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Perspectives on oligopoly theory MacLeod, William Bentley 1983

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PERSPECTIVES ON OLIGOPOLY THEORY  by  WILLIAM BENTLEY MACLEOD B.A., MSc,  Queen's Queen's  University, University,  1975 1979  A T H E S I S SUBMITTED IN P A R T I A L , F U L F I L M E N T THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n  THE FACULTY OF GRADUATE STUDIES Department o f  We a c c e p t  this  to the  Economics  thesis  required  as  standard  THE UNIVERSITY OF B R I T I S H JANUARY 0  conforming  COLUMBIA  1984  W i l l i a m Bentley MacLeod,  1983  OF  In  presenting  requirements  this for  an  of  British  it  freely available  agree for  that  I  by  understood  that  his  that  or  be  her or  shall  Date  T«-  3 / ^  the  University shall  and  study.  I  copying  granted  by  the  of  publication  not  be  allowed  Columbia  of  make  further this  head  representatives.  of  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l Vancouver, Canada V6T 1Y3  at  the  Library  permission.  Department  f u l f i l m e n t of  the  extensive  may  copying  f i n a n c i a l gain  degree  reference  for  purposes  or  partial  agree  for  permission  scholarly  in  advanced  Columbia,  department  for  thesis  It  this  without  thesis  of  my  is  thesis my  written  -  ii"  -  PERSPECTIVES ON OLIGOPOLY THEORY  Abstract  This  thesis  is  a collection  gopoly problem from v a r i o u s  perspectives.  s t r a t e d t h a t t h o u g h demand c u r v e s tiated generic  product models, property.  Nash e q u i l i b r i u m  in quantities  is  the o l i g o p o l y  p r o b l e m t o show t h e  of e q u i l i b r i a  locally  for  Finally  o f both conscious  parallelism  with free  An a x i o m a t i c  provided which i s fined,  is  always  costs.  foundation  we mean subgame p e r f e c t  Nash e q u i l i b r i a  in  the  games  sta-  a unique rationality model  parallelism  pricing, widely entry.  to  sufficient  i n a dynamic  conscious  potential  the  a Cournot-  local  often y i e l d  pricing  for  local  and  Chapter 4 studies  face of  is  classical  between t h e  Necessary  and p r e d a t o r y  i n the  differen-  t o t h e use o f  t h e n u s e d t o show t h a t p r e d a t o r y rational  in  equivalent  are provided which w i l l  stable equilibrium.  demon-  the  the theory of  oli-  is  t h e use o f  relationship  and a d j u s t m e n t  continuous  the  o f a Nash e q u i l i b r i a  Chapter 3 applies  stability  entry.  often  studying  In Chapter 1 i t  generally  C h a p t e r 2 shows t h a t  concept.  conditions  are  the n o n - e x i s t e n c e  core s o l u t i o n  bility  of four essays  By  the sense o f S e l t e n  is  de-  rational [1975].  - i i i  -  Table of  Contents  Section  Page  Abstract  ii  Table of List  of  Contents  iii  Figures  v  Acknowledgements  vi  Introduction  1  Chapter  1:  On t h e N o n - e x i s t e n c e  of  Equilibria  in  Differentiated  Product Models 1. 2. 3. 4. 5. Chapter 1. 2. 3. 4. 5. 6. Chapter 1. 2. 3. 4. 5. 6. 7. 8.  Introduction . The Model The F i r m ' s P r o b l e m C o n c l u d i n g Comments Notes 2:  5 7 11 17 20  ..  The C o r e and D u o p o l y T h e o r y  21  Introduction ... The M o d e l The C o m p e t i t i v e E q u i l i b r i u m The C o r e A l l o c a t i o n s D i s c u s s i o n o f t h e Model Notes 3:  On A d j u s t m e n t  Costs  21 23 25 26 29 31  and t h e S t a b i l i t y  of  Equilibria  33  Introduction The B a s i c M o d e l The D i r e c t i o n a l C o r e An E x a m p l e The Nash B a r g a i n i n g C o r e 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 C o n c l u d i n g Remarks Notes  Chapter 4:  1. 2. 3. 4. 5. 6. 7.  5  Conscious P a r a l l e l i s m t e s t a b l e Market  and P r e d a t o r y  Introduction The I n d u s t r y Model The A n n o u n c e m e n t Game An A x i o m a t i c A p p r o a c h t o C o n s c i o u s The E f f e c t o f E n t r y C o n c l u d i n g Comments Notes.  Pricing  33 36 41 47 52 58 61 64 in  a Con73 >  Parallelism  73 77 82 87 92 96 97  -  Section  iv  -  Page  References  99  Appendix  to Chapter 1  104  Appendix  to Chapter  3  114  Appendix  to Chapter  4  120  - V -  List  of  Figures Page 32  Chapter 2 -  Figure 1  Chapter 3 -  Figure  1  65  Figure  2  66  Figure  3  67  Figure  4  68  Figure 5  69  Figure  6  70  Figure  7  71  Figure 8  72  - vi  -  Acknowledgements  This  thesis  ment and s t i m u l a t i n g many p e o p l e .  At  Charles Tracy  comments and c r i t i c i s m s  and f e l l o w  of  British  as w e l l  that  graduate students  Queen's  University,  particularly  Richard Arnott  of the t h e s i s  would  version  t h e comments site  received  Catholique  Donsimoni,  Jean  from d i s c u s s i o n s  at  de L o u v a i n , p a r t i c u l a r l y  David  and R i c h a r d  and  Donaldson,  Margaret  w i t h my c o l l e a g u e s  at  Harris. without  CORE and ISE a t t h e  Claude d'Aspremont,  Louis^Ge.v.ers,  help.  Hugh N e a r y  n o t have b e e n p o s s i b l e  f r o m my c o l l e a g u e s  Gabszewicz,  like to  M i c h e l P a t r y , A n d r e P l o u r d e , and  benefited  from  invaluable  J o h n Weymark,  I have a l s o  encourage-  received  as'Mukesh Eswaran, C u r t i s Eaton,  Ke'izo N a g a t a n i ,  the  I would  for t h e i r  Slade.  The p r e s e n t  I have  Columbia  I am i n d e b t e d t o my c o m m i t t e e  Blackorby  Lewis,  have b e e n p o s s i b l e w i t h o u t  the U n i v e r s i t y  t h a n k my t e a c h e r s In p a r t i c u l a r  would not  Univer-  Marie-Paule  Paul G e r o s k i , Jacques  Thisse  and Tom T h u r s t o n .  Finally  I would l i k e  their excel lent typing.  to  thank  S h a r o n A l t o n and  Jennine  Ball  for  INTRODUCTION  The c l a s s i c a l of markets  characterized  ing with a large question, tion  oligopoly  the  by a r e l a t i v e l y  group o f  status  of  buyers.  lers  will  has  determinant  in  price  Despite  needs  arguelthat while  Levitan  and S h u b i k  prices  the expected outcome. the  the situa-  sela  between be-  is  [1982]  Still  for a  Friedman of the  P a n z a r and W i l l i g  Despite  solution,  perpetually  the apparent  foilow essentially  variety  t h e same  by  [1982]  [ 1 9 7 8 ] , S h a p l e y and S h u b i k or  prob-  o t h e r work  the a p p r o p r i a t e  dis-  [1969] varying  of  pro-  methodology,  approach.  By game t h e o r e t i c f o r w a r d by S c h e l l i n g  the  at all  indeterminacy  t h a t mixed s t r a t e g i e s  results  namely a g a m e - t h e o r e t i c  price  w h i l e t h e work o f  equilibria.  equilibrium  [1980] suggests  posed s o l u t i o n s ,  of  price,  a n d no p r i c e  the b a s i c  i n many p o s s i b l e  or Varian  put  price,  of collusion,  competitive  t h e work o f  is  a competitive  [ 1 9 8 1 ] , G r o s s m a n [ 1 9 8 1 ] and B a u m o l , the  clarity  b e t w e e n two  t o s e e t h e w o r k by K u r z  [1977] and Green [ 1 9 8 0 ] d e m o n s t r a t e  Bresnahan  deal-  impossible".  of the p o s s i b i l i t y  can r e s u l t  price,  oscillating  T o d a y one o n l y  lem t h a t  the apparent  between t h e m , an i n d e t e r m i n a n t  cause the problem i s  study  number o f s e l l e r s  the competition  a monopoly  them, a p e r p e t u a l l y  cussion  small  the  [1933]:  been h e l d t h a t  result  concerned with  t h e p r o b l e m i s much t h e same t o d a y a s  d e s c r i b e d by C h a m b e r l i n "It  problem i s  our n o t i o n  is  probably closest  [ 1 9 6 0 ] , though i n terms  -  1:.-  of  to the  industrial  ideas  organiza-  -  t i o n Shubik  [1959] i s  consists  first  of  2 -  the c l a s s i c  carefully  reference.  specifying  Simply  put the  a model a n d t h e  information  which agents  have a c c e s s .  Giventhis  is  assuming t h a t  not o n l y are t h e y m a x i m i z i n g  explained  but t h e y a l s o  have c o n s i s t e n t  competitors'  behaviour.  In t h i s oligopolistic We b e g i n i n Hotelling price of  thesis  models  Chapter  [1929].  and o u t p u t  felt  that  his  by E d g e w o r t h [1979] duction tiated  location.  [1925].  This  satisfactory  o f the  hence i t  thesis  issue  is  to e x p l i c i t l y  model  issues.  model  to choose the  of  type also  problem  raised  t h e work o f d ' A s p r e m o n t not e x i s t  et  an e q u i l i b r i u m  similar.  the  Hotelling  Though  al.  pro-  differen-  r e a l i s m t h e r e does not seem a t the  existence  we move away f r o m t h i s do e x i s t . are e i t h e r  Intuitively buyers  free  of  chose not o n l y  observation  way t o a d d r e s s  assumes t h a t b u y e r s  seems t h a t  of  firms  a non-existence  sufficiently  to those f o r which e q u i l i b r i a  w i t h o u t much j u s t i f i c a t i o n .  this  there w i l l  p r o d u c t m o d e l s a d d an e l e m e n t  t h e o r y one s i m p l y  larger  but are a l s o  chapter extends  p l a n when p r o d u c t s a r e  rest  was t h a t  agents  their  to a s e r i e s  known s p a t i a l  to  payoffs...  some o u t s t a n d i n g  the well  model w o u l d a d d r e s s  their  of  concerning  applied  In m a k i n g t h i s  t h a t in general  For the  is  to address  insight  product  data the behaviour  conjectures  methodology  1 by g e n e r a l i z i n g  of t h e i r  time a completely  models  this  i n an a t t e m p t  spatial  to.show  or c o r r e c t  Hotelling's  p r o d u c t and i t s  basic  methodology  Typically price  h a v e no m a r k e t the trade  power.  problem.  class  in  of  oligopoly  or quantity  the argument i s  that  takers  firms  A way t o  between s e l l e r s  this  arid  are  address  buyers,  - 3 -  which,following papers  Aumann [ 1 9 6 4 . ] ' s  by Aumann [ 1 9 7 3 ] , G a b s z e w i c z  and S h i t o v i t z  [1982]  free,solution  concept.  they  that  the market  power t h e y  of S h i t o v i t z ,  [1838]  view t h a t  librium  firms  through t h e i r  can m u t u a l l y  exist  many e q u i l i b r i a .  will  Matthews  often select  firm's  observe  assumption with  a unique  market shares  c o s t s , a l 1 o w i n g us t o of  bargaining  ganization  will  models  using a  and  result  markets  open.  the c l a s s i c a l the  to  Cournot  competitive  equi-  the  is  actions  intuitively  rather  shown t h a t  be i n v e r s e l y  then  that  some i d e a s  equilibrium.  indeterminacy in  general  quantity  setting  of  adjustment  costs. par-  stability  equilibrium  p r o p o r t i o n a l to t h e i r  interpret  firms  game t h e o r y ,  criterion  Further, at t h i s  If  there  face adjustment  from l o c a l the  issue.  costs  adjustment as a m e a s u r e  strength.  Finally framework  addresses  each o t h e r s  it  defined  decisions.  a d j u s t o u t p u t but  [1982],  institution  have t h e a b i l i t y  supportto  We-suppose however,  instantaneously  ticularly  formal  [1973]  of agents  o f a c t i o n before  h a v e power by m a n i p u l a t i n g supply  firms  when f i r m s  courses  gives  Chapter 3 e x p l i c i t l y  By c o m b i n i n g t h i s  I n C h a p t e r 2 we s h o w ,  to certain  our a n a l y s i s  as t h e a p p r o p r i a t e ,  between t h e s i z e  power a r i s e s  of  [1971], S h i t o v i t z  in these a b s t r a c t l y  relationship  t h a t market  precommit themselves In p a r t i c u l a r  used t h e . c o r e  possess.  done i n a s e q u e n c e  and Mertens  Unfortunately  f o u n d no c o n s i s t e n t  cannot  suggestion,was  i n C h a p t e r 4 we move away f r o m t h e q u a n t i t y  t o s t u d y some i s s u e s literature  and  relevant  "antitrust  to the informal  issues;  In  the  setting  industrial  or-  - 4 -  work o f entry  Baumol,  and e x i t  firms.  In t h i s  contrary, and e x i t  P a n z a r and W i l l i g that is  conscious  for  it  is  argued t h a t  completely  information  conditions  consistent with collusive  profit.  parallelism,  is  costless  pricing  t o show,  costless  a unique t a c i t l y  by  to  the  entry  p r i c i n g behaviour  I n t h e c h a p t e r we a l s o p r e s e n t that yields  it  ensuring competitive  c h a p t e r we u s e a r e p e a t e d game f r a m e w o r k  long run p o s i t i v e of  responsible  under the a p p r o p r i a t e is  [1982]  a formal  collusive  and theory  equili-  brium.  What t h e s e r e s u l t s testability trial  is  not  organization  a fundamental as i t s  many p o s s i b l e m o d e l s  authors  that the theory of p e r f e c t  starting  the f o u n d a t i o n stone f o r  is  s i m p l y an e x a m p l e o f  point  suggest.  assumption t h a t  is  models.  is  t h a t o n e may u s e i n  standard microeconomic  but relevant  suggest  Rather  firms  of  the study of it  represents  studying o l i g o p o l i e s .  indusone It  are profit-maximizers  t h e modern s t u d y o f  an a p p l i c a t i o n  for  this  large firms. theory to f o u r  con-  This  is  of the  that thesis  different,  CHAPTER 1 ON THE NON-EXISTENCE OF E Q U I L I B R I A IN DIFFERENTIATED  1.  Introduction  Much o f roots  i n the  acteristics Typically, cess:  is  (locations)  pertains  second stage  f o r s u c h a model  f o r any s e t o f  the payoff  of  firms  for  teristics.  The o b j e c t i v e  of spatial  one f i r s t  The a i m  its of char-  choose both the  one t h e n c o m p u t e s  models  of  that  solve  the  first  w i t h no w e l l  this  the f i r s t  [1925].  He showed t h a t  Bertrand  [1883] prices.  introduction  of  price  the product  choice.  To c o m p u t e  is  This  choice  the  computation  in  the f i r s t  t h e Nash e q u i l i b r i u m is  t o show f o r  a significant  does n o t e x i s t .  problem s i n c e  in  defines  stage.  in  charac-  a very  general  set of  locations  Hence i t  there w i l l  is  exist  not  for pos-  strategy  outcome.  non-existence  the a d d i t i o n  of  results  I t was H o t e l l i n g ' s  aim t o s o l v e  differentiation:  - 5 -  i  is  due t o  Edgeworth  capacity constraints  s e t t i n g model w o u l d r e s u l t  product  charac-  c o m p u t e s t h e Nash e q u i l i b r i u m  paper  there  stage  defined  of  characteristics.  second stage e q u i l i b r i u m  One o f  tuating  to the s e l e c t i o n  any c h a r a c t e r i s t i c  payoffs  choices  when f i r m s  to the p r i c e  product  Given these  to  [1929].  has  and t h e p r i c e of t h e product they are t o p r o d u c e .  stage  and t h e  equilibria  sible  structure  competition  the'.-prjbblem.faced by f i r m s can be. modelled as a two stage p r o -  teristics  which the  d e v e l o p e d by H o t e l l i n g  to study market  The f i r s t  prices  t h e modern t h e o r y o f m o n o p o l i s t i c  l o c a t i o n model  these models  class  PRODUCT MODELS  in  to  indefinitely  this  problem  "The a s s u m p t i o n ,  the fluc-  through  implicit  in  the  - 6 t h e i r work  (of Cournot,  the cheapest the  seller  in  price". *  differentiation tial  model  to a type of  Gabszewicz  problem.  and T h i s s e  not continuous, close  However,  w i t h the  demand c o n t i n u i t y intuition  preferring  continuity. worth's. when f i r m s retail  the l e v e l  of  in  is  then applied  if,  the n a t u r a l  t h e n one s h o u l d a t  our s i m p l e e x t e n s i o n .  spaof  the  d'Aspremont,  locate  fact  sufficiently  exist.  spatial It  model is  and show  shown t h a t  on c o n s u m e r s a l w a y s  and sufficient  that Hotelstrict-  condition  t o a problem s i m i l a r - t o  simultaneously.  for  Edge-  exists  a Nash e q u i l i b r i u m  If  firms  the output  the  variable  are' viewed is  simply  store.  as A r c h i b a l d , framework  the  product  demand c u r v e s w e r e i n  depends  space,then  h e l d by t h e  the  with when  of  resolution  shown i n  we a s k w h e t h e r t h e r e  geographical  inventory  ket s t r u c t u r e ,  a general  necessary  a  p r i c e and o u t p u t  Certainly models a r e  provides  firm locations  stores  the  does n o t  demand c o n t i n u i t y  T h i s model  choose  demands and t h e  c a n be e x p e c t e d i n m o s t c a s e s . that  function  was hoped t h a t  was r e c e n t l y  price  deal  which disappears  c o n s e q u e n c e t h a t when f i r m s  e s s a y we p r e s e n t  one f i r m  Given  it  it  buyers  u s e o f t h e one d i m e n s i o n a l  (AGT) t h a t  t o g e t h e r an e q u i l i b r i u m  In t h i s  ling's  continuous  [1979]  all  c o n s i d e r e d as a c o n t i n u o u s  introduced through the in  that  instability  In o t h e r ' w o r d s ,  would r e s u l t  non-existence  as  leads  q u a n t i t y s o l d by each i s  differences  ly  A m o r o s e and E d g e w o r t h ) ,  for  E a t o n and L i p s e y  [1983] suggest,  the study of product  least  have e x i s t e n c e  choice  such  and m a r -  o f an e q u i l i b r i u m i n  However, we show t h a t whenever o n e . f i r m l o c a t e s  s u f f i c i e n t l y c l o s e t o . a n o t h e r , an equi 1 ibriumi wi 1.1. not e x i s t ,  - 7 -  demand. E a t o n a n d L i p s e y [1978] a n d N o v s h e k  even with continuous suggest  that  concept  is'employed"wh"ich  cutting.  this  problem w i l l  .However,-we  existence  be r e s o l v e d - i f  i n c o r p o r a t e s • an a s s u m p t i o n  also  show t h a t  approach  we p r e s e n t  o f demand c o n t i n u i t y . - S e c t i o n  presents 2.  this  equilibrium  o f no m i l l  does  price  not a l t e r  the non-existence  the non-  result.  t h e model  3 examines  Section  along with the  t h e f i r m , p r o b l e m and  4 has o u r c o n c l u s i o n s .  The Model  Let  t h e market  The d i s t r i b u t i o n  function  different,  locations  prices^ from p o i n t price  and p  be r e p r e s e n t e d  o f consumers  measurable  over  v ( x ) , x e U. x^, x  2  x to y will  from f i r m  be g i v e n  There w i l l will  The u n i t  be t ( x , y ) .  o f t h e good b o u g h t  U will  e U that  respectively.  2  by a c o n v e x ,  Hence i  will  def  1  It  will  following  1  by a b o u n d e d  at f i x e d , but  t h e same p r o d u c t of transporting  f o r a consumer  Borel  at mill  t h e good  located  at x the  be  (1)  1  b e assumed t h a t  the t r a n s p o r t a t i o n  cost  f u n c t i o n has  properties.  Assumption  Tl:  t(x,y)  Assumption  T2:  t(x,y)  rn  |  be t w o f i r m s  sell cost  compact s e t U c=iR .  P,-(x) = p, + t ( x , x ) .  the  under-  result. In t h e f o l l o w i n g , s e c t i o n  result  an a l t e r n a t i v e  [1980]  is continuous > 0  for all  for ( x , y ) e U x U. x f- y , x , y e U and t ( x , x )  for all x e U . Assumption T3: t ( x , y ) + t ( y . , z ) > t ( x , z ) , V x , y e U .  =0  - 8 Assumption ensures  that  costs.  Finally,  T l i s the standard  a l l locations  are differentiated  T3 corresponds  Consumers w i l l  continuity  a s s u m p t i o n w h i l e T2  by p o s i t i v e  to the triangle  inequality.  a l l h a v e t h e same demand c u r v e  be a s s u m e d t o have t h e f o l l o w i n g  Assumption DI: D ( p ) is  transport  D(p) which  will  properties.  differentiable.  3 k and p > 0 such that 0 < D ( p ) < k V p e IR and  Assumption D2:  +  +  D(p)  =0  for p > p * .  Assumption DZ: 3 a > 0 such that - a < ^ j j ^ < 0 for p e [ 0 , p ] . +  These assumptions downward s l o p i n g T h i s model  type with  can easily  ensure bounds  be e x t e n d e d  that  t h e demand i s o f t h e s t a n d a r d  o n b o t h demand a n d i t s d e r i v a t i v e s . to allow  demand t o d e p e n d on l o c a t i o n  2  without  altering  any o f t h e p r o o f s .  Consumers w i l l delivered  price.  _P = ( p j , p ) 2  w i s h t o buy from t h e f i r m o f f e r i n g  Given  e [0,p ] +  the firm  x [0,p ],  locations, t h e lowest  +  the lowest  X. - ( x ^ X g ) e " U x U a n d p r i c e s delivered  price  to location  x e U i s given by:  p(x)  where  = min { p ( x ) , x  (2)  2  p^(x) i s g i v e n b y ( 1 ) .  Firm i ' s market quotes  p (x)},  the lowest  price.  area  i s formed by t h o s e  I t therefore  locations  i s defined by:  a t which  it  - 9 U.(X,P) 1  = {x e U | p . ( x ) def  In g e n e r a l ,  there  areas to s i g n i f i c a n t l y i n \i^{X,P) {] U ( X , P ) 2  overlapping  creates  is  tiated nates  i  = 1,2.  no r e a s o n n o t t o e x p e c t t h e f i r m  overlap,  i n which case a l l  w o u l d be i n d i f f e r e n t  difficulties.  s u c h a r e a s m u s t be a l l o c a t e d However,  = p(x)},  1  difficulty  in a necessarily  ad'hoc  go a g a i n s t  is  that  lowing  "indifferent".  Such  demand f r o m  f a s h i o n between  the s p i r i t  of  firms.  a differen-  p r o d u c t m o d e l , ; w h i c h a r e meant t o p r o v i d e a c o n s u m e r w i t h o v e r w h i c h he i s not  located  b e t w e e n t h e two f i r m s .  The f i r s t  such a r b i t r a r y a l l o c a t i o n s  the consumers  market  alter-  C o n s e q u e n t l y we make t h e  fol-  assumption.  Ho telling  Assumption:  V _P e [ 0 , p ]  x [ 0 , p ] and X E U x U suah  +  x  1  +  that  + x , 2  Lu^x.pjnu^p)  D ( P ( X ) ) v ( x ) dx = 0.  In e s s e n c e H o t e l l i n g a r g u e d t h a t  s u c h an a s s u m p t i o n w o u l d e n -  3 sure' that that  t h e demand f a c i n g  in practice  discontinuity f o r the  is  model. and x  ?  