%Paper: ewp-io/9607002
%From: Eric Rasmusen <erasmuse@rasmusen.bus.indiana.edu>
%Date: Sat, 20 Jul 96 13:49:07 -0600

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                     \begin{center}
            \begin{large}
    {\bf Bertrand Competition Under  Uncertainty  

} \\
             \end{large}
                     \vskip 15pt
June 24, 1996 \\
                    \bigskip
                    Eric Rasmusen \\
                    \vskip 1in
                    {\it Abstract} 

                    \end{center}
     Consider a Bertrand model in which each firm may be inactive  
with a known probability, so the number of active firms is uncertain.    
This simple model has a mixed-strategy equilibrium in which  industry  
profits are positive and decline with the number of firms, the  same  
features which  make the Cournot model attractive.   Unlike in a  
Cournot model with similar incomplete information,  Bertrand profits  
always increase  in  the probability other firms are inactive.         
Profits do decline more sharply than in the Cournot model,  and the  
pattern is similar to that found by Bresnahan \& Reiss (1991). 

   		   

          \vskip .3in \begin{small}
          \noindent 

\hspace*{20pt}	  	  Indiana University
School of Business, Rm. 456,   

  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.  Fax: 812-855-3354. Email:  
Erasmuse@Indiana.edu. 

Web:  http://silver.ucs.indiana.edu/$\sim$erasmuse. 

       



Footnotes beginning with xxx are notes to myself for redrafting. 

   I  intend to retain copyright on this article, granting  only   
non-exclusive rights  to any journal that may wish to publish it.    


            \end{small}
 

%%-----------------------------%-------------------------------------- 
----

\newpage

  

\noindent
1.  INTRODUCTION

  An enormous number of papers have been written on the Cournot and  
Bertrand models of oligopoly, and it is very easy to discover  
unpublished  and uninteresting elaborations of them.    Any addition  
to the literature   needs  considerable justification,  which I will    
attempt to provide.    My addition  will be  a Bertrand model  that  
includes  the possibility   that  each firm  in the industry might be  
inactive in a particular  transaction. Any  reader  immediately  
persuaded that this is interesting  may skip directly to Section 2.  



The  dichotomy   between competition in quantity and competition in  
price has existed for over a hundred years.    Cournot  (1838)   
proposed a model in which $N$ firms simultaneously choose quantities  
and let the market determine the price.   Bertrand (1883)   pointed  
out that entirely different conclusions result if the firms choose  
prices simultaneously instead.   



   Students   try to decide whether to use the Cournot or the  
Bertrand model based on descriptive realism, but the more  
sophisticated position is that both are simply models,  to be used  
not because of  simplicity and realistic properties.    The Cournot  
model is  preferred for looking at oligopoly, but not because  
economists think oligopolists choose quantities  rather than prices.  
Rather, the model   is easy to use and  has properties we think  
realistic:  the oligopoly industry  profits are  between the monopoly  
and perfectly competitive levels,  and  they fall as $N$ increases.    
This makes  the model  a convenient  building block for studying  
other features of industries with positive profits and varying  
numbers of firms.    At the same time, it is well-known that  the  
modeller must be careful not to let  accidental special features of  
the Cournot model drive his results.    If  market demand is not  
linear,  perverse results can occur, and the model is not suitable  
for studying mergers because it can happen that the merger of two  
firms reduces their joint profits rather than increasing  
them.\footnote{See  Gaudet \& Salant (1991) or   the papers collected  
in Daughety (1988)   for illustration of the properties and  
usefulness of the Cournot model.   Suzumura's 1995 book is a  
particularly comprehensive study,   with special emphasis on the  
welfare aspects of free entry.       Salant , Switzer  \&  Reynolds   
(1983)  point out the  peculiar  implications of the model for  
mergers. }
  

  The Bertrand model of competition by choice of price is  
descriptively more realistic, but has the perverse outcome that two  
firms are enough to achieve the perfectly competitive price.  It is  
therefore often used as a building block in models where that result  
is desired.   The model also has the unfortunate property of   
yielding   mixed strategy equilibria  when firms have standard  
upward-sloping marginal cost curves,  the Edgeworth  
Paradox.\footnote{Nonexistence of a pure strategy equilibrium when  
capacities are limited in a Cournot model was  discovered by  
Edgeworth (1893).   The mixed strategy equilibrium can be found in   
Levitan \& Shubik (1972) or  Dasgupta \& Maskin (1986).}     More  
moderate results can be obtained with a  Bertrand model of  
differentiated products with constant marginal cost  but such a model  
is harder to use as a building block.\footnote{Hotelling (1929) is  
the classic reference. See   p. 317ff of  Rasmusen (1994)   for a  
textbook explanation.}   Also, Kreps \& Scheinkman (1983)  show that    
a two-stage game in which firms first choose capacities and then  
choose prices  has an outcome very similar to Cournot competition.    
Since Cournot competition is much simpler than the two-stage game,    
it seems sensible to use the Cournot model instead. 


  

Despite the considerable work on these models,  little attention has  
been paid to the Bertrand model with incomplete information.    If  
$N$ Bertrand firms have different levels of constant marginal cost,  
the firm with the lowest marginal cost will serve the entire market  
at a price equal to the marginal cost of the next  lowest firm.   But  
what happens if firms do not know each other's costs?  


  In some situations one can view this as simply an auction.  Auction  
theory is the the theory of competition in price. One  difference, as     

Spulber  (1995) points out, is that  in a  normal  auction the  
quantity to be sold is specified in advance, whereas in  most markets  
demand is downward sloping.      A bidder in an auction faces just  
one tradeoff:  between  winning and  collecting a higher price. A  
bidder in a Bertrand game with downward-sloping demand faces an extra  
tradeoff: between high price with low quantity and low price with  
high quantity.  This makes the Bertrand game with  downward-sloping  
demand resemble an auction with risk-averse bidders. 


 Another difference between auctions and Bertrand competition is that  
one side of the market generally controls the rules of an auction.    
If sellers bid  for the right to sell to a buyer, the buyer  
structures the auction rules to make the price as low as possible.       
Thus,  articles on auction theory  focus  on auction design, rather  
than on    what happens for given, suboptimal designs,    though this  
usually must be  examined  to some extent on the way to finding the  
optimal design.      In addition,  the sellers in a Bertrand model  
can often realistically be modelled as risk-neutral, being large   
firms with diversified ownership, whereas  risk aversion is a major  
complication in auction models. 


