( s e e V a r i a n , 1 9 92 ). H e n c e , t h e a p p l i c a b il i tyo f F a r r e l l a n d S h a p i r o ' s P r o p o s i t i o n 3 i s r a t h e r l i m i t e d .H o w e v e r , i t i s p o s s i b l e t o e x t e n d F a r r e l l a n d S h a p i r o ' s r e s u l t b y a p p l y i n g t h e i r ' c r o s s -s e c t i o n a l d i f f e r e n t i a t i o n ' t e c h n i q u e . T h i s t e c h n i q u e d e t e r m i n e s w h e t h e r a f u n c t i o n f ( x , k )i n c r e a s e s o r d e c r e a s e s w i t h k , l o o k i n g a c r o s s f i r m s i n a g i v e n e q u i l i b r i u m a n d a c c o u n t i n g f o rt h e c r o s s - s e c t i o n a l v a r i a t i o n i n x t h a t a c c o m p a n i e s t h e v a r i a t i o n i n k ( F a r r e l l a n d S h a p i r o ,1 9 9 0 , p . 2 9 1 ) . H e n c e , i n a c r o s s - s e c t i o n a l d i f f e r e n t i a t i o n , w e h o l d t h e a g g r e g a t e o u t p u t f i x e da n d e x a m i n e , f o r e x a m p l e , r e g a r d l e s s o f w h e t h e r o r n o t a f ir m w i t h a l a r g e s to c k o f c a p i t a lp r o d u c e s m o r e o u t p u t t h a n a f i r m w i t h a s m a l l s t o c k o f c a p i t a l i n a g i v e n e q u i l i b r i u m . U s i n gt h is c r o s s - s e c t io n a l d i f f e r e n t i a t i o n t e c h n i q u e , t h e f o l l o w i n g r e s u l t e x t e n d s F a r r e l l a n d S h a p i r o ' sP r o p o s i t i o n 3 t o p r o d u c t i o n f u n c t i o n s t h a t a r e h o m o g e n e o u s o f d e g r e e t , 0 < t ~< 1.P r o p o s i t i o n 3 ' . S u p p o s e t h at a ll f i r m s h a v e a cc e ss t o t h e s a m e p r o d u c t i o n f u n c t i o n , w h i c h i sh o m o g e n e o u s o f d e g r e e t , 0 < t


4/6

440 s.-s . Yi / Economics Letters 50 (1996) 437- 44 2desd k 6 -

- 1 [ [ " deS ) / d e ' ) ]( C x x _ p , ) : L ,C x *- d-E X + C x ** ( C x x - p ' ) - C x , \C x x- d-E x + c * * * , ( 7 )w h e r e t h e s u p e r s c r i p t ' c s ' s t a n d s f o r t h e c r o s s - s e c t i o n a l d e r i v a t i v e .

S u b s t i t u t i n g f o rdcs - C x kd k x - ( = 6 )Cxx - p ' ( X )

a n d r e a r r a n g i n g , w e h a v ed C S - - 1 t 2 Cdk 6 - (Cxx _ p , ) 3 [~9 - 2p '12 + (p ) ~kk] , ( 8 )

2 2w h e r e ~9 = C~xxC~k - 2 C x x k C x x C x k + CxkkCxx an d 12 = CxkkCxx - - CxxkCxk. T h e c l a i m h o l d s if b o t h Oa n d 12 a r e n o n - n e g a t i v e w h e n t ~< 1 .F r o m ( 3 ) , Cxx > 0 w h e n t ~< 1 . S u b s t i t u t i n g fo r Cxxk,

0 ~ - C x k k C x x "~- ( t C x k d - k C x k k ) C x k / t X= Cxkk(Cxx + kC xk/tX ) + C2k/X

2= (1 - t )CxkkCx/ tX + Cxk/X > 0 ,a n d

0 = cx (CxxxCxk - CxxkC xx ) += --C~k[(kcxx k + ( 2 t - 1)Cxx)C~k/tX + C~xkCxx + C~xO= - ( 2 t 2- 1)C~xC~ k/tx - (Cx~ + kCxk/tX )C~ kC~ k + C~xO= - ( 2 / - 2 21)CxxCxk/tX -- (1 - - t)CxCxxkC~k/tX + Cxx[(1 -- t )C~kkCx/ tx + Cxk/X ]= (1 - - t ) ( c ~ C 2 k + C x 1 2 ) / t x ~ O . Q . E . D .