continuous.  o f demand i n homogeneous  Edgeworth  N o t o n l y d i d he but also  p r o d u c t models  that  t h a t was  it  feel  was  the  responsible  cycle. ironically,  not s a t i s f i e d For example, = 3/4.  is  demand was n o r m a l l y c o n t i n u o u s  Somewhat tion  firms  as p o i n t e d o u t b y A G T , t h e H o t e l l i n g  in Hotelling's let  original  U = [ 0 , 1 ] and suppose  Let transport costs  one d i m e n s i o n a l firm locations  b e g i v e n by t ( x , y )  = |x-y|,  assump-  spatial a r e x^ = 1/4  -  10  -  v(x) = 1 and D(p) = 1. For p. = 1 and p9•= U tion will  a simple  computa-  show  U (X,P)  =  U (X,P)  = [f  r  2  [0,1]  ,  1].  Therefore  v(x)  xeMU  dx =  \f  0.  Given the H o t e l l i n g assumption, uniquely  defined  t h e demand f a c i n g  firm  l i s  by:  (3)  Formally is  not s a t i s f i e d .  be o v e r s t a t e d ,  d . (_X.JP) However,  is  d e f i n e d even i f  in that  b u t t h e demand c u r v e s  the  Hotelling  case not o n l y w i l l will  n o t be  assumption  a firm's  demand  continuous.  PROPOSITION 1  If, assumptions Dl and Tl - T3 are satisfied, demand function ling  d.(X_,_P) is continuous  in (_K,P_) if and J only if the Hotel-  assumption holds.  The p r o o f o f  Although costs  then the  this  t h e one d i m e n s i o n a l  has d i s c o n t i n u o u s  sumpiton  is  proposition  satisfied  in  demands,  is  contained  spatial model :  i t "will, be-the  a wide v a r i e t y  i n the  with  case that  of'cases.  :  appendix.  linear  transport  the. H o t e l T i n g  as-  The - f o i l owing;" e x a m p l e s  -  i11ustrate  this  11 -  claim.  Example_l  In t h e p a p e r model  ...  with  by A'GT,  it  is  shown t h a t  p  t(x,y)  = - c | x - y | - / demands  f o l l o w s from the f a c t  for  the  one-dimensional  are.continuousThis  t h a t . U^. n  will  always  conclusion  be a s i n g l e  point.  Example 2 It is  at  least  is  easy to v e r i f y  two,  3.  o f "dimension  assumption  The F i r m ' s  firms  will  of  t h e consumer  space  c o s t s , " t ( x , y ) ' = . |jx-y|| w i l l F o r m _> 2 ,  will  n  albe a  t h a n rr,.  o f t h e p a p e r we w i l l  assume t h a t  the  Hotel-  holds.  Problem  s e c t i o n , we a d d r e s s  are  Nash e q u i l i b r i u m . restoration  less  remainder  In t h i s firm locations  the dimension  h a v i n g measure z e r o .  n  For the ling  if  then Euclidean, t r a n s p o r t  ways r e s u l t ' i n manifold  that  fixed  and p r i c e s  and o u t p u t c h o i c e s  Since  one o f t h e i s s u e s  o f demand c o n t i n u i t y w i l l choose p r i c e s  one t o s e e t h e e f f e c t More g e n e r a l l y ,  if  of  and o u t p u t s allowing  one a c c e p t s  s u c h as t h i s  form a l a r g e s u b s e t ,  the  a r e m o d e l l e d as  This  to^choose c a p a c i t y  models, are  the  put  which  i s whether  a the  Edgeworth e x i s t e n c e  simultaneously.  the arguments  [1983] t h a t address  problem i n  t o be a d d r e s s e d  resolve  firms  E a t o n and L i p s e y one  the second s t a g e  problem,  assumption-allows levels  ex  ante.  f o r w a r d by A r c h i b a l d ,  of which s p a t i a l natural  vehicle  models for  demand  - 12 models, t h e n t h e r e a l i s t i c ante to  assumption that  firm production takes  p r o d u c e an i n v e n t o r y s h o u l d r e s u l t  in well  defined  place  ex  equi1i-  4 bria".  Certainly,  stricting costs  oneself-  firm i's  the  strategy  noted s.  1  inventory of If  in spatial  marginal  analysis. be _X = ( x ^ ^ )  ; x^ f x^.  be a p r i c e - o u t p u t  pair  Then de-  1  goods  o c c u r s we w i l l  suppose t h a t  the r a t i o n e d  consumers  will  competitor.  have t h e same c o s t  satisfy  the  ^ dq" C2:  our r e s u l t s ,  does  mi  eatisfiee  function  denoted  C(-)-  conditions.  <Jm > o, v „ > o. dq  "  = 3 > 0. assumption,  t h o u g h n o t t h e most  i n c l u d e as s p e c i a l  b o t h i n c r e a s i n g and d e c r e a s i n g cond a s s u m p t i o n ensures  Let  following  5  q>0 inf  The f i r s t  strategy  demand and z e r o  = S. The o u t p u t q . w i l l s i m p l y be an def p r o d u c e d ex a n 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  be a s s u m e d t o  Assumption  theory by re-  e [ 0 , p ] x IR  1  The two f i r m s w i l l It will  inelastic  i n the second stage w i l l  a stock-out  go t o t h e  t o have a v e r y g e n e r a l  pair of firm locations  = (p.,q.)  1  unlikely  t o t h e c a s e of  so u b i q u i t o u s Let  P-.  one i s  that  us now s p e c i f y  pair,  cases,  returns  production  possible  constant marginal  to scale is  general  cost  always  the behaviour o f  costs  functions.  The  for and se-  costly.  r a t i o n e d consumers  given  a  S = ( s , , s ) e 5 x S. Suppose a t any g i v e n time f i r m i def f a c e s demand f r o m a l l p a r t s o f i t s m a r k e t a r e a , IK U , £ ) , i n p r o p o r t i o n ?  1  to the  density  v(x).  Thus  if  a firm  stocks  out,  consumers  from a l l  parts  - 13 of  the market  area w i l l  same, p r o p o r t i o n  of  be a f f e c t e d  consumers' r a t i o n e d  t h e n go t o t h e c o m p e t i t o r  This  evenly.  to purchase  We w i l l  everwhere  therefore  i n U-('.X,P_).,  have  the  who-will  t h e good.-  proportion of unsatisfied  demand w i l l  be g i v e n  by  /  0 U x . s )  = def  1  for  d i  (X P)  i  where S = [0,p  x S  consumers w i l l  f  d  f  due t o t h e c o n t i n u i t y o f  (_X,Sj e U  q  t h e n be X . ( X , S )  v(x)  price.  Hence t h e demand c u r v e f o r  by f i r m  i  given  DjU.S)  where j  f  > 0  l ^ . P ) > q  > 0 .  1  ] x (0,°°).  The d e n s i t y  f o r x e U.{X,P).  firm j  continuous  f i  of  These  consumers  at-the. higher  o n c e we a l l o w f o r  rationed  delivered a stock  out  by: =  d ^ X :,p) .P)  A ., (( XX,, SS )) - >/ + X  x e U  .(  X 5  p)  D( .(x)) P j  v(x)  dx  (5)  i.  Given the 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  demand c u r v e s , we c a n now d e f i n e  n (X,S) 1  where  > d.(X,P)  d . ( X , P j , A.(_X,_S) i s  a r e assumed t o demand t h e good f r o m f i r m . ' j  is  i  d.(X,P.):.- q [  Notice that  if  = def  P. 1  Q.(X,S)  the p r o f i t  - C(q )  1  1  Q-(X_,S_) = m i n { q . , D . ( X , S ) } . def 1  1  1  function  for  contingent  firm  i:  -  Here  {X,S) s i m p l y  o f c o u r s e c a n n o t be g r e a t e r  14  -  represents  t h e q u a n t i t y o f good s o l d  than the l e v e l  of  inventory.  From o u r 2  continuous  for  b r i u m 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 e n s u r e t h a t q . f 0, t h u s  If petitor always  (_X,_S) e U  x 3" .  the d i s c o n t i n u i t y one f i r m  would  act  in fact  for  is  defined  for  U,S_)  e U  x S  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  p r o b l e m when q . =0.  does not  as a m o n o p o l i s t ;  be p r o f i t a b l e  monopoly  t h a t n.:(_X,_S)  2  it  avoiding  clear  as-  sumptions,  also  is  which  produce  We w i l l  equili-  .V r  any o u t p u t ,  now r e q u i r e  t h e smal1 f i r m t o - s t a r t  and  then  that  it  up o p e r a t i o n s  its  com-  will  -  at  the  equilibrium.  Assumption M:  There exists > g | (0)  p(x)  an a-> 0 such that  + a , V x e U,  where p ( x ) solves max p D ( x , p )  -  C(D(x,p))  P  and  L  D (:X,P) x,p) = = J  xcU  def  This any l o c a t i o n when t h e r e  is  D(p + t ( x , x ) ) v ( x )  assumption simply  is no  strictly output.  greater  says  that  dx.  the monopoly  than the marginal  price  cost of  choice  production  for  -  Firms w i l l librium  15 -  now e v a l u a t e  price-quantity  pair  lem f o r two e q u i 1 i b r i u m  their  s^ £ 5 .  location  We w i l l  c h o i c e s , _X, at the e q u i -  study the e x i s t e n c e  prob-  concepts.  DEFINITION 1 2 Given  a pair  a Nash Equilibrium  the strategy  =  max n ( X , S.eS  1  the non-existence  i  S ,, S * ) ;  1  Recently,  will  be  = 1, 2  j  and  t  i.  J  E a t o n and L i p s e y  [ 1 9 7 6 ] and Novshek  [1980] argue  p r o b l e m c a n be r e s o l v e d by m a k i n g t h e a s s u m p t i o n  do n o t u n d e r c u t e a c h o t h e r a t t h e mi 1 1 - d o o r .  following  S *£ S  pair  (NE) i f and only i f  JI,(X,S*)  firms  _X,  of locations  This motivates  that  that the  definition.  DEFINITION 2 0 Given a no mill-price  a pair  undercutting  n . ( X , S ° )  =  max  > Pj° - t ( x  Unfortunately, general  expect  r  X, the strategy  equilibrium  n.(X,  S E S  1  Pi  of locations  1  S . , 1  S ° )  2  S_ £ S  pair  will  be  (ME) i f and only i f for  i  =  1,  2;  j  f  i  and  0  x.).  our f i n a l  result  states  t h a t one c a n n o 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  in  solution  concept.  - 16 -  PROPOSITION 2  For  e U there  every  of locations,  ^satisfying  a ME in price  and  outline.of  tCx^^)  a n d show t h a t exists,then'  the proof  this  assumption r e s u l t s  be n e i t h e r  • of equilibria  as n  '  both  excess  f o r b o t h N E and M E .  be i m p o s s i b l e  always  will  of rationing,  and use t h e f a c t  benefit.  an NE nor  This w i l l  that  again  The  equilibrium I f a NE o r ME  such t h a t  (vx^.Xj)  output. T h e n , f o r any  t(x^,x^)  ->•  0  p r i c e must a p p r o a c h t h e m a r g i n a l i f an  as output fall  cost  equilibrium/exists,": pricing  by f i r m s .  reasons.  increases,  then i f  b e l o w t h e AVC ( a v e r a g e  not to produce.  price variable  B u t by a s s u m p t i o n M,  H e n c e , we have a c o n t r a d i c t i o n .  In t h e o t h e r case,  the p o s s i b i l i t y price  prefer  an  nor r a t i o n i n g .  l e a d to c o m p e t i t i v e  I f we have MC f a l l i n g  be i m p o s s i b l e .  exist  have p o s i t i v e  f o r one o f t h e f o l l o w i n g  H e n c e some f i r m w i l l  (2)  will  Hence,  a p p r o a c h e s MC i t m u s t a t some p o i n t  this  not exist  pairs  G i v e n a l o c a t i o n x^ e U, we  supply  for locations  competition w i l l  (1)  costs).  will  in a contradiction.  firms  i t c a n t h e n be shown t h a t  production  This w i l l  i s as f o l l o w s .  by a s s u m p t i o n M  there w i l l  increasing  < WQ there  for  o f t h e p r o o f a r e t o be f o u n d i n t h e a p p e n d i x .  sequence  of  a Wg > 0 such that  f o r e v e r y x^ e U, x^ f x^> t h e r e w i l l  assume t h a t  Further  exist  output.  The d e t a i l s basic  will  as p r i c e  approaches  one f i r m w i l l  always wish to r a i s e  i t s competitor w i l l  result  M C , due t o  ration  in a contradiction.  consumers  its to  its  -  In both cases m i l l  17 -  price  proof w i l l  apply  i n c a s e 1;  h e n c e t h e a r g u m e n t .^wi 11 c o n t i n u e  only  prices  type of  ex  t o ME as w e l l .  undercutting  post.  which  N o t i c e as w e l l not r e s t r i c t i v e  if  any two l o c a t e  4.  Concluding  is  that  the  close  together,  ex p o s t .  that  raises  power o f o u r t h e o r y . they  that  the  choose  Edgeworth  rationing.  analysis  t o two  firms market,  exist.  is  do n o t e x i s t questions  particularly  this  fact  one c a n a l w a y s  place  sufficiently  t h e y become l o c a l  large  theory  that are  concerning  monopolists.  deci-  the  product  predictive models  homogeneous  demands c a n be  ex-  resolve; non-existence  models.  restrictions  In t h i s that  of  consistent  the case w i t h these  alone cannot  Of c o u r s e  restore existence.  the  i n models w i t h  although  product  to  to  t o make o u r s t a n d a r d  As we h a v e s h o w n ,  behaved,  central  with expectations  equilibria  fundamental  is  n o t i o n t h a t a g e n t s make  i n homogeneous  by s u p p o s i n g U i s that  the  firms  ;  arise  o f t h e model  apart  payoffs  This  realistic.  p e c t e d t o be w e l l problems  of-the  if  used  f o r - a n y number o f f i r m s ' - i n t h e  a r e m o t i v a t e d by an a t t e m p t  m o d e l s more  of  no e q u i l i b r i u m w i l l  It captures  to maximize  differentiation  since  to  never  the  Comments  The f a c t  6  is  t o h o l d even  restriction  i n the sense t h a t  imperfect competition. ex a n t e  rationing  due t o t h e p r e s e n c e  The n o t i o n o f a Nash e q u i l i b r i u m  sions  n e v e r u s e d and  C a s e 2 c o r r e s p o n d more C l o s e l y  non-existence,  is  Observe t h a t  is  model  the  this  on t h e may be  f i r m can l o c a t e  Given  that  the  parameters achieved far  demand  enough distri-  - 18 -  bution  is  uniform,  would e x i s t  regardless  an e q u i l i b r i u m .  new e n t r a n t s  of  the s o l u t i o n , concept  However f o r s u f f i c i e n t l y  w o u l d a l w a y s be i n a p o s i t i o n o f  outcome o f l o c a t i n g modelling this  close  situation  to a n o t h e r one w i l l  firm.  not  trying  Without  chosen  there  low f i x e d  costs,  to evaluate  the  a consistent  way  have a c o m p l e t e l y  of  satisfactory  model.  The m o s t common way t o of'mixed  restore  o r random s t r a t e g i e s . ^  i n m i x e d o r random s t r a t e g i e s  For  will  an e q u i l i b r i u m  x^ f.x.^;  always  is  through the  i n o u r model  exist  since  an  the  use  equilibrium  profit-function  Q  is  continuous.  Further  u s e d t o show e x i s t e n c e level  the r e s u l t s  when f i r m s  f o r such a procedure  graph p r o p e r t y haved,it model  for  D a s g u p t a and M a s k i n  locate  together.  Nash e q u i l i b r i a .  equi1ibria  that  That  is  equilibria  for  for  [1982] can  On a p u r e l y  t o make s e n s e we s h o u l d a t  s h o u l d be t h e c a s e  approximate  of  least  t h e model  t h e homogeneous  product  technical  have  the  closed  t o be w e l l  f o r the s l i g h t l y  be  be-  differentiated  case,'"otherwise  g are  left  with another  pathology  make t h e m o s t s e n s e i n b e r s c a n be a p p l i e d . shot  It  is  libria  follow  some d y n a m i c s t r u c t u r e ,  increase  a possibility  hence p r e f e r  e n o r m o u s l y due t o  is  locating  solution  situations  not c l e a r  d e c i s i o n would n e c e s s a r i l y  one a d m i t s  firms  repeated  to e x p l a i n .  to  quite  close  together  agents  generally  faced with a  the set  the p o s s i b i l i t y  for differentiated  rather  strategies  where t h e l a w o f l a r g e  a mixed s t r a t e g y .  t o e a c h o t h e r may f i n d  locate  concept.^  serious  that  Secondly, mixed  it  single  However  once  possible' equi-  of cooperation.^ p r o d u c t models  easier  than apart  of  num-  Such  since  t o c o o p e r a t e , and  due t o a s h i f t  in  the  we  - 19 -  Another Palma e t  al.  S u c h a model between f i r m s  then s i m p l y s t u d i e s firms  Finally  of  of  that  assumes  enough t o  that  characteristics problem,  by R o b e r t s  the f i r m choice  by  that-there  in  are  space with  left  One the  equilibria.  product  o f markets, then d e s p i t e still  absolute  being uncertain.  generate  models the  non-  wjth the question  of  s u c h s i t u a t i o n s . ' We. w o u l d s u g g e s t , t h a t  and a l s o  go t o  a centre of  general  and S o n n e n s c h e i n  [1977].  the-non-existence  e q u i l i b r i u m models, 12  out  de  differentiated  one i s  e q u i 1 i b r i uiti p r o b l e m a s s o c i a t e d . w i t h  pointed  r e c e n t work o f  p r o b l e m i n some o b j e c t i v e  different  fundamental  into  are best m o d e l l e d ' a s  agents'.' b e h a v i o u r  problems"are  uncertainty  one d o e s f e e l  equilibrium  how t o e x p l a i n these  if  some e s s e n t i a l  existence  t o be f o u n d i n t h e  essentially  the l o c a t i o n  being s u b j e c t i v e l y  capture  is  [1983] who i n t r o d u c e  individuals. differences  possibility  as  -  20  -  NOTES  1.  Hotelling  [ 1 9 2 9 ] , page 4 7 1 .  2.  See MacLeod  3.  This  s o as t o p. 4.  "the s t a b i l i z i n g  have a n a t u r a l  precisely effect  preference  See L e v i t a n  and S h u b i k  [1978]  to H o t e l l i n g  [1929];  o f masses o f consumers for  one s e l l e r  f o r an a n a l y s i s  model.  5.  It  not continuous  6.  To be p r e c i s e we a r e  is  quite clearly  Selten  [ 1 9 7 5 ] on t h i s  7.  S e e Nash  8.  This  o r the  he placed  other",  such models  in  a  a t q^ = 0 .  subgame p e r f e c t  Nash e q u i l i b r i u m .  See  point.  be s e e n a s a n o t h e r model w h i c h ,  shows t h e e x i s t e n c e  The r e s u l t s  of Roberts  not o b t a i n such 10.  See F r i e d m a n  11.  Gabszewicz  of  [1980]  like  Varian  sales. indicate  in  l a r g e economies  one  does  results.  [1977].  [1983] argues  new way t o model Also  studying  of  [1951].  model m i g h t a l s o  [1980],  12.  added.  470.  homogeneous p r o d u c t  9.  p h r a s e has been  [1982]..  assumption corresponds  argued f o r  The b r a c k e t e d  that  imperfect  s e e t h e s u r v e y by H a r t  s u c h i d e a s may f o r m t h e b a s i s  competition. [1983].  for  a  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 [ 1 9 6 4 ] t h a t m i x e d m a r k e t i.e.  m o d e l s w i t h atoms as w e l l  most a p p r o p r i a t e tion  for  spawned a l i n e  being large  the study of of  there  equilibrium. is  no n e c e s s a r y  results  significantly are several left  that  pretation that  traders  relationship in  the core  traders  not only  is  it  o f these models  t h e y do n o t a d d r e s s  in the off  but also issues  that a monopolist  demonstrate  might  equiliobtain.  [ 1 9 7 1 ] and S h i t o v i t z  theorem t o  results  relevant  an  t h a n a t t h e com-  t h e c a s e when We a r e  there  then  economic  from t h e i r  to the  [1973]  [ 1 9 7 4 ] and Aumann  to provide a natural  the  which  core of  o f Aumann [ 1 9 7 3 ]  of Hildenbrand  difficult  under  "advantageous".  w i t h t h e same c h a r a c t e r i s t i c s . conclusions  the  sugges-  between t h e c o m p e t i t i v e  extend the core equivalence  large  was  better  o f G a b s z e w i c z and M e r t e n s  with the p e s s i m i s t i c  [1973]  large  This  conditions  allocations  However t h e e x a m p l e s  b r i u m and t h e maximum p a y o f f Further, the  are  w o u l d be  competition.  o r an atom i n a c o n t i n u u m o f t r a d e r s  e x c h a n g e economy t h a t make t h e  that there  imperfect  research which looked f o r  By a d v a n t a g e o u s we mean t h a t  petitive  as a c o n t i n u u m o f t r a d e r s ,  models,  study  theory of  inter-  suggest  imperfect  com-  petition.  In t h i s different  e s s a y we w i l l  suggest  and p o s s i b l y more i m p o r t a n t  competition.  We b e g i n by e x p l i c i t l y  [1973] to the c l a s s i c a l  duopoly  that  role  in  adapting  these models the theory t h e model  problem o f Cournot  -  21  -  play quite of  of  a  imperfect Shitovitz  [ 1 8 3 8 ] and  Bertrand  -.22 [1883].  The f i r s t  point  the core as a s o l u t i o n ket competition it  tiated  by B e r t r a n d  priate  strategic  ducts, will  starting  o f t e n use a p r i c e  go f u r t h e r  concept s i n c e ,  q u o t e d by one s e l l e r  institution.  As s u c h  o f the debate  or quantities  ini-  are the appro-  and a r g u e t h a t  the core  f o r many c l a s s e s  bargain with s e l l e r s .  a procedure  of  In p a r t i c u l a r ,  pro-  buyers  t o o b t a i n a b e t t e r deal  akin to the b l o c k i n g concept  used to  with  define  core.  In p a r t i a l that  One m i g h t  in fact  trading  for the analysis  [ 1 8 8 3 ] on w h e t h e r p r i c e s  the c o r r e c t s o l u t i o n  buyers w i l l  the use o f  i s d e s i g n e d t o a n a l y z e t h e outcome o f mar-  point  variable.  his.iCompetitor; the  concept  i n d e p e n d e n t o f any s p e c i f i c  forms a n a t u r a l  is actually  i s , a s a r g u e d by Aumann [ 1 9 6 4 ] , t h a t  buyers  take  an a s s u m p t i o n  e q u i l i b r i u m o l i g o p o l y m o d e l s one t y p i c a l l y  the s e l l e r s  that,  strategies  However t h i s  g i v e n t h e Aumann [ 1 9 7 3 ] e x a m p l e s ,  from m e r e l y having a l a r g e  stocks  [1973]  the s e l l e r s  in the Bertrand important theory  trading their  o f money h e l d b y a c o n t i n u u m o f b u y e r s .  interpretation of Shitovitz  buyers.  [1973] that  for the rather  inventories  n o t e we a r g u e t h a t  have  for  In c o n t r a s t w i t h t h e  h i s core equivalence  In the f i n a l  theorem  provides  complacent assumption of  that the buyers a c t c o m p e t i t i v e l y .  that  We c a n t h e n u s e  [ 1 8 8 3 ] t r a d i t i o n , we a r g u e t h a t t h i s model  foundation  merely  follow  I t w.iTl b e s u p p o s e d t h a t we h a v e two s e l l e r s  t o model  is  s e p a r a t i o n between p r o d u c t i o n and  p r e v i o u s l y bought o r p r o d u c e d an i n v e n t o r y o f goods. Shitbvitz  does n o t  t r a d e r f a c i n g a continuum o f small  The model we p r e s e n t makes a d i s t i n c t sales a c t i v i t i e s .  as g i v e n .  