 One article outside the auction literature which studies price   
competition under asymmetric information is 

Spulber (1995).   Spulber   analyzes a market in which $N$ firms with  
constant   marginal cost  and zero or positive fixed costs compete by  
simultaneous choice of price  but do not know each other's cost  
curves. \footnote{His model allows increasing marginal cost  under  
the additional assumption that  a firm   supplies every customer that  
wishes to buy from it. That is optimizing behavior only under  
regulation or under non-increasing marginal cost, however. }   Cost  
is parametrized by a single variable, which is continuously  
distributed.  The equilibrium   is in pure strategies, and the   
expected price is between the perfectly competitive and the monopoly  
levels. 


 

       The present paper will take a different approach to the same  
problem.   I will  assume that firms are either active, with marginal  
cost $c$, or inactive, with infinite  marginal cost, and  that firms  
do not know which other firms are active in a given transaction.   
Being inactive could have a number of interpretations--- the firm has  
reached its capacity, it has gone out of business, it has not yet  
entered,  it has priced grossly high by mistake,  it  has not  
discovered that it could bid on this particular transaction, it has  
closed down because of strike, illness, or vacation , and so forth.     

   

The result   will be  a relatively simple model which has the same  
desirable features as the Cournot model.    

The  oligopoly industry  profits  will be   between the monopoly and  
perfectly competitive levels, and   they will  fall as $N$ increases.    
The model is  more complex than Cournot in that the equilibrium is in  
mixed strategies and the  outcomes are expected rather than  
deterministic, but it is simpler in that demand can be specified as  
for one unit  at less than a reservation price rather than as a   
linear function.  It neither includes nor is included by the Spulber  
model,  but  it is easier to analyze.     I hope, therefore, that  
this may be a useful building block as well as  telling us something  
about the properties of price competition under uncertainty.  


 Section 2 will lay out the basic model. Section 3   analyzes  
simultaneous Bertrand competition by symmetric firms, and contains  
the main results. Section 4  compares the Bertrand and Cournot  
relationships between profits and concentration, with and without  
uncertainty.  Section 5  contains an asymmetric duopoly model, in  
which one firm is known to be active but the other is not.  Section 6  
shows what happens if price choice is sequential in either the  
symmetric or asymmetric model.   Section 7 concludes.  



 \bigskip
\noindent
2.   THE  BASIC  MODEL


     Each firm $i$ of     $N+1$  risk-neutral   firms   submits a   
bid  $p_i$  to supply the customer.\footnote{xxx Consider making this  
N instead of N+1. }  Each firm  has  a marginal cost of either $c$,  
or, with probability  $\theta \in (0,1)$   independent for each firm,   
infinity, which is incurred only if the firm wins the contract.   We  
will say that a firm with a marginal cost of infinity is  
``inactive''.   The     customer  will buy one unit,  at the lowest  
price, paying up to reservation price $R$.  If    firm  $i$  uses a  
mixed strategy,  denote its cumulative distribution  of prices  by  
$F_i(p)$.  



\noindent
 INTERPRETATION.
               The variable $\theta$ represents the probability that  
a given firm  is inactive and does not make a  bid that could  
possibly win the contract. As  the introduction said, this could have  
a number of interpretations: 

\begin{enumerate}
 \item
 The firm has reached its capacity.
 \item
 The firm   has gone out of business. 

 \item
 The firm  has not yet entered the business, though competitors  
thought it might have.   

 \item 

     The firm    priced grossly high, above $R$  because of a   
mistake  in judgement or mistaken information about the customer. 

 \item
  The firm  has not discovered that it could bid on this particular  
transaction. 

 \item
  The firm  has closed down temporarily because of strike, illness,  
or vacation.
 \item
 The customer has not realized that he  could invite the firm to bid  
on this transaction. 

 \end{enumerate}

   One way to view the situation being modelled is as  an auction.   
McAfee and Macmillan (1987)  analyze an  auction with independent  
private values,  possibly risk-averse bidders, and an unknown number  
of active bidders.   They find that the number of bidders being  
stochastic does not matter in the optimal auction when bidders are  
risk neutral---  the seller auctioning off an item can do as well as  
if the number of bidders were known.   The optimal auction is most  
definitely not   the first-price sealed-bid auction which is   
equivalent to the Bertrand model, however.  It is, rather, an English  
auction, in which bidders bid sequentially, publicly,  and as many  
times as they  wish.  In such an auction, it is clear that the winner  
will bid at the second-highest valuation of  the active  bidders,   
and it will become clear in the course of the auction who is active.    
In a sealed-bid auction this does not happen.\footnote{An  
intermediate form of auction, which has not been analyzed as far as I  
know, is the first-price sealed-bid auction where bidders know how  
many bids have been submitted, but not the size of the bids.}      


               Optimal or not, first-price sealed bid auctions are  
commonly  used,   and this paper's model represents them.     It also  
represents Bertrand competition, in which a number of firms offer  
prices to  one or more  customers, without knowing how many other  
firms are active.   Section 3 analyzes a model with the step function  
demand described above, while Section 4 will show the  modification   
needed to change to linear demand.   


 

\bigskip
 \noindent
3. THE SIMULTANEOUS SYMMETRIC BERTRAND  MODEL
 

 The first and most important model to be examined is Bertrand  
competition under the following clarification to the basic model.  


{\it Assumption:}   Firms submit bids simultaneously.   


 

  Under this assumption there 

  exists no equilibrium in pure strategies unless $\theta$ equals 0  
or 1.   If    $\theta$ is zero, all  firms charge $c$, in   the  
ordinary Bertrand equilibrium.   If    $\theta$ is 1,  then    no  
firm is active.   If, somehow,  a firm were active anyway (which has  
zero probability but   is conceivable), then it would charge $R$, the  
ordinary monopoly equilibrium.      