I n t h e p a r a g r a p h f o l l o w i n g t h e i r P r o p o s i t i o n 3 , F a r r e l l a n d S h a p i r o a r g u e t h a t , f o r t h eC o b b - D o u g l a s p r o d u c t i o n f u nc t io n x = k ~ v 1 -~ , w h e r e k i s c a p i t a l a n d v i s a v a r i a b l e i n p u t , c xis c o n v e x i n ( x, k ) i f a n d o n l y i f a < 1 / 3 . T h e i r c l a i m is in c o r r e c t . S i n c e th e C o b b - D o u g l a sp r o d u c t i o n f u n c t i o n e x h ib i ts C R S , t h e c o n v e x i t y c o n d i t io n i s v i o l a te d f o r a ll v a l u e s o f a .

I t i s t r u e , h o w e v e r , t h a t a g e n e r a l i z e d C o b b - D o u g l a s f u n c t i o n x = k ~ v ~ s a t i s f i e s t h ec o n v e x i t y a s s u m p t i o n i f a = f l < 1 / 3 . T o s e e w h y , n o t i c e t h a t , f o r t h e g e n e r a l i z e d C o b b -D o u g l a s f u n c t i o n ,[1 - o~][1 - 2 a ] c x

C x x x ~ . 2 2Ot X- [ 1 - a l f l c xC x x k o t 2 x k

+C x k k - - o t 2 k 2


5/6

s.-s . Yi / Economics Letters 50 (1996) 437-4 42 441a n d

2 [1 -a] /3[13x2 k22a /3] c2C x x x C x k k - - C x k k - - O lT h u s , c x i s c o n v e x i n ( x , k ) i f a n d o n l y i f a I-- 1 o r 2 a + / 3 ~< 1 . W e o b t a i n F a r r e l l a n d S h a p i r o ' sc o n d i t i o n f r o m t h e s e c o n d i n e q u a l i t y b y s e t t i n g a = / 3 .T h i s e x a m p l e i l l u s t r a t e s t h a t F a r r e l l a n d S h a p i r o ' s c o n d i t i o n a n d m y c o n d i t i o n a r e n o ts u b s et s o f e a c h o th e r : t h e g e n e r a l i z e d C o b b - D o u g l a s f u n c t io n is h o m o g e n e o u s o f d e g r e e t,0 < t ~< 1 , i f an d o n ly i f a + /3 ~< 1 . M y co n d i t io n d o e s n o t ap p ly w h e n ot 1-- 1 , b u t F a r re U an dS h a p i r o ' s c o n d i t i o n d o e s . N o t i c e , h o w e v e r , t h a t Cxxk 4 0 ( an d c ~ I --0 ) i f an d o n l y i f a ~< 1 .H e n c e , i f w e r e q u i r e t h e p l a u s i b le c o n d i t i o n t h a t a d d i t i o n a l c a p it a l l o w e r s t h e s l o p e o f t h em a r g i n a l c o s t c u r v e (Cxxk ~ 0 ) a s w e l l a s t h e m a r g i n a l c o s t c u r v e i t s e l f (C~k < 0 ) , m y c o n d i t i o ns u b s u m e s F a r r e l l a n d S h a p i r o ' s , e x c e p t w h e n a = 1. W h e n a = 1 , cx~ i s i d e n t i c a l l y e q u a l t oz e r o . T h e r e f o r e , d " 8 / d k < 0 i f a n d o n l y i f Cxkk ~ O.L a s t l y , I e x t e n d F a r r e l l a n d S h a p i r o ' s P r o p o s i t i o n 4 w h i c h s h ow s t h a t , w h e n a ll f ir m s h a v ea c c e s s to t h e s a m e C R S t e c h n o l o g y (t = 1 ) , t h e d i r e c t e f fe c t o f a l a r g e r f ir m b u y i n g s o m e o f as m a l l e r f i r m ' s a s s