assumes  is an  oligopoly  s e c t i o n o f the  t h e " m i x e d m a r k e t " m o d e l s c a n be q u i t e n a t u r a l l y  inter-  - 23 -  p r e t e d i n the C o u r n o t [1838] q u a n t i t y s e t t i n g approach to  imperfect  com-  petition.  2.  The M o d e l  1  C o n s i d e r a two good e x c h a n g e economy w i t h a s e t g i v e n by T = [ 0 , 1 ] u { a , b } w h e r e [ 0 , 1 ] {a,b}  represents  two a t o m s .  represents  An a l l o c a t i o n  of  a continuum  traders while  o f t h e two goods w i l l  be  the  map,  x : T -*IRJ def where x ( t )  = (y(t),m(t)). The p r e f e r e n c e s  a utility  function U  (i) strictly  t  U (y,m) t  increasing  in  of traders  o v e r t h e goods a r e  represented  by  satisfying:  for t  e [0,1]  (y,m)  e  is  assumed t o be c o n t i n u o u s  and  M; +  def (ii)  U (y,m) t  The i n i t i a l  = 6y + m  allocation  for  t e {a,b>  i  goods  of  and some  6 > 0.  h e l d by t h e t r a d e r s  is  given  by (i)  i(t)  = (0,m),  for t  (ii)  i(t)  = (1,0),  for  Finally define  to complete  e [0,1]  and  t e {a,b}.  the s p e c i f i c a t i o n  the measure o r s i z e o f t h e t r a d e r s .  o f t h e model we need  Let A denote the  smallest  to  - 24 -  a-algebra  containing  { a } and { b } .  the Lebesgue measurable  Define  (i)  y(.)  sets  a m e a s u r e p ( . ) on t h e s e t s  is  on [ 0 , 1 ]  and t h e  atoms  i n A by  L e b e s g u e m e a s u r e when r e s t r i c t e d  to  [0,1];  def (i.i)  y(a)  = q def  y(b)  "  = q  T h i s model p r e t e d as a f o r m a l  a  is  W  w o u l d on t h e i r  r  V  e  special  [0,1],  good y . part  case  wish  of  > °-  of S h i t o v i t z  [1973].  a duopolistic  It  market  to maximize t h e i r  is  inter-  i n which  utility  by  to maximize  The 6y t e r m i n t h e u t i l i t y  function  It will  represented their  by t h e  profits  normally  be s m a l l  money  two atoms  o r money  of the duopolists  a  trading  m, a c o m p o s i t e c o m m o d i t y d e n o m i n a t e d i n  The s e l l e r s ,  Tike  value of unsold stock.  %  b  some o f t h e i r h o l d i n g s ' o f f o r the  e  representation  continuum of buyers,  units,  h  {a,b},  holdings.  represents  the  b u t c a n be a s s u m e d  to  2 be p o s i t i v e  by s u p p o s i n g  measure o r s i z e o f holdings amount o f  the s e l l e r s  o f good y .  good y a v a i l a b l e  £  There are the s e l l e r s  when one a p p l i e s  =  i(t)  = (1,0)  %  +  two r e a s o n s  terminology,  his  for  t  Finally  their  the  inventory  e {a,b}  the  total  is  V for  proceeding t h i s  have t h e same p r e f e r e n c e s  t o use S h i t o v i t z s 1  ( t )  has a s c r a p v a l u e .  { a } and { b } r e p r e s e n t s  By s u p p o s i n g  t {a,b}^  since  that y always  results.  of  it  The f i r s t  and endowments  t h e same type.  Secondly  way.  provides  they w i l l  This w i l l another  be  is be,  important  interpreta-  -  t i o n of Notice ante  3.  the s i z e that,  in contrast  The C o m p e t i t i v e  with Shitovitz  y,  normalized takers.  good y b y t  e [0,1]  is  Let to  t h e p r i c e o f good y i s  the agents  in  of y for  f o u n d by  [0,1]  p while  all  the  agents  have no i n i t i a l  any p r i c e .  ex  price  in T act  holdings  The demand D ( p ) t  of  for  solving  py - m < m.  demand f o r y w i l l  us assume t h a t  generate a continuous,  correspondence,  that  1  = 4[O,I]¥P>  }  occurs  thus  be:  f  def  P  holdings.  goods.  such t h a t  t  The t o t a l  [1982], production  Now, 'further suppose  be b u y e r s  max U ( y , m ) y,m  D (  that  to 1. Since  then they w i l l  of  inventory  Equilibrium  L e t us s u p p o s e  as p r i c e  -  o f an atom i n t e r m s o f a t r a d e r ' s  to produce the i n v e n t o r y  of m is  25  S (p), t  D  ^ ) -  the preferences  downward s l o p i n g  t e {a,b},  for  are s u f f i c i e n t l y  demand c u r v e .  good y i s  ing: max 6 y + m y,m  1  the s o l u t i o n o f which  such t h a t  py + m < p • 1,  is 1  if  p > 6  [0,1]  if  p = 6  0  if  p < 6.  of course  The  regular supply  g i v e n by  solv-  -  26 -  Hence t h e a g g r e g a t e s u p p l y o f good y  f  def S (  P>  ^e{a,b} t P>  =  S  q [0,  + q  a  q  (  b  + q ]  a  b  0  competitive  The  the market c l e a r s Figure  is:  or S(p*)  if  p >  6  if  p =  6  if  p <  6  equilibrium  is  = D(p*).  This  g i v e n by t h e p r i c e , is  p*, a t  which  shown d i a g r a m m a t i c a l l y  in  1.  Notice  that  demand c u r v e D^(p)  there are  we have  basically  0-^(6) > q  two c a s e s  to c o n s i d e r .  + q^., h e n c e a t  f l  the  With  competitive  * equilibrium  p^ t h e two atoms s e l l *  all  of  their  inventory  at a  price  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  is  such t h a t  a  + q^.  This  a case o f excess  capacity  D (6) < q 2  from t h e s e l l e r s P  2  =  that  <5,  this  seller 4.  the scrap  is  point of  view;'  value o f y ,  equilibrium  is  is  with the e q u i l i b r i u m  while  not u n i q u e  the t o t a l  sales  is  i n the sense t h a t  characterized  D (6).  by  Notice  2  the sales  of  each  indeterminate.  The C o r e  Allocations  Aumann [ 1 9 6 4 ] a r g u e d t h a t and p r i c e s  essentially  a r e no more t h a n a d e v i c e  the c o r e concept  is  only  free  institution  "Intuitively, to s i m p l i f y  seen as a fundamental but i s  also well  one f e e l s  t h a t money  trading."  In  s o l u t i o n concept  that  defined'regardless  contrast is  not  o f the s i z e  of  - 27 -  the t r a d e r s . sellers  Further  set prices  choices.  in the oligopoly  or quantities  problem i t  w h i l e -buyers  Y e t a s Aumann [ 1 9 7 3 ] a r g u e s ,  and d o e s n o t e x p l a i n why a s y m m e t r i e s  this  i s assumed t h a t  passively  is  accept  fundamentally  the  these  asymmetric  i n s i z e a l o n e c a n endow f i r m s  with  4 market which  power.  Accordingly  institution  now f o r m a l l y  specific  define  the  feasible  dy(t) =  A coalition a feasible  t  i(t)  teS  S can improve  tions  ,for almost  t  S £ A.  e a s y t o compute s i n c e  o f Theorem B o f S h i t o v i t z  competitive  u  all  if  there  t e S.  allocations  t h e model  which  cannot  the core  satisfies  Hence~we  The oove allocations  x(.)  that  F o r o u r model  [1973].  all  be  allocation the  assump-  have:  are exactly  the set of  allocations.  Using the computations citly  x(.),  d (t).  such  ;  > U (x(t))  PROPOSITION'(Shitovitz).  L e t us  An a l l o c a t i o n ,  upon an a l l o c a t i o n  f o r S, x ( . ) ,  i m p r o v e d upon by a n y c o a l i t i o n particularly  against  if  The oove i s t h e s e t o f f e a s i b l e  is  c a n be c o m p a r e d .  a coalition.  i  allocation  U (x(t))  concepts  be c a l l e d  f o r S i f and o n l y  x(t)  exists  solution  can a c t as a benchmark  core.  A set S e A w i l l is  the core concept  compute t h e c o r e a l l o c a t i o n  of the previous  s e c t i o n we c a n e x p l i -  f o r t h e two c a s e s .  -  Case 1:  The e q u i l i b r i u m  P* where  =  D^(.)  l  D  is  If  e  a  ( q  is  V '  +  the inverse  demand  curve.  x ( . ) is a core a l l o c a t i o n ,  x(t)  = (D (p*),  x(t)  = (0,,.p*),  t  Notice t  1  price  28 -  that  then  m - p*D (p*)),  t e [0,1]  t  t e  {a,b}.  the t o t a l  amount o f money s t o c k  atom,  {a,b},  j  def •V a'  V  q  =  K=t  P* ^ ) d  T  =  P* t q  = Dj  1  Hence t h e r e v e n u e o f t h e s e l l e r s the Cournot as o u t p u t  Case 2: x(.),  h e l d by each  [1838] p r o f i t  functions  (q  a  + q )q . b  t  has e x a c t l y  t h e same f o r m as  when t h e i n v e n t o r i e s  are  interpreted  choices.  In t h i s  case the competitive  price  is 6;  hence t h e c o r e  are given by:  x(t)  = (D (6)„  x(t)  = (1 - y ( t ) , 6 y ( t ) ) ,  t  m-  6D (6)),  where y ( a ) , y ( b ) a r e any s a l e s  q y(a) a  + q y(b) b  t  levels  = D(6)  t e [0,1]  t e {a,b}  by s e l l e r s  {a,b}  satisfying  allocations  - 29 -  and  0 < y(t)  tory at  t e {a,b}.  In t h i s  case,  given that  the scrap  price  6,the  M (q , t  5.  < 1,  q )  a  Discussion  of  total  - 6q ,  b  the  firms  sell  revenue  unused  inven-  f o r each s e l l e r w i l l  now b e :  t e {a,b:}..  t  t h e Model  Though we h a v e p r e s e n t e d a r a t h e r Shitovitz  their  [1973] model,  it  is  specific  however a q u i t e  example o f  general  the  representation  of  5 the duopoly agents trary  problem.  selling  The model  their  inventory  demand c u r v e .  simply  t o many b u y e r s  I f we now s u p p o s e  t h e good y a t a c o n s t a n t m a r g i n a l each s e l l e r  is  given %  If  the  represents  that  two p r o f i t  maximizing  r e p r e s e n t e d by an  the agents  c o s t c > 6 then the  buy o r total  arbi-  produce  profits  of  by:  }  =  M  t  (  V  %  inventories  ~  C  V  t  e  {  a  '  b  K  m u s t be c h o s e n i n d e p e n d e n t l y  t h e o u t c o m e c a n be d e s c r i b e d Hence t h e e q u i l i b r i u m  r  as a Nash [ 1 9 5 1 ]  *  *  ( q , a  q,)  is  a solution  equilibrium  ex a n t e  then  in:outputs.  6  to  D  * max  q  TT,(q,q, ) 3  b  * max q This  is  duopoly  TT.  (q  b  of course problem.  ,q). 3  simply the standard Cournot-Nash e q u i l i b r i u m  for  the  - 30  T h i s model fits  into  butions  thus  shows t h a t  of  this  work t h e n  is  the assumption  behave c o m p e t i t i v e l y . ^  setting  an e x p l i c i t  The c o r e a p p r o a c h a l s o  firms,  One o f  i n much o f o l i g o p o l y  the models w i t h ; p r i c e , s e t t i n g  price  [1973] q u i t e  framework.  to provide  firms,  whenever a c o m p e t i t i v e e q u i l i b r i u m with  Shitovitz  the C o u r n o t q u a n t i t y - s e t t i n g  foundation for  over  -  though  institution theory  competitive,  contri-  free  that  buyers advantage  the core.wi11  On t h e o t h e r  intuitively  the major  has a d i s t i n c t  in that  exists.  naturally  exist,  hand,  markets  in fact  often  have a  natural  g suffer  from the n o n - e x i s t e n c e  We c o n c l u d e economic view o f  that  interpretation imperfect  i n which economic advantage. when f i r m s  t r i c t e d output  an  but they;..also  the B e r t r a n d ,  the market  models  highlight  i n which the  agents can a f f e c t  levels,  equilibrium.  "mixed market"  competition  Finally in  of  not o n l y  the c l a s s i c a l  fundamental  the c o m p e t i t i v e  a result  corresponding  is  equilibrium  "competitive", result  have no way t o p r e c o m m i t  issue  "Cournot"  will  themselves  to our case  only to  t h e way to  their  arise res-  2 above.  - 31  -  NOTES  1.  See S h i t o v i t z  2.  F o r e x a m p l e an u n s o l d c a r a l w a y s tent.  for  the complete  The main r e a s o n f o r  assumptions  made by S h i t o v i t z  3.  Aumann [ 1 9 6 4 ] , p a g e 4 0 .  4.  Aumann [ 1 9 7 3 ] , page  5.  Of c o u r s e [1973],  6.  to the o l i g o p o l y  level  Nash e q u i l i b r i u m  7.  its  model.  metal  con-  to s a t i s f y  the  [1973].  the model, given  problem i s  being able  is  even to  the  hotel  perfect  information  quite  theorem B o f  to observe  Shitovitz  simple.  e a c h f i r m must c h o o s e  the case i n which  the  its  competitor's  f i r m s maximize  costs  are i n c l u d e d t h a t  example o f S h i t o v i t z then the monopolist  level..  at t h e monopoly  See E d g e w o r t h  of 6 is  the  inven-  choice.  their  payoffs  this  applies  A  expectations.  to the monopoly  8.  of  One c a n a r g u e o n c e p r o d u c t i o n  librium  of  v a l u e due t o  inclusion  c h o i c e we mean t h a t  without  g i v e n correct  has  details  10.  the e x t e n s i o n  By'independent tory  the  technical  [1925]  for  [1973].  Given that  would b u i l d  capacity  Hence ex p o s t we w o u l d h a v e a price. the f i r s t  proof of  this.  one  has  equal  competitive'equi-  - 32 -  Figure 1  CHAPTER 3 ON ADJUSTMENT COSTS AND THE S T A B I L I T Y OF E Q U I L I B R I A  1.  Introduction  It can observe  is well  known t h a t when f i r m s  directly  Nash e q u i l i b r i a  each o t h e r s  that y i e l d  actions  higher  i n an o l i g o p o l i s t i c  there  profits  exists  for all  a large  firms  industry class  than those  as-  sociated with the Cournot-Nash e q u i l i b r i u m .  As F r i e d m a n  [ 1 9 7 7 ] has  shown, t h e s e e q u i l i b r i a  use t h r e a t s  against  titors  that  unless  some f i r m d e v i a t e s  all  firms  shown t h a t  [1975].  In t h i s bria  from t h i s  produce  their  little  basis  p a p e r we s u g g e s t  t o have t h e p r o p e r t y  of equilibria  for a description  of  local  outputs.  are also p e r f e c t  concerning  a solution  If deviation  Cournot-Nash  these e q u i l i b r i a  predictions  an a g r e e d upon l e v e l  agreement.  This m u l t i p l i c i t y  t o make q u a n t i t a t i v e and p r o v i d e s  when f i r m s  the form o f m a i n t a i n i n g  subsequently  has f u r t h e r of Selten  take  can a r i s e  to t h i s  compe-  of  occurs Green  ones  i n such  of disequilibrium  when f i r m s  then  [1980]  ability markets behaviour.  problem by r e q u i r i n g  stability  output  i n the sense  limits  behaviour  of  face  equili-  a cost  of  adjustment.  In p r a c t i c e  one w o u l d e x p e c t  t h a t would l i m i t  their  question  the r e l a t i o n s h i p  then i s  behaviour of firms. strict will  firms  We w i l l  to small,  i n a sense  ability  to adjust  postulate  output  a cost  33 -  costs  instantaneously. costs  and t h e  o f adjustment  o u t p u t changes  the p e r s p e c t i v e  -  to face adjustment  between t h e s e  infinitesimal  restrict  firms  of firms  The  strategic  that w i l l  at each time  to evaluating  t. the  reThis ef-  - 34 -  feet firms  of small  changes  o u t p u t on p r o f i t s  till  location Poussin Matthews  approach  is modelled  now has been a l m o s t literature  [1971],  in,  for  Roberts  exclusively  example,  o f change o f  as a l o c a l  used i n  papers  us t o assume  profits.  game, a  the p u b l i c  theory  goods  by D r e z e and de l a [1979] a n d , most  that  al-  Valine  recently,  [1982].  ing a d i f f e r e n t i a t e  rationality.  equilibrium  is  solution  A solution  as b e i n g a v e c t o r  by a f i r m w i l l  n o t make i t  p o n s e s by c o m p e t i t o r s .  lem t o a n y a r b i t r a r y  of outputs  for  The l o c a l  In the essay the  due t o M a t t h e w s  the  allows;one  identify  of  f o r which any s m a l l  disequilibrium analysis of  four solution The f i r s t  output  anticipated  of  change  res-  be v i e w e d a s  a  prob-  identifying  our a n a l y s i s of  a  stable  approach to the o l i g o p o l y  stability  tak-  notions  a locally  Instead of just  local  than  to study i n  t h a t embody v a r i o u s  game a p p r o a c h c a n a l s o  variations  interaction  literature.  also  given c o r r e c t l y  vector of outputs.  b a s e d o n t h e strategic  a r e new t o  concepts  much more t r a c t a b l e  concept w i l l  through the c r i t e r i o n  the basis  It  better off  way t o e x t e n d t h e c o n j e c t u r a l  equilibria  analytically  game a p p r o a c h . *  s i m p l e manner d i f f e r e n t  provides  rate  formally  [ 1 9 7 9 ] , Schoumaker  S u c h an a p p r o a c h  local  and a l l o w  choose output changes t o maximize the  This that  in  firm  also  behaviour  firms.  concepts  are s t u d i e d ,  of these i s  the  two o f  directional  which core  [1982].  The d i r e c t i o n a l  core  is  i n many r e s p e c t s  the  natural  definition  -  of  local  rationality  group or  coalition  Matthews,  this  for i t  solution  point  equilibrium c o r e does payoffs  will  firms  joint-profit  maximizing  core.  industry  directional  core  for  the  vector  other  To model  the  simply  f r o m an i n d u s t r y Formally  existence  it  strategies  is for  z a t i o n o f the  some v e c t o r is  equilibria  stable  that  perspective  for  of  to  outputs  the  the  directional  regions  solution  ultimate  that  and  yield  the  the second s o l u t i o n  with  industry's  firms  perspective  the  Nash  to  since  the  for  Nash  is well  there w i l l  known gen-  that will'make every firm better  that  firms  have t o  h a v e no i n c e n t i v e core  of those output and a r e b e t t e r solution  a c t as a group to play  introduced  changes  that  t h a n t h e Nash  the  game. < . Our m a i n r e s u l t equilibria  is  is  this  stable  respect  The C o u r n o t - N a s h e q u i l i b r i u m  shown t h a t global  call  concept,  of Cournot-Nash e q u i l i b r i a  outputs  of outputs  consists  locally  be shown  that  game, w h i c h we w i l l  n o t i o n o f a Nash bargaining It  It will  i n a r e g i o n where  restrict  incentives  same t i m e e n s u r e  theory.  lie  local  a r e t h e same.  erally  the  for  The r e s u l t  We show t h a t t h e s e t  from t h e  Nash,  will  ensure  n o t t o be r a t i o n a l  at the  stable.  exist.  in  solution.  and t h e l o c a l l y  exist  does e x i s t  c h a n g e s w h i c h no  H o w e v e r , as..;shown  somewhere b e t w e e n t h e C o u r n o t - N a s h  t h e Nash e q u i l i b r i u m  the  core  This w i l l  We c a n a l w a y s  directional  of those output  not i n general  the o l i g o p o l y  not e x i s t .  to  consists  is not l o c a l l y  for  -  of f i r m s would wish to a l t e r .  t h a t when t h e d i r e c t i o n a l then t h i s  35  the  local is  under t h i s  to are  off.  and  Cournotlocal  game  rational  equilibrium.  equivalent a complete  solution  of  threat  characteri-  concept.  -  These e q u i l i b r i a being  inversely  will  proportional  to t h e i r  of adjustment costs  equilibria  but a l s o  power o f  has  will  not e x i s t  costs  not o n l y  a natural  of  solution  concept  sophisticated  in general,  coalition  to  that'might  i d e n t i f i e d with  the bargaining form t h e y  the  portant  are c h a r a c t e r i z e d .  illustrate  bargaining core  is  introduced  the c h a r a c t e r i z a t i o n these  results  section  for  the s t r u c t u r e  of  the  in-  of bar-  by f i r m s .  locally  Though  stable  be  regardless  locally  of  stable  the out-  core  strategy is  model  s p a c e t o be u s e d .  introduced  4 goes  2 the  and i t s  im-  through a duopoly  The n o t i o n o f  a Nash  5 a l o n g w i t h o u r main r e s u l t  outputs/ setting  Section 6  on  specializes  oligopoly while  the  final  remarks.  Model  Consider  an o l i g o p o l i s t i c  it  equili-  also  In S e c t i o n  of the model.  stable  captures  core.  Section  the standard q u a n t i t y  contains our concluding  The B a s i c  the s e t  These o u t p u t s w i l l  of the  in Section  locally  at the  as f o l l o w s .  3 the notion of a d i r e c t i o n a l  properties  example to  2.  is  formation  i m p r o v e on t h e  Nash b a r g a i n i n g  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  In S e c t i o n  firms  the  core which  core proving t h a t  cannot  The a g e n d a o f t h e e s s a y is  Thus  i n terms o f  the b a r g a i n i n g  core'.  respect  it  is  b r i u m g i v e n by t h e Nash b a r g a i n i n g  puts  restricts  interpretation  exist  coalitions  of adjustment.  greatly  will  stable with  shares of the  firms.  The f i n a l the p o s s i b i l i t y  -  o f t e n be u n i q u e w i t h t h e m a r k e t  troduction  gaining  36  industry consisting  of n firms  in-  -  d e x e d by t h e s e t a single q(t)  is  +  period  N = {l,...,n}.  undifferential  e 1R  a vector  be g i v e n  37  Each  product of  to  outputs  = TT.(q(t))  Here C..(q. ( t ) , q . ( t ) )  -  change o f o u t p u t  period.  represents  tion  q^(t)  time  a cost  number o f t,  then  a n amount  of  consumers.  let  the  If  profits  per  If  were  firms  not  for  can o b s e r v e  many e q u i l i b r i a  that  of adjustment  where  the  rate  of  (t)  = — ^ —  an e q u i l i b r i u m ,  that  exist  is  q* i s it  a large at  e N sells  i  C.(q.(t),q.(t)).  dq  if  firm  by  V.(t)  hence  -  .  It  is  t h e n iT|(q*)  represents  the adjustment each o t h e r ' s  supposed t h a t  costs,  actions  can be s u p p o r t e d  C.(q^,0) = 0  the  profits  then under  the  immediately  by t h e  per assump-  there  appropriate  would  threat  3 strategies. costs will  that  In t h i s will  restrict  be s u p p o s e d  o u t p u t change  paper  that  firms  at time  game g e n e r a t e d firms  will  their  change  pate  their  the f i r m ' s  at  by t h i s each time  in  profits  competitors  g.(t),  the  problem w i l l t  i  the  1ocal  V.(t),  i  reactions.  if  a long  for  to  the property  be p l a c e d o n t h e  adjustment  output  Hence  interaction strategic  given  that  There are  there  be t o c h o o s e  exists  of  the  firm  as  behaviour of the  By t h i s  x.(t)",  i  they  several  run e q u i l i b r i u m ,  that  will  It its  e N.  be a n a l y z e d .  e N,  changes.  strategy  choose a s t r a t e g y  Firstly have  q* i s  game,  this. it  t  than analyze  differential  will  to continuous  d e n o t e d by x . ( t ) ' =  However r a t h e r full-fledged  assumptions  then  we mean  N to  it  no a c t i o n  is  local  that  maximize  correctly reasons  a  antici-  for  doing  reasonable  by a f i r m  that  - .38 -  will  result  V.