  

 What if $\theta  \in (0,1)$?  No set of     prices of which   one   
price    is  greater than $c$ can be an   equilibrium.  If     the  
two lowest prices were not  equal,  whichever firm has the lowest  
price  would deviate to increase its price.   If the two lowest  
prices were equal, one firm could reduce its price and win the entire  
market with certainty.  

  Nor is it an equilibrium  for all firms to charge $c$.    This  
would lead to zero profits,  whereas a firm that deviated and charged  
$R$ would gain profits of $R-c$ with probability $\theta^N$.   


      

 

Instead, the equilibrium must be in mixed strategies.   Each firm  
randomizes,   choosing prices of at least $c$ and  no more than $R$.   
And, since a firm  can guarantee itself a positive expected profit  
from the pure strategy of charging  $R$ and winning with probability  
$\theta^N$, it must be that  the range of mixing is not all the way  
down to $p=c$, but rather is in the interval $[B, R]$, where   $B>c$. 

There exists a symmetric equilibrium for $N+1 >1$ in which 

  \begin{equation} \label{a0}
   F(p) =    \sqrt[N]{   1 -  \left(\frac{\theta^N}{1-   
\theta^N}\right)    \left( \frac{R-p }{ p-c} \right)  } .
 \end{equation}
 over the price range   $[\theta^N R + (1-  \theta^N) c, R]$.  At  
both the individual  firm and industry level, expected prices and  
profits are falling in $N$.\footnote{A caveat: industry profits fall  
in $N$ when these are defined conditionally on at least one firm  
being active, as will be explained below.  Greater $N$ increases the  
chance of at least one firm being active.}  The rest of the section  
will demonstrate this. 

  

 Start by  hypothesizing that such an equilibrium does exist  with  
mixing over the range $[B, R]$.   

 The expected payoff to   firm $i$  from the pure strategy of  $p_i  
=p$ is then 

    \begin{equation} \label{a1}
     \pi_i(p) =  \left[    \theta^N  +  (1-  \theta^N)(1-F(p))^N    
\right]        [p-c].     

 \end{equation}
     Over the mixing range, this profit is equal for any price.  If  
$p=R$, it must be  that
   \begin{equation} \label{a2}
     \pi_i(R) =     \theta^N         [R-c],      

 \end{equation}
 because Firm $i$   will  win the auction only if no other firm bids  
(except for the infinitesimal  probability that the   other  prices  
also equal   $R$, which has probability zero in the conjectured  
equilibrium.)  Equating (\ref{a1}) and 

 (\ref{a2}) yields
  \begin{equation} \label{a3}
     \left[    \theta^N  +  (1-  \theta^N)(1-F(p))^N   \right]         
[p-c] = \theta^N         [R-c],     

 \end{equation}
 which can be solved to yield
 \begin{equation} \label{a4}
   F(p) =    \sqrt[N]{1 -  \left(\frac{\theta^N}{1-  \theta^N}\right)     
\left(\frac{R-p }{ p-c}\right)}.
 \end{equation}

$F(B)=0$ by definition of $B$, so  from  (\ref{a4}), 

  \begin{equation} \label{a5}
     0 =   1-   \left(\frac{\theta^N }{1-  \theta^N}\right) \left(    
\frac{R-p }{ p-c} \right), 

  \end{equation}
 which can be solved out to get 

   \begin{equation} \label{a6}
     B =    \theta^N R + (1-  \theta^N) c
   \end{equation}
  This completes the derivation of the equilibrium.  

 

\noindent 

EQUILIBRIUM OUTCOMES. 

 The expected price a firm charges is $p'$ such that
  \begin{equation} \label{a7a}
 0.5 =    F(p') =    \sqrt[N]{1 -  \left(\frac{\theta^N}{1-   
\theta^N}\right)   \left( \frac{R-p' }{ p'-c}   \right) }, 

   \end{equation}
which when solved out yields
  \begin{equation} \label{a7b}
p'     =     c+  \frac{  (R - c) \theta^N    }{   1- .5^N   
(1+\theta^N)        }  

   \end{equation}

 

Expected profit  for one firm is, since it is in the industry with  
probability $(1-\theta)$, 

   \begin{equation} \label{a7}
     \pi_i =    (1-\theta)  \theta^N (R-c).
   \end{equation}
Note that individual profit is declining in $N$. 



Expected profit for the industry is\footnote{Note that although the  
profits of the different firms are   not  independent,  the expected  
profits are, so this summation is legitimate. }
   \begin{equation} \label{a8}
   \sum_{i=1}^{N+1} \pi_i =   (N+1) (1-\theta)  \theta^N (R-c), 

   \end{equation}
  Industry profit is a bit tricky.    It can increase with $N$ simply  
because a greater $N$ reduces the chance that no firm is active and  
profits are zero.      


  


We can  back out the expected price from the expected industry  
profit.  The probability of an industry profit of 0 is   
$\theta^{N+1}$.  Denote the average  winning bid when at least one  
firm bids by $p''$. It is true that
  \begin{equation} \label{a10}
 \sum_{i=1}^{N+1} \pi_i =   (N+1) (1-\theta)  \theta^N (R-c)  

=  \theta^{N+1} (0) + (1-  \theta^{N+1})(p''-c). 

     \end{equation}
   From this we can deduce that
   \begin{equation} \label{a11}
 \begin{array}{ll}
p''  &  =   c + \frac{ (N+1) (1-\theta)  \theta^N (R-c)}{(1-   
\theta^{N+1})}\\
 & \\
      &=c + \frac{ (N+1)   \theta^N}{(1-  \theta^{N+1})}   (1-\theta)  
(R-c)\\
 \end{array}
      \end{equation}
 


   Expected industry profit conditioning on there being at least one  
active firm is then
  \begin{equation} \label{a12}
p''  - c  =      \frac{ (n+1) (1-\theta)  \theta^n (R-c)}{(1-   
\theta^{n+1})}
      \end{equation}
  To see how industry profit  changes with $N$, note that after some  
algebra, 

    \begin{equation} \label{a13}
 \frac{d p''}{d N}    =    \left[ \frac{\theta^N  (1-\theta^{N+1} +  
log(\theta)     + N log(\theta) )  }{  (\theta^{N+1} - 1)^2 }\right]   
\left[ (1-\theta)    (R-c) \right], 

