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Economic s Let ters 41 (1993) 139-143 1390165-1765/93/$06.00 1993 Elsevier Science Publishers B.V. All rights reserved

P r i c e a d j u s t m e n t a n d m a r k e t s t r u c t u r eChris topher Mart in *Department of Economics, Queen Mary and Wesrfield College, University of London, Mile End Road,London E1 4NS, UKReceived 1 September 1992Accepted 29 September 1992

Ab s t r a c t

This paper shows how the extent of price inertia depends on the form of market competition. We first establish that thespeed of price adjustment dep ends on the curvature of the profit function in the region around the optimum price. We thenshow how this depends on market structure. We prove the following results:(1) the s peed of price adjustment depends positively on the price elasticity of demand;(2) the speed of price adjustment depend s positively on the number of firms in a market; and(3) the sp eed of price adjustment depends negatively on the degree of collusion between firms.

I . I n t r o d u c t i o n

N o m i n a l r i g id i ti e s i n pr ic e s e t t in g w e r e a t t h e c e n t r e o f e a rl y K e y n e s i a n m o d e l s a n d h a v er e g a i n e d p r o m i n e n c e w i th t he e m e r g e n c e o f N e w K e y n e s i a n E c o n o m i c s . T h e e x i s te n c e o fw i d e s p r e a d p r i c e i n e r t i a s e e m s t o b e c o n f i r m e d b y a s t e a d y a c c u m u l a t i o n o f e m p i r i c a l e v i d e n c e a tb o t h t h e m i c r o e c o n o m i c a n d t h e m a c r o e c o n o m i c le v e l [ s ee , in t e r a l ia , C a r l t o n ( 1 9 8 6 ) , A l o g o s -k o u f is e t a l . (1 9 9 0) a n d G o r d o n ( 1 99 0 )] . I n a d d i t i o n , i t i s b e c o m i n g a p p a r e n t t h a t t h e s p e e d o fp r ic e a d j u s t m e n t d e p e n d s o n i m p e r f e c t c o m p e t i t i o n . W h i l e n o c l e a r c o n s e n s u s h a s e m e r g e d , t h eb a l a n c e o f t h e e v i d e n c e s u g g e s t s 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 i n le s s c o n c e n t r a t e d i n d u s t r i e s[ C a r l t o n ( 1 98 6 , 1 9 8 9 ), E n c a o u a a n d G e r o s k i ( 1 9 8 6) , H a y a n d M o r r i s ( 1 9 91 ) ; b u t a l s o s eeD o m b e r g e r ( 1 9 7 9 ) ] .

T h i s p a p e r s e e k s t o e x p l o r e t h e t h e o r e t i c a l r e l a t i o n s h i p s b e t w e e n n o m i n a l i n e r t i a a n d m a r k e ts t r u c t u r e . W e s h o w h o w t h e s p e e d o f p r ic e a d j u s t m e n t d e p e n d s o n t h e s e c o n d d e r i v a t i v e , t h a t ist h e c u r v a t u r e , o f th e p r o f it fu n c t i o n . W e t h e n s h o w h o w t h is d e p e n d s o n m a r k e t s t r u c t u r e . W eb e g i n w i t h m o n o p o l i s t i c c o m p e t i t i o n . W e s h o w t h a t t h e s p e e d o f pr i ce a d j u s t m e n t i n c r e a s e s a s t h ee l a s ti c it y o f d e m a n d r is e s . T h i s s u g g es t s th a t p r i ce s a d j u s t m o r e q u i c k l y m o r e q u i c k l y in a m o r ec o m p e t i t i v e e n v i r o n m e n t . W e t h e n c o n s i d e r o li g o p o li s ti c c o m p e t i t i o n . W e f u r t h e r s h o w t h a t p r i c e sa d j u s t m o r e s l o w l y i n m o r e c o n c e n t r a t e d i n d u s t r i e s a n d t h a t m o r e c o l l u s iv e b e h a v i o u r r e s u l t s in as l o w e r r a t e o f p r ic e a d j u s t m e n t .

T h e s t r u c t u r e o f t h e p a p e r i s a s f o ll o w s . I n s e c ti o n 2 w e m o d e l p r i c e a d j u s t m e n t . I n s e c t i o n 3 w er e l a t e t h e d e g r e e o f n o m i n a l i n e r t i a t o t h e d e g r e e o f i m p e r f e c t c o m p e t i t i o n . S e c t i o n 4 p r e s e n t s t h er e s u l t s o f a s i m p l e s i m u l a t i o n t h a t i l l u s t r a t e s o u r r e s u l t s .