(t)  > 0.  def x(t) =  Secondly,this  for  x'(t)  analysis  recently  achieve  every q e K  of the oligopoly  a Pareto-efficient  Let  provide  of the local  us now c o m p l e t e  the analysis  will  problem.  process  outcome.  result  in  solutions,  that  This  incentive  a basis  Further  local  be o f t h e f o l l o w i n g  It will  such  will  can p r o -  schemes.  o f the framework.  be assumed t h a t  r a t h e r extreme  [1980]  by f i r m s  game f r a m e w o r k  properties'of  the d e s c r i p t i o n  for the dis-  Shapiro  i f followed  occur a t a given time t f u r t h e r  p e n d e n c e w i l l ' be s u p p r e s s e d . will  will  or formally  a n d hence p r o v i d e  +  s u g g e s t e d a dynamic  v i d e an a n a l y s i s  all  analysis  in profits,  Xj(t)  equilibrium has  i n an i m m e d i a t e i n c r e a s e  explicit  Since time de-  the adjustment  costs  form:  x. c [-a.Cq.J.b^q.)]  0,  if  °°,  i f not  Vvx,)  where a ^ . ) , b . . ( . ) . a r e m e a s u r a b l e assumed t h a t  firms  can always  all  Since  negative  q. > 0.  non-negative  v a l u e d maps.  It will  be  a d j u s t o u t p u t ; h e n c e s u p p o s e b^(q^) > 0 f o r  outputs  will  n o t be p e r m i t t e d  t h e n we need  o n l y assume a ^ O ) = 0 whil:e>.a.. (q.|) > 0 f o r q. > 0 .  Since  firms  one m i g h t a s w e l l  will  not choose a c t i o n s  assume t h a t  the vector o f s t r a t e g i e s ,  from t h e s e t  X(q)  def =  X [-a.(q.),b ieN 1  1  with i n f i n i t e  (q ) ) ] . 1  1  costs  x, is  then  chosen  -  Under t h e s e given  cost  assumptions  39 -  the  r a t e o f change o f  profits  will  be  by def U,(q,x)  where i t  is  = V TT.(q)x = V . ( t )  assumed t h a t  ir. Cq)  is  differentiable  with  its  gradient  given  by  and x e X ( q ) .  Thus  (i) (ii)  the l o c a l  the s t r a t e g y the  payoffs  space to  game a t a v e c t o r  cepts  have s u f f i c i e n t  for  the main  that w i l l  associate  feasible  strategies.  studied  for  librium  behaviour  by  e N , d e n o t e d by  for  is  of a rather  the discussion  More g e n e r a l complicate  U.(q,x).  strategy  special  of the  spaces  the a n a l y s i s  form  solution  c a n be  without  it  con-;  introduced,  substantially  results.  A solution  In t h i s  i  space X(q)  structure  however t h a t w o u l d g r e a t l y altering  given  n  firm i  t o be i n t r o d u c e d .  q is  X(q)_=IR ,  Though t h e s t r a t e g y will  of outputs  for for  the l o c a l  game w i l l  a vector of outputs,  paper the e x i s t e n c e  several is  solution  concepts.  described  simply  be a  q, a set  and s t r u c t u r e  of  S ( q ) <= X ( q )  S(q) w i l l  Given a s o l u t i o n  by consistent  correspondence  trajectories.  then A  of  be  disequiconsistent  - 40 -  trajectory that  is  a time  i n output space  d e n o t e d by  q(t)  satisfies:  e S(q(t)),  M^-  This  paper w i l l  characterization tion  dependent path  concepts  q(0)  o f such t r a j e c t o r i e s  a n d how t h e y  Dreze and Henry  In p a r t i c u l a r local  Vt > 0.  Q  n o t be c o n c e r n e d w i t h t h e e x i s t e n c e  it  but w i t h t h e n a t u r e  define S(q).  t y p e o f s y s t e m c a n be f o u n d i n Champsaur,  = q ,  Existence  Castaing,  and  of the  results  for  C . a n d M. V a l a d i e r  solu-  this  [1969]  and  [1977].  will  be o f  interest  vector  q* i s  locally  to  describe  the notion  of  stability.  An o u t p u t  DEFINITION 1.  if  stable  and o n l y  if  0 e S(q*).  Local exists  stability  c a n be i n t e r p r e t e d  no i n c e n t i v e s .to d e v i a t e  solution  concept  defining  By r e q u i r i n g in considerable  from the v e c t o r  that  of outputs  there  q* u n d e r  the  S(q*).  that equilibria  restrictions  The f i r s t  as i m p l y i n g  solution  are  on t h e s e t o f  concept  locally possible  t o be s t u d i e d  stable will  result  equilibria.  is  the  directional  4 core. vector  When i t i n the  o r group o f actions.  exists  it  directional  firms.  has t h e a p p e a l i n g  property that  no  strategy  c o r e c a n be i m p r o v e d upon by any s i n g l e  Such s o l u t i o n s  can c e r t a i n l y  qualify  as  firm  "rational"  - 41  3.  The D i r e c t i o n a l  Core  To d e f i n e  this  A coalition  of  firms  solution  will  direction  a number o f  definitions  be d e n o t e d by a s u b s e t M c ± N .  has c h o s e n a v e c t o r o f o u t p u t change t h i s  -  changes x e X(q)  t o any p o i n t  in  {y e X(q)  = x  then the  are If  required.  the  industry  coalition  M can  def F(q,x,M)  F(q,x,M)  is  =  the s e t  This  set  of f e a s i b l e  is  directions  shown f o r t h e  a square  in  ing at x that these  In the  the s e t  firm  strategy  their  petitors. x is  choices  open to  Hence i f  firm  strategies  reactions  exercises  given  in  firm  Figure  the  will  1 when  X(q)  those s t r a t e g i e s  One s h o u l d p r o b a b l y  not  rather  responses  that  firm  of  its  strategy  represents  strategy  a coalition's  and p r o v i d e s  l's  the basis  x  2 >  explicit for  the  directions  the  is  start-  these  light  of anticipated  stra-  view  as t h e y t r y  then F(2,x,{2})  represent  all  t o some c h o i c e x b u t  2 believes  solution,  open t o i t  by  in  M.  a "proposed" or considered  contains  directional core. DEFINITION 2. The s e t o f M - u n d o m i n a t e d q is  coalition  c a r r i e d o u t by f i r m s  2 through the choice  duce a s e t t h a t  x is  F(q,x,{2})  as  actions  a potential  figure,  2 can b r i n g about.  i n a sense conceptual evaluate  for  t M}.  —  IR .  tegy choice while  i  two f i r m c a s e  2 is  if  f  are  to  by c o m such  that  alternatives  We W i l l  now  evaluation  of  d e f i n i t i o n of  at a vector of  introthe the outputs  - 42 -  K(q,M)  = {x e X ( q ) | ^ y e F ( q , x , M )  U.(q,y)  > U.(q,x)  such  Vi e M}.  that  ,  A v e c t o r o f changes x i s undominated w i t h t i o n M i f there all  firms  does n o t e x i s t  i n the c o a l i t i o n  DEFINITION  3.  any f e a s i b l e  better  The d i r e c t i o n a l  respect  alternative  to a c o a l i -  that w i l l  make  off.  core  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 self  i m p r o v e on t h i s  vides  an a p p e a l i n g  change i s  strategy.  definition  x e K ( q ) , then not o n l y w i l l lot  but also  strategy  cept s a t i s f i e s Our f i r s t  When i t  result  PROPOSITION 1.  is that  K(q,M)  behaviour.  directions  pro-  For i f  exist  its  no o t h e r  Hence s u c h a s o l u t i o n  o f both group and i n d i v i d u a l  t h e M-undoiiiinated  core  be a b l e t o i m p r o v e  p o i n t o f view there w i l l off.  c a n by i t -  the d i r e c t i o n a l  rational  no f i r m by i t s e l f  make e v e r y o n e b e t t e r  the c r i t e r i a  exists  of locally  from the i n d u s t r y ' s  that w i l l  i n K ( q ) , t h e n no c o a l i t i o n  con-  rationality.  always  exist.  f <> j for all M <=• N .  The p r o o f o f t h i s  as w e l l  as the remaining  propositions  are to  c a n be q u i t e  easy  be f o u n d i n t h e A p p e n d i x .  I n many c a s e s to compute. directions  Before by  t h e M-undominated  d o i n g so d e f i n e  directions  the cone o f l o c a l l y  Pareto-improving  - 43 +  d  e  C (q,M) This  set  bers o f the  n {x e ]R |v IT. ( q ) x  f  =  represents  all  the c o a l i t i o n  initial  output  on w h i c h t h i s using t h i s  set  set  those changes  M strictly  vector is  q.  for  i n output  better  off  The r e a d e r  is  every  F(q,x,M)  such t h a t  C (q,M)}.  e M}. mem-  compared w i t h  remaining  at  referred  to Figure  1  the M-undominated  = {x e X ( q ) | # y £  i  t h a t w o u l d make a l l  shown f o r t h e two d i m e n s i o n a l  one c a n d e f i n e  K(q,M)  > 0  back  case.  Notice  directions  that  by  (1)  This  characterization  dimensional tions  example o f  follow  PROPOSITION  almost  If  immediately  t o compute K ( q , M ) Also  from t h i s  C (q,M)  point  for  of outputs  the c o a l i t i o n  directions  are  M.  undominated.  (q)  3q.  proposi-  That  q is is,  a locally  strict-  it  pos-  is  i n a way t o make e v e r y member o f M b e t t e r  If j e.M and for all i £ M we have  —4—  two  and q £ ft" then  = cb t h e n t h e v e c t o r  +  feasible  3TT  following  the  characterization.  PROPOSITION 3. (i)  the  for  = X(q).  to vary outputs  hence a l l  easier  the next section;.  +  Pareto e f f i c i e n t  sible  £  make i t  If C ( q , M ) =  :2.  K(q,M)  ly  will  (y-x)  +  > o.  then if x £ K ( q , M ) we must have x . = b . ( q . ) .  not  off,  - 44 (ii)  j  If  £ M and  for  all  K ( q , M ) we  must  i  £ M we  have  3Tf,(q)  then  i / x  £  This  simple  x. J  have  =  -a.(q.). J J  proposition states  that  if  all  firms  w o u l d w i s h t o c h a n g e an o u t p u t q . i n t h e same d i r e c t i o n ,  in a  coalition  then they  will  J  w i s h t o do s o a s q u i c k l y as  possible.  Having c h a r a c t e r i z e d  the M-undominated  s i m p l e m a t t e r t o compute t h e d i r e c t i o n a l ever,  the d i r e c t i o n a l  core w i l l  flict  between i n d i v i d u a l  t i o n o f an M-undomi n a t e d  This tion.  fact w i l l  We may s t i l l  are l o c a l l y existence  stable.  proof,  it  is  core using D e f i n i t i o n  3.  not always e x i s t  and group r a t i o n a l i t y  due t o  a How-  an e s s e n t i a l  con-  as d e f i n e d u s i n g t h e  no-  direction.  be d e m o n s t r a t e d i n t h e e x a m p l e o f t h e n e x t  wonder i f  there are outputs  In such a s i t u a t i o n ,  we w o u l d s t i l l  would wish to d e v i a t e .  directions  despite  have o u t p u t s  Unfortunately,  f o r which the  K(q)  sec-  exist  and  lack of a general  f r o m w h i c h no g r o u p o f  the answer to t h i s  question  firms is  generally  negative. I f q* i s l o c a l l y s t a b l e , then 0 £ K ( q ) . It w i l l Sirfollow that - J C — ( q * ) = 0 f o r a l l i £ N. But these are s i m p l y the f i r s t ;  order conditions yield This  locally is  f o r the Cournot-Nash e q u i l i b r i u m which g e n e r a l l y  Pareto  efficient  outputs  and h e n c e 0 E K ( q )  summarized i n the f o l l o w i n g ' p r o p o s i t i o n .  is  do  not  impossible.  - 45 9TT.  PROPOSITION  4:  Suppose  that  q* s a t i s f y i n g  for every  - (q*) q  = 0, V i  e N  i  3TT.  - r - (q*) < 0 for i 3q.  (i)  1  £ j , i ,j  e N;  9TT.  CiiJ  there  Then there respect  exist  e N an i ' ^ j  no vector  Notice proposition  that  simply  requires  is  firms  one c a n i n g e n e r a l  if  Assumption  a competitor  that this  are a c t u a l l y  inequality  for  the c h a r a c t e r i z a t i o n  obtain existence  equilibrium  given  is  for  4.  the  expect (i)  < 0. with  of  output.  be s t r i c t  locally  to  for  core  stable outputs. requirement  profits.  for  This  as  is  n ieN  PROPOSITION  5.  proposition  x e K ( q ) for q e IR" i f and only  that  One way  by h a v i n g the  to firms  Nash-  follows:  the l o c a l  completely  (ii)  solution  game a t  q will  K(q,{i}).  The f o l l o w i n g  is  other.  by  K(q) =  this  some c o m p e t i t o r ,  t o be t o o s t r o n g a  game w h i c h we d e f i n e  The Nash d i r e c t i o n a l  of  Assumption  competition w i t h each  t o weaken o u r r a t i o n a l i t y  local  the assumptions  s i m p l y means^that a f i r m  core appears  o n l y t h e i r own c o n t r i b u t i o n  DEFINITION  (q*)  are l o c a l l y stable  increases  in effective  Hence the d i r e c t i o n a l  evaluate  q that  of outputs  t o be s a t i s f i e d .  n o t made b e t t e r o f f  concept  suc/z t/zat ^ —  K(q).  to  the  Vj  exist  characterizes  i f x.  i s given  K(q).  by  be  - 46 3TT.  (i)  x. e C - a ^ q . ) ,  (ii)  x. = b. ( q . ) when  3-rr.  l  l  i  when ~  b ^ . ) ]  dq  ( q ) = 0;  ( q ) > 0;  i  aw. x. = -a.(q.)  (Hi)  wfren T — -  The Nash d i r e c t i o n a l the  Cournot-Nash  core  are evaluated  only. to  equilibrium  Under  this  by f i r m s  core  captures  concept. taking  behavioural  the f i r s t - o r d e r  ( q ) < 0.  conditions  the e s s e n t i a l  Strategies  into  account  assumption  local  characterizing  features of  i n t h e Nash  their  directional  individual  stability  is  actions  equivalent  the Cournot-Nash  equili-  brium.  PROPOSITION 6.  A vector of outputs q * is locally stable with respect to  the Nash directional 3iT.(q*) — \  core if and only if  = 0 V i  e N.  3q. This directional  core  one t o argue sense to  that  their  result  and t h e Cournot-Nash  that  t h e Cournot-Nash  i t results  profits  shows t h e d i r e c t  from f i r m s  and ignores  will  be s e e n w i t h i n  help  illustrate  equilibrium.  equilibrium  evaluating  the benefits  the context  t h e concepts  relationship  that  b e t w e e n t h e Nash Further  allows  i s not "rational"  only  their  from a c t i n g  of the following h a v e been  it  own  contribution  as a group.  example,  introduced.  in the  This  which w i l l  also  - 47 -  An  4.  Example  Consider pose t h a t  costs  an i n d u s t r y w i t h  are  two f i r m s  z e r o a n d t h e demand c u r v e  and N = { 1 , 2 } . is  linear  Let  us  and g i v e n  by  sup-  def P(Q)  where i  =  1 - Q  Q = industry  (without  output  adjustment  and P i s  costs)  price.  The p r o f i t s  per  period of  firm  are:  def ^ (q ,q ) 1  We jwill  assume  :  = (1 - •(q +q )>q -  2  that  a.^.)  1  2  the adjustment  = b.tq.)  = 1,  i  i  cost  parameters  are  given  by  e {1,2}  for q.  Thus  the  of the  0.  >  set of  profit  feasible  functions  is  X = [-1,1]  = (l-q -2q ,  VTT (q ,q )  = (-q ,  l-q -2q ).  firm  Cournot  1  2  2  1  Recall  2  that  2  2  by  3TT. 8qT  (  q  l' 2> q  -  x [ - 1 , 1 ] _=]R .  The  gradients  are  VTr (q ,q ) 1  defined  changes  0  i's  -q^  1  1  2  reaction  function  is  implicitly  - 48 -  from which i t  is  an e a s y m a t t e r  brium i s  (-j ,  P(q^+q )  = 0 and t h e c o n t r a c t  2  VTT^ = 0 . will  .  This  These l i n e s  G  i  be u s e d t o c o m p l e t e l y  m  1 :  = {-1}  K(q,{2})  = [-1,1]  2 along with  t h e two f i r m s  the  into  local  the  g i v e n by  five game  regions  equililine VTT^ + that  structure.  x  of outputs  3 it  r e g i m e we h a v e :  {-1}.  core  is  =  Figure  (q)  also  follow  that  = (=1,-1).  core  is  u n i q u e and i s  given  by  (-1,-1)  any q i n Regime 1.  dq-^  this  [-1,1]  x  will  Hence t h e d i r e c t i o n a l  K(q)  q in  = (-1,-1).  K(q,N)  2:  the Cournot-Nash  ?  t h e Nash d i r e c t i o n a l  From P r o p o s i t i o n  REGIME  Figure  the output space  vector  K(q,{l})  K(q)  in  curve for  that  3 ^ <q> < 0. 3 ^ <<> < °'  Consequently  picted  in  characterize  3TT  For a t y p i c a l  for  depicted  divide  9TT, m  is  t o compute  In t h i s  regime  3.  >  0,  dq  (q) 2  <  0.  K(q)  = K(q).  These s e t s  are  de-  - 49 -  For a t y p i c a l  K(q,{l})  vector of outputs  = {1} x  [-1,1].  K(q,{2}) = [ - 1 , 1 ]  K(q,N)  x {-1}.  = K(q,{2}).  Hence t h e Nash d i r e c t i o n a l  K(q)  and the  =  These s e t s  are  =  core  is  given  illustrated  in  of  firms  •REGIME 4: ^ dq^  given  by  Figure  are  quite  (i)  4.  (q) > 0.  analogous  1 and 2 a r e s i m p l y  to  those obtained  i n Regime  2;  reversed.  (q) > 0, -r-^ (q) > 0. dq  2  H e r e we have two s u b c a s e s tions  by  37Tp  (q) < 0, -~  H e r e we g e t r e s u l t s the r o l e s  is  (1,-1).  BIT.  REGIME 3: j±  core  (1,-D  directional  K(q)  we h a v e :  i n which the Pareto  different.  ^GIME_4a:  For a t y p i c a l  Vir^q) + VTr (q)  vector  2  of outputs  _ 5 > 0. we have  improving  direc-  -  K(q,{l})  = {1} x  [-1,1]  K(q,{2})  = [-1,1]  x {1}  K(q,N)  = K(q,{l}) u  H e n c e t h e Nash and d i r e c t i o n a l  K(q)  as d e p i c t e d  in  = K(q)  = (1,1)  Figure  5.  (ii)  K(q,{2}).  c o r e s a r e g i v e n by  VfTj(q)  REGIMEJb:  For a t y p i c a l  = {1} x  K(q,{2})  = [ , 1 , 1 ] x {1}  However i n t h i s  t(q)  < 0.  [-1,1]  x  c a s e t h e Nash d i r e c t i o n a l  [-1,1].  core  is  given  by  = (1,1).  in  Figure 6 the d i r e c t i o n a l  = K(q) n  Note t h a t  K(q,N)  core  is  empty:  = <J>,  i n R e g i m e s 4a and 4b t h e Nash d i r e c t i o n a l  We l e a v e a s an e x e r c i s e  the d i r e c t i o n a l  2  = [-1,1] x { - 1 } U { - 1 }  H e n c e , as d e p i c t e d  K(q)  + V7T (q)  v e c t o r o f o u t p u t s we have  K(q,{l})  K(q,N)  identical.  50 -  cores  for  t o the  cores  reader the d e t e r m i n a t i o n  o u t p u t s on t h e boun'daries o f  these  are of  regions.  -  An i m p l i c a t i o n that  no v e c t o r  Further  of outputs  any t r a j e c t o r i e s  Thus t h i s  solution  puts,  it  up i n  region 4b.  as  of  that  This  analysis  regimes  consistent  concept  does s u g g e s t  this  in  is  though if  with  3 or  appealing  region 4b.  Consider firm  locally  since  this  is  to  identify  the  i n the  behaviour of  Certainly  firm  one w o u l d e x p e c t  clearly  worse  One must a s k  than if  that  it  in this  2 could  example in  remaining  f a c t rational assumptions  for  firms  I f we w e r e t o  stable.  stable  out-  they w i l l  c a n be  4b.  end  identified equilibrium  an o l i g o p o l i s t i c  equi-  have  the' n o t i o n  argument s u g g e s t s  c a n n o t be e x p e c t e d In t h e  to  provide  following  how t o  each o t h e r ' s  that  region  behaviour 4b.  2 and c h o o s e s f i r m 2 to = (1,1).  firms of  a t q and f i r m  is  of  in  Suppose  x-^ = 1 so  do l i k e w i s e T h i s outcome  to choose x .  "cheating"  immediately  = 1 since  The p r o b l e m t h e n  concerning  the  can always  start  rational  reis  a t q and c h o o s i n g x = ( 0 , 0 ) .  for  n o t g a i n by c h o o s i n g  This  gime 4 b .  is  analysis  w i t h x^= 1 . firms  for  informational  tegy changes.  react  region  interior  i n a s t r a t e g y x £ K ( q , { l } ) Pi K ( q , { 2 } )  firm  national  locally  sulting  role  any l o c a l l y  c u r v e and t h e C o u r n o t - N a s h  some o u t p u t  1 ignores  x e K(q,{l}).  Under t h e  locally  is  exist.  The p r o b l e m t h e n  then t h a t  core  K ( q ) must end up i n r e g i o n  not y i e l d i n g are  4a a r e  s u g g e s t e d as t h e a r e a w h e r e one m o s t e x p e c t s  1ibriurn to  that  of the d i r e c t i o n a l  1, 2,  firms  the a r e a between the c o n t r a c t  often  -  51  plays  respond  1 chooses  f i r m 1 would formalize  the  = 1.  to  no stra-  x-^ = 0 t h e n immediately conjectures  actions.  the  Nash d i r e c t i o n a l  a description  of  s e c t i o n we s u g g e s t  core,  K(q),  rational  behaviour  in  a solution  concept  that  re-  -  not only that  is rational  firms  K(q,{i}),  5.  from the i n d u s t r y ' s  can c r e d i b l y i  choose responses  view b u t a l s o in their  In t h i s  not suffer  the d i r e c t i o n a l  a solution  ever  rational  c o r e y e t has s t r o n g e r  start  the d i r e c t i o n a l for firms  without  expect  invoking  will  a  tivates  DEFINITION 5. ~ B(q)  The  f o r the whole  to e x i s t  solution  that  do l i k e w i s e  core.  Since  than is loin  industry.  precisely  in K(q,N).  x'. £ K ( q , { i } )  f i r m s must c o n c l u d e  of competitors  the following  properties  I f the industry  S u p p o s e we i.  I f a firm tries  this  to  firm i s acting non-  resulting  firms w i l l  in a vector deviate  only  t h e y a r e made b e t t e r o f f , t h i s mo-  Core i s g i v e n b y  def = {x £ X ( q ) | x  This  y £ K(q,{i})  concept.^  Nash Bargaining  How-  when t h e r e a r e  f o r some f i r m  response f r o m t h e o t h e r f i r m s .  and hence t h e y w i l l  the response  core.  game  associated  i t s output to achieve a d i r e c t i o n  £ i ( q ) , t h e Nash d i r e c t i o n a l  after  idea  sets,  to l o c a l  problem  rationality  from elements  such t h a t  to adjust  concept  w i s h t o c h o o s e some d i r e c t i o n  K ( q ) , ceases  to d e v i a t e  x then the other  cooperatively z  core  w i t h a n x-'e K ( q , N )  firm, cannot  block  then a l l firms  the s e t o f undominated d i r e c t i o n s  incentives  own u n d o m i n a t e d  from t h e n o n - e x i s t e n c e  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  K(q,N),  the  Core  s e c t i o n we i n t r o d u c e  t h e o r y t h a t does  cally  captures  e N.  The N a s h " B a r g a i n i n g  with  52 -  £ K(q,N)  and  3y £ K ( q )  IKtq.x)  f o r which  > U^q.y)  V i £ N}.  if  - 53 -  A direction be made b e t t e r generally playing local  off  the case  Nash'.  threat  x e B(q)  is  one i n w h i c h f i r m s  by d e f e c t i n g . if  K(q)  I n f a c t when K ( q ) n K ( q , N )  = <j), t h e n t h e y c a n e x p e c t  N o t i c e t h a t each element I f x e B(q)  strategies.  cannot expect  i n B(q)  is  the t h r e a t  to  = cj>, a s  is  t o be worse o f f  by  an e q u i l i b r i u m  strategy  in  for.firm  i  is  simply:  where y E K(q)  if  x . = x . J J  for  y.,  if  x . f x\ J J  f o r some j  such t h a t  IK ( q , x )  S i n c e B(q) e x i s t s analysis  of  the  > U..