        \end{equation}
  a  derivative which is well-defined even though only integer values  
of $N$ have an economic interpretation.   The  sign of  expression    
(\ref{a13})  is  the sign of   

    \begin{equation} \label{a14}
   1-\theta^{N+1} + log(\theta)     + N log(\theta),  

     \end{equation}
    which most nearly is positive when $N=1$ and $\theta=1$.  Since  
$\theta$ is a fraction,   expression  (\ref{a14})   falls  with $N$,  
and so is greatest when $N=1$ and it equals   $1 - \theta^2 + 2 log  
(\theta) $. Its derivative with respect to $\theta$  when $N=1$ is   
$-2\theta + 2/\theta$, which is positive, so it is increasing in  
$\theta$.  But its maximum, at $N=1$ and $\theta=1$, is only      
$(1-1^{1+1} + log(1)     + 1 log(1) )=0$.  Thus, $p''$ must be  
decreasing with $N$.   The expected lowest price and industry profit  
conditional on at least one firm serving the market is falling with  
the number of firms.  


 Thus, this simple Bertrand model  lacks the  discontinuous behavior  
of the  original Bertrand model. Profits are positive, but the   
expected price and profits decrease smoothly in the number of firms,  
as in the Cournot model.   Note also that the Betrand model easily  
handles the situation where demand is given by a step-function, which  
creates difficulties for the Cournot model.  


This   model  can also  illustrate in a simple way the  intuition for  
the result  of  McAfee and Macmillan (1987) that   the seller in an  
auction  can benefit from the risk aversion  of the buyers and  their  
lack of precise knowledge of how many bidders are active.   Note that  
in the present model,  a seller who charges below  $B$, the lower  
bound of the support of the mixing distribution, will  win the  
customer with certainty, and still earn a profit.  A risk-averse   
seller would wish to take advantage of this, and would tend to push  
down the prices  charged.    A high price is a gamble in the hope  
that other firms are inactive or are themselves charging high prices,   
so risk aversion tends to reduce prices.   This also suggests  that  
prices might rise when firms are in financial distress.  A firm near  
bankruptcy is often risk-loving,  because its additional losses are  
borne by its creditors but its gains are partly kept by its owners.     
Such a firm should be more willing to set out a high  price of $R$ in  
a gamble that no other firm is active.  

 




 \bigskip
 \noindent
4.   COMPARING    BERTRAND AND  COURNOT  


To compare  Bertrand and Cournot oligopoly, we cannot use a  
step-function demand curve, since Cournot equilibrium applies  rather  
uncomfortably to that situation.  Continue to let marginal cost be  
constant at $c$ with probability $1-\theta$ and infinite with  
probability $\theta$, but now   assume that demand is linear, so 

  \begin{equation} \label{b1}
     p\left(\sum_{i=1}^{N+1}  q_{i}\right)  =        \alpha - \beta   
\sum_{i=1}^{N+1}  q_{i}. 

 \end{equation}
   Let us define $q(p)$ as the   demand  facing a monopolist at a   
price of $p$, so 

  \begin{equation} \label{b2}
    q(p)   =    \frac{ \alpha}{\beta}  -  \frac{ p}{\beta}, 

 \end{equation}
   and  redefine $R$ to be the monopoly price, so  

  \begin{equation} \label{b3}
  R   =      \frac{\alpha +c}{2}.   

 \end{equation}
 (Note that $q(R) = \frac{ \alpha -c}{2 \beta}$.)

  What  I will do in this section  is to compute the expected profits  
from Cournot and Bertrand for different levels of $N$, to obtain   
some idea of  the effects of concentration in each.     (The profit  
path, however, is  no longer the same shape as   the price path  now,  
because quantity changes with price. )


\bigskip
\noindent
 BERTRAND EQUILIBRIUM  


    The expected payoff to   firm $i$  from the pure strategy of   
$p_i =p$ is      \begin{equation} \label{b4}
     \pi_i(p) =  \left[    \theta^N  +  (1-  \theta^N)(1-F(p))^N    
\right]        [p-c]    q(p).     

 \end{equation}
     Over the mixing range, this profit is equal for any price.  If  
$p=R$, it must be  that individual firm profit is
   \begin{equation} \label{b5}
     \pi_i(R) =     \theta^N         [R-c] q(R).      

 \end{equation}
 and   industry profit   is 

    \begin{equation} \label{b6}
  \sum_{i=1}^{N+1} \pi_i =   (N+1) (1-\theta)  \theta^N (R-c)   q(R)
=  \theta^{N+1} (0) + (1-  \theta^{N+1}) \pi_{bertrand},   

     \end{equation}
 where  $\pi_{bertrand}$ is the industry profit conditional upon at  
least one firm being active.  We can solve out for this to get: 

   \begin{equation} \label{b7}
\pi_{bertrand}=   \frac{   (N+1) (1-\theta)  \theta^N (R-c)    
q(R)}{(1-  \theta^{N+1})}. 

      \end{equation}
 

\pagebreak
 

 

\noindent
 COURNOT EQUILIBRIUM  


  

  Now let us compute the Cournot equilibrium. Let $q^*$ be the  
Cournot output we are trying to determine.  Then, 

   \begin{equation} \label{b8}
 \begin{array}{ll}
     \pi_i(q_i) & =      (1-\theta)^{N}  [p(q_i + Nq^*)   -c]q_i+    
\theta     (1-\theta)^{N-1}  [p(q_i +  (N-1)q^*)   -c]q_i    \\ 

      & \\  

     &  +   \theta^2   (1-\theta)^{N-2}  [p(q_i +  (N-2)q^*)   -c]q_i  
+ ... \\
    & \\  

   & =    \sum_{j=0}^n    \theta^j    (1-\theta)^{N-j}  [p(q_i +   
(N-j)q^*)   -c]q_i \\ 

\end{array}
  \end{equation}
  The first order condition is 

  \begin{equation} \label{b9}
 \begin{array}{ll}
    \frac{d \pi_i(q_i)}{d q_i}    & =    \sum_{j=0}^n    \theta^j     
(1-\theta)^{N-j}  [-\beta q_i     