* I wish to thank Carlo Favero and Jonathan Haskel for some very helpful comments.

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14 0 C . M a r t in / E c o n o m i c s L e t te r s 4 1 ( 1 9 9 3 ) 1 3 9 - 1 4 32 . A d j u s t m e n t c o s t s a n d s t ic k y p r i c e s

W e s u p p o s e t h a t t h e f i r m c h o o s e s a p ri c e o f p b u t w o u l d c h o o s e a p ri c e p * i n t h e a b s e n c e o fa d j u s t m e n t c o s ts , w h e r e w e a s s u m e t h a t p * is e x o g e n o u s . W e a l s o s u p p o s e t h e f i r m f ac e s c os t sw h e n i t a d j u s t s i ts p ri c e b y m o r e t h a n p , w h e r e p c a n b e r e l a t e d t o t h e p r e v a i l i n g i n f la t i o n r a t e[ L a y a r d e t a l . ( 1 9 9 1 ) ] . T o t a l p r o f i t a t t i m e t i s t h e n g i v e n b y V , = 1 r ( p , ) - O / 2 ( p , - p , l - p ) 2 ,w h e r e 7 r( .) g i v e s p ro f i t e x c e p t f o r t h e e f f e c t o f a d j u s t m e n t c o s t s. W e a s s u m e t h a t f i r m s c h o o s e t h e i rp r i c e t o m a x i m i z e t h e p r e s e n t v a l u e o f t h e s t r e a m o f fu t u r e p r o f i ts

i 0 _ p)2}V = i ~ _ o a { r r ( p t + i ) - - ~ ( P , + i - - P , + i l ( 1 )W e p r o c e e d b y e x p a n d i n g ~ ( p ) a r o u n d p * . W e c a n t h e re f o r e w r it e r r ( p ) = 7 r ( p * ) + ( 1 /

2 ) T r " ( p * ) ( p _ p , ) 2 , w h e r e r r " (p * ) i s t h e s e c o n d d e r iv a t i v e o f 7r (p ) e v a l u a t e d a t p * . I n d o i n g t h i s ,w e h a v e u s e d t h e f a c t t h a t ~ " ( p * ) = 0 b y d e f in i t io n . A s a r e s u l t , p r o fi t m a x i m i z a t i o n i s e q u i v a l e n tt o t h e m i n i m i z a t i o n o f t h e f o l l o w i n g l os s f u n c t i o n :

A = ~ o 6 s { ~ ( p , + i * 2 0 _ p ) 2 }.= - - P , + i ) + 2 ( P t + i - - P , + , - I , ( 2 )

w h e r e q5 = + ~ ' " ( p * ) .M o s t a n a l y s e s o f p r ic e a d j u s t m e n t h a v e s t a r t e d f r o m a l o ss fu n c t i o n s i m i la r to ( 2 ) [ R o t e m b e r g( 1 9 8 2 a ) , A l o g o s k o u f i s e t a l . ( 1 9 90 ) a n d L a y a r d e t a l . ( 19 9 1 ) ]. 1 O u r a n a l y s i s s h o w s h o w t h e

w e i g h t s o n t h e l o s s f u n c t i o n d e p e n d o n t h e p r o f it f u n c t i o n . W e m i g h t f u r t h e r e x p e c t t h a t t h i s w i lld e p e n d o n t h e m a r k e t s t r u c t u r e i n w h i c h t h e f i r m f in d s i ts e lf . G i v e n ( 2 ) , t h e p r i c e c h o s e n b y t h ef i r m c a n b e s h o w n t o b e

p , = A p , _ l + ( 1 - A ) ( 1 A a ) L i ,( a a ) p,+, +i= 0a (1 - a )pa ( 1 - a 6 ) ' (3)

w h e r e t h e d e g r e e o f s l ug g i s hn e s s , A is g iv e n b y t h e s t a b l e r o o t o f t h e e q u a t i o n0 6 A 2 - ( fr o + 0 ( 1 + 8 ) ) A + 0 = O . ( 4 )