('q,y),  The set B ( q ) always  PROPOSITION 7.  oligopoly  zation of the l o c a l l y  •PROPOSITION 8.  al 1 j  x., i  it  stable  i  1  $ i  V i  e N.  exists.  can p r o v i d e  problem.  f  a basis  The q u e s t i o n  for a  then  disequilibrium  is'the  characteri-  outputs.  Suppose at q e K "  ^ - M ' q ) < 0 V i f j ; i , j E N then q  + J)q. stable with respect to B ( q ) , that is 0 e B ( q ) , if and only if  is locally (i)  C (q,N)  (ii)  x ='( bj(qj),. . . ,b (q  +  !  n  This pect  = <> f  to B(q)  o f v i e w and  proposition  then i t (ii)  ) ) satisfies  states  must be ( i )  the gradients  that  if  VTT^ ( q ) « x q is  locally  Pareto e f f i c i e n t  o f the  profit  = 0 Vi  £ N.  stable with  from t h e f i r m s  functions  m u s t be  res-  point  orthogonal  -  to x.  Condition  implies  that  (i)  is  no x w o u l d make a l l  hold since  B(q)<= K ( q , N ) .  fact  that  did  if  that  it  not  for.v. some i  -  interpreted  must  case  54  hold,  as  Pareto  firms  efficiency  simultaneously  The s e c o n d  condition  t h e n by P a r e t o  In F i g u r e output that  in  B(q)  a locally  unique.  length  on t h e  stable  on t h e  should  overly  That  core w i l l less  also is  often  restrictive  A criticism supposed t h a t  if,  other  However  firms.  main b e t t e r  off  say  acting  This  than  B(q)  it  will  this  the  firm  i,  may be t h e  this  or  in  that  considerable of  B(q)  case,  1,  at  con-  or  be  equilibrium,  the  previous  at  greater  section.  2,  core  that  by  exist  a Nash  next  3 , 4a we  have  a n d t h e Nash  this  solution  bar-  is  not  core.  solution  independently, case  requirement  characterized  for  direction  suggesting  c a n be made o f acted  the  3Tr..(q) — > 0  this  be d i s c u s s e d  regimes  the  place  the  that  in  that  not always  is  directional  as a g r o u p ,  is  in  that  it  be  arbitrary  the s t r u c t u r e  problem  same,  the  does  issue w i l l  exists  be t h e  the  dq^  Notice  conditions  f o r which  be n o t e d  when i t  must  shown f o r an  8 shows  Since  first-order  oligopoly  is  to' K(q)  ir.-C'q) t h e n  possibility.  the standard  = K(:q).  gaining  the  B(q)  Figure  of outputs. of  from  0.  example.  respect  many e x a m p l e s  b e i n g one  It B(q)  like  to  of  previous  K(q,N).  derivatives  exist  for  set  vector  However,  there w i l l example  the  be u n d o m i n a t e d w i t h  restrictions  ditions  7 the s t r u c t u r e  r e g i m e 4b o f  and  dq.j  be p r e f e r r e d  = <> j  +  off  results it  C (q,N)  - - — — - < 0 implies  l and hence x e K(q) w i l l  better  efficiency  e N, V i T . ( q ) x > 0 s i n c e  since  the  is  that  we  t h e n so w o u l d a l l firms  o t h e r words  it  in  N / {i}  is  not  have the re-  credible  - 55 -  f o r them t o a c t  i n a Nash f a s h i o n .  of the conjectured behaviour of to block  solutions  of outputs.  This  in B(q) will  We w i l l  firms,  now show t h a t ,  they w i l l  not f i n d  it  i n a.';neighborhood o f a l o c a l l y  be done u s i n g t h e n o t i o n  regardless profitable  stable  of a c o a l i t i o n  vector  struc-  ture.  Let N = { l , . . . , n } of  partitions  3(N)  denote  the s e t  of firms  and d e f i n e  the  set  o f N by  ( 3 e P(N) I ( i )  =  u  M = N and  (ii)  if  e 3,M f  M,L  L,  Me3 then M O L  where  P(N)  = power s e t o r s e t  A partition, To s e e w h a t t h i s Within  interpretation  this  coalition notion  with  of a c o a l i t i o n  this  Given t h a t tion  freely  cannot  this-is  what  use s t r a t e g i e s  the c o a l i t i o n  only.  However,  action  affects  all  problem and a l s o  among t h e m s e l v e s  agents  clarify  as a g r o u p  and h e n c e may use  by o t h e r members o f  meant b y a c o a l i t i o n  other  in  We p r o p o s e a n o t i o n o f  L e t us v i e w a c o a l i t i o n  that.are  coalition. natural  possible.  upon a c t i o n s is  of a  has t h e  interdependence  structure.  t h a t are contingent  not  the n o t i o n  structure.  a coalition  game w h e r e e a c h a g e n t ' s  a g e n t s who c o m m u n i c a t e gies  first  trading within  is  N.  be c a l l e d a c o a l i t i o n  us c o n s i d e r  interpretation  to deal  of  o f a p u r e t r a d e model  of-agents  a non-cooperative  of subsets  3 e 3(N), will  means l e t  the c o n t e x t  payoffs,  = <j>}  the  the of  strate-  coalition.  then agents o u t s i d e  c o n t i n g e n t on a c t i o n s  a  of agents  the in  coali-  - 56  the  coalition.  For t h i s  tions  of agents  tion,  then d i r e c t i o n s  firms  not  jecture  of  change w i l l  still  necessary  choose  structure  actions  other  then c o n j e c t u r e s the  industry  M is  a  ac-  coali-  M-undomi-  i n M form e x p e c t a t i o n s  and t h e y m u s t a l s o  than M w i l l  are  if  form.  actions  B(q,3) defined  B(q,3) =  H  con-  firm  containing  f o r m e d on t h e b e h a v i o u r o r  and n o t t h e p a r t i c u l a r  on how  form a  Since every  only a coalition  3 e 3 ( N ) , t h e n we w o u l d e x p e c t  Given a c o n j e c t u r e the s e t  firms  must be i n some c o a l i t i o n ,  of  Hence i f  be c h o s e n f r o m t h e s e t o f  that  their  on w h i c h c o a l i t i o n s  firm,  as given.  the  K(q,M).  in M w i l l  definition single  is  r e a s o n , members o f a c o a l i t i o n m u s t t a k e  o u t s i d e the c o a l i t i o n  nated d i r e c t i o n s  It  -  a  coalition  t a k e n by  t h e outcome  by  firms.  to l i e  in  by  K(q,M).  Me3 That  is  x e B(q,3) i f  coalition  in 3.  We n o t e  The s e t  jectures  M forms  that  it  this  B(q,3) c o n s i s t s  structure  a coalition  that  if  is  undominated w i t h  set w i l l  always  respect  to  each  b e non-empty.  B ( q , 3 ) t <$>.  PROPOSITION 9.  coalition  and o n l y  is  A way o f v i e w i n g  3.  then i t  the c o a l i t i o n  respond so t h a t  the f i n a l  merely the c r e d i b l e  of those a c t i o n s  will  direction  Hence a d i r e c t i o n  set  choose x e K(q,M),  structure  responses  this  3 will  t h a t would occur is  however i f  it  form then the f i r m s  chosen w i l l  lie  in B(q,3).  t o an a c t i o n x e K ( q , M )  x e B(q)  to suppose  if  the  that  if  conwill  These  are  t a k e n by c o a l i t i o n  c a n be b l o c k e d by a c o a l i t i o n  M only  M.  - 57  if  this  direction  c a n be b l o c k e d by some c o a l i t i o n  define our f o u r t h s o l u t i o n ' c o n c e p t ,  DEFINITION  The Bargaining  6:  -  the b a r g a i n i n g  structure.  We now  core.  Core a t a v e c t o r o f o u t p u t s  q is  given  by:  def B(q)  = {x £ X ( q ) | V 3 £ B ( N ) , By £ B(q,g) s u c h  U^q.x)  Notice that since tion structures, the s o l u t i o n in  > U\(q,y),  Vi  {N} and { ( 1 ) , ( 2 ) , . . . , ( n ) } a r e  then c l e a r l y  concept  that  B ( q ) •<= B ( q ) .  since the notion  is  possible  coali-  We d i d n o t s t a r t w i t h B ( q )  far  too s t r o n g and w i l l  not  as  exist  generalJ  F u r t h e r t h e 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 b e i n g a f a i r l y weak n o t i o n used w h i l e s t i l l We w i l l in  £ N}.  i n t h e s e n s e o n l y Nash b l o c k i n g  providing a strong characterization  now show t h e n e v e n t h e s t r o n g n o t i o n s  the s e t B(q) w i l l  not upset the l o c a l  PROPOSITION 10;  Suppose  and i f V i  have  £ N we  that  q is locally  stable  of  behaviour  is  of local  of blocking  stability  property  stability.  incorporated  at a vector of  with  respect  outputs.  to B ( q )  3TT.  d)  (q) < 0  for  (ii)  VTT.. ( q ) and b ^ q )  then B ( q )  = B(q)  j  f i  are continuous  in a neighborhood  Condition  {i) s i m p l y  t i t i o n w i t h each o t h e r .  of q .  requires  The r e s u l t  at q  that  follows  firms  are i n  from the f a c t  direct that  in  compethe  - 58 -  neighborhood incentives  of a l o c a l l y  for  stable  coalitions  smaller  pare t h e outcome from a c t i n g  Notice applicable sumption results  6.  to  the o l i g o p o l y  section  on t h e  usual  simple  characterization  ship  form f o r  between f i r m s '  where Q =  = q.  in  now has been q u i t e  Setting  Oligopoly  we w i l l  assume  no  K(q).  general  adjustment  and  cost  s e c t i o n we s p e c i a l i z e  as-  the  of  local  profits  are  the  profit  oligopolist  stability.  of adjustment  P(Q)  Problem  that  setting  We a l s o  and t h e i r  given  functions  and p r o v i d e  show t h e  market  take a  relation-  shares.  by  C.(q.)  -  X q.- i s ieN  aggregate  P(Q)  is  a differentiable  C^'(q^)  is  a twice  Under  this  c a n be f u r t h e r  exist  Hence f i r m s must com-  t h e outcomes  following  the q u a n t i t y  cost  form.  game f o r w h i c h t h e  In the  Suppose then t h a t  TT.(q)  until  there w i l l  problem.  The S t a n d a r d Q u a n t i t y  In t h i s  outputs  as a g r o u p w i t h  the a n a l y s i s  reasonable.  of  than N to  any n o n - c o o p e r a t i v e  seems for  that  vector  output,  downward s l o p i n g  differentiate  functional  cost  demand  curve,  function.  form t h e c o n d i t i o n s  for  local  stability  simplified.  PROPOSITION 11. given above, then  Suppose the profit  functions  are of tne standard form  - 59 -  = 0 Vi e N  if  VTT.. ( q ) x ( q )  where  x(q)•=  then  C (q,N)  = <j>.  Thus  the p r o f i t  +  condition Hence w i t h cient  (bj(q)..,b  if  (B) w i l l  imply  Proposition  condition  (call  this  (q)),  functions  are of the standard  that q is a locally  8 condition  characterizing  The r e s t r i c t i o n s and h e n c e one c a n e x p e c t  (B) w i l l  local  to f i n d .  I n s t e a d we w i l l  form,  efficient  then  point.  and s u f f i -  stability.  stability  restrictions  S i n c e we h a v e 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 and u n i q u e n e s s  Pareto  be a n e c e s s a r y  i m p l i e d by l o c a l  significant  existence  (B)J  condition  are quite  on t h e s e t o f  t o depend on q , a  theorem o f s i g n i f i c a n t consider existence  interest  strong equilibria.  general  w o u l d be h a r d  and'uniqueness  for a special  case.  PROPOSITION 12.  Suppose  (i)  Firms  have  (ii)  The adjustment  that  identical cost  marginal parameters  c;  costs:  are independent  of q and given  by  b = { b p . . . , b > and a = {a^ , . . . , a } ; n  P ( Q ) is twice  (iii)  n  differentidble  with  P ' ( Q ) < 0, P " ( Q ) < 0,  < °° and 1 im P ( Q ) = 0.  c < P(0)  Q-*». Then there respect  exists  a unique  vector  to the Nash bargaining In F i g u r e  of outputs  that -is locally  stable  with  core.  8 we have shown t h e s t r u c t u r e  of the local  game f o r  - 60 o u r d u o p o l y e x a m p l e when t h e o u t p u t case  the  Pareto  such p o i n t s  efficient  gradients  = {\  VTT (q)  = (-q ,  Since output  2  (b b ) 1}  2  l b~fb^ b  =  profit  q is  be g i v e n  functions  locally by  stable.  In  = Q = q^ + q .  For  2  will  this  be:  -qj)  p  j - q ). 2  for  = -VIT^Cq)  Q  =  j, t h e n t o i d e n t i f y t h e  locally  solve  = 0.  J  yield: 1 * 7 •  In o u r e x a m p l e , If  will  v e c t o r we need o n l y  Simple manipulations  l  -  ViTj(q-)  VTTjCq)  q  the  q  VT'j(q)  2  stable  of  points  vector  b^ = b  we had s t a r t e d  at  2  = 1,Whence t h e  locally  t h e Nash e q u i l i b r i u m ^  is  easily  seen to  ,  stable  point  then  be a t r a j e c t o r y  the  is straight  consistent  w i t h t h e Nash B a r g a i n i n g C o r e . 11 ne f r o m I 3 > 3 J ( 7j » 4 J t  Notice be d e t e r m i n e d This  result  in r e l a t i o n  o  that  by t h e  is  quite  in  this  relative  market  share.  cost,  then t h e  larger  sizes  appealing  to a c o m p e t i t o r ,  one's  example of  in that  as  We c a n a l s o its  the market  market  the  share  adjustment  the lower  is  share.  the  These  cost  one's  g i v e n by a h i g h e r b^, show t h a t  o f each f i r m  then  parameter. adjustment the  lower a f i r m ' s results  will  cost  larger  is  marginal  a r e summarized  in  - 61  the f o l l o w i n g  proposition.  PROPOSITION IS. tiable  profit  i t will  (1)  Suppose  functions  q is locally  stable  in the standard respect  q.  (2)  If for two firms  i  and  > q . i f and only  b,(q,)  also  market  7.  differen-  Further  Core B ( q )  suppose J  then  j  i  and  > b.(q.).  j  -bjUj) i f C.(q.)  proposition  c a n be i n t e r p r e t e d  as e x p e c t e d ,  > C . (q . ) .  demonstrates t h a t as  that  the  representing  Firms With lower adjustment  has,  above.  to the Nash Bargaining  i f b.(q.)  then q . > q . i f and only  firms.  form given  with  = Cj(q)  then  This  industry  that:  If for two firms  Cjtq)  we have an oligopolistic  with  be the case  parameters  -  costs  :  t h e b a r g a i n i n g power o f  will  a lower marginal  adjustment cost,  have  larger markets.  cost w i l l ' r e s u l t  in a  the One  larger  share.  Concluding  Remarks  Through the i n t r o d u c t i o n oligopoly haviour of tractable  of e x p l i c i t  p r o b l e m we have b e e n a b l e t o a n a l y z e firms.  N o t o n l y does  way o f s t u d y i n g  the framework  industry  dynamics,  it  adjustment the l o c a l provide also  costs  in  the  strategic  be-  an  allows  analytically one t o  exa-  - 62  mine t h e c o n s e q u e n c e s ting.  of  using d i f f e r e n t  cepts  that are  are h i g h l y  locally  stable with  restricted. to the  when f i r m s  the b e n e f i t s  result  is  recognize that  and a d j u s t m e n t in  problem o f  such e q u i l i b r i a  have m a r k e t s h a r e s  that  costs.  concepts  the  in a.dynamic  of  costs  relative  bargaining  strengths o f  Nash e q u i l i b r i a  not taken i n t o  this  "rational" locally  conjectural  (1981)  or  Perry  p r o a c h t o make s e n s e  variations  it  is  Hence i t  that expectations  are always  variations  (CCV)  Firstly  as for  assumed t h a t  petitors' actions.  whether e x p e c t a t i o n s  locally  is  reasonable  consistent.  are c o n s i s t e n t  is  as  were  pre-  using  more  that  were  f r o m t h e r e c e n t work  The r e a l  of  It  actions  solutions  on  f o r example by B r e s h variations  issue  then i s  behaviour.  equilibria  com-  a s we h a v e ,  r a t h e r , g i v e n the rational  ap-  immediately  to s i m p l y assume,  made t o c h a r a c t e r i z e  costs  important  conditions.  can observe  or not but  firms  as g e n e r a l i z a t i o n  a conjectural  a p p r o a c h , how d o e s one c h a r a c t e r i z e  CCV a p p r o a c h an a t t e m p t  stable  is  firms.  to a f i r m ' s  presented  firms  and  same s o l u t i o n  Our a p p r o a c h d i f f e r s  (1982).  is  r e s p o n s e s were c h a r a c t e r i z e d  t h e n we o b t a i n e d  Pareto e f f i c i e n t .  consistent  competitors  core  Our m a j o r  p l a y an  by f i r s t - o r d e r  model y i e l d e d t h e  Once t h e s e  criterion  model  set  con-  to t h e i r marginal  adjustment  responses o f  d i c t e d by C o u r n o t .  behaviour. efficient  proportional  set-  equilibrium  way,  was shown t h a t when t h e account,  solution  a unique  Pareto  of t h i s  c o r e , .the  t h e Nash b a r g a i n i n g  selecting  are l o c a l l y  interpretation  the c h a r a c t e r i z a t i o n  to these  from c o o p e r a t i v e  inversely  In t h i s  characterizing  A possible  are  respect  In p a r t i c u l a r ,  s e e n as one s o l u t i o n  nahan  solution  W h e t h e r one u s e s t h e Nash c o r e o r t h e Nash b a r g a i n i n g  of outputs  role  -  not local In  under the  the as-  - 63 sumption t h a t profit  should e i t h e r  maximizing.  Quite  firm experiment  clearly  As an a l t e r n a t i v e  the'approach  that  incorporates  an e x p l i c i t  mine  rational  trast  output  sharply with  (1982) a n a l y z e s in  contrast  ing  some o f  s u c h an e x e r c i s e  to the  i s , assumed to  is  open to  presented here d e f i n e s notion of  rationality  f o r any v e c t o r  Pareto-efficient  the  predicted  it  concept to  deter-  The r e s u l t s  For example,  4.and gets  result  and u s e s  be  question.  a solution  of outputs.  t h e CCV r e s u l t s .  the example o f S e c t i o n  con-  Breshnahan  competitive by t h e Nash  result bargain-  core.  To c a r r y tice  firms w i l l  does not gests is  changes  the other  imply  point  not always that  this  f u r t h e r one s h o u l d r e c o g n i z e  be a t t h e  model  is  introduction  of  Pareto-optimal  unreasonable.  t h a t when t h e number o f f i r m s  instantaneous,  wishes  this  is  costs  t o u n d e r s t a n d why f i r m s  seems t h a t t h e assumptions  in  prac-  however  The a n a l y s i s  outcome s h o u l d  simply yields  are  point;  f i x e d and i n f o r m a t i o n  then a P a r e t o - e f f i c i e n t  adjustment  that  here  sug-  transmission  be e x p e c t e d .  a unique s o l u t i o n .  n o t on t h e P a r e t o  this  frontier,  o f t h e model m u s t change t o a l l o w ,  If  then for  The one  it  example,  g for entry  and e x i t  be i n t e r e s t i n g lysis  presented  and i m p e r f e c t  information.  t o s e e how s u c h a s s u m p t i o n s here.  will  In f u t u r e work affect  it  would  the s t a b i l i t y  ana-  - 64  -  NOTES  1.  In Case tial  (1979) there  games.  is  some a n a l y s i s  of oligopoly  However,Case does e x p r e s s  using  reservations  differen-  on t h e  efficacy  o f s u c h an a p p r o a c h .  2.  IR" = {x e F | x .  Let  1R" 3.  > 0 , 1 < i < n}  n  = {x e I R | x  > 0,  n  +  See F r i e d m a n  (1977)  i  for  1 < i  < n}.  a general  discussion  of  this  point  and S p e n c e  (1978). 4.  This  n o t i o n was f i r s t  5.  For x , y e IR ,  6.  The i m p o r t a n c e o f  introduced to the l i t e r a t u r e  x > y means t h a t x^ > y ^ , V i  n  e v a l u a t i n g outcomes a f t e r  has been p o i n t e d o u t  by M a r s c h a k  the advantage t h a t i t 7.  As an e x a m p l e o f  and  VTT  x  = (1,  VTT  2  =  (-2,  VTT  3  =  (1,  Note t h a t or 3 8.  1  every  = {(1),  See S t i g l e r  considerably  non-existence,  -2,  0)  1,  0)  1,  X = [-1,  is  and S e l t e n  let  N -  [1982].  e N. all  players  [1978]. to  {1,2,3}  have  responded  Our a n a l y s i s  has  apply. and  let  1)  1] x [ - 1 ,  1] x [ - 1 ,  1]  feasible  direction  is  (2),  easier  by M a t t h e w s  (3)}.  [ 1 9 6 4 ] on t h i s  point.  b l o c k e d by e i t h e r  3^ = { ( 1 , 2 ) , ( 3 ) }  -  65  X = [-1,1]  -  x [-1,1]  / 4-  / F(q,X,{l»  V^ (q)  +  /  2  t  F(q,x,{2})  C (q,N) \  \  1  Nftr^q)  ^  t N = {1,2}  F(q,x,N)  Figure Feasible Typical  1  Directions Local  Game  For  a  = X  - 66 -  Figure 2  -  67  -  Figure 3 The  Local  Game i n Regime 1  - 68 -  rl.O  x  2  K(q,{l»  -'7  -1  \ V T T  2  1.0 X  ( q )  -C (q,N) +  ^  K(q,{2})=K(q,N)  ( l , - l ) = K(q) = K(q). 1. 0  Figure The L o c a l  4  Game i n Regime 2  l  - 69  -  Figure The L o c a l  5  Game i n Regime  4a  - 70 -  !  /  J  K(q,{2})  [  V7T (q)  1.0 x  2  * (l,D=K(q)  /  2  K(q{l})-^^  \  r' '  '  C (q,N)  /  +  :  N/  \  V^(q)  ,  K(q,N)  /  L___V _-/...  -1.0  Figure The L o c a l  6  Game i n Regime 4b  "  -  71  -  Figure The B a r g a i n i n g  7  C o r e i n Regime 4b  Figure 8 The B a r g a i n i n g  C o r e a t an  Equilibrium  CHAPTER 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 IN A C O N T E S T A B L E MARKET 1.  Introduction The 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 and p r e d a t o r y  have long been p a r t o f t h e f o l k l o r e o f t h e 1 paradigm of i n d u s t r i a l organization. a t t e n t i o n as c o n s t r u c t s  of formal  structure-conduct-performance  However they  theory  (1959) 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 idiocy of the charge of conscious  paper i s to f i t these notions  have r e c e i v e d  little  and as a r e s u l t t h e r a t i o n a l i t y  of such p r a c t i c e s i s often c a l l e d i n t o question.  the  pricing  For example, Shubik  o f e c o n o m i s t s has p o i n t e d  parallelism".  The p u r p o s e o f t h i 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  e l u c i d a t e t h e r e l a t i o n s h i p between informal  out  descriptions of firm  will behaviour  and 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 precisely defined  with conscious  p a r a l l e l i s m , note that t h i s i s not a  concept, but r e f e r s to the general  s i t u a t i o n i n which  f i r m s seek 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 by a d o p t i n g matching each other's for  such  behaviour.  a strategy  of  Our f i r s t o b j e c t i v e i s t o seek a r a t i o n a l e  behaviour. I t i s w e l l known 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  can  lead  LO h i g h e r  f i r m p r o f i t s than does n o n - c o o p e r a t i v e behaviour.  t h e r , t h e r e l a t i o n between the i n f o r m a t i o n •  of  •  Fur-  a v a i l a b l e t o f i r m s and t h e s e t 3  ' 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  draw-  back o f such r e s u l t s i s t h e l a r g e number 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 allow.  