  + \alpha - \beta ( q_i + (N-j)q^* )  -c]  =0 \\ 

    & \\  

 & =     \sum_{j=0}^n    \theta^j    (1-\theta)^{N-j}  [  \alpha -  
\beta  (N-j+2)q^*   -c]\\
   & \\  

 &  =     \left(\sum_{j=0}^n    \theta^j    (1-\theta)^{N-j} \right)       
\left(\alpha -c   \right)          -      \left(\sum_{j=0}^n     
\theta^j    (1-\theta)^{N-j} (N-j+2)     \right)  \beta q^*, \\
 \end{array}
 \end{equation}
 so
 \begin{equation} \label{b10}
   q^* = \frac{  \left(\sum_{j=0}^N    \theta^j    (1-\theta)^{N-j}  
\right)  (\alpha -c)}  {   \left(\sum_{j=0}^N    \theta^j     
(1-\theta)^{N-j}  (N-j+2) \right)  \beta}  \\
   \end{equation}
     Note that if $\theta=0$, this boils down to $q^* = \frac{ \alpha  
-c}  {(N+2)\beta}$.  Adding incomplete information makes no great  
difference in the Cournot model. If some firms might not be active,  
each active  firm produces somewhat more than it would have  
otherwise.  

 

    From the quantity  we can get the expected  profit    conditional  
upon there being at least one firm in the market. 

    \begin{equation} \label{b11}
 \pi_{Cournot}    = 

           \sum_{j=1}^{N +1}    \left( \frac{  \theta^{N+1-j}     
(1-\theta)^{j} }{ 1 - \theta^{N+1}} \right)   [ p( jq^*)  -c]  jq^*  

    \end{equation}
      Note that  equation (\ref{b11})  is conditional upon  
$(N+1)*q^*$ not being so large as to drive the price to zero, which  
might rationally happen, since a firm   would  be willing to accept a  
price of zero occasionally as the result of  all $(N+1)$ firms     
coincidentally being active and  producing  a large amount. 



\bigskip
\noindent
 COMPARISONS

 Profits are positive but fall with the number of firms in both the  
Bertrand and Cournot equilibria.   The question is how fast profits  
fall.  This is best seen with a numerical example. 

Let  $\alpha=110$, $c=100$,  $\beta =1$, and   $n= [0, 7]$  for  
$\theta = 0$ and $\theta=.1$. Table 1 and Figures 1 throught 3  show  
the levels of profits.      Table 1 consists of industry profits  
under the Bertrand and Cournot models with certainty and with $\theta  
=1$, and its numbers are repeated graphically in Figure 1.  Figures 2  
and 3 show how the profit/concentration relationship changes for  
different values of $\theta$ in the two models.\footnote{In every  
case,  expected industry profits are conditional upon at least one  
firm being active.  When $\theta=1$, this is to be interpreted as the  
probability zero  (but possible) event that  one firm is active and  
the expected number of other firms is zero. }    Tables 2 and 3    
pertain to the Bresnahan-Reiss empirical results, which we shall come  
to shortly.\footnote{xxx Consider adding the expected proportion of  
firms that are active as an item of discussion. Rescale so that  
profits are 100 at monopoly.}    


Consider first the Cournot model.       Table 1 and Figure 1 show  
that   a small amount of uncertainty makes little difference in the  
Cournot model, though,  oddly enough,    industry profits actually  
fall when the expected percentage of active firms declines.   Under  
Cournot competition, a firm expands its output when  it expects fewer  
rivals to be helping push down the price. Uncertainty  over the  
number of rivals ends up increasing average output and reducing  
profits, a peculiar result.   Figure 3 shows that   this is a  very  
delicate conclusion.        Profits     fall when $\theta$ rises from  
0.0 to 0.1, but fall when  $\theta$ rises from  0.3 to 0.9,   for   
$N+1>4$, thought the reverse is true for  smaller $N+1$.      
Conflicting forces are at work in Cournot equilibrium, and the net  
result is sensitive to  the  particular assumptions of the model.       


 Uncertainty is much more important in the Bertrand model, and the  
comparative statics are more consistent and intuitive.    Table 1 and  
Figure 1 show that   a small amount of uncertainty   changes the  
Bertrand model  in a small but crucial way, because profits do become  
positive and monotonic in the number of firms.  The sharp fall in  
profits moving from monopoly to duopoly  under certainty is not so  
unreasonable as it looks.  It is extreme, but  it is a limiting  
result as $\theta$ goes to zero, as Figure 2  illustrates.  
\footnote{xxx Switch figure 2 and 3.} 

  

           


 

   \epsfysize=3in 

   

\epsffile{Bert1.eps} 


\begin{center}
     {\bf  Figure 1:  Bertrand and Cournot Profits} 

\end{center}

\vspace*{36pt}



 

 

  \epsfysize=3in 

   

\epsffile{Bert2.eps} 


\begin{center}
     {\bf  Figure 2:  Bertrand   Profits For Different Probabilities  
of    Inactivity $\theta$ and Numbers of Firms $(N+1)$ }\\
 (conditional on  at least one firm being active)  \end{center}


\vspace*{36pt}


  \epsfysize=3in 

   

\epsffile{Bert3.eps} 


\begin{center}
     {\bf  Figure 3:  Cournot Profits For Different Probabilities of     
Inactivity $\theta$ and Numbers of Firms $(N+1)$ } \\
 (conditional on at least one firm being active)  \end{center}

\vspace*{36pt}



\newpage

\bigskip
\begin{tabular}{     l   |   rrrrr rr   }   

\hline
\hline
  Number of Firms   $(N+1)$ & 1 & 2 & 3 & 4 &5 &6 &7\\
\hline
   &   &   &  &   &  &  &   \\
    Bertrand, $\theta=0$ & 25.0 & 0.0 & 0.0 & 0.0 &0.0 &0.0 & 0.0\\
  Bertrand, $\theta=0.1$ (eq. (\ref{b7}))&25.0&5.6& 0.9& 0.1& 0.02&  
0.003  & 0.0003 \\
  &   &   &  &   &  &  &   \\
 Cournot, $\theta=0$  &25.0& 22.2&18.8&16.0&13.9&12.2&10.9\\
  Cournot, $\theta=0.1$  (eq. (\ref{b11})) & 25.0&19.6&15.0&11.6 &  
9.1& 7.3 &    5.8 \\
  &   &   &  &   &  &  &   \\
  \hline
\hline
  \end{tabular}