T h i s e q u a t i o n s s h o w s t h a t t h e s p e e d o f p r ic e a d j u s t m e n t w i ll d e p e n d o n t h e c u r v a t u r e o f t h ep r o f i t f u n c t i o n t h r o u g h t h e t e r m ~b. S i n c e w e c a n s h o w t h a t A d e c r e a s i n g i n ~ b, t h e g r e a t e r i s t h es e c o n d d e r i v a t i v e o f t h e p r o f i t f u n c t i o n , t h e l e s s e r i s i n e r t i a . T h u s , i f a f i r m l o s e s g r e a t l y b yd e v i a t i n g f r o m t h e o p t i m a l p r i c e , f o r e x a m p l e b y p r i c i n g a b o v e m a r g i n a l c o s t i n a c o m p e t i t i v em a r k e t , t h e d e g r e e o f c u r v a t u r e o f th e p r o f i t f u n c t i o n w i ll b e s u b s t a n t ia l a n d a s a r e s u lt p r i c ea d j u s t m e n t w i ll b e s w i f t . B y c o n t r a s t a f i rm w h o s e p r o f i t f u n c t i o n i s f l a t c l o s e t o t h e r e g i o n a r o u n dt h e o p t i m a l p r i c e w i l l l o s e r e l a t iv e l y l i tt l e b y n o t f o l l o w i n g t h e c u r r e n t l y o p t i m a l p r i c e a n d s o i tsp r i c e w i l l e x h i b i t m o r e s t i c k i n e s s .

I t is usua l f o r t he we i gh t o f t he t e r m ( p - p * ) t o be un i t y so t ha t 0 r epr esen t s t he r e l a t ive cos ts o f p r ice ad j u s t me nt . S eea l so Rot ember g ( 1982b) .

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C . M a r t i n / E c o n o m i c s L e t t e r s 4 1 ( 1 9 9 3 ) 1 3 9 - 1 4 3 141

3 . Mark et s t ru ctu re an d p r ice ad ju s tmen tI n t h i s s e c t i o n w e w i l l d r a w o u t t h e i m p l i c a t i o n s o f t h e s e r e s u l t s u s i n g t h e f a m i l i a r e x a m p l e s 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 a n d o l i g o p o l i s t i c b e h a v i o u r .

3 . I . M o n o p o l i s t i c c o m p e t i t i o nC o n s i d e r i n g m o n o p o l i s t i c c o m p e t i t i o n fi rs t, s u p p o s e t h a t t h e f i r m h a s c o n s t a n t m a r g i n a l c o s t

a n d f a c e s t h e d e m a n d c u r v eq = A p - ~ . ( 5 )

W r i t i n g p r o f i t s a s A p * [ p - M C ] , t h e s l o p e o f t h e p r o f i t f u n c t i o n i s g i v e n b y

~ ' ( p ) = A p - ~ 1 - e p p - = A p - ' w . (6 )U s i n g t h e f a c t t h a t o ) = 0 a t t h e o p t i m a l p r i c e p * , w e c a n s h o w t h a t t h e s e c o n d d e r i v a t i v e o f t h ep r o f i t f u n c t i o n i s s i m p l y

4 ' = ( e - 1 )A p* (~ + ' ) . (7 )S i nc e t h i s i s i nc re a s i ng i n e , i t fo l l ows t ha t 6 A / 6 ~ < 0. T h e s p e e d o f p r i ce a d j u s t m e n t i n c r e a s e s a st h e e l a s ti c it y o f d e m a n d r is e s. T h i s e s t a b l is h e s 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 w h e n f i rm s h a v e l es sm a r k e t p o w e r t o e x p l o i t .3 . 2 . O l i g o p o l y

W e n o w t u r n t o t h e m o r e g e n e r a l c a s e o f a f i rm i n a n o l ig o p o l i s ti c i n d u s t r y . W e s u p p o s e t h e r ea r e n i d e n t i c a l f i r m s i n t h e i n d u s t r y , e a c h p r o d u c i n g a n a m o u n t Y i . E a c h f i r m h a s c o n s t a n tm a r g i n a l c o s t a n d c o n j e c t u r e s t h a t t h e o u t p u t r e s p o n s e o f o t h e r f i rm s t o a c h a n g e i n i ts o w n o u t p u tis y = d ( y - y i ) / d y i , w h e r e y is i n d u s tr y o u t p u t . F i n a ll y , w e s u p p o s e t h a t t h e i n d u s tr y d e m a n dc u r v e is a g a i n g i v e n b y ( 5 ) . I t is a s t a n d a r d r e s u l t t h a t t h e r a t i o o f p r i c e t o m a r g i n a l c o s t i n t h i sc a s e is (1 - m ) - 1 , w h e r e m = k ~(1 + y ) / e a n d ~ is t h e m a r k e t s h a r e o f e a c h f i r m ] C o w l i n g a n dW a t e r s o n ( 1 9 7 6 ) ]. U s i n g t h e s e r e s u lt s , w e c an s h o w t h a t t h e s e c o n d d e r i v a t i v e o f th e p r o f i tf u n c t i o n u n d e r o l i g o p o l y is g iv e n b y

& = e ( 1 - m ) A p * - ( ~ + 1 ) (8 )E q u a t i o n ( 8 ) s i m p l i f ie s to ( 7 ) i f t h e r e i s o n l y o n e f i r m i n t h e m a r k e t .