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 ibrium. One f e a t u r e o f i m p o r t a n c e firms' expectations  i n t h i s respect i s t h e process  on each o t h e r ' s b e h a v i o u r  are formed.  whereby  Experimental  s t u d i e s o f t h e type done by Roth and Schoumaker [1983] d e m o n s t r a t e , that consistency of expectations l u s i v e outcomes, and tency by a d o p t i n g iables  secondly,  rule-of-thumb  i s instrumental  i n the attainment  that players tend t o acheive behaviour  of colconsis-  with respect to observable  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.  vations suggest  such  first,  t h a t , on t h e one h a n d , t h e e x p e c t a t i o n s  These  formation  varobser-  process  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 hand, 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 b e a n e c e s s a r y  achieving determinate behaviour  outcomes i n such models.  r e f e r r e d t o as c o n s c i o u s  e a s i l y v e r i f i a b l e rule-of-thumb  We s u g g e s t  construct i n  that the matching-  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 as an  t h a t f i r m s can t a c i t l y agree on, and t h e  function o f which i s t o render mutually consistent the expectations o f f i r m s o n how t h e y w i l l To d e v e l o p  r e a c t t o each o t h e r ' s p r o f i t - i n c r e a s i n g moves.  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 a n a b s t r a c t  of expectations-formation,  process  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 communicate w i t h each other through  t h e medium o f announced p r i c e  changes.  no way t o d e f i n e u n i q u e l y t h e  Then, s i n c e there i s i n general  announcement s t r a t e g i e s o f f i r m s i n such following a rule-of-thumb social convention.  a game we m o d e l s u c h  that i s derived from  The d e r i v e d r u l e - o f - t h u m b  price changes, o r conscious  behaviour  as  an a x i o m a t i c a l l y s p e c i f i e d i s t h e matching o f announced  p a r a l l e l i s m . We s h o w t h a t o u r m o d e l o f e x p e c t -  -  75  -  ations formation, together with the s o c i a l convention, unique c o l l u s i v e outcome f o r the i n d u s t r y . thus  results in a  Conscious  parallelism is  s e e n as 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 unique c o l l u s i v e outcome.  We  t u r n now  to the question of  achieve  predatory  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  entry.  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 comes c o n c e r n s The  general  the robustness  perception  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  i s that b a r r i e r s to entry are a necessary  f o r p o s i t i v e long run p r o f i t s .  T h i s p e r c e p t i o n has  f u r t h e r i n the r e c e n t work of Baumol, Panzar  entry.  condition  been c a r r i e d even  and W i l l i g [ 1 9 8 2 ] .  t h a t w i t h c o s t l e s s e n t r y and 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 tures are e s s e n t i a l l y the only s u s t a i n a b l e p o s s i b i l i t y .  We  They argue  industry strucaddress  this  i s s u e by 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 model o f c o l l u s i v e behaviour.  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 and e x i t c o s t l e s s  but 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  In c o n t r a s t w i t h Baumol, Panzar i n d u s t r y members w i l l  a n d W i l l i g [ 1 9 8 2 ] we  show t h a t  set.  current  have 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 are  entry  d e t e r r i n g ; these e q u i l i b r i u m s t r a t e g i e s are a form of predatory p r i c i n g . The  consequence i s that the e x i s t e n c e of long-run  contestable market i s a l o g i c a l l y d i f f e r e n t r e s u l t i s due able to firms.  possibility.  The  positive profits in a reason  for this  striking-  t o t h e a s s u m p t i o n m a d e on t h e i n f o r m a t i o n  I f p o t e n t i a l e n t r a n t s c a n be i d e n t i f i e d b e f o r e t h e y  ally start production  then the e q u i l i b r i u m response  availactu-  for existing firms is :  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 .  Hence i t can  be  -  concluded  76  -  that t h e nature o fmarket i n s t i t u t i o n s  i s as important as 4  the c o s t o f e n t r y and e x i t i n d e t e r m i n i n g As a f i n a l dustry structure.  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 o n i n -  I n t h e work o f Baumol, Panzar  is argued that t h e presence and  supernormal  profits.  posite conclusion. of committing  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 -  To a p o t e n t i a l e n t r a n t sunk c o s t s has t h e e f f e c t  a c a p i t a l t ot h e market, thus m i t i g a t i n g t h e e f f e c t o f The consequence i s t h a t i n d u s t r y p r o f i t s  w h i l e t h e number o f e n t r a n t s i n c r e a s e s a s sunk c o s t s The  plan o fthe chapter  model i s p r e s e n t e d equilibrium.  i s as f o l l o w s .  Insection 2 the basic  a l o n g w i t h a d e f i n i t i o n o f a subgame p e r f e c t Nash  parallelisms.  ordination problem.  which i s best modelled  as an  announcement..game  p a r a l l e l i s m i s v i e w e d a s a social as r e s u l t i n g from  axiomatically.  e q u i l i b r i u m concept  i t should be seen as a c o -  This i s f o r m a l l y modelled  In s e c t i o n 4, c o n s c i o u s  t h e doc-  I t i s argued that since i t i s gener-  a l l y viewed as a s i g n a l l i n g phenomena then  Finally,  rise.  S e c t i o n s 3 and 4 study t h e problem o fm o d e l l i n g  trine o fconscious  presented  and W i l l i g [1982] i t  o f sunk 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  entry deterring strategies. decrease  market s t r u c t u r e .  convention  simple p r i n c i p l e s that are r e -  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  o f section 2 define a unique c o l l u s i v e e q u i l i b r i u m .  i n section 5 the effects o fentry are studied.  p r i c i n g i s d e f i n e d i n t h e broad  Here  predatory  sense as r a d i c a l p r i c e drops i n t h e face  -  of the  entry.  It  appropriate  bining  this  shown t h a t using  2.  is  argued solution  with firms  The I n d u s t r y  Friedman  index  in  differentiated IT.(•£)  c a n be i n entry  be t h e  -  Cournot-Nash  concept  for  equilibrium  d e f i n i n g " the  behaviour  of  entry  the  long-run  equilibrium  deterring  strategy!  in  forms  equilibrium.  previous  at  prices  Com-  sections,  "collusive"'  it  prices  section  adynamic [1980]  oligopoly is  presented.  an o l i g o p o l i s t i c  industry,  product  sold  profits  that  is  per period  of  model  for  firm  similar  Let  the  each o f w h i c h a price  i  to t h a t  set  N =  produce  p. > 0 ,  i  a  e N.  found {l,...,n}  single Let  while  def P =  is  the  vector  prices  from  profit  at  time  is  a specified  For starting  prices  time T to  this  where & i s  of  at  the  time  is  Model  [ 1 9 7 7 ] o r Green  firms  the  the cooperative  an e q u i l i b r i u m  In t h i s in  that  77  charged.  the  t=0,  indefinite  given  only  {P_ }^ r e p r e s e n t s future  then  the  the  time  firm's  path  of  discounted  by  constant  present  If  discount  analysis adjust  it  rate.  will  their  be a s s u m e d t h a t  prices  firms  every Y periods.  can, This  - 78  p e r i o d can e i t h e r to a d j u s t time  prices  required  -  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 or,  like  to a c q u i r e  Spence  [1978a],  it  enough i n f o r m a t i o n  in a firm's  ability  c a n be s e e n as t h e amount in order  to j u s t i f y  a  of  price  5 change.  As a n o t a t i o n a l  Y periods. remain  Thus  in the  convenience  rescale  t so t h a t  t=l  p e r i o d t e [ m , m + l ) , m an i n t e g e r ,  represents  prices  will  fixed. The d i s c o u n t e d  where T i s  an  profits  c a n now be w r i t t e n  i n summation  form,  integer. rt+1  1  z  t=T t  1  1  t=T d  e  f  where 3 =  e  -5Y  t and J__ i s  Notice 3 approaches  that  to observe a l l  choose past  price choices.  tory  past  I  Let  H  t  time t .  prices  in  period  required  [t,t+l).  to change p r i c e s ,  Y,  decreases,  it  prices  will  but cannot observe  Hence t h e i r  denoted  be assumed t h a t  information  t h e y have  their  set  been  competitors'  consists  of  the  his-  by:  = {P |0 < x < t}. T  t  represent A  the p r i c e  as t h e t i m e  current of  0  1 *  When f i r m s able  1  decision  the set o f at  possible  time t  by f i r m  histories i will  or  information  simply  sets  be t h e map:  at  - 79 -  S:  H. + IR t +  That  is  l  i  price  choice  by f i r m  i,  decisions  of firm i .  given  Let  °°  t  t=T  Let  d  6  (SPE).  firms  acting  always  >  by t h e  |l  e  The n o t a t i o n  at  n  that w i l l  give  of possible  time T w i l l  the  decisions  be a s e q u e n c e  of  of s t r a t e g i e s  be u s e d i s ensure  for  the  in-  t h e Subgame  that  for every  Perfect time  t  "rationally".  is  T  a subgame p e r f e c t  equilibrium  if  and  only  5*11*)  i  e H , t  Vt > T, and Vi  ) = V . ({P_ }^) and P_ i s T  strategy  the s e t  This concept w i l l  vj(a*  max  VI*  where V.{a  i  = P|, w i l l  t  be t h e v e c t o r  concept  A strategy a  i  of firm  S*(l )  t  represent  (aj,...,a^)  Nash Equilibrium  a  e H ,  *•  =  1.  t  T  f  The e q u i l i b r i u m  DEFINITION  I  by  1  are  history  t h e n t h e strategy  T  dustry.  f o r each  a* g i v e n an i n i t i a l  (a^a^)  represents  e N.  the sequence o f p r i c e s  informationrset with  firm  i's  1^ a t  strategy  generated  time  t.  replaced  by  a*. 1 In g e n e r a l ,  as shown i n G r e e n [ 1 9 8 0 ] , t h e r e w i l l  exist  many S P E .  - 80 -  We now p o i n t used i n the fits  o u t two c l a s s e s following  satisfy  the  Assumption Al.  produce  is  TT. ( • )  This firms  sections.  following  JTT.(P>|  o f such e q u i l i b r i a  < A,V  £  will  d o i n g so suppose  be  extensively  that  flow  pro-  assumptions.  continuously  assumption  e IR*,  is  p r o d u c e d by t h e same i n d u s t r y , following  Vi  differentiate  product.'  the  with our o r i g i n a l Further  7  by d e f i n i t i o n  signs  with  bound  N.  e  consistent  a differentiated  H e n c e we have t h e  Before  that  on t h e  since  assertion  the  t h e y m u s t be  partial  products  that are  substitutes.  derivatives:  3TT.  Assumption A2.  (i)  — P  (_P) > 0,  j 3TT.  (ii)  I f TT. ( P ) k-  > 0, V k £ N, t h e n ^ 3  V i,j  £ N,  The s e c o n d c o n d i t i o n , . ; r e q u i r e s stitutes  if  all  firms  i  that  are making p o s i t i v e  t  P j  (P) -  > 0  j.  the  products  profits.  Finally  be s t r i c t it  will  subbe  g assumed t h a t  a Cournot-Nash e q u i l i b r i u m  Assumption A3. fined  There e x i s t s  a unique  in prices  exists.  Cournot-Nash e q u i l i b r i u m ,  P^,  de-  by  •n.{P°) = max 1  P.>0  T T . ( P . , P ° 1  . ),  Vi  l -  For the remainder o f the essay profits  satisfy  £ N.  - 1  these  assumptions.  it  will  always  be s u p p o s e d  that  - 81  The f o l l o w i n g lationship  well  Suppose  that  t That is S . ( I . )  t S|(l ) t  d  e  f  =  0 P",  where P^ is the unique  ment, if  including  Y is  1  "cooperative" can a d j u s t  prices  known r e s u l t  (i)  P^ =  (ii)  If 6 > 0.  result  regame.  past  L  Then the  t  by the following  decision  se-  e N.  equilibrium  given  and subsequent  then that  interest  now l i e s  are p o s s i b l e .  quickly,  of  and V I . e H. .  T  t o Green  of uncertainty.  large,  on r e p e a t e d  PROPOSITION 2.  i  Cournot-Nash  equilibria  are independent  1  V t > T,  the effects  Our m a j o r  and S P E f o r t h e dynamic  ' decisions  The r e a d e r may r e f e r  sufficiently  the important  f  The p r o o f o f t h i s in the a p p e n d i x .  firms  shows  t t = P., for some P. e IR  e  set at time T is given  SPE in this  quence:  d  w  I  unique  known p r o p o s i t i o n  between C o u r n o t - N a s h e q u i l i b r i a  PROPOSITION 1. history.  -  this  in Assumption  results  a r e t o be f o u n d  [1980] f o r a d e t a i l e d  I t should also solution  in studying  be n o t e d  treatthat  is the only SPE.  the case i n  For Y s m a l l ,  we h a v e t h e f o l l o w i n g  A3.  that  special  is  which if  firms  case of a well  games.  Let { P J^._Q be a sequence  of prices  'and suppose  that  at most f i n i t e l y many times  f _P° then ir. (P ) 1  > TI\ ( P ^ ) + 6, V i £ N and for some  constant  - 82 -  Let  the  strategy  r  Then  o  is  a  o  i»  be  defined  if  Pt - 1  if  P  SPE  for  t-1  by  the  decisions:  ~t-1  =  f  1  P  t-1  sufficiently  small  Y.  Hence 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 can t a c i t l y upon and t h a t l e a v e them 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 cf  T  i s a SPE.  Condition  issue of predatory  3.  agree  equilibrium,  ( i ) w i l l be used i n a l a t e r s e c t i o n when t h e  p r i c i n g i s addressed.  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  viewed as 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 i s s u e t h a t has r e c e i v e d is the question  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  The 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  firms are individual profit-maximizers unique equilibrium.  repeated  game, a n  relatively -little attention in oligopoly  which.of  as t h e " s o l u t i o n " .  defined  outcome can be  theory  be selected  the assumption  that  i s not s u f f i c i e n t t o s e l e c t an  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  there  have  been 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-person cooperative  games.  I f one a l l o w s  pre-play  the problem can a l s o be viewed as a bargaining most s o l u t i o n s a r e found b y u s i n g  appropriate  communication  problem, i n t h i s selection  then case  criteria,  - 83 -  with  intuitively  proaches  is  appealling  that  properties.  they r e q u i r e  illegal  i n many c o n c e n t r a t e d  suppose  that  variables  profits  The d i f f i c u l t y  coordination  industries.  or payoffs are  are often d i f f i c u l t  In t h i s is  explicit  9  Further,  and t h e f o l l o w i n g  Here i t thus  is  t h e y do t r a n s m i t  assumed t h a t  section  provide  duopolistic  a basis  competitor  all.  I f the  tacit  its  planned  and r e s p o n s e s  justed.  behaviour w i l l  This  Since  point  time  for  we w i l l should  cooperative  second f i r m does n o t  announcements  the current  announcements  responds w i t h e i t h e r  rescinds  suppress  in  behaviour.  price  for-the  the announcement  It  that  and a r e n o t a c t u a l l y be shown l a t e r  that  the  proposed  do n o t  observable  price,  prices. and  can  in  s h o u l d be n o t e d t h a t  a  usually  then the f i r s t  at firm  such  announcement game.  be t h e s t a r t i n g  L e t _P_ s i m p l y point  initiates  "cheating" w i l l  not  occur.  represent quo  complications,  a price  are merely announcing  we-will  or status  change.  planned  committed to c a r r y i n g out these changes. in general  which  o r no i n c r e a s e  Further to avoid excess  players  lag.  directly  increase,  increase  present discussion.  game.  these  before p r i c e s have a c t u a l l y b e e n a d -  s u p p o s e o n l y one f i r m e a c h p e r i o d be s t r e s s e d  a price  its  is  game.can be--played each p e r i o d  v e c t o r which w i l l  pre-  For example,  be f o r m a l l y m o d e l e d a s a n  t h e announcement  indices  increase  can o c c u r  is  way t h e y a d j u s t  are e a s i l y  a similar  increase.  that  in practice  firms  the  i n d u s t r y when one f i r m a n n o u n c e s  the  usually  for  ap-  these approaches  a solution  Even i f  information  price  these  o r known o n l y w i t h a l a r g e  b a s e d 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 .  communicate,  of a sort  common k n o w l e d g e ;  to observe  with  It  It  prices will  - 84 -  Suppose t h a t f i r m i announces a p r i c e change o f AP^.  The other  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  will  T h e r e a c t i o n f u n c t i o n rl(_P,AP^)  s e t A P j = rj(IP,AP.j).  represents  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 been p o i n t e d  o u t , once f i r m s begin c o o p e r a t i v e  t h e i r " o p t i m a l " responses a r e no longer u n i q u e l y d e f i n e d . r . (P_,AP.) w i l l 1  cognized  represent  a convention  by t h e industry.  Before  the notion o f a strategy w i l l  o fbehaviour  Rather  that i st a c i t l y r e -  f u r t h e r s p e c i f y i n g these  be discussed.  behaviour  responses  By d e f i n i t i o n i t w i l l be  s u p p o s e d t h a t rJ(_P_,AP.j) = A P a n d r l ( P , 0 ) = 0 V j e N. T h e s e c o n d a s i  sumption i spurely formal will  occur without  s i n c e i t simply r e q u i r e s t h a t no p r i c e changes  an announcement.  An announcement strategy  i  i  =  As was p o i n t e d  those  / rj(P,AP.);  i fj e K  {  a n d R^ =  lb;  i f not.  i  d e f  R J ( _ P , A P ,K)  changes should  f o r f i r m j , j e N i sgiven by:  i  f o r k eK/j  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  they n o t be followed by other f i r m s .  price  The s e t K represents  f i r m s t h a t must a l l change t h e i r p r i c e s i n order t o s u s t a i n an a n -  nounced p r i c e change.  The 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  later  we m a y 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 . ^ L e t R"" ( P _ , A P - , K ) r e p r e s e n t 1  strategies.  t h e v e c t o r o f announcement  -  Definition  The s e t  2.  announcement  if  (i)  -  { A P . . R (_P , A P . , K ) } w i l l  1  >  i  max  T T . ( P + ( A P . , R ^ . ( P , A P .  A P . e { o V . ( P , A P ^ . ( P + R ^ P . A P ^ K ) )  An equilibrium ment s t r a t e g i e s flow the  profits firm  it  brium any  simply  is  in  (i)  the  1  quo o r  current  of  the  Let A(R)  describes  a set  each f i r m w i t h other  and  better  off.  initiate all  price  F_ w i l l  always  (ii)  the  announceto  which  This  a price  its make  second  change  un-  equili-  possible  n o t be u n i q u e .  ( _ P , 0 , K ) ) e A ( P ) ; by d e f i n i t i o n  1  of  respect  firms  represent  A(F_) w i l l  V  ?Q.  I  simply  no f i r m w i l l  In general  K <= N, we h a v e " ( O . R  A P  for  ,K))),  )}  change s t r i c t l y  that  interest.  announcements.  status  price  requires  its  if  are optimal  the s t r a t e g i e s  initiating  condition less  given  > TT.(P)  announcement  which  equilibrium  an  if  Tr (P+R {P,AP ,K))  (ii)  be c a l l e d  1  and o n l y  J  85  For  remaining  be an e q u i l i b r i u m  at  the  announce-  ment.  Let dynamic  game o f  histories librium,  let  the previous  be t h e - f i r s t  the  { A P .  now d e s c r i b e  t o d r a w upon i t  announcement sively  us  period  game i s  sequence  only  Given  natural  price.  that  of  prices.  t  t  ~t i Now d e f i n e _ P +  L  def =  Let  Define  equilibrium  ~. * ~+ t P + R (F_ , A P . T , K L  game f i t s firms  t h a t _ P ^ , the  Suppose t h a t  e A(J__ ) be t h e  ,R '(_P ',AP*,K- )} T  section.  p l a y e d once each p e r i o d .  resulting 1  is  how t h e a n n o u n c e m e n t  at  into  t=0  the  have  Cournot-Nash  no equi-  the us now d e f i n e =  and a t  announcement  recurperiod  selected.  t  ).  It  will  be  t  assumed  j  e  N  - 86  that actual prices will i n p r o p o s i t i o n 2. as o t h e r  -  be chosen according  That i s each f i r m w i l l  f i r m s have done s o i n p r e v i o u s  t othe strategies s e t p r i c e equal  periods.  presented  t o P* a s l o n g  The consequence i s that  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 announced p r i c e change.  deviate  o r cheat from i t s  Although firms are not forced t o follow stated  changes, t h e f a c t that t h e announcements a s s i s t i n making expectations sistent i n the following period creates Before  going  an i n c e n t i v e f o r "good  price con-  behaviour".  on t o r p e c i s e l y s p e c i f y t h e r ^ . , . ) used i n t h e announcement  game, l e t u s d e f i n e The  an 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 .  industry will  have 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 s o m e T > 0. will  B y 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  g e n e r a l l y be better o f f than p l a y i n g Cournot-Nash andthus cause a  change i n p r i c e s over time.  There w i l l  cease t o be an i n c e n t i v e t o change  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 n o longer  e x i s t non t r i v i a l e q u i l i b r i u m announcements.  motivates DEFINITION  the following 3.  This  discussion  definition.  A--price. v e c t o r  P* e  will  b e a n equilibrium  price  inthe  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 (P_*,AP.,K)} e A ( P * ) , 1  AP  i  = 0.  