{\bf  Table 1:  Industry Profits for Different Concentration  
Levels}\footnote{Numerical calculations and  figure-drawing used {\it  
Mathematica}.  Values are rounded. } 


\bigskip


\begin{tabular}{  l   |  rr rrr    }   

\hline
\hline
  Number of Firms  $(N+1)$ & 1 & 2 & 3 & 4 &5  \\
\hline
   &   &   &  &   &         \\
 Doctors & 0.88 & 1.75 & 1.93 & 1.93 & 1.83  \\
Dentists &  0.71 & 1.27 & 1.39 & 1.36 & 1.28  \\
Druggists &0.53 & 1.06 & 1.68 & 1.92 & 1.88  \\
Plumbers & 1.43 & 1.51 & 1.51 & 1.55 & 1.49  \\
Tire Dealers & 0.49 &0.89 & 1.14 &  1.19 & 1.22  \\
  &   &   &  &   &         \\
 \hline
\hline
   \end{tabular}

{\bf  Table 2:    Bresnahan-Reiss Entry Thresholds $s_i$: Original   
(1,000's of inhabitants)}\footnote{Calculated from Table 5A of  
Bresnahan \& Reiss (1991).  Note that the  entry of .79 in the second  
row of their original paper is a mistake and should be 1.09, and  
their Figure 4 illustrates  $s_i/s_5$, not   the   $s_5/s_i$  in the  
legend.}

 \bigskip
\begin{tabular}{  l   |   rr rrr    }   

\hline
\hline
  Number of Firms $(N+1)$  & 1 & 2 & 3 & 4 &5  \\
\hline
  &   &   &  &   &         \\
  Doctors & 25.0 & 4.3 & 0.0  &0.0  & 0.0  \\
Dentists &  25.0 &  4.4 & 0.0  & 0.0  &0.0  \\
Druggists &25.0 & 15.5 & 4.3 & 0.0  & 0.0  \\
Plumbers & 25.0 & 8.3 & 8.3 & 0.0  & 0.0  \\
Tire Dealers &  25.0 & 11.3 &2.7 &  1.0 & 0.0  \\
  &   &   &  &   &         \\
\hline
Average & 25.0  & 9.6 & 2.3 &  0.2 & 0.0  \\
\hline
\hline
     \end{tabular}

{\bf  Table 3:    Bresnahan-Reiss Entry Thresholds: Rescaled  
($\frac{25   (s_m-s_i)}{ (s_m-s_1) }$) }  



  Let us also consider the shape of the profit-concentration paths.     
All the curves in Figures 1 through  3  have convex shapes, if only  
weakly in the limiting cases, but  the curvatures, and therefore the  
empirical implications, are different.  As Figure 1 and Table 1, in  
particular, show,   profits decline much more rapidly  in   Bertrand    
than in Cournot.  For the parameters chosen,  industry profits fall  
from the monopoly level of 25 to duopoly profits of 5.6, triopoly  
profits of 0.9, and  negligible levels thereafter.  Cournot profits  
show a much more uniform decline as concentration falls.  


 Comparison of Figures 2 and 3 shows that  for larger values of the  
inactivity probability $\theta$  the Bertrand profit path becomes  
flatter and the Cournot path more curved, but even at extreme values      
Cournot   does not generate such sharp differences from the addition  
of one firm to the market.  

 

      For most modelling purposes,   these models are building  
blocks, and  such subtle differences in the profit-concentration path   
are unimportant.  They are interesting, however,  if one wishes to  
consider Bertrand and Cournot as serious oligopoly models in their  
own right.    Empirically, then, how  do profits react to the number  
of firms?   Do they decline to zero with duopoly and then stay  
constant, as in  the original Bertrand model?   Do they decline  
smoothly, as  either version of the Cournot model would suggest? Or  
do they  decline rapidly, as the Bertrand model with uncertainty  
would suggest?     


     Measuring the relationship between  profits and concentration is  
an old exercise now in some disrepute.\footnote{xxx Find a refrence.}      
The difficulty is that  the usual unit of observation  has been the  
industry.  This is natural enough, since  one needs a measurement of  
concentration for each observation.   Comparing   accounting  profits  
across  industries is fraught with danger, however, since accounting  
profits differ from economic profits in  ways that depend  on the  
industry chosen and which are very likely to be correlated with   
technology, and hence with concentration.  Moreover, it is not clear  
that the concentration-profits  path is even  the same across  
industries.  



A clever recent approach  to the same problem is that of Bresnahan \&  
Reiss (1991).    They took the unit of observation to be   the market  
for a particular profuct  in a  particular small town, rather than  
for  many products over  the entire United States,  and they looked  
at market size rather than directly at profits.    They collected  
data on the size of  a town  and the number of   dentists there, for  
example.   If a town is very small---say, 500 people----   it will  
have no dentist, since  a dentist   incurs a   fixed cost   and   
could not make any profit there even as a monopoly.  If  it grows to  
800  people, it will have one dentist, since the profits are enough  
for monopoly, but entry by a second dentist would drive them  
negative.      If the town grows to 1,600 people, however, it may  
still have only one dentist--- if entry by the second dentist would  
not just split the industry  profits, but reduce them.   


Bresnahan and Reiss used this approach to estimate the thresholds for  
entry in small markets for a number of industries.  Table 2 shows  
these thresholds in thousands of inhabitants per firm.    Table 3  
rescales the same numbers  to be very roughly comparable with the  
numerical example used earlier in this paper.\footnote{Table 3's  
rescaling uses the following procedure. 

    Define  the monopoly  level of profits in an industry to be 25,  
and the competitive level to be 0.  Assume that when $s_i$ reaches  
its maximum level  $s_m$ over $[1,5]$, the competitive level of  
profits is reached and any further changes are measurement error.    
Apply the conversion formula  $s_i^*= \frac{25   (s_m-s_i)}{  
(s_m-s_1) }$, and Table 3 results. } The rescaling is somewhat  
arbitrary, since the theory of Bresnahan and Reiss is that  some   
quasi-rents remain to  cover fixed cost even when the minimum scale  
for entry flattens out,  but it  creates a   comparison measure  for   
how the intensity of competition changes with the number of firms.  