T h i s r e s u l t a g a in s h o w s t h a t t h e s p e e d o f p ri c e a d j u s t m e n t i n c r e a s e s w i th t h e p r i c e e l a s t ic i t y o fi n d u s t r y d e m a n d . I n a d d i t i o n , w e s e e th a t p r i c es a d j u s t m o r e q u i c k l y as th e n u m b e r o f fi r m s r i se s ,s i n ce t hi s r e d u c e s m a r k e t s h a r e . W e a l s o se e t h a t t h e s p e e d o f a d j u s t m e n t d e p e n d s o n t h e f i r m s 'o u t p u t c o n j e c t u r e , y . C o u r n o t c o m p e t i t i o n c o r r e s p o n d s t o Y = 0 , w h i le B e r t r a n d c o m p e t i t i o n i np r i c e s e m e r g e s i f y = - 1 . W e c a n s e e f r o m t h i s t h a t p r i c e i n e r t i a r i s e s a s Y i n c r e a s e s . F i n a l l y , w er e l a t e t h e c o n j e c t u r a l v a r i a t i o n t e r m t o c o l lu s i v e b e h a v i o u r , f o l l o w i n g C l a r k e a n d D a v i e s ( 1 9 8 2 ) .W e s u p p o s e t h a t t h e c o n j e c t u r a l v a r i a t i o n , y , c a n b e e x p r e s s e d a s y = ~ r ( 1 - P 0 / ~ . I n t h i s, t h eo u t p u t r e s p o n s e o f o t h e r f ir m s is a ss u m e d t o d e p e n d o n m a r k e t s h a r e s. T h e d e g r e e o f co l l u si o n ism e a s u r e d b y ~ r; i f cr = 1 w e h a v e f u l l c o ll u s i o n a n d s o t h e o u t p u t o f e a c h f i rm i n c r e a s e s b y t h e

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C . M a r t i n / E c o n o m i c s L e t t e r s 4 1 ( 1 9 9 3 ) 1 3 9 - 1 4 3 14 3Co wl in g , K . a n d M. W a te r s o n , 1 9 7 6 , P r i c e - c o s t m a r g in s a n d ma r k e t s t r u c tu r e , E c o n o m ic a 4 3 , 2 67 - 2 7 4 .Do m b e r g e r , S . , 1 9 7 9 , I n d u s tr i a l s t r u c tu r e , p r i c in g a n d in f l a tio n ( Ma r t in R o b e r t s o n , L o n d o n ) .E n c a o u a , D . a n d P . Ge r o s k i , 1 9 8 6 , P r i ce d y n a mic s a n d c o mp e t i t io n in f ive O E C D c o u n t r i e s , OE CD E c o n o mic S tu d ie s 6 ,

4 7 - 7 4 .Go r d o n , R . , 1 9 9 0 , W h a t i s Ne w- K e y n e s ia n E c o n o mic s ? , J o u r n a l o f E c o n o mic L i t e r a tu r e .Ha y , D . a n d D . M o r r i s , 1 9 9 1 , I n d u s tr i a l e c o n o mic s an d o r g a n i s a t io n : T h e o r y a n d e v id e n c e ( Ox f o r d Un iv e r s i ty P r e s s,Ox f o r d ) .L a y a r d , R . , S . N ic k e l l a n d R . J a c k ma n , 1 9 9 1 , Un e m p lo y m e n t : Ma c r o e c o n o mic p e r f o r ma n c e a n d th e l a b o u r ma r k e t( Ox f o r d Un iv e r s i ty P re s s , Ox f o r d ) .Rotemberg , J . , 1982a , S t icky pr ices in the United Sta tes , Journa l o f Po l i t ica l Economy 90(6) , 1187-1211 .Ro te m b e r g , J . , 1 9 82 b , Mo n o p o l i s t i c p r i c e a d ju s tme n t a n d a g g r e g a te o u tp u t , Re v ie w o f E c o n o m ic S tu d ie s 4 6 , 5 1 7 -5 3 1 .