We apply a spatial model that includes both circular-city and linear-city models as special cases to the analysis of location-quantity model in mixed oligopoly. We find that the equilibrium pattern continuously moves from that of the circular-city to that of the linear-city and that the linear-city result is more likely in our setting as the equilibrium location.Linear-city, Circular-city, Location-quantity, Mixed oligopoly

Daisuke Shimizu

Takeshi Ebina

Toshihiro Matsumura

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 4 
  
Mixed oligopoly and spatial agglomeration in quasi-linear city 
  
 
 
Takeshi Ebina  
Department of Social Engineering, Tokyo Institute of Technology 
Toshihiro Matsumura  
Institute of Social Science, University of Tokyo 
Daisuke Shimizu  
Faculty of Economics, Gakushuin University
Abstract 
We apply a spatial model that includes both circular-city and linear-city models as special cases to the analysis of 
location-quantity model in mixed oligopoly. We find that the equilibrium pattern continuously moves from that of the 
circular-city to that of the linear-city and that the linear-city result is more likely in our setting as the equilibrium 
location.
We would like to thank the seminar participants at the University of Tokyo, Kwansei Gakuin University, and Kyoto University. The financial 
supports of the Grant-in-Aid from JSPS and MEXT are greatly appreciated. 
Citation: Takeshi Ebina and Toshihiro Matsumura and Daisuke Shimizu, (2009) ''Mixed oligopoly and spatial agglomeration in quasi-linear 
city'', Economics Bulletin, Vol. 29 no.4 pp. 2722-2729. 
Submitted: Aug 05 2009.   Published: October 28, 2009. 
 
     1. Introduction
                                                                                                
                                                                                              
                                            ﬀ                                             
                                                                                            
                                                                                             
                                                                                                
                                                     1
                                                                                        
                                                                                             
                                                                                              
                           2   ;                                                              
                                                                                               
                     ﬃ                                                                      
                                                
                                                                                               
                                                                          2               
                                                                                         
                                                                                                   
                                                                                          3   
                                                                                              
                                                                                               
                            ;                                                                   
                                                          
2. The model
                                          n                                               
                                              n       n              N    f ; ; ;ng       
                            N+    f ; ; ; ;ng                                             
                                   
1For discussions on location-quantity models in oligopoly, see Gupta (2004), Gupta et al. (2004, 2006),
Matsumura et al. (2005), Matsushima (2001), Shimizu and Matsumura (2003), and Yu and Lai (2003). All
models are delivered pricing models (shipping models). We can interpret that the shipping models as models
of exible manufacturing system (FMS). See Eaton and Schmitt (1994). For the discussion of FMS in mixed
oligopoly, see Gil-Molt o and Poyago-Theotoky (2008).
2For the recent discussions of mixed oligopoly, see Cato (2008a,b), Ishida and Matsushima (2009), Ogawa and
Kato (2006), Ogawa (2006), Tomaru (2006) and works cited by them.
3See Li (2006) and Matsushima and Matsumura (2006) for other discussions on the location-quantity model
in mixed oligopoly. Location models are intensively used in the analysis of mixed oligopoly. See, Cremer et al.
(1991), Heywood and Ye (2009a, 2009b), Inoue et al. (2009), Nilssen and Srgard (2002), and Ogawa and Sanjo
(2007). Delivered pricing (shipping) models are widely adopted for analyzing competition and public policies.
See, Chen and Lai (2008), Gupta et al. (1994), Gupta and Venkatu (2002), Matsumura (2003), Matsushima
(2001), and Nikae and Ikeda (2006).
                                                                                               
                                                                                                    
                                                                                            
  ;                                                                                            
                                                   
                            i  i 2 N+                                      xi 2   ;          
                                                                                  qi x  2   ;1 
                        x 2   ;        p x                                 x     q x    
∑n
i=0 qi x 
                                  x                                                     p x   
a   q x         a >        t x;xi                                                             
             xi     x                                                                       
                         t                                                                       
                                                                                               
                                                                   2   ;   4      t x;xi     
t x;xi   
{
jx   xij    xi  x  xi         =     xi         =  < x < xi;
    jx   xij       xi         =  < x    x         =   xi 
                                       =          =                                       
                                                                               
                                                                                         
                                                          i 2 N+                             
x              
i x     a   q x    t x;xi  qi x ; w x   
∫ q(x)
0
 a   r dr  
∑
i2N+
t x;xi qi x :
                                                                               W  
∫ 1
0 w x dx
                                                           i  
∫ 1
0 i x dx               
                                                          5                                 
                                                                                   a   n  
        =                                                                                  
                           