T h a t i s P* i s a n 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 nouncements a r e p o s s i b l e . m e n t game t h i s w i l l  I n terms o f t h e implementation o f t h e announce-  r e s u l t i n constant  p r i c e s s e t a t P* o v e r t i m e .  Ob-  - 87  serve  -  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 ,  m e n t s and e q u i l i b r i u m  prices.  namely e q u i l i b r i u m  The e q u i l i b r i u m a n n o u n c e m e n t s  announce-  is  merely  11 the  formal  model  o f t h e way one may move t o an e q u i l i b r i u m  4.  An A x i o m a t i c A p p r o a c h t o C o n s c i o u s  In the p r e v i o u s  Parallelism  s e c t i o n an a b s t r a c t  d e s c r i b e d whose r o l e was t o c o m m u n i c a t e intent In d o i n g  so i t  pectations sistent  p r o v i d e d a way t o e n s u r e  with  respect  expectations  to next  has  the attainment payoffs  b a s e d on w i d e l y sults fact  between a g e n t s  of  consistent  easily  observable  proxy v a r i a b l e s  i n the  that  con-  experimental  which dominate the  for  default  T h e s e were  often  one e x a m i n e s  the  [ 1 9 8 2 ] , one i s s t r u c k  division  ex-  a key i n g r e d i e n t  For example i f  t h a t agreements were o f t e n equal  h e l d t h e same  The i m p o r t a n c e o f  expectations.  w o r k b y R o t h and M u r n i g h a n  game" was  industry..members.  firms  prices.  They f i n d  accepted p r i n c i p l e s .  of e a r l i e r  among  that a l l  periods'  [1983].  o f agreements  was t h e e x i s t e n c e  "announcement  r e c e n t l y been h i g h l i g h t e d  w o r k o f R o t h and S c h o u m a k e r  price.  by  outcomes w i t h r e s p e c t  when p a y o f f s w e r e n o t d i r e c t l y  rethe to  observ-  able.  It  has b e e n e x p l i c i t l y  most e a s i l y o b s e r v a b l e perimental  i n d u s t r y wide parameter  work s u g g e s t s  nouncements u s i n g e a s i l y game t h e  function  out these changes.  r  1  assumed t h a t  that  firms  p r i c e : forms and a s  the  such t h e  above  would match each o t h e r ' s p r i c e  identifiable  principles.  In the  represents • the  way t h e  In t h i s  section  be s u p p o s e d t h a t  will  an-  announcement  (P_,AP.)  it  firms  ex-  would  carry  these  func-  - 88  tions  satisfy  widely  accepted  presented a x i o m a t i c a l l y . since  the outcome o f  social  tacit  collusion  Once we have c h a r a c t e r i z e d equilibria  will  The f i r s t the axioms,  :  is  or  norms t h a t  indeterminant,  "socially"  r^PjAP.)  profit the  g e n e r a t e d common  are  re-  maximizers,  final  equi-  expectations.  a x i o m a t i c a l l y , the r e s u l t i n g  "coopera-  t h e n be e x a m i n e d .  axiom i s  the set  conventions  Although firms are i n d i v i d u a l  l i b r i u m must be m o d e l e d u s i n g  tive"  -  a s t r o n g form o f  (Pj w i l l  represent  continuity.  thefeasible  When  stating  p r i c e , changes.. A P . j ,  i.e.  .  P + r^P.AP.)  continuously  This the  VAP.  For a l l  Axiom 1 (Continuity): is  CIR",  dif ferentiable  assumption  is  Note t h a t  in  in  i n t h i s model  will  not e x p e c t  phenomena s u c h as i n f l a t i o n  i  2 (Independence  rather  of Seale  e B.. ( P ) and i e N;  than  real  with  that small  one does changes  are p r i m a r i l y prices.  to bias  Change):  rj(P,AP..)  e N.  the sense  radically  in general  Axiom  be n o t i o n a l  i  (P^.AP^.), V j  natural  prices  B.(P).  P elR-", A P  r e s p o n s e c o n v e n t i o n t o change  parameters.  e  not in  the response  should  convention.  F o r a l l P_ e I R " , A P . e B ^ ( £ ) ,  e N and a > 0 t h e n  r!(aP,oiAP.) j "i  —  Finally., labels  of  firms  = ar!(P,AP.), j i  —  Vj  e N.  we i m p o s e t h e a s s u m p t i o n  should  have no e f f e c t  the that  signals  H e n c e one  wish  o f symmetry,  on t h e i r  responses.  i.e.  the  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 b e t h e Rather,  same.  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 symmetry assumption w i l l  have the e f f e c t o f main-  t a i n i n g such d i f f e r e n c e s . Axiom  3 (Symmetry):  F o r a l l £ e K", AP^ e  permutation,of  N, g i v e n b y 0 ( * ) »  where  T (P)  = (P  and  AP.e B.(T (P)),  Q  Q ( 2 )  ,...,P  0 ( n )  r  PROPOSITION  e(j)(P» i) A P  3.  =  r  j(  T 0  (P)'  If the response  i t i s uniquely  r](P,AP.)  )  defined  =A P  A P  i),  convention  e N.  s a t i s f i e s Axiom  changes, a behaviour a n a l y s i s thus  1 to Axiom  3,  by  r  This preposition states that firms w i l l  per  and f o r any  0  then  then  ,P  e ( 1 }  (_P), d e N,  completely  views conscious  r e s p o n d w i t h equal  analogous t oconscious  parallelism.  p a r a l l e l i s m not a s c o l l u s i v e  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  price This  behaviour  that serves  t o ensure  12  the consistency  o f expected  p r i c e changes.  I t r e m a i n s t o show how t h i s  p r i n c i p l e c a nr e s u l t i nt a c i t c o l l u s i o n . F i r s t l e t us adapt our convention.  previous  n o t a t i o n f o r the above response  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  - 90 -  AP  i f  i  j e K for  0  and  R  k  = AP  i  k e K/j  if not.  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 as a g r o u p o r do n o t , t h e n with t h e Cournot-Nash  e q u i l i b r i u m as 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  will  be g i v e n b y :  =  P.(a)  for  1  def  P  0  + a  s o m e a e JR;., In g e n e r a l  p r e f e r a n a > 0. Assumption  one can a l s o expect f i r m s , i f r e s t r i c t e d t o P ( a ) , t o  This  A4.  i s given formally i n the f o l l o w i n g assumption.  ( i ) There e x i s t s a unique  TT-; ( P . ( - j ) ) a  1  (ii) Condition  =  1  a  Tr,(_P(a))  m a x  ae F  du. -^T ( P ( ) ) >  0  > 0 s o l v i n g Tr.(P_(a^)) =  1  f o r  a  e  [°> ) a 1  ( i ) ensures t h a t each f i r m has a unique most  p o i n t o n P ( a ) , a e ]R.  Condition  preferred  ( 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 increasing i n a f o ra e [0,a^).  Neither condition places  very  s e v e r e r e s t r i c i t o n s on t h e p r o f i t f u n c t i o n s s i n c e as a i n c r e a s e s , demand for  a l l products will  sumption  A2  i n general  be d e c r e a s i n g .  dTT (P(a)) i  da  a=0  > 0.  Finally  define  Notice  as w e l l t h a t by as-  -  91 -  = mm a . ieN  a*  1  and l e t  e N|a. = a*}.  M = {j  PROPOSITION 4.  For all i e N , a e  if firm  [0,a*)  announces a price  i  change A P . e ( 0 , a * - a ] then R(P_(a) , A P . ,N) is an equilibrium This along  P(a)  is  easy p r o p o s i t i o n always  We now show t h a t  an e q u i l i b r i u m  P(a*)  AP.,  simply  points  price  out  change  c a n be an e q u i l i b r i u m .  j  e K  0, j  e K  that  announcement.  increasing  prices  i n the announcement  Define  R(-P,AP.,K)  game.  by  R.(P,AP.,K) =  PROPOSITION 5. Suppose Va > a * , VK = N , K f N and V j e H/K;the condition  holds Tt (P(a)+R(P_,AP ,K)) J  whenever  i  e K  and  Condition  firms  from a p r i c e  AP.  other  (*)  obtain  vector  firms  to  higher foil  i  e [P^-P^(a),0).  price  (*)  in the announcement game. that  should  some f i r m o r s e t  P ( a ) , then i t  will  The p o i n t  of course that  profits  such  t h e n be an e q u i l i b r i u m  J  requires  to match such c u t s .  tempt to  < TT (P(a)+R(P,AP ,K-u{j}))  i  Then _P(a*) is an equilibrium  prices  following  by p r i c e  initiatives.  price.  is  Notice  always  cutting With t h i s  that  it  pay f o r  of  the  firms  remaining  s h o u l d any f i r m  will  always  condition  the equilibrium  pay  P(a*) is  k i n k e d demand c u r v e t h e o r y  o f Sweezy.  At  P(a*)  firms  will  similar  will  at-  the  13 the  cut  match  to  - 92  price  decreases  firm's  but not  perspective  game h a s p r o v i d e d brium s t a r t i n g brium  5.  is  -  i n c r e a s e s ; thus the i n d u s t r y  kinked.  In c o n t r a s t  a way t o e x p l i c i t l y  at the t h r e a t  point  with  model  defined  demand c u r v e  Sweezy,  the  from a  announcement  t h e movement t o  an  equili-  by t h e C o u r n o t - N a s h  equili-  P^.  The E f f e c t  of  Entry  To c o m p l e t e consider  the d e s c r i p t i o n  the p o s s i b i l i t y  m a j o r weakness typically  of  an i n d u s t r y  o f e n t r y by new f i r m s .  previous  assumed e n t r y  of  models  is  of  cooperative  equilibrium  This  point  has  one m u s t been a  b e h a v i o u r where  b l o c k a d e d and h e n c e t h e number o f  it  is  firms  is  14 fixed.  In p r a c t i c e  For example, cartel  in  Porter's  the  out,  [1982]  19th century  l u d e even i n . t h e points  few m a r k e t s  s t a n d these outcomes  able  to enter  the  c= N d e n o t e t h e t TT.J(£>N  shows t h a t  f i r m s may o f t e n  Suppose  analysis  face of entry.  ing a period of cutthroat  competition.  industry. set of  firms  relation  Let  firms  from  the face of  It  is  clearly  to c o l l u s i v e o f each p e r i o d  producing  in  to  col-  h a n d , as S c h e r e r  in  N be t h e s e t  entry.  railroad  often continue  On t h e o t h e r  the beginning  free  o f an A m e r i c a n  drop p r i c e s  and t h e i r  now a t  are completely  of  all  p e r i o d "t.  [1980]  new e n t r y , i n s t i g a t important  to  under-  behaviour. t  new f i r m s  potential Further  are  firms  and  let  t ),  i  e N  the vector  of  posed t h a t  all  be t h e  profits , with £  representing-  industry  by N .  It w i l l  t h e a s s u m p t i o n s we have made c o n t i n u e  to hold.  prices  for  flow the  indexed  t  be s u p Now  let  let  - 93 -  fP  denote the Cournot-Nash e q u i l i b r i u m  By a s s u m p t i o n a l l h e n c e ir. = 0 f o r del  i  [t,t+l),  sunk c o s t  t  < -E then i  t  i  Notice  t h a t we a r e  E.  If  any f i r m ' s  it  goes  though t h i s  is  now,  to the c o n s t r a i n t  profit  only  t + 1  the  is  explicitly  sunk c o s t s .  of e x i t  less  i n some s e n s e  in addition  If  in  terms  t h a n sunk c o s t s the f o l l o w i n g  naive  (since  see i t  to choosing  i m p o s e d by A s s u m p t i o n A5  t h e n be shown how t h i s  industry  structure  using a s t r a t e g y analogous  is  it not  of  per-  the will  period.  does  not  be Al-  take  particularly  price,  will  be f r e e  structure  wi11  predatory  industry  TTJCP  ,N*)  >  - E ,  Vi  e  pricing.  structure  N*  £ N/N*  the Cournot-Nash  ©, N *  TT.(P  J  equilibrium.  U{j})  where P ^ i s t h e C o u r n o t - N a s h  < 0  equilibrium  industry.  f o r N* U  It  cooperative  © P_ i s  subject  be d e f i n e d .  c a n be s u s t a i n e d a s a to  the  if  Vj  mo-  i n any  it  to enter or e x i t  A s e t N* e= N d e f i n e s a stable  4.  where (ii)  ^.  industry,  © (i)  e  .  and m u s t e x i t  The n o t i o n o f a s t a b l e  DEFINITION  i  t  .  Firms  equilibrium  ft N  of  p e r s p e c t i v e ) , a s we s h a l l  restrictive.  will  level  imposing a cost  bankrupt  assumption  an i n t e r t e m p o r a l  outside  .ir. ( £ , 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 > 0 denote the  ^ (P ,N )  supposed that  profits  payoffs  condition.  Let  A5.  can e a r n zero  e N/.N*.  the e n t r y / e x i t  Assumption iod,  firms  f o r the  {j}.  i f and  - 94  Observe t h a t try  costs  number o f  requires possible  increases.  negative stable  It  will  r e p e a t e d game o f dustry  serve  2.  section  all  = t  At  {  time  -^ |° T  T  up i t s  -  t  -  1  }  u  {  P^, as b e f o r e ,  plant  ^ | T  Q  that  -  is  T  "-  as E to  the  sustainable firms  the  for  set w i l l  before  the  in-  period.  incumbents  thus  in  i n the  for this  and e q u i p m e n t  N^=N* i s  r e p r e s e n t a sequence  is  are set  requires  information  Now s u p p o s e a t t=0 t h a t Let  it  increase  of  [1982].  structure  prices  that  t the  will  en-  the  paribus  which corresponds  L e t us s u p p o s e  t h e new f i r m s e t t i n g  duction begins.  this  the presence  Thus ceteris  (and f i r m s ) exit,  the sense t h a t  while  P a n z a r and W i l l i g  known before  since  (ii))  to e x i t .  structures  now be shown t h a t  reasonable  l  profits  in  asymmetric  (condition  o f Baumol,  in period t are  quite  is  E = 0, we h a v e c o s t l e s s  If  contestability=notion  is  the d e f i n i t i o n  requires nonnegative p r o f i t s  sunk  -  to  actual  This obpro-  be  t }  a stable  of prices  industry  resulting  structure. from t h e  an-  * nouncement game. following  Consider  decisions  at time  the s t r a t e g y  for  firm  i,  a.,'defined  =  P  using  t. /  (i)  If  i  e N*, S * ( I ) t  = <  PT, i  (ii)  If  i  e N / N * , do n o t  enter.  if  {P  and  N  or  {N  i f  not  t  t _ 1  t _ 1  = N*} t _ 1  f N*  and  N^ = N*}  the  - 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 enter while go t o  firms  i n N* d e t e r  the Cournot-Nash  6.  Suppose  proposition  2, then a* defines  has s e v e r a l  the market  However,  o f e n t r y means t h a t  the market  Cournot-Nash e q u i l i b r i u m of  is  is  defines  cal"  i n the  face of e n t r y .  little cuts is  empirical  entry  is  Moving to  is  firms  can a d j u s t  contestable  irrational points  of  E=0  are  not the  pricing  case  for  face  the  Further  despite  provides  price there  t h a t may  e n j o y e d by an i n d u s t r y  in  form of p r i c i n g  e q u i l i b r i u m when  at-  practicing  collusion.  costs.  Baumol,  P a n z a r and W i l l i g  a b a r r i e r to entry fact  a larger  interesting  and h e n c e g e n e r a t e  h a v e t h e opposite  number o f  [1982]  firms  result.  in a stable  result  is  the e f f e c t  argue t h a t  rents  for  sunk c o s t s  the i n d u s t r y  Namely as E i n c r e a s e s industry  of  structure,  sunk act  as  members. t h e r e can  which  an  to."classi-  below c o s t  the type of  to deter entry  free  i n the and  o u t , the l a t t e r  the Cournot-Nash  benefits  if  t h e t h r e a t o f a move t o  completely "rational"and serves  P r o b a b l y t h e most  We i n  Firstly  T h i s t h r e a t " does not c o r r e s p o n d  sense o f  which  tempt t o reap,some o f the tacit  to  the conditions  in that  prices  not p e r f e c t l y  As S c h e r e r . . [ l 9 8 0 ]  support,  d e f i n e d b y a*.  that  f r e e e n t r y and e x i t ,  pricing  not  by a t h r e a t  the entry e q u i l i b r i u m . ^  behaviour.  the  implications.  contestable  the f a c t  incentive .for c o l l u s i v e predatory  and e n t r y  N* do  a SPE.  to e n t e r and e x i t .  the p o s s i b i l i t y  outside  {P_ )^._Q s a t i s f i e s  the sequence  proposition  then one can say t h a t  cuts  firms  equilibrium.  PROPOSITION  This  both p r i c e  that  be  canonly  has  - 96 -  imply lower  p r o f i t s per f i r m .  guous s i n c e the excess may  be o f f s e t  6.  Concluding  The e x p l i c i t w e l f a r e e f f e c t s are ambi-  c a p a c i t y generated  by the i m p l i c i t  by a l a r g e r number o f f i r m s  i n c r e a s e i n product  variety.  Comments  In t h i s paper a simple dynamic o l i g o p o l y model sented to demonstrate t h a t the mainly p a r a l l e l i s m and  predatory p r i c i n g  on the p a r t o f f i r m s . scious p a r a l l e l i s m  has  been pre-  i n t u i t i v e n o t i o n s of c o n s c i o u s  are c o n s i s t e n t with r a t i o n a l  behaviour  However, our main p o i n t i s not to argue t h a t con-  i s necessary f o r c o l l u s i v e behaviour but r a t h e r the  announcement game i s meant to demonstrate the importance e x p e c t a t i o n s i n any  tacit  agreement.  of consistent  In many r e s p e c t s t h i s work can  seen as a f o r m a l i z a t i o n o f F e l l n e r [ I 9 6 0 ] s seminal 1  be  c o n t r i b u t i o n s to the  problem.  C e r t a i n l y our model a l s o demonstrates, [1956],  that b a r r i e r s  to e n t r y are i n no way  the a t t a i n m e n t o f non-competitive does not imply a lack  and  Hence such a model suggests  R&D.  At the same time the model The  u l t i m a t e l y depends on the Cournot-Nash  f o r f i r m s t o use some o f t h e i r p r o f i t s advertising  a necessary c o n d i t i o n f o r  o f competition in concentrated i n d u s t r i e s .  c o o p e r a t i v e e q u i l i b r i u m , P(a*) equilibrium^.  profits.  i n c o n t r a s t with Bain  t h a t there e x i s t  incentives  f o r n o n - p r i c e c o m p e t i t i o n such  as  Such e x p e n d i t u r e s can l e a d to l a r g e r market shares  at the Cournot-Nash e q u i l i b r i u m the c o o p e r a t i v e e q u i l i b r i u m .  and thus u l t i m a t e l y to higher p r o f i t s  at  - 97 NOTES  1.  See Scherer  [1980] f o r a d e t a i l e d survey  of these  2.  Shubik [1959],  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 .  issues.  page 285-86.  Friedman [1977],  Green [1980],  See f o r example,  Green and P o r t e r [1981],  and S t i g l e r  [1964]. 4.  Several  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  criticisms of  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 have here.  See f o r example, K n i e p s and V o g e l s a n g [1982],  Schwartz and Reynolds 5.  Though S e l t e n  Green and P o r t e r [1982],  [1975] o r i g i n a t e d t h e notion  and Green o f subgame  perfectness, [1980].  See 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 and boundedness o f t h e profit  8.  [1964],  [1980].  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 7.  i n price  i s a w e l l s t u d i e d phenomena, f o r example S t i g l e r  Friedman [1977], 6.  [1983].  The 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 observations  and  function.  As shown 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 suppose that t h e underlying  assumption.  L e t us  model i s l i k e de Palma e t . a l . [1983]  where a l l these assumptions a r e s a t i s f i e d . 9.  See Nash [ 1 9 5 0 ] and Nash [ 1 9 5 3 ] f o r t h e seminal problem. Recently  Roth [1979] provides  c o n t r i b u t i o n s on t h i s  an e x c e l l e n t s u r v e y  of the literature.  O s b o r n e and 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  of bargaining  theory  t o the duopoly  problem.  application  - 98 -  10.  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 all  f i r m s do o r do n o t f o l l o w .  This  address t h e issue o f whether conscious  s t r a t e g i e s than  i s important  simply  i f we w i s h t o  p a r a l l e l i s m i s an i n d u s t r y -  wide phenomena. 11.  Also see Cyert  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 . problem then  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 .  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 all 12.  f i r m s , t h e process  will  The  In our a n a l y s i s , improvements f o r  q u i c k l y c o n v e r g e t o P*.  See S t a n b u r y and R e s c h e n t h a l e r  [1977] f o r a nice d i s c u s s i o n o f con-  s c i o u s p a r a l l e l i s m and s o c i o l o g i c a l approaches t o c o l l u s i o n . 13.  See Sweezy [1939].  I t should  be noted  t h a t t h e t h e o r y we  present  here a d d r e s s e s most 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 example i n Scherer 14.  [1980].  F o r example 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 model based on 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  15.  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[1980],  "A M o d e l  o f S a l e s " , American  of P o l i t i c a l  Eco-  Journal  of  L o c a l Games i n D y n a m i c Studies, 46, 3 0 5 - 3 1 4 .  Economic  Review,  70, 6 5 1 -  -  104 -  APPENDIX FOR CHAPTER 1  Proof  of Proposition  Sufficiency:  1.  Suppose t h e H o t e l l i n g assumption  choose any h e l R  such t h a t  2 m + 2  (X,P) + h e U  holds.  Given  x [0,p ] . +  2  (X,P) e U  x [0,p ] ,  L e t Z = ( X , P ) and  2  Z(e) = Z + eh.  Define:  u(e) This  = ( u ^ z ) rti u ( z ( e ) ) } u i u ^ i U ) )  i s i n some s e n s e t h e d i s p u t e d m a r k e t  the p e r t u r b a t i o n  nu (z)}  2  h.  The i n d i c a t o r  boundary between f i r m s  function  1  ,  i f  x e U(e)  0  ,  i f  x t U(e).  2  1 and 2 under  f o r U(e) i s g i v e n b y :  I (x) £  F o r f i r m 1 we c a n w r i t e  down t h e f o l l o w i n g  inequalities,  where p^(x) i s  t h e 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 .  IdjCZ)  - d^Zje))!  =  U  x e U  (  z  D ( P l ( x ) ) v ( x ) dx  )  " 'xeU^ZU))  D  (Pl(x))  < ^ x e U ^ n u ^ e ) )  l xeU(c)  +  J  S  u  p  {  D  (  p  1  - 'xeU^Z^tfU))  K  + /  e  U  Pl  D (  x  )  i  D (  )  '  ( x ) )  D  ~  x  Pl  D (  (Pl( ))  }  ( x  ^  v  (  (  x  >  d  x  v(x) dx|  P l x> > " ° ( P l ( ) ) l  I ( x ) s u p {D(p 1 (x)) > £  x  (  v ( x ) dx |  x  v  < ) x  d  x  D(pJ(x))} v ( x ) d x .  l  -  Since  t(-,-)  and D ( - )  lim  are  |D(p,(x))  105  -  continuous,  -  D(pf(x))|  = 0  e-K) Also  lim  I (x) e  = I°(x)  where  I  x e u (z)  0.,  if  x ^ UjtZ) n  n u (z)  x  2  taking  limits  u (z) 2  a s e->0 we g e t  (by t h e Lebesgue dominated  con-  theorem):  Id^Z)  lim  But  if  (x) =  Hence, vergence  1 ,  1  by t h e H o t e l l i n g  assumption  |d,(Z* -  lim  « 0 + J"^  - d (l(e))\  I°(x)  t h e t e r m on t h e  cMZ(e))|  D ( ( x ) ) v(x) P l  right  is  zero,  dx.  hence  = 0,  £+0 and t h e demand c u r v e s  Necessity:  Suppose  sumption does  are  the  continuous.  