e t s i s t o i n c r e a s e t o t a l p r o d u c t i o n c o s t s .C o n s i d e r a t r a n s f e r o f c a p i t a l f r o m a s m a l l f i r m 2 t o a l a r g e f ir m 1 : d k = ( d k , - d k , O , . . . , O )a n d k I > k 2. H o w d o e s t h e a b o v e t r a n s f e r o f c a p it a l a f fe c t t h e c o m b i n e d c o s ts o f p r o d u c t i o n o ft h e t w o f i r m s i n e q u i l i b r i u m ? T h a t i s t o s a y , w h a t i s t h e s i g n o f d ( c I + c 2 ) / d k = c I - c 2 ? T h is i sp o s i t i v e i f a n d o n l y i f d " C k / d k > 0 : c k i n c r e a s e s w i t h k i n t h e c r o s s - s e c t i o n o f f i r m s i n a g i v e ne q u i l i b r i u m . A s F a r r e l l a n d S h a p i r o p o i n t o u t , t h e s a l e o f c a p i t a l f r o m a s m a l l f i r m t o a l a r g ef i r m h a s t w o c o n f l i c t i n g e f f e c t s o n c o s t s , s i n c e d ~ c k / d k = Ck k + Cxk , d " x / d k = Ck k " 1 - ~ C x k . O nt h e o n e h a n d , l a r g e r f i r m s a l r e a d y h a v e m o r e c a p i t a l , a n d Ckk > 0 f o r t > 0 b y E q . ( 4 ) . O n t h eo t h e r h a n d , a d d i t i o n a l c a p i t a l r e d u c e s m a r g i n a l c o s t s (C~k < 0 ) a n d 8 > 0 . F a r r e l l a n d S h a p i r os h o w t h a t t h e f i r s t e f f e c t ( C k k ~ 0 ) d o m i n a t e s t h e s e c o n d e f f e c t (SCxk < 0 ) f o r C R S t e c h n o l o g y( t = 1 ). I e x t e n d t h e i r a n a l y s is t o p r o d u c t i o n f u n c t i o n s th a t a r e h o m o g e n e o u s o f d e g r e e t ,0 < t ~< 1 . E q . ( 4 ) p r o v i d e s a n i n t u i t i v e e x p l a n a t i o n f o r m y r e s u l t : a s t b e c o m e s s m a l l e r , t h em a g n i t u d e o f Ckk r e l a t i v e t o c~k i n c r e a s e s i n a b s o l u te v a l u e , m a k i n g i t m o r e l i k e ly t h a t t h e s a l eo f c a p i t a l f r o m a s m a l l f i r m t o a l a r g e f i r m i n c r e a s e s t h e c o s t s o f p r o d u c t i o n .P r o p o s i t i o n 4 ' . S u p p o s e t h a t a ll f i r m s h a v e a cc e s s t o t h e s a m e p r o d u c t i o n f u n c t i o n , w h i c h ish o m o g e n e o u s o f d e g r e e t , 0 < t Ckk -- C~k/Cx~ = (Ckk/Cxx)[Cxx -- Cxk2Cxx - Cxk/C kk = C~x + kC ~k /tX ( b y ( 4 ) )

= ( t xcx~ + kC~k) / t x= (1 -- t ) c ~ / t x ( b y ( 3 ) )> / 0 Q . E . D .


6/6

4 4 2 S . - S . Y i / E c o n o m i c s L e t t er s 5 0 ( 1 9 9 6 ) 4 3 7 - 4 4 2R e f e r e n c e sFarre l l , J . and Shap iro , C . , 1990 , Asse t ownersh ip and ma rket s t ruc ture in o l igopoly , Ran d Journa l o f Econ omic s21 , 275-292 .Var ian , H . , 1992 , Microec onom ic analysis , 3 rd edn . ( No r ton , N ew York) .

Documents

Asset Ownership and Market Structure in Oligopoly - Further Results