3. Equilibrium
                                                                                     
3.1 Quantity competition
4Note that  = 0 reverts to the circular-city setting, while  = 1 reverts to the linear-city setting, as no one
will nd it benecial to transport across point 0.
5This assumption is not essential. Unless transport costs for consumers are strictly smaller than those of
rms, consumer arbitrage plays no role in our model. For this discussion, see Hamilton et al. (1989).
                                                                                              
                                                                                            
       x     
@w x 
@q0
    () a   q x    t x;x0                      ;    
@i x 
@qi
    () a   q x    qi x    t x;xi     ; i    ; ;n               :    
                                                                                           
                                                                                                  
                                                    
Lemma 1: In equilibrium, p x    t x;x0 :
Proof:                                                       
                                                                                             
                                                                                               
                                                               
Lemma 2: The output for private rm i (i 2 N) is given by
qi x   
{
t x;x0    t x;xi  if t x;x0   t x;xi ;
  if t x;x0  < t x;xi .
Proof:                                                        qi x                  
                                                x                   i                    
                                                                                           
                                            i x    qi x 2                                    
                                       
Lemma 3: The total prot for a private rm i,  i, does not depend on the location of other
private rms.
3.2 Location Equilibrium
                                                                                         
                                                                                                
                                                                                             
                                                                              =        =  
                                                                                               
                                                                                              
                 
Proposition: Let T             
√
         2 = . The location pattern where the public
 rm locates at  = , n=  private rms locate at xL, and the other n=  private rms locate at xR
constitutes an equilibrium, where
 x
L;x
R   
{
 T  ;    T   ; if     <  = 
  =  ; =    if  =      :
Proof:              
          T       ;T  =      =       T                      2   ; =  :                
                                          xL                                   xL    =  
                                                                                               
                                                                                          
                         =        =                                                          
                                                                        T  =      =    xL
                                                                                         
                                           
                                                                                 xL    =       
                  2   = ;                                                  :                 
                                                 6
4. Concluding Remarks
                                                                                               
                                                                                                 
                                                                                               
                                                                                              
                                                                                                 
                                                 
                                                                                             
                    ﬀ                                            ﬀ                             
                                                                                               
                                                  
6Although the analysis is much more complicated, we can show that this holds true in private oligopoly, too.
However, in private oligopoly, the continuity result of equilibrium location with respect to  does not hold.
 Appendix
Proof of Proposition:                                                               x0    =  
                                       i                         xi      xj  i 6  j :      
                                                                                               
                                                       x1 2   ; =                              
                 xL 2   ; =                                    xR       xL                   
                   
 1  
 
                   
                   
∫ x1
0 f x0   x     x1   x g2dx  
∫ (x0+x1)=2
x1 f x0   x     x   x1 g2dx
   x1 2        = ; =  
∫ x1
0 f x0   x     x1   x g2dx  
∫ (x0+x1)=2
x1 f x0   x     x   x1 g2dx
   x1 2   =    ;      =  
∫ x1
0 f x0   x     x1   x g2dx  
∫ (x0+x1)=2
x1 f x0   x     x   x1 g2dx
 
∫ 1
(x0+x1+1+)=2f x   x0         x      x1 g2dx    x1 <  =    :
   
             x0    =        ﬀ                               x1        
@ 1
@x1
 
{
  =     x1   =    x1 =     x1 2   =    ; =  
 x2
1         x1         =      x1   T    x1   T       x1 <  =    ;
   
      T                                      T               2   ;  :     x1 <  =     1   
                           x1     x1 <  >  T        x1   =     1                           
   x1     x1 <  >   =                 x1   T    x1    =        x1    =              1:
                                1          
x
L  
{
T          <  = 
 =       =       
   
                                                     n=                 xL     n=      
          xR                        x0 2   ; =                                             
                                                                :                          
       >  :   ﬀ                                                     x0                      
xL     xR                  
@W
@x0
 
{
  a   n   n           x0 =       <  <  = 
   a    n           x0 =       =       
   
                     a      ﬃ               a   n           =                                
                        x0    =                                                               
                     
 References
                                                                                             
                                        
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Mixed oligopoly and spatial agglomeration in quasi-linear city

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Mixed oligopoly and spatial agglomeration in quasi-linear city

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