demand c u r v e s  not hold at  are  continuous,  but the H o t e l l i n g  as-  (_X,J__).  Let  3  =  /  X£U (X,P)^U (X,P) 1  2  Then by a s s u m p t i o n T2 f o r Thus a l o n g w i t h a s s u m p t i o n  D  (  P l  (  x  )  )  v  (  x  any £ > 0 we w i l l D3 we have  )  d  x  have  > °" ( X , ( p ^ + e , p ) ) n UgCX.P.) = 0. 2  -  d^X.^+e.pg))  and thus d^{X P)  < d (X,P) 1  +3  i s not continuous at (_X,P).  t  Proof  106 -  2.  of Proposition  The p r o o f w i l l  proceed by c o n t r a d i c t i o n .  0 k w e x i s t s , then f o r a sequence w > 0 will  X = (Xj.x*).  Let S  1  - (s^,s )  k  2  k < w and a NE(ME) e x i s t s f o r  denote the NE(ME).  2  I f no  k such t h a t w -> 0, as k -* °° there  ~k ~k always e x i s t an x such t h a t ' • t ' ( x , x ) 2  Take X j e U.  Now d e f i n e a subse-  ~k ~k quence o f {X ,S_ ) as f o l l o w s ,  (i)  X  (ii)  = X  1  1  D e f i n e _X r e c u r s i v e l y from X ^ £  k then choose X_. , k k t(x ,x )  -  1  .  For some k, X _ - • = _X Jlr  k  1  > k such t h a t  1  k 1 < min { t ( x , x ) , j} .  1  1  Let X  = X  2  1  .  Thus t ( x ^ , x ) 2  NE(ME) by S  2  £  < -j f o r a l l I > 1.  Denote the c o r r e s p o n d i n g  {s[,s\).  =  Si  We w i l l In p a r t i c u l a r ,  now d e r i v e some o f t h e p r o p e r t i e s  our arguments w i l l  case o f ME as well  as NE.  It w i l l  f o r l a r g e I.  be c o n s t r u c t e d so as to a p p l y to the be shown t h a t f o r some l a r g e % one  gets a c o n t r a d i c t i o n  and our o r i g i n a l  be f a l s e .  complete  That w i l l  of S  assumption t h a t no w^ e x i s t e d must  the p r o o f .  We now proceed with a s e r i e s  o f lemmas.  When we use the word  -.107  "equilibrium", the result  -  applies  t o b o t h t h e c a s e o f a NE a n d ME.  neither  rationing  LEMMA 1 In equilibrium occur.  nor excess  production  will  That is  = d.(X ,P ),  4  A  (a)  Proof:  A  V I > 0 , i• = 1 , 2 .  S u p p o s e q^ < d . ( X , P ) . A  n,(X ,S ) £  Profit  A  = p^q^ - C ( q * ) .  £  S i n c e demand i s c o n t i n u o u s ,  price  t h e r e e x i s t s , a p^ > p^ s u c h  that  > q*  d.i^Ap^pp) Thus a t t h i s  is then:  firm i ' s  profit  is,  p^q| - C(q^) > I I . ( X , S ) . A  Hence S^ was n o t a n e q u i l i b r i u m , curs at equilibrium  (b) d^X^.P ) 3 1  production  a contradiction.  and hence  Now s u p p o s e q^-> d . ( X , P ) . A  £  1  = D.(X ,S ). A  Thus no r a t i o n i n g o c -  A  From ( a ) q j > d . ( X , P ) , A  T h e n , by a s s u m p t i o n C 2 , f i r m i  to lower costs without  decreasing  was n o t a n e q u i l i b r i u m a n d a c o n t r a d i c t i o n  revenue.  A  can o f f e r Hence S  so lower  again  results.  LEMMA 2 The sequence be S  n  with  limit  S*.  S  has a convergent  subsequence.  Let this  sequence  . - 108 £ Proof:  We need o n l y show t h a t  [0,p ]  and hence bounded.  +  q{ =  and s i n c e  S_  is  bounded.  Prices  are  taken  from  By lemma 1  d.(x\p ) £  demand i s  bounded,  the outputs  must  also  be b o u n d e d .  Hence  £  S_  is  b o u n d e d a b o v e and b e l o w , f r o m w h i c h one c o n c l u d e s  convergent  there  exists  a  subsequence.  LEMMA 3  i.:e. q " > 0,  The outputs of both firms are always positive, i  = 1, 2, V n > 0.  Proof: charge  S u p p o s e q " = 0.  p {x^)  a price  Then f i r m  q " such  acting  as a m o n o p o l i s t  > C ' ( 0 ) + a ( b y - a s s u m p t i o n M)..  n  1 could charge a p r i c e duce, output  is  2  p" e ( C ( 0 ) , p ( x ) ) n  2  face  But  positive  then  and firm  demand and  that  n n  p ^ ' - C(q^) > -C(0). Hence q? = 0 i s  impossible.  We w i l l Since  there  JI.  now s t u d y some o f t h e  is  no  rationing,  (X  ,S  ) = p.q.  1  Notice  1  that the  lim n^  profits n  * *  n" = p q 1  1  1  1  - C(q. ) = def 1  have a l i m i t , *\ - C(q ) = def 1  n 1  *  properties  IT, . 1  as  well,  of  the  limit  will  S*.  pro-  -  109 -  LEMMA 4  In the limit,'prices * Pi  * =  Po  P*.  =  ^def  1  Proof:  are the same, i.e.  S i n c e q ? > 0, i  =  1  1, 2 a n d t h e r e i s no r a t i o n i n g , we must  have Pi  - p |  < t(x ,x )  2  1  < w ,  2  p  Hence -i • lim  I n ni „ |p, - p J = 0.  Notice satisfies mill  these  the properties  price  apply  that  undercutting  to both  results  required will  show t h a t  f o r a ME.  if  a NE e x i s t s , i t  For the r e s t  also  o f the  n e v e r be u s e d and h e n c e t h e / r e s u l t s  proof will  cases.  LEMMA 5  *  In the limit  both outputs are positive,  Proof:  Due t o a s s u m p t i o n M a n d t h e f a c t  we must  have  q  x  + q  2  t  Suppose q l o w s t h a t we m u s t and  that  q . > 0 , i = 1, 2 .  there  is  no  rationing,  o.  2  = O.and q  1  V 0.  have p* > C ' ( 0 ) .  By a s s u m p t i o n _  Choose q  M, i t .  2  easily  > 0 such t h a t  _  q  2  e  fol*  (0,%q^)  110  n  =  2  P  *q  - c(q )  2  2  >-C(0) = n*.  def  _  This  is  since  For s u f f i c i e n t l y p. l r  - e  2  n s  This  for  n, p  small  value  p* > C ' ( 0 ) a n d C ' ( q )  is continuous.  satisfies  n  d (X  large  *  Let Y = q^/q^. —n l a r g e n we c a n c h o o s e a s e q u e n c e c -+ 0 s u c h . t h a t p  possible  ( ;,p  n  P  is  2  ))  =  Yd^xV")  due t o t h e c o n t i n u i t y  c a n be c h o s e n s o t h a t  2  e.  = YqJ. o f demand a n d t h e f a c t  d (X ,P_ ) n  n  2  Hence t h e c o n s t r u c t i o n o f  t a k e s on v a l u e s  pj , f o l l o w s  in  from the  0  that  for  (0,q"-e)  intermediate  theorem.  Note  lim Yp" qj n-*»  - C(YqJ) = fj  Hence _5 c a n n o t be a NE(ME) f o r * * q 2 > 0, and s i m i l a r l y q^ > 0.  > JT*  some l a r g e  n  n, a c o n t r a d i c t i o n .  We now show t h a t t h e s e q u e n c e o f e q u i l i b r i u m p r i c e s  LEMMA  niust a p p r o a c h MC.  6  * Pi Proof:  Therefore  =  dC Hq  (  q  * i  )  j  1  =  *  U  *  S u p p o s e p^ > G ' ( q ^ ) .  n  i  =  p  i q i  "  C  (  Let 3 = 6 / q 2 - •  q  i  )  2  -  Then t h e r e e x i s t s  <  Pi^V ) 6  -  c  (qi+<$)  By t h e c o n t i n u i t y  *  a 6 e ( 0 , q 2 ) such  =  n  def  that  i -  o f - d e m a n d and t h e  intermediate  - 111 value  theorem, t h e r e e x i s t s • /vn  / n  a sequence e  n>,  d ^ X .(pj-e ,p2)) S i nee e  =q  n , „ n  > 0 such  ~n  + 6q  x  and  n  = q  .  that  ^  n > H  for  def  y  -»- 0 as n -* °°, '  n  lim  ((pj-e )q  - C(q" ))  n  n  n  Hence 3 n s u c h  = H  > H*  that  / n x~n /~n ^ n (p -£ )q - C(q ) > n . r  1  Therefore  x  n  n  S^ was n o t a n 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 .  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 .the l i m i t  even f o r  b r i u m wi11 y i e l d now show t h a t  i n Lemma 3 t h a t not used u n t i l  o u r model  p.. = p  +  result  this  i f q.  is  production takes  e x i s t e n c e .of  as d i f f e r e n t i a t i o n - b e c o m e s  impossible.  = 0 as a m a t t e r of  now a n y p r o p e r t i e s  for  demand we g e t p r i c e  M E : / Thus m e r e l y t h e a s s u m p t i o n o f  the c o m p e t i t i v e  for  of  Similarly  o f the  Notice.-if  one  convention,  p^,  = MC an  q  in  equili-  s m a l l . We  supposes  t h e n we h a v e  rationing  scheme o r t h e  to have  * dC *\ p^ = (q^), i  fact  p l a c e ex ante."  LEMMA 7 It  i s not possible  We have two  Proof:  in our model  cases  * (1)  If  p* < A V C ( q ) i  AVC(q)  = def  for either C(q)  q  C(0)  i  = 1,  2 where  2  = 1,  2.  - .112 t h e n t h e r e must be an n s u c h t h a t a contradiction  p? < A V C ( q " ? ) , i n w h i c h c a s e q? = 0 ,  o f Lemma 3.  (2)  p. > A V C ( q . ) ,  i  = 1 , 2 .  tion  on demand we h a v e  From d e f i n i t i o n  o f TJ on p.  14 and t h e assump-  9D 9p Define *  n^p) =  D(x p)  P  15  * q  Since  U(x p*) =' ls  /  * - clrKxpp)  q i  + q  :  *  *  \  2  q  + q.  2  + q ,  q  2  x  Sjtp*) = n*. Further,  p* = C (q*)  Si.(p*)  l ldpL i  thus  ci(q*)] - C p * - •C ^! (l  a  [  L  p  P  *  _  q  ;  q  ( -P*) 3p ^ ) ] ^  9 P j i  X  n q  Hence t h e r e  q  w h e r e s"  exists  n  a p-^ > p* s u c h t h a t n ^ p j )  fied.  it  is  Thus  +  +  q  * > 0 l q  q  > n^.  Now c h o o s e  n  x  - P  2  As p  > w  For t h e s e since  „* 2  +  = D (X. ,(sJ,s2))  = (p^q^).  Pj  ^ * l  2  -> p* t h e n f o r some N  > t(Xj,x )  strategies  2  for  2  > 0 we must  n > N,,.  firm 2 is undercutting  f i r m 1 who i s d e v i a t i n g , t h e  have  condition  f i r m 1.  Note  f o r ME a r e s t i l l  that satis-  -113 d (x ,(sj,s5)) n  2  Therefore  f i r m 1 faces  * TJ(x^p^)  o n l y r a t i o n e d demand o f t h e Tr/  ~n q =  / n ,=n n A f X , ( , S )j -  n D  Y  l  P( •  c  S l  By demand c o n t i n u i t y ,  we.must  ~n Tim q = D ( x , p * ) 1  x  n n\ n * 2 " 2 ^ ~ * , • D(x ) p  2  form  )  n  q  n  l i P l  have q  •  * l  *  Thus  l i m n ^ x ^ s " s"))  = n ^ p j ) >n  n-x» Hence f o r Thus  p.j = C (q^) 1  These WQ  > 0 such t h a t  is  some l a r g e  n , S_ i s n  n o t an e q u i l i b r i u m ,  NE o r ME  impossible.  lemmas  imply,-by  contradiction,  no-NE o r ME " e x i s t s when t  (  ,  that  there-must  ) <_ WQ.  exist  a  - 114 APPENDIX  Proof of Proposition s i n c e X(q) x is  is  1.  FOR CHAPTER 3  Let x solve  max xeX(q)  c o m p a c t and t h e o b j e c t i v e  Such a x  ViT.(q)x.  I  ieM  function  is continuous.  Clearly  u n d o m i n a t e d ; h e n c e x -e K ( q , M ) .  Proof of Proposition  Follows  2.  K(q>M) g i v e n by t h e e x p r e s s i o n  Proof of Proposition  3.  immediately  in equation  Straightforward  by P r o p o s i t i o n  Therefore 0 e K(q)  4.  (i) is  Proof of Proposition  1.  application  of  the  definition.  If  and ( i i )  ViT^qJx  5.  and o n l y  1  =  max — s — - y. yc[-a (q ),b (.q )] i 8  i  1  i  q  n  Proof of Proposition  6.  Direct  Proof of Proposition  7.  From P r o p o s i t i o n 5 K ( q ) a l w a y s e x i s t s .  then i f  an x e K ( q ) n K ( q , N ) , x e K(q)  y e K(q,N).  it  is,  application  of Proposition  5.  t h e n x e B ( q ) and we a r e d o n e .  by t h e d e f i n i t i o n o f  T h a t i s VTT. ( q ) ( y - x )  > 0 Vi  £ N.  If  If  Proof of Proposition  8.  Suppose  = <f>, t h e n 0 £ K ( q , N ) .  (i)  and ( i i )  there  not,  K ( q , N ) , d o m i n a t e d by some Hence y £ B ( q ) and we a r e  done.  C (q,N)  if  37T-(q)  x. i  £ N- H e n c e : TJ £ K ( q , N ) .  S i m p l y n o t e t h a t x e K (q , { i J.) i f  i  —^  > 0 Vi  impossible.  3 7 F (q>  exists  of  n  t h e n by a s s u m p t i o n s  3 q  from t h e d e f i n i t i o n  0 £ K ( q ) , t h e n "0 e K ( q , { i } ) , V i e N. Hence 3-rr, 3 we must have ~ - ( q ) = 0 V i e N. L e t x = "(.-.a^ (q.y),'... ,-a -('q )),  Proof of Proposition  +  exists  1  are s a t i s f i e d .  Since  We have by t h e a s s u m p t i o n s on TT. ( q ) ,  n  3TT  3TT  1  T^J-  115  < 0 for  (q)  by P r o p o s i t i o n  i  f j .  C (q,N) +  0 a n d we w o u l d  (q)  5 we have x e K ( q ) .  Conversely, quence  1  Thus j^-  0 e K ( q , N ) , we have 0 e  with  -  let  = <b f o r  > 0 because VTK ( q ) - x  = 0.  = 0 Vi  e N,  Since  V"n\(q)-x  0 e B(q)  which  0 e K(q,N).  implies  a n y x e X ( q ) .f\ C ( q , N )  show t h a t  there  dominate  n £ A . > 0 such i=l  A . > 0, " have  exist  = 0.  Suppose . n o t ,  t h e n we m u s t  that  1  1  = {x|x  =  Note t h a t  C^(q)  is  vector  p such t h a t  of the  or VTT.-p > 0 Vi  e N.  contradiction.  Therefore  Thus  set  that  separating  > 0 for  p-x  all  is  +  +  Since ^ — < 0 i t aq. will  ViT.(q)x  must  follow  < 0 for  all  i  e N.  ~  must  S u p p o s e VTT. ( q ) x  be a k s u c h t h a t  Hence V i r ^ C q J x = 0 V i  e N.  Proof  9.  of Proposition  s u c h t h a t y. = x. f o r  i  Z  Notice e M.  e N  is  a = 0".  A.Vn-.(q) By  Proposition  1  < 0.  Since  1  there  Vi  a  ieN  case t h a t  1  and A . > 0,  exists  f cj> w h i c h  A . > 0 a n d T C — - > 0. r dq. def^ = ( b ^ q ) , . . . , b ( q ) 0 = xv. S i n c e  be t h e  ~ t h a t K(q)  that  0.  from  there  .. e C ^ ( q )  Hence C ( q , N )  ~  separated  theorem  But  A . > 0 such  exist  £ > 0.  for  '  1  strictly  hyperplane  x e C^('q).  p e C (q,N). there  Z A . > £}, ieN  1  a c l o s e d convex  Hence b y an e x t e n s i o n  > 0,  Z A.VTr.(q),A. ieN  J  conse-  have TJ f. K ( q , N ) .  •0 t C^(q)  5 it  As a  would  +  1  1  then  B(q).  otherwise  We now w i s h t o Z A.VTT.(q) ieN  Hence  that  if  Thus s i n c e  VTT. ( q ) x K  > 0,  x e K(q,M) K(q,M)  0 e B(q), A.V-rr.(q)  Z' jeN  J  which  is  not  t h e n so  is  any y e  f <> j for  all  x = 0  J  possible.  M = N and  X(q)  116 since  3 is  a partition  of  N,  it  then  follows  that  n  K(q,M)  $ c|>.  MeS Proof that  x(qj  First  10.  of Proposition  we d e f i n e  = (bj ( q ' p , . . . b (q' )) e K(q) i  r )  for  n  a neighbourhood rij o f q~ e n .  This  is  q such  possible  since  3TT  by P r o p o s i t i o n  6 we know - r — ( q )  > 0,  Vi  £ N a n d h e n c e we c a n c h o o s e  dq. dir. such t h a t K(q)  i  1  —  (q)  > 0,  Vq c r i j »  = {(b (q ),...,b-(q )}for 1  m  1  n  i  £ N.  e K(q~,M)' f o r  0 Vi  £ N.  any H 9  Since  we a l s o  neighbourhood  of  another  N, M ^ N.  neighbourhood  By P r o p o s i t i o n  have T T — - ( q )  q such  2 we w i l l  have  q e ^ .  We now c l a i m we c a n f i n d x(qj  By P r o p o s i t i o n  < 0 Vi  VM 5= N, M f  j ;  f  N and i  such  that  8 we have V i T . ( q ) . x ( q ) i,  £ M we  j  =  e N t h e n we c a n f i n d  have  3TT,(q)  £ J£M  3  aj  We c l a i m H e  N, M / I .  If  x £ F(q,x,M),  x f  • b,(q.) > 0, "  q  3  that, this  for  were  x(qj,  Vq £  n  J  q £ n  ?  .  2  1  n n  2 >  not the case,  such t h a t  x(qj  = ( b ( q ) . . . ,b 1  then there  (x-x(qj)  e C (q,M). +  1  J  (q^)) £  would e x i s t Choose  i  an £ N such  that  x-  -  1  b.(q.)  J  1  W Since  x £ F(q,x,M)  X  j  $ M.  Let a = —  i  x. < -J  -  b.(q\)  "  b.( )  Vj  "  b  £ N.  qj  t h e n we m u s t h a v e  i i> ' — - < 0. (  q  i  x £ M and —  - b  K(q,M),  (q.) * — - = 0 for  all  a  - 117 Now  VTT ( q )  • (x-x(q))  1  =  3TT . E -^-l (q) jeN j  (x.-b  8 q  X _J  3TT  _ L jeM j z  d  q  J  (q.))  J  J  - b.(q ) _ L i .(q .). b (q.) b  J  J  J  J  3TT. B u t by a s s u m p t i o n T — - ( q ) < 0 f o r  i  f j ;  hence  3q. _  „  VTT ( q )  • (x - x ( q ) )  1  Hence  +  (x-x(q))  3TT. b. j  _1  A  9 q  for  ~  But B(q)  =  of Proposition  11.  Vir (q)x  J  for  q e n_  = B(q^) f o r  e  b.)  have  q^ £ n •  q.P'(Q)  = 0.  b. Define  > 0, s o  A. = — 1  q  n I j=l  i X J  3  3TT, ^ (q) i q  , = A.(P(Q) - C (q J  = ^  J  J  (P(Q) - C ] (  = 0 from above  )) + ( E \j=l  q i  ))  +f ^  U J  P'(Q) J  bj)  L  Hence  Note' t h a t +^  f  =  P, B ( q , 3 ) , s o we a l s o Beg(N)  = b. (P'(-.Q) - C . ( q . ) )  i  3 q  e v e r y .3 t {N} 'and q~£ n.  B ( q ) ^ B(q") f r o m w h i c h we c o n c l u d e B ( q )  Proof  3TT. E — 1 b . < 0. jeM j " d  x(q) £ B(q,3) f o r  q £ n.  = a  J  f. C ( q , M ) a n d t h u s x E K ( q , M )  Therefore B ( q ) <=• B ( q )  < E " jeM  P'(Q)  .118 n Z j=l  Hence  J  A.VTT.(q).t  j=l Hence e i t h e r  J  follows  of  = 0.  = 0, V j  In e i t h e r that  t t C (q,N).  Since  +  a k e N such  that  t was a r b i t r a r y ,  it  of  locally  stable  outputs  q are  by  expression  n I j=l  + q.  c a n be w r i t t e n  i  q  v  L  v  y  = 0  for  i  e N.  J  -  "\  cj 1  for  i  e N.  b.  Summing b o t h s i d e s  QP'(Q)  yields  _  From t h e  i  assumptions  Q* s a t i s f y i n g given  v  b.P'(Q)  as  i/V(Q)  Q '\ P'( Q )  n  uniquely  exists  The v e c t o r  12.  1  unique  there  = <$>.  +  b-(P(Q) - c )  b  e N or  case  C (q,N)  Proposition  characterized  This  have  J  Vir.(q)-t  V7T^(q)• t < 0.  Proof  n  J  n X  then  _ e l R , t f- 0. We m u s t  Let t  X.Vir.(q) = 0.  -  by  this  on P ( Q )  condition.  it  easily  The l o c a l l y  follows stable  that  there  outputs  are  is  a  then  - 119 Proof  (1)  of  Proposition  13.  Since 0 e B ( q ) , we have  Vi  n E  e N,  3TT,  = 0  b.  or b.(q.P'(Q) + P(Q) - C:(q.)) + 1  Let b =  1  1  E b.q.P'(Q) = 0. jYi J  1  E b., so ieN 1  b^PCQ) - C.(q.)) + bq.P'(Q) = 0.  For f i rms i , j we have  b  i  h.  qj P(Q) -  =  Define f ^ Q , ^ )  c;.(q.)  = .. ) J p(  qj  Q  1  1  P(Q) - C'.( ) qj  ^ . ) .  By assumption f.(Q,q) = f .(Q,q).  '  J  Since C.(-q) > 0, 1  3f. 3qr  (Q,q) > 0.  Hence  q. > q.  (2)  i f and o n l y i f b. > b..  In t h i s case b. = b.. q.  From above we get  P(Q) - C'(q.) =  P(Q) " C j (  q j  )  Thus C.(q) > C,(q) i f and o n l y i f q. > q..  -  120 -  APPENDIX  Proof it  of Proposition  Since  1.  must be a c o n s t a n t  FOR" CHAPTER 4  function  i s independent  + g V*  ^ ( P . ^ . )  P.>0 Since is  future prices  history  then  s a y P^.  Firm i ' s problem i s t o s o l v e  max  o f past  are  +1  a t time  ({P*}"  =t+1  t  )  independent  of history  then  the  above  equivalent to  max T r . ( P . , P . ) . P.>0 t  1  1  By a s s u m p t i o n . A 3 , i  i s P^,  Proof  the Cournot-Nash  of Proposition  the unique  are  L e t m be t h e number o f t i m e s  2.  following  optimal  p j = p^ \/- i e N w h i c h the case firms  i n which  I  T  a SPE we m u s t show t h a t decisions.  = {P  co t=T firm i  T i s A , as g i v e n  Suppose  i s by p r o p o s i t i o n f c |  does  problem f o r a l l  P^ = P^.  for  any  history  lj f { p * | . t < T} t h e n  1 .a SPE f o r  0 <_ I < T } .  f o l l o w a"'", t h e n t h e p a y o f f  If  to this  equilibrium.  To show t h a t o ^ - ' i s firms  solution  t > T.->  Consider  I f f i r m 1. p l a y s " s { ( I ) . and T  oth.er  is: \  0  not f o l l o w a j , t h e n  the best  i n assumption A l . Subsequent  periods  i t c a n do i n p e r i o d will  have  payoffs  - 121 given fit  by 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 .  when f i r m  i  does  not f o l l o w aj,we  H e n c e , i f Vi  is  t h e maximum p r o -  have t h e f o l l o w i n g  inequalities:  V i - Vl" > ( T T ( P ) - A ) T  1  00  +  S t=T+l  ~  (since  [7r,(P )-TT.(P°)]3 ~ ~ T  1  1  small for  y.  Hence sj  sufficiently  6 Y  A  )  lli3)  |  +  6  m_  3 -* 1 as y -> 0 w h i c h  ,  is optimal  small  6  times)  (...(pt)..  Since 3 = e  6  t=T+m  6  p} = P_° a t m o s t - m  t - T  1  for firm  i  implies  for.small  y.  y.  > 0 for  v| - ?T Thus c .  T  .'  is  * -  ar!  Proof exists  of Proposition and i s  L e t Fl(P_,w)  3.  continuous.  3r! a  and hence  =  (P,w) which  by-Axiom--1  By a x i o m 2  ar . 1  (aP,aw)  FlfaP.aw) J  = a  = F . (JP,w). J 1  (P,w)  Since  F ^-,-) 1  3  is continuous  a SPE  then  -122  -  F(P_,w) = l i m Fl(aP_,aw)  = F^O.O). J  By t h e s y m m e t r y  r  0(j)(£» ) w  But r l ( 0 , w ) Hehte  axiom  =  r  j(T (O)>w)  = w. Hence r l ( 0 , w )  1 tf  f]0\AP,)"'=  = w . f o r j e ,M.  e R, n  P  A ? ,  By d e f i n i t i o n , . r . l . ( P _ , 0 ) " r l ( P , A P . )  Proof is  -  A  P  ,  ,P  c  of Proposition  strictly  K J . A P .  A P .  Proof  N  A l a n d A2 a * > 0 .  Vi  1  a l l firms  F .(P_,w)  we g e t  _  i  S i n c e TT..(P_(CO)  of R  play  1  ('•.,•,•)  A P , =  e  N.  > i t " any f i r m does n o t f o l l o w  0. ' C l e a r l y  R  1  ( P ( a ) , A P . N )  J  i s an  1  announcement.  that  5. their  are  two c a s e s :  AP^  < 0 then c o n d i t i o n  cut.  so i n t e g r a t i n g  < TT (P(a)+R (_P(a),AP.-,N))  of Proposition  demonstrated  = 0*  = 1 V w > 0,  ;  ;  B ^ P ) .  I  equilibrium  B(P_) .  Thus F ^ O . w )  for a < a*,  By t h e d e f i n i t i o n t h e change  e  e  By a s s u m p t i o n s  4'.  quasi-concave  Tr.j(P(a))  f o r a l l permutations 8.  0  To show t h a t _P(a*) exist  no n o n t r i v i a l  ( i ) I f some f i r m  i decides  ('*) e n s u r e s  S i n c e TT^(_P(a)) i s i n c r e a s i n g  scind i t s price c u t .  i s an e q u i l i b r i u m equilibrium  f o r a e [0,a*],  ( i i ) Suppose t h a t  announcements.  t o announce a.'!price  that a l l f i r m s ' w i l l  i t i s firm  i t must be  change  match t h e p r i c e  firm k that  i will resatisfies  There  ^ 123 -i  Tr^(_P(a*))  AP..  > 0,  = max T r ^ ( £ ( a ) ) . aelR then  Then a g a i n  it  is  Then i f  never  by c o n d i t i o n  in  (*)  firm firm  finn  k's  announces  interest  i will  Proof  of Proposition  6.  By p r o p o s i t i o n  case  of a potential  entry  b y some f i r m  to  i  a;price  t o match  rescind  the  s u c h an  planned  2 one need o n l y j  e N/N*.  increase  increase.  increase.  consider  There are  two  the cases  consider. (i)  Suppose  Tr.(_Fp,N* \3• { j } ) of entry Nash.  firm  > "  j  enters  Since  then f i r m j ' s  S i n c e "TT. > - E ,  all  optimal at  these  at  time T > 0,  firms  price  play  further  Cournot-Nash  response  prices  and  is  also  the f i r m w i l l  to  not  i n the play  face  Cournot-  exit.  By  pro-  3 position  1 _P  is  a'SPE.  Hence g i v e n  Nash" s o l u t i o n  is  a. S P E .  But  TT. < 0 and t h e r e f o r e (ii)  choose next  V  T  Clearly  Hence o * i s  price  the  j  {j})  is  firm  j  N* i s  enters.,  As  the  a'stable  better-off  < -E.  upon e n t r y  pricing  a SPE.  industry  so f i r m  J S u p p o s e TT.(-F^,N* J  the Cournot-Nash  period.  <"0,  the  that  structure,  outside  before  firm  and c o n s e q u e n t l y  strategies  given  j's  Cournot-  is  the j  industry.  must  out  actions  hence  the  are  SPE.  

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