  


 What is significant is how   profits flatten out, even though the  
choice of 0  as the flat level  in Table 3 is   arbitrary.\footnote{    
Add 9.1 to each entry in Table 3, and the profit at $N+1=5$ is   9.1,  
as with Cournot competition and $\theta=0.1$ in Table 1,  but the    
shape is still more like that of Bertrand competition. }  The  
empirical result that  going from one firm to two is much more  
important than going from two to three, and that  full-fledged  
competition  kicks in very quickly matches the Bertrand model  with  
uncertainty very well, and is inconsistent with the Cournot model.  


 


 \vspace*{36pt}
  \noindent
5. THE ASYMMETRIC   BERTRAND  MODEL WITH TWO FIRMS

  The purpose of this section of the paper is   technical.   It is a  
warning that  the apparently simpler  Bertrand  duopoly  model in  
which only firm might be inactive   is actually more difficult.      
It  is not the way to go  to try to find a  simpler building-block  
model. 

   


{\it Assumption:   There are two firms. Firm 1   is always active.   
Firm 2 is inactive  with probability $\theta$.  }

    

Letting  the firms have possibly different mixing cumulative  
distributions $F_i(p)$ over $[B, R]$. 

 The equilibrium is
  $$
 F_1(p) =  \left|
 \begin{array}{lll}
   0 &   {\rm for  } & p  \leq   (1-\theta)c +  \theta R\\
  1 - \theta \left( \frac{R-c}{p-c}  \right)&   {\rm for  } & p  \in  
[  (1-\theta)c +  \theta R, R)\\
   1  &   {\rm for  } & p \geq   R\\
 \end{array}
 \right.
$$
    $$
 F_2(p) =  \left|
 \begin{array}{lll}
   0 &   {\rm for  } & p  \leq  (1-\theta)c +  \theta R\\
  1 - \left(   \frac{ \theta }{ 1-\theta } \right)      \left(  
\frac{R-p}{p-c }  \right)
 &   {\rm for  } & p  \in [  (1-\theta)c +  \theta R, R]\\
   1  &   {\rm for  } & p \geq   R\\
 \end{array}
 \right.
$$
 The    equilibrium of the asymmetric model is similar to the  
symmetric one  of Section 3,  but    Firm 1's equilibrium  mixing  
probability has an atom of probability at $p_1=R$.   This shows up  
very subtly in the equilibrium description, as the open-set bracket   
in   $[  (1-\theta)c +  \theta R, R)$. 

$F_1(R- \epsilon)  \approx   1-\theta$ for small $\epsilon$, but  
$F(R) =1$. 


    I will now justify the equilibrium.   In equilibrium, the  
expected payoff from each pure strategy mixed over must be equal.   
The expected payoff to Firm 1 from the pure strategy of  $p_1=R$ is 

 \begin{equation} \label{e1}
  \pi_1 (R) =  \theta (R-c)
 \end{equation}
 because Firm 1 only wins then if Firm 2's cost is infinite. (With  a  
continuous   mixing distribution for Firm 2, the probability of  
$p_2=R$ in a mixed strategy equals zero.)      

 

 The expected payoff to Firm 1 from the pure strategy of $p_1=p$  is    
\begin{equation} \label{e2}
  \pi_1 (p) =  (\theta  + (1-\theta) (1-F_2(p))      (p-c), 

 \end{equation}
  since Firm 1 wins the bid if Firm 2 has infinite costs or if Firm  
2's mixing has led to a price of   $ p_2>p$, which has probability   
$(1-F_2(p))$.  Equating (\ref{e1}) and ({\ref{e2}) yields 

 \begin{equation} \label{e3}
   \theta (R-c) =(\theta  + (1-\theta) (1-F_2(p))      (p-c), 

 \end{equation}
 which can be solved out for $F_2(p)$ to give
  \begin{equation} \label{e4}
   F_2(p) =    1 - \left(   \frac{ \theta }{ 1-\theta } \right)       
\left( \frac{R-p}{p-c }  \right)
 \end{equation}
    By definition, $F_2(B )=0$, so 

   \begin{equation} \label{e5}
     1 - \left(   \frac{ \theta }{ 1-\theta } \right)      \left(  
\frac{R-B } {B-c }  \right)  =0. 

 \end{equation}
 Solving that out yields  $B  =   (1-\theta)c +  \theta R  $. 

  


 Now we do the same sort of thing starting with Firm 2's payoffs. 

 The expected payoff to Firm 2 from the pure strategy of $p_2=p$  is    
\begin{equation} \label{e6}
  \pi_2 (p) =   (1-\theta) (1-F_1(p))      (p-c), 

 \end{equation}
  since Firm 2 wins the bid if  Firm 1's mixing has led to a price of    
$ p_1>p$, which has probability  $(1-F_1(p))$. Since  $F_1(B)=0$, 

   \begin{equation} \label{e7}
        \pi_2 (p) = \pi_2(B)=    (1-\theta) (1-0)      (B -c),
 \end{equation}
 or 

   \begin{equation} \label{e8}
        \pi_2 (p) =      (1-\theta) (1-0)      ( (1-\theta)c +   
\theta R  -c) = (1-\theta) \theta (R-c).  

   \end{equation}
    Substituting  for   $\pi_2 (p)$ in equation (\ref{e6}) using  
equation  (\ref{e8}) yields
 \begin{equation} \label{e9}
 (1-\theta) \theta (R-c) =   (1-\theta) (1-F_1(p))      (p-c), 

 \end{equation}
 which when solved yields
  \begin{equation} \label{e10}
 F_1(p) =  1 - \theta \left( \frac{R-c}{p-c} \right).
  \end{equation}

  One complication     on looking at (\ref{e9})  is  that  $F_2(R) =  
1-\theta$, rather than 1.   Equation (\ref{e9}) only holds for $p<R$,  
and there is an atom of probability for Firm 1 at $p_1 =R$. 


 This establishes that the equilibrium is indeed Nash. 

 


 The  expected payoffs are 

 $$
 \pi_1 =  \theta(R-c), \;\;\;\;\;\;    \pi_2 = (1-\theta)   
\theta(R-c),
 $$
 which sum to 

 $$
 (2-\theta) \theta (R-c). 

 $$ 

  This sum lies between  0 and $(R-c)$, the competitive and monopoly  
profits, because  $\theta <1$ implies  $(1-\theta)(\theta-1) <0$,  
which implies that  $2\theta - \theta^2 <1$.   If $\theta$ grows,  
then  industry profits rise smoothly.   



 \bigskip
 \noindent
 6. THE SEQUENTIAL  BERTRAND  MODEL

Competition when two duopolists pick  quantities in sequence is known  
as the Stackelberg model.   It is worth thinking about the Bertrand  
analog, which although simple is not widely understood even under  
certainty.  

  

 

{\it Assumption :   Firms submit bids publicly in sequence from Firm  
1 to Firm $(N+1)$.      }

  

First, consider what happens in the Bertrand model with no   
uncertainty--- the special case of $\theta =0$.   There are two   
classes of equilibria. 


In  the first  class of equilibrium,   at least one  of the first $N$   
firms   chooses $p=c$, and  consumers  buy  from   firms charging  
$p=c$.   Profits are zero, and the outcome is the same as in the  
simultaneous Bertrand model. 

 

  

 In the  second class  of equilibria, the first    $N$   firms choose  
prices in a set   with minimum     $p_{min}>C$   and the last  firm  
chooses $p_{N+1}  =  Min \{ p_{min}, R  \}$. The consumers    all  
choose to buy from firm $(N+1)$.    Profits are zero for all firms  
except Firm $(N+1)$, who has positive profit. 



 The second class of equilibrium is  counter-intuitive.  For  
concreteness, consider the particular member of the class in which  
all firms offer the price $R$.   None of the first $N$ firms have any  
incentive to deviate. If a firm deviates to $p=c$, his profit remains  
zero. If a firm deviates to any price between $c$ and $R$, Firm  
$(N+1)$ will respond with the same price and capture the market, so  
the deviating  firm's profit remains zero.  Firm $(N+1)$ clearly has  
no incentive to deviate. And the consumers have no incentive to  
deviate because all firms charge the same price.  


  This is, to be sure, a weak Nash equilibrium,  which is why it  is   
counterintuitive.   No Nash  equilibrium exists,  however, in which  
consumers are not indifferent about where they buy, and in which   
more than one firm earns positive profits.  This is the standard  
open-set problem; if consumers did not follow this behavior, the last  
firm would choose to undercut its lowest competitor by an  
infinitesimal amount and gain the entire market.\footnote{See   p.   
103 of Rasmusen (1994) for  a discussion of the open-set problem.   
Note that the first edition of that book does not contain any  
discussion of it. }  It seems reasonable, however, to prefer an  
equilibrium in which players  behave symmetrically, when such an  
equilibrium exists, and with that assumption, the only equilibrium is  
the symmetric one in the first class with $p=c$ for all firms, and  
consumers evenly divided among them.  


 Next,   consider the symmetric Bertrand model  in which each firm is  
inactive with probability $\theta$.  The last active firm will  
undercut any previous firm in the sequence that has offered  $p>c$.     
Thus,  if any  later firm in the sequence is  active, the earlier  
firms' payoffs will all equal zero regardless of their bid.  If no  
later firm is active,   an early firm will win the market, and might  
as well bid $p=R$.   Since  if $\theta >0$ there is a positive  
probability that all later firms will be inactive,     every   firm    
bids $p=R$.  Consumers buy from the last firm bidding, again, to  
resolve the open-set problem.  


   

The equilibrium in the sequential model is somewhat bizarre.   Even a  
tiny amount of uncertainty   reduces  a continuum of equilibria  to a  
unique equilibrium.   Moreover,  the most plausible level of profits  
rises from zero to the monopoly level.   


What this illustrates is the  tremendous power of    open-cry  
auctions in revealing information.     When there is no uncertainty,    
this does not make much difference.  When there is uncertainty about  
the number of firms, however,  the open-cry auction resolves that  
uncertainty, giving the last bidder, in particular, a tremendous  
advantage.   Earlier bidders know they cannot overcome that  
advantage, so their only hope is that no later bidders will be  
active.  


          The sequential Bertrand model is, of course, not  a typical  
open-cry auction, because the sequence of bidding is predetermined  
and each firm only gets one bid.   In the classic English auction,   
each bidder can bid as often as he wishes.  In the present context,  
this would result in a winning bid of $p=c$ if at least two firms are  
active, whatever the value of $\theta$ may be. 


    The caveat  ``if at least two firms are active'',  however, is  
important.    With probability $ (N+1)(1-\theta) \theta^{N}$, only  
one firm is active and the winning bid will be $p=R$.  The expected  
industry profit is therefore
 $ (N+1)(1-\theta) \theta^{N}  (R-c)$, exactly the same as the profit   
given by  equation  (\ref{a8}) in the simultaneous game!    This is  
the same result  found in  McAfee \& Macmillan (1987).    From the  
point of view of the buyer, the   English auction  has the  advantage  
of pitting bidders against each other head to head, but the  
disadvantage of  letting a bidder  know if he has no competition.  As  
a result, the English auction has much greater risk, and a  
risk-averse buyer would prefer  simultaneous bids.   

 

   


 \bigskip
 \noindent
7. CONCLUDING REMARKS

 The Bertrand model with incomplete information about the number of  
firms is simple,  but its properties are both interesting and useful.    
The extreme transition from monopoly to competition found in the  
standard Bertrand model disappears.  Profits are positive,   but  
decline with the number of firms in the industry, and   decline in a  
way that empirical work suggests is more realistic than in  the  
Cournot  model.      Expected profits also decline as the expected  
fraction of firms that are actively seeking business increases,  in  
contrast to the Cournot model.       

  I hope that the model may be useful both as a simple description of  
oligopoly  and as a building block for  other topics in industrial  
organization.       

  



  \